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\begin{document}
\title{Quantum simulation of quantum field theory using continuous variables}
\author{Kevin Marshall}
\affiliation{Department of Physics, University of Toronto, Toronto, M5S 1A7, Canada} \author{Raphael Pooser}
\affiliation{Quantum Information Science Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, U.S.A} \affiliation{Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996-1200, U.S.A.} \author{George Siopsis} \email{[email protected]} \affiliation{Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996-1200, U.S.A.} \author{Christian Weedbrook}
\affiliation{QKD Corp, 60 St.~George St., Toronto, M5S 1A7, Canada}
\date{\today} \pacs{42.50.Ex, 03.70.+k, 42.50.Dv, 03.67.Lx}
\begin{abstract}
Much progress has been made in the field of quantum computing using continuous variables over the last couple of years. This includes the generation of extremely large entangled cluster states (10,000 modes, in fact) as well as a fault tolerant architecture. This has led to the point that continuous-variable quantum computing can indeed be thought of as a viable alternative for universal quantum computing. With that in mind, we present a new algorithm for continuous-variable quantum computers which gives an exponential speedup over the best known classical methods. Specifically, this relates to efficiently calculating the scattering amplitudes in scalar bosonic quantum field theory, a problem that is believed to be hard using a classical computer. Building on this, we give an experimental implementation based on cluster states that is feasible with today's technology. \end{abstract}
\maketitle
\section{Introduction}
For more than a decade now, continuous-variable (CV) quantum information~\cite{Braunstein2005,Weedbrook2011} has been a prominent substrate in implementing quantum technologies. Primarily this can be attributed to its largely Gaussian nature which invites simple and convenient mathematical calculations, as well as accessible experimental demonstrations, which are often deterministic in nature. Furthermore, one can also use CVs as a key element in another promising architecture, known as hybrid quantum information~\cite{Furusawa2011}.
The field of quantum computing~\cite{Ladd2010} using CVs~\cite{Lloyd1999,Weedbrook2011} has also progressed significantly in the last few years. From its original conception in 1999~\cite{Lloyd1999}, progress began to accelerate after a cluster state~\cite{Raussendorf2001} version was established in 2006~\cite{Zhang2006,Menicucci2006}, leading to something significantly more tangible for experimentalists. This resulted in numerous proof-of-principle demonstrations~\cite{Yokoyama2014,Miyata2014,Pysher2011,Takeda2013}, currently culminating in a 10,000 node cluster~\cite{Yokoyama2013} created `on-the-go' along with a 60 node cluster created simultaneously~\cite{Chen2014}. From a theoretical perspective, much progress has been made \cite{Marshall2014,Lau2013,Loock2007,Gu2009,Alexander2014,Demarie2014,Menicucci2015,Wang2014,Menicucci2011}, including recently, an important fault tolerant architecture~\cite{Menicucci2014}, achieved by leveraging the Gottesman-Kitaev-Preskill (GKP) encoding~\cite{Gottesman2001}. However, one area that is significantly underdeveloped is that of algorithms for a CV quantum computer. Thus far there only exists CV versions of quantum searching~\cite{Pati2000} and the Deutsch-Jozsa algorithm~\cite{Pati2003,Adcock2009,Zwierz2010,Adcock2013}.
In this paper, we present an algorithm that simulates~\cite{Georgescu2014} the scattering amplitudes in scalar bosonic quantum field theory (QFT) using a continuous-variable quantum computer. In fact, we show one can obtain an exponential speedup over the best known classical algorithms. A discrete version of the algorithm was originally shown in Refs.~\cite{Jordan2012,Jordan2014a} for a quantum computer based on qubits. Further work extended this result to fermionic QFTs~\cite{Jordan2014}, as well as using wavelets for multi-scale simulations~\cite{Brennen2014}.
Typically, $q$ and $p$ are the CVs spreading across all real numbers. To encode them in qubits, one needs a whole register of qubits at each point in space. However, with CVs, there is a 1-to-1 mapping to qumodes (the CV equivalent of a qubit). In fact it is arguable that a CV quantum computer is the natural choice for such a QFT problem given that​ the fields are continuous variables. Thus, the value of the field at a given point in space can be mapped onto a qumode naturally. If qubits are used, instead, the qumode needs to be replaced by a register of $M$ qubits which only allows the field to take on $2^M$ discrete values. Brennen \textit{et al.} describe both possibilities in Ref.~\cite{Brennen2014}, although they do not explain how to implement the quartic phase gate with CVs, which we do here. Furthermore, the quartic vertex in wavelets becomes very complicated. Implementing it would require gates acting on more than two modes (resulting in logarithmic overhead in complexity). Another benefit to our approach is in the development of the initial cluster state. Here we show how to create the initial CV cluster state as well as suggesting an experimental implementation based on standard linear optics. Furthermore, we also note that in the preparation of the initial state we see a slight improvement over the original qubit approach of Ref.~\cite{Jordan2012}. There they require $O(N^{2.376})$ gates to engineer the ground cluster state; whereas in our scheme, we require slightly less than that, specifically, $O(N^2)$ gates.
Our paper is structured in the following way. In Sec.~\ref{sec2}, we discretize space for a one-dimensional scalar bosonic QFT while leaving the field and time as continuous parameters. Next, we show how to generate the initial cluster state using only Gaussian operations in Sec.~\ref{prep}. In Sec.~\ref{compute} we outline the steps necessary to compute a scattering amplitude including the required measurement. We provide an explicit experimental implementation in Sec.~\ref{experiment}. Finally, the benefits of our approach over classical methods are discussed in Sec.~\ref{conclusion}. \section{Discretization in one-dimension} \label{sec2}
We consider a relativistic scalar field $\phi$ in one spatial dimension including a quartic self-interaction. We shall outline the discretization specifically in the one-dimensional case so as not to clutter the notation unnecessarily, but generalization to higher dimensions is straightforward and is discussed in the supplementary material. We note that the field $\phi$ is a function of $x$ and $t$ (time), $\phi(x,t)$. All three parameters are continuous. In our approach, we discretize $x$, but not $\phi$ or $t$. In the case of qubits, one would discretize $x$ and $\phi$, but not $t$. In classical lattice calculations, one discretizes all three $\phi$, $x$, and $t$.
In the continuum, the one-dimensional free scalar QFT is given by the Hamiltonian \begin{equation} H_0 = \frac{1}{2} \int_0^L dx \left[ \pi^2 + \left( \frac{\partial\phi}{\partial x} \right)^2 + m^2 \phi^2 \right] \end{equation} where $\phi$ is the scalar field and $\pi$ the conjugate momentum field. They obey commutation relations $[ \phi (x),\pi (x') ] = i\delta (x-x')$ where we choose units in which $\hbar =1$.
We discretize space by letting $x = na$, $n=0,1,\dots, N-1$, where $a$ is the lattice spacing and $L=Na$ is the finite length of the spatial dimension ($L\gg a$). We choose units in which $a=1$, for simplicity, and denote $Q_n = \phi (x)$, $P_n = \pi(x)$. The discretized variables obey standard commutator relations, $[ Q_n,P_m ] = i \delta_{nm}$. The Hamiltonian becomes \begin{equation}\label{eq18} H_0 = \sum_{n=0}^{N-1} \frac{ P_n^2 + m^2 Q_n^2}{2} + \frac{1}{2} \sum_{n=0}^{N-1} (Q_n- Q_{n+1})^2 \end{equation} where we employed periodic boundary conditions and defined $Q_{N} \equiv Q_0$.
It is useful to define creation and annihilation operators, $A_n^\dagger$ and $A_n$, respectively, by ${A}_n = (Q_n + iP_n)/\sqrt{2}$. They obey the commutation relations $[ A_n,A_m^\dagger ] = \delta_{nm}$ and the Hamiltonian can then be written as \begin{equation} \label{eq8} H_0 = \frac{1}{2} \mathbf{P}^T \mathbf{P} + \frac{1}{2} \mathbf{Q}^T \mathbf{V} \mathbf{Q} \end{equation} where $\mathbf{P} \equiv [ P_0,P_1, \dots, P_{N-1}]^T$ and $\mathbf{Q} \equiv [ Q_0, Q_1, \dots, Q_{N-1}]^T$ . The eigenvalues of the matrix $\mathbf{V}$ and the components of the corresponding normalized eigenvectors $\mathbf{e}^n$ are, respectively, $\omega_n^2 = m^2 + 4\sin^2 \frac{n\pi}{N}$, and $\mathbf{e}_{k}^{n} = \frac{1}{\sqrt{N}} e^{2\pi i kn/N},\ k=0,\dots, N-1$. Notice that the massless case is special because it contains a zero mode (for $m=0$, $\omega_0 =0$), so the matrix $\mathbf{V}$ is not invertible. To avoid the problems that arise, we can shift the mass by a small amount $\sim 1/N$, which vanishes in the continuum limit ($N\to\infty$).
We also wish to add a quartic interaction, $ H_{int} = \frac{\lambda}{4!} \int_0^L dx \phi^4 \to \frac{\lambda}{4!} \sum_n Q_n^4 $ which necessitates the addition of a mass counter term $ H_{c.t.} = \frac{\delta_m}{2} \int_0^L dx \phi^2 \to \frac{\delta_m}{2} \sum_n Q_n^2 $ due to renormalization, as explained in the supplementary material. We find that for weak coupling, the physically interesting case is stable for $\lambda>0$.
To diagonalize the Hamiltonian, we introduce new creation and annihilation operators, $a_k^\dagger$ and $a_k$, respectively, defined by ${a}_k = \sqrt{\frac{\omega_k}{2}} (\mathbf{e}^\dagger \mathbf{Q})_k + \frac{i}{\sqrt{2\omega_k}} (\mathbf{e}^\dagger \mathbf{P})_k$ where $\mathbf{e}$ is the matrix of the eigenvectors. Notice that $ \mathbf{e}$ is unitary, $\mathbf{e}^\dagger \mathbf{e} = \mathbf{I}$. These operators obey standard commutation relations, $[ a_k,a_l^\dagger ] = \delta_{kl}$ and the free Hamiltonian reads \begin{equation} H_0 = \sum_{k=0}^{N-1} \omega_k \left( a_k^\dagger a_k + \frac{1}{2} \right) .\end{equation} In this form, it is straightforward to construct the states in the Hilbert space.
\section{Initial cluster state preparation}\label{prep}
For the initial cluster state, in Refs.~\cite{Jordan2012,Brennen2014} the excited state was created \textit{after} creating the ground state. This is difficult because it involves manipulating a large number of qubits. In our approach, we create a single photon state in a single mode \textit{before} creating the cluster state. This is more accessible, as it involves creating the state $\ket{1}$ for a single mode. It can be done in a variety of ways, via a heralded single photon source, for instance. At the end of the computation, the field modes are all measured and the distribution of single photons across them determines the result.
To begin with, we build the system with $N$ oscillators representing the variables $(Q_n, P_n)$, $n=0,1,2,\dots$. The $n$th oscillator has a Hilbert space constructed by successive application of the creation operator $A_n^\dagger$ on the vacuum $|0\rangle_n$, which is annihilated by $A_n$. Here $|0\rangle_n$ is shorthand for a product state of vacuum fields
\begin{equation}\label{eq16} |0\rangle = |0\rangle_0 \otimes |0\rangle_1 \otimes \cdots \otimes |0\rangle_{N-1} \ , \end{equation} with $ A_n |0\rangle = 0$. For a scattering process, we are given an initial state typically consisting of a fixed number of particles, usually two, which undergoes evolution and then a measurement is performed (detection of particles) on the final state. Both initial and final states asymptote to eigenstates of the free Hamiltonian $H_0$. Thus quantum computation starts with preparation of an eigenstate of $H_0$.
\label{sec:groundstate}
First, we consider the ground state of $H_0$. It is the cluster state $|\Omega\rangle$ annihilated by all $a_k$, i.e., $ a_k |\Omega\rangle =0$ for $k=0,1,\dots, N-1 $. It can be constructed from the vacuum state \eqref{eq16} by acting with the Gaussian unitary $U^\dagger$, where $a_n = U^\dagger A_n U $. Noticing the relationship between the operators $a_k$ and $A_k$ we can use the Bloch-Messiah reduction \cite{Braunstein2005decomp} to determine $U=VSW^\dagger$ as a decomposition involving a multiport interferometer ($V$) followed by single mode squeezing ($S$) followed by a final multiport interferometer ($W$). These unitary operators can be realized with $O(N^2)$ gates \cite{Reck1994}. This is in contrast to the qubit version \cite{Jordan2012} where they require $O(N^{2.376})$ gates.
To implement $U$ we first perform the rotation \begin{eqnarray}\label{eq23} A_0 &\to & A_0' = \sum_{k=0}^{N-1} A_k \nonumber\\ A_n &\to& A_n' = \sum_{k=0}^{N-1} \cos \frac{2\pi nk}{N} A_k \nonumber\\ A_{N-n} &\to& A_{N-n}' = \sum_{k=0}^{N-1} \sin \frac{2\pi nk}{N} A_k \end{eqnarray} where $1\le n\le N/2$, which can be expressed in terms of rotations each involving only a couple of oscillators. Notice that if $N$ is even, $A_{N/2}$ does not have a partner; we obtain $A_{N/2} \to \sum_k (-)^k A_k$. Next, we squeeze each mode as $A_n' \to A_n'' = \cosh r_n A_n' + \sinh r_n {A_n'}^\dagger$ where $e^{2r_n} = \omega_n$ for $n\le N/2$, and $e^{-2r_n} = \omega_n$, for $n>N/2$. Finally, we untangle the pairs by rotating them, $A''_k\to a_k$ where $a_0=A_0''$, $a_n =( A_n'' + iA_{N-n}'' )/\sqrt 2$, and $a_{N-n} = ( iA_n'' + A_{N-n}'' )/\sqrt 2$.
Excited states can be constructed with the same number of gates, e.g., the single-particle state $ |k\rangle \equiv a_k^\dagger |\Omega\rangle$ can be constructed by acting upon the vacuum with $A_k^\dagger$. This turns the initial state of the $k$th mode into a one-photon state, $A_k^\dagger |0\rangle_k$, which can be accomplished in a variety of ways; see supplementary material. Having engineered $ A_k^\dagger |0\rangle_k$, we then apply the Gaussian unitary $U^\dagger$, to obtain the one-particle state
\begin{equation} {a_k}^\dagger |\Omega\rangle = U^\dagger A_k^\dagger |0\rangle \end{equation} Extending the above to the engineering of multi-particle states, $|k_1,k_2,\dots \rangle \propto {a_{k_1}}^\dagger {a_{k_2}}^\dagger \cdots |0\rangle$, is straightforward.
\section{Quantum Computation}\label{compute}
We wish to calculate a general scattering amplitude, which can be written as
\begin{equation} \mathcal{A} = \langle out | T \exp\left\{ i \int_{-T}^T dt (H_{int} (t) + H_{c.t.} (t) ) \right\} | in \rangle \end{equation} in the limit $T\to\infty$, where time evolution is defined with respect to the non-interacting Hamiltonian.
We start by preparing the initial state $|in\rangle$ as in the previous section and define initial time as $t=-T$. Then we act successively with evolution operators of the form \begin{equation}\label{eq33} U(t) = \exp\left\{ i \delta t (H_{int} (t) + H_{c.t.} (t) ) \right\} \end{equation} Time dependence is obtained via the free Hamiltonian, \begin{equation} Q_i (t) = e^{it H_0} Q_i (0) e^{-it H_0} \end{equation}
Therefore, the evolution \eqref{eq33} can be implemented as \begin{equation}\label{eq33a} U(t) = e^{it H_0}e^{ i \delta t (H_{int} + H_{c.t.} ) } e^{-it H_0} \end{equation} We deduce
\begin{equation} \mathcal{A} = \langle out | \left[ e^{i \delta t H_0} e^{ i \delta t (H_{int} + H_{c.t.} ) } \right]^N |in\rangle \end{equation} where we divided the time interval into $N = \frac{2T}{\delta t}$ segments.
The coupling constants in \eqref{eq33} are turned on and off adiabatically. This is achieved by splitting the time interval $[-T,T]$ into three segments, $[-T, -T_1]$, $[-T_1, T_1]$, and $[T_1,T]$. For $t\in [-T,-T_1]$, we turn the coupling constants on by replacing $\lambda \to \lambda(t)$, $\delta m \to \delta m (t)$, so that $\lambda (-T) = \delta m (-T) =0$, and $\lambda (-T_1) = \lambda$, $\delta m (-T_1) = \delta m$. Then for $t\in [-T_1,T_1]$ the coupling constants are held fixed. Finally, for $t\in [T_1,T]$, they are turned off adiabatically by reversing the process in the first time interval. In the case of small $\lambda$, the time dependence of the coupling constants can be chosen efficiently by making use of perturbative renormalization. Eqs.\ \eqref{eqA13} and \eqref{eqA14} inform the choice $\lambda(t) = \frac{T+t}{T-T_1} \lambda$, $\delta m (t) = \frac{\lambda(t)}{8\pi} \log \frac{64}{m^2}$, for $-T \le t \le -T_1$.
The unitary operators $e^{i\delta t H_0}$ and $e^{i\delta t H_{c.t.}}$ are Gaussian and can be implemented with second order nonlinear optical interactions and linear optics beam splitter networks. The interaction is implemented through a \emph{quartic} phase gate for each mode, \begin{equation}\label{eq41} e^{i\delta t H_{int} } = \prod_n e^{i\gamma Q_n^4 } \ \ , \ \ \ \ \gamma = \delta t \frac{\lambda}{4!} \end{equation} The quartic phase gate may be implemented in a similar manner to the cubic phase gate previously proposed~\cite{Marshall2014}.
\begin{figure*}
\caption{(Color Online) Sketch of an experimental setup for electromagnetic field modes used as qudits in a QFT calculation involving four field modes. The modes are encoded into electric field modes (colored red, blue, yellow, green), which are then prepared via beam splitters, swap gates, and squeezers for the compute stage. The compute stage consists of an interferometer, a quartic phase gate (black box, see Ref.~\cite{Marshall2014}), and free propagation. An uncompute stage, which is the inverse of the preparation stage, and a detection stage in the Fock basis, yield the scattering amplitudes into the four QFT field modes.}
\label{setup}
\end{figure*}
After evolution, we must project onto the state $|out\rangle$. This is similar to the state $|in\rangle$, and its construction depends on the number of desired particles. The latter are excitations created with $a_n^\dagger$, so in general,
\begin{equation}\label{eq95} |out\rangle = a_{n_1}^\dagger a_{n_2}^\dagger \cdots |\Omega\rangle = U^\dagger A_{n_1}^\dagger A_{n_2}^\dagger \cdots |0\rangle\end{equation} It follows that the next step is to \emph{uncompute} by applying the Gaussian unitary $U$ (which is the inverse operation to the preparation of the initial state), and then measure the number of photons in each mode. The final uncompute step projects the set of output modes onto the Fock basis. Thus, the scattering amplitude calculation is a mapping from one set of field modes on the input to a separate set of field modes on the output, as expected. That is, for each click on the photodetector for mode $n$, there is an operator $a_n^\dagger$ present in the final state \eqref{eq95}. If the QFT calculation involved an initial input state with two excitations spread across 100 field modes, say, then the entire calculation would involve two photons, for instance. We note that the calculation has made use of a quartic phase gate up to this point, and thus technically speaking a non-Gaussian operation would not be necessary during this measurement step in order to achieve an exponential speedup over the classical QFT algorithm. However, in order to achieve high accuracy in the final result, photon number resolving detectors with high efficiency~\cite{Nam2008} would be desirable for the measurment phase.
\section{Experimental implementation}\label{experiment}
An example of the experimental implementation is given in Fig.~\ref{setup}. For brevity the setup for calculating four space time points is given. For the electromagnetic field, the initial unitary rotation involves weighted beam splitters with the appropriate splitting to achieve the desired sums over the field operators (see appendix A.2). A swap gate is involved in the input state preparation stage. We note that a swap gate contains essentially the CV version of the CNOT operator along with parity operators~\cite{Wang2001}, but in some cases the gate can be simplified to a beam splitter interaction~\cite{Braunstein2005} such as for the electromagnetic field. Here we use a mode label swap operator, which is possible in systems with movable qubits, such as CV optical fields. Next, $H_{c.t.}$ is quadratic in position quadrature operators, which can be implemented with a series of phase shifts~\cite{Braunstein2005}. The non-Gaussian piece of the computation is then the quartic phase gate contained in $H_{int}$, which can be implemented via repeated application of the photon number-dependent phase gate~\cite{Marshall2014}. Lastly, the free propagation $H_0$ can be implemented by a calibrated free propagation before the uncompute stage. We note that the QFT field modes are encoded into the qudits which are themselves electromagnetic field modes, meaning that the free propagation contained in $H_0$ is not arbitrary. It must conform to the calculated QFT free propagation distance, and phase stability must be maintained throughout.
\section{Conclusion}\label{conclusion}
In conclusion, we developed a new algorithm for a continuous-variable quantum computer which gave an exponential speedup over the best known classical algorithms. This algorithm was the calculation of the scattering amplitudes in scalar bosonic quantum field theory, and as previously mentioned, arguably a natural choice for a continuous variable quantum computer to solve. At weak coupling, analytic calculations are possible, however, at strong coupling no such calculations are generally available, and one has to rely on numerical techniques. A widely used framework is lattice field theory which is based on the discretization of space into a finite set of points. The complexity of classical computations on a lattice increases exponentially with the number of lattice sites \cite{Creutz1985}.
Quantum computations offer a distinct advantage (first shown in Ref.~\cite{Jordan2014a} for qubits, and here for qumodes), since complexity only grows polynomially. Using continuous variables we also see an advantage over the original qubit proposal; specifically, in the preparation of the initial cluster state. There they required $O(N^{2.376})$ gates to synthesize the ground state. However, in our scheme, we required slightly less, $O(N^2)$ gates. Finally, we also gave an example of an experimental implementation on a continuous-variable cluster state quantum computer that calculated four space time points. We noted that such a scheme is feasible with current linear optical technology and consisted of a set of Gaussian operations along with the non-Gaussian quartic phase gate.
\acknowledgments We are grateful to Peter Rohde for valuable feedback. R. C. P. performed portions of this work at Oak Ridge National Laboratory, operated by UT-Battelle for the US Department of Energy under Contract No. DE-AC05-00OR22725. Work performed by the US government is not subject to copyright restrictions.
\appendix
\section{Renormalization}
Define the Green function $\mathbf{G} (t_1,t_2)$ as
\begin{equation} G_{ij}(t_1,t_2) = \langle 0 | \mathcal T(Q_i (t_1) Q_j(t_2)) |0\rangle, \end{equation} where $\mathcal T$ denotes the time-ordering operator. It obeys \begin{equation}\left[ \partial_{t_1}^2 + \mathbf{V} \right] \mathbf{G} (t_1,t_2) = -i \mathbf{I} \delta (t_1-t_2).\end{equation} Using the Fourier transform, \begin{equation} \mathbf{G} (t_1,t_2) = \int \frac{d\omega}{2\pi} e^{i\omega (t_1-t_2)} \tilde{\mathbf{G}} (\omega)\end{equation} we obtain \begin{equation} \tilde{\mathbf{G}} (\omega) = i\left[ -\omega^2 \mathbf{I} + \mathbf{V} \right]^{-1} = \sum_n \frac{-i}{\omega^2 - \omega_n^2} \mathbf{e}_n \mathbf{e}_n^\dagger, \end{equation} exhibiting poles at $\omega^2 = \omega_{n}^2$.
When we switch on the interaction term, \begin{equation} H_{int} = \frac{\lambda}{4!} \int_0^L dx \phi^4 \to \frac{\lambda}{4!} \sum_n Q_n^4,\end{equation} we have that at $\mathcal{O} (\lambda)$ the Green function is corrected by
\begin{equation}\delta G_{ij}(t_1,t_2) = \langle 0 | \mathcal T\left[ Q_i (t_1) Q_j(t_2) \int dt H_{int} (t) \right] |0\rangle.\end{equation} For the Fourier transform, we obtain \begin{equation}\delta \tilde{\mathbf{G}} (\omega) = \lambda [\tilde{\mathbf{G}} (\omega)]^2 \int \frac{d\omega'}{2\pi} \mathrm{Tr}\, \tilde{\mathbf{G}} (\omega')\end{equation} which leads to a shift of the poles, \begin{equation} \tilde{\mathbf{G}} (\omega) + \delta \tilde{\mathbf{G}} (\omega) = \sum_n \frac{-i}{\omega^2 - \omega_n^2 - \Sigma} \mathbf{e}_n \mathbf{e}_n^\dagger + \mathcal{O} (\lambda^2), \end{equation} where \begin{equation} \Sigma = \frac{\lambda}{2N} \int \frac{d\omega'}{2\pi} \sum_n \frac{-i}{{\omega'}^2 -\omega_i^2} = \frac{\lambda}{4N} \sum_n \frac{1}{\omega_n} \end{equation} The shift can be corrected by the addition of the counter term \begin{equation} H_{c.t.} = \frac{\delta_m}{2} \int_0^L dx \phi^2 \to \frac{\delta_m}{2} \sum_n Q_n^2,\end{equation} with $\delta_m = -\Sigma + \mathcal{O} (\lambda^2)$, i.e., the mass parameter in the Hamiltonian is not physical, but bare, \begin{equation} m_0^2 = m^2 + \delta_m = m^2 - \frac{\lambda}{4N} \sum_n \frac{1}{\omega_n} + \mathcal{O} (\lambda^2). \end{equation} For large $N$, the sum can be approximated by an integral, \begin{equation} \Sigma = \frac{\lambda}{4} \int_0^1 \frac{dk}{\sqrt{m^2 + 4\sin^2 k\pi}} \end{equation} which has a logarithmic divergence at small $m^2$ (i.e., length scale $1/m$ large in units of lattice spacing, which is the physically interesting limit). We easily obtain \begin{equation} \label{eqA13}\Sigma = \frac{\lambda}{8\pi} \log \frac{64}{m^2} + \mathcal{O} (m^2) \end{equation} The bare mass is \begin{equation} \label{eqA14} m_0^2 = m^2 -\Sigma + \mathcal{O} (\lambda^2) = m^2 - \frac{\lambda}{8\pi} \log \frac{64}{m^2} + \mathcal{O} (\lambda^2, m^2) \end{equation} Notice that for weak coupling (small $\lambda$), the physically interesting case has $m_0^2 < 0$ (a stable system, as long as $\lambda >0$).
\subsection{Ground State Construction} To find the required transformation $U$, we work as follows. Notice that for $n=0$, \begin{equation} a_0 = \frac{1}{2\sqrt{N}} \sum_{k=0}^{N-1}\left[ \left( \sqrt{m} + \frac{1}{\sqrt{m}} \right) A_k + \left( \sqrt{m} - \frac{1}{\sqrt{m}} \right) A_k^\dagger \right] \end{equation} where we used $\omega_0 =m$. For $n\ne 0$, we consider pairs $(a_n , a_{N-n})$. We have \begin{eqnarray} a_n + a_{N-n} &=& \frac{1}{2\sqrt{N}} \sum_{k=0}^{N-1} \cos \frac{2\pi nk}{N} \left[ \left( \sqrt{\omega_n} + \frac{1}{\sqrt{\omega_n}} \right) A_k \right. \nonumber\\ && \left. + \left( \sqrt{\omega_n} - \frac{1}{\sqrt{\omega_n}} \right) A_k^\dagger \right] \nonumber\\ a_n - a_{N-n} &=& \frac{i}{2\sqrt{N}} \sum_{k=0}^{N-1} \sin \frac{2\pi nk}{N} \left[ \left( \sqrt{\omega_n} + \frac{1}{\sqrt{\omega_n}} \right) A_k \right. \nonumber\\ && \left. - \left( \sqrt{\omega_n} - \frac{1}{\sqrt{\omega_n}} \right) A_k^\dagger \right] \end{eqnarray} where we used $\omega_n = \omega_{N-n}$.
The above expressions suggest that we transform $A_n$ into $a_n$ in three steps as detailed in Sec. \ref{sec:groundstate}
\subsection{Example: $N=4$}
To illustrate the above algorithm, we consider the case in which space has been discretized to four points. The rotation ($\mathbf{A}' = \mathbf{O} \mathbf{A}$) is described by the orthogonal matrix \begin{equation} \mathbf{O} = \frac{1}{2} \left[ \begin{array}{cccc} 1 & 1 & 1 & 1 \\ \sqrt{2} & 0 & -\sqrt{2} & 0 \\ 1 & -1 & 1 & -1 \\ 0 & \sqrt{2} & 0 & -\sqrt{2} \end{array} \right] \end{equation} We have \begin{equation} \mathbf{O} = R_{02} \left( \frac{\pi}{4} \right) S_{01} R_{13} \left( \frac{\pi}{4} \right) R_{02} \left( \frac{\pi}{4} \right) \end{equation} where $R_{ij} (\theta)$ is a rotation in the $ij$-plane of angle $\theta$ and $S_{ij}$ is the swap $i\leftrightarrow j$. Therefore the rotation $\mathbf{O}$ can be implemented with four two-mode unitaries.
Next, we squeeze each mode as $ A_n' \to A_n'' = \cosh r_n A_n' + \sinh r_n {A_n'}^\dagger$, where $e^{2r_0} = \omega_0$, $e^{2r_1} = \omega_1$, $e^{2r_2} = \omega_2$, and $e^{-2r_3} = \omega_3$. Notice that $r_3 = - r_1$, because $\omega_3 = \omega_1$.
Finally, we perform the rotation, $A_1'' \to \frac{1}{\sqrt{2}} ( A_1'' +iA_3'')$, $A_3'' \to \frac{1}{\sqrt{2}} (iA_1'' + A_3'')$, to arrive at the desired modes, \begin{eqnarray} a_0 &=& \frac{1}{2} \sum_n \left[ \cosh r_0 A_n + \sinh r_0 \sum_n A_n^\dagger \right] \nonumber\\
a_1 &=& \frac{1}{2} \sum_n i^n \left[ \cosh r_1 A_n + \sinh r_1 \sum_n A_n^\dagger \right] \nonumber\\
a_2 &=& \frac{1}{2} \sum_n (-1)^n \left[ \cosh r_2 A_n + \sinh r_2 \sum_n A_n^\dagger \right] \nonumber\\
a_3 &=& \frac{1}{2} \sum_n (-i)^n \left[ \cosh r_3 A_n + \sinh r_3 \sum_n A_n^\dagger \right] \end{eqnarray}
Each of the above steps is implemented with a Gaussian unitary involving at most two modes.
\section{Excited States} To generate the required one-photon state, two methods can be used. One can first squeeze the vacuum of the $k$th mode with an optical parametric amplifier to
\begin{equation} S_k(s) |0\rangle_k \ \ , \ \ \ \ S_k(s) = e^{\frac{s}{2} ({A_k^\dagger}^2 - A_k^2 )} \end{equation} Then pass the squeezed state through a (highly transmitting) beam splitter of transmittance $T$, and place a photodetector on the auxiliary output port. A click of the detector heralds a successful photon subtraction, which is described by the non-unitary operator \begin{equation} \sqrt{1-T}\, T^{A_k^\dagger A_k/2} A_k \end{equation} The transmittance has to be high so that the probability of detecting two or more photons is negligible. If no photon is detected, the process is repeated until a photon is detected. Finally, apply anti-squeezing $S_k^\dagger (s')$.
We obtain the state (unnormalized)
\begin{equation}\label{eq72} S_k^\dagger (s') T^{A_k^\dagger A_k/2} A_k S_k(s) |0\rangle_k \end{equation} If the squeezing parameters are chosen so that \begin{equation} T = \frac{\tanh s'}{\tanh s} \end{equation} then it is straightforward to show that \eqref{eq72} is the desired state,
\begin{equation} S_k^\dagger (s') T^{A_k^\dagger A_k/2} A_k S_k(s) |0\rangle_k \propto A_k^\dagger |0\rangle_k. \end{equation}
Optionally, one may also use a heralded single photon source. Such a source would consist of a parametric downconverter with a high efficiency heralding detector. To obtain exactly one photon when operating the source with high brightness (but on average less than one pair per pulse), the heralding detector would consist of a high efficiency photon number resolving detector, such as a transition edge sensor.
\section{Generalization to Arbitrary Dimensions}
Generalization to arbitrary spatial dimension $d$ is straightforward. The free-scalar Hamiltonian in the continuum reads \begin{equation}\label{eq61} H_0 = \frac{1}{2} \int d^dx \left[ \pi^2 + (\mathbf{\nabla} \phi )^2 + m^2 \phi^2 \right] \end{equation} where $\mathbf{x} \in [0,L]^d$, with the fields obeying standard commutation relations, \begin{equation} [ \phi ( \mathbf{x} ) \ , \ \pi(\mathbf{x}') ] = i \delta^d (\mathbf{x} - \mathbf{x}' ) \end{equation} Each coordinate $x_i$ ($i=1,\dots, d$) is discretized as before, $x_i = n_i a$, $n_i = 0,1,\dots, N-1$, and we define $Q_{\mathbf{n}} \equiv \phi (\mathbf{x})$, $P_{\mathbf{n}} \equiv \pi (\mathbf{x})$, $A_{\mathbf{n}} = \frac{1}{\sqrt{2}} (Q_{\mathbf{n}} + iP_{\mathbf{n}})$, where $\mathbf{n} \in \mathbb{Z}_N^d$.
The Hamiltonian \eqref{eq61} can then be rendered in the form \eqref{eq8}, where $\mathbf{V}$ has eigenvalues and corresponding normalized eigenvectors, \begin{eqnarray} \omega_{\mathbf{k}}^2 &=& m^2 + 4 \sum_{i=1}^d \sin^2 \frac{k_i}{2} \ , \nonumber\\ \mathbf{e}_{\mathbf{k}}^{\mathbf{n}} &=& \frac{1}{N^{d/2}} e^{i\mathbf{k} \cdot \mathbf{n}} \end{eqnarray} where $\mathbf{k} \in \frac{2\pi}{N} \mathbb{Z}_N^d$ (the dual lattice). The eigenvectors form a unitary matrix.
The discretized Hamiltonian is diagonalized as \begin{equation} H_0 = \sum_{\mathbf{k} \in \Gamma} \omega_{\mathbf{k}} \left( a_{\mathbf{k}}^\dagger a_{\mathbf{k}} + \frac{1}{2} \right) \end{equation} where $a_{\mathbf{k}}$ is the annihilation operator defined in Sec.~\ref{sec2} (extended to $d$ dimensions in an obvious way).
Introducing an interaction term, $H_{int} = \frac{\lambda}{4!} \sum_{\mathbf{n}} Q_{\mathbf{n}}^4$, and the attendant counter term, $H_{c.t.} = \frac{\delta_m}{2} \sum_{\mathbf{n}} Q_{\mathbf{n}}^2$, and working as in the one-dimensional case, we obtain a shift in the poles of the Green function, \begin{equation} \Sigma = \frac{\lambda}{4} \sum_{\mathbf{k} \in \Gamma} \frac{1}{\omega_{\mathbf{k}}} + \mathcal{O} (\lambda^2) \end{equation} which is related to the counter-term parameter $\delta_m$ via $\delta_m = -\Sigma + \mathcal{O} (\lambda^2)$. For large $N$, the sum is approximated by an integral over the hypercube $[0,2\pi]^d$. For $d=1$, it reduces to the previous result, whereas for $d>1$, we obtain at lowest order in $m$ and $\lambda$, \begin{equation} \Sigma = C_d \lambda + \dots \end{equation} Numerically, $C_2 \approx 0.16$, and $C_3 \approx 0.11$.
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Some inequalities for interval-valued functions on time scales}
\author{Dafang Zhao$^{1,2}$ \and Guoju Ye$^{1}$ \and Wei Liu$^{1}$ \and Delfim F. M. Torres$^{3}$}
\authorrunning{D. F. Zhao et al.}
\institute{Dafang Zhao \at \email{[email protected]} \and Guoju Ye \at \email{[email protected]} \and Wei Liu \at \email{[email protected]} \and Delfim F. M. Torres \at \email{[email protected]}\\ \and $^{1}$College of Science, Hohai University, Nanjing, Jiangsu 210098, P. R. China.\\ $^{2}$School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China.\\ $^{3}$Center for Research and Development in Mathematics and Applications (CIDMA),\\ Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal.\\}
\date{Submitted: 20-Dec-2017 / Revised: 31-Aug-2018 / Accepted: 11-Sept-2018}
\maketitle
\begin{abstract} We introduce the interval Darboux delta integral (shortly, the $ID$ $\Delta$-integral) and the interval Riemann delta integral (shortly, the $IR$ $\Delta$-integral) for interval-valued functions on time scales. Fundamental properties of $ID$ and $IR$ $\Delta$-integrals and examples are given. Finally, we prove Jensen's, H\"{o}lder's and Minkowski's inequalities for the $IR$ $\Delta$-integral. Also, some examples are given to illustrate our theorems.
\keywords{interval-valued functions \and time scales \and Jensen's inequality \and H\"{o}lder's inequality \and Minkowski's inequality} \end{abstract}
\section{Introduction} \label{intro}
Interval analysis was initiated by Moore for providing reliable computations \cite{M66}. Since then, interval analysis and interval-valued functions have been extensively studied both in mathematics and its applications: see, e.g., \cite{B13,C13,C15,CB,CC,G17,J01,L15,M79,M09,O15,S09,WG00,Z17}. Rece-\\ntly, several classical integral inequalities have been extended to the context of interval-valued functions by Chalco-Cano et al. \cite{C12,CL15}, Costa \cite{C17}, Costa and Rom\'{a}n-Flores \cite{CF}, Flores-Franuli\v{c} et al. \cite{FC}, Rom\'{a}n-Flores et al. \cite{R16,R13}.
Motivated by \cite{C81,C17,R16}, we introduce the $ID$ and $IR$ $\Delta$-integrals, and present some integral inequalities on time scales. A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of the real numbers $\mathbb{R}$ with the subspace topology inherited from the standard topology of $\mathbb{R}$. The theory of time scales was born in 1988 with the Ph.D. thesis of Hilger \cite{H4}. The aim is to unify various definitions and results from the theories of discrete and continuous dynamical systems, and to extend them to more general classes of dynamical systems. It has undergone tremendous expansion and development on various aspects by several authors over the past three decades: see, e.g., \cite{BCT1,BCT2,BP1,BP2,FB15,FT16,V16,W05,YZ,YZT,ZL16}.
In 2013, Lupulescu introduced the Riemann $\Delta$-inte-\\gral for interval-valued functions on time scales and presented some of its basic properties \cite{L13}. Nonetheless, to our best knowledge, there is no systematic theory of integration for interval-valued functions on time scales. In this work, in order to complete the theory of $IR$ $\Delta$-integration and improve recent results given in \cite{C81,C17,R16}, we introduce the $ID$ $\Delta$-integral and the $IR$ $\Delta$-integral on time scales. We show that the $ID$ $\Delta$-integral ia a generalization of the $IR$ $\Delta$-integral. Also, some basic properties for the $ID$ and $IR$ $\Delta$-integrals, and some examples, are given. Finally, we present Jensen's inequality, H\"{o}lder's inequality and Minkowski's inequality for the $IR$ $\Delta$-integral. Some celebrated inequalities are derived as consequences of our results.
The paper is organized as follows. After a Section~\ref{sec:2} of preliminaries, in Section~\ref{sec:3} the $ID$ and $IR$ $\Delta$-integrals for interval-valued functions are introduced. Moreover, some basic properties and examples are given. In Section~\ref{sec:4}, we prove Jensen's, H\"{o}lder's and Minkowski's inequalities for the general $IR$ $\Delta$-integral. We end with Section~\ref{sec:5} of conclusions.
\section{Preliminaries} \label{sec:2}
In this section, we recall some basic definitions, notations, properties and results on interval analysis and the time scale calculus, which are used throughout the paper. A real interval $[u]$ is the bounded, closed subset of $\mathbb{R}$ defined by $$
[u]=[\underline{u},\overline{u}]=\{x\in\mathbb{R}|\ \underline{u}\leq x\leq\overline{u}\}, $$ where $\underline{u}, \overline{u}\in \mathbb{R}$ and $\underline{u}\leq\overline{u}$. The numbers $\underline{u}$ and $\overline{u}$ are called the left and the right endpoints of $[\underline{u},\overline{u}]$, respectively. When $\underline{u}$ and $\overline{u}$ are equal, the interval $[u]$ is said to be degenerate. In this paper, the term interval will mean a nonempty interval. We call $[u]$ positive if $\underline{u}>0$ or negative if $\overline{u}<0$. The partial order ``$\leq$" is defined by $$ [\underline{u},\overline{u}]\leq[\underline{v},\overline{v}]\Longleftrightarrow \underline{u}\leq\underline{v},\overline{u}\leq\overline{v}. $$ The inclusion ``$\subseteq$" is defined by $$ [\underline{u},\overline{u}]\subseteq[\underline{v},\overline{v}]\Longleftrightarrow \underline{v}\leq\underline{u},\overline{u}\leq\overline{v}. $$ For an arbitrary real number $\lambda$ and $[u]$, the interval $\lambda [u]$ is given by \begin{equation*} \lambda[\underline{u},\overline{u}]= \begin{cases} [\lambda\underline{u},\lambda\overline{u}]& \text{if $\lambda>0$},\\ \{0\}& \text{if $\lambda=0$},\\ [\lambda\overline{u},\lambda\underline{u}]& \text{if $\lambda<0$}. \end{cases} \end{equation*} For $[u]=[\underline{u},\overline{u}]$ and $[v]=[\underline{v},\overline{v}]$, the four arithmetic operators (+,-,$\cdot$,/) are defined by $$ [u]+[v]=[\underline{u}+\underline{v},\overline{u}+\overline{v}], $$ $$ [u]-[v]=[\underline{u}-\overline{v},\overline{u}-\underline{v}], $$ $$ [u]\cdot[v]=\big[\min\{\underline{u}\underline{v},\underline{u}\overline{v}, \overline{u}\underline{v},\overline{u}\overline{v}\}, \max\{\underline{u}\underline{v},\underline{u}\overline{v}, \overline{u}\underline{v},\overline{u}\overline{v}\}\big], $$ \begin{equation*} \begin{split} [u]/[v]=\big[&\min\{\underline{u}/\underline{v},\underline{u}/\overline{v}, \overline{u}/\underline{v},\overline{u}/\overline{v}\},\\ &\max\{\underline{u}/\underline{v},\underline{u}/\overline{v}, \overline{u}/\underline{v},\overline{u}/\overline{v}\}\big], {\rm where}\ \ 0\notin[\underline{v},\overline{v}]. \end{split} \end{equation*} We denote by $\mathbb{R}_{\mathcal{I}}$ the set of all intervals of $\mathbb{R}$, and by $\mathbb{R}^{+}_{\mathcal{I}}$ and $\mathbb{R}^{-}_{\mathcal{I}}$ the set of all positive intervals and negative intervals of $\mathbb{R}$, respectively. The Hausdorff--Pompeiu distance between intervals $[\underline{u},\overline{u}]$ and $[\underline{v},\overline{v}]$ is defined by $$ d\big([\underline{u},\overline{u}],[\underline{v},\overline{v}]\big)
=\max\Big\{|\underline{u}-\underline{v}|,|\overline{u}-\overline{v}|\Big\}. $$ It is well known that $(\mathbb{R}_{\mathcal{I}}, d)$ is a complete metric space.
Let $\mathbb{T}$ be a time scale. We define the half-open interval $[a,b)_{\mathbb{T}}$ by $$ [a,b)_{\mathbb{T}} =\left\{t\in \mathbb{T}: a \leq t < b\right\}. $$ The open and closed intervals are defined similarly. For $t\in \mathbb{T}$, we denote by $\sigma$ the forward jump operator, i.e., $\sigma(t):=\inf\{s>t: s\in \mathbb{T}\}$, and by $\rho$ the backward jump operator, i.e., $\rho(t):=\sup\{s<t: s\in \mathbb{T}\}$. Here, we put $\sigma(\sup\mathbb{T})=\sup\mathbb{T}$ and $\rho(\inf\mathbb{T})=\inf\mathbb{T}$, where $\sup\mathbb{T}$ and $\inf\mathbb{T}$ are finite. In this situation, $\mathbb{T}^{\kappa}:=\mathbb{T}\backslash \{\sup\mathbb{T}\}$ and $\mathbb{T}_{\kappa}:=\mathbb{T}\backslash\{\inf\mathbb{T}\}$, otherwise, $\mathbb{T}^{\kappa}:=\mathbb{T}$ and $\mathbb{T}_{\kappa}:=\mathbb{T}$. If $\sigma(t)>t$, then we say that $t$ is right-scattered, while if $\rho(t)<t$, then we say that $t$ is left-scattered. If $\sigma(t)=t$ and $t< \sup\mathbb{T}$, then $t$ is called right-dense, and if $\rho(t)=t$ and $t> \inf\mathbb{T}$, then $t$ is left-dense. The graininess functions $\mu$ and $\eta$ are defined by $\mu(t):=\sigma(t)-t$ and $\eta(t):=t-\rho(t)$, respectively.
A function $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}$ is called right-dense continuous ($rd$-continuous) if it is right continuous at each right-dense point and there exists a finite left limit at all left-dense points. The set of $rd$-continuous function $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}$ is denoted by $C_{rd}([a,b]_{\mathbb{T}},\mathbb{R})$.
A function $f$ is said to be an interval function of $t$ on $[a,b]_{\mathbb{T}}$ if it assigns a nonempty interval $$ f(t)=\big[\underline{f}(t), \overline{f}(t)\big] $$ to each $t\in[a,b]_{\mathbb{T}}$. We say that $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ is continuous at $t_{0}\in[a,b]_{\mathbb{T}}$ if for each $\epsilon>0$ there exists a $\delta>0$ such that $$ d(f(t),f(t_{0}))<\epsilon $$
whenever $|t-t_{0}|<\delta$. The set of continuous function $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ is denoted by $C([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}})$. It is clear that $f$ is continuous at $t_{0}$ if and only if $\underline{f}$ and $\overline{f}$ are continuous at $t_{0}$.
A division of $[a,b]_{\mathbb{T}}$ is any finite ordered subset $D$ having the form $$ \mathcal{D}=\{a=t_{0}<t_{1}<\cdots <t_{n}=b\}. $$ We denote the set of all divisions of $[a,b]_{\mathbb{T}}$ by $\mathcal{D}=\mathcal{D}([a,b]_{\mathbb{T}})$.
\begin{lemma}[Bohner and Peterson \cite{BP2}] \label{lem1} For every $\delta>0$ there exists some division $D\in\mathcal{D}([a,b]_{\mathbb{T}})$ given by $$ a=t_{0}<t_{1}<\cdots <t_{n-1}<t_{n}=b $$ such that for each $i\in\{1,2,\ldots,n\}$ either $t_{i}-t_{i-1}\leq\delta$ or $$ t_{i}-t_{i-1}>\delta \ \ {\rm and}\ \ \rho(t_{i})=t_{i-1}. $$ \end{lemma}
Let $\mathcal{D}(\delta,[a,b]_{\mathbb{T}})$ be the set of all $D\in\mathcal{D}([a,b]_{\mathbb{T}})$ that possess the property indicated in Lemma \ref{lem1}. In each interval $[t_{i-1},t_{i})_{\mathbb{T}}$, where $1\leq i\leq n$, choose an arbitrary point $\xi_{i}$ and form the sum $$ S(f,D,\delta)=\displaystyle \sum^{n}_{i=1}f(\xi_{i})(t_{i}-t_{i-1}), $$ where $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}(or \ \mathbb{R}_{\mathcal{I}})$. We call $S(f,D,\delta)$ a Riemann $\Delta$-sum of $f$ corresponding to $D\in \mathcal{D}(\delta,[a,b]_{\mathbb{T}})$.
\begin{definition}[Bohner and Peterson \cite{BP2}] \label{defn2.1} A function $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}$ is called Riemann $\Delta$-integrable on $[a,b]_{\mathbb{T}}$ if there exists an $A\in \mathbb{R}$ such that for each $\epsilon>0$ there exists a $\delta>0$ for which
$$\big|S(f,\mathcal{D},\delta), A\big|<\epsilon$$ for all $D\in \mathcal{D}(\delta,[a,b]_{\mathbb{T}})$. In this case, $A$ is called the Riemann $\Delta$-integral of $f$ on $[a,b]_{\mathbb{T}}$ and is denoted by $A=(R)\int_{a}^{b}f(t)\Delta t$ or $A=\int_{a}^{b}f(t)\Delta t$. The family of all Riemann $\Delta$-integrable functions on $[a,b]_{\mathbb{T}}$ is denoted by $\mathcal{R}_{(\Delta,\ [a,b]_{\mathbb{T}})}$. \end{definition}
\section{The Interval Darboux and Riemann delta integrals} \label{sec:3}
Let $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ be such that $f(t)=\big[\underline{f}(t), \overline{f}(t)\big]$ for all $t\in [a,b]_{\mathbb{T}}$. We denote $$ M=\sup\{\overline{f}(t):t\in [a,b)_{\mathbb{T}}\},\ \ m =\inf\{\underline{f}(t):t\in [a,b)_{\mathbb{T}}\}, $$ and for $1\leq i\leq n$, $$ M_{i}=\sup\{\overline{f}(t):t\in [t_{i-1},t_{i})_{\mathbb{T}}\}, $$ $$ m_{i}=\inf\{\underline{f}(t):t\in [t_{i-1},t_{i})_{\mathbb{T}}\}. $$ The lower Darboux $\Delta$-sum $L(\underline{f},D)$ of $\underline{f}$ with respect to $D\in\mathcal{D}([a,b]_{\mathbb{T}})$ is the sum $$ L(\underline{f},D)=\sum_{i=1}^{n}m_{i}(t_{i}-t_{i-1}), $$ and the upper Darboux $\Delta$-sum $U(\overline{f},D)$ is $$ U(\overline{f},D)=\sum_{i=1}^{n}M_{i}(t_{i}-t_{i-1}). $$
\begin{definition}[The Interval Darboux delta integral] \label{defn3} Let $I=[a,b]_{\mathbb{T}}$, where $a,b \in \mathbb{T}$. The lower Darboux $\Delta$-integral of $\underline{f}$ on $[a,b]_{\mathbb{T}}$ is defined by $$ (D)\infint_{a}^{b}\underline{f}(t)\Delta t =\sup_{D\in \mathcal{D}([a,b]_{\mathbb{T}})}\Big\{L(\underline{f},D)\Big\} $$ and the upper Darboux $\Delta$-integral of $\overline{f}$ on $[a,b]_{\mathbb{T}}$ is defined by $$ (D)\supint_{a}^{b}\overline{f}(t)\Delta t =\inf_{D\in \mathcal{D}([a,b]_{\mathbb{T}})}\Big\{U(\overline{f},D)\Big\}. $$ Then, we define the $ID$ $\Delta$-integral of $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ on $[a,b]_{\mathbb{T}}$ as the interval $$ (ID)\int_{a}^{b}f(t)\Delta t=\Bigg[(D)\infint_{a}^{b}\underline{f}(t)\Delta t, (D)\supint_{a}^{b}\overline{f}(t)\Delta t\Bigg]. $$ The family of all $ID$ $\Delta$-integrable functions on $[a,b]_{\mathbb{T}}$ is denoted by $\mathcal{ID}_{(\Delta,\ [a,b]_{\mathbb{T}})}$. \end{definition}
\begin{theorem} \label{thm2} Let $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ be such that $$ f(t)=\big[\underline{f}(t), \overline{f}(t)\big] $$ for all $t\in [a,b]_{\mathbb{T}}$. Then $f\in \mathcal{ID}_{(\Delta,\ [t,\sigma(t)]_{\mathbb{T}})}$ and $$ (ID)\int_{t}^{\sigma(t)}f(s)\Delta s=\big[\mu(t)\underline{f}(t),\mu(t)\overline{f}(t)\big]. $$ \end{theorem}
\begin{proof} If $\sigma(t)=t$, then the result is obvious. If $\sigma(t)>t$, then $\mathcal{D}([a,b]_{\mathbb{T}})$ only contains one single element given by $$ t=s_{0}<s_{1}=\sigma(t). $$ Since $[s_{0},s_{1})=[t,\sigma(t))=\{t\}$, we have $$ L(f,D)=\underline{f}(t)(\sigma(t)-t)=\mu(t)\underline{f}(t), $$ $$ U(f,D)=\overline{f}(t)(\sigma(t)-t)=\mu(t)\overline{f}(t). $$ Consequently, we obtain $$ (ID)\int_{t}^{\sigma(t)} f(s)\Delta s =\big[\mu(t)\underline{f}(t),\mu(t)\overline{f}(t)\big]. $$ The result is proved. \end{proof}
\begin{remark} \label{rmk3.1} It is clear that if $f$ is a real-valued function, then our Definition~\ref{defn3} implies the definition of Darboux $\Delta$-integral introduced by \cite{BP2}. We also have the following:
\noindent(1) If $\mathbb{T}=\mathbb{R}$, then Definition~\ref{defn3} implies the definition of Darboux interval integral introduced by Caprani et al. \cite{C81,R82}.
\noindent(2) If $\mathbb{T}=\mathbb{Z}$, then each function $f:\mathbb{Z}\rightarrow \mathbb{R}_{\mathcal{I}}$ is $ID$ $\Delta$-integrable on $[a,b]_{\mathbb{T}}$. Moreover, $$ (ID)\int_{a}^{b}f(t)\Delta t=\Bigg[\sum_{t=a}^{b-1} \underline{f}(t), \sum_{t=a}^{b-1}\overline{f}(t)\Bigg]. $$
\noindent(3) If $\mathbb{T}=h\mathbb{Z}$, then each function $f:h\mathbb{Z}\rightarrow \mathbb{R}_{\mathcal{I}}$ is $ID$ $\Delta$-integrable on $[a,b]_{\mathbb{T}}$. Moreover, $$ (ID)\int_{a}^{b}f(t)\Delta t=\Bigg[ \sum_{k=\frac{a}{h}}^{\frac{b}{h}-1}\underline{f}(kh)h, \sum_{k=\frac{a}{h}}^{\frac{b}{h}-1}\overline{f}(kh)h\Bigg]. $$ \end{remark}
\begin{example} \label{ex1} Suppose that $[a,b]_{\mathbb{T}}=[0,1]$, $\mathbb{Q}$ is the set of rational numbers in $[0,1]$, and $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ is defined by \begin{equation*} f(t)= \begin{cases} [-1,0], & \text{if $t\in \mathbb{Q}$},\\ [1,2], & \text{if $t\in[0,1]\backslash \mathbb{Q}$}. \end{cases} \end{equation*} Then, \begin{equation*} \begin{split} (ID)\int_{0}^{1}f(t)\Delta t&=\Bigg[(D)\infint_{0}^{1} \underline{f}(t)dt, (D)\supint_{0}^{1}\overline{f}(t)dt\Bigg]\\ &=[-1,2]. \end{split} \end{equation*} \end{example}
\begin{example} \label{ex2} Suppose that $[a,b]_{\mathbb{T}}=\big\{0,\frac{1}{3},\frac{1}{2},1\big\}$ and \\ $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ is defined by \begin{equation*} f(t)= \begin{cases} [-1,0], & \text{if $t=0$},\\ [-\frac{1}{3},\frac{1}{3}], & \text{if $t=\frac{1}{3}$},\\ [-\frac{1}{2},\frac{1}{2}], & \text{if $t=\frac{1}{2}$},\\ [1,2], & \text{if $t=1$}. \end{cases} \end{equation*} Then, \begin{equation*} \begin{split} (D)\infint_{0}^{1}\underline{f}(t)\Delta t &=(-1)\cdot\frac{1}{3}+\left(-\frac{1}{3}\right)\cdot\frac{1}{6} +\left(-\frac{1}{2}\right)\cdot\frac{1}{2}\\ &=-\frac{23}{36}, \end{split} \end{equation*} \begin{equation*} (D)\supint_{0}^{1}\overline{f}(t)\Delta t=0\cdot\frac{1}{3} +\frac{1}{3}\cdot\frac{1}{6}+\frac{1}{2}\cdot\frac{1}{2}=\frac{11}{36}, \end{equation*} and therefore $$ (ID)\int_{0}^{1}f(t)\Delta t=\Bigg[-\frac{23}{36}, \frac{11}{36}\Bigg]. $$ \end{example}
\begin{theorem} \label{thm3} Let $f,\ g\in \mathcal{ID}_{(\Delta,\ [a,b]_{\mathbb{T}})}$, and $\lambda$ be \\an arbitrary real number. Then,
\noindent(1) $\lambda f \in \mathcal{ID}_{(\Delta,\ [a,b]_{\mathbb{T}})}$ and $$ (ID)\int_{a}^{b}\lambda f(t)\Delta t=\lambda (ID)\int_{a}^{b}f(t)\Delta t; $$
\noindent(2) $f+g \in \mathcal{ID}_{(\Delta,\ [a,b]_{\mathbb{T}})}$ and \begin{equation*} \begin{split} &(ID)\int_{a}^{b}(f(t)+g(t))\Delta t\\ &\subseteq(ID)\int_{a}^{b}f(t)\Delta t+(ID)\int_{a}^{b}g(t)\Delta t; \end{split} \end{equation*}
\noindent(3) for $c\in [a,b]_{\mathbb{T}}$ and $a<c<b$, $$(ID)\int_{a}^{c}f(t)\Delta t +(ID)\int_{c}^{b} f(t)\Delta t=(ID)\int_{a}^{b} f(t)\Delta t; $$
\noindent(4) if $f\subseteq g$ on $[a,b]_{\mathbb{T}}$, then $$ (ID)\int_{a}^{b} f(t)\Delta t\subseteq (ID)\int_{a}^{b} g(t)\Delta t. $$ \end{theorem}
\begin{proof} We only prove that part (2) of Theorem~\ref{thm3} holds. The other relations are obvious. Suppose that $$ f(t)=\big[\underline{f}(t),\overline{f}(t)\big], \ g(t) =\big[\underline{g}(t),\overline{g}(t)\big]. $$ Select any division $D\in\mathcal{D}([a,b]_{\mathbb{T}})$ having the form $$ D=\{a=t_{0}<t_{1}<\cdots <t_{n}=b\}. $$ Then, \begin{equation*} \begin{split} &\inf_{t\in [t_{i-1},t_{i})_{\mathbb{T}}}\{\underline{f}(t)\} +\inf_{t\in [t_{i-1},t_{i})_{\mathbb{T}}}\{\underline{g}(t)\}\\ &\leq \inf_{t\in [t_{i-1},t_{i})_{\mathbb{T}}}\{\underline{f}(t) +\underline{g}(t)\}, \end{split} \end{equation*} \begin{equation*} \begin{split} &\sup_{t\in [t_{i-1},t_{i})_{\mathbb{T}}}\{\overline{f}(t)+\overline{g}(t)\}\\ &\leq\sup_{t\in [t_{i-1},t_{i})_{\mathbb{T}}}\{\overline{f}(t)\} +\sup_{t\in [t_{i-1},t_{i})_{\mathbb{T}}}\{\overline{g}(t)\},
\end{split} \end{equation*} and it follows that $$ L(\underline{f},D)+L(\underline{g},D)\leq L(\underline{f}+\underline{g},D), $$ $$ U(\underline{f},D)+U(\underline{g},D)\geq U(\underline{f}+\underline{g},D). $$ The intended result follows. \end{proof}
\begin{example} \label{ex3} Suppose that $[a,b]_{\mathbb{T}}=[0,1]$, $\mathbb{Q}$ is the set of rational numbers in $[0,1]$, and $f,g:[a,b]_{\mathbb{T}} \rightarrow \mathbb{R}_{\mathcal{I}}$ are defined by \begin{equation*} f(t)= \begin{cases} [-1,0], & \text{if $t\in \mathbb{Q}$},\\ [1,2], & \text{if $t\in[0,1]\backslash \mathbb{Q}$}, \end{cases} \end{equation*} \begin{equation*} g(t)= \begin{cases} [0,1], & \text{if $t\in \mathbb{Q}$},\\ [-2,-1], & \text{if $t\in[0,1]\backslash \mathbb{Q}$}. \end{cases} \end{equation*} Then $$ f(t)+g(t)=[-1,1] $$ for all $t\in[0,1]$. It follows that \begin{equation*} \begin{split} &(ID)\int_{0}^{1}f(t)\Delta t+(ID)\int_{0}^{1}g(t)\Delta t\\ &=[-1,2]+[-2,1]\\ &=[-3,3], \end{split} \end{equation*} \begin{equation*}
(ID)\int_{0}^{1}(f(t)+g(t))\Delta t =[-1,1]. \end{equation*} Therefore, we have \begin{equation*} \begin{split} &(ID)\int_{a}^{b}(f(t)+g(t))\Delta t\\ &\subseteq (ID)\int_{a}^{b}f(t)\Delta t+(ID)\int_{a}^{b}g(t)\Delta t. \end{split} \end{equation*} \end{example}
We now give Riemann's definition of integrability, which is equivalent to the Riemann $\Delta$-integral given in \cite[Definition 13]{L13}.
\begin{definition}[The Interval Riemann delta integral] \label{defn4} A function $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ is called $IR$ $\Delta$-integrable on $[a,b]_{\mathbb{T}}$ if there exists an $A\in \mathbb{R}_{\mathcal{I}}$ such that for each $\epsilon>0$ there exists a $\delta>0$ for which $$d\big(S(f,\mathcal{D},\delta), A\big)<\epsilon$$ for all $D\in \mathcal{D}(\delta,[a,b]_{\mathbb{T}})$. In this case, $A$ is called the $IR$ $\Delta$-integral of $f$ on $[a,b]_{\mathbb{T}}$ and is denoted by $A=(IR)\int_{a}^{b}f(t)\Delta t$. The family of all $IR$ $\Delta$-integrable functions on $[a,b]_{\mathbb{T}}$ is denoted by $\mathcal{IR}_{(\Delta,\ [a,b]_{\mathbb{T}})}$. \end{definition}
\begin{remark} \label{rmk3.2} Definitions \ref{defn3} and \ref{defn4} are not equivalent. If $f\in\mathcal{IR}_{(\Delta,\ [a,b]_{\mathbb{T}})}$, then $f\in\mathcal{ID}_{(\Delta,\ [a,b]_{\mathbb{T}})}$. However, the converse is not always true (see Example \ref{ex1}). It is clear that $f\in\mathcal{ID}_{(\Delta,\ [a,b]_{\mathbb{T}})}$, but $f\notin\mathcal{IR}_{(\Delta,\ [a,b]_{\mathbb{T}})}$. In fact, all bounded interval functions are $ID$ $\Delta$-integrable, but boundedness of $f$ is not a sufficient condition for $IR$ $\Delta$-integrability. If $f$ is a continuous function, then $f\in\mathcal{IR}_{(\Delta,\ [a,b]_{\mathbb{T}})}$ if and only if $f\in\mathcal{ID}_{(\Delta,\ [a,b]_{\mathbb{T}})}$, in which case the value of the integrals agree. \end{remark}
The following two theorems can be easily verified and so the proofs are omitted.
\begin{theorem} \label{thm4} If $f\in C([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}})$, then $f\in\mathcal{IR}_{(\Delta,\ [a,b]_{\mathbb{T}})}$ and $$ (IR)\int_{a}^{b}f(t)\Delta t=\Bigg[\int_{a}^{b} \underline{f}(t)\Delta t,\int_{a}^{b}\overline{f}(t)\Delta t\Bigg]. $$ \end{theorem}
\begin{theorem} \label{thm5} Let $f,\ g\in \mathcal{IR}_{(\Delta,\ [a,b]_{\mathbb{T}})}$, and $\lambda$ be an arbitrary real number. Then,
\noindent(1) $\lambda f \in \mathcal{IR}_{(\Delta,\ [a,b]_{\mathbb{T}})}$ and $$ (IR)\int_{a}^{b}\lambda f(t)\Delta t=\lambda (IR)\int_{a}^{b}f(t)\Delta t; $$
\noindent(2) $f+g \in \mathcal{IR}_{(\Delta,\ [a,b]_{\mathbb{T}})}$ and \begin{equation*} \begin{split} &(IR)\int_{a}^{b}(f(t)+g(t))\Delta t\\ &=(IR)\int_{a}^{b}f(t)\Delta t+(IR)\int_{a}^{b}g(t)\Delta t; \end{split} \end{equation*}
\noindent(3) for $c\in [a,b]_{\mathbb{T}}$ and $a<c<b$, $$ (IR)\int_{a}^{c}f(t)\Delta t+(IR)\int_{c}^{b} f(t)\Delta t =(IR)\int_{a}^{b} f(t)\Delta t; $$
\noindent(4) if $f\subseteq g$ on $[a,b]_{\mathbb{T}}$, then $$ (IR)\int_{a}^{b} f(t)\Delta t\subseteq (IR)\int_{a}^{b} g(t)\Delta t. $$ \end{theorem}
\begin{example} \label{ex4} Suppose that $\mathbb{T}=[-1,0]\cup 3^{\mathbb{N}_{0}}$, where $[-1,0]$ is a real-valued interval and $\mathbb{N}_{0}$ is the set of nonnegative integers. Let $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ be defined by \begin{equation*} f(t)= \begin{cases} [t,t+1], & \text{if $t\in [-1,0)$},\\ [1,2], & \text{if $t=0$},\\ [t,t^{2}+1], & \text{if $t\in 3^{\mathbb{N}_{0}}$}. \end{cases} \end{equation*} If $[a,b]_{\mathbb{T}}=[-1,3]_{\mathbb{T}}$, then \begin{equation*} \begin{split} &(IR)\int_{-1}^{3}f(t)\Delta t\\ &=\Bigg[\int_{-1}^{3}\underline{f}(t)\Delta t,\int_{-1}^{3}\overline{f}(t)\Delta t\Bigg]\\ &=\Bigg[\int_{-1}^{0}tdt+(R)\int_{0}^{1}\Delta t+\int_{1}^{3}t\Delta t,\\ &\ \ \ \ \ \ \int_{-1}^{0}(t+1)dt+\int_{0}^{1}2\Delta t+\int_{1}^{3}(t^{2}+1)\Delta t\Bigg]\\
&=\Bigg[\frac{1}{2}t^{2}\Big|_{-1}^{0}+1+2t^{2}\Big|_{1},\\
&\ \ \ \ \ \ \ \frac{1}{2}(t^{2}+t)\Big|_{-1}^{0}+2+2t(t^{2}+1)\Big|_{1}\Bigg]\\ &=\Big[2\frac{1}{2},6\frac{1}{2}\Big]. \end{split} \end{equation*} \end{example}
\section{Some inequalities for the interval Riemann delta integral} \label{sec:4}
We begin by recalling the notions of convexity on time scales.
\begin{definition}[Dinu \cite{D08}] \label{defn4.1} We say that $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}$ is a convex function if for all $x,y\in[a,b]_{\mathbb{T}}$ and $\alpha\in[0,1]$ we have \begin{equation} \label{1} f(\alpha x+(1-\alpha)y)\leq \alpha f(x)+(1-\alpha)f(y) \end{equation} for which $\alpha x+(1-\alpha)y\in[a,b]_{\mathbb{T}}$. If inequality \eqref{1} is reversed, then $f$ is said to be concave. If $f$ is both convex and concave, then $f$ is said to be affine. The set of all convex, concave and affine interval-valued functions are denoted by $SX([a,b]_{\mathbb{T}},\mathbb{R})$, $SV([a,b]_{\mathbb{T}},\mathbb{R})$, and $SA([a,b]_{\mathbb{T}},\mathbb{R})$, respectively. \end{definition}
We can now introduce the concept of interval-valued convexity.
\begin{definition} \label{defn4.2} We say that $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ is a convex interval-valued function if for all $x,y\in[a,b]_{\mathbb{T}}$ and $\alpha\in(0,1)$ we have \begin{equation} \label{4} \alpha f(x)+(1-\alpha)f(y) \subseteq f(\alpha x+(1-\alpha)y) \end{equation} for which $\alpha x+(1-\alpha)y\in[a,b]_{\mathbb{T}}$. If the set inclusion \eqref{4} is reversed, then $f$ is said to be concave. If $f$ is both convex and concave, then $f$ is said to be affine. The set of all convex, concave and affine interval-valued functions are denoted by $SX([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}})$, $SV([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}})$ and $SA([a,b]_{\mathbb{T}}, \mathbb{R}_{\mathcal{I}})$, respectively. \end{definition}
\begin{remark} \label{rmk4.0} It is clear that if $\mathbb{T}=\mathbb{R}$, then Definition~\ref{defn4.2} implies the definition of convexity introduced by Breckner \cite{B}. \end{remark}
\begin{theorem} \label{thm4.1} Let $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ be such that $$ f(t)=[\underline{f}(t),\overline{f}(t)] $$ for all $t\in [a,b]_{\mathbb{T}}$. Then,
\noindent(1) $f\in SX([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}})$ if and only if $\underline{f}\in SX([a,b]_{\mathbb{T}},\mathbb{R})$ and $\overline{f}\in SV([a,b]_{\mathbb{T}},\mathbb{R})$,
\noindent(2) $f\in SV([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}})$ if and only if $\underline{f}\in SV([a,b]_{\mathbb{T}},\mathbb{R})$ and $\overline{f}\in SX([a,b]_{\mathbb{T}},\mathbb{R})$,
\noindent(3) $f\in SA([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}})$ if and only if $\underline{f}, \overline{f} \in SA([a,b]_{\mathbb{T}},\mathbb{R})$. \end{theorem}
\begin{proof} We only prove that part (1) of Theorem~\ref{thm4.1} holds. Suppose that $f\in SX([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}})$ and consider $x,y\in[a,b]_{\mathbb{T}}$, $\alpha\in[0,1]$. Then, $$ \alpha f(x)+(1-\alpha)f(y) \subseteq f(\alpha x+(1-\alpha)y), $$ that is, \begin{equation} \label{5} \begin{split} &\big[\alpha\underline{f}(x)+(1-\alpha)\underline{f}(y), \alpha\overline{f}(x)+(1-\alpha)\overline{f}(y)\big]\\ &\subseteq \big[\underline{f}(\alpha x+(1-\alpha)y), \overline{f}(\alpha x+(1-\alpha)y)\big]. \end{split} \end{equation} It follows that $$ \alpha\underline{f}(x)+(1-\alpha)\underline{f}(y) \geq \underline{f}(\alpha x+(1-\alpha)y) $$ and $$ \alpha\overline{f}(x)+(1-\alpha)\overline{f}(y) \leq \overline{f}(\alpha x+(1-\alpha)y). $$ This shows that $$ \underline{f}\in SX([a,b]_{\mathbb{T}},\mathbb{R})\ {\rm and }\ \overline{f}\in SV([a,b]_{\mathbb{T}},\mathbb{R}). $$ Conversely, if $$ \underline{f}\in SX([a,b]_{\mathbb{T}},\mathbb{R})\ {\rm and }\ \overline{f}\in SV([a,b]_{\mathbb{T}},\mathbb{R}), $$ by Definition~\ref{defn4.2} and the set inclusion \eqref{5}, we have \\$f\in SX([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}})$. \end{proof}
\begin{theorem}[Dinu \cite{D08}] \label{thm4.2} A convex function on \\$[a,b]_{\mathbb{T}}$ is continuous on $(a,b)_{\mathbb{T}}$. \end{theorem}
\begin{theorem} \label{thm4.4} Let $f:[a,b]_{\mathbb{T}}\rightarrow \mathbb{R}_{\mathcal{I}}$ be such that $$ f(t)=[\underline{f}(t),\overline{f}(t)] $$ for all $t\in [a,b]_{\mathbb{T}}$. If $$ f\in SX([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}) \cup SV([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}) \cup SA([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}), $$ then $f\in\mathcal{IR}_{(\Delta,\ [a,b]_{\mathbb{T}})}$. \end{theorem}
\begin{proof} Suppose that $$ f\in SV([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}) \cup SV([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}) \cup SA([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}). $$ Due to Theorems~\ref{thm4.1} and \ref{thm4.2}, it follows that $\underline{f}$ and $\overline{f}$ are continuous. Then, from Theorem~5.19 of \cite{BP2}, we have that $$ \overline{f}(t), \underline{f}(t)\in\mathcal{R}_{(\Delta,\ [a,b]_{\mathbb{T}})}. $$ Hence, $f\in\mathcal{IR}_{(\Delta,\ [a,b]_{\mathbb{T}})}$. \end{proof}
\begin{theorem}[Wong et al. \cite{W06}] \label{thm4.5} Let $a,b\in[a,b]_{\mathbb{T}}$ and $c,d\in\mathbb{R}$. Suppose that $g\in C_{rd}([a,b]_{\mathbb{T}},(c,d))$ and $h\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R})$ with $$
\int_{a}^{b}|h(s)|\Delta s>0. $$ If $f\in C((c,d),\mathbb{R})$ is convex, then \begin{equation}\label{6} \begin{split}
&f\Bigg(\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg)
\leq \frac{\int_{a}^{b}|h(s)|f(g(s))\Delta s}{\int_{a}^{b}|h(s)|\Delta s}. \end{split} \end{equation} If $f$ is concave, then inequality \eqref{6} is reversed. \end{theorem}
\begin{theorem}[Jensen's inequality] \label{thm4.6} \\Let $g\in C_{rd}([a,b]_{\mathbb{T}},(c,d))$ and $h\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R})$ with $$
\int_{a}^{b}|h(s)|\Delta s>0. $$ If $f\in C((c,d),\mathbb{R}^{+}_{\mathcal{I}})$ is a convex function, then \begin{equation*} \begin{split}
&\frac{(IR)\int_{a}^{b}|h(s)|f(g(s))\Delta s}{\int_{a}^{b}|h(s)|\Delta s}
\subseteq f\Bigg(\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg). \end{split} \end{equation*} \end{theorem}
\begin{proof} By hypothesis, we have $$
|h|\overline{f(g)},\ \
|h|\underline{f(g)}\in\mathcal{R}_{(\Delta,\ [a,b])}. $$
Hence, $|h|f(g)\in\mathcal{IR}_{(\Delta,\ [a,b])}$ and \begin{equation*} \begin{split}
&(IR)\int_{a}^{b}|h(s)|f(g(s))\Delta s\\
&=\Bigg[\int_{a}^{b}|h(s)|\underline{f(g)}(s)\Delta s,\int_{a}^{b}|h(s)|\overline{f(g)}(s)\Delta s\Bigg]. \end{split} \end{equation*} From Theorem~\ref{thm4.5}, it follows that $$
\underline{f}\Bigg(\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{(\int_{a}^{b}|h(s)|\Delta s}\Bigg)\leq\frac{\int_{a}^{b}|h(s)|\underline{f(g)}(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s} $$ and $$
\overline{f}\Bigg(\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg)\geq\frac{\int_{a}^{b}|h(s)|\overline{f(g)}(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}, $$ which implies \begin{equation*} \begin{split}
&\Bigg[\frac{\int_{a}^{b}|h(s)|\underline{f(g)}(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}, \frac{\int_{a}^{b}|h(s)|\overline{f(g)}(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg]\\
&\ \ \subseteq \Bigg[\underline{f}\Bigg(\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg),\overline{f}\Bigg(\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg)\Bigg], \end{split} \end{equation*} that is, \begin{equation*} \begin{split}
&\frac{\Bigg[\int_{a}^{b}|h(s)|\underline{f(g)}(s)\Delta s,
\int_{a}^{b}|h(s)|\overline{f(g)}(s)\Delta s\Bigg]}{\int_{a}^{b}|h(s)|\Delta s}\\
&\ \ \subseteq \Bigg[\underline{f}\Bigg(\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg),\overline{f}\Bigg(\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg)\Bigg]. \end{split} \end{equation*} Finally, we obtain \begin{equation*}
\frac{(IR)\int_{a}^{b}|h(s)|f(g(s))\Delta s}{\int_{a}^{b}|h(s)|\Delta s}
\subseteq f\Bigg(\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg). \end{equation*} The proof is complete. \end{proof}
\begin{example} \label{ex5} Suppose that $[a,b]_{\mathbb{T}}=[0,1]\cup \{\frac{3}{2}\}$, where $[0,1]$ is a real-valued interval. Let $g(s)=s^{2}$, $h(s)=e^{s}$, and $f(s)=[s^{2},4\sqrt{s}]$. Then \begin{equation*} \begin{split}
&\frac{(IR)\int_{a}^{b}|h(s)|f(g(s))\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\\ &=\frac{(IR)\int_{0}^{\frac{3}{2}}\big[s^{4}e^{s},4se^{s}\big] \Delta s}{\int_{0}^{\frac{3}{2}}e^{s}\Delta s}\\ &=\frac{\bigg[\int_{0}^{\frac{3}{2}}s^{4}e^{s}\Delta s, \int_{0}^{\frac{3}{2}}4se^{s}\Delta s\bigg]}{\int_{0}^{\frac{3}{2}}e^{s}\Delta s}\\ &=\frac{\bigg[\int_{0}^{1}s^{4}e^{s}ds+\int_{1}^{\frac{3}{2}}s^{4}e^{s}\Delta s,\int_{0}^{1}4se^{s}ds+\int_{1}^{\frac{3}{2}}4se^{s}\Delta s\bigg]}{ \int_{0}^{1}e^{s}ds+\int_{1}^{\frac{3}{2}}e^{s}\Delta s}\\ &=\frac{\bigg[9\frac{1}{2}e-24,4+2e\bigg]}{\frac{3}{2}e-1}\\ &=\Bigg[\frac{19e-48}{3e-2},\frac{8+4e}{3e-2}\Bigg], \end{split} \end{equation*} and \begin{equation*} \begin{split}
&f\Bigg(\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg)\\ &=f\Bigg(\frac{\int_{0}^{\frac{3}{2}}s^{2}e^{s}\Delta s}{\int_{0}^{\frac{3}{2}}e^{s}\Delta s}\Bigg)\\ &=f\Bigg(\frac{\int_{0}^{1}s^{2}e^{s}ds+\int_{1}^{\frac{3}{2}}s^{2}e^{s}\Delta s}{\frac{3}{2}e-1}\Bigg)\\ &=f\Bigg(\frac{\frac{3}{2}e-2}{\frac{3}{2}e-1}\Bigg)\\ &=\Bigg[\bigg(\frac{3e-4}{3e-2}\bigg)^{2},4\sqrt{\frac{3e-4}{3e-2}}\Bigg]. \end{split} \end{equation*} It follows that \begin{equation*} \begin{split} &\Bigg[\frac{19e-48}{3e-2},\frac{8+4e}{3e-2}\Bigg] \subseteq \Bigg[\bigg(\frac{3e-4}{3e-2}\bigg)^{2},4\sqrt{\frac{3e-4}{3e-2}}\Bigg]. \end{split} \end{equation*} \end{example}
It is clear that if $[a,b]_{\mathbb{T}}=[0,1]$ and $h(s)\equiv1$, then we get a similar result given in \cite[Theorem~3.5]{C17} by T. M. Costa. Similarly, we can get the following results that generalize \cite[Theorem~3.4]{C17} and \cite[Corollary 3.3]{C17}.
\begin{theorem} \label{thm4.7} Let $g\in C_{rd}([a,b]_{\mathbb{T}},(c,d))$ and \\ $h\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R})$ with $$
\int_{a}^{b}|h(s)|\Delta s>0. $$ If $f\in C((c,d),\mathbb{R}^{+}_{\mathcal{I}})$ is a concave function, then \begin{equation*} \begin{split}
&\frac{(IR)\int_{a}^{b}|h(s)|f(g(s))\Delta s}{\int_{a}^{b}|h(s)|\Delta s}
\supseteq f\Bigg(\frac{\int_{a}^{b}|h(s)|\underline{f(g)}(s)\Delta s}{
\int_{a}^{b}|h(s)|\Delta s}\Bigg). \end{split} \end{equation*} \end{theorem}
\begin{theorem} \label{thm4.8} Let $g\in C_{rd}([a,b]_{\mathbb{T}},(c,d))$ and \\ $h\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R})$ with $$
\int_{a}^{b}|h(s)|\Delta s>0. $$ If $f\in C((c,d),\mathbb{R}^{+}_{\mathcal{I}})$ is an affine function, then \begin{equation*} \begin{split}
&\frac{(IR)\int_{a}^{b}|h(s)|f(g(s))\Delta s}{\int_{a}^{b}|h(s)|\Delta s}
= f\Bigg(\frac{\int_{a}^{b}|h(s)|
\underline{f(g)}(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}\Bigg). \end{split} \end{equation*} \end{theorem}
\begin{theorem}[Agarwal et al. \cite{A14}] \label{thm4.9} \\Let $f,g,h\in C_{rd}([a,b]_{\mathbb{T}},(0,\infty))$. If $\frac{1}{p}+\frac{1}{q}=1$, with $p>1$, then \begin{equation*} \begin{split} &\int_{a}^{b}h(s)f(s)g(s)\Delta s\\ &\leq \Bigg(\int_{a}^{b}h(s)f^{p}(s)\Delta s\Bigg)^{\frac{1}{p}}\Bigg( \int_{a}^{b}h(s)g^{q}(s)\Delta s\Bigg)^{\frac{1}{q}}. \end{split} \end{equation*} \end{theorem}
Next we present a H\"{o}lder type inequality for interval-valued functions on time scales.
\begin{theorem}[H\"{o}lder's inequality] \label{thm4.10} \\Let $h\in C_{rd}([a,b]_{\mathbb{T}},(0,\infty))$, $f,g\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}^{+})$.\\ If $\frac{1}{p}+\frac{1}{q}=1$, with $p>1$, then \begin{equation*} \begin{split} &\int_{a}^{b}h(s)f(s)g(s)\Delta s\\ &\leq \Bigg(\int_{a}^{b}h(s)f^{p}(s)\Delta s\Bigg)^{\frac{1}{p}}\Bigg( \int_{a}^{b}h(s)g^{q}(s)\Delta s\Bigg)^{\frac{1}{q}}. \end{split} \end{equation*} \end{theorem}
\begin{proof} By hypothesis, we have \begin{equation*} \begin{split} &\int_{a}^{b}h(s)f(s)g(s)\Delta s\\ &=\int_{a}^{b}h(s)\big[\underline{f}(s)\underline{g}(s), \overline{f}(s)\overline{g}(s)\big]\Delta s\\ &=\Bigg[\int_{a}^{b}h(s)\underline{f}(s)\underline{g}(s)\Delta s, \int_{a}^{b}h(s)\overline{f}(s)\overline{g}(s)\Delta s\Bigg]\\ &\leq \Bigg[\bigg(\int_{a}^{b}h(s)\underline{f}^{p}(s)\Delta s \bigg)^{\frac{1}{p}}\bigg(\int_{a}^{b}h(s)\underline{g}^{q}(s)\Delta s\bigg)^{\frac{1}{q}},\\ &\ \ \ \ \bigg(\int_{a}^{b}h(s)\overline{f}^{p}(s) \Delta s\bigg)^{\frac{1}{p}}\bigg(\int_{a}^{b}h(s) \overline{g}^{q}(s)\Delta s\bigg)^{\frac{1}{q}}\Bigg]\\ &= \Bigg[\bigg(\int_{a}^{b}h(s)\underline{f}^{p}(s)\Delta s \bigg)^{\frac{1}{p}},\bigg(\int_{a}^{b}h(s)\overline{f}^{p}(s)\Delta s\bigg)^{\frac{1}{p}}\Bigg]\\ &\ \ \ \ \cdot\Bigg[\bigg(\int_{a}^{b}h(s)\underline{g}^{q}(s)\Delta s\bigg)^{\frac{1}{q}},\bigg(\int_{a}^{b}h(s)\overline{g}^{q}(s)\Delta s\bigg)^{\frac{1}{q}}\Bigg]\\ &= \Bigg[\int_{a}^{b}h(s)\underline{f}^{p}(s)\Delta s,\int_{a}^{b}h(s) \overline{f}^{p}(s)\Delta s\Bigg]^{\frac{1}{p}}\\ &\ \ \ \ \cdot\Bigg[\int_{a}^{b}h(s)\underline{g}^{q}(s)\Delta s, \int_{a}^{b}h(s)\overline{g}^{q}(s)\Delta s\Bigg]^{\frac{1}{q}}\\ &=\Bigg(\int_{a}^{b}h(s)\Big[\underline{f}(s),\overline{f}(s)\Big]^{p}\Delta s\Bigg)^{\frac{1}{p}}\Bigg(\int_{a}^{b}h(s)\Big[\underline{g}(s), \overline{g}(s)\Big]^{q}\Delta s\Bigg)^{\frac{1}{q}}\\ &=\Bigg(\int_{a}^{b}h(s)f^{p}(s)\Delta s\Bigg)^{\frac{1}{p}}\Bigg( \int_{a}^{b}h(s)g^{q}(s)\Delta s\Bigg)^{\frac{1}{q}}. \end{split} \end{equation*} This concludes the proof. \end{proof}
For the particular case $p=q=2$ in Theorem~\ref{thm4.10}, we obtain the following Cauchy--Schwarz inequality.
\begin{theorem}[Cauchy--Schwarz inequality] \label{thm4.11} \\Let $h\in C_{rd}([a,b]_{\mathbb{T}},(0,\infty))$, $f,g\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}^{+})$. Then, \begin{equation*} \begin{split} &\int_{a}^{b}h(s)f(s)g(s)\Delta s\\ &\leq \sqrt{\Bigg(\int_{a}^{b}h(s)f^{2}(s)\Delta s\Bigg)\Bigg( \int_{a}^{b}h(s)g^{2}(s)\Delta s\Bigg)}. \end{split} \end{equation*} \end{theorem}
\begin{example} \label{ex6} Suppose that $[a,b]_{\mathbb{T}}=[0,\frac{\pi}{2}]$. Let $h(s)=s$, $f(s)=[s,s+1]$, and $g(s)=[\sin s,s]$ for $s\in[0,\frac{\pi}{2}]$. Then \begin{equation*} \begin{split} \int_{a}^{b}&h(s)f(s)g(s)\Delta s\\ &=\int_{0}^{\frac{\pi}{2}}\big[s^{2}\sin s,s^{3}+s^{2}\big]\Delta s\\ &=\bigg[\int_{0}^{\frac{\pi}{2}}s^{2}\sin s\Delta s, \int_{0}^{\frac{\pi}{2}}(s^{3}+s^{2})\Delta s\bigg]\\ &=\bigg[\pi-2,\frac{\pi^{4}}{64}+\frac{\pi^{3}}{24}\bigg], \end{split} \end{equation*} and \begin{equation*} \begin{split} &\sqrt{\Bigg(\int_{a}^{b}h(s)f^{2}(s)\Delta s\Bigg)\Bigg(\int_{a}^{b}h(s)g^{2}(s)\Delta s\Bigg)}\\ &=\sqrt{\bigg(\int_{0}^{\frac{\pi}{2}}\big[s^{3},s^{3}+2s^{2}+s\big]\Delta s \bigg)\bigg(\int_{0}^{\frac{\pi}{2}}\big[s\sin ^{2}s,s^{3}\big]\Delta s\bigg)}\\ &=\sqrt{\bigg[\int_{0}^{\frac{\pi}{2}}s^{3}ds,\int_{0}^{\frac{\pi}{2}}(s^{3}+2s^{2} +s)ds\bigg]\cdot \bigg[\int_{0}^{\frac{\pi}{2}}s\sin ^{2}sds,\int_{0}^{\frac{\pi}{2}}s^{3}ds\bigg]}\\ &=\sqrt{\bigg[\frac{\pi^{4}}{64},\frac{\pi^{4}}{64}+\frac{\pi^{3}}{12} +\frac{\pi^{2}}{8}\bigg]\cdot\bigg[\frac{\pi^{2}}{16}+\frac{1}{4},\frac{\pi^{4}}{64}\bigg]}\\ &=\sqrt{\bigg[\frac{\pi^{6}}{1024}+\frac{\pi^{4}}{256},\frac{\pi^{8}}{4096} +\frac{\pi^{7}}{768}+\frac{\pi^{6}}{512}\bigg]}\\ &=\Bigg[\sqrt{\frac{\pi^{6}}{1024}+\frac{\pi^{4}}{256}},\sqrt{\frac{\pi^{8}}{4096} +\frac{\pi^{7}}{768}+\frac{\pi^{6}}{512}}\Bigg]. \end{split} \end{equation*}
Consequently, we obtain \begin{equation*} \begin{split} &\bigg[\pi-2,\frac{\pi^{4}}{64}+\frac{\pi^{3}}{24}\bigg]\\ &\leq \Bigg[\sqrt{\frac{\pi^{6}}{1024}+\frac{\pi^{4}}{256}}, \sqrt{\frac{\pi^{8}}{4096}+\frac{\pi^{7}}{768}+\frac{\pi^{6}}{512}}\Bigg]. \end{split} \end{equation*} \end{example}
\begin{example} \label{ex7} Suppose that $[a,b]_{\mathbb{T}}=\{0,1,2,3\}$. Let $h(s)=s$, $f(s)=[s,s+1]$, and $g(s)=[\frac{s}{2},s]$ for $s\in\{0,1,2,3\}$. Then \begin{equation*} \begin{split} \int_{a}^{b}&h(s)f(s)g(s)\Delta s\\ &=\int_{0}^{3}\Big[\frac{s^{3}}{2},s^{3}+s^{2}\Big]\Delta s\\ &=\bigg[\int_{0}^{3}\frac{s^{3}}{2}\Delta s,\int_{0}^{3}s^{3}+s^{2}\Delta s\bigg]\\ &=\bigg[\frac{9}{2},14\bigg], \end{split} \end{equation*} and \begin{equation*} \begin{split} &\sqrt{\Bigg(\int_{a}^{b}h(s)f^{2}(s)\Delta s\Bigg)\Bigg( \int_{a}^{b}h(s)g^{2}(s)\Delta s\Bigg)}\\ &=\sqrt{\bigg(\int_{0}^{3}\big[s^{3},s^{3}+2s^{2}+s\big]\Delta s\bigg) \bigg(\int_{0}^{3}\Big[\frac{s^{3}}{4},s^{3}\Big]\Delta s\bigg)}\\ &=\sqrt{[9,22]\cdot\Big[\frac{9}{4},9\Big]}\\ &=\bigg[\frac{9}{2},3\sqrt{22}\bigg]. \end{split} \end{equation*} Consequently, we obtain $$ \bigg[\frac{9}{2},14\bigg]\leq \bigg[\frac{9}{2},3\sqrt{22}\bigg]. $$ \end{example}
\begin{theorem}[Agarwal et al.\cite{A14}; Wong et al.\cite{W05}] \label{thm4.12} Let $f,g,h\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R})$ and $p>1$. Then, \begin{equation*} \begin{split}
&\bigg(\int_{a}^{b}|h(s)||f(s)+g(s)|^{p}\Delta s\bigg)^{\frac{1}{p}}\\
&\leq \bigg(\int_{a}^{b}|h(s)||f(s)|^{p}\Delta s\bigg)^{\frac{1}{p}}
+\bigg(\int_{a}^{b}|h(s)||g(s)|^{p}\Delta s\bigg)^{\frac{1}{p}}. \end{split} \end{equation*} \end{theorem}
By the same technique used in the proof of Theorem~4 in \cite{R16}, we get a more general result.
\begin{theorem}[Minkowski's inequality] \label{thm4.13} \\Let $h\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R})$, $f,g\in C([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}^{+})$ and \\$p>1$. Then, \begin{equation*} \begin{split}
&\bigg(\int_{a}^{b}|h(s)|(f(s)+g(s))^{p}\Delta s\bigg)^{\frac{1}{p}}\\
&\leq \bigg(\int_{a}^{b}|h(s)|f^{p}(s)\Delta s\bigg)^{\frac{1}{p}}
+\bigg(\int_{a}^{b}|h(s)|g^{p}(s)\Delta s\bigg)^{\frac{1}{p}}. \end{split} \end{equation*} \end{theorem}
\begin{proof} By hypothesis, we have \begin{equation*} \begin{split}
&\bigg(\int_{a}^{b}|h(s)|(f(s)+g(s))^{p}\Delta s\bigg)^{\frac{1}{p}}\\
&=\bigg(\int_{a}^{b}|h(s)|\big[\underline{f}(s)+\underline{g}(s), \overline{f}(s)+\overline{g}(s)\big]^{p}\Delta s\bigg)^{\frac{1}{p}}\\
&=\bigg(\int_{a}^{b}|h(s)|\big[(\underline{f}(s) +\underline{g}(s))^{p},(\overline{f}(s)+\overline{g}(s))^{p}\big]\Delta s\bigg)^{\frac{1}{p}}\\
&=\Bigg[\bigg(\int_{a}^{b}|h(s)|(\underline{f}(s)+\underline{g}(s))^{p}\Delta s\bigg)^{\frac{1}{p}},\\
&\ \ \ \ \ \ \ \ \ \bigg(\int_{a}^{b}|h(s)|(\overline{f}(s) +\overline{g}(s))^{p}\Delta s\bigg)^{\frac{1}{p}}\Bigg]\\
&\leq \Bigg[\bigg(\int_{a}^{b}|h(s)|\underline{f}^{p}(s)\Delta s\bigg)^{\frac{1}{p}}
+\bigg(\int_{a}^{b}|h(s)|\underline{g}^{p}(s)\Delta s\bigg)^{\frac{1}{p}},\\
&\ \ \ \bigg(\int_{a}^{b}|h(s)|\overline{f}^{p}(s)\Delta s\bigg)^{\frac{1}{p}}
+\bigg(\int_{a}^{b}|h(s)|\overline{g}^{p}(s)\Delta s\bigg)^{\frac{1}{p}}\Bigg]\\
&= \Bigg[\bigg(\int_{a}^{b}|h(s)|\underline{f}^{p}(s)\Delta s\bigg)^{\frac{1}{p}},
\bigg(\int_{a}^{b}|h(s)|\overline{f}^{p}(s)\Delta s\bigg)^{\frac{1}{p}}\Bigg]\\
&\ \ +\Bigg[\bigg(\int_{a}^{b}|h(s)|\underline{g}^{p}(s)\Delta s\bigg)^{\frac{1}{p}},
\bigg(\int_{a}^{b}|h(s)|\overline{g}^{p}(s)\Delta s\bigg)^{\frac{1}{p}}\Bigg]\\
&= \Bigg[\int_{a}^{b}|h(s)|\underline{f}^{p}(s)\Delta s,\int_{a}^{b}|h(s)| \overline{f}^{p}(s)\Delta s\Bigg]^{\frac{1}{p}}\\
&\ \ \ \ \ \ +\Bigg[\int_{a}^{b}|h(s)|\underline{g}^{p}(s)\Delta s,
\int_{a}^{b}|h(s)|\overline{g}^{p}(s)\Delta s\Bigg]^{\frac{1}{p}}\\
&=\bigg(\int_{a}^{b}|h(s)|\big[\underline{f}(s), \overline{f}(s)\big]^{p}\Delta s\bigg)^{\frac{1}{p}}\\
&\ \ \ \ \ +\bigg(\int_{a}^{b}|h(s)|\big[\underline{g}(s), \overline{g}(s)\big]^{p}\Delta s\bigg)^{\frac{1}{p}}\\
&=\bigg(\int_{a}^{b}|h(s)|f^{p}(s)\Delta s\bigg)^{\frac{1}{p}}
+\bigg(\int_{a}^{b}|h(s)|g^{p}(s)\Delta s\bigg)^{\frac{1}{p}}. \end{split} \end{equation*} The proof is complete. \end{proof}
\begin{example} \label{ex8} Suppose that $[a,b]_{\mathbb{T}}=[0,1]\cup \{2\}$. Let\\ $h(s)=s$, $f(s)=[s,2s]$, $g(s)=[s,e^{s}]$ and $p=2$. Then, \begin{equation*} \begin{split}
&\bigg(\int_{a}^{b}|h(s)|(f(s)+g(s))^{p}\Delta s\bigg)^{\frac{1}{p}}\\ &=\sqrt{\int_{0}^{2}\big[4s^{3},se^{2s}+4s^{2}e^{s}+4s^{3}\big]\Delta s}\\ &=\sqrt{\bigg[\int_{0}^{2}4s^{3}\Delta s,\int_{0}^{2}se^{2s}+4s^{2}e^{s}+4s^{3}\Delta s\bigg]}\\ &=\Bigg[\sqrt{5},\frac{\sqrt{5e^{2}+32e-11}}{2}\Bigg], \end{split} \end{equation*} and \begin{equation*} \begin{split}
&\bigg(\int_{a}^{b}|h(s)|f^{p}(s)\Delta s\bigg)^{\frac{1}{p}}
+\bigg(\int_{a}^{b}|h(s)|g(^{p}s)\Delta s\bigg)^{\frac{1}{p}}\\ &=\sqrt{\int_{0}^{2}\big[s^{3},4s^{3}\big]\Delta s} +\sqrt{\int_{0}^{2}\big[s^{3},se^{2s}\big]\Delta s}\\ &=\bigg[\frac{\sqrt{5}}{2},\sqrt{5}\bigg]+\bigg[ \frac{\sqrt{5}}{2},\frac{\sqrt{5e^{2}+1}}{2}\bigg]\\ &=\bigg[\sqrt{5},\sqrt{5}+\frac{\sqrt{5e^{2}+1}}{2}\bigg]. \end{split} \end{equation*} Consequently, we obtain $$ \Bigg[\sqrt{5},\frac{\sqrt{5e^{2}+32e-11}}{2}\Bigg] \leq \bigg[\sqrt{5},\sqrt{5}+\frac{\sqrt{5e^{2}+1}}{2}\bigg]. $$ \end{example}
The next results follow directly from Theorems~\ref{thm4.10} and \ref{thm4.13}, respectively.
\begin{corollary} \label{cor4.1} Let $h\in C_{rd}([a,b]_{\mathbb{T}},(0,\infty))$, and\\ $f,g\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}^{-})$. If $\frac{1}{p}+\frac{1}{q}=1$, with $p>1$, then \begin{equation*} \begin{split} &\int_{a}^{b}h(s)f(s)g(s)\Delta s\\ &\leq \Bigg(\int_{a}^{b}h(s)(-f)^{p}(s)\Delta s\Bigg)^{\frac{1}{p}}\Bigg( \int_{a}^{b}h(s)(-g)^{q}(s)\Delta s\Bigg)^{\frac{1}{q}}. \end{split} \end{equation*} \end{corollary}
\begin{corollary} \label{cor4.2} Let $h\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R})$, $f,g\in C([a,b]_{\mathbb{T}},\mathbb{R}_{\mathcal{I}}^{-})$ and $p\in2^{\mathbb{N}}$. Then, \begin{multline*}
\bigg(\int_{a}^{b}|h(s)|(f(s)+g(s))^{p}\Delta s\bigg)^{\frac{1}{p}}\\
\leq \bigg(\int_{a}^{b}|h(s)|f^{p}(s)\Delta s\bigg)^{\frac{1}{p}}
+\bigg(\int_{a}^{b}|h(s)|g^{p}(s)\Delta s\bigg)^{\frac{1}{p}}. \end{multline*} \end{corollary}
\section{Conclusion} \label{sec:5}
We investigated Darboux and Riemann interval delta integrals for interval-valued functions on time scales. Inequalities for interval-valued functions were proved. Our results generalize previous inequalities presented by Costa \cite[Corollary 3.3, Theorem~3.4, Theorem~3.5]{C17} and Rom\'{a}n-Flores \cite[Theorem~4]{R16}.
\end{document}
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arXiv
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\begin{document}
\begin{frontmatter}{}
\title{A $k$-additive Choquet integral-based approach to approximate the SHAP values for local interpretability in machine learning}
\author[aff1,aff2]{Guilherme Dean Pelegrina\corref{fn1}} \ead{[email protected]} \author[aff1]{Leonardo Tomazeli Duarte} \ead{[email protected]} \author[aff2,aff3]{Michel Grabisch} \ead{[email protected]}
\cortext[fn1]{Corresponding author}
\address[aff1]{School of Applied Sciences - University of Campinas, Limeira, Brazil} \address[aff2]{Centre d’Économie de la Sorbonne - Université Paris I Panthéon-Sorbonne, Paris, France} \address[aff3]{Paris School of Economics - Université Paris I Panthéon-Sorbonne, Paris, France}
\begin{abstract}
Besides accuracy, recent studies on machine learning models have been addressing the question on how the obtained results can be interpreted. Indeed, while complex machine learning models are able to provide very good results in terms of accuracy even in challenging applications, it is difficult to interpret them. Aiming at providing some interpretability for such models, one of the most famous methods, called SHAP, borrows the Shapley value concept from game theory in order to locally explain the predicted outcome of an instance of interest. As the SHAP values calculation needs previous computations on all possible coalitions of attributes, its computational cost can be very high. Therefore, a SHAP-based method called Kernel SHAP adopts an efficient strategy that approximate such values with less computational effort. In this paper, we also address local interpretability in machine learning based on Shapley values. Firstly, we provide a straightforward formulation of a SHAP-based method for local interpretability by using the Choquet integral, which leads to both Shapley values and Shapley interaction indices. Moreover, we also adopt the concept of $k$-additive games from game theory, which contributes to reduce the computational effort when estimating the SHAP values. The obtained results attest that our proposal needs less computations on coalitions of attributes to approximate the SHAP values.
\end{abstract}
\begin{keyword} Local interpretability; Choquet integral; Machine learning; Shapley values \end{keyword}
\end{frontmatter}{}
\section{Introduction} \label{sec:intro}
In the last decade, Machine Learning (ML) models have been used to deal with problems that directly affect people's life, such as consumer credit scoring~\citep{Kruppa2013}, cybersecurity~\citep{Xin2018}, disease detection~\citep{Ahsan2022} and patient care evaluation~\citep{BenIsrael2020}. Aiming at dealing with such problems, complex ML models have been proposed to achieve good solutions in terms of accuracy. Examples include random forests~\citep{Fawagreh2014,Biau2016}, deep neural networks~\citep{LeCun2015,Goodfellow2016} and gradient boosting algorithms~\citep{Bentejac2021}. Despite the good performance in terms of accuracy, these models act as black box models, since the obtained results (predictions and/or classifications) are difficult to be interpreted. Therefore, there is an inherent trade-off between adopting an accurate model, whose structure is frequently complex, or an interpretable model, such as linear/logistic regression~\citep{Molnar2021}.
Interpretability plays an important role in machine learning-based automatic decisions and has been discussed in several recent works in the ML community~\citep{Lipton2018,Gilpin2018,Carvalho2019,Molnar2021,Setzu2021}. As stated by~\citet{Miller2019}, interpretability can be defined as ``\textit{the degree to which an observer can understand the cause of a decision}''. Therefore, we can argue that interpretability is as important as accuracy in automatic decisions since it can show if the model can or cannot be trusted. For example, suppose a situation in which a person asks for a credit to his/her bank manager. Moreover, suppose that, after an internal analysis based on a machine learning model, the bank system classifies that person as a possible default and, as a consequence, he/she would not receive the credit. He/she will naturally ask to the bank manager why such a classification was achieved. If the machine is a black box, the manager would not be able to explain such a classification and, therefore, the client may not to trust the algorithm. Therefore, in this situation, a local interpretation would be suitable to understand how each characteristic (e.g., salary, presence or absence of previous default, etc...) contributed to the default credit classification.
There are practically two main types of interpretability in machine learning: global and local ones (see~\citep{Molnar2021} for a further discussion on them). The aim of global interpretability methods consists in explaining the trained model as a whole. In other words, one attempts to derive the average behavior of a trained machine by taking all samples. An example of such a method is the partial dependence plot~\citep{Molnar2021}, whose goal is to provide the marginal effects that each feature has in the predicted outcome. On the other hand, methods for local interpretability attempts to explain, for a specific instance of interest (e.g., a person asking for a credit), how each attribute's value contributes to achieve the associated prediction or classification. In this paper, we deal with local interpretability. Moreover, we consider a model-agnostic approach, i.e., a method that can be applied to interpret the prediction or classification of any machine learning model.
Among the model-agnostic methods proposed in the literature, two are of interest in this paper: LIME (Local Interpretable Model-agnostic Explanations)~\citep{Ribeiro2016} and SHAP (SHapley Additive exPlanations)~\citep{Lundberg2017}. In summary, the idea in LIME to understand the prediction of a specific instance consists in, locally, adjusting an interpretable function (e.g., a linear model) based on a set of perturbed samples in the neighborhood of the instance of interest. When adjusting this linear function, one considers an exponential kernel that ensures that closer are the perturbed samples from the instance of interest, greater are their importance in the learning procedure. Although this function may not be complex enough to explain the model as a whole, it can locally provide a good understanding of the contribution of each attribute in the model prediction. The other approach, called SHAP, brings concepts from game theory to provide local interpretability. The idea is to explain a prediction by means of the Shapley value~\citep{Shapley1953} associated with each attribute value. An interesting aspect in such an approach, which leads to the SHAP values\footnote{In this paper, since the SHAP values are referred to as the Shapley values obtained by means of the SHAP formulation, we will frequently adopt SHAP values or Shapley values interchangeably in the context of machine learning interpretability.}, is that it satisfies desired properties in interpretability, such as local accuracy, missingness and consistency~\citep{Lundberg2017}. For that reason, the classical SHAP and its extended versions have been largely used in the literature~\citep{Lundberg2020,Chen2021,Aas2021}.
Although the Shapley value (as well as the SHAP value in SHAP method) appears as an interesting solution for model-agnostic machine learning interpretability, there is a drawback in its calculation. Since it lies on the marginal contribution of each attribute by taking into account all possible coalitions of attributes, the number of evaluations increases exponentially with the number of attributes. Precisely, if we have $m$ attributes, we need $2^m$ evaluations to calculate the Shapley values. This makes the calculation impracticable in situations where $m$ is large. In order to soften this inconvenience, one may adopt some approaches that approximate the Shapley values, such as the Shapley sampling values strategy~\citep{Strumbelj2010,Strumbelj2014} or the Kernel SHAP~\citep{Lundberg2017}. We address the latter in this paper.
The Kernel SHAP was also proposed in the original SHAP paper~\citep{Lundberg2017}. It provides a link between LIME and the use of SHAP values for local machine learning interpretability. Although the authors provide this link by assuming an additive function as the interpretable model and a specific kernel, the formulation is not straightforward and there is a lack of details in the proof. With respect to the SHAP values calculation, in order to reduce the computational effort, the authors adopted a clever strategy that selects the evaluations that are most promising to approximate such values. However, this strategy does not reduce the number of evaluations needed for an exact SHAP values calculation. Moreover, the authors do not assume further game theory concepts that could speed up the convergence.
Aiming at providing a straightforward formulation of a Kernel SHAP-based method and speeding up the SHAP values approximation, in this paper, we propose to adopt game theory-based concepts frequently used in multicriteria decision making: the Choquet integral~\citep{Choquet1954,Grabisch1996,Grabisch2010} and $k$-additive games~\citep{Grabisch1997b}. Instead of assuming an additive function as the interpretable model, as a first contribution of this paper, we show that the use of the non-additive function called Choquet integral also leads to the same desired properties for local interpretability. Indeed, we can directly associate the Choquet integral parameters to the Shapley indices, which include both Shapley values and Shapley interaction indices. While the Shapley values indicate the marginal contribution of each attribute, individually, the Shapley interaction indices provide the understanding about how they interact between them (positively or negatively). This is of interest in ML interpretability since it indicates if the simultaneous presence of two characteristics has a higher (or lower) contribution than both of them isolated. It is worth mentioning that~\citet{Lundberg2020} also discuss how the Shapley inter SHAP method could be adapted to find the Shapley interaction indices. However, in our proposal, they are obtained automatically.
Besides the aforementioned formulation, we can also assume some degree of additivity about the Choquet integral which contributes to reduce its number of parameters. Therefore, as a second contribution, we propose to adopt a $k$-additive Choquet integral. For instance, $2$-additive models have proved flexible enough to achieve good results in terms of generalization~\citep{Grabisch2002,Grabisch2006,Pelegrina2020}. As attested by numerical experiments, by reducing the number of parameters, we also decrease the number of evaluations needed to approximate the SHAP values.
The rest of this paper is organized as follow. Section~\ref{sec:theor} contains the theoretical aspects of Shapley values and the adopted Choquet integral. We also provide a description of LIME and SHAP as model-agnostic methods for local interpretability. In Section~\ref{sec:prop}, we present our Choquet integral-based formulation that leads to the Shapley interaction indices and how the concept of $k$-additive games can be used to reduce the effort when estimating the SHAP values. Thereafter, in Section~\ref{sec:exp}, we conduct some numerical experiments in order to attest our proposal. Finally, in Section~\ref{concl}, we present our concluding remarks and discuss future perspectives.
\section{Background} \label{sec:theor}
In this section, we present some theoretical aspects that will be used along this work. We start by some concepts frequently used in game theory and multicriteria decision making. Thereafter, we discuss both LIME and SHAP as well as the SHAP values approximation strategy used in Kernel SHAP.
It is worth recalling that, in this paper, we deal with local interpretability. Therefore, we consider a classical machine learning scenario where a model $f(\cdot)$ (e.g., a black box model) has been trained based on a set of $n_{tr}$ training data $\left(\mathbf{X},\mathbf{y} \right)$, where $\mathbf{X} = \left[\mathbf{x}_1, \ldots, \mathbf{x}_{n_{tr}} \right]$ and $\mathbf{y} = \left[y_1, \ldots, y_{n_{tr}} \right]$ represent the inputs and the output (e.g., a predicted value or a class), respectively. Our aim consists in explaining the predicted outcome $f(\mathbf{x}^*)$ of the instance of interest $\mathbf{x}^* = \left[x_1^*, \ldots, x_m^* \right]$, where $m$ is the number of attributes. Therefore, how $f(\cdot)$ was trained is not important in this paper. We only consider that we are able to use the model $f(\cdot)$ in order to predict the outcome of any instance.
\subsection{Shapley values and $k$-additive games} \label{subsec:shapley}
In cooperative game theory, a coalitional game is defined by a set $M=\left\{1, 2, \ldots, m\right\}$ of $m$ players and a function $\upsilon: \mathcal{P}(M) \rightarrow \mathbb{R}$, where $\mathcal{P}(M)$ is the power set of $M$, that maps subsets of players to real numbers. For a coalition of players $A$, $\upsilon(A)$ represents the payoff that this coalition can obtain by cooperation. By definition, one assumes $\upsilon(\emptyset) = 0$, i.e., there is no payoff when there is no coalition.
The Shapley value (or Shapley power index) is a well-known solution concept in cooperative game theory~\citep{Shapley1953}. In summary, the Shapley value of a player $j$ indicates its (positive or negative) marginal contribution in the game payoff when taking into account all possible coalitions of players in $M$. It is defined as follows: \begin{equation} \label{eq:power_ind_s}
\phi_{j} = \sum_{A \subseteq M\backslash \left\{j\right\}} \frac{\left(m-\left|A\right|-1\right)!\left|A\right|!}{m!} \left[\upsilon(A \cup \left\{j\right\}) - \upsilon(A) \right], \end{equation}
where $\left| A \right|$ represents the cardinality of subset $A$. An interesting property of the Shapley value (called efficiency, which will be further discussed in this paper) is that $\sum_{j=1}^m \phi_j = \upsilon(M) - \upsilon(\emptyset)$. For this reason, the Shapley value is a convenient way of sharing the payoff of the grand coalition between the players.
Similarly as in Equation~\eqref{eq:power_ind_s}, one may also measure the marginal effect of a coalition $\left\{j,j'\right\}$ in the payoffs. In this case, one obtains the Shapley interaction index, which is defined by~\citep{Murofushi1993,Grabisch1997a} \begin{equation} \label{eq:int_pair_ind_s}
I_{j,j'} = \sum_{A \subseteq M\backslash \left\{j,j'\right\}} \frac{\left(m-\left|A\right|-2\right)!\left|A\right|!}{\left(m-1\right)!} \left[\upsilon(A \cup \left\{j,j'\right\}) - \upsilon(A \cup \left\{j\right\}) - \upsilon(A \cup \left\{j'\right\}) + \upsilon(A)\right] \end{equation} and can be interpreted as the interaction degree of coalition $\left\{j,j'\right\}$ by taking into account all possible coalitions of players in $M$. The sign of $I_{j,j'}$ indicates the type of interaction between players $j,j'$: \begin{itemize}
\item If $I_{j,j'} < 0$, there is a negative interaction (also called redundant effect) between players $j,j'$.
\item If $I_{j,j'} > 0$, there is a positive interaction (also called complementary effect) between players $j,j'$.
\item If $I_{j,j'} = 0$, there is no interaction players $j,j'$.
\end{itemize}
Besides $\phi_{j}$ and $I_{j,j'}$, one may also define the interaction index for any $A \subseteq M$. In this case, the (generalized) interaction index is defined by~\citep{Grabisch1997a} \begin{equation} \label{eq:int_ind_s}
I(A) = \sum_{D \subseteq M\backslash A} \frac{\left(m-\left|D\right|-\left|A\right|\right)!\left|D\right|!}{\left(m-\left|A\right|+1\right)!} \left( \sum_{D' \subseteq A} \left(-1\right)^{\left| A \right| - \left| D' \right|}\upsilon(D \cup D') \right). \end{equation} However, one does not have a clear interpretation as for $\phi_{i}$ and $I_{i,i'}$.
It is important to remark that, given the interaction indices $I(A)$, one may recover the payoffs $\upsilon(A)$ through the linear transformation \begin{equation} \label{eq:iitomu_s}
\upsilon(A) = \sum_{D \subseteq M} \gamma^{\left|D \right|}_{\left| A \cap D \right|}I(D), \end{equation}
where $\gamma^{\left|D \right|}_{\left| A \cap D \right|}$ is defined by \begin{equation} \label{eq:gamma} \gamma^{r'}_{r} = \sum_{l=0}^{r}\binom{r}{l}\eta_{r'-l}, \end{equation} with \begin{equation} \eta_{r} = -\sum_{r'=0}^{r-1}\frac{\eta_{r'}}{r-r'+1}\binom{r}{r'} \end{equation} being the Bernoulli numbers and $\eta_0=1$. Since the relation between the game and the interaction indices is linear, it is common to represent the aforementioned transformations using matrix notation. Assume, for instance, that the vectors $\upsilon = [\upsilon(\emptyset), \upsilon(\left\{ 1 \right\}), \ldots, \upsilon(\left\{ m \right\}), \upsilon(\left\{ 1,2 \right\}), \ldots, \upsilon(\left\{ m-1,m \right\}), \ldots,$ $\upsilon(\left\{ 1, \ldots, m \right\}) ]$ and $\mathbf{I} = \left[I(\emptyset), \phi_1, \ldots, \phi_m, I_{1,2}, \ldots, I_{m-1,m}, \ldots, I(\left\{ 1, \ldots, m \right\}) \right]$ are represented in a cardinal-lexicographic order (i.e., the elements are sorted according to their cardinality and, for each cardinality, based on the lexicographic order). The transformation from the interaction indices to $\upsilon$ can be represented by $\upsilon = \mathbf{T} \mathbf{I}$, where $\mathbf{T} \in \mathbb{R}^{2^{m} \times 2^{m}}$ is the transformation matrix. For example, in a game with 3 players, we have \begin{equation*} \mathbf{T} =\left[\begin{matrix} 1 & -1/2 & -1/2 & -1/2 & 1/6 & 1/6 & 1/6 & 0 \\ 1 & 1/2 & -1/2 & -1/2 & -1/3 & -1/3 & 1/6 & 1/6 \\ 1 & -1/2 & 1/2 & -1/2 & -1/3 & 1/6 & -1/3 & 1/6 \\ 1 & -1/2 & -1/2 & 1/2 & 1/6 & -1/3 & -1/3 & 1/6 \\ 1 & 1/2 & 1/2 & -1/2 & 1/6 & -1/3 & -1/3 & -1/6 \\ 1 & 1/2 & -1/2 & 1/2 & -1/3 & 1/6 & -1/3 & -1/6 \\ 1 & -1/2 & 1/2 & 1/2 & -1/3 & -1/3 & 1/6 & -1/6 \\ 1 & 1/2 & 1/2 & 1/2 & 1/6 & 1/6 & 1/6 & 0 \\ \end{matrix}\right]. \end{equation*}
Another concept in game theory directly associated with the interaction indices is the concept of $k$-additive games. We say that a game is $k$-additive if $I(A) = 0$ for all $A$ such that $\left| A \right| > k$. As it will be further detailed in the next section, an advantage of such games is that one reduces the number of parameters to be defined (e.g., from $2^m$ to $m(m+1)/2$ when $k=2$). In the example with 3 players, for instance, if one assumes a $2$-additive game, the last column of $\mathbf{T}$ can be removed since $I({1,2,3}) = 0$.
\subsection{The Choquet integral} \label{subsec:choquet}
The (discrete) Choquet integral~\citep{Choquet1954} is a non-additive (more precisely, a piecewise linear) aggregation function that models interactions among attributes. It is defined on a set of parameters associated with all possible coalitions of attributes. It has been largely used in multicriteria decision making problems~\citep{Grabisch1996,Grabisch2010} and, in such situations, the parameters associated with the Choquet integral are called capacity coefficients. A capacity is a set function $\mu:2^{M} \rightarrow \mathbb{R}_{+}$ satisfying the axioms of normalization ($\mu(\emptyset) = 0$ and $\mu(M) = 1$) and monotonicity (if $A \subseteq D \subseteq M$, $\mu(A) \leq \mu(D) \leq \mu(M)$). However, the Choquet integral is not restricted to capacities~\citep{Grabisch2016}. Indeed, it can be defined by means of a game $\upsilon:2^{M} \rightarrow \mathbb{R}$ satisfying $\upsilon(\emptyset) = 0$. The Choquet integral definition based on a game $\upsilon$ is given as follows: \begin{equation} \label{eq:model_ci} f_{CI}(\mathbf{x}) = \sum_{j=1}^m (x_{(j)} - x_{(j-1)})\upsilon(\{(j), \ldots, (m)\}), \end{equation} where $\cdot_{(j)}$ indicates a permutation of the indices $j$ such that $0 \leq x_{(1)} \leq x_{(j)} \leq \ldots \leq x_{(m)} \leq 1$ (with $x_{(0)}=0$).
Since the Choquet integral is defined by means of a game, one may define it in terms of Shapley values and interaction indices. Therefore, one has a clear interpretation about the marginal contribution of each feature in the aggregation procedure as well as the interaction degree between them. For instance, if two attributes have a positive (resp. negative) interaction, the payoff of such a coalition is (resp. is not) greater than the sum of its individual payoffs. Moreover, one may also consider the case of a $k$-additive game and, therefore, a $k$-additive Choquet integral~\citep{Grabisch1997b}. For example, if one assumes a $2$-additive game,~\eqref{eq:model_ci} can be formulated as follows: \begin{equation} \label{eq:choquet_2add}
f_{CI}(\mathbf{x}) = \sum_j x_j \left( \phi_j - \frac{1}{2} \sum_{j'} \left| I_{j,j'} \right| \right) + \sum_{I_{j,j'} < 0} (x_j \vee x_{j'}) \left| I_{j,j'} \right| + \sum_{I_{j,j'} > 0} (x_j \wedge x_{j'}) I_{j,j'}, \end{equation} where $\vee$ and $\wedge$ represent the maximum and the minimum operators, respectively. Note that, when learning the Choquet integral parameters, if one assumes a $2$-additive model, one reduces the number of parameters from $2^m$ to $m(m+1)/2$. Therefore, $2$-additive and, more generally, $k$-additive models emerge as a strategy that reduces the computational complexity in optimization tasks and provides a more interpretable model (since one has less parameters to interpret). Moreover, it is also known from multicriteria decision making applications~\citep{Grabisch2002,Grabisch2006,Pelegrina2020} that, even if one adopts a $2$-additive model, the Choquet integral is still being flexible enough to model inter-attributes relations and can achieve a high level of generalization.
It is important to remark that, if one assumes that the game is $1$-additive, the Choquet integral becomes a weighted arithmetic mean.
\subsection{Model-agnostic methods for local interpretability} \label{subsec:lime_shap}
We describe in this section two famous model-agnostic methods for local interpretability: LIME and SHAP. At first, we briefly present the idea behind tabular LIME (i.e., LIME for tabular data). Then, we further discuss the SHAP method, specially the Kernel SHAP strategy. It is worth mentioning that, differently from~\citep{Ribeiro2016,Lundberg2017}, we here adopt a notation based on set theory in order to clearly define the elements used in the considered approaches.
\subsubsection{LIME} \label{subsubsec:lime}
The main idea of LIME~\citep{Ribeiro2016} for local explanations is to locally approximate a (generally) complex function $f(\cdot)$ (frequently obtained by a black box model) by an interpretable model $g(\cdot)$. For this purpose, in order to explain the outcome $f(\mathbf{x}^*)$ of an instance $\mathbf{x}^*$, one firstly generates a set of $q$ perturbed samples $\mathbf{z}_l$, $l=1, \ldots, q$, in the neighborhood of $\mathbf{x}^*$. For each sample $\mathbf{z}_l$, one also defines a binary vector $\mathbf{z}_l'$ such that $z_{l,j}' = 1$ if $z_{l,j}$ is close enough\footnote{In order to define how close $z_{l,j}$ is from $x_{j}^*$, for each attribute, LIME equally splits the training data (by taking the quantiles of the training data) into predefined bins. Therefore, if $z_{l,j}$ is on the same bin as $x_{j}^*$, $z_{l,j}' = 1$, or $z_{l,j}' = 0$ otherwise. For further details about this procedure, the interested reader may refer to~\citep{Garreau2020}} to $x_{j}^*$, or $z_{l,j}' = 0$ otherwise. Once all samples have been generated, LIME deals with the following optimization problem: \begin{equation}
\label{eq:lime_form}
\min_{g \in G} \mathcal{L}(f,g,\pi_{\mathbf{x}^*}) + \Omega(g), \end{equation} where $\mathcal{L}(f,g,\pi_{\mathbf{x}^*})$ is the loss function, $\pi_{\mathbf{x}^*}$ is a proximity measure between the instance to be explained and the perturbed samples and $\Omega(g)$ is a measure of complexity of the interpretable model $g(\cdot)$. In tabular LIME, the authors used the exponential kernel for the proximity measure, which leads to the expression \begin{equation}
\pi_{\mathbf{x}^*}(\mathbf{z}_l') = \exp \left(\frac{-\| \mathbbm{1} - \mathbf{z}_l' \|^2}{\alpha^2} \right), \end{equation}
where $\| \cdot \|$ is the Euclidean norm, $\mathbbm{1}$ is a vector of 1's and $\alpha$ is a positive bandwidth parameter (as default, the authors assumed $\alpha = \sqrt{0.75m}$). By assuming a weighted least squared function for $\mathcal{L}(f,g,\pi_{\mathbf{x}^*})$, a linear function $g(\mathbf{z}') = \beta_0 + \beta^T \mathbf{z}'$ (where $\beta =\left( \beta_1, \ldots, \beta_m \right)$) and letting $\Omega(\beta) = \lambda \| \beta \|^2$ represent a regularization term with $\lambda > 0$, LIME can be formulated as follows: \begin{equation}
\label{eq:lime_opt}
\min_{\beta_0, \beta_1, \ldots, \beta_m} \sum_{l=1}^q \pi_{\mathbf{x}^*}(\mathbf{z}_l') \left( f(\mathbf{z}_l) - \left(\beta_0 + \beta^T \mathbf{z}_l' \right) \right)^2 + \lambda \| \beta \|^2. \end{equation} After solving~\eqref{eq:lime_opt}, one can visualize the obtained parameters $\beta$ and, therefore, interpret the (positive or negative) contribution of each attribute in the predicted outcome in the vicinity of $\mathbf{x}^*$.
\subsubsection{SHAP} \label{subsubsec:shap}
Differently from LIME, the purpose of SHAP is to use the Shapley values in order to locally explain a prediction. The idea is to associate to each attribute its marginal contribution in the predicted outcome. In this section, we present a summary of the idea behind SHAP. Moreover, we discuss the Kernel SHAP, which is a kernel-based approach for approximating the SHAP values by using the LIME formulation. For further details, the interested reader may refer to~\citep{Lundberg2017,Lundberg2018,Lundberg2020,Aas2021}.
The idea that brings Shapley values into interpretability methods in machine learning associates players and payoffs in game theory to attributes and values of a subset of attributes in the model prediction, respectively. Before presenting the idea behind SHAP, let us define the characteristic vector of $A$. Recall that $M = \left\{1, \ldots, m \right\}$ represents the set of $m$ attributes. For any $A \subseteq M$, $\mathbf{1}_A \in \{0,1\}^m$ denotes the characteristic vector of $A$, i.e., a binary vector such that the $j$-th coordinate is 1, if $j \in A$, and 0, otherwise. For example, for $M = \left\{1, 2, 3 \right\}$, $\mathbf{1}_{\left\{2,3\right\}} = \left[0,1,1 \right]$ means a coalition of attributes $\left\{2, 3 \right\}$.
Based on the aforementioned definition, in order to explain the predicted outcome $f(\mathbf{x}^*)$ of an instance $\mathbf{x}^*$, the authors decompose $f(\mathbf{x}^*)$ by assuming the additive feature attribution function given by \begin{equation}
\label{eq:shap_g}
f(\mathbf{x}^*) = g(\mathbf{1}_M) = \phi_0 + \sum_{j \in M} \phi_j. \end{equation} Moreover, they argue that the only possible explanation model $g(\cdot)$ that follows Equation~\eqref{eq:shap_g} and satisfies the local accuracy, missingness and consistency properties (see Appendix A for the definitions) consists in defining $\phi_0 = \mathbb{E}\left[f(\mathbf{x})\right]$, i.e., the (overall) expected prediction when one does not know any attribute value from $\mathbf{x}^*$, and the (exact) SHAP values $\phi_{j}$, $j=1, \ldots, m$, given by \begin{equation} \label{eq:shap_values}
\phi_{j}(f,\mathbf{x}^*) = \sum_{A \subseteq M \backslash \left\{j\right\}} \frac{\left(m-\left| A \right|-1\right)! \left| A \right|!}{m!} \left[ \hat{f}_{\mathbf{x}^*}( A \cup \left\{j\right\}) - \hat{f}_{\mathbf{x}^*}( A) \right], \end{equation} where $\hat{f}_{\mathbf{x}^*}( A)$ is the expected model prediction given the knowledge on the attributes values of $\mathbf{x}^*$ that are present in coalition $A$, that is: \begin{equation} \label{eq:exp_pred}
\hat{f}_{\mathbf{x}^*}( A) = \mathbb{E}\left[f\left( \mathbf{x} \right) | x_j = x_j^* \, \, \forall \, \, j \in A \right]. \end{equation} Note in Equation~\eqref{eq:exp_pred} that one has missing values for all attributes $j' \in \overline{A}$, where $\overline{A}$ is the complement set of $A$ (if $A = M$, then $\hat{f}_{\mathbf{x}^*}( M) = \mathbb{E}\left[f\left( \mathbf{x}^* \right) \right] = f\left( \mathbf{x}^* \right)$ and there are no missing values). In this case, in order to calculate the expected prediction, one randomly samples these missing values from the training data. In this paper, as well as in the Kernel SHAP method, we assume independence among attributes. Therefore, the expected prediction can be calculated as follows: \begin{equation} \label{eq:exp_pred_indep} \hat{f}_{\mathbf{x}^*}( A) = \frac{1}{q} \sum_{l=1}^q f\left(\mathbf{x}_{A}^*,\mathbf{x}_{l,\overline{A}} \right), \end{equation} where $\mathbf{x}_{l,\overline{A}}$, $l=1, \ldots, q$, are samples from the training data. Note that, in comparison with the game theory formulation presented in Equation~\eqref{eq:power_ind_s}, $\hat{f}_{\mathbf{x}^*}( A)$ represents the payoff $\upsilon(A)$. Moreover, when all attributes are missing, i.e., $A = \emptyset$, one has $\hat{f}_{\mathbf{x}^*}( \emptyset ) = \mathbb{E}\left[f\left( \mathbf{x} \right)\right] = \phi_0$.
Among the properties satisfied by the SHAP values, the local accuracy plays an important role in local interpretability and differentiate SHAP from the original LIME formulation (as presented in Section~\ref{subsubsec:lime}). It states that one can decompose the predicted outcome $f(\mathbf{x}^*)$ by the sum of the SHAP values and the overall expected prediction $\phi_0$, i.e., $f(\mathbf{x}^*) = \phi_0 + \sum_{j=1}^m \phi_j$. Therefore, one may interpret the SHAP values as the contribution of each attribute when one moves from the overall expected prediction when all attributes are missing to the actual outcome $f(\mathbf{x}^*)$.
\subsubsection{Kernel SHAP} \label{subsubsec:kernel_shap}
An important remark in the exact SHAP values calculation is that one needs to sample all $2^m$ possible coalitions of attributes and calculate its expected model prediction. Therefore, this procedure may be computationally heavy for a large number of attributes. In order to overcome this inconvenience, the authors proposed a SHAP value-based formulation called Kernel SHAP~\citep{Lundberg2017}. Kernel SHAP emerges as the formulation of LIME method that leads to the SHAP values. For instance, the authors claimed that if one assumes \begin{itemize}
\item $\Omega(g) = 0$,
\item $\pi(A) = \frac{(m-1)}{\binom{m}{\left|A\right|} \left|A\right| (m - \left|A\right|)}$,
\item $\mathcal{L}(f,g,\pi) = \sum_{A \in \mathcal{M}} \pi(A) \left( \hat{f}_{\mathbf{x}^*}( A) - g(\mathbf{1}_A) \right)^2$,
where $g(\mathbf{1}_A) = \phi_0 + \sum_{j \in A} \phi_j$ and $\mathcal{M} \subseteq \mathcal{P}(M)$ (recall that $\mathcal{P}(M)$ is the power set of $M$), \end{itemize} the solution of the weighted least square problem \begin{equation}
\label{eq:shap_opt}
\min_{\phi_0, \phi_1, \ldots, \phi_m} \sum_{A \in \mathcal{M}} \frac{(m-1)}{\binom{m}{\left|A\right|} \left|A\right| (m - \left|A\right|)} \left( \hat{f}_{\mathbf{x}^*}( A) - \left( \phi_0 + \sum_{j \in A} \phi_j \right)\right)^2 \end{equation} leads to the SHAP values. Note that, differently from the LIME formulation, $\pi(A)$ in Kernel SHAP only depends on coalition $A$. Moreover, $\pi(A)$ tends to infinity when $A = M$. Therefore, in the optimal solution, $\hat{f}_{\mathbf{x}^*}( M) = f(\mathbf{x}^*) = g(\mathbf{1}_M) = \phi_0 + \sum_{j=1}^m \phi_j$. This ensures that $f(\mathbf{x}^*)$ is explained by the sum of the SHAP values and the overall expected prediction $\mathbb{E}\left[f(\mathbf{x})\right]$. Similarly, when $A = \emptyset$, the associated $\pi(\emptyset)$ also tends to infinity. This ensures that $\hat{f}_{\mathbf{x}^*}( \emptyset ) = \mathbb{E}\left[f(\mathbf{x})\right] = g(\mathbf{1}_{\emptyset}) = \phi_0$. In practice, we replace these infinite values by a big constant (e.g., $10^6$).
As~\eqref{eq:shap_opt} is a weighted least square problem, one may easily represent it (as well as its solution) by means of matrices and vectors (we borrow such a formulation from~\citep{Aas2021}). Suppose that $n_{\mathcal{M}}$ represents the number of elements in $\mathcal{M}$ (i.e., the number of coalitions considered in the optimization problem~\eqref{eq:shap_opt}). Let us also define $\phi = \left[ \phi_0, \phi_1, \ldots, \phi_m \right]$ and $\mathbf{Z} \in \left\{0,1\right\}^{n_{\mathcal{M}} \times (m+1)}$ as the matrix such that the first column is 1 for every row and the remaining $m+1$ columns are composed, in each row, by all $\mathbf{1}_A$, $A \in \mathcal{M}$. Moreover, assume that $\mathbf{f} \in \mathbb{R}^{n_{\mathcal{M}} \times 1}$ and $\mathbf{W} \in \mathbb{R}^{n_{\mathcal{M}} \times n_{\mathcal{M}}}$ are the vector of evaluations $\hat{f}_{\mathbf{x}^*}( A)$ and the diagonal matrix whose elements are given by $\pi(A)$, respectively, associated with all $A \in \mathcal{M}$. For example, in a problem with 3 attributes ($M=\left\{1,2,3 \right\}$) and using $\emptyset$, $\left\{1 \right\}$, $\left\{2 \right\}$, $\left\{1,3 \right\}$ and $M$ as the coalitions of attributes, we have the following: \begin{equation*} \phi = \left[\begin{matrix} \phi_0 \\ \phi_1 \\ \phi_2 \\ \phi_3 \end{matrix}\right], \, \, \mathbf{Z} = \left[\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 \\ \end{matrix}\right], \, \, \mathbf{f} = \left[\begin{matrix} \hat{f}_{\mathbf{x}^*}( \emptyset) \\ \hat{f}_{\mathbf{x}^*}( \left\{1 \right\} ) \\ \hat{f}_{\mathbf{x}^*}( \left\{2 \right\} ) \\ \hat{f}_{\mathbf{x}^*}( \left\{1,3 \right\} ) \\ \hat{f}_{\mathbf{x}^*}( M) \\ \end{matrix}\right]\text{ and }\mathbf{W} = \left[\begin{matrix} 10^6 & 0 & 0 & 0 & 0 \\ 0 & \pi(\left\{1 \right\}) & 0 & 0 & 0 \\ 0 & 0 & \pi(\left\{2 \right\}) & 0 & 0 \\ 0 & 0 & 0 & \pi(\left\{1,3 \right\}) & 0 \\ 0 & 0 & 0 & 0 & 10^6 \\ \end{matrix}\right]. \end{equation*} Based on the vector/matrix notation, one may represent the optimization problem~\eqref{eq:shap_opt} as \begin{equation}
\label{eq:shap_opt_matrix}
\min_{\phi} \left(\mathbf{f} - \mathbf{Z}\phi \right)^T \mathbf{W} \left(\mathbf{f} - \mathbf{Z}\phi \right), \end{equation} whose solution is given by \begin{equation}
\label{eq:shap_opt_matrix_sol}
\phi = \left(\mathbf{Z}^T\mathbf{W}\mathbf{Z} \right)^{-1} \mathbf{Z}^T\mathbf{W}\mathbf{f}. \end{equation}
Remark that $\mathbf{S} = \left(\mathbf{Z}^T\mathbf{W}\mathbf{Z} \right)^{-1} \mathbf{Z}^T\mathbf{W}$ can be calculated independently of the instance of interest $\mathbf{x}^*$. Therefore, an interesting aspect in Kernel SHAP is that, even if one would like to explain the outcome of several instances of interest, one only needs to calculate $\mathbf{S}$ once. The only element that varies in Equation~\eqref{eq:shap_opt_matrix_sol} is the vector of evaluations $\mathbf{f}$, which is dependent on the instance of interest under analysis.
Another remark in Kernel SHAP is that, if $\mathcal{M} = \mathcal{P}(M)$, Equation~\eqref{eq:shap_opt_matrix_sol} leads to the exact SHAP values (as in Equation~\eqref{eq:shap_values}). Therefore, in this exact calculation, one needs the expected predictions $\hat{f}_{\mathbf{x}^*}( A )$ for all possible coalitions $A$, which can be infeasible for a large number of attributes. However, the clever strategy used in Kernel SHAP aims at selecting the most promising expected predictions to approximate the SHAP values. For instance, if one considers the weighting kernel $\pi(A)$, one may note that the majority of $A$ has a low contribution in the SHAP value calculation. Therefore, the aim in Kernel SHAP consists in defining a subset $\mathcal{M}$ from $\mathcal{P}(M)$ such that the elements $A \in \mathcal{M}$ are sampled\footnote{In order to avoid double selecting the same $A$, in the experiments conducted in this paper, we adopted a sampling procedure without replacement. Therefore, after sampling a coalition, we update the probability distribution by removing the associated kernel weight and normalizing the probabilities.} from a probability distribution following the weighting kernel $\pi(A)$. Greater is the weight associated with $A$, greater is the chance that $A$ is sampled from $\mathcal{P}(M)$.
\section{A more general model for local interpretability based on Shapley values} \label{sec:prop}
As highlighted in Section~\ref{sec:intro}, we have two main contributions in this paper: to provide a straightforward formulation of the Kernel SHAP method based on the Choquet integral, and to adopt the concept of $k$-additive games in order to reduce the number of evaluations needed to approximate the SHAP values. Both contributions are presented in the sequel.
\subsection{The Choquet integral as an interpretable model for Kernel SHAP formulation}
We here show that we need not consider an additive function as the interpretable model in order to explain a prediction based on the Shapley values. Indeed, if we adopt the non-additive function called Choquet integral, we also achieve such values. Recall the Choquet integral function defined in Equation~\eqref{eq:model_ci}. The idea is to define the local interpretable model $g(\cdot)$ as \begin{equation}
\label{eq:choquet_lime}
g(\mathbf{1}_A) = \phi_0 + f_{CI}(\mathbf{1}_A), \end{equation} where $\phi_0$ is the intercept parameter. In order to simplify the notation, let us also define $\bar{f}_{\mathbf{x}^*}( A) = \hat{f}_{\mathbf{x}^*}( A) - \phi_0$. In this case and based on the LIME formulation for local interpretability, one obtains the following loss function: \begin{equation}
\label{eq:choquet_lime_loss}
\mathcal{L}(f,g,\pi) = \sum_{A \in \mathcal{M}}\pi'(A) \left( \bar{f}_{\mathbf{x}^*}( A) - f_{CI}(\mathbf{1}_A) \right)^2, \end{equation} where the weights $\pi'(A)$ have the same values for all $A$ (e.g., 1) except for the empty set and the grand coalition $M$, whose associated weights are big numbers (e.g., $10^6$). We clarify these choices soon.
An interesting aspect on the Choquet integral and that can be easily checked from Equation~\eqref{eq:model_ci} is that, when we only have binary data (which is our case since $\mathbf{1}_A$ is a binary vector), $f_{CI}(\mathbf{1}_A) = \upsilon(A)$. Therefore, we may redefine the loss function presented in~\eqref{eq:choquet_lime_loss} as \begin{equation}
\label{eq:choquet_lime_loss_game}
\mathcal{L}(f,g,\pi) = \sum_{A \in \mathcal{M}} \pi'(A) \left( \bar{f}_{\mathbf{x}^*}( A) - \upsilon(A) \right)^2. \end{equation} Remark that, for $A = \emptyset$, we minimize $\bar{f}_{\mathbf{x}^*}( \emptyset) - \upsilon(\emptyset) = \hat{f}_{\mathbf{x}^*}( \emptyset) - \phi_0 - \upsilon(\emptyset) = \upsilon(\emptyset)$, since $\hat{f}_{\mathbf{x}^*}( \emptyset) = \phi_0$ by definition. In order to ensure that $\upsilon(\emptyset) = 0$ (according to the definition of a game), we assume a big number for $\pi'(\emptyset)$ when solving the optimization problem. Similarly, when $A=M$, we minimize the difference between $\hat{f}_{\mathbf{x}^*}( M)$ and $\phi_0 + \upsilon(M)$. In this case, since $\upsilon(M) = \upsilon(\emptyset) + \sum_{j=1}^m \phi_j$, the big weight $\pi'(M)$ ensures that $\phi_0 + \sum_{j=1}^m \phi_j = \hat{f}_{\mathbf{x}^*}( M) = f(\mathbf{x}^*)$ (the local accuracy property).
Furthermore, if one considers the linear transformation presented in Equation~\eqref{eq:iitomu_s}, the loss function can be directly defined in terms of the generalized Shapley interaction indices. In this case, we have the following optimization problem: \begin{equation}
\label{eq:shap_opt_ci}
\min_{\mathbf{I}} \sum_{A \in \mathcal{M}} \pi'(A) \left( \bar{f}_{\mathbf{x}^*}( A) - \sum_{D \subseteq M} \gamma^{\left|D \right|}_{\left| A \cap D \right|}I(D) \right)^2, \end{equation} where $\gamma$ is defined as in Equation~\eqref{eq:gamma}. As in the Kernel SHAP, our proposal also leads to the exact SHAP values if $\mathcal{M}=\mathcal{P}(M)$. We prove it in the sequel.
\begin{theor}{} \label{theo1} If $\mathcal{M} = \mathcal{P}(M)$, the solution of~\eqref{eq:shap_opt_ci} leads to the exact SHAP values as calculated in Equation~\eqref{eq:shap_values}. \end{theor}
\begin{proof}
Assume $\mathcal{M} = \mathcal{P}(M)$. In this scenario, the optimization problem~\eqref{eq:shap_opt_ci} has a unique solution such that $\sum_{D \subseteq M} \gamma^{\left|D \right|}_{\left| A \cap D \right|}I(D) = \upsilon(A) = \bar{f}_{\mathbf{x}^*}( A)$. From the obtained game and the linear transformation presented in Equation~\eqref{eq:power_ind_s}, we have that $\phi_{j} = \sum_{A \subseteq M\backslash \left\{j\right\}} \frac{\left(m-\left|A\right|-1\right)!\left|A\right|!}{m!} \left[\upsilon(A \cup \left\{j\right\}) - \upsilon(A) \right]$. It remains to show that $\phi_{j} \equiv \phi_{j}(f,\mathbf{x}^*)$.
Recall that we defined $\bar{f}_{\mathbf{x}^*}( A) = \hat{f}_{\mathbf{x}^*}( A) - \phi_0$ and, then, $\upsilon(A) = \hat{f}_{\mathbf{x}^*}( A) - \phi_0$ in the optimal solution. Therefore, we have the following: \begin{equation} \label{eq:proof_equiv}
\begin{split}
\phi_{j} & = \sum_{A \subseteq M\backslash \left\{j\right\}} \frac{\left(m-\left|A\right|-1\right)!\left|A\right|!}{m!} \left[\hat{f}_{\mathbf{x}^*}( A \cup \left\{j\right\}) - \phi_0 - \hat{f}_{\mathbf{x}^*}( A) + \phi_0 \right] \\
& = \sum_{A \subseteq M\backslash \left\{j\right\}} \frac{\left(m-\left|A\right|-1\right)!\left|A\right|!}{m!} \left[\hat{f}_{\mathbf{x}^*}( A \cup \left\{j\right\}) - \hat{f}_{\mathbf{x}^*}( A) \right] \\
& = \phi_{j}(f,\mathbf{x}^*),
\end{split} \end{equation} which proves that our proposal also converges to the exact SHAP values when $\mathcal{M} = \mathcal{P}(M)$. \end{proof}
Similarly as in the Kernel SHAP, we may here also rewrite the optimization problem in vector/matrix notation. For this purpose, let us represent $\hat{\mathbf{f}} \in \mathbb{R}^{n_{\mathcal{M}} \times 1}$ as the vector $\mathbf{f}$ (as defined in Section~\ref{subsubsec:kernel_shap}) discounted by $\phi_0$ and $\bar{\mathbf{W}} \in \mathbb{R}^{n_{\mathcal{M}} \times n_{\mathcal{M}}}$ as the diagonal matrix whose elements are 1's except for the elements associated with the empty set and the grand coalition $M$, whose weights are a big number (e.g., $10^6$). Moreover, we define $\upsilon_{\mathcal{M}}$ as the vector of payoffs for all coalitions $A$ such that $A \in \mathcal{M}$. In addition, we consider $\mathbf{T}_{\mathcal{M}}$ as the transformation matrix whose rows are composed by the rows of $\mathbf{T}$ (as defined in Section~\ref{subsec:shapley}) associated with all coalitions $A$ such that $A \in \mathcal{M}$. For example, in the same problem when $M=\left\{1,2,3 \right\}$ and using $\emptyset$, $\left\{1 \right\}$, $\left\{2 \right\}$, $\left\{1,3 \right\}$ and $M$ as the coalitions of attributes, we have the following: \begin{equation*} \upsilon_{\mathcal{M}} = \left[\begin{matrix} \upsilon(\emptyset) \\ \upsilon(\left\{1 \right\}) \\ \upsilon(\left\{2 \right\}) \\ \upsilon(\left\{1,3 \right\}) \\ \upsilon(M) \end{matrix}\right] = \left[\begin{matrix} 1 & -1/2 & -1/2 & -1/2 & 1/6 & 1/6 & 1/6 & 0 \\ 1 & 1/2 & -1/2 & -1/2 & -1/3 & -1/3 & 1/6 & 1/6 \\ 1 & -1/2 & 1/2 & -1/2 & -1/3 & 1/6 & -1/3 & 1/6 \\ 1 & 1/2 & -1/2 & 1/2 & -1/3 & 1/6 & -1/3 & -1/6 \\ 1 & 1/2 & 1/2 & 1/2 & 1/6 & 1/6 & 1/6 & 0 \end{matrix}\right] \left[\begin{matrix} I(\emptyset) \\ \phi_1 \\ \phi_2 \\ \phi_3 \\ I_{1,2} \\ I_{1,3} \\ I_{2,3} \\ I(\left\{1,2,3 \right\}) \end{matrix}\right] = \mathbf{T}_{\mathcal{M}} \mathbf{I}. \end{equation*} \begin{equation*} \hat{\mathbf{f}} = \left[\begin{matrix} \hat{f}_{\mathbf{x}^*}( \emptyset ) - \phi_0 \\ \hat{f}_{\mathbf{x}^*}( \left\{1 \right\} ) - \phi_0 \\ \hat{f}_{\mathbf{x}^*}( \left\{2 \right\} ) - \phi_0 \\ \hat{f}_{\mathbf{x}^*}( \left\{1,3 \right\} ) - \phi_0 \\ \hat{f}_{\mathbf{x}^*}( M ) - \phi_0 \\ \end{matrix}\right]\text{ and }\bar{\mathbf{W}} = \left[\begin{matrix} 10^6 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 10^6 \\ \end{matrix}\right]. \end{equation*} The vector/matrix notation leads to the following optimization problem: \begin{equation}
\label{eq:shap_opt_matrix_ci}
\min_{\mathbf{I}} \left(\bar{\mathbf{f}} - \mathbf{T}_{\mathcal{M}} \mathbf{I} \right)^T \bar{\mathbf{W}} \left(\bar{\mathbf{f}} - \mathbf{T}_{\mathcal{M}} \mathbf{I} \right), \end{equation} whose solution is given by \begin{equation}
\label{eq:shap_opt_matrix_sol_ci}
\mathbf{I} = \left(\mathbf{T}_{\mathcal{M}}^T\bar{\mathbf{W}}\mathbf{T}_{\mathcal{M}} \right)^{-1} \mathbf{T}_{\mathcal{M}}^T\bar{\mathbf{W}}\bar{\mathbf{f}}. \end{equation}
It is important to note that, differently from the Kernel SHAP formulation discussed in Section~\ref{subsubsec:kernel_shap}, in our proposal we obtain all Shapley interaction indices (which, obviously, include the SHAP values). Therefore, this can also be infeasible for a large number of attributes, since the number of parameters is given by $2^m$. However, one may exploit some degree of additivity about the Choquet integral which contributes to reduce its number of parameters. We discuss this aspect in the next section.
\subsection{$k$-additive games for local interpretability} \label{subsec:kadd}
As a second contribution, we propose to adopt the concept of $k$-additive games in the Choquet integral-based formulation for local interpretability. Called here $k_{ADD}$-SHAP, our proposal consists in dealing with the following weighted least square problem: \begin{equation}
\label{eq:shap_opt_ci_kadd}
\min_{\phi_0,\mathbf{I}_k} \sum_{A \in \mathcal{M}} \pi'(A) \left( \bar{f}_{\mathbf{x}^*}( A) - \sum_{\substack{D \subseteq M, \\ \left| D \right| \leq k}} \gamma^{\left|D \right|}_{\left| A \cap D \right|}I(D) \right)^2, \end{equation}
where $\mathbf{I}_k = \left[I(\emptyset), \phi_1, \ldots, \phi_m, I_{1,2}, \ldots, I(\left\{m-k, \ldots, m\right\} \right]$ is the vector of Shapley interaction indices, in a cardinal-lexicographic order, for all $I(D)$ such that $\left| D \right| \leq k$. By using the vector/matrix notation, we may rewrite~\eqref{eq:shap_opt_ci_kadd} as follows: \begin{equation}
\label{eq:shap_opt_matrix_ci_kadd}
\min_{\mathbf{I}_k} \left(\bar{\mathbf{f}} - \mathbf{T}_{\mathcal{M},k} \mathbf{I}_k \right)^T \bar{\mathbf{W}} \left(\bar{\mathbf{f}} - \mathbf{T}_{\mathcal{M},k} \mathbf{I}_k \right), \end{equation} whose solution is given by \begin{equation}
\label{eq:shap_opt_matrix_sol_ci_kadd}
\mathbf{I}_k = \left(\mathbf{T}_{\mathcal{M},k}^T\bar{\mathbf{W}}\mathbf{T}_{\mathcal{M},k} \right)^{-1} \mathbf{T}_{\mathcal{M},k}^T\bar{\mathbf{W}}\bar{\mathbf{f}}, \end{equation}
where $\mathbf{T}_{\mathcal{M},k}$ is equal to $\mathbf{T}_{\mathcal{M}}$ up to the columns associated with all $I(D')$ such that $\left| D' \right| \leq k$ ($I(D') = 0$ for all coalitions $D'$ such that $\left| D' \right| > k$).
Note that, as in~\eqref{eq:shap_opt_ci_kadd} (or\eqref{eq:shap_opt_matrix_ci_kadd}) we restrict the feasible domain to Shapley indices whose cardinalities are at most $k$, we can not guarantee to achieve the exact SHAP values even if $\mathcal{M} = \mathcal{P}(M)$. In other words, Theorem~\ref{theo1} is not valid for the proposed $k_{ADD}$-SHAP. However, as already mentioned in Section~\ref{subsec:choquet}, an advantage of such a model is that one both drastically reduces the number of parameters to be determined while still having a flexible model to generalize the relation between inputs and outputs. Therefore, in order to approximate the exact SHAP values, we avoid over-parametrization and we may need less evaluations when adopting~\eqref{eq:shap_opt_matrix_ci_kadd} compared to~\eqref{eq:shap_opt_matrix}. It is important to note that, even if in the Kernel SHAP formulation one only searches for the Shapley values, implicitly, one needs the expected predicted evaluations on all coalitions of attributes in order to calculate the exact SHAP values.
With respect to how to select the subset $\mathcal{M}$ of evaluations, we consider the same strategy as in Kernel SHAP. We sample the elements $A \in \mathcal{M}$ according to the probability distribution defined by $p_A = \frac{\pi(A)}{\sum_{A \subseteq M} \pi(A)}$. As we adopt in this paper a sampling procedure without replacement, after sampling a coalition, we update the probability distribution and normalize it. Moreover, as $p_\emptyset$ and $p_M$ are much greater than the other probabilities, it is very likely that both the empty set and the grand coalition $M$ are sampled to compose the subset $\mathcal{M}$.
Equivalently as in the Kernel SHAP formulation, $\mathbf{S}_{\mathcal{M},k} = \left(\mathbf{T}_{\mathcal{M},k}^T\bar{\mathbf{W}}\mathbf{T}_{\mathcal{M},k} \right)^{-1} \mathbf{T}_{\mathcal{M},k}^T\bar{\mathbf{W}}$ (or $\mathbf{S}_{\mathcal{M}} = \left(\mathbf{T}_{\mathcal{M}}^T\bar{\mathbf{W}}\mathbf{T}_{\mathcal{M}} \right)^{-1} \mathbf{T}_{\mathcal{M}}^T\bar{\mathbf{W}}$) can also be calculated independently of the instance of interest $\mathbf{x}^*$. Therefore, in order to explain the outcome of several instances of interest, one only needs to calculate $\mathbf{S}_{\mathcal{M},k}$ (or $\mathbf{S}_{\mathcal{M}}$) once.
\section{Numerical experiments} \label{sec:exp}
In this section, we present some numerical experiments in order to check the validity and interest of our model\footnote{All codes can be accessed in \url{https://github.com/GuilhermePelegrina/k_addSHAP}.}. The experiments are based on two datasets frequently used in the literature: Diabetes~\citep{Efron2004} and Red Wine Quality~\citep{Cortez2009}. In the sequel, we provide a brief description of both datasets: \begin{itemize} \item Diabetes dataset: This dataset contains $m=10$ attributes (age, sex, body mass index, average blood pressure and six blood serum measurements) that describe $n=442$ diabetes patients. All collected data are centralized (with zero mean) and with standard deviation equal to $0.0476$. For each patient, one also has as the predicted value a measure of the diabetes progression. The mean and the standard deviation for the diabetes progression measure are $152.13$ and $77.00$, respectively. In our experiments, we split the dataset into training (80\%, i.e., $n_{tr}=353$ samples) and test (20\%, i.e., $n_{te}=89$ samples).
\item Red Wine Quality dataset: In this dataset, one has $m=11$ attributes describing $n=1599$ red wines. Both mean and standard deviation (std) of attributes are described in Table~\ref{tab:wine}. For each wine, one also has a score (between 0 and 10) indicating its quality. In our experiments, we use this data for the purpose of classification and, therefore, we assume that a good (resp. a bad) wine has a score greater than 5 (resp. at most 5). In total, one has 855 good wine (class value 1) and 744 bad wine (class value 0). Moreover, we split the dataset into training (80\%, i.e., $n_{tr}=1279$ samples) and test (20\%, i.e., $n_{te}=320$ samples). \end{itemize}
\begin{table}[ht]
\begin{center}
\caption{Summary of the Wine dataset.}\label{tab:wine}
{
\renewcommand{1.0}{1.0}
\small
\begin{tabular}{cccccccc}
\cline{1-2}\cline{4-5}\cline{7-8} \textbf{Attributes} & \makecell{ \textbf{Mean} \\ \textbf{($\pm$ std)} } & & \textbf{Attributes} & \makecell{ \textbf{Mean} \\ \textbf{($\pm$ std)} } & & \textbf{Attributes} & \makecell{ \textbf{Mean} \\ \textbf{($\pm$ std)} } \\
\cline{1-2}\cline{4-5}\cline{7-8}
fixed acidity & \makecell{ $8.320$ \\ ($\pm 1.740$) } & & chlorides & \makecell{ $0.087$ \\ ($\pm 0.047$) } & & pH & \makecell{ $3.311$ \\ ($\pm 0.154$) } \\
\cline{1-2}\cline{4-5}\cline{7-8}
volatile acidity & \makecell{ $0.528$ \\ ($\pm 0.179$) } & & free sulfur dioxide & \makecell{ $15.875$\\ ($\pm 10.457$) } & & sulphates & \makecell{ $0.658$ \\ ($\pm 0.169$) } \\
\cline{1-2}\cline{4-5}\cline{7-8}
citric acid & \makecell{ $0.271$ \\ ($\pm 0.195$) } & & total sulfur dioxide & \makecell{ $46.468$ \\ ($\pm 32.885$) } & & alcohol & \makecell{ $10.423$ \\ ($\pm 1.065$) } \\
\cline{1-2}\cline{4-5}\cline{7-8}
residual sugar & \makecell{ $2.539$ \\ ($\pm 1.409$) } & & density & \makecell{ $0.997$ \\ ($\pm 0.002$) } & & & \\
\cline{1-2}\cline{4-5}\cline{7-8}
\end{tabular}
}
\end{center} \end{table}
Besides different datasets, we also evaluate our proposal by assuming two training models: Neural Network and Random Forest\footnote{We borrowed these methods from the Scikit-learn library~\citep{Pedregosa2011} in Python and adopted the following parameters: \begin{itemize}
\item Neural Network: $max_{iter} = 10^6$ for both MLPRegressor and MLPClassifier.
\item Random Forest: $n\_estimators=1000$, $max\_depth=None$ and $min\_samples\_split=2$ for both RandomForestRegressor and RandomForestClassifier. \end{itemize}}. Recall that the purpose of this paper is to address interpretability in any trained machine learning models. We do not work on improving the model itself. So we attempt to explain the contributions of attributes regardless how the model is accurate.
\subsection{Experiment varying the number of expected prediction evaluations until the exact SHAP values convergence} \label{subsec:exp1}
In the first experiment, we verify the convergence of the proposed $k_{ADD}$-SHAP and the Kernel SHAP to the exact SHAP values. For each dataset and test sample (recall that we use the training data to calculate the expected predictions given the coalitions in $\mathcal{M}$), we vary the number of expected prediction evaluations, apply both $k_{ADD}$-SHAP and Kernel SHAP and calculate the squared error when estimating the exact SHAP values. Let us represent, for a given test sample $i'$, the SHAP values obtained by the Equation~\eqref{eq:shap_values} (the exact SHAP values), the $k_{ADD}$-SHAP and the Kernel SHAP as $\phi^{exact,i'}$, $\phi^{k_{ADD},i'}$ and $\phi^{Kernel,i'}$, respectively. The squared error between $\phi^{exact,i'}$ and $\phi^{k_{ADD},i'}$ is given as follows: \begin{equation} \label{eq:error}
\varepsilon_{k_{ADD},i'} = \sum_{j=1}^m \left(\phi_j^{exact,i'} - \phi_j^{k_{ADD},i'} \right)^2. \end{equation} In order to calculate the squared error with respect to the Kernel SHAP, one only needs to replace $\phi_j^{k_{ADD},i'}$ by $\phi^{Kernel,i'}$. By increasing $n_{\mathcal{M}}$, i.e., the number of coalitions selected to calculate the expected prediction evaluations used in the estimation procedure, the aim is to verify the convergence to the exact SHAP values. We show the obtained results by taking the median (50th percentile or $q_{0.5}$ - a central tendency measure), the 90th percentile and the 10th percentile ($q_{0.9}$ and $q_{0.5}$, respectively, both used to indicate the dispersion around the median) over $s = 501$ simulations. For each simulation, we calculate the errors when estimating the exact SHAP values of all test samples. For the proposed $k_{ADD}$-SHAP, the percentile $q_a$, $a=0.1,0.5,0.9$, is calculated as follows: \begin{equation} \label{eq:average_error}
\bar{\varepsilon}_{a,k_{ADD}} = q_a \left( \frac{1}{n_{te}}\sum_{i'=1}^{n_{te}} \varepsilon_{k_{ADD},i'}^{1}, \ldots, \frac{1}{n_{te}}\sum_{i'=1}^{n_{te}} \varepsilon_{k_{ADD},i'}^{s} \right) \end{equation} where $\varepsilon_{k_{ADD},i'}^{r}$, $r=1, \ldots, s$ represents the squared error for test sample $i'$ in simulation $r$. Equation~\eqref{eq:average_error} can be easily adapted to calculate the metrics when adopting the Kernel SHAP.
The results are presented in Figures~\ref{fig:convergence_diabetes} and~\ref{fig:convergence_wine}. The central line represents the average median and the shaded area indicates the averaged dispersion between the 10th and 90th percentiles. For both datasets and trained models, the $3_{ADD}$-SHAP leads to a faster approximation to the exact SHAP values in comparison with the Kernel SHAP. Moreover, the dispersion was lower for the $3_{ADD}$-SHAP even with reduced numbers of expected prediction evaluations. For Kernel SHAP, one achieves a high dispersion for low number of expected prediction evaluations (see, especially, Figure~\ref{fig:convergence_diabetes}), which decreases as one includes more samples. With respect to the $2_{ADD}$-SHAP, it has a good performance (better than the Kernel SHAP) for few evaluations, however, it diverges as more samples are include in the SHAP values estimation. An explanation for these results is that the $2_{ADD}$-SHAP could rapidly approximate the exact SHAP values when less evaluations are used because it can avoid over-parametrization when only few data are considered. However, when increasing the number of expected prediction evaluations, the $2_{ADD}$-SHAP has not enough flexibility to model the data and the adjusted parameters could not converge to the correct ones. As can be also seen in Figure~\ref{fig:convergence_wine}, the $3_{ADD}$-SHAP also diverges for a high number of evaluations (recall from Section~\ref{subsec:kadd} that we can not guarantee to achieve the exact SHAP values even if $\mathcal{M} = \mathcal{P}(M)$), however, it still achieves a very low error. Clearly, as we increase $k$, the parameters become more flexible to model the data and estimate the exact SHAP values.
\begin{figure}\label{fig:convergence_diabetes_2add_mlp}
\label{fig:convergence_diabetes_2add_rf}
\label{fig:convergence_diabetes_3add_mlp}
\label{fig:convergence_diabetes_3add_rf}
\label{fig:convergence_diabetes}
\end{figure}
\begin{figure}\label{fig:convergence_wine_2add_mlp}
\label{fig:convergence_wine_2add_rf}
\label{fig:convergence_wine_3add_mlp}
\label{fig:convergence_wine_3add_rf}
\label{fig:convergence_wine}
\end{figure}
\subsection{Experiment comparing the obtained SHAP values} \label{subsec:exp2}
In this experiment, we compare the obtained SHAP values with the exact ones. For an instance of interest among the test data, we use the previous experiment and select the SHAP values that lead to the median error over all the simulations. For ease of visualization, we only plotted the five attributes that contribute the most (either positively or negatively) according to the exact SHAP values. As an illustrative example and without loss of generality, we selected a test sample $\mathbf{x}^*$ from the Diabetes dataset that has the attributes values described in Table~\ref{tab:diabetes_samp} (recall that this dataset is already centered with zero mean). The predicted measure of diabetes progression is equal to 84, which is less than the overall expected prediction provided by both Neural Networks and Random Forest (154.92 and 153.81, respectively). This means that the SHAP values help to explain, for the instance of interest $\mathbf{x}^*$, how each attribute value contributes to decrease the diabetes progression measure from the overall prediction until the actual 84.
\begin{table}[!h]
\begin{center}
\caption{Summary of the selected test sample - Diabetes dataset.}\label{tab:diabetes_samp}
{
\renewcommand{1.0}{1.0}
\small
\begin{tabular}{cccccccc}
\cline{1-2}\cline{4-5}\cline{7-8} \textbf{Attributes} & \textbf{Values} & & \textbf{Attributes} & \textbf{Values} & & \textbf{Attributes} & \textbf{Values} \\
\cline{1-2}\cline{4-5}\cline{7-8}
age & $0.009$ & & blood serum 1 & $0.099$ & & blood serum 5 & $-0.021$ \\
\cline{1-2}\cline{4-5}\cline{7-8}
sex & $-0.045$ & & blood serum 2 & $0.094$ & & blood serum 6 & $0.007$ \\
\cline{1-2}\cline{4-5}\cline{7-8}
body mass index & $-0.024$ & & blood serum 3 & $0.071$ & & & \\
\cline{1-2}\cline{4-5}\cline{7-8}
average blood pressure & $-0.026$ & & blood serum 4 & $-0.002$ & & & \\
\cline{1-2}\cline{4-5}\cline{7-8}
\end{tabular}
}
\end{center} \end{table}
Figure~\ref{fig:shapley_diabetes} presents the estimated SHAP values when using $n_{\mathcal{M}}=290$, $n_{\mathcal{M}}=590$, $n_{\mathcal{M}}=890$ different coalitions of attributes to calculate the expected prediction evaluations. As a first remark, we note that the estimated SHAP values (specially the illustrated five ones) for the Neural Network (Figures~\ref{fig:shapley_diabetes_5_290_mlp},~\ref{fig:shapley_diabetes_5_590_mlp} and~\ref{fig:shapley_diabetes_5_890_mlp}) practically do not change regardless the number of predicted evaluations. All approaches led to very small errors, i.e., they could rapidly approximate the exact SHAP values associated with the Neural Networks model. For the Random Forest, we see that the contributions provided by the $3_{ADD}$-SHAP are close to the exact ones even with small number of predicted evaluations (see Figure~\ref{fig:shapley_diabetes_5_290_rf}). As one increases the number of evaluations, the Kernel SHAP converges to the exact SHAP values.
\begin{figure}
\caption{Comparison between the estimated SHAP values provided by the $2_{ADD}$-SHAP, $3_{ADD}$-SHAP and Kernel SHAP for different machine learning models and varying the number of coalitions used to calculate the expected prediction evaluations (Diabetes dataset).}
\label{fig:shapley_diabetes_5_290_mlp}
\label{fig:shapley_diabetes_5_290_rf}
\label{fig:shapley_diabetes_5_590_mlp}
\label{fig:shapley_diabetes_5_590_rf}
\label{fig:shapley_diabetes_5_890_mlp}
\label{fig:shapley_diabetes_5_890_rf}
\label{fig:shapley_diabetes}
\end{figure}
Regarding the Red Wine dataset, we selected as an illustrative example a test sample classified as a good wine. The attributes values described in Table~\ref{tab:wine_samp}. The overall expected probability prediction for class 1 (good wine) for both Neural Networks and Random Forest is approximately 0.53. In this case, the SHAP values indicates the contributions of attributes that increase the probability of being classified as a good wine from the overall expected probability until the actual classification (class value equals to 1).
\begin{table}[!h]
\begin{center}
\caption{Summary of the selected test sample - Red Wine dataset.}\label{tab:wine_samp}
{
\renewcommand{1.0}{1.0}
\small
\begin{tabular}{cccccccc}
\cline{1-2}\cline{4-5}\cline{7-8} \textbf{Attributes} & \textbf{Values} & & \textbf{Attributes} & \textbf{Values} & & \textbf{Attributes} & \textbf{Values} \\
\cline{1-2}\cline{4-5}\cline{7-8}
fixed acidity & $9.4$ & & chlorides & $0.08$ & & pH & $3.15$ \\
\cline{1-2}\cline{4-5}\cline{7-8}
volatile acidity & $0.3$ & & free sulfur dioxide & $6$ & & sulphates & $0.92$ \\
\cline{1-2}\cline{4-5}\cline{7-8}
citric acid & $0.56$ & & total sulfur dioxide & $17$ & & alcohol & $11.7$ \\
\cline{1-2}\cline{4-5}\cline{7-8}
residual sugar & $2.8$ & & density & $0.9964$ & & & \\
\cline{1-2}\cline{4-5}\cline{7-8}
\end{tabular}
}
\end{center} \end{table}
Figure~\ref{fig:shapley_wine} presents the estimated SHAP values when using $n_{\mathcal{M}}=420$, $n_{\mathcal{M}}=1020$ and $n_{\mathcal{M}}=1800$ coalitions of attributes. As in the previous dataset, we can see that, even for a reduced number of samples, the $3_{ADD}$-SHAP converges faster to the exact SHAP values. When the number of expected prediction evaluations increases, the Kernel SHAP converges to the exact SHAP values while the $2_{ADD}$-SHAP slightly diverges.
\begin{figure}
\caption{Comparison between the estimated SHAP values provided by the $2_{ADD}$-SHAP, $3_{ADD}$-SHAP and Kernel SHAP for different machine learning models and varying the number of coalitions used to calculate the expected prediction evaluations (Red Wine dataset).}
\label{fig:shapley_wine_18_420_mlp}
\label{fig:shapley_wine_18_420_rf}
\label{fig:shapley_wine_18_1020_mlp}
\label{fig:shapley_wine_18_1020_rf}
\label{fig:shapley_wine_18_1800_mlp}
\label{fig:shapley_wine_18_1800_rf}
\label{fig:shapley_wine}
\end{figure}
\subsection{Illustrative example and results visualization} \label{subsec:exp3}
The purpose of this last experiment is to apply our proposal to visualize the attributes contribution towards the actual predicted outcome. We use as an illustrative example the Red Wine dataset and applied the $3_{ADD}$-SHAP. We also consider the test sample used in the previous experiment, which is classified as a good wine. Based on 1500 predicted evaluations and using the Random Forest, the contributions of attributes are presented in Figure~\ref{fig:example_wine_shapley}. Note that there are three attributes that contribute the most into the predicted outcome: alcohol, sulphates and volatile acidity. They are all positively contributing to predict the sample as a good wine.
\begin{figure*}
\caption{Attributes contribution towards the predicted outcome - $3_{ADD}$-SHAP and Red Wine dataset. }
\label{fig:example_wine_shapley}
\end{figure*}
Recall that, more than the contribution of features, our proposal automatically provides the interaction degree between them. We highlight that these interaction effects do not come up with the original Kernel SHAP formulation. Indeed, further adaptations must be made in Kernel SHAP in order to retrieve the interaction effects~\citep{Lundberg2020}. Figure~\ref{fig:example_wine_interaction} shows the interaction degree between attributes for the considered test sample. It indicates that, although volatile acidity, sulphates and alcohol (attributes 1, 9 and 10, respectively) contributes the most to the predicted outcome, there are negative interactions between alcohol and both volatile acidity and sulphates. This suggests that there are some redundancies between alcohol and the other two attributes when predicting the sample as a good wine.
\begin{figure*}
\caption{Interaction degree between attributes - $3_{ADD}$-SHAP and Red Wine dataset. }
\label{fig:example_wine_interaction}
\end{figure*}
\section{Conclusions and future perspectives} \label{concl}
Interpretability in machine learning has become as important as accuracy in real problems. For instance, even if there is a correct classification (e.g., a denied credit), the explanation about how this result was achieved is required to ensure the model trustfulness. A very famous model-agnostic algorithm for machine learning interpretability is the SHAP method. Based on the Shapley values, the SHAP method indicates the contribution of each attribute in the predicted outcome. For this purpose, we look at the machine learning task as a cooperative game theory problem and calculate the marginal contribution of each attribute by taking the predicted outcomes of all possible coalitions of attributes. A point of attention in this calculation is that, as the number of predicted outcomes evaluations exponentially increases with the number of attributes, one may not be able to obtain the exact SHAP values.
In order to reduce the computational effort of SHAP method, the Kernel SHAP emerges as a clever strategy to approximate the SHAP values. However, its formulation is not easy to follow and any further considerations about the modeled game is assumed when approximating the SHAP values. In this paper, we first proposed a straightforward Choquet integral-based formulation for local interpretability. As the parameters used in the Choquet integral are directly associated with the Shapley values, our formulation also leads to the SHAP values. Therefore, we can also exploit the benefits of the SHAP values when interpreting local predictions. Moreover, our formulation also provides the interaction effects between attributes without further adaptations in the algorithm. Therefore, we can interpret the marginal contribution of each attribute towards the predicted outcome and how they interact between them.
As a second contribution, we exploit the concept of $k$-additive games. The use of $k$-additive models has revealed to be useful in multicriteria decision making problems in order to reduce the number of parameters in capacity-based aggregation functions (such as the Choquet integral) while keeping a good level of flexibility in data modeling. Therefore, as attested in the numerical experiments, when adopting $k$-additive games (specially the $3$-additive, which leads to the proposed $3_{ADD}$-SHAP), we could approximate the SHAP values using less predicted outcomes evaluations in comparison with the Kernel SHAP. As one reduced the number of parameters in the Choquet integral formulation, one avoided over-parametrization in scenarios with a low number of predicted outcomes evaluations. On the other hand, as we restricted the modeling data domain, in the scenario with all evaluations the proposed $k_{ADD}$-SHAP may slightly diverge from the exact SHAP values. However, as could be seen in the experiments, this difference is very low (mainly for the $3_{ADD}$-SHAP) and it does not affect the interpretability.
Future works include to extend the proposed approach when assuming that the attributes are dependent. In such a scenario, the formulation could be adjusted in order to better approximate the Shapley values~\citep{Aas2021}. Another perspective consists in evaluating the use of other game-based aggregation functions to deal with local interpretability. However, as some of them do not ensure the efficiency property, one must be careful in how one can apply them in the context of machine learning in a way that the feature attribution makes sense for local or global interpretability.
\section*{Acknowledgments} Work supported by S\~{a}o Paulo Research Foundation (FAPESP) under the grants \#2020/09838-0 (BI0S - Brazilian Institute of Data Science), \#2020/10572-5 and \#2021/11086-0.
\section*{Appendix A}
We here describe the desired properties satisfied by SHAP values, which are derived from the Shapley values properties~\citep{Shapley1953,Young1985}. Recall that $f(\mathbf{x})$ is the predicted outcome of a trained model $f(\cdot)$, $\mathbf{x}$ is the instance to be explained and $\mathbf{z}'$ is a binary vector. The proofs are provided in the original SHAP paper~\citep{Lundberg2017}.
\begin{Properties}
\item \textbf{Local accuracy (or efficiency)} \\
\begin{equation}
f(\mathbf{x}) = \phi_0 + \sum_{j=1}^m \phi_j(f,\mathbf{x})
\end{equation}
The local accuracy property states that the predicted outcome $f(\mathbf{x})$ can be decomposed by the sum of the SHAP values and the overall expected prediction $\phi_0$.
\item \textbf{Missingness} \\
If, for all subset of attributes represented by the coalition $\mathbf{z}'$,
\begin{equation}
f\left( h_{\mathbf{x}}(\mathbf{z}')\right) = f\left( h_{\mathbf{x}}(\mathbf{z}'\backslash j)\right),
\end{equation}
then $\phi_j(f,\mathbf{x}) = 0$. This property states that, if adding attribute $j$ into the coalition the expected prediction remains the same, the marginal contribution of such an attribute is null.
\item \textbf{Consistency (or monotonicity)} \\
For any two models $f(\cdot)$ and $f'(\cdot)$, if
\begin{equation}
f'\left( h_{\mathbf{x}}(\mathbf{z}')\right) - f'\left( h_{\mathbf{x}}(\mathbf{z}' \backslash j)\right) \geq f\left( h_{\mathbf{x}}(\mathbf{z}')\right) - f\left( h_{\mathbf{x}}(\mathbf{z}'\backslash j)\right)
\end{equation}
for any binary vector $\mathbf{z}' \in \left\{0,1\right\}^m$, then $\phi_j(f',\mathbf{x}) \geq \phi_j(f,\mathbf{x})$. The consistency property states that, if one changes the trained model and the contribution of an attribute $j$ increases or stays the same regardless of the other inputs, the marginal contribution of such an attribute should not decrease. \end{Properties}
\biboptions{authoryear}
\end{document}
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\begin{document}
\makeatletter
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\let\@oddhead\@empty
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\begin{frontmatter}
\title{System Identification and $H_\infty$-based Control of Quadrotor Attitude}
\author[mainaddress]{Ali Noormohammadi-Asl \corref{correspondingauthor}} \cortext[correspondingauthor]{Corresponding author} \ead{[email protected]}
\author[mainaddress]{Omid Esrafilian } \ead{[email protected]} \author[mainaddress]{Mojtaba Ahangar Arzati } \ead{[email protected]} \author[mainaddress]{Hamid D. Taghirad} \ead{[email protected]} \address[mainaddress]{\textbf{A}dvanced \textbf{R}obotics and
\textbf{A}utomated \textbf{S}ystems (ARAS)\fntext[arassite]{website: aras.kntu.ac.ir}\\Faculty of Electrical Engineering, K. N. Toosi University of Technology}
\begin{abstract} The attitude control of a quadrotor is a fundamental problem, which has a pivotal role in a quadrotor stabilization and control. What makes this problem more challenging is the presence of uncertainty such as unmodelled dynamics and unknown parameters. In this paper, to cope with uncertainty, an $H_\infty$ control approach is adopted for a real quadrotor. To achieve $H_\infty$ controller, first a continuous-time system identification is performed on the experimental data to encapsulate a nominal model of the system as well as a multiplicative uncertainty. By this means, $H_\infty$ controllers for both roll and pitch angles are synthesized. To verify the effectiveness of the proposed controllers, some real experiments and simulations are carried out. Results verify that the designed controller does retain robust stability, and provide a better tracking performance in comparison with a well-tuned PID and a $\mu$ synthesis controller. \end{abstract}
\begin{keyword} Quadrotor\sep Attitude control\sep Continuous-time system identification\sep Linear uncertain system\sep Robust $H_\infty$ control. \end{keyword}
\end{frontmatter}
\section{Introduction}
Quadrotors are widely considered due to their remarkable features and applications in a variety of areas such as monitoring \cite{wang2016detecting,berra2017commercial}, surveillance \cite{acevedo2014decentralized, kingston2008decentralized}, search and rescue \cite{silvagni2017multipurpose,khan2015information}, agriculture \cite {tokekar2016sensor,tetila2017identification}, and delivery of goods \cite{gawel2017aerial,arbanas2016aerial}.
Quadrotors are also of paramount interest owing to the advantages of low cost, maneuverable, small size, simple design structure. The quadrotor is a nonlinear system with coupled states. In general, there are six states including the robot position $\left(x,y,z\right)$ and the angles (yaw, pitch, roll) with only four inputs, resulting in an underactuated system. Despite the mentioned advantages of quadrotors, the open-loop instability, coupled states, and underactuation make their control challenging. A quadrotor control is accompanied with several other serious challenges, particularly when it turns from a simple model in the theory to a practical problem in which many other key factors such as parameters uncertainty, unmodelled dynamics, and input constraints need to be considered. The main focus of this paper is the stabilization and attitude control of a quadrotor considering such uncertainties and input constraints.
Modeling and control of a quadrotor have been extensively studied in the literature. PID control is the most common controller widely used in commercial quadrotoros \cite{hoffmann2007quadrotor,szafranski2011different,garcia2012robust,zhou2017robust}. In \cite{szafranski2011different} three different PID structures are employed for the attitude control. In \cite{garcia2012robust} a robust PID controller is designed using affine parameterization. In \cite{zhou2017robust} a cascade control for the attitude control is employed and a robust compensator is added to reduce the wind disturbance effect. Linear Quadratic Regulator (LQR) is another linear-based controller used for quadrotors \cite{bouabdallah2004pid,liu2013robust}. In LQR controllers, setting proper weights is difficult and needs trial and error. Linear-based controllers have simple designs and have been tested successfully on experimental and simulated platforms. Linear controllers, however, are based on a linear approximation of the actual system model by neglecting the nonlinear behavior of the system which may increase the probability of the system failure. Moreover, they are limited to low velocities as well as small angles of deviation, and their stability is guaranteed only near the selected operating point.
To transcend the limitations of the linear controllers, many nonlinear control approaches have been proposed.
In \cite{flores2013lyapunov}, a lyapunov-based controller is designed using singular perturbation theory for a quadrotor stabilization. Backstepping-based control has also been developed by many researchers \cite{huo2014attitude,das2009backstepping,djamel2016attitude,shao2018robust}. A backstepping controller using a lagrangian form of the quadrotor dynamics is designed in \cite{das2009backstepping}. In \cite{djamel2016attitude} an optimal backstepping controller using $\text{H}_\infty$ is proposed. Sliding mode control is another nonlinear control method which is capable of dealing with some classes of uncertainties and external disturbances \cite{chen2016robust,jia2017integral,yang2016attitude,zheng2014second,wang2019disturbance}. In \cite{chen2016robust,jia2017integral}, a combination of the sliding mode control and backstepping control techniques are used for the control of quadrotors. Despite the mentioned capabilities, sliding mode control suffers from chattering, and it is necessary to know the upper bounds of uncertainties.
In order to reduce the chattering effect, different approaches like fuzzy gain scheduling \cite{yang2016attitude} and higher order sliding mode have been proposed\cite{zheng2014second}. A fuzzy state observer is proposed in \cite{mallavalli2018fault} to estimate the unknown nonlinear functions of the uncertain system model, then an integral terminal sliding mode controller is used. Many other robust nonlinear control techniques have also been proposed to cope with the quadrotor uncertainties. In \cite{liu2017robust}, a robust compensating technique is used to control quadrotors with time-varying uncertainties and delays. The design of a nonlinear $\text{H}_\infty$ controller is studied in \cite{raffo2010integral} to achieve the robustness. In \cite{kerma2012nonlinear}, a nonlinear $\text{H}_\infty$ output feedback controller coupled with a high order sliding mode estimator is utilized to control the quadrotor in the presence of parameters uncertainty and external disturbances. The adaptive control is another approach for controlling systems with unknown model parameters and have been widely used for controlling various practical systems\cite{hu2018adaptive,zhang2016active,jiang2019hydrothermal}. An adaptive controller is employed in \cite{tran2018adaptive} for trajectory tracking of a quadrotor with considering the input constraints and uncertain parameters in its nonlinear model.
In order to design a nonlinear-based controller, the quadrotor nonlinear model is required, which is hard to obtain. For instance, inertia matrix, center of gravity and some other parameters of a quadrotor are not easily accessible and it is difficult to obtain their accurate values. In most of the previous studies, a symmetric model for the quadrotor is assumed, however in real robots this assumption does not hold. Furthermore, some factors, like wind, cause environmental disturbances. Especially in an actual flight near obstacles or low attitude, undesirable wind effects on the stability and performance of a quadrotor is more obvious. Considering the wind model makes designing a nonlinear controller arduous and challenging. An approach to overcome the model uncertainty, such as unmodelled dynamics and parameter uncertainty, is using system identification techniques. In \cite{liu2018parameter}, a closed-loop multivariable extremum seeking algorithm (MESA) is suggested for parameter identification of a quadrotor. The black-box model identification using a continuous-time approach is adopted in \cite{bergamasco2014identification} to obtain a linear model for the dynamics of a quadrotor.
Linear robust control is a popular approach in the control theory that can help to mitigate the effect of unmodeled dynamics in linear-based controllers. This approach has been used in many practical problems such as surgery robots\cite{agand2017decentralized}, parallel robots \cite{bataleblu2016robust}, harmonic drive systems \cite{taghirad2001h}, and vibration control of elastic gantry crane and 3D bar structure \cite{golovin2019robust,mystkowski2016mu}; and the results confirm its remarkable ability to deal with uncertainty. In \cite{safaee2013system,wang2013robust}, the linear robust $\text{H}_\infty$ controller is employed for attitude and altitude control of a helicopter. In order to utilize the $\text{H}_\infty$ controller, a comprehensive identification phase is required, which provides information about the linear model of the system and the associated uncertainty.
In this paper, there is no information available about the model and parameters (e.g. CAD model) of the quadrotor. Thus, in order to overcome the aforementioned limitations of linear and nonlinear controllers, embracing lack of system model and different sources of uncertainty, a linear robust controller has been derived and implemented on a real quadrotor. In this case, a continuous time system identification from the sampled data of the real robot is required to gather information about both the system model and uncertainty. The main contributions and phases of this paper are as follows: \begin{itemize} \item The first phase is performing a system identification based on the experimental frequency response estimates to obtain a nominal linear model aligned with a multiplicative uncertainty block. This phase is the principle and most challenging part of control design because of system identification difficulties and its importance in designing an effective controller. In this phase, the first step is designing appropriate experiments which provide us with informative and usable data for system identification. Then, by applying existing tools and methods such as MATLAB identification and CONTSID toolbox, the nominal models and uncertainty weighting functions are acquired. \item Having obtained information about the system in the previous phase, a linear $H_\infty$ control is synthesized. In this phase, it is important to choose proper sensitivity and input weighting functions to achieve desired tracking and regulating performance. \item Finally, the controller is implemented on the robot and its high performance and robustness are shown by simulation and experimental results. In addition, a PID and a $\mu$ analysis controllers are designed to compare their performance with $\text{H}_\infty$ controller. \end{itemize}
\section{System and uncertainty encapsulation}\label{sec2} In many systems, uncertainty emerges due to the dynamical perturbation, such as unmodelled and high frequency dynamics, in various parts of a system. To capture uncertainty in the system, we utilize the classical multiplicative perturbation model. This model helps to encapsulate the various sources of uncertainty, e.g. parametric and unmodelled dynamics, in a full block of multiplicative uncertainty. Considering the nominal system transfer function as $G_0$, the family of uncertain systems may be shown as follows: \begin{equation}\label{eq21} \mathcal{G}=\left\{G\left(s\right) \mid G=\left(1+\Delta W\right)G_0 \right\} , \end{equation} where $W$ is the uncertainty weighting function, and $\Delta$ is a stable perturbation satisfying $\lVert\Delta\rVert_\infty<1$.
Hence, for designing a robust control for a nonlinear system, an approximate linear model along with a weighting function providing information about the uncertainty profile, are required. In order to obtain a controller having proper performance in practice, it is necessary to perform a system identification on the experimental data generated from the real system. There are some challenges, however, for a quadrotor system identification: \begin{itemize} \item The quadrotor robots are inherent unstable, thus a closed loop identification method is recommended. \item For the robust control design a continuous-time model is needed. \item The quadrotor system is a multi-input multi-output (MIMO) system, but due to its open-loop instability and piloting difficulty, the experimental data is better to be obtained for each channel separately, then perform the system identification. \end{itemize} In what follows, we dwell on the closed loop system identification. For this, first we derive a theoretical model for the system, then identify the system based on the gathered experimental data.
\subsection{Theoretical model} Modeling the dynamic of quadrotors has been studied extensively in the literature. Many models attempt to consider parameters and factors that are difficult to calculate or their values may vary in different situations such as the presence of wind. The goal of this paper is to design a controller that is largely independent of modeling parameters. To this end, first we model the dynamic of the robot. As shown in Fig. \ref{fig21} for a drone with four rotors, which rotate in opposite directions (two rotors rotate in a clockwise direction and two other rotate counterclockwise), the angular accelerations pertain to the pitch, roll, and yaw angles are modeled by the following nonlinear models: \begin{equation}\label{eq22} \begin{aligned} &\ddot{\phi}=\frac{J_r\dot{\theta}\left(\Omega_1+\Omega_3-\Omega_2-\Omega_4\right)}{I_{xx}} +\frac{I_{yy}-I_{zz}}{I_{xx}}\dot{\psi}\dot{\theta}+\frac{bl\left(\Omega_2^2-\Omega_4^2\right)}{I_{xx}}, \\ &\ddot{\theta}=\frac{J_r\dot{\phi}\left(-\Omega_1-\Omega_3+\Omega_2+\Omega_4\right)}{I_{yy}} +\frac{I_{zz}-I_{xx}}{I_{yy}}\dot{\psi}\dot{\phi}+\frac{bl\left(\Omega_3^2-\Omega_1^2\right)}{I_{yy}}, \\ &\ddot{\psi}=\frac{d\left(\Omega_1^2+\Omega_3^2-\Omega_2^2-\Omega_4^2\right)}{I_{zz}} +\frac{I_{xx}-I_{yy}}{I_{zz}}\dot{\theta}\dot{\phi}, \end{aligned} \end{equation} where $\phi$, $\theta$, and $\psi$ stand for the roll, pitch and yaw angles, respectively. The remaining parameters are listed in table \ref{table21}. $\left(I_{yy}-I_{zz}\right)\dot{\psi}\dot{\theta}$, $\left(I_{zz}-I_{xx}\right)\dot{\psi}\dot{\phi}$ and $\left(I_{xx}-I_{yy}\right)\dot{\theta}\dot{\phi}$ are body gyro effects. Considering $\Omega_r=\left(\Omega_1+\Omega_3-\Omega_2-\Omega_4\right)$, $J_r\dot{\theta}\Omega_r$ and $J_r\dot{\phi}\Omega_r$ represent propeller gyro effects.
\begin{figure}
\caption{Quadrotor body frame, rotor arrangement and corresponding forces}
\label{fig21}
\end{figure}
\begin{table}[t] \caption{List of symbols in Eq. \ref{eq22}} \label{table21} \begin{center}
\begin{tabular}{|c||c|}
\hline
Symbol & Description \\
\hline\hline $I_{xx,yy,zz}$ & Inertia moments\\ \hline $J_r$ & Rotor inertia\\ \hline $\Omega_i$ & Propeller angular rate\\ \hline $d$ & Drag factor\\ \hline $b$ & Thrust factor\\ \hline $l$ & Horizontal distance: propeller center to CoG{\textsuperscript{\tiny {*}}}\\
\hline \multicolumn{2}{l}{\small {Center of Gravity}} \end{tabular} \end{center} \end{table}
In order to analyze the quadrotor model in the presence of a PID controller, the gyroscopic effects can be ignored compared to the motors' action. Thus, Eq. (\ref{eq22}) is rewritten as: \begin{equation}\label{eq23} \begin{aligned} &\ddot{\phi}=\frac{bl\left(\Omega_2^2-\Omega_4^2\right)}{I_{xx}}, \\ &\ddot{\theta}=\frac{bl\left(\Omega_3^2-\Omega_1^2\right)}{I_{yy}}, \\ &\ddot{\psi}=\frac{d\left(\Omega_1^2+\Omega_3^2-\Omega_2^2-\Omega_4^2\right)}{I_{zz}}. \end{aligned} \end{equation} The dynamics of the rotor is considered as $\frac{T_1}{s+T_2}$, in which $T_1,T_2\in \mathbb{R}^+$ and $T_2$ is the pole of the system and $\sfrac{T_1}{T_2}$ is the DC gain. Thus, the model of the robot is obtained as follows: \begin{equation}\label{eq24} \begin{aligned} &\phi\left(s\right)=\frac{T_1^2bl}{s^2\left(s+T_2\right)^2I_{xx}}\left(u_4^2\left(s\right)-u_2^2\left(s\right) \right),\\ &\theta\left(s\right)=\frac{T_1^2bl}{s^2\left(s+T_2\right)^2I_{yy}}\left(u_3^2\left(s\right)-u_1^2\left(s\right) \right),\\ &\psi\left(s\right)=\frac{T_1^2d}{s^2\left(s+T_2\right)^2I_{zz}}\left(u_1^2\left(s\right)+u_3^2\left(s\right)-u_2^2\left(s\right) -u_4^2\left(s\right)\right), \end{aligned} \end{equation} where $u_1$, $u_2$, $u_3$, and $u_4$ are motors inputs. Assuming the motors have fast response, the Eq. (\ref{eq24}) can be rewritten as follows, which is a double-integrator unstable system: \begin{equation}\label{eq25} \begin{aligned} &\phi\left(s\right)=\frac{A_1}{s^2}U_1;\qquad U_1=u_4^2\left(s\right)-u_2^2\left(s\right),\\ &\theta\left(s\right)=\frac{A_2}{s^2}U_2; \qquad U_2=u_3^2\left(s\right)-u_1^2\left(s\right),\\ &\psi\left(s\right)=\frac{A_3}{s^2}U_3;\qquad U_3=u_1^2\left(s\right)+u_3^2\left(s\right)-u_2^2\left(s\right) -u_4^2\left(s\right), \end{aligned} \end{equation} where $A_1, A_2$, and $A_3 \in \mathbb{R}^+$. The system model in the presence of a PID controller, as shown in Fig. \ref{fig22}, is obtained as: \begin{equation}\label{eq26} \begin{aligned} &G_{roll}\left(s\right)=\frac{A_1\left(k_{d_r}s^2+k_{p_r}s+k_{i_r}\right)}{s^3+A_1\left(k_{d_r}s^2+k_{p_r}s+k_{i_r}\right)}U_1,\\ &G_{pitch}\left(s\right)=\frac{A_2\left(k_{d_p}s^2+k_{p_p}s+k_{i_p}\right)}{s^3+A_2\left(k_{d_p}s^2+k_{p_p}s+k_{i_p}\right)}U_2, \end{aligned} \end{equation} where $K_{p_{r,p}}$, $K_{d_{r,p}}$, and $K_{i_{r,p}}$ are the nonnegative proportional, derivative, and integral gains, respectively. These equations give a useful insight into the number of poles and zeros of the system in the identification phase. \begin{figure}
\caption{PID control structure for quadrotor}
\label{fig22}
\end{figure}
\subsection{Control problem} Attitude control is the integral part of the stabilization problem and helps to reach the desired 3d orientation. In this paper, we focus on the roll and pitch angles because they play the main role in the stability of the quadrotor. Due to the coupling of the system's states, the performance of roll and pitch has considerable effect on other states. For example, when the quadrotor moves forward, the pitch angle changes and then returns to zero. Thus, a fast and reliable control on roll and pitch angles is crucial. In addition, the disturbance exerts significant influence on the performance and stability of roll and pitch angles and consequently the overall performance of the robot.
By considering $X=\left(\phi,\dot{\phi},\theta,\dot{\theta}\right)$, the control problem is to reach $X_d=\left(\phi_d,0,\theta_d,0\right)$ in which $\phi_d$ and $\theta_d$ are desired roll and pitch angles. In our case, the $\phi_d$ and $\theta_d$ are set to zero to stabilize the quadrotor. However, as mentioned before, overcoming model uncertainty is the main goal of this paper which should be achieved with considering two following conditions. \begin{enumerate}
\item Having a high tracking performance and disturbance rejection
To achieve this goal, we need that the controller provides a short settling time (about $0.3s$) and a small overshoot (about $10^{-6} rad$).
\item Avoiding actuator saturation
Due to the limitations of the motor, the control effort should be in a feasible range. In addition, actuator saturation may cause instability of the system. \end{enumerate} Briefly, the problem is finding a controller providing the stability for the quadrotor by striking a balance between robustness and performance of the system, in the presence of uncertainty and the actuator saturation.
\subsection{Experimental setup} The quadrotor used in the experiments (Fig \ref{fig2a}) is based on a typical quadrotor design with some structural modifications. The power management circuit board and the carbon fiber tubes are used as the frame hub and motors arm, respectively. The main goal in the design of this quadrotor is achieving a compact integration of the mechanical structure and electronic devices in order to reduce the weight while maintaining a symmetric mass distribution with a center of gravity close to the center of the cross. The vehicle propeller-tip to propeller-tip distance is $38 \,cm$. This robot, embracing all the modules, sensors, battery, motors, and props has a mass of $570\,g$, and provides a total thrust of 850 g. The flight time is approximately 9 minutes with a 3-cell $1300\, mAh$ Li-Po battery.
Three main units which play significant roles in the control of the robot are explained below, and the block diagram of their structure and interconnections are illustrated in Fig. \ref{fig2b}.
\textbf{\em{Embedded electronics:}} The robot configuration consists of a flight control, Inertial Measurement Unit (IMU), radio control unit, telemetry module, and external IO. The flight control board, which is the main board of the robot, runs on an ARM STM32F407 micro-controller, which can operate up to $168\, MHz$. Other units interface with this controller via I2C and UART protocols. Each brushless motor has its own ESC (Electronic Speed Control) units, which is connected to the main board. PWM (Pulse-Width Modulation) is used to communicate with ESC at the speed of $400\, Hz$, and the main board sends control signals to ESC to adjust the rotational speed of the motor.
\textbf{\em{Inertial Measurement Unit (IMU):}} The IMU utilized in this quadrotor is MPU-6050, which contains a 3-axis gyroscope, 3-axis accelerometer, 3-axis magnetometer. The main board communicates with the MPU-6050 via I2C bus and runs a sensor fusion algorithm with the frequency of $500 \,Hz$. The quadrotor attitude is computed by using Mahony filter algorithm \cite{mahony2008nonlinear}, which estimates the absolute roll and pitch angles via fusing the acceleration vector and angular speeds measured by the gyroscopes. The angular speed which is measured by the gyroscope associated with yaw axis is used for estimating the yaw angle.
\textbf{\em{Wireless communication systems:}} We established two types of system to support real time communications between the flying quadrotor and our personal computer as a ground control system (GCS): a digital radio telemetry unit ($915\,MHz$) and an analog radio link ($2.4\,GHz$). We use NRF modules as a digital radio telemetry for data acquisition. The telemetry wireless interfaces with the main board using a UART serial protocol. Necessary flight data can be either saved on the internal memory of the micro-controller or sent to the GCS at rates of up to $115200\, bps$ using telemetry communication. Futaba remote controller is used as the analog radio link. The radio control (RC) receiver is connected to the main board through the SBUS protocol that allows to receive flight commands at the speed of $100 \,K bps$.
\begin{figure}
\caption{Quadrotor used in this paper}
\label{fig2a}
\caption{Block diagram of quadrotor’s units Interconnection}
\label{fig2b}
\caption{Experimental setup }
\label{fig2}
\end{figure}
\subsection{System identification} The problem of system identification of an aircraft or rotorcraft is a topic of interest and have been widely studied. In most of the studies, the system identification methods are used to obtain parameters value in the model of the system. However, in this paper, a black-box system identification, merely based on input and output experimental data, is performed.
The first and basic step in the identification phase is selecting a proper input for applying to the system. In the frequency-domain identification, which is desired in this paper, a periodic excitation input is much preferred because it helps to cover and sweep the desired frequency spectrum effectively. The excitation inputs are better to be applied automatically (e.g programmed on the microcontroller) rather than manually (e.g radio controller). Thus, in order to obtain a continuous time model, a frequency-domain identification method is performed in which periodic chirp functions with different frequencies and amplitudes are implemented on the on-board micro-controller, and are used as exciting inputs. During the experiments, the identification for the pitch and roll angles are executed separately, and the quadrotor is supposed to track the reference chirp input. For example, if the chirp signal is selected as a reference input for the roll angle, the quadrotor uses the implemented simple PID controller for the roll angle to track the reference. Setting proper frequencies and amplitudes for chirp functions is important, because an improper frequency or amplitude, which are beyond the quadrotor ability to track, may cause inappropriate tracking and produces misleading information for identification. In this identification, the frequency and amplitude range vary in $\left[0.05 Hz,5 Hz\right]$ and $\left[5^\circ,11^\circ\right]$, respectively, which are feasible and reasonable pitch and roll angles.
Prior to the identification phase, the probable existence of delay is studied using correlation analysis method, and according to its results, the delay of the system is negligible. Secondly, the effect of exciting the roll angle on the pitch angle, and vice versa, is analyzed. In Fig. \ref{fig231}, the pitch angle, the desired reference pitch angle, and the roll angle are plotted, for one of the experiences. As shown in this figure, the coupling is negligible and the roll angle is almost independent of the pitch angle, thus their coupling can be considered as an unmodelled system uncertainty. The same condition holds for the pitch angle as well (Fig. \ref{fig232}). As a result, off-diagonal elements of the system model can be considered zero, which implies that the system can be represented by two SISO subsystems. Although this assumption may make the controller task more difficult at high frequencies, the controller design becomes much easier.
\begin{figure}
\caption{Coupling effect of the pitch on the roll}
\label{fig231}
\caption{Coupling effect of the roll on the pitch}
\label{fig232}
\caption{The coupling effects}
\label{fig23}
\end{figure}
After performing several experiments for both pitch and roll angles, a linear transfer function should be fitted for each of them to obtain the stable closed-loop system model. We use MATLAB identification toolbox for obtaining continuous-time transfer function using time-domain data. In this step, it is important to select a proper number of poles and zeros. For this purpose, in each identification, we use different combinations of zeros and poles. However, to reduce the number of these combinations, the number of poles and zeros can be assumed close to the approximate analytical transfer function of the system, obtained in Eq. (\ref{eq26}). Finally, for the model identification, the number of poles and zeros are considered in the ranges of $\left[2,5\right]$ and $\left[0,3\right]$, respectively. Consequently, for each experiment, fifteen proper transfer functions are obtained, and one of them is selected as the best-fitted model. In order to select a transfer function, fitness percentage of different fitted models has a decisive role. However, there are two other factors which must be considered. First, it is preferred that the selected transfer functions have similar frequency responses, meaning that the zero/pole numbers should be close to each other. Second, the transfer functions with right hand-side zeros are not selected because the quadrotor system does not show the characteristics of a system with right half-plane zeros.
Based on the obtained transfer functions, for the pitch angle, the real part of poles mainly lie in the range of $\left[-6,0\right]$, and for the roll angle, real part of poles are in the ranges of $\left[-7,0\right]$ and $\left[-10,-8\right]$. In both cases, the zeros lie in the range of $\left[-1,1\right]$. These zeros, which are located near the imaginary axis, mostly appear in pairs with poles very close to them, and therefore, they may be ignored due to the zero-pole cancellation.
By analyzing the obtained transfer functions according to the three aforementioned criteria, it is observed that in the most cases, transfer functions with three poles and no zero can suitably describe the quadrotor system behavior. Figure \ref{fig261} and \ref{fig262} show the histogram of the real part of poles in the transfer functions of the pitch and roll angles, with three poles and no zero, respectively. According to these figures, it is expected that poles lie in the range of $\left[-7, -3\right]$ and $\left[-10, -8\right]$. In Fig. \ref{fig27}, the Bode diagram of the transfer functions obtained for the pitch and roll angle are depicted. It is clear from Fig. \ref{fig27} that the behavior of the transfer functions is similar, especially in frequencies less than $15 Hz$ where the robot mostly operates in. Now, by having the system transfer functions pertain to both axes of pitch and roll, we select one of them as the nominal transfer function. Hence, the transfer function which lies near to the median of all responses, is considered as the nominal model. The selected nominal model transfer function for the pitch and roll angle are as follows: \begin{align} G_{pitch}\left(s\right)=\frac{1547.4}{\left(s+5.373\right)\left(s^2+10.12s+390.4\right)},\label{eq27}\\ G_{roll}\left(s\right)=\frac{2049.8}{\left(s+6.764\right)\left(s^2+19.03s+426.2\right)}. \end{align}
\begin{figure}
\caption{Pitch}
\label{fig261}
\caption{Roll}
\label{fig262}
\caption{Distribution of poles in transfer functions with 3 poles and no zero}
\label{fig26}
\end{figure}
\begin{figure}
\caption{Pitch}
\label{fig271}
\caption{Roll}
\label{fig272}
\caption{Bode diagram of selected transfer functions in the experiments}
\label{fig27}
\end{figure} In order to verify the nominal models, the real experimental data (dashed) and the time-domain response (solid) are depicted in Fig. \ref{fig281} and \ref{fig282}. For the sake of brevity, only two of the experiment results are shown. In these figures, the applied inputs to the nominal model are the same as the real experiments. \begin{figure}
\caption{Pitch}
\label{fig281}
\caption{Roll}
\label{fig282}
\caption{Comparison of transfer functions time-response and real data (for the same input)}
\label{fig28}
\end{figure}
\subsection{Uncertainty weighting function} As mentioned before, based on Eq. (\ref{eq21}), $W$ is a fixed transfer function for normalizing $\Delta$. In other words, $W\Delta$ expresses the normalized system variation away from unity at each frequency, and based on Eq. (\ref{eq21}), may be obtained by $G\left(jw\right)/G_0\left(jw\right)-1=\Delta\left(jw\right) W\left(jw\right)$. Since $\lVert\Delta\rVert_\infty<1$, the following can be concluded, which provide a practical approach for obtaining the uncertainty weighting function: \begin{equation} \lvert\frac{G\left(jw\right)}{G_0\left(jw\right)}-1\rvert \leq\lvert W\left(jw\right)\rvert \qquad \forall w. \end{equation}
After selecting a nominal model between all best-fitted transfer functions, other systems can be considered as a perturbed model of the nominal system, and $\lvert G\left(jw\right)/G_0\left(jw\right)-1 \rvert$ is plotted for each experiment (as illustrated in Fig. \ref{fig29}). As a result, by estimating a transfer function, which is an upper bound for those variations, the uncertainty weighting function (the dashed red line in Fig. \ref{fig291} and \ref{fig292}) is acquired. The uncertainty weighting functions for pitch and roll angles are obtained as follows: \begin{align} W_{pitch}=\frac{1659.6\left(s^2+2.868s+60.44\right)}{\left(s+2.477e04\right)\left(s+9.678\right)},\\ W_{roll}=\frac{1.9017\left(s^2+3.813s+91.61\right)}{s^2+43.53s+545.3}. \end{align}
The above-mentioned steps for obtaining the nominal models of the systems and the uncertainty weighting functions are summarized as below: \begin{enumerate}[
leftmargin=*,
label={\textbf{\textit{Step \arabic*.}}}
]
\item Designing appropriate experiments and obtaining 8 and 9 data packages for pitch and roll angles, respectively.
\item Fitting 15 transfer functions for each data package.
\item Selecting the best-fitted transfer function for each data package.
\item Selecting the nominal models among 8 and 9 transfer functions obtained for pitch and roll angles.
\item Considering the other transfer functions as perturbed models, obtaining the $\lvert G\left(jw\right)/G_0\left(jw\right)-1 \rvert$s, and plotting them for roll and pitch angles.
\item Obtaining the uncertainty weighting functions, which are an upper bound for the variations obtained in {\textbf{\textit{Step 5}}}, for
each roll and pitch angle. \end{enumerate}
\begin{figure}
\caption{Pitch}
\label{fig291}
\caption{Roll}
\label{fig292}
\caption{Multiplicative uncertainty profiles}
\label{fig29}
\end{figure}
\section{Robust ${H}_\infty$ Control ÙSynthesis}
The goal of ${H}_\infty$ controller is providing a robust stable closed-loop system with high performance tracking and disturbance rejection, in the presence of uncertainty and the actuator saturation. Based on Fig. \ref{fig31}, in order to achieve this goal the following objectives shall be met simultaneously: \begin{enumerate} \item $\rVert T_{z_1y_d}\lVert_\infty=\rVert W_sS\lVert_\infty\le 1$, where $T_{z_1y_d}=\frac{W_s}{1+CP}=W_sS$ is the transfer function from $y_d$ to $z_1$, which is the nominal tracking performance in an asymptotic sense. $W_s$ is a frequency dependent sensitivity weighting function for normalizing and shaping the closed loop performance specification. \item $\rVert T_{z_2y_d}\lVert_\infty=\rVert W_uU\lVert_\infty\le 1$, in which $T_{z_2y_d}=\frac{W_uC}{1+CP}=W_uCS=W_uU$ is the transfer function from $y_d$ to $z_2$, and expresses the nominal performance of the control effort. $W_u$ is a frequency dependent weighting function for normalizing and shaping the control input. \item $\rVert T_{z_3y_d}\lVert_\infty=\rVert WT\lVert_\infty\le 1$, which is the result of the small gain theorem. $T_{z_3y_d}=\frac{WCP}{1+CP}=WT$ is the transfer function from $y_d$ to $z_3$, and $W$ represents the uncertainty weighting function. \end{enumerate} Considering this system with input, $y_d$, and output vector $\mathbf{z}=[z_1, z_2, z_3]^T$, the aforementioned conditions can be merged in an induced norm of the transfer function from $y_d$ to $\mathbf{z}$, in which the goal is to find a controller to minimize this norm. This problem is called a mixed-sensitivity problem and is formulated as follows: \begin{equation} \gamma_{opt}= \min_{C}{\left\lVert T_zy_d \right\rVert}=\min_{C}{\left\lVert\begin{matrix}
W_sS \\
W_uU \\
WT
\end{matrix}\right\rVert_\infty} . \end{equation} \begin{figure}
\caption{Block diagram of closed-loop system as a generalized regulator problem in $H_\infty$ framework}
\label{fig31}
\end{figure} In order to define the sensitivity weighting function, first an ideal closed-loop transfer function is designed. In this case, based on the standard second order systems, $\omega_n$ and $\zeta$ are chosen such that the following overshoot and the settling time are satisfied \begin{align} &T_{cl}=\frac{\omega_n^2}{s^2+2\zeta \omega_ns+\omega_n^2},\\ &t_s\approx\frac{4}{\zeta\omega_n}=0.3,\\ &M_p=e^{\frac{-\pi\zeta} {\sqrt{1-\zeta^2}}}=10^{-6}, \qquad 0<\zeta<1 . \end{align} To achieve $W_s$, we use the inverse of the desired sensitivity function: \begin{align} &T_{id}=\frac{247.3}{s^2+30.67s+247.3}\label{tid},\\ &W_s=a\frac{1}{S_{id}}=a\frac{1}{1-T_{id}}=a\frac{s^2+30.67s+247.3}{s\left(s+30.67\right)} \label{eq317}, \end{align} where $a\le1$ is a tuning parameter. Eq. (\ref{eq317}) is modified by inserting a non-dominant pole and by slightly transferring its purely imaginary pole to the left half plane in order to convert $W_s$ to a strictly proper and stable weighting function, respectively. Hence, $W_s$ is corrected as: \begin{equation}\label{eq318} W_s=a\frac{\left(s^2+30.67s+247.3\right)}{\left(s+1000\right)\left(s+30.67\right)\left(s+0.001\right)}. \end{equation}
In the obtained nominal system, Eq. (\ref{eq27}), the input of the system is the desired angle, which should be less than $20^\circ$. Therefore, $W_u$ is set to $0.05$. Having set the weighting functions, the mixed sensitivity problem can be solved using the robust toolbox of MATLAB. After a few correction steps, the controller for the pitch angle by tuning $a=0.88$ with $\gamma_{opt}=0.9929$ is synthesized as follows: \begin{equation*}\resizebox{0.98\hsize}{!}{$ C_{pitch}\left(s\right)=\frac{3.3227e05\left(s+ 2.477e04\right)\left(s+ 461.2\right)\left(s +50.77\right)\left(s+ 9.678\right)\left(s +5.373\right)\left(s^2+10.12s+ 390.4\right)} {\left(s+2.477e04\right) \left(s+1.673e04\right) \left(s+1000 \right)\left(s+30.67\right) \left(s+8.082\right) \left(s+0.001\right) \left(s^2+80.13s + 3556\right)}. $}\end{equation*}
The singular values of the closed-loop system, and the frequency response of each I/O pair are depicted in Fig. \ref{fig32}. The solid blue line shows the maximum singular value of the closed loop system, which is flat within a large frequency range, and is less than one in all range of the frequency domain. The dotted orange line shows the Bode diagram of $WT$ in which the maximum value is 0.42 indicating the robustness of the closed loop system with a margin of greater than two. The performance transfer function and the control effort transfer function are shown by the red and green lines, respectively. Based on Fig. \ref{fig32}, the magnitude of $WT$ starts to reduce significantly in frequencies higher than about 30 Hz due to the limited bandwidth of the system. About this frequency, a slight decrease also appears in the magnitude of $W_sS$ until roughly 900 Hz where a dramatic drop occurs. Conversely, the magnitude of $ W_uU$ increases in these frequencies to compensate for the lack of the system stability and performance. However, since the robot working frequency is less than approximately 20 Hz, high frequencies performance of the system can be ignored. In low frequencies, as expected, the magnitude of $W_sS$ increases when that of $WT$ decreases. \begin{figure}
\caption{The closed loop system singular values and Bode plot of $W_sS$, $ W_uU$, $WT$ (Pitch)}
\label{fig32}
\end{figure}
As mentioned in section \ref{sec2}, a PD controller has been employed to identify the system. Therefore, to implement the final controller a cascade architecture must be used in which the PD controller is in the inner loop, and the $H_\infty$ controller is functioning in the outer loop as illustrated in Fig. \ref{fig36}. The closed-loop step response of the nominal (red) and some uncertain samples of the system are shown in Fig. \ref{fig331}. It can be seen that the controller can robustly stabilize the system and provide a fast response with a reasonable control effort, which is depicted in Fig. \ref{fig332}. \begin{figure}
\caption{Block diagram of the control structure}
\label{fig36}
\end{figure} \begin{figure}
\caption{Time response}
\label{fig331}
\caption{Controll effort}
\label{fig332}
\caption{Closed-loop control of nominal and uncertain systems for the unit-step input (Pitch)}
\label{fig33}
\end{figure}
The same procedure is applied to obtain the $H_\infty$ controller for the roll angle. $W_u$ is set to 0.05, and $W_s$ is same as that of the pitch angle, in Eq. (\ref{eq318}). After solving the mixed-sensitivity problem, $\gamma_{opt}$ value is obtained 0.9925, for $a = 0.92$ and the following controller: \begin{equation*}\resizebox{0.98\hsize}{!}{$ C_{roll}\left(s\right)=\frac {4.3173e06 \left(s+371.5\right) \left(s+59.89\right) \left(s+6.764\right) \left(s^2 + 43.53s + 545.3\right)\left(s^2 + 19.03s + 426.2\right)} {\left(s+2.175e05\right) \left(s+1000\right) \left(s+30.67\right) \left(s+27.95\right) \left(s+19.41\right) \left(s+0.001\right)\left(s^2 + 80.99s + 3749\right)} . $}\end{equation*} The singular values of the closed-loop system and the frequency response of each I/O pair are illustrated in Fig. \ref{fig34}. The analysis of this plot is the same as that of the pitch angle, which is omitted for the sake of brevity. The step responses of the nominal and perturbed systems in the presence of $H_\infty$ controller are depicted in Fig. \ref{fig351}. The control efforts are also shown in Fig. \ref{fig352}. \begin{figure}
\caption{The closed loop system singular values and Bode plot of $W_sS$, $ W_uU$, $WT$ (Roll)}
\label{fig34}
\end{figure} \begin{figure}
\caption{Time response}
\label{fig351}
\caption{Controll effort}
\label{fig352}
\caption{Closed-loop control of nominal and uncertain systems for the unit-step input (Roll)}
\label{fig35}
\end{figure}
\section{Experimental results} \subsection{Real Implementation} In this section, we verify the performance and the applicability of the controllers which were designed in the preceding section by implementing on the real robot. The control structure is the same as Fig. \ref{fig36} for both pitch and roll angles. By setting the time-step of the system to 0.001, the continuous controllers are discretized with the frequency of $1\, kHz$ for the practical implementation. To make this task easier, an order reduction is performed that decreases the order of the controllers from 8 to 6. Finally, the controllers for both axes after applying the order reduction are given by: \begin{align*} &C_{pitch}=\frac{3.3227e05 (s+461.5) (s+45.87) (s+5.903) (s^2 + 9.836s + 388.1)}{(s+1.673e04) (s+1000) (s+25.13) (s+0.001)(s^2 + 79.66s + 3577)},\\ &C_{roll}=\frac{4.3173e06 (s+371.8) (s+53.33) (s+7.035) (s^2 + 18.34s + 428.1)}{(s+2.175e05) (s+1000) (s+27.87) (s+0.001) (s^2 + 80.84s + 3820)}. \end{align*} Based on the frequency response of each I/O pair, this reduction has a negligible effect.
Having implemented the controllers, some tests are performed in the outdoor and indoor environments to evaluate the $H_{\infty}$ controller ability in coping with uncertainties. In the first experiment, the operator drives the quadrotor using a radio-controller to assess the qualitative performance of the robot in comparison with a well-tuned PID controller. The tuning parameters of the PID controller for both roll and pitch angles are selected as follows: \begin{equation*} K_p=2.6, K_i=0.2, K_d=0.65 \end{equation*}
According to this experience, the maneuverability has been enhanced, and the quadrotor can quickly regulate the roll and the pitch angle in the presence of the external disturbances such as the unexpected wind effects (i.e. caused by the drone itself while flying near the walls or the weather) or some manual disturbances applied by the operator. Since the states of the quadrotor are coupled, a refinement on the altitude control of the robot is also observed. The video of this experiment is also available online\footnote{\href{https://youtu.be/vAi4x2_XSQQ}{https://youtu.be/vAi4x2\_XSQQ}}.
To evaluate the robustness, stability and tracking performance of the proposed controllers, an experiment is done in which the quadrotor should track the reference input in the outdoor environment, in the presence of light wind. Fig. \ref{fig411} shows the tracking performance of the pitch controller and the ability of the roll controller in regulating the roll angle close to zero. In addition, as it is shown by data-tip in Fig. \ref{fig411}, the performance of the system (e.g. settling time and overshoot) is close to that of the desired transfer function obtained in Eq. (\ref{tid}). The performance of the well-tuned PID controller is also depicted in Fig \ref{fig412}. According to Fig. \ref{fig41}, the $H_{\infty}$ controller has significantly enhanced the performance of the system.
\begin{figure}
\caption{Robust $H_{\infty}$ controller}
\label{fig411}
\caption{PID controller}
\label{fig412}
\caption{Tracking and regualting performance of $H_{\infty}$ and PID controllers }
\label{fig41}
\end{figure}
\subsection{Simulation Results} After validating the practicability and efficiency of the proposed controllers through real experiments, in what follows, the performance of the controllers are evaluated in the simulation and is compared with a well-tuned PID and another robust controller obtained by $\mu$-synthesis method. The robust stability and robust performance of the controllers are also analyzed using the structural singular values. The gains of PID controller are as below: \begin{equation*} \begin{aligned} &Roll: \; &&K_p=1.18, K_i=10.6, K_d=0.0329\\ &Pitch: \; &&K_p=1.01, K_i=10.2, K_d=0.0132 \end{aligned} \end{equation*}
Fig. \ref{fig421} illustrates the step-response of an uncertain system in the presence of the disturbances and uniform random sensor noise. The amplitude of the disturbances is $0.1$, and the maximum value of the random sensor noise is $0.02$. Fig. \ref{fig422}, depicts the roll angle performance in a similar experiment. Based on these figures, the $H_{\infty}$ controller's overshoots for the pitch and roll systems are about $1\%$ and $0.1\%$, respectively, which are less than that of PID and $\mu$ synthesis controllers which are approximately $10\%$. In addition, the $H_{\infty}$ controller regulates the system faster, and its settling time is roughly $0.35\, sec$. The settling times of both PID and $\mu$ synthesis controllers are about $1\,sec$. $H_{\infty}$ controller also has better performance when disturbances occur.
\begin{figure}
\caption{Pitch}
\label{fig421}
\caption{Roll}
\label{fig422}
\caption{$H_{\infty}$, PID and $\mu$ synthesis controllers performance in the presence of disturbance and sensor noise}
\label{fig42}
\end{figure}
In Fig. \ref{fig43}, the robust stability of the $H_{\infty}$, $\mu$ synthesis, and PID controllers are evaluated by structural singular values. As expected, the $H_{\infty}$ and $\mu$ synthesis controllers fulfill the robust stability condition and their $\mu$ values are less than $1$. The robust stability condition is also met for the PID controller, however, in frequencies about $20 \, rad/sec$ gets close to $1$. Moreover, when the PID parameters are set in such a way that the system has a faster response, the robust stability will be lost. The robust performance analysis of the roll and pitch systems using structural singular values are depicted in Fig. \ref{fig44}. Predictably, the $\mu$ synthesis controller offers the robust performance in contrast to $H_{\infty}$ and PID controllers. The $\mu$ synthesis controller is a conservative approach to have the robust performance, which results in poor time-domain performance in comparison with the $H_{\infty}$ controller. Moreover, $\mu$ synthesis method produces a high-order controller which makes its implementation on a real robot challenging \cite{mystkowski2013robust}.
\begin{figure}
\caption{Pitch}
\label{fig431}
\caption{Roll}
\label{fig432}
\caption{Robust stability of $H_{\infty}$, PID and $\mu$ synthesis controllers }
\label{fig43}
\end{figure}
\begin{figure}
\caption{Pitch}
\label{fig441}
\caption{Roll}
\label{fig442}
\caption{Robust performance of $H_{\infty}$, PID and $\mu$ synthesis controllers }
\label{fig44}
\end{figure}
\section{Conclusion} This research has studied the robust attitude control of a quadrotor with taking into account the uncertainty and inputs constraints. Knowing that the model and parameters of the system do not exist, a continuous-time black-box model identification based on the real sampled data is adopted to acquire both nominated linear model of the system and the uncertainty model encapsulating the deviation of the nonlinear system from the nominal model. Now, the essential prerequisites for applying the robust $H_{\infty}$ controller is provided. By solving the mixed-sensitivity problem, a robust stabilizing $H_{\infty}$ controller is obtained which satisfies tracking, disturbance attenuation and input saturation objectives. Having accomplished that, the pitch and roll controllers are implemented on the robot, and qualitative and quantitative experiments are performed to verify the performance of $H_{\infty}$ controllers. According to both experimental and simulation results, the controllers successfully fulfill the aforementioned objectives. Finally, by comparing the performance of the proposed controllers with PID and $\mu$ synthesis controllers, it is demonstrated that the $H_{\infty}$ controller has a better performance.
\end{document}
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\begin{document}
\title{Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation } \author{ G\"unther H\"ormann\thanks{Faculty of Mathematics, University of Vienna, A-1090 Wien, Austria} ~~\&~~ Hisashi Okamoto\thanks{ Dept. of Math., Gakushuin University, Tokyo, 171-8588. Partially supported by JSPS Kakenhi 18H01137. The present work was initiated during HO's stay in the Erwin Schr\"odinger
Institute in Vienna, Austria. Its support is highly appreciated. He also acknowledges the support of JSPS A3 foresight program: Modeling and Simulation of Hierarchical and
Heterogeneous Flow Systems with Applications to Materials Science. } }
\maketitle
\begin{abstract} Spatially periodic solutions of the Fornberg-Whitham equation are studied to
illustrate the mechanism of wave breaking and the formation of shocks for a large class of initial data.
We show that these solutions can be considered to be weak solutions satisfying the entropy condition. By numerical experiments, we show that the breaking waves become shock-wave type in the time evolution. \end{abstract}
\section{Introduction}
We denote by $\mathbb{T} = \mathbb{R} / \mathbb{Z}$ the one-dimensional torus group and identify functions on $\mathbb{T}$ with $1$-periodic functions on $\mathbb{R}$. The Fornberg-Whitham
equation for the wave height $u \colon \mathbb{T} \times [0,\infty[ \to \mathbb{R}$ as a
function of a spatial variable $x \in \mathbb{T}$ and time $t \geq 0$ reads \begin{equation} u_t + uu_x + \left( 1 - \partial_x^2 \right)^{-1} u_x = 0 \qquad ( x \in \mathbb{T}, t > 0) \label{eq:fw01} \end{equation} and is supplied with the initial condition \begin{equation}\label{eq:fw02} u(x,0) = u_0(x) \qquad (x \in \mathbb{T}). \end{equation}
Well-posedness results, local in time, with strong solutions in Sobolev and Besov
spaces have been obtained for both cases, periodic and non-periodic, in \cite{Holmes16, HolTho17}. We cannot always expect globally defined strong solutions. This was predicted with a sketch of proof by \cite{seliger} and later proved rigorously in \cite{CE, Haziot17, hoermann}. These results proved the existence of wave-breaking, i.e., that the solution $u$ remains bounded but $u_x$ becomes singular,
if the initial data displays sufficient asymmetry in its slope. However two questions seem to have remained unanswered: (i) What can we say about the solution after a singularity has emerged? (ii) What is the nature of the singularity?
Our first goal is to prove existence of globally defined weak solutions which extend beyond the time of wave-breaking and singularity formation. The second goal of the present paper is to show numerically that the singularity
developing in such a solution looks very similar to a shock-wave solution of
the inviscid Burgers equation. We emphasize that for the case of the real line in place of the torus, similar investigations have been carried out in \cite{FS},
where the Fornberg-Whitham equation was formulated as and named Burgers-Poisson system (hence we have been unaware of that paper until almost completion of our current paper).
\section{Weak entropy solutions}
A well-known machinery exists for weak solution concepts for nonlinear partial differential equations in the form of hyperbolic conservation laws, but our Equation \eqref{eq:fw01} involves a non-local term and the question is
whether or not this can be harmful to global existence. As we will prove, this is not a big hurdle.
In the literature on nonlinear conservation laws, one of the fundamental references is Kru\v{z}kov's classic paper \cite{K}, where he considered equations of the form $$
u_t + \left( \varphi(u) \right)_x + \psi(u) = 0 \qquad (x \in \mathbb{R}, t > 0). $$ He proved, under mild assumptions on the functions $\varphi$, $\psi$ and on the initial data $u_0$, that a weak solution exists globally in time and that it is unique in an appropriate class of functions. His setting is different from ours in two respects: First, we have $x$ belonging to the one-dimensional torus, while Kru\v{z}kov described the case with $x$ on the real line. Second, in Kru\v{z}kov's paper
$\phi$ and $\psi$ are ordinary functions, while our equation involves a non-local dependence on $u$ in $\psi$. The first difference causes no problem.
The second issue indeed forces us to alter some of the technicalities along the way, but a close examination of the very lucid presentation of the proofs in \cite{K} reveals that the essence of most arguments can be applied to
our equation with only slight modifications, which we will indicate in the sequel.
Note first that we may write $(1 - \partial_x^2)^{-1} u_x = K \ast u_x = K' \ast u$, where the convolution is in the $x$-variable only and with kernel function given by $K \colon \mathbb{T} \to \mathbb{R}$ is given by $K(x) = (e^x + e^{1-x})/(2 (e-1)) = \frac{\sqrt{e}}{e-1}
\cosh(x - \frac{1}{2}) $ for
$0 \leq x \leq 1$ (see, e.g., \cite[Section 3]{hoermann}). Note that $K$ is continuous but is not $C^1$ on $\mathbb{T}$. Note also that the derivative $K'$ is not continuous but is bounded.
We will relate to Kru\v{z}kov's notation in \cite{K} as closely as possible, but will switch the function arguments from $(t,x)$ to $(x,t)$ and occasionally write $u(t)$ to denote the function $x \mapsto u(x,t)$ with a frozen $t$. Compared with the main terms of the equation as labeled by Kru\v{z}kov we set $$
\varphi(u(x,t)) = u(x,t)^2/2 $$ and $$
(\psi u)(x,t) = \big((1 - \partial_x^2)^{-1} u_x(.,t)\big)(x) = (K' \ast u(.,t))(x). $$ The term involving $\varphi$ conforms perfectly with the specifications from \cite{K} and requires no extra consideration at all. For the linear, but non-local, term given by the operator $\psi$ we will describe below suitable adaptations in the proof of uniqueness and make note of an alternative a priori estimate in the proof of existence. Overall in our
situation, spatial periodicity, i.e., the compactness of $\mathbb{T}$, simplifies several
estimates along the way in following the various proofs of key results in \cite{K}. Moreover, we have $L^\infty(\mathbb{T}) \hookrightarrow L^1(\mathbb{T})$.
We use the convolution representation $K' \ast u$ in place of Kru\v{z}kov's term $\psi(u)$ in the following definition of the weak solution concept (where we also incorporate the initial condition into the basic inequality). \begin{definition} Let $u_0 \in L^\infty(\mathbb{T})$ and $T > 0$.
A function $u \in L^\infty(\mathbb{T} \times [0,T])$ is called a weak entropy solution of \eqref{eq:fw01}--\eqref{eq:fw02} if the following \eqref{eqn:entropy-sol} holds true:
\begin{align}
0 &\leq \int_{0}^{T} \int_{\mathbb{T}} \bigg(
|u(x,t) - \lambda| \partial_t \phi(x,t) + \sgn( u(x,t)-\lambda) \frac{u^2(x,t) - \lambda^2}{2}\partial_x \phi(x,t)\nonumber\\
&-\sgn(u(x,t)-\lambda)K'*(u(\cdot,t)-\lambda) (x) \phi(x,t)
\bigg) \,dx \,dt
+ \int_{\mathbb{T}} |u_0(x) - \lambda|\phi(x,0)\,dx
\label{eqn:entropy-sol}
\end{align}
for any $\lambda \in \mathbb{R}$ and for any nonnegative $C^1$-function $\phi$ of compact support in $\mathbb{T} \times \mathbb{R}$. \end{definition}
As is well-known, upon putting $\lambda = \pm \sup |u(x,t)|$ we may deduce that a weak
entropy solution is also a weak (distributional) solution in the sense that in
the integro-differential equation \eqref{eq:fw01} the term $u u_x$ may be interpreted as $\partial_x (u^2)/2$, since $u \in L^\infty$. Thus, we have $$
\text{\ div}_{(x,t)} \begin{pmatrix} u \\ u^2/2 \end{pmatrix} =\partial_t u + \partial_x \big(\frac{u^2}{2}\big) = - K' \ast u + u_0 \otimes \delta
\qquad \text{in } \mathcal{D}'(\mathbb{T} \times ]-T,T[), $$ if we extend $u$ by setting it to $0$ in $\mathbb{T} \times ]-T,0[$. We may therefore
call on Lemma 1.3.3 and the discussion of the weak solution concept in Section 4.3 in \cite{Dafermos}: Upon possibly changing $u$ on a null-set
we may assume that $t \mapsto u(t)$ is continuous $[0,T] \to L^\infty(\mathbb{T})$ with respect to the weak$^*$ topology on $L^\infty(\mathbb{T})$ and we have, in particular, $$
\lim_{t \downarrow 0} \| u(t) - u_0 \|_{L^1} = 0, $$ as required originally by Kru\v{z}kov in \cite[Definition 1]{K}.
\subsection{Uniqueness} We first show that the solution of \eqref{eqn:entropy-sol} is unique. Existence will be proved later.
To show uniqueness of weak entropy solution we need an adaptation of \cite[Theorem 1]{K}---noting that for large $R$ we simply have $K = [0,T_0]
\times \mathbb{T}$ and $S_\tau = \mathbb{T}$ in that statement---and of its proof to our
situation with the non-local linear term $\psi u = K' \ast u$. This requires an appropriate replacement of the constant $\gamma$ in \cite[Equation (3.1)]{K} and an alternative argument in the course of the
proof of \cite[Theorem 1]{K}, namely on the lines following \cite[Equation (3.12)]{K} concerning the term $I_4$ in Kru\v{z}kov's notation (defined there in \cite[Equation (3.4)]{K}), since in our case we cannot directly have a pointwise estimate calling on
the mean value theorem for a classical differentiable function $\psi$. Instead, with two weak solutions $u$ and $v$ with initial values $u_0$ and $v_0$, respectively, we obviously have $$
\sup_{x \in \mathbb{T}} |K'\ast( u(\cdot, t) - v(\cdot, t))(x)| \leq \|K'\|_{L^\infty(\mathbb{T})}
\int_{\mathbb{T}} |u(y,t) - v(y,t)| dy
= \|K'\|_{L^\infty(\mathbb{T})} \| u(t) - v(t)\|_{L^1(\mathbb{T})}, $$ which implies the following replacement of the next to last inequality on page 228 in \cite{K} (with mollifier $\delta_h$ and cut-off $\chi_\varepsilon$
as chosen by Kru\v{z}kov) \begin{multline*}
\int_0^{T_0} \int_{\mathbb{T}} \Big[ \big( \delta_h(t - \rho) - \delta_h(t - \tau) \big) \chi_\varepsilon(x,t)
|u(x,t) - v(x,t)|
\\ + \|K'\|_{L^\infty(\mathbb{T})} \chi_\varepsilon(x,t) \| u(t) - v(t)\|_{L^1(\mathbb{T})} \Big] dx dt \geq 0. \end{multline*} Therefore, we replace the inequality stretching in \cite{K} from the bottom of page 228 to the top of page 229 by \begin{multline*}
\mu(\tau) := \int_{\mathbb{T}} |u(x,\tau) - v(x,\tau)| dx \leq\\ \int_{\mathbb{T}} |u(x,\rho) - v(x,\rho)| dx +
\|K'\|_{L^\infty(\mathbb{T})} \int_\rho^\tau \int_\mathbb{T} \| u(t) - v(t)\|_{L^1(\mathbb{T})} dx dt \\
= \mu(\rho) + \|K'\|_{L^\infty(\mathbb{T})} \int_\rho^\tau \| u(t) - v(t)\|_{L^1(\mathbb{T})} dt. \end{multline*} Then, sending $\rho \to 0$, we arrive at $$
\| u(\tau) - v(\tau)\|_{L^1(\mathbb{T})} \leq \| u_0 - v_0\|_{L^1(\mathbb{T})} +
\|K'\|_{L^\infty(\mathbb{T})} \int_0^\tau \| u(t) - v(t)\|_{L^1(\mathbb{T})}. $$ (Here, the $L^1$-continuity of the weak solution is used.) Now Gronwall's inequality implies the following uniqueness result. \begin{theorem} For any $T > 0$, the weak solution
to \eqref{eq:fw01}--\eqref{eq:fw02} is unique in $\mathbb{T} \times [0,T]$.
More precisely, if $u$ and $v$ are weak solutions with initial values $u_0$ and $v_0$, respectively, then for every $t \in [0,T]$, $$
\int_{\mathbb{T}} |u(x,t) - v(x,t)| dx \leq e^{t \| K' \|_{\infty}} \int_{\mathbb{T}} |u_0(x) - v_0(x)| dx. $$ \end{theorem}
\subsection{Existence}
We will establish the global existence here under the condition that
$ u_0 \in L^{\infty}(\mathbb{T})$.
The key idea in \cite{K} is to consider a parabolic regularization of \eqref{eq:fw01} in the form \begin{equation} u_t + u u_x + \left( 1 - \partial_x^2 \right)^{-1} u_x = \varepsilon u_{xx} \label{eq:vis01} \end{equation} with a small parameter $ \varepsilon > 0$. Our aim is to show that, at least for a
subsequence of $\varepsilon \to 0$, the corresponding solutions (all with the same initial value $u_0$) converge strongly in $L^1$ and are uniformly bounded in $L^{\infty}$.
For any $\varepsilon > 0$, we will show that a strong solution $u$ to \eqref{eq:vis01} with initial value \begin{equation}\label{eq:vis01a}
u(0) = u_0 \in L^2(\mathbb{T}) \end{equation} exists uniquely and globally in time. Due to the nonlocal term, we cannot directly call on the same references that Kru\v{z}kov uses in \cite{K} on
page 231, but the desired result can be shown by a careful iteration of a standard contraction argument, which we describe in Appendix. The solution $u$ of
\eqref{eq:vis01} and \eqref{eq:vis01a}
belongs to $ C([0,T]; L^2(\mathbb{T})) \cap C(]0,T] ; H^1(\mathbb{T}))$ and, in fact, is smooth if $t > 0$. Moreover, if $u_0 \in L^{\infty}(\mathbb{T}) \subset L^{2}(\mathbb{T})$, then
$ t \mapsto \| u(t) \|_{L^\infty}$ is continuous on $[0,T]$.
Let $T > 0$ be arbitrary and consider the unique solution of \eqref{eq:vis01}
and \eqref{eq:vis01a}. For every $\varepsilon > 0$ we denote now the solution of
\eqref{eq:vis01}--\eqref{eq:vis01a} by $u^{\varepsilon}$.
We now have to find an alternative way to obtain
Kru\v{z}kov's basic estimate (4.6) in \cite{K}, which he got from a direct application of the maximum principle. To arrive at our analogue of the basic estimate in \eqref{eq:maxprinc} below, we proceed as follows: Multiplying \eqref{eq:vis01} by $u^{\varepsilon}$ and integrating over the spatial variable $x$ yields $$
\int_\mathbb{T} u^{\varepsilon} u_t^{\varepsilon} dx + \int_\mathbb{T} (u^{\varepsilon})^2 u_x^{\varepsilon} dx +
\int_\mathbb{T} (\psi u^{\varepsilon}) u^{\varepsilon} dx = \varepsilon \int_\mathbb{T} u^{\varepsilon} u_{xx}^{\varepsilon} dx. $$ Noting that $u^{\varepsilon} \mapsto \psi u^{\varepsilon} = (1- \partial_x^2)^{-1} \partial_x u^{\varepsilon} = K' \ast u^{\varepsilon}$
is a skew-symmetric linear operator with respect to the standard inner product on $L^2(\mathbb{T})$, writing $u^{\varepsilon} u_t^{\varepsilon} = \partial_t (u^{\varepsilon})^2/2$
and $(u^{\varepsilon})^2 u_x^{\varepsilon} = \partial_x((u^{\varepsilon})^3)/3$, and integrating by parts on the right-hand side, yields (thanks to periodicity) \begin{equation} \frac{1}{2} \frac{d}{dt} \int_{\mathbb{T}} (u^{\varepsilon})^2 dx = - \epsilon \int_{\mathbb{T}} (u_x^{\varepsilon})^2 dx. \label{eq:vis02} \end{equation} In particular, we have for every $t \in [0,T]$ the a priori estimate \begin{equation}
\| u^{\varepsilon}(t) \|_{L^2(\mathbb{T})} \le \| u_0 \|_{L^2(\mathbb{T})}, \label{eq:vis03} \end{equation} which is independent of $\varepsilon$.
From \eqref{eq:vis03} and since $( 1 - \partial_x^2)^{-1}$ is a pseudodifferential operator of order $-2$, the $H^1$-norm of $v^\varepsilon(t) := ( 1 - \partial_x^2)^{-1} u^\varepsilon_x(t)$
is bounded by $c \| u_0 \|_{L^2(\mathbb{T})}$ for some constant $c$ independent of $\varepsilon$ and of $t$. Via the continuous imbedding $H^1(\mathbb{T})
\hookrightarrow L^\infty(\mathbb{T})$ we thus have $\| v^\varepsilon (t) \|_{L^{\infty}(\mathbb{T})}
\leq c' \| u_0 \|_{L^2(\mathbb{T})}$ for some constant $c'$ independent of
$\varepsilon$ and of $t$. We now interpret the original equation in the form $$
\partial_t u^\varepsilon (x,t) - \varepsilon \partial_x^2 u^\varepsilon(x,t) + u^\varepsilon(x,t)
\partial_x u^\varepsilon(x,t) = - v^\varepsilon(x,t) $$ and apply the theorem on page 230 in \cite{J} to obtain $$
|u^\varepsilon(x,t) | \le \| u^\varepsilon(\cdot,s) \|_{L^{\infty}} + (T-s) \sup_{s\le \tau \le T, y \in
\mathbb{T}} | v^\varepsilon(y,\tau)| $$ for all $ 0 < s \le t \le T$. (Here we regard the factor $u^\varepsilon(x,t)$ in the term $ u^\varepsilon(x,t) \partial_x u^\varepsilon(x,t)$ as a coefficient to the first order derivative and also note that in our case
the zero order coefficient vanishes. Although the text in \cite{J}
assumes more regularity of the coefficients and data, continuity is sufficient.)
We then let $ s \rightarrow 0$ to obtain \begin{equation}\label{eq:maxprinc}
\forall x \in \mathbb{T}, \forall t \in [0,T] : \quad
|u^\varepsilon(x,t)| \leq \| u_0 \|_{L^{\infty}(\mathbb{T})} + c'\, T \| u_0 \|_{L^2(\mathbb{T})}. \end{equation} This inequality is our analogue of Kru\v{z}kov's basic estimate (4.6) in \cite{K}
and we may now return to follow along his lines again more closely (note that we are closest to what he classifies as `Case A' in his paper on page 230 in next
to the last paragraph) to establish a uniform modulus of $L^1$-continuity. In fact, \cite[Equation (4.7)]{K} for $w(x,t) := u^\varepsilon(x+z,t) - u^\varepsilon(x,t)$ holds in our case with $e_i =0$ and replacing the term $c w$ by $K' \ast w$, preserving the Lipschitz continuity properties noted on top of page 232
in \cite{K} (in our case even globally on the compact torus). Lemma 5 in \cite{K} is applicable in essentially the same way in establishing the key
modulus of continuity estimate \cite[Equation (4.15)]{K} (with an appropriate
proof variant taking into account the convolution term and showing
(4.13) directly from (1.3) there). Thus, we have at least sketched how
everything is in place and sufficient to apply Kru\v{z}kov's method in
proving existence from compactness in $L^1$ (via boundedness and equicontinuity of the $L^1$-norm, \cite[Theorem 4.26]{Brezis2011}) and
uniform $L^\infty$-bounds. In combination with the above uniqueness result, we have the following statement. \begin{theorem} Let $T > 0$ be arbitrary. For any $u_0 \in L^\infty(\mathbb{T})$, there exists a unique weak entropy solution of \eqref{eq:fw01} and \eqref{eq:fw02} on the time interval $[0,T]$. \end{theorem}
\section{Numerical experiments by finite differences}
We now study numerical solutions and will observe that many of these exhibit singularities of shock-wave type. The solutions have been computed numerically under periodic boundary conditions and employing Godunov's finite difference method.
We employ the following finite difference method with the uniform grid sizes $ h = \Delta x, \tau = \Delta t$. The nonlinear term is discretized by $$ u^{n+1}_k = u_k^n - \frac{\tau}{h} \big( g(u_k^n, u_{k+1}^n) - g(u_{k-1}^n, u_k^n) \big). $$ where the numerical flux is defined by $$ g(u_{k-1}^n, u_k^n) = \left\{ \begin{array}{ll} f\left( u_{k-1}^n \right) \qquad & \hbox{if} \quad u_{k-1}^n \ge u_k^n ~~ \hbox{and} ~~ f\left( u_{k-1}^n\right) \ge
f\left( u_k^n\right), \\ f\left( u_{k}^n \right) \qquad & \hbox{if} \quad u_{k-1}^n \ge u_k^n ~~ \hbox{and} ~~ f\left( u_{k-1}^n\right) \le
f\left( u_k^n\right), \\ f\left( u_{k-1}^n \right) \qquad & \hbox{if} \quad u_{k-1}^n \le u_k^n ~~ \hbox{and} ~~ f'\left( u_{k-1}^n\right) \ge 0, \\ f\left( u_{k}^n \right) \qquad & \hbox{if} \quad u_{k-1}^n \le u_k^n ~~ \hbox{and} ~~ f'\left( u_{k-1}^n\right) \le 0, \\ 0 \qquad & \hbox{otherwise}. \end{array} \right. $$ Here $u_k^n$ is the approximation for $ u(kh,n\tau)$, and $f(u) = u^2/2$. This is the Godunov method as explicated in \cite{CM}.
The nonlocal term is discretized by the following central difference scheme: We put $ v = (1 - \partial_x^2)^{-1} u_x$, whence we have $ v - v_{xx} = u_x$. The function $v$ is determined by solving the linear equation $$ v_k - \frac{ v_{k+1} -2 v_k + v_{k-1}}{h^2} = \frac{u_{k+1} - u_{k-1}}{2h}, $$ or, $$ ( 1 + 2h^{-2}) v_k - h^{-2} ( v_{k+1} + v_{k-1} )= \frac{u_{k+1} - u_{k-1}}{2h}. $$
We set $ N = 1000$, $ h = 1/N$, and $\tau = 0.4h/q$, where $q$ is the typical size of the initial data. We first test the following two initial data: \begin{align*} & data 1 \hskip 1cm u_0(x) = \cos (2\pi x + 0.5) + 1, \\ & data 2 \hskip 1cm u_0(x) = 0.2 \cos (2\pi x) + 0.1\cos (4\pi x) - 0.3 \sin (6\pi x) + 0.5 \end{align*} The corresponding profiles of $u$ from \emph{data}1 are shown in Figure \ref{zu:1}. As the profile moves to the right, the formation of a shock is clearly visible. After the emergence of the shock, the jump height at the discontinuity decreases as $t$ increases, namely, if $ \xi(t)$ denotes the position of the discontinuity at time $t$, the difference of the one-sided limits $u(\xi(t)-0) - u(\xi(t)+0)$ is a decreasing function of $t$.
The solution with \emph{data}2 is shown in Figure \ref{zu:2}. In this case, even multiple shocks are formed and merging of some of the shocks over time can be observed.
\begin{figure}
\caption{ The solution from \emph{ data}1. $ 0 \le t \le 0.65$. }
\label{zu:1}
\end{figure}
\begin{figure}
\caption{ data2 }
\label{zu:2}
\end{figure}
These and other experiments suggest the development of wave-breaking singularities into shock discontinuities. In fact, as far as we have computed, only shock-type singularities in wave solutions have been found. Moreover, the computations also suggest that global solutions exist, if the initial data are small---the recent result in \cite[Theorem 1.5]{Itasaka} on Fornberg-Whitham-type equations with nonlinear term $\partial_x (u^p/p)$, $p \geq 5$, seems to support our observation.
\subsection{Remarks on shock conditions and asymptotic properties}
In our experiments all initial functions of the following form $u_0 = \sum_{1 \le k \le 3} [ a_k \cos(2k\pi x) + b_k \sin(2k\pi x) ] $
which are periodic in $x$, but not necessarily symmetric, produce a shock spontaneously, if their $L^{\infty}$-norms are large. If the initial data is small, the numerical solution exists for quite a long time. For instance, if $ u(x,0) = q\cos(2\pi x)$, a shock wave was observed for $ q > 0.015$. For $ q < 0.01$ the solution seems to exist forever. For $ 0.01 < q < 0.015$, our experiments are indecisive. It may exist forever, or it may have a shock after quite a long, numerically undetectable time has passed. Note that the guaranteed time of existence according
to \cite[Theorem 1]{Holmes16} is inverse proportional to $\| u_0 \|_{H^2}$. The paper mentioned above, \cite{Itasaka} discussing a whole class of Fornberg-Whitham-type equations, contains also detailed information
on the existence time.
The propagation speed of the shock is the same as in the case of the inviscid Burgers equation \begin{equation} u_t + uu_x = 0 \qquad (x \in \mathbb{T}, t > 0). \label{eq:bur} \end{equation} Indeed,
for any $w$ in $L^2$, the function $ \left( 1 - \partial_x^2 \right)^{-1} w_x$ belongs to $H^2({\mathbb T})$, hence is continuous, and therefore does not contribute to the Rankine-Hugoniot jump relation $$ c = \frac{u^+ + u^-}{2}. $$ Here, $c$ is the propagation speed of the shock and $u^+$ and $ u^-$ denote the limit values of $u$ from the right and the left at the discontinuity,
respectively.
Although the formation of a shock is quite similar in both equations, the asymptotic behavior may be different. In the case of the Burgers equation, the jump discontinuity vanishes in the limit and the solution converges to a constant function as $ t \rightarrow \infty$ (for a proof see \cite{lax} or \cite[Theorems 6.4.9 and 11.12.5]{Dafermos}). On the other hand, the Fornberg-Whitham equation possesses many traveling wave solutions that obviously do not decay and it is not yet conclusively shown whether
or not for the discontinuous solutions displayed above the jump heights definitely
decay to zero. We are not aware of a decisive result whether discontinuous
\emph{periodic} traveling waves exist, although we discuss below a negative result in the case of a single shock. (Existence of non-periodic discontinuous wave solutions has been shown in \cite{FS,hoermann2}.)
Another feature of smooth solutions $u$ to the inviscid Burgers equation is that
it preserves the values of maxima and minima of $ u(t,\cdot)$ as $t$ progresses. The Fornberg-Whitham equation does not keep those values constant due to the presence of the nonlocal term. However, our experiments suggest that, although $\max_x u(x,t)$ and $ \min_x u(x,t)$ is not a constant function of $t$, they are nearly constant, or, at least they seem to stay bounded as $t \to \infty$.
The number of peaks of any solution to the inviscid Burgers equation is non-increasing, while we observe that in case of our equation they may increase (and decrease) as $t$ varies.
Although several of the minor discrepancies exist as indicated, the solutions of \eqref{eq:fw01} have some
similarity with solutions of the inviscid Burgers equation in any time interval until and little after the formation of a shock discontinuity.
\subsection{Traveling waves} In this subsection we first investigate the stability of traveling wave
solutions and then the non-existence of such with a single shock.
Consider a continuously differentiable traveling wave $u(x,t) = U(x-ct)$ with speed $c > 0$. Inserting this into \eqref{eq:fw01} and integrating once we
deduce that the profile function $U$ satisfies the equation \begin{equation} -cU + \frac{U^2}{2} + \left( 1 - \partial_x^2 \right)^{-1} U = 0. \label{eq:U} \end{equation} (The constant of integration may be set to zero without losing generality, if $U$ and $c$ are suitably normalized.)
If we define $V$ by $$ V = -cU + \frac{U^2}{2}, $$ then $2 V + c^2 = (U - c)^2 \geq 0$ and $U(x) = c \pm \sqrt{ c^2 + 2V(x)}$, where we choose the negative sign for the root, since only this connects
$U(x) = 0$ with $V(x) = 0$. We obtain the differential equation \begin{equation} V - V_{xx} + c - \sqrt{ c^2 + 2V} = 0, \label{eq:V} \end{equation} which possesses $V \equiv 0$ as trivial solution for all $c$. Linearization of \eqref{eq:V} at $V \equiv 0$ yields $$ V - V_{xx} - \frac{V}{c} = 0. $$ In terms of the Fourier coefficients $(\hat{V}(m))_{m \in \mathbb{Z}}$ for the
$1$-periodic function $V$, this means $\left(1 + (2 \pi m)^2 - \frac{1}{c}\right) \cdot \hat{V}(m) = 0$ for every
$m \in \mathbb{Z}$. We obtain for every $n \in \mathbb{N}$ a nontrivial solution to the linearized equation in the form $V_n(x) := \cos (2\pi n x)$, if the speed attains the bifurcation value $$
c = c_n := \frac{1}{ 1 + (2\pi n)^2}, $$ otherwise we are left with $V \equiv 0$ as the only solution.
In the case of $n=1$, nontrivial bifurcating solutions to the nonlinear
differential equation exist in the interval $ c_1 \approx 0.0247 < c < 0.02695$. As $c$ tends to the upper limit, $\min ( c^2 + 2V)$ tends to zero, and the profile $U = c - \sqrt{c^2 + 2V}$
tends to a function with a corner singularity. This Lipshitz continuous traveling wave, called peakon, has already been known to exist for a long time (cf.\ \cite{FW, W}) and occurs also as part of the analysis in \cite{ZT} . Here we have computed these traveling waves in the context above,
i.e., as the solutions of the boundary value problem \eqref{eq:V} and illustrate some of them in Figure \ref{zu:travel}.
\begin{figure}
\caption{ traveling wave $U$; $c = 0.025, 0.0255, 0.026, 0.0269$.
}
\label{zu:travel}
\end{figure}
In order to investigate stability, we picked the solution corresponding to
$ c = 0.0255$, input it as initial data, and computed the time dependent
solution shown in Figure \ref{zu:trav}, which illustrates that the fixed wave profile travels at constant speed. In the time interval used by us, $ 0 < t < 30$, the effect of numerical viscosity is invisible, but for long time intervals, it is expected to influence the numerical solution.
\begin{figure}
\caption{
The time dependent solution with the traveling wave as the initial data. }
\label{zu:trav}
\end{figure}
We consider a perturbation of the initial data in the form $ U(x) + \delta d(x)$, where $\delta$ is $5\%$ of the amplitude of $U$ and $d(x)$ is given as follows: \begin{equation} \label{eq:dis2} d(x) = \cos(2k\pi x) \quad ( k = 2,3,4) \quad \text{ or } \quad d(x) = \left\{ \begin{array}{ll} 1 - \cos(4\pi x) \qquad & ( 0 \le x < 1/2), \\ 0 & (\hbox{otherwise}). \end{array} \right. \end{equation} Note that the latter disturbance is asymmetric. It turns out that in the time interval
$ 0 \le t \le 300$ the solution stays in a small neighborhood of the original
solution $U$ and only a small oscillation could be observed as is shown in Figure \ref{zu:trav4}, where the points $( u_{300}^n,u_{600}^n)$ with $n$ corresponding to $ 0 \le t \le 300$ were plotted. If the initial perturbation is null, these describe a closed curve; once the initial disturbance is added, the corresponding curve departs from the closed orbit, but remains in a certain neighborhood. These and similar experiments may be interpreted as support for the conjecture of stability of the traveling wave with speed $ c = 0.0255$. More experiments carried out for the case $ c = 0.0265$ gave similar results.
\begin{figure}
\caption{
The time dependent solution with $d$ as in the second case of \eqref{eq:dis2}. The points $( u_{300}^n,u_{600}^n)$ with $n$ corresponding to $ 0 \le t \le 300$ are plotted. }
\label{zu:trav4}
\end{figure}
\subsubsection{Nonexistence of traveling waves with a single shock}
In view of the solutions shown earlier that created shocks after wave breaking, with jump height decreasing as time progresses, it is natural to ask whether
periodic traveling wave solutions with jump discontinuities exist. For the non-periodic
case it has been shown that discontinuous traveling waves with single jumps exist (see \cite{FS,hoermann2}). Contrary to this, there is no periodic traveling wave with a single jump, as we will argue in the following.
Suppose $U$ is the profile function for a \emph{discontinuous} traveling wave solution that is piecewise $C^1$ on the torus $\mathbb{T}$, i.e., $C^1$ except for a
single point $x_0 \in \mathbb{T}$ where $U$ as well as $U'$ possess one-sided limits. Since the Fornberg-Whitham equation is invariant under translations in the $x$-variable, we may restrict to the case $x_0 = 0$, thus we may think of the profile function as a $C^1$-function $U \colon [0,1] \to \mathbb{R}$ with $U(0) \neq U(1)$.
The Rankine-Hugoniot shock condition requires $$
U(0) + U(1) = 2c. $$
We come back to Equation \eqref{eq:U} for the profile function $U$, but this time keep track of the constant of integration, for later convenience in the form $$
-cU + \frac{U^2}{2} + \left( 1 - \partial_x^2 \right)^{-1} U = \beta - \frac{c^2}{2} + c, $$ with $\beta \in \mathbb{R}$. We now put $Y := U - c$ and $ Z := Y Y' = ( Y^2 /2 )'$, note that $\left( 1 - \partial_x^2 \right)^{-1} 1
= K \ast 1 = \int_{\mathbb{T}} K(x) \, dx = 1$, and insert into the equation for $U$ to obtain $$
\frac{Y^2}{2} + \left( 1 - \partial_x^2 \right)^{-1} Y = \beta, $$ which also shows that $Z = (Y^2/2)' = - K' \ast Y$ is continuous as a function on the torus and piecewise $C^1$, i.e.\ can be thought of as $C^1$- function $Z \colon [0,1] \to \mathbb{R}$ with \begin{equation}
Z(0) = Z(1). \label{eq:Zcont} \end{equation}
We may therefore apply $(1 - \partial_x^2)$ and arrive at $$
\beta = \frac{Y^2}{2} - \left( \frac{Y^2}{2} \right)'' + Y = \frac{Y^2}{2} - Z' + Y. $$ We collect the equations for $Y$ and $Z$ in the following first-order system \begin{align}
Y Y' &= Z, \label{eq:Y}\\
Z' &= \frac{Y^2}{2} + Y - \beta = \frac{1}{2} \left( (Y + 1)^2 - (2 \beta +1) \right), \label{eq:Z} \end{align} for the $C^1$-functions $Y$ and $Z$ on $[0,1]$. The requirements on $U$,
including the Rankine-Hugoniot condition, now read \begin{equation}
Y(0) \neq Y(1) \quad\text{and}\quad Y(0) + Y(1) = 0, \label{eq:Ycond} \end{equation} while for $Z$ we have the periodic continuity condition \eqref{eq:Zcont}.
We split the further analysis into two cases depending on the value of $\beta$:
\emph{Case $2 \beta + 1 \leq 0$}: Equation \eqref{eq:Z} implies that $Z' \geq 0$. By \eqref{eq:Zcont}, this leaves only the option that $Z' = 0$, hence
$0 \leq (Y+1)^2 = 2 \beta + 1 \leq 0$, so that $Y$ would have to be constant
(equal to $-1$), which is a contradiction to \eqref{eq:Ycond}.
\emph{Case $2 \beta + 1 > 0$}: As a preliminary observation, we note the following: The conditions \eqref{eq:Ycond} imply that either $Y(0) < 0 < Y(1)$ or $Y(0) > 0 > Y(1)$, hence there exists $s \in \,]0,1[$ such that $Y(s) = 0$; by \eqref{eq:Y}, we have also $Z(s) = 0$, thus the trajectory in the $(Y,Z)$-phase diagram passes
through the origin $(0,0)$ and, due to \eqref{eq:Ycond} and \eqref{eq:Zcont},
has as end point $(Y(1),Z(1)$ precisely the reflection at the $Z$-axis of
its starting point $(Y(0),Z(0))$.
Since $2 \beta + 1 > 0$, the vector field defining the right-hand sides of
\eqref{eq:Y} and \eqref{eq:Z} has equilibirum points in $(-1 \pm \sqrt{2 \beta + 1 },0)$
and the qualitative analysis in \cite[Section 3]{FS} may be applied to show that no
trajectory satisfying all the above specifications can exist (the quantities $u$, $E$, $d$ used there correspond here to $Y$, $Z$, $\beta$, respectively).
In summary, we have shown that there is no periodic traveling wave with a single shock.
The question remains whether there exist traveling wave solutions with two or more shock discontinuities (and being piecwise $C^1$). Reviewing the above line of arguments, the reasoning in the first case, $2 \beta + 1 \leq 0$, seems to essentially stay valid (with monotonicity arguments on subintervals instead), whereas in the second case, $2 \beta + 1 > 0$, the crucial consequence that $Y$ (and hence $Z$) has to vanish somewhere is lost, if $Y$ is allowed to have an additional jump discontinuity. We have to leave this issue open for potential
future analysis and note that, even in that case, $Z$ still has to be a continuous function on all of $\mathbb{T}$ thanks to the relation $Z = (Y^2/2)' = - K' \ast Y$.
\section{Concluding remarks}
The Godunov method is of first order and in further numerical studies one might
want to employ a method of higher order such as the ENO (Essentially Non-Oscillatory)
scheme for better accuracy. Furthermore, the computation of $(1-\partial_x^2)^{-2}\partial_x u$ from $u$ in our approach may contain a significant truncation error. Therefore, there admittedly is a lot of room for improvement in the numerical experiments. Nevertheless, we do expect our numerical solutions to correctly show several
main qualitative features. For instance, the almost generic emergence of
shocks in the wave solution $u$ appears to be undoubted and our experiments
strongly indicate that many of the traveling wave solutions are stable. On the other hand, the large time behavior of solutions in general cannot be assessed quantitatively by our method and we clearly lack theoretical insight.
In summary, we are left with (at least) the following questions which we were unable to answer in terms of a rigorous analysis so far: \begin{enumerate} \item Existence of periodic traveling waves with jump discontinuities
(i.e, non-decreasing shocks), although the case of a single jump could be ruled out. \item Boundedness of the spatial minimum and maximum of $u(t)$ as a function of $t$ and independent of the existence time T. \item Wave breaking in more generic cases than those covered by the asymmetry condition in the wave breaking theorems. \item Global (in time) existence of strong solutions for (generic) initial data with small Sobolev norm. \end{enumerate}
\appendix
\section{Appendix: Proof of the existence of a global solution to the regularized equation}
We point out that Fujita and Kato's theorem on the local existence of a strong solution of the Navier-Stokes equations (\cite{FK}) can be used to prove the existence of a strong solution of \eqref{eq:vis01} and \eqref{eq:vis01a}. Their method is explained and put into a more general context in \cite{H} and \cite{pazy}. The same techniques have also been applied in \cite{CO}, which indeed involves
a nonlinearity similar to that in the Fornberg-Whitham equation. Given some familiarity with the theory of analytic semigroups, one would quickly see that the proof described
below is only a slight variation of the classical methods. However, we think that
there will be some benefit or at least convenience for the reader in outlining the proof here.
We first note that $ A := - \epsilon \partial_x^2$ generates an analytic semigroup of operators in $L^2(\mathbb{T})$ (see, e.g., \cite{EN, kato, pazy}). In terms of the semigroup, the Cauchy problem \eqref{eq:vis01} and \eqref{eq:vis01a}
is equivalent to the following fixed point problem \begin{equation} u(t) = e^{-tA} u_0 + \int_0^t e^{-(t-s)A} \left[ - u(s) u_x(s) - (1-\partial_x^2)^{-1} u_x(s) \right] ds =: F(u)(t) \label{eq:fu} \end{equation} We will show that this equation has a unique solution for every $ u_0 \in L^2(\mathbb{T})$. The proof is carried out by a successive approximation as follows: $u^{(0)}(t) = e^{-tA} u_0$,
$u^{(n+1)}(t ) = F( u^{(n)})(t)$ for $ n = 0,1,\ldots$ and the solution is then obtained by showing that $F$ is a contraction mapping with respect to a suitable metric.
In the sequel we will denote by $c$ or $c'$ various constants that may depend
on $\epsilon$ but not on $t$. Furthermore, let $\gamma_0$ be a constant such that $$
\left\| \partial_x e^{-tA} v \right\| \le \gamma_0 \| v\| t^{-1/2} \qquad (v \in L^2), $$
where here and hereafter $\| ~~\|$ denotes the $L^2$-norm.
Let $R > 0$ and suppose that we are given a continuous map $t \mapsto w(t)$,
$[0,T] \to L^2({\mathbb T})$, which takes its values for $t > 0$ in $H^1(\mathbb{T})$ and satisfies \begin{equation}
\| w(t) \| \le R \qquad ( 0 \le t \le T),
\qquad \| w_x(t) \| \le Rt^{-1/2} \qquad (0 < t \le T). \label{eq:uuu} \end{equation}
We then have $ \| w(t) \|_{L^{\infty}} \le c \| w(t) \|^{1/2} \| \partial_x w(t) \|^{1/2} \le c R t^{-1/4}$ and it is not difficult to deduce \begin{align*}
\| F(w)(t)\| & \le \| u_0 \| + \int_0^t \| e^{-(t-s)A} \|
\left[ \| w(s) \|_{L^{\infty}} \| w_x(s) \| + \| (1-\partial_x^2)^{-1} w_x(s) \| \right] ds \\
& \le \| u_0 \| + \int_0^t \left[ cR^2s^{-3/4} + cR \right] ds
= \| u_0 \| + 4 cR^2 t^{1/4} + cR t. \end{align*} Similarly, \begin{align*}
\| \partial_x F(w)(t)\| & \le \gamma_0\| u_0 \| t^{-1/2} + \int_0^t \| \partial_x e^{-(t-s)A} \|
\left[ \| w(s) \|_{L^{\infty}} \| w_x(s) \| + \| (1-\partial_x^2)^{-1} w_x(s) \| \right] ds \\
& \le \gamma_0 \| u_0 \|t^{-1/2} + \int_0^t \gamma_0 (t-s)^{-1/2} \left[ c R^2 s^{-3/4} + cR \right] ds \\ &
= \gamma_0 \| u_0 \| t^{-1/2} + \gamma_0 c R^2 t^{-1/4}B(1/2,1/4) +
2 \gamma_0 c R t^{1/2}, \end{align*} where $\gamma_0 $ is as above and $B(\cdot,\cdot)$ denotes Euler's beta function.
Now we may take any $R > \max\{ \| u_0 \|, \gamma_0 \| u_0 \| \}$. Then we may choose $T$ small enough such that
any $w$ satisfying \eqref{eq:uuu} implies that \eqref{eq:uuu} holds also with $F(w)$ in place of $w$.
If we equip $Y_T := C([0,T] ; L^2) \cap C(]0,T] ; H^1)$ with the (complete) norm $$
\| u \|_* := \max \left\{
\max_{0 \le t \le T} \| u(t) \|, \sup_{0 < t \le T} t^{1/2} \| u_x(t) \| \right\}, $$ then the corresponding closed ball $B_R$ of radius $R$ around $0$ in $Y_T$ is mapped into itself (upon noting that $t \mapsto \partial_x F(u)(t)$ is continuous $]0,T] \to L^2$ for every $u \in Y_T$).
Our next task is to show that the map $F$ is a contraction with respect to
the norm $\| ~~ \|_*$ (for some $T > 0$ and on bounded subsets). The proof is similar to the above estimates. Indeed, \begin{align*}
\| F(w)(t) - F(z)(t) \| \le & \int_0^t
\| e^{-(t-s)A} \|
\bigg[ \| w(s) \|_{L^{\infty}} \| w_x(s) - z_x(s) \| \\ & +
\| w(s) - z(s) \|_{L^{\infty}} \| z_x(t) \| +
\left\| (1-\partial_x^2)^{-1} \big( w_x(s) - z_x(s) \big) \right\| \bigg] ds \\
& \le \int_0^t \Big( c\| w\|_* \| w-z \|_* s^{-3/4} +
c\| z\|_* \| w-z \|_* s^{-3/4} + c \| w-z \|_* \Big) ds \\
& \le c' t^{1/4} \big( \| w \|_* + \| z\|_* \big) \| w-z \|_* + ct \| w- z\|_*. \end{align*}
and similarly for $ \| \partial_x F(u) - \partial_x F(z)\| $. Thus, for $T$ sufficiently small, $F$ becomes a contractive mapping on $B_R$ and we
obtain a unique solution $u \in Y_T$ of \eqref{eq:fu}.
\begin{theorem} Let $R>0$ and $\varepsilon > 0$, then there exists $T >0$ such that
\eqref{eq:vis01} and \eqref{eq:vis01a} possesses a unique strong solution in $[0,T]$ for every $u_0 \in L^2({\mathbb T})$ such
that $\max\{ \| u_0 \|, \gamma_0 \| u_0 \| \} < R$. \end{theorem}
We emphasize that here $T$ depends on $R$ and $\epsilon$, but not on any individual $u_0$ as far as $u_0$ satisfies $\max\{ \| u_0 \|, \gamma_0 \| u_0 \| \} < R$. Since the spatial $L^2$-norm in the solution is conserved over time,
the solution may be continued any number of times, thus we obtain a solution globally in time. Therefore, we conclude the global unique existence of a solution to the parabolic equation \eqref{eq:vis01}. The following additional features of this solution are used
in the proof of the weak solution: Due to the properties of the heat kernel, $$ u \in C^\infty(\mathbb{T} \times\, ]0,T]) $$ and moreover, if $u_0 \in L^\infty$, although $t \mapsto u(t)$ may be discontinuous at $ t = 0$ as a map into $L^{\infty}$, the real-valued map
$t \mapsto \| u(t) \|_{L^{\infty}}$ is continuous on the closed interval $[0,T]$.
\end{document}
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arXiv
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\begin{document}
\title{Honeycomb arrays}
\author{Simon R.\ Blackburn\thanks{Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom. \texttt{\{s.blackburn, a.panoui\}@rhul.ac.uk}}\\ Anastasia Panoui$^*$\\ Maura B. Paterson\thanks{Department of Economics, Mathematics and Statistics, Birkbeck, University of London, Malet Street, London WC1E 7HX, United Kingdom. \texttt{[email protected]}}\\ and\\ Douglas R. Stinson\thanks{David R.\ Cheriton School of Computer Science, University of Waterloo, Waterloo Ontario, N2L 3G1, Canada. \texttt{[email protected]}}}
\maketitle
\begin{abstract} A honeycomb array is an analogue of a Costas array in the hexagonal grid; they were first studied by Golomb and Taylor in 1984. A recent result of Blackburn, Etzion, Martin and Paterson has shown that (in contrast to the situation for Costas arrays) there are only finitely many examples of honeycomb arrays, though their bound on the maximal size of a honeycomb array is too large to permit an exhaustive search over all possibilities.
The present paper contains a theorem that significantly limits the number of possibilities for a honeycomb array (in particular, the theorem implies that the number of dots in a honeycomb array must be odd). Computer searches for honeycomb arrays are summarised, and two new examples of honeycomb arrays with 15 dots are given. \end{abstract}
\section{Introduction}
Honeycomb arrays were introduced by Golomb and Taylor~\cite{GolombTaylor} in 1984, as a hexagonal analogue of Costas arrays. Examples of honeycomb arrays are given in Figures~\ref{honey137_figure} to~\ref{honey45_figure} below. \begin{figure}
\caption{A Lee sphere, and three natural directions}
\label{fig:Lee_example}
\end{figure}A honeycomb array is a collection of $n$ dots in a hexagonal array with two properties: \begin{itemize} \item \textbf{(The hexagonal permutation property)} There are three natural directions in a hexagonal grid (see Figure~\ref{fig:Lee_example}). Considering `rows' in each of these three directions, the dots occupy $n$ consecutive rows, with exactly one dot in each row. \item \textbf{(The distinct differences property)} The $n(n-1)$ vector differences between pairs of distinct dots are all different. \end{itemize}
Golomb and Taylor found $10$ examples of honeycomb arrays (up to symmetry), and conjectured that infinite families of honeycomb arrays exist. Blackburn, Etzion, Martin and Paterson~\cite{BlackburnEtzion} recently disproved this conjecture: there are only a finite number of honeycomb arrays. Unfortunately, the bound on the maximal size of a honeycomb array that Blackburn et al.\ provide is far too large to enable an exhaustive computer search over all open cases. In this paper, we prove a theorem that significantly limits the possibilities for a honeycomb array with $n$ dots. (In particular, we show that $n$ must be odd.) We report on our computer searches for honeycomb arrays, and give two previously unknown examples with $15$ dots.
We now introduce a little more notation, so that we can state the main result of our paper more precisely.
We say that a collection of dots in the hexagonal grid is a \emph{hexagonal permutation} if it satisfies the hexagonal permutation property. A collection of dots is a \emph{distinct difference configuration} if it satisfies the distinct difference property. So a honeycomb array is a hexagonal permutation that is a distinct difference configuration.
We say that hexagons are \emph{adjacent} if they share an edge, and we say that two hexagons $A$ and $B$ are \emph{at distance $d$} if the shortest path from $A$ to $B$ (travelling along adjacent hexagons) has length $d$. A \emph{Lee sphere of radius $r$} is a region of the hexagonal grid consisting of all hexagons at distance $r$ or less from a fixed hexagon (the \emph{centre} of the sphere). The region in Figure~\ref{fig:Lee_example} is a Lee sphere of radius $2$. Note that a Lee sphere of radius $r$ intersects exactly $2r+1$ rows in each of the three natural directions in the grid. A \emph{honeycomb array of radius $r$} is a honeycomb array with $2r+1$ dots contained in a Lee sphere of radius $r$.
There are many other natural regions of the hexagonal grid that have the property that they intersect $n$ rows in each direction. One example, the tricentred Lee sphere of radius $r$, is shown in Figure~\ref{fig:tricentred_example}: it is the union of three Lee spheres of radius $r$ with pairwise adjacent centres, and intersects exactly $2r+2$ rows in any direction.\begin{figure}
\caption{A tricentred Lee sphere}
\label{fig:tricentred_example}
\end{figure}
Does there exist a honeycomb array with $2r+2$ dots contained in a tricentred Lee sphere of radius~$r$? Golomb and Taylor did not find any such examples: they commented~\cite[Page~1156]{GolombTaylor} that all known examples of honeycomb arrays with $n$ dots were in fact honeycomb arrays of radius $r$, but stated ``we have not proved that this must always be the case''. We prove the following:
\begin{theorem} \label{thm:main} Let $n$ be an integer, and suppose there exists a hexagonal permutation $\pi$ with $n$ dots. Then $n$ is odd, and the dots of $\pi$ are contained in a Lee sphere of radius $(n-1)/2$. \end{theorem}
Since any honeycomb array is a hexagonal permutation, the following result follows immediately from Theorem~\ref{thm:main}:
\begin{corollary} \label{cor:honeycomb} Any honeycomb array is a honeycomb array of radius $r$ for some integer $r$. In particular, a honeycomb array must consist of an odd number of dots. \end{corollary}
So if we are looking for honeycomb arrays, we may restrict ourselves to searching for honeycomb arrays of radius $r$.
The structure of the remainder of the paper is as follows. In Section~\ref{sec:anticodes}, we state the results on the hexagonal grid that we need. In Section~\ref{sec:brooks}, we remind the reader of the notion of a brook, or bee-rook, and state a theorem on the maximum number of non-attacking brooks on a triangular board. In Section~\ref{sec:honeycomb} we prove Theorem~\ref{thm:main}, we summarise our computer searches for honeycomb arrays, and we provide a list of all known arrays. This last section also contains a conjecture, and some suggestions for further work.
\section{The hexagonal grid} \label{sec:anticodes}
Because the hexagonal grid might be difficult to visualise, we use an equivalent representation in the square grid (see Figure~\ref{fig:hex_to_square}). In this representation, we define each square to be adjacent to the four squares it shares an edge with, and the squares sharing its `North-East' and `South-West' corner vertices. The map $\xi$ in Figure~\ref{fig:hex_to_square} distorts the centres of the hexagons in the hexagonal grid to the centres of the squares in the square grid. The three types of rows in the hexagonal grid become the rows, the columns and the diagonals that run from North-East to South-West. For brevity, we define a \emph{standard diagonal} to mean a diagonal that runs North-East to South-West. \begin{figure}
\caption{From the hexagonal to the square grid}
\label{fig:hex_to_square}
\end{figure}
\begin{figure}
\caption{The region $S_{i}(n)$}
\label{fig:anticode}
\end{figure} For non-negative integers $n$ and $i$ such that $0\leq i\leq n-1$, define the region $S_i(n)$ of the square grid as in Figure~\ref{fig:anticode}. Note that $S_{i}(n)$ and $S_{n-1-i}(n)$ are essentially the same region: one is obtained from the other by a reflection in a standard diagonal. The regions $\xi^{-1}(S_i(n))$ are important in the hexagonal grid, as they are the maximal anticodes of diameter $n-1$; see Blackburn et al.~\cite[Theorem~5]{BlackburnEtzion}. Note that the region $\xi^{-1}(S_{r}(2r+1))$ is a Lee sphere of radius $r$. Regions of the form $\xi^{-1}(S_{r}(2r+2))$ or $\xi^{-1}(S_{r+1}(2r+2))$ are tricentred Lee spheres of radius $r$. Also note that the regions $S_i(n)$ as $i$ varies are precisely the possible intersections of an $n\times n$ square region with $n$ adjacent standard diagonals, where each diagonal intersects the $n\times n$ square non-trivially.
In the lemma below, by a `region of the form $X$', we mean a region that is a translation of $X$ in the square grid.
\begin{lemma} \label{lem:anticode} Let $\pi$ be a hexagonal permutation with $n$ dots, and let $\xi(\pi)$ be the image of $\pi$ in the square grid. Then the dots in $\xi(\pi)$ are all contained in a region of the form $S_{i}(n)$ for some $i$ in the range $0\leq i\leq n-1$. \end{lemma} \textbf{Proof:} Let $R$ be the set of squares that share a row with a dot of $\xi(\pi)$. Similarly, let $C$ and $D$ be the sets squares sharing respectively a column or a standard diagonal with a dot of $\xi(\pi)$. The dots in $\xi(\pi)$ are contained in $R\cap C\cap D$.
Since $\pi$ is a hexagonal permutation, $R$ consists of $n$ adjacent rows and $C$ consists of $n$ adjacent columns. Hence $R\cap C$ is an $n\times n$ square region. (Since there is exactly one dot in each row and column of the square $R\cap C$, the dots in $\xi(\pi)$ correspond to a permutation; this justifies the terminology `hexagonal permutation'.)
Now, $D$ consists of $n$ adjacent standard diagonals; each diagonal contains a dot in $\xi(\pi)$, and so each diagonal intersects $R\cap C$ non-trivially. Hence $R\cap C\cap D$ is a region of the form $S_i(n)$, as required.
$\Box$
\section{Brooks on a triangular board} \label{sec:brooks}
A \emph{brook} is a chess piece in the square grid that moves like a rook plus half a bishop: it can move any distance along a row, a column or a standard (North-East to South-West) diagonal. Brooks were first studied by Bennett and Potts~\cite{BennettPotts}, who pointed out connections to constant-sum arrays and hexagonal lattices. A set of brooks in a square grid is \emph{non-attacking} if no two brooks lie in a row, a column or a standard diagonal.
Under the correspondence $\xi$ between the square and hexagonal grids mentioned in the previous section, brooks in the square grid correspond to \emph{bee-rooks} in the hexagonal grid: pieces that can move any distance along any row, where a row can go in each of the three natural directions. A set of bee-rooks is therefore \emph{non-attacking} if no two bee-rooks lie in the same row of the hexagonal grid. In particular, bee-rooks placed on the dots in a hexagonal permutation $\pi$ are non-attacking, and so the corresponding set $\xi(\pi)$ of brooks in the square grid is non-attacking.
A \emph{triangular board of width $w$} is the region $S_{0}(w)$ in the square grid depicted in Figure~\ref{fig:triangular_board}. Let $b(w)$ be the maximum number of non-attacking brooks that can be placed in the triangular board of width $w$. The following theorem is proved by Nivasch and Lev~\cite{NivaschLev} and in Vaderlind, Guy and Larson~\cite[P252 and R252]{VaderlindGuy}: \begin{figure}
\caption{A triangular board of width $w$}
\label{fig:triangular_board}
\end{figure}
\begin{theorem} \label{thm:brooks} For any positive integer $w$, $b(w)=\lfloor (2w+1)/3\rfloor$. \end{theorem}
Three of the present authors have found an alternative proof for this theorem, using linear programming techniques~\cite{BlackburnPaterson}. See Bell and Stevens~\cite{BellStevens} for a survey of similar combinatorial problems.
\section{Honeycomb arrays} \label{sec:honeycomb}
We begin this section with a proof of Theorem~\ref{thm:main}. We then describe our searches for honeycomb arrays. We end the section by describing some avenues for further work.
\noindent \textbf{Proof of Theorem~\ref{thm:main}:} Let $\pi$ be a hexagonal permutation with $n$ dots. By Lemma~\ref{lem:anticode}, the dots of $\xi(\pi)$ are contained in a region of the form $S_i(n)$ where $0\leq i\leq n-1$. When $i=(n-1)/2$ (so $n$ is odd and $\xi^{-1}(S_i(n))$ is a Lee sphere) the theorem follows. Suppose, for a contradiction, that $i\not=(n-1)/2$.
By reflecting $\pi$ in a horizontal row in the hexagonal grid, we produce a hexagonal permutation $\pi'$ such that $\xi(\pi')$ is contained in a region of the form $S_{(n-1)-i}(n)$. By replacing $\pi$ by $\pi'$ if necessary, we may assume that $i<(n-1)/2$.
\begin{figure}
\caption{A triangular board covering $\pi$}
\label{fig:covering_triangle}
\end{figure} Consider the triangular board of width $n+i$ in Figure~\ref{fig:covering_triangle} containing $S_i(n)$. Since no two dots in $\xi(\pi)$ lie in the same row, column or standard diagonal, the dots in $\xi(\pi)$ correspond to $n$ non-attacking brooks in this triangular board. But this contradicts Theorem~\ref{thm:brooks}, since \[ \frac{2(n+i)+1}{3}<\frac{2n+(n-1)+1}{3}= n. \] This contradiction completes the proof of the theorem.
$\Box$
Theorem~\ref{thm:main} tells us that the only honeycomb arrays are those of radius $r$ for some non-negative integer $r$. A result of Blackburn et al~\cite[Corollary~12]{BlackburnEtzion} shows that $r\leq 643$. We now report on our computer searches for examples of honeycomb arrays. The known honeycomb arrays are drawn in Figures~\ref{honey137_figure}, \ref{honey9_figure}, \ref{honey15_figure}, \ref{honey2127_figure} and~\ref{honey45_figure}. This list includes two new examples not known to Golomb and Taylor~\cite{GolombTaylor}, namely the second and third examples of radius $7$; we found these examples as follows.
A \emph{Costas array} is a set of $n$ dots in an $n\times n$ region of the square grid, with the distict difference property and such that every row and column of the array contains exactly one dot. Golomb and Taylor observed that some Costas arrays produce honeycomb arrays, by mapping the dots in the Costas array into the hexagonal grid using the map $\xi^{-1}$ given by Figure~\ref{fig:hex_to_square}. Indeed, it is not difficult to see that all honeycomb arrays must arise in this way. We searched for honeycomb arrays by taking each known Costas array with $200$ or fewer dots, and checking whether the array gives rise to a honeycomb array. For our search, we made use of a database of all known Costas arrays with 200 or fewer dots that has been made available by James K. Beard~\cite{Beard}. This list is known to be complete for Costas arrays with $27$ or fewer dots; see Drakakis et al.~\cite{DrakakisRickard} for details. So our list of honeycomb arrays of radius $13$ or less is complete.
\begin{figure}
\caption{Honeycomb arrays of radius $0$, $1$ and $3$}
\label{honey137_figure}
\end{figure}
\begin{figure}
\caption{Honeycomb arrays of radius 4}
\label{honey9_figure}
\end{figure}
\begin{figure}
\caption{Honeycomb arrays of radius $7$}
\label{honey15_figure}
\end{figure}
\begin{figure}
\caption{Honeycomb arrays of radius $10$ and $13$}
\label{honey2127_figure}
\end{figure}
\begin{figure}
\caption{A honeycomb array of radius $22$}
\label{honey45_figure}
\end{figure}
It is a remarkable fact that all known honeycomb arrays possess a non-trivial symmetry (a horizontal reflection as we have drawn them). Indeed, apart from a single example of radius $3$ (the first radius $3$ example in Figure~\ref{honey137_figure}) all known honeycomb arrays possess a symmetry group of order $6$: the group generated by the reflections along the three lines through opposite `corners' of the hexagonal sphere. We implemented an exhaustive search for honeycomb arrays with $r\leq 31$ having this $6$-fold symmetry: we found no new examples. We also checked all constructions of honeycomb arrays from Costas arrays in Golomb and Taylor~\cite{GolombTaylor} (whether symmetrical or not) for $r\leq 325$, and again we found no new examples.
After these searches, we feel that we can make the following conjecture:
\begin{conjecture} The list of known honeycomb arrays is complete. So there are are exactly $12$ honeycomb arrays, up to symmetry. \end{conjecture}
Theorem~\ref{thm:main} shows that hexagonal permutations are always contained in some Lee sphere. But such permutations have been prevously studied in several contexts: Bennett and Potts~\cite{BennettPotts} study them as non-attacking configurations of bee-rooks and as the number of zero-sum arrays; Kotzig and Laufer~\cite{Kotzig} study them as the number of $\sigma$-permutations; Bebeacua, Mansour, Postnikov and Severini~\cite{BebeacuaMansour} study them as X-rays of permutations with maximum degeneracy. Let $h_n$ be the number of hexagonal permutations with $2n-1$ dots. The On-Line Encyclopedia of Integer Sequences~\cite[Sequence A002047]{OEIS} quotes a computation due to Alex Fink that computes the first few terms of the sequence $h_n$: \[
\begin{array}{c|cccccccccc} n&1&2&3&4&5&6&7&8&9&10\\\hline h_n&1&2&6&28&244&2544&35600&659632&15106128&425802176 \end{array} \] Kotzig and Laufer ask: How big can $h_n$ be? It seems that the sequence grows faster than exponentially with $n$. We ask a more precise question: Is it true that $(\log h_n)/n\log n$ tends to a constant as $n\rightarrow\infty$?
\paragraph{Acknowledgements} Part of this work was completed under EPSRC Grant EP/D053285/1. The authors would like to thank Tuvi Etzion for discussions, funded by a Royal Society International Travel Grant, which inspired this line of research.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\maketitle \begin{abstract}
We present quasi-Banach spaces which are surprisingly similar to certain dual Banach spaces. In particular we show a quasi-Banach space which shares many properties of the dual of the Schreier space: it is $\ell_1$-saturated and does not have the
Schur property, its dual is isometrically isomorphic to the double dual of the Schreier space and it has 'the same' extreme
points of the unit ball as the dual of the Schreier space. \end{abstract} \section{Introduction}
A combinatorial Banach space is the completion of $c_{00}$, the space of all finitely supported sequences of real numbers, with respect to the following norm
\begin{equation}\label{dd1}
\lVert x \rVert_\mathcal{F} = \sup_{F\in \mathcal{F}} \sum_{k\in F} |x(k)|, \end{equation}
where $\mathcal{F}$ is a family of finite subsets of $\omega$ (which is hereditary and covers $\omega$, see Section \ref{schreier-like} for the details). We denote such completion by $X_\mathcal{F}$. One of the most 'famous' examples of a Banach space of this form is the \emph{Schreier space} induced by so called Schreier family $\mathcal{S}$ (see Section \ref{schreier-like}) and so combinatorial Banach spaces are sometimes called Schreier-like spaces. They were studied mainly for $\mathcal{F}$ being compact families, e.g. by Castillo and Gonzales (\cite{Castillo-Gonzales}, \cite{Castillo93}), Lopez Abad and Todorcevic (\cite{Jordi-Stevo}). Recall that a family $\mathcal{F} \subseteq \mathcal{P}(\omega)$ is compact if it is compact as a subset of $2^\omega$, via the natural identification of $\mathcal{P}(\omega)$ and $2^\omega$. In \cite{NaFa} Borodulin-Nadzieja and Farkas considered Schreier-like spaces for families which are not necessarily compact and which were motivated by the theory of analityc P-ideals.
The main motivation of our article is an attempt to study Banach spaces dual to combinatorial Banach spaces. Even in the case of Schreier space, not much seems to be known about its dual (see \cite{Lipieta}). Perhaps the reason lies in the lack of a nice description of the dual norm. Seeking such a description we came up with the following function $\lVert \cdot\rVert^{\mathcal{F}} \colon c_{00} \to \mathbb{R}$:
\begin{equation} \label{dd2}
\lVert x \rVert^\mathcal{F} = \inf\{ \sum_{F \in \mathcal{P}} \sup_{i\in F} |x(i)|\colon \mathcal{P}\subseteq \mathcal{F} \mbox{ is a partition of }\omega\}. \end{equation} Perhaps this formula does not look tempting at first glance, but in a 'combinatorial' sense it is dual to $\lVert \cdot \rVert_\mathcal{F}$. Indeed, we can think about evaluating $\lVert x \rVert_{\mathcal{F}}$ as partititioning $\omega$ into pieces from
$\mathcal{F}$, suming up $|x(i)|$ for $i$ from one piece of the partition and then maximizing the result, for all partitions and all pieces. On the other hand, evaluating $\lVert x \rVert^\mathcal{F}$ comes down to partitioning $\omega$ into pieces from $\mathcal{F}$, taking maximum of $|x(i)|$ for $i$ from one piece of the partition, summing up those maxima and then minimazing the result for all possible partitions.
For certain families $\mathcal{F}$ the completion of $c_{00}$ in $\lVert \cdot \rVert^\mathcal{F}$ (which we will denote by $X^\mathcal{F}$) is a Banach space which is isometrically isomorphic to $X^*_\mathcal{F}$ (see Proposition \ref{for-partitions}). E.g. notice that if $\mathcal{F}$ consists of singletons, then $X_\mathcal{F}$ is isometrically isomorphic to $c_0$ and $X^\mathcal{F}$ is isometrically isomorphic to $\ell_1$.
In general, if a family $\mathcal{F} \subseteq [\omega]^{<\omega}$ is compact, hereditary and covering $\omega$, then $X^\mathcal{F}$ shares many properties with $X^*_\mathcal{F}$. In fact it is difficult to find a property distinguishing those spaces apart of the fact that for many families $\mathcal{F}$ the space $X^\mathcal{F}$ is not a Banach space! More precisely, $\lVert \cdot \rVert^\mathcal{F}$ does not need to satisfy the triangle inequality (and it does not e.g. for $\mathcal{F}$ being the Schreier family).
The lack of triangle inequality is not something welcomed in the theory of Banach spaces. However, we will argue that $X^\mathcal{F}$ is an interesting object anyway. First of all, if $\mathcal{F}\subseteq [\omega]^{<\omega}$ is compact, herediatary and covering $\omega$, then $\lVert \cdot \rVert^\mathcal{F}$ enjoys some perverted form of triangle inequality. Namely, for every $x,y \in c_{00}$ \[ \lVert x+y \rVert^\mathcal{F} \leq 2\lVert x \rVert^\mathcal{F} + \lVert y \rVert^\mathcal{F}. \] Consequently, $\lVert x+y \rVert^\mathcal{F} \leq \dfrac{3}{2} (\lVert x\rVert^\mathcal{F}+ \lVert y \rVert^\mathcal{F})$ for each $x,y\in c_{00}$ and so $\lVert \cdot \rVert^\mathcal{F}$ is a quasi-norm (see Section \ref{quasi-banach} for the definition and basic facts concerning quasi-norms). Thus, by Aoki-Rolewicz theorem (Theorem \ref{aoki_rolewicz}), it is metrizable and it makes sense to define the completion of $c_{00}$ with respect to $\lVert \cdot \rVert^\mathcal{F}$, which we denote by $X^\mathcal{F}$. The theory of quasi-Banach spaces is quite well developed and it seems that most of the important results and techniques known in Banach space theory, works well enough in the realm of quasi-Banach spaces (see \cite{Kalton_quasi}).
As we have mentioned, despite the fact that $X^\mathcal{F}$ is typically not a Banach space, it very much resembles $X^*_\mathcal{F}$, at least if $\mathcal{F}$ is compact. In particular we prove that \begin{itemize}
\item The space $X^\mathcal{F}$ and $X^*_\mathcal{F}$ have isometrically isomorphic duals (Theorem \ref{isomorphism}).
\item The unit balls of $X^\mathcal{F}$ and $X^*_\mathcal{F}$ have 'the same' extreme points, although if $\lVert \cdot \rVert^\mathcal{F}$ is not a norm, then the unit ball in $X^\mathcal{F}$ is not convex (Proposition \ref{extreme_to_extreme}).
\item Both $X^\mathcal{F}$ and $X^*_\mathcal{F}$ have CSRP, i.e. their unit balls can be nicely approximated by their extreme points (Theorem \ref{CSRPthm}). \end{itemize}
Then we concentrate on the particular case of the Schreier family. It seems that the space $X^*_\mathcal{S}$, the dual to the Scheier space, is a rather mysterious object and not much of it is known. Recently, Galego, Gonz\'{a}lez and Pello (\cite{Galego}) proved that it is $\ell_1$-saturated (i.e. every closed infinite-dimensional subspace contains an isomorphic copy of $\ell_1$) and it is known that it does not enjoy the Schur property. In fact, for years it was an open problem whether such spaces, $\ell_1$-saturated but without Schur property, exists at all. The first example was constructed by Bourgain (\cite{Bourgain}) and then several other involved constructions of such spaces have been presented (see e.g. \cite{Hagler}, \cite{Popov}). The result of Galego, Gonz\'{a}lez and Pello says that you do not have to \emph{construct} an $\ell_1$-saturated space without a Schur property: the dual to Schrier space is already a good example (although the proof contained in \cite{Galego} is rather difficult).\footnote{The authors of \cite{Galego} prove this result en passant. They do not mention it neither in the abstract nor in the introduction, nor they relate it to the Bourgain construction. We are grateful to Jes\'us Castillo for informing us about this result.}
We show that the space $X^\mathcal{S}$ is also $\ell_1$-saturated (Theorem \ref{ell1}) and it does not have Schur property (Proposition \ref{Schur}). The proof is considerably simpler that those from \cite{Galego} and the previous constructions.
All the above results indicate that the spaces $X^\mathcal{F}$ are kind of quasi-Banach alter-egos of $X^*_\mathcal{F}$. Since the quasi-norm $\lVert \cdot \rVert^\mathcal{F}$ is much easier to deal with than the dual norm of $\lVert \cdot \rVert_\mathcal{F}$, the spaces $X^\mathcal{F}$ can be used as a kind of 'test spaces' where some concrete properties can be checked to state appropriate conjectures concerning $X^*_\mathcal{F}$. Perhaps there is some general reason why $X^\mathcal{F}$ and $X^*_\mathcal{F}$ are so similar. Finding a concrete theorem explaining the connection $X^\mathcal{F}$ and $X^*_\mathcal{F}$ might allow to prove results about $X^*_\mathcal{F}$ dealing with the quasi-norm $\lVert \cdot \rVert^\mathcal{F}$ instead of bother oneself with the dual norm to $\lVert \cdot \rVert_\mathcal{F}$.
The article is organized as follows. In Section 3 we introduce the basic notions and recall some facts concerning quasi-Banach spaces. In Section 4 we present basic definitions and facts about combinatorial Banach spaces. In Section 5 we deal with the function defined by (\ref{dd2}). We prove that $\lVert \cdot \rVert^{\mathcal{F}}$ is a nice quasi-norm and that $X^{\mathcal{F}}$ is a quasi-Banach space for every compact family $\mathcal{F}$ (which is hereditary and which covers $\omega$). In Section 6 we present similarities between $X^{\mathcal{F}}$ and the dual to $X_{\mathcal{F}}$. Finally, in Section 7 we prove that $X^{\mathcal{S}}$ is $\ell_{1}$-saturated and that it does not have the Schur property, for the Schreier family $\mathcal{S}$.
\section{Preliminaries} \label{preliminaries}
In this section we present the standard facts and notations, used in the further sections of this article.
By $\omega$ we denote the set of natural numbers. For various technical reasons for the sake of this article we will not recognize 0 as a natural number. If $k \in \omega$, then $[\omega]^{\leq k} = \lbrace A \subseteq \omega: \lvert A \rvert \leq k \rbrace$ and $[\omega]^{< \omega}$ is the family of all finite subsets of $\omega$.
Recall that the Cantor set $2^{\omega}$ is a compact and complete metric space equipped with the metric
$$
d(x,y) = \dfrac{1}{2^{k}},
$$
where $k$ is the smallest natural number for which $x(k) \neq y(k)$ (and $d(x,y)=0$ in case $x=y$).
By $\mathcal{P}(\omega)$ we denote the power set of $\omega$ which we identify with a Cantor set $2^{\omega}$ via the map
$$\mathcal{P}(\omega) \ni A \mapsto \chi_{A} \in 2^{\omega},$$
where $\chi_{A}$ denotes the characteristic function of a set $A$.
Henceforth we do not distinguish subset of $\omega$ from its characteristic function, so if we write that a family is open, closed, compact etc. we mean openness, closedness, and compactness of a subset of $2^{\omega}$. In addition, we use interchangebly the following notations
$$
j \in A \hspace{0.1cm} (\notin A)
$$
and
$$
A(j) = 1 \hspace{0.1cm} (=0).
$$
If $A,B \subseteq \omega$ are two finite sets, then by $A < B$ we mean that $\max(A) < \min(B)$. In what follows $\mathcal{F}$ will always denote a family of finite subsets of $\omega$ which is hereditary (i.e. it is closed under taking subsets)
and which covers
$\omega$. If $\mathcal{A}$ is a family of subsets of $\omega$ then by its \emph{hereditary closure} we mean the smallest hereditary family containing $\mathcal{A}$.
\begin{df} A family $\mathcal{P}\subseteq \mathcal{P}(\omega)$ is a \emph{partition} if
\begin{itemize}
\item $\emptyset\in \mathcal{P}$,
\item $\bigcup \mathcal{P} = \omega$,
\item $\mathcal{P}$ consists of pairwise disjoint element.
\end{itemize}
\end{df}
Notice that our definition of partition is non-standard. We want $\emptyset$ to be an element of a partition, since then it forms a compact subset of $2^\omega$ and we are going to use this fact later on.
We denote by $\mathbb{P}_{\mathcal{F}}$ the family of all partitions $\mathcal{P}\subseteq \mathcal{F}$.
In this paper we consider spaces of sequences of real numbers. \\ For $x \in \mathbb{R}^{\omega}$ by its \textit{support} we mean the set $supp(x) = \lbrace n \in \omega: x(n) \neq 0 \rbrace$ and $c_{00}$ is the set of all sequences $x$ for which $supp(x)$ is finite.
For $A \subseteq \omega$ the function $P_{A}: \mathbb{R}^{\omega} \rightarrow \mathbb{R}^{\omega}$ given by
$$
P_{A}(x)(k) =
\begin{cases}
x(k), & \text{if } k \in A \\
0, & \text{otherwise,}
\end{cases}
$$
is called the \textit{projection} of $x$ onto coordinates from $A$. In particular if $n \in \omega$, then by $P_{n}$ we denote the projection onto the initial segment $\lbrace 1,2,...,n \rbrace$ and by $P_{\omega \setminus n}$ - the projection onto
the complement of this segment.
\subsection{Quasi-Banach spaces} \label{quasi-banach} \begin{df} \label{quasi_norm_def} A \emph{quasi-norm} on a vector space $X$ is a function $\lVert \cdot \rVert: X \rightarrow \mathbb{R}$ satysfing \begin{itemize} \item $\lVert x \rVert = 0 \Leftrightarrow x = 0$
\item For every $\lambda \in \mathbb{R}$ $\lVert \lambda x \rVert = | \lambda | \lVert x \rVert$ \item There is $c \geq 1$ such that $\lVert x + y \rVert \leq c ( \lVert x \rVert + \lVert y \rVert )$. \end{itemize} \end{df}
The minimal constant $c$ working above is sometimes called the \emph{modulus of concavity} of the quasi-norm. In particular, for $c=1$ we get the definition of a norm. In what follows sometimes we will allow quasi-norms to take possibly infinite values. If $\lVert \cdot \rVert$ is a quasi-norm (taking only finite values) on a vector space $X$, then the pair $(X, \lVert \cdot \rVert)$ is called \emph{quasi-normed space}.
An important class of quasi-norms are so called $p$-norms. A quasi-norm is a $p$-norm if it satisfies the following condition: \begin{equation} \label{p_norm} \left \vvvert x + y \right \vvvert^{p} \leq \left \vvvert x \right \vvvert^{p} + \left \vvvert y \right \vvvert^{p} \end{equation}
The classical Aoki-Rolewicz theorem (\cite{Aoki}, \cite{Rolewicz}, see also \cite[Theorem 1.3]{Kalton-F}), says that every quasi-norm can be renormed to a $p$-norm:
\begin{thm}[Aoki-Rolewicz] \label{aoki_rolewicz}
Let $(X, \lVert \cdot \rVert)$ be a quasi-normed space. Then there exists $p \in (0, 1] $ and $p$-norm $\left \vvvert \cdot \right \vvvert$ on $X$ which is equivalent to $\lVert \cdot \rVert$. If $c$ is a modulus of concavity of $\lVert \cdot
\rVert$, then $c = 2^{\frac{1}{p}-1}$. \end{thm}
The appropriate $p$-norm $\left \vvvert \cdot \right \vvvert$ is given by the formula \begin{equation} \label{equi_q_norm} \left \vvvert x \right \vvvert = \inf \bigg \{ \Big ( \sum \limits_{i=1}^{n} \lVert x_i \rVert^{p} \Big )^{\frac{1}{p}}: n \in \omega, x_1,...,x_n \in X, x = \sum \limits_{i=1}^{n} x_i \bigg \}. \end{equation}
The important consequence of Aoki-Rolewicz theorem is that every quasi-normed space is metrizable. The compatible metric $d$ can be defined by $d(x,y) = \left \vvvert x - y \right \vvvert^{p}$ (see \cite{Kalton_quasi}). A quasi-normed space is called a \emph{quasi-Banach} space, if the quasi-norm induces complete metrizable topology on $X$.
Note that a quasi-Banach space $X$ which is not a Banach space cannot be locally convex, i.e. the unit ball, $B_X$, is not a convex set. By this, it turns out that results which hold in Banach spaces and are based somehow on the local convexity (e.g. Hahn-Banach extension property or Krein-Milman theorem) are no longer true, in general, in quasi-Banach spaces (see \cite{Kalton_quasi}). However, the standard results of Banach space theory such as Open Mapping Theorem, Uniform Boundedness Principle, and Closed Graph Theorem can be applied in quasi-Banach spaces since they depend only on the completeness of the space. Later in the article we use the Open Mapping Theorem in the context of quasi-Banach spaces, so we recall it below.
\begin{thm}[Open Mapping Theorem] \label{open_mapping} Let $X$ and $Y$ be quasi-Banach spaces and $T: X \rightarrow Y$ be continuous linear operator onto $Y$. Then $T$ is open. \end{thm}
\section{Combinatorial spaces} \label{schreier-like}
In \cite{NaFa} the authors presented an approach towards Banach spaces with unconditional bases motivated by the theory of analityc P-ideals on $\omega$. We will briefly overview the basic definitions and results which we are going to use in more general case, namely for quasi-Banach spaces.
\begin{df}\label{nicenorm} \cite{NaFa}
We say that a function $\varphi\colon \mathbb{R}^{\omega} \rightarrow [0,\infty]$ is \textit{nice}, if
\begin{enumerate}[(i)]
\item (\textit{Non-degeneration}) $\varphi(x) < \infty$ for every $x \in c_{00}$.
\item (\textit{Monotonicity}) For every $x,y \in \mathbb{R}^\omega$, if $\lvert x(n) \rvert \leq \lvert y(n) \rvert$ for ach each $n \in \omega$, then $\varphi(x) \leq \varphi(y)$.
\item (\textit{Lower semicontinuity}) $\lim \limits_{n \rightarrow \infty} \varphi(P_{n}(x)) = \varphi(x)$ for every $x\in \mathbb{R}^\omega$.
\end{enumerate} \end{df} For an extended quasi-norm $\varphi: \mathbb{R}^{\omega} \rightarrow [0,\infty]$ define
\begin{center}
$FIN(\varphi) = \lbrace x \in \mathbb{R}^{\omega}: \varphi(x) < \infty \rbrace,$\\
$EXH(\varphi) = \lbrace x \in \mathbb{R}^{\omega}: \lim \limits_{n \rightarrow \infty} \varphi(P_{\omega \setminus n}(x)) = 0 \rbrace. $
\end{center}
\begin{thm}\label{5.1}(\cite[Proposition 5.1]{NaFa}) If $\varphi: \mathbb{R}^{\omega} \rightarrow [0,\infty]$ is a nice extended \emph{norm}, then $FIN(\varphi)$ and $EXH(\varphi)$ (normed with $\varphi$) are Banach spaces with unconditional bases.
\end{thm}
Note that the above theorem can be reversed: every Banach space with an unconditional basis is isometrically isomorphic to $EXH(\varphi)$ for some nice extended norm $\varphi$ (\cite[Proposition 5.1]{NaFa}). Now we will present two structure
theorems about spaces the spaces $FIN(\varphi)$ and $EXH(\varphi)$.
\begin{thm}\label{5.4}(\cite[Theorem 5.4]{NaFa})\label{Banach} Suppose that $\varphi: \mathbb{R}^{\omega} \rightarrow [0,\infty]$ is an extended norm. Then the following conditions are equivalent:
\begin{itemize}
\item $EXH(\varphi) = FIN(\varphi)$,
\item $EXH(\varphi)$ does not contain an isomorphic copy of $c_0$.
\end{itemize}
\end{thm}
\begin{thm}\label{5.5}(\cite[Proposition 5.5]{NaFa}) Suppose that $\varphi: \mathbb{R}^{\omega} \rightarrow [0,\infty]$ is an extended norm. Then the following conditions are equivalent:
\begin{itemize}
\item $FIN(\varphi)$ is isometrically isomorphic to $EXH(\varphi)^{**}$,
\item $EXH(\varphi)$ does not contain an isomorphic copy of $\ell_1$.
\end{itemize}
\end{thm}
In this article we consider particular norms induced by families of subsets of $\omega$. More precisely, if $\mathcal{F} \subseteq \mathcal{P}(\omega)$ is hereditary and covering $\omega$, then for a sequence $x$ consider the following expression
\begin{equation} \label{Lower_norm}
\lVert x \rVert_{\mathcal{F}} = \sup_{F \in \mathcal{F}} \sum_{k \in F} \lvert x(k) \rvert.
\end{equation}
It is not difficult to check that this is a nice extended norm and so $EXH(\lVert \cdot \rVert_{\mathcal{F}})$ is a Banach space (which we will denote, for simplicity, by $X_\mathcal{F}$). See \cite{NaFa} for examples of spaces of this form.
The spaces of the form $X_\mathcal{F}$ are quite well studied for compact families $\mathcal{F}$ (see e.g. \cite{Jordi-Stevo}, \cite{Castillo-Gonzales}). The most notable representative here is the Schreier space,
$X_\mathcal{S}$, where $\mathcal{S}$ is so-called \textit{Schreier family}: $$ \mathcal{S} = \lbrace A \subseteq \omega: \lvert A \rvert \leq \min(A) \rbrace. $$
\section{Dual quasi-norms} \label{dual-norms}
Now we are going to introduce the main notion of this article.
\begin{equation} \label{Upper_norm}
\lVert x \rVert^{\mathcal{F}} = \inf_{\mathcal{P} \in \mathbb{P}_{\mathcal{F}}} \sum_{F \in \mathcal{P}} \sup_{k \in F} \lvert x(k) \rvert.
\end{equation}
If $\mathcal{P}$ is a partition of $\omega$, then by $||x||^\mathcal{P}$ we will denote $\| x\|^\mathcal{F}$ where $\mathcal{F}$ is the hereditary closure of $\mathcal{P}$. Note that in this case
$$
\lVert x \rVert^{\mathcal{P}} = \sum_{P \in \mathcal{P}} \sup_{k \in P} \lvert x(k) \rvert.
$$
So, for a family $\mathcal{F} \subseteq \mathcal{P}(\omega)$ we have
\[ \lVert x \rVert^\mathcal{F} = \inf_{\mathcal{P}\in \mathbb{P}_\mathcal{F}} \lVert x \rVert^\mathcal{P}. \]
Note that in general the expression (\ref{Upper_norm}) does not define a norm.
\begin{eg}\label{notnorm} Let $\mathcal{S}$ be the Schreier family. Consider the finitely supported sequences $x = (0,1,1,0,0,0,...)$, $y = (0,0,1,1,1,0,0,0,...)$. We can easily check that $\lVert x \rVert^{\mathcal{S}} = \lVert y \rVert^{\mathcal{S}} = 1$, but $\lVert x+y \rVert^{\mathcal{S}} = 3$, so the triangle inequality is not satisfied. \end{eg}
However, it turns out that $\lVert \cdot \rVert^{\mathcal{F}}$ is a quasi-norm, at least if $\mathcal{F}$ is a compact family. Moreover, it is a nice quasi-norm (in the sense of Definition \ref{nicenorm}).
Perhaps quite surprisingly, before showing that $\lVert \cdot \rVert^\mathcal{F}$ is a quasi-norm and then showing that it is nice, we will do the opposite: first we will check that $\lVert \cdot \rVert^\mathcal{F}$ satisfies all the conditions of Definition \ref{nicenorm}. The reason is that lower semicontinuity will make our life easier allowing to focus on finitely supported sequences.
It is easy to check that if $\mathcal{F}$ is a family covering $\omega$, then $\lVert \cdot \rVert^\mathcal{F}$ is monotone and non-degenerated (in the sense of Definition \ref{nicenorm}). However, it is not necessarily lower semicontinuous. Consider e.g. the
family $\mathcal{F}$ of all finite subsets of $\omega$ and $x$ defined by $x(k)=1$ for each $k$. Then $\lVert P_n(x) \rVert^\mathcal{F} = 1$ for each $n$ but $\lVert x \rVert^\mathcal{F} = \infty$. We will show that if we additionally assume that
$\mathcal{F}$ is compact, then $\lVert \cdot \rVert^\mathcal{F}$ is lower semicontinuous and so it is nice.
\begin{thm} \label{nice_condition} If $\mathcal{F} \subseteq \mathcal{P}(\omega)$ is compact, hereditary and covering $\omega$, then
$\lVert \cdot \rVert^{\mathcal{F}}$ is a nice quasi-norm.
\end{thm}
Before we start the proof we recall some definitions and facts about the Vietoris topology.
Fix a compact $\mathcal{F}\subseteq \mathcal{P}(\omega)$. Every partition can be considered as a subset of $2^{\omega}$ and thus we can treat a set $\mathbb{P}_{\mathcal{F}}$ as a subset of the power set of
$2^{\omega}$. We can endow this set with a Vietoris topology.
\begin{df}
Let $X$ be a compact topological space. By $\mathcal{K}(X)$ denote the family of all closed subsets of $X$. The \textit{Vietoris topology} is the one generated by sets of the form
\begin{equation} \label{Vietoris}
\big <U_{1},U_{2},...,U_{n} \big > = \lbrace K \in \mathcal{K}(X): K \subseteq \bigcup_{i \leq n} U_{i} \wedge \forall i \leq n \ K \cap U_{i} \neq \emptyset \rbrace,
\end{equation}
where $U_{i}$ are open subsets of $X$.
\end{df}
Note that if $X$ is a compact space, then $K(X)$ endowed with the Vietoris topology is compact as well. Also, $K(X)$ is metrizable (by the Hausdorff metric).
In our case, the role of $X$ is played by $\mathcal{F}$. According to the above, we would like to consider $\mathbb{P}_{\mathcal{F}}$ as a subspace of $\mathcal{K}(\mathcal{F})$. Notice that according to our definition of partition, it contains
$\emptyset$ and so it is a closed subset of $\mathcal{F}$ (in fact it forms a sequence converging to $\emptyset$).
\begin{lem}
$\mathbb{P}_{\mathcal{F}}$ is closed in $\mathcal{K}(\mathcal{F})$. Consequently, $\mathbb{P}_{\mathcal{F}}$ is a compact subspace of $\mathcal{K}(\mathcal{F})$.
\end{lem}
\begin{proof}
Let $\mathcal{G} \in \overline{\mathbb{P}_{\mathcal{F}}}$. We need to prove that $\mathcal{G}$ is a partition, i.e.
\begin{enumerate}[(i)]
\item $\emptyset \in \mathcal{G}$,
\item $\bigcup \mathcal{G} = \omega$,
\item All elements of $\mathcal{G}$ are pairwise disjoint.
\end{enumerate}
Of those (i) is straightforward.
Suppose now that there exists $n \in \omega$ such that $n \notin \mathcal{G}$. Put $U = \lbrace x \in 2^{\omega}: x(n) = 0 \rbrace$. $U$ is an open subset of $2^{\omega}$. Take a basic (in Vietoris topology) set $\mathcal{K}_{U}$ = $\lbrace K \in
\mathcal{K}(\mathcal{F}): K \subseteq U \rbrace$, being an open neighborhood of $\mathcal{G}$. Then $\mathcal{K}_{U} \cap \mathbb{P}_{\mathcal{F}} = \emptyset$. Indeed, otherwise, there would be partition $\mathcal{P}$ such that $\mathcal{P}
\subseteq U$, which is impossible, because there is $A \in \mathcal{P}$ such that $n \in A$. The set $\mathcal{K}_{U}$, therefore, testifies that $\mathcal{G} \notin \overline{\mathbb{P}_{\mathcal{F}}}$, which is a contradiction. It proves $(ii)$.
To prove (iii) suppose that there are $A,B \in \mathcal{G}$ such that $A \cap B \neq \emptyset$ and $A \setminus B \ne \emptyset$. Let $n \in A \cap B$ and $m \in A \setminus B$. Consider the following open subsets in $2^{\omega}$
\begin{center}
$U_{1} = \lbrace x \in 2^{\omega}: x(n) = 1 \wedge x(m) = 1 \rbrace$,\\
$U_{2} = \lbrace x \in 2^{\omega}: x(n) = 1 \wedge x(m) = 0 \rbrace$,\\
$U_{3} = 2^{\omega}$,
\end{center}
and the basic set $\big < U_{1}, U_{2}, U_{3} \big>$. Then $\mathcal{G} \cap U_{1} \neq \emptyset$, because $A \in U_{1}$ and $\mathcal{G} \cap U_{2} \neq \emptyset$, since $B \in U_{2}$. So the set $\big <U_{1}, U_{2}, U_{3} \big >$ is an open
neighborhood of $\mathcal{G}$. If $\mathbb{P}_{\mathcal{F}} \cap \big < U_{1}, U_{2}, U_{3} \big> \neq \emptyset$, then there is a partition $\mathcal{P}$ and sets $K,L \in \mathcal{P}$ such that $n,m \in K$, $n \in L$, and $m \notin L$. But it
is impossible, since elements of $\mathcal{P}$ are pairwise disjoint. It implies that $\mathbb{P}_{\mathcal{F}} \cap \big <U_{1}, U_{2}, U_{3} \big > = \emptyset$, which is a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem \ref{nice_condition}] As we have mentioned it is enough to show lower semicontinuity. Fix $x \in \mathbb{R}^{\omega}$.
Assume that $\lVert x \rVert^{\mathcal{F}} = D$ (possibly $D=\infty$). Then for every partition $\mathcal{P}$ we have $\lVert x \rVert^{\mathcal{P}} \geq D$. For each $n \in \omega$ put $x_{n} = P_{n}(x)$. Suppose that there exists $M < D$ such that
$\lVert x_{n} \rVert^{\mathcal{F}} < M$ for each $n$. Then for every $n$ there is a partition $\mathcal{P}_{n}$ such that $\lVert x_{n} \rVert^{\mathcal{P}_{n}} < M$. By compactness, we may assume (passing to a subsequence if needed) that
$(\mathcal{P}_{n})_{n \in \omega}$ converges (in the Vietoris topology) to a partition $\mathcal{P}$. Since $\lVert x \rVert^\mathcal{P} \geq D$, there is $N \in \omega$ such that $\lVert x_{N} \rVert^{\mathcal{P}}\geq D$. There are only finitely many elements $R_{1},R_{2},...,R_{j}$ of $\mathcal{P}$ having non-empty intersection with $\lbrace 1,2,...,N \rbrace$. For $k \leq j$ put
\[ U_{k} = \lbrace x \in 2^{\omega}: \forall i \in R_{k} \ x(i) = 1 \rbrace \]
and consider the basic open set $\big < U_{1},U_{2},...,U_{j}, U_{j+1} \big>$, where $U_{j+1} = 2^{\omega}.$ Then \[ \mathcal{P} \in \big < U_{1},U_{2},...,U_{j},U_{j+1} \big>.\] Indeed, trivially $\mathcal{P} \cap U_{j+1} = \mathcal{P}$
and for $k \leq j$ we have $ R_k \in \mathcal{P} \cap U_{k} \neq \emptyset$. Since $\mathcal{P}_{n}$ converges to $\mathcal{P}$, there is $k>N$ such that $\mathcal{P}_{k}
\in \big < U_{1},U_{2},...,U_{j}, U_{j+1} \big>$. It means that \[ \{P\cap \{1,\dots,N\}\colon P \in \mathcal{P}_k\} = \{P\cap \{1,\dots,N\} \colon P \in \mathcal{P}\}.\]
So, \[ \lVert x_k \rVert^{\mathcal{P}_k} \geq \lVert x_N \rVert^{\mathcal{P}_k} = \lVert x_N \rVert^{\mathcal{P}} > M, \]
a contradiction. \end{proof}
Now we can prove that $\lVert \cdot \rVert^\mathcal{F}$ is indeed a quasi-norm.
\begin{thm}\label{quasi-quasi} Let $\mathcal{F}$ be a compact hereditary family. Then for every $x,y\in \mathbb{R}^\omega$
\begin{itemize}
\item[(a)] if $x,y$ have disjoint supports, then $\lVert x+y \rVert^\mathcal{F} \leq \lVert x \rVert^\mathcal{F} + \lVert y \rVert^\mathcal{F}$,
\item[(b)] $\lVert x+y \rVert^{\mathcal{F}} \leq 2\lVert x \rVert^{\mathcal{F}} + \lVert y \rVert^{\mathcal{F}}$,
\item[(c)] $\lVert x+y \rVert^{\mathcal{F}} \leq \dfrac{3}{2} (\lVert x \rVert^{\mathcal{F}} + \lVert y \rVert^{\mathcal{F}})$ and so $\lVert \cdot \rVert^\mathcal{F}$ is a quasi-norm.
\end{itemize} \end{thm}
\begin{proof}
Of the above (a) is clear and (c) is a direct consequence of (b).
So, we will check (b).
Let $x, y \in \mathbb{R}^\omega$. By lower semicontinuity of $\lVert \cdot \rVert^\mathcal{F}$ it is enough to consider the case when $x$ and $y$ are finitely supported. Moreover, we will assume that $x(k), y(k)\geq 0$ for every $k$ (since $\lVert
x+y\rVert \leq \lVert |x| + |y| \rVert$ and $\lVert x \rVert = \lVert |x| \rVert$ for each $x,y\in X^\mathcal{F}$) (by $|x|$ we mean a sequence defined by $|x|(k) = |x(k)|$ for each $k \in \omega$).
Assume that $x$ is of the form $x = a\chi_F$, where $F\in \mathcal{F}$ and let $A = \mathrm{supp}(y)$. Then, using (a), we have
\[ \lVert x + y \rVert^\mathcal{F} \leq \lVert P_A(x+y) \rVert^\mathcal{F} + \lVert P_{\omega\setminus A}(x)\rVert^\mathcal{F} \leq (\lVert y \rVert^\mathcal{F} + a) + a = 2\lVert x\rVert^\mathcal{F} + \lVert y \rVert^\mathcal{F}. \]
Next, suppose that $x$ is of the form $x = a_0\chi_{F_0} + \dots + a_n \chi_{F_n}$, where $F_i\in \mathcal{F}$ and $F_i \cap F_j = \emptyset$ for every $i,j\leq n$. Then, by the above inequality
\begin{multline}\label{ppp}
\lVert x+y \rVert^\mathcal{F} \leq 2 \lVert a_0\chi_{F_0}\rVert^\mathcal{F} + \lVert a_1\chi_{F_1} + \dots + a_n\chi_{F_n} + y \rVert^\mathcal{F} \leq \dots \\ \leq 2(\lVert a_0\chi_{F_0}\rVert^\mathcal{F} + \dots + \lVert a_n\chi_{F_n}\rVert^\mathcal{F}) +
\lVert y \rVert^\mathcal{F}
= 2(a_0 + \dots + a_n) + \lVert y \rVert^\mathcal{F} \leq 2\lVert x \rVert^\mathcal{F}+\lVert y \rVert^\mathcal{F}.
\end{multline}
Finally, assume $x$ is finitely supported and $\mathcal{P}$ is a partition of $\omega$ such that $\lVert x \rVert^\mathcal{F} = \lVert x \rVert^\mathcal{P}$. Of course only finitely many elements of $\mathcal{P}$, say $F_0, \dots, F_n$, have nonempty intersection with
the support of $x$. Let $a_i = \max \limits_{k\in F_i} x(k)$. Then $\lVert x \rVert^\mathcal{F} = a_0 + \dots + a_n$. Let $x' = a_0 \chi_{F_0} + \dots + a_n \chi_{F_n}$ and notice that $\lVert x' \rVert^\mathcal{F} = \lVert x \rVert^\mathcal{F}$ (as we
may assume that all $F_i$'s are maximal). Then, by monotonicity of $\lVert \cdot \rVert^\mathcal{F}$ and by (\ref{ppp})
\[ \lVert x+y \rVert^\mathcal{F} \leq \lVert x' + y \rVert^\mathcal{F} \leq 2\lVert x' \rVert^\mathcal{F} + \lVert y\rVert^\mathcal{F} = 2\lVert x \rVert^\mathcal{F} + \lVert y\rVert^\mathcal{F}. \] \end{proof}
Below we prove a natural counterpart of Theorem \ref{Banach} (mimicking the proof from \cite{NaFa}).
\begin{thm}\label{quasi-Banach} If $\mathcal{F}$ is a compact hereditary family and $\mathcal{F}$ covers $\omega$, then $FIN(\lVert \cdot \rVert^\mathcal{F})$ and $EXH(\lVert \cdot \rVert^\mathcal{F})$ are quasi-Banach spaces. \end{thm}
\begin{proof} That $\lVert \cdot \rVert^\mathcal{F}$ is a nice quasi-norm follows directly from Theorem \ref{quasi-quasi} and of Theorem \ref{nice_condition}.
We are going to show that $FIN(\lVert \cdot \rVert^\mathcal{F})$ is complete and then that $X^\mathcal{F}$ is a closed subset of $FIN(\lVert \cdot \rVert^\mathcal{F})$. The
For simplicity denote $\varphi = \lVert \cdot \rVert^\mathcal{F}$.
First we will prove that $FIN(\varphi)$ is complete. Let $(x_n)$ be a Cauchy sequence in $FIN(\varphi)$. Applying monotonicity, $\varphi(P_{\{k\}}(x_n-x_m))\leq \varphi(x_n-x_m)$ for every $k,n,m$, and hence $(P_{\{k\}}(x_n))_{k\in\omega}$ is a Cauchy sequence in the $k$th $1$-dimensional coordinate space of $\mathbb{R}^\omega$ (which is a quasi-Banach space, as $\varphi$ is finite on $c_{00}$), $P_{\{k\}}(x_n)\xrightarrow{n\to\infty}y_k$ for some $y_k$. Put $y=(y_k)$. We will first show that $y\in FIN(\varphi)$. The sequence
$\{x_n\colon n\in\omega\}$ is bounded, let say $\varphi(x_n)\leq B$ for every $n$. We show that $\varphi(y)\leq 3B$, i.e. (by the lower semicontinuity of $\varphi$) $\varphi(P_M(y))\leq 3B$ for every $M\in\omega$. Fix an $M>0$. If $n$ is large enough, say $n\geq n_0$, then
$\varphi(P_{\{k\}}(y-x_n))\leq \dfrac{B}{M}$ for every $k<M$ and hence \[\varphi(P_M(y))\leq \dfrac{3}{2}(\varphi(P_M(y-x_n))+\varphi(P_M(x_n))) \leq \dfrac{3}{2}(\sum_{k<M}\varphi(P_{\{k\}}(y-x_n))+\varphi(x_n))\leq 3B.\] The first inequality follows from Theorem \ref{quasi-quasi}(c) and the second from Theorem \ref{quasi-quasi}(a).
Now we will prove that $x_n\to y$. If not, then there are $\varepsilon>0$ and $n_0<n_1<\dots<n_j<\dots$ such that $\varphi(x_{n_j}-y)>\varepsilon$, that is, $\varphi(P_{M_j}(x_{n_j}-y))>\varepsilon$ for some $M_j\in \omega\setminus\{0\}$ for every $j$. Pick
$j_0$ such that $\varphi(x_{n_{j_0}}-x_n)< \dfrac{\varepsilon}{2}$ for every $n\geq n_{j_0}$ and then pick $j_1>j_0$ such that $\varphi(P_{\{k\}}(x_{n_{j_1}}-y))\leq \dfrac{\varepsilon}{2 M_{j_0}}$ for every $k<M_{j_0}$. Then, using Theorem \ref{quasi-quasi}(a) \[
\varepsilon<\varphi(P_{M_{j_0}}(x_{n_{j_0}}-y))\leq \varphi(P_{M_{j_0}}(x_{n_{j_0}}-x_{n_{j_1}}))+\sum_{k<M_{j_0}}\varphi(P_{\{k\}}(x_{n_{j_1}}-y)) < \varepsilon,\] a contradiction.
Now we will show that $EXH(\varphi)=\overline{c_{00}}$. The space $c_{00}$ is dense in $EXH(\varphi)$ because $\varphi(x-P_n(x))=\varphi(P_{\omega\setminus n}(x))\xrightarrow{n\to\infty}0$ for every $x\in EXH(\varphi)$. We have to show that $EXH(\varphi)$ is closed. Let
$x\in FIN(\varphi)$ be an accumulation point of $EXH(\varphi)$. For any $\varepsilon>0$ we can find $y\in EXH(\varphi)$ such that $\varphi(x-y)<\varepsilon$, and then $n_0$ such that $\varphi(P_{\omega\setminus n}(y))<\varepsilon$ for every
$n\geq n_0$. If $n\geq n_0$ then $\varphi(P_{\omega\setminus n}(x))\leq \dfrac{3}{2}(\varphi(P_{\omega\setminus n}(x-y))+\varphi(P_{\omega\setminus n}(y)))< 3\varepsilon$.
\end{proof}
We will be mainly interested in $EXH(\lVert \cdot \rVert^\mathcal{F})$ and so it will be convenient to denote $X^\mathcal{F} = EXH(\lVert \cdot \rVert^\mathcal{F})$. The main corollary of this section is the following reformulation of (the part of) Theorem \ref{quasi-Banach}:
\begin{thm} If $\mathcal{F}$ is compact, hereditary and covering $\omega$, then $X^\mathcal{F}$ is a quasi-Banach space. \end{thm}
The following is a simple consequence of (a) of Theorem \ref{quasi-quasi}.
\begin{cor} If a family $\mathcal{F}$ is a hereditary closure of a partition $\mathcal{P}$, then the formula (\ref{Upper_norm}) defines a norm. \end{cor}
In what follows we will be mainly interested in the families of finite subsets of $\omega$. In this case $FIN(\lVert \cdot \rVert)^\mathcal{F} = EXH(\lVert \cdot \rVert^\mathcal{F})$:
\begin{prop}\label{exh=fin} If $\mathcal{F} \subseteq [\omega]^{<\omega}$ is a compact hereditary family covering $\omega$, then \[ X^\mathcal{F} = FIN(\lVert \cdot \rVert^\mathcal{F}) \] \end{prop} \begin{proof} For each $x \in \mathbb{R}^{\omega}$ and for fixed $n$ we can write $x = x_{n} + x'_{n}$, where $x_{n} = P_{n}(x)$ and $x'_{n} = P_{\omega \setminus n}(x)$. Thus if $x \in X^{\mathcal{F}}$ then $\lVert x'_{n} \rVert^{\mathcal{F}} \rightarrow 0$ and \[ \lVert x \rVert^{\mathcal{F}} \leq \dfrac{3}{2} (\lVert x_{n} \rVert^{\mathcal{F}} + \lVert x'_{n} \rVert^{\mathcal{F}}) < \infty, \] because $x_{n}$ is finitely supported. It shows that $X^{\mathcal{F}} \subseteq FIN(\lVert \cdot \rVert^{\mathcal{F}})$. On the other hand, if $x \in FIN(\lVert \cdot \rVert^{\mathcal{F}})$, then there is a partition $\mathcal{G} = \lbrace G_{k}: k \in \omega \rbrace \subseteq \mathcal{F}$ such that $\mathlarger{\sum}_{k \in \omega} \sup \limits_{j \in G_{k}} \lvert x(j) \rvert < \infty$. It implies that \[\mathlarger{\sum}_{k \geq n} \sup \limits_{j \in G_{k}} \lvert x(j) \rvert \xrightarrow{n \rightarrow \infty} 0. \]
Let $\varepsilon>0$ and fix $m$ such that $\mathlarger{\sum}_{k\geq m} \sup \limits_{j\in G_k} |x(j)| < \varepsilon$. Let $n > \max (\bigcup \limits_{i<m} G_i)$. Then
\[ \lVert x'_n \rVert^\mathcal{F} \leq \lVert x'_n \rVert^\mathcal{G} \leq \sum_{k\geq m} \sup_{j\in G_k} |x(j)| < \varepsilon. \] It finishes the proof. \end{proof}
\section{$X^\mathcal{F}$ and the dual of $X_\mathcal{F}$}\label{how-close}
The most interesting aspect of the spaces of the form $X^\mathcal{F}$ is the fact that they very much resemble the dual spaces to $X_\mathcal{F}$. In fact they are so much similar that the only distinguishing property we have managed to find is the fact that $X^*_\mathcal{F}$ is always a Banach space whereas $X^\mathcal{F}$ typically is not.
We will first prove two general results. At first, we show that if $\mathcal{F}$ is sufficiently simple, $X^\mathcal{F}$ is a Banach space and it is isometrically isomorphic to $X^*_\mathcal{F}$ (see Proposition \ref{for-partitions}). Moreover, for every (reasonable) family $\mathcal{F}$ the dual space to $X^\mathcal{F}$ is a Banach space which is isometrically isometric to $X^{**}_\mathcal{F}$ (see Theorem \ref{isomorphism}). So, $X^*_\mathcal{F}$ and $X^\mathcal{F}$ share the same duals.
First of all we show that for some particular families $\mathcal{F}$ spaces $X^{\mathcal{F}}$ and $X^*_{\mathcal{F}}$ are indeed the same
\begin{prop} \label{for-partitions}
Suppose $\mathcal{P}$ is a partition of $\omega$ (into finite sets) and $\mathcal{F}$ is its hereditary closure. Then $X_\mathcal{F}^*$ is isometrically isomorphic to $X^\mathcal{F}$.
\end{prop}
\begin{proof}
Enumerate $\mathcal{P} = \{P_1, P_2, \dots\}$. Let $f$ be a functional on $X_{\mathcal{F}}$. It is given by
\[ f(x) = \sum\limits_{n \in \omega} x(n)y(n) \] for $y(n) = f(e_{n})$. For each $n \in \omega$ let $k_{n}$ be such index that $y(k_{n}) = \sup \lbrace \lvert y(k) \rvert: k \in P_{n} \rbrace$. Then for every $x$ from the unit ball of $X_{\mathcal{F}}$ we have \[ \big \lvert \sum\limits_{n \in \omega} x(n)y(n) \big \rvert \leq \sum \limits_{n \in \omega} \sum_{k \in P_{n}} \lvert x(k)y(k) \rvert \leq \sum \limits_{n \in \omega} \lvert y(k_{n}) \rvert \sum_{k \in P_{n}} \lvert x(k) \rvert \leq \sum \limits_{n \in \omega} \lvert y(k_{n}) \rvert = \lVert y \rVert^{\mathcal{F}}. \] It implies that the norm of the functional $f$ (denoted further by $\lVert y \rVert^{*}$) is bounded by $\lVert y \rVert^{\mathcal{F}}$.\\ Now, for $l \in \omega$ define the sequence $x_{l}$ given by $$ x_{l}(m) = \begin{cases}
1, & \text{if } m \in \lbrace k_{1}, ..., k_{l} \rbrace,\\
0, & \text{otherwise.} \end{cases} $$ It is easy to see that $\lVert x_{l} \rVert_{\mathcal{F}} = 1$ and then we have \[ \lVert y \rVert^{*} = \sup \limits_{\lVert x \rVert_{\mathcal{F}} \leq 1} \big \lvert \sum\limits_{n \in \omega} x(n)y(n) \big \rvert \geq \big \lvert \sum\limits_{n \in \omega} x_{l}(n)y(n) \big \rvert = \big \lvert \sum\limits_{m \leq l} y(k_{m}) \big \rvert = \sum\limits_{m \leq l} y(k_{m}). \] Since the above inequality holds for every $l$, we have $$ \sup \limits_{\lVert x \rVert_{\mathcal{F}} \leq 1} \big \lvert \sum\limits_{n \in \omega} x(n)y(n) \big \rvert \geq \sum\limits_{m \in \omega} y(k_{m}) = \lVert y \rVert^{\mathcal{F}}. $$ This inequality finishes the proof.
\end{proof}
\begin{rem} The above proposition can be reversed. Namely, if there are two maximal (in the sense of inclusion) elements $F_0, F_1$ of $\mathcal{F}$ such that $F_0 \cap F_1 \ne \emptyset$, then we can easily modify Example \ref{notnorm}:
\[ \lVert \chi_{F_0} + \chi_{F_1}\rVert^\mathcal{F} \geq 3 > 2 = \lVert \chi_{F_0} \rVert^\mathcal{F} + \lVert \chi_{F_1} \rVert^\mathcal{F}. \]
By the above and by Theorem \ref{quasi-quasi} for every compact, hereditary family $\mathcal{F}$ which covers $\omega$ and which is not a hereditary closure of a partition, the modulus of concavity of $\lVert \cdot \rVert^\mathcal{F}$ equals $\dfrac{3}{2}$. Thus, by Aoki-Rolewicz
Theorem
(Theorem \ref{aoki_rolewicz}) for $\mathcal{F}\subseteq \mathcal{P}(\omega)$ being compact, hereditary and covering $\omega$ we have dichotomy:
\begin{itemize}
\item either $\mathcal{F}$ is a hereditary closure of a partition and then $\lVert \cdot \rVert^\mathcal{F}$ is a norm,
\item or $\lVert \cdot \rVert^\mathcal{F}$ is equivalent to a $p$-norm, where $p=\log_3 2$.
\end{itemize}
\end{rem}
For general case of compact, hereditary family $\mathcal{F}$ covering $\omega$ we are able to prove a slightly weaker result. Let $T\colon X^{\mathcal{F}} \rightarrow X^{*}_{\mathcal{F}}$ be a linear operator given by \begin{equation} \label{identity_operator} T(y)(x) = \sum \limits_{k \in \omega} x(k)y(k) \end{equation} for $x \in X_{\mathcal{F}}$.
\begin{prop}\label{identityoperator} The mapping $T$ defined above is injective and continuous. Consequently, $X^\mathcal{F}$ can be embedded to $X^*_{\mathcal{F}}$. \end{prop}
\begin{proof} It is plain to check that it is injective. In addition we have the following sequence of (in)equalities
\[ \sup_{x \in X_{\mathcal{F}}} \dfrac{\big \lvert \sum \limits_{k \in \omega} x(k)y(k) \big \rvert }{ \lVert x \rVert_{\mathcal{F}}} = \sup_{x \in X_{\mathcal{F}}} \dfrac{ \big \lvert \sum \limits_{k \in \omega} x(k)y(k) \big \rvert }{ \sup \limits_{\mathcal{P} \in \mathbb{P}_{\mathcal{F}}} \lVert x \rVert_{\mathcal{P}}} = \sup_{x \in X_{\mathcal{F}}} \inf_{\mathcal{P} \in \mathbb{P}_{\mathcal{F}}} \dfrac{\big \lvert \sum \limits_{k \in \omega} x(k)y(k) \big \rvert }{\lVert x \rVert_{\mathcal{P}}} \stackrel{\text{(*)}}{\leq}$$ $$ \stackrel{\text{(*)}}{\leq} \inf_{\mathcal{P} \in \mathbb{P}_{\mathcal{F}}} \sup_{x \in X_{\mathcal{F}}} \dfrac{\big \lvert \sum \limits_{k \in \omega} x(k)y(k) \big \rvert }{\lVert x \rVert_{\mathcal{P}}} \stackrel{\text{(**)}}{=} \inf_{\mathcal{P} \in \mathbb{P}_{\mathcal{F}}} \lVert y \rVert^{\mathcal{P}} = \lVert y \rVert^{\mathcal{F}}, \] where the inequality (*) follows from the general fact saying that \[ \sup_{a\in A} \inf_{b\in B} f(a,b) \leq \inf_{b\in B} \sup_{a\in A} f(a,b)\] whatever $A,B$ and $f\colon A\times B \to \mathbb{R}$ are, and equality (**) is a consequence of
Proposition \ref{for-partitions}. It implies that $\lVert T(y) \rVert \leq \lVert y \rVert^{\mathcal{F}}$ for every $y \in X^{\mathcal{F}}$, so $T$ is continuous. In particular $T[B_{X^{\mathcal{F}}}]$ is a subspace of the dual unit ball
$B_{X^{*}_{\mathcal{F}}}$ and by Theorem \ref{open_mapping} the space $X^{\mathcal{F}}$ is isomorphic with a subspace of $X^*_{\mathcal{F}}$, which finishes the proof. \end{proof}
For each compact family $\mathcal{F}$ the space $X^\mathcal{F}$ is a (quasi-Banach) pre-dual of
$(X_{\mathcal{F}})^{**}$. In other words $X^\mathcal{F}$ and $X^*_\mathcal{F}$ have the same dual spaces.
\begin{thm} \label{isomorphism}
If $\mathcal{F}\subseteq [\omega]^{<\omega}$ is a compact hereditary family covering $\omega$, then $(X^{\mathcal{F}})^{*}$ is isometrically isomorphic to $(X_{\mathcal{F}})^{**}$.
\end{thm}
\begin{proof}
By Theorem \ref{5.5} (and the fact that $X_\mathcal{F}$ does not contain an isomorphic copy of $\ell_1$ if $\mathcal{F}$ is as above) the space $(X_{\mathcal{F}})^{**}$ is isometrically isomorphic to $FIN(\lVert \cdot \rVert_{\mathcal{F}})$. We need to prove that $\lVert y \rVert_{*}^{\mathcal{F}} = \lVert y \rVert_{\mathcal{F}}$, where $\lVert \cdot \rVert_{*}^{\mathcal{F}}$ denotes the functional norm on $X^{\mathcal{F}}$.
Take any $y \in c_{00}$. Then there is a set $F_{0} \in \mathcal{F}$ such that $\lVert y \rVert_{\mathcal{F}} = \mathlarger{\sum}_{n \in F_{0}} \lvert y(n) \rvert$. Consider $x_{0} \in X^\mathcal{F}$ given by
$$
x_{0}(n) =
\begin{cases}
sgn(y(n)), & \text{if } n \in F_{0},\\
0, & \text{otherwise.}
\end{cases}
$$
This is a unit vector in $X^{\mathcal{F}}$ and thus
$$
\lVert y \rVert_{*}^{\mathcal{F}} \geq \lvert \sum_{n \in F_{0}} x_{0}(n)y(n) \rvert = \sum_{n \in F_{0}} \lvert y(n) \rvert = \lVert y \rVert_{\mathcal{F}}.
$$
To prove the second inequality, fix any $x \in c_{00}$ such that $\lVert x \rVert_\mathcal{F} = 1$. There exists partition $\mathcal{P} = \lbrace F_{1}, F_{2}, ..., F_{j} \rbrace$ of the support of $x$ for which the infimum in the definition of quasi-norm is obtained, namely
$$
\lVert x \rVert^{\mathcal{F}} = \sum_{i=1}^{j} \sup \limits_{k \in F_{i}} \lvert x(k) \rvert.
$$
Let $x'$ be defined by $x'(j)=a_i \cdot sgn(y(j))$ if $j\in F_i$,
where $a_{i} = \sup \limits_{k \in F_{i}} \lvert x(k) \rvert$ (if $j\notin \bigcup_i F_i$, then let $x'(j)=0$). Then $\lVert x' \rVert^\mathcal{F} = \lVert x \rVert^\mathcal{F} = 1$ and
$$
\big \lvert \sum_{n \in \omega}x(n)y(n) \big \rvert \leq \sum_{n \in \omega} \lvert x(n)y(n) \rvert \leq \sum_{n \in \omega} \lvert x'(n)y(n) \rvert.
$$ Moreover $$
\sum_{n \in \omega} \lvert x'(n)y(n) \rvert = \sum_{i=1}^{j} \sum_{n \in F_{i}} \lvert x'(n)y(n) \rvert = \sum_{i=1}^{j} a_{i} \sum_{n \in F_{i}} \lvert y(n) \rvert \leq \sum_{i=1}^{j} a_{i} \lVert y \rVert_{\mathcal{F}} = \lVert y \rVert_{\mathcal{F}}
$$
which implies that $\lVert y \rVert_{*}^{\mathcal{F}} \leq \lVert y \rVert_{\mathcal{F}}$ and finishes the proof.
\end{proof}
\begin{rem} In this article we are interested in the spaces induced by families of finite sets, but perhaps one can prove a more general version of the above theorem, working for families containing also infinite sets, but with $FIN(\lVert \cdot \rVert^\mathcal{F})$ instead of
$X^\mathcal{F}$ (for $\mathcal{F}$ as in Theorem \ref{isomorphism}, those spaces coincide, see Theorem \ref{exh=fin}). E.g. notice that if $\mathcal{F}$ is the family of all finite subsets of $\omega$,
then $X_\mathcal{F}$ is isometrically isomorphic to $\ell_1$ and $FIN(\lVert \cdot \rVert^\mathcal{F})$ is isometrically isomorphic to $\ell_\infty$. \end{rem}
\subsection{Geometric properties of $X^{\mathcal{F}}$}
\begin{df} Let $K$ be a subset of a vector space $X$ we say that $e \in K$ is an \emph{extreme point} of $K$ if there do not exist $x,y \in K$ and $t \in (0,1)$ such that $e = (1-t)x + ty$. Equivalently, $e$ is an extreme point of $K$ if and only if the following condition is satisfied \begin{equation} \label{extreme} e+x, e-x \in K \Rightarrow x = 0 \end{equation} The set of all extreme points of $K$ we denote by $E(K)$. \end{df}
Although the notion of extreme point is usually considered in the context of convex sets, the definition itself does not require a priori the convexity. Thus we can consider extreme points in the case of non-convex sets as well.
In the case of (quasi)-Banach spaces $X$ it is common to denote by $E(X)$ the set of all extreme points of the unit ball of $X$ and we will follow this notation.
In this section we work with general compact, hereditary family $\mathcal{F}$. The combinatorial spaces and their duals were discovered geometrically in the context of extreme points: see e.g. \cite{Antunes}. The authors show that the set of extreme points of the unit ball in $X^{*}_{\mathcal{F}}$ is of the form
\begin{equation} \label{dual_extreme} \Big \lbrace \sum \limits_{i \in F} \varepsilon_i e^{*}_{i}: F \in \mathcal{F}^{MAX}, \varepsilon_{i} \in \lbrace -1,1 \rbrace \Big \rbrace \end{equation}
where \begin{itemize} \item $e^{*}_{i}$ are functionals given by $e^{*}_{i}(e_{j}) = 1$ if $i=j$ and $e^{*}_{i}(e_{j}) = 0$ otherwise for a Schauder basis $(e_{i})$, $i,j \in \omega$. \item $\mathcal{F}^{MAX}$ is a family of \emph{maximal} sets from $\mathcal{F}$, i.e. these sets $F$ for which $F \cup \lbrace k \rbrace \notin \mathcal{F}$ for every $k \in \omega$. \end{itemize}
Actually, the fact that $E(X^{*}_{\mathcal{F}})$ is given by (\ref{dual_extreme}) was proven only for Schreier space and for \emph{higher order Schreirer spaces}. However, that result holds also for general compact, hereditary family $\mathcal{F}\subseteq [\omega]^{<\omega}$ (see Remark 4.4 from \cite{Antunes}). We will show that $X^{\mathcal{F}}$ has basically \emph{the same} extreme points (interpreting elements of $X^\mathcal{F}$ as elements of $X^*_\mathcal{F}$ in the sense of Proposition \ref{identityoperator}):
\begin{prop} \label{extreme_to_extreme}
Assume that $\mathcal{F}\subseteq [\omega]^{<\omega}$ is a compact, hereditary family covering $\omega$.
A vector $y\in X^\mathcal{F}$ is an extreme point of the unit ball of $X^\mathcal{F}$ if and only if it is of the form \begin{equation} \label{seq_1} y(i) = \begin{cases} \varepsilon_{i}, & \text{if } i \in F\\ 0 & \text{otherwise,} \end{cases} \end{equation}
for some $F\in \mathcal{F}^{MAX}$ and $\varepsilon_i\in \{-1,1\}$. \end{prop}
\begin{proof}
First, assume that $y$ equals 1, up to an absolute value, on some maximal set $F \in \mathcal{F}$. Now suppose that for $u \in X^{\mathcal{F}}$ we have $y+u, y-u \in B_{X^{\mathcal{F}}}$. Then for every $k \in supp(y) \cap supp(u)$ we have $|1 \pm
u(k)| \leq 1$ or $|-1 \pm u(k)| \leq 1$. In both cases it implies that $u(k) = 0$, hence $supp(y) \cap supp(u) = \emptyset$. Thus $y(m) + u(m) = \pm 1$ for some $m \in \omega$. In particular, by maximality of $F$, $\lVert y \pm u \rVert^{\mathcal{F}} \geq 1$, and thus $u = 0$.
Now suppose that $y \in E(X^{\mathcal{F}})$. Then $\lVert y \rVert^{\mathcal{F}} = 1$.
Let $\mathcal{P} \in \mathbb{P}_{\mathcal{F}}$ for which $\lVert y \rVert^{\mathcal{F}} = \lVert y \rVert^{\mathcal{P}}$. Notice that for every $P\in \mathcal{P}$ we have $|y(i)| = |y(j)|$ for every $i,j\in P$. Suppose otherwise. Then there is
$P\in \mathcal{P}$ and $j,i\in P$ such that $|y(j)|<|y(i)|$ and so for $\eta< |y(i)|-|y(j)|$ we would have $\lVert y \pm \eta e_j \rVert^\mathcal{F} \leq \lVert y \pm \eta e_j \rVert^\mathcal{P} \leq 1$, hence $y$ would not be an extreme point.
It follows, that if $supp(y)\in \mathcal{F}$, then $y$ is of the promised form. If $supp(y)\notin \mathcal{F}$, then we may find distinct $P_0, P_1\in \mathcal{P}$ and $a_0, a_1 \ne 0$ such that $y(i)=a_j$ for $i\in P_j$, $j\in \{0,1\}$. Since
$\lVert y \rVert^\mathcal{F} = 1$, $|a_0|, |a_1|<1$. But then for sufficiently small $\eta>0$ and for $u$ defined by \begin{equation} u(i) = \begin{cases} a_0 + \eta, & \text{if } i \in P_0\\
a_1 - \eta, & \text{if } i \in P_1\\ 0 & \text{otherwise,} \end{cases} \end{equation} we would have \[ \lVert y \pm u \rVert^\mathcal{F} \leq \lVert y \pm u \rVert^\mathcal{P} = \lVert y\rVert^\mathcal{P} = 1. \]
So, $y$ has to be of the form as in the proposition. \end{proof}
One of the few properties of the spaces dual to combinatorial spaces which are exposed in the literature is that their balls can be well approximated by extreme points.
\begin{df} \label{CSRP} We say that a quasi-Banach space $X$ has \emph{convex series representation property (CSRP)} if for every $x \in B_{X}$ there exists a sequence $(\lambda_{n})$ of positive real numbers with $\sum \limits_{n \in \omega} \lambda_{n} = 1$ and a sequence $(u_{n})$ of extreme points of $B_{X}$ such that \begin{equation} \label{CSRP_eq} x = \sum \limits_{n \in \omega} \lambda_{n} u_{n} \end{equation} \end{df}
It is known (see \cite{Antunes}) that $X^{*}_{\mathcal{F}}$ has CSRP, for $\mathcal{F}$ as above. Using the shape of $E(X^{\mathcal{F}})$, we will construct sequences as in Definition \ref{CSRP} and show that $X^{\mathcal{F}}$ has CSRP as well.
\begin{thm}\label{CSRPthm}
For any compact, hereditary family $\mathcal{F}\subseteq [\omega]^{<\omega}$ covering $\omega$, the space $X^{\mathcal{F}}$ has CSRP \end{thm}
\begin{proof} We have to show that for every $x\in B_X$ there exists an appropriate sequence of extreme points and coefficients. First, we will prove it assuming that $supp(x) \in \mathcal{F}$. Then we will generalize it for the
case $x\in c_{00}$ and at the end we will show the final result.
\textbf{1) $supp(x)\in \mathcal{F}$}
Assume $supp(x) \subseteq F_0$ for some $F_0 \in \mathcal{F}^{MAX}$. Put $\alpha = \min \{ |x(k)|\colon k \in supp(x) \}$. Define $\lambda_0 = \min \{ \alpha, 1 - \alpha \}$ and $\lambda_n = \dfrac{1-\lambda_0}{2^n}$ for $n \geq 1$. Let $u_0$ be an extreme point defined by $$ u_0(k) = \begin{cases} sgn(x(k)), &\text{if } k \in supp(x)\\ 1, &\text{if } k \in F_0 \setminus supp(x) \\ 0 &\text{otherwise.} \end{cases} $$ Put $v_0 = \lambda_0 u_0$ and define $S_0 = \{ k \in \omega: x(k) = v_0(k) \}$. Note that, a priori, it is possible for $S_0$ to be empty. If not, let $G_0\subseteq \omega$ be such that $F_0 < G_0$ and $F_1:= (F_0 \setminus S_0) \cup G_0 \in \mathcal{F}^{MAX}$ (for $S_0 = \emptyset$ we take $G_0 = \emptyset$ as well).
We iterate this construction for $n \geq 1$, i.e. we put \begin{equation} \label{inductive_construction} u_n(k) = \begin{cases} sgn(x(k) - \sum \limits_{j < n} v_j(k)), &\text{if } k \in F_{n-1}\\ 1, &\text{if } k \in F_n \setminus F_{n-1}\\ 0, &\text{otherwise,} \end{cases} \end{equation}
we let $v_n = \lambda_n u_n$, $S_n = \{ k \in \omega\colon x(k) - \sum \limits_{j < n} v_j(k) = v_n(k) \}$ and let $G_n$ be such a (possibly empty) set that $F_n < G_n$ and $F_{n+1}:= (F_n \setminus S_n) \cup G_n$ is maximal. Note that on each step of the construction the sequence $x - \sum \limits_{j \leq n} v_{j}$ is supported on a subset of the maximal set $F_n$.
Now we show that the sequences $(\lambda_n)$ and $(u_n)$ are as desired by Definition \ref{CSRP}. It is rather clear that $\sum \limits_{n \in \omega} \lambda_n = 1$ and for every $n$ the vector $u_n$ is an extreme point in $X^{\mathcal{F}}$ (by Lemma \ref{extreme_to_extreme}). It remains to check that the series $\sum \limits_{n \in \omega} \lambda_n u_n$ is convergent to $x$ in a quasi-norm $\lVert \cdot \rVert^{\mathcal{F}}$.
\textbf{Claim.} For every $k \in \omega$ $r_n(k) := \Big \lvert x(k) - \sum \limits_{j \leq n} v_j(k) \Big \rvert \leq \dfrac{1-\lambda_0}{2^n}$.
\begin{proof}[Proof of claim] We prove it by induction with respect to $n$. If $n = 0$ and $k \in supp(x)$ then we have $ - \lambda_0 < x(k) - \lambda_0 \leq 1 - \lambda_0$ for $x(k) > 0$. Since $\lambda_0 \leq 1 - \lambda_0$ by the definition, the inequality holds for $x(k) > 0$. The definition of $\lambda_0$ implies also immediately that $r_0(k) \leq 1-\lambda_0$ for $k \notin supp(x)$. Finally, if $x(k) < 0$, then $v_0(k) = - \lambda_0$ and then $ -1 + \lambda_0 \leq x(k) - v_0(k) < \lambda_0 \leq 1 - \lambda_0$. \\ Now suppose that $r_n(k) \leq \dfrac{1-\lambda_0}{2^n}$ for some $n$. If $x(k) - \sum \limits_{j \leq n} v_j(k) > 0$, then $v_{n+1}(k) = \lambda_{n+1}$, if $k \in F_{n}$ and thus $$ -\dfrac{1-\lambda_0}{2^{n+1}} = -\lambda_{n+1} \leq x(k) - \sum \limits_{j \leq n+1} v_j(k) \leq \dfrac{1-\lambda_0}{2^n} - \dfrac{1-\lambda_0}{2^{n+1}} = \dfrac{1-\lambda_0}{2^{n+1}} $$ If $x(k) - \sum \limits_{j \leq n} v_j(k) < 0$ then $v_{n+1}(k) = - \lambda_{n+1}$ and the case is symmetric. Thus for $k \in F_{n}$ $r_{n+1}(k) \leq \dfrac{1-\lambda_0}{2^{n+1}}$. If $k \in F_{n+1} \setminus F_n$ then $ x(k) - \sum \limits_{j \leq n+1} v_j(k) = -v_{n+1}(k) = - \lambda_{n+1}$ which finishes the proof of the claim. \end{proof} Note that the above claim implies that $\sum \limits_{n \in \omega} \lambda_n u_n$ is convergent to $x$ since $$ \lVert x - \sum \limits_{j \leq n} \lambda_n u_n \rVert^{\mathcal{F}} = \max_{k \in F_n} \Big \lvert x(k) - \sum \limits_{j \leq n}v_n(k) \Big \rvert \leq \dfrac{1 - \lambda_0}{2^n} \xrightarrow{n \rightarrow \infty} 0 $$ It finishes the proof for $x$ having a support in $\mathcal{F}$.
\textbf{2) $x\in c_{00}$.}
If $x$ is a finitely supported sequence, then $x = \sum \limits_{i=1}^{m} x_i $ for some $m \in \omega$, where $x_i$ are sequences with supports contained in some $F_i \in \mathcal{F}^{MAX}$. Then, we make a similar construction as in the previous
case for each $x_i$ separately. Put $M = \lVert x \rVert^{\mathcal{F}}$, and for each $1 \leq i \leq m$ let $\alpha_i = \min \limits_{k \in F_i} |x(k)|$, $\beta_i = \max \limits_{k \in F_i} |x(k)|$. Next, define a sequence $(\lambda^{i}_{n})_n$ by taking $\lambda^{i}_{0} = \min \{\alpha_i, \dfrac{\beta_i}{M}-\alpha_i \}$ and $\lambda^{i}_{n} = \dfrac{\frac{\beta_{i}}{M} - \lambda^{i}_{0}}{2^n}$ for $n \geq 1$. For each $i$ $\sum \limits_{n \in \omega} \lambda^{i}_{n} = \dfrac{\beta_i}{M}$ and thus $\sum \limits_{i=1}^{m} \sum \limits_{n \in \omega} \lambda^{i}_{n} = 1$.
The sequence of extreme points $(u^{i}_{n})_n$ define as in the first case. Then, repeating arguments from the previous case for every $i$, we obtain $$ \big \lVert x_i - \sum \limits_{j \leq n} \lambda^{i}_{j} u^{i}_{j} \big \rVert^{\mathcal{F}} \leq \dfrac{\frac{\beta_{i}}{M} - \lambda^{i}_{0}}{2^n}. $$
Thus for each $n$ we get
\begin{equation} \label{total convergence} \big \lVert x - \sum \limits_{i=1}^{m} \sum \limits_{j \leq n} \lambda^{i}_{j}u^{i}_{j} \big \rVert^{\mathcal{F}} \leq \big(\dfrac{3}{2} \big)^{m} \sum \limits_{i=1}^{m} \big \lVert x_i - \sum \limits_{j \leq n} \lambda^{i}_{j} u^{i}_{j} \big \rVert^{\mathcal{F}} \leq \big(\dfrac{3}{2} \big)^{m} \sum \limits_{i=1}^{m} \dfrac{\frac{\beta_i}{M} - \lambda^{i}_{0}}{2^n} \end{equation} and the last expression tends to $0$ when $n \rightarrow \infty$. Hence every finitely supported sequence can be expressed as a convex series of extreme points.
\textbf{3) The general case.}
For $x \in X^{\mathcal{F}}$ the result follows from lower semicontinuity of $\lVert \cdot \rVert^{\mathcal{F}}$ and the previous cases. Indeed, for any $\varepsilon > 0$ find $N \in \omega$ such that $\lVert x - P_{N}(x) \rVert^{\mathcal{F}} < \dfrac{\varepsilon}{3}$. For that $N$ we can find a convex combination as in the second case, converging to $P_{N}(x)$. Namely, for sufficiently big $n \in \omega$ $$ \big \lVert P_{N}(x) - \sum \limits_{i=1}^{m} \sum \limits_{j \leq n} \lambda^{i}_{j}u^{i}_{j} \big \rVert^{\mathcal{F}} < \dfrac{\varepsilon}{3} $$ Thus using \emph{quasi-triangle inequality} we have $$ \lVert x - \sum \limits_{i=1}^{m} \sum \limits_{j \leq n} \lambda^{i}_{j}u^{i}_{j} \big \rVert^{\mathcal{F}} \leq \dfrac{3}{2} \Big ( \big \lVert x - P_{N}(x) \big \rVert^{\mathcal{F}} + \big \lVert P_{N}(x) - \sum \limits_{i=1}^{m} \sum \limits_{j \leq n} \lambda^{i}_{j}u^{i}_{j} \big \rVert^{\mathcal{F}} \Big ) < \varepsilon $$ It finishes the proof. \end{proof}
\section{$X^\mathcal{S}$ is $\ell_1$-saturated and does not have the Schur property} \label{l1-schur}
In this section we will focus on the relations between $X^\mathcal{F}$ and $X^*_\mathcal{F}$ for the particular case of the Schreier family $\mathcal{S}$. Among combinatorial spaces, $X_\mathcal{S}$ is the most studied (apart of $c_0$ and $\ell_1$, of course). However, the literature concerning the dual to the Schreier space is not rich. One of its main properties is that it is an example of a Banach space which is $\ell_1$-saturated but which does not enjoy the Schur property.
\begin{df} \label{schur_def}
We say that a space $X$ has a \textit{Schur property} if every weakly null sequence is convergent to zero in the norm.
\end{df}
\begin{df} \label{l1-def}
A Banach space $X$ is \textit{$\ell_{1}$-saturated} if its every closed, infinitely dimensional subspace $E$ contains an isomorphic copy of $\ell_{1}$. Equivalently, if $E = \overline{span(x_{n})}$ for some sequence $(x_{n})$ in $X$, then there is a sequence $(y_{n})$ in $E$, which is equivalent to the standard $\ell_{1}$-norm.
\end{df}
In theory of Banach spaces the classical theorem of Rosenthal \cite{Rosenthal-l1} implies that every Banach space with Schur property is $\ell_{1}$-saturated. First example of an $\ell_{1}$-saturated Banach space without Schur property was constructed by Bourgain (\cite{Bourgain})). Then Azimi and Hagler (\cite{Hagler}) and Popov (\cite{Popov}) presented another examples. Yet another example of such space is a dual to the Schreier space. The lack of Schur property is the result of \cite{Lipieta}. On the other hand, satisfying of $\ell_{1}$-saturation property follows form \cite{Galego}.
We will show that Definition \ref{schur_def} and Definition \ref{l1-def} are also valid for quasi-Banach spaces and that $X^{\mathcal{S}}$ is an $\ell_{1}$-saturated quasi-Banach space which does not have Schur property.
The part of Schur property we prove using Theorem \ref{isomorphism}.
\begin{prop} \label{Schur}
Suppose that $\mathcal{F}$ is a family of finite subsets of $\omega$ with the following property: for each infinite $M\subseteq \omega$ and each $k\in \omega$ there is $F\in \mathcal{F}$ such that $F\subseteq M$ and $|F|>k$. Then $X^\mathcal{F}$
does not have the Schur property. In particular, $X^\mathcal{S}$ does not have the Schur property.
\end{prop}
\begin{proof}
Consider the sequence $(e_{n})$, the standard Schauder basis. We claim that this sequence is weakly null, but it is not convergent to zero in the quasi-norm.
Indeed, fix $\varphi \in (X^{\mathcal{F}})^{*}$ and denote $y(n) = \varphi(e_{n})$. By Theorem \ref{isomorphism} we have $y = (y(n)) \in FIN(\lVert \cdot \rVert_{\mathcal{F}})$, so $\lVert y \rVert_{\mathcal{F}} < \infty$. If $ \lim
\limits_{n \rightarrow \infty} y(n) \neq 0$, then there is an infinite $M\subseteq \omega$ and $c>0$ such that $\lvert y(m) \rvert \geq c$ for each $m\in M$. By the assumption, for each $k\in \omega$ there is $F\in \mathcal{F}$ such that $F\subseteq M$,
$|F|>k$ and so
$\mathlarger{\sum}_{i \in F} \lvert y(i) \rvert \geq c \cdot k$. Hence, $\lVert y \rVert_\mathcal{F} = \sup \limits_{F \in \mathcal{F}} \mathlarger{\sum}_{i \in F} \lvert y(i) \rvert = \infty$, a contradiction.
On the other hand, for every $n \in \omega$ $\lVert e_{n} \rVert^{\mathcal{F}} = 1$, so $(e_{n})$ is not convergent to zero in norm.
The Schreier family $\mathcal{S}$ satisfies the above condition.
\end{proof}
As we can see not only Schreier space does not enjoy the Schur property but it is more general phenomenon for spaces of this type.
The proof that $X^{\mathcal{S}}$ is $\ell_{1}$-saturated is, however, slightly more technical and uses more of the particular shape of the Schreier family $\mathcal{S}$. First, we introduce the definition which is necessary to prove that fact.
\begin{df} We say that the sequence $x$ is \textit{$k$-stable} if for every set $A \in [\omega]^{\leq k}$ $$\lVert P_{\omega \setminus A}(x) \rVert^{\mathcal{S}} \geq \frac{1}{2} \lVert x \rVert^{\mathcal{S}}.$$
\end{df}
\begin{lem} \label{simple_lemma}
Let $x_{1},x_{2} \in c_{00}$ be such that $supp(x_1) < supp(x_2)$. If $\mathcal{P}$ is a partition such that for every $P \in \mathcal{P}$ $P \cap supp(x_1) = \emptyset$ or $P \cap supp(x_2) = \emptyset$, then $$ \lVert x_{1} + x_{2}
\rVert^{\mathcal{P}} = \lVert x_{1} \rVert^{\mathcal{P}} + \lVert x_{2} \rVert^{\mathcal{P}}.$$
\end{lem}
\begin{proof}
Let $P \in \mathcal{P}$. Then, either $x_{1}$ or $x_{2}$ vanishes on $P$, so for each $k \in P$ $\lvert x_{1}(k) + x_{2}(k) \rvert = \lvert x_{1}(k) \rvert + \lvert x_{2}(k) \rvert$. It implies immediately that $\lVert x_{1} + x_{2} \rVert^{\mathcal{P}} = \lVert x_{1} \rvert^{\mathcal{P}} + \lVert x_{2} \rVert^{\mathcal{P}}$
\end{proof}
\begin{prop} \label{stable-proposition}
Let $x,y \in c_{00}$ be such that
\begin{enumerate}[(i)]
\item $supp(x) < supp(y)$
\item $y$ is $k_{0}$-stable, where $k_{0} = (\max supp(x))^{2}$.
\end{enumerate}
Then for each $\lambda$ $$\lVert x+ \lambda y \rVert^{\mathcal{S}} \geq \lVert x \rVert^{\mathcal{S}} + \frac{\lambda}{2}\lVert y \rVert^{\mathcal{S}}.$$
\end{prop}
\begin{proof} First, notice that if $y$ is $k$-stable, then $\lambda y$ is $k$-stable and so we may assume that $\lambda=1$.
Since $x, y, x+y$ are finitely supported, there exist partitions $\mathcal{P}_{0}, \mathcal{P}_{1}$ and $\mathcal{P}_{2}$ such that
\begin{center}
$\lVert x \rVert^{\mathcal{S}} = \lVert x \rVert^{\mathcal{P}_{0}}$,\\
$\lVert y \rVert^{\mathcal{S}} = \lVert y \rVert^{\mathcal{P}_{1}}$,\\
$\lVert x+y \rVert^{\mathcal{S}} = \lVert x+y \rVert^{\mathcal{P}_{2}}.$
\end{center}
Let $A = \lbrace n \in supp(y): \forall P \in \mathcal{P}_{2} \ (n \in P \Rightarrow P \cap supp(x) \neq \emptyset) \rbrace$. Note that the sequences $x$, $P_{\omega \setminus A}(y)$ and the partition $\mathcal{P}_{2}$ satisfy the assumption of Lemma \ref{simple_lemma}. In addition,
\[ |A| \leq \sum_{k \in supp(x)}k \leq (\max supp(x))^2 = k_0 \]
So, using the assumption of $k_{0}$-stability we obtain
$$
\lVert x+y \rVert^{\mathcal{S}} = \lVert x+y \rVert^{\mathcal{P}_{2}} \geq \lVert x + P_{\omega \setminus A}(y) \rVert^{\mathcal{P}_{2}} = \lVert x \rVert^{\mathcal{P}_{2}} + \lVert P_{\omega \setminus A}(y) \rVert^{\mathcal{P}_{2}} \geq \lVert x
\rVert^{\mathcal{P}_{0}} + \lVert y \rVert^{\mathcal{P}_{2}} \geq \lVert x \rVert^{\mathcal{S}} + \dfrac{\lVert y \rVert^{\mathcal{S}}}{2}. $$ \end{proof}
\begin{thm}\label{ell1} $X^{S}$ is $\ell_{1}$-saturated \end{thm}
\begin{proof}
Let $(x_{n})$ be a sequence in $X^{\mathcal{S}}$ and let $E$ be its closed subspace. We are going to show that $E$ contains an isomorphic copy of $\ell_1$. By the standard arguments (see e.g. \cite{BePe}) we may assume that $E =
\overline{span(x_{n})}$, where for each $n\in \omega$ we have $x_{n} \in c_{00}$, $\lVert x_{n} \rVert^{\mathcal{S}} = 1$ and there exists a finite subset $F_{n}$ such that $supp(x_{n}) \subseteq F_{n}$. Additionally, we assume that $F_{n} < F_{n+1}$.
It is enough to construct a sequence $(y_n)$ of unit vectors in $E$ which will be equivalent to the standard $\ell_1$-basis, i.e. such that for each sequence $(\lambda_i)_{i\leq n}$ of scalars \[ \lVert \sum_{i=1}^{n}\lambda_{i}y_{i} \rVert^{\mathcal{S}} \geq \dfrac{1}{2} \mathlarger{\sum}_{i=1}^{n} \lvert \lambda_{i} \rvert. \]
The sequence $(y_n)$ will be of a form of \emph{block sequence} of $(x_n)$, i.e. $y_{n} = \sum_{i=1}^{k} \alpha_{i}x_{n_{i}}$ for some $k \in \omega$ and an increasing sequence of natural numbers $(n_{i})$.
We define by induction sequences of natural numbers $(k_{n})$, $(l_{n})$ and the sequence of vectors $(y_{n})$. Let $l_{1} = 1$, $y_{1} = x_{1}$ and $k_{1} = (\max supp(x_{1}))^{2}$. Next, let \begin{itemize}
\item[(1)] $l_{n+1}\in \omega$ be such that $\mathlarger{\sum}_{i=l_{n}+1}^{l_{n+1}}x_{i}$ is $k_{n}$-stable, \item[(2)] $y_{n+1} = \dfrac{\mathlarger{\sum}_{i=l_{n}+1}^{l_{n+1}}x_{i}}{\lVert \mathlarger{\sum}_{i=l_{n}+1}^{l_{n+1}}x_{i}\rVert}$ and \item[(3)] $k_{n+1} = \big(\max supp(y_{n+1})\big)^{2}$. \end{itemize}
We will show that we are able to perform such construction. Only condition (1) needs some explanations.
\textbf{Claim.} For each $l\in \omega$ and each $k\in \omega$ there is $L>l$ such that $\mathlarger{\sum}_{i=l}^{L}x_{i}$ is $k$-stable.
Indeed, denote $x = \sum \limits_{i=l}^\infty x_i$ and notice that $\lVert x \rVert^\mathcal{S} = \infty$. Otherwise, $x \in FIN(\lVert \cdot \rVert^\mathcal{S})$ and so $x\in X^\mathcal{S}$, by Proposition \ref{exh=fin}. But this is impossible since $\lVert x_i \rVert^\mathcal{S} = 1$ for each $i$. So, we may find $L$ big enough so that \[ \lVert \sum_{i=l}^L x_i \rVert^\mathcal{S} > 2k. \]
But if $|A|\leq k$, then (since $x_i$'s have disjoint supports)
\[ \lVert P_A(\sum_{i=l}^L x_i) \rVert^\mathcal{S} \leq |A|\lVert x_i\rVert^\mathcal{S} \leq k \] and so $L$ is as desired.
Having this construction, fix $n \in \omega$ and a sequence $(\lambda_{i})_{i=1}^{n}$. Of course $\lVert y_{i} \rVert^{\mathcal{S}} = 1$ for each $i$ and subsequently using Proposition \ref{stable-proposition} we have $$ \lVert \sum_{i=1}^{n}\lambda_{i}y_{i} \rVert^{\mathcal{S}} = \lVert \sum_{i=1}^{n-1}\lambda_{i}y_{i} + \lambda_{n}y_{n} \rVert^{\mathcal{S}} \geq \lVert \sum_{i=1}^{n-1}\lambda_{i}y_{i} \rVert^{\mathcal{S}} + \frac{\lvert \lambda_{n}\rvert}{2} \geq ... \geq \lvert \lambda_{1} \rvert + \frac{1}{2}\mathlarger{\sum}_{i=2}^{n} \lvert \lambda_{i} \rvert \geq \frac{1}{2} \mathlarger{\sum}_{i=1}^{n} \lvert \lambda_{i} \rvert $$ and so $(y_n)$ is equivalent to $\ell_1$-basis. \end{proof}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{abstract}
We study the problem of finding the Artin $L$-functions with the
smallest conductor for a given Galois type. We adapt standard
analytic techniques to our novel situation of fixed Galois type and
get much improved lower bounds on the smallest conductor. For small
Galois types we use complete tables of number fields to determine
the actual smallest conductor. \end{abstract}
\title{Artin $L$-functions of small conductor}
\setcounter{tocdepth}{1} \tableofcontents
\section{Overview} \label{overview}
Artin $L$-functions $L(\Chi,s)$ are remarkable analytic objects built from number fields. Let $\overline{\Q}$ be the algebraic closure of the rational number
field $\Q$ inside the field of complex numbers $\C$. Then Artin $L$-functions are indexed by continuous complex characters $\Chi$ of the absolute Galois group $\G = \Gal(\overline{\Q}/\Q)$, with the unital character $1$ giving the Riemann zeta function $L(1,s) = \zeta(s)$. An important problem in modern number theory is to obtain a fuller understanding of these higher analogs of the Riemann zeta function. The analogy is expected to be very tight: all Artin $L$-functions are expected by the Artin conjecture to be entire except perhaps for a pole at $s=1$; they are all expected to satisfy the Riemann hypothesis that all zeros with $\mbox{Re}(s) \in (0,1)$ satisfy $\mbox{Re}(s)=1/2$.
The two most basic invariants of an Artin $L$-function $L(\Chi,s)$
are defined via the two explicit elements of $\G$, the
identity $e$
and the complex conjugation element $\sigma$.
These invariants are the degree $n = \Chi(e)$ and the
signature $r = \Chi(\sigma)$ respectively.
A measure of the complexity
of $L(\Chi,s)$ is its conductor $D \in \Z_{\geq 1}$,
which can be computed from the discriminants of related number fields.
It is best for purposes such as ours to focus instead
on the root conductor $\delta = D^{1/n}$.
In this paper, we aim to find the simplest Artin $L$-functions exhibiting a given Galois-theoretic behavior. To be more precise,
consider triples $(G,c,\chi)$ consisting of a finite group $G$,
an involution $c \in G$, and a faithful character $\chi$. We say that $\Chi$
has {\em Galois type} $(G,c,\chi)$ if there is a surjection $h : \G \rightarrow
G$ with $h(\sigma) = c$, and
$\Chi = \chi \circ h$. Let $\cL(G,c,\chi)$ be the
set of $L$-functions of type $(G,c,\chi)$, and
let $\cL(G,c,\chi;B)$ be the subset
consisting of $L$-functions with root conductor
at most $B$. Two natural problems for any given Galois type $(G,c,\chi)$ are
\begin{itemize}
\item[\textbf{1}:] Use known and the above
conjectured properties of $L$-functions to get a lower bound $\frak{d}(G,c,\chi)$ on the root conductors
of $L$-functions in $\cL(G,c,\chi)$.
\item[\textbf{2}:] Explicitly
identify the sets $\cL(G,c,\chi;B)$ with $B$ as large as possible.
\end{itemize}
This paper gives answers to both problems, although for
brevity we often fix only $(G,\chi)$ and work instead
with the sets $\cL(G,\chi;B) := \cup_c \cL(G,c,\chi;B)$.
There is a large literature on a special case of the situation
we study. Namely let $(G,c,\phi)$ be a Galois type
where $\phi$ is the character of a transitive permutation
representation of $G$. Then the set
$\cL(G,c,\phi;B)$ is exactly
the set of Dedekind zeta functions $\zeta(K,s)$
arising from a corresponding
set $\cK(G,c,\phi;B)$ of arithmetic equivalence classes of
number fields. In this context, root conductors
are just root discriminants, and lower bounds date back to
Minkowski's work on the geometry of numbers.
Use of Dedekind zeta functions as in {\bf 1} above began with
work of Odlyzko \cite{od-disc1,od-disc1a,od-disc2}, Serre
\cite{serre-minorations}, Poitou
\cite{poitou-minorations,poitou-petits}, and Martinet \cite{martinet}. Extensive responses to {\bf 2} came shortly
thereafter, with papers often focusing on a single degree
$n=\phi(e)$. Early
work for quartics, quintics, sextics,
and septics include respectively
\cite{bf-quartic2,f-quartic1,bfp-quartic3},
\cite{spd}, \cite{pohst,BMO,olivier1,olivier2,olivier3}, and
\cite{letard}.
Further results towards {\bf 2} in higher degrees are
extractable from the websites
associated to \cite{jr-global-database},
\cite{kluners-malle}, and \cite{LMFDB}.
The full situation that we are studying here was identified
clearly by Odlyzko in \cite{odlyzko-durham}, who responded to {\bf 1} with
a general lower bound. However this more general
case of Artin $L$-functions has almost no subsequent presence
in the literature. A noticeable exception is a
recent paper of Pizarro-Madariaga \cite{PM}, who improved
on Odlyzko's results on {\bf 1}. A novelty of our
paper is the separation into Galois types. For many
Galois types this separation allows us to go
considerably further on {\bf 1}. This paper is also the first
systematic study of {\bf 2} beyond the case
of number fields.
Sections~\ref{Artin} and \ref{Signature} review background on Artin $L$-functions and tools used to bound their conductors. Sections~\ref{Type}--\ref{otherchoices} form the new material on the lower bound problem {\bf 1}, while Sections~\ref{S5}--\ref{discussion} focus on the tabulation problem {\bf 2}. Finally, Section~\ref{asymp} returns to {\bf 1} and considers asymptotic lower bounds for root conductors of Artin $L$-functions in certain families. In regard to {\bf 1}, Figure~\ref{amalia-plot} and Corollary~\ref{limitcor} give a quick indication of how our type-based lower bounds compare with the earlier degree-based lower bounds. In regard to both {\bf 1} and {\bf 2}, Tables~\ref{tablelabel1}--\ref{tablelabel8} show how the new lower bounds compare with actual first conductors for many types.
\cmmt{
$**********$
Section~\ref{Artin} reviews some of the basic
formalism behind Artin $L$-functions,
with emphasis on the
direct connection with
number fields. In particular,
suppose $\phi$ is the
character of a transitive
permutation representation of $\G$;
then the set
$\cL(G,c,\phi;B)$
is naturally identified
with the set of Dedekind zeta functions $\zeta(K,s)$
arising from a corresponding
set $\cK(G,c,\phi;B)$ of arithmetic equivalence classes of
number fields.
Section~\ref{Signature} summarizes the
literature on lower bounds
for root conductors.
We restrict attention to
bounds which are
conditional on standard analytic hypotheses,
namely the Artin conjecture and
the Riemann hypothesis for the relevant
auxiliary $L$-functions.
If $\chi$ takes only
nonnegative values,
like permutation characters do,
there
is a direct method for
obtaining a conditional lower bound
for the smallest root conductor.
For general $\chi$, there is
an indirect method
that uses the very general
conductor relation \eqref{gencondrel}
based on the auxiliary character $\phi_S = \chi \bar{\chi}$.
Sections~\ref{Type}--\ref{otherchoices} form
the new material on analytic lower
bounds. Instead
of the general
conductor relation involving $\phi_S$,
we use type-based conductor
relations that depend on
the choice of an auxiliary nonnegative character $\phi$.
The form
of these relations is given in \eqref{maincondrel} and
the relation is made more explicit in Section~\ref{choices}
for four simple and useful types of $\phi$.
Section~\ref{otherchoices} describes the polytope of possible choices for $\phi$.
Our formalism of triples $(G,c,\chi)$ captures the strong tradition in
the literature of paying close attention to the placement of complex conjugation.
However in the remaining sections we keep things relatively brief by
working simply with $(G,\chi)$, which we
also call a type. We pursue the problem of identifying
the sets $\cL(G,\chi;B) := \cup_c \cL(G,c,\chi;B)$ with
a focus on finding at least the the minimal root conductor
$\delta_1(G,\chi)$.
Sections~\ref{S5}--\ref{discussion} focus on this tabulation problem.
Conductor relations similar
to \eqref{maincondrel} let one
use known typically large complete
lists coming from number field tables
to determine new typically smaller complete lists
of $L$-functions.
Section~\ref{S5} explains
the process and illustrates it
by working out the case $G=S_5$
in detail. Section~\ref{Tables}
considers $G$ arising as
transitive subgroups of
$S_n$,
restricting to $n \leq 9$ for
solvable groups and $n \leq 7$ for
nonsolvable groups.
It presents conditional
lower bounds
$\mathfrak{d}(G,\chi)$
and then initial segments
$\cL(G,\chi;B)$,
almost always non-empty.
As illustrated by Figure~\ref{amalia-plot}, our
type-based lower bounds $\mathfrak{d}(G,\chi)$ are
usually substantially larger then
the best degree-based lower bounds coming
from \cite{PM}. We find always
\begin{equation}
\mathfrak{d}(G,\chi)< \delta_1(G,\chi),
\end{equation}
thus no contradiction to the Artin
conjecture or Riemann hypothesis.
In fact, typically $\delta_1(G,\chi)$
is considerably larger than
$\mathfrak{d}(G,\chi)$.
Section~\ref{discussion}
discusses several aspects of
the information presented in the
tables.
Section~\ref{asymp} shows in Corollary~\ref{limitcor}
that restricting the type in simple ways gives
much increased
asymptotic lower bounds. It speculates that these larger
lower bounds may hold even with no restriction
on the type.
}
Artin $L$-functions
have recently become much more
computationally accessible through
a package implemented in {\em Magma}
by Tim Dokchitser. Thousands
are now collected
in a section on the LMFDB \cite{LMFDB}.
The present work
increases our understanding of all this
information in several ways, including by providing completeness
certificates for certain ranges.
\section{Artin $L$-functions} \label{Artin}
In this section we provide some background. An important
point is that our problems allow us to restrict consideration
to Artin characters which take rational values only. In this setting,
Artin $L$-functions can be expressed as products and quotients
of roots of Dedekind zeta functions, minimizing the background needed. General references on Artin $L$-functions include \cite{Mar77,Mur01}.
\subsection{Number fields} A number field $K$ has many invariants relevant for our study. First of all, there is the degree $n = [K:\Q]$. The other invariants we need are local in that they are associated with a place $v$ of $\Q$ and can be read off from the corresponding completed algebra $K_v = K \otimes \Q_v$, but not from other completions. For $v=\infty$, the complete invariant is the signature $r$, defined by $K_\infty \cong \R^{r} \times \C^{(n-r)/2}$. It is more convenient sometimes to work with the eigenspace dimensions for complex conjugation, $a = (n+r)/2$ and $b = (n-r)/2$. For an ultrametric place $v=p$, the full list of invariants is complicated. The most basic one is the positive integer $D_p = p^{c_p}$ generating the discriminant ideal of $K_p/\Q_p$. We package the $D_p$ into the single invariant $D = \prod_p D_p \in \Z_{\geq 1}$, the absolute discriminant of $K$.
\subsection{Dedekind zeta functions} Associated with a number field is its Dedekind zeta function \begin{equation} \label{prodser} \zeta(K,s) = \prod_{p} \frac{1}{P_p(p^{-s})} = \sum_{m=1}^\infty \frac{a_m}{m^s}. \end{equation} Here the polynomial $P_p(x) \in \Z[x]$ is a $p$-adic invariant. It has degree $\leq n$ with equality if and only if $D_p=1$. The integer
$a_m$ is the number of ideals of index $m$ in the ring of integers $\OK$.
\subsection{Analytic properties of Dedekind zeta functions}
Let $\Gamma_\R(s) = \pi^{-s/2} \Gamma(s/2)$, where
$\Gamma(s) = \int_0^\infty x^{s-1} e^{-x} dx$ is the standard
gamma function. Let
\begin{equation}
\label{completed}
\what{\zeta}(K,s) = D^{s/2} \Gamma_\R\left(s\right)^a \Gamma_\R\left(s+1\right)^b \zeta(K,s).
\end{equation} Then this completed Dedekind zeta function
$\what{\zeta}(K,s)$ meromorphically continues to the whole complex plane,
with simple poles at $s=0$ and $s=1$ being its only singularities. It satisfies the functional equation $\what{\zeta}(K,s) = \what{\zeta}(K,1-s)$.
\subsection{Permutation characters} We recall from the introduction that throughout this paper we are taking
$\overline{\Q}$ to be the algebraic closure of $\Q$ in $\C$ and
$\G = \Gal(\overline{\Q}/\Q)$ its absolute Galois group. A degree $n$ number field
$K$ then corresponds to the transitive $n$-element
$\G$-set $X = \mbox{Hom}(K,\overline{\Q})$.
A number field thus has a permutation character $\Phi = \Phi_K = \Phi_X$
with $\Phi(e)=n$. Also signature has the character-theoretic
interpretation $\Phi(\sigma) = r$, where $\sigma$
as before is the complex conjugation element.
\subsection{General characters and Artin $L$-functions} Let $\Chi$ be a character of $\G$.
Then one has an associated Artin $L$-function $L(\Chi,s)$ and conductor
$D_\Chi$, agreeing
with the Dedekind zeta function $\zeta(K,s)$ and the discriminant $D_K$
if $\Chi$ is the permutation
character of $K$. The function $L(\Chi,s)$ has both an Euler product and
Dirichlet series representation as in \eqref{prodser}.
In general, if
$\Phi = \sum_\Chi m_{\lChi} \Chi$ then \begin{align} \label{decomp} L(\Phi,s) & = \prod_{\Chi} L(\Chi,s)^{m_{\lChi}} & D_\Phi & = \prod_{\Chi} D_\Chi^{m_{\lChi}}. \end{align} One is often interested in \eqref{decomp} where the $\Chi$ are irreducible characters.
For a finite set of primes $S$, let $\overline{\Q}_S$
be the compositum of all number fields in $\overline{\Q}$
with discriminant divisible only by primes in $S$.
Let $\G_S = \Gal(\overline{\Q}_S/\Q)$ be the corresponding
quotient of $\G$. Then for primes $p \not \in S$ one has
a well-defined Frobenius conjugacy class $\Fr_p$ in $\G_S$.
The local factor $P_p(x)$ in \eqref{prodser} is the characteristic polynomial
$\det(1 - \rho(\Fr_p) x)$, where $\rho$ is a representation
with character $\Chi$.
\subsection{Relations with other objects}
Artin $L$-functions of degree $1$ are exactly Dirichlet
$L$-functions, so that $\Chi$ can be identified with a
faithful character of the quotient group $(\Z/D\Z)^\times$
of $\G$, with $D$ the conductor of $\Chi$.
Artin $L$-functions coming from irreducible degree $2$ characters
and conductor $D$ are expected to come from
cuspidal modular forms on $\Gamma_1(D)$, holomorphic if $r=0$ and
nonholomorphic otherwise. This expectation
is proved in all cases, except for those with $r = \pm 2$
and projective image the nonsolvable group $A_5$.
In general, to understand how an Artin $L$-function $L(\Chi,s)$
qualitatively relates to other objects, one needs to
understand its Galois theory, including
the placement of complex conjugation; in other words,
one needs to identify its Galois type.
To be more quantitative, one brings in the conductor.
\subsection{Analytic Properties of Artin $L$-functions}
An Artin $L$-function
has a meromorphic continuation and functional
equation, although each with an extra complication
in comparison with the special case of Dedekind zeta functions.
For the
meromorphic continuation, the behavior at $s=1$ is known:
the pole order is the multiplicity $(1,\Chi)$ of $1$
in $\Chi$. The complication is that one has
poor
control over other possible poles. The Artin conjecture for $\Chi$
says however that there are no poles other than $s=1$.
The completed $L$-function \[ \what{L}(\Chi,s) = D_\Chi^{s/2} \Gamma_\R(s)^a \Gamma_\R(s+1)^b L(\Chi,s)\] satisfies the functional equation
\[
\what{L}(\Chi,1-s) = w \what{L}(\overline{\Chi},s)
\] with root number $w$.
Irreducible characters of any compact group come
in three types, orthogonal, non-real, and symplectic.
The type is identified by the Frobenius-Schur
indicator, calculated with respect to the Haar probability
measure $dg$:
\[
FS(\chi) =
\int_{G} \chi(g^2) \, dg \in \{-1,0,1\}.
\]
For orthogonal characters $\Chi$ of $\G$, one has $\Chi = \overline{\Chi}$
and moreover $w = 1$. The complication in
comparison with permutation characters
is that for the other two types,
the root number $w$ is
not necessarily $1$. For symplectic characters,
$\Chi = \overline{\Chi}$ and $w$
can be either of the two possibilities $1$ or
$-1$. For non-real characters, $\Chi \neq \overline{\Chi}$
and $w$ is some algebraic number of norm $1$.
Recall from the introduction that an Artin $L$-function is said to satisfy the Riemann
hypothesis if all its zeros in the critical strip $0<\mbox{Re}(s)<1$ are actually on the critical line $\mbox{Re}(s)= 1/2$.
We will be using the Riemann hypothesis through
Lemma~\ref{lowboundlem}. If we replaced the function \eqref{Odlyzko} with the appropriately
scaled version of (5.17) from \cite{PM},
then our lower bounds would be only conditional on the Artin
conjecture, which is
completely known for some Galois types $(G,c,\chi)$. However
the bounds obtained would be much smaller, and the comparison
with first conductors as presented in Tables~\ref{tablelabel1}--\ref{tablelabel8} below would be
less interesting.
\subsection{Rational characters and rational Artin $L$-functions} \label{rat-chars} The abelianized Galois group $\G^{\rm ab}$ acts on continuous complex characters of profinite groups through its action on their values. If $\Chi'$ and $\Chi''$ are conjugate via this action then their conductors agree: \begin{equation} \label{discequal} D_{\Chi'} = D_{\Chi''}. \end{equation} Our study is simplified by this equality because it allows us to study a given irreducible character $\Chi'$ by studying instead the rational character $\Chi$ obtained by summing its conjugates.
By the Artin induction theorem \cite[Prop.~13.2]{feit}, a rational character $\Chi$ can be expressed as a rational linear combination of permutation characters: \begin{equation} \label{chiexpress} \Chi = \sum k_\Phi \Phi. \end{equation} For general characters $\Chi'$, computing the Frobenius traces $a_p = \Chi'(\Fr_p)$ requires the results of \cite{dok-dok}. Similarly the computation of bad Euler factors and the root number $w$ present difficulties. For Frobenius traces and bad Euler factors, these complications are not present for rational characters $\Chi$ because of \eqref{chiexpress}.
\section{Signature-based analytic lower bounds} \label{Signature} Here and in the
next section we aim to be brief, with the main point being to explain how type-based
lower bounds are usually much larger than signature-based lower bounds. We employ the standard framework for establishing lower bounds for conductors and discriminants, namely Weil's explicit formula. General references for the material
in this section are \cite{odlyzko-durham,PM}.
\subsection{Basic quantities} \label{basic-quantities}
The theory allows general test functions that satisfy certain axioms. We work only with a function introduced by Odlyzko (see \cite[(9)]{poitou-petits}), \begin{equation} \label{Odlyzko} f(x) = \left\{ \begin{array}{ll} {\displaystyle (1-x) \cos(\pi x) + \frac{\sin(\pi x)}{\pi}}, & \mbox{ if $0 \leq x \leq 1$,} \\ 0, & \mbox{ if $x>1$}. \end{array} \right. \end{equation} For $z \in [0,\infty)$ let \begin{align*} N(z) & = \log(\pi) + \int_0^{\infty} \frac{e^{-x/4}+e^{-3x/4}}{2(1-e^{-x})} f(x/(2z))
- \frac{e^{-x}}{x} \,dx, \\ &= \gamma+ \log(8\pi) + \int_0^\infty \frac{f(x/z)-1}{2\sinh(x/2)} \, dx \\ &= \gamma+\log(8\pi) + \int_0^z \frac{f(x/z)-1}{2\sinh(x/2)} \, dx -\log\left(\frac{e^{z/2}+1}{e^{z/2}-1} \right), \\ R(z) & = \int_0^{\infty}\frac{e^{-x/4}-e^{-3x/4}}{2(1-e^{-x})} f(x/(2z))\, dx, \\ &= \int_0^z \frac{f(x/z)}{2\cosh(x/2)}\, dx, \\ P(z) & = 4 \int_0^{\infty} f(x/z) \cosh(x/2)\, dx \\ &= \frac{256 \pi^2 z \cosh^2(z/4)}{(z^2+4\pi^2)^2}. \end{align*} The simplifications in the integrals for $N(z)$ and $R(z)$ are fairly standard and apply to any test function, with the exception of the final steps which make use of the support for $f(x)$. Evaluation of $P(z)$ depends on the choice of $f(x)$. The integrals for $N(z)$ and $R(z)$ cannot be evaluated in closed form like the third, but, as indicated in \cite[\S2]{poitou-petits}, they do have simple limits $N(\infty) = \log(8 \pi) + \gamma$ and $R(\infty) = \pi/2$ as $z \rightarrow \infty$. Here $\gamma \approx 0.5772$ is the Euler $\gamma$-constant. The constants $\Omega = e^{N(\infty)} \approx 44.7632$ and $e^{R(\infty)} \approx 4.8105$, as well as their product $\Theta = e^{N(\infty)+R(\infty)} \approx 215.3325$, will play important roles in the sequel.
\subsection{The quantity $M(n,r,u)$} Consider triples $(n,r,u)$ of real numbers with $n$ and $u$ positive and $r \in [-n,n]$. For such a triple, define \[ M(n,r,u) = \mbox{Max}_z \left( \exp \left( N(z) + \frac{r}{n} R(z) - \frac{u}{n} P(z) \right) \right). \] It is clear that $M(n,r,u) = M(n/u,r/u,1)$. Accordingly we regard $u=1$ as the essential case and abbreviate $M(n,r)=M(n,r,1)$. For fixed $\epsilon \in [0,1]$ and $u>0$, one has the asymptotic behavior \begin{equation} \label{asymptotic1} \lim_{n \rightarrow \infty} M(n,\epsilon n) = \Omega^{1-\epsilon} \Theta^{\epsilon} \approx 44.7632^{1-\epsilon} 215.3325^{\epsilon}. \end{equation} Figure~\ref{contourM} gives one a feel for the fundamental function $M(n,r)$. Particularly important are the univariate functions $ M(n,0)$ and $
M(n,n)$ corresponding to the left and right edges of this figure. \begin{figure}
\caption{ A contour plot of $M(n,\epsilon n)$ in the
window $[0,1] \times [1,1 \, 000 \, 000]$ of the $\epsilon$-$n$ plane, with a vertical logarithmic scale and contours at $2$, $4$, $6$, $8$, ${\bf 10}$, \dots, {\bf 170}, 172, 174, 176.
Some limits for $n \rightarrow \infty$ are shown on the upper boundary.}
\label{contourM}
\end{figure}
\subsection{Lower bounds for root discriminants} Suppose that
$\Phi$ is a nonzero Artin character which takes real values only. We say
that $\Phi$ is nonnegative if \begin{equation} \label{nonnegativity} \Phi(g) \geq 0 \mbox{ for all $g \in \G$.} \end{equation} This nonnegativity ensures that the inner product $(\Phi,1)$ of $\Phi$ with the unital character $1$ is positive. A central result of the theory, a special case of the statement in \cite[(7)]{poitou-petits}, then serves us as a lemma. \begin{lemma} \label{lowboundlem} The lower bound \[ \delta_{\Phi} \geq M(n,r,u). \] is valid for all nonnegative characters $\Phi$ with $(\Phi(e),\Phi(\sigma),(\Phi,1)) = (n,r,u)$ and $L(\Phi,s)$ satisfying the Artin conjecture and the Riemann hypothesis. \end{lemma} If $\Phi$ is a permutation character, then the nonnegativity condition \eqref{nonnegativity} is automatically satisfied. This makes the application of the analytic theory to lower bounds of root discriminants of fields relatively straightforward.
\subsection{Lower bounds for general Artin conductors} To pass from nonnegative characters to general characters, the classical method uses the following lemma. \begin{lemma}[Odlyzko \cite{odlyzko-durham}] \label{lemma1} The conductor relation \begin{equation} \label{gencondrel} \delta_\Chi \geq \delta_{\Phi}^{n/(2n-2)} \end{equation} holds for any degree $n$ character
$\Chi$ and its absolute square $\Phi = |\Chi|^2$. \end{lemma} \noindent A proof of this lemma from first principles is given in \cite{odlyzko-durham}.
Combining Lemma~\ref{lowboundlem} with Lemma~\ref{lemma1} one gets the following immediate consequence
\begin{thm} The lower bound \label{thm1} \[ \delta_\Chi \geq M(n^2,r^2,w)^{n/(2n-2)} \] is valid for all characters $\Chi$ with
$(\Chi(e),\Chi(\sigma),(\Chi,\overline{\Chi})) = (n,r,w)$ such that $L(|\Chi|^2,s)$ satisfies the Artin conjecture and the Riemann hypothesis. \end{thm} This theorem is essentially the main result in the literature on lower bounds for Artin conductors. It appears in \cite{odlyzko-durham, PM} with the right side replaced by explicit bounds. For fixed $\epsilon \in [-1,1]$ and $w>0$, one has the asymptotic behavior \begin{equation} \label{asymptotic2} \lim_{n \rightarrow \infty} M(n^2,\epsilon^2 n^2,w) = \Omega^{(1-\epsilon^2)/2} \Theta^{\epsilon^2/2} \approx 6.6905^{1-\epsilon^2} 14.6742^{\epsilon^2}. \end{equation} The bases $\Omega \approx 44.7632$ and $\Theta \approx 215.3325$ of \eqref{asymptotic1} serve as limiting lower bounds for root discriminants via Lemma~\ref{lowboundlem}. However it is only their square roots $\sqrt{\Omega} \approx 6.6905 $ and $\sqrt{\Theta} \approx 14.6742$ which Theorem~\ref{thm1} gives as limiting lower bounds for root conductors. This
discrepancy will be addressed in Section~\ref{asymp}.
\section{Type-based analytic lower bounds} \label{Type}
In this section we establish Theorem~\ref{thm2}, which is
a family of lower bounds on the root conductor
$\delta_\Chi$ of a given Artin character, dependent
on the choice of an auxiliary character $\phi$.
\subsection{Conductor relations} Let $G$ be a finite group, $c$ an involution in $G$, $\chi$ a faithful character of $G$, and $\phi$ a non-zero real-valued character of $G$. Say that a pair of Artin characters
($\Chi$,$\Phi$) has joint type $(G,c,\chi,\phi)$ if there is a surjection $h : \G \rightarrow G$ with $h(\sigma)=c$, $\Chi = \chi \circ h$, and $\Phi = \phi \circ h$.
Write the conductors respectively as \begin{align*} D_\Chi & = \prod_p p^{c_p(\Chi)}, & D_\Phi & = \prod_p p^{c_p(\Phi)}. \end{align*} Just as in the last section, we need lower bounds on $D_\Chi$ in terms of $D_{\Phi}$. Our paper \cite{jr-tame-wild} produces bounds of this sort in the context of many characters. Here we present some of these results restricted to the setting of two characters, but otherwise following the notation of \cite{jr-tame-wild}.
For $\tau \in G$, let $\bar{\tau}$ be its order. Let $\psi$ be a rational character of $G$. Define two similar numbers, \begin{align} \label{twosimilar} \what{c}_\tau(\psi)&= \psi(e)-\psi(\tau), & c_\tau(\psi)& =
\psi(e) - \frac{1}{\bar{\tau}}
\sum_{k|\bar{\tau}} \varphi(\bar\tau/k) \psi(\tau^k). \end{align}
Here $\varphi$ is the Euler totient function given by $\varphi(k) = |(\Z/k)^\times|$. For the identity element $e$, one clearly has $\what{c}_e(\psi) = c_e(\psi)=0$. When $\bar{\tau}$ is prime, the functions on rational characters defined in \eqref{twosimilar} are proportional: $(\bar{\tau}-1) \what{c}_\tau(\psi) =\bar{\tau}{c}_\tau(\psi)$.
The functions $\what{c}_\tau$ and ${c}_\tau$ are related to ramification as follows. Let $\Psi$ be an Artin character corresponding to $\psi$ under $h$. If $\Psi$ is tame at $p$ then \begin{equation} \label{tameidentity} c_p(\Psi) = c_\tau(\psi), \end{equation} for $\tau$ corresponding to a generator of tame inertia. The identity \eqref{tameidentity} holds because $c_\tau(\psi)$ is the number of non-unital eigenvalues of $\rho(\tau)$ for a representation $\rho$ with character $\psi$. For general $\Psi$, there is a canonical expansion \begin{equation} \label{wildbound} c_p(\Psi) = \sum_{\tau} k_\tau \what{c}_\tau(\psi), \end{equation} with always $k_\tau \geq 0$, coming from the filtration by higher ramification groups on the $p$-adic inertial subgroup of $G$.
Because \eqref{twosimilar}--\eqref{wildbound} are only correct for $\psi$ rational, when we apply them to characters $\chi$ and $\phi$ of interest, we are always assuming that $\chi$ and $\phi$ are rational. As explained in \S\ref{rat-chars}, the restriction to rational characters still allows obtaining general lower bounds. Also, as will be illustrated by an example in \S\ref{spectralwidth}, focusing on rational characters does not reduce the quality of these lower bounds.
For the lower bounds we need, we define the parallel quantities \begin{align} \label{alphaprod} \what{\alpha}(G,\chi,\phi) & = \min_{\tau \in G-\{e\}} \frac{\what{c}_\tau(\chi)}{\what{c}_\tau(\phi)}, & \alpha(G,\chi,\phi) & = \min_{\tau \in G-\{e\}} \frac{c_\tau(\chi)}{c_\tau(\phi)}. \end{align} Let $B(G,\chi,\phi)$ be the best lower bound, valid for all primes $p$, that one can make on $c_p(\Chi)/c_p(\Phi)$ by purely local arguments. As emphasized in \cite[\S2]{jr-tame-wild}, $B(G,\chi,\phi)$ can in principle be calculated by individually inspecting all possible $p$-adic ramification behaviors. The above discussion says \begin{equation} \label{aba} \what{\alpha}(G,\chi,\phi) \leq B(G,\chi,\phi) \leq \alpha(G,\chi,\phi). \end{equation} The left inequality holds because of the nonnegativity of the $k_\tau$ in
\eqref{wildbound}. The right inequality holds because of \eqref{tameidentity}.
A central theme of \cite{jr-tame-wild}
is that one is often but not always
in the extreme situation
\begin{equation}
\label{ba}
B(G,\chi,\phi) = \alpha(G,\chi,\phi).
\end{equation}
For example, it often occurs in practice that
the minimum in the expression \eqref{alphaprod} for $\what{\alpha}(G,\chi,\phi)$
occurs at a $\tau$ of prime order. Then the proportionality
remark above shows that in fact
all three quantities in \eqref{aba} are
the same, and so in particular \eqref{ba} holds.
As a quite different example, Theorem~7.3 of \cite{jr-tame-wild}
says that if $\phi$ is the regular character of $G$ and $\chi$ is a
permutation character, then
\eqref{ba} holds. Some other examples of \eqref{ba} are worked out in
\cite{jr-tame-wild} by explicit analysis of wild ramification;
a few examples show that $B(G,\chi,\phi) < \alpha(G,\chi,\phi)$ is possible too.
\subsection{Root conductor relations} To switch the focus from conductors to root conductors, we multiply all three quantities in \eqref{aba} by $\phi(e)/\chi(e)$ to obtain \begin{equation} \label{aba2} \walp(G,\chi,\phi) \leq b(G,\chi,\phi) \leq \alp(G,\chi,\phi). \end{equation} Here the elementary purely group-theoretic quantity $\walp(G,\chi,\phi)$ is improved to the best bound $b(G,\chi,\phi)$ which in turn often agrees with a second more complicated but still purely group-theoretic quantity $\alp(G,\chi,\phi)$. The notations $\what{\alpha}$, $\alpha$, $\what{\underline{\alpha}}$, $\underline{\alpha}$ are all taken from Section~7 of \cite{jr-tame-wild} while the notations $B$ and $b$ correspond to quantities not named in \cite{jr-tame-wild}.
Our discussion establishes the following lemma. \begin{lemma} \label{lemma2} The conductor relation \begin{equation} \label{maincondrel} \delta_\Chi \geq \delta_\Phi^{b(G,\chi,\phi)} \end{equation} holds for all pairs of Artin characters $(\Chi,\Phi)$ with joint type of the form $(G,c,\chi,\phi)$. \end{lemma}
\subsection{Bounds via an auxiliary Artin character $\Phi$.} \label{bounds} For $u \in \{\walp,b,\alp\}$, define \begin{equation} \label{mdef} m(G,c,\chi,\phi,u) = M(\phi(e),\phi(c),(\phi,1))^{u(G,\chi,\phi)}. \end{equation} Just like Lemma~\ref{lowboundlem} combined with Lemma~\ref{lemma1} to give Theorem~\ref{thm1}, so too Lemma~\ref{lowboundlem} combines with Lemma~\ref{lemma2} to give the following theorem. \begin{thm} \label{thm2} The lower bound \begin{equation} \label{thm2bound} \delta_\Chi \geq m(G,c,\chi,\phi,b) \end{equation} is valid for all character pairs $(\Chi,\Phi)$ of joint type $(G,c,\chi,\phi)$ such that $\Phi$ is non-negative and $L(\Phi,s)$ satisfies the Artin conjecture and the Riemann hypothesis. \end{thm} Computing the right side of \eqref{thm2bound} is difficult because the base in \eqref{mdef} requires evaluating the maximum of a complicated function, while the exponent $b(G,\chi,\phi)$ involves an exhaustive study of wild ramification. Almost always in the sequel, $\chi$ and $\phi$ are rational-valued and we replace $b(G,\chi,\phi)$ by $\walp(G,\chi,\phi)$; in the common case that all three quantities of \eqref{aba2} are equal, this is no loss.
\section{Four choices for $\phi$} \label{choices} This section fixes a type $(G,c,\chi)$ where the faithful character $\chi$ is rational-valued and uses the notation $(n,r) = (\chi(e),\chi(c))$. The section introduces four nonnegative characters $\phi_i$ built from $(G,\chi)$. For the first character $\phi_L$, it makes $m(G,c,\chi,\phi_L,b)$, the lower bound appearing in Theorem~\ref{thm2}, more explicit. For the remaining three characters $\phi_i$, it makes the perhaps smaller quantity $m(G,c,\chi,\phi_i,\walp)$ more explicit.
Two simple quantities enter into the constructions as follows. Let $X$ be the set of values of $\chi$, so that $X \subset \Z$ by our rationality assumption. Let $-\widecheck{\chi}$ be the least element of $X$. The greatest element of $X$ is of course $\chi(e)=n$, and we let $\widehat{\chi}$ be the second greatest element. Thus, $-\widecheck{\chi} <0 \leq \widehat{\chi} \leq n-1$.
\subsection{Linear auxiliary character} A simple nonnegative character associated to $\chi$ is $\phi_L = \chi+\widecheck{\chi}$. Both $m(G,c,\chi,\phi_L,\walp)$ and $m(G,c,\chi,\phi_L,\alp)$ easily evaluate to \begin{equation} \label{formlinear} m(G,c,\chi,\phi_L,b) = M(n+\widecheck{\chi},r+\widecheck{\chi},\widecheck{\chi})^{(n + \widecheck{\chi})/n}. \end{equation} The character $\phi_L$ seems most promising as an auxiliary character when $\widecheck{\chi}$ is very small.
In \cite[\S3]{PM} the auxiliary character $\chi+n$ is used, which has the advantage of being nonnegative for any rational character $\chi$. Odlyzko also uses $\chi+n$ in \cite{odlyzko-durham}, and suggests using the auxiliary character $\phi_L = \chi+\widecheck{\chi}$ since it gives a better bound whenever $\widecheck{\chi}<n$. This strict inequality holds exactly when the center of $G$ has odd order.
\subsection{Square auxiliary character} Another very simple nonnegative character associated to $\chi$ is $\phi_S = \chi^2$. This character gives \begin{equation} \label{formsquare} m(G,c,\chi,\phi_S,\walp) = M(n^2,r^2,(\chi,\chi))^{n/(n+\widehat{\chi})}. \end{equation} The derivation of \eqref{formsquare} uses the simple formula in \eqref{twosimilar} for $\what{c}_\tau$. The formula for $c_\tau$ in \eqref{twosimilar} is more complicated, and we do not expect a simple general formula for $m(G,c,\chi,\phi_S,{\alp})$, nor for the best bound $m(G,c,\chi,\phi_S,b)$ in Theorem~\ref{thm2}.
The character $\phi_S$ is used prominently in \cite{odlyzko-durham, PM}. When $\widehat{\chi}=n-2$, the lower bound $m(G,c,\chi,\phi_S,\walp)$ coincides with that of Lemma~\ref{lemma1}. Thus for $\widehat{\chi}=n-2$, Theorem~\ref{thm2} with $\phi = \phi_S$ gives the same bound as Theorem~\ref{thm1}. On the other hand, as soon as $\widehat{\chi}<n-2$, Theorem~\ref{thm2} with $\phi = \phi_S$ is stronger. The remaining case $\widehat{\chi}=n-1$ occurs only three times among the $195$ characters we consider in Section~\ref{Tables}. In these three cases, the bound in Theorem~\ref{thm1} is stronger because the exponent is larger. However, in each of these cases, the tame-wild principle applies \cite{jr-tame-wild} and we can use exponent $m(G,c,\chi,\phi_S,{\alp})$, which gives the same bound as Theorem~\ref{thm1} in two cases, and a better bound in the third.
\subsection{Quadratic auxiliary character} Let $-\widetilde{\chi}$ be the greatest negative element of the set $X$ of values of $\chi$. A modification of the given character $\chi$ is $\chi^* = \chi+\widetilde{\chi}$, with degree $n^* = n+\widetilde{\chi}$ and signature $r^* = r + \widetilde{\chi}$. A modification of $\phi_S$ is $\phi_Q = \chi \chi^*$. The function $\phi_Q$ takes only nonnegative values because the interval $(-\tilde{\chi},0)$ in the $x$-line where $x(x+\tilde{\chi})$ is negative is disjoint from the set $X$ of values of $\chi$. The lower bound associated to $\phi_Q$ is \begin{equation} \label{formquad} m(G,c,\chi,\phi_Q,\walp) = M(nn^*,rr^*,(\chi,\chi))^{(n^*)/(n^*+\widehat{\chi})}. \end{equation} Comparing formulas \eqref{formsquare} and \eqref{formquad}, $n^2$ strictly increases to $nn^*$ and $n/(n+\widehat{\chi})$ increases to $n^*/(n^* + \widehat{\chi})$. In the totally real case $n=r$, the monotonicity of the function $M(n,n)$ as exhibited in the right edge of Figure~\ref{contourM} then implies that $m(G,c,\chi,\phi_S,\walp)$ strictly increases to $m(G,c,\chi,\phi_Q,\walp)$. Even outside the totally real setting, one can expect that $\phi_Q$ almost always yields a better lower bound than $\phi_S$. The character $\phi_Q$ seems promising as an auxiliary character when $\widehat{\chi}$ is very small so that the exponent is near $1$ rather than its lower limit of $1/2$. As for the square case, we do not expect a simple formula for the best bound $m(G,c,\chi,\phi_Q,b)$ in Theorem~\ref{thm2}.
\subsection{Galois auxiliary character} Finally there is a strong candidate for a good auxiliary character that does not depend on $\chi$, namely the regular character $\phi_G$. By definition,
$\phi_G(e) = |G|$ and else $\phi_G(g)=0$. In this case one has \begin{equation} \label{formula3a}
m(G,c,\chi,\phi_G,\walp) = M(|G|,\delta_{ce}|G|,1)^{(n-\widehat{\chi})/n}. \end{equation} Here $\delta_{ce}$ is defined to be $1$ in the totally real case and $0$ otherwise. This auxiliary character again seems most promising when $\widehat{\chi}$ is small. As in the square and quadratic cases, we do not expect a simple formula for $m(G,c,\chi,\phi_G,b)$.
\subsection{Spectral bounds and rationality} \label{spectralwidth} To get large lower bounds on root conductors, one wants $\widecheck{\chi}/n$ to be small for \eqref{formlinear} or $\widehat{\chi}/n$ to be small for \eqref{formsquare}--\eqref{formula3a}. The analogous quantities $\widecheck{\chi}_1/n_1$
and $\widehat{\chi}/n_1$ are well-defined for a general real character $\chi_1$, and replacing $\chi_1$ by the sum $\chi$ of its conjugates can substantially reduce them.
For example, let $p$ be a prime congruent to 1 modulo 4, and let $G$ be the simple group $\PSL_2(p)$. Then $G$ has two irrational irreducible characters, say $\chi_1$ and $\chi_2$, both of degree $(p+1)/2$. For each, its set of values is $$\left\{\frac{-\sqrt{p}-1}{2},-1,0,1,\frac{\sqrt{p}-1}{2}, \frac{p+1}{2}\right\}$$ (except that $1$ is missing if $p=5$). However for $\chi = \chi_1+\chi_2$, the set of values is just $\{-2,0,2,p+1\}$. Thus in passing from $\widecheck{\chi}_1/n_1$ to $\widecheck{\chi}/n$, one saves a factor of $\sqrt{p}+1$. Similarly in passing from $\widehat{\chi}_1/n_1$ to $\widehat{\chi}/n$, one saves a factor of $\sqrt{p}-1$.
\section{Other choices for $\phi$} \label{otherchoices} To apply Theorem~\ref{thm2} for a given Galois type $(G,c,\chi)$, one needs to choose an auxiliary character $\phi$. We presented four choices in Section~\ref{choices}. We discuss all possible choices here, using $G=A_4$ and $G=A_5$ as illustrative examples.
\subsection{Rational character tables} As a preliminary, we review the notion of rational character table. Let $G^\sharp = \{C_j\}_{j \in J}$ be the set of power-conjugacy classes in $G$. Let $G^{\rm rat} = \{\chi_i\}_{i \in I}$ be the set of rationally irreducible characters. These sets have the same size $k$ and one has a $k\times k$ matrix $\chi_i(C_j)$, called the rational character table.
\begin{table}[htb] \[
\begin{array}{c|rrr c c|rrrr}
A_4 &1A & 2A & 3AB & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; & A_5 & 1A & 2A & 3A & 5AB \\ \cline{1-4} \cline{6-10} \chi_1 & 1 & 1 & 1 & & \chi_1 & 1 & 1 & 1 & 1 \\ \chi_2 & 2 & 2 & -1 & &\chi_4 & 4 & 0 & 1 & - 1 \\ \chi_3 & 3 & - 1& 0 && \chi_5 & 5 & 1 & -1 & 0 \\
\multicolumn{2}{c}{\;} & &&&\chi_6 & 6 & -2 & 0 & 1 \\ \end{array} \] \caption{\label{ratchartables} Rational character tables for $A_4$ and $A_5$}
\end{table}
Two examples are given in Table~\ref{ratchartables}.
We index characters by their degree, with $I = \{1,2,3\}$ for $A_4$ and $I = \{1,4,5,6\}$ for $A_5$. All characters are absolutely irreducible except for $\chi_2$ and $\chi_{6}$, which each break as a sum of two conjugate irreducible complex characters. We likewise index power-conjugacy classes by the order of a representing element, always adding letters as is traditional. Thus $J = \{1A,2A,3AB\}$ for $A_4$ and $J = \{1A,2A,3A,5AB\}$ for $A_5$, with $3AB$ and $5AB$ each consisting of two conjugacy classes.
\subsection{The polytope $P_G$ of normalized nonnegative characters} \label{polytope-pg} A general real-valued function $\phi \in \R(G^\sharp)$ has an expansion $\sum x_i \chi_i$ with $x_i \in \R$. The coefficients are recovered via inner products, $x_i = (\phi,\chi_i)/(\chi_i,\chi_i)$. Alternative coordinates are given by $y_j = \phi(C_j)$. The $\phi$ allowed for Theorem~\ref{thm2} are the nonzero $\phi$ with the $x_i$ and $y_j$ non-negative integers.
An allowed $\phi$ gives the same lower bound in Theorem~\ref{thm2} as any of its positive multiples $m \phi$. Without getting any new bounds, we can therefore give ourselves the convenience of allowing the $x_i$ and $y_j$ to be nonnegative rational numbers. Similarly, we can extend by continuity to allow the $x_i$ and $y_j$ to be nonnegative real numbers. The allowed $\phi$ then become the cone in $k$-dimensional Euclidean space given by $x_i \geq 0$ and $y_j \geq 0$, excluding the tip of the cone at the origin.
Writing the identity character as $\chi_1$, we can normalize via scaling to $x_1=1$. Writing the identity class as $C_{1A}$, the inequality $y_{1A} \geq 0$ is implied by the other $y_j \geq 0$ and so the variable $y_{1A}$ can be ignored. The polytope $P_G$ of normalized nonnegative characters is then defined by $x_1=1$, the inequalities $x_i \geq 0$ for $i \neq 1$, and inequalities $y_j \geq 0$ for $j \neq 1A$. The point where all the $x_i$ are zero is the unital character $\phi_1$. The point where all the $y_j$ are zero is the regular character $\phi_G$. Thus the $(k-1)$-dimensional polytope $P_G$ is determined by $2k-2$ linear inequalities, with $k-1$ corresponding to non-unital characters and intersecting at $\phi_1$, and $k-1$ corresponding to non-identity classes and intersecting at $\phi_G$.
\begin{table}[htb] \centering \includegraphics{twopolytopes} \caption{\label{twopolytopes} The polytopes $P_{A_4}$ and $P_{A_5}$} \end{table}
Figure~\ref{twopolytopes} continues our two examples. On the left, $P_{A_4}$ is drawn in the $x_2$-$x_3$ plane. The character faces give the coordinate axes and are dashed. The class faces are calculated from columns in the rational character table and are solid.
On the right, a view of $P_{A_5}$ is given in $x_{4}$-$x_{5}$-$x_{6}$ space. The three pairwise intersections of character faces give coordinate axes and are dashed, while all other edges are solid. In this view, the point $\phi_G = \phi_{60} = (4,5,3)$ should be considered as closest to the reader, with the solid lines visible and the dashed lines hidden by the polytope. Note that $P_{A_4}$ has the combinatorics of a square and $P_{A_5}$ has the combinatorics of a cube. While the general $P_G$ is the intersection of an orthant with tip $\phi_1$ and an orthant with tip $\phi_G$, its combinatorics are typically more complicated than $[0,1]^{(k-1)}$. For example, the groups $G=A_6$, $S_5$, $A_7$, and $S_6$, have $k=6$, $7$, $8$, and $11$ respectively; but instead of having $32$, $64$, $128$ and $1024$ vertices, their polytopes $P_G$ have $28$, $40$, $115$, and $596$ vertices respectively.
\subsection{Points in $P_G$} In the previous subsection, we have mentioned already the distinguished vertices $\phi_1$ and $\phi_G$. For every rationally irreducible character, we also have $\phi_{\chi,L} = \chi + \widecheck{\chi}$, $\phi_{\chi,S} = \chi^2$, and $\phi_{\chi,Q} = \chi \chi^*$, as in Section~\ref{choices}.
For every subgroup $H$ of $G$, another element of $P_G$ is the permutation character $\phi_{G/H}$. For $H = G$, this character is just the $\phi_1$ considered before, which is a vertex. Otherwise, a theorem of Jordan, discussed at length in \cite{Ser03}, says that $\phi_{G/H}(C_j)=0$ for at least one $j$; in other words, $\phi_{G/H}$ is on at least one character face. For $A_4$ and $A_5$, there are respectively five and nine conjugacy classes of subgroups, distinguished by their orders. Figures~\ref{twopolytopes} draws the corresponding points, labeled by
$\phi_{|G/H|}$. All four vertices of $P_{A_4}$ and six of the eight vertices of $P_{A_5}$ are of the form $\phi_{N}$. The remaining one $\phi_N$ in $P_{A_4}$ is on an edge, while the remaining three $\phi_N$ in $P_{A_5}$
are on edges as well.
\subsection{The best choice for $\phi$} Given $(G,c,\chi)$ and $u \in \{\walp,b,\alp\}$, let $m(G,c,\chi,u) = \max_{\phi \in P_G} m(G,c,\chi,\phi,u)$. Computing these maxima seems difficult. Instead we vary $\phi$ over a modestly large finite set, denoting the largest bound appearing as $\mathfrak{d}(G,c,\chi,u)$. For most $G$, the set of $\phi$ we inspect consists of all $\phi_{\chi,L}$, $\phi_{\chi,S}$, and $\phi_{\chi,Q}$, all $\phi_{G/H}$ including the regular character $\phi_G$, and all vertices. For some $G$, like $S_7$, there are too many vertices and we exclude them from the list of $\phi$ we try.
For each $(G,\chi)$, we work either with $u=\walp$ or with $u=\alp$, as explained in the ``middle four columns" part of \S\ref{remainingrows}. We then report $\mathfrak{d}(G,\chi) = \min_c \mathfrak{d}(G,c,\chi,u)$ in Section~\ref{Tables}.
\section{The case $G=S_5$} \label{S5}
Our focus in the next two sections is on finding initial segments
$\cL(G,\chi; B)$ of complete lists of Artin $L$-functions, and
in particular on finding the first root conductor $\delta_1(G,\chi)$. It is a question of transferring completeness statements for number fields to completeness statements for Artin $L$-functions via conductor relations. In this section, we explain the process by presenting the case $G=S_5$ in some detail.
\subsection{Different orders on the same set of fields}
Consider the set $\cK$ of isomorphism classes of
quintic fields $K$ over $\Q$ with splitting field
$L/\Q$ having Galois group $\gal(L/\Q) \cong S_5$.
The group
$S_5$ has seven irreducible characters which we index by
degree and an auxiliary label: $\chi_{1a} = 1$, $\chi_{1b}$,
$\chi_{4a}$, $\chi_{4b}$, $\chi_{5a}$, $\chi_{5b}$, and $\chi_{6a}$.
For $\phi$ a permutation character, let $D_\phi(K) = D(K_\phi)$ be the absolute
discriminant of the associated resolvent algebra $K_\phi$ of $K$.
Extending by multiplicativity, functions $D_\chi : \cK \rightarrow \R_{>0}$
are defined for general $\chi = \sum m_n \chi_n$. They do not depend on the
coefficient $m_{1a}$. We follow our practice of often shifting attention to
the corresponding root conductors $\delta_\chi(K) = D_\chi(K)^{1/\chi(e)}$.
\begin{table}[htb] {\renewcommand{4pt}{3pt} \[
\begin{array}{r|rrrrrrr|rrrrrrr|rr} \lambda_5 & 1^5 & 2^2 1 & 31^2 & 5 & 21^3 & 41 & 32 & 1^5 & 2^2 1 & 31^2 & 5 & 21^3 & 41 & 32 & \\ \lambda_6 & 1^6 & 2^2 1^2 & 33 & 51 & 2^3 & 41^2 & 6 & 1^6 & \!\! 2^2 1^2 & 33 & 51 & 2^3 & 41^2 & 6 & \walp(n) & \alp(n) \\ \hline \chi_{1a} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\ \chi_{1b} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & \\ \chi_{4a} & 4 & 0 & 1 & -1 & {\bf 2} & 0 & -1 & 0 & 2 & 2 & 4 & {\bf 1} & 3 & 3 & 0.50 & 0.50 \\ \chi_{4b} & 4 & 0 & {\bf 1} & -1 & -2 & 0 & {\bf 1} & 0 & 2 & {\bf 2} & 4 & 3 & 3 & 3 & 0.75 & 0.75 \\ \chi_{5a} & 5 & {\bf 1} & -1 & 0 & {\bf 1} & -1 & {\bf 1} & 0 & {\bf 2} & 4 & 4 & {\bf 2} & 4 & 4 & 0.80 & 0.80 \\ \chi_{5b} & 5 & {\bf 1} & -1 & 0 & -1 & {\bf 1} & -1 & 0 & {\bf 2} & 4 & 4 & 3 & 3 & 5 & 0.80 & 0.80 \\ \chi_{6a} & 6 & -2 & 0 & {\bf 1} & 0 & 0 & 0 & 0 & 4 & 4 & {\bf 4} & 3 & 5 & 5 & 0.8\overline{3} & 0.8\overline{3} \\ \hline \phi_{120} & 120 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 60 & 80 & 96 & 60 & 90 & 100 & & \\
\end{array} \] } \caption{\label{chartabs5} Standard character table of $S_5$ on the left, with entries $\chi_n(\tau)$; tame table \cite[\S4.3]{jr-tame-wild}, on the right, with entries $c_\tau(\chi_n)$ as defined in \eqref{twosimilar}. } \end{table}
Let $\cK(\chi; B) = \{K \in \cK : \delta_\chi(K) \leq B\}$. Suppose now all
the $m_{n}$ are nonnegative with at least one coefficient
besides $m_{1a}$ and $m_{1b}$ positive. Then $\delta_\chi$ is a
height function in the sense that all the $\cK(\chi; B)$
are finite. Suppressing the secondary phenomenon
that ties among a finite number of fields can occur,
we think of each $\delta_\chi$ as giving an ordering on
the set $\cK$.
The orderings coming from different $\delta_\chi$ can be very different. For example, consider
the field $K \in \cK$ defined by the polynomial $x^5 - 2x^4 + 4x^3 - 4x^2 + 2x - 4$.
This field is the first field in $\cK$ when ordered by the regular character
$\phi_{120} = \sum_n \chi_n(n) \chi_n$. However it is
the $22^{\rm nd}$ field when
ordered by $\phi_6 = 1 + \chi_{5b}$
only the $2298^{\rm th}$ field when ordered by
$\phi_5 = 1+ \chi_{4a}$.
This phenomenon of different orderings on the same set of number fields plays
a prominent role in asymptotic studies \cite{wood}. Here we are interested instead
in initial segments and how they depend on $\chi$. Our formalism lets
us treat any $\chi$. Following the conventions for general $G$ of the next section,
we focus on the five irreducible $\chi$ with
$\chi(e)>1$, thus $\chi_n$ for $n \in \{4a,4b,5a,5b,6a\}$.
\subsection{Computing Artin conductors} To compute general $D_\chi(K)$, one needs to work with enough resolvents of
$K=K_5 = \Q[x]/f_5(x)$. For starters, we have the quadratic resolvent $K_2 = \Q[x]/(x^2-D(K_5))$ and the
Cayley-Weber resolvent $K_6 = \Q[x]/f_6(x)$ \cite{generic,jr-tame-wild}. The other resolvents
we will need are $K_{10} = K_5 \otimes K_2$,
$K_{12} = K_2 \otimes K_6$, and $K_{30} = K_5 \otimes K_6$. Defining
polynomials are obtained for $K_a \otimes K_b$ by the general formula \[ f_{ab}(x) = \prod_{i=1}^a \prod_{j=1}^b (x-\alpha_i-\beta_j), \] where $f_a(x)$ has roots $\alpha_i$ and $f_b(x)$ has roots $\beta_j$. So discriminants $D_2$, $D_5$, $D_6$, $D_{10}$, $D_{12}$, $D_{30}$ are easily computed.
From the character table, the permutation characters $\phi_N$ in question are expressed in the basis $\chi_n$ as on the left in the following display. Inverting, one gets the $\chi_n$ in terms of the $\phi_N$ as on the right. \[ \begin{array}{r@{\:}l@{\:}l@{\qquad}r@{\:}l}
\phi_2 && = 1 + \chi_{1b}, &\chi_{1b} & = -1 + \phi_2,\\
\phi_5 && = 1 + \chi_{4a}, & \chi_{4a} & = -1 + \phi_{5}, \\
\phi_6 && = 1 + \chi_{5b}, & \chi_{4a} & = 1 - \phi_2 - \phi_{5} + \phi_{10}, \\
\phi_{10} & = \phi_{5} \phi_2 & = 1 + \chi_{1b} + \chi_{4a} + \chi_{4b}, &
\chi_{5a} & = 1 - \phi_2 - \phi_6 + \phi_{12}, \\
\phi_{12} & = \phi_6 \phi_2 & = 1 + \chi_{1b} + \chi_{5a} + \chi_{5b}, & \chi_{5b} & = -1 + \phi_6, \\
\phi_{30} & = \phi_5 \phi_6 & = 1 + 2 \chi_{4a} + 2 \chi_{5a} + \chi_{5b} + \chi_{6a}, \!\!\!\! & \chi_{6a} & = 2 \phi_2 - 2 \phi_5 + \phi_6 - 2 \phi_{12} + \phi_{30}. \end{array} \] Conductors $D_n$ belonging to the $\chi_n$ are calculable through these formulas, as e.g.\ $D_{6a} = D_2^2 D_5^{-2} D_6 D_{12}^{-2} D_{30}$.
For all the groups $G$ considered in the next section, we proceeded similarly. Thus we started with rational character tables from {\em
Magma}. We used linear algebra to express rationally irreducible characters in terms of permutation characters. We used {\em Magma} again to compute resolvents and then {\em Pari} to evaluate their discriminants. In this last step, we often confronted large degree polynomials with large coefficients. The discriminant computation was only feasible because we knew {\em a priori} the set of primes dividing the discriminant, and could then easily compute the $p$-parts of the discriminants of these resolvent fields for relevant primes $p$ using {\em Pari/gp} without fully factoring the discriminants of the resolvent polynomials.
{\em Magma}'s Artin representation package computes conductors of Artin representations in a different and more local manner. Presently, it does not compute all conductors in our range because some decomposition groups are too large.
\subsection{Transferring completeness results.} As an initial complete list of fields, we take $\cK(\phi ; 85)$ with $\phi=\phi_G=\phi_{120}$. We know from \cite{jr-global-database} that this set consists of $2080$ fields. We list these fields by increasing discriminant, $K^1$, \dots, $K^{2080}$, with the resolution of ties conveniently not affecting the explicit results appearing in Table~\ref{tablelabel1}.
The quantities of Section~\ref{Type} reappear here, and we will use the abbreviations $\walp(n) = \walp(S_5,\chi_n,\phi)$ and $\alp(n) = \alp(S_5,\chi_n,\phi)$. Since $\phi$ is zero outside of the identity class, the formulas simplify substantially: \begin{align*} \walp(n) & =\frac{\phi(e)}{\chi_n(e)} \min_\tau \frac{\chi_n(e)-\chi_n(\tau)}{\phi(e)-\phi(\tau)} = 1-\max_{\tau} \frac{\chi_n(\tau)}{n}, \\ \alp(n)& = \frac{\phi(e)}{\chi_n(e)} \min_{\tau} \frac{c_\tau(\chi_{n})}{c_\tau(\phi)} = \frac{1}{n} \min_{\tau}
\frac{c_\tau(\chi_{n}) \overline{\tau}}{\overline{\tau} - 1}. \end{align*} For each of the five $n$, the classes contributing to the minima are in bold on Table~\ref{chartabs5}. So, extremely simply, for computing $\walp(n)$ on the left, the largest $\chi_n(\tau)$ besides $\chi_n(e)$ are in bold. For computing $\alp(n)$ on the right, the $c_\tau(\chi_n)$ with $c_\tau(\chi_n)/c_\tau(\phi)$ minimized are put in bold. For the group $S_5$, one has agreement $\walp(n) = \alp(n)$ in all five cases. This equality occurs for 170 of the lines in Tables~\ref{tablelabel1}--\ref{tablelabel8}, with the other possibility $\walp(n) < \alp(n)$ occurring for the remaining 25 lines.
For any cutoff $B$, conductor relations give \[ \cK(\chi_n;B^{\walp(n)}) \subseteq \cK(\phi; B). \] One has an analogous inclusion for general $(G,\chi)$, with $\phi$ again the regular character for $G$. When $G$ satisfies the tame-wild principle of \cite{jr-tame-wild}, the $\walp$ in exponents can be replaced by $\alp$. The group $S_5$ does satisfy the tame-wild principle, but in this case
the replacement has no effect.
The final results are on Table~\ref{tablelabel1}. In particular for $n = 4a$, $4b$, $5a$, $5b$, $6a$ the unique minimizing fields are $K^{103}$, $K^{21}$, $K^{14}$, $K^{6}$, and $K^{12}$, with root conductors approximately $6.33$, $18.72$, $17.78$, $16.27$, and $18.18$. The lengths of the initial segments identified are $45$, $15$, $186$, $592$, and $110$. Note that because of the relations $\phi_5 = 1+ \chi_{4a}$ and $\phi_6 = 1 + \chi_{5b}$, the results for $4a$ and $5b$ are just translation of known minima of discriminants of number fields with Galois groups $5T5$ and $6T14$ respectively. For $4b$, $5b$, $6a$, and the majority of the characters on the tables of the next section, the first root conductor and the entire initial segment are new.
\section{Tables for $84$ groups $G$} \label{Tables} In this section, we present our computational results for small Galois types. For simplicity, we focus on results coming from complete lists of Galois number fields. Summarizing statements are given in \S\ref{twotheorems} and then many more details in \S\ref{table-org}.
\subsection{Lower bounds and initial segments} \label{twotheorems} We consider all groups with a faithful transitive permutation representation in some degree from two to nine, except we exclude the nonsolvable groups in degrees eight and nine. There are $84$ such groups, and we consider all associated Galois types $(G,\chi)$ with $\chi$ a rationally irreducible faithful character. Our first result gives conditional lower bounds:
\begin{thm} \label{bounds-are-right}
For each of the $195$ Galois types $(G,\chi)$ listed in
Tables~\ref{tablelabel1}--\ref{tablelabel8}, the listed value
$\mathfrak{d}$ gives a lower bound for the root conductor of all
Artin representations of type $(G,\chi)$,
assuming the Artin conjecture and Riemann hypothesis for relevant
$L$-functions. \end{thm}
The bounds in Tables~\ref{tablelabel1}--\ref{tablelabel8} are graphed with the best previously known \begin{figure}
\caption{Points $(\chi_1(e),\mathfrak{d}(G,\chi))$ present lower bounds from Tables~\ref{tablelabel1}--\ref{tablelabel8}. The piecewise-linear curve plots lower bounds from \cite{PM}. Both the points and the curve assume the Artin conjecture and Riemann hypothesis for the relevant $L$-functions. }
\label{amalia-plot}
\end{figure} bounds from \cite{PM} in Figure~\ref{amalia-plot}. The horizontal axis represents the dimension $n_1=\chi_1(e)$ of any irreducible constituent $\chi_1$ of $\chi$. The vertical axis corresponds to lower bounds on root conductors. The piecewise-linear curve connects bounds from \cite{PM}, and there is one dot at height $\mathfrak{d}(G,\chi)$ for each $(G,\chi)$ from Tables~\ref{tablelabel1}--\ref{tablelabel8} with $\chi_1(e)\leq 20$. Here we are freely passing back and forth between a rational character $\chi$ and an irreducible constituent $\chi_1$ via $\delta_1(G,\chi) = \delta_1(G,\chi_1)$, which is a direct consequence of \eqref{discequal}.
Not surprisingly, the type-based bounds are larger. In low dimensions $n_1$, some type-based bounds are close to the general bounds, but by dimension $5$ there is a clear separation which widens as the dimension grows. This may in part be explained by the fact that we are only seeing a small number of representations for each of these dimensions. However, as we explain in \S\ref{speculation}, we also expect that the asymptotic lower bound
of $\sqrt{\Omega} \approx 6.7$ \cite{PM} is not optimal, and that this bound is more likely to be at least $\Omega\approx 44.8$.
Our second result unconditionally identifies initial segments: \begin{thm} \label{minima-and-segments-are-right} For $144$ Galois
types $(G,\chi)$, Tables~\ref{tablelabel1}--\ref{tablelabel8} identify
a non-empty initial segment $\cL(G,\chi;B^\beta)$, and in particular
identify the minimal root conductor $\delta_1(G,\chi)$. \end{thm}
\subsection{Tables detailing results on lower bounds and initial segments} \label{table-org} Our tables are organized by the standard doubly-indexed lists of transitive permutation groups $mTj$, with degrees $m$ running from $2$ through $9$. Within a degree, the blocks of rows are indexed by increasing $j$. There is no block to print if $mTj$ has no faithful irreducible characters. For example, there is no block to print for groups having noncyclic center, such as $4T2 = V = C_2 \times C_2$ or $8T9 = D_4 \times C_2$. Also the block belonging to $mTj$ is omitted if the abstract group $G$ underlying $mTj$ has appeared earlier. For example $G=S_4$ has four transitive realization in degrees $m \leq 8$, namely $4T5$, $6T7$, $6T8$, and $8T14$; there is correspondingly a $4T5$ line on our tables, but no $6T7$, $6T8$, or $8T14$ lines.
\begin{table}[htbp] \centering \caption{\label{tablelabel1}Artin $L$-functions with small conductor from groups in degrees $2$, $3$, $4$, and $5$}
\begin{tabular}{l@{\;}r@{\;}c@{\;}c@{\;}|@{\;}r@{\;}r@{\;}c@{}r@{\;}|@{\;}c@{\;}r@{\;}r} $G$& $n_1$ & $z$ & $\Imnaught$ & \multicolumn{1}{c}{$\mathfrak{d}$} & \multicolumn{1}{c}{$\delta_1$} & $\Delta_1$ & pos'n & $\beta$ & $B^\beta$ & \# \\ \hline $C_2$ & 2 & TW & 2, 2 & & 1.73 & $3^*$& & & 100 & 6086\\ 2T1 & $1$ & & $[-1, -1]$ & $2.97_{\ell}$ & 3.00 & $3$ & 1 & $2.00_{\phantom{\text{$\bullet$}}}$ & 10000.00 & 6086\\ \hline $C_3$ & 3 & TW & 2, 2 & & 3.66 & $7^*$& & & 500 & 1772\\ 3T1 & $1$ & $\sqrt{-3}$ & $[-1, -1]$ & $6.93_{\ell}$ & 7.00 & $7$ & 1 & $1.50_{\phantom{\text{$\bullet$}}}$ & 11180.34 & 1772\\ \hline $S_3$ & 6 & TW & 3, 4 & & 4.80 & $23^*$& & & 250 & 24484\\ 3T2 & $2$ & & $[-1, 0]$ & $4.74_{\ell}$ & 4.80 & $23$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 250.00 & 13329\\ \hline $C_4$ & 4 & TW & 3, 4 & & 3.34 & $5^*$& & & 150 & 2668\\ 4T1 & $1$ & $i$ & $[-2, 0]$ & $4.96_{S}$ & 5.00 & $5$ & 1 & $1.33_\bullet$ & 796.99 & 489\\ \hline $D_4$ & 8 & TW & 5, 10 & & 6.03 & $3^*7^*$& & & 150 & 31742\\ 4T3 & $2$ & & $[-2, 0]$ & $5.74_{q}$ & 6.24 & $3\!\cdot\! 13$ & 2 & $1.00_{\phantom{\text{$\bullet$}}}$ & 150.00 & 9868\\ \hline $A_4$ & 12 & TW & 3, 4 & & 10.35 & $2^*7^*$& & & 150 & 846\\ 4T4 & $3$ & & $[-1, 0]$ & $7.60_{q}$ & 14.64 & $2^{6} 7^{2}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 150.00 & 270\\ \hline $S_4$ & 24 & TW & 5, 12 & & 13.56 & $2^*11^*$& & & 150 & 14587\\ 6T8 & $3$ & & $[-1, 1]$ & $8.62_{G}$ & 11.30 & $2^{2} 19^{2}$ & 4 & $0.89_\bullet$ & 85.96 & 779\\ 4T5 & $3$ & & $[-1, 1]$ & $5.49_{p}$ & 6.12 & $229$ & 9 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 1603\\ \hline $C_5$ & 5 & TW & 2, 2 & & 6.81 & $11^*$& & & 200 & 49\\ 5T1 & $1$ & $\zeta_{5}$ & $[-1, -1]$ & $10.67_{\ell}$ & 11.00 & $11$ & 1 & $1.25_{\phantom{\text{$\bullet$}}}$ & 752.12 & 49\\ \hline $D_5$ & 10 & TW & 3, 4 & & 6.86 & $47^*$& & & 200 & 3622\\ 5T2 & $2$ & $\sqrt{5}$ & $[-1, 0]$ & $6.73_{q}$ & 6.86 & $47$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 200.00 & 3219\\ \hline $F_5$ & 20 & TW & 4, 8 & & 11.08 & $2^*5^*$& & & 200 & 3010\\ 5T3 & $4$ & & $[-1, 0]$ & $10.28_{q}$ & 13.69 & $2^{4} 13^{3}$ & 2 & $1.00_{\phantom{\text{$\bullet$}}}$ & 200.00 & 2066\\ \hline $A_5$ & 60 & TW & 4, 8 & & 18.70 & $2^*17^*$& & & 85 & 473\\ 5T4 & $4$ & & $[-1, 1]$ & $8.18_{g}$ & 11.66 & $2^{6} 17^{2}$ & 1 & $0.75_{\phantom{\text{$\bullet$}}}$ & 27.99 & 46\\ 6T12 & $5$ & & $[-1, 1]$ & $10.18_{p}$ & 12.35 & $2^{6} 67^{2}$ & 3 & $0.80_{\phantom{\text{$\bullet$}}}$ & 34.96 & 216\\ 12T33 & $3$ & $\sqrt{5}$ & $[-2, 1]$ & $10.34_{g}$ & 26.45 & $2^{6} 17^{2}$ & 1 & $0.83_{\phantom{\text{$\bullet$}}}$ & 40.54 & 18\\ \hline $S_5$ & 120 & TW & 7, 40 & & 24.18 & $2^*3^*5^*$& & & 85 & 2080\\ 5T5 & $4$ & & $[-1, 2]$ & $6.28_{\ell}$ & 6.33 & $1609$ & 103 & $0.50_{\phantom{\text{$\bullet$}}}$ & 9.22 & 45\\ 10T12 & $4$ & & $[-2, 1]$ & $10.28_{V}$ & 18.72 & $5^{2} 17^{3}$ & 21 & $0.75_{\phantom{\text{$\bullet$}}}$ & 27.99 & 15\\ 10T13 & $5$ & & $[-1, 1]$ & $12.13_{V}$ & 16.27 & $2^{4} 3^{2} 89^{2}$ & 6 & $0.80_{\phantom{\text{$\bullet$}}}$ & 34.96 & 592\\ 6T14 & $5$ & & $[-1, 1]$ & $11.09_{g}$ & 17.78 & $2^{6} 3^{4} 7^{3}$ & 14 & $0.80_{\phantom{\text{$\bullet$}}}$ & 34.96 & 186\\ 20T35 & $6$ & & $[-2, 1]$ & $12.26_{g}$ & 18.18 & $2^{4} 3^{3} 17^{4}$ & 12 & $0.83_{\phantom{\text{$\bullet$}}}$ & 40.54 & 110\\ \end{tabular} \end{table} \begin{table}[htbp] \centering \caption{\label{tablelabel2}Artin $L$-functions of small conductor from sextic groups}
\begin{tabular}{l@{\;}r@{\;}c@{\;}c@{\;}|@{\;}r@{\;}r@{\;}c@{}r@{\;}|@{\;}c@{\;}r@{\;}r} $G$& $n_1$ & $z$ & $\Imnaught$ & \multicolumn{1}{c}{$\mathfrak{d}$} & \multicolumn{1}{c}{$\delta_1$} & $\Delta_1$ & pos'n & $\beta$ & $B^\beta$ & \# \\ \hline $C_6$ & 6 & TW & 4, 6 & & 5.06 & $7^*$& & & 200 & 9609\\ 6T1 & $1$ & $\sqrt{-3}$ & $[-2, 1]$ & $6.93_{P}$ & 7.00 & $7$ & 1 & $1.20_\bullet$ & 577.08 & 617\\ \hline $D_6$ & 12 & TW & 6, 14 & & 8.06 & $3^*5^*$& & & 150 & 46197\\ 6T3 & $2$ & & $[-2, 1]$ & $7.60_{G}$ & 9.33 & $3\!\cdot\! 29$ & 6 & $1.00_\bullet$ & 150.00 & 10242\\ \hline $S_3C_3$ & 18 & & 6, 17 & & 10.06 & $2^*3^*7^*$& & & 200 & 9420\\ 6T5 & $2$ & $\sqrt{-3}$ & $[-2, 1]$ & $5.69_{q*}$ & 7.21 & $2^{2} 13$ & 4 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 503\\ \hline $A_4C_2$ & 24 & & 6, 16 & & 12.31 & $2^*7^*$& & & 150 & 6676\\ 6T6 & $3$ & & $[-3, 1]$ & $7.60_{p}$ & 8.60 & $7^{2} 13$ & 3 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 98\\ \hline $S_3^2$ & 36 & & 9, 69 & & 15.53 & $2^*19^*$& & & 200 & 45117\\ 6T9 & $4$ & & $[-2, 1]$ & $7.98_{q*}$ & 14.83 & $2^{4} 5^{2} 11^{2}$ & 27 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 824\\ \hline $C_3^2{\rtimes}C_4$ & 36 & TW & 5, 16 & & 23.57 & $3^*5^*$& & & 150 & 331\\ 6T10 & $4^{\rlap{\scriptsize{2}}}$ & & $[-2, 1]$ & $7.98_{q*}$ & 17.80 & $2^{11} 7^{2}$ & 2 & $0.75_{\phantom{\text{$\bullet$}}}$ & 42.86 & 33\\ \hline $S_4C_2$ & 48 & & 10, 96 & & 16.13 & $2^*23^*$& & & 150 & 70926\\ 6T11 & $3^{\rlap{\scriptsize{2}}}$ & & $[-3, 1]$ & $6.14_{g}$ & 6.92 & $2^{2} 83$ & 7 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 3694\\ \hline $C_3^2{\rtimes}D_4$ & 72 & TW & 9, 105 & & 21.76 & $3^*11^*$& & & 150 & 8536\\ 6T13 & $4^{\rlap{\scriptsize{2}}}$ & & $[-2, 2]$ & $7.60_{p}$ & 7.90 & $3^{2} 433$ & 52 & $0.50_{\phantom{\text{$\bullet$}}}$ & 12.25 & 41\\ 12T36 & $4^{\rlap{\scriptsize{2}}}$ & & $[-2, 1]$ & $11.29_{P}$ & 23.36 & $3^{5} 5^{2} 7^{2}$ & 18 & $0.75_{\phantom{\text{$\bullet$}}}$ & 42.86 & 106\\ \hline $A_6$ & 360 & TW & 6, 28 & & 31.66 & $2^*3^*$& & & 60 & 26\\ 6T15 & $5^{\rlap{\scriptsize{2}}}$ & & $[-1, 2]$ & $7.71_{\ell}$ & 12.35 & $2^{6} 67^{2}$ & 8 & $0.60_{\phantom{\text{$\bullet$}}}$ & 11.67 & 0\\ 10T26 & $9$ & & $[-1, 1]$ & $17.69_{g}$ & 28.20 & $2^{18} 3^{16}$ & 1 & $0.89_{\phantom{\text{$\bullet$}}}$ & 38.07 & 7\\ 30T88 & $10$ & & $[-2, 1]$ & $18.34_{g}$ & 30.61 & $2^{24} 3^{16}$ & 1 & $0.90_{\phantom{\text{$\bullet$}}}$ & 39.84 & 4\\ 36T555 & $8$ & $\sqrt{5}$ & $[-2, 1]$ & $20.70_{g}$ & 42.81 & $2^{18} 3^{16}$ & 1 & $0.94_{\phantom{\text{$\bullet$}}}$ & 46.45 & 3\\ \hline $S_6$ & 720 & & 11, 596 & & 33.50 & $2^*3^*5^*$& & & 60 & 99\\ 12T183 & $5^{\rlap{\scriptsize{2}}}$ & & $[-3, 2]$ & $8.21_{v}$ & 11.53 & $11^{2} 41^{2}$ & 6 & $0.60_{\phantom{\text{$\bullet$}}}$ & 11.67 & 1\\ 6T16 & $5^{\rlap{\scriptsize{2}}}$ & & $[-1, 3]$ & $6.23_{\ell}$ & 6.82 & $14731$ & 53 & $0.40_{\phantom{\text{$\bullet$}}}$ & 5.14 & 0\\ 10T32 & $9$ & & $[-1, 3]$ & $10.77_{v}$ & \textit{16.60} & $2^{15} 11^{3} 13^{3}$ & 74 & $0.67_{\phantom{\text{$\bullet$}}}$ & 15.33 & 0\\ 20T145 & $9$ & & $[-3, 1]$ & $19.33_{g}$ & 31.25 & $2^{6} 5^{6} 73^{4}$ & 16 & $0.89_{\phantom{\text{$\bullet$}}}$ & 38.07 & 4\\ 30T176 & $10^{\rlap{\scriptsize{2}}}$ & & $[-2, 2]$ & $16.88_{v}$ & 24.22 & $11^{4} 41^{6}$ & 6 & $0.80_{\phantom{\text{$\bullet$}}}$ & 26.46 & 1\\ 36T1252 & $16$ & & $[-2, 1]$ & $22.73_{g}$ & 35.46 & $2^{36} 3^{8} 7^{12}$ & 5 & $0.94_{\phantom{\text{$\bullet$}}}$ & 46.45 & 11\\ \end{tabular} \end{table} \begin{table}[htbp] \centering \caption{\label{tablelabel3}Artin $L$-functions of small conductor from septic groups}
\begin{tabular}{l@{\;}r@{\;}c@{\;}c@{\;}|@{\;}r@{\;}r@{\;}c@{}r@{\;}|@{\;}c@{\;}r@{\;}r} $G$& $n_1$ & $z$ & $\Imnaught$ & \multicolumn{1}{c}{$\mathfrak{d}$} & \multicolumn{1}{c}{$\delta_1$} & $\Delta_1$ & pos'n & $\beta$ & $B^\beta$ & \# \\ \hline $C_7$ & 7 & TW & 2, 2 & & 17.93 & $29^*$& & & 200 & 15\\ 7T1 & $1$ & $\zeta_{7}$ & $[-1, -1]$ & $14.03_{\ell}$ & 29.00 & $29$ & 1 & $1.17_{\phantom{\text{$\bullet$}}}$ & 483.65 & 15\\ \hline $D_7$ & 14 & TW & 3, 4 & & 8.43 & $71^*$& & & 200 & 2078\\ 7T2 & $2$ & $\zeta_7^+$ & $[-1, 0]$ & $8.38_{q}$ & 8.43 & $71$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 200.00 & 1948\\ \hline $C_7{\rtimes}C_3$ & 21 & TW & 3, 4 & & 31.64 & $2^*73^*$& & & 100 & 11\\ 7T3 & $3$ & $\sqrt{-7}$ & $[-1, 0]$ & $25.50_{q}$ & 34.93 & $2^{3} 73^{2}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 100.00 & 8\\ \hline $F_7$ & 42 & TW & 5, 12 & & 15.99 & $2^*7^*$& & & 75 & 342\\ 7T4 & $6$ & & $[-1, 0]$ & $14.47_{q}$ & 18.34 & $11^{3} 13^{4}$ & 2 & $1.00_{\phantom{\text{$\bullet$}}}$ & 75.00 & 287\\ \hline $\textrm{GL}_3(2)$ & 168 & TW & 5, 14 & & 32.25 & $2^*3^*11^*$& & & 45 & 19\\ 42T37 & $3$ & $\sqrt{-7}$ & $[-2, 2]$ & $15.55_{G}$ & 26.06 & $7^{2} 19^{2}$ & 7 & $0.89_\bullet$ & 29.48 & 1\\ 7T5 & $6$ & & $[-1, 2]$ & $9.36_{p}$ & 11.23 & $13^{2} 109^{2}$ & 4 & $0.67_{\phantom{\text{$\bullet$}}}$ & 12.65 & 1\\ 8T37 & $7$ & & $[-1, 1]$ & $14.10_{g}$ & 32.44 & $3^{8} 7^{8}$ & 11 & $0.86_{\phantom{\text{$\bullet$}}}$ & 26.12 & 0\\ 21T14 & $8$ & & $[-1, 1]$ & $14.90_{g}$ & 23.16 & $2^{6} 3^{6} 11^{6}$ & 1 & $0.88_{\phantom{\text{$\bullet$}}}$ & 27.96 & 1\\ \hline $A_7$ & 2520 & & 8, 115 & & 39.52 & $2^*3^*7^*$& & & 45 & 1\\ 7T6 & $6$ & & $[-1, 3]$ & $9.13_{\ell}$ & 12.54 & $3^{6} 73^{2}$ & 26 & $0.50_{\phantom{\text{$\bullet$}}}$ & 6.71 & 0\\ 15T47 & $14$ & & $[-1, 2]$ & $19.39_{g}$ & \textit{36.05} & $3^{24} 53^{6}$ & 4 & $0.86_{\phantom{\text{$\bullet$}}}$ & 26.12 & 0\\ 21T33 & $14$ & & $[-1, 2]$ & $19.39_{g}$ & \textit{31.07} & $3^{18} 17^{10}$ & 2 & $0.86_{\phantom{\text{$\bullet$}}}$ & 26.12 & 0\\ 42T294 & $15$ & & $[-1, 3]$ & $18.18_{v}$ & \textit{35.73} & $2^{12} 3^{20} 7^{12}$ & 1 & $0.80_{\phantom{\text{$\bullet$}}}$ & 21.02 & 0\\ 70 & $10$ & $\sqrt{-7}$ & $[-4, 2]$ & $22.49_{g}$ & \textit{41.21} & $2^{9} 3^{14} 7^{8}$ & 1 & $0.90_{\phantom{\text{$\bullet$}}}$ & 30.75 & 0\\ 42T299 & $21$ & & $[-3, 1]$ & $26.95_{g}$ & \textit{38.33} & $2^{18} 3^{30} 7^{16}$ & 1 & $0.95_{\phantom{\text{$\bullet$}}}$ & 37.54 & 0\\ 70 & $35$ & & $[-1, 1]$ & $\mathit{28.79_{g}}$ & \textit{41.28} & $2^{30} 3^{50} 7^{28}$ & 1 & $0.97_\circ$ & 40.36 & 0\\ \hline $S_7$ & 5040 & & 15, -- & & 40.49 & $2^*3^*5^*$& & & 35 & 0\\ 7T7 & $6$ & & $[-1, 4]$ & $7.50_{\ell}$ & 7.55 & $184607$ & &
$0.33_{\phantom{\text{$\bullet$}}}$ & 3.27 & 0\\ 14T46 & $6$ & & $[-4, 3]$ & $\mathit{7.66_{p}}$ & \textit{17.02} & $2^{2} 7^{5} 19^{2}$ & 194 & $0.50_{\phantom{\text{$\bullet$}}}$ & 5.92 & 0\\ 30T565 & $14$ & & $[-2, 4]$ & $16.32_{p}$ & \textit{26.02} & $2^{20} 53^{8}$ & 2 & $0.71_{\phantom{\text{$\bullet$}}}$ & 12.67 & 0\\ 30T565 & $14$ & & $[-4, 2]$ & $20.24_{g}$ & \textit{30.98} & $2^{14} 71^{9}$ & 46 & $0.86_{\phantom{\text{$\bullet$}}}$ & 21.06 & 0\\ 42T413 & $14$ & & $[-6, 2]$ & $20.24_{g}$ & \textit{38.27} & $2^{20} 3^{12} 11^{10}$ & 6 & $0.86_{\phantom{\text{$\bullet$}}}$ & 21.06 & 0\\ 21T38 & $14$ & & $[-1, 6]$ & $13.12_{p}$ & \textit{22.02} & $2^{24} 3^{12} 29^{4}$ & 170 & $0.57_{\phantom{\text{$\bullet$}}}$ & 7.63 & 0\\ 42T412 & $15$ & & $[-3, 5]$ & $16.96_{p}$ & \textit{32.90} & $3^{12} 5^{5} 11^{13}$ & 24 & $0.67_{\phantom{\text{$\bullet$}}}$ & 10.70 & 0\\ 42T411 & $15$ & & $[-5, 3]$ & $16.56_{g}$ & \textit{29.92} & $2^{30} 3^{12} 17^{6}$ & 3 & $0.80_{\phantom{\text{$\bullet$}}}$ & 17.19 & 0\\ 70 & $20$ & & $[-4, 2]$ & $23.53_{g}$ & \textit{35.18} & $2^{34} 53^{12}$ & 2 & $0.90_{\phantom{\text{$\bullet$}}}$ & 24.53 & 0\\ 42T418 & $21$ & & $[-3, 3]$ & $20.24_{g}$ & \textit{33.42} & $2^{41} 3^{18} 17^{9}$ & 3 & $0.86_{\phantom{\text{$\bullet$}}}$ & 21.06 & 0\\ 84 & $21$ & & $[-3, 1]$ & $28.27_{g}$ & \textit{39.59} & $2^{38} 3^{18} 7^{16}$ & 4 & $0.95_{\phantom{\text{$\bullet$}}}$ & 29.55 & 0\\ 70 & $35$ & & $[-1, 5]$ & $\mathit{25.92_{p}}$ & \textit{40.71} & $2^{61} 3^{30} 7^{28}$ & 4 & $0.86_{\phantom{\text{$\bullet$}}}$ & 21.06 & 0\\ 126 & $35$ & & $[-5, 1]$ & $\mathit{30.23_{g}}$ & \textit{43.26} & $2^{54} 3^{42} 5^{30}$ & &
$0.97_\circ$ & 31.62 & 0\\ \end{tabular} \end{table} \begin{table}[htbp] \centering \caption{\label{tablelabel4}Artin $L$-functions of small conductor from octic groups}
\begin{tabular}{l@{\;}r@{\;}c@{\;}c@{\;}|@{\;}r@{\;}r@{\;}c@{}r@{\;}|@{\;}c@{\;}r@{\;}r} $G$& $n_1$ & $z$ & $\Imnaught$ & \multicolumn{1}{c}{$\mathfrak{d}$} & \multicolumn{1}{c}{$\delta_1$} & $\Delta_1$ & pos'n & $\beta$ & $B^\beta$ & \# \\ \hline $C_8$ & 8 & TW & 4, 8 & & 11.93 & $17^*$& & & 125 & 198\\ 8T1 & $1$ & $\zeta_{8}$ & $[-4, 0]$ & $8.84_{S}$ & 17.00 & $17$ & 1 & $1.14_\bullet$ & 249.15 & 41\\ \hline $Q_8$ & 8 & TW & 5, 10 & & 18.24 & $2^*3^*$& & & 100 & 72\\ 8T5 & $2$ & & $[-2, 0]$ & $26.29_{S}$ & 48.00 & $2^{8} 3^{2}$ & 2 & $1.33_\bullet$ & 464.16 & 41\\ \hline $D_8$ & 16 & TW & 6, 20 & & 9.75 & $5^*19^*$& & & 125 & 6049\\ 8T6 & $2$ & $\sqrt{2}$ & $[-4, 0]$ & $9.07_{q}$ & 9.75 & $5\!\cdot\! 19$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 125.00 & 2296\\ \hline $C_8{\rtimes}C_2$ & 16 & & 7, 24 & & 9.32 & $3^*5^*$& & & 125 & 672\\ 8T7 & $2$ & $i$ & $[-4, 0]$ & $9.07_{q}$ & 15.00 & $3^{2} 5^{2}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 125.00 & 75\\ \hline $QD_{16}$ & 16 & & 6, 20 & & 10.46 & $2^*3^*$& & & 125 & 1664\\ 8T8 & $2$ & $\sqrt{-2}$ & $[-4, 0]$ & $9.07_{q}$ & 16.97 & $2^{5} 3^{2}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 125.00 & 155\\ \hline $Q_8{\rtimes}C_2$ & 16 & & 9, 32 & & 9.80 & $2^*3^*$& & & 100 & 3366\\ 8T11 & $2$ & $i$ & $[-4, 0]$ & $9.07_{q}$ & 10.95 & $2^{3} 3\!\cdot\! 5$ & 3 & $1.00_{\phantom{\text{$\bullet$}}}$ & 100.00 & 825\\ \hline $\textrm{SL}_2(3)$ & 24 & TW & 5, 14 & & 29.84 & $163^*$& & & 250 & 681\\ 24T7 & $2$ & & $[-2, 1]$ & $65.51_{P}$ & 163.00 & $163^{2}$ & 1 & $1.20_\bullet$ & 754.27 & 94\\ 8T12 & $2$ & $\sqrt{-3}$ & $[-4, 1]$ & $8.09_{p}$ & 12.77 & $163$ & 1 & $0.75_{\phantom{\text{$\bullet$}}}$ & 62.87 & 78\\ \hline
& 32 & & 11, 74 & & 13.79 & $2^*5^*$& & & 125 & 11886\\ 8T15 & $4$ & & $[-4, 0]$ & $12.92_{q}$ & 16.12 & $2^{4} 5^{2} 13^{2}$ & 4 & $1.00_{\phantom{\text{$\bullet$}}}$ & 125.00 & 3464\\ \hline
& 32 & & 9, 58 & & 13.56 & $5^*11^*$& & & 125 & 766\\ 8T16 & $4$ & & $[-4, 0]$ & $12.92_{q}$ & 16.58 & $5^{4} 11^{2}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 125.00 & 129\\ \hline $C_4\wr C_2$ & 32 & & 10, 90 & & 13.37 & $2^*5^*$& & & 125 & 2748\\ 8T17 & $2^{\rlap{\scriptsize{2}}}$ & $i$ & $[-4, 2]$ & $\mathit{5.74_{p}}$ & 8.25 & $2^{2} 17$ & 6 & $0.50_\circ$ & 11.18 & 3\\ \hline
& 32 & & 9, 58 & & 14.05 & $2^*$& & & 125 & 2720\\ 8T19 & $4$ & & $[-4, 0]$ & $12.92_{q}$ & 19.03 & $2^{17}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 125.00 & 1282\\ \hline
& 32 & & 17, 806 & & 18.42 & $2^*3^*5^*$& & & 100 & 3284\\ 8T22 & $4$ & & $[-4, 0]$ & $12.92_{q}$ & 20.49 & $2^{4} 3^{2} 5^{2} 7^{2}$ & 3 & $1.00_{\phantom{\text{$\bullet$}}}$ & 100.00 & 1162\\ \hline $\textrm{GL}_2(3)$ & 48 & & 7, 41 & & 16.52 & $2^*43^*$& & & 100 & 2437\\ 24T22 & $2$ & $\sqrt{-2}$ & $[-4, 2]$ & $\mathit{5.74_{v}}$ & \textit{16.82} & $283$ & 2 & $0.50_\circ$ & 10.00 & 0\\ 8T23 & $4$ & & $[-4, 1]$ & $9.07_{p}$ & 9.95 & $3^{4} 11^{2}$ & 4 & $0.75_{\phantom{\text{$\bullet$}}}$ & 31.62 & 99\\ \hline $C_2^3{\rtimes}C_7$ & 56 & TW & 3, 4 & & 17.93 & $29^*$& & & 200 & 28\\ 8T25 & $7$ & & $[-1, 0]$ & $16.10_{q}$ & 17.93 & $29^{6}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 200.00 & 27\\ \hline
& 64 & & 16, -- & & 20.37 & $2^*5^*$& & & 125 & 10317\\ 8T26 & $4^{\rlap{\scriptsize{2}}}$ & & $[-4, 2]$ & $9.07_{p}$ & \textit{12.85} & $3^{2} 5^{2} 11^{2}$ & 7 & $0.50_\circ$ & 11.18 & 0\\ \hline $C_2\wr C_4$ & 64 & & 11, 206 & & 19.44 & $2^*$& & & 125 & 2482\\ 8T27 & $4^{\rlap{\scriptsize{2}}}$ & & $[-4, 2]$ & $9.07_{p}$ & 10.60 & $5^{3} 101$ & 19 & $0.50_{\phantom{\text{$\bullet$}}}$ & 11.18 & 1\\ \hline $C_2 \wr C_2^2$ & 64 & & 16, -- & & 19.41 & $2^*7^*$& & & 125 & 11685\\ 8T29 & $4^{\rlap{\scriptsize{2}}}$ & & $[-4, 2]$ & $9.07_{p}$ & 10.13 & $2^{4} 3^{2} 73$ & 28 & $0.50_{\phantom{\text{$\bullet$}}}$ & 11.18 & 1\\ \end{tabular} \end{table} \begin{table}[htbp] \centering \caption{\label{tablelabel5}Artin $L$-functions of small conductor from octic groups}
\begin{tabular}{l@{\;}r@{\;}c@{\;}c@{\;}|@{\;}r@{\;}r@{\;}c@{}r@{\;}|@{\;}c@{\;}r@{\;}r} $G$& $n_1$ & $z$ & $\Imnaught$ & \multicolumn{1}{c}{$\mathfrak{d}$} & \multicolumn{1}{c}{$\delta_1$} & $\Delta_1$ & pos'n & $\beta$ & $B^\beta$ & \# \\ \hline
& 64 & & 11, 206 & & 19.44 & $2^*$& & & 125 & 1217\\ 8T30 & $4^{\rlap{\scriptsize{2}}}$ & & $[-4, 2]$ & $9.07_{p}$ & \textit{14.57} & $5^{3} 19^{2}$ & 3 & $0.50_\circ$ & 11.18 & 0\\ \hline
& 96 & & 9, 49 & & 34.97 & $2^*5^*13^*$& & & 250 & 5520\\ 8T32 & $4$ & & $[-4, 1]$ & $9.12_{g}$ & 22.80 & $2^{6} 5^{2} 13^{2}$ & 2 & $0.75_{\phantom{\text{$\bullet$}}}$ & 62.87 & 180\\ 24T97 & $4$ & $\sqrt{-3}$ & $[-8, 1]$ & $\mathit{13.19_{g}}$ & 43.30 & $2^{6} 5^{2} 13^{3}$ & 2 & $0.88_\circ$ & 125.37 & 112\\ \hline
& 96 & & 8, 44 & & 30.01 & $2^*5^*7^*$& & & 150 & 791\\ 8T33 & $6^{\rlap{\scriptsize{2}}}$ & & $[-2, 2]$ & $11.29_{p}$ & 25.14 & $5^{3} 7^{4} 29^{2}$ & 12 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 3\\ \hline
& 96 & & 10, 92 & & 27.28 & $2^*3^*31^*$& & & 110 & 1915\\ 8T34 & $6$ & & $[-2, 2]$ & $11.29_{p}$ & 22.61 & $31^{3} 67^{2}$ & 64 & $0.67_{\phantom{\text{$\bullet$}}}$ & 22.96 & 1\\ \hline $C_2\wr D_4$ & 128 & & 20, -- & & 22.91 & $2^*3^*13^*$& & & 125 & 14369\\ 8T35 & $4^{\rlap{\scriptsize{4}}}$ & & $[-4, 2]$ & $9.07_{p}$ & 9.45 & $5^{2} 11\!\cdot\! 29$ & 110 & $0.50_{\phantom{\text{$\bullet$}}}$ & 11.18 & 9\\ \hline $C_2^3{\rtimes}F_{21}$ & 168 & & 5, 14 & & 31.64 & $2^*73^*$& & & 200 & 342\\ 8T36 & $7$ & & $[-1, 1]$ & $16.06_{p}$ & 21.03 & $2^{6} 73^{4}$ & 1 & $0.86_{\phantom{\text{$\bullet$}}}$ & 93.82 & 120\\ 24T283 & $7$ & $\sqrt{-3}$ & $[-2, 1]$ & $\mathit{20.23_{p}}$ & 38.55 & $2^{6} 7^{11}$ & 2 & $0.93_\circ$ & 136.98 & 81\\ \hline $C_2\wr A_4$ & 192 & & 12, 700 & & 37.27 & $2^*5^*7^*$& & & 250 & 13649\\ 8T38 & $4^{\rlap{\scriptsize{2}}}$ & & $[-4, 2]$ & $8.56_{v}$ & 15.20 & $2^{6} 7^{2} 17$ & 11 & $0.50_{\phantom{\text{$\bullet$}}}$ & 15.81 & 1\\ 24T288 & $4^{\rlap{\scriptsize{2}}}$ & $\sqrt{-3}$ & $[-8, 4]$ & $\mathit{10.84_{p}}$ & \textit{23.95} & $2^{6} 3\!\cdot\! 5\!\cdot\! 7^{3}$ & 66 & $0.50_{\phantom{\text{$\bullet$}}}$ & 15.81 & 0\\ \hline
& 192 & & 13, 559 & & 32.35 & $2^*23^*$& & & 100 & 1193\\ 8T39 & $4^{\rlap{\scriptsize{2}}}$ & & $[-4, 2]$ & $5.74_{p}$ & 8.71 & $2^{4} 359$ & 49 & $0.50_{\phantom{\text{$\bullet$}}}$ & 10.00 & 8\\ 24T333 & $8$ & & $[-8, 1]$ & $\mathit{15.28_{g}}$ & 39.94 & $2^{10} 43^{6}$ & 2 & $0.88_\circ$ & 56.23 & 16\\ \hline
& 192 & & 13, 559 & & 29.71 & $2^*23^*$& & & 100 & 2001\\ 8T40 & $4^{\rlap{\scriptsize{2}}}$ & & $[-4, 2]$ & $\mathit{5.74_{p}}$ & \textit{13.04} & $2^{2} 5^{2} 17^{2}$ & 9 & $0.50_\circ$ & 10.00 & 0\\ 24T332 & $8$ & & $[-8, 1]$ & $\mathit{15.28_{g}}$ & 29.71 & $2^{12} 23^{6}$ & 1 & $0.88_\circ$ & 56.23 & 47\\ \hline
& 192 & & 14, 1210 & & 28.11 & $2^*11^*$& & & 100 & 4723\\ 12T108 & $6^{\rlap{\scriptsize{2}}}$ & & $[-2, 2]$ & $\mathit{11.29_{p}}$ & 20.78 & $2^{12} 3^{9}$ & 5 & $0.67_{\phantom{\text{$\bullet$}}}$ & 21.54 & 2\\ 8T41 & $6^{\rlap{\scriptsize{2}}}$ & & $[-2, 2]$ & $11.29_{p}$ & 13.01 & $5^{3} 197^{2}$ & 13 & $0.67_{\phantom{\text{$\bullet$}}}$ & 21.54 & 40\\ \hline $A_4\wr C_2$ & 288 & & 10, 178 & & 32.18 & $2^*37^*$& & & 135 & 1362\\ 8T42 & $6$ & & $[-2, 3]$ & $11.29_{p}$ & 11.58 & $5^{3} 139^{2}$ & 76 & $0.50_{\phantom{\text{$\bullet$}}}$ & 11.62 & 1\\ 18T112 & $9$ & & $[-3, 1]$ & $\mathit{17.10_{g}}$ & 35.11 & $2^{24} 13^{6}$ & 2 & $0.89_{\phantom{\text{$\bullet$}}}$ & 78.28 & 66\\ 12T128 & $9$ & & $[-3, 3]$ & $13.59_{p}$ & 22.52 & $7^{6} 233^{3}$ & 56 & $0.67_{\phantom{\text{$\bullet$}}}$ & 26.32 & 6\\ 24T703 & $6$ & $\sqrt{-3}$ & $[-4, 4]$ & $\mathit{13.79_{p}}$ & \textit{32.18} & $2^{4} 37^{5}$ & 1 & $0.67_{\phantom{\text{$\bullet$}}}$ & 26.32 & 0\\ \hline $C_2 \wr S_4$ & 384 & & 20, -- & & 31.38 & $5^*197^*$& & & 100 & 6400\\ 8T44 & $4^{\rlap{\scriptsize{4}}}$ & & $[-4, 2]$ & $5.74_{p}$ & 7.53 & $5\!\cdot\! 643$ & 391 & $0.50_{\phantom{\text{$\bullet$}}}$ & 10.00 & 26\\ 24T708 & $8^{\rlap{\scriptsize{2}}}$ & & $[-8, 4]$ & $\mathit{13.19_{p}}$ & \textit{25.55} & $2^{12} 5^{2} 11^{6}$ & 4 & $0.50_{\phantom{\text{$\bullet$}}}$ & 10.00 & 0\\ \end{tabular} \end{table} \begin{table}[htbp] \centering \caption{\label{tablelabel6}Artin $L$-functions of small conductor from octic groups}
\begin{tabular}{l@{\;}r@{\;}c@{\;}c@{\;}|@{\;}r@{\;}r@{\;}c@{}r@{\;}|@{\;}c@{\;}r@{\;}r} $G$& $n_1$ & $z$ & $\Imnaught$ & \multicolumn{1}{c}{$\mathfrak{d}$} & \multicolumn{1}{c}{$\delta_1$} & $\Delta_1$ & pos'n & $\beta$ & $B^\beta$ & \# \\ \hline
& 576 & & 16, -- & & 29.35 & $2^*3^*$& & & 100 & 2664\\ 12T161 & $6$ & & $[-2, 3]$ & $\mathit{7.60_{p}}$ & \textit{19.04} & $2^{8} 3^{3} 83^{2}$ & 1179 & $0.50_{\phantom{\text{$\bullet$}}}$ & 10.00 & 0\\ 8T45 & $6$ & & $[-2, 3]$ & $7.60_{p}$ & 15.48 & $2^{14} 29^{2}$ & 110 & $0.50_{\phantom{\text{$\bullet$}}}$ & 10.00 & 0\\ 18T179 & $9$ & & $[-3, 1]$ & $18.84_{g}$ & 29.69 & $2^{12} 5^{6} 23^{4}$ & 16 & $0.89_{\phantom{\text{$\bullet$}}}$ & 59.95 & 121\\ 12T165 & $9$ & & $[-3, 3]$ & $10.60_{p}$ & 17.20 & $2^{6} 19^{3} 67^{3}$ & 161 & $0.67_{\phantom{\text{$\bullet$}}}$ & 21.54 & 12\\ 18T185 & $9^{\rlap{\scriptsize{2}}}$ & & $[-3, 3]$ & $\mathit{13.74_{p}}$ & \textit{22.69} & $2^{12} 3^{11} 13^{3}$ & 6 & $0.67_{\phantom{\text{$\bullet$}}}$ & 21.54 & 0\\ 24T1504 & $12$ & & $[-4, 4]$ & $\mathit{16.40_{p}}$ & \textit{35.03} & $2^{8} 5^{6} 31^{8}$ & 51 & $0.67_{\phantom{\text{$\bullet$}}}$ & 21.54 & 0\\ \hline
& 576 & & 11, 522 & & 49.75 & $3^*5^*7^*$& & & 100 & 153\\ 8T46 & $6$ & & $[-2, 3]$ & $9.66_{p}$ & 19.51 & $3^{6} 5^{4} 11^{2}$ & 42 & $0.50_{\phantom{\text{$\bullet$}}}$ & 10.00 & 0\\ 12T160 & $6$ & & $[-2, 3]$ & $\mathit{7.60_{p}}$ & \textit{27.27} & $2^{23} 7^{2}$ & 11 & $0.50_{\phantom{\text{$\bullet$}}}$ & 10.00 & 0\\ 16T1030 & $9$ & & $[-3, 1]$ & $18.84_{g}$ & 35.40 & $2^{22} 3^{6} 13^{4}$ & 6 & $0.89_{\phantom{\text{$\bullet$}}}$ & 59.95 & 50\\ 18T184 & $9$ & & $[-3, 1]$ & $18.84_{g}$ & 35.50 & $2^{12} 5^{7} 23^{4}$ & 7 & $0.89_{\phantom{\text{$\bullet$}}}$ & 59.95 & 32\\ 24T1506 & $12$ & & $[-4, 4]$ & $\mathit{16.40_{p}}$ & \textit{55.16} & $2^{20} 3^{18} 5^{9}$ & 4 & $0.67_{\phantom{\text{$\bullet$}}}$ & 21.54 & 0\\ 36T766 & $9$ & $i$ & $[-6, 2]$ & $\mathit{18.84_{g}}$ & 58.55 & $2^{36} 7^{6}$ & 3 & $0.89_{\phantom{\text{$\bullet$}}}$ & 59.95 & 2\\ \hline $S_4\wr C_2$ & 1152 & & 20, -- & & 35.05 & $2^*5^*41^*$& & & 150 & 23694\\ 12T200 & $6$ & & $[-2, 4]$ & $\mathit{7.60_{p}}$ & \textit{15.62} & $5^{3} 11^{2} 31^{2}$ & 20668 & $0.33_\circ$ & 5.31 & 0\\ 8T47 & $6$ & & $[-2, 4]$ & $7.60_{p}$ & 10.51 & $2^{9} 2633$ & 20566 & $0.33_{\phantom{\text{$\bullet$}}}$ & 5.31 & 0\\ 12T201 & $6$ & & $[-4, 3]$ & $\mathit{7.60_{p}}$ & \textit{19.19} & $3^{7} 151^{2}$ & 21 & $0.50_{\phantom{\text{$\bullet$}}}$ & 12.25 & 0\\ 12T202 & $6$ & & $[-4, 3]$ & $\mathit{7.60_{p}}$ & \textit{16.51} & $3^{10} 7^{3}$ & 12 & $0.50_{\phantom{\text{$\bullet$}}}$ & 12.25 & 0\\ 18T272 & $9$ & & $[-3, 3]$ & $\mathit{9.70_{p}}$ & 19.73 & $3^{6} 853^{3}$ & 450 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 44\\ 18T274 & $9$ & & $[-3, 3]$ & $15.95_{p}$ & 27.07 & $2^{12} 5^{7} 29^{3}$ & 105 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 3\\ 18T273 & $9$ & & $[-3, 3]$ & $\mathit{12.88_{p}}$ & \textit{30.86} & $2^{16} 3^{18}$ & 5 & $0.67_\circ$ & 28.23 & 0\\ 16T1294 & $9$ & & $[-3, 3]$ & $10.60_{p}$ & 13.16 & $43^{3} 53^{3}$ & 39 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 295\\ 36T1946 & $12$ & & $[-4, 4]$ & $\mathit{14.77_{p}}$ & \textit{32.80} & $2^{10} 13^{7} 17^{6}$ & 16 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 0\\ 24T2821 & $12$ & & $[-4, 4]$ & $\mathit{16.03_{p}}$ & 26.48 & $2^{16} 5^{6} 41^{5}$ & 1 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 1\\ 36T1758 & $18$ & & $[-6, 2]$ & $\mathit{20.29_{g}}$ & 36.08 & $2^{24} 5^{9} 41^{9}$ & 1 & $0.89_{\phantom{\text{$\bullet$}}}$ & 85.96 & 1222\\ \end{tabular} \end{table} \begin{table}[htbp] \centering \caption{\label{tablelabel7}Artin $L$-functions of small conductor from nonic groups}
\begin{tabular}{l@{\;}r@{\;}c@{\;}c@{\;}|@{\;}r@{\;}r@{\;}c@{}r@{\;}|@{\;}c@{\;}r@{\;}r} $G$& $n_1$ & $z$ & $\Imnaught$ & \multicolumn{1}{c}{$\mathfrak{d}$} & \multicolumn{1}{c}{$\delta_1$} & $\Delta_1$ & pos'n & $\beta$ & $B^\beta$ & \# \\ \hline $C_9$ & 9 & TW & 3, 4 & & 13.70 & $19^*$& & & 200 & 48\\ 9T1 & $1$ & $\zeta_{9}$ & $[-3, 0]$ & $17.02_{Q}$ & 19.00 & $19$ & 1 & $1.13_\bullet$ & 387.85 & 26\\ \hline $D_9$ & 18 & TW & 4, 8 & & 12.19 & $2^*59^*$& & & 200 & 699\\ 9T3 & $2$ & $\zeta_9^+$ & $[-3, 0]$ & $9.70_{q}$ & 14.11 & $199$ & 3 & $1.00_{\phantom{\text{$\bullet$}}}$ & 200.00 & 638\\ \hline $3^{1+2}_{-}$ & 27 & & 6, 12 & & 31.18 & $7^*13^*$& & & 200 & 32\\ 9T6 & $3$ & $\sqrt{-3}$ & $[-3, 0]$ & $30.32_{q}$ & 38.70 & $7^{3} 13^{2}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 200.00 & 14\\ \hline $3^{1+2}_{+}$ & 27 & TW & 6, 12 & & 50.20 & $3^*19^*$& & & 200 & 16\\ 9T7 & $3$ & $\sqrt{-3}$ & $[-3, 0]$ & $30.32_{q}$ & 64.08 & $3^{6} 19^{2}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 200.00 & 12\\ \hline $3^{1+2}.2$ & 54 & & 7, 34 & & 17.01 & $2^*3^*$& & & 200 & 981\\ 9T10 & $6$ & & $[-3, 0]$ & $15.90_{q}$ & 17.49 & $31^{5}$ & 2 & $1.00_{\phantom{\text{$\bullet$}}}$ & 200.00 & 741\\ \hline
& 54 & & 7, 34 & & 16.83 & $3^*7^*$& & & 200 & 880\\ 9T11 & $6$ & & $[-3, 0]$ & $15.90_{q}$ & 19.01 & $3^{9} 7^{4}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 200.00 & 805\\ \hline
& 54 & & 8, 42 & & 16.72 & $2^*3^*5^*$& & & 200 & 2637\\ 9T12 & $3$ & $\sqrt{-3}$ & $[-3, 2]$ & $7.75_{p}$ & 10.71 & $2^{2} 307$ & 7 & $0.67_{\phantom{\text{$\bullet$}}}$ & 34.20 & 256\\ 18T24 & $3$ & $\sqrt{-3}$ & $[-3, 1]$ & $\mathit{10.03_{g}}$ & 20.08 & $2^{2} 3^{4} 5^{2}$ & 1 & $0.83_\circ$ & 82.70 & 77\\ \hline $M_9$ & 72 & & 6, 20 & & 29.72 & $2^*3^*$& & & 100 & 27\\ 9T14 & $8$ & & $[-1, 0]$ & $17.50_{q}$ & 31.59 & $2^{24} 3^{10}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 100.00 & 26\\ \hline $C_3^2{\rtimes}C_8$ & 72 & TW & 5, 16 & & 25.41 & $2^*3^*$& & & 100 & 19\\ 9T15 & $8$ & & $[-1, 0]$ & $17.50_{q}$ & 25.41 & $2^{31} 3^{4}$ & 1 & $1.00_{\phantom{\text{$\bullet$}}}$ & 100.00 & 16\\ \hline $C_3\wr C_3$ & 81 & & 9, 59 & & 75.41 & $3^*19^*$& & & 500 & 131\\ 9T17 & $3^{\rlap{\scriptsize{3}}}$ & $\sqrt{-3}$ & $[-3, 3]$ & $12.92_{q}$ & 30.14 & $7^{2} 13\!\cdot\! 43$ & 22 & $0.50_{\phantom{\text{$\bullet$}}}$ & 22.36 & 0\\ \hline $C_3^2{\rtimes}D_6$ & 108 & & 11, 262 & & 22.06 & $3^*23^*$& & & 150 & 12002\\ 18T55 & $6$ & & $[-3, 1]$ & $\mathit{11.98_{g}}$ & 24.38 & $2^{8} 3^{8} 5^{3}$ & 4 & $0.83_\circ$ & 65.07 & 229\\ 9T18 & $6$ & & $[-3, 2]$ & $9.70_{p}$ & 13.46 & $3^{3} 7^{2} 67^{2}$ & 53 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 216\\ \hline $C_3^2{\rtimes}QD_{16}$ & 144 & & 8, 62 & & 23.41 & $3^*7^*$& & & 100 & 488\\ 9T19 & $8$ & & $[-1, 2]$ & $\mathit{12.13_{v}}$ & 25.65 & $3^{13} 7^{6}$ & 1 & $0.75_{\phantom{\text{$\bullet$}}}$ & 31.62 & 14\\ 18T68 & $8$ & & $[-2, 1]$ & $\mathit{14.44_{g}}$ & 25.65 & $3^{13} 7^{6}$ & 1 & $0.88_\circ$ & 56.23 & 48\\ \hline $C_3\wr S_3$ & 162 & & 13, 2004 & & 29.89 & $3^*$& & & 200 & 1617\\ 9T20 & $3^{\rlap{\scriptsize{3}}}$ & $\sqrt{-3}$ & $[-3, 3]$ & $7.75_{p}$ & 11.17 & $7\!\cdot\! 199$ & 23 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 20\\ 18T86 & $3^{\rlap{\scriptsize{3}}}$ & $\sqrt{-3}$ & $[-3, 3]$ & $\mathit{8.23_{v}}$ & \textit{19.48} & $2^{2} 43^{2}$ & 5 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ \hline
& 162 & & 11, 223 & & 24.90 & $2^*3^*5^*$& & & 100 & 597\\ 9T21 & $6^{\rlap{\scriptsize{3}}}$ & & $[-3, 3]$ & $9.70_{p}$ & 15.58 & $5^{2} 83^{3}$ & 2 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ \hline
& 162 & & 10, 205 & & 26.46 & $3^*$& & & 100 & 180\\ 9T22 & $6^{\rlap{\scriptsize{3}}}$ & & $[-3, 3]$ & $9.70_{p}$ & 17.21 & $2^{6} 7^{4} 13^{2}$ & 6 & $0.50_{\phantom{\text{$\bullet$}}}$ & 10.00 & 0\\ \hline $C_3^2{\rtimes}\textrm{SL}_2(3)$ & 216 & & 7, 44 & & 49.57 & $349^*$& & & 100 & 37\\ 9T23 & $8$ & & $[-1, 2]$ & $11.29_{p}$ & 23.39 & $547^{4}$ & 10 & $0.75_{\phantom{\text{$\bullet$}}}$ & 31.62 & 3\\ 24T569 & $8$ & $\sqrt{-3}$ & $[-2, 1]$ & $18.99_{g}$ & 38.84 & $349^{5}$ & 1 & $0.94_{\phantom{\text{$\bullet$}}}$ & 74.99 & 20\\ \hline
& 324 & & 17, -- & & 30.64 & $2^*3^*11^*$& & & 100 & 1816\\ 18T129 & $6^{\rlap{\scriptsize{3}}}$ & & $[-3, 3]$ & $\mathit{9.34_{p}}$ & \textit{22.88} & $2^{9} 23^{4}$ & 16 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ 9T24 & $6^{\rlap{\scriptsize{3}}}$ & & $[-3, 3]$ & $9.70_{p}$ & 15.84 & $3^{7} 5^{2} 17^{2}$ & 399 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ \end{tabular} \end{table} \begin{table}[htbp] \centering \caption{\label{tablelabel8}Artin $L$-functions of small conductor from nonic groups}
\begin{tabular}{l@{\;}r@{\;}c@{\;}c@{\;}|@{\;}r@{\;}r@{\;}c@{}r@{\;}|@{\;}c@{\;}r@{\;}r} $G$& $n_1$ & $z$ & $\Imnaught$ & \multicolumn{1}{c}{$\mathfrak{d}$} & \multicolumn{1}{c}{$\delta_1$} & $\Delta_1$ & pos'n & $\beta$ & $B^\beta$ & \# \\ \hline
& 324 & & 9, 116 & & 29.96 & $2^*3^*$& & & 100 & 107\\ 9T25 & $6$ & & $[-3, 3]$ & $12.73_{p}$ & 22.25 & $2^{6} 3^{8} 17^{2}$ & 59 & $0.50_{\phantom{\text{$\bullet$}}}$ & 10.00 & 0\\ 18T141 & $6$ & & $[-3, 3]$ & $\mathit{9.34_{p}}$ & \textit{30.81} & $3^{8} 19^{4}$ & 9 & $0.50_{\phantom{\text{$\bullet$}}}$ & 10.00 & 0\\ 12T133 & $4$ & $\sqrt{-3}$ & $[-4, 2]$ & $11.15_{g}$ & 19.34 & $2^{6} 3^{7}$ & 1 & $0.75_{\phantom{\text{$\bullet$}}}$ & 31.62 & 10\\ 12T132 & $4^{\rlap{\scriptsize{2}}}$ & $\sqrt{-3}$ & $[-4, 2]$ & $\mathit{11.15_{g}}$ & \textit{33.50} & $2^{6} 3^{9}$ & 1 & $0.75_{\phantom{\text{$\bullet$}}}$ & 31.62 & 0\\ 18T142 & $12$ & & $[-3, 3]$ & $19.68_{p}$ & 30.57 & $2^{18} 3^{26}$ & 3 & $0.75_{\phantom{\text{$\bullet$}}}$ & 31.62 & 2\\ \hline $C_3^2{\rtimes}\textrm{GL}_2(3)$ & 432 & & 10, 206 & & 27.88 & $3^*11^*$& & & 76 & 453\\ 9T26 & $8$ & & $[-1, 2]$ & $11.54_{g}$ & 17.59 & $2^{6} 523^{3}$ & 14 & $0.75_{\phantom{\text{$\bullet$}}}$ & 25.74 & 17\\ 18T157 & $8$ & & $[-2, 2]$ & $\mathit{12.14_{p}}$ & 19.04 & $3^{7} 53^{4}$ & 16 & $0.75_{\phantom{\text{$\bullet$}}}$ & 25.74 & 3\\ 24T1334 & $16$ & & $[-2, 1]$ & $21.27_{g}$ & 26.68 & $3^{26} 11^{10}$ & 1 & $0.94_{\phantom{\text{$\bullet$}}}$ & 57.98 & 134\\ \hline $S_3\wr C_3$ & 648 & & 14, 3706 & & 33.56 & $2^*5^*13^*$& & & 150 & 1677\\ 18T197 & $6^{\rlap{\scriptsize{2}}}$ & & $[-3, 3]$ & $\mathit{9.70_{v}}$ & \textit{19.71} & $7^{4} 29^{3}$ & 1 & $0.50_{\phantom{\text{$\bullet$}}}$ & 12.25 & 0\\ 18T202 & $6$ & & $[-4, 3]$ & $\mathit{9.70_{v}}$ & \textit{27.73} & $2^{6} 3^{9} 19^{2}$ & 49 & $0.50_{\phantom{\text{$\bullet$}}}$ & 12.25 & 0\\ 9T28 & $6$ & & $[-3, 4]$ & $9.70_{p}$ & 12.20 & $3^{8} 503$ & 335 & $0.33_{\phantom{\text{$\bullet$}}}$ & 5.31 & 0\\ 12T176 & $8$ & & $[-4, 2]$ & $12.05_{g}$ & 21.75 & $11^{4} 43^{4}$ & 4 & $0.75_{\phantom{\text{$\bullet$}}}$ & 42.86 & 57\\ 36T1102 & $12$ & & $[-4, 3]$ & $\mathit{15.58_{v}}$ & 29.90 & $2^{6} 5^{10} 13^{8}$ & 1 & $0.75_{\phantom{\text{$\bullet$}}}$ & 42.86 & 2\\ 18T206 & $12$ & & $[-3, 4]$ & $15.23_{v}$ & 21.99 & $2^{6} 7^{10} 29^{4}$ & 10 & $0.67_{\phantom{\text{$\bullet$}}}$ & 28.23 & 8\\ 24T1539 & $8$ & $\sqrt{-3}$ & $[-8, 4]$ & $\mathit{17.07_{v}}$ & \textit{45.71} & $2^{14} 3^{19}$ & 3 & $0.75_{\phantom{\text{$\bullet$}}}$ & 42.86 & 0\\ \hline
& 648 & & 13, 2206 & & 40.81 & $2^*3^*17^*$& & & 200 & 838\\ 9T29 & $6$ & & $[-3, 3]$ & $12.73_{p}$ & 16.62 & $2^{8} 7^{2} 41^{2}$ & 31 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ 18T223 & $6$ & & $[-3, 3]$ & $\mathit{9.70_{v}}$ & \textit{30.14} & $2^{4} 5^{2} 37^{4}$ & 71 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ 24T1527 & $4$ & $\sqrt{-3}$ & $[-4, 2]$ & $12.05_{g}$ & 32.34 & $2^{2} 3^{7} 5^{3}$ & 18 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 43\\ 12T175 & $4$ & $\sqrt{-3}$ & $[-4, 4]$ & $7.60_{p}$ & 9.23 & $11\!\cdot\! 659$ & 164 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 14\\ 36T1131 & $6$ & $\sqrt{-3}$ & $[-6, 6]$ & $\mathit{11.95_{v}}$ & \textit{36.04} & $2^{2} 3^{8} 17^{4}$ & 6 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ 36T1237 & $12$ & & $[-3, 3]$ & $\mathit{17.78_{p}}$ & 44.72 & $2^{22} 3^{7} 17^{8}$ & 1 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 6\\ 18T219 & $12$ & & $[-3, 3]$ & $\mathit{14.05_{v}}$ & 33.23 & $3^{17} 107^{5}$ & 5 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 65\\ 24T1540 & $8$ & $\sqrt{-3}$ & $[-8, 4]$ & $\mathit{17.07_{v}}$ & 49.37 & $2^{10} 3^{15} 7^{4}$ & 2 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 1\\ \hline
& 648 & & 13, 1322 & & 30.37 & $2^*269^*$& & & 200 & 4001\\ 9T30 & $6$ & & $[-3, 3]$ & $9.70_{p}$ & 10.67 & $11^{2} 23^{3}$ & 3 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 1\\ 18T222 & $6$ & & $[-3, 3]$ & $\mathit{9.70_{v}}$ & \textit{23.27} & $31^{3} 73^{2}$ & 57 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ 12T178 & $8$ & & $[-4, 2]$ & $12.05_{g}$ & 18.27 & $2^{10} 59^{4}$ & 10 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 327\\ 12T177 & $8^{\rlap{\scriptsize{2}}}$ & & $[-4, 2]$ & $\mathit{12.05_{g}}$ & 24.98 & $2^{10} 23^{6}$ & 22 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 173\\ 36T1121 & $6$ & $\sqrt{3}$ & $[-6, 6]$ & $\mathit{11.95_{v}}$ & \textit{31.61} & $2^{8} 7^{2} 43^{3}$ & 91 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ 36T1123 & $12$ & & $[-3, 3]$ & $\mathit{17.78_{p}}$ & 28.87 & $2^{35} 5^{10}$ & 2 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 71\\ 18T218 & $12$ & & $[-3, 3]$ & $14.05_{v}$ & 18.33 & $2^{10} 269^{5}$ & 1 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 453\\ \hline $S_3 \wr S_3$ & 1296 & & 22, -- & & 36.26 & $2^*3^*$& & & 200 & 12152\\ 18T320 & $6$ & & $[-3, 4]$ & $\mathit{8.80_{p}}$ & \textit{14.80} & $5\!\cdot\! 23^{3} 173$ & 8562 & $0.33_{\phantom{\text{$\bullet$}}}$ & 5.85 & 0\\ 18T312 & $6$ & & $[-4, 3]$ & $\mathit{8.79_{p}}$ & \textit{17.45} & $3^{5} 11^{2} 31^{2}$ & 343 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ 9T31 & $6$ & & $[-3, 4]$ & $9.70_{p}$ & 10.38 & $31^{2} 1303$ & 10036 & $0.33_{\phantom{\text{$\bullet$}}}$ & 5.85 & 0\\ 18T303 & $6$ & & $[-4, 3]$ & $\mathit{8.79_{p}}$ & \textit{18.34} & $5^{5} 23^{3}$ & 4 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ 12T213 & $8$ & & $[-4, 4]$ & $11.29_{p}$ & 13.38 & $5^{2} 7^{4} 131^{2}$ & 397 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 3\\ 24T2895 & $8$ & & $[-4, 2]$ & $\mathit{12.79_{g}}$ & 30.65 & $2^{8} 3^{4} 5^{6} 7^{4}$ & 77 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 230\\ 18T315 & $12$ & & $[-3, 4]$ & $13.59_{p}$ & 22.81 & $2^{10} 23^{4} 37^{5}$ & 72 & $0.67_{\phantom{\text{$\bullet$}}}$ & 34.20 & 105\\ 36T2216 & $12$ & & $[-3, 4]$ & $\mathit{13.79_{p}}$ & 29.13 & $2^{10} 3^{17} 41^{4}$ & 78 & $0.67_{\phantom{\text{$\bullet$}}}$ & 34.20 & 11\\ 36T2305 & $12$ & & $[-4, 3]$ & $\mathit{15.14_{p}}$ & 31.73 & $2^{18} 331^{5}$ & 58 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 151\\ 36T2211 & $12$ & & $[-6, 6]$ & $\mathit{14.64_{p}}$ & \textit{37.29} & $2^{16} 5^{8} 7^{10}$ & 55 & $0.50_{\phantom{\text{$\bullet$}}}$ & 14.14 & 0\\ 36T2214 & $12$ & & $[-4, 3]$ & $\mathit{15.14_{p}}$ & 32.07 & $2^{22} 3^{24}$ & 11 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 136\\ 24T2912 & $16$ & & $[-8, 4]$ & $\mathit{18.82_{p}}$ & 35.12 & $5^{12} 23^{12}$ & 4 & $0.75_{\phantom{\text{$\bullet$}}}$ & 53.18 & 40\\ \end{tabular} \end{table}
\subsubsection{Top row of the $G$-block.} The top row in the $G$-block is different from the other rows, as it gives information corresponding to the abstract group $G$. Instead of referring to a faithful irreducible character, as the other lines do, many of its entries are the corresponding quantities for the regular character $\phi_G$. The first four entries are a common name for the group $G$ (if there is one), the order
$\phi_G(e) = |G|$, the symbol TW if $G$ is known to have the universal tame-wild property as defined in \cite{jr-tame-wild}, and finally $k,N$. Here, $k$ is the size of the rational character table, and $N$ is number of vertices of the polytope $P_G$ discussed in \S\ref{polytope-pg}, or a dash if we did not compute $N$. The last four entries are the smallest root discriminant of a Galois $G$ field, the factored form of the corresponding discriminant, a cutoff $B$ for which the set $\cK(G;B)$ is known, and the
size $|\cK(G;B)|$.
\subsubsection{Remaining rows of the $G$-block}
\label{remainingrows}
Each remaining line of the $G$-block
corresponds to a type $(G,\chi)$.
However the number of rows in the $G$-block is typically substantially less than the number of faithful irreducible characters of $G$, as we list only one representative of each $\Gal(\overline{\Q}/\Q) \times \mbox{Out}(G)$ orbit of such characters. As an example, $S_6$ has eleven characters, all rational. Of the nine which are faithful, there are three which are fixed by the nontrivial element of $\mbox{Out}(S_6)$ and the others form three two-element orbits. Thus the $S_6$-block has six rows. In general, the information on a $(G,\chi)$ row comes in three parts, which we now describe in turn.
{\em First four columns.} The first column gives the lexicographically first permutation group $mTj$ for which the corresponding permutation character has $\chi$ as a rational constituent. Then $n_1=\chi_1(e)$ is the degree of an absolutely irreducible character $\chi_1$ such that $\chi$ is the sum of its conjugates. The number $n_1$ is superscripted by the size of the $\Out(G)$ orbit of $\chi$, in the unusual case when this orbit size is not $1$. Next, the complex number $z$ is a generator for the field generated by the values of the character $\chi_1$, with no number printed in the common case that $\chi_1$ is rational-valued. The last entry gives the interval $\Imnaught$, where $\widecheck{\chi}$ and $\widehat{\chi}$ are the numbers introduced in the beginning of Section~\ref{choices}.
In the range presented, the data of $mTj$, $n_1$, $z$, and $\Imnaught$
suffice to distinguish Galois types $(G,\chi)$ from each other.
{\em Middle four columns.} The next four columns focus on
minimal root conductors. In the first entry, $\frak d$ is the best
conditional lower bound we obtained for root conductors, and the subscript $i \in \{\ell,s,q,g,p,v\}$ gives information on
the corresponding auxiliary character $\phi$. The first four possibilities refer to the methods of
Section~\ref{choices}, namely {\em linear}, {\em square}, {\em quadratic},
and {\em Galois}. The last two, $p$ and $v$, indicate a {\em permutation}
character and a character coming from a {\em vertex} of the polytope $P_G$.
The best $\phi$ of the ones
we inspect is always at a vertex, except in the three cases on
Table~\ref{tablelabel2}
where $*$ is appended to the subscript.
Capital letters
$S$, $Q$, $G$, $P$, and $V$ also appear as subscripts.
These occur only for groups marked with TW, and indicate
that the tame-wild principle improved the lower bound.
For most groups with fifteen or more classes, it was prohibitive
to calculate all vertices, and the best of the other methods
is indicated.
When the second entry is in roman type, it is the minimal root conductor and the third entry is the minimal conductor in factored form. When the second entry is in italic type, then it is the smallest currently known root conductor. The fourth entry gives the position of the source number field on the complete list ordered by Galois root discriminant. This information lets readers obtain further information from \cite{jr-global-database}, such as a defining polynomial and details on ramification.
{\em Last three columns.} The quantity $\beta$ is
the exponent we are using to pass from Galois number fields to Artin representations.
Writing $\walp = \walp(G,\chi,\phi_G)$
and $\alp = \alp(G,\chi,\phi_G)$,
one has the universal relation $\walp \leq \alp$.
When equality holds then the common number is printed.
To indicate that inequality holds, an extra symbol is printed.
When we know that $G$ satisfies TW then we can use larger
exponent and $\alp_\bullet$ is printed.
Otherwise we use the smaller exponent and $\walp_\circ$ is printed.
The column
$B^\beta$ gives the corresponding upper bound on our
complete list of root conductors. Finally
the column $\#$ gives $|\cL(G,\chi; B^\beta)|$,
the length of the complete list of Artin $L$-functions
we have identified. For the $L$-functions themselves,
we refer to \cite{LMFDB}.
\section{Discussion of tables}
\label{discussion}
In this section, we discuss four topics, each of which makes
specific reference to parts of the tables of the
previous section. Each of the topics also serves the
general purpose of making the
tables more readily understandable.
\subsection{Comparison of first Galois root discriminants and root conductors}
Suppose first, for notational simplicity, that $G$ is a group for which
all irreducible complex characters take rational values only.
When one fixes $K^{\rm gal}$ with $\Gal(K^{\rm gal}/\Q) \cong G$
and lets $\chi$ runs over all the irreducible characters of $G$, the root discriminant $\delta_{\rm Gal}$ is just the
weighted multiplicative average $\prod_\chi = {\delta_\chi^{\chi(e)^2/|G|}}$.
Deviation of a root conductor $\delta_\chi$ from $\delta_{\rm Gal}$ is
caused by nonzero values of $\chi$.
When $\chi(e)$ is large and $\Imnaught$ is small, $\delta_\chi$
is necessarily close to $\delta_{\rm Gal}$. One
can therefore think generally of $\delta_{\rm Gal}$ as a first
approximation to $\delta_{\rm \chi}$. The general principle
of $\delta_{\rm Gal}$ approximating $\delta_{\rm \chi}$
applies to groups $G$ with irrational characters as well.
Our first example of $S_5$ illustrates both how the principle
$\delta_{\rm Gal} \approx \delta_{\chi}$ is reflected in the tables,
and how it tends to be somewhat off in the direction that
$\delta_{\rm Gal} > \delta_{\chi}$.
For a given $K^{\rm gal}$, the variance of its $\delta_\chi$ about its
$\delta_{\rm Gal}$ is substantial and depends on the details
of the ramification in $K^{\rm gal}$. There are many $K^{\rm gal}$
with root discriminant near the minimal root discriminant, all of which
are possible sources of minimal root conductors. It
is therefore expected that the minimal conductors $\delta_1(S_5,\chi) = \min \delta_\chi$
printed in the table, $6.33$, $18.72$, $16.27$, $17.78$, and $18.18$,
are substantially less than the printed
minimal root discriminant $\delta_1(S_5,\phi_{120}) \approx 24.18$. As groups $G$
get larger, one can generally expect tighter clustering
of the $\delta_1(G,\chi)$ about $\delta_1(G,\phi_G)$. One can see
the beginning of this trend in our partial results for
$S_6$ and $S_7$.
\subsection{Known and unknown minimal root conductors} Our method of starting with a complete list of
Galois fields is motivated by the principle from the previous
subsection that the Galois root discriminant $\delta_{\rm Gal}$ is
a natural first approximation to $\delta_\chi$. Indeed, as the tables show via nonzero entries in the $\#$
column, this general method suffices to obtain a non-empty initial segment for most $(G,\chi)$.
As our focus is primarily on the first root conductor $\delta_1 = \delta_1(G,\chi)$, we do not pursue larger
initial segments in these cases.
When the initial segment from our general method is empty, as reported by a $0$ in the $\#$
column, we aim to nonetheless present the minimal root conductor $\delta_1$.
Suppose there are subgroups $H_m \subset H_k \subseteq G$, of the
indicated indices, such that a multiple of the character $\chi$ of interest
is a difference of the corresponding permutation characters: $c \chi = \phi_m - \phi_k$.
Suppose one has the complete list of all degree $m$ fields
corresponding to the permutation representation of $G$ on $G/H_m$ and
root discriminant $\leq B$. Then one can extract the complete
list of $\cL(G,\chi;B^{m/(m-k)})$ of desired Artin $L$-functions.
For example, consider $\chi_5$, the absolutely irreducible $5$-dimensional character of $A_6$. The permutation character for a corresponding sextic field decomposes $\phi_6 = 1+\chi_5$, and so the discriminant of the sextic field equals the conductor of $\chi_5$. As an example with $k>1$,
consider the $6$-dimension character $\chi_6$ for $C_3\wr C_3 = 9T17$, which is the sum of a three-dimensional character and its conjugate. The nonic field has a cubic subfield, and the characters are related by $\phi_9 = \phi_3 + \chi_6$. In terms of conductors, $D_9 = D_3 \cdot D_{\chi_6}$, where $D_9$ and $D_3$ are field discriminants. So, we can determine the minimal conductor of an $L$-function with type $(C_3\wr C_3,\chi_6)$ from a sufficiently long complete list of nonic number fields with Galois group $C_3\wr C_3$.
This method, applied to both old and newer lists presented in \cite{jr-global-database},
accounts for all but one of the $\delta_1$ reported
in Roman type on the same line as a $0$ in the $\#$ column.
The remaining case of an established $\delta_1$ is for the type $(\GL_3(2),\chi_7)$. The
group $\GL_3(2)$ appears on our tables as $7T5$. The permutation
representation $8T37$ has character $\chi_7+1$. Here
the general method says that $\cL(\GL_3(2),\chi_7; 26.12)$ is empty.
It is prohibitive to compute the first octic discriminant by
searching among octic polynomials. In \cite{jr-psl27} we carried out a long search of septic polynomials,
examining all local possibilities giving an octic discriminant at most $30$.
This computation shows that $|\cL(\GL_3(2),\chi_7; 48.76)| = 25$ and in particular identifies
$\delta_1 = 21^{8/7} \approx 32.44$.
The complete lists of Galois fields for a group first appearing in degree $m$ were likewise computed
by searching polynomials in degree $m$, targeting for small $\delta_{\rm Gal}$. This single
search can give many first root conductors at once. For example, the largest groups
on our octic and nonic tables are $S_4 \wr S_2 = 8T47$ and $S_3 \wr S_3 = 9T31$.
In these cases, minimal root conductors were obtained for $5$ of the $10$ and
$7$ of the $12$ faithful $\chi$ respectively. Searches adapted to a particular character
$\chi$ as in the previous paragraph
can be viewed as a refinement of our method, with targeting being
not for small $\delta_{\rm Gal}$ but instead for small $\delta_\chi$.
Many of the italicized entries in the column $\delta_1$ seem
improvable to known minimal root conductors via this refinement.
\subsection{The ratio $\delta_1/{\frak d}$} In all cases on the table, $\delta_1>{\frak d}$. Thus, as expected, we did not encounter a contradiction to the Artin conjecture or the Riemann hypothesis. In some cases on the table, the ratio $\delta_1/{\frak d}$ is quite close to $1$. As two series of examples, consider $S_m$ with its reflection character $\chi_{m-1} = \phi_m-1$, and $D_m$ and the sum $\chi$ of all its faithful $2$-dimensional characters. Then these ratios are as follows:
\[
\begin{array}{r|llllllll}
m & \;\; 2 & \;\; 3 & \; \;4 & \; \; 5 & \; \; 6 & \; \; 7 & \; \;8 & \; \;9 \\
\hline
\delta_1/{\frak d}\mbox{ for } (S_m,\chi_{m-1}) & 1.00 & 1.02 & 1.2 & 1.005 & 1.1 & 1.007 \\ \delta_1/{\frak d} \mbox{ for } (D_m,\chi) & & & 1.09 & 1.02 & 1.23 & 1.006 & 1.07 & 1.45 \\ \end{array}
\]
In the cases with the smallest ratios,
the transition
from no $L$-functions to many $L$-functions is commonly abrupt. For example,
in the case $(S_7,\chi_6)$ the lower bound is ${\frak d} \approx 7.50$ and the first seven
rounded root conductors are $7.55$, $7.60$, $7.61$, $7.62$, $7.64$, $7.66$, and $7.66$.
When the translation from no $L$-functions to many $L$-functions is not
abrupt, but there is an $L$-function with outlyingly small conductor, again
$\delta_1/{\frak d}$ may be quite close to $1$. As an example,
for $(8T25,\chi_7)$, one has ${\frak d} \approx 16.10$ and $\delta_1 = 29^{6/7} \approx 17.93$
yielding $\delta_1/{\frak d} \approx 1.11$. However in this
case the next root conductor is $\delta_2 = 113^{6/7} \approx 57.52$, yielding
$\delta_2/{\frak d} \approx 3.57$. Thus the close agreement is entirely dependent
on the $L$-function with outlyingly small conductor. Even the second
root conductor is somewhat of an outlier as the next three conductors
are $71.70$, $76.39$, and $76.39$, so that already $\delta_3/{\frak d} \approx 4.45$.
There are many $(G,\chi)$ on the table for which the ratio $\delta_1/{\frak d}$ is
around $2$ or $3$.
There is some room for improvement in our analytic lower bounds, for example changing the test function \eqref{Odlyzko}, varying $\phi$ over all of $P_G$, or replacing the exponent $\walp$ with the best possible exponent $b$. However examples like the one in the previous paragraph suggest to us that in many cases the resulting increase in $\frak d$ towards $\delta_1$ would be very small.
\subsection{Multiply minimal fields} Tables~\ref{tablelabel1}--\ref{tablelabel8} make implicit reference to many
Galois number fields, and all necessary complete lists are accessible on
the database \cite{jr-global-database}. Table~\ref{multmin} presents
a small excerpt from this database by giving six polynomials $f(x)$. For each $f(x)$,
Table~\ref{multmin} first gives the Galois group $G$ and the root discriminant $\delta$ of the
splitting field $K^{\rm gal}$. We are highlighting these particular Galois number fields $K^{\rm gal}$ here
because they are {\em multiply minimal}: they each give rise to the minimal root conductor for at least
two different rationally irreducible characters $\chi$. The
degrees of these characters are given in the last column of Table~\ref{multmin}.
\begin{table}[htb]
\[
{\renewcommand{4pt}{4pt}
\begin{array}{crclll}
G & & \!\! \delta \!\! & & \mbox{Polynomial} & \chi(e) \\
\hline
A_5 & 2^{3/2} 17^{2/3} & \!\! \approx \!\! & 18.70 & x^5-x^4+2 x^2-2 x+2 &4,6 \\
A_6 & 2^{13/6} 3^{16/9} & \!\! \approx \!\! & 31.66 & x^6-3 x^4-12 x^3-9 x^2+1 &9,10,16 \\
S_6 & 11^{1/2} 41^{2/3} & \approx & 39.44 & x^6 - x^5 - 4x^4 + 6 x^3 - 6x + 5 & 5,10 \\
\SL_2(3) & 163^{2/3} & \!\! \approx \!\! & 29.83 & x^8+9 x^6+23 x^4+14 x^2+1 & 2,4 \\
8T47 & 2^{31/24} 5^{1/2} 41^{1/2} & \!\! \approx \!\! & 35.05 & x^8-2 x^7+6 x^6-2 x^5+26 x^4&12,18\\
& & & & \; -24 x^3-24 x^2+16 x+4\\
9T19 & 3^{37/24} 7^{3/4} & \!\! \approx \!\! & 23.41 & x^9-3 x^8-3 x^7+12 x^6-21 x^5 &8,8\\
& & & & \;+ 36 x^4-48 x^3+45 x^2-24 x+7 \\ \end{array} }
\]
\caption{\label{multmin} Invariants and defining polynomials for Galois
number fields giving rise to minimal root discriminants for
at least two rationally irreducible characters $\chi$ }
\end{table}
Further information on the characters $\chi$ is given in Tables \ref{tablelabel1}--\ref{tablelabel8}.
An interesting point, evident from repeated $1$'s in the $G$-block on these tables,
is that five of the six fields $K^{\rm gal}$ are also first on the list of $G$ fields
ordered by root discriminant. The exception is
the $S_6$ field on Table~\ref{multmin}, which is only sixth on the
list of Galois $S_6$ fields ordered by root discriminant.
\section{Lower bounds in large degrees} \label{asymp} In this section, we continue our practice of assuming the Artin conjecture and Riemann hypothesis for the relevant $L$-functions. For $n$ a positive integer, let $\Delta_1(n)$ be the smallest root discriminant of a degree $n$ field. As illustrated by Figure~\ref{contourM}, one has, \begin{equation} \label{lowfield} \liminf_{n \to \infty} \Delta_1(n) \geq \Omega \approx 44.7632. \end{equation} Now let $\delta_1(n)$ be the smallest root conductor of an absolutely irreducible degree $n$ Artin representation. Theorem~4.2 of \cite{PM} uses the quadratic method to conclude that
$\delta_1(n) \geq 6.59 e^{(-13278.42/n)^2}$. If one repeats the argument there without concerns for effectivity, one gets
\begin{equation}
\label{lowrep} \liminf_{n \to \infty} \delta_1(n) \geq \sqrt{\Omega} \approx 6.6905. \end{equation} The contrast between \eqref{lowfield} and \eqref{lowrep} is striking, and raises the question of whether $\sqrt{\Omega}$ in \eqref{lowrep} can be increased at least part way to $\Omega$.
\subsection{The constant $\Omega$ as a limiting lower bound.} The next corollary makes use of the extreme character values $\widecheck{\chi}$ and $\widehat{\chi}$ introduced at the beginning of Section~\ref{choices}. It shows that if one restricts the type, then one can indeed increase $\sqrt{\Omega}$ all the way to $\Omega$. We formulate the corollary in the context of rationally irreducible characters, to stay in the main context we have set up. However via \eqref{discequal}, it can be translated to a statement about absolutely irreducible characters.
\begin{cor}
\label{limitcor} Let $(G_k,\chi_k)$ be a sequence of rationally
irreducible Galois types of degree $n_k = \chi_k(e)$ . Suppose that
the number of irreducible constituents $(\chi_k,\chi_k)$ is bounded,
$n_k \rightarrow \infty$, and either \begin{description} \item[A] $\widecheck{\chi}_k/n_k \rightarrow 0$, or \item[B] $\widehat{\chi}_k/n_k \rightarrow 0$. \end{description} Then, assuming the Artin conjecture and Riemann hypothesis for relevant $L$-functions, \begin{equation} \label{limlowerbound} \liminf_{k \to \infty} \delta_1(G_k,\chi_k) \geq \Omega. \end{equation} \end{cor} \begin{proof} For Case A, Theorem~\ref{thm2} using a linear auxiliary character as in \eqref{formlinear} says \[ \delta_1(G_k,\chi_k) \geq M \left( \frac{n_k}{\widecheck{\chi}_k} + 1, \frac{r_k}{\widecheck{\chi}_k} + 1,(\chi_k,\chi_k) \right)^{{1+\widecheck{\chi}_k/{n_k}} }. \] For Case B, Theorem~\ref{thm2} using a Galois auxiliary character as in \eqref{formula3a} says \[
\delta_1(G_k,\chi_k) \geq M \left( |G_k| , 0 ,(\chi_k,\chi_k) \right)^{1 - {\widehat{\chi}}/{n_k}}. \] In both cases, the first argument of $M$ tends to infinity, the second argument does not matter, the third argument does not matter either by boundedness, and the exponent tends to $1$. By \eqref{asymptotic1}, these right sides thus have an infimum limit of at least $\Omega$, giving the conclusion \eqref{limlowerbound}. \end{proof}
For the proof of Case~B, the square auxiliary character would work equally well through \eqref{formsquare}. Also~\eqref{lowfield}, \eqref{lowrep}, and Corollary~\ref{limitcor} could all be strengthened by considering the placement of complex conjugation. For example, when restricting to the totally real case $c=e$, the $\Omega$'s in \eqref{lowfield}, \eqref{lowrep}, and \eqref{limlowerbound} are simply replaced by $\Theta \approx 215.3325$.
\subsection{Four contrasting examples.} Many natural sequences of types are covered by either Hypothesis A or Hypothesis B, but some are not.
Table~\ref{limitcor} summarizes four sequences which we discuss together with some related sequences next.
\begin{table}[htb] { \[
\begin{array}{c| l |cccc|cc}
G_k & \multicolumn{1}{c|}{\chi_k} & \widecheck{\chi}_k & \widehat{\chi}_k & n & |G| & \mbox{A}& \mbox{B} \\ \hline
\PGL_2(k) &\mbox{Steinberg} & 1 & 1 & k & k^3-k & \checkmark & \checkmark \\
S_{k} & \mbox{Reflection} & 1 & k-3 & k-1 & k! & \checkmark \\
2_\epsilon^{1+2k} & \mbox{Spin} & 2^k & 0 & 2^k & 2^{1+2k} & & \checkmark \\
2^k.S_k & \mbox{Reflection} & k & k-2 & k & 2^k k! & &
\end{array} \] } \caption{\label{fourasymp} Four sequences of types, with Corollary~\ref{limitcor} applicable to the first three.} \end{table}
\subsubsection{The group $\PGL_2(k)$ and its characters of degree $k-1$, $k$, and $k+1$.} \label{pglk} In the sequence $(\PGL_2(k),\chi_k)$ from the first line of Table~\ref{fourasymp}, the index $k$ is restricted to be a prime power. The permutation character $\phi_{k+1}$ arising from the natural action of $\PGL_2(k)$ on $\mathbb{P}^1(\mathbb{F}_k)$ decomposes as $1+\chi_k$ where $\chi_k$ is the Steinberg character. Table~\ref{fourasymp} says that the ratios $\widecheck{\chi}_k/n_k$ and $\widehat{\chi}_k/n_k$ are both $1/k$, so Corollary~\ref{limitcor} applies through both Hypotheses A and B.
The conductor of $\chi_k$ is the absolute discriminant of the degree $k+1$ number field with character $\phi_{k+1}$. Thus, in this instance, \eqref{limlowerbound} is already implied by the classical \eqref{lowfield}. However, the other nonabelian irreducible characters $\chi$ of $\PGL_2(k)$ behave very similarly to $\chi_k$. Their dimensions are in $\{k-1,k,k+1\}$ and their values besides $\chi(e)$ are all in $[-2,2]$. So suppose for each $k$, an arbitrary nonabelian rationally irreducible character $\chi_k$ of $\PGL_2(k)$ were chosen, in such a way that the sequence $(\chi_k,\chi_k)$ is bounded. Then Corollary~\ref{limitcor} would again apply through both Hypotheses A and B. But now the $\chi_k$ are not particularly closely related to permutation characters.
\subsubsection{The group $S_{k}$ and its canonical characters.} \label{sk} As with the last example, the permutation character $\phi_{k}$ arising from the natural action of $S_{k}$ on $\{1,\dots,k\}$ decomposes as $1 + \chi_k$ where $\chi_k$ is the reflection character with degree $k-1$. The second line of Table~\ref{fourasymp} shows that Corollary~\ref{limitcor} applies through Hypothesis A. In fact, using the linear auxiliary character underlying Hypothesis A here is essential; the limiting lower bound coming from the square or quadratic auxiliary characters is $\sqrt{\Omega}$, and this lower bound is just $1$ from the Galois auxiliary character.
Again in parallel to the previous example, the familiar sequence $(S_k,\chi_k)$ of types needs to be modified to make it a good illustration of the applicability of Corollary~\ref{limitcor}.
Characters of $S_k$ are most commonly indexed by partitions of $k$, with $\chi_{(k)} =1$, $\chi_{(k-1,1)}$ being the reflection character, and $\chi_{(1,1,\dots,1,1)}$ being the sign character. However an alternative convention is to include explicit reference to the degree $k$ and then omit the largest part of the partition, so that the above three characters have the alternative names $\chi_{k,()}$, $\chi_{k,(1)}$, and $\chi_{k,(1,\dots,1,1)}$. With this convention, one can prove that for any fixed partition $\mu$ of a positive integer $m$, the sequence of types $(G_k,\chi_{k,\mu})$ satisfies Hypothesis A but not B.
The case of general $\mu$ is well represented by the two cases where $m=2$. In these two cases, information in the same format as Table~\eqref{fourasymp} is \[ {\def1.3{1.3}
\begin{array}{c| l |cll}
G_k & \multicolumn{1}{c|}{\chi_k} & \multicolumn{1}{c}{\widecheck{\chi}_k} & \multicolumn{1}{c}{\widehat{\chi}_k} & \multicolumn{1}{c}{n} \\ \hline
S_{k} & \chi_{k,(1,1)} & {\lfloor \frac{k-1}{2} \rfloor }& { \frac{1}{2} (k-2)(k-5)} & {\frac{1}{2} (k-1)(k-2)} \\
S_{k} & \chi_{k,(2)} & 1 & { \frac{1}{2} (k-2)(k-5)} + 1& {\frac{1}{2} (k-1)(k-2)} -1 \\
\end{array}.
} \] Let $X_{k,m}$ be the $S_k$-set consisting of $m$-tuples of distinct elements of $\{1,\dots,k\}$. Then its permutation character $\phi_{k,m}$ decomposes into $\chi_{k,\mu}$ with $\mu$ a partition of an integer $\leq m$. These formulas are uniform in $k$, as in \[ \phi_{k,2} = \chi_{k,(1,1)} + \chi_{k,(2)} + 2 \chi_{k,(1)} + \chi_{k,()}. \] For $\mu$ running over partitions of a large integer $m$, the characters $\chi_{k,\mu}$ can be reasonably regarded as quite far from permutation characters, and they thus serve as a better illustration of Corollary~\ref{limitcor}. The sequences $(S_k,\chi_{k,\mu})$ satisfy Hypothesis A but not B, because $n_k$ and $\widehat{\chi}_k$ grow polynomially as $k^m$, while $\widecheck{\chi}_k$ grows polynomially with degree $<m$.
\subsubsection{The extra-special group $2_\epsilon^{1+2k}$ and its degree $2^k$ character} \label{esk} Fix $\epsilon \in \{+,-\}$. Let $G_k$ be the extra-special $2$-group of type $\epsilon$ and order $2^{1+2k}$, so that $2^{1+2}_+$ and $2^{1+2}_{-}$ are the dihedral and quaternion groups respectively. These groups each have exactly one irreducible character of degree larger than $1$, this degree being $2^k$. There are just three character values, $-2^k$, $0$, and $2^k$. For these two sequences, Corollary~\ref{limitcor} again applies, but now only through Hypothesis B.
\subsubsection{The Weyl group $2^k.S_k$ and its degree $k$ reflection character} The Weyl group $W(B_k) \cong 2^k.S_k$ of signed permutation matrices comes with its defining degree $k$ character $\chi_k$. Here, as indicated by the fourth line of Table~\ref{fourasymp}, neither hypothesis of Corollary~\ref{limitcor} applies.
However the conclusion \eqref{limlowerbound} of Corollary~\ref{limitcor} continues to hold as follows. Relate the character $\chi_k$ in question to the two standard permutation characters of $2^k.S_k$ via $\phi_{2k} = \phi_k + \chi_k$. For a given $2^k.S_k$ field, $D_{\Phi_{2k}}=D_{\Phi_k}D_{\Chi_k}$. But, since $\Phi_k$ corresponds to an index $2$ subfield of the degree $2k$ number field for $\Phi_{2k}$, we have $D_{\Phi_k}^2\mid D_{\Phi_{2k}}$. Combining these we get $D_{\Phi_k} \mid D_{\Chi_k}$ and hence $\delta_{\Phi_k} < \delta_{\Chi_k}$. So \eqref{lowfield} implies \eqref{limlowerbound}.
\subsection{Concluding speculation} \label{speculation} As we have illustrated in \S\ref{pglk}--\ref{esk}, both Hypothesis A and Hypothesis B are quite broad. This breadth, together with the fact that the conclusion \eqref{limlowerbound} still holds for our last sequence, raises the question of whether \eqref{limlowerbound} can be formulated more universally. While the evidence is far from definitive, we expect a positive answer. Thus we expect that the first accumulation point of the numbers $\delta_1(G,\chi)$ is at least $\Omega$, where $(G,\chi)$ runs over all types with $\chi$ irreducible. Phrased differently, we expect that the first accumulation point of the root conductors of all irreducible Artin $L$-functions is at least $\Omega$.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Tilting modules in category $\mathcal{O}$ and sheaves on moment graphs}
\author{Johannes K\"ubel*}
\address{Department of Mathematics, University of Erlangen, Germany}
\curraddr{Cauerstr. 11, 91058 Erlangen, Germany} \email{[email protected]}
\thanks{*supported by the DFG priority program 1388}
\date{\today}
\keywords{representation theory, category $\mathcal{O}$}
\begin{abstract} We describe tilting modules of the deformed category $\mathcal{O}$ over a semisimple Lie algebra as certain sheaves on a moment graph associated to the corresponding block of $\mathcal{O}$. We prove that they map to Braden-MacPherson sheaves constructed along the reversed Bruhat order under Fiebig's localization functor. By this means, we get character formulas for tilting modules and explain how Soergel's result about the Andersen filtration gives a Koszul dual proof of the semisimplicity of subquotients of the Jantzen filtration. \end{abstract}
\maketitle
\section{Introduction}
Let $\mathfrak{g} \supset \mathfrak{b} \supset \mathfrak{h}$ be a complex semisimple Lie algebra with a Borel and a Cartan subalgebra. Let $A$ be the localization of the symmetric algebra $S=S(\mathfrak{h})$ at the maximal ideal of zero. The deformed category $\mathcal{O}_A$ is the full subcategory of $\mathfrak{g}$-$A$-bimodules that are finitely generated over $\mathfrak{g} \otimes_\mathbb{C} A$, semisimple over $\mathfrak{h}$ and locally finite over $\mathfrak{b}$. $\mathcal{O}_A$ decomposes into blocks which are parameterized by antidominant weights. For a given antidominant weight ${\lambda} \in \mathfrak{h}^*$ the weights involved in the corresponding block are given by the orbit $\mathcal{W}_{\lambda} \cdot {\lambda}$ of ${\lambda}$ under the dot-action of the integral Weyl group corresponding to ${\lambda}$. This combinatorial data defines a graph with $\mathcal{W}_{\lambda} \cdot {\lambda}$ the set of vertices being partially ordered by the Bruhat order on $\mathcal{W}_{\lambda}$ divided by the stabilizer of ${\lambda}$. Two different vertices are linked by an edge if there is a reflection of $\mathcal{W}_{\lambda}$ mapping one vertex to the other. In addition, every edge has a labeling given by the coroot corresponding to the positive root of the according reflection. Denote by $\mathcal{M}_{A,{\lambda}}$ the subcategory of the block corresponding to the antidominant weight ${\lambda}$ consisting of modules which have a Verma flag, i.e., a filtration with subquotients isomorphic to deformed Verma modules. Now the usual duality on category $\mathcal{O}$ extends to the deformed version $\mathcal{O}_A$. The modules which are self-dual and admit a Verma flag are called deformed tilting modules. The indecomposable tilting modules are parameterized by their highest weight and $\mathcal{M}_{A,{\lambda}}$ contains those with a highest weight lying in the orbit $\mathcal{W}_{\lambda} \cdot {\lambda}$.\\ While Fiebig shows that indecomposable deformed projective modules of $\mathcal{M}_{A,{\lambda}}$ correspond to Braden-MacPherson sheaves constructed along the Bruhat order of the associated moment graph, we prove in a very similar way that indecomposable deformed tilting modules correspond to Braden-MacPherson sheaves constructed along the reversed order on the moment graph. This approach implies a character formula for tilting modules which was already discovered in \cite{A}. There, Soergel uses a tilting equivalence to trace back character formulas for tilting modules to the known ones for projective modules. Our approach, however, doesn't use the tilting functor but has the disadvantage that it doesn't generalize to Kac-Moody algebras without using the tilting functor.\\ Another application of our result about tilting modules as sheaves on a moment graph is the connection between the Andersen and Jantzen filtration. The Jantzen filtration on a Verma module induces a filtration on the space of homomorphisms from a projective to the Verma module. The Andersen filtration on the space of homomorphisms from a Verma module to a tilting module is constructed in a very similar way as the Jantzen filtration. In \cite{11} we already proved that there is an isomorphism between both spaces of homomorphisms which interchanges the mentioned filtrations.\\ In this paper we describe both Hom-spaces on the level of sheaves on moment graphs. Since the construction of those involves the symmetric algebra $S(\mathfrak{h})$ we discover an inherited grading on both Hom-spaces. Now the advantage of this approach compared to \cite{11} is that we are able to construct an isomorphism which respects the grading and lifts to an isomorphism on the Hom-spaces which also interchanges both filtrations. In \cite{14} it is proved that the Andersen filtration coincides with the grading filtration on this Hom-space. Soergel's approach, however, is Koszul dual to \cite{1} and in combination with our result, leads to another proof of the semisimplicity of the Jantzen filtration layers. \section{Preliminaries}
In this section we repeat results of \cite{7} and \cite{3} about certain sheaves on moment graphs. We mostly follow the lecture notes \cite{10} and \cite{15} which are more introductory to this subject.
\subsection{Moment graphs}
For a vector space $V$ we denote by $S:= S(V)$ the symmetric algebra of $V$ with the usual grading doubled, i.e., $\mathrm{deg} V=2$. Let $\mathcal{V}$ be the set of vertices and $\mathcal{E}$ the set of edges of a finite graph $(\mathcal{V},\mathcal{E})$. I.e., $\mathcal{V}$ is a finite set and $\mathcal{E} \subset \mathcal{P}(\mathcal{V})$ a subset of the power set of $\mathcal{V}$ with the following property:\\ If $E$ is an element of $\mathcal{E}$, then the cardinality of $E$ is two.
\begin{definition} An unordered $V$-moment graph $\mathcal{G}=(\mathcal{V},\mathcal{E}, \alpha)$ is a finite graph $(\mathcal{V},\mathcal{E})$ without loops and double edges, which is equipped with a map $\alpha: \mathcal{E} \rightarrow \mathbb{P}(V)$ that associates to any edge $E$ a line $\alpha_E :=\alpha(E)$ in $V$. \end{definition}
\begin{remark} The subsets $Y \subset \mathcal{V} \cup \mathcal{E}$ with the property:
$$x \in Y \cap \mathcal{V} \Rightarrow \{ E \in \mathcal{E} \,| \, x \in E\} \subset Y$$ form the open sets of a topology on $\mathcal{V} \cup \mathcal{E}$. By this means, we view $\mathcal{G}$ as a topological space. \end{remark}
\subsection{Sheaves on moment graphs}
Let $A$ be a commutative $S$-algebra. For $x \in \mathcal{V}$ define $x^{\circ}:= \{x\} \cup \{E \in \mathcal{E} \,| \, x\in E\}$. For a sheaf $\mathscr{M}$ of $A$-modules on the topological space $\mathcal{G}$ the stalks are given by\\ $\mathscr{M}_x = \mathscr{M}(x^{\circ})$ for $x\in \mathcal{V}$ and $\mathscr{M}_E = \mathscr{M}(\{E\})$ for $E \in \mathcal{E}$.\\ We denote by $\rho_{x,E}: \mathscr{M}_x \rightarrow \mathscr{M}_E$ the restriction map for $x \in E$. The sheaf $\mathscr{M}$ is uniquely determined by this data and we can define a sheaf of rings $\mathscr{A}$, namely the \textit{structure sheaf} of $\mathcal{G}$ over $A$, by setting \begin{itemize} \item $\mathscr{A}_x =A$ $\forall x \in \mathcal{V}$ \item $\mathscr{A}_E = A / \alpha_E A$ $\forall E \in \mathcal{E}$ \item $\rho_{x,E}: \mathscr{A}_x \rightarrow \mathscr{A}_E$ the quotient map for $x \in E$. \end{itemize}
By \cite{3} Proposition 1.1, an $\mathscr{A}$-module $\mathscr{M}$ is characterized by a tuple \\ $(\{\mathscr{M}_x\},\{\mathscr{M}_E\},\{\rho_{x,E}\})$ with the properties \begin{itemize} \item $\mathscr{M}_x$ is an $A$-module for any $x \in \mathcal{V}$ \item $\mathscr{M}_E$ is an $A$-module for all $E \in \mathcal{E}$ with $\alpha_E \mathscr{M}_E =0$ \item $\rho_{x,E}: \mathscr{M}_x \rightarrow \mathscr{M}_E$ is a homomorphism of $A$-modules for $x \in \mathcal{V}$, $E \in \mathcal{E}$ with $x \in E$. \end{itemize}
\begin{remark} In what follows, we will always work with this characterization of sheaves on the moment graph. If the $S$-algebra $A$ is $S$ itself, we consider all modules as graded $S$-modules and all maps between them as graded homomorphisms of degree zero.\\ To distinguish between the $S$-algebras we are working with we sometimes call the sheaf $\mathscr{M}$ an $A$-sheaf. \end{remark}
\subsection{Global sections}
Now let $A$ be a localization of $S$ at a prime ideal $\mathfrak{p} \subset S$. Denote by $\mathcal{SH}_A(\mathcal{G})^{f}$ the subcategory of $\mathscr{A}$-modules, such that $\mathscr{M}_x$ is torsion free and finitely generated over $A$ for all $x \in \mathcal{V}$. We denote by $\mathcal{Z} = \mathcal{Z}_A (\mathcal{G})$ the global sections $\Gamma(\mathscr{A})$ of the structure sheaf and call it the \textit{structure algebra of $\mathcal{G}$ over $A$}. By \cite{7} section 2.5. we get $\mathcal{Z} :=\mathcal{Z}_A(\mathcal{G}) = \{(z_x) \in \prod_{x \in \mathcal{V}} A \, | \, (z_x \equiv z_y \, \mathrm{mod} \, \alpha_E) \, \mathrm{for} \, \{x,y\}=E\}$.
\begin{remark} In case $A=S$, $\mathcal{Z}_S(\mathcal{G})$ carries a grading induced by $S$. In this case we consider all $\mathcal{Z}$-modules as graded modules. \end{remark}
The functor of global sections
\begin{equation*} \Gamma: \mathscr{A}- \mathrm{mod} \longrightarrow \mathcal{Z}-\mathrm{mod} \end{equation*} has a left adjoint, namely the localization functor $\mathcal{L}$. Denote by $\mathcal{Z}_A-\mathrm{mod}^f$ the subcategory of $\mathcal{Z}$-modules that are finitely generated and torsion free over $A$.
\begin{lemma}(\cite{7}, Proposition 3.5.) \label{adj} The functors $\Gamma$ and $\mathcal{L}$ induce a pair of adjoint functors
\[ \begin{xy} \xymatrix{
\mathcal{SH}_A(\mathcal{G})^f \ar@<2pt>[r]^{\Gamma} & \mathcal{Z}_A-\mathrm{mod}^f \ar@<2pt>[l]^{\mathcal{L}} \\ } \end{xy} \] and the canonical maps $\Gamma(\mathscr{M}) \rightarrow \Gamma \mathcal{L} \Gamma (\mathscr{M})$ and $\mathcal{L}(M) \rightarrow \mathcal{L} \Gamma \mathcal{L} (M)$ are isomorphisms. \end{lemma}
\begin{remark} Lemma \ref{adj} implies that we get a pair of mutually inverse equivalences between the images of both functors
\[ \begin{xy} \xymatrix{
\mathcal{L}(\mathcal{Z}_A-\mathrm{mod}^f ) \ar@<2pt>[r]^{\Gamma} & \Gamma(\mathcal{SH}_A(\mathcal{G})^f) \ar@<2pt>[l]^{\mathcal{L}} \\ } \end{xy} \] \end{remark}
We now follow \cite{10} to give a concrete description of $\mathcal{L}$. Let $M \in \mathcal{Z}_A-\mathrm{mod}^f$ and denote by $Q$ the quotient field of $A$. Since $M$ is torsion free over $A$ we get an inclusion $M \hookrightarrow M \otimes_A Q$.\\ Let $\sum_{x \in \mathcal{V}} {e_x} = 1 \otimes 1 \in \mathcal{Z} \otimes_A Q \cong \prod_{x \in \mathcal{V}} Q$ be a decomposition of $1 \in \mathcal{Z} \otimes_A Q$ into idempotents. For $x \in \mathcal{V}$ set $$\mathcal{L}(M)_x = e_xM \subset M \otimes_A Q$$ For an edge $E=\{x,y\}$ with $\alpha:= \alpha(E)$ we set $$M(E) = (e_x + e_y) M + \alpha e_x M \subset e_x(M\otimes_A Q) \oplus e_y (M \otimes_A Q)$$ and form the push-out diagram
\begin{eqnarray*}
\begin{CD}
M(E) @>\pi_x >> \mathcal{L}(M)_x\\
@VV\pi_y V @VV\rho_{x,E} V\\
\mathcal{L}(M)_y @>\rho_{y,E} >> \mathcal{L}(M)_E
\end{CD} \end{eqnarray*}
where $\pi_x$, $\pi_y$ are defined by $\pi_x(z)=e_x z$ and $\pi_y(z) = e_y z$.
This gives the sought after stalk $\mathcal{L}(M)_E$ with restriction maps $\rho_{x,E}$, $\rho_{y,E}$ coming from the push-out diagram.
\subsection{Sheaves on ordered moment graphs}
\begin{definition} An ordered moment graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\alpha, \leq)$ is a moment graph $(\mathcal{V},\mathcal{E},\alpha)$ with a partial order $\leq$ on the set $\mathcal{V}$ of vertices, such that for any $E = \{x,y\}$ the vertices $x, y \in \mathcal{V}$ are comparable. \end{definition}
\begin{definition} An F-open subgraph $\mathcal{H} = (\mathcal{V}',\mathcal{E}', \alpha',\leq')$ of $\mathcal{G}$ is a subgraph with $\alpha'$ and $\leq'$ the restrictions of $\alpha$ and $\leq$, respectively, such that \begin{itemize} \item If $E= \{x,y\}$ and $x,y \in \mathcal{V}'$, then $E \in \mathcal{E}'$ \item If $x \in \mathcal{V}'$ and $y \in \mathcal{V}$ with $y \leq x$, then $y \in \mathcal{V}'$ \end{itemize} \end{definition}
\begin{definition}
An $A$-sheaf $M$ on $\mathcal{G}$ is called F-flabby if for any F-open subgraph $\mathcal{H}$ of $\mathcal{G}$ the restriction map $\Gamma(\mathscr{M}) \rightarrow \Gamma(\mathcal{H}, \mathscr{M})$ is surjective, where $\Gamma(\mathcal{H}, \mathscr{M}):= \{(m_x) \in \prod_{x \in \mathcal{V}'} \mathscr{M}_x \, | \, \rho_{x,E} (m_x) = \rho_{y,E}(m_y) \,\, \mathrm{for}\,\, E=\{x,y\}\}$ \end{definition}
\begin{definition} A moment graph $(\mathcal{G}, \alpha, \leq)$ has the GKM-property if for every pair $E, E' \in \mathcal{E}$ with $E \neq E'$ and $E \cap E' \neq \emptyset$ we have $\alpha(E) \neq \alpha(E')$. \end{definition}
Recall that we decomposed $1 \in \mathcal{Z} \otimes_A Q \cong \prod_{x \in \mathcal{V}} Q$ into idempotents $1= \sum_{x \in \mathcal{V}}e_x$. For an F-open subset $\mathcal{H}$ of $\mathcal{G}$ with vertices $\mathcal{V}'$, we set $e_{\mathcal{H}} := \sum_{x \in \mathcal{V}'} e_x$.
\begin{prop}[\cite{10}, Proposition 3.14] \label{B} Suppose that $\mathcal{G}$ is a GKM-graph. Let $M$ be a finitely generated $\mathcal{Z}$-module that is torsion free over A. Suppose in addition, that $e_{\mathcal{H}}M$ is a reflexive $A$-module for any F-open subgraph $\mathcal{H}$ of $\mathcal{G}$. Then $\mathcal{L}(M)$ is F-flabby on $\mathcal{G}$ and we have an isomorphism $$e_{\mathcal{H}}M \stackrel{\sim}{\longrightarrow} \Gamma({\mathcal{H}}, \mathcal{L}(M))$$ \end{prop}
\subsection{Braden-MacPherson sheaves}
In this section we will repeat the notion of F-projective sheaves on a moment graph and introduce Braden-MacPherson sheaves which form the indecomposable F-projective sheaves. \begin{definition} A sheaf $\mathscr{M}$ on $\mathcal{G}$ is \textit{generated by global sections} if the map $\Gamma(\mathscr{M}) \rightarrow \mathscr{M}_x$ is surjective for every $x \in \mathcal{V}$. \end{definition}
\begin{notation} \label{not1}
For any $x \in \mathcal{V}$ set $D_x := \{E \in \mathcal{E} \,| \, E=\{x,y\}$, $y \in \mathcal{V}$, $y\leq x\}$ and $U_x := \{E \in \mathcal{E} \,|\, E=\{x,y\}$, $y\in \mathcal{V}$, $x \leq y\}$. \end{notation}
\begin{definition} An $A$-sheaf $\mathscr{P}$ on $\mathcal{G}$ is called F-projective if \begin{itemize} \item $\mathscr{P}$ is F-flabby and generated by global sections \item Each $\mathscr{P}_x$ with $x \in \mathcal{V}$ is a free (graded free for $A=S$) $A$-module \item Any $\rho_{x,E}$ with $x \in \mathcal{V}$, $E \in U_x$ induces an isomorphism $\mathscr{P}_x / \alpha_E\mathscr{P}_x \rightarrow \mathscr{P}_E$ of (graded) $A$-modules. \end{itemize} \end{definition}
Next, we cite some results about Braden-MacPherson (BMP) sheaves from \cite{B} and \cite{10}. For this we take $A=S_{(0)}$ to be the localization of $S$ at the maximal ideal generated by $V$.
\begin{thm}[\cite{10}, section 3.5 and \cite{B}, Theorem 6.3] \begin{enumerate} \item For any $x \in \mathcal{V}$ there is an up to isomorphism unique graded $S$-sheaf $\mathscr{B}(x)$ on $\mathcal{G}$ with the following properties:
\begin{itemize}
\item $\mathscr{B}(x)$ is F-projective
\item $\mathscr{B}(x)$ is indecomposable (even as a non-graded sheaf)
\item $\mathscr{B}(x)_x \cong S$ and $\mathscr{B}(x)_y = 0$ unless $ x \leq y$
\end{itemize}
\item Let $\mathscr{P}$ be an F-projective $A$-sheaf of finite type on $\mathcal{G}$. Then there exists an isomorphism of $A$-sheaves
$$\mathscr{P} \cong \mathscr{B}(z_1) \otimes_S A \oplus ... \oplus \mathscr{B}(z_n) \otimes_S A$$
with suitable vertices $z_1,..., z_n$.
\item Let $\mathscr{P}$ be a graded F-projective $S$-sheaf of finite type on $\mathcal{G}$. Then there exists an isomorphism of graded $S$-sheaves
$$\mathscr{P} \cong \mathscr{B}(z_1)[l_1] \oplus ... \oplus \mathscr{B}(z_n)[l_n]$$
with suitable vertices $z_1,..., z_n$ and suitable shifts $l_1,...,l_n$. \end{enumerate} \end{thm}
\section{Deformed category $\mathcal{O}$}
In this section we recall results about the deformed category $\mathcal{O}$ of a semisimple complex Lie algebra $\mathfrak{g}$ with Borel $\mathfrak{b}$ and Cartan $\mathfrak{h}$, which one can find in \cite{5}, \cite{14} and \cite{11}. Let $S$ denote the universal enveloping algebra of the Cartan $\mathfrak{h}$ which is equal to the ring of polynomial functions $\mathbb{C}[\mathfrak{h}^{*}]$. We call a commutative, associative, noetherian, unital, local $S$-algebra $A$ with structure morphism $\tau: S \rightarrow A$ a \textit{local deformation algebra}.\\
Let $A$ be a local deformation algebra with structure morphism $\tau: S\rightarrow A$ and let $M \in \mathfrak{g}$-mod-$A$. For ${\lambda} \in \mathfrak{h}^{*}$ we set
$$M_{\lambda} = \{m \in M| hm=({\lambda} + \tau)(h)m \, \, \forall h \in \mathfrak{h}\}$$ where $({\lambda}+\tau)(h)$ is meant to be an element of $A$. We call the $A$-submodule $M_{\lambda}$ the \textit{deformed ${\lambda}$-weight space} of $M$.\\
The \textit{deformed category} $\mathcal{O}_{A}$ is the full subcategory of all bimodules $M \in \mathfrak{g}$-mod-$A$ with the properties \begin{itemize} \item $M= \bigoplus\limits_{{\lambda} \in \mathfrak{h}^{*}} M_{\lambda}$, \item for every $m \in M$ the $\mathfrak{b}$-$A$-sub-bimodule generated by $m$ is finitely generated as an $A$-module, \item $M$ is finitely generated as a $\mathfrak{g}$-$A$-bimodule. \end{itemize}
Taking $A=S/S\mathfrak{h} \cong \mathbb{C}$, $\mathcal{O}_{A}$ is just the usual BGG-category $\mathcal{O}$.\\ For ${\lambda} \in \mathfrak{h}^{*}$ the \textit{deformed Verma module} is defined by $$\Delta_A({\lambda}) = U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} A_{\lambda}$$ where $A_{\lambda}$ denotes the $U(\mathfrak{b})$-$A$-bimodule $A$ with $\mathfrak{b}$-structure given by the composition $U(\mathfrak{b}) \rightarrow S \stackrel{{\lambda} + \tau}{\longrightarrow} A$.\\
The Lie algebra $\mathfrak{g}$ possesses an involutive anti-automorphism $\sigma:\mathfrak{g} \rightarrow \mathfrak{g}$ with $\sigma|_{\mathfrak{h}} = -\mathrm{id}$. This gives the $A$-module $\mathrm{Hom}_A(M,A)$ a $\sigma$-twisted $\mathfrak{g}$-module structure. Denoting by $dM$ the sum of all deformed weight spaces in $\mathrm{Hom}_A(M,A)$, we get a functor
$$d=d_\sigma : \mathcal{O}_{A} \longrightarrow \mathcal{O}_{A}$$
which is a duality on $\mathfrak{g}$-$A$-bimodules which are free over $A$. We now set $\nabla_A({\lambda})=d\Delta_A({\lambda})$ for ${\lambda} \in \mathfrak{h}^{*}$ and call this the \textit{deformed nabla module}.
\begin{prop}[\cite{14}, Proposition 2.12.]\label{Soe1} \begin{enumerate} \item For all ${\lambda}$ the restriction to the deformed weight space of ${\lambda}$ together with the two canonical identifications $\Delta_A({\lambda})_{\lambda} \stackrel{\sim}{\rightarrow} A$ and $\nabla_A({\lambda})_{\lambda} \stackrel{\sim}{\rightarrow} A$ induces an isomorphism $$\mathrm{Hom}_{\mathcal{O}_{A}}(\Delta_A({\lambda}),\nabla_A({\lambda})) \stackrel{\sim}{\longrightarrow}A$$
\item For ${\lambda} \neq \mu$ in $\mathfrak{h}^{*}$ we have $\mathrm{Hom}_{\mathcal{O}_{A}}(\Delta_A({\lambda}),\nabla_A(\mu))=0$.
\item For all ${\lambda},\mu \in \mathfrak{h}^{*}$ we have $\mathrm{Ext}^{1}_{\mathcal{O}_{A}}(\Delta_A({\lambda}),\nabla_A(\mu))=0$. \end{enumerate} \end{prop}
\begin{cor}[\cite{14}, Corollary 2.13.]\label{Soe2} Let $M,N \in \mathcal{O}_{A}$. If $M$ has a $\Delta_A$-flag and $N$ a $\nabla_A$-flag, then the space of homomorphisms $\mathrm{Hom}_{\mathcal{O}_{A}}(M,N)$ is a finitely generated free $A$-module and for any homomorphism $A\rightarrow A'$ of local deformation algebras the obvious map defines an isomorphism $$\mathrm{Hom}_{\mathcal{O}_{A}}(M,N)\otimes_{A} A' \stackrel{\sim}{\longrightarrow} \mathrm{Hom}_{\mathcal{O}_{A'}}(M\otimes_{A} A',N\otimes_{A} A')$$ \end{cor}
\begin{proof} This follows from Proposition \ref{Soe1} by induction on the length of the $\Delta_A$- and $\nabla_A$-flag. \end{proof}
If $\mathfrak{m}\subset A$ is the unique maximal ideal in the local deformation algebra $A$ we set $\mathbb{K}=A/\mathfrak{m}A$ for its residue field.
\begin{thm}[\cite{5}, Propositions 2.1 and 2.6] \label{Fie1}
\begin{enumerate} \item The base change $\cdot \otimes_A \mathbb{K}$ gives a bijection\\ \begin{displaymath}
\begin{array}{ccc}
\left\{\begin{array}{c}
\textrm{simple isomorphism}\\
\textrm{classes of $\mathcal{O}_{A}$}
\end{array}\right\}
&
\longleftrightarrow
&
\left\{\begin{array}{c}
\textrm{simple isomorphism}\\
\textrm{classes of $\mathcal{O}_{\mathbb{K}}$}
\end{array}\right\}
\end{array}
\end{displaymath}
\item The base change $\cdot \otimes_A \mathbb{K}$ gives a bijection\\ \begin{displaymath}
\begin{array}{ccc}
\left\{\begin{array}{c}
\textrm{projective isomorphism}\\
\textrm{classes of $\mathcal{O}_{A}$}
\end{array}\right\}
&
\longleftrightarrow
&
\left\{\begin{array}{c}
\textrm{projective isomorphism}\\
\textrm{classes of $\mathcal{O}_{\mathbb{K}}$}
\end{array}\right\}
\end{array}
\end{displaymath} \end{enumerate} \end{thm}
The category $\mathcal{O}_{\mathbb{K}}$ is a direct summand of the category $\mathcal{O}$ over the Lie algebra $\mathfrak{g} \otimes \mathbb{K}$. It consists of all objects whose weights lie in the complex affine subspace $\tau + \mathfrak{h}^{*} = \tau + \mathrm{Hom}_\mathbb{C}(\mathfrak{h},\mathbb{C}) \subset \mathrm{Hom}_\mathbb{K}(\mathfrak{h}\otimes \mathbb{K},\mathbb{K})$ for $\tau$ the restriction to $\mathfrak{h}$ of the map that makes $\mathbb{K}$ to an $S$-algebra. Thus the simple objects of $\mathcal{O}_{A}$ are parameterized by their highest weight in $\mathfrak{h}^{*}$. Denote by $L_A({\lambda})$ the simple object with highest weight ${\lambda}$. We also use the usual partial order on $\mathfrak{h}^{*}$ to partially order $\tau + \mathfrak{h}^{*}$.
\begin{thm}[\cite{5}, Propositions 2.4 and Theorem 2.7] \label{Fie2} Let $A$ be a local deformation algebra and $\mathbb{K}$ its residue field. Let $L_A({\lambda})$ be a simple object in $\mathcal{O}_{A}$.
\begin{enumerate} \item There is a projective cover $P_A({\lambda})$ of $L_A({\lambda})$ in $\mathcal{O}_{A}$ and every projective object in $\mathcal{O}_{A}$ is isomorphic to a direct sum of projective covers. \item $P_A({\lambda})$ has a Verma flag, i.e., a finite filtration with subquotients isomorphic to Verma modules, and for the multiplicities we have the BGG-reciprocity formula $$(P_A({\lambda}):\Delta_A(\mu)) = [\Delta_{\mathbb{K}}(\mu):L_{\mathbb{K}}({\lambda})]$$ for all Verma modules $\Delta_A(\mu)$ in $\mathcal{O}_{A}$. \item Let $A \rightarrow A'$ be a homomorphism of local deformation algebras and $P$ projective in $\mathcal{O}_{A}$. Then $P\otimes_A A'$ is projective in $\mathcal{O}_{A'}$ and the natural transformation $$\mathrm{Hom}_{\mathcal{O}_{A}}(P,\cdot) \otimes_A A' \longrightarrow \mathrm{Hom}_{\mathcal{O}_{A'}}(P \otimes_A A', \cdot \otimes_A A')$$ is an isomorphism of functors from $\mathcal{O}_{A}$ to $A'$-mod. \end{enumerate} \end{thm}
\subsection{Block decomposition}
Let $A$ again denote a local deformation algebra and $\mathbb{K}$ its residue field.
\begin{definition} Let $\sim_A$ be the equivalence relation on $\mathfrak{h}^{*}$ generated by ${\lambda} \sim_A \mu$ if $[\Delta_{\mathbb{K}}({\lambda}):L_{\mathbb{K}}(\mu)] \neq 0$. \end{definition}
\begin{definition} Let $\Lambda \in \mathfrak{h}^{*}/\sim_{A}$ be an equivalence class. Let $\mathcal{O}_{A,\Lambda}$ be the full subcategory of $\mathcal{O}_{A}$ consisting of all modules $M$ such that every highest weight of a subquotient of $M$ lies in $\Lambda$. \end{definition}
\begin{prop}[\cite{5}, Proposition 2.8] (Block decomposition)\label{Fie3} The functor \begin{eqnarray*}
\begin{array}{ccc}
\bigoplus\limits_{\Lambda \in \mathfrak{h}^{*}/\sim_A} \mathcal{O}_{A,\Lambda} &\longrightarrow& \mathcal{O}_{A}\\
(M_\Lambda)_{\Lambda \in \mathfrak{h}^{*}/\sim_A}& \longmapsto & \bigoplus\limits_{\Lambda \in \mathfrak{h}^{*}/\sim_A} M_\Lambda
\end{array}
\end{eqnarray*} is an equivalence of categories. \end{prop}
\begin{remark} For $R=S_{(0)}$ the localization of $S$ at the maximal ideal generated by $\mathfrak{h}$, the block decomposition of $\mathcal{O}_R$ corresponds to the block decomposition of the BGG-category $\mathcal{O}$ over $\mathfrak{g}$.\\ \end{remark}
Let $\tau:S \rightarrow \mathbb{K}$ be the induced map that makes $\mathbb{K}$ into an $S$-algebra. Restricting to $\mathfrak{h}$ and extending with $\mathbb{K}$ yields a $\mathbb{K}$-linear map $\mathfrak{h} \otimes \mathbb{K} \rightarrow \mathbb{K}$ which we will also call $\tau$. Let $\mathcal{R} \supset \mathcal{R}^{+}$ be the root system with positive roots according to our data $\mathfrak{g}\supset \mathfrak{b} \supset \mathfrak{h}$. For ${\lambda} \in \mathfrak{h}_{\mathbb{K}}^{*}=\mathrm{Hom}_\mathbb{K}(\mathfrak{h} \otimes \mathbb{K},\mathbb{K})$ and $\check{\alpha} \in \mathfrak{h}$ the dual root of a root $\alpha \in \mathcal{R}$ we set $\left\langle {\lambda}, \check{\alpha}\right\rangle_{\mathbb{K}} = {\lambda}(\check{\alpha}) \in \mathbb{K}$. Let $\mathcal{W}$ be the Weyl group of $(\mathfrak{g},\mathfrak{h})$ and denote by $s_\alpha$ the reflection corresponding to $\alpha \in \mathcal{R}$.
\begin{definition} For $\mathcal{R}$ the root system of $\mathfrak{g}$ and $\Lambda \in \mathfrak{h}^{*}/\sim_A$ we define
$$\mathcal{R}_A(\Lambda)=\{\alpha \in \mathcal{R} | \left\langle {\lambda} + \tau, \check{\alpha}\right\rangle_{\mathbb{K}} \in \mathbb{Z} \subset \mathbb{K} \text{ for some } {\lambda} \in \Lambda\}$$
and call it the integral roots corresponding to $\Lambda$. Let $\mathcal{R}_A^{+}(\Lambda)$ denote the positive roots in $\mathcal{R}_A(\Lambda)$ and set $$\mathcal{W}_A(\Lambda)=\langle \{s_\alpha \in \mathcal{W} | \alpha \in \mathcal{R}_A^{+}(\Lambda)\}\rangle \subset \mathcal{W}$$ We call it the integral Weyl group with respect to $\Lambda$. \end{definition} From \cite{5} Corollary 3.3 it follows that $$\Lambda=\mathcal{W}_A(\Lambda)\cdot {\lambda} \text{ for any } {\lambda} \in \Lambda$$ where we denote by $\cdot$ the $\rho$-shifted dot-action of the Weyl group.\\ Since most of the following constructions commute with base change, we are particularly interested in the case when $A=R_{\mathfrak{p}}$ is a localization of $R$ at a prime ideal $\mathfrak{p}$ of height one. The functor $\cdot \otimes_R R_{\mathfrak{p}}$ will split the deformed category $\mathcal{O}_{A}$ into generic and subgeneric blocks:
\begin{lemma}[\cite{6}, Lemma 3] \label{Fie4} Let $\Lambda \in \mathfrak{h}^{*}/\sim_{R}$ and let $\mathfrak{p} \in R$ be a prime ideal. \begin{enumerate} \item If $\check{\alpha} \notin \mathfrak{p}$ for all roots $\alpha \in \mathcal{R}_{R}(\Lambda)$, then $\Lambda$ splits under $\sim_{R_\mathfrak{p}}$ into generic equivalence classes. \item If $\mathfrak{p} = R\check{\alpha}$ for a root $\alpha \in \mathcal{R}_{R}(\Lambda)$, then $\Lambda$ splits under $\sim_{R_\mathfrak{p}}$ into subgeneric equivalence classes of the form $\{{\lambda},s_{\alpha}\cdot {\lambda}\}$. \end{enumerate} \end{lemma} We recall that we denote by $P_A({\lambda})$ the projective cover of the simple object $L_A({\lambda})$. It is indecomposable and up to isomorphism uniquely determined. For an equivalence class $\Lambda \in \mathfrak{h}^{*}/\sim_A$ which contains ${\lambda}$ and is generic, i.e., $\Lambda=\{{\lambda}\}$, we get $P_A({\lambda})=\Delta_A({\lambda})$. If $\Lambda=\{{\lambda},\mu\}$ and $\mu< {\lambda}$, we have $P_A({\lambda})=\Delta_A({\lambda})$ and there is a non-split short exact sequence in $\mathcal{O}_{A}$ $$0\rightarrow \Delta_A({\lambda}) \rightarrow P_A(\mu) \rightarrow \Delta_A(\mu)\rightarrow 0$$ In this case, every endomorphism $f: P_A(\mu) \rightarrow P_A(\mu)$ maps $\Delta_A({\lambda})$ to $\Delta_A({\lambda})$ since ${\lambda}>\mu$. So $f$ induces a commutative diagram \begin{eqnarray*} \begin{CD}
0 @>>> \Delta_{A}({\lambda}) @>>> P_{A}(\mu) @>>> \Delta_{A}(\mu) @>>> 0\\
@VVV @V f_{\lambda} VV @VV f V @VV f_\mu V @VVV\\
0 @>>> \Delta_{A}({\lambda}) @>>> P_{A}(\mu) @>>> \Delta_{A}(\mu) @>>> 0 \end{CD}\end{eqnarray*} Since endomorphisms of Verma modules correspond to elements of $A$, we get a map \begin{displaymath}
\begin{array}{ccc}
\chi: \mathrm{End}_{\mathcal{O}_{A}}(P_{A}(\mu))& \longrightarrow & A \oplus A \\
f & \longmapsto &(f_{\lambda} , f_\mu)
\end{array} \end{displaymath}
For $\mathfrak{p}=R\check{\alpha}$ we define $R_\alpha :=R_\mathfrak{p}$ for the localization of $R$ at the prime ideal $\mathfrak{p}$.
\begin{prop}[\cite{5}, Corollary 3.5] \label{Fie5} Let $\Lambda \in \mathfrak{h}^{*}/\sim_{R_\alpha}$. If $\Lambda=\{{\lambda},\mu\}$ and ${\lambda}=s_{\alpha}\cdot \mu >\mu$, the map $\chi$ from above induces an isomorphism of $R_\alpha$-modules
$$\mathrm{End}_{\mathcal{O}_{R_\alpha}}(P_{R_\alpha}(\mu)) \cong \left\{(t_{\lambda},t_\mu) \in {R_\alpha}\oplus {R_\alpha} \middle| t_{\lambda} \equiv t_\mu \text{ mod } \check{\alpha}\right\}$$ \end{prop}
\subsection{Deformed tilting modules}
In this chapter, $A$ will be a localization of $R=S_{(0)}$ at a prime ideal $\mathfrak{p} \subset R$ and $\mathbb{K}$ its residue field. Let ${\lambda} \in \mathfrak{h}^{*}$ be such that $\Delta_{\mathbb{K}}({\lambda})$ is a simple object in $\mathcal{O}_{\mathbb{K}}$. Thus, we have $\Delta_{\mathbb{K}}({\lambda}) \cong \nabla_{\mathbb{K}}({\lambda})$ and the canonical inclusion $\Delta_A({\lambda}) \hookrightarrow \nabla_A({\lambda})$ becomes an isomorphism after applying $\cdot \otimes_A \mathbb{K}$. So by Nakayama's lemma, we conclude that this inclusion is bijective.
\begin{definition} By $\mathcal{K}_A$ we denote the full subcategory of $\mathcal{O}_{A}$ which \begin{enumerate} \item includes the self-dual deformed Verma modules, \item is stable under tensoring with finite dimensional $\mathfrak{g}$-modules, \item is stable under forming direct sums and summands. \end{enumerate} \end{definition}
\begin{prop}(\cite{11}, Proposition 3.2.) \label{tilt1} The base change $\cdot \otimes_A \mathbb{K}$ gives a bijection\\ \begin{displaymath}
\begin{array}{ccc}
\left\{\begin{array}{c}
\textrm{isomorphism classes}\\
\textrm{of $\mathcal{K}_A$}
\end{array}\right\}
&
\longleftrightarrow
&
\left\{\begin{array}{c}
\textrm{isomorphism classes}\\
\textrm{of $\mathcal{K}_\mathbb{K}$}
\end{array}\right\}
\end{array}
\end{displaymath} \end{prop}
\begin{remark} For $A=S/S\mathfrak{h}=\mathbb{C}$ the category $\mathcal{K}_A$ is just the usual subcategory of tilting modules of the category $\mathcal{O}$ over $\mathfrak{g}$. The definition also implies that deformed tilting modules have a Verma flag and are self-dual. Furthermore, the indecomposable tilting modules are classified by their highest weight and we denote by $K_A({\lambda})$ the deformed tilting module with highest weight ${\lambda} \in \mathfrak{h}^*$. \end{remark}
\section{Tilting modules as sheaves on moment graphs}
In this section we repeat the connection between representation theory and sheaves on moment graphs via the structure functor $\mathbb{V}$ as it is described in \cite{7}. We prove without using the tilting functor that tilting modules of a deformed block in category $\mathcal{O}$ become certain BMP-sheaves on a certain moment graph associated to this block. As a corollary of this we get character formulas of tilting modules without using the tilting functor.
\subsection{The functor $\mathbb{V}$}
Again, $A$ denotes a localization of $R$ at a prime ideal. We first want to get a functor from a block of deformed category $\mathcal{O}$ to sheaves on a certain moment graph. Given an antidominant weight ${\lambda} \in \mathfrak{h}^*$ denote by $\Lambda \in \mathfrak{h}^* /\sim_A$ its equivalence class. We now set $\mathcal{W}_{\lambda} = \mathcal{W}_A (\Lambda)$ and $\mathcal{W}':= \mathrm{Stab}_{\mathcal{W}_{\lambda}} ({\lambda})$. We then get a bijection $$\mathcal{W}_{\lambda} \cdot {\lambda} \cong \mathcal{W}_{\lambda} / \mathcal{W}'$$ Now we define the ordered $\mathfrak{h}^*$-moment graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\alpha, \leq)$ associated to the block $\mathcal{W}_{\lambda} \cdot {\lambda}$ by letting
$$\mathcal{V} := \mathcal{W}_{\lambda} / \mathcal{W}'$$
and two different vertices $x,y \in \mathcal{V}$ are joined by an edge $E=\{x,y\}$ if there is a positive root $\alpha \in \mathcal{R}_A^+(\Lambda)$ with $x=s_\alpha \cdot y$. The labeling of $E$ is $\alpha(E)= \check{\alpha}$. For $w,w' \in \mathcal{V}$ we define the order $\leq$ by $$w \leq w' \Leftrightarrow w \cdot {\lambda} \leq w'\cdot {\lambda}$$ This ordered moment graph has the GKM-property. Note that this order is not the Bruhat order in general. But for two adjacent vertices both orderings coincide.
\begin{thm}[\cite{5}, Theorem 3.6.] Let $\mathcal{Z}= \mathcal{Z}_A (\mathcal{G})$ be the structure algebra of the above moment graph. Then there is an isomorphism $$\mathcal{Z} \cong \mathrm{End}_{\mathcal{O}_A}(P_A({\lambda}))$$ \end{thm}
We now get a functor
$$\mathbb{V} := \mathrm{Hom}_{\mathcal{O}_A}(P_A({\lambda}), \cdot) : \mathcal{O}_{A, {\lambda}} \longrightarrow \mathcal{Z}-mod$$
\begin{definition} Denote by $\mathscr{V}_A(w)$ the skyscraper sheaf on $\mathcal{G}$ at the vertex $w \in \mathcal{V}$, i.e., the $A$-sheaf with $\mathscr{V}_A(w)_w \cong A$ and whose stalks at every other vertex and at every edge are zero. \end{definition}
\begin{prop} \begin{enumerate} \item Let $w \in \mathcal{V}$. Then $\mathcal{L}(\mathbb{V} \Delta_A(w \cdot {\lambda})) \cong \mathcal{L}(\mathbb{V} \nabla_A(w \cdot {\lambda})) \cong \mathscr{V}_A(w)$.
\item Let $M \in \mathcal{O}_{A,{\lambda}}$ admit a Verma- or a Nabla-flag. Then $\mathbb{V} M$ is a free $A$-module of finite rank. \end{enumerate} \end{prop}
\begin{proof} The proof is analogous to \cite{10} Proposition 4.13. \end{proof}
\subsection{Deformed tilting modules and BMP-sheaves}\label{BMP}
Following \cite{10} section 4.14., we construct certain submodules and quotients of a module $M \in \mathcal{O}_{A}$. Let $D$ be a subset of $\mathfrak{h}^*$ with the property:\\ If ${\lambda} \in D$ and $\mu \in \mathfrak{h}^*$ with $\mu \leq {\lambda}$, then $\mu \in D$.\\ Set for $M \in \mathcal{O}_A$ $$ O^D M:= \sum_{\mu \notin D} U(\mathfrak{g}_A) M_\mu \, \, and \,\, M[D] := M/O^D M$$
\begin{prop} [\cite{10}, section 4.14.]´
\begin{enumerate}
\item If $M$ has a Verma flag, so do $O^D M$ and $M[D]$
\item
\begin{displaymath}
\begin{array}{ccc}
O^D\Delta_A({\lambda}) \cong
\left\{ \begin{array}{c}
\Delta_A ({\lambda}) \, \textrm{if}\, {\lambda} \notin D\\
0 \,\,\,\,\,\,\,\,\, \textrm{else}
\end{array}\right.
&
and
&
\Delta_A({\lambda})[D] \cong
\left \{\begin{array}{c}
0 \,\,\,\,\,\,\,\, \textrm{if}\, {\lambda} \notin D\\
\Delta_A({\lambda}) \,\,\,\textrm{else}
\end{array}\right.
\end{array}
\end{displaymath}
\end{enumerate} \end{prop}
We now change notation: For the moment graph $\mathcal{G}=(\mathcal{V}, \mathcal{E}, \alpha , \leq)$ associated to the block $\mathcal{O}_{A,{\lambda}}$ we write $\uparrow$-open for an F-open subgraph. For a subgraph which is F-open according to the moment graph with the reversed order, we write $\downarrow$-open. A subgraph $\mathcal{H}$ with set of vertices $\mathcal{V}'$ is $\uparrow$-open if and only if $\mathcal{H}^c$ is $\downarrow$-open, where $\mathcal{H}^c$ is the full subgraph of $\mathcal{G}$ with vertices $\mathcal{V}^c:= \mathcal{V} - \mathcal{V}'$. For $\mathcal{H}$ an $\downarrow$-open sugraph, set $D$ equal to the set of all $\nu \in \mathfrak{h}^*$, such that there exists $x \in \mathcal{V}^c$ with $\nu \leq x \cdot {\lambda}$.\\ Set $O^{\mathcal{H}^c} M := O^D M$ and $M[\mathcal{H}^c] =M [D]$.
\begin{lemma} \label{A} Let $M$ be a tilting module. There is a natural isomorphism $$e_{\mathcal{H}} \mathbb{V} M \cong \mathbb{V}(d(O^{\mathcal{H}^c} M))$$ \end{lemma}
\begin{proof} Dualising the short exact sequence $$O^{\mathcal{H}^c} M \hookrightarrow M \twoheadrightarrow M[\mathcal{H}^c]$$
we get by self-duality of $M$ and the freeness of all modules over $A$ a short exact sequence
$$d(O^{\mathcal{H}^c} M) \twoheadleftarrow M \hookleftarrow d(M[\mathcal{H}^c])$$
Since $\mathbb{V}$ is exact we get a short exact sequence
$$\mathbb{V}(d(O^{\mathcal{H}^c} M)) \twoheadleftarrow \mathbb{V}(M) \hookleftarrow \mathbb{V}(d(M[\mathcal{H}^c]))$$
Since $d(M[\mathcal{H}^c])\otimes_A Q$ is the direct sum of Verma modules of the form $\Delta_Q (x \cdot {\lambda})\cong \nabla_Q (x \cdot {\lambda})$ with $x \notin \mathcal{V}'$ we get $e_{\mathcal{H}} \mathbb{V} M \cong \mathbb{V}(d(O^{\mathcal{H}^c} M))$. \end{proof}
\begin{definition} A sheaf $\mathscr{M}$ on $\mathcal{G}$ is called $\downarrow$-flabby (resp. $\downarrow$-projective) if it is F-flabby (resp. F-projective) according to the moment graph $\mathcal{G}$ with reversed order. \end{definition}
\begin{prop} \label{D} Let $M$ be a tilting module. Then $\mathcal{L}(\mathbb{V} M)$ is $\downarrow$-flabby. \end{prop}
\begin{proof} For any $\downarrow$-open subgraph $\mathcal{H}$ we get that $e_{\mathcal{H}}M$ is a free $A$-module by lemma \ref{A}. Now proposition \ref{B} tells us that $\mathcal{L}(\mathbb{V} M)$ is $\downarrow$-flabby. \end{proof}
\begin{notation} For $x \in \mathcal{V}$ we denote by $\mathscr{B}^{\uparrow}(x)$ the BMP-sheaf $\mathscr{B}(x)$ for our moment graph with the original order. For the moment graph with reversed order we denote the BMP-sheaf at the vertex $x$ by $\mathscr{B}^{\downarrow}(x)$. \end{notation}
In the following we want to show that the indecomposable deformed tilting module $K_A(x \cdot {\lambda})$ with highest weight $x\cdot {\lambda}$ corresponds to $\mathscr{B}^{\downarrow}(x)\otimes_S A$ under $\mathcal{L} \circ \mathbb{V}$. For this we need some preparation.
\begin{lemma}(\cite{7}, Lemma 7.4.)\label{inj} Let $M \in \mathcal{O}$ admit a Verma flag, $\mu \in \mathfrak{h}^*$ and $k \in \mathbb{N}$. If $g: (\Delta_\mathbb{C}(\mu))^k \rightarrow M$ is a morphism which induces an injective map $g_\mu : (\Delta_\mathbb{C}(\mu))_\mu ^k \rightarrow M_\mu$ on the $\mu$-weight spaces, then $g$ is injective. \end{lemma}
\begin{lemma} \label{C} Let $K \in \mathcal{O}_{A,{\lambda}}$ be a tilting module. Then for any $w \in \mathcal{V}$ $\mathcal{L}(\mathbb{V} K)_w$ is free over $A$ of rank $r:= (K: \Delta_A (w\cdot {\lambda}))$. \end{lemma}
\begin{proof} We want to construct an isomorphism $e_w (\mathbb{V} K) = \mathcal{L}(\mathbb{V} K)_w \xrightarrow{\sim} \mathbb{V}(\nabla_A (w\cdot {\lambda})^r)$. Using corollary \ref{Soe2} and the fact that over the residue field $\mathbb{K}$ we have $\mathrm{dim}_\mathbb{K} \mathrm{Hom}_{\mathcal{O}_\mathbb{K}}(\Delta_\mathbb{K}(w \cdot {\lambda}) , K \otimes_A \mathbb{K}) = \mathrm{dim}_\mathbb{K} \mathrm{Hom}_{\mathcal{O}_\mathbb{K}}(K \otimes_A \mathbb{K}, \nabla_\mathbb{K}(w\cdot {\lambda})) = (K\otimes_A \mathbb{K} : \Delta_\mathbb{K}(w\cdot {\lambda}))$, we can deduce that $\mathrm{Hom}_{\mathcal{O}_A}(\Delta_A(w\cdot {\lambda}),K)$ is a free $A$- module of rank $r$. Now choose an $A$-basis $f_1,...,f_r$ of $\mathrm{Hom}_{\mathcal{O}_A}(\Delta_A(w\cdot {\lambda}),K)$. Dualising yields a basis $df_1,..., df_r$ of $\mathrm{Hom}_{\mathcal{O}_A}(K,\nabla_A(w\cdot {\lambda}))$. Consider now the map \begin{displaymath}
\begin{array}{cc}
f: \Delta_A(w\cdot {\lambda})^r \longrightarrow K
&
(v_1,...,v_r) \mapsto \sum f_i(v_i)
\end{array} \end{displaymath}
Dualising this map yields by self-duality of $K$
\begin{displaymath}
\begin{array}{cc}
df: K \longrightarrow \nabla_A(w\cdot {\lambda})^r
&
v \mapsto \sum (df_i(v))
\end{array} \end{displaymath}
Applying the functor $\mathbb{V}$ gives a map $$\mathbb{V} df :\mathbb{V} K \longrightarrow \mathbb{V} \nabla_A(w\cdot {\lambda})^r$$ which factors over $e_{w}(\mathbb{V} K)$, since $\mathbb{V} \nabla_A (w \cdot {\lambda})^r$ has support $\{w\}$. So this gives a well-defined map \begin{displaymath}
\begin{array}{cc}
(\mathbb{V} df)^w: e_w \mathbb{V} K \longrightarrow \mathbb{V}\nabla_A(w\cdot {\lambda})^r
\end{array} \end{displaymath} which is injective, since it becomes an isomorphism after applying $\cdot \otimes_A Q$. We now want to prove that $(\mathbb{V} df)^w$ is surjective. For this, by Nakayama's lemma and by exactness of $\mathbb{V}$, it is enough to prove that \begin{displaymath}
\begin{array}{cc}
df\otimes_A \mathrm{id_\mathbb{C}}: K \otimes_A \mathbb{C} \longrightarrow \nabla_A(w\cdot {\lambda})^r \otimes_A \mathbb{C}
\end{array} \end{displaymath} is surjective. But since $f: \Delta_\mathbb{C} (w \cdot {\lambda})^r \rightarrow K\otimes_A \mathbb{C}$ is injective, by lemma \ref{inj} we get the surjectivity of $df: K\otimes_A \mathbb{C} \rightarrow \nabla_\mathbb{C}(w \cdot {\lambda})^r$. Since $\mathbb{V} \nabla_A (w\cdot {\lambda})^r$ is free of rank $r$ over $A$ we get the result. \end{proof}
\begin{thm} Let $K \in \mathcal{O}_{A,{\lambda}}$ be a tilting module. Then $\mathcal{L}(\mathbb{V} K)$ is $\downarrow$-projective as an $A$-sheaf on $\mathcal{G}$. \end{thm}
\begin{proof} By \cite{10} section 2.12.(A) $\mathcal{L}(\mathbb{V} K)$ is generated by global sections and $\downarrow$-flabby by proposition \ref{D}. Furthermore, the lemma above shows that $(\mathcal{L} (\mathbb{V} K))_w$ is free over $A$ for every $w \in \mathcal{V}$.\\ So we only have to prove for any edge $E:=\{x,y\}$ $x> y$ and $\alpha(E)= \check{\alpha}$ where $x,y \in \mathcal{W}_{\lambda} / \mathcal{W}'$ and $\check{\alpha}$ the coroot of a root $\alpha \in \mathcal{R}_{\lambda}$, that the map $\rho_{x,E} : \mathcal{L} (\mathbb{V} K)_x \rightarrow \mathcal{L} (\mathbb{V} K)_E$ induces an isomorphism $$\mathcal{L} (\mathbb{V} K)_x / \check{\alpha} \mathcal{L} (\mathbb{V} K)_x \stackrel{\sim}\longrightarrow \mathcal{L} (\mathbb{V} K)_E$$
Since $\check{\alpha} \mathcal{L} (\mathbb{V} K)_x \subset \mathrm{ker} \rho_{x,E}$, we have to show $\mathrm{ker} \rho_{x,E} \subset \check{\alpha} \mathcal{L} (\mathbb{V} K)_x$. For this it suffices to show that \begin{equation}\label{E} \mathrm{ker} \rho_{x,E} \subset \check{\alpha} (\mathcal{L} (\mathbb{V} K)_x \otimes_A A_\mathfrak{p}) = \check{\alpha} \cdot e_x(\mathcal{L} (\mathbb{V} K) \otimes_A A_\mathfrak{p}) \end{equation} for every prime ideal $\mathfrak{p} \subset A$ of hight 1. For $\check{\alpha} \notin \mathfrak{p}$ (\ref{E}) follows since $\check{\alpha}$ is invertible in $A_\mathfrak{p}$.\\ So we have to prove (\ref{E}) for $\mathfrak{p}=A \check{\alpha}$.
Since $\rho_{x,E}$ is a push-out map, we can identify $\mathrm{ker} \rho_{x,E}$ with the set $\{e_x u | u \in \mathbb{V} K(E), \, e_yu =0\}$. Since an element $u \in \mathbb{V} K(E)$ is of the form $u = (e_x +e_y) v + \check{\alpha} e_x w$ for $v,w \in \mathbb{V} K$, we have to prove
$$\{e_x u| u \in (e_x+e_y) \mathbb{V} K, \, e_y u =0\} \subset \check{\alpha} e_x (\mathbb{V} K \otimes_A A_\mathfrak{p})$$ Since $x > y$, we get $K_{A_\mathfrak{p}}(y \cdot {\lambda}) \cong \Delta_{A_\mathfrak{p}}(y \cdot {\lambda})$ and $K_{A_\mathfrak{p}}(x \cdot {\lambda})\cong P_{A_\mathfrak{p}}( y \cdot {\lambda})$. Now we can identify $(e_x +e_y) (\mathbb{V} K \otimes_A A_\mathfrak{p})$ with a direct sum of $A_\mathfrak{p}$-modules of the form $M:= \mathrm{Hom}_{\mathfrak{g}_{A_\mathfrak{p}}}(P_{A_\mathfrak{p}}(y\cdot {\lambda}) , \Delta_{A_\mathfrak{p}}(y \cdot {\lambda})) \cong A_\mathfrak{p}$ and $N:= \mathrm{Hom}_{\mathfrak{g}_{A_\mathfrak{p}}}(P_{A_\mathfrak{p}}(y\cdot {\lambda}) , K_{A_\mathfrak{p}}(x \cdot{\lambda})) \cong A_\mathfrak{p} (e_x+e_y) + A_\mathfrak{p} \check{\alpha} e_x$ by proposition \ref{Fie5}.\\ If $f \in M$, we get $e_x f =0$ which is equal to $\check{\alpha} e_x f$. If $f \in N$, $e_y f =0$ implies $e_xf \in \check{\alpha} A_\mathfrak{p} e_x$. But with our identification we get $e_x f \in \check{\alpha} e_x (\mathbb{V} K \otimes_A A_\mathfrak{p})$. \end{proof}
\begin{cor} We have $\mathcal{L} (\mathbb{V} K_A (w \cdot {\lambda})) \cong \mathscr{B}^\downarrow(w) \otimes_S A$ for all $w \in \mathcal{W}_{\lambda}$. \end{cor}
\begin{proof} The proof is essentially the same as in \cite{10} Theorem 4.22. and relies on the facts that $\mathcal{L} (\mathbb{V} K_A (w \cdot {\lambda}))$ is $\downarrow$-projective and indecomposable. Now the description of the indecomposable $\downarrow$-projective sheaves by BMP-sheaves gives the claim. \end{proof}
\begin{cor} We have $(K(w \cdot {\lambda}):\Delta(x \cdot {\lambda}))= \mathrm{rk}_S \mathscr{B}^\downarrow(w)_x$. \end{cor}
\begin{proof} Lemma \ref{C} shows that $\textrm{rk}_A (\mathcal{L} (\mathbb{V} K_A (w \cdot {\lambda})_x) = (K_A(w \cdot {\lambda}):\Delta_A(x \cdot {\lambda}))= (K(w \cdot {\lambda}):\Delta(x \cdot {\lambda}))$. Now apply the above corollary. \end{proof}
Let $w_{\circ} \in \mathcal{W}_{\lambda}$ be the longest element according to the Bruhat order. Then the multiplication from the left $$w_{\circ}: \mathcal{G} \longrightarrow \mathcal{G}^{\circ}, \,\,\, x\mathcal{W}' \mapsto w_{\circ} x \mathcal{W}'$$ is an isomorphism of moment graphs in the sense of \cite{12}. Thus, it induces a pull-back functor (\cite{12} definition 3.3.)
$$w_{\circ}^*: \mathcal{SH}_A(\mathcal{G}^{\circ})^f \longrightarrow \mathcal{SH}_A(\mathcal{G})^f$$
\begin{prop} $w_{\circ} ^*(\mathscr{B}^\downarrow (x)) = \mathscr{B}^\uparrow (w_{\circ}x)$ \end{prop}
\begin{proof} \cite{12} Lemma 5.1. \end{proof}
\begin{cor} $(K(w \cdot {\lambda}):\Delta(x \cdot {\lambda}))= P_{x,w}(1)$ where $P_{x,w}$ denotes the Kazhdan-Lusztig polynomial. \end{cor}
\begin{proof} Since $w_{\circ}^* \mathscr{B}^\downarrow(w) =\mathscr{B}^\uparrow(w_{\circ}w)$ we get $\mathrm{rk}_S \mathscr{B}^\downarrow(w)_x =\mathrm{rk}_S \mathscr{B}^\uparrow(w_{\circ} w)_{w_\circ x}$ and the result follows since the stalks of BMP-sheaves describe the local equivariant intersection cohomology of the according Schubert variety by \cite{3} Theorem 1.6. \end{proof}
\begin{remark} Character formulae for tilting modules were already discovered in \cite{A} for Kac-Moody algebras by using a tilting equivalence on category $\mathcal{O}$ which interchanges projective with tilting modules. Our approach, however, uses sheaves on moment graphs but only works for the finite dimensional case. \end{remark}
\section{The Jantzen and Andersen filtrations}
We fix a deformed tilting module $K \in \mathcal{K}_A$ and let ${\lambda} \in \mathfrak{h}^{*}$. The composition of homomorphisms induces an $A$-bilinear pairing \begin{eqnarray*}
\begin{array}{ccc}
\mathrm{Hom}_{\mathcal{O}_{A}}(\Delta_A({\lambda}),K)\times \mathrm{Hom}_{\mathcal{O}_{A}}(K,\nabla_A({\lambda})) & \longrightarrow & \mathrm{Hom}_{\mathcal{O}_{A}}(\Delta_A({\lambda}),\nabla_A({\lambda}))\cong A\\
\\
(\varphi,\psi) & \longmapsto & \psi \circ \varphi
\end{array} \end{eqnarray*}
For any $A$-module $H$ we denote by $H^{*}$ the $A$-module $\mathrm{Hom}_A(H,A)$. As in \cite{14} Section 4 one shows that for $A$ a localization of $S$ at a prime ideal $\mathfrak{p}$ our pairing is non-degenerated and induces an injective map $$E=E_A^{{\lambda}}(K): \mathrm{Hom}_{\mathcal{O}_{A}}(\Delta_A({\lambda}),K) \longrightarrow \left(\mathrm{Hom}_{\mathcal{O}_{A}}(K,\nabla_A({\lambda}))\right)^{*}$$ of finitely generated free $A$-modules.\\ If we take $A=\mathbb{C} \llbracket v \rrbracket$ the ring of formal power series around the origin on a line $\mathbb{C}\delta \subset \mathfrak{h}^{*}$ not contained in any hyper plane corresponding to a reflection of the Weyl group (e.g. $\mathbb{C} \rho$, where $\rho$ is the half sum of positive roots), we get a filtration on $\mathrm{Hom}_{\mathcal{O}_{A}}(\Delta_A({\lambda}),K)$ by taking the preimages of $\left (\mathrm{Hom}_{\mathcal{O}_{A}}(K,\nabla_A({\lambda})) \right )^{*}\cdot v^{i}$ for $i=0,1,2,...$ under $E$.
\begin{definition}[\cite{14}, Definition 4.2.] Given $K_\mathbb{C} \in \mathcal{K}_\mathbb{C}$ a tilting module of $\mathcal{O}$ and $K \in \mathcal{K}_{\mathbb{C} \llbracket v \rrbracket}$ a preimage of $K_\mathbb{C}$ under the functor $\cdot \otimes_{\mathbb{C} \llbracket v \rrbracket} \mathbb{C}$, which is possible by Proposition \ref{tilt1} with $S\rightarrow \mathbb{C} \llbracket v \rrbracket$ the restriction to a formal neighborhood of the origin in the line $\mathbb{C} \rho$. Then the image of the filtration defined above under specialization $\cdot \otimes_{\mathbb{C} \llbracket v \rrbracket} \mathbb{C}$ is called the \textit{Andersen filtration} on $\mathrm{Hom}_\mathfrak{g} (\Delta({\lambda}), K_\mathbb{C})$. \end{definition}
The Jantzen filtration on a Verma module $\Delta({\lambda})$ induces a filtration on the vector space $\mathrm{Hom}_\mathfrak{g}(P , \Delta({\lambda}))$, where $P$ is a projective object in $\mathcal{O}$. Now consider the embedding $\Delta_{\mathbb{C} \llbracket v \rrbracket}({\lambda}) \hookrightarrow \nabla_{\mathbb{C} \llbracket v \rrbracket}({\lambda})$. Let $P_{\mathbb{C} \llbracket v \rrbracket}$ denote the up to isomorphism unique projective object in $\mathcal{O}_{\mathbb{C} \llbracket v \rrbracket}$ that maps to $P$ under $\cdot \otimes_{\mathbb{C} \llbracket v \rrbracket}\mathbb{C}$, which is possible by theorem \ref{Fie1}. We then get the same filtration if we first take the preimages of $\mathrm{Hom}_{\mathcal{O}_{\mathbb{C} \llbracket v \rrbracket}}(P_{\mathbb{C} \llbracket v \rrbracket},\nabla_{\mathbb{C} \llbracket v \rrbracket}({\lambda})) \cdot v^{i}$, $i=0,1,2,...$, under the induced inclusion $$J=J_A^{{\lambda}}(P):\mathrm{Hom}_{\mathcal{O}_{A}}(P_A,\Delta_A({\lambda})) \longrightarrow \mathrm{Hom}_{\mathcal{O}_{A}}(P_A,\nabla_A({\lambda}))$$ for $A=\mathbb{C} \llbracket v \rrbracket$ and then the image of this filtration under the map\\
$\mathrm{Hom}_{\mathcal{O}_{A}}(P_A,\Delta_A({\lambda}))\twoheadrightarrow \mathrm{Hom}_{\mathfrak{g}}(P,\Delta({\lambda}))$ induced by $\cdot \otimes_A \mathbb{C}$.
\subsection{Sheaves with a Verma flag}
For $M \in \mathcal{Z}_A-\mathrm{mod}^f$ and $\mathcal{I} \subset \mathcal{V}$, we define $$M_\mathcal{I} := M \cap \bigoplus_{x\in \mathcal{I}} e_x (M \otimes_A Q)$$ and $$M^{\mathcal{I}}:= \mathrm{Im} \left( M \rightarrow M\otimes_A Q \rightarrow \bigoplus_{x\in \mathcal{I}} e_x (M \otimes_A Q) \right)$$
\begin{definition} We say that $M \in \mathcal{Z}_A -\mathrm{mod}^f$ admits a Verma flag if the module $M^{\mathcal{I}}$ is free (graded free in case $A=S$) for each F-open subset $\mathcal{I}$. \end{definition} Denote the full subcategory of $\mathcal{Z}_A(\mathcal{G})-\mathrm{mod}^f$ consisting of all modules admitting a Verma flag by $\mathcal{Z}_A(\mathcal{G})-\mathrm{mod}^{VF}$. Now for any vertex $x\in \mathcal{V}$ and an $A$-sheaf $\mathscr{M} \in \mathcal{SH}_A(\mathcal{G})^f$ define $$\mathscr{M}^{[x]}:= \mathrm{ker}(\mathscr{M}_x \rightarrow \bigoplus_{E \in U_x} \mathscr{M}_E)$$ where $U_x$ is from Notation \ref{not1}. Furthermore, denote by $$\mathscr{M}^{x}:= \bigcap_E \mathrm{ker}(\rho_{x,E})$$ the \textit{costalk} of $\mathscr{M}$ at the vertex $x$. Here the intersection runs over all edges $E \in \mathcal{E}$ with $x \in E$.\\ Denote the image of $\mathcal{Z}_A(\mathcal{G})-\mathrm{mod}^{VF}$ under $\mathcal{L}$ by $\mathcal{C}_A(\mathcal{G})$. The next proposition gives an explicit description of $\mathcal{C}_A(\mathcal{G})$
\begin{prop}(\cite{C}, Proposition 2.9) For $M \in \mathcal{Z}_A(\mathcal{G})-\mathrm{mod}^f$, set $\mathscr{M}= \mathcal{L}(M)$. Then $M$ admits a Verma flag if and only if $\mathscr{M}$ is flabby and $\mathscr{M}^{[x]}$ is (graded) free for all $x \in \mathcal{V}$. \end{prop}
In \cite{8} section 2.6 Fiebig introduces a duality $D$ on $\mathcal{Z}_S-\mathrm{mod}^f$. For $M \in \mathcal{Z}_S-\mathrm{mod}^f$ we set $$D(M)= \bigoplus_{i\in \mathbb{Z}} \mathrm{Hom}_S^i(M,S)$$ where $\mathrm{Hom}_S^i(M,S) = \mathrm{Hom}_S(M,S[i])$ ($[\cdot]$ grading shift). This induces an equivalence $D : \mathcal{C}_S(\mathcal{G}) \stackrel{\sim}{\rightarrow} \mathcal{C}_S^{op}(\mathcal{G}^{\circ})$, where $\mathcal{G}^{\circ}$ denotes the moment graph $\mathcal{G}$ with reversed order.\\ For $A$ a localization of $S$ at a prime ideal we define $D_A:= \mathrm{Hom}_A(\cdot, A): \mathcal{C}_A(\mathcal{G}) \stackrel{\sim}{\rightarrow} \mathcal{C}_A^{op}(\mathcal{G}^{\circ})$.
\begin{thm}(\cite{8}, Theorem 6.1 and \cite {D}, Proposition 3.20) The BMP-sheaves are self-dual: $D \mathscr{B}^\uparrow (x) \cong \mathscr{B}^\uparrow (x)$ for all $x\in \mathcal{V}$. \end{thm}
Recall the pull-back functor of section \ref{BMP} $$w_{\circ}^*: \mathcal{SH}_A(\mathcal{G}^{\circ})^f \longrightarrow \mathcal{SH}_A(\mathcal{G})^f$$
\begin{lemma} We have $w_{\circ}^*(\mathcal{C}_A(\mathcal{G}^{\circ}))=\mathcal{C}_A(\mathcal{G})$. \end{lemma}
\begin{proof} Let $\mathscr{M} \in \mathcal{C}_A(\mathcal{G}^{\circ})$. We have to show that $w_{\circ}^*(\mathscr{M})$ is $\uparrow$-flabby and $(w_{\circ}^*(\mathscr{M}))^{[x]}$ is free over $A$ for $x\in \mathcal{V}$.\\ Let $\mathcal{I}$ be $\downarrow$-open. Then $w_{\circ}\mathcal{I}$ is $\uparrow$-open and we get that $$\Gamma(\mathscr{M})\cong \Gamma(w_{\circ}^*(\mathscr{M})) \rightarrow \Gamma(\mathcal{I}, w_{\circ}^*(\mathscr{M})) \cong \Gamma(w_{\circ} \mathcal{I}, \mathscr{M})$$ is surjective since $\mathscr{M}$ is flabby.\\ As $(w_{\circ}^*(\mathscr{M}))^{[x]}=\mathscr{M}^{[w_{\circ}x]}$ the claim follows. \end{proof}
We get an equivalence of categories $$w_\circ ^*:\mathcal{C}_A(\mathcal{G}^{\circ}) \longrightarrow \mathcal{C}_A(\mathcal{G})$$ with the properties: $w_\circ ^*(\mathscr{B}^\uparrow (x)\otimes_S A) \cong \mathscr{B}^\downarrow (w_\circ x)\otimes_S A$ and $w_\circ ^*(\mathscr{V}_A (x)) \cong \mathscr{V}_A (w_\circ x)$. Thus the composition $$F_A= (w_\circ ^*)^{op} \circ D_A : \mathcal{C}_A(\mathcal{G}) \longrightarrow \mathcal{C}_A(\mathcal{G})^{op}$$ is an equivalence with $F_A(\mathscr{B}^\uparrow (x)\otimes_S A) \cong \mathscr{B}^\downarrow (w_\circ x)\otimes_S A$ and $F(\mathscr{V}_A (x)) \cong \mathscr{V}_A (w_\circ x)$ which are isomorphisms of graded sheaves if $A=S$.\\
\begin{thm}(\cite{7}, Theorem 7.1.) The functor $\mathbb{V} : \mathcal{M}_{A,{\lambda}} \rightarrow \mathcal{C}_A(\mathcal{G})$ is an equivalence of categories for $A=S_{(0)}$. \end{thm}
Now we can lift the functor $F_A$ via Fiebig's equivalence to a functor $T_A$ on the representation theoretic side such that the following diagram of functors commutes:
\begin{eqnarray*}
\begin{CD}
\mathcal{M}_{A,{\lambda}} @> \mathbb{V} >> \mathcal{C}_A(\mathcal{G})\\
@VT_A VV @VVF_A V\\
\mathcal{M}^{op}_{A,{\lambda}} @> \mathbb{V} ^{op} >> \mathcal{C}_A(\mathcal{G})^{op}
\end{CD} \end{eqnarray*}
\begin{thm} Let ${\lambda}\in \mathfrak{h}^{*}$ be antidominant, $x,y \in \mathcal{W}$ and $w_\circ \in \mathcal{W}$ the longest element. Denote by $A=S_{(0)}$ the localization of $S$ at $0$. There exists an isomorphism $L=L_A(x,y)$ which makes the diagram
{\small \begin{eqnarray*}\label{eqn:DiaS}
\begin{CD}
\mathrm{Hom}_{\mathcal{O}_A}(P_A(x \cdot {\lambda}),\Delta_A(y \cdot {\lambda})) @>J>> \mathrm{Hom}_{\mathcal{O}_A}(P_A(x \cdot {\lambda}),\nabla_A(y \cdot {\lambda}))\\
@VVT V @VVL V\\
\mathrm{Hom}_{\mathcal{O}_A}(\Delta_A(w_\circ y \cdot {\lambda}),K_A(w_\circ x \cdot {\lambda})) @>E >> \left(\mathrm{Hom}_{\mathcal{O}_A}(K_A(w_\circ x \cdot {\lambda}),\nabla_A(w_\circ y \cdot {\lambda}))\right)^{*}
\end{CD} \end{eqnarray*}} commutative. Here $J=J_A^{y\cdot {\lambda}}(P_A(x \cdot{\lambda}))$ and $E=E_A^{w_\circ y \cdot {\lambda}}(K_A(w_\circ x \cdot {\lambda}))$ denote the inclusions defined above and $T=T_A$ denotes the isomorphism induced by the functor $T_A$ from above. \end{thm}
\begin{proof} The proof is essentially the same as the one for Theorem 4.2. in \cite{11}, where we prove a similar result for $T_A$ the tilting functor. \end{proof} Denote by $T_\mathbb{C}:\mathrm{Hom}_\mathfrak{g}(P(x \cdot {\lambda}),\Delta(y \cdot {\lambda})) \stackrel{\sim}{\rightarrow} \mathrm{Hom}_\mathfrak{g}(\Delta(w_\circ y \cdot {\lambda}),K(w_\circ x \cdot {\lambda}))$ the isomorphism we get from $T_A \otimes_A \mathrm{id}_\mathbb{C}$ after base change. The next corollary now follows in the same way as Corollary 4.3. in \cite{11}. \begin{cor} The isomorphism $$T_\mathbb{C}:\mathrm{Hom}_\mathfrak{g}(P(x \cdot {\lambda}),\Delta(y \cdot {\lambda})) \stackrel{\sim}{\rightarrow} \mathrm{Hom}_\mathfrak{g}(\Delta(w_\circ y \cdot {\lambda}),K(w_\circ x \cdot {\lambda}))$$ identifies the filtration induced by the Jantzen filtration with the Andersen filtration. \end{cor}
Now we consider $\mathbb{C} \cong S/S\mathfrak{h}$ as a simple graded $S$-module living in degree $0$. The map $$T_\mathbb{C}:\mathrm{Hom}_\mathfrak{g}(P(x \cdot {\lambda}),\Delta(y \cdot {\lambda})) \stackrel{\sim}{\rightarrow} \mathrm{Hom}_\mathfrak{g}(\Delta(w_\circ y \cdot {\lambda}),K(w_\circ x \cdot {\lambda}))$$ can then be identified with $$F_S \otimes \mathrm{id}_\mathbb{C}: \mathrm{Hom}_{\mathcal{C}_S(\mathcal{G})}(\mathscr{B}^\uparrow (x), \mathscr{V}_S (y))\otimes_S \mathbb{C} \stackrel{\sim}{\rightarrow} \mathrm{Hom}_{\mathcal{C}_S(\mathcal{G})} (\mathscr{V}_S (w_{\circ} y),\mathscr{B}^\downarrow (w_{\circ} x))\otimes_S \mathbb{C}$$ which is now an isomorphism of graded $\mathbb{C}$-vector spaces. But using the proof of proposition 7.1. (3) in \cite{6} this isomorphism becomes a graded isomorphism between certain costalks of the Braden-MacPherson sheaves, namely an isomorphism $$\varphi: \mathscr{B}^\uparrow (x)^y \otimes_S \mathbb{C} \stackrel{\sim}{\rightarrow} \mathscr{B}^\downarrow (w_{\circ} x)^{w_{\circ}y}\otimes_S \mathbb{C}$$
In \cite{14} Soergel shows that the filtration on $\mathscr{B}^\downarrow (w_{\circ} x)^{w_{\circ}y}\otimes_S \mathbb{C}$ induced by the Andersen filtration coincides with the grading filtration we get from the grading on the Braden-MacPherson sheaf $\mathscr{B}^\downarrow (w_{\circ} x)$. Since the graded isomorphism $\varphi$ interchanges the filtration on $\mathscr{B}^\uparrow (x)^y \otimes_S \mathbb{C}$ induced by the Jantzen filtration with the filtration on $\mathscr{B}^\downarrow (w_{\circ} x)^{w_{\circ}y}\otimes_S \mathbb{C}$ induced by the Andersen filtration, we get that the Jantzen filtration coincides with the grading filtration coming from the grading on the Braden-MacPherson sheaf $\mathscr{B}^\uparrow (x)$.
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\thispagestyle{empty} \title{On colouring point visibility graphs}
\begin{abstract} In this paper we show that it can be decided in polynomial time whether or not the visibility graph of a given point set is $4$-colourable, and such a $4$-colouring, if it exists, can also be constructed in polynomial time. We show that the problem of deciding whether the visibility graph of a point set is $5$-colourable, is NP-complete. We give an example of a point visibility graph that has chromatic number $6$ while its clique number is only $4$. \end{abstract} \section{Introduction} The visibility graph is a fundamental structure studied in the field of computational geometry and geometric graph theory \cite{bcko-cgaa-08,g-vap-07}.
Some of the early applications of visibility graphs included computing Euclidean shortest paths in the presence of obstacles \cite{lw-apcf-79} and decomposing two-dimensional shapes into clusters \cite{sh-ddsg-79}. Here, we consider problems concerning the colouring of visibility graphs. $\\ \\$ Let $P$ be a set of points $\{p_1, p_2, \ldots, p_n \}$ in the plane. Two points $p_i$ and $p_j$ of $P$ are said to be \emph{mutually visible} if there is no third point $p_k$ on the line segment joining $p_i$ and $p_j$. Otherwise, $p_i$ and $p_j$ are said to be mutually \emph{invisible}. The \emph {point visibility graph} (denoted as PVG) $G(V,E)$ of $P$ is defined as follows. The set $V$ of vertices contains a vertex $v_i$ for every
point $p_i$ in $P$. The set $E$ contains an undirected edge $v_iv_j$ if and only if the corresponding points $p_i$ and $p_j$ are mutually visible \cite{prob-ghosh}. Point visibility graphs have been studied in the contexts of construction \cite{cgl-pgd-85,Edelsbrunner:1986:CAL}, recognition \cite{pvg-card,prob-ghosh,pvg-tcs,pvg-np-hard}, partitioning \cite{pvg_part}, connectivity \cite{viscon-wood-2012}, chromatic number and clique number \cite{penta-2013,kpw-ocnv-2005,p-vgps-2008}. $\\ \\$ A graph is said to be \emph{$k$-colourable} if each vertex of the graph can be assigned a colour, so that no two adjacent vertices are assigned the same colour, and the total number of distinct colours assigned to the vertices is at most $k$. K\'{a}ra et al characterized PVGs that are $2$-colourable and $3$-colourable \cite{kpw-ocnv-2005}.
It was not known whether the
chromatic number of a PVG can be found in polynomial time.
In Section \ref{sec5col} we show that the
problem of deciding whether a PVG is $k$-colourable, for $k \geq 5$, is NP-complete. $\\ \\$ K\'{a}ra et al also asked whether there is a function $f$ such that for every point visibility graph $G$, $\chi (G) \leq f(\omega (G))$ \cite{kpw-ocnv-2005}? \label{prob3} They presented a family of PVGs that have their chromatic number lower bounded by an exponential function of their clique number. Their question was answered by Pfender, showing that for a PVG with $\omega (G) = 6$, $\chi (G)$ can be arbitrarily large \cite{p-vgps-2008}. However, it is not known whether the chromatic number of a PVG is bounded, if its clique number is only $4$ or $5$. In another related paper, Cibulka et al showed that PVGs of point sets $S$ such that there is no convex pentagon with vertices in $S$ and no other point of $S$ lying in the pentagon, might have arbitrarily large clique numbers \cite{penta-2013}. In this direction, K\'{a}ra et al showed that there is a PVG $G$ with $\omega (G) = 4$ and $\chi (G) = 5$ \cite{kpw-ocnv-2005}. In Section \ref{secexample2} we construct a PVG $G'$ with $\omega (G') = 4$ and $\chi (G') = 6$. \section{Four-colouring} \label{sec4col} In this section, we provide a polynomial-time algorithm to decide if the PVG of a given point set is 4-colourable, and construct a 4-colouring if it exists. Consider a finite set $P$ of $n$ points in the Euclidean plane. We start with a brief overview of our algorithm: \begin{enumerate} [(i)]
\item Check if $P$ is 3-colourable. If $P$ is 3-colourable then construct the 3-colouring and terminate.
\item Find a convex hull vertex of $P$ that forms a $K_4$ with three other vertices. Delete this convex hull vertex from $P$.
Repeat this step until there is no such convex hull vertex.
\item There are at most eight possible colourings of the reduced $P$. Check if any of these colourings is valid. If none of
then are valid then output ``NO'' and terminate.
\item Consider each of the valid colourings and progressively add the deleted points to $P$, in the reversed order of their deletion.
With each addition, colour the added point and check if the colouring is valid. If it is not valid then output ``NO'' and terminate. \end{enumerate}
In the next section we provide a proof of correctness and analysis of our algorithm. \subsection{Correctness of the algorithm} The algorithm begins with checking for 3-colourability. This can be done in polynomial time due to K\'{a}ra et al \cite{kpw-ocnv-2005}. We present the following lemma and theorem from K\'{a}ra et al verbatim without proof. Note that in Lemma \ref{kara1}, $V(P)$ being planar actually means that $V(P)$ drawn on $P$ is plane. \begin{figure}
\caption{(a) }
\label{figocta}
\end{figure}
\lemma \label{kara1} \textbf{(K\'{a}ra et al \cite{kpw-ocnv-2005})}
Let $P$ be a point set. Then $V(P )$ is planar if and only if at least one of the following conditions hold: \begin{enumerate} [(a)] \item all the points in $P$ are collinear, \item all the points in $P$ , except for one, are collinear, \item all the points in $P$ are collinear, except for two non-visible points, \item all the points in $P$ are collinear, except for two points $v$, $w \in P$ , such that the line-segment $vw$ does not intersect the line-segment that contains $P \setminus \{ v, w \}$, \item $V(P)$ is the drawing of the octahedron shown in Figure \ref{figocta}. \end{enumerate}
\theorem
\label{karatheo} \textbf{(K\'{a}ra et al \cite{kpw-ocnv-2005})}
Let $P$ be a finite point set. Then the following are equivalent:
\begin{enumerate} [(i)] \item $\chi (V(P )) \leq 3$, \item $P$ satisfies conditions (a), (b), (c) or (e) in Lemma \ref{kara1}, \item $V(P )$ has no $K_4$ subgraph. \end{enumerate}
\emph{ If the algorithm finds $P$ to be 3-colourable, then it produces a 3-colouring and terminates. Suppose that the algorithm finds that $P$ is not 3-colourable. Then the algorithm proceeds to the next step. It deletes any convex hull vertex that sees three mutually visible points in the rest of $P$. It continues this process till no such convex hull vertex is left. We call the resultant point set the \emph{reduced set} $P_r$. The set $P_r$ can be obtained from $P$ in $O(n^4)$ time. We have the following lemma.} $\\ \\$
\lemma \label{lemrestr}
$P$ is 4-colourable only if $P_r$ is 4-colourable, and given a 4-colouring of $P_r$, it can be found in polynomial time
if it is a 4-colouring of $P$ restricted to $P_r$
\proof The contrapositive of the first part is easy to see, since the PVG of $P_r$ is an induced subgraph of the PVG of $P$. $\\ \\$ Consider the deleted points of $P$ in the reverse order of their deletion. Since each deleted point sees a $K_3$ in the remaining points of $P$, its colour must be uniquely determined by the remaining points of $P$. Given a 4-colouring of $P_r$, we can add the deleted points in the reverse order of their deletion, colour them and check if the colouring is valid. It takes $O(n^3)$ time for each point to locate its corresponding $K_3$, and $O(n^2)$ time to check if the colouring is valid. So, the total procedure takes $O(n^4)$ time. \qed $\\ \\$ Now the algorithm checks if $P_r$ is 4-colourable. First it checks if $P_r$ is 3-colourable. According to the characterization in Theorem \ref{karatheo}, this can be achieved in polynomial time. Furthermore, we have the following lemmas.
\lemma \label{k3}
A reduced set must contain a $K_3$.
\proof A reduced set is obtained only after progressively deleting all convex hull vertices which see a $K_3$. So, after the final deletion step, the $K_3$ which the deleted point saw must remain. \qed
\lemma \label{unq3col}
If a reduced set is 3-colourable, then it requires three colours, and has a unique 3-colouring.
\proof By Theorem \ref{karatheo}, any 3-colourable PVG must be of the forms (a), (b), (c) or (e) of Lemma \ref{kara1}. Among these, the PVGs of the forms (b), (c) or (e), require 3-colours and have unique 3-colourings. A reduced set can never be of the form (a), i.e. all collinear points, because by Lemma \ref{k3} every reduced set must contain a $K_3$. \qed \begin{figure}
\caption{Three 4-colourings of point $p$ added to reduced set of type (b) in Lemma \ref{kara1}. }
\label{figcol1}
\end{figure} \begin{figure}
\caption{Four colourings of point $p$ added to reduced set of type (c) in Lemma \ref{kara1}. }
\label{figcol2}
\end{figure}
\lemma \label{cont4col}
A reduced 3-colourable set is no more 3-colourable if its last deleted point is added to it. It then requires four colours and can have no more than a constant number of
4-colourings.
\proof Suppose that the last deleted point $p$ is added back to the reduced set. Due to Lemma \ref{unq3col} the reduced set can only be of three types. If the reduced set is of type (b) in Lemma \ref{kara1}, then the three colourings in Figure \ref{figcol1} are the only possibilities. If the reduced set is of type (c) in Lemma \ref{kara1}, then the four colourings in Figure \ref{figcol2} are the only possibilities. If the reduced set is of type (e) in Lemma \ref{kara1}, then if has only a constant number of points and hence a constant number of 4-colourings. \qed $\\ \\$ If the algorithm finds $P_r$ to be 3-colourable, then it adds the last deleted point to $P_r$, and due to Lemma \ref{cont4col} constructs a constant number of 4-colourings in polynomial time. For each 4-colouring of $P_r$, the algorithm then reintroduces the deleted points of $P$ progressively in the reversed order of their deletion, and by Lemma \ref{lemrestr}, constructs a 4-colouring of $P$ if it exists, in $O(n^3)$ time. Suppose that the algorithm finds that $P_r$ is not 3-colourable. Then it checks whether $P_r$ is 4-colourable. Now, we describe the structure of reduced sets that are 4-colourable but not 3-colourable. $\\ \\$ Let $p_x$ be a vertex of the convex hull of $P_r$. All the other points of $P_r$ lie on rays emanating from $p_x$. Here we consider only \emph{open rays} emanating from $p_x$, i.e. rays that do not contain their initial point $p_x$. Let $r_1, r_2, \ldots r_k$ be the rays emanating from $p_x$ in the clockwise order, with the rays $r_1$ and $r_k$ respectively being tangents from $p_x$ to the convex hull of $P \setminus \{ p_x \}$. Let $q_i$ be the closest point on $r_i$ to $p_x$. We call the path $(q_1, q_2, \ldots, q_k)$ the \emph{frontier} of $p_x$. The points $q_1, q_2, \ldots, q_k$ are called the \emph{frontier points} of $p_x$. The points $q_2, q_3, \ldots, q_{k-1}$ are called the \emph{internal frontier points} of $p_x$. A continuous subpath $(q_i, q_{i+1}, q_{i+2})$ of the frontier is said to be a \emph{convex triple} (or, \emph{concave}) when $q_{i+2}$,
lies to the right (respectively, left) of the ray $\overrightarrow{q_iq_{i+1}}$. If the continuous subpath is a straight line-segment then it is said to be a \emph{straight triple}. The frontier might have concave, straight and convex triples. We have the following lemmas. \begin{figure}
\caption{ (a) The convex hull points $p_x$ and $p_y$ have more than one point between them. (b) The convex hull points $p_x$ and $p_y$ have exactly one point $q_x$ between them. }
\label{fignopoint}
\end{figure}
\lemma \label{noconv}
The following holds for each convex hull vertex of $P_r$:
\begin{enumerate} [(a)]
\item There is no concave triple in its frontier.
\item If there is a convex triple in its frontier then the ray containing the convex vertex has at least two points.
\end{enumerate}
\proof If three consecutive points of its frontier are concave with respect to $p_x$, then they together form a $K_4$. If three consecutive points of its frontier are convex, then they do not form a $K_3$ if and only if there is a blocker on the ray containing the convex vertex that prevents the first points of its two neighbouring rays from seeing each other. \qed
\lemma \label{lemintfront}
If a reduced set is not 3-colourable then each of its convex hull vertices has an internal frontier point.
\proof Suppose that a convex hull vertex $p_x$ of the reduced set $P_r$ does not have an internal frontier point. Then the rest of the points of $P_r$ must be lying on only two rays emanating from $p_x$. Denote these two rays as $r_1$ and $r_2$ and the points of $P_r$ on them farthest from $p_x$ as $p_1$ and $p_2$ respectively. If both $r_1$ and $r_2$ each have two or more points excluding $p_x$, then $p_1$, $p_2$ and the two points preceding them form a $K_4$. This is a contradiction, since $P_r$ is reduced and both $p_1$ and $p_2$ are convex hull vertices of $P_r$. If at least one of $r_1$ and $r_2$ has only one point excluding $p_x$ then $P_r$ is of the form (b) in Lemma \ref{kara1} and hence 3-colourable, a contradiction. \qed
\lemma \label{lemtype} If a reduced set is not 3-colourable, then all of its convex hull points are vertices.
\proof Assume on the contrary that the point set is not 3-colourable and not all the convex hull points are convex hull vertices. This means that there are two consecutive convex hull vertices $p_x$ and $p_y$ such that there is at least one convex hull point in the interior of the line segment joining them. Wlog assume that $p_x$ precedes $p_y$ in the clockwise order of points on the convex hull. There can be either more than one or exactly one point in the interior of $\overline{p_xp_y}$. $\\ \\$ First consider the case where there is more than one point in the interior of $\overline{p_xp_y}$ (Figure \ref{fignopoint}(a)). Let $q_x$ and $q_y$ be the points among them closest to $p_x$ and $p_y$ respectively. By Lemma \ref{lemintfront}, $p_x$ has an internal frontier point. Let $q_z$ be the first internal frontier point of $p_x$ following $q_x$. Suppose that $q_z$ is not an internal frontier point of $p_y$. Then there must be a point (say, $q_u$) on $\overline{p_yq_z}$ that is an internal frontier point of $p_y$. But then there must also be an internal frontier point $q_w$ of $p_x$ on $\overline{p_xq_u}$, contradicting the assumption that $q_z$ is the first internal frontier point of $p_x$. So, $q_z$ must be an internal frontier point of $p_y$. By Lemma \ref{noconv}, the frontier of $p_x$ cannot have concave triples, so that the rest of the frontier points of $p_x$ must lie on or to the left of $\overline{q_xq_z}$. Similarly, the frontier of $p_y$ cannot have concave triples, so that the rest of the frontier points of $p_y$ must lie on or to the right of $\overline{q_yq_z}$. But if there is a frontier point $q_t$ of $p_x$ on or to the left of $\overline{q_xq_z}$, then $p_y$ must have a frontier point on $\overline{p_yq_t}$ that is to the right of $\overline{q_yq_z}$, a contradiction. $\\ \\$ Now consider the case where there is exactly one point in the interior of $\overline{p_xp_y}$ and denote it as $q_x$ (Figure \ref{fignopoint}(b)). As before, we consider $q_z$ which is a frontier point of both $p_x$ and $p_y$. As before, by Lemma \ref{noconv}, the frontier of $p_x$ cannot have concave triples, so that the rest of the frontier points of $p_x$ must lie on or to the left of $\overline{q_xq_z}$. Similarly, the frontier of $p_y$ cannot have concave triples, so that the rest of the frontier points of $p_y$ must lie on or to the right of $\overline{q_xq_z}$. This means that all of the points of $P_r$ other than $p_x$ and $p_y$ must lie on $\overline{q_xq_z}$. But then $P_r$ is of the form (c) in Lemma \ref{kara1} and hence 3-colourable, a contradiction. \qed \cor \label{cor1} If a reduced set is not 3-colourable, and its convex hull has at least four vertices, then the interior of its convex hull is not empty.
\proof Suppose on the contrary the interior of the convex hull of such a reduced set $P_r$ is empty. By Lemma \ref{lemtype} $P_r$ has no convex hull points that are not convex hull vertices. So, all the convex hull vertices of $P_r$ see each other. Since $P_r$ has at least four convex hull points, all of them are a part of a $K_4$. Hence, $P_r$ is not reduced, a contradiction. \qed \begin{figure}
\caption{(a) The intersection of $\overline {p_1p_7}$ and $\overline {p_4p_5}$ forces another point on $\overline {p_4p_5}$. (b) A non 3-colourable reduced set with only three convex hull vertices}
\label{figstruct}
\end{figure}
\lemma \label{lemstruct}
If a reduced set is not 3-colourable, then
its convex hull has only three vertices.
\proof Suppose on the contrary that a reduced set $P_r$ is not 3-colourable and its convex hull has at least four vertices. Consider the lowest vertex of the convex hull of $P_r$, and its two adjacent vertices on the convex hull of $P_r$. Call these three points $p_1$, $p_2$ and $p_3$ in the clockwise order (Figure \ref{figstruct}(a)). Denote as $P'$ the set of points other than the convex hull vertices of $P_r$. $\\ \\$ Since by our assumption the convex hull of $P_r$ has at least four vertices, by Corollary \ref{cor1}, $P'$ must be nonempty. Suppose that all the points of $P'$ lie on $\overline {p_1p_3}$. If there are at least two convex hull vertices above $\overline {p_1p_3}$, then they can see two mutually visible points of $P'$, and hence $P_r$ is not reduced, a contradiction. If there is only one convex hull vertex above $\overline {p_1p_3}$, then either it sees $p_2$ and two mutually visible points of $P'$, which means $P_r$ is not reduced, or it is blocked from $p_2$, thereby making $P_r$ 3-colourable, a contradiction. So, not all the points of $P'$ lie on $\overline {p_1p_3}$. $\\ \\$ If all the points of $P'$ lie above $\overline {p_1p_3}$, then the lowermost point of $P'$ forms a $K_4$ with $p_1$, $p_2$ and $p_3$, a contradiction. Similarly, not all the points of $P'$ can lie below $\overline {p_1p_3}$ as well. So, the convex hull of $P'$ must intersect $\overline {p_1p_3}$. Suppose that the convex hull of $P'$ intersects $\overline {p_1p_3}$ at only one point, say $p_i$. If there are points of $P'$ to the left or right of the ray $\overrightarrow{p_2p_i}$, then among such points let $p_j$ be a point closest to $\overline {p_1p_3}$. Wlog if $p_j$ lies to the right of $\overrightarrow{p_2p_i}$ then it sees both $p_i$ and $p_1$. But $p_2$ sees $p_1$, $p_i$ and $p_j$, so $P_r$ is not reduced, a contradiction. Otherwise, all the points of $P'$ lie on the ray $\overrightarrow{p_2p_i}$. If a fourth convex hull vertex of $P_r$ above $\overline {p_1p_3}$
lies on $\overrightarrow{p_2p_i}$, then $P_r$ is 3-colourable, a contradiction. Otherwise, wlog let this fourth convex hull vertex lie to the left of $\overrightarrow{p_2p_i}$. Let $p_j$ be the point of $P'$ immediately after $p_i$ on $\overrightarrow{p_2p_i}$. Then $p_3$ forms a $K_4$ with $p_i$, $p_j$ and the fourth convex hull vertex, so $P_r$ is not reduced, a contradiction. So, the convex hull of $P'$ must intersect $\overline {p_1p_3}$ at two points. $\\ \\$ Let $\overline {p_4p_5}$ and $\overline {p_6p_7}$ be the segments of the convex hull of $P'$ intersecting $\overline {p_1p_3}$, where $p_4$ and $p_5$ (respectively, $p_6$ and $p_7$) are consecutive points on the convex hull of $P'$. Also assume that $p_4$, $p_5$, $p_6$ and $p_7$ are in the clockwise order on the convex hull of $P'$, with none of the segments $\overline {p_1p_4}$ $\overline {p_1p_5}$ $\overline {p_3p_6}$ and $\overline {p_3p_7}$ intersecting the convex hull of $P'$. $\\ \\$ Consider the segment $\overline {p_3p_4}$. Suppose that $\overline {p_3p_4}$ and $\overline {p_6p_7}$ intersect. To prevent a concave frontier of $p_3$ from forming, there must be a point of $P'$ on the intersection of $\overline {p_3p_4}$ and $\overline {p_6p_7}$. But that is not possible because $p_6$ and $p_7$ are consecutive points on the convex hull of $P'$. Thus, $\overline {p_6p_7}$ must lie to the right of $\overrightarrow {p_3p_4}$. But then, $\overline {p_4p_5}$ and $\overline {p_1p_7}$ must intersect. So, there must be another point of $P'$ on $\overline {p_4p_5}$, which is a contradiction to $p_4$ and $p_5$ being consecutive points on the convex hull of $P'$ (Figure \ref{figstruct}(a)). Hence, the convex hull of $P$ can have at most three points (Figure \ref{figstruct}(b)). \qed $\\ \\$ Let $P_r$ be a reduced set and $p_x$ be a convex hull vertex of $P_r$. We will henceforth refer to the four colours as red, blue, yellow and green. If a ray emanating from a convex hull vertex of a reduced set has only one point, we call it a \emph{small ray}. Otherwise, we call it a \emph{big ray}. On a ray, the point closest to $p_x$ is called its \emph{first point}, next closest point to $p_x$ is called its \emph{second point} and so on. We have the following lemmas.
\lemma \label{noblock}
The first point of any ray emanating from a convex hull vertex of a reduced set can block only first points of other rays from each other.
\proof Consider three rays $r_i$, $r_j$ and $r_k$ emanating from a convex hull vertex $p_x$ lying in clockwise order around it. Let $p_2$ be the first point of $r_j$. Suppose that $p_2$ blocks $p_1$ and $p_3$ lying on $r_i$ and $r_k$ respectively, and wlog $p_1$ is not the first point of $r_i$. Let $p_i$ be the first point of $r_i$. If both of the triangles $\bigtriangleup p_ip_2p_x$ and $\bigtriangleup p_2p_3p_x$ are empty, then $p_x$ forms a $K_4$ with $p_i$, $p_2$ and $p_r$, hence $P_r$ is not reduced. So, at least one of the two triangles must be nonempty. Wlog suppose that $\bigtriangleup p_1p_2p_x$ is nonempty. Let $p_4$ be a point contained in $\bigtriangleup p_ip_2p_x$ such that no other point contained in $\bigtriangleup$ is closer to $\overline {p_1p_2}$. Then if $\bigtriangleup p_2p_3p_x$ is empty then $p_4$ forms a $K_4$ with $p_x$, $p_2$ and $p_3$. If $\bigtriangleup p_2p_3p_x$ too is nonempty then let analogously $p_5$ be a point contained in $\bigtriangleup p_2p_3p_x$ that is a point closest to $\overline {p_1p_2}$. Again, $p_3$, $p_4$, $p_5$ and $p_i$ form a $K_4$, so $P_r$ is not reduced, a contradiction. \qed
\begin{figure}
\caption{ (a) The break is even and $r_x$ has a red point. (b) The break is even and $r_x$ does not have a red point. (c) The break is odd and $r_x$ has a red point. (d) The break is odd and $r_x$ does not have a red point.}
\label{fig1col}
\end{figure}
\lemma \label{2must} A reduced set that is not 3-colourable must have at least two big rays emanating from each of its convex hull vertices.
\proof Consider a reduced set $P_r$ that is not 3-colourable.
Suppose that there is a convex hull vertex $p_x$ of $P_r$ such that only one big ray $r_x$ emanates from it. First suppose that $r_x$ has only two points. By Lemma \ref{noconv}, the frontier of $p_x$ is either convex or a straight line. If the frontier of $p_x$ is a straight line, then $P_r$ is 3-colourable, a contradiction. So, the frontier of $p_x$ must be convex. Suppose the first point of some small ray is a convex point in the frontier of $p_x$. Then, the last and second-last points of the frontier of $p_x$ in the same side of $r_x$, form a $K_4$ with the two points of $r_x$. Among the four points forming the $K_4$, the last point of the frontier or the second point of $r_x$ is a convex hull vertex of $P_r$. Then $P_r$ is not reduced, a contradiction. So, the first point of $r_x$ must be the only convex point in the frontier of $p_x$ Denote the two neighours of the first point of $r_x$ on the frontier of $p_a$ as $p_1$ and $p_b$. If $p_a$ and $p_b$ see each other then they form a $K_4$ with $p_x$ and the first point of $r_x$. Then since $p_x$ sees a $K_3$, $P_r$ is not reduced, a contradiction. Hence, $p_a$ and $p_b$ must be blocked from each other by the second point of $r_x$. The frontier of $p_x$ must have more points, for otherwise $P_r$ is 3-colourable. But now, each point of the frontier sees both the points of $r_x$. One of the end points of the frontier must be a convex hull vertex. This end point forms a $K_4$ with both points of $r_x$ and the second-last point of the frontier in the same side. Hence $P_r$ is not reduced, a contradiction. $\\ \\$ Now suppose that $r_x$ has at least three points. If the frontier of $p_x$ has two or more points in the same side of $r_x$, then the last and second-last points of the frontier in that side form a $K_4$ with the last and second-last points of $r_x$. Since the last point of the frontier or the last point of $r_x$ is a convex hull vertex, $P_r$ is not a reduced set, a contradiction. Then the frontier of $p_x$ can have at most one point on each side of $r_x$. If there is no point on one side of $r_x$, then $P_r$ is 3-colourable. So the frontier of $p_x$ must have exactly one point on each side of $r_x$. If these two end points of the frontier do not see each other then again $P_r$ is 3-colourable. So, the two end points of the frontier see each other. But they also see the last and second-last points of $r_x$, which means that $P_r$ is not reduced, a contradiction.
\qed
\lemma \label{lem3colray} In any 4-colouring of a reduced set that is not 3-colourable, any big ray emanating from a convex hull vertex has exactly two colours. \proof Let $p_x$ be a convex hull vertex and $\{r_1, r_2, \ldots, r_k \}$ be the open rays emanating from it in the clockwise angular order. Suppose that a big ray $r_x$ emanating from $p_x$ has three points that are assigned three different colours. By Lemma \ref{2must}, there is another big ray with respect to $p_x$. Consider a big ray $r_y$ such that there is no other big ray between $r_x$ and $r_y$, and $y<x$, without loss of generality. If $r_x$ and $r_y$ are neighbouring rays, i.e. $y=x-1$, then both of the first and second points of $r_y$ must be assigned the fourth colour, which is not possible. Suppose that there is at least one small ray between $r_x$ and $r_y$. Then the first point of the ray $r_{y+1}$ forms a $K_5$ with the first two points of $r_y$ and the second and third points of $r_x$, a contradiction. \qed $\\ \\$ We call the occurrence of a small ray after a big ray, or vice versa, a \emph{break}. If there are a consecutive odd number of small rays before or after the break, it is called an \emph{odd break}. If there are a consecutive even number of small rays before or after the break, it is called an \emph{even break}. We have the following lemmas.
\lemma \label{fixcol}
A reduced set that is not 3-colourable can have at most a constant number of 4-colourings. \proof By Lemma \ref{2must}, in such sets, at least two big rays emanate from every convex hull vertex. We consider a reduced set $P_r$. Suppose that $P_r$ is 4-colourable. Consider a convex hull vertex $p_x$ of $P_r$. By our assumption, at least two big rays must emanate from $p_x$. Wlog assign red to $p_x$. Consider any big ray $r_x$ and suppose that it contains the colour red. Due to Lemma \ref{lem3colray}, each ray can be assigned at most two colours. Wlog let the other colour of $r_x$ be blue. Observe that since $p_x$ is red, the first point of $r_x$ must be blue, the second point red and so on. If a neighbouring ray of $r_x$ is also big, then it must have yellow and green alternatingly assigned to its points, but either of yellow and green can be assigned to its first point. In general, till a break occurs, every alternate big ray must contain red points. Furthermore, the second point of every such big ray must be red and red points should occur alternatingly on it. Hence, our initial choice of $r_x$ as a ray containing red points fixes the assignment of red till a break occurs. $\\ \\$ Suppose that we have one or more consecutive big rays in which the assignment of red is fixed, and a break occurs after a big ray $r_x$. Suppose that this break is even. Call the first big ray occuring after the break $r_y$. Suppose that $r_x$ contains red points. Wlog let the other colour of $r_x$ be blue. (Figure \ref{fig1col} (a)). Then the second point of $r_y$ must be assigned either yellow or green. Suppose that it is yellow. Then the colour of the only point in the first small ray occuring in the break is determined by the first and second points of $r_x$ and the second point of $r_y$, and it must be green. Similarly, the next small ray gets blue. The first point of $r_y$ gets green. Thus, yellow and green alternate throughout $r_y$ and if a neighbouring ray of $r_y$ is big, then it must contain a red point. $\\ \\$ Suppose that $r_x$ does not contain any red point (Figure \ref{fig1col} (b)), and wlog the first and second points of $r_x$ are yellow and green respectively. Then the only points of the small rays between $r_x$ and $r_y$ must be assigned blue and yellow alternately. The first and second points of $r_y$ must be assigned blue and red respectively. So, the assigment of red to points in rays containing an even break depends only on whether or not $r_x$ contains a red point. $\\ \\$ Suppose that a break is odd. Suppose that the break starts after $r_x$, and ends at $r_y$, both being big rays. Supose that $r_x$ contains a red point (Figure \ref{fig1col} (c)). Wlog let the other colour of $r_x$ be blue. Then the first and second points of $r_x$ must be blue and red respectively. Then the second point of $r_y$ can be either yellow or green. Wlog let it be yellow. Then green and blue must be alternately assigned to the only points of the small rays in between, and the first point of $r_y$ must be blue. Thus, $r_y$ does not contain a red point. $\\ \\$ Now suppose that $r_x$ does not contain any red point (Figure \ref{fig1col} (d)). Wlog suppose that the first and second points of $r_x$ are yellow and green respectively. Then the second point of $r_y$ must be red, and the first points of all rays till $r_y$ must be assigned blue and yellow alternately. $\\ \\$ In all cases, the points that are assigned red are fixed. Thus, the assignment of red has only two possibilities, which depend on our initial choice of whether or not $r_x$ contains a red point. Now, by Lemma \ref{lemtype}, all the convex hull points of a reduced set are also its convex hull vertices, and by Lemma \ref{lemstruct} it can have at most three convex hull vertices. In a 4-colouring, these three convex hull vertices must be assigned three distinct colours. For each of these three colours, there are at most two possible assignments to the rest of the points of $P_r$. Each assignment of three colours also fixes the assignment of the fourth colour. This means that there are at most eight possible four colourings. \qed $\\ \\$ Thus, our algorithm checks if any of the eight colourings are valid 4-colourings, and then adds the deleted points back to $P_r$, assigning them their unique colours. Finally, we sum up our algorithm in the following theorem. \theorem It can be determined in polynomial time if a point set has a 4-colouring. Such a 4-colouring, if it exists, can be constructed in polynomial time.
\proof If the given point set $P$ is 3-colourable then it is be identified and 3-coloured in $O(n^2)$ time due to Theorem \ref{karatheo}. If $P$ is not 3-colourable then it is reduced to $P_r$ in $O(n^4)$ time by the method of Lemma \ref{lemrestr}. $\\ \\$ If the reduced set $P_r$ is 3-colourable by Lemma \ref{unq3col}, then the last deleted point is added to it and a constant number of possible 4-colourings are constructed and checked due to Lemma \ref{cont4col}. Each 4-colouring is considered one by one, and each of the deleted point is added in the reversed order of its deletion, and assigned the unique colour determined by the $K_3$ it sees in the remaining point set. At each step, it is checked whether the 4-colouring is valid or not. If at any step two mutually visible points are forced to have the same colour, then $P$ does not have a 4-colouring, with the chosen 4-colouring of $P_r$. Otherwise, after adding all the deleted points, a 4-colouring is obtained in $O(n^3)$ time. $\\ \\$ If the reduced set $P_r$ is not 3-colourable, then three distinct colours are assigned to its convex hull vertices, and all eight possible 4-colourings are found in $O(n^2)$ time due to Lemma \ref{fixcol}. It is checked whether any of these colourings is a valid 4-colouring or not. If some valid 4-colourings are obtained, then each of them is considered one by one, and as before, the deleted points are added and coloured. If at any step two mutually visible points are forced to have the same colour, then $P$ does not have a 4-colouring, with the chosen 4-colouring of $P_r$. Otherwise, after adding all the deleted points, a 4-colouring is obtained in $O(n^3)$ time. The whole algorithm takes $O(n^4)$ time. $\\ \\$ \qed \section{$5$-colouring point visibility graphs} \label{sec5col}
In this section we prove that deciding whether a PVG with a given embedding is $5$-colourable, is NP-hard. We provide a reduction of 3-SAT to the PVG $5$-colouring problem. We use the reduction of 3-SAT to the $3$-colouring problem of general graphs. \subsection{$3$-colouring a general graph} \begin{figure}
\caption{(a) A variable gadget. (b) A clause gadget.}
\label{gadgets}
\end{figure} For convenience, we first briefly describe the reduction for the $3$-colouring of general graphs, considering the graph as an embedding $\xi$ of points and line segments in the plane \cite{cai-79}. Consider a 3-SAT formula $\theta$ with variables $x_1, x_2, \ldots, x_n$ and clauses $C_1, C_2, \ldots, C_m$. Suppose the corresponding graph is to be coloured with red, green and blue. Consider Figure \ref{gadgets}(a). It shows the variable-gadgets. The points representing a variable $x_i$ and its negation (say, $p(x_i)$ and $p(\overline{x_i})$, respectively) are adjacent to each other, making $n$ pairs altogether. No two points in different pairs are adjacent to each other. A separate point $p_b$ is wlog assumed to be blue and made adjacent to all the other points in the variable gadgets. So, each variable point must have exactly one red and one green point. For variable points, let green and red represent an assignment of true and false, respectively. The point $p_b$ is also adjacent to a separate point $p_r$ assumed to be red. $\\ \\$ Now consider Figure \ref{gadgets}(b). It shows a clause gadget. Suppose that the points $p_1$, $p_2$ and $p_6$ can be coloured only with green and red. Then $p_9$ can be coloured with green if and only if at least one of $p_1$, $p_2$ and $p_6$ are coloured with green. To prevent $p_9$ from being coloured red or blue, $p_9$ is made adjacent to $p_r$ and $p_b$. $\\ \\$ The whole embedding corresponding to the 3-SAT formula is shown in Figure \ref{gengraph}. The points $p_r$ and $p_b$ are the same for all variables and clauses. For each clause gadget, the points corresponding to $p_1$, $p_2$ and $p_6$ are in the respective variable gadgets. Thus, for $n$ variables and $m$ clauses, $\xi$ has $2n + 6m + 2$ points in total. \begin{figure}
\caption{The full embedding for a given 3-SAT formula.}
\label{gengraph}
\end{figure} \subsection{Transformation to a point visibility graph} Consider the following transformation of $\xi$ into a new embedding $\zeta$ (Figure \ref{finalemb}(a)) for a given 3-SAT formula $\theta$. We use some extra points called \emph{dummy points} to act as blockers during the transformation. \begin{enumerate} [(a)]
\item All points of $\xi$ are embedded on two vertical lines $l_1$ and $l_3$.
\item Two points $p_r$ and $p_b$ are placed on $l_3$ above all other points of $\xi$, followed by a dummy point.
\item Each pair of variable gadget points are embedded as consecutive points on $l_1$.
\item Separating a variable gadget from the next variable gadget is a dummy point.
\item For every clause gadget, the points corresponding to $p_5$ and $p_9$ are on $l_1$,
separated by a dummy point.
\item For every clause gadget, the points corresponding to $p_3$, $p_4$, $p_7$ and $p_8$
are on $l_3$, in the vertical order from top to bottom. The points $p_3$ and $p_4$ are consecutive.
The points $p_7$ and $p_8$ are consecutive. There is a dummy point between $p_4$ and $p_7$.
\item The points of consecutive clause gadgets are separated by a dummy point each.
\item Let $l_2$ be a vertical line lying between $l_1$ and $l_3$. On $l_2$, embed points to block all visibility relationships other than those corresponding to edges in $\xi$.
Perturb the points of $l_1$ and $l_3$ so that each point in $l_2$ blocks exactly one pair of points. \end{enumerate} The total number of points needed in the new embedding is as follows: \begin{itemize}
\item $p_r$ and $p_b$ are $2$ points.
\item Variable gadgets are $2n$ points.
\item Clause gadgets are $2m$ points on $l_1$ and $4m$ points on $l_3$.
\item There are $n + 2m - 1$ dummy points on $l_1$ and $2m$ dummy points on $l_3$.
\item Thus, there are $3n + 4m - 1$ points on $l_1$ and $6m +2$ points on $l_3$.
\item There are $9m + 2n$ edges from $\xi$ between $l_1$ and $l_3$.
\item Thus, there are $(3n + 4m - 1)(6m +2) - (9m + 2n)$ points on $l_2$ to block the visibility of the rest of the pairs. \end{itemize} \begin{figure}
\caption{(a) The new embedding $\zeta$ on three vertical lines. The dummy points are shown in gray. (b) A 3-colouring of $\zeta$ representing $x_1, x_2, \overline{x_3}$ and $\overline{x_4}$ assigned $1$ in $\theta$.}
\label{finalemb}
\end{figure}
\lemma \label{poly}
The above construction can be achieved in polynomial time.
\proof
As shown above, the number of points used is polynomial. All the points of $l_1$ and $l_3$ are embedded on lattice points. The intersections of the line segments joining points of $l_1$ and $l_3$ are computed, and $l_2$ is chosen such that none of these intersection point lies on $l_2$.
To block visibilities,
and the intersection of the line segments joining points of $l_1$ and $l_3$ with $l_2$ are computed and blockers are placed on the intersection
points. All of this is achievable in polynomial time. \qed
\lemma \label{iff}
The PVG of $\zeta$ can be $5$-coloured if and only if $\theta$ has a satisfying assignment.
\proof Suppose that $\theta$ has a satisfying assignment. Then the points of $\zeta$ obtained from $\xi$ can be coloured with red, blue
and green.
The two neighbours of a dummy point on $l_1$ or $l_3$ can have at most two colours, so the dummy point can always be assigned the third colour.
The points on $l_2$ are coloured alternately with two different colours
(Figure \ref{finalemb}(b)).
$\\ \\$
Now suppose that $\zeta$ has a 5-colouring. All points of $l_2$ are visible from all points of $l_1$ and $l_3$. The points of
$l_2$ must be coloured at least with two colours. This means, the points of the graph induced by $\xi$ are coloured with
at most $3$ colours, which is possible only when $\theta$ is satisfiable. \qed We have the following theorem. \theorem The problem of deciding whether the
visibility graph of a given point set is $5$-colourable, is NP-complete.
\proof
A $5$-colouring of a point visibility graph can be verified in polynomial time. Thus, the problem is in NP.
On the other hand, 3-SAT can be reduced to the problem.
Given a 3-SAT formula $\theta$, by Lemmas \ref{poly} and \ref{iff}, a point set $\zeta$ can be constructed in time polynomial in the size of $\theta$
such that $\zeta$ can be $5$-coloured if and only if $\theta$ has a satisfying assignment. Thus, the problem is NP-complete. \qed
\section{Colouring a point set with small clique number} In general, graphs with small clique numbers can have arbitrarily large chromatic numbers. In fact, there exist triangle free graphs with arbitrarily high chromatic numbers due to the construction of Mycielski \cite{myc}. Pfender showed that for a PVG with $\omega (G) = 6$, $\chi (G)$ can be arbitrarily large \cite{p-vgps-2008}. But it is not known whether the chromatic number of a PVG is bounded, if its clique number is only $4$ or $5$. Here, we address this question. \subsection{A graph with $\omega (G) = 4$ and $\chi (G) = 6$} \label{secexample2}
K\'{a}ra et al \cite{kpw-ocnv-2005} showed that a PVG with clique number $4$ can have chromatic number $5$. They then generalized their example to prove that there is an exponential function $f$ such that for a family of PVGs, the identity $\chi (G) = f(\omega (G))$ holds for all graphs in the family.
The main question remaining is whether PVGs with maximum clique size $4$ have bounded chromatic number. Here we construct a visibility graph $G'$ with $\omega (G') = 4$ and $\chi (G') = 6$ (Figure \ref{6graph}). We construct $G'$ directly as a visibility embedding, as follows. \begin{enumerate}
\item Consider three horizontal lines $l_1$, $l_2$ and $l_3$ parallel to each other. \item On the first line, embed ten points $\{ p_1, p_2, \ldots, p_{10}\}$ from left to right. \item From left to right embed the points $q_1, \ldots, q_4$ on $l_3$. \item Join with line segments the pairs $(q_1,p_1)$, $(q_1,p_4)$, $(q_2,p_2)$, $(q_2,p_5)$. \label{step1} \item Join with line segments the pairs $(q_3,p_6)$, $(q_3,p_9)$, $(q_4p_7)$, $(q_4,p_{10})$. \item Starting from the right of $q_4$, embed the points $r_1, b_1, r_2, b_2, \ldots, r_{10}$ on $l_3$. \item Join each $r_i$ with $1 \leq i \leq 5$ with $p_1$, $p_3$ and $p_{i+5}$. \item Join each $r_i$ with $6 \leq i \leq 10$ with $p_1$, $p_4$ and $p_{i}$. \label{step2} \item Embed points on $l_2$ such that only the adjacencies described from steps \ref{step1} to \ref{step2} hold. \end{enumerate} We have the following lemmas.
\lemma
The clique number of $G'$ is $4$.
\proof By construction, the points on $l_1$ and $l_3$ together induce a triangle free graph. The points of $l_2$ can contribute at most two more points to cliques induced by the points on $l_1$ and $l_3$. So, $G'$ has cliques of size at most four. \qed
\lemma
The chromatic number of $G'$ is $6$.
\proof
Each point on $l_2$ is adjacent to every point on $l_1$ and $l_3$, so the points on $l_2$ require two colours which are absent from the points
of $l_1$ and $l_3$. So, it suffices to show that the graph induced by points on $l_1$ and $l_3$ is not three colourable. Suppose that the points
$p_1, \ldots p_5$ have only two colours (say, $C_1$ and $C_2$). This means that they are coloured with $C_1$ and $C_2$ alternately,
and $q_1$ and $q_2$ must be coloured with two extra colours, $C_3$ and $C_4$ respectively.
$\\ \\$
Now suppose that all three of $C_1$, $C_2$ and $C_3$ occur among $p_1, \ldots p_5$. Similarly,
all three of $C_1$, $C_2$ and $C_3$ occur among $p_6, \ldots p_{10}$, for otherwise the previous argument is applicable to $q_3$, $q_4$
and $p_6, \ldots p_{10}$. Also, $p_1$ and $p_3$, or $p_1$ and $p_4$ must have two distinct colours. On the other hand, three distinct colours
must also occur among $p_6, \ldots p_{10}$. But for every $p_i$, $6 \leq i \leq 10$, there are two points among $r_1, r_2, \ldots, r_{10}$
that are adjacent to $p_i$ and one pair among $\{p_1, p_3\}$ and $\{p_1, p_4\}$. So, at least one of the points among
$r_1, r_2, \ldots, r_{10}$ must have the fourth colour. \qed
\begin{figure}
\caption{A point visibility graph with clique number four but chromatic number six.}
\label{6graph}
\end{figure}
\section{Concluding Remarks} We have settled the question of colouring PVGs, showing that the $4$-colour problem on them is solvable in polynomial time while the $5$-colour problem is NP-complete. We have shown that there is a PVG with clique number four but chromatic number six. However, it is still open to show whether a PVG with clique number four can have a greater chromatic number or not.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\renewcommand{\arabic{footnote}}{\fnsymbol{footnote}}\setcounter{footnote}{0}
\begin{center} {\Large Construction of Locally Conservative Fluxes for High Order Continuous Galerkin Finite Element Methods } \end{center}
\renewcommand{\arabic{footnote}}{\fnsymbol{footnote}} \renewcommand{\arabic{footnote}}{\arabic{footnote}}
\begin{center} Q.~Deng and V.~Ginting\\ Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071, USA\footnote{Complete mailing address: Department of Mathematics, 1000 E. University Ave. Dept. 3036, Laramie, WY 82071} \end{center}
\renewcommand{1.5}{1.5} \begin{abstract}
We propose a simple post-processing technique for linear and high order continuous Galerkin Finite Element Methods (CGFEMs) to obtain locally conservative flux field. The post-processing technique requires solving an auxiliary problem on each element independently which results in solving a linear algebra system whose size is $\frac{1}{2}(k+1)(k+2)$ for $k^\text{th}$ order CGFEM. The post-processing could have been done directly from the finite element solution that results in locally conservative flux on the element. However, the normal flux is not continuous at the element's boundary. To construct locally conservative flux field whose normal component is also continuous, we propose to do the post-processing on the nodal-centered control volumes which are constructed from the original finite element mesh. We show that the post-processed solution converges in an optimal fashion to the true solution in an $H^1$ semi-norm. We present various numerical examples to demonstrate the performance of the post-processing technique.
\end{abstract}
\paragraph*{Keywords} CGFEM; FVEM; conservative flux; post-processing
\section{Introduction} \label{sec:intro}
\renewcommand{1.5}{1.5}
Both finite volume element method (FVEM) and continuous Galerkin finite element method (CGFEM) are widely used for solving partial differential equations and they have both advantages and disadvantages. Both methods share a property in how the approximate solutions are represented through linear combinations of the finite element basis functions. They have a main advantage of the ability to solve the partial differential equations posed in complicated geometries. However the methods differ in the variational formulations governing the approximate solutions. CGFEMs are defined as a global variational formulation while FVEM relies on local variational formulation, namely one that imposes local conservation of fluxes. In the case of linear finite element, it is well-known that the bilinear form of FVEM is closely related to its CGFEM counterpart, and this closeness is exploited to carry out the error analysis of FVEM \cite{bank1987some, hackbusch1989first}. In 2002, Ewing \textit{et al.} showed that the stiffness matrix derived from the linear FVEM is a small perturbation of that of the linear CGFEM for sufficiently small mesh size of the triangulation \cite{ewing2002accuracy}. In 2009, an identity between stiffness matrix of linear FVEM and the matrix of linear CGFEM was established; see Xu \textit{et al.} \cite{xu2009analysis}. A significant amount of work has been done to investigate the closeness of linear FVEM and linear CGFEM. However, the current understanding and implementation of higher order FVEMs are still at its infancy and are not as satisfactory as linear FVEM. For one-dimensional elliptic equations, high order FVEMs have been developed in \cite{plexousakis2004construction}. Other relevant high order FVEM work can be found in \cite{liebau1996finite, li2000generalized, cai2003development, chen2012higher, chen2015construction}.
As mentioned, FVEM produces locally conservative fluxes while, due to the global formulation, CGFEMs do not. Robustness of the CGFEMs for any order has been established through extensive and rigorous error analysis, while this is not the case for FVEM. Development of linear algebra solvers for CGFEMs has reached an advanced stage, mainly driven from a solid understanding of the variational formulations and their properties, such as coercivity (and symmetry) of the bilinear form in the Galerkin formulation. On the other hand, the resulting linear algebra systems derived from FVEMs, especially high order FVEMs, are not that easy to solve. Typically, the matrices resulting from FVEMs are not symmetric even if the original boundary value problem is. Furthermore, at most FVEM discretization with linear finite element basis yields M-matrix, while with quadratic finite element basis it is not (see \cite{liebau1996finite}).
Preservation of numerical local conservation property of approximate solutions are imperative in simulations of many physical problems especially those that are derived from law of conservation. In order to maintain the advantages of CGFEM as well as to obtain locally conservative fluxes, post-processing techniques are developed; see \cite{arbogast1995characteristics, bush2015locally, bush2013application, chou2000conservative, cockburn2007locally, cordes1992continuous,
deng2015construction, gmeiner2014local, hughes2000continuous,
kees2008locally, larson2004conservative, loula1995higher, nithiarasu2004simple, srivastava1992three, sun2009locally, thomas2008element, thomas2008locally, toledo1989mixed, zhang2013locally}. The post-processing techniques proposed in the aforementioned references are mainly techniques for post-processing linear finite element related methods and they include finite element methods for solving pure elliptic equations, advection diffusion equations, advection dominated diffusion equations, elasticity problems, Stokes problem, etc. Among them, some of the proposed post-processing techniques require solving global systems. We will focus on a brief review on the post-processing techniques for high order CGFEMs. Generally, those post-processing techniques that works for high order CGFEMs also work for lower order CGFEMs, but may not vice versa.
There are very limited work on the post-processing for high order CGFEMs to obtain locally conservative fluxes. An interesting work on post-processing for high order CGFEMs is in Zhang \textit{et al} \cite{zhang2012flux}. In their work, they showed that the elemental fluxes directly calculated from any order of CGFEM solutions converge to the true fluxes in an optimal order but the fluxes are not naturally locally conservative. They proposed two post-processing techniques to obtain the locally conservative fluxes at the boundaries of the each element. The post-processed solutions are of optimal both $L^2$ norm and $H^1$ semi-norm convergence orders. Very interestingly, one of the post-processed solution still satisfies the original finite element equations. Other work on post-processing to gather locally conservative flux that include high order finite elements is recorded in \cite{cockburn2007locally}. The post-processing involves two steps: solving a set of local systems followed by solving a global system.
A uniform approach to local reconstruction of the local fluxes from various finite element method (FEM) solutions was presented in \cite{becker2015robust}. These methods includes any order of conforming, nonconforming, and discontinuous FEMs. They proposed a hybrid formulation by utilizing Lagrange multipliers, which can be computed locally. However, the reconstructed fluxes are not locally conservative. They used the reconstructed fluxes to derive a posteriori error estimator \cite{becker2015stopping}.
In this paper, we propose a post-processing technique for any order of CGFEMs to obtain fluxes that are locally conservative on a dual mesh consisting of control volumes. The dual mesh is constructed from the original mesh in a different way for different order of CGFEMs. The technique requires solving an auxiliary problem which results in a low dimensional linear algebra system on each element independently. Thus, the technique can be implemented in a parallel environment and it produces locally conservative fluxes wherever it is needed. The technique is developed on triangular meshes and it can be naturally extended to rectangular meshes.
The rest of the paper is organized as follows. The CGFEM formulation of the model problem is presented in Section \ref{sec:cgfem} followed by the description of the methodology of the post-processing technique in Section \ref{sec:pp}. Analysis of the post-processing technique is presented in Section \ref{sec:ana} and numerical examples are presented to demonstrate the performance of the technique in Section \ref{sec:num}.
\section{Continuous Galerkin Finite Element Method} \label{sec:cgfem}
For simplicity, we consider the elliptic boundary value problem \begin{equation}\label{pde} \begin{cases} \begin{aligned} - \nabla \cdot ( \kappa \nabla u ) & =f \quad \text{in} \quad \Omega, \\ u&= g \quad \text{on} \quad \partial\Omega, \\ \end{aligned} \end{cases} \end{equation} where $\Omega$ is a bounded open domain in $\mathbb{R}^2$ with Lipschitz boundary $\partial\Omega$, $\kappa = \kappa(\boldsymbol{x})$ is the elliptic coefficient, $u=u(\boldsymbol{x})$ is the solution to be found, $f=f(\boldsymbol{x})$ is a forcing function. Assuming $0 < \kappa_{\min} \leq \kappa(\boldsymbol{x}) \leq \kappa_{\max} < \infty $ for all $\boldsymbol{x} \in\Omega$ and $f \in L^2(\Omega)$, Lax-Milgram Theorem guarantees a unique weak solution to \eqref{pde}. For the polygonal domain $\Omega$, we consider a partition $\mathcal{T}_h$ consisting of triangular elements $\tau$ such that $\overline\Omega = \bigcup_{\tau\in\mathcal{T}_h} \tau.$ We set $h=\max_{\tau\in\mathcal{T}_h} h_\tau$ where $h_\tau$ is defined as the diameter of $\tau$. The continuous Galerkin finite element space is defined as $$
V^k_h = \big\{w_h\in C(\overline\Omega): w_h|_\tau \in P^k(\tau), \ \forall \ \tau\in\mathcal{T}_h \ \text{and} \ w_h|_{\partial\Omega} = 0 \big\}, $$ where $P^k(\tau)$ is a space of polynomials with degree at most $k$ on $\tau$. The CGFEM formulation for \eqref{pde} is to find $u_h$ with $(u_h - g_h) \in V^k_h$, such that \begin{equation} \label{eq:fem} a(u_h, w_h) = \ell(w_h) \quad \forall \ w_h \in V^k_h, \end{equation} where \begin{equation*} a(v, w) = \int_\Omega \kappa \nabla v \cdot \nabla w \ \text{d} \boldsymbol{x}, \qquad \text{and} \qquad \ell(w) = \int_\Omega f w \ \text{d} \boldsymbol{x}, \end{equation*} and $g_h \in V_h^k$ can be thought of as the interpolant of $g$ using the usual finite element basis.
\begin{figure}
\caption{Setting for $V_h^1$: the dots represent the degrees of freedom and $\Omega^z = \cup_{i=1}^5 \tau_i^z$ is the support of $\phi_z$.}
\caption{Setting for $V_h^2$: The dots represent the degrees of freedom. $\Omega^z= \cup_{i=1}^5 \tau_i^z$ (left) is support of $\phi_z$ and $\Omega^y = \cup_{i=1}^2 \tau_i^y$ (right) is the support of $\phi_y$.}
\caption{Setting for $V_h^3$: The dots represent the degrees of freedom. $\Omega^z= \cup_{i=1}^5 \tau_i^z$ (left) is the support of $\phi_z$, and $\Omega^y = \cup_{i=1}^2 \tau_i^y$ (middle) is the support of $\phi_y$, and $\Omega^x = \tau_1^x$ is the support of $\phi_x$.}
\label{fig:lelemsupp}
\label{fig:qelemsupp}
\label{fig:celemsupp}
\end{figure}
We now present a fundamental but obvious fact for this CGFEM formulation. To make it as general as possible, let $Z$ be the set of nodes in $\Omega$ resulting from the partition $\mathcal{T}_h$ and placing the degrees of freedom owned by $V_h^k$. In particular, $Z$ consists of vertices for $V_h^1$ (see Figure \ref{fig:lelemsupp}), vertices and degrees of freedom on the edges for $V_h^2$ (see Figure \ref{fig:qelemsupp}), vertices and degrees of freedom on the edges and in the elements' barycenters for $V_h^3$ (see Figure \ref{fig:celemsupp}). Furthermore, $Z = Z_{\text{in}} \cup Z_\text{d}$, where $Z_{\text{in}}$ is the set of interior degrees of freedom and $Z_\text{d}$ is the set of corresponding points on $\partial\Omega$. Denoting the usual Lagrange nodal basis of $V^k_h$ as $\{ \phi_\xi \}_{\xi \in Z_\text{in}}$, \eqref{eq:fem} yields \begin{equation} \label{eq:femphi} a(u_h, \phi_\xi) = \ell (\phi_\xi) \quad \forall \ \xi \in Z_{\text{in}}. \end{equation}
For a $\xi \in Z_{\text{in}}$, let $\Omega^\xi$ be the support of the basis function $\phi_\xi$. Then in the partition $\mathcal{T}_h$, $\Omega^\xi = \cup_{i=1}^{N_\xi} \tau_i^\xi$, where $\tau_i^\xi$ is an element that has $\xi$ as one of its degrees of freedom, and
$N_\xi$ is the total number of such elements. In the linear case, $N_\xi$ is the number of elements sharing vertex $\xi$ (see Figure \ref{fig:lelemsupp}); in the quadratic case, $N_\xi$ is the number of elements sharing vertex $\xi$ or a middle point on an edge in which case $N_\xi = 2$ (see Figure \ref{fig:qelemsupp}); in the cubic case, $N_\xi$ can be those in quadratic case plus that it can be one if $\xi$ is inside the element (see Figure \ref{fig:celemsupp}). With this in mind, \eqref{eq:femphi} is expressed as
\begin{equation} \label{eq:localfem} \sum_{i=1}^{N_\xi} a_{\tau^\xi_i}(u_h, \phi_\xi) = \sum_{i=1}^{N_\xi} \ell_{\tau^\xi_i} (\phi_\xi), \end{equation} where $a_\tau(v_h, w_h)$ is $a(v_h, w_h)$ restricted to element $\tau$ and $\ell_\tau (w_h)$ is $\ell(w_h)$ restricted to element $\tau$.
Equation \eqref{eq:localfem} is fundamental and we will use this fact to derive the post-processing technique in Section \ref{sec:pp}.
\section{A Post-processing Technique} \label{sec:pp}
A naive derivative calculation of $u_h$ does not yield locally conservative fluxes. For this reason, in this section we propose a post-processing technique to construct locally conservative fluxes over control volumes from CGFEM solutions. We will focus on the construction for $k^{\text{th}}$ order CGFEM, where $k=1, 2, 3$, i.e., linear, quadratic, and cubic CGFEMs. Construction for orders higher than these CGFEMs can be conducted in a similar fashion.
\subsection{Auxiliary Elemental Problem} \label{sec:bvp}
Based upon the original finite element mesh and $V_h^k$, a dual mesh that consists of control volumes is generated over which the post-processed fluxes is to satisfy the local conservation. For $V_h^1$, we connect the barycenter and middle points of edges of a triangular element; see Figure \ref{fig:lelemcv}. For $V_h^2$, we firstly discretize the triangular element into \textcolor{red}{four} sub-triangles and then connect the barycenters and middle points of each sub-triangle; see plots in Figure \ref{fig:qelemcv}, and similarly $V_h^3$, see plots in Figure \ref{fig:celemcv}. We can also see the construction of the dual mesh on a single element in Figure \ref{fig:elem}. Each control volume corresponds to a degree of freedom in CGFEMs. We post-process the CGFEM solution $u_h$ to obtain $\widetilde {\boldsymbol{\nu}}_h = -\kappa \nabla \widetilde{u}_h $ such that it is continuous at the boundaries of each control volume and satisfies the local conservation property in the sense \begin{equation} \label{eq:cvconservation} \int_{\partial C^\xi} \widetilde {\boldsymbol{\nu}}_h \cdot \boldsymbol{n} \ \text{d} l = \int_{C^\xi} f \ \text{d} \boldsymbol{x}, \end{equation} where $C^\xi$ can be a control volume surrounding a vertex as $C^z$ in Figure \ref{fig:lelemcv}, \ref{fig:qelemcv}, and \ref{fig:celemcv}, or a control volume surrounding a degree of freedom on an edge as $C^y$ in Figure \ref{fig:qelemcv} and \ref{fig:celemcv}, or $C^x$ in Figure \ref{fig:celemcv}.
\begin{figure}
\caption{ $C^z$ is the control volume corresponding to $\phi_z$ in $V_h^1$. }
\caption{ $C^z, C^y$ are control volumes corresponding to $\phi_z$ (left) and $\phi_y$ (right), respectively, in $V_h^2$. }
\caption{ $C^z, C^y, C^x$ are control volumes corresponding to $\phi_z$ (left), $\phi_y$ (middle), and $\phi_x$ (right), respectively, in $V_h^3$. }
\label{fig:lelemcv}
\label{fig:qelemcv}
\label{fig:celemcv}
\end{figure}
In order to obtain the locally conservative fluxes on each control volume, we set and solve an elemental/local problem on $\tau$. Let $N_k = \frac{1}{2}(k+1)(k+2)$ be the total number of degrees of freedom on a triangular element for $V_h^k$. We denote the collection of those degrees of freedom by $s(\tau, k) = \{ z_j \}_{j=1}^{N_k}$; see Figure \ref{fig:elem}. We partition each element $\tau$ into $N_k$ non-overlapping polygonals $\{ t_{z_j} \}_{j=1}^{N_k}$; see Figure \ref{fig:elem}. For $t_\xi$ with $\xi \in s(\tau,k)$, we make decomposition $\partial t_\xi = ( \partial \tau \cap \partial t_\xi ) \cup ( \partial C^\xi \cap \partial t_\xi ).$ We also define the average on an edge or part of the edge which is the intersection of two elements $\tau_1$ and $\tau_2$ for vector $\boldsymbol{v}$ as \begin{equation} \label{def:ave}
\{ \boldsymbol{v} \} = \frac{\boldsymbol{v}|_{\tau_1} + \boldsymbol{v}|_{\tau_2} }{2}. \end{equation}
\begin{figure}
\caption{ Control volume construction and degrees of freedom on an element for $V_h^k$: $k=1$ (left), $k=2$ (middle), and $k=3$ (right). }
\label{fig:elem}
\end{figure}
Let $V^0(\tau)$ be the space of piecewise constant functions on element $\tau$ such that $V^0(\tau) = \text{span} \{ \psi_\eta \}_{\eta \in s(\tau,k)}$, where $\psi_\eta$ is the characteristic function of the polygonal $t_\eta$, i.e., \begin{equation} \psi_\eta(\boldsymbol{x}) = \Big\lbrace\begin{array}{c} 1 \quad \text{if} \quad \boldsymbol{x} \in t_\eta \\ 0 \quad \text{if} \quad \boldsymbol{x} \notin t_\eta \end{array}. \end{equation} We define a map $I_\tau: H^1(\tau) \rightarrow V^0(\tau)$ with $I_\tau w = \displaystyle \sum_{\xi \in s(\tau,k)} w_\xi \psi_\xi$, where $w_\xi = w(\xi)$ for $w\in H^1(\tau)$. We define the following bilinear forms \begin{equation} \label{eq:bebiforms} b_\tau( v, w) = -\sum_{\xi \in s(\tau,k)} \int_{\partial C^\xi \cap \partial t_\xi} \kappa \nabla v \cdot \boldsymbol{n} I_\tau w \ \text{d} l, \qquad e_\tau(v, w) = \int_{\partial \tau} \{ \kappa \nabla v \} \cdot \boldsymbol{n} w \ \text{d} l. \end{equation}
Let $ V^k_h(\tau) = \text{span}\{ \phi_\eta \}_{ \eta \in s(\tau,k) }$ where $ \phi_\eta$ can be thought as the usual nodal $\eta$ basis function restricted to element $\tau$. The elemental calculation for the post-processing is to find $\widetilde u_{\tau, h} \in V^k_h(\tau) $ satisfying \begin{equation} \label{eq:bvpvf} b_\tau(\widetilde u_{\tau, h}, w) = \ell_\tau ( I_\tau w - w ) + a_\tau(u_h, w) + e_\tau(u_h, I_\tau w - w), \quad \forall \ w \in V^k_h(\tau). \end{equation}
\newtheorem{lem}{Lemma}[section] \begin{lem} \label{lem:ebvp} The variational formulation \eqref{eq:bvpvf} has a unique solution up to a constant. \end{lem}
\begin{proof} We have $V_h^k(\tau) = \text{span}\{ \phi_\xi \}_{ \xi \in s(\tau,k) }$, where $\phi_\xi(\eta) = \delta_{\xi \eta}$ with $\delta_{\xi \eta}$ being the Kronecker delta, for all $\xi, \eta \in s(\tau,k)$. By replacing the test function $w$ with $\phi_\xi$ for all $\xi \in s(\tau,k)$,
\eqref{eq:bvpvf} is reduced to \begin{equation} \label{eq:cz10} - \int_{\partial C^\xi \cap \partial t_\xi } \kappa \nabla \widetilde u_{\tau, h} \cdot \boldsymbol{n} \ \text{d} l = \int_{t_\xi} f \ \text{d} \boldsymbol{x} - \ell_\tau ( \phi_\xi ) + a_\tau(u_h, \phi_\xi ) + e_\tau( u_h, I_\tau \phi_\xi - \phi_\xi ), \quad \forall \ \xi \in s(\tau,k). \end{equation} This is a fully Neumann boundary value problem in $\tau$ with boundary condition satisfying \begin{equation} \label{eq:bvpb} - \int_{\partial \tau \cap \partial t_\xi } \kappa \nabla \widetilde u_{\tau, h} \cdot \boldsymbol{n} \ \text{d} l = \ell_\tau ( \phi_\xi ) - a_\tau(u_h, \phi_\xi ) - e_\tau( u_h, I_\tau \phi_\xi - \phi_\xi ), \quad \forall \ \xi \in s(\tau,k). \end{equation} To establish the existence and uniqueness of the solution, one needs to verify the compatibility condition \cite{evans2010partial}. We calculate \begin{equation*} -\int_{\partial \tau} \kappa \nabla \widetilde u_{\tau, h} \cdot \boldsymbol{n} \ \text{d} l = \sum_{\xi \in s(\tau,k)} \Big( \ell_\tau ( \phi_\xi ) - a_\tau(u_h, \phi_\xi ) - e_\tau( u_h, I_\tau \phi_\xi - \phi_\xi ) \Big). \end{equation*} Using the fact that $\sum_{\xi \in s(\tau,k)} \phi_\xi = 1$ and linearity, we obtain \begin{equation*} \sum_{\xi\in s(\tau,k)} \ell_\tau( \phi_\xi) = \sum_{\xi\in s(\tau,k)} \int_\tau f \phi_\xi \ \text{d} \boldsymbol{x} = \int_\tau f \sum_{\xi\in s(\tau,k)} \phi_\xi \ \text{d} \boldsymbol{x} = \int_\tau f \ \text{d} \boldsymbol{x}. \end{equation*} Using the fact that $\nabla \big( \sum_{\xi\in s(\tau,k)} \phi_\xi \big) = \boldsymbol{0} $ and linearity, we obtain \begin{equation*} \sum_{\xi\in s(\tau,k)} a_\tau(u_h, \phi_\xi) = \sum_{\xi\in s(\tau,k)} \int_\tau \kappa \nabla u_h \cdot \nabla \phi_\xi \ \text{d} \boldsymbol{x} = \int_\tau \kappa \nabla u_h \cdot \nabla \big( \sum_{\xi\in s(\tau,k)} \phi_\xi \big) \text{d} \boldsymbol{x} = 0. \end{equation*} Also, we notice that \begin{equation*} \sum_{\xi\in s(\tau,k)} e_\tau(u_h, I_\tau \phi_\xi - \phi_\xi ) = \sum_{\xi\in s(\tau,k)} \int_{\partial \tau \cap \partial t_\xi } \{ \kappa \nabla u_h \} \cdot \boldsymbol{n} \ \text{d} l - \int_{\partial \tau} \{ \kappa \nabla u_h \} \cdot \boldsymbol{n} \sum_{\xi\in s(\tau,k)} \phi_\xi \ \text{d} l = 0. \end{equation*} Combining these equalities, compatibility condition $\int_{\partial \tau} - \kappa \nabla \widetilde u_{\tau, h} \cdot \boldsymbol{n} \ \text{d} l = \int_\tau f \ \text{d} \boldsymbol{x}$ is verified. This completes the proof. \end{proof}
\begin{remark} The technique proposed here can be naturally generalized to rectangular elements. In the proof of Lemma \ref{lem:ebvp}, for $\xi = z_{10}$ in cubic CGFEM, \eqref{eq:cz10} is naturally reduced to a local conservation equation $- \int_{\partial C^\xi } \kappa \nabla \widetilde u_{\tau, h} \cdot \boldsymbol{n} \ \text{d} l = \int_{C^\xi} f \ \text{d} \boldsymbol{x}$. This can be proved by using \eqref{eq:localfem}, $\partial \tau \cap \partial t_\xi = \emptyset$, and $\phi_{z_{10}} = 0$ on $\partial \tau.$ \end{remark}
\begin{lem} \label{lem:elemlc} The piecewise boundary fluxes defined in \eqref{eq:bvpb} satisfy local conservation on each element, i.e., \begin{equation} \label{eq:elemlc} - \int_{\partial\tau } \kappa \nabla \widetilde u_{\tau, h} \cdot \boldsymbol{n} \ \text{d} l = \int_\tau f \ \text{d} \boldsymbol{x}, \end{equation} and \begin{equation} \label{eq:fluxed} - \int_{\partial\tau } \kappa \nabla \widetilde u_{\tau, h} \cdot \boldsymbol{n} \ \text{d} l = - \int_{\partial\tau } \kappa \nabla u \cdot \boldsymbol{n} \ \text{d} l. \end{equation} \end{lem}
\begin{proof} Equation \eqref{eq:elemlc} is established in the proof of Lemma \ref{lem:ebvp}. Identity \eqref{eq:fluxed} is verified by using \eqref{pde} and Divergence theorem: \begin{equation*} \begin{aligned} \int_{\partial \tau} - \kappa \nabla \widetilde u_{\tau, h} \cdot \boldsymbol{n} \ \text{d} l = \int_\tau f \ \text{d} \boldsymbol{x} = \int_\tau \nabla \cdot ( -\kappa \nabla u) \ \text{d} \boldsymbol{x} = \int_{\partial\tau } -\kappa \nabla u \cdot \boldsymbol{n} \ \text{d} l. \end{aligned} \end{equation*}
\end{proof}
\begin{remark} Lemma \ref{lem:elemlc} implies that the proposed way of imposing elemental boundary condition in \eqref{eq:bvpb} is a rather simple post-processing technique: we set flux at $\partial \tau \cap \partial t_\xi $ as $\ell_\tau ( \phi_\xi ) - a_\tau(u_h, \phi_\xi ) - e_\tau( u_h, I_\tau \phi_\xi - \phi_\xi )$. It does not require solving any linear system but provides locally conservative fluxes at the element boundaries. In two-dimensional case, $\partial \tau \cap \partial t_\xi $ consists of two segments in the setting as shown in Figure \ref{fig:elem}. If we want to provide a flux approximation on each segment, we can for example set the flux for each segment $\Gamma_\xi$ as \begin{equation} \frac{\ell_\tau ( \phi_\xi ) - a_\tau(u_h, \phi_\xi ) + e_\tau( u_h, \phi_\xi )}{2} - \int_{\Gamma_\xi} \{ \kappa \nabla u_h \} \cdot \boldsymbol{n} \ \text{d} l. \end{equation} If $\partial \tau \cap \partial t_\xi $ consists of more segments, we can set the flux in the similar way. For instance, we assign weights to $\ell_\tau ( \phi_\xi ) - a_\tau(u_h, \phi_\xi ) + e_\tau( u_h, \phi_\xi )$ according to the length ratio of the segment over $\partial \tau \cap \partial t_\xi $. A drawback of this technique, however, is that the post-processed fluxes in general is not continuous at the element boundaries, except only when $\ell_\tau ( \phi_\xi ) - a_\tau(u_h, \phi_\xi )$ from two neighboring elements are equal. This reveals the merits of the main post-processing technique proposed in this paper. The main post-processing technique provides a way to obtain locally conservative fluxes on control volumes and the fluxes are continuous at the boundaries of each control volume.
\end{remark}
\begin{lem} \label{lem:ppsol} The true solution $u$ of \eqref{pde} satisfies \begin{equation} \label{eq:bvpvft} b_\tau(u, w) = \ell_\tau ( I_\tau w - w ) + a_\tau(u, w) + e_\tau(u, I_\tau w - w), \quad \forall \ w \in H^1(\tau), \end{equation} and this further implies \begin{equation} \label{eq:bvpvfe} b_\tau(u - \widetilde u_{\tau, h}, w) = a_\tau(u - u_h, w) + e_\tau(u - u_h, I_\tau w - w), \quad \forall \ w \in V^k_h(\tau). \end{equation} \end{lem}
\begin{proof} This can be easily proved by simple calculations. \end{proof}
\begin{remark}
The first result in Lemma \ref{lem:ppsol} tells us that accurate boundary data gives us a chance to obtain accurate fluxes. This is the very reason that the post-processing technique proposed in \cite{bush2013application} and extended in \cite{bush2014application, bush2015locally, deng2015construction} could not be generalized to a post-processing technique for high order CGFEMs. The boundary condition imposed in \cite{bush2013application} is \begin{equation}
\int_{\partial \tau \cap \partial t_\xi } - \kappa \nabla \widetilde u_{\tau, h} \cdot \boldsymbol{n} \ \text{d} l = \ell_\tau( \phi_\xi) - a_\tau(u_h, \phi_\xi), \end{equation} and clearly $\ell_\tau(w) - a_\tau(u, w) \ne 0$ for the true solution $u$ of \eqref{pde}. Specifically for high order CGFEM solutions, only imposing $\ell_\tau(w) - a_\tau(u_h, w)$ as a boundary condition for $\widetilde{u}_{\tau,h}$ is not sufficient to guarantee optimal accuracy. The convergence order and accuracy of the post-processed solution strongly depend on the boundary data. By imposing the boundary data in the way shown in \eqref{eq:bvpb}, we can get optimal convergence order for the post-processed solution $\widetilde u_h$ for any high order CGFEMs. \end{remark}
\begin{remark} Equation \eqref{eq:bvpvfe} in Lemma \ref{lem:ppsol} resembles Galerkin orthogonality and plays a crucial role in establishing the post-processing error.
\end{remark}
\subsection{Elemental Linear System } \label{sec:llas} We note that the dimension of $V^k_h(\tau)$ is $N_k$ and hence the variational formulation \eqref{eq:bvpvf} yields an $N_k$-by-$N_k$ linear algebra system. Since $\widetilde u_{\tau, h} \in V^k_h(\tau)$, \begin{equation} \label{eq:ppsol} \widetilde u_{\tau, h} = \sum_{\eta \in s(\tau,k)} \alpha_\eta \phi_\eta, \end{equation} so by inserting this representation to \eqref{eq:bvpvf} and replacing the test function by $\phi_\xi$ give us the linear algebra system \begin{equation} \label{eq:axb} A \boldsymbol{\alpha} = \boldsymbol{\beta}, \end{equation} where $\boldsymbol{\alpha} \in \mathbb{R}^{N_k} $ whose entries are the nodal solutions in \eqref{eq:ppsol}, $\boldsymbol{\beta} \in \mathbb{R}^{N_k} $ with entries \begin{equation} \beta_\xi = \ell_\tau ( I_\tau \phi_\xi - \phi_\xi ) + a_\tau(u_h, \phi_\xi) + e_\tau (u_h, I_\tau \phi_\xi - \phi_\xi ), \quad \forall \ \xi \in s(\tau,k), \end{equation} and \begin{equation} A_{\xi \eta} = b_\tau( \phi_\eta, \phi_\xi), \quad \forall \ \xi, \eta \in s(\tau,k), \end{equation}
The linear system \eqref{eq:axb} is singular and there are infinitely many solutions since the solution to \eqref{eq:bvpvf} is unique up to a constant by Lemma \ref{lem:ebvp}. However, this does not cause any issue since
to obtain locally conservative fluxes, the desired quantity from the post-processing is $\nabla \widetilde{u}_{\tau,h}$, which is unique.
\subsection{Local Conservation} \label{sec:loccons}
At this stage, we verify the local conservation property \eqref{eq:cvconservation} on control volumes for the post-processed solution. It is stated in the following lemma.
\begin{lem} \label{lem:localconserv} The desired local conservation property \eqref{eq:cvconservation} is satisfied on the control volume $C^\xi$ where $\xi \in Z_{\text{in}}$. \end{lem}
\begin{proof} Obviously, for $\xi = z_{10}$ in the case of $V_h^3$, the polygonal $t_{z_{10}} = C^{10}$, \eqref{eq:cvconservation} is directly satisfied from solving \eqref{eq:bvpvf}. Similar situation occurs for $k>3$ in $V_h^k$. Thus, we only need to prove this lemma in the case that a control volume is associated with $\xi$ that is either on the edge of $\tau$ or the vertex of $\tau$. For a basis function $\phi_\xi$, let $\Omega^\xi = \cup_{i=1}^{N_\xi} \tau_i^\xi$ be its support. Noting that the gradient component is averaged, it is obvious that \begin{equation*} \sum_{j=1}^{N_\xi} \int_{\partial \tau_j^\xi} \{ \kappa \nabla u_{\tau_j, h} \} \cdot \boldsymbol{n} \phi_\xi \ \text{d} l = 0 \qquad \text{and} \qquad \sum_{j=1}^{N_\xi} \int_{\partial \tau_j^\xi \cap \partial t_\xi } \{ \kappa \nabla u_{\tau_j, h} \} \cdot \boldsymbol{n} \ \text{d} l = 0. \end{equation*} This implies that $\sum_{j=1}^{N_\xi} e_{\tau_j^\xi}(u_h, \phi_\xi) = 0.$ Straightforward calculation and \eqref{eq:localfem} gives \begin{equation*}
\int_{ \partial C^\xi } - \kappa \nabla \widetilde u_{\tau, h} \cdot \boldsymbol{n} \ \text{d} l = \int_{C^\xi} f \ \text{d} \boldsymbol{x} + \sum_{j=1}^{N_\xi} \Big( a_{\tau_j^\xi}(u_h, \phi_\xi) - \ell_{\tau_j^\xi}(\phi_\xi) - e_{\tau_j^\xi}(u_h, \phi_\xi) \Big) = \int_{C^\xi} f \ \text{d} \boldsymbol{x}, \end{equation*} which completes the proof. \end{proof}
\section{An Error Analysis for the Post-processing} \label{sec:ana}
In this section, we focus on establishing an optimal convergence property of the post-processed solution $\widetilde u_{\tau, h}$ in $H^1$ semi-norm. We denote $\| \cdot \|_{L^2}$ the usual $L^2$ norm and $| \cdot |_{W}$ the usual semi-norm in a Sobolev space $W$. We start with proving a property of $I_\tau$ defined in Section \ref{sec:bvp}.
\begin{lem} \label{lem:map} Let $I_\tau$ be as defined in Section \ref{sec:bvp}. Then \begin{equation} \label{eq:interr}
\| w - I_\tau w \|_{L^2(\tau)} \leq Ch^2_\tau | w |_{H^2(\tau)} + C h_\tau | w |_{H^1(\tau)}, \quad \text{for} \quad w \in H^2(\tau). \end{equation} \end{lem}
\begin{proof} For $w \in H^2(\tau)$, suppose $\Pi w \in V_h^1(\tau)$ is the standard linear interpolation of $w$. Then we have $I_\tau w = I_\tau (\Pi w)$. By adding and subtracting $\Pi w$, and invoking triangle inequality we get \begin{equation*} \begin{aligned}
\| w - I_\tau w \|_{L^2(\tau)} \leq \| w - \Pi w \|_{L^2(\tau)} + \| \Pi w - I_\tau (\Pi w) \|_{L^2(\tau)}. \end{aligned} \end{equation*} Standard interpolation theory (see for example Theorem 4.2 in \cite{johnson2009numerical}) states that \begin{equation}
\| w - \Pi w \|_{L^2(\tau)} \leq Ch^2_\tau | w |_{H^2(\tau)}. \end{equation} Since $I_\tau w = \displaystyle \sum_{\xi \in s(\tau,k)} w_\xi \psi_\xi$, we divide $\tau$ equally into $k^2$ sub-triangles $\tau_j, j=1, \cdots, k^2$. By Lemma 6.1 in \cite{chatzipantelidis2002finite}, we have \begin{equation}
\| \Pi w - I_\tau (\Pi w) \|^2_{L^2(\tau)} = \sum_{j=1}^{k^2} \| \Pi w - I_\tau (\Pi w) \|^2_{L^2(\tau_j)} \leq \sum_{j=1}^{k^2} C h^2_{\tau_j} | \Pi w |^2_{H^1(\tau_j)} \leq Ch^2_\tau | \Pi w |^2_{H^1(\tau)}. \end{equation} Taking square root gives \begin{equation}
\| \Pi w - I_\tau (\Pi w) \|_{L^2(\tau)} \leq Ch_\tau | \Pi w |_{H^1(\tau)}. \end{equation}
By using triangle inequality and interpolation theory again, we have $$
| \Pi w |_{H^1(\tau)} \leq | w |_{H^1(\tau)} + | w - \Pi w |_{H^1(\tau)} \leq | w |_{H^1(\tau)} + Ch_\tau | w |_{H^2(\tau)}. $$ Combining these inequalities gives the desired result. \end{proof}
\begin{lem} \label{lem:loccoe} The bilinear form defined in \eqref{eq:bvpvf} is bounded, i.e., for all $w \in H^2(\tau), v \in V^k_h(\tau),$ \begin{equation} \label{eq:bbounded}
b_\tau(w, v) \leq C |w|_{H^1(\tau)} |v|_{H^1(\tau)}. \end{equation} Furthermore, for $v\in V^k_h(\tau)$ with $k=1, 2$, $b_\tau(\cdot,\cdot)$ is coercive, namely, \begin{equation} \label{eq:coerc}
b_\tau(v, v) \geq C_b |v|^2_{H^1(\tau)}, \end{equation} for some positive constant $C_b$.
\end{lem}
\begin{proof} The boundedness of $b_\tau(\cdot, \cdot)$ has been established in Theorem 1 in \cite{xu2009analysis}. The local coercivity is also established for linear (Theorem 2) and quadratic (Theorem 5) CGFEM in \cite{xu2009analysis}. \end{proof}
\begin{lem} \label{lem:eb} Fix a triangle $\tau = \tau_0.$ Suppose $\{ \tau_i \}_{i=1}^3$ are the neighbors (sharing edges) of $\tau$, i.e., $\partial \tau \cap \partial \tau_i \neq \emptyset.$ Then for $w, v\in H^2(\tau)$ \begin{equation} \label{eq:ebounded}
e_\tau(w, I_\tau v - v) \leq C h_\tau^{1/2} \Big( |v|_{H^1(\tau)} + h_\tau |v|_{H^2(\tau)} \Big) \sum_{i=0}^3 \Big( h_{\tau_i}^{-1/2} | w |_{H^1(\tau_i)} + h_{\tau_i}^{1/2} | w |_{H^2(\tau_i)} \Big), \end{equation} where $C$ is a constant independent on $h_\tau$ and $h_{\tau_i}.$ \end{lem}
\begin{proof} By definition and Cauchy-Schwarz inequality, \begin{equation} \label{eq:ede0} \begin{aligned} e_\tau(w, I_\tau v - v) & = \int_{\partial \tau} \{ \kappa \nabla w \} \cdot \boldsymbol{n} ( I_\tau v - v ) \ \text{d} l \\
& \leq \Big( \int_{\partial \tau} | \{ \kappa \nabla w \} \cdot \boldsymbol{n} |^2 \ \text{d} l \Big)^{1/2} \Big( \int_{\partial \tau} | I_\tau v - v |^2 \ \text{d} l \Big)^{1/2} \\
& \leq \kappa_{\tau, \text{max}} \| \{ \nabla w \} \cdot \boldsymbol{n} \|_{L^2(\partial \tau)} \| I_\tau v - v \|_{L^2(\partial \tau)}, \end{aligned} \end{equation} where $\kappa_{\tau, \text{max}}$ is the maximum of $\kappa$ on $\tau.$
By trace inequality, we have \begin{equation} \label{eq:ede1}
\| \{ \nabla w \} \cdot \boldsymbol{n} \|_{L^2(\partial \tau)} \leq \frac{1}{2} \sum_{i=0}^3 \| \nabla w \cdot \boldsymbol{n} \|_{L^2(\partial \tau_i)} \leq \frac{1}{2} \sum_{i=0}^3 \Big( C h_{\tau_i}^{-1/2} | w |_{H^1(\tau_i)} + C h_{\tau_i}^{1/2} | w |_{H^2(\tau_i)} \Big). \end{equation}
Similarly by trace inequality and Lemma \ref{lem:map}, we have \begin{equation} \label{eq:ede2} \begin{aligned}
\| I_\tau v - v \|_{L^2(\partial \tau)} & \leq \sum_{\xi \in s(\tau,k)} \| I_\tau v - v \|_{L^2(\partial t_\xi)} \\
& \leq \sum_{\xi \in s(\tau,k)} C h_\tau^{-1/2} || I_\tau v - v ||_{L^2(t_\xi)} + C h_\tau^{1/2} | v |_{H^1(t_\xi)} \\
& \leq C \Big( h_\tau^{-1/2} || I_\tau v - v ||_{L^2(\tau)} + h_\tau^{1/2} | v |_{H^1(\tau)} \Big) \\
& \leq C h_\tau^{-1/2} \Big( C h^2_\tau |v|_{H^2(\tau)} + C h_\tau |v|_{H^1(\tau)} \Big)+ C h_\tau^{1/2} | v |_{H^1(\tau)} \\
& \leq C h_\tau^{1/2} \Big( |v|_{H^1(\tau)} + h_\tau |v|_{H^2(\tau)} \Big), \end{aligned} \end{equation} where $t_\xi$ are the polygonals defined in Figure \ref{fig:elem}. Putting \eqref{eq:ede1} and \eqref{eq:ede2} into \eqref{eq:ede0} gives us the desired result.
\end{proof}
\begin{lem} \label{lem:locerr} We have the following local error estimate \begin{equation}
|u - \widetilde u_{\tau, h} |_{H^1{(\tau)}} \leq C | u - u_h |_{H^1{(\tau)}} + Ch_\tau^{1/2} \sum_{i=0}^3 \Big( h_{\tau_i}^{-1/2} | u - u_h |_{H^1(\tau_i)} + h_{\tau_i}^{1/2} | u - u_h |_{H^2(\tau_i)} \Big). \end{equation} \end{lem}
\begin{proof} Triangle inequality gives \begin{equation} \label{eq:j1}
| u - \widetilde u_{\tau, h} |_{H^1{(\tau)}} \leq | u - u_h |_{H^1{(\tau)}} + | u_h - \widetilde u_{\tau, h} |_{H^1{(\tau)}}. \end{equation}
By Lemma \ref{lem:loccoe}, we have \begin{equation} \label{eq:j2} \begin{aligned}
C_b | u_h - \widetilde u_{\tau, h} |^2_{H^1{(\tau)}}& \leq b_\tau(u_h - \widetilde u_{\tau, h}, u_h - \widetilde u_{\tau, h} ) \\ & = b_\tau(u_h - u, u_h - \widetilde u_{\tau, h} ) + b_\tau(u - \widetilde u_{\tau, h}, u_h - \widetilde u_{\tau, h} ) \\
& \leq C | u - u_h |_{H^1{(\tau)}} | u_h - \widetilde u_{\tau, h} |_{H^1{(\tau)}} + b_\tau(u - \widetilde u_{\tau, h}, u_h - \widetilde u_{\tau, h} ). \\ \end{aligned} \end{equation}
For simplicity, we set $\delta_{\tau, h} = u_h - \widetilde u_{\tau, h}$. By \eqref{eq:bvpvfe}, we have \begin{equation}\label{eq:j3} b_\tau(u - \widetilde u_{\tau, h}, \delta_{\tau, h} ) = a_\tau(u - u_h, \delta_{\tau, h} ) + e_\tau(u - u_h, \delta_{\tau, h} ) \end{equation}
Now, by boundedness of the bilinear form of $a_\tau(\cdot, \cdot)$, \begin{equation}\label{eq:j4}
a_\tau(u - u_h, \delta_{\tau, h}) \leq \kappa_{\tau, \text{max}} | u - u_h |_{H^1{(\tau)}} | \delta_{\tau, h} |_{H^1{(\tau)}}, \end{equation} where $\kappa_{\tau, \text{max}}$ is the maximum of $\kappa$ on $\tau.$ By Lemma \ref{lem:eb} and inverse inequality, we obtain \begin{equation} \label{eq:j5} \begin{aligned}
e_\tau(u - u_h, \delta_{\tau, h} ) & \leq C h_\tau^{1/2} \Big( |\delta_{\tau, h}|_{H^1(\tau)} + h_\tau |\delta_{\tau, h}|_{H^2(\tau)} \Big) \sum_{i=0}^3 \Big( h_{\tau_i}^{-1/2} | u - u_h |_{H^1(\tau_i)} + h_{\tau_i}^{1/2} | u - u_h |_{H^2(\tau_i)} \Big) \\
& \leq Ch_\tau^{1/2} | \delta_{\tau, h} |_{H^1(\tau)} \sum_{i=0}^3 \Big( h_{\tau_i}^{-1/2} | u - u_h |_{H^1(\tau_i)} + h_{\tau_i}^{1/2} | u - u_h |_{H^2(\tau_i)} \Big).
\end{aligned} \end{equation}
Combining \eqref{eq:j2}, \eqref{eq:j3}, \eqref{eq:j4}, and \eqref{eq:j5}, dividing both side $ | u_h - \widetilde u_{\tau, h} |_{H^1{(\tau)}}$ and then putting it into \eqref{eq:j1} gives the desired result. \end{proof}
\newtheorem{thm}{Theorem}[section] \begin{thm} \label{thm:perr} Assume $u$ is the solution of \eqref{pde} and it is sufficiently smooth. Let $\widetilde u_h = \sum_{\tau \in \mathcal{T}_h } \widetilde u_{\tau, h} \chi_\tau$, where $\widetilde u_{\tau, h} \in V^k_h(\tau)$ is the post-proccessed solution \eqref{eq:ppsol} and $\chi_\tau$ is the usual characteristic function for $\tau$, then we have $$
| u - \widetilde u_h |_{H^1(\Omega)} \leq Ch^k |u|_{H^{k+1}(\Omega)}, $$ where $C$ is a constant independent of $h$. \end{thm}
\begin{proof} Noticing that $h_\tau \leq h$, this can be proved by using Lemma \ref{lem:locerr} and arithmetic-geometric mean inequality: \begin{equation*} \begin{aligned}
| u - \widetilde u_h |^2_{H^1(\Omega)} & = \sum_{\tau} | u - \widetilde u_h |^2_{H^1(\tau)} \\
& \leq C \sum_{\tau} \Big( | u - u_h |^2_{H^1{(\tau)}} + h^2 | u - u_h |^2_{H^2{(\tau)}} \Big) \\
& \leq C | u - u_h |^2_{H^1(\Omega)} + C h^2 | u - u_h |^2_{H^2(\Omega)} . \end{aligned} \end{equation*} By the property of CGFEM approximation (see Theorem 14.3.3 in \cite{brenner2008mathematical} for example), we have \begin{equation}
| u - u_h |_{H^1(\Omega)} \leq C h^k |u|_{H^{k+1}(\Omega)}, \qquad | u - u_h |_{H^2(\Omega)} \leq C h^{k-1} |u|_{H^{k+1}(\Omega)}. \end{equation} Substituting these inequalities back gives the desired result.
\end{proof}
\begin{comment} Lots of work has been done to investigate the closeness of the linear FVEM and linear CGFEM. It is shown in \cite{ewing2002accuracy} that linear FVEM is a small perturbation of linear CGFEM. An identity between linear FVEM and linear CGFEM was established in \cite{xu2009analysis}. However, for higher order FVEM and CGFEM, closeness are not investigated as satisfactory as in the linear case. Now, as a by-product of our post-processing technique, we present and prove that for higher order case, the local FVEM bilinear form is close to its corresponding local CGFEM bilinear form plus an extra term for sufficiently small mesh discretization size and this extra term vanishes in the linear FVEM case.
\begin{thm} \label{thm:fvem2fem} When $\kappa$ is sufficiently smooth, we have \begin{equation} \label{eq:localfvem2fem}
\Big| b_\tau(v, w) - a_\tau(v, w) - \int_\tau \kappa \Delta v (w - I_\tau w) \text{d} \boldsymbol{x} \Big| \leq C h_\tau | v |_{H^1(\tau)} | w |_{H^1(\tau)}, \end{equation} for $v, w \in V^k_h(\tau)$. \end{thm}
\begin{proof} Using definition and Divergence theorem, we have \begin{equation} b_\tau(v, w) = -\sum_{\xi\in v(\tau)} \int_{\partial C^\xi \cap \partial t_\xi} \kappa \nabla v \cdot \boldsymbol{n} I_\tau w \ \text{d} l = - \int_\tau \nabla \cdot ( \kappa \nabla v ) I_\tau w \ \text{d} \boldsymbol{x} + \int_{\partial \tau} \kappa \nabla v \cdot \boldsymbol{n} I_\tau w \ \text{d} l. \end{equation} Integration by parts gives \begin{equation} a_\tau(v, w) = - \int_\tau \nabla \cdot ( \kappa \nabla v ) w \ \text{d} \boldsymbol{x} + \int_{\partial \tau} \kappa \nabla v \cdot \boldsymbol{n} w \ \text{d} l. \end{equation} Taking the difference of the two bilinear form yields \begin{equation} \begin{aligned} b_\tau(v, w) & - a_\tau(v, w) = \int_\tau \nabla \cdot (\kappa \nabla v ) (w - I_\tau w) \ \text{d} \boldsymbol{x} + \int_{\partial \tau} \kappa \nabla v \cdot \boldsymbol{n} ( I_\tau w - w ) \ \text{d} l \\ & = \int_\tau \nabla \kappa \cdot \nabla v (w - I_\tau w) \ \text{d} \boldsymbol{x} + \int_\tau \kappa \Delta v (w - I_\tau w) \ \text{d} \boldsymbol{x} + \int_{\partial \tau} \kappa \nabla v \cdot \boldsymbol{n} ( I_\tau w - w ) \ \text{d} l, \\ \end{aligned} \end{equation} which in turn gives
\begin{equation} b_\tau(v, I_\tau w) - a_\tau(v, w) - \int_\tau\kappa \Delta v (w - I_\tau w) \text{d} \boldsymbol{x} = I_1 + I_2, \end{equation}
where
\begin{equation} I_1 = \int_\tau \nabla \kappa \cdot \nabla v (w - I_\tau w) \ \text{d} \boldsymbol{x} \hspace*{0.3cm} \text{and} \hspace*{0.3cm} I_2 = \int_{\partial \tau} \kappa \nabla v \cdot \boldsymbol{n} ( I_\tau w - w ) \ \text{d} l. \end{equation}
By Cauchy-Schwarz inequality followed by application of Lemma \ref{lem:map}, $I_1$ is estimated as
\begin{equation}
|I_1| \le \| \nabla \kappa \|_{L^\infty(\tau)} \, | v |_{H^1(\tau)} \| w - I_\tau w \|_{L^2(\tau)}
\le C \| \nabla \kappa \|_{L^\infty(\tau)} h_\tau | v |_{H^1(\tau)} \, | w |_{H^1(\tau)}. \end{equation}
\begin{equation} \begin{aligned}
\Big| b_\tau(v, I_\tau w) & - a_\tau(v, w) - \int_\tau\kappa \Delta v (w - I_\tau w) \text{d} \boldsymbol{x} \Big| \\
& = \Big| \int_\tau \nabla \kappa \cdot \nabla v (w - I_\tau w) \ \text{d} \boldsymbol{x} + \int_{\partial \tau} \kappa \nabla v \cdot \boldsymbol{n} ( I_\tau w - w ) \ \text{d} l \Big| \\
& \leq \Big| \int_\tau \nabla \kappa \cdot \nabla v (w - I_\tau w) \ \text{d} \boldsymbol{x} \Big| + \Big| \int_{\partial \tau} \kappa \nabla v \cdot \boldsymbol{n} ( I_\tau w - w ) \ \text{d} l \Big| \\
& \leq \| \nabla \kappa \|_{L^2(\tau)} | v |_{H^1(\tau)} \| w - I_\tau w \|_{L^2(\tau)} + C h_\tau | v |_{H^1(\tau)} | w |_{H^1(\tau)} \\
& \leq C h_\tau | v |_{H^1(\tau)} | w |_{H^1(\tau)}. \end{aligned} \end{equation}
Here, we used the same argument in the proof of Lemma \ref{lem:eb} and inverse inequality for approximation of the second integration, and used Lemma \ref{lem:map} for the approximation of $\| w - I_\tau w \|_{L^2(\tau)}$. This completes the proof. \end{proof}
\begin{remark} For linear FVEM, $\Delta v = 0$, \eqref{eq:localfvem2fem} tells us that for sufficiently small $h_\tau$, the local bilinear forms of FVEM and CGFEM are equivalent in the sense that the error is of higher order.
\end{remark}
\end{comment}
\section{Numerical Experiments}\label{sec:num}
In this section we present various numerical examples to illustrate the performance of the proposed post-processing technique for CGFEM using $V_h^k$, $k=1,2,3$. For the numerical examples, we consider mainly the local conservation property of the post-processed fluxes and the convergence behavior of the post-processed solutions. For these purposes, we consider the following test problems in the unit domain $[0, 1]\times[0, 1]$.
\textbf{Example 1.} Elliptic equation \eqref{pde} with $\kappa =1, u = (x-x^2) (y-y^2)$ with fully Dirichlet boundary condition $g = 0$. $f$ is the function derived from \eqref{pde}.
\textbf{Example 2.} Elliptic equation \eqref{pde} with $\kappa =e^{2x-y^2}, f = -e^{x}, u = e^{-x + y^2}$ with the fully Dirichlet boundary condition $g$ satisfying the true solution.
\textbf{Example 3.} Elliptic equation \eqref{pde} with $$ \kappa = \frac{1}{1-0.8\sin(6\pi x)} \cdot \frac{1}{1-0.8\sin(6\pi y)}, $$ $$ u = 1 - \frac{2\cos(6\pi x) + 15\pi x - 2}{15 \pi}, $$ and $f = 0$. We impose boundary conditions as Dirichlet $1$ at the left boundary and $0$ at the right boundary with homogeneous Neumann boundary conditions at the top and bottom boundaries.
For each test problem, we will present the numerical results of linear, quadratic, and cubic CGFEMs. Now we start with a study of local conservation property of the post-processed fluxes.
\subsection{Conservation Study} \label{sec:cns} To numerically illustrate the behavior of the post-processed fluxes, we run the examples by verifying that the post-processed fluxes satisfies the desired local conservation property \eqref{eq:cvconservation}. For this purpose, we define a local conservation error (LCE) as \begin{equation} \text{LCE}_z = \int_{\partial C^z} - \kappa \nabla \hat u_h \cdot \boldsymbol{n} \ \text{d} l - \int_{C^z} f \ \text{d} \boldsymbol{x}, \end{equation} where $\hat u_h = u_h$ for CGFEMs solution and $\hat u_h = \widetilde u_h$ for the post-processed solution. Naturally, $\text{LCE}_z=0$ means local conservation property \eqref{eq:cvconservation} is satisfied while $\text{LCE}_z \ne 0$ means local conservation property \eqref{eq:cvconservation} is not satisfied on the control volume $C^z$.
Without a post-processing, for instance, LCEs of $u_h$ solved by quadratic CGFEMs are shown by red plots in the left column for Example 1 and right column for Example 2 in Figure \ref{fig:ex12lce}, respectively. We see that these errors are non-zeros, which means that the local conservation property \eqref{eq:cvconservation} is not satisfied. The control volume indices in the figures are arranged as follows: firstly indices from vertices of the mesh, secondly the indices of the degrees of freedom on edges of elements, and lastly (for the cubic case) indices of degrees of freedom inside the elements. Now with the post-processing, LCEs of $\widetilde u_h$ in the quadratic case are shown by dotted green plots in the left column for Example 1 and right column for Example 2 in Figure \ref{fig:ex12lce}, respectively. These errors are practically negligible, which is mainly attributed to the errors in the application of numerical integration and the machine precision. Theoretically, these errors should be zeros as discussed in Section \ref{sec:loccons}. The LCEs for both $u_h$ and $\widetilde u_h$ for Example 3 are shown in Figure \ref{fig:ex3lce}.
\begin{figure}
\caption{ LCEs for $u_h$ (red plots) and for $\widetilde u_h$ (dotted green plots) for Example 1 (left column) and Example 2 (right column) using CGFEM with $V_h^2$.}
\label{fig:ex12lce}
\end{figure}
\begin{figure}
\caption{ LCEs for $u_h$ (red plots) and for $\widetilde u_h$ (dotted green plots) for Example 3 using $V_h^k$ with $k=1$ (top), $k=2$ (middle), and $k=3$ (bottom). }
\label{fig:ex3lce}
\end{figure}
\subsection{Convergence Study} \label{sec:cnv}
Now we show the numerical convergence rates for Example 1, 2, and 3. We collect the $H^1$ semi-norm errors of the CGFEM solution, post-processed solution, and also the difference between these two solutions defined as $| u_h - \widetilde u_h |_{H^1(\Omega)}$. The results for Example 1, 2, and 3 are shown in Figure \ref{fig:ex123h1err}. The $H^1$ semi-norm errors of $k^{\text{th}}$ order CGFEM solutions and post-processed solutions are of optimal convergence orders, which confirm convergence analysis in Theorem \ref{thm:perr} in Section \ref{sec:ana}.
From Figure \ref{fig:ex123h1err}, we can also see that for quadratic and cubic CGFEM, $| u_h - \widetilde u_h |_{H^1(\Omega)}$ tends to be good error estimators. In the linear case, $| u_h - \widetilde u_h |_{H^1(\Omega)}$ is of order 2, which is higher than the optimal error convergence order of CGFEM solution.
\begin{figure}\label{fig:ex123h1err}
\end{figure}
\section{Conclusion}
In this work, we proposed a post-processing technique for any order CGFEM for elliptic problems. This technique builds a bridge between any order of finite volume element method (FVEM) and any order of CGFEM. FVEM has the advantage of its local conservation property but the disadvantages in analysis especially for higher order FVEMs, while CGFEM has its fully established analysis but it lacks the local conservation property. This technique is naturally proposed in this less than ideal situation to serve as a great tool if one would like to use CGFEM to solve PDEs while maintaining a local conservation property, for instance in two-phase flow simulations.
Since the problems that the post-processing technique requires to solve are localized, they are independent of each other. For linear CGFEM, the technique requires solving a 3-by-3 system for each element while for quadratic and cubic CGFEMs, the technique requires solving a 6-by-6 system and 10-by-10 system for each element, respectively. It thus can be easily, naturally, and efficiently implemented in a parallel computing environment.
As for future work, one can use this technique for other differential equations, such as advection diffusion equations, two-phase flow problems, and elasticity models; also one can apply this technique for other numerical methods, such as SUPG. One interesting direction we would like to work on is to develop a post-processing technique which requires solving only 3-by-3 systems for each element and for any order of CGFEMs.
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{frontmatter}
\title{Cancellability and Regularity of Operator Connections}
\author{Pattrawut Chansangiam} \ead{[email protected]}
\address{Department of Mathematics, Faculty of Science,
King Mongkut's Institute of Technology Ladkrabang,
Bangkok 10520, Thailand.}
\begin{abstract}
An operator connection is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above and the transformer inequality. In this paper, we introduce and characterize the concepts of cancellability and regularity of operator connections with respect to operator monotone functions, Borel measures and certain operator equations. In addition, we investigate the existence and the uniqueness of solutions for such operator equations.
\end{abstract}
\begin{keyword} operator connection \sep operator monotone function \sep operator mean \MSC[2010]47A63 \sep 47A64 \end{keyword}
\end{frontmatter}
\section{Introduction}
A general theory of connections and means for positive operators was given by Kubo and Ando \cite{Kubo-Ando}. Let $B(\mathcal{H})$ be the algebra of bounded linear operators on a Hilbert space $\mathcal{H}$. The set of positive operators on $\mathcal{H}$ is denoted by $B(\mathcal{H})^+$. Denote the spectrum of an operator $X$ by $\Sp(X)$. For Hermitian operators $A,B \in B(\mathcal{H})$, the partial order $A \leqslant B$ means that $B-A \in B(\mathcal{H})^+$. The notation $A>0$ suggests that $A$ is a strictly positive operator. A \emph{connection} is a binary operation $\,\sigma\,$ on $B(\mathcal{H})^+$ such that for all positive operators $A,B,C,D$: \begin{enumerate}
\item[(M1)] \emph{monotonicity}: $A \leqslant C, B \leqslant D \implies A \,\sigma\, B \leqslant C \,\sigma\, D$
\item[(M2)] \emph{transformer inequality}: $C(A \,\sigma\, B)C \leqslant (CAC) \,\sigma\, (CBC)$
\item[(M3)] \emph{continuity from above}: for $A_n,B_n \in B(\mathcal{H})^+$,
if $A_n \downarrow A$ and $B_n \downarrow B$,
then $A_n \,\sigma\, B_n \downarrow A \,\sigma\, B$.
Here, $A_n \downarrow A$ indicates that $A_n$ is a decreasing sequence
and $A_n$ converges strongly to $A$. \end{enumerate} Two trivial examples are the left-trivial mean $\omega_l : (A,B) \mapsto A$ and the right-trivial mean $\omega_r: (A,B) \mapsto B$. Typical examples of a connection are the sum $(A,B) \mapsto A+B$ and the parallel sum \begin{align*}
A \,:\,B = (A^{-1}+B^{-1})^{-1}, \quad A,B>0, \end{align*} the latter being introduced by Anderson and Duffin \cite{Anderson-Duffin}. From the transformer inequality, every connection is \emph{congruence invariant} in the sense that for each $A,B \geqslant 0$ and $C>0$ we have \begin{align*}
C(A \,\sigma\, B)C \:=\: (CAC) \,\sigma\, (CBC). \end{align*}
A \emph{mean} is a connection $\sigma$ with normalized condition $I \,\sigma\, I = I$ or, equivalently, fixed-point property $A \,\sigma\, A =A$ for all $A \geqslant 0$. The class of Kubo-Ando means cover many well-known operator means in practice, e.g.
\begin{itemize}
\item $\alpha$-weighted arithmetic means: $A \triangledown_{\alpha} B = (1-\alpha)A + \alpha B$
\item $\alpha$-weighted geometric means:
$A \#_{\alpha} B = A^{1/2}
({A}^{-1/2} B {A}^{-1/2})^{\alpha} {A}^{1/2}$
\item $\alpha$-weighted harmonic means: $A \,!_{\alpha}\,
B = [(1-\alpha)A^{-1} + \alpha B^{-1}]^{-1}$
\item logarithmic mean: $(A,B) \mapsto A^{1/2}f(A^{-1/2}BA^{-1/2})A^{1/2}$
where $f: \mathbb{R}^+ \to \mathbb{R}^+$, $f(x)=(x-1)/\log{x}$, $f(0) \equiv 0$ and $f(1) \equiv 1$.
Here, $\mathbb{R}^+=[0, \infty)$.
\end{itemize}
It is a fundamental that there are one-to-one correspondences between the following objects: \begin{enumerate}
\item[(1)] operator connections on $B(\mathcal{H})^+$
\item[(2)] operator monotone functions from $\mathbb{R}^+$ to $\mathbb{R}^+$
\item[(3)] finite (positive) Borel measures on $[0,1]$
\item[(4)] monotone (Riemannian) metrics on the smooth manifold of positive definite matrices. \end{enumerate}
Recall that a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ is said to be \emph{operator monotone} if \begin{align*}
A \leqslant B \implies f(A) \leqslant f(B) \end{align*} for all positive operators $A,B \in B(\mathcal{H})$ and for all Hilbert spaces $\mathcal{H}$. This concept was introduced in \cite{Lowner}; see also \cite{Bhatia,Hiai,Hiai-Yanagi}. A remarkable fact is that (see \cite{Hansen-Pedersen}) a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ is operator monotone if and only if it is \emph{operator concave}, i.e. \begin{align*}
f((1-\alpha)A + \alpha B) \:\geqslant\: (1-\alpha) f(A) + \alpha f(B), \quad \alpha \in (0,1) \end{align*} holds for all positive operators $A,B \in B(\mathcal{H})$ and for all Hilbert spaces $\mathcal{H}$.
A connection $\sigma$ on $B(\mathcal{H})^+$ can be characterized via operator monotone functions as follows:
\begin{thm}[\cite{Kubo-Ando}] \label{thm: Kubo-Ando f and sigma}
Given a connection $\sigma$, there is a unique operator monotone function $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying
\begin{align*}
f(x)I = I \,\sigma\, (xI), \quad x \geqslant 0.
\end{align*}
Moreover, the map $\sigma \mapsto f$ is a bijection.
\end{thm}
We call $f$ the \emph{representing function} of $\sigma$. A connection also has a canonical characterization with respect to a Borel measure via a meaningful integral representation as follows.
\begin{thm}[\cite{Pattrawut}] \label{thm: 1-1 conn and measure} Given a finite Borel measure $\mu$ on $[0,1]$, the binary operation
\begin{align}
A \,\sigma\, B = \int_{[0,1]} A \,!_t\, B \,d \mu(t), \quad A,B \geqslant 0
\label{eq: int rep connection}
\end{align}
is a connection on $B(\mathcal{H})^+$.
Moreover, the map $\mu \mapsto \sigma$ is bijective,
in which case the representing function of $\sigma$ is given by
\begin{align}
f(x) = \int_{[0,1]} (1 \,!_t\, x) \,d \mu(t), \quad x \geqslant 0. \label{int rep of OMF}
\end{align} \end{thm} We call $\mu$ the \emph{associated measure} of $\sigma$.
A connection is a mean if and only if $f(1)=1$ or its associated measure is a probability measure. Hence every mean can be regarded as an average of weighted harmonic means.
From \eqref{eq: int rep connection} and \eqref{int rep of OMF}, $\sigma$ and $f$ are related by \begin{align}
f(A) \:=\: I \,\sigma\, A, \quad A \geqslant 0. \label{eq: f(A) = I sm A} \end{align} A connection $\sigma$ is said to be \emph{symmetric} if $A \,\sigma\, B = B \,\sigma\, A$ for all $A,B \geqslant 0$.
The notion of monotone metrics arises naturally in quantum mechanics. A metric on the differentiable manifold of $n$-by-$n$ positive definite matrices is a continuous family of positive definite sesqilinear forms assigned to each invertible density matrix in the manifold. A monotone metric is a metric with contraction property under stochastic maps. It was shown in \cite{Petz} that there is a one-to-one correspondence between operator connections and monotone metrics. Moreover, symmetric metrics correspond to symmetric means.
In \cite{Hansen}, the author defined a symmetric metric to be \emph{nonregular} if $f(0)=0$ where $f$ is the associated operator monotone function. In \cite{Gibilisco}, $f$ is said to be \emph{nonregular} if $f(0) = 0$, otherwise $f$ is \emph{regular}. It turns out that the regularity of the associated operator monotone function guarantees the extendability of this metric to the complex projective space generated by the pure states (see \cite{Petz-Sudar}).
In the present paper, we introduce the concept of cancellability for operator connections in a natural way. Various characterizations of cancellability with respect to operator monotone functions, Borel measures and certain operator equations are provided. It is shown that a connection is cancellable if and only if it is not a scalar multiple of trivial means.
Applications of this concept go to certain operator equations involving operator means. We investigate the existence and the uniqueness of such equations. It is shown that such equations are always solvable if and only if $f$ is unbounded and $f(0)=0$ where $f$ is the associated operator monotone function. We also characterize the condition $f(0)=0$ for arbitrary connections without assuming the symmetry. Such connection is said to be nonregular.
\section{Cancellability of connections}
Since each connection is a binary operation, we can define the concept of cancellability as follows.
\begin{defn}
A connection $\sigma$ is said to be
\begin{itemize}
\item \emph{left cancellable} if for each $A>0, B \geqslant 0$ and $C \geqslant 0$,
\begin{align*}
A \,\sigma\, B = A \,\sigma\, C \implies B=C
\end{align*}
\item \emph{right cancellable} if for each $A>0, B \geqslant 0$ and $C \geqslant 0$,
\begin{align*}
B \,\sigma\, A = C \,\sigma\, A \implies B=C
\end{align*}
\item \emph{cancellable} if it is both left and right cancellable.
\end{itemize} \end{defn}
\begin{lem} \label{lem: nonconstant OM}
Every nonconstant operator monotone function from $\mathbb{R}^+$ to $\mathbb{R}^+$ is injective. \end{lem} \begin{proof}
Let $f: \mathbb{R}^+ \to \mathbb{R}^+$ be a nonconstant operator monotone function.
Suppose there exist $b>a \geqslant 0$ such that $f(a)=f(b)$.
Since $f$ is monotone increasing (in usual sense),
$f(x)=f(a)$ for all $a \leqslant x \leqslant b$ and
$f(y) \geqslant f(b)$ for all $y \geqslant b$.
Since $f$ is operator concave,
$f$ is concave in usual sense and hence $f(x)=f(b)$ for all $x \geqslant b$.
The case $a=0$ contradicts the fact that $f$ is nonconstant.
Consider the case $a>0$.
The monotonicity of $f$ yields $f(x) \leqslant f(a)$ for all $0 \leqslant x \leqslant a$.
The differentiability of $f$ implies that $f(x)=f(a)$ for all $x \geqslant 0$,
again a contradiction. \end{proof}
A similar result for this lemma under the restriction that $f(0)=0$ was obtained in \cite{Molnar}. The left-cancellability of connections is characterized as follows.
\begin{thm} \label{thm: cancallability 1}
Let $\sigma$ be a connection with representing function $f$ and associated measure $\mu$.
Then the following statements are equivalent:
\begin{enumerate}
\item[(1)] $\sigma$ is left cancellable ;
\item[(2)] for each $A \geqslant 0$ and $B \geqslant 0$, $I \sigma A = I \sigma B \implies A=B$ ;
\item[(3)] $\sigma$ is not a scalar multiple of the left-trivial mean ;
\item[(4)] $f$ is injective, i.e., $f$ is left cancellable in the sense that
\begin{align*}
f \circ g = f \circ h \:\implies\: g=h \quad ;
\end{align*}
\item[(5)] $f$ is a nonconstant function ;
\item[(6)] $\mu$ is not a scalar multiple of the Dirac measure $\delta_0$ at $0$.
\end{enumerate} \end{thm} \begin{proof}
Clearly, (1) $\Rightarrow$ (2) $\Rightarrow$ (3) and (4) $\Rightarrow$ (5).
For each $k \geqslant 0$, it is straightforward to show that
the representing function of the connection
\begin{align*}
k \omega_l \;:\; (A,B) \mapsto kA
\end{align*}
is the constant function $f \equiv k$
and its associated measure is given by $k\delta_0$.
Hence, we have the implications (3) $\Leftrightarrow$ (5) $\Leftrightarrow$ (6).
By Lemma \ref{lem: nonconstant OM}, we have (5) $\Rightarrow$ (4).
(4) $\Rightarrow$ (2): Assume that $f$ is injective.
Consider $A \geqslant 0$ and $B \geqslant 0$ such that
$I \sigma A = I \sigma B$. Then $f(A) = f(B)$ by \eqref{eq: f(A) = I sm A}.
Since $f^{-1} \circ f (x) =x$
for all $x \in \mathbb{R}^+$, we have $A=B$.
(2) $\Rightarrow$ (1): Let $A>0, B\geqslant 0$ and $C \geqslant 0$ be such that $A \,\sigma\, B = A \,\sigma\, C$.
By the congruence invariance of $\sigma$, we have
\begin{align*}
A^{\frac{1}{2}} (I \,\sigma\, A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) A^{\frac{1}{2}}
\:=\: A^{\frac{1}{2}} (I \,\sigma\, A^{-\frac{1}{2}} C A^{-\frac{1}{2}}) A^{\frac{1}{2}}
\end{align*}
and thus
$I \,\sigma\, A^{-\frac{1}{2}} B A^{-\frac{1}{2}} = I \,\sigma\, A^{-\frac{1}{2}} C A^{-\frac{1}{2}} $.
Now, the assumption (2) implies
$$ A^{-\frac{1}{2}} B A^{-\frac{1}{2}} \:=\: A^{-\frac{1}{2}} C A^{-\frac{1}{2}}, $$ i.e., $B=C$. \end{proof}
Recall that the \emph{transpose} of a connection $\sigma$ is the connection \begin{align*}
(A,B) \;\mapsto B \,\sigma\, A. \end{align*} If $f$ is the representing function of $\sigma$, then the representing function of the transpose of $\sigma$ is given by the \emph{transpose} of $f$ (see \cite{Kubo-Ando}), defined by \begin{align*}
x \mapsto xf(1/x), \quad x>0. \end{align*} A connection is \emph{symmetric} if it coincides with its transpose.
\begin{thm}
Let $\sigma$ be a connection with representing function $f$ and associated measure $\mu$.
Then the following statements are equivalent:
\begin{enumerate}
\item[(1)] $\sigma$ is right cancellable ;
\item[(2)] for each $A \geqslant 0$ and $B \geqslant 0$, $A \sigma I = B \sigma I \implies A=B$ ;
\item[(3)] $\sigma$ is not a scalar multiple of the right-trivial mean ;
\item[(4)] the transpose of $f$ is injective ;
\item[(5)] $f$ is not a scalar multiple of the identity function $x \mapsto x$ ;
\item[(6)] $\mu$ is not a scalar multiple of the Dirac measure $\delta_1$ at $1$.
\end{enumerate} \end{thm} \begin{proof}
It is straightforward to see that, for each $k\geqslant 0$,
the representing function of the connection
\begin{align*}
k \omega_r \;:\; (A,B) \mapsto kB
\end{align*}
is the function $x \mapsto kx$ and its associated measure is given by
$k\delta_1$.
The proof is done by replacing $\sigma$ with its transpose in Theorem \ref{thm: cancallability 1}. \end{proof}
\begin{remk}
The injectivity of the transpose of $f$ does not imply the surjectivity of $f$.
To see that, take $f(x)=(1+x)/2$. Then the transpose of $f$ is $f$ itself. \end{remk}
The following results are characterizations of cancellability of connections.
\begin{cor}
Let $\sigma$ be a connection with representing function $f$ and associated measure $\mu$.
Then the following statements are equivalent:
\begin{enumerate}
\item[(1)] $\sigma$ is cancellable ;
\item[(2)] $\sigma$ is not a scalar multiple of the left/right-trivial mean ;
\item[(3)] $f$ and its transpose are injective ;
\item[(4)] $f$ is neither a constant function nor a scalar multiple of the identity function;
\item[(5)] $\mu$ is not a scalar multiple of $\delta_0$ or $\delta_1$.
\end{enumerate}
In particular, every nontrivial mean is cancellable. \end{cor}
\begin{remk}
The ``order cancellability" does not hold for general connections,
even if we restrict them to the class of means.
For each $A,B>0$, it is not true that the condition $I \,\sigma\, A \leqslant I \,\sigma\, B$
or the condition $A \,\sigma\, I \leqslant B \,\sigma\, I$ implies $A \leqslant B$.
To see this, take $\sigma$ to be the geometric mean.
It is not true that $A^{1/2} \leqslant B^{1/2}$ implies $A \leqslant B$ in general. \end{remk}
\section{Applications to certain operator equations}
Cancellability of connections can be restated in terms of the uniqueness of certain operator equations as follows. A connection $\sigma$ is left cancellable if and only if \begin{quote}
for each given $A>0$ and $B \geqslant 0$, if the equation $A \,\sigma\, X =B$ has a solution, then it has a unique solution. \end{quote} The similar statement for right-cancellability holds. In this section, we investigate the existence and the uniqueness of the operator equation $A \,\sigma\, X =B$.
\begin{thm} \label{thm: operator eq 1}
Let $\sigma$ be a connection which is not a scalar multiple of the left-trivial mean.
Let $f$ be its representing function.
Given $A>0$ and $B \geqslant 0$, the operator equation
\begin{align*}
A \, \sigma X \, \:=\: B
\end{align*}
has a (positive) solution if and only if $\Sp (A^{-\frac{1}{2}} B A^{-\frac{1}{2}})
\subseteq \Range (f)$.
In fact, such a solution is unique and given by
\begin{align*}
X \:=\: A^{\frac{1}{2}} f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) A^{\frac{1}{2}}.
\end{align*} \end{thm} \begin{proof}
Suppose that there is a positive operator $X$ such that $A \,\sigma\, X =B$.
The congruence invariance of $\sigma$ yields
\begin{align*}
A^{\frac{1}{2}} (I \,\sigma\, A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) A^{\frac{1}{2}} \:=\: B.
\end{align*}
The property \eqref{eq: f(A) = I sm A} now implies
\begin{align*}
f(A^{-\frac{1}{2}} X A^{-\frac{1}{2}})
\:=\: I \,\sigma\, A^{-\frac{1}{2}} X A^{-\frac{1}{2}}\:=\: A^{-\frac{1}{2}} B A^{-\frac{1}{2}}.
\end{align*}
By spectral mapping theorem,
\begin{align*}
\Sp (A^{-\frac{1}{2}} B A^{-\frac{1}{2}} )
=\Sp\left(f(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) \right)
= f\left(\Sp(A^{-\frac{1}{2}} X A^{-\frac{1}{2}})\right)
\subseteq \Range (f).
\end{align*}
Conversely, suppose that $\Sp (A^{-\frac{1}{2}} B A^{-\frac{1}{2}})
\subseteq \Range (f)$.
Since $\sigma \neq k \omega_l$ for all $k \geqslant 0$, we have that $f$ is nonconstant
by Theorem \ref{thm: cancallability 1}.
It follows that $f$ is injective by Lemma \ref{lem: nonconstant OM}.
The assumption yields the existence of the operator
$X \equiv A^{\frac{1}{2}} f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) A^{\frac{1}{2}}$.
We obtain from the property \eqref{eq: f(A) = I sm A} that
\begin{align*}
A \,\sigma\, X
\:&=\: A \,\sigma\, A^{\frac{1}{2}} f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) A^{\frac{1}{2}} \\
\:&=\: A^{\frac{1}{2}} \left[I \,\sigma\, f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) \right]
A^{\frac{1}{2}} \\
\:&=\: A^{\frac{1}{2}} f\left(f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}\right)
A^{\frac{1}{2}} \\
\:&=\: B
\end{align*}
The uniqueness of a solution follows from the left-cancellability of $\sigma$.
\end{proof}
Similarly, we have the following theorem.
\begin{thm}
Let $\sigma$ be a connection which is not a scalar multiple of the right-trivial mean.
Given $A>0$ and $B \geqslant 0$, the operator equation
\begin{align*}
X \,\sigma A \, \:=\: B
\end{align*}
has a (positive) solution if and only if $\Sp (A^{-1/2} B A^{-1/2}) \subseteq \Range (g)$,
here $g$ is the representing function of the transpose of $\sigma$.
In fact, such a solution is unique and given by
\begin{align*}
X \:=\: A^{1/2} g^{-1}(A^{-1/2}BA^{-1/2}) A^{1/2}.
\end{align*}
\end{thm}
\begin{thm} \label{cor: f is unbounded and f(0)=0}
Let $\sigma$ be a connection
with representing function $f$.
Then the following statements are equivalent:
\begin{enumerate}
\item[(1)] $f$ is unbounded and $f(0)=0$ ;
\item[(2)] $f$ is surjective, i.e., $f$ is right cancellable in the sense that
\begin{align*}
g \circ f = h \circ f \:\implies\: g=h \quad ;
\end{align*}
\item[(3)] the operator equation
\begin{align}
A \,\sigma X \, \:=\: B \label{eq: A sm X =B}
\end{align}
has a unique solution for any given $A>0$ and $B \geqslant 0$.
\end{enumerate}
Moreover, if (1) holds, then
the solution of \eqref{eq: A sm X =B} varies continuously in each given $A>0$
and $B \geqslant 0$, i.e. the map $(A,B) \mapsto X$ is separately continuous with respect to
the strong-operator topology. \end{thm} \begin{proof}
(1) $\Rightarrow$ (2): This follows directly from the intermediate value theorem.
(2) $\Rightarrow$ (3): It is immediate from Theorem \ref{thm: operator eq 1}.
(3) $\Rightarrow$ (1): Assume (3). The uniqueness of solution for the equation $A \,\sigma\, X =B$
implies the left-cancellability of $\sigma$.
By Theorem \ref{thm: cancallability 1}, $f$ is injective.
The assumption (3) implies the existence of a positive operator $X$ such that
\begin{align*}
f(X) \:=\: I \,\sigma\, X \:=\: 0.
\end{align*}
The spectral mapping theorem implies that $f(\ld)=0$ for all $\lambda\in \Sp(X)$.
Since $f$ is injective, we have $\Sp(X) =\{\ld\}$ for some $\lambda\in \mathbb{R}^+$.
Suppose that $\ld>0$. Then there is $\epsilon >0$ such that $X > \epsilon I >0$.
It follows from the monotonicity of $\sigma$ that
\begin{align*}
0 \:=\: I \,\sigma\, X \:\geqslant\: I \,\sigma\, \epsilon I \:=\: f(\epsilon)I,
\end{align*}
i.e. $f(\epsilon)=0$. Hence, $\epsilon = \ld$. Similarly, since $X > (\epsilon/2) I >0$, we have
$\epsilon = 2 \ld$, a contradiction. Thus, $\ld=0$ and $f(0)=0$.
Now, let $k>0$. The assumption (3) implies the existence of $X \geqslant 0$ such that
$I \,\sigma\, X =kI$. Since $f(X)=kI$, we have $f(\ld)=k$ for all $\lambda\in \Sp(X)$.
Since $\Sp(X)$ is nonempty, there is $\lambda\in \Sp(X)$ such that $f(\ld)=k$.
Therefore, $f$ is unbounded.
Assume that (1) holds. Then the map $(A,B) \mapsto X$ is well-defined.
Recall that if $A_n \in B(\mathcal{H})^+$ converges strongly to $A$,
then $\phi(A_n)$ converges strongly to $\phi(A)$
for any continuous function $\phi$.
It follows that the map
$$(A,B) \mapsto X = A^{\frac{1}{2}} f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) A^{\frac{1}{2}} $$
is separately continuous in each variable. \end{proof}
\begin{ex}
Consider the quasi-arithmetic power mean $\#_{p,\alpha}$ with exponent $p \in [-1,1]$ and weight $\alpha \in (0,1)$, defined by \begin{align*}
A \,\#_{p,\alpha}\, B \:=\: \left[(1-\alpha)A^p + \alpha B^p \right]^{1/p}. \end{align*} Its representing function of this mean is given by \begin{align*}
f_{p,\alpha}(x) \:=\: (1-\alpha+\alpha x^p)^{1/p}. \end{align*} The special cases $p=1$ and $p=-1$ are the $\alpha$-weighted arithmetic mean and the $\alpha$-weighted harmonic mean, respectively. The case $p=0$ is defined by continuity and, in fact, $\#_{0,\alpha} = \#_{\alpha}$ and $f_{0,\alpha}(x)=x^{\alpha}$. Given $A>0$ and $B \geqslant 0$, consider the operator equation
\begin{align}
A \,\#_{p,\alpha}\, X \:=\: B. \label{eq weighted GM}
\end{align}
\underline{The case $p=0$ :}
Since the range of $f_{0,\alpha}(x)=x^{\alpha}$ is $\mathbb{R}^+$,
the equation \eqref{eq weighted GM} always has a unique solution given by
\begin{align*}
X
\:=\: A^{1/2} (A^{-1/2}BA^{-1/2})^{1/\alpha} A^{1/2}
\:\equiv \: A \,\#_{1/\alpha}\, B.
\end{align*}
\underline{The case $0<p\leqslant 1$ :} The range of $f_{p,\alpha}$ is the interval $[(1-\alpha)^{1/p}, \infty)$. Hence, the equation \eqref{eq weighted GM} is solvable if and only if $\Sp(A^{-1/2} B A^{-1/2}) \subseteq [(1-\alpha)^{1/p}, \infty)$, i.e., $B \geqslant (1-\alpha)^{1/p} A$.
\underline{The case $-1 \leqslant p< 0$ :}
The range of $f_{p,\alpha}$ is the interval $[0, (1-\alpha)^{1/p})$. Hence, the equation \eqref{eq weighted GM} is solvable if and only if $\Sp(A^{-1/2} B A^{-1/2}) \subseteq [0, (1-\alpha)^{1/p})$, i.e., $B < (1-\alpha)^{1/p} A$.
For each $p \in [-1,0) \cup (0,1]$ and $\alpha \in (0,1)$, we have \begin{align*}
f_{p,\alpha}^{-1} (x) \:=\: \left(1-\frac{1}{\alpha} + \frac{1}{\alpha} x^p \right)^{1/p}. \end{align*} Hence, the solution of \eqref{eq weighted GM} is given by \begin{align*}
X \:=\: \left[(1-\frac{1}{\alpha})A^p + \frac{1}{\alpha} B^p \right]^{1/p}
\:\equiv\: A \;\#_{p, \frac{1}{\alpha}}\; B. \end{align*} \end{ex}
\begin{ex}
Let $\sigma$ be the logarithmic mean with representing function
\begin{align*}
f(x) \:=\: \frac{x-1}{\log x}, \quad x>0.
\end{align*}
Here, $f(0) \equiv 0$ by continuity. We see that $f$ is unbounded.
Thus, the operator equation $A \,\sigma X \, =B$ is solvable for all $A>0$ and $B \geqslant 0$. \end{ex}
\begin{ex}
Let $\eta$ be the dual of the logarithmic mean, i.e.,
\begin{align*}
\eta \,:\, (A,B) \mapsto \left(A^{-1} \, \sigma \, B^{-1}\right)^{-1}
\end{align*}
where $\sigma$ denotes the logarithmic mean.
The representing function of $\eta$ is given by
\begin{align*}
f(x) \:=\: \frac{x}{x-1} \log{x}, \quad x>0.
\end{align*}
We have that $f(0) \equiv 0$ by continuity and $f$ is unbounded.
Therefore, the operator equation $A \,\eta X \, =B$ is solvable for all $A>0$ and $B \geqslant 0$. \end{ex}
\section{Regularity of connections}
In this section, we consider the regularity of connections.
\begin{thm} \label{thm: regularity}
Let $\sigma$ be a connection with representing function $f$ and associated measure $\mu$.
Then the following statements are equivalent.
\begin{enumerate}
\item[(1)] $f(0)=0$ ;
\item[(2)] $\mu(\{0\}) = 0$ ;
\item[(3)] $I \,\sigma\, 0 = 0$ ;
\item[(4)] $A \,\sigma\, 0 =0$ for all $A \geqslant 0$ ;
\item[(5)] for each $A \geqslant 0$, $0 \in \Sp(A) \Longrightarrow 0 \in \Sp(I \,\sigma\, A)$ ;
\item[(6)] for each $A,X \geqslant 0$, $0 \in \Sp(A) \Longrightarrow 0 \in \Sp(X \,\sigma\, A)$.
\end{enumerate} \end{thm} \begin{proof}
From the integral representation \eqref{int rep of OMF}, we have
\begin{align}
f(x) \:=\: \mu(\{0\}) + \mu(\{1\})x + \int_{(0,1)} (1\,!_t\, x)\,d\mu(t), \quad x\geqslant 0,
\label{eq: int rep of OMR expand}
\end{align}
i.e. $f(0)=\mu(\{0\})$. From the property \eqref{eq: f(A) = I sm A}, we have
$I \,\sigma\, 0 =f(0)I$. Hence, (1)-(3) are equivalent.
It is clear that (4) $\Rightarrow$ (3) and (6) $\Rightarrow$ (5).
(3) $\Rightarrow$ (4): Assume that $I \,\sigma\, 0=0$. For any $A>0$,
we have by the congruence invariance that
\begin{align*}
A \,\sigma\, 0 \:=\: A^{\frac{1}{2}} (I \,\sigma\, 0) A^{\frac{1}{2}} \:=\: 0.
\end{align*}
For general $A \geqslant 0$,
we have $(A + \epsilon I) \,\sigma\, 0 =0$ for all $\epsilon>0$ by the previous claim
and hence $A \,\sigma\, 0 =0$ by the continuity from above.
(5) $\Rightarrow$ (1): We have $0 \in \Sp(I \,\sigma\, 0) = \Sp(f(0)I) = \{f(0)\}$, i.e. $f(0)=0$.
(1) $\Rightarrow$ (6): Assume $f(0)=0$. Consider $A \geqslant 0$ such that $0 \in \Sp(A)$,
i.e. $A$ is not invertible.
Assume first that $X>0$. Then
\begin{align*}
X \,\sigma\, A
\:=\: X^{\frac{1}{2}} (I \,\sigma\, X^{-\frac{1}{2}} A X^{-\frac{1}{2}}) X^{\frac{1}{2}}
\:=\: X^{\frac{1}{2}} f(X^{-\frac{1}{2}} A X^{-\frac{1}{2}}) X^{\frac{1}{2}}.
\end{align*}
Since $X^{-\frac{1}{2}} A X^{-\frac{1}{2}}$ is not invertible, we have
$0 \in \Sp(X^{-\frac{1}{2}} A X^{-\frac{1}{2}})$ and hence by spectral mapping theorem
\begin{align*}
0 \:=\: f(0)
\in f \left(\Sp(X^{-\frac{1}{2}} A X^{-\frac{1}{2}}) \right)
\:=\: \Sp\left(f(X^{-\frac{1}{2}} A X^{-\frac{1}{2}}) \right).
\end{align*}
This implies that $X \,\sigma\, A$ is not invertible.
Now, consider $X \geqslant 0$. The previous claim shows that $(X+I)\,\sigma\, A$ is not invertible.
Since $X \,\sigma\, A \leqslant (X+I) \,\sigma\, A$, we conclude that $X \,\sigma\, A$ is not invertible. \end{proof}
We say that a connection $\sigma$ is \emph{nonregular} if one (thus, all) of the conditions in Theorem \ref{thm: regularity} holds, otherwise $\sigma$ is \emph{regular}. Hence, regular connections correspond to regular operator monotone functions and regular monotone metrics.
\begin{remk} Recall from \cite{Pattrawut} that every connection $\sigma$ can be written as the sum of three connections.
The \emph{singularly discrete part} is a countable sum of weighted harmonic means with nonnegative coefficients. The \emph{absolutely continuous part} is a connection admitting an integral representation with respect to Lebesgue measure $m$ on $[0,1]$. The \emph{singularly continuous part} is associated to a continuous measure mutually singular to $m$. Hence, a connection, whose associated measure has no singularly discrete part, is nonregular. \end{remk}
\begin{remk}
Let $\sigma$ be a connection with representing function $f$ and associated measure $\mu$.
Let $g$ be the representing function of the transpose of $\sigma$.
From \eqref{eq: int rep of OMR expand},
\begin{align*}
g(0) = \lim_{x \to 0^+} xf(\dfrac{1}{x}) = \lim_{x \to \infty} \frac{f(x)}{x}
= \mu(\{1\}).
\end{align*}
Thus, the transpose of $\sigma$ is nonregular if and only if $\mu(\{1\})=0$. \end{remk}
\begin{thm} \label{thm: regularity of mean}
The following statements are equivalent for a mean $\sigma$.
\begin{enumerate}
\item[(1)] $\sigma$ is nonregular ;
\item[(2)] $I \,\sigma\, P = P$ for each projection $P$.
\end{enumerate} \end{thm} \begin{proof}
(1) $\Rightarrow$ (2): Assume that $f(0)=0$ and consider a projection $P$.
Since $f(1)=1$, we have $f(x)=x$ for all $x \in \{0,1\}\supseteq \Sp(P)$.
Thus $I \,\sigma\, P = f(P)=P$.
(2) $\Rightarrow$ (1): We have $0=I \,\sigma\, 0=f(0)I$, i.e. $f(0)=0$. \end{proof}
To prove the next result, recall the following lemma.
\begin{lem}[\cite{Kubo-Ando}]
If $f: \mathbb{R}^+ \to \mathbb{R}^+$ is an operator monotone function such that $f(1)=1$
and $f$ is neither the constant function $1$ nor the identity function, then
\begin{enumerate}
\item[1)] $0<x<1 \implies x<f(x)<1$
\item[2)] $1<x \implies 1<f(x)<x$.
\end{enumerate} \end{lem}
\begin{thm}
Let $\sigma$ be a nontrivial mean.
For each $A \geqslant 0$, if $I \,\sigma\, A =A$, then $A$ is a projection.
Hence, the following statements are equivalent: \begin{enumerate}
\item[(1)] $\sigma$ is nonregular ;
\item[(2)] for each $A \geqslant 0$, $A$ is a projection if and only if $I \,\sigma\, A = A$. \end{enumerate} \end{thm} \begin{proof}
Since $I \,\sigma\, A=A$, we have $f(A)=A$ by \eqref{eq: f(A) = I sm A}.
Hence $f(x)=x$ for all $x \in \Sp(A)$ by the injectivity of the continuous functional calculus.
Since $\sigma$ is a nontrivial mean, the previous lemma forces that $\Sp(A) \subseteq \{0,1\}$,
i.e. $A$ is a projection. \end{proof}
\begin{thm}
Under the condition that $\sigma$ is a left-cancellable connection
with representing function $f$, the following
statements are equivalent:
\begin{enumerate}
\item[(1)] $\sigma$ is nonregular.
\item[(2)] The equation $f(x)=0$ has a solution $x$.
\item[(3)] The equation $f(x)=0$ has a unique solution $x$.
\item[(4)] The only solution to $f(x)=0$ is $x=0$.
\item[(5)] For each $A>0$, the equation $A \,\sigma\, X =0$ has a solution $X$.
\item[(6)] For each $A>0$, the equation $A \,\sigma\, X =0$ has a unique solution $X$.
\item[(7)] For each $A>0$, the only solution to the equation $A \,\sigma\, X =0$ is $X=0$.
\item[(8)] The equation $I \,\sigma\, X =0$ has a solution $X$.
\item[(9)] The equation $I \,\sigma\, X =0$ has a unique solution $X$.
\item[(10)] The only solution to the equation $I \,\sigma\, X =0$ is $X=0$.
\end{enumerate}
Similar results for the case of right-cancellability hold. \end{thm} \begin{proof}
It is clear that (1) $\Rightarrow$ (2), (6) $\Rightarrow$ (5) and (10) $\Rightarrow$ (8).
Since $f$ is injective by Theorem \ref{thm: cancallability 1},
we have (2) $\Rightarrow$ (3) $\Rightarrow$ (4).
(4) $\Rightarrow$ (10): Let $X \geqslant 0$ be such that $I \,\sigma\, X =0$.
Then $f(X)=0$ by \eqref{eq: f(A) = I sm A}.
By spectral mapping theorem, $f(\Sp(X)) =\{0\}$. Hence, $\Sp(X)=\{0\}$, i.e. $X=0$.
(8) $\Rightarrow$ (9): Consider $X \geqslant 0$ such that $I \,\sigma\, X =0$. Then $f(X)=0$.
Since $f$ is injective with continuous inverse, we have $X=f^{-1}(0)$.
(9) $\Rightarrow$ (6): Use congruence invariance.
(5) $\Rightarrow$ (7): Let $A>0$ and consider $X \geqslant 0$ such that $A \,\sigma\, X=0$.
Then $A^{\frac{1}{2}} (I \,\sigma\, A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) A^{\frac{1}{2}} =0$,
i.e. $f(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) = I \,\sigma\, A^{-\frac{1}{2}} X A^{-\frac{1}{2}} =0$.
Hence,
\begin{align*}
f\left( \Sp(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) \right)
= \Sp\left( f(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) \right)
=\{0\}.
\end{align*}
Suppose there exists $\lambda\in \Sp(A^{-\frac{1}{2}} X A^{-\frac{1}{2}})$
such that $\ld>0$.
Then $f(0)<f(\ld)=0$, a contradiction.
Hence, $\Sp(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) =\{0\}$, i.e.
$A^{-\frac{1}{2}} X A^{-\frac{1}{2}} =0$ or $X=0$.
(7) $\Rightarrow$ (1): We have $f(0)I = I \,\sigma\, 0 =0$, i.e. $f(0)=0$. \end{proof}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title {Entropy and the Combinatorial Dimension} \author {S. Mendelson\footnote{Research School of Information Sciences and Engineering, The Australian National University, Canberra, ACT 0200, Australia, e-mail: [email protected]} \and
R. Vershynin\footnote{
Department of Mathematical Sciences,
University of Alberta,
Edmonton, Alberta T6G 2G1, Canada,
e-mail: [email protected]}} \date{}
\maketitle
\begin{abstract} We solve Talagrand's entropy problem: the $L_2$-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley's theorem on classes of $\{0,1\}$-valued functions, for which the shattering dimension is the Vapnik-Chervonenkis dimension.
In convex geometry, the solution means that the entropy of a convex body $K$ is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of $K$. This has a number of consequences, including the optimal Elton's Theorem and estimates on the uniform central limit theorem in the real valued case. \end{abstract}
\section{Introduction}
The fact that the covering numbers of a set are exponential in its linear algebraic dimension is fundamental and simple. Let $A$ be a class of functions bounded by $1$, defined on a set $\Omega$. If $A$ is a finite dimensional class then for every probability measure on $\mu$ on $\Omega$, \begin{equation} \label{volumetric}
N(A, t, L_2(\mu)) \le \Big( \frac{3}{t} \Big)^{\dim(A)},
\ \ \ \ 0 < t < 1, \end{equation} where $\dim(A)$ is the linear algebraic dimension of $A$ and the left-hand side of \eqref{volumetric} is the covering number of $A$, the minimal number of functions needed to approximate any function in $A$ within an error $t$ in the $L_2(\mu)$-norm. This inequality follows by a simple volumetric argument (see e.g. \cite{Pi} Lemma 4.10) and is, in a sense, optimal: the dependence both on $t$ and on the dimension is sharp (except, perhaps, for the constant $3$).
The linear algebraic dimension of $A$ is often too large for \eqref{volumetric} to be useful, as it does not capture the ``size'' of $A$ in different directions but only determines in how many directions $A$ does not vanish. The aim of this paper is to replace the linear algebraic dimension by a combinatorial dimension originated from the classical works of Vapnik and Chervonenkis \cite{VC 71}, \cite{VC 81}.
We say that a subset $\s$ of $\Omega$ is $t$-shattered by a class $A$ if there exists a level function $h$ on $\s$ such that, given any subset $\s'$ of $\s$, one can find a function $f \in A$ with $f(x) \le h(x)$ if $x \in \s'$ and $f(x) \ge h(x) + t$ if $x \in \s \setminus \s'$. The {\em shattering dimension} of $A$, denoted by $\vc(A,t)$ after Vapnik and Chervonenkis, is the maximal cardinality of a set $t$-shattered by $A$. Clearly, the shattering dimension does not exceed the linear algebraic dimension, and is often much smaller. Our main result states that the linear algebraic dimension in \eqref{volumetric} can be essentially replaced by the shattering dimension.
\begin{theorem} \label{main}
Let $A$ be a class of functions bounded by $1$,
defined on a set $\Omega$.
Then for every probability measure $\mu$ on $\Omega$,
\begin{equation} \label{combinatorial}
N(A, t, L_2(\mu))
\le \Big( \frac{2}{t} \Big)^{K \cdot \vc(A,\, c t)},
\ \ \ \ 0 < t < 1,
\end{equation}
where $K$ and $c$ are positive absolute constants. \end{theorem} There also exists a (simple) reverse inequality complementing \eqref{combinatorial}: for some measure $\mu$, one has $N(A, t, L_2(\mu)) \ge 2^{K \cdot \vc(A,\, c t)}$, where $K$ and $c$ are some absolute constants, see e.g. \cite{T 02}.
The origins of Theorem \ref{main} are rooted in the work of Vapnik and Chervonenkis, who first understood that entropy estimates are essential in determining whether a class of functions obeys the uniform law of large numbers. The subsequent fundamental works of Koltchinskii \cite{K} and Gin\`{e} and Zinn \cite{GZ} enhanced the link between entropy estimates and uniform limit theorems (see also \cite{T 96}).
In 1978, R.~Dudley proved Theorem \ref{main} for classes of $\{0,1\}$-valued functions (\cite{Du}, see \cite{LT} 14.3). This yielded that a $\{0,1\}$-class obeys the uniform law of large numbers (and even the uniform Central Limit Theorem) if and only if its shattering dimension is finite for $0 < t < 1$. The main difficulty in proving such limit theorems for general classes has been the absence of a uniform entropy estimate of the nature of Theorem \ref{main} (\cite{T 88}, \cite{T 92}, \cite{T 96}, \cite{ABCH}, \cite{BL}, \cite{T 02}). However, proving Dudley's result for general classes is considerably more difficult due to the lack of the obvious property of the $\{0,1\}$-valued classes, namely that if a set $\s$ is $t$-shattered for some $0 < t < 1$ then it is automatically $1$-shattered.
In 1992, M.~Talagrand proved a weaker version of Theorem \ref{main}: under some mild regularity assumptions, $\log N(A, t, L_2(\mu)) \le K \cdot \vc(A, \,ct) \log^M (\frac{2}{t})$, where $K$, $c$ and $M$ are some absolute constants (\cite{T 92}, \cite{T 02}). Theorem \ref{main} is Talagrand's inequality with the best possible exponent $M = 1$ (and without regularity assumptions).
Talagrand's inequality was motivated not only by limit theorems in probability, but to a great extent by applications to convex geometry. A subset $B$ of $\R^n$ can be viewed as a class of real valued functions on $\{1, \ldots, n\}$. If $B$ is convex and, for simplicity, symmetric, then its shattering dimension $\vc(B,t)$ is the maximal cardinality of a subset $\s$ of $\{1, \ldots, n\}$ such that $P_\s (B) \supset [-\frac{t}{2}, \frac{t}{2}]^\s$, where $P_\s$ denotes the orthogonal projection in $\R^n$ onto $\R^\s$. In the general, non-symmetric, case we allow translations of the cube $[-\frac{t}{2}, \frac{t}{2}]^\s$ by a vector in $\R^\s$.
The following entropy bound for convex bodies is then an immediate consequence of Theorem \ref{main}. Recall that $N(B,D)$ is the covering number of $B$ by a set $D$ in $\R^n$, the minimal number of translates of $D$ needed to cover $B$.
\begin{corollary} \label{main corollary}
There exist positive absolute constants $K$ and $c$ such that
the following holds.
Let $B$ be a convex body contained in $[0,1]^n$,
and $D_n$ be the unit Euclidean ball in $\R^n$.
Then for $0 < t < 1$
$$
N(B, t \sqrt{n} D_n)
\le \Big( \frac{2}{t} \Big)^{K d},
$$
where $d$ is the maximal cardinality of a subset $\s$
of $\{1, \ldots, n\}$ such that
$$
P_\s (B) \supseteq h + [0, ct]^\s
\ \ \ \text{for some vector $h$ in $\R^n$.}
$$ \end{corollary}
As M.~Talagrand notices in \cite{T 02}, Theorem \ref{main} is a ``concentration of pathology'' phenomenon. Assume one knows that a covering number of the class $A$ is large. All this means is that $A$ contains many well separated functions, but it tells nothing about the structure these functions form. The conclusion of \eqref{combinatorial} is that $A$ must shatter a large set $\s$, which detects a very accurate pattern: one can find functions in
$A$ oscillating on $\s$ in all possible $2^{|\s|}$ ways around fixed levels. The ``largeness'' of $A$, {\em a priori} diffused, is {\em a fortiori} concentrated on the set $\s$.
The same phenomenon is seen in Corollary \ref{main corollary}: given a convex body $B$ with large entropy, one can find an entire cube in a coordinate projection of $B$, the cube that certainly {\em witnesses} the entropy's largeness.
When dualized, Corollary \ref{main corollary} solves the problem of finding the best asymptotics in Elton's Theorem. Let $x_1, \ldots, x_n$ be vectors in the unit ball of a Banach space, and $\e_1, \ldots, \e_n$ be Rademacher random variables (independent Bernoulli random variables taking values $1$ and $-1$ with probability $1/2$). By the triangle inequality, the expectation
$\E \| \sum_{i=1}^n \e_i x_i\|$ is at most $n$, and assume that
$\E \| \sum_{i=1}^n \e_i x_i \| \ge \d n$ for some number $\d > 0$.
In 1983, J.~Elton \cite{E} proved an important result that there exists a subset $\s$ of $\{1, \ldots, n\}$ of size proportional to $n$ such that the set of vectors $(x_i)_{i \in \s}$ is equivalent to the $\ell_1$ unit-vector basis. Specifically, there exist numbers $s, t > 0$, depending only on $\d$, such that \begin{equation} \label{st}
|\s| \ge s^2 n \ \ \ \text{and} \ \ \ \Big\| \sum_{i \in \s} a_i x_i \Big\|
\ge t \sum_{i \in \s} |a_i| \ \ \text{for all real numbers $(a_i)$}. \end{equation} Several steps have been made towards finding the best possible $s$ and $t$ in Elton's Theorem. A trivial upper bound is $s, t \le \d$ which follows from the example of identical vectors and by shrinking the usual $\ell_1$ unit-vector basis. As for the lower bounds, J.~Elton proved \eqref{st} with $s \sim \d / \log(1/\d)$ and $t \sim \d^3$. A.~Pajor \cite{Pa} removed the logarithmic factor from $s$. M.~Talagrand \cite{T 92}, using his inequality discussed above, improved $t$ to $\d / \log^M (1/\d)$. In the present paper, we use Corollary \ref{main corollary} to solve this problem by proving the optimal asymptotics: $s, t \sim \d$.
\begin{theorem} \label{intro elton}
Let $x_1, \ldots, x_n$ be vectors in the unit
ball of a Banach space, satisfying
$$
\E \Big\| \sum_{i=1}^n \e_i x_i \Big\| \ge \d n
\ \ \ \text{for some number $\d > 0$}.
$$
Then there exists a subset $\s \subset \{ 1, \ldots, n \}$
of cardinality $|\s| \ge c \d^2 n$ such that
$$
\Big\| \sum_{i \in \s} a_i x_i \Big\|
\ge c \d \sum_{i \in \s} |a_i|
\ \ \text{for all real numbers $(a_i)$},
$$
where $c$ is a positive absolute constant. \end{theorem} Furthermore, there is an interplay between the size of $\s$ and the isomorphism constant -- they can not attain their worst possible values together. Namely, we prove that $s$ and $t$ in \eqref{st} satisfy in addition to $s,t \gtrsim \d$ also the lower bound $s \cdot t \log^{1.6}(2/t) \gtrsim \d$, which, as an easy example shows, is optimal for all $\d$ within the logarithmic factor. The power 1.6 can be replaced by any number greater than 1.5. This estimate improves one of the main results of the paper \cite{T 92} where this phenomenon in Elton's Theorem was discovered and proved with a constant (unspecified) power of logarithm.
The paper is organized as follows. In the remaining part of the introduction we sketch the proof of Theorem \ref{main}; the complete proof will occupy Section \ref{s:proof}. Section \ref{s:convexity} is devoted to applications to Elton's Theorem and to empirical processes.
Here is a sketch of the proof of Theorem \ref{main}. Starting with a set $A$ which is separated with respect to the $L_2(\mu)$-norm, it is possible find a coordinate $\omega \in \Omega$ (selected randomly) on which $A$ is diffused, i.e. the values $\{f(\omega),
\; f \in A\}$ are spread in the interval $[-1,1]$. Then there exist two nontrivial subsets $A_1$ and $A_2$ of $A$ with their set of values $\{f(\omega), \; f \in A_1\}$ and $\{f(\omega), \; f \in A_2\}$ well separated from each other on the line. Continuing this process of separation for $A_1$ and $A_2$, etc., one can construct a dyadic tree of subsets of $A$, called a separating tree, with at least $|A|^{1/2}$ leaves. The ``largeness'' of the class $A$ is thus captured by its separating tree.
The next step evoked from a beautiful idea in \cite{ABCH}. First, there is no loss of generality in discretizing the class: one can assume that $\Omega$ is finite (say $|\Omega|=n$) and that the functions in $A$ take values in $\frac{t}{6}\Z \cap [-1,1]$. Then, instead of producing a large set $\s$ shattered by $A$ with a certain level function $h$, one can count the number of different pairs $(\s,h)$ for which $\s$ is shattered by $A$ with the level function $h$. If this number exceeds $\sum_{k=0}^d
\binom{n}{k}(\frac{12}{t})^k$ then there must exist a set $\s$ of size $|\s|>d$ shattered by $A$ (because there are $\binom{n}{k}$ possible sets $\s$ of cardinality $k$, and for such a set there are at most $(\frac{12}{t})^k$ possible level functions).
The only thing remaining is to bound below the number of pairs $(\s,h)$ for which $\s$ is shattered by $A$ with a level function
$h$. One can show that this number is bounded below by the number of the leaves in the separating tree of $A$, which is $|A|^{1/2}$. This implies that $|A|^{1/2} \leq\sum_{k=0}^d \binom{n}{k}(\frac{12}{t})^k \sim (\frac{n}{td})^d$, where $d=\vc(A,ct)$. The ratio $\frac{n}{d}$ can be eliminated from this estimate by a probabilistic extraction principle which reduces the cardinality of $\Omega$.
\noindent ACKNOWLEDGEMENTS
\noindent The first author was supported by an Australian Research Council Discovery grant. The second author thanks Nicole Tomczak-Jaegermann for her constant support. He also acknowledges a support from the Pacific Institute of Mathematical Sciences, and thanks the Department of Mathematical Sciences of the University of Alberta for its hospitality. Finally, we would like to thank the referee for his valuable comments and suggestions.
\section{The Proof of Theorem \ref{main}} \label{s:proof}
For $t > 0$, a pair of functions $f$ and $g$ on $\Omega$ is {\em
$t$-separated in $L_2(\mu)$} if $\|f - g\|_{L_2(\mu)} > t$. A set of functions is called $t$-separated if every pair of distinct points in the set is $t$-separated. Let $N_{\rm sep} (A, t, L_2(\mu))$ denote the maximal cardinality of a $t$-separated subset of $A$. It is standard and easily seen that \begin{equation*}
N (A, t, L_2(\mu))
\le N_{\rm sep} (A, t, L_2(\mu))
\le N (A, \frac{t}{2}, L_2(\mu)). \end{equation*} This inequality shows that in the proof of Theorem \ref{main} we may assume that $A$ is $t$-separated in the $L_2(\mu)$ norm, and replace its covering number by its cardinality.
We will need two probabilistic results, the first of which is straightforward.
\begin{lemma} \label{l:xx'}
Let $X$ be a random variable
and $X'$ be an independent copy of $X$.
Then
$$
\E |X - X'|^2 = 2 \E |X - \E X|^2 = 2 \inf_a \E |X - a|^2.
$$ \end{lemma}
The next lemma is a small deviation principle. Denote by $\s(X)^2
= \E |X - \E X|^2$ the variance of the random variable $X$.
\begin{lemma} \label{l:small deviation}
Let $X$ be a random variable with nonzero variance.
Then there exist numbers $a \in \R$ and
$0 < \b \le \frac{1}{2}$, so that letting
\begin{align*}
p_1 &= \P \{ X > a + {\textstyle \frac{1}{6}} \s(X) \}
\ \ \ \text{and} \\
p_2 &= \P \{ X < a - {\textstyle \frac{1}{6}} \s(X) \},
\end{align*}
one has either $p_1 \ge 1-\b$ and $p_2 \ge \frac{\b}{2}$,
or $p_2 \ge 1-\b$ and $p_1 \ge \frac{\b}{2}$. \end{lemma}
\proof Recall that a median of $X$ is a number $M_X$ such that $\P \{ X \ge M_X \} \ge 1/2$ and $\P \{ X \le M_X \} \ge 1/2$; without loss of generality we may assume that $M_X = 0$. Therefore $\P \{ X > 0 \} = 1 - \P \{ X \le 0 \} \le 1/2$ and similarly $\P \{ X < 0 \} \le 1/2$.
By Lemma \ref{l:xx'}, \begin{align} \label{integrals} \s(X)^2
&\le \E |X|^2
= \int_0^\infty \P \{ |X| > \l \} \; d\l^2 \nonumber \\
&= \int_0^\infty \P \{ X > \l \} \; d\l^2
+ \int_0^\infty \P \{ X < -\l \} \; d\l^2 \end{align} where $d\lambda^2 = 2\lambda \;d\lambda$.
Assume that the conclusion of the lemma fails, and let $c$ be any number satisfying $\frac{1}{3} < c < \frac{1}{\sqrt{8}}$. Divide $\R_+$ into intervals $I_k$ of length $c \s(X)$ by setting $$ I_k = \Big( c \s(X) k, \; c \s(X) (k + 1) \Big], \ \ \ k = 0, 1, 2, \ldots $$ and let $\b_0, \b_1, \b_2, \ldots$ be the non-negative numbers defined by $$ \P \{ X > 0 \} = \b_0 \le 1/2, \ \ \ \ \ P \{X \in I_k \} = \b_k - \b_{k+1}, \ \ \ k = 0, 1, 2, \ldots $$ We claim that \begin{equation} \label{bk}
\text{for all $k \ge 0$,} \ \ \
\b_{k+1} \le \frac{1}{2} \b_k. \end{equation} Indeed, assume that $\b_{k+1} > \frac{1}{2} \b_k$ for some $k$ and consider the intervals $J_1 = \left(-\infty, c \s(X) k \right]$ and $J_2 = \left(c \s(X) (k+1), \infty \right)$. Then $J_1 = (-\infty, 0] \cup ( \bigcup_{0 \le l \le k-1} I_l)$, so $$ \P \{ X \in J_1 \} = (1 - \b_0) + \sum_{0 \le l \le k-1} (\b_l - \b_{l+1}) = 1- \b_k. $$ Similarly, $J_2 = \bigcup_{l \ge k+1} I_l$ and thus $$ \P \{ X \in J_2 \} = \sum_{l \ge k+1} (\b_l - \b_{l+1}) = \b_{k+1} > \frac{1}{2} \b_k. $$ Moreover, since the sequence $(\b_k)$ is non-increasing by its definition, then $\b_k \ge \b_{k+1} > \frac{1}{2} \b_k \ge 0$ and $\b_k \le \b_0 \le \frac{1}{2}$. Then the conclusion of the lemma would hold with $a$ being the middle point between the intervals $J_1$ and $J_2$ and with $\b = \b_k$, which contradicts the assumption that the conclusion of the lemma fails. This proves \eqref{bk}.
Now, one can apply \eqref{bk} to estimate the first integral in \eqref{integrals}. Note that whenever $\l \in I_k$, $$ \P \{ X > \l \} \le \P \{ X > c \s(X) k \} = \P \Big( \bigcup_{l \ge k} I_l \Big) = \b_k. $$ Then \begin{align} \int_0^\infty \P \{ X > \l \} \; d\l^2
&\le \sum_{k \ge 0} \int_{I_k} \b_k \cdot 2 \l \;d\l \nonumber\\
&\le \sum_{k \ge 0} \b_k \cdot
2 c \s(X) (k+1) \;{\rm length}(I_k). \label{series} \end{align} Applying \eqref{bk} inductively, it is evident that $\b_k \le (\frac{1}{2})^k \b_0 \le \frac{1}{2^{k+1}}$, and since ${\rm length }(I_k) = c \s(X)$, \eqref{series} is bounded by $$ 2 c^2 \s(X)^2 \sum_{k \ge 0} \frac{k+1}{2^{k+1}} = 4 c^2 \s(X)^2 < \frac{1}{2} \s(X)^2. $$
By an identical argument one can show that the second integral in \eqref{integrals} is also bounded by $\frac{1}{2} \s(X)^2$. Therefore $$ \s(X)^2 < \frac{1}{2} \s(X)^2 + \frac{1}{2} \s(X)^2 = \s(X)^2, $$ and this contradiction completes the proof. \endproof
\subsection*{Constructing a separating tree}
Let $A$ be a finite class of functions on a probability space
$(\Omega, \mu)$, which is $t$-separated in $L_2 (\mu)$. Throughout the proof we will assume that $|A| > 1$. One can think of the class $A$ itself as a (finite) probability space with the uniform measure on it, that is, each element $x$ in $A$ is assigned probability $\frac{1}{|A|}$.
\begin{lemma} \label{separationlemma}
Let $A$ be a $t$-separated subset of $L_2(\mu)$. Then, there
exist a coordinate $i$ in $\Omega$
and numbers $a \in \R$ and $0<\beta \leq 1/2$, so that setting
\begin{align*}
N_1 &= | \{ x \in A : \; x(i) > a + {\textstyle \frac{1}{12}} t \}
|
\ \ \ \text{and} \\
N_2 &= | \{ x \in A : \; x(i) < a - {\textstyle \frac{1}{12}} t \}
|,
\end{align*}
one has either $N_1 \ge (1 - \b) |A|$
and $N_2 \ge \frac{\b}{2} |A|$, or vice versa. \end{lemma}
\begin{proof} Let $x, x'$ be random points in $A$ selected independently according to the uniform (counting) measure on $A$. By Lemma \ref{l:xx'}, \begin{align} \label{exys}
\E \|x - x'\|_{L_2(\mu)}^2
&= \E \int_\Omega |x(i) - x'(i)|^2 \; d\mu(i)
= \int_\Omega \E |x(i) - x'(i)|^2 \; d\mu(i) \nonumber \\
&= 2 \int_\Omega \E |x(i) - \E x(i)|^2 \; d\mu(i) \\ \nonumber
&= 2 \int_\Omega \s(x(i))^2 \; d\mu(i) \end{align} where $\s(x(i))^2$ is the variance of the random variable $x(i)$ with respect to the uniform measure on $A$.
On the other hand, with probability $1 - \frac{1}{|A|}$ we have $x
\ne x'$ and, whenever this event occurs, the separation assumption on $A$ implies that $\|x - x'\|_{L_2(\mu)} \ge t$. Therefore $$
\E \|x - x'\|^2_{L_2(\mu)} \ge \Big( 1 - \frac{1}{|A|} \Big) t^2 \ge \frac{t^2}{2} $$
provided that $|A| > 1$.
Together with \eqref{exys} this proves the existence of a coordinate $i \in \Omega$, on which \begin{equation} \label{sxi}
\s(x(i)) \ge \frac{t}{2}, \end{equation} and the claim follows from Lemma \ref{l:small deviation} applied to the random variable $x(i)$. \end{proof}
This lemma should be interpreted as a separation lemma for the set $A$. It means that one can always find two nontrivial subsets of $A$ and a coordinate in $\Omega$, on which the two subsets are separated with a ``gap" proportional to $t$.
Based on Lemma \ref{separationlemma}, one can construct a large separating tree in $A$. Recall that a {\em tree of subsets} of a set $A$ is a finite collection $T$ of subsets of $A$ such that, for every pair $B, D \in T$ either $B$ and $D$ are disjoint or one of them contains the other. We call $D$ a {\em son} of $B$ if $D$ is a maximal (with respect to inclusion) proper subset of $B$ that belongs to $T$. An element of $T$ with no sons is called a {\em leaf}.
\begin{definition}
Let $A$ be a class of functions on $\Omega$ and $t > 0$.
A {\em $t$-separating tree $T$ of $A$} is a tree of subsets of $A$
such that every element $B \in T$ which is not a leaf has
exactly two sons $B_+$ and $B_{-}$ and, for some
coordinate $i \in \Omega$,
$$
f(i) > g(i) + t
\ \ \ \text{for all $f \in B_+$, $g \in B_-$.}
$$ \end{definition}
\begin{proposition} \label{thm:separatingtree} Let $A$ be a finite class of functions on a probability space $(\Omega,\mu)$. If $A$ is $t$-separated with respect to the
$L_2(\mu)$ norm, then there exists a $\frac{1}{6} t$-separating tree of $A$ with at least $|A|^{1/2}$ leaves. \end{proposition}
\begin{proof} By Lemma \ref{separationlemma}, any finite class $A$ which is $t$-separated with respect to the $L_2(\mu)$ norm has two subsets $A_+$ and $A_{-}$ and a coordinate $i \in \Omega$ for which $f(i) > g(i) + \frac{1}{6} t$ for every $f \in A_+$ and $g \in A_{-}$. Moreover, there exists some number $0<\beta \leq 1/2$ such that \begin{equation*}
|A_+|\geq (1-\beta)|A| \ \ \ \text{and} \ \ \ |A_{-}| \geq \frac{\beta}{2}, \ \ \ \text{or vice versa}. \end{equation*} Thus, $A_+$ and $A_{-}$ are sons of $A$ which are both large and well separated on the coordinate $i$.
The conclusion of the proposition will now follow by induction on the cardinality of $A$. The proposition clearly holds for $|A|=2$. Assume it holds for every $t$-separated class of cardinality bounded by $N$, and let $A$ be a $t$-separated class of cardinality $N+1$. Let $A_+$ and $A_{-}$ be the sons of $A$ as above; since $\b > 0$, we have $|A_+|,|A_{-}| \leq N$. Moreover, if $A_+$ has a $\frac{1}{6} t$-separating tree with $N_+$ leaves and $A_{-}$ has a $\frac{1}{6} t$-separating tree with $N_{-}$ leaves then, by joining these trees, $A$ has a $\frac{1}{6}
t$-separating tree with $N_+ + N_{-}$ leaves, the number bounded below by $|A_+|^{1/2}+|A_-|^{1/2}$ by the induction hypothesis. Since $\b \le 1/2$, \begin{align*}
|A_+|^{\frac{1}{2}}+|A_{-}|^{\frac{1}{2}}
&\geq \bigl((1-\beta)|A|\bigr)^{\frac{1}{2}}
+\bigl(\frac{\beta}{2}|A|\bigr)^{\frac{1}{2}}\\
&= \Bigl[(1-\beta)^{\frac{1}{2}}
+\bigl(\frac{\beta}{2}\bigr)^{\frac{1}{2}}\Bigr]
|A|^{\frac{1}{2}}
\geq |A|^{\frac{1}{2}} \end{align*} as claimed. \end{proof}
The exponent $1/2$ has no special meaning in Proposition \ref{thm:separatingtree}. It can be improved to any number smaller that $1$ at the cost of reducing the constant $\frac{1}{6}$.
\subsection*{Counting shattered sets}
As explained in the introduction, our aim is to construct a large set shattered by a given class. We will first try to do this for classes of integer-valued functions.
Let $A$ be a class of integer-valued functions on a set $\Omega$. We say that a couple $(\s, h)$ is a {\em center} if $\s$ is a finite subset of $\Omega$ and $h$ is an integer-valued function on $\s$. We call the cardinality of $\s$ the dimension of the center. For convenience, we introduce (the only) $0$-dimensional center $(\emptyset, \emptyset)$, which is the {\em trivial center}.
\begin{definition} \label{shatteredcenter} The {\em set $A$ shatters a center $(\s, h)$} if the following holds: \begin{itemize}
\item{either $(\s, h)$ is trivial and $A$ is nonempty,}
\item{or, otherwise, for every choice of signs
$\theta \in \{-1,1\}^\s$
there exists a function $f \in A$ such that for $i \in \s$
\begin{equation} \label{discrete shatter}
\begin{cases}
f(i) > h(i) & \text{when $\theta(i) = 1$},\\
f(i) < h(i) & \text{when $\theta(i) = -1$}.
\end{cases}
\end{equation}
} \end{itemize} \end{definition} It is crucial that both inequalities in \eqref{discrete shatter} are strict: they ensure that whenever a $d$-dimensional center is shattered by $A$, one has $\vc(A, 2) \ge d$. In fact, it is evident that $\vc(A,2)$ is the maximal dimension of a center shattered by $A$.
\begin{proposition} \label{centers vs leaves}
The number of centers shattered by $A$ is
at least the number of leaves in any $1$-separating tree of $A$. \end{proposition}
\proof Given a class $B$ of integer-valued functions, denote by $s(B)$ the number of centers shattered by $B$. It is enough to prove that if $B_+$ and $B_-$ are the sons of an element $B$ of a $1$-separating tree in $A$ then \begin{equation} \label{HHH}
s(B) \ge s(B_+) + s(B_-). \end{equation} By the definition of the $1$-separating tree, there is a coordinate $i_0 \in \Omega$, such that $f(i_0) > g(i_0) + 1$ for all $f \in B_+$ and $g \in B_-$. Since the functions are integer-valued, there exists an integer $t$ such that $$ f(i_0) > t \ \ \text{for $f \in B_+$} \ \ \ \text{and} \ \ \ g(i_0) < t \ \ \text{for $g \in B_-$}. $$
If a center $(\s, h)$ is shattered either by $B_+$ or by $B_-$, it is also shattered by $B$. Next, assume that $(\s, h)$ is shattered by both $B_+$ and $B_-$. Note that in this case $i_0 \not\in \s$. Indeed, if the converse holds then $\s$ contains $i_0$ and hence is nonempty. Thus the center $(x,\s)$ is nontrivial and there exist $f \in B_+$ and $g \in B_-$ such that $t < f(i_0) < h(i_0)$ (by \eqref{discrete shatter} with $\theta(i_0) = -1$) and $t > g(i_0) > h(i_0)$ (by \eqref{discrete shatter} with $\theta(i_0) = 1$), which is impossible. Consider the center $(\s', h') = (\s \cup \{i_0\}, h \oplus t)$, where $h \oplus t$ is the extension of the function $h$ onto the set $\s \cup \{i_0\}$ defined by $(h \oplus t) (i_0) = t$.
Observe that $(\s', h')$ is shattered by $B$. Indeed, since $B_+$ shatters $(\s, h)$, then for every $\theta \in \{-1,1\}^\s \times \{1\}^{\{i_0\}}$ there exists a function $f \in B_+$ such that \eqref{discrete shatter} holds for $i \in \s$. Also, since $f \in B_+$, then automatically $f(i_0) > t = h'(i_0)$. Similarly, for every $\theta \in \{-1,1\}^\s \times \{-1\}^{\{i_0\}}$, there exists a function $f \in B_-$ such that \eqref{discrete shatter} holds for $i \in \s$ and automatically $f(i_0) < t = h'(i_0)$.
Clearly, $(\s', h')$ is shattered by neither $B_+$ nor by $B_-$, because $f(i_0) > t = h'(i_0)$ for all $f \in B_+$, so \eqref{discrete shatter} fails if $\theta(i_0) = -1$; a similar argument holds for $B_-$.
Summarizing, $(\s, h) \to (\s', h')$ is an injective mapping from the set of centers shattered by both $B_+$ and $B_-$ into the set of centers shattered by $B$ but not by $B_+$ or $B_-$, which proves our claim. \endproof
Combining Propositions \ref{thm:separatingtree} and \ref{centers vs leaves}, one bounds from below the number of shattered centers.
\begin{corollary} \label{c: lowerbound}
Let $A$ be a finite class of integer-valued functions on
a probability space $(\Omega,\mu)$.
If $A$ is $6$-separated with respect to the $L_2(\mu)$ norm
then it shatters at least $|A|^{1/2}$ centers. \end{corollary}
To show that there exists a large dimensional center shattered by $A$, one must assume that the class $A$ is bounded in some sense, otherwise one could have infinitely many low dimensional centers shattered by the class. A natural assumption is the uniform boundedness of $A$, under which we conclude a preliminary version of Theorem \ref{main}.
\begin{proposition} \label{p: upperbound}
Let $(\Omega,\mu)$ be a probability space,
where $\Omega$ is a finite set of cardinality $n$.
Assume that $A$ is a class of functions on
$\Omega$ into $\{0, 1, \ldots, p \}$,
which is $6$-separated in $L_2(\mu)$.
Set $d$ to be the maximal dimension of a center
shattered by $A$.
Then
\begin{equation} \label{A discrete}
|A| \le \Big( \frac{p n}{d} \Big)^{C d},
\end{equation}
where $C$ is an absolute constant.
In particular, the same assertion holds for $d=\vc(A,2)$. \end{proposition}
\proof By Corollary \ref{c: lowerbound}, $A$ shatters at least
$|A|^{1/2}$ centers. On the other hand, the total number of centers whose dimension is at most $d$ that a class of $\{0, 1,
\ldots, p \}$-valued functions on $\Omega$ can shatter is bounded by $\sum_{k=0}^d \binom{n}{k} p^k$. Indeed, for every $k$ there exist at most $\binom{n}{k}$ subsets $\s \subset \Omega$ of cardinality $k$ and, for each $\s$ with $|\s| = k$ there are at most $p^k$ level functions $h$ for which the center $(\s, h)$ can be shattered by such a class. Therefore $|A|^{1/2} \le \sum_{k=0}^d \binom{n}{k} p^k$ (otherwise there would exist a center of dimension larger than $d$ shattered by $A$, contradicting the maximality of $d$). The proof is completed by approximating the binomial coefficients using Stirling's formula. \endproof
Actually, the ratio $n/d$ can be eliminated from \eqref{A discrete} (perhaps at the cost of increasing the separation parameter $6$). To this end, one needs to reduce the size of $\Omega$ without changing the assumption that the class is ``well separated". This is achieved by the following probabilistic extraction principle.
\begin{lemma} \label{extraction}
There is a positive absolute constant $c$ such that the
following holds.
Let $\Omega$ be a finite set with the uniform probability measure
$\mu$ on it.
Let $A$ be a class of functions bounded by $1$, defined
on $\Omega$.
Assume that for some $0 < t < 1$
$$
\text{$A$ is $t$-separated with respect to the $L_2(\mu)$ norm.}
$$
If $|A| \leq \frac{1}{2} \exp(c t^4 k)$
for some positive number $k$,
there exists a subset $\s \subset \Omega$
of cardinality at most $k$ such that
$$
\text{$A$ is $\frac{t}{2}$-separated with respect to the $L_2(\mu_\s)$
norm,}
$$
where $\mu_\s$ is the uniform probability measure on $\s$. \end{lemma}
As the reader guesses, the set $\s$ will be chosen randomly in $\Omega$. We will estimate probabilities using a version of Bernstein's inequality (see e.g. \cite{VW}, or \cite{LT} 6.3 for stronger inequalities).
\begin{lemma}[Bernstein's inequality] \label{thm:bernstein} Let $X_1, \ldots, X_n$ be independent random variables with zero mean. Then, for every $u>0$, \begin{equation*}
\P \Big\{ \big| \sum_{i=1}^n X_i \big| > u \Big\}
\leq 2 \exp \Big(-\frac{u^2}{2(b^2 + a u/3)} \Big), \end{equation*}
where $a=\sup_i \|X_i\|_\infty$ and $b^2=\sum_{i=1}^n \E |X_i|^2$. \end{lemma}
\noindent {\bf Proof of Lemma \ref{extraction}. } For the sake of simplicity we identify $\Omega$ with $\{1,2, \ldots, n\}$. The difference set $S = \{f - g |\; f \not = g, \ f,g \in A\}$ has cardinality $|S| \leq |A|^2$. For each $x \in S$ we have $|x(i)|
\leq 2$ for all $i \in \{1,...,n\}$ and $\sum_{i=1}^n |x(i)|^2 \geq t^2 n$. Fix an integer $k$ satisfying the assumptions of the lemma and let $\delta_1, \ldots, \d_n$ be independent $\{0,1\}$-valued random variables with $\E \delta_i = \frac{k}{2n} =: \delta$. Then for every $z \in S$ \begin{align*}
\P \Big\{\sum_{i=1}^n \delta_i |x(i)|^2 \leq \frac{t^2\delta n}{2} \Big\}
& \leq \P \Big\{ \Big| \sum_{i=1}^n \delta_i |x(i)|^2
- \delta \sum_{i=1}^n |x(i)|^2 \Big |
> \frac{t^2 \delta n}{2} \Big\} \\
&= \P \Big\{ \Big|\sum_{i=1}^n (\delta_i -\delta)|x(i)|^2 \Big|
> \frac{t^2 \d n}{2} \Big\} \\
&\le 2 \exp \Big(-\frac{c t^4 \d n}{1+t^2} \Big)
\le 2 \exp (-c t^4 k), \end{align*} where the last line follows from Bernstein's inequality for
$a=\sup_i \|X_i\| \leq 2$ and $$
b^2 = \sum_{i=1}^n \E |X_i|^2
= \sum_{i=1}^n |x(i)|^4 \; \E(\delta_i - \delta)^2 \leq 16 \d n. $$ Therefore, by the assumption on $k$ $$ \P \Big\{ \exists x \in S :
\Big( \frac{1}{k} \sum_{i = 1}^n \d_i |x(i)|^2 \Big)^{1/2}
\leq \frac{t}{2} \Big\}
\le |S| \cdot 2 \exp(- c t^4 k) < 1/2. $$
Moreover, if $\s$ is the random set $\{i \,|\; \delta_i =1\}$ then by Chebyshev's inequality, $$
\P \{ |\s| > k \} = \P \Big\{ \sum_{i=1}^n \d_i > k \Big\} \le 1/2, $$ which implies that $$
\P \big \{ \exists x \in S : \|x\|_{L_2(\mu_\s)} \le \frac{t}{2} \big\} < 1. $$ This translates into the fact that with positive probability the class $A$ is $\frac{t}{2}$-separated with respect to the $L_2(\mu_\s)$ norm. \endproof
\qquad
\noindent {\bf Proof of Theorem \ref{main}. } One may clearly assume that $|A| > 1$ and that the functions in $A$ are defined on a finite domain $\Omega$, so that the probability measure $\mu$ on $\Omega$ is supported on a finite number of atoms. Next, by splitting these atoms (by replacing an atom $\w$ by, say, two atoms $\w_1$ and $\w_2$, each carrying measure $\frac{1}{2} \mu(\w)$ and by defining $f(\w_1) = f(\w_2) = f(\w)$ for $f \in A$), one can make the measure $\mu$ almost uniform without changing neither the covering numbers nor the shattering dimension of $A$. Therefore, assume that the domain $\Omega$ is $\{1, 2, \ldots, n\}$ for some integer $n$, and that $\mu$ is the uniform measure on $\Omega$.
Fix $0 < t \leq 1/2$ and let $A$ be a $2 t$-separated in the
$L_2(\mu)$ norm. By Lemma \ref{extraction}, there is a set of coordinates $s \subset \{1,...,n\}$ of size $|\s| \leq
\frac{C\log|A|}{t^4}$ such that $A$ is $t$-separated in $L_2(\mu_\s)$, where $\mu_\s$ is the uniform probability measure on $\s$.
Let $p = \lfloor 7/t \rfloor$, define $\tilde{A} \subset \{0,1,...,p\}^\s$ by $$ \tilde{A}=\Bigl\{ \Bigl( \Bigl \lfloor\frac{7f(i)}{t} \Big \rfloor
\Bigr)_{i \in \s} \, | \; f \in A \Bigr\}, $$ and observe that $\tilde{A}$ is $6$-separated in $L_2(\mu_\s)$. By Proposition \ref{p: upperbound},
$$
|A|=|\tilde{A}| \le \Big( \frac{p |\sigma|}{d} \Big)^{C d}
$$ where $d=\vc(\tilde{A},2)$, implying that
$$
|A| \leq \Big(\frac{C\log |A|}{dt^5}\Big)^{Cd}.
$$ By a straightforward computation,
$$
|A| \leq \Bigl(\frac{1}{t}\Bigr)^{Cd},
$$ and our claim follows from the fact that $\vc(\tilde{A},2) \leq \vc(A, t/7)$. \endproof
\qquad
\remark Theorem \ref{main} also holds for the $L_p(\mu)$ covering numbers for all $0 < p < \infty$, with constants $K$ and $c$ depending only on $p$. The only minor modification of the proof is in Lemma \ref{l:xx'}, where the equations would be replaced by appropriate inequalities.
\qquad
\section{Applications: Gaussian Processes and Convexity} \label{s:convexity}
The first application is a bound on the expectation of the supremum of a Gaussian processes indexed by a set $A$. Such a bound is provided by Dudley's integral in terms of the $L_2$ entropy of $A$; the entropy, in turn, can be majorized through Theorem \ref{main} by the shattering dimension of $A$. The resulting integral inequality improves the main result of M.~Talagrand in \cite{T 92}.
If $A$ be a class of functions on the finite set $I$, then a natural Gaussian process $(X_a)_{a \in A}$ indexed by elements of $A$ is $$ X_a = \sum_{i \in I} g_i \, a(i) $$ where $g_i$ are independent standard Gaussian random variables.
\begin{theorem} \label{thm:talagrand}
Let $A$ be a class of functions bounded by $1$,
defined on a finite set $I$ of cardinality $n$.
Then $E = \E \sup_{a \in A} X_a$
is bounded as
$$
E \le K \sqrt{n}
\int_{cE/n}^{1} \sqrt{\vc(A,t) \cdot \log (2/t)}\; dt,
$$
where $K$ and $c$ are absolute positive constants. \end{theorem}
The nonzero lower limit in the integral will play an important role in the application to Elton's Theorem.
The first step in the proof is to view $A$ as a subset of $\R^n$. Dudley's integral inequality can be stated as $$ E \le K \int_0^\infty \sqrt{\log N(A, t D_n)} \; dt, $$ where $D_n$ is the unit Euclidean ball in $\R^n$, see \cite{Pi} Theorem 5.6. The lower limit in this integral can be improved by a standard argument. This fact was first noticed by A. Pajor.
\begin{lemma} \label{dudley}
Let $A$ be a subset of $\R^n$.
Then $E = \E \sup_{a \in A} X_a$
is bounded as
$$
E \le K \int_{cE/\sqrt{n}}^\infty \sqrt{\log N(A, t D_n)} \; dt,
$$
where $K$ is an absolute constant. \end{lemma}
\proof Fix positive absolute constants $c_1, c_2$ whose values will be specified later. There exists a subset $\NN$ of $A$, which is a $(\frac{c_1 E}{\sqrt{n}})$-net of $A$ with respect to the Euclidean norm and has cardinality $|\NN| \le N(A, \frac{c_1 E}{2 \sqrt{n}} D_n)$. Then $A \subset \NN + \frac{c_1 E}{2 \sqrt{n}} D_n$, and one can write \begin{equation} \label{E}
E = \E \sup_{a \in A} X_a
\le \E \max_{a \in \NN} X_a
+ \E \sup_{a \in \frac{c_1 E}{\sqrt{n}} D_n} X_a. \end{equation} The first summand is estimated by Dudley's integral as \begin{equation} \label{first NN}
\E \max_{a \in \NN} X_a
\le K \int_0^\infty \sqrt{\log N(\NN, t D_n)} \; dt. \end{equation} On the interval $(0, \frac{c_2 E}{\sqrt{n}})$, \begin{align*} K \int_0^{ \frac{c_2 E}{\sqrt{n}} } \sqrt{\log N(\NN, t D_n)} \; dt
&\le K \frac{c_2 E}{\sqrt{n}} \cdot \sqrt{\log|\NN|} \\
&\le K \frac{c_2 E}{\sqrt{n}} \cdot
\sqrt{\log N(A, {\textstyle \frac{c_1 E}{2 \sqrt{n}} } D_n)}. \end{align*} The latter can be estimated using Sudakov's inequality \cite{D,Pi}, which states that $\e \sqrt{\log(N, \e D_n)} \le K \,\E \sup_{a \in A} X_a$ for all $\e > 0$. Indeed, $$ K \frac{c_2 E}{\sqrt{n}} \cdot
\sqrt{\log N(A, {\textstyle \frac{c_1 E}{2 \sqrt{n}} } D_n)}
\leq K_1 (2c_2 / c_1) \,\E \sup_{a \in A} X_a
= K_1 (2c_2 / c_1) E
\le \frac{1}{4} E, $$ if we select $c_2$ as $c_2 = c_1 / 8 K_1$. Combining this with \eqref{first NN} implies that \begin{equation} \label{first sum}
\E \max_{x \in \NN} X_a
\le \frac{1}{4} E + K \int_{ \frac{c_2 E}{\sqrt{n}} }^\infty
\sqrt{\log N(A, t D_n)} \; dt \end{equation} because $\NN$ is a subset of $A$.
To bound the second summand in \eqref{E}, we apply the Cauchy-Schwarz inequality to obtain that for any $t>0$, $$ \E \sup_{a \in tD_n} X_a \le t \cdot \E \Big( \sum_{i \in I} g_i^2 \Big)^{1/2} \le t \sqrt{n}. $$
In particular, if $c_1 < 1/4$ then $$ \E \sup_{a \in \frac{c_1E}{\sqrt{n}}D_n} X_a \leq c_1E \leq \frac{1}{4}E. $$ This, \eqref{E} and \eqref{first sum} imply that $$ E \le K_2 \int_{ \frac{c_2 E}{\sqrt{n}} }^\infty
\sqrt{\log N(A, t D_n)} \; dt, $$ where $K_2$ is an absolute constant. \endproof
\qquad
\noindent{\bf Proof of Theorem \ref{thm:talagrand}. } By Lemma \ref{dudley}, $$ E \le K \int_{cE/\sqrt{n}}^\infty \sqrt{\log N(A, t D_n)} \; dt. $$ Since $A \subset [-1,1]^n \subset \sqrt{n} D_n$, the integrand vanishes for $t \ge \sqrt{n}$. Hence, by Theorem \ref{main} \begin{align*} E &\le K \int_{cE/\sqrt{n}}^{\sqrt{n}}
\sqrt{\log N(A, t D_n)} \; dt \\
&= K \sqrt{n} \int_{cE/n}^1
\sqrt{\log N(A, t \sqrt{n} D_n)} \; dt \\
&\le K_1\sqrt{n} \int_{cE/n}^1
\sqrt{\vc(A, c_1 t) \cdot \log (2/t)} \; dt. \end{align*} The absolute constant $0 < c_1 < 1/2$ can be made $1$ by a further change of variable. \endproof
\qquad
The main consequence of Theorem \ref{thm:talagrand} is Elton's Theorem with the optimal dependence on $\d$.
\begin{theorem} \label{thm:elton}
There is an absolute constant $c$ for which the following
holds.
Let $x_1, \ldots, x_n$ be vectors in the unit
ball of a Banach space.
Assume that
$$
\E \Big\| \sum_{i = 1}^n g_i x_i \Big\| \ge \d n
\ \ \ \text{for some number $\d > 0$}.
$$
Then there exist numbers $s, t \in (c \d, 1)$,
and a subset $\s \subset \{ 1, \ldots, n \}$
of cardinality $|\s| \ge s^2 n$,
such that
\begin{equation} \label{lower l1}
\Big\| \sum_{i \in \s} a_i x_i \Big\|
\ge t \sum_{i \in \s} |a_i|
\ \ \ \text{for all real numbers $(a_i)$}.
\end{equation}
In addition, the numbers $s$ and $t$ satisfy the inequality
$s \cdot t \log^{1.6} (2 / t) \ge c\d$. \end{theorem}
Before the proof, recall the interpretation of the shattering dimension of convex bodies. If a set $B \subset \R^n$ is convex and symmetric then $\vc(B,t)$ is the maximal cardinality of a subset $\s$ of $\{1, \ldots, n\}$ such that $P_\s(B) \supset [-\frac{t}{2}, \frac{t}{2}]^\s$. Indeed, every convex symmetric set in $\R^n$ can be viewed as a class of functions on $\{1,...,n\}$. If $\s$ is $t$-shattered with a level function $h$ then for every $\s' \subset \s$ there is some $f_{\s'}$ such that $f_{\s'}(i) \geq h(i)+t$ if $i \in \s'$ and $f_{\s'} \leq h$ on $\s \backslash \s'$. By selecting for every such $\s'$ the function $(f_{\s'}-f_{\s \backslash \s'})/2$ and since the class is convex and symmetric, it follows that $P_\s(B) \supset [-\frac{t}{2},\frac{t}{2}]^\s$, as claimed.
Taking the polars, this inclusion can be written as $\frac{t}{2}
(B^\circ \cap \R^\s) \subset B_1^n$, where $B_1^n$ is the unit ball of $\ell_1^n$. Denoting by $\|\cdot\|_{B^\circ}$ the Minkowski functional (the norm) induced by the body $B^\circ$, one can rewrite this inclusion as the inequality $$
\Big\| \sum_{i \in \s} a_i e_i \Big\|_{B^\circ}
\ge \frac{t}{2} \sum_{i \in \s} |a_i|
\ \ \ \text{for all real numbers $(a_i)$}, $$ where $(e_i)$ is the standard basis of $\R^n$. Therefore, to prove Theorem \ref{thm:elton}, one needs to bound below the shattering dimension of the dual ball of a given Banach space.
\qquad
\noindent {\bf Proof of Theorem \ref{thm:elton}.} By a perturbation argument, one may assume that the vectors $(x_i)_{i
\le n}$ are linearly independent. Hence, using an appropriate linear transformation one can assume that $X = (\R^n, \|\cdot\|)$ and that $(x_i)_{i \le n}$ are the unit coordinate vectors
$(e_i)_{i \le n}$ in $\R^n$. Let $B=(B_X)^\circ$ and note that the assumption $\|e_i\|_X \leq 1$ implies that $B \subset [-1,1]^n$.
Set $$
E = \E \Big\|\sum_{i=1}^n g_i x_i \Big\|_X
= \E \sup_{b \in B} \sum_{i=1}^n g_i \,b(i). $$ By Theorem \ref{thm:talagrand}, \begin{equation*} \d n \leq E \leq K\sqrt{n} \int_{c \delta}^{1}
\sqrt{\vc(B,t) \cdot \log(2/t)} \; dt. \end{equation*} Consider the function $$ h(t) = \frac{c_0}{t \log^{1.1} (2 / t)} $$ where the absolute constant $c_0 > 0$ is chosen so that $\int_0^{1} h(t) \; dt = 1$. It follows that there exits some $c\d \le t \le 1$ such that $$ \sqrt{\vc(B, t) / n \cdot \log(2 / t)} \ge \d h(t). $$ Hence $$ \vc(B, t) \ge \frac{c_0 \d^2}{t^2 \log^{3.2} (2 / t)} n. $$ Therefore, letting $s^2 = \vc(B, t) / n$, it follows that $s \cdot t \log^{1.6} (2 / t) \ge \sqrt{c_0} \d$ as required, and by the discussion preceding the proof there exists a subset $\s$ of $\{1,
\ldots, n\}$ of cardinality $|\s| \ge s^2 n$ such that \eqref{lower l1} holds with $t/2$ instead of $t$. The only thing remaining is to check that $s \gtrsim \d$. Indeed, $s \ge \frac{\sqrt{c_0} \d}{t \log^{1.6} (2 / t)} \ge c_1 \d$, because $t \le 1$. \endproof
\noindent {\bf Remarks. } 1. As the proof shows, the exponent $1.6$ can be reduced to any number larger than $3/2$.
2. The relation between $s$ and $t$ in Theorem \ref{thm:elton} is optimal up to a logarithmic factor for all $0 < \d < 1$. This is seen from by the following example, shown to us by Mark Rudelson. For $0 < \d < 1 / \sqrt{n}$, the constant vectors $x_i = \d \sqrt{n} \cdot e_1$ in $X = \R$ show that $s t$ in Theorem \ref{thm:elton} can not exceed $\d$. For $1 / \sqrt{n} \le \d \le 1$, we consider the body $D = \conv( B_1^n \cup \frac{1}{\d
\sqrt{n}} D_n )$ and let $X = (\R^n, \|\cdot\|_D)$ and $x_i =
e_i$, $i = 1, \ldots, n$. Clearly, $\E \| \sum g_i x_i \|_X \ge
\E \| \sum \e_i e_i \|_D = \d n$. Let $0 < s, t < 1$ be so that \eqref{lower l1} holds for some subset $\s \subset \{ 1, \ldots, n
\}$ of cardinality $|\s| \ge s^2 n$. This means that $\|x\|_D
\ge t \|x\|_1$ for all $x \in \R^\s$. Dualizing, $\frac{t}{\d
\sqrt{n}} \|x\|_2 \le t \|x\|_{D^\circ} \le \|x\|_\infty$ for all $x \in \R^\s$. Testing this inequality for $x = \sum_{i \in
\s} e_i$, it is evident that $\frac{t}{\d \sqrt{n}} \sqrt{|\s|} \le 1$ and thus $s t \le \d$.
\qquad
We end this article with an application to empirical processes. A key question is when a class of functions satisfies the central limit theorem uniformly in some sense. Such classes of functions are called {\it uniform Donsker classes}. We will not define these classes formally but rather refer the reader to \cite{D,VW} for an introduction on the subject. It turns out that the uniform Donsker property is related to uniform estimates on covering numbers via the Koltchinskii-Pollard entropy integral.
\begin{theorem} \cite{D} \label{dud} Let $F$ be a class of functions bounded by $1$. If \begin{equation*}
\int_0^\infty \sup_{n}\sup_{\mu_n} \sqrt{ \log N \bigl(F,L_2(\mu_n),\e\bigr) } \;d\eps < \infty, \end{equation*} then $F$ is a uniform Donsker class. \end{theorem} Having this condition in mind, it is natural to try to seek entropy estimates which are ``dimension free", that is, do not depend on the size of the sample. In the $\{0,1\}$-valued case, such bounds where first obtained by Dudley who proved Theorem \ref{main} for these classes (see \cite{LT} Theorem 14.13) which implied through Theorem \ref{dud} that every VC class is a uniform Donsker class.
Theorem \ref{main} solves the general case: the following corollary extends Dudley's result on the uniform Donsker property from $\{0,1\}$ classes to classes of real valued functions.
\begin{corollary}
Let $F$ be a class of functions bounded by $1$
and assume that the integral
$$
\int_0^1 \sqrt{ \vc(F,t)\log \frac{2}{t}} \; dt
$$
converges.
Then $F$ is a uniform Donsker class. \end{corollary}
In particular this shows that if $\vc(F,t)$ is ``slightly better" than $1/t^2$, then $F$ is a uniform Donsker class.
This result has an advantage over Theorem \ref{dud} because in many cases it is easier to compute the shattering dimension of the class rather than its entropy (see, e.g. \cite{AB}).
{\small
}
\end{document}
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\begin{document}
\begin{abstract} The number of standard Young tableaux possible of shape corresponding to a partition $\lambda$ is called the dimension of the partition and is denoted by $f^{\lambda}$. Partitions with odd dimensions were enumerated by McKay and were further classified by Macdonald. Let $a_i(n)$ be the number of partitions of $n$ with dimension congruent to $i$ modulo 4. In this paper, we refine Macdonald's and McKay's results by calculating $a_1(n)$ and $a_3(n)$ when $n$ has no consecutive 1s in its binary expansion or when the sum of binary digits of $n$ is 2 and providing values for $a_2(n)$ for all $n$. We also present similar results for irreducible representations of alternating groups. \end{abstract} \title{Enumeration of partitions modulo 4}
\section{Introduction} \subsection{Foreword} A partition of $n$ is a tuple of natural numbers, $\lambda:= (\lambda_1, \ldots, \lambda_k)$, with non-increasing entries that add up to $n$. The \textit{dimension of the partition $\lambda$}, $f^\lambda$, is the number of standard Young tableaux of shape $\lambda$ (ref. \cref{introsec}) and can be calculated by the famous hook-length formula (\cref{HLF}). Denote by $m_p(n)$, the number of partitions whose dimensions are not divisible by a given natural number, $p$.\\ Macdonald, in his landmark paper \cite{mcd}, gave a complete answer for $m_p(n)$ which can be understood quite elegantly using the $p$-core tower approach (ref. \cref{sec:2mod4}). The case for $p=2$ has a wonderful form: for $n = 2^{k_1}+\ldots + 2^{k_\ell}$ with $k_1 > \ldots > k_\ell$, we have, \[m_2(n) = 2^{k_1 + \ldots + k_\ell},\] which is entirely dependent on the binary expansion of $n$.\\ Note that $m_2(n)$ counts all partitions of $n$ which have odd dimension. We call these \textit{odd partitions} and will denote their count by $a(n) := m_2(n)$.
Recently, there has been some interest in extending Macdonald's results. In their unpublished notes, Amrutha P and T. Geetha \cite{geetha} tackle the problem of computing $m_{2^k}(n)$. They provide general recursive results and in the case of ``hook-partitions" of $2^\ell$, they find $m_4(2^\ell)$ and $m_8(2^\ell)$. As we shall see, the enumeration results for $m_4(n)$ in the preprint follow from our analysis of ``sparse numbers", i.e., numbers with no consecutive 1s in their binary expansions.\\ This study of partitions has a representation theoretic context as partitions of $n$ index the irreducible representations of the symmetric group, $S_n$. There are results regarding the behaviour of irreducible representations corresponding to odd partitions under restriction (\cite{gian}, \cite{aps}). There are results (\cite{evensarah}, \cite{evensteven}) which show that the density of odd character values (not just degrees) goes to zero. Although, they add to the literature of odd partitions, none specifically focus on analysing the dimensions further. Some papers (\cite{rosa}, \cite{lass}) give explicit results for character values of symmetric groups but the formulae do not aid us in enumeration.\\ We aim to fill this gap by providing explicit enumeration results which extend the odd dimensional partition enumeration formula of Macdonald.
\subsection{Main Results} Let $a_i(n)$ denote the number of partitions of $n$ with dimension congruent to $i$ modulo 4. Then, the number of odd partitions, $a(n)= m_2(n) = a_1(n) + a_3(n)$. We compute the values of $a_1(n)$ and $a_3(n)$ for some specific values of $n$. The case where $n$ has more than two 1s in its binary expansion and the leftmost two digits are 1 still remains unsolved.\\ Define $\delta(n) := a_1(n) - a_3(n)$. In this paper, we present the following main theorem: \begin{thm}\label{mainthm} Let $n, m, R \in \mathbb{N}$ with $R\geq 2$ and $m > 0$. Suppose, $n = 2^{R} + m$ with $2^{R-1} > m$. Then, we have \[ \delta(n) = \begin{cases} 0, & \text{if }n \text{ is even}\\ 4 \delta(m), &\text{if } n \text{ is odd}. \end{cases} \] Equivalently, for $k_1 > \ldots > k_\ell$, we have \[ a_1(2^{k_1}+\ldots + 2^{k_\ell}) = \begin{cases} 2^{k_1+\ldots + k_\ell - 1}, & \text{if }k_\ell >0 \text{}\\ 4a_1(\sum\limits_{i=2}^{l} 2^{k_i}) + (2^{k_1-1}-2)2^{k_2+\ldots + k_\ell}, &\text{if } k_\ell = 0 \text{}, \end{cases} \] and \[ a_3(2^{k_1}+\ldots + 2^{k_\ell}) = \begin{cases} 2^{k_1+\ldots + k_\ell - 1}, & \text{if }k_\ell >0 \text{}\\ 4a_3(\sum\limits_{i=2}^{l} 2^{k_i}) + (2^{k_1-1}-2)2^{k_2+\ldots + k_\ell}, &\text{if } k_\ell = 0 \text{} \end{cases} \] \end{thm}
We call a positive integer \textit{sparse} if it does not have any consecutive ones in its binary expansion, i.e., $k_i > k_{i+1} + 1$ in our notation. For instance, $(42)_{10}= (101010)_2$ is sparse. This case has the following pleasing corollary: \begin{cor}\label{maincor} If $n$ is a sparse number, then \[ \delta(n) = \begin{cases} 2\text{,} & \text{if }n = 2\\ 0, &\text{if } n > 2 \text{ is even}\\ 4^{\nu(n) - 1}, & \text{if } n\text{ is odd}, \end{cases} \] where $\nu(n)$ denotes the number of 1s (or equivalently the sum of digits) in the binary expansion of $n$. For $k_1 > \ldots > k_\ell\geq 0$ with $k_i > k_{i+1}+1$ such that $\ell \geq 2$, we have \[ a_1(2^{k_1}+\ldots + 2^{k_\ell}) = \begin{cases} 2^{k_1+\ldots + k_\ell - 1}, &\text{if } k_\ell > 0\\ 2^{k_1+\ldots + k_\ell - 1} + 4^{\ell - 2}, & \text{if }k_\ell = 0\text{}, \end{cases} \] and \[ a_3(2^{k_1}+\ldots + 2^{k_\ell}) = \begin{cases} 2^{k_1+\ldots + k_\ell - 1}, &\text{if } k_\ell > 0\\ 2^{k_1+\ldots + k_\ell - 1} - 4^{\ell - 2}, & \text{if }k_\ell = 0\text{}, \end{cases} \] \end{cor} Note that we recover explicit and recursion formulae for $a_1(n)$ and $a_3(n)$ by using $\delta(n) = a_1(n) - a_3(n)$ and $a(n) = a_1(n) + a_3(n)$.
In the case where the binary expansion starts with $\mathtt{11}$, we have the following partial result: \begin{thm}\label{11thm} Let $n = 2^R + 2^{R-1}$ with $R\geq 1$, then \[ \delta(n) = \begin{cases} 2, & \text{if }n = 3\\ 8, & \text{if } n = 6\\ 0, & else. \end{cases} \] Explicitly, we have\[ a_1(2^R + 2^{R-1}) = \begin{cases} 2, & \text{if }R = 1\\ 8, & \text{if } R = 2\\ 4^{R-1}, & else, \end{cases} \] and \[ a_3(2^R + 2^{R-1}) = \begin{cases} 0, & \text{if }R = 1\\ 0, & \text{if } R = 2\\ 4^{R-1}, & else. \end{cases} \] \end{thm}
For $a_2(n)$, which counts even dimensional partitions with $f^{\lambda}$ not divisible by 4, we have recursive results for all natural numbers and closed form results for sparse $n$: \begin{thm}\label{2mod4thm} Let $n = 2^R + m$ such that $m < 2^R$. Then we have \[a_2(n) = \begin{cases} 2^R\cdot a_2(m) + \binom{2^{R-1}}{2}\cdot a(m), & \text{ if }m < 2^{R-1}\\ 2^R\cdot a_2(m) + \left(\binom{2^{R-1}}{3} + 2^{R-1}\right) \cdot \displaystyle{\frac{a(m)}{2^{R-1}}}, & \text{ if } 2^{R-1} \leq m < 2^R. \end{cases}\] \end{thm} \begin{cor}\label{2mod4cor} When $n$ is sparse, we have \[a_2(n) = \begin{cases} \displaystyle{\frac{a(n)}{8}(n - 2 \nu(n))},& \text{if }n \text{ is even}\\ a_2(n-1), &\text{if } n \text{ is odd}. \end{cases}\] \end{cor}
\subsection{Structure of the paper} In \cref{sec:notation}, we recall the basic notions relating to partitions such as tableaux, cores and quotients, and the hook-length formula. We present a characterization of odd partitions due to Macdonald \cite{mcd} and introduce the notion of parents. We also define a function, $\Od$, which extracts the odd part of the input and returns it modulo 4. In \cref{sec:workhorse}, we prove ``the workhorse formula" that relates the $\Od$ values of the cores and parents which allows us to enumerate $\delta(n)$ recursively. In \cref{sec:proofmain} and \cref{sec:proof11}, we analyse the workhorse formula to obtain enumeration results and prove \cref{mainthm} and \cref{11thm} respectively. In \cref{sec:2mod4}, we use the theory of 2-core towers to prove \cref{2mod4thm}. In \cref{sec:problems}, we present some relevant problems that remain unsolved.
\section{Notations and Definitions}\label{sec:notation} We denote by $\mathbb{N}$ the set of natural numbers beginning at 1. \subsection{Partitions, Ferrers diagrams and Young Tableaux}\label{introsec} Let \\${\Lambda \subset \bigcup\limits_{i=1}^\infty \mathbb{N}^i}$ denote the set of all tuples $\lambda = (\lambda_1, \ldots, \lambda_k)$ such that \\$\lambda_1 \geq \ldots \geq \lambda_k$. \begin{defn}[Partition] An element of $\Lambda$ is known as a \textit{partition}. We call $\lambda \in \Lambda$ a \textit{partition of $n$} if $\sum\limits_{i=1}^k \lambda_i = n$. \end{defn} \begin{notation}
We use the notation $\lambda\vdash n$ to denote $\lambda$ is a partition of $n$. Denote the \textit{size of $\lambda$} by $|\lambda| := \sum\limits_{i=1}^k \lambda_i = n$. \end{notation} We can represent a partition in the Cartesian plane by constructing a top-left justified array of boxes with the \nth{i} row containing $\lambda_i$ many boxes. \begin{example}The partition $\lambda = (4, 3, 3, 1)$ can be represented as \begin{center} \ydiagram{4, 3, 3, 1}. \end{center}\end{example} This is called the \textit{Ferrers diagram} of $\lambda$ and will be denoted by $\sh(\lambda)$, short for shape on $\lambda$.
The boxes of the Ferrers diagram $\sh(\lambda)$ can be filled with distinct numbers from $1$ to $n:= |\lambda|$. We impose the constraint that the numbers increase from top to bottom and from left to right. Such a filling is called a \textit{standard Young tableau (SYT)} on the shape $\sh(\lambda)$. \begin{example}Continuing with the above example, the following filling \begin{center} \ytableaushort{1 2 7 8, 3 5 {10}, 4 9 {11}, 6} \end{center} is an SYT on $\sh((4, 3, 3, 1))$. \end{example}
We can have multiple such SYT of a given shape, and the total number of SYT on $\sh(\lambda)$ is denoted by $f^{\lambda}$. We call this number, the \textit{dimension of the partition $\lambda$}.
\subsection{Hooks and hook-lengths} We call the boxes in Ferrers diagrams \textit{cells}. We label the cell in the \nth{i} row from top and \nth{j} column from left by $(i,j)$. Write $(i,j) \in \sh(\lambda)$ if the $(i,j)$ cell exists.\\ A very useful tool in the study of partitions is the notion of a \textit{hook}. For a cell at $(i,j)$ consider all the cells to the right, to the bottom of it and the cell itself. This constitutes the \textit{hook at $(i,j)$}. The number of cells is known as its \textit{hook-length} and is denoted by $h_{i,j}$. \begin{example} In the previous example, the cell at $(1,2)$ has the hook-length, $h_{1,2} = 5$. The hook is denoted in pink. \begin{center} \ydiagram{4, 3, 3, 1}*[*(pink)]{1 + 3, 1 +1, 1+ 1} \end{center} \end{example}
\subsection{Hook removal and cores} A hook of length $t$ is called a \textit{$t$-hook}. If the Ferrers diagram has a $t$-hook, then remove all the cells contained in the hook. This gives us two (possibly empty) Ferrers diagrams. Slide the bottom diagram to the left and then upwards to reconnect and form a Ferrers diagram.
\begin{example} Consider the partition $(5,5, 5, 4, 2)$ and its Ferrers diagram \begin{center} \ydiagram{5, 5, 5, 4, 2}. \end{center} In our case, let $t = 5$, that is, we wish to remove a 5-hook from the diagram. We undergo the following procedure: \begin{tiny} \begin{center} \ytableausetup{boxsize = 0.3cm, aligntableaux = center} \ydiagram[*(cyan)]{5, 2, 2, 2, 2}*[*(pink)]{0, 0, 3 + 2, 3 + 1}*[*(lightgray)]{0, 2 + 3, 2 + 1, 2 + 1}$\rightarrow$ \ydiagram[*(cyan)]{5, 2, 2, 2, 2}*[*(pink)]{0, 0, 3 + 2, 3 + 1}$\rightarrow$ \ytableaushort{\none, \none, \none \none {\none[\leftarrow]} , \none \none {\none[\leftarrow]}, \none}*[*(cyan)]{5, 2, 2, 2, 2}*[*(pink)]{0, 0, 3 + 2, 3 + 1} \quad $\rightarrow$ \ytableaushort{\none, \none, \none \none \none , \none \none \none, \none}*[*(cyan)]{5, 2, 2, 2, 2}*[*(pink)]{0, 0, 2+ 2, 2 + 1}$\rightarrow$ \ytableaushort{\none, \none\none {\none[\uparrow]}{\none[\uparrow]}, \none \none \none , \none \none \none, \none}*[*(cyan)]{5, 2, 2, 2, 2}*[*(pink)]{0, 0, 2+ 2, 2 + 1}$\rightarrow$ \ytableaushort{\none, \none, \none \none \none , \none \none \none, \none}*[*(cyan)]{5, 2, 2, 2, 2}*[*(pink)]{0, 2+ 2, 2 + 1} \ytableausetup{boxsize = normal, aligntableaux = top} \end{center} \end{tiny}
to obtain the partition $(5, 4, 3, 2, 2)$ \begin{center} \ydiagram{5, 4, 3, 2, 2}. \end{center} \end{example} \begin{defn}[$t$-core of a partition] If $\sh(\mu)$ does not have a $t$-hook, then we call $\mu$ a \textit{$t$-core}. \end{defn} For a partition $\lambda$, if we keep applying this $t$-hook removal process on the subsequent partitions we obtain, we will eventually reach a $t$-core. The $t$-core of $\lambda$ is independent of the order of hook-removal and is thus unique (cf. Example 8(c) \cite{SymmF} p. 12). We denote it by $\core_t(\lambda)$.
\begin{notation} The set of all $t$-cores is denoted by $\widetilde{\Lambda}_t$. \end{notation}
\subsection{\texorpdfstring{$\beta$}-sets} The set of first column hook-lengths of $\sh(\lambda)$ for $\lambda = (\lambda_1, \ldots, \lambda_k)$ will be denoted by \[H(\lambda) = \{h_{i, 1}\mid 1\leq i \leq k\},\] which we conventionally order in a decreasing fashion. Also, we have the relation, $h_{i,1} = \lambda_i + k - i$.
For any finite $X\subset \N$, define the \textit{$r$-shift of $X$} to be \[X^{+r} = \{x + r\mid x\in X\} \cup \{0, \ldots, r-1\}.\] We fix $X^{+0} = X$. \begin{defn}[$\beta$-set] For a partition $\lambda$, all sets of the form $H(\lambda)^{+r}$ for $r\geq 0$ are known as the \textit{$\beta$-sets of $\lambda$}. \end{defn} We can impose the relation, $\sim_{\beta}$, on finite sets $X, Y\subset \mathbb{N}$ such that $X\sim_{\beta} Y$ if and only if $X = Y^{+r}$ or $Y = X^{+r}$, for some $r \in \N$. This is an equivalence relation on the set of $\beta$-sets, i.e., $\N$.
There is a natural way to understand $t$-hooks and $t$-cores through $\beta$-sets. \begin{prop}\label{hookremprop} Let $\lambda$ be a partition and $X$ be a $\beta$-set of $\lambda$. We have that $\sh(\lambda)$ contains a $t$-hook if and only if there exists an $h$ in $X$ such that $h >t$ and $h - t$ is not in $X$. Furthermore, if $\mu$ is the partition obtained after removing the $t$-hook, then $H(\mu) \sim_{\beta} (H(\lambda)\cup \{h-t\})\backslash \{h\}$. \end{prop} \begin{proof} The proof can be found in Example 8(a) of \cite{SymmF} and Corollary 1.5 on page 7 of \cite{olsson}. \end{proof} \begin{remark}\label{remark:aff} The element $h$, as above, will be referred to as the \textit{affected hook-length}. With the above notation, we write $h^\lambda_\mu:= h$. \end{remark} The recipe for hook removal is quite clear. Pick an element $h$ of the $\beta$-set, $X$, and subtract $t$ from it. If $h-t\not\in X$, then replace $h$ by $h-t$; otherwise choose another element from $X$. If no ``$x \rightarrow x-t$" replacements can be performed, declare the partition as a $t$-core. \begin{example} Let $H(\lambda) = \{10, 8, 7, 5, 2\}$. In this case, we have that $8 \in H(\lambda)$ but $3 \not\in H(\lambda)$. Thus, there exists a 5-hook in $\sh(\lambda)$ which we can remove to obtain $\lambda'$. Following the above rules, we get $H(\lambda') = \{10, 7, 5, 3, 2\}$. Now, we can further replace $5$ by 0 to obtain $H(\lambda'') = \{10, 7, 5, 3, 2, 0\} \sim_{\beta} \{9,6,4,2,1\}$. No further replacements can be performed. Thus, $\lambda'' = (5,3,2,1,1)$ is the 5-core of $\lambda$. \end{example}
We can reconstruct the original partition from a $\beta$-set by setting $\lambda_i = h_i + i - k$ and considering only the values where $\lambda_i >0$. \begin{notation} If $X$ is a $\beta$-set of $\lambda$, then we define $\Part(X)= \lambda$. \end{notation}
\subsection{Odd dimensional partitions}\label{hlfsec} Using the notation, $f^{\lambda}$, for the dimension of a partition $\lambda$ (refer \cref{introsec}), we have the following proposition due to Frobenius: \begin{prop}[Hook-length formula]\label{HLF} Let $\lambda\in \Lambda$ and $X$ be a $\beta$-set of $\lambda$. If explicitly, $X = \{h_1, \ldots, h_k\}$ such that $h_1 > \ldots > h_k$, then we have \[
f^{\lambda} = \dfrac{|\lambda|!\prod\limits_{1\leq i<j\leq k} (h_i - h_j)}{\prod\limits_{i=1}^k h_i!}. \] \end{prop}
\begin{remark}\label{frthlf} The more famous hook-length formula \cite{frthook} \[ f^\lambda = \dfrac{n!}{\prod\limits_{h} h} \] has a product of all hook-lengths of $\lambda$ in the denominator. This can be used to derive the formula in \cref{HLF}. \end{remark}
We call $\lambda$ an \textit{odd partition} if $f^{\lambda}$ is odd. Macdonald~\cite{mcd} gave the following characterization of odd partitions which forms the backbone of our analysis:
\begin{prop}\label{mcdprop} If $\lambda$ is a partition of $n = 2^R + m$ with $m < 2^R$, then $\lambda$ is an odd partition if and only if $\lambda$ contains exactly one $2^R$-hook and $\core_{2^R}(\lambda)$ is also an odd partition. \end{prop}
\begin{defn}\label{parentsrem} If $\core_{2^R}(\lambda) = \mu$, then we call $\lambda$ a \textit{$2^R$-parent of $\mu$}. \end{defn}
We can now give an explicit form for the $2^R$-parents of odd partitions: \begin{prop}\label{parents} Let $2^R>m$ and $\mu$ be a partition of $m$. If $\lambda$ is a partition such that $\core_{2^R}(\lambda) = \mu$, then exactly one of the following holds: \begin{enumerate} \item[Type I.] $H(\lambda) = \left(H(\mu)\cup \{x+2^R\}\right)\backslash \{x\}$ for some $x\in H(\mu)$, \item[Type II.] $H(\lambda) = \left(H(\mu)^{+r}\cup \{2^R\}\right)\backslash\{0\}$ for some $1\leq r\leq 2^R$ such that $2^R\not\in H(\mu)^{+r}$. \end{enumerate} \end{prop}
\begin{proof} By \cref{hookremprop}, we get $(H(\lambda)\cup\{h - 2^R\})\backslash \{h\}\sim_{\beta}H(\mu)$ which implies $(H(\lambda)\cup\{h - 2^R\})\backslash \{h\} = H(\mu)^{+r}$.\\ The reader can show that for non-empty finite sets $A$, $B$, $C \subset \N$, we have \begin{enumerate} \item $A\backslash B = C \implies A = B \cup C$ if and only if $B \subset A$ \item $ A = B \cup C \implies A\backslash B = C$ if and only if $B\cap C = \varnothing$. \end{enumerate} Using this, we get \textit{Type I} by letting $r = 0$, putting $h - 2^R = x$. Similarly, we get \textit{Type II} by putting $h = 2^R$. \end{proof} \begin{example} Let $\mu = (2,2,2)$ with $2^R = 8$. We have $H(\mu) = \{4,3,2\}$. If we let $x = 3$, then we obtain a Type I parent, $\lambda$, with $H(\lambda) = \{11, 4, 2\}$. For a Type II parent, we choose $r= 2$, which gives us $\lambda'$ with $H(\lambda') = \{8,6,5,4,1\}$. The corresponding Ferrers diagrams for $\lambda$ (left) and $\lambda'$ (right) with hooks added to $\mu$ are: \begin{center} \ydiagram{2,2,2}*[*(pink)]{2 + 7, 2 + 1} \qquad \ydiagram{2,2,2}*[*(pink)]{2 + 2, 2 + 1, 2 + 1, 3, 1}. \end{center} \end{example} Using this, we can recover the enumeration result for odd partitions: \begin{prop}\label{count} Let $n = 2^{k_1} + \ldots + 2^{k_r}$ with $k_1 > \ldots > k_r$. Then, the number of odd partitions of $n$ is given by $a(n) := 2^{k_1 + \ldots + k_r}$. \end{prop}
\begin{proof} Let $n = 2^{k_1} + m$ and $\mu\vdash m$ be an odd partition. There are exactly $2^{k_1}$ many $2^{k_1}$-parents of $\mu$ and all of them are odd partitions. This gives us the recursion, $a(n) = 2^{k_1}a(m)$. \end{proof} \subsection{\textnormal{Od} function}\label{odsubs}
\begin{notation} For $n\in \mathbb{N}$, let $v_2(n)$ denote the largest power of 2 that divides $n$. \end{notation} \begin{defn} Let $\Od:\mathbb{N} \rightarrow \{\pm1\}$ be defined as follows \[ \Od(n) = \begin{cases} 1, & \text{if }n/2^{v_2(n)}\equiv 1 \mod 4\\ -1, &\text{if } n/2^{v_2(n)}\equiv 3 \mod 4. \end{cases} \] \end{defn} \begin{lem} For all $m, n\geq 1$, we have $\Od(mn) = \Od(m)\Od(n)$. \end{lem} \begin{proof} The proof is elementary and follows from $o(n):= n/2^{v_2(n)}$ being multiplicative. \end{proof}
\begin{notation}[Binary expansion]\label{not:binary} Let $n\in \N$. \begin{enumerate} \item Let $n = \sum\limits_{i=0}^k b_i 2^i$ such that $b_i = \{0,1\}$. We write $n = b_k\ldots b_0$ if $b_k = 1$ and $b_i =0$ for $i>k$. This is the \textit{binary expansion of $n$.} \item Let $\bin(n) :=\{i \mid b_i = 1\}$, i.e., the ``positions" of 1s in the binary expansion of $n$. \item Denote the sum of digits by $\nu(n):= \sum\limits_{i\geq 0} b_i$. \item Let $s_2(n) := b_k + b_{k-1}$, i.e., the sum of the leftmost two digits of $n$. \end{enumerate}
\end{notation}
\begin{remark} Let $n$ be as above. Denote $j := \min(\bin(n))$. Then, $\Od(n) \equiv b_{j+1}b_{j}\mod 4$. This corresponds to taking the binary expansion of $n$, removing all the rightmost zeros and returning the rightmost two digits of the newly obtained string modulo 4. \end{remark}
\section{The Workhorse Formula}\label{sec:workhorse} In this section, we construct a relationship between partitions and their $2^R$-parents, which will allow us to analyse their dimensional behaviour modulo 4. We first define a statistic of consequence inspired by \cite{Bin4}: \begin{defn} For a natural number $n$ (with binary expansion $b_k\ldots b_0$), let the number of pairs of consecutive 1s (disjoint or overlapping) be denoted by $D(n)$. Notationally, $D(n)$ is the number of $i$ such that the product $b_i\cdot b_{i+1} = 1$. \end{defn} \begin{example} We have $D(7) = 2$, $D(42) =0$ and $D(367) = 4$. \end{example} We start with the following lemma: \begin{lem}\label{lem:odnfac} For any natural number $n$, we have \[\Od(n!) = (-1)^{D(n) + \nu(\lfloor n/4 \rfloor)},\] where $\nu(n)$ is the number of 1s in the binary expansion of $n$ as in \cref{not:binary} and $\lfloor \cdot \rfloor$ denotes the integral part. \end{lem} \begin{proof} As $\Od$ is multiplicative, we have \[ \Od(n!) = \prod\limits_{r=1}^n \Od(r) = \prod\limits_{\substack{1 \leq r \leq n\\ r \text{ odd}}} \Od(r) \prod\limits_{\substack{1 \leq r \leq n\\ r \text{ even}}} \Od(r). \] The above rearrangement allows us to use the following two facts to obtain a recursion. Firstly, $\Od(2r) = \Od(r)$ which gives \[\prod\limits_{\substack{1 \leq r \leq n\\ r \text{ even}}} \Od(r) = \Od(\lfloor n/2 \rfloor!).\] We have \begin{align*} \prod\limits_{\substack{1 \leq r \leq n\\ r \text{ odd}}} \Od(r) &= \prod\limits_{\substack{1 \leq r \leq n\\ r \in 4\mathbb{N}+1}} \Od(r) \prod\limits_{\substack{1 \leq r \leq n\\ r \in 4\mathbb{N}-1}} \Od(r)\\ &= \prod\limits_{\substack{1 \leq r \leq n\\ r \in 4\mathbb{N}-1}} (-1). \end{align*}
As there are exactly $\lfloor (n+1)/4\rfloor$ terms of the arithmetic progression $3, 7, 11, \ldots$ less than or equal to $n$, we get
\[ \Od(n!) = (-1)^{\lfloor (n+1)/4\rfloor}\Od(\lfloor n/2 \rfloor!). \] Let $n$ have the binary expansion $b_k\ldots b_0$ as in \cref{not:binary}. By using $\lfloor \frac{b_\ell\ldots b_0}{2}\rfloor = b_\ell\ldots b_1$, we can rewrite our expression as \[ \Od(n!) = (-1)^{\lfloor \frac{b_k\ldots b_{0} + 1}{4}\rfloor + \lfloor \frac{b_k\ldots b_{1} + 1}{4}\rfloor + \lfloor \frac{b_k\ldots b_{2} + 1}{4}\rfloor + \ldots+ \lfloor \frac{b_k b_{k-1} + 1}{4}\rfloor} \Od(b_k!). \]
Notice that $\lfloor \frac{b_k\ldots b_{j+1}b_j + 1}{4}\rfloor = \lfloor \frac{b_k\ldots b_{j+1}b_j}{4}\rfloor + 1$ if and only if both $b_j$ and $b_{j+1}$ are 1. For all other cases, we have the equality, $\lfloor \frac{b_k\ldots b_{j+1}b_j + 1}{4}\rfloor = \lfloor \frac{b_k\ldots b_{j+1}b_j}{4}\rfloor$.\\
The inequality occurs exactly $D(n)$ times, giving us \[ \Od(n!) = (-1)^{D(n)}(-1)^{\lfloor \frac{b_k\ldots b_{0} }{4}\rfloor + \lfloor \frac{b_k\ldots b_{1}}{4}\rfloor + \ldots+ \lfloor \frac{b_kb_{k-1}b_{k-2}}{4}\rfloor + \lfloor \frac{b_k b_{k-1}}{4}\rfloor} \Od(b_k!). \] Using $\lfloor \frac{b_\ell\ldots b_0}{4}\rfloor = b_\ell\ldots b_2$, and the fact that $b_k \in \{0,1\}$, we get \[ \Od(n!) = (-1)^{D(n)}(-1)^{{b_k\ldots b_{2} } + {b_k\ldots b_{3}} + \ldots+ b_k + 0}. \] When $(-1)$ is raised to a number, the parity of the number entirely decides the result, thus $(-1)^{b_k\ldots b_j} = (-1)^{b_j}$. Using this, we obtain, \begin{align*} \Od(n!) &= (-1)^{D(n) + b_2 + b_3 + \ldots + b_k}\\ &= (-1)^{D(n) + \nu(\lfloor n/4 \rfloor)}. \end{align*} \end{proof} Before stating the main formula, we define an important quantity: \begin{defn}\label{defn:inversions}
Let $\lambda$ be a $2^R$-parent of $\mu$ with $|\mu| < 2^R$. Let $h^\lambda_\mu \in H(\lambda)$ be the affected hook-length (\cref{remark:aff}). We define $\eta^\lambda_\mu \in \mathbb{Z}/2\mathbb{Z}$ as follows: \[
(-1)^{\eta^{\lambda}_\mu} = \prod\limits_{\substack{x \in H(\lambda)\\ x \neq h^\lambda_\mu}} \dfrac{\Od(|h^\lambda_\mu- x|)}{\Od(|h^\lambda_\mu - 2^R - x|)}. \] \end{defn}
Now, we have the arsenal to approach the hook-length formula and apply our $\Od$ function to it. This gives us the following proposition: \begin{prop}[Workhorse formula]\label{workhorse} Let $n = 2^R +m > 3$ with $m<2^R$. Let $\mu\vdash m$ and $\lambda$ be a $2^R$-parent of $\mu$. Let $h^\lambda_\mu\in H(\lambda)$ be the affected hook-length. Then, \[ \Od(f^{\lambda}) = (-1)^{s_2(n) + s_2(h^\lambda_\mu) + \eta^\lambda_\mu}\Od(f^{\mu}), \] where $\eta^{\lambda}_{\mu}$ is as defined above and $s_2$ is as in \cref{not:binary}. \end{prop} \begin{proof} By applying $\Od$ on \cref{HLF} for $\lambda$, we get \[
\Od(f^{\lambda}) = (-1)^{D(n) + \nu(\lfloor \frac{n}{4} \rfloor)}\prod\limits_{h\in H(\lambda)} (-1)^{D(h) + \nu(\lfloor \frac{h}{4} \rfloor)} \prod\limits_{\substack{x > y\\ x,y\in H(\lambda)}} \Od(|x-y|). \]
Suppose, $|H(\mu)^{+r}| = |H(\lambda)|$ for some $r$. Then, we obtain, \[
\Od(f^{\mu}) = (-1)^{D(m) + \nu(\lfloor \frac{m}{4} \rfloor)}\prod\limits_{h\in H(\mu)^{+r}} (-1)^{D(h) + \nu(\lfloor \frac{h}{4} \rfloor)} \prod\limits_{\substack{x> y\\ x,y\in H(\mu)^{+r}}} \Od(|x-y|).\] Let $n$ have the binary expansion $b_R\ldots b_0$. Then, $m = n-2^R = b_{R-1}\ldots b_0$. Clearly, $D(m) = D(n)-1$ if and only if $b_{R-1} = 1$. Also, $\nu(\lfloor m/4 \rfloor) = \nu(\lfloor n/4 \rfloor) - 1$. These two facts, give us \[(-1)^{D(n) + \nu(\lfloor \frac{n}{4} \rfloor) - D(m) - \nu(\lfloor \frac{m}{4} \rfloor)} = (-1)^{s_2(n)}.\]\\ We have that $h^\lambda_\mu\in H(\lambda)$ and $h^\lambda_\mu-2^R \in H(\mu)^{+r}$. All the other elements of the sets $H(\lambda)$, $H(\mu)^{+r}$ are same. Using the same analysis as above, we get \begin{align*} &\dfrac{\prod\limits_{h\in H(\lambda)} (-1)^{D(h) + \nu(\lfloor \frac{h}{4} \rfloor)}}{\prod\limits_{h\in H(\mu)^{+r}} (-1)^{D(h) + \nu(\lfloor \frac{h}{4} \rfloor)}}\\ &= (-1)^{D(h^\lambda_\mu) + \nu(\lfloor \frac{h^\lambda_\mu}{4} \rfloor) - {D(h^\lambda_\mu - 2^R) - \nu(\lfloor \frac{h^\lambda_\mu - 2^R}{4}\rfloor)}} \\&= (-1)^{s_2(h^\lambda_\mu)}. \end{align*} Extending this argument to the remaining productand using \cref{defn:inversions} gives us the formula. \end{proof}
Although the above recursion is quite compact, it must be untangled to aid in enumeration. The $s_2$ terms are simple to deal with and so we must investigate the $\eta^\lambda_\mu$ term. \begin{lem} For any natural number $a$, we have: \begin{enumerate}\label{lem:inv} \item for $2^{R} < a < 2^{R+1}$, $\Od(a - 2^R) \neq \Od(a)\iff a = 2^R + 2^{R-1}$, \item for $0<a < 2^R$, $\Od(a +2^R) \neq \Od(a) \iff a = 2^{R-1}$, and \item for $0<a < 2^R$, $\Od(2^R - a) = \Od(a)\iff a = 2^{R-1}$. \end{enumerate} \end{lem} \begin{proof} Let $a$ have the binary expansion $b_R\ldots b_j\ldots 0$. The results can be shown by noticing that $\Od(a) \equiv b_{j+1}b_{j} \mod 4$ and performing elementary algebraic manipulations. \end{proof} \begin{defn}[Indicator function] Let $\mathbb{I}_{X}(x)$ be equal to 1 if $x\in X$ otherwise 0. We call $\mathbb{I}_{X}$, \textit{the indicator function} \textit{on $X$}. \end{defn} \begin{notation} For a partition $\lambda$ of $n = 2^R + m$ with $m < 2^R$ and $h\in H(\lambda)$, let $N_{\lambda}(h)= \#\{y \in H(\lambda)\mid h - 2^R< y < h\}$. \end{notation}
\cref{lem:inv} enables us to prove the following proposition which forms the crux of our enumeration endeavour: \begin{prop}\label{prop:inversions}
Let $\lambda$ be a $2^R$-parent of $\mu$ with $|\mu| < 2^R$. Then, we have \[ \eta^{\lambda}_{\mu} = N_{\lambda}(h^\lambda_\mu) - \mathbb{I}_{H(\lambda)}(h^\lambda_\mu - 2^{R-1}) + \mathbb{I}_{H(\lambda)}(h^\lambda_\mu + 2^{R-1}) + \mathbb{I}_{H(\lambda)}(h^\lambda_\mu - 3\cdot 2^{R-1}). \] \end{prop} \begin{proof} We consider the ratio
\[\rho_h(x) := \Od(|h-x|)/\Od(|h - 2^R - x|). \] In this proof, we will put $h= h^\lambda_\mu$ and $x\in H(\lambda)$ except for $x = h$. The idea of this proof is to find values of $x$ such that $\rho_h(x) = -1$, which allows us to compute \[ (-1)^{\eta^\lambda_\mu} = \prod\limits_{\substack{x\in H(\lambda)\\ x\neq h}} \rho_h(x). \] We split the proof into three cases: \begin{enumerate} \item Let $x < h-2^R < h$. The ratio $\rho_h(x)$ simplifies to $\frac{\Od(h-x)}{\Od(h - 2^R - x)}$. Put $h-x$ as $a$ and use (1) in \cref{lem:inv}, to give $x = h - 2^R - 2^{R-1}$. Thus, \[ \rho_h(h - 3\cdot 2^{R-1}) = -1. \] \item Let $h-2^R < h < x$. Similarly we use (2) from the lemma to get \[ \rho_h(h + 2^{R-1}) = -1. \] \item Let $h -2^R < x < h$. Now, $\rho_h(x)$ becomes $\dfrac{\Od(h-x)}{\Od(- h +2^R + x)}$. Using (3) from the lemma, we get that in this range, $\rho_h(x) = 1$ if and only if $x = h- 2^{R-1}$. Thus, for exactly \[{N_{\lambda}(h) - \mathbb{I}_{H(\lambda)}(h- 2^{R-1})}\] many values, we have $\rho_h(x) = -1$. \end{enumerate} Combining these three together gives us the proposition. \end{proof}
We are now ready to enumerate odd dimensional partitions.
\section{Odd Partitions of Sparse Numbers}\label{sec:proofmain} In this section, we prove \cref{mainthm} and \cref{maincor} through heavy use of \cref{workhorse} and \ref{prop:inversions}. \begin{defn}[Sparse number] A natural number $n$ with the binary expansion $b_k\ldots b_0$ is called \textit{sparse} if for all $0\leq i \leq k-1$, we have the product $b_{i+1}\cdot b_i = 0$. \end{defn} Recall that $a_i(n)$ denotes the number of partitions of $n$ whose dimensions are congruent to $i$ modulo 4. Further, $\delta(n):= a_1(n) - a_3(n)$, and the total number of odd partitions of $n$ is denoted by $a(n) = a_1(n) + a_3(n)$. We start by first defining \textit{hook partitions} and looking into their dimensions, which simplify to binomial coefficients. \begin{defn}\label{def:hooks} A partition $\lambda\vdash n$ is called a \textit{hook partition} if it is of the form $(a+1,1, \ldots, 1)$ for $0 \leq a \leq n-2$ or $(n)$. Equivalently, $H(\lambda) = \{n, b,b-1, \ldots, 1\}$ or $H(\lambda) = \{n\}$ where $a + b + 1 = n$. \end{defn} \begin{lem}\label{lem:hookdims} We have $\delta(1) = 1$, $\delta(2) = 2$ and $\delta(2^R) = 0$ for $R \geq 2$. \end{lem} \begin{proof} The cases of $\delta(1)$ and $\delta(2)$ are easy to see as they have 1 and 2 partitions respectively, all of dimension 1. By \cref{mcdprop}, we deduce that all odd partitions of $2^R$ are hook-partitions. Let $\lambda\vdash 2^R$ such that $H(\lambda) = \{2^R, b, b-1, \ldots, 1\}$. Putting this into the hook-length formula (\cref{HLF}), we get $f^{\lambda} = \binom{2^R-1}{b}$. By using \cref{davisandwebb} (stated below) and noticing that $2^R-1$ contains only 1s in its binary expansion, we get the lemma. \end{proof} \begin{prop}[Davis and Webb, \cite{Bin4}]\label{davisandwebb} Let $n\in \mathbb{N}$ be \textit{not} sparse, then \[ \#\{0 \leq k \leq n\mid\binom{n}{k}\equiv 1 \textnormal{ mod } 4\} = \#\{0 \leq k\leq n\mid \binom{n}{k}\equiv 3\textnormal{ mod } 4\}. \] \end{prop} \begin{proof} Refer to Theorem 6 of \cite{Bin4} for the proof. \end{proof}
Let $n = 2^R + m$ with $m < 2^{R-1} < 2^R$. We proceed with the following strategy: consider an odd partition $\mu\vdash m$ as our $2^R$-core. Consider all partitions of $n$ which are the $2^R$-parents of $\mu$. These are all odd partitions by \cref{mcdprop} and by \cref{workhorse}, we can determine $\delta(n)$ in terms of $\delta(m)$. Firstly, we see that \cref{workhorse} simplifies nicely in the $m < 2^{R-1}$ case. \begin{cor} Let $n = 2^R + m$ with $m < 2^{R-1}$. For odd partitions, $\lambda \vdash n$ and $\mu\vdash m$ such that $\lambda$ is a $2^R$-parent of $\mu$, we have \[ \Od(f^{\lambda}) = (-1)^{\eta^\lambda_\mu}\Od(f^\mu). \] \end{cor} \begin{proof} Let $n$ have the binary expansion $b_Rb_{R-1}\ldots b_0$. As $n$ is sparse, it must be that $b_{R-1} = 0$ and $s_2(n) = b_R + b_{R-1} = 1$. Also, we have $2^R \leq h^\lambda_\mu < n < 2^R + 2^{R-1}$ as all elements of $H(\lambda)$ are less than or equal to $n$. By the same logic as above, we get $s_2(h^\lambda_\mu) = 1$. \end{proof}
This makes finding $\eta^\lambda_\mu$ our primary concern. In our particular case of $m < 2^{R-1}$, even this calculation is simplified. We obtain a corollary of \cref{prop:inversions}: \begin{cor}\label{cor:inversions} Let $n = 2^R + m$ with $m < 2^{R-1}$. For odd partitions, $\lambda \vdash n$ and $\mu\vdash m$ such that $\lambda$ is a $2^R$-parent of $\mu$, we have \[ \eta^{\lambda}_{\mu} = N_{\lambda}(h^\lambda_\mu) - \mathbb{I}_{H(\lambda)}(h^\lambda_\mu - 2^{R-1}) \] with notation as in \cref{prop:inversions}. \end{cor} \begin{proof} We have $2^R \leq h^\lambda_\mu < n < 3\cdot2^{R-1}$. Clearly, $\mathbb{I}_{H(\lambda)}(h^\lambda_\mu - 3\cdot 2^{R-1}) = 0$. Along similar lines, $h^\lambda_\mu + 2^{R-1} \geq 2^R + 2^{R-1} > n$ and so it cannot be an element of $H(\lambda)$. Thus, $\mathbb{I}_{H(\lambda)}(h^\lambda_\mu + 2^{R-1}) = 0$. \end{proof} We enumerate Type I and Type II parents (\cref{parents}) separately. \begin{notation} When $2^R$ is understood, we denote the set of all $2^R$-parents of $\mu$ by $\p(\mu)$. Denote the set of Type I and Type II $2^R$-parents of $\mu$ by $\p_1(\mu)$ and $\p_2(\mu)$ respectively. \end{notation}
Clearly, we have $\p(\mu) = \p_1(\mu) \cup \p_2(\mu)$. Also, $|\p_1(\mu)| = H(\lambda)$ and $|\p_2(\mu)| = 2^R - H(\lambda)$. \begin{notation}[Signed-sum] For any finite $\Lambda' \subset \Lambda$, and a partition $\mu$, define the \textit{signed-sum}, $S_{\Lambda'}(\mu) = \frac{1}{\Od(f^\mu)}\sum\limits_{\lambda\in \Lambda'} \Od(f^\lambda)$. \end{notation} We see that $S_{\Lambda'}(\mu)$ counts the number of partitions in $\Lambda'$ with same value of $\Od$ as $\mu$ minus the number of partitions in $\Lambda'$ with a different value of $\Od$. For the following discussion, we will consider natural numbers $m$ and $R$ such that $m < 2^{R-1}$.\\ We now count Type I parents. \begin{prop}\label{prop:phistuff} Let $\mu$ be an odd partition of $m$. Then, for $\lambda\in \p_1(\mu)$, we have $\mathbb{I}_{H(\lambda)}(h^\lambda_\mu - 2^{R-1}) = 0$. Further, \[ S_{\p_1(\mu)}(\mu) = \begin{cases}
0, & \text{if }|H(\mu)| \text{ is even}\\
1, & \text{if }|H(\mu)| \text{ is odd}.\\ \end{cases} \] \end{prop} \begin{proof} By definition, we have $H(\lambda) = (H(\mu)\cup \{h^{\lambda}_\mu\})\backslash \{h^{\lambda}_\mu - 2^R\}$ where $h^{\lambda}_\mu > 2^R$. Thus, if $h^{\lambda}_\mu - 2^{R-1} \in H(\lambda)$ then $h^{\lambda}_\mu - 2^{R-1} \in H(\mu)$, which is not possible as $h^{\lambda}_\mu - 2^{R-1} > m$. This shows $\mathbb{I}_{H(\lambda)}(h^\lambda_\mu - 2^{R-1}) = 0$. We also have \begin{align*} S_{\p_1(\mu)}(\mu) &= \sum\limits_{\lambda\in \p_1(\mu)} \frac{\Od(f^\lambda)}{\Od(f^\mu)}\\ &= \sum\limits_{\lambda\in \p_1(\mu)}(-1)^{\eta^\lambda_\mu}\\ &= \sum\limits_{\lambda\in \p_1(\mu)}(-1)^{N_{\lambda}(h^\lambda_\mu)}. \end{align*} If, explicitly, $H(\mu) = \{h_1, \ldots, h_k\}$ with $h_1 > \ldots > h_k$, then the map $\phi: H(\mu)\rightarrow \p_1(\mu)$ defined by \[ \phi(h_i) = \Part((H(\mu)\cup\{h_i + 2^R\})\backslash \{h_i\}) \] is a bijection such that $h^{\phi(h_i)}_\mu = h_i + 2^R$. As all elements of $H(\mu)$ are strictly smaller than $2^R$, the element $h^{\phi(h_i)}_\mu$ is the largest element in $H(\phi(h_i))$. Thus, \begin{align*} N_{\phi(h_i)}(h^{\phi(h_i)}_\mu) &= \#\{y\in H(\phi(h_i))\mid h_i + 2^R > y > h_i\} \\ &= \#\{y\in H(\mu)\mid y > h_i\}\\ &= i - 1. \end{align*} This gives us, \[ S_{\p_1(\mu)}(\mu) = \sum\limits_{i=1}^k (-1)^{i-1}. \]
By considering the parity of $k = |H(\mu)| = |H(\lambda)|$, we get the result. \end{proof} The similar calculation for Type II parents is more involved. We will eventually prove the following proposition: \begin{prop}\label{prop:type2} Let $\mu$ be an odd partition of $m$ and $2^{R-1}>m$. Then, \[ S_{\p_2(\mu)}(\mu) = \begin{cases}
2 - 2(-1)^{m}, &\text{if }|H(\mu)| \text{ is even}\\
1 - 2(-1)^{m}, & \text{if }|H(\mu)| \text{ is odd}.\\ \end{cases} \] \end{prop}
Fix $\mu\vdash m$ and $n = 2^R + m$ with $m < 2^{R-1}$ as before. If $\lambda$ is a Type II $2^R$-parent, i.e., $\lambda\in\p_2(\mu)$, we have $h^\lambda_\mu = 2^R$. Thus, \cref{cor:inversions} simplifies to give $\eta^\lambda_\mu = N_{\lambda}(h^\lambda_\mu) - \mathbb{I}_{H(\lambda)}(2^{R-1})$. Although it looks simpler on the surface, this calculation comes with its own caveat that $r$-shifts do not give a $2^R$-parent. \begin{notation}\label{notn:defined} Define \[ \D := \{1\leq r \leq 2^R\mid 2^R \not\in H(\mu)^{+r}\}, \] wherein we assume that $R$ and $\mu$ are understood. For a set $X\subset [1, 2^R]\subset \mathbb{N}$, we write $\D_X := \D\cap X$.\\ For $r\in \D$, let \[\lambda_{[r]} = \Part\left((H(\mu)^{+r}\cup\{2^R\})\backslash \{0\}\right).\] \end{notation}
We define a new quantity, which may seem arbitrary at this point, but will be quite useful in our analysis: \begin{defn}[Parity gap] For a finite $X\subset \mathbb{N}$, define the \textit{parity gap of $X$}, $\g(X)$, to be the number of even elements of $X$ minus the number of odd elements of $X$. In notation,\[ \g(X) = \#\{x\in X\mid x \equiv 0\text{ mod } 2\} - \#\{x\in X\mid x\equiv 1\text{ mod } 2 \}. \] \end{defn} \begin{example} If $X = \{13, 12, 8, 5, 3, 1, 0\}$ then $\g(X) = 3 - 4 = -1$. \end{example} \begin{notation}\label{not:typeii} Let \[\p_2^\uparrow(\mu) := \{ \lambda_{[r]}\mid r\in \D_{[1, 2^{R-1}]}\}\] and \[\p_2^\downarrow(\mu) := \{\lambda_{[r]}\mid r\in \D_{[2^{R-1}+ 1, 2^{R}]}\}.\] \end{notation} \begin{lem}\label{lem:toprow} Following the above notation, we claim that $D_{[1, 2^{R-1}]} = [1, 2^{R-1}]$. Further, \[
S_{\p_2^\uparrow(\mu)}(\mu) = 2(-1)^{|H(\mu)|}\g(H(\mu)). \] \end{lem} \begin{proof}
The maximum possible element of $H(\mu)^{+r}$ is ${\max (|H(\mu)| + r)}$ which itself is \textit{strictly} smaller than $2^R$ as $|H(\mu)|< m< 2^{R-1}$ and $r \leq 2^{R-1}$. Thus, for $1\leq r \leq 2^{R-1}$, we have $2^R \not\in H(\mu)^{+r}$.\\
For the signed-sum part, it is easy to see that $N_{\lambda_{[r]}}(2^R) = |H(\mu)|+r - 1$. Thus, we get the following summation: \[
S_{\p_2^\uparrow(\mu)}(\mu) = \sum\limits_{r=1}^{2^{R-1}} (-1)^{|H(\mu)| + r - 1}(-1)^{\mathbb{I}_{H(\lambda_{[r]})}(2^{R-1})}. \] For every $h\in H(\mu)$, we can choose an $r_h := 2^{R-1} - h$ which ensures that $2^{R-1} \in H(\mu)^{+r_h}$. Conversely, if $2^{R-1} \in H(\mu)^{+r}$, then as $r\leq 2^{R-1}$, we must have $h + r = 2^{R-1}$ for some $h\in H(\mu)$. Thus, there is an injective map $H(\mu)\rightarrow\p_2^{\uparrow}(\mu)$ given by $h\mapsto \lambda_{[r_h]}$. Further, if $h\in H(\mu)$ and $h+r_h = 2^{R-1}$, then $h$ and $r_h$ have the same parity. We can now compute the signed-sum as follows: \begin{align*}
S_{\p_2^\uparrow(\mu)}(\mu) &= (-1)^{|H(\mu)| - 1}\sum\limits_{r=1}^{2^{R-1}} (-1)^{r +\mathbb{I}_{H(\lambda_{[r]})}(2^{R-1})} \end{align*} We break the sum into two parts depending on whether $2^{R-1}$ belongs to $H(\lambda_{[r]})$. \begin{align*}
&S_{\p_2^\uparrow(\mu)}(\mu)\\ &= (-1)^{|H(\mu)|-1}\left(\sum\limits_{\substack{r_h\\ h\in H(\mu)}} (-1)^{r_h +\mathbb{I}_{H(\lambda_{[r_h]})}(2^{R-1})} +\sum\limits_{\substack{\text{other } r}} (-1)^{r +\mathbb{I}_{H(\lambda_{[r]})}(2^{R-1})}\right)\\
&= (-1)^{|H(\mu)|-1}\left(\sum\limits_{\substack{h\in H(\mu)}} (-1)^{r_h + 1} +\sum\limits_{\substack{\text{other } r}} (-1)^{r}\right) \\
&= (-1)^{|H(\mu)|-1}\left(\sum\limits_{\substack{h\in H(\mu)}} \left((-1)^{r_h + 1} - (-1)^{r_h}\right) +\sum\limits_{r=1}^{2^{R-1}} (-1)^{r}\right)\\
&= 2(-1)^{|H(\mu)|-1}\left(\sum\limits_{\substack{h\in H(\mu)}} \left((-1)^{r_h+1}\right)+0\right)\\
&= 2(-1)^{|H(\mu)|}\left(\sum\limits_{\substack{h\in H(\mu)}} (-1)^{r_h}\right)\\
&= 2(-1)^{|H(\mu)|}\g(H(\mu)). \end{align*} \end{proof}
We now hand the reader the following proposition which explicitly states what values $\g(X)$ can take where $X$ is a $\beta$-set of an odd partition of $n$. \begin{lem}\label{prop:parity} Let $\lambda$ be an odd partition of $n$ and $X$ be its $\beta$-set. Then, we have \[ \g(X) = \begin{cases}
1 - (-1)^n, &\text{if }|X| \text{ is even}\\
(-1)^n, & \text{if }|X| \text{ is odd}.\\ \end{cases} \] \end{lem} \begin{proof} Recall that if $\lambda\in \p(\mu)$, then there exists an $0\leq s\leq 2^R$ such that $H(\lambda) = (H(\mu)^{+s}\cup \{h^\lambda_\mu\})\backslash\{h^\lambda_\mu - 2^R\}$ for $h^\lambda_\mu\in H(\lambda)$. The case $s = 0$ corresponds to Type I parents.
Using the fact that $h^\lambda_\mu$ and $h^\lambda_\mu - 2^R$ have the same parity, we can deduce that $\g(H(\lambda)) = \g(H(\mu)^{+s})$. With elementary computations, one can check that $\g(X^{+1}) = 1 - \g(X)$. This shows that \[ \g(X^{+s}) = \begin{cases} \g(X), &\text{if } s \text{ is even}\\ 1-\g(X),&\text{if }s \text{ is odd.} \end{cases} \] We can now work our way down hook-by-hook and calculate $\g(H(\lambda))$ by finding $\g(H(\alpha))$ where $\alpha = \varnothing$ when $n$ is even and $\alpha = (1)$ when $n$ is odd. Applying the above relation repeatedly, we get, \[\g(H(\lambda)) = \begin{cases}
\g(H(\alpha)), & \text{if }|H(\lambda)| - |H(\alpha)| \text{ is even}\\
1 -\g(H(\alpha)),&\text{if }|H(\lambda)| - |H(\alpha)|\text{ is odd.} \end{cases} \]
In a sense, $|H(\lambda)| - |H(\alpha)|$ counts the total number of $r$-shifts required to reach $\lambda$.\\ Trivially, $\g(H(\varnothing)) = 0$. If $n$ is even,we obtain \[\g(H(\lambda)) = \begin{cases}
0 & |H(\lambda)| \text{ is even}\\
1 &|H(\lambda)|\text{ is odd.} \end{cases} \] On the other hand, we have $\g(\{1\}) = -1$ and if $n$ is odd, then \[\g(H(\lambda)) = \begin{cases}
2, &\text{if }|H(\lambda)|\text{ is even.}\\
-1, & \text{if }|H(\lambda)| \text{ is odd}
\end{cases} \] The claim follows immediately from this computation. \end{proof} \begin{example} Let $\lambda$, $\mu$, $\nu$ and $\pi$ be partitions such that \begin{itemize} \item $H(\lambda) = (H(\mu^{+5})\cup\{2^R\})\backslash \{0\}$ \item $H(\mu)$ is a Type I $2^S$-parent of $\nu$ \item $H(\nu) = (H(\pi^{+7})\cup\{2^T\})\backslash \{0\}$ \end{itemize} By above, we have $\g(H(\nu)) = 1 - \g(H(\pi))$. Further, $\g(H(\mu)) = \g(H(\nu))$ and $\g(H(\lambda)) = 1 - \g(H(\mu))$. Thus, $\g(H(\lambda)) = 1 - \g(H(\nu)) = 1 - (1 - \g(H(\pi))) = \g(H(\pi))$. \end{example}
\begin{lem}\label{lem:bottomrow} For $2^{R-1}+ 1 \leq r \leq 2^R$, we have $\lambda_{[r]} \in \p_2^\downarrow(\mu)$ if and only if $2^{R-1}\not\in H(\lambda_{[r-2^{R-1}]})$. Further, \vspace*{-5pt} \[ S_{\p_2^\downarrow(\mu)}(\mu) = \begin{cases}
0, &\text{if }|H(\mu)| \text{ is even}\\
1, &\text{if } |H(\mu)| \text{ is odd}.\\ \end{cases} \] \end{lem} \begin{proof}
It is clear that $2^{R-1}\in \lambda_{[r-2^{R-1}]}$ if and only if $2^R \in H(\mu)^{+r}$ for $2^{R-1}+1 \leq r \leq 2^R$. In this range, the values of $r\not\in \D$ are exactly the values of $r$ such that $2^{R-1}\in H(\lambda_{[r]})$. By considering these values as $r_h$ (refer \cref{lem:toprow}), we get that $|\p_2^\downarrow(\mu)| = 2^{R-1} - |H(\mu)|$.\\ For $r > 2^{R-1}$, we have $2^{R-1}\in H(\lambda_{[r]})$ and thus, $\eta^{\lambda}_\mu = N_{\lambda_{[r]}}(2^R) + 1$.\\ Suppose, $\lambda_{[r]}, \lambda_{[r+s]}\in \p_2^\downarrow(\mu)$ and $r+1,\ldots,r+s-1\not\in \D$. This gives us that $H(\mu) = \{h_1, \ldots, h_k, h_{k+1}, \ldots, h_{k+s-1}\ldots h_{l}\}$ where for $1\leq i \leq s-1$, we have $h_{k + i} = h_{k+1} - i +1$ and $h_{k+i} + r + i = 2^R$. It follows that $h_k + r >2^R$ and $h_{k+s} + r + s < 2^R$.
We have $N_{\lambda_{[r]}}(2^R) = |H(\mu)| + r - 1 - j_r$ where $j_r$ is the number of elements in $H(\lambda_{[r]})$ strictly greater than $2^R$. As $h_{k+i} + r + s > 2^R$ for $1\leq i \leq s-1$, we have $j_{r+s} = j_r + s - 1 $. This gives us $N_{\lambda_{[r+s]}}(2^R) = |H(\mu)| + (r + s - 1) - (j_r + s - 1) = N_{\lambda_{[r]}}(2^R) + 1$.\\
Let $r_1 < \ldots < r_{2^{R-1} - |H(\mu)|}$ such that $\lambda_{[r_1]}, \ldots, \lambda_{[r_{2^{R-1} - |H(\mu)|}]}\in \p_2^\downarrow(\mu)$. By the above discussion, we have $N_{\lambda_{[r_i]}}(2^R) = N_{\lambda_{[r_1]}}(2^R) + i - 1$. Thus, \begin{align*}
S_{\p_2^\downarrow(\mu)}(\mu) &= \sum\limits_{i = 1}^{2^{R-1}-|H(\mu)|} (-1)^{N_{\lambda_{[r_i]}(2^R)} + 1}\\
&= (-1)^{N_{\lambda_{[r_1]}}(2^R) + 1}\sum\limits_{i = 1}^{2^{R-1}-|H(\mu)|}(-1)^{i-1}\\
&= (-1)^{|H(\mu)| + r_1 - j_{r_1}}\sum\limits_{i = 1}^{2^{R-1}-|H(\mu)|}(-1)^{i-1}\\
&= (-1)^{|H(\mu)| + 2^{R-1}}\sum\limits_{i = 1}^{2^{R-1}-|H(\mu)|}(-1)^{i}. \end{align*}
The last equality follows by noticing that if $r_1 = 2^{R-1} + x$, then $j_{r_1} = x-1$ as $2^R \in H(\mu)^{+(2^{R-1} + 1)}, \ldots, H(\mu)^{+(2^{R-1} + x - 1)}$. Putting the proper parity of $|H(\mu)|$ and evaluating the expression gives us our result. \end{proof} We are now ready to prove the result for Type II parents. \begin{proof}[Proof of \cref{prop:type2}] We use the value of $\g(H(\mu))$ from \cref{prop:parity} and apply it to the result in \cref{lem:toprow} to obtain \[ S_{\p_2^\uparrow(\mu)}(\mu) = \begin{cases}
2 - 2(-1)^m, &\text{if }|H(\mu)| \text{ is even}\\
2(-1)^{m+1}, & \text{if }|H(\mu)| \text{ is odd}.\\ \end{cases} \] As $S_{\p_2(\mu)}(\mu) = S_{\p_2^\uparrow(\mu)}(\mu) + S_{\p_2^\downarrow(\mu)}(\mu)$, we add the result of \cref{lem:bottomrow}, to obtain the result of \cref{prop:type2} which is \[ S_{\p_2(\mu)}(\mu) = \begin{cases}
2 - 2(-1)^{m}, &\text{if }|H(\mu)| \text{ is even}\\
1 - 2(-1)^{m}, & \text{if }|H(\mu)| \text{ is odd}.\\ \end{cases} \] \end{proof} We now have all the tools in our hands to give a proof of \cref{mainthm} and \cref{maincor}. \begin{proof}[Proof of \cref{mainthm}] If $\p(\mu)$ denotes the set of $2^R$-parents of $\mu$, then $S_{\p(\mu)}(\mu) = S_{\p_1(\mu)}(\mu) + S_{\p_2(\mu)}(\mu)$. Clearly, $S_{\p(\mu)}(\mu) = 2 - 2(-1)^m$. The theorem follows from the following computation and the observation that $n$ and $m$ have the same parity: \begin{align*} \delta(n) &= \sum\limits_{\substack{\\\lambda\vdash n\\\lambda \text{ is odd}}} \Od(f^\lambda)\\ & = \sum\limits_{\substack{\mu\vdash m\\\mu \text{ is odd}}} S_{\p(\mu)}({\mu})\Od(f^{\mu})\\ &= (2 - 2(-1)^m) \sum\limits_{\substack{\mu\vdash m\\\mu \text{ is odd}}} \Od(f^{\mu})\\ &= (2-2(-1)^m )\delta(m). \end{align*} This gives us the result: \[ \delta(n) = \begin{cases} 0, & \text{if }n \text{ is even}\\ 4 \delta(m), &\text{if } n \text{ is odd}. \end{cases} \] \end{proof} \begin{proof}[Proof of \cref{maincor}] Repeatedly apply \cref{mainthm} and use \cref{lem:hookdims}. The $\nu(n)-1$ comes from $\delta(1) = 1$. This gives us the result in the sparse case and we find $n = 2$ separately: \[ \delta(n) = \begin{cases} 2\text{,} & \text{if }n = 2\\ 0, &\text{if } n > 2 \text{ is even}\\ 4^{\nu(n) - 1}, & \text{if } n\text{ is odd}. \end{cases} \] \end{proof}
\section{Odd Partitions of $n = 2^R + 2^{R-1}$}\label{sec:proof11} We wish to extend the results of the previous section beyond the sparse case. Although, a general formula remains elusive; we can take confident steps towards it. We consider all numbers whose binary expansions contain a pair of adjacent 1s with all other bits zero. In notation, we consider $n$ with $\nu(n) = 2$ and $D(n) = 1$. We now prove \cref{11thm}.\\ We use similar methods as we did in the previous section. Recall that for $\Lambda'\subset \Lambda$ (the set of all partitions), we have \[ S_{\Lambda'}(\mu) := \frac{1}{\Od(f^\mu)}\sum\limits_{\lambda\in \Lambda'} \Od(f^\lambda).\] We also state a corollary of \cref{workhorse} which will be pertinent in this case. \begin{cor} Let $n = 2^R + 2^{R-1}$ for some $R \geq 2$. Let $\lambda\vdash n$ and $\mu\vdash 2^{R-1}$ be odd partitions such that $\lambda$ is a $2^R$-parent of $\mu$. Then, \[ \Od(f^{\lambda}) = (-1)^{s_2(h^\lambda_\mu) + \eta^\lambda_{\mu}}\Od(f^\mu). \] \end{cor}
The proof of the above corollary is trivial as the sum of first two digits of $n$ is 2. Notice that we start from $R = 2$ as the case of $R = 1$ is slightly different and can be handled independently.\\ Throughout our discussion, we will take $\mu$ to be a hook partition such that $H(\mu) = \{2^{R-1}, b, b-1,\ldots, 1\}$, where $b = 0$ implies $H(\mu) = \{2^{R-1}\}$. Now, we first count the $2^R$-parents of Type I which, as before, we denote by $\p_1(\mu)$. \begin{lem}\label{lem:7} With the above notation, we have \[ S_{\p_1(\mu)}(\mu) = \begin{cases} 1 ,& \text{if }b \text{ is even}\\ 2, & \text{if }b \text{ is odd}.\\ \end{cases} \] \end{lem} \begin{proof} As in the proof of \cref{prop:phistuff}, define the bijection $\phi:H(\mu)\rightarrow \p_1(\mu)$ such that \[ \phi(h) = \Part((H(\mu)\cup\{h+2^R\})\backslash\{h\}). \] Firstly, consider $\lambda:= \phi(2^{R-1})$. In this case, $s_2(h^\lambda_\mu) = 2$. Further, using \cref{prop:inversions} regarding $\eta^\lambda_\mu$, we have \[ \eta^\lambda_\mu = N_\lambda(2^R + 2^{R-1}) - \mathbb{I_{H(\lambda)}}(2^R) + \mathbb{I_{H(\lambda)}}(2^{R+1}) + \mathbb{I_{H(\lambda)}}(0). \] As $2^{R-1}$ is the largest entry in $H(\mu)$, it is easy to check that that $N_\lambda(2^R + 2^{R-1}) = 0$. Further, $H(\lambda) = \{2^R + 2^{R-1}, b, \ldots, 1\}$ and we can see that it does not contain 0, $2^{R}$ or $2^{R+1}$. Thus, $\eta^\lambda_\mu$ in this case is equal to 0, which gives us $(-1)^0 = 1$.\\ On the other hand, for $1\leq x \leq b$, define $\lambda_{[x]} := \phi(x)$ . In this case, $s_2(h^{\lambda_{[x]}}_\mu) = 1$ and \[ \eta^{\lambda_{[x]}}_\mu = N_{\lambda_{[x]}}(2^R + x) - \mathbb{I}_{H({\lambda_{[x]})}}(x + 2^{R-1}) + \mathbb{I}_{H(\lambda_{[x]})}(x + 2^R + 2^{R-1}) + \mathbb{I}_{H(\lambda_{[x]})}(x - 2^{R-1}). \]
We have $H(\lambda_{[x]}) = \{x + 2^R, 2^{R-1},\ldots \}$ and $N_{\lambda_{[x]}}(2^R + x) = b+1-x$. As $1 \leq x < 2^{R-1}$ and $|\lambda| = 2^{R} + 2^{R-1}$, $H(\lambda_{[x]})$ cannot contain, $ x - 2^{R-1}$ or $x + 2^{R} + 2^{R-1}$. From the explicit form of $H(\lambda_{[x]})$, it is easy to see the only element greater than $2^{R-1}$ is $x + 2^R$. Thus, $\eta^{\lambda_{[x]}}_\mu = b + 1 - x$. This gives us,
\begin{align*} S_{\p_1(\mu)}({\mu}) &= 1 + \sum\limits_{x=1}^{b} (-1)^{b + 1 - x + 1}\\ &= 1 + \sum\limits_{i=0}^{b-1} (-1)^x \end{align*} which with appropriate values of $b$ simplifies to give the intended result. \end{proof}
Notice that we are using the parity of $b$ which is opposite to the parity of $H(\mu)$. As one gives the other, we use $b$ for convenience.\\ Now, we move on to the results for Type II $2^R$-parents. \begin{lem}\label{lem:8} With the above notation, we have \[ S_{\p_2(\mu)}({\mu}) = \begin{cases} 3 ,& \text{if }b \text{ is even}\\ 2, & \text{if }b \text{ is odd}.\\ \end{cases} \] \end{lem} \begin{proof} As before, let $1\leq r \leq 2^R$ and $\lambda_{[r]}$ denote the $2^R$-parent corresponding to that $r$-shift. In notation, \[ \lambda_{[r]} = \Part\left((H(\mu)^{+r}\cup\{2^R\})\backslash \{0\}\right) \textnormal{ if and only if } r\in \D, \] where $\D$ is as in \cref{notn:defined}. For $r\in \D$, we have $\lambda_{[r]}\in\p_2(\mu)$.\\ As for all $\lambda\in\p_2(\mu)$, $h^\lambda_\mu = 2^R$, we have $s_2(h^\lambda_\mu) = 1$. Thus, \begin{align*} S_{\p_2(\mu)}({\mu}) &= \sum\limits_{\lambda\in \p_2(\mu)} (-1)^{\eta^\lambda_\mu + 1}\\ &= (-1)\cdot\sum\limits_{\lambda\in \p_2(\mu)} (-1)^{\eta^\lambda_\mu} \end{align*}
where \[ \eta^\lambda_\mu = N_{\lambda}(2^R) - \mathbb{I}_{H(\lambda)}(2^{R-1}) + \mathbb{I}_{H(\lambda)}(2^R +2^{R-1}). \]
We divide the values taken by $r$ into 6 (possibly empty, depending on $b$) intervals, which although overkill, would help illuminate the process better. It is recommended that the reader works out the assertions in the upcoming discussion using the basic definitions. As before, $H(\mu) = \{2^{R-1}, b, b-1,\ldots, 1\}$. Here, $b$ can take values from 0 to $2^{R-1}$ where $b=0$ implies $H(\mu) =\{2^{R-1}\}$. \begin{enumerate}
\item \underline{$1 \leq r\leq 2^{R-1} - b - 1$:}\\ In this case, $\eta^{\lambda_{[r]}}_\mu = N_{\lambda_{[r]}}(2^R) = |H(\mu)^{+r}|-1 = b + r$. \item\underline{$2^{R-1} - b \leq r \leq 2^{R-1}-1$:}\\ We have $\eta^{\lambda_{[r]}}_\mu = N_{\lambda_{[r]}}(2^R) - 1$ as $2^{R-1}\in H(\mu)^{+r}$. Thus, $\eta^{\lambda_{[r]}}_\mu = b + r - 1$. \item \underline{$r = 2^{R-1}:$}\\ As $2^{R-1}+2^{R-1} = 2^R\in H(\mu)^{+2^{R-1}}$, $2^{R-1}\not\in\D$. \item\underline{$2^{R-1} + 1 \leq r \leq 2^{R}-b-1$.}\\ For this, we have $\eta^{\lambda_{[r]}}_\mu = (b + r - 1) - 1 = b + r$. \item\underline{$2^R - b \leq r \leq 2^{R}-1$.}\\ For all these values, $r\not\in\D$. \item\underline{$r = 2^R$:}\\ In this case, $\eta^{\lambda_{\left[2^R\right]}}_\mu= N_{\lambda_{\left[2^R\right]}}(2^R) = 2^{R-1} - 1$ as $\mathbb{I}_{H(\lambda_{\left[2^R\right]})}(2^{R-1}) = \mathbb{I}_{H(\lambda_{\left[2^R\right]})}(2^R +2^{R-1})=1$. \end{enumerate}
We can now combine the above information to give \[ S_{\p_2(\mu)}({\mu}) = (-1)\cdot\left(\sum\limits_{r = 1}^{2^{R-1}-b-1} (-1)^{b+r} + \sum\limits_{r = 2^{R-1}-b}^{2^{R-1}-1} (-1)^{b+r-1} + \sum\limits_{r=2^{R-1} +1}^{2^{R}-b-1} (-1)^{b+r} + (-1)^{2^R - 1}\right). \] With some sleight-of-hand, this can be simplified to give \[ S_{\p_2(\mu)}({\mu}) = 1 + 2(-1)^b\sum\limits_{r = 1}^{2^{R-1}-b-1} (-1)^{r+1} + \sum\limits_{r=0}^{b-1} (-1)^r. \] When $b$ is even, we get $S_{\p_2(\mu)}({\mu}) = 1 + 2\cdot 1 + 0 = 3$. When $b$ is odd, we get $S_{\p_2(\mu)}({\mu}) = 1 + 0 + 1 = 2$. \end{proof}
With all of these ingredients in our hands, we are ready to give a \begin{proof}[Proof of \cref{11thm}] Combining the results of \cref{lem:7} and \cref{lem:8}, we see that $S_{\p(\mu)}(\mu) = 4$ when $\mu\vdash 2^{R-1}$. Thus, for $n = 2^R + 2^{R-1}$, we can do the following computation: \begin{align*} \delta(n) &= \sum\limits_{\substack{\\\lambda\vdash n\\\lambda \text{ is odd}}} \Od(f^\lambda)\\ & = \sum\limits_{\substack{\mu\vdash 2^{R-1}\\\mu \text{ is odd}}} S_{\p(\mu)}({\mu})\Od(f^{\mu})\\ &= 4 \sum\limits_{\substack{\mu\vdash 2^{R-1}\\\mu \text{ is odd}}} \Od(f^{\mu})\\ &= 4\delta(2^{R-1}). \end{align*} By the results of \cref{lem:hookdims}, we have our theorem, except for $R = 1$, which can be done by hand. \end{proof} \section{Partitions with Dimension 2 modulo 4}\label{sec:2mod4} We now consider partitions whose dimensions are congruent to 2 modulo 4, i.e., $v_2(f^{\lambda}) = 1$ for all such partitions $\lambda$.\\ To tackle the problem of enumeration, we use the machinery of \textit{2-core towers} which requires us to introduce the notion of a \textit{2-quotient}.\\ Recall that $\widetilde{\Lambda}_t$ is the set of all $t$-cores. It is well-known (refer Proposition 3.7 of \cite{olsson} p. 20) that there exists a bijection between the sets, $\Lambda$ and $\Lambda^2 \times \widetilde{\Lambda}_2$, given by $\lambda\mapsto (\quo_2(\lambda), \core_2(\lambda))$. Here, $\quo_2(\lambda)$ (known as the \textit{2-quotient} of $\lambda$) is a pair $(\lambda^{(0)}, \lambda^{(1)})$ with the property, \[
|\lambda| = 2(|\lambda^{(0)}| + |\lambda^{(1)}|) + |\core_2(\lambda)|. \]
\begin{notation} We simplify the notation by writing $\lambda^{(ij)}$ for $(\lambda^{(i)})^{(j)}$ for any binary string $i$ and $j\in \{0,1\}$. Further, $\lambda^{(\varnothing)} = \lambda$ and $(\lambda^{(\varnothing)})^{(j)} = \lambda^{(j)}$. \end{notation}
We recall the construction of 2-core towers as presented in \cite{olsson}.\\ For $\lambda\in \Lambda$, construct an infinite rooted binary tree with nodes labelled by 2-cores as follows: \begin{itemize} \item Label the root node with $\core_2(\lambda) = \core_2(\lambda^{(\varnothing)})$. \item If the length of the unique path from the root node to a node $v$ is $i$, then we say that the \textit{node $v$ is in the \nth{i} row.} \item Every node in the \nth{i} row is adjacent to two nodes in the $(i+1)^{\text{st}}$ row. Define a recursive labelling as follows: if the label of some node, $v$, in the \nth{i} row, is $\core_2(\lambda^{(b)})$ for some binary string $b$, then the two nodes in the $(i+1)^{\text{st}}$ row adjacent to $v$ have labels $\core_2(\lambda^{(b0)})$ and $\core_2(\lambda^{(b1)})$ respectively. \end{itemize} This tree is known as the 2-core tower of $\lambda$. \begin{example} The partition $(6,5,4,2,1,1)$ has the 2-core tower:\\ \adjustbox{scale = {0.5}{0.65}, center}{ \begin{tikzcd}
&&&&&&& {(2,1)} \\
\\
&&& \varnothing &&&&&&&& \varnothing \\
\\
& {(1)} &&&& \varnothing &&&& \varnothing &&&& {(1)} \\
\varnothing && \varnothing && \varnothing && \varnothing && {(1)} && \varnothing && \varnothing && \varnothing
\arrow[from=5-2, to=6-1]
\arrow[from=5-2, to=6-3]
\arrow[from=5-6, to=6-5]
\arrow[from=5-6, to=6-7]
\arrow[from=5-10, to=6-9]
\arrow[from=5-10, to=6-11]
\arrow[from=5-14, to=6-13]
\arrow[from=3-4, to=5-2]
\arrow[from=3-4, to=5-6]
\arrow[from=5-14, to=6-15]
\arrow[from=1-8, to=3-4]
\arrow[from=1-8, to=3-12]
\arrow[from=3-12, to=5-14]
\arrow[from=3-12, to=5-10] \end{tikzcd} }. \end{example}
Every partition has a unique 2-core tower and every 2-core tower comes from a unique partition. This bijection is inherited from the core-quotient bijection above.\\ We state a well-known classification result for 2-cores which we encourage the reader to prove on their own. \begin{lem}\label{lem:2cores} A partition $\lambda$ is a 2-core if and only if $\lambda = (n , n-1, \ldots 1)$ for some $n\geq 0$. \end{lem}
\begin{notation}
Let $\T_k(w)$ denote the number of solutions $(\mu_{[i]})_{i=1}^{2^k}$ to \[\sum\limits_{i=1}^{2^k}|\mu_{[i]}| = w,\] where each $\mu_{[i]}$ is a 2-core. \end{notation} \begin{lem}\label{lem:weight} With the above notation, we have, \[ \T_k(w) = \begin{cases} 1, & \text{if }w = 0\\ 2^k, &\text{if } w = 1\\ \binom{2^k}{2},&\text{if } w = 2\\ \binom{2^k}{3} + 2^k, & \text{if }w = 3. \end{cases} \] \end{lem} \begin{proof} When $w = 0$, the only solution is $\mu_{[i]}= \varnothing$ for all $i$. Thus, $\T_k(0) = 1$. \\ When $w = 1$, choose $\mu_{[i_0]} = (1)$ for some $1\leq i_0\leq 2^k$ and $\varnothing$ for the rest. As there are $2^k$ possible options for $i_0$, we get $\T_k(1) = 2^k$.\\ For $w = 2$, we must choose $\mu_{[i]} = \mu_{[j]} = (1)$, for some $i$ and $j$, and the rest $\varnothing$. Thus, $\T_{k}(2) = \binom{2^k}{2}$.\\ There does exist a 2-core of size 3, which is $(2,1)$. If $w=3$, we have two options, either we choose $i, j, k$ such that $\mu_{[i]} = \mu_{[j]} = \mu_{[k]} = (1)$ or $p$ such that $\mu_{[p]} = (2,1)$. For the former, we get $\binom{2^k}{3}$ ways and the latter can be done in $2^k$ ways, thus giving $\T_k(3) = \binom{2^k}{3} + 2^k$. \end{proof}
\begin{defn}[Weight in a 2-core tower]\label{defn:weight} The \textit{weight of the \nth{k} row of the 2-core tower of a partition $\lambda$ }is given by \[
w_k(\lambda) := \sum\limits_{b\in\{0,1\}^k} |\core_2(\lambda^{(b)})|, \]
where we define $w_0(\lambda) = |\core_2(\lambda)|$. \end{defn} We state some important properties of 2-core towers and how they relate to dimensions. For the subsequent discussion, we will let $n = \sum\limits_{i\geq 0} b_i2^i$, where $b_i\in\{0,1\}$. Further, recall that $\bin(n) = \{i\mid b_i = 1\}$. \begin{notation} Let $\bin'(n) = \{i>0\mid b_i = 1\}$. \end{notation} Note the strict inequality in the definition. Furthermore, we have the relation, $\bin'(n) = \bin(n)\backslash \{0\}$. \begin{prop}[Macdonald]\label{prop:mcdmain} Let $\lambda$ be a partition of $n$. We write $w_i:= w_i(\lambda)$. In this case, the following hold: \begin{enumerate} \item The 2-core tower of $\core_{2^k}(\lambda)$ is given by labelling the rows $0$ to $k-1$ as in the 2-core tower of $\lambda$ and by labelling every node in \nth{i} row, for $i\geq k$, with $\varnothing$, the empty partition. \item $\lambda$ is an odd partition if and only $w_i = b_i$ for all $i\geq 0$. \item $v_2(f^{\lambda}) = 1$ if and only if for some $R \in \bin'(n)$, we have $w_{R -1} = b_{R-1} + 2$, $w_{R} = 0$ and $w_i = b_i$ otherwise. \end{enumerate} \end{prop} \begin{proof} The proofs of (1) and (2) are given in \cite{mcd}. Through section 4 of \cite{mcd}, we know that there exists a sequence of non-negative integers $(z_i)_{i\geq 0}$ with $z_0 = 0$ such that \[ w_i + z_i = b_i + 2z_{i+1}. \] From Section 3 of \cite{mcd}, we know that if $v_2(f^\lambda) = 1$, then we have $\sum\limits_{i\geq 0} w_i = \sum\limits_{i\geq 0}b_i + 1$. Combining these, we get that we must have $z_i = 1$ for exactly one $i > 0$ and zero for the rest. Putting $z_k = 1$ gives us the equations, \begin{align*} w_{k-1} &= b_{k-1} + 2\\ w_k + 1 &= b_k. \end{align*} Notice that the second equation above tells us that $z_k = 1$ is possible only when $k \in \bin(n)$. Thus, $w_k = 0$ and $w_{k-1} = b_{k-1} + 2$. The rest is clear as $z_i = z_{i+1} = 0$. \end{proof}
\begin{notation}\label{notation:wk} For $k>0$, define a sequence of non-negative integers, $\mathbf{w}_k := (w^k_i)_{i\geq 0}$ with the following properties: \begin{enumerate} \item $w^k_{k-1} = b_{k-1}+2$, \item $w^k_{k} = 0$, and \item $w^k_i = b_i$ for all other values of $i$. \end{enumerate} Let $\T(\mathbf{w}_k) := \prod\limits_{i\geq 0} \T_i(w^k_i)$. \end{notation} Note that the above product is the number of ways of constructing 2-core towers such that $w_i(\lambda) = w^k_i$, for some given $k$.
\begin{proof}[Proof of \cref{2mod4thm}] By the above discussion, we have that the number of partitions with dimensions congruent to 2 modulo 4 is given by \begin{align*} a_2(n) &= \sum\limits_{k\in \bin'(n)} \T(\mathbf{w}_k)\\ &= \sum\limits_{k\in \bin'(n)} \prod\limits_{i\geq 0} \T_i(w^k_i). \end{align*} Denote the binary expansion of $n$ as we did before. If we suppose $n = 2^R + m$ with $m<2^R$, then $b_R = 1$ and $b_i = 0$ for $i > R$. We can break the sum up as, \[ a_2(n) = \left(\sum\limits_{k\in \bin'(n)\backslash \{R\}} \prod\limits_{i\geq 0} \T_i(w^k_i) \right)+ \T(\mathbf{w}_R). \] For the term on the left of the plus sign, we have \begin{align*} \sum\limits_{k\in \bin'(n)\backslash \{R\}} \prod\limits_{i\geq 0} \T_i(w^k_i) &= \sum\limits_{k\in \bin'(n)\backslash \{R\}} \T_R(w^k_R)\prod\limits_{i \neq R} \T_i(w^k_i)\\ &= 2^R\sum\limits_{k\in \bin'(m)} \prod\limits_{i\neq R} \T_i(w^k_i)\\ &= \frac{2^R}{\T_R(0)}\sum\limits_{k\in \bin'(m)} \prod\limits_{i\geq 0} \T_i(w^k_i)\\ &= 2^R a_2(m). \end{align*} Here, the last equality follows from point (1) in \cref{prop:mcdmain}.\\ Now, we deal with the term on the right of the plus sign. We see that \begin{align*} \T(\mathbf{w}_R) &= \T_{R-1}(w^R_{R-1})\T_{R}(w^R_R)\prod\limits_{i\neq R, R-1} \T_i(w^R_i)\\ &= \T_{R-1}(w^R_{R-1})\T_{R}(w^R_R)\prod\limits_{i\neq R, R-1} \T_i(b_i)\\ &= \frac{\T_{R-1}(w^R_{R-1})\T_{R}(w^R_R)}{\T_{R-1}(b_{R-1})\T_{R}(b_R)} a(n). \end{align*}
The $a(n)$ appearing is a consequence of (2) in \cref{prop:mcdmain}. We have $b_R = 1$, $w_R = 0$ and $w_{R-1} = b_{R-1} + 2$. This simplifies the above expression to \[ \frac{\T_{R-1}(b_{R-1} + 2)}{2^{R}\T_{R-1}(b_{R-1})} a(n). \] By \cref{lem:weight} and $a(n) = 2^Ra(m)$, we can complete the proof to get \[a_2(n) = \begin{cases} 2^R\cdot a_2(m) + \binom{2^{R-1}}{2}\cdot a(m), & \text{ if }m < 2^{R-1}\\ 2^R\cdot a_2(m) + \left(\binom{2^{R-1}}{3} + 2^{R-1}\right) \cdot \displaystyle{\frac{a(m)}{2^{R-1}}}, & \text{ if } 2^{R-1} < m < 2^R. \end{cases}\]
\end{proof}
In the sparse case, this theorem takes a nice form as evident in \cref{2mod4cor}. Although, we can prove the corollary using the recursive relations in \cref{11thm}, we choose to do perform direct computations as it is more illuminating. \begin{proof}[Proof of \cref{2mod4cor}] Let $n$ be a sparse number, i.e., it has no consecutive 1s in its binary expansion. The summation as before is: \begin{align*} a_2(n) &= \sum\limits_{k\in \bin'(n)} \T(\mathbf{w}_k)\\ &= \sum\limits_{k\in \bin'(n)} \frac{\T_{k-1}(w^k_{k-1})\T_{k}(w^k_k)}{\T_{k-1}(b_{k-1})\T_{k}(b_k)} a(n). \end{align*} As $n$ is sparse, for $k\in \bin'(n)$, we have $b_{k-1} = 0$. Further, $w^k_{k} = 0$ and $w^k_{k-1} = b_{k-1} +2 = 2$.\\ This gives us the following summation, \[ a_2(n) = \sum\limits_{k\in \bin'(n)} \frac{\T_{k-1}(2)}{\T_{k}(1)} a(n). \] By using the results from \cref{lem:weight}, this simplifies to \begin{align*} a_2(n) &= a(n)\sum\limits_{k\in \bin'(n)} \frac{\binom{2^{k-1}}{2}}{2^k}\\ &= a(n)\sum\limits_{k\in \bin'(n)} \frac{(2^{k-1})(2^{k-1}-1)}{2\cdot 2^k}\\ &= \frac{a(n)}{8}\sum\limits_{k\in \bin'(n)} 2^k - 2.\\ \end{align*} If $n$ is even, then this summation becomes $n - 2\nu(n)$. If $n$ is odd, the summation becomes $(n-1) + 2(\nu(n) - 1) = (n-1) + 2(\nu(n-1))$. This gives us the final answer as, \[a_2(n) = \begin{cases} \displaystyle{\frac{a(n)}{8}(n - 2 \nu(n))},& \text{if }n \text{ is even}\\ a_2(n-1), &\text{if } n \text{ is odd}. \end{cases}\] \end{proof} \begin{remark} The theorem above immediately leads us to the value of $m_4(n)$. We have $m_4(n) = a_1(n) + a_2(n) + a_3(n) = a(n) + a_2(n)$ by definition. \end{remark}
\section{Irreducible Representations of Alternating Groups} We can show similar results about degrees of irreducible representations in the case of $A_n$, the alternating group on $n$ letters, which is defined to be the unique subgroup of index 2. \begin{defn} Let $a_i^\circ(n)$ be the number of irreducible representations of $A_n$ (upto isomorphism) whose dimensions are congruent to $i$ modulo 4. Let $\delta^\circ(n) := a_1^\circ(n) - a_3^\circ(n)$. Let $a^\circ(n):= a_1^\circ(n) + a_3^\circ(n)$ be the number of odd dimensional irreducible representations of $A_n$. \end{defn} In \cite{geetha}, the value of $a^\circ(n)$ was computed in terms of $a(n)$ for which we present a proof sketch here. Recall that a partition, $\hat{\lambda}$, is called the \textit{conjugate} of $\lambda$ if there is a cell in position $(j,i)$ in $\sh(\hat{\lambda})$ if and only if there is a cell in position $(i,j)$ in $\sh(\lambda)$. If $\hat{\lambda} = \lambda$, then $\lambda$ is called a \textit{self-conjugate partition}. \begin{remark}\label{rem:towerflip} The 2-core tower of $\hat{\lambda}$ can be obtained by flipping the 2-core tower of $\lambda$ along the central axis of symmetry. That is, the node of the 2-core tower of $\hat{\lambda}$ indexed by $\core_2(\hat{\lambda}^{(b)})$ is indexed by $\core_2(\lambda^{(a)})$ where $a$ and $b$ are complementary binary strings. \end{remark} \begin{notation} Let $\hat{m}_2(n)$ denote the number of self-conjugate partitions with dimension congruent to 2 modulo 4. \end{notation} \begin{lem}[\cite{geetha}]\label{lemmaan} For $n\in \mathbb{N}$, we have \[ \hat{m}_2(n) = \begin{cases} 1, & \text{if } n = 3 \text{}\\ 2^{k-2}, & \text{if } n = 2^k, 2^k + 1, k>1 \text{}\\ 0, & \text{otherwise} \text{}. \end{cases} \] \end{lem} \begin{proof} For a given $n$, define $w^k_i$ as in \cref{notation:wk}. By point 3 of \cref{prop:mcdmain}, we know that such sequences of weights of rows in the 2-core tower, $\mathbf{w}_k$, correspond to partitions with dimensions congruent to 2 modulo 4.\\ Firstly, consider the case when the root node is labelled by $\varnothing$. We know that $w^k_i \in \{0,1,2,3\}$ and for $w^k_i = 1, 3$, we cannot flip the 2-core tower about the central axis to preserve symmetry (\cref{rem:towerflip}). Thus, $w^k_i = 0$ or $2$ for all $i>0$ in this case and $w_0^k = 0$ as the root node is labelled by $\varnothing$.\\ Similarly, in the case when the root node is labelled by $(1)$, we have $w^k_0=1$ and $w^k_i = 0$ or $2$ for all $i>0$ and $k\in \bin(n)$. In both these cases, we cannot have two rows $j > i > 0$ such that both $i$ and $j$ in $\bin(n)$. If this was the case, then we will have $w^i_i=0$ but $w^i_j = 1$ which violates our condition on $w_i^k$ taking only the values 0 and 2. Thus, we must have $\nu(n) = 1$ or when $0\in\bin(n)$, $\nu(n) = 2$. Thus, self-conjugate partitions with dimension congruent to 2 modulo 4 exist only for $n = 2^k, 2^k+1$.\\ In these cases, we have $w^k_k = 0$, $w^k_{k-1} = 2$ and the $(k-1)^{\text{st}}$ row is labelled by two $(1)$s arranged symmetrically about the centre. Notationally, $\core_2(\lambda^{(x)}) = \core_2(\lambda^{(y)}) = (1)$ where $x$ and $y$ are binary complementary strings in $\{0,1\}^{k-1}$ and the other nodes are labelled by $\varnothing$. Clearly, there are $2^{k-2}$ many ways to achieve such an arrangement, which gives us our result. \end{proof}
The partition $(3,3,3)\vdash 9$ has dimension congruent to 2 modulo 4 and is self-conjugate. Its 2-core tower is the following and satisifes the symmetric arrangement: This partition has the 2-core tower:\\ \adjustbox{scale = {0.5}{0.65}, center}{ \begin{tikzcd}
&&&&&&& {(1)} \\
\\
&&& \varnothing &&&&&&&& \varnothing \\
\\
& {(1)} &&&& \varnothing &&&& \varnothing &&&& {(1)} \\
\varnothing && \varnothing && \varnothing && \varnothing && \varnothing && \varnothing && \varnothing && \varnothing
\arrow[from=5-2, to=6-1]
\arrow[from=5-2, to=6-3]
\arrow[from=5-6, to=6-5]
\arrow[from=5-6, to=6-7]
\arrow[from=5-10, to=6-9]
\arrow[from=5-10, to=6-11]
\arrow[from=5-14, to=6-13]
\arrow[from=3-4, to=5-2]
\arrow[from=3-4, to=5-6]
\arrow[from=5-14, to=6-15]
\arrow[from=1-8, to=3-4]
\arrow[from=1-8, to=3-12]
\arrow[from=3-12, to=5-14]
\arrow[from=3-12, to=5-10] \end{tikzcd} }.
\begin{remark}\label{rem:cliff} By Clifford theory (refer Theorem 5.12.5 in \cite{amribook} and Chapter 6 in \cite{olsson}), we know that a self-conjugate partition, $\lambda\vdash n$, corresponds to two irreducible representations of $A_n$ with dimension $f^\lambda/2$. Furthermore, a conjugate pair of distinct partitions of $n$, $\mu$ and $\hat{\mu}$, corresponds to one irreducible representation of $A_n$ with dimension $f^\mu$. \end{remark} \begin{prop}[\cite{geetha}] For $n\in \mathbb{N}$, the number of odd dimensional irreducible representations of $A_n$ is given by \[ a^\circ(n) = \begin{cases} 1, & \text{if } n=1,2 \text{}\\ 3, & \text{if } n=3 \text{}\\ 2^k, & \text{if } n = 2^k, 2^k + 1, k>1\text{}\\ a(n)/2, & \text{for other values of } n > 4\text{}\\ \end{cases} \] where $a(n)$ is the number of odd partitions. \end{prop} \begin{proof} By \cref{rem:cliff}, we get that \[ a^\circ(n) = 2\hat{m}_2(n) + \frac{a(n)}{2}. \] Using \cref{lemmaan} and $a(2^k) = a(2^k + 1) = 2^k$, we have our result. \end{proof} We now wish to extend this result to find $a_1^\circ(n)$ and $a_3^\circ(n)$. We do so by calculating $\delta^\circ(n)$. Pick $\lambda$ such that $\lambda = \widehat{\lambda}$. If we want it to correspond to an irreducible representation of $A_n$ with odd dimension, then $f^\lambda$ must be congruent to 2 modulo 4 by \cref{rem:cliff}. As the dimension gets halved, we only need to find what $\Od(f^\lambda)$ is.\\ In order to do so, we require what $\lambda\vdash 2^k, 2^k +1$ with $f^\lambda$ congruent to 2 modulo 4 look like. For this, we cite two propositions from Geetha and Amruta's paper. \begin{prop}[\cite{geetha}]\label{prop:geethapowersof2} If $\lambda\vdash 2^k$ such that $f^\lambda \equiv 2$ mod 4, then \begin{enumerate} \item $\lambda$ is a hook partition of $2^k$ or, \item $\lambda$ is of the form $(2^{k-1}-i, 2^{k-1}-j, \underbrace{2, \ldots, 2}_{i \text{ times}}, \underbrace{1, \ldots, 1}_{j-i\text{ times}})$ where $0 \leq i \leq j \leq 2^{k-1}-2$. \end{enumerate} \end{prop} \begin{prop}[\cite{geetha}]\label{prop:geetha2kplusone} If $\lambda\vdash 2^k+1$ such that $f^\lambda \equiv 2$ mod 4 and $\lambda$ is self-conjugate, then \begin{enumerate} \item $\lambda$ has $h_{1,1} = 2^k + 1$ or \item $\lambda$ has $h_{1,1} = p_1$, $h_{2,2} = p_2$ and $h_{3,3} = 1$ for some odd $p_1 > p_2 > 1$. \end{enumerate} \end{prop} \begin{thm} For $n\geq 3$, we have \[ \delta^\circ(n) = \begin{cases} 3, & \text{if } n = 3 \text{}\\ 2^{k-1}, & \text{if } n = 2^k, k>1 \text{}\\ 2^{k-1}-2, & \text{if } n = 2^k+1, k>1 \text{}\\ \delta(n)/2, & \text{otherwise.} \text{} \end{cases}\] \end{thm} \begin{proof} For $n\neq 2^k, 2^{k}+1$, there are no self-conjugate partitions corresponding to irreducible representations of $A_n$ with dimension congruent to 2 modulo 4, thus $a_i^\circ(n) = a_i(n)/2$ for $i=1,3$ by \cref{rem:cliff} and $\delta^\circ(n) = \delta(n)/2$.\\ Notice that $\Od(a)^2 = 1$. Thus, for a self conjugate partition as we have $h_{i,j} = h_{j,i}$ for all $(i,j)\in\sh(\lambda)$, it follows that \[ \Od\left(\prod\limits_{(i,j)\in\sh(\lambda)} h_{i,j}\right) = \Od\left(\prod\limits_{\substack{i\\(i,i)\in\sh(\lambda)}} h_{i,i}\right). \] We saw above that for only $n = 2^k, 2^k + 1$ for $k >1$, we have $\hat{m}_2(n)\neq 0$. By using the formula in \cref{frthlf} and \[\Od(2^k!) = \Od((2^k+1)!) = -1,\text{(\cref{lem:odnfac})}\] we can show that for the case when $\lambda\vdash 2^k$, $k>1$,and $\lambda$ is self conjugate with dimension congruent to 2 mod 4, we have \[ \Od(f^\lambda) = -\Od\left(\prod\limits_{\substack{i\\(i,i)\in\sh(\lambda)}} h_{i,i}\right). \] To continue with our computations, we use \cref{prop:geethapowersof2}. Following the notation from \cite{geetha}, we deduce that that for $\lambda\vdash 2^k$ with $f^\lambda\equiv 2 \mod 4$, we have $h_{1,1} = 2^{k-1} - i + j + 1$ and $h_{2,2} = 2^{k-1} + i - j - 1$.\\ Further imposing the condition that $\lambda = \hat{\lambda}$, we get $h_{1,1} = 2^k - (2\alpha + 1)$ and $h_{2,2} = 2\alpha+1$, for $0\leq \alpha \leq 2^{k-2}-1$. It is easy to see that $\Od(2^k - (2\alpha + 1)) = -\Od(2\alpha+1)$ and thus, \[ \Od(f^\lambda) = -\Od(h_{1,1})\Od(h_{2,2}) = -\Od(2\alpha+1)\cdot (-\Od(2\alpha+1)) = 1 \] for this case. Thus, by \cref{rem:cliff}, each irreducible representation of $S_n$ with degree congruent to 2 modulo 4 which is indexed by a self-conjugate representation restricts to two irreducible representations of $A_n$ with degrees which are always 1 modulo 4.\\ An irreducible representation of $A_n$ has degree congruent to 1 modulo 4 if it corresponds to a non-self conjugate pair with dimensions congruent to 1 modulo 4 or it corresponds to a self-conjugate partition with dimension congruent to 2 modulo 4. This gives us that \[ a_1^\circ(n) = 2\hat{m}_2(n) + \frac{a_1(n)}{2}. \] On the other hand, $a_3^\circ(n)$ gets no contribution from $\hat{m}_2(n)$. Thus, we have \begin{enumerate} \item $a_1(2^k) = a_3(2^k) = 2^{k-1}$ where each \textit{pair} gives \textit{one} irreducible representation. \item There are $2^{k-2}$ possible values of $\alpha$, each of which corresponds to \textit{two} irreducible representations of $A_n$ with dimension congruent to 1 modulo 4 coming from representations of $S_n$ with dimension congruent to 2 modulo 4. \end{enumerate} We get that $a^\circ_1(2^k) = 2^{k-1} + 2^{k-2}$ and $a_3^\circ(2^k) = 2^{k-2}$. Thus, $\delta^\circ(2^k) = 2^{k-1}$.\\
Now, we consider the case when $\lambda\vdash 2^k+1$. We deal with the two cases in \cref{prop:geetha2kplusone}: \begin{enumerate} \item When $\lambda$ has $h_{1,1} = 2^k+1$, we get $\Od(f^\lambda) = -\Od(2^k+1) = -1$. \item In the other case, we have $p_1 + p_2 = 2^k$. As $p_1 \equiv -p_2\mod 4$, we have $\Od(f^\lambda) = -\Od(p_1)\Od(p_2)\Od(1)= 1$. Note that because of the conditions $p_1 > p_2$ and $p_1$ an odd integer, $p_1$ takes exactly $2^{k-2}-1$ values. \end{enumerate} We again use \cref{rem:cliff}. We add up the contributions from odd partitions of $2^k+1$ and partitions of $2^k+1$ with dimension congruent to 2 modulo 4 which we computed above. We have \[ a_1^\circ(2^k+1) = \frac{2^k-1 + 2}{2} + 2\cdot(2^{k-2}-1)\] and \[ a_3^\circ(2^k+1) = \frac{2^k-1 - 2}{2} + 2\cdot 1.\]
Thus, $\delta^\circ(2^k+1) = 2^{k-1}-2$. \end{proof} \section{Closing Remarks and Open Problems}\label{sec:problems} It is natural to ask why the case of $n = 2^R + m$ for $2^{R-1}<m<2^R$ is so difficult that it evades our methods. As we saw before, the values of $S_{\p(\mu)}(\mu)$ are constants, thus, independent of the core $\mu$ that we are building upon. In the harder case, we do not have this luxury. The values of $S_{\p(\mu)}(\mu)$ vary wildly and resist coalescing into nice summations. Along with this roadblock, we present some relevant problems that remain unsolved. \begin{enumerate} \item What is $\delta(n)$ in the case when $n = 2^R + m$ for $2^{R-1} < m < 2^{R}$? \item Is it possible to provide a reasonable bound for $S_{\p(\mu)}(\mu)$? \item Can we say something about the dimension of a partition by just looking at its 2-core tower? \item Is there a characterization of odd partitions modulo 4 in terms of $\beta$-sets? \end{enumerate} It is possible to answer (3) and (4) in the case of hook partitions as the analysis simplifies to looking at binomial coefficients.
\end{document}
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\begin{document}
\markboth{Sl. Shtrakov and I. Damyanov} {On the complexity of finite valued functions}
\title{On the complexity of finite valued functions}
\author{Slavcho Shtrakov}
\address{Department of Computer Science,\\ South-West University, Blagoevgrad, Bulgaria\\
}
\author{Ivo Damyanov}
\address{Department of Computer Science,\\ South-West University, Blagoevgrad, Bulgaria\\
}
\begin{abstract} The essential variables in a finite function $f$ are defined as variables which occur in $f$ and weigh with the values of that function.
The number of essential variables is an important measure of complexity for discrete functions.
When replacing some variables in a function with constants the resulting functions are called subfunctions, and when replacing all essential variables in a function with constants we obtain an implementation of this function.
Such an implementation corresponds with a path in an ordered decision diagram (ODD) of the function which connects the root with a leaf of the diagram. The sets of essential variables in subfunctions of $f$ are called separable in $f$. In this paper we study several properties of separable sets of variables in functions which directly impact on the number of implementations and subfunctions in these functions.
We define equivalence relations which classify the functions of $k$-valued logic into classes with same number of implementations, subfunctions or separable sets. These relations induce three transformation groups which are compared with the lattice of all subgroups of restricted affine group (RAG). This allows us to solve several important computational and combinatorial problems.
\end{abstract}
\keywords{Ordered decision diagram; implementation; subfunction; separable set.} \maketitle \section{Introduction}\label{sec1}
Understanding the complexity of $k$-valued functions is still one of the fundamental tasks in the theory of computation. At present, besides classical methods like substitution or degree arguments a bunch of combinatorial and algebraic techniques have been introduced to tackle this extremely difficult problem. There has been significant progress analysing the power of randomness and quantum bits or multiparty communication protocols that help to capture the complexity of switching functions. For tight estimations concerning the basic, most simple model switching circuits there still seems a long way to go (see \cite{comp_sem}).
In Section \ref{sec2} we introduce the basic notation and give definitions of separable sets, subfunctions, etc. The properties of distributive sets of variables with their s-systems are also discussed in Section \ref{sec2}. In Section \ref{sec3} we study the ordered decomposition trees (ODTs), ordered decision diagrams (ODDs), and implementations of discrete functions. We also discuss several problems with the complexity of representations of functions with respect to their ODDs, subfunctions and separable sets. In Section \ref{sec5} we classify discrete functions by transformation groups and equivalence relations concerning the number of implementations, subfunctions and separable sets in functions. In Section \ref{sec6} we use these results to classify all boolean (switching) functions with "small" number of its essential variables. Here we calculate the number of equivalence classes and cardinalities of equivalence classes of boolean functions depending on 3, 4 and 5 variables. \section{Separable and Distributive Sets of Variables}\label{sec2}
We start this section with basic definitions and notation. A discrete function is defined as a mapping: $f:A\to B$ where the domain $A={\times}_{i=1}^n A_i$ and range $B$ are non-empty finite or countable sets.
To derive the means and methods to represent, and calculate with finite valued functions, some algebraic structure is imposed on the domain $A$ and the range $B$. Both $A$ and $B$ throughout the present paper will be finite ring of integers.
Let $X=\{x_1,x_2,\ldots \}$ be a countable set of variables and $X_n=\{x_1,x_2,\ldots,x_n\}$ be a finite subset of $X$. Let $k$, $k\geq 2$ be a natural number
and let us denote by $Z_k=\{0,1,\ldots,k-1\}$ the set (ring) of
remainders modulo $k$. The set $Z_k$ will identify the ring of residue classes $mod\ k$, i.e. $Z_k=Z/_{kZ}$, where $Z$ is the ring of all integers. An {\it $n$-ary $k$-valued function (operation) on $Z_k$ } is a mapping $f: Z_k^n\to Z_k$ for some natural number $n$, called {\it the arity} of $f$. $P_k^n$ denotes the set of all $n$-ary $k$-valued functions on $Z_k$. It is well known fact that there are $k^{k^n}$ functions in $P_k^n$. The set of all $k$-valued functions $P_k=\bigcup_{n=1}^\infty P_k^n$ is called {\it the algebra of $k$-valued logic}.
All results obtained in the present paper can be easily extended to arbitrary algebra of finite operations.
For a given variable $x$ and $\alpha\in Z_k$, $x^\alpha$ is defined as follows: $$
x^\alpha=\left\{\begin{array}{ccc}
1 \ &\ if \ &\ x=\alpha \\
0 & if & x\neq\alpha.
\end{array}
\right. $$
We use {\it sums of products (SP)} to represent the functions from $P_k^n$. This is the most natural representation and it is based on so called operation tables of the functions. Thus each function $f\in P_k^n$ can be uniquely represented of SP-form as follows \[ f=a_0.x_1^{0}\ldots x_n^{0}\oplus\ldots\oplus
a_{m}.x_1^{\alpha_1}\ldots
x_n^{\alpha_n}\oplus\ldots\oplus a_{k^n-1}.x_1^{k-1}\ldots
x_n^{k-1}\] with ${\mathbf{\alpha}}={({\alpha_1,\ldots,\alpha_n})}\in Z_k^n$, where $m=\sum_{i=0}^n\alpha_ik^i\leq k^n-1$. $"\oplus"$ and $"."$ denote the operations addition (sum) and multiplication (product) modulo $k$ in the ring $Z_k$. Then $(a_0,\ldots,a_{k^n-1})$ is the vector of output values of $f$ in its table representation.
Let $f\in P_k^n$ and $var(f)=\{x_1,\ldots,x_n\}$ be the set of all variables, which occur in $f$.
We say that the $i$-th variable $x_i\in var(f)$ is {\it essential} in $f$, or $f$ {\it essentially} {\it depends} on $x_i$, if there exist values $a_1,\ldots,a_n,b\in Z_k$, such that \[
f(a_1,\ldots,a_{i-1},a_{i},a_{i+1},\ldots,a_n)\neq f(a_1,\ldots,a_{i-1},b,a_{i+1},\ldots,a_n). \]
The set of all essential variables in the function $f$ is denoted by $Ess(f)$ and the number of essential variables in $f$ is denoted by
$ess(f)=|Ess(f)|$.
The variables from $var(f)$ which are not essential in
$f$ are called {\it inessential} or {\it fictive}.
The set of all output values of a function $f\in P_k^n$ is called {\it the range} of $f$, which is denoted as follows:
\[range(f)=\{c\in Z_k\ |\ \exists (a_1,\ldots,a_n)\in Z_k^n, \quad such\ that\quad f(a_1,\ldots,a_n)=c\}.\] \begin{definition}\label{d1.2} Let $x_i$ be an essential variable in $f$ and $c\in Z_k$ be a constant from $Z_k$. The function $g=f(x_i=c)$ obtained from $f\in P_{k}^{n}$ by replacing the variable $x_i$ with $c$ is called a {\it simple subfunction of $f$}.
When $g$ is a simple subfunction of $f$ we shall write $g\prec f$. The transitive closure of $\prec$ is denoted by $\preceq$. $Sub(f)=\{g\ | \ g\preceq f\}$ is the set of all subfunctions of $f$ and $sub(f)=|Sub(f)|$. \end{definition}
For each $m=0,1,\ldots, n$ we denote by $sub_m(f)$ the number of subfunctions in $f$ with $m$ essential variables, i.e. $sub_m(f)=|\{g\in Sub(f)\ |\ ess(g)=m\}|$.
Let $g\preceq f$, $\mathbf{c}=(c_1,\ldots,c_m)\in Z_k^m$ and $M=\{x_1,\ldots,x_m\}\subset X$ with \[g\prec g_1\prec\ldots\prec g_m=f,\quad g=g_1(x_1=c_1)\quad and \quad g_i=g_{i+1}(x_{i+1}=c_{i+1})\] for $i=1,\ldots,m-1$. Then we shall write $g=f(x_1=c_1,\ldots,x_m=c_m)$ or equivalently, $g\preceq_M^{\mathbf{c}} f$ and we shall say that the vector $\mathbf{c}$ {\it determines} the subfunction $g$ in $f$.
\begin{definition}\label{d1.3} Let $M$ be a non-empty set of essential variables in the function $f$. Then $M$ is called {\it a separable set} in $f$ if there exists a subfunction $g$, $g\preceq f$ such that $M=Ess(g)$.
$Sep(f)$ denotes the set of the all separable sets in $f$ and $sep(f)=|Sep(f)|$. \end{definition}
The sets of essential variables in $f$ which are not separable are called {\em inseparable} or {\em non-separable}.
For each $m= 1,\ldots, n$ we denote by $sep_m(f)$ the number of separable sets in $f$ which consist of $m$ essential variables, i.e. $sep_m(f)=|\{M\in Sep(f)\ |\ |M|=m\}|$. The numbers $sub(f)$ and $sep(f)$ characterize the computational complexity of the function $f$ when calculating its values. Our next goal is to classify the functions from $P_k^n$ under these complexities. The initial and more fundamental results concerning essential variables and separable sets were obtained in the work of Y. Breitbart \cite{bre}, K. Chimev \cite{ch51}, O. Lupanov \cite{lup}, A. Salomaa \cite{sal}, and others. ~
\begin{remark}\label{r1.1} Note that if $g\preceq f$ and $x_i\notin Ess(f)$ then $x_i\notin Ess(g)$ and also if $M\notin Sep(f)$ then $M\notin Sep(g)$. \end{remark}
\begin{definition}\label{d2.1} Let $M$ and $J$ be two non-empty sets of essential variables in the function $f$ such that $M\cap J=\emptyset.$ The set $J$ is called {\it a distributive set of $M$ in $f$,
} if for every $|J|$-tuple of constants $\mathbf{c}$ from $Z_k$ it holds $M\not\subseteq Ess(g)$, where $g\preceq_J^{\mathbf{c}} f$
and $J$ is minimal with respect to the order $\subseteq$.
$Dis(M,f)$ denotes the set of the all distributive sets of $M$ in $f.$ \end{definition}
It is clear that if $M\notin Sep(f)$ then $Dis(M,f)\neq\emptyset$. So, the distributive sets ``dominate'' on the inseparable sets of variables in a function. We are interested in the relationship between the structure of the distributive sets of variables and complexity of functions concerning $sep(f)$ and $sub(f)$, respectively, which is illustrated by the following example.
\begin{example}\label{ex2.1}
Let $k=2$, $f=x_1x_2\oplus x_1x_3$ and $g=x_1x_2\oplus x_1^0x_3.$ It is easy to verify that the all three pairs of variables $\{x_1,x_2\}$, $\{x_1,x_3\}$ and $\{x_2,x_3\}$ are separable in $f$, but $\{x_2,x_3\}$ is inseparable in $g$. {Figure} \ref{f1} presents graphically, separable pairs in $f$ and $g$. The set $\{x_1\}$ is distributive of $\{x_2,x_3\}$ in $g$. \end{example} \begin{figure}
\caption{Separable pairs.}
\label{f1}
\end{figure}
\begin{definition}\label{d2.2} Let $\mathcal F=\{P_1,\ldots,P_m\}$ be a family of non-empty sets. A set $\beta=\{x_1,\ldots,x_p\}$ is called {\it an $s$-system} of $\mathcal F$, if for all $P_i\in \mathcal F$, $1\leq i\leq m$ there exists $x_j\in\beta$ such that $x_j\in P_i$ and for all $x_s\in\beta$ there exists $P_l\in \mathcal F$ such that $\{x_s\}=P_l\cap\beta.$ $Sys(\mathcal F)$ denotes the set of the all $s$-systems of the family $\mathcal F$. \end{definition}
Applying the results concerning $s$-systems of distributive sets is one of the basic tools for studying inseparable pairs and inseparable sets in functions. These results are deeply discussed in \cite{ch51,s27,s23}.
\begin{theorem}\label{t2.1}\ Let $M\subseteq Ess(f)$ be a non-empty inseparable set of essential variables in $f\in P_k^n$ and $\beta\in Sys(Dis(M,f))$. Then the following statements hold: \begin{enumerate}
\item[(i)] $M\cup \beta \in Sep(f)$;
\item[(ii)] $(\ \forall \alpha,\ \alpha\subseteq\beta,\ \alpha\neq\beta)\quad M\cup \alpha\notin Sep(f)$. \end{enumerate} \end{theorem} \begin{proof} $(i)$\
First, note that $M\notin Sep(f)$ implies $|M|\geq 2$. Without loss of generality assume that $\beta=\{x_1,\ldots,x_m\}\in Sys(Dis(M,f))$ and $M=\{x_{m+1},\ldots,x_{p}\}$ with $1\leq m<p\leq n$. Let us denote by $L$ the following set of variables $L=Ess(f)\setminus(M\cup\beta )=\{x_{p+1},\ldots,x_{n}\}.$
Since $\beta\in Sys(Dis(M,f))$ it follows that for each $Q\subseteq L$ we have $Q\notin Dis(M,f)$. Hence there is a vector $\mathbf{c}=(c_{p+1},\ldots,c_n)\in Z_k^{n-p}$ such that $M\subseteq Ess(g)$ where $g\preceq_L^{\mathbf{c}} f$.
Next, we shall prove that
$\beta\subset Ess(g).$
For suppose this were not the case and without loss of generality, assume $x_1\notin Ess(g)$, i.e. $g=g(x_1=d_1)$ for each $d_1\in Z_k.$ Let $J\in Dis(M,f)$ be a distributive set of $M$ such that $J\cap\beta=\{x_1\}.$ The existence of the set $J$ follows because $\beta$ is an $s$-system of $Dis(M,f)$ (see Definition \ref{d2.2}). Since $J\cap M=\emptyset$ and
$x_1\notin Ess(g)$ it follows that $J\cap Ess(g)=\emptyset.$ Now $M\subset Ess(g)$ implies that $J\notin Dis(M,f)$, which is a contradiction. Thus we have $x_1\in Ess(g)$ and $\beta\subset Ess(g)$. Then $Ess(f)\setminus L=M\cup\beta$ shows that $M\cup\beta=Ess(g)$ and hence $M\cup\beta\in Sep(f).$
$(ii)$\ Let $\alpha$, $\alpha\subseteq\beta,\ \alpha\neq\beta$ be a proper subset of $\beta$. Let $x_i\in \beta\setminus \alpha$. Then $\beta\in Sys(Dis(M,f))$ implies that there
is a distributive set $P\in Dis(M,f)$ of $M$ such that $P\cap\beta=\{x_i\}$. Hence $P\cap\alpha=\emptyset$ which shows that there is an non-empty distributive set $P_1$ for $M\cup\{\alpha\}$ with $P_1\subseteq P$. Hence
$M\cup\alpha\notin Sep(f)$.
\end{proof} \begin{corollary}\label{c2.1} Let $\emptyset\neq M\subset Ess(f)$ and $M\notin Sep(f)$. If $\beta\in Sys(Dis(M,f))$ and $x_i\in\beta$ then $M\setminus Ess(f(x_i=c))\neq\emptyset$ for all $c\in Z_k$. \end{corollary}
\begin{theorem}\label{t2.2}\cite{s23} For each finite family $\mathcal F$ of non-empty sets there exists at least one $s-$system of $\mathcal F$. \end{theorem}
\begin{theorem}\label{t2.3} Let $M$ be an inseparable set in $f$. A set $\beta\subset Ess(f)$ is an $s$-system of $Dis(M,f)$ if and only if $\beta\cap J\neq\emptyset$ for all $J\in Dis(M,f)$ and $\alpha\subseteq\beta,\ \alpha\neq\beta$ implies $\alpha\cap P=\emptyset$ for some $P\in Dis(M,f)$.
\end{theorem} \begin{proof} "$\Leftarrow$" Let $\beta\cap J\neq \emptyset$ for all $J\in Dis(M,f)$ and $\alpha\varsubsetneqq \beta$ implies $\alpha\cap P=\emptyset$ for some $P\in Dis(M,f)$. Since $\beta\cap J\neq \emptyset$ it follows that there is a set $\beta'$, $\beta'\subset \beta\subset Ess(f)$ and $\beta'\in Sys(Dis(M,f))$. If we suppose that $\beta'\neq\beta$ then there is $P\in Dis(M,f)$ with $\beta'\cap P=\emptyset$. Hence $M\cup\beta'\notin Sep(f)$ because of $P\in Dis(M\cup\beta',f)$ which contradicts Theorem \ref{t2.1}.
"$\Rightarrow$" Let $\beta$ be an $s$-system of $Dis(M,f)$ and $\alpha\varsubsetneqq \beta$. Let $x\in \beta\setminus\alpha$ and $P\in Dis(M,f)$ be a distributive set of $M$ for which $\beta\cap P=\{x\}$. Hence $\alpha\cap P=\emptyset$ and we have $P\in Dis(M\cup \alpha,f)$ and $M\cup\alpha\notin Sep(f)$ which shows that $\alpha\notin Sys(Dis(M,f))$. ~~~~~
\end{proof}
\section{Ordered Decision Diagrams and Complexity of Functions}\label{sec3}
The distributive sets are also important when constructing efficient procedures for simplifying in analysis and synthesis of functional schemas.
In this section we discuss {\it ordered decision diagrams} (ODDs) for the functions obtained by restrictions on their {\it ordered decomposition trees} (ODTs).
{Figure} \ref{f2} shows an ordered decomposition tree for the function $g=x_1x_2\oplus x_1^0x_3\in P_2^3$ from Example \ref{ex2.1}, which essentially depends on all its three variables $x_1, x_2$ and $x_3$. The node at the top, labelled $g$ - is the {\it function} node. The nodes drawn as filled circles labelled with variable names are the {\it internal (non-terminal)} nodes, and the rectangular nodes (leaves of the tree) are the {\it terminal} nodes. The terminal nodes are labelled by the numbers from $Z_k$. Implementation of $g$ for a given values of $x_1, x_2$ and $x_3$ consists of selecting a path from the function node to a terminal node. The label of the terminal node is the sought value. At each non-terminal node the path follows the solid edge if the variable labelling the node evaluates to $1$, and the dashed edge if the variable evaluates to $0$. In the case of $k>2$ we can use colored edges with $k$ distinct colors.
The ordering in which the variables appear is the same along all paths of an ODT. {Figure} \ref{f2} shows the ODT for the function $g$ from Example \ref{ex2.1}, corresponding to the variable ordering $x_1,x_2,x_3$ (denoted briefly as $\langle 1; 2; 3\rangle$). It is known that for a given function $g$ and a given ordering of its essential variables there is a unique ODT.
We extend our study to ordered decision diagrams for the functions from $P_k^n$ which were studied by D. Miller and R. Drechsler \cite{mil1,mil2}.
\begin{figure}
\caption{Decomposition tree for $g=x_1x_2\oplus x_1^0x_3.$}
\label{f2}
\end{figure}
An {\it ordered decision diagram} of a function $f$ is obtained from the corresponding ODT by {\it reduction} of its nodes applying of the following two rules starting from the ODT and continuing until neither rule can be applied:
{\bf Reduction rules} \begin{enumerate} \item[$\bullet$] If two nodes are terminal and have the same label, or are non-terminal and have the same children, they are merged. \item[$\bullet$] If an non-terminal node has identical children it is removed from the graph and its incoming edges are redirected to the child. \end{enumerate}
When $k=2$ ODD is called {\it a binary decision diagram} (BDD). BDDs are extensively used in the theory of {\it switching circuits} to represent and manipulate Boolean functions and to measure the complexity of binary terms.
The size of the ODD is determined both by the function being represented and the chosen ordering of the variables. It is of crucial importance to care about variable ordering when applying ODDs in practice. The problem of finding the best variable ordering is NP-complete (see \cite{bra}).
{Figure} \ref{f3} shows the BDDs for the functions from Example \ref{ex2.1} obtained from their decomposition trees under the natural ordering of their variables - $\langle 1; 2; 3\rangle$. The construction of the ODT for $f$ under the natural ordering of the variables is left to the reader.
\begin{figure}
\caption{BDD for $f$ and $g$ under the natural ordering of variables.}
\label{f3}
\end{figure}
The BDD of $f$ is more complex than the BDD of $g$. This reflects the fact that $f$ has more separable pairs.
Thus we have $M=\{x_2,x_3\}\notin Sep(g)$, $\{x_1\}\in Dis(M,g)$ and $\{x_1\}\in Sys(Dis(M,g))$. Additionally, the diagram of $g$ starts with $x_1$ - a variable which belongs to an $s$-system of $Dis(M,g)$. In this simple case we have $Sys(Dis(M,g))= Dis(M,g)=\{x_1\}$.
Figure \ref{f3} shows that when constructing the ODD of a function, it is better to start with the variables from an $s$-system of the family of distributive sets of an inseparable set $M$ in this function. In \cite{ivo2} it is shown that the BDDs of functions have to be most simple when starting with variables from $Sys(Dis(M,f))$.
Consequently, the inseparable sets with their distributive sets are important in theoretical and applied computer science concerning the computational complexity of the functions.
Next, we define and explore complexity measures of the functions in $P_k^n$ which are directly connected with the computational complexity of functions. We might think that the complexity of a function $f$ depends on the complexity of its ODDs.
Let ${f\in P_k^n}$ and let $DD(f)$ be the set of the all ODDs for $f$ constructed under different variable orderings in $f$.
\begin{definition}\label{d3.2} Each path starting from the function node and finishing into a terminal node is called an {\it implementation} of the function $f$ under the given variable ordering.
The set of the all implementations of $D_f$ we denote by $Imp(D_f)$ and \[Imp(f)=\bigcup_{D_f\in DD(f)}Imp(D_f).\]
\end{definition}
Each implementation of the function ${f\in P_k^n}$, obtained from the diagram $D_f$ of $f$ by the non-terminal nodes $x_{i_1},\ldots,x_{i_r}$ and corresponding constants $c_{1},\ldots,c_{r},c\in Z_k$ with $f(x_{i_1}=c_{1},\ldots,x_{i_r}=c_{r})=c,\quad r\leq ess(f)$,
can be represented as a pair $\mathbf{(i,c)}$ of two words (strings) over $\mathbf{n}=\{1,\ldots,n\}$ and $Z_k$ where $\mathbf{i}=i_1i_2\ldots i_r\in \mathbf{n}^*$ and $\mathbf{c}=c_1c_2\ldots c_rc\in Z_k^*$.
There is an obvious way to define a measure of complexity of a given ordered decision diagram $D_f$, namely as the number ${imp}(D_f)$ of all paths in $D_f$ which starts from the function node and finish in a terminal node of the diagram.
The {\it implementation complexity} of a function ${f\in P_k^n}$ is defined as the number of all implementations of $f$, i.e.
$imp(f)=|Imp(f)|.$
We shall study also two other measures of computational complexity of functions as $sub(f)$ and $sep(f)$.
\begin{example}\label{ex3.1} Let us consider again the functions $f$ and $g$ from Example \ref{ex2.1}, namely $f=x_1x_2\oplus x_1x_3\quad and\quad g=x_1x_2\oplus x_1^0x_3.$
Then $(123, 1011)$ is an implementation of $f$ obtained by the diagram $D_f$ presented in {Figure} \ref{f3}, following the path $\pi= (f; x_1: 1; x_2: 0; x_3: 1;terminal\ node: 1)$.
It is easy to see that there are six distinct BDDs for $f$ and five distinct BDDs for $g$. We shall calculate the implementations of $f$ and $g$ for the variable orderings $\langle 1; 2; 3\rangle$ (see Figure \ref{f3}) and $\langle 2; 1; 3\rangle$, only. Thus for $f$ we have:
\noindent
\begin{tabular}{|l|c|} \hline
ordering & implementations\\ \hline
$\langle 1; 2; 3\rangle$ & $(1,00);\ (123,1000);\ (123,1011);\ (123,1101);\ (123, 1110)$ \\ \hline $\langle 2; 1; 3\rangle$& $(21,000);\ (213,0100);\ (213,0111);\ (21,100); (213,1101);\ (213,1110)$ \\ \hline \end{tabular}
For the function $g$ we obtain:
\noindent
\begin{tabular}{|l|c|} \hline
ordering &implementations\\ \hline
$\langle 1; 2; 3\rangle$ & $(13,000);\ (13,011);\ (12,100);\ (12,111)$\\ \hline $\langle 2; 1; 3\rangle$& $(21,010);\ (213,0000);\ (213,0011);\ (213,1000); (213,1011);\ (21,111)$\\ \hline \end{tabular}
For the diagrams in
{Figure} \ref{f3} we have ${imp}(D_f)=5$ and ${imp}(D_g)=4$.
Since $f$ is a symmetric function with respect to $x_2$ and $x_3$ one can count that $imp(f)=33$. Note that the implementation $(1,00)$ occurs in two distinct diagrams of $f$, namely under the orderings $\langle 1; 2; 3\rangle$ and $\langle 1; 3; 2\rangle$. Hence, it has to be counted one time and we obtain that $imp(f)$ is equal to $33$ instead of $34$.
For the function $g$, the diagrams under the orderings $\langle 1; 2; 3\rangle$ and $\langle 1; 3; 2\rangle$ have the same implementations, i.e. the diagrams are identical (isomorphic). This fact is a consequence of inseparability of the set $\{x_2,x_3\}$. Hence $g$ has five (instead of six for $f$) distinct ordered decision diagrams. Then, one might calculate that $imp(g)=28$.
For the other two measures of complexity we obtain: $sub(f)=13$ because of $Sub(f)=\{0,1,x_1,x_2,x_3,x_2^0,x_3^0,x_2\oplus x_3,x_1x_2,$ $ x_1x_2^0,x_1x_3,x_1x_3^0, f\}$ and $sub(g)=11$ because of $Sub(g)=\{0,1,x_1,x_2,x_3,x_1^0,x_1 x_2,x_1^0x_3,x_1\oplus x_1^0x_3,x_1x_2\oplus x_1^0,g\}$. Furthermore, $sep(f)=7$ because of
$M\in Sep(f)$ for all $M$, $\emptyset\neq M\subseteq \{x_1,x_2,x_3\}$ and $sep(g)=6$ because of $Sep(g)=\{\{x_1\},\{x_2\},\{x_3\},\{x_1,x_2\},\{x_1,x_3\},\{x_1,x_2,x_3\}\}$. \end{example}
\begin{lemma}\label{l17} A variable $x_i$ is essential in $f\in P_k^n$ if and only if $x_i$ occurs as a label of an non-terminal node in any ODD of $f$. \end{lemma} \begin{proof} $"\Rightarrow"$ Let us assume that $x_i$ does not occur as a label of any non-terminal node in an ordered decision diagram $D_f$ of $f$. Since all values of the function $f$ can be obtained by traversal walk-trough all paths in $D_f$ from function node to leaf nodes this will mean that $x_i$ will not affect the function value and hence $x_i$ is an inessential variable in $f$.
$"\Leftarrow"$ Let $x_i\notin Ess(f)$ be an inessential variable in $f$. It is obvious that for each subfunction $g$ of $f$ we have $x_i\notin Ess(g)$. Then we have $f(x_i=c)=f(x_i=d)$ for all $c,d\in Z_k$. Consequently, if there is a non-terminal node labelled by $x_i$ in an ODT of $f$ then its children have to be identical, which shows that this node has to be removed from the ODT, according to the reduction rules, given above. \end{proof}
An essential variable $x_i$ in a function $f$ is called {\it a strongly essential variable } in $f$ if there is a constant $c\in Z_k$ such that $Ess(f(x_i=c))=Ess(f)\setminus\{x_i\}$.
\begin{fact}\label{fc1} If $ess(f)\geq 1$ then there is at least one strongly essential variable in $f$. \end{fact} This fact was proven by O. Lupanov \cite{lup} in case of Boolean functions and by A. Salomaa \cite{sal} for arbitrary functions.
Later, Y. Breitbart \cite{bre} and K. Chimev \cite{ch51} proved that if $ess(f)\geq 2$ then there exist at least two strongly essential variables in $f$.
We need Fact \ref{fc1} to prove the next important theorem.
\begin{theorem}\label{t3.2} A non-empty set $M$ of essential variables is separable in $f$ if and only
if there exists an implementation $\mathbf{(j,c)}$ of the form \[\mathbf{(j,c)}=(j_1j_2\ldots j_{r-m}j_{r-m+1}\ldots j_r, c_1c_2\ldots c_{r-m}c_{r-m+1}\ldots c_r c)\in Imp(f)\] where $M=\{x_{j_{r-m+1}},\ldots, x_{j_r}\}$ and $1\leq m\leq r\leq ess(f)$. \end{theorem} \begin{proof} "$\Leftarrow$" Let \[\mathbf{(j,c)}=(j_1\ldots j_{r-m}j_{r-m+1}\ldots j_r, c_1\ldots c_{r-m}c_{r-m+1}\ldots c_rc)\in Imp(f)\] be an implementation of $f$ and let $M=\{x_{j_{r-m+1}},\ldots, x_{j_r}\}$. Hence the all variables from $\{x_{j_{r-m+1}},\ldots,x_{j_r}\}$ are essential in the following subfunction of $f$ \[g=f(x_{j_1}=c_{1},\ldots,x_{j_{r-m}}=c_{{r-m}})\] which shows that $M\in Sep(f)$.
"$\Rightarrow$" Without loss of generality let us assume that $M=\{x_1,\ldots,x_m\}$ is a non-empty separable set in $f$ and $n=ess(f)$. Then there are constants $d_{m+1},\ldots,d_n\in Z_k$ such that $M=Ess(h)$ where $h=f(x_{m+1}=d_{m+1},\ldots,x_{n}=d_{n})$. From Fact \ref{fc1} it follows that there is a variable $x_{i_1}\in M$ and a constant $d_{1}\in Z_k$ such that $Ess(h_1)=M\setminus \{x_{i_1}\}$ where $h_1=h(x_{i_1}=d_{1})$. Consequently, we might inductively obtain that there are variables $x_{i_r}\in M$ and constants $d_{r}\in Z_k$ for $r=2,\ldots,m$, such that $Ess(h_r)=M\setminus \{x_{i_1},\ldots,x_{i_r}\}$ where $h_r=h_{r-1}(x_{i_r}=d_{r})$. Hence, the string $m+1m+2\ldots n$ has a substring $j_1\ldots j_s$ such that $(j_1\ldots j_si_1\ldots i_m, d_{j_1}\ldots d_{j_s}d_{1}\ldots d_{m}d)$ is an implementation of $f$ with $M=\{x_{i_1},\ldots,x_{i_m}\}$ and $d=h_m$. \end{proof} \begin{corollary}\label{c3.2} For each variable $x_i\in Ess(f)$ there is an implementation $\mathbf{(j,c)}$ of $f$ whose last letter of $\mathbf{j}$ is $i$, i.e. $\mathbf{(j,c)}=(j_1\ldots j_{m-1}i, c_{j_1}\ldots c_{j_{m-1}}c_ic)\in Imp(f)$, $m\leq ess(f)$. \end{corollary} Note that there exists an ODD of a function whose non-terminal nodes are labelled by the variables from a given set, but this set might not be separable. For instance, the implementation $(231,0101)\in Imp(g)$ of the function $g$ from Example \ref{ex3.1} shows that the variables from the set $M=\{x_2,x_3\}$ occur as labels of the starting two non-terminal nodes in the BDD of $g$ under the ordering $\langle 2; 3; 1 \rangle$, but $M\notin Sep(g)$.
\begin{lemma}\label{l2.1} If $ess(f)=n$, $g\preceq f$ with $ess(g)= m<n$ then there exists a variable $x_t\in Ess(f)\setminus Ess(g)$ such that $Ess(g)\cup\{x_t\}\in Sep(f)$.
\end{lemma} \begin{proof} Let $M=Ess(g)$. Then $M\in Sep(f)$ and from Theorem \ref{t3.2} it follows that there is an implementation $\mathbf{(j,c)}$ of the form $\mathbf{(j,c)}=(j_1j_2\ldots j_{r-m}j_{r-m+1}\ldots j_r,$ $c_1c_2\ldots c_{r-m}c_{r-m+1}\ldots c_r c)\in Imp(f)$ where $M=\{x_{j_{r-m+1}},\ldots, x_{j_r}\}$ and $1\leq m\leq r\leq ess(f)$. Since $m<n$ it follows that $r-m>0$ and Lemma \ref{l17} shows that there is $x_{j_i}\in Ess(h)$ where \[h=f(x_{j_1}=c_1,\ldots,x_{j_{i-1}}=c_{i-1},x_{j_{i+1}}=c_{i+1},\ldots x_{j_{r-m}}=c_{r-m}).\] It is easy to see that $Ess(h)=M\cup\{x_{j_i}\}$. \end{proof}
Now, as an immediate consequence of the above lemma we obtain Theorem \ref{t2.4} which was inductively proven by K. Chimev. \begin{theorem}\label{t2.4} \cite{ch51} If $ess(f)=n$, $g\preceq f$ with $ess(g)= m\leq n$ then there exist $n-m$ subfunctions $g_1,\ldots,g_{n-m}$ such that \[g \prec g_1 \prec g_2\prec \ldots\prec g_{n-m}= f\] and $ess(g_i)=m+i$ for $i=1,\ldots,n-m$. \end{theorem}
The {\it depth}, (denoted by $Depth(D_f)$) of an ordered decision diagram $D_f$ for a function $f$ is defined as the number of the edges in a longest path from the function node in $D_f$ to a leaf of $D_f$.
Thus for the diagrams in Figure \ref{f3} we have $Depth(D_f)=4$ and $Depth(D_g)=3$.
Clearly, if $ess(f)=n$ then $Depth(D_f)\leq n+1$ for all ODDs of $f$.
\begin{theorem}\label{l3.2} If $ess(f)=n\geq 1$ then there is an ordered decision diagram $D_f$ of $f$ with $Depth(D_f)=n+1$. \end{theorem} \begin{proof} Let $Ess(f)=\{x_1,\ldots,x_n\}$, $n\geq 1$. Since $x_1$ is an essential variable it follows that $\{x_1\}\in Sep(f)$. Theorem \ref{t2.4} implies that there is an ordering $\langle i_1; i_2; \ldots; i_{n-1}\rangle$ of the rest variables $x_{2}, \ldots, x_{{n}}$ such that for each $j$, $1\leq j\leq n-1$ we have $g_j\prec_J^{\mathbf{c}} f$ where $J=\{x_{i_1},\ldots,x_{i_j}\}$, $\mathbf{c}\in Z_k^{J}$ and $Ess(g_j)=\{x_1,x_{i_{j+1}},\ldots,x_{i_{n-1}}\}$. This shows that the all variables from $J$ have to be labels of non-terminal nodes in a path $\pi$ of the ordered decision diagram $D_f$ of $f$ under the variable ordering $\langle i_1; i_2; \ldots; i_{n-1}; 1\rangle$. Hence $\pi$ has to contain all essential variables in $f$ as labels at the non-terminal nodes of $\pi$. Hence $Depth(D_f)=n+1$. \end{proof}
\begin{theorem}\label{t3.3} Let $f\in P_k^n$ and $Ess(f)=\{x_1,\ldots,x_n\}$, $n\geq 1$. If $M\neq \emptyset$, $M\subset Ess(f)$ and $M\notin Sep(f)$ then there is a decision diagram $D_f$ of $f$ with $Depth(D_f)<n+1$. \end{theorem} \begin{proof} Without loss of generality, let us assume that $M=\{x_1,\ldots,x_m\}$, $m<n$. Since $M$ is inseparable in $f$, the family $Dis(M,f)$ of the all distributive sets of $M$ is non-empty. According to Theorem \ref{t2.2} there is a non-empty $s$-system $\beta=\{x_{i_1},\ldots,x_{i_t}\}$ of $Dis(M,f)$. Since $f(x_{i_1}=c_1)\neq f(x_{i_1}=c_2)$ for some $c_1,c_2\in Z_k$ it follows that there exists an ODD $D_f$ for $f$ under a variable ordering with $x_{i_1}$ as the label of the first non-terminal node of $D_f$. According to Corollary \ref{c2.1} for all $c\in Z_k$ there is a variable $x_j\in M$ which is inessential in $f(x_{i_1}=c)$. Hence, each path of $D_f$ does not contain at least one variable from $M$ among its labels of non-terminal nodes. Hence $Depth(D_f)<n+1$. \end{proof}
\section{Equivalence Relations and Transformation Groups in $P_k^n$}\label{sec5}
Many of the problems in applications of the $k$-valued functions are compounded because of the large number of the functions, namely $k^{k^n}$. Techniques which involve enumeration of functions can only be used if $k$ and $n$ are trivially small. A common way for extending the scope of such enumerative methods is to classify the functions into equivalence classes under some natural equivalence relation.
In this section we define equivalence relations in $P_k^n$ which classify functions with respect to number of their implementations, subfunctions and separable sets. We are intended to determine several numerical invariants of the transformation groups generated by these relations. The second goal is to compare these groups with so called classical subgroups of the Restricted Affine Group(RAG) \cite{lech} which have a variety of applications such as coding theory, switching theory, multiple output combinational logic, sequential machines and other areas of theoretical and applied computer sciences.
Let us denote by $S_A$ the symmetric group of all permutations of a given no-empty set $A$. $S_m$ denotes the symmetric group $S_{\{1,\ldots,m\}}$ for a natural number $m$, $m\geq 1$.
Let us define the following three equivalence relations: $\simeq_{imp}$, $\simeq_{sub}$ and $\simeq_{sep}$. \begin{definition}\label{d5.1} Let $f,g\in P_k^n$ be two functions. \begin{enumerate} \item[(i)] If $ess(f)=ess(g)\leq 1$ then $f\simeq_{imp} g$; \item[(ii)] Let $ess(f)=n>1$. We say that $f$ is $imp$-equivalent to $g$ (written $f\simeq_{imp} g$) if there are $\pi\in S_n$ and $\sigma_i\in S_{Z_k}$ such that $f(x_i=j)\simeq_{imp} g(x_{\pi(i)}=\sigma_i(j))\quad\mbox{for all}\quad i=1,\ldots,n\quad\mbox{and}\quad j\in Z_k.$ \end{enumerate} \end{definition} Hence two functions are ${imp}$-equivalent if they produce same number of implementations, i.e.
$imp(f)=imp(g)$ and there are $\pi\in S_n$, and $\sigma$, $\sigma_i\in S_{Z_k}$ such that $(i_1\ldots i_m,c_{1},\ldots,c_{m}c)\in Imp(f)\iff $ $ (\pi(i_1)\ldots \pi(i_m), \sigma_1(c_{1})\ldots\sigma_m(c_{m})\sigma(c))\in Imp(g).$
Table \ref{tb1} shows the classification of Boolean functions of two variables into four classes, called {\it imp-classes} under the equivalence relation $\simeq_{imp}$. The second column shows the number of implementations of the functions from the $imp$-classes given at the first column. The third column presents the number of functions per each $imp$-class.
\begin{table}[h] \caption{$Imp$-classes in $P_2^2$.} \label{tb1}
\centering
\begin{tabular}{l|l|l} \hline\hline
$[\ 0,\ 1\ ]$ & \ 1& 2\\ \hline $[\ x_1,\ x_2,\ x_1^0,\ x_2^0\ ]$&\ 2& 4\\ \hline $[\ x_1x_2,\ x_1x_2^0,\ x_1^0x_2,\ x_1^0x_2^0,\ x_1\oplus x_1x_2,$ &&\\ $ x_2^0\oplus x_1x_2,\ x_1^0\oplus x_1x_2,\ x_1^0\oplus x_1x_2^0\ ]$& \ 6 & 8\\ \hline $[\ x_1\oplus x_2,\ x_1\oplus x_2^0\ ]$&\ 8& 2\\ \hline\hline \end{tabular} \end{table}
\begin{definition}\label{d5.2} Let $f,g\in P_k^n$ be two functions. \begin{enumerate} \item[(i)] If $ess(f)=ess(g)=0$ then $f\simeq_{sub} g$; \item[(ii)] If $ess(f)=ess(g)=1$ then $f\simeq_{sub} g\iff range(f)=range(g)$; \item[(iii)] Let $ess(f)=n>1$. We say that $f$ is $sub$-equivalent to $g$ (written $f\simeq_{sub} g$) if $sub_m(f)=sub_m(g)$ for all $m=0,1,\ldots, n$. \end{enumerate} \end{definition} It is easy to see that the equivalence relation $\simeq_{sub}$ partitions the algebra of Boolean functions of two variables in the same equivalence classes (called {\it the sub-classes}) as the relation $\simeq_{imp}$ (see Table \ref{tb1}). \begin{definition}\label{d5.3} Let $f,g\in P_k^n$ be two functions. \begin{enumerate} \item[(i)] If $ess(f)=ess(g)\leq 1$ then $f\simeq_{sep} g$; \item[(ii)]Let $ess(f)=n>1$. We say that $f$ is $sep$-equivalent to $g$ (written $f\simeq_{sep} g$) if $sep_m(f)=sep_m(g)$ for all $m=1,\ldots, n$. \end{enumerate} \end{definition} The equivalence classes under $\simeq_{sep}$ are called {\it sep-classes}.
Since $P_k^n$ is a finite algebra of $k$-valued functions each equivalence relation $\simeq$ on $P_k^n$ makes a partition of the algebra in the set of disjoint equivalence classes $Cl(\simeq)=\{P_1^\simeq,\ldots,P_r^\simeq\}$. Then, in the set of all equivalence relations a partial order is defined as follows: $\simeq_1\ \leq\ \simeq_2$ if for each $P\in Cl(\simeq_1)$ there is a $Q\in Cl(\simeq_2)$ such that $P\subseteq Q$. Thus $\simeq_1\ \leq\ \simeq_2$ if and only if $f\simeq_1 g\ \Rightarrow f\simeq_2 g$, for $f,g\in P_k^n$.
\begin{theorem}\label{t5.1}
~~~ \begin{enumerate} \item[(i)] $ \simeq_{imp}\ \leq\ \simeq_{sep}$; \quad (iii) $\simeq_{imp}\ \not\leq\ \simeq_{sub}$; \item[(ii)] $ \simeq_{sub}\ \leq\ \simeq_{sep}$; \quad (iv) $ \simeq_{sub}\ \not\leq\ \simeq_{imp}$. \end{enumerate}
\end{theorem}
\begin{proof} (i)\ Let $f,g\in P_k^n$ be two ${imp}$-equivalent functions, i.e. $f\simeq_{imp} g$. We shall proceed by induction on the number $n=ess(f)$ of essential variables in $f$ and $g$.
Clearly, if $n\leq 1$ then $f\simeq_{sub} g$, which is our inductive basis. Let us assume that $f\simeq_{imp} g$ implies $f\simeq_{sep} g$ if $n< r$ for some natural number $r$, $r\geq 2$.
Let $f$ and $g$ be two functions with $f\simeq_{imp}g$ and $ess(f)=ess(g)=r$. Then there are $\pi\in S_r$ and $\sigma_i\in S_{Z_k}$ for $i=1,\ldots,r$ such that $f(x_i=j)\simeq_{imp} g(x_{\pi(i)}=\sigma_i(j))$. Let $M$, $\emptyset\neq M\in Sep(f)$ be a separable set of essential variables in $f$ with $|M|=m$, $1\leq m\leq r$. Theorem \ref{t3.2} implies that there is an implementation \[\mathbf{(j,c)}=(j_1\ldots j_{r-m} i_1\ldots i_m, c_{j_1}\ldots c_{j_{r-m}}c_{i_1}\ldots c_{i_m}c)\] of $f$ obtained under an ODD whose variable ordering finishes with the variables from $M$, i.e. $M=\{x_{i_1}\ldots,x_{i_m}\}$. Then $f(x_{j_1}=c_{j_1})\simeq_{imp} g(x_{\pi(j_1)}=\sigma_{j_1}(c_{j_1}))$ implies that \[({\pi({j_1})}\ldots {\pi({j_{r-m}})}{\pi({i_1})}\ldots {\pi({i_m})},\sigma_{j_1}(c_{j_1})\ldots \sigma_{j_{r-m}}(c_{j_{r-m}})\sigma_{i_1}(c_{i_1})\ldots \sigma_{i_m}(c_{i_m})\sigma(c))\]
is an implementation of $g$, for some $\sigma\in S_{Z_k}$. Again, from Theorem \ref{t3.2} it follows that $\pi(M)=\{x_{\pi(i_1)},\ldots,x_{\pi(i_m)}\}\in Sep(g).$ Since $\pi$ is a permutation of $S_r$ it follows that $sep_m(f)=sep_m(g)$ for $m=1,\ldots,r$ and hence $\simeq_{imp}\ \leq\ \simeq_{sep}.$
(ii) \
Definition \ref{d1.3} shows that $M\in Sep(f)$ if and only if there is a subfunction $g\in Sub(f)$ with $g\prec_Q^{\mathbf{c}}f$ where $Q=Ess(f)\setminus M$ and $\mathbf{c}\in Z_k^{n-|M|}$. Hence \[\forall f,g\in P_k^n,\quad Sub(f)=Sub(g)\implies Sep(f)=Sep(g),\] which implies that $sub_m(f)=sub_m(g)\implies sep_m(f)=sep_m(g)$ and $\simeq_{sub}\leq \simeq_{sep}$.
(iii)\ Let us consider the functions \[f=x_1^0x_2x_3\oplus x_1x_2^0x_3^0\ (mod\ 2)\quad\mbox{and}\quad g=x_2x_3\oplus x_1x_2^0x_3\oplus x_1x_2x_3^0\ (mod\ 2).\]
The set of the all simple subfunctions in $f$ is: $\{x_1x_2^0, x_1^0x_2, x_1x_3^0, x_1^0x_3, x_2x_3, x_2^0x_3^0\}$ and in $g$ is: $\{x_1x_2, x_1x_3, x_2x_3, x_2\oplus x_1x_2^0, x_3\oplus x_1x_3^0, x_2^0x_3^0\oplus 1\}$.
Hence $f$ and $g$ have six simple subfunctions, which depends essentially on two variables. Table \ref{tb1} shows that all these subfunctions belong to same $imp$-class and the number of their implementations is $6$. Thus we might calculate that $imp(f)=imp(g)=36$ and $f\simeq_{imp}g$.
The set of the all subfunctions with one essential variable in the function $f$ is: $\{ x_1, x_2, x_3, x_1^0, x_2^0, x_3^0\}$ and in $g$ is: $\{x_1, x_2, x_3\}$.
Then we have $sub_0(f)=sub_0(g)=2$, $sub_1(f)=6$, $sub_1(g)=3$ and $sub_2(f)=sub_2(g)=6$ and hence $f\not\simeq_{sub}g$. It is clear that $sub(f)=15$, $sub(g)=12$ and $\simeq_{imp}\ \not\leq\ \simeq_{sub}$.
(iv)\ Let us consider the functions \[f=x_1x_2^0x_3^0\oplus x_1\ (mod\ 2)\quad\mbox{and}\quad g= x_1x_2x_3\ (mod\ 2).\]
The simple subfunctions in $f$ and $g$ are:\\ \begin{tabular}{lll} $f(x_1=0)=0,$&$f(x_3=0)=x_1x_2^0\oplus x_1,$& $g(x_2=0)=0,$ \\ $f(x_1=1)=x_2^0x_3^0\oplus 1,$& $f(x_3=1)=x_1,$ & $g(x_2=1)=x_1x_3,$\\ $f(x_2=0)=x_1x_3^0\oplus x_1,$ & $g(x_1=0)=0,$&$g(x_3=0)=0,$\\
$f(x_2=1)=x_1,$ & $g(x_1=1)=x_2x_3,$&$g(x_3=1)=x_1x_2$.\\
\end{tabular}
Now, using Table \ref{tb1}, one can easily calculate that $imp(f)=23$ and $imp(g)=21$, and hence $ f \not\simeq_{imp} g$. On the other side we have $Sub(f)=\{0, 1, x_1, x_2, x_3, x_2^0x_3^0\oplus 1, x_1x_3^0\oplus x_1, x_1x_2^0\oplus x_1, f\}$ and $Sub(g)=\{0, 1, x_1, x_2, x_3, x_2x_3, x_1x_3, x_1x_2, g\}$ which show that $sub_m(f)=sub_m(g)\quad\mbox{for}\quad m=0,1,2,3$ and $f\simeq_{sub} g$. Hence $ \simeq_{sub}\ \not\leq\ \simeq_{imp}$. \end{proof}
A {\it transformation} $\psi:P_k^n\longrightarrow P_k^n$ can be viewed as an $n$-tuple of functions \[\psi=(g_1,\ldots,g_n),\quad g_i\in P_k^n,\quad i=1,\ldots,n\] acting on any function $f=f(x_1,\ldots,x_n)\in P_k^n$ as follows $\psi(f)=f(g_1,\ldots,g_n)$. Then the composition of two transformations $\psi$ and $\phi=(h_1,\ldots,h_n)$ is defined as follows \[\psi\phi=(h_1(g_1,\ldots,g_n),\ldots,h_n(g_1,\ldots,g_n)).\]
Thus the set of all transformations of $P_k^n$ is the {\it universal monoid $\Omega_k^n$} with unity - the identical transformation. When taking only invertible transformations we obtain the {\it universal group} $C_k^n$ isomorphic to the symmetric group $S_{Z_k^n}$. Throughout this paper we shall consider invertible transformation, only. The groups consisting of invertible transformations of $P_k^n$ are called {\it transformation groups}.
Let $\simeq$ be an equivalence relation in $P_k^n$. A mapping $\varphi:P_k^n\longrightarrow P_k^n$ is called {\it a transformation, preserving $\simeq$} if $f\simeq \varphi(f)$ for all $f\in P_k^n$. Taking only invertible transformations which preserve $\simeq$, we get the group $G$ of all transformations preserving $\simeq$, whose {\it orbits} (also called {\it $G$-types}) are the equivalence classes $P_1,\ldots,P_r$ under $\simeq$. The number of orbits of a group $G$ of transformations in finite algebras of functions is denoted by $t(G)$.
Next, we relate groups to combinatorial problems trough the following obvious, but important definition:
\begin{definition}\label{d5.4}~Let $G$ be a transformation group acting on the algebra of functions $P_k^n$and suppose that $f,g\in P_k^n$. We say that $f$ is $G$-equivalent to $g$ (written $f\simeq_G g$) if there exists $\psi\in G$ so that $g=\psi(f)$. \end{definition} Clearly, the relation $\simeq_G $ is an equivalence relation. We summarize and extend the results for the "classical" transformation groups, following \cite{har2,lech,str3}, where these notions are used to study classification and enumeration in the algebra of boolean functions. Such groups are induced under the following notions of equivalence: complementation and/or permutation of the variables; any linear or affine function of the variables. Since we want to classify functions from $P_k^n$ into equivalence classes, three natural problems occur.
\begin{enumerate}
\item[$\bullet$] We ask for the number $t(G)$ of such equivalence classes. This problem will be partially discussed for the family of ``natural'' equivalence relations in the algebra of boolean functions. \item[$\bullet$] We ask for the cardinalities of the equivalence classes. This problem is important in applications as functioning the switching gates, circuits etc. For boolean functions of 3 and 4 variables we shall solve these two problems, also concerning $imp$-, $sub$- and $sep$-classes.
\item[$\bullet$] We want to give a method which will decide the class to which an arbitrary function belongs. In some particular cases this problem will be discussed below. We also develop a class of algorithms for counting the complexities $imp$, $sub$ and $sep$ for each boolean function which allow us to classify the algebras $P_2^n$ for $n=2,3,4$ with respect to these complexities as group invariants. \end{enumerate} These problems are very hard and for $n\geq 5$ they are practically unsolvable.
We use the denotation $\leq$ also, for order relation ``subgroup''. More precisely, $H\leq G$ if there is a subgroup $G'$ of $G$ which is isomorphic to $H$.
Let us denote by $IM_k^n$, $SB_k^n$ and $SP_k^n$ the transformation groups induced by the equivalence relations $\simeq_{imp}$, $\simeq_{sub}$ and $\simeq_{sep}$, respectively.
Now, as a direct consequence of Theorem \ref{t5.1} we obtain the following proposition.
\begin{proposition}\label{c5.1}~
\begin{enumerate}\item[(i)] $IM_k^n\leq SP_k^n$; \quad (iii) $IM_k^n\not\leq SB_k^n$; \item[(ii)] $SB_k^n\leq SP_k^n$; \quad (iv) $SB_k^n\not\leq IM_k^n$. \end{enumerate} \end{proposition}
We deal with "natural" equivalence relations which involve variables in some functions. Such relations induce permutations on the domain $Z_k^n$ of the functions. These mappings form a transformation group whose number of equivalence classes can be determined.
The restricted affine group (RAG) is defined as a subgroup of the symmetric group on the direct sum of the vector space $Z_k^n$ of arguments of functions and the vector space $Z_k$ of their outputs. The group RAG permutes the direct sum $Z_k^n+Z_k$ under restrictions which preserve single-valuedness of all functions from $P_k^n$. The equivalence relation induced by RAG is called {\it prototype equivalence relation}.
In the model of RAG an affine transformation operates on the domain or space of inputs $\mathbf{x}=(x_1,\ldots,x_n)$ to produce the output $\mathbf{y}=\mathbf{xA}\oplus \mathbf{c}$, which might be used as an input in a function $g$. Its output $g(\mathbf{y})$ together with the function variables $x_1,\ldots,x_n$ are linearly combined by a range transformation which defines the image $f(\mathbf{x})$ as follows: \begin{equation}\label{eq2} f(\mathbf{x})=g(\mathbf{y})\oplus a_1x_1\oplus\ldots\oplus a_nx_n\oplus d=g(\mathbf{xA}\oplus \mathbf{c})\oplus \mathbf{a^tx}\oplus d \end{equation} where $d$ and $a_i$ for $i=1,\ldots,n$ are constants from $Z_k$.
Such a transformation belongs to RAG if $\mathbf{A}$ is a non-singular matrix. The name RAG was given to this group by R. Lechner in 1963 (see \cite{lech1}) and it was studied by Ninomiya (see \cite{nin}) who gave the name "prototype equivalence" to the relation it induces on the function space $P_k^n$.
We want to extract basic facts about some of the subgroups of RAG which are "neighbourhoods" or "relatives" of our transformation groups - $IM_k^n$, $SB_k^n$ and $SP_k^n$.
First, we consider a group which is called $CA_k^n$ (complement arguments) and each transformation $\mathbf{j}\in CA_k^n$ is determined by an $n$-tuple from $Z_k^n$, i.e. $CA_k^n=\{(j_1,\ldots,j_n)\in Z_k^n\}.$ Intuitively, $CA_k^n$ will complement some of the variables of a function. If $\mathbf{j}=(j_1,\ldots,j_n)$ is in $CA_k^n$, define $\mathbf{j}(x_1,\ldots,x_n)=(x_1\oplus j_1,\ldots,x_n\oplus j_n).$ The group operation is sum mod $k$ and written $\oplus$. For example if $n=k=3$ and $\mathbf{j}=(2,1,0)$ then $\mathbf{j}(x_1,x_2,x_3)=(x_1\oplus 2,x_2\oplus 1,x_3)$ and $\mathbf{j}$ induces a permutation on $Z_3^3=\{0,1,2\}^3$. Then the following sequence of images: $\mathbf{j}: 000\rightarrow 210\rightarrow 120\rightarrow 000$ determines the cycle $(0,21,15)$ and if we agree to regard each triple from $Z_3^3$ as a ternary number, then the permutation induced by $\mathbf{j}$ can be written in cyclic notation as $(0,21,15)(1,22,16)(2,23,17)(3,24,9)(4,25,10)(5,26,11)$ $(6,18,12)(7,19,13)(8,20,14).$ In \cite{har2} M. Harrison showed that the boolean functions of two variables are grouped into seven classes under the group $CA_2^2$.
Another classification occurs when permuting arguments. If $\pi\in S_n$ then $\pi$ acts on variables by: $\pi(x_1,\ldots,x_n)=(x_{\pi(1)},\ldots,x_{\pi(n)}).$ Each permutation induces a map on the domain $Z_k^n$. For instance the permutation $\pi=(1,2)$ induces a permutation on $\{0,1,2\}^3$ when considering the algebra $P_3^3$. Then we have $\pi: 010\rightarrow 100 \rightarrow 010$ and in cyclic notation it can be written as \[(3,9)(4,10)(5,11)(6,18)(7,19)(8,20)(15,21)(16,22)(17,23).\] $S_k^n$ denotes the transformation group induced by permuting of variables. It is clear that $S_k^n$ is isomorphic to $S_n$.
If we allow both complementations and permutations of the variables, then a transformation group, called $G_k^n$, is induced. The group action on variables is represented by $((j_1,\ldots,j_n),\pi)(x_1,\ldots,x_n)=(x_{\pi(1)}\oplus j_1,\ldots, x_{\pi(n)}\oplus j_n)$ where $j_m\in Z_k$ for $1\leq m\leq n$ and $\pi\in S_n$. The group $G_2^n$ is especially important in switching theory and other areas of discrete mathematics, since it is the symmetry group of the $n$-cube. The classification of the boolean functions under $G_2^2$ into six classes is shown in \cite{har2}.
Let us allow a function to be equivalent to its complements as well as using equivalence under $G_k^n$. Then the transformation group which is induced by this equivalence relation is called the {\it genera} of $G_k^n$ and it is denoted by $GE_k^n$. Thus the equivalence relation $\simeq_{gen}$ which induces genera of $G_k^n$ is defined as follows $f\simeq_{gen}g\iff f\simeq_{G_k^n}g$ or $f=g\oplus j$ for some $j\in Z_k$. Then there exist only four equivalence classes in $P_2^2$, induced by $GE_2^2$.
These classes are the same as the classes induced by the group $IM_2^2$ in the algebra $P_2^2$ (see \cite{har2} and Table \ref{tb1}, given above).
Next important classification is generated by equivalence relations which allow adding linear or affine functions of variables. In order to preserve the group property we shall consider invertible linear transformations and assume that $k$ is a prime number such that $LG_k^n$ the general linear group on an $n$-dimensional vector space is over the field $Z_k$. The transformation groups $LG_2^n$ and $A_2^n$ of linear and affine transformations in the algebra of boolean functions are included in the lattice of the subgroups of RAG. We extend this view to the functions from $P_k^n$. The algebra of boolean functions in the simplest case of two variables is classified in eight classes under $LG_2^2$ and in five classes under $A_2^2$. Table \ref{tb1_1} presents both equivalence classes of boolean functions from $P_2^2$ under the transformation group $RAG$. \begin{table} \caption{Classes in $P_2^2$ under $RAG$.} \label{tb1_1}
\centering \begin{tabular}{l} \hline\hline
$[\ 0,\
1,\
x_1,\ x_2,\ x_1^0,\ x_2^0,\ x_1\oplus x_2,\ x_1\oplus x_2^0\ ]$\\ \hline $[\ x_1x_2,\ x_1x_2^0,\ x_1^0x_2,\ x_1^0x_2^0,\
x_1\oplus x_1x_2,\ x_2^0\oplus x_1x_2,\
x_1^0\oplus x_1x_2,\ x_1^0\oplus x_1x_2^0\ ]$\\ \hline\hline \end{tabular} \end{table}
The subgroups of RAG defined above are determined by equivalence relations as it is shown in Table \ref{tb2}, where $\mathbf{P}$ denotes a permutation matrix, $\mathbf{I}$ is the identity matrix, $\mathbf{b\mbox{ and }c}$ are vectors from $Z_k^n$ and $d\in Z_k$.
\begin{table} \caption{Subgroups of RAG}\label{tb2}
\begin{tabular}{||l|l|l||}\hline\hline Subgroup& Equivalence relations& Determination\\ \hline RAG & Prototype equivalence& $\mathbf{A}$-non-singular\\ $GE_k^n$ & genus & $\mathbf{A}=\mathbf{P}$, $\mathbf{a}=\mathbf{0}$\\
$CF_k^n$ & complement function & $\mathbf{A}=\mathbf{I}$, $\mathbf{a}=\mathbf{0}$, $\mathbf{c}=\mathbf{0}$\\
$A_k^n$ &affine transformation & $\mathbf{a}=\mathbf{0}$, $d=0$\\
$G_k^n$ & permute \& complement & \\
& variables (symmetry types) & $\mathbf{A}=\mathbf{P}$, $\mathbf{a}=\mathbf{0}$, $d=0$\\
$LF_k^n$ & add linear function & $\mathbf{A=I}$, $\mathbf{c=0}$, $d=0$\\ $CA_k^n$ & complement arguments & $\mathbf{A}=\mathbf{I}$, $\mathbf{a}=\mathbf{0}$, $d=0$\\
$LG_k^n$ & linear transformation & $\mathbf{c}=\mathbf{0}$, $\mathbf{a}=\mathbf{0}$, $d=0$\\
$S_k^n$ & permute variables & $\mathbf{A}=\mathbf{P}$, $\mathbf{c}=\mathbf{0}$, $\mathbf{a}=\mathbf{0}$, $d=0$\\ \hline\hline \end{tabular} \end{table} It is naturally to ask which subgroups of RAG are subgroups of the groups $IM_k^n$ or $SB_k^n$. The answer of this question is our next goal. \begin{example}\label{ex5.1} Let $f=x_1x_2^0x_3\oplus x_1^0\quad\mbox{and}\quad g=x_1x_2^0x_3\oplus x_1x_2$ be two boolean functions. Then \[sub_1(f)=sub_1(g)=3,\ sub_2(f)=sub(g)=3\quad\mbox{and}\quad sub_3(f)=sub_3(g)=1.\] Hence $f\simeq_{sub} g$. In a similar way, it can be shown that $f\simeq_{imp} g$.
The details are left to the reader.
On the other side, one can prove that there is no transformation $\varphi\in RAG$ such that $\varphi(x_1^0)=x_1x_2$ (see Table \ref{tb1_1}) and hence there is no affine transformation $\varphi\in RAG$ for which $g=\varphi(f)$.
Consequently, each group among $IM_k^n$, $SB_k^n$ and $SP_k^n$ can not be a subgroup of $RAG$. \end{example}
Table \ref{tb2} allows us to establish the following fact. \begin{fact}\label{fc2} If $f$ and $g$ satisfy (\ref{eq2}) with $\mathbf{A\notin \{0,P,I\}}$ or $\mathbf{a\neq 0}$ then $f\not\simeq_{imp} g$, $f\not\simeq_{sub} g$ and $f\not\simeq_{sep} g$. \end{fact}
\begin{proposition}\label{p5.1}~~ \begin{enumerate} \item[(i)] $LG_k^n\not\leq SP_k^n$;\quad (ii) $LF_k^n\not\leq SP_k^n$; \item[(iii)] $IM_k^n\not\leq RAG$;\quad (iv) $SB_k^n\not\leq RAG$. \end{enumerate} \end{proposition} \begin{proof} Immediate from Fact \ref{fc2} and Example \ref{ex5.1}. \end{proof} Let $\sigma:Z_k\longrightarrow Z_k$ be a mapping and $\psi_\sigma:P_k^n\longrightarrow P_k^n$ be a transformation of $P_k^n$ determined by $\sigma$ as follows $\psi_\sigma(f)(\mathbf{a})=\sigma(f(\mathbf{a}))$ for all $\mathbf{a}=(a_1,\ldots,a_n)\in Z_k^n$.
\begin{theorem}\label{t5.2} $\psi_\sigma\in IM_k^n$ and $\psi_\sigma\in SB_k^n$ if and only if $\sigma$ is a permutation of $Z_k$, $k>2$. \end{theorem} \begin{proof} "$\Leftarrow$" Let $\sigma\in S_{Z_k}$ be a permutation of $Z_k$ and let $f$ be an arbitrary function with $ess(f)=n\geq 0$. We shall proceed by induction on $n$, the number of essential variables in $f$.
If $n=0$ then clearly $\psi_\sigma(f)$ is a constant and hence $f\simeq_{imp} \psi_\sigma(f)$ and $f\simeq_{sub} \psi_\sigma(f)$.
Assume that if $n<p$ then $f\simeq_{imp} \psi_\sigma(f)$ and $f\simeq_{sub} \psi_\sigma(f)$ for some natural number $p, p>0$. Hence $f(x_i=j)\simeq_{imp}\psi_\sigma(f(x_i=j))$ and $sub_m(f(x_i=j))=sub_m(\psi_\sigma(f(x_i=j)))$ for all $i\in\{1,\ldots,n\}$, $m\in\{1,\ldots,n-1\}$ and $j\in Z_k$.
Let $n=p$. Let $x_i\in\{x_1,\ldots,x_n\}=Ess(f)$ and $j\in Z_k$, and let us set $g=f(x_i=j)$. Then $\psi_\sigma(g)=\psi_\sigma(f(x_i=j)$ and $ess(g)=n-1<p$. Hence our inductive assumption implies $g\simeq_{imp}\psi_\sigma(g)$ and $g\simeq_{sub}\psi_\sigma(g)$. Consequently, we have \[f(x_i=j)\simeq_{imp}\psi_\sigma(f(x_i=j)) \quad\mbox{and}\quad sub_m(f(x_i=j))=sub_m(\psi_\sigma(f(x_i=j)))\] for all $x_i\in\{x_1,\ldots,x_n\}$ and $j\in Z_k$, which shows that $f\simeq_{imp}\psi_\sigma(f)$ and $f\simeq_{sub}\psi_\sigma(f)$.
"$\Rightarrow$" Let us assume that $\sigma$ is not a permutation of $Z_k$. Hence there exist two constants $a_1$ and $a_2$ from $Z_k$ such that $a_1\neq a_2$ and $\sigma(a_1)=\sigma(a_2)$. Let us fix the vector $\mathbf{b}=(b_1,\ldots,b_n)\in Z_k^n$. Then we define the following function from $P_k^n$: \[ f(x_1,\ldots,x_n)=\left\{\begin{array}{ccc}
a_1 \ &\ if \ &\ x_i=b_i\ for\ i=1,\dots,n \\
a_2 & & otherwise.
\end{array}
\right. \] Clearly, $Ess(f)=X_n$. On the other hand the range of $f$ is $range(f)=\{a_1,a_2\}$ and $\sigma(range(f))=\{\sigma(a_1)\}$, which implies that $\psi_\sigma(f)(c_1,\ldots,c_n)=\sigma(a_1)$ for all $(c_1,\ldots,c_n)\in Z_k^n$. Hence $\psi_\sigma(f)$ is the constant $\sigma(a_1)\in Z_k$ and $Ess(\psi_\sigma(f))=\emptyset$. Thus we have $f\not\simeq_{imp} \psi_\sigma(f)$ and $f\not\simeq_{sub} \psi_\sigma(f)$. \end{proof}
\begin{theorem}\label{t5.3} Let $\pi\in S_n$ and $\sigma_i\in S_{Z_k}$ for $i=1,\ldots,n$. Then $f(x_1,\ldots,x_n)\simeq_{imp} f(\sigma_1(x_{\pi(1)}),\ldots,\sigma_n(x_{\pi(n)}))$ and $f(x_1,\ldots,x_n)\simeq_{sub} f(\sigma_1(x_{\pi(1)}),\ldots,\sigma_n(x_{\pi(n)}))$. \end{theorem} \begin{proof} Let $f\in P_k^n$ be an arbitrary function and assume $Ess(f)=X_n$.
First, we shall prove that \[f(x_1,\ldots,x_n)\simeq_{imp} f(x_{\pi(1)},\ldots,x_{\pi(n)})\] {and} \[f(x_1,\ldots,x_n)\simeq_{sub} f(x_{\pi(1)},\ldots,x_{\pi(n)}).\] Let $g=f(x_{\pi(1)},\ldots,x_{\pi(n)})$. Clearly, if $n\leq 1$ then $f\simeq_{imp} g$ and $f\simeq_{sub} g$. Assume that if $n<p$ then $f\simeq_{imp} g$ and $f\simeq_{sub} g$ for some natural number $p$, $p\geq 1$.
Let us suppose $n=p$. Let $x_i\in Ess(f)$ be an arbitrary essential variable in $f$ and let $c\in Z_k$ be an arbitrary constant from $Z_k$. Then we have \[f(x_i=c)(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_p)=\]
\[=g(x_{\pi^{-1}(i)}=c)(x_{\pi^{-1}(1)},\ldots,x_{\pi^{-1}({i-1})},x_{\pi^{-1}({i+1})},\ldots,x_{\pi^{-1}(p)}).\]
Our inductive assumption implies $f(x_i=c)\simeq_{imp}g(x_{\pi(i)}=c)$
{and} $sub_m(f(x_i=c))=sub_m(g(x_{\pi(i)}=c))$ for all $x_i\in X_n$, $m\in\{1,\ldots,p-1\}$ and $c\in Z_k$. Hence $f\simeq_{imp}g \ \mbox{and}\ f\simeq_{sub}g$.
Second, let us prove that \[f(x_1,\ldots,x_n)\simeq_{imp} f(\sigma_1(x_{1}),\ldots,\sigma_n(x_{n}))\]
{and} \[f(x_1,\ldots,x_n)\simeq_{sub} f(\sigma_1(x_{1}),\ldots,\sigma_n(x_{n})).\] Let $h=f(\sigma_1(x_{1}),\ldots,\sigma_n(x_{n}))$. Then we have \[f(a_1,\ldots,a_n)=h(\sigma_1^{-1}(a_1),\ldots,\sigma_n^{-1}(a_n)).\] Hence, if $(i_1\ldots i_r, a_{i_1}\ldots a_{i_r}c)\in Imp(f)$ then $(i_1\ldots i_r, \sigma_{i_1}^{-1}(a_{i_1})\ldots \sigma_{i_r}^{-1}(a_{i_r})c)\in Imp(h)$ for some $r$, $1\leq r\leq n$. Since $\sigma_i$ is a permutation of $Z_k$ for $i=1,\ldots,n$ it follows that $f\simeq_{imp}h$. By similar arguments it follows that $f\simeq_{sub}h$. \end{proof}
\begin{corollary}\label{c5.4} (i) $GE_k^n\leq IM_k^n$; \quad (ii) $GE_k^n\leq SB_k^n$; \quad (iii) $GE_k^n\leq SP_k^n$. \end{corollary}
\section{Classification of Boolean Functions}\label{sec6}
In this section we compare a collection of subgroups of RAG with the groups of transformations preserving the relations $\simeq_{imp}$, $\simeq_{sub}$ and $\simeq_{sep}$ and to obtain estimations for the number of equivalence classes, and for the cardinalities of these classes in the algebra of Boolean functions. Our results are based on Proposition \ref{p5.1}, Theorem \ref{t5.2} and Theorem \ref{t5.3}. Thus we have \begin{equation}\label{eq3} GE_2^n\leq IM_2^n,\quad GE_2^n\leq SB_2^n, \quad LG_2^n\not\leq SP_2^n \quad\mbox{and}\quad LF_2^n\not\leq SP_2^n. \end{equation}
These relationships determine the places of the groups $IM_2^n$, $SB_2^n$ and $SP_2^n$ with respect to the subgroups of RAG. Figure \ref{f4} shows the location of these groups together with the subgroups of RAG.
M. Harrison \cite{har2} and R. Lechner \cite{lech} counted the number of equivalence classes and the cardinalities of the classes under some transformation subgroups of RAG for Boolean functions of 3 and 4 variables.
The relations (\ref{eq3}) show that if we have the values of $t(GE_2^n)$ then we can count the numbers $t(IM_2^n)$, $t(SB_2^n)$ and $t(SP_2^n)$ because the equivalence classes under these transformation groups are union of equivalence classes under $GE_2^n$ and hence we have $t(IM_2^n)\leq t(GE_2^n)$ and $t(SB_2^n)\leq t(GE_2^n)$. Moreover, if we know the factor-set $P_2^n/_{\simeq_{gen}}$ of representative functions under $\simeq_{gen}$ then we can effectively calculate the sets $P_2^n/_{\simeq_{imp}}$, $P_2^n/_{\simeq_{sub}}$ and $P_2^n/_{\simeq_{sep}}$ because of $P_2^n/_{\simeq_{imp}}\subseteq P_2^n/_{\simeq_{gen}}$ and $P_2^n/_{\simeq_{sub}}\subseteq P_2^n/_{\simeq_{gen}}$.
The next theorem allows us to count the number $imp(f)$ of the implementations of any function $f$ by a recursive procedure. Such a procedure is realized and its execution is used when calculating the number of the implementations and classifying the functions under the equivalence $\simeq_{imp}$. \begin{theorem} \label{t35} Let $f\in P_2^n$ be a boolean function. The number of all implementations in $f$ is determined as follows: \[imp(f) = \left\{\begin{array}{ccc}
1 \ &\ if \ &\ ess(f)=0 \\
&&\\
2 & if\ & ess(f)=1\\
&&\\
\sum_{x\in Ess(f)}[imp(f(x=0)) + imp(f(x=1))] & if\ & ess(f)\geq 2.
\end{array}
\right. \] \end{theorem} \begin{proof} We shall proceed by induction on $n=ess(f)$ - the number of essential variables in $f$. The lemma is clear if $ess(f)=0$. If $f$ depends essentially on one variable $x_1$, then there is a unique BDD of $f$ with one non-terminal node which has two outcoming edges. These edges together with the labels of the corresponding terminal nodes form the set $Imp(f)$ of all implementations of $f$, i.e. $imp(f)=2$.
Let us assume that \[imp(f)= \sum_{i=1}^n[imp(f(x_i=0)) + imp(f(x_i=1))]\] if $n< s$ for some natural number $s$, $1\leq s$.
Next, let us consider a function $f$ with $ess(f)=s$. Without loss of generality, assume that $Ess(f)=\{x_1,\ldots,x_n\}$ with $n=s$. Since $x_i\in Ess(f)$ for $i=1,\ldots,n$ it follows that $f(x_i=0)\neq f(x_i=1)$ and there exist BDDs of $f$ whose label of the first non-terminal node is $x_i$. Let $D_f$ be a such BDD of $f$ and let $(ij_2\ldots j_m,c_1c_2\ldots c_mc)\in Imp(f)$ with $m\leq n$. Hence \[(j_2\ldots j_m,c_2\ldots c_mc)\in Imp(g)\] where $g=f(x_i=c_1)$. On the other side it is clear that if $(j_2\ldots j_m,d_2\ldots d_md)\in Imp(g)$ then $(ij_2\ldots j_m,c_1d_2\ldots d_md)\in Imp(f)$. Consequently, there is an one-to-one mapping between the set of implementations of $f$ with first variable $x_i$ and first edge labelled by $c_1$, and $Imp(g)$, which completes the proof. \end{proof} We also develop recursive algorithms to count $sub_m(f)$ and $sep_m(f)$ for $f\in P_2^n$, presented below.
Table \ref{tb4} shows the number of equivalence classes under the equivalence relations induced by the transformation groups $G_2^n$, $IM_2^n$, $SB_2^n$ and $SP_2^n$ for $n=1,2,3,4$. M. Harrison found from applying Polya's counting theorem (see \cite{har2}) the numbers $t(G_2^5)$ and $t(G_2^6)$, which are upper bounds of $t(IM_2^n)$, $t(SB_2^n)$ and $t(SP_2^n)$ for $n=5,6$.
Figure \ref{f4} and Table \ref{tb3} show that for the algebra $P_2^3$ there are only 14 different generic equivalent classes, 13 imp-classes, 11 sub-classes and 5 sep-classes. Hence three mappings that converts each generic class into an imp-class, into a sub-class and into a sep-class are required. Each generic class is a different row of Table \ref{tb3}. For example, the generic class \textnumero 12 (as it is numbered in Table VIII, \cite{lech}) is presented by 10-th row of Table \ref{tb3}. It consists of 8 functions obtained by complementing function $f$ and/or permuting and/or complementing input variables in all possible ways, where $f=x_1x_2^0x_3\oplus x_1x_2x_3^0\oplus x_2x_3$. This generic class \textnumero 12 is included in imp-class \textnumero 9, sub-class \textnumero 8 and sep-class \textnumero 5 which shows that $imp(f)=36$, $sub(f)=12$ and $sep(f)=7$. The average cardinalities of equivalence classes and complexities of functions are also shown in the last row of Table \ref{tb3}.
Table \ref{tb5} shows the $sep$-classes of boolean functions depending on at most five variables. Note that there are $2^{32}=4294967296$ functions in $P_2^5$. All calculations were performed on a computer with two Intel Xeon E5/2.3 GHz CPUs. The execution with total exhaustion took 244 hours.
\begin{figure}
\caption{Transformation groups in $P_2^n$ ($n=3/n=4$)}
\label{f4}
\end{figure}
\begin{table} \centering \caption{Number of classes under $symmetry$ $type$, $\simeq_{imp}$, $\simeq_{sub}$ and $\simeq_{sep}$}\label{tb4}
\begin{tabular}{rrrrr} $n$ & $t(G_2^n)$ & $t(IM_2^n)$ &$t(SB_2^n)$ &$t(SP_2^n)$ \\ \hline\hline 1&3&2&2&2\\ 2&6&4&4&3 \\ 3&22&13&11&5 \\ 4&402&104&74&11 \\ 5&1\ 228\ 158&1606&{$<$ 1228158}& 38 \\ 6&400\ 507\ 806\ 843\ 728&\multicolumn{3}{c}{$<$ 400\ 507\ 806\ 843\ 728} \\ \hline\hline \end{tabular} \end{table}
\begin{sidewaystable}[p]\centering
\caption{Classification of $P_2^3$ under $\simeq_{sep}$, $\simeq_{sub}$, $\simeq_{imp}$ and genus.} \label{tb3}
\begin{tabular}{||c|c|c||c|c|c||c|c|c||c|c||c||} \hline \hline sep- & $sep(f)$ & func. & sub- & $sub(f)$ & func. & imp- & $imp(f)$ & func.& Generic & func. & representative \\ class & & per & class & & per & class & & per &class \cite{lech}& per & function $f$ \\ \textnumero &&class&\textnumero &&class&\textnumero &&class&\textnumero &class&\\
\hline\hline 1 & 0 & 2 & 1 & 1 & 2 & 1 & 1 & 2&1&2 & $0$ \\ \hline 2 & 1 & 6 & 2 & 3 & 6 & 2 & 2 & 6&9&6& $x_1$ \\ \hline \multirow{2}{*}{3} & \multirow{2}{*}{3} & \multirow{2}{*}{30} & 3 & 5 & 24 & 3 & 6 & 24&3&24& $x_1x_2$ \\ \cline{4-12}
& & & 4 & 7 & 6 & 4 & 8 & 6&10&6& $x_1\oplus x_2$ \\ \hline 4 & 6 & 24 & 5 & 11 & 24 & 5 & 28 & 24&13&24& $x_1\oplus x_1x_3\oplus$ \\
& & & & & & & & &&& $ x_2x_3$ \\
\hline \multirow{9}{*}{5} & \multirow{9}{*}{7} & \multirow{9}{*}{194} & \multirow{2}{*}{6} & \multirow{2}{*}{9} & \multirow{2}{*}{64} & 6 & 21 & 16&2&16 & $x_1x_2x_3$ \\ \cline{7-12}
& & & & & & 7 & 23 & 48&6&48& $x_1x_2^0x_3^0\oplus x_1$ \\ \cline{4-12}
& & & 7 & 12 & 48 & 8 & 30 & 48&7&48 & $x_1x_2^0x_3^0\oplus $ \\ & & & & & & & & & & & $x_2x_3$ \\ \cline{4-12}
& & & 8 & 12 & 8 & \multirow{2}{*}{} & \multirow{2}{*}{} & \multirow{2}{*}{}&12&8& $x_1x_2^0x_3\oplus$ \\ & & & & & & \multirow{2}{*}{9} & \multirow{2}{*}{36} & \multirow{2}{*}{16}&& & $ x_1x_2x_3^0 \oplus x_2x_3$ \\ \cline{4-6} \cline{10-12}
& & & \multirow{3}{*}{9} & \multirow{3}{*}{15} & \multirow{3}{*}{26} & & & &5&8& $x_1^0x_2x_3\oplus x_1x_2^0x_3^0$ \\ \cline{7-12}
& & & & & & 10 & 42 & 16&8&16& $x_1x_2^0x_3\oplus $ \\
& & & & & & & & & && $ x_1x_2x_3^0\oplus x_1^0x_2x_3$ \\
\cline{7-12}
& & & & & & 11 & 48 & 2& 11&2& $x_1\oplus x_2\oplus x_3$ \\ \cline{4-12}
& & & 10 & 13 & 24 & 12 & 32 & 24&14&24 & $x_1\oplus x_2x_3$ \\ \cline{4-12}
& & & 11 & 13 & 24 & 13 & 33 & 24&4&24 & $x_1x_2^0x_3\oplus x_1x_2x_3^0$ \\ \hline aver.&6.2&51.2&&10.6&23.3& &26.0&19.7&&18.3&\\ \hline\hline \end{tabular} \end{sidewaystable}
\begin {table} \centering \caption{Classes in $P_2^5$ under $\simeq_{sep}$ }\label{tb5}
\noindent
\begin{tabular}{||c|c|c|c|c|c|c|c||} \hline\hline
sep- & \multirow{3}{*}{\rotatebox{00}{$sep_5(f)$}} & \multirow{3}{*}{\rotatebox{0}{$sep_4(f)$}} & \multirow{3}{*}{\rotatebox{0}{$sep_3(f)$}} & \multirow{3}{*}{\rotatebox{0}{$sep_2(f)$}} & \multirow{3}{*}{\rotatebox{0}{$sep_1(f)$}} & \multirow{3}{*}{\rotatebox{90}{$sep(f)$}} & functions \\ class & & & & & & & per \\ \textnumero & & & & & & & class \\ \hline
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Transitionless quantum drivings for the harmonic oscillator}
\author{J. G. Muga$^{1}$, X. Chen$^{1,2}$, S. Ib\'a\~nez$^{1}$, I. Lizuain$^{1}$, A. Ruschhaupt$^{3}$ }
\address{$^{1}$ Departamento de Qu\'{\i}mica-F\'{\i}sica, UPV-EHU, Apdo 644, 48080 Bilbao, Spain}
\address{$^{2}$ Department of Physics, Shanghai University, 200444 Shanghai, P. R. China}
\address{$^{3}$ Institut f\"ur Theoretische Physik, Leibniz Universit\"{a}t Hannover, Appelstra$\beta$e 2, 30167 Hannover, Germany}
\begin{abstract} Two methods to change a quantum harmonic oscillator frequency without transitions in a finite time are described and compared. The first method, a transitionless-tracking algorithm, makes use of a generalized harmonic oscillator and a non-local potential. The second method, based on engineering an invariant of motion, only modifies the harmonic frequency in time, keeping the potential local at all times. \end{abstract} \pacs{37.10.De, 42.50.-p, 37.10.Vz}
\section{Introduction}
Changing the external parameters of the Hamiltonian is a fundamental and standard operation to probe, control, or prepare a quantum system. In many cases it is desirable to go from an initial parameter configuration to a final one without inducing transitions, as in the expansions performed in fountain clocks \cite{Bize}. In fact most of the current experiments with cold atoms are based on a cooling stage and then an adiabatic drive of the system to some desired final trap or regime \cite{Polkov}. These ``transitionless'' \cite{Berry09}, or ``frictionless'' \cite{Ronnie} adiabatic processes may require exceedingly large times and become impractical, even impossible \cite{Polkov}, or quite simply a faster process is desirable, e.g. to increase the repetition rate of a cycle, or a signal-to-noise ratio. This motivates the generic objective of achieving the same final state as the slow adiabatic processes, possibly up to phase factors, but in a much shorter time. One may try to fulfill that goal in two different ways: (a) designing appropriate ``parameter trajectories'' of the Hamiltonian from the initial to the final times, or (b) applying entirely new interactions that modify the Hamiltonian beyond a simple parameter evolution of the original form, for example by adding different terms to it. In this paper we shall analyze and discuss, for the harmonic oscillator, two recently proposed methods whose relation had not been investigated. It turns out that they actually implement these two different routes. While most of the treatment is applicable to an ``abstract'' harmonic oscillator, we shall discuss physical implementations specific of ultracold atoms or ions. Indeed, harmonic traps and their manipulation are basic working horses of this field.
For the harmonic oscillator the parameter we consider is the trap frequency, which should go from $\omega_0$ to $\omega_f$ in a time $t_f$, preserving the populations of the levels, $P_n(t_f)=P_n(0)$. ``$n$'' labels the instantaneous $n$-th eigenstate of the initial and final harmonic oscillator Hamiltonians,
\begin{eqnarray} H_0(0)|n(0)\rangle&=&\hbar\omega_0(n+1/2)|n(0)\rangle, \nonumber\\
H_0(t_f)|n({t_f})\rangle&=&\hbar\omega_f(n+1/2)|n({t_f})\rangle. \label{hot} \end{eqnarray}
One of the methods we shall discuss here relies on a general framework set by Kato in a proof of the adiabatic theorem \cite{Kato}, and has been formulated recently by Berry \cite{Berry09}. We shall term it ``transitionless-tracking'' approach, or TT for short; the other one \cite{harmo,becs} engineers the Lewis-Riesenfeld invariant \cite{LR69} by an inverse method \cite{Palao} to satisfy the desired boundary conditions; we shall call this method ``inverse-invariant'', or II for short.
In the basic version of TT the dynamics is set to follow at all
intermediate times the adiabatic path defined by an auxiliary Hamiltonian $H_0(t)$ (in our case a regular harmonic oscillator with frequency $\omega(t)$ and boundary conditions $\omega(0)=\omega_0$ and $\omega_f=\omega(t=t_f)$), and its instantaneous eigenvectors $|n(t)\rangle$, up to phase factors. Instead, in the II approach the auxiliary object is an engineered Lewis-Riesenfeld invariant $I(t)$ set to commute with $H_0(0)$ at $t=0$ and with $H_0(t_f)$ at $t_f$. In both cases intermediate states may be highly non-adiabatic with respect to the instantaneous eigenstates of the Hamiltonians actually applied, $H_{TT}(t)$ and $H_{II}(t)$.
We shall provide first the equations characterizing the two approaches and then comment on possible physical implementations.
\section{Transitionless tracking algorithm}
\subsection{General formalism}
For the general formalism we follow \cite{Berry09} closely. Assume a time-dependent Hamiltonian $H_0(t)$ with initial and final values (\ref{hot}), instantaneous eigenvectors
$|n(t)\rangle$ and eigenvalues $E_n(t)$,
\begin{equation} H_0(t)|n(t)\rangle = E_n(t)|n(t)\rangle. \end{equation}
A slow change would preserve the eigenvalue and eigenvector along the dynamical evolution times a phase factor,
\begin{equation}
|\psi_n(t)\rangle = \exp\left\{
-\frac{i}{\hbar}\int_0^{t} dt' E_n(t') -\int_0^t dt' \langle n(t')|\partial_{t'}n(t')\rangle\right\}|n(t)\rangle. \label{22} \end{equation}
We now seek a Hamiltonian $H(t)$ such that the adiabatic approximation
$|\psi_n(t)\rangle$ represents the {\it exact} dynamics,
\begin{equation} i\hbar\partial_t|\psi_n(t)\rangle=H(t)|\psi_n(t)\rangle. \end{equation}
$H(t)$ (which is $H_{TT}$ if distinction with the other method is needed) is related to the corresponding unitary operator by
\begin{eqnarray} i\hbar\partial_t U(t)&=&H(t)U(t), \\ \label{ht} H(t)&=&i\hbar(\partial_t U(t))U^\dagger(t). \end{eqnarray}
Choosing
\begin{equation} U(t)=\sum_n\exp\left\{-\frac{i}{\hbar}\int_0^t dt' E_n(t')- \int_0^t dt'\langle n(t')|\partial_{t'} n(t')\rangle\right\}
|n(t)\rangle\langle n(0)|, \end{equation}
we find from (\ref{ht}),
\begin{equation}
\hat{H}(t)=\sum_n|n\rangle E_n\langle n|
+i\hbar\sum_n(|\partial_t n\rangle\langle n|-\langle n|\partial_t n\rangle|n\rangle\langle n|) \equiv \hat{H}_0+\hat{H}_1, \end{equation}
where we have simplified the notation, $|n\rangle=|n(t)\rangle$. It is also possible to choose other phases in (\ref{22}) \cite{Berry09}. The simplest case is
$U(t)=\sum |n(t)\rangle\langle n(0)|$, without phase factors, which leads to
$H(t)=i\hbar\sum |\partial_t n\rangle\langle n|$. Note that with this choice
$H_0(t)$ has been formally suppressed in $H(t)$ but still plays a role through its eigenfunctions $|n(t)\rangle$.
\subsection{Application to the harmonic oscillator}
We now apply the above to the harmonic oscillator
\begin{equation} \hat{H}_0(t)=\hat{p}^2/2m+\omega(t)^2 \hat{x}^2/2m
=\hbar\omega(t)(\hat{a}^\dagger_t \hat{a}_t +1/2), \label{ho} \end{equation}
where $\hat{a}_t$ and $\hat{a}_t^+$ are the (Schr\"odinger picture!) annihilation and creation operators at time $t$,
\begin{eqnarray} \hat{x}&=&\sqrt{\frac{\hbar}{2m\omega(t)}}(a_t^\dagger +a_t), \\ \hat{p}&=&i\sqrt{\frac{\hbar m\omega(t)}{2}}(a_t^\dagger -a_t), \\ \hat{a}_t&=&\sqrt{\frac{m\omega(t)}{2\hbar}} \left(\hat{x}+\frac{i}{m\omega(t)}\hat{p}\right), \\ \hat{a}^\dagger_t&=&\sqrt{\frac{m\omega(t)}{2\hbar}} \left(\hat{x}-\frac{i}{m\omega(t)}\hat{p}\right). \end{eqnarray}
This time dependence may be misleading and a bit unusual at first so we insist: since the frequency depends on time the ``instantaneous'' ladder operators $\hat{a}_t$, $\hat{a}^\dagger_t$ create or annihilate different ``instantaneous'' states, adapted to the corresponding frequency. Thus, ladder operators with different time labels do not commute in general, although some combinations, e.g. those equivalent to powers of $\hat{x}$ and/or $\hat{p}$, do commute, as we shall see later.
The instantaneous eigenstates $|n(t)\rangle$ can be written in coordinate representation as
\begin{equation}
\langle x|n(t)\rangle =\frac{1}{\sqrt{2^n n!}}\left(\frac{m \omega (t)}{\pi \hbar}\right)^{1/4}\exp{\left(-\frac{1}{2}\frac{m \omega (t)}{\hbar} x^2\right)}H_{n}\left(\sqrt{\frac{m \omega (t)}{\hbar}}x\right), \end{equation}
and their derivative with respect to $t$ is
\begin{eqnarray}
\langle x |\partial_t n(t) \rangle = \left(\frac{1}{4} - \frac{m \omega (t)}{2 \hbar} x^2 \right)\frac{\dot{\omega}}{\omega(t)} |n \rangle
+ \sqrt{\frac{m \omega (t)}{2 \hbar}}x \frac{\dot{\omega}}{\omega(t)} \sqrt{n} | n-1 \rangle. \end{eqnarray}
We find, using the recursion relation of Hermite polynomials and their orthogonality,
\begin{eqnarray}
\langle k|\partial_t n \rangle= \left\{ \begin{array}{ll} \frac{1}{4}\sqrt{n(n-1)}\frac{\dot{\omega}}{\omega(t)} &~~~ k=n-2 \\ \\ -\frac{1}{4}\sqrt{(n+1)(n+2)}\frac{\dot{\omega}}{\omega(t)} &~~~ k=n+2 \\ \\ 0 &~~~ (\mbox{otherwise}) \end{array} \right., \end{eqnarray}
so that $\hat{H}_1 (t)$ can be written as
\begin{eqnarray}
\hat{H}_1 (t)&=& i \hbar \sum_n |\partial_t n \rangle \langle n| \equiv i \hbar \frac{\dot{\omega}}{\omega(t)}\sum_n \Bigg[\left(\frac{1}{4} - \frac{m \omega (t)}{2 \hbar} \hat{x}^2 \right)|n \rangle \langle n| \nonumber \\
&+& \sqrt{\frac{m \omega (t)}{2 \hbar}}\hat{x} \sqrt{n} | n-1 \rangle \langle n|\Bigg]. \end{eqnarray}
Using $a_t=\sum_n \sqrt{n} |n-1(t) \rangle\langle n(t)|$, and the relations between $\hat{x}$, $\hat{p}$, $\hat{a}_t$ and $\hat{a}_t^\dagger$ written above,
\begin{eqnarray} \hat{H}_1 (t)&=& i \hbar \frac{\dot{\omega}}{\omega(t)}\sum_n \left[\frac{1}{4} - \frac{m \omega (t)}{2 \hbar} \hat{x}^2 + \sqrt{\frac{m \omega (t)}{2 \hbar}}\hat{x} \hat{a}_t \right] \nonumber \\
&=&\frac{i \hbar}{4}\frac{\dot{\omega}}{\omega(t)} - \frac{1}{2}\frac{\dot{\omega}}{\omega(t)} \hat{x} \hat{p}. \end{eqnarray}
Using $[\hat{x}, \hat{p}]= i \hbar$, we finally write the Hamiltonian $\hat{H}_1(t)$ in the following simple forms
\begin{equation} \hat{H}_1(t)=-\frac{\dot{\omega}}{4\omega}(\hat{x}\hat{p}+\hat{p}\hat{x})
=i\hbar\frac{\dot{\omega}}{4\omega}[{\hat{a}}^2-({\hat{a}}^\dagger)^2]. \label{h1a} \end{equation}
In the last expression the subscript $t$ in $\hat{a}$ and $\hat{a}^\dagger$ has been dropped because the squeezing combination ${\hat{a}}^2-(\hat{a}^\dagger)^2$ is actually independent of time, so one may evaluate it at any convenient time, e.g. at $t=0$. The connection with squeezing operators is worked out in the appendix.
$H_1$ is therefore a non-local operator, and does not have the form of a regular harmonic oscillator potential with an $x^2$ term. Nevertheless the final Hamiltonian $H=H_0+H_1$ is still quadratic in $\hat{x}$ and $\hat{p}$, so it may be considered a generalised harmonic oscillator \cite{gho}.
\subsection{Physical realization}
The nonlocality of $\hat{H}_1$, with a constant prefactor, can be realized in a laboratory by means of 2-photon Raman transitions for trapped ions \cite{Itano,Zeng}. Since we have to evaluate as well the possibility of making the prefactor in $\hat{H}_1$ time dependent we need to provide the derivation with some detail, first for a time-independent $\omega$.
\subsubsection{Raman two-photon transition in a trapped ion}
Let us consider a harmonically trapped two-level system in 1D driven by two different lasers (with coupling strengths $\Omega_j$ and frequencies $\omega_j$, $j=1,2$), see Fig. \ref{level_scheme_fig} and Refs. \cite{zeng95,zeng98}. The time dependent ``Raman'' Hamiltonian in the Schr\"odinger picture will be given by
\begin{eqnarray} \hat{H}_{R}(t)&=&\hat{H}_T+\hat{H}_A+\hat{H}_{int}, \end{eqnarray}
with ``trap'' ($T$), ``atomic'' ($A$), and interaction ($int$) terms
\begin{eqnarray} \hat{H}_T&=&\hbar\omega \hat{a}^\dag \hat{a},\\
\hat{H}_A&=&\hbar\omega_e|e\rangle\langle e|,\\
\hat{H}_{int}&=&\sum_{j=1}^2\hbar\Omega_j\cos\left(\omega_jt-k_jx+\phi_j\right) (|g\rangle\langle e|+|e\rangle\langle g|), \end{eqnarray}
where $\hbar\omega_e$ is the energy of the excited state $|e\rangle$ and ${\bf k}_j=k_j{\bf \hat x}$ the wavevector of each laser which are assumed to be pointing along the principal trap direction, the $x$-direction.
\subsubsection{Interaction picture\label{ip}}
Let us now write the above Hamiltonian in an interaction picture defined by the Hamiltonian $\hat{h}_0=\hat{H}_T+\hbar\tilde\omega_L|e\rangle\langle e|$, where $\tilde\omega_L=(\omega_1+\omega_2)/2$ has been introduced. The interaction Hamiltonian $\hat{H}_I=e^{i\hat{h}_0t/\hbar}(\hat{H}_R-\hat{h}_0)e^{-i\hat{h}_0t/\hbar}$ reads
\begin{eqnarray} \label{H_int_time_dep}
\hat{H}_I(t)&=&-\hbar\tilde\Delta|e\rangle\langle e| \nonumber\\ &+&\sum_{j=1}^2\frac{\hbar\Omega_j}{2} \left( e^{i\eta_j\left[\hat{a}(t)+\hat{a}^\dag(t)\right]}e^{-i(\omega_j-\tilde\omega_L) t}
e^{-i\phi_j}|e\rangle\langle g| + H.c.\right), \end{eqnarray}
where $\tilde\Delta=\tilde\omega_L-\omega_e$, and now $\hat{a}(t)=\hat{a} e^{-i\omega t}$, $\hat{a}^\dag(t)=\hat{a}^\dag e^{i\omega t}$ are the time dependent Heisenberg annihilation and creation operators respectively. Note also that fast oscillating off-resonant $e^{\pm i(\omega_j+\tilde\omega_L)t}$ terms have been neglected in the rotating wave approximation (RWA). The parameter $\eta_j=k_jx_0$ is known as the Lamb-Dicke (LD) parameter, where $x_0=\sqrt{\hbar/2m\omega}$ is the extension (square root of the variance) of the ion's ground state, i. e., $\hat{x}=x_0(\hat{a}+\hat{a}^\dag)$.
\begin{figure}
\caption{Schematic electronic and vibrational level structure for a two-photon transition in an ion trapped with frequency $\omega$. $\omega_1$ and $\omega_2$ are the laser frequencies, and $\omega_e$ the transition frequency between ground and excited states. See the text for further details.}
\label{level_scheme_fig}
\end{figure}
\subsubsection{Adiabatic elimination and effective Hamiltonian}
For a general wavefunction (in the corresponding interaction picture) such as
\begin{equation}
|\psi_I(t)\rangle=\sum_{n=0}^\infty \left[g_n(t)|g,n\rangle+e_n(t)|e,n\rangle\right] \end{equation}
the differential equations of motion for the probability amplitudes $g_n(t)$ and $e_n(t)$ are obtained from the Schr\"odinger equation $i\hbar\partial_t|\psi_I(t)\rangle=\hat{H}_I|\psi_I(t)\rangle$,
\begin{eqnarray} \label{system_gn} i\dot g_n(t)&=&\frac{1}{2}\sum_{j=1}^2\sum_{n'=0}^\infty
\Omega_je^{i(\theta_jt+\phi_j)}\langle n|e^{-i\eta_j\left[\hat{a}(t)+\hat{a}^\dag(t)\right]}|n'\rangle e_{n'}(t),\\
i\dot e_n(t)&=&-\tilde\Delta e_n(t)+\frac{1}{2}\sum_{j=1}^2\sum_{n'=0}^\infty
\Omega_je^{-i(\theta_jt+\phi_j)}\langle n|e^{i\eta_j\left[\hat{a}(t)+\hat{a}^\dag(t)\right]}|n'\rangle g_{n'}(t), \label{system_en} \end{eqnarray}
where $\theta_j=\omega_j-\tilde\omega_L$. For large detunings, i. e., for $|\tilde\Delta|\gg\Omega_j,\omega$, see Fig. \ref{level_scheme_fig}, and for an ion initially in the ground state one may assume that the excited state $|e\rangle$ is scarcely populated and it may be adiabatically eliminated. Then, setting $\dot e(t)=0$, $e_n(t)$ may be written as a function of the $g_{n'}(t)$ from Eq. (\ref{system_en}), and substituting this result into (\ref{system_gn}) there results a differential equation for the ground state probability amplitude,
\begin{equation}
i\dot g_n(t)=s g_n(t)+\frac{\tilde\Omega}{2}\sum_{n'=0}^\infty \mathcal{F}_{n,n'}(t)g_{n'}(t), \end{equation}
where
\begin{eqnarray} s&=&\frac{\Omega_1^2+\Omega_2^2}{4\tilde\Delta}, \\
\mathcal{F}_{n,n'}(t)&=&\langle n|e^{-i\tilde\eta\left[\hat{a}(t)+\hat{a}^\dag(t)\right]}|n'\rangle e^{i(\tilde\delta t+\tilde\phi)}
+\langle n|e^{i\tilde\eta\left[\hat{a}(t)+\hat{a}^\dag(t)\right]}|n'\rangle e^{-i(\tilde\delta t+\tilde\phi)}, \end{eqnarray}
and where the effective two-photon Raman parameters, denoted by tildes, are given by
\begin{eqnarray} \tilde\delta&=&\omega_1-\omega_2, \nonumber\\ \tilde\eta&=&\eta_1-\eta_2, \nonumber\\ \tilde\phi&=&\phi_1-\phi_2, \nonumber\\ \frac{\tilde\Omega}{2}&=&\frac{\Omega_1\Omega_2}{4\tilde\Delta}. \end{eqnarray}
The equation for the ground state probability amplitude corresponds to an effective Hamiltonian
\begin{eqnarray} \label{effective_ham_g}
\hat{H}_{eff}&=&\hbar s|g\rangle\langle g|+
\frac{\hbar\tilde\Omega}{2}\left(e^{i\tilde\eta[\hat{a}(t)+\hat{a}^\dag(t)]}e^{-i(\tilde\delta t+\tilde\phi)}+H.c\right)|g\rangle\langle g|. \end{eqnarray}
Note that the Stark-Shift produced by off resonant driving is included in $s$, which is a constant of motion and produces no effect on the Raman coupling between sidebands. We have thus adiabatically eliminated the excited state $|e\rangle$ ending with a Hamiltonian of the same form as (\ref{H_int_time_dep}) where the transitions between electronic levels are not present.
\subsubsection{Two-photon Jaynes-Cummings Hamiltonian in the Raman Scheme: Vibrational RWA}
Using the Baker-Campbell-Hausdorff (BCH) identity, the exponential in the effective Hamiltonian (\ref{effective_ham_g}) may be expanded in power series of $\tilde\eta$ \cite{Orszag,LME08},
\begin{eqnarray} \label{time_dep_ham_02} \hat{H}_{eff}=\frac{\hbar\tilde\Omega}{2} \left[e^{-\tilde\eta^2/2}\sum_{nn'} \frac{\left(i \tilde\eta\right)^{n+n'}}{n!n'!} \hat{a}^{\dag n}\! \hat{a}^{n'} e^{i (n-n') \omega t}e^{-i\tilde\delta t}e^{-i\tilde\phi}+H.c\right]. \end{eqnarray}
If the effective detuning is $\tilde\delta=\omega_1-\omega_2=2\omega$, the second blue sideband becomes resonant, and we may neglect rapidly oscillating terms in a second or vibrational RWA \cite{LME08}. The above Hamiltonian is then simplified to a two-photon Jaynes-Cummings-like Hamiltonian without electronic transitions. To leading order in $\tilde\eta$ it takes the form
\begin{equation} \hat{H}_{2B}=\tilde\eta^2\frac{\hbar\tilde\Omega}{4}\left(\hat{a}^{\dag 2} e^{i\tilde\phi}+\hat{a}^2 e^{-i\tilde\phi}\right) =i\hbar\frac{\tilde\eta^2\tilde\Omega}{4}\left(\hat{a}^2 -\hat{a}^{\dag 2}\right), \label{2b} \end{equation}
where for the last step a relative phase between the applied fields $\tilde\phi=\phi_1-\phi_2=-\pi/2$ has been assumed.
\subsubsection{Validity for time-dependent $\omega$}
Unfortunately the above formal manipulations and approximations cannot be carried out in general for a time dependent $\omega$. The interaction picture performed in \ref{ip}, in particular, assumes a constant $\hat{h}_0$. A time dependent one would require a more complex approach with time-ordering operators \cite{Glauber}. Similarly, the vibrational rotating wave approximation requires the stability of the frequency for times larger than a period to avoid off-resonant couplings. One may still obtain (\ref{2b}) for a sufficiently slowly varying $\omega$, the criterion being that the change of the time-dependent trapping frequency in one time period $T$ has to be much smaller than the frequency itself. We can write this condition as $\dot\omega(t) T\ll\omega(t)$ or
\begin{equation} \frac{\dot\omega(t)}{\omega(t)^2}\ll1, \end{equation}
which turns out to be the adiabaticity condition for the harmonic oscillator. Of course, if satisfied, the whole enterprise of applying the TT method would be useless. These arguments are far from constituting a proof that the TT method cannot be implemented for the harmonic oscillator. They simply leave this as an open question.
\section{Engineering the Lewis-Riesenfeld invariant}
In this section we describe a different method for transitionless dynamics of the harmonic oscillator \cite{harmo}. A harmonic oscillator such as $H_0(t)$ in Eq. (\ref{ho}) has the following time dependent invariant \cite{LR69}
\begin{equation} I(t)=\frac{1}{2}\left(\frac{\hat{x}^2}{b^2} m \omega_0^2+\frac{1}{m} \hat{\pi}^2\right), \end{equation}
where $\hat{\pi}=b(t)\hat{p}-m\dot{b}\hat{x}$ plays the role of a momentum conjugate to $\hat{x}/b$, the dots are derivatives with respect to time, and $\omega_0$ is in principle an arbitrary constant. The scaling, dimensionless function $b=b(t)$ satisfies the subsidiary condition
\begin{equation}\label{subsi} \ddot{b}+\omega(t)^2 {b}=\omega_0^2/b^3, \end{equation}
an Ermakov equation where real solutions must be chosen to make $I$ Hermitian. $\omega_0$ is frequently rescaled to unity by a scale transformation of $b$ \cite{LR69}. Other common and convenient choice, which we shall adopt here, is $\omega_0=\omega(0)$. The eigenstates of $I(t)$ become, with appropriate phase factors, solutions of the time-dependent Schr\"odinger equation,
\begin{eqnarray} \Psi_n (t,x) &=& \left(\frac{m\omega_0}{\pi\hbar}\right)^{1/4} \!\frac{1}{(2^n n! b)^{1/2}} \fexp{-i (n+1/2) \int_0^t dt'\, \frac{\omega_0}{b(t')^2}} \\ &\times&\fexp{i \frac{m}{2\hbar}\left(\frac{\dot{b}}{b(t)} + \frac{i\omega_0}{b^2}\right)x^2} H_n\left[\left(\frac{m\omega_0}{\hbar}\right)^{1/2}\frac{x}{b}\right], \label{emode} \end{eqnarray}
and form a complete basis to expand any time-dependent state, $\psi(x,t)=\sum_n c_n \Psi_n(x,t)$, with the amplitudes $c_n$ constant. A method to achieve frictionless, population preserving processes is to leave $\omega(t)$ undetermined first, and then set $b$ so that $I(0)=H_0(0)$ and $[I(t_f),H_0(t_f)]=0$. This guarantees that the eigenstates of $I$ and $H_0$ are common at initial and finite times. We can do this by setting
\begin{eqnarray} b(0)=1, \dot{b}(0)=1, \ddot{b}=0 \nonumber\\ b(t_f)=\gamma=[\omega_0/\omega_f]^{1/2}, \dot{b}(t_f)=0, \ddot{b}(t_f)=0, \label{bes} \end{eqnarray}
and interpolating $b(t)$ with some real function that satisfies these boundary condition. The simplest choice is a polynomial,
\begin{equation}\label{ans} b(t)=\sum_{j=0}^5 a_j t^j. \end{equation}
Once the $a_j$ are determined from (\ref{bes}), $\omega(t)$ is obtained from the Ermakov equation (\ref{subsi}), and one gets directly a transitionless Hamiltonian $H_{II}(t)=H_0(t)$ with a local, ordinary, harmonic potential, but note that $\omega(t)^2$ may become negative for some time interval, making the potential an expulsive parabola \cite{harmo,Salomon}.
The II method is thus clearly distinct from from TT and implements a different Hamiltonian. Note also, by comparison of the coefficients, that the invariant operator $I$ corresponding to $H_{II}$ is different from $H_{TT}$, although they are both generalized harmonic oscillators.
\subsection{Physical realization}
The TT method only requires the time variation of a parabolic potential. Effective harmonic optical traps for neutral atoms may be formed by magnetic and/or optical means and their frequencies are routinely varied in time as part of many cold atom experiments. In magnetic traps, for example the frequency is modulated harmonically to look for collective excitation modes of a condensate \cite{Cornell96}, and ramped down adiabatically to change its conditions (critical temperature, particle number, spatial extension) \cite{Ketadiab,Cornell96}, or as a preliminary step to superimpose an optical lattice \cite{Cas}. Some experiments involve both time-dependent magnetic and optical traps or antitraps \cite{ch}. Purely optical traps are also manipulated in time, e.g. for adiabatic cooling of single neutral atoms \cite{acoo}.
In particular laser beams detuned with respect to the atomic transition form effective potentials for the ground state depending on Rabi frequency $\Omega$ and detuning $\Delta$ as $\Omega^2/4\Delta$ by adiabatic elimination of the excited state, thus forming attractive or repulsive potentials. This effective interaction can be made time dependent by varying the laser intensity, the frequency, or both \cite{Bize}, since the optical frequencies are many orders of magnitude larger than Rabi frequencies or detunings, and the changes will be slowly varying in the scale of optical periods. The intensity of a dipole trap can be changed by three or four orders of magnitude in $100$ ns using acousto-optics or electro-optics modulators. To monitor the sign of the square frequencies, one can superimpose two dipole beams locked respectively on the blue and red side of the line. By controlling their relative intensity, one can shape the square frequencies and their signs at will.
\section{Discussion}
We have compared and distinguished two different methods: a ``transitionless-tracking'' (TT) algorithm, and an ``inverse-invariant'' (II) method, to achieve transitionless dynamics for a fast frequency change of a quantum harmonic oscillator. They imply different driving Hamiltonians. The one in the II method can be implemented for ultracold atoms or ions in the laboratory by varying the trap frequency in time along a certain trajectory, and a generalization to Bose Einstein condensates has been worked out \cite{becs}, but its extension to other potentials or systems may be difficult and remains an open question. By contrast, we have found some difficulties to realize the TT Hamiltonian for the harmonic oscillator, but the TT method has the advantage of being, at least formally, more generally applicable. The feasibility of the actual realization is quite another matter and has to be studied in each case. An example of application is provided in \cite{Berry09} for two-level systems.
\ack{We acknowledge funding by Projects No. GIU07/40, FIS2009-12773-C02-01, 60806041, 08QA14030, 2007CG52, S30105,
and Juan de la Cierva Program.}
\appendix \section{Relation to the squeezing operator}
The evolution operator takes a particularly simple form when using the simplified case $E_n(t)=0$, so that $\hat{H}(t)=\hat{H}_1(t)$. Taking into account that $[\hat{H}_1(t),\hat{H}_1(t')]=0$ we can write
\begin{equation} \hat{U}(t)=e^{-i\int_0^t \hat{H}_1(t)dt/\hbar}. \end{equation}
This may be evaluated explicitly with (\ref{h1a}) fixing the time of the creation and annihilation operators to 0,
\begin{equation} \hat{U}(t)=e^{\frac{1}{2}\ln\left(\sqrt{\frac{\omega(t)}{\omega(0)}}\right) \left[a_0^2-(a_0^\dagger)^2\right]}=\hat{S}[r(t)], \end{equation}
which is a sqeezing operator with real argument $r(t)=\ln\left(\sqrt{\frac{\omega(t)}{\omega(0)}}\right)$. It is unitary with inverse $[\hat{S}(r)]^{-1}=\hat{S}(-r)$. Using the relations
\begin{eqnarray} \hat{a}_t^\dagger+\hat{a}_t&=&\sqrt{\frac{\omega(t)}{\omega(0)}}(\hat{a}_0^\dagger+\hat{a}_0) \nonumber \\ \hat{a}_t^\dagger-\hat{a}_t&=&\sqrt{\frac{\omega(0)}{\omega(t)}}(\hat{a}_0^\dagger-\hat{a}_0) \end{eqnarray}
and the formal properties of $\hat{S}$, see e.g. \cite{bar}, it is easy to prove that
\begin{eqnarray} \hat{S}(r)\hat{a}_0\hat{S}(-r)&=&\hat{a}_t, \nonumber \\ \hat{S}(r)\hat{a}^\dagger_0\hat{S}(-r)&=&\hat{a}_t. \end{eqnarray}
In fact any combination of powers of $\hat{a}_0$ and $\hat{a}^\dagger_0$ is mapped to the same combination of powers of $\hat{a}_t$ and $\hat{a}^\dagger_t$ by this unitary transformation.
To show that $|0_t\rangle\equiv \hat{S}|0_0\rangle$ is indeed the vacuum at time $t$, note that
\begin{equation}
\hat{a}_t|0_t\rangle=\hat{S}(r)\hat{S}(-r)a\hat{S}(r)|0_0\rangle=\hat{S}(r)\hat{a}_0|0_0\rangle=0. \end{equation}
Similarly we note that, consistently,
\begin{eqnarray}
\hat{S}(r)|n(0)\rangle&=&\hat{S}(r)\frac{1}{\sqrt{n!}}(\hat{a}_0^\dagger)^n|0_0\rangle
=\frac{1}{\sqrt{n!}}\hat{S}(r)(\hat{a}_0^\dagger)^n\hat{S}(-r)\hat{S}(r)|0_0\rangle \nonumber \\
&=&\frac{1}{\sqrt{n!}}(\hat{a}_t^\dagger)^n|0_t\rangle=|n(t)\rangle. \end{eqnarray}
\section*{References}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\author[a]{Galit Ashkenazi-Golan} \author[b]{J\'{a}nos Flesch} \author[c]{Arkadi Predtetchinski} \author[d]{Eilon Solan}
\affil[a]{London School of Economics and Political Science, Houghton Street London WC2A 2AE, UK} \affil[b]{Department of Quantitative Economics, Maastricht University, P.O.Box 616, 6200 MD, The Netherlands} \affil[c]{Department of Economics, Maastricht University, P.O.Box 616, 6200 MD, The Netherlands} \affil[d]{School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel, 6997800}
\title{Regularity of the minmax value and equilibria in multiplayer Blackwell games\thanks{Ashkenazi-Golan acknowledges the support of the Israel Science Foundation, grants \#217/17 and \#722/18, and the NSFC-ISF Grant \#2510/17. Solan acknowledges the support of the Israel Science Foundation, grant \#217/17. This work has been partly supported by COST Action CA16228 European Network for Game Theory.}} \maketitle
\begin{abstract} \noindent
A real-valued function $\varphi$ that is defined over all Borel sets of a topological space is \emph{regular} if for every Borel set $W$, $\varphi(W)$ is the supremum of $\varphi(C)$, over all closed sets $C$ that are contained in $W$, and the infimum of $\varphi(O)$, over all open sets $O$ that contain $W$.
We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player's minmax value is regular.
We then use the regularity of the minmax value to establish the existence of $\varepsilon$-equilibria in two distinct classes of Blackwell games. One is the class of $n$-player Blackwell games where each player has a finite action space and an analytic winning set, and the sum of the minmax values over the players exceeds $n-1$. The other class is that of Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values.
For the latter class, we obtain a characterization of the set of equilibrium payoffs. \end{abstract}
\noindent\textbf{Keywords:} Blackwell games, determinacy, value, equilibrium, regularity.
\noindent\textbf{AMS classification code:} Primary: \textsc{91A44} (Games involving topology, set theory, or logic). Secondary: \textsc{91A20} (Multistage and repeated games).
\section{Introduction} Blackwell games (Blackwell \cite{Blackwell69}) are dynamic multiplayer simultaneous-move games where the action sets of the players may be history dependent, and the payoff function is an arbitrary Borel-measurable function of the play. When the payoff function of a player is given by the characteristic function of a given set $W$, we say that $W$ is the \emph{winning set} of the player. These games subsume several familiar classes of dynamic games: repeated games with the discounted payoff or the limiting average payoff (e.g., Sorin \cite{Sorin92}, Mailath and Samuelson \cite{{Mailath06}}), games with perfect information (e.g., Gale and Stewart \cite{Gale53}), and graph games arising in the computer science applications (e.g., Apt and Gr\"{a}del \cite{AptGradel12}, Bruy\`{e}re \cite{Bruyere17, Bruyere21}, Chatterjee and Henzinger \cite{Chatterjee12}).
While two-player zero-sum Blackwell games and Blackwell games with perfect information are quite well understood (see, e.g., Martin \cite{Martin75, Martin98}, Mertens \cite{Mertens86}, Kuipers, Flesch, Schoenmakers, and Vrieze \cite{Kuipers21}), general multiplayer nonzero-sum Blackwell games have so far received relatively little attention.
The goal of this paper is to introduce a new technique to the study of multiplayer Blackwell games: regularity of the minmax value, along with a number of related approximation results. In a nutshell, the technique amounts to the approximation of the minmax value of a winning Borel set using a closed subset. This approach allows us to establish existence of $\varepsilon$-equilibria in two distinct classes of Blackwell games.
\noindent\textsc{Regularity and approximation results:} A real-valued function $\varphi$ that is defined over all Borel sets of a certain space is \emph{inner regular} if for every Borel set $W$, $\varphi(W)$ is the supremum of $\varphi(C)$, over all closed sets $C$ that are contained in $W$. The function $\varphi$ is \emph{outer regular} if for every Borel set $W$ it is the infimum of $\varphi(O)$, over all open sets $O$ that contain $W$. The function $\varphi$ is \emph{regular} if it is both inner regular and outer regular. Borel probability measures on metric spaces are one example of a regular function (see, e.g., Kechris \cite[Theorems 17.10 and 17.11]{Kechris95}).
When restricted to two-player zero-sum Blackwell games with finite action sets and Borel-measurable winning set for Player~1, the value function is known (Martin \cite{Martin98}) to be regular. This result was extended to two-player zero-sum stochastic games by Maitra, Purves, and Sudderth \cite{Maitra92}.
We show that in multiplayer Blackwell games with countable action sets and Borel winning sets, the minmax value of all players is regular. We thus extend the regularity result of Martin \cite{Martin98} in terms of both the number of actions (countable versus finite) and the number of players (finite versus two).
A related approximation result concerns the case when a player's objective is represented by a bounded Borel-measurable payoff function. Denote by $v_i(f)$ player~$i$'s minmax value when her payoff function is $f$. We show that $v_i(f)$ is the supremum of $v_i(g)$ over all bounded limsup functions $g \leq f$, and the infimum of $v_i(g)$ over all bounded limsup function $g \geq f$. A \emph{limsup function} is a function that can be written as the limit superior of a sequence of rewards assigned to the nodes of the game tree. This too, is an extension of results by Maitra, Purves, and Sudderth \cite{Maitra92} and Martin \cite{Martin98} for two-player games to multiplayer games. If, moreover, the player's minmax value is the same in every subgame, one obtains an approximation from below by an upper semi-continuous function, and an approximation from above by a lower semi-continuous function.
\noindent\textsc{Existence of $\varepsilon$-equilibria:} The main contribution of the paper is the application of the regularity of the minmax value to the problem of existence of an $\varepsilon$-equilibrium in multiplayer Blackwell games. We establish the existence in two distinct classes of Blackwell games.
One is the class of $n$-player Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values. The latter assumption means that every player's minmax value is the same in each subgame. Under these assumptions, for each $\varepsilon > 0$, there is an $\varepsilon$-equilibrium with a pure path of play.
A prominent sufficient condition for the minmax value to be history-independent the is that payoff be tail-measurable. Roughly speaking, tail-measurability amounts to the requirement that the payoff is unaffected by a change of the action profile in any finite number of stages. We thus obtain the existence of $\varepsilon$-equilibria in Blackwell games with history-independent finite action spaces and bounded, upper semi-analytic, and tail-measurable payoff functions.
The second class of games for which we derive an existence result is $n$-player Blackwell games where each player has a finite action space at each history, her objective is represented by an analytic winning set, and the sum of the minmax values over the players exceeds $n-1$. Under these conditions we show that there exists a play that belongs to each player's winning set; any such play induces a $0$-equilibrium. At the heart of the proof is an approximation of each player's minmax value by the minmax value of a closed subset of the player's winning set.
The key idea of the proof of the first result is to consider an auxiliary Blackwell game with winning sets, where the winning set of player $i$ is the set of player $i$'s $\varepsilon$-individually rational plays: the plays that yield player $i$ a payoff no smaller then her minmax value minus $\varepsilon$. We show that, in the thus-defined auxiliary Blackwell game, each player's minmax value equals 1, and apply the second\textbf{} result.
The question whether $\varepsilon$-equilibria exist in multiplayer Blackwell games is a largely uncharted territory. An important benchmark is the result of Mertens and Neyman (see Mertens \cite{Mertens86}): all games of perfect information with bounded Borel-measurable payoff functions admit an $\varepsilon$-equilibrium for every $\varepsilon > 0$. Zero-sum Blackwell games (where at least one of the two players has a finite set of actions) are known to be determined since the seminal work of Martin \cite{Martin98}. Shmaya \cite{Shmaya11} extends the latter result by showing the determinacy of zero-sum games with eventual perfect monitoring, and Arieli and Levy \cite{Arieli15} extend Shmaya's result to stochastic signals.
Only some special classes of multiplayer dynamic games have been shown to have an $\varepsilon$-equilibrium. These include stochastic games with discounted payoffs (see, e.g., the survey by Ja\'{s}kiewicz and Nowak \cite{Nowak16}), two-player stochastic games with the limiting average payoff (Vieille \cite{Vieille00I,Vieille00II}), and graph games with classical computer science objectives (e.g., Secchi and Sudderth \cite{Sechi}, Chatterjee \cite{Chatterjee04,Chatterjee05}, Bruy\`{e}re \cite{Bruyere21}, Ummels, Markey, Brenguier, and Bouyer \cite{Ummels15}).
A companion paper (\cite{AFPS}) establishes the existence of $\varepsilon$-equilibria in Blackwell games with countably many players, finite action sets, and bounded, Borel-measurable, and tail-measurable payoff functions. The present paper departs from \cite{AFPS} in two dimensions. Firstly, it invokes a new proof technique, the regularity of the minmax value. Secondly, it makes different assumptions on the primitives. The second of our two existence results (Theorem \ref{theorem:sumofprob}) has, in fact, no analogue in \cite{AFPS}. The first (Theorem \ref{theorem:minmax_indt}) applies to a larger class of payoff functions than does the main result in \cite{AFPS}: it only requires players' minmax values to be history-independent. While tail-measurability of the payoff functions is a sufficient condition for history-independence of the minmax values, it is by no means a necessary condition. Furthermore, Borel-measurability imposed in \cite{AFPS} is relaxed here to upper semi-analyticity. On the other hand, \cite{AFPS} has a countable rather than a finite set of players, something that the methods developed here do not allow for.
\noindent\textsc{Characterisation of equilibrium payoffs:} An equilibrium payoff is an accumulation point of the expected payoff vectors of $\varepsilon$-equilibria, as $\varepsilon$ tends to $0$. We establish a characterisation of equilibrium payoffs in games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values.
In repeated games with patient players the folk theorem asserts that under proper conditions, the set of limiting average equilibrium payoffs (or the limit set of equilibrium payoffs, as the discount factor goes to 0 or the horizon increases to infinity) is the set of all vectors that are individually rational and lie in the convex hull of the range of the stage payoff function (see, e.g., Aumann and Shapley \cite{Aumann94}, Sorin \cite{Sorin92}, or Mailath and Samuelson \cite{{Mailath06}}). Our result identifies the set of equilibrium payoffs of a Blackwell game as the set of all vectors that lie in the convex hull of the set of feasible and individually rational payoffs. The intuition for this discrepancy is that in standard repeated games, a low payoff in one stage can be compensated by a high payoff in another stage, therefore payoff vectors that are convex combinations of the stage payoff function can be equilibrium payoffs as long as this convex combination of payoffs is individually rational. In particular, these combinations can place some positive weight on payoff vectors that are not individually rational. In Blackwell games, however, the payoff is obtained only at the end of the game, hence only plays that generate individually rational payoffs can be taken into account when constructing equilibria.
Our characterization of the set of equilibrium payoffs is related to the rich literature on the folk theorem, and the study of the minmax value is instrumental to this characterizaion (see, e.g., the folk theorems in Fudenberg and Maskin \cite{Fudenberg86}, Mailath and Samuelson \cite{Mailath06}, or H\"{o}rner, Sugaya, Takahashi, and Vieille \cite{Horner11}). The minmax value of a player would often be used in the proofs of equilibrium existence to construct suitable punishments for a deviation from the supposed equilibrium play (as is done, for instance, in Aumann and Shapley \cite{Aumann94}, Rubinstein \cite{Rubinstein94}, Fudenberg and Maskin \cite{Fudenberg86}, and Solan \cite{Solan01}).
The paper is structured as follows. Section \ref{secn.games} describes the class of Blackwell games. Section \ref{secn.approx} is devoted to the regularity of the minmax value and related approximation theorems. Section \ref{secn.appl} applies these tools to the problem of existence of equilibrium. Section~\ref{secn.folk} is devoted to the characterisation of equilibrium payoffs. Section \ref{secn.tail} discusses the implications of the results for games with tail-measurable payoffs. Section \ref{secn.disc} contains a discussion, concluding remarks, and open questions.
\section{Blackwell games}\label{secn.games} \noindent\textbf{Blackwell games:} An $n$-\textit{player Blackwell game} is a tuple $\Gamma = (I, A, H, (f_{i})_{i \in I})$. The elements of $\Gamma$ are as follows.
The set of players is $I$, a finite set of cardinality $n$. For a player $i \in I$ we write $-i$ to denote the set of $i$'s opponents, $I \setminus \{i\}$.
The set $A$ is a countable set and $H \subseteq \cup_{t \in \mathbb{N}}A^{t}$ is the game tree (throughout the paper $\mathbb{N} = \{0,1,\ldots\}$). Elements of $H$ are called histories. The set $H$ is assumed to have the following properties: (a) $H$ contains the empty sequence, denoted $\oslash$; (b) a prefix of an element of $H$ is an element of $H$; that is, if for some $h \in \cup_{t \in \mathbb{N}}A^{t}$ and $a \in A$ the sequence $(h,a)$ is an element of $H$, so is $h$; (c) for each $h \in H$ there is an element $a \in A$ such that $(h, a) \in H$; we define $A(h) := \{a \in A: (h,a) \in H\}$; and (d) for each $h \in H$ and each $i \in I$ there exists a set $A_{i}(h)$ such that $A(h) = \prod_{i \in I}A_{i}(h)$. The set $A_{i}(h)$ is called player $i$'s set of actions at history $h$, and $A(h)$ the set of action profiles at $h$.
Conditions (a), (b), and (c) above say that $H$ is a pruned tree on $A$. Condition (c) implies that the game has infinite horizon. Let $H_{t} := H \cap A^{t}$ denote the set of histories in stage $t$.
An infinite sequence $(a_0,a_1,\ldots) \in A^{\mathbb{N}}$ such that $(a_0,\ldots,a_t) \in H$ for each $t \in \mathbb{N}$ will be called a \emph{play}. The set of plays is denoted by $[H]$. This is the set of infinite branches of $H$. For $h \in H$ let $O(h)$ denote the set of all plays of $\Gamma$ having $h$ as a prefix. We endow $[H]$ with the topology generated by the basis consisting of the sets $\{O(h):h \in H\}$. The space $[H]$ is Polish. For $t \in \mathbb{N}$ let $\mathcal{F}_{t}$ be the sigma-algebra on $[H]$ generated by the sets $\{O(h):h \in H_{t}\}$. The Borel sigma-algebra of $[H]$ is denoted by $\mathscr{B}$. It is the minimal sigma-algebra containing the topology. A subset $S$ of $[H]$ is \emph{analytic} if it is the image of a continuous function from the Baire space $\mathbb{N}^\mathbb{N}$ to $[H]$. Each Borel set is analytic.
Each analytic set is universally measurable. Recall that a set $S \subseteq [H]$ is said to be universally measurable if (Kechris \cite[Section 17.A]{Kechris95}), for every Borel probability measure $\mathbb{P}$ on $[H]$, there exist Borel sets $B,Z \in \mathscr{B}$ such that $S \bigtriangleup B \subseteq Z$ and $\mathbb{P}(Z) = 0$; here $S \bigtriangleup B = (S \setminus B) \cup (B \setminus S)$ is the symmetric difference of the sets $S$ and $B$.
The last element of the game is a vector $(f_i)_{i \in I}$, where $f_i : [H] \to \mathbb{R}$ is player $i$'s \emph{payoff function}. The most general class of payoff functions we allow for are bounded upper semi-analytic functions. A function $f_i : [H] \to \mathbb{R}$ is said to be \emph{upper semi-analytic} if, for each $r \in \mathbb{R}$, the set $\{p \in [H]: r \leq f(p)\}$ is analytic. In particular, the indicator function $1_{S}$ of a subset $S \subseteq [H]$ of plays is upper semi-analytic if and only if $S$ is an analytic set. Each Borel-measurable function in upper semi-analytic. Note that a bounded upper semi-analytic function is universally measurable, i.e., for each open set $U \subseteq \mathbb{R}$, the set $f^{-1}(U) \subseteq [H]$ is universally measurable (see, e.g., Chapter 7 in Bertsekas and Shreve \cite{Bertsekas96}).
The play of the game starts at the empty history $h_0 = \oslash$. Suppose that by a certain stage $t \in \mathbb{N}$ a history $h_t \in H_t$ has been reached. Then in stage $t$, the players simultaneously choose their respective actions; thus player $i \in I$ chooses an action $a_{i,t} \in A_{i}(h_{t})$. This results in the stage $t$ action profile $a_{t} = (a_{i,t})_{i \in I} \in A(h_{t})$. Once chosen, the actions are revealed to all players, and the history $h_{t+1} = (h_{t},a_{t})$ is reached. The result of the infinite sequence of choices is the play $p = (a_0,a_1,\ldots)$, an element of $[H]$. Each player $i \in I$ receives the corresponding payoff $f_i(p)$.
Given a Blackwell game $\Gamma$ and a history $h \in H$, the \textit{subgame} of $\Gamma$ starting at $h$ is the Blackwell game $\Gamma_h = (I, A, H_h, (f_{i,h})_{i \in I})$. The set $H_h$ of histories of $\Gamma_h$ consists of finite sequences $g \in \bigcup_{t \in \mathbb{N}} A^{t}$ such that $hg \in H$, where $hg$ is the concatenation of $h$ and $g$. The payoff function $f_{i,h} : [H_h] \to \mathbb{R}$ is the composition $f_i \circ s_h$, with $s_h : [H_h] \to [H]$ given by $p \mapsto hp$, where $hp$ is the concatenation of $h$ and $p$. Note that $\Gamma_\oslash$ is just the game $\Gamma$ itself.
The Blackwell game $\Gamma$ is said to have \textit{history-independent action sets} if $A_{i}(h) = A_{i}(\oslash)$ for each history $h \in H$ and each player $i \in I$; the common action set is simply denoted by $A_i$. If $\Gamma$ has history-independent action sets, then the set of its histories is $H = \cup_{t \in \mathbb{N}}A^{t}$, and the set of plays in $\Gamma$ is $[H] = A^{\mathbb{N}}$. A Blackwell game with history-independent action sets can be described as a tuple $(I,(A_{i},f_{i})_{i \in I})$.
\noindent\textbf{Strategies and expected payoffs:} A strategy for player $i\in I$ is a function $\sigma_i$ assigning to each history $h \in H$ a probability distribution $\sigma_{i}(h)$ on the set $A_{i}(h)$. The set of player $i$'s strategies is denoted by $\Sigma_i$. We also let $\Sigma_{-i} := \prod_{j \in -i} \Sigma_{j}$ and $\Sigma := \prod_{i \in I} \Sigma_{i}$. Each strategy profile $\sigma=(\sigma_i)_{i\in I}$ induces a unique probability measure on the Borel sets of $[H]$, denoted $\mathbb{P}_{\sigma}$. The corresponding expectation operator is denoted $\mathbb{E}_{\sigma}$. In particular, $\mathbb{E}_{\sigma}[f_{i}]$ denotes an expected payoff to player $i$ in the Blackwell game under the strategy profile $\sigma$. It is well defined under the maintained assumptions, namely boundedness and upper semi-analyticity of $f_{i}$.
Take a history $h \in H_t$ in stage $t$. A strategy profile $\sigma \in \Sigma$ in $\Gamma$ induces the strategy profile $\sigma_h$ in $\Gamma_h$ defined as $\sigma_h(g)=\sigma(hg)$ for each history $g \in H_h$. Let us define $\mathbb{E}_{\sigma}(f_i \mid h)$ as the expected payoff to player $i$ in the Blackwell game $\Gamma_h$ under the strategy profile $\sigma_h$: that is, $\mathbb{E}_{\sigma}(f_i \mid h) := \mathbb{E}_{\sigma_h}(f_{i,h})$. Note that $\mathbb{E}_{\sigma}(f_i \mid h)$, when viewed as an $\mathcal{F}_{t}$-measurable function on $[H]$, is a conditional expectation of $f_i$ with respect to the measure $\mathbb{P}_{\sigma}$ and the sigma-algebra $\mathcal{F}_{t}$; whence our choice of notation.
\noindent\textbf{Minmax value:} Consider a Blackwell game $\Gamma$, and suppose that player $i$'s payoff function $f_i$ is bounded and upper semi-analytic. Player $i$'s \textit{minmax value} is defined as \[v_i(f_i) := \inf_{\sigma_{-i} \in \Sigma_{-i}}\sup_{\sigma_{i} \in \Sigma_{i}} \mathbb{E}_{\sigma_{-i},\sigma_{i}}(f_i).\] Whenever $f_i = 1_{W_i}$ is an indicator of an analytic set $W_i \subseteq [H]$ we write $v_i(W_i)$ for $v_i(1_{W_i})$.
Player $i$'s minmax value is said to be \textit{history-independent} if her minmax value in the subgame $\Gamma_h$ equals that in the game $\Gamma$, for each history $h \in H$.
\section{Regularity and approximation theorems}\label{secn.approx} In this section we state the regularity property of the minmax: the minmax value of a Borel winning set can be approximated from below by the minmax value of closed subset and from above by the minmax value of an open superset. We also describe two related approximation results: the minmax value of a bounded Borel-measurable payoff function can be approximated from below and from above by limsup functions. If, in addition, the minmax values are history-independent, then one can choose the approximation from below to be upper semicontinuous, and the approximation from above to be lower semicontinuous. The proofs of all results are detailed in the appendix.
\begin{theorem} {\rm (Regularity of the minmax value)}\label{thrm:reg} Consider a Blackwell game. Suppose that player $i$'s objective is given by a winning set $W_i \subseteq [H]$. Suppose that $W_i$ is Borel. Then \begin{align*} v_i(W_i) &= \sup\{v_i(C):C\subseteq W_i, C \text{ is closed}\}=\inf\{v_i(O):O\supseteq W_i, O \text{ is open}\}. \end{align*} \end{theorem}
One implication of Theorem~\ref{thrm:reg} concerns the complexity of strategies of player~$i$ that ensures that her probability of winning is close to her minmax value. Suppose, for example, that $v_i(W_i) = \frac{1}{2}$. Then for every strategy profile $\sigma_{-i}$ of the opponents of player~$i$ and every $\varepsilon > 0$, she has a response $\sigma_i$ such that $\mathbb{P}_{\sigma_{-i},\sigma_i}(W_i) \geq \frac{1}{2}-\varepsilon$. The strategy profile $\sigma_{-i}$ and the winning set $W_i$ may be complex, and accordingly the good response $\sigma_i$ may be complex as well. However, take now a closed subset $C \subseteq W_i$ such that $v_i(C) > v_i(W_i) - \varepsilon = \frac{1}{2}-\varepsilon$. The complement of $C$, denoted $C^c$, is open, hence it is the union of basic open sets; that is, it can be presented as a union $C^c = \bigcup_{h \in H'} O(h)$, for some subset $H' \subseteq H$ of histories. A strategy $\sigma'_i$ that satisfies $\mathbb{P}_{\sigma_{-i},\sigma_{i}'}(C_i) \geq \frac{1}{2}-\varepsilon$ must aim at avoiding $C^c$, that is, at avoiding histories in $H'$. In that sense, $\sigma'$ may have a simple structure.
\begin{exl}\label{exl.io}\rm Here we consider a Blackwell game where the same stage game is being played at every stage. The stage game specifies a stage winning set for each player. A player's objective in the Blackwell game is to win the stage game infinitely often.
Thus let $\Gamma = (I,(A_{i},1_{W_{i}})_{i \in I})$ be a Blackwell game with history-independent countable action sets, where player $i$'s winning set is \[W_i = \{(a_0,a_1,\ldots) \in A^\mathbb{N}:a_t \in U_i\text{ for infinitely many }t \in \mathbb{N}\};\] here $U_{i}$, called player $i$'s \textit{stage winning set}, is a given subset of $\prod_{i \in I}A_{i}$. If $a_t \in U_i$, we say that player $i$ \textit{wins stage $t$}. Thus, player~$i$'s objective is to win infinitely many stages of the Blackwell game. The set $W_{i}$ is a $G_\delta$-set, i.e., an intersection of countably many open subsets of $A^\mathbb{N}$.
Fix a player $i \in I$. Let \begin{equation}\label{eqn.stageminmax} d_{i} := \inf_{x_{-i} \in X_{-i}} \sup_{x_i \in X_i} \mathbb{P}_{x_{-i},x_i}(U_i) \end{equation} be player~$i$'s minmax value in the stage game. As follows from the arguments below, $v_{i}(W_{i})$ is either $0$ or $1$, and it is $1$ exactly when $d_{i} > 0$. In either case, there are intuitive approximations of player $i$'s wining sets by a closed set from below and an open set from above.
First assume that $d_{i} > 0$. Take an $\varepsilon > 0$. Let us imagine that player $i$'s objective is not merely to win infinitely many stages in the course of the Blackwell game, but to make sure that she wins at least once in every block of stages $t_{n},\ldots,t_{n+1}-1$, where the sequence of stages $t_0 < t_1 < \cdots$ is chosen to satisfy \[(1 - \tfrac{1}{2}d_{i})^{t_{n+1} - t_{n}} < 2^{-n-1}\cdot\varepsilon\] for each $n \in \mathbb{N}$. This, more demanding condition, defines an approximating set. Formally, define \[C_{i} := \bigcap_{n \in \mathbb{N}}\bigcup_{t_{n} \leq k < t_{n+1}} \{(a_0,a_1,\ldots) \in A^{\mathbb{N}}: a_k \in U_i\}.\]
As the intersection of closed sets, $C_{i}$ is a closed subset of $W_{i}$. Moreover, $1 - \varepsilon \leq v_{i}(C_{i})$. To see this, fix any strategy $\sigma_{-i}$ for $i$'s opponents. At any history $h$, player $i$ has a mixed action $\sigma_{i}(h)$ that, when played against $\sigma_{-i}(h)$, guarantees a win at history $h$ with probability of at least $\tfrac{1}{2}d_{i}$. Thus, under the measure $\mathbb{P}_{\sigma_{-i},\sigma_{i}}$ the probability for player $i$ not to win at least once in a block of stages $t_{n},\ldots,t_{n+1}-1$ is at most $2^{-n-1}\cdot\varepsilon$, for any history of play up to stage $t_{n}$. And hence the probability that there is a block within which player $i$ does not win once is at most $\varepsilon$.
Suppose that $d_{i} = 0$. Let us imagine that player $i$'s objective is merely to win the stage game at least once. This modest objective defines the approximating set: \[O_{i} = \bigcup_{t \in \mathbb{N}}\{(a_{0},a_{1},\ldots)\in A^\mathbb{N}: a_{t} \in U_i\}.\] As the union of open sets, $O_{i}$ is an open set containing $W_{i}$. Moreover, $v_{i}(O_{i}) \leq \varepsilon$. To see this, let $\sigma_{-i}$ be the strategy for $i$'s opponents such that, at any stage $t \in \mathbb{N}$ and any history $h \in A^{t}$ of stage $t$, the probability that the action profile $a_{t}$ is an element of $U_{i}$ is not greater than $2^{-t-1}\cdot\varepsilon$ regardless of the action of player $i$. Then, for any player $i$'s strategy $\sigma_{i}$, the probability that $i$ wins at least once is not greater than $\varepsilon$. $\Box$
\end{exl}
We turn to two related approximation results for Blackwell games with Borel payoff functions. A function $f : [H] \to \mathbb{R}$ is said to be a \textit{limsup function} if there exists a function $u : H \to \mathbb{R}$ such that for each play $(a_0,a_1,\ldots) \in [H]$, \[f(a_0,a_1,\ldots) = \limsup_{t \to \infty} u(a_0,\dots,a_t).\] The function $f : [H] \to \mathbb{R}$ is a \textit{liminf function} if $-f$ is a limsup function.
Limsup and liminf payoff functions are ubiquitous in the literature on infinite dynamic games. At least since the work of Gillette \cite{Gillette57}, the so-called limiting average payoff (that is, the limit superior or the limit inferior of the average of the stage payoffs) is a standard specification of the payoffs in a stochastic game (see for example Mertens and Neyman \cite{Mertens81}, or Levy and Solan \cite{Levy20}). Stochastic games with limsup payoff functions have been studied in Maitra and Sudderth \cite{Maitra93}.
Limsup functions have relatively ``low" set-theoretic complexity. Various characterizations of the limsup functions can be found in Hausdorff \cite{Hausdorff05}. In particular, $f$ is a limsup function if and only if, for each $r \in \mathbb{R}$, the set $\{p \in [H]: r \leq f(p)\}$ is a $G_{\delta}$-set.
We now state a result on the approximation of the minmax value for Blackwell games where a player's objective is represented by a bounded Borel-measurable payoff function.
\begin{theorem}\label{thrm:regfunc} Consider a Blackwell game. Suppose that player $i$'s payoff function $f_i : [H] \to \mathbb{R}$ is bounded and Borel-measurable. Then: \[\begin{aligned} v_i(f_i) &= \sup\{v_i(g): g\text{ is a bounded limsup function and }g \leq f_i\}\\ &= \inf\{v_i(g): g \text{ is a bounded limsup function and }f_i \leq g\}. \end{aligned}\] \end{theorem}
Theorems~\ref{thrm:reg} and~\ref{thrm:regfunc} have been proven by Martin \cite{Martin98} for the case $n=2$, see \cite[Theorem 5, and Remark (b)]{Martin98} and \cite[Remark (c)]{Martin98}. They have been extended to two-player stochastic games by Maitra and Sudderth \cite{Maitra98}. Theorems~\ref{thrm:reg} and~\ref{thrm:regfunc} extend the known results in two respects. First, they allow for more than two players, and second, they allow for countably many actions.
The proof of Theorem \ref{thrm:reg} combines and fine-tunes the arguments in Martin \cite{Martin98} and Maitra and Sudderth \cite{Maitra98}. The key element of the proof is a zero-sum perfect information game, denoted $G_i(f_i,c)$, where the aim of Player I is to ``prove" that the minmax value of $f_i$ is at least $c$. Roughly speaking, the game proceeds as follows. Player~I commences the game by proposing a fictitious continuation payoff, which one could think of as a payoff player $i$ hopes to attain, contingent on each possible stage $0$ action profile. The number $c$ serves as the initial threshold: player $i$'s minmax value of the proposed continuation payoffs is required to be at least $c$. Player II then chooses a stage $0$ action profile, and the corresponding continuation payoff serves as the new threshold. Player I then proposes a fictitious continuation payoff contingent on each possible stage $1$ action profile, and Player II chooses the stage $1$ action profile, etc. Player I wins if the sequence of continuation payoffs is ``justified" by the actual payoff on a play produced by Player II. Ultimately the proof rests on the determinacy of the game $G_i(f_i,c)$, which follows by Martin \cite{Martin75}.
The perfect information game $G_i(f_i,c)$ is a version of the games used in Martin \cite{Martin98}. The main difference is in the use of player $i$'s minmax value that constrains Player I's choice of fictitious continuation payoffs. The details of our proof are slightly closer to those in Maitra and Sudderth \cite{Maitra98}. Like them we invoke martingale convergence and the Fatou lemma.
Finally, we state an approximation result for a Blackwell game with history-independent minmax values. Recall that a function $g : A^\mathbb{N} \to \mathbb{R}$ is \emph{upper semicontinuous} if, for each $r \in \mathbb{R}$, the set $\{p \in [H]: r \leq f(p)\}$ is a closed set, and $g$ is \emph{lower semicontinuous} if $-g$ is upper semicontinuous. When $g = 1_B$ for some $B \subseteq A^\mathbb{N}$, $g$ is upper semicontinuous (resp.~lower semicontinuous) if and only if $B$ is closed (resp.~open).
\begin{theorem}\label{thrm:tailapprox} Consider a Blackwell game. Suppose that player $i$'s payoff function $f_i$ is bounded and Borel-measurable, and player $i$'s minmax values are history-independent. Then \[\begin{aligned} v_i(f_i) &= \sup\{v_i(g): \text{g is a bounded upper semicontinuous function and }g \leq f_i\}\\ &= \inf\{v_i(g): \text{g is a bounded lower semicontinuous function and }f_i \leq g\}. \end{aligned}\] \end{theorem}
As the proof reveals, both the upper semicontinuous and the lower semicontinuous functions can be chosen to be two-valued. Recall that (Hausdorff \cite{Hausdorff05}) an upper semicontinuous function and a lower semicontinuous function are both a limsup and a liminf function. Consequently, in comparison to Theorem \ref{thrm:regfunc}, an additional assumption of history-independence of the minmax values in Theorem \ref{thrm:tailapprox} leads to a stronger approximation result. The latter condition cannot be dropped; see Section~\ref{secn.disc} for an example of a game with a limsup payoff function such that the minmax value cannot be approximated from below by an upper semicontinuous function.
\section{Existence of equilibria}\label{secn.appl} In this section, we employ the results of the previous section to establish existence of $\varepsilon$-equilibria in two distinct classes of Blackwell games. Theorem~\ref{theorem:sumofprob} concerns $n$-player Blackwell games where each player has a finite action space at each history, her objective is represented by an analytic winning set, and the sum of the minmax values over the players exceeds $n-1$. Theorem~\ref{theorem:minmax_indt} concerns for Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values.
Consider a Blackwell game $\Gamma$ and let $\varepsilon \geq 0$. A strategy profile $\sigma \in \Sigma$ is \emph{an $\varepsilon$-equilibrium} of $\Gamma$ if for each player $i \in I$ and each strategy $\eta_i \in \Sigma_i$ of player $i$, \[\mathbb{E}_{\sigma_{-i},\eta_{i}}(f_i) \leq \mathbb{E}_{\sigma_{-i},\sigma_{i}}(f_i) + \varepsilon.\]
We state our first existence result.
\begin{theorem}\label{theorem:sumofprob} Consider an $n$-player Blackwell game $\Gamma = (I, A, H, (1_{W_{i}})_{i \in I})$. Suppose that for each player $i \in I$ player $i$'s action set $A_i(h)$ at each history $h \in H$ is finite, and that her winning set $W_i$ is analytic. If $v_1(W_1)+ \cdots +v_n(W_n) > n-1$, then the set $W_1 \cap \cdots \cap W_n$ is not empty. Consequently, $\Gamma$ has a 0-equilibrium. \end{theorem}
Note that any play $p \in W_1 \cap \cdots \cap W_n$ is in fact a 0-equilibrium, or more precisely, any strategy profile that requires all the players to follow $p$ is a 0-equilibrium, because it yields all players the maximal payoff 1.
The key step of the proof is the approximation of the minmax value of a player using a closed subset of her winning set. To prove Theorem~\ref{theorem:sumofprob} we need the following technical observation.
\begin{lemma}\label{lemma:intersect} Let $(X,\mathscr{B},P)$ be a probability space, and let $Q_1,\ldots,Q_n\in \mathscr{B}$ be $n$ events. Then \[P(Q_1\cap\cdots\cap Q_n)\geq P(Q_1)+\cdots+P(Q_n)-n+1.\] \end{lemma}
\begin{proof} For $n=1$ the statement is obvious, and for $n=2$ we have \begin{equation}\label{indineq} P(Q_1\cap Q_2)=P(Q_1)+P(Q_2)-P(Q_1\cup Q_2)\geq P(Q_1)+P(Q_2)-1. \end{equation} Assume that the statement holds for some $n-1$. Then for $n$ we have \begin{align*} P(Q_1\cap\cdots\cap Q_n)\,&=\,P((Q_1\cap\cdots\cap Q_{n-1})\cap Q_n)\\ &\geq\, P(Q_1\cap\cdots\cap Q_{n-1}) + P(Q_n)-1\\ &\geq\, \big(P(Q_1)+\cdots+P(Q_{n-1})-n+2\big)+ P(Q_n)-1\\ &=\,P(Q_1)+\cdots+P(Q_n)-n+1, \end{align*} where the first inequality follows from Eq.~\eqref{indineq} and the second by the induction hypothesis. \end{proof}
\noindent\textbf{Proof of Theorem \ref{theorem:sumofprob}:} We first establish the theorem in the special case of Borel winning sets, and then generalize it to analytic winning sets.
\noindent\textsc{Part I:} Suppose that for each $i \in I$ the set $W_{i} \subseteq [H]$ is Borel.
By Theorem \ref{thrm:reg} there are closed sets $C_1\subseteq W_1,\ldots,C_n\subseteq W_n$ such that $v_1(C_1)+\cdots+v_n(C_n) > n-1$. We show that the intersection $C_1 \cap\cdots\cap C_n$ is not empty.
Given $m \in \mathbb{N}$ consider the $n$-player Blackwell game $\Gamma^{m} = (I, A, H, (1_{C_{i}^{m}})_{i \in I})$, where player $i$'s winning set is defined by \[C_i^{m} := \bigcup\{O(h): h \in H_{m}\text{ such that }O(h) \cap C_{i} \neq \oslash\}.\]
The game $\Gamma^{m}$ essentially ends after $m$ stages: by stage $m$ each player $i$ knows whether the play is an element of her winning set $C_{i}^{m}$ or not. In $\Gamma^{m}$, player $i$ wins if after $m$ stages there is a continuation play that leads to $C_i$. Note that this continuation play might be different for different players.
The set $C_i^{m}$ is a clopen set. For each $m \in \mathbb{N}$ and $i \in I$ we have the inclusion $C_i^{m} \supseteq C_i^{m+1}$ (winning in $\Gamma^{m+1}$ is more difficult than winning in $\Gamma^{m}$). Moreover, $\bigcap_{m \in \mathbb{N}} C_i^{m} = C_i$. Indeed, the inclusion $C_i^{m} \supseteq C_i$ is evident from the definition. Conversely, take an element $q$ of the set $[H] \setminus C_i$. Since $[H] \setminus C_i$ is an open set, there exists a history $h \in H$ such that $q \in O(h)$ and $O(h) \subseteq [H] \setminus C_{i}$. But then $q \in [H] \setminus C_i^{m}$, where $m$ is the length of the history $h$.
Define $C^m := C_1^m \cap\cdots\cap C_n^m$. Thus $\{C^{m}\}_{m \in \mathbb{N}}$ is a nested sequence of closed sets converging to $C_1 \cap\cdots\cap C_n$. Note that, since by the assumption of the theorem $H$ is a finitely branching tree, the space $[H]$ is compact. Thus $C^{m}$ is a compact set. Consequently, to prove that $C_{1} \cap \cdots \cap C_{n}$ is not empty, we only need to argue that $C^{m}$ is not empty for each $m \in \mathbb{N}$.
The game $\Gamma^m$ being finite, it has a $0$-equilibrium (Nash \cite{Nash50}), say $\sigma^m$. By the definition of $0$-equilibrium, the equilibrium payoff is not less than the minmax value: \[\mathbb{P}_{\sigma_{-i}^m,\sigma_{i}^{m}} (C_i^m) \,=\, \sup_{\sigma_{i} \in \Sigma_{i}} \mathbb{P}_{\sigma_{-i}^m,\sigma_{i}} (C_i^m)\, \geq\, \inf_{\sigma_{-i} \in \Sigma_{-i}} \sup_{\sigma_{i} \in \Sigma_{i}} \mathbb{P}_{\sigma_{-i},\sigma_{i}} (C_i^m) = v_i(C_i^m).\] Moreover, since $C_i^{m} \supseteq C_{i}$, it holds that $v_i(C_i^m) \geq v_i(C_i)$. We conclude that \[\mathbb{P}_{\sigma^m}(C_1^m)+ \cdots +\mathbb{P}_{\sigma^m}(C_n^m) > n-1.\] Finally, we apply Lemma \ref{lemma:intersect} to conclude that $\mathbb{P}_{\sigma^m}(C^m) > 0$, hence $C^{m}$ is not empty.
\noindent\textsc{Part II:} Now let $\Gamma$ be any game as in the statement of the theorem. Suppose by way of contradiction that $W_1 \cap \cdots \cap W_n$ is empty. By Novikov's separation theorem (Kechris \cite[Theorem 28.5]{Kechris95}) there exist Borel sets $B_1, \ldots, B_n$ such that $W_{i} \subseteq B_{i}$ for each $i \in I$ and $B_1 \cap \cdots \cap B_n = \oslash$. But since $v_{i}(W_{i})\leq v_{i}(B_{i})$ for each $i \in I$, the game $\Gamma = (I, A, H, (1_{B_{i}})_{i \in I})$ satisfies the assumptions of the theorem, and Part I of the proof yields a contradiction. $\Box$
We state our second and main existence result.
\begin{theorem}\label{theorem:minmax_indt} Consider a Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$. Suppose that for each player $i \in I$, player $i$'s action set $A_i(h)$ at each history $h \in H$ is finite, her payoff function $f_i$ is bounded and upper semi-analytic, and her minmax value is history-independent. Then for every $\varepsilon>0$ the game admits an $\varepsilon$-equilibrium. \end{theorem}
The key idea behind the proof is to consider an auxiliary Blackwell game with winning sets, the winning set of a player consisting of that player's $\varepsilon$-individually rational plays. We show that in the thus-defined auxiliary Blackwell game each player's minmax value equals 1, and apply Theorem \ref{theorem:sumofprob}.
Given $\varepsilon > 0$ we define the set of \textit{player $i$'s $\varepsilon$-individually rational plays}: \[Q_{i,\varepsilon}(f_i) := \{p\in [H] : f_i(p) \geq v_i(f_i) - \varepsilon\}.\] Also define the set \[U_{i,\varepsilon}(f_i) := \{p\in [H] : f_i(p) \geq v_i(f_i) + \varepsilon\}.\] Note that under the assumptions of Theorem \ref{theorem:minmax_indt} both sets are analytic.
\begin{proposition}\label{prop:v(Q)=1} Consider a Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$ and a player $i \in I$. Suppose that player $i$'s payoff function $f_i$ is bounded and upper semi-analytic, and that her minmax values are history-independent. Let $\varepsilon > 0$. Then \begin{enumerate} \item $v_i(Q_{i, \varepsilon}(f_i)) = 1$. In fact, for each strategy profile $\sigma_{-i} \in \Sigma_{-i}$ of players $-i$ there is a strategy $\sigma_i \in \Sigma_{i}$ for player $i$ such that $\mathbb{P}_{\sigma_{-i},\sigma_{i}}(Q_{i, \varepsilon}(f_i)) = 1$. \item $v_i(U_{i, \varepsilon}(f_i)) = 0$. In fact, there exists a strategy profile $\sigma_{-i} \in \Sigma_{-i}$ of players $-i$ such that for each strategy $\sigma_i \in \Sigma_{i}$ for player $i$ it holds that $\mathbb{P}_{\sigma_{-i},\sigma_{i}}(U_{i, \varepsilon}(f_i)) = 0$. \end{enumerate}
\begin{proof} \noindent\textsc{Claim 1:} It suffices to prove the second statement. Take a strategy profile $\sigma_{-i}$ of players $-i$. It is known that player $i$ has a strategy $\sigma_i$ that is an $\varepsilon/2$-best response to $\sigma_{-i}$ in each subgame (see, for example, Mashiah-Yaakovi \cite[Proposition 11]{Ayala15}, or Flesch, Herings, Maes, and Predtetchinski \cite[Theorem 5.7]{JJJJ}), and therefore \[\mathbb{E}_{\sigma_{-i},\sigma_i}(f_i \mid h) \geq v_i(f_{i,h}) -\varepsilon/2\,=\,v_i(f_i)-\varepsilon/2,\] for each history $h\in H$. Since the payoff function $f_i$ is bounded, it follows that there is $d>0$ such that \[\mathbb{P}_{\sigma_{-i},\sigma_i}(Q_{i, \varepsilon}(f_i) \mid h) \geq d\] for each $h \in H$. Indeed, it is easy to verify that one can choose \[d\,=\,\frac{\varepsilon}{2(\sup_{p \in [H]}f_i(p)-v_i(f_i)+\varepsilon)}.\]
Since $Q_{i,\varepsilon}(f_i)$ is an analytic set, there is a Borel set $B$ such that $\mathbb{P}_{\sigma_{-i},\sigma_i}(Q_{i, \varepsilon}(f_i) \bigtriangleup B) = 0$, where $\bigtriangleup$ stands for the symmetric difference of two sets. It follows that $\mathbb{P}_{\sigma_{-i},\sigma_i}(Q_{i, \varepsilon}(f_i) \bigtriangleup B \mid h) = 0$, and consequently $\mathbb{P}_{\sigma_{-i},\sigma_i}(B \mid h) \geq d$ for each history $h \in H$ that is reached under $\mathbb{P}_{\sigma_{-i},\sigma_i}$ with positive probability. L\'{e}vy's zero-one law implies that $\mathbb{P}_{\sigma_{-i},\sigma_i}(B) = 1$, and hence $\mathbb{P}_{\sigma_{-i},\sigma_i}(Q_{i, \varepsilon}(f_i)) = 1$.
\noindent\textsc{Claim 2:} By an argument similar to that
in Mashiah-Yaakovi \cite[Proposition 11]{Ayala15} or Flesch, Herings, Maes, and Predtetchinski \cite[Theorem 5.7]{JJJJ}, one shows that there is a strategy profile $\sigma_{-i} \in \Sigma_{-i}$ such that \[\mathbb{E}_{\sigma_{-i},\sigma_i}(f_{i} \mid h) \leq v_{i}(f_{i,h}) + \varepsilon/2 = v_{i}(f_{i}) + \varepsilon/2,\] for each history $h \in H$ and each strategy $\sigma_{i} \in \Sigma_{i}$. Fix any $\sigma_{i} \in \Sigma_{i}$. The rest of the proof of the claim is similar to that of Claim 1. \end{proof} \end{proposition}
\noindent\textbf{The proof of Theorem \ref{theorem:minmax_indt}:} Fix an $\varepsilon > 0$. By Proposition \ref{prop:v(Q)=1}, $v_{i}(Q_{i,\varepsilon}(f_{i})) = 1$.
Let $\Gamma^\varepsilon = (I, A, H, (1_{Q_{i, \varepsilon}(f_i)})_{i \in I})$ be an auxiliary Blackwell game where player $i$'s winning set is $Q_{i, \varepsilon}(f_i)$, the set of player $i$'s $\varepsilon$-individually rational plays in $\Gamma$. Each player's minmax value in the game $\Gamma^\varepsilon$ equals $1$. Therefore, the auxiliary game $\Gamma^\varepsilon$ satisfies the hypothesis of Theorem \ref{theorem:sumofprob}. We conclude that the intersection $\bigcap_{i \in I}Q_{i, \varepsilon}(f_i)$ is not empty, and hence there is a play $p^* \in [H]$ such that $f_i(p^*) \geq v_i(f_i) - \varepsilon$, for every $i \in I$.
The following strategy profile is a $2\varepsilon$-equilibrium of $\Gamma$ (see also Aumann and Shapley \cite{Aumann94}): \begin{itemize} \item The players follow the play $p^*$, until the first stage in which one of the players deviates from this play. Denote by $i$ the minimal index of a player who deviates from $p^*$ at that stage. \item From the next stage and on, the players in $-i$ switch to a strategy profile that reduces player~$i$'s payoff to $v_i(f_i) + \varepsilon$. A strategy profile with this property does exist by the assumption of history-independence of the minmax values. \end{itemize} This completes the proof of the theorem. $\Box$
We illustrate the construction of the $\varepsilon$-equilibrium with the following example.
\begin{exl}\label{exl.eq}\rm We consider a 2-player Blackwell game with history-independent action sets where the same stage game is being played at each stage, and a player's objective is to maximize the long-term frequency of the stages she wins. Specifically, $\Gamma = (\{1,2\}, A_{1}, A_{2}, f_{1}, f_{2})$, where $A_1$ and $A_2$ are finite, and \[f_{i}(a_{0},a_{1},\ldots) = \limsup_{t \to \infty}\tfrac{1}{t} \cdot \#\{k < t : a_{k} \in U_{i}\},\] for each $(a_{0},a_{1},\ldots) \in A^{\mathbb{N}}$. Here $U_i$ is player $i$'s stage winning set. We assume that $U_1$ and $U_2$ are disjoint, and let $d_i$ denote player $i$'s minmax value in the stage game.
Note that $f_i$ is a tail function (see Section \ref{secn.tail}), and it is a limsup function (in the sense of the definition in Section \ref{secn.approx}). We have $d_{i} = v_{i}(f_{i})$, i.e., player~$i$'s minmax value in the stage game is also player~$i$'s minmax value in the Blackwell game.
Take any Nash equilibrium $x \in \prod_{i \in I}\Delta(A_{i})$ of the stage game. Playing $x$ at each stage is certainly a $0$-equilibrium of the Blackwell game $\Gamma$, but typically it is \textbf{not} of the type that appears in the proof of Theorem \ref{theorem:minmax_indt}. An important feature of the $\varepsilon$-equilibrium constructed in the proof is that the equilibrium play is pure; only off the equilibrium path might a player be requested to play a mixed action. In this particular example we can even choose the equilibrium play to be periodic. This can be done as follows.
First note that $d_{1} + d_{2} \leq 1$. This follows since $\mathbb{P}_{x}(U_{1}) + \mathbb{P}_{x}(U_{2}) \leq 1$ (because $U_1$ and $U_2$ are disjoint by supposition) and since $d_i \leq \mathbb{P}_{x}(U_{i})$ for $i=1,2$ (because the Nash equilibrium payoff is at least the minmax value). Let $\varepsilon > 0$. Choose natural numbers $m$, $m_{1}$, and $m_{2}$ such that $d_{i} - \varepsilon \leq \tfrac{m_{i}}{m} \leq d_{i}$, for $i=1,2$. Note that $m_{1} + m_{2} \leq m$. Pick a point $a_{1} \in U_{1}$ and a point $a_{2} \in U_{2}$, and let $p^*$ be the periodic play with period $m_1+m_2$ obtained by repeating $a_{1}$ for the first $m_{1}$ stages, and repeating $a_{2}$ for the next $m_{2}$ stages. We have \[d_{i} - \varepsilon \leq \tfrac{m_{i}}{m} \leq \tfrac{m_{i}}{m_{1} + m_{2}} = f_{i}(p^*),\] for $i \in \{1,2\}$. One can support $p^*$ as an $\varepsilon$-equilibrium play by a threat of punishment: in case of a deviation by player $1$, player $2$ will switch to playing the minmax action profile from the stage game for the rest of the game, thus reducing $i$'s payoff to $d_{i}$. A symmetric punishment is imposed on player 2 in case of a deviation.
Under the periodic play $p^*$, the sum of the players' payoffs is $1$. There are alternative plays where the payoff to \textit{both} players is 1, which can support 0-equilibria. For example, consider the non-periodic play $p$ that is played in blocks of increasing size: for each $k \in \mathbb{N}$, the length of block $k$ is $2^{2^k}$. In even (resp.~odd) blocks the players play the action profile $a_1$ (resp.~$a_2$). The reader can verify that since the ratio between the length of block $k$ and the total length of the first $k$ blocks goes to $\infty$, the payoff to both players at $p$ is 1. \end{exl}
\section{Regularity and the folk theorem}\label{secn.folk}
A payoff vector $w\in \mathbb{R}^{|I|}$, assigning a payoff to each player, is called \emph{an equilibrium payoff} of the Blackwell game $\Gamma$ if for every $\varepsilon>0$ there exists an $\varepsilon$-equilibrium $\sigma^{\varepsilon}$ of $\Gamma$ such that $\|w-\mathbb{E}_{\sigma^{\varepsilon}}(f)\|_{\infty}\leq \varepsilon$. In other words, an equilibrium payoff is an accumulation point of $\varepsilon$-equilibrium payoff vectors as $\varepsilon$ goes to $0$. We let $\mathcal{E}$ denote the set of equilibrium payoffs. Our goal here is to provide a description of $\mathcal{E}$.
In repeated games with stage payoffs, where the total payoff is some average (discounted average with low discounting, liminf of average, limsup of average, etc.) of the stage game payoffs, the folk theorem states that the set of equilibrium payoffs coincide with the set of all individually rational vectors that are in the convex hull of the feasible payoff vectors, see, e.g., Aumann and Shapley \cite{Aumann94}, Sorin \cite{Sorin92}, and Mailath and Samuelson \cite{{Mailath06}}. As we will see, when the payoff functions are general, the set of equilibrium payoffs is the convex hull of the set of feasible payoff vectors that are individually rational. The reason for the difference is that in repeated games with stage payoffs, getting a low payoff in one stage can be compensated by getting a high payoff in the following stage; when the payoff is obtained only at the end of the game, there is no opportunity to compensate low payoffs.
Define \begin{align*} Q^\varepsilon(f)&:=\bigcap_{i\in I} Q_{i, \varepsilon}(f_i),\\ W^\varepsilon(f)&:=\{f(p):p\in Q^\varepsilon(f)\}. \end{align*}
The set $Q^\varepsilon(f)$ is the set of $\varepsilon$-individually rational plays, and $W^\varepsilon(f)$ is the set of feasible and $\varepsilon$-individually rational payoffs vectors. Whenever convenient, we write simply $Q^\varepsilon$ and $W^\varepsilon$. For every set $X$ in a Euclidean space we denote its closure by $\textnormal{cl}(X)$ and its convex hull by $\textnormal{conv}(X)$.
\begin{theorem}\label{thrm:folk} Consider a Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$. Suppose that for each player $i \in I$, player $i$'s action set $A_i(h)$ at each history $h \in H$ is finite, her payoff function $f_i$ is bounded and upper semi-analytic, and her minmax value is history-independent. Then \[\mathcal{E}=\bigcap_{\varepsilon>0}\textnormal{conv}(\textnormal{cl}(W^\varepsilon(f))).\] \end{theorem}
To prove Theorem~\ref{thrm:folk} we need the following result, which states that every $\varepsilon$-equilibrium assigns high probability to plays in $Q^{\varepsilon^{1/3}}(f)$.
\begin{lemma}\label{lemma:largeprob} Consider an $n$-player Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$. Suppose that for each player $i \in I$, player $i$'s action set $A_i(h)$ at each history $h \in H$ is finite, her payoff function $f_i$ is bounded and upper semi-analytic, and her minmax value is history-independent. Let $\varepsilon > 0$ be sufficiently small, and let $\sigma^\varepsilon$ be an $\varepsilon$-equilibrium. Then \[\mathbb{P}_{\sigma^{\varepsilon}}(A^{\mathbb{N}} \setminus Q^{\varepsilon^{1/3}}(f)) < n\varepsilon^{1/3}.\] \end{lemma} \begin{proof}
Set $\eta := \varepsilon^{1/3}$. It suffices to show that for every $i \in I$, \begin{equation}\label{equ:231} \mathbb{P}_{\sigma^{\varepsilon}}(A^{\mathbb{N}} \setminus Q_{i, \eta} (f_i)) < \eta. \end{equation}
Fix a player $i \in I$ and suppose to the contrary that Eq.~\eqref{equ:231} does not hold. We derive a contradiction by showing that player~$i$ has a deviation from $\sigma^{\varepsilon}$ that yields her a gain higher than $\varepsilon$.
For $t \in \mathbb{N}$, denote by $X_t := \mathbb{P}_{\sigma^{\varepsilon}} (Q_{i,\eta}(f_i) |\mathcal{F}_t)$ the conditional probability of the event $Q_{i,\eta}(f_i)$ under the strategy profile $\sigma^\varepsilon$ given the sigma-algebra $\mathcal{F}_t$. By Doob's martingale convergence theorem, $(X_{t})_{t \in \mathbb{N}}$ converges to the indicator function of the event $Q_{i,\eta}(f_i)$, almost surely under $\mathbb{P}_{\sigma^{\varepsilon}}$. Since by supposition $\mathbb{P}_{\sigma^{\varepsilon}}(A^{\mathbb{N}} \setminus Q_{i, \eta} (f_i)) > \eta$, we know that $\mathbb{P}_{\sigma^\varepsilon}(X_{t} \to 0) > \eta$.
Let $K$ be a bound on the game's payoffs, and let $\rho := \varepsilon^2/K$. Let us call a history $h \in H_{t}$ a \textit{deviation history} if under $h$, stage $t$ is the first one such that $X_t < \rho$. On the event $\{X_{t} \to 0\}$, a deviation history arises at some point during play. Consequently, under $\mathbb{P}_{\sigma^{\varepsilon}}$, a deviation history arises with probability of at least $\eta$.
Consider the following strategy $\sigma_{i}'$ of player $i$: play according to $\sigma_i^\varepsilon$ until a deviation history, say $h$, occurs (and forever if a deviation history never occurs). At $h$, switch to playing a strategy which guarantees player $i$ a payoff of at least $v_i(f_i) - \varepsilon$ against $\sigma_{-i}^{\varepsilon}$ in $\Gamma_h$. Such a strategy exists by our supposition of history-independence of the minmax values. To conclude the argument, we compute the gain from the deviation to $\sigma_{i}'$.
For every deviation history $h \in H_{t}$, \begin{eqnarray}\label{equ:232} \mathbb{E}_{\sigma_{-i}^{\varepsilon},\sigma_{i}'}(f_{i}\mid h) &\geq& v_{i}(f_i) - \varepsilon,\\ \mathbb{E}_{\sigma_{-i}^{\varepsilon},\sigma_{i}^{\varepsilon}}(f_{i}\mid h) &\leq& \rho K + (1- \rho)(v_i(f_i) - \eta). \label{equ:233} \end{eqnarray} Eq.~\eqref{equ:232} holds by the choice of $\sigma_{i}'$. To derive Eq.~\eqref{equ:233}, suppose that, following the history $h$, player $i$ conforms to $\sigma_{i}^{\varepsilon}$. Then, conditional on $h$, with probability at most $\rho$ the play belongs to $Q_{i,\eta}(f_i)$, and player $i$'s payoff is at most $K$, and with probability at least $1-\rho$ the play does not belong to $Q_{i,\eta}(f_i)$, and player~$i$'s payoff is at most $v_i(f_i)-\eta$.
We can now compute the gain from the deviation to $\sigma_{i}'$: If a deviation history never arises, $\sigma_{i}'$ recommends the same actions as $\sigma_{i}^{\varepsilon}$, and therefore the gain is 0. A deviation history occurs with a probability of at least $\eta$, and thus \begin{align*} \mathbb{E}_{\sigma_{-i}^{\varepsilon},\sigma_{i}'}(f_{i}) - \mathbb{E}_{\sigma_{-i}^{\varepsilon},\sigma_{i}^{\varepsilon}}(f_{i}) &\geq \eta \bigl(v_i(f_i)-\varepsilon - \rho K - (1 - \rho)(v_i(f_i)-\eta)\bigr)\\ &= \eta(-\varepsilon - \varepsilon^2 + \rho v_i(f_i) + \eta - \rho \eta)\\ &= \varepsilon^{\frac{2}{3}}(1 - \varepsilon^{\frac{2}{3}} - \varepsilon^{\frac{5}{3}} + \tfrac{v_i(f_i)}{K} \varepsilon^{\frac{5}{3}} - \tfrac{1}{K} \varepsilon^{2}), \end{align*} which behaves like $\varepsilon^{\frac{2}{3}}$ when $\varepsilon$ is small, and therefore exceeds $\varepsilon$. \end{proof}
\noindent\textbf{Proof of Theorem~\ref{thrm:folk}}: Let $|I| = n$. Let $w \in \mathbb{R}^{n}$ be an equilibrium payoff. Assume by contradiction that there is an $\alpha > 0$ such that $w \not\in \textnormal{conv}(\textnormal{cl}(W^{\alpha}))$. For a vector $z \in \mathbb{R}^{n}$ write ${\rm dist}(z)$ to denote the distance from $z$ to the set $\textnormal{conv}(\textnormal{cl}(W^{\alpha}))$ under the $\|\cdot\|_{\infty}$ metric on $\mathbb{R}^{n}$. By assumption, $\delta := \tfrac{1}{4}{\rm dist}(w) > 0$. Denote $\varepsilon: = \min(\delta, \alpha^3, (\frac{\delta}{Kn})^3) > 0$, where $K$ is a bound on the game payoff.
From $w$ being an equilibrium payoff, there exits an $\varepsilon$-equilibrium, say $\sigma^\varepsilon$, such that $\|w-\mathbb{E}_{\sigma^\varepsilon}(f)\|_{\infty} \leq \varepsilon \leq \delta$. We have the following chain of inequalities:
\[{\rm dist}(\mathbb{E}_{\sigma^\varepsilon}(f)) \leq \mathbb{E}_{\sigma^\varepsilon}({\rm dist}(f)) \leq 2K \cdot \mathbb{P}_{\sigma^\varepsilon}(A^{\mathbb{N}} \setminus Q^{\alpha}(f)) \leq 2K \cdot n \cdot \varepsilon^{\frac{1}{3}} \leq 2\delta,\] where the first inequality follows from the fact that ${\rm dist}:\mathbb{R}^{n} \to \mathbb{R}$ is a convex function, the second from the fact that $f(p) \in W^{\alpha}$ whenever $p \in Q^{\alpha}(f)$, the third follows since $Q^{\varepsilon^{1/3}}(f) \subseteq Q^{\alpha}(f)$ and by Lemma~\ref{lemma:largeprob}, and the last holds by the choice of $\varepsilon$. But then
\[{\rm dist}(w) \leq \|w-\mathbb{E}_{\sigma^\varepsilon}(f)\|_{\infty} + {\rm dist}(\mathbb{E}_{\sigma^\varepsilon}(f)) \leq 3\delta,\] contradicting the choice of $\delta$.
We turn to prove the other direction. Let $w\in \bigcap_{\varepsilon>0}\textnormal{conv}(\textnormal{cl}(W^\varepsilon))$. We need to show that $w$ is an equilibrium payoff. Fix an $\varepsilon>0$.
Carath\'eodory's Theorem (Carath\'eodory, \cite{Caratheodory07}) implies that $\textnormal{cl} (\textnormal{conv}(W^{\varepsilon})) = \textnormal{conv} (\textnormal{cl}(W^{\varepsilon}))$, hence $w$ is an element of $\textnormal{cl} (\textnormal{conv}(W^{\varepsilon}))$, and thus we can choose a vector $w_{\varepsilon}\in \textnormal{conv}(W^{\varepsilon})$ such that $\|w-w_{\varepsilon}\|_\infty \leq \varepsilon$. We argue that $w_{\varepsilon}$ is a vector of expected payoffs in some $3\varepsilon$-equilibrium.
The payoff $w_{\varepsilon}$ can be presented as a convex combination of $n + 1$ vector payoffs, say \linebreak $f(p^1), \ldots, f(p^{n+1})$, with each $p^{k}$ an element of $Q^{\varepsilon}(f)$. Using jointly controlled lotteries as done, e.g., in Forges \cite{Forges}, Lehrer \cite{Lehrer1996}, or Lehrer and Sorin \cite{LehrerSorin1997}, the players can generate the required randomization over the plays $p^1, \dots, p^{n+1}$ during the first stages of the game. Once a specific play $p^k$ has been chosen, the construction of the $3\varepsilon$-equilibrium is standard: the players play $p^k$, and if player $i$ deviates, her opponents revert to playing a strategy profile that gives player $i$ at most $v_i(f_i)+\varepsilon$. Such a strategy exists by the assumption of history-independence of the minmax values. $\Box$
\begin{exl}\label{exl.folk}\rm Consider the 2-player Blackwell game $\Gamma = (\{1,2\}, A_{1}, A_{2}, f_{1}, f_{2})$, where the action sets are $A_1=\left\{{\rm T},{\rm M},{\rm B}\right\}$ and $A_2=\left\{{\rm L},{\rm C},{\rm R}\right\}$, and for a play $p = (a_0,a_1,\ldots)$ the payoffs are \[(f_1(p),f_2(p)) = \begin{cases} (1,1)&\text{if }\displaystyle\liminf_{n\to \infty} \tfrac{1}{t} \cdot \#\{k < t: a_k = ({\rm T},{\rm L})\text{ or } a_k = ({\rm M},{\rm C})\} >\frac{1}{2},\\ (4,-1)&\text{if }\displaystyle\liminf_{n\to \infty} \tfrac{1}{t} \cdot \#\{k < t: a_k = ({\rm B},{\rm L})\} = 1,\\ (-1,4)&\text{if }\displaystyle\liminf_{n\to \infty} \tfrac{1}{t} \cdot \#\{k < t: a_k = ({\rm T},{\rm R})\} = 1,\\ (0,0)&\text{otherwise}. \end{cases}\] Thus the payoff is $(1,1)$ if the (liminf) frequency of the stages where either (T,L) or (M,C) is played is larger than $\tfrac{1}{2}$. It is $(4,-1)$ if (B,L) is played with frequency of 1, and $(-1,4)$ if (T,R) is played with the frequency of 1. All other cases result in a payoff of $(0,0)$.
Observe that when player 1 plays B repeatedly, the maximal payoff that player 2 can achieve is 0, and this is player 2's minmax value. Similarly, player 1's minmax value is 0. For each $\varepsilon \in (0,1)$, the set $W^{\varepsilon}(f)$ consists of the two points $(0,0)$ and $(1,1)$. By Theorem~\ref{thrm:folk}, the set of equilibrium payoffs $\mathcal{E}$ is the line segment connecting $(0,0)$ and $(1,1)$, see Figure \ref{figure}.
Naturally, all equilibrium payoffs $w$ are (a) convex combinations of the feasible payoffs vectors $(0,0)$, $(1,1)$, $(4,-1)$, and $(-1,4)$, and (b) individually rational, i.e., they satisfy $w_{1} \geq 0$ and $w_{2} \geq 0$. The set of all payoff vectors satisfying (a) and (b) is represented in Figure~\ref{figure} by the shaded triangle. The point we wish to make here is that the properties (a) and (b) are not sufficient for a payoff vector to be an equilibrium payoff.
Take for concreteness the point $(3,0)$. This payoff vector is in the convex hull of the feasible payoff vectors and is individually rational. Yet, for $\varepsilon < \tfrac{2}{3}$, there is no $\varepsilon$-equilibrium with the payoff (close to) the vector $(3,0)$. We give a heuristic argument.
Suppose to the contrary that $\sigma$ is such an $\varepsilon$-equilibrium. The strategy profile $\sigma$ necessarily assigns a probability of at least $\tfrac{2}{3}$ to the set of plays that yield the payoff vector $(4,-1)$. But this implies that Player 2 has a deviation that would improve her payoff over the candidate $\varepsilon$-equilibrium by at least $\tfrac{2}{3}$. Player 2 needs to deviate to playing R forever (for example), at any history of the game where her conditional expected payoff under $\sigma$ is close enough to $-1$. Since playing R would yield at least $0$, by such a deviation, she would improve her conditional expected payoff by at least $1$. Levy's zero-one law guarantees that the histories where player 2 is called to deviate in this way arise with a probability close to $\tfrac{2}{3}$, so that the expected gain from the deviation is also close to $\tfrac{2}{3}$.
\begin{figure}
\caption{The set of equilibrium payoffs (the segment connecting $(0,0)$ and $(1,1)$) vs. the set of convex combinations of feasible payoffs that are individually rational (the dark triangle).}
\label{figure}
\end{figure} \end{exl}
The above discussion of Example~\ref{exl.folk} leads to a slightly more general conclusion: if the set of feasible payoffs is finite, then the set of equilibrium payoffs is the convex-hull of the feasible payoffs that are individually rational (equal or larger than the minmax). For each player the minmax value is within the finite set of feasible payoffs, and placing any probability on a payoff that is not individually rational enables profitable deviations.
\section{Blackwell games with tail-measurable payoffs}\label{secn.tail}
An important class of games with history-independent minmax values are those where the payoff functions are tail-measurable. In this section we concentrate on games with tail-measurable payoffs.
Consider a Blackwell game with history-independent action sets, $\Gamma = (I,(A_{i},f_{i})_{i \in I})$. A set $Q \subseteq A^{\mathbb{N}}$ is said to be a \textit{tail set} if whenever a play $p = (a_{0},a_{1},\ldots)$ is an element of $Q$ and $q = (b_{0},b_{1},\ldots)$ is such that $a_t = b_t$ for all $t \in \mathbb{N}$ sufficiently large, then $q$ is also an element of $Q$. Let $\mathscr{T}$ denote the sigma-algebra of the tail subsets of $A^{\mathbb{N}}$. We note that the tail sigma-algebra $\mathscr{T}$ and the Borel sigma-algebra $\mathscr{B}$ are not nested. For constructions of tail sets that are not Borel, see Rosenthal \cite{Rosenthal75} and Blackwell and Diaconis \cite{Diaconis96}.
Examples of tail sets are: (1) the winning sets of Example \ref{exl.io}, (2) the set of plays in which a certain action profile $a\in A$ is played with limsup-frequency at most $\tfrac{1}{2}$, and (3) the set of plays in which a certain action profile $a^*\in A$ is played at most finitely many times at even stages (with no restriction at odd stages).
An important class of tail sets are the shift invariant sets. A set $Q \subseteq A^{\mathbb{N}}$ is a \textit{shift invariant set} if for each play $p = (a_0,a_1,\ldots)$, $p \in Q$ if and only if $(a_1,a_2,\ldots) \in Q$. Equivalently, shift invariant sets are the sets that are invariant under the backward shift operator on $A^{\mathbb{N}}$. Shift invariant sets are tail sets. The converse is not true: while the sets in examples (1) and (2) above are shift invariant, that of example (3) is not.
A function $f:A^{\mathbb{N}} \to \mathbb{R}$ is called tail-measurable if, for each $r\in\mathbb{R}$, the set $\{p\in A^{\mathbb{N}} : r \leq f(p)\}$ is an element of $\mathscr{T}$. Intuitively, a payoff function is tail measurable if an action taken in any particular stage of the game has no impact on the payoff. The payoff function in Example \ref{exl.eq} is tail-measurable.
\begin{remark}\rm The assumption that the set of actions of each player is history-independent is required so that the tail-measurability of the payoff functions has a bite. If the sets of actions were history-dependent, then by having a different set of actions at each history, any function could be turned into tail-measurable. \end{remark}
We now state one key implication of tail-measurability, namely the history-independence of minmax values.
\begin{proposition}\label{prop:minmaxtail} Let $\Gamma = (I,(A_{i},f_{i})_{i \in I})$ be a Blackwell game with history-independent action sets, and let $i \in I$ be a player. If player $i$'s payoff function is bounded, upper semi-analytic, and tail-measurable, then her minmax value is history-independent. \end{proposition} \begin{proof} It suffices to show that $v_{i}(f_{i,a}) = v_{i}(f_{i})$ for each $a \in A$, where, with a slight abuse of notation, we write $a$ for a history in stage $1$. Since $f_i$ is tail-measurable, all the functions $f_{i,a}$ for $a \in A$ are identical to each other. Hence, fixing any particular action profile $\bar{a} \in A$, letting $X_i := \Delta(A_i)$ and $X_{-i} := \prod_{j \in -i}X_{j}$, we have \begin{align*} v_{i}(f_{i}) &= \inf_{x_{-i} \in X_{-i}}\sup_{x_{i} \in X_{i}} \sum_{a \in A} \prod_{j \in I}x_{j}(a_{j}) \cdot \Big(\inf_{\sigma_{-i} \in \Sigma_{-i}}\sup_{\sigma_{i} \in \Sigma_{i}} \mathbb{E}_{\sigma_{-i},\sigma_{i}}(f_{i,a})\Big)\\ &= \inf_{x_{-i} \in X_{-i}}\sup_{x_{i} \in X_{i}} \sum_{a \in A} \prod_{j \in I}x_{j}(a_{j}) \cdot v_{i}(f_{i,\bar{a}}) = v_{i}(f_{i,\bar{a}}). \end{align*} \end{proof}
If the payoff functions of all the players in a game $\Gamma$ are tail-measurable, then, for each fixed stage $t \in \mathbb{N}$, all the subgames of $\Gamma$ starting at stage $t$ are identical. On the other hand, the subgames starting, say, at stage $1$, are not identical to the game itself (see example (3) of a tail-measurable payoff function above). Nonetheless, as Proposition \ref{prop:minmaxtail} implies, the players' minmax values \textit{are} the same in every subgame.
The condition of history-independence of the minmax values is more inclusive than that of tail-measurability of the payoffs; the examples that follow illustrate the point.
\begin{exl}\rm Consider a one-player Blackwell game where the player's payoff function is $1_{S}$, the indicator of a set $S \subseteq [H]$. If $S$ is dense in $[H]$, then the minmax value of the player is $1$ in each subgame. A dense set may or may not be a tail set. \end{exl}
\begin{exl}\rm We consider a Blackwell game similar to that of Example \ref{exl.io}, but where the stage game may depend on the history, as long as each player's stage minmax value is the same.
Specifically, let $\Gamma = (I, A, H, (1_{W_i})_{i \in I})$. Suppose that at each history $h \in H$, each player $i \in I$ has a stage winning set $U_{i}(h) \subseteq A(h)$, and her winning set in the Blackwell game $\Gamma$ is \[W_i = \{(a_0,a_1,\ldots) \in [H]:a_t \in U_i(a_0,\ldots,a_{t-1})\text{ for infinitely many }t \in \mathbb{N}\}.\]
Assume that the stage minmax value of player $i$ is the same at each history: there is a number $d_{i}$ such that \[d_{i} = \inf_{x_{-i} \in \Delta(A_{-i}(h))} \sup_{x_i \in \Delta(A_i(h))} \mathbb{P}_{x_{-i},x_i}(U_i(h))\] for every $h \in H$. Then player $i$'s minmax value in each subgame of $\Gamma$ is $0$ if $d_{i} = 0$, and is $1$ if $d_{i} > 0$. Thus player $i$'s minmax value is history-independent.
Note that the game $\Gamma$ need not have history-independent action sets. Even when the action sets \textit{are} history-independent, the winning sets need not necessarily be tail-measurable.
To illustrate the last claim, suppose that there are two players playing matching pennies at each stage. At stage 0, player 1 wants to match the choice of player 2 (and player 2 wants to mismatch the choice of player~1). Subsequently the roles of the two players swap as follows: the player to win stage $t$ wants to match her opponent's action at stage $t+1$, while the loser at stage $t$ wants to mismatch the action of her opponent at stage $t+1$. Formally, we let $\Gamma = (\{1,2\}, A_1, A_2, 1_{W_{1}}, 1_{W_{2}})$ be the 2-player Blackwell game with history-independent action sets, where $A_1 = A_2 = \{{\rm H},{\rm T}\}$, the winning sets $W_1$ and $W_2$ are as above, and the stage winning sets are defined recursively as follows: \[U_1(\oslash) = \{({\rm H},{\rm H}), ({\rm T},{\rm T})\}\quad\text{and}\quad U_2(\oslash) = \{({\rm H},{\rm T}), ({\rm T},{\rm H})\},\] and \[U_1(h,a) = \begin{cases} U_1(\oslash) &\text{if }a \in U_1(h),\\ U_2(\oslash) &\text{if }a \in U_2(h), \end{cases}\quad\text{and}\quad U_2(h,a) = \begin{cases} U_2(\oslash) &\text{if }a \in U_1(h),\\ U_1(\oslash) &\text{if }a \in U_2(h), \end{cases}\] for each $h \in H$ and $a \in A$. The sets $W_1$ and $W_2$ are not tail: out of the two plays \begin{align*} &(({\rm H},{\rm H}), ({\rm H},{\rm H}),({\rm H},{\rm H}),\ldots) \hbox{ and}\\ &(({\rm H},{\rm T}), ({\rm H},{\rm H}), ({\rm H},{\rm H}),\ldots), \end{align*} the first is an element of $W_1 \setminus W_2$, while the second is an element of $W_2 \setminus W_1$. \end{exl}
\begin{exl}\rm Consider a Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$, where player $i$'s objective is (as in Example \ref{exl.eq}) to maximize the long-term frequency of the stages she wins: \[f_{i}(a_{0},a_{1},\ldots) = \limsup_{t \to \infty}\tfrac{1}{t}\cdot\#\{k < t : a_{k} \in U_{i}(a_0,\ldots,a_{k-1})\}.\] As in the previous example, $U_{i}(h) \subseteq A(h)$ is player $i$'s stage winning set at history $h \in H$. Assume, as above, that player $i$'s minmax value in each stage game is $d_i$. Then also her minmax value in each subgame of $\Gamma$ is $d_{i}$. \end{exl}
\begin{exl}\rm Start with a Blackwell game with tail-measurable payoff functions. Suppose that the minmax values of all the players in the game are $0$. Take any history $h$, and redefine the payoff functions so that any play having $h$ as a prefix has a payoff of $0$. In the resulting game, the minmax value of each player in each subgame remains $0$, but the payoff functions are no longer tail-measurable (unless the original payoff functions are constant). A similar modification can be performed with any subset of histories, not just one. \end{exl}
From the results above we now deduce a number of implications for Blackwell games with tail-measurable payoffs.
\begin{corollary}\label{cor.0-1law} Consider a Blackwell game $\Gamma = (I, (A_{i}, 1_{W_{i}})_{i \in I})$ with history-independent action sets. If player $i$'s winning set $W_i$ is an analytic tail set, then $v_i(W_i)$ is either $0$ or $1$. \end{corollary} \begin{proof} Suppose that $v_i(W_i) > 0$. Let $\varepsilon := v_i(W_i)/2$. In view of Proposition \ref{prop:minmaxtail}, player $i$'s minmax value in $\Gamma$ is history-independent. Applying Proposition \ref{prop:v(Q)=1}, we conclude that $v_i(Q_{i, \varepsilon}(1_{W_{i}})) \linebreak = 1$. But $Q_{i, \varepsilon}(1_{W_{i}}) = W_{i}$ by the choice of $\varepsilon$. \end{proof}
The following conclusion follows directly from Proposition \ref{prop:minmaxtail} and Theorem \ref{theorem:minmax_indt}.
\begin{corollary}\label{cor:eq} Suppose that the game $\Gamma = (I, (A_{i}, 1_{W_{i}})_{i \in I})$ has history-independent action sets. Suppose, furthermore, that for each player $i \in I$, player $i$'s action set $A_i$ is finite and her payoff function $f_i$ is bounded, upper semi-analytic, and tail-measurable. Then for every $\varepsilon>0$ the game admits an $\varepsilon$-equilibrium. \end{corollary}
\section{Concluding remarks}\label{secn.disc} \noindent\textbf{Approximations by compact sets.} Any Borel probability measure on $[H]$ (recall that $[H]$ is Polish under the maintained assumptions), is not merely regular, but is tight: the probability of a Borel set $B \subseteq [H]$ can be approximated from below by the probability of a compact subset $K \subseteq B$ (Kechris \cite[Theorem 17.11]{Kechris95}). The minmax value is not tight in this sense. To see this, consider any 2-player Blackwell game where player 1's winning set $W_{1}$ is the entire set of plays $[H]$, so that $v_1(W_1) = 1$, and where $A_{2}(\o)$, player 2's action set at the beginning of the game, is $\mathbb{N}$. We argue that $v_1(K) = 0$ for every compact set $K \subseteq W_1$. Indeed, the projection of a compact set $K \subseteq W_1$ on $A_{2}(\o)$ is a compact, and hence a finite set. Therefore, player 2 can guarantee that the realized play is outside $K$ by choosing a sufficiently large action at stage $0$. Thus $v_1(K) = 0$, as claimed.
\noindent\textbf{Approximations by semicontinuous functions.} The conclusion of Theorem \ref{thrm:tailapprox} would no longer be true without the assumption of history-indepenence of the minmax values. Here we give an example of a game with a limsup payoff function where the minmax value cannot be approximated from below by an upper semicontinuous function.
Consider a zero-sum game $\Gamma$ where $A_{1} = A_{2} = \{0,1\}$, and player 1's payoff function is \[f(a_0,a_1,\ldots) = \begin{cases} \displaystyle\limsup_{t \to \infty}\tfrac{1}{t}\#\{k < t: a_{2,k} = 0\},&\text{if }\tau = \infty,\\ 2, &\text{if } \tau < \infty\text{ and }a_{2,\tau} = 1,\\ 0, &\text{if } \tau < \infty\text{ and }a_{2,\tau} = 0, \end{cases}\] where $\tau = \tau(a_0,a_1,\ldots) \in \mathbb{N} \cup \{\infty\}$ is the first stage where player 1 chooses action $1$. The game was analyzed in Sorin \cite{Sorin86}, who showed that $v_1(f) = 2/3$.
Let $g \leq f$ be a bounded upper semicontinuous function. We argue that $v_1(g) \leq1/2$. For $t \in \mathbb{N}$, let $S_t$ denote the set of plays $p$ such that $t \leq \tau(p)$. Note that $S_{t}$ is closed. We argue that \[\inf_{t \in \mathbb{N}} \sup\{g(p):p \in S_{t}\} \leq 1.\] Suppose this is not the case. Take an $\varepsilon > 0$ such that $1 + \varepsilon < \sup\{g(p):p \in S_{t}\}$ for each $t \in \mathbb{N}$. Let $U_{0} := \{1 + \varepsilon \leq g\}$, and for each $t \geq 1$ let $U_{t} := U_{0} \cap S_{t}$. The set $U_{t}$ is not empty for each $t \in \mathbb{N}$. Moreover, it is a closed, and hence a compact subset of $A^{\mathbb{N}}$. Thus $U_{0} \supseteq U_{1} \supseteq \cdots$ is a nested sequence of non-empty compact sets. Therefore, there is a play $p \in \bigcap_{t \in \mathbb{N}} U_{t}$. It holds that $\tau(p) = \infty$, and consequently $f(p) \leq 1 < g(p)$, a contradiction.
Take an $\varepsilon > 0$. Find a $t \in \mathbb{N}$ such that $\sup\{g(p):p \in S_{t}\} \leq 1 + \varepsilon$. Suppose that player 2 plays $0$ for the first $t$ stages, and thereafter plays $0$ with probability $1/2$ at each stage. This guarantees that the payoff under the function $g$ is at most $(1 + \varepsilon)/2$.
\noindent\textbf{On the assumption of finiteness of the action sets.} The hypothesis of Theorem \ref{theorem:sumofprob} requires that the action sets at each history be finite, and its conclusion is not true without this assumption. Indeed, consider the 2-player Blackwell game $\Gamma = (\{1,2\}, A_{1}, A_{2}, W_{1}, W_{2})$ with history-independent action sets $A_1 = A_2 = \mathbb{N}$. Player 1's winning set $W_1$ consists of all plays $(a_{1,t}, a_{2,t})_{t\in\mathbb{N}}$ such that $a_{1,t} > a_{2,t}$ holds for all sufficiently large $t \in \mathbb{N}$, and player 2's winning set $W_2$ consists of all plays $(a_{1,t},a_{2,t})_{t\in\mathbb{N}}$ such that $a_{1,t} < a_{2,t}$ holds for all sufficiently large $t \in \mathbb{N}$. Then $W_1$ and $W_2$ are Borel-measurable and tail-measurable, and $v_1(W_1) = v_2(W_2)=1$, but $W_1\cap W_2=\emptyset$. Hence, the game has no $\varepsilon$-equilibrium for any $\varepsilon < 1/2$. Indeed, an $\varepsilon$-equilibrium $\sigma$ would need to satisfy $\mathbb{P}_{\sigma}(W_i) \geq v_i(W_i) - \varepsilon > 1/2$ for both $i \in \{1,2\}$.
As discussed above, the assumption that the sets of actions are history-dependent is intertwined with the assumption that the payoffs are tail-measurable.
\noindent\textbf{Continuity of the minmax.} Unlike Borel probability measures, the minmax value is in general not continuous in the following sense: there is an increasing sequence of Borel sets $C_0 \subseteq C_1\subseteq \ldots$ such that $\lim_{n\to \infty} v_i(C_n) < v_i(\bigcup_{n \in \mathbb{N}} C_n)$. In fact, one can construct an example of this kind where $C_{n}$ is both a $G_{\delta}$ and an $F_{\sigma}$ set, as follows. Consider a 2-player Blackwell game with history-independent action sets where $A_1$ is a singleton (player 1 is a dummy) while $A_2$ contains at least two distinct elements. Let $\{p_{0},p_{1},\ldots\}$ be a converging (with respect to any compatible metric on $A^{\mathbb{N}}$) sequence of plays, no two members of which are the same. Let $C_{n} := A^{\mathbb{N}}\setminus\{p_{n},p_{n+1},\ldots\}$. Then $v_1(C_n) = 0$ for each $n \in \mathbb{N}$ while $v_1(\bigcup_{n \in \mathbb{N}}C_n) = v_1(A^{\mathbb{N}}) = 1$.
\noindent\textbf{Maxmin value.} Consider a Blackwell game $\Gamma$, and suppose that player $i$'s payoff function $f_i$ is bounded and upper semi-analytic. Player $i$'s \textit{maxmin value} is defined as \[z_i(f_i) = \sup_{\sigma_i \in \Sigma_{i}}\inf_{\sigma_{-i} \in \Sigma_{-i}}\mathbb{E}_{\sigma_{-i},\sigma_i}(f_i).\]
The minmax value is not smaller than the maxmin value: $z_i(f_i) \leq v_i(f_i)$. If $I = \{1,2\}$, player 1's payoff function $f_1$ is bounded and Borel-measurable, and for every $h \in H$ either the set $A_1(h)$ of player $1$'s actions or the set $A_2(h)$ of player 2's actions at $h$ is finite, then in fact $z_1(f_1) = v_1(f_1)$, as follows from the determinacy of zero-sum Blackwell games (Martin \cite{Martin98}). Strict inequality might arise for at least two reasons.
The first is the failure of determinacy. The results of Section \ref{secn.approx} are established under the assumption that the action sets be countable, an assumption that is insufficient to guarantee determinacy of a two-player zero-sum Blackwell game even if player 1's winning set is clopen. Wald's game provides an illustration. Suppose that each of the two players chooses a natural number; player 1 wins provided that his choice is at least as large as player 2's. Formally, consider a Blackwell game with $I = \{1,2\}$, where the action sets at $\o$ are $A_1(\o) = A_2(\o) = \mathbb{N}$, and player 1's winning set $W_1$ consists of plays such that player 1's stage $0$ action is at least as large as player 2's stage 0 action: $a_{1,0} \geq a_{2,0}$. Then player 1's minmax value is $v_1(W_1) = 1$ while her maxmin value is $z_1(W_1) = 0$.
The second possibility for a maxmin and the minmax values to be different arises in games with three or more players. The reason is that the definitions of both the maxmin and the minmax values impose that the opponents of player $i$ choose their actions independently after each history. The point is illustrated by Maschler, Solan, and Zamir \cite[Example 5.41]{Maschler13}, which can be seen as a 3-player Blackwell game with binary action sets, where the player's payoff function only depends on the stage 0 action profile.
Analogues of Theorems ~\ref{thrm:reg}, \ref{thrm:regfunc}, and \ref{thrm:tailapprox} could be established for the maxmin values using the same approach.
\noindent\textbf{Open problems.} Existence of an $\varepsilon$-equilibrium in dynamic games with general \linebreak (Borel-measurable) payoffs has been, and still is, one of the Holy Grails of game theory. A more modest approach, also pursued in this paper, is to establish existence in some special classes of games. Blackwell games, as they are defined here, do not include moves of nature. An interesting avenue for a follow up research is to extend the methods developed in this paper to the context of stochastic games with general Borel-measurable payoff functions.
Theorems \ref{thrm:reg} and \ref{thrm:regfunc} provide two distinct approximation results, and neither seems to be a consequence of the other. This raises the question of whether there is a natural single generalization that would encompass both these results as two special cases.
\section{Appendix: The proof of Theorems~\ref{thrm:reg}, \ref{thrm:regfunc}, and \ref{thrm:tailapprox}}\label{subsecn.proof} The proofs of Theorems~\ref{thrm:reg} and~\ref{thrm:regfunc} are adaptations of the corresponding arguments in Maitra and Sudderth \cite{Maitra98} and in Martin \cite{Martin98} and are provided here for completeness. Theorem \ref{thrm:tailapprox} follows easily from Theorem~\ref{thrm:reg} and Proposition \ref{prop:v(Q)=1}.
Consider a Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$, fix a player $i \in I$, and suppose that player $i$'s payoff function $f_{i}$ is bounded and Borel-measurable. Also assume w.l.o.g. that $0 \leq f_{i} \leq 1$. When we will consider Theorem~\ref{thrm:reg} we will substitute $f_i = 1_{W_i}$.
Given $h \in H$, let $R(h)$ denote the set of one-shot payoff functions $r: A(h) \to [0,1]$. Let $X_i(h) := \Delta(A_i(h))$ denote player $i$'s set of mixed actions at history $h$, and let $X_{-i}(h) := \prod_{j \in -i}X_{j}(h)$. For $x \in \prod_{i \in I}X_{i}(h)$ we write $r(x)$ to denote $\mathbb{E}_{x}(r)$, the expectation of $r$ with respect to $x$. Player $i$'s minmax value of the function $r \in R(h)$ is \[d_i(r) := \inf_{x_{-i} \in X_{-i}(h)}\sup_{x_i \in X_i(h)} r(x_{-i},x_i).\]
We next introduce the main tool of the proof, an auxiliary two-player game of perfect information denoted by $G_i(f_i,c)$. This is a variation of the games $G_v$ and $G'_v$ in Martin \cite[pp. 1575]{Martin98}.
Given $c \in (0,1]$ and a Borel measurable function $f_i \colon [H] \to [0,1]$, define the game $G_i(f_i,c)$ as follows: \begin{itemize}
\item Let $h_0 := \oslash$. Player~I chooses a one-shot payoff function $r_0:A(h_0)\to[0,1]$ such that $d_i(r_0)\geq c$.
\item Player~II chooses an action profile $a_0 \in A(h_0)$ such that $r_0(a_0) > 0$.
\item Let $h_1 := (a_0)$. Player~I chooses a one-shot payoff function $r_1:A(h_1)\to[0,1]$ such that $d_i(r_1)\geq r_0(a_0)$.
\item Player~II chooses an action profile $a_1 \in A(h_1)$ such that $r_1(a_1)>0$.
\item Let $h_2 := (a_0,a_1)$. Player~I chooses a one-shot payoff function $r_2:A(h_2)\to[0,1]$ such that $d_i(r_2)\geq r_1(a_1)$. And so on. \end{itemize} This results in a run\footnote{To distinguish histories and plays of $\Gamma$ from those of $G_i(f_i,c)$, we refer to the latter as \textit{positions} and \textit{runs}. To distinguish the players of $\Gamma$ from those of $G_i(f_i,c)$, we refer to the latter as Player I and Player II, using the initial capital letters.} $(r_0,a_0,r_1,a_1,\ldots)$. Player I wins the run if \[\limsup_{t \to \infty}r_{t}(a_{t}) \leq f_{i}(a_0,a_1,\ldots) \quad\text{and}\quad 0 < f_{i}(a_0,a_1,\ldots).\]
Let $T$ be the set of all legal positions in the game $G_i(f_i,c)$. This is a tree on the set $R \cup A$ where $R := \cup_{h \in H}R(h)$. Sequences of even (odd) length in $T$ are Player I's (Player II's) positions. The tree $T$ is pruned: an active player has a legal move at each legal position of the game. Indeed, consider Player I's legal position in the game $G_i(f_i,c)$ and let $h_t$ denote, as above, the sequence of action profiles produced, to date, by Player II. Then the function $r_t$ which is identically equal to $1$ on the set $A(h_t)$ is a legal move for Player I. Consider now Player II's legal position in $G_i(f_i,c)$, let $h_{t}$ denote the sequence of action profiles produced to date by Player II, and let $r_{t}$ be Player I's latest move. Then $d_i(r_t) > 0$. Therefore, there exists an action profile $a_{t} \in A(h_{t})$ such that $r_{t}(a_{t}) > 0$, and thus $a_{t}$ is Player II's legal move at the given position.
The set $[T]$ is the set of all runs of the game $G_i(f_i,c)$, a subset of $(R \cup A)^{\mathbb{N}}$.
A run is \emph{consistent} with a pure strategy $\sigma_{\rm I}$ of Player~I if it is generated by the pair $(\sigma_{\rm I},\sigma_{\rm II})$, for some pure strategy $\sigma_{\rm II}$ of Player~II. Runs that are consistent with pure strategies of Player~II are defined analogously.
Player I's pure strategy $\sigma_{{\rm I}}$ in $G_i(f_i,c)$ is said to be \emph{winning} if Player I wins all runs of the game that are consistent with $\sigma_{{\rm I}}$.
\begin{proposition}\label{prop:PlayerI} Let $c \in (0,1]$ and let $f_i : [H] \to [0,1]$ be a Borel-measurable function. If Player {\rm I}~ has a winning strategy in the game $G_i(f_i,c)$, then there exists a closed set $C \subseteq [H]$ and a limsup function $g : [H] \to [0,1]$ such that $g \leq f_i$, $\{g > 0\} \subseteq C \subseteq \{f_i > 0\}$, and $c \leq v_{i}(g)$. In particular, $c \leq v_i(C)$; and if $f_i = 1_{W_i}$, then $C \subseteq W_i$. \end{proposition}
\begin{proof} Fix Player I's winning strategy $\sigma_{{\rm I}}$ in $G_i(f_i,c)$.
\noindent\textsc{Step 1:} Defining $C \subseteq [H]$ and $g : [H] \to [0,1]$.
Let $T_{\rm I} \subseteq T$ denote the set of positions in the game $G_i(f_i,c)$ of even length (i.e., Player I's positions) that are consistent with $\sigma_{\rm I}$, i.e., those positions that can be reached under a strategy profile $(\sigma_{{\rm I}},\sigma_{{\rm II}})$ for some pure strategy of $\sigma_{{\rm II}}$ of Player II. Let $\pi_{\rm I} : T_{\rm I} \to H$ be the projection that maps a position of length $2t$ in $G_i(f_i,c)$ to a history of length $t$ in $\Gamma$: Formally, $\pi_{\rm I}(\oslash) := \oslash$, $\pi_{\rm I}(r_0,a_0) := (a_0)$, etc. Let $H_{\rm I} \subseteq H$ be the image of $T_{\rm I}$ under $\pi_{\rm I}$. Since in the tree $T_{\rm I}$ Player I's moves are uniquely determined by $\sigma_{\rm I}$, the map $\pi_{\rm I}$ is in fact a bijection between $T_{\rm I}$ and $H_{\rm I}$. We write $\phi : H_{\rm I} \to T_{\rm I}$ for the inverse of $\pi_{\rm I}$. The map $\phi$ induces a continuous bijection $[H_{\rm I}] \to [T_{\rm I}]$, which we also denote by $\phi$. We say that positions in $H_{\rm I}$ are \emph{$\sigma_{\rm I}$-acceptable}, and define $C$ to be the set $[H_{\rm I}]$.
For each $t \in \mathbb{N}$, define the function $\rho_t : H_{t} \to \mathbb{R}$ as follows: $\rho_0(\oslash) := c$. Let $t \in \mathbb{N}$ and consider a history $h_{t} \in H_{t}$. If $h_{t}$ is not $\sigma_{\rm I}$-acceptable, we define $\rho_{t+1}(h_{t},a_{t}) := 0$ for each $a_{t} \in A(h_{t})$. Suppose that $h_{t}$ is $\sigma_{\rm I}$-acceptable, and let $r_{t} := \sigma_{\rm I}(\phi(h_{t}))$. For each $a_{t} \in A(h_{t})$ define $\rho_{t+1}(h_{t},a_{t}) := r_{t}(a_{t})$. Note that if $h_{t}$ is $\sigma_{\rm I}$-acceptable while $(h_{t},a_{t})$ is not, we have $\rho_{t+1}(h_{t},a_{t}) = r_{t}(a_{t}) = 0$.
Also define $g : [H] \to [0,1]$ by letting \[g(a_{0},a_{1},\ldots) := \limsup_{t \to \infty}\rho_t(a_{0},\dots,a_{t-1}).\]
\noindent\textsc{Step 2:} Verifying that $g \leq f_i$ and $\{g > 0\} \subseteq C \subseteq \{f_i > 0\}$.
Since $\sigma_{\rm I}$ is Player I's winning strategy in $G_i(f_i,c)$, all runs in $[T_{\rm I}]$ are won by Player I, and hence $[H_{\rm I}] \subseteq \{f_i > 0\}$. For a play $p = (a_{0},a_{1},\ldots)$ in $[H_{\rm I}]$, if $\phi(p) = (r_{0},a_{0},r_{1},a_{1},\ldots)$, then $g(p)$ equals $\limsup_{t \to \infty}r_{t}(a_{t})$. Since the run $\phi(p)$ is won by Player I, we conclude that $g(p) \leq f_i(p)$. Thus $g \leq f_i$ on $[H]$.
\noindent\textsc{Step 3:} Verifying that $c \leq v_{i}(g)$. Since $g \leq 1_{C}$ it will then follow that $c \leq v_i(C)$.
Fix a strategy profile $\sigma_{-i} \in \Sigma_{-i}$ for the players in $-i$ in the game $\Gamma$. Take any $\epsilon > 0$. We define a strategy $\sigma_i$ for player $i$ in the game $\Gamma$ with the property that $\mathbb{E}_{\sigma_{-i},\sigma_{i}}(g) \geq c - 2\epsilon$.
\noindent\textsc{Step 3.1:} Defining player $i$'s strategy $\sigma_{i}$.
Let $r_0 := \sigma_{\rm I}(\oslash)$, Player I's first move in $G_i(f_i,c)$ according to her strategy $\sigma_{\rm I}$. Define $\sigma_{i}(\oslash)$ to be a mixed action on $A_i(\oslash)$ such that \[r_0(\sigma_{-i}(\oslash),\sigma_{i}(\oslash)) \geq c - \epsilon.\] Let $t \geq 1$ and consider a history $h_{t} = (a_{0},\ldots,a_{t-1}) \in H_{t}$ of $\Gamma$. If $h_{t}$ is not $\sigma_{\rm I}$-acceptable, then $\sigma_{i}(h_{t})$ is arbitrary. If $h_{t}$ is $\sigma_{\rm I}$-acceptable, let $\phi(h_{t}) := (r_{0}, a_{0}, \ldots, r_{t-1}, a_{t-1})$ and $r_{t} := \sigma_{\rm I}(\phi(h_{t}))$. Define $\sigma_{i}(h_{t})$ to be a mixed action on $A_i(h_{t})$ such that \[r_t(\sigma_{-i}(h_{t}),\sigma_{i}(h_{t})) \geq r_{t-1}(a_{t-1})-\epsilon \cdot 2^{-t}.\]
\noindent\textsc{Step 3.2:} Verifying that $\mathbb{E}_{\sigma_{-i},\sigma_{i}}(g) \geq c - 2\epsilon$.
For each $t \in \mathbb{N}$ let us define $\rho_t^{\epsilon} := \rho_t - \epsilon \cdot 2^{-t+1}$. One can think of the functions $\rho_0^{\epsilon}, \rho_1^{\epsilon}, \dots$ as a stochastic process on $[H]$ that is measurable with respect to the filtration $\{\mathcal{F}_{t}\}_{t \in \mathbb{N}}$. We now argue that this process is a submartingale with respect to the measure $\mathbb{P}_{\sigma_{-i},\sigma_{i}}$.
Letting $r_0 := \sigma_{\rm I}(\oslash)$ we have \[\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\rho_{1}^{\epsilon}) = \mathbb{E}_{\sigma_{-i},\sigma_{i}}(r_{0}(a_{0})) - \epsilon= r_{0}(\sigma_{-i}(\oslash),\sigma_{i}(\oslash)) - \epsilon \geq c-2\epsilon = \rho_{0}^{\epsilon}(\oslash).\] Consider a $\sigma_{\rm I}$-acceptable history $h_{t} = (a_{0},\dots,a_{t-1}) \in H_t$ of length $t \geq 1$. Let $(r_{0},a_{0},\ldots,r_{t-1},a_{t-1}) := \phi(h_{t})$ and $r_{t} := \sigma_{\rm I}(\phi(h_{t}))$. We have \begin{align*}
\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\rho_{t+1}^{\epsilon} | h_{t}) &= \mathbb{E}_{\sigma_{-i},\sigma_{i}}(r_{t}(a_{t}) | h_{t}) - \epsilon \cdot 2^{-t}\\ &= r_{t}(\sigma_{-i}(h_{t}),\sigma_{i}(h_{t})) - \epsilon \cdot 2^{-t}\\ &\geq r_{t-1}(a_{t-1}) - \epsilon \cdot 2^{-t} - \epsilon \cdot 2^{-t}\\ &= \rho_t^{\epsilon}(h_{t}). \end{align*} On the other hand, if $h_{t}$ is not $\sigma_{\rm I}$-acceptable, then
\[\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\rho_{t+1}^{\epsilon} | h_{t}) = -\epsilon \cdot 2^{-t} > -\epsilon \cdot 2^{-t+1} = \rho_t^{\epsilon}(h_{t}).\] This establishes the submartingale property for $\rho_0^{\epsilon}, \rho_1^{\epsilon}, \dots$.
The submartingale property implies that $\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\rho_t^{\epsilon}) \geq \rho_0^{\epsilon}(\oslash) = c - 2\epsilon$ for each $t \in \mathbb{N}$. Using Fatou lemma we thus obtain \[\mathbb{E}_{\sigma_{-i},\sigma_{i}}(g) =\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\limsup_{t \to \infty}\rho_t) \geq \mathbb{E}_{\sigma_{-i},\sigma_{i}}(\limsup_{t \to \infty}\rho_t^{\epsilon}) \geq \limsup_{t \to \infty}\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\rho_t^{\epsilon}) \geq c - 2\epsilon,\] as desired. \end{proof}
\begin{proposition}\label{prop:PlayerII} Let $c \in (0,1]$ and let $f_i : [H] \to [0,1]$ be a Borel-measurable function. If Player {\rm II}~has a winning strategy in the game $G_i(f_i,c)$, then for every $\epsilon > 0$ there exists an open set $O \subseteq [H]$ and a limsup function $g : [H] \to [0,1]$ such that $f_i \leq g$, $\{f_i = 1\} \subseteq O \subseteq \{g = 1\}$, and $v_{i}(g) \leq c + \epsilon$. In particular, $v_i(O) \leq c + \epsilon$; and if $f_i = 1_{W_i}$, then $W_i \subseteq O$. \end{proposition}
\begin{proof} Fix Player II's winning strategy $\sigma_{\rm II}$ in $G_i(f_i,c)$.
\noindent\textsc{Step 1:} Defining $O \subseteq [H]$ and $g : [H] \to [0,1]$.
We recursively define (a) the notion of a \textit{$\sigma_{\rm II}$-acceptable} history in the game $\Gamma$, (b) for each $\sigma_{\rm II}$-acceptable history $h$ in $\Gamma$, Player~I's position $\psi(h)$ in the game $G_i(f_i,c)$, and (c) for each $\sigma_{\rm II}$-acceptable history $h$ of $\Gamma$, a function $u_{h} : A(h) \to [0,1]$.
The empty history $\oslash$ of $\Gamma$ is $\sigma_{\rm II}$-acceptable. We define $\psi(\oslash) := \oslash$, the empty history in $G_i(f_i,c)$. Let $t \in \mathbb{N}$ and consider a history $h_{t} \in H_{t}$ of the game $\Gamma$. If $h_{t}$ is not $\sigma_{\rm II}$-acceptable, so is the history $(h_{t},a_{t})$ for each $a_{t} \in A(h_{t})$. Suppose that $h_{t}$ is $\sigma_{\rm II}$-acceptable and that Player I's position $\psi(h_{t})$ in $G_i(f_i,c)$ has been defined. Take $a_{t} \in A(h_{t})$. Let $R^{*}(h_{t},a_{t})$ denote the set of Player~I's legal moves at position $\psi(h_{t})$ to which $\sigma_{\rm II}$ responds with $a_{t}$: \[R^{*}(h_{t},a_{t}) := \{r_{t} \in R(h_t): (\psi(h_{t}),r_{t}) \in T\text{ and }\sigma_{\rm II}(\psi(h_{t}),r_{t}) = a_{t}\}.\] The history $(h_{t},a_{t})$ is defined to be $\sigma_{\rm II}$\emph{-acceptable} if $R(h_{t},a_{t})$ is not empty. In this case we define \[u_{h_{t}}(a_{t}) := \inf\{r_{t}(a_{t}): r_{t} \in R^{*}(h_{t},a_{t})\}.\] Choose $r_{t} \in R^{*}(h_{t},a_{t})$ with the property that \begin{equation}\label{eqn:proprt} u_{h_{t}}(a_{t}) \leq r_{t}(a_{t}) \leq u_{h_{t}}(a_{t}) + \epsilon \cdot 3^{-t-2}, \end{equation} and define $\psi(h_{t},a_{t}) := (\psi(h_{t}),r_{t},a_{t})$.
Finally, extend the definition of $u_{h}$ to all histories $h$ of $\Gamma$ by setting $u_{h}(a) := 1$ whenever $(h,a)$ is not $\sigma_{\rm II}$-acceptable.
Let $H_{{\rm II}}$ be the set of $\sigma_{\rm II}$-acceptable histories of $\Gamma$. We define the set $O$ to be the complement of $[H_{\rm II}]$, that is $O := [H] \setminus [H_{\rm II}]$. Since $[H_{\rm II}]$ is a closed subset of $[H]$ (e.g. Kechris \cite[Proposition 2.4]{Kechris95}), $O$ is an open subset of $[H]$. Let $T_{{\rm II}} \subseteq T$ be the image of $H_{\rm II}$ under $\psi$. The function $\psi_{{\rm II}} : H_{{\rm II}} \to T_{{\rm II}}$ induces a continuous function $\psi_{{\rm II}} : [H_{{\rm II}}] \to [T_{{\rm II}}]$. Note that all runs in $[T_{{\rm II}}]$ are consistent with Player II's winning strategy $\sigma_{\rm II}$.
For $t \in \mathbb{N}$ define a function $\upsilon_{t} : H_{t} \to \mathbb{R}$ by letting $\upsilon_0(\oslash) := c$; and for each $t \in \mathbb{N}$ and each history $(h_{t},a_{t}) \in H_{t+1}$, by letting $\upsilon_{t+1}(h_{t},a_{t}) := u_{h_{t}}(a_{t})$. Note that, for $t \in \mathbb{N}$ and $h_{t} \in H_{t}$, we have $\upsilon_{t}(h_{t}) = 1$ whenever $h_{t}$ is not $\sigma_{\rm II}$-acceptable.
Also define $g : [H] \to [0,1]$ by letting \[g(a_0,a_1,\ldots) := \limsup_{t \to \infty}\upsilon_{t}(a_0,\ldots,a_{t-1}).\]
\noindent\textsc{Step 2:} Verifying that $f_i \leq g \leq 1$ and $\{f_i = 1\} \subseteq O \subseteq \{g = 1\}$.
The function $g$ is equal to $1$ on the set $O$; thus $O \subseteq \{g = 1\}$. Consider a play $p = (a_0,a_1,\ldots) \in [H_{\rm II}]$, and let $\psi(p) := (r_{0}, a_{0}, r_{1}, a_{1}, \ldots)$. It follows by \eqref{eqn:proprt} that \[g(p) := \limsup_{t \to \infty}r_{t}(a_{t}).\] Since the run $\psi(p)$ is won by Player II, it must hold that either $f_i(p) < g(p)$ or $0 = f_i(p)$; in either case $f_i(p) < 1$ and $f_i(p) \leq g(p)$. We conclude that $[H_{{\rm II}}] \subseteq \{f_i < 1\}$, or equivalently that $\{f_i = 1\} \subseteq O$, and that $f_i \leq g$ on $[H]$.
\noindent\textsc{Step 3:} Verifying that $v_i(g) \leq c + \epsilon$. Since $1_{O} \leq g$, it then follows that $v_i(O) \leq c + \epsilon$.
\noindent\textsc{Step 3.1:} Defining a strategy profile for player~$i$'s opponents.
First we argue that \begin{equation}\label{eqn argue} d_i(u_{\oslash}) \leq c. \end{equation} Suppose to the contrary that $c \leq d_i(u_{\oslash}) - \lambda$ for some $\lambda > 0$. Define $r_0 \in R(\oslash)$ by letting $r_0(a) := \max\{u_{\oslash}(a) - \lambda,0\}$. Since $u_{\oslash} - \lambda \leq r_0$, it holds that $c \leq d_{i}(u_{\oslash}) - \lambda \leq d_{i}(r_0)$. Consequently, $r_0$ is a legal move of Player~I in the game $G_i(f_i,c)$ at position $\oslash$. Denote $a_0 := \sigma_{\rm II}(r_{0})$. As $a_0$ is Player~II's legal move in $G_i(f_i,c)$ at position $(r_0)$, it must be the case that $r_0(a_0) > 0$, and hence $r_0(a_0) = u_{\oslash}(a_{0}) - \lambda$. On the other hand, $r_{0} \in R^{*}(\oslash,a_{0})$, so the definition of $u_{\oslash}$ implies that $u_{\oslash}(a_{0}) \leq r_{0}(a_{0})$, a contradiction.
Take $t \geq 1$, let $h_{t} := (h_{t-1},a_{t-1}) \in H_{t}$ be a $\sigma_{\rm II}$-acceptable history, and let $r_{t-1}$ be such that $\psi(h_{t}) = (\psi(h_{t-1}),r_{t-1},a_{t-1})$. Then \begin{equation}\label{eqn:argue1} d_i(u_{h_{t}}) \leq r_{t-1}(a_{t-1}). \end{equation}
Indeed, suppose to the contrary that $r_{t-1}(a_{t-1}) \leq d_i(u_{h_{t}}) - \lambda$ for some $\lambda > 0$. Define $r_{t} \in R(h_{t})$ by letting $r_{t}(a) := \max\{u_{h_{t}}(a) - \lambda,0\}$. Since $u_{h_{t}} - \lambda \leq r_{t}$, it holds that $r_{t-1}(a_{t-1}) \leq d_{i}(u_{h_{t}}) - \lambda \leq d_{i}(r_{t})$. Consequently, $r_{t}$ is a legal move of Player~I at position $\psi(h_{t})$. Let $a_{t} := \sigma_{\rm II}(\psi(h_{t}), r_{t})$. As $a_{t}$ is Player~II's legal move at position $(\psi(h_{t}), r_{t})$, it must be the case that $r_{t}(a_{t}) > 0$, and hence $r_{t}(a_{t}) = u_{h_{t}}(a_{t}) - \lambda$. On the other hand, $r_{t} \in R^{*}(h_{t}, a_{t})$, so the definition of $u_{h_{t}}$ implies that $u_{h_{t}}(a_{t}) \leq r_{t}(a_{t})$, a contradiction.
We now define a strategy profile $\sigma_{-i}$ of $i$'s opponents in $\Gamma$ as follows: For a history $h_{t} \in H_{t}$ of $\Gamma$ let $\sigma_{-i}(h_{t}) \in X_{-i}(h)$ be such that \begin{equation}\label{eqn str opponent} u_{h_{t}}(\sigma_{-i}(h_{t}),x_{i}) \leq d_{i}(u_{h_{t}}) + \epsilon \cdot 3^{-t-1}\text{ for each }x_{i} \in \Delta(A_i(h_{t})). \end{equation}
\noindent\textsc{Step 3.2:} Verifying that $\mathbb{E}_{\sigma_{-i},\sigma_{i}}(g) \leq c + \epsilon$ for each strategy $\sigma_{i} \in \Sigma_{i}$ of player $i$ in $\Gamma$.
Fix a strategy $\sigma_{i} \in \Sigma_{i}$. For $t \in \mathbb{N}$ define a function $\upsilon_{t}^\epsilon := \upsilon_{t} + \epsilon \cdot 3^{-t}$. The sequence $\upsilon_0^\epsilon, \upsilon_1^\epsilon, \dots$ could be thought of as a process on $[H]$, measurable with respect to the filtration $\{\mathcal{F}_{t}\}_{t \in \mathbb{N}}$. We next show that the process is a supermartingale w.r.t $\mathbb{P}_{\sigma_{-i},\sigma_{i}}$.
By Eqs. \eqref{eqn str opponent} and \eqref{eqn argue}, \begin{align*} \mathbb{E}_{\sigma_{-i},\sigma_{i}}(\upsilon_1^\epsilon) &= \mathbb{E}_{\sigma_{-i},\sigma_{i}}(u_{\oslash}(a_0)) + \epsilon\cdot 3^{-1}\\ &= u_{\oslash}(\sigma_{-i}(\oslash),\sigma_{i}(\oslash)) + \epsilon\cdot 3^{-1}\\ &\leq d_{i}(u_{\oslash}) + \epsilon\cdot 2 \cdot 3^{-1}\\ &\leq c + \epsilon = \upsilon_{0}^\epsilon(\oslash). \end{align*} Take $t \geq 1$, let $h_{t} = (h_{t-1},a_{t-1}) \in H_{t}$ be a $\sigma_{\rm II}$-acceptable history, and let $r_{t-1}$ be such that $\psi(h_{t}) = (\psi(h_{t-1}),r_{t-1},a_{t-1})$. We have by Eqs. \eqref{eqn str opponent}, \eqref{eqn:argue1}, and \eqref{eqn:proprt}: \begin{align*} \mathbb{E}_{\sigma_{-i},\sigma_{i}}(\upsilon_{t+1}^\epsilon \mid h_{t}) &= \mathbb{E}_{\sigma_{-i},\sigma_{i}}(u_{h_{t}}(a_t) \mid h_{t}) + \epsilon\cdot3^{-t-1}\\ &= u_{h_{t}}(\sigma_{-i}(h_{t}),\sigma_{i}(h_{t})) + \epsilon\cdot3^{-t-1}\\ &\leq d_{i}(u_{h_{t}}) + \epsilon \cdot 2 \cdot 3^{-t-1}\\ &\leq r_{t-1}(a_{t-1}) + \epsilon \cdot 2 \cdot 3^{-t-1}\\ &\leq u_{h_{t-1}}(a_{t-1}) + \epsilon \cdot 3 \cdot 3^{-t-1}\\ &= \upsilon_{t}^\epsilon(h_{t-1},a_{t-1}) = \upsilon_{t}^\epsilon(h_{t}). \end{align*} If, on the other hand, the history $h_{t}$ is not $\sigma_{\rm II}$-acceptable, then \[\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\upsilon_{t+1}^\epsilon \mid h_{t}) = 1+\epsilon\cdot 3^{-t-1} \leq 1+\epsilon\cdot 3^{-t} = \upsilon_{t}^\epsilon(h_{t}).\]
Since the process $\upsilon_0^\epsilon,\upsilon_1^\epsilon,\dots$ is bounded below (by 0), by the Martingale Convergence Theorem, it converges pointwise $\mathbb{P}_{\sigma_{-i},\sigma_{i}}$-almost surely; whenever the process converges, its limit is $g$. Hence $\mathbb{E}_{\sigma_{-i},\sigma_{i}}(g) = \mathbb{E}_{\sigma_{-i},\sigma_{i}}(\lim_{t \to \infty}\upsilon_t^\epsilon) \leq \upsilon_0^\epsilon(\oslash) = c + \epsilon$, as desired. \end{proof}
We now invoke the result of Martin \cite{Martin75} on Borel determinacy of perfect information games. To do so, we endow $[T]$ with its relative topology as a subspace of the product space $(R \cup A)^{\mathbb{N}}$, where $R \cup A$ is given its discrete topology. One can then check that Player I's winning set in $G_i(f_i,c)$ is a Borel subset of $[T]$. It follows that for each $c \in (0,1]$ the game $G_i(f_i,c)$ is determined: either Player~I has a winning strategy in the game or Player~II does. We arrive at the following conclusion.
\begin{proposition}\label{prop:det} If $v_i(f_i) < c$, then Player II has a winning strategy in $G_i(f_i,c)$. If $c< v_i(f_i)$, then Player I has a winning strategy in $G_i(f_i,c)$. \end{proposition}
Theorems \ref{thrm:reg} and \ref{thrm:regfunc} follow from Propositions \ref{prop:PlayerI}, \ref{prop:PlayerII}, and \ref{prop:det}.
\noindent\textbf{Proof of Theorem \ref{thrm:tailapprox}:} Take an $\varepsilon > 0$. Without loss of generality, suppose that $f_i$ takes values in $[0,1]$.
By Proposition \ref{prop:v(Q)=1} we know that $v_{i}(Q_{i,\varepsilon}(f_i)) = 1$. To obtain an approximation from below, use Theorem \ref{thrm:reg} to choose a closed set $C \subseteq Q_{i, \varepsilon}(f_i)$ such that $1 - \varepsilon \leq v_{i}(C)$, and define the function $g := (v_i(f_i) - \varepsilon) \cdot 1_{C}$. Then $g \leq f_i$ and $v_i(f_i) - 2\varepsilon \leq (v_i(f_i) - \varepsilon)\cdot(1-\varepsilon) \leq v_{i}(g)$. Since $C$ is closed, $g$ is upper semicontinuous.
By Proposition \ref{prop:v(Q)=1} we know that $v_{i}(U_{i,\varepsilon}(f_i)) = 0$. To obtain an approximation from above, use Theorem \ref{thrm:reg} to choose an open set $O \supseteq U_i^{\varepsilon}(f_i)$ such that $v_{i}(O) \leq \varepsilon$, and define the function $g := v_i(f_i) + \varepsilon + (1 - v_i(f_i) - \varepsilon) \cdot 1_{O}$. Then $f_i \leq g \leq 1$ and $v_{i}(g) \leq v_i(f_i) + 2\varepsilon$. Since $O$ is open, $g$ is lower semicontinuous. $\Box$
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{abstract}
We prove that if $\sigma \in S_m$ is a pattern of $w \in S_n$, then we can express the Schubert polynomial $\mathfrak{S}_w$ as a monomial times $\mathfrak{S}_\sigma$ (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar's orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on $\mathfrak{S}_w$ being zero-one. In this case, the Schubert polynomial $\mathfrak{S}_w$ is equal to the integer point transform of a generalized permutahedron. \end{abstract}
\title{Zero-one Schubert polynomials}
\section{Introduction}
Schubert polynomials, introduced by Lascoux and Sch\"utzenberger in \cite{LS}, represent cohomology classes of Schubert cycles in the flag variety. Knutson and Miller also showed them to be multidegrees of matrix Schubert varieties \cite{multidegree}. There are a number of combinatorial formulas for the Schubert polynomials \cite{laddermoves, BJS, FK1993, nilcoxeter, thomas, lenart, manivel, prismtableaux},
yet only recently has the structure of their supports been investigated:
the support of a Schubert polynomial $\mathfrak{S}_w$ is the set of all integer points of a certain generalized permutahedron $\mathcal{P}(w)$ \cite{FMS, MTY}.
The question motivating this paper is to characterize when $\mathfrak{S}_w$ equals the integer point transform of $\mathcal{P}(w)$,
in other words, when all the coefficients of $\mathfrak{S}_w$ are equal to $0$ or $1$.
One of our main results is a pattern-avoidance characterization of the permutations corresponding to these polynomials:
\begin{theorem} \label{thm:01}
The Schubert polynomial $\mathfrak{S}_w$ is zero-one if and only if $w$ avoids the patterns $12543$, $13254$, $13524$, $13542$, $21543$, $125364$, $125634$, $215364$, $215634$, $315264$, $315624$, and $315642$.
\end{theorem}
In Theorem~\ref{thm:011} we provide further equivalent conditions on the Schubert polynomial $\mathfrak{S}_w$ being zero-one. One implication of Theorem~\ref{thm:01} follows from our other main result, which relates the Schubert polynomials $\mathfrak{S}_\sigma$ and $\mathfrak{S}_w$ when $\sigma$ is a pattern of $w$:
\begin{theorem}
\label{thm:pattern}
Fix $w \in S_n$ and let $\sigma \in S_{n-1}$ be the pattern with Rothe diagram $D(\sigma)$ obtained by removing row $k$ and column $w_k$ from $D(w)$. Then
\begin{align*}
\mathfrak{S}_{w}(x_1, \ldots, x_n)=M(x_1, \ldots, x_n) \mathfrak{S}_{\sigma}(x_{1}, \ldots, \widehat{x_k}, \ldots, x_{n})+F(x_1, \ldots, x_n),
\end{align*}
where $F\in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n]$ and
\[M(x_1, \ldots, x_n) = \left(\prod_{(k,i)\in D(w)}{x_k}\right)\left(\prod_{(i,w_k)\in D(w)}{x_i} \right).\]
\end{theorem}
Theorem~\ref{thm:pattern} is a special case of Theorem~\ref{thm:pattern2}, which applies to the dual character of the flagged Weyl module of any diagram.
\subsection*{Outline of this paper}
Section~\ref{sec:magyar} gives an expression of Magyar for Schubert polynomials in terms of orthodontic sequences $(\bm{i}, \bm{m})$. In Section~\ref{sec:suff}, we give a condition ``multiplicity-free'' on the orthodontic sequence $(\bm{i}, \bm{m})$ of $w$ which implies that $\mathfrak{S}_w$ is zero-one. In Section~\ref{sec:mult-patt} we show that multiplicity-freeness can equivalently be phrased in terms of pattern avoidance. We then prove in Section~\ref{sec:mult-patt} that multiplicity-freeness is also a necessary condition for $\mathfrak{S}_w$ to be zero-one. In the latter proof we assume Theorem~\ref{thm:pattern}, whose generalization (Theorem~\ref{thm:pattern2}) and proof is the subject of Section~\ref{sec:trans}.
\section{Magyar's orthodontia for Schubert polynomials}
\label{sec:magyar}
In this section we explain the results we use to show one direction of Theorem~\ref{thm:01}. We include the classical definition of Schubert polynomials here for reference.
The \emph{Schubert polynomial} of the longest permutation $w_0=n \hspace{.1cm} n\!-\!1 \hspace{.1cm} \cdots \hspace{.1cm} 2 \hspace{.1cm} 1 \in S_n$ is
\[\mathfrak{S}_{w_0}\coloneqq x_1^{n-1}x_2^{n-2}\cdots x_{n-1}.\]
For $w\in S_n$, $w\neq w_0$, there exists $i\in [n-1]$ such that $w_i<w_{i+1}$.
For any such~$i$, the Schubert polynomial $\mathfrak{S}_{w}$ is defined as
\[\mathfrak{S}_{w}(x_1, \ldots, x_n)\coloneqq \partial_i \mathfrak{S}_{ws_i}(x_1, \ldots, x_n),\]
where $s_i$ is the transposition swapping $i$ and $i+1$, and $\partial_i$ is the $i$th divided difference operator
\[\partial_i (f):=\frac{f(x_1,\ldots,x_n)-f(x_1,\ldots,x_{i-1},x_{i+1},x_i,x_{i+2},\ldots,x_n)}{x_i-x_{i+1}}.\]
Since the operators $\partial_i$ satisfy the braid relations, the Schubert polynomials $\mathfrak{S}_{w}$ are well-defined.
We will not be using the above definition of Schubert polynomials in this work. Instead, we will make use of several results due to Magyar in \cite{magyar}. We start by summarizing Proposition 15 and Proposition 16 of \cite{magyar} and supplying the necessary background, closely following the exposition of \cite{magyar}.
By a \emph{diagram}, we mean a sequence $D=(C_1,C_2,\ldots,C_n)$ of finite subsets of $[n]$, called the \emph{columns} of $D$. We interchangeably think of $D$ as a collection of boxes $(i,j)$ in a grid, viewing an element $i\in C_j$ as a box in row $i$ and column $j$ of the grid. When we draw diagrams, we read the indices as in a matrix: $i$ increases top-to-bottom and $j$ increases left-to-right.
Two diagrams $D$ and $D'$ are called \emph{column-equivalent} if one is obtained from the other by reordering nonempty columns and adding or removing any number of empty columns. For a column $C\subseteq [n]$, let the \emph{multiplicity} $\mathrm{mult}_D(C)$ be the number of columns of $D$ which are equal to $C$. The sum of diagrams, denoted $D\oplus D'$, is constructed by concatenating the lists of columns; graphically this means placing $D'$ to the right of $D$.
The \emph{Rothe diagram} $D(w)$ of a permutation $w\in S_n$ is the diagram
\[ D(w)=\{(i,j)\in [n]\times [n] \mid i<(w^{-1})_j\mbox{ and } j<w_i \}. \]
Note that Rothe diagrams have the \emph{northwest property}: If $(r,c'),(r',c)\in D(w)$ with $r<r'$ and $c<c'$, then $(r,c)\in D(w)$.
\begin{example}
If $w=31542$, then
\begin{center}
\begin{tikzpicture}[scale=.55]
\draw (0,0)--(5,0)--(5,5)--(0,5)--(0,0);
\filldraw[draw=black,fill=lightgray] (0,4)--(1,4)--(1,5)--(0,5)--(0,4);
\filldraw[draw=black,fill=lightgray] (1,4)--(2,4)--(2,5)--(1,5)--(1,4);
\filldraw[draw=black,fill=lightgray] (1,2)--(2,2)--(2,3)--(1,3)--(1,2);
\filldraw[draw=black,fill=lightgray] (1,1)--(2,1)--(2,2)--(1,2)--(1,1);
\filldraw[draw=black,fill=lightgray] (3,2)--(4,2)--(4,3)--(3,3)--(3,2);
\node at (-1.5,2.5) {$D(w)=$};
\node at (8.9,2.5) {$=(\{1\},\{1,3,4\},\emptyset,\{3\},\emptyset).$};
\end{tikzpicture}
\end{center}
\end{example}
We next recall Magyar's orthodontia. Let $D$ be the Rothe diagram of a permutation $w\in S_n$ with columns $C_1,C_2,\ldots,C_n$. We describe an algorithm for constructing a reduced word $\bm{i}=(i_1,\ldots,i_l)$ and a multiplicity list $\bm{m}=(k_1,\ldots,k_n;\,m_1,\ldots,m_l)$ such that the diagram $D_{\bm{i},\bm{m}}$ defined by
\[D_{\bm{i},\bm{m}} = \bigoplus_{j=1}^n {k_j\cdot [j]} \quad \oplus \quad \bigoplus_{j=1}^l m_j\cdot (s_{i_1}s_{i_2}\cdots s_{i_j}[i_j]), \]
is column-equivalent to~$D$. In the above, $m\cdot C$ denotes $C\oplus \cdots\oplus C$ with $m$ copies of~$C$; in particular $0\cdot C$ should be interpreted as a diagram with no columns, not the empty column.
The algorithm to produce $\bm{i}$ and $\bm{m}$ from $D$ is as follows. To begin the first step, for each $j\in [n]$ let $k_j=\mathrm{mult}_D([j])$, the number of columns of $D$ of the form $[j]$. Replace all such columns by empty columns for each $j$ to get a new diagram $D_-$.
Given a column $C\subseteq [n]$, a \emph{missing tooth} of $C$ is a positive integer $i$ such that $i\notin C$, but $i+1\in C$. The only columns without missing teeth are the empty column and the intervals $[i]$. Hence the first nonempty column of $D_-$ (if there is any) contains a smallest missing tooth $i_1$. Switch rows $i_1$ and $i_1+1$ of $D_-$ to get a new diagram $D'$.
In the second step, repeat the above with $D'$ in place of $D$. That is, let $m_1=\mathrm{mult}_{D'}([i_1])$ and replace all columns of the form $[i_1]$ in $D'$ by empty columns to get a new diagram $D_-'$. Find the smallest missing tooth $i_2$ of the first nonempty column of $D_-'$, and switch rows $i_2$ and $i_2+1$ of $D_-'$ to get a new diagram $D''$.
Continue in this fashion until no nonempty columns remain. It is easily seen that the sequences $\bm{i}=(i_1,\ldots,i_l)$ and $\bm{m}=(k_1,\ldots,k_n;\,m_1,\ldots,m_l)$ just constructed have the desired properties.
\begin{definition}
\label{def:imsequence}
The pair $(\bm{i},\bm{m})$ constructed from the preceding algorithm is called the \emph{orthodontic sequence} of $w$.
\end{definition}
\begin{example}
If $w=31542$, then the orthodontic sequence algorithm produces the diagrams
\begin{center}
\includegraphics[scale=.85]{orthodontia.pdf}
\end{center}
The sequence of missing teeth gives $\bm{i}=(2,3,1)$ and $\bm{m}=(1,0,0,0,0;\,0,1,1)$, so
\begin{center}
\begin{tikzpicture}[scale=.5]
\draw (9.5,0)--(12.5,0)--(12.5,5)--(9.5,5)--(9.5,0);
\filldraw[draw=black,fill=lightgray] (9.5,4)--(10.5,4)--(10.5,5)--(9.5,5)--(9.5,4);
\filldraw[draw=black,fill=lightgray] (10.5,4)--(11.5,4)--(11.5,5)--(10.5,5)--(10.5,4);
\filldraw[draw=black,fill=lightgray] (10.5,2)--(11.5,2)--(11.5,3)--(10.5,3)--(10.5,2);
\filldraw[draw=black,fill=lightgray] (10.5,1)--(11.5,1)--(11.5,2)--(10.5,2)--(10.5,1);
\filldraw[draw=black,fill=lightgray] (11.5,2)--(12.5,2)--(12.5,3)--(11.5,3)--(11.5,2);
\node at (8,2.5) {$D_{\bm{i},\bm{m}}=$};
\node at (12.7,2.5) {.};
\end{tikzpicture}
\end{center}
\end{example}
\begin{theorem}[{\cite[Proposition~15]{magyar}}]
\label{thm:magyaroperatortheorem}
Let $w\in S_n$ have orthodontic sequence $(\bm{i},\bm{m})$. If $\pi_j=\partial_j x_j$ denotes the $j$th Demazure operator and $\omega_j=x_1x_2\cdots x_j$, then
\[\mathfrak{S}_w = \omega_1^{k_1}\cdots\omega_n^{k_n}\pi_{i_1}(\omega_{i_1}^{m_1} \pi_{i_2}(\omega_{i_2}^{m_2}\cdots \pi_{i_l}(\omega_{i_l}^{m_l})\cdots )). \]
\end{theorem}
\begin{example}
For $w=31542$, it is easily checked that
\[\mathfrak{S}_w=x_1\pi_2\pi_3(x_1x_2x_3\pi_1(x_1)). \]
\end{example}
Theorem~\ref{thm:magyaroperatortheorem} can also be realized on the level of tableaux, analogous to Young tableaux in the case of Schur polynomials. A \emph{filling} (with entries in $\{1,...,n\}$) of a diagram $D$ is a map $T$ assigning to each box in $D$ an integer in $[n]$. A filling $T$ is called \emph{column-strict} if $T$ is strictly increasing down each column of $D$. The \emph{weight} of a filling $T$ is the vector $wt(T)$ whose $i$th component $wt(T)_i$ is the number of times $i$ occurs in $T$.
Given a permutation $w\in S_n$ with orthodontic sequence $(\bm{i},\bm{m})$, we will define a set $\mathcal{T}_w$ of fillings of the diagram $D_{\bm{i},\bm{m}}$ which satisfy
\[\mathfrak{S}_w=\sum_{T\in\mathcal{T}_w}x_1^{wt(T)_1}x_2^{wt(T)_2}\cdots x_n^{wt(T)_n}. \]
We start by recalling the \emph{root operators}, first defined in \cite{rootoperators}. These are operators $f_i$ which either take a filling $T$ of a diagram $D$ to another filling of $D$, or are undefined on $T$. To define root operators, we first encode a filling $T$ in terms of its reading word. The \emph{reading word} of a filling $T$ of a diagram $D=D_{\bm{i},\bm{m}}$ is the sequence of the entries of $T$ read in order, down each column going left-to-right along columns; that is the sequence
\[T(1,1),T(2,1),\ldots,T(n,1),T(1,2),T(2,2),\ldots,T(n,2),\ldots, T(n,n) \]
ignoring any boxes $(i,j)\notin D$.
If it is defined, the operator $f_i$ changes an entry of $i$ in $T$ to an entry of $i+1$ according to the following rule. First, ignore all the entries in $T$ except those which equal $i$ or $i+1$. Now ``match parentheses'':
if, in the list of entries not yet ignored, an $i$ is followed by an $i+1$, then henceforth ignore that pair of entries as well; look again for an $i$ followed (up to ignored entries) by an $i+1$, and henceforth ignore this pair; continue doing this until all no such pairs remain unignored. The remaining entries of $T$ will be a subword of the form $i+1,i+1,\ldots,i+1,i,i,\ldots,i$. If $i$ does not appear in this word, then $f_i(T)$ is undefined. Otherwise, $f_i$ changes the leftmost $i$ to an $i+1$. Reading the image word back into $D$ produces a new filling. We can iteratively apply $f_i$ to a filling $T$.
\begin{example}
If $T=3\,1\,2\,2\,2\,1\,3\,1\,2\,4\,3\,2\,4\,1\,3\,1$, applying $f_1$ iteratively to $T$ yields:
\begin{center}
\begin{tabular}{rccccccccccccccccc}
$\phantom{f_1(}T\phantom{)}$=&3&1&2&2&2&1&3&1&2&4&3&2&4&1&3&1&\\
$\phantom=$&$\cdot$&1&2&2&2&1&$\cdot$&1&2&$\cdot$&$\cdot$&2&$\cdot$&1&$\cdot$&1&\\
$\phantom=$&$\cdot$&$\cdot$&$\cdot$&2&2&1&$\cdot$&$\cdot$&$\cdot$&$\cdot$&$\cdot$&2&$\cdot$&1&$\cdot$&1&\\
$\phantom=$&$\cdot$&$\cdot$&$\cdot$&2&2&$\cdot$&$\cdot$&$\cdot$&$\cdot$&$\cdot$&$\cdot$&$\cdot$&$\cdot$&1&$\cdot$&1&\\
$f_1(T)=$&3&1&2&2&2&1&3&1&2&4&3&2&4&\textbf{2}&3&1&\\
$f_1^2(T)=$&3&1&2&2&2&1&3&1&2&4&3&2&4&2&3&\textbf{2}&\\
$f_1^3(T)\mathrel{\phantom=}$&&&&&&&&&&&&&&&&&\hspace{-45ex}\mbox{is undefined}
\end{tabular}
\end{center}
\end{example}
Define the set-valued \emph{quantized Demazure operator} $\widetilde{\pi}_i$ by $\widetilde{\pi}_i(T)=\{T,f_i(t),f_i^2(T),\ldots\}$. For a set $\mathcal{T}$ of tableaux, let
\[\widetilde{\pi}_{i}(\mathcal{T})=\bigcup_{T\in\mathcal{T}}\widetilde{\pi}_i(T).\]
Next, consider the column $[j]$ and its minimal column-strict filling $\widetilde{\omega}_j$ ($j$th row maps to $j$). For a filling $T$ of a diagram $D$ with columns $(C_1,C_2,\ldots,C_n)$, define in the obvious way the composite filling of $[j]\oplus D$, corresponding to concatenating the reading words of $[j]$ and $D$. Define $[j]^r\oplus D$ analogously by adding $r$ columns $[j]$ to $D$, each with filling $\widetilde{\omega}_j$.
\begin{definition}
\label{def:magyartableauxset}
Let $w\in S_n$ be a permutation with orthodontic sequence $(\bm{i},\bm{m})$. Define the set $\mathcal{T}_w$ of tableaux by
\[\mathcal{T}_w=\widetilde{\omega}_1^{k_1}\oplus\cdots\oplus\widetilde{\omega}_n^{k_n}\oplus\widetilde{\pi}_{i_1}(\widetilde{\omega}_{i_1}^{m_1}\oplus \widetilde{\pi}_{i_2}(\widetilde{\omega}_{i_2}^{m_2}\oplus\cdots \oplus\widetilde{\pi}_{i_l}(\widetilde{\omega}_{i_l}^{m_l})\cdots )). \]
\end{definition}
\begin{theorem}[{\cite[Proposition~16]{magyar}}]
\label{thm:magyartableauxtheorem}
Let $w\in S_n$ be a permutation with orthodontic sequence $(\bm{i},\bm{m})$. Then,
\[\mathfrak{S}_w=\sum_{T\in\mathcal{T}_w}x_1^{wt(T)_1}x_2^{wt(T)_2}\cdots x_n^{wt(T)_n}. \]
\end{theorem}
\begin{example}
Consider again $w=31542$, so the orthodontic sequence of $w$ is $\bm{i}=(2,3,1)$ and $\bm{m}=(1,0,0,0,0; 0,1,1)$.
The set $\mathcal{T}_w$ is built up as follows:
\begin{align*}
\{\}&\xrightarrow{\widetilde{\omega}_1}\{1\}\xrightarrow{\widetilde{\pi}_1}\{1,2\}\xrightarrow{\widetilde{\omega}_3}\{1231,1232\}\xrightarrow{\widetilde{\pi}_3}\{1231,1241,1232,1242\}\\
&\xrightarrow{\widetilde{\pi}_2}\{1231,1241,1341,1232,1233,1242,1342,1343\}\\
&\xrightarrow{\widetilde{\omega}_1}\{11231,11241,11341,11232,11233,11242,11342,11343\}
\end{align*}
which agrees with
\[\mathfrak{S}_{w}=x_1^3x_2x_3+x_1^3x_2x_4+x_1^3x_3x_4+x_1^2x_2^2x_3+x_1^2x_2x_3^2+x_1^2x_2^2x_4+x_1^2x_2x_3x_4+x_1^2x_3^2x_4.\]
\end{example}
We now describe a way to view each step of the construction of $\mathcal{T}_w$ as producing a set of fillings of a diagram.
\begin{definition}
\label{def:partialfilling}
Let $w$ be a permutation with orthodontic sequence $(\bm{i},\bm{m})$, $\bm{i}=(i_1,\ldots,i_l)$. For each $r\in [l]$, define
\[\mathcal{T}_w(r)=\widetilde{\omega}_{i_r}^{m_r}\oplus\widetilde{\pi}_{i_{r+1}}(\widetilde{\omega}_{i_{r+1}}^{m_{r+1}}\oplus\cdots\oplus \widetilde{\pi}_{i_l}(\widetilde{\omega}_{i_l})\cdots). \]
Set $\mathcal{T}_w(0)=\mathcal{T}_w$.
\end{definition}
\begin{definition}
\label{def:partialdiagram}
Let $w$ be a permutation with orthodontic sequence $(\bm{i},\bm{m})$, $\bm{i}=(i_1,\ldots,i_l)$. For any $r \in [l]$, let $O(w,r)$ be the diagram obtained from $D(w)$ in the construction of $(\bm{i},\bm{m})$ at the time when the row swaps of the missing teeth $i_1,\ldots,i_{r}$ have all been executed on $D(w)$, but after executing the row swap of the missing tooth $i_r$, columns without missing teeth have not yet been removed ($m_r$ has not yet been recorded). Set $O(w,0)=D(w)$. For each $r$, give $O(w,r)$ the same column indexing as $D(w)$, so any columns replaced by empty columns in the execution of the missing teeth $i_1,\ldots,i_{r-1}$ retain their original index in $D(w)$.
\end{definition}
The motivation behind Definition~\ref{def:partialfilling} and Definition~\ref{def:partialdiagram} is that the elements of $\mathcal{T}_w(r)$ can be viewed as column-strict fillings of $O(w,r)$ for each $r$. To do this, the choice of filling order for $O(w,r)$ is crucial. Let $w\in S_n$ and consider $D=D(w)$ and $D_{\bm{i},\bm{m}}$. Suppose $D$ has $z$ nonempty columns. There is a unique permutation $\tau$ of $[n]$ taking the column indices of $D$ to the column indices of $D_{\bm{i},\bm{m}}\oplus \emptyset^{n-z}$ with the following properties:
\begin{itemize}
\item Column $c$ of $D$ is the same as column $\tau(c)$ of $D_{\bm{i},\bm{m}}$.
\item If column $c$ and column $c'$ of $D$ are equal with $c<c'$, then $\tau(c)<\tau(c')$.
\end{itemize}
Recall that the columns of $O(w,r)$ have the same column labels as $D$. To read an element $T\in \mathcal{T}_w(r)$ into $O(w,r)$, read $T$ left-to-right and fill in top-to-bottom columns $\tau^{-1}(n),\,\tau^{-1}(n-1),\ldots,\tau^{-1}(1)$ (ignoring any column indices referring to empty columns).
\begin{lemma}
\label{lem:fillings}
Let $w\in S_n$ have orthodontic sequence $(\bm{i},\bm{m})$, $\bm{i}=(i_1,\ldots,i_l)$. In the filling order specified above, the elements of $\mathcal{T}_w(r)$ are column-strict fillings of $O(w,r)$ for each $0\leq r\leq l$.
\end{lemma}
\begin{example}
Take again $w=31542$ with orthodontic sequence $\bm{i}=(2,3,1)$ and $\bm{m}=(1,0,0,0,0;0,1,1)$.
Recall that
\begin{center}
\begin{tikzpicture}[scale=.5]
\draw (0,0)--(5,0)--(5,5)--(0,5)--(0,0);
\filldraw[draw=black,fill=lightgray] (0,4)--(1,4)--(1,5)--(0,5)--(0,4);
\filldraw[draw=black,fill=lightgray] (1,4)--(2,4)--(2,5)--(1,5)--(1,4);
\filldraw[draw=black,fill=lightgray] (1,2)--(2,2)--(2,3)--(1,3)--(1,2);
\filldraw[draw=black,fill=lightgray] (1,1)--(2,1)--(2,2)--(1,2)--(1,1);
\filldraw[draw=black,fill=lightgray] (3,2)--(4,2)--(4,3)--(3,3)--(3,2);
\node at (-1.6,2.5) {$D(w)=$};
\node at (6,2.5) {and};
\draw (9.5,0)--(12.5,0)--(12.5,5)--(9.5,5)--(9.5,0);
\filldraw[draw=black,fill=lightgray] (9.5,4)--(10.5,4)--(10.5,5)--(9.5,5)--(9.5,4);
\filldraw[draw=black,fill=lightgray] (10.5,4)--(11.5,4)--(11.5,5)--(10.5,5)--(10.5,4);
\filldraw[draw=black,fill=lightgray] (10.5,2)--(11.5,2)--(11.5,3)--(10.5,3)--(10.5,2);
\filldraw[draw=black,fill=lightgray] (10.5,1)--(11.5,1)--(11.5,2)--(10.5,2)--(10.5,1);
\filldraw[draw=black,fill=lightgray] (11.5,2)--(12.5,2)--(12.5,3)--(11.5,3)--(11.5,2);
\node at (8,2.5) {$D_{\bm{i},\bm{m}}=$};
\node at (12.7,2.5) {,};
\end{tikzpicture}
\end{center}
so $\tau=12435=\tau^{-1}$. Consider the elements $1\in\mathcal{T}_w(3)$, $1232\in\mathcal{T}_w(2)$, $1242\in\mathcal{T}_w(1)$, and $11342\in\mathcal{T}_w(0)$. The column filling order of each $O(w,r)$ is given by reading $\tau^{-1}$ in one-line notation right to left: in the indexing of $D(w)$, fill down column 4, then down column 2, then down column 1. The elements of each set $\mathcal{T}_w(r)$ are column-strict fillings in the corresponding diagrams $O(w,r)$:
\begin{center}
\includegraphics[scale=1]{tableauxinterpretation.pdf}
\end{center}
\end{example}
\begin{lemma}
Let $w$ be a permutation with orthodontic sequence $(\bm{i},\bm{m})$, $\bm{i}=(i_1,\ldots,i_l)$. For each $0\leq r \leq l$, $O(w,r)$ has the northwest property.
\end{lemma}
\begin{definition}
A filling $T$ of a diagram $D$ is called \emph{row-flagged} if $T(p,q)\leq p$ for each box $(p,q)\in D$.
\end{definition}
\begin{lemma}
\label{lem:rowflagged}
For each $0\leq r\leq l$, the elements of $\mathcal{T}_w(r)$ are row-flagged fillings of $O(w,r)$.
\end{lemma}
\begin{proof}
Clearly, the singleton $\mathcal{T}_w(l)$ is a row-flagged filling of $O(w,l)$. Assume that for some $l\geq s>0$, the result holds with $r=s$. We show that the result also holds with $r=s-1$. Let $T\in\mathcal{T}_w(s)$. We must show that for each $u$, if $f_{i_s}^u(T)$ is defined, then $\widetilde{\omega}_{i_{s-1}}^{m_{s-1}}\oplus f_{i_s}^u(T)$ is a row-flagged filling of $O(w,s-1)$. By the orthodontia construction, $O(w,s)$ is obtained from $O(w,s-1)$ by removing the $m_{s-1}$ columns with no missing tooth, and then switching rows $i_s+1$ and $i_s$.
Since $T$ is a row-flagged filling of $O(w,s)$, each box in $O(w,s)$ containing an entry of $T$ equal to $i_s$ lies in a row with index at least $i_s$. Any box in $O(w,s)$ containing an entry of $T$ equal to $i_s$ and lying in row $i_s$ of $O(w,s)$ will have row index $i_s+1$ in $O(w,s-1)$. Any box in $O(w,s)$ containing an entry in $T$ equal to $i_s$ and lying in a row $d>i_s$ of $O(w,s)$ will still have row index $d$ in $O(w,s-1)$. Then if $f_{i_s}^u(T)$ is defined, $\widetilde{\omega}_{i_{s-1}}^{m_{s-1}}\oplus f_{i_s}^u(T)$ will be a row-flagged filling of $O(w,s-1)$.
\end{proof}
\section{Zero-one Schubert polynomials}
\label{sec:suff}
This section is devoted to giving a sufficient condition on the orthodontic sequence $(\bm{i},\bm{m})$ of $w$ for the Schubert polynomial $\mathfrak{S}_w$ to be zero-one. We give such a condition in Theorem~\ref{thm:multfree}. We will see in Theorem~\ref{thm:011} that this condition turns out to also be a necessary condition for $\mathfrak{S}_w$ to be zero-one.
We start with a less ambitious result:
\begin{proposition}
\label{prop:norepeats}
Let $w\in S_n$ and $(\bm{i},\bm{m})$ be the orthodontic sequence of $w$. If $\bm{i}=(i_1,\ldots,i_l)$ has distinct entries, then $\mathfrak{S}_w$ is zero-one.
\end{proposition}
\begin{proof}
Let $T,T'\in\mathcal{T}_w$ with $wt(T)=wt(T')$. By Definition~\ref{def:magyartableauxset}, we can find $p_1,\ldots,p_l$ so that
\[T=\widetilde{\omega}_1^{k_1}\oplus\cdots\oplus \widetilde{\omega}_n^{k_n}\oplus f_{i_1}^{p_1}(\widetilde{\omega}_{i_1}^{m_1}\oplus\cdots\oplus f_{i_l}^{p_l}(\widetilde{\omega}_{i_l}^{m_l})\cdots). \]
Then if $e_1,\ldots,e_n$ denote the standard basis vectors of $\mathbb{R}^n$,
\[wt(T) = \sum_{j=1}^{n}wt(\widetilde{\omega}_{j}^{k_j}) + \sum_{j=1}^{l}wt(\widetilde{\omega}_{i_j}^{m_j})+\sum_{j=1}^{l}p_j(e_{i_j+1}-e_{i_j}). \]
Similarly, we can find $q_1,\ldots,q_l$ so that
\[T'=\widetilde{\omega}_1^{k_1}\oplus\cdots\oplus \widetilde{\omega}_n^{k_n}\oplus f_{i_1}^{q_1}(\widetilde{\omega}_{i_1}^{m_1}\oplus\cdots\oplus f_{i_l}^{q_l}(\widetilde{\omega}_{i_l}^{m_l})\cdots), \]
which implies
\[wt(T') = \sum_{j=1}^{n}wt(\widetilde{\omega}_{j}^{k_j}) + \sum_{j=1}^{l}wt(\widetilde{\omega}_{i_j}^{m_j})+\sum_{j=1}^{l}q_j(e_{i_j+1}-e_{i_j}). \]
As $wt(T)=wt(T')$,
\begin{align}
\label{eqn:vectorexpansion}
0=wt(T)-wt(T')=(p_1-q_1)(e_{i_1+1}-e_{i_1})+\cdots+(p_l-q_l)(e_{i_l+1}-e_{i_l})\tag{$*$}.
\end{align}
Since the vectors $\{e_{i_j+1}-e_{i_j}\}_{j=1}^{l}$ are independent and $\bm{i}$ has distinct entries, $p_j=q_j$ for all $j$. Thus $T=T'$. This shows that all elements of $\mathcal{T}_w$ have distinct weights, so $\mathfrak{S}_w$ is zero-one.
\end{proof}
We now strengthen Proposition~\ref{prop:norepeats} to allow $\bm{i}$ to not have distinct entries. To do this, we will need a technical definition related to the orthodontic sequence. Recall the construction of the orthodontic sequence $(\bm{i},\bm{m})$ of a permutation $w\in S_n$ (Definition~\ref{def:imsequence}) and the intermediate diagrams $O(w,r)$ (Definition~\ref{def:partialdiagram}).
Let $\bm{i}=(i_1,\ldots,i_l)$, and define $O(w,r)_-$ to be the diagram $O(w,r)$ with all columns of the form $[i_{r}]$ replaced by empty columns.
\begin{definition}
Define the \emph{orthodontic impact function} $\mathcal{I}_w:[l]\to 2^{[n]}$ by
\[\mathcal{I}_w(j)=\{c\in [n] \mid (i_j+1,c)\in O(w,j-1)_-\}. \]
\end{definition}
\noindent That is, $\mathcal{I}_w(j)$ is the set of indices of columns of $O(w,j-1)_-$ that are changed when rows $i_j$ and $i_j+1$ are swapped to form $O(w,j)$.
\begin{definition}
Let $w\in S_n$ have orthodontic sequence $(\bm{i},\bm{m})$, $\bm{i}=(i_1,\ldots,i_l)$. We say $w$ is \emph{multiplicity-free} if for any $r,s\in [l]$ with $r\neq s$ and $i_r=i_s$, we have $\mathcal{I}_w(r)=\mathcal{I}_w(s)=\{c\}$ for some $c\in [n]$.
\end{definition}
\begin{example}
If $w=457812693$, then
\begin{center}
\begin{tikzpicture}[scale=.5]
\draw (0,0)--(9,0)--(9,9)--(0,9)--(0,0);
\filldraw[draw=black,fill=lightgray] (0,8)--(1,8)--(1,9)--(0,9)--(0,8);
\filldraw[draw=black,fill=lightgray] (1,8)--(2,8)--(2,9)--(1,9)--(1,8);
\filldraw[draw=black,fill=lightgray] (2,8)--(3,8)--(3,9)--(2,9)--(2,8);
\filldraw[draw=black,fill=lightgray] (0,7)--(1,7)--(1,8)--(0,8)--(0,7);
\filldraw[draw=black,fill=lightgray] (1,7)--(2,7)--(2,8)--(1,8)--(1,7);
\filldraw[draw=black,fill=lightgray] (2,7)--(3,7)--(3,8)--(2,8)--(2,7);
\filldraw[draw=black,fill=lightgray] (0,6)--(1,6)--(1,7)--(0,7)--(0,6);
\filldraw[draw=black,fill=lightgray] (1,6)--(2,6)--(2,7)--(1,7)--(1,6);
\filldraw[draw=black,fill=lightgray] (2,6)--(3,6)--(3,7)--(2,7)--(2,6);
\filldraw[draw=black,fill=lightgray] (0,5)--(1,5)--(1,6)--(0,6)--(0,5);
\filldraw[draw=black,fill=lightgray] (1,5)--(2,5)--(2,6)--(1,6)--(1,5);
\filldraw[draw=black,fill=lightgray] (2,5)--(3,5)--(3,6)--(2,6)--(2,5);
\filldraw[draw=black,fill=lightgray] (5,6)--(6,6)--(6,7)--(5,7)--(5,6);
\filldraw[draw=black,fill=lightgray] (5,5)--(6,5)--(6,6)--(5,6)--(5,5);
\filldraw[draw=black,fill=lightgray] (2,2)--(3,2)--(3,3)--(2,3)--(2,2);
\filldraw[draw=black,fill=lightgray] (2,1)--(3,1)--(3,2)--(2,2)--(2,1);
\node at (-1.5,4.5) {$D(w)=$};
\node at (13.5,4.5) {and $\bm{i}=(6,5,7,6,2,1,3,2).$};
\end{tikzpicture}
\end{center}
The only entries of $\bm{i}$ occurring multiple times are $i_1=i_4=6$ and $i_5=i_8=2$. Their respective impacts are $\mathcal{I}_w(1)=\mathcal{I}_w(4)=\{3\}$ and $\mathcal{I}_w(5)=\mathcal{I}_w(8)=\{6\}$, so $w$ is multiplicity-free.
\end{example}
The proof of the generalization of Proposition~\ref{prop:norepeats} will require the following technical lemma. Before proceeding, recall Lemma~\ref{lem:fillings} and Lemma~\ref{lem:rowflagged}: for every $0\leq j\leq l$, elements of $\mathcal{T}_w(j)$ can be viewed as row-flagged, column-strict fillings of $O(w,j)$ (via the column filling order of $O(w,j)$ specified prior to Lemma~\ref{lem:fillings}). Applying $\widetilde{\omega}_{i_{j-1}}^{m_{j-1}}\oplus f_{i_j}$ to an element of $\mathcal{T}_w(j)$ gives an element of $\mathcal{T}_w(j-1)$, a filling of $O(w,j-1)$. Thus, when we speak below of the application of $f_{i_j}$ to an element $T\in\mathcal{T}_w(j)$ ``changing an $i_j$ to an $i_j+1$ in column $c$'', we specifically mean that when we view $T$ as a filling of $O(w,j)$ and $\widetilde{\omega}_{i_{j-1}}^{m_{j-1}}\oplus f_{i_j}(T)$ as a filling of $O(w,j-1)$, $T$ and $\widetilde{\omega}_{i_{j-1}}^{m_{j-1}}\oplus f_{i_j}(T)$ differ (in the stated way) in their entries in column $c$.
\begin{lemma}
\label{lem:rootoperatorproperty}
Let $w$ be a multiplicity-free permutation with orthodontic sequence $(\bm{i},\bm{m})$, $\bm{i}=(i_1,\ldots,i_l)$. Suppose $i_r=i_s$ with $r<s$ and $\mathcal{I}_w(r)=\mathcal{I}_w(s)=\{c\}$.
Then for each $j$ with $r\leq j \leq s$, $\mathcal{I}_w(j)=\{c\}$ and the application of $f_{i_j}$ to an element of
$\mathcal{T}_w(j)$
is either undefined or changes an $i_j$ to an $i_{j}+1$ in column $c$.
\end{lemma}
\begin{proof}
We handle first the case that $j=r$. In the diagram $O(w,r-1)$, column $c$ is the leftmost column containing a missing tooth, and $i_r$ is the smallest missing tooth in column $c$. Reading column $c$ of $O(w,r-1)$ top-to-bottom, one sees a (possibly empty) sequence of boxes in $O(w,r-1)$, followed by a sequence of boxes not in $O(w,r-1)$. The sequence of boxes not in $O(w,r-1)$ has length at least two since $i_r$ occurs at least twice in $\bm{i}$, and terminates with the box $(i_r+1,c)\in O(w,r-1)$. Note that since $(i_r-1,c),(i_r,c)\notin O(w,r-1)$, the northwest property of $O(w,r-1)$ implies that there can be no box $(i_r-1,c')$ or $(i_r,c')$ in $O(w,r-1)$ with $c'>c$.
Note also that since $\mathcal{I}_w(r)=\{c\}$, we have $(i_r+1,c')\notin O(w,r-1)$ for each $c'>c$.
Lastly, observe that for any $c'>c$ and $d>i_r+1$, there can be no box $(d,c')\in O(w,r-1)$. Otherwise there would be some $t\in [l]$ with $i_t=i_r$ and $t\neq r$ such that $c'\in \mathcal{I}_w(t)$, violating that $w$ is multiplicity-free.
As a consequence of the previous observations, the largest row index that a column $c'>c$ of $O(w,r-1)$ can contain a box in is $i_r-2$. In particular, Lemma~\ref{lem:rowflagged} implies that the application of $f_{i_r}$ to an element of $\mathcal{T}_w(r)$ either is undefined or changes an $i_r$ to an $i_r+1$ in column $c$. This concludes the case that $j=r$.
When $j=s$, an entirely analogous argument works. The only significant difference in the observations is that when column $c$ of $O(w,s-1)$ is read top-to-bottom, the (possibly empty) initial sequence of boxes in $O(w,s-1)$ is followed by a sequence of boxes not in $O(w,s-1)$ with length at least 1, ending with the box $(i_s+1,c)$. Consequently, the largest row index that a column $c'>c$ of $O(w,s-1)$ can contain a box in is $i_s-1$. In particular, Lemma~\ref{lem:rowflagged} implies that the application of $f_{i_s}$ to an element of $\mathcal{T}_w(s)$ either is undefined or changes an $i_s$ to an $i_s+1$ in column $c$. This concludes the case that $j=s$.
Now, let $r<j<s$. Since $\mathcal{I}_w(r)=\mathcal{I}_w(s)=\{c\}$, we have $c\in \mathcal{I}_w(j)$.
If $i_j$ occurs multiple times in $\bm{i}$, then multiplicity-freeness of $w$ implies $\mathcal{I}_w(j)=\{c\}$. In this case, we can find $j'\neq j$ with $i_j=i_{j'}$ and apply the previous argument (with $r$ and $s$ replaced by $j$ and $j'$) to conclude that the application of $f_{i_j}$ to an element of $\mathcal{T}_w(j)$ is either undefined or changes an $i_j$ to an $i_j+1$ in column $c$.
Thus, we assume $i_j$ occurs only once in $\bm{i}$. Recall that it was shown above that $O(w,r-1)$ has no boxes $(d,c')$ with $d>i_r$ and $c'>c$. Read top-to-bottom, let column $c$ of $O(w,r-1)$ have a (possibly empty) initial sequence of boxes ending with a missing box in row $u$, so clearly $u\leq i_r-1$. Since the first missing tooth in column $c$ of $O(w,r-1)$ is in row $i_r$, none of the boxes $(u,c),(u+1,c),\ldots,(i_r,c)$ are in $O(w,r-1)$, but $(i_r+1,c)\in O(w,r-1)$. Then, the northwest property implies that there is no box in $O(w,r-1)$ in any column $c'>c$ in any of rows $u,u+1,\ldots,i_r$. In particular, the largest row index such that a column $c'>c$ of $O(w,r-1)$ can contain a box in is $u-1$.
As $r<j<s$ and $\mathcal{I}_w(r)=\mathcal{I}_w(s)=\{c\}$, we have that $c\in\mathcal{I}_w(j)$.
Also since $r<j<s$, the leftmost nonempty column in $O(w,j-1)$ is column $c$, and $i_j\geq u$. Then in $O(w,j-1)$, the maximum row index a box in a column $c'>c$ can have is $u-1$. In particular, $\mathcal{I}_w(j)=\{c\}$, and Lemma~\ref{lem:rowflagged} implies that the application of $f_{i_j}$ to an element of $\mathcal{T}_w(j)$ is either undefined or changes an $i_j$ to an $i_j+1$ in column $c$.
\end{proof}
\begin{theorem}
\label{thm:multfree}
If $w$ is multiplicity-free, then $\mathfrak{S}_w$ is zero-one.
\end{theorem}
\begin{proof}
Assume $wt(T)=wt(T')$ for some $T,T'\in \mathcal{T}_w$. If we can show that $T=T'$, then we can conclude that all elements of $\mathcal{T}_w$ have distinct weights, so $\mathfrak{S}_w$ is zero-one. To begin, write
\[T=\widetilde{\omega}_1^{k_1}\oplus\cdots\oplus \widetilde{\omega}_n^{k_n}\oplus f_{i_1}^{p_1}(\widetilde{\omega}_{i_1}^{m_1}\oplus\cdots\oplus f_{i_l}^{p_l}(\widetilde{\omega}_{i_l}^{m_l})\cdots) \]
and
\[T'=\widetilde{\omega}_1^{k_1}\oplus\cdots\oplus \widetilde{\omega}_n^{k_n}\oplus f_{i_1}^{q_1}(\widetilde{\omega}_{i_1}^{m_1}\oplus\cdots\oplus f_{i_l}^{q_l}(\widetilde{\omega}_{i_l}^{m_l})\cdots), \]
for some $p_1,\ldots,p_l,q_1,\ldots,q_l$. The basic idea of the proof is to show that as $T$ and $T'$ are constructed step-by-step from $\widetilde{\omega}_{i_l}^{m_l}$, the resulting intermediate tableaux are intermittently equal. At termination of the construction, this will imply that $T=T'$.
By the expansion (\ref{eqn:vectorexpansion}) of $wt(T)-wt(T')$ used in the proof of Proposition~\ref{prop:norepeats}, we observe that $p_u=q_u$ for all $u$ such that $i_u$ occurs only once in $\bm{i}$. Let $s$ be the largest index such that $p_s\neq q_s$. Suppose $\mathcal{I}_w(s)=\{c\}$.
Let $r_1$ be the smallest index such that $i_{r_1}$ occurs multiple times in $\bm{i}$ and $\mathcal{I}_w(r_1)=\{c\}$.
We know $r_1<s$, because (\ref{eqn:vectorexpansion}) implies that $p_{s'}\neq q_{s'}$ for another $s'<s$ with $i_{s'}=i_s$, and by multiplicity-freeness $\mathcal{I}_w(s')=\{c\}$.
We wish to find an interval $[r,s]\subseteq [r_1,s]$ such that $r<s$ and the following two conditions hold:
\begin{itemize}
\item[(i)] \label{itm:i} If $v\geq r$ and $i_v$ occurs multiple times in $\bm{i}$, then any other $v'$ with $i_v=i_{v'}$ will satisfy $v'\geq r$.
\item[(ii)] \label{itm:ii} For every $j$ with $r<j<s$ and $i_j$ occurring only once in $\bm{i}$, there are $t$ and $u$ with $r\leq t<j<u\leq s$ such that $i_t=i_u$.
\end{itemize}
We first show that (i) holds for $[r_1,s]$. Note that if $i_v$ occurs multiple times in $\bm{i}$ and $r_1\leq v\leq s$, then it must be that $\mathcal{I}_w(v)=\{c\}$
by the fact that the orthodontia construction records all missing teeth needed to eliminate one column before moving on to the next column.
If $i_{v'}=i_{v}$, then $\mathcal{I}_w(v')=\{c\}$ also, by multiplicity-freeness of $w$. The choice of $r_1$ implies $r_1\leq v'$.
If $i_v$ occurs multiple times in $\bm{i}$ with $s<v$ and $\mathcal{I}_w(v)=\{c\}$, then the choice of $r_1$ again implies that $r_1\leq v'$ for any $i_{v'}=i_v$.
If $i_v$ occurs multiple times in $\bm{i}$ with $s<v$ and $\mathcal{I}_w(v)\neq \{c\}$, then the orthodontia construction implies that
any $v'$ with $i_v=i_{v'}$ must satisfy $s<v'$.
In particular, $r_1<v'$ as needed. Thus, (i) holds for $[r_1,s]$. If $[r_1,s]$ also satisfies (ii), then we are done.
Otherwise, assume $[r_1,s]$ does not satisfy (ii). Then there is some $j$ with $r_1<j<s$ such that $i_j$ occurs only once in $\bm{i}$ and
there are no $t$ and $u$ with $r_1\leq t<j<u\leq s$ and $i_t=i_u$. Consequently for every pair $i_u=i_t$ with $r_1\leq t<u\leq s$, it must be that either $t<u<j$ or $j<t<u$. Let $r_2$ be the smallest index such that $j<r_2$ and $i_{r_2}$ occurs multiple times in $\bm{i}$. By the choice of $j$, it is clear that the interval $[r_2,s]$ still satisfies (i). If $[r_2,s]$ also satisfies (ii), then we are done.
Otherwise, $[r_2,s]$ satisfies (i) but not (ii), and we can argue exactly as in the case of $[r_1,s]$ to find an $r_3$ such that $r_2<r_3<s$ and $[r_3,s]$ satisfies (i).
Continue working in this fashion. We show that this process terminates with an interval $[r,s]$ satisfying $r<s$, (i), and (ii).
As mentioned above, there exists $s'<s$ such that $i_{s'}=i_s$. Let $s'$ be the maximal index less than $s$ with this property.
Since all of the intervals $[r_*,s]$ will satisfy (i), it follows that $r_1<r_2<\cdots\leq s'$. At worst, the process will terminate after finitely many steps with the interval $[s',s]$. The interval $[s',s]$ will then satisfy (i) since the process reached it, and will trivially satisfy (ii) since $i_s=i_{s'}$.
Hence, we can assume that we have found an interval $[r,s]$ satisfying $r<s$, (i), and (ii). Consider the tableaux
\begin{align*}
&T_r=\widetilde{\omega}_{i_{r-1}}^{m_{r-1}}\oplus f_{i_r}^{p_r}(\widetilde{\omega}_{i_r}^{m_r}\oplus\cdots\oplus f_{i_l}^{p_l}(\widetilde{\omega}_{i_l}^{m_l})\cdots),\quad &T_s=\widetilde{\omega}_{i_{s}}^{m_{s}}\oplus f_{i_{s+1}}^{p_{s+1}}(\widetilde{\omega}_{i_{s+1}}^{m_{s+1}}\oplus\cdots\oplus f_{i_l}^{p_l}(\widetilde{\omega}_{i_l}^{m_l})\cdots),\\
&T'_r=\widetilde{\omega}_{i_{r-1}}^{m_{r-1}}\oplus f_{i_r}^{q_r}(\widetilde{\omega}_{i_r}^{m_r}\oplus\cdots\oplus f_{i_l}^{q_l}(\widetilde{\omega}_{i_l}^{m_l})\cdots),\quad &T'_s=\widetilde{\omega}_{i_{s}}^{m_{s}}\oplus f_{i_{s+1}}^{q_{s+1}}(\widetilde{\omega}_{i_{s+1}}^{m_{s+1}}\oplus\cdots\oplus f_{i_l}^{q_l}(\widetilde{\omega}_{i_l}^{m_l})\cdots).
\end{align*}
By definition, $T_r,T_r'\in\mathcal{T}_w(r-1)$, so we can view $T_r$ and $T_r'$ as fillings of $O(w,r-1)$. Similarly, $T_s,T_s'\in\mathcal{T}_w(s)$, so we can view $T_s$ and $T_s'$ as fillings of $O(w,s)$. Since we chose $s$ to be the largest index such that $p_s\neq q_s$, it follows that $T_s=T'_s$. By property (i) of $[r,s]$, $i_u\neq i_v$ for any $u<r\leq v$. Hence, it must be that $wt(T_r)=wt(T'_r)$. Finally, property (ii) of $[r,s]$ allows us to apply Lemma~\ref{lem:rootoperatorproperty} and conclude that for any $a_r,a_{r+1},\ldots,a_{s}\geq 0$,
when $\widetilde{\omega}_{i_{r-1}}^{m_{r-1}}\oplus f_{i_r}^{a_r}(\widetilde{\omega}_{m_r}^{i_r}\oplus\cdots\oplus \widetilde{\omega}_{i_{s-1}}^{m_{s-1}}f_{i_s}^{a_s}(\mbox{---})\cdots)$
is applied to an element of $\mathcal{T}_w(s)$, only the entries in column $c$ are affected by the root operators $f_{i_r}^{a_r},\ldots,f_{i_s}^{a_s}$. Since
\[
T_r= \widetilde{\omega}_{i_{r-1}}^{m_{r-1}}\oplus f_{i_r}^{p_r}(\widetilde{\omega}_{m_r}^{i_r}\oplus\cdots\oplus \widetilde{\omega}_{i_{s-1}}^{m_{s-1}}f_{i_s}^{p_s}(T_s)\cdots)
\quad\mbox{and}\quad
T_r'= \widetilde{\omega}_{i_{r-1}}^{m_{r-1}}\oplus f_{i_r}^{q_r}(\widetilde{\omega}_{m_r}^{i_r}\oplus\cdots\oplus \widetilde{\omega}_{i_{s-1}}^{m_{s-1}}f_{i_s}^{q_s}(T_s')\cdots),
\]
$T_r$ and $T_r'$ must coincide outside of column $c$. Since we already deduced that $wt(T_r)=wt(T'_r)$, it follows that column $c$ of $T_r$ and $T_r'$ have the same weight. By column-strictness of $T_r$ and $T_r'$, column $c$ of $T_r$ and $T_r'$ must coincide, so $T_r=T_r'$.
To complete the proof, let $\hat{s}$ be the largest index $\hat{s}<r$ such that $p_{\hat{s}}\neq q_{\hat{s}}$. If no such index exists, then $T=T'$. Otherwise, set $\hat{r}_1$ to be the smallest index such that $i_{\hat{r}_1}$ occurs multiple times in $\bm{i}$ and $\mathcal{I}_w(\hat{r}_1)=\mathcal{I}_w(\hat{s})$.
We have $\hat{r}_1<\hat{s}$ because some other $\hat{s}'$ distinct from~$\hat{s}$ such that $p_{\hat{s}'}\neq q_{\hat{s}'}$ and $i_{\hat{s}'}=i_{\hat{s}}$ must exist as before,
and $\hat{s}'$ is also less than~$r$ by property (i) of $[r,s]$.
Use the previous algorithm to find an interval $[\hat{r},\hat{s}]\subseteq [\hat{r}_1,\hat{s}]$ satisfying $\hat{r}<\hat{s}$, (i), and (ii). Construct $T_{\hat{r}},T_{\hat{r}}',T_{\hat{s}},T'_{\hat{s}}$, and argue exactly as in the case of $[r,s]$ that $T_{\hat{r}}=T_{\hat{r}}'$.
Continuing in this manner for a finite number of steps will show that $T=T'$.
\end{proof}
As we will show in Theorem~\ref{thm:011},
it is not only sufficient but also necessary that $w$ be multiplicity-free for the Schubert polynomial $\mathfrak{S}_w$ to be zero-one.
\section{Pattern avoidance conditions for multiplicity-freeness}
\label{sec:mult-patt}
This section is devoted to showing that $w$ being multiplicity-free is equivalent to a certain pattern avoidance condition. We then prove our full characterization of zero-one Schubert polynomials.
We start with several definitions.
\begin{definition}
\label{def:confA}
We say a Rothe diagram $D=D(w)$ contains an instance of configuration $\mathrm{A}$ if there are $r_1,c_1,r_2,c_2,r_3$ such that $1\leq r_3<r_1<r_2$, $1<c_1<c_2$, $(r_1,c_1),(r_2,c_2)\in D$, $(r_1,c_2)\notin D$, and $w_{r_3}<c_1$.
\end{definition}
\begin{definition}
\label{def:confB}
We say a Rothe diagram $D=D(w)$ contains an instance of configuration $\mathrm{B}$ if there are $r_1,c_1,r_2,c_2,r_3,r_4$ such that $1\leq r_4\neq r_3<r_1<r_2$, $2<c_1<c_2$, $(r_1,c_1),(r_1,c_2),(r_2,c_2)\in D$, $w_{r_3}<c_1$, and $w_{r_4}<c_2$.
\end{definition}
\begin{definition}
\label{def:confBprime}
We say a Rothe diagram $D=D(w)$ contains an instance of configuration $\mathrm{B}'$ if there are $r_1,c_1,r_2,c_2,r_3,r_4$ such that $1\leq r_4<r_3<r_1<r_2$, $2<c_1<c_2$, $(r_1,c_1),(r_1,c_2),(r_2,c_1)\in D$, $w_{r_3}<c_1$, and $w_{r_4}<c_1$.
\end{definition}
Given a Rothe diagram $D(w)$, we will call a tuple $(r_1,c_1,r_2,c_2,r_3)$ meeting the conditions of Definition~\ref{def:confA} an instance of configuration $\mathrm{A}$ in $D(w)$. Similarly, we will call a tuple $(r_1,c_1,r_2,c_2,r_3,r_4)$ meeting the conditions of Definition~\ref{def:confB} (resp. \ref{def:confBprime}) an instance of configuration $\mathrm{B}$ (resp. $\mathrm{B}'$) in $D(w)$.
\begin{figure}
\caption{Examples of instances of the configurations $\mathrm{A}$, $\mathrm{B}$, and $\mathrm{B}'$ in Rothe diagrams. Both the hooks removed from the $n\times n$ grid to form each Rothe diagram and the remaining boxes are shown.}
\end{figure}
\begin{figure}
\caption{The Rothe diagrams of the twelve multiplicitous patterns.}
\label{fig:badpatterns}
\end{figure}
\begin{theorem}\label{thm:mult-pattern}
If $w\in S_n$ is a permutation such that $D(w)$ does not contain any instance of configuration $\mathrm{A}$, $\mathrm{B}$, or $\mathrm{B}'$,
then $w$ is multiplicity-free.
\end{theorem}
Theorem~\ref{thm:011} will also imply the converse of this theorem.
\begin{proof}
We prove the contrapositive.
Assume $w$ is not multiplicity-free and let $(\bm{i},\bm{m})$ be the orthodontic sequence of $w$. Then, we can find entries $i_{p_1}=i_{p_2}$ of $\bm{i}$ with $p_1<p_2$ such that either $\mathcal{I}_w(p_1)\neq \mathcal{I}_w(p_2)$, or $\mathcal{I}_w(p_1)=\mathcal{I}_w(p_2)$ with $|\mathcal{I}_w(p_1)|>1$. We show that $D(w)$ must contain at least one instance of configuration $\mathrm{A}$, $\mathrm{B}$, or $\mathrm{B}'$.\\
\noindent\emph{Case 1:} Assume that $\mathcal{I}_w(p_1)\nsubseteq\mathcal{I}_w(p_2)$ and $\mathcal{I}_w(p_2)\nsubseteq\mathcal{I}_w(p_1)$.
Take $c_1\in \mathcal{I}_w(p_1)\backslash \mathcal{I}_w(p_2)$ and $c_2\in \mathcal{I}_w(p_2)\backslash \mathcal{I}_w(p_1)$.
We show that columns $c_1$ and $c_2$ of $D(w)$ contain an instance of configuration $\mathrm{A}$.
In step $p_1$ of the orthodontia on $D(w)$, a box in column $c_1$ is moved (by the missing tooth $i_{p_1}$) to row $i_{p_1}$. Let this box originally be in row $r_1$ of $D(w)$. Analogously, let the box in column $c_2$ moved to row $i_{p_2}$ in step $p_2$ of the orthodontia (by the missing tooth $i_{p_2}$) originally be in row $r_2$ of $D(w)$.
Observe that $r_1<r_2$.
If $c_2<c_1$, then the northwest property would imply that $(r_1,c_2)\in D(w)$, contradicting that $c_2\notin \mathcal{I}_w(p_1)$. Thus $c_1<c_2$. Since $c_2\notin \mathcal{I}_w(p_1)$, $(r_1,c_2)\notin D(w)$.
Lastly, since the box $(r_1,c_1)$ is moved by the orthodontia, there is some box $(r_3,c_1)\notin D(w)$ with $r_3<r_1$. Consequently, $w_{r_3}<c_1$. Thus, $(r_1,c_1,r_2,c_2,r_3)$ is an instance of configuration $\mathrm{A}$.\\
\noindent\emph{Case 2:}
Assume $\mathcal{I}_w(p_2)$ is a proper subset of $\mathcal{I}_w(p_1)$.
Let $c_1=\max(\mathcal{I}_w(p_2))$
and $c_2=\min(\mathcal{I}_w(p_1)\backslash\mathcal{I}_w(p_2))$.
Let the box in column $c_1$ moved to row $i_{p_1}=i_{p_2}$ in step $p_1$ (resp. $p_2$) of the orthodontia originally be in row $r_1$ (resp. $r_2$) of $D(w)$. Observe that $r_1<r_2$.
Assume first that $c_1<c_2$. Since $c_1\in\mathcal{I}_w(p_1)\cap \mathcal{I}_w(p_2)$, the boxes $(r_1,c_1)$ and $(r_2,c_2)$ both move weakly above row $i_{p_1}$ in the orthodontia. Then,
we can find indices $r_3,r_4$ with $r_4<r_3<r_1$ such that $(r_3,c_1),(r_4,c_1)\notin D(w)$. Hence, $w_{r_3}<c_1$ and $w_{r_4}<c_1$, so $(r_1,c_1,r_2,c_2,r_3,r_4)$ is an instance of configuration $\mathrm{B}'$.
Otherwise $c_1>c_2$. Since the box $(r_1,c_2)$ is moved by the orthodontia, we can find $r_3<r_1$ with $(r_3,c_2)\notin D(w)$. Then $w_{r_3}<c_2$. As we are assuming $c_2<c_1$, $(r_3,c_1)\notin D(w)$ also. Since the boxes $(r_1,c_1)$ and $(r_2,c_1)$ in $D(w)$ are moved weakly above row $i_{p_1}$ by the orthodontia, we can find some $r_4<r_1$ with $r_4\neq r_3$ such that $(r_4,c_1)\notin D(w)$. Then, $w_{r_4}<c_1$, so $(r_1,c_2,r_2,c_1,r_3,r_4)$ is an instance of configuration $\mathrm{B}$.\\
\noindent\emph{Case 3:}
Assume $\mathcal{I}_w(p_1)$ is a proper subset of $\mathcal{I}_w(p_2)$.
This case is handled similarly to Case 2.
Let $c_1=\max(\mathcal{I}_w(p_1))$ and $c_2=\min(\mathcal{I}_w(p_2)\backslash\mathcal{I}_w(p_1))$.
Let the box in column $c_1$ moved to row $i_{p_1}=i_{p_2}$ in step $p_1$ (resp. $p_2$) of the orthodontia originally be in row $r_1$ (resp. $r_2$) of $D(w)$. Observe that $r_1<r_2$.
Assume $c_1<c_2$. Since the boxes $(r_1,c_1)$ and $(r_2,c_1)$ of $D(w)$ are moved weakly above row $i_{p_1}$ by the orthodontia, we can find indices $r_3,r_4$ with $r_4<r_3<r_1$ such that $(r_3,c_1),(r_4,c_1)\notin D(w)$. Then, $w_{r_3}<c_1$ and $w_{r_4}<c_1$.
Since $c_2\notin \mathcal{I}_w(p_1)$,
$(r_1,c_2)\notin D(w)$. Then, $(r_1,c_1,r_2,c_2,r_3)$ is an instance of configuration $\mathrm{A}$.
Otherwise $c_1>c_2$.
As $c_2\notin \mathcal{I}_w(p_1)$,
$(r_1,c_2)\notin D(w)$. Since $(r_2,c_2),(r_1,c_1)\in D(w)$, this is a contradiction of the northwest property of $D(w)$.\\
\noindent\emph{Case 4:}
Assume $\mathcal{I}_w(p_1)=\mathcal{I}_w(p_2)$ is not a singleton.
Let $c_1,c_2\in \mathcal{I}_w(p_1)$ with $c_1<c_2$.
Let the box in column $c_1$ moved to row $i_{p_1}=i_{p_2}$ in step $p_1$ (resp. $p_2$) of the orthodontia originally be in row $r_1$ (resp. $r_2$) of $D(w)$. Observe that $r_1<r_2$. Since the boxes $(r_1,c_1)$ and $(r_2,c_1)$ in $D(w)$ are moved weakly above row $i_{p_1}$ by the orthodontia, we can find indices $r_3,r_4$ with $r_4<r_3<r_1$ such that $(r_3,c_1),(r_4,c_1)\notin D(w)$. Then, $w_{r_3}<c_1$ and $w_{r_4}<c_1$. Thus, $(r_1,c_1,r_2,c_2,r_3,r_4)$ is an instance of configuration $\mathrm{B}'$.
\end{proof}
We now relate multiplicity-freeness to pattern avoidance of permutations. We begin by clarifying our pattern avoidance terminology.
A \emph{pattern} $\sigma$ of \emph{length}~$n$ is a permutation in~$S_n$.
The length $n$ is a crucial part of the data of a pattern;
we make no identifications between patterns of different lengths, unlike what is usual when handling permutations in the Schubert calculus.
A permutation $w$ \emph{contains} $\sigma$ if $w$ has $n$ entries $w_{j_1},\ldots, w_{j_n}$ with $j_1<j_2<\cdots<j_n$ that are in the same relative order as $\sigma_1,\sigma_2,\ldots,\sigma_n$. In this case, the indices $j_1<j_2<\cdots<j_n$ are called a \emph{realization} of $\sigma$ in $w$. We say that $w$ \emph{avoids} the pattern $\sigma$ if $w$ does not contain $\sigma$.
To illustrate the dependence of these definitions on $n$, note that $w=154623$ contains the pattern 132, but not the pattern 132456.
The following easy lemma gives a diagrammatic interpretation of pattern avoidance.
\begin{lemma}
\label{lem:patterndiagram}
Let $w\in S_n$ be a permutation and $\sigma$ a pattern of length~$m$ contained in $w$. Choose a realization
$j_1<j_2<\cdots<j_{m}$ of $\sigma$ in $w$.
Then $D(\sigma)$ is obtained from $D(w)$ by deleting the rows $[n]\backslash\{j_1,\ldots,j_{m} \}$ and the columns $[n]\backslash\{w_{j_1},\ldots,w_{j_{m}} \}$,
and reindexing the remaining rows and columns by~$[m]$, preserving their order.
\end{lemma}
\begin{definition}
The \emph{multiplicitous patterns} are those in the set
\[\mathrm{MPatt}=\{12543, 13254, 13524, 13542, 21543, 125364, 125634, 215364, 215634, 315264, 315624, 315642\}.\]
\end{definition}
\begin{theorem}
\label{thm:badpatterns}
Let $w\in S_n$. Then $D(w)$ does not contain any instance of configuration $\mathrm{A}$, $\mathrm{B}$, or $\mathrm{B}'$ if and only if $w$ avoids all of the multiplicitous patterns.
\end{theorem}
\begin{proof}
It is easy to check (see Figure~\ref{fig:badpatterns}) that each of the twelve multiplicitous patterns contains an instance of configuration $\mathrm{A}$, $\mathrm{B}$, or $\mathrm{B}'$. Lemma~\ref{lem:patterndiagram} implies that if $w$ contains $\sigma\in\mathrm{MPatt}$, then deleting some rows and columns from $D(w)$ yields $D(\sigma)$. Since $D(\sigma)$ contains at least one instance of configuration $\mathrm{A}$, $\mathrm{B}$, or $\mathrm{B}'$, so does $D(w)$.
Conversely, assume $D(w)$ contains at least one instance of configuration $\mathrm{A}$, $\mathrm{B}$, or $\mathrm{B}'$. We must show that $w$ contains some multiplicitous pattern.
Let $\tau^1,\tau^2,\ldots,\tau^n$ be the $n$ patterns of length~$n-1$ contained in $w$;
say $\tau^j$ is realized in $w$ by forgetting $w_j$.
Without loss of generality, we may assume none of $D(\tau^1),\ldots,D(\tau^n)$ contain an instance of configuration $\mathrm{A}$, $\mathrm{B}$, or $\mathrm{B}'$: if $D(\tau^j)$ does contain an instance of one of these configurations, replace $w$ by $\tau^j$ and iterate.
For each $j$, $D(\tau^j)$ is obtained from $D(w)$ by deleting row $j$ and column $w_j$.
Since $D(\tau^j)$ does not contain any instance of any of our three configurations,
each cross $\{(j,q) \mid (j,q)\in D(w) \}\cup \{(p,w_j) \mid (p,w_j)\in D(w)\}$ intersects each instance of every configuration contained in $D(w)$. However, an instance of configuration $\mathrm{A}$ involves only three rows and two columns, and an instance of $\mathrm{B}$ or $\mathrm{B}'$ involves only four rows and two columns. Thus, it must be that $w\in S_n$ for some $n\leq 6$.
It can be checked by exhaustion that the only permutations in $S_n$ with $n\leq 6$ that are minimal (with respect to pattern avoidance) among those whose Rothe diagrams contain an instance of configuration $\mathrm{A}$, $\mathrm{B}$, or $\mathrm{B}'$ are the twelve multiplicitous patterns.
\end{proof}
We are now ready to state our full characterization of zero-one Schubert polynomials,
and most of the elements of the proof are at hand.
\begin{theorem} \label{thm:011}
The following are equivalent:
\begin{itemize}
\item[(i)] The Schubert polynomial $\mathfrak{S}_w$ is zero-one.
\item[(ii)] The permutation $w$ is multiplicity-free,
\item[(iii)] The Rothe diagram $D(w)$ does not contain any instance of configuration $\mathrm{A}$, $\mathrm{B}$, or $\mathrm{B}'$,
\item[(iv)] The permutation $w$ avoids the multiplicitous patterns, namely $12543$, $13254$, $13524$, $13542$, $21543$, $125364$, $125634$, $215364$, $215634$, $315264$, $315624$, and $315642$.
\end{itemize}
\end{theorem}
\begin{proof}
Theorem~\ref{thm:multfree} shows $(ii)\Rightarrow(i)$. Theorem~\ref{thm:mult-pattern} shows $(iii)\Rightarrow(ii)$. Theorem~\ref{thm:badpatterns} shows $(iii)\Leftrightarrow(iv)$. The implication $(i)\Rightarrow(iv)$ will follow immediately from Corollary~\ref{cor:pattern}, since the Schubert polynomials associated to the permutations $12543$, $13254$, $13524$, $13542$, $21543$, $125364$, $125634$, $215364$, $215634$, $315264$, $315624$, and $315642$ each have a coefficient equal to 2. We prove Corollary~\ref{cor:pattern} in the next section.
\end{proof}
\section{A coefficient-wise inequality for dual characters of flagged Weyl modules of diagrams}
\label{sec:trans}
The aim of this section is to prove a generalization of Theorem~\ref{thm:pattern}, namely, Theorem~\ref{thm:pattern2}.
We now explain the necessary background and terminology for Theorem~\ref{thm:pattern2} and its proof.
Let $G=\mathrm{GL}(n,\mathbb{C})$ be the group of $n\times n$ invertible matrices over $\mathbb{C}$ and $B$ be the subgroup of $G$ consisting of the $n\times n$ upper-triangular matrices. The flagged Weyl module is a representation $\mathcal{M}_D$ of $B$ associated to a diagram $D$. The dual character of $\mathcal{M}_D$ has been shown in certain cases to be a Schubert polynomial \cite{KP} or a key polynomial \cite{flaggedLRrule}. We will use the construction of $\mathcal{M}_D$ in terms of determinants given in \cite{magyar}.
Denote by $Y$ the $n\times n$ matrix with indeterminates $y_{ij}$ in the upper-triangular positions $i\leq j$ and zeros elsewhere. Let $\mathbb{C}[Y]$ be the polynomial ring in the indeterminates $\{y_{ij}\}_{i\leq j}$. Note that $B$ acts on $\mathbb{C}[Y]$ on the right via left translation: if $f(Y)\in \mathbb{C}[Y]$, then a matrix $b\in B$ acts on $f$ by $f(Y)\cdot b=f(b^{-1}Y)$. For any $R,S\subseteq [n]$, let $Y_S^R$ be the submatrix of $Y$ obtained by restricting to rows $R$ and columns $S$.
For $R,S\subseteq [n]$, we say $R\leq S$ if $\#R=\#S$ and the $k$\/th least element of $R$ does not exceed the $k$\/th least element of $S$ for each $k$. For any diagrams $C=(C_1,\ldots, C_n)$ and $D=(D_1,\ldots, D_n)$, we say $C\leq D$ if $C_j\leq D_j$ for all $j\in[n]$.
\begin{definition}
For a diagram $D=(D_1,\ldots, D_n)$, the \emph{flagged Weyl module} $\mathcal{M}_D$ is defined by
\[\mathcal{M}_D=\mathrm{Span}_\mathbb{C}\left\{\prod_{j=1}^{n}\det\left(Y_{D_j}^{C_j}\right)\ \middle|\ C\leq D \right\}. \]
$\mathcal{M}_D$ is a $B$-module with the action inherited from the action of $B$ on $\mathbb{C}[Y]$.
\end{definition}
Note that since $Y$ is upper-triangular, the condition $C\leq D$ is technically unnecessary since $\det\left(Y_{D_j}^{C_j}\right)=0$ unless $C_j\leq D_j$. Conversely, if $C_j\leq D_j$, then $\det\left(Y_{D_j}^{C_j}\right)\neq 0$.
For any $B$-module $N$, the \emph{character} of $N$ is defined by $\mathrm{char}(N)(x_1,\ldots,x_n)=\mathrm{tr}\left(X:N\to N\right)$ where $X$ is the diagonal matrix $\mathrm{diag}(x_1,x_2,\ldots,x_n)$ with diagonal entries $x_1,\ldots,x_n$, and $X$ is viewed as a linear map from $N$ to $N$ via the $B$-action. Define the \emph{dual character} of $N$ to be the character of the dual module $N^*$:
\begin{align*}
\mathrm{char}^*(N)(x_1,\ldots,x_n)&=\mathrm{tr}\left(X:N^*\to N^*\right) \\
&=\mathrm{char}(N)(x_1^{-1},\ldots,x_n^{-1}).
\end{align*}
A special case of dual characters of flagged Weyl modules of diagrams are Schubert polynomials:
\begin{theorem}[\cite{KP}]
\label{thm:kp}
Let $w$ be a permutation, $D(w)$ be the Rothe diagram of $w$, and $\mathcal{M}_{D(w)}$ be the associated flagged Weyl module. Then,
\[\mathfrak{S}_w = \mathrm{char}^*\mathcal{M}_{D(w)}. \]
\end{theorem}
Another special family of dual characters of flagged Weyl modules of diagrams, for so-called skyline diagrams of compositions, are key polynomials \cite{keypolynomials}.
\begin{definition}
For a diagram $D\subseteq [n]\times [n]$, let $\chi_D=\chi_D(x_1,\ldots,x_n)$ be the dual character
\[\chi_D=\mathrm{char}^*\mathcal{M}_D. \]
\end{definition}
We now work towards proving Theorem~\ref{thm:pattern2}.
We start by reviewing some material from \cite{FMS} for the reader's convenience.
We then derive several lemmas that simplify the proof of Theorem~\ref{thm:pattern2}.
\begin{theorem}[cf. {\cite[Theorem 7]{FMS}}]
For any diagram $D\subseteq [n]\times [n]$, the monomials appearing in $\chi_D$ are exactly
\[\left\{\prod_{j=1}^{n}\prod_{i\in C_j}x_i\ \middle|\ C\leq D \right\}.\]
\end{theorem}
\begin{proof}
(Following that of \cite[Theorem 7]{FMS}) Denote by $X$ the diagonal matrix $\mathrm{diag}(x_1,x_2,\ldots,x_n)$. First, note that $y_{ij}$ is an eigenvector of $X$ with eigenvalue $x_i^{-1}$. Take a diagram $C=(C_1,\ldots,C_n)$ with $C\leq D$. Then, the element $\prod_{j=1}^{n}\det\left(Y_{D_j}^{C_j}\right)$ is an eigenvector of $X$ with eigenvalue $\prod_{j=1}^{n}\prod_{i\in C_j}x_i^{-1}$. Since $\mathcal{M}_D$ is spanned by elements $\prod_{j=1}^{n}\det\left(Y_{D_j}^{C_j}\right)$ and each is an eigenvector of $X$, the monomials appearing in the dual character $\chi_D$ are exactly
$\left\{\prod_{j=1}^{n}\prod_{i\in C_j}x_i\ \middle|\ C\leq D \right\}$.
\end{proof}
\begin{corollary}
\label{cor:fms}
Let $D\subseteq [n]\times [n]$ be a diagram. Fix any diagram $C^{(1)}\leq D$ and set \[\bm{m}=\prod_{j=1}^{n}\prod_{i\in C^{(1)}_j}x_i.\]
Let $C^{(1)}, \ldots, C^{(r)}$ be all the diagrams $C$ such that $C\leq D$ and $\prod_{j=1}^{n}\prod_{i\in C_j}x_i=\bm{m}$. Then, the coefficient of $\bm{m}$ in $\chi_D$ is equal to
\[\dim \left(\mathrm{Span}_\mathbb{C}\left\{\prod_{j=1}^{n}\det\left(Y_{D_j}^{C^{(i)}_j}\right) \ \middle|\ i\in [r] \right\}\right).\]
\end{corollary}
\begin{proof}
The coefficient of $\bm{m}$ in $\chi_D$ equals the dimension of the eigenspace of $\bm{m}^{-1}$ in $\mathcal{M}_D$ ($\bm{m}^{-1}$ occurs here instead of $\bm{m}$ since $\chi_D$ is the dual character of $\mathcal{M}_D$). This eigenspace equals
\[\mathrm{Span}_{\mathbb{C}}\left\{\prod_{j=1}^{n}\det\left(Y_{D_j}^{C^{(i)}_j}\right) \ \middle|\ i\in [r] \right\},\] so the result follows.
\end{proof}
The understanding of the coefficients of the monomials of $\chi_D$ given in Corollary~\ref{cor:fms} is key to our proof of Theorem~\ref{thm:pattern2}. We set up some notation now.
Given diagrams $C,D\subseteq [n]\times[n]$ and $k,l\in [n]$, let ${\widehat{C}}$ and ${\widehat{D}}$ denote the diagrams obtained from $C$ and $D$ by removing any boxes in row $k$ or column $l$.
Fix a diagram $D$. For each diagram ${\widehat{C}}$, let
\[{{\widehat{C}}}_{\rm aug}={\widehat{C}} \cup \{(k,i) \mid (k,i)\in D\} \cup \{(i,l) \mid (i,l) \in D \}\subseteq [n]\times [n].\]
The following lemma is immediate and its proof is left to the reader.
\begin{lemma}
\label{lem:c}
Let $C, D \subseteq [n]\times[n]$ be diagrams and $k,l\in [n]$. If ${\widehat{C}}\leq {\widehat{D}}$, then ${{\widehat{C}}}_{\rm aug}\leq D$.
In particular, every diagram $C'\leq {\widehat{D}}$ with no boxes in row $k$
can be obtained from some diagram $C\leq D$ by removing any boxes in row $k$ or column $l$ from $C$.
\end{lemma}
The following result is our key lemma.
For a polynomial $f\in\mathbb{Z}[x_1,\ldots,x_n]$ and a monomial $\bm{m}$, let $[\bm{m}]f$ denote the coefficient of $\bm{m}$ in $f$.
\begin{lemma}
\label{lem:keylemma}
Fix a diagram $D$ and $k,l\in [n]$. Let $\{\widehat{C}^{(i)}\}_{i\in [m]}$ be a set of diagrams with $\widehat{C}^{(i)}\leq \widehat{D}$ for each $i$, and denote $\widehat{C}^{(i)}_{\mathrm{aug}}$ by $C^{(i)}$ for $i\in[m]$. If the polynomials $\displaystyle\left\{\prod_{j\in [n]}\det\left(Y_{D_j}^{{C^{(i)}_j}}\right)\right\}_{i \in [m]}$ are linearly dependent, then so are the polynomials $\displaystyle\left\{\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{{\widehat{D}}_j}^{\widehat{C}^{(i)}_j}\right)\right\}_{i \in [m]}$.
\end{lemma}
\begin{proof}
We are given that
\begin{align}\label{eq:proof-1} \sum_{i\in [m]}{c_{i}}\prod_{j\in [n]}\det\left(Y_{D_j}^{C^{(i)}_j}\right)=0\end{align} for some constants $(c_{i})_{i \in [m]}\in \mathbb{C}^m$ not all zero. Since ${C^{(i)}}=\widehat{C}^{(i)}_{\rm aug}$ for $\widehat{C}^{(i)}\leq {\widehat{D}}$ we have that ${C_l^{(i)}}=D_l$ for every $i \in [m]$. Thus, \eqref{eq:proof-1} can be rewritten as
\begin{align*}\det\left(Y_{D_l}^{{D_l}}\right)\left(\sum_{i\in [m]}{c_{i}}\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right)\right)=0.\end{align*}
However, since $\det\left(Y_{D_l}^{{D_l}}\right)\neq 0$, we conclude that
\begin{align}\label{eq:proof-2} \sum_{i\in [m]}{c_{i}}\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right)=0.\end{align}
First consider the case that the only boxes of $D$ in row $k$ or column $l$ are those in $D_l$. If this is the case then
\begin{align*}
\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{{\widehat{D}}_j}^{{\widehat{C}^{(i)}}_j}\right)=\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right)
\end{align*}
for each $i \in [m]$.
Therefore,
\begin{align}\label{eq:proof-3} \sum_{i\in [m]}{c_{i}}\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{{\widehat{D}}_j}^{{\widehat{C}^{(i)}}_j}\right)=\sum_{i\in [m]}{c_{i}}\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right).\end{align}
Combining \eqref{eq:proof-2} and \eqref{eq:proof-3} we obtain that the polynomials $\left\{\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{{\widehat{D}}_j}^{\widehat{C}^{(i)}_j}\right)\right\}_{i \in [m]}$ are linearly dependent, as desired.
Now, suppose that there are boxes of $D$ in row $k$ that are not in $D_l$. Let $j_1< \ldots< j_p$ be all indices $j \neq l$ such that $D_j={\widehat{D}}_j \cup \{k\}$.
Then also $C^{(i)}_{j_q}=\widehat{C}^{(i)}_{j_q} \cup \{k\}$ for each $i \in [m]$ and $q\in [p]$.
View the left-hand side of \eqref{eq:proof-2} as a polynomial in $y_{kk}$. Then, \eqref{eq:proof-2} implies that the coefficient of $y_{kk}^p$ is $0$:
\begin{align}\label{eq:proof-4} [y_{kk}^p]\sum_{i\in [m]}{c_{i}}\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right)=0.\end{align}
On the other hand, the determinants $\det\left(Y_{D_j}^{C^{(i)}_j}\right)$ involve $y_{kk}$ exactly when $j=j_q$ for some $q\in[p]$. In this case, applying Laplace expansion along the row of $Y_{D_{j_q}}^{C^{(i)}_{j_q}}$ containing $y_{kk}$ implies
\begin{align*}
[y_{kk}]\det\left(Y_{D_{j_q}}^{C^{(i)}_{j_q}}\right) = \xi_{i,q}\det\left(Y_{\widehat{D}_{j_q}}^{\widehat{C}^{(i)}_{j_q}}\right)
\end{align*}
with $\xi_{i,q}\in\{1,-1 \}$. Thus,
\begin{align*}
[y_{kk}^p]\sum_{i\in [m]}{c_{i}}\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right)
&=\sum_{i\in [m]}{c_{i}}\left([y_{kk}^p]\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right)\right)\\
&=\sum_{i\in [m]}{c_{i}}\left(\prod_{j\in [n]\backslash \{l,j_1,\ldots,j_p\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right)\right)\left([y_{kk}^p]\prod_{q\in [p]}\det\left(Y_{D_{j_q}}^{C^{(i)}_{j_q}}\right)\right)\\
&=\sum_{i\in [m]}{c_{i}}\left(\prod_{j\in [n]\backslash \{l,j_1,\ldots,j_p\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right)\right)\left(\prod_{q\in [p]}[y_{kk}]\det\left(Y_{D_{j_q}}^{C^{(i)}_{j_q}}\right)\right)\\
&=\sum_{i\in [m]}{c_{i}}\left(\prod_{j\in [n]\backslash \{l,j_1,\ldots,j_p\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right)\right)\left(\prod_{q\in [p]}\xi_{i,q}\det\left(Y_{\widehat{D}_{j_q}}^{\widehat{C}^{(i)}_{j_q}}\right)\right).
\end{align*}
Since $\widehat{C}_j^{(i)}=C_j^{(i)}$ and $\widehat{D}_j=D_j$ whenever $j\neq l,j_1,\ldots,j_p$, we obtain
\begin{align*}
[y_{kk}^p]\sum_{i\in [m]}{c_{i}}\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{D_j}^{C^{(i)}_j}\right)
&=\sum_{i\in [m]}{c_{i}}\left(\prod_{j\in [n]\backslash \{l,j_1,\ldots,j_p\}}\det\left(Y_{\widehat{D}_j}^{\widehat{C}^{(i)}_j}\right)\right)\left(\prod_{q\in [p]}\xi_{i,q}\det\left(Y_{\widehat{D}_{j_q}}^{\widehat{C}^{(i)}_{j_q}}\right)\right)\\
&=\sum_{i\in [m]}{c_{i}}\left(\prod_{q\in [p]}\xi_{i,q}\right)
\left(\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{\widehat{D}_j}^{\widehat{C}^{(i)}_j}\right)\right).\\
\end{align*}
Setting $c_i'=c_i\prod_{q\in [p]}\xi_{i,q}$ and applying (\ref{eq:proof-4}) yields the dependence relation
\begin{align*}
\sum_{i\in [m]}c_{i}'
\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{\widehat{D}_j}^{\widehat{C}^{(i)}_j}\right)=0.
\end{align*}
This implies the polynomials $\displaystyle\left\{\prod_{j\in [n]\backslash \{l\}}\det\left(Y_{{\widehat{D}}_j}^{\widehat{C}^{(i)}_j}\right)\right\}_{i \in [m]}$ are linearly dependent, concluding the proof.
\end{proof}
We now state and prove Theorem~\ref{thm:pattern} and its generalization Theorem~\ref{thm:pattern2}.
\begin{theorem}
\label{thm:pattern2}
Fix a diagram $D\subseteq [n]\times [n]$ and let ${\widehat{D}}$ be the diagram obtained from $D$ by removing any boxes in row $k$ or column $l$. Then
\begin{align*}
\chi_D(x_1, \ldots, x_n)=M(x_1, \ldots, x_n) \chi_{{\widehat{D}}}(x_{1}, \ldots,x_{k-1},0,x_{k+1},\ldots, x_{n})+F(x_1, \ldots, x_n),
\end{align*}
where $F(x_1,\ldots,x_n) \in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n]$ and
\[M(x_1, \ldots, x_n) = \left(\prod_{(k,i)\in D}{x_k}\right)\left(\prod_{(i,l)\in D}{x_i} \right).\]
\end{theorem}
\begin{proof}
Let $M=M(x_1,\ldots,x_n)$.
We must show that $[M\bm{m}]\chi_D\geq [\bm{m}]\chi_{{\widehat{D}}}$ for each monomial $\bm{m}$ of $\chi_{{\widehat{D}}}$ not divisible by $x_k$.
Let $C^{(1)}, \ldots, C^{(r)}$ be all the diagrams $C$ such that $C\leq D$ and $\prod_{j=1}^{n}\prod_{i\in C_j}x_i=M\bm{m}$. By Corollary~\ref{cor:fms},
\[[M\bm{m}]\chi_D=\dim\left(\mathrm{Span}_{\mathbb{C}} \left\{\prod_{j=1}^{n}\det\left(Y_{D_j}^{C^{(i)}_j}\right) \ \middle|\ i\in [r] \right\}\right).\]
Let $1,2,\ldots,q$ be the indices of the distinct diagrams among $\widehat{C}^{(1)}, \ldots, \widehat{C}^{(r)}$ such that $\widehat{C}^{(j)}\leq \widehat{D}$ for $j\in [q]$.
By Lemma~\ref{lem:c}, $\widehat{C}^{(1)}, \ldots, \widehat{C}^{(q)}$ are all the diagrams $C$ such that $C\leq \widehat{D}$ and $\prod_{j=1}^{n}\prod_{i\in C_j}x_i=\bm{m}$,
as no diagram with this dual eigenvalue can have a box in row~$k$.
So Corollary~\ref{cor:fms} implies that
\[[\bm{m}]\chi_{\widehat{D}}=\dim\left(\mathrm{Span}_{\mathbb{C}} \left\{\prod_{j=1}^{n}\det\left(Y_{\widehat{D}_j}^{\widehat{C}^{(i)}_j}\right) \ \middle|\ i\in [q] \right\}\right).\]
Finally, Lemma~\ref{lem:keylemma} implies that
\[
\dim\left(\mathrm{Span}_{\mathbb{C}} \left\{\prod_{j=1}^{n}\det\left(Y_{D_j}^{C^{(i)}_j}\right) \ \middle|\ i\in [r] \right\}\right)\geq
\dim\left(\mathrm{Span}_{\mathbb{C}} \left\{\prod_{j=1}^{n}\det\left(Y_{\widehat{D}_j}^{\widehat{C}^{(i)}_j}\right) \ \middle|\ i\in [q] \right\}\right),
\]
so $[M\bm{m}]\chi_D\geq [\bm{m}]\chi_{{\widehat{D}}}$ for each monomial $\bm{m}$ of $\chi_{{\widehat{D}}}$ not divisible by $x_k$; that is
\[\chi_D(x_1, \ldots, x_n)-M\chi_{{\widehat{D}}}(x_{1}, \ldots,x_{k-1},0,x_{k+1},\ldots, x_{n})\in \mathbb{Z}_{\geq 0}[x_1,\ldots,x_n].\]
\end{proof}
\begin{namedtheorem}[\ref{thm:pattern}]
Fix $w \in S_n$ and let $\sigma \in S_{n-1}$ be the pattern with Rothe diagram $D(\sigma)$ obtained by removing row $k$ and column $w_k$ from $D(w)$. Then
\begin{align*}
\mathfrak{S}_{w}(x_1, \ldots, x_n)=M(x_1, \ldots, x_n) \mathfrak{S}_{\sigma}(x_{1}, \ldots, \widehat{x_k}, \ldots, x_{n})+F(x_1, \ldots, x_n),
\end{align*}
where $F\in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n]$ and
\[M(x_1, \ldots, x_n) = \left(\prod_{(k,i)\in D(w)}{x_k}\right)\left(\prod_{(i,w_k)\in D(w)}{x_i} \right).\]
\end{namedtheorem}
\begin{proof}
Specialize Theorem~\ref{thm:pattern2} to the case that $D$ is a Rothe diagram $D(w)$ and $l=w_k$. The dropping of $x_k$ is due to reindexing, since the entirety of row $k$ and column $w_k$ of $D(w)$ are removed from to obtain $D(\sigma)$, not just the boxes in row $k$ and column $w_k$.
\end{proof}
\begin{corollary}
\label{cor:pattern}
Fix $w \in S_n$ and let $\sigma\in S_m$ be any pattern contained in $w$. If $k$ is a coefficient of a monomial in $\mathfrak{S}_\sigma$, then $\mathfrak{S}_w$ contains a monomial with coefficient at least $k$.
\end{corollary}
\begin{proof}
Immediate consequence of repeated applications of Theorem~\ref{thm:pattern}.
\end{proof}
\section*{Acknowledgments} We are grateful to Sara Billey and Allen Knutson for many discussions about Schubert polynomials. We thank the Institute for Advanced Study for providing a hospitable environment for our collaboration. Many thanks to Arthur Tanjaya for his careful reading.
\end{document}
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\begin{document}
\allowdisplaybreaks
\newcommand{2110.07042}{2110.07042}
\renewcommand{113}{113}
\FirstPageHeading
\ShortArticleName{Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP$(2j)$}
\ArticleName{Orthogonal Polynomial Stochastic Duality Functions\\ for Multi-Species SEP$\boldsymbol{(2j)}$ and Multi-Species IRW}
\Author{Zhengye ZHOU}
\AuthorNameForHeading{Z.~Zhou}
\Address{Department of Mathematics, Texas A$\&$M University, College Station, TX 77840, USA} \Email{\href{mailto:[email protected]}{[email protected]}}
\ArticleDates{Received October 16, 2021, in final form December 24, 2021; Published online December 26, 2021}
\Abstract{We obtain orthogonal polynomial self-duality functions for multi-species version of the symmetric exclusion process (SEP$(2j)$) and the independent random walker process (IRW) on a finite undirected graph. In each process, we have $n>1$ species of particles. In addition, we allow up to $2j$ particles to occupy each site in the multi-species SEP$(2j)$. The duality functions for the multi-species SEP$(2j)$ and the multi-species IRW come from unitary intertwiners between different $*$-representations of the special linear Lie algebra $\mathfrak{sl}_{n+1}$ and the Heisenberg Lie algebra $\mathfrak{h}_n$, respectively. The analysis leads to multivariate Krawtchouk polynomials as orthogonal duality functions for the multi-species SEP$(2j)$ and homogeneous products of Charlier polynomials as orthogonal duality functions for the multi-species IRW.}
\Keywords{orthogonal duality; multi-species SEP$(2j)$; multi-species IRW}
\Classification{60K35}
\section{Introduction} In recent years, stochastic duality has been used as a powerful tool in the study of stochastic processes (see, e.g., \cite{Borodin_2014,corwin2016asepqj,kuan2019stochastic,zhou2020hydrodynamic}). More recently, orthogonal stochastic dualities were derived for some classical interacting particle systems. For instance, the independent random walker process (IRW) is self-dual with respect to Charlier polynomials \cite{Carinci_2019,Groenevelt_2018}, and the symmetric exclusion process (SEP$(2j)$) is self-dual with respect to Krawtchouk polynomials \cite{Carinci_2019,Groenevelt_2018}. Orthogonal polynomial duality functions turn out to be useful in applications as they form a convenient orthogonal basis in a suitable space of the systems’ observables. There are many applications of orthogonal duality functions (see, e.g., \cite{Ayala_2018,ayala2020higher,chen2021higher}). In a series of previous works, several ways to find orthogonal dualities were introduced. In \cite{franceschini2017stochastic}, the two-terms recurrence relations of orthogonal polynomials were used, method via generating functions was used in \cite{2018}, while Lie algebra representations and unitary intertwiners were used in~\cite{Groenevelt_2018}. In addition, two more approaches were described in \cite{Carinci_2019}. The first approach is based on unitary symmetries, another one is based on scalar products of classical duality functions. In this paper, we make use of the method introduced in~\cite{Groenevelt_2018}.
We first study a model of interacting particle systems with symmetric jump rates. The multi-species symmetric exclusion process is a generalization of the SEP$(2j)$ to multi-species systems, where we have up to $2j\in\mathbb{N}$ particles allowed for each site and we have $n>1$ species particles in the system. It is worth mentioning that the multi-species SEP$(2j)$ we consider is closely related to other multi-species (multi-color) exclusion processes studied over the past decades. For example, when $j=\frac{1}{2}$, it degenerates to a special case of the multi-color exclusion processes studied in \cite{Caputo2008ONTS,Dermoune2008SpectralGF}. This model also arises naturally as a special case of the multi-species ASEP$(q,j)$ studied in \cite{Kuan_2017} when $q=1$. Given the fact that the single-species SEP$(2j)$ is self-dual with respect to Krawtchouk polynomials, it's expected that similar results could be found for the multi-species SEP$(2j)$. We prove that the multi-species SEP$(2j)$ is self-dual with multivariate Krawtchouk polynomials as duality functions.
Another process we study is the multi-species independent random walker, which can be thought of as $n>1$ independent copies of IRW evolving simultaneously. Although it is straightforward to obtain the duality functions using the independence property, it is still interesting to show how the duality functions arise from representations of the nilpotent Heisenberg Lie algebra $\mathfrak{h}_n$.
The organization of this paper is as follows. In Section~\ref{section2} we give an overview of the method that we use to construct orthogonal duality functions. In Section~\ref{section3} we obtain the orthogonal duality functions for the multi-species SEP$(2j)$ and in Section~\ref{section4} for the multi-species IRW.
\section{Background}\label{section2} In this section we describe the method to obtain the orthogonal dualities which was introduced in \cite{Groenevelt_2018}. We start by recalling the definition of stochastic duality.
\begin{Definition} Two Markov processes $\mathfrak{s}_t$ and $\mathfrak{s}_t'$ on state spaces $\mathfrak{S}$ and $\mathfrak{S}'$ are dual with respect to duality function $D(\cdot,\cdot)$ on $\mathfrak{S}\times\mathfrak{S}'$ if \begin{gather*}
E_{\mathfrak{s}}[D(\mathfrak{s}_t,\mathfrak{s}')]=E'_{\mathfrak{s}'}[D(\mathfrak{s},\mathfrak{s}_t')] \qquad \text{for all} \ \mathfrak{s} \in \mathfrak{S},\ \mathfrak{s}'\in \mathfrak{S}', \ \text{and} \ t>0, \end{gather*} where $E_{\mathfrak{s}}$ denotes expectation with respect to the law of $\mathfrak{s}_t$ with $\mathfrak{s}_0=\mathfrak{s}$ and similarly for $E'_{\mathfrak{s}'}$. If~$\mathfrak{s}_t'$ is a copy of $\mathfrak{s}_t$, we say that the process $\mathfrak{s}_t$ is self-dual.
In most relevant examples, duality could also be stated at the level of Markov generators. We say that generator $L_1$ is dual to $L_2$ with respect to duality function $D(\cdot,\cdot)$ if for all~$\mathfrak{s}$ and~$\mathfrak{s}'$, \begin{gather*}
[L_1D(\cdot,\mathfrak{s}')](\mathfrak{s})=[L_2D(\mathfrak{s},\cdot)](\mathfrak{s}'). \end{gather*} If $L_1=L_2$, we have self-duality. \end{Definition}
Let $\mathfrak{g}=(\mathfrak{g},[\cdot,\cdot])$ be a complex Lie algebra with a $*$-structure, i.e., there exists an involution $*\colon X\xrightarrow[]{}X^*$ such that for any $X,Y\in \mathfrak{g}$, $a,b\in \mathbb{C}$, \begin{gather*}
(aX+bY)^*=\overline{a}X^*+\overline{b}Y^*,\qquad [X,Y]^*=[Y^*,X^*]. \end{gather*} Let $U(\mathfrak{g})$ be the universal enveloping algebra of~$\mathfrak{g}$.
Given a Hilbert space $(H,\langle \cdot,\cdot\rangle)$ and a representation $\rho$ of $\mathfrak{g}$ on $H$, we call $\rho$ a $*$-representation if for any $f,g\in H$ and any $X\in\mathfrak{g} $, \begin{gather*} \langle \rho(X)f,g\rangle =\langle f,\rho(X^*)g\rangle . \end{gather*} Suppose we have state spaces $\Omega_1$ and $\Omega_2$ of configurations on $L$ sites given by $\Omega_1=E_1\times\dots\times E_L$ and $\Omega_2=F_1\times\dots\times F_L$. Let $\mu=\mu_1\otimes\dots\otimes \mu_L$ and $\nu=\nu_1\otimes\dots\otimes \nu_L$ be product measures on~$\Omega_1$ and~$\Omega_2$.
For $1\le x\le L$, let $\rho_x$ and $\sigma_x$ be unitarily equivalent $*$-representations of a Lie algebra $\mathfrak{g}$ on $ L^2(E_x,\mu_x)$ and $ L^2(F_x,\nu_x)$, respectively. Then $\rho=\rho_1\otimes\dots\otimes \rho_L$ and $\sigma=\sigma_1\otimes\dots\otimes\sigma_L$ are $*$-representations of $\mathfrak{g}$. We assume that the corresponding unitary intertwiner $\Lambda_x\colon L^2(E_x,\mu_x)\xrightarrow{}L^2(F_x,\nu_x)$ has the following form: \begin{gather*}
(\Lambda_x f)(z_2)=\int_{E_x} f(z_1)K_x(z_1,z_2)\,{\rm d}\mu_x(z_1),\qquad \text{for $\nu_x$-almost all $z_2\in F_x$}, \end{gather*} for some kernel $K_x\in L^2(E_x\times F_x,\mu_x\otimes \nu_x)$ satisfying the relation \begin{gather*}
[\rho_x(X^*)K_x(\cdot,z_2)](z_1)=[\sigma_x(X)K_x(z_1,\cdot)](z_2), \qquad (z_1,z_2)\in E_x\times F_x,\quad X\in\mathfrak{g}. \end{gather*}
With all the above structures, the following theorem provides a way to construct duality functions. \begin{Theorem}[\cite{Groenevelt_2018}]\label{th:2.1} Suppose $L_1$ and $L_2$ are self-adjoint operators on $L^2(\Omega_1,\mu)$ and $L^2(\Omega_2,\nu)$, respectively, given by \begin{gather*}
L_1=\rho(Y),\qquad L_2=\sigma(Y), \end{gather*} for some self-adjoint $Y\in U(\mathfrak{g})^{\otimes L}$. Then $L_1$ and $L_2$ are in duality, with duality function \begin{gather*}
D(z_1,z_2)=\prod_{x=1}^L K_x(z_{1x},z_{2x}), \qquad z_1=(z_{11},\dots,z_{1L})\in \Omega_1,\qquad z_2=(z_{21},\dots,z_{2L})\in \Omega_2. \end{gather*} \end{Theorem}
\section[Multi-species SEP(2j) and Lie algebra sl\_\{n+1\}]{Multi-species SEP$\boldsymbol{(2j)}$ and Lie algebra $\boldsymbol{\mathfrak{sl}_{n+1}}$}\label{section3}
In this section, we study the multi-species version of the SEP$(2j)$ with $2j\in\mathbb{N}$ on a finite undirected graph $G=(V,E)$, where $V=\{1,\dots,L\}$ with $L\in\mathbb{N}$ and $L>2$ is the set of sites (vertices) and $E$ is the set of edges. In what follows, we write site $x\in G$ instead of mentioning~$V$ for ease of notation.
The state space $\mathcal{S}(n,2j,G)$ of particle configurations consists of variables $\xi=\big(\xi_i^x\colon 0\le i\le n$, $x\in G\big)$, where $\xi_i^x$ denotes the number of particles of species $i$ at site $x$, and \begin{gather*}
\xi^x=\big(\xi^x_0,\dots,\xi^x_n\big)\in \Omega_{2j}:=\left\{\xi=(\xi_0,\dots,\xi_n)\,\Bigg|\,\sum_{i=0}^n \xi_i=2j,\, \xi_i\ge 0\right\} \end{gather*} for any site $x\in G $. One can think of $\xi^x_0$ as the number of holes at site~$x$.
\begin{Definition}\label{def:3.1} The generator of the multi-species SEP$(2j)$ on a finite undirected graph $G=(V,E)$ is given by \begin{gather} \begin{split}
&\mathcal{L}f(\xi)=\sum_{\text{edge}\{x,y\}\in E} \mathcal{L}_{x,y} f(\xi), \\ & \mathcal{L}_{x,y}f(\xi)=\sum_{0\le k<l\le n}\xi_l^{x}\xi_k^{y}\big[f\big(\xi^{x,y}_{l,k}\big)-f(\xi)\big]+\xi_l^{y}\xi^{x}_k\big[f\big(\xi^{y,x}_{l,k}\big)-f(\xi)\big], \end{split}\label{eq:6} \end{gather} where $\xi_{l,k}^{x,y}$ denotes the particle configuration obtained by switching a particle of the $l^{\rm th}$ species at site $x$ with a particle of the $k^{\rm th}$ species at site $y$ if $\xi_{l,k}^{x,y}\in\mathcal{S}(n,2j,G)$. \end{Definition}
Note that when $n=1$, this process reduces to the single-species SEP(2j) defined in \cite{Giardin__2009}.
Suppose $p=(p_0,\dots,p_n)$ is a probability distribution, the multinomial distribution on $\Omega_{2j}$ is defined as \begin{gather*}
w_p(\xi)=\binom{2j}{\xi}\prod_{i=0}^np_i^{\xi_i}, \end{gather*} where $ \binom{2j}{\xi}$ denotes the multinomial coefficient $\frac{(2j)!}{\prod_{i=0}^n\xi_i!}$. Following a simple detailed balance computation, we can show that the product measure with marginals $w_p(\xi)$ for any fixed $p$ being a distribution is a reversible measure of the multi-species SEP$(2j)$, i.e., $\otimes_G w_p$ is a reversible measure of the multi-species SEP$(2j)$ when $p$ is the same for all sites.
\subsection[Multivariate Krawtchouk polynomials and Lie algebra sl\_\{n+1\}]{Multivariate Krawtchouk polynomials and Lie algebra $\boldsymbol{\mathfrak{sl}_{n+1}}$}\label{section3.1}
First, we introduce the $n$-variable Krawtchouk polynomials defined by Griffiths \cite{https://doi.org/10.1111/j.1467-842X.1971.tb01239.x}. We shall adopt the notation of Iliev~\cite{iliev_2012} in the following.
\begin{Definition}\label{def:1} Let $\mathcal{K}_n$ be the set of 4-tuples $\big(\nu,P,\hat{P},U\big)$ such that
$P$, $\hat{P}$, $U$ are $(n+1)\times(n+1)$ complex matrices satisfying the following conditions: \begin{enumerate}\itemsep=0pt \item[(1)] $P=\operatorname{diag}(p_0,\dots,p_n)$, $\hat{P}=\operatorname{diag}(\hat{p}_0,\dots,\hat{p}_n)$ and $p_0=\hat{p}_0=\frac{1}{\nu}\neq 0$, \item[(2)] $U=(u_{kl})_{0\le k,l\le n}$ with $U_{k0}=U_{0k}=1$ for all $0\le k \le n$, \item[(3)] $\nu PU\hat{P}U^{\rm T}=I_{n+1}$. \end{enumerate} \end{Definition}
It follows from the above definition that $p=(p_0,\dots,p_n)$ and $\hat{p}=(\hat{p}_0,\dots,\hat{p}_n)$ satisfy that $\sum_{k=0}^n p_k=\sum_{k=0}^n \hat{p}_k=1$ and $p_k,\hat{p}_k\neq 0$ for any $k$.
For all points $\kappa\in \mathcal{K}_n$, Griffith constructed multivariate Krawtchouk polynomials using a~generating function as follows.
\begin{Definition}[\cite{https://doi.org/10.1111/j.1467-842X.1971.tb01239.x}]
For $\xi,\eta\in \Omega_{2j}$ and $\kappa\in \mathcal{K}_n$, the multivariate Krawtchouk polynomial $K(\xi,\eta,\kappa,j)$ is defined by
\begin{gather*}
\sum_{\xi\in\Omega_{2j}}\binom{2j}{\xi}K(\xi,\eta,\kappa,j)z_1^{\xi_1}\cdots z_n^{\xi_n}=\prod_{k=0}^n\bigg(1+\sum_{l=1}^nu_{kl}z_l\bigg)^{\eta_k}.
\end{gather*}
\end{Definition}
Although it's not obvious to tell from the generating function, $K(\xi,\eta,\kappa,j)$ depends on~$P$ and~$\hat{P}$ because the matrix $U\in\kappa$ satisfies the condition~(3) in Definition~\ref{def:1}.
In what follows, we fix a 4-tuple $\kappa\in \mathcal{K}_n$ as in Definition~\ref{def:1}, we also write $K(\xi,\eta,\kappa,j) $ as~$K(\xi,\eta)$ for simplicity.
In \cite{iliev_2012}, Iliev interpreted multivariate Krawtchouk polynomials with representations of the Lie algebra~$\mathfrak{sl}_{n+1}$. We recall some of the essential results. Let's start by introducing some basic notations. Let $z_0,\dots,z_n$ be mutually commuting variables, we set $z=(z_0,\dots,z_n)$. For each $\xi=(\xi_0,\dots,\xi_n)\in\Omega_{2j}$, we denote
$z^\xi=z_0^{\xi_0}z_1^{\xi_1}\cdots z_n^{\xi_n}$ and $\xi!=\xi_0!\cdots\xi_n!$. Also define $V_{2j}=\operatorname{span}\big\{z^\xi|\xi\in\Omega_{2j}\big\}\subset \mathbb{C}[z]$, which is the space consisting of all homogeneous complex polynomials of total degree $2j$.
Let $I_{n+1}$ denote the $(n+1)\times(n+1)$ identity matrix. For $0\le k,l\le n$, let $e_{k,l}$ denote the $(n+1)\times(n+1)$ matrix with $(k,l)^{\rm th}$ entry $1$ and other entries~$0$. The special linear Lie algebra of order $n+1$ denoted by $\mathfrak{sl}_{n+1}$ consists of $(n+1)\times (n+1) $ matrices with trace zero and has the Lie bracket $[X,Y]=XY-YX$. It has basis $\{e_{kl}\}_{0\le k\neq l\le n}$ and $\{h_l\}_{0< l\le n}$, where $h_l=e_{ll}-\frac{1}{n+1}I_{n+1}$.
The $*$-structure of $\mathfrak{sl}_{n+1}$ is given by \begin{gather}\label{eq:1}
e_{kl}^*=e_{lk}, \quad k\neq l , \qquad h_l^*=h_l. \end{gather}
Let ${\mathfrak {gl}}(V_{2j})$ denote the space of endomorphisms of ${V_{2j}}$, we consider the representation $\rho\colon \mathfrak{sl}_{n+1}\allowbreak \xrightarrow{} \mathfrak{gl}(V_{2j})$ defined by \begin{gather}\label{eq:3.2}
\rho e_{kl} =z_k\partial z_l,\quad k\neq l,\qquad \rho h_l=z_l\partial z_l-\frac{2j}{n+1}. \end{gather}
Next, define an antiautomorphism $\mathfrak{a}$ on $\mathfrak{sl}_{n+1}$ by $\mathfrak{a}(X)=\hat{P}X^{\rm T}\hat{P}^{-1}$. It follows easily that \begin{gather}\label{eq:3.1}
\mathfrak{a}(e_{kl})=\frac{\hat{p}_l}{\hat{p}_k}e_{lk},\quad k\neq l, \qquad
\mathfrak{a}(h_{l})=h_l. \end{gather}
We define a symmetric bilinear form $\langle\,,\,\rangle_{\kappa}$ on $V_{2j}$ by \begin{gather*} \big\langle z^\xi,z^\eta\big\rangle _{\kappa}=\delta_{\xi,\eta}\frac{\xi!}{\hat{p}^\xi}\theta^{2j}. \end{gather*}
Then it is easy to check that for any $X\in \mathfrak{sl}_{n+1}$ and $v_1,v_2\in V_{2j}$ \begin{gather}\label{eq:2} \langle \rho Xv_1,v_2\rangle _{\kappa}=\langle v_1,\rho \mathfrak{a}(X)v_2\rangle_{\kappa}. \end{gather}
Let $R$ be the matrix \begin{gather}\label{eq:4}
R=\hat{\theta}\hat{P}U^{\rm T}, \end{gather} where $\hat{\theta}\in \mathbb{C}$ such that $\det(R)=1$. Next, we define $\hat{z}=(\hat{z}_0,\dots,\hat{z}_n)$ by $\hat{z}=zR$. \begin{Lemma}\label{lemma:3.1} Define operator $\operatorname{Ad}_R$ on $\mathfrak{sl}_{n+1}$ by $\operatorname{Ad}_R(X)=R^{-1}XR$, where $R=(r_{kl})_{0\le k,l\le n}$ is defined in equation~\eqref{eq:4}. Then $\operatorname{Ad}_R$ is a Lie algebra automorphism of $\mathfrak{sl}_{n+1}$. \end{Lemma} \begin{proof}
It can be checked directly. \end{proof}
Last, we list some properties of the multivariate Krawtchouk polynomial whose proof could be found in~\cite{iliev_2012}. \begin{Proposition}[{\cite[Corollary 5.2]{iliev_2012}}]\label{prop:3.1} For $\xi,\eta\in \Omega_{2j}$, the multivariate polynomial~$K$ has the following bilinear form, \begin{gather*}
K(\xi,\eta)=\frac{p_0^{2j}}{(2j)!}\big\langle z^\xi,\hat{z}^\eta\big\rangle_{\kappa}. \end{gather*} \end{Proposition}
\begin{Proposition}[{\cite[Corollary 5.3]{iliev_2012}}]\label{prop:3.2} For $\xi,\eta,\zeta\in \Omega_{2j}$ we have the following relations: \begin{gather*}
\sum_{\xi\in\Omega_{2j}}K(\xi,\eta)K(\xi,\zeta)w_{\hat{p}}(\xi)=\frac{p_0^{2j}\delta_{\eta,\zeta}}{w_p(\eta)}, \\
\sum_{\xi\in\Omega_{2j}}K(\eta,\xi)K(\zeta,\xi)w_p(\xi)=\frac{p_0^{2j}\delta_{\eta,\zeta}}{w_{\hat{p}}(\eta)}. \end{gather*} \end{Proposition} \begin{Remark} If $U\in\kappa$ is a real matrix, then it follows from the generating function that the multivariate Krawtchouk polynomial is real valued. In this case, Proposition~\ref{prop:3.2} is the orthogonality relation for the multivariate Krawtchouk polynomial in $l^2(w_p)$ and $l^2(w_{\hat{p}})$. \end{Remark}
\subsection[Self-duality of the multi-species SEP(2j)]{Self-duality of the multi-species SEP$\boldsymbol{(2j)}$}\label{section3.2}
In this subsection, we show that the multi-species SEP($2j$) is self dual with respect to duality functions given by homogeneous products of multivariate Krawtchouk polynomials. Suppose $p$ and $\hat{p}$ in the 4-tuple $\kappa\in \mathcal{K}_n$ as in Definition~\ref{def:1} are both probability measures.
Let $l^2(w_p)$ be a Hilbert space with inner product $(f,g)_p=\sum_{\xi\in\Omega_{2j}}f(\xi)\overline{g(\xi)}w_p(\xi)$. Now we define a $*$-representations $\rho_p$ of $\mathfrak{sl}_{n+1}$ on $l^2(w_p)$ by \begin{alignat*}{3}
&\rho_p(e_{kl})f(\xi)=\sqrt{\frac{p_k}{p_l}}\xi_lf\big(\xi_{l,k}^{-1,+1}\big) \qquad &&\text{for} \ 0\le k\neq l\le n,& \\
&\rho_p(h_l)f(\xi)=\left(\xi_l-\frac{2j}{n+1}\right)f(\xi)\qquad&& \text{for} \ 0< l\le n,& \end{alignat*} where $\xi_{l,k}^{+1,-1}$ represents the variable with $\xi_l$ increased by 1 and $\xi_k$ decreased by 1. Recalling the $*$-structure defined in equation~\eqref{eq:1}, it is straightforward to check that $(\rho_p(X)f,g)_p=(f,\rho_p(X^*)g)_p$ for all $X\in \mathfrak{sl}_{n+1} $.
Next, we introduce another non-trivial $*$-representation $\sigma_p$ of $\mathfrak{sl}_{n+1}$ on $l^2(w_{p})$ that is unitarily equivalent to $\rho_{\hat{p}}$. \begin{Definition}\label{def:2} For each $\rho_{\hat{p}}$, define a corresponding representation by $\sigma_p=\rho_{\hat{p}}\circ \operatorname{Ad}_R $, where~$\operatorname{Ad}_R$ is the automorphism defined in Lemma~\ref{lemma:3.1}. \end{Definition}
\begin{Proposition}\label{prop:3.4} If the matrix $U$ in the $4$-tuple $\kappa\in \mathcal{K}_n$ is real, then the representation $\sigma_p$ defined in Definition~{\rm \ref{def:2}} is a $*$-representation of~$\mathfrak{sl}_{n+1}$ on~$l^2(w_{p})$. \end{Proposition}
\begin{proof} By definitions of the matrices $U$ and $R$, when $U$ is a real matrix, then $R$ and $R^{-1}$ are all real matrices. For ease of notation, we write $Q=R^{-1}=(q_{i,m})_{0\le i,m\le n}$. Then, computing $\sigma_p$ explicitly, we have \begin{gather}\label{eq:5.3}
\sigma_p(e_{im})f(\eta)=\sqrt{\frac{\hat{p}_i}{\hat{p}_m}}\sum_{k,l=0}^n q_{ki}r_{ml}\eta_lf\big(\eta_{k,l}^{+1,-1}\big). \end{gather}
Next we verify that for any $X\in \mathfrak{sl}_{n+1} $, $\left(\sigma_p(X) f(\eta),g(\eta)\right)_{p}= ( f(\eta),\sigma_p(X^*)g(\eta) )_{p}$. First, we plug equation~\eqref{eq:5.3} in the inner products, when $i\neq m$, \begin{gather}
\big(\sigma_p(e_{im}) f(\eta),g(\eta)\big)_{p}=\sqrt{\frac{\hat{p}_i}{\hat{p}_m}}\sum_{k,l=0}^n q_{ki}r_{ml}\big(\eta_lf\big(\eta_{k,l}^{+1,-1}\big),g(\eta)\big)_{p},\nonumber\\ \label{eq:5.1} \big( f(\eta),\sigma_p(e_{mi})g(\eta)\big)_{p}=\sqrt{\frac{\hat{p}_m}{\hat{p}_i}}\sum_{k,l=0}^n q_{km}r_{il}\big(f(\eta),\eta_lg\big(\eta_{k,l}^{+1,-1}\big)\big)_{p}. \end{gather} By switching $k$ and $l$ in equation~\eqref{eq:5.1}, we have that \begin{gather*}
\big( f(\eta),\sigma_p(e_{mi})g(\eta)\big)_{p}=\sqrt{\frac{\hat{p}_m}{\hat{p}_i}}\sum_{k,l=0}^n q_{lm}r_{ik}\big(\eta_kf(\eta),g\big(\eta_{l,k}^{+1,-1}\big)\big)_{p}. \end{gather*} Now define $\tilde{\eta}=\eta_{l,k}^{+1,-1}$, we get \begin{gather*} \big( f(\eta),\sigma_p(e_{mi})g(\eta)\big)_{p}=\sqrt{\frac{\hat{p}_m}{\hat{p}_i}}\sum_{k,l=0}^n q_{lm}r_{ik}\frac{p_k}{p_l}\big(\tilde{\eta}_lf\big(\tilde{\eta}_{l,k}^{-1,+1}\big),g(\tilde{\eta})\big)_{p}. \end{gather*}
Computing the entries of matrices $R$ and $Q$ explicitly, we have $r_{ki}=\hat{\theta}\hat{p}_{k}u_{ik}$ and $q_{ki}=\frac{1}{p_0\hat{\theta}}p_ku_{ki}$. Plugging in $r$ and $q$, we have that $\frac{\hat{p}_i}{\hat{p}_m}q_{ki}r_{ml}=\frac{p_k}{p_l}q_{lm}r_{ik}$, which gives that \[ \big(\sigma_p(e_{im}) f(\eta),g(\eta)\big)_{p}=\big( f(\eta),\sigma_p(e_{mi})g(\eta)\big)_{p}. \]
The proof for $h_l$ is similar. \end{proof} \begin{Remark} Proposition~\ref{prop:3.4} is nontrivial since the automorphism $\operatorname{Ad}_R$ does not preserve the $*$-structure of $\mathfrak{sl}_{n+1}$, i.e., there exists $X\in \mathfrak{sl}_{n+1}$ such that $\operatorname{Ad}_R(X^*)\neq \operatorname{Ad}_R(X)^*$. For example, \[ \operatorname{Ad}_R(e_{12}^*)=\operatorname{Ad}_R(e_{21})=\sum_{k,l}q_{k2}r_{1l}e_{kl}, \] while \[ \operatorname{Ad}_R(e_{12})^*=\bigg(\sum_{k,l}q_{k1}r_{2l}e_{kl}\bigg)^*=\sum_{k,l}q_{k1}r_{2l}e_{lk}=\sum_{k,l}q_{l1}r_{2k}e_{kl}, \] which are not equal for general~$\kappa$. \end{Remark}
\begin{Proposition} When the matrix $U$ in the $4$-tuple $\kappa$ is real, $\rho_{\hat{p}}$ and $\sigma_p$ satisfy the following property: $[\rho_{\hat{p}}(X^*)K(\cdot,\eta)](\xi)=[\sigma_p(X)K(\xi,\cdot)](\eta)$ for any $X\in\mathfrak{sl}_{n+1}$ and $(\xi,\eta)\in \Omega_{2j}\times \Omega_{2j}$. \end{Proposition}
\begin{proof} Recall from Proposition~\ref{prop:3.1}, the multivariate Krawtchouk polynomials can be written in bilinear form. Also recall that the antiautomorphism $\mathfrak{a}$ defined in \eqref{eq:3.1} has property~\eqref{eq:2}. Notice that for function~$z^\xi$, we can write~$\rho_{p}$ in terms of the representation $\rho$ defined in~\eqref{eq:3.2}, \begin{gather*}
\rho_{\hat{p}}(e_{im})z^{\xi}=\sqrt{\frac{\hat{p}_i}{\hat{p}_m}}\rho(e_{im})z^{\xi}. \end{gather*} Thus, for $i\neq m$, \begin{align*}
[\rho_{\hat{p}}(e_{im})K(\cdot,\eta)](\xi)& =\frac{p_0^{2j}}{(2j)!}\sqrt{\frac{\hat{p}_i}{\hat{p}_m}}\big\langle \rho(e_{im})z^{\xi},\hat{z}^\eta\big\rangle_{\kappa} =\frac{p_0^{2j}}{(2j)!}\sqrt{\frac{\hat{p}_i}{\hat{p}_m}}\big\langle z^{\xi},\rho\mathfrak{a}(e_{im})\hat{z}^\eta\big\rangle _{\kappa}\\ & =\frac{p_0^{2j}}{(2j)!}\sqrt{\frac{\hat{p}_m}{\hat{p}_i}}\big\langle z^{\xi},\rho(e_{mi})\hat{z}^\eta\big\rangle _{\kappa}=[\sigma_p(e_{mi})K(\xi,\cdot)](\eta), \end{align*} where the last equality follows form the fact that $e_{im}=\operatorname{Ad}_R(\hat{e}_{im})$ and $\rho(\hat{e}_{im})\hat{z}^\eta=\hat{z}_i\partial_{\hat{z}_m}\hat{z}^\eta$~\cite{iliev_2012}, thus \[ \sqrt{\frac{\hat{p}_m}{\hat{p}_i}}\rho(e_{mi})\hat{z}^\eta=\sqrt{\frac{\hat{p}_m}{\hat{p}_i}}\rho\circ \operatorname{Ad}_R(\hat{e}_{mi})\hat{z}^\eta=\sigma_p(e_{mi})\hat{z}^\eta.\]
The proof for $h_l$ follows from the same argument. \end{proof}
\begin{Proposition}
If the matrix $U$ in the $4$-tuple $\kappa$ is real, define the operator $\Lambda\colon l^2(w_{\hat{p}})\xrightarrow[]{} l^2(w_p)$ by \begin{gather*}
(\Lambda f)(\eta)=p_0^{-j}\sum_{\xi\in \Omega_{2j}}w_{\hat{p}}(\xi)f(\xi)K(\xi,\eta). \end{gather*}
Then $\Lambda$ is an unitary operator and intertwines $\rho_{\hat{p}}$ with $\sigma_p$. The kernel $K(\xi,\eta)$ satisfies \begin{gather}\label{eq:3}
[\rho_{\hat{p}}(X^*)K(\cdot,\eta)](\xi)=[\sigma_p(X)K(\xi,\cdot)](\eta). \end{gather}
\end{Proposition}
\begin{proof}It follows directly from equation~\eqref{eq:3} that $\Lambda[\rho_{\hat{p}}(X)f]=\sigma_p(X)\Lambda(f)$ for all $X\in\mathfrak{sl}_{n+1}$, thus $\Lambda$ intertwines $\rho_{\hat{p}}$ with $\sigma_p$. Recall that if a process has reversible measure, then it's self dual with respect to the cheap duality function, which comes from the reversible measure. For the multi-species SEP$(2j)$, the cheap duality function is given by $\delta_{\zeta}(\xi)=\frac{\delta_{\zeta,\xi}}{w_{\hat{p}}(\xi)}$, which has squared norm $\frac{1}{w_{\hat{p}}(\zeta)}$ in $l^2(w_{\hat{p}})$.
On the other hand, $\Lambda(\delta_\zeta)(\eta)=p_0^{-j}K(\zeta,\eta)$ has squared norm $\frac{1}{w_{\hat{p}}(\zeta)}$ in $l^2(w_p)$. Thus $\Lambda$ maps an orthogonal basis to another orthogonal basis preserving the norm, hence $\Lambda$ is unitary. \end{proof}
Last we show the generator defined in~\eqref{eq:6} is the image of some self-adjoint element in $\mathcal{U}(\mathfrak{sl}_{n+1})^{\otimes L}$ under the $*$-representations $\rho_{\hat{p}}^{\otimes L}$ and $\sigma_p^{\otimes L}$. We generalize the construction of the Markov generator in terms of the co-product of a Casimir element (see, e.g., \cite{Giardin__2009,Groenevelt_2018}) to the multi-species cases.
We start by constructing a Casimir element of $\mathcal{U}(\mathfrak{sl}_{n+1})$. Under the non-degenerate bilinear form $B(X,Y)=\operatorname{tr}(XY)$, the dual basis of $\mathfrak{sl}_{n+1}$ is given by \begin{gather*}
e_{lk}^\star=e_{kl}, \quad k\neq l, \qquad h_l^\star=e_{ll}-e_{00}=h_l+\sum_{k=1}^nh_k. \end{gather*} The Casimir element $\Omega$ of $\mathcal{U}(\mathfrak{sl}_{n+1})$ is given by \begin{gather*}
\Omega=\sum_{0\le k<l\le n}(e_{kl}e_{lk}+e_{lk}e_{kl})+\sum_{0< l\le n} h_lh_l^\star . \end{gather*} It is easy to verify that $\Omega$ is self-adjoint, i.e., with the $*$-structure given in~\eqref{eq:1}, $\Omega^*=\Omega$. Next, define the coproduct for the basis $\{e_{kl}\}_{0\le k\neq l\le n}$ and $\{h_l\}_{0< l\le n}$ as $\Delta(X)=1\otimes X+X\otimes 1$, and define an element \begin{gather*}
Y=\Delta(\Omega)-\Omega\otimes 1-1\otimes \Omega. \end{gather*}
\begin{Lemma} $\mathcal{L}_{x,y}$ is the image of a self-adjoint element in $\mathcal{U}(\mathfrak{sl}_{n+1})^{\otimes 2}$ under the representation $\rho_{\hat{p}}\otimes\rho_{\hat{p}}$ and $\sigma_p\otimes\sigma_p$. Specifically, there exists a constant $c\in \mathbb{R}$ such that \begin{align}\label{eq:5}
\mathcal{L}_{x,y}& =\frac{1}{2}\rho_{\hat{p}}\otimes\rho_{\hat{p}} (Y_{x,y} )-c \\ \label{eq:5.2} & = \frac{1}{2}\sigma_p\otimes\sigma_p (Y_{x,y} )-c. \end{align} \end{Lemma}
\begin{proof}To prove \eqref{eq:5}, we make use of the following identity: \begin{gather*}
\sum_{0\le k<l\le n}\xi_k^x\xi_l^y+\xi_k^x\xi_l^y=\bigg(\sum_{0\le l\le n}\xi_l^x\bigg)\bigg(\sum_{0\le l\le n}\xi_l^y\bigg)-\sum_{0\le l \le n} \xi^x_l\xi^y_l=(2j)^2-\sum_{0\le l \le n} \xi^x_l\xi^y_l. \end{gather*} Expanding $\Omega$ in $Y$, we have \begin{gather*}
Y=2\sum_{0\le k< l\le n} ( e_{lk}\otimes e_{kl}+e_{kl}\otimes e_{lk} )+\sum_{1\le l\le n} (h_l\otimes h_l^{\star}+h_l^{\star}\otimes h_l ). \end{gather*} Now we can compute the right-hand side of~\eqref{eq:5} using the above identities and the representation $\rho_{\hat{p}}$ to see that it agrees with $\mathcal{L}_{x,y}$. Note that $\mathcal{L}_{x,y}$ does not depend on~$\hat{p}$ since all terms with~$\hat{p}$ get cancelled.
To prove \eqref{eq:5.2}, it suffices to show $\operatorname{Ad}_R\otimes \operatorname{Ad}_R(Y)=Y$. First, we can check that $\operatorname{Ad}_R(\Omega)=\Omega$ by direct calculation. Using the fact that $\operatorname{Ad}_R\otimes \operatorname{Ad}_R \circ \Delta=\Delta\circ \operatorname{Ad}_R$, we have $\operatorname{Ad}_R\otimes \operatorname{Ad}_R(Y)\allowbreak =Y$, thus \begin{gather*} \sigma_p \otimes\sigma_p(Y_{x,y})= (\rho_{\hat{p}}\otimes\rho_{\hat{p}} )\circ (\operatorname{Ad}_R\otimes \operatorname{Ad}_R ) (Y_{x,y}) =\rho_{\hat{p}}\otimes\rho_{\hat{p}} (Y_{x,y}).\tag*{\qed} \end{gather*}
\renewcommand{\qed}{} \end{proof}
Applying Theorem~\ref{th:2.1} yields the self duality for the multi-species SEP$(2j)$. \begin{Theorem}
The multi-species ${\rm SEP}(2j)$ defined in Definition~{\rm \ref{def:3.1}} is self dual with respect to duality functions \begin{gather*}
\prod_{x\in G} K\big(\xi^x,\eta^x,\kappa,2j\big), \end{gather*}
for any $\kappa\in \mathcal{K}_n$ such that $U$ in $\kappa$ is real and~$p$, $\hat{p}$ in $\kappa$ are probability measures. \end{Theorem}
\section[Multi-species IRW and Heisenberg Lie algebra h\_n]{Multi-species IRW and Heisenberg Lie algebra $\boldsymbol{\mathfrak{h}_n}$}\label{section4}
In this section, we find a family of self-duality functions of the multi-species independent random walk (multi-species IRW) using the Heisenberg Lie algebra $\mathfrak{h}_n$.
The $n$-species independent random walk is a generalization of the usual IRW to~$n$ species on a finite undirected graph $G=(V,E)$, where $V=\{1,\dots,L\}$ with $L\in\mathbb{N}$ and $L>2$ is the set of sites (vertices) and $E$ is the set of edges. It is a Markov process where $n$ species of particles move independently between $L$ sites. The jump rate for a particle of species $i$ from a site is proportional to the number of species $i$ particles at that site.
The state space $\mathcal{S}(n,G)$ of particle configurations consists of variables $\xi=\big(\xi_i^x\colon 1\le i\le n$, $x\in G\big)$, where $\xi_i^x\in\mathbb{N}_0$ (non-negative integers) denotes the number of species~$i$ particles at site~$x$. \begin{Definition}\label{def:4.1} The generator of $n$-species IRW on $G=(V,E)$ is given by \begin{gather*}
\mathcal{L}f(\xi)=\sum_{\text{edge}\{x,y\}\in E} \mathcal{L}_{x,y},\\
\mathcal{L}_{x,y}= \sum_{i=1}^n\big [\xi_i^x\big(f\big(\xi_i^{x,y}\big)-f(\xi)\big)+\xi_i^y\big(f\big(\xi_i^{y,x}\big)-f(\xi)\big)\big], \end{gather*} where $\xi_i^{x,y}$ denotes the particle configuration obtained by moving a particle of species~$i$ from site~$x$ to site~$y$ if $\xi_i^{x,y}\in\mathcal{S}(n,G)$. \end{Definition}
Define measure $ \mu_\lambda(\xi)=\prod_{i=1}^n \frac{\lambda^{\xi_i}}{\xi_i!}{\rm e}^{-\lambda}$ with $ \lambda>0$. Following a simple detailed balance computation, we can show that the product measure $\otimes_G\mu_\lambda$ is a reversible measure of the $n$-species IRW when $\lambda$ is the same for all sites.
Next, we mention here that the space $l^2(\mu_\lambda)$ is equipped with inner product \begin{gather*}
(f,g)_\lambda=\sum_{\xi\in\mathbb{N}_0^{ n}}\mu_\lambda(\xi)f(\xi)\overline{g(\xi)}. \end{gather*}
\subsection[The Charlier polynomials and Heisenberg Lie algebra h\_n]{The Charlier polynomials and Heisenberg Lie algebra $\boldsymbol{\mathfrak{h}_n}$}\label{section4.1}
\begin{Definition} The Heisenberg Lie algebra $\mathfrak{h}_n$ is the $2n+1$ dimensional complex Lie algebra with generators $\{P_1,\dots,P_n,Q_1,\dots,Q_n,Z\}$ and commutation relations: for $1\le i,l \le n$, \begin{gather*}
[P_i,P_l]=[Q_i,Q_l]=[P_i,Z]=[Q_i,Z]=0,\qquad
[P_i,Q_l]=\delta_{i,l}Z. \end{gather*} \end{Definition} The Heisenberg Lie algebra $\mathfrak{h}_n$ is nilpotent but not semisimple. It has a $*$-structure given by \begin{gather*}
P_i^*=Q_i,\qquad Q_i^*=P_i,\qquad Z^*=Z. \end{gather*}
The Charlier polynomials are given by \begin{gather*}
C_m(z,\lambda)={}_2F_0\left(\left.\begin{matrix}
-m,\ -z \\
-
\end{matrix}\,\right| -\frac{1}{\lambda}\right). \end{gather*} Here we list some properties of Charlier polynomials that will be used later on. First, Charlier polynomials are Orthogonal,
\begin{gather*}
\sum_{z\in \mathbb{N}_0}C_m(z,\lambda)C_{\tilde{m}}(z,\lambda)\frac{\lambda^{z}}{z!}{\rm e}^{-\lambda}=\delta_{m,\tilde{m}}\lambda^{-\tilde{m}}\tilde{m}!.
\end{gather*}
They have the following raising and lowering property,
\begin{gather*}
mC_{m-1}(z,\lambda)=\lambda C_m(z,\lambda)-\lambda C_m(z+1,\lambda),
\\
\lambda C_{m+1}(z,\lambda)=\lambda C_m(z,\lambda)-z C_m(z-1,\lambda).
\end{gather*}
To construct unitary operator later, we define function $C(\xi,\eta,\lambda)$ for $\xi,\eta\in \mathbb{N}_0^{ n}$ by \begin{gather}\label{eq: 4.9}
C(\xi,\eta,\lambda)=\prod_{i=1}^n {\rm e}^{\lambda} C_{\xi_i}(\eta_i,\lambda). \end{gather}
\subsection{Self duality of the multi-species IRW}\label{section4.2} Now we define the $*$-representation $\rho_\lambda$ of $\mathfrak{h}_n$ on $l^2(\mu_\lambda)$ by \begin{gather*}
[\rho_\lambda(Q_i)f](\xi)=\xi_if\big(\xi_i^{-1}\big),\\
[\rho_\lambda(P_i)f](\xi)=\lambda f\big(\xi_i^{+1}\big),\\
[\rho_\lambda(Z)f](\xi)=\lambda f(\xi), \end{gather*} where $\xi_i^{+1}$ \big($\xi_i^{-1}$\big) means that $\xi_i$ is increased (decreased) by~$1$.
Next, we define the map $\theta$ by \begin{gather*}
\theta(P_i)=Z-P_i,\qquad \theta(Q_i)=Z-Q_i,\qquad \theta(Z)=Z, \end{gather*} then $\theta$ extends to a Lie algebra isomorphism of $\mathfrak{h}_n$, preserving the $*$-structure.
\begin{Proposition}\label{prop:4.1}
For any $X\in \mathfrak{h}_n$, we have
\begin{gather*}
\rho_\lambda(X^*)C(\cdot,\eta,\lambda)(\xi)=\rho_\lambda(\theta(X))C(\xi,\cdot,\lambda)(\eta).
\end{gather*} \end{Proposition}
\begin{proof} It's easy to verify by definition and the raising and lowering property, \begin{gather*}
\rho_\lambda(Q_i)C(\cdot,\eta,\lambda)(\xi)=\xi_i C\big(\xi_i^{-1},\eta,\lambda\big)\\
\hphantom{\rho_\lambda(Q_i)C(\cdot,\eta,\lambda)(\xi)}{} =\lambda C(\xi,\eta,\lambda)-\lambda C\big(\xi,\eta_i^{+1},\lambda\big)
=\rho_\lambda(\theta(P_i))C(\xi,\cdot,\lambda)(\eta),\\
\rho_\lambda(P_i)C(\cdot,\eta,\lambda)(\xi)=\lambda C\big(\xi_i^{+1},\eta,\lambda\big)\\
\hphantom{\rho_\lambda(P_i)C(\cdot,\eta,\lambda)(\xi)}{}
=\lambda C(\xi,\eta,\lambda)-\eta_i C\big(\xi,\eta_i^{-1},\lambda\big)
=\rho_\lambda(\theta(Q_i))C(\xi,\cdot,\lambda)(\eta) .\tag*{\qed} \end{gather*} \renewcommand{\qed}{} \end{proof}
\begin{Proposition}
Define the operator $\Lambda\colon l^2(\mu_\lambda)\xrightarrow[]{} l^2(\mu_\lambda)$ by \begin{gather*}
(\Lambda f)(\eta)=\sum_{\xi\in \mathbb{N}_0^{ n}}\mu_\lambda(\xi)f(\xi)C(\xi,\eta,\lambda), \end{gather*} then $\Lambda$ is an unitary operator and intertwines $\rho_\lambda$ with $\rho_\lambda\circ\theta$. \end{Proposition} \begin{proof} It follows directly from Proposition~\ref{prop:4.1} that $\Lambda[\rho_\lambda(X)f]=\rho_\lambda\circ\theta(X)\Lambda(f)$ for all $X\in\mathfrak{h}_n$, thus $\Lambda$ intertwines $\rho_{\lambda}$ with $\rho_\lambda\circ\theta$. The cheap duality functions for the $n$-species IRW given by $\delta_{\zeta}(\xi)=\frac{\delta_{\zeta,\xi}}{\mu_\lambda(\xi)}$ form an orthogonal basis for $l^2(\mu_\lambda)$ with squared norm $\frac{1}{\mu_\lambda(\zeta)}$, while $\Lambda(\delta_\zeta)(\eta)=C(\zeta,\eta,\lambda)$ also has squared norm $\frac{1}{\mu_\lambda(\zeta)}$ in $l^2(\mu_\lambda)$. By the fact that all $C(\zeta,\eta,\lambda)$ form an orthogonal basis for $l^2(\mu_\lambda)$, $\Lambda$ is unitary. \end{proof}
Finally, to show self duality, we define $Y\in \mathcal{U}(\mathfrak{h}_n)^{\otimes 2}$ by \begin{gather*}
Y=\sum_{i=1}^n (1\otimes Q_i-Q_i\otimes 1)(P_i\otimes 1-1\otimes P_i). \end{gather*}
\begin{Lemma} The generator for the multi-species IRW can be written as the following: \begin{align}\label{eq:7}
\mathcal{L}_{x,y}& =\lambda^{-1} \rho_\lambda\otimes\rho_\lambda (Y_{x,y}) \\ \label{eq:8}
& =\lambda^{-1} (\rho_\lambda\circ\theta)\otimes(\rho_\lambda \circ\theta)(Y_{x,y}). \end{align} \end{Lemma} \begin{proof}
\eqref{eq:7} is obtained by plugging in definitions, and to prove \eqref{eq:8}, we show that $(\rho_\lambda\circ\theta)\otimes(\rho_\lambda \circ\theta)(Y)=\rho_\lambda\otimes\rho_\lambda (Y)$, which follows from \cite[Lemma~3.5]{Groenevelt_2018}. \end{proof}
Again, applying Theorem~\ref{th:2.1}, we obtain the self duality for the multi-species IRW. \begin{Theorem} The multi-species IRW defined in Definition~{\rm \ref{def:4.1}} is self dual with respect to duality functions \begin{gather}\label{eq: 4.20}
\prod_{x\in G} C\big(\xi^x,\eta^x,\lambda\big),\qquad \lambda>0. \end{gather} \end{Theorem} \begin{Remark} These duality functions could be obtained by the independence of the evolution of each species of particles and the fact that the duality functions given by \eqref{eq: 4.9} and~\eqref{eq: 4.20} suitably factorize over species. \end{Remark}
\subsection*{Acknowledgments} The author is very grateful to Jeffrey Kuan and anonymous referees for helpful discussions and insightful comments.
\pdfbookmark[1]{References}{ref}
\LastPageEnding
\end{document}
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\begin{document}
\sf
\title[Calabi-Yau objects in triangulated categories]{Calabi-Yau objects in triangulated categories} \author[C. Cibils, P. Zhang] {Claude Cibils$^a$ and Pu Zhang$^{b, *}$} \thanks{The second named author is supported by the CNRS of France, the NSF of China and of Shanghai City (Grant No. 10301033 and ZR0614049).} \thanks{$^*$ The corresponding author} \keywords{Serre functor, Calabi-Yau object, Auslander-Reiten triangle, stable module category, self-injective Nakayama algebra} \maketitle
\begin{center} $^A$D\'epartement de Math\'ematiques, \ \ Universit\'e de Montpellier 2\\ F-34095, Montpellier Cedex 5, France\ \ \ Claude.Cibils$\symbol{64}$math.univ-montp2.fr\\ $^B$Department of Mathematics, \ \ Shanghai Jiao Tong University\\ Shanghai 200240, P. R. China\ \ \ \ pzhang$\symbol{64}$sjtu.edu.cn \end{center} \begin{abstract} We introduce the Calabi-Yau (CY) objects in a Hom-finite Krull-Schmidt triangulated $k$-category, and notice that the structure of the minimal, consequently all the CY objects, can be described. The relation between indecomposable CY objects and Auslander-Reiten triangles is provided. Finally we classify all the CY modules of self-injective Nakayama algebras, determining this way the self-injective Nakayama algebras admitting indecomposable CY modules. In particular, this result recovers the algebras whose stable categories are Calabi-Yau, which have been obtained in [BS]. \end{abstract}
\vskip10pt
\section {\bf Introduction}
Calabi-Yau (CY) categories have been introduced by Kontsevich [Ko]. They provide a new insight and a wide framework for topics as in mathematical physics ([Co]), non-commutative geometry ([B], [Gin1], [Gin2]), and representation theory of Artin algebras ([BS], [ES], [IR], [Ke], [KR1], [KR2]).
Triangulated categories with Serre dualities ([BK], [RV]) and CY categories have important global naturality. On the other hand, even in non CY categories, inspired by [Ko], one can introduce CY objects. It turns out that they arise naturally in non CY categories and enjoy ``local naturality'' and interesting properties (Prop. 4.4, Theorems 3.2, 4.2, 5.5 and 6.1).
\vskip10pt
The first aim of this paper is to study the properties of such objects in a Hom-finite Krull-Schmidt triangulated $k$-category with Serre functor $F$. We give the relation between indecomposable CY objects and the Auslander-Reiten triangles ($\S 3$), and describe all the $d$-th CY objects via the minimal ones, which are exactly the direct sum of all the objects in finite $\langle [-d]\circ F\rangle$-orbits of $\operatorname{Ind}(\mathcal A)$ ($\S 4$). We classify all the $d$-th CY modules of self-injective Nakayama algebras for any integer $d$ ($\S 5$). Finally, we determine all the self-injective Nakayama algebras which admit indecomposable CY modules. In particular, this recovers the algebras whose stable categories are Calabi-Yau ($\S 6$), included in the work of Bialkowski and Skowro\'nski [BS]. Note that the CY modules are invariant under stable equivalences between self-injective algebras, with a very few exceptions (Prop.3.1). Consequently our results on self-injective Nakayama algebras extend to the one on the wreath-like algebras ([GR]), which contains the Brauer tree algebras ([J]).
This also raises an immediate question. Let $\mathcal A$ be a Hom-finite Krull-Schmidt triangulated $k$-category with a Serre functor. If all objects are $d$-th CY with the same $d$, whether $\mathcal A$ is a Calabi-Yau category?
\vskip10pt
\section {\bf Backgrounds and Preliminaries}
\subsection{} Let $k$ be a field and $\mathcal{A}$ a Hom-finite $k$-category. Recall from Bondal and Kapranov [BK] that a $k$-linear functor $F \colon \mathcal{A} \to \mathcal{A}$ \ is {\em a right Serre functor} if there exist $k$-isomorphisms $$\eta_{A, B}: \ \ \operatorname{Hom}_{\mathcal{A}}(A, B) \longrightarrow D \operatorname{Hom}_{\mathcal{A}}(B, FA), \ \ \forall \ \ A, \ B\in \mathcal{A},$$ which are natural both in $A$ and $B$, where $D = \operatorname{Hom}_{k}(-, k)$. Such an $F$ is unique up to a natural isomorphism, and fully-faithful; if it is an equivalence, then a quasi-inverse $F^{-1}$ is a left Serre functor; in this case we call $F$ a Serre functor. Note that $\mathcal{A}$ has a Serre functor if and only if it has both right and left Serre functor. See Reiten and Van den Bergh [RV].
\vskip10pt
For triangulated categories we refer to [Har], [V], and [N]. Let $\mathcal{A}$ be a Hom-finite triangulated $k$-category. Following Happel [Hap1], {\em an Auslander-Reiten triangle} $X\stackrel{f} {\longrightarrow} Y \stackrel{g} {\longrightarrow} Z \stackrel{h} {\longrightarrow} X[1]$ of $\mathcal{A}$ is a distinguished triangle satisfying:
(AR1) \ $X$ and $Z$ are indecomposable;
(AR2) \ $h\ne 0$;
(AR3) \ If $t: Z'\longrightarrow Z$ is not a retraction, then there exists $t': Z'\longrightarrow Y$ such that $t = gt'$.
\vskip10pt
Note that (AR3) is equivalent to
(AR4) \ If $Z'$ is indecomposable and $t: Z'\longrightarrow Z$ is a non-isomorphism, then $ht = 0$.
Under (AR1) and (AR2), (AR3) is equivalent to
(AR3') \ If $s: X\longrightarrow X'$ is not a section, then there exists $s': Y\longrightarrow X'$ such that $s = s'f$.
Also, (AR3') is equivalent to
(AR4') \ If $X'$ is indecomposable and $s: X\longrightarrow X'$ is a non-isomorphism, then $s\circ h[-1] = 0$.
\vskip10pt
In an Auslander-Reiten triangle $X {\longrightarrow} Y {\longrightarrow} Z {\longrightarrow} X[1]$, the object $X$ is uniquely determined by $Z$. Write $X = \tau_\mathcal A Z$. In general $\tau_\mathcal A$ is {\em not} a functor. By definition $\mathcal A$ has right Auslander-Reiten triangles if there exists an Auslander-Reiten triangle $X {\longrightarrow} Y {\longrightarrow} Z {\longrightarrow} X[1]$ for any indecomposable $Z$; and $\mathcal A$ has Auslander-Reiten triangles if $\mathcal{A}$ has right and left Auslander-Reiten triangles. We refer to [Hap1], [XZ] and [A] for the Auslander-Reiten quiver of a triangulated category.
\vskip10pt
A Hom-finite $k$-category is {\em Krull-Schmidt} if the endomorphism algebra of any indecomposable is local. In this case any object is uniquely decomposed into a direct sum of indecomposables, up to isomorphisms and up to the order of indecomposable direct summands (Ringel [R], p.52).
Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated $k$-category. Theorem I.2.4 in [RV] says that $\mathcal{A}$ has a right Serre functor $F$ if and if $\mathcal{A}$ has right Auslander-Reiten triangles. In this case, $F$ coincides with $[1]\circ \tau_\mathcal A$ on objects, up to isomorphisms.
\vskip10pt
\subsection{}Let $\mathcal{A}$ be a Hom-finite triangulated $k$-category with Serre functor $F$. Denote by $[1]$ the shift functor of $\mathcal A$. Following Kontsevich [Ko], $\mathcal{A}$ is {\em a Calabi-Yau category} if there is a natural isomorphism $F \cong [d]$ of functors for some $d\in\Bbb Z$.
Denote by $o([1])$ the order of $[1]$. If $o([1]) = \infty$ then the integer $d$ above is unique, and is called {\em the CY dimension} of $\mathcal{A}$; if $o([1])$ is finite then we call the minimal non-negative integer $d$ such that $F\cong [d]$ {\em the CY dimension} of $\mathcal{A}$. Denote by $\operatorname{CYdim}(\mathcal A)$ the CY dimension of $\mathcal A$.
\vskip10pt
For example, if $A$ is a symmetric algebra and $\mathcal P$ is the category of projective modules, then the homotopy category $K^b(\mathcal P)$ is of CY dimension $0$. Moreover, if $\mathcal A$ is of CY dimension $d$, then $\operatorname {Ext}_\mathcal A^i(X, Y)\cong D\circ \operatorname {Ext}_\mathcal A^{d-i}(Y, X), \ X, Y\in\mathcal A, \ i\in\Bbb Z$, where $\operatorname {Ext}_\mathcal A^i(X, Y): =\operatorname {Hom}_\mathcal A(X, Y[i])$. Thus, if $A$ is a CY algebra ([B], [Gin2]), i.e. the bounded derived category $D^b(A\mbox{-mod})$ is Calabi-Yau of CY dimension $d$, then $\operatorname {gl.dim}(A\mbox{-mod})=d$ (see [B]).
\vskip10pt
\subsection{} Let $\mathcal{A}$ and $\mathcal{B}$ be triangulated categories. {\em A triangle functor} from $\mathcal{A}$ to $\mathcal{B}$ is a pair $(F, \eta^F)$, where $F\colon \mathcal{A} \to \mathcal{B}$ is an additive functor and $\eta^F: \ F\circ [1] \longrightarrow [1]\circ F$ is a natural isomorphism, such that if $X \stackrel{f} {\longrightarrow} Y \stackrel{g} {\longrightarrow} Z \stackrel{h} {\longrightarrow} X[1]$ is a distinguished triangle of $\mathcal{A}$ then $FX \stackrel{Ff} {\longrightarrow} FY \stackrel{Fg} {\longrightarrow} FZ \stackrel{\eta^F_X \circ Fh} {\longrightarrow} (FX)[1]$ is a distinguished triangle of $\mathcal{B}$. Triangle functors $(F, \ \eta^F)$ and $(G, \ \eta^G)$ are {\em natural isomorphic} if there is a natural isomorphism $\xi: \ F\longrightarrow G$ such that the following diagram commutes for any $A\in\mathcal{A}$ \[\xymatrix{ F(A[1]) \ar[rr]^{\eta^F_A} \ar[d]_{\xi_{A[1]}} && F(A)[1]
\ar[d]^-{\xi_A[1]}\\ G(A[1]) \ar[rr]^{\eta^G_A} && G(A)[1].}\]
As Keller pointed out, the pair $([n], \ (-1)^n{\rm Id}_{[n+1]}): \ \mathcal{A}\longrightarrow\mathcal{A}$ is a triangle functor for $n\in\Bbb Z$. However, $([n], \ {\rm Id}_{[n+1]})$ may be {\em not}.
We need the following important result. A nice proof given by Van den Bergh is in the Appendix of [B].
\begin {lem} \ \ (Bondal-Kapranov {\em [BK]}; Van den Bergh {\em [B]}) \ \ Let $F$ be a Serre functor of a Hom-finite triangulated $k$-category $\mathcal{A}$. Then there exists a natural isomorphism $\eta^F: \ F\circ [1] \longrightarrow [1]\circ F$ such that $(F, \ \eta^F): \ \mathcal{A}\longrightarrow\mathcal{A}$ is a triangle functor. \end{lem}
>From [Ke, 8.1] and [B, A.5.1] one has the following
\begin{prop} (Keller; Van den Bergh)\ Let $\mathcal{A}$ be a Hom-finite triangulated $k$-category with Serre functor $F$. Then $\mathcal{A}$ is a Calabi-Yau category if and only if there exists a natural isomorphism $\eta^F: F\circ [1] \longrightarrow [1]\circ F,$ such that $(F, \ \eta^F)$ is a triangle functor and $(F, \eta^F) \longrightarrow ([d], (-1)^d\operatorname {Id}_{[d+1]})$\ is a natural isomorphism of triangle functors, for some integer $d$. \end{prop}
\noindent {\bf Proof.} For convenience we justify the ``only if'' part. By assumption we have a natural isomorphism $\xi: F \cong [d]$. Define $\eta^F_A \colon F(A[1]) \longrightarrow (FA)[1]$ for $A\in\mathcal{A}$ by \ $\eta^F_A:=(-1)^d(\xi_A)^{-1}[1]\circ \xi_{A[1]}.$\ Then $\xi_A[1] \circ \eta_A^F = (-1)^d \xi_{A[1]}$. The naturality of $\eta^F: \ F\circ [1] \longrightarrow [1]\circ F$ follows from the one of $\xi$. It remains to show that $(F, \eta^F)\colon \mathcal{A}\to \mathcal{A}$ is a triangle functor. Let $X \stackrel{f} {\longrightarrow} Y \stackrel{g} {\longrightarrow} Z \stackrel{h} {\longrightarrow} X[1]$ be a distinguished triangle. Since \ $X[d]\stackrel{f[d]} {\longrightarrow}Y[d]\stackrel{g[d]} {\longrightarrow} Z[d] \stackrel{(-1)^d h[d]} {\longrightarrow}X[d+1]$\ \ is a distinguished triangle, it suffices to prove that the following diagram is commutative \[\xymatrix{ F(X)\ar[r]^-{F(f)} \ar[d]_{\xi_X}^-{\wr} & F(Y) \ar[r]^-{F(g)}\ar[d]_{\xi_Y}^-{\wr}& F(Z)\ar[rr]^-{\eta_X^F \circ F(h)}\ar[d]_{\xi_Z}^-{\wr}&& (F(X))[1]\ar[d]\ar[d]_{\xi_X[1]}^-{\wr}\\ X[d]\ar[r]^-{f[d]} & Y[d]\ar[r]^-{g[d]}& Z[d]\ar[rr]^-{(-1)^d h[d]}&& X[d+1].}\] By the naturality of $\xi$ the first and the second square are commutative. We also have $$\xi_X[1]\circ \eta_X^F \circ F(h) = (-1)^d \xi_{X[1]} \circ F(h) = (-1)^d h[d]\circ \xi_Z. \ \ \ \blacksquare $$
\vskip10pt
\subsection{} Let $A$ be a self-injective $k$-algebra, $A$-mod the category of finite-dimensional left $A$-modules, and $A\underline {\mbox{-mod}}$ the stable category of $A$-mod modulo projective modules. Then the Nakayama functor $\mathcal N: = D( A)\otimes_A-$, Heller's syzygy functor $\Omega$, and the Auslander-Reiten translate $\tau \cong \Omega^2\circ \mathcal N\cong\mathcal N\circ \Omega^2$ ([ARS], p.126), and are endo-equivalences of $A\underline {\mbox{-mod}}$ ([ARS], Chap. IV). Note that $A\underline {\mbox{-mod}}$ is a Hom-finite Krull-Schmidt triangulated $k$-category with $[1] = \Omega^{-1}$ ([Hap1], p.16). By the bi-naturality of the Auslander-Reiten isomorphisms ([AR]) $$ \underline {\operatorname {Hom}}(X, Y) \cong D\circ\operatorname {Ext}_A^1(Y, {\tau} X) \cong D\circ\underline {\operatorname {Hom}}(Y, [1]\circ {\tau} X),$$ where $\underline {\operatorname {Hom}}(X, Y): = \operatorname {Hom} _{A\underline {\mbox{-mod}}}(X, Y)$, one gets the Serre functor $F:=[1]\circ {\tau} \cong \Omega\circ \mathcal N$ of $A\underline {\mbox{-mod}}$. It follows that $A\underline {\mbox{-mod}}$ is Calabi-Yau if and only if $\mathcal N \cong \Omega^{-(d+1)}$ for some $d$ ([Ke, 8.3]). In this case denote by $\operatorname{CYdim}(A)$ the CY dimension of $A\underline {\mbox{-mod}}$. Note that $\Omega, \ F, \ \mathcal N, \ \tau$ are pairwise commutative as functors of $A\underline {\mbox{-mod}}$. This follows from Lemma 2.1. \vskip10pt
\subsection{} Let $A$ be a finite-dimensional $k$-algebra. Recall that $A$ is a Nakayama algebra if any indecomposable is uniserial, i.e. it has a unique composition series ([ARS], p.197). In this case $A$ is representation-finite. If $k$ is algebraically closed then any connected self-injective Nakayama algebra is Morita equivalent to $\Lambda(n, t),$ $n\ge 1, \ t\ge 2$ ([GR], p.243), which is defined below.
Let $\Bbb Z_n$ be the cyclic quiver with vertices indexed by the cyclic group $\Bbb Z/n\Bbb Z$ of order $n$, and with arrows $a_i: \ i \longrightarrow i+1, \ \forall \ i\in \Bbb Z/n\Bbb Z$. Let $k\Bbb Z_n$ be the path algebra of the quiver $\Bbb Z_n$, $J$ the ideal generated by all arrows, and $\Lambda = \Lambda(n, t): =k\Bbb Z_n/J^t$ with $t\ge 2$. Denote by $\gamma^l_i$ the path starting at vertex
$i$ and of length $l$, and $e_i: = \gamma^0_i$. We write the conjunction of paths from right to left. Then $\{\gamma^l_i \ | \ 0\le i\le n-1, \ 0\le l\le t-1\}$ is a basis of $\Lambda$; while
$\{P(i): = \Lambda e_i\ | \ 0\le i\le n-1\}$ is the set of pairwise non-isomorphic indecomposable projective modules, and $\{I(i): =
D(e_{i}\Lambda)\ | \ 0\le i\le n-1\}$ is the set of pairwise non-isomorphic indecomposable injective modules, with $P(i) \cong I(i+t-1)$. Note that $\Lambda$ is a Frobenius algebra, and $\Lambda$ is symmetric if and only if $n\mid (t-1)$. Write $S(i): = P(i)/\operatorname{rad}P(i)$, and $S^l_i: =
\Lambda\gamma_{i+l-t}^{t-l}$. Then $S_i^l$ is the indecomposable with top $S(i)$ and the Loewy length $l$, and $\{ S^l_i \ | \ 0\le i\le n-1, \ 1\le l\le t \}$ is the set of pairwise non-isomorphic indecomposable modules, with $S^{t}_i = P(i)$ and $\operatorname{soc}(S^l_i) = S(i+l-1)$. For the Auslander-Reiten quiver of $\Lambda$ see [GR], Section 2, and [ARS], p.197. In particular, the stable Auslander-Reiten quiver of $\Lambda$ is $\Bbb Z A_{t-1}/\langle\tau^n\rangle.$
\vskip10pt
\section{\bf Indecomposable Calabi-Yau objects}
The purpose of this section is to introduce the Calabi-Yau objects and to give the relation between indecomposable Calabi-Yau objects and Auslander-Reiten triangles.
\vskip10pt
\subsection{} Let $\mathcal{A}$ be a Hom-finite triangulated $k$-category. A non-zero object $X$ is called \emph {a Calabi-Yau object} if there exists a natural isomorphism \begin{align}\operatorname{Hom}_{\mathcal A}(X, -)\cong D\circ \operatorname{Hom}_{\mathcal A}(-, X[d])\end {align} for some integer $d$.
By Yoneda Lemma, such a $d$ is unique up to a multiple of the relative order $o([1]_X)$ of $[1]$ respect to $X$. Recall that $o([1]_X)$ is the minimal positive integer such that $X[o([1]_X)]\cong X$, otherwise $o([1]_X) = \infty$. If $o([1]_X)=\infty$ then $d$ in $(3.1)$ is unique and is called {\em the CY dimension} of $X$. If $o([1]_X)$ is finite then the minimal non-negative integer $d$ in $(3.1)$ is called {\em the CY dimension} of $X$. We denote $\operatorname{CYdim}(X)$ the CY dimension. Thus, if $o([1]) < \infty$ then $o([1]_X)\mid o([1])$ and $0\le \operatorname{CYdim}(X) < o([1]_X).$
Let $A$ be a finite-dimensional self-injective algebra. An $A$-module $M$ without projective direct summands is called {\em a Calabi-Yau module} of CY dimension $d$, if it is a Calabi-Yau object of $A\underline{\mbox{-mod}}$ with $\operatorname{CYdim}(M) = d$.
\vskip10pt
Note that $\operatorname{CYdim}(X)$ is usually not easy to determine. In case $(3.1)$ holds for some $d$, we say that $X$ is a {\em $d$-th CY object}. Of course, if $o([1]_X) < \infty$ then $o([1]_X)\mid (d - \operatorname{CYdim}(X))\ge 0.$
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If $\mathcal A$ has right Serre functor $F$, then by Yoneda Lemma a non-zero object $X$ is a $d$-th CY object if and only if $F(X)\cong X[d]$, or equivalently, $F(X)[-d]\cong X$. Thus, a non-zero $A$-module $M$ without projective direct summands is a $d$-th CY module if and only if $\mathcal N(M)\cong \Omega^{-(d+1)}(M)$ in $A\underline{\mbox{-mod}}$ (in fact, this isomorphism can be taken in $A$-mod).
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\subsection{} We have the following basic property.
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\begin {prop} \ $(i)$ \ The Calabi-Yau property for a category or an object, is invariant under triangle-equivalences.
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$(ii)$ \ The Calabi-Yau property for a module is {\em ``usually"} invariant under stable equivalences between self-injective algebras. Precisely, let $A$ and $B$ be self-injective algebras, $G: A\underline {\mbox{-mod}}\longrightarrow B\underline {\mbox{-mod}}$ a stable equivalence, and $X$ a CY $A$-module of dimension $d$. If $A\ncong \Lambda(n, 2),$ or if $A$ and $B$ are symmetric algebras, then $G(X)$ is a CY $B$-module of dimension $d$. \end{prop}
\noindent{\bf Proof.} \ $(i)$ \ Let $\mathcal A$ be a Calabi-Yau category with $F_\mathcal A\cong [d]$, where $F_\mathcal A$ is the Serre functor. Clearly $F_\mathcal B: = G \circ F_\mathcal A\circ G^{-1}$ is a Serre functor of $\mathcal B$ (if $\mathcal B$ has already one, then it is natural isomorphic to $F_\mathcal B$). By the natural isomorphism $(\xi^G)^d: G\circ [d] \longrightarrow [d]\circ G$, which is the composition $G\circ [d] \longrightarrow [1]\circ G\circ [d-1] \longrightarrow \cdots \longrightarrow [d]\circ G$ (A.2 in [B]), we see that $\mathcal B$ is a Calabi-Yau category with $F_\mathcal B\cong [d].$ If $X$ is a calabi-Yau object with a natural isomorphism $\eta$ as in $(3.1)$, then we have natural isomorphism \ $\operatorname{Hom}_\mathcal B(-, (\xi^G)^d_X)\circ G\circ \eta\circ G^{-1}: \ \operatorname{Hom}_{\mathcal B}(G(X), -)\cong D\circ \operatorname{Hom}_{\mathcal B}(-, (GX)[d]),$ \ which implies that $G(X)$ is a Calabi-Yau object of $\mathcal B$.
$(ii)$ \ Recall that an equivalence $G: A\underline {\mbox{-mod}}\longrightarrow B\underline {\mbox{-mod}}$ of categories is called a stable equivalence. Note that in general $G$ is not induced by an exact functor (cf. [ARS], p.339), hence $G$ may be not a triangle-equivalence (cf. [Hap1], Lemma 2.7, p.22. Note that the converse of Lemma 2.7 is also true). One may assume that $A$ is connected. If $A\ncong \Lambda(n, 2),$ or if $A$ and $B$ are symmetric algebras, then by Corollary 1.7 and Prop. 1.12 in [ARS], p.344, we know that $G$ commutes with $\tau$ and $\Omega$ on modules, hence we have isomorphism \begin{align*}\Omega_B^{-1}\circ \tau_B (G(X)) \cong G(\Omega_A^{-1}\circ \tau_A) (X)) \cong G(\Omega_A^{-d}(X))\cong \Omega_B^{-d}(G(X)),\end{align*} which implies that $G(X)$ is a Calabi-Yau $B$-module of CY dimension $d$.
$\blacksquare$
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It seems that the Calabi-Yau property for the stable category is also invariant under stable equivalence $G$ between self-injective algebras. However, this need natural isomorphiams between $G$ and $\tau$, and $G$ and $\Omega$, which are not clear to us.
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\subsection{} The main result of this section is as follows.
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\begin{thm} Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated $k$-category, and $X$ an indecomposable object of $\mathcal A$. Then $X$ is a $d$-th CY object if and only if there exists an Auslander-Reiten triangle of the form \begin{align}X[d-1]\stackrel f \longrightarrow Y \stackrel g \longrightarrow X\stackrel h\longrightarrow X[d].\end{align}
Moreover, $Y$ is also a $d$-th CY object. \end{thm}
\vskip10pt \subsection{}The proof of the first part of Theorem 3.2 follows an argument of Reiten and Van den Bergh in [RV]. For the convenience we include a complete proof.
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\begin{lem} \ Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated $k$-category, and $X$ a non-zero object of $\mathcal A$. Then $X$ is a $d$-th CY object if and only if for any indecomposable $Z$ there exists a non-degenerate bilinear form \begin{align}(-,-)_{Z}: \ \operatorname{Hom}_{\mathcal A}(X, Z)\times \operatorname{Hom}_{\mathcal A}(Z, X[d]) \longrightarrow k\end{align} such that for any \ $u\in \operatorname{Hom}_{\mathcal A}(X, Z), \ \ v\in \operatorname{Hom}_{\mathcal A}(Z, W)$, \ and \ $w\in \operatorname{Hom}_{\mathcal A}(W, X[d]),$ there holds \begin{align}(u, wv)_Z=(vu, w)_W.\end{align}
\end{lem} \noindent{\bf Proof.} \ If $X$ is a $d$-th CY object, then we have $\eta_Z: \ \operatorname{Hom}_{\mathcal A}(X, Z)\cong D\circ \operatorname{Hom}_{\mathcal A}(Z, X[d]), \ \forall \ Z\in\mathcal A$, which are natural in $Z$. Each isomorphism $\eta_Z$ induces a non-degenerate bilinear form $(-,-)_{Z}$ in $(3.3)$ by $(u, z)_Z: = \eta_Z(u)(z),$ and $(3.4)$ follows from the naturality of $\eta_Z$ in $Z$. Conversely, if we have $(3.3)$ and $(3.4)$ for any indecomposable $Z$, then we have isomorphism $\eta_Z: \ \operatorname{Hom}_{\mathcal A}(X, Z)\cong D\circ \operatorname{Hom}_{\mathcal A}(Z, X[d])$ given by $\eta_Z(u)(z): = (u, z)_Z, \ \forall \ z\in \operatorname{Hom}_{\mathcal A}(Z, X[d])$. By $(3.4)$ $\eta_Z$ are natural in $Z$. Since $\mathcal A$ is Krull-Schmidt, it follows that we have isomorphisms $\eta_Z$ for any $Z$ which are natural in $Z$. This means that $X$ is a $d$-th CY object.
$\blacksquare$
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The following Lemma in [RV] will be used.
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\begin{lem} \ \ ({\em [RV]}, Sublemma I.2.3)\ Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated $k$-category, $\tau_{\mathcal A}(X) \longrightarrow Y
\longrightarrow X\stackrel h\longrightarrow \tau_\mathcal A(X)[1]$ an Auslander-Reiten triangle of $\mathcal A$, and $Z$ an indecomposable in $\mathcal A$. Then
$(i)$ \ For any non-zero $z\in \operatorname{Hom}_{\mathcal A}(Z, \tau_{\mathcal A}(X)[1])$ there exists $u\in \operatorname{Hom}_{\mathcal A}(X, Z)$ such that $zu = h$.
$(ii)$ For any non-zero $u\in \operatorname{Hom}_{\mathcal A}(X, Z)$ there exists $z\in \operatorname{Hom}_{\mathcal A}(Z,
\tau_{\mathcal A}(X)[1])$ such that $zu = h$. \end{lem}
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In a Hom-finite Krull-Schmidt triangulated $k$-category without Serre functor (e.g., by [Hap2] and [RV] if $\operatorname{gl.dim}(A) = \infty$ then $D^b(A\mbox{-mod})$ has no Serre functor), one may use {\em the generalized Serre functor} introduced by Chen [Ch].
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\begin{lem} (Chen [Ch]) Let $\mathcal A$ be a Hom-finite Krull-Schmidt triangulated $k$-category. Consider the full subcategories of $\mathcal A$
given by $$\mathcal{\mathcal A}_r:=\{X \in \mathcal{A}\; |\; D\circ\operatorname{Hom}_{\mathcal A}(X, -) \mbox{ is representable} \}$$ and $$\mathcal{\mathcal A}_l:=\{X \in
\mathcal{A}\;|\; D\circ \operatorname{Hom}_\mathcal A(-, X) \mbox{ is representable}\}.$$ Then both $\mathcal{A}_r$ and $\mathcal{A}_l$ are thick triangulated subcategories of $\mathcal A$. Moreover, one has
$(i)$ \ There is a unique $k$-functor $S: \mathcal{A}_r \longrightarrow \mathcal{A}_l$ which is an equivalence, such that there are natural isomorphisms \begin{align}\operatorname{Hom}_\mathcal A(X, -) \simeq D\circ\operatorname{Hom}_\mathcal A(-, S(X)), \ \ \forall \ X\in \mathcal {A}_r,\end{align} which are natural in $X$. $S$ is called the generalized Serre functor, with range $\mathcal A_r$ and domain $\mathcal A_l$.
$(ii)$ \ There exists a natural isomorphism $\eta^S: S\circ [1]\longrightarrow [1]\circ S$ such that the pair $(S, \eta^S): \mathcal{A}_r \longrightarrow \mathcal{A}_l$ is an triangle-equivalence. \end{lem}
In this terminology, a non-zero object $X$ is a $d$-th CY object if and only if $X\in\mathcal A_r$ and $S(X)\cong X[d]$, by $(3.5)$ and Yoneda Lemma.
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\subsection{} {\bf Proof of Theorem 3.2.}\quad Let $X$ be an indecomposable $d$-th CY object. By Lemma 3.3 we have a non-degenerate bilinear from $(-,-)_X: \ \operatorname{Hom}_{\mathcal A}(X, X)\times \operatorname{Hom}_{\mathcal A}(X, X[d]) \longrightarrow k.$ It follows that there exists $0\ne h\in \operatorname{Hom}_{\mathcal A}(X, X[d])$ such that $(\operatorname{radHom}_{\mathcal A}(X, X), h)_X=0.$ Embedding $h$ into a distinguished triangle as in $(3.2)$. We claim that it is an Auslander-Reiten triangle. For this it remains to prove (AR4) in 2.1. Let $X'$ be indecomposable and $t: X'\longrightarrow X$ a non-isomorphism. Then by $(3.4)$ for any $u\in \operatorname{Hom}_{\mathcal A}(X, X')$ we have $(u, ht)_{X'} = (tu, h)_X = 0.$ Since $(-,-)_{X'}$ is non-degenerate, it follows that $ht=0$.
Conversely, let $(3.2)$ be an Auslander-Reiten triangle. In order to prove that $X$ is a $d$-th CY object, by Lemma 3.3 it suffices to prove that for any indecomposable $Z$ there exists a non-degenerate bilinear form $(-,-)_{Z}$ as in $(3.3)$ satisfying $(3.4)$. For this, choose an arbitrary linear function $\operatorname {tr}\in D\circ \operatorname{Hom}_{\mathcal A}(X, X[d])$ such that $\operatorname {tr}(h)\ne 0$, and define $(u, z)_Z = \operatorname{tr}(zu).$ Then $(3.4)$ is automatically satisfied. It remains to prove that $(-, -)_Z$ is non-degenerate. In fact, for any $0\ne z\in \operatorname{Hom}_{\mathcal A}(Z, X[d])$, by Lemma 3.4 there exists $u\in \operatorname{Hom}_{\mathcal A}(X, Z)$ such that $zu = h$. So $(u, z)_Z = \operatorname{tr}(zu) = \operatorname{tr}(h)\ne 0$. Similarly, for any $0\ne u\in \operatorname{Hom}_{\mathcal A}(X, Z)$ we have $z\in \operatorname{Hom}_{\mathcal A}(Z, X[d])$ such that $(u, z)_Z \ne 0$. This proves the non-degenerateness of $(-, -)_Z$.
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Now we prove that $Y$ in $(3.2)$ is also a $d$-th CY object. We make use of the generalized Serre functor in [Ch] (For the reader prefer Serre functor, one can assume the existence, and use Lemma 2.1). Since $X, \ X[d-1]\in\mathcal A_r$, it follows from Lemma 3.5 that $Y\in\mathcal A_r$. Applying the generalized Serre functor $(S, \eta^S)$ to $(3.2)$ we get the distinguished triangle (by Lemma 3.5$(ii)$) $$S(X[d-1])\stackrel {S(f)}\longrightarrow S(Y)\stackrel {S(g)}\longrightarrow S(X)\stackrel {\eta^S_{X[d-1]}\circ S(h)}\longrightarrow S(X[d-1])[1].\eqno(*)$$ Also, we have the Auslander-Reiten triangle $$X[2d-1]\stackrel {f[d]} {\longrightarrow}Y[d] \stackrel {g[d]} {\longrightarrow}X[d]\stackrel {(-1)^d h[d]}{\longrightarrow} X[2d].$$ Since $X$ is a $d$-th CY object it follows that we have an isomorphism $w: S(X)\longrightarrow X[d]$. Note that $S(h)\ne 0$ means that $S(g)$ is not a retraction ([Hap1], p.7). Thus, by (AR3) there exists $v: S(Y)\longrightarrow Y[d]$ such that $w\circ S(g) = g[d]\circ v$. By the definition of a triangulated category we get $u: S(X[d-1])\longrightarrow X[2d-1]$ such that the following diagram is commutative \[\xymatrix{ S(X[d-1])\ar[r]^-{S(f)} \ar[d]_{u} & S(Y) \ar[r]^-{S(g)}\ar[d]_{v}& S(X)\ar[rr]^-{\eta_{X[d-1]}^S \circ S(h)}\ar[d]_{w}&& S(X[d-1])[1]\ar[d]\ar[d]_{u[1]}\\ X[2d-1]\ar[r]^-{f[d]} & Y[d]\ar[r]^-{g[d]}& X[d]\ar[rr]^-{(-1)^d h[d]}&& X[2d].}\eqno(**)\] We claim that $u: S(X[d-1])\longrightarrow X[d]$ is an isomorphism, hence by the property of a triangulated category we know that $v: S(Y)\longrightarrow Y[d]$ is also an isomorphism, i.e. $Y$ is a $d$-th CY object.
Otherwise $u: S(X[d-1])\longrightarrow X[2d-1]$ is not an isomorphism. Note that $S$ is {\em only} defined on $\mathcal A_r$, it follows that we do not know if $(*)$ is an Auslander-Reiten triangle.
Since $X[d-1]$ is a $d$-th CY object, it follows that we have an isomorphism $\alpha: X[2d-1]\longrightarrow S(X[d-1])$, hence $\alpha\circ u\in \operatorname{Hom}_\mathcal A(S(X[d-1]), S(X[d-1]))$. So we have $u'\in \operatorname{Hom}_\mathcal A(X[d-1], X[d-1])$ such that $ \alpha\circ u = S(u')$, and $u'$ is also a non-isomorphism. Since $X[d-1]\stackrel f \longrightarrow Y \stackrel g \longrightarrow X\stackrel h\longrightarrow X[d]$ is an Auslander-Reiten triangle it follows from (AR4') that $u'\circ h[-1] = 0$, or equivalently, $S(u')\circ S(h[-1]) = 0$ (note that $h[-1]\in \mathcal A_r$). Thus we have $$u[1] \circ S(h[-1])[1]\circ \eta^S_{X[-1]} = 0$$ where $\eta^S_{X[-1]}: S(X)\longrightarrow S(X[-1])[1]$ is an isomorphism. By the naturality of $\eta^S$ we have the commutative diagram \[\xymatrix{ S(X) \ar[rr]^{\eta_{X[-1]}^S} \ar[d]_{S(h)}&& S(X[-1])[1] \ar[d]^-{S(h[-1])[1]}\\ S(X[d]) \ar[rr]^{\eta_{X[d-1]}^S} && S(X[d-1])[1].}\] It follows that we have $u[1]\circ \eta^S_{X[d-1]}\circ S(h) = 0,$ and hence by the commutative diagram $(**)$ we get a contradiction \ $(-1)^dh[d]\circ w = u[1]\circ \eta^S_{X[d-1]}\circ S(h) = 0.$ This completes the proof.
$\blacksquare$
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\subsection{} {\bf Remark 3.6.} Let $\mathcal A$ be a Hom-finite Krull-Schmidt triangulated $k$-category. If every indecomposable $X$ in $\mathcal A$ is a $d_X$-th CY object, then $\mathcal A$ has a Serre functor $F$ with $F(X)\cong X[d_X]$.
In fact, by Theorem 3.2 $\mathcal A$ has right and left Auslander-Reiten triangles, and then by Theorem I.2.4 in [RV] $\mathcal A$ has Serre functor $F$. By Prop.I.2.3 in [RV] and $(3.2)$ we have $F(X)\cong \tau_\mathcal A(X)[1] = X[d_X-1][1] = X[d_X]$ for any indecomposable $X$.
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However, even if all indecomposables are $d$-th CY objects with the same $d$, we do not know whether $\mathcal A$ is a Calabi-Yau category, although $F$ and $[d]$ coincide on objects. The examples we know have a positive answer to this question.
$\blacksquare$
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\section{\bf Minimal Calabi-Yau objects}
The purpose of this section is to describe all the Calabi-Yau objects of a Hom-finite Krull-Schmidt triangulated $k$-category with a Serre functor.
\subsection{} Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated $k$-category. A $d$-th CY object $X$ is said to be {\em minimal} if any {\em proper} direct summand of $X$ is {\em not} a $d$-th CY object.
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\begin{lem} \ Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated $k$-category with right Serre functor $F$. Then a non-zero object $X$ is a minimal $d$-th CY object if and only if the following are satisfied:
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1. The indecomposable direct summands of $X$ can be ordered as $X = X_1\oplus \cdots \oplus X_r$ such that \begin{align} F(X_1) \cong X_2[d], \ F(X_2) \cong X_3[d], \ \cdots, \ F(X_{r-1}) \cong X_r[d], \ F(X_r)\cong X_1[d].\end{align}
We call the cyclic order arising from this property {\em a canonical order} of $X$ (with respect to $F$ and $[d]$).
2. \ $X$ is multiplicity-free, i.e. its indecomposable direct summands are pairwise non-isomorphic. \end{lem}
\noindent {\bf Proof.}\quad In the following we often use that a non-zero object $X$ is a $d$-th CY object if and only if $F(X)\cong X[d]$.
Let $X= X_1\oplus \cdots \oplus X_r$ be a minimal $d$-th CY object, with each $X_i$ indecomposable and $r\ge 2$. Then $F(X_1) \oplus\cdots\oplus F(X_r)\cong X_1[d]\oplus\cdots\oplus X_r[d]$. Since $\mathcal{A}$ is Krull-Schmidt, it follows that there exists a permutation $\sigma$ of $1, \cdots, r$, such that $F(X_i)\cong X_{\sigma(i)}[d]$ for each $i$. Write $\sigma$ as a product of disjoint cyclic permutations. Since $X$ is minimal, it follows that $\sigma$ has to be a cyclic permutation of length $r$. By reordering the indecomposable direct summands of $X$, one may assume that $\sigma = (12\cdots r)$. Thus, $X$ satisfies the condition 1.
Now, we consider a canonical order $X= X_1\oplus \cdots \oplus X_r$. If $X_i\cong X_j$ for some $1\le i < j\le r$, then $F(X_{j-1})\cong X_j[d]\cong X_i[d]$, it follows that $X_i\oplus \cdots\oplus X_{j-1}$ is already a $d$-th CY object, which contradicts the minimality of $X$. This proves that $X$ is multiplicity-free.
\vskip10pt Conversely, assume that a multiplicity-free object $X= X_1\oplus \cdots \oplus X_r$ is in a canonical order. By $(4.1)$ we have $F(X)\cong X[d]$. So $X$ is a $d$-th CY object. It remains to show the minimality. If not, then there exists a proper direct summand $X_{i_1}\oplus \cdots\oplus X_{i_t}$ of $X$ which is a minimal $d$-th CY object, so $1\le t< r$. By what we have proved above we may assume that this is a canonical order. Then $$F(X_{i_1})\cong X_{i_2}[d],\ \cdots, \ F(X_{i_{t-1}})\cong X_{i_t}[d], \ F(X_{i_t})\cong X_{i_1}[d].$$ While $X= X_1\oplus \cdots \oplus X_r$ is also in a canonical order, it follows that (note that we work on indices modulo $r$, e.g. if $i_1 = r$ then $i_1+1$ is understood to be $1$) $$F(X_{i_1})\cong X_{i_1+1}[d], \ \cdots, \ F(X_{i_{t-1}})\cong X_{i_{t-1}+1}[d], \ F(X_{i_t})\cong X_{i_t+1}[d].$$ Since $X$ is multiplicity-free, it follows that (considering indices modulo $r$) $$i_2 = i_1+1, \ \cdots, \ i_t = i_{t-1}+1, \ i_1 = i_t + 1,$$ hence $i_1 = i_1+t$, which means $r\mid t$. This is impossible since $1\le t< r.$
$\blacksquare$
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\subsection{} Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated $k$-category with Serre functor $F$. For each $d\in\Bbb Z$, consider the triangle-equivalence $G:=[-d]\circ F\cong F\circ [-d]: \mathcal A\longrightarrow \mathcal A$. For each indecomposable $M\in\mathcal A$, denote by $o(G_M)$ the relative order of $G$ respect to $M$, that is, $r:=o(G_M)$ is the minimal positive integer such that $G^r(M)\cong M$, otherwise $o(G_M) = \infty$. Denote by $\operatorname{Aut}(\mathcal A)$ the group of the triangle-equivalences of $\mathcal A$, and by $\langle G\rangle$ the cyclic subgroup of $\operatorname{Aut}(\mathcal A)$ generated by $G$. Then $\langle G\rangle$ acts naturally on $\operatorname{Ind}(\mathcal A)$, the set of the isoclasses of indecomposables of $\mathcal A$. Denote by $\mathcal O_M$ the
$G$-orbit of an indecomposable $M$. Then $|\mathcal O_M| = o(G_M)$. If $|\mathcal O_M| < \infty$ then the set $\mathcal O_M$ is a finite $G$-orbit.
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Denote by $\operatorname{Fin}\mathcal{O}(\mathcal A, d)$ the set of all the finite $G$-orbits of $\operatorname{Ind}(\mathcal A)$, and by $\operatorname{MinCY}(\mathcal A, d)$ the set of isoclasses of minimal $d$-th CY objects. We have the following
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\begin {thm} \ \ Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated $k$-category with Serre functor $F$. Then
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$(i)$ \ Every $d$-th CY object is a direct sum of finitely many minimal $d$-th CY objects.
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$(ii)$ \ With the notations above, for each $d\in\Bbb Z$ the map \begin{align}\mathcal O_M \ \mapsto \ \bigoplus\limits_{X\in \mathcal O_M} X = M\oplus G(M)\oplus\cdots \oplus G^{o(G_M)-1}(M)\end{align} \noindent gives a one-to-one correspondence between the sets $\operatorname{Fin}\mathcal{O}(\mathcal A, d)$ and $\operatorname{MinCY}(\mathcal A, d)$, where $G: = [-d]\circ F$.
Thus, a minimal $d$-th CY object is exactly the direct sum of all the objects in a finite $G$-orbit of $\operatorname{Ind}(\mathcal A)$.
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$(iii)$ Non-isomorphic minimal $d$-th CY objects are disjoint, i.e. they have no isomorphic indecomposable direct summands. \end {thm}
\noindent {\bf Proof.}\quad Let $X$ be a $d$-th CY object. If $X$ is not minimal, then $X = Y\oplus Z$ with $Y\ne 0\ne Z$ such that $F(Y)\oplus F(Z) \cong Y[d]\oplus Z[d]$ and $F(Y)\cong Y[d]$. Since $\mathcal{A}$ is Krull-Schmidt, it follows that $F(Z) \cong Z[d]$, i.e. $Z$ is also a $d$-th CY object. Then $(i)$ follows by induction.
Thanks to Lemma 2.1, $(4.1)$ becomes $X_i = G^{i-1}(X_1), \ 1\le i\le r.$ Then $(ii)$ is a reformulation of Lemma 4.1. Moreover $(iii)$ follows from $(ii)$.
$\blacksquare$
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\begin {cor} \ \ Let $A$ be a finite-dimensional self-injective algebra. Then $X$ is a minimal $d$-th CY module if and only if $X$ is of the form $X \cong \bigoplus \limits_{0\le i\le r-1}G^i(M)$, where $M$ is an indecomposable non-projective $A$-module with $r: = o(G_M)<\infty$, and $G: = \Omega^{d+1}\circ \mathcal N$. \end {cor}
\noindent {\bf Proof.}\quad Note that in this case $[-d]\circ F = \Omega^{d+1}\circ \mathcal N$.
$\blacksquare$
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\subsection{} As an example, we describe all the Calabi-Yau objects in $D^b(kQ\mbox{-mod})$, the bounded derived category of $kQ\mbox{-mod}$, where $Q$ is a finite quiver without oriented cycles.
Note that indecomposable objects of $D^b(kQ\mbox{-mod})$ are exactly stalk complexes of indecomposable $kQ$-modules. The category $D^b(kQ\mbox{-mod})$ has Serre functor $F =[1]\circ \tau_D$, where $\tau_D$ is the Auslander-Reiten translation of $D^b(kQ\mbox{-mod})$. Recall that $\tau_D$ is given by $$\tau_D(M) = \begin{cases}\tau(M), & \ \mbox{if} \ M \ \mbox{is an indecomposable non-projective}; \\ I[-1], & \ \mbox{if} \ P \ \mbox{is an indecomposable projective}, \end{cases}$$ where $I$ is the indecomposable injective with \ $\operatorname{soc}(I) = P/\operatorname{rad} P$, and $\tau$ is the Auslander-Reiten translation of $kQ\mbox{-mod}$ ([Hap1], p.51). Note that $D^b(kQ\mbox{-mod})$ is {\em not} a Calabi-Yau category except that $Q$ is the trivial quiver with one vertex and no arrows. However, the cluster category $\mathcal C_{kQ}$ introduced in [BMRRT], which is the orbit category of $D^b(kQ\mbox{-mod})$ respect to the functor $\tau_D^{-1}\circ [1]$, is a Calabi-Yau category of CY dimension $2$ ([BMRRT]).
Let $M$ be a minimal Calabi-Yau object of $D^b(kQ\mbox{-mod})$ of CY dimension $d$ (in this case $o([1]_M) = \infty$, hence $d$ is unique). By shifts we may assume that $M$ is a $kQ$-module. By $F(M)=\tau_D(M[1]) = \tau_D(M)[1] = M[d]$ we see $d = 1$ or $0$. Note that $kQ$ admits an indecomposable projective-injective module if and only if $Q$ is of type $A_n$ with the linear orientation. However, in this case the unique indecomposable projective-injective module $P = I$ does not satisfy the relation $\operatorname{soc}(I) = P/\operatorname{rad} P$. It follows that $d\ne 0$. Thus $d=1$ and $\tau(M) = M$. Consequently, $Q$ is an affine quiver and $M$ is a $\tau$-periodic (regular) module of period $1$. All such modules are well-known, by the classification of representations of affine quivers (see Dlab-Ringel [DR]). Thus we have
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\begin{prop} \ \ $D^b(kQ\mbox{-mod})$ admits a Calabi-Yau object $M$ if and only if $Q$ is an affine quiver. In this case, $M$ is minimal if and only if $M$ is an indecomposable in a homogeneous tube of the Auslander-Reiten quiver of $kQ$-mod, or $M$ is the direct sum of all the indecomposables of same quasi-length in a non-homogeneous tube of the Auslander-Reiten quiver of $kQ$-mod, up to shifts. Moreover, all such $M$'s have CY dimension $1$. \end{prop}
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\section{\bf Calabi-Yau modules of self-injective Nakayama algebras}
This purpose of this section is to classify all the $d$-th CY modules of self-injective Nakayama algebras $\Lambda(n, t), \ n\ge 1, \ t\ge 2$, where $d$ is any given integer. By Theorem 4.2$(i)$ it suffices to consider the minimal $d$-th CY $\Lambda$-modules. By Corollary 4.3 this reduces to computing the relative order $o(G_M)$ of $G: = \Omega^{d+1}\circ \mathcal N$ respect to any indecomposable $\Lambda(n, t)$-module $M=S_i^l, \ 1\le l\le t-1$, for any integer $d$.
\vskip10pt
\subsection{} Recall that $\Lambda(n, t)$ is the quotient of the path algebra of the cyclic quiver with $n$ vertices by the truncated ideal $J^t$, where $J$ is the two-sided ideal generated by the arrows. From now on we write $\Lambda$ instead of $\Lambda(n, t)$.
We keep the notations introduced in 2.5. Note that the indecomposable module $S^l_i$ \ ($i\in\Bbb Z/n\Bbb Z,$ \ $1\le l\le t-1$) \ has a natural $k$-basis, consisting of all the paths of quiver $\Bbb Z_n$ starting at the vertex $i+l-t$ and of lengths at least $t-l$: $$\gamma^{t-l}_{i+l-t}, \ \gamma^{t-l+1}_{i+l-t}, \ \cdots, \ \gamma^{t-1}_{i+l-t}.$$ For $1\le l\le t-1$ denote by $\sigma_{i}^{l}: S_{i}^{l}\longrightarrow S_{i-1}^{l+1}$ the inclusion by embedding the basis above; and for $2\le l\le t$ denote by $p_{i}^{l}: S_{i}^{l}\longrightarrow S_{i}^{l-1}$ the $\Lambda$-epimorphism given by the right multiplication by arrow $a_{i+l-t-1}$. These $\sigma_i^l$'s and $p_i^l$'s are all the irreducible maps of $\Lambda$-mod, up to scalars.
\vskip10pt
We need the explicit actions of functors $\mathcal N$ and $\Omega^{-(d+1)}$ of $\Lambda\underline {\mbox{-mod}}$. By the exact sequences $0 \longrightarrow S_{i}^l \longrightarrow I(i+l-1)\longrightarrow S_{i+l-t}^{t-l}\longrightarrow 0$ (with the canonical maps), via the basis above one has the actions of functor $\Omega^{-1}$ for $i\in\Bbb Z/n\Bbb Z, \ \ 1\le l \le t-1$: \begin{align*}\Omega^{-1}(S^l_i) = S_{i+l-t}^{t-l}, \ \ \ \ \Omega^{-1}(\sigma^l_i) = p_{i+l-t}^{t-l}, \ \ \ \ \Omega^{-1}(p^l_i) = \sigma_{i+l-t}^{t-l}.\end{align*} By induction one has in $\Lambda\underline {\mbox{-mod}}$ for any integer $m$ (even negative): \begin{align} \Omega^{-(2m-1)}(S_i^l) = S_{i+l-mt}^{t-l}; \ \ \ \ \Omega^{-2m}(S_i^l) = S_{i-mt}^{l};\end{align} and $$\Omega^{-(2m-1)}(\sigma_i^l) = p_{i+l-mt}^{t-l}, \ \ \ \ \Omega^{-(2m-1)}(p_i^l) = \sigma_{i+l-mt}^{t-l};\eqno(*)$$ and $$\Omega^{-2m}(\sigma_i^l) = \sigma_{i-mt}^{l}, \ \ \ \ \Omega^{-2m}(p_i^l) = p_{i-mt}^{l}.\eqno(**)$$ In particular, we have $$o([1]) = \begin{cases} n, & \ t = 2;\\ 2m, & \ t\ge 3,\end{cases}\eqno(***)$$ where $m$ is the minimal positive integer such that $n\mid mt.$
\vskip10pt
\subsection{} Again using the natural basis of $S_i^l$ one has the following commutative diagrams in $\Lambda\underline {\mbox{-mod}}$: \[\xymatrix{\mathcal N(S_{i}^l)\ar[rr]^{\mathcal N(\sigma_{i}^l)} \ar[d]_{\theta_i^l}^-{\wr} && \mathcal N(S_{i-1}^{l+1})\ar[d]_{\theta_{i-1}^{l+1}}^-{\wr}\\ S_{i+1-t}^l \ar[rr]^{\sigma_{i+1-t}^l} && S_{i-t}^{l+1};} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \xymatrix{ \mathcal N(S_{i}^l)\ar[rr]^{\mathcal N(p_{i}^l)} \ar[d]_{\theta_i^l}^-{\wr} && \mathcal N(S_{i}^{l-1})\ar[d]_{\theta_i^{l-1}}^-{\wr}\\ S_{i+1-t}^l \ar[rr]^{p_{i+1-t}^l} && S_{i+1-t}^{l-1}.}\]
\vskip10pt We justify the commutative diagrams above. Note that for any finite quiver $Q$ the bimodule structure of $D(kQ)$ is given by (using the dual basis) $$p^*a = \begin{cases} b^*, \ & \mbox{if} \ ab = p, \\ 0, \ & \mbox{otherwise;}\end{cases} \ \ \ \ \ \ \ \ \ \ \ \mbox{and} \ \ \ \ \ \ \ \ \ \ \ ap^* = \begin{cases} b^*, \ & \mbox{if} \ ba = p, \\ 0, \ & \mbox{otherwise.}\end{cases}$$ for any paths $p$ and $a$. Note that $N(S_{i}^l) = D(\Lambda)\otimes_\Lambda S_{i}^l$ is spanned by $(\gamma_j^{l'})^*\otimes_\Lambda \gamma_{i+l-t}^{t-u}$, where $j\in\Bbb Z/n\Bbb Z, \ 1\le l'\le t-1, \ 1\le u\le l.$ By $(\gamma_j^{l'})^*\otimes_\Lambda \gamma_{i+l-t}^{t-u}= (\gamma_j^{l'})^*\gamma_{i+l-t}^{t-u}\otimes_\Lambda e_{i+l-t}$ we see that if $(\gamma_j^{l'})^*\otimes_\Lambda \gamma_{i+l-t}^{t-u}\ne 0$ then $j = i+l-l'-u$; and in this case we have $$(\gamma_{i+l-l'-u}^{l'})^*\otimes_\Lambda \gamma_{i+l-t}^{t-u}= (\gamma_{i+l-l'-u}^{l'})^* \gamma_{i+l-t}^{t-u}\otimes_\Lambda e_{i+l-t} =(\gamma_{i+l-(l'+u)}^{l'+u-t})^*\otimes_\Lambda e_{i+l-t}.$$ This makes sense only if $l'+u\ge t$. So we have a basis of $N(S_{i}^l)$: $$(\gamma_{i+l-t-v}^v)^*\otimes_\Lambda e_{i+l-t}, \ \ 0\le v\le l-1.$$ Using the natural basis of $S_{i+1-t}^l$ given in 5.1 we have a $\Lambda$-isomorphism $\theta_i^l: \ \mathcal N(S_{i}^l)\longrightarrow S_{i+1-t}^l$ for any $i\in\Bbb Z/n\Bbb Z$ and $1\le l\le t-1$: $$\theta_i^l: \ (\gamma_{i+l-t-v}^v)^*\otimes_\Lambda e_{i+l-t}\mapsto \gamma_{i+1+l-2t}^{t-(v+1)}, \ \ 0\le v\le l-1.$$ (One checks that this is indeed a left $\Lambda$-map.) Note that $N(\sigma_{i}^l): \ \mathcal N(S_{i}^l)\longrightarrow \mathcal N(S_{i-1}^{l+1})$ is a natural embedding given by $$(\gamma_{i+l-t-v}^v)^*\otimes_\Lambda e_{i+l-t}\mapsto (\gamma_{i+l-t-v}^v)^*\otimes_\Lambda e_{i+l-t}, \ \ 0\le v\le l-1;$$ and that $N(p_{i}^l): \ \mathcal N(S_{i}^l)\longrightarrow \mathcal N(S_{i}^{l-1})$ is a $\Lambda$-epimorphism given by $$(\gamma_{i+l-t-v}^v)^*\otimes_\Lambda e_{i+l-t}\mapsto (\gamma_{i+l-t-v}^{v-1})^*\otimes_\Lambda e_{i+l-1-t}, \ \ 0\le v\le l-1$$ where $\gamma_{i+l-1-t}^{-1}$ is understood to be $0$. Then one easily checks the following $$\sigma_{i+1-t}^l\circ \theta_i^l = \theta_{i-1}^{l+1}\circ \mathcal N(\sigma_i^l); \ \ \ \ p_{i+1-t}^l\circ \theta_i^l = \theta_{i}^{l-1}\circ \mathcal N(p_i^l).$$ This justifies the commutative diagrams.
Since all these $\theta_i^l$ depend only on $i$ and $l$, which means that they do not depend on whatever the maps $\sigma_i^l$ or $p_i^l$ are (this is important for the bi-naturality of a Calabi-Yau category), it follows, without loss of the generality, that we can specialize these maps to identities. Thus we have \begin{align} \mathcal N(S^l_i) = S_{i+1-t}^{l}, \ \ \ \mathcal N(\sigma^l_i) = \sigma_{i+1-t}^{l}, \ \ \ \mathcal N(p^l_i) = p_{i+1-t}^{l}.\end{align}
\vskip10pt
\subsection{} By $(***)$ we have $o([1]) < \infty$, it follows, without loss of generality, that we can assume $d\ge 0$. For convenience, set $d(t): = 1 + \frac{(d-1)t}{2}\in \frac {1}{2}\Bbb Z$, whatever $d$ is even or odd; denote by $N=N(d, n, t)$ the minimal positive integer such that \begin{align} \begin{cases} n\mid N d(t), \ &\mbox{if} \ (d-1)t \ \mbox{is even}; \\ n\mid N(2d(t)), \ &\mbox{if} \ (d-1)t \ \mbox{is odd}. \end{cases}\end{align} (When $2d(t)$ is odd, we will write it together in the following.) \vskip10pt By $(5.1)$ we have \ $\Omega^{(2m-1)}(S_i^l) = S_{i+l+(m-1)t}^{t-l}$ \ and \ $ \Omega^{2m}(S_i^l) = S_{i+mt}^{l}$, hence by $(5.2)$ we have (remember $G: = \Omega^{d+1}\circ \mathcal N$ and $d\ge 0$) \begin{align*} G(S_i^l) = S_{i+1+(m-1)t}^{l}, \ \mbox{if} \ d = 2m-1\end{align*} and \begin{align*} G(S_i^l) = S_{i+l+1+(m-1)t}^{t-l}, \ \mbox{if} \ d = 2m.\end{align*} Thus, by induction we have \begin{align} G^{m'}(S_i^l) = S_{i+m'(1+(m-1)t)}^{l} = S_{i+m'd(t)}^{l}, \ \mbox{if} \ d = 2m-1;\end{align} and \begin{align} G^{2m'}(S_i^l) = S_{i+m'(2+(2m-1)t)}^{l} = S_{i+m'(2d(t))}^{l}, \ \mbox{if} \ d = 2m, \end{align} and \begin{align} G^{2m'+1}(S_i^l) = S_{i+(m'+1)(2d(t))+l-1-mt}^{t-l}, \ \mbox{if} \ d = 2m, \end{align} for $m'\ge 0$.
\subsection{} If $d= 2m-1\ge1$, then by $(5.4)$ we see that $o(G_{S_i^l}) = N$, where $N$ is as given in $(5.3)$, i.e. $N$ is the minimal positive integer such that $n\mid N(1+(m-1)t)$. It follows from Corollary 4.3 and $(5.4)$ that we have
\vskip10pt
\begin{lem} Let $d = 2m-1\ge 1$. Then $M$ is a minimal $d$-th CY $\Lambda$-module if and only if $M$ is isomorphic to one of the following
\begin{align} S_i^l\oplus S_{i+d(t)}^l\oplus S_{i+2d(t)}^l\oplus \cdots \oplus S_{i + (N-1)d(t)}^l, \ \ 1\le l\le t-1, \ \ i\in\Bbb Z/n\Bbb Z.\end{align}
\vskip10pt In particular, all the minimal $d$-th CY modules have the same number $N = N(d, n, t)$ of indecomposable direct summands. \end{lem}
\vskip10pt
\subsection{} If $d = 2m\ge 0$ and $t$ is odd, then by $(5.6)$ we see $G^{2m'+1}(S_i^l)\ne S_i^l$ for any $i, l$ (since $t-l\ne l$). Note that in this case $(d-1)t$ is odd. It follows from $(5.5)$ that $o(G_{S_i^l}) = 2N$, where $N$ is as given in $(5.3)$, i.e. $N$ is the minimal positive integer such that $n\mid N(2d(t))$. It follows from Corollary 4.3, $(5.5)$ and $(5.6)$ that we have
\vskip10pt
\begin{lem} Let $t\ge 3$ be an odd integer and $d = 2m\ge 0$. Then $M$ is a minimal $d$-th CY $\Lambda$-module if and only if $M$ is isomorphic to one of the following \begin{align} S_i^l\oplus S_{i+l' + 2d(t)}^{t-l} \oplus S_{i+2d(t)}^l\oplus S_{i+l'+ 4d(t)}^{t-l} \oplus \cdots \oplus S_{i + 2d(t)(N-1)}^l\oplus S_{i + l'+ 2d(t)N}^{t-l}, \end{align} where $l': = l-1-mt,$ \ $1\le l\le t-1$ \ and \ $i\in\Bbb Z/n\Bbb Z$.
In particular, any minimal $d$-th CY modules has $2N = 2N(d, n, t)$ indecomposable direct summands.\end{lem}
\vskip10pt
\subsection{} Let $d = 2m\ge 0$ and $t=2s$. Then $d(t) = 1+ (2m-1)s\in\Bbb Z$.
First, we consider $o(G_{S^s_i})$. In this case $(5.5)$ and $(5.6)$ can be written in a unified way: \begin{align} G^{m'}(S_i^s) = S_{i+m'd(t)}^{s}, \ m'\ge 0, \ \mbox{if} \ d = 2m, \ t=2s.\end{align} So we have $o(G_{S_i^s}) = N$, where $N$ is as given in $(5.3)$, i.e. $N$ is the minimal positive integer such that $n\mid N(1+(2m-1)s)$.
\vskip10pt
Now, we consider $o(G_{S^l_i})$ with $l\ne s, \ 1\le l\le t-1$. In this case $(5.5)$ and $(5.6)$ are written respectively as: \begin{align} G^{2m'}(S_i^l) = S_{i+2m'd(t)}^{l}, \ \mbox{if} \ d = 2m, \ t=2s, \ l\ne s,\end{align} and \begin{align} G^{2m'+1}(S_i^l) = S_{i+(2m'+1)d(t)+l-s}^{t-l}, \ \mbox{if} \ d = 2m, \ t=2s, \ l\ne s\end{align} for $m'\ge 0.$ Since $l\ne t-l$ for $l\ne s$, it follows that $G^{2m'+1}(S_i^l) \ne S_i^l$. So by $(5.10)$ we see $o(G_{S_i^l}) = 2N',$ where $N'$ is the minimal positive integer such that $n\mid 2N'd(t)$. In order to determine $N'$, we divided into two cases.
{\em Case} 1. \ If $N = N(d, n, t) = N(2m, n, 2s)$ is even, then $o(G_{S_i^l}) = N$ for any $1\le l\le t-1$. It follows from Corollary 4.3, $(5.9)$, $(5.10)$, and $(5.11)$ that we have the following (note that in this case $(5.9)$ is exactly $(5.10)$ together with $(5.11)$, by taking $l=s$)
\vskip10pt
\begin{lem} Let $t = 2s$ and $d = 2m\ge 0$. Assume that $N = N(d, n, t)$ is even. Then $M$ is a minimal $d$-th CY $\Lambda$-module if and only if $M$ is isomorphic to one of the following \begin{align} S_i^l\oplus S_{i+d(t)+l-s}^{t-l} \oplus S_{i+2d(t)}^l\oplus S_{i+3d(t)+l-s}^{t-l}\oplus \cdots \oplus S_{i + (N-2)d(t)}^l\oplus S_{i + (N-1)d(t)+l-s}^{t-l}\end{align} where $1\le l\le t-1, \ i\in\Bbb Z/n\Bbb Z$.
\vskip10pt In particular, all the minimal $d$-th CY modules have the same number $N$ of indecomposable direct summands. \end{lem}
\vskip10pt
It remains to deal with
{\em Case} 2. Let $N = N(d, n, t) = N(2m, n, 2s)$ be odd. Since by definition $N$ is the minimal positive integer such that $n\mid Nd(t)$, it follows that $N < o(G_{S_i^l}) = 2N' \le 2N.$ It is easy to see $N'=N$: otherwise $1\le 2N' - N\le N-1$ and $n\mid (2N' - N)d(t)$, which contradicts the minimality of $N$. It follows from Corollary 4.3, $(5.9)$, $(5.10)$, and $(5.11)$ that we have
\vskip10pt
\begin{lem} Let $t = 2s$ and $d = 2m\ge 0$. Assume that $N=N(d, n, t)$ is odd. Then $M$ is a minimal $d$-th CY $\Lambda$-module if and only if $M$ is isomorphic to one of the following
\begin{align} S_i^s\oplus S_{i+d(t)}^{s} \oplus S_{i+2d(t)}^s\oplus \cdots \oplus S_{i + (N-1)d(t)}^{s}\end{align} where $i\in\Bbb Z_n$, and \begin{align}\begin{matrix} S_i^l \oplus & S_{i+d(t)+l-s}^{t-l} \oplus & S_{i+2d(t)}^l \oplus & S_{i+3d(t)+l-s}^{t-l} \oplus \cdots \oplus & S_{i + (N-1)d(t)}^{l}\\ \oplus S_{i+l-s}^{t-l} \oplus & S_{i+d(t)}^{l} \oplus & S_{i+2d(t)+l-s}^{t-l} \oplus & S_{i+3d(t)}^{l} \oplus \cdots \oplus & S_{i + (N-1)d(t)+l-s}^{t-l}\end{matrix}\end{align} where $l\ne s$, \ $1\le l\le t-1$ and $i\in\Bbb Z_n$.
\vskip10pt In particular, all the minimal $d$-th CY modules have either $N$, or $2N$ indecomposable direct summands. \end{lem}
\vskip10pt
\subsection{} By Lemmas 5.1-5.4 all the minimal $d$-th CY modules of self-injective Nakayama algebras have been classified, where $d$ is any given integer. The main result of this section is as follows.
\vskip10pt
\begin{thm} For any $n\ge 1, \ t\ge 2, \ d\ge 0$, let $N=N(d, n, t)$ be as in $(5.3)$. Then $M$ is a minimal $d$-th CY \ $\Lambda$-module if and only if $M$ is isomorphic to one of the following
\vskip10pt
$(i)$ \ \ The modules in $(5.7)$, when $d=2m-1$;
$(ii)$ \ \ The modules in $(5.8)$, when $d=2m$ and $t$ is odd;
$(iii)$ \ \ The modules in $(5.12)$, when $d=2m$, $t=2s$, and $N(d, n, t)$ is even;
$(iv)$ \ \ The modules in $(5.13)-(5.14)$, when $d=2m$, $t=2s$, and $N(d, n, t)$ is odd.
\vskip10pt
In particular, any minimal $d$-th CY $\Lambda$-module has either $N$ or $2N$ indecomposable direct summands; and
\begin{align}\operatorname{min} \{d\ge 0 \ |\ N = c(M), \mbox{or} \ 2N = c(M) \} \ \le \operatorname{CYdim}(M) < o([1]_M) \le 2n \end{align} for any minimal Calabi-Yau $\Lambda$-modules $M$, where $c(M)$ is the number of indecomposable direct summands of $M$. \end{thm} \noindent {\bf Proof.}\ \ By Lemmas 5.1-5.4 the CY dimension $d$ of any minimal Calabi-Yau module $M$ satisfies $N(d, n, t) = c(M) \ \mbox{or} \ \ 2N(d, n, t) = c(M).$
$\blacksquare$
\vskip10pt
\begin{rem} The modules in $(5.7)-(5.8)$ and $(5.12)-(5.14)$ have overlaps. This is because $d$ is not uniquely determined by a minimal $d$-th CY module. A general formula of the CY dimensions of the minimal Calabi-Yau modules seems to be difficult to obtain.
Note that the inequality on the left hand side in $(5.15)$ can not be an equality in general. For example, take $n = 2, \ t=4, \ m = 2, \ d=2m-1=3$. Then $d(t) = 5$, \ $N = N(3, 2, 4) = 2$, and $S_i^l\oplus S_{i+1}^l, \ 1\le l\le 3$, are all the minimal $3$-th CY modules of $CY$ dimension $0$ if $l = 2$, and $1$ if $l=1, \ 3$. However, the left hand side in $(5.15)$ is $0$ since $N(0, 2, 4) = 2$. \end{rem}
\vskip10pt
\section{\bf Self-injective Nakayama algebras with indecomposable Calabi-Yau modules}
In this section we determine all the self-injective Nakayama algebras $\Lambda = \Lambda(n, t), \ n\ge 1, \ t\ge 2$, which admit indecomposable Calabi-Yau modules.
\vskip10pt
Note that Erdmann and Skowro\'nski have proved in $\S 2$ of [ES] that self-injective algebras $A$ such that $A\underline {\mbox{-mod}}$ is Calabi-Yau of CY dimension $0$ (resp. $1$) are the algebras Morita equivalent to $\Lambda(n, 2)$ for some $n\ge 1$ (resp. $\Lambda(1, t)$ for some $t\ge 3$). So we assume that $t\ge 3$.
\vskip10pt
\begin{thm} \ \ Let $t\ge 3$. Then $\Lambda$ has an indecomposable Calabi-Yau module if and only if $n$ and $t$ satisfy one of the following conditions
$(i)$ \ $g.c.d. \ (n, \ t) = 1$. This is exactly the case where $\Lambda\underline {\mbox{-mod}}$ is a Calabi-Yau category. In this case we have $\operatorname{CYdim}(\Lambda) = 2m-1$, where $m$ is the minimal positive integer such that $n\mid (m-1)t + 1$.
\vskip10pt
$(ii)$ \ $g.c.d. \ (n,\ t)\ne 1$, \ $t = 2s$, \ and \ $g.c.d. \ (n,\ s) = 1$. This is exactly the case where $\Lambda\underline {\mbox{-mod}}$ is not a Calabi-Yau category but admits indecomposable Calabi-Yau modules.
In this case, we have $g.c.d.\ (n,\ t) = 2$ and
$(a)$\ \ $S_i^s, \ i\in \Bbb Z/n\Bbb Z,$ \ are all the indecomposable Calabi-Yau modules;
$(b)$ \ \ $S_i^l\oplus S_{i+l-s}^{t-l}, \ 1\le l\le s-1, \ i\in\Bbb Z/n\Bbb Z,$ \ are all the decomposable minimal $2m$-th CY modules, where $m$ is the minimal non-negative integer such that $n\mid (2m-1)s + 1$;
$(c)$ \ All of these modules in $(a)$ and $(b)$ have the same CY dimension $2m$. \end{thm}
\vskip10pt
{\bf Remark.} Bialkowski and Skowro\'nski [BS] have classified representation-finite self-injective algebras whose stable categories are Calabi-Yau. This includes the assertion $(i)$ of Theorem 6.1.
\vskip10pt
\noindent {\bf Proof.}\ \ If $\Lambda$ has an indecomposable $d$-th CY module $S_i^l$, then $\mathcal N(S^l_i) \cong \Omega^{-(d+1)}(S^l_i)$. By $(5.2)$ and $(5.1)$ we have \begin{align}1-t\equiv -mt \ (\mbox{mod} \ n), \ \mbox{if} \ d+1 =2m,\end{align} or \begin{align} t= t-l, \ 1-t\equiv l-mt \ (\mbox{mod} \ n), \ \mbox{if} \ d+1 =2m-1.\end{align} In the first case we get $g.c.d.\ (n, \ t) = 1.$ In the second case we get $t=2s, \ l=s$ and $g.c.d.\ (n, \ s) = 1.$ Excluding the overlap situations we conclude that either $g.c.d.\ (n, \ t) = 1;$ \ or \ $g.c.d.\ (n, \ t) \ne 1, \ t= 2s, \ g.c.d.\ (n, \ s) = 1$.
\vskip10pt
Assume that $g.c.d. \ (n, \ t) = 1$. Then there exists an integer $m$ such that $n\mid (m-1)t+1$. We chose a positive (otherwise, add $n(-m+2)t$), and minimal $m$. Set $d: = 2m-1$. Then the same computation shows that every indecomposable is a $d$-th CY module. We claim that $\Lambda\underline {\mbox{-mod}}$ is a Calabi-Yau category. For this, it remains to show $\mathcal N(f) = \Omega^{-(d+1)}(f) =\Omega^{-2m}(f)$ for any morphism $f$ between indecomposables of $\Lambda\underline {\mbox{-mod}}$. Since $n\mid (m-1)t +1$, it follows from $(**)$ in $\S 5$ that \begin{align}\mathcal N(\sigma_i^l) = \Omega^{-2m}(\sigma^l_i), \ \ \mathcal N(p_i^l) = \Omega^{-2m}(p^l_i).\end{align} Since $\Lambda$ is representation-finite, it follows that $f$ is a $k$-combination of compositions of irreducible maps $\sigma^l_i$'s and $p^l_i$'s, hence $\mathcal N(f) = \Omega^{-2m}(f)$ by $(6.3)$. We stress here that, this argument relies on the fact that all the isomorphisms $\theta_i^l$ in 5.2 depend only on $i$ and $l$, which means that they do not depend on whatever the maps $\sigma_i^l$ or $p_i^l$ are. Otherwise we can not take them as identities, and then we can not get the naturality for the Calabi-Yau category of this case.
This proves the claim, hence $\Lambda\underline {\mbox{-mod}}$ is a Calabi-Yau category of CY dimension $D$, with $0\le D \le d = 2m-1$. We claim $D=d$. In fact, since every indecomposable is a $D$-th CY module, and since we have $l$ such that $t\ne t-l$, it follows from $(6.1)$ and $(6.2)$ that $D = 2m'-1$ with $n\mid (m'-1)t+1$, hence $D = d$, by the minimality of $m$.
The argument above also proves that if $\Lambda\underline {\mbox{-mod}}$ is a Calabi-Yau category then $g.c.d. \ (n, \ t) = 1$.
\vskip10pt
Assume that $g.c.d. \ (n,\ t)\ne 1$, \ $t = 2s$, \ and $g.c.d. \ (n, \ s) = 1$. Then there exists an integer $m'$ such that $n\mid (m'-1)s+1$. We choose a positive $m'$. Since $g.c.d.\ (n, \ t) \ne 1$, it follows that $m'$ is even, say $m' = 2m$ with $m\ge 1$. Let $m$ be the the minimal positive integer such that $n\mid (2m-1)s+1$, and set $d: = 2m$. Then the same computation shows that \ $S_i^s, \ i\in \Bbb Z/n\Bbb Z,$ \ are all the indecomposable Calabi-Yau modules. This proves $(a)$.
By applying Lemma 5.4 to $d$ given above (note that the corresponding $N = N(n, d, t) = 1$ in this case), we know that $S_i^l\oplus S_{i+l-s}^{t-l}, \ 1\le l\le t-1, \ l\ne s, \ i\in\Bbb Z/n\Bbb Z,$ \ are all the decomposable minimal $2m$-th CY modules. By symmetry one can consider $1\le l\le s-1:$ if $l>s$ then one can replace $l$ by $t-l$, since $i = (i+l-s) + (t-l)-s.$ This proves $(b)$.
It remains to prove $(c)$. Let $\operatorname{CYdim}(S^s_i) = d'$. Since $g.c.d. \ (n,\ t)\ne 1$, it follows that $d'$ has to be an even integer $2m'\ge 0$ with $n\mid (2m'-1)s +1$. it follows that $d' = d$, by the minimality of $m$. Let $\operatorname{CYdim}(S_i^l\oplus S_{i+l-s}^{t-l}) = d'$. Then we have $\mathcal N(S^l_i) = \Omega^{-(d'+1)}(S_{i+l-s}^{t-l})$. Since $l\ne s$ it follows from $(5.2)$ and $(5.1)$ that $d'$ has to be $2m'\ge 0$ with $n\mid (2m'-1)s+1$. Again by the minimality of $m$ we have $d' = d$. This completes the proof.
$\blacksquare$
\vskip10pt
\begin{rem} $(i)$ $\operatorname{CYdim}(X)$ usually differs from $\operatorname{CYdim}(\mathcal A)$ in a Calabi-Yau category $\mathcal A$.
For example, if $n=3, \ t=4$, then $o([1]) = 6$, $\operatorname{CYdim}(\Lambda) = 5$, while $\operatorname{CYdim}(S_i^2) = 2$ and $o([1]_{S_i^2}) = 3, \ \forall \ i\in\Bbb Z/3\Bbb Z$.
However, for $t\ge 3$ and \ $g.c.d.\ (n,\ t) = 1$, if $X$ is indecomposable and $\operatorname{CYdim}(X)$ is odd then $\operatorname{CYdim}(X) = \operatorname{CYdim}(\Lambda)$. In fact, since $o([1]) < \infty$ it follows that $\operatorname{CYdim}(X) = 2m'-1\ge 1$, and $n\mid 1+(m'-1)t$. By Theorem 6.1$(i)$ $\operatorname{CYdim}(\Lambda) = 2m-1$, where $m$ is the minimal positive integer such that $n\mid 1+(m-1)t$. It follows that $m'\ge m$, hence $\operatorname{CYdim}(X) \ge \operatorname{CYdim}(\Lambda)$. On the other hand we have $\operatorname{CYdim}(X) \le \operatorname{CYdim}(\Lambda)$ by definition.
$(ii)$ \ Consider the algebra $A(t): = kA_\infty ^\infty/J^t$ where $A_\infty^\infty$ is the infinite quiver $$ \cdots \longrightarrow \bullet \longrightarrow \bullet \longrightarrow \bullet \longrightarrow \bullet \longrightarrow \bullet\longrightarrow \cdots.$$ Then $A(t)\underline {\mbox{-mod}}$ has a Serre functor, and there is a natural covering functor $A(t)\underline {\mbox{-mod}}$ $\longrightarrow \Lambda(n, t)\underline {\mbox{-mod}}$ ([Gab], 2.8). But one can prove that in any case $A(t)\underline {\mbox{-mod}}$ is not a Calabi-Yau category. \end{rem}
\vskip10pt
{\bf Acknowledgements.} This work is done during a visit of the second named author at Universit\'e de Montpellier 2, supported by the CNRS of France. He thanks the first named author and his group for the warm hospitality, the D\'epartement de Math\'ematiques of Universit\'e de Montpellier 2 for the working facilities, and the CNRS for the support. We thank Bernhard Keller for helpful conversations.
\vskip30pt
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title[Some cohomologically similar manifolds and special generic maps]{New families of manifolds with similar cohomology rings admitting special generic maps} \author{Naoki Kitazawa} \keywords{Special generic maps. (Co)homology rings. Closed and simply-connected manifolds. \\ \indent {\it \textup{2020} Mathematics Subject Classification}: Primary~57R45. Secondary~57R19.} \address{Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku Fukuoka 819-0395, Japan\\
TEL (Office): +81-92-802-4402 \\
FAX (Office): +81-92-802-4405 \\ } \email{[email protected]} \urladdr{https://naokikitazawa.github.io/NaokiKitazawa.html}
\begin{abstract}
As Reeb's theorem shows, Morse functions with exactly two singular points on closed manifolds are very simple and important. They characterize spheres whose dimensions are not $4$ topologically and the $4$-dimensional unit sphere.
{\it Special generic} maps are generalized versions of these maps. Canonical projections of unit spheres are special generic. Studies of Saeki and Sakuma since the 1990s, followed by Nishioka and Wrazidlo, show that the differentiable structures of the spheres and the homology groups of the manifolds (in several classes) are restricted. We see special generic maps are attractive.
Our paper studies the cohomology rings of manifolds admitting such maps. As our new result, we find a new family of manifolds whose cohomology rings are similar and find that the (non-)existence of special generic maps are closely related to the topologies. More explicitly, we have previously found related families and our new manifolds add to these discoveries.
\end{abstract}
\maketitle \section{Introduction.} \label{sec:1}
According to the so-called Reeb's theorem, Morse functions with exactly two singular points on closed manifolds characterize spheres whose dimensions are not $4$ topologically and the $4$-dimensional unit sphere.
{\it Special generic} maps are, in short, their higher dimensional versions.
We first define a {\it special} generic map between smooth manifolds with no boundaries.
Before that, we introduce fundamental and important terminologies, notions and notation.
Hereafter, for an integer $k>0$, ${\mathbb{R}}^k$ denotes the $k$-dimensional Euclidean space, which is a smooth manifold canonically and also a Riemannian manifold endowed with the standard Euclidean metric.
For ${\mathbb{R}}^1$, $\mathbb{R}$ is used in a natural way. This is also a commutative ring and $\mathbb{Z} \subset \mathbb{R}$ denotes the subring of all integers.
$||x|| \geq 0$ is for the distance between $x$ and the origin $0$ in ${\mathbb{R}}^k$.
$S^k\ {\rm (}D^{k+1}{\rm )}:=\{x \in {\mathbb{R}}^{k+1} \mid ||x||=1\ {\rm (}resp.\ ||x|| \leq 1{\rm )}\}$ is the $k$-dimensional unit sphere (resp. ($k+1$)-dimensional unit disk). A canonical projection of the unit sphere $S^k \subset {\mathbb{R}}^{k+1}$ into ${\mathbb{R}}^{k^{\prime}}$ with $k \geq k^{\prime}$ is defined as a map mapping $(x_1,x_2) \in {\mathbb{R}}^{k^{\prime}} \times {\mathbb{R}}^{k+1-k^{\prime}}$ to $x_1 \in {\mathbb{R}}^{k^{\prime}}$. This is also a simplest special generic map.
Hereafter, $\dim X$ is the dimension of a topological space regarded as a CW complex, which we can define uniquely. A manifold is always regarded as a CW complex. A smooth manifold always has the structure of a so-called {\it PL} manifold in a canonical way and this is a polyhedron.
For a smooth map $c:X \rightarrow Y$ between smooth manifold, a point $p \in X$ is a {\it singular} point of $c$ if the rank of the differential ${dc}_p$ here is smaller than $\min{\dim X,\dim Y}$. $S(c)$ denotes the set of all singular points of $c$ and we call this the {\it singular set} of $c$.
A {\it diffeomorphism} means a homeomorphism which is a smooth map with no singular points.
Two smooth manifolds are defined to be {\it diffeomorphic} if a diffeomorphism between the two manifolds exists.
A smooth manifold homeomorphic to a sphere is said to be a {\it homotopy sphere}. A homotopy sphere is a {\it standard} (an {\it exotic}) sphere if it is diffeomorphic to a unit sphere (resp. not diffeomorphic to any unit sphere).
\begin{Def}
A smooth map $c:X \rightarrow Y$ between manifolds with no boundaries satisfying $\dim X \geq \dim Y$ is said to be {\it special generic} if at each singular point $p$, there exists suitable local coordinates and $f$ is locally
represented by $(x_1,\cdots,x_{\dim X}) \rightarrow (x_1,\cdots,x_{\dim Y-1},{\Sigma}_{j=1}^{\dim X-\dim Y+1} {x_{\dim Y+j-1}}^2)$.
\end{Def}
Morse functions in the Reeb's theorem are special generic. As a kind of exercises on smooth manifolds and maps and the theory of Morse functions, we can see that canonical projections of unit spheres are special generic.
\begin{Prop}
The singular set $S(c)$ of the special generic map $c:X \rightarrow Y$ is a {\rm (}$\dim Y-1${\rm )}-dimensional smooth closed submanifold of $X$ and has no boundary. Furtheremore, the restriction $c {\mid}_{S(c)}$ is a smooth immersion,
\end{Prop}
Other properties are presented in the next section. Studies of Saeki and Sakuma (\cite{saeki1,saeki2,saekisakuma,saekisakuma2,sakuma}) since the 1990s,
followed by Nishioka (\cite{nishioka}) and Wrazidlo (\cite{wrazidlo,wrazidlo2,wrazidlo3}), show that the differentiable structures of the spheres and the homology groups of the manifolds are restricted in several cases. This explicitly shows that special generic maps are attractive. Some of these results are also presented later.
Our paper studies the cohomology rings of manifolds admitting such maps. We also introduce terminologies and notation on homology groups, cohomology rings, characteristic classes of manifolds such as {\it Stiefel-Whitney classes} and {\it Pontrjagin classes} and other notions from algebraic topology and differential topology in the next section. We also expect that readers have some knowledge on them. \cite{hatcher} and \cite{milnorstasheff} explain about them systematically.
\begin{MainThm}
\label{mthm:1}
Let $m \geq 8$ be an arbitrary integer. Let $m^{\prime} \geq 6$ be another integer satisfying the condition $m-m^{\prime} \geq 2$. Assume also the existence of some integer $a$. Furthermore, the following conditions hold.
\begin{itemize}
\item $a=2$ or $a=m^{\prime}-3$.
\item $a+m^{\prime}-1 \leq m$
\item $2m^{\prime}-m-a=1$.
\end{itemize}
Let $G$ be an arbitrary finite commutative group which is not the trivial group.
Then we have a closed and simply-connected manifold $M$ enjoying the following properties.
\begin{enumerate}
\item \label{mthm:1.1} There exists an $m^{\prime}$-dimensional closed and simply-connected manifold $M^{\prime}$ and $M$ is diffeomorphic to $M^{\prime} \times S^{m-m^{\prime}}$.
\item \label{mthm:1.2} The $j$-th homology group of $M^{\prime}$ whose coefficient ring is $\mathbb{Z}$ is isomorphic to $G$ if $j=2,m^{\prime}-3$, $\mathbb{Z}$ if $j=0,m^{\prime}$, and the trivial group, otherwise.
\item \label{mthm:1.3} $M$ admits a special generic map into ${\mathbb{R}}^n$ for $m^{\prime}<n\leq m$ whereas it admits no special generic maps into ${\mathbb{R}}^n$ for $1 \leq n \leq 4$ and $n=m^{\prime}$. Furthermore, the special generic maps can be obtained as ones whose restrictions to the singular sets are embeddings and {\rm product organized} ones, defined in Definition \ref{def:2}, later.
\end{enumerate}
\end{MainThm}
\begin{MainThm}
\label{mthm:2}
Let $\{G_j\}_{j=1}^5$ be a sequence of free commutative groups such that $G_{j}$ and $G_{6-j}$ are isomorphic for $1 \leq j \leq 2$, that $G_{1}$ is not the trivial group and that the rank of $G_{3}$ is even. Let $G_{{\rm F},1}$ be an arbitrary finite commutative group and $G_{{\rm F},2}$ a finite commutative group which is not the trivial group.
Let $\{M_i\}_{i=1}^2$ be a pair of $8$-dimensional closed and simply-connected manifolds enjoying the following properties.
\begin{enumerate}
\item The $j$-th homology groups of these manifolds whose coefficient rings are $\mathbb{Z}$ are mutually isomorphic to $G_{1} \oplus G_{{\rm F},1}$ for $j=2$, the direct sum of $G_{2} \oplus G_{{\rm F},2}$ for $j=3$, the direct sum of $G_{3} \oplus G_{{\rm F},2}$ for $j=4$, the direct sum of $G_{4} \oplus G_{{\rm F},1}$ for $j=5$, and $G_{5}$ for $j=6$.
\item Consider the restrictions to the subgroups generated by all elements whose orders are infinite for these manifolds. They are regarded as subalgebras and isomorphic to the cohomology rings of manifolds represented as connected sums of finitely many manifolds diffeomorphic to the products of two homotopy spheres where the coefficient ring is $\mathbb{Z}$.
\item For an arbitrary positive integer $j>0$, the $j$-th Stiefel-Whitney classes and the $j$-th Pontrjagin classes of these manifolds are the zero elements of the $j$-th cohomology groups whose coefficient rings are $\mathbb{Z}/2\mathbb{Z}$, the ring of order $2$, and the $4j$-th cohomology groups whose coefficient rings are $\mathbb{Z}$, respectively.
\item $M_i$ does not admit special generic maps into ${\mathbb{R}}^n$ for $1 \leq n < i+5$ whereas it admits one into ${\mathbb{R}}^n$ for $i+5 \leq n \leq 8$.
Furthermore, the special generic maps can be obtained as ones whose restrictions to the singular sets are embeddings and {\rm product organized} ones, defined in Definition \ref{def:2}, later.
\end{enumerate}
\end{MainThm}
Main Theorem \ref{mthm:2} is regarded as a variant of Main Theorem 4 of \cite{kitazawa2}.
In the next section, we present additional fundamental properties of special generic maps. The third section is devoted to our proof of Main Theorems. We also present existing studies on the the homology groups and the cohomology rings of the manifolds. \\
\ \\
{\bf Conflict of Interest.} \\
The author was a member of the project supported by JSPS KAKENHI Grant Number JP17H06128 "Innovative research of geometric topology and singularities of differentiable mappings"
(Principal investigator: Osamu Saeki) and
is also a member of the project JSPS KAKENHI Grant Number JP22K18267 "Visualizing twists in data through monodromy" (Principal Investigator: Osamu Saeki). The present study is also supported by these projects. \\
\ \\
{\bf Data availability.} \\
Data essentially supporting the present study are all in the present paper.
\section{Fundamental properties and existing studies on special generic maps and the manifolds.}
First, we review elementary algebraic topology and other notions, terminologies and notation we need.
Let $(X,X^{\prime})$ be a pair of topological spaces where these two spaces satisfy the relation $X^{\prime} \subset X$ and may be empty. The ({\it $k$-th}) {\it homology group} of this pair $(X,X^{\prime})$ whose coefficient ring is $A$ is denoted by $H_{k}(X,X^{\prime};A)$. In the case $X^{\prime}$ is empty, we may omit ",$X^{\prime}$" in the notation and we call the homology group (cohomology group) of $(X,X^{\prime})$ the {\it homology group} (resp. {\it cohomology group}) of $X$ in general.
For a topological space $X$ of suitable classes, the {\it $k$-th homotopy group} of $X$ is denoted by ${\pi}_k(X)$. The class of arcwise-connected cell complex, containing the class of arcwise-connected CW complexes for example, is one of such classes.
Let $(X,X^{\prime})$ and $(Y,Y^{\prime})$ be pairs of topological spaces where these spaces satisfy the relations $X^{\prime} \subset X$ and $Y^{\prime} \subset Y$ and may be empty. Given a continuous map $c:X \rightarrow Y$ satisfying $c(X^{\prime}) \subset Y^{\prime}$, then $c_{\ast}:H_{k}(X,X^{\prime};A) \rightarrow H_{k}(Y,Y^{\prime};A)$, $c^{\ast}:H^{k}(Y,Y^{\prime};A) \rightarrow H^{k}(X,X^{\prime};A)$ and $c_{\ast}:{\pi}_k(X) \rightarrow {\pi}_k(Y)$ denote the canonically induced homomorphisms where in considering homotopy groups the spaces satisfy the suitable conditions for example.
Let $X$ be a topological space and $A$ a commutative ring. Let $H^{\ast}(X;A)$ denote the direct sum ${\oplus}_{j=0}^{\infty} H^j(X;A)$ for all integers greater than or equal to $0$. For any sequence $\{a_j\}_{j=1}^l \subset H^{\ast}(X;A)$ of elements of length $l>0$, we can define the {\it cup product} ${\cup}_{j=1}^l a_j \in H^{\ast}(X;A)$ for general $l$. $a_1 \cup a_2$ is also used in the case $l=2$. We can have the structure of a graded commutative algebra $H^{\ast}(X;A)$. This is the {\it cohomology ring} of $X$ whose {\it coefficient ring} is $A$.
The {\it fundamental class} of a compact, connected and oriented manifold $Y$ is defined as the canonically and uniquely defined element of the ($\dim Y$)-th homology group, which is also a generator of the group $H_{\dim Y}(Y,\partial Y;\mathbb{Z})$, isomorphic to the group $\mathbb{Z}$. Let $i_{Y,X}:Y \rightarrow X$ be a smooth immersion satisfying $i_{Y,X}(\partial Y) \subset \partial X$ and $i_{Y,X}({\rm Int}\ Y) \subset {\rm Int}\ X$. In other words, $Y$ is smoothly and {\it properly} immersed or embedded into $X$.
Note that in considering other categories such as the PL category and the topology category, we consider suitable maps playing roles as the smooth immersions play. If for an element $h \in H_j(X,\partial X;A)$, the value of the homomorphism ${i_{Y,X}}_{\ast}$ induced by the map $i_{Y,X}:Y \rightarrow X$ at the fundamental class of $Y$ is $h$, then $h$ is {\it represented} by $Y$.
We also need several fundamental methods and theorems. For example, homology exact sequences for pairs of topological spaces, Mayer-Vietoris sequences, Poincar\'e duality (theorem) for compact and connected (orientable) manifolds, universal coefficient theorem and K\"unneth theorem for the products of topological spaces.
For such explanations, see \cite{hatcher} again for example.
The {\it diffeomorphism group} of a smooth manifold is the group of all diffeomorphisms from the manifold to itself endowed with the so-called {\it Whitney $C^{\infty}$ topology}.
A {\it smooth} bundle means a bundle whose fiber is a smooth manifold whose structure group is the diffeomorphism group. A {\it linear} bundle means a bundle whose fiber is a Euclidean space, unit sphere, or a unit disk and whose structure group consists of linear transformations. Note that a linear transformation here is defined in a matural and canonical way.
For general theory of bundles, see \cite{steenrod}. \cite{milnorstasheff} is for linear bundles. \begin{Prop}
\label{prop:2}
For a special generic map on an $m$-dimensional closed and connected manifold $M$ into ${\mathbb{R}}^n$, we have the following properties.
\begin{enumerate}
\item \label{prop:2.1}
There exists an $n$-dimensional compact and connected smooth manifold $W_f$ such that a smooth surjection $q_f:M \rightarrow W_f$ and a smooth immersion $\bar{f}:W_f \rightarrow {\mathbb{R}}^n$ enjoying the relation $f=\bar{f} \circ q_f$ exist.
\item \label{prop:2.2}
There exists a small collar neighborhood $N(\partial W_f)$ of the boundary $\partial W_f \subset W_f$ and the composition of the restriction of $f$ to the preimage with the canonical projection to $\partial W_f$ gives a linear bundle whose fiber is the {\rm (}$m-n+1${\rm )}-dimensional unit disk $D^{m-n+1}$.
\item \label{prop:2.3}
The restriction of $f$ to the preimage of $W_f-{\rm Int}\ N(\partial W_f)$ gives a smooth bundle whose fiber is an {\rm (}$m-n${\rm )}-dimensional standard sphere. It is regarded as a linear bundle in specific cases. For example, in the case $m-n=0,1,2,3$, the bundle is regarded as a linear bundle.
\end{enumerate} \end{Prop}
\begin{Prop}
\label{prop:3}
In Proposition \ref{prop:2}, we can have an {\rm (}m+1{\rm )}-dimensional compact and connected topological {\rm (}PL{\rm )} manifold $W$ and a continuous {\rm (}resp. piesewise smooth{\rm )} surjection $r:W \rightarrow W_f$ enjoying the following properties where we abuse the notation.
\begin{enumerate}
\item $M$ is the boundary of $W$ and the restriction of $r$ to the boundary $M$ is $q_f$.
\item $r$ gives the structure of a bundle over $W_f$ whose fiber is homeomorphic {\rm (}resp. PL homeomorphic{\rm )} to the {\rm (}$m-n+1${\rm )}-dimensional unit disk $D^{m-n+1}$ and whose structure group consists of piesewise smooth homeomorphisms.
\item $W$ is decomposed into two {\rm (}m+1{\rm )}-dimensional topological {\rm (}resp. PL{\rm )} manifolds $W_1$ and $W_2$.
\item We can consider the composition of the restriction of $r$ to $W_1$ with a canonical projection to $\partial W_f$. This gives a linear bundle whose fiber is the {\rm (}$m-n+2${\rm )}-dimensional unit disk $D^{m-n+2}$. Furthermore, this bundle is realized as a bundle whose subbundle obtained by considering the boundary of the unit disk and a suitable smoothly embedded copy of the unit disk $D^{m-n+1}$ there as its fiber is the linear bundle of Proposition \ref{prop:2} {\rm (}\ref{prop:2.2}{\rm )}.
\item The restriction of $r$ to $W_2$ gives a bundle whose fiber is diffeomorphic to the {\rm (}$m-n+1${\rm )}-dimensional unit disk $D^{m-n+1}$ {\rm (}resp. and whose structure group consists of piesewise smooth homeomorphisms{\rm )} . Furthermore, this bundle is realized as a bundle whose subbundle obtained by considering the boundary of the unit disk as its fiber is the smooth bundle of Proposition \ref{prop:2} {\rm (}\ref{prop:2.3}{\rm )}.
\item In the case $m-n=0,1,2,3$ for example, $W$ can be chosen as a smooth one and we can also do in such a way that $r$ gives a linear bundle over $W_f$.
\end{enumerate} \end{Prop} \begin{Ex}
\label{ex:1}
Let $l>0$ be an integer. Let $m \geq n \geq 2$ be integers.
Other than canonical projections of units spheres, we present simplest special generic maps. We consider a connected sum of $l>0$ manifolds in the family $\{S^{n_j} \times S^{m-n_j}\}_{j=1}^l$ in the smooth category where $1 \leq n_j \leq n-1$ is an integer for $1 \leq j \leq l$. We have a special generic map $f:M \rightarrow {\mathbb{R}}^n$ on the resulting manifold $M$ such that in Proposition \ref{prop:2}, the following properties are enjoyed.
\begin{enumerate}
\label{ex:1.1}
\item $\bar{f}$ is an embedding.
\item
\label{ex:1.2}
$W_f$ is represented as a boundary connected sum of $l>0$ manifolds each of which is diffeomorphic to each of $\{S^{n_j} \times D^{n-n_j}\}_{j=1}^l$ in the family. The boundary connected sum is considered in the smooth category.
\item \label{ex:1.3}
The bundles in Proposition \ref{prop:2} (\ref{prop:2.2}) and (\ref{prop:2.3}) are trivial.
\end{enumerate} \end{Ex} We introduce a result of \cite{saeki1} related to Example \ref{ex:1}.
\begin{Thm}[\cite{saeki1}]
\label{thm:1}
An $m$-dimensional closed and connected manifold $M$ admits a special generic map into ${\mathbb{R}}^2$ if and only if either of the following holds.
\begin{enumerate}
\item $M$ is a homotopy sphere which is not an $4$-dimensional exotic sphere.
\item A manifold
represented as a connected sum of smooth manifolds considered in the smooth category where each of the manifolds here is represented as either of the following manifolds. \begin{enumerate}
\item The total space of a smooth bundle over $S^1$ whose fiber is a homotopy sphere for $m \neq 5$.
\item The total space of a smooth bundle over $S^1$ whose fiber is a standard sphere for $m=5$. \end{enumerate} \end{enumerate} Furthermore, for each manifold here, we can construct a special generic map so that the properties {\rm (}\ref{ex:1.1}{\rm )} and {\rm (}\ref{ex:1.2}{\rm )} in Example \ref{ex:1} are enjoyed with $n_j=1$. \end{Thm} \begin{Prop}
\label{prop:4}
Let $A$ be a commutative ring.
In Proposition \ref{prop:2}, the homomorphisms ${q_f}_{\ast}:H_j(M;A) \rightarrow H_j(W_f;A)$, ${q_f}^{\ast}:H^j(W_f;A) \rightarrow H^j(M;A)$ and ${q_f}_{\ast}:{\pi}_j(M) \rightarrow {\pi}_j(W_f)$ are isomorphisms for $0 \leq j \leq m-n$.
\end{Prop} We can show the following theorem by applying Proposition \ref{prop:4} with Proposition \ref{prop:2}, some fundamental theory on $3$-dimensional compact manifolds and some previously presented facts. For a proof, see the
original paper \cite{saeki1} for example. \begin{Thm}[\cite{saeki1}]
\label{thm:2}
Let $m>3$ be an arbitrary integer.
\begin{enumerate}
\item If an $m$-dimensional closed and simply-connected manifold admits a special generic map into ${\mathbb{R}}^3$, then it is either of the following.
\begin{enumerate}
\item A homotopy pshere which is not a $4$-dimensional exotic sphere.
\item A manifold represented as a connected sum of smooth manifolds considered in the smooth category where each of the manifolds here is represented as the total space of a smooth bundle over $S^2$ whose fiber is a homotopy sphere. Moreover, the homotopy sphere of the fiber is also not a $4$-dimensional exotic sphere.
\end{enumerate}
\item
Furthermore, for each manifold here admitting a special generic map into ${\mathbb{R}}^3$, we can construct a special generic map so that the properties {\rm (}\ref{ex:1.1}{\rm )} and {\rm (}\ref{ex:1.2}{\rm )} in Example \ref{ex:1} are enjoyed with $n_j=2$.
\end{enumerate} \end{Thm}
The following is shown in \cite{nishioka} to show Theorem \ref{thm:3} for $(m,n)=(5,4)$ by applying a classification result of $5$-dimensional closed and simply-connected manifolds of \cite{barden}, Propositon \ref{prop:4} and some additional arguments on homology groups. \begin{Prop}[\cite{nishioka}]
\label{prop:5}
Let $k \geq 3$ be an integer. For a $k$-dimensional compact and connected manifold $X$ such that $H_1(X;\mathbb{Z})$ is the trivial group, $X$ is orientable and the homology group $H_{j}(X;\mathbb{Z})$ is free for $j=k-2,k-1$.
\end{Prop}
\begin{Thm}[\cite{saeki1} for $(m,n)=(4,3),(5,3),(6,3)$,\cite{nishioka} for $(m,n)=(5,4)$, and \cite{kitazawa5} for $(m,n)=(6,4)$] \label{thm:3}
For integers $(m,n)=(4,3),(5,3),(5,4),(6,4)$, An $m$-dimensional closed and simply-connected manifold $M$ admits a special generic map into ${\mathbb{R}}^n$ if and only if $M$ is as follows.
\begin{enumerate}
\item A standard sphere.
\item A manifold represented as a connected sum of smooth manifolds considered in the smooth category where each of the manifolds here is as follows.
\begin{enumerate}
\item The total space of a linear bundle over $S^2$ whose fiber is the unit sphere $S^{m-2}$.
\item Only in the case $(m,n)=(6,4)$, a manifold diffeomorphic to $S^3 \times S^3$.
\end{enumerate}
\end{enumerate}
Furthermore, for each manifold here, we can construct a special generic map so that the properties {\rm (}\ref{ex:1.1}{\rm )} and {\rm (}\ref{ex:1.2}{\rm )} in Example \ref{ex:1} are enjoyed with $n_j=2$ unless $(m,n)=(6,4)$. In the case $(m,n)=(6,4)$, we can do similarly with $n_j=2,3$. \end{Thm} \begin{Prop}
\label{prop:6}
Let $A$ be a commutative ring.
In Proposition \ref{prop:2}, the cup product for a finite sequence $\{u_j\}_{j=1}^{l} \subset H^{\ast}(M;A)$ of length $l>0$ consisting of elements whose degrees are at most $m-n$ is the zero element if the sum of all $l$ degrees is greater than or equal to $n$. \end{Prop} We introduce a proof of Propositions \ref{prop:4} and \ref{prop:6} referring to \cite{kitazawa1}. For terminologies from the PL category and similar categories, see \cite{hudson} for example. \begin{proof}[A proof of Propositions \ref{prop:4} and \ref{prop:6}]
This is based on an argument of \cite{saekisuzuoka} for example. We abuse the notation of some Propositions such as Proposition \ref{prop:2} and \ref{prop:3}.
$W$ is an ($m+1$)-dimensional compact and connected (PL) manifold whose boundary is $M$ and collapses to $W_f$, which is an $n$-dimensional compact smooth manifold immersed smoothly into ${\mathbb{R}}^n$. This has the (simple) homotopy type of an ($n-1$)-dimensional polyhedron and more precisely, collapses to an ($n-1$)-dimensional polyhedron. We can see that for $0 \leq j < m+1-(n-1)-1=m-n+1$, the three kinds of homomorphisms in Proposition \ref{prop:4} are isomorphisms.
In Proposition \ref{prop:6}, the cup product for the finite sequence is regarded as the value of ${q_f}^{\ast}:H^{\ast}(W_f;A) \rightarrow H^{\ast}(M;A)$ at the cup product of a sequence of length $l$ where ${q_f}^{\ast}:H^{\ast}(W_f;A) \rightarrow H^{\ast}(M;A)$ is defined in a canonical way from ${q_f}^{\ast}:H^{i}(W_f;A) \rightarrow H^{i}(M;A)$. This sequence of elements of $H^{\ast}(W_f;A)$ is defined in a canonical and unique way by Proposition \ref{prop:4}. The cup product is the zero element of $H^{\ast}(W_f;A)$. This follows from the fact that $W_f$ has the homotopy type of an ($n-1$)-dimensional polyhedron.
This completes the proof.
\end{proof} Hereafter, in the present section, we introduce arguments in \cite{kitazawa7} and present some new arguments.
\begin{Prop}[Partially discussed in \cite{kitazawa7}]
\label{prop:7} Let $m \geq n \geq 1$ be integers. Let ${\bar{f}}_N:\bar{N} \rightarrow {\mathbb{R}}^n$ be a smooth immersion of an $n$-dimensional compact, connected and orientable manifold $\bar{N}$. Then there exist a suitable closed and connected smooth manifold $M$ and a special generic map $f:M \rightarrow {\mathbb{R}}^n$ enjoying the following properties where we abuse the notation in Proposition \ref{prop:2}.
\begin{enumerate}
\item $W_f=\bar{N}$ and $\bar{f}={\bar{f}}_N$.
\item Two bundles in Proposition \ref{prop:2} {\rm (}\ref{prop:2.2}{\rm )} and {\rm (}\ref{prop:2.3}{\rm )} are trivial bundles.
\item Let $f_0:=f$. Then there exist a special generic map $f_j:M \rightarrow {\mathbb{R}}^{n+j}$ for $0 \leq j \leq m-n$ and a smooth immersion $f_j:M \rightarrow {\mathbb{R}}^{n+j}$ for $j \geq m-n$. If $f_N$ is an embedding, then each immersion can be obtained as an embedding. Furthermore, they can be constructed enjoying the relation $f_{j_1}={\pi}_{n+j_2,n+j_1} \circ f_{j_2}$ for any distinct integers $j_1<j_2$ and a canonical projection ${\pi}_{n+j_2,n+j_1};{\mathbb{R}}^{n+j_2} \rightarrow {\mathbb{R}}^{n+j_1}$. \end{enumerate} \end{Prop} \begin{proof}
Most of our proof is based on a proof in the original paper.
By considering the restriction of the canonical projection of a unit sphere to $\mathbb{R}$ to a suitable hemisphere, we have a Morse function on a copy of the unit disk. This function can be obtained enjoying the following properties.
\begin{itemize}
\item It has exactly one singular point and there the function has maximal value (minimal value).
\item The preimage of the minimal (resp. maximal) value is the boundary of the disk.
\end{itemize}
We can prepare the product map of such a function and the identity map on the boundary $\partial \bar{N}$ and we have a surjection onto $N(\partial W_f)$, a suitable collar neighborhood of $\partial \bar{N}$, by restricting the manifold of the target of the product map and composing a suitable embedding.
We have a smooth trivial bundle over $\bar{N}-{\rm Int}\ N(\partial W_f)=W_f-{\rm Int}\ N(\partial W_f)$ whose fiber is an ($m-n$)-dimensional standard sphere. By gluing these maps in a suitable way we have a special generic map $f=f_0$ enjoying the first two properties.
\begin{itemize}
\item The diffeomorphism agreeing with the diffeomorphism used for identification between the connected component in the boundary of $\partial N(\partial W_f)$ and the boundary $\partial W_f -{\rm Int}\ N(\partial W_f)$.
\item The identity map on the fiber, which is the unit sphere $S^{m-n}$. Note that the fibers of the both trivial bundles can be regarded as the unit sphere $S^{m-n}$.
\end{itemize}
Let $0 \leq j \leq m-n$.
We replace the canonical projection of the unit sphere to $\mathbb{R}$ in the former product map by a suitable one onto ${\mathbb{R}}^{1+j}$. For the projection of the latter trivial smooth bundle, we replace the projection by the product map of a canonical projection of the unit sphere to ${\mathbb{R}}^j$ and the identity map on the base space. We glue in a natural and suitable way to have a desired map $f_j$.
In the case $j>m-n$, for construction of a desired map $f_j$, we consider the canonically defined embeddings of the unit spheres instead. This argument is a new one, presented first in the present paper.
This completes the proof of Proposition \ref{prop:7}, and by the construction, Proposition \ref{prop:8}, which is presented later. \end{proof} \begin{Def}[Partially defined in \cite{kitazawa2}]
\label{def:2}
We define the special generic map $f=f_0$ obtained in this way a {\it product-orginized} special generic map. \end{Def}
\begin{Prop}
\label{prop:8}
For $0 \leq j \leq m-n$, $f_j$ in Proposition \ref{prop:7} can be also constructed as a product-organized special generic map. \end{Prop} We can generalize ${\mathbb{R}}^n$ here to a general connected smooth manifold with no boundary. However, we concentrate on the cases of ${\mathbb{R}}^n$ essentially.
For a linear bundle over a base space $X$, we can define the {\it $j$-th Stiefel-Whitney class} as an element of $H^j(X;\mathbb{Z}/2\mathbb{Z})$ and the {\it $j$-th Pontrjagin class} as an element of $H^{4j}(X;\mathbb{Z})$.
The tangent bundles of smooth manifolds are important linear bundles whose fibers are Euclidean spaces.
The {\it $j$-th Stiefel-Whitney class} of a smooth manifold $X$ is defined as that of its tangent bundle and an element of $H^j(X;\mathbb{Z}/2\mathbb{Z})$. The {\it $j$-th Pontrjagin class} of a smooth manifold $X$ is defined as that of its tangent bundle and an element of $H^{4j}(X;\mathbb{Z})$.
The following follows from elementary arguments related to this.
\begin{Prop}
\label{prop:9}
Let $j>0$ be an arbitrary integer.
For the manifold $M$ and any positive integer $j$, the $j$-th Stiefel-Whitney class and the $j$-th Pontrjagin class of $M$ are the zero elements of $H^j(M;\mathbb{Z}/2\mathbb{Z})$ and $H^{4j}(M;\mathbb{Z})$, respectively.
\end{Prop}
See \cite{milnorstasheff} for Stiefel-Whitney classes and Pontrjagin classes.
\section{Main Theorems and related arguments.} We prove Main Theorems. We need some arguments and results of the author. For example, ones in \cite{kitazawa2,kitazawa3,kitazawa4,kitazawa5,kitazawa6} are important.
\begin{Prop}[\cite{kitazawa2,kitazawa3}]
\label{prop:10}
Let $n \geq 5$ be an integer.
Let $G$ be a finite commutative group. We have an $n$-dimensional compact and simply-connected manifold ${\bar{N}}_{G,n}$ enjoying the following properties. \begin{enumerate}
\item ${\bar{N}}_{G,n}$ is smoothly embedded in ${\mathbb{R}}^n$.
\item The boundary $\partial {\bar{N}}_{G,n}$ is connected.
\item $H_{n-3}({\bar{N}}_{G,n};\mathbb{Z})$ is isomorphic to $G$.
\item $H_j({\bar{N}}_{G,n};\mathbb{Z})$ is the trivial group for $j \neq 0,n-3$. \end{enumerate} \end{Prop} \begin{proof}
We prove this referring to \cite{kitazawa3} and in a different way. However, the method is essentially same.
First Let $G$ be a cyclic group.
We can choose a 3-dimensional closed and connected manifold $_G$ smoothly embedded in ${\mathbb{R}}^n$ such that ${\pi}_1(Y_G)$ and $H_1(Y_G;\mathbb{Z})$ are isomorphic to $G$ and $H_2(Y_G;\mathbb{Z})$ is the trivial group. This is due to \cite{wall} and for example, we can choose a so-called {\it Lens space}.
We remove the interior of a small closed tubular neighborhood $N(Y_G)$ of $Y_G$ and ${S^n}_{N(Y_G)}$ denotes the resulting manifold. $N(Y_G)$ is regarded as the total space of a trivial linear bundle over $Y_G$ whose fiber is the ($n-3$)-dimensional unit disk $D^{n-3}$.
$\partial {S^{n}}_{N(Y_G)}$ is regarded as the total space of the subbundle of the previous bundle whose fiber is $\partial D^{n-3} \subset D^{n-3}$.
We have a Mayer-Vietoris sequence
$$\rightarrow H_j(\partial {S^n}_{Y_G};\mathbb{Z})=H_j(\partial N(Y_G);\mathbb{Z}) \rightarrow H_j(N(Y_G);\mathbb{Z}) \oplus H_j({S^n}_{N(Y_G)};\mathbb{Z}) \rightarrow H_j(S^n;\mathbb{Z}) \rightarrow$$
\ \\
and the homomorphism from the first group to the second group is an isomorphism for $1 \leq j \leq n-1$.
This is due to the fundamental fact that $H_j(S^n;\mathbb{Z})$ is isomorhic to $\mathbb{Z}$ for $j=0,n$ and the trivial group for $1 \leq j \leq n-1$ and that the homomorphism from $H_n(S^n;\mathbb{Z})$ into $H_{n-1}(\partial {S^n}_{N(Y_G)};\mathbb{Z})$ is an isomorphism between groups isomorphic to $\mathbb{Z}$.
${S^n}_{N(Y_G)} \supset \partial {S^n}_{N(Y_G)}$ are a trivial linear bundle over $Y_G$ whose fiber is the unit disk $D^{n-3}$ and its subbundle obtained by considering the fiber $S^{n-4}=\partial D^{n-3} \subset D^{n-3}$, respectively.
${S^n}_{N(Y_G)}$ is a connected manifold.
$H_j({S^n}_{N(Y_G)};\mathbb{Z})$ is isomorphic to $H_{n-1}(Y_G;\mathbb{Z})$ for $1 \leq j \leq n-2$ and the trivial group for $j=n-1,n$.
${\pi}_1(\partial {S^n}_{N(Y_G)})$ is isomorphic to the direct sum $\mathbb{Z} \oplus G$ in the case $n=5$ and $G$ in the case $n>5$ and commutative. $S^n$ is simply-connected. For $S^n$ and the same submanifolds $N(Y_G)$ and ${S^n}_{N(Y_G);\mathbb{Z}}$, we can apply Seifert van-Kampen theorem. ${\pi}_1({S^n}_{N(Y_G)})$ is shown to be isomorphic to $\mathbb{Z}$.
We define
${\bar{N}}_G$ as a manifold in the following way. First attach a copy of $D^3 \times D^{n-3}$ so that this is the total space of the restriction of the linear bundle $N(Y_G)$ to a copy of the $3$-dimensional unit disk $D^3$ smoothly embedded in the base space $Y_G$. After that eliminate the corner.
We have a Mayer-Vietoris sequence for ${\bar{N}}_G$ and its decomposition into the two manifolds, one of which is ${S^n}_{N(Y_G)}$ and the other of which is the copy of $D^3 \times D^{n-3}$. More precisely, the resulting manifold is decomposed by an ($n-1$)-dimensional smoothly embedded manifold diffeomorphic to $D^3 \times \partial D^{n-3}$ into the two manifold.
We can also apply Seifert van-Kampen theorem for these manifolds.
Investigating our Mayer-Vietoris sequence and our argument using Seifert van-Kampen theorem, this completes our proof in the case of a cyclic group $G$.
For a general finite commutative group $G$, we have a desired manifold by considering a boundary connected sum of finitely many such manifolds forming the family $\{{\bar{N}}_{G_j}\}_{j=1}^l$ such that each $G_j$ is a cyclic group and that the direct sum ${\oplus}_{j=1}^l G_j$ is isomorphic to $G$. We add that the boundary connected sum is taken in the smooth category.
This completes the proof. \end{proof}
We have the following by applying Proposition \ref{prop:10} and other arguments such as`Propositions \ref{prop:7}, \ref{prop:8} and \ref{prop:9}.
\begin{Thm}[\cite{kitazawa5} ($(m,n)=(6,5)$)]
\label{thm:4}
Let $m \geq n+1$ be an integer. Let $G$ be a finite commutative group. We have an $m$-dimensional closed and simply-connected manifold $M_{G,n,m-n}$ and a product-organized special generic map $f_{G,n,m-n}:M_{G,n,m-n} \rightarrow {\mathbb{R}}^n$ enjoying the following properties.
\begin{enumerate}
\item \label{thm:4.1}
For any positive integer $j$, the $j$-th Stiefel-Whitney class and the $j$-th Pontrjagin class of $M_{G,n,m-n}$ are the zero elements of $H^j(M_{G,n,m-n};\mathbb{Z}/2\mathbb{Z})$ and $H^{4j}(M_{G,n,m-n};\mathbb{Z})$, respectively.
\item \label{thm:4.2}
The restriction of the map to the singular set is an embedding. The image is diffeomorphic to ${\bar{N}}_G$ in Proposition \ref{prop:10} and $W_{f_{G,n,m-n}}$ where $W_{f_{G,n,m-n}}$ abuse "$W_f$ in Proposition \ref{prop:2}".
\item \label{thm:4.3}
$H_j(M_{G,n,m-n};\mathbb{Z})$ is isomorphic to $G$ for $j=n-3,m-n+2$, $\mathbb{Z}$ for $j=0,m$, and the trivial group otherwise.
\end{enumerate} Furthermore, according to \cite{jupp,wall2,zhubr,zhubr2} for example, in a suitable case, the topology and the differentiable structure of the manifold $M_{G,5,1}$ is uniquely defined. \end{Thm} \begin{proof}
We have a product-organized special generic map $f_{G,n,m-n}:M_{G,n,m-n} \rightarrow {\mathbb{R}}^n$ on a suitable closed and simply-connected manifold $M_{G,n,m-n}$ enjoying (\ref{thm:4.1}) and (\ref{thm:4.2}) here by Propositions \ref{prop:7}, \ref{prop:8} and \ref{prop:9}
with Proposition \ref{prop:4}.
It is sufficient to show (\ref{thm:4.3}) here.
$W_{f_{G,n,m-n}}$ is simply-connected and by Proposition \ref{prop:10} collapses to an ($n-2$)-dimensional polyhedron.
By revising the proof of Propositions \ref{prop:4} and \ref{prop:6}, we can see that ${q_{f_{G,n,m-n}}}_{\ast}:H_j(M_{G,n,m-n};\mathbb{Z}) \rightarrow H_j(W_{f_{G,n,m-n}};\mathbb{Z})$ is an isomorphism for $0 \leq j \leq m-n+1$.
In the present proof, "Proposition \ref{prop:4}" means this isomorphism. \\
\\
Hereafter, $W_{G,n,m-n}$ denotes "$W$ in Propositions \ref{prop:2} and \ref{prop:3}" where $M_{G,n,m-n}$ is used instead of "$M$ in these propositions". This is ($m+1$)-dimensional, closed and simply-connected.
We also have a homology exact sequence
$$\rightarrow H_{j+1}(W_{G,n,m-n},M_{G,n,m-n};\mathbb{Z}) \rightarrow H_j(M_{G,n,m-n};\mathbb{Z}) \rightarrow H_j(W_{G,n,m-n};\mathbb{Z}) \rightarrow $$ \quad $H_j(W_{G,n,m-n},M_{G,n,m-n};\mathbb{Z}) \rightarrow $ \\ \ \\
for $(W_{G,n,m-n},M_{G,n,m-n})$.\\
\ \\
Case 1 The case $m-n+1 \geq \frac{m}{2}$. \\
The resulting manifold $M_{G,n,m-n}$ is a desired manifold
by applying Proposition \ref{prop:4} and Poincar\'e duality theorem. \\
\ \\
Case 2 The case $m$ is even and $m-n+2=\frac{m}{2}$.\\
\\
We also have a homology exact sequence
$$\rightarrow H_{n-2}(W_{G,n,m-n},M_{G,n,m-n};\mathbb{Z}) \rightarrow H_{n-3}(M_{G,n,m-n};\mathbb{Z}) \rightarrow H_{n-3}(W_{G,n,m-n};\mathbb{Z}) \rightarrow $$
$H_{n-3}(W_{G,n,m-n},M_{G,n,m-n};\mathbb{Z}) \rightarrow $ \\
\ \\
for $(W_{G,n,m-n},M_{G,n,m-n})$. $H_{n-2}(W_{G,n,m-n},M_{G,n,m-n};\mathbb{Z})$ is isomorphic to $H^{m-n+3}(W_{G,n,m-n};\mathbb{Z})$, $H^{m-n+3}(W_{f_{G,n,m-n}};\mathbb{Z})$, and $H_{2n-m-3}(W_{f_{G,n,m-n}},\partial W_{f_{G,n,m-n}};\mathbb{Z})$ by Proposition \ref{prop:3} and by virtue of Poincar\'e duality theorem for $W_{f_{G,n,m-n}}$. We have the relation $2n-m-4=0$ and the previous groups are isomorphic to $H_{1}(W_{f_{G,n,m-n}},\partial W_{f_{G,n,m-n}};\mathbb{Z})$. This is the trivial group since $W_{f_{G,n,m-n}}$ is simply-connected and $\partial W_{f_{G,n,m-n}}$ is connected. This is shown by the homology exact sequence for $(W_{f_{G,n,m-n}},\partial W_{f_{G,n,m-n}})$.
Similarly, $H_{n-1}(W_{G,n,m-n},M_{G,n,m-n};\mathbb{Z})$ is isomorphic to $H_{2}(W_{f_{G,n,m-n}},\partial W_{f_{G,n,m-n}};\mathbb{Z})$ and $H^{n-2}(W_{f_{G,n,m-n}};\mathbb{Z})$ by Proposition \ref{prop:3} and by virtue of Poincar\'e duality theorem for $W_{f_{G,n,m-n}}$. This group is shown to be finite by Proposition \ref{prop:10} and universal coefficient theorem. Similarly, $H_{n-2}(W_{G,n,m-n},M_{G,n,m-n};\mathbb{Z})$ is isomorphic to $H_{0}(W_{f_{G,n,m-n}},\partial W_{f_{G,n,m-n}};\mathbb{Z})$ and the trivial group. We have the relation $n-2=\frac{m}{2}$. By Proposition \ref{prop:4}, Proposition \ref{prop:10} and Poincar\'e duality theorem for $M_{G,n,m-n}$, this completes the proof for this case. \\
\ \\
Case 3 The case $m$ is odd and $m-n+\frac{3}{2}=\frac{m}{2}$.\\
\\ We also have a homology exact sequence
$$\rightarrow H_{n-2}(W_{G,n,m-n},M_{G,n,m-n};\mathbb{Z}) \rightarrow H_{n-3}(M_{G,n,m-n};\mathbb{Z}) \rightarrow H_{n-3}(W_{G,n,m-n};\mathbb{Z}) \rightarrow $$ $H_{n-3}(W_{G,n,m-n},M_{G,n,m-n};\mathbb{Z}) \rightarrow $ \\ \ \\
for $(W_{G,n,m-n},M_{G,n,m-n})$. $H_{n-2}(W_{G,n,m-n},M_{G,n,m-n};\mathbb{Z})$ is isomorphic to $H^{m-n+3}(W_{G,n,m-n};\mathbb{Z})$, $H^{m-n+3}(W_{f_{G,n,m-n}};\mathbb{Z})$, and $H_{2n-m-3}(W_{f_{G,n,m-n}},\partial W_{f_{G,n,m-n}};\mathbb{Z})$ by Proposition \ref{prop:3} and by virtue of Poincar\'e duality theorem for $W_{f_{G,n,m-n}}$. We have the relation $2n-m-3=0$ and the previous groups are isomorphic to $H_{0}(W_{f_{G,n,m-n}},\partial W_{f_{G,n,m-n}};\mathbb{Z})$ and the trivial group. We have the relation $n-2=\frac{m-1}{2}$. By Proposition \ref{prop:4}, Proposition \ref{prop:10} and Poincar\'e duality theorem for $M_{G,n,m-n}$, this completes the proof for this case. \\
\ \\
This completes the proof of (\ref{thm:4.3}). \\
This completes the proof.
\end{proof} \begin{Rem}
\cite{nishioka} has studied Case 3 of the proof of Theorem \ref{thm:4}. More precisely, for a positive integer $k>0$ and a ($2k+1$)-dimensional closed and connected manifold $M$ such that the group $H_1(M;\mathbb{Z})$ is the trivial group admitting a special generic map into ${\mathbb{R}}^{k+2}$, the group $H_k(M;\mathbb{Z})$ has been shown to be free. The case $(m,n)=(5,4)$ of Theorem \ref{thm:3} concerns the case $k=2$ here. \end{Rem}
\begin{proof} [A proof of Main Theorem \ref{mthm:1}] First consider an $m^{\prime}$-dimensional closed and simply-connected manifold $M_{G,5,m^{\prime}-5}$ in Theorem \ref{thm:4} where $m^{\prime}$ denotes "$m$ of Theorem \ref{thm:4}". We can embed this smoothly into ${\mathbb{R}}^{m^{\prime}+1}$ by Proposition \ref{prop:7} for example.
We define $M$ as $M:=M_{G,5,m^{\prime}-5} \times S^{m-m^{\prime}}$. This completes our exposition on the properties (\ref{mthm:1.1}) and (\ref{mthm:1.2}).
By Propositions \ref{prop:2}, \ref{prop:4} and \ref{prop:5} for example, this does not admit special generic maps into ${\mathbb{R}}^n$ for $n=1,2,3,4$. By the previous argument, we can consider the product map of a Morse function with exactly two singular points on $S^{m-m^{\prime}}$ and the identity map on $M_{G,5,m^{\prime}-5}$ and embed the image smoothly in a suitable way into ${\mathbb{R}}^{m^{\prime}+1}$. This can be constructed as product-organized one. This implies the existence of special generic maps into ${\mathbb{R}}^n$ for $m^{\prime}<n \leq m$.
We prove the non-existence into ${\mathbb{R}}^{m^{\prime}}$. Suppose that such a special generic map $f$ on $M$ into ${\mathbb{R}}^{m^{\prime}}$ exists. From Proposition \ref{prop:4}, the homomorphism ${q_f}_{\ast}$ maps the homology group $H_{m-m^{\prime}}(M;\mathbb{Z})$ onto $H_{m-m^{\prime}}(W_f;\mathbb{Z})$ as an isomorphism. $m^{\prime}-1$ is the dimension of the boundary $\partial W_f \subset W_f$ and the relation $a+(m^{\prime}-1) \leq m$ is assumed as a condition. We can apply Poincar\'e duality or intersection theory for $M$. From this, the torsion subgroup of $H_{a}(M;\mathbb{Z})$, which is not the trivial group from the assumption and K\"unneth theorem for the product $M_{G,5,m^{\prime}-5} \times S^{m-m^{\prime}}$, is mapped onto $H_{a}(W_f;\mathbb{Z})$ by the homomorphism ${q_f}_{\ast}$ and this is an isomorphism.
Here, we apply fundamental methods used first in \cite{kitazawa4}, generalizing some methods used in \cite{saeki1}, and used also in \cite{kitazawa5,kitazawa6} for example.
We choose an element of $h_1 \in H_{m-m^{\prime}}(M;\mathbb{Z})$ and an element of $h_2 \in H_{a}(M;\mathbb{Z})$ which are not the zero elements. Furthermore, we choose $h_2$ as an element of a summand of a direct sum decomposition of the torsion subgroup of $H_{a}(M;\mathbb{Z})$ into cyclic groups. This decomposition is due to fundamental theorem on the structures of finitely generated commutative groups. Note that $a=2,m^{\prime}-3$ and that $M$ is the product of a manifold $M_{G,5,m^{\prime}-5}$ in Theorem \ref{thm:4} and $S^{m-m^{\prime}}$. By the isomorphisms before and Poincar\'e duality theorem for $W_f$, we have an element of $H_{m^{\prime}-(m-m^{\prime})}(W_f,\partial W_f;\mathbb{Z})$ which is not the zero element corresponding to the element ${q_f}_{\ast}(h_1)$, the value of the isomorphism presented before at $h_1$, in a canonical way uniquely. In a similar reason, there exists a suitable commutative group $G_{h_2}$ which is finite and cyclic and which is not the trivial group and we have an element of $H_{m^{\prime}-a}(W_f,\partial W_f;G_{h_2})$ which is not the zero element corresponding to the element of $H_{a}(W_f;G_{h_2})$, obtained by mapping $h_2$ by the isomorphism defined canonically by ${q_f}_{\ast}$ before and mapping by the canonically defined homomorphism defined naturally from the canonical quotient map from $\mathbb{Z}$ onto $G_{h_2}$, in a canonical way uniquely. For $h_1$, we consider the value of the canonically defined homomorphism defined naturally from the canonical quotient map from $\mathbb{Z}$ onto $G_{h_2}$ at the obtained element of $H_{m^{\prime}-(m-m^{\prime})}(W_f,\partial W_f;\mathbb{Z})$. We put these two elements by ${h_{1,G_{h_2}}}^{\prime} \in H_{m^{\prime}-(m-m^{\prime})}(W_f,\partial W_f;G_{h_2})$ and ${h_{2,G_{h_2}}}^{\prime} \in H_{m^{\prime}-a}(W_f,\partial W_f;G_{h_2})$. By considering a (so-called {\it generic}) {\it intersection}, we have an element of $H_1(W_f;\partial W_f;G_{h_2})$. The degree $1$ is due to the relation $(m^{\prime}-(m-m^{\prime}))+(m^{\prime}-a)-m^{\prime}=2m^{\prime}-m-a=1$ with the assumption on the integers. This is the sum of elements represented by closed intervals embedded smoothly and properly in $W_f$. Furthermore, two boundary points of the each interval is mapped into distinct connected components of $\partial W_f$ and the interior is embedded into the interior of $W_f$. Related to this, circles in (the interior of) $W_f$ are null-homotopic since $W_f$ is simply-connected. By respecting the bundle whose projection is $r:W \rightarrow M$ in Proposition \ref{prop:3} where we abuse the notation and considering a variant of so-called {\it Thom isomorphisms} or {\it prism-operators}, we have an element of $H_{m-m^{\prime}+2}(W,M;G_{h_2})$. By considering the boundary, we have an element of $H_{m-m^{\prime}+1}(M;G_{h_2})$. For this see \cite{saeki1} and for algebraic topological notions see \cite{hatcher,milnorstasheff} for example.
We go back to our arguments for the proof. If this element is not the zero element, then by fundamental arguments on Poincar\'e duality and intersection theory, this element must be obtained as an element which is the value of the canonically defined homomorphism induced naturally from the canonical quotient map from $\mathbb{Z}$ to $G_{h_2}$. By the structure of $M=M_{G,5,m^{\prime}-5} \times S^{m-m^{\prime}}$ and conditions on homology groups together with Poincar\'e duality and K\"unneth theorem for example, this is not the zero element and this is not an element which is the value of the canonically defined homomorphism induced naturally from the canonical quotient map from $\mathbb{Z}$ to $G_{h_2}$. We add that this homomorohism is one from $H_{m-m^{\prime}+1}(M;\mathbb{Z})$ into $H_{m-m^{\prime}+1}(M;G_{h_2})$ as a precise exposition.
We find a contradiction. We have (\ref{mthm:1.3}). This completes the proof.
\end{proof} \begin{proof}[A proof of Main Theorem \ref{mthm:2}]
$M_1$ is obtained as a connected sum of the following three manifolds taken in the smooth category.
\begin{itemize}
\item A $8$-dimensional manifold
admitting a product-organized special generic map into ${\mathbb{R}}^5$ in Theorem \ref{thm:4} by considering "$(m,n,G)=(8,5,G_{{\rm F},1})$ in the theorem".
\item A $8$-dimensional manifold
admitting a product-organized special generic map into ${\mathbb{R}}^6$ in Theorem \ref{thm:4} by considering "$(m,n,G)=(8,6,G_{{\rm F},2})$ in the theorem".
\item A manifold represented as a suitably chosen connected sum of manifolds diffeomorphic to $S^2 \times S^6$, $S^3 \times S^5$ or $S^4 \times S^4$, which admit special generic maps into ${\mathbb{R}}^6$ in Example \ref{ex:1}. The special generic maps discussed in Example \ref{ex:1} are also constructed as product-organized maps by fundamental arguments on special generic maps.
\end{itemize}
We can see the non-existence of special generic map on $M_1$ into ${\mathbb{R}}^n$ for $n=1,2,3,4,5$ from Propositions \ref{prop:2}, \ref{prop:4}, \ref{prop:5} and Theorem \ref{thm:1} for example.
By a fundamental argument for construction of special generic maps in \cite{saeki1}, we can construct a product-organized special generic map on $M_1$ into ${\mathbb{R}}^6$.
We give $M_2$. This is obtained as a connected sum of the following three manifolds taken in the smooth category.
\begin{itemize}
\item A $8$-dimensional manifold
admitting a product-organized special generic map into ${\mathbb{R}}^5$ in Theorem \ref{thm:4} by considering "$(m,n,G)=(8,5,G_{{\rm F},1})$ in the theorem".
\item
A $8$-dimensional closed and simply-connected manifold "$M^{\prime} \times S^2$ in Main Theorem \ref{mthm:1}", having the 3rd homology group $H_3(M^{\prime} \times S^3;\mathbb{Z})$ isomorphic to $G_{{\rm F},2}$. More precisely, we consider the case "$(m,m^{\prime},a,G)=(8,6,3,G_{{\rm F},2})$ in Main Theorem \ref{mthm:1}".
\item A manifold represented as a suitably chosen connected sum of manifolds diffeomorphic to $S^2 \times S^6$, $S^3 \times S^5$ or $S^4 \times S^4$, which admit special generic maps into ${\mathbb{R}}^7$ in Example \ref{ex:1}. These maps can be constructed as product-organized maps as before.
\end{itemize}
We can see the non-existence of special generic maps on $M_2$ into ${\mathbb{R}}^n$ for $n=1,2,3,4,5,6$. For $n=1,2,3,4,5$, for example, Propositions \ref{prop:2}, \ref{prop:4}, \ref{prop:5}, \ref{prop:6} and Theorem \ref{thm:1} complete the proof. For $n=6$ here, we apply Main Theorem \ref{mthm:1} directly or revise arguments in the proof of Main Theorem \ref{mthm:1} suitably if we need. In the case ($m=8$ and) $n=6$, $m-n=8-6=2$ and we do not have any sequence of elements of a finite length of $H^j(M;A)$ with $0 \leq j \leq m-n=2$ the cup product for which is not the zero element whose degree is greater than or equal to $6$. This means that we cannot apply Proposition \ref{prop:6} to give a proof for the case $n=6$. We can show the existence of special generic maps into ${\mathbb{R}}^n$ for $n=7,8$ similarly.
This completes the proof. \end{proof}
This adds to Main Theorem 4 of \cite{kitazawa2} where situations are a bit different. Last, for example, our new examples are discovered especially motivated by \cite{kitazawa6}. Compare these results to our new results.
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\markboth{J. Jansson, C. Johnson, and A. Logg}{Computational Modeling of Dynamical Systems}
\title{COMPUTATIONAL MODELING OF DYNAMICAL SYSTEMS}
\author{JOHAN JANSSON} \address{Department of Computational Mathematics,
Chalmers University of Technology, SE--412 96 G\"{o}teborg, Sweden,
\emph{email}: johanjan{\@@}math.chalmers.se.}
\author{CLAES JOHNSON} \address{Department of Computational Mathematics,
Chalmers University of Technology, SE--412 96 G\"{o}teborg, Sweden,
\emph{email}: claes{\@@}math.chalmers.se.}
\author{ANDERS LOGG} \address{Department of Computational Mathematics,
Chalmers University of Technology, SE--412 96 G\"{o}teborg, Sweden,
\emph{email}: logg{\@@}math.chalmers.se.}
\maketitle
\begin{abstract}
In this short note, we discuss the basic approach to computational
modeling of dynamical systems. If a dynamical system contains
multiple time scales, ranging from very fast to slow, computational
solution of the dynamical system can be very costly.
By resolving the fast time scales in a short time simulation, a
model for the effect of the small time scale variation on large time
scales can be determined, making solution possible on a long time interval.
This process of computational modeling can be completely automated.
Two examples are presented, including a simple model problem
oscillating at a time scale of $10^{-9}$ computed over the
time interval $[0,100]$, and a lattice
consisting of large and small point masses. \end{abstract}
\keywords{Modeling, dynamical system, reduced model, automation}
\section{Introduction}
We consider a dynamical system of the form \begin{equation}
\label{eq:u'=f}
\begin{array}{rcl}
\dot{u}(t) &=& f(u(t),t), \quad t \in (0,T], \\
u(0) &= u_0,
\end{array} \end{equation} where $u : [0,T] \rightarrow \mathbb{R}^N$ is the solution to be computed, $u_0 \in \mathbb{R}^N$ a given initial value, $T>0$ a given final time, and $f : \mathbb{R}^N \times (0,T] \rightarrow \mathbb{R}^N$ a given function that is Lipschitz-continuous in $u$ and bounded. We consider a situation where the exact solution $u$ varies on different time scales, ranging from very fast to slow. Typical examples include meteorological models for weather prediction, with fast time scales on the range of seconds and slow time scales on the range of years, protein folding represented by a molecular dynamics model of the form (\ref{eq:u'=f}), with fast time scales on the range of femtoseconds and slow time scales on the range of microseconds, or turbulent flow with a wide range of time scales.
To make computation feasible in a situation where computational resolution of the fast time scales would be prohibitive because of the small time steps required, the given model (\ref{eq:u'=f}) containing the fast time scales needs to be replaced with a \emph{reduced model} for the variation of the solution $u$ of (\ref{eq:u'=f}) on resolvable time scales. As discussed below, the key step is to correctly model the effect of the variation at the fast time scales on the variation on slow time scales.
The problem of model reduction is very general and various approaches have been taken\cite{RuhSko98,Kre91}. We present below a new approach to model reduction, based on resolving the fast time scales in a short time simulation and determining a model for the effect of the small time scale variation on large time scales. This process of computational modeling can be completely \emph{automated} and the validity of the reduced model can be evaluated a posteriori.
\section{A simple model problem}
We consider a simple example illustrating the basic aspects: Find $u=(u_1,u_2):[0,T] \rightarrow \mathbb{R}^2$, such that \begin{equation}
\label{eq:model}
\begin{array}{rcl}
\ddot{u}_1 + u_1 - u_2^2/2 &=& 0 \quad \mbox{on } (0,T],\\
\ddot{u}_2 + \kappa u_2 &=& 0 \quad \mbox{on } (0,T],\\
u(0) = (0,1) &&\quad \dot{u}(0) = (0,0),
\end{array} \end{equation} which models a moving unit point mass $M_1$ connected through a soft spring to another unit point mass $M_2$, with $M_2$ moving along a line perpendicular to the line of motion of $M_1$, see Figure \ref{fig:model}. The second point mass $M_2$ is connected to a fixed support through a very stiff spring with spring constant $\kappa = 10^{18}$ and oscillates rapidly on a time scale of size $1/\sqrt{\kappa} = 10^{-9}$. The oscillation of $M_2$ creates a force $\sim u_2^2$ on $M_1$ proportional to the elongation of the spring connecting $M_2$ to $M_1$ (neglecting terms of order $u_2^4$).
The short time scale of size $10^{-9}$ requires time steps of size $\sim 10^{-10}$ for full resolution. With $T = 100$, this means a total of $\sim 10^{12}$ time steps for solution of (\ref{eq:model}). However, by replacing (\ref{eq:model}) with a reduced model where the fast time scale has been removed, it is possible to compute the (averaged) solution of (\ref{eq:model}) with time steps of size $\sim 0.1$ and consequently only a total of $10^3$ time steps.
\begin{figure}
\caption{A simple mechanical system with large time scale $\sim 1$ and
small time scale $\sim 1/\sqrt{\kappa}$.}
\label{fig:model}
\end{figure}
\section{Taking averages to obtain the reduced model}
Having realized that point-wise resolution of the fast time scales of the exact solution $u$ of (\ref{eq:u'=f}) may sometimes be computationally very expensive or even impossible, we seek instead to compute a time average $\bar{u}$ of $u$, defined by \begin{equation}
\label{eq:average}
\bar{u}(t)= \frac{1}{\tau}\int_{-\tau/2}^{\tau/2}u(t+s)\, ds, \quad
t \in [\tau/2,T - \tau/2], \end{equation} where $\tau > 0$ is the size of the average. The average $\bar{u}$ can be extended to $[0,T]$ in various ways. We consider here a constant extension, i.e., we let $\bar{u}(t) = \bar{u}(\tau/2)$ for $t \in [0,\tau/2)$, and let $\bar{u}(t) = \bar{u}(T-\tau/2)$ for $t \in (T - \tau/2,T]$.
We now seek a dynamical system satisfied by the average $\bar{u}$ by taking the average of (\ref{eq:u'=f}). We obtain \begin{displaymath}
\dot{\bar{u}}(t) = \bar{\dot{u}}(t) = \overline{f(u,\cdot)}(t)
= f(\bar{u}(t),t) + (\overline{f(u,\cdot)}(t) - f(\bar{u}(t),t)), \end{displaymath} or \begin{equation}
\label{eq:u'=f,average}
\dot{\bar{u}}(t) = f(\bar{u}(t),t) + \bar{g}(u,t), \end{equation} where the \emph{variance} $\bar{g}(u,t) = \overline{f(u,\cdot)}(t) - f(\bar{u}(t),t)$ accounts for the effect of small scales on time scales larger than $\tau$. (Note that we may extend (\ref{eq:u'=f,average}) to $(0,T]$ by defining $\bar{g}(u,t) = - f(\bar{u}(t),t)$ on $(0,\tau/2] \cup (T-\tau/2,T]$.)
We now seek to model the variance $\bar{g}(u,t)$ in the form $\bar{g}(u,t)\approx \tilde{g}(\bar{u}(t),t)$ and replace (\ref{eq:u'=f,average}) and thus (\ref{eq:u'=f}) by \begin{equation}
\label{eq:reduced}
\begin{array}{rcl}
\dot{\tilde{u}}(t) &=& f(\tilde{u}(t),t) + \tilde{g}(\tilde{u}(t),t),
\quad t \in (0,T], \\
\tilde{u}(0) &=& \bar{u}_0,
\end{array} \end{equation} where $\bar{u}_0 = \bar{u}(0) = \bar{u}(\tau/2)$. We refer to this system as the \emph{reduced model} with \emph{subgrid model} $\tilde{g}$ corresponding to (\ref{eq:u'=f}).
To summarize, if the solution $u$ of the full dynamical system (\ref{eq:u'=f}) is computationally unresolvable, we aim at computing the average $\bar{u}$ of $u$. However, since the variance $\bar{g}$ in the averaged dynamical system (\ref{eq:u'=f,average}) is unknown, we need to solve the reduced model (\ref{eq:reduced}) for $\tilde{u} \approx \bar{u}$ with an approximate subgrid model $\tilde{g} \approx \bar{g}$. Solving the reduced model (\ref{eq:reduced}) using e.g. a Galerkin finite element method, we obtain an approximate solution $U \approx \tilde{u} \approx \bar{u}$. Note that we may not expect $U$ to be close to $u$ point-wise in time, while we hope that $U$ is close to $\bar{u}$ point-wise.
\section{Modeling the variance}
There are two basic approaches to the modeling of the variance $\bar{g}(u,t)$ in the form $\tilde{g}(\tilde{u}(t),t)$; (i) scale-extrapolation or (ii) local resolution. In (i), a sequence of solutions is computed with increasingly fine resolution, but without resolving the fastest time scales. A model for the effects of the fast unresolvable scales is then determined by extrapolation from the sequence of computed solutions\cite{Hof02}. In (ii), the approach followed below, the solution $u$ is computed accurately over a short time period, resolving the fastest time scales. The reduced model is then obtained by computing the variance \begin{equation}
\bar{g}(u,t) = \overline{f(u,\cdot)}(t) - f(\bar{u}(t),t) \end{equation} and then determining $\tilde{g}$ for the remainder of the time interval such that $\tilde{g}(\tilde{u}(t),t) \approx \bar{g}(u,t)$.
For the simple model problem (\ref{eq:model}), which we can write in the form (\ref{eq:u'=f}) by introducing the two new variables $u_3 = \dot{u}_1$ and $u_4 = \dot{u}_2$ with \begin{displaymath}
f(u,\cdot) = (u_3, u_4, -u_1 + u_2^2/2, -\kappa u_2), \end{displaymath} we note that $\bar{u}_2 \approx 0$ (for $\sqrt{\kappa} \tau$ large) while $\overline{u_2^2} \approx 1/2$. By the linearity of $f_1$, $f_2$, and $f_4$, the (approximate) reduced model takes the form \begin{equation}
\label{eq:model,reduced}
\begin{array}{rcl}
\ddot{\tilde{u}}_1 + \tilde{u}_1 - 1/4 &=& 0 \quad \mbox{on } (0,T],\\
\ddot{\tilde{u}}_2 + \kappa \tilde{u}_2 &=& 0 \quad \mbox{on } (0,T],\\
\tilde{u}(0) = (0,0), &&\quad \dot{\tilde{u}}(0) = (0,0),
\end{array} \end{equation} with solution $\tilde{u}(t) = (\frac{1}{4}(1 - \cos t),0)$.
In general, the reduced model is constructed with subgrid model $\tilde{g}$ varying on resolvable time scales. In the simplest case, it is enough to model $\tilde{g}$ with a constant and repeatedly checking the validity of the model by comparing the reduced model (\ref{eq:reduced}) with the full model (\ref{eq:u'=f}) in a short time simulation. Another possibility is to use a piecewise polynomial representation for the subgrid model $\tilde{g}$.
\section{Solving the reduced system}
Although the presence of small scales has been decreased in the reduced system (\ref{eq:reduced}), the small scale variation may still be present. This is not evident in the reduced system (\ref{eq:model,reduced}) for the simple model problem (\ref{eq:model}), where we made the approximation $\tilde{u}_2(0) = 0$. In practice, however, we compute $\tilde{u}_2(0) = \frac{1}{\tau}\int_0^\tau u_2(t) \, dt = \frac{1}{\tau}\int_0^\tau \cos(\sqrt{\kappa} t) \, dt \sim 1/(\sqrt{\kappa}\tau)$ and so $\tilde{u}_2$ oscillates at the fast time scale $1/\sqrt{\kappa}$ with amplitude $1/(\sqrt{\kappa}\tau)$.
To remove these oscillations, the reduced system needs to be stabilized by introducing damping of high frequencies. Following the general approach\cite{HofJoh04b}, a least squares stabilization is added in the Galerkin formulation of the reduced system (\ref{eq:reduced}) in the form of a modified test function. As a result, damping is introduced for high frequencies without affecting low frequencies.
Alternatively, components such as $u_2$ in (\ref{eq:model,reduced}) may be \emph{inactivated}, corresponding to a subgrid model of the form $\tilde{g}_2(\tilde{u},\cdot) = -f_2(\tilde{u},\cdot)$. We take this simple approach for the example problems presented below.
\section{Error analysis}
The validity of a proposed subgrid model may be checked a posteriori. To analyze the modeling error introduced by approximating the variance $\bar{g}$ with the subgrid model $\tilde{g}$, we introduce the \emph{dual problem} \begin{equation}
\label{eq:dual}
\begin{array}{rcl}
- \dot{\phi}(t) &=& J(\bar{u},U,t)^{\top} \phi(t), \quad t \in [0,T), \\
\phi(T) &=& \psi,
\end{array} \end{equation} where $J$ denotes the Jacobian of the right-hand side of the dynamical system (\ref{eq:u'=f}) evaluated at a mean value of the average $\bar{u}$ and the computed numerical (finite element) solution $U \approx \tilde{u}$ of the reduced system (\ref{eq:reduced}), \begin{equation}
J(\bar{u},U,t) = \int_0^1 \frac{\partial f}{\partial u}
(s \bar{u}(t) + (1-s) U(t),t) \, ds, \end{equation} and where $\psi$ is initial data for the backward dual problem.
To estimate the error $\bar{e} = U - \bar{u}$ at final time, we note that $\bar{e}(0) = 0$ and $\dot{\phi} + J(\bar{u},U,\cdot)^{\top} \phi = 0$, and write \begin{displaymath}
\begin{array}{rcl}
(\bar{e}(T),\psi)
&=& (\bar{e}(T),\psi) -
\int_0^T (\dot{\phi} + J(\bar{u},U,\cdot)^{\top} \phi,
\bar{e}) \, dt \\
&=& \int_0^T (\phi, \dot{\bar{e}} - J\bar{e}) \, dt
= \int_0^T (\phi, \dot{U} - \dot{\bar{u}} -
f(U,\cdot) + f(\bar{u},\cdot)) \, dt \\
&=& \int_0^T (\phi, \dot{U} - f(U,\cdot) - \tilde{g}(U,\cdot)) \, dt +
\int_0^T (\phi, \tilde{g}(U,\cdot) - \bar{g}(u,\cdot)) \, dt \\
&=& \int_0^T (\phi, \tilde{R}(U,\cdot)) \, dt +
\int_0^T (\phi, \tilde{g}(U,\cdot) - \bar{g}(u,\cdot)) \, dt.
\end{array} \end{displaymath} The first term, $\int_0^T (\phi, \tilde{R}(U,\cdot)) \, dt$, in this \emph{error representation} corresponds to the \emph{discretization error} $U - \tilde{u}$ for the numerical solution of (\ref{eq:reduced}). If a Galerkin finite element method is used\cite{EriEst95,EriEst96}, the \emph{Galerkin orthogonality} expressing the orthogonality of the residual $\tilde{R}(U,\cdot) = \dot{U} - f(U,\cdot) - \tilde{g}(U,\cdot)$ to a space of test functions can be used to subtract a test space interpolant $\pi \phi$ of the dual solution $\phi$. In the simplest case of the $\mathrm{cG}(1)$ method for a partition of the interval $(0,T]$ into $M$ subintervals $I_j = (t_{j-1},t_j]$, each of length $k_j = t_j - t_{j-1}$, we subtract a piecewise constant interpolant to obtain \begin{displaymath}
\begin{array}{rcl}
\int_0^T (\phi, \tilde{R}(U,\cdot)) \, dt
&=& \int_0^T (\phi - \pi \phi, \tilde{R}(U,\cdot)) \, dt
\leq \sum_{j=1}^M k_j \max_{I_j} \|\tilde{R}(U,\cdot)\|_{l_2}
\int_{I_j} \|\dot{\phi}\|_{l_2} \, dt \\
&\leq& S^{[1]}(T) \max_{[0,T]} \|k\tilde{R}(U,\cdot)\|_{l_2},
\end{array} \end{displaymath}
where the \emph{stability factor} $S^{[1]}(T) = \int_0^T \|\dot{\phi}\|_{l_2} \, dt$ measures the sensitivity to discretization errors for the given output quantity $(\bar{e}(T),\psi)$.
The second term, $\int_0^T (\phi, \tilde{g}(U,\cdot) - \bar{g}(u,\cdot)) \, dt$, in the error representation corresponds to the \emph{modeling error} $\tilde{u} - \bar{u}$. The sensitivity to modeling errors is measured by the stability factor $S^{[0]}(T) = \int_0^T \|\phi\|_{l_2} \, dt$. We notice in particular that if the stability factor $S^{[0]}(T)$ is of moderate size, a reduced model of the form (\ref{eq:reduced}) for $\tilde{u} \approx \bar{u}$ may be constructed.
We thus obtain the error estimate \begin{equation}
\vert (\bar{e}(T),\psi) \vert
\leq S^{[1]}(T) \max_{[0,T]} \|k\tilde{R}(U,\cdot)\|_{l_2} +
S^{[0]}(T) \max_{[0,T]} \|\tilde{g}(U,\cdot) - \bar{g}(u,\cdot)\|_{l_2}, \end{equation} including both discretization and modeling errors. The initial data $\psi$ for the dual problem (\ref{eq:dual}) is chosen to reflect the desired output quantity, e.g. $\psi = (1,0,\ldots,0)$ to measure the error in the first component of $U$.
To estimate the modeling error, we need to estimate the quantity $\tilde{g} - \bar{g}$. This estimate is obtained by repeatedly solving the full dynamical system (\ref{eq:u'=f}) at a number of control points and comparing the subgrid model $\tilde{g}$ with the computed variance $\bar{g}$. As initial data for the full system at a control point, we take the computed solution $U \approx \bar{u}$ at the control point and add a perturbation of appropriate size, with the size of the perturbation chosen to reflect the initial oscillation at the fastest time scale.
\section{Numerical results}
We present numerical results for two model problems, including the simple model problem (\ref{eq:model}), computed with DOLFIN\cite{logg:www:01} version 0.4.10. With the option \emph{automatic modeling} set, DOLFIN automatically creates the reduced model (\ref{eq:reduced}) for a given dynamical system of the form (\ref{eq:u'=f}) by resolving the full system in a short time simulation and then determining a constant subgrid model $\bar{g}$. Components with constant average, such as $u_2$ in (\ref{eq:model}), are automatically marked as inactive and are kept constant throughout the simulation. The automatic modeling implemented in DOLFIN is rudimentary and many improvements are possible, but it represents a first attempt at the automation of modeling, following the recently presented\cite{logg:thesis:03} directions for the \emph{automation of computational mathematical modeling}.
\subsection{The simple model problem}
The solution for the two components of the simple model problem (\ref{eq:model}) is shown in Figure \ref{fig:model,solution} for $\kappa = 10^{18}$ and $\tau = 10^{-7}$. The value of the subgrid model $\bar{g}_1$ is automatically determined to $0.2495 \approx 1/4$.
\begin{figure}
\caption{The solution of the simple model problem (\ref{eq:model})
on $[0,100]$ (above) and on $[0,4\cdot{10}^{-7}]$ (below). The automatic
modeling is activated at time $t = 2\tau = 2\cdot{10}^{-7}$.}
\label{fig:model,solution}
\end{figure}
\subsection{A lattice with internal vibrations}
The second example is a lattice consisting of a set of $p^2$ large and $(p-1)^2$ small point masses connected by springs of equal stiffness $\kappa = 1$, as shown in Figure \ref{fig:latticedetail} and Figure \ref{fig:lattice}. Each large point mass is of size $M = 100$ and each small point mass is of size $m = 10^{-12}$, giving a large time scale of size $\sim 10$ and a small time scale of size $\sim 10^{-6}$.
The fast oscillations of the small point masses make the initially stationary structure of large point masses contract. Without resolving the fast time scales and ignoring the subgrid model, the distance $D$ between the lower left large point mass at $x = (0,0)$ and the upper right large point mass at $x = (1,1)$ remains constant, $D = \sqrt{2}$. In Figure \ref{fig:lattice,solution2}, we show the computed solution with $\tau = 10^{-4}$, which manages to correctly capture the oscillation in the diameter $D$ of the lattice as a consequence of the internal vibrations at time scale $10^{-6}$.
With a constant subgrid model $\bar{g}$ as in the example, the reduced model stays accurate until the configuration of the lattice has changed sufficiently. When the change becomes too large, the reduced model can no longer give an accurate representation of the full system, as shown in Figure \ref{fig:lattice,solution1}. At this point, the reduced model needs to be reconstructed in a new short time simulation.
\begin{figure}
\caption{Detail of the lattice. The arrows indicate the direction
of vibration perpendicular to the springs connecting the small
mass to the large masses.}
\label{fig:latticedetail}
\end{figure}
\begin{figure}
\caption{Lattice consisting of $p^2$ large masses and $(p-1)^2$ small masses.}
\label{fig:lattice}
\end{figure}
\begin{figure}
\caption{The diameter $D$ of the lattice as function of time on
$[0,20]$ (left) and on $[0,100]$ (right) for $m = 10^{-4}$ and
$\tau = 1$. The solid line represents the diameter for the solution of the
reduced system (\ref{eq:reduced}) and the dashed line
represents the solution of the full system (\ref{eq:u'=f}).}
\label{fig:lattice,solution1}
\end{figure}
\begin{figure}
\caption{Distance $D$ between the lower left large mass and the upper
right large mass (above) and the distance $d$ between the lower
left large mass and the lower left small mass (below) as function of
time on $[0,10]$ and on $[0,4\cdot 10^{-4}]$, respectively.}
\label{fig:lattice,solution2}
\end{figure}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Transformed CNNs: \recasting pre-trained convolutional layers with self-attention} \begin{abstract} Vision Transformers (ViT) have recently emerged as a powerful alternative to convolutional networks (CNNs). Although hybrid models attempt to bridge the gap between these two architectures, the self-attention layers they rely on induce a strong computational bottleneck, especially at large spatial resolutions. In this work, we explore the idea of reducing the time spent training these layers by initializing them as convolutional layers. This enables us to transition smoothly from any pre-trained CNN to its functionally identical hybrid model, called Transformed CNN (T-CNN). With only 50 epochs of fine-tuning, the resulting T-CNNs demonstrate significant performance gains over the CNN (+2.2\% top-1 on ImageNet-1k for a ResNet50-RS) as well as substantially improved robustness (+11\% top-1 on ImageNet-C). We analyze the representations learnt by the T-CNN, providing deeper insights into the fruitful interplay between convolutions and self-attention. Finally, we experiment initializing the T-CNN from a partially trained CNN, and find that it reaches better performance than the corresponding hybrid model trained from scratch, while reducing training time. \end{abstract}
\section*{Introduction}
Since the success of AlexNet in 2012~\cite{krizhevsky2017imagenet}, the field of Computer Vision has been dominated by Convolutional Neural Networks (CNNs)~\cite{lecun1998gradient,lecun1989backpropagation}. Their local receptive fields give them a strong inductive bias to exploit the spatial structure of natural images~\cite{scherer_evaluation_2010,schmidhuber_deep_2015,goodfellow_deep_2016}, while allowing them to scale to large resolutions seamlessly. Yet, this inductive bias limits their ability to capture long-range interactions.
In this regard, self-attention (SA) layers, originally introduced in language models~\cite{bahdanau2014neural,vaswani2017attention,devlin2018bert}, have gained interest as a building block for vision~\citet{ramachandran2019stand,zhao2020exploring}. Recently, they gave rise to a plethora of Vision Transformer (ViT) models, able to compete with state-of-the-art CNNs in various tasks~\citet{dosovitskiy2020image,touvron2020training,wu_visual_2020,touvron2021going,liu2021swin,heo2021rethinking} while demonstrating better robustness~\cite{bhojanapalli2021understanding,mao2021rethinking}. However, capturing long-range dependencies necessarily comes at the cost of quadratic complexity in input size, a computational burden which many recent directions have tried to alleviate~\cite{bello2021lambdanetworks,wang2020self,choromanski2020rethinking,katharopoulos2020transformers}. Additionally, ViTs are generally harder to train~\cite{zhang2019adaptive,liu2020understanding}, and require vast amounts of pre-training~\cite{dosovitskiy2020image} or distillation from a convolutional teacher~\cite{hinton2015distilling,jiang2021token,graham2021levit} to match the performance of CNNs.
Faced with the dilemma between efficient CNNs and powerful ViTs, several approaches have aimed to bridge the gap between these architectures. On one side, hybrid models append SA layers onto convolutional backbones ~\cite{chen20182,bello2019attention,graham2021levit,chen2021visformer,srinivas2021bottleneck}, and have already fueled successful results in a variety of tasks~\cite{carion2020end,hu2018relation,chen2020uniter,locatello2020object,sun2019videobert}. Conversely, a line of research has studied the benefit of introducing convolutional biases in Transformer architectures to ease learning~\cite{d2021convit,wu2021cvt,yuan2021incorporating}. Despite these interesting compromises, modelling long-range dependencies at low computational cost remains a challenge for practitioners.
\begin{figure}
\caption{\textbf{Transformed ResNets strike a strong accuracy-robustness balance.} Our models (red) significantly outperform the original ResNet-RS models (dark blue) they were initialized from when evaluated on ImageNet-1k. On various robustness benchmarks (ImageNet-C, A and R, from left to right), they narrow or close the gap with Transformer architectures.}
\label{fig:intro}
\end{figure}
\paragraph{Contributions} At a time when pre-training on vast datasets has become common practice, we ask the following question: does one need to train the SA layers during the whole learning process? Could one instead learn cheap components such as convolutions first, leaving the SA layers to be learnt at the end? In this paper, we take a step in this direction by presenting a method to fully reparameterize a pre-trained convolutional layer as a \textit{Gated Positional Self-Attention} (GPSA) layer~\cite{d2021convit}. The latter is initialized to reproduce the mapping of the convolutional layer, but is then encouraged to learn more general mappings which are not accessible to the CNN by adjusting positional gating parameters.
We leverage this method to reparametrize pre-trained CNNs as functionally equivalent hybrid models. After only 50 epochs of fine-tuning, the resulting Transformed CNNs (T-CNNs) boast significant performance and robustness improvements as shown in Fig.~\ref{fig:intro}, demonstrating the practical relevance of our method. We analyze the inner workings of the T-CNNs, showing how they learn more robust representations by combining convolutional heads and SA heads in a complementary way. Finally, we investigate how performance gains depend on the reparametrization epoch. Results suggest that reparametrizing at intermediate times is optimal in terms of speed-performance trade-offs.
\paragraph{Related work}
Our work mainly builds on two pillars. First, the idea that SA layers can express any convolution, introduced by~\citet{cordonnier2019relationship}. This idea was recently leveraged in~\citet{d2021convit}, which initialize the SA layers of the ViT as \textit{random} convolutions and observe performance gains compared to the standard initialization, especially in the low-data regime where inductive biases are most useful. Our approach is a natural follow-up of this idea: what happens if the SA layers are instead initialized as \textit{trained} convolutions?
Second, we exploit the following learning paradigm: train a simple and fast model, then reparameterize it as a more complex model for the final stages of learning. This approach was studied from a scientific point of view in~\citet{d2019finding}, which shows that reparameterizing a CNN as a fully-connected network (FCN) halfway through training can lead the FCN to outperform the CNN. Yet, the practical relevance of this method is limited by the vast increase in number of parameters required by the FCN to functionally represent the CNN. In contrast, our reparameterization hardly increases the parameter count of the CNN, making it easily applicable to any state-of-the-art CNN. Note that these reparameterization methods can be viewed an informed version of dynamic architecture growing algorithms such as AutoGrow~\cite{wen2020autogrow}.
In the context of hybrid models, various works have studied the performance gains obtained by introducing MHSA layers in ResNets with minimal architectural changes~\cite{srinivas2021bottleneck,graham2021levit,chen2021visformer}. However, the MHSA layers used in these works are initialized randomly and need to be trained from scratch. Our approach is different, as it makes use of GPSA layers, which can be initialized to represent the same function as the convolutional layer it replaces. We emphasize that the novelty in our work is not in the architectures used, but in the unusual way they are blended together.
\section{Background}
\paragraph{Multi-head self-attention} \label{sec:mhsa}
The SA mechanism is based on a trainable associative memory with (key, query) vector pairs. To extract the semantic interpendencies between the $L$ elements of a sequence $\boldsymbol X\in\mathbb{R}^{L\times D_{in}}$, a sequence of ``query'' embeddings $\boldsymbol Q = \boldsymbol W_{qry} \boldsymbol X\in \mathbb{R}^{L\times D_{h}}$ is matched against another sequence of ``key'' embeddings $\boldsymbol K = \boldsymbol W_{key} \boldsymbol X\in\mathbb{R}^{L\times D_{h}}$ using inner products. The result is an attention matrix whose entry $(ij)$ quantifies how semantically relevant $\boldsymbol Q_i$ is to $\boldsymbol K_j$: \begin{equation}
\boldsymbol A = \operatorname{softmax}\left(\frac{\boldsymbol Q \boldsymbol K^\top}{\sqrt {D_{h}}}\right) \in \mathbb{R }^{L\times L}.
\label{eq:attention} \end{equation}
Multi-head SA layers use several SA heads in parallel to allow the learning of different kinds of dependencies: \begin{align}
\operatorname{MSA}(\boldsymbol{X}):=\sum_{h=1}^{N_h} \left[\text{SA}_{h}(\boldsymbol{X})\right] \boldsymbol{W}^h_{out}, \quad\quad \text{SA}_h(\boldsymbol{X}) := \boldsymbol{A}^h \boldsymbol{X} \boldsymbol{W}_{val}^h, \end{align} where $\boldsymbol W_\text{val}^h \in R^{D_{in}\times D_{v}}$ and $\boldsymbol W_\text{out}^h \in R^{D_{v}\times D_{out}}$ are two learnable projections.
To incorporate positional information, ViTs usually add absolute position information to the input at embedding time, before propagating it through the SA layers. Another possibility is to replace the vanilla SA with positional SA (PSA), including a position-dependent term in the softmax~\cite{ramachandran2019stand, shaw2018self}. Although there are several way to parametrize the positional attention, we use encodings $\boldsymbol r_{ij}$ of the relative position of pixels $i$ and $j$ as in~\cite{cordonnier2019relationship,d2021convit}: \begin{align}
\boldsymbol{A}^h_{ij}:=\operatorname{softmax}\left(\boldsymbol Q^h_i \boldsymbol K^{h\top}_j+\boldsymbol{v}_{pos}^{h\top} \boldsymbol{r}_{ij}\right). \label{eq:local-attention} \end{align} Each attention head learns an embedding $\boldsymbol{v}_{pos}^h \in \mathbb R^{D_{pos}}$, and the relative positional encodings $\boldsymbol{r}_{ij}\in \mathbb R^{D_{pos}}$ only depend on the distance between pixels $i$ and $j$, denoted denoted as a two-dimensional vector $\boldsymbol \delta_{ij}$.
\paragraph{Self-attention as a generalized convolution}
\citet{cordonnier2019relationship} shows that a multi-head PSA layer (Eq. \ref{eq:local-attention}) with $N_h$ heads and dimension $D_{pos}\geq 3$ can express any convolutional layer of filter size $\sqrt N_h$, with $D_{in}$ input channels and $\min(D_v, D_{out})$ output channels, by setting the following: \begin{align}
\begin{cases}
&\boldsymbol{v}_{pos}^{h}:=-\alpha^{h}\left(1,-2 \Delta_{1}^{h},-2 \Delta_{2}^{h}, 0,\ldots 0\right)\\ &\boldsymbol{r}_{\boldsymbol{\delta}}:=\left(\|\boldsymbol{\delta}\|^{2}, \delta_{1}, \delta_{2},0,\ldots 0\right)\\
&\boldsymbol{W}_{q r y}=\boldsymbol{W}_{k e y}:=\mathbf{0}
\end{cases}
\label{eq:local-init} \end{align}
In the above, the \emph{center of attention} $\boldsymbol \Delta^h\in\mathbb{R}^2$ is the position to which head $h$ pays most attention to, relative to the query pixel, whereas the \emph{locality strength} $\alpha^h>0$ determines how focused the attention is around its center $\boldsymbol \Delta^h$. When $\alpha^h$ is large, the attention is focused only on the pixel located at $\boldsymbol \Delta^h$; when $\alpha^h$ is small, the attention is spread out into a larger area. Thus, the PSA layer can achieve a convolutional attention map by setting the centers of attention $\boldsymbol \Delta^h$ to each of the possible positional offsets of a $\sqrt{N_h}\times \sqrt{N_h}$ convolutional kernel, and sending the locality strengths $\alpha^h$ to some large value.
\section{Approach}
In this section, we introduce our method for mapping a convolutional layer to a functionally equivalent PSA layer with minimal increase in parameter count. To do this, we leverage the GPSA layers introduced in~\citet{d2021convit}.
\paragraph{Loading the filters}
We want each head $h$ of the PSA layer to functionally mimic the pixel $h$ of a convolutional filter $\boldsymbol W_\text{filter} \in \mathbb{R}^{N_h \times D_{in} \times D_{out}}$, where we typically have $D_{out}\geq D_{in}$. Rewriting the action of the MHSA operator in a more explicit form, we have \begin{equation}
\operatorname{MHSA}(\boldsymbol{X})=\sum_{h =1}^{N_h} \boldsymbol{A}^{h} \boldsymbol{X} \underbrace{\boldsymbol{W}_{\text {val }}^{h} \boldsymbol{W}_{\text {out }}^h}_{\boldsymbol{W}^{h}\in\mathbb{R}^{D_{in}\times D_{out}}} \end{equation} In the convolutional configuration of Eq.~\ref{eq:local-init}, $\boldsymbol A^h \boldsymbol X$ selects pixel $h$ of $\boldsymbol X$. Hence, we need to set $\boldsymbol W^h=\boldsymbol W_\text{filter}^h$. However, as a product of matrices, the rank of $\boldsymbol W_h$ is bottlenecked by $D_v$. To avoid this being a limitation, we need $D_v \geq D_{in}$ (since $D_{out}\geq D_{in}$). To achieve this with a minimal number of parameters, we choose $D_v = D_{in}$, and simply set the following initialization: \begin{align}
\boldsymbol W_\text{val}^h = \boldsymbol I, \quad\quad\quad \boldsymbol W_\text{out}^h = \boldsymbol W_\text{filter}^h. \end{align} Note that this differs from the usual choice made in SA layers, where $D_v = \lfloor D_{in}/N_h \rfloor$. However, to keep the parameter count the same, we share the same $\boldsymbol W_{val}^h$ across different heads $h$, since it plays a symmetric role at initialization.
Note that this reparameterization introduces three additional matrices compared to the convolutional filter: $\boldsymbol W_{qry}, \boldsymbol W_{key}, \boldsymbol W_{val}$, each containing $ D_{in} \times D_{in}$ parameters. However, since the convolutional filter contains $N_h \times D_{in} \times D_{out}$ parameters, where we typically have $N_h=9$ and $D_{out}\in\{D_{in},2D_{in}\}$, these additional matrices are much smaller than the filters and hardly increase the parameter count. This can be seen from the model sizes in Tab.~\ref{tab:robustness}.
\paragraph{Gated Positional self-attention} Recent work~\cite{d2021convit} has highlighted an issue with standard PSA: the fact that the content and positional terms in Eq.~\ref{eq:local-attention} are potentially of very different magnitudes, in which case the softmax ignores the smallest of the two. This can typically lead the PSA to adopt a greedy attitude: choosing the form of attention (content or positional) which is easiest at a given time then sticking to it.
To avoid this, the ConViT~\citet{d2021convit} uses GPSA layers which sum the content and positional terms \emph{after} the softmax, with their relative importances governed by a learnable \emph{gating} parameter $\lambda_h$ (one for each attention head). In GPSA layers, the attention is parametrized as follows: \begin{align}
\boldsymbol{A}^h_{ij}:=&\left(1-\sigma(\lambda_h)\right) \operatorname{softmax}\left(\boldsymbol Q^h_i \boldsymbol K^{h\top}_j\right) + \sigma(\lambda_h) \operatorname{softmax}\left(\boldsymbol{v}_{pos}^{h\top} \boldsymbol{r}_{ij}\right),
\label{eq:gating-param} \end{align} where $\sigma:x\mapsto \nicefrac{1}{\left(1+e^{-x}\right)}$ is the sigmoid function. In the positional part, the encodings $\boldsymbol r_{ij}$ are fixed rather than learnt (see Eq.~\ref{eq:local-init}), which makes changing input resolution straightforward (see SM.~\ref{app:resolution}) and leaves only 3 learnable parameters per head: $\boldsymbol \Delta_1, \boldsymbol \Delta_2$ and $\alpha$\footnote{Since $\alpha$ represents the temperature of the softmax, its value must stay positive at all times. To ensure this, we instead learn a rectified parameter $\tilde \alpha$ using the softplus function: $\alpha = \frac{1}{\beta} \log (1+e^{-\beta \tilde \alpha})$, with $\beta=5$.}.
\paragraph{How convolutional should the initialization be?}
The convolutional initialization of GPSA layers involves two parameters, determining how strictly convolutional the behavior is: the initial value of the \emph{locality strength} $\alpha$, which determines how focused each attention head is on its dedicated pixel, and the initial value of the \emph{gating parameters} $\lambda$, which determines the importance of the positional information versus content. If $\lambda_h\gg 0$ and $\alpha\gg 1$, the T-CNN will perfectly reproduce the input-output function of the CNN, but may stay stuck in the convolutional configuration. Conversely, if $\lambda_h\ll 0$ and $\alpha\ll 1$, the T-CNN will poorly reproduce the input-output function of the CNN. Hence, we choose $\alpha =1$ and $\lambda = 1$ to lie in between these two extremes. This puts the T-CNN ``on the verge of locality'', enabling it to escape locality effectively throughout training.
\paragraph{Architectural details} To make our setup as canonical as possible, we focus on ResNet architectures~\cite{he2016deep}, which contain 5 stages, with spatial resolution halfed and number of channels doubled at each stage. Our method involves reparameterizing $3\times 3$ convolutions as GPSA layers with 9 attention heads. However, global SA is too costly in the first layers, where the spatial resolution is large. We therefore only reparameterize the last stage of the architecture, while replacing the first stride-2 convolution by a stride-1 convolution, exactly as in~\cite{srinivas2021bottleneck}. We also add explicit padding layers to account for the padding of the original convolutions.
\section{Performance of the Transformed CNNs} \label{sec:finetuning}
In this section, we apply our reparametrization to state-of-the-art CNNs, then fine-tune the resulting T-CNNs to learn better representations. This method allows to fully disentangle the training of the SA layers from that of the convolutional backbone, which is of practical interest for two reasons. First, it minimizes the time spent training the SA layers, which typically have a slower throughput. Second, it separates the algorithmic choices of the CNN backbone from those of the SA layers, which are typically different; for example, CNNs are typically trained with SGD whereas SA layers perform much better with adaptive optimizers such as Adam~\cite{zhang2019adaptive}, an incompatibility which may limit the performance of usual hybrid models.
\paragraph{Training details} To minimize computational cost, we restrict the fine-tuning to 50 epochs\footnote{We study how performance depends on the number of fine-tuning epochs in SM.~\ref{app:epochs}.}. Following~\cite{zhang2019adaptive}, we use the AdamW optimizer, with a batch size of 1024\footnote{Confirming the results of~\cite{zhang2019adaptive}, we obtained worse results with SGD.}. The learning rate is warmed up to $10^{-4}$ then annealed using a cosine decay. To encourage the T-CNN to escape the convolutional configuration and learn content-based attention, we use a larger learning rate of 0.1 for the gating parameters of Eq.~\ref{eq:gating-param} (one could equivalently decrease the temperature of the sigmoid function).
We use the same data augmentation scheme as the DeiT~\cite{touvron2020training}, as well as rather large stochastic depth coefficients $d_r$ reported in Tab.~\ref{tab:finetune}. Hoping that our method could be used as an alternative to the commonly used practice of fine-tuning models at higher resolution, we also increase the resolution during fine-tuning~\cite{touvron2019fixing}. In this setting, a ResNet50 requires only 6 hours of fine-tuning on 16 V100 GPUs, compared to 33 hours for the original training. For our largest model (ResNet350-RS), the fine-tuning lasts 50 hours.
\paragraph{Performance gains}
\begin{figure}
\caption{ImageNet-1k}
\caption{Robustness benchmarks}
\caption{\textbf{T-CNNs present better speed-accuracy trade-offs than the CNNs they stem from.} Total training time (original training + finetuning) is normalized by the total training time of the ResNet50-RS. Inference throughput is the number of images processed per second on a V100 GPU at batch size 32.}
\label{fig:pareto}
\end{figure}
We applied our method to pre-trained ResNet-RS~\cite{bello2021revisiting} models, using the weights provided by the timm package~\cite{rw2019timm}. These models are derived from the original ResNet~\cite{he2016deep}, but use improved architectural features and training strategies, enabling them to reach better speed-accuracy trade-offs than EfficientNets. Results are presented in Tab.~\ref{tab:finetune}, where we also report the baseline improvement of fine-tuning in the same setting but without SA. In all cases, our fine-tuning improves top-1 accuracy, with a significant gap over the baseline. To demonstrate the wide applicability of our method, we report similar improvements for ResNet-D architectures in SM.~\ref{app:resnetd}.
Despite the extra fine-tuning epochs and their slower throughput, the resulting T-CNNs match the performance of the original CNNs at equal throughput, while significantly outperforming them at equal total training time, as shown in the Pareto curves of Fig.~\ref{fig:pareto}(a)\footnote{We estimated the training times of the original ResNet-RS models based on their throughput, for the same hardware as used for the T-ResNet-RS.}. However, the major benefit of the reparametrization is in terms of robustness, as shown in Fig.~\ref{fig:pareto}(b) and explained below.
\begin{table}[tb]
\centering
\begin{tabular}{{c|cc|cc|cc|cc|cc}}
\toprule
\multirow{3}{*}{\textbf{Backbone}} & \multicolumn{4}{c|}{Training} & \multicolumn{6}{c}{Fine-tuning} \\\cline{2-11}
\rule{0pt}{3ex}
&&&&&&& \multicolumn{2}{c|}{Without SA} & \multicolumn{2}{c}{With SA} \\
& Res. & $d_r$ & TTT & Top-1 & Res. & $d_r$ & TTT & Top-1 & TTT & Top-1 \\
\midrule
ResNet50-RS & 160 & 0.0 & 1 (ref.) & 78.8 & 224 & 0.1 & 1.16 & 80.4 & 1.30 & \textbf{81.0}\\
ResNet101-RS & 192 & 0.0 & 1.39 & 80.3 & 224 & 0.1 & 1.65 & 81.9 & 1.79 & \textbf{82.4}\\
ResNet152-RS & 256 & 0.0 & 3.08 & 81.2 & 320 & 0.2 & 3.75 & 83.4 & 4.13 & \textbf{83.7}\\
ResNet200-RS & 256 & 0.1 & 4.15 & 82.8 & 320 & 0.2 & 5.04 & 83.7 & 5.42 & \textbf{84.0}\\
ResNet270-RS & 256 & 0.1 & 6.19 & 83.8 & 320 & 0.2 & 7.49 & 83.9 & 7.98 & \textbf{84.3}\\
ResNet350-RS & 288 & 0.1 & 10.49 & 84.0 & 320 & 0.2 & 12.17 & 84.1 & 12.69& \textbf{84.5}\\
\bottomrule
\end{tabular}
\caption{\textbf{Statistics of the models considered, trained from scratch on ImageNet.} Top-1 accuracy is measured on ImageNet-1k validation set. ``TTT'' stands for total training time (including fine-tuning), normalized by the total training time of the ResNet50-RS. $d_r$ is the stochastic depth coefficient used for the various models.
}
\label{tab:finetune} \end{table}
\begin{figure}
\caption{\textbf{Robustness is most improved for strong and blurry corruption categories.} We report the relative improvement between the top-1 accuracy of the T-ResNet50-RS and that of the ResNet50-RS on ImageNet-C, averaging over the different corruption categories (left) and corruption severities (right).}
\label{fig:robustness}
\end{figure}
\newcolumntype{?}{!{\vrule width 1pt}} \begin{table}
\centering
\begin{tabular}{c|cccc|cccc}
\toprule
\textbf{Model} & Res. & Params & Speed & Flops & ImNet-1k & ImNet-C & ImNet-A & ImNet-R\\\midrule
\multicolumn{9}{c}{Transformers}\\\midrule
ViT-B/16 & 224 & 86 M & 182 & 16.9 & 77.9 & 52.2 & 7.0 & 21.9 \\
ViT-L/16 & 224 & 307 M & 55 & 59.7 & 76.5 & 49.3 & 6.1 & 17.9 \\\midrule
DeiT-S & 224 & 22 M & 544 & 4.6 & 79.9 & 55.4 & 18.9 & 31.0 \\
DeiT-B & 224 & 87 M & 182 & 17.6 & 82.0 & 60.7 & 27.4 & 34.6 \\\midrule
ConViT-S & 224 & 28 M & 296 & 5.4 & 81.5 & 59.5 & 24.5 & 34.0 \\
ConViT-B & 224 & 87 M & 139 & 17.7 & 82.4 & \textbf{61.9} & 29.0 & 36.9 \\\midrule
\multicolumn{9}{c}{CNNs}\\\midrule
ResNet50 & 224 & 25 M & 736 & 4.1 & 76.8 & 46.1 & 4.2 & 21.5 \\
ResNet101 & 224 & 45 M & 435 & 7.85 & 78.0 & 50.2 & 6.3 & 23.0 \\
ResNet101x3 & 224 & 207 M & 62 & 69.6 & 80.3 & 53.4 & 9.1 & 24.5 \\
ResNet152x4 & 224 & 965 M & 18 & 183.1& 80.4 & 54.5 & 11.6 & 25.8 \\
\midrule
ResNet50-RS & 160 & 36 M & 938 & 4.6& 78.8 & 36.8 & 5.7 & 39.1 \\
ResNet101-RS & 192 & 64 M & 674 & 12.1& 80.3 & 44.1 & 11.8 & 44.8 \\
ResNet152-RS & 256 & 87 M & 304 & 31.2& 81.2 & 49.9 & 23.4 & 45.9 \\
ResNet200-RS & 256 & 93 M & 225 & 40.4& 82.8 & 49.3 & 25.4 & 48.1 \\
ResNet270-RS & 256 & 130 M & 152 & 54.2& 83.8 & 53.6 & 26.6 & 48.7 \\
ResNet350-RS & 288 & 164 M & 89 & 87.5& 84.0 & 53.9 & 34.9 & 49.7 \\
\midrule
\multicolumn{9}{c}{Our transformed CNNs}\\\midrule
T-ResNet50-RS & 224 & 38 M & 447 & 17.6 & 81.0 & 48.0 & 18.7 & 42.9 \\
T-ResNet101-RS & 224 & 66 M & 334 & 25.1 & 82.4 & 52.9 & 27.7 & 47.8 \\
T-ResNet152-RS & 320 & 89 M & 128 & 65.8 & 83.7 & 54.5 & 39.8 & 50.6 \\
T-ResNet200-RS & 320 & 96 M & 105 & 80.2 & 84.0 & 57.0 & 41.2 & 51.1 \\
T-ResNet270-RS & 320 & 133 M & 75 & 107.2 & 84.3 & 58.6 & 43.7 & 51.4 \\
T-ResNet350-RS & 320 & 167 M & 61 & 130.5 & \textbf{84.5}& 59.2 & \textbf{44.8} & \textbf{53.8}\\
\bottomrule
\end{tabular}
\caption{\textbf{Accuracy of our models on various benchmarks.} Throughput is the number of images processed per second on a V100 GPU at batch size 32. The ViT and ResNet results are reported in~\cite{bhojanapalli2021understanding}. For ImageNet-C, we keep a resolution of 224 at test time to avoid distorting the corruptions.
}
\label{tab:robustness} \end{table}
\paragraph{Robustness gains}
Recent work~\cite{bhojanapalli2021understanding,mao2021rethinking} has shown that Transformer-based architectures are more robust to input perturbations than convolutional architectures. We therefore investigate whether our fine-tuning procedure brings robustness gains to the original CNNs. To do so, we consider three benchmarks. First, ImageNet-C~\cite{hendrycks2019robustness}, a dataset containing 15 sets of randomly generated corruptions, grouped into 4 categories: ‘noise’, ‘blur’, ‘weather’, and ‘digital’. Each corruption type has five levels of severity, resulting in 75 distinct corruptions. Second, ImageNet-A~\cite{hendrycks2021nae}, a dataset containing naturally ``adversarial'' examples from ImageNet. Finally, we evaluate robustness to distribution shifts with ImageNet-R \cite{hendrycks2020many}, a dataset with various stylized “renditions” of ImageNet images ranging from paintings to embroidery, which strongly modify the local image statistics.
As shown in Tab.~\ref{tab:robustness} and illustrated in Fig.~\ref{fig:intro}, the T-ResNet-RS substantially outperforms the ResNet-RS on all three benchmarks. For example, our T-ResNet101-RS reaches similar or higher top-1 accuracy than the ResNet200-RS on each task, despite its lower top-1 accuracy on ImageNet-1k. This demonstrates that SA improves robustness more than it improves classification accuracy.
To better understand where the benefits come from, we decompose the improvement of the T-ResNet50-RS over the various corruption severeties and categories of ImageNet-C in Fig.~\ref{fig:robustness}. We observe that improvement increases almost linearly with corruption severity. Although performance is higher in all corruption categories, there is a strong variability: the T-CNN shines particularly in tasks where the objects in the image are less sharp due to lack of contrast, bad weather or blurriness. We attribute this to the ability of SA to distinguish shapes in the image, as investigated in Sec~\ref{sec:dissection}.
\section{Dissecting the Transformed CNNs} \label{sec:dissection}
In this section, we analyze various observables to understand how the representations of a T-ResNet270-RS evolve from those of the ResNet270-RS throughout training.
\begin{figure*}
\caption{\textbf{The later layers effectively escape the convolutional configuration.} \textbf{A:} top-1 accuracy throughout the 50 epochs of fine-tuning of a T-ResNet270-RS. \textbf{B:} size of the receptive field of the various heads $h$ (thin lines), calculated as $\alpha_h^{-1}$ (see Eq.~\ref{eq:local-attention}). Thick lines represent the average over the heads. \textbf{C:} depicts how much attention the various heads $h$ (thin lines) pay to positional information, through the value of $\sigma(\lambda_h)$ (see Eq.~\ref{eq:gating-param}). Thick lines represent the average over the heads.}
\label{fig:dynamics}
\end{figure*}
\begin{figure}
\caption{Input image}
\caption{Attention maps}
\caption{\textbf{GPSA layers combine local and global attention in a complementary way.} We depicted the attention maps of the four GPSA layers of the T-ResNet270-RS, obtained by feeding the image on the left through the convolutional backbone, then selecting a query pixel in the center of the image (red box). For each head $h$, we indicate the value of the gating parameter $\sigma(\lambda_h)$ in red (see Eq.~\ref{eq:gating-param}). In each layer, at least one of the heads learns to perform content-based attention ($\sigma(\lambda_h)=0$).}
\label{fig:attention}
\end{figure}
\paragraph{Unlearn to better relearn} In Fig.~\ref{fig:dynamics}A, we display the train and test accuracy throughout training\footnote{The train accuracy is lower than the test accuracy due to the heavy data augmentation used during fine-tuning.}. The dynamics decompose into two distinct phases: accuracy dips down during the learning rate warmup phase (first 5 epochs of training), then increases back up as the learning rate is decayed.
Interestingly, as shown in SM.~\ref{app:lr}, the depth of the dip depends on the learning rate. For too small learning rates, the dip is small, but the test accuracy increases too slowly after the dip; for too large learning rates, the test accuracy increases rapidly after the dip, but the dip is too deep to be compensated for. This suggests that the T-CNN needs to ``unlearn'' to some extent, a phenomenon reminiscent of the ``catapult'' mechanism of~\citet{lewkowycz2020large} which propels models out of sharp minima to land in wider minima.
\paragraph{Escaping the convolutional representation} In Fig.~\ref{fig:dynamics}B, we show the evolution of the ``attention span'' $1/\alpha_h$ (see Eq.~\ref{eq:local-init}), which reflects the size of the receptive field of attention head $h$. On average (thick lines), this quantity increases in the first three layers, showing that the attention span widens, but variability exists among different attention heads (thin lines): some broaden their receptive field, whereas others contract it.
In Fig.~\ref{fig:dynamics}C, we show the evolution of the gating parameters $\lambda^h$ of Eq.~\ref{eq:gating-param}, which reflect how much attention head $h$ pays to position versus content. Interestingly, the first layer stays strongly convolutional on average, as $\mathbb{E}_h \sigma(\lambda_h)$ rapidly becomes close to one (thick blue line). The other layers strongly escape locality, with most attention heads focusing on content information at the end of fine-tuning.
In Fig.~\ref{fig:attention}, we display the attention maps after fine-tuning. A clear divide appears between the ``convolutional'' attention heads, which remain close to their initialization, and the ``content-based'' attention heads, which learn more complex dependencies. Notice that the attention head initially focusing on the query pixel (head 5) stays convolutional in all layers. Throughout the layers, the shape of the central object is more and more clearly visible, as observed in~\cite{caron2021emerging}. This supports the hypothesis that robustness gains obtained for blurry corruptions (see Fig.~\ref{fig:robustness}) are partly due to the ability of the SA layers to isolate objects from the background.
\section{When should one start learning the self-attention layers?}
Previous sections have demonstrated the benefits of initializing T-CNNs from pre-trained CNNs, a very compelling procedure given the wide availability of pretrained models. But one may ask: how does this compare to training a hybrid model from scratch? More generally, given a computational budget, how long should the SA layers be trained compared to the convolutional backbone?
\paragraph{Transformed CNN versus hybrid models} To answer the first question, we consider a ResNet-50 trained on ImageNet for 400 epochs. We use SGD with momentum 0.9 and a batch size of 1024, warming up the learning rate for 5 epochs before a cosine decay. To achieve a strong baseline, we use the same augmentation scheme as in~\cite{touvron2020training} for the DeiT. Results are reported in Tab.~\ref{tab:tw}. In this modern training setting, the vanilla ResNet50 reaches a solid performance of 79.04\% on ImageNet, well above the 77\% usually reported in litterature.
\begin{table}
\centering
\begin{tabular}{c|c|c|c|c}
\toprule
\textbf{Name} & $t_1$ & $t_2$ & Train time & Top-1 \\\midrule
Vanilla CNN & 400 & 0 & 2.0k mn & 79.04 \\
Vanilla CNN$\uparrow$320 & 450 & 0 & 2.4k mn & \textbf{79.78}\\\midrule
T-CNN & 400 & 50 & 2.3k mn & 79.88 \\
T-CNN$\uparrow$320 & 400 & 50 & 2.7k mn & \textbf{80.84} \\\midrule
Vanilla hybrid & 0 & 400 & 2.8k mn & 79.95 \\%\midrule
T-CNN$^\star$ & 100 & 300 & 2.6k mn & \textbf{80.44} \\
T-CNN$^\star$ & 200 & 200 & 2.4k mn & 80.28 \\
T-CNN$^\star$ & 300 & 100 & 2.2k mn & 79.28 \\
\bottomrule
\end{tabular}
\caption{\textbf{The benefit of late reparametrization.} We report the top-1 accuracy of a ResNet-50 on ImageNet reparameterized at various times $t_1$ during training. $\uparrow$320 stands for fine-tuning at resolution 320. The models with a $\star$ keep the same optimizer after reparametrization, in contrast with the usual T-CNNs.}
\label{tab:tw}
\end{table}
The T-CNN obtained by fine-tuning the ResNet for 50 epochs at same resolution obtains a top-1 accuracy of 79.88\%, with a 15\% increase in training time, and 80.84 as resolution 320, with a 35\% increase in training time. In comparison, the hybrid model trained for 400 epochs in the same setting only reaches 79.95\%, in spite of a 40\% increase in training time. Hence, fine-tuning yields better results than training the hybrid model from scratch.
\paragraph{What is the best time to reparametrize?}
We now study a scenario between the two extreme cases: what happens if we reparametrize halfway through training? To investigate this question in a systematic way, we train the ResNet50 for $t_1$ epochs, then reparametrize and resume training for another $t_2$ epochs, ensuring that $t_1+t_2=400$ in all cases. Hence, $t_1=400$, amounts to the vanilla ResNet50, whereas $t_1=0$ corresponds to the hybrid model trained from scratch. To study how final performance depends on $t_1$ in a fair setting, we keep the same optimizer and learning rate after the reparametrization, in contrast with the fine-tuning procedure which uses fresh optimizer.
Results are presented in Tab.~\ref{tab:tw}. Interestingly, the final performance evolves non-monotonically, reaching a maximum of $80.44$ for $t_1=100$, then decreasing back down as the SA layers have less and less time to learn. This non-monotonicity is remarkably similar to that observed in~\cite{d2019finding}, where reparameterizing a CNN as a FCN in the early stages of training enables the FCN to outperform the CNN. Crucially, this result suggests that reparametrizing during training not only saves time, but also helps the T-CNN find better solutions.
\section*{Discussion}
In this work, we showed that complex building blocks such as self-attention layers need not be trained from start. Instead, one can save in compute time while gaining in performance and robustness by initializing them from pre-trained convolutional layers. At a time where energy savings and robustness are key stakes, we believe this finding is important.
On the practical side, our fine-tuning method offers an interesting new direction for practitioners. One clear limitation of our method is the prohibitive cost of reparametrizing the early stages of CNNs. This cost could however be alleviated by using linear attention methods~\cite{wang2020self}, an important direction for future work. Note also that while our T-CNNs significantly improve the robustness of CNNs, they do not systematically reach the performance of end-to-end Transformers such as the DeiT (for example on ImageNet-C, see Fig.~\ref{fig:intro}). Bridging this gap is an important next step for hybrid models.
On the theoretical side, our results spark several interesting questions. First, why is it better to reparametrize at intermediate times? One natural hypothesis, which will be explored in future work, is that SA layers benefit from capturing meaningful dependencies between the features learnt by the CNN, rather than the random correlations which exist at initialization. Second, why are the representations learnt by the SA layers more robust? By inspecting the attention maps and the most improved corruption categories of ImageNet-C, we hypothesized that SA helps isolating objects from the background, but a more thorough analysis is yet to come. \paragraph{Acknowledgements} We thank Matthew Leavitt, Hugo Touvron, Hervé Jégou and Francisco Massa for helpful discussions. SD and GB acknowledge funding from the French government under management of Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute).
\printbibliography
\appendix
\numberwithin{equation}{section}
\appendix \setlength\intextsep{10pt}
\section{Changing of learning rate} \label{app:lr}
As shown in Fig.~\ref{fig:dynamics} of the main text, the learning dynamics decompose into two phases: the learning rate warmup phase, where the test loss drops, then the learning rate decay phase, where the test loss increases again. This could lead one to think that the maximal learning rate is too high, and the dip could be avoided by choosing a lower learning rate. Yet this is not the case, as shown in Fig.~\ref{fig:lr}. Reducing the maximal learning rate indeed reduces the dip, but it also slows down the increase in the second phase of learning. This confirms that the model needs to ``unlearn'' the right amount to find better solutions.
\begin{figure}
\caption{\textbf{The larger the learning rate, the lower the test accuracy dips, but the faster it climbs back up.} We show the dynamics of the ResNet50, fine-tuned for 50 epochs at resolution 224, for three different values of the maximal learning rate.}
\label{fig:lr}
\end{figure}
\section{Changing the test resolution} \label{app:resolution}
One advantage of the GPSA layers introduced by~\cite{d2021convit} is how easily they adapt to different image resolutions. Indeed, the positional embeddings they use are fixed rather than learnt. They simply consist in 3 values for each pair of pixels: their euclidean distance $\Vert \boldsymbol \delta\Vert$, as well as their coordinate distance $\boldsymbol \delta_1, \boldsymbol \delta_2$ (see Eq.~\ref{eq:local-init}). Our implementation automatically adjusts these embeddings to the input image, allowing us to change the test resolution seamlessly.
In Fig.~\ref{fig:resolution}, we show how the top-1 accuracies of our T-ResNet-RS models compares to those of the ResNet-RS models finetuned at same resolution but without SA. At test resolution 416, our T-ResNetRS-350 reaches an impressive top-1 accuracy of 84.9\%, beyond those of the best EfficientNets and BotNets~\cite{srinivas2021bottleneck}.
\begin{figure}
\caption{\textbf{Performance at different test-time resolutions, for the finetuned models with and without SA.} The ResNet50-RS and ResNet101-RS models are finetuned at resolution 224, and all other models are finetuned at resolution 320.}
\label{fig:resolution}
\end{figure}
\section{Changing the number of epochs} \label{app:epochs}
In Tab.~\ref{tab:epochs}, we show how the top-1 accuracy of the T-ResNet-RS model changes with the number of fine-tuning epochs. As expected, performance increases significantly as we fine-tune for longer, yet we chose to set a maximum of 50 fine-tuning epochs to keep the computational cost of fine-tuning well below that of the original training.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c}
\toprule
\textbf{Model} & Epochs & Top-1 acc \\\midrule
ResNet50-RS & 0 & 79.91 \\
T-ResNet50-RS & 10 & 80.11 \\
T-ResNet50-RS & 20 & 80.51 \\
T-ResNet50-RS & 50 & \textbf{81.02} \\\midrule
ResNet101-RS & 0 & 81.70 \\
T-ResNet101-RS & 10 & 81.54 \\
T-ResNet101-RS & 20 & 81.90 \\
T-ResNet101-RS & 50 & \textbf{82.39} \\
\bottomrule
\end{tabular}
\caption{\textbf{Longer fine-tuning increases final performance.} We report the top-1 accuracies of our models on ImageNet-1k at resolution 224.}
\label{tab:epochs} \end{table}
\section{Changing the architecture} \label{app:resnetd}
Our framework, which builds on the timm package, makes changing the original CNN architecture very easy. We applied our fine-tuning procedure to the ResNet-D models~\cite{he2019bag} with the exact same hyperparameters, and observed substantial performance gains, similar to the ones obtained for ResNet-RS, see Tab.~\ref{tab:finetune-resnetd}. This suggests the wide applicability of our method.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
\textbf{Model} & Original res. & Original acc. & Fine-tune res. & Fine-tune acc. & Gain\\
\midrule
T-ResNet50-D & 224 & 80.6 & 320 & 81.6 & +1.0\\
T-ResNet101-D & 320 & 82.3 & 384 & 83.1 & +0.8\\
T-ResNet152-D & 320 & 83.1 & 384 & 83.8 & +0.7\\
T-ResNet200-D & 320 & 83.2 & 384 & \textbf{83.9} & +0.7\\
\midrule
T-ResNet50-RS & 160 & 78.8 & 224 & 81.0 & +2.8\\
T-ResNet101-RS & 192 & 81.2 & 224 & 82.4 & +1.2\\
T-ResNet152-RS & 256 & 83.0 & 320 & 83.7 & +0.7\\
T-ResNet200-RS & 256 & 83.4 & 320 & \textbf{84.0} & +0.6\\
\bottomrule
\end{tabular}
\caption{\textbf{Comparing the performance gains of the ResNet-RS and ResNet-D architectures.} Top-1 accuracy is measured on ImageNet-1k validation set. The pre-trained models are all taken from the timm library~\cite{rw2019timm}.
}
\label{tab:finetune-resnetd} \end{table}
\section{More attention maps}
\begin{figure}
\caption{Attention maps}
\caption{Attention maps}
\caption{\textbf{GPSA layers combine local and global attention in a complementary way.} We depicted the attention maps of the four GPSA layers of the T-ResNet270-RS, obtained by feeding the image on the left through the convolutional backbone, then selecting a query pixel in the center of the image (red box). For each head $h$, we indicate the value of the gating parameter $\sigma(\lambda_h)$ in red (see Eq.~\ref{eq:gating-param}). ($\sigma(\lambda_h)=0$).}
\label{fig:more-attention}
\end{figure}
\end{document}
|
arXiv
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\begin{document}
\title{Faster and more accurate computation of the $\Hinf$ norm via optimization} \begin{abstract} In this paper, we propose an improved method for computing the ${\hinfsym_\infty}$ norm of linear dynamical systems that results in a code that is often several times faster than existing methods. By using standard optimization tools to rebalance the work load of the standard algorithm due to Boyd, Bala\-krish\-nan, Bru\-insma, and Stein\-buch, we aim to minimize the number of expensive eigenvalue computations that must be performed. Unlike the standard algorithm, our modified approach can also calculate the ${\hinfsym_\infty}$ norm to full precision with little extra work, and also offers more opportunity to further accelerate its performance via parallelization. Finally, we demonstrate that the local optimization we have employed to speed up the standard globally-convergent algorithm can also be an effective strategy on its own for approximating the ${\hinfsym_\infty}$ norm of large-scale systems. \end{abstract}
\section{Introduction} \label{sec:intro}
Consider the continuous-time linear dynamical system \begin{subequations} \label{eq:lti_cont} \begin{align} E\dot{x} &= Ax + Bu \\ y &= Cx + Du, \end{align} \end{subequations} where $A \in \Cmn{n}{n}$, $B \in \Cmn{n}{m}$, $C \in \Cmn{p}{n}$, $D \in \Cmn{p}{m}$, and $E \in \Cmn{n}{n}$. The system defined by \eqref{eq:lti_cont} arises in many engineering applications and as a consequence, there has been a strong motivation for fast methods to compute properties that measure the sensitivity of the system or its robustness to noise. Specifically, a quantity of great interest is the ${\hinfsym_\infty}$ norm, which is defined as \begin{equation}
\label{eq:hinf_cont}
\|G\|_{{\hinfsym_\infty}} \coloneqq \sup_{\omega \in \mathbb R} \| G({\bf i} \omega)\|_2, \end{equation} where \begin{equation}
G(\lambda) = C(\lambda E - A)^{-1} B + D \end{equation} is the associated \emph{transfer function} of the system given by \eqref{eq:lti_cont}. The ${\hinfsym_\infty}$ norm measures the maximum sensitivity of the system; in other words, the higher the value of the ${\hinfsym_\infty}$ norm, the less robust the system is, an intuitive interpretation considering that the ${\hinfsym_\infty}$ norm is in fact the reciprocal of the \emph{complex stability radius}, which itself is a generalization of the \emph{distance to instability} \cite[Section 5.3]{HinP05}. The ${\hinfsym_\infty}$ norm is also a key metric for assessing the quality of reduced-order models that attempt to capture/mimic the dynamical behavior of large-scale systems, see, e.g., \cite{morAnt05,morBenCOW17}. Before continuing, as various matrix pencils of the form $\lambda B - A$ will feature frequently in this work, we use notation $(A,B)$ to abbreviate them.
When $E = I$, the ${\hinfsym_\infty}$ norm is finite as long as $A$ is stable, whereas an unstable system would be considered infinitely sensitive. If $E\ne I$, then \eqref{eq:lti_cont} is called a \emph{descriptor system}. Assuming that $E$ is singular but $(A,E)$ is regular and at most index 1, then \eqref{eq:hinf_cont} still yields a finite value, provided that all the controllable and observable eigenvalues of $(A,E)$ are finite and in the open left half plane, where for $\lambda$ an eigenvalue of $(A,E)$ with right and left eigenvectors $x$ and $y$, $\lambda$ is considered \emph{uncontrollable} if $B^*y = 0$ and \emph{unobservable} if $Cx=0$. However, the focus of this paper is not about detecting when \eqref{eq:hinf_cont} is infinite or finite, but to introduce an improved method for computing the ${\hinfsym_\infty}$ norm when it is finite. Thus, for conciseness in presenting our improved method, we will assume in this paper that any system provided to an algorithm has a finite ${\hinfsym_\infty}$ norm, as checking whether it is infinite can be considered a preprocessing step.\footnote{We note that while our proposed improvements are also directly applicable for computing the ${\linfsym_\infty}$ norm, we will restrict the discussion here to just the ${\hinfsym_\infty}$ norm for brevity.}
While the first algorithms \cite{BoyBK89, BoyB90,BruS90} for computing the ${\hinfsym_\infty}$ norm date back to nearly 30 years ago, there has been continued interest in improved methods, particularly as the state-of-art methods remain quite expensive with respective to their dimension $n$, meaning that computing the ${\hinfsym_\infty}$ norm is generally only possible for rather small-dimensional systems. In 1998, \cite{GenVV98} proposed an interpolation refinement to the existing algorithm of \cite{BoyB90,BruS90} to accelerate its rate of convergence. In the following year, for the special case of the distance to instability where $B=C=E=I$ and $D=0$, \cite{HeW99} used an inverse iteration to successively obtain increasingly better locally optimal approximations as a way of reducing the number of expensive Hamiltonian eigenvalue decompositions; as we will discuss in Section~\ref{sec:hybrid_opt}, this method shares some similarity with the approach we propose here. More recently, in 2011, \cite{BelP11} presented an entirely different approach to computing the ${\hinfsym_\infty}$ norm, by finding isolated common zeros of two certain bivariate polynomials. While they showed that their method was much faster than an implementation of \cite{BoyB90,BruS90} on two SISO examples (single-input, single-output, that is, $m=p=1$), more comprehensive benchmarking does not appear to have been done yet. Shortly thereafter, \cite{BenSV12} extended the now standard algorithm of \cite{BoyB90,BruS90} to descriptor systems. There also has been a very recent surge of interest in efficient ${\hinfsym_\infty}$ norm approximation methods for large-scale systems. These methods fall into two broad categories: those that are applicable for descriptor systems with possibly singular $E$ matrices but require solving linear systems \cite{BenV14,FreSV14,AliBMetal17} and those that don't solve linear systems but require that $E=I$ or that $E$ is at least cheaply inverted \cite{GugGO13,MitO16}. Our contribution in this paper is twofold. First, we improve upon the exact algorithms of \cite{BoyB90,BruS90,GenVV98} to not only compute the ${\hinfsym_\infty}$ norm significantly faster but also obtain its value to machine precision with negligible additional expense (a notable difference compared to these earlier methods). This is accomplished by incorporating local optimization techniques within these algorithms, a change that also makes our new approach more amenable to additional acceleration via parallelization. Second, that standard local optimization can even be used on its own to efficiently obtain locally optimal approximations to the ${\hinfsym_\infty}$ norm of large-scale systems, a simple and direct approach that has surprisingly not yet been considered and is even embarrassingly parallelizable.
The paper is organized as follows. In Sections~\ref{sec:bbbs} and \ref{sec:costs}, we describe the standard algorithms for computing the ${\hinfsym_\infty}$ norm and then give an overview of their computational costs. In Section~\ref{sec:hybrid_opt}, we introduce our new approach to computing the ${\hinfsym_\infty}$ norm via leveraging local optimization techniques. Section~\ref{sec:discrete} describes how the results and algorithms are adapted for discrete-time problems. We present numerical results in Section~\ref{sec:numerical} for both continuous- and discrete-time problems. Section~\ref{sec:hinf_approx} provides the additional experiments demonstrating how local optimization can also be a viable strategy for approximating the ${\hinfsym_\infty}$ norm of large-scale systems. Finally, in Section~\ref{sec:parallel} we discuss how significant speedups can be obtained for some problems when using parallel processing with our new approach, in contrast to the standard algorithms, which benefit very little from multiple cores. Concluding remarks are given in Section~\ref{sec:wrapup}.
\section{The standard algorithm for computing the ${\hinfsym_\infty}$ norm} \label{sec:bbbs} We begin by presenting a key theorem relating the singular values of the transfer function to purely imaginary eigenvalues of an associated matrix pencil. For the case of simple ODEs, where $B=C=E=I$ and $D=0$, the result goes back to \cite[Theorem 1]{Bye88}, and was first extended to linear dynamical systems with input and output with $E=I$ in \cite[Theorem 1]{BoyBK89}, and then most recently generalized to systems where $E\ne I$ in \cite[Theorem 1]{BenSV12}. We state the theorem without the proof, since it is readily available in \cite{BenSV12}.
\begin{theo} \label{thm:eigsing_cont} Let $\lambda E - A$ be regular with no finite eigenvalues on the imaginary axis, $\gamma > 0$ not a singular value of $D$, and $\omega \in \mathbb R$. Consider the matrix pencil $(\Mc,\Nc)$, where \begin{equation}
\label{eq:MNpencil_cont}
\mathcal{M}_\gamma \coloneqq \begin{bmatrix} A - BR^{-1}D^*C & -\gamma BR^{-1}B^* \\
\gamma C^*S^{-1}C & -(A - BR^{-1}D^*C)^* \end{bmatrix}
~\text{and}~
\mathcal{N} \coloneqq \begin{bmatrix} E & 0\\ 0 & E^*\end{bmatrix}
\end{equation} and $R = D^*D - \gamma^2 I$ and $S = DD^* - \gamma^2 I$. Then ${\bf i} \omega$ is an eigenvalue of matrix pencil $(\Mc,\Nc)$ if and only if $\gamma$ is a singular value of $G({\bf i} \omega)$. \end{theo}
Theorem~\ref{thm:eigsing_cont} immediately leads to an algorithm for computing the ${\hinfsym_\infty}$ norm based on computing the imaginary eigenvalues, if any, of the associated matrix pencil \eqref{eq:MNpencil_cont}. For brevity in this section, we assume that $\max \|G({\bf i} \omega)\|_2$ is not attained at $\omega = \infty$, in which case the ${\hinfsym_\infty}$ norm would be $\|D\|_2$.
Evaluating the norm of the transfer function for any finite frequency along the imaginary axis immediately gives a lower bound to the ${\hinfsym_\infty}$ norm while an upper bound can be obtained by successively increasing $\gamma$ until the matrix pencil given by \eqref{eq:MNpencil_cont} no longer has any purely imaginary eigenvalues. Then, it is straightforward to compute the ${\hinfsym_\infty}$ norm using bisection, as first proposed in \cite{BoyBK89} and was inspired by the breakthrough result of \cite{Bye88} for computing the \emph{distance to instability} (i.e. the reciprocal of the ${\hinfsym_\infty}$ norm for the special case of $B=C=E=I$ and $D=0$).
As Theorem~\ref{thm:eigsing_cont} provides a way to calculate all the frequencies where $\|G({\bf i} \omega)\|_2 = \gamma$, it was shortly thereafter proposed in \cite{BoyB90, BruS90} that instead of computing an upper bound and then using bisection, the initial lower bound could be successively increased in a monotonic fashion to the value of the ${\hinfsym_\infty}$ norm. For convenience, it will be helpful to establish the following notation for the transfer function and its largest singular value, both as parameters of frequency $\omega$: \begin{align}
\label{eq:tf_cont}
G_\mathrm{c}(\omega) {}& \coloneqq G({\bf i} \omega) \\
\label{eq:ntf_cont}
g_\mathrm{c}(\omega) {}& \coloneqq \| G({\bf i} \omega) \|_2 = \| G_\mathrm{c}(\omega) \|_2. \end{align} Let $\{ \omega_1,\ldots,\omega_l\}$ be the set of imaginary parts of the purely imaginary eigenvalues of \eqref{eq:MNpencil_cont} for the initial value $\gamma$, sorted in increasing order. Considering the intervals $I_k = [\omega_k,\omega_{k+1}]$, \cite{BruS90} proposed increasing $\gamma$ via: \begin{equation}
\label{eq:gamma_mp}
\gamma_\mathrm{mp} = \max g_\mathrm{c}(\hat\omega_k)
\qquad \text{where} \qquad
\hat\omega_k ~\text{are the midpoints of the intervals}~ I_k. \end{equation} Simultaneously and independently, a similar algorithm was proposed by \cite{BoyB90}, with the additional results that (a) it was possible to calculate which intervals $I_k$ satisfied $g_\mathrm{c}(\omega) \ge \gamma$ for all $\omega \in I_k$, thus reducing the number of evaluations of $g_\mathrm{c}(\omega)$ needed at every iteration, and (b) this midpoint scheme actually had a local quadratic rate of convergence, greatly improving upon the linear rate of convergence of the earlier, bisection-based method. This midpoint-based method, which we refer to as the BBBS algorithm for its authors Boyd, Bala\-krish\-nan, Bru\-insma, and Stein\-buch, is now considered the standard algorithm for computing the ${\hinfsym_\infty}$ norm and it is the algorithm implemented in the {MATLAB}\ Robust Control Toolbox, e.g. routine \texttt{hinfnorm}. Algorithm~\ref{alg:bbbs} provides a high-level pseudocode description for the standard BBBS algorithm while Figure~\ref{fig:bbbs_std} provides a corresponding pictorial description of how the method works.
\begin{algfloat} \begin{algorithm}[H] \floatname{algorithm}{Algorithm} \caption{The Standard BBBS Algorithm} \label{alg:bbbs} \begin{algorithmic}[1]
\REQUIRE{
$A \in \Cmn{n}{n}$, $B \in \Cmn{n}{m}$, $C \in \Cmn{p}{n}$, $D \in \Cmn{p}{m}$,
$E \in \Cmn{n}{n}$ and $\omega_0 \in \mathbb R$.
}
\ENSURE{
$\gamma = \| G \|_{\hinfsym_\infty}$ and $\omega$ such that
$\gamma = g_\mathrm{c}(\omega)$.
\\ \quad
}
\STATE $\gamma = g_\mathrm{c}(\omega_0)$
\WHILE {not converged}
\STATE \COMMENT{Compute the intervals that lie under $g_\mathrm{c}(\omega)$ using
eigenvalues of the pencil:}
\STATE Compute $\Lambda_\mathrm{I} =
\{ \Im \lambda : \lambda \in \Lambda(\Mc,\Nc) ~\text{and}~ \Re \lambda = 0\}$.
\STATE Index and sort $\Lambda_\mathrm{I} = \{\omega_1,...,\omega_l\}$ s.t.
$\omega_j \le \omega_{j+1}$.
\STATE Form all intervals $I_k = [\omega_k, \omega_{k+1}]$ s.t.
each interval at height $\gamma$ is below $g_\mathrm{c}(\omega)$.
\label{algline:bbbs_ints}
\STATE \COMMENT{Compute candidate frequencies of the level-set intervals $I_k$:}
\STATE Compute midpoints $\hat\omega_k = 0.5(\omega_k +\omega_{k+1})$ for each interval $I_k$.
\label{algline:bbbs_points}
\STATE \COMMENT{Update to the highest gain evaluated at these candidate frequencies:}
\STATE $\omega = \argmax g_\mathrm{c}(\hat\omega_k)$.
\STATE $\gamma = g_\mathrm{c}(\omega)$.
\ENDWHILE \end{algorithmic} \end{algorithm} \algnote{ The quartically converging variant proposed by \cite{GenVV98} replaces the midpoints of $I_k$ with the maximizing frequencies of Hermite cubic interpolants, which are uniquely determined by interpolating the values of $g_\mathrm{c}(\omega)$ and $g_\mathrm{c}^\prime(\omega)$ at both endpoints of each interval $I_k$. } \end{algfloat}
\begin{figure}
\caption{The blue curves show the value of $g_\mathrm{c}(\omega) = \|G({\bf i} \omega)\|_2$ while the red circles mark the frequencies $\{\omega_1,\ldots,\omega_l\}$ where $g_\mathrm{c}(\omega) = \gamma = 1.4$, computed by taking the imaginary parts of the purely imaginary eigenvalues of \eqref{eq:MNpencil_cont} on a sample problem. The left plot shows the midpoint scheme of the standard BBBS algorithm to obtain an increased value for $\gamma$, depicting by the dashed horizontal line. The right plot shows the cubic interpolation refinement of \cite{GenVV98} for the same problem and initial value of $\gamma$. The dashed black curves depict the cubic Hermite interpolants for each interval while the dotted vertical lines show their respective maximizing frequencies. As can be seen, the cubic interpolation scheme results in a larger increase in $\gamma$ (again represented by the dashed horizontal line) compared to the standard BBBS method on this iterate for this problem. }
\label{fig:bbbs_std}
\label{fig:bbbs_interp}
\label{fig:bbbs_plots}
\end{figure}
In \cite{GenVV98}, a refinement to the BBBS algorithm was proposed which increased its local quadratic rate of convergence to quartic. This was done by evaluating $g_\mathrm{c}(\omega)$ at the maximizing frequencies of the unique cubic Hermite interpolants for each level-set interval, instead of at the midpoints. That is, for each interval $I_k = [\omega_k,\omega_{k+1}]$, the unique interpolant $c_k(\omega) = c_3\omega^3 + c_2\omega^2 + c_1\omega + c_0$ is constructed so that \begin{equation} \label{eq:interp_cont} \begin{aligned}
c_k(\omega_k) &= g_\mathrm{c}(\omega_k) \\
c_k(\omega_{k+1}) &= g_\mathrm{c}(\omega_{k+1}) \end{aligned} \qquad \text{and} \qquad \begin{aligned}
c^\prime_k(\omega_k) &= g_\mathrm{c}^\prime(\omega_k) \\
c^\prime_k(\omega_{k+1}) &= g_\mathrm{c}^\prime(\omega_{k+1}). \end{aligned} \end{equation} Then, $\gamma$ is updated via \begin{equation}
\label{eq:gamma_cubic}
\gamma_\mathrm{cubic} = \max g_\mathrm{c}(\hat\omega_k)
\qquad \text{where} \qquad
\hat\omega_k = \argmax_{\omega \in I_k} c_k(\omega), \end{equation} that is, $\hat\omega_k$ is now the maximizing value of interpolant $c_k(\omega)$ on its interval $I_k$, which of course can be cheaply and explicitly computed. In \cite{GenVV98}, the single numerical example shows the concrete benefit of this interpolation scheme, where the standard BBBS algorithm required six eigenvalue decompositions of \eqref{eq:MNpencil_cont} to converge, while their new method only required four. As only the selection of the $\omega_k$ values is different, the pseudocode for this improved version of the BBBS algorithm remains largely the same, as mentioned in the note of Algorithm~\ref{alg:bbbs}. Figure~\ref{fig:bbbs_interp} provides a corresponding pictorial description of the cubic-interpolant-based refinement.
As it turns out, computing the derivatives in \eqref{eq:interp_cont} for the Hermite interpolations of each interval can also be done with little extra work. Let $u(\omega)$ and $v(\omega)$ be the associated left and right singular vectors corresponding to $g_\mathrm{c}(\omega)$, recalling that $g_\mathrm{c}(\omega)$ is the largest singular value of $G_\mathrm{c}(\omega)$, and assume that $g_\mathrm{c}(\hat\omega)$ is a simple singular value for some value $\hat\omega \in \mathbb R$. By standard perturbation theory (exploiting the equivalence of singular values of a matrix $A$ and eigenvalues of $\left[ \begin{smallmatrix} 0 & A \\ A^* & 0\end{smallmatrix} \right]$ and applying \cite[Theorem 5]{Lan64}), it then follows that \begin{equation}
\label{eq:ntfcprime_cont}
g_\mathrm{c}^\prime(\omega) \Big\rvert_{\omega=\hat\omega} = \Real{ u(\hat\omega)^* G_\mathrm{c}^\prime(\hat\omega) v(\hat\omega) }, \end{equation} where, by standard matrix differentiation rules with respect to parameter $\omega$, \begin{equation}
G_\mathrm{c}^\prime(\omega) = - {\bf i} C \left({\bf i} \omega E - A \right)^{-1} E \left({\bf i} \omega E - A \right)^{-1} B. \end{equation} As shown in \cite{SreVT95}, it is actually fairly cheap to compute \eqref{eq:ntfcprime_cont} if the eigenvectors corresponding to the purely imaginary eigenvalues of \eqref{eq:MNpencil_cont} have also been computed, as there is a correspondence between these eigenvectors and the associated singular vectors for $\gamma$. For $\gamma = g_\mathrm{c}(\hat\omega)$, if $\left[ \begin{smallmatrix} q \\ s \end{smallmatrix} \right]$ is an eigenvector of \eqref{eq:MNpencil_cont} for imaginary eigenvalue ${\bf i} \hat\omega$, then the equivalences \begin{equation}
\label{eq:qs_vecs}
q = \left( {\bf i} \hat\omega E - A \right)^{-1}Bv(\hat\omega)
\quad \text{and} \quad
s = \left( {\bf i} \hat\omega E - A \right)^{-*}C^*u(\hat\omega) \end{equation} both hold, where $u(\hat\omega)$ and $v(\hat\omega)$ are left and right singular vectors associated with singular value $g_\mathrm{c}(\hat\omega)$. (To see why these equivalences hold, we refer the reader to the proof of Theorem~\ref{thm:eigsing_disc} for the discrete-time analog result.) Thus, \eqref{eq:ntfcprime_cont} may be rewritten as follows: \begin{align}
\label{eq:ntfcprime_defn_cont}
g_\mathrm{c}^\prime(\omega)\Big\rvert_{\omega=\hat\omega} &=
\Real{ u(\hat\omega)^* G_\mathrm{c}^\prime(\hat\omega) v(\hat\omega)} \\
\label{eq:ntfcprime_direct_cont}
&= -\Real{ u(\hat\omega)^* {\bf i} C \left({\bf i} \hat\omega E - A \right)^{-1} E \left({\bf i} \hat \omega E - A \right)^{-1} B v(\hat\omega)} \\
\label{eq:ntfcprime_eig_cont}
&= -\Real{{\bf i} s^* E q}, \end{align} and it is thus clear that \eqref{eq:ntfcprime_cont} is cheaply computable for all the endpoints of the intervals $I_k$, provided that the eigenvalue decomposition of \eqref{eq:MNpencil_cont} has already been computed.
\section{The computational costs involved in the BBBS algorithm} \label{sec:costs} The main drawback of the BBBS algorithm is its algorithmic complexity, which is ${\cal O}(n^3)$ work per iteration. This not only limits the tractability of computing the ${\hinfsym_\infty}$ norm to rather low-dimensional (in $n$) systems but can also make computing the ${\hinfsym_\infty}$ norm to full precision an expensive proposition for even moderately-sized systems. In fact, the default tolerance for \texttt{hinfnorm} in {MATLAB}\ is set quite large, 0.01, presumably to keep its runtime as fast as possible, at the expense of sacrificing accuracy. In Table~\ref{table:hinfnorm_tol}, we report the relative error of computing the ${\hinfsym_\infty}$ norm when using \texttt{hinfnorm}'s default tolerance of $0.01$ compared to $10^{-14}$, along with the respective runtimes for several test problems, observing that computing the ${\hinfsym_\infty}$ norm to near full precision can often take between two to three times longer. While computing only a handful of the most significant digits of the ${\hinfsym_\infty}$ norm may be sufficient for some applications, this is certainly not true in general. Indeed, the source code for HIFOO \cite{BurHLetal06}, which designs ${\hinfsym_\infty}$ norm fixed-order optimizing controllers for a given open-loop system via nonsmooth optimization, specifically contains the comment regarding \texttt{hinfnorm}: ``default is .01, which is too crude". In HIFOO, the ${\hinfsym_\infty}$ norm is minimized by updating the controller variables at every iteration but the optimization method assumes that the objective function is continuous; if the ${\hinfsym_\infty}$ norm is not calculated sufficiently accurately, then it may appear to be discontinuous, which can cause the underlying optimization method to break down. Thus there is motivation to not only improve the overall runtime of computing the ${\hinfsym_\infty}$ norm for large tolerances, but also to make the computation as fast as possible when computing the ${\hinfsym_\infty}$ norm to full precision.
\begin{table} \centering
\begin{tabular}{ l | rrr | c | c | SS } \toprule \multicolumn{8}{c}{\texttt{$h$ = hinfnorm($\cdot$,1e-14)} versus \texttt{$\hat h$ = hinfnorm($\cdot$,0.01)}}\\ \midrule \multicolumn{1}{c}{} & \multicolumn{3}{c}{Dimensions} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{Relative Error} & \multicolumn{2}{c}{Wall-clock time (sec.)}\\ \cmidrule(lr){2-4} \cmidrule(lr){6-6} \cmidrule(lr){7-8} \multicolumn{1}{l}{Problem} &
\multicolumn{1}{c}{$n$} &
\multicolumn{1}{c}{$m$} &
\multicolumn{1}{c}{$p$} &
\multicolumn{1}{c}{$E=I$} &
\multicolumn{1}{c}{$ \frac{\hat h - h}{h}$} &
\multicolumn{1}{c}{\texttt{tol=1e-14}} &
\multicolumn{1}{c}{\texttt{tol=0.01}} \\ \midrule \texttt{CSE2} & 63 & 1 & 32 & Y & $-2.47 \times 10^{-4}$ & 0.137 & 0.022 \\ \texttt{CM3} & 123 & 1 & 3 & Y & $-2.75 \times 10^{-3}$ & 0.148 & 0.049 \\ \texttt{CM4} & 243 & 1 & 3 & Y & $-4.70 \times 10^{-3}$ & 1.645 & 0.695 \\ \texttt{ISS} & 270 & 3 & 3 & Y & $-1.04 \times 10^{-6}$ & 0.765 & 0.391 \\ \texttt{CBM} & 351 & 1 & 2 & Y & $-4.20 \times 10^{-5}$ & 3.165 & 1.532 \\ \texttt{randn 1} & 500 & 300 & 300 & Y & 0 & 21.084 & 30.049 \\ \texttt{randn 2} & 600 & 150 & 150 & N & $-6.10 \times 10^{-8}$ & 31.728 & 16.199 \\ \texttt{FOM} & 1006 & 1 & 1 & Y & $-1.83 \times 10^{-5}$ & 128.397 & 36.529 \\ \midrule \texttt{LAHd} & 58 & 1 & 3 & Y & $-7.36 \times 10^{-3}$ & 0.031 & 0.015 \\ \texttt{BDT2d} & 92 & 2 & 4 & Y & $-7.67 \times 10^{-4}$ & 0.070 & 0.031 \\ \texttt{EB6d} & 170 & 2 & 2 & Y & $-5.47 \times 10^{-7}$ & 0.192 & 0.122 \\ \texttt{ISS1d} & 280 & 1 & 273 & Y & $-1.53 \times 10^{-3}$ & 16.495 & 3.930 \\ \texttt{CBMd} & 358 & 1 & 2 & Y & $-2.45 \times 10^{-6}$ & 1.411 & 0.773 \\ \texttt{CM5d} & 490 & 1 & 3 & Y & $-7.38 \times 10^{-3}$ & 10.802 & 2.966 \\ \bottomrule \end{tabular} \caption{For various problems, the relative error of computing the ${\hinfsym_\infty}$ norm using \texttt{hinfnorm} with its quite loose default tolerance is shown. The first eight are continuous time problems while the last six, ending in \texttt{d}, are discrete time. As can be seen, using \texttt{hinfnorm} to compute the ${\hinfsym_\infty}$ norm to near machine accuracy can often take between two to four times longer, and that this penalty is not necessarily related to dimension: for example, the running times are increased by factors of 3.02 and 3.51 for \texttt{CM3} and \texttt{FOM}, respectively, despite that \texttt{FOM} is nearly ten times larger in dimension. } \label{table:hinfnorm_tol} \end{table}
The dominant cost of the BBBS algorithm is computing the eigenvalues of \eqref{eq:MNpencil_cont} at every iteration. Even though the method converges quadratically, and quartically when using the cubic interpolation refinement, the eigenvalues of $(\Mc,\Nc)$ will still generally be computed for multiple values of $\gamma$ before convergence, for either variant of the algorithm. Furthermore, pencil $(\Mc,\Nc)$ is $2n \times 2n$, meaning that the ${\cal O}(n^3)$ work per iteration also contains a significantly larger constant factor; computing the eigenvalues of a $2n \times 2n$ problem typically takes at least eight times longer than a $n \times n$ one. If cubic interpolation is used, computing the derivatives \eqref{eq:ntfcprime_cont} via the eigenvectors of $(\Mc,\Nc)$, as proposed by \cite{GenVV98} using the equivalences in \eqref{eq:qs_vecs} and \eqref{eq:ntfcprime_eig_cont}, can sometimes be quite expensive as well. If on a particular iteration, the number of purely imaginary eigenvalues of $(\Mc,\Nc)$ is close to $n$, say $\hat n$, then assuming 64-bit computation, an additional $4 \hat{n}^2$ doubles of memory would be required to store these eigenvectors.\footnote{ Although computing eigenvectors with \texttt{eig} in {MATLAB}\ is currently an all or none affair, LAPACK does provide the user the option to only compute certain eigenvectors, so that all $2n$ eigenvectors would not always need to be computed.} Finally, computing the purely imaginary eigenvalues of $(\Mc,\Nc)$ using the regular QZ algorithm can be ill advised; in practice, rounding error in the real parts of the eigenvalues can make it difficult to detect which of the computed eigenvalues are supposed to be the purely imaginary ones and which are merely just close to the imaginary axis. Indeed, purely imaginary eigenvalues can easily be perturbed off of the imaginary axis when using standard QZ; \cite[Figure 4]{BenSV16} illustrates this issue particularly well. Failure to properly identify the purely imaginary eigenvalues can cause the BBBS algorithm to return incorrect results. As such, it is instead recommended \cite[Section II.D]{BenSV12} to use the specialized Hamiltonian-structure-preserving eigensolvers of \cite{BenBMetal02,BenSV16} to avoid this problem. However, doing so can be even more expensive as it requires computing the eigenvalues of a related matrix pencil that is even larger: $(2n+m+p)\times(2n+m+p)$.
On the other hand, computing \eqref{eq:ntf_cont}, the norm of the transfer function, is typically rather inexpensive, at least relative to computing the imaginary eigenvalues of the matrix pencil \eqref{eq:MNpencil_cont}; Table~\ref{table:ntf_vs_pencil} presents for data on how much faster computing the singular value decomposition of $G({\bf i} \omega)$ can be compared to computing the eigenvalues of $(\Mc,\Nc)$ (using regular QZ), using randomly-generated systems composed of dense matrices of various dimensions. In the first row of Table~\ref{table:ntf_vs_pencil}, we see that computing the eigenvalues of \eqref{eq:MNpencil_cont} for tiny systems ($n=m=p=20$) can take up to two-and-a-half times longer than computing the SVD of $G({\bf i} \omega)$ on modern hardware and this disparity quickly grows larger as the dimensions are all increased (up to 36.8 faster for $n=m=p=400$). Furthermore, for moderately-sized systems where $m,p \ll n$ (the typical case in practice), the performance gap dramatically widens to up to 119 times faster to compute the SVD of $G({\bf i} \omega)$ versus the eigenvalues of $(\Mc,\Nc)$ (the last row of Table~\ref{table:ntf_vs_pencil}). Of course, this disparity in runtime speeds is not surprising. Computing the eigenvalues of \eqref{eq:MNpencil_cont} involves working with a $2n \times 2n$ (or larger when using structure-preserving eigensolvers) matrix pencil while the main costs to evaluate the norm of the transfer function at a particular frequency involve first solving a linear system of dimension $n$ to compute either the $({\bf i} \omega E - A)^{-1}B$ or $C({\bf i} \omega E - A)^{-1}$ term in $G({\bf i} \omega)$ and then computing the maximum singular value of $G({\bf i} \omega)$, which is $p \times m$. If $\max(m,p)$ is small, the cost to compute the largest singular value is negligible and even if $\max(m,p)$ is not small, the largest singular value can still typically be computed easily and efficiently using sparse methods. Solving the $n$-dimensional linear system is typically going to be much cheaper than computing the eigenvalues of the $2n \times 2n$ pencil, and more so if $A$ and $E$ are not dense and $({\bf i} \omega E - A)$ permits a fast (sparse) LU decomposition.
\begin{table} \setlength{\tabcolsep}{8pt} \centering
\begin{tabular}{ccc|cc} \toprule
\multicolumn{5}{c}{Computing $\|G({\bf i} \omega)\|_2$ versus \texttt{eig}($\Mc,\Nc$)}\\ \midrule \multicolumn{3}{c}{} & \multicolumn{2}{c}{Times faster}\\ \cmidrule(lr){4-5} $n$ & $m$ & $p$ & min & max \\ \midrule 20 & 20 & 20 & 0.71 & 2.47 \\ 100 & 100 & 100 & 6.34 & 10.2 \\ 400 & 400 & 400 & 19.2 & 36.8 \\ 400 & 10 & 10 & 78.5 & 119.0\\ \bottomrule \end{tabular} \caption{ For each set of dimensions (given in the leftmost three columns),
five different systems were randomly generated and the running times to compute $\|G({\bf i} \omega)\|_2$ and \texttt{eig}($\Mc,\Nc$) were recorded to form the ratios of these five pairs of values.
Ratios greater than one indicate that it is faster to compute $\|G({\bf i} \omega)\|_2$ than \texttt{eig}($\Mc,\Nc$), and by how much, while ratios less than one indicate the opposite. The rightmost two columns of the table give the smallest and largest of the five ratios observed per set of dimensions. } \label{table:ntf_vs_pencil} \end{table}
\section{The improved algorithm} \label{sec:hybrid_opt} Recall that computing the ${\hinfsym_\infty}$ norm is done by maximizing $g_\mathrm{c}(\omega)$ over $\omega \in \mathbb R$ but that the BBBS algorithm (and the cubic interpolation refinement) actually converges to a global maximum of $g_\mathrm{c}(\omega)$ by iteratively computing the eigenvalues of the large matrix pencil $(\Mc,\Nc)$ for successively larger values of $\gamma$. However, we could alternatively consider a more direct approach of finding maximizers of $g_\mathrm{c}(\omega)$, which as discussed above, is a much cheaper function to evaluate numerically. Computing such maximizers could allow larger increases in $\gamma$ to be obtained on each iteration, compared to just evaluating $g_\mathrm{c}(\omega)$ at the midpoints or maximizers of the cubic interpolants. This in turn should reduce the number of times that the eigenvalues of $(\Mc,\Nc)$ must be computed and thus speed up the overall running time of the algorithm; given the performance data in Table~\ref{table:ntf_vs_pencil}, the additional cost of any evaluations of $g_\mathrm{c}(\omega)$ needed to find maximizers seems like it should be more than offset by fewer eigenvalue decompositions of
$(\Mc,\Nc)$. Of course, computing the eigenvalues of $(\Mc,\Nc)$ at each iteration
cannot be eliminated completely, as it is still necessary for asserting
whether or not any of the maximizers was a global maximizer
(in which case, the ${\hinfsym_\infty}$ norm has been computed), or
if not, to provide the remaining level set intervals where a global maximizer lies
so the computation can continue.
As alluded to in the introduction, for the special case of the distance to instability, a similar cost-balancing strategy has been considered before in \cite{HeW99} but the authors themselves noted that the inverse iteration scheme they employed to find locally optimal solutions could sometimes have very slow convergence and expressed concern that other optimization methods could suffer similarly. Of course, in this paper we are considering the more general case of computing the ${\hinfsym_\infty}$ norm and, as we will observe in our later experimental evaluation, the first- and second-order optimization techniques we now propose do in fact seem to work well in practice.
Though $g_\mathrm{c}(\omega)$ is typically nonconvex, standard optimization methods should generally still be able to find local maximizers, if not always global maximizers, provided that $g_\mathrm{c}(\omega)$ is sufficiently smooth. Since $g_\mathrm{c}(c)$ is the maximum singular value of $G({\bf i}\omega)$, it is locally Lipschitz (e.g. \cite[Corollary 8.6.2]{GolV13}). Furthermore, in proving the quadratic convergence of the midpoint-based BBBS algorithm, it was shown that at local maximizers, the second derivative of $g_\mathrm{c}(\omega)$ not only exists but is even locally Lipschitz \cite[Theorem~2.3]{BoyB90}. The direct consequence is that Newton's method for optimization can be expected to converge quadratically when it is used to find a local maximizer of $g_\mathrm{c}(\omega)$. Since there is only one optimization variable, namely $\omega$, there is also the benefit that we need only work with first and second derivatives, instead of gradients and Hessians, respectively. Furthermore, if $g_\mathrm{c}^{\prime\prime}(\omega)$ is expensive to compute, one can instead resort to the secant method (which is a quasi-Newton method in one variable) and still obtain superlinear convergence. Given the large disparity in costs to compute the eigenvalues of $(\Mc,\Nc) $ and $g_\mathrm{c}(\omega)$, it seems likely that even just superlinear convergence could still be sufficient to significantly accelerate the computation of the ${\hinfsym_\infty}$ norm. Of course, when $g_\mathrm{c}^{\prime\prime}(\omega)$ is relatively cheap to compute, additional acceleration is likely to be obtained when using Newton's method. Note that many alternative optimization strategies could also be employed here, potentially with additional efficiencies. But, for sake of simplicity, we will just restrict the discussion in this paper to the secant method and Newton's method, particularly since conceptually there is no difference.
Since $g_\mathrm{c}(\omega)$ will now need to be evaluated at any point requested by an optimization method, we will need to compute its first and possibly second derivatives directly; recall that using the eigenvectors of the purely imaginary eigenvalues of \eqref{eq:MNpencil_cont} with the equivalences in \eqref{eq:qs_vecs} and \eqref{eq:ntfcprime_eig_cont} only allows us to obtain the first derivatives at the end points of the level-set intervals. However, as long as we also compute the associated left and right singular vectors $u(\omega)$ and $v(\omega)$ when computing $g_\mathrm{c}(\omega)$, the value of the first derivative $g_\mathrm{c}^\prime(\omega)$ can be computed via the direct formulation given in \eqref{eq:ntfcprime_direct_cont} and without much additional cost over computing $g_\mathrm{c}(\omega)$ itself. For each frequency $\omega$ of interest, an LU factorization of $({\bf i} \omega E - A)$ can be done once and reused to solve the linear systems due to the presence of $({\bf i} \omega E - A)^{-1}$, which appears once in $g_\mathrm{c}(\omega)$ and twice in $g_\mathrm{c}^\prime(\omega)$.
To compute $g_\mathrm{c}^{\prime\prime}(\omega)$, we will need the following result for second derivatives of eigenvalues, which can be found in various forms in \cite{Lan64}, \cite{OveW95}, and \cite{Kat82}. \begin{theo} \label{thm:eig2ndderiv} For $t \in \mathbb R$, let $H(t)$ be a twice-differentiable $n \times n$ Hermitian matrix family with distinct eigenvalues at $t=0$ with $(\lambda_k,x_k)$ denoting the $k$th such eigenpair and where each eigenvector $x_k$ has unit norm and the eigenvalues are ordered $\lambda_1 > \ldots > \lambda_n$. Then: \[
\lambda_1''(t) \bigg|_{t=0}= x_1^* H''(0) x_1 + 2 \sum_{k = 2}^{n} \frac{| x_1^* H'(0) x_k |^2}{\lambda_1 - \lambda_k}. \] \end{theo}
Since $g_\mathrm{c}(\omega)$ is the largest singular value of $\|G({\bf i} \omega)\|_2$, it is also the largest eigenvalue of the matrix: \begin{equation}
\label{eq:eigderiv_mat}
H(\omega) =
\begin{bmatrix} 0 & G_\mathrm{c}(\omega) \\
G_\mathrm{c}(\omega)^* & 0
\end{bmatrix}, \end{equation} which has first and second derivatives \begin{equation}
\label{eq:eigderiv12_mat}
H^\prime(\omega) =
\begin{bmatrix} 0 & G_\mathrm{c}^\prime(\omega) \\
G_\mathrm{c}^\prime(\omega)^* & 0
\end{bmatrix}
\quad \text{and} \quad
H^{\prime\prime}(\omega) =
\begin{bmatrix} 0 & G_\mathrm{c}^{\prime\prime}(\omega) \\
G_\mathrm{c}^{\prime\prime}(\omega)^* & 0
\end{bmatrix}. \end{equation} The formula for $G_\mathrm{c}^\prime(\omega)$ is given by \eqref{eq:ntfcprime_direct_cont} while the corresponding second derivative is obtained by straightforward application of matrix differentiation rules: \begin{equation}
\label{eq:tfc2_cont}
G_\mathrm{c}^{\prime\prime}(\omega) =
-2 C({\bf i} \omega E - A)^{-1}E({\bf i} \omega E - A)^{-1}E({\bf i} \omega E - A)^{-1} B. \end{equation} Furthermore, the eigenvalues and eigenvectors of \eqref{eq:eigderiv_mat} needed to apply Theorem~\ref{thm:eig2ndderiv} are essentially directly available from just the full SVD of $G_\mathrm{c}(\omega)$. Let $\sigma_k$ be the $k$th singular value of $G_\mathrm{c}(\omega)$, along with associated right and left singular vectors $u_k$ and $v_k$, respectively. Then $\pm\sigma_k$ is an eigenvalue of \eqref{eq:eigderiv_mat} with eigenvector $\left[ \begin{smallmatrix} u_k \\ v_k \end{smallmatrix} \right]$ for $\sigma_k$ and eigenvector $\left[ \begin{smallmatrix} u_k \\ -v_k \end{smallmatrix}\right]$ for $-\sigma_k$. When $\sigma_k = 0$, the corresponding eigenvector is either $\left[\begin{smallmatrix} u_k \\ \mathbf{0} \end{smallmatrix} \right]$ if $p > m$ or $\left[\begin{smallmatrix} \mathbf{0} \\ v_k \end{smallmatrix} \right]$ if $p < m$, where $\mathbf{0}$ denotes a column of $m$ or $p$ zeros, respectively. Given the full SVD of $G_\mathrm{c}(\omega)$, computing $g_\mathrm{c}^{\prime\prime}(\omega)$ can also be done with relatively little additional cost. The stored LU factorization of $({\bf i} \omega E - A)$ used to obtain $G_\mathrm{c}(\omega)$ can again be reused to quickly compute the $G_\mathrm{c}^\prime(\omega)$ and $G_\mathrm{c}^{\prime\prime}(\omega)$ terms in \eqref{eq:eigderiv12_mat}. If obtaining the full SVD is particularly expensive, i.e. for systems with many inputs/outputs, as mentioned above, sparse methods can still be used to efficiently obtain the largest singular value and its associated right/left singular vectors, in order to at least calculate $g_\mathrm{c}^{\prime}(\omega)$, if not $g_\mathrm{c}^{\prime\prime}(\omega)$ as well.
\begin{rema} On a more theoretical point, by invoking Theorem~\ref{thm:eig2ndderiv} to compute $g_\mathrm{c}^{\prime\prime}(\omega)$, we are also assuming that the singular values of $G_\mathrm{c}(\omega)$ are unique as well. However, in practice, this will almost certainly hold numerically, and to adversely impact the convergence rate of Newton's method, it would have to frequently fail to hold, which seems an exceptionally unlikely scenario. As such, we feel that this additional assumption is not of practical concern. \end{rema}
Thus, our new proposed improvement to the BBBS algorithm is to not settle for the increase in $\gamma$ provided by the standard midpoint or cubic interpolation schemes, but to increase $\gamma$ \emph{as far as possible} on every iteration using standard optimization techniques applied to $g_\mathrm{c}(\omega)$. Assume that $\gamma$ is still less than the value of the ${\hinfsym_\infty}$ norm and let \[
\hat\omega_j = \argmax g_\mathrm{c}(\hat \omega_k), \] where the finite set of $\hat\omega_k$ values are the midpoints of the level-set intervals $I_k$ or the maximizers of the cubic interpolants on these intervals, respectively defined in \eqref{eq:gamma_mp} or \eqref{eq:gamma_cubic}. Thus the solution $\hat\omega_j \in I_j$ is the frequency that provides the updated value $\gamma_\mathrm{mp}$ or $\gamma_\mathrm{cubic}$ in the standard algorithms. Now consider applying either Newton's method or the secant method (the choice of which one will be more efficient can be made more or less automatically depending on how $m,p$ compares to $n$) to the following optimization problem with a simple box constraint: \begin{equation}
\label{eq:gamma_opt}
\max_{\omega \in I_j} g_\mathrm{c}(\omega). \end{equation} If the optimization method is initialized at $\hat\omega_j$, then even if $\omega_\mathrm{opt}$, a \emph{computed} solution to \eqref{eq:gamma_opt}, is actually just a \emph{local} maximizer (a possibility since \eqref{eq:gamma_opt} could be nonconvex), it is still guaranteed that \[
g_\mathrm{c}(\omega_\mathrm{opt}) >
\begin{cases}
\gamma_\mathrm{mp} & \text{initial point $\hat \omega_j$ is a midpoint of $I_j$} \\
\gamma_\mathrm{cubic} & \text{initial point $\hat \omega_j$ is a maximizer of interpolant $c_j(\omega)$}
\end{cases} \] holds, provided that $\hat\omega_j$ does not happen to be a stationary point of $g_\mathrm{c}(\omega)$. Furthermore, \eqref{eq:gamma_opt} can only have more than one maximizer when the current estimate $\gamma$ of the ${\hinfsym_\infty}$ norm is so low that there are multiple peaks above level-set interval $I_j$. Consequently, as the algorithm converges, computed maximizers of \eqref{eq:gamma_opt} will be assured to be globally optimal over $I_j$ and in the limit, over all frequencies along the entire imaginary axis. By setting tight tolerances for the optimization code, maximizers of \eqref{eq:gamma_opt} can also be computed to full precision with little to no penalty,
due to the superlinear or quadratic rate of convergence we can expect from the secant method or Newton's method, respectively. If the computed optimizer of \eqref{eq:gamma_opt} also happens to be a global maximizer of $g_\mathrm{c}(\omega)$, for all $\omega \in \mathbb R$, then the ${\hinfsym_\infty}$ norm has indeed been computed to full precision, but the algorithm must still verify this by computing the imaginary eigenvalues of $(\Mc,\Nc)$ just one more time. However, if a global optimizer has not yet been found, then the algorithm must compute the imaginary eigenvalues of $(\Mc,\Nc)$ at least two times more: one or more times as the algorithm increases $\gamma$ to the globally optimal value, and then a final evaluation to verify that the computed value is indeed globally optimal. Figure~\ref{fig:bbbs_opt} shows a pictorial comparison of optimizing $g_\mathrm{c}(\omega)$ compared to the midpoint and cubic-interpolant-based updating methods.
\begin{figure}
\caption{For the same example as Figure~\ref{fig:bbbs_plots}, the larger increase in $\gamma$ attained by optimizing \eqref{eq:gamma_opt}
is shown by the red dashed line going through the red circle at the top of the leftmost peak of $\| G({\bf i} \omega)\|_2$. By comparison, the BBBS midpoint (red dotted vertical lines and x's) and the cubic-interpolant-based schemes (black dotted vertical lines and x's) only provide suboptimal increases in $\gamma$.}
\label{fig:opt_comparison}
\label{fig:bbbs_opt}
\end{figure}
In the above discussion, we have so far only considered applying optimization over the single level-set interval $I_j$ but we certainly could attempt to solve \eqref{eq:gamma_opt} for other level-set intervals as well. Let $\phi = 1,2,\ldots$ be the max number of level-set intervals to optimize over per iteration and $q$ be the number of level-set intervals for the current value of $\gamma$. Compared to just optimizing over $I_j$, optimizing over all $q$ of the level-set intervals could yield an even larger increase in the estimate $\gamma$ but would be the most expensive option computationally. If we do not optimize over all the level-set intervals, i.e. $\phi < q$, there is the question of which intervals should be prioritized for optimization. In our experiments, we found that prioritizing by first evaluating $g_\mathrm{c}(\omega)$ at all the $\hat\omega_k$ values and then choosing the intervals $I_k$ to optimize where $g_\mathrm{c}(\hat\omega_k)$ takes on the largest values seems to be a good strategy. However, with a serial MATLAB code, we have observed that just optimizing over the most promising interval, i.e. $\phi = 1$ so just $I_j$, is generally more efficient in terms of running time than trying to optimize over more intervals. For discussion about optimizing over multiple intervals in parallel, see Section~\ref{sec:parallel}.
Finally, if a global maximizer of $g_\mathrm{c}(\omega)$ can potentially be found before ever computing eigenvalues of $(\Mc,\Nc)$ even once, then only one expensive eigenvalue decomposition $(\Mc,\Nc)$ will be incurred, just to verify that the initial maximizer is indeed a global one. Thus, we also propose initializing the algorithm at a maximizer of $g_\mathrm{c}(\omega)$,
obtained via applying standard optimization techniques to \begin{equation}
\label{eq:gamma_init}
\max g_\mathrm{c}(\omega), \end{equation} which is just \eqref{eq:gamma_opt} without the box constraint. In the best case, the computed maximizer will be a global one, but even a local maximizer will still provide a higher initial estimate of the ${\hinfsym_\infty}$ norm compared to initializing at a guess that may not even be locally optimal. Of course, finding maximizers of \eqref{eq:gamma_init} by starting from multiple initial guesses can also be done in parallel; we again refer to Section~\ref{sec:parallel} for more details.
\begin{algfloat} \begin{algorithm}[H] \floatname{algorithm}{Algorithm} \caption{The Improved Algorithm Using Local Optimization} \label{alg:hybrid_opt} \begin{algorithmic}[1]
\REQUIRE{
Matrices $A \in \Cmn{n}{n}$, $B \in \Cmn{n}{m}$, $C \in \Cmn{p}{n}$, $D \in \Cmn{p}{m}$,
and $E \in \Cmn{n}{n}$,
initial frequency guesses $\{\omega_1,\ldots,\omega_q\} \in \mathbb R$ and
$\phi$, a positive integer indicating the number of intervals/frequencies to optimize per round.
}
\ENSURE{
$\gamma = \| G \|_{\hinfsym_\infty}$ and $\omega$ such that
$\gamma = g_\mathrm{c}(\omega)$.
\\ \quad
}
\STATE \COMMENT{Initialization:}
\STATE Compute $[\gamma_1,\ldots,\gamma_q] = [g_\mathrm{c}(\omega_1),\ldots,g_\mathrm{c}(\omega_q)]$.
\label{algline:init_start}
\STATE Reorder $\{\omega_1,\ldots,\omega_q\}$ s.t. $\gamma_j \ge \gamma_{j+1}$.
\STATE Find maximal values $[\gamma_1,\ldots,\gamma_\phi]$ of \eqref{eq:gamma_init}, using initial points
$\omega_j$, $j=1,\ldots,\phi$.
\label{algline:init_opt}
\STATE $\gamma = \max([\gamma_1,\ldots,\gamma_\phi])$.
\label{algline:gamma_init}
\STATE $\omega = \omega_j$ for $j$ s.t. $\gamma = \gamma_j$.
\label{algline:init_end}
\STATE \COMMENT{Convergent Phase:}
\WHILE {not converged}
\STATE \COMMENT{Compute the intervals that lie under $g_\mathrm{c}(\omega)$ using
eigenvalues of the pencil:}
\STATE Compute $\Lambda_\mathrm{I} =
\{ \Im \lambda : \lambda \in \Lambda(\Mc,\Nc) ~\text{and}~ \Re \lambda = 0\}$.
\label{algline:pencil}
\STATE Index and sort $\Lambda_\mathrm{I} = \{\omega_1,...,\omega_l\}$ s.t.
$\omega_j \le \omega_{j+1}$.
\label{algline:frequencies}
\STATE Form all intervals $I_k = [\omega_k, \omega_{k+1}]$ s.t.
each interval at height $\gamma$ is below $g_\mathrm{c}(\omega)$.
\label{algline:hyopt_ints}
\STATE \COMMENT{Compute candidate frequencies of the level-set intervals $I_k$ ($q$ of them):}
\STATE Compute all $\hat\omega_k$ using either \eqref{eq:gamma_mp} or \eqref{eq:gamma_cubic}.
\label{algline:hyopt_points}
\STATE \COMMENT{Run box-constrained optimization on the $\phi$ most promising frequencies:}
\STATE Compute $[\gamma_1,\ldots,\gamma_q] = [g_\mathrm{c}(\hat\omega_1),\ldots,g_\mathrm{c}(\hat\omega_q)]$.
\STATE Reorder $\{\hat\omega_1,\ldots,\hat\omega_q\}$ and intervals $ I_k$ s.t. $\gamma_j \ge \gamma_{j+1}$.
\label{algline:hyopt_reorder}
\STATE Find maximal values $[\gamma_1,\ldots,\gamma_\phi]$ of \eqref{eq:gamma_opt} using initial points $\hat\omega_j$, $j = 1,\ldots,\phi$.
\label{algline:hyopt_opt}
\STATE \COMMENT{Update to the highest gain computed:}
\STATE $\gamma = \max([\gamma_1,\ldots,\gamma_\phi])$.
\STATE $\omega = \omega_j$ for $j$ s.t. $\gamma = \gamma_j$.
\ENDWHILE
\STATE \COMMENT{Check that the maximizing frequency is not at infinity (continuous-time only)}
\IF { $\gamma < \|D\|_2$}
\STATE $\gamma = \|D\|_2$.
\STATE $\omega = \infty$.
\ENDIF \end{algorithmic} \end{algorithm} \algnote{ For details on how embarrassingly parallel processing can be used to further improve the algorithm, see Section~\ref{sec:parallel}. } \end{algfloat}
Algorithm~\ref{alg:hybrid_opt} provides a high-level pseudocode description of our improved method. As a final step in the algorithm, it is necessary to check whether or not the value of the ${\hinfsym_\infty}$ norm is attained at $\omega = \infty$. We check this case after the convergent phase has computed a global maximizer over the union of all intervals it has considered. The only possibility that the ${\hinfsym_\infty}$ norm may be attained at $\omega = \infty$ in Algorithm~\ref{alg:hybrid_opt} is when the initial value of $\gamma$ computed in
line~\ref{algline:gamma_init} of is less than $\|D\|_2$. As the assumptions of Theorem~\ref{thm:eigsing_cont} require
that $\gamma$ not be a singular value of $D$, it is not valid to use $\gamma = \|D\|_2$ in $(\Mc,\Nc)$ to check if this pencil has any imaginary eigenvalues. However, if the optimizer computed by the convergent phase of Algorithm~\ref{alg:hybrid_opt} yields a
$\gamma$ value less than $\|D\|_2$, then it is clear that the optimizing frequency is at $\omega = \infty$.
\section{Handling discrete-time systems} \label{sec:discrete} Now consider the discrete-time linear dynamical system \begin{subequations} \label{eq:lti_disc} \begin{align} Ex_{k+1} &= Ax_k + Bu_k \\ y_k & = Cx_k + Du_k, \end{align} \end{subequations} where the matrices are defined as before in \eqref{eq:lti_cont}. In this case, the ${\hinfsym_\infty}$ norm is defined as \begin{equation}
\label{eq:hinf_disc}
\|G\|_{{\hinfsym_\infty}} \coloneqq \max_{\theta \in [0,2\pi)} \| G(e^{\imagunit \theta}) \|_2, \end{equation} again assuming that pencil $(A,E)$ is at most index one. If all finite eigenvalues are either strictly inside the unit disk centered at the origin or are uncontrollable or unobservable, then \eqref{eq:hinf_disc} is finite and the ${\hinfsym_\infty}$ norm is attained at some $\theta \in [0,2\pi)$. Otherwise, it is infinite.
We now show the analogous version of Theorem~\ref{thm:eigsing_cont} for discrete-time systems. The $D=0$ and $E=I$ case was considered in \cite[Section 3]{HinS91} while the more specific $B=C=E=I$ and $D=0$ case was given in \cite[Theorem 4]{Bye88}.\footnote{Note that equation (10) in \cite{Bye88} has a typo: $A^H - e^{\imagunit \theta} I$ in the lower left block of $K(\theta)$ should actually be $A^H - e^{-\imagunit \theta} I$.} These results relate singular values of the transfer function for discrete-time systems to eigenvalues with modulus one of associated matrix pencils. Although the following more general result is already known, the proof, to the best of our knowledge, is not in the literature so we include it in full here. The proof follows a similar argumentation as the proof of Theorem~\ref{thm:eigsing_cont}. \begin{theo} \label{thm:eigsing_disc} Let $\lambda E - A$ be regular with no finite eigenvalues on the unit circle, $\gamma > 0$ not a singular value of $D$, and $\theta \in [0,2\pi)$. Consider the matrix pencil $(\Md,\Nd)$, where \begin{equation}
\label{eq:MNpencil_disc} \begin{aligned}
\mathcal{S}_\gamma \coloneqq {}& \begin{bmatrix} A - BR^{-1}D^*C & -\gamma BR^{-1}B^* \\
0 & E^* \end{bmatrix}, \\
\mathcal{T}_\gamma \coloneqq {}& \begin{bmatrix} E & 0\\
-\gamma C^*S^{-1}C & (A - BR^{-1}D^*C)^*
\end{bmatrix}, \end{aligned} \end{equation} $R = D^*D - \gamma^2 I$ and $S = DD^* - \gamma^2 I$. Then $e^{{\bf i} \theta}$ is an eigenvalue of matrix pencil $(\Md,\Nd)$ if and only if $\gamma$ is a singular value of $G(e^{{\bf i} \theta})$. \end{theo}
\begin{proof} Let $\gamma$ be a singular value of $G(e^{\imagunit \theta})$ with left and right singular vectors $u$ and $v$, that is, so that $G(e^{\imagunit \theta})v = \gamma u$ and $G(e^{\imagunit \theta})^*u = \gamma v$. Using the expanded versions of these two equivalences \begin{equation}
\label{eq:tfsv_equiv_disc}
\left( \tfs{e^{\imagunit \theta}} \right) v = \gamma u
\quad \text{and} \quad
\left( \tfs{e^{\imagunit \theta}} \right)^* u = \gamma v, \end{equation} we define \begin{equation}
\label{eq:qs_disc}
q = \left( e^{\imagunit \theta} E - A \right)^{-1}Bv
\quad \text{and} \quad
s = \left( e^{-\imagunit \theta} E^* - A^* \right)^{-1}C^*u. \end{equation} Rewriting \eqref{eq:tfsv_equiv_disc} using \eqref{eq:qs_disc} yields the following matrix equation: \begin{equation}
\label{eq:uv_disc}
\begin{bmatrix} C & 0 \\ 0 & B^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix}
=
\begin{bmatrix} -D & \gamma I \\ \gamma I & -D^* \end{bmatrix}
\begin{bmatrix} v \\ u \end{bmatrix}
~\Longrightarrow~
\begin{bmatrix} v \\ u \end{bmatrix}
=
\begin{bmatrix} -D & \gamma I \\ \gamma I & -D^* \end{bmatrix}^{-1}
\begin{bmatrix} C & 0 \\ 0 & B^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix} , \end{equation} where \begin{equation}
\label{eq:Dgamma_inv}
\begin{bmatrix} -D & \gamma I \\ \gamma I & -D^* \end{bmatrix}^{-1}
=
\begin{bmatrix} -R^{-1}D^* & -\gamma R^{-1} \\ -\gamma S^{-1} & -DR^{-1} \end{bmatrix}
\quad \text{and} \quad
\begin{bmatrix} q \\ s \end{bmatrix} \ne 0. \end{equation} Rewriting \eqref{eq:qs_disc} as: \begin{equation}
\label{eq:EAqs1_disc}
\left(
\begin{bmatrix} e^{\imagunit \theta} E & 0 \\ 0 & e^{-\imagunit \theta} E^* \end{bmatrix} -
\begin{bmatrix} A & 0 \\ 0 & A^* \end{bmatrix}
\right)
\begin{bmatrix} q \\ s \end{bmatrix}
=
\begin{bmatrix} B & 0 \\ 0 & C^* \end{bmatrix}
\begin{bmatrix} v \\ u \end{bmatrix} , \end{equation} and then substituting in \eqref{eq:uv_disc} for the rightmost term of \eqref{eq:EAqs1_disc} yields \begin{equation}
\label{eq:EAqs2_disc}
\setlength\arraycolsep{3pt}
\begingroup
\left(
\begin{bmatrix} e^{\imagunit \theta} E & 0 \\ 0 & e^{-\imagunit \theta} E^* \end{bmatrix} -
\begin{bmatrix} A & 0 \\ 0 & A^* \end{bmatrix}
\right)
\begin{bmatrix} q \\ s \end{bmatrix}
=
\begin{bmatrix} B & 0 \\ 0 & C^* \end{bmatrix}
\begin{bmatrix} -D & \gamma I \\ \gamma I & -D^* \end{bmatrix}^{-1}
\begin{bmatrix} C & 0 \\ 0 & B^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix} .
\endgroup \end{equation} Multiplying the above on the left by \[
\begin{bmatrix} I & 0 \\ 0 & -e^{\imagunit \theta} I \end{bmatrix} \] and then rearranging terms, we have \begin{equation*}
\label{eq:EAqs3_disc}
e^{\imagunit \theta} \begin{bmatrix} E & 0 \\ 0 & A^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix}
=
\begin{bmatrix} A & 0 \\ 0 & E^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix}
+
\begin{bmatrix} B & 0 \\ 0 & -e^{\imagunit \theta} C^* \end{bmatrix}
\begin{bmatrix} -D & \gamma I \\ \gamma I & -D^* \end{bmatrix}^{-1}
\begin{bmatrix} C & 0 \\ 0 & B^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix} . \end{equation*} Substituting the inverse in \eqref{eq:Dgamma_inv} for its explicit form and multiplying terms yields: \begin{equation*}
e^{\imagunit \theta} \begin{bmatrix} E & 0 \\ 0 & A^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix}
=
\begin{bmatrix} A & 0 \\ 0 & E^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix}
+
\begin{bmatrix} B & 0 \\ 0 & -e^{\imagunit \theta} C^* \end{bmatrix}
\begin{bmatrix} -R^{-1}D^*C & -\gamma R^{-1}B^* \\
-\gamma S^{-1}C & -DR^{-1}B^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix} . \end{equation*} Finally, multiplying terms further, separating out the $-e^{\imagunit \theta}$ terms to bring them over to the left hand side, and then recombining, we have that \begin{equation*}
e^{\imagunit \theta} \begin{bmatrix} E & 0 \\ -\gamma C^*S^{-1}C & A^* - C^*DR^{-1}B^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix}
=
\begin{bmatrix} A - BR^{-1}D^*C & -\gamma BR^{-1}B^* \\ 0 & E^* \end{bmatrix}
\begin{bmatrix} q \\ s \end{bmatrix} . \end{equation*} It is now clear that $e^{\imagunit \theta}$ is an eigenvalue of pencil $(\Md,\Nd)$.
Now suppose that $e^{\imagunit \theta}$ is an eigenvalue of pencil $(\Md,\Nd)$ with eigenvector given by $q$ and $s$ as above. Then it follows that \eqref{eq:EAqs2_disc} holds, which can be rewritten as \eqref{eq:EAqs1_disc} by defining $u$ and $v$ using the right-hand side equation of \eqref{eq:uv_disc}, noting that neither can be identically zero. It is then clear that the pair of equivalences in \eqref{eq:qs_disc} hold. Finally, substituting \eqref{eq:qs_disc} into the left-hand side equation of \eqref{eq:uv_disc}, it is clear that $\gamma$ is a singular value of $G(e^{\imagunit \theta})$, with left and right singular vectors $u$ and $v$. \end{proof}
Adapting Algorithm~\ref{alg:hybrid_opt} to the discrete-time case is straightforward. First, all instances of $g_\mathrm{c}(\omega)$ must be replaced with \[
g_\mathrm{d}(\theta) \coloneqq \|G(e^{\imagunit \theta})\|_2. \] To calculate its first and second derivatives, we will need the first and second derivatives of $G_\mathrm{d}(\theta) \coloneqq G(e^{\imagunit \theta})$ and for notational brevity, it will be convenient to define $Z(\theta) \coloneqq (e^{\imagunit \theta} E - A)$. Then \begin{equation}
\label{eq:tfd1st}
G_\mathrm{d}^\prime(\theta) = - {\bf i} e^{\imagunit \theta} C Z(\theta)^{-1} E Z(\theta)^{-1} B \end{equation} and \begin{equation}
\label{eq:tfd2nd}
G_\mathrm{d}^{\prime\prime}(\theta) =
e^{\imagunit \theta} C Z(\theta)^{-1} E Z(\theta)^{-1} B
- 2 e^{2{\bf i} \theta} C Z(\theta)^{-1} E Z(\theta)^{-1} E Z(\theta)^{-1} B. \end{equation} The first derivative of $g_\mathrm{d}(\theta)$ can thus be calculated using \eqref{eq:ntfcprime_defn_cont}, where $\omega$, $g_\mathrm{c}(\omega)$, and $G_\mathrm{c}^\prime(\omega)$ are replaced by $\theta$, $g_\mathrm{d}(\theta)$, and $G_\mathrm{d}^\prime(\theta)$ using \eqref{eq:tfd1st}. The second derivative of $g_\mathrm{d}(\theta)$ can be calculated using Theorem~\ref{thm:eig2ndderiv} using \eqref{eq:tfd1st} and \eqref{eq:tfd2nd} to define $H(\theta)$, the analog of \eqref{eq:eigderiv_mat}. Line~\ref{algline:pencil} must be changed to instead compute the eigenvalues of unit modulus of \eqref{eq:MNpencil_disc}. Line~\ref{algline:frequencies} must instead index and sort the angles $\{\theta_1,\ldots,\theta_l\}$ of these unit modulus eigenvalues in ascending order. Due to the periodic nature of \eqref{eq:hinf_disc}, line~\ref{algline:hyopt_ints} must additionally consider the ``wrap-around" interval $[\theta_l,\theta_0+2\pi]$.
\section{Numerical experiments} \label{sec:numerical} We implemented Algorithm~\ref{alg:hybrid_opt} in {MATLAB}, for both continuous-time and discrete-time cases. Since we can only get timing information from \texttt{hinfnorm} and we wished to verify that our new method does indeed reduce the number of times the eigenvalues of $(\Mc,\Nc)$ and $(\Md,\Nd)$ are computed, we also designed our code so that it can run just using the standard BBBS algorithm or the cubic-interpolant scheme. For our new optimization-based approach, we used \texttt{fmincon} for both the unconstrained optimization calls needed for the initialization phase and for the box-constrained optimization calls needed in the convergent phase; \texttt{fmincon}'s optimality and constraint tolerances were set to $10^{-14}$ in order to find maximizers to near machine precision. Our code supports starting the latter optimization calls from either the midpoints of the BBBS algorithm \eqref{eq:gamma_mp} or the maximizing frequencies calculated from the cubic-interpolant method \eqref{eq:gamma_cubic}. Furthermore, the optimizations may be done using either
the secant method (first-order information only) or with Newton's method using second derivatives, thus leading
to four variants of our proposed method to test. Our code has a user-settable parameter that determines
when $m,p$ should be considered too large relative to $n$, and thus when it is likely that using secant method
will actually be faster than Newton's method, due to the additional expense of computing the second derivative
of the norm of the transfer function.
For initial frequency guesses, our code simply tests zero and the imaginary part of the rightmost eigenvalue of $(A,E)$, excluding eigenvalues that are either infinite, uncontrollable, or unobservable.
Eigenvalues are deemed uncontrollable or unobservable if $\|B^*y\|_2$ or $\|Cx\|_2$ are respectively below a user-set tolerance, where $x$ and $y$ are respectively the right and left eigenvectors for a given eigenvalue of $(A,E)$. In the discrete-time case, the default initial guesses are zero, $\pi$, and the angle for the largest modulus eigenvalue.\footnote{ For producing a production-quality implementation, see \cite{BruS90} for more sophisticated initial guesses that can be used, \cite[Section III]{BenV11} for dealing with testing properness of the transfer function, and \cite{Var90a} for filtering out uncontrollable/unobservable eigenvalues of $(A,E)$ when it has index higher than one.}
For efficiency of implementing our method and conducting these experiments, our code does not yet take advantage of structure-preserving eigensolvers. Instead, it uses the regular QZ algorithm (\texttt{eig} in {MATLAB}) to compute the eigenvalues of $(\Mc,\Nc)$ and $(\Md,\Nd)$. To help mitigate issues due to rounding errors, we consider any eigenvalue $\lambda$ imaginary or of unit modulus if it lies within a margin of width $10^{-8}$ on either side of the imaginary axis or unit circle. Taking the imaginary parts of these nearly imaginary eigenvalues forms the initial set of candidate frequencies, or the angles of these nearly unit modulus eigenvalues for the discrete-time case. Then we simply form all the consecutive intervals, including the wrap-around interval for the discrete-time case, even though not all of them will be level-set intervals, and some intervals may only be a portion of a level-set interval (e.g. if the use of QZ causes spurious candidate frequencies). The reason we do this is is because we can easily sort which of the intervals at height $\gamma$ are below $g_\mathrm{c}(\omega)$ or $g_\mathrm{d}(\omega)$ just by evaluating these functions at the midpoint or the maximizer of the cubic interpolant for each interval. This is less expensive because we need to evaluate these interior points regardless, so also evaluating the norm of the transfer function at all these endpoints just adds additional cost. However, for the cubic interpolant refinement, we nonetheless still evaluate $g_\mathrm{c}(\omega)$ or $g_\mathrm{d}(\omega)$ at the endpoints since we need the corresponding derivatives there to construct the cubic interpolants; we do not use the eigenvectors of $(\Mc,\Nc)$ or $(\Md,\Nd)$ to bypass this additional cost as \texttt{eig} in {MATLAB}\ does not currently provide a way to only compute selected eigenvectors, i.e. those corresponding to the imaginary (unit-modulus) eigenvalues. Note that while this strategy is sufficient for our experimental comparisons here, it certainly does not negate the need for structure-preserving eigenvalue solvers.
We evaluated our code on several continuous- and discrete-time problems up to moderate dimensions, all listed with dimensions in Table~\ref{table:hinfnorm_tol}. For the continuous-time problems, we chose four problems from the test set used in \cite{GugGO13} (\texttt{CBM}, \texttt{CSE2}, \texttt{CM3}, \texttt{CM4}), two from the SLICOT benchmark examples\footnote{Available at \url{http://slicot.org/20-site/126-benchmark-examples-for-model-reduction}} (\texttt{ISS} and \texttt{FOM}), and two new randomly-generated examples using \texttt{randn()} with a relatively large number of inputs and outputs. Notably, the four problems from \cite{GugGO13} were generated via taking open-loop systems from $COMPL_eib$\ \cite{compleib} and then designing controllers to minimize the ${\hinfsym_\infty}$ norm of the corresponding closed-loop systems via {\sc hifoo}\ \cite{BurHLetal06}. Such systems can be interesting benchmark examples because $g_\mathrm{c}(\omega)$ will often have several peaks, and multiple peaks may attain the value of the ${\hinfsym_\infty}$ norm, or at least be similar in height. Since the discrete-time problems from \cite{GugGO13} were all very small scale (the largest order in that test set is only 16) and SLICOT only offers a single discrete-time benchmark example, we instead elected to take additional open-loop systems from $COMPL_eib$\ and obtain usable test examples by minimizing the discrete-time ${\hinfsym_\infty}$ norm of their respective closed-loop systems, via optimizing controllers using {\sc hifoo}d\ \cite{PopWM10}, a fork of {\sc hifoo}\ for discrete-time systems. On all examples, the ${\hinfsym_\infty}$ norm values computed by our local-optimization-enhanced code (in all its variants) agreed on average to 13 digits with the results provided by \texttt{hinfnorm}, when used with the tight tolerance of $10^{-14}$, with the worst discrepancy being only 11 digits of agreement. However, our improved method often found slightly larger values, i.e. more accurate values, since it optimizes $g_\mathrm{c}(\omega)$ and $g_\mathrm{d}(\omega)$ directly.
All experiments were performed using {MATLAB}\ R2016b running on a Macbook Pro with an Intel i7-6567U dual-core CPU, 16GB of RAM, and Mac OS X v10.12.
\subsection{Continuous-time examples}
\begin{table}[!t] \center
\begin{tabular}{ l | cccc | cc } \toprule \multicolumn{7}{c}{Small-scale examples (continuous time)}\\ \midrule \multicolumn{1}{c}{} &
\multicolumn{4}{c}{Hybrid Optimization} & \multicolumn{2}{c}{Standard Algs.} \\ \cmidrule(lr){2-5} \cmidrule(lr){6-7} \multicolumn{1}{c}{} &
\multicolumn{2}{c}{Newton} & \multicolumn{2}{c}{Secant} & \multicolumn{2}{c}{} \\ \cmidrule(lr){2-3} \cmidrule(lr){4-5} \multicolumn{1}{l}{Problem} &
\multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{MP} & \multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{MP} &
\multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{BBBS} \\ \midrule \multicolumn{7}{c}{Number of Eigenvalue Computations of $(\Mc,\Nc)$}\\ \midrule \texttt{CSE2} & 2 & 3 & 1 & 1 & 2 & 3 \\ \texttt{CM3} & 2 & 3 & 2 & 2 & 3 & 5 \\ \texttt{CM4} & 2 & 2 & 2 & 2 & 4 & 6 \\ \texttt{ISS} & 1 & 1 & 1 & 1 & 3 & 4 \\ \texttt{CBM} & 2 & 2 & 2 & 2 & 5 & 7 \\ \texttt{randn 1} & 1 & 1 & 1 & 1 & 1 & 1 \\ \texttt{randn 2} & 1 & 1 & 1 & 1 & 2 & 2 \\ \texttt{FOM} & 1 & 1 & 1 & 1 & 2 & 2 \\ \midrule \multicolumn{7}{c}{Number of Evaluations of $g_\mathrm{c}(\omega)$}\\ \midrule \texttt{CSE2} & 10 & 7 & 10 & 10 & 9 & 8 \\ \texttt{CM3} & 31 & 26 & 53 & 45 & 31 & 24 \\ \texttt{CM4} & 19 & 17 & 44 & 43 & 46 & 36 \\ \texttt{ISS} & 12 & 12 & 22 & 22 & 39 & 27 \\ \texttt{CBM} & 34 & 28 & 59 & 55 & 46 & 36 \\ \texttt{randn 1} & 1 & 1 & 1 & 1 & 1 & 1 \\ \texttt{randn 2} & 4 & 4 & 17 & 17 & 6 & 4 \\ \texttt{FOM} & 4 & 4 & 16 & 16 & 7 & 5 \\ \bottomrule \end{tabular} \caption{The top half of the table reports the number of times the eigenvalues of $(\Mc,\Nc)$ were computed in order to compute the ${\hinfsym_\infty}$ norm to near machine precision. From left to right, the methods are our hybrid optimization approach using Newton's method and the secant method, the cubic-interpolant scheme (column `Interp') and the standard BBBS method, all implemented by our single configurable code. The subcolumns `Interp.' and 'MP' of our methods respectively indicate that the optimization routines were initialized at the points from the cubic-interpolant scheme and the BBBS midpoint scheme. The bottom half of the table reports the number of times it was necessary to evaluate the norm of the transfer function (with or without its derivatives). The problems are listed in increasing order of their state-space sizes $n$; for their exact dimensions, see Table~\ref{table:hinfnorm_tol}. } \label{table:evals_dense} \end{table}
In Table~\ref{table:evals_dense}, we list the number of times the eigenvalues of $(\Mc,\Nc)$ were computed and the number of evaluations of $g_\mathrm{c}(\omega)$ for our new method compared to our implementations of the existing BBBS algorithm and its interpolation-based refinement. As can be seen, our new method typically limited the number of required eigenvalue computations of $(\Mc,\Nc)$ to just two, and often it only required one (in the cases where our method found a global optimizer of $g_\mathrm{c}(\omega)$ in the initialization phase). In contrast, the standard BBBS algorithm and its interpolation-based refinement had to evaluate the eigenvalues $(\Mc,\Nc)$ more times; for example, on problem \texttt{CBM}, the BBBS algorithm needed seven evaluations while its interpolation-based refinement still needed five. Though our new method sometimes required more evaluations of $g_\mathrm{c}(\omega)$ than the standard algorithms, often the number of evaluations of $g_\mathrm{c}(\omega)$ was actually less with our new method, presumably due its fewer iterations and particularly when using the Newton's method variants. Even when our method required more evaluations of $g_\mathrm{c}(\omega)$ than the standard methods, the increases were not too significant (e.g. the secant method variants of our method on problems \texttt{CM4}, \texttt{CBM}, \texttt{randn 2}, and \texttt{FOM}). Indeed, the larger number of evaluations of $g_\mathrm{c}(\omega)$ when employing the secant method in lieu of Newton's method was still generally quite low.
\begin{table}[!t] \setlength{\tabcolsep}{3pt} \robustify\bfseries \center
\begin{tabular}{ l | SSSS | SS | SS } \toprule \multicolumn{9}{c}{Small-scale examples (continuous time)} \\ \midrule \multicolumn{1}{c}{} &
\multicolumn{4}{c}{Hybrid Optimization} & \multicolumn{2}{c}{Standard Algs.} & \multicolumn{2}{c}{\texttt{hinfnorm($\cdot$,tol)}}\\ \cmidrule(lr){2-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9} \multicolumn{1}{c}{} &
\multicolumn{2}{c}{Newton} & \multicolumn{2}{c}{Secant} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{\texttt{tol}}\\ \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){8-9} \multicolumn{1}{l}{Problem} &
\multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{MP} & \multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{MP} &
\multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{BBBS} &
\multicolumn{1}{c}{\texttt{1e-14}} & \multicolumn{1}{c}{\texttt{0.01}} \\ \midrule \multicolumn{9}{c}{Wall-clock running times in seconds}\\ \midrule \texttt{CSE2} & 0.042 & 0.060 & 0.036 & 0.032 & 0.043 & \bfseries 0.031 & 0.137 & 0.022 \\ \texttt{CM3} & \bfseries 0.125 & 0.190 & 0.198 & 0.167 & 0.170 & 0.167 & 0.148 & 0.049 \\ \texttt{CM4} & \bfseries 0.318 & 0.415 & 0.540 & 0.550 & 0.712 & 0.811 & 1.645 & 0.695 \\ \texttt{ISS} & 0.316 & 0.328 & 0.379 & \bfseries 0.303 & 0.709 & 0.757 & 0.765 & 0.391 \\ \texttt{CBM} & 0.744 & \bfseries 0.671 & 1.102 & 1.071 & 1.649 & 1.757 & 3.165 & 1.532 \\ \texttt{randn 1} & 0.771 & 0.868 & 1.006 & 0.871 & \bfseries 0.700 & 0.756 & 21.084 & 30.049 \\ \texttt{randn 2} & \bfseries 9.551 & 9.746 & 9.953 & 11.275 & 14.645 & 15.939 & 31.728 & 16.199 \\ \texttt{FOM} & \bfseries 3.039 & 3.426 & 4.418 & 4.176 & 5.509 & 5.182 & 128.397 & 36.529 \\ \midrule \multicolumn{9}{c}{Running times relative to hybrid optimization (Newton with `Interp.')}\\ \midrule \texttt{CSE2} & 1 & 1.42 & 0.86 & 0.75 & 1.02 & 0.75 & 3.24 & 0.53 \\ \texttt{CM3} & 1 & 1.52 & 1.59 & 1.34 & 1.36 & 1.34 & 1.19 & 0.39 \\ \texttt{CM4} & 1 & 1.31 & 1.70 & 1.73 & 2.24 & 2.55 & 5.18 & 2.19 \\ \texttt{ISS} & 1 & 1.04 & 1.20 & 0.96 & 2.24 & 2.39 & 2.42 & 1.24 \\ \texttt{CBM} & 1 & 0.90 & 1.48 & 1.44 & 2.22 & 2.36 & 4.26 & 2.06 \\ \texttt{randn 1} & 1 & 1.13 & 1.31 & 1.13 & 0.91 & 0.98 & 27.36 & 38.99 \\ \texttt{randn 2} & 1 & 1.02 & 1.04 & 1.18 & 1.53 & 1.67 & 3.32 & 1.70 \\ \texttt{FOM} & 1 & 1.13 & 1.45 & 1.37 & 1.81 & 1.71 & 42.25 & 12.02 \\ \midrule Average & 1 & 1.18 & 1.33 & 1.24 & 1.67 & 1.72 & 11.15 & 7.39 \\ \bottomrule \end{tabular} \caption{ In the top half of the table, the running times (fastest in bold) are reported in seconds for the same methods and configurations as in Table~\ref{table:evals_dense}, with the running times of \texttt{hinfnorm} additionally listed in the rightmost two columns, for a tolerance of $10^{-14}$ (as used by the other methods) and its default value of $0.01$. The bottom of half the table normalizes all the times relative to the running times for our hybrid optimization method (Newton and `Interp.'), along with the overall averages relative to this variant. } \label{table:times_dense} \end{table}
In Table~\ref{table:times_dense}, we compare the corresponding wall-clock times, and for convenience, we replicate the timing results of \texttt{hinfnorm} from Table~\ref{table:hinfnorm_tol} on the same problems. We observe that our new method was fastest on six out of the eight test problems, often significantly so. Compared to our own implementation of the BBBS algorithm, our new method was on average 1.72 times as fast and on three problems, 2.36-2.55 times faster. We see similar speedups compared to the cubic-interpolation refinement method as well. Our method was even faster when compared to \texttt{hinfnorm}, which had the advantage of being a compiled code rather than interpreted like our code. Our new method was over eleven times faster than \texttt{hinfnorm} overall, but this was largely due to the two problems (\texttt{FOM} and \texttt{randn 1}) where our code was 27-42 times faster. We suspect that this large performance gap on these problems was not necessarily due to a correspondingly dramatic reduction in the number of times that the eigenvalues of $(\Mc,\Nc)$ were computed but rather that the structure-preserving eigensolver \texttt{hinfnorm} employed sometimes has a steep performance penalty compared to standard QZ. However, it is difficult to verify this as \texttt{hinfnorm} is not open source. We also see that for the variants of our method, there was about a 24-33\% percent penalty on average in the runtime when resorting to the secant method instead of Newton's method. Nonetheless, even the slower secant-method-based version of our hybrid optimization approach was still typically much faster than BBBS or the cubic-interpolation scheme. The only exception to this was problem \texttt{CSE2}, where our secant method variants were actually faster than our Newton's method variants; the reason for this was because during initialization, the Newton's method optimization just happened to find worse initial local maximizers than the secant method approach, which led to more eigenvalue computations of $(\Mc,\Nc)$.
The two problems where the variants of our new method were not fastest were \texttt{CSE2} and \texttt{randn 1}. However, for \texttt{CSE2}, our secant method variant using midpoints was essentially as fast as the standard algorithm. As mentioned above, the Newton's method variants ended up being slower since they found worse initial local maximizers. For \texttt{randn 1}, all methods only required a single evaluation of $g_\mathrm{c}(\omega)$ and computing the eigenvalues of $(\Mc,\Nc)$; in other words, their respective initial guesses were all actually a global maximizer. As such, the differences in running times for \texttt{randn 1} seems likely attributed to the variability of interpreted {MATLAB}\ code.
\subsection{Discrete-time examples} We now present corresponding experiments for the six discrete-time examples listed in Table~\ref{table:hinfnorm_tol}. In Table~\ref{table:evals_dense_disc}, we see that our new approach on discrete-time problems also reduces the number of expensive eigenvalue computations of $(\Md,\Nd)$ compared to the standard methods and that in the worst cases, there is only moderate increase in the number of evaluations of $g_\mathrm{d}(\omega)$ and often, even a reduction, similarly as we saw in Table~\ref{table:evals_dense} for the continuous-time problems.
Wall-clock running times are reported in Table~\ref{table:times_dense_disc}, and show similar results, if not identical, to those in Table~\ref{table:times_dense} for the continuous-time comparison. We see that our Newton's method variants are, on average, 1.66 and 1.41 times faster, respectively, than the BBBS and cubic-interpolation refinement algorithms. Our algorithms are often up to two times faster than these two standard methods and were even up to 25.2 times faster on \texttt{ISS1d} compared to \texttt{hinfnorm} using \texttt{tol=1e-14}. For three of the six problems, our approach was not fastest but these three problems (\texttt{LAHd}, \texttt{BDT2d}, \texttt{EB6d}) also had the smallest orders among the discrete-time examples ($n=58,92,170$, respectively). This underscores that our approach is likely most beneficial for all but rather small-scale problems, where there is generally an insufficient cost gap between computing $g_\mathrm{d}(\omega)$ and the eigenvalues of $(\Md,\Nd)$. However, for \texttt{LAHd} and \texttt{EB6d}, it was actually \texttt{hinfnorm} that was fastest, where we are comparing a compiled code to our own pure {MATLAB}\ interpreted code. Furthermore, on these two problems, our approach was nevertheless not dramatically slower than \texttt{hinfnorm} and for \texttt{EB6d}, was actually faster than our own implementation of the standard algorithms. Finally, on \texttt{BDT2}, the fastest version of our approach essentially matched the performance of our BBBS implementation, if not the cubic-interpolation refinement.
\begin{table}[!t] \center
\begin{tabular}{ l | cccc | cc } \toprule \multicolumn{7}{c}{Small-scale examples (discrete time)} \\ \midrule \multicolumn{1}{c}{} &
\multicolumn{4}{c}{Hybrid Optimization} & \multicolumn{2}{c}{Standard Algs.} \\ \cmidrule(lr){2-5} \cmidrule(lr){6-7} \multicolumn{1}{c}{} &
\multicolumn{2}{c}{Newton} & \multicolumn{2}{c}{Secant} & \multicolumn{2}{c}{} \\ \cmidrule(lr){2-3} \cmidrule(lr){4-5} \multicolumn{1}{l}{Problem} &
\multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{MP} & \multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{MP} &
\multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{BBBS} \\ \midrule \multicolumn{7}{c}{Number of Eigenvalue Computations of $(\Md,\Nd)$}\\ \midrule \texttt{LAHd} & 2 & 2 & 1 & 1 & 3 & 4 \\ \texttt{BDT2d} & 2 & 3 & 2 & 2 & 3 & 4 \\ \texttt{EB6d} & 1 & 1 & 2 & 1 & 3 & 5 \\ \texttt{ISS1d} & 1 & 1 & 1 & 1 & 2 & 2 \\ \texttt{CBMd} & 1 & 1 & 1 & 1 & 3 & 2 \\ \texttt{CM5d} & 2 & 2 & 2 & 2 & 3 & 4 \\ \midrule \multicolumn{7}{c}{Number of Evaluations of $g_\mathrm{d}(\omega)$}\\ \midrule \texttt{LAHd} & 13 & 11 & 24 & 24 & 17 & 15 \\ \texttt{BDT2d} & 17 & 18 & 43 & 40 & 18 & 17 \\ \texttt{EB6d} & 22 & 22 & 37 & 34 & 32 & 32 \\ \texttt{ISS1d} & 5 & 5 & 24 & 24 & 7 & 6 \\ \texttt{CBMd} & 5 & 5 & 26 & 26 & 12 & 6 \\ \texttt{CM5d} & 20 & 16 & 27 & 27 & 22 & 18 \\ \bottomrule \end{tabular} \caption{The column headers remain as described for Table~\ref{table:evals_dense}. } \label{table:evals_dense_disc} \end{table}
\begin{table}[!t] \setlength{\tabcolsep}{3pt} \robustify\bfseries \center
\begin{tabular}{ l | SSSS | SS | SS } \toprule \multicolumn{9}{c}{Small-scale examples (discrete time)} \\ \midrule \multicolumn{1}{c}{} &
\multicolumn{4}{c}{Hybrid Optimization} & \multicolumn{2}{c}{Standard Algs.} & \multicolumn{2}{c}{\texttt{hinfnorm($\cdot$,tol)}}\\ \cmidrule(lr){2-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9} \multicolumn{1}{c}{} &
\multicolumn{2}{c}{Newton} & \multicolumn{2}{c}{Secant} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{\texttt{tol}}\\ \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){8-9} \multicolumn{1}{l}{Problem} &
\multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{MP} & \multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{MP} &
\multicolumn{1}{c}{Interp.} & \multicolumn{1}{c}{BBBS} &
\multicolumn{1}{c}{\texttt{1e-14}} & \multicolumn{1}{c}{\texttt{0.01}} \\ \midrule \multicolumn{9}{c}{Wall-clock running times in seconds}\\ \midrule \texttt{LAHd} & 0.051 & 0.038 & 0.056 & 0.056 & 0.034 & 0.040 & \bfseries 0.031 & 0.015 \\ \texttt{BDT2d} & 0.075 & 0.123 & 0.146 & 0.191 & \bfseries 0.057 & 0.076 & 0.070 & 0.031 \\ \texttt{EB6d} & 0.271 & 0.312 & 0.409 & 0.296 & 0.469 & 0.732 & \bfseries 0.192 & 0.122 \\ \texttt{ISS1d} & 0.654 & \bfseries 0.636 & 0.828 & 0.880 & 1.168 & 1.291 & 16.495 & 3.930 \\ \texttt{CBMd} & 0.898 & \bfseries 0.795 & 0.999 & 1.640 & 2.015 & 1.420 & 1.411 & 0.773 \\ \texttt{CM5d} & \bfseries 7.502 & 8.022 & 9.887 & 8.391 & 9.458 & 14.207 & 10.802 & 2.966 \\ \midrule \multicolumn{9}{c}{Running times relative to hybrid optimization (Newton with `Interp.')}\\ \midrule \texttt{LAHd} & 1 & 0.74 & 1.09 & 1.11 & 0.67 & 0.78 & 0.60 & 0.29 \\ \texttt{BDT2d} & 1 & 1.65 & 1.96 & 2.55 & 0.76 & 1.01 & 0.93 & 0.41 \\ \texttt{EB6d} & 1 & 1.15 & 1.51 & 1.09 & 1.73 & 2.70 & 0.71 & 0.45 \\ \texttt{ISS1d} & 1 & 0.97 & 1.27 & 1.34 & 1.79 & 1.97 & 25.21 & 6.01 \\ \texttt{CBMd} & 1 & 0.89 & 1.11 & 1.83 & 2.24 & 1.58 & 1.57 & 0.86 \\ \texttt{CM5d} & 1 & 1.07 & 1.32 & 1.12 & 1.26 & 1.89 & 1.44 & 0.40 \\ \midrule Avg. & 1 & 1.08 & 1.38 & 1.51 & 1.41 & 1.66 & 5.08 & 1.40 \\ \bottomrule \end{tabular} \caption{The column headers remain as described for Table~\ref{table:times_dense}. } \label{table:times_dense_disc} \end{table}
\section{Local optimization for ${\hinfsym_\infty}$ norm approximation} \label{sec:hinf_approx} Unfortunately, the ${\cal O}(n^3)$ work necessary to compute all the imaginary eigenvalues of $(\Mc,\Nc)$ restricts the usage of the level-set ideas from \cite{Bye88,BoyBK89} to rather small-dimensional problems. The same computational limitation of course also holds for obtaining all of the unit-modulus eigenvalues of $(\Md,\Nd)$ in the discrete-time case. Currently there is no known alternative technique that would guarantee convergence to a global maximizer of $g_\mathrm{c}(\omega)$ or $g_\mathrm{d}(\theta)$, to thus ensure exact computation of the ${\hinfsym_\infty}$ norm, while also having more favorable scaling properties. Indeed, the aforementioned scalable methods of \cite{GugGO13,BenV14,FreSV14,MitO16,AliBMetal17} for approximating the ${\hinfsym_\infty}$ norm of large-scale systems all forgo the expensive operation of computing all the eigenvalues of $(\Mc,\Nc)$ and $(\Md,\Nd)$, and consequently, the most that any of them can guarantee in terms of accuracy is that they converge to a local maximizer of $g_\mathrm{c}(\omega)$ or $g_\mathrm{d}(\theta)$. However, a direct consequence of our work here to accelerate the exact computation of the ${\hinfsym_\infty}$ norm is that the straightforward application of optimization techniques to compute local maximizers of either $g_\mathrm{c}(\omega)$ of $g_\mathrm{d}(\theta)$ can itself be considered an efficient and scalable approach for approximating the ${\hinfsym_\infty}$ norm of large-scale systems. It is perhaps a bit staggering that such a simple and direct approach seems to have been until now overlooked, particularly given the sophistication of the existing ${\hinfsym_\infty}$ norm approximation methods.
In more detail, recall that the initialization phase of Algorithm~\ref{alg:hybrid_opt}, lines~\ref{algline:init_start}-\ref{algline:init_end}, is simply just applying unconstrained optimization to find one or more maximizers of $g_\mathrm{c}(\omega)$. Provided that $({\bf i} \omega E - A)$ permits fast linear solves, e.g. a sparse LU decomposition, there is no reason why this cannot also be done for large-scale systems. In fact, the methods of \cite{BenV14,FreSV14,AliBMetal17} for approximating the ${\hinfsym_\infty}$ norm all require that such fast solves are possible (while the methods of \cite{GugGO13,MitO16} only require fast matrix-vector products with the system matrices). When $m,p \ll n$, it is still efficient to calculate second derivatives of $g_\mathrm{c}(\omega)$ to obtain a quadratic rate of convergence via Newton's method. Even if $m,p \ll n$ does not hold, first derivatives of $g_\mathrm{c}(\omega)$ can still be computed using sparse methods for computing the largest singular value (and its singular vectors) and thus the secant method can be employed to at least get superlinear convergence. As such, the local convergence and superlinear/quadratic convergence rate guarantees of the existing methods are at least matched by the guarantees of direct optimization. For example, while the superlinearly-convergent method of \cite{AliBMetal17} requires that $m,p \ll n$, our direct optimization approach remains efficient even if $m,p \approx n$, when it also has superlinear convergence, and it has quadratic convergence in the more usual case of $m,p \ll n$.
Of course, there is also the question of whether there are differences in approximation quality between the methods. This is a difficult question to address since beyond local optimality guarantees, there are no other theoretical results concerning the quality of the computed solutions. Actual errors can only be measured when running the methods on small-scale systems, where the exact value of the ${\hinfsym_\infty}$ norm can be computed, while for large-scale problems, only relative differences between the methods' respective approximations can be observed. Furthermore, any of these observations may not be predictive of performance on other problems. For nonconvex optimization, the quality of a locally optimal computed solution is often dependent on the starting point, which will be a strong consideration for the direct optimization approach. On the other hand, it is certainly plausible that the sophistication of the existing ${\hinfsym_\infty}$ norm algorithms may favorably bias them to converge to better (higher) maximizers more frequently than direct optimization would, particularly if only randomly selected starting points were used. With such complexities, in this paper we do not attempt to do a comprehensive benchmark with respect to existing ${\hinfsym_\infty}$ norm approximation methods but only attempt to demonstrate that direct optimization is a potentially viable alternative.
We implemented lines~\ref{algline:init_start}-\ref{algline:init_end} of Algorithm~\ref{alg:hybrid_opt} in a second, standalone routine, with the necessary addition for the continuous-time case that
the value of $\|D\|_2$ is returned if the computed local maximizers of $g_\mathrm{c}(\omega)$ only yield lower function values than $\|D\|_2$. Since we assume that the choice of starting points will be critical, we initialized our sparse routine using starting frequencies computed by \texttt{samdp}, a {MATLAB}\ code that implements the subspace-accelerated dominant pole algorithm of \cite{RomM06}. Correspondingly, we compared our approach to the {MATLAB}\ code \texttt{hinorm}, which implements the spectral-value-set-based method using dominant poles of \cite{BenV14} and also uses \texttt{samdp} (to compute dominant poles at each iteration). We tested \texttt{hinorm} using its default settings, and since it initially computes 20 dominant poles to find a good starting point, we also chose to compute 20 dominant poles via \texttt{samdp} to obtain 20 initial frequency guesses for optimization.\footnote{Note that these are not necessarily the same 20 dominant poles, since \cite{BenV14} must first transform a system if the original system has nonzero $D$ matrix.} Like our small-scale experiments, we also ensured zero was always included as an initial guess and reused the same choices for \texttt{fmincon} parameter values. We tested our optimization approach by optimizing $\phi=1,5,10$ of the most promising frequencies, again using a serial MATLAB code. Since we used LU decompositions to solve the linear systems, we tested our code in two configurations: with and without permutations, i.e. for some matrix given by variable \texttt{A}, \texttt{[L,U,p,q] = lu(A,'vector')} and \texttt{[L,U] = lu(A)}, respectively.
Table~\ref{table:problems_large} shows our selection or large-scale test problems, all continuous-time since \texttt{hinorm} does not support discrete-time problems (in contrast to our optimization-based approach which supports both). Problems \texttt{dwave} and \texttt{markov} are from the large-scale test set used in \cite{GugGO13} while the remaining problems are freely available from the website of Joost Rommes\footnote{Available at \url{https://sites.google.com/site/rommes/software}}. As $m,p \ll n$ holds in all of these examples, we just present results for our code when using Newton's method. For all problems, our code produced ${\hinfsym_\infty}$ norm approximations that agreed to at least 12 digits with \texttt{hinorm}, meaning that the additional optimization calls done with $\phi=5$ and $\phi=10$ did not produce better maximizers than what was found with $\phi=1$ and thus, only added to the serial computation running time.
\begin{table}[!t] \centering
\begin{tabular}{ l | rrr | c } \toprule \multicolumn{5}{c}{Large-scale examples (continuous time)}\\ \midrule \multicolumn{1}{l}{Problem} &
\multicolumn{1}{c}{$n$} &
\multicolumn{1}{c}{$p$} &
\multicolumn{1}{c}{$m$} &
\multicolumn{1}{c}{$E=I$} \\ \midrule \texttt{dwave} & 2048 & 4 & 6 & Y \\ \texttt{markov} & 5050 & 4 & 6 & Y \\ \texttt{bips98\_1450} & 11305 & 4 & 4 & N \\ \texttt{bips07\_1693} & 13275 & 4 & 4 & N \\ \texttt{bips07\_1998} & 15066 & 4 & 4 & N \\ \texttt{bips07\_2476} & 16861 & 4 & 4 & N \\ \texttt{descriptor\_xingo6u} & 20738 & 1 & 6 & N \\ \texttt{mimo8x8\_system} & 13309 & 8 & 8 & N \\ \texttt{mimo28x28\_system} & 13251 & 28 & 28 & N \\ \texttt{ww\_vref\_6405} & 13251 & 1 & 1 & N \\ \texttt{xingo\_afonso\_itaipu} & 13250 & 1 & 1 & N \\ \bottomrule \end{tabular} \caption{The list of test problems for the large-scale ${\hinfsym_\infty}$-norm approximation comparing direct local optimization against \texttt{hinorm}, along with the corresponding problem dimensions and whether they are standard state-space systems ($E=I$) or descriptor systems $(E \ne I)$.} \label{table:problems_large} \end{table}
\begin{table}[!t] \setlength{\tabcolsep}{3pt} \robustify\bfseries \center
\begin{tabular}{ l | SSS | SSS | S } \toprule \multicolumn{8}{c}{Large-scale examples (continuous time)}\\ \midrule \multicolumn{1}{c}{} &
\multicolumn{6}{c}{Direct optimization: $\phi=1,5,10$ } & \multicolumn{1}{c}{\texttt{hinorm}} \\ \cmidrule(lr){2-7} \cmidrule(lr){8-8} \multicolumn{1}{c}{} & \multicolumn{3}{c}{\texttt{lu} with permutations} & \multicolumn{3}{c}{\texttt{lu} without permutations} & \multicolumn{1}{c}{} \\ \cmidrule(lr){2-4} \cmidrule(lr){5-7} \multicolumn{1}{l}{Problem} &
\multicolumn{1}{c}{1} & \multicolumn{1}{c}{5} & \multicolumn{1}{c}{10} &
\multicolumn{1}{c}{1} & \multicolumn{1}{c}{5} & \multicolumn{1}{c}{10} & \multicolumn{1}{c}{} \\ \midrule \multicolumn{8}{c}{Wall-clock running times in seconds (initialized via \texttt{samdp})}\\ \midrule \texttt{dwave} & 1.979 & 1.981 & 1.997 & 5.536 & 5.154 & 5.543 & \bfseries 1.861 \\ \texttt{markov} & \bfseries 3.499 & 3.615 & 3.593 & 26.734 & 26.898 & 27.219 & 3.703 \\ \texttt{bips98\_1450} & \bfseries 6.914 & 8.333 & 10.005 & 14.559 & 17.157 & 19.876 & 31.087 \\ \texttt{bips07\_1693} & \bfseries 8.051 & 9.155 & 11.594 & 18.351 & 21.367 & 24.322 & 75.413 \\ \texttt{bips07\_1998} & \bfseries 10.344 & 11.669 & 13.881 & 50.059 & 56.097 & 59.972 & 51.497 \\ \texttt{bips07\_2476} & \bfseries 14.944 & 16.717 & 18.942 & 65.227 & 70.920 & 73.206 & 76.697 \\ \texttt{descriptor\_xingo6u} & 13.716 & 15.328 & 16.997 & \bfseries 7.907 & 9.225 & 11.133 & 36.775 \\ \texttt{mimo8x8\_system} & 7.566 & 8.934 & 11.211 & \bfseries 6.162 & 7.562 & 9.321 & 30.110 \\ \texttt{mimo28x28\_system} & 12.606 & 17.767 & 20.488 & \bfseries 10.815 & 16.591 & 21.645 & 33.107 \\ \texttt{ww\_vref\_6405} & 7.353 & 6.785 & 7.552 & \bfseries 4.542 & 5.076 & 5.437 & 18.553 \\ \texttt{xingo\_afonso\_itaipu} & 5.780 & 6.048 & 7.772 & \bfseries 4.676 & 4.975 & 5.573 & 16.928 \\ \midrule \multicolumn{8}{c}{Running times relative to direct optimization (\texttt{lu} with permutations and $\phi =1$)}\\ \midrule \texttt{dwave} & 1 & 1.00 & 1.01 & 2.80 & 2.60 & 2.80 & \bfseries 0.94 \\ \texttt{markov} & \bfseries 1 & 1.03 & 1.03 & 7.64 & 7.69 & 7.78 & 1.06 \\ \texttt{bips98\_1450} & \bfseries 1 & 1.21 & 1.45 & 2.11 & 2.48 & 2.87 & 4.50 \\ \texttt{bips07\_1693} & \bfseries 1 & 1.14 & 1.44 & 2.28 & 2.65 & 3.02 & 9.37 \\ \texttt{bips07\_1998} & \bfseries 1 & 1.13 & 1.34 & 4.84 & 5.42 & 5.80 & 4.98 \\ \texttt{bips07\_2476} & \bfseries 1 & 1.12 & 1.27 & 4.36 & 4.75 & 4.90 & 5.13 \\ \texttt{descriptor\_xingo6u} & 1 & 1.12 & 1.24 & \bfseries 0.58 & 0.67 & 0.81 & 2.68 \\ \texttt{mimo8x8\_system} & 1 & 1.18 & 1.48 & \bfseries 0.81 & 1.00 & 1.23 & 3.98 \\ \texttt{mimo28x28\_system} & 1 & 1.41 & 1.63 & \bfseries 0.86 & 1.32 & 1.72 & 2.63 \\ \texttt{ww\_vref\_6405} & 1 & 0.92 & 1.03 & \bfseries 0.62 & 0.69 & 0.74 & 2.52 \\ \texttt{xingo\_afonso\_itaipu} & 1 & 1.05 & 1.34 & \bfseries 0.81 &0.86 & 0.96 & 2.93 \\ \midrule Average & 1 & 1.12 & 1.30 & 2.52 & 2.74 & 2.97 & 3.70 \\ \bottomrule \end{tabular} \caption{In the top half of the table, the running times (fastest in bold) are reported in seconds for our direct Newton-method-based optimization approach in two configurations (\texttt{lu} with and without permutations) and \texttt{hinorm}. Each configuration of our approach optimizes the norm of the transfer function for up to $\phi$ different starting frequencies ($\phi=1,5,10$), done sequentially. The bottom half the table normalizes all the times relative to the running times for our optimization method using \texttt{lu} with permutations and $\phi=1$, along with the overall averages relative to this variant. } \label{table:times_large} \end{table}
In Table~\ref{table:times_large}, we present the running times of the codes and configurations. First, we observe that for our direct optimization code, using \texttt{lu} with permutations is two to eight times faster than without permutations; on average, using \texttt{lu} with permutations is typically 2.5 times faster. Interestingly, on the last five problems, using \texttt{lu} without permutations was actually best, but using permutations was typically only about 25\% slower and at worse, about 1.7 times slower (\texttt{descriptor\_xingo6u}). We found that our direct optimization approach, using just one starting frequency ($\phi=1$) was typically 3.7 times faster than \texttt{hinorm} on average and almost up to 10 times faster on problem \texttt{bips07\_1693}. Only on problem \texttt{dwave} was direct optimization actually slower than \texttt{hinorm} and only by a negligible amount. Interestingly, optimizing just one initial frequency versus running optimization for ten frequencies ($\phi=10$) typically only increased the total running time of our code by 20-30\%. This strongly suggested that the dominant cost of running our code is actually just calling \texttt{samdp} to compute the 20 initial dominant poles to obtain starting guesses. As such, in Table~\ref{table:times_samdp}, we report the percentage of the overall running time for each variant/method that was due to their initial calls to \texttt{samdp}. Indeed, our optimization code's single call to \texttt{samdp} accounted for 81.5-99.3\% of its running-time (\texttt{lu} with permutations and $\phi=1$). In contrast, \texttt{hinorm}'s initial call to \texttt{samdp} usually accounted for about only a quarter of its running time on average, excluding \texttt{dwave} and \texttt{markov} as exceptional cases. In other words, the convergent phase of direct optimization is actually even faster than the convergent phase of \texttt{hinorm} than what Table~\ref{table:times_large} appears to indicate. On problem \texttt{bips07\_1693}, we see that our proposal to use Newton's method to optimize $g_\mathrm{c}(\omega)$ directly is actually over 53 times faster than \texttt{hinorm}'s convergent phase.
\begin{table}[!t] \center
\begin{tabular}{ l | SSS | SSS | S } \toprule \multicolumn{8}{c}{Large-scale examples (continuous time)}\\ \midrule \multicolumn{8}{c}{Percentage of time just to compute 20 initial dominant poles (first call to \texttt{samdp})}\\ \midrule \multicolumn{1}{c}{} &
\multicolumn{6}{c}{Direct optimization: $\phi=1,5,10$} & \multicolumn{1}{c}{\texttt{hinorm}} \\ \cmidrule(lr){2-7} \cmidrule(lr){8-8} \multicolumn{1}{c}{} & \multicolumn{3}{c}{\texttt{lu} with permutations} & \multicolumn{3}{c}{\texttt{lu} without permutations} & \multicolumn{1}{c}{} \\ \cmidrule(lr){2-4} \cmidrule(lr){5-7} \multicolumn{1}{l}{Problem} &
\multicolumn{1}{c}{1} & \multicolumn{1}{c}{5} & \multicolumn{1}{c}{10} &
\multicolumn{1}{c}{1} & \multicolumn{1}{c}{5} & \multicolumn{1}{c}{10} & \multicolumn{1}{c}{} \\ \midrule \texttt{dwave} & 99.3 & 99.3 & 99.2 & 99.3 & 99.6 & 99.6 & 98.0 \\ \texttt{markov} & 99.2 & 99.1 & 99.1 & 99.6 & 99.6 & 99.6 & 98.1 \\ \texttt{bips98\_1450} & 84.1 & 72.7 & 60.4 & 84.9 & 73.0 & 61.8 & 21.7 \\ \texttt{bips07\_1693} & 84.2 & 75.1 & 61.6 & 85.6 & 77.1 & 64.4 & 10.1 \\ \texttt{bips07\_1998} & 85.2 & 77.1 & 66.0 & 88.5 & 81.5 & 72.8 & 18.3 \\ \texttt{bips07\_2476} & 88.0 & 80.2 & 71.4 & 89.2 & 83.6 & 73.0 & 18.6 \\ \texttt{descriptor\_xingo6u} & 90.2 & 82.8 & 75.0 & 89.3 & 79.9 & 66.8 & 38.4 \\ \texttt{mimo8x8\_system} & 84.6 & 74.4 & 59.1 & 84.1 & 67.7 & 55.0 & 24.2 \\ \texttt{mimo28x28\_system} & 81.5 & 59.4 & 51.2 & 75.5 & 46.9 & 39.1 & 40.9 \\ \texttt{ww\_vref\_6405} & 95.1 & 79.6 & 63.9 & 91.9 & 78.3 & 75.1 & 33.7 \\ \texttt{xingo\_afonso\_itaipu} & 90.2 & 84.4 & 76.8 & 89.3 & 83.7 & 74.7 & 33.6 \\ \bottomrule \end{tabular} \caption{The column headings are the same as in Table~\ref{table:times_large}.} \label{table:times_samdp} \end{table}
\section{Parallelizing the algorithms} \label{sec:parallel} The original BBBS algorithm, and the cubic-interpolation refinement, only provide little opportunity for parallelization at the \emph{algorithmic} level, i.e. when not considering that the underlying basic linear algebra operations may be parallelized themselves when running on a single shared-memory multi-core machine. Once the imaginary eigenvalues\footnote{ For conciseness, our discussion in Section~\ref{sec:parallel} will be with respect to the continuous-time case but note that it applies equally to the discrete-time case as well.} have been computed, constructing the level-set intervals (line~\ref{algline:bbbs_ints} of Algorithm~\ref{alg:bbbs}) and calculating $g_\mathrm{c}(\omega)$ at their midpoints or cubic-interpolant-derived maximizers (line~\ref{algline:bbbs_points} of Algorithm~\ref{alg:bbbs}) can both be done in an embarrassingly parallel manner, e.g. across nodes on a cluster. However, as we have discussed to motivate our improved algorithm, evaluating $g_\mathrm{c}(\omega)$ is a rather cheap operation compared to computing the eigenvalues of $(\Mc,\Nc)$. Crucially, parallelizing these two steps does not result in an improved (higher) value of $\gamma$ found per iteration and so the number of expensive eigenvalue computations of $(\Mc,\Nc)$ remains the same.
For our new method, we certainly can (and should) also parallelize the construction of the level-set intervals and the evaluations of their midpoints or cubic-interpolants-derived maximizers (lines~\ref{algline:hyopt_ints} and \ref{algline:hyopt_points} in Algorithm~\ref{alg:hybrid_opt}), despite that we do not expect large gains to be had here. However, optimizing over the intervals (line~\ref{algline:hyopt_opt} in Algorithm~\ref{alg:hybrid_opt}) is also an embarrassingly parallel task and here significant speedups can be obtained. As mentioned earlier, with serial computation (at the algorithmic level), we typically recommend only optimizing over a single level-set interval ($\phi=1$) out of the $q$ candidates (the most promising one, as determined by line~\ref{algline:hyopt_reorder} in Algorithm~\ref{alg:hybrid_opt}); otherwise, the increased number of evaluations of $g_\mathrm{c}(\omega)$ can start to outweigh the benefits of performing the local optimization. By optimizing over more intervals in parallel, e.g. again across nodes on a cluster, we increase the chances on every iteration of finding even higher peaks of $g_\mathrm{c}(\omega)$, and possibly a global maximum, \emph{without any increased time penalty} (besides communication latency).\footnote{Note that distributing the starting points for optimization and taking the max of the resulting optimizers involves very little data being communicated on any iteration.} In turn, larger steps in $\gamma$ can be taken, potentially reducing the number of expensive eigenvalue computations of $(\Mc,\Nc)$ incurred. Furthermore, parallelization can also be applied to the initialization stage to optimize from as many starting points as possible without time penalty (lines~\ref{algline:init_start} and \ref{algline:init_opt} in Algorithm~\ref{alg:hybrid_opt}), a standard technique for nonconvex optimization problems. Finding a global maximum of $g_\mathrm{c}(\omega)$ during initialization means that the algorithm will only need to compute the eigenvalues of $(\Mc,\Nc)$ just once, to assert that maximum found is indeed a global one.
When using direct local optimization techniques for ${\hinfsym_\infty}$ approximation, as discussed in Section~\ref{sec:parallel}, optimizing from as many starting points as possible of course also increases the chances of finding the true value of the ${\hinfsym_\infty}$ norm, or at least better approximations than just starting from one point. With parallelization, these additional starting points can also be tried without any time penalty (also lines~\ref{algline:init_start} and \ref{algline:init_opt} in Algorithm~\ref{alg:hybrid_opt}), unlike the experiments we reported in Section~\ref{sec:hinf_approx} where we optimized using $\phi=1,5,10$ starting guesses with a serial MATLAB code and therefore incurred longer running times as $\phi$ was increased.
For final remarks on parallelization, first note that there will generally be less benefits when using more than $n$ parallel optimization calls, since there are at most $n$ peaks of $g_\mathrm{c}(\omega)$. However, for initialization, one could simply try as many starting guesses as there are parallel nodes available (even if the number of nodes is greater than $n$) to maximize the chances of finding a high peak of $g_\mathrm{c}(\omega)$ or a global maximizer. Second, the number of level-set intervals encountered by the algorithm at each iteration may be significantly less than $n$, particularly if good starting guesses are used. Indeed, it is not entirely uncommon for the algorithm to only encounter one or two level-set intervals on each iteration. On the other hand, for applications where $g_\mathrm{c}(\omega)$ has many similarly high peaks, such as controller design where the ${\hinfsym_\infty}$ norm is minimized, our new algorithm may consistently benefit from parallelization with a higher number of parallel optimization calls.
\section{Conclusion and outlook} \label{sec:wrapup} We have presented an improved algorithm that significantly reduces the time necessary to compute the ${\hinfsym_\infty}$ norm of linear control systems compared to existing algorithms. Furthermore, our proposed hybrid optimization approach also allows the ${\hinfsym_\infty}$ norm to be computed to machine precision with relatively little extra work, unlike earlier methods. We have also demonstrated that approximating the ${\hinfsym_\infty}$ norm of large-scale problems via directly optimizing the norm of the transfer function is not only viable but can be quite efficient. In contrast to the standard BBBS and cubic-interpolation refinement algorithms, our new approaches for ${\hinfsym_\infty}$ norm computation and approximation also can benefit significantly more from parallelization. Work is ongoing to add implementations of our new algorithms to a future release of the open-source library ROSTAPACK: RObust STAbility PACKage.\footnote{Available at \url{http://www.timmitchell.com/software/ROSTAPACK}} This is being done in coordination with our efforts to also add implementations of our new methods for computing the spectral value set abscissa and radius, proposed in \cite{BenM17a} and which use related ideas to those in this paper. The current v1.0 release of ROSTAPACK contains implementations of scalable algorithms for approximating all of these aforementioned measures \cite{GugO11,GugGO13,MitO16}, as well as variants where the uncertainties are restricted to be real valued \cite{GugGMetal17}.
Regarding our experimental observations, the sometimes excessively longer compute times for \texttt{hinfnorm} compared to all other methods we evaluated possibly indicates that the structure-preserving eigensolver that it uses can sometimes be much slower than QZ. This certainly warrants further investigation, and if confirmed, suggests that optimizing the code/algorithm of the structure-preserving eigensolver could be a worthwhile pursuit. In the large-scale setting, we have observed that the dominant cost for our direct optimization approach is actually due to obtaining the starting frequency guesses via computing dominant poles. If the process of obtaining good initial guesses can be accelerated, then approximating the ${\hinfsym_\infty}$ norm via direct optimization could be significantly sped up even more.
\end{document}
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\begin{document}
\title{Learning Deterministic Finite Automata Decompositions from Examples and Demonstrations \thanks{This work was partially supported by NSF grants 1545126 (VeHICaL) and 1837132, by the DARPA contracts FA8750-18-C-0101 (Assured Autonomy) and FA8750-20-C-0156 (SDCPS), by Berkeley Deep Drive, by Toyota under the iCyPhy center, and by Toyota Research Institute.}}
\author{
\IEEEauthorblockN{
Niklas Lauffer\thanks{\textsuperscript{\IEEEauthorrefmark{1}}Equal contribution}\textsuperscript{\IEEEauthorrefmark{1}}\IEEEauthorrefmark{2},
Beyazit Yalcinkaya\textsuperscript{\IEEEauthorrefmark{1}}\IEEEauthorrefmark{2},
Marcell Vazquez-Chanlatte\IEEEauthorrefmark{2},
Ameesh Shah\IEEEauthorrefmark{2} and
Sanjit A. Seshia\IEEEauthorrefmark{2}
}\\
\IEEEauthorblockA{\IEEEauthorrefmark{2}University of California, Berkeley} }
\maketitle
\begin{abstract} The identification of a \emph{deterministic finite automaton} (DFA) from labeled examples is a well-studied problem in the literature; however, prior work focuses on the identification of monolithic DFAs. Although monolithic DFAs provide accurate descriptions of systems’ behavior, they lack simplicity and interpretability; moreover, they fail to capture sub-tasks realized by the system and introduce inductive biases away from the inherent decomposition of the overall task. In this paper, we present an algorithm for learning conjunctions of DFAs from labeled examples. Our approach extends an existing SAT-based method to systematically enumerate Pareto-optimal candidate solutions. We highlight the utility of our approach by integrating it with a state-of-the-art algorithm for learning DFAs from demonstrations. Our experiments show that the algorithm learns sub-tasks realized by the labeled examples, and it is scalable in the domains of interest. \end{abstract}
\section{Introduction}
Grammatical inference is a mature and well-studied field with many application domains ranging from various computer science fields, e.g., machine learning, to areas of natural sciences, e.g. computational biology~\cite{de2005bibliographical}. The identification of a minimum size \emph{deterministic finite automaton} (DFA) from labeled examples is one of the most well-investigated problems in this field. Furthermore, with the increase in computational power in recent years, the problem can be solved efficiently by various tools available in the literature (e.g.,~\cite{verwer2017flexfringe,zakirzyanov2019efficient}).
Existing work on DFA identification primarily focuses on the monolithic case, i.e., learning a single DFA from examples. Although such DFAs capture a language consistent with the examples, they may lack simplicity and interpretability. Furthermore, complex tasks often decompose into independent sub-tasks; hence, the system traces implicitly reflect this behavior. However, monolithic DFA identification fails to capture the natural decomposition of the system behavior, introducing an inductive bias away from the inherent decomposition of the overall task. In this paper, we present an algorithm for learning \emph{DFA decompositions} from examples by reducing the problem to graph coloring in SAT and a Pareto-optimal solution search over candidate solutions. A DFA decomposition is a set of DFAs such that \emph{intersection} of their language is the language of the system, which implicitly defines a conjunction of simpler specifications realized by the overall system. We present an application of our algorithm to a state-of-the-art method for learning task specifications from unlabeled demonstrations~\cite{marcell2021DISS} to showcase a domain of interest for DFA decompositions.
\textbf{Related Work.} Existing work considers the problem of minimal DFA identification from labeled examples~\cite{de2005bibliographical}. It is shown that the DFA identification problem with a given upper bound on the number of states is an NP-complete problem~\cite{gold1978complexity}. Another work shows that this problem cannot be efficiently approximated~\cite{pitt1993minimum}. Fortunately, practical methods exist in the literature. A common approach is to apply the evidence driven state-merging algorithm~\cite{lang1998results,lang1999faster,bugalho2005inference}, which is a greedy algorithm that aims to find a good local optimum. Other works for learning DFAs use evolutionary computation~\cite{dupont1994regular,luke1999genetic}, later improved by multi-start random hill climbing~\cite{lucas2003learning}.
A different approach to the monolithic DFA identification is to leverage highly-optimized modern SAT solvers by encoding the problem in SAT~\cite{heule2010exact}. In follow up works, several symmetry breaking predicates are proposed for the SAT encoding to reduce the search space \cite{zakirzyanov2019efficient,ulyantsev2015bfs,ulyantsev2016symmetry,zakirzyanov2017finding}. However, to the best of our knowledge, no work considers directly learning DFA decompositions from examples and demonstrations.
This work also relates to the problem of decomposing a known automaton. Ashar et al.~\cite{ashar1992finite} explore computing cascade and general decomposition of finite state machines. The Krohn–Rhodes theorem \cite{rhodes2010applications} reduces a finite automaton into a cascade of irreducible automata.
Kupferman \& Mosheiff~\cite{kupferman2015prime} present various complexity results for DFA decomposability.
Finally, the problem of learning objectives from demonstrations of an expert dates back to the problem of Inverse Optimal Control~\cite{kalman1964linear} and, more recently in the artificial intelligence community, the problem of Inverse Reinforcement Learning (IRL)~\cite{ng2000irl}. The goal in IRL is to recover the unknown reward function that an expert agent is trying to maximize based on observations of that expert. Recently, several works have considered a version of the IRL problem in which the expert agent is trying to maximize the satisfaction of a Boolean task specification~\cite{kasenberg2017apprenticeship,chou2020explaining, marcell2021DISS}. However, no work considers learning \emph{decompositions} of specifications from demonstrations.
\section{Problem Formulation}
Let $\mathcal{D}$ denote the set of DFAs over some fixed alphabet $\Sigma$. An $(m_1, \ldots, m_n)$-\emph{DFA decomposition} is a tuple of $n$ DFAs $({\mathcal{A}}_1, \dots, {\mathcal{A}}_n) \in \mathcal{D}^n$ where ${\mathcal{A}}_i$ has $m_i$ states and $m_1 \leq m_2 \leq \dots \leq m_n$. We associate a partial order $\prec$ on DFA decompositions using the standard product order on the number of states. That is, $({\mathcal{A}}_1', \dots, {\mathcal{A}}_n') \prec ({\mathcal{A}}_1, \dots, {\mathcal{A}}_n)$, if $m_i' \leq m_i$ for all $i \in [n]$ and $m_j' < m_j$ for some $j \in [n]$. In this case, we say $({\mathcal{A}}_1', \dots, {\mathcal{A}}_n')$ \emph{dominates} $({\mathcal{A}}_1, \dots, {\mathcal{A}}_n)$. A DFA decomposition $({\mathcal{A}}_1, \ldots, {\mathcal{A}}_n)$ \emph{accepts} a string $w$ iff all ${\mathcal{A}}_i$ accept $w$. A string that is not accepted is \emph{rejected}. The \emph{language} of a decomposition, ${\mathcal{L}}({\mathcal{A}}_1, \ldots, {\mathcal{A}}_n)$, is the set of accepting strings, i.e., the intersection of all DFA languages.
We study the problem of finding a DFA decomposition from a set of positive and negative labeled examples such that the decomposition accepts the positive examples and rejects the negative examples. Next, we formally define \emph{the DFA decomposition identification problem} (\texttt{DFA-DIP}\xspace), and then present an overview of the proposed approach.
\begin{problembox}{The Deterministic Finite Automaton Decomposition Identification Problem (\texttt{DFA-DIP}\xspace).} Given positive examples, $D_+$ and negative examples, $D_-$, and a natural number $n \in \mathbb{N}$, find a $(m_1, \ldots, m_n)$-DFA decomposition $({\mathcal{A}}_1, \dots, {\mathcal{A}}_n)$ satisfying the following conditions. \begin{enumerate}[label=\textbf{(C\arabic*)}]
\item\label{problem:language_condition} The decomposition is consistent with $(D_+, D_-)$:
\[
\begin{split}
&D_+ \subseteq {\mathcal{L}}({\mathcal{A}}_1, {\mathcal{A}}_2, \dots, {\mathcal{A}}_n),\\
&D_- \subseteq \Sigma^* \setminus {\mathcal{L}}({\mathcal{A}}_1, {\mathcal{A}}_2, \dots, {\mathcal{A}}_n).
\end{split}
\]
\item\label{problem:numbers_of_states_condition} There does not exist a DFA decomposition that \emph{dominates} $({\mathcal{A}}_1, \dots, {\mathcal{A}}_n)$ and satisfies \ref{problem:language_condition}.
\end{enumerate} \end{problembox}
We refer to the set of DFA decompositions that solve an instance of \texttt{DFA-DIP}\xspace as the Pareto-optimal frontier of solutions. Note that for $n=1$, \texttt{DFA-DIP}\xspace reduces to monolithic DFA identification. We propose finding the set of DFA decompositions that solve \texttt{DFA-DIP}\xspace by reduction to graph coloring in SAT and a breadth first search in solution space. Specifically, we extend the existing work on SAT-based monolithic DFA identification~\cite{heule2010exact,ulyantsev2016symmetry} to finding $n$ DFAs with $m_1, \dots, m_n$ states such that the intersection of their languages is consistent with the given examples. On top of this SAT-based approach, we develop a search strategy over the numbers of states passed to the SAT solver as these values are not known a priori.
\section{Learning DFAs from Examples}
In this section, we present the proposed approach. We start with the SAT encoding of the DFA decomposition problem and continue with the Pareto frontier search in the solution space. We then showcase an example of learning conjunctions of DFAs from labeled examples. Finally, we present experimental results and evaluate the scalability of our method.
\subsection{Encoding \texttt{DFA-DIP}\xspace in SAT}
We extend the SAT encoding for monolithic DFA identification presented in~\cite{heule2010exact,ulyantsev2016symmetry}, which solves a graph coloring problem, to finding $n$ DFAs with $m_1, m_2, \dots, m_n$ states.
The extension relies on the observation that for conjunctions of DFAs, we need to enforce that a positive example must be accepted by \emph{all} DFAs, and a negative example must be rejected by \emph{at least} one of the DFAs. Due to space limitations, we only present the modified clauses of the encoding, and invite reader to \Cref{sec:appendix} for further details.
The encoding works on an \emph{augmented prefix tree acceptor} (APTA), a tree-shaped automaton constructed from given examples, which has paths for each example leading to accepting or rejecting states based on the example's label; therefore, an APTA defines $D_+$ and $D_-$ which then constrains the accepting states, rejecting states, and the transition function of the unknown DFAs. For each DFA, ${\mathcal{A}}_i$, the encoding will associate the APTA states with one of the $m_i$ colors for DFA ${\mathcal{A}}_i$, subject to the constraints imposed by $D_+$ and $D_-$. APTA states with the same (DFA-indexed) color will be the same state in the corresponding DFA. We refer to states of an APTA as $V$, its accepting states as $V_{+}$, and its rejecting states as $V_{-}$. Given $n$ for the number of DFAs and $m_1, \dots, m_n$ for the number of states of DFAs, the SAT encoding uses three types of variables: \begin{enumerate}
\item \emph{color} variables $x^{k}_{v,i} \equiv 1$ ($k \in [n]$; $v \in V$; $i \in [m_k]$) iff APTA state $v$ has color $i$ in DFA $k$,
\item \emph{parent relation} variables $y^{k}_{l,i,j} \equiv 1$ ($k \in [n]$; $l \in \Sigma$, where $\Sigma$ is the alphabet; $i, j \in [m_k]$) iff DFA $k$ transitions with symbol $l$ from state $i$ to state $j$, and
\item \emph{accepting color} variables $z^{k}_{i} \equiv 1$ ($k \in [n]$; $i \in [m_k]$) iff state $i$ of DFA $k$ is an accepting state. \end{enumerate} The encoding for the monolithic DFA identification also uses the same variable types; however, in our encoding, we also index variables over $n$ DFAs instead of a single DFA. With this extension, one can trivially instantiate the encoding presented in \cite{heule2010exact,ulyantsev2016symmetry}. Below, we list the new rules we define for our problem. For the complete list of rules, see~\Cref{sec:appendix}. \begin{enumerate}[label=\textbf{(R\arabic*)},leftmargin=23pt]
\item\label{rule1} A negative example must be rejected by \emph{at least} one DFA:
\[
\bigwedge_{v \in V_{-}} \bigvee_{k \in [n]} \bigwedge_{i \in [m_k]} x^{k}_{v, i} \implies \neg z^{k}_{i}.
\]
\item\label{rule2} Accepting and rejecting states of APTA cannot be merged:
\[
\bigwedge_{v_{-} \in V_{-}} \bigwedge_{v_{+} \in V_{+}} \bigwedge_{k \in [n]} \bigwedge_{i \in [m_k]} (x^{k}_{v_{-}, i} \wedge \neg z^{k}_{i}) \implies \neg x^{k}_{v_{+}, i}.
\] \end{enumerate} In the encoding of \cite{heule2010exact,ulyantsev2016symmetry}, we replace the rule stating that the resulting DFA must reject all negative examples with \ref{rule1}, and \ref{rule2} is used instead of the original rule stating that accepting and rejecting states of APTA cannot be merged. Notice that since a rejecting state of APTA is not necessarily a rejecting state of a DFA $k$, we need to use the new rule~\ref{rule2}.
\begin{theorem}\label{thm:sat} Given labeled examples with $n$ and $m_1, \dots, m_n$, a solution to our SAT encoding is a solution to \texttt{DFA-DIP}\xspace. \end{theorem}
See \Cref{appendix:proof_main} for the proof of \Cref{thm:sat}.
\begin{figure*}
\caption{Experiment results evaluating the scalability of our algorithm w.r.t. (a) number of DFAs implied by the examples and (b) number of labeled examples.}
\label{fig:vary_dfas}
\label{fig:vary_examples}
\label{fig:exps}
\end{figure*}
\subsection{Pareto Frontier Search}
\texttt{DFA-DIP}\xspace requires finding a conjunction of $n$ DFAs that identify a language. There may exist multiple DFA decompositions that solve the problem with varying number of states $m_1, m_2, \dots, m_n$. With only a single DFA, the notion of minimal size is well-captured by the number of states. However, with multiple DFAs in the decomposition, the notion of a minimal solution is less clear. For example, there may exist a decomposition of two size three DFAs, and a separate decomposition of a size two and a size four DFA, both identifying the given set of labeled examples. Neither solution is strictly smaller than the other, and either solution might be preferred in different scenarios. Therefore, the set of solutions to \texttt{DFA-DIP}\xspace form a Pareto-optimal frontier in solution space.
Our proposed Pareto frontier enumeration algorithm is a breadth first search (BFS) over DFA decomposition size tuples that skips tuples that are dominated by an existing solution. This BFS is over a directed acyclic graph $G = (V,E)$ formed in the following way. There is a vertex in the graph for every ordered tuple of states sizes. There is an edge from $(m_1, m_2, \dots, m_n)$ to $(m'_1, m'_2, \dots, m'_n)$ if there exists some $j \in [n]$ such that:
\begin{equation*}
m'_i =
\begin{cases}
m_i + 1 & \text{if $i = j$;} \\
m_i & \text{otherwise.}
\end{cases} \end{equation*} A size tuple $(m_1, \dots, m_n)$ is a sink, i.e., the search does not continue past this vertex, if there exists a $(m_1, \dots, m_n)$-decomposition that solves \texttt{DFA-DIP}\xspace or the size tuple is dominated by a previously traversed solution. In the prior case, the associated DFA decomposition is also returned as a solution on the Pareto-optimal frontier. The BFS starts from $m_1 = m_2 = \dots = m_n = 1$, and performs the search as explained. See \Cref{appendix:pareto_search} for the details of the algorithm.
\begin{theorem}\label{thm:pareto} The described BFS is sound and complete; it outputs the full Pareto-optimal frontier of solutions without returning any dominated solutions. \end{theorem}
See \Cref{appendix:proof_pareto} for the proof of \Cref{thm:pareto}.
\subsection{Example: Learning Partially-Ordered Tasks}\label{subsec:toy_example}
We continue with a toy example showcasing the capabilities of the proposed approach. Later, we use the same class of decompositions to evaluate the scalability of our algorithm. \setlength{\columnsep}{10pt} \begin{wrapfigure}[13]{l}{0.25\textwidth} \centering
\subfloat[Learned DFA recognizing the ordering between ${\color{yellow}\blacksquare}$ and ${\color{redTileColor}\blacksquare}$.\label{fig:toy_example_dfa1}]{
\includegraphics[width=\linewidth]{figures/dfa1.pdf}}
\subfloat[Learned DFA recognizing the ordering between ${\color{blueTileColor}\blacksquare}$ and ${\color{brownTileColor}\blacksquare}$.\label{fig:toy_example_dfa2}]{
\includegraphics[width=\linewidth]{figures/dfa2.pdf}}
\caption{Learned DFA decomposition.} \label{fig:toy_example} \end{wrapfigure} Inspired from the multi-task reinforcement learning literature~\cite{vaezipoor2021ltl2action}, our example focuses on partially-ordered temporal tasks executed in parallel. Specifically, consider a case where an agent is performing two ordering tasks in parallel: \begin{enumerate*}[label=(\roman*)]
\item observe ${\color{yellow}\blacksquare}$ before ${\color{redTileColor}\blacksquare}$, and
\item observe ${\color{blueTileColor}\blacksquare}$ before ${\color{brownTileColor}\blacksquare}$. \end{enumerate*} A positive example of such behavior is simply any sequence of observations ensuring both of the given orderings, e.g. ${\color{yellow}\blacksquare}{\color{redTileColor}\blacksquare}{\color{blueTileColor}\blacksquare}{\color{brownTileColor}\blacksquare}$, and a negative example is any sequence that fails to satisfy both orderings, e.g. ${\color{yellow}\blacksquare}{\color{redTileColor}\blacksquare}{\color{brownTileColor}\blacksquare}{\color{blueTileColor}\blacksquare}$. We generate such positive and negative examples and feed them to our algorithm. \Cref{fig:toy_example} presents the learned DFAs recognizing ordering sub-tasks of the example. The intersection of their languages is consistent with the given observations, and their conjunction is the overall task realized by the system generating the traces. The monolithic DFA recognizing the same language has nine states, and is more complicated, see \Cref{fig:monolithic_dfa} in \Cref{appendix:figs}.
\subsection{Experimental Evaluation}\label{subsec:dfa_eval}
We evaluate the scalability of our algorithm through experiments with changing sizes of partially-ordered tasks introduced in \Cref{subsec:toy_example}. In our evaluation, we aim to answer two questions: \begin{enumerate*}[label=\textbf{(Q\arabic*)}]
\item\label{q1} ``How does solving time scale with the number of ordering tasks?'', and
\item\label{q2} ``How does solving time scale with the number of labeled examples?''.
\end{enumerate*} We implement our algorithm in Python with PySAT~\cite{imms-sat18}, and we use Glucose4~\cite{een2003extensible} as the SAT solver. Our baseline is an implementation of the monolithic DFA identification encoding from~\cite{heule2010exact,ulyantsev2016symmetry} with the same software as our implementation. Experiments are performed on a Quad-Core Intel i7 processor clocked at 2.3 GHz and a 32 GB main memory.
To evaluate the scalability, we randomly generate positive and negative examples with varying problem sizes. For \ref{q1}, we generate 10 (half of which is positive and half of which is negative) partially-ordered task examples with (i) 2 symbols, and (ii) 4 symbols, and we vary the number of DFAs from 2 to 12. For \ref{q2}, we generate 10 to 20 partially-ordered task examples with (i) 2 symbols and 4 DFAs, and (ii) 4 symbols and 2 DFAs. Half of these examples are positive and the other half is negative. Since the examples are generated randomly, we run the experiments for 10 different random seeds and report the average. We set the timeout limit to 10 minutes, and stop when our algorithm timeouts for all random seeds.
\begin{figure*}
\caption{\Cref{fig:gridworld} shows the stochastic grid world environment. \Cref{fig:diss_exp:examples} shows the positive and negative examples of the expert's behavior conjectured by DISS and \Cref{fig:dfa_diss1,fig:dfa_diss2,fig:dfa_diss3} showcases the associated DFA decomposition identified by our algorithm. \Cref{fig:diss} shows the monolithic DFA learned in~\cite{marcell2021DISS}.}
\label{fig:diss_exp}
\label{fig:gridworld}
\label{fig:diss_exp:examples}
\label{fig:dfa_diss1}
\label{fig:dfa_diss2}
\label{fig:dfa_diss3}
\label{fig:diss}
\end{figure*}
\Cref{fig:vary_dfas} presents the experiment results answering \ref{q1}, where we vary the number of DFAs implied by the given examples. For partially-ordered tasks with 2 symbols, green solid line is the (monolithic DFA) baseline and the blue solid is our algorithm. Similarly, for partially-ordered tasks with 4 symbols, pink dashed line is the baseline and the red dashed line is our algorithm. \Cref{fig:vary_examples} presents the experiment results answering \ref{q2}, where we vary the number of examples. For partially-ordered tasks with 2 symbols and 4 DFAs, green solid line is the baseline and the blue solid is our algorithm; for partially-ordered tasks with 4 symbols and 2 DFAs, pink dashed line is the baseline and the red dashed line is our algorithm. As expected, the baseline scales better than our algorithm as we also search for the Pareto frontier and solve an inherently harder problem. Notice that given 10 examples, our algorithm is able to scale up to 11 DFAs for tasks with 2 symbols, and 8 DFAs for tasks with 4 symbols; for 2 symbols and 4 DFAs, it is able to scale up to 60 examples, and for 4 symbols and 2 DFAs, it is able to scale up to 190 examples. As we demonstrate in the next section, these limits for scalability are practically useful in certain domains.
\section{Learning DFAs from Demonstrations}
Next, we show how our algorithm can be incorporated into Demonstration Informed Specification Search (DISS) - a framework for learning languages from expert demonstrations~\cite{marcell2021DISS}. For our purposes a \emph{demonstration} is an unlabeled path through a workspace that maps to a string and is biased towards being accepting by some unknown language. For example, we ran our implementation of DISS using demonstrations produced by an expert attempting to accomplish a task in a stochastic grid world environment, the same example used in \cite{marcell2021DISS} and shown in~\Cref{fig:gridworld}. At each step, the agent can move in any of the four cardinal directions, but because of wind blowing from the north to the south, with some probability, the agent will transition to the space south of it in spite of its chosen action. Two demonstrations of the task ``Reach ${\color{yellow}\blacksquare}$ while avoiding ${\color{redTileColor}\blacksquare}$. If it ever touches ${\color{blueTileColor}\blacksquare}$, it must then touch ${\color{brownTileColor}\blacksquare}$ before reaching ${\color{yellow}\blacksquare}$.'' are shown in~\Cref{fig:gridworld}.
In order to efficiently search for tasks, DISS reduces the learning from demonstrations problem into a series of identification problems to be solved by a black-box identification algorithm. The goal of DISS is to find a task that minimizes the joint description length, called the energy, of the task and the demonstrations assuming the agent were performing said task. The energy is measured in bits to encode an object.
Below, we reproduce the results from~\cite{marcell2021DISS}, but using our algorithm as the task identifier rather than the monolithic DFA identifier provided. The use of DFA decompositions biases DISS to conjecture concepts that are \emph{simpler} to express in terms of a DFA decomposition. To define the description length of DFA decompositions, we adapt the DFA encoding used in~\cite{marcell2021DISS} by expressing a decomposition as the concatenation of the encodings of the individual DFAs. To remove unnecessary redundancy two optimizations were performed. First common headers, e.g. indicating the alphabet size, were combined. Second, as the DFAs in a decomposition are ordered by size, we expressed changes in size rather than absolute size, see \Cref{appendix:encode_sizes} for details.
\subsection{Experimental Evaluation}\label{subsec:diss_eval}
In~\Cref{fig:dfa_diss1,fig:dfa_diss2,fig:dfa_diss3} we present the learned DFA decomposition along with the corresponding~\Cref{fig:diss_exp:examples} labeled examples conjectured by DISS to explain the expert behavior. Importantly, this decomposition exactly captures the demonstrated task. We note that this is in contrast to the DFA learned in~\cite{marcell2021DISS}, shown in~\Cref{fig:diss}, which allows visiting ${\color{redTileColor}\blacksquare}$ after visiting ${\color{yellow}\blacksquare}$. Further, we remark that the time required to learn the monolithic and decomposed DFAs was comparable. In particular, the number of labeled examples was less than 60 and as with the monolithic baseline, most of the time is not spent in task identification, but instead conjecturing the labeled examples. As we saw with in~\Cref{subsec:dfa_eval}, this number of examples is easily handled by our SAT-based identification algorithm. Finally, the number of labeled examples that needed to be conjectured to find low energy tasks was similar for both implementations (see~\Cref{fig:diss_exp:plot,fig:diss_exp:inc_plot} in~\Cref{appendix:figs}). Thus, our variant of DISS performed similar to the monolithic variant, while finding DFAs that exactly represented the task.
\section{Conclusion} To the best of our knowledge, this work presents the first approach for solving \texttt{DFA-DIP}\xspace. Our algorithm works by reducing the problem to a Pareto-optimal search of the space the number of states in a DFA decomposition with a SAT call in the inner loop. The SAT-based encoding is based on an efficient reduction to graph coloring. We demonstrated the scalability of our algorithm on a class of problems inspired by the multi-task reinforcement learning literature and show that the additional computational cost for identifying DFA decompositions over monolithic DFAs is not prohibitive. Finally, we showed how identifying DFA decompositions can provide a useful inductive bias while learning from demonstrations.
\appendix\label{sec:appendix}
\subsection{Complete SAT Encoding of \texttt{DFA-DIP}\xspace} Below, we list the complete SAT encoding of the \texttt{DFA-DIP}\xspace. Observe that the encoding extends the SAT encoding for monolithic DFA identification presented in~\cite{heule2010exact,ulyantsev2016symmetry}. We refer to the root node (i.e., the initial state) of an APTA as $v_r$, for $v \in V \setminus \{v_r\}$, $l(v)$ denotes the symbol on the incoming transition of $v$, and $p(v)$ is the parent node (i.e., the previous state) of $v$, as APTA is a tree-like automaton, $p(v)$ is unique.
\begin{enumerate}
\item A positive example must be accepted by \emph{all} DFAs:
\[
\bigwedge_{v \in V_{+}} \bigwedge_{k \in [n]} \bigwedge_{i \in [m_k]} x^{k}_{v, i} \implies z^{k}_{i}.
\]
\item A negative example must be rejected by \emph{at least} one DFA:
\[
\bigwedge_{v \in V_{-}} \bigvee_{k \in [n]} \bigwedge_{i \in [m_k]} x^{k}_{v, i} \implies \neg z^{k}_{i}.
\]
\item Each state of APTA has at least one color for each DFA:
\[
\bigwedge_{v \in V} \bigwedge_{k \in [n]} \bigwedge_{i \in [m_k]} x^{k}_{v, i}.
\]
\item A transition of a DFA is set when a state and its parent are both colored:
\[
\bigwedge_{v \in V \setminus \{v_r\}} \bigwedge_{k \in [n]} \bigwedge_{i,j \in [m_k]} (x^{k}_{p(v), i} \wedge x^{k}_{v, j}) \implies y^{k}_{l(v), i, j}.
\]
\item A transition of a DFA targets at most one state:
\[
\bigwedge_{l \in \Sigma} \bigwedge_{k \in [n]} \bigwedge_{\substack{i, j, t \in [m_k]\\j < t}} y^{k}_{l, i, j} \implies \neg y^{k}_{l, i, t}.
\]
\item Each state of APTA has at most one color for each DFA:
\[
\bigwedge_{v \in V} \bigwedge_{k \in [n]} \bigwedge_{i, j \in [m_k]} \neg x^{k}_{v, i} \vee \neg x^{k}_{v, j}.
\]
\item A transition of a DFA targets at least one state:
\[
\bigwedge_{l \in \Sigma} \bigwedge_{k \in [n]} \bigwedge_{i,j \in [m_k]} y^{k}_{l, i, j}.
\]
\item For each DFA, a node color is set when the color of the parent node and the transition between them are set:
\[
\bigwedge_{v \in V \setminus \{v_r\}} \bigwedge_{k \in [n]} \bigwedge_{i, j \in [m_k]} (x^{k}_{p(v), i} \wedge y^{k}_{l(v), i, j}) \implies x^{k}_{v, j}.
\]
\item Accepting-rejecting nodes of APTA cannot be merged:
\[
\bigwedge_{v_{-} \in V_{-}} \bigwedge_{v_{+} \in V_{+}} \bigwedge_{k \in [n]} \bigwedge_{i \in [m_k]} (x^{k}_{v_{-}, i} \wedge \neg z^{k}_{i}) \implies \neg x^{k}_{v_{+}, i}.
\] \end{enumerate}
The next set of constraints encode the symmetry breaking clauses intruduced in~\cite{ulyantsev2016symmetry} to avoid consideration of isomorphich DFAs. The main idea of the symmetry breaking clauses is to enforce individual DFA states to be enumerated in a depth-first search (DFS) order. See~\cite{ulyantsev2016symmetry} for more details. The symmetry breaking clauses make use of new auxilliary variables $p^k_{j,i}$ and $t^k_{i,j}$ for $k \in [n], i,j \in [m_k]$ and $m_{l,i,j}$ for $l \in \Sigma, i,j \in [m_k]$. Let $\Sigma = \{l_1, \dots, l_L\}$. \begin{enumerate}
\item Each state must have a smaller parent in the DFS order:
\[
\bigwedge_{k \in [n]} \bigwedge_{i \in [2,m_k]} ( p^{k}_{i, 1} \lor \dots \lor p^{k}_{i,i-1} ).
\]
\item Define $p^k_{j,i}$ in terms of auxilliary variable $t^k_{i,j}$:
\[
\bigwedge_{k \in [n]} \bigwedge_{\substack{i,j \in [m_k] \\ i < j}} (p^k_{j,i} \iff t^k_{i,j} \land t^k_{i+1,j} \land \dots \land t^k_{j-1,j} )
\]
\item Define $t^k_{i,j}$ in terms of $y_{l,j,j}$:
\[
\bigwedge_{k \in [n]} \bigwedge_{\substack{i,j \in [m_k] \\ i < j}} (t^k_{i,j} \iff y^k_{l_1,i,j} \lor \dots \lor y^k_{l_L,i,j})
\]
\item The parent relationship follows the DFS order
\[
\bigwedge_{k \in [n]} \bigwedge_{\substack{i,j,p,q \in [m_k] \\ i < p < j < q}} (p^k_{j,i} \implies \lnot t^k_{p,q})
\]
\item Define $m^k_{l,i,j}$ in terms of $y^k_{l,i,j}$:
\[
\bigwedge_{k \in [n]} \bigwedge_{\substack{i,j \in [m_k] \\ i < j}} \bigwedge_{l_r \in \Sigma} (m^k_{l_r,i,j} \iff y^k_{l_r,i,j}
\dots \land y^k_{l_1,i,j})
\]
\item Enforce DFAs to be DFS-enumerated in the order of symbols on transitions:
\[
\bigwedge_{k \in [n]} \bigwedge_{\substack{i,j,q \in [m_k] \\ i < j < q}} \bigwedge_{\substack{l_r, l_s \in \Sigma \\ r < s}} (p^k_{j,i} \land p^k_{q,i} \land m^k_{l_r,i,}j \implies \lnot m^k_{l_s,i,k})
\] \end{enumerate}
\subsection{Proof of \Cref{thm:sat}} \label{appendix:proof_main}
\begin{theorem_repeat} Given labeled examples with $n$ and $m_1, \dots, m_n$, a solution to our SAT encoding is a solution to \texttt{DFA-DIP}\xspace. \end{theorem_repeat}
\begin{IEEEproof}
We assume that the SAT-based reduction to graph coloring for monolithic DFA identification given in \cite{heule2010exact} is correct.
Constraint \ref{rule1} and \ref{rule2} replace similar constraint in the monolithic encoding given in \cite{heule2010exact}: \begin{enumerate}[label=\textbf{(R\arabic*')},leftmargin=26pt]
\item\label{rule1_prime} a negative example must be rejected by the DFA:
\[
\bigwedge_{v \in V_{-}} \bigwedge_{i \in [m_k]} x_{v, i} \implies \neg z_{i}, \text{ and}
\]
\item\label{rule2_prime} accepting and rejecting states of the APTA cannot be merged:
\[
\bigwedge_{v_{-} \in V_{-}} \bigwedge_{v_{+} \in V_{+}} \bigwedge_{i \in [m_k]} x_{v_{-}, i} \implies \neg x_{v_{+}, i}.
\] \end{enumerate} In the monolithic DFA case, there is only a single DFA so for ease of notation, we drop the index $k$. First notice that constraints \ref{rule1_prime} and \ref{rule2_prime} have no bearing on whether the DFA accepts each positive example. Therefore, our encoding automatically requires that each DFA in the DFA decomposition accepts all of the positive examples and is not constrained to unecessarily accept any unspecified examples.
Constraint \ref{rule1_prime} ensures that the resulting monolithic DFA rejects every negative example by making the color of the node in the APTA associated with the negative example rejecting. Constraint \ref{rule1} replaces this and ensures that at least one of the DFAs in the DFA decomposition rejects a negative example by making the color of the node in the APTA associated with the negative example rejecting in at least one of the $n$ DFAs in the decomposition. Thus, the language intersection of the resulting decomposition correctly rejects negative examples.
Constraint \ref{rule2_prime} ensures that all pairs of rejecting and accepting nodes of the APTA cannot be assigned the same color (i.e., merged) in the resulting DFAs. Constraint \ref{rule2}, which replaces \ref{rule2_prime}, ensures that for each DFA in the decomposition, the pair $(x_{v_{-},i}^k, x_{v_{+},i}^k)$ of accepting and rejecting nodes of the APTA cannot be assigned the same color only if DFA $k$ is rejecting the negative example associated with $x_{v_{-},i}^k$ (which is handled by constraint \ref{rule1}). This allows all but one DFA in the DFA decomposition to accept negative examples. Therefore, no DFA in the decomposition is constrained to unnecessarily reject a negative example if some other DFA in the DFA decomposition already does so. Therefore, the language intersection of the DFAs in the DFA decomposition is not constrained to reject any unspecified examples.
So, if there exists a DFA decomposition with the specified number of states such that all DFAs accept the positive examples and at least one DFA in the decomposition rejects each rejecting example, our encoding will find it. \end{IEEEproof}
\subsection{Details of the Pareto Frontier Search Algorithm} \label{appendix:pareto_search}
\Cref{alg:pareto_frontier} presents the details of the BFS performed in the solution space for finding the Pareto-optimal frontier.
\begin{algorithm}[h]
\caption{Pareto frontier enumeration algorithm.
\label{alg:pareto_frontier}}
\begin{algorithmic}[1]
\Require{Positive $D_{+}$ and negative $D_{-}$ labeled examples and positive integer $n$.}
\State $Q^{\star} \gets \emptyset$ \Comment{Maintains the Pareto frontier}
\State $S \gets \{(2,\dots,2)\}$ \Comment{Initialize the queue}
\While{$S \neq \emptyset$}
\State $m \gets Q.dequeue()$
\If{$\nexists \hat{m} \in P^{\star} \text{ s.t. } \hat{m} \prec m$}
\State $SAT, \mathcal{A} \gets \textproc{Solve}(n, m, D_{+}, D_{-})$
\If{$SAT$}
\State $P^{\star} = P^{\star} \cup \mathcal{A}$ \Comment{Add to the Pareto frontier}
\Else
\For{$k=1,\dots,n$}
\State $m' \gets m$
\State $m'_k \gets m'_k + 1$
\If{$\text{ordered}(m')$}
\State $Q.enqueue(m')$
\EndIf
\EndFor
\EndIf
\EndIf
\EndWhile
\State \Return{$P^{\star}$}
\end{algorithmic} \end{algorithm}
\subsection{Proof of \Cref{thm:pareto}} \label{appendix:proof_pareto}
\begin{theorem_repeat} The described BFS is sound and complete; it outputs the all Pareto-optimal frontier of solutions without returning any dominated solutions. \end{theorem_repeat}
\begin{IEEEproof} The described BFS enumerates the number of states of DFA decomposition in product order. Therefore, before reaching a vertex with number of states $(m_1, m_2, \dots, m_n)$, it explores all number of states of DFA decompositions that dominate the DFA decomposition with $(m_1, m_2, \dots, m_n)$ states. If any of these number of states admit a solution to \texttt{DFA-DIP}\xspace, then the DFA decomposition associated with number of states $(m_1, m_2, \dots, m_n)$ will be marked as a sink and not returned on the Pareto-optimal frontier. Therefore, the described BFS is sound.
If none of those number of states admit a solution to \texttt{DFA-DIP}\xspace, then none of them are sinks, so $(m_1, m_2, \dots, m_n)$ will be reached, and if $(m_1, m_2, \dots, m_n)$ does admit a solution to \texttt{DFA-DIP}\xspace, it will correctly be returned as a solution on the Pareto-optimal frontier. Thus, the described BFS is complete. \end{IEEEproof}
\subsection{DFA Encoding Sizes for DISS} \label{appendix:encode_sizes} When sampling a concept identifying a set of labeled examples, DISS algorithms are exponentially more likely to sample concepts with smaller \textit{size complexity}, i.e., number of bits required to represent the concept. We define the \textit{size} of a DFA decomposition $\mathcal{A} = (\mathcal{A}_1, \dots, \mathcal{A}_n)$ with number of states $m_1, \dots, m_n$ based on the size of the underlying DFAs as: \begin{equation}
\sum_{i=1}^n\text{size}(\mathcal{A}_i) - (n - 1)(2\ln(\Sigma) + 1) - 2(n - 1)\ln(m_1) \end{equation}
where the size of each underlying DFA is given by $\text{size}(\mathcal{A}_i) = 3 + 2 \ln(m_i) + 2 \ln(\Sigma) + (|F_i| + 1) \ln(m_i) + z ( \ln(\Sigma) + 2 \ln(m_i) )$ where $F_i$ are the accepting states of $\mathcal{A}_i$ and $z$ is the number of non-stuttering transitions of $\mathcal{A}_i$.
\subsection{Extra Figures} \label{appendix:figs}
\begin{figure}
\caption{Monolithic DFA for the example presented in \Cref{subsec:toy_example}.}
\label{fig:monolithic_dfa}
\end{figure}
\begin{figure}
\caption{How quickly DISS finds explanatory DFA decompositions compared to the enumeration baselines for a setting without prior partial knowledge. The temperature $\beta$ controls the degree of likelihood to which better explaining DFA decompositions are sampled.
}
\label{fig:diss_exp:plot}
\end{figure}
\begin{figure}
\caption{How quickly DISS finds explanatory DFA decompositions compared to the enumeration baselines for a setting with prior partial knowledge. The temperature $\beta$ controls the degree of likelihood to which better explaining DFA decompositions are sampled.
}
\label{fig:diss_exp:inc_plot}
\end{figure}
\begin{figure}
\caption{This plot compares the scalability of our algorithm to the monolithic DFA identification baseline. Each point in the plot represents a problem instance that is in the experiments presented in \Cref{subsec:dfa_eval}. The dashed black line is the $x = y$ line. As expected, our algorithm requires more time to solve same problem instances. However, we should note that the relationship between the solution times is not far away from the $x = y$ line.}
\label{fig:all}
\end{figure}
\end{document}
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\begin{document}
\title{The duality of computation under focus} \begin{abstract} We review the close relationship between abstract machines for (call-by-name or call-by-value) $\lambda$-calculi (extended with Felleisen's $\cal C$) and sequent calculus, reintroducing on the way Curien-Herbelin's syntactic kit expressing the duality of computation. We use this kit to provide a term language for a presentation of $\mathsf{LK}$ (with conjunction, disjunction, and negation), and to transcribe cut elimination as (non confluent) rewriting. A key slogan here, which may appear here in print for the first time, is that commutative cut elimination rules are explicit substitution propagation rules. We then describe the focalised proof search discipline (in the classical setting), and narrow down the language and the rewriting rules to a confluent calculus (a variant of the second author's focalising system $\mathsf{L}$).
We then define a game of patterns and counterpatterns, leading us to a fully focalised finitary syntax for a synthetic presentation of classical logic, that provides a quotient on (focalised) proofs, abstracting out the order of decomposition of negative connectives.\footnote{A slighlty shorter version appears in the Proceedings of the Conference IFIP TCS, Brisbane, Sept. 2010, published as a Springer LNCS volume., With respect to the published conference version, the present version corrects some minor mistakes in the last section, and develops a bit further the material of Section 5.}
\end{abstract}
\section{Introduction} \label{introduction}
This paper on one hand has an expository purpose and on the other hand pushes further the syntactic investigations on the duality of computation undertaken in \cite{CH2000}.
Section \ref{intro-sec} discusses the relation between familiar {\em abstract machines} for the $\lambda$-calculus (extended with control) and (classical) {\em sequent calculus}. Section \ref{LK-proofs-sec} presents a faithful language (with a one-to-one correspondence between well-typed terms and proof trees) for a presentation of LK that anticipates a key ingredient of focalisation, by choosing a dissymetric presentation for the conjunction on one side and the disjunction on the other side of sequents. We recall the non-confluence of unconstrained classical cut-elimination.
In Section \ref{LKQ-sec}, we present the {\em focalised proof search discipline} (for {\em classical logic}), and adapt the syntactic framework of Section \ref{LK-proofs-sec} to get a confluent system whose normal forms are precisely the terms denoting (cut-free) focalised proofs. The system we arrive at from these proof-search motivations is (a variant of) the second author's {\em focalising system $\mathsf{L}$} ($\mathsf{L_{\textrm{foc}}}$) \cite{Munch2009}
We prove the completeness of $\mathsf{L_{\textrm{foc}}}$ with respect to $\mathsf{LK}$ for provability.
In Section \ref{encodings-sec}, we define some simple encodings having $\mathsf{L_{\textrm{foc}}}$ as source or target, indicating its suitability as an intermediate language (between languages and their execution or compilation).
Finally, in Section \ref{LKQS-sec}, we further reinforce the focalisation discipline, which leads us to {\em synthetic system $\mathsf{L}$} ($\mathsf{L_{\textrm{synth}}}$), a logic of synthetic connectives in the spirit of Girard's ludics and Zeilberger's CU, for which we offer a syntactic account based on a simple game of patterns and counterpatterns that can be seen as another manifestation of dualities of computation. We show that the synthetic system $\mathsf{L}$ is complete with respect to focalising system $\mathsf{L}$.
\noindent {\em Notation.} We shall write $\Subimpl{t}{x}{v}$ the result of substituting $v$ for $x$ at all (free) occurrences of $x$ in $t$, and $\Sub{t}{x}{v}$ for an explicit operator \cite{ACCL} added to the language together with rules propagating it. Explicit substitutions are relevant here because they account for the commutative cut rules (see Section \ref{LK-proofs-sec}).
\section{Abstract machines and sequent calculus} \label{intro-sec}
In this section, we would like to convey the idea that sequent calculus could have arisen from the goal of providing a typing system for the states of an abstract machine for the ``mechanical evaluation of expressions'' (to quote the title of Peter Landin's pioneering paper \cite{Landin64}).
Here is a simple device for executing a (closed) $\lambda$-term in call-by-name (Krivine machine \cite{KrivineMach}): \begin{center} \fbox{$\begin{array}{lllllllll} \coupe{MN}{E} & \longrightarrow& \coupe{M}{N\cdot E} \quad &&& \quad \coupe{\lambda x.M}{N\cdot E} & \longrightarrow & \coupe{\Subimpl{M}{x}{N}}{E} \end{array}$} \end{center} A state of the machine is thus a pair $\coupe{M}{E}$ where
$M$ is ``where the computation is currently active'', and
$E$ is the stack of things that are waiting to be done in the future, or the continuation, or the evaluation context. In $\lambda$-calculus litterature, contexts are more traditionally presented as terms with a hole: with this tradition, $\coupe{M}{E}$ (resp. $M\cdot E$) reads as $E[M]$ (resp. $E[[]M]$), or ``fill the hole of $E$ with $M$ (resp. $[]M$)''.
How can we type the components of this machine? We have three categories of terms and of typing judgements: $$\begin{array}{ccccccccc} \mbox{Expressions} &&& \mbox{Contexts} &&& \mbox{Commands} \\
M::= x \,\mbox{\large\boldmath$\mid$}\, \lambda x.M \,\mbox{\large\boldmath$\mid$}\, MM \quad&&&\quad
E::= [\:] \,\mbox{\large\boldmath$\mid$}\, M\cdot E\quad &&& \quad
c::=\coupe{M}{E}\\
(\lkc{\Gamma}{}{}{M:A})\quad&&&\quad (\lke{\Gamma}{E:A}{}{R})
\quad &&& \quad c:(\lkc{\Gamma}{}{}{R}) \end{array}$$
where $R$ is a (fixed) type of {\em final results}. The type of an expression (resp. a context) is the type of the value that it is producing (resp. expecting). The typing rules for contexts and commands are as follows: $$\seq{}{\lke{\Gamma}{ [\:]:R}{}{R}}\quad\quad \seq{ \lkc{\Gamma}{}{}{ M:A}\quad \lke{\Gamma}{ E:B}{}{R}} {\lke{\Gamma}{M\cdot E:A\rightarrow B}{}{R}}\quad\quad \seq{\lkc{\Gamma}{}{}{ M:A}\quad\lke{\Gamma}{ E:A}{}{R}} {\coupe{M}{E}:(\lkc{\Gamma}{}{}{R})}$$ and the typing rules for expressions are the usual ones for simply typed $\lambda$-calculus. Stripping up the term information, the second and third rules are rules of {\em sequent calculus} (left introduction of implication and cut).
We next review Griffin's typing of Felleisen's control operator ${\cal C}$. As a matter of fact, the behaviour of this constructor is best expressed at the level of an abstract machine: \begin{center} \fbox{$\begin{array}{lllllllll} \coupe{{\cal C}(M)}{E} & \longrightarrow &\coupe{M}{\reify{E}\cdot[\:]} \quad &&& \quad \coupe{\reify{E}}{N\cdot E'} &\longrightarrow& \coupe{N}{E} \end{array}$} \end{center} The first rule explains how the continuation $E$ gets {\em captured}, and the second rule how it gets {\em restored}.
Griffin \cite{Griffin90} observed that the typing constraints induced by the well-typing of these four commands are met when ${\cal C}(M)$ and $\reify{E}$ are typed as follows:
$$\begin{array}{cccc}
\seq{\lkc{\Gamma}{}{}{M: (A\rightarrow R)\rightarrow R}} {\lkc{\Gamma}{}{}{{\cal C}(M): A}} \quad&&& \quad\seq{\lke{\Gamma}{E: A}{}{R}}{\lkc{\Gamma}{}{}{\reify{E}: A\rightarrow R}} \end{array}$$ These are the rules that one adds to intutionistic natural deduction to make it classical, if we interpret $R$ as $\bot$ (false), and if we encode $\neg A$ as $A\rightarrow R$.
Hence, Griffin got no less than {Curry-Howard for classical logic! But how does this sound in sequent calculus style?
In classical sequent calculus, sequents have several formulas on the right and $\lkc{\Gamma}{}{}{\Delta}$ reads as ``if all formulas in $\Gamma$ hold, then at least one formula of $\Delta$ holds''. Then it is natural to associate continuation variables with the formulas in $\Delta$: a term will depend on its input variables, and on its output continuations. With this in mind, we can read the operational rule for ${\cal C}(M)$ as `` ${\cal C}(M)$ is a map $E \mapsto \coupe{M}{\reify{E}\cdot[\:]}$'', and write it with a new binder (that comes from \cite{Parigot92}): $${\cal C}(M) =\mu\beta.\coupe{M}{\reify{\beta}\cdot[\:]}$$ where $[\:]$ is now a continuation variable (of ``top-level'' type $R$). Likewise, we synthesise $\reify{E}=\lambda x.\mu\alpha.\coupe{x}{E}$, with $\alpha,x$ fresh, from the operational rules for $\reify{E}$ and for $\lambda x.M$.
The typing judgements are now: $
(\lkv{\Gamma}{}{M:A}{\Delta})$, $(\lke{\Gamma}{E:A}{}{\Delta})$, and $c:(\lkc{\Gamma}{}{}{\Delta}) $. The two relevant new typing rules are (axiom, right {\em activation}): \begin{center} $\begin{array}{llll}
\seq{}{\lke{\Gamma}{\alpha:A}{}{\alpha:A,\Delta}} &&& \seq{c:(\lkc{\Gamma}{}{}{\alpha:A,\Delta})}{ \lkv{\Gamma}{}{\mu\alpha.c:A}{\Delta}} \end{array}$ \end{center} plus a reduction rule: $\coupe{\mu\alpha.c}{E}\longrightarrow\Subimpl{c}{\alpha}{E}$.
Note that in this setting, there is no more need to ``reify'' a context $E$ into an expression $\reify{E}$, as it can be directly substituted for a continuation variable.
Similarly, we can read off a (call-by-name) definition of $MN$ from its operational rule:
$MN=\mu\beta.\coupe{M}{N.\beta}$. Hence we can
remove application from the syntax and arrive at a system in sequent calculus style {\em only}} (no more elimination rule). This yields Herbelin's $\overline{\lambda}\mu$-calculus \cite{Her95}: $$ \mbox{Expressions} \;\; M::= x \,\mbox{\large\boldmath$\mid$}\, \lambda x.M \,\mbox{\large\boldmath$\mid$}\, \mu\alpha.c\quad\quad \mbox{Contexts} \;\; E::= \alpha \,\mbox{\large\boldmath$\mid$}\, M\cdot E \quad\quad \mbox{Commands} \;\; c::=\coupe{M}{E} $$
which combines the first two milestones above: ``sequent calculus'', ``classical".
Let us step back to the $\lambda$-calculus. The following describes a call-by-value version of Krivine machine:
\begin{center} \fbox{$\begin{array}{llllllll} \coupe{MN}{e} & \longrightarrow & \coupe{N}{M\odot e} \quad&&&\quad \coupe{V}{M\odot e}& \longrightarrow & \coupe{M}{V\cdot e} \end{array}$} \end{center} (the operational rule for $\lambda x.M$ is unchanged)\footnote{The reason for switching notation from $E$ to $e$ will become clear in Section \ref{encodings-sec}.}. Here, $V$ is a {\em value}, defined as being either a variable or an abstraction (this goes back to \cite{PlotkinCBNV}). Again, we can read $M\odot e$ as ``a map $V\mapsto \coupe{M}{V\cdot e}$'', or, introducing a new binder $\tilde\mu$ (binding now ordinary variables): $$M\odot e = \tilde\mu x. \coupe{M}{x\cdot e}$$ The typing rule for this operator is (left activation): \begin{center} $\begin{array}{c}
\seq{ c:(\lkc{\Gamma,x:A}{}{}{\Delta})}{\lke{\Gamma}{\tilde\mu x.c:A}{}{\Delta}} \end{array}$ \end{center} and the operational rule is $\coupe{V}{\tilde\mu x.c}\longrightarrow \Subimpl{c}{x}{V}\;\;(V\mbox{ value})$.
Finally, we get from the rule for $MN$ a call-by-value definition of application: $MN=\mu\alpha.\coupe{N}{\tilde\mu x.\coupe{M}{x\cdot \alpha}}$.
We have arrived at Curien and Herbelin's $\overline\lbd\mu\tilde\mu_Q$-calculus \cite{CH2000}: \begin{center}
\begin{tabular}{|cc|cc|cc|c|} \hline Expressions $ M::= \Vtov{V} \,\mbox{\large\boldmath$\mid$}\, \mu\alpha.c $ && \:Values $\; V::= x \,\mbox{\large\boldmath$\mid$}\, \lambda x.M$ && \:Contexts $\; e::= \alpha \,\mbox{\large\boldmath$\mid$}\, V\cdot e \,\mbox{\large\boldmath$\mid$}\, \tilde\mu x.c$ && \:Commands $\; c::=\coupe{M}{e}$\\
$ \lkv{\Gamma}{}{M:A}{\Delta}$ && $\lkV{\Gamma}{}{V:A}{\Delta}$ &&
$\lke{\Gamma}{e:A}{}{\Delta} $
&&
$c:(\lkc{\Gamma}{}{}{\Delta})$\\
\hline \end{tabular} \end{center} with a new judgement for values (more on this later) and an
explicit coercion from values to expressions. The syntax for contexts is both extended ($\tilde\mu x.c$) and restricted ($V\cdot e$ instead of $M\cdot e$). The reduction rules are as follows: \begin{center} \fbox{$\begin{array}{lllllllll} \coupe{\Vtov{(\lambda x.M)}}{V\cdot e}\longrightarrow \coupe{\Subimpl{M}{x}{V}}{e} \quad&&\quad \coupe{\mu\alpha.c}{e}\longrightarrow \Subimpl{c}{\alpha}{e}\quad &&\quad \coupe{\Vtov{V}}{\tilde\mu x.c}\longrightarrow \Subimpl{c}{x}{V} \end{array}$} \end{center} \section{A language for $\mathsf{LK}$ proofs} \label{LK-proofs-sec} In this section, we use some of the kit of the previous section to give a term language for classical sequent calculus $\mathsf{LK}$, with negation, conjunction, and disjunction as connectives. Our term language is as follows: $$\begin{array}{lllll} \mbox{Commands} &&&& c::=\coupe{x}{\alpha} \,\mbox{\large\boldmath$\mid$}\, \coupe{v}{\alpha} \,\mbox{\large\boldmath$\mid$}\, \coupe{x}{e} \,\mbox{\large\boldmath$\mid$}\, \coupe{\mu\alpha.c}{\tilde\mu x.c}\\ \mbox{Expressions} &&&& v::= (\tilde\mu x.c)^\bullet \,\mbox{\large\boldmath$\mid$}\, (\mu\alpha.c,\mu\alpha.c) \,\mbox{\large\boldmath$\mid$}\, \inl{\mu\alpha.c} \,\mbox{\large\boldmath$\mid$}\, \inr{\mu\alpha.c}\\
\mbox{Contexts} &&&& e::= \tilde\mu\alpha^\bullet.c \,\mbox{\large\boldmath$\mid$}\, \tilde\mu (x_1,x_2).c \,\mbox{\large\boldmath$\mid$}\, \tilde\mu\copaireP{\inl{x_1}.c_1|\inr{x_2}.c_2} \end{array}$$ (In $\coupe{v}{\alpha}$ (resp. $\coupe{x}{e}$), we suppose $\alpha$ (resp. $x$) fresh for $v$ (resp. $e$).) A {\em term} $t$ is a command, an expression, or a context. As in section \ref{intro-sec}, we have three kinds of sequents: $(\Gamma\vdash\Delta)$, $(\lkv{\Gamma}{}{A}{\Delta})$, and $(\lke{\Gamma}{A}{}{\Delta})$. We decorate $\mathsf{LK}$'s inference rules with terms, yielding the following typing system (one term construction for each rule of $\mathsf{LK}$):
\begin{center} {\small (axiom and cut/contraction) $\quad\quad \seq{}{\coupe{x}{\alpha}:(\Gamma,x:A\vdash \alpha:A,\Delta)}\quad\quad \seq{c:(\Gamma\vdash \alpha: A,\Delta)\quad\quad d:(\Gamma, x:A\vdash\Delta)} {\coupe{\mu\alpha.c}{\tilde\mu x.d}:(\Gamma\vdash\Delta)}$
$(\mbox{right})\quad\quad\quad\quad\quad\seq{c:(\Gamma,x:A\vdash\Delta)}{\lkv{\Gamma}{}{(\tilde\mu x.c)^\bullet:\neg A}{\Delta}}\quad\quad\seq{c_1:(\Gamma\vdash \alpha_1:A_1,\Delta) \quad\quad c_2:(\Gamma\vdash \alpha_2:A_2,\Delta)} {\lkv{\Gamma}{}{(\mu\alpha_1.c_1,\mu\alpha_2.c_2):A_1\wedge A_2}{\Delta}}$
$ \seq{c_1:(\Gamma\vdash \alpha_1:A_1,\Delta)}{\lkv{\Gamma}{}{\inl{\mu\alpha_1.c_1}:A_1\vee A_2}{\Delta}}\quad\quad \seq{c_2:(\Gamma\vdash \alpha_2:A_2,\Delta)}{\lkv{\Gamma}{}{\inr{\mu\alpha_2.c_2}:A_1\vee A_2}{\Delta}}$
$(\mbox{left})\quad \seq{c:(\Gamma\vdash \alpha:A,\Delta)}{\lke{\Gamma}{\tilde\mu\alpha^\bullet.c:\neg A}{}{\Delta}}\quad\quad \seq{c:(\Gamma,x_1:A_1,x_2:A_2\vdash\Delta)}{\lke{\Gamma}{\tilde\mu(x_1,x_2).c:A_1\wedge A_2}{}{\Delta}}\quad\quad\seq{c_1:(\Gamma,x_1:A_1\vdash\Delta)\quad\quad c_2:(\Gamma,x_2:A_2\vdash\Delta)}
{\lke{\Gamma}{\tilde\mu\copaireP{\inl{x_1}.c_1|\inr{x_2}.c_2}:A_1\vee A_2}{}{\Delta}}$
$(\mbox{deactivation})\quad\quad\seq{\lkv{\Gamma}{}{v:A}{\Delta}}{\coupe{v}{\alpha}:(\Gamma\vdash\alpha:A,\Delta)}\quad\quad \quad\quad\seq{\lke{\Gamma}{e:A}{}{\Delta}}{\coupe{x}{e}:(\Gamma,x:A\vdash\Delta)}$} \end{center} Note that the activation rules are packaged in the introduction rules and in the cut rule. As for the underlying sequent calculus rules, we have made the following choices: \begin{enumerate} \item We have preferred {\em additive} formulations for the cut rule and for the right introduction of conjunction (to stay in tune with the tradition of typed $\lambda$-calculi) over a multiplicative one where the three occurrences of $\Gamma$ would be resp. $\Gamma_1$, $\Gamma_2$, and $\Gamma_1,\Gamma_2$ (idem for $\Delta$). An important consequence of this choice is that contraction is a derived rule of our system, whence the name of {\em cut/contraction} rule above\footnote{In usual syntactic accounts of contraction, one says that if, say $t$ denotes a proof of $\Gamma,x:A,y:A\vdash\Delta$, then $\Subm{t}{\Subi{x}{z},\Subi{y}{z}}$ denotes a proof of $\Gamma,z:A\vdash\Delta$. Note that if this substitution is explicit, then we are back to an overloading of cut and contraction.}: \begin{center} $\seq{\seq{}{\lkc{\Gamma,A}{}{}{A,\Delta}}\quad\quad \lkc{\Gamma,A,A}{}{}{\Delta}}{\lkc{\Gamma,A}{}{}{\Delta}}\quad\quad\quad\seq{\lkc{\Gamma}{}{}{A,A,\Delta}\quad\quad \seq{}{\lkc{\Gamma,A}{}{}{A,\Delta}}}{\lkc{\Gamma}{}{}{A,\Delta}}$ \end{center} \item Still in the $\lambda$-calculus tradition, weakening is ``transparent''. If $c:\Gamma\vdash\Delta$ is well-typed, then $c:(\Gamma,\Gamma'\vdash\Delta,\Delta')$ is well-typed (idem $v,e$). (Also, we recall that all free variables of $c$ are among the ones declared in $\Gamma,\Delta$.) \item More importantly, we have adopted {\em irreversible} rules for right introduction of disjunction. On the other hand, we have given a {\em reversible} rule for left introduction of conjunction: the premise is derivable from the conclusion. {\em This choice prepares the ground for the next section on focalisation.}\footnote{For the same reason, we have chosen to take three connectives instead of just two, say, $\vee$ and $\neg$, because in the focalised setting $\neg(\neg A\vee \neg B)$ is only equivalent to $A\wedge B$ at the level of {\em provability}.} \end{enumerate} The relation between our typed terms and $\mathsf{LK}$ proofs is as follows.
\noindent - Every typing proof induces a proof tree of $\mathsf{LK}$ (one erases variables naming assumptions and conclusions, terms, the distinction between the three kinds of sequents, and the application of the deactivation rules).
\noindent - If bound variables are explicitly typed (which we shall refrain from doing in the sequel),
then every provable typing judgement, say $\lke{\Gamma}{e:A}{}{\Delta}$, has a unique typing proof, i.e. all information is in $\Gamma$, $A$, $\Delta$, $e$.
\noindent - If $\Pi$ is an $\mathsf{LK}$ proof tree of $(A_1,\ldots,A_m\vdash B_1,\ldots,B_n)$, and if names $x_1,\ldots,x_m$, $\alpha_1,\ldots,\alpha_n$ are provided, then there exists a unique command $c:(\lkc{x_1:A_1,\ldots,x_m:A_m}{}{}{\alpha_1:B_1,\ldots,\alpha_n:B_n})$, whose (unique) typing proof gives back $\Pi$ by erasing.
With this syntax, we can express the cut-elimination rules of $\mathsf{LK}$ as {\em rewriting rules}:
Logical rules (redexes of the form $\coupe{\mu\alpha.\coupe{v}{\alpha}}{\tilde\mu x.\coupe{x}{e}}$): \begin{center} $\begin{array}{l} \coupe{\mu\alpha.\coupe{(\tilde\mu x.c)^\bullet}{\alpha}} {\tilde\mu y.\coupe{y}{\tilde\mu\alpha^\bullet.d}} \longrightarrow \coupe{\mu\alpha.d}{\tilde\mu x.c}\quad\quad (\mbox{similar rules for conjunction and disjunction})
\end{array}$ \end{center}
Commutative rules (going ``up left'', redexes of the form $\coupe{\mu\alpha.\coupe{v}{\beta}}{\tilde\mu x.c}$): $$\begin{array}{l}
\coupe{\mu\alpha.\coupe{(\tilde\mu y.c)^\bullet}{\beta}}{\tilde\mu x.d} \longrightarrow \coupe{\mu\beta'.\coupe{(\tilde\mu y.\coupe{\mu\alpha.c}{\tilde\mu x.d})^\bullet}{\beta'}}{\tilde\mu y.\coupe{y}{\beta}} \quad(\neg\mbox{ right})\\ (\mbox{similar rules of commutation with the other right introduction rules and with the left introduction rules})\\
\coupe{\mu\alpha.\coupe{\mu\beta.\coupe{y}{\beta}}{\tilde\mu y'.c}}{\tilde\mu x.d}\longrightarrow \coupe{\mu\beta.\coupe{y}{\beta}}{\tilde\mu y'.\coupe{\mu\alpha.c}{\tilde\mu x.d}} \quad(\mbox{contraction right})\\ \coupe{\mu\alpha.\coupe{\mu\beta'.c}{\tilde\mu y.\coupe{y}{\beta}}}{\tilde\mu x.d} \longrightarrow \coupe{\mu\beta'.\coupe{\mu\alpha.c}{\tilde\mu x.d}}{\tilde\mu y.\coupe{y}{\beta}} \quad (\mbox{contraction left})\\ \coupe{\mu\alpha.\coupe{\mu\alpha'.c}{\tilde\mu x'.\coupe{x'}{\alpha}}}{\tilde\mu x.d} \longrightarrow \coupe{\mu\alpha.\coupe{\mu\alpha'.c}{\tilde\mu x.d}}{\tilde\mu x.d} \quad(\mbox{duplication})\\ \coupe{\mu\alpha.\coupe{y}{\beta}}{\tilde\mu x.d} \longrightarrow \coupe{y}{\beta} \quad (\mbox{erasing}) \end{array}$$
Commutative rules (going ``up right'', redexes of the form $\coupe{\mu\alpha.c}{\tilde\mu x.\coupe{y}{e}}$ ): similar rules.
The (only?) merit of this syntax is its tight fit with proof trees and traditional cut elimination defined as transformations of undecorated proof trees. If we accept to losen this,
we arrive at the following more ``atomic'' syntax: $$\begin{array}{lllll} \mbox{Commands} &&&& c::=\coupe{v}{e} \,\mbox{\large\boldmath$\mid$}\, \Subm{c}{\sigma}\\ \mbox{Expressions} &&&& v::= x \,\mbox{\large\boldmath$\mid$}\, \mu\alpha.c \,\mbox{\large\boldmath$\mid$}\, e^\bullet \,\mbox{\large\boldmath$\mid$}\, (v,v) \,\mbox{\large\boldmath$\mid$}\, \inl{v} \,\mbox{\large\boldmath$\mid$}\, \inr{v} \,\mbox{\large\boldmath$\mid$}\, \Subm{v}{\sigma}\\
\mbox{Contexts} &&&& e::= \alpha \,\mbox{\large\boldmath$\mid$}\, \tilde\mu x.c \,\mbox{\large\boldmath$\mid$}\, \tilde\mu\alpha^\bullet.c \,\mbox{\large\boldmath$\mid$}\, \tilde\mu (x_1,x_2).c \,\mbox{\large\boldmath$\mid$}\, \tilde\mu\copaireP{\inl{x_1}.c_1|\inr{x_2}.c_2} \,\mbox{\large\boldmath$\mid$}\, \Subm{e}{\sigma} \end{array}$$ where $\sigma$ is a list $\Subi{x_1}{v_1},\ldots,\Subi{x_m}{v_m},\Subi{\alpha_1}{e_1},\ldots, \Subi{\alpha_n}{e_n}$.
In this syntax, activation becomes ``first class'', and two versions of the axiom are now present ($x$, $\alpha$, which give back the axiom of the previous syntax by deactivation).
The typing rules are as follows (we omit the rules for $\tilde\mu x.c$, $\tilde\mu\alpha^\bullet.c$, $\tilde\mu (x_1,x_2).c$, $\tilde\mu\copaireP{\inl{x_1}.c_1|\inr{x_2}.c_2}$, which are unchanged): {\small $$\seq{}{\lkv{\Gamma\:,\:x:A}{}{x:A}{\Delta}}\quad\quad \seq{}{\lke{\Gamma}{\alpha:A}{}{\alpha:A\:,\:\Delta}}\quad\quad \seq{\lkv{\Gamma}{}{v:A}{\Delta}\quad\quad \lke{\Gamma}{e:A}{}{\Delta}} {\coupe{v}{e}:(\lkc{\Gamma}{}{}{\Delta})} $$ $$ \seq{c:(\lkc{\Gamma\:,\:x:A}{}{}{\Delta})}{\lke{\Gamma}{\tilde\mu x.c:A}{}{\Delta}} \quad\quad \seq{c:(\lkc{\Gamma}{}{}{\alpha:A\:,\:\Delta})}{\lkv{\Gamma}{}{\mu\alpha.c:A}{\Delta}} $$ $$\seq{\lke{\Gamma}{e:A}{}{\Delta}}{\lkv{\Gamma}{}{e^\bullet:\neg A}{\Delta}}\quad\quad\seq{\lkv{\Gamma}{}{v_1:A_1}{\Delta} \quad\quad \lkv{\Gamma}{}{v_2:A_2}{\Delta}} {\lkv{\Gamma}{}{(v_1,v_2):A_1\wedge A_2}{\Delta}}\quad\quad \seq{\lkv{\Gamma}{}{v_1:A_1}{\Delta}}{\lkv{\Gamma}{}{\inl{v_1}:A_1\vee A_2}{\Delta}}\quad\quad \seq{\lkv{\Gamma}{}{v_2:A_2}{\Delta}}{\lkv{\Gamma}{}{\inr{v_2}:A_1\vee A_2}{\Delta}}$$ $$\seq{c:(\lkc{\Gamma,x_1:A_1,\ldots,x_m:A_m}{}{}{\alpha_1:B_1,\ldots,\alpha_n:B_n})\;\ldots\;\lkv{\Gamma}{}{v_i:A_i}{\Delta}\;\ldots\;\ldots\;\lke{\Gamma}{e_j:B_j}{}{\Delta}\;\ldots}{\Subm{c}{\Subi{x_1}{v_1},\ldots,\Subi{x_m}{v_m},\Subi{\alpha_1}{e_1},\ldots, \Subi{\alpha_n}{e_n}}:(\lkc{\Gamma}{}{}{\Delta})} \quad (\mbox{idem } \Subm{v}{\sigma}, \Subm{e}{\sigma})$$}
Note that we also have now {\em explicit substitutions}
$\Subm{t}{\sigma}$, which feature a form of (multi-)cut where the receiver $t$'s active formula, if any, is not among the cut formulas, in contrast with the construct $\coupe{v}{e}$ where the cut formula is active on both sides.
It is still the case that, by erasing, a well-typed term of this new syntax induces a proof of $\mathsf{LK}$, and that all proofs of $\mathsf{LK}$ are reached (although not injectively anymore), since all terms of the previous syntax are terms of the new syntax.
The rewriting rules divide now in {\em three} groups: $$\begin{array}{lllll} (\mbox{control}) && \coupe{\mu\alpha.c}{e} \longrightarrow \Sub{c}{\alpha}{e} \quad\quad\quad\quad\quad\quad
\coupe{v}{\tilde\mu x.c} \longrightarrow \Sub{c}{x}{v}\\
(\mbox{logical}) &&\coupe{e^\bullet}{\tilde\mu \alpha^\bullet.c} \longrightarrow \Sub{c}{\alpha}{e} \quad\quad\quad\quad\quad \coupe{(v_1,v_2)}{\tilde\mu(x_1,x_2).c} \longrightarrow \Subm{c}{\Subi{x_1}{v_1},\Subi{x_2}{v_2}}\\
&& \coupe{\inl{v_1}}{\tilde\mu\copaireP{\inl{x_1}.c_1|\inr{x_2}.c_2}} \longrightarrow \Sub{c_1}{x_1}{v_1}\quad (\mbox{idem }{\it inr})\\ (\mbox{commutation}) && \Subm{\coupe{v}{e}}{\sigma}\longrightarrow \coupe{\Subm{v}{\sigma}}{\Subm{e}{\sigma}}\\ && \Subm{x}{\sigma}\longrightarrow x \;\; (x\mbox{ not declared in }\sigma)\quad\quad \Subm{x}{\Subi{x}{v},\sigma}\longrightarrow v \quad (\mbox{idem }\Subm{\alpha}{\sigma})\\ && \Subm{(\mu\alpha.c)}{\sigma} \longrightarrow \mu\alpha.(\Subm{c}{\sigma}) \quad(\mbox{idem } \Subm{(\tilde\mu x.c)}{\sigma}) \quad(\mbox{capture avoiding})\\ && (\mbox{etc, no rule for composing substitutions})
\end{array}$$ The {\em control rules} mark the decision to launch a substitution (and, in this section, of the direction in which to go, see below). The {\em logical rules} provide the interesting cases of cut elimination, corresponding to cuts where the active formula has been just introduced on both sides. The {\em commutative cuts} are now accounted for ``trivially'' by means of the {\em explicit substitution machinery} that carries substitution progressively inside terms towards their variable occurrences. Summarising, by liberalising the syntax, we have gained considerably in readability of the cut elimination rules\footnote{The precise relation with the previous rules is as follows: for all $s_1,s_2$ such that $s_1\longrightarrow s_2$ in the first system, there exists $s$ such that $s_1\longrightarrow^* s {}^*\longleftarrow s_2$ in the new system, e.g.,
for ($\neg$ right) $\coupe{\mu\alpha.\coupe{(\tilde\mu y.c)^\bullet}{\beta}}{e}$ $\longrightarrow^*$ $\coupe{(\tilde\mu y.(\Sub{c}{\alpha}{e}))^\bullet}{\beta}$ $ {}^*\!\!\longleftarrow$ $\coupe{\mu\beta'.\coupe{(\tilde\mu y.\coupe{\mu\alpha.c}{e})^\bullet}{\beta'}}{\tilde\mu y.\coupe{y}{\beta}}$.}.
\begin{remark} In the ``atomic'' syntax, contractions are transcribed as terms of the form $\coupe{v}{\beta}$ where $\beta$ occurs free in $v$, or of the form $\coupe{x}{e}$ where $x$ occurs freely in $e$. If $\beta$ (resp. $x$) does not occur free in $v$ (resp. $e$), then the command expresses a simple deactivation.
\end{remark}
The problem with classical logic viewed as a computational system is its wild non confluence, as captured by Lafont's critical pair \cite{Girard89,DJSDec}, for which the $\mu\tilde\mu$ kit offers a crisp formulation. For any $c_1,c_2$ both of type $(\Gamma\vdash\Delta)$, we have (with $\alpha,x$ fresh for $c_1,c_2$, respectively): \begin{center} $ c_1\quad{}^*\!\longleftarrow \quad \coupe{\mu\alpha.c_1}{\tilde\mu x.c_2}\quad\longrightarrow^*\quad c_2$ \end{center} So, all proofs are identified... {\em Focalisation}, discussed in the next section, will guide us to solve this dilemma.
\section{A syntax for focalised classical logic} \label{LKQ-sec} In this section, we adapt the
{\em focalisation discipline} (originally introduced by \cite{Andreoli92} in the setting of linear logic) to $\mathsf{LK}$.
A focalised proof search
alternates between right and left phases, as follows:
\noindent - {\it Left phase}: Decompose (copies of) formulas on the left, in any order. Every decomposition of a negation on the left feeds the right part of the sequent. At any moment, one can change the phase from left to right.
\noindent - {\it Right phase}: Choose a formula $A$ on the right, and {\em hereditarily} decompose a copy of it in all branches of the proof search. This {\em focusing} in any branch can only end with an axiom (which ends the proof search in that branch), or with a decomposition of a negation, which prompts a phase change back to the left. Etc\ldots
Note the irreversible (or {\em positive, active}) character of the whole right phase, by the choice of $A$, by the choice of the left or right summand of a disjunction. One takes the risk of not being able to eventually end a proof search branch with an axiom. In contrast, all the choices on the left are reversible (or {\em negative, passive}). This strategy is not only complete (see below), it also guides us to design a disciplined logic whose behaviour will not collapse all the proofs.
To account for right focalisation, we introduce a fourth kind of judgement and a fourth syntactic category of terms: the {\em values}, typed as $(\lkV{\Gamma}{}{V:A}{\Delta})$ (the zone between the turnstyle and the semicolon is called the {\em stoup}, after \cite{GirardLC}). We also make official the existence of two disjunctions (since the behaviours of the conjunction on the left and of the disjunction on the right are different) and two conjunctions, by renaming $\wedge,\vee,\neg$ as $\otimes,\oplus,\notP{}$, respectively. Of course, this choice of linear logic like notation is not fortuitous. Note however that the source of distinction is not based here on the use of resources like in the founding work on linear logic, which divides the line between {\em additive} and {\em multiplicative} connectives. In contrast, our motivating dividing line here is that between {\em irreversible} and {\em reversible} connectives, and hopefully this provides additional motivation for the two conjunctions and the two disjunctions. Our formulas are thus defined by the following syntax: \begin{center} $P::=X \,\mbox{\large\boldmath$\mid$}\, P\otimes P \,\mbox{\large\boldmath$\mid$}\, P\oplus P \,\mbox{\large\boldmath$\mid$}\, \notP{P}$ \end{center} These formulas are called positive. We can define their De Morgan duals as follows: \begin{center} $\overline{P_1\otimes P_2}=\overline{P_1}\parr\overline{P_2}\quad\quad \overline{P_1\oplus P_2}=\overline{P_1}\with\overline{P_2}\quad\quad \overline{\notP{P}}=\notN{\overline{P}}$ \end{center} These duals are {\em negative} formulas: $N::=\overline{X} \,\mbox{\large\boldmath$\mid$}\, N\parr N \,\mbox{\large\boldmath$\mid$}\, N\with N \,\mbox{\large\boldmath$\mid$}\, \notN{N}$. They restore the duality of connectives, and are implicit in the presentation that follows (think of $P$ on the left as being a $\overline{P}$ in a unilateral sequent $\vdash \overline{\Gamma},\Delta$).
We are now ready to give the syntax of our calculus, which is a variant of the one given by the second author in \cite{Munch2009}\footnote{The main differences with the system presented in \cite{Munch2009} is that we have here an explicit syntax of values, with an associated form of typing judgement, while focalisation is dealt with at the level of the reduction semantics in \cite{Munch2009} (see also Remark \ref{LK-LKQ-not-red-refl}). Also, the present system is bilateral but limited to positive formulas on both sides, it thus corresponds to the positive subsystem of the bilateral version of $\mathsf{L}_{\textrm{foc}}$ as presented in \cite{Munch2009}[long version, Appendix A].}.
\begin{center} \fbox{$\begin{array}{lllllll} \mbox{Commands} && c::=\coupe{v}{e} \,\mbox{\large\boldmath$\mid$}\, \Subm{c}{\sigma}&&\\ \mbox{Expressions} && v::= \Vtov{V} \,\mbox{\large\boldmath$\mid$}\, \mu\alpha.c \,\mbox{\large\boldmath$\mid$}\, \Subm{v}{\sigma}\\ \mbox{Values} && V::= x \,\mbox{\large\boldmath$\mid$}\, (V,V) \,\mbox{\large\boldmath$\mid$}\, \inl{V} \,\mbox{\large\boldmath$\mid$}\, \inr{V} \,\mbox{\large\boldmath$\mid$}\, e^\bullet \,\mbox{\large\boldmath$\mid$}\, \Subm{V}{\sigma}&&
\\
\mbox{Contexts} && e::=\alpha \,\mbox{\large\boldmath$\mid$}\, \tilde\mu x.c \,\mbox{\large\boldmath$\mid$}\, \tilde\mu\alpha^\bullet.c \,\mbox{\large\boldmath$\mid$}\, \tilde\mu (x_1,x_2).c \,\mbox{\large\boldmath$\mid$}\, \tilde\mu\copaireP{\inl{x_1}.c_1|\inr{x_2}.c_2} \,\mbox{\large\boldmath$\mid$}\, \Subm{e}{\sigma}&&
\end{array}$} \end{center} The typing rules are given in Figure 1. Henceforth, we shall refer to the calculus of this section (syntax + rewriting rules) as $\mathsf{L}_{\textrm{foc}}$, and to the typing system as $\mathsf{LKQ}$ (after \cite{DJSDec}).
Here are examples of proof terms in $\mathsf{LKQ}$.
\begin{example} \label{LKQ-proofs-ex} \begin{itemize} \item[] $(\lkV{}{}{(\tilde\mu(x,\alpha^\bullet).\coupe{\Vtov{x}}{\alpha})^\bullet:\notP{(P\otimes \notP{P})}}{}$), where $\tilde\mu(x,\alpha^\bullet).c$ is an abbreviation for $\tilde\mu(x,y).\coupe{\Vtov{y}}{\tilde\mu\alpha^\bullet.c}$.
\item[]
$\coupe{\Vtov{\inr{(\tilde\mu x.\coupe{\Vtov{\inl{x}}}{\alpha})^\bullet}}}{\alpha}:(\lkc{}{}{}{\alpha:P\oplus\notP{P}})$. \item[]
$(\lke{}{\tilde\mu(x_2,x_1).\coupe{\Vtov{(x_1,x_2)}}{\alpha}:P_2\otimes P_1}{}{\alpha:P_1\otimes P_2}$). \end{itemize} \end{example} \begin{figure}
\caption{System $\mathsf{LKQ}$}
\label{LKQ-fig}
\end{figure} \begin{proposition} \label{LKQ-complete}
If $\Gamma\vdash\Delta$ is provable in $\mathsf{LK}$, then it is provable in $\mathsf{LKQ}$. \end{proposition}
\noindent {\sc Proof}. Since we have defined a syntax for $\mathsf{LK}$ proofs in section \ref{LK-proofs-sec}, all we have to do is to translate this syntax into the focalised one. All cases are obvious (only inserting the coercion from values to expressions where appropriate) except for the introduction of $\otimes$ and $\oplus$ on the right, for which we can define $\inl{\mu\alpha_1.c_1}$ as \begin{center} $ \lkv{\Gamma}{}{\mu\alpha.\coupe{\mu\alpha_1.c_1}{\tilde\mu x_1.\coupe{\Vtov{(\inl{x_1})}} {\alpha}}:P_1\oplus P_2}{\Delta} \quad\quad(\mbox{idem {\it inr}})$ \end{center} and $(\mu\alpha_1.c_1,\mu\alpha_2.c_2)$ as $(\lkv{\Gamma}{}{\mu\alpha.\coupe{\mu\alpha_2.c_2} {\tilde\mu x_2.\coupe{\mu\alpha_1.c_1} {\tilde\mu x_1.\coupe{\Vtov{(x_1,x_2)}} {\alpha}}}:P_1\otimes P_2}{\Delta})$. \qed
We make two observations on the translation involved in the proof of Proposition \ref{LKQ-complete}.
\noindent \begin{remark} \label{LK-LKQ-choice} The translation {\em introduces cuts}: in particular, a cut-free proof is translated to a proof with (lots of) cuts. It also {\em fixes an order of evaluation}: one should read the translation of right introduction as a protocol prescribing the evaluation of the second element of a pair and then of the first (the pair is thus in particular {\em strict}, as observed in \cite{Munch2009}) (see also
\cite{ZeilbergerCU,Levy2004}). An equally reasonable choice would have been to permute the two $\tilde\mu$s: that would have encoded a left-to-right order of evaluation. This non-determinism of the translation has been known ever since Girard's seminal work \cite{GirardLC}. \end{remark} \begin{remark} \label{LK-LKQ-not-red-refl} The translation is not reduction-preserving, which is expected (since focalisation induces restrictions on the possible reductions), but it is not reduction-reflecting either, in the sense that new reductions are possible on the translated terms. Here is an example (where, say $\mu\_.c$ indicates a binding with a dummy (i.e., fresh) variable). The translation of $\coupe{(\mu\_.c_1,\mu\_.c_2)}{\tilde\mu x.c_3}$ rewrites to (the translation of) $c_2$: \begin{center} $\coupe{\mu\alpha.\coupe{\mu\_.c_2} {\tilde\mu x_2.\coupe{\mu\_.c_1} {\tilde\mu x_1.\coupe{\Vtov{(x_1,x_2)}} {\alpha}}}}{\tilde\mu x.c_3}\quad\longrightarrow^*\quad \coupe{\mu\alpha.c_2}{\tilde\mu x.c_3}\quad\longrightarrow^*\quad c_2 $ \end{center} while the source term is blocked. If we wanted to cure this, we could turn Proposition \ref{LKQ-complete}'s encodings into additional rewriting rules in the source language. We refrain to do so, since we were merely interested in the source syntax as a stepping stone for the focalised one, and we are content that on one hand the rewriting system of Section \ref{LK-proofs-sec} was good enough to eliminate cuts, and that on the other hand the focalised system is complete with respect to provability. But we note that the same additional rules {\em do} appear in the {\em target} language (and are called $\varsigma$-rules, after \cite{Wadlerdual}) in \cite{Munch2009}. This is because in the focalised syntax proposed in \cite{Munch2009} there is no restriction on the terms of the language, hence $(\mu\_.c_1,\mu\_.c_2)$ is a legal term. \end{remark}
We move on to cut elimination, which (cf. Section \ref{LK-proofs-sec}) is expressed by means of three sets of rewriting rules, given in Figure 2.
Note that we now have only one way to reduce $\coupe{\mu\alpha.c_1}{\tilde\mu x.c_2}$ (no more critical pair).
As already stressed in Section \ref{LK-proofs-sec}), the commutation rules
are the usual rules defining (capture-avoiding) substitution.
The overall operational semantics features call-by-value by the fact that variables $x$ receive values,
and features also call-by-name (through symmetry, see the logic $\mathsf{LKT}$ in Section \ref{encodings-sec}) by the fact that continuation variables $\alpha$ receive contexts.
\begin{figure}\label{LKQ-cut-elim-rules}
\end{figure}
The reduction system presented in Figure 2 is {\em confluent}, as it is an orthogonal system in the sense of higher-order rewriting systems (left-linear rules, no critical pairs) \cite{NipkHORS}.
\begin{remark} \label{mu-cosmetic} {\it About $\mu$}: we note that an expression $\mu\beta.c$ is used only in a command $\coupe{\mu\beta.c}{e}$, and in such a context it can be expressed as $\coupe{\Vtov{(e^\bullet)}}{\tilde\mu\beta^\bullet.c}$, which indeed reduces to $\Sub{c}{\beta}{e}$. However, using such an encoding would mean to shift from a direct to an indirect style for terms of the form $\mu\alpha.c$. \end{remark} \begin{proposition} \label{LKQ-cut-elim-prop} Cut-elimination holds in $\mathsf{LKQ}$. \end{proposition} \noindent {\sc Proof}. This is an easy consequence of the following three properties:
\noindent 1) {\it Subject reduction.} This is checked as usual rule by rule.
\noindent 2) {\it Weak normalisation.} One first gets rid of the redexes $\coupe{\mu\alpha.c}{e}$ by reducing them all (no such redex is ever created by the other reduction rules). As usual, one measures cuts by the size of the cut formula, called the degree of the redex, and at each step of normalisation, one chooses a redex of maximal degree all of whose subredexes have striclty lower degree. We then package reductions by considering $\coupe{\Vtov{V}}{\tilde\mu x.c}\longrightarrow \Subwn{c}{\Subi{x}{V}}$
(idem for the logical rules) as a single step, where $\Subwn{c}{\sigma}$ is an augmented (implicit) substitution, defined by induction as usually except for $\coupe{v}{e}$: $$\begin{array}{l} \Subwn{\coupe{\Vtov{x}}{\tilde\mu\alpha^\bullet.c}}{\Subi{x}{e^\bullet},\sigma} = \Subwn{c}{\Subi{\alpha}{e},\Subi{x}{e^\bullet},\sigma}\\ \Subwn{\coupe{\Vtov{x}}{\tilde\mu (x_1,x_2).c}}{\Subi{x}{(V_1,V_2)},\sigma}= \Subwn{c}{\Subi{x_1}{V_1},\Subi{x_2}{V_2},\Subi{x}{(V_1,V_2)},\sigma}\\
\Subwn{\coupe{\Vtov{x}}{\tilde\mu[\inl{x_1}.c_1|\inr{x_2}.c_2]}}{\Subi{x}{\inl{V_1}},\sigma}= \Subwn{c}{\Subi{x_1}{V_1},\Subi{x}{\inl{V_1}},\sigma} \quad(\mbox{idem {\it inr}})\\ \Subwn{\coupe{v}{e}}{\sigma} = \coupe{\Subwn{v}{\sigma}}{\Subwn{e}{\sigma}} \quad\mbox{otherwise} \end{array}$$ This is clearly a well-founded definition, by induction on the term in which substitution is performed (whatever the substitution is). This new notion of reduction ensures the following property: is $t_1 \longrightarrow t_2$ is obtained by reducing $R_1$ in $t_1$ and if $R_2$ is a redex created in $t_2$ by this (packaged) reduction, then $R_1$ is of the form $\coupe{e^\bullet}{\tilde\mu\alpha^\bullet.c}$, where $c$ contains a subterm $\coupe{\Vtov{V}}{\alpha}$, which becomes $R_2$ in $t_2$. The key property is then that the degree of the created redex (the size of some formula $P$) is strictly smaller than the degree of the creating one (the size of $\notP{P}$)\footnote{If we had not packaged reduction, we would have had to deal with the creation of redexes, say by susbstitution of some $V$ for $x$, where the substitution could have been launched by firing a redex of the same degree as the created one.}. The other useful property is that residuals of redexes preserve their degree. Then the argument is easily concluded by associating to each term as global measure the multiset of the degrees of its redexes. This measure strictly decreases at each step (for the multiset extension of the ordering on natural numbers).
\noindent 3) {\it Characterisation of normal forms.} A command in normal form has one of the following shapes (all contractions): $$\coupe{\Vtov{V}}{\alpha} \quad\quad\quad \coupe{\Vtov{x}}{\tilde\mu\alpha^\bullet.c}\quad\quad \coupe{\Vtov{x}}{\tilde\mu(x_1,x_2).c}\quad\quad
\coupe{\Vtov{x}}{\tilde\mu[\inl{x_1}.c_1|\inr{x_2}.c_2]} \quad\quad\quad\quad\qed$$
\begin{corollary} Every sequent $\Gamma\vdash\Delta$ that is provable in $\mathsf{LK}$ admits a (cut-free) proof respecting the focalised discipline. \end{corollary} \noindent {\sc Proof}. Let $\pi$ be a proof of $\Gamma\vdash\Delta$. By Proposition \ref{LKQ-complete}, $\pi$ translates to a command $c:(\lkc{\Gamma}{}{}{\Delta})$, which by Proposition \ref{LKQ-cut-elim-prop} reduces to a term denoting a cut-free proof. The $\mathsf{LK}$ proof obtained by erasing meets the requirement\footnote{This argument of focalisation via normalisation goes back to \cite{GirardLC} (see also \cite{LaurentFoc} for a detailed proof in the case of linear logic).}. \qed
Also, by confluence and weak normalisation, $\mathsf{LKQ}$ is computationally coherent: $(\lkV{x:P,y:P}{}{x:P}{})$ and $(\lkV{x:P,y:P}{}{y:P}{})$ are not provably equal, being normal forms.
Our syntactic choices in this paper have been guided by the phases of focalisation. Indeed, with our syntax, the focalised proof search cycle can be represented as follows (following a branch from the root): {\small $$\begin{array}{lllllll} (\mbox{right phase}) && \coupe{\Vtov{V}}{\alpha}:(\lkc{\Gamma}{}{}{\alpha:P,\Delta}) \;\rightsquigarrow_{-/+} \;\lkV{\Gamma}{}{V:P}{\alpha:P,\Delta}\; \rightsquigarrow_{+}^*\; \lkV{\Gamma}{}{(\tilde\mu x.c)^\bullet:\notP{Q}}{\Delta}\\ &&\quad \rightsquigarrow_{+/-}\; \lke{\Gamma}{\tilde\mu x.c:Q}{}{\Delta}\; \rightsquigarrow_{-} \; c:(\lkc{\Gamma,x:Q}{}{}{\Delta}) \quad\quad(\mbox{idem other $\tilde\mu$ binders})\\ (\mbox{left phase}) && \coupe{\Vtov{x}}{\tilde\mu\alpha^\bullet.c}:(\lkc{\Gamma,x:\notP{P}}{}{}{\Delta})
\; \rightsquigarrow_{-}^* \; c:(\lkc{\Gamma,x:\notP{P}}{}{}{\alpha:P,\Delta}) \\ && \coupe{\Vtov{x}}{\tilde\mu(x_1,x_2).c}:(\lkc{\Gamma,x:P_1\otimes P_2}{}{}{\Delta})
\; \rightsquigarrow_{-}^* \; c:(\lkc{\Gamma,x_1:P_1,x_2:P_2,x:P_1\otimes P_2}{}{}{\Delta})\\ &&
\coupe{\Vtov{x}}{\tilde\mu[\inl{x_1}.c_1|\inr{x_2}.c_2]}:(\lkc{\Gamma,x:P_1\oplus P_2}{}{}{\Delta})
\; \rightsquigarrow_{-}^* \; c_1:(\lkc{\Gamma,x_1:P_1,x:P_1\oplus P_2}{}{}{\Delta})\\
&&
\coupe{\Vtov{x}}{\tilde\mu[\inl{x_1}.c_1|\inr{x_2}.c_2]}:(\lkc{\Gamma,x:P_1\oplus P_2}{}{}{\Delta})
\; \rightsquigarrow_{-}^* \; c_2:(\lkc{\Gamma,x_2:P_2,x:P_1\oplus P_2}{}{}{\Delta}) \end{array} $$} Note that values and commands correspond to positive and negative phases, respectively. The other two categories of terms act as intermediates.
We can also add $\eta$-equivalences (or expansion rules, when read from right to left) to the system, as follows (where all mentioned variables are fresh for the mentioned terms): \begin{center} $\begin{array}{lll} \mu\alpha.\coupe{v}{\alpha}=v &\quad\quad\quad & \tilde\mu(x_1,x_2).\coupe{\Vtov{(x_1,x_2)}}{e}=e\\
\tilde\mu x.\coupe{\Vtov{x}}{e}=e &\quad\quad\quad & \tilde\mu[\inl{x_1}.\coupe{\Vtov{\inl{x_1}}}{e}|\inr{x_2}.\coupe{\Vtov{\inr{x_2}}}{e}]=e\\ &\quad\quad\quad& \tilde\mu\alpha^\bullet\coupe{\Vtov{(\alpha^\bullet)}}{e} =e \end{array}$ \end{center} The rules on the left column allow us to cancel a deactivation followed by an activation (the control rules do the job for the sequence in the reverse order), while the rules in the right column express the reversibility of the negative rules.
\begin{example} \label{LKQ-exp-ex} We relate $(\notP{P_1})\otimes(\notP{P_2})$ and $\notP{(P_1\oplus P_2)}$ (cf. the well-know isomorphism of linear logic, reading $\notP{P}$ as $!\overline{P}$).
There exist $$\begin{array}{l} c_1:(\lkc{y:\notP{(P_1\oplus P_2)}}{}{}{\alpha:\notP{P_1}\otimes\notP{P_2}})\quad\quad c_2:(\lkc{x:\notP{P_1}\otimes\notP{P_2}}{}{}{\gamma:\notP{(P_1\oplus P_2)}}) \end{array}$$ such that, say\footnote{In the $\lambda$-calculus a provable isomorphism is a pair $(x:A\vdash v:B)$, $(y:B\vdash w:A)$ such that $\Subimpl{w}{y}{v}$ reduces to $x$ (and conversely). Here, we express this substitution (where $v,w$ are not values) as $\mu\alpha.\coupe{v}{\tilde\mu y.\coupe{w}{\alpha}}$, and the reduction as $\coupe{v}{\tilde\mu y.\coupe{w}{\alpha}}\longrightarrow^*\coupe{\Vtov{x}}{\alpha}$. } $\coupe{\mu\gamma.c_2}{\tilde\mu y.c_1}$, reduces
$\coupe{\Vtov{x}}{\alpha}:(\lkc{x:\notP{P_1}\otimes\notP{P_2}}{}{}{\alpha:\notP{P_1}\otimes\notP{P_2}})$ . We set $$\begin{array}{lll} V_1=((\tilde\mu y'_1.\coupe{\Vtov{\inl{y'_1}}}{\beta})^\bullet,(\tilde\mu y'_2.\coupe{\Vtov{\inr{y'_2}}}{\beta})^\bullet) &\quad\quad\quad& \lkV{}{}{V_1:\notP{P_1}\otimes\notP{P_2}}{\beta:P_1\oplus P_2}\\
V_2=(\tilde\mu\copaireP{\inl{y_1}.\coupe{\Vtov{y_1}}{\alpha_1}|\inr{y_2}.\coupe{\Vtov{y_2}}{\alpha_2}})^\bullet &\quad\quad&
\lkV{}{}{V_2:\notP{(P_1\oplus P_2)}}{\alpha_1:P_1,\alpha_2:P_2} \end{array}$$ We take $c_1=\coupe{\Vtov{y}}{\tilde\mu\beta^\bullet. \coupe{\Vtov{V_1}}{\alpha}}$ and $c_2=\coupe{\Vtov{x}}{\tilde\mu(\alpha_1^\bullet,\alpha_2^\bullet). \coupe{\Vtov{V_2}}{\gamma}}$, where $\tilde\mu(\alpha_1^\bullet,\alpha_2^\bullet).c$ is defined as a shorthand for, say $\tilde\mu(x_1,x_2).\coupe{\Vtov{x_2}}{\tilde\mu\alpha_2^\bullet.\coupe{\Vtov{x_1}}{\tilde\mu\alpha_1^\bullet .c}}$. We have: $$\begin{array}{lll} \coupe{\mu\gamma.c_2}{\tilde\mu y.c_1} & \longrightarrow^* & \coupe{\Vtov{x}}{\tilde\mu(\alpha_1^\bullet,\alpha_2^\bullet). \coupe{((\tilde\mu y'_1.\coupe{\Vtov{(y')_1}}{\alpha_1})^\bullet,(\tilde\mu y'_2.\coupe{\Vtov{(y')_2}}{\alpha_2}^\bullet)}{\alpha}}\\ && = \coupe{\Vtov{x}}{\tilde\mu(\alpha_1^\bullet,\alpha_2^\bullet). \coupe{(\alpha_1^\bullet,\alpha_2^\bullet)}{\alpha}}\\ && = \coupe{\Vtov{x}}{\alpha} \end{array}$$
\end{example}
We end the section with a lemma that will be useful in Section \ref{LKQS-sec}. \begin{lemma} \label{e-subst-lemma} \begin{itemize} \item If $\lke{\Gamma,x:\notP{P}}{e:Q}{}{\Delta}$, then $\lke{\Gamma}{\Subimpl{e}{x}{\alpha^\bullet}:Q}{}{\alpha:P,\Delta}$. \item If $\lke{\Gamma,x:P_1\otimes P_2}{e:Q}{}{\Delta}$, then $\lke{\Gamma,x_1:P_1,x_2:P_2}{\Subimpl{e}{x}{(x_1,x_2)}:Q}{}{\Delta}$. \item If $\lke{\Gamma,x:P_1\oplus P_2}{e:Q}{}{\Delta}$, then $\lke{\Gamma,x_1:P_1}{\Subimpl{e}{x}{\inl{x_1}}:Q}{}{\Delta}$ and $\lke{\Gamma,x_2:P_2}{\Subimpl{e}{x}{\inr{x_2}}:Q}{}{\Delta}$. \end{itemize} (and similarly for $c,V,v$), where $\Subimpl{t}{x}{V}$ (resp. $\Subimpl{t}{\alpha}{e}$) denotes the usual substitution (cf. Section \ref{introduction}). \end{lemma}
\section{Encodings} \label{encodings-sec}
\noindent{\bf Encoding CBV $\lambda$($\mu$)-calculus into $\mathsf{LKQ}$.} We are now in a position to hook up with the material of Section \ref{intro-sec}. We
can encode the call-by-value $\lambda$-calculus, by defining the following derived CBV implication and terms: $$\begin{array}{c} P\rightarrow^v Q=\notP{(P\:\otimes\:\notP{Q})}\\ \lambda x.v=\Vtov{((\tilde\mu(x,\alpha^\bullet).\coupe{v}{\alpha})^\bullet)}\quad\quad\quad v_1v_2=\mu\alpha.\coupe{v_2}{\tilde\mu x.\coupe{v_1}{\Vtoe{(\Vtov{(x,\alpha^\bullet)})}}} \end{array}$$ where $\tilde\mu(x,\alpha^\bullet).c$ is the abbreviation used in Example \ref{LKQ-proofs-ex} and where $\Vtoe{V}$ stands for $\tilde\mu\alpha^\bullet.\coupe{\Vtov{V}}{\alpha}$. These definitions provide us with a translation, which extends to (call-by-value) $\lambda\mu$-calculus \cite{OngStew,Rocheteau}, and factors though $\overline\lbd\mu\tilde\mu_Q$-calculus (cf. Section \ref{intro-sec}), defining $V\cdot e$ as $\Vtoe{(V,e^\bullet)}$\footnote{In \cite{CH2000} we also had a difference operator $B-A$ (dual to implication), and two associated introduction operations, whose encodings in the present syntax are $\beta\lambda.e=\tilde\mu(\beta^\bullet,x).\coupe{\Vtov{x}}{e}$ and $e\cdot V=(e^\bullet,V)$.}. The translation makes also sense in the untyped setting, as the following example shows. \begin{example} \label{Delta-Delta} Let $\Delta=\lambda x.xx$. We have $\CBVP{\Delta\Delta}=\mu\gamma.c$, and $c\longrightarrow^* c$, with
$$c=\coupe{\Vtov{(e^\bullet)}}{\tilde\mu z.\coupe{\Vtov{(e^\bullet)}}{\Vtoe{(z,\gamma^\bullet)}}}\quad\mbox{ and }\quad e=\tilde\mu(x,\alpha^\bullet).\coupe{\Vtov{x}}{\tilde\mu y.\coupe{\Vtov{x}}{\Vtoe{(y,\alpha^\bullet)}}}$$ \end{example}
\noindent {\bf Encoding CBN $\lambda$($\mu$)-calculus.} What about CBN? We can translate it to $\mathsf{LKQ}$, but at the price of translating terms to contexts, which is a violence to our goal of giving an intuitive semantics to the first abstract machine presented in Section \ref{intro-sec}. Instead, we spell out the system dual to $\mathsf{LKQ}$, which is known as $\mathsf{LKT}$, in which expressions and contexts will have negative types, and in which we shall be able to express CBN $\lambda$-terms as expressions. Our syntax for $\mathsf{LKT}$ is a mirror image of that for $\mathsf{LKQ}$: it exchanges the $\mu$ and $\tilde\mu$, the $x$'s and the $\alpha$'s, etc..., and renames {\it inl}, {\it inr} as {\it fst}, {\it snd}, which are naturally associated with $\with$ while the latter were naturally associated with $\oplus$: $$\begin{array}{lll} \mbox{Commands} && c::=\coupe{v}{e}\\ \mbox{Covalues} && E::= \alpha \,\mbox{\large\boldmath$\mid$}\, [E,E] \,\mbox{\large\boldmath$\mid$}\, \fst{E} \,\mbox{\large\boldmath$\mid$}\, \snd{E} \,\mbox{\large\boldmath$\mid$}\, v^\bullet\\ \mbox{Contexts} && e::= \Etoe{E} \,\mbox{\large\boldmath$\mid$}\, \tilde\mu x.c\\ \mbox{Expressions} && v::=x \,\mbox{\large\boldmath$\mid$}\, \mu \alpha.c \,\mbox{\large\boldmath$\mid$}\, \mu x^\bullet.c \,\mbox{\large\boldmath$\mid$}\, \ldots \end{array}$$ Note that focalisation is now on the left, giving rise to a syntactic category of {\em covalues} (that were called applicative contexts in \cite{CH2000}).\footnote{Note also that the duality sends a command $\coupe{v}{e}$ to a command $\coupe{e'}{v'}$ where $v'$, $e'$ are the mirror images of $v$, $e$.}
The rules are all obtained from $\mathsf{LKQ}$ by duality: $$\begin{array}{c} \seq{}{\lkE{\Gamma}{\alpha:N}{}{\Delta\:,\:\alpha:N}}\quad\quad \seq{\lkE{\Gamma}{E_1:N_1}{}{\Delta}}{\lkE{\Gamma}{\fst{E_1}:N_1\with N_2}{}{\Delta}} \\\\ \seq{}{\lkv{\Gamma\:,\:x:N}{}{x:N}{\Delta}}\quad\quad\seq{\lkv{\Gamma}{}{v:N}{\Delta}\quad\quad \lke{\Gamma}{e:N}{}{\Delta}} {\coupe{v}{e}:(\lkc{\Gamma}{}{}{\Delta})} \quad\quad\quad \ldots \end{array}$$
We would have arrived to this logic naturally if we had chosen in Section \ref{LK-proofs-sec} to present $\mathsf{LK}$ with a reversible disjunction on the right and an irreversible conjunction on the left, and in Section \ref{LKQ-sec} to present a focalisation discipline with focusing on formulas on the left.
In $\mathsf{LKT}$ we can define the following derived CBN implication and terms: $$\begin{array}{c} M\rightarrow^n N=(\notN{M})\:\parr\:N\\ \lambda x.v=\mu( x^\bullet,\alpha).\coupe{v}{\Vtov{\alpha}}\quad\quad\quad v_1v_2=\mu\alpha.\coupe{v_1}{\Vtov{(v_2^\bullet,\alpha)}} \end{array}$$ The translation extends to $\lambda\mu$-calculus \cite{Parigot92} and factors though the $\overline\lbd\mu\tilde\mu_T$-calculus of \cite{CH2000}, defining $v \cdot E$ as $(v^\bullet,E)$. Note that the covalues involved in executing call-by-name $\lambda$-calculus are just {\em stacks} of expressions (cf. Section \ref{intro-sec}).
With these definitions, we have: $$\begin{array}{lll}
\coupe{\lambda x.v_1}{\Vtov{(v_2\cdot E)}} =
\coupe{\mu( x^\bullet,\alpha).\coupe{v_1}{\Vtov{\alpha}}}{\Vtov{(v_2^\bullet,E)}}
\longrightarrow \coupe{\Sub{v_1}{x}{v_2}}{\Vtov{E}}\\
\coupe{v_1v_2}{\Vtov{E}} = \coupe{\mu\alpha.\coupe{v_1}{\Vtov{(v_2^\bullet,\alpha)}}}{\Vtov{E}} \longrightarrow \coupe{v_1}{\Vtov{(v_2^\bullet,E)}} =
\coupe{v_1}{\Vtov{(v_2\cdot E)}} \end{array}$$ We are thus back on our feet (cf. section 1)!
\noindent {\bf Translating $\mathsf{LKQ}$ into $\mathsf{NJ}$.}
Figure 3
presents a translation from $\mathsf{LKQ}$ to intuitionistic natural deduction $\mathsf{NJ}$, or, via Curry-Howard, to $\lambda$-calculus extended with products and sums. In the translation, $R$ is a fixed target formula (cf. Section \ref{intro-sec}). We translate $(\notP{\_})$ as ``$\_$ implies $R$'' (cf. \cite{Krivine91,LRS93}).
We write $B^A$ for function types / intuitionistic implications. The rules of $\mathsf{L}_{\textrm{foc}}$ are simulated by $\beta$-reductions. One may think of the source $\mathsf{L}_{\textrm{foc}}$ terms as a description of the target ones ``in direct style'' (cf. \cite{DanvyBack}).
\begin{figure}
\caption{Translation of $\mathsf{LKQ}$ into the $\lambda$-calculus / $\mathsf{NJ}$}
\label{LKQ-to-NJ}
\end{figure} \begin{proposition} We set $\CPSP{\Gamma}=\setc{x:\CPSP{P}}{x:P\in\Gamma}\quad\quad R^{\CPSP{\Delta}}=\setc{\contvartovar{\alpha}:R^{\CPSP{P}}}{\alpha:P\in\Delta}$. We have:
$$\begin{array}{|cc|cc|cc|c|} \hline c:(\lkc{\Gamma}{}{}{\Delta}) && \lkV{\Gamma}{}{V:P}{\Delta} && \lkv{\Gamma}{}{v:P}{\Delta} && \lke{\Gamma}{e:P}{}{\Delta}\\ \Downarrow &&\Downarrow &&\Downarrow &&\Downarrow \\ \CPSP{\Gamma}\:,\:R^{\CPSP{\Delta}}\vdash \CPSP{c}:R && \CPSP{\Gamma}\:,\:R^{\CPSP{\Delta}}\vdash \CPSP{V}:\CPSP{P} && \CPSP{\Gamma}\:,\:R^{\CPSP{\Delta}}\vdash \CPSP{v}:R^{R^{\CPSP{P}}} && \CPSP{\Gamma}\:,\:R^{\CPSP{\Delta}}\vdash \CPSP{e}:R^{\CPSP{P}}\\ \hline \end{array}$$
Moreover, the translation preserves reduction: if $t\longrightarrow t'$, then $\CPSP{t}\longrightarrow^*\CPSP{(t')}$. \end{proposition} Composing the previous translations, from CBN $\lambda\mu$-calculus to $\mathsf{LKT}$ then through duality to $\mathsf{LKQ}$ then to $\mathsf{NJ}$, what we obtain is the CPS translation due to Lafont, Reus, and Streicher \cite{LRS93} (LRS translation, for short).
\noindent {\bf Translating $\mathsf{LKQ}$ into $\mathsf{LLP}$.} The translation just given from $\mathsf{LKQ}$ to $\mathsf{NJ}$ does actually two transformations for the price of one: {\em from classical to intuitionistic}, and {\em from sequent calculus style to natural deduction style}. The intermediate target and source of this decomposition is nothing but a subsystem of Laurent's polarised linear logic $\mathsf{LLP}$ \cite{LaurentTh} \footnote{Specifically, no positive formula is allowed on the right in the rules for $\parr$ and $\with$ and in the right premise of the cut rule.}. We adopt a presentation of $\mathsf{LLP}$ in which all negative formulas are handled as positive formulas on the left, and hence in which $!N$ and $?P$ are replaced by $\notP{}$ with the appropriate change of side. With these conventions, $\mathsf{LLP}$ is nothing but the system called $\mathsf{LJ}_0$ in \cite{Laurent2009}. For the purpose of giving a system $\mathsf{L}$ term syntax, we distinguish three kinds of sequents for our subsystem of $\mathsf{LLP}$: $$(\lkc{\Gamma}{}{}{})\quad\quad(\lkV{\Gamma}{}{P}{})\quad\quad (\lke{\Gamma}{P}{}{})$$ The syntax, the computation rules, and the typing rules are as follows (omitting explicit substitutions): $$\begin{array}{l}
c::=\coupe{V}{e}\quad V::= x \,\mbox{\large\boldmath$\mid$}\, e^\bullet \,\mbox{\large\boldmath$\mid$}\, (V,V) \,\mbox{\large\boldmath$\mid$}\, \inl{V} \,\mbox{\large\boldmath$\mid$}\, \inr{V}\\ e::=\derel{V} \,\mbox{\large\boldmath$\mid$}\, \tilde\mu x.c \,\mbox{\large\boldmath$\mid$}\, \tilde\mu(x_1,x_2).c \,\mbox{\large\boldmath$\mid$}\, \tilde\mu[\inl{x_1}.c_1|\inr{c_2}.c_2] \end{array} $$
$$\begin{array}{lllll} \coupe{V}{\tilde\mu x.c} \longrightarrow \Sub{c}{x}{V}\\ \coupe{e^\bullet}{\derel{V}} \longrightarrow \coupe{V}{e} \\ \coupe{(V_1,V_2)}{\tilde\mu(x_1,x_2).c} \longrightarrow \Subm{c}{\Subi{x_1}{V_1},\Subi{x_2}{V_2}}\\
\coupe{\inl{V_1}}{\tilde\mu\copaireP{\inl{x_1}.c_1|\inr{x_2}.c_2}} \longrightarrow \Sub{c_1}{x_1}{V_1}
\quad\quad \coupe{\inr{V_2}}{\tilde\mu\copaireP{\inl{x_1}.c_1|\inr{x_2}.c_2}} \longrightarrow \Sub{c_2}{x_2}{V_2}
\end{array}$$
$$\seq{}{\lkV{\Gamma\:,\:x:P}{}{x:P}{}}\quad\quad \seq{\lkV{\Gamma}{}{V:P}{}\quad\quad \lke{\Gamma}{e:P}{}{}} {\coupe{V}{e}:(\lkc{\Gamma}{}{}{})}\quad\quad \seq{c:(\lkc{\Gamma\:,\:x:P}{}{}{})}{\lke{\Gamma}{\tilde\mu x.c:P}{}{}} $$ $$\seq{\lke{\Gamma}{e:P}{}{}}{\lkV{\Gamma}{}{e^\bullet:\notP{P}}{}}\quad\quad\seq{\lkV{\Gamma}{}{V_1:P_1}{} \quad\quad \lkV{\Gamma}{}{V_2:P_2}{}} {\lkV{\Gamma}{}{(V_1,V_2):P_1\otimes P_2}{}}\quad\quad \seq{\lkV{\Gamma}{}{V_1:P_1}{}}{\lkV{\Gamma}{}{\inl{V_1}:P_1\oplus P_2}{}}\quad\quad \seq{\lkV{\Gamma}{}{V_2:P_2}{}}{\lkV{\Gamma}{}{\inr{V_2}:P_1\oplus P_2}{}}$$ $$\quad\seq{\lkV{\Gamma}{}{V:P}{}}{\lke{\Gamma}{\derel{V}:\notP{P}}{}{}}\quad\quad \seq{c:(\Gamma,x_1:P_1,x_2:P_2\vdash)}{\lke{\Gamma}{\tilde\mu(x_1,x_2).c:P_1\otimes P_2}{}{}}\quad\quad \seq{c_1:(\Gamma,x_1:P_1\vdash)\quad\quad c_2:(\Gamma,x_2:P_2\vdash)}
{\lke{\Gamma}{\tilde\mu\copaireP{\inl{x_1}.c_1|\inr{x_2}.c_2}:P_1\oplus P_2}{}{}}$$
\noindent
The constructs $e^\bullet$ and $\derel{V}$ transcribe $\mathsf{LLP}$'s {\em promotion} and {\em dereliction} rule, respectively. The compilation from $\mathsf{LKQ}$ to (the subsystem of) $\mathsf{LLP}$
turns every $\alpha:P$ on the right to a $\contvartovar{\alpha}:\notP{P}$ on the left. We write
$\notP{(\ldots,\alpha:P,\ldots)}= (\ldots,k_\alpha:\notP{P},\ldots)$.
The translation is as follows (we give only the non straightforward cases):
$$\begin{array}{l}
\coupe{v}{e}_{\mathsf{LLP}}=\coupe{(e_{\mathsf{LLP}})^\bullet}{v_{\mathsf{LLP}}}\\
(\mu\alpha.c)_{\mathsf{LLP}}=\tilde\mu\contvartovar{\alpha}.c_{\mathsf{LLP}}\quad\quad
(\Vtov{V})_{\mathsf{LLP}}= \derel{(V_{\mathsf{LLP}})}\\
\alpha_{\mathsf{LLP}}=\tilde\mu x.\coupe{\contvartovar{\alpha}}{\derel{x}}\quad (\tilde\mu\alpha^\bullet.c)_{\mathsf{LLP}}=\tilde\mu\contvartovar{\alpha}.(c_{\mathsf{LLP}})
\end{array}$$
Note that we can optimise the translation of $\coupe{\Vtov{V}}{e}$, and (up to an expansion rule) of $\Vtoe{V}=\tilde\mu\alpha^\bullet.\coupe{\Vtov{V}}{\alpha}$:
$$\begin{array}{lll}
\coupe{\Vtov{V}}{e}_{\mathsf{LLP}}=\coupe{(e_{\mathsf{LLP}})^\bullet}{\derel{(V_{\mathsf{LLP}})}}
\longrightarrow\coupe{V_{\mathsf{LLP}}}{e_{\mathsf{LLP}}}\\
(\Vtoe{V})_{\mathsf{LLP}} =
\tilde\mu \contvartovar{\alpha}.\coupe{V_{\mathsf{LLP}}}{\alpha_{\mathsf{LLP}}}
\;= \tilde\mu \contvartovar{\alpha}.\coupe{V_{\mathsf{LLP}}}{\tilde\mu x.\coupe{\contvartovar{\alpha}}{\derel{x}}} \longrightarrow \tilde\mu \contvartovar{\alpha}.\coupe{\contvartovar{\alpha}}{\Vtov{(V_{\mathsf{LLP}})}} = \derel{(V_{\mathsf{LLP}})} \end{array} $$ These optimisations allow us to define a right inverse to ${}_{\mathsf{LLP}}$ (that maps $\Vtov{V}$ to $\Vtoe{V}$)), i.e.: \begin{center} {\em $\mathsf{LLP}$ (restricted as above) appears as a retract of $\mathsf{LKQ}$}. \end{center}
The translation simulates reductions and is well typed:
$$\begin{array}{lll}
c:(\lkc{\Gamma}{}{}{\Delta}) & \quad\Rightarrow\quad & c_{\mathsf{LLP}}: (\lkc{\Gamma,\notP{\Delta}}{}{}{})\\
\lkv{\Gamma}{}{v:P}{\Delta} & \quad\Rightarrow\quad & \lke{\Gamma,\notP{\Delta}}{v_{\mathsf{LLP}}:\notP{P}}{}{}\\
\lkV{\Gamma}{}{V:P}{\Delta} & \quad\Rightarrow\quad & \lkV{\Gamma,\notP{\Delta}}{}{V_{\mathsf{LLP}}:P}{}\\ \lke{\Gamma}{e:P}{}{\Delta} & \quad\Rightarrow\quad & \lke{\Gamma,\notP{\Delta}}{e_{\mathsf{LLP}}:P}{}{}
\end{array}$$
We note that this compilation blurs the distinction between a continuation variable and an ordinary variable (like in the classical CPS translations).
\begin{example} The classical proof $(\lke{}{\tilde\mu\beta^\bullet.\coupe{\Vtov{(\alpha^\bullet)}}{\beta}:\notP{\notP{P}}}{}{\alpha:P})$
is translated (using the above optimisation) to the intuitionistic proof
$(\lke{\contvartovar{\alpha}:\notP{P}}{\derel{((\tilde\mu x.\coupe{\contvartovar{\alpha}}{\derel{x}})^\bullet)}:\notP{\notP{P}}}{}{})$. Without term decorations, we have turned a proof of the classically-only provable sequent
$(\lke{}{\notP{\notP{P}}}{}{P})$ into an intuitionistic proof of $(\lke{\notP{P}}{\notP{\notP{P}}}{}{})$. \end{example}
All what is left to do in order to reach then $\mathsf{NJ}$\footnote{In fact, the target of the translation uses only implications of the form $R^P$. Seeing $R$ as ``false'', this means that the target is in fact intuitionistic logic with conjunction, disjunction and negation in natural deduction style.} from (our subsystem of) $\mathsf{LLP}$ is to turn contexts $e:P$ into values of type $\notP{P}$, and to rename $\notP{\_}$, $\otimes$, and $\oplus$ as $R^{\_},\times$, and $+$, respectively. More precisely, we describe the target syntax (a subset of a $\lambda$-calculus with sums and products) as follows: $$\begin{array}{l} c::= VV\quad\quad V::= x \,\mbox{\large\boldmath$\mid$}\, (V,V) \,\mbox{\large\boldmath$\mid$}\, \inl{V} \,\mbox{\large\boldmath$\mid$}\, \inr{V} \,\mbox{\large\boldmath$\mid$}\, \lambda x.c \,\mbox{\large\boldmath$\mid$}\, \lambda(x_1,x_2).c \,\mbox{\large\boldmath$\mid$}\, \lambda z.\fcase{z}{\fcasei{\inl{x_1}}{c_1,\fcasei{\inr{x_2}}{c_2}}} \end{array}$$ Again, we give only the non trivial cases of the translation: $$\begin{array}{l} \coupe{V}{e}_{\mathsf{NJ}}= (e_{\mathsf{NJ}})(V_{\mathsf{NJ}})\quad\quad (e^\bullet)_{\mathsf{NJ}}=e_{\mathsf{NJ}}\quad\quad (\derel{V})_{\mathsf{NJ}}=\lambda k.k(V_{\mathsf{NJ}})\\(\tilde\mu x.c)_{\mathsf{NJ}}=\lambda x.(c_{\mathsf{NJ}})\quad (\tilde\mu(x_1,x_2).c)_{\mathsf{NJ}}=\lambda(x_1,x_2).(c_{\mathsf{NJ}})\\
(\tilde\mu[\inl{x_1}.c_1|\inr{c_2}.c_2])_{\mathsf{NJ}}=\lambda z.\fcase{z}{\fcasei{\inl{x_1}}{((c_1)_{\mathsf{NJ}}),\fcasei{\inr{x_2}}{(c_2)_{\mathsf{NJ}}}}} \end{array}$$
\begin{proposition} \label{factor-cps} For all $\mathsf{L}_{\textrm{foc}}$ term $t$ (where $t::= c \,\mbox{\large\boldmath$\mid$}\, V \,\mbox{\large\boldmath$\mid$}\, v \,\mbox{\large\boldmath$\mid$}\, e$), we have: $$\CPSP{t}=_{\beta\eta} (t_{\mathsf{LLP}})_{\mathsf{NJ}}\;.$$ \end{proposition} \noindent {\sc Proof}. We treat the non trivial cases: $$\begin{array}{lll} \coupe{v}{e}_{\mathsf{LLP},\mathsf{NJ}} = \coupe{(e_{\mathsf{LLP}})^\bullet}{v_{\mathsf{LLP}}}_{\mathsf{NJ}} = (v_{\mathsf{LLP},\mathsf{NJ}})(((e_{\mathsf{LLP}})^\bullet)_{\mathsf{NJ}}) = (v_{\mathsf{LLP},\mathsf{NJ}})(e_{\mathsf{LLP},\mathsf{NJ}})\\
(\Vtov{V})_{\mathsf{LLP},\mathsf{NJ}} = ( \derel{(V_{\mathsf{LLP}})})_{\mathsf{NJ}} =\lambda k.k(V_{\mathsf{LLP},\mathsf{NJ}} )\\ \alpha_{\mathsf{LLP},\mathsf{NJ}} = (\tilde\mu x.\coupe{\contvartovar{\alpha}}{\derel{x}})_{\mathsf{NJ}} = \lambda x.(\coupe{\contvartovar{\alpha}}{\derel{x}}_{\mathsf{NJ}})=\lambda x.((\derel{x})_{\mathsf{NJ}})\contvartovar{\alpha}=\lambda x.(\lambda k.kx)\contvartovar{\alpha}=_\beta\lambda x.\contvartovar{\alpha}x=_\eta \contvartovar{\alpha}\;\qed \end{array}$$ Note that, in the proof of the above proposition, the $\beta$ step is a typical ``administrative reduction'', so that morally the statement of the proposition holds with $\eta$ only.
\noindent {\bf A short foray into linear logic.} We end this section by placing the so-called Girard's translation of the call-by-name $\lambda$-calculus to linear logic in perspective. The target of this translation is in fact the polarised fragment $\mathsf{LL}_{\textrm{pol}}$ of linear logic, obtained by restriction to the polarised formulas: $$P::= X \,\mbox{\large\boldmath$\mid$}\, P\otimes P \,\mbox{\large\boldmath$\mid$}\, P\oplus P \,\mbox{\large\boldmath$\mid$}\, !N\quad\quad\quad N::= X^\bot \,\mbox{\large\boldmath$\mid$}\, N\parr N \,\mbox{\large\boldmath$\mid$}\, N\with N \,\mbox{\large\boldmath$\mid$}\, ?P$$ This fragment is also a fragment of $\mathsf{LLP}$, (cf. \cite{LaurentTh}), up to the change of notation for the formulas: we write here $\overline{P}$ for $P^\bot$ and $\notP{\overline{N}}$ for $!N$. Girards translation encodes call-by-name implication as follows:
$$(A\rightarrow B)^*=!(A^*)\multimap B^*=(?(A^*)^\bot)\parr B^*$$
and then every
$\lambda$-term $\Gamma\vdash M:A$ into a proof of $!(\Gamma)^*\vdash A^*$.
Up to the inclusion of $\mathsf{LL}_{\textrm{pol}}$ into $\mathsf{LLP}$, up to the definition of the $\lambda$-calculus inside $\mathsf{LKT}$ given above, up to the change of notation, and up to the duality between $\mathsf{LKT}$ and $\mathsf{LKQ}$, Girard's translation coincides with the (restriction of) our translation above from $\mathsf{LKT}$ to $\mathsf{LLP}$.
On the other hand, the restriction to the $\lambda\mu$-calculus of our translation from $\mathsf{LKT}$ to $\mathsf{NJ}$ is the CPS translation of Lafont-Reus-Streicher. Thus, restricted to the $\lambda$-calculus, Proposition \ref{factor-cps} reads as follows: \begin{center} {\em Lafont-Reus-Streicher's CPS factors through Girard's translation}. \end{center} Explicitly, on types, we have that $A^*$ coincides with $A$ as expanded in $\mathsf{LKT}$, and (cf. \cite{CH2000}), starting from the simply-typed $\lambda$-term $(\Gamma\vdash M:A)$, \begin{itemize} \item we view $M$ as an expression $(\lkv{\Gamma}{}{M:A}{})$ of $\mathsf{LKT}$ (using the CBN encoding of implication), \item and then as a context $(\lke{}{M:\overline{A}}{}{\overline{\Gamma}})$ of $\mathsf{LKQ}$, \item then by our above translation we find back the result of Girard's translation $(\lke{\notP{(\overline{\Gamma})}}{M_{\mathsf{LLP}}:\overline{A}}{}{})$, \item and we arrive finally at the Hofmann-Streicher CPS-transform $(\notP{(\overline{\Gamma})}\vdash \CPSP{M}:\notP{(\overline{A})})$ of $M$, through the translation ${}_{\mathsf{NJ}}$.\footnote{The LRS translation of implication goes as follows: $(A\rightarrow B)_{\textrm{LRS}}= R^{A_{\textrm{LRS}}}\times B_{\textrm{LRS}}$, and we have $\CPSP{(\overline{A})}=A_{\textrm{LRS}}$ .} \end{itemize} But the above reading is actually stronger, because it is not hard to describe a translation ${}_{\mathsf{LJ}}$ inverse to ${}_{\mathsf{NJ}}$, so that up to this further isomorphism, we have that: \begin{center} {\em The LRS translation of the CBN $\lambda$-calculus coincides with Girard's translation}. \end{center} This nice story does not extend immediately to the $\lambda\mu$-calculus, for which the simplest extension of Girard's translation, taking $\lkv{\Gamma}{}{M:A}{\Delta}$ to a proof of $!(\Gamma^*)\vdash A^*,?(\Delta)^*$ is not polarised. In fact, Laurent \cite{LaurentTh} has shown that the natural target for an extension of Girard's translation to CBN $\lambda\mu$-calculus is $\mathsf{LLP}$, in which we can spare the ?'s on the right, i.e., we can translate $\lkv{\Gamma}{}{M:A}{\Delta}$ into a proof of $!(\Gamma^*)\vdash A^*,\Delta^*$ (contractions on negative formulas are free in $\mathsf{LLP}$). So, the extension of the picture to call-by-name $\lambda\mu$-calculus is\footnote{See also \cite{LR2003} for further discussion.}: \begin{center} {\em The LRS translation of the CBN $\lambda\mu$-calculus coincides with Laurent-Girard's translation into $\mathsf{LLP}$}. \end{center}
\section{A synthetic system} \label{LKQS-sec}
In this section we pursue two related goals.
\begin{enumerate} \item We want to account for the full (or strong) focalisation (cf. \cite{QuaTor96}), which consists in removing the use of contractions in the negative phases and carrying
these phases maximally, up to having only atoms on the left of the sequent. The positive phases are made also ``more maximal'' by allowing the use of the axiom only on positive atoms $X$. This is of interest in a proof search perspective, since the stronger discipline
further reduces the search space. \item We would like our syntax to quotient proofs over the order of decomposition of negative formulas. The use of structured pattern-matching
(cf. Examples \ref{LKQ-proofs-ex}, \ref{LKQ-exp-ex}) is relevant, as we can describe the construction of a proof of $(\Gamma, x:(P_1\otimes P_2)\otimes(P_3\otimes P_4)\vdash \Delta)$ out of a proof of $c:(\Gamma, x_1:P_1,x_2:P_2,x_3:P_3,x_4:P_4\vdash\Delta)$ ``synthetically'', by writing $\coupe{\Vtov{x}}{\tilde\mu((x_1,x_2),(x_3,x_4)).c}$, where $\tilde\mu((x_1,x_2),(x_3,x_4)).c$ stands for an abbreviation of either of the following two commands: $$\begin{array}{lll} \coupe{\Vtov{x}}{\tilde\mu(y,z).\coupe{\Vtov{y}}{\tilde\mu(x_1,x_2).\coupe{\Vtov{z}}{\tilde\mu(x_3,x_4).c}}} & \quad\quad & \coupe{\Vtov{x}}{\tilde\mu(y,z).\coupe{\Vtov{z}}{\tilde\mu(x_3,x_4).\coupe{\Vtov{y}}{\tilde\mu(x_1,x_2).c}}} \end{array}$$ \end{enumerate} The two goals are connected, since applying strong focalisation will forbid the formation of these two terms (because $y,z$ are values appearing with non atomic types), keeping the synthetic form only... provided we make it first class.
We shall proceed in {\em two steps}. The first, intermediate one consists in introducing first-class {\em counterpatterns} and will serve goal 1 but not quite goal 2: $$\begin{array}{lllllllll} \mbox{Simple commands} && c::=\coupe{v}{e} &\quad\quad\quad\quad& \mbox{Commands} && C::= c \,\mbox{\large\boldmath$\mid$}\, \copaireC{C}{q}{q}{C}\\ \mbox{Expressions} && v::= \Vtov{V} \,\mbox{\large\boldmath$\mid$}\, \mu\alpha.C &\quad\quad\quad\quad& \mbox{Values} && V::= x \,\mbox{\large\boldmath$\mid$}\, (V,V) \,\mbox{\large\boldmath$\mid$}\, \inl{V} \,\mbox{\large\boldmath$\mid$}\, \inr{V} \,\mbox{\large\boldmath$\mid$}\, e^\bullet \\ \mbox{Contexts} && \fbox{$e::=\alpha \,\mbox{\large\boldmath$\mid$}\, \tilde\mu q.C$} &\quad\quad\quad\quad&
\mbox{Counterpatterns} && \fbox{$q::= x \,\mbox{\large\boldmath$\mid$}\, \alpha^\bullet \,\mbox{\large\boldmath$\mid$}\, (q,q) \,\mbox{\large\boldmath$\mid$}\, \copaireP{q,q}$} \end{array}$$ The counterpatterns are to be thought of as constructs that match patterns (see below).
In this syntax, we have gained a unique $\tilde\mu$ binder, but the price to pay (provisionally) is that now commands are trees of copairings $\copaireC{\_}{q_1}{q_2}{\_}$ whose leaves are simple commands.
The typing discipline is restricted with respect to that of Figure 1 (and adapted to the setting with explicit counterpatterns). Let $\Xi=x_1:X_1,\ldots,x_n:X_n$ denote a left context consisting of {\it atomic formulas only}. The rules are as follows: {\small $$\begin{array}{c}\seq{}{\lkV{\Xi\:,\:x:X}{}{x:X}{\Delta}}\quad\quad \seq{}{\lke{\Xi}{\alpha:P}{}{\alpha:P\:,\:\Delta}}\quad\quad \seq{\lkv{\Xi}{}{v:P}{\Delta}\quad\quad \lke{\Xi}{e:P}{}{\Delta}} {\coupe{v}{e}:(\lkc{\Xi}{}{}{\Delta})} \\\\ \seq{C:(\lkc{\Xi\:,\:q:P}{}{}{\Delta})}{\lke{\Xi}{\tilde\mu q.C:P}{}{\Delta}} \quad\quad \seq{C:(\lkc{\Xi}{}{}{\alpha:P\:,\:\Delta})}{\lkv{\Xi}{}{\mu\alpha.C:P}{\Delta}}\quad\quad \seq{\lkV{\Xi}{}{V:P}{\Delta}}{\lkv{\Xi}{}{\Vtov{V}:P}{\Delta}} \\\\ \seq{\lke{\Xi}{e:P}{}{\Delta}}{\lkV{\Xi}{}{e^\bullet:\notP{P}}{\Delta}}\quad\quad\seq{\lkV{\Xi}{}{V_1:P_1}{\Delta} \quad\quad \lkV{\Xi}{}{V_2:P_2}{\Delta}} {\lkV{\Xi}{}{(V_1,V_2):P_1\otimes P_2}{\Delta}}\quad\quad \seq{\lkV{\Xi}{}{V_1:P_1}{\Delta}}{\lkV{\Xi}{}{\inl{V_1}:P_1\oplus P_2}{\Delta}}\quad\quad \seq{\lkV{\Xi}{}{V_2:P_2}{\Delta}}{\lkV{\Xi}{}{\inr{V_2}:P_1\oplus P_2}{\Delta}} \\\\\seq{C:(\lkc{\Gamma}{}{}{\alpha:P\:,\:\Delta})}{C:(\lkc{\Gamma\:,\:\alpha^\bullet:\notP{P}}{}{}{\Delta})}\quad\quad \seq{C:(\lkc{\Gamma\:,\:q_1:P_1\:,\:q_2:P_2}{}{}{\Delta})}{C:(\lkc{\Gamma\:,\:(q_1,q_2):P_1\otimes P_2}{}{}{\Delta})}\quad\quad \seq{C_1:(\lkc{\Gamma\:,\:q_1:P_1}{}{}{\Delta})\quad\quad C_2:(\lkc{\Gamma\:,\:q_2:P_2}{}{}{\Delta})} {\copaireC{C_1}{q_1}{q_2}{C_2}:(\lkc{\Gamma\:,\:\copaireP{q_1,q_2}:P_1\oplus P_2}{}{}{\Delta})} \end{array}$$}
Our aim now ({\em second step}) is to get rid of the tree structure of a command. Indeed, towards our second goal, if $c_{ij}:(\Gamma,x_i:P_i,x_j:P_j\vdash_S\Delta)$ ($i=1,2, j=3,4$), we want to identify $\copaireC{\copaireC{c_{13}}{x_3}{x_4}{c_{14}}}{x_1}{x_2}{\copaireC{c_{23}}{x_3}{x_4}{c_{24}}}$ and $\copaireC{\copaireC{c_{13}}{x_1}{x_2}{c_{23}}}{x_3}{x_4}{\copaireC{c_{14}}{x_1}{x_2}{c_{24}}} $. To this effect, we need a last ingredient. We introduce a syntax of {\it patterns}, and we redefine the syntax of values, as follows: \begin{center} \fbox{${\cal V}::=x \,\mbox{\large\boldmath$\mid$}\, e^\bullet\quad\quad\quad V::=p\:\patc{\Subi{i}{{\cal V}_i}}{i\in p} \quad\quad\quad\quad p::= x \,\mbox{\large\boldmath$\mid$}\, \alpha^\bullet \,\mbox{\large\boldmath$\mid$}\, (p,p) \,\mbox{\large\boldmath$\mid$}\, \inl{p} \,\mbox{\large\boldmath$\mid$}\, \inr{p}$} \end{center} where $i\in p$ is defined by: $$\seq{}{x\in x}\quad\seq{}{\alpha^\bullet\in\alpha^\bullet}\quad \seq{i\in p_1}{i\in (p_1,p_2)}\quad\seq{i\in p_2}{i\in(p_1,p_2)}\quad \seq{i\in p_1}{i\in\inl{p_1}}\quad\seq{i\in p_2}{i\in \inr{p_2}}$$ Moreover, ${\cal V}_i$ must be of the form $y$ (resp. $e^\bullet$) if $i=x$ (resp. $i=\alpha^\bullet$).
Patterns are required to be linear, as well as the counterpatterns, for which the definition of ``linear'' is adjusted in the case $\copaireP{q_1,q_2}$, in which a variable can occur (but recursively linearly so) in both $q_1$ and $q_2$.
Note also that the reformulation of values is up to $\alpha$-conversion: for example, it is understood that $\alpha^\bullet\langle\Subi{\alpha^\bullet}{e^\bullet}\rangle = \beta^\bullet\langle\Subi{\beta^\bullet}{e^\bullet}\rangle$
We can now rephrase the logical reduction rules in terms of pattern/counterpattern interaction (whence the terminology), resulting in the following packaging of rules: \begin{center} $\seq{V=p\:\langle\ldots \Subi{x}{y},\ldots,\Subi{\alpha^\bullet}{e^\bullet},\ldots\rangle\quad\quad\Sub{C}{q}{p}\longrightarrow^* c}{ \coupe{\Vtov{V}}{\tilde\mu q.C} \longrightarrow \Submimp{c}{\ldots, \Subi{x}{y},\ldots,\Subi{\alpha}{e},\ldots} }$ \end{center} where $c\{\sigma\}$ is the usual, implicit substitution, and where $c$ (see the next proposition) is the normal form of $\Sub{C}{q}{p}$ with respect to the following set of rules:
$$\begin{array}{l} \Subm{C}{\Subi{(q_1,q_2)}{(p_1,p_2)},\sigma}\longrightarrow \Subm{C}{\Subi{q_1}{p_1},\Subi{q_2}{p_2},\sigma}\\ \Subm{\copaireC{C_1}{q_1}{q_2}{C_2}}{\Subi{\copaireP{q_1,q_2}}{\inl{p_1}},\sigma}\longrightarrow \Subm{C_1}{\Subi{q_1}{p_1},\sigma}\quad\quad \Subm{\copaireC{C_1}{q_1}{q_2}{C_2}}{\Subi{\copaireP{q_1,q_2}}{\inr{p_2}},\sigma}\longrightarrow \Subm{C_2}{\Subi{q_2}{p_2},\sigma}\\ \Subm{C}{\Subi{\alpha^\bullet}{\alpha^\bullet},\sigma}\longrightarrow \Subm{C}{\sigma}\quad\quad\quad \Subm{C}{\Subi{x}{x},\sigma}\longrightarrow \Subm{C}{\sigma}\\ \end{array}$$ Logically, this means that we now consider each formula as made of blocks of {\em synthetic} connectives. \begin{example} \begin{itemize} \item Patterns for $P=X\otimes(Y\oplus\notP{Q})$. Focusing on the right yields two possible proof searches: $$\seq{\lkV{\Gamma}{}{x'\set{{\cal V}_{x'}}:X}{\Delta} \quad\lkV{\Gamma}{}{y'\set{{\cal V}_{y'}}:Y}{\Delta}}{\lkV{\Gamma}{}{{ (x',\inl{y'})}\set{{\cal V}_{x'},{\cal V}_{y'}}:X\otimes(Y\oplus\notP{Q})}{\Delta}} \quad\quad \seq{\lkV{\Gamma}{}{x'\set{{\cal V}_{x'}}:X}{\Delta}\quad\lkV{\Gamma}{}{{\alpha'}^\bullet\set{{\cal V}_{{\alpha'}^\bullet}}:\notP{Q}}{\Delta}}{\lkV{\Gamma}{}{{ (x',\inr{{\alpha'}^\bullet})}\set{{\cal V}_{x'},{\cal V}_{{\alpha'}^\bullet}}:X\otimes(Y\oplus\notP{Q})}{\Delta}}$$ \item Counterpattern for $P=X\otimes(Y\oplus\notP{Q})$. The counterpattern describes the tree structure of $P$:
$$\seq{{ c_1:(\lkc{\Gamma\:,\:x:X\:,\:y:Y}{}{}{\Delta})}\quad\quad { c_2:(\lkc{\Gamma\:,\:x:X\:,\:\alpha^\bullet:\notP{Q}}{}{}{\Delta})}} {\copaireC{c_1}{y}{\alpha^\bullet}{c_2}:(\lkc{\Gamma\:,\:{(x,\copaireP{y,\alpha^\bullet})}:X\otimes(Y\oplus\notP{Q})}{}{}{\Delta})}$$ \end{itemize} We observe that the { leaves} of the decomposition are in one-to-one correspondence with the patterns $p$ for the (irreversible) decomposition of $P$ on the right: $$\begin{array}{l}\Sub{\copaireC{c_1}{y}{\alpha^\bullet}{c_2}}{{ q}}{{ p_1}} \longrightarrow^* c_1\quad\quad \Sub{\copaireC{c_1}{y}{\alpha^\bullet}{c_2}}{{ q}}{{ p_2}}\longrightarrow^* c_2 \end{array}$$ where ${ q=(x,\copaireP{y,\alpha^\bullet})}\;, \;{ p_1=(x,\inl{y})}\;, \;{ p_2=(x,\inr{\alpha^\bullet})}$. \end{example} This correspondence is general. We define two predicates $c\in C$ and $\suits{q}{p}$ (``$q$ is orthogonal to $p$'') as follows: $$\seq{}{c\in c}\quad\quad\seq{c\in C_1}{c\in\copaireC{C_1}{q_1}{q_2}{C_2}}\quad\quad \seq{c\in C_2}{c\in\copaireC{C_1}{q_1}{q_2}{C_2}}$$ \begin{center} \fbox{$\seq{}{\suits{x}{x}} \quad\quad \seq{}{\suits{\alpha^\bullet}{\alpha^\bullet}}\quad\quad \seq{\suits{q_1}{p_1}\;\;\suits{q_2}{p_2}}{\suits{(q_1,q_2)}{(p_1,p_2)}}\quad\quad \seq{\suits{q_1}{p_1}}{\suits{\copaireP{q_1,q_2}}{\inl{p_1}}}\quad\quad \seq{\suits{q_2}{p_2}}{\suits{\copaireP{q_1,q_2}}{\inr{p_2}}} \quad\quad\quad$} \end{center}
We can now state the correspondence result. \begin{proposition} \label{pattern-counterpatten-corr} Let { $C:(\lkc{\Xi\,,\,q:P}{}{}{\Delta})$} (as in the assumption of the typing rule for $\tilde\mu q.C$), and let $p$ be such that $q$ is orthogonal to $p$. Then the normal form $c$ of $\Sub{C}{q}{p}$ is a simple command, and the mapping $p\mapsto c$ ($q,C$ fixed) from $\setc{p}{\suits{q}{p}}$ to $\setc{c}{c\in C}$ is one-to-one and onto. \end{proposition} \noindent {\sc Proof}. The typing and the definition of orthogonality entail that in all intermediate $\Subm{C}{\sigma}$'s the substitution $\sigma$ has an item for each counterpattern in the sequent, and that reduction progresses. The rest is easy. (Note that a more general statement is needed for the induction to go through, replacing $\Xi\,,\,q:P$, ``$q$ orthogonal to $p$'', and $\Sub{C}{q}{p}$ with $\Xi\,,\,q_1:P_1\,,\,\ldots\,,\,q_n:P_n$, ``$q_i$ orthogonal to $p_i$ for $i=1,\ldots,n$'', and $\Subm{C}{\Subi{q_1}{p_1},\ldots\Subi{q_n}{p_n}}$, respectively.) \qed
\begin{figure}\label{big-step-mu-tilde}
\end{figure}
Thanks to this correspondence, we can quotient over the ``bureaucracy'' of commands, and we arrive at the calculus
described in Figure 4, together with its typing rules,
which we call {\em synthetic system }$\mathsf{L}$, or $\mathsf{L}_{\textrm{synth}}$.
The $\tilde\mu$ construct of $\mathsf{L}_{\textrm{synth}}$ is closely related to Zeilberger's higher-order abstract approach to focalisation in \cite{ZeilbergerCU}: indeed we can view $\setc{p\mapsto c}{\suits{q}{}{p}}$ as a function from patterns to commands. We actually prefer to see here a {\em finite} record whoses fields are the $p$'s orthogonal to $q$. There are only two reduction rules in $\mathsf{L}_{\textrm{synth}}$. the $\mu$-rule now expressed with implicit substitution and the $\tilde\mu^+$-rule, which
combines two familiar operations: select a field $p$ (like in object-oriented programming), and substitute (like in functional programming). The next proposition relates $\mathsf{L}_{\textrm{synth}}$ to $\mathsf{L}_{\textrm{foc}}$. \begin{proposition} \label{synth-foc-comp} The typing system of $\:\mathsf{L}_{\textrm{synth}}$ is complete\footnote{It is also easy to see that the underlying translation from $\mathsf{L}_{\textrm{foc}}$ to $\mathsf{L}_{\textrm{synth}}$ is reduction-reflecting (cf. Remark \ref{LK-LKQ-not-red-refl}).} with respect to $\mathsf{LKQ}$. \end{proposition}
\noindent {\sc Proof}. The completeness of $\mathsf{L}_{\textrm{synth}}$ with respect to the intermediate system above is an easy consequence of Proposition \ref{pattern-counterpatten-corr}. We are thus left with proving the completeness of the intermediate system. We define a rewriting relation between sets of sequents as follows: \begin{center} $\begin{array}{l} (\lkc{\Gamma,x:\notP{P}}{}{}{\Delta}),{\bf S} \rightsquigarrow (\lkc{\Gamma}{}{}{\alpha: P,\Delta}),{\bf S} \\ (\lkc{\Gamma,x:P_1\otimes P_2}{}{}{\Delta}),{\bf S} \rightsquigarrow (\lkc{\Gamma,x_1:P_1,x_2:P_2}{}{}{\Delta}),{\bf S} \\ (\lkc{\Gamma,x:P_1\oplus P_2}{}{}{\Delta}),{\bf S} \rightsquigarrow (\lkc{\Gamma,x_1:P_1}{}{}{\Delta}),(\lkc{\Gamma,x_2:P_2}{}{}{\Delta}),{\bf S} \end{array}$ \end{center}
(where $\alpha,x_1,x_2$ are fresh). A normal form for this notion of reduction is clearly a set of sequents of the form $\lkc{\Xi}{}{}{\Delta}$. It is also easy to see that
$\rightsquigarrow$ is confluent and (strongly) normalising.
In what follows, $\vdash_S$ (resp. $\vdash$) will signal the intermediate proof system (resp. $\mathsf{L}_{\textrm{foc}}$). The following property is easy to check.
If $(\Xi_1\vdash \Delta_1),\ldots,(\Xi_n\vdash\Delta_n)$ is the normal form of $(x_1:P_1,\ldots,x_m:P_m\vdash\Delta)$ for $\rightsquigarrow$ and if $c_i:(\Xi_i\vdash_S\Delta_i)$, then there exist $q_1,\ldots,q_m$ and a command $C:(q_1:P_1,\ldots,q_m:P_m\vdash_S\Delta)$ whose leaves are the $c_i$'s.
We prove the following properties together:
\noindent 1) If $c:(x_1:P_1,\ldots,x_m:P_m\vdash\Delta)$, then there exist $q_1,\ldots,q_m$ and $C$ such that $C:(q_1:P_1,\ldots,q_m:P_m\vdash_S\Delta)$.
\noindent 2) If $\lke{\Xi}{e:P}{}{\Delta}$, then there exists $e'$ such that $\lkes{\Xi}{e':P}{}{\Delta}$ (and similarly for expressions $v$).
The proof is by induction on a notion of size which is the usual one except that the size of a variable $x$ is not 1 but the size of its type. It is easy to check that the substitutions involved in Lemma \ref{e-subst-lemma} do not increase the size. The interesting case is $c=\coupe{v}{e}$. Let $(\Xi_1\vdash \Delta_1),\ldots,(\Xi_n\vdash\Delta_n)$ be the normal form of $(x_1:P_1,\ldots,x_m:P_m\vdash\Delta)$. Then, by repeated application of Lemma \ref{e-subst-lemma}, we get $v_1,\ldots,v_n$ and $e_1,\ldots,e_n$, and then by induction $v'_1,\ldots,v'_n$ and $e'_1,\ldots,e'_n$ which we assemble pairwise to form $\coupe{v'_1}{e'_1},\ldots, \coupe{v'_n}{e'_n}$, which in turn, as noted above, can be assembled in a tree $C:(q_1:P_1,\ldots,q_m:P_m\vdash_S\Delta)$. \qed
Putting together Propositions \ref{LKQ-complete} and \ref{synth-foc-comp}, we have proved that $\mathsf{L}_{\textrm{synth}}$ is complete with respect to $\mathsf{LK}$ for provability.
\begin{remark} \label{Boehm-ludique}
\begin{itemize} \item In the multiplicative case (no $C$, $\inl{V}$, $\inr{V}$, $\copaireP{q_1,q_2}$), there is a unique $p$ such that $\suits{q}{}{p}$, namely $q$, and the syntax boils down to \begin{center} $ {\cal V}::=x \,\mbox{\large\boldmath$\mid$}\, e^\bullet\quad
V::=p\:\patc{\Subi{i}{{\cal V}_i}}{i\in p}\quad { v::=x \,\mbox{\large\boldmath$\mid$}\, \tilde\mu q.\set{c}}\quad { c::=\coupe{\Vtov{V}}{\alpha}} $ \end{center} Compare with {\it B\"ohm trees}: $\begin{array}{lll} M::=\overbrace{\lambda\vec{x}.\underbrace{P}_{{ c}}}^{{ e}} &\quad\quad& P::=y\underbrace{\overbrace{M_1}^{{ {\cal V}}}\ldots \overbrace{M_n}^{{ {\cal V}}}}_{{ V}} \end{array}$. For example (cf. the CBN translation in Section \ref{encodings-sec}),
if $M_j$ translates to $e_j$, then $\lambda x_1x_2.xM_1M_2M_3$ translates to $\tilde\mu(\tilde{x_1}^\bullet,\tilde{x_2}^\bullet,y).\coupe{\Vtov{p\patc{\Subi{i}{{\cal V}_i}}{i\in p}}}{\tilde{x}}$, where $p=(\alpha_1^\bullet,\alpha_2^\bullet,\alpha_3^\bullet,z)$, ${\cal V}_{z_j}=(e_j)^\bullet$ , and ${\cal V}_z=y$. \item (for readers familiar with \cite{GirardLS})
Compare with the syntax for ludics presented in \cite{CuLLintroII}:
$\begin{array}{l} \overbrace{M}^{{ e}}::= \setc{\overbrace{J}^p \mapsto \lambda\setc{x_j}{j\in J}.P_J}{\overbrace{J\in{\cal N}}^{\suits{q}{}{p}}}\quad\quad \underbrace{P}_{ c}::=(x\cdot \underbrace{\overbrace{I}^p)\setc{M_i}{i\in I}}_{ V} \,\mbox{\large\boldmath$\mid$}\, \Omega \,\mbox{\large\boldmath$\mid$}\, \maltese \end{array}$
\end{itemize} \end{remark}
\section{Conclusion} \label{conclusion-sec}
We believe that Curien-Herbelin's syntactic kit, which we could call {\em system $\mathsf{L}$} for short, provides us with a robust infrastructure for proof-theoretical investigations, and for applications in formal studies in operational semantics. Thus, the work presented here is faithful to the spirit of Herbelin's Habilitation Thesis \cite{HerbelinHabil}, where he advocated an incremental approach to connectives, starting from a pure control kernel.
On the proof-theoretical side, we note, with respect to the original setting of ludics \cite{GirardLS}, that a pattern $p$ is more precise than a ramification $I$ (finite tree of subaddresses vs a set of immediate subaddresses). We might use this additional precision to design a version of ludics where axioms are first-class rather than treated as infinite expansions.
On the side of applications to programming language semantics, the good fit between abstract machines and our syntax $\mathsf{L}_{\textrm{foc}}$ makes it a good candidate for being used as an intermediate language appropriate to reason about the correctness of abstract machines (see also \cite{Levy2004}). In this spirit, in order to account for languages with mixed call-by-value / call-by-name features, one may give a truly bilateral presentation of $\mathsf{L}_{\textrm{foc}}$ that freely mixes positive and negative formulas like in Girard's $\mathsf{LC}$ \cite{GirardLC}.\footnote{See also \cite{MurthyLC} for an early analysis of the computational meaning of $\mathsf{LC}$ from a programming language perspective.} Such a system is presented in the long version of \cite{Munch2009}.
Finally, we wish to thank Bob Harper, Hugo Herbelin, and Olivier Laurent for helpful discussions.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\begin{abstract}
{\em Ropelength} and {\em embedding thickness} are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of $n$-component links in terms of the Milnor linking numbers. The main goal of the current paper is to provide such estimates, and thus generalizing the known linking number bound. In the process, we collect several facts about finite type invariants and ropelength/crossing number of knots. We give examples of families of knots, where such estimates behave better than the well known knot--genus estimate.
\end{abstract}
\maketitle
\section{Introduction}\label{S:intro}
Given an $n$--component link (we assume class $C^1$ embeddings) in $3$--space
\begin{equation}\label{eq:n-link}
L:S^1\sqcup\ldots \sqcup S^1\longrightarrow \mathbb{R}^3,\qquad L=(L_1, L_2,\ldots, L_n),\quad L_i=L|_{\text{the $i$'th circle}},
\end{equation} its {\em ropelength} $\operatorname{\mathsf{rop}}(L)$ is the ratio $\operatorname{\mathsf{rop}}(L)=\frac{\ell(L)}{r(L)}$ of length $\ell(L)$, which is a sum of lengths of individual components of $L$, to {\em reach} or {\em thickness}: $r(L)$, i.e. the largest radius of the tube embedded as a normal neighborhood of $L$. The {\em ropelength within the isotopy class} $[L]$ of $L$ is defined as
\begin{equation}\label{eq:Rop-def}
\operatorname{\mathsf{Rop}}(L)=\inf_{L'\in [L]} \operatorname{\mathsf{rop}}(L'),\qquad \operatorname{\mathsf{rop}}(L')=\frac{\ell(L')}{r(L')},
\end{equation}
(in \cite{Cantarella-Kusner-Sullivan:2002} it is shown that the infimum is achieved within $[L]$ and the minimizer is of class $C^{1,1}$).
A related measure of complexity, called {\em embedding thickness} was introduced recently in \cite{Freedman-Krushkal:2014}, in the general context of embeddings' complexity. For links, the embedding thickness $\tau(L)$ of $L$ is given by a value of its reach $r(L)$ assuming that $L$ is a subset of the unit ball $B_1$ in $\mathbb{R}^3$ (note that any embedding can be scaled and translated to fit in $B_1$). Again, the embedding thickness of the isotopy class $[L]$ is given by
\begin{equation}\label{eq:thck-def}
\mathcal{T}(L)=\sup_{L'\in [L]} \tau(L').
\end{equation}
For a link $L\subset B_1$, the volume of the embedded tube of radius $\tau(L)$ is $\pi \ell(L)\tau(L)^2$, \cite{Gray:2004} and the tube is contained in the ball of radius $r=2$, yielding
\begin{equation}\label{eq:t(L)-rop(L)}
\operatorname{\mathsf{rop}}(L)=\frac{\pi\ell(L)\tau(L)^2}{\pi\tau(L)^3}\leq \frac{\frac 43 \pi 2^3}{\pi\tau(L)^3},\quad \Rightarrow\quad \tau(L)\leq \Bigl( \frac{11}{\operatorname{\mathsf{rop}}(L)}\Bigr)^{\frac 13}.
\end{equation}
In turn a lower bound for $\operatorname{\mathsf{rop}}(L)$ gives an upper bound for $\tau(L)$ and vice versa. For other measures of complexity of embeddings such as distortion or Gromov-Guth thickness see e.g. \cite{Gromov:1983}, \cite{Gromov-Guth:2012}.
Bounds for the ropelength of knots, and in particular the lower bounds, have been studied by many researchers, we only list a small fraction of these works here \cite{Buck-Simon:1999,Buck-Simon:2007,Cantarella-Kusner-Sullivan:2002,Diao-Ernst-Janse-van-Rensburg:1999,Diao-Ernst:2007,Diao:2003,Ernst-Por:2012,Litherland-Simon-Durumeric-Rawdon:1999,Rawdon:1998, Ricca-Maggioni:2014, Maggioni-Ricca:2009, Ricca-Moffatt:1992}. Many of the results are applicable directly to links, but the case of links is treated in more detail by Cantarella, Kusner and Sullivan \cite{Cantarella-Kusner-Sullivan:2002} and in the earlier work of Diao, Ernst, and Janse Van Rensburg \cite{Diao-Ernst-Janse-Van-Rensburg:2002} concerning the estimates in terms of the pairwise linking number. In \cite{Cantarella-Kusner-Sullivan:2002}, the authors introduce a cone surface technique and show the following estimate, for a link $L$ (defined as in \eqref{eq:n-link}) and a given component $L_i$ \cite[Theorem 11]{Cantarella-Kusner-Sullivan:2002}:
\begin{equation}\label{eq:len(L_i)-lk} \operatorname{\mathsf{rop}}(L_i)\geq 2\pi+2\pi\sqrt{\operatorname{\mathsf{Lk}}(L_i,L)}, \end{equation} where $\operatorname{\mathsf{Lk}}(L_i,L)$ is the maximal total linking number between $L_i$ and the other components of $L$. A stronger estimate was obtained in \cite{Cantarella-Kusner-Sullivan:2002} by combining the Freedman and He \cite{Freedman-He:1991} asymptotic crossing number bound for energy of divergence free fields and the cone surface technique as follows
\begin{equation}\label{eq:len(L_i)-Ac-gen} \operatorname{\mathsf{rop}}(L_i)\geq 2\pi+2\pi\sqrt{\operatorname{\mathsf{Ac}}(L_i,L)},\qquad \operatorname{\mathsf{rop}}(L_i)\geq 2\pi+2\pi\sqrt{2g(L_i,L)-1}, \end{equation} where $\operatorname{\mathsf{Ac}}(L_i,L)$ is the {\em asymptotic crossing number} (c.f. \cite{Freedman-He:1991}) and the second inequality is a consequence of the estimate $\operatorname{\mathsf{Ac}}(L_i,L)\geq 2g(L_i,L)-1$, where $g(L_i,L)$ is a minimal genus among surfaces embedded in $\mathbb{R}^3\setminus\{L_1\cup\ldots\cup\widehat{L_i}\cup\ldots\cup L_n\}$, \cite[p. 220]{Freedman-He:1991} (in fact, Estimate \eqref{eq:len(L_i)-Ac-gen} subsumes Estimate \eqref{eq:len(L_i)-lk} since $\operatorname{\mathsf{Ac}}(L_i,L)\geq \operatorname{\mathsf{Lk}}(L_i,L)$).
A relation between $\operatorname{\mathsf{Ac}}(L_i,L)$ and the higher linking numbers of Milnor, \cite{Milnor:1954, Milnor:1957} is unknown and appears difficult. The following question, concerning the embedding thickness, is stated in \cite[p. 1424]{Freedman-Krushkal:2014}:
\begin{prob}\label{q:F-K}
Let $L$ be an $n$-component link which is Brunnian (i.e. almost trivial in the sense of Milnor \cite{Milnor:1954}). Let $M$ be the maximum value among Milnor's $\bar{\mu}$-invariants with distinct indices i.e. among $|\bar{\mu}_{\mathtt{I};j}(L)|$. Is there a bound
\begin{equation}\label{eq:thk-FK}
\operatorname{\tau}(L) \leq c_n M^{-\frac{1}{n}},
\end{equation}
for some constant $c_n > 0$, independent of the link $L$? Is there a bound on the crossing number $\operatorname{\mathsf{Cr}}(L)$ in terms of $M$? \end{prob} \noindent Recall that the Milnor $\bar{\mu}$--invariants $\{\bar{\mu}_{\mathtt{I};j}(L)\}$ of $L$, are indexed by an ordered tuple $(\mathtt{I};j)=(i_1,i_2,\ldots,i_k;$ $j)$ from the index set $\{1,\ldots,n\}$, there the last index $j$ has a special role (see below). If all the indexes in $(\mathtt{I};j)$ are distinct, $\{\bar{\mu}_{\mathtt{I};j}\}$ are link homotopy invariants of $L$ and are often referred to simply as {\em Milnor linking numbers} or {\em higher linking numbers}, \cite{Milnor:1954, Milnor:1957}. The definition $\{\bar{\mu}_{\mathtt{I};j}\}$ begins with coefficients $\mu_{\mathtt{I};j}$ of the Magnus expansion of the $j$th longitude of $L$ in $\pi_1(\mathbb{R}^3-L)$. Then
\begin{equation}\label{eq:bar-mu-Delta(I)} \overline{\mu}_{\mathtt{I};j}(L) \equiv \mu_{\mathtt{I};j}(L) \mod\ \Delta_\mu(\mathtt{I};j),\qquad\quad \Delta_\mu(\mathtt{I};j)=\gcd(\Gamma_\mu(\mathtt{I};j)). \end{equation} where $\Gamma_\mu(\mathtt{I};j)$ is a certain subset of lower order Milnor invariants, c.f. \cite{Milnor:1957}. Regarding $\bar{\mu}_{\mathtt{I};j}(L)$ as an element of $\mathbb{Z}_d=\{0,1,\ldots,d-1\}$, $d=\Delta_\mu(\mathtt{I};j)$ (or $\mathbb{Z}$, whenever $d=0$) let us set
\begin{equation}\label{eq:[mu]}
\Big[\bar{\mu}_{\mathtt{I};j}(L)\Big]:=\begin{cases}
\min\Bigl(\bar{\mu}_{\mathtt{I};j},d-\bar{\mu}_{\mathtt{I};j}\Bigr) &\ \text{for $d> 0$},\\
|\bar{\mu}_{\mathtt{I};j}| & \ \text{for $d= 0$}.
\end{cases} \end{equation} \noindent Our main result addresses Question \ref{q:F-K} for general $n$--component links (deposing of the Brunnian assumption) as follows.
\begin{thm}\label{thm:main}
Let $L$ be an $n$-component link $n\geq 2$ and $\bar{\mu}(L)$ one of its top Milnor linking numbers, then
\begin{equation}\label{eq:mthm-rop-cr}
\operatorname{\mathsf{rop}}(L)^{\frac 43}\geq \sqrt[3]{n} \Bigl(\Big[\bar{\mu}(L)\Big]\Bigr)^{\frac{1}{n-1}},\qquad \operatorname{\mathsf{Cr}}(L) \geq \tfrac{1}{3}(n-1) \Bigl(\Big[\bar{\mu}(L)\Big]\Bigr)^{\frac{1}{n-1}}.
\end{equation} \end{thm} \noindent In the context of Question \ref{q:F-K}, the estimate of Theorem \ref{thm:main} transforms, using \eqref{eq:t(L)-rop(L)}, as follows \[
\tau(L)\leq \Bigl(\frac{11}{\sqrt[4]{n}}\Bigr)^{\frac 13} M^{-\frac{1}{4(n-1)}} . \] Naturally, Question \ref{q:F-K} can be asked for knots and links and lower bounds in terms of finite type invariants in general. Such questions have been raised for instance in \cite{Cantarella:2009}, where the Bott-Taubes integrals \cite{Bott-Taubes:1994, Volic:2007} have been suggested as a tool for obtaining estimates. \begin{prob}\label{q:F-K-general}
Can we find estimates for ropelength of knots/links, in terms of their finite type invariants? \end{prob} In the remaining part of this introduction let us sketch the basic idea behind our approach to Question \ref{q:F-K-general}, which relies on the relation between the finite type invariants and the crossing number.
Note that since $\operatorname{\mathsf{rop}}(K)$ is scale invariant, it suffices to consider {\em unit thickness knots}, i.e. $K$ together with the unit radius tube neighborhood (i.e. $r(K)=1$). In this setting, $\operatorname{\mathsf{rop}}(K)$ just equals the {\em length} $\ell(K)$ of $K$. From now on we assume unit thickness, unless stated otherwise. In \cite{Buck-Simon:1999}, Buck and Simon gave the following estimates for $\ell(K)$, in terms of the crossing number $\operatorname{\mathsf{Cr}}(K)$ of $K$:
\begin{equation}\label{eq:L(K)-BS} \ell(K)\geq \Bigr(\frac{4\pi}{11} \operatorname{\mathsf{Cr}}(K)\Bigl)^{\frac 34},\qquad \ell(K)\geq 4\sqrt{\pi \operatorname{\mathsf{Cr}}(K)}. \end{equation} Clearly, the first estimate is better for knots with large crossing number, while the second one can be sharper for low crossing number knots (which manifests itself for instance in the case of the trefoil). Recall that $\operatorname{\mathsf{Cr}}(K)$ is a minimal crossing number over all possible knot diagrams of $K$ within the isotopy class of $K$. The estimates in \eqref{eq:L(K)-BS} are a direct consequence of the ropelength bound for the {\em average crossing number}\footnote{i.e. an average of the crossing numbers of diagrams of $K$ over all projections of $K$, see Equation \eqref{eq:aov-acr}.} $\operatorname{\mathsf{acr}}(K)$ of $K$ (proven in \cite[Corollary 2.1]{Buck-Simon:1999}) i.e.
\begin{equation}\label{eq:rop-BS-acr} \ell(K)^{\frac 43}\geq \frac{4\pi}{11} \operatorname{\mathsf{acr}}(K),\qquad \ell(K)^{2}\geq 16\pi \operatorname{\mathsf{acr}}(K). \end{equation}
In Section \ref{S:links_ropelength}, we obtain an analog of \eqref{eq:L(K)-BS} for $n$--component links ($n\geq 2$) in terms of the {\em pairwise crossing number}\footnote{see \eqref{eq:PCr(L)} and Corollary \ref{cor:rop-PCr}, generally $\operatorname{\mathsf{PCr}}(L)\leq \operatorname{\mathsf{Cr}}(L)$, as the individual components can be knotted.} $\operatorname{\mathsf{PCr}}(L)$, as follows \begin{equation}\label{eq:PCr-rop-3/4-2} \ell(L)\geq \frac{1}{\sqrt{n-1}}\Bigl(\tfrac 32 \operatorname{\mathsf{PCr}}(L)\Bigr)^{\frac 34},\qquad \ell(L)\geq \frac{n\sqrt{16\pi}}{\sqrt{n^2-1}}\Bigl(\operatorname{\mathsf{PCr}}(L)\Bigr)^{\frac 12}. \end{equation}
For low crossing number knots, the Buck and Simon bound \eqref{eq:L(K)-BS} was further improved by Diao\footnote{More precisely: $16\pi \operatorname{\mathsf{Cr}}(K) \leq \ell(K)(\ell(K)-17.334)$ \cite{Diao:2003}.} \cite{Diao:2003}, \begin{equation}\label{eq:L(K)-D} \ell(K)\geq \tfrac{1}{2} \bigl(d_0 +\sqrt{d^2_0+64\pi \operatorname{\mathsf{Cr}}(K)}\bigr),\qquad d_0=10-6(\pi+\sqrt{2})\approx 17.334. \end{equation}
On the other hand, there are well known estimates for $\operatorname{\mathsf{Cr}}(K)$ in terms of finite type invariants of knots. For instance, \begin{equation}\label{eq:c_2-cr-LW-PV}
\tfrac{1}{4}\operatorname{\mathsf{Cr}}(K)(\operatorname{\mathsf{Cr}}(K)-1)+\tfrac{1}{24}\geq |c_2(K)|, \qquad \tfrac{1}{8}(\operatorname{\mathsf{Cr}}(K))^2\geq |c_2(K)|. \end{equation} Lin and Wang \cite{Lin-Wang:1996} considered the second coefficient of the Conway polynomial $c_2(K)$ (i.e. the first nontrivial type $2$ invariant of knots) and proved the first bound in \eqref{eq:c_2-cr-LW-PV}. The second estimate of \eqref{eq:c_2-cr-LW-PV} can be found in Polyak--Viro's work \cite{Polyak-Viro:2001}. Further, Willerton, in his thesis \cite{Willerton:2002} obtained estimates for the ``second'', after $c_2(K)$, finite type invariant $V_3(K)$ of type $3$, as \begin{equation}\label{eq:V_3-cr}
\tfrac{1}{4}\operatorname{\mathsf{Cr}}(K)(\operatorname{\mathsf{Cr}}(K)-1)(\operatorname{\mathsf{Cr}}(K)-2)\geq |V_3(K)|. \end{equation}
In the general setting, Bar-Natan \cite{Bar-Natan:1995} shows that if $V(K)$ is a type $n$ invariant then $|V(K)| = O(\operatorname{\mathsf{Cr}}(K)^n)$. All these results rely on the arrow diagrammatic formulas for Vassiliev invariants developed in the work of Goussarov, Polyak and Viro \cite{Goussarov-Polyak-Viro:2000}.
Clearly, combining \eqref{eq:c_2-cr-LW-PV} and \eqref{eq:V_3-cr} with \eqref{eq:L(K)-BS} or \eqref{eq:L(K)-D}, immediately yields lower bounds for ropelength in terms of a given Vassiliev invariant. One may take these considerations one step further and extend the above estimates to the case of the $2n$\textsuperscript{th} coefficient of the Conway polynomial $c_{2n}(K)$, with the help of arrow diagram formulas for $c_{2n}(K)$, obtained recently in \cite{Chmutov-Duzhin-Mostovoy:2012, Chmutov-Khoury-Rossi:2009}. In Section \ref{S:knots}, we follow the Polyak--Viro's argument of \cite{Polyak-Viro:2001} to obtain
\begin{thm}\label{thm:conway}
Given a knot $K$, we have the following crossing number estimate
\begin{equation}\label{eq:crn_conway}
\operatorname{\mathsf{Cr}}(K) \geq \bigl((2n)! |c_{2n}(K)|\bigl)^{\frac{1}{2n}}\geq\frac{2n}{3} |c_{2n}(K)|^{\frac{1}{2n}}.
\end{equation}
\end{thm}
\noindent Combining \eqref{eq:crn_conway} with Diao's lower bound \eqref{eq:L(K)-D} one obtains
\begin{cor}
For a unit thickness knot $K$,
\begin{equation}\label{eq:rop_conway}
\ell(K) \geq \tfrac{1}{2} \Bigl(d_0 +\bigl(d^2_0+ \tfrac{128 n \pi}{3} |c_{2n}(K)|^{\frac{1}{2n}}\bigr)^{\frac 12}\Bigr).
\end{equation}
\end{cor}
Recall that a somewhat different approach to ropelength estimates is presented in \cite{Cantarella-Kusner-Sullivan:2002}, where the authors introduce a cone surface technique, which combined with the asymptotic crossing number, $\operatorname{\mathsf{Ac}}(K)$, bound of Freedman and He, \cite{Freedman-He:1991} gives
\begin{equation}\label{eq:L(K)-AC-g(K)}
\ell(K)\geq 2\pi+2\pi\sqrt{\operatorname{\mathsf{Ac}}(K)},\qquad \ell(K)\geq 2\pi+2\pi\sqrt{2 g(K)-1},
\end{equation}
where the second bound follows from the knot genus estimate of \cite{Freedman-He:1991}: $\operatorname{\mathsf{Ac}}(K)\geq 2 g(K)-1$.
When comparing Estimate \eqref{eq:L(K)-AC-g(K)} and \eqref{eq:rop_conway}, in favor of Estimate \eqref{eq:rop_conway}, we may consider a family of {\em pretzel knots}: $P(a_1,\ldots,a_n)$ where $a_i$ is the number of signed crossings in the $i$th tangle of the diagram, see Figure \ref{fig:pretzel_knots}. Additionally, for a diagram $P(a_1,\ldots,a_n)$ to represent a knot one needs to assume either both $n$ and all $a_i$ are odd or one of the $a_i$ is even, \cite{Kawauchi:1996}.
\begin{wrapfigure}{l}{7cm}
\centering
\includegraphics[width=.35\textwidth]{pretzel_knot.pdf}
\caption{$P(a_1,\ldots,a_n)$ pretzel knots.}\label{fig:pretzel_knots}
\end{wrapfigure}
Genera of selected subfamilies of pretzel knots are known, for instance \cite[Theorem 13]{Manchon:2012} implies
\begin{align*}
\qquad & \qquad & \qquad & \qquad & g(P(a,b,c)) =1,\\
\qquad & \qquad & \qquad & \qquad & c_2(P(a,b,c))=\frac 14 (ab+ac+bc+1),
\end{align*}
where $a$, $b$, $c$ are odd integers with the same sign
(for the value of $c_2(P(a,b,c))$ see a table in \cite[p. 390]{Manchon:2012}). Concluding, the lower bound in
\eqref{eq:rop_conway} can be made arbitrary large by letting $a,b,c\to+\infty$, while the lower bound
in \eqref{eq:L(K)-AC-g(K)} stays constant for any values of $a$, $b$, $c$, under consideration. Yet another\footnote{out of a few such examples given in \cite{Manchon:2012}.} example of a family of pretzel knots with constant genus one and arbitrarily large $c_2$--coefficient is $D(m,k)=P(m,\varepsilon,\stackrel{|k|-\text{times}}{\ldots},\varepsilon)$, $m>0$, $k$, where $\varepsilon=\frac{k}{|k|}$ is the sign of $k$ (e.g. $D(3,-2)=P(3,-1,-1)$). For any such $D(m,k)$, we have $c_2(D(m,k))=\frac{mk}{4}$. \begin{rem} {\rm A natural question can be raised about the the reverse situation: Can we find a family of knots with constant $c_{2n}$-coefficient (or any finite type invariant, see Remark \ref{rem:method-extended}), but arbitrarily large genus? For instance, there exist knots with $c_2=0$ and nonzero genus (such as $8_2$), in these cases \eqref{eq:L(K)-AC-g(K)} still provides a nontrivial lower bound.} \end{rem}
The paper is structured as follows: Section \ref{S:knots} is devoted to a review of arrow polynomials for finite type invariants and Kravchenko-Polyak tree invariants in particular, it also contains the proof Theorem \ref{thm:conway}. Section \ref{S:links_ropelength} contains information on the average overcrossing number for links and link ropelength estimates analogous to the ones obtained by Buck and Simon \cite{Buck-Simon:1999} (see Equation \eqref{eq:rop-BS-acr}). The proof of Theorem \ref{thm:main} is presented in Section \ref{S:proof-main}, together with final comments and remarks.
\begin{figure}
\caption{$5_2$ knot and its Gauss diagram (all crossings are positive).}
\label{fig:5_2-gauss}
\end{figure}
\section{Arrow polynomials and finite type invariants}\label{S:knots}
Recall from \cite{Chmutov-Duzhin-Mostovoy:2012}, the {\em Gauss diagram} of a knot $K$ is a way of representing signed overcrossings in a knot diagram, by arrows based on a circle ({\em Wilson loops}, \cite{Bar-Natan:1991}) with signs encoding the sign of the crossings (see Figure \ref{fig:5_2-gauss} showing the $5_2$ knot and its Gauss diagram). More precisely, the Gauss diagram $G_K$ of a knot $K:S^1\longrightarrow \mathbb{R}^3$ is constructed by marking pairs of points in the domain $S^1$, endpoints of a correspoinding arrow in $G_K$, which are mapped to crossings in a generic planar projection of $K$. The arrow always points from the under-- to the over--crossing and the orientation of the circle $S^1$ in $G_K$ agrees with the orientation of the knot.
Given a Gauss diagram $G$, the {\em arrow polynomials} of \cite{Goussarov-Polyak-Viro:2000, Polyak-Viro:1994} are defined simply as a signed count of selected subdiagrams in $G$. For instance the second coefficient of the Conway polynomial $c_2(K)$ is given by the signed count of $\vvcenteredinclude{.2}{c2-arrow.pdf}$ in $G$, denoted as \begin{equation}\label{eq:c_2-arrow}
c_2(K)=\langle\vvcenteredinclude{.2}{c2-arrow.pdf}, G\rangle=\sum_{\phi:\vvcenteredinclude{.1}{c2-arrow.pdf}\longrightarrow G} \text{\rm sign}(\phi),\qquad \text{\rm sign}(\phi)=\prod_{\alpha\in \vvcenteredinclude{.1}{c2-arrow.pdf}} \text{\rm sign}(\phi(\alpha)), \end{equation} where the sum is over all basepoint preserving graph embeddings $\{\phi\}$ of $\vvcenteredinclude{.2}{c2-arrow.pdf}$ into $G$, and the sign is a product of signs of corresponding arrows in $\phi(\vvcenteredinclude{.2}{c2-arrow.pdf})\subset G$. For example in the Gauss diagram of $5_2$ knot in Figure \ref{fig:5_2-gauss}, there are two possible embeddings of $\vvcenteredinclude{.2}{c2-arrow.pdf}$ into the diagram. One matches the pair of arrows $\{a,d\}$ and another pair $\{c,d\}$, since all crossings are positive we obtain $c_2(5_2)=2$. \begin{figure}
\caption{Turning a one-component chord diagram with a base point into an arrow diagram}
\label{fig:chord_to_arrow}
\end{figure}
For other even coefficients of the Conway polynomial, the work in \cite{Chmutov-Khoury-Rossi:2009} provides the following recipe for their arrow polynomials. Given $n>0$, consider any chord diagram $D$, on a single circle component with $2n$ chords, such as $\vvcenteredinclude{.25}{chord-1.pdf}$, $\vvcenteredinclude{.25}{chord-2.pdf}$, $\vvcenteredinclude{.25}{chord-3.pdf}$. A chord diagram $D$ is said to be a {\em $k$-component diagram}, if after parallel doubling of each chord according to \vcenteredinclude{k-component_chord.png}, the resulting curve will have $k$--components. For instance $\vvcenteredinclude{.25}{1-component.pdf}$ is a $1$--component diagram and $\vvcenteredinclude{.25}{3-component.pdf}$ is a $3$--component diagram. For the coefficients $c_{2n}$, only one component diagrams will be of interest and we turn a one-component chord diagram with a base point into an arrow diagram according to the following rule \cite{Chmutov-Khoury-Rossi:2009}: \begin{quote}
{\em Starting from the base point we move along the diagram with doubled chords. During this journey we pass both copies of each chord in opposite directions. Choose an arrow on each chord which corresponds to the direction of the first passage of the copies of the chord (see Figure \ref{fig:chord_to_arrow} for the illustration).} \end{quote}
We call, the arrow diagram obtained according to this method, the {\em ascending arrow diagram} and denote by $C_{2n}$ the sum of all based one-component ascending arrow diagrams with $2n$ arrows. For example $\vvcenteredinclude{.3}{C_2.pdf}$ and $C_{4}$ is shown below (c.f. \cite[p. 777]{Chmutov-Khoury-Rossi:2009}). \[ \vvcenteredinclude{.31}{C_4.pdf} \] In \cite{Chmutov-Khoury-Rossi:2009}, the authors show for $n \geq 1$, that the $c_{2n}(K)$ coefficient of the Conway polynomial of $K$ equals
\begin{equation}\label{eq:conway_coefficients}
c_{2n}(K) = \langle C_{2n}, G_K \rangle.
\end{equation}
\begin{repthm}{thm:conway}
Given a knot $K$, we have the following crossing number estimate
\begin{equation}\label{eq2:crn_conway}
\operatorname{\mathsf{Cr}}(K) \geq \bigl((2n)! |c_{2n}(K)|\bigl)^{\frac{1}{2n}}\geq\tfrac{2n}{3} |c_{2n}(K)|^{\frac{1}{2n}}.
\end{equation}
\end{repthm}
\begin{proof}
\noindent Given $K$ and its Gauss diagram $G_K$, let $X=\{1,2,\ldots, \operatorname{\mathsf{cr}}(K)\}$ index arrows of $G_K$ (i.e. crossings of a diagram of $K$ used to obtain $G_K$). For diagram term $A_i$ in the sum $C_{2n}=\sum_i A_i$, an embedding $\phi:A_i\longmapsto G_K$ covers a certain $2n$ element subset of crossings in $X$ that we denote by $X_{\phi}(i)$. Let $\mathcal{E}(i;G_K)$ be the set of all possible embeddings $\phi:A_i\longmapsto G_K$, and
\[
\mathcal{E}(G_K)=\bigsqcup_i \mathcal{E}(i;G_K).
\]
Note that $X_\phi(i)\neq X_\xi(j)$ for $i\neq j$ and $X_\phi(i)\neq X_\xi(i)$ for $\phi\neq \xi$, thus for each $i$ we have an injective map
\[
F_i:\mathcal{E}(i;G_K)\longmapsto \mathcal{P}_{2n}(X),\qquad F_i(\phi)=X_\phi(i),
\]
where $\mathcal{P}_{2n}(X)=\{\text{$2n$--element subsets of $X$}\}$. $F_i$ extends in an obvious way to the whole disjoint union $\mathcal{E}(G_K)$, as $F:\mathcal{E}(G_K)\longrightarrow \mathcal{P}_{2n}(X)$, $F=\sqcup_i F_i$ and remains injective. In turn, for every $i$ we have
\[
|\langle A_i,G_K\rangle|\leq \# \mathcal{E}(i;G_K)
\]
and therefore
\[
|\langle C_{2n},G_K\rangle|\leq \# \mathcal{E}(G_K)< \#\mathcal{P}_{2n}(X)={\operatorname{\mathsf{cr}}(K) \choose 2n}.
\]
If $\operatorname{\mathsf{cr}}(K)<2n$ then the left hand side vanishes.
Since ${\operatorname{\mathsf{cr}}(K) \choose 2n}\leq \frac{\operatorname{\mathsf{cr}}(K)^{2n}}{(2n)!}$, we obtain
\[
|c_{2n}(K)|\leq \frac{\operatorname{\mathsf{cr}}(K)^{2n}}{(2n)!}\quad \Rightarrow\quad \bigl((2n)! |c_{2n}(K)|\bigl)^{\frac{1}{2n}}\leq \operatorname{\mathsf{cr}}(K),
\] which gives the first part of \eqref{eq2:crn_conway}. Using the following upper lower bound for $m!$ (Stirling's approximation, \cite{Abramowitz:1964aa}) \[ m!\geq \sqrt{2\pi} m^{m+\frac 12} e^{-m}, \] applying , $e^{-1}\geq \frac{1}{3}$, $\bigl(\sqrt{2\pi}\bigr)^{\frac{1}{m}}\geq 1$, $\bigl(m^{m+\frac 12}\bigr)^{\frac{1}{m}}\geq m \bigl(\sqrt{m}\bigr)^{\frac{1}{m}}\geq m$, yields \begin{equation}\label{eq:stirling2} \bigl(m!\bigr)^{\frac{1}{m}}\geq \bigl(\sqrt{2\pi} (m)^{m+\frac 12} e^{-m}\bigr)^{\frac{1}{m}} \geq \frac{m}{3}, \end{equation} for $m=2n$ one obtains the second part of \eqref{eq2:crn_conway}. \end{proof}
\begin{figure}
\caption{Elementary trees $e$ and $\bar{e}$ and the $Z_{2;1}$ arrow polynomial. }
\label{fig:elem-tree}
\end{figure}
Next, we turn to arrow polynomials for Milnor linking numbers. In \cite{Kravchenko-Polyak:2011}, Kravchenko and Polyak introduced tree invariants of string links and established their relation to Milnor linking numbers via the skein relation of \cite{Polyak:2000}. In the recent paper, the authors\footnote{consult \cite{Kotorii:2013} for a related result.} \cite{Komendarczyk-Michaelides:2016} showed that the arrow polynomials of Kravchenko and Polyak, applied to Gauss diagrams of closed based links, yield certain $\bar{\mu}$--invariants (as defined in \eqref{eq:bar-mu-Delta(I)}). For a purpose of the proof of Theorem \ref{thm:main}, it suffices to give a recursive definition, provided in \cite{Komendarczyk-Michaelides:2016}, for the arrow polynomial of $\bar{\mu}_{23\ldots n;1}(L)$ denoted by $Z_{n;1}$. Changing the convention, adopted for knots, we follow \cite{Kravchenko-Polyak:2011}, \cite{Komendarczyk-Michaelides:2016} and use vertical segments (strings) oriented downwards in place of circles (Wilson loops) as components. \begin{figure}
\caption{Obtaining a term in $Z_{3;1}$ via stacking $e$ on the second component of $Z_{2;1}$, i.e. $Z_{2;1}\prec_2 e$. }
\label{fig:Z_2;1-to-term-Z_3;1}
\end{figure} The polynomial $Z_{n;1}$ is obtained inductively from~$Z_{n-1;1}=\sum_k \pm A_k$ by expanding each term $A_k$ of $Z_{n;1}$ through stacking elementary tree diagrams $e$ and $\bar{e}$, shown in Figure \ref{fig:elem-tree}, the sign of a resulting term is determined accordingly. The stacking operation is denoted by $\prec_i$, where $i=1,\ldots, n$ tells which component is used for stacking. Figure \ref{fig:Z_2;1-to-term-Z_3;1} shows $Z_{2;1}\prec_2 e$. The inductive procedure is defined as follows: \begin{itemize}
\item[$(i)$] $Z_{2;1}$ is shown in Figure \ref{fig:elem-tree}(right).
\item[$(ii)$] For each term $A_k$ in $Z_{n-1;1}$ produce terms in $Z_{n;1}$ by stacking\footnote{Note that $\bar{e}$ is not allowed to be stacked on the first component.} $e$ and $\bar{e}$ on each component i.e. $A_k\prec_i e$ for $i=1,\ldots, n$ and $A_k\prec_i \bar{e}$ for $i=2,\ldots, n$, see Figure \ref{fig:Z_2;1-to-term-Z_3;1}. Eliminate isomorphic (duplicate) diagrams.
\item[$(iii)$] The sign of each term in $Z_{n;1}$ equals to $(-1)^q$, $q=$number of arrows pointing to the right. \end{itemize}
As an example consider $Z_{3;1}$; we begin with the initial tree $Z_{2;1}$, and expand by stacking $e$ and $\bar{e}$ on the strings of $Z_{2;1}$, this is shown in Figure \ref{fig:Z_2;1-to-Z_3;1}, we avoid stacking $\bar{e}$ on the first component (called the {\em trunk}, \cite{Komendarczyk-Michaelides:2016}). Thus $Z_{3;1}$ is obtained as $A+B-C$, where $A=Z_{2;1}\prec_2 e$, $B=Z_{2;1}\prec_1 e$, and $C=Z_{2;1}\prec_2 \bar{e}$.
\begin{figure}
\caption{$Z_{3;1}=A+B-C$ obtained from $Z_{2;1}$ via $(i)$--$(iii)$. }
\label{fig:Z_2;1-to-Z_3;1}
\end{figure}
\noindent Given $Z_{n;1}$, the main result of \cite{Komendarczyk-Michaelides:2016} (see also \cite{Kotorii:2013} for a related result) yields the following formula \begin{equation}\label{eq:bar-mu-Z_n;1} \overline{\mu}_{n;1}(L) \equiv \langle Z_{n;1}, G_L\rangle \mod\ \Delta_\mu(n;1). \end{equation} where $\overline{\mu}_{n;1}(L):=\overline{\mu}_{2\ldots n;1}(L)$, $G_L$ a Gauss diagram of an $n$--component link $L$, and the indeteminacy $\Delta_\mu(n;1)$ is defined in \eqref{eq:bar-mu-Delta(I)}. Recall that $\langle Z_{n;1}, G_L\rangle=\sum_k \pm \langle A_k, G_L\rangle$ ($Z_{n;1}=\sum_k \pm A_k$) where $\langle A_k, G_L\rangle=\sum_{\phi:A_k\longrightarrow G_L} \text{\rm sign}(\phi)$ is a signed count of subdiagrams isomorphic to $A_k$ in $G_L$.
For $n = 2$, we obtain the usual linking number \begin{equation}\label{eq:lk-gauss} \bar{\mu}_{2;1}(L)=\langle Z_{2;1}, G_L\rangle = \Bigl\langle\vvcenteredinclude{0.23}{Z_2_1.pdf}, G_L\Bigr\rangle. \end{equation} For $n=3$ and $n=4$ the arrow polynomials can be obtained following the stacking procedure \[ \begin{split}
\bar{\mu}_{3;1}(L) & =\langle Z_{3;1}, G_L\rangle \mod \gcd\{\bar{\mu}_{2;1}(L), \bar{\mu}_{3;1}(L), \bar{\mu}_{3;2}(L)\},\\
&\quad Z_{3;1} = \vvcenteredinclude{0.3}{Z_23_1.pdf}, \end{split} \] and \[ \begin{split} \bar{\mu}_{4;1}(L) & =\langle Z_{4;1}, G_L\rangle \mod \Delta_\mu(4;1),\\
&\quad Z_{4;1} = \vvcenteredinclude{0.3}{Z_234_1-a.pdf} \\ &\qquad\qquad\vvcenteredinclude{0.3}{Z_234_1-b.pdf}\ . \end{split} \]
Given a formula for $\bar{\mu}_{n;1}(L)=\bar{\mu}_{23\ldots n;1}(L)$ all remaining $\bar{\mu}$--invariants with distinct indices can be obtained from the following permutation identity (for $\sigma\in \Sigma(1,\ldots,n)$) \begin{equation}\label{eq:mu-symmetry} \bar{\mu}_{\sigma(2)\sigma(3)\ldots\sigma(n);\sigma(1)}(L)=\bar{\mu}_{23\ldots n;1}(\sigma(L)),\qquad \sigma(L)=(L_{\sigma(1)},L_{\sigma(2)},\ldots, L_{\sigma(n)}). \end{equation} By \eqref{eq:bar-mu-Z_n;1}, \eqref{eq:mu-symmetry} and \eqref{eq:bar-mu-Delta(I)} we have \begin{equation}\label{eq:bar-mu-sigma-Z}
\bar{\mu}_{\sigma(2)\sigma(3)\ldots\sigma(n);\sigma(1)}(L)= \langle \sigma(Z_{n;1}), G_L\rangle \mod\ \Delta_\mu(\sigma(2)\sigma(3)\ldots\sigma(n);\sigma(1)), \end{equation} where $\sigma(Z_{n;1})$ is the arrow polynomial obtained from $Z_{n;1}$ by permuting the strings according to $\sigma$. \begin{rem}\label{rem:mu-cyclic}
{\em
One of the properties of $\bar{\mu}$--invariants is their cyclic symmetry, \cite[Equation (21)]{Milnor:1957}, i.e. given a cyclic permutation $\rho$, we have
\[
\bar{\mu}_{\rho(2)\rho(3)\ldots\rho(n);\rho(1)}(L)=\bar{\mu}_{23\ldots n;1}(L).
\]
} \end{rem}
\section{Overcrossing number of links}\label{S:links_ropelength}
We will denote by $D_L$ a regular diagram of a link $L$, and by $D_L(v)$, the diagram obtained by the projection of $L$ onto the plane normal to a vector\footnote{unless otherwise stated we assume that $v$ is generic and thus $D_L(v)$ to be a regular diagram.} $v\in S^2$. For a pair of components $L_i$ and $L_j$ in $L$, define the {\em overcrossing number} in the diagram and the {\em pairwise crossing number} of components $L_i$ and $L_j$ in $D_L$ i.e.
\begin{equation}\label{eq:ov-cr} \begin{split} \mathsf{ov}_{i,j}(D_L) & =\{\text{number of times $L_i$ overpasses $L_j$ in $D_L$}\}.\\ \mathsf{cr}_{i,j}(D_L) & =\{\text{number of times $L_i$ overpasses and underpasses $L_j$ in $D_L$}\}\\ & =\mathsf{ov}_{i,j}(D_L)+\mathsf{ov}_{j,i}(D_L)=\mathsf{cr}_{j,i}(D_L). \end{split} \end{equation}
In the following, we also use the {\em average overcrossing number} and {\em average pairwise crossing number} of components $L_i$ and $L_j$ in $L$, defined as an average over all $D_L(v)$, $v\in S^2$, i.e.
\begin{equation}\label{eq:aov-acr} \mathsf{aov}_{i,j}(L)=\frac{1}{4\pi}\int_{S^2} \mathsf{ov}_{i,j}(v)\, dv,\qquad \mathsf{acr}_{i,j}(L)=\frac{1}{4\pi}\int_{S^2} \mathsf{cr}_{i,j}(v)\, dv=2\, \mathsf{aov}_{i,j}(L) \end{equation}
In the following result is based on the work in \cite{Cantarella:2009} and \cite{Buck-Simon:1999}, the idea of using the rearrangement inequality comes from \cite{Cantarella:2009}. \begin{lem}\label{lem:aov-bound}
Given a unit thickness link $L$, and any $2$--component sublink $(L_i,L_j)$:
\begin{equation}\label{eq:aov-bound}
\min\bigl(\ell_i\ell^{\frac 13}_j,\ell_j\ell^{\frac 13}_i\bigr)\geq 3\,\mathsf{aov}_{i,j}(L),\qquad \ell_i\ell_j\geq 16\pi\,\mathsf{aov}_{i,j}(L),
\end{equation}
for $\ell_i=\ell(L_i)$, $\ell_j=\ell(L_j)$, the
length of $L_i$ and $L_j$ respectively. \end{lem}
\begin{proof}
Consider the Gauss map of $L_i=L_i(s)$ and $L_j=L_j(t)$:
\begin{equation*}
F_{i,j}:S^1\times S^1\longmapsto \text{\rm Conf}_2(\mathbb{R}^3)\longmapsto S^2,\quad F_{i,j}(s,t)=\frac{L_i(s)-L_j(t)}{||L_i(s)-L_j(t)||}.
\end{equation*}
\noindent If $v\in S^2$ is a regular value of $F_{i,j}$ (which happens for the set of full measure on $S^2$) then
\begin{equation*}
\mathsf{ov}_{i,j}(v)=\#\{\text{points in $F^{-1}_{i,j}(v)$} \}.
\end{equation*}
i.e. $\mathsf{ov}_{i,j}(v)$ stands for number of times the $i$--component of $L$ passes over the $j$--component, in the projection of $L$ onto the plane in $\mathbb{R}^3$ normal to $v$. As a direct consequence of Federer's coarea formula \cite{Federer:1969} (see e.g. \cite{Michaelides:2015} for a proof)
\begin{align} \label{eq:gauss-linking-int}
\int_{L_i\times L_j} |F^\ast_{i,j} \omega| & =\frac{1}{4\pi}\int_{S^1\times S^1} \frac{|\langle L_i(s)-L_j(t),L'_i(s), L'_j(t) \rangle|}{||L_i(s)-L_j(t)||^3} ds \, dt\\
& =\frac{1}{4\pi}\int_{S^2} \mathsf{ov}_{i,j}(v)\, dv, \label{eq:avg-overcrossing}
\end{align}
where $\omega = \frac{1}{4\pi}\bigl(x \,dy \wedge dz - y \, dx \wedge dz + z\, dx \wedge dy\bigr)$ is the normalized area form on the unit sphere in $\mathbb{R}^3$ and
\[
\langle v, w, z \rangle:=\det\bigl(v,w,z\bigr),\qquad\text{for}\quad v,w,z\in \mathbb{R}^3.
\]
Assuming the arc--length parametrization by $s \in [0, \ell_i]$ and $t \in [0, \ell_j]$ of the components we have $\|L'_i (s)\|=\|L'_j(t)
\|=1$ and therefore:
\begin{equation}\label{eq:quarter}
\Bigl|\frac{\langle L_i(s)-L_j(t), L'_i(s),L'_j(t) \rangle}{||L_i(s)-L_j(t)||^3}\Bigl|\leq \frac{1}{||L_i(s)-L_j(t)||^2}
\end{equation}
\noindent Combining Equations \eqref{eq:avg-overcrossing} and \eqref{eq:quarter} yields
\begin{equation}\label{eq:overcrossing}
\int^{\ell_j}_0\int^{\ell_i}_0 \frac{1}{||L_i(s) - L_j(t)||^2}ds\,dt=\int^{\ell_j}_0 I_i(L_j(t))dt,
\end{equation}
where
\[
I_i(p)=\int^{\ell_i}_0 \frac{1}{||L_i(s) - p||^2}ds=\int^{\ell_i}_0 \frac{1}{r(s)^2}ds,\quad r(s)=\|L_i(s) - p\|,
\]
is often called {\em illumination} of $L_i$ from the point $p\in \mathbb{R}^3$, c.f. \cite{Buck-Simon:1999}. Following the approach of \cite{Buck-Simon:1999}, and \cite{Cantarella:2009} we estimate $I_i(t)=I_i(p)$ for $p=L_j(t)$. Denote by $B_a(p)$ the ball at $p=L_j(t)$ of radius $a$, and $s(z)$ the length of a portion of $L_i$ within the spherical shell: $Sh(z)=B_z(p)\setminus B_1(p)$, $z\geq 1$. Note that, because the distance between $L_i$ and $L_j$ is at least $2$, the unit thickness tube about $L_i$ is contained entirely in $Sh(z)$ for big enough $z$. Clearly, $s(z)$ is nondecreasing. Since the volume of a unit thickness tube of length $a$ is $\pi a$, comparing the volumes we obtain
\begin{equation}\label{eq:vol-tube-shell}
\begin{split}
\pi s(z) & \leq \operatorname{Vol}(Sh(z))=\tfrac{4}{3}\pi\bigl(z^3-1^3\bigr), \ \text{and}\\
s(z) & \leq \tfrac{4}{3}\, z^3, \qquad \text{for}\quad z\geq 1.
\end{split}
\end{equation}
Next, using the monotone rearrangement $\bigl(\frac{1}{r^2}\bigr)^\ast$ of $\frac{1}{r^2}$, (Remark \ref{rem:rearrangement}):
\begin{equation}\label{eq:rng-ineq}
\Bigl(\frac{1}{r^2}\Bigr)^\ast(s)\leq (\tfrac{4}{3})^{\frac 23}\,s^{-\frac 23},
\end{equation}
and the monotone rearrangement inequality \cite{Lieb-Loss:2001}:
\begin{equation}\label{eq:I_i-monotone}
I_i(p)= \int^{\ell_i}_0 \frac{1}{r^2(s)}ds \leq \int^{\ell_i}_0 \Bigl(\frac{1}{r^2}\Bigr)^\ast(s)ds \leq \int^{\ell_i}_0 (\tfrac{4}{3})^{\frac 23}\,s^{-\frac 23} ds= 3(\tfrac{4}{3})^{\frac 23}\,\ell_i^{\frac 13}.
\end{equation}
\noindent Integrating \eqref{eq:I_i-monotone} with respect to the $t$--parameter, we obtain
\[
\mathsf{aov}(L)\leq \frac{1}{4\pi}\int^{\ell_j}_0\int^{\ell_i}_0 \frac{1}{||L_i(s) - L_j(t)||^2}ds\,dt\leq 3(\tfrac{4}{3})^{\frac 23}\tfrac{1}{4\pi}\, \ell_j\,\ell_i^{\frac 13}< \tfrac{1}{3} \ell_j\,\ell_i^{\frac 13}.
\]
Since the argument works for any choice of $i$ and $j$ estimates in Equation \eqref{eq:aov-bound} are proven. The second estimate in \eqref{eq:aov-bound} follows immediately from the fact that $\frac{1}{\|L_i(s)-L_j(t)\|^2}\leq \frac 14$. \end{proof}
\begin{rem}\label{rem:rearrangement}
{\rm Recall, that for a nonnegative real valued function $f$, (on $\mathbb{R}^n$) vanishing at infinity, the {\em rearrangement, $f^\ast$ of $f$} is given by
\[
f^\ast(x)=\int^\infty_0 \chi^\ast_{\{f > u\}}(x)\, du,
\]
where $\chi^\ast_{\{f > u\}}(x)=\chi_{B_\rho}(x)$ is the characteristic function of the ball $B_\rho$ centered at the origin, determined by the volume condition: $\operatorname{Vol}(B_\rho)=\operatorname{Vol}(\{x\ |\ f(x)>u\})$, see \cite[p. 80]{Lieb-Loss:2001} for further properties of the rearrangements. In particular, the rearrangement inequality states \cite[p. 82]{Lieb-Loss:2001}:
$\int_{\mathbb{R}^n} f(x)\,dx\leq \int_{\mathbb{R}^n} f^\ast(x)\,dx$. For one variable functions, we may use the interval $[0,\rho]$ in place of the ball $B_\rho$, then $f^\ast$ is a decreasing function on $[0,+\infty)$. Specifically, for $f(s)=\frac{1}{r^2(s)}=\frac{1}{\|L_i(s)-p\|^2}$, we have \[
\Bigl(\frac{1}{r^2}\Bigr)^\ast(s)=u,\qquad\text{for}\quad \operatorname{length}\bigl(\{x\ |\ u<\frac{1}{r^2(x)}\leq 1\}\bigr)=s, \]
where $\operatorname{length}\bigl(\{x\ |\ u<\frac{1}{r^2(x)}\leq 1\}\bigr)$ stands for the length of the portion of $L_i$ satisfying the given condition. Further, by the definition of $s(z)$, from the previous paragraph, and \eqref{eq:vol-tube-shell}, we obtain \[
s=\operatorname{length}\bigl(\{x\ |\ \frac{1}{r^2(x)}>u\}\bigr)=\operatorname{length}\bigl(\{x\ |\ 1\leq r(x)<\frac{1}{\sqrt{u}}\}\bigr)=s\bigl(\frac{1}{\sqrt{u}}\bigr)\leq \tfrac{4}{3}\, \bigl(\frac{1}{\sqrt{u}}\bigr)^3, \] and \eqref{eq:rng-ineq} as a result.
}
\end{rem}
\noindent From the Gauss linking integral \eqref{eq:gauss-linking-int}: \[
|\operatorname{\mathsf{Lk}}(L_i,L_j)|\leq \operatorname{\mathsf{aov}}_{i,j}(L), \]
thus we immediately recover the result of \cite{Diao-Ernst-Janse-Van-Rensburg:2002} (but with a specific constant): \begin{equation}\label{eq:aov-lk-bound}
3 |\operatorname{\mathsf{Lk}}(L_i,L_j)|\leq \min\bigl(\ell_i\ell^{\frac 13}_j,\ell_j\ell^{\frac 13}_i\bigr),\qquad 16\pi|\operatorname{\mathsf{Lk}}(L_i,L_j)|\leq \ell_i\ell_j. \end{equation} Summing up over all possible pairs: $i$, $j$ and using the symmetry of the linking number we have \[
6\sum_{i< j} |\operatorname{\mathsf{Lk}}(L_i,L_j)|=3\sum_{i\neq j} |\operatorname{\mathsf{Lk}}(L_i,L_j)|\leq \sum_{i\neq j} \ell_i\ell^{\frac 13}_j= (\sum_i \ell_i)(\sum_j \ell^{\frac 13}_j)-\sum_i \ell^{\frac 43}_i. \] From Jensen's Inequality \cite{Lieb-Loss:2001}, we know that $\frac{1}{n}(\sum_i \ell^{\frac 13}_i)\leq \bigl(\frac{1}{n}\sum_i \ell_i\bigr)^{\frac 13}$ and $\frac{1}{n}(\sum_i \ell^{\frac 43}_i)\geq \bigl(\frac{1}{n}\sum_i \ell_i\bigr)^{\frac 43}$, therefore \begin{equation}\label{eq:lk-jensen} (\sum_i \ell_i)(\sum_j \ell^{\frac 13}_j)-\sum_i \ell^{\frac 43}_i\leq n^{\frac 23} \operatorname{\mathsf{rop}}(L)\operatorname{\mathsf{rop}}(L)^{\frac 13}-n^{-\frac 13} \operatorname{\mathsf{rop}}(L)^{\frac 43}=\frac{n-1}{n^\frac 13}\operatorname{\mathsf{rop}}(L)^{\frac 43}. \end{equation} Analogously, using the second estimate in \eqref{eq:aov-lk-bound} and Jensen's Inequality, yields \[
32\pi \sum_{i< j} |\operatorname{\mathsf{Lk}}(L_i,L_j)|=16\pi \sum_{i\neq j} |\operatorname{\mathsf{Lk}}(L_i,L_j)|\leq \sum_{i\neq j} \ell_i\ell_j\leq (1-\frac{1}{n^2})\bigl(\sum_i \ell_i\bigr)^2. \] \begin{cor}\label{cor:rop-lk}
Let $L$ be an $n$-component link ($n\geq 2$), then \begin{equation}\label{eq:lk-rop-3/4-2}
\operatorname{\mathsf{rop}}(L)^{\frac 43}\geq \frac{6 n^{\frac{1}{3}}}{(n-1)}\sum_{i<j} |\operatorname{\mathsf{Lk}}(L_i,L_j)|,\qquad \operatorname{\mathsf{rop}}(L)^{2}\geq \frac{32\pi n^2 }{n^2-1}\sum_{i<j} |\operatorname{\mathsf{Lk}}(L_i,L_j)|. \end{equation} \end{cor}
In terms of growth of the pairwise linking numbers $|\operatorname{\mathsf{Lk}}(L_i,L_j)|$, for a fixed $n$, the above estimate performs better than the one in \eqref{eq:len(L_i)-lk}. One may also replace $\sum_{i<j} |\operatorname{\mathsf{Lk}}(L_i,L_j)|$ with the isotopy invariant \begin{equation}\label{eq:PCr(L)} \operatorname{\mathsf{PCr}}(L)=\min_{D_L} \Bigl(\sum_{i\neq j} \operatorname{\mathsf{cr}}_{i,j}(D_L)\Bigr), \end{equation} (satisfying $\operatorname{\mathsf{PCr}}(L)\leq \operatorname{\mathsf{Cr}}(L)$), we call the {\em pairwise crossing number} of $L$. This conclusion can be considered as an analog of the Buck and Simon estimate \eqref{eq:L(K)-BS} for knots.
\begin{cor}\label{cor:rop-PCr}
Let $L$ be an $n$-component link ($n\geq 2$), and $\operatorname{\mathsf{PCr}}(L)$ its pairwise crossing number, then
\begin{equation}\label{eq2:PCr-rop-3/4-2}
\operatorname{\mathsf{rop}}(L)^{\frac{4}{3}}\geq \frac{3 n^{\frac{1}{3}}}{(n-1)}\operatorname{\mathsf{PCr}}(L),\qquad \operatorname{\mathsf{rop}}(L)^2\geq \frac{16\pi n^2}{n^2-1}\operatorname{\mathsf{PCr}}(L).
\end{equation} \end{cor}
\section{Proof of Theorem \ref{thm:main}}\label{S:proof-main}
\noindent The following auxiliary lemma will be useful. \begin{lem}\label{lem:a_i-ineq}
Given nonnegative numbers: $a_1$, \ldots, $a_{N}$ we have for $k\geq 2$:
\begin{equation}\label{eq:prod-est}
\sum_{1\leq i_1<i_2<\ldots<i_{k}\leq N} a_{i_1}a_{i_2}\ldots a_{i_{k}}\leq \frac{1}{N^{k}} {N \choose k}\Bigl(\sum^{N}_{i=1} a_i\Bigr)^{k}.
\end{equation} \end{lem} \begin{proof}
It suffices to observe that for $a_i\geq 0$ the ratio $\Bigl(\sum_{1\leq i_1<i_2<\ldots<i_{k}\leq N} a_{i_1}a_{i_2}\ldots a_{i_{k}}\Bigr)/\Bigl(\sum^{N}_{i=1} a_i\Bigr)^{k}$ achieves its maximum for $a_1=a_2=\ldots=a_N$. \end{proof} \noindent Recall from \eqref{eq:[mu]} that $\bar{\mu}_{n;1}:=\bar{\mu}_{23\ldots n;1}$, and
\begin{equation}\label{eq2:[mu]} \Big[\bar{\mu}_{n;1}(L)\Big]:=\begin{cases} \min\Bigl(\bar{\mu}_{n;1}(L),d-\bar{\mu}_{n;1}(L)\Bigr) &\ \text{for $d> 0$},\\
|\bar{\mu}_{n;1}(L)| & \ \text{for $d= 0$} \end{cases}\qquad d=\Delta_\mu(n;1). \end{equation} For convenience, recall the statement of Theorem \ref{thm:main}.
\begin{repthm}{thm:main}
Let $L$ be an $n$-component link of unit thickness, and $\bar{\mu}(L)$ one of its top Milnor linking numbers, then
\begin{equation}\label{eq2:mthm-rop-cr}
\ell(L)\geq \sqrt[4]{n} \Bigl(\sqrt[n-1]{\Big[\bar{\mu}(L)\Big]}\Bigr)^{\frac 34},\qquad \operatorname{\mathsf{Cr}}(L) \geq \frac{1}{3}(n-1) \sqrt[n-1]{\Big[\bar{\mu}(L)\Big]}.
\end{equation} \end{repthm} \begin{proof} Let $G_L$ be a Gauss diagram of $L$ obtained from a regular link diagram $D_L$. Consider, any term $A$ of the arrow polynomial: $Z_{n;1}$ and index the arrows of $A$ by $(i_k,j_k)$, $k=1,\ldots,n-1$ in such a way that $i_k$ is the arrowhead and $j_k$ is the arrowtail, we have the following obvious estimate: \begin{equation}\label{eq:estimate_term-n}
|\bigl\langle A, G_L\bigr\rangle|\leq \prod^{n-1}_{k=1}\mathsf{ov}_{i_k,j_k}(D_L)\leq \prod^{n-1}_{k=1}\mathsf{cr}_{i_k,j_k}(D_L). \end{equation} Let $N={n \choose 2}$, since every term (a tree diagram) of $Z_{n;1}$ is uniquely determined by its arrows indexed by string components, ${N\choose n-1}$ gives an upper bound for the number of terms in $Z_{n;1}$. Using Lemma \ref{lem:a_i-ineq}, with $k=n-1$, $N$ as above and $a_k=\mathsf{cr}_{i_k,j_k}(D_L)$, $k=1,\ldots, N$, one obtains from \eqref{eq:estimate_term-n} \begin{equation}\label{eq:Z_n;1-cr}
|\langle Z_{n;1},G_L\rangle|\leq \frac{1}{N^{n-1}} {N \choose n-1}\Bigl(\sum_{i<j} \mathsf{cr}_{i,j}(D_L) \Bigr)^{n-1}. \end{equation} \begin{rem}\label{rem:general-arrow-poly}
{\em The estimate \eqref{eq:Z_n;1-cr} is valid for any arrow polynomial, in place of $Z_{n;1}$, which has arrows based on different components and no parallel arrows on a given component.} \end{rem} By \eqref{eq:bar-mu-Z_n;1}, we can find $k\in \mathbb{Z}$ such that $\langle Z_{n;1},G_L\rangle=\bar{\mu}_{n;1}+k\,d$. Since \[
\Big[\bar{\mu}_{n;1}(D_L)\Big]\leq |\bar{\mu}_{n;1}(D_L)+k\,d|=|\langle Z_{n;1},G_L\rangle|,\qquad\text{for all}\ k\in\mathbb{Z}, \] replacing $D_L$ with a diagram obtained by projection of $L$ in a generic direction $v\in S^2$, we rewrite the estimate \eqref{eq:Z_n;1-cr} as follows \begin{equation}\label{eq:alpha_n-cr} \alpha_n\sqrt[n-1]{\Big[\bar{\mu}_{n;1}(D_L(v))\Big]}\leq \sum_{i<j} \mathsf{cr}_{i,j}(v),\qquad \alpha_n=\Bigl(\frac{1}{ N^{n-1}} {N \choose n-1}\Bigr)^{\frac{-1}{n-1}}. \end{equation} Integrating over the sphere of directions and using invariance\footnote{both $\bar{\mu}_{n;1}$ and $d$ are isotopy invariants.} of $\Big[\bar{\mu}_{n;1}\Big]$ yields \[ 4\pi \alpha_n \sqrt[n-1]{\Big[\bar{\mu}_{n;1}(L)\Big]}\leq \sum_{i<j} \int_{S^2}\mathsf{cr}_{i,j}(v) dv. \] By Lemma \ref{lem:aov-bound}, we obtain \[ \alpha_n \sqrt[n-1]{\Big[\bar{\mu}_{n;1}(L)\Big]}\leq \sum_{i<j} \mathsf{acr}_{i,j}(L)= 2\sum_{i<j} \mathsf{aov}_{i,j}(L)\leq 2\sum_{i< j} \tfrac{1}{3}\min\bigl(\ell_i\ell^{\frac 13}_j,\ell_j\ell^{\frac 13}_i\bigr)\leq \tfrac{1}{3}\sum_{i \neq j} \ell_i\ell^{\frac 13}_j, \] since $\sum_{i< j} 2\min\bigl(\ell_i\ell^{\frac 13}_j,\ell_j\ell^{\frac 13}_i\bigr)\leq \sum_{i \neq j} \ell_i\ell^{\frac 13}_j$. As in derivation of \eqref{eq:lk-jensen} (see Corollary \ref{cor:rop-lk}), by Jensen Inequality: \begin{equation}\label{eq:rop-end} \operatorname{\mathsf{rop}}(L)^{\frac 43}\geq \frac{3\, n^{\frac 13}\, \alpha_n}{(n-1)} \sqrt[n-1]{\Big[\bar{\mu}_{n;1}(L)\Big]}. \end{equation} Now, let us estimate the constant $\alpha_n$. Note that \[ \frac{N^{n-1}}{{N \choose n-1}}=\frac{N^{n-1}}{N(N-1)\ldots (N-(n-1)+1)} (n-1)!\geq (n-1)!\ . \] Again, by Stirling's approximation (letting $m=n-1$ in \eqref{eq:stirling2}) we obtain for $n\geq 2$: \begin{equation}\label{eq:alpha_n-stirling} \alpha_n\geq \bigl((n-1)!\bigr)^{\frac{1}{n-1}}\geq \frac{n-1}{3}, \end{equation} thus \eqref{eq:rop-end} can be simplified to \begin{equation}\label{eq:rop-end2} \operatorname{\mathsf{rop}}(L)^{\frac 43}\geq \sqrt[3]{n} \sqrt[n-1]{\Big[\bar{\mu}_{n;1}(L)\Big]}, \end{equation} as claimed in the first inequality of Equation \eqref{eq2:mthm-rop-cr}. For a minimal diagram $D^{\min}_L$ of $L$: \[ \operatorname{\mathsf{Cr}}(L)\geq \sum_{i<j} \mathsf{cr}_{i,j}(D^{\min}_L), \]
thus the second inequality of \eqref{eq2:mthm-rop-cr} is an immediate consequence of \eqref{eq:alpha_n-cr}(with $D_L(v)$ replaced by $D^{\min}_L$) and \eqref{eq:alpha_n-stirling}. Using the permutation identity \eqref{eq:mu-symmetry} and the fact that $\operatorname{\mathsf{rop}}(\sigma(L))=\operatorname{\mathsf{rop}}(L)$ for any $\sigma\in\Sigma(1,\ldots,n)$, we may replace $\bar{\mu}_{n;1}(L)$ with any other\footnote{there are $(n-2)!$ different top Milnor linking numbers \cite{Milnor:1954}.} top $\bar{\mu}$--invariant of $L$. \end{proof}
In the case of almost trivial (Borromean) links $d=0$, and we may slightly improve the estimate in \eqref{eq:Z_n;1-cr} of the above proof, by using cyclic symmetry of $\bar{\mu}$--invariants pointed in \eqref{rem:mu-cyclic}. We have in particular
\begin{equation}\label{eq:mu-sum-cyclic}
n\,\bar{\mu}_{23\ldots n;1}(L)=\sum_{\rho, \text{$\rho$ is cyclic}} \bar{\mu}_{\rho(2)\rho(3)\ldots\rho(n);\rho(1)}(L)=\sum_{\rho, \text{$\rho$ is cyclic}} \langle \rho(Z_{n;1}),G_L\rangle.
\end{equation}
Since cyclic permutations applied to the terms of $Z_{n;1}$ produce distinct arrow diagrams\footnote{since the trunk of a tree diagram is unique, c.f. \cite{Kravchenko-Polyak:2011}, \cite{Komendarczyk-Michaelides:2016}.}, by Remark \ref{rem:general-arrow-poly}, we obtain the following bound
\begin{equation}\label{eq:rho-Z_n;1-cr}
n\,|\bar{\mu}_{n;1}(L)|\leq \sum_{\rho, \text{$\rho$ is cyclic}} |\langle \rho(Z_{n;1}),G_L\rangle|\leq \frac{1}{N^{n-1}} {N \choose n-1}\Bigl(\sum_{i<j} \mathsf{cr}_{i,j}(D_L) \Bigr)^{n-1}.
\end{equation}
Disregarding the Stirling's approximation, we have
\begin{equation}
\operatorname{\mathsf{rop}}(L)^{\frac 43}\geq \frac{3\sqrt[3]{n}\, \tilde{\alpha}_n}{(n-1)} \sqrt[n-1]{|\bar{\mu}_{n;1}(L)|},\quad \tilde{\alpha}_n=\Bigl(\frac{1}{n N^{n-1}} {N \choose n-1}\Bigr)^{\frac{-1}{n-1}},
\end{equation}
or using the second bound in \eqref{eq:aov-bound}
\[
\operatorname{\mathsf{rop}}(L)^2\geq 4^3\pi \tilde{\alpha}_n\Bigl(\frac{n^2}{n^2-1}\Bigr) \sqrt[n-1]{|\bar{\mu}_{n;1}(L)|}.
\]
In particular, for $n=3$, we have $N=3$ and $\tilde{\alpha}_3=3$ and the estimates read \begin{equation}\label{eq:rop-mu123}
\operatorname{\mathsf{rop}}(L)\geq \Bigl(5\sqrt[3]{3}\sqrt{|\bar{\mu}_{23;1}(L)|}\Bigr)^{\frac 34},\quad \operatorname{\mathsf{rop}}(L)\geq 6\sqrt{6\pi}\sqrt[4]{|\bar{\mu}_{23;1}(L)|}. \end{equation}
\noindent Since $6\sqrt{6\pi}\approx 26.049$, the second estimate is better for Borromean rings ($\mu_{23;1}=1$) and improves the linking number bound of \eqref{eq:len(L_i)-lk}: $6\pi\approx 18.85$, but fails short of the genus bound \eqref{eq:len(L_i)-Ac-gen}: $12\pi\approx 37.7$. Numerical simulations suggest that the ropelength of Borromean rings $\approx 58.05$, \cite{Cantarella-Kusner-Sullivan:2002, Buniy-Cantarella-Kephart-Rawdon:2014}.
\begin{rem}\label{rem:method-extended} {\em
This methodology can be easily extended to other families of finite type invariants of knots and links. For illustration, let us consider the third coefficient of the Conway polynomial i.e. $c_3(L)$ of a two component link $L$. The arrow polynomial $C_3$ of $c_3(L)$ is given as follows \cite[p. 779]{Chmutov-Khoury-Rossi:2009}
\[
\vvcenteredinclude{.31}{C_3.pdf}
\]
Let $G_L$ be the Gauss diagram obtained from a regular link diagram $D_L$, and $D_{L_k}$ the subdiagram of the $k$th--component of $L$, $k=1,2$. The absolute value of the first term $\langle \vvcenteredinclude{.25}{C_3-term-1.pdf}, G_L\rangle$ of $\langle C_3, G_L\rangle$ does not exceed ${\operatorname{\mathsf{cr}}_{1,2}(D_L) \choose 3}$, the absolute value of the sum $\langle \vvcenteredinclude{.25}{C_3-term-2.pdf}, G_L\rangle$ does not exceed $\operatorname{\mathsf{cr}}(D_{L_1}){\operatorname{\mathsf{cr}}_{1,2}(D_{L}) \choose 2}$ and for the remaining terms a bound is ${\operatorname{\mathsf{cr}}(D_{L_1}) \choose 2}\operatorname{\mathsf{cr}}_{1,2}(D_{L})$. Therefore, a rough upper bound for $|\langle C_3, G_L\rangle|$ can be written as
\[
|\langle C_3, G_L\rangle|\leq (\operatorname{\mathsf{cr}}_{1,2}(D_{L})+\operatorname{\mathsf{cr}}(D_{L_1}))^3
\]
\noindent Similarly, as in \eqref{eq:alpha_n-cr}, replacing $D_L$ with $D_L(v)$ and integrating over the sphere of directions we obtain \[
|c_3(L)|^{\frac 13}\leq \mathsf{acr}_{1,2}(L)+\mathsf{acr}(L_1). \] For a unit thickness link $L$, \eqref{eq:rop-BS-acr} and \eqref{eq:aov-bound} give $\mathsf{acr}_{1,2}(L)+\mathsf{acr}(L_1)\leq \frac{1}{3}\ell^{\frac 13}_1\ell_2+\frac{1}{3}\ell^{\frac 13}_2\ell_1+\frac{4}{11}\ell^{\frac 13}_1\ell_1$, and $\mathsf{acr}_{1,2}(L)+\mathsf{acr}(L_1)\leq \frac{\ell^2_1}{16\pi}+\frac{\ell_1\ell_2}{8\pi}$. Thus, for some constants: $\alpha$, $\beta>0$, we have \[
\ell(L)^2\geq A\, |c_3(L)|^{\frac 13},\qquad \ell(L)^{\frac{4}{3}}\geq B\, |c_3(L)|^{\frac 13}. \] \noindent In general, given a finite type $n$ invariant $V_n(L)$ and a unit thickness $m$--link $L$, we may expect constants $\alpha_{m,n}$, $\beta_{m,n}$; such that \[
\ell(L)^2\geq \alpha_{m,n}\, |V_n(L)|^{\frac{1}{n}},\qquad \ell(L)^{\frac{4}{3}}\geq \beta_{m,n}\, |V_n(L)|^{\frac{1}{n}}. \] } \end{rem}
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\begin{center}
\Large\bf{A covariant Stinespring type theorem for $\tau$-maps} \end{center}
\begin{center} HARSH TRIVEDI \end{center}
\begin{abstract} Let $\tau$ be a linear map from a unital $C^*$-algebra $\CMcal A$ to a von Neumann algebra $\CMcal B$ and let $\CMcal C$ be a unital $C^*$-algebra. A map $T$ from a Hilbert $\CMcal A$-module $E$ to a von Neumann $\CMcal C$-$\CMcal B$ module $F$ is called a $\tau$-map if $$\langle T(x),T(y)\rangle=\tau(\langle x, y\rangle)~\mbox{for all}~x,y\in E.$$ A Stinespring type theorem for $\tau$-maps and its covariant version are obtained when $\tau$ is completely positive. We show that there is a bijective correspondence between the set of all $\tau$-maps from $E$ to $F$ which are $(u',u)$-covariant with respect to a dynamical system $(G,\eta,E)$ and the set of all $(u',u)$-covariant $\widetilde{\tau}$-maps from the crossed product $E\times_{\eta} G$ to $F$, where $\tau$ and $\widetilde{\tau}$ are completely positive.
\noindent {\bf AMS 2010 Subject Classification:} Primary: 46L08,~46L55; Secondary: 46L07,~46L53.\\ \noindent {\bf Key words:} Stinespring representation; completely positive maps; von Neumann modules; dynamical systems.
\end{abstract}
\section{Introduction}
A linear mapping $\tau$ from a (pre-)$C^*$-algebra $\CMcal A$ to a (pre-)$C^*$-algebra $\CMcal B$ is called {\it completely positive} if $$\sum_{i,j=1}^n b_j^{*} \tau(a_j^{*}a_i)b_i\geq 0$$ for each $n\in \mathbb{N}$, $b_1,b_2,\ldots,b_n\in\CMcal B$ and $a_1,a_2,\ldots,a_n\in\CMcal A$. The completely positive maps are used significantly in the theory of measurements, quantum mechanics, operator algebras etc. Paschke's Gelfand-Naimark-Segal (GNS) construction (cf. Theorem 5.2, \cite{Pas73}) characterizes completely positive maps between unital $C^*$-algebras, which is an abstraction of the Stinespring's theorem for operator valued completely positive maps (cf. Theorem 1, \cite{St55}). Now we define Hilbert $C^*$-modules which are a generalization of Hilbert spaces and $C^*$-algebras, were introduced by Paschke in the paper mentioned above and were also studied independently by Rieffel in \cite{Ri74}. \begin{definition} Let $\CMcal B$ be a (pre-)$C^*$-algebra and ${E}$ be a vector space which is a right $\CMcal B$-module satisfying $\alpha(xb)=(\alpha x)b=x(\alpha b)$ for $x\in {E},b\in \CMcal B,\alpha\in\mathbb{C}$. The space ${E}$ is called an {\rm inner-product $\CMcal B$-module} or a {\rm pre-Hilbert $\CMcal B$-module} if there exists a mapping $\langle \cdot,\cdot \rangle : E \times E \to \CMcal{B}$ such that \begin{itemize}
\item [(i)] $\langle x,x \rangle \geq 0 ~\mbox{for}~ x \in {E} $ and $\langle x,x \rangle = 0$ only if $x = 0 ,$ \item [(ii)] $\langle x,yb \rangle = \langle x,y \rangle b ~\mbox{for}~ x,y \in {E}$ and $~\mbox{for}~ b\in \CMcal B, $
\item [(iii)]$\langle x,y \rangle=\langle y,x \rangle ^*~\mbox{for}~ x ,y\in {E} ,$ \item [(iv)]$\langle x,\mu y+\nu z \rangle = \mu \langle x,y \rangle +\nu \langle x,z \rangle ~\mbox{for}~ x,y,z \in {E} $ and for $\mu,\nu \in \mathbb{C}$. \end{itemize}
An inner-product $\CMcal B$-module ${E}$
which is complete with respect to the norm $$\| x\| :=\|\langle x,x\rangle\|^{1/2} ~\mbox{for}~ x \in {E}$$ is called a {\rm Hilbert $\CMcal B$-module} or {\rm Hilbert $C^{*}$-module over $\CMcal B$}. It is said to be {\rm full} if the closure of the linear span of $\{\langle x,y\rangle:x,y\in{E}\}$ equals $\CMcal B$. \end{definition}
Hilbert $C^*$-modules are important objects to study the classification theory of $C^*$-algebras, the dilation theory of semigroups of completely positive maps, and so on. If a completely positive map takes values in any von Neumann algebra, then it gives us a von Neumann module by Paschke's GNS construction (cf. \cite{Sk01}). The von Neumann modules were recently utilized in \cite{BSu13} to explore Bures distance between two completely positive maps. Using the following definition of adjointable maps we define von Neumann modules: Let $E$ and $ F$ be (pre-)Hilbert $\CMcal A$-modules, where $\CMcal A$ is a (pre-)$C^*$-algebra. A map $S:E\to F$ is called {\it adjointable} if there exists a map $S': F\to E$ such that
\[
\langle S (x),y\rangle =\langle x,S'(y) \rangle~\mbox{for all}~x\in E, y\in F.
\] $S'$ is unique for each $S$, henceforth we denote it by $S^{*}$. We denote the set of all adjointable maps from $E$ to $ F$ by $\mathcal B^a (E,F)$ and we use $\mathcal B^a (E)$ for $\mathcal B^a (E,E)$. Symbols $\mathcal B(E, F)$ and $\mathcal B^r (E,F)$ represent the set of all bounded linear maps from $E$ to $ F$ and the set of all bounded right linear maps from $E$ to $F$, respectively.
\begin{definition}(cf. \cite{Sk00}) Let $\CMcal B$ be a von Neumann algebra acting on a Hilbert space $\CMcal H$, i.e., strongly closed $C^*$-subalgebra of $\mathcal B(\CMcal H)$ containing the identity operator. Let $E$ be a (pre-)Hilbert $\CMcal B$-module. The Hilbert space $E\bigodot \CMcal H$ is the interior tensor product of $E$ and $\CMcal H$. For each $x\in E$ we get a bounded linear map from $\CMcal H$ to $E\bigodot \CMcal H$ defined as $$L_x (h):=x\odot h~\mbox{for all}~ h\in \CMcal H.$$ Note that $L^*_{x_1} L_{x_2} =\langle x_1,x_2\rangle~\mbox{for}~ x_1,x_2\in E.$ So we identify each $x\in E$ with $L_x$ and consider $E$ as a concrete submodule of $\mathcal B(\CMcal H,E\bigodot \CMcal H)$. The module $E$ is called a {\rm von Neumann $\CMcal B$-module} or a {\rm von Neumann module over $\CMcal B$} if $E$ is strongly closed in $\mathcal B(\CMcal H,E\bigodot \CMcal H)$. Let $\CMcal A$ be a unital (pre-)$C^*$-algebra. A von Neumann $\CMcal B$-module $E$ is called a {\rm von Neumann $\CMcal A$-$\CMcal B$ module} if there exists an adjointable left action of $\CMcal A$ on $E$. \end{definition}
An alternate approach to the theory of von Neumann modules is introduced recently in \cite{BMSS12} and an analogue of the Stinespring's theorem for von Neumann bimodules is discussed. The comparison of results coming from these two approach is provided by \cite{Ske12}.
Let $G$ be a locally compact group and let $M(\CMcal A)$ denote the multiplier algebra of any $C^*$-algebra $\CMcal A$. An {\it action of $G$ on $\CMcal A$} is defined as a group homomorphism $\alpha:G\to Aut(\CMcal A)$. If $t\mapsto \alpha_{t}(a)$ is continuous for all $a\in\CMcal A$, then we call $(G,\alpha,\CMcal A)$ a {\it $C^*$-dynamical system}. \begin{definition}\label{def3}(cf. \cite{Kap93}) Let $\CMcal A$, $\CMcal B$ be unital (pre-)$C^*$-algebras and $G$ be a locally compact group. Let $(G,\alpha,\CMcal A)$ be a $C^*$-dynamical system and $u:G\to \CMcal U\CMcal B$ be a unitary representation where $\CMcal U \CMcal B$ is the group of all unitary elements of $\CMcal B$. A completely positive map $\tau:\CMcal A\to\CMcal B$ is called {\rm $u$-covariant} with respect to $(G,\alpha,\CMcal A)$ if \[ \tau(\alpha_{t}(a))=u_{t}\tau(a)u^{*}_t~\mbox{for all}~a\in\CMcal A~\mbox{and}~t\in G. \]
\end{definition}
The existence of covariant completely positive liftings (cf. \cite{CE76}) and a covariant version of the Stinespring's theorem for operator-valued $u$-covariant completely positi- ve maps were obtained by Paulsen in \cite{Pau82}, and they were used to provide three groups out of equivalence classes of covariant extensions. Later Kaplan (cf. \cite{Kap93}) extended this covariant version and as an application analyzed the completely positive lifting problem for homomorphisms of the reduced group $C^*$-algebras.
A map ${T}$ from a (pre-)Hilbert $\CMcal A$-module ${E}$ to a (pre-)Hilbert $\CMcal B$-module ${F}$ is called {\it $\tau$-map} (cf. \cite{SSu14}) if $$\langle T(x),T(y)\rangle=\tau(\langle x,y\rangle)~\mbox{for all} ~x,y\in{E}.$$ Recently a Stinespring type theorem for $\tau$-maps was obtained by Bhat, Ramesh and Sumesh (cf. \cite{BRS12}) for any operator valued completely positive map $\tau$ defined on a unital $C^*$-algebra. There are two covariant versions of this Stinespring type theorem see Theorem 3.4 of \cite{Jo11} and Theorem 3.2 of \cite{HJ11}. In Section \ref{sec1.2}, we give a Stinespring type theorem for $\tau$-maps, when $\CMcal B$ is any von Neumann algebra and $F$ is any von Neumann $\CMcal B$-module.
In \cite{DH14} the notion of $\mathfrak K$-families is introduced, which is a generalization of the $\tau$-maps, and several results are derived for covariant $\mathfrak K$-families. In \cite{SSu14} different characterizations of the $\tau$-maps were obtained and as an application the dilation theory of semigroups of the completely positive maps was discussed. Extending some of these results for $\mathfrak K$-families, application to the dilation theory of semigroups of completely positive definite kernels is explored in \cite{DH14}.
In this article we get a covariant version of our Stinespring type theorem which requires the following notions: Let $\CMcal A$ and $\CMcal B$ be $C^*$-algebras, $E$ be a Hilbert $\CMcal A$-module, and let $ F$ be a Hilbert $\CMcal B$-module. A map $\Psi:E\to F$ is said to be a {\it morphism of Hilbert $C^*$-modules} if there exists a $C^*$-algebra homomorphism $\psi:\CMcal A\to\CMcal B$ such that $$\langle \Psi(x),\Psi(y)\rangle=\psi(\langle x,y\rangle)~\mbox{for all}~ x,y\in E.$$ If $E$ is full, then $\psi$ is unique for $\Psi$. A bijective map $\Psi:E\to F$ is called an {\it isomorphism of Hilbert $C^*$-modules} if $\Psi$ and $\Psi^{-1}$ are morphisms of Hilbert $C^*$-modules. We denote the group of all isomorphisms of Hilbert $C^*$-modules from $E$ to itself by $Aut(E)$.
\begin{definition}\label{def1.8} Let $G$ be a locally compact group and let $\CMcal A$ be a $C^*$-algebra. Let $E$ be a full Hilbert $\CMcal A$-module. A group homomorphism $t\mapsto\eta_{t}$ from $G$ to $Aut({E})$ is called a {\em continuous action of $G$ on ${E}$} if $t\mapsto \eta_{t}(x)$ from $G$ to ${E}$ is continuous for each $x\in E$. In this case we call the triple $(G,\eta,E)$ a {\rm dynamical system on the Hilbert $\CMcal A$-module $E$}. Any $C^*$-dynamical system $(G,\alpha,\CMcal A)$ can be regarded as a dynamical system on the Hilbert $\CMcal A$-module $\CMcal A$. \end{definition}
Let $E$ be a full Hilbert $C^*$-module over a unital $C^*$-algebra $\CMcal A$. Let $ F$ be a von Neumann $\CMcal C$-$\CMcal B$ module, where $\CMcal C$ is a unital $C^*$-algebra and $\CMcal B$ is a von Neumann algebra. We define covariant $\tau$-maps with respect to $(G,\eta,E)$ in Section \ref{sec1.2}, and develop a covariant version of our Stinespring type theorem. If $(G,\eta,E)$ is a dynamical system on $E$, then there exists a crossed product Hilbert $C^*$-module $E\times_{\eta} G$ (cf. \cite{EKQR00}). In Section \ref{sec1.3}, we prove that any
$\tau$-map from $E$ to $F$ which is $(u',u)$-covariant with respect to the dynamical system $(G,\eta,E)$ extends to a $(u',u)$-covariant $\widetilde{\tau}$-map from $E\times_{\eta} G$ to $F$, where $\tau$ and $\widetilde{\tau}$ are completely positive. As an application we describe how covariant $\tau$-maps on $(G,\eta,E)$ and covariant $\widetilde{\tau}$-maps on $E\times_{\eta} G$ are related, where $\tau$ and $\widetilde{\tau}$ are completely positive maps. The approach in this article is similar to \cite{BRS12} and \cite{Jo11}. \section{A Stinespring type theorem and its covariant version}\label{sec1.2} \begin{definition}\label{def1.1} Let $\CMcal A$ and $\CMcal B$ be (pre-)$C^*$-algebras. Let $E$ be a Hilbert $\CMcal A$-module and let $ F$, $ F'$ be inner product $\CMcal B$-modules. A map $\Psi:E\to \mathcal B^r( F, F')$ is called {\rm quasi-representation} if there exists a $*$-homomorphism $\pi:\CMcal A\to \mathcal B^a ( F)$ satisfying $$ \langle\Psi(y) f_1,\Psi(x) f_2\rangle=\langle\pi(\langle x,y\rangle)f_1,f_2\rangle~\mbox{for all}~x,y\in E~\mbox{and}~f_1,f_2\in F.$$ In this case we say that $\Psi$ is a quasi-representation of $E$ on $ F$ and $ F'$, and $\pi$ is associated to $\Psi$. \end{definition}
It is clear that Definition \ref{def1.1} generalizes the notion of representations of Hilbert $C^*$-modules on Hilbert spaces (cf. p.804 of \cite{Jo11}). The following theorem provides a decomposition of $\tau$-maps in terms of quasi-representations. We use the symbol sot-$\lim$ for the limit with respect to the strong operator topology. Notation $[S]$ will be used for the norm closure of the linear span of any set $S.$
\begin{theorem}\label{prop1.3} Let $\CMcal A$ be a unital $C^*$-algebra and let $\CMcal B$ be a von Neumann algebra acting on a Hilbert space $\CMcal H$. Let $E$ be a Hilbert $\CMcal A$-module, $ E'$ be a von Neumann $\CMcal B$-module and let $\tau:\CMcal A\to\CMcal B$ be a completely positive map. If $T:E\to E'$ is a $\tau$-map, then there exist
\begin{itemize} \item [(i)] \begin{enumerate} \item [(a)] a von Neumann $\CMcal B$-module $F$ and a representation $\pi$ of $\CMcal A$ to $\mathcal B^a (F)$, \item [(b)] a map $V\in \mathcal B^a (\CMcal B,F)$ such that $\tau(a)b=V^{*}\pi(a)Vb~\mbox{for all}~a\in\CMcal A~\mbox{and}~b\in \CMcal B,~$ \end{enumerate} \item [(ii)] \begin{enumerate} \item [(a)] a von Neumann $\CMcal B$-module $ F'$ and a quasi-representation $\Psi:E\to \mathcal B^a (F, F')$ such that $\pi$ is associated to $\Psi$, \item [(b)] a coisometry $S$ from $ E'$ onto $ F'$ satisfying
$$T(x)b=S^{*}\Psi(x)Vb~\mbox{for all}~x\in E~\mbox{and}~b\in \CMcal B.$$ \end{enumerate} \end{itemize} \end{theorem} \begin{proof}
Let $\langle~,~\rangle$ be a $\CMcal B$-valued positive definite semi-inner product on $\CMcal A\bigotimes_{alg} \CMcal B$ defined by $$\langle a\otimes b, c\otimes d\rangle:=b^*\tau(a^* c)d~\mbox{for}~a,c\in \CMcal A~\mbox{and}~b,d\in\CMcal B.$$ Using Cauchy-Schwarz inequality we deduce that $K=\{x\in {\CMcal A\bigotimes_{alg} \CMcal B}:\langle x, x\rangle=0\}$ is a submodule of $\CMcal A\bigotimes_{alg} \CMcal B$. Therefore $\langle~,~\rangle$ extends naturally on the quotient module $\left({\CMcal A\bigotimes_{alg} \CMcal B}\right)/ K$ as a $\CMcal B$-valued inner product. We get a Stinespring triple $(\pi_0, V, F_0)$ associated to $\tau$, construction is similar to Proposition 1 of \cite{Kap93}, where $F_0$ is the completion of the inner-product $\CMcal B$-module $\left({\CMcal A\bigotimes_{alg} \CMcal B}\right)/ K$, $\pi_0:\CMcal A\to \mathcal B^a (F_0)$ is a $*$-homomorphism defined by $$\pi_0(a')(a\otimes b+ K):= a' a\otimes b+ K~\mbox{for all}~a,a'\in\CMcal A~\mbox{and}~b\in \CMcal B,$$ and a mapping $V\in \mathcal B^a(\CMcal B, F_0)$ is defined by \[ V(b)=1\otimes b+ K~\mbox{for all}~b\in \CMcal B.\] Indeed, $[\pi_0(\CMcal A)V \CMcal B]=F_0$. Let $F$ be the strong operator topology closure of $F_0$ in $\mathcal B(\CMcal H,F_0\bigodot\CMcal H)$. Without loss of generality we can consider $V\in \mathcal B^a(\CMcal B, F)$. Adjointable left action of $\CMcal A$ on $F_0$ extends to an adjointable left action of $\CMcal A$ on $F$ as follows: \[
\pi(a)(f):=\mbox{sot-}\displaystyle\lim_{\alpha} \pi_0(f^0_{\alpha})~\mbox{where $a\in \CMcal A$,
$f$=sot-$\displaystyle\lim_{\alpha} f^0_{\alpha}\in F$ with $f^0_{\alpha}\in F_0$.} \] For all $a\in \CMcal A$; $f$=sot-$\displaystyle\lim_{\alpha} f^0_{\alpha}$, $g$=sot-$\displaystyle\lim_{\beta} g^0_{\beta}\in F$ with $f^0_{\alpha},g^0_{\beta}\in F_0$ we have
\begin{align*}
\langle \pi(a) f,g\rangle &=\mbox{sot-}\displaystyle\lim_{\beta}\langle \pi(a) f,g^0_{\beta}\rangle
=\mbox{sot-}\displaystyle\lim_{\beta}(\mbox{sot-}\displaystyle\lim_{\alpha}\langle g^0_{\beta},\pi_0(a) f^0_{\alpha}\rangle)^*
\\ &=\mbox{sot-}\displaystyle\lim_{\beta}(\mbox{sot-}\displaystyle\lim_{\alpha}\langle \pi_0(a)^*g^0_{\beta}, f^0_{\alpha}\rangle)^*
= \langle f,\pi(a^*)g\rangle. \end{align*} The triple $(\pi, V, F)$ satisfies all the conditions of the statement (i).
Let $ F''$ be the Hilbert $\CMcal B$-module $[T(E)\CMcal B]$. For $x\in E$, define $\Psi_0(x):F_0\to F''$ by \begin{align*}\Psi_0(x)(\displaystyle\sum_{j=1}^{n}\pi_0(a_j)Vb_j):=\displaystyle\sum_{j=1}^{n}T(xa_j)b_j~\mbox{ for all}~a_j\in \CMcal A, b_j\in\CMcal B. \end{align*} It follows that \begin{align*}&\langle\Psi_0(y) (\displaystyle\sum_{j=1}^n\pi_0(a_j)Vb_j),\Psi_0(x)(\displaystyle\sum_{i=1}^m\pi_0(a'_i)Vb'_i)\rangle
= \displaystyle\sum_{j=1}^n \displaystyle\sum_{i=1}^m b_j^{*}\langle T(ya_j),T(xa'_i)\rangle b'_i\\ &=\displaystyle\sum_{i=1}^m \displaystyle\sum_{j=1}^n b_j^{*}\tau(\langle ya_j,xa'_i\rangle) b'_i = \displaystyle\sum_{j=1}^n \displaystyle\sum_{i=1}^m\langle \pi_0(a'_i)^*\pi_0(\langle x,y\rangle)\pi_0(a_j)Vb_j,Vb'_i\rangle \\& = \langle \pi_0(\langle x,y\rangle)(\displaystyle\sum_{j=1}^n\pi_0(a_j)Vb_j), \displaystyle\sum_{i=1}^m\pi_0(a'_i)Vb'_i\rangle \end{align*} for all $x,y\in E, a'_i,a_j\in\CMcal A, b'_i,b_j\in\CMcal B~\mbox{where}~ 1\leq j\leq n~\mbox{and}~1\leq i\leq m$. This computation proves that $\Psi_0(x) \in \mathcal B^r(F_0, F'')$ for each $x\in E$ and also that $\Psi_0:E\to \mathcal B^r(F_0, F'')$ is a quasi-representation. We denote by $ F'$ the strong operator topology closure of $F''$ in $\mathcal B(\CMcal H, E'\bigodot \CMcal H)$. Let $x\in E$, and let $\Psi(x):F\to F'$ be a mapping defined by \[ \Psi(x)(f):=\mbox{sot-}\lim_{\alpha} \Psi_0 (x) f^0_{\alpha}~\mbox{where $f$=sot-$\displaystyle\lim_{\alpha} f^0_{\alpha}\in F$ for $f^0_{\alpha}\in F_0$.}~ \]
For all $f$=sot-$\displaystyle\lim_{\alpha} f^0_{\alpha}\in F$ with $f^0_{\alpha} \in F_0$ and for all $x,y\in E$ we have \begin{align*} \langle \Psi(x) f,\Psi(y) f\rangle= \mbox{sot-}\lim_{\alpha}\{\mbox{sot-}\lim_{\beta}\langle \Psi_0(y)f^0_{\alpha}, \Psi_0(x)f^0_{\beta}\rangle\}^*=\langle f,\pi(\langle x,y\rangle)f\rangle. \end{align*} Since $F$ is a von Neumann $\CMcal B$-module, this proves that $\Psi:E\to \mathcal B^a (F, F')$ is a quasi-representation. Since, $ F'$ is a von Neumann $\CMcal B$-submodule of $ E'$, there exists an orthogonal projection
from $ E'$ onto $ F'$ (cf. Theorem 5.2 of \cite{Sk00}) which we denote by $S$. Eventually \begin{eqnarray*} S^*\Psi(x) Vb=\Psi(x)Vb=\Psi(x)(\pi(1)Vb)=T(x)b~\mbox{for all}~x\in E, b\in\CMcal B. \end{eqnarray*}
\end{proof}
Let $E$ be a (pre-)Hilbert $\CMcal A$-module, where $\CMcal A$ is a (pre-)$C^*$-algebra $\CMcal A$. A map $u\in \mathcal B^a (E)$ is said to be {\it unitary} if $u^{*} u=u u^{*}=1_{E}$ where $1_{E}$ is the identity operator on $E$. We denote the set of all unitaries in $\mathcal B^a (E)$ by $\CMcal U \mathcal B^a (E)$.
\begin{definition}\label{def1.4} Let $\CMcal B$ be a (pre-)$C^*$-algebra, $(G,\alpha,\CMcal A)$ be a $C^*$-dynamical system of a locally compact group $G$, and let $ F$ be a (pre-)Hilbert $\CMcal B$-module. A representation $\pi:\CMcal A\to \mathcal B^a ( F)$ is called {\rm $v$-covariant} with respect to $(G,\alpha,\CMcal A)$ and with respect to a unitary representation $v:G\to \CMcal U \mathcal B^a ( F)$ if \[ \pi(\alpha_{t}(a))=v_{t}\pi(a)v^{*}_{t}~\mbox{for all}~a\in\CMcal A,t\in G. \] In this case we write $(\pi,v)$ is a covariant representation of $(G,\alpha, \CMcal A)$. \end{definition}
Let $E$ be a full Hilbert $\CMcal A$-module and let $G$ be a locally compact group. If $(G,\eta,E)$ is a dynamical system on $E$, then there exists a unique $C^*$-dynamical system $(G,\alpha^{\eta},\CMcal A)$ (cf. p.806 of \cite{Jo11}) such that $$\alpha^{\eta}_t(\langle x,y \rangle)=\langle {\eta}_{t}(x), {\eta}_{t}(y)\rangle~\mbox{for all}~x,y\in E~\mbox{and}~t\in G.$$ We denote by $(G,\alpha^{\eta},\CMcal A)$ the $C^*$-dynamical system coming from the dynamical system $(G,\eta,E)$. For all $x\in E$ and $a\in \CMcal A$ we infer that $\eta_t (xa)=\eta_t(x) \alpha^{\eta}_t (a)$, for \begin{align*}
\|\eta_t (xa)-\eta_t(x)\alpha^{\eta}_t (a)\|^2 =&\| \langle \eta_t (xa), \eta_t (xa)\rangle
- \langle\eta_t (xa), \eta_t (x)\alpha^{\eta}_t (a)\rangle\\ &-\langle \eta_t (x)\alpha^{\eta}_t (a),\eta_t (xa)\rangle
+\langle\eta_t (x)\alpha^{\eta}_t (a),\eta_t (x)\alpha^{\eta}_t (a)\rangle\|\\ =&\| \alpha^{\eta}_t(\langle xa, xa\rangle)
- \langle\eta_t (xa), \eta_t (x)\rangle\alpha^{\eta}_t (a)\\ &-\alpha^{\eta}_t (a^*)\langle \eta_t (x),\eta_t (xa)\rangle
+\alpha^{\eta}_t (a^*)\langle\eta_t (x),\eta_t (x)\rangle\alpha^{\eta}_t (a)\| =& 0. \end{align*}
\begin{definition} Let $\CMcal B$ and $\CMcal C$ be unital (pre-)$C^*$-algebras. A {\rm (pre-)$C^*$-correspondence from $\CMcal C$ to $\CMcal B$} is defined as a (pre-)Hilbert $\CMcal B$-module $ F$ together with a $*$-homomorphism $\pi':\CMcal C\to \mathcal B^a ( F)$. The adjointable left action of $\CMcal C$ on $ F$ induced by $\pi'$ is defined as \[ cy:=\pi'(c)y~\mbox{for all}~c\in \CMcal C,y\in F. \] \end{definition}
In the remaining part of this section a covariant version of Theorem \ref{prop1.3} is derived, which finds applications in the next section. For that we first define covariant $\tau$-maps using the notion of (pre-)$C^*$-correspondence. Every von Neumann $\CMcal B$-module $E$ can be considered as a (pre-)$C^*$-correspondence from $\mathcal B^a (E)$ to $\CMcal B$.
\begin{definition} (cf. \cite{Jo11}) Let $\CMcal A$ be a unital $C^*$-algebra and let $\CMcal B$, $\CMcal C$ be unital (pre-)$C^*$-algebras. Let $E$ be a Hilbert $\CMcal A$-module and let $ F$ be a (pre-)$C^*$-correspondence from $\CMcal C$ to $\CMcal B$. Let $u:G\to \CMcal U \CMcal B$ and $u':G\to \CMcal U \CMcal C$ be unitary representations on a locally compact group $G$. A $\tau$-map, $T:E\to F$, is called {\rm $(u',u)$-covariant} with respect to the dynamical system $(G,\eta,E)$ if \[ T(\eta_{t}(x))=u'_tT(x)u^{*}_t~\mbox{for all}~x\in E~\mbox{and}~t\in G. \] \end{definition}
If $E$ is full and $T:E\to F$ is a $\tau$-map which is $(u',u)$-covariant with respect to $(G,\eta,E)$, then the map $\tau$ is $u$-covariant with respect to the induced $C^*$-dynamical system $(G,\alpha^{\eta},\CMcal A)$, because \begin{align*} \tau(\alpha^{\eta}_t(\langle x,y \rangle))&=\tau(\langle {\eta}_{t}(x), {\eta}_{t}(y)\rangle)=\langle T({\eta}_{t}(x)), T({\eta}_{t}(y))\rangle=\langle u'_tT(x)u^{*}_t,u'_tT(y)u^{*}_t\rangle \\&=\langle T(x)u^{*}_t,T(y)u^{*}_t\rangle = u_t \langle T(x),T(y)\rangle u^{*}_t=u_t\tau(\langle x,y\rangle)u^{*}_t \end{align*} for all $x,y\in E$ and $t\in G$.
\begin{definition} Let $(G,\eta,E)$ be a dynamical system on a Hilbert $\CMcal A$-module $E$, where $\CMcal A$ is a $C^*$-algebra. Let $ F$ and $ F'$ be Hilbert $\CMcal B$-modules over a (pre-)$C^*$-algebra $\CMcal B$. $w:G\to \CMcal U\mathcal B^a ( F')$ and $v:G\to \CMcal U\mathcal B^a ( F)$ are unitary representations on a locally compact group $G$. A quasi-representation of $E$ on $ F$ and $ F'$ is called {\rm $(w,v)$-covariant with respect to $(G,\eta,E)$} if \[ \Psi(\eta_{t}(x))=w_t\Psi(x)v^{*}_t~\mbox{for all}~x\in E~\mbox{and}~t\in G. \] In this case we say that $(\Psi,v,w, F, F')$ is a covariant quasi-representation of $(G,\eta,E)$. Any $v$-covariant representation of a $C^*$-dynamical system $(G,\alpha,\CMcal A)$ can be regarded as a $(v,v)$-covariant representation of a dynamical system on the Hilbert $\CMcal A$-module $\CMcal A$. \end{definition}
Let $\CMcal A$ be a $C^*$-algebra and let $G$ be a locally compact group. Let $E$ be a full Hilbert $\CMcal A$-module, and let $ F$ and $ F'$ be Hilbert $\CMcal B$-modules over a (pre-)$C^*$-algebra $\CMcal B$. If $(\Psi,v,w, F, F')$ is a covariant quasi-representation with respect to $(G,\eta,E)$, then the representation of $\CMcal A$ associated to $\Psi$ is $v$-covariant with respect to $(G,\alpha^{\eta},\CMcal A)$. Moreover, if $\pi$ is the representation associated to $\Psi$, then \begin{align*} \langle\pi(\alpha^{\eta}_t(\langle x,y \rangle))f,f'\rangle &=\langle\pi(\langle {\eta}_{t}(x), {\eta}_{t}(y)\rangle)f,f'\rangle=\langle \Psi({\eta}_{t}(y))f,\Psi({\eta}_{t}(x)) f'\rangle\\&= \langle w_t\Psi(y)v^{*}_t f, w_t\Psi(x)v^{*}_t f'\rangle=\langle v_t\pi(\langle x,y\rangle)v^{*}_t f,f'\rangle \end{align*} for all $x,y\in E$, $t\in G$ and $f,f'\in F$.
\begin{theorem}\label{prop1.6} Let $\CMcal A$, $\CMcal C$ be unital $C^*$-algebras and let $\CMcal B$ be a von Neumann algebra acting on $\CMcal H$. Let $u:G\to \CMcal U \CMcal B$, $u':G\to \CMcal U \CMcal C$ be unitary representations of a locally compact group $G$. Let $E$ be a full Hilbert $\CMcal A$-module and $E'$ be a von Neumann $\CMcal C$-$\CMcal B$ module. If $T:E\to E'$ is a $\tau$-map which is $(u',u)$-covariant with respect to $(G,\eta,E)$ and if $\tau:\CMcal A\to \CMcal B$ is completely positive, then there exists
\begin{itemize} \item [(i)] \begin{enumerate} \item [(a)] a von Neumann $\CMcal B$-module $F$ with a covariant representation $(\pi, v)$ of $(G,\alpha^{\eta},\CMcal A)$ to $\mathcal B^a (F)$, \item [(b)] a map $V\in \mathcal B^a (\CMcal B,F)$ such that \begin{enumerate} \item [(1)] $\tau(a)b=V^{*}\pi(a)Vb~\mbox{for all}~a\in\CMcal A,~b\in \CMcal B,~$ \item [(2)] $v_{t}Vb=Vu_{t}b$ for all $t\in G$, $b\in \CMcal B$, \end{enumerate} \end{enumerate} \item [(ii)] \begin{enumerate} \item [(a)] a von Neumann $\CMcal B$-module $ F'$ and a covariant quasi-representation \newline $(\Psi,v, w,F, F')$ of $(G,\eta,E)$ such that $\pi$ is associated to $\Psi$, \item [(b)] a coisometry $S$ from $ E'$ onto $ F'$ such that \begin{enumerate} \item [(1)] $T(x)b=S^{*}\Psi(x)Vb~\mbox{for all}~x\in E,~b\in \CMcal B,$ \item [(2)] $w_{t}Sy=Su'_{t}y$ for all $t\in G$, $y\in E'$. \end{enumerate} \end{enumerate} \end{itemize} \end{theorem} \begin{proof} By part (i) of Theorem \ref{prop1.3} we obtain the triple $(\pi,V, F)$ associated to $\tau$. Here $F$ is a von Neumann $\CMcal B$-module,
$V\in \mathcal B^a (\CMcal B,F)$, and $\pi$ is a representation of $\CMcal A$ to $\mathcal B^a (F)$ such that $$\tau(a)b=V^{*}\pi(a)Vb~\mbox{for all}~a\in\CMcal A,~b\in \CMcal B.~$$ Recall the proof, using the submodule $K$ we have constructed the triple $(\pi_0, V, F_0)$ with $[\pi_0(\CMcal A)V\CMcal B]=F_0$. Define $v^0: G\to \mathcal B^a (F_0)$ (cf. Theorem 3.1, \cite{He99}) by $$v^0_t(a\otimes b+ K):=\alpha_t(a)\otimes u_{t}(b) + K~\mbox{for all $a\in\CMcal A$, $b\in \CMcal B$ and $t\in G.$}~$$
Since $\tau$ is $u$-covariant with respect to $(G,\alpha^{\eta},\CMcal A)$, for $a,a'\in\CMcal A$, $b,b'\in \CMcal B$ and $t\in G$ it follows that
\begin{align*}
\langle v^0_t a\otimes b+ K, v^0_t a'\otimes b'+ K\rangle
&=\langle \alpha_t (a)\otimes u_t b,\alpha_t (a')\otimes u_t b'\rangle=(u_t b)^* \tau(\alpha_t(a^*a'))u_t b'
\\ &=b^*\tau(a^*a')b'=\langle a\otimes b+ K, a'\otimes b'+ K\rangle.
\end{align*} This map $v^0_t$ extends as a unitary on $F_0$ for each $t\in G$ and further we get a group homomorphism $v^0:G\to \CMcal U \mathcal B^a (\CMcal F_0)$. The continuity of $t\mapsto\alpha^{\eta}_{t}(b)$ for each $b\in\CMcal B$, the continuity of $u$ and the fact that $v^0_t$ is a unitary for each $t\in G$ together implies the continuity of $v^0$. Thus $v^0:G\to \CMcal U \mathcal B^a (\CMcal F_0)$ becomes a unitary representation. For each $t\in G$ define $v_t :F\to F$ by \[
v_t (\mbox{sot-}\lim_{\alpha} f^0_{\alpha}):=\mbox{sot-}\lim_{\alpha} v^0_t (f^0_{\alpha})~\mbox{where $f$
=sot-$\displaystyle\lim_{\alpha} f^0_{\alpha}\in F$ for $f^0_{\alpha}\in F_0$.}~ \] It is clear that $v:G\to \mathcal B^a (F)$ is a unitary representation of $G$ on $F$ and moreover it satisfies the condition (i)(b)(2) of the statement.
Notation $F''$ will be used for $[T(E)\CMcal B]$ which is a Hilbert $\CMcal B$-module. Let $ F'$ be the strong operator topology closure of $ F''$ in $\mathcal B(\CMcal H, E'\bigodot \CMcal H)$. For each $x\in E$, define $\Psi_0(x):F_0\to F''$ by \[\Psi_0(x)(\displaystyle\sum_{j=1}^{n}\pi(a_j)Vb_j):=\displaystyle\sum_{j=1}^{n}T(xa_j)b_j~\mbox{ for all}~a_j\in \CMcal A, b_j\in\CMcal B \] and define $\Psi(x):F\to F'$ by \[ \Psi(x)(f):=\mbox{sot-}\lim_{\alpha} \Psi_0 (x) f^0_{\alpha}~\mbox{where $f$=sot-$\displaystyle\lim_{\alpha} f^0_{\alpha}\in F$ for $f^0_{\alpha}\in F_0$.}~ \] $\Psi_0:E\to \mathcal B^r(F_0, F'')$ and $\Psi:E\to \mathcal B^a (F, F')$ are quasi-representations (see part (ii) of Theorem \ref{prop1.3}). Indeed, there exists an orthogonal projection $S$ from $ E'$ onto $ F'$ such that $$T(x)b=S^{*}\Psi(x)Vb~\mbox{for all}~x\in E~\mbox{and}~b\in \CMcal B.$$ Since $T$ is $(u',u)$-covariant, we have
\[u'_t(\displaystyle\sum_{i=1}^{n}T(x_i)b_i)=\displaystyle\sum_{i=1}^{n}T(\eta_{t}(x_i))u_{t}b_i ~\mbox{for all}~t\in G,x_i\in E,b_i\in \CMcal B, i=1,2,\ldots,n.\] From this computation it is clear that $ F''$ is invariant under $u'$. For each $t\in G$ define $w^0_t:=u'_t|_{ F''}$, the restriction of $u'_t$ to $ F''$. In fact, $t\mapsto w^0_t$ is a unitary representation of $G$ on $ F''$. Further \begin{align*} &\Psi_0(\eta_{t}(x))(\displaystyle\sum_{i=1}^{n}\pi_0(a_i)Vb_i)= \displaystyle\sum_{i=1}^{n}T(\eta_{t}(x)\alpha^{\eta}_{t}\alpha^{\eta}_{t^{-1}}(a_i))b_i = \displaystyle\sum_{i=1}^{n}T(\eta_{t}(x\alpha^{\eta}_{t^{-1}}(a_i)))b_i \\ &=\displaystyle\sum_{i=1}^{n}u^{'}_tT(x\alpha^{\eta}_{t^{-1}}(a_i))u_{t^{-1}}b_i =w^0_{t}\Psi_0(x)(\displaystyle\sum_{i=1}^{n}\pi_0(\alpha^{\eta}_{t^{-1}}(a_i))Vu_{t^{-1}}b_i)\\ &= w^0_{t}\Psi_0(x)v_{t^{-1}}(\displaystyle\sum_{i=1}^{n}\pi_0(a_i)Vb_i) \end{align*} for all $a_1,a_2,\ldots,a_n\in \CMcal A$, $b_1,b_2,\ldots,b_n\in \CMcal B, x\in E,t\in G$. Therefore $(\Psi_0,v^0, w^0,F_0, F'')$ is a covariant quasi-representation of $(G,\eta,E)$ and $\pi_0$ is associated to $\Psi_0$. For each $t\in G$ define $w_t: F'\to F'$ by \[
w_t (\mbox{sot-}\displaystyle\lim_{\alpha} f''_{\alpha}):=\mbox{sot-}\displaystyle\lim_{\alpha} u'_t f''_{\alpha}
~\mbox{where all $f''_{\alpha}\in F''$.}~ \]
It is evident that the map $t\mapsto w_t$ is a unitary representation of $G$ on $ F'$. $S$ is the orthogonal projection of $ E'$ onto $ F'$ so we obtain $w_tS=Su'_t$ on $ F$ for all $t\in G$. Finally \begin{align*} &\Psi(\eta_{t}(x))f=\mbox{sot-}\lim_{\alpha} \Psi_0 (\eta_{t}(x)) f^0_{\alpha} =\mbox{sot-}\lim_{\alpha} w^0_{t}\Psi_0(x)v^0_{t^{-1}} f^0_{\alpha}=w_{t}\Psi(x)v_{t^{-1}}f \end{align*} for all $x\in E$, $t\in G$ and $f$=sot-$\displaystyle\lim_{\alpha} f^0_{\alpha}\in F$ for $f^0_{\alpha}\in F_0$. Whence $(\Psi,v, w,F, F')$ is a covariant quasi-representation of $(G,\eta,E)$ and observe that $\pi$ is associated to $\Psi$. \end{proof}
\section{$\tau$-maps from the crossed product of Hilbert $C^*$-modules}\label{sec1.3}
Let $(G,\eta,E)$ be a dynamical system on $E$, which is a full Hilbert $C^*$-module over $\CMcal A$, where $G$ is a locally compact group. The {\it crossed product} Hilbert $C^*$-module $E\times_{\eta}G$ (cf. Proposition 3.5, \cite{EKQR00}) is the completion of an inner-product $\CMcal A\times_{\alpha^{\eta}}G$-module $C_c(G,E)$ such that the module action and the $\CMcal A\times_{\alpha^{\eta}}G$-valued inner product are given by \begin{align*}\label{eqn1} lg(s)&=\int_{G} l(t)\alpha^{\eta}_{t}(g(t^{-1}s))dt,\\ \langle l,m\rangle_{\CMcal A\times_{\alpha^{\eta}}G }(s)&=\int_{G} \alpha^{\eta}_{t^{-1}}(\langle l(t),m(ts)\rangle)dt \end{align*} respectively for $s\in G$, $g\in C_c (G,\CMcal A)$ and $l,m\in C_c (G,E)$. The following lemma shows that any covariant quasi-representation $(\Psi_0,v^0,w^0,F_0, F')$ with respect to $(G,\eta,E)$ provides a quasi-representation $\Psi_0\times v^0$ of $E\times_{\eta} G$ on $F_0$ and $F'$ satisfying \[ (\Psi_0\times v^0)(l)=\int_{G} \Psi_0(l(t))v^0_t dt~\mbox{for all $l\in C_c(G,E)$.} \] Moreover, it says that if $\pi_0$ is associated to $\Psi_0$, then the integrated form of the covariant representation $(\pi_0,v^0,F_0)$ with respect to $(G,\alpha^{\eta},\CMcal A)$ is associated to $\Psi_0\times v^0$.
\begin{lemma}\label{lem1.3} Let $(G,\eta,E)$ be a dynamical system on a full Hilbert $\CMcal A$-module $E$, where $\CMcal A$ is a unital $C^*$-algebra and $G$ is a locally compact group. Let $F_0$ and $ F'$ be Hilbert $\CMcal B$-modules, where $\CMcal B$ is a von Neumann algebra acting on a Hilbert space $\CMcal H$. If $(\Psi_0,v^0,w^0,F_0, F')$ is a covariant quasi-representation with respect to $(G,\eta,E)$, then $\Psi_0\times v^0$ is a quasi-representation of $E\times_{\eta} G$ on $F_0$ and $ F'$. \end{lemma} \begin{proof} For $l\in C_c(G,E)$ and $g\in C_c(G,\CMcal A)$, we get \begin{align*} (\Psi_0\times v^0)(lg)&=\int_{G}\int_{G}\Psi_0(l(t)\alpha^{\eta}_{t} (g(t^{-1}s))v^0_{s}ds dt \\&=\int_{G}\int_{G}\Psi_0(l(t))\pi_0(\alpha^{\eta}_{t} (g(t^{-1}s))v^0_{s}ds dt\\ &=\int_{G}\int_{G}\Psi_0(l(t))v^0_{t}\pi_0 (g(t^{-1}s)v^{0*}_{t}v^0_{s}ds dt \\ &=(\Psi_0\times v^0)(l)(\pi_0\times v^0)(g). \end{align*} For $l,m\in C_c(G,E)$ and $f_0,f'_0\in F_0$ we have \begin{align*} \langle(\pi_0\times v^0)(\langle l,m\rangle)f_0,f'_0\rangle &=\left< \int_{G}\pi_0(\langle l,m\rangle(s))v^{0}_{s}f_0 ds,f'_0 \right> \\ &= \left< \int_{G}\int_{G}v^{0*}_t\pi_0 (\langle l(t),m(ts)\rangle)v^{0}_{ts} f_0 dt ds, f'_0 \right> \\&= \int_{G}\int_{G} \langle \Psi_0(m(ts))v^0_{ts}f_0,\Psi_0(l(t))v^0_t f'_0\rangle dt ds \\ &= \left< \int_{G}\Psi_0(m(s))v^0_{s}f_0 ds,\int_{G}\Psi_0(l(t))v^0_t f'_0dt\right> \end{align*} \begin{align*} &= \langle(\Psi_0\times v^0)(m)f_0,(\Psi_0\times v^0)(l)f'_0\rangle.\qedhere \end{align*} \end{proof}
\begin{definition}(cf. \cite{Jo11})
Let $G$ be a locally compact group with the modular function $\bigtriangleup$. Let $u:G\to \CMcal U \CMcal B$ and $u':G\to \CMcal U \CMcal C$ be unitary representations of $G$ on unital
(pre-)$C^*$-algebras $\CMcal B$ and $\CMcal C$, respectively. Let $ F$ be a (pre-)$C^*$-correspondence from $\CMcal C$ to $\CMcal B$ and let $(G,\eta,E)$ be a dynamical system on a Hilbert $\CMcal A$-module $E$, where $\CMcal A$ is a unital $C^*$-algebra. A $\tau$-map, $T:E\times_{\eta} G\to F$, is called {\rm $(u',u)$-covariant} if \begin{enumerate}
\item [(a)] $T(\eta_t \circ m^{l}_t)=u'_t
T(m)$ where
$m^{l}_t(s)=m(t^{-1} s)$ for all $s,t\in G$, $m\in C_c(G,E)$;
\item [(b)] $T(m^{r}_t)=T(m)u_t$ where
$m^{r}_t (s)=\bigtriangleup(t)^{-1}m(s t^{-1})$ for all $s,t\in G$, $m\in C_c(G,E)$. \end{enumerate}
\end{definition}
\begin{proposition}\label{prop3.3} Let $\CMcal B$ be a von Neumann algebra acting on a Hilbert space $\CMcal H$, $\CMcal C$ be a unital $C^*$-algebra, and let $ F$ be a von Neumann $\CMcal C$-$\CMcal B$ module. Let $(G,\eta,E)$ be a dynamical system on a full Hilbert $\CMcal A$-module $E$, where $\CMcal A$ is a unital $C^*$-algebra and $G$ is a locally compact group. Let $u:G\to \CMcal U \CMcal B$, $u':G\to \CMcal U \CMcal C$ be unitary representations and let $\tau:\CMcal A\to \CMcal B$ be a completely positive map. If $T:E\to F$ is a $\tau$-map which is $(u',u)$-covariant
with respect to $(G,\eta,E)$, then there exist a completely positive map $\widetilde{\tau}:\CMcal A\times_{\alpha^{\eta}}G\to \CMcal B$ and a $(u',u)$-covariant map $\widetilde{T}:E\times_{\eta} G\to F$ which is a $\widetilde{\tau}$-map. Indeed, $\widetilde{T}$ satisfies $$\widetilde{T}(l)=\int_G T(l(s))u_s ds~\mbox{for all}~l\in C_c (G,E).$$ \end{proposition} \begin{proof} By Theorem \ref{prop1.6} there exists the Stinespring type construction $(\Psi,\pi,v,w,V,$ $S,F,F')$, associated to $T$, based on the construction $(\Psi_0,\pi_0,v^0,T,F_0, F'')$. Define a map $\widetilde{T}:E\times_{\eta} G\to
F$ by \[ \widetilde{T}(l):=S^{*}(\Psi_0\times v^0)(l)V,~\mbox{for all}~l\in C_c(G,E). \] Indeed, for all $l\in C_c (G,E)$ we obtain \begin{align*}\widetilde{T}(l)&=S^{*}(\Psi_0\times v^0)(l)V=S^{*}\int_{G}\Psi_0(l(s))v^0_s dsV=\int_{G}S^{*}\Psi_0(l(s))V u_s ds \\ &=\int_G T(l(s))u_s ds. \end{align*} It is clear that $(\pi_0\times v^0,V,F_0)$ is the Stinespring triple (cf. Theorem \ref{prop1.3}) associated to the completely positive map $\widetilde{\tau}:\CMcal A\times_{\alpha^{\eta}}G\to \CMcal B$ defined by \[\widetilde{\tau}(h):=\int_{G} \tau(f(t))v^0_{t}dt~\mbox{for all}~ f\in C_c (G,\CMcal A); b,b'\in \CMcal B. \] We have \begin{eqnarray*} \langle \widetilde{T}(l),\widetilde{T}(m)\rangle b &=&\langle S^{*}(\Psi_0\times v^0)(m)V,S^{*}(\Psi_0\times v^0)(l)V\rangle b =\widetilde{\tau}(\langle l,m\rangle)b \end{eqnarray*} for all $l,m\in E\times_{\eta} G,~b\in\CMcal B.$ Hence $\widetilde{T}$ is a $\widetilde{\tau}$-map. Further, \begin{align*} \widetilde{T}(\eta_t \circ m^{l}_t)&=S^*\int_G \Psi_0(\eta_t(m(t^{-1} s)))v^0_s ds V=S^*\int_G w^0_t \Psi_0(m(t^{-1} s))v^0_{t^{-1} s} ds V \\ &= u'_t
\widetilde{T}(m);\\
\widetilde{T}(m^{r}_t)&=S^*\int_G \bigtriangleup (t)^{-1}\Psi_0(m(s t^{-1}))v^0_s ds V= S^*\int_G \Psi_0(m(g))v^0_g v^0_t dg V\\ &=\widetilde{T}(m)u_t~\mbox{where $t\in G$, $m\in C_c(G,E)$.}~ \end{align*}
\end{proof} Proposition \ref{prop3.3} gives us a map $T\mapsto \widetilde{T}$ where $T:E\to F$ is a $\tau$-map which is $(u',u)$-covariant
with respect to $(G,\eta,E)$ and $\widetilde{T}:E\times_{\eta} G\to F$ is $(u',u)$-covariant $\widetilde{\tau}$-map such that $\tau$ and $\widetilde{\tau}$ are completely positive. This map is actually a one-to-one correspondence. To prove this result we need the following terminologies:
We identify $M(\CMcal A)$ with $\mathcal B^a (\CMcal A)$ (cf. Theorem 2.2 of \cite{La95}), here $\CMcal A$ is considered as a Hilbert $\CMcal A$-module in the natural way.
The {\it strict topology} on $\mathcal B^a ({E})$ is the topology given by the seminorms $a\mapsto \|ax\| $, $a\mapsto \|a^*y\| $ for each $x,y\in E$. For each $C^*$-dynamical system $(G,\alpha,\CMcal A)$ we get a non-degenerate faithful homomorphism $i_{\CMcal A}:\CMcal A\to M(\CMcal A\times_{\alpha} G)$ and an injective strictly continuous homomorphism $i_G : G\to \CMcal UM(\CMcal A \times_{\alpha} G)$ (cf. Proposition 2.34 of \cite{Wi07}) defined by \[
i_{\CMcal A}(a)(f)(s):=af(s)~\mbox{for}~a\in\CMcal A,~s\in G,~f\in C_c (G,\CMcal A); \] \[
i_{G} (r) f(s):=\alpha_r (f(r^{-1} s))~\mbox{for}~r,s\in G,~f\in C_c (G,\CMcal A). \] Let $E$ be a Hilbert $C^*$-module over a $C^*$-algebra $\CMcal A$.
Define the {\it multiplier module} $M(E):=\mathcal B^a (\CMcal A,E)$. $M(E)$ is a Hilbert $C^*$-module over $M(\CMcal A)$ (cf. Proposition 1.2 of \cite{RT03}). For a dynamical system $(G,\eta,E)$ on $E$ we get a non-degenerate morphism of modules $i_{E}$ from $E$ to $M(E\times_{\eta} G)$ (cf. Theorem 3.5 of \cite{Jo12b}) as follows: For each $x\in E$ define $i_{E}(x):C_c(G,\CMcal A)\to C_c(G,E)$ by $$ i_{E}(x)(f)(s):=xf(s)~\mbox{for all}~f\in C_c(G,\CMcal A),~s\in G.$$ Note that $i_{E}$ is an $i_{\CMcal A}$-map.
\begin{theorem}
Let $\CMcal A$, $\CMcal C$ be unital $C^*$-algebras, and let $\CMcal B$ be a von Neumann algebra acting on a Hilbert space $\CMcal H$. Let $u:G\to \CMcal U \CMcal B$, $u':G\to \CMcal U \CMcal C$ be unitary representations of a locally compact group $G$. If $(G,\eta,E)$ is a dynamical system on a full Hilbert $\CMcal A$-module $E$, and if $ F$ is a von Neumann $\CMcal C$-$\CMcal B$ module, then there exists a bijective correspondence $\mathfrak{I}$ from the set of all $\tau$-maps, $T:E\to F$, which are $(u',u)$-covariant with respect to $(G,\eta,E)$ onto the set of all maps $\widetilde{T}:E\times_{\eta} G\to F$ which are $(u',u)$-covariant $\widetilde{\tau}$-maps such that $\tau:\CMcal A\to \CMcal B$ and $\widetilde{\tau}:\CMcal A\times_{\alpha^{\eta}}G\to \CMcal B$ are completely positive maps. \end{theorem} \begin{proof} Proposition \ref{prop3.3} ensures that the map $\mathfrak{I}$ exists and is well-defined. Let $T:E\times_\eta G\to F$ be a $(u',u)$-covariant $\tau$-map, where $\tau:\CMcal A\times_{\alpha^{\eta}}G\to \CMcal B$ is a completely positive map. Suppose $(\Psi_0,\pi_0,V,F_0, F'')$ and $(\Psi,\pi,V,S,F, F')$ are the Stinespring type constructions associated to $T$ as in the proof of Theorem \ref{prop1.3}. Let $\{e_i\}_{i\in \CMcal I}$ be an approximate identity for $\CMcal A\times_{\alpha^{\eta}} G$. Then there exists a representation $\overline{\pi_0}:M(\CMcal A\times_{\alpha^{\eta}} G)\to \mathcal B^a (F_0)$ (cf. Proposition 2.39 of \cite{Wi07}) defined by \[
\overline{\pi_0}(a)x:=\displaystyle\lim_i \pi_0(a e_i)x~\mbox{for all $a\in M(\CMcal A\times_{\alpha^{\eta}} G)$ and $x\in F_0$.}~ \] A mapping $\overline{\Psi_0}:M(E \times_{\eta} G)\to \mathcal B^r(F_0, F'')$ defined by \[ \overline{\Psi_0}(h)x:=\lim_i \Psi_0(h e_i)x~\mbox{for all $h\in M(E \times_{\eta} G)$ and $x\in F_0$,} \] is a quasi representation and $\overline{\pi_0}$ is associated to $\overline{\Psi_0}$. If $\widetilde{\pi_0}:=\overline{\pi_0}\circ i_{\CMcal A}$, then we further get a quasi-representation $\widetilde{\Psi_0}:E \to \mathcal B^r(F_0, F'')$ defined as $\widetilde{\Psi_0}:=\overline{\Psi_0}\circ i_{E}$ such that $\widetilde{\pi_0}$ is associated to $\widetilde{\Psi_0}$. Define maps $T_0:E\to F$ and $\tau_0:\CMcal A\to \CMcal B$ by \[
T_0(x)b:=S^*\widetilde{\Psi_0}(x)Vb~\mbox{for}~b\in \CMcal B,~x\in E~{and} \] \[
\tau_0(a):=V^*\widetilde{\pi_0}(a)V~\mbox{for all}~a\in\CMcal A. \] It follows that $\tau_0$ is a completely positive map and $T_0$ is a $\tau_0$-map.
Let $v^0:G\to \CMcal U\mathcal B^a (F_0)$ be a unitary representation defined by $v^0:=\overline{\pi_0}\circ i_G$ where $$i_G (t)(f)(s):=\alpha_t (f(t^{-1} s))~\mbox{for all}~t,s\in G,~f\in C_c (G,\CMcal A).$$ Observe that $\widetilde{\pi_0}:\CMcal A\to \mathcal B^a (F_0)$ is a $v^0$-covariant and $\widetilde{\pi_0} \times v^0=\pi_0$ (cf. Proposition 2.39, \cite{Wi07}). We extend $v^0$ to a unitary representation $v:G\to \CMcal U\mathcal B^a (F)$ as in the proof of Theorem \ref{prop1.6}. It is easy to verify that $$\alpha^\eta_t\circ \langle m,m'\rangle^l_t=\langle m^r_{t^{-1}},m'\rangle~\mbox{for all}~m,m'\in C_c (G,E).$$ Using the fact that $T$ is $(u',u)$-covariant we get \[ \tau(\alpha^\eta_t\circ \langle m,m'\rangle^l_t)=\tau(\langle m^r_{t^{-1}},m'\rangle)=\langle T(m) u_{t^{-1}} ,T(m')\rangle=u_t \tau (\langle m,m'\rangle), \] $\mbox{for all $m,m'\in C_c (G,E)$.}$ Therefore we have \begin{eqnarray*}
\langle v_t(\pi_0 (f)V b), Vb'\rangle &=& \langle v^0_t ((\widetilde{\pi_0} \times v^0)(f)V b, Vb'\rangle
=\left< \int_G \widetilde{\pi_0} (\alpha^\eta_t (f(s))) v^0_{ts} V b ds,Vb'\right>
\\ &=&\langle (\widetilde{\pi_0}\times v^0) (\alpha^\eta_t \circ f^l_t)V b, Vb'\rangle
=\langle\tau(\alpha^\eta_t \circ f^l_t)b, b'\rangle\\ &=& \langle(\pi_0(f)V b),Vu_{t^{-1}}b'\rangle \end{eqnarray*} for all $t\in G$, $b,b'\in \CMcal B$ and $f\in C_c(G, \CMcal A)$. This implies that $v_t V=Vu_t$ for each $t\in G$. For each $t\in G$ define $w^0_t :[\widetilde{\Psi_0}(E)V\CMcal B]\to [\widetilde{\Psi_0}(E)V\CMcal B]$ by \[
w^0_t (\widetilde{\Psi_0}(x)Vb):=\widetilde{\Psi_0}(\eta_t(x))Vu_t b~\mbox{for all $x\in E$, $b\in \CMcal B$.}~ \] Let $t\in G$, $x,y\in E$ and $b,b'\in \CMcal B$. Then \begin{align*}
&\langle \widetilde{\Psi_0}(\eta_t (x)) Vu_t b, \widetilde{\Psi_0}(\eta_t (y)) Vu_t b'\rangle
\\=&\langle \widetilde{\pi_0}(\langle \eta_t (y),\eta_t(x)\rangle)Vu_t b, Vu_t b'\rangle
=\langle \widetilde{\pi_0}(\alpha^{\eta}_t(\langle y,x\rangle))Vu_t b, Vu_t b'\rangle \\ =&\langle v^0_t\widetilde{\pi_0}(\langle y,x\rangle)v^0_{t^{-1}}Vu_t b, Vu_t b'\rangle=\langle \widetilde{\pi_0}(\langle y,x\rangle)V b, V b'\rangle \\ =& \langle \widetilde{\Psi_0}(x) V b, \widetilde{\Psi_0} (y) V b'\rangle. \end{align*} Indeed, for fix $t\in G$, the continuity of the maps $t\mapsto \eta_t (x)$ and $t\mapsto u_t b$ for $b\in \CMcal B,~x\in E$ provides the fact that the map $t\mapsto w^0_t(z)$ is continuous for each $z\in \widetilde{\Psi_0}(E)V\CMcal B$. Therefore $w^0$ is a unitary representation of $G$ on $[\widetilde{\Psi_0}(E)V\CMcal B]$ and hence it naturally extends to a unitary representation of $G$ on the strong operator topology closure of $[\widetilde{\Psi_0}(E)V\CMcal B]$ in $\mathcal B(\CMcal H,F\bigodot \CMcal H)$, which we denote by $w$.
Note that $E\otimes C_c (G)$ is dense in $E\times_{\eta} G$ (cf. Theorem 3.5 of \cite{Jo12b}). For $x\in E$ and $f\in C_c(G)$ we have \begin{eqnarray*} (\widetilde{\Psi_0}\times v^0)(x\otimes f)&=&\int_G \widetilde{\Psi_0} (x f(t))v^0_t dt =\int_G \overline{\Psi_0}(i_{E}(x f(t))) \overline{\pi_0}(i_G (t))dt\\ &=& \overline{\Psi_0}(i_{E}(x)\int_G f(t) i_G (t)dt )=\overline{\Psi_0}(i_{E}(y)i_{\CMcal A}(\langle y,y \rangle)\int_G f(t) i_G (t)dt)\\ &=& \overline{\Psi_0} (i_{E}(y)(\langle y,y\rangle\otimes f))=\overline{\Psi_0}(y\langle y,y\rangle\otimes f)={\Psi_0}(x\otimes f) \end{eqnarray*} where $x=y\langle y,y\rangle~\mbox{for some}~y\in E~(\mbox{cf. Proposition 2.31 \cite{RW98}})$. Also the 3rd last equality follows from Corollary 2.36 of \cite{Wi07}. This proves $\widetilde{\Psi_0}\times v^0=\Psi_0$ on $E\times_{\eta} G$. Also for all $m\in C_c(G,E)$ and $b\in \CMcal B$ we get \begin{eqnarray*}
S u'_t (T(m)b) &=& S T(\eta_t \circ m^l_t)b=SS^*\Psi_0 (\eta_t\circ m^l_t)Vb=\Psi_0 (\eta_t\circ m^l_t)Vb
\\ &=&\int_G \widetilde{\Psi_0}(\eta_t (m(t^{-1}s))) v^0_s V b ds = w_t\int_G \widetilde{\Psi_0}(m(t^{-1}s)) v^0_{t^{-1}s}V b ds \\ &=& w_t \Psi_0 (m) Vb=w_t ST(m)b. \end{eqnarray*} As $T$ is $(u',u)$-covariant, it satisfies $T(\eta_t \circ m^{l}_t)=u'_t
T(m)$, where
$m^{l}_t(s)=m(t^{-1} s)$ for all $s,t\in G$, $m\in C_c(G,E)$. Thus the strong operator topology closure of $[T(E\times_{\eta} G)\CMcal B]$ in $\mathcal B(\CMcal H, F \bigodot \CMcal H)$, say $F_T$, is invariant under $u'$. This together with the fact that $S$ is an orthogonal projection onto $F_T$ provides $Su'_tz=w_tSz$ for all $z\in F^{\perp}_T$. So we obtain the equality $S u'_t y=w_t S y~\mbox{for all}~y\in F.$ Hence \begin{eqnarray*}
T_0(\eta_t(x))b=S^*\widetilde{\Psi_0}(\eta_t(x))V b=S^* w_t \widetilde{\Psi_0}(x)V u_{t^{-1}}b=u'_t T_0(x)u^*_t b \end{eqnarray*} for all $t\in G$, $x\in E$ and $b\in \CMcal B$. Moreover, \begin{eqnarray*} \widetilde{T_0}(m)b&=&S^*\int_G \widetilde{\Psi_0}(m(t))Vu_t bdt=S^*\Psi_0(m)Vb=T(m)b \end{eqnarray*} for all $m\in C_c (G,E)$, $b\in \CMcal B$. This gives $\widetilde{T_0}=T$ and proves that the map $\mathfrak{I}$ is onto.
Let $\tau_1:\CMcal A\to\CMcal B$ be a completely positive map and let $T_1:E\to F$ be a $(u',u)$-covariant $\tau_1$-map satisfying $\widetilde{T}_1=T$. If $(\Psi_1,\pi_1,V_1,S_1,F_1, F'_1)$ is the $(w_1,v_1)$-covariant Stinespring type construction associated to $T_1$ coming from Theorem \ref{prop1.6}, then we show that $(\Psi_1\times v_1, V_1, S_1,F_1, F'_1)$ is unitarily equivalent to the Stinespring type construction associated to $T$. Indeed, from Proposition \ref{prop3.3}, there exists a decomposition $$\widetilde{T_1}(m)=S^*_1 (\Psi_1\times v_1)(m)V_1~\mbox{for all}~m\in C_c(G,E).$$ This implies that for all $m,m'\in C_c(G,E)$ we get \begin{align*}
\tau(\langle m,m'\rangle)&=\langle T(m),T(m')\rangle=\langle \widetilde{T_1}(m),\widetilde{T_1}(m')\rangle \\ & =\langle S^*_1 (\Psi\times v_1)(m)V_1,S^*_1 (\Psi\times v_1)(m')V_1\rangle \\ & =\langle (\pi_1\times v_1)(\langle m,m'\rangle)V_1,V_1\rangle.
\end{align*}
$E$ is full gives $E\times_{\eta} G$ is full (cf. the proof of Proposition 3.5, \cite{EKQR00}) and hence $\tau(f)=\langle (\pi_1\times v_1)(f)V_1,V_1\rangle$ for all $f\in C_c (G,\CMcal A)$. Using this fact we deduce that
\begin{align*}
\langle \pi(f)Vb,\pi(f')Vb'\rangle &=\langle \pi(f^{\prime*}f)Vb,Vb'\rangle=b^*\tau(f^{\prime*}f) b'\\ &=\langle \pi_1\times v_1(f)V_1b,
\pi_1\times v_1(f')V_1b'\rangle
\end{align*} for all $f,f'\in C_c(G,\CMcal A)$ and $b,b'\in \CMcal B$. Thus we get a unitary $U_1:F\to F_1$ defined by \[ U_1(\pi(f)V b):=\pi_1\times v_1 (f)V_1b ~\mbox{for}~f \in C_c(G,\CMcal A),~b\in\CMcal B \] and which satisfies $V_1=U_1 V$, $\pi_1\times v_1 (f)=U_1\pi(f)U^*_1$ for all $f \in C_c(G,\CMcal A)$. Another computation
\begin{align*}
\| \Psi(m)Vb\|^2 &=\|\langle\Psi(m)Vb,\Psi(m)Vb\rangle\|=\|\langle\pi(\langle m,m\rangle)Vb,Vb\rangle\|=\|b^*\tau(\langle m,m\rangle) b\|
\\ &= \|b^*\langle \pi_1\times v_1(\langle m,m\rangle)V_1,V_1\rangle b\|=\|\langle\Psi_1\times v_1(m)V_1b,\Psi_1\times v_1(m)V_1b\rangle\|
\\ &=\| \Psi_1\times v_1(m)V_1b\|^2 \end{align*} for all $m\in C_c (G,E)$, $b\in \CMcal B$ provides a unitary $U_2: F'\to F'_1$ defined as \[
U_2 (\Psi(m)Vb):=\Psi_1\times v_1 (m)V_1 b~\mbox{for}~m \in C_c(G,E),~b\in\CMcal B. \] Further, it satisfies conditions $S_1=U_2S$ and $U_2\Psi(m)=\Psi_1\times v_1(m)U_1$ for all $m\in C_c(G,E)$. This implies $U_2\overline{\widetilde{\Psi} \times v}(z')=\overline{\Psi_1 \times v_1}(z')U_1$ for all $z' \in M(E\times_{\eta} G)$ and so $U_2\widetilde{\Psi}(x)=\Psi_1\times v_1 (x)U_1$ for all $x\in E$. Using it we have \begin{eqnarray*}
T_0(x)=S^*\widetilde{\Psi}(x)V=S^*_1 U_2\widetilde{\Psi}(x)U^*_1V_1 =S^*_1 U_2 U^*_2 (\Psi_1\times v_1)(x)U_1U^*_1V_1 =T_1(x) \end{eqnarray*} for all $x\in E$ and $b\in\CMcal B$. Hence $\mathfrak{I}$ is injective. \end{proof}
\noindent \textbf{Acknowledgement.} The author would like to express thanks of gratitude to Santanu Dey for several discussions. This work was supported by CSIR, India.
{
}
{\footnotesize Department of Mathematics, Indian Institute of Technology Bombay,}
{\footnotesize Powai, Mumbai-400076,}
{\footnotesize India.}
{\footnotesize e-mail: [email protected]}
\end{document}
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\begin{document}
\title{ {Complete convergence of message passing algorithms for
some satisfiability problems } \begin{abstract} In this paper we analyze the performance of \textsf{Warning Propagation}, a popular message passing algorithm. We show that for 3CNF formulas drawn from a certain distribution over random satisfiable 3CNF formulas, commonly referred to as the planted-assignment distribution, running \textsf{Warning Propagation} in the standard way (run message passing until convergence, simplify the formula according to the resulting assignment, and satisfy the remaining subformula, if necessary, using a simple ``off the shelf" heuristic) results in a satisfying assignment when the clause-variable ratio is a sufficiently large constant.
\end{abstract}
\section{Introduction} A CNF formula over the variables $x_{1},x_{2},...,x_{n}$ is a conjunction of clauses $C_{1},C_{2},... ,C_{m}$ where each clause is a disjunction of one or more literals. Each literal is either a variable or its negation. A formula is said to be in $k$-CNF form if every clause contains exactly $k$ literals. A CNF formula is satisfiable if there is a boolean assignment to the variables such that every clause contains at least one literal which evaluates to true. 3SAT, the language of all satisfiable 3CNF formulas, is well known to be NP-complete \cite{Cook71}.
The plethora of worst-case NP-hardness results for many interesting optimization problems motivates the study of heuristics that give ``useful'' answers for ``typical'' subset of the problem instances. In this paper we seek to evaluate those two measures rigorously and for that we shall use random models and average case analysis.
In this paper we study random satisfiable 3CNF formulas with an arbitrary density. For this we use the {\em planted distribution} \cite{flaxman,AlonKrivSudCliqe,OnTheGreedy,ExpectedPoly3SAT,AlonKahale97,ChenFrieze} denoted throughout by ${\cal{P}}^{{\rm plant}}_{n,p}$. A random 3CNF in this distribution is obtained by first picking an assignment $\varphi$ to the variables, and then including every clause satisfied by $\varphi$ with probability $p=p(n)$, thus guaranteeing that the resulting instance is satisfiable.
We briefly note that there exists another model of random 3CNF formulas which consists of $m$ clauses chosen uniformly at random from the set of all $8\binom{n}{3}$ possible ones ($m$ is a parameter of the distribution, and $m/n$ is referred to as the clause-variable ratio, or the density, of the random instance). This distribution shows a sharp threshold with respect to satisfiability \cite{Friedgut}. Specifically, a random 3CNF with clause-variable ratio below the threshold is satisfiable $whp$ (with high probability, meaning with probability tending to 1 as $n$ goes to infinity) and one with ratio above the threshold is unsatisfiable $whp$. Experimental results predict the threshold to be around 4.2 \cite{ExperimentalUpperBoundOnThresh}.
\\\\ To describe our main result, we formally define the \textsf{Warning Propagation (WP)} algorithm.
\subsection{Warning Propagation}\label{WPSubs} $\textsf{WP}$ is a simple iterative message passing algorithm similar to \textsf{Belief Propagation} \cite{Pearl82} and \textsf{Survey Propagation} \cite{SurveyPropagation}.
Messages in the $\textsf{WP}$ algorithm can be interpreted as "warnings", telling a clause the values that variables will have if the clause "keeps quite" and does not announce its wishes, and telling a variable which clauses will not be satisfied if the variable does not commit to satisfying them. We now present the algorithm in a formal way.
Let ${\cal{F}}$ be a CNF formula. For a variable $x$, let $N^+(x)$ be the set of clauses in ${\cal{F}}$ in which $x$ appears positively (namely, as the literal $x$), and $N^{-}(x)$ be the set of clauses in which $x$ appears negatively. For a clause $C$, let $N^+(C)$ be the set of variables that appear positively in $C$, and respectively $N^-(C)$ for negative ones.
There are two types of messages involved in the \textsf{WP} algorithm. Messages sent from a variable $x_i$ to a clause $C_j$ in which it appears, and vice a versa. Consider a clause $C_j$ that contains a variable $x_i$. A message from $x_i$ to $C$ is denoted by $x_i \rightarrow C_j$, and it has an integer value. A positive value indicates that $x_i$ is tentatively set to true, and a negative value indicates that $x_i$ is tentatively set to false. A message from $C_j$ to $x_i$ is denoted by $C_j \rightarrow x_i$, and it has a Boolean value. A value of 1 indicates that $C_j$ tentatively wants $x_i$ to satisfy it, and a 0 value indicates that $C_j$ is tentatively indifferent to the value of $x_i$. We now present the update rules for these messages. $$ x_i \rightarrow C_j = \left(\sum_{C_k \in N^+(x_i),k\neq j} C_k \rightarrow x_i \right) - \left(\sum_{C_k\in N^-(x_i),k\neq j}C_k \rightarrow x_i\right).$$ If $x_i$ appears only in $C_j$ then we set the message to 0. $$ C_j \rightarrow x_i = \left(\prod_{x_k \in N^+(C_j),k\neq i} I_{<0}(x_k \rightarrow C_j) \right) \cdot \left( \prod_{x_k \in N^-(C_j),k\neq i} I_{>0}(x_k \rightarrow C_j) \right),$$ where $I_{< 0}(b)$ is an indicator function which is `1' iff $b<0$ (respectively $I_{>0}$). If $C_j$ contains only $x_i$ (which cannot be the case in 3CNF formulas) then the message is set to 1.
Lastly, we define the current assignment of a variable $x_i$ to be $$ B_i = \left(\sum_{C_j \in N^+(x_i)} C_j \rightarrow x_i \right)- \left(\sum_{C_j\in N^-(x_i)}C_j \rightarrow x_i\right).$$ If $B_i>0$ then $x$ is assigned TRUE, if $B_i<0$ then $x_i$ is assigned FALSE, otherwise $x_i$ is UNASSIGNED. Assume some order on the clause-variable messages (e.g. the lexicographical order on pairs of the form $(j,i)$ representing the message $C_{j} \rightarrow x_i$). Given a vector $\alpha \in \{0,1\}^{3m}$ in which every entry is the value of the corresponding $C_{j}\rightarrow x_i$ message, a partial assignment $\psi\in \{TRUE,FALSE,UNASSIGNED\}^n$ can be generated according to the corresponding $B_i$ values (as previously explained).
Given a 3CNF formula ${\cal{F}}$ on $n$ variables and $m$ clauses, the \textbf{factor graph} of ${\cal{F}}$, denoted by $FG({\cal{F}})$, is the following graph representation of ${\cal{F}}$. The factor graph is a bipartite graph, $FG({\cal{F}})=(V_1\cup V_2,E)$ where $V_{1}=\{x_{1},x_2,...,x_n\}$ (the set of variables) and $V_2=\{C_1,C_2,...,C_m\}$ (the set of clauses). $(x_i,C_j)\in E$ iff
$x_i$ appears in $C_j$. For a 3CNF ${\cal{F}}$ with $m$ clauses it holds that $\#E=3m$, because every clause contains exactly 3 different variables. (Here and elsewhere, $\#A$ denotes the cardinality of a set $A$. The notation $|a|$ will denote the absolute value of a real number $a$.)
It would be convenient to think of the messages in terms of the corresponding factor graph. Every undirected edge $(x_i,C_j)$ of the factor graph is replaced with 2 anti-parallel directed edges, $(x_i\rightarrow C_j)$ associated with the message $x_i\rightarrow C_j$ and respectively the edge $(C_j\rightarrow x_i)$.
Let us now formally describe the algorithm. \\\\ $\verb"Warning Propagation(3CNF formula"$ ${\cal{F}}):$ \\ $\verb"1. construct the corresponding factor graph"$ $FG({\cal{F}}).$\\ $\verb"2. randomly initialize the clause-variable messages to 0 or 1."$\\ $\verb"3. repeat until no clause-variable message changed from the"$\\ $\verb" previous iteration:"$\\ $\verb" 3.a randomly order the edges of" $ $FG({\cal{F}}).$\\ $\verb" 3.b update all clause-variable messages"$ $C_j\rightarrow x_i$ $\verb"according"$\\ $\verb" to the random edge order."$\\ $\verb"4. compute a partial assignment"$ $\psi$ $\verb"according to the"$ $B_i$ $\verb"messages".$\\ $\verb"5. return "$ $\psi$.\\
The variable-clause message updates are implicit in line 3.b: when evaluating the clause-variable message along the edge $C \rightarrow x$, $C=(x\vee y\vee z)$, the variable-clause messages concerning this calculation ($z,y \rightarrow C$) are evaluated on-the-fly using the last updated values $C_i \rightarrow y$, $C_j \rightarrow z$ (allowing feedback from the same iteration). We allow the algorithm not to terminate (the clause-variable messages may keep changing every iteration). If the algorithm does return an assignment $\psi$ then we say that it converged. In practice it is common to limit in advance the number of iterations, and if the algorithm does not converge by then, return a failure.
\subsection{Main Results}\label{ResutlsSubs} Our contribution is analyzing the performance of \textsf{Warning Propagation} (\textsf{WP} for brevity), a popular message passing algorithm, when applied to satisfiable formulas drawn from a certain random distribution over satisfiable 3CNF formulas, commonly called the planted distribution. We show that the standard way of running message passing algorithms -- run message passing until convergence, simplify the formula according to the resulting assignment, and satisfy the remaining subformula, if possible, using a simple ``off the shelf" heuristic -- works for planted random satisfiable formulas with a sufficiently large constant clause-variable ratio. As such, our result is the first to rigorously prove the effectiveness of a message passing algorithm for the solution of a non-trivial random SAT distribution.
We note that a recent work \cite{AminAchi} demonstrated the usefulness of analytical tools developed for the planted distribution for ``hard" uniform instances.
To formally state our result we require a few additional definitions.
Given a 3CNF ${\cal{F}}$, \textbf{simplify} ${\cal{F}}$ according to $\psi$, when $\psi$ is a partial assignment, means: in every clause substitute every assigned variable with the value given to it by
$\psi$. If a clause contains a literal which evaluates to true, remove the clause. From the remaining clauses, remove all literals which evaluate to false. The resulting instance is not necessarily in 3CNF form, as clauses may have any number of literals between~0 and~3. Denote by ${\cal{F}}|_{\psi}$ the 3CNF ${\cal{F}}$ simplified according to $\psi$. Note that ${\cal{F}}|_{\psi}$ may contain empty clauses, in which case it is not satisfiable. For a set of variables $A\subseteq V$, denote by ${\cal{F}}[A]$ the set of clauses in which all variables belong to $A$.
We call a 3CNF formula \textbf{simple}, if it can be satisfied using simple well-known heuristics (examples include very sparse random 3CNF formulas which are solvable $whp$ using the pure-literal heuristic \cite{BorderFriezeUpfal}, formulas with small weight terminators -- to use the terminology of \cite{BenSassonAlke} -- solvable $whp$ using RWalkSat, etc). This is a somewhat informal notion, but it suffices for our needs.
\begin{thm}\label{ConvergenceThmMed} Let ${\cal{F}}$ be a 3CNF formula randomly sampled according to ${\cal{P}}^{{\rm plant}}_{n,p}$, where $p \geq d/n^2$, $d$ a sufficiently large constant. Then the following holds $whp$ (the probability taken over the choice of ${\cal{F}}$, and the random choices in lines 2 and 4 of the \textsf{WP} algorithm). There exists a satisfying assignment $\varphi^*$ (not necessarily the planted one) such that:
\begin{enumerate}
\item $\textsf{WP}({\cal{F}})$ converges after at most $O(\log n)$
iterations.
\item Let $\psi$ be the partial assignment returned by $\textsf{WP}({\cal{F}})$, let
$V_A$ denote the variables assigned to either TRUE or FALSE in
$\psi$, and $V_U$ the variables left UNASSIGNED. Then for every
variable $x\in V_A$,
$\psi(x)=\varphi^*(x)$. Moreover, $\#{V_A} \geq (1-e^{-\Theta(d)})n$.
\item
${\cal{F}}|_{\psi}$ is a simple formula which can be satisfied in time $O(n)$. \end{enumerate} \end{thm}
\begin{rem}\rm Theorem~\ref{ConvergenceThmMed} relates to the planted 3SAT model, but \cite{UniformSAT} implies that it also true to the uniform random 3SAT distribution, in which a \emph{satisfiable} formula with $m$ clauses is chosen uniformly at random among all such formulas. \end{rem}
\begin{prop}\label{ConvergencePropDense} Let ${\cal{F}}$ be a 3CNF formula randomly sampled according to ${\cal{P}}^{{\rm plant}}_{n,p}$, where $p\geq c\log n /n^2$, with $c$ a sufficiently large constant, and let $\varphi$ be its planted assignment. Then $whp$ after at most 2 iterations, $\textsf{WP}({\cal{F}})$ converges, and the returned $\psi$ equals $\varphi$. \end{prop}
It is worth noting that formulas in ${\cal{P}}^{{\rm plant}}_{n,p}$, with $n^2p$ some large constant, are \emph{not} known to be simple (in the sense that we alluded to above). For example, it is shown in \cite{BenSassonAlke} that RWalkSat is very unlikely to hit a satisfying assignment in polynomial time when running on a random ${\cal{P}}^{{\rm plant}}_{n,p}$ instance in the setting of Theorem \ref{ConvergenceThmMed}. Nevertheless, planted 3CNF formulas with sufficiently large (constant) density were shown to be solvable $whp$ in $\cite{flaxman}$ using a spectral algorithm. Though in our analysis we use similar techniques to \cite{flaxman} (which relies on \cite{AlonKahale97}), our result is conceptually different in the following sense. In \cite{AlonKahale97,ChenFrieze,flaxman}, the starting point is the planted distribution, and then one designs an algorithm that works well under this distribution. The algorithm may be designed in such a way that makes its analysis easier. In contrast, our starting point is a given algorithm, $\textsf{WP}$, and then we ask for which input distributions it works well. We cannot change the algorithm in ways that would simplify the analysis. Another difference between our work and that of \cite{AlonKahale97,ChenFrieze,flaxman} is that unlike the algorithms analyzed in those other papers, \textsf{WP} is a randomized algorithm, a fact which makes its analysis more difficult. We could have simplified our analysis had we changed \textsf{WP} to be deterministic (for example, by initializing all clause-variable messages to~1 in step~2 of the algorithm), but there are good reasons why \textsf{WP} is randomized. For example, it can be shown that (the randomized version) \textsf{WP} converges with probability~1 on 2CNF formulas that form one cycle of implications, but might not converge if step~4 does not introduce fresh randomness in every iteration of the algorithm.
The planted 3SAT model is also similar to LDPC codes in many ways. Both constructions are based on random factor graphs. In codes, the received corrupted codeword provides noisy information on a single bit or on the parity of a small number of bits of the original codeword. In ${\cal{P}}^{{\rm plant}}_{n,p}$, $\varphi$ being the planted assignment, the clauses containing a variable $x_i$ provide noisy information on the polarity of $\varphi(x_i)$. Comparing our results with the coding setting, the effectiveness of message passing algorithms for amplifying local information in order to decode codes close to channel capacity was established in a number of papers, e.g.~\cite{LMSS98,RSU01}. Our results are similar in flavor, however the combinatorial analysis provided here allows to recover an assignment satisfying \emph{all} clauses, whereas in the random LDPC codes setting, message passing allows to recover only $1-o(1)$ fraction of the codeword correctly. In \cite{LMSS01} it is shown that for the erasure channel, all bits may be recovered correctly using a message passing algorithm, however in this case the LDPC code is designed so that message passing works for it.
It is natural to ask whether our analysis can be extended to show that \textsf{Belief Propagation} (\textsf{BP}) finds a satisfying assignment to ${\cal{P}}^{{\rm plant}}_{n,p}$ in the setting of Theorem \ref{ConvergenceThmMed}. Experimental results predict the answer to be positive. However our analysis of \textsf{WP} does not extend as is to \textsf{BP}. In \textsf{WP}, all warnings received by a variable (or by a clause) have equal weight, but in \textsf{BP} this need not be the case (there is a probability level associated with each warning). In particular, this may lead to the case where a small number of ``very" wrongly assigned variables ``pollute" the entire formula, a possibility that our analysis managed to exclude for the \textsf{WP} algorithm.
\subsection{Paper's organization} The remainder of the paper is structured as follows. Section \ref{sec:intuition} provides an overview that may help the reader follow the more technical parts of the proofs. In Section \ref{PropOfRandomInstSection} we discuss some properties that a typical instance in ${\cal{P}}^{{\rm plant}}_{n,p}$ possesses. Using these properties, we prove in Section \ref{ResultsProofSection} Theorem \ref{ConvergenceThmMed} and Proposition \ref{ConvergencePropDense}.
\section{Proof Outline} \label{sec:intuition}
Let us first consider some possible fixed points of the Warning Propagation (WP) algorithm. The {\em trivial} fixed point is the one in which all messages are \textbf{0}. One may verify that this is the unique fixed point in some cases when the underlying 3CNF formula is very easy to satisfy, such as when all variables appear only positively, or when every clause contains at least two variables that do not appear in any other clause. A {\em local maximum} fixed point is one that corresponds to a strict local maximum of the underlying MAX-3SAT instance, namely to an assignment $\tau$ to the variables in which flipping the truth assignment of any single variable causes the number of satisfied clauses to strictly decrease. The reader may verify that if every clause $C$ sends a \textbf{1}\ message to a variable if no other variable satisfies $C$ under $\tau$, and a \textbf{0}\ message otherwise, then this is indeed a fixed point of the WP algorithm. Needless to say, the WP algorithm may have other fixed points, and might not converge to a fixed point at all.
Recall the definition of ${\cal{P}}^{{\rm plant}}_{n,p}$. First a truth assignment $\varphi$ to the variables $V=\{x_{1},x_{2},...,x_{n}\}$ is picked uniformly at random. Next, every clause satisfied by $\varphi$ is included in the formula with probability $p$ (in our case $p\geq d/n^2$, $d$ a sufficiently large constant). There are $(2^3-1)\cdot\binom{n}{3}$ clauses satisfied by $\varphi$, hence the expected size of ${\cal{F}}$ is $p\cdot7\cdot\binom{n}{3} = 7 d n/6+o(n)$ (when $d$ is constant, then this is linear in $n$, and therefore such instances are sometimes referred to as \emph{sparse} 3CNF formulas). To simplify the presentation, we assume w.l.o.g. (due to symmetry) that the planted assignment $\varphi$ is the all-one vector.
To aid intuition, we list some (incorrect) assumptions and analyze the performance of WP on a ${\cal{P}}^{{\rm plant}}_{n,p}$ instance under these assumptions.
\begin{enumerate}
\item In expectation, a variable appears in $4\binom{n}{2}p=2d+o(1)$ clauses positively, and in $3d/2+o(1)$ clauses negatively. Our first assumption is that for every variable, its number of positive and negative appearances is equal to these expectations.
\item We say that a variable {\em supports} a clause with respect to the planted assignment (which was assumed without loss of generality to be the all \textbf{1}\ assignment) if it appears positively in the clause, and the other variables in the clause appear negatively. Hence the variable is the only one to satisfy the clause under the planted assignment. For every variable in expectation there are roughly $d/2$ clauses that it supports. Our second assumption is that for every variable, the number of clauses that it supports is equal to this expectation.
\item Recall that in the initialization of the \textsf{WP} algorithm, every clause-variable message $C\to x$ is 1 w.p. $\frac{1}{2}$, and 0 otherwise. Our third assumption is that with respect to every variable, half the messages that it receives from clauses in which it is positive are initialized to~1, and half the messages that it receives from clauses in which it is negative are initialized to~1.
\item Recall that in step 3b of \textsf{WP}, clause-variable messages are updated in a random order. Our fourth assumption is that in each iteration of step~3, the updates are based on the values of the other messages from the previous iteration, rather than on the last updated values of the messages (that may correspond either to the previous iteration or the current iteration, depending on the order in which clause-variable messages are visited). Put differently, we assume that in step 3b all clause-variable messages are evaluated in \emph{parallel}.
\end{enumerate}
Observe that under the first two assumptions, the planted assignment is a local maximum of the underlying MAX-3SAT instance. We show that under the third and fourth assumption, \textsf{WP} converges to the corresponding local maximum fixed point in two iterations: Based on the initial messages as in our third assumption, the messages that variables send to clauses are all roughly $(2d-3d/2)/2=d/4$. Following the initialization, in the first iteration of step~3 every clause $C$ that $x$ supports will send $x$ the message 1, and all other messages will be 0. Here we used our fourth assumption. (Without our fourth assumption, \textsf{WP} may run into trouble as follows. The random ordering of the edges in step 3 may place for some variable $x$ all messages from clauses in which it appears positively before those messages from clauses in which it appears negatively. During the iteration, some of the messages from the positive clauses may change from 1 to 0. Without our fourth assumption, this may at some point cause $x$ to signal to some clauses a negative rather than positive value.) The set of clause-variable messages as above will become a fixed point and repeat itself in the second iteration of step 3. (For the second iteration, the fourth assumption is no longer needed.) Hence the algorithm will terminate after the second iteration.
Unfortunately, none of the four assumptions that we made are correct. Let us first see to what extent they are violated in the context of Proposition~\ref{ConvergencePropDense}, namely, when $d$ is very large, significantly above $\log n$. Standard concentration results for independent random variables then imply that the first, second and third assumptions simultaneously hold for all variables, up to small error terms that do not effect the analysis. Our fourth assumption is of course never true, simply because we defined \textsf{WP} differently. This complicates the analysis to some extent and makes the outcome depend on the order chosen in the first iteration of step 3a of the algorithm. However, it can be shown that for most such orders, the algorithm indeed converges to the fixed point that corresponds to the planted assignment.
The more difficult part of our work is the case when $d$ is a large constant. In this case, already our first two assumptions are incorrect. Random fluctuations with respect to expected values will $whp$ cause a linear fraction of the variables to appear negatively more often than positively, or not to support any clause (with respect to the planted assignment). In particular, the planted assignment would no longer be a local maximum with respect to the underlying MAX-3SAT instance. Nevertheless, as is known from previous work~\cite{flaxman}, a large fraction of the variables will behave sufficiently close to expectation so that the planted assignment is a local maximum with respect to these variables. Slightly abusing notation, these set of variables are often called the {\em core} of the 3CNF formula. Our proof plan is to show that $\textsf{WP}$ does converge, and that the partial assignment in step 4 assigns all core variables their correct planted value. Moreover, for non-core variables, we wish to show that the partial assignment does not make any unrecoverable error -- whatever value it assigns to some of them, it is always possible to assign values to those variables that are left unassigned by the partial assignment so that the input formula is satisfied. The reason why we can expect such a proof plan to succeed is that it is known to work if one obtains an initial partial assignment by means other than $\textsf{WP}$, as was already done in~\cite{flaxman,TechReport}.
Let us turn now to our third assumption. It too is violated for a linear fraction of the variables, but is nearly satisfied for most variables. This fact marks one point of departure for our work compared to previous work~\cite{flaxman,TechReport}. Our definition of the core variables will no longer depend only on the input formula, but also on the random choice of initialization messages. This adds some technical complexity to our proofs.
The violation of the fourth assumption is perhaps the technical part in which our work is most interesting. It relates to the analysis of $\textsf{WP}$ on factor graphs that contain cycles, which is often a stumbling point when one analyzes message passing algorithms. Recall that when $d$ is very large (Proposition~\ref{ConvergencePropDense}), making the fourth assumption simplifies the proof of convergence of WP. Hence removing this assumption in that case becomes a nuisance. On the other hand, when $d$ is smaller (as in Theorem~\ref{ConvergenceThmMed}), removing this assumption becomes a necessity. This will become apparent when we analyze convergence of WP on what we call {\em free cycles}. If messages in step 3b of \textsf{WP} are updated based on the value of other messages in the {\em previous} iteration (as in our fourth assumption), then the random choice of order in step 3a of \textsf{WP} does not matter, and one can design examples in which the messages in a free cycle never converge. In contrast, if messages in step 3b of \textsf{WP} are updated based on the latest value of other messages (either from the previous iteration or from the current iteration, whichever one is applicable), free cycles converge with probability 1 (as we shall later show).
To complete the proof plan, we still need to show that simplifying the input formula according to the partial assignment returned by \textsf{WP} results in a formula that is satisfiable, and moreover, that a satisfying assignment for this sub-formula can easily be found. The existential part (the sub-formula being satisfiable) will follow from a careful analysis of the partial assignment returned by \textsf{WP}. The algorithmic part (easily finding an assignment that satisfies the sub-formula) is based on the same principles used in~\cite{AlonKahale97,flaxman}, showing that the sub-formula breaks into small connected components.
\section{Properties of a Random ${\cal{P}}^{{\rm plant}}_{n,p}$
Instance}\label{PropOfRandomInstSection}
In this section we discuss relevant properties of a random ${\cal{P}}^{{\rm plant}}_{n,p}$ instance. This section is rather technical in nature. The proofs are based on probabilistic arguments that are standard in our context. In the rest of the paper, for simplicity of presentation, we assume w.l.o.g. that the planted assignment is the all TRUE assignment.
\subsection{Preliminaries}
The following well-known concentration result (see, for example
\cite[p. 21]{JRL2000}) will be used several times in the proof. We denote by $B(n,p)$ the binomial random variable with parameters $n$ and $p$, and expectation $\mu=np$. \begin{thm}\label{Thm:Chernoff2}(Chernoff's inequality) If $X \sim B(n,p)$ and $t \geq 0$ is some number, then \[ Pr\big(X \geq \mu + t) \leq e^{-\mu \cdot f(t/\mu)}, \] \[ Pr\big(X \leq \mu - t) \leq e^{-\mu \cdot f(-t/\mu)},\]
where $f(x) = (1+x)\ln (1+x) - x$. \end{thm} \noindent We use the following Azuma-Hoeffding inequality for martingales:
\begin{thm}\label{Thm:RegMartingale} Let $\{X_i\}_{i=0}^{N}$ be a martingale with $|X_k - X_{k-1}| \leq c_k$, then
\[Pr \left[|X_N - X_0| \geq t\right] \leq 2 e^{-t^2 / \sum_{k=1}^N c_k^2}\] \end{thm} We shall also use the following version of the martingale argument, which can be found in \cite[Page 101]{TheProbMethod}. Assume the probability space is generated by a finite set of mutually independent Yes/No choices, indexed by $i\in I$. Given a random variable $Y$ on this space, let $p_i$ denote the probability that choice $i$ is Yes. Let $c_i$ be such that changing choice $i$ (keeping all else the same) can change $Y$ by at most $c_i \leq c$. We call $p_i (1 - p_i )c^2$ the variance of choice $i$. Now consider a solitaire game in which one finds the value of $Y$ by making queries of an always truthful oracle. The choice of query can depend on previous responses. All possible questioning lines can be naturally represented in a decision tree form. A ``line of questioning'' is a path from the root to a leaf of this tree, a sequence of questions and responses that determine $Y$. The total variance of a line of questioning is the sum of the variances of the queries in it.
\begin{thm}\label{thm_martingale} For every $\varepsilon > 0$ there exists a $\delta > 0$ so that if for every line of questioning, with parameters $p_1,p_2,\ldots,p_n$ and $c_1,c_2,\ldots,c_n$, that determines $Y$, the total variance is at most $\sigma^2$, then
$$Pr[|Y -E[Y ]| > \alpha\sigma] \leq 2e^{-\alpha^2/(2(1+\varepsilon))},$$ for all positive $\alpha$ with $\alpha c < \sigma(1 + \varepsilon)\delta$, where $c$ is such that $\forall i \phantom{i} c_i \leq c$. \end{thm}
\subsection{Stable Variables}\label{StableVarsSubs} \begin{defn} A variable $x$ \textbf{supports} a clause $C$ with respect to a partial assignment $\psi$, if it is the only variable to satisfy $C$ under $\psi$, and the other two variables are assigned by $\psi$. \end{defn}
\begin{prop}\label{SupportSuccRate} Let ${\cal{F}}$ be a 3CNF formula randomly sampled according to ${\cal{P}}^{{\rm plant}}_{n,p}$, where $p = d/n^2$, $d \geq d_0$, $d_0$ a sufficiently large constant. Let $F_{SUPP}$ be a random variable counting the number of variables in ${\cal{F}}$ whose support w.r.t. $\varphi$ is less than $d/3$. Then $whp$ $F_{SUPP}\leq e^{-\Theta(d)}n$. \end{prop} \begin{Proof} Every variable is expected to support $\frac{d}{n^2}\cdot\binom{n}{2}=\frac{d}{2}+O(\frac{1}{n})$ clauses, thus using Chernoff's inequality the probability of a single variable supporting less than $d/3$ clauses is at most $e^{-\Theta(d)}$. Using linearity of expectation, $\mathbb{E}[F_{SUPP}]\leq e^{-\Theta(d)}n$. To prove concentration around the expected value we use Chernoff's bound once more as the support of one variable is independent of the others (since it concerns different clauses which are included independently of each other). The claim then follows. \end{Proof}
Following the definitions in Section \ref{WPSubs}, given a 3CNF ${\cal{F}}$ with a satisfying assignment $\varphi$, and a variable $x$, we let $N^{++}(x)$ be the set of clauses in ${\cal{F}}$ in which $x$ appears positively but doesn't support w.r.t. $\varphi$. Let $N^{s}(x)$ be the set of clause in ${\cal{F}}$ which $x$ supports w.r.t. $\varphi$. Let $\pi=\pi({\cal{F}})$ be some ordering of the clause-variable message edges in the factor graph of ${\cal{F}}$. For an index $i$ and a literal $\ell_x$ (by $\ell_x$ we denote a literal over the variable $x$) let $\pi^{-i}(\ell_x)$ be the set of clause-variable edges $(C \rightarrow x)$ that appear before index $i$ in the order $\pi$ and in which $x$ appears in $C$ as $\ell_x$. For a set of clause-variable edges $\cal{E}$ and a set of clauses $\cal{C}$ we denote by ${\cal{E}}\cap{\cal{C}}$ the subset of edges containing a clause from $\cal{C}$ as one endpoint.
\begin{defn}\label{StableDefn} Let $\pi$ be an ordering of the clause-variable messages of a 3CNF formula ${\cal{F}}$. Let $\varphi$ be a satisfying assignment of ${\cal{F}}$. A variable $x$ is \textbf{$d$-stable} in ${\cal{F}}$ w.r.t. $\pi$ and $\varphi$ if for every location $i$ in $\pi$ that contains a message $C\rightarrow x$, $C=(\ell_x\vee \ell_y \vee \ell_z)$, the following holds:
\begin{enumerate} \item $|\#\pi^{-i}(y)\cap N^{++}(y)-\#\pi^{-i}(\bar{y})\cap N^{-}(y)|\leq d/30$.
\item $|\#N^{++}(y)-\#N^{-}(y)|\leq d/30$. \item $\#N^{s}(y)\geq d/3$ \end{enumerate} and the same holds for $z$. \end{defn} When $d$ is clear from context, which will usually be the case, we will suppress the $d$ in the ``$d$-stable". \begin{prop}\label{StableSuccRate} Let ${\cal{F}}$ be a 3CNF formula randomly sampled according to ${\cal{P}}^{{\rm plant}}_{n,p}$, where $p = d/n^2$, $d \geq d_0$, $d_0$ a sufficiently large constant. Let $\pi$ be a random ordering of the clause-variable messages, and $F_{UNSTAB}$ be a random variable counting the number of variables in ${\cal{F}}$ which are \emph{not} stable. Then $whp$ $F_{UNSTAB}\leq e^{-\Theta(d)}n$. \end{prop} \begin{Proof} We start by bounding $\mathbb{E}[F_{UNSTAB}]$. Consider a clause-variable message edge $C \rightarrow x$ in location $i$ in $\pi$, $C=(\ell_x\vee \ell_y \vee \ell_z)$. Now consider location $j \leq i$. The probability of an edge $C' \rightarrow \bar{y}$ in location $j$ is $\left(3\binom{n}{2}\right)/\left(7\binom{n}{3}\right)=\frac{9}{7n}+O(\frac{1}{n^2})$ which is exactly the probability of an edge $C'' \rightarrow y$, $C'' \in N^{++}(y)$. This implies
$$E[|\#\pi^{-i}(y)\cap N^{++}(y)-\#\pi^{-i}(\bar{y})\cap N^{-}(y)|]=0.$$ If however
$$|\#\pi^{-i}(y)\cap N^{++}(y)-\#\pi^{-i}(\bar{y})\cap N^{-}(y)| > d/30$$ then at least one of the quantities deviates from its expectation by $d/60$.
Look at $\#\pi^{-i}(y)\cap N^{++}(y)$ -- this is the number of successes in draws without replacement. It is known that this quantity is more concentrated than the corresponding quantity if the draws were made with replacement~\cite{Hoeffding:63}. In particular, since the expectation of $\#\pi^{-i}(y)\cap N^{++}(y)$ is $O(d)$ it follows from Chernoff's bound that the probability that it deviates from its expectation by more than $d/60$ is $e^{-\Theta(d)}$. A similar statement holds for $\#\pi^{-i}(\bar{y})\cap N^{-}(y)$. Properties $(b)$ and $(c)$ are bounded similarly using concentration results.
\noindent The calculations above hold in particular for the first $5d$ appearances of messages involving $x$. As for message $5d+1$, the probability of this message causing $x$ to become unstable is bounded by the event that $x$ appears in more than $5d$ clauses. As $x$ is expected to appear in $3.5d$ clauses, the latter event happens w.p. $e^{-\Theta(d)}$ (again using Chernoff's bound). To sum up, $$Pr[x\text{ is unstable }]\leq 5d\cdot e^{-\Theta(d)}+e^{-\Theta(d)}=e^{-\Theta(d)}.$$ The bound on $E[F_{UNSTAB}]$ follows by linearity of expectation.
We are now left with proving that $F_{UNSTAB}$ is concentrated around its expectation, we do so using a martingale argument. Define two new random variables, $F_1$ counting the number of unstable variables $x$ s.t. there exists a clause $C$, containing $x$, and another variable $y$, s.t. $y$ appears in more than $\log n$ clauses, and $F_2$ to be the unstable variables s.t. in all clauses in which they appear, all the other variables appear in at most $\log n$ clauses. Observe that $F_{UNSTAB}=F_1+F_2$. To bound $F_1$, observe that if $F_1\geq 1$, then in particular this implies that there exists a variable which appears in more than $\log n$ clauses in ${\cal{F}}$. This happens with probability $o(1)$ since every variable is expected to appear only in $O(d)$ clauses (using Chernoff's bound). To bound $F_2$ we use a martingale argument in the constellation of \cite{TheProbMethod}, page 101. We use the clause-exposure martingale (the clause-exposure martingale implicitly includes the random ordering $\pi$, since one can think of the following way to generate the random instance -- first randomly shuffle all possible clauses, and then toss the coins). The exposure of a new clause $C$ can change $F_2$ by at most $6\log n$ since every variable in $C$ appears in at most $\log n$ clauses, namely with at most $2\log n$ other variables that might become (un)stable due to the new clause. The martingale's total variance, in the terminology of Theorem \ref{thm_martingale}, is $\sigma^2=\Theta(dn\log ^2 n)$. Theorem \ref{thm_martingale}, with $\alpha=e^{-\Theta(d)}\sqrt{n}/\log n$, and the fact that $E[F_2]\leq E[F_{UNSTAB}]$, implies concentration around the expectation of $F_2$. \end{Proof}
Let $\alpha\in\{0,1\}^{3\#{\cal{F}}}$ be a clause-variable message vector. For a set of clause-variable message edges $\cal{E}$ let $\textbf{1}_\alpha(\cal{E})$ be the set of edges along which the value is 1 according to $\alpha$. For a set of clauses $\cal{C}$, $\textbf{1}_\alpha(\cal{C})$ denotes the set of clause-variable message edges in the factor graph of ${\cal{F}}$ containing a clause from $\cal{C}$ as one endpoint and along which the value is 1 in $\alpha$.
\begin{defn}\label{ViolatedDefn} Let $\pi$ be an ordering of the clause-variable messages of a 3CNF formula ${\cal{F}}$, and $\alpha$ a clause-variable message vector. Let $\varphi$ be a satisfying assignment of ${\cal{F}}$. We say that a variable $x$ is \textbf{$d$-violated} by $\alpha$ in $\pi$ if there exists a message $C \rightarrow x$, $C=(\ell_x\vee \ell_y\vee \ell_z)$, in place $i$ in $\pi$ s.t. one of the following holds: \begin{enumerate}
\item $|\#\textbf{1}_\alpha(\pi^{-i}(y)\cap N^{++}(y))-\#\textbf{1}_\alpha(\pi^{-i}(\bar{y})\cap N^{-}(y))|> d/30$
\item $|\#\textbf{1}_\alpha(N^{++}(y))-\#\textbf{1}_\alpha(N^{-}(y))|> d/30$ \item $\#\textbf{1}_\alpha(N^s(y))< d/7$. \end{enumerate} Or one of the above holds for $z$. \end{defn}
We say that a variable is \emph{$r$-bounded} in ${\cal{F}}$ if it appears in at most $r$ clauses. We say that a variable $x$ has an \emph{$r$-bounded neighborhood} in ${\cal{F}}$ if every clause $C=(\ell_x\vee \ell_y \vee \ell_z)$ in ${\cal{F}}$ that contains $x$ is such that $y$ and $z$ are $r$-bounded ($x$ itself is not limited)
\begin{lem}\label{GeneralViolatedSuccRate} Let F be an arbitrary satisfiable 3CNF formula, let $\psi$ be an arbitrary satisfying assignment for $F$, and let $\pi$ be an arbitrary ordering of clause-variable messages. For a given value $d$, let $X$ be the set of variables that have a $20d$-bounded neighborhood in $F$ and are $d$-stable with respect to $\psi$ and $\pi$.
Let $\alpha$ be a random clause-variable message vector, and $F_{VIO}$ a random variable counting the number of $d$-violated variables in $X$. Then $whp$ $F_{VIO}\leq e^{-\Theta(d)}n$. \end{lem} \begin{Proof} First observe that the probability in the statement is taken only over the coin tosses in the choice of $\alpha$, as all other parameters are fixed. As in the proof of Proposition \ref{StableSuccRate}, we first bound $\mathbb{E}[F_{VIO}]$, and then prove concentration using a martingale argument. Fix $x \in X$ and a clause-variable message $C \rightarrow x$ at location $i$ in $\pi$, $C=(\ell_x\vee \ell_y \vee \ell_z)$. Let $A^+=\pi^{-i}(y)\cap N^{++}(y)$ (the set of messages preceding location $i$ where $y$ appears positively but doesn't support the clause) and $A^-=\pi^{-i}(y)\cap N^{-}(y)$ (the set of messages preceding location $i$ where $y$ appears negatively). Let us assume w.l.o.g that $\#A^+ \geq \#A^-$. Stability implies $$\#A^+-\#A^-\leq d/30.$$ Since $\alpha$ is a random assignment to the clause-variable messages, $$\mathbb{E}[\#\textbf{1}_\alpha(A^+)]-\mathbb{E}[\#\textbf{1}_\alpha(A^-)]\leq d/60.$$ If however \begin{equation}\label{equa}
|\#\textbf{1}_\alpha(A^+)-\#\textbf{1}_\alpha(A^-)|> d/30, \end{equation} then at least one of $\#\textbf{1}_\alpha(A^+),\#\textbf{1}_\alpha(A^-)$ deviated from its expectation by at least $(d/30-d/60)/2=d/120$. Both quantities are binomially distributed with expectation $O(d)$ ($x$ has a $20d$-bounded neighborhood, and therefore $y$ appears in at most $20d$ clauses), and therefore the probability of the latter happening is $e^{-\Theta(d)}$ (using standard concentration results). Properties $(b)$ and $(c)$ are bounded similarly, and the same argument takes care of $z$. Using the union bound and the linearity of expectation, one obtains that $\mathbb{E}[F_{VIO}] \leq e^{-\Theta(d)}\#X \leq e^{-\Theta(d)}n$. Let us use Theorem \ref{Thm:RegMartingale} to obtain concentration around $\mathbb{E}[F_{VIO}]$. Let $Y$ be the set of variables that share some clause with a variable in $X$. Expose the value of messages $C \to y$ for $y \in Y$. Since all $y \in Y$ are $20d$-bounded, the length of the martingale, $N = O(dn)$. The boundedness condition also gives that the martingale difference is $O(d)$. Taking $t=e^{-\Theta(d)}n$, and plugging all the parameters in Theorem \ref{Thm:RegMartingale}, the result follows.
\end{Proof}
\begin{prop}\label{ViolatedSuccRate} Let ${\cal{F}}$ be a 3CNF formula randomly sampled according to ${\cal{P}}^{{\rm plant}}_{n,p}$, where $p = d/n^2$, $d \geq d_0$, $d_0$ is a sufficiently large constant. Let $\pi$ be a random ordering of the clause-variable messages and $\alpha$ a random clause-variable message vector. Then $whp$ the number of stable variables which are violated in $\alpha$ is at most $e^{-\Theta(d)}n$. \end{prop} \begin{Proof} Let $X'$ be the set of stable variables in ${\cal{F}}$ w.r.t.~$\pi$, and $X \subseteq X'$ be the set of variables with a $20d$-bounded neighborhood in ${\cal{F}}$. Lemma \ref{GeneralViolatedSuccRate} guarantees that the number of violated variables in $X$ is at most $e^{-\Theta(d)}n$. It suffices to show that $\#(X' \setminus X) \leq e^{-\Theta(d)}n$. Let $Z_t$ be the set of variables that appear in $t$ clauses in ${\cal{F}}$. The number of clauses in which a variable appears is binomially distributed with expected value $\mu = 7\binom{n}{2}d/n^2 \leq 7d/2$. Using Theorem \ref{Thm:Chernoff2}, it holds that $$Pr[x \in Z_t] \leq e^{-\mu \cdot f((t-\mu)/\mu)},$$ where $f(x) = (1+x)\ln (1+x) - x$. For $t \geq 20d$, $f((t-\mu)/\mu) \geq t/(2\mu)$, and therefore $$Pr[x \in Z_t \, , t \geq 20d] \leq e^{-t/2}.$$ Using the linearity of expectation, and standard concentration results, it holds that $whp$ for every $t \geq 20d$, $\# Z_t \leq e^{-t/3}n$. The number of variables in $X' \setminus X$ is then $whp$ at most $$\sum_{t \geq 20d} 2t \cdot \#Z_t \leq \sum_{t \geq 20d} 2t \cdot e^{-t/3}n \leq e^{-\Theta(d)}n.$$ (Every variable in $Z_t$ ``spoils" at most two other variables in every clause in which it appears, which possibly end up in $X' \setminus X$). \end{Proof}
\subsection{Dense Subformulas}\label{DesnseSubSubs} The next property we discuss is analogous to a property proved in \cite{AlonKahale97} for random graphs. Loosely speaking, a random graph typically doesn't contain a small induced subgraph with a large average degree. A similar proposition for 3SAT can also be found in \cite{flaxman}. \begin{prop}\label{NoDenseSubformulas} Let $c > 1$ be an arbitrary constant. Let ${\cal{F}}$ be a 3CNF formula randomly sampled according to ${\cal{P}}^{{\rm plant}}_{n,p}$, where $p = d/n^2$, $d \geq d_0$, $d_0$ a sufficiently large constant. Then $whp$ there exists \emph{no} subset of variables $U$, s.t. $\#U \leq e^{-\Theta(d)} n$ and there are at least $c\#U$ clauses in ${\cal{F}}$ containing two variables from $U$. \end{prop} \begin{Proof} For a fixed set $U$ of variables, $\#U=k$, the number of clauses containing two variables from $U$ is $$\binom{k}{2}(n-2)2^3 \leq 4k^2n.$$\\ Each of these clauses is included independently w.p. $\frac{d}{n^2}$. Thus, the probability that $ck$ of them are included is at most $$\binom{4k^2n}{ck}\left(\frac{d}{n^2}\right)^{ck}\leq \left(\frac{4k^2ne}{ck}\cdot \frac{d}{n^2}\right)^{ck} \leq \left(\frac{12
kd }{cn}\right)^{ck}.$$ Using the union bound, the probability there exists a ``dense" set $U$ is at most $$\sum_{k=2}^{e^{-\Theta(d)} n}\binom{n}{k}\left(\frac{12
kd }{cn}\right)^{ck}=O(d^{2c}/n^{2c-2}).$$ The last equality is obtained using standard calculations, and the standard estimate on the binomial coefficient: $\binom{n}{k} \leq (en/k)^k$. \end{Proof}
\subsection{The Core Variables}\label{CoreSubsection} We describe a subset of the variables, denoted throughout by ${\cal{H}}$ and referred to as the \emph{core variables}, which plays a crucial role in the analysis. The notion of a stable variable is not enough to ensure that the algorithm will set a stable variable according to the planted assignment, as it may happen that a stable variable $x$ appears in many of its clauses with unstable variables. Thus, $x$ can be biased in the wrong direction (by wrong we mean disagreeing with the planted assignment). However, if most of the clauses in which $x$ appears contain only stable variables, then this is already a sufficient condition to ensure that $x$ will be set correctly by the algorithm. The set ${\cal{H}}$ captures the notion of such variables. There are several ways to define a set of variables with these desired properties, we present one of them, and give a constructive way of obtaining it (though it has no algorithmic implications, at least not in our context).
\noindent Formally, ${\cal{H}}={\cal{H}}({\cal{F}},\varphi,\alpha,\pi)$ is constructed using the following iterative procedure: \begin{figure*}\end{figure*} \begin{prop}\label{SizeOfHBarPr} If both $\alpha$ and $\pi$ are chosen uniformly at random then with probability $1-n^{-\Omega(d)}$, $\#{\cal{H}} = (1-e^{-\Omega(d)})n$. \end{prop} \begin{Proof} Let $\bar{{\cal{H}}}=V\setminus {\cal{H}}$. Set $\delta = e^{-\Theta(d)}$. Partition the variables in $\bar{{\cal{H}}}$ into variables that belong to $A_1 \cup A_2 \cup A_3$, and variables that were removed in the iterative step, $\bar{H}^{it}=H_{0}\setminus {\cal{H}}$. If $\#\bar{{\cal{H}}}\geq \delta n$, then at least one of $A_1 \cup A_2 \cup A_3$, $\bar{H}^{it}$ has cardinality at least $\delta n/2$. Consequently, \begin{align*} Pr[\#\bar{{\cal{H}}}\geq \delta n] \leq \underbrace{Pr[\#A_1 \cup A_2 \cup A_3\geq \delta n/2]}_{(a)}+\underbrace{Pr[\#\bar{H}^{it} \geq
\delta n/2\text{ }\big|\text{ }\#A_1 \cup A_2 \cup A_3 \leq\delta n/2]}_{(b)}. \end{align*} Propositions \ref{SupportSuccRate}, \ref{StableSuccRate}, and \ref{ViolatedSuccRate} are used to bound $(a)$. To bound $(b)$, observe that every variable that is removed in iteration $i$ of the iterative step (step 2), supports at least $(d/3-d/4)=d/12$ clauses in which at least another variable belongs to $\{a_{1},a_{2},...,a_{i-1}\}\cup A_1 \cup A_2 \cup A_3$, or appears in $d/30$ clauses each containing at least one of the latter variables. Consider iteration $\delta n/2$. Assuming $\#A_1 \cup A_2 \cup A_3 \leq\delta n/2$, by the end of this iteration there exists a set containing at most $\delta n$ variables, and there are at least $d/30\cdot\delta n/2\cdot 1/3$ clauses containing at least two variables from it (we divide by 3 as every clause might have been counted 3 times). Plugging $c=d/180$ in Proposition \ref{NoDenseSubformulas}, $(b)$ is bounded. Finally observe that $(a)$ occurs with exponentially small probability, and $(b)$ occurs with probability $n^{-\Omega(d)}$. \end{Proof} \subsection{The Factor Graph of the non-Core Variables}\label{NonCoreStructSec} Proposition \ref{SizeOfHBarPr} implies that for $p=c\log n/n^2$, $c$ a sufficiently large constant, $whp$ ${\cal{H}}$ contains already all variables. Therefore the following propositions are relevant for the setting of Theorem \ref{ConvergenceThmMed} (namely, $p=O(1/n^2)$). In what follows we establish the typical structure of the factor graph induced on the non-core variables.
\begin{prop}\label{prop_NonCoreFactorGraph} Let ${\cal{F}}$ be a 3CNF formula randomly sampled according to ${\cal{P}}^{{\rm plant}}_{n,p}$, where $p = d/n^2$, $d \geq d_0$, $d_0$ a sufficiently large constant, let $\pi$ be the initial random ordering of the clause-variable messages, and $\alpha$ the initial random clause-variable message vector. Let $T$ be the factor graph induced on the non-core variables. $T$ enjoys the following properties: \begin{enumerate}
\item Every connected component contains $whp$ $O(\log n)$ variables,
\item every connected component contains $whp$ at most one cycle,
\item the probability of a cycle of length at least $k$ in $T$ is at most $e^{-\Theta(dk)}$. \end{enumerate} \end{prop}
A proposition of similar flavor to $(a)$ was proven in \cite{flaxman} though with respect to a different notion of core. This alone would not have sufficed to prove our result, and we need $(b)$ and $(c)$ as well.
\begin{cor}\label{cor:NoLongCycle} Let $f=f(n)$ be such that $f(n)\rightarrow \infty$ as $n\to \infty$. Then $whp$ there is no cycle of length $f(n)$ in the non-core factor graph. \end{cor}
The proof of Proposition \ref{prop_NonCoreFactorGraph} is quite long and technical. To avoid distraction, we go ahead and prove Theorem \ref{ConvergenceThmMed} and Proposition \ref{ConvergencePropDense}, and defer the proof of Proposition \ref{prop_NonCoreFactorGraph} to Section \ref{sec:NoTwoCycleProof}.
\section{Proof of Theorem \ref{ConvergenceThmMed} and Proposition \ref{ConvergencePropDense}}\label{ResultsProofSection} We start by giving an outline of the proof of Theorem \ref{ConvergenceThmMed}. Proposition \ref{ConvergencePropDense} is derived as an easy corollary of that proof. To prove Theorem \ref{ConvergenceThmMed} we need to establish three properties:
\begin{enumerate}
\item {\em Convergence}: the WP algorithm converges to a fixed point.
\item {\em Consistency}: the partial assignment implied by this fixed point is consistent with some satisfying assignment.
\item {\em Simplicity}: after simplifying the input formula by substituting in the values of the assigned variables, the remaining subformula is not only satisfiable (this is handled by consistency), but also simple.
\end{enumerate}
We assume that the formula ${\cal{F}}$ and the execution of $\textsf{WP}$ are {\em typical} in the sense that Propositions \ref{SizeOfHBarPr} and \ref{prop_NonCoreFactorGraph} hold. First we prove that after one iteration \textsf{WP} sets the core variables ${\cal{H}}$ correctly ($B_i$ agrees with $\varphi$ in sign) and this assignment does not change in later iterations. The proof of this property is rather straightforward from the definition of a core. This establishes convergence and consistency for the core variables. From iteration 2 onwards $\textsf{WP}$ is basically running on ${\cal{F}}$ in which variables belonging to ${\cal{H}}$ are substituted with their planted assignment. This subformula is satisfiable. Moreover, its factor graph contains small (logarithmic size) connected components, each containing at most one cycle. This last fact serves a dual purpose. It shows that if WP will eventually converge, the simplicity property will necessarily hold. Moreover, it will assist us in proving convergence and consistency for the subformula. Consider a connected component composed of a cycle and trees ``hanging" on the cycle. Proving convergence on the trees is done using a standard inductive argument. The more interesting part is proving convergence on the cycle. The difficulty there is that messages on a cycle may have more than one fixed point to which they may possibly converge, which makes it more difficult to prove that they converge at all. Our proof starts with a case analysis that identifies those cases that have multiple fixed points. For these cases we prove that almost surely random fluctuations caused by step~3.a of the WP algorithm will lead to convergence to some fixed point. This is similar in flavor to the fact that a random-walk on a line eventually reaches an endpoint of the line (even though one cannot tell a-priori which endpoint this will be). Hand-in-hand with establishing convergence for the trees and cycle, we shall also prove consistency.
The set $V_A$ of Theorem~\ref{ConvergenceThmMed} is composed of all variables from ${\cal{H}}$ and those variables from the non-core factor graph that get assigned. The set $V_U$ is composed of the UNASSIGNED variables from non-core factor graph. We now proceed with the formal proof.
\subsection{Analysis of \textsf{WP} on the core factor graph} We start by proving that the messages concerning the factor graph induced by the core-variables converge to the correct value, and remain the same until the end of the execution.
We say that a message $C \rightarrow x$, $C=(\ell_x \vee \ell_y \vee \ell_z)$, is \emph{correct} if its value is the same as it is when $y \rightarrow C$ and $z\rightarrow C$ are 1 (that is agree in sign with their planted assignment). In other words, $C \rightarrow x$ is 1 iff $C=(x\vee \bar{y}\vee\bar{z})$ ($x$ supports $C$).
\begin{prop} \label{prop:invariant} If $x_i\in {\cal{H}}$ and all messages $C \rightarrow x_i$, $C \in {\cal{F}}[{\cal{H}}]$ are correct at the beginning of an iteration (line 3 in the \textsf{WP} algorithm), then this invariant is kept by the end of that iteration. \end{prop} \begin{Proof} By contradiction, let $C_0 \rightarrow x$ be the first wrongly evaluated message in the iteration. W.l.o.g. assume $C_0=(\ell_x\vee \ell_y \vee \ell_z)$. Then at least one of $y,z$ sent a wrong message to $C_0$. $$y \rightarrow C_0 = \sum_{C \in N^+(y),C\neq C_0} C \rightarrow y - \sum_{C'\in N^-(y),C'\neq C_0}C' \rightarrow y.$$ Every message $C'' \rightarrow y$, $C''\in F[{\cal{H}}]\cap \{N^{++}(y)\cup N^{-}(y)\}$ is 0 (since it was correct at the beginning of the iteration and that didn't change until evaluating $C_0 \rightarrow x$). On the other hand, $y \in {\cal{H}}$ and therefore it supports at least $d/4$ clauses in ${\cal{F}}[{\cal{H}}]$. Thus at least $(d/4-1)$ messages in the left hand sum are `1' (we subtract 1 as $y$ might support $C_0$). $y$ appears in at most $d/30$ clauses with non-core variables (all of which may contribute a wrong '1' message to the right hand sum). All in all, $y \rightarrow C_0 \geq (d/4-d/30-1) > d/5$, which is correct (recall, we assume $\varphi = \textbf{1}^n$). The same applies for $z$, contradicting our assumption. \end{Proof}
\begin{prop}\label{PropNoProof} If $x_i\in {\cal{H}}$ and all messages $C \rightarrow x_i$, $C \in {\cal{F}}[{\cal{H}}]$ are correct by the end of a \textsf{WP} iteration, then $B_i$ agrees in sign with $\varphi(x_i)$ by the end of that iteration. \end{prop} \noindent Proposition \ref{PropNoProof} follows immediately from the definition of ${\cal{H}}$ and the message $B_i$. It suffices to show then that after the first iteration all messages $C \rightarrow x_i$, $C \in {\cal{F}}[{\cal{H}}]$ are correct. \begin{prop}\label{CoreSetCorrectProp} If ${\cal{F}}$ is a typical instance in the setting of Theorem \ref{ConvergenceThmMed}, then after one iteration of $\textsf{WP({\cal{F}})}$, for every variable $x_i\in {\cal{H}}$, every message $C \rightarrow x_i$, $C \in {\cal{F}}[{\cal{H}}]$ is correct. \end{prop} \begin{Proof} The proof is by induction on the order of the execution in the first iteration. Consider the first message $C \rightarrow x$, $C=(\ell_x\vee \ell_y \vee \ell_z)$, $C \in {\cal{F}}[{\cal{H}}]$, to be evaluated in the first iteration. Now consider the message $y \rightarrow C$ at the time $C \rightarrow x$ is evaluated. All messages $C' \rightarrow y$, $C'\in{\cal{F}}[H]$ have their initial random value (as $C \rightarrow x$ is the first core message to be evaluated). Furthermore, $y \in {\cal{H}}$, and therefore there are at most $d/30$ messages of the form $C'' \rightarrow y$, $C'' \notin {\cal{F}}[{\cal{H}}]$. $x \in {\cal{H}}$ hence it is stable w.r.t. $\pi$ and not violated by the initial clause-variable random messages. Therefore $$y \rightarrow C \geq \underbrace{d/7}_{\text{property (c) in defn. \ref{ViolatedDefn}}} -\underbrace{d/30}_{\text{property $(b)$ in defn. \ref{ViolatedDefn}}}-\underbrace{d/30}_{\text{non-core messages}}>d/14.$$ The same applies to $z$, to show that $C \rightarrow x$ is correct. Now consider a message $C \rightarrow x$ at position $i$, and assume all core messages up to this point were evaluated correctly. Observe that every core message $C' \rightarrow y$ that was evaluated already, if $C'\in \{N^{++}(y)\cup N^{-}(y)\}\cap {\cal{F}}[{\cal{H}}]$ then its value is '0' by the induction hypothesis. Since $x$ is not violated by $\alpha$, property $(b)$ in definition \ref{ViolatedDefn} ensures that to begin with
$|\#\textbf{1}_\alpha(N^{++}(y))-\#\textbf{1}_\alpha(N^{-}(y))|\leq d/30$. $y \in {\cal{H}}$, therefore it appears in at most $d/30$ non-core messages, all of which could have been already wrongly evaluated, changing the above difference by additional $d/30$. As for the core messages of $y$ which were already evaluated, since they were evaluated correctly, property $(a)$ in definition \ref{ViolatedDefn} ensures that the above difference changes by at most additional $d/30$. All in all, by the time we evaluate $C \rightarrow x$, $$\sum_{C' \in N^{++}(y),C'\neq C} C' \rightarrow y - \sum_{C''\in N^-(y),C''\neq C}C'' \rightarrow y \geq -3\cdot d/30.$$ As for messages that $y$ supports, property $(c)$ in definition \ref{ViolatedDefn} ensures that their contribution is at least $d/7$ to begin with. Every core message in $N^{s}(y)$ that was evaluated turned to '1', every non-core message was already counted in the above difference. Therefore $y \rightarrow C \geq d/7-3\cdot d/30>d/25$. The same applies to $z$ showing that $C \rightarrow x$ is correct. \end{Proof}
To prove Proposition \ref{ConvergencePropDense}, observe that when $p=c\log n/n^2$, with $c$ a sufficiently large constant, Proposition \ref{SizeOfHBarPr} implies ${\cal{H}} = V$. Combining this with Proposition \ref{CoreSetCorrectProp}, Proposition \ref{ConvergencePropDense} readily follows.
\subsection{The effect of messages that already converged}
It now remains to analyze the behavior of \textsf{WP} on the non-core factor graph, given that the messages involving the core factor graph have converged correctly. A key observation is that once the messages in the factor graph induced by the core variables converged, we can think of $\textsf{WP}$ as if running on the formula resulting from replacing every core variable with its planted assignment and simplifying (which may result in a 1-2-3CNF). The observation is made formal by the following proposition:
\begin{prop}\label{prop:AsIfSimplified} Consider a run of $\textsf{WP}$ that has converged on the core. Starting at some iteration after $\textsf{WP}$ has converged on the core, consider two alternative continuations of the warning propagation algorithm. $\textsf{WP}_1$ denotes continuing with $\textsf{WP}$ on the original input formula. $\textsf{WP}_2$ denotes continuing with $\textsf{WP}$ on the formula obtained by replacing each core variable with its planted assignment and simplifying. Then for every iteration $t$, the sequence of messages in the $t$'th iteration of $\textsf{WP}_2$ is identical to the respective subsequence in $\textsf{WP}_1$. (This subsequence includes those messages not involving the core variables, and includes messages of type $x \to C$ and of the type $C \to x$.) \end{prop}
\begin{Proof} First note that all messages $x \to C$, $x\in {\cal{H}}$, do not change (sign) from the second iteration onwards (by the analysis in the proof of Proposition \ref{CoreSetCorrectProp}). Furthermore, if $\ell_x$ satisfies $C$ in $\varphi$, then $x\to C$ is positive (if $x$ is a true literal in $C$, or negative otherwise), and therefore all messages $C\to y$, $y\neq x$ are constantly 0. Namely, they don't effect any calculation, and this is as if we replaced $\ell_x$ with TRUE, and in the simplification process $C$ disappeared. If $\ell_x$ is false in $C$ under $\varphi$, then $x \to C$ is constantly negative (if $\ell_x=x$, or constantly positive if $\ell_x=\bar{x}$), and this is exactly like having $\ell_x$ removed from $C$ (which is the result of the simplification process).
\end{Proof}
\subsection{Analysis of \textsf{WP} on the \emph{non}-core factor graph} \label{sec:noncoreoutline}
Note that to prove the convergence of the algorithm we need also to prove that messages of the sort $C \to x$ where $C$ is not in the core and $x$ is in the core converge. However, if we prove that all messages in the factor graph induced by the non-core variables converge, then this (with the fact that the core factor graph messages converge) immediately implies the convergence of messages of this type. Therefore, our {\em goal reduces to proving convergence of \textsf{WP} on the factor graph induced by
${\cal{F}}|_\psi$, where $\psi$ assigns the core variables their planted assignment, and the rest are UNASSIGNED.}
We say that $\textsf{WP}$ converged correctly in a connected component $\cal{C}$ of the non-core factor graph if there exists a satisfying assignment $\psi$ of the entire formula which is consistent with the planted assignment on the core, and with the assignment of $\textsf{WP}$ to $\cal{C}$.
Consider a connected component in the non-core factor graph consisting of a cycle with trees hanging from it. Our analysis proceeds in three steps:
\begin{enumerate} \item We first prove that clause-variable and variable-clause messages of the form $\alpha \to \beta$ where $\alpha \to \beta$ lead from the trees to the cycle, converge weakly correctly w.r.t. the planted assignment. In the case that the component has no cycles, this concludes the proof. \item Then, using a refined case analysis, we show that the
messages along the cycle also converge $whp$, this time not necessarily to the planted assignment, but to some satisfying assignment which agrees with the already converged messages. \item We conclude by showing that messages in the direction from the cycle to the trees converge. Finally we show that together, all messages (including parts $(a)$ and $(b)$) in the connected component converge correctly according to some satisfying assignment. \end{enumerate}
Consider the factor graph $F$ induced by the simplified formula. A {\em cycle} in $F$ is a collection $x_1,C_2,x_3,C_4,\ldots,x_r = x_1$ where $x_i$ and $x_{i+2}$ belong to $C_{i+1}$ for all $i$ (in our description we consider only odd values of $i$) and $x_i \neq x_{i+2}$, $C_{i+1} \neq C_{i+3}$ for all $i$. A factor graph $F$ is a {\em tree} if it contains no cycles. It is {\em unicyclic} if it contains exactly one cycle. Let $x \to C$ be a directed edge of $F$. We say that $x \to C$ {\em belongs} to the cycle, if both $x$ and $C$ belong to the cycle. For an edge $x \to C$ that does not belong to the cycle, we say that $x \to C$ {\em is directed towards} the cycle if $x$ doesn't belong to the cycle and $C$ lies on the simple path from $x$ to the cycle. We say that the edge $x \to C$ is {\em directed away} from the cycle if $C$ doesn't belong to the cycle and $x$ lies on the simple path from the cycle to $C$. Similarly we define what it means for an edges $C \to x$ to belong to the cycle, to be directed towards the cycle and to be directed away from the cycle. \begin{prop} Let $F$ be a unicyclic factor graph. Then every directed edge of the form $x \to C$ or $C \to x$ either belongs to the cycle, or is directed towards it or directed away from it. \end{prop} \begin{Proof} Recall that the factor graph is an undirected graph, and the direction is associated with the messages. Take an edge $x\to C$ (similarly for $C\to x$), if it lies on the cycle, then we are done. Otherwise, since the factor graph is connected, consider the path in the tree leading from some element of the cycle to $C$. This path is either contained in the path to $x$ or contains it (otherwise there is another cycle). In the first case $x\to C$ is directed towards the cycle, and in the latter $x\to C$ is directed away from the cycle. \end{Proof}
Our analysis proceeds in two parts: first we shall analyze $\textsf{WP}$ on the trees, then $\textsf{WP}$ on the cycle and connect the two (which is relevant for the uni-cyclic components).
\subsection{$\textsf{WP}$ on the trees} As we already mentioned before, there are two directions to consider: messages directed towards the cycle and away from the cycle. In this section we shall consider a rooted tree, and partition the messages according to messages which are oriented away from the root (they will correspond in the sequel to messages going away from the cycle) and messages towards the root (messages from the leaves towards the root -- later to be identified with messages going into the cycle). The first lemma concerns messages going towards the root.
\begin{rem}\rm\label{rem:ConvergenceOnTrees} Lemma \ref{MessageIntoCycleConvergeLem} is a special case of the known fact (see~\cite{SurveyPropagation} for example) that for every tree induced by a satisfiable formula, $\textsf{WP}$ converges and there exists a satisfying assignment $\psi$ such that every $B_i$ is either 0 or agrees with $\psi$. In Lemma \ref{MessageIntoCycleConvergeLem} we assume that the formula is satisfiable by the all~1 assignment (the planted assignment), and consider only messages towards the root. \end{rem}
\begin{lem}\label{MessageIntoCycleConvergeLem} Let $C \rightarrow x$ be an edge in the non-core factor graph belonging to a connected component of size $s$, and in particular to a rooted tree $T$. If $C\rightarrow x$ is directed towards the root then the message $C \rightarrow x$ converges after at most $O(s)$ iterations. Furthermore, if $C \rightarrow x = 1$ then $x$ appears positively in $C$. \end{lem} \begin{Proof} We consider the case $C=(\ell_x \vee \ell_y)$ -- the case $C=(\ell_x \vee \ell_y \vee \ell_z)$ where all three literals belong to non-core variables is proved similarly. For an edge $(C,x)$ in the factor graph, we define $\verb"level"(C,x)$ to be the number of edges in a path between $C$ and the leaf most distant from $C$ in the factor graph from which the edge $(C,x)$ is removed. The lemma is now proved using induction on the level $i$. Namely, after the $i^{th}$ iteration, all messages $C \rightarrow x$ associated with an edge $(C,x)$ at level $i$ converge, and if $C \rightarrow x = 1$ then $x$ appears positively in $C$.
The base case is an edge $(C,x)$ at level 0. If $\verb"level"(C,x)=0$ then $C$ is a unit clause containing only the variable $x$. By the definition of the messages, in this case $C\rightarrow x = 1$ and indeed it must be the case that $x$ is positive in $C$ (as the other two variables evaluate to FALSE under the planted). Now consider an edge $(C,x)$ at level $i$, and consider iteration $i$. Since $i>0$, it must be that there is another non-core variable $y$ in $C$ (or two more variables $y,z$). Consider an edge $(C',y)$, $y\in C'$ (if no such $C'$ exists that we are done as $C \to x$ will be constantly 0 in this case).
$\verb"level"(C',y)$ is strictly smaller than $i$ since every path from $C$ to a leaf (when deleting the edge $(C,x)$) passes through some edge $(C',y)$. By the induction hypothesis, all messages $C' \rightarrow y$ already converged, and therefore also $y\to C$ and in turn $C\to x$. It is only left to take care of the case $C\to x=1$. In this case, there must be a clause $C'$ s.t. $C'
\rightarrow y = 1$ and $y$ appears positively in $C'$ (by the induction hypothesis). If $C \to x=1$ it must be that $y$ appears negatively in $C$ and therefore $x$ must appear positively (otherwise $C$ is not satisfied by the planted assignment). \end{Proof}
Next we consider several scenarios that correspond to messages going from the root towards the leaves. Those scenarios correspond to step~(c) of our analysis, referred to in Section~\ref{sec:noncoreoutline}.
\begin{clm}\label{1MsgIffUnsatClm} Let $F$ be a unicyclic formula, and assume that WP has converged on $F$. Let $x \to C$ be directed away from the cycle, and define $F_C$ to be the subformula inducing the tree rooted at $C$ ($x$ itself is removed from $C$). Then $C \to x = 0$ in the fixed point if and only if $F_C$ is satisfiable. \end{clm} \begin{Proof} The structure of the proof is similar to that of Lemma \ref{MessageIntoCycleConvergeLem}. For convenience we extend the definition of level above as to include edges on the cycle. We say that an edge $(C,x)$ in the factor graph has $\verb"level"(C,x)$ equal $\infty$ if $(C,x)$ lies on a cycle and $\verb"level"(C,x) = t < \infty$ if $t$ is the maximal length of a path between $C$ and a leaf in the factor graph from which the edge $(C,x)$ is removed. The lemma is now proved using induction on $t$.
The base case is an edge $(C,x)$ at level 0. If $\verb"level"(C,x)=0$ then $C$ is a unit clause containing only the variable $x$, and then $F_C$ is the empty formula. Indeed $C \to x =1$ by definition and $F_C$ is unsatisfiable (by definition again, the empty formula is not satisfiable).
Now consider an edge $(C,x)$ at level $t>0$ ($t$ is still finite because we are not interested in cycle edges). Assume w.l.o.g. that $C=(x\vee y \vee z)$ (maybe only $y$). Let$C_1,C_2,\ldots,C_r$ be the clauses in $F_C$ that contain the variable $y$, and let $D_1,D_2,\ldots,D_s$ be the clauses that contain the variable $z$. Similarly to $F_C$ we can define $F_{C_i}$ and $F_{D_j}$. Observe that every edge $(C_i,y)$ and $(D_j,z)$ in $F_C$ has $\verb"level"$ value which is strictly smaller than $t$ -- since every path from $C$ to a leaf (when deleting the edge $(C,x)$) passes through some such edge.
First consider the case where $C \to x = 1$. For that to happen, it must be that there are two clauses $C_i,D_j$ in $F_C$ so that $C_i \to y = 1$ and $D_j \to z = 1$ and $C_i$ contains $\bar{y}$, $D_j$ contains $\bar{z}$. By the induction hypothesis, $F_{C_i}$ and $F_{D_j}$ are not satisfiable. The only way that $F_C$ can now be possibly satisfied is by assigning $z$ and $y$ to FALSE, but then $F_C$ is not satisfied as the clause $C$ (without $x$) evaluates to FALSE.
Next we prove that if $F_C$ is not satisfiable then it must be that $C\to x=1$. If $F_C$ is not satisfiable then it must be that $C=(\ell_x\vee \bar{y} \vee \bar{z})$ (otherwise $\varphi$ satisfies $F_C$). Further, observe that there must exist at least one clause $C_i$ containing $y$ positively, and at least one $D_j$ containing $z$ positively s.t. $F_{C_{i}}$ and $F_{D_{j}}$ are unsatisfiable. Otherwise, we can define $\varphi'$ to be $\varphi$ except that $y$ or $z$ are assigned FALSE (depending which of $C_i$ or $D_j$ doesn't exist). It is easy to see that $\varphi'$ satisfies $F_C$, contradicting our assumption. By the induction hypothesis $C_i\to y=1$ and $D_j\to z=1$. Further, by Lemma \ref{MessageIntoCycleConvergeLem}, there cannot be a message $C' \to y = 1$ where $y$ is negative, or $D' \to z =1$ where $z$ is negative (because all of these messages are directed towards the cycle). This in turn implies that $C\to x=1$. \end{Proof}\\ Recall our definition for $\textsf{WP}$ converges correctly on a subformula $F'$ of $F$ if there exists a satisfying assignment $\psi$ to $F$ which is consistent with the planted assignment $\varphi$ on the core, and with the assignment of $\textsf{WP}$ to $F'$. \begin{clm}\label{spinalDecomp} Assume that $F$ is a unicyclic formula. Assume further that WP has converged on $F$. Let $C \to y$ be directed away from the cycle. Consider a subformula $F_C$ which induces a tree rooted at a clause $C$. This formula contains the clause $C$ and all other clauses whose path to the cycle goes via $y$. If in the fixed point for $F$ it holds that \begin{itemize}
\item $C\to y=1$,
\item $y$ appears negatively in $C$,
\item $y\to C \geq 1$, \end{itemize} then $\textsf{WP}$ converges correctly on $F_C$. \end{clm} \begin{Proof} We split the variables of $F_C$ into two sets. The first is the set of spinal variables (to be defined shortly), and the remaining variables. The \emph{spine} (rooted at $y$) of a tree-like formula is defined using the following procedure: Let us track the origin of the message $y\to C$ in the tree (which is directed towards the root). For $y\to C \geq 1$ to occur, there must be a clause $D_1$ in the tree that contains $y$ positively and a message $D_1 \to y = 1$ in the direction of the root (as messages in the direction of the root are only effected by other messages in that direction). Let us backtrack one more step.
$D_1=(y \vee \bar{k} \vee \bar{w})$ for some variables $w,k$; $k$ and $w$ must appear negatively in $D_1$ by Lemma \ref{MessageIntoCycleConvergeLem},
and the fact that $D_1 \to y = 1$, that is both $k$ and $w$ were issued warnings having them not satisfy $D_1$. Let us consider the clauses $D_2$ and $D_3$ that issues warnings to $k$ and $w$ respectively. $D_2=(k\vee ...),D_3=(w\vee ...)$, $D_2 \to k =1$ and $D_3 \to w = 1$, and both messages are directed towards the root. Obviously, one can inductively continue this backtracking procedure which terminates at the leaves of the tree (since there are no cycles the procedure is well defined and always terminates). Let us call the clauses and variables that emerge in this backtrack the {\em spine} of the tree. The figure below illustrates this procedure.
\begin{figure*}
\caption{The spine of a tree}
\label{fig:a}
\end{figure*}
Let us start with the $B_i$-messages for the non-spinal variables (the $B_i$ message was defined in Section \ref{WPSubs}). The spine splits $F_C$ into sub-trees hanging from the spinal variables (the root of each such tree is a clause, e.g. $C_1,C_2,C_3$ in our example). Every non-spinal variable belongs to one of those trees. It suffices to prove that for every subtree rooted at $C_i$, hanging from a spinal variable $s$, it holds that $s \to C_i \geq 0$ in the fixed point (that is, $s$ transmits its assignment according to the planted assignment). This ensures, by Remarks \ref{rem:ConvergenceOnTrees}, convergence for that tree to a satisfying assignment (consistent with the planted). Let us prove that $s \to C_i \geq 0$. Let $F_s$ be the subformula corresponding to the tree hanging from $s$ rooted at some clause $C_i$. We prove that $s \to C_i \geq 0$ by induction on the distance in $F_C$ between $y$ and $s$. The base case is distance 0, which is $y$ itself. The messages that we need to verify are of the form $y \to C_i$, $C_i \neq D_1$, which are pointing away from the cycle. For every message $y \to C_i$, the wrong message $C \to y$ is evened by the correct warning $D_1 \to y$. Since $y\to C_i$ depends only on one message which is directed away from the cycle (or else
there is a second cycle), and all the other messages that contribute to the calculation of $y\to C_i$ are correct (Lemma \ref{MessageIntoCycleConvergeLem}), we conclude that $y \to C_i \geq 0$. The induction step follows very similarly.
Let us now consider the spinal variables. For every such variable there is always at most one message in the direction away from the cycle (otherwise there is more than one cycle); this message may be wrong. Using a very similar inductive argument as in the previous paragraph, one can show that there is always at least one correct warning (in the direction of the cycle) for each spinal variable. Therefore $B_s \geq 0$ for every spinal variable $s$. \end{Proof}
\subsection{$\textsf{WP}$ on cycles} \label{subsec:cycle}
We will denote a cycle by $x_1, C_2 , x_3, C_4 ...x_{2r-1},C_{2r},x_1$ where by this we mean that $x_i$ appears in the clauses before/after it and that $C_i$ contains the two variables before/after it. We consider two different types of cycles. \begin{itemize} \item \emph{Biased} cycles: cycles that have at least one warning message $C \to x_i = 1$ coming into the cycle, where $C \to x_i$ directs into the cycle and the value of $C \to x_i$ is the value after the edge has converged. \item \emph{Free} cycles: cycles that do not have such messages coming in, or all messages coming in are 0 messages. \end{itemize}
\subsubsection{Convergence of \textsf{WP} when the cycle is biased}
\noindent First we observe that we may assume w.l.o.g. that edges that enter the cycle enter it at a variable rather than at a clause (hence that every clause on the cycle contains exactly two non-core variables). This is because of a simple argument similar to Proposition \ref{prop:AsIfSimplified}: consider an edge going into the cycle, $z\to C$, and w.l.o.g. assume that $z$ appears positively in $C$. After all the edges going into the cycle have converged, if $z\to C\geq 0$ it follows that $C\to x=0$ for cycle edges $(C,x)$, and thus execution on the cycle is the same as if $C$ was removed from the formula, only now we are left with a tree, for which convergence to a correct assignment is guaranteed (Remark \ref{rem:ConvergenceOnTrees}). If $z\to C<0$, then the execution is exactly as if $z$ was removed from $C$ (and $C$ is in 2-CNF form).
\begin{prop}\label{prop:ConnvergenceInBiasedCaseClm} Let $\cal{C}$ be a connected component of the factor graph of size $s$ containing one cycle s.t. there exists an edge directed into the cycle $C \to x_i$ where $x_i$ belongs to the cycle and such that the message converges to $C \to x_i = 1$. Then $\textsf{WP}$ converges on $\cal{C}$ after at most $O(s)$ rounds. Moreover for the fixed point, if the message $C' \to x = 1$ then $x$ appears positively in $C'$. \end{prop}
\begin{Proof} A message of the cycle $C_j \to x_{j+1}$ depends only on cycle messages of the type $C_{j'} \to x_{j'+1}, x_{j'+1} \to C_{j'+2}$ and on messages coming into the cycle. In other words during the execution of $\textsf{WP}$ the values of all messages $C_{j'} \to x_{j'-1},x_{j'-1} \to C_{j'-2}$ do not effect the value of the message $C_j \to x_{j+1}$. Recall that we are in the case where there exists a message $C \to x_i = 1$ going into the cycle (after the convergence of these messages). Also $x_i$ must appear positively in $C$ (Lemma \ref{MessageIntoCycleConvergeLem}). We consider the following cases:
\begin{itemize} \item There exists a variable $x_j$ that appears positively in both $C_{j-1}$ and $C_{j+1}$ (the case $j=i$ is allowed here). We note that in this case the message $x_j \to C_{j+1}$ must be non-negative which implies that the message $C_{j+1} \to x_{j+2}$ converges to the value $0$. This in turn implies that the value of all messages $x_r \to C_{r+1}$ and $C_{r+1} \to x_{r+2}$ for $r \neq j$ will remain the same if the clause $C_{j+1}$ is removed from the formula. However, this case reduces to the case of tree formula. \item $x_i$ appears negatively in $C_{i+1}$ and positively in $C_{i-1}$. We note that in this case the value of the message $x_i \to C_{i+1}$ is always at least 1, which implies that the message $C_{i+1} \to x_{i+2}$ always take the value $1$. Thus in this case we may remove the clause $C_{i+1}$ from the formula and replace it by the unit clause $\ell_y$ where $C_{i+1} = \ell_y \vee \bar{x_i}$. Again, this reduces to the case of a tree formula. \item The remaining case is the case where $x_i$ appears negatively in both $C_{i-1}$ and $C_{i+1}$ and there is no $j$ such that $x_j$ appears positively in both $C_{j-1}$ and $C_{j+1}$. We claim that this leads to contradiction. An easy corollary of Lemma \ref{MessageIntoCycleConvergeLem} is that all the messages that go into the cycle have converged according to the planted assignment. Therefore w.l.o.g one can ignore messages of the form $y \to C_i$, $y$ is a non-cycle variable, and $C_i$ is a cycle clause (we can assume that every such message says that the literal of $y$ is going to be false in $C_i$, otherwise the cycle structure is again broken and we reduce to the tree case). Therefore the cycle reduces to a 2SAT formula which has to be satisfiable by the planted assignment, in which in particular $x_i = 1$. Write $C_{i+1} = \bar{x_i} \vee \ell_{i+2}$. Then for the satisfying assignment we must have $\ell_{i+2} = 1$, similarly $\ell_{i+4} = 1$, etc, until we reach $\bar{x}_i = 1$, a contradiction. \end{itemize} To summarize, by Lemma \ref{MessageIntoCycleConvergeLem} the messages going into the rooted tree at $x_i$ converge after $O(s)$ steps, and at least one warning is issued. By the above discussion, for every clause $D$ in the connected component it holds that $x_i \to D\geq 0$ (as $x_i$ appears in at most one message which may be wrong -- a cycle message). Since there is always a satisfying assignment consistent with $x_i$ assigned TRUE, then after reducing the cycle to a tree we are left with a satisfiable tree. Remark \ref{rem:ConvergenceOnTrees} guarantees convergence in additional $O(s)$ iterations. \end{Proof}
\subsubsection {Convergence of \textsf{WP} when the cycle is free} The main result of this subsection is summarized in the following claim: \begin{clm}\label{ConnvergenceInNonBiasedCaseMainClm} Let $\cal{C}$ be a connected component of the factor graph of size $s$ containing one cycle of size $r$ s.t. the fixed point contains no messages $C \to x = 1$ going into the cycle (the cycle is free). Then $whp$ \textsf{WP} converges on $\cal{C}$ after at most $O(r^2 \cdot \log n+s)$ rounds. Moreover for the fixed point, if we simplify the formula which induces $\cal{C}$ according to the resulting $B_i$'s, then the resulting subformula is satisfiable. \end{clm}
\begin{rem}\label{DiffRem}\rm Observe that the free case is the only one where convergence according to the planted assignment is not guaranteed. Furthermore, the free cycle case is the one that may not converge ``quickly" (or not at all), though this is extremely unlikely. The proof of Proposition \ref{ConnvergenceInNonBiasedCaseMainClm} is the only place in the analysis where we use the fact that in line $3.a$ of \textsf{WP} we use fresh randomness in every iteration. \end{rem}
We consider two cases: the easy one is the case in which the cycle contains a pure variable w.r.t the cycle (though this variable may not be pure w.r.t to the entire formula).
\begin{clm}\label{ConnvergenceOfNonBiasedCyclePure} If the cycle contains a variable $x_i$ appearing in the same polarity in both $C_{i+1},C_{i-1}$, then the messages $C \to x$ along cycle edges converge. Moreover for the fixed point, if $C \to x = 1$ then $x$ satisfies $C$ according to $\varphi$. \end{clm} The proof is essentially the first case in the proof of Proposition \ref{prop:ConnvergenceInBiasedCaseClm}. We omit the details.
\noindent We now move to the harder case, in which the cycle contains no pure variables (which is the case referred to in Remark \ref{DiffRem}). \begin{prop}\label{FreeCycleNoPureConvergenceClm} Consider a free cycle of size $r$ with no pure literal, and one of the two directed cycles of messages. Then the messages along the cycle converge $whp$ to either all $0$ or all $1$ in $O(r^2\log n)$ rounds. \end{prop} Convergence in $O(r^2\log n)$ rounds suffices due to Corollary \ref{cor:NoLongCycle} (which asserts that $whp$ the length of every cycle is constant). The proof of Proposition \ref{FreeCycleNoPureConvergenceClm} is given at the end of this section. We proceed by analyzing \textsf{WP} assuming that Proposition \ref{FreeCycleNoPureConvergenceClm} holds, which is the case $whp$.
\begin{prop}\label{prop:free_cycle_comp_convergence} Suppose that the cycle messages have converged (in the setting of Proposition \ref{FreeCycleNoPureConvergenceClm}), then the formula resulting from substituting every $x_i$ with the value assigned to it by $B_i$ (according to the fixed point of $\textsf{WP}$), and simplifying, is satisfiable. \end{prop} \begin{Proof} Let $F$ be the subformula that induces the connected component $\cal{C}$, and decompose it according to the trees that hang on the cycle's variables and the trees that hang on the cycle's clauses. Observe that the formulas that induce these trees are variable and clause disjoint (since there is only one cycle in $\cal{C}$).
Let us start with the cycle clauses. The key observation is that setting the cycle variables according to one arbitrary orientation (say, set $x_i$ to satisfy $C_{i+1}$) satisfies the cycle and doesn't conflict with any satisfying assignment of the hanging trees: if the tree hangs on a variable $x_i$, then since the cycle is free, the tree is satisfiable regardless of the assignment of $x_i$ (Proposition \ref{1MsgIffUnsatClm}). In the case that the tree hangs on a cycle-clause $C$, then the cycle variables and the tree variables are disjoint, and $C$ is satisfied already by a cycle-variable regardless of the assignment of the tree-variables. Now how does this coincide with the result of $\textsf{WP}$. Recall that we are in the case where the cycle is free. Therefore only messages $C \to x_i$ where both $C$ and $x_i$ belong to the cycle effect $B_i$. If in the fixed point one cycle orientation is 0 and one orientation is 1, then the $B_i$ messages of the cycle variables implement exactly this policy. If both cycle orientations converged to 1 or to 0, then the corresponding $B_i$ messages of all cycle variables are UNASSIGNED (since the cycle is free), but then the same policy can be used to satisfy the clauses of the cycle in a manner consistent with the rest of the formula.
It remains to show that $\textsf{WP}$ converges on every tree in a manner that is consistent with some satisfying assignment of the tree. We consider several cases.
Consider a tree hanging on a cycle variable $x_i$. Let $C$ be some non-cycle clause that contains $x_i$, and $F_C$ the subformula that induces the tree rooted at $C$. Observe that once the cycle has converged, then the message $x_i\to C$ does not change anymore. If $x_i \to C$ agrees with $\varphi$ there are two possibilities. Either $x_i$ satisfies $C$ under $\varphi$, in which case $C$ always sends 0 to $F_C$, and then $\textsf{WP}$ executes on $F_C$ as if $C$ is removed. Remark \ref{rem:ConvergenceOnTrees} guarantees correct convergence (as $F_C\setminus C$ is satisfiable), and as for $C$, $B_i \geq 0$ and we can set $x_i$ to TRUE so that it satisfies $C$ and is consistent with the assignment of the cycle ($B_i \geq 0$ since $x_i \geq 0$ and $C \to x_i = 0$ as we are in the free cycle case). If $x_i$ appears negatively in $C$, then $\textsf{WP}$ executes as if $x_i$ was deleted from $C$. Still $F_C$ is satisfiable and correct convergence is guaranteed.
Now consider the case where $x_i \to C$ disagrees with $\varphi$. Recall that we assume $\varphi(x_i)=TRUE$, and therefore $x_i \to C$ is negative in the fixed point. If $x_i$ appears negatively in $C$ then $C\rightarrow y=0$ for every $y\in C$ (since $x_i$ signals $C$ that it satisfies it), and therefore $C$ doesn't effect any calculation from this point onwards, and the correct convergence of $F_C$ is again guaranteed by Remark \ref{rem:ConvergenceOnTrees} on the convergence for satisfiable trees. The more intricate case is if $C$ contains $x_i$ positively. Since we are in the free case, it must hold that $C \to x=0$. Therefore using Proposition \ref{1MsgIffUnsatClm} one obtains that $F_C$ is satisfiable (regardless of the assignment of $x_i$), and $\textsf{WP}$ will converge as required (again Remark \ref{rem:ConvergenceOnTrees}).
Now consider a tree hanging on a cycle clause. Namely, $C_{i+1}=(x_i\vee x_{i+2} \vee y)$, where $x_i,x_{i+2}$ are cycle variables, and $(C_{i+1},y)$ is a tree edge. If one of the cycle orientations converged to 0, then $C_{i+1}\to y$ converges to 0, and then Remark \ref{rem:ConvergenceOnTrees} guarantees correct convergence. The same applies to the case where $C_{i+1} \to y$ converges to 1 and $y$ is positive in $C_{i+1}$ (since then we can remove $y$ from the tree, use Remark \ref{rem:ConvergenceOnTrees} for the remaining part, then add back $y$, and set it to TRUE without causing any conflict with the tree assignment, but satisfying $C_{i+1}$ according to the planted assignment).
The delicate case remains when $C_{i+1} \to y$ converges to 1 but $y$'s polarity in $C_{i+1}$ disagrees with $\varphi$, that is, $y$ is negative in $C_{i+1}$. The key observation is that the message $y\to C_{i+1}$ (which is directed towards the cycle) must have converged to a positive value (otherwise, $C_{i+1}\to x_i$ and $C_{i+1}\to x_{i+2}$ would have converged to 0). However this complies with the scenario of Proposition \ref{spinalDecomp}, and again correct convergence is guaranteed. \end{Proof}
In Theorem \ref{ConvergenceThmMed} the unassigned variables are required to induce a ``simple" formula, which is satisfiable in linear time. Observe that the factor graph induced by the UNASSIGNED variables consists of connected components whose structure is a cycle with trees hanging on it, or just a tree. A formula whose factor graph is a tree can be satisfied in linear time by starting with the leaves (which are determined uniquely in case that the leaf is a clause -- namely, a unit clause, or if the leaf is a variable then it appears only in one clause, and can be immediately assigned) and proceeding recursively. Regarding the cycle, consider an arbitrary variable $x$ on the cycle. By assigning $x$ and simplifying accordingly, we remain with a tree. Since there are only two ways to assign $x$, the whole procedures is linear in the size of the connected component. This completes the proof of Theorem \ref{ConvergenceThmMed}.
\subsubsection{Proof of Proposition \ref{FreeCycleNoPureConvergenceClm}} Since the cycle has no pure literal it must be of the following form: $C_1 = (\ell_{x_1} \vee \overline{\ell}_{x_2}), C_2 = (\ell_{x_2} \vee \overline{\ell}_{x_3}), \ldots, C_L = (\ell_{x_L} \vee \overline{\ell}_{x_1})$ (recall the definition of cycles at the beginning of subsection \ref{subsec:cycle}).
Consider one of the directed cycles, say: $x_1 \to C_1 \to x_2 \to \cdots$ and note that when the message $x_i \to C_i$ is updated it obtains the current value of $C_{i-1} \to x_i$ and when the message $C_i \to x_{i+1}$ is updated, it obtains the current value of $x_i \to C_i$.
It thus suffices to show that the process above converges to all $0$ or all $1$ in time polynomial in the cycle length. This we prove in the lemma below.
\begin{clm} \label{clm:perm_process} Consider the process $(\Gamma^i : i \geq 0)$ taking values in $\{0,1\}^{L}$. The process is a Markov process started at $\Gamma^0 \in \{0,1\}^{L}$.
Given the state of the process $\Gamma^i$ at step $i$, the distribution of $\Gamma^{i+1}$ at round $i+1$ is defined by picking a permutation $\sigma \in S_L$ uniformly at random and independently. Then $\Gamma^{i+1}$ is defined via the following process: let $\Delta_0 = \Gamma^i$ and for $1 \leq j \leq L$: \begin{itemize} \item Let $\Delta_j$ be obtained from $\Delta_{j-1}$ by setting $\Delta_j(\sigma(j-1)) = \Delta_{j-1}((\sigma(j-1)+1) \mod L)$ and $\Delta_j(r) = \Delta_{j-1}(r)$ for all $r \neq \sigma(j-1)$. \end{itemize} Then set $\Gamma^{i+1} = \Delta_L$.
Let $T$ be the stopping time where the process hits the state all $0$ or all $1$. Then for all $\Gamma^0$: \begin{equation} \label{eq:tail_perm} Pr[T \geq 4 a L^2] \leq L 2^{-a}. \end{equation} for all $a \geq 1$ integer. \end{clm}
The proof of Proposition~\ref{clm:perm_process} is based on the fact that the process defined in this lemma is a martingale. This is established in the following two lemmas.
\begin{lem} \label{lem:perm_process_mart} Consider the following variant $\tilde{\Gamma^i}$ of the process $\Gamma^i$ defined in Proposition~\ref{clm:perm_process}. In the variant, different intervals of $0/1$ are assigned different colors and the copying procedure is as above. Fix one color and let $X_i$ denote the number of elements of the cycle of that color in
$\tilde{\Gamma}^i$. Then $X_i$ is a martingale with respect to the filtration defined by $\tilde{\Gamma}^i$. \end{lem}
\begin{Proof}
From the definition of the copying process it is easy to see that \[
\mathbb{E}[X_{i+1} | \tilde{\Gamma}^i,\tilde{\Gamma}^{i-1},\ldots] =
\mathbb{E}[X_{i+1} | X_i]. \]
We will show below that $\mathbb{E}[X_{i+1} | X_i] = 0$ and thus that $X_{i}$ is a martingale with respect to the filtration $\tilde{\Gamma}^i$.
Assume that $X_i = k$. Then w.l.o.g. we may assume that
the configuration $\tilde{\Gamma}^i$ consists of an interval of $1$'s of length $k$ and an interval of $0$'s of length $L-k$. Indeed if the configuration consists of a number of intervals of length $Y_{i,1},\ldots,Y_{i,r}$ where $X_i = \sum_{t} Y_{i,t}$ then we may think of the different sub-intervals as having different colors. Then proving that each of the $Y_{i,t}$ is a martingale implies that $X_i$ is an interval as needed.
We calculate separately the expected shifts in the locations of left end-points of the $0$ and $1$ interval respectively. We denote the two shift random variables by $L_0$ and $L_1$. Clearly $L_0 = I_{0,1} + I_{0,2} + \ldots + I_{0,k-1}$ where $I_{0,j}$ is the indicator of the event that the $0$ left-end point shifted by at least $j$ and similarly for $L_1$. Note that \[ \mathbb{E}[I_{0,j}] = \frac{1}{j!} - \frac{1}{(L-k+j)!} \] and that \[ \mathbb{E}[I_{1,j}] = \frac{1}{j!} - \frac{1}{(k+j)!}. \] The last equation follows from the fact that in order for the $1$ interval to extend by at least $j$, the $j$ copying has to take place in the correct order and it is forbidden that they all took place in the right order and the interval has become a $0$ interval. The previous equation is derived similarly. Thus \[
\mathbb{E}[(X_{i+1} - X_i) | X_i] = \mathbb{E}[L_1] - \mathbb{E}[L_0] = \sum_{j=1}^{L-k} \left(\frac{1}{j!} - \frac{1}{(k+j)!}\right) - \sum_{j=1}^{k} \left(\frac{1}{j!} - \frac{1}{(L-k+j)!}\right) = 0 \] This concludes the proof that $X_i$ is a martingale. The proof follows. \end{Proof}\\\\ The proof of Proposition~\ref{clm:perm_process} follows by a union bound from the following lemma where the union is taken over all intervals.
\begin{lem} \label{lem:perm_process_tail} Consider the process $\tilde{\Gamma^i}$ defined in Lemma~\ref{lem:perm_process_mart}. Fix one interval and let $X_i$ denote its length. Let $T$ be the stopping time where $ X_i$ equals either $0$ or $L$. Then \[ Pr[T \geq 4 a L^2] \leq 2^{-a}. \] \end{lem}
\begin{Proof} In order to bound the hitting probability of $0$ and $L$, we need some bounds on the variance of the martingale differences. In particular, we claim that unless $k=0$ or $k=L$ it holds that \[
\mathbb{E}[(X_{t+1}-X_t)^2 | X_t = k] \geq 1/2. \] If $k=1$ or $k=L-1$ this follows since with probability at least $1/2$ the end of the interval will be hit. Otherwise, it is easy to see that the probability that $X_{t+1}-X_t$ is at least $1$ is at least $1/4$ and similarly the probability that it is at most $-1$ is at least $1/4$. This can be verified by considering the event that one end-point moves by at least $2$ and the other one by at most $1$.
Let $T$ be the stopping time when $X_T$ hits $0$ or $n$. Then by a Wald kind of calculation we obtain: \begin{eqnarray*} L^2 &\geq& \mathbb{E}[(X_T-X_0)^2] = \mathbb{E}[ (\sum_{t=1}^{\infty} 1(T \geq t) (X_t-X_{t-1}))^2] \\ &=& \mathbb{E}[\sum_{t,s=1}^{\infty} (X_t-X_{t-1})(X_s-X_{s-1}) 1(T \geq \max{t,s})] \\ &=& \mathbb{E}[\sum_{t=1}^{\infty} (X_t-X_{t-1})^2 1(T \geq t)] \geq \frac{1}{2} \sum_{t=1}^{\infty} P[T \geq t] = \mathbb{E}[T]/2, \end{eqnarray*} where the first equality in the last line follows from the fact that if $s < t$ say then: \[ \mathbb{E}[(X_t-X_{t-1})(X_s-X_{s-1}) 1(T \geq \max{t,s})] \] \[ = \mathbb{E}[(X_t-X_{t-1})(X_s-X_{s-1}) (1 - 1(T < t))] \] \[
= \mathbb{E}[\mathbb{E}[(X_t - X_{t-1}) (X_s - X_{s-1}) (1 - 1(T < t)) | X_1,\ldots,X_{t-1}]] \] \[ =
\mathbb{E}[(X_s - X_{s-1}) (1 - 1(T < t)) E [X_t - X_{t-1} | X_1,\ldots,X_{t-1}]] = 0. \]
We thus obtain that $\mathbb{E}[T] \leq 2 L^2$. This implies in turn that $Pr[T \geq 4 L^2] \leq 1/2$ and that $P[T \geq 4 a L^2] \leq 2^{-a}$ for $a \geq 1$ since $X_t$ is a Markov chain. The proposition follows.
\end{Proof}
\section{Proof of Proposition \ref{prop_NonCoreFactorGraph}}\label{sec:NoTwoCycleProof} We shall start with the proof of Item $(b)$. Then we shall remark how to adjust this proof to prove $(c)$. Item $(a)$ was proven, though with respect to a slightly different definition of a core, in \cite{flaxman}. The proof of Item $(a)$ is identical to that in \cite{flaxman}, with the adjustments made along the proof of Item $(b)$. Details of the proof of $(a)$ are omitted.
In order to prove Proposition \ref{prop_NonCoreFactorGraph} $(b)$ it suffices to prove that $whp$ there are no two cycles with a simple path (maybe of length 0) connecting the two. To this end, we consider all possible constellations of such prohibited subgraphs and prove the proposition using a union bound over all of them.
Every simple $2k$-cycle in the factor graph consists of $k$ variables, w.l.o.g. say $x_{1},...,x_{k}$ (all different), and $k$ clauses $C_{1},...,C_{k}$, s.t. $x_{i},x_{i+1}\in C_{i}$. The cycle itself consists of $2k$ edges.
As for paths, we have 3 different types of paths: paths connecting a clause in one cycle with a variable in the other (type 1), paths connecting two clauses (type 2), and paths connecting two variables (type 3). Clause-variable paths are always of odd length, and clause-clause, variable-variable paths are always of even length. A $k$-path $P$ consists of $k$ edges. If it is a clause-variable path, it consists of $(k-1)/2$ clauses and the same number of variables. If it is a variable-variable path, it consists of $k/2-1$ variables and $k/2$ clauses and symmetrically for the clause-clause path (we don't take into account the clauses/variables that participate in the cycle, only the ones belonging exclusively to the path).
Our prohibited graphs consist of two cycles $C_1,C_2$ and a simple path $P$ connecting them. We call a graph containing exactly two simple cycles and a simple path connecting them a \emph{bi-cycle}. The path $P$ can be of either one of the three types described above. Similarly to the bi-cycle case, one can have a cycle $C$ and a chord $P$ in it. We call such a cycle a \emph{chord-cycle}. For parameters $i,j,k\in [1,n]$, and $t\in \{1,2,3\}$, we denote by $B_{2i,2j,k,t}$ a bi-cycle consisting of a $2i$-cycle connected by a $k$-path of type $t$ to a $2j$-cycle. Similarly, we denote by $B_{2i,k,t}$ a chord-cycle consisting of a $2i$-cycle with a $k$-path of type $t$ as a chord.
Our goal is then to prove that $whp$ the graph induced by the non-core variables contains no bi-cycles and no chord-cycles.
For a fixed factor graph $H$ we let $F_{H}\subseteq {\cal{F}}$ be a fixed minimal set of clauses inducing $H$, and $V(H)$ be the set of variables in $H$. In order for a fixed graph $H$ to belong to the factor graph induced by the non-core variables it must be that there exists some $F_H$ s.t. $F_{H} \subseteq {\cal{F}}$ and that $V(H)\subseteq \bar{{\cal{H}}}$ (put differently, $V(H)\cap {\cal{H}}=\emptyset$).
Let $B=B_{2i,2j,k,t}$ (or $B=B_{2i,k,t}$ if $B$ is a chord-cycle) be a fixed bi-cycle and $F_B$ a fixed minimal-set of clauses inducing $B$. We start by bounding $Pr[F_{B} \subseteq {\cal{F}} \text{ and } V(B)\cap {\cal{H}} = \emptyset]$ and then use the union bound over all possible bi-cycles (chord-cycles) and inducing minimal sets of clauses. As the two events -- $\{F_B \subseteq {\cal{F}}\}$ and $\{V(B)\cap {\cal{H}} = \emptyset\}$ -- are not independent, the calculations are more involved. Loosely speaking, to circumvent the dependency issue, one needs to defuse the effect that the event $\{F_B \subseteq {\cal{F}}\}$ might have on ${\cal{H}}$. To this end we introduce a set ${\cal{H}}^*$, defined very similarly to ${\cal{H}}$ only ``cushioned" in some sense to overcome the dependency issues (the ``cushioning" depends on $F_B$). This is done using similar techniques to \cite{AlonKahale97,flaxman}.
We start by defining the new set of core variables ${\cal{H}}^*$ (again w.r.t. an ordering $\pi$ of the clause-variable messages and an initial values vector $\alpha$). The changes compared to ${\cal{H}}$ are highlighted in bold. \begin{figure*}\end{figure*}\\ Propositions \ref{StableSuccRate} and \ref{ViolatedSuccRate} could be easily adjusted to accommodate the 6-gap in the new definition in $B_2$ and $B_3$. Therefore Proposition \ref{SizeOfHBarPr} can be safely restated in the context of ${\cal{H}}^*$: \begin{prop}\label{SizeOfHBarTagPr} If both $\alpha$ and $\pi$ are chosen uniformly at random then $whp$ $\#{\cal{H}}^*\geq (1-e^{-\Theta(d)})n$. \end{prop} \begin{prop} Let $b=\#V(B)$, then the set $J$ defined above satisfies $\#J\geq b/4$ \end{prop} \begin{Proof} Observe that if $F_B$ is minimal then $\#F_B\leq b+1$. This is because in every cycle the number of variables equals the number of clauses, and in the worst case, the path contains at most one more clause than the number of variables, and the same goes for the chord-cycle. Now suppose in contradiction that $\#J<b/4$, then there are more than $3b/4$ variables in $V(B)$, each appearing in at least 6 different clauses in $F_B$. Thus, $\#F_B>(6\cdot 3b/4)/3=1.5b\underbrace{>}_{b\geq 3}b+1$ (we divided by three as every clause might have been counted 3 times), contradicting $\#F_B\leq b+1$. \end{Proof}\\ The following proposition ``defuses" the dependency between the event that a bi-cycle (chord-cycle) was included in the graph and the fact that it doesn't intersect the core variables. In the following proposition we fix an arbitrary $\pi$ and $\alpha$ in the definition of ${\cal{H}}^*$, therefore the probability is taken only over the randomness in the choice of ${\cal{F}}$. \begin{prop}\label{CycleProbProp}$Pr[F_{B} \subseteq {\cal{F}} \text{ and } V(B) \cap {\cal{H}} = \emptyset]\leq Pr[F_{B} \subseteq {\cal{F}}]\cdot Pr[J \cap {\cal{H}}^* = \emptyset].$ \end{prop} \noindent Before proving Proposition \ref{CycleProbProp}, we establish the following fact. \begin{lem}\label{BasicIncLemma} For every bi-cycle (chord-cycle) $B$ and every minimal inducing set $F_{B}$, ${\cal{H}}^*({\cal{F}},\varphi,\alpha,\pi) \subseteq {\cal{H}}({\cal{F}}\cup F_{B},\varphi,\alpha,\pi)$. \end{lem} \begin{Proof} The lemma is proved using induction on $i$ ($i$ being the iteration counter in the construction of ${\cal{H}}$). For the base case $H'_{0}({\cal{F}}) \subseteq H_{0}({\cal{F}} \cup F_B)$, since every variable in $H'_{0}({\cal{F}})$ appears in at most 6 clauses in $F_B$ it holds that $A_i({\cal{F}}\cup F_B) \subseteq B_i ({\cal{F}})$, $i=2,3$. $A_1({\cal{F}}\cup F_B) \subseteq B_1({\cal{F}})$ holds at any rate as more clauses can only increase the support, and the set $J$ was not even considered for $H_0$. Suppose now that $H_{i}'({\cal{F}}) \subseteq H_{i}({\cal{F}} \cup F_{B})$, and prove the lemma holds for iteration $i+1$. If $x \in H'_{i+1}({\cal{F}})$ then $x$ supports at least $d/3$ clauses in which all variables are in $H'_{i}({\cal{F}})$. Since $H'_{i}({\cal{F}}) \subseteq H_{i}({\cal{F}} \cup F_B)$, then $x$ supports at least this number of clauses with only variables of $H_i({\cal{F}} \cup F_{B})$. Also, $x$ appears in at most $d/30-6$ clauses with some variable outside of $H'_{i}({\cal{F}})$, again since $H'_{i}({\cal{F}}) \subseteq H_{i}({\cal{F}} \cup F_B)$ and $F_{B}$ contains at most 6 clauses containing $x$, $x$ will appear in no more than $d/30$ clauses each containing some variable not in $H_{i}({\cal{F}} \cup F_B)$. We conclude then that $x \in H_{i}({\cal{F}} \cup F_B)$. \end{Proof}\\ This lemma clarifies the motivation for defining ${\cal{H}}^*$. It is not necessarily true that ${\cal{H}}({\cal{F}})\subseteq {\cal{H}}({\cal{F}} \cup F_{B})$. For example, a variable which appears in ${\cal{H}}({\cal{F}})$ could disappear from ${\cal{H}}({\cal{F}} \cup F_{B})$ since the clauses in $F_{B}$ make it unstable. Loosely speaking, ${\cal{H}}^*$ is cushioned enough to prevent such a thing from happening.\\\\
\begin{Proof}(Proposition \ref{CycleProbProp}) $$Pr[F_B \subseteq {\cal{F}} \text{ and } V(B) \cap {\cal{H}} = \emptyset] \leq Pr[F_B \subseteq {\cal{F}} \text{ and } J \cap {\cal{H}} = \emptyset]
= Pr[J \cap {\cal{H}} = \emptyset | F_B \subseteq {\cal{F}}] Pr[F_B \subseteq {\cal{F}}].$$ Therefore, it suffices to prove
$$Pr[J \cap {\cal{H}} = \emptyset | F_B \subseteq {\cal{F}}] \leq Pr[J \cap {\cal{H}}^* = \emptyset].$$ $$Pr[J \cap {\cal{H}}^* = \emptyset] = \sum_{F:J \cap {\cal{H}}^*(F) = \emptyset}Pr[{\cal{F}}=F]\underbrace{\geq }_{Lemma~\ref{BasicIncLemma}}\sum_{F:J \cap {\cal{H}}(F\cup F_B) = \emptyset}Pr[{\cal{F}}=F]$$ Break each set of clauses $F$ into $F'=F \setminus F_B$ and $F''=F \cap F_B$, and the latter equals $$\sum_{F':F' \cap F_B=\emptyset,J \cap {\cal{H}}(F'\cup F_B)=\emptyset} \sum_{F'':F''\subseteq F_B} Pr[{\cal{F}}\setminus F_B=F' \text{ and } {\cal{F}} \cap F_B = F'']$$ Since the two sets of clauses, ${\cal{F}}\setminus F_B$, and ${\cal{F}} \cap F_B$, are disjoint, and clauses are chosen independently, the last expression equals, \begin{align*} &\sum_{F':F' \cap F_B=\emptyset,J \cap {\cal{H}}(F'\cup F_B)=\emptyset}\sum_{F'':F''\subseteq F_B} Pr[{\cal{F}}\setminus F_B=F']Pr[{\cal{F}} \cap F_B = F'']= \\ & \sum_{F':F' \cap F_B=\emptyset,J \cap {\cal{H}}(F'\cup F_B)=\emptyset} Pr[{\cal{F}}\setminus F_B=F'] \underbrace{\sum_{F'':F''\subseteq F_B} Pr[{\cal{F}} \cap F_B = F'']}_{1} = \\ & \sum_{F':F' \cap F_B=\emptyset,J \cap {\cal{H}}(F'\cup F_B)=\emptyset}Pr[{\cal{F}}\setminus F_B = F'] \end{align*}
Since $({\cal{F}} \setminus F_B) \cap F_B =\emptyset$, and clauses are chosen independently, the event $\{F_B\subseteq {\cal{F}}\}$ is independent of the event $\{{\cal{F}}\setminus F_B = F'\}$. Therefore, the latter expression can be rewritten as $$\sum_{F':F' \cap F_B=\emptyset,J \cap {\cal{H}}(F'\cup F_B)=\emptyset}Pr[{\cal{F}}\setminus F_B = F'|F_B \subseteq {\cal{F}}]=
Pr[J \cap {\cal{H}} = \emptyset|F_B \subseteq {\cal{F}}].$$ \end{Proof}\\
\begin{lem}\label{lem_chordCycleProbCor} Let $B=B_{2i,k,t}$ be a chord-cycle, then
$Pr[V(B) \cap {\cal{H}}^* = \emptyset | \# {\cal{H}}^* = (1-\lambda) n]\leq p(i,k)$ where:\begin{enumerate}
\item $p(i,k)\leq \lambda^{(i+\frac{k}{2}-1)/4}$ if $B$
consists of a $2i$-cycle and a variable-variable $k$-path as a chord.
\item $p(i,k)\leq \lambda^{(i+\frac{k}{2})/4}$ if $B$
consists of $2i$-cycle and a clause-clause $k$-path as a chord.
\item $p(i,k)\leq \lambda^{(i+\frac{k-1}{2})/4}$ if $B$
consists of $2i$-cycle and a variable-clause $k$-path as a chord. \end{enumerate} \end{lem}
\begin{Proof} In $(a)$, we have $i+\frac{k}{2}-1$ variables and $i+\frac{k}{2}$ clauses. To bound the event $\{J \cap {\cal{H}}^* = \emptyset\}$, given the size of ${\cal{H}}^*$, observe that $F_B$ is fixed in the context of this event, and there is no pre-knowledge whether $F_B$ is included in ${\cal{F}}$ or not. Therefore, $J$ can be treated as a fixed set of variables, thus the choice of ${\cal{H}}^*$ is uniformly distributed over $J$. Recalling that $\#J \geq (i+\frac{k}{2}-1)/4$, it follows that \begin{align*}
Pr[J \cap {\cal{H}}^* = \emptyset | \# {\cal{H}}^* = (1-\lambda) n] \leq \frac{\binom{n-\#{\cal{H}}^*}{\#J}}{\binom{n}{\#J}}=\frac{\binom{\lambda n}{(i+\frac{k}{2}-1)/4}}{\binom{n}{(i+\frac{k}{2}-1)/4}} \leq \lambda^{(i+\frac{k}{2}-1)/4}. \end{align*} The last inequality follows from standard bounds on the binomial coefficients. This proves $(a)$. In the same manner items $b,c$ are proven (just counting how many variables and clauses $B$ contains, depending on the type of its path). \end{Proof}
\begin{lem}\label{lem_BiCycleProbCor} Let $B=B_{2i,2j,k,t}$ be a bi-cycle, then
$Pr[V(B) \cap {\cal{H}}^* = \emptyset | \# {\cal{H}}^* = (1-\lambda) n]\leq p(i,j,k)$ where:\begin{enumerate}
\item $p(i,j,k)\leq \lambda^{(i+j+\frac{k}{2}-1)/4}$ if $B$
consists of a $2i$,$2j$-cycles and a variable-variable $k$-path.
\item $p(i,j,k)\leq \lambda^{(i+j+\frac{k}{2})/4}$ if $B$
consists of $2i$,$2j$-cycles and a clause-clause $k$-path.
\item $p(i,j,k)\leq \lambda^{(i+j+\frac{k-1}{2})/4}$ if $B$
consists of $2i$,$2j$-cycles and a variable-clause $k$-path. \end{enumerate} \end{lem} \noindent Lemma \ref{lem_BiCycleProbCor} is proven in a similar way to Lemma \ref{lem_chordCycleProbCor}.
To complete the proof of Proposition \ref{prop_NonCoreFactorGraph}, we use the union bound over all possible bi/chord-cycles. We present the proof for the bi-cycle case with a variable-variable path; the proof for all other cases is identical. $s=s_{i,j,k}=i+j+\frac{k}{2}-1$ (namely, $\#V(B)=s$ and $\#F_B=s+1$). We also let $p_\lambda=Pr[\#{\cal{H}}^*=(1-\lambda) n]$. The probability of $B$ is then at most \begin{align*} &\sum_{\lambda=0}^{n} p_\lambda\cdot \sum_{i,j,\frac{k}{2}=1}^{n}\binom{n}{s_{i,j,k}}\cdot (s_{i,j,k})! \cdot (7n)^{s_{i,j,k}+1}\cdot \left(\frac{d}{n^2}\right)^{s_{i,j,k}+1} \cdot \lambda^{s_{i,j,k}/4} \leq\\& \sum_{\lambda=0}^{n} p_\lambda\cdot\sum_{i,j,\frac{k}{2}=1}^{n} 7d\cdot \left(\frac{7en}{s}\right)^{s} \cdot s^{s} \cdot n^{s+1} \cdot \left(\frac{d}{n^2}\right)^{s+1} \cdot \lambda^{s/4} \leq \sum_{\lambda=0}^{n} p_\lambda\cdot\sum_{i,j,\frac{k}{2}=1}^{n}(7e \cdot d \cdot \lambda^{1/4})^{s}\cdot \frac{7d}{n} . \end{align*} Let us now break the first sum according to the different values of $\lambda$. Proposition \ref{SizeOfHBarPr} implies that $p_\lambda < n^{-5}$ for all $\lambda > d^{-8}$. Therefore the contribution of this part to the (double) summation is $O(1/n)$. For $\lambda < d^{-8}$, the latter simplifies to
\begin{equation}\label{eq:lambda}\sum_{\lambda=0}^{n/d^8}p_\lambda\sum_{i,j,\frac{k}{2}=1}^{n}\left(\frac{1}{2}\right)^{s}\cdot \frac{7d}{n}. \end{equation} Finally observe that $$\sum_{i,j,\frac{k}{2}=1}^{n}\left(\frac{1}{2}\right)^{s}\cdot \frac{7d}{n} \leq \sum_{i+j+\frac{k}{2} \leq 4\log n}\frac{7d}{n}+\sum_{i+j+\frac{k}{2}\geq 4\log n} \left(\frac{1}{2}\right)^{s}\leq (4\log n)^3\cdot \frac{7d}{n}+n^3\cdot \frac{1}{n^4}=o(1). $$ Therefore, (\ref{eq:lambda}) sums up to $o(1)$.
\subsection{Proof of Proposition \ref{prop_NonCoreFactorGraph} $(c)$} The proof is basically the same as that of Proposition \ref{prop_NonCoreFactorGraph} $(b)$. One defines the same notion of ``cushioned" core ${\cal{H}}^*$, and proceeds similarly. We therefore reprove only the last part -- the union bound over all possible cycles.
First let us bound the number of cycles of length $k$. There are $\binom{n}{k}$ ways to choose the variables inducing the cycle, and $k!/2$ ways to order them on the cycle. As for the set of clauses that induces the cycle, once the cycle is fixed, we have at most $(7n)^k$ ways of choosing the third variable and setting the polarity in every clause. We also let $p_\lambda=Pr[\#{\cal{H}}^*=(1-\lambda) n]$.
Using the union bound, the probability of a cycle of length at least $k$ in the non-core factor graph is at most \begin{align*} &\sum_{\lambda=0}^{n} p_\lambda\cdot\sum_{t=k}^{n}\binom{n}{t}\cdot t! \cdot (7n)^t\cdot \left(\frac{d}{n^2}\right)^t \cdot \lambda^{t/2} \leq \sum_{\lambda=0}^{n} p_\lambda\cdot\sum_{t=k}^{n} \left(\frac{7en}{t}\right)^t \cdot t^t \cdot n^t \cdot \left(\frac{d}{n^2}\right)^t \cdot \lambda^{t/2} \\& = \sum_{\lambda=0}^{n}p_\lambda \cdot\sum_{t=k}^{ n}(7e \cdot d \cdot \sqrt{\lambda})^t. \end{align*} Let us now break the first sum according to the different values of $\lambda$. Proposition \ref{SizeOfHBarPr} implies that there exists a constant $c$ s.t. $p_\lambda < n^{-3}$ for $\lambda > e^{-cd}$. Therefore the contribution of this part to the (double) summation is $O(1/n)$. For $\lambda \leq e^{-cd}$, $7e \cdot d \cdot \sqrt{e^{-cd}}=e^{-\Theta(d)}$. In this case, the last summation is simply the sum of a decreasing geometric series with quotient $e^{-\Theta(d)}$, which sums up to at most twice the first term, namely $e^{-\Theta(dk)}$. The proposition then follows.
\end{document}
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\begin{document}
\title{The Expressive Power of $\dlliterc$}
\begin{abstract} Description logics are knowledge representation formalisms that provide the formal underpinning of the semantic web and in particular of the OWL web ontology language. In this paper we investigate the expressive power of $\g{DL-Lite}_{R,\sqcap}\xspace$, and some of its computational properties. We rely on simulations to characterize the absolute expressive power of $\g{DL-Lite}_{R,\sqcap}\xspace$ as a {\em concept language}, and to show that disjunction is not expressible. We also show that no simulation-based closure property exists for $\g{DL-Lite}_{R,\sqcap}\xspace$ assertions. Finally, we show that query answering of unions of conjunctive queries is \textsc{NP}\xspace-complete. \end{abstract}
\section{Introduction}\label{intro}
Description logics (DLs) are knowledge representation formalisms that provide the formal underpinning of the semantic web and in particular of the OWL web ontology language\footnote{http://www.w3.org/TR/owl-features/}. In this paper we are interested in investigating the expressive power of the DL known as \g{DL-Lite}$_{R,\sqcap}$ \cite{Calvanese2006B}. The \g{DL-Lite} family of logics, of which $\g{DL-Lite}_{R,\sqcap}\xspace$ makes part, has been proposed by Calvanese et al. as a foundation of ontology-based data access systems. They are intended \cite{Calvanese2005A,Calvanese2007D} as the least expressive DLs capable of capturing the main features of conceptual modelling languages such as UML\footnote{http://www.omg.org/uml/}. By the expressive power of a DL we understand \textit{(i)} the computational complexity of its reasoning problems and \textit{(ii)} its model-theoretic properties. As most DLs, $\g{DL-Lite}_{R,\sqcap}\xspace$ is contained in $\logic{Fo}^2$, the 2-variable fragment of \logic{Fo} and is therefore decidable \cite{Borgida1996,Hudstadt2004,DLHandbook}. However, its expressive power is still not known completely.
DLs model domains in terms of concepts (representing classes of objects), and binary relations known as roles (representing relations and attributes of objects) \cite{DLHandbook}, all of which are structured into hierarchies by concept and role inclusion assertions. Extensional information (the data), by contrast, is conveyed by membership assertions. This information can be accessed by posing suitable \logic{Fo} formulas, viz., unions of conjunctive queries. This crucial reasoning problem is known as the knowledge base query answering problem.
The main contributions of this paper consist, on the one hand, in determining the (so-called) combined complexity of $\g{DL-Lite}_{R,\sqcap}\xspace$'s query answering problem and, on the other hand, to define what we call $\g{DL-Lite}_{R,\sqcap}\xspace$ simulations. This relation stems from the notion of bisimulations (see e.g. \cite{vanBenthem2006}) for modal logics, known to hold for the DL $\f{ALC}$ \cite{DLHandbook}, that has been proposed \cite{deRijke1997} as a means of characterizing the (absolute) expressivity of arbitrary DLs as concept languages.
The structure of this paper is as follows. Section~\ref{two} recalls \textit{(i)} $\g{DL-Lite}_{R,\sqcap}\xspace$'s syntax and semantics and \textit{(ii)} those of unions of conjunctive queries. In section~\ref{two} we characterize the combined complexity of answering unions of conjunctive queries over $\g{DL-Lite}_{R,\sqcap}\xspace$ knowledge bases. In section~\ref{three} we introduce the notion of $\g{DL-Lite}_{R,\sqcap}\xspace$ simulations and show that a \logic{Fo} formula is equivalent to a $\g{DL-Lite}_{R,\sqcap}\xspace$ concept when and only when it is closed under $\g{DL-Lite}_{R,\sqcap}\xspace$ simulations. In section~\ref{four} we show that no such closure property exists for assertions. Finally, in section~\ref{five} we sum up our conclusions.
\section{Preliminaries}\label{one}
The syntax of $\g{DL-Lite}_{R,\sqcap}\xspace$ is defined by the grammar: \begin{itemize} \item $R ::= P \mid P^-$, \item $D ::= A \mid \exists R \mid D \sqcap D'$ (left concepts), \item $E ::= C \mid \neg A \mid \neg \exists R \mid \exists R . E$ (right concepts), \end{itemize} where $A$ stands for an atomic concept symbol (a unary predicate), $P$ for an atomic role symbol (a binary predicate) and $R^-$ for its inverse.
Concepts combine into {\em concept inclusion assertions} of the form $D \sqsubseteq E$, where $D$ is a left concept, $E$ is a right concept and $\sqsubseteq$ is the {\em subsumption} relation. Roles into {\em role inclusion assertions} of the form $R \sqsubseteq R'$. A {\em teminology} $\f{T}$ (TBox) is a set of such assertions. A {\em membership assertion} is an assertion of the form $A(c)$ or $P(c,c')$, where $c,c'$ are object (or individual) constants. We denote $\f{A}$ any set of membership assertions (ABox). The integer $\card{\f{A}}$ denotes the number of (distinct) tuples occuring among the atoms in $\f{A}$. The integer $\card{\f{T}}$ the number of axioms in the terminology. A {\em knowledge base} is a pair $\tup{\f{T},\f{A}}$.
Let $\e{Dom}$ denote a countable infinite set of constants. The semantics of $\g{DL-Lite}_{R,\sqcap}\xspace$ is based on \logic{Fo}\xspace {\em interpretations} $\f{I} := \tup{\Delta^{\f{I}}, .^{\f{I}}}$, where $\Delta^{\f{I}} \subsetneq \e{Dom}$ is a non-empty {\em domain}. Interpretations map each constant $c$ to itself, each atomic concept $A$ to $A^{\f{I}} \subseteq \Delta^{\f{I}}$ and each atomic role $P$ to $P^{\f{I}} \subseteq \Delta^{\f{I}} \times \Delta^{\f{I}}$ such that the following conditions hold: \begin{itemize} \item $(P^-)^{\f{I}} := \set{\tup{d, e} \in \Delta^{\f{I}} \times \Delta^{\f{I}} \mid \tup{e, d} \in P^{\f{I}}}$, \item $(\exists R)^{\f{I}} := \set{d \in \Delta^{\f{I}} \mid \text{exists } e \in \Delta^{\f{I}} \text{ s.t. } \tup{d, e} \in R^{\f{I}}}$, \item $(D \sqcap D')^{\f{I}} := D^{\f{I}} \cap D'^{\f{I}}$, \item $(\neg A)^{\f{I}} := \Delta^{\f{I}} - A^{\f{I}}$, \item $(\neg \exists R)^{\f{I}} := \Delta^{\f{I}} - (\exists R)^{\f{I}}$, and \item $(\exists R . E)^{\f{I}} := \set{d \in \Delta^{\f{I}} \mid \text{exists } e \in \Delta^{\f{I}} \text{ s.t. } \tup{d, e} \in R^{\f{I}} \text{ and } e \in E^{\f{I}}}$. \end{itemize}
We say that $\f{I}$ {\em models} an assertion $D \sqsubseteq E$ (resp. $R \sqsubseteq R'$), and write $\f{I} \models D \sqsubseteq E$ (resp. $\f{I} \models R \sqsubseteq R'$), whenever $D^{\f{I}} \subseteq E^{\f{I}}$ (resp. $R^{\f{I}} \subseteq R'^{\f{I}}$) and a TBox $\f{T}$, and write $\f{I} \models \f{T}$, whenever it is a model of all of its assertions. We say that it {\em models} a membership assertion $A(c)$ (resp. $R(c,c')$), and write $\f{I} \models A(a)$ (resp. $\f{I} \models R(c,c')$), whenever $c^{\f{I}} \in A^{\f{I}}$ (resp. $\tup{c^{\f{I}},c^{\f{I}}} \in R^{\f{I}}$) and an ABox $\f{A}$, and write $\f{I} \models \f{A}$, when it {\em models} all of its membership assertions. Finally, we say that it is a {\em model} of a KB $\tup{\f{T}, \f{A}}$, and write $\f{I} \models \tup{\f{T}, \f{A}}$, if it is a model of both $\f{T}$ and $\f{A}$.
The semantics \logic{Fo} formulas is defined, we recall, in the usual terms of satisfaction w.r.t. interpretations $\f{I}$. Let $\phi$ be a \logic{Fo} formula and let $\Var{\phi}$ denote the set of its variables. An {\em assignment} for $\phi$ relative to $\f{I}$ is a function $v \colon \Var{\phi} \to \Delta^{\f{I}}$, that can be recursively extended in the standard way to complex formulas (see, e.g., \cite{CoriLascar}). It is said to {\em satisfy} an atom $R(x_1,...,x_n)$ w.r.t. $\f{I}$ iff $\tup{v(x_1),...,v(x_n)} \in R^{\f{I}}$. This definition is recursively extended to complex formulas \cite{CoriLascar}. If $v$ satisfies $\phi$ w.r.t. $\f{I}$, we write $\f{I} \models_v \phi$. An interpretation $\f{I}$ is said to be a {\em model} of $\phi$, written $\f{I} \models \phi$, if there exists an assignment $v$ s.t. $\f{I} \models_v \phi$.
A {\em union of conjunctive queries} (UCQ) of arity $n$ is a (positive existential) \logic{Fo} formula of the form $\phi := \psi_{1}(\vec{x},\vec{y}_1) \lor ... \lor \psi_{k}(\vec{x},\vec{y}_k)$ where $\vec{x}$ is a sequence of $n \geq 0$ {\em distinguished variables} and the $\psi_i$s, for $i \in [1,k]$, are conjunctions of atoms. A UCQ is said to be {\em boolean} if $\vec{x}$ is an empty sequence. The integer $\textit{size}(\phi)$ denotes the number of symbols of $\phi$.
Let $\tup{\f{T},\f{A}}$ be a KB and $\phi$ a UCQ of arity $n$. KB $\tup{\f{T},\f{A}}$ is said to {\em entail} $\phi$, written $\tup{\f{T},\f{A}} \models \phi$, iff for all interpretations $\f{I}$, $\f{I} \models \tup{\f{T},\f{A}}$ implies that $\f{I} \models \phi$. The {\em certain answers} of a UCQ $\phi$ over KB $\tup{\f{T},\f{A}}$ are defined as the set $\textit{cert}(q,\f{O},\f{D}) := \set{\vec{c} \in \e{Dom}^n \mid \f{T}, \f{A} \models \phi(\vec{c})}$, where $\phi(\vec{c})$ denotes the instantiation of $\vec{x}$ in $\phi$ by a sequence of constants $\vec{c}$. The associated decision problem is known as the KB {\em query answering} problem (\g{QA}) and is defined as follows: \begin{itemize} \item given $\vec{c} \in \e{Dom}^n$, a UCQ $\phi$ of arity $n$ and a KB $\tup{\f{T}, \f{A}}$, \item does $\f{T}, \f{A} \models \phi(\vec{c})$? \end{itemize}
When $\card{T}$ and $\textit{size}(\phi)$ are fixed we speak about the {\em data complexity} of \g{QA}, when only $\textit{size}(\phi)$ about its {\em KB complexity}, when $\card{T}$ and $\card{A}$ are fixed about its {\em query complexity} and finally, when none is fixed, about its {\em combined complexity}. It is known \cite{Calvanese2007C} that $\g{DL-Lite}_{R,\sqcap}\xspace$ is in \textsc{LogSpace}\xspace in data complexity, \textsc{PTime}\xspace-complete in KB complexity and \textsc{NP}\xspace-complete in query complexity, but its combined complexity remains unknown.
\section{Combined Complexity of \g{QA}}\label{two}
A {\em perfect reformulation} is an algorithm that takes as input a DL TBox $\f{T}$ and a UCQ $\phi$ and rewrites $\phi$ w.r.t. $\f{T}$ into a UCQ $\phi_{\f{T}}$ s.t., for every DL ABox $\f{A}$ and every $\vec{c} \in \e{Dom}$ it holds that: $\f{T} , \f{A} \models \phi(\vec{c})$ iff $\f{I}(\f{A}) \models \phi_{\f{T}}(\vec{c})$, where $\f{I}(\f{A})$ denotes the interpretation built out of $\f{A}$ (i.e., $\f{A}$ seen as a \logic{Fo} interpretation).
\begin{proposition} {\bf (Calvanese et al. 2006)} A perfect reformulation exists for $\g{DL-Lite}_{R,\sqcap}\xspace$. \end{proposition}
\begin{theorem} QA for $\g{DL-Lite}_{R,\sqcap}\xspace$ is \textsc{NP}\xspace-complete in combined complexity. \end{theorem}
\proof (\e{Membership}) Let $\tup{\f{T},\f{A}}$ be a KB and let $\phi(\vec{c})$ be the grounding of a UCQ $\phi$. First, consider: $\f{T} , \f{A} \models \phi(\vec{c}).$ We know that $\f{T}$ can be "compiled" into $\phi$ by a perfect reformulation, yielding a UCQ $\phi_{\f{T}}(\vec{c}) := \psi^{\f{T}}_1(\vec{c},\vec{y}_1) \lor ... \lor \psi^{\f{T}}_k(\vec{c},\vec{y}_k)$. Guess, therefore, a disjunct $\psi^{\f{T}}_i(\vec{c},\vec{y}_i)$, for some $i \in [1,k]$. This can be done in time constant in $\card{\f{T}}$ and $\textit{size}(q)$. Clearly, $\f{T} , \f{A} \models \phi(\vec{c})$ iff $\f{I}(\f{A}) \models_v \psi^{\f{T}}_i(\vec{c},\vec{y_i})$, for some assignment $v$. Guess now an assignment $v \colon \Var{\psi_i} \to \Delta^{\f{I}(\f{A})}$. This can be done in time constant in, ultimately, $\textit{size}(\phi)$. Finally, check in time polynomial on $\card{\f{A}}$ and $\textit{size}(\phi)$ whether $\f{I}(\f{A})\models_{v} \psi_i(\vec{c},\vec{y}_i)$.
(\e{Hardness}) By reduction from the graph homomorphism problem, where, given two graphs $G_1 = \tup{V_1, E_1}$ and $G_2 = \tup{V_2, E_2}$ we ask whether there exists an homomorphism $h$ from $G_1$ to $G_2$. A graph homomorphism, we recall, is a function $h \colon V_1 \to V_2$ s.t. for all $\tup{u,v} \in V_1$, $\tup{h(u),h(v)} \in V_2$. This problem is known to the \textsc{NP}\xspace-complete. We will consider $\g{DL-Lite}_{R,\sqcap}\xspace$ KBs with empty TBoxes. Polynomially encode $G_1$ and $G_2$ as follows: \begin{itemize} \item for each $\langle u, v \rangle \in E_1$, add the fact $R(c_u,c_v)$ to the ABox $\f{A}_{G_1}$, \item for each $\langle u', v' \rangle \in E_2$, add the ground atom $R(c_{u'},c_{v'})$ to the boolean UCQ $\phi_{G_2}$, which is the conjunction of such atoms. \end{itemize}
We now claim that there exists an homomorphism $h$ from graph $G_2$ to graph $G_1$ iff $\f{A}_{G_1} \models \phi_{G_2}$.
Since there is a perfect reformulation for $\g{DL-Lite}_{R,\sqcap}\xspace$, then $\f{A}_{G_1} \models \phi_{G_2}$ iff $\f{I}(\f{A}_{G_1}) \models \phi_{G_2}$. Now, clearly, $\f{I}(\f{A}_{G_1}) = G_1$. Thus, the interpretation function $.^{\f{I}(\f{A}_{G_1})}$ can be seen as an homomorphism mapping $\phi_{G_2}$ to $G_1$. Finally, given that $\phi_{G_2}$ encodes $G_2$, the claim follows. \qed
\section{$\g{DL-Lite}_{R,\sqcap}\xspace$ Simulations}\label{three}
Given two interpretations $\f{I}$ and $\f{J}$, a {\em $\g{DL-Lite}_{R,\sqcap}\xspace$ left} $\f{B}_l$ or {\em right simulation} $\f{B}_r$ is a relation $\f{B}_l, \f{B}_r \subseteq \f{P}(\Delta^{\f{I}}) \times \Delta^{\f{J}}$ s.t., for every $X \subseteq \Delta^{\f{I}}$, every $d' \in \Delta^{\f{J}}$\footnote{Observe that the clause for $D \sqcap D'$ follows implicitly from the first two.}: \begin{itemize} \item if $\tup{X,d'} \in \f{B}_l$ and $X \subseteq A^{\f{I}}$, then $d' \in \Delta^{\f{J}} \, (A)$. \item if $\tup{X,d'} \in \f{B}_l$ and forall $d \in X$ there is some $e \in Y \subseteq \Delta^{\f{I}}$ s.t. $\tup{d,e} \in R^{\f{I}}$, then there exists an $e' \in \Delta^{\f{J}}$ s.t. $\tup{d',e'} \in R^{\f{J}} \, (\exists R)$. \item if $\tup{X,d'} \in \f{B}_r$ and $X \subseteq \neg B^{\f{I}}$, then $d' \not\in B^{\f{J}} \, (\neg B)$. \item if $\tup{X,d'} \in \f{B}_r$ and forall $d \in X$ there exists no $e \in Y \subseteq \Delta^{\f{I}}$ s.t. $\tup{d,e} \in R^{\f{I}}$, then there is no $e' \in \Delta^{\f{J}}$ s.t. $\tup{d',e'} \in R^{\f{J}} \, (\neg \exists R)$. \item if $\tup{X,d'} \in \f{B}_r$ and forall $d \in X$ there exists an $e \in Y \subseteq \Delta^{\f{I}}$ s.t. $\tup{d,e} \in R^{\f{I}}$, then there is an $e' \in \Delta^{\f{J}}$ s.t. $\tup{d',e'} \in R^{\f{J}}$ and $\tup{Y,e'} \in \f{B} \, (\exists R . C)$. \end{itemize}
A {\em $\g{DL-Lite}_{R,\sqcap}\xspace$ simulation} $\f{B}$ is either a left, a right or a combination of both simulations (i.e., their union). If a $\g{DL-Lite}_{R,\sqcap}\xspace$ simulation $\f{B}$ exists among two interpretations ${\f{I}}$ and ${\f{J}}$ we say that they are {\em DL-similar} and write ${\f{I}} \sim_{DL} {\f{J}}$.
We say that a \logic{Fo} formula $\phi$ is {\em closed under $\g{DL-Lite}_{R,\sqcap}\xspace$ simulations} iff for every two interpretations $\f{I}$ and $\f{J}$, if $\f{I} \models \phi$ and $\f{I} \sim_{DL} \f{J}$, then $\f{J} \models \phi$.
We say that a \logic{Fo} formula $\phi$ {\em entails} a $\g{DL-Lite}_{R,\sqcap}\xspace$ concept $C$, written $\phi \models C$, iff for all $\f{I}$, $\f{I} \models \phi$ implies that $C^{\f{I}} \neq \emptyset$, and conversely, that $C$ {\em entails} $\phi$, written $C \models \phi$, whenever, for all $\f{I}$, $C^{\f{I}} \neq \emptyset$ implies $\f{I} \models \phi$. If both entailments hold, we say that they are {\em equivalent}.
\begin{lemma} If A \logic{Fo} formula $\phi$ is closed under \g{DL-Lite} simulations, then it is equivalent to a \g{DL-Lite} right hand or left hand side concept. \end{lemma}
\proof Let $\phi$ be a \g{FOL} formula closed under $\g{DL-Lite}_{R,\sqcap}\xspace$ simulations. Let $\textit{Con}(\phi)$ denote the set of consequences in $\g{DL-Lite}_{R,\sqcap}\xspace$ of a \logic{Fo} formula $\phi$, i.e., $\textit{Con}(\phi) := \set{C \mid \phi \models C}$. By compactness for DLs \cite{DLHandbook} the set of concepts $\textit{Con}(\phi)$ has a model iff every finite $\Sigma \subseteq \textit{Con}(\phi)$ has a model, whence the concept $C_{\phi} := \bigsqcap \set{C \mid C \in \Sigma}$ should have a model too. We claim that $\phi$ is equivalent to $C_{\phi}$. Clearly, $\phi \models C_{\phi}$. We claim now that \begin{equation}\label{eq:e} C_{\phi} \models \phi. \end{equation} Assume that $C^{\f{I}}_{\phi} \neq \emptyset$, for an arbitrary intrepretation $\f{I}$. Then, there exists a $d \in \Delta^{\f{I}}$ s.t. $d \in C^{\f{I}}_{\phi}$. Put now $\Gamma := \set{C \mid d \not\in C^{\f{I}}}$. Then, for every $C \in \Gamma, \phi \not\models C$. Hence for every $C \in \Gamma$ there exists an interpretation ${\f{I}}_C$ s.t. ${\f{I}}_C \models \phi$ and $C^{{\f{I}}_C}=\emptyset$. The idea now is to build an interpretation $\f{J} := \tup{\Delta^{\f{J}}, .^{\f{J}}}$ from the $\f{I}_C$s: \begin{itemize} \item $\Delta^{\f{J}} := \bigcup \set{\Delta^{\f{I}_C} \mid C \in \Gamma}$, \item $.^{\f{J}}$ extends each $.^{\f{I}_C}$, for $C \in \Gamma$. \end{itemize} Define now a \g{DL-Lite} simulation $\f{B} \subseteq \f{P}(\Delta^{\f{J}}) \times \Delta^{\f{I}}$ by putting: \begin{center} \begin{tabular}{c} $\tup{X,d'} \in \f{B}$ iff for every concept $C$, $X \subseteq C^{\f{J}}$ implies $d' \in C^{\f{I}}$. \end{tabular} \end{center} We now claim that $\f{B}$ is a $\g{DL-Lite}_{R,\sqcap}\xspace$ simulation between $\f{J}$ and $\f{I}$ and a fortiori that $\f{J} \sim_{DL} \f{I}$. We prove this by induction on $C$: \begin{itemize} \item Basis: \begin{itemize} \item The property trivially holds for basic concepts. \item $C := \neg A$. Let $X \subseteq \neg A^{\f{J}}$, $\tup{X,d'} \in \f{B}$. By definition of $\f{B}$, $d' \in (\neg A)^{\f{I}}$, that is, $d' \in \Delta^{\f{I}} - A^{\f{I}}$. \item $C := \exists R$. Let $\tup{X,d'} \in \f{B}$ and $d \in X$ such that there is some $e \in Y \subseteq \Delta^{\f{J}}$ such that $\tup{d,e} \in R^{\f{J}}$. Now, $X \subseteq (\exists R)^{\f{J}}$, so $d' \in (\exists R)^{\f{I}}$ and hence there is some $e' \in \Delta^{\f{I}}$ such that $\tup{d',e'} \in R^{\f{J}}$. \item $C := \neg \exists R$. This is proven by combining the two previous cases. \end{itemize} \item Inductive step: \begin{itemize} \item $C := \exists R . E$. Let $\tup{X,d'} \in \f{B}$ s.t. exists $e \in Y \subseteq \Delta^{\f{J}}$ and $\tup{d,e} \in R^{\f{J}}$. $X \subseteq (\exists R . E)^{\f{J}}$, therefore, $d' \in (\exists R \colon D)^{\f{I}}$ by definition and so there is an $e' \in \Delta^{\f{I}}$ such that $\tup{d',e'} \in R^{\f{I}}$ and $e' \in E^{\f{I}}$. Suppose that $Y \subseteq E^{\f{J}}$. By induction hypothesis, $e' \in E^{\f{I}}$. Thus, by definition of $\f{B}$, $\tup{Y,e'} \in \f{B}$. \item $C := D \sqcap D'$ (trivial). \end{itemize} \end{itemize}
Therefore, $\f{J} \sim_{DL} \f{I}$ and since by assumption $\phi$ is closed under $\g{DL-Lite}_{R,\sqcap}\xspace$ simulations, $\f{I} \models \phi$. This means that claim (\ref{eq:e}) holds. \qed
\begin{lemma} If a \logic{Fo} formula $\phi$ is equivalent to a \g{DL-Lite} right hand or left hand side concept, then it is closed under $\g{DL-Lite}_{R,\sqcap}\xspace$ simulations. \end{lemma}
\proof Let $\f{I}$ be s.t. $\f{I} \models \phi$. Let $\f{J}$ be an interpretation DL-similar to $\f{I}$. Let $X \subseteq \Delta^{\f{I}}, d \in X, d' \in \Delta^{\f{J}}, \f{B} \subseteq \f{P}(\Delta^{\f{I}}) \times \Delta^{\f{J}}$ and assume that $\tup{X,d'} \in \f{B}$. We prove now, by induction on $C$, that $C^{\f{J}} \neq \emptyset$: \begin{itemize} \item Basis: \begin{itemize} \item $C := A$. Let $d \in A^{\f{I}}$. Then, $X \subseteq A^{\f{I}}$, whence (by definition) $d' \in A^{\f{J}}$. \item $C := \neg A$ (analogous argument). \item $C := \exists R$. Let $d \in (\exists R)^{\f{I}}$. Then there exists $e \in \Delta^{\f{I}}$ s.t. $\tup{d,e} \in R^{\f{I}}$, whence, by definition of \g{DL-Lite} simulations $\f{B}$, there is an $e' \in \Delta^{\f{J}}$ s.t. $\tup{d',e'} \in R^{\f{J}}$, that is, s.t. $d' \in (\exists R)^{\f{J}}$. \item $C := \neg \exists R$ (analogous argument). \end{itemize} \item Inductive step: \begin{itemize} \item $C := \exists R . E$. Suppose that $d \in (\exists R \colon E)^{\f{I}}$. Therefore there is some $e \in \Delta^{\f{I}}$ s.t. $e \in E^{\f{I}}$ and $\tup{d,e'} \in R^{\f{I}}$. By induction hypothesis this implies that $e \in E^{\f{J}}$, whence $d \in (\exists R . E)^{\f{J}}$ as well. \item $C := D \sqcap D'$. By induction hypothesis the property holds for $D$ and $D'$. Now: \begin{equation*} \begin{array}{ccl} d \in (D \sqcap D')^{\f{I}} & \text{ iff } & d \in D^{\f{I}} \text{ and } d \in D'^{\f{I}}\\
& \text{ implies } & d' \in D^{\f{J}} \text{ and } d' \in D'^{\f{J}}\\
& \text{ iff } & d' \in (D \sqcap D')^{\f{J}}. \end{array} \end{equation*} \end{itemize} Therefore, since $\phi$ is equivalent to $C$, $\f{J} \models \phi$, as desired. \qed \end{itemize}
\begin{theorem} A \logic{Fo} formula $\phi$ is equivalent to a $\g{DL-Lite}_{R,\sqcap}\xspace$ right hand or left hand side concept iff it is closed under \g{DL-Lite} simulations. \end{theorem}
\begin{example} The \logic{Fo} formula $\phi := \forall y P(x,y) \to A(y)$ is not equivalent to any $\g{DL-Lite}_{R,\sqcap}\xspace$ concept, because it is not closed under $\g{DL-Lite}_{R,\sqcap}\xspace$ simulations.
\begin{center} \begin{pspicture}(10,4.5) \psline{->}(2.8,2.5)(0.5,3.3) \psline{->}(2.8,2.5)(0.5,1.7) \psellipse(0.5,2.5)(0.4,1.3) \put(0,4.1){$\Delta^{\f{I}}$} \put(9.5,4.1){$\Delta^{\f{J}}$} \put(5,2.5){$\f{B}$} \put(1.3,2.5){$P^{\f{I}}$} \put(7.8,2.5){$P^{\f{J}}$} \put(0.8,3.6){$A^{\f{I}}$} \cput[doubleline=false](3,2.5){$d$} \put(2.8,1.8){$X$} \put(0.3,3.35){$e_1$} \put(0.3,1.5){$e_2$} \put(6.7,2.6){$e'$} \psline[linearc=.25]{<-}(6.8,2.5)(8,3)(9.5,2.5) \put(9.5,2.6){$d'$} \psframe(0,1)(3.5,4) \psframe(6.5,1)(10,4) \psline[linestyle=dashed,linearc=.5](0.8,3.5)(5,3.5)(6.7,2.5) \psline[linestyle=dashed,linearc=.5](3.4,2.5)(5,1.5)(9.5,2.5) \end{pspicture} \end{center} As the reader can see, $\f{B}$ is a $\g{DL-Lite}_{R,\sqcap}\xspace$ simulation there \textit{(i)} $\tup{\set{d},d'} \in \f{B}$, \textit{(ii)} $\tup{\set{e_1,e_2},e'} \in \f{B}$ and \textit{(iii)} $\f{I} \sim_{DL} \f{J}$. Now, clearly, $\f{I} \models_{v[x:=d]} \forall y P(x,y) \to A(y)\}$, but $\f{J} \not\models_{v'[x:=d']} \forall y P(x,y) \to A(y)$, since $A^{\f{J}} = \emptyset$. \hspace*{\fill}$\clubsuit$ \end{example}
\section{Some Negative Results}\label{four}
\begin{proposition} Disjunction is not expressible in $\g{DL-Lite}_{R,\sqcap}\xspace$. \end{proposition}
\proof $\g{DL-Lite}_{R,\sqcap}\xspace$ is contained in \g{HORN} (the set of \logic{Fo} horn clauses)\cite{Calvanese2007C,Calvanese2007E}, which cannot express disjunctions of the form $\phi := A(c) \lor A'(c')$. Otherwise, let $\f{H} := \set{A(c)}$ and $\f{H'} := \set{A'(c')}$ be two Herbrand models of $\phi$. Clearly, $\f{H}$ and $\f{H'}$ are minimal (w.r.t. set inclusion) models of $\phi$ s.t. $\f{H} \neq \f{H'}$. But this is impossible, since \g{HORN} verifies the least (w.r.t. set inclusion) Herbrand model property \cite{CoriLascar}. \qed
\begin{theorem} There is no relation $\sim$ over interpretations such that, for every \logic{Fo} sentence $\phi$, $\phi$ is equivalent to a $\g{DL-Lite}_{R,\sqcap}\xspace$ assertion iff it is closed under the relation $\sim$. \end{theorem}
\proof Recall that a \logic{Fo} sentence is a \logic{Fo} formula with no free variables. Suppose the contrary and consider the sentence $A(c)$. Let $\f{I}$ and $\f{J}$ be two structures s.t. $\f{I} \sim \f{J}$ and suppose that $\f{I} \models A(c)$. Then, obviously, $\f{J} \models A(c)$ too. But then: \begin{center} \begin{tabular}{l} $\f{I} \models A(c)$ implies $\f{I} \models A(c) \lor A'(c)$, and\\ $\f{J} \models A(c)$ implies $\f{J} \models A(c) \lor A'(c)$. \end{tabular} \end{center} That is, $A(c) \lor A'(c)$ is closed under $\sim$ and is a fortiori equivalent to some $\g{DL-Lite}_{R,\sqcap}\xspace$ assertion. But this is impossible, because disjunction is not expressible in $\g{DL-Lite}_{R,\sqcap}\xspace$. \qed
\section{Conclusions}\label{five}
In this paper we have shown four things: \textit{(i)} Answering UCQs over $\g{DL-Lite}_{R,\sqcap}\xspace$ KBs is \textsc{NP}\xspace-complete in combined complexity. \textit{(ii)} A simulation relation among interpretations, viz., a $\g{DL-Lite}_{R,\sqcap}\xspace$ simulation, can be used to characterize the expressive power of $\g{DL-Lite}_{R,\sqcap}\xspace$ as a concept language. \textit{(iii)} \logic{Fo} formulas that are closed under $\g{DL-Lite}_{R,\sqcap}\xspace$ simulations are equivalent to a (left or right) $\g{DL-Lite}_{R,\sqcap}\xspace$ concept. \textit{(iv)} This closure property holds only w.r.t. concepts, but not w.r.t. assertions. Simulations, in particular, can be generalized, with minor adjustments, to the whole \g{DL-Lite} family of DLs, although, since all of them are in \g{HORN}, no such closure property exists for their assertions.
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\thispagestyle{empty} \baselineskip=28pt \vskip 5mm \begin{center} {\Large{\bf Partial Tail-Correlation Coefficient \\Applied to Extremal-Network Learning}} \end{center} \baselineskip=12pt \vskip 5mm
\begin{center} \large Yan Gong$^1$, Peng Zhong$^1$, Thomas Opitz$^2$, Rapha\"el Huser$^1$ \end{center}
\footnotetext[1]{ \baselineskip=10pt Statistics Program, Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia. E-mails: [email protected]; [email protected]; [email protected].} \footnotetext[2]{ \baselineskip=10pt Biostatistics and Spatial Processes, INRAE, Avignon 84914, France. E-mail: [email protected].}
\baselineskip=17pt \vskip 4mm \centerline{\today} \vskip 6mm
\begin{center} {\large{\bf Abstract}} \end{center} We propose a novel extremal dependence measure called the partial tail-correlation coefficient (PTCC), in analogy to the partial correlation coefficient in classical multivariate analysis. The construction of our new coefficient is based on the framework of multivariate regular variation and transformed-linear algebra operations. We show how this coefficient allows identifying pairs of variables that have partially uncorrelated tails given the other variables in a random vector. Unlike other recently introduced conditional independence frameworks for extremes, our approach requires minimal modeling assumptions and can thus be used in exploratory analyses to learn the structure of extremal graphical models. Similarly to traditional Gaussian graphical models where edges correspond to the non-zero entries of the precision matrix, we can exploit classical inference methods for high-dimensional data, such as the graphical LASSO with Laplacian spectral constraints, to efficiently learn the extremal network structure via the PTCC. We apply our new method to study extreme risk networks in two different datasets (extreme river discharges and historical global currency exchange data) and show that we can extract meaningful extremal structures with meaningful domain-specific interpretations.
\baselineskip=16pt
\par
\noindent {\bf Keywords:} Extreme event; Graphical Lasso; Multivariate regular variation; Network structure learning; Partial tail dependence.\\
\pagenumbering{arabic} \baselineskip=26pt
\section{Introduction}\label{sec:introduction}
Characterizing the extremal dependence of complex stochastic processes (e.g., in spatial, temporal, and spatio-temporal settings) is fundamental for both statistical modeling and applications, such as risk assessment in environmental and financial contexts. Important applications include the modeling of precipitation extremes \citep{Huser.Davison:2014a,opitz2018inla,bacro2020hierarchical,saunders2021regionalisation,richards2022modelling}, heatwaves \citep{winter2016modelling,zhong2022modeling}, and air pollution \citep{Vettori.etal:2019, Vettori.etal:2020}, as well as financial risk assessment \citep{bassamboo2008portfolio,ferro2011extremal,marcon2016bayesian,bekiros2017extreme,yan2019cross,gong2019asymmetric}. Often, applications illustrate the benefits of methodological innovations, such as the application of the extremal dependence measure in \citet{larsson2012extremal}, which is a key ingredient of the present work, in the analysis of financial data.
Models for extremal dependence traditionally rely on asymptotic frameworks, such as max-stable processes for block maxima or $r$-Pareto processes for threshold exceedances of a summary functional of the process over a high threshold. Recently, more advanced models have been proposed to further improve flexibility, especially towards modeling of asymptotically independent data with dependence vanishing at the most extreme levels, such as inverted max-stable processes \citep{wadsworth2012dependence}, max-mixture models \citep{ahmed2020spatial}, random scale mixtures of Gaussian processes \citep{Opitz2016,Huser2017} or of more general processes \citep{wadsworth2017modelling,engelke2019extremal,huser2019modeling}, max-infinitely divisible processes \citep{bopp2021hierarchical}, and conditional spatial extremes models \citep{wadsworth2019higher}; for a comprehensive review, see \citet{huser2022advances}. Specifics of serial extremal dependence have been studied by \citet{davis2013measures}, among others.
In the study of stochastic dependence structures, networks and graphs are natural tools to represent dependence relationships in multivariate data. Conditional independence, sparsity, and parsimonious representations are key concepts in graph-based approaches for random vectors. Recently, graph-based tools have also been developed for extremal dependence, where variants of conditional independence apply to variables not directly connected by the edges of the graph. For example, \citet{huang2019new} provide an exploratory tool, called the \emph{$\chi\text{-}$network,} for modeling extremal dependence, and they use it to analyze maximum precipitation during the hurricane season in the United States (US) Gulf Coast and in surrounding areas. In their approach, however, the $\chi$-network does not remove the effect of confounding variables, so it does not naturally lead to sparse extremal dependence representations. More recently, \citet{engelke2020graphical} introduce a notion of conditional independence adapted to multivariate Pareto distributions arising for limiting multivariate threshold exceedances, and they use it to develop {parametric graphical models for extremes based on the H\"usler--Reiss distribution}. Similarly, \citet{gissibl2018graphical} and \citet{kluppelberg2021estimating} propose max-linear constructions for modeling maxima on tree-like supports, and \citet{tran2021estimating} propose {QTree}, a simple and efficient algorithm to solve the ``Latent River Problem" for the important case of extremes on trees. In the same vein, \citet{engelke2020structure} develop a data-driven methodology for learning the graphical structure in the setting of \citet{engelke2020graphical}, whereas \citet{rottger2021total} further propose H\"usler--Reiss graphical models under the assumption of multivariate total positivity of order two (MTP$_2$), which allows estimating sparse graphical structures. Finally, \citet{engelke2021sparse} review the recent developments in sparse representations, dimension reduction approaches, and graphical models for extremes. Overall, existing graphical representations for extremes from the literature often rely on rather stringent asymptotically justified models, sometimes leading to issues when dealing with relatively high-dimensional problems or when specific graph structure assumptions (e.g., trees) are required. A recent exception is \citet{Engelke.etal:2022}, who develop theories and methods for learning extremal graphical structures in high dimensions based on $L_1$ regularized optimization, though their methodology still assumes a parametric extremal dependence structure of H\"usler--Reiss type.
By contrast, rather than restricting ourselves to a strict parametric modeling framework, we adopt a more pragmatic and empirical approach. Specifically, our goal is to extend and enrich existing approaches by defining the new concept of \emph{partial tail correlation} as an extreme-value analog of the notion of partial correlation widely used in classical multivariate analysis, and by introducing a new coefficient that enables estimation of general extremal networks under minimal modeling assumptions. In the same way that correlation does not imply independence in general, our concept of \emph{partial tail-uncorrelatedness} is a weaker assumption than \emph{conditional tail independence}. However, we shall show that it still provides relevant insights into various forms of extremal dependence structures and helps in guiding modeling choices at a data exploratory stage.
As a novel extremal dependence measure, we propose the \emph{partial tail correlation coefficient (PTCC)} as an equivalent of the partial correlation coefficient in the non-extreme setting. In the classical setting, the Pearson correlation coefficient between two random variables can give misleading interpretations when there are confounding variables that influence both variables, whereas the partial correlation coefficient measures the residual degree of association between two variables after the linear effects of a set of other variables have been removed. To compute the partial correlation between two variables of interest, we regress each of these variables onto the set of covariates given by all the other variables in the multivariate random vector, and then compute the correlation between the residuals from the two fitted linear regressions. In the Gaussian setting, a partial correlation of zero is equivalent to conditional independence between two variables \citep{lawrance1976conditional, baba2004partial}, and the elements of the inverse of the covariance matrix (i.e., the \emph{precision matrix}) of the full vector are known to characterize this conditional (in)dependence structure. In this paper, we adopt a similar strategy to define the PTCC, namely by computing a suitable measure of tail dependence between residuals obtained by regressing variables using transformed-linear operations that do not alter tail properties. While classical linear regression only makes sense for Gaussian-like data, such transformed-linear operations can be used for tail regression of multivariate regularly-varying random vectors, which is a fundamental assumption characterizing asymptotically dependent extremes.
To be more precise, we here define the PTCC by building upon the framework of \citet{cooley2019decompositions}, who developed a customized transformed-linear algebra on the positive orthant, preserving multivariate regular variation and thus being well adapted to ``linear'' methods for joint extremes. \citet{cooley2019decompositions} used this framework for principal component analysis of extremes based on decompositions of the so-called tail pairwise dependence matrix (TPDM), which conveniently summarizes information about extremal dependence in random vectors and possesses appealing properties for such decompositions. The TPDM can be thought of as an analogy of the classical covariance matrix but tailored for multivariate extremes. In some follow-up work, \citet{mhatre2020transformed} then developed non-negative regularly varying time series models with autoregressive moving average (ARMA) structure using the transformed-linear operations for time series extremes. For spatial extremes, \citet{fix2021simultaneous} extended the simultaneous autoregressive (SAR) model under the transformed-linear framework and developed an estimation method to minimize the discrepancy between the TPDM of the fitted model and an empirically estimated TPDM. Furthermore, \citet{lee2021transformed} recently introduced transformed-linear prediction methods for extremes. In the aforementioned papers, the TPDM always plays a central role. Similar to covariances, the entries of the TPDM are tail dependence measures giving insights into the direct extremal dependence structure without removing the influence of other confounding variables. However, just as the covariance matrix does not reflect partial correlations, the TPDM does not directly inform us about partial associations among extremes. In this work, we fill this gap with our new proposed PTCC. Thanks to its definition in terms of transformed-linear operations, we show that the PTCC inherits several appealing features of the classical partial correlation coefficient. In particular, the PTCC between two components $X_i$ and $X_k$ from a random vector $\bm X$ is such that the $(i,k)$th entry of the inverse TPDM matrix of $\bm X$ equals zero if and only if the corresponding PTCC for these two variables is also equal to zero. In other words, \emph{partial tail-uncorrelatedness} can be conveniently read off from the zero elements of the inverse TPDM, similar to classical Gaussian graphical models. We then exploit this property to define a new class of extremal graphical models based on the PTCC and then use efficient inference methods to learn the extremal network structure from high-dimensional data based on state-of-the-art techniques from graph theory (e.g., the graphical LASSO with or without Laplacian spectral constraints). We note that here, our focus is on studying undirected graph structures, which is different from causal inference, where causal relationships can be encoded using directed graph edges.
The remainder of this article is organized as follows. In Section~\ref{sec:ptcc}, we first review the necessary background on multivariate regular variation and transformed-linear algebra, as introduced in \citet{cooley2019decompositions}. Then, we define the new PTCC and the related notion of partial tail-uncorrelatedness.
In Section~\ref{sec:extnet}, we present methods for learning general extremal network structures from the PTCC in a high-dimensional data setting, and we discuss two particularly appealing approaches, namely the graphical LASSO and Laplacian spectral constraint-based methods. Section~\ref{sec:simulation} presents a simulation study for general structured undirected graphs using the above two inference methods. In Section~\ref{sec:applications}, we apply these new tools to explore the risk networks formed by river discharges observed at a collection of monitoring stations in the upper Danube basin, and by historical global currency exchange rate data from different historical periods, covering different economic cycles, the COVID-19 pandemic, and the 2022 military conflict in Ukraine.
\section{Transformed-linear algebra for multivariate extremes}\label{sec:ptcc}
Before introducing the partial tail correlation coefficient (PTCC) and the related notion of partial-tail uncorrelatedness, we first briefly review the multivariate regular variation framework, which is our main assumption for defining the PTCC, and we also summarize the foundations of transformed-linear algebra.
\subsection{Regular variation framework and transformed linear algebra}\label{sec:regvar}
A random vector is multivariate regularly varying \citep{resnick2007heavy} (i.e., jointly heavy-tailed) if its joint tail decays like a power function. Precisely, we say that a $p$-dimensional random vector $\boldsymbol X\in \mathbb{R}_+^p = [0,\infty)^p$ with $p\in\mathbb{N}$ is \emph{regularly varying} if there exists a sequence $b_n \to\infty$ such that \begin{equation}\label{eq:mrv} n\,\pr(b_n^{-1} \boldsymbol X\in\cdot) \xrightarrow{v} \nu_{\boldsymbol X}(\cdot),\quad n\to \infty, \end{equation}
where $\xrightarrow{v}$ denotes vague convergence to the non-null limit measure $\nu_{\boldsymbol X}$, a Radon measure defined on the space $[0,\infty]^p\setminus \{0\}$. This measure has the scaling property $r^\alpha\nu_{\boldsymbol X}(rB)=\nu_{\boldsymbol X}(B)$ for $r>0$ and Borel sets $B \subset [0,\infty]^p\setminus \{0\}$, where $\alpha>0$ controls the tail decay (with $1/\alpha$ commonly called the tail index). For this reason, the measure can be further decomposed into a radial measure and an angular measure $H_{\boldsymbol X}$ on the unit sphere $\mathbb{S}_{p-1}^+ = \{\boldsymbol w \in \mathbb{R}_+^p: ||\boldsymbol w||_2 = 1\}$, such that $\nu_{\boldsymbol X}(\{\boldsymbol x \in [0,\infty]^p\setminus \{0\}: \|\boldsymbol x\|_2 \geq r, \ \boldsymbol x/\|\boldsymbol x\|_2 \in B_H)=r^{-\alpha}\times H_{\boldsymbol X}(B_H)$ for $r>0$ and Borel subsets $B_H$ of $\mathbb{S}_{p-1}^+$. The normalizing sequence $b_n$ is not uniquely determined but must satisfy $b_n = L(n)n^{1/\alpha}$, where $L(n)$ is a slowly varying function (at infinity), i.e., $L(n)>0$ and $L(rn)/L(n)\rightarrow 1$ for any $r>0$, as $n\rightarrow\infty$. We use the short-hand notation $\boldsymbol X \in$ RV$_+^{p}(\alpha)$ for a regularly varying vector $\boldsymbol X$ with tail index $1/\alpha$.
\citet{cooley2019decompositions} introduced the \emph{transformed-linear algebra} framework to construct an inner product space on an open set (the so-called \emph{target space}) via a suitable transformation, where the distribution of the random vector $\bm X$ has support within this set. Our use will mainly concern transformation towards the target space $\mathbb{R}_+^p$ from the space $\mathbb{R}^p$, but we first present the general approach. Let $t$ be a bijective transformation from $\mathbb{R}$ onto some open set $\mathbb{X}\subset\mathbb{R}$, and let $t^{-1}$ be its inverse. For a $p$-dimensional vector $\boldsymbol y \in \mathbb{R}^p$, we define $\boldsymbol x = t(\boldsymbol y)\in \mathbb{X}^p$ componentwise. Then, arithmetic operations among elements of the target space are carried out in the space $\mathbb{R}^p$ before transforming back to the target space. We define vector addition in $\mathbb{X}^p$ as $\boldsymbol x_1\oplus \boldsymbol x_2 = t\{t^{-1}(\boldsymbol x_1) + t^{-1}(\boldsymbol x_2)\}$, and scalar multiplication with a factor $a\in\mathbb{R}$ as $a\circ \boldsymbol x = t\{at^{-1}(\boldsymbol x)\}.$ The additive identity in $\mathbb{X}^p$ is set to $\boldsymbol 0_{\mathbb{X}^p}=t(\boldsymbol 0)$, and the additive inverse of $\boldsymbol x\in\mathbb{X}^p$ is given as $\ominus \boldsymbol x=t\{-t^{-1}(\boldsymbol x)\}$. A valid inner product between two elements $\boldsymbol x_1=(x_{1,1},\ldots,x_{1,p})^T,\boldsymbol x_2=(x_{2,1},\ldots,x_{2,p})^T\in\mathbb{X}^p$ from the target space is then obtained by applying the usual scalar product in $\mathbb{R}^p$, i.e., we set $\langle \boldsymbol x_1,\boldsymbol x_2\rangle =\sum_{j=1}^p t^{-1}(x_{1,j})t^{-1}(x_{2,j})$. To obtain an inner product space on the positive orthant for which arithmetic operations preserve multivariate regular variation, thus having a negligible effect on large values, we follow \citet{cooley2019decompositions} and define the specific transformation $t: \mathbb{R}\mapsto (0, \infty)$ given by $$ t(y) = \log\{1+\exp(y)\}, $$ though they are other possibilities. We have $y/t(y)\rightarrow 1$ as $y\rightarrow \infty$, such that the upper tail behavior of a random vector $\boldsymbol Y=t(\boldsymbol X)$ is preserved through $t$. For lower tails, we have $\exp(y)/t(y)\rightarrow 1$ as $y\rightarrow -\infty$. The inverse transformation is $t^{-1}(x) = \log(\exp(x)-1)$, $x>0$. Algebraic operations done in the vector space induced by the above transformation $t$ are commonly called \emph{transformed-linear operations}, and we can exploit this framework to extend classical linear algebra methods (e.g., principal component analysis, etc.) to the multivariate extremes setting, where heavy-tailed vectors and models are often conveniently expressed on the positive orthant, $\mathbb R^p_+$. We note that our main assumption, multivariate regular variation, implies asymptotic tail dependence (as well as homogeneity of the limit measure in \eqref{eq:mrv}), but it does not impose further parametric structural assumptions such as with Pareto models of H\"usler--Reiss type.
\subsection{Inner product space of regularly varying random variables}
With transformed-linear operations, we can use a vector of independent and identically distributed (i.i.d.) random variables to construct new regularly varying random vectors on the positive orthant that possess tail dependence. Suppose that $\boldsymbol Z = (Z_1,\ldots,Z_q)^T \geq 0$ is a vector of $q\in\mathbb{N}$ i.i.d.\ regularly varying random variables with tail index $1/\alpha$, such that there exists a sequence $\{b_n\}$ that yields $$ n\,\pr(Z_j>b_n z)\to z^{-\alpha}, \quad n\,\pr\{Z_j\leq \exp(-k b_n)\}\to 0, \quad k>0, \quad j=1,\ldots,q, $$ where the first condition is equivalent to regular variation \eqref{eq:mrv} in dimension $p=1$. The random vector $\bm Z$ with independent components has a limit measure of multivariate regular variation characterized by $\nu_{\bm Z}\left\{[\bm 0, \bm z]^C\right\}=\sum_{j=1}^q z_j^{-\alpha}$ for $\bm z=(z_1,\ldots,z_q)^T > \bm 0$. Then, we can construct new regularly varying $p$-dimensional random vectors $\boldsymbol X = (X_1, \ldots,X_p)^T$ by exploiting transformed-linear operations, via a matrix product with a deterministic matrix $A = (\boldsymbol a_1, \ldots, \boldsymbol a_q) \in \mathbb{R}_+^{p\times q},$ with columns $\boldsymbol a_j\in \mathbb{R}_+^p$, as follows: \begin{equation}\label{eq:transformed-linear construction}
\boldsymbol X = \overset{q}{\underset{j=1}{\oplus}}\boldsymbol a_j\circ Z_j. \end{equation} We write $\boldsymbol X = A \circ \boldsymbol Z \in$ RV$_+^{p}(\alpha)$. This construction ensures that the multivariate regular variation property is preserved with the same index $\alpha$ \citep[Corollary 1,][]{cooley2019decompositions}. Furthermore, we require $A$ to have a full row-rank. Based on the construction \eqref{eq:transformed-linear construction}, it is possible to define a (different) inner product space spanned by the random variables obtained by transformed-linear operations on $\boldsymbol Z$, where some but not all of the components of $\boldsymbol a_j$ are further allowed to be non-positive.
Following \citet{lee2021transformed}, an inner product of $\langle X_i,X_k\rangle $ on the space spanned by all possible transformed-linear combinations of the elements of the random vector $\boldsymbol X$ constructed as in \eqref{eq:transformed-linear construction}, may be defined as follows:
\[\langle X_i, X_k\rangle = \sum_{j=1}^q a_{ij} a_{kj},\]
where $a_{ij}$ refers to the entry in row $i$ of the column $j$ of the matrix $A$ for $i\in\{1,\ldots,p\}$ and $j\in\{1,\ldots,q\}$, and the corresponding norm becomes $||X|| = \sqrt{\langle X, X\rangle }$. The metric induced by the inner product is $d(X_i, X_k) = ||X_i\ominus X_k||=[\sum_{j=1}^q(a_{ij}-a_{kj})^2]^{1/2},$ for $i,k = 1,\ldots, p.$
\subsection{Generality of the framework}
In practice, given a random vector $\bm X$ for which we assume $\boldsymbol X \in$ RV$_+^{p}(\alpha)$, we will further assume that it allows for a stochastic representation as in \eqref{eq:transformed-linear construction}. Since the constructions of type \eqref{eq:transformed-linear construction} form a dense subclass of the class of multivariate regularly varying vectors (if $q$ is not fixed but allowed to tend to infinity, i.e., $q\rightarrow\infty$), this assumption is not restrictive; see \citet{fougeres2013dense} and \citet{cooley2019decompositions}.
Thanks to their flexibility, representations akin to the transformed-linear random vectors in \eqref{eq:transformed-linear construction} have recently found widespread interest in statistical learning for extremes. The fundamental model structure used in the causal discovery framework for extremes developed by \citet{gnecco2021} is essentially based on a variant of \eqref{eq:transformed-linear construction}. In the setting of max-linear models, in particular the graphical models of \citet{gissibl2018graphical}, we can use \eqref{eq:transformed-linear construction} to construct random vectors $\boldsymbol X$ possessing the same limit measure $\nu_{\boldsymbol X}$ as the max-linear vectors. Finally, low-dimensional representations of extremal dependence in random vectors obtained through variants of the $k$-means algorithm can be shown to be equivalent to the extremal dependence induced by construction \eqref{eq:transformed-linear construction}; see \citet{janssen2020}.
\subsection{Tail pairwise dependence matrix}
The tail pairwise dependence matrix \citep[TPDM,][]{cooley2019decompositions} is defined to summarize the pairwise extremal dependence of a regularly varying random vector using the second-order properties of its angular measure. Let $\alpha = 2$, which ensures desirable properties; in practice, this condition can be ensured through appropriate marginal pre-transformation of data \citep{cooley2019decompositions}. Then, the TPDM $\Sigma$ of $\boldsymbol X\in$ RV$_+^{p}(2)$ is defined as follows: $$ \Sigma=(\sigma_{ik})_{i,k=1,\ldots,p},\qquad \text{with}\qquad \sigma_{{ik}} := \int_{\mathbb{S}_{p-1}^+}w_i w_k\text{d}H_{\boldsymbol X}(\boldsymbol w), $$
where $H_{\boldsymbol X}$ is the angular measure on $\mathbb{S}_{p-1}^+ = \{\boldsymbol w \in \mathbb{R}_+^p: ||\boldsymbol w||_2 = 1\}$ as introduced in Section~\ref{sec:regvar}. The matrix $\Sigma$ is an extreme-value analog of the covariance matrix, and it has similar useful properties. It is positive semi-definite and completely positive, i.e., there exists a finite $p\times q$ matrix $A$ with nonnnegative entries such that the TPDM can be factorized as $\Sigma=AA^T$ \citep[][Proposition~5]{cooley2019decompositions}. The matrix $A$ is not unique, in particular if we do not impose nonnegative entries.
Specifically, for random vectors $\bm X$ obtained by the transformed-linear construction \eqref{eq:transformed-linear construction}, the entries of $\Sigma$ correspond to the values of the inner product $\sigma_{ik}=\langle X_i,X_k\rangle $. In the following, we further assume that $\Sigma$ is positive definite, which guarantees the existence of the inverse matrix of the TPDM.
We emphasize that the special case where $\sigma_{ik}=0$ is equivalent to asymptotic tail independence of the components $X_i$ and $X_k$ \citep[see][]{cooley2019decompositions}, meaning that the conditional exceedance probability $\mathrm{Pr}(F_{X_i}(X_i)>u \mid F_{X_k}(X_k)>u)$ tends to $0$ as $u\to1$ \citep{sibuya1960bivariate, ledford1996statistics}.
By exploiting the property that the TPDM is completely positive, we can construct new transformed-linear random vectors that have the same TPDM as a given random vector $\boldsymbol X$. Since we can always factorize the TPDM as $\Sigma=AA^T$ for some matrix $A$ of dimension $p\times q$, we can then use the construction \eqref{eq:transformed-linear construction} by multiplying $A$ with a (new) random vector $\boldsymbol Z\in \text{RV}_+^{p}(2)$ of independent regularly varying random variables, and the resulting vector will still have TPDM $\Sigma$. It is also worth noting that as $q\to\infty$, the angular measure of the new random vector can be arbitrarily close to that of $\boldsymbol X$ thanks to the denseness property of discrete angular measures. While $A$ is not unique, we note that it is very important that the inner product depends only on the entries of the matrix $\Sigma = AA^T$, such that the specific choice of $A$ does not matter.
An estimator of the TPDM was proposed by \citet[][Section~7.1]{cooley2019decompositions}. For an i.i.d.\ sequence of vectors $\boldsymbol x_t$, $t=1,\ldots,n_{\text{samp}}$, i.e., independent realizations from a random vector $\boldsymbol X\in$ RV$_+^{p}(2)$, define
\begin{equation}\label{eq:estimator} \hat{\sigma}_{{ik}} = \hat{m}\int_{\mathbb{S}_{p-1}^+}w_i w_k\text{d}\hat{N}_{\boldsymbol X}(w)=\hat{m}\, n^{-1}_{\text{ext}}\sum^{n_{\text{samp}}}_{t=1}w_{ti}w_{tk}\mathbbm{1}(r_t>r_0),
\end{equation}
where $r_t = ||\boldsymbol x_t||_2$, $w_t=\boldsymbol x_t/r_t$, $r_0$ is a high threshold for the radial component, $n_{\text{ext}}=\sum_{t=1}^{n_{\text{samp}}}\mathbbm{1}(r_t>r_0)$ refers to the number of threshold exceedances, and the probability measure $N_{\boldsymbol X}(\cdot)=m^{-1}H_{\boldsymbol X}(\cdot)$ (with $m = H_{\boldsymbol X}(\mathbb{S}_{p-1}^+)$) is obtained by normalizing $H_{\boldsymbol X}$. Moreover, $\hat{m}$ denotes an estimate of $H_{\boldsymbol X}(\mathbb{S}_{p-1}^+)$, and $\hat{N}_{\boldsymbol X}$ is the empirical counterpart of $N_{\boldsymbol X}$. We note that when the data are preprocessed to have a common unit scale, we can set $m=p$ and there is no need to estimate the normalizing factor. The estimator \eqref{eq:estimator} was discussed by \citet{larsson2012extremal} in the bivariate case.
\subsection{Partial tail correlation coefficient}
In this section, we introduce our new measure, the partial tail correlation coefficient (PTCC), which is analogous to the partial correlation coefficient but tailored to heavy-tailed random vectors.
Let $\boldsymbol X = A \circ \boldsymbol Z \in$ RV$_+^{p}(\alpha)$ be a $p$-dimensional vector constructed as in \eqref{eq:transformed-linear construction}, with TPDM $\Sigma$. We write $\boldsymbol X_{ik}=(X_i,X_k)^T$; $\boldsymbol X_{{\rm rest}}$ for the $(p-2)$-dimensional random vector obtained by removing the two components $X_i$ and $X_k$ from $\boldsymbol X$; $A_{ik}$ for the matrix comprising the $i$-th and $k$-th columns of $A$; and $A_{{\rm rest}}$ for the matrix $A$ without its $i$-th and $k$-th columns. Moreover, we define the $p$-dimensional random vector $\boldsymbol X'$ by re-ordering the columns of $\boldsymbol X$ as $\boldsymbol X' = (\boldsymbol X_{ik}^T, \boldsymbol X_{{\rm rest}}^T)^T= (A_{ik}, A_{{\rm rest}})\circ \boldsymbol Z.$ It is straightforward to show that the best transformed-linear predictor of $\boldsymbol X_{ik}$ given $\boldsymbol X_{{\rm rest}}$ can be obtained as $\hat{\boldsymbol X}_{ik}=\boldsymbol B\circ\boldsymbol X_{{\rm rest}} = (\hat{X}_{ik}^i, \hat{ X}_{ik}^k)^T$, where $\boldsymbol B = (\boldsymbol b_1, \boldsymbol b_2)^T$ is a $2\times(p-2)$ matrix with $\boldsymbol b_1, \boldsymbol b_2\in\mathbb R_+^{p-2}$ chosen such that $d(X_{i}, \hat{X}_{ik}^i)$ and $d(X_{k}, \hat{X}_{ik}^k)$ attain their minimum, respectively. Suppose that the TPDM of $\boldsymbol X'$ is of the block-matrix form $$ \Sigma_{\boldsymbol X'} = \begin{bmatrix} \Sigma_{ik,ik} & \Sigma_{ik,{\rm rest}}\\ \Sigma_{{\rm rest},ik} & \Sigma_{{\rm rest},{\rm rest}} \end{bmatrix} $$ where $\Sigma_{ik,ik}\in\mathbb R_+^{2\times 2}$ is the TPDM restricted to $\boldsymbol X_{ik}$, i.e., $\Sigma_{ik,ik}=\text{TPDM}(\boldsymbol X_{ik})=\begin{bmatrix} \Sigma_{ii} & \Sigma_{ik} \\ \Sigma_{ki} & \Sigma_{kk} \end{bmatrix}$; $\Sigma_{{\rm rest},{\rm rest}}\in\mathbb R_+^{(p-2)\times (p-2)}$ is the TPDM restricted to $\boldsymbol X_{{\rm rest}}$, and $\Sigma_{ik,{\rm rest}} = \Sigma_{{\rm rest},ik}^T = (\Sigma_{i,{\rm rest}}^T, \Sigma_{k,{\rm rest}}^T)^T\in\mathbb R_+^{2\times (p-2)}$ is the cross-TPDM between $\boldsymbol X_{ik}$ and $\boldsymbol X_{{\rm rest}}$. Then, based on the {projection theorem} for vector spaces with inner products, we have that $$\hat{\boldsymbol B} = \Sigma_{ik,{\rm rest}}\Sigma_{{\rm rest},{\rm rest}} ^{-1}.$$ Straightforward calculations show that the {prediction error} $\boldsymbol e = \boldsymbol X_{{ik}}\ominus\hat{\boldsymbol X}_{\text{}ik}$ has the following TPDM:
\begin{align}\label{eq:prediction_error} \text{TPDM}(\boldsymbol e) &= \Sigma_{ik,ik} - \Sigma_{ik,{\rm rest}}\Sigma_{{\rm rest},{\rm rest}}^{-1}\Sigma_{{\rm rest},ik}\nonumber\\ &=\begin{bmatrix} \Sigma_{ii} - \Sigma_{i,{\rm rest}}\Sigma_{{\rm rest},{\rm rest}}^{-1}\Sigma_{{\rm rest},i} & \Sigma_{ik} - \Sigma_{i,{\rm rest}}\Sigma_{{\rm rest},{\rm rest}}^{-1}\Sigma_{{\rm rest},k} \\ \Sigma_{ki} - \Sigma_{k,{\rm rest}}\Sigma_{{\rm rest},{\rm rest}}^{-1}\Sigma_{{\rm rest},i} & \Sigma_{kk} - \Sigma_{k,{\rm rest}}\Sigma_{{\rm rest},{\rm rest}}^{-1}\Sigma_{{\rm rest},k} \end{bmatrix}.
\end{align}
\begin{defn}\label{def:ptcc} The \emph{partial tail correlation coefficient (PTCC)} of two random variables $X_i$ and $X_k$ is defined as the off-diagonal {TPDM coefficient of the bivariate residual vector} $\boldsymbol e$ in \eqref{eq:prediction_error}, such that transformed-linear dependence with respect to all other random variables is removed. \end{defn}
\begin{defn}
Let $\boldsymbol X_{ik} = (X_i, X_k)^T$ and $\boldsymbol X_{{\rm rest}}$ be defined as above. Given $\boldsymbol X_{{\rm rest}}$, we say that ${X_i}$ and $X_k$ are \emph{partially tail-uncorrelated} if the PTCC of $X_i$ and $X_k$ (given $\boldsymbol X_{{\rm rest}}$) is equal to {zero}, i.e., if $\Sigma_{ki}-\Sigma_{k,{\rm rest}}\Sigma_{{\rm rest},{\rm rest}}^{-1}\Sigma_{{\rm rest},i}=0$ as defined in \eqref{eq:prediction_error}. \end{defn} \noindent \textbf{Remark 1}: {Thanks to the properties of TPDMs, the residuals of two partially tail-uncorrelated random variables are necessarily asymptotically tail independent.}
The following proposition links tail-uncorrelatedness to the entries of the inverse of the TPDM of $\bm X$.
\begin{prop}\label{prop:partial} Given the representation of the TPDM of $\boldsymbol X'$ as a $3\times3$ block matrix as follows,
\begin{equation}\label{eq:sigma}
\Sigma_{\boldsymbol X'}=\text{TPDM}\{(X_i, X_k,\bm X_{{\rm rest}}^T)^T\}=
\begin{bmatrix}
\Sigma_{ii} & \Sigma_{ik} & \Sigma_{i,{\rm rest}}\\
\Sigma_{ki} & \Sigma_{kk} & \Sigma_{k,{\rm rest}}\\
\Sigma_{{\rm rest},i} & \Sigma_{{\rm rest},k} & \Sigma_{{\rm rest},{\rm rest}}
\end{bmatrix},
\end{equation} where the dimensions of submatrices are as above, then the following two statements are equivalent: $$
(1) \text{ }\Sigma_{ki} - \Sigma_{k,{\rm rest}}\Sigma_{{\rm rest},{\rm rest}}^{-1}\Sigma_{{\rm rest},i} =0,\quad\quad
(2) \text{ }(\Sigma^{-1})_{ik}=0 $$ where $\Sigma$ is the TPDM of the original vector $\boldsymbol X$. \end{prop} Since $\Sigma_{\boldsymbol X'}$ is a positive definite and therefore an invertible covariance matrix, this result is a direct consequence of the equivalence of statements $(i')$ and $(ii)$ of Proposition~$1$ of \citet{speed1986gaussian}. In the notation of \citet{speed1986gaussian}, we have the following index sets: $a=\{k,{\rm rest}\}$, $b=\{i,{\rm rest}\}$, and $ab = a\cap b = {\rm rest}$.
The following corollary is a direct consequence of this result.
\begin{cor}\label{prop:q} Denote the inverse matrix of the TPDM of a random vector $\boldsymbol X$ by $Q = \Sigma^{-1}$. Then, $$ Q_{ik}=0 \quad \text{if and only if}\quad \text{PTCC}_{ik}=0, $$ where PTCC$_{ik}$ is the PTCC of components $X_i$ and $X_k$. {Recall that a PTCC equal to zero corresponds to partial tail-uncorrelatedness.}
\end{cor}
\section{Learning extremal networks for high-dimensional extremes}\label{sec:extnet}
Using the PTCC, we can now explore the partial tail correlation structure of multivariate random variables under the framework of multivariate regular variation. In this section, we define new graphical models to represent extremal dependence for extremes. Thanks to the transformed-linear framework exposed in the previous section, we can proceed as for classical graphical models by replacing the classical covariance matrix with the TPDM.
\subsection{Graphical models for extremes}
Let $G = (V, E)$ be a graph, where $V = \{1, \ldots,p\}$ represents the node set and $E \in V\times V$ the edge set. We call $G$ an undirected graph if for two nodes $i,k\in V$, the edge $(i, k)$ is in $E$ if and only if the edge $(k,i)$ is also in $E$. We show how to estimate graphical structures for extremes for any type of undirected graph in which we have no edge $(i, k)$ if and only if the variables $X_i$ and $X_k$ are partially tail-uncorrelated given all the other variables in the graph, which we write as $$X_i \perp_p X_k \mid \boldsymbol{X}_{{\rm rest}}.$$
Our methods work for general undirected graph structures, including trees, decomposable graphs and non-decomposable graphs, see the example illustrations in Figure~\ref{fig:graphs}. Note, however, that our general method cannot restrict the estimated graph to be of a specific type, such as a tree.
\begin{figure}
\caption{Examples of undirected graph structures: a tree (left), a decomposable graph (middle), and a non-decomposable graph (right).}
\label{fig:graphs}
\end{figure}
\subsection{Sparse representation of high-dimensional extreme networks}
For high-dimensional extremes with a relatively large number of components $p$ (e.g., up to tens and hundreds of variables), a graphical representation of the extremal dependence structure is desirable for reasons of parsimony and interpretability. We now introduce two efficient inference methods to learn extremal networks from high-dimensional data via the PTCC based on two state-of-the-art graphical methods: the extremal graphical Lasso, and the structured graph learning method via Laplacian spectral constraints. These two methods both work efficiently in high-dimensional settings and return an estimate of the underlying extremal dependence with sparse structure, i.e., with the cardinality of $E$ being of the same order as the one of $V$.
\subsubsection{Extremal graphical Lasso}\label{subsec:lasso}
Given an empirical estimator $\hat{\Sigma}$ of the TPDM and a tuning parameter $\lambda\geq 0$, the optimization carried out by the extremal graphical Lasso method is expressed as follows \citep{friedman2008sparse}: $$
\hat{Q}_{\lambda}= \arg \max_{\Theta\succeq 0} \big[ \log\det\Theta - \text{tr}(\hat{\Sigma}\Theta) - \lambda \sum_{i\neq k}|\Theta_{ik}|\big] $$ where $\succeq$ indicates positive-semidefiniteness and $\hat{Q}_\lambda$ is a $L_1$-regularized estimate of the precision matrix $Q=\Sigma^{-1}$. Note that thanks to the $L_1$ regularization, the estimate $\hat{Q}_{\lambda}$ will tend to be sparse (with exact zero entries) and thus contains information on the extremal graph structure. A larger $\lambda$ enforces a larger proportion of zeros in $\hat{Q}_{\lambda}$ and hence fewer edges in the graph. Choosing an appropriate value for $\lambda$ is thus critical. On the one hand, we want to enforce sparsity in the graph, where only significant connections are maintained in the network. On the other hand, $\hat{Q}_{\lambda}$ should be well-defined, with estimation being stable, and with meaningful dependence structures in the estimated model. In our river discharge application in Section~\ref{sec:appldanube} we use a voting procedure to select the best value for $\lambda$, while in our global currency application in Section~\ref{sec:applcurrency} we set the sparsity level to a pre-defined level for interpretation purposes.
\subsubsection{Structured graph learning via Laplacian spectral constraints}\label{SGL}
As an alternative to the graphical Lasso approach, we can seek to include more structural information into the graph by using the structured graph Laplacian (SGL) method of \citet{kumar2019structured}, which assumes that the signal residing on the graph changes ``smoothly'' between connected nodes. This method allows us to better balance the sparsity and connectedness of the estimated precision matrix, thanks to additional constraints on the eigenvalues of the graph Laplacian operator that encodes the graph structure. For instance, if exactly one eigenvalue is zero and all other eigenvalues are positive, then the graph is connected. Laplacian matrix estimation can be formulated as the estimation problem for a precision matrix $Q$, which is therefore linked to our framework that uses the TPDM and its inverse. For any vector of eigenvalues $\boldsymbol\lambda\in S_{\boldsymbol\lambda}$ with appropriate a priori constraints for the desired graph structure defined through the set of admissible eigenvalues $S_{\boldsymbol\lambda}$, we set {$Q = \mathcal{L} \boldsymbol w$} with $\mathcal{L}$ the linear operator that maps a non-negative set of edge weights $\boldsymbol w\in\mathbb R_+^{p(p-1)/2}$ to the matrix $Q$ with Laplacian constraints. The Laplacian matrix $Q$ can be factorized as {$Q = U{\rm Diag}(\boldsymbol \lambda)U^T$} (with an orthogonal matrix $U$) to enforce the constraints on $\boldsymbol \lambda$. Then the optimization problem can be formulated as follows: $$
(\hat{\boldsymbol\lambda},\hat{U}) = \arg \max_{\boldsymbol\lambda,U}\max_{\boldsymbol w} \left(
\log \text{gdet}(\text{Diag}(\boldsymbol\lambda)) - \text{tr}(\hat{\Sigma} \mathcal{L} \boldsymbol w) + \textcolor{black}{\alpha}||\mathcal{L}\boldsymbol w||_1 +\dfrac{\textcolor{black}{\beta}}{2}||\mathcal{L}\boldsymbol w - U\text{Diag}(\boldsymbol\lambda)U^T||_{F}^2 \right), $$ $$ \text{subject to } \boldsymbol w\geq 0, \boldsymbol\lambda \in S_{\boldsymbol\lambda}, \text{and } U^TU = I, $$
where $S_{\boldsymbol\lambda}$ denotes the set of constrained eigenvalues, $||\cdot||_F$ is the Frobenius norm, and $\text{gdet}$ is the generalized determinant defined as the product of all positive values in $\boldsymbol \lambda$. The optimization problem can be viewed as penalized likelihood if data have been generated from a Gaussian Markov random field; in more general cases such as ours, it still provides meaningful graphical structures since it can be viewed as a so-called penalized log-determinant Bregman divergence problem. Therefore, this method can be seen as an extension of the graphical Lasso that allows us to set useful spectral constraints with respect to the structure of the graph. A larger value of $\alpha$ increases the sparsity level of the graph. The hyperparameter $\beta \geq 0$ additionally controls the level of connectedness, and a larger value of $\beta$ enforces a higher level of connectedness of the estimated graph structure.
\section{Simulation study}\label{sec:simulation}
We present a simulation study with three examples where the corresponding true structures of the extremal dependence graphs are as in Figure~\ref{fig:graphs}, i.e., a tree (Case 1), a decomposable graph (Case 2), and a non-decomposable graph (Case 3). The simulated models are constructed as follows. We simulate a dataset of $n = 10^5$ i.i.d.\ random variables $R_1, R_2, R_3, R_4\sim $ Fr\'echet$(2)$. We then construct $n$ replicates of the random vector $\boldsymbol X = (X_1, X_2, X_3, X_4)^T$ according to the following three cases, for which we also specify the true TPDM $\Sigma$ and its inverse $Q=\Sigma^{-1}$:\\ {\bf Case 1:} \[ \begin{cases} \notag X_1 &= R_1\\ \notag X_2 &= R_1 \oplus R_2\\ \notag X_3 &= R_1 \oplus R_3\\ \notag X_4 &= R_1 \oplus R_4 \end{cases}, \quad \Sigma= \begin{bmatrix} 1 &1 &1 &1\\ 1 &2 &1 &1\\ 1 &1 &2 &1\\ 1 &1 &1 &2 \end{bmatrix}, \quad Q = \begin{bmatrix} 4 &-1 &-1 &-1\\ -1 &1 &0 &0\\ -1 &0 &1 &0\\ -1 &0 &0 &1 \end{bmatrix}. \] {\bf Case 2:} \[ \begin{cases} \notag X_1 = R_1\\ \notag X_2 =R_1 \oplus R_2\\ \notag X_3 = R_1 \oplus R_3\\ \notag X_4 = R_1 \oplus 2R_3 \oplus R_4 \end{cases}, \quad \Sigma= \begin{bmatrix} 1 &1 &1 &1\\ 1 &2 &1 &1\\ 1 &1 &2 &3\\ 1 &1 &3 &6 \end{bmatrix}, \quad Q = \begin{bmatrix} -4 &-1 &1-3 &1\\ -1 &1 &0 &0\\ -3 &0 &5 &-2\\ 1 &0 &-2 &1 \end{bmatrix}. \] {\bf Case 3:} \[ \begin{cases} \notag X_1 = R_1 \\ \notag X_2 = R_1\oplus 3/{\sqrt{6}}R_2\\ \notag X_3 = R_1\oplus 1/\sqrt{6}R_2 \oplus 2/\sqrt{3}R_3\\ \notag X_4 = R_1 \oplus \sqrt{6}/3R_2 \oplus 1/\sqrt{3}R_3\oplus R_4 \end{cases}, \quad \Sigma= \begin{bmatrix} 1 &1 &1 &1\\ 1 &2.5 &1.5 &2\\ 1 &1.5 &2.5 &2\\ 1 &2 &2 &3 \end{bmatrix}, \quad Q = \begin{bmatrix} 2 &-0.5 &-0.5 &0\\ -0.5 &1 &0 &-0.5\\ -0.5 &0 &1 &-0.5\\ 0 &-0.5 &-0.5 &1 \end{bmatrix}. \]
We proceed as follows to infer the extremal graph structure in each case. First, we estimate the TPDM of $\boldsymbol X$, $\Sigma$, based on the estimator $\hat\Sigma$ specified through \eqref{eq:estimator} using the $99\%$ quantile for the threshold $r_0$ (i.e., there are $1000$ threshold exceedances to estimate the TPDM). Then, we apply the extremal graphical Lasso and SGL methods. In each setting, we test $m_1 = 300$ different values for the regularization parameter $\lambda$ when using extremal graphical Lasso and $m_2 = 400$ different settings for the combination of $\alpha$ and $\beta$ for the SGL method. The range of $\lambda$ and $\{\alpha,\beta\}$ values is chosen to span a wide range of graphical structures, from fully connected to fully sparse (no connection). In all experimental results, we have found that when the true number of edges is achieved, both methods can retrieve 100\% of the true extremal graph structure, i.e., all connections are correctly identified and the estimated graph has no wrong connections.
\begin{figure}
\caption{Estimated extremal graph structures using the extremal graphical Lasso method based on the PTCC for Case 3, as a function of the tuning parameter $\lambda$ (shown at the top of each display).}
\label{fig:simu3}
\end{figure}
This is illustrated in Figure~\ref{fig:simu3}, which displays the estimated extremal graph structure for Case 3 (general non-decomposable ``square graph'') when using the extremal graphical Lasso method. The heading of each display shows the range of $\lambda$ values that leads to the estimated graph shown below. The tuning parameter $\lambda$ controls the number of edges in the graph, i.e., the sparsity level: when $\lambda$ decreases, the number of edges increases, and vice versa. Interestingly, the proposed method always retrieves true connections (i.e., it never yields wrong connections) whenever the estimated graph is as sparse, or sparser than the true graph. This simple experiment shows that our method is able to retrieve the true extremal dependence graph structure, provided the tuning parameter $\lambda$ (or $\{\alpha,\beta\}$ for the SGL method) is well specified. While our numerical experiments were performed in dimension $p=4$, we expect similar results to hold in higher dimensions provided enough data replicates are available. Our higher-dimensional data applications in Section~\ref{sec:applications} demonstrate that the estimated graph structures indeed make sense and yield interpretable results.
We also note that with our distribution-free approach, there is no universally optimal way of setting the tuning parameters. However, we can use problem-specific criteria to achieve the desired outcome (and therefore to set the penalty parameters); see the data applications in Section~\ref{sec:appldanube} and \ref{sec:applcurrency}.
\section{Applications}\label{sec:applications}
Risk networks are useful in quantitative risk management to elucidate complex extremal dependence structures in collections of random variables. We show two examples of both environmental and financial risk analysis. First, we study river discharge data of the upper Danube basin \citep{asadi2015extremes}, which has become a benchmark dataset for learning extremal networks in the recent literature. The true underlying physical river flow network is available, which can be used as a benchmark to compare the performance of our method with other existing approaches. Second, we apply our method to historical global currency exchange rate data from different historical periods, including different economic cycles, the COVID-19 period, and the period of the 2022 military conflict opposing Russia and Ukraine (2022.02.24--2022.09.26).
\subsection{Extremal network estimation for a river network}\label{sec:appldanube} We apply our method to study the dependence structure of extreme discharges on the river network of the upper Danube basin (see the left panel of Figure~\ref{fig:danube} for the topographic map). This region has been regularly affected by severe flooding events in its history, which have caused losses of human lives and damage to material goods. The original daily discharge data from 1960 to 2009 were provided by the Bavarian Environmental Agency (http://www.gkd.bayern.de), and \citet{asadi2015extremes} preprocessed the data, which now include $n=428$ approximately independent events $\boldsymbol X_1,\ldots, \boldsymbol X_n \in \mathbb{R}^d$ recorded at $d=31$ gauging stations located on the river network from three summer months (June, July, and August), obtained using declustering methods. The data were later also studied by \citet{engelke2020graphical} among others, using graphical models for extremes based on a conditional independence notion adapted to multivariate Pareto distributions. The true physical river flow connections and directions are represented by a directed graph shown in the right panel of Figure~\ref{fig:danube}, where the arrows indicate the flow directions. This can serve as an accurate benchmark of the ``true'' conditional independence structure, against which we can compare the results from our proposed extremal graphical structure learning methods based on the PTCC.
\begin{figure}
\caption{Left: Topographic map of the upper Danube basin \citep[from][]{asadi2015extremes}, showing 31 sites of gauging stations (red circles) and the altitudes of the region. Right: The true physical river flow connections; the arrows show the flow directions.}
\label{fig:danube}
\end{figure}
\subsubsection{Graph structure learning using the extremal graphical Lasso} To learn the extremal dependence structure of the river network, we first perform a nonparametric empirical transformation of the data to satisfy Fr\'echet($\alpha = 2$) margins, for each station separately. Next, we estimate the TPDM, $\Sigma$, using the proposed estimator, $\hat\Sigma$, defined through \eqref{eq:estimator}. In particular, we choose $m=d = 31$ because the margins are preprocessed to have a common unit Fr\'echet scale and $r_0 = 11.4$, which corresponds to the empirical 90\% quantile. Therefore, $n_{\text{exc}} = 43$ is the number of extreme observations (i.e., threshold exceedances) which are used to estimate $\Sigma$. The left panel of Figure~\ref{fig:tpdm} displays the estimated TPDM of the river discharge data from the upper Danube basin, while the right panel displays the votes (in percentage) of the edges selected based on the extremal graphical Lasso method, obtained from multiple fits with a range of $\lambda$ values producing different dependence structures, from fully connected to fully sparse graphs.
\begin{figure}
\caption{Left: Estimated TPDM of the river discharge data from the upper Danube basin. Right: Votes ($\%$) of the edges selected based on the extremal graphical Lasso method. Darker red cells indicate that the corresponding edge has been selected more often by the graphical Lasso.}
\label{fig:tpdm}
\end{figure}
\subsubsection{Graph structure learning using the SGL method} To enhance connectedness, we further explore the SGL method, which learns sparse graph structures under additional spectral constraints. In particular, as described in Section~\ref{SGL}, we can control both the sparsity level and the graph connectedness by modulating the two tuning parameters $\alpha$ and $\beta$, respectively. As shown in the left panel of Figure~\ref{fig:SGL}, the overall graph sparsity varies for different combinations of $\alpha$ and $\beta$. The right panel of Figure~\ref{fig:SGL} displays the votes (in percentage) of the edges selected based on the SGL method, obtained by fitting a large number of models for each of the $\{\alpha,\beta\}$ combinations shown in the left panel.
\begin{figure}
\caption{Left: Number of edges selected by the SGL method as a function of the tuning parameters $\alpha$ and $\beta$. Lighter blue cells correspond to parameter combinations producing sparser graphs. Right: Votes ($\%$) of the edges selected based on the SGL method. Darker red cells indicate that the corresponding edge has been selected more often by the SGL method.}
\label{fig:SGL}
\end{figure}
\subsubsection{Results: estimated extremal river discharge networks}
For both the extremal graphical Lasso and the SGL method, it is important to carefully select the tuning parameters, $\lambda$ and $\{\alpha,\beta\}$, respectively. These tuning parameters impact both the sparsity level and the connectedness of the resulting graph structure. Ideally, we would like to obtain a sparse graph while keeping it connected. However, there is a tradeoff between these two requirements. Our approach is to control the overall sparsity level while imposing a soft connectedness condition: we start from the fully sparse graph (no edges) and sequentially add edges between nodes according to the ranking of the votes shown in Figures~\ref{fig:tpdm} and \ref{fig:SGL}, until no node is left alone (i.e., each node has at least one connection with another node). The estimated graph structure thus prioritizes edges that are most often selected and is obtained by blending the results from several model fits, which also makes it less sensitive to specific values of the tuning parameters.
Figure~\ref{fig:river_result} displays the estimated extremal river discharge network using our approach for both the extremal graphical Lasso and the SGL method, respectively. The edge thickness is proportional to the votes shown in Figures~\ref{fig:tpdm} and \ref{fig:SGL}. \begin{figure}
\caption{Extremal river discharge network estimated using the extremal graphical Lasso (left) and the SGL method (right). The edge thickness is proportional to the votes shown in Figures~\ref{fig:tpdm} and \ref{fig:SGL}.}
\label{fig:river_result}
\end{figure} Recall that when two nodes are not connected, it means that they are partially tail-uncorrelated in our framework. The estimation based on the extremal graphical Lasso method has a few more edges than the true physical river flow network. The extra links could be interpreted as being due to extremal dependence induced by regional weather events, though this is not very clear. By contrast, the estimation result based on the SGL method matches most of the true river flow connections, while the votes (shown in terms of the edge thickness) represent the strength of the extremal dependence connections. Interestingly, this dependence strength seems well-aligned with the actual physical strength of the river flow. Compared with existing methods \citep{engelke2020graphical, kluppelberg2021estimating}, our estimated extremal network based on the SGL method looks more realistic (thus more easily interpretable), as it is closer to the true flow structure of the river network, though the recent results from \citet{Engelke.etal:2022} are quite similar to ours (with a few extra connections).
\subsection{Extremal network estimation for global currency exchange rate network}\label{sec:applcurrency}
We now apply our method to explore historical global currency exchange rate data for 20 currencies from different historical periods, including two different global economic cycles (2009--2014 and 2015--2019, where the segmentation is determined from the world GDP cycles illustrated in Figure~\ref{fig:gdp} from Appendix~\ref{appd:currency}), COVID-19 (2020.01.01--2022.02.23), and the period from the beginning of the 2022 military conflict between Russia and Ukraine until a most recent date when we downloaded the data (2022.02.24--2022.09.26). Historical data were downloaded from \href{https://finance.yahoo.com/?guccounter=1&guce_referrer=aHR0cHM6Ly93d3cuZ29vZ2xlLmNvbS8&guce_referrer_sig=AQAAAArrJ51ii0LwZtFYwOr7XyQJJJefzGD3DqIN_OCi-CjH2j981I5L7tgeoZv4eTXCY0OSAxXw09alMxpKV7ygEdoDki02tMTic03tGlae7MXXVHzLszZQgvx5L9EFfqzlYo-S6UGPY7IHtZbO6fi1ZFbs_5hcGAkjzmw24f-EE8KR}{Yahoo Finance}. We chose the currencies from all G20 countries, and also added the Ukrainian and Kazakhstani currencies. The list of selected currencies and their corresponding symbols can be found in Table~\ref{table:currency} in Appendix~\ref{appd:currency}. Since the unit of the currencies is the US Dollar (USD), USD is not considered in our list of currencies under study.
First, we preprocess the historical daily closing prices of the currencies. An ARMA(1,1)-GARCH(1,1) time series model is fitted to the negative log return time series of each currency, and then standardized residuals are extracted and transformed marginally to Frech\'et margins with shape parameter $\alpha=2$. The extremal dependence graph structure governing negative log returns therefore represents partial associations among extreme losses, shedding light on the integration and/or vulnerability of major economies in periods of stress. To estimate this risk network, we follow the same procedure as before, estimating first the TPDM using the estimator in \eqref{eq:estimator} with $r_0$ as the empirical $90\%$ quantile, and then applying the SGL method.
In this analysis, we use only the SGL method, because it includes the extremal graphical Lasso method as a special case when $\beta = 0$, and it has shown better performance in the Danube river application. Furthermore, for comparability among different historical periods, we here fix the sparsity level to 80\% (i.e., with only about 38 edges), rather than using the method based on votes. This approach yields a unique combination of $\{\alpha,\beta\}$ tuning parameters, which then yields the final estimated graph structure. To assess the estimation uncertainty of the graph structure, we have further conducted 300 bootstrap simulations, whereby standardized residuals are resampled with replacement and the TPDM is then re-estimated, as well as the graph structure fixing the same sparsity level (i.e., potentially with different selected $\{\alpha,\beta\}$ values for each bootstrap simulation). The bootstrap results are shown in Figures~\ref{fig:eco} and \ref{fig:events} for the first two, and last two periods, respectively, where different edge types represent the ``significance'' of the displayed connections: the thickest edges (in red) indicate a frequency of at least $90\%$ to be included among the 300 bootstrap fitted models; thick edges (in blue) indicate a frequency between $70\%$ and $90\%$ to be selected; thin edges (in grey) indicate a frequency between $50\%$ and $70\%$ to be selected; absent edges have been selected less than $50\%$ of the time. Moreover, the size of nodes in the displayed graphs is proportional to their degree, i.e., to their number of connections. The bigger a node, the more connected it is, giving an idea of the centrality of a currency in the risk network.
\begin{figure}
\caption{Extremal currency exchange rate risk network estimated for 2009--2014 (left) and 2015--2019 (right). Different edge types (respectively thickest, thick, and thin) with different colors (respectively red, blue, and grey) indicate a frequency of $>90\%$, $70\%$--$90\%$, and $50\%$--$70\%$, respectively, to be selected among the $300$ bootstrap fitted models, while absent edges indicate that this frequency is less than $50\%$.}
\label{fig:eco}
\end{figure}
We now provide some interpretation of the estimated risk networks, assuming that more strongly connected nodes tend to be more vulnerable/exposed to network risks, or are important currencies that strongly determine the behavior of other currencies in the monetary system. We can back up some of our findings using historical events and economic development regimes for different countries. From the left panel in Figure~\ref{fig:eco}, representing the economic cycle 2009--2014, the strongest-connected currencies within the estimated risk network are GBP (United Kingdom) and ARS (Argentina), whereas KZT (Kazakstan) and UAH (Ukraine) are less connected from the major clusters. By contrast, in the right panel of Figure~\ref{fig:eco}, representing the economic cycle 2015--2019, the strongest-connected currencies from the estimated risk network are KRW (Republic of Korea), AUD (Australia) and EUR (EU), whereas ARS, KZT, and MXN (Mexico) are relatively isolated. ARS and GBP are two examples of currencies that were much weaker connected during the second period. Historical economic data show that the Argentinian economy went through a major crisis around 2000, establishing strong economic links to other countries and international financial markets in the following years. This seems to clearly transpire from the estimated graph for the 2009--2014 period following the global financial crises of 2008, but the monetary interconnections of ARS to other currencies have drastically been reduced in the later 2015--2019 period. As to GBP, the exit of Great Britain from the European Union (Brexit), and the political and economic decisions accompanying it, could be the origin of this change. \begin{figure}
\caption{Extremal currency exchange rate risk network estimated for the COVID-19 period (left) and the period since the 2022 military conflict between Russia and Ukraine (right). Different edge types (respectively thickest, thick, and thin) with different colors (respectively red, blue, and grey) indicate a frequency of $>90\%$, $70\%$--$90\%$, and $50\%$--$70\%$, respectively, to be selected among the $300$ bootstrap fitted models, while absent edges indicate that this frequency is less than $50\%$.}
\label{fig:events}
\end{figure} The left panel of Figure~\ref{fig:events} shows that during the COVID-19 pandemic, the ARS currency is again the strongest-connected one, by far, in the risk network, whereas TRY (Turkey) and CAD (Canada) are isolated and most of the other currencies are also less interconnected. As to ARS, Argentina was struck by COVID-19 during an already fragile economic situation, and its aforementioned strong international monetary ties might have led to strong extremal connectedness to other countries. Finally, the right panel of Figure~\ref{fig:events} shows the risk network for currency exchange rates from the beginning of the 2022 military conflict between Russia and Ukraine until 2022.09.26. Since the length of the time period is shorter, there is higher estimation uncertainty, and no edges have been selected more than $90\%$ of the time among the $300$ bootstrap model fits. Nevertheless, we can still make the interesting observation that RUB (Russia) is the only isolated currency from the rest of the network, which might be due to the antagonism and strong economic sanctions imposed by western countries, as well as the dramatic changes in monetary policies.
\section{Conclusions}\label{sec:conclusion}
We have proposed the partial tail correlation as a novel notion of extremal dependence measure that removes the effect of confounding variables through transformed-linear operations. Unlike other approaches from the recent literature, our new partial tail correlation coefficient (PTCC) assumes multivariate regular variation but does not rely on any further strict parametric assumptions. Furthermore, the PTCC has appealing theoretical properties and it can be used to define a new class of extremal graphical models, where the absence of edges indicates partial tail-uncorrelatedness between variables (i.e., when the PTCC of the corresponding edges equals zero). We have shown that the zero PTCC values between variable pairs can be retrieved by identifying the zero entries in the inverse tail pairwise dependence matrix (TPDM). This convenient property, which is akin to classical Gaussian graphical models, allows us to efficiently learn high-dimensional extremal networks defined in terms of the PTCC by exploiting state-of-the-art methods from graph theory, such as the graphical Lasso and structured graph learning via Laplacian spectral constraints.
Our graph-inference approach is flexible, can be applied to general undirected graphs, and easily scales to high dimensions. We demonstrate the effectiveness of our method as an exploratory tool for interpretable extremal network estimation. In our first application to river discharge data from the upper Danube basin, we show that the proposed method outperforms other existing methods by realistically capturing most physical flow connections, together with the strength of the connections, while largely avoiding spurious connections. In our second application to historical currency exchange rate data, we obtain interesting findings based on the estimated risk network for four recent periods, which can be backed up by real historical evidence. While our interpretations remain fairly basic, it would be interesting to get further insights from economists.
We have identified some theoretical and methodological challenges for future research. First, we have not obtained theoretical guarantees that our graph learning method can consistently estimate the unknown graph structure, and it would be interesting to see if this could be proven under general assumptions. However, our simulations and applications have provided convincing evidence that the method works well to extract useful structural information from extreme observations. Moreover, it is also worth noting that in some cases, there is no interest in recovering the exact true graph structure, but rather a sparser representation containing a fixed percentage with the ``most important'' edges, thus facilitating interpretations. In such cases, consistency is not a criterion that is well-adapted to the problem at hand. Second, we have focused in this paper on graph learning through the graphical Lasso, and the SGL method that imposes further Laplacian constraints. It would be interesting to extend these methods, in order to let the graphical structure depend on well-chosen covariates (e.g., temperature in our river network application, or the time period index in our currency network application), so that the estimated (non-stationary) risk network can then be interpreted through the glasses of these covariates. Moreover, unlike our currency application, where separate model fits were obtained for each time period under consideration, introducing covariates in the graph learning procedure would allow estimating the networks simultaneously from the combined dataset, thus gaining efficiency for higher accuracy of the estimated graph structure. A possibility could be to extend the Gaussian graphical regression approach proposed by \citet{Zhang.Li:2022} to our PTCC-based extremes framework. Third, by analogy with the widely used Gaussian Markov random field framework, one could imagine constructing Markov random fields for extremes where the distribution of connected variables may be fitted with an asymptotically justified model for extremes (e.g., of multivariate Pareto type), whereas the factorization of the likelihood could be pre-specified by the extremal network learned from the data in a preliminary step using our proposed method. Finally, another direction to investigate concerns the geometric representation of multivariate extremes \citep{nolde2020linking, simpson2021geometric}. It would be interesting to see if the new PTCC can be defined through this geometric approach and to explore its links with other popular measures of extremal dependence.
\section*{Acknowledgments} We point out that there has been independent and parallel work by Lee \& Cooley, who also investigate partial tail correlation for extremes. Our understanding is that inference in their work focuses rather on hypothesis testing (with the goal of checking if the partial tail correlations for given pairs of variables are significantly different from zero), and not on learning extremal networks with structural constraints. This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No. OSR-CRG2020-4394.
\section*{Disclosure statement} The authors report there are no competing interests to declare.
\section*{Appendix} \appendix
\section{Data details} \label{appd:currency} \begin{table}[h] \centering \begin{tabular}{ccc} \hline Countries & Currency symbol & Currency name\\ \hline EU & EUR & EURO \\ United Kingdom & GBP & Pound Sterling \\ India & INR & Indian Rupee \\ Australia & AUD & Australian Dollar \\ Canada & CAD & Canadian Dollar \\ South Africa & ZAR & Rand \\ Japan & JPY & Yen \\ Singapore & SGD & Singapore Dollar \\ China & CNY & Yuan\ \\ Switzerland & CHF & Swiss Franc \\ Republic of Korea & KRW & Won \\ Turkey & TRY & Turkish Lira \\ Mexico & MXN & Mexican Peso \\ Brazil & BRL & Real \\ Indonesia & IDR & Rupiah \\ Saudi Arabia & SAR & Saudi Riyal \\ Russia & RUB & Ruble \\ Argentina & ARS & Argentine Peso \\ Ukraine & UAH & Ukrainian Hryvnia \\ Kazakstan & KZT & Kazakhstani Tenge\\ \hline \end{tabular} \caption{Currency symbol list.} \label{table:currency} \end{table}
\begin{figure}
\caption{World GDP (current trillion US\$) from 2000 to 2020. Data source: \href{https://data.worldbank.org/indicator/NY.GDP.MKTP.CD}{The World Bank}.}
\label{fig:gdp}
\end{figure}
\end{document}
|
arXiv
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| null | null |
\begin{document}
\title{The Path Partition Conjecture is True and its Validity Yields
Upper Bounds for Detour Chromatic Number and Star Chromatic Number}
\author{G. Sethuraman\\ Department of Mathematics, Anna University\\ Chennai 600 025, INDIA\\ [email protected]}
\maketitle
\begin{abstract} The detour order of a graph $G$, denoted $\tau(G)$, is the order of a longest path in $G$. A partition $(A, B)$ of $V(G)$ such that $\tau(\langle A \rangle) \leq a$ and $\tau(\langle B \rangle) \leq b$ is called an $(a, b)$-partition of $G$. A graph $G$ is called $\tau$-partitionable if $G$ has an $(a, b)$-partition for every pair $(a, b)$ of positive integers such that $a + b = \tau(G)$. The well-known Path Partition Conjecture states that every graph is $\tau$-partitionable. In \cite{df07} Dunber and Frick have shown that if every 2-connected graph is $\tau$-partitionable then every graph is $\tau$-partitionable. In this paper we show that every 2-connected graph is $\tau$-partitionable. Thus, our result settles the Path Partition Conjecture affirmatively. We prove the following two theorems as the implications of the validity of the Path Partition Conjecture.\\ {\bf Theorem 1:} For every graph $G$, $\chi_s(G) \leq \tau(G)$, where $\chi_s(G)$ is the star chromatic number of a graph $G$.
The $n^{th}$ detour chromatic number of a graph $G$, denoted $\chi_n(G)$, is the minimum number of colours required for colouring the vertices of $G$ such that no path of order greater than $n$ is mono coloured. These chromatic numbers were introduced by Chartrand, Gellar and Hedetniemi\cite{cg68} as a generalization of vertex chromatic number $\chi(G)$.\\ {\bf Theorem 2:} For every graph $G$ and for every $n \geq 1$, $\chi_n(G) \leq \left\lceil \frac{\tau_n(G)}{n} \right\rceil$, where $\chi_n(G)$ denote the $n^{th}$ detour chromatic number.\\ Theorem 2 settles the conjecture of Frick and Bullock \cite{fb01} that $\chi_n(G) \leq \left\lceil \frac{\tau(G)}{n} \right\rceil$, for every graph $G$, for every $n \geq 1$, affirmatively.
\end{abstract}
{\bf Keywords:}Path Partition;Path Partition Conjecture;Star Chromatic Number;Detour Chromatic Number;Upper bound of chromatic number;Upper bound of Star Chromatic Number;Upper bound of Detour Chromatic Number.\\
\section{Introduction}
All graphs considered here are simple, finite and undirected. Terms not defined here can be referred from the book \cite{we02}. A longest path in a graph $G$ is called a detour of $G$. The number of vertices in a detour of $G$ is called the detour order of $G$ and is denoted by $\tau(G)$. A partition $(A, B)$ of $V(G)$ such that $\tau(\langle A \rangle) \leq a$ and $\tau(\langle B \rangle) \leq b$ is called an $(a, b)$-partition of $G$. If $G$ has an $(a, b)$-partition for every pair $(a, b)$ of positive integers such that $a + b = \tau(G)$, then we say that $G$ is $\tau$-partitionable. The following conjecture is popularly known as the Path Partition Conjecture.
\noindent\textbf{Path Partition Conjecture:} {\it Every graph is
$\tau$-partitionable}.
The Path Partition Conjecture was discussed by Lovasz and Mihok in 1981 in Szeged and treated in the theses \cite{ha84} and \cite{vr86}. The Path Partition Conjecture first appeared in the literature in 1983, in a paper by Laborde et al. \cite{lp82}. In 1995 Bondy \cite{bo95} posed the directed version of the Path Partition Conjecture. In 2004, Aldred and Thomassen \cite{at04} disproved two stronger versions of the Path Partition Conjecture, known as the Path Kernel Conjecture \cite{bh97,mi85} and the Maximum $P_n$-free Set Conjecture \cite{df04}. Similar partitions were studied for other graph parameters too. Lovasz proved in \cite{lo66} that every graph is $\Delta$-partitionable, where $\Delta$ denotes the maximum degree (A graph $G$ is $\Delta$-partitionable if, for every pair $(a, b)$ of positive integers satisfying $a + b = \Delta(G) - 1$, there exists a partition $(A, B)$ of $V(G)$ such that $\Delta(\langle A \rangle) \leq a$ and $\Delta(\langle B \rangle) \leq b$). For the results pertaining to the Path Partition Conjecture and related conjectures refer \cite{bd98,bh97,df99,df07,df04,fb01,fr13,ha84,lp82,mi85,se11,vr86,niel}. An $n$-detour colouring of a graph $G$ is a colouring of the vertices of $G$ such that no path of order greater than $n$ is monocoloured. The $n^{th}$ detour chromatic number of graph $G$, denoted by $\chi_n$, is the minimum number of colours required for an $n$-detour colouring of a graph $G$. It is interesting to note that for a graph $G$, when $n=1$, $\chi_1(G)=\chi(G)$. These chromatic numbers were introduced by Chartrand, Gellor and Hedetnimi \cite{cg68} in 1968 as a generalization of vertex chromatic number.
If the Path Partition Conjecture is true, then the following conjecture of Frick and Bullock \cite{fb01} is also true.
\noindent\textbf{Frick-Bullock Conjecture:} $\chi_n(G) \leq \left\lceil \frac{\tau(G)} {n} \right\rceil$ {\it for every graph $G$ and for
every $n \geq 1$.}\\ Recently, Dunbar and Frick \cite{df07} proved the following theorem.
\begin{theorem}[Dunber and Frick \cite{df07}] \label{thm1.1} If every 2-connected graph is \break $\tau$-partitionable then every graph is $\tau$-partitionable. \end{theorem}
In this paper we show that the Path Partition Conjecture is true for every 2-connected graph. Thus, Theorem \ref{thm1.1} and our result imply that the Path Partition Conjecture is true. The validity of the Path Partition Conjecture would imply the following Path Partition Theorem.
\noindent{\bf Path Partition Theorem.} {\it For every graph $G$ and for
every $t$-tuple \break $(a_1, a_2, \dots, a_t)$ of positive
integers with $a_1 + a_2 + \cdots + a_t = \tau(G)$ and $t \geq 1$,
there exists a partition $(V_1, V_2, \dots, V_t)$ of $V(G)$
such that $\tau(G(\langle V_i \rangle) \leq a_i$, for every $i$, $1 \leq i
\leq t$.}\\ The Path Partition Theorem immediately implies that the Conjecture of Frick and Bullock is true. The validity of Frick and Bullock Conjecture naturally implies the classical upper bound for the chromatic number of a graph $G$ that $\chi(G)=\chi_1(G) \leq \tau(G)$ proved by Gallai\cite{gall}.
A star colouring of a graph $G$ is a proper vertex colouring in which every path on four vertices uses at least three distinct colours. The star chromatic number of $G$ denoted by $\chi_s(G)$ is the least number of colours needed to star color $G$. As a consequence of the Path Partition Theorem, we have obtained an upper bound for the star chromatic number. More precisely, we show that $\chi_s(G) \leq \tau(G)$ for every graph $G$.
\section{Main Result}
In this section we prove our main result that every 2-connected graph is $\tau$-partitionable.
We use Whitney's Theorem on the characterization of 2-connected graph in the proof of our main result given in Theorem \ref{thm2.2}.
An ear of a graph $G$ is a maximal path whose internal vertices have degree 2 in $G$. An ear decomposition of $G$ is a decomposition $P_0, P_1, \dots, P_k$ such that $P_0$ is a cycle and $P_i$ for $i \geq 1$ is an ear of $P_0 \cup P_1 \cup \dots \cup P_i$.
\begin{theorem}[Whitney \cite{wh}] A graph is 2-connected if and only if it has an ear decomposition. Furthermore, every cycle in a 2-connected graph is the initial cycle in some ear decomposition. \end{theorem}
\begin{theorem}\label{thm2.2} Every 2-connected graph is $\tau$-partitionable. \end{theorem}
\begin{proof} Let $G$ be a 2-connected graph. By Whitney's Theorem there exists an ear decomposition $S = \{P_0, P_1, \dots, P_n\}$, where $P_0$ is a cycle and $P_i$ for $i \geq 1$ is an ear of $P_0 \cup P_1 \cup \dots \cup P_i$. We prove that $G$ is $\tau$-partitionable by induction on
$|S|$.
When $|S| = 1$, $S = \{P_0\}$. Then $G = P_0$. Thus, $G$ is a cycle. As every cycle is $\tau$-partitionable, $G$ is
$\tau$-partitionable. By induction, we assume that if $G$ is any 2-connected graph having an ear decomposition $S = \{P_0, P_1, \dots, P_{k-1}\}$, that is, with $|S| = k$, then $G$ is $\tau$-partitionable.
Let $H$ be a 2-connected graph with an ear decomposition \break $S =
\{P_0, P_1, \dots, P_{k-1}, P_k\}$. That is, $|S| = k + 1$. We claim that $H$ is\\ $\tau$-partitionable. Let $(a,b)$ be a pair of positive integers with $a+b=\tau(H)$. Since $H$ is having the ear decomposition $S=\{P_0,P_1,\dots,P_{k-1},P_k\}$, $H$ can be considered as a 2-connected graph obtained from the 2-connected graph $G$ having the ear decomposition $S'=\{P_0,P_1,\dots,P_{k-1}\}$ by adding a new path (ear) $P_k:xv_1v_2\dots v_ry$ to $G$, where $x,y \in V(G)$ and $v_1,v_2,\dots,v_r$ are new vertices to $G$. As $G$ is a 2-connected graph having the ear decomposition $S'=\{P_0,P_1,\dots,P_{k-1}\}$ with $|S|=k$, by induction $G$ is $\tau$-partitionable. Let $(a_1,b_1)$ be a pair of positive integers such that $a_1 \leq a$, $b_1 \leq b$ with $\tau(G)=a_1+b_1$. Since $G$ is $\tau$-partitionable, there exists an $(a_1,b_1)$ partition $(A',B')$ of $V(G)$ such that $\tau(G(\langle A' \rangle)) \leq a_1$ and $\tau(G(\langle B' \rangle)) \leq b_1$. In order to prove our claim that $H$ is $\tau$-partitionable, we define an $(a,b)$-partition $(A,B)$ of $V(H)$ from the $(a_1,b_1)$ partition $(A',B')$ of $V(G)$ as well as using the path $P_k: xv_1v_2\dots v_ry$. The construction of an $(a,b)$-partition $(A,B)$ of $V(H)$ is given under three cases, depending on $r=0$, $r=1$ and $r \geq 2$, where $r$ is the number of new vertices in the path $P_k$.
\noindent\textbf{Case 1.} $r = 0$
Then $P_k : xy$, where $x$ and $y$ are the vertices of $G$.\\ Thus, $H = G + xy$. This implies, $V(H) = V(G)$.
\noindent\textbf{Case 1.1.} Suppose $x$ and $y$ are in different parts of the partition $(A^\prime, B^\prime)$ of
\hspace{1.2cm} $V(G)$.
Then, as $x$ and $y$ are in different parts of the partition $(A^\prime, B^\prime)$ of $V(G)$, the introduction of the new edge $xy$ between the vertices $x$ and $y$ does not increase the length of any path either in $G(\langle A^\prime \rangle)$ or in $G(\langle B^\prime \rangle)$. Further, as $V(H) = V(G)$, we have $\tau(H(\langle A^\prime \rangle)) = \tau(G(\langle A^\prime \rangle)) \leq a_1 \leq a \ \text{ and } \ \tau(H(\langle B^\prime \rangle)) = \tau(G(\langle B^\prime \rangle)) \leq b_1 \leq b.$ Thus, $(A^\prime, B^\prime)$ is a required $(a, b)$-partition of $V(H)$. \noindent\textbf{Case 1.2.} Suppose $x$ and $y$ are in the same part of the partition $(A^\prime, B^\prime)$ of
\hspace{1.2cm} $V(G)$.
Without loss of generality, we assume that $x$ and $y$ are in $A^\prime$. \\ Suppose $\tau(H(\langle A^\prime \rangle)) \leq a$. Then, as $\tau(H(\langle B^\prime \rangle)) \leq b_1 \leq b$, the $(A^\prime, B^\prime)$ is a required $(a, b)$-partition of $V(H)$.\\ Suppose $\tau(H(\langle A^\prime \rangle)) > a$, then observe that the addition of the edge $xy$ to $G$ has increased the order of some of the longest paths (at least one longest path) in $H(\langle A^\prime \rangle)$ from $a_1$ to $t = a_1 + k > a$, where $k \geq 1$. On the other hand, any path of order $t > a$ in $H(\langle A^\prime \rangle)$ must contain the edge $xy$ also.
Let $P : u_1u_2u_3 \dots u_iu_{i+1} \dots u_{a_1} u_{a_{1+1}} \dots u_t$ be any path of order $t > a$. Then, note that the edge $xy = u_j u_{j+1}$ for some $j$, $1 \leq j \leq t-1$ and $t \leq 2a_1$.
\begin{observation} If we remove the vertex $u_{a+1}$ from the path $P$, then we obtain two subpaths $u_1 u_2 \dots u_i u_{i+1} \dots u_{a-1} u_a$, say $P^\prime$ and $u_{a+2} u_{a+3} \dots u_{t-1} u_t$, say $P^{\prime\prime}$ of $P$. The number of vertices in $P^\prime$ is exactly $a$ and the number of vertices in $P^{\prime\prime}$ is $t-(a+1) \leq t-(a_1+1) \leq 2a_1-a_1-1 = a_1-1 < a_1 \leq a$. \end{observation}
\begin{observation} Consider the subpath $Q : u_1 u_2 \dots u_{a-1} u_a u_{a+1}$ of $P$. Then observe that the end vertex $u_{a+1}$ of $Q$ cannot be adjacent to any of the end vertices of any path of order $b$ in the induced subgraph $H(\langle B^\prime \rangle) = G(\langle B^\prime \rangle)$ in $H$. \end{observation}
For, suppose $u_{a+1}$ is adjacent to an end vertex of a path, say $Z$ of order $b$ in $H(\langle B^\prime \rangle) = G(\langle B^\prime \rangle)$. Let $Z = v_1 v_2 \dots v_b$. Without loss of generality, let $u_{a+1}$ be adjacent to $v_1$. Then, there exists a path $Q \cup Z : u_1 u_2 \dots u_{a-1} u_a u_{a+1} v_1 v_2 \dots v_b$ of order $a+b+1 > a+b = \tau(H)$, a contradiction (Similar contradiction hold good if $u_{a+1}$ is adjacent $v_b$).
Let $\{R_0, R_1, \dots, R_t\}$ be the set of all paths in $H(\langle A^\prime \rangle)$ of order at least $a+1$. For $1 \leq i \leq t$, let $u_{a+1}^i$ denote the terminus vertex of the subpath of $R_i$ of order $a+1$ and having its origin as the origin of $R_i$. Let $\{u_{a+1}^{\alpha_1}, u_{a+1}^{\alpha_2}, \dots, u_{a+1}^{\alpha_h}\}$ be the set of distinct vertices from the vertices $u_{a+1}^1, u_{a+1}^2, \dots, u_{a+1}^t$, where $h \leq t$. Suppose $\{u_{a+1}^{\alpha_1}, u_{a+1}^{\alpha_2}, \dots, u_{a+1}^{\alpha_h}\}$ induces any path in $H(\langle A^\prime \rangle)$. Consider any such path $X : u_{a+1}^{\beta_1} u_{a+1}^{\beta_2} \dots u_{a+1}^{\beta_c}$, where $\{\beta_1, \beta_2, \dots, \beta_c\} \subseteq \{\alpha_1, \alpha_2, \dots, \alpha_h\}$. Then, for $1 \leq i \leq c$, any vertex $u_{a+1}^{\beta_i}$ divides the path $X$ into three subpaths $u_{a+1}^{\beta_1} u_{a+1}^{\beta_2} \dots u_{a+1}^{\beta_{i-1}}$, $u_{a+1}^{\beta_i}$, and $u_{a+1}^{\beta_{i+1}} u_{a+1}^{\beta_{i+2}} \dots u_{a+1}^{\beta_c}$.
\begin{figure}
\caption{Structures of various paths of order $t \geq a+1$ in $A'$}
\label{fig1}
\end{figure}
\noindent\textbf{Claim 1.} For every $i$, $1 \leq i \leq h$, the vertex $u_{a+1}^{\beta_i}$ cannot be adjacent to any of the end vertices of any path of order greater than or equal to $b-q$ in $H(\langle B^\prime \rangle)$, where $q = i-1$ or $c-i$.\\ First we ascertain $b<q+1$ in Observation 2.3 then we prove the Claim 1.
\begin{observation} $b \geq q+1$ \end{observation} For, suppose $b<q+1$. If $q = c-i$, then consider the path, $$K = u_{a+1}^{\beta_i} u_{a+1}^{\beta_{i+1}} \dots u_{a+1}^{\beta_c} u_a^{\beta_c} u_{a-1}^{\beta_c} \dots u_2^{\beta_c} u_1^{\beta_c}$$ in $H(\langle A^\prime \rangle)$ having $1+c-i+a=1+q+a $ vertices. As $q+1 > b$, the path $K$ has at least $a+b+1$ vertices. This implies there exists a path of order at least $a+b+1$ in $H(\langle A^\prime \rangle)$. A contradiction to the fact that $\tau(H) = a+b$. Similarly, if $q = i-1$, then consider the path, $$K^\prime = u_{a+1}^{\beta_i} u_{a+1}^{\beta_{i-1}} \dots u_{a+1}^{\beta_1} u_a^{\beta_1} u_{a-1}^{\beta_1} \dots u_2^{\beta_1} u_1^{\beta_1}$$ in $H(\langle A^\prime \rangle)$ having $i+a = 1+q+a$ vertices. As $q+1 > b$, the path $K^\prime$ has at least $a+b+1$ vertices. This implies that there exists a path of order at least $a+b+1$ in in $H(\langle A^\prime \rangle)$. A contradiction to the fact that $\tau(H) = a+b$. Hence, $b \geq q+1$.\\
To prove Claim 1, we suppose $u_{a+1}^{\beta_i}$, for some $i$, $1 \leq i \leq h$ is adjacent to an end vertex of a path of order $l \geq b-q$ in $H(\langle B^\prime \rangle)$. Let $Y = w_1 w_2 w_3 \dots w_{l}$ be a path of order $l \geq b-q$ in $H(\langle B^\prime \rangle)$ such that (without loss of generality) $w_{l}$ is adjacent to the vertex $u_{a+1}^{\beta_i}$.
\noindent\textbf{Case 1.2a.} $q = c-i$\\ Then consider the path $S = w_1 w_2 \dots w_{l} u_{a+1}^{\beta_i} u_{a+1}^{\beta_{i+1}} \dots u_{a+1}^{\beta_c} u_a^{\beta_c} u_{a-1}^{\beta_c} \dots u_2^{\beta_c} u_1^{\beta_c}$, where $u_1^{\beta_c} u_2^{\beta_c} \dots u_{a}^{\beta_c} u_{a+1}^{\beta_c}$ is a subpath of $R_{\beta_c}$ of order $a+1$ having the vertex $u_1^{\beta_c}$, the origin of $R_{\beta_c}$ as its origin. As $Y : w_1 w_2 w_3 \dots w_{l}$ is the path in $H(\langle B^\prime \rangle)$ such that $w_{l}$ is adjacent to $u_{a+1}^{\beta_i}$, it follows that $S$ is a path in $H$ having the order $l+1+c-i+a \geq b-q+1+q+a=b+a+1 > \tau(H)$, a contradiction.
\noindent\textbf{Case 1.2b.} $q = i-1$
Then consider the path $S^\prime = w_1 w_2 \dots w_{l} u_{a+1}^{\beta_i} u_{a+1}^{\beta_{i-1}} \dots u_{a+1}^{\beta_2} u_{a+1}^{\beta_1} u_a^{\beta_1} u_{a-1}^{\beta_1}$ $\dots$ $u_2^{\beta_1} u_1^{\beta_1}$, where $u_1^{\beta_1} u_2^{\beta_1} \dots u_a^{\beta_1} u_{a+1}^{\beta_1}$ is the subpath of $R_{\beta_1}$ of order $a+1$ having the vertex $u_1^{\beta_1}$, the origin of $R_{\beta_1}$ as its origin. As $Y : w_1 w_2 w_3 \dots w_{l}$ is the path in $H(\langle B^\prime \rangle)$ such that $w_{l}$ is adjacent to $u_{a+1}^{\beta_i}$, it follows that $S^\prime$ is a path in $H$ having the order $l+1+i-1+a \geq b-q+1+q+a=b+a+1 > \tau(H)$, a contradiction.\\ Hence the Claim 1.
Thus, it follows from the Claim 1 that \begin{align}\label{eq1} \tau(H(\langle B^\prime \cup \{u_{a+1}^{\alpha_1}, u_{a+1}^{\alpha_2}, \dots, u_{a+1}^{\alpha_h}\} \rangle)) \leq b \end{align} From Observation 1, it follows that \begin{align}\label{eq2} \tau(H(\langle A^\prime \backslash \{u_{a+1}^{\alpha_1}, u_{a+1}^{\alpha_2}, \dots, u_{a+1}^{\alpha_h}\} \rangle)) \leq a \end{align} Let $A = A^\prime \backslash \{u_{a+1}^{\alpha_1}, u_{a+1}^{\alpha_2}, \dots, u_{a+1}^{\alpha_h}\}$ and $B = B^\prime \cup \{u_{a+1}^{\alpha_1}, u_{a+1}^{\alpha_2}, \dots, u_{a+1}^{\alpha_h}\}$. Then, from (\ref{eq1}) and (\ref{eq2}) it follows that $\tau(H(\langle A \rangle) \leq a$ and $\tau(H(\langle B \rangle) \leq b$.\\ Hence $(A, B)$ is a required $(a, b)$-partition of $H$.
\noindent\textbf{Case 2.} $r = 1$\\ Then $P_k : xv_1y$.
\noindent\textbf{Case 2.1.} Both $x$ and $y$ belong to the same partition $A^\prime$ or $B^\prime$. \\ Without loss of generality, we assume that $x, y \in B^\prime$. That is, $x, y \not\in A^\prime$. Then $(A^\prime \cup \{v_1\}, B^\prime)$ is a required $(a, b)$-partition of $V(H)$.\\ \noindent\textbf{Case 2.2.} The vertices $x$ and $y$ belong to different partitions $A^\prime$ and $B^\prime$.
Without loss of generality, we assume that $x \in A^\prime$ and $y \in B^\prime$. If $x$ is not an end vertex of a path of order $a$ in $H(\langle A^\prime \rangle)$, then $(A^\prime \cup \{v_1\}, B^\prime)$ is a required $(a, b)$-partition of $V(H)$. If $x$ is an end vertex of a path of order $a$ in $H(\langle A^\prime \rangle)$, then $y$ cannot be an end-vertex of a path of order $b$ in $H(\langle B^\prime \rangle)$ (otherwise, $H$ would have a path of order $a+b+1 > \tau(H)$). Therefore $(A^\prime, B^\prime \cup \{v_1\})$ is a required $(a, b)$-partition of $V(H)$.
\noindent\textbf{Case 3.} $r \geq 2$\\ Colour all vertices of $A^\prime$ with red colour and colour all the vertices of $B^\prime$ with blue colour. Since the vertices $x, y \in V(G)$, they are coloured with either blue or red colour. Without loss of generality, we assume that $x \in A^\prime$. Give $v_r$ the alternate colour to that of the vertex $y$. As $x$ is coloured with red colour, colour the vertex $v_1$ with blue colour. In general, for $2 \leq i \leq r-1$, sequentially colour the vertex $v_i$ with the alternate colour to the colour of the vertex $v_{i-1}$. Then observe that $P_k$ contains no induced monochromatic subgraph of order greater than 2 and no monochromatic path in $A^\prime$ or in $B^\prime$ can be extended to include any of the vertices $v_1, v_2, \dots, v_r$ of $P_k$.
Let $X_1$ be the set of all red coloured vertices of $P_k - \{x, y\}$ and let $X_2$ be the set of all blue coloured vertices of $P_k -\{x, y\}$. Then $H(\langle A^\prime \cup X_1 \rangle) \leq a_1 \leq a$ and $H(\langle B^\prime \cup X_2 \rangle) \leq b_1 \leq b$. Hence $(A^\prime \cup X_1, B^\prime \cup X_2)$ is a required $(a, b)$-partition of $H$.\\ Thus, $H$ is $\tau$-partitionable. This completes the induction. Hence every 2-connected graph is $\tau$-partitionable. \end{proof}
The following Corollary \ref{cor2.1} is an immediate consequence of Theorem \ref{thm1.1} and Theorem \ref{thm2.2}.
\begin{corollary}\label{cor2.1} Every graph is $\tau$-partitionable. \end{corollary}
It is clear that Corollary \ref{cor2.1} settles the Path Partition Conjecture affirmatively. Thus, ``{\bf the Path Partition Conjecture is true}''.
The following Theorem \ref{thm2.3} called ``Path Partition Theorem'' is a simple implication of Corollary \ref{cor2.1}.
\begin{theorem}[Path Partition Theorem] \label{thm2.3} For every graph $G$ and for every $t$-tuple $(a_1, a_2, \dots, a_t)$ of positive integers with $a_1 + a_2 + \cdots + a_t = \tau(G)$ and $t \geq 1$, there exists a partition $(V_1, V_2, \dots, V_t)$ of $V(G)$ such that $\tau(G(\langle V_i \rangle)) \leq a_i$, for every $i$, $1 \leq i \leq t$. \end{theorem} \begin{proof} Let $G$ be a graph. Consider any $t$-tuple $(a_1,a_2,\dots,a_t)$ of positive integers with $a_1+a_2+\dots+a_t=\tau(G)$, and $t \geq 1$. Then by Corollary 2.1, for the pair of positive integers $(a,b)$ with $a+b=\tau(G)$, where $a=a_1$ and $b=a_2+\cdots+a_t$, there exists a partition $(U_1,U_2)$ of $V(G)$ such that $\tau(G(\langle U_1\rangle)) \leq a = a_1$ and $\tau(G(\langle U_2 \rangle)) \leq b = a_2+a_3+\cdots+a_t$. Consider the graph $H=G(\langle U_2 \rangle)$. Then for the pair of positive integers $(c,d)$ with $c+d=\tau(H)=\tau(G(\langle U_2 \rangle))$, where $c=a_2$ and $d=a_3+a_4+\dots+a_t$, by Corollary 2.1, there exists a partition $(U_{21},U_{22})$ of $V(H)$ such that $\tau(H(\langle U_{21}\rangle )) \leq c =a_2$ and $\tau(H(\langle U_{22}\rangle )) \leq d=a_3+a_4+\dots+a_t$. As $H(\langle U_{21} \rangle)=G(\langle U_{21} \rangle)$ and $H(\langle U_{22} \rangle)=G(\langle U_{22} \rangle)$, we have $\tau(G(\langle U_{21} \rangle))\leq c = a_2$ and $\tau(G(\langle U_{22} \rangle))\leq d = a_3+a_4+\cdots+a_t$. Similarly, if we consider the pair of positive integers $(x,y)$ with $x+y=\tau(Q)$, where $Q=G(\langle U_{22} \rangle)$, $x=a_3$ and $y=a_4+a_5+\cdots+a_t$, by Corollary 2.1, we get a partition $(U_{31},U_{32})$ such that $\tau(G(\langle U_{31} \rangle)) \leq x =a_3$ and $\tau(G(\langle U_{32} \rangle)) \leq y =a_4+a_5+\cdots+a_t$. Continuing this process, finally we get a partition $(V_1,V_2,\cdots,V_t)$ of $V(G)$ such that $\tau(G(\langle V_i \rangle)) \leq a_i$, for every $i$, $1 \leq i \leq t$, where $V_1=U_1$, $V_2=U_{21}$, $V_3=U_{31}$ and so on. This completes the proof. \end{proof} \begin{corollary}\label{cor2.2} The $n^{th}$ detour chromatic number $\chi_n(G) \leq \left\lceil \frac{\tau(G)} {n} \right\rceil$ for every graph $G$ and for every $n \geq 1$. \end{corollary}
\begin{proof} Let $G$ be any graph. For every $n \geq 1$, consider the $\frac{\tau(G)} {n}$-tuple $(n, n, \dots, n)$ if $\tau(G)$ is a multiple of $n$, while if $\tau(G)$ is not a multiple of $n$, then consider the $\left\lceil \frac{\tau(G)} {n} \right\rceil$-tuple $(n, n, \dots, n, \nu)$, where $\nu = \tau(G)$ (mod $n$). Then, by Path Partition Theorem, there exist a partition $(V_1, V_2, \dots, V_t)$, where \[t = \begin{cases} \frac{\tau(G)} {n} & \text{if } \tau(G) \text{ is
a multiple of } n \\ \left\lceil \frac{\tau(G)} {n} \right\rceil &
\text{if } \tau(G) \text{ is not a multiple of } n \end{cases}\] such that $\tau(G(\langle V_i \rangle)) \leq n$, for every $i$, $1 \leq i \leq t$. For each $i$, $1 \leq i \leq t$, assign the (distinct) colour $i$ to all the vertices in each $G(\langle V_i \rangle)$. Then every monochromatic path in $G$ has the order at most $n$. Thus, $\chi_n(G) \leq \left\lceil \frac{\tau(G)} {n} \right\rceil$. \end{proof}
Corollary \ref{cor2.2} essentially ascertains that ``{\bf Frick-Bullock Conjecture is true}''.\\ {\bf Remark 1:} It is clear from the definition of $\chi_n(G)$, when $n=1$, $\chi_1(G) = \chi(G)$. Thus, by Corollary \ref{cor2.2}, for a graph $G$, $\chi(G) = \chi_1(G) \leq \tau(G)$. This upper bound for the chromatic number of a graph $G$ that $\chi(G) \leq \tau(G)$ is the well known Gallai's Theorem \cite{gall}. \section{An Upper Bound for Star Chromatic Number}
In this section we obtain an upper bound for star chromatic number as a consequence of path partition theorem.
\begin{theorem}\label{star} Let $G$ be a graph. Then the star chromatic number of $G$, $\chi_s(G) \leq \tau(G)$. \end{theorem}
\begin{proof} First we prove the result for connected graphs, then the result follows naturally for the disconnected graphs. Let $G$ be a connected graph.
\noindent\textbf{Claim 1:} There exists a proper $\tau(G)$-vertex colouring for $G$.
\noindent Consider $\tau(G)$. If $\tau(G)$ is even, say $2k$, for some $k \geq 1$, then consider the $k$-tuple $(2, 2, \dots, 2)$ with $2 + 2 + 2 + \cdots + 2 = 2k = \tau(G)$. By Path Partition Theorem, there exists a partition $(V_1, V_2, \dots, V_k)$ such that $\tau(G(\langle V_i \rangle)) \leq 2$, for every $i$, $1 \leq i \leq k$. Therefore, every induced subgraph $G(\langle V_i \rangle)$, for $i$, $1 \leq i \leq k$ is the union of a set of independent vertices and/or a set of independent edges. Thus, it is clear that, for $i$, $1 \leq i \leq k$, each $G(\langle V_i \rangle)$ is proper 2-vertex colourable. Properly colour the vertices of each $G(\langle V_i \rangle)$ with a distinct pair of colours $c_{i_1}$ and $c_{i_2}$, for $i$, $1 \leq i \leq k$. Consequently, this proper 2-vertex colouring of $G(\langle V_i \rangle)$, for all $i$, $1 \leq i \leq k$ induces a proper $\tau(G)$-vertex colouring for the graph $G$. If $\tau(G)$ is odd, say $2k+1$, for some $k \geq 1$, then consider the $k+1$-tuple $(2, 2, \dots, 2, 1)$ with $2 + 2 + 2 + \cdots + 1 = 2k+1 = \tau(G)$. Then by Path Partition Theorem there exists a partition $(V_1, V_2, \dots, V_k, V_{k+1})$ such that $\tau(G(\langle V_i \rangle)) \leq 2$, for every $i$, $1 \leq i \leq k$ and $\tau(G(\langle V_{k+1} \rangle)) \leq 1$. Consequently, the vertices of each $G(\langle V_i \rangle)$ can be properly coloured with a distinct pair of colours $c_{i_1}$ and $c_{i_2}$, for $i$, $1 \leq i \leq k$ and the vertices of $G(\langle V_{k+1} \rangle)$ are colored properly with a distinct color $c_{(k+1)_1}$. Thus, this proper 2-vertex colouring of $G(\langle V_i \rangle)$, for all $i$, $1 \leq i \leq k$ and the proper 1 colouring of $G(\langle V_{k+1} \rangle)$ induce a proper $\tau(G)$-vertex colouring for the graph $G$. Hence the Claim 1.
\noindent\textbf{Claim 2:} $\chi_s(G) \leq \tau(G)$
\noindent To prove Claim 2, we show that the vertices of every path of order four is either coloured with 3 or 4 different colours by the above proper $\tau(G)$-vertex colouring of $G$ or if there exists a bicoloured path of order four in $G$ by the above proper $\tau(G)$-vertex colouring of $G$, then those vertices of such a bicoloured path of order four are properly recoloured so that those vertices are coloured with at least three different colours after the recolouring.
\begin{observation} As \[\tau(G(\langle V_i \rangle)) \leq \begin{cases} 2, & \text{for } 1
\leq i \leq k \\ 1, & \text{for } i = k+1 \text{ and } \tau(G)
\text{ is odd} \end{cases}\] any path of order four in $G$ must contain vertices from at least two of induced subgraphs $G(\langle V_i \rangle)$'s, where $1 \leq i \leq \alpha$, and $\alpha = k$ when $\tau(G)$ is even, while when $\tau(G)$ is odd, $\alpha = k+1$ \mbox{[}Hereafter $\alpha$ is either $k$ or $k+1$ depending on $\tau(G)$ is even or odd respectively\mbox{]}. If any path of order four of $G$ contains vertices from three or four of the induced subgraphs $G(\langle V_i \rangle)$'s then such a path has vertices coloured with three or four colours by the proper $\tau(G)$-vertex colouring of $G$. Thus, we consider only those paths of order four in $G$ having vertices from exactly two of the induced subgraphs $G(\langle V_i \rangle)$'s, where $1 \leq i \leq \alpha$, for recolouring if it is bicoloured. \end{observation}
Consider any path $P$ of order four in $G$ having at least one vertex (at most three vertices) in $G(\langle V_i \rangle)$ for each $i$, $1 \leq i \leq \alpha$ and at least one vertex (at most three vertices) in $G(\langle V_j \rangle)$, for every $j$, $1 \leq i < j \leq \alpha$.
\noindent\textbf{Case 1.} Suppose a path $P$ of order four in $G$ has one vertex in $G(\langle V_i \rangle)$
\hspace{0.8cm} and three vertices in $G(\langle V_j \rangle)$, for $i,j$, $1 \leq i < j \leq \alpha$
\noindent Then without loss of generality we assume that $u_{i_1}$ is one of the vertices of $P$ which is in $G(\langle V_i \rangle)$ and we assume $w_{j_1}, w_{j_2}$ and $w_{j_3}$ are the other three vertices of $P$ which are in $G(\langle V_j \rangle)$. Under this situation, in order that the path $P$ is to be a path of order four with the vertices $u_{i_1}, w_{j_1}, w_{j_2}, w_{j_3}$, two of the vertices from the three vertices $w_{j_1}, w_{j_2}$ and $w_{j_3}$ in $G(\langle V_j \rangle)$ must be adjacent in $G(\langle V_j \rangle)$. Since $V_{k+1}$ is an independent set of vertices, $j \leq k$. As the vertices of each induced subgraph $G(\langle V_j \rangle)$ are properly coloured with 2 colours $c_{j_1}, c_{j_2}$, for $j$, $1 \leq j \leq k$ by the proper $\tau(G)$-vertex colouring, those two adjacent vertices from the three vertices $w_{j_1}, w_{j_2}$ and $w_{j_3}$ in $G(\langle V_j \rangle)$ should have been coloured with two different colours $c_{j_1}, c_{j_2}$ by the $\tau(G)$-vertex colouring. In $G(\langle V_i \rangle)$ each vertex is coloured with either $c_{i_1}$ or $c_{i_2}$ by the proper $\tau(G)$ vertex colouring, the vertex $u_{i_1}$ is coloured with either $c_{i_1}$ or $c_{i_2}$ in $G(\langle V_i \rangle)$ by the proper $\tau(G)$-vertex colouring. This implies that the path $P$ of order four having the vertices $u_{i_1}, w_{j_1}, w_{j_2}$ and $w_{j_3}$ are coloured with at least three different colours by the proper $\tau(G)$-vertex colouring of $G$.
\noindent\textbf{Case 2} Suppose a path $P$ of order four in $G$ has exactly two vertices in
\hspace{0.8cm} $G(\langle V_i \rangle)$ and has exactly two vertices in $G(\langle V_j \rangle)$.
\noindent Let $u_{i_1}$ and $u_{i_2}$ be the two vertices of $P$ in $G(\langle V_i \rangle)$ and let $w_{j_1}$ and $w_{j_2}$ be the two vertices of $P$ in $G(\langle V_j \rangle)$.
\noindent\textbf{Case 2.1.} Suppose either $u_{i_1}, u_{i_2}$ are coloured with two different colours
\hspace{1.3cm} $c_{i_1}, c_{i_2}$ in $G(\langle V_i \rangle)$ or $w_{j_1}, w_{j_2}$ are coloured with two different colours
\hspace{1.3cm} $c_{j_1}, c_{j_2}$ in $G(\langle V_j \rangle)$ by the proper $\tau(G)$-vertex colouring.
\noindent Then the vertices of the path $P$ of order four having the vertices $u_{i_1}, u_{i_2}, w_{j_1}$ and $w_{j_2}$ are coloured with three or four different colours by the proper $\tau(G)$-vertex colouring of $G$.
\noindent\textbf{Case 2.2.} Suppose neither the vertices $u_{i_1}, u_{i_2}$ received different colours in
\hspace{1.3cm} $G(\langle V_i \rangle)$ nor the vertices $w_{j_1}, w_{j_2}$ received different colours in
\hspace{1.3cm} $G(\langle V_j \rangle)$ by the proper $\tau(G)$-vertex colouring.
\noindent Then without loss of generality, we assume that $u_{i_1}, u_{i_2}$ received the same colour $c_{i_1}$ in $G(\langle V_i \rangle)$ and without loss of generality, we assume that $w_{j_1}, w_{j_2}$ received the same colour $c_{j_1}$ in $G(\langle V_i \rangle)$ by the $\tau(G)$-vertex colouring. As the vertices of $G$ are properly coloured, the vertices $u_{i_1}$ and $u_{i_2}$ should be non-adjacent in $G(\langle V_i \rangle)$ as well as the vertices $w_{j_1}$ and $w_{j_2}$ should also be non-adjacent in $G(\langle V_j \rangle)$. Since for every $h$, $1 \leq h \leq \alpha$, $\tau(G(\langle V_h \rangle)) \leq 2$, $G(\langle V_h \rangle)$ is the union of independent vertices and / or independent edges, every vertex in each $G(\langle V_h \rangle)$ is of degree either 0 or 1. Suppose either $u_{i_1}$ or $u_{i_2}$ is of degree 0 in $G(\langle V_i \rangle)$. Then without loss of generality, we assume that $u_{i_1}$ is of degree 0 in $G(\langle V_i \rangle)$. Since $u_{i_1}$ is not adjacent to any vertex in $G(\langle V_i \rangle)$, recolour the vertex $u_{i_1}$ with the colour $c_{i_2}$ [Since vertices of $G(\langle V_i \rangle)$
are properly coloured with either $c_{i_1}$ or $c_{i_2}$ colours, this recolouring is possible]. Thus, after this recolouring, the vertices $u_{i_1}, u_{i_2}$, $w_{j_1}$ and $w_{j_2}$ of the path $P$ have received three different colours. Hence, we assume neither $u_{i_1}$ nor $u_{i_2}$ is of degree 0 in $G(\langle V_i \rangle)$. Therefore, the degree of each of the vertices $u_{i_1}$ and $u_{i_2}$ must be of degree 1 in $G(\langle V_i \rangle)$. As vertices of each $G(\langle V_i \rangle)$ are properly coloured for $i$, $1 \leq i \leq \alpha$ and as the vertices $u_{i_1}$ and $u_{i_2}$ are coloured with the same colour $c_{i_1}$ in $G(\langle V_i \rangle)$, the vertices $u_{i_1}$ and $u_{i_2}$ must be non-adjacent in $G(\langle V_i \rangle)$. Since the $deg(u_{i_1}) = 1$ in $G(\langle V_i \rangle)$, the vertex $u_{i_1}$ should have an adjacent vertex $u_{i_1}^\prime$ in $G(\langle V_i \rangle)$ and it should have been coloured with the colour $c_{i_2}$ in $G(\langle V_i \rangle)$ by the proper $\tau(G)$-vertex colouring. For each $h$, $1 \leq h \leq k$, $\tau(G(\langle V_h \rangle) \leq 2$, the edge $u_{i_1} u_{i_1}^\prime$ must be an independent edge in $G(\langle V_i \rangle)$. Exchange the colours of $u_{i_1}$ and $u_{i_1}^\prime$. Thus, after this recolouring (this exchange), the vertex $u_{i_1}$ is coloured with $c_{i_2}$. Therefore, after the recolouring the vertices $u_{i_1}$ and $u_{i_2}$ received two different colours $c_{i_2}$ and $c_{i_1}$ respectively in $G(\langle V_i \rangle)$. As a result, the vertices $u_{i_1}$, $u_{i_2}$, $w_{j_1}$ and $w_{j_2}$ have received three different colours in $G(\langle V_i \cup V_j \rangle)$. Hence the path $P$ of order four having the four vertices $u_{i_1}$, $u_{i_2}$, $w_{j_1}$, $w_{j_2}$ are coloured with three different colours in $G$ after the recolouring.
\noindent\textbf{Case 3} Suppose the path $P$ of order four in $G$ has three vertices in $G(\langle V_i \rangle)$
\hspace{0.8cm} and the remaining one vertex in $G(\langle V_j \rangle)$, for $i,j$, $1 \leq i < j \leq \alpha$.
\noindent Without loss of generality, we assume that $w_{j_1}$ is one of the vertices of $P$ which is in $G(\langle V_j \rangle)$ and we assume $u_{i_1}$, $u_{i_2}$ and $u_{i_3}$ are the other three vertices of $P$ which are in $G(\langle V_i \rangle)$. Then as seen in Case 1, two of the vertices from the three vertices $u_{i_1}$, $u_{i_2}$ and $u_{i_3}$ should have received two different colours $c_{i_1}$, $c_{i_2}$ by the proper $\tau(G)$-vertex colouring. Consequently, the vertices of the Path $P$ should have received three or four different colours in $G$.
Thus, every path $P$ of order four in $G$ is either coloured with at least three different colours by the proper $\tau(G)$-vertex colouring of $G$ or else if they are bicoloured by the proper $\tau(G)$-vertex colouring, then the vertices of such a path $P$ can be recoloured as done in the above recolouring process so that the vertices of $P$ are coloured with at least three different colours.\\ Thus there exist a $\tau(G)$-star colouring for $G$. Hence, $\chi_s(G) \leq \tau(G)$. Hence Claim 2.
If $G$ is a disconnected graph with $t \geq 2$ components $G_1,G_2,\dots,G_t$. Then by Claim 2, $\chi_s(G_i) \leq \tau(G_i)$, for $i$, $1 \leq i \leq t$. Let $\smash{\displaystyle\max_{1 \leq i \leq t}}\,\, \chi_s(G_i) = \chi_s(G_k)$ for some $k$, $1 \leq k \leq t$. Since $\chi_s(G) = \smash{\displaystyle\max_{1 \leq i \leq t}}\,\,\chi_s(G_i)$, we have $\chi_s(G) = \chi_s(G_k) \leq \tau(G_k) \leq \smash{\displaystyle\max_{1 \leq i \leq t}}\,\, \tau(G_i) = \tau(G)$. Thus, $\chi_s(G) \leq \tau(G)$. This completes the proof. \end{proof}
An acyclic colouring of $G$ is a proper vertex colouring of $G$ such that no cycle of $G$ is bicoloured. Acyclic chromatic number of a graph $G$, denoted $a(G)$ is the minimum of colours which are necessary to acyclically colour $G$.
\begin{corollary} Let $G$ be any graph. Then the acyclic chromatic number of $G$, $a(G) \leq \tau(G)$. \end{corollary}
\begin{proof} For every graph $G$, $a(G) \leq \chi_s(G)$. By Theorem \ref{star}, we have $\chi_s(G) \leq \tau(G)$ for any graph $G$. Thus, $a(G) \leq \tau(G)$, for any graph $G$. \end{proof}
\section{Discussion} Path Partition Theorem is a beautiful and natural theorem and it significantly helped to get the upper bounds for chromatic number, star chromatic number and detour chromatic number. We believe that Path Partition Theorem can be significantly used for obtaining upper bounds of other different coloring related parameters too. In a general approach, understanding the following question will be interesting and significant too. \begin{quotation} What are the other graph parameters for which such partitions(like $\tau$-partition) can be obtained? \end{quotation}
\section*{References}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{The One-Phase Bifurcation For The $p$-Laplacian} \author{Alaa Akram Haj Ali \& Peiyong Wang\footnote{Peiyong Wang is partially supported by a Simons Collaboration Grant.}\\ \footnotesize Department of Mathematics\\ \footnotesize Wayne State University\\
\footnotesize Detroit, MI 48202\\
\normalsize} \date{} \maketitle \begin{abstract} A bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the $p$-Laplacian, subject to given boundary condition is proved in this paper. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second part, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense. \end{abstract}
\textbf{AMS Classifications:} 35J92, 35J25, 35J62, 35K92, 35K20, 35K59
\textbf{Keywords:} bifurcation, phase transition, $p$-Laplacian, Mountain Pass Theorem, Palais-Smale condition, critical point, critical boundary data, convergence of evolution.
\section{Introduction}\label{introduction} In this paper, one considers the phase transition problem of minimizing the $p$-functional \begin{equation}\label{p-functional}
J_{p,\varepsilon}(u) = \int_{\Omega}\frac{1}{p}|\nabla u(x)|^p + Q(x)\Gamma_{\varepsilon}(u(x))\,dx\ \ \ (1<p<\infty) \end{equation} which is a singular perturbation of the one-phase problem of minimizing the functional associated with the $p$-Laplacian \begin{equation}\label{p-functional_original}
J_p(u) = \int_{\Omega}\frac{1}{p}|\nabla u(x)|^p + Q(x)\chi_{\{u(x)>0\}}\,dx, \end{equation} where $\Gamma_{\varepsilon}(s) = \Gamma(\frac{s}{ \varepsilon})$ for $\varepsilon > 0$ and for a $C^{\infty}$ function $\Gamma$ defined by $$\Gamma(s) = \left\{\begin{array}{ll} 0 &\ \text{\ if\ }s\leq 0\\ 1 &\ \text{\ if\ }s\geq 1, \end{array}\right.$$ and $0\leq\Gamma(s)\leq 1$ for $0<s<1$, and $Q\in W^{2,2}(\Omega)$ is a positive continuous function on $\Omega$ such that $\inf_{\Omega}Q(x) > 0$. Let $\beta_{\varepsilon}(s) = \Gamma'_{\varepsilon}(s) = \frac{1}{ \varepsilon}\beta(\frac{s}{\varepsilon})$ with $\beta = \Gamma'$. The domain $\Omega$ is always assumed to be smooth in this paper for convenience. As in the following we will fix the value of $\varepsilon$ unless we specifically examine the influence of the value of $\varepsilon$ on the critical boundary data and will not use the notation $J_p$ for a different purpose, we are going to abuse the notation by using $J_p$ for the functional $J_{p,\varepsilon}$ from now on.
The Euler-Lagrange equation of (\ref{p-functional}) is \begin{equation}\label{eulereq} -\bigtriangleup_p u + Q(x)\beta_{\varepsilon}(u) = 0\ \ x\in\Omega \end{equation} One imposes the boundary condition \begin{equation}\label{bdrycondition} u(x) = \sigma(x),\ \ x\in\partial\Omega \end{equation} on $u$, for $\sigma\in C(\partial\Omega)$ with $\min_{\partial\Omega}\sigma > 0$, to form a boundary value problem.
In this paper, we take on the task of establishing in the general case when $p\ne 2$ the results proved in \cite{CW} for the Laplacian when $p=2$. The main difficulty in this generalization lies in the lack of sufficient regularity and the singular-degenerate nature of the $p$-Laplacian when $p\ne 2$. A well-known fact about $p$-harmonic functions is the optimal regularity generally possessed by them is $C^{1,\alpha}$ (e.\,g.\,\cite{E} and \cite{Le}). Thus we need to employ more techniques associated with the $p$-Laplacian, and in a case or two we have to make our conclusion slightly weaker. Nevertheless, we follow the overall scheme of approach used in \cite{CW}. In the second section, we prove the bifurcation phenomenon through the Mountain Pass Theorem. In the third section, we establish a parabolic comparison principle. In the last section, we show the convergence of an evolution to a stable steady state in accordance with respective initial data.
\section{A Third Solution}\label{thirdsolution} We first prove if the boundary data is small enough, then the minimizer is nontrivial. More precisely, let $u_0$ be the trivial solution of (\ref{eulereq}) and (\ref{bdrycondition}), being $p$-harmonic in the weak sense, and $u_2$ be a minimizer of the $p$-functional (\ref{p-functional}), and set $$\sigma_M = \max_{\partial\Omega}\sigma(x)\ \ \text{ and\ }\ \ \sigma_m = \min_{\partial\Omega}\sigma(x).$$ If $\sigma_M$ is small enough, then $u_0\neq u_2$.
In fact, we pick $u\in W^{1,p}(\Omega)$ so that \begin{equation} \left\{\begin{array}{ll} u = 0 &\ \ \text{ in $\Omega_{\delta}$}\\ u = \sigma &\ \ \text{ on $\partial\Omega$,\ \ \ \ and }\\ -\bigtriangleup_p u = 0 &\ \ \text{ in $\Omega\backslash\bar{\Omega}_{\delta}$,}\end{array}\right. \end{equation} where $\Omega_{\delta} = \{x\in\Omega\colon dist(x,\partial\Omega) > \delta\}$ and $\delta > 0$ is a small constant independent of $\varepsilon$ and $\sigma$ so that $\int_{\Omega_{\delta}}Q(x)\,dx$ has a positive lower bound which is also independent of $\varepsilon$ and $\sigma$. Using an approximating domain if necessary, we may assume $\Omega_{\delta}$ possesses a smooth boundary. Clearly, \begin{equation*}
J_p(u_0) = \int_{\Omega}\frac{1}{p}|\nabla u_0|^p + Q(x)\,dx \geq \int_{\Omega}Q(x)\,dx. \end{equation*} It is well-known that \begin{equation*}
\int_{\Omega\backslash\Omega_{\delta}}|\nabla u|^p \leq C\sigma^{\,p}_M\delta^{1-p}\ \ \text{ for $C = C(n,p,\Omega)$}, \end{equation*} so that \begin{alignat*}{1}
&J_p(u) = \int_{\Omega\backslash \Omega_{\delta}}\frac{1}{p}|\nabla u|^p + \int_{\Omega\backslash\Omega_{\delta}}Q(x)\,dx\\ &\leq C\sigma^{\,p}_M\delta^{1-p} + \int_{\Omega\backslash\Omega_{\delta}}Q(x)\,dx. \end{alignat*} So, for all small $\varepsilon > 0$, \begin{equation*} J_p(u) - J_p(u_0) \leq C\sigma^{\,p}_M\delta^{1-p} - \int_{\Omega_{\delta}} Q(x)\,dx < 0 \end{equation*} if $\sigma_M\leq \sigma_0$ for some small enough $\sigma_0 = \sigma_0(\delta, \Omega, Q)$.
Let $\mathfrak{B}$ denote the Banach space $W^{1,p}_0(\Omega)$ we will work with. For every $v\in\mathfrak{B}$, we write $u = v + u_0$ and adopt the norm $\|v\|_{\mathfrak{B}} = \left(\int_{\Omega}|\nabla v|^p\right)^{\frac{1}{p}} = \left(\int_{\Omega}|\nabla u - \nabla u_0|^p\right)^{\frac{1}{p}}$. We define the functional \begin{equation}
I[v] = J_p(u) - J_p(u_0) = \int_{\Omega}\frac{1}{p}|\nabla u|^p - \int_{\{u < \varepsilon\}}Q(x)\left(1 - \Gamma_{\varepsilon}(u)\right) - \int_{\Omega}\frac{1}{p} |\nabla u_0|^p \end{equation} Set $v_2 = u_2 - u_0$. Clearly, $I[0] = 0$ and $I[v_2] \leq 0$ on account of the definition of $u_2$ as a minimizer of $J_p$. If $I[v_2] < 0$ which is the case if $\sigma_M$ is small, we will apply the Mountain Pass Lemma to prove there exists a critical point of the functional $I$ which is a weak solution of the problem (\ref{eulereq}) and (\ref{bdrycondition}).
The Fr\'{e}chet derivative of $I$ at $v\in \mathfrak{B}$ is given by \begin{equation}
I'[v]\varphi = \int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi + Q(x)\beta_{\varepsilon}(u)\varphi\ \ \ \ \varphi\in\mathfrak{B} \end{equation} which is obviously in the dual space $\mathfrak{B}^*$ of $\mathfrak{B}$ in light of the H\"{o}lder's inequality. Equivalently \begin{equation} I'[v] = -\bigtriangleup_p (v+u_0) + Q(x)\beta_{\varepsilon}(v + u_0)\in\mathfrak{B}^*. \end{equation} We see that $I'$ is Lipschitz continuous on any bounded subset of $\mathfrak{B}$ with Lipschitz constant depending on $\varepsilon$, $p$, and $\sup Q$. In fact, for any $v$, $w$, and $\varphi\in \mathfrak{B}$, \begin{alignat*}{1}
&\ \ \left|I'[v]\varphi - I'[w]\varphi\right| = |\int_{\Omega}|\nabla v + \nabla u_0|^{p-2}(\nabla v + \nabla u_0)\cdot\nabla\varphi + Q(x)\beta_{\varepsilon}(v+u_0)\\
&- |\nabla w + \nabla u_0|^{p-2}(\nabla w + \nabla u_0)\cdot\nabla\varphi - Q(x)\beta_{\varepsilon}(w+u_0)| \\
&\leq \left|\int_{\Omega}|\nabla v + \nabla u_0|^{p-2}(\nabla v + \nabla u_0)\cdot\nabla\varphi -
|\nabla w + \nabla u_0|^{p-2}(\nabla w + \nabla u_0)\cdot\nabla\varphi\right| \\
&+ \left|\int_{\Omega}Q(x)\beta_{\varepsilon}(v+u_0) - Q(x)\beta_{\varepsilon}(w+u_0)\right| \end{alignat*} Furthermore, \begin{alignat*}{1}
&\ \ \ \ \left|\int_{\Omega}Q(x)\beta_{\varepsilon}(v+u_0) - Q(x)\beta_{\varepsilon}(w + u_0)\right|\\
&= \left|\int_{\Omega}Q(x)\int^1_0\beta'_{\varepsilon}((1-t)w + tv + u_0)\,dt\,(v(x) - w(x))\,dx \right|\\
&\leq \sup|\beta'_{\varepsilon}|\int_{\Omega}\left|Q(x)\left(v(x) - w(x)\right)\right|\,dx \\
&\leq \frac{C}{\varepsilon^2}\left(\int_{\Omega}Q^{p'}(x)\right)^{\frac{1}{p'}}\left(\int_{\Omega}|v(x) - w(x)|^p\,dx\right)^{\frac{1}{p}} \end{alignat*} and \begin{alignat*}{1}
&\ \ \ \ \left|\int_{\Omega}|\nabla v + \nabla u_0|^{p-2}(\nabla v + \nabla u_0)\cdot \nabla \varphi - |\nabla w + \nabla u_0|^{p-2}(\nabla w + \nabla u_0)\cdot \nabla \varphi\right|\\
&\leq \left|\int_{\Omega}|\nabla v + \nabla u_0|^{p-2}(\nabla v - \nabla w)\cdot \nabla \varphi\right| \\
&\ \ \ \ + \left|\int_{\Omega}\left(|\nabla v + \nabla u_0|^{p-2} - |\nabla w + \nabla u_0|^{p-2}\right)(\nabla w + \nabla u_0)\cdot \nabla \varphi\right|. \end{alignat*} In addition, \begin{alignat*}{1}
&\ \ \ \ \left|\int_{\Omega}|\nabla v + \nabla u_0|^{p-2}(\nabla v - \nabla w)\cdot \nabla \varphi\right|\\
&\leq \left(\int_{\Omega}|\nabla v + \nabla u_0|^p\right)^{\frac{p-2}{p}}\left(\int_{\Omega}|\nabla\varphi|^p\right)^{\frac{1}{p}}\left(\int_{\Omega}|\nabla v - \nabla w|^p\right)^{\frac{1}{p}}, \end{alignat*} and \begin{alignat*}{1}
&\ \ \ \ \left|\int_{\Omega}\left(|\nabla v + \nabla u_0|^{p-2} - |\nabla w + \nabla u_0|^{p-2}\right)\left(\nabla w + \nabla u_0\right)\cdot\nabla\varphi\right|\\
&\leq C(p)\int_{\Omega}\left(|\nabla v + \nabla u_0|^{p-3} + |\nabla w + \nabla u_0|^{p-3}\right)|\nabla v - \nabla w||\nabla w + \nabla u_0||\nabla\varphi|\\
&\leq C(p)\left(\|\nabla v\|_{L^p} + \|\nabla w\|_{L^p} + \|\nabla u_0\|_{L^p}\right)^{p-2}\|\nabla v - \nabla w\|_{L^p(\Omega)}\|\nabla\varphi\|_{L^p(\Omega)}. \end{alignat*} Therefore $I'$ is Lipschitz continuous on bounded subsets of $\mathfrak{B}$.
We note that $f\in\mathfrak{B}^*$ if and only if there exist $f^0$, $f^1$, $f^2$, ..., $f^n\in L^{p'}(\Omega)$, where $\frac{1}{p} + \frac{1}{p'} = 1$, such that \begin{alignat}{1} &<f,u>\ = \int_{\Omega}f^0u + \sum^n_{i=1}f^iu_{x_i} \ \ \text{ holds for all $u\in\mathfrak{B}$; and}\label{repre}\\
&\|f\|_{\mathfrak{B}^*} = \inf\left\{\left(\int_{\Omega}\sum^n_{i=0}|f^i|^{p'}\,dx\right)^{\frac{1}{p'}}\colon \text{(\ref{repre}) holds.}\right\} \end{alignat}
Next we justify the Palais-Smale condition on the functional $I$. Suppose $\{v_k\}\subset\mathfrak{B}$ is a Palais-Smale sequence in the sense that \begin{equation*}
\left|I[v_k]\right|\leq M\ \ \ \ \text{and\ \ }\ \ I'[v_k]\rightarrow 0\ \ \ \ \text{in $\mathfrak{B}^*$} \end{equation*} for some $M > 0$. Let $u_k = v_k + u_0\in W^{1,p}(\Omega)$, $k = 1, 2, 3, ...$.
That $Q(x)\beta_{\varepsilon}(v + u_0)\in W^{1,p}_0(\Omega)$ implies that the mapping $v\mapsto Q(x)\beta_{\varepsilon}(v + u_0)$ from $W^{1,p}_0(\Omega)$ to $\mathfrak{B}^*$ is compact due to the fact $W^{1,p}_0(\Omega)\subset\subset L^p(\Omega)\subset \mathfrak{B}^*$ following from the Rellich-Kondrachov Compactness Theorem. Then there exists $f\in L^p(\Omega)\subset\mathfrak{B}^*$ such that for a subsequence, still denoted by $\{v_k\}$, of $\{v_k\}$, it holds that \begin{equation*} Q(x)\beta_{\varepsilon}(u_k)\rightarrow -f\ \ \text{ in $L^p(\Omega)$.} \end{equation*} Recall that \begin{equation*}
\left|I'[v_k]\varphi\right| = \sup_{\|\varphi\|_{\mathfrak{B}}\leq 1}\left|\int_{\Omega}|\nabla u_k|^{p-2}\nabla u_k\cdot\nabla\varphi + Q(x)\beta_{\varepsilon}(u_k) \varphi\right|\rightarrow 0. \end{equation*} As a consequence, \begin{equation}\label{test1}
\sup_{\|\varphi\|_{\mathfrak{B}}\leq M}\left|\int_{\Omega}|\nabla u_k|^{p-2}\nabla u_k\cdot\nabla \varphi - f\varphi\right| \rightarrow 0\ \ \ \ \text{for any $M\geq 0$.} \end{equation} Obviously, that $\{I[v_k]\}$ is bounded implies that a subsequence of $\{v_k\}$, still denoted by $\{v_k\}$ by abusing the notation without confusion, converges weakly in $\mathfrak{B} = W^{1, p}_0(\Omega)$. In particular, \begin{equation*} \int_{\Omega}fv_k - fv_m\rightarrow 0\ \ \ \ \text{as $k$, $m\rightarrow\infty$.} \end{equation*} Then by setting $\varphi = v_k - v_m = u_k - u_m$ in (\ref{test1}), one gets \begin{equation}\label{conv}
\left|\int_{\Omega}\left(|\nabla u_k|^{p-2}\nabla u_k - |\nabla u_m|^{p-2}\nabla u_m\right)\cdot \nabla (u_k - u_m)\right| \rightarrow 0\ \ \ \ \text{as $k$, $m\rightarrow\infty$,} \end{equation} since \begin{equation*}
\|u_k - u_m\|^p_{\mathfrak{B}} = \|v_k - v_m\|^p_{\mathfrak{B}} \leq 2pM + 2J_p[u_0]. \end{equation*} In particular, if $p = 2$, $\{v_k\}$ is a Cauchy sequence in $W^{1,2}_0(\Omega)$ and hence converges. We will apply the following elementary inequalities associated with the $p$-Laplacian, \cite{L}, to the general case $p\neq 2$: \begin{alignat}{1}
&<|b|^{p-2}b - |a|^{p-2}a,\,b - a> \geq (p-1)|b-a|^2(1 + |a|^2 + |b|^2)^{\frac{p-2}{2}},\ \ 1\leq p\leq 2;\label{ele1}\\
&\text{and}\ \ \ \ <|b|^{p-2}b - |a|^{p-2}a,\,b - a> \geq 2^{2-p}|b-a|^p,\ \ p\geq 2.\label{ele2} \end{alignat} We assume first $1 < p < 2$. Let $K = 2pM + 2J_p[u_0]$. Then the first elementary inequality (\ref{ele1}) implies \begin{alignat*}{1}
&\ \ \ \ \ (p-1)\int_{\Omega}|\nabla u_k - \nabla u_m|^2\left(1+|\nabla u_k|^2 + |\nabla u_m|^2\right)^{\frac{p-2}{2}}\\ &\leq \int_{\Omega}\left(|\nabla u_k|^{p-2}\nabla u_k - |\nabla u_m|^{p-2}\nabla u_m\right)\cdot \nabla (u_k - u_m) \rightarrow 0 \end{alignat*} Meanwhile H\"{o}lder's inequality implies \begin{alignat*}{1}
&\ \ \ \ \ \int_{\Omega}|\nabla v_k - \nabla v_m|^p = \int_{\Omega}|\nabla u_k - \nabla u_m|^p \\
&\leq \left(\int_{\Omega}|\nabla u_k - \nabla u_m|^2\left(1 + |\nabla u_k|^2 + |\nabla u_m|^2\right)^{\frac{p-2}{2}}\right)^{\frac{p}{2}}
\left(\int_{\Omega}\left(1 + |\nabla u_k|^2 + |\nabla u_m|^2\right)^{\frac{p}{2}}\right)^{\frac{2-p}{2}} \\
&\leq C(p)\left(|\Omega| + K\right)^{\frac{2-p}{2}} \left(\int_{\Omega}|\nabla u_k - \nabla u_m|^2\left(1 + |\nabla u_k|^2 + |\nabla u_m|^2\right)^{\frac{p-2}{2}}\right)^{\frac{p}{2}} \end{alignat*} Therefore, $\{v_k\}$ is a Cauchy sequence in $\mathfrak{B}$ and hence converges.
Suppose $p > 2$. The second elementary inequality (\ref{ele2}) implies \begin{alignat*}{1}
&\ \ \ \ \ \int_{\Omega}|\nabla v_k - \nabla v_m|^p = \int_{\Omega}|\nabla u_k - \nabla u_m|^p \\
&\leq 2^{p-2}\int_{\Omega}\left(|\nabla u_k|^{p-2}\nabla u_k - |\nabla u_m|^{ p-2}\nabla u_m\right)\cdot \left(\nabla u_k - \nabla u_m\right), \end{alignat*} which in turn implies $\{v_k\}$ is a Cauchy sequence in $\mathfrak{B}$ and hence converges, on account of (\ref{conv}). The Palais-Smale condition is verified for $1 < p < \infty$ for the functional $I$ on the Banach space $W^{1,p}_0(\Omega)$.
Before we continue the main proof, let us state an elementary result closely related to the $p$-Laplacian, which follows readily from the Fundamental Theorem of Calculus. \begin{lemma}\label{p-inequalities} For any $a$ and $b\in\mathbb{R}^n$, it holds \begin{equation}\label{ele3}
|b|^p \geq |a|^p + p<|a|^{p-2}a, b-a> +\, C(p)|b - a|^p\ \ \ \ (p\geq 2) \end{equation} where $C(p) > 0$.
If $1 < p < 2$, then \begin{equation}\label{ele4}
|b|^p \geq |a|^p + p<|a|^{p-2}a, b-a> +\, C(p)|b-a|^2\int^1_0\int^t_0\left|(1-s)a+sb\right|^{p-2}\,dsdt, \end{equation} where $C(p) = p(p-1)$. \end{lemma}
We are now in a position to show there is a closed mountain ridge around the origin of the Banach space $\mathfrak{B}$ that separates $v_2$ from the origin with the energy $I$ as the elevation function, which is the content of the following lemma. \begin{lemma}
For all small $\varepsilon > 0$ such that $C\varepsilon \leq \frac{1}{2}\sigma_m$ for a large universal constant $C$, there exist positive constants $\delta$ and $a$ independent of $\varepsilon$, such that, for every $v$ in $\mathfrak{B}$ with $\|v\|_{\mathfrak{B}} = \delta$, the inequality $I[v] \geq a$ holds. \end{lemma} \begin{pf}
It suffices to prove $I[v] \geq a > 0$ for every $v\in C^{\infty}_0(\Omega)$ with $\|v\|_{\mathfrak{B}} = \delta$ for $\delta$ small enough, as $I$ is continuous on $\mathfrak{B}$, and $C^{\infty}_0(\Omega)$ is dense in $\mathfrak{B}$.
Let $\Lambda = \{x\in\Omega\colon u(x)\leq\varepsilon\}$, where $u = v + u_0$. We claim that $\Lambda = \emptyset$ if $\delta$ is small enough. If not, one may pick $z\in\Lambda$. Let $\mathcal{AC}([a,b], S)$ be the set of absolutely continuous functions $\gamma\colon [a,b]\rightarrow S$, where $S\subseteq\mathbb{R}^n$. For each $\gamma\in\mathcal{AC}([a,b], S)$, we define its length to be $L(\gamma) = \int^b_a|\gamma'(t)|\,dt$. For $x_0\in\partial\Omega$, we define the distance from $x_0$ to $z$ to be \begin{equation*} d(x_0,z) = \inf\{L(\gamma): \gamma\in\mathcal{AC}([0,1],\bar{\Omega}), \ \text{s.t.\ }\gamma(0) = x_0, \ \text{and\ }\gamma(1)=z\} \end{equation*} As shown in \cite{CW}, there is a minimizing path $\gamma_{x_0}$ for the distance $d(x_0, z)$.
Suppose the domain $\Omega$ is convex or star-like about $z$. For any $x_0\in\partial\Omega$, let $\gamma = \gamma_{x_0}$ be a minimizing path of $d(x_0, z)$. Then it is clear that $\gamma$ is a straight line segment and $\gamma(t)\neq z$ for $t\in [0,1)$. Furthermore, for any two distinct points $x_1$ and $x_2\in\partial\Omega$, the corresponding minimizing paths do not intersect in $\Omega\backslash\{z\}$. For this reason, we can carry out the following computation. Clearly $v(x_0) = 0$ and $v(\gamma(1)) = \varepsilon - u_0(\gamma(1))\leq \varepsilon - \sigma_m < 0$. So the Fundamental Theorem of Calculus \begin{equation*} v(\gamma(1)) - v(\gamma(0)) = \int^1_0\nabla v(\gamma(t))\cdot\gamma'(t)dt \end{equation*} implies \begin{equation}\label{ineq-ftc}
\sigma_m - \varepsilon \leq \int^1_0|\nabla v(\gamma(t))||\gamma'(t)|dt. \end{equation} For each $x_0\in\partial\Omega$, let $e(x_0)$ be the unit vector in the direction of $x_0 - z$ and $\nu(x_0)$ the outer normal to $\partial\Omega$ at $x_0$. Then $\nu(x_0)\cdot e(x_0) > 0$ everywhere on $\partial\Omega$. Hence the above inequality (\ref{ineq-ftc}) implies \begin{alignat*}{1} &\ \ \ \ (\sigma_m - \varepsilon)\int_{\partial\Omega}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0)\\
&\leq \int_{\partial\Omega}\int^1_0|\nabla v(\gamma(t))| |\gamma'(t)|\,dt\,\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \\
&\leq \int_{\partial\Omega}\left(\int^1_0|\gamma'(t)|\,dt\right)^{\frac{1}{p'}}\left(\int^1_0|\nabla v(
\gamma(t))|^p|\gamma'(t)|\,dt\right)^{\frac{1}{p}}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0),\\ &\hspace{3.5in}\text{ where $\frac{1}{p} + \frac{1}{p'} = 1$,} \\
&= \int_{\partial\Omega} L(\gamma_{x_0})^{\frac{1}{p'}}\left(\int^1_0|\nabla v(\gamma(t))|^p|\gamma'(t)| \,dt\right)^{\frac{1}{p}}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \\ &\leq \left(\int_{\partial\Omega}L(\gamma_{x_0})\nu(x_0)\cdot e(x_0)\,dH^{n-1}\right)^{\frac{1}{p'}}\left(\int_{\partial\Omega}
\int^1_0|\nabla v(\gamma(t))|^p|\gamma'(t)|\nu \cdot e\,dt\,dH^{n-1}\right)^{\frac{1}{p}} \\
&= C|\Omega|^{\frac{1}{p'}}\left(\int_{\Omega}|\nabla v|^p\,dx\right)^{\frac{1}{p}}\\
&\leq C|\{u > \varepsilon\}|^{\frac{1}{p'}}\delta \leq C|\{u>0\}|^{\frac{1}{p'}}\delta, \end{alignat*} where the second and third inequalities are due to the application of the H\"{o}lder's inequality, and the constant $C$ depends on $n$ and $p$. The second equality follows from the two representation formulas \begin{equation*}
\left|\Omega\right| = C(n)\int_{\partial\Omega}L(\gamma_{x_0})\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \end{equation*} and \begin{equation*}
\int_{\Omega}\left|\nabla v(x)\right|^p\,dx = C(n)\int_{\partial\Omega}\int^1_0\left|\nabla v(\gamma_{x_0}(t))\right|^p\,\left|\gamma'_{x_0}(t)\right|\nu(x_0) \cdot e(x_0)\,dt\,dH^{n-1}(x_0). \end{equation*} If we take $\delta$ sufficiently small and independent of $\varepsilon$ in the preceding inequality \begin{equation*}
(\sigma_m - \varepsilon)\int_{\partial\Omega}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \leq C|\{u>0\}|^{\frac{1}{p'}}\delta, \end{equation*}
the measure $|\{u > 0\}|$ of the positive domain would be greater than that of $\Omega$, which is impossible, provided that \begin{equation}\label{direction-normal-ineq} \int_{\partial\Omega}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \geq C, \end{equation}
for a constant $C$ which depends on $n$, $p$ and $|\Omega|$, but not on $z$ or $v$. Hence $\Lambda$ must be empty. So we need to justify the inequality (\ref{direction-normal-ineq}). To fulfil that condition, for $e = e(x_0)$, we set $l(e,z) = l(e) = L(\gamma_{x_0})$. Then \begin{equation*} \int_{\partial\Omega}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) = \int_{e\in \partial B}\left(l(e)\right)^{n-1}\,d\sigma(e), \end{equation*} where $B$ is the unit ball about $z$ and $d\sigma(e)$ is the surface area element on the unit sphere $\partial B$ which is invariant under rotation and reflection. Clearly, \begin{equation*} \left(\int_{\partial B}\left(l(e)\right)^{n-1}\,d\sigma(e)\right)^{\frac{2}{n-1}}\geq C(n)\int_{\partial B}l^2(e)\,d\sigma(e) \end{equation*} Consequently, in order to prove (\ref{direction-normal-ineq}), one needs only to prove \begin{equation}\label{equiv-integ}
\int_{\partial B}l^2(e)\,d\sigma(e) \geq C(n, p, |\Omega|). \end{equation}
Next, we show the integral on the left-hand-side of (\ref{equiv-integ}) is minimal if $\Omega$ is a ball while its measure is kept unchanged. In fact, this is almost obvious if one notices the following fact. Let $\pi$ be any hyperplane passing through $z$, and $x_1$ and $x_2$ be the points on $\partial\Omega$ which lie on a line perpendicular to $\pi$. Let $x^*_1$ and $x^*_2$ be the points on the boundary $\partial\Omega_{\pi}$, where $\Omega_{\pi}$ is the symmetrized image of $\Omega$ about the hyperplane $\pi$, which lie on the line $\overline{x_1x_2}$. Let $2a = |\overline{x_1x_2}| = |\overline{x^*_1x^*_2}|$ and $d$ be the distance from $z$ to the line $\overline{x_1x_2}$. Then for some $t$ in $-a \leq t \leq a$, it holds that \begin{equation*} L^2(\gamma_{x_1}) + L^2(\gamma_{x_2}) = \left(d^2+(a-t)^2\right) + \left(d^2+(a+t)^2\right) \geq 2(d^2 + a^2) = 2\left(L^*(\gamma_{x^*_1})\right)^2. \end{equation*} As a consequence, if $\Omega^*$ is the symmetrized ball with measure equal to that of $\Omega$, then \begin{equation*}
\int_{\partial B}l^2(e)\,d\sigma(e) \geq \int_{\partial B}\left(l^*(e)\right)^2\,d\sigma(e) = C(n,|\Omega|), \end{equation*} where $l^*$ is the length from $z$ to a point on the boundary $\partial\Omega^*$ which is constant. This finishes the proof of the fact that $\Lambda = \emptyset$.
In case the domain $\Omega$ is not convex, the minimizing paths of $d(x_1, z)$ and $d(x_2, z)$ for distinct $x_1$, $x_2\in\partial\Omega$ may partially coincide. We form the set $\mathcal{DA}(\partial\Omega)$ of the points $x_0$ on $\partial\Omega$ so that a minimizing path $\gamma$ of $d(x_0, z)$
satisfies $\gamma(t)\in\Omega\backslash\{z\}$ for $t\in (0,1)$. We call a point in $\mathcal{DA}(\partial\Omega)$ a \textbf{directly accessible} boundary point. Let $\Omega_1$ be the union of these minimizing paths for the directly accessible boundary points. It is not difficult to see that $|\Omega_1| > 0$ and hence $H^{n-1}(\mathcal{DA}(\partial\Omega)) > 0$. Then we may apply the above computation to the star-like domain $\Omega_1$ with minimal modification. We have \begin{equation}
(\sigma_m - C\varepsilon)\int_{\partial\Omega}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \leq C|\Omega_1|^{\frac{1}{p'}}\delta \leq C|\Omega|^{\frac{1}{p'}}\delta. \end{equation}
For small enough $\delta$, this raises a contradiction $|\Omega| > |\Omega|$. So $\Lambda = \emptyset$.
Finally we prove that $\|v\|_{\mathfrak{B}} = \delta$ implies \begin{equation}
I[v]= \int_{\Omega}\frac{1}{p}|\nabla v + \nabla u_0|^p - \frac{1}{p}|\nabla u_0|^p \geq a\ \ \text{for a certain $a > 0$.} \end{equation}
If $p\geq 2$, then the elementary inequality (\ref{ele3}) implies that \begin{alignat*}{1}
I[v] &= \int_{\Omega}\frac{1}{p}\left|\nabla v+ \nabla u_0\right|^p - \frac{1}{p}\left|\nabla u_0\right|^p \\
&\geq \int_{\Omega}<\left|\nabla u_0\right|^{p-2}\nabla u_0, \nabla v> + C(p)\left|\nabla v\right|^p \\
&= C(p)\int_{\Omega}\left|\nabla v\right|^p = C(p)\delta^p > 0, \end{alignat*} while if $1 < p < 2$, then the elementary inequality (\ref{ele4}) implies \begin{alignat*}{1}
I[v] &\geq p(p-1)\int_{\Omega}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left|\nabla u_0 + s\nabla v\right|^{2-p}}\,dsdtdx \\
&\geq p(p-1)\int_{\Omega}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(\left|\nabla u_0\right| + s\left|\nabla v\right|\right)^{2-p}}\,dsdtdx.
\end{alignat*}
If $\int_{\Omega}|\nabla u_0|^p = 0$, then $I[v] = \frac{1}{p}\delta^p > 0$. So in the following, we assume $\int_{\Omega}|\nabla u_0|^p > 0$.
Let $S = S_{\lambda} = \{x\in\Omega\colon |\nabla v| > \lambda\delta\}$, where the constant $\lambda = \lambda(p,|\Omega|)$ is to be taken. Then \begin{alignat*}{1}
\delta^p &= \int_{\Omega}|\nabla v|^p = \int_{\{|\nabla v|\leq \lambda\delta\}}|\nabla v|^p + \int_S|\nabla v|^p \\
&\leq (\lambda\delta)^p|\Omega| + \int_S|\nabla v|^p \end{alignat*} and hence \begin{equation*}
\int_S|\nabla v|^p \geq \delta^p\left(1 - \lambda^p|\Omega|\right) \geq \frac{1}{2}\delta^p,\ \ \text{if $\lambda$ satisfies\ }\frac{1}{4} < \lambda^p|\Omega| \leq \frac{1}{2}. \end{equation*} Meanwhile, for $1 < p < 2$, it holds that \begin{alignat*}{1}
I[v] &\geq C(p)\int_{S}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(\left|\nabla u_0\right| + s\left|\nabla v\right|\right)^{2-p}}\,dsdtdx \\
&=C(p)\left(\int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(|\nabla u_0| + s|\nabla v|\right)^{2-p}}\,dsdtdx \right. \\
&\ \ \ \ + \left.\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(\left|\nabla u_0\right| + s\left|\nabla v\right|\right)^{2-p}}\,dsdtdx\right). \end{alignat*} The first integral on the right satisfies \begin{alignat*}{1}
&\ \ \ \ \int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(|\nabla u_0| + s|\nabla v|\right)^{2-p}}\,dsdtdx \\
&\geq \int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}\left|\nabla v\right|^p\int^1_0\int^t_0\frac{1}{\left(1 + s\right)^{2-p}}\,dsdtdx \\
&= C(p)\int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}\left|\nabla v\right|^p\,dx, \end{alignat*} while the second integral on the right satisfies \begin{alignat*}{1}
&\ \ \ \ \int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(\left|\nabla u_0\right| + s\left|\nabla v\right|\right)^{2-p}}\,dsdtdx \\
&\geq \int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\frac{\left|\nabla v\right|^2}{|\nabla u_0|^{2-p}}\int^1_0\int^t_0\frac{ds\,dt}{(1+s)^{2-p}}\,dx \\
&= C(p) \int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\frac{\left|\nabla v\right|^2}{|\nabla u_0|^{2-p}}\,dx. \end{alignat*} The H\"{o}lder's inequality applied with exponents $\frac{2}{p}$ and $\frac{2}{2-p}$ implies that \begin{equation*}
\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^p \leq \left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\frac{|\nabla v|^2}{|\nabla u_0|^{2-p}}\right)^{\frac{p}{2}}\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}|\nabla u_0|^p\right)^{\frac{2-p}{2}}, \end{equation*} or equivalently \begin{alignat*}{1}
\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\frac{|\nabla v|^2}{|\nabla u_0|^{2-p}} &\geq \frac{\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^p\right)^{\frac{2}{p}}}{\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}|\nabla u_0|^p\right)^{\frac{2-p}{p}}} \\
&\geq \frac{\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^p\right)^{\frac{2}{p}}}{\left(\int_{\Omega}|\nabla u_0|^p\right)^{\frac{2-p}{p}}}. \end{alignat*} Consequently, \begin{alignat*}{1}
I[v] &\geq C(p)\int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}|\nabla v|^p + C(p)\frac{\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^p\right)^{\frac{2}{p}}}{\left(\int_{\Omega}|\nabla u_0|^p\right)^{\frac{2-p}{p}}} \\
&\geq C(p)\left(\int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}|\nabla v|^p\right)^{\frac{2}{p}} + C(p)\frac{\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^p\right)^{\frac{2}{p}}}{\left(\int_{\Omega}|\nabla u_0|^p\right)^{\frac{2-p}{p}}},\ \ \text{as $\delta$ is small} \\
&\geq C(p)A(u_0)\left(\left(\int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}|\nabla v|^p\right)^{\frac{2}{p}} + \left(\int_{S\cap\{|\nabla u_0|> |\nabla v|\}}|\nabla v|^p\right)^{\frac{2}{p}}\right) \\
&\geq C(p)A(u_0)\left(\int_{S}|\nabla v|^p\right)^{\frac{2}{p}} = C(p)A(u_0)\delta^2, \end{alignat*} where the last inequality is a consequence of the elementary inequality \begin{equation*} a^{\frac{2}{p}} + b^{\frac{2}{p}} \geq C(p)\left(a+b\right)^{\frac{2}{p}}\ \ \text{for\ }\ a, b\geq 0, \end{equation*} and the constant \begin{equation*}
A(u_0) = \min\left\{1,\frac{1}{\left(\int_{\Omega}|\nabla u_0|^p\right)^{\frac{2-p}{p}}}\right\}. \end{equation*}
So we have proved $I[v] \geq a > 0$ for some $a> 0$ whenever $v\in C^{\infty}_0(\Omega)$ satisfies $\|v\|_{\mathfrak{B}} = \delta$, for any $p\in (1, \infty)$. \end{pf}
Let \begin{equation*} \mathcal{G} = \{\gamma\in C([0,1],H): \gamma(0) = 0\ \text{and\ }\gamma(1) = v_2\} \end{equation*} and \begin{equation*} c = \inf_{\gamma\in\mathcal{G}}\max_{0\leq t\leq 1}I[\gamma(t)]. \end{equation*} The verified Palais-Smale condition and the preceding lemma allow us to apply the Mountain Pass Theorem as stated, for example, in \cite{J} to conclude that there is a $v_1\in \mathfrak{B}$ such that $I[v_1] = c$, and $I'[v_1] = 0$ in $\mathfrak{B}^*$. That is \begin{equation*}
\int_{\Omega}\left|\nabla u_1\right|^{p-2}\nabla u_1\cdot\nabla \varphi + Q(x)\beta_{\varepsilon}(u_1)\varphi dx = 0 \end{equation*} for any $\varphi\in \mathfrak{B} = W^{1,p}_0(\Omega)$, where $u_1 = v_1 + u_0$. So $u_1$ is a weak solution of the problem (\ref{eulereq}) and (\ref{bdrycondition}). In essence, the Mountain Pass Theorem is a way to produce a saddle point solution. Therefore, in general, $u_1$ tends to be an unstable solution in contrast to the stable solutions $u_0$ and $u_2$.
In this section, we have proved the following theorem. \begin{theorem} If $\varepsilon << \sigma_m$ and $J_p(u_2) < J_p(u_0)$, then there exists a third weak solution $u_1$ of the problem (\ref{eulereq}) and (\ref{bdrycondition}). Moreover, $J_p(u_1) \geq J_p(u_0) + a$, where $a$ is independent of $\varepsilon$. \end{theorem}
\section{A Comparison Principle for Evolution}\label{comparison} In this section, we prove a comparison theorem for the following evolution problem. \begin{equation}\label{evolution2} \left\{\begin{array}{ll} w_t - \bigtriangleup_p w + \alpha(x,w) = 0 &\ \text{in\ }\Omega\times(0, T)\\ w(x,t) = \sigma(x) &\ \text{on\ }\partial \Omega\times (0, T)\\ w(x,0) = v_0(x) &\ \text{for\ }x\in\bar{\Omega},\end{array}\right. \end{equation} where $T>0$ may be finite or infinite, and $\alpha$ is a continuous function satisfying $0 \leq \alpha(x,w) \leq Kw$ and \begin{equation*}
\left|\alpha(x,r_2) - \alpha(x,r_1)\right| \leq K\left|r_2 - r_1\right| \end{equation*} for all $x\in\Omega$, $r_1$ and $r_2\in\mathbb{R}$, and some $K \geq 0$. Let us introduce the notation $H_pw = w_t - \bigtriangleup_p w + \alpha(x,w)$. We recall a weak sub-solution $w\in L^2(0,T; W^{1,p}(\Omega))$ satisfies \begin{equation*}
\left.\int_Vw\varphi\ \right|_{t_1}^{t_2} + \int^{t_2}_{t_1}\int_V-w\varphi_t + |\nabla w|^{p-2}\nabla w\cdot\nabla\varphi + \alpha(x,w)\varphi \leq 0 \end{equation*} for any region $V\subset\subset\Omega$ and any test function $\varphi\in L^2(0,T; W^{1,p}(\Omega))$ such that $\varphi_t\in L^2(\Omega\times\mathbb{R}_T)$ and $\varphi\geq 0$ in $\Omega\times\mathbb{R}_T$, where $L^2_0(0,T; W^{1,p}(\Omega))$ is the subset of $L^2(0,T; W^{1,p}(\Omega))$ that contains functions which is equal zero on the boundary of $\Omega\times\mathbb{R}_T$, where $\mathbb{R}_T = [0, T]$. For convenience, we let $\mathfrak{T}_+$ denote this set of test functions in the following.
In particular, it holds that \begin{equation*}
\int^T_0\int_{\Omega}-w\varphi_t + <|\nabla w|^{p-2}\nabla w, \nabla\varphi> + \alpha(x,w)\varphi \leq 0 \end{equation*} for any test function $\varphi\in L^2_0(0,T; W^{1,p}(\Omega))$ such that $\varphi_t\in L^2(\Omega\times\mathbb{R}_T)$ and $\varphi\geq 0$ in $\Omega\times\mathbb{R}_T$.
The comparison principle for weak sub- and super-solutions is stated as follows. \begin{theorem}\label{paraboliccomparison} Suppose $w_1$ and $w_2$ are weak sub- and super-solutions of the evolutionary problem (\ref{evolution2}) respectively with $w_1\leq w_2$ on the parabolic boundary $(\bar{\Omega}\times\{0\}) \cup(\partial \Omega\times (0, +\infty))$. Then $w_1\leq w_2$ in $\mathcal{D}$. \end{theorem} Uniqueness of a weak solution of (\ref{evolution2}) follows from the comparison principle, Theorem \ref{paraboliccomparison}, immediately. \begin{lemma} For $T > 0$ small enough, if $H_pw_1 \leq 0 \leq H_pw_2$ in the weak sense in $\Omega\times \mathbb{R}_T$ and $w_1 < w_2$ on $\partial_p(\Omega \times \mathbb{R}_T)$, then $w_1\leq w_2$ in $\Omega\times\mathbb{R}_T$. \end{lemma} \begin{pf} For any given small number $\delta > 0$, we define a new function $\tilde{w}_1$ by $$\tilde{w}_1(x,t) = w_1(x,t) - \frac{\delta}{T-t},$$ where $x\in\bar{\Omega}$ and $0\leq t < T$. In order to prove $w_1\leq w_2$ in $\Omega\times\mathbb{R}_T$, it suffices to prove $\tilde{w}_1\leq w_2$ in $\Omega\times\mathbb{R}_T$ for all small $\delta > 0$. Clearly, $\tilde{w}_1 < w_2$ on $\partial_p(\Omega\times \mathbb{R}_T)$, and $\lim_{t\rightarrow T}\tilde{w}_1(x,t) = -\infty$ uniformly on $\Omega$. Moreover, the following holds for any $\varphi\in\mathfrak{T}_+$: \begin{alignat*}{1}
&\ \ \ \ \ \ \int^T_0\int_{\Omega}-\tilde{w}_1\varphi_t + <|\nabla\tilde{w}_1|^{p-2}\nabla\tilde{w}_1, \nabla\varphi> + \alpha(x,\tilde{w}_1)\varphi \\
&= \int^T_0\int_{\Omega}-w_1\varphi_t + <|\nabla w_1|^{p-2}\nabla w_1, \nabla\varphi> + \frac{\delta}{T-t}\varphi_t + \left(\alpha(x,\tilde{w}_1) - \alpha(x, w_1)\right)\varphi \\ &\leq \int^T_0\int_{\Omega}\frac{\delta}{T-t}\varphi_t + K\frac{\delta}{T-t}\varphi, \ \ \text{as $w_1$ is a weak sub-solution}\\ &= \int^T_0\int_{\Omega}\left(-\frac{\delta}{(T-t)^2} + K\frac{\delta}{T-t}\right)\varphi \\ &\leq \int^T_0\int_{\Omega} -\frac{\delta}{2(T-t)^2}\varphi,\ \ \text{for $T\leq\frac{1}{2K}$ so that $2K\leq \frac{1}{T-t}$} \\ &< 0, \end{alignat*} i.\,e.\, \begin{equation*} H_p\tilde{w}_1 \leq -\frac{\delta}{2(T-t)^2} \leq -\frac{\delta}{2T^2} < 0\ \ \text{in the weak sense.} \end{equation*}
That is, if we abuse the notation a little by denoting $\tilde{w}_1$ by $w_1$ in the following for convenience, it holds for any $\varphi\in\mathfrak{T}_+$, \begin{equation*}
\int^T_0\int_{\Omega}-w_1\varphi_t + <|\nabla w_1|^{p-2}\nabla w_1, \nabla\varphi> + \alpha(x,w_1)\varphi \leq \int^T_0\int_{\Omega}-\frac{\delta}{2T^2}\varphi < 0. \end{equation*} Meanwhile, for any $\varphi\in \mathfrak{T}_+$, $w_2$ satisfies \begin{equation*}
\int^T_0\int_{\Omega}-w_2\varphi_t + <|\nabla w_2|^{p-2}\nabla w_2, \nabla\varphi> + \alpha(x,w_2)\varphi \geq 0. \end{equation*}
Define, for $j = 1, 2$, $v_j(x,t) = e^{-\lambda t}w_j(x, t)$, where the constant $\lambda > 2K$. Then $w_j(x,t) = e^{\lambda t}v_j(x, t)$, and it is clear that $w_1\leq w_2$ in $\Omega\times\mathbb{R}_T$ is equivalent to $v_1\leq v_2$ in $\Omega\times\mathbb{R}_T$. In addition, for any $\varphi\in \mathfrak{T}_+$, the following inequalities hold: \begin{alignat*}{1}
&\int^T_0\int_{\Omega}-e^{\lambda t}v_1\varphi_t + e^{\lambda(p-1)t}<|\nabla v_1|^{p-2}\nabla v_1, \nabla\varphi> + \alpha(x,e^{\lambda t}v_1)\varphi \leq -\int^T_0\int_{\Omega}\frac{\delta}{2T^2}\varphi \\
&\text{and\ }\ \int^T_0\int_{\Omega}-e^{\lambda t}v_2\varphi_t + e^{\lambda(p-1)t}<|\nabla v_2|^{p-2}\nabla v_2, \nabla\varphi> + \alpha(x,e^{\lambda t}v_2)\varphi \geq 0. \end{alignat*} Consequently, it holds for any $\varphi\in \mathfrak{T}_+$ \begin{alignat*}{1}
&\int^T_0\int_{\Omega}-e^{\lambda t}(v_1 - v_2)\varphi_t + e^{\lambda(p-1)t}<|\nabla v_1|^{p-2}\nabla v_1 - |\nabla v_2|^{p-2}\nabla v_2, \nabla\varphi> \\
&\ \ \ \ \ \ \ \ + \left( \alpha(x, e^{\lambda t}v_1) - \alpha(x, e^{\lambda t}v_2)\right)\varphi \leq -\int^T_0\int_{\Omega}\frac{\delta}{2T^2}\varphi. \end{alignat*} We take $\varphi = \left(v_1 - v_2\right)^+ = \max\{v_1 - v_2, 0\}$ as the test function, since it vanishes on the boundary of $\Omega\times\mathbb{R}_T$. Then \begin{alignat*}{1}
&\int^T_0\int_{\{v_1>v_2\}}-e^{\lambda t}(v_1 - v_2)(v_1 - v_2)_t + e^{\lambda(p-1)t}<|\nabla v_1|^{p-2}\nabla v_1 - |\nabla v_2|^{p-2}\nabla v_2, \nabla v_1 - \nabla v_2> \\ &\ \ \ \ \ \ \ \ + \left(\alpha(x,e^{\lambda t}v_1) - \alpha(x,e^{\lambda t}v_2)\right)(v_1 - v_2) \leq -\frac{\delta}{2T^2}\int^T_0\int_{\{v_1>v_2\}}(v_1-v_2). \end{alignat*} Since \begin{equation*} \{v_1>v_2\}\subset\Omega\times(0,T)\ \ \text{due to the facts $v_1\leq v_2$ on $\partial_p(\Omega\times\mathbb{R}_T)$ and $v_1\rightarrow -\infty$ as $t\uparrow T$,} \end{equation*} the divergence theorem implies \begin{equation*} \int^T_0\int_{\{v_1 > v_2\}}-e^{\lambda t}(v_1-v_2)(v_1-v_2)_t = \int^T_0\int_{\{v_1>v_2\}}\lambda e^{\lambda t}\frac{1}{2}(v_1-v_2)^2. \end{equation*} On the other hand, \begin{equation*} \left(\alpha(x,e^{\lambda t}v_1) - \alpha(x,e^{\lambda t}v_2)\right)(v_1 - v_2)\geq -Ke^{\lambda t}(v_1-v_2)^2\ \ \text{on $\{v_1 > v_2\}$.} \end{equation*} As a consequence, it holds that \begin{alignat*}{1}
&\int^T_0\int_{\{v_1>v_2\}}\left(\frac{\lambda}{2} - K\right)e^{\lambda t}(v_1 - v_2)^2 + e^{\lambda(p-1)t}<|\nabla v_1|^{p-2}\nabla v_1 - |\nabla v_2|^{p-2}\nabla v_2, \nabla v_1 - \nabla v_2> \\ & \leq -\frac{\delta}{2T^2}\int^T_0\int_{\{v_1>v_2\}}(v_1-v_2). \end{alignat*} We call into play two elementary inequalities (\cite{L}) associated with the $p$-Laplacian: \begin{equation*}
<|b|^{p-2}b - |a|^{p-2}a, b-a> \geq (p-1)|b-a|^2\left(1 + |b|^2 + |a|^2\right)^{\frac{p-2}{2}}\ \ (1\leq p\leq 2), \end{equation*} and \begin{equation*}
<|b|^{p-2}b - |a|^{p-2}a, b-a> \geq 2^{2-p}|b-a|^p\ \ (p\geq 2)\ \ \text{for any $a$, $b\in\mathbb{R}^n$.} \end{equation*} By applying them with $b = \nabla v_1$ and $a = \nabla v_2$ in the preceding inequalities, we obtain \begin{alignat*}{1}
&\int^T_0\int_{\{v_1>v_2\}}\left(\frac{\lambda}{2} - K\right)e^{\lambda t}(v_1 - v_2)^2 + (p-1)e^{\lambda(p-1)t}\left|\nabla v_1 - \nabla v_2\right|^2\left( 1 + |\nabla v_1|^2 + |\nabla v_2|^2\right)^{\frac{p-2}{2}} \\ & \leq -\frac{\delta}{2T^2}\int^T_0\int_{\{v_1>v_2\}}(v_1-v_2)\ \ \ \ \text{for $1 < p <2$} \end{alignat*} and \begin{alignat*}{1}
&\int^T_0\int_{\{v_1>v_2\}}\left(\frac{\lambda}{2} - K\right)e^{\lambda t}(v_1 - v_2)^2 + 2^{2-p}e^{\lambda(p-1)t}\left|\nabla v_1 - \nabla v_2\right|^p \\ & \leq -\frac{\delta}{2T^2}\int^T_0\int_{\{v_1>v_2\}}(v_1-v_2)\ \ \ \ \text{for $p\geq 2$.} \end{alignat*} One can easily see in either case the respective inequality is true only if the measure of the set $\{v_1>v_2\}$ is zero. The proof is complete. \end{pf}
In the next lemma, we show the strict inequality on the boundary data can be relaxed to a non-strict one. \begin{lemma} For $T > 0$ sufficiently small, if $H_pw_1 \leq 0 \leq H_pw_2$ in the weak sense in $\Omega\times\mathbb{R}_T$ and $w_1\leq w_2$ on $\partial_p(\Omega \times \mathbb{R}_T)$, then $w_1\leq w_2$ on $\overline{\Omega\times\mathbb{R}_T}$. \end{lemma} \begin{pf} For any $\delta > 0$, take $\tilde{\delta} > 0$ such that $\tilde{\delta} \leq\frac{\delta}{4K}$ and define \begin{equation*} \tilde{w}_1(x,t) = w_1(x,t) - \delta t - \tilde{\delta}\ \ (x,t)\in\bar{\Omega}\times\mathbb{R}^n. \end{equation*} Then $\tilde{w}_1 < w_1 \leq w_2$ on $\partial_p(\Omega\times\mathbb{R}^n)$, and for any $\varphi\in\mathfrak{T}_+$, the following holds: \begin{alignat*}{1}
&\ \ \ \ \int^T_0\int_{\Omega}-\tilde{w}_1\varphi_t + <|\nabla\tilde{w}_1|^{p-2}\nabla\tilde{w}_1,\nabla\varphi> + \alpha(x,\tilde{w})\varphi \\
&= \int^T_0\int_{\Omega}-w_1\varphi_t + <|\nabla w_1|^{p-2}\nabla w_1,\nabla\varphi> + \alpha(x,w_1)\varphi\\ &\ \ \ \ \ \ \ \ \ \ - \delta \varphi + \left(\alpha(x,w_1 - \delta t - \tilde{\delta}) - \alpha(x,w_1)\right)\varphi \\ &\leq \int^T_0\int_{\Omega} -\delta\varphi + K\left(\delta t + \tilde{\delta}\right)\varphi \leq \int^T_0\int_{\Omega} -\delta\varphi + K\left(\delta T + \tilde{\delta}\right)\varphi \\ &\leq \int^T_0\int_{\Omega}\left(-\delta + \frac{\delta}{2} + \frac{\delta}{4}\right)\varphi\ \ \text{for $T$ small} \\ &= -\frac{\delta}{4}\int^T_0\int_{\Omega}\varphi. \end{alignat*} The preceding lemma implies $\tilde{w}_1\leq w_2$ in $\overline{\Omega\times\mathbb{R}_T}$ for small $T$ and for any small $\delta > 0$, and whence the conclusion of this lemma. \end{pf}
Now the parabolic comparison theorem (\ref{paraboliccomparison}) follows from the preceding lemma quite easily as shown by the following argument: Let $T_0 > 0$ be any small value of $T$ in the preceding lemma so that the conclusion of the preceding lemma holds. Then $w_1\leq w_2$ on $\overline{\Omega\times (0, T_0)}$. In particular, $w_1 \leq w_2$ on $\partial_p(\Omega\times (T_0,2T_0))$. The preceding lemma may be applied again to conclude that $w_1 \leq w_2$ on $\overline{\Omega\times (T_0, 2T_0)}$. And so on. This recursion allows us to conclude that $w_1 \leq w_2$ on $\overline{\Omega\times\mathbb{R}_T}$.
\section{Convergence of Evolution}
Define $\mathfrak{S}$ to be the set of weak solutions of the stationary problem (\ref{eulereq}) and (\ref{bdrycondition}). The $p$-harmonic function $u_0$ is the maximum element in $\mathfrak{S}$, while $u_2$ denotes the least solution which may be constructed as the infimum of super-solutions. We also use the term \textit{non-minimal solution} with the same definition in \cite{CW}. That is, $u$ a non-minimal solution of the problem (\ref{eulereq}) and (\ref{bdrycondition}) if it is a viscosity solution but not a local minimizer in the sense that for any $\delta > 0$, there exists $v$ in the admissible set of the functional $J_p$ with $v = \sigma$ on $\partial\Omega$ such that
$\|v - u\|_{L^{\infty}} < \delta$, and $J_p(v) < J_p(u)$.
In this section, we consider the evolutionary problem \begin{equation}\label{evolution} \left\{\begin{array}{ll} w_t - \bigtriangleup_p w + Q(x)\beta_{\varepsilon}(w) = 0 &\ \text{in\ }\Omega\times(0,+\infty)\\ w(x,t) = \sigma(x) &\ \text{on\ }\partial \Omega\times (0, +\infty)\\ w(x,0) = v_0(x) &\ \text{for\ }x\in\bar{\Omega},\end{array}\right. \end{equation} and will apply the parabolic comparison principle (\ref{paraboliccomparison}) proved in Section \ref{comparison} to prove the following convergence of evolution theorem. One just notes that the parabolic problem (\ref{evolution2}) includes the above problem (\ref{evolution}) as a special case so that the comparison principle (\ref{paraboliccomparison}) applies in this case. \begin{theorem}\label{convergence} If the initial data $v_0$ falls into any of the categories specified below, the corresponding conclusion of convergence holds. \begin{enumerate} \item If $v_0 \leq u_2$ on $\bar{\Omega}$, then $\lim_{t\rightarrow +\infty}w(x,t) = u_2(x)$ locally uniformly for $x\in\bar{\Omega}$; \item Define \begin{equation*} \bar{u}_2(x) = \inf_{u\in\mathfrak{S}, u \geq u_2, u \neq u_2}u(x),\ x\in\bar{\Omega}. \end{equation*} If $\bar{u}_2 > u_2$, then for $v_0$ such that $u_2 < v_0 < \bar{u}_2$, $\lim_{t\rightarrow +\infty}w(x,t) = u_2(x)$ locally uniformly for $x\in\bar{\Omega}$; \item Define $\bar{u}_0(x) = \sup_{u\in\mathfrak{S}, u \leq u_0, u\neq u_0}u(x)$, $x\in\bar{\Omega}$. If $\bar{u}_0 < u_0$, then for $v_0$ such that $\bar{u}_0 < v_0 < u_0$, $\lim_{t\rightarrow +\infty}w(x,t) = u_0(x)$ locally uniformly for $x\in\bar{\Omega}$; \item If $v_0 \geq u_0$ in $\bar{\Omega}$, then $\lim_{t\rightarrow +\infty}w(x,t) = u_0(x)$ locally uniformly for $x\in\bar{\Omega}$;
\item Suppose $u_1$ is a non-minimal solution of (\ref{eulereq}) and (\ref{bdrycondition}). For any small $\delta > 0$, there exists $v_0$ such that $\|v_0 - u_1\|_{L^{\infty}(\Omega)} < \delta$ and the solution $w$ of the problem (\ref{evolution}) does not satisfy $$\lim_{t\rightarrow \infty} w(x,t) = u_1(x)\ \text{\ in\ } \Omega.$$ \end{enumerate} \end{theorem} \begin{pf}
We first take care of case 4. We may take new initial data a smooth function $\tilde{v}_0$ so that $D^2\tilde{v}_0 < -KI$ and $|\nabla \tilde{v}_0| \geq \delta > 0$ on $\bar{\Omega}$. According to the parabolic comparison principle (\ref{paraboliccomparison}), it suffices to prove the solution $\tilde{w}$ generated by the initial data $\tilde{v}_0$ converges locally uniformly to $u_0$ if we also take $\tilde{v}_0$ large than $v_0$, which can easily be done. So we use $v_0$ and $w$ for the new functions $\tilde{v}_0$ and $\tilde{w}$ without any confusion.
For any $V\subset\subset\Omega$ and any nonnegative function $\varphi$ which is independent of the time variable $t$ and supported in $V$, it holds that \begin{equation*} \begin{split}
\int_V|\nabla v_0|^{p-2}\nabla v_0\cdot\nabla\varphi &= \int_V - div\left(|\nabla v_0|^{p-2}\nabla v_0\right)\varphi \\ &\geq \int_V M\varphi\ \ \ \ \text{for some $M = M(n,p,K,\delta) > 0$.} \end{split} \end{equation*} The H\"{o}lder continuity of $\nabla w$ up to $t = 0$ as stated in \cite{DiB}, then implies \begin{equation*}
\int_V|\nabla w|^{p-2}\nabla w\cdot\nabla\varphi \geq \frac{M}{2}\int_V\varphi \end{equation*} for any small $t$ in $(0,t_0)$, and any nonnegative function $\varphi$ which is independent of $t$, supported in $V$ and subject to the condition \begin{equation}\label{conda}
\frac{\int_V|\nabla \varphi|}{\int_V\varphi}\leq A \end{equation} for a fixed constant $A > 0$ and some $t_0 > 0$ dependent on $A$. Then the sub-solution condition on $w$ \begin{equation*}
\left.\int_Vw\varphi\right|_{t=t_2} - \left.\int_Vw\varphi\right|_{t=t_1} + \int^{t_2}_{t_1}\int_V|\nabla w|^{p-2}\nabla w\cdot\nabla\varphi \leq 0 \end{equation*} implies that \begin{equation*}
\left.\int_Vw\varphi\right|_{t=t_2} - \left.\int_Vw\varphi\right|_{t=t_1} \leq -\frac{M}{2}(t_2-t_1)\int_V\varphi \end{equation*}
for any small $t_2 > t_1$ in $(0, t_0)$, and any nonnegative function $\varphi$ which is independent of $t$, supported in $V$ and subject to (\ref{conda}). In particular, $\left.\int_Vw\varphi\right|^{t_2}_{t_1} \leq 0$ for any nonnegative function $\varphi$ independent of $t$, supported in $V$ and subject to (\ref{conda}). So $$w(x, t_2) \leq w(x, t_1)$$ for any $x\in\Omega$ and $0\leq t_1\leq t_2$. Then the parabolic comparison principle readily implies $w$ is decreasing in $t$ for $t$ in $[0, \infty)$. Therefore $w(x,t)\rightarrow u^{\infty}(x)$ locally uniformly as $t\rightarrow\infty$ and hence $u^{\infty}$ is a solution of (\ref{eulereq}) and (\ref{bdrycondition}). Furthermore, the parabolic comparison principle also implies $w(x,t)\geq u_0(x)$ at any time $t > 0$. Consequently, $u^{\infty} = u_0$ as $u_0$ is the greatest solution of (\ref{eulereq}) and (\ref{bdrycondition}).
Next, we briefly explain the proof for case 1. We may take a new smooth initial data $\tilde{v}_0$ such that $\tilde{v}_0$ is very large negative, $D^2\tilde{v}_0 \geq KI$ and $|\nabla \tilde{v}_0| \geq\delta$ on $\bar{\Omega}$ for large constant $K>0$ and constant $\delta > 0$. It suffices to prove the solution $\tilde{w}$ generated by the initial data $\tilde{v}_0$ converges to $u_2$ locally uniformly on $\bar{\Omega}$ as $t\rightarrow\infty$. Following a computation exactly parallel to that in case 4, we can prove $w$ is increasing in $t$ in $[0, \infty)$. So $w$ converges locally uniformly to a solution $u^{\infty}$ of (\ref{eulereq}) and (\ref{bdrycondition}). As $u^{\infty}\leq u_2$ and $u_2$ is the least solution of (\ref{eulereq}) and (\ref{bdrycondition}), we conclude $u^{\infty} = u_2$.
In case 2, we may replace $v_0$ by a strict super-solution of $\bigtriangleup_p v - Q\beta_{\varepsilon}(v) = 0$ in $\bar{\Omega}$ between $u_2$ and $\bar{u}_2$, by employing the fact that $u_2$ is the infimum of super-solutions of (\ref{eulereq}) and (\ref{bdrycondition}). Using $v_0$ as the initial data, we obtain a solution $w(x,t)$ of (\ref{evolution}). Then one argues as in case 4 that for any $V\subset\subset\Omega$, there exist constants $A > 0$ and $t_0 > 0$ such that for $t_1 < t_2$ with $t_1$, $t_2\in [0, t_0)$, $\int_Vw\varphi\,|^{t_2}_{t_1} \leq 0$ for any nonnegative function $\varphi$ independent of $t$, supported in $V$ and subject to the condition $\frac{\int_V|\nabla\varphi|}{\int_V\varphi} \leq A$. As a consequence, $w(x,t_1) \geq w(x,t_2)\ \ (x\in\Omega)$. Then the parabolic comparison principle implies $w$ is decreasing in $t$ over $[0, +\infty)$. Therefore $w(x,t)$ converges locally uniformly to some function $u^{\infty}$ as $t\rightarrow\infty$ which solves (\ref{eulereq}) and (\ref{bdrycondition}). Clearly $u_2(x) \leq w(x,t) \leq \bar{u}_2(x)$ from which $u_2(x) \leq u^{\infty}(x) \leq \bar{u}_2(x)$ follows. As $w$ is decreasing in $t$ and $v_0\neq \bar{u}_2$, $u^{\infty} \neq \bar{u}_2$. Hence $u^{\infty} = u_2$.
The proof of case 3 is parallel to that of case 2 with the switch of sub- and super-solutions. Hence we skip it.
In case 5, we pick $v_0$ with $\|v_0 - u_1\|_{L^{\infty}} < \delta$ and $J_p(v_0) < J_p(u_1)$. Let $w$ be the solution of (\ref{evolution}) with $v_0$ as the initial data. Clearly, we may change the value of $v_0$ slightly if necessary so that it is not a solution of the equation
$$-\nabla\cdot\left(\left(\varepsilon + |\nabla u|^2\right)^{p/2-1}\nabla u\right) + Q(x)\beta(u) = 0$$ for any small $\varepsilon > 0$.
Let $w^{\varepsilon}$ be the smooth solution of the uniformly parabolic boundary-value problem \begin{equation*} \left\{\begin{array}{ll}
w_t - \nabla\cdot\left(\left(\varepsilon + |\nabla w|^2\right)^{p/2-1}\nabla w\right) + Q\beta(w) = 0 &\ \ \text{in $\Omega\times (0, +\infty)$}\\ w(x,t) = \sigma(x) &\ \ \text{on $\partial\Omega\times (0, +\infty)$}\\ w(x,0) = v_0(x) &\ \ \text{on $\bar{\Omega}$.} \end{array}\right. \end{equation*} $w^{\varepsilon}$ converges to $w$ in $W^{1,p}(\Omega)$ for every $t\in [0, \infty)$ as $\varepsilon\rightarrow 0$.
We define the functional
$$J_{\varepsilon, p}(u) = \frac{1}{p}\int_{\Omega}\left(\varepsilon + |\nabla u|^2\right)^{p/2} + Q(x)\Gamma(u)\,dx.$$ It is easy to see that
$$\int^t_0\int_{\Omega}\left(w^{\varepsilon}_t\right)^2 - \nabla\cdot\left(\left(\varepsilon + |\nabla w^{\varepsilon}|^2\right)^{p/2-1}\nabla w^{\varepsilon}\right)w^{\varepsilon}_t + Q\beta(w^{\varepsilon})w^{\varepsilon}_t = 0.$$ As $w^{\varepsilon}_t = 0$ on $\partial\Omega\times (0,\infty)$, one gets
$$\int^t_0\int_{\Omega}\left(w^{\varepsilon}_t\right)^2 + \left(\varepsilon + |\nabla w^{\varepsilon}|^2\right)^{p/2-1}\nabla w^{\varepsilon}\cdot \nabla w^{\varepsilon}_t + Q(x)\Gamma(w^{\varepsilon})_t = 0,$$ which implies
$$\int^t_0\int_{\Omega}\left(w^{\varepsilon}_t\right)^2 + \frac{1}{p}\left(\left(\varepsilon + |\nabla w^{\varepsilon}|^2\right)^{p/2}\right)_t + Q(x)\Gamma(w^{\varepsilon})_t = 0.$$ Consequently, it holds \begin{equation*} \begin{split}
&\ \ \ \ \int^t_0\int_{\Omega}\left(w^{\varepsilon}_t\right)^2 + \frac{1}{p}\int_{\Omega}\left(\varepsilon + |\nabla w^{\varepsilon}(x,t)|^2\right)^{p/2} + Q\Gamma(w^{\varepsilon}(x,t)) \\
&= \frac{1}{p}\int_{\Omega}\left(\varepsilon + |\nabla w^{\varepsilon}(x,0)|^2\right)^{p/2} + Q\Gamma(w^{\varepsilon}(x,0)) \end{split} \end{equation*} i.\,e.\, \begin{equation*} \int^t_0\int_{\Omega}\left(w^{\varepsilon}_t\right)^2 + J_{\varepsilon, p}(w^{\varepsilon}(\cdot,t)) = J_{\varepsilon, p}(w^{\varepsilon}(\cdot,0)). \end{equation*} Therefore $$J_{\varepsilon, p}(w^{\varepsilon}(\cdot, t) \leq J_{\varepsilon, p}(v_0),$$ which in turn implies $$J_p(w(\cdot, t) \leq J_p(v_0) < J_p(u_1).$$ In conclusion, $w$ does not converge to $u_1$ as $t\rightarrow\infty$. \end{pf}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document} \sloppy \title{Counting Words Avoiding a Short Increasing Pattern and the Pattern 1k\dots2}
\begin{abstract} We find finite-state recurrences to enumerate the words on the alphabet $[n]^r$ which avoid the patterns 123 and $1k(k-1)\dots2$, and, separately, the words which avoid the patterns 1234 and $1k(k-1)\dots2$. \end{abstract}
\section{Introduction} A word $W = w_1w_2 \dots w_n$ on an ordered alphabet \emph{contains} the pattern $p_1p_2\dots p_k$ if there exists a (strictly) increasing sequence $i_1,i_2,\dots, i_k$ such that $w_{i_r} < w_{i_s}$ if and only if $p_r < p_s$ and $w_{i_r} > w_{i_s}$ if and only if $p_r > p_s$. If both $W$ and $p_1p_2 \dots p_k$ are permutations, then $w_{i_r} < w_{i_s}$ if and only if $p_r < p_s$ is an equivalent and more common definition. If $W$ does not contain $p_1p_2 \dots p_k$, then $W$ \emph{avoids} it. The study of pattern-avoiding permutations began with Donald Knuth in \textit{The Art of Computer Programming}, and has become an active area of combinatorial research. See \cite{Vatter} for an in-depth survey of the major results in this field.
The study of pattern-avoiding words other than permutations is comparatively recent, being inaugurated in \cite{Regev} and greatly expanded in \cite{Burstein}. Much is known about avoidance properties for specific patterns and families of patterns; for instance, Burstein counted the number of length-$n$ words with letters in $[k]=\{1,2,\dots,k\}$ which avoid all patterns in $S$ for all $S \subseteq S_3$. Meanwhile, \cite{Mansour} found generating functions for the number of such words which avoid both 132 and one of a large family of other patterns including $12\dots l$ and $l12\dots(l-1)$.
Other authors have looked at words with letters in $[n]$ where each letter must appear exactly $r$ times (we will say that these are the words on $[n]^r$). These words are a direct generalization of permutations, which are given by the $r=1$ case. In \cite{Shar}, the authors created an algorithm to find the ordinary generating functions enumerating words on $[n]^r$ which avoid 123, while \cite{Zeil1d} found that the generating functions enumerating words on $[n]^r$ avoiding $12\dots l$ are D-finite.
We study this second type of word. Our contribution is to find finite, linear recurrences for the numbers of words on $[n]^r$ that avoid 123 and $1k(k-1)\dots2$ as well as the ones that avoid 1234 and $1k(k-1)\dots2$. It is well known (see \cite{Cfinite} for instance) that this fact implies that these quantities have rational generating functions, and, moreover, gives a way to compute them (in principle if not always in practice - see Section \ref{compute}). While generating functions were previously found in \cite{Kratten} for the permutations avoiding 123 and $1k(k-1)\dots2$, this is the first time that such a result has been extended to these more general words. In the 1234 and $1k(k-1)\dots2$ case, the result was, to the best of our knowledge, previously not even known for permutations with $k$ as small as 5.
\section{Words Avoiding 123}\label{123} We begin this section with an algorithm for counting 123 avoiding permutations from \cite{DrZ}. For $L = [l_1,l_2,\dots l_n]$, let $A(L)$ be the number of words containing $l_i$ copies of $i$ for $1 \le i \le n$ which avoid 123. The following result allows us to quickly compute $A(L)$. \begin{thm} The following recurrence holds: \begin{displaymath} A(L) = \sum_{i=1}^n A([l_1,l_2,\dots,l_{i-1}, l_i-1, l_{i+1}+l_{i+2}+\dots+l_n]). \end{displaymath} \end{thm} \begin{proof} Let $A_i(L)$ be the number of words with letter counts $l_1,l_2,\dots,l_n$ which avoid 123 and begin with the letter $i$. We will biject the words counted by $A_i(L)$ with those counted by $A([l_1,l_2,\dots,l_{i-1}, l_i-1, l_{i+1}+l_{i+2}+\dots+l_n])$. Let $W=iw_2,\dots,w_t$ have letter counts in $L$ and let $f(W)$ be given by removing the initial $i$ from $W$ and then replacing all letters greater than $i$ with $i+1$. Also, for some word $V = v_1v_2,\dots,v_{t-1}$ with letter counts in $[l_1,l_2,\dots,l_{i-1},l_i-1,l_{i+1}+\dots+l_n]$, let $f^{-1}$ be given by replacing the sequence of $i+1$'s with $l_{n} \text{ }n$'s, $l_{n-1} \text{ }n-1$'s, and so on in that order, and then prepending $i$ to this word.
We claim that $f^{-1}$ is the inverse of $f$. To find $f(f^{-1}(V))$, we would replace the sequence of $i+1$s with $l_{n} \text{ }n$'s, $l_{n-1} \text{ }n-1$'s, and so on, and then prepend an $i$, before removing that $i$ and replacing all those letters larger than $i$ with $i+1$ again, giving us back $V$. To find $f^{-1}f(W)$, we would replace all the letters larger than $i$ with $i+1$ and remove the initial $i$, before replacing that $i$ and putting back all the letters larger than $i$ (note that they had to be in descending order to begin with or else $W$ would contain a 123 pattern). Thus, $A_i(L)=A([l_1,l_2,\dots,l_{i-1}, l_i-1, l_{i+1}+l_{i+2}+\dots+l_n])$ and summing over all $i$ gives the promised equality. \end{proof}
This technique can be extended to many more avoidance classes. In this section we use it to count words avoiding both $123$ and $1k(k-1)\dots2$ simultaneously. We first fix $k \ge 3$, choose integers $n$ and $r$, and consider the number of words on the alphabet $[n]^r$ which avoid both $123$ and $1k(k-1)\dots2$. This time, however, we will need to keep track of more information than just the letter counts. To that end, we consider the set of words $A(r,a,b,L)$ where $r,a,$ and $b$ are integers, and $L=[l_1,l_2,\dots,l_t]$. This is the number of words with $r$ copies of the letters $1, \dots, a$, $b$ copies of the letter $a+1$, and $l_i$ copies of the letter $a+1+i$ which not only avoid both 123 and $1k(k-1)\dots2$, but would still avoid both those patterns if $a+1$ were prepended to the word. Note that this condition implies $t \le k-2$ because any sequence of $k-1$ distinct letters will either contain an increasing subsequence of length 2 (and hence create a 123 pattern) or will be entirely decreasing (and hence create a $1k(k-1)\dots2$ pattern).
Before we state the next theorem, we describe in more human-friendly language the algorithm that it suggests. Suppose that we are building a word $W$. Up to this point, the smallest letter which has been used is $a+1$, and $r-b$ copies of it have been used. We have a list $L=[l_1,l_2,\dots l_t]$ indicating how many copies of each letter greater than $a+1$ remain to be added, and we note that all these letters must be added in reverse order. To complete $W$, we need to add $r$ copies each of $1,2,\dots,a$, $b$ copies of $a+1$, and $l_i$ copies of $a+1+i$ for all $1 \le i \le t$. Examine $W$'s next letter $w_1$; considering only the requirement that $w_1$ be succeeded by at most $k-2$ distinct letters larger than it, we find that $w_1$ can be any element of $\{a+2, a+3,\dots,a+1+t\}$ or else it can be an element of $\{a-(k-2)+t+1, a-(k-2)+t+2, \dots, a+1\}$. But, we also need to consider the requirement that prepending $a+1$ to the new word will not create a 123 pattern. Therefore, $w_1 \in \{a-(k-2)+t+1, a-(k-2)+t+2, \dots, a+1, a+1+t\}$. If $w_1 = a+1+t$ or $a+1$, then removing it gives a word counted by $A(r,a,b,L')$ or $A(r,a,b-1,L)$ respectively where $L' = [l_1,\dots,l_{t-1},l_t -1]$. Otherwise, we need to add all the letters larger than $w_1$ to $L$ in order to ensure that future letters don't create 123 patterns.
Since we want $L$ to contain only letter counts for letters which will be added to $W$, i.e. we don't want it to contain 0, define the operator $R$ which removes all the zeroes from the list $L$.
\begin{thm} If $b \ge 1$, then \begin{align*} A(r,a,b,L) &= \sum_{i = a-(k-2)+t+1}^{a} A(r,i-1,r-1,[\underbrace{r,r,\dots,r,}_{a-i \text{ copies}} b, l_1,\dots,l_t]) \\ &+ A(r,a,b-1,L) + A(r,a,b,R([l_1,l_2,\dots, l_t -1])). \end{align*} If $b =0$, then \begin{align*} A(r,a,b,L) &= \sum_{i = a-(k-2)+t+1}^{a} A(r,i-1,r-1,[\underbrace{r,r,\dots,r,}_{a-i \text{ copies}} l_1,\dots,l_t]) \\ &+A(r,a,b,R([l_1,l_2,\dots, l_t -1])). \end{align*} \end{thm}
\begin{proof} As noted in the previous paragraph, $w_1 \in \{a - (k-2) +t +1, a-(k-2)+t+2,\dots,a+1,a+1+t\}$; each member of this set corresponds to a term of the summation. Fix $i$ with $a-(k-2)+t+1 \le i \le a$, and consider those words $W$ with $w_1 =i$. Suppose we remove $w_1$ from one of these words to form a word $W'$. We are left with $r$ copies of the letters 1 through $i-1$, $r-1$ copies of $i$ (because $i \le a$ there were $r$ copies of it including $w_1$), and $l'_j$ copies of $(i-1)+1+j$ where $L'=[l'_1,\dots,l'_u]=[\underbrace{r,r,\dots,r}_{a-i \text{ copies}},b,l_1,\dots,l_t]$. We are left with a word which avoids 123 and $1k(k-1)\dots2$, and, moreover, avoids 123 even when $i$ is prepended. Furthermore, prepending an $i$ to any word fitting this description gives a word counted by $A(r,a,b,L)$, and so the number of words counted by $A(r,a,b,L)$ which begin with $i$ is $A(r,i-1,r-1,[\underbrace{r,r,\dots,r,}_{a-i \text{ copies}} b, l_1,\dots,l_t])$ for all $a-(k-1)+t+1 \le i \le a$.
This leaves two other possibilities for $w_1$: $a+1$ and $a+t+1$. If $w_1 = a+1$, then the only difference between the letter counts of $W$ and $W'$ is that $W$ has $b$ copies of $a+1$ and $W'$ has only $b-1$. In terms of avoidance, both $W$ and $W'$ avoid 123 and $1k(k-1)\dots2$ even with $a+1$ prepended. Thus, the number of $W$ with $w_1=a+1$ is $A(r,a,b-1,L)$ as long as $b \ge 1$, and, 0 if $b=0$.
Similarly, if $w_1=a+t+1$, then the only difference between the letter counts of $W$ and $W'$ is that $W$ has $l_t$ copies of $a+t+1$ and $W'$ has $a+t$. Just as in the previous case, the avoidance properties are identical and so the number of $W$ with $w_1=a+t+1$ is $A(r,a,b,R([l_1,l_2,\dots,l_t-1]))$ where we needed to remove $l_t-1$ if it is zero so that we know that the next letter is allowed to be $l_{t-1}$.
Summing over all possible $w_1$ now gives the promised result.
\end{proof}
\section{Words Avoiding 1234}\label{1234} Just as we can find recurrences, and therefore generating functions, for words avoiding 123 and $1k(k-1)\dots2$, we can (in principle at least) find a similar system of recurrences and generating functions for words on $[n]^r$ avoiding 1234 and $1k(k-1)\dots2$. The idea is to construct a word $W$ one letter at a time, and with each letter see if we have made a forbidden pattern. Unfortunately, doing this naively would require keeping track of all previous letters in $W$, denying us a finite recurrence. By only paying attention to the letters that could actually contribute to a forbidden pattern, though, we find that we actually only need to retain a bounded quantity of information regarding $W$.
\subsection{The Existence of a Finite Recurrence} In order to discuss the structure of words avoiding 1234 and $1k(k-1)\dots2$, we recall one common definition and introduce some new ones. A \emph{left-to-right minimum} (LTR min) is a letter of a word which is (strictly) smaller than all the letters which precede it. To an LTR min, we associate an \emph{activated sequence} which consists of all the letters following the LTR min which are (again strictly) larger. Notice that, since 1234 is forbidden, anytime a letter $w$ is preceded by some smaller letter, all the letters larger than and following $w$ must occur in reverse order. We call these letters \emph{fixed}. If all the letters greater than an LTR min are fixed, then it is either guaranteed or impossible that the LTR min and its activated sequence form a $1k(k-1)..2$ pattern; in this case we say that the activated sequence has been \emph{deactivated} and we no longer consider it an activated sequence. If an LTR min with an empty activated sequence is followed by another LTR min (or another copy of itself), then any forbidden pattern using the first LTR min could also be made using the second LTR min; we say that the first LTR min is \emph{superceded} and no longer consider it an LTR min.
With these definitions, we are nearly ready to state the actual set we will be recursively enumerating. Let $r,k,$ and $a$ be integers, let $\mathcal{S}=[S_1=[s_{1,1},\dots,s_{1,q_1}],\dots, S_u=[s_{u,1},\dots,s_{q_u}]]$ be a list of lists whose elements are in $[t]$, let $M=[m_1,\dots,m_u]$ be a list with elements in $[t]$, and let $L=[l_1,\dots,l_t]$ be a list with elements in $\{0\} \cup [r]$. Suppose we are building a word, and so far the letters $1,\dots,a$ have never been used, while the letters greater than $a+t+1$ have been entirely used up and, moreover, are not LTR mins or in any activated sequence. Suppose this word has LTR mins $m_1+a,\dots,m_s+a$ with corresponding activated sequences $[s_{1,1}+a,\dots,s_{1,q_1}+a],\dots,[s_{u,1}+a,\dots,s_{u,q_u}+a]$ (excluding LTR mins which have been superceded or whose sequences have been deactivated). Finally, assume that the word so far avoids 1234 and $1k(k-1)\dots2$, and that $L_i$ copies of $a+i$ remain to be placed for all $1 \le i \le t$ (recall that $r$ copies of $1,\dots,a$ and 0 copies of $a+t+1,a+t+2,\dots$ remain to be placed). Then, the number of ways to completing the word is defined to be $A(r,k,a,M,\mathcal{S},L)$.
Our plan is to show that (i) $A$ is well defined, (ii) all arguments of $A$ besides $a$ take on finitely many values, and (iii) $A$ satisfies a recurrence in which a particular non-negative function of its arguments is always reduced (until the base case). Before carrying out this plan, though, we provide an example to make sure our definitions are clear.
\begin{eg}\label{ex} Suppose we are building a word on the alphabet $[9]^2$ to avoid 1234 and 15432, and so far have 69945. The LTR mins are 6 and 4 with corresponding activated strings 99 and 5. However, 99 has been deactivated because all the letters greater than its LTR min are fixed and must occur in decreasing order. Thus, the number of ways to complete this word is given by $A(2,5,3,[1],[[2]],[1,1,1,2,2]).$ \end{eg}
Notice that the number of ways is also given by $A(2,5,2,[2],[[3]],[2,1,1,1,2,2])$. While this is not a problem in principle, it would be nice to have a canonical way of expressing this quantity, and so we will eventually insist that $L$ have a particular length given by a function of $r$ and $k$.
\begin{thm}\label{well-defined} $A$ is well defined. \end{thm} \begin{proof} Suppose that $W_1$ and $W_2$ are two partial words that give the same arguments to $A$.
Given the LTR mins of a partial word, it is easy to see in which order they occurred. It is similarly easy to see in which order the elements of activated sequences occurred, since they are all listed in order in the activated sequence corresponding to the first LTR min (any element of some other activated sequence lower than or equal to the first LTR min would fix that LTR min's activated sequence and thus deactivate it). Finally, we can see how the sequence of LTR mins and the sequence of other elements are interweaved by noting that a non-LTR min occurs after an LTR min if and only if it appears in that min's activated sequence. Therefore, the subwords formed by the LTR mins and activated strings of $W_1$ and $W_2$ are identical.
Suppose $A$ is not well-defined; then there is some string which can be added to (without loss of generality) $W_1$ without creating a forbidden pattern, but which does create a forbidden pattern when added to $W_2$. By the argument of the previous paragraph, there is an element of $W_2$ which is neither an LTR min nor part of an activated sequence, but which does participate in this pattern.
But, this is not possible. Every element in $W_2$ is an LTR min, part of an activated sequence, fixed, or a copy of the LTR min immediately preceding it. We have assumed that the first two cases do not hold. The third case similarly cannot hold because when an element is fixed, so are all the elements larger than its LTR min, which is to say all the elements which could conceivably be part of a forbidden pattern with it. Therefore, every fixed element either must participate in a forbidden pattern or it cannot possibly do so. Finally the fourth case cannot hold because any forbidden pattern involving a copy of the immediately preceding LTR min could also be formed with that LTR min. Thus we have a contradiction. \end{proof}
Next, we want to establish bounds on $s$ and $t$ as well on the number of elements in any $S_i$. These bounds should depend only on $r$ and $k$.
\begin{thm}
Bounds for $t$, $u$, and all $|S_i|$ are as follows: $t \le 6(k-2)+2$, $u \le 2r(k-2)+1$, and $|S_i| \le 2r(k-2)$ for all $1 \le i \le u$. \end{thm}
\begin{proof} Recall that the Erd\H os-Szekeres Theorem states that any sequence of distinct real numbers of length $(p-1)(q-1)+1$ must contain either a length$-p$ increasing sequence or a length$-q$ decreasing sequence (see \cite{ESz}). Since every activated sequence must avoid both 123 and $(k-1)(k-2)\dots1$, it follows that no activated sequence can have more than $2(k-2)$ distinct letters. Since there are at most $r$ copies of any single letter, the longest an activated sequence could possibly be is $2r(k-2)$. Each activated sequence must either have some element that the next one lacks or correspond to the most recent LTR min (or else its LTR min would be superseded), and in the proof of Theorem \ref{well-defined} we showed that the first activated sequence must contain all the elements of every other activated sequence. Therefore, there can be at most $2r(k-2)+1$ activated sequences, and we have successfully bounded both $u$ and the size of any $S_i$.
To find a bound on $t$, note that as soon as we have used all $r$ copies of a letter and none of those copies remain as either LTR mins or in activated sequences, we can ignore that letter entirely, secure in the knowledge that if it is not already part of a forbidden pattern, it never will be. Thus, we only need to keep track of letters which are LTR mins, are in activated sequences, or are among the largest $2(k-2)+1$ letters still available to be used. By the reasoning of the previous paragraph, at most $2(k-2)+1$ distinct letters are LTR mins, at most $2(k-2)$ are in activated sequences, and so we need to keep track of $6(k-2)+2$ letters all together; all other letters either have never been used and so have $r$ copies remaining or else have had all copies used and can no longer participate in forbidden patterns. \end{proof}
After Example \ref{ex}, we commented that there may be several ways to describe a given partial word using $a,L,M,$ and $\mathcal{S}$. To allow for unique descriptions, we adopt the convention that $|L| = t = 6(k-2)+2$.
\subsection{Finding the Recurrence}
At this point, we have finished parts (i) and (ii) of our program; all that's left to show is that $A$ can be computed using a recurrence. To this end, we introduce three new functions. $\Fix(m,S,L)$ returns $S$ with all available letters (with counts determined by $L$) which exceed $m$ appended in decreasing order, $\Red(L,i)$ returns $L$ with the $i\tss{th}$ element decreased by one, and $\Rem(r,M,\mathcal{S},L,i)$ returns the tuple $M,\mathcal{S},L$ with the following changes: all elements of $M$ and all elements of every $S$ in $\mathcal{S}$ that are below $i$ are incremented by one, the $i\tss{th}$ element of $L$ is deleted and $r$ is prepended to $L$.
\begin{eg}\label{fix} $\Fix(1,[2,3],[1,1,1,2,2]) = [2,3,5,5,4,4,3,2].$ \end{eg}
\begin{eg}\label{reduce} $\Red([1,1,1,2,2],2) = [1,0,1,2,2].$ \end{eg}
\begin{eg}\label{remove} $\Rem(2,[1],[[2,4]],[1,1,0,1,2],3) = ([2],[[3,4]],[2,1,1,1,2]).$ \end{eg}
For fixed $r$ and $k$, the base cases are $A(r,k,a,M,\mathcal{S},L)$ for all $M,\mathcal{S}, L$ and $0\le a \le t$. Otherwise, there are no more than $2(k-2)+1$ possibilities for the next letter $i$ (corresponding to the largest $2(k-2)+1$ nonzero entries of $L$) which we divide into $u+1$ cases: when $i \le m_u$, when $m_h < i \le m_{h-1}$ for $2 \le h \le u-1$, and when $m_1 < i$. Suppose that $i \le m_u$; then we calculate the number of ways to complete the word after adding an $i$ as follows.
Suppose that $S_u = []$, then adding an element less than or equal to the current LTR min will supercede that LTR min. Symbolically, we have $\mathcal{S}' = [S_1,\dots,S_{u-1},[]]$, $M' = [m_1,\dots,m_{u-1},i]$, and $L' = \Red(L,i)$. If $S_u \neq []$, then we are adding a new LTR min while leaving all existing ones in place. This gives arguments $\mathcal{S}' = [S_1,\dots,S_{u},[]]$, $M'=[m_1,\dots,m_u,i]$, and $L' = \Red(L,i)$. Now, suppose that $J = \{j_1,\dots,j_w\}$ is a set of all the the integers $j \in [t]$ such that $L'_j = 0$, and $j$ fails to appear in $M'$ or in any $S \in \mathcal{S}'$. For all $j \in J$ from smallest to largest, update $\mathcal{S}',M',$ and $L'$ by setting $M',\mathcal{S}',L' = \Rem(r,M',\mathcal{S}',L',j)$. We finally have that the number of ways to complete the word after adding an $i$ is $A(r,k,a-|J|,M',\mathcal{S}',L')$.
Alternatively, we may add $i$ such that $i > m_h$ with $h$ chosen as small as possible. Either this $i$ is no larger than the smallest element of $S_1$, or else it is the largest letter that still remains to be added (otherwise a 1234 pattern is inevitable once the largest remaining letter is added). For all $m_j \ge i$, check to see if $\Fix(m_j,S_j,L)$ contains a $(k-1)(k-2)\dots1$ pattern. If so, this choice of $i$ contributes nothing to $A(r,k,a,M,\mathcal{S},L)$. If this is not true for any $j$, then we can add $i$ to our word, but doing so deactivates $S_1,S_2,\dots,S_{h-1}$, and so we forget about those activated sequences and their LTR mins. Thus, we take $\mathcal{S}' = [S_h,\dots,S_u]$, $M' = [M_h,\dots,M_u]$, and $L' = \Red(L,i)$. As before, suppose that $J = \{j_1,\dots,j_w\}$ is a set of all the the integers $j \in [t]$ such that $L'_j = 0$, and $j$ fails to appear in $M'$ or in any $S \in \mathcal{S}'$. For all $j \in J$ from smallest to largest, update $M',\mathcal{S}',$ and $L'$ by setting $M',\mathcal{S}',L' = \Rem(r,M',\mathcal{S}',L',j)$. Again we have that the number of ways to complete the word after adding an $i$ is $A(r,k,a-|J|,\mathcal{S}',M',L')$.
We have expressed $A(r,k,a,M,\mathcal{S},L)$ as a sum of other terms. Notice that in each of these other terms, the number of letters left to be added (given by $r\cdot a +\sum_{i=1}^t L_i$) decreases by 1; eventually it will decrease below $r\cdot t$ and a base case will apply.
While all the base cases could in principle be computed individually, this would probably be a long and unpleasant task. Fortunately, our recurrence can be easily tweaked to calculate base cases. To do so, simply run the recurrence as given, but anytime $A$ would be called with a negative $a$, replace the first $|a|$ nonzero entries of $L$ with 0 and change $a$ to 0. As it turns out, the only base case that we really need is $A(r,k,0,M,\mathcal{S},[0,0,\dots,0])=1$. A full implementation of this recurrence is available in an accompanying Maple package -- see Section \ref{maple}.
\subsection{From Recurrences to Generating Functions}\label{rigor} This subsection contains an algorithm for turning the recurrences found in this section and Section \ref{123} into generating functions. Readers interested in a more complete exposition should consult Chapter 4 in \cite{Kauers}.
Suppose the different terms in our system of recurrences are given by $A_1(n), A_2(n),\dots A_m(n)$. While we could choose $n$ like before and let it be the number of distinct unused letters whose counts do not appear in $L$, the rest of this process will be easier if each $A_i(n)$ depends only on $A_j(n-1)$. To make this happen, we interpret $n$ as the total number of unused letters. For example, we might fix $r=2,k=5$ (note that $t=|L|$ is then chosen to be 20), and let $A_1(n) = A(2,5,n,[],[],[2,2,\dots,2]).$ If we let $A_2(n) = A(2,5,n,[1],[[]],[1,2,\dots,2])$, $A_3(n) = A(2,5,n,[2],[[]],[2,1,2,\dots,2])$ and so on until $A_{21}(n) = A(2,5,n,[20],[[]],[2,\dots,2,1])$, we find the recurrence relation $A_1(n) = \sum_{i=2}^{21} A_i(n-1)$.
Let $M$ be the matrix whose $i,j$ entry is the coefficient of $A_j(n-1)$ in the recurrence for $A_i(n)$, and let $f_i(x)$ be the generating function $\sum_{n=0}^\infty A_i(n) x^n$. It follows that $f_i(x)$ is a rational function with denominator $\text{det}(xI-M)$ for all $i$. The numerator of each generating function has degree less than the number of rows of $M$, and the coefficients of each one can be determined using the system of recurrences' initial conditions.
Since we are treating $n$ as the total number of letters in a word, we must make the substitution $x^r \mapsto x$ in order to obtain the generating function for the number of words on the alphabet $[n]^r$ avoiding the two patterns.
\section{Computational Results}\label{compute} The first algorithm presented in this paper, the one which enumerates words avoiding 123 and $1k(k-1)\dots2$, runs very quickly. With $r = 2$, we are able to get generating functions for $k$ as large as 8 (and we could go even further if we chose to). With $r = 3$, we are able to get generating functions for $k$ as large as 7.
Unfortunately, the algorithm in Section \ref{1234} is much slower. In the simplest open case of $r=1, k=5$, we are able to conjecture the generating function to be $\ds \frac{-2x^3+7x^2-6x+1}{2x^4-11x^3+17x^2-8x+1}$, but rigorously deriving it seems to be out of the question without carefully pruning the recurrence. We can also use the recurrence to just generate terms without worrying about finding generating functions. With $r =1$, i.e. in the permutation case, we find ten terms apiece in the enumeration sequences for $k=3,4 \dots, 10$, and could easily get more terms; in fact in the particular case of $k=5$ we found 16 in 20 minutes.
All these results can be found in the output files on this paper's webpage (see Section \ref{maple}).
\section{Future Work} The driving force behind the argument in this paper is the Erdo\H s-Szekeres theorem; it ensures that we only have finitely many possible letters to add to a word at any point in time. For any pair of patterns which are not of the form $12\dots l, 1k(k-1)\dots 2$, this theorem will not apply, and so it is difficult to see how strategies like those in this paper could work.
It does seems reasonable to hope that they would work for other patterns of the form $12\dots l, 1k(k-1)\dots 2$. The only problem with applying them to the pair 12345, $1k(k-1)\dots 2$ is that we lose the fact that any element greater than an LTR min immediately fixes all elements above it. As a result, it is possible to have multiple activated strings, neither of which is a subset of the other. However, we are hopeful that some clever idea can get around this obstacle.
\section{Maple Implementation}\label{maple} This paper is accompanied by three Maple packages available from the paper's website: \url{http://sites.math.rutgers.edu/~yb165/SchemesForWords/SchemesForWords.html}. The packages are {\tt 123Avoid.txt} which implements the recurrence described in Section \ref{123}, {\tt 123Recurrences.txt} which uses this recurrence to rigorously find the generating functions enumerating the words on $[n]^r$ avoiding 123 and $1k(k-1)\dots2$, and {\tt 1234Avoid.txt} which implements the recurrence described in Section \ref{1234}. It also uses Doron Zeilberger's package Cfinite to automatically conjecture generating functions for the sequences of numbers of words on $[n]^r$ avoiding 1234 and $1k(k-1)\dots2$.
After loading any of these packages, type {\tt Help();} to see a list of available functions. You can get more details about any function by calling Help again with the function's name as an argument. This will also give an example of the function's usage.
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\tolerance2500
\title{\Large{\textbf{On some groupoids of small orders with Bol-Moufang type of identities}}} \author{\normalsize {Vladimir Chernov, Alexander Moldovyan, Victor Shcherbacov} }
\maketitle
\begin{abstract} We count number of groupoids of order 3 with some Bol-Moufang type identities.
\noindent \textbf{2000 Mathematics Subject Classification:} 20N05 20N02
\noindent \textbf{Key words and phrases:} groupoid, Bol-Moufang type identity. \end{abstract}
\section{Introduction}
A binary groupoid $(G, \cdot)$ is a non-empty set $G$ together with a binary operation \lq\lq $\cdot$\rq\rq. This definition is very general, therefore usually groupoids with some identities are studied. For example, groupoids with identity associativity (semi-groups) are researched.
We continue the study of groupoids with some Bol-Moufang type identities \cite{NOVIKOV_08, VD, 2017_Scerb}. Here we present results published in \cite{CHErnov, CHErnov_2018}.
{\bf Definition.} \label{Bol_Moufang_TYpe_Id} Identities that involve three variables, two of which appear once on both sides of the equation and one of which appears twice on both sides are called Bol-Moufang type identities. \index{identity!Bol-Moufang type}
Various properties of Bol-Moufang type identities in quasigroups and loops are studied in \cite{Fenyves_1, Ph_2005, Cote, AKHTAR}.
Groupoid $(Q, \ast)$ is called a quasigroup, if the following conditions are true \cite{VD}: $(\forall u, v \in Q) (\exists ! \, x, y \in Q) (u * x = v \, \& \, y * u = v)$.
For groupoids the following natural problems are researched: how many groupoids with some identities of small order there exist? A list of numbers of semigroups of orders up to 8 is given in \cite{Satoh}; a list of numbers of quasigroups up to 11 is given in \cite{HOP, WIKI_44}.
\section{Some results}
Original algorithm is elaborated and corresponding program is written for generating of groupoids of small (2 and 3) orders with some Bol-Moufang identities, which are well known in quasigroup theory.
To verify the correctness of the written program the number of semigroups of order 3 was counted. Obtained result coincided with well known, namely, there exist 113 semigroups of order 3.
The following identities have the property that any of them define a commutative Moufang loop \cite{BRUCK_46, VD, HOP, 2017_Scerb} in the class of loops: left (right) semimedial identity, Cote identity and its dual identity, Manin identity and its dual identity or in the class of quasigroups (identity (\ref{Comm_Muf_quas_Id}) and its dual identity).
\subsection{Groupoids with left semi-medial identity}
Left semi-medial identity in a groupoid $(Q, \ast)$ has the following form: $xx*yz=xy*xz$. Bruck \cite{BRUCK_46, VD, 2017_Scerb} uses namely this identity to define commutative Moufang loops in the class of loops.
There exist 10 left semi-medial groupoids of order 2. There exist 7 non-isomorphic left semi-medial groupoids of order 2. The first five of them are semigroups \cite{WIKI_44}.
\[ \begin{array}{lcrr}
\begin{array}{l|ll} \ast&1&2\\ \hline 1&1&1\\ 2&1&1\\ \end{array} &
\begin{array}{l|ll} \star&1&2\\ \hline 1&1&1\\ 2&1&2\\ \end{array} &
\begin{array}{l|ll} \circ&1&2\\ \hline 1&1&1\\ 2&2&2\\ \end{array} &
\begin{array}{l|ll} \cdot&1&2\\ \hline 1&1&2\\ 2&1&2\\ \end{array} \end{array} \]
\[ \begin{array}{lcr}
\begin{array}{l|ll} \diamond&1&2\\ \hline 1&1&2\\ 2&2&1\\ \end{array} &
\begin{array}{l|ll} \odot&1&2\\ \hline 1&2&1\\ 2&2&1\\ \end{array} &
\begin{array}{l|ll} \bullet&1&2\\ \hline 1&2&2\\ 2&1&1\\ \end{array} \end{array} \]
There exist 399 left semi-medial groupoids of order 3.
The similar results are true for groupoids with right semi-medial identity $xy*zz=xz*yz$. It is clear that the identities of left and right semi-mediality are dual. In other language they are (12)-parastrophes of each other \cite{VD, 2017_Scerb}.
It is clear that groupoids with dual identities have similar properties, including the number of groupoids of a fixed order.
\subsection{Groupoids with Cote identity}
Identity $x(xy*z) = (z*xx)y$ is discovered in \cite{Cote}. Here we name this identity Cote identity.
There exist 6 groupoids of order 2 with Cote identity. There exist 3 non-isomorphic in pairs groupoids of order 2 with Cote identity.
There exist 99 groupoids of order 3 with Cote identity.
The similar results are true for groupoids with the following identity $(z\ast yx)x = y(xx \ast z)$. The last identity is (12)-parastrophe of Cote identity.
\subsection{Groupoids with Manin identity}
The identity $x(y*xz) = (xx*y)z$ we call Manin identity \cite{MANIN}. The following identity is dual identity to Manin identity: $(zx\ast y)x = z(y\ast xx)$.
There exist 10 groupoids of order 2 with Manin identity. There exist 7 non-isomorphic in pairs groupoids of order 2 with Manin identity.
There exist 167 groupoids of order 3 with Manin identity.
\subsection{Groupoids with identity $(xy\ast x)z = (y\ast xz) x$ \label{Comm_Muf_quas_Id} (identity (\ref{Comm_Muf_quas_Id}))}
Some properties of identity (\ref{Comm_Muf_quas_Id}) are given in \cite{VS_2014_Kiev, 2017_Scerb}.
The following identity is dual identity to identity (\ref{Comm_Muf_quas_Id}): $z(x\ast yx) = x(zx\ast y)$.
There exist 6 groupoids of order 2 with identity (\ref{Comm_Muf_quas_Id}).
There exist 3 non-isomorphic in pairs groupoids of order 2 with (\ref{Comm_Muf_quas_Id}) identity. Any of these groupoids is a semigroup.
There exist 117 groupoids of order 3 with identity (\ref{Comm_Muf_quas_Id}).
\subsection{Number of groupoids of order 3 with some identities}
We count number of groupoids of order 3 with some identities. We use list of Bol-Moufang type identities given in \cite{Cote}. In Table 1 we present number of groupoids of order 3 with the respective identity.
\begin{table} \centering \caption{Number of groupoids of order 3 with some identities.} \footnotesize{ \[
\begin{array}{|c||c| c| c| c|} \hline
Name & Abbreviation & Identity & Number \\ \hline\hline Semigroups & SGR & x(yz) = (xy)z & 113\\ \hline
Extra & EL & x(y(zx)) = ((xy)z)x & 239\\ \hline Moufang & ML & (xy)(zx) = (x(yz))x & 196\\ \hline Left Bol & LB & x(y(xz)) = (x(yx))z & 215\\ \hline Right Bol & RB & y((xz)x) = ((yx)z)x & 215\\ \hline C-loops & CL & y(x(xz)) = ((yx)x)z & 133\\ \hline LC-loops & LC & (xx)(yz) = (x(xy))z & 220\\ \hline RC-loops & RC & y((zx)x) = (yz)(xx) & 220\\ \hline Middle Nuclear Square & MN & y((xx)z) = (y(xx))z & 350\\ \hline Right Nuclear Square & RN & y(z(xx)) = (yz)(xx) & 932\\ \hline Left Nuclear Square & LN & ((xx)y)z = (xx)(yz) & 932\\ \hline Comm. Moufang & CM & (xy)(xz) = (xx)(zy) & 297\\ \hline Abelian Group & AG & x(yz) = (yx)z & 91\\ \hline Comm. C-loop & CC & (y(xy))z = x(y(yz)) & 169\\ \hline Comm. Alternative & CA & ((xx)y)z = z(x(yx)) & 110\\ \hline Comm. Nuclear square & CN & ((xx)y)z = (xx)(zy) & 472\\ \hline Comm. loops & CP & ((yx)x)z = z(x(yx)) & 744\\ \hline Cheban \, 1 & C1 & x((xy)z) = (yx)(xz) & 219\\ \hline Cheban \, 2 & C2 & x((xy)z) = (y(zx))x & 153\\ \hline Lonely \, I & L1 & (x(xy))z = y((zx)x) & 117\\ \hline Cheban\, I\, Dual & CD & (yx)(xz) = (y(zx))x & 219\\ \hline Lonely \, II & L2 & (x(xy))z = y((xx)z) & 157\\ \hline Lonely \, III & L3 & (y(xx))z = y((zx)x) & 157\\ \hline Mate \, I & M1 & (x(xy))z = ((yz)x)x & 111\\ \hline Mate \, II & M2 & (y(xx))z = ((yz)x)x & 196\\ \hline Mate \, III & M3 & x(x(yz)) = y((zx)x) & 111\\ \hline Mate \, IV & M4 & x(x(yz)) = y((xx)z) & 196\\ \hline Triad \, I & T1 & (xx)(yz) = y(z(xx)) & 162\\ \hline Triad \, II & T2 & ((xx)y)z = y(z(xx)) & 180\\ \hline Triad \, III & T3 & ((xx)y)z = (yz)(xx) & 162\\ \hline Triad \, IV & T4 & ((xx)y)z = ((yz)x)x & 132\\ \hline Triad \, V & T5 & x(x(yz)) = y(z(xx)) & 132\\ \hline Triad \, VI & T6 & (xx)(yz) = (yz)(xx) & 1419\\ \hline Triad \, VII & T7 & ((xx)y)z = ((yx)x)z & 428\\ \hline Triad \, VIII & T8 & (xx)(yz) = y((zx)x) & 120\\ \hline Triad \, IX & T9 & (x(xy))z = y(z(xx)) & 102\\ \hline Frute & FR & (x(xy))z = (y(zx))x & 129\\ \hline Crazy Loop & CR & (x(xy))z = (yx)(xz) & 136\\ \hline Krypton & KL & ((xx)y)z = (x(yz))x & 268\\ \hline \end{array} \]} \end{table}
\textbf{Acknowledgments.} Authors thank Dr. V.D. Derech for his information on semigroups of small orders.
\begin{center} \begin{parbox}{118mm}{\footnotesize Vladimir Chernov$^{1}$, Nicolai Moldovyan$^{2}$, Victor Shcherbacov$^{3}$
\noindent $^{1}$Master/Shevchenko Transnistria State University
\noindent Email: [email protected]
\noindent $^{2}$Professor/St. Petersburg Institute for Informatics and Automation of Russian Academy of Sciences
\noindent Email: [email protected]
\noindent $^{3}$Principal Researcher/Institute of Mathematics and Computer Science of Moldova
\noindent Email: [email protected]
}
\end{parbox} \end{center}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Noether's Theorem and the Willmore Functional} \author{Yann Bernard\footnote{Departement Mathematik, ETH-Zentrum, 8093 Z\"urich, Switzerland.}} \date{ } \maketitle
{\bf Abstract :} {\it Noether's theorem and the invariances of the Willmore functional are used to derive conservation laws that are satisfied by the critical points of the Willmore energy subject to generic constraints. We recover in particular previous results independently obtained by R. Capovilla and J. Guven, and by T. Rivi\`ere. Several examples are considered in details.}
\reset
\section{Introduction}
Prior to establishing herself as a leading German mathematician of the early 20$^\text{th}$ century through her seminal work in abstract algebra, Emmy Noether had already made a significant contribution to variational calculus and its applications to physics. Proved in 1915 and published in 1918 \cite{Noe}, what was to become known as {\it Noether's theorem}, is a fundamental tool in modern theoretical physics and in the calculus of variations \cite{GM, Kos, Run}. Generalizing the idea of constants of motion found in classical mechanics, Noether's theorem provides a deep connection between symmetries and conservation laws. It is a recipe to construct a divergence-free vector field from a solution of a variational problem whose corresponding action (i.e. energy) is invariant under a continuous symmetry. For example, in 1-dimensional problems where the independent variable represents time, these vector fields are quantities which are conserved in time, such as the total energy, the linear momentum, or the angular momentum. We now precisely state one version of Noether's theorem\footnote{further generalizations may be found {\it inter alia} in \cite{Kos, Run}.}.
Let $\Omega$ be an open subset of $\mathcal{D}\subset\mathbb{R}^s$, and let $\mathcal{M}\subset\mathbb{R}^m$. Suppose that $$
L\,:\,\Big\{(x,q,p)\,\big|\,(x,q)\in\mathcal{D}\times\mathcal{M}\;,\;p\in T_q\mathcal{M}\otimes T^*_x\mathcal{D} \Big\}\;\longmapsto\;\mathbb{R} $$ is a continuously differentiable function. Choosing a $C^1$ density measure $d\mu(x)$ on $\Omega$, we can define the {\it action functional} $$ \mathcal{L}(u)\;:=\;\int_\Omega L(x,u(x),du(x))\,d\mu(x) $$ on the set of maps $u\in C^1(\Omega,\mathcal{M})$. A tangent vector field $X$ on $\mathcal{M}$ is called an {\it infinitesimal symmetry} for $\mathcal{L}$ if it satisfies $$ \dfrac{\partial L}{\partial q^i}(x,q,p)X^i(q)+\dfrac{\partial L}{\partial p^i_\alpha}(x,q,p)\dfrac{\partial X^i}{\partial q^j}(q)p^j_\alpha\;=\;0\:. $$ \begin{Th} Let $X$ be a Lipschitz tangent vector field on $\mathcal{M}$ which is an infinitesimal symmetry for the action $\mathcal{L}$. If $u:\Omega\rightarrow\mathcal{M}$ is a critical point of $\mathcal{L}$, then \begin{equation}\label{noether} \sum_{\alpha=1}^{s}\,\dfrac{\partial}{\partial x^\alpha}\bigg(\rho(x)X^j(u)\dfrac{\partial L}{\partial p^j_\alpha}(x,u,du) \bigg)\;=\;0\:, \end{equation} where $\{x^\alpha\}_{\alpha=1,\ldots,s}$ are coordinates on $\Omega$ such that $d\mu(x)=\rho(x)dx^1\cdot\cdot\cdot dx^s$. \end{Th} Equation (\ref{noether}) is the conservation law associated with the symmetry represented by $X$. The quantity $$ \rho(x)X^j(u)\dfrac{\partial L}{\partial p^j_\alpha}(x,u,du) $$ is often called {\it Noether current}, especially in the physics literature. \\
Whether in the form given above or in analogous forms, Noether's theorem has long been recognized as a fundamental tool in variational calculus. In the context of harmonic map theory, Noether's theorem was first used by \cite{Raw} in the mid 1980s. A few years later, several authors \cite{YMC, KRS, Sha} have independently used it to replace the harmonic map equation into spheres, where derivatives of the solution appear in a quadratic way, by an equation in divergence form, where derivatives of the solution appear in a linear way. This gives a particularly helpful analytical edge when studying harmonic maps with only very weak regularity hypotheses. Fr\'ed\'eric H\'elein made significant contributions to the analysis of harmonic maps using conservation laws via Noether's theorem \cite{Hel}. In the same vein, Tristan Rivi\`ere used conservation laws to study conformally invariant variational problems \cite{Riv1}. \\
We will in this paper also make use of Noether's theorem\footnote{not directly in the form (\ref{noether}), but the spirit behind our derivations is the same.}, this time in the context of fourth-order geometric problems in connection with the {\it Willmore functional}. We now briefly recall the main historical landmarks that led to the discovery -- and rediscovery, indeed -- of the Willmore functional. \\
Imagine that you had at your disposal the bow of a violin and a horizontal thin metallic plate covered with grains of sand. What would you observe if you were to rub the bow against the edge of the plate? In 1680, the English philosopher and scientist Robert Hooke was the first to try to answer this question (then posed in slightly different experimental terms). Some 120 years later, the German physicist and musician Ernst Chladni repeated the experiment in a systematic way \cite{Chl}. Rubbing the bow with varying frequency, he observed that the grains of sand arrange themselves in remarkable patterns -- nowadays known as {\it Chladni figures}. Those who witnessed Chladni's experiment were fascinated by the patterns, as was in 1809 the French emperor Napol\'eon I. Eager to understand the physical phenomenon at the origin of the Chladni figures, the emperor mandated Pierre-Simon de Laplace of the Acad\'emie des Sciences to organize a competition whose goal would be to provide a mathematical explanation for the figures. The winner would receive one kilogram of solid gold. Joseph-Louis Lagrange discouraged many potential candidates as he declared that the solution of the problem would require the creation of a new branch of mathematics. Only two contenders remained in the race: the very academic Sim\'eon-Denis Poisson and one autodidactic outsider: Sophie Germain. It is unfortunately impossible to give here a detailed account of the interesting events that took place in the following years (see \cite{Dah}). In 1816, Sophie Germain won the prize -- which she never claimed. Although Germain did not answer Napol\'eon's original question, and although she did not isolate the main phenomenon responsible for the Chladni figures, namely resonance, her work proved fundamental, for, as predicted by Lagrange, she laid down the foundations of a whole new branch of applied mathematics: the theory of elasticity of membranes. For the sake of brevity, one could synthesize Germain's main idea by isolating one single decisive postulate which can be inferred from her work \cite{Ger}. Having found her inspiration in the works of Daniel Bernoulli \cite{DBer} and Leonhard Euler \cite{Eul} on the elastica (flexible beams), Sophie Germain postulates that the density of elastic energy stored in a thin plate is proportional to the square of the mean curvature\footnote{Incidentally, the notion of mean curvature was first defined and used in this context; it is a creation which we owe to Germain.} $H$. In other words, the elastic energy of a bent thin plate $\Sigma$ can be expressed in the form $$ \int_{\Sigma}H^2(p)d\sigma(p)\:, $$ where $d\sigma$ denotes the area-element. In the literature, this energy is usually referred to as {\it Willmore energy}. It bears the name of the English mathematician Thomas Willmore who rediscovered it in the 1960s \cite{Wil1}. Prior to Willmore and after Germain, the German school of geometers led by Wilhelm Blaschke considered and studied the Willmore energy in the context of conformal geometry. Blaschke observed that minimal surfaces minimize the Willmore energy and moreover that the Willmore energy is invariant under conformal transformations of $\mathbb{R}^3\cup\{\infty\}$. In his nomenclature, critical points of the Willmore energy were called {\it conformal minimal surfaces} \cite{Bla}. Gerhard Thomsen, a graduate student of Blaschke, derived the Euler-Lagrange equation corresponding to the Willmore energy \cite{Tho} (this was further generalized to higher codimension in the 1970s by Joel Weiner \cite{Wei}). It is a fourth-order nonlinear partial differential equation for the immersion. Namely, let $\vec{\Phi}:\Sigma\rightarrow\mathbb{R}^{m\ge3}$ be a smooth immersion of an oriented surface $\Sigma$. The pull-back metric $g:=\vec{\Phi}^*g_{\mathbb{R}^3}$ is represented in local coordinates with components $g_{ij}$. We let $\nabla_j$ denote the corresponding covariant derivative. The second fundamental form is the normal valued 2-tensor with components $\vec{h}_{ij}:=\nabla_i\nabla_j\vec{\Phi}$. Its half-trace is the mean curvature vector $\vec{H}:=\dfrac{1}{2}\vec{h}^{j}_{j}$. The {\it Willmore equation} reads \begin{equation}\label{will0}
\Delta_\perp\vec{H}+\big(\vec{h}^{i}_{j}\cdot\vec{H}\big)\vec{h}^{j}_{i}-2|\vec{H}|^2\vec{H}\;=\;\vec{0}\:, \end{equation} where $\Delta_\perp$ is the negative covariant Laplacian for the connection $\nabla$ in the normal bundle derived from the ambient scalar product in $\mathbb{R}^m$. Note, in passing, that it is not at all clear how one could define a weak solution of (\ref{will0}) using only the requirement that $\vec{H}$ be square-integrable (i.e. that the Willmore energy be finite). \\
The Willmore energy appears in various areas of science: general relativity, as the main contributor to the Hawking mass \cite{Haw} ; in cell biology (see below) ; in nonlinear elasticity theory \cite{FJM} ; in optical design and lens crafting \cite{KR} ; in string theory, in the guise of a string action \`a la Polyakov \cite{Pol}. As mentioned earlier, the Willmore energy also plays a distinguished role in conformal geometry, where it has given rise to many interesting problems and where it has stimulated too many elaborate works to be cited here. We content ourselves with mentioning the remarkable tours de force of Fernando Marques and Andr\'e Neves \cite{MN} to solve the celebrated Willmore conjecture stating that, up to M\"obius transformations, the Clifford torus\footnote{obtained by rotating a circle of radius 1 around an axis located at a distance $\sqrt{2}$ of its center.} minimizes the Willmore energy amongst immersed tori in $\mathbb{R}^3$. \\
Aiming at solving the Willmore conjecture, Leon Simon initiated the ``modern" variational study of the Willmore functional \cite{Sim} when proving the existence of an embedded torus into $\mathbb{R}^{m\ge3}$ minimizing the $L^2$ norm of the second fundamental form. As this norm does not provide any control of the $C^1$ norm of the surface, speaking of ``immersion" is impossible. Simon thus had to weaken the geometric notion of immersion, and did so by using varifolds and their local approximation by biharmonic graphs. In the following years, this successful ``ambient approach" was used by various authors \cite{BK, KS1, KS2, KS3} to solve important questions about Willmore surfaces. \\
Several authors \cite{CDDRR, Dal, KS2, Pal, Rus} have observed that the Willmore equation in codimension 1 is cognate with a certain divergence form. We will prove below (Theorem \ref{Th1}) a pointwise equality to that effect in any codimension. The versions found in the aforementioned works are weaker in the sense that they only identify an integral identity. \\ In 2006, Tristan Rivi\`ere \cite{Riv2} showed that the fourth-order Willmore equation (\ref{will0}) can be written in divergence form and eventually recast as a system two of second-order equations enjoying a particular structure useful to the analysis of the critical points of the Willmore energy. This observation proved to be decisive in the resolution of several questions pertaining to Willmore surfaces \cite{YBer, BR1, BR2, BR3, KMR, MR, Riv2, Riv3, Riv4}. It also led to the so-called ``parametric approach" of the problem. In contrast with the ambient approach where surfaces are viewed as subsets of $\mathbb{R}^m$, the parametric approach favors viewing surfaces as images of (weak) immersions, and the properties of these immersions become the analytical point of focus. \\
We briefly review the results in \cite{Riv2} (in codimension 1), adapting slightly the original notation and statements to match the orientation of our paper. With the same notation as above, the first conservation law in \cite{Riv2} states that a smooth immersion $\vec{\Phi}:\Sigma\rightarrow\mathbb{R}^3$ is a critical point of the Willmore functional if and only if \begin{equation}\label{ri1}
\nabla_j\big(\nabla^j\vec{H}-2(\vec{n}\cdot\nabla^j\vec{H})\vec{n}+|\vec{H}|^2\nabla^j\vec{\Phi}\big)\;=\;\vec{0}\:, \end{equation} where $\vec{n}$ is the outward unit normal. \\ Locally about every point, (\ref{ri1}) may be integrated to yield a function $\vec{L}\in\mathbb{R}^3$ satisfying $$
|g|^{-1/2}\epsilon^{kj}\nabla_k\vec{L}\;=\;\nabla^j\vec{H}-2(\vec{n}\cdot\nabla^j\vec{H})\vec{n}+|\vec{H}|^2\nabla^j\vec{\Phi}\:, $$
where $|g|$ is the volume element of the pull-back metric $g$, and $\epsilon^{kj}$ is the Levi-Civita symbol. The following equations hold: \begin{equation}\label{ri2} \left\{\begin{array}{rcl}
\nabla_j\big(|g|^{-1/2}\epsilon^{kj}\vec{L}\times\nabla_k\vec{\Phi}-\vec{H}\times\nabla^j\vec{\Phi}\big)&=&\vec{0}\\[1ex]
\nabla_j\big(|g|^{-1/2}\epsilon^{kj}\vec{L}\cdot\nabla_k\vec{\Phi}\big)&=&0\:. \end{array}\right. \end{equation} These two additional conservation laws give rise (locally about every point) to two potentials $\vec{R}\in\mathbb{R}^3$ and $S\in\mathbb{R}$ satisfying \begin{equation*} \left\{\begin{array}{rcl}
\nabla_k\vec{R}&=&\vec{L}\times\nabla_k\vec{\Phi}-|g|^{1/2}\epsilon_{kj}\vec{H}\times\nabla^j\vec{\Phi}\\[1ex] \nabla_k S&=&\vec{L}\cdot\nabla_k\vec{\Phi}\:. \end{array}\right. \end{equation*} A computation shows that these potentials are related to each other via the {system \begin{equation}\label{ri3} \left\{\begin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\epsilon^{jk}\partial_j\vec{n}\cdot\partial_k\vec{R}\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\epsilon^{jk}\Big[\partial_j\vec{n}\,\partial_kS+\partial_j\vec{n}\times\partial_k\vec{R}\Big]\:.
\end{array}\right. \end{equation} This system is linear in $S$ and $\vec{R}$. It enjoys the particularity of being written in flat divergence form, with the right-hand side comprising Jacobian-type terms. The Willmore energy is, up to a topological constant, the $W^{1,2}$-norm of $\vec{n}$. For an immersion $\vec{\Phi}\in W^{2,2}\cap W^{1,\infty}$, one can show that $S$ and $\vec{R}$ belong to $W^{1,2}$. Standard Wente estimates may thus be performed on (\ref{ri3}) to thwart criticality and regularity statements ensue \cite{BR1, Riv2}. Furthermore, one verifies that (\ref{ri3}) is stable under weak limiting process, which has many nontrivial consequences \cite{BR1,BR3}. \\
In 2013, the author found that the divergence form and system derived by Rivi\`ere can be obtained by applying Noether's principle to the Willmore energy\footnote{The results were first presented at Oberwolfach in July 2013 during the mini-workshop {\it The Willmore functional and the Willmore conjecture}.}. The translation, rotation, and dilation invariances of the Willmore energy yield via Noether's principle the conservations laws (\ref{ri1}) and (\ref{ri2}).
\begin{Th}\label{Th1} Let $\vec{\Phi}:\Sigma\rightarrow\mathbb{R}^m$ be a smooth immersion of an oriented surface $\Sigma$. Introduce the quantities $$ \left\{\begin{array}{lcl}
\vec{\mathcal{W}}&:=&\Delta_\perp\vec{H}+\big(\vec{h}^{i}_{j}\cdot\vec{H}\big)\vec{h}^{j}_{i}-2|\vec{H}|^2\vec{H}\\[1ex]
\vec{T}^j&:=&\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi}\:, \end{array}\right. $$ where $\pi_{\vec{n}}$ denotes projection onto the normal space. \\ Via Noether's theorem, the invariance of the Willmore energy by translations, rotations, and dilations in $\mathbb{R}^m$ imply respectively the following three conservation laws: \begin{equation}\label{laws} \left\{\begin{array}{rcl} \nabla_j\vec{T}^j&=&-\,\vec{\mathcal{W}}\\[1ex] \nabla_j\big(\vec{T}^j\wedge\vec{\Phi}+\vec{H}\wedge\nabla^j\vec{\Phi}\big)&=&-\,\vec{\mathcal{W}}\wedge\vec{\Phi}\\[1ex] \nabla_j\big(\vec{T}^j\cdot\vec{\Phi}\big)&=&-\,\vec{\mathcal{W}}\cdot\vec{\Phi}\:. \end{array}\right. \end{equation} In particular, the immersion $\vec{\Phi}$ is Willmore if and only if the following conservation holds: \begin{equation*}
\nabla_j\big(\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi} \big)\;=\;\vec{0}\:. \end{equation*} \end{Th}
For the purpose of local analysis, one may apply Hodge decompositions in order to integrate the three conservation laws (\ref{laws}). Doing so yields ``potential" functions related to each other in a very peculiar way, which we state below. The somewhat unusual notation -- the price to pay to work in higher codimension -- is clarified in Section \ref{nota}. \begin{Th}\label{Th2} Let $\vec{\Phi}:D^2\rightarrow\mathbb{R}^m$ be a smooth\footnote{in practice, this strong hypothesis is reduced to $\vec{\Phi}\in W^{2,2}\cap W^{1,\infty}$ without modifying the result.} immersion of the flat unit disk $D^2\subset\mathbb{R}^2$. We denote by $\vec{n}$ the Gauss-map, by $g:=\vec{\Phi}^*g_{\mathbb{R}^m}$ the pull-back metric, and by $\Delta_g$ the associated negative Laplace-Beltrami operator. Suppose that $\vec{\Phi}$ satisfies the fourth-order equation $$
\Delta_\perp\vec{H}+\big(\vec{h}^{i}_{j}\cdot\vec{H}\big)\vec{h}^{j}_{i}-2|\vec{H}|^2\vec{H}\;=\;\vec{\mathcal{W}}\:, $$ for some given $\vec{\mathcal{W}}$. Let $\vec{V}$, $\vec{X}$, and $Y$ solve the problems $$ \Delta_g\vec{V}\;=\;-\,\vec{\mathcal{W}}\qquad,\qquad \Delta_g\vec{X}\;=\;\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\qquad,\qquad\Delta_gY\;=\;\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\:. $$ Then $\vec{\Phi}$ is a solution of the second-order equation \begin{equation}\label{eqphi}
|g|^{1/2}\Delta_g\vec{\Phi}\;=\;-\,\epsilon^{jk}\Big[\partial_jS\partial_k\vec{\Phi}+\partial_j\vec{R}\bullet\partial_k\vec{\Phi}\Big]+|g|^{1/2}\big(\nabla^jY\nabla_j\vec{\Phi}+\nabla^j\vec{X}\bullet\nabla_j\vec{\Phi} \big)\:, \end{equation} where $S$ and $\vec{R}$ satisfy the system \begin{equation}\label{thesys000} \left\{\begin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\epsilon^{jk}\partial_j(\star\,\vec{n})\cdot\partial_k\vec{R}
\,+\,|g|^{1/2}\nabla_j\big((\star\,\vec{n})\cdot\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\epsilon^{jk}\Big[\partial_j(\star\,\vec{n})\partial_kS+\partial_j(\star\,\vec{n})\bullet\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\big((\star\,\vec{n})\nabla^jY+(\star\,\vec{n})\bullet\nabla^j\vec{X}\big)\:. \end{array}\right. \end{equation} \end{Th} In the special case when $\vec{\Phi}$ is Willmore, we have $\mathcal{\vec{W}}\equiv\vec{0}$, and we may choose $\vec{V}$, $\vec{X}$, and $Y$ to identically vanish. Then (\ref{thesys000}) becomes the conservative Willmore system found originally in \cite{Riv2}. \\
Although perhaps at first glance a little cryptic, Theorem \ref{Th2} turns out to be particularly useful for local analytical purposes. If the given $\mathcal{\vec{W}}$ is sufficiently regular, the non-Jacobian terms involving $Y$ and $\vec{X}$ on the right-hand side of (\ref{thesys000}) form a subcritical perturbation of the Jacobian terms involving $S$ and $\vec{R}$ (see \cite{BWW1} for details). From an analytic standpoint, one is left with studying a linear system of Jacobian-type. Wente estimates provide fine regularity information on the potential functions $S$ and $\vec{R}$, which may, in turn, be bootstrapped into (\ref{eqphi}) to yield regularity information on the immersion $\vec{\Phi}$ itself. \\
In \cite{DDW}, the authors study Willmore surfaces of revolution. They use the invariances of the Willmore functional to recast the Willmore ODE in a form that is a special case of (\ref{ri1}). Applying Noether's principle to the Willmore energy had already been independently done and used in the physics community \cite{CG, Mue}. As far as the author understands, these references are largely unknown in the analysis and geometry community. One goal of this paper is to bridge the gap, as well as to present results which do not appear in print. The author hopes it will increase in the analysis/geometry community the visibility of results, which, he believes, are useful to the study of fourth-order geometric problems associated with the Willmore energy. \\ For the sake of brevity, the present work focuses only on computational derivations and on examples. A second work \cite{BWW1} written jointly with Glen Wheeler and Valentina-Mira Wheeler will shortly be available. It builds upon the reformulations given in the present paper to derive various local analytical results.
\paragraph{Acknowledgments.} The author is grateful to Daniel Lengeler for pointing out to him \cite{CG} and \cite{Mue}. The author would also like to thank Hans-Christoph Grunau, Tristan Rivi\`ere, and Glen Wheeler for insightful discussions. The excellent working conditions of the welcoming facilities of the Forschungsinstitut f\"ur Mathematik at the ETH in Z\"urich are duly acknowledged.
\section{Main Result}
After establishing some notation in Section \ref{nota}, the contents of Theorem \ref{Th1} and of Theorem \ref{Th2} will be proved simultaneously in Section \ref{proof}.
\subsection{Notation}\label{nota}
In the sequel, $\vec{\Phi}:\Sigma\rightarrow\mathbb{R}^{m\ge3}$ denotes a smooth immersion of an oriented surface $\Sigma$ into Euclidean space. The induced metric is $g:=\vec{\Phi}^*g_{\mathbb{R}^m}$ with components $g_{ij}$ and with volume element $|g|$. The components of the second fundamental form are denoted $\vec{h}_{ij}$. The mean curvature is $\vec{H}:=\dfrac{1}{2}\,g^{ij}\vec{h}_{ij}$. At every point on $\Sigma$, there is an oriented basis $\{\vec{n}_\alpha\}_{\alpha=1,\ldots,m-2}$ of the normal space. We denote by $\pi_{\vec{n}}$ the projection on the space spanned by the vectors $\{\vec{n}_\alpha\}$, and by $\pi_T$ the projection on the tangent space (i.e. $\pi_T+\pi_{\vec{n}}=\text{id}$). The Gauss map $\vec{n}$ is the $(m-2)$-vector defined via $$
\star\,\vec{n}\;:=\;\dfrac{1}{2}\,|g|^{-1/2}\epsilon^{ab}\nabla_a\vec{\Phi}\wedge\nabla_b\vec{\Phi}\:, $$ where $\star$ is the usual Hodge-star operator, and $\epsilon^{ab}$ is the Levi-Civita symbol\footnote{Recall that the Levi-Civita is {\it not} a tensor. It satisfies $\epsilon_{ab}=\epsilon^{ab}$.} with components $\epsilon^{11}=0=\epsilon^{22}$ and $\epsilon^{12}=1=-\epsilon^{21}$. Einstein's summation convention applies throughout. We reserve the symbol $\nabla$ for the covariant derivative associated with the metric $g$. Local flat derivatives will be indicated by the symbol $\partial$. \\
\noindent As we work in any codimension, it is helpful to distinguish scalar quantities from vector quantities. For this reason, we append an arrow to the elements of $\Lambda^p(\mathbb{R}^m)$, for all $p>0$. The scalar product in $\mathbb{R}^m$ is denoted by a dot. We also use dot to denote the natural extension of the scalar product in $\mathbb{R}^m$ to multivectors (see \cite{Fed}).\\ Two operations between multivectors are useful. The interior multiplication $\res$ maps the pair comprising a $q$-vector $\gamma$ and a $p$-vector $\beta$ to the $(q-p)$-vector $\gamma\res\beta$. It is defined via \begin{equation*} \langle \gamma\res\beta\,,\alpha\rangle\;=\;\langle \gamma\,,\beta\wedge\alpha\rangle\:\qquad\text{for each $(q-p)$-vector $\alpha$.} \end{equation*} Let $\alpha$ be a $k$-vector. The first-order contraction operation $\bullet$ is defined inductively through \begin{equation*} \alpha\bullet\beta\;=\;\alpha\res\beta\:\:\qquad\text{when $\beta$ is a 1-vector}\:, \end{equation*} and \begin{equation*} \alpha\bullet(\beta\wedge\gamma)\;=\;(\alpha\bullet\beta)\wedge\gamma\,+\,(-1)^{pq}\,(\alpha\bullet\gamma)\wedge\beta\:, \end{equation*} when $\beta$ and $\gamma$ are respectively a $p$-vector and a $q$-vector.
\subsection{Variational Derivations}\label{proof}
Consider a variation of the form: \begin{equation*} \vec{\Phi}_t\;:=\;\vec{\Phi}\,+\,t\big(A^j\nabla_j\vec{\Phi}+\vec{B}\big)\:, \end{equation*} for some $A^j$ and some normal vector $\vec{B}$. We have \begin{equation*} \nabla_i\nabla_j\vec{\Phi}\;=\;\vec{h}_{ij}\:. \end{equation*} Denoting for notational convenience by $\delta$ the variation at $t=0$, we find: \begin{equation*} \delta\nabla_j\vec{\Phi}\;\equiv\;\nabla_j\delta\vec{\Phi}\;=\;(\nabla_jA^s)\nabla_s\vec{\Phi}+A^s\vec{h}_{js}+\nabla_j\vec{B}\:. \end{equation*} Accordingly, we find \begin{eqnarray*} \pi_{\vec{n}}\nabla^j\delta\nabla_j\vec{\Phi}&=&2(\nabla_jA^s)\vec{h}^j_{s}+A^s\pi_{\vec{n}}\nabla^j\vec{h}_{js}+\pi_{\vec{n}}\nabla^j\pi_{\vec{n}}\nabla_j\vec{B}+\pi_{\vec{n}}\nabla^j\pi_T\nabla_j\vec{B}\\ &=&2(\nabla_jA^s)\vec{h}^j_{s}+2A^s\nabla_s\vec{H}+\Delta_\perp\vec{B}+\pi_{\vec{n}}\nabla^j\pi_T\nabla_j\vec{B}\:, \end{eqnarray*} where we have used the definition of the normal Laplacian $\Delta_\perp$ and the contracted Codazzi-Mainardi equation \begin{equation*} \nabla^j\vec{h}_{js}\;=\;2\nabla_s\vec{H}\:. \end{equation*} Since $\vec{B}$ is a normal vector, one easily verifies that \begin{equation*} \pi_T\nabla_j\vec{B}\;=\;-\,(\vec{B}\cdot\vec{h}_j^s)\,\nabla_s\vec{\Phi}\:, \end{equation*} so that \begin{equation*} \pi_{\vec{n}}\nabla^j\pi_T\nabla_j\vec{B}\;=\;-\,(\vec{B}\cdot\vec{h}_j^s)\,\vec{h}_s^j\:. \end{equation*} Hence, the following identity holds \begin{equation}\label{eq1} \pi_{\vec{n}}\nabla^j\delta\nabla_j\vec{\Phi}\;=\;2(\nabla_jA^s)\vec{h}_{js}+2A^s\nabla_s\vec{H}+\Delta_\perp\vec{B}-(\vec{B}\cdot\vec{h}_j^s)\,\vec{h}_s^j\:. \end{equation} Note that \begin{equation}\label{eq2} \delta g^{ij}\;=\;-2\nabla^jA^i+2\vec{B}\cdot\vec{h}^{ij}\:, \end{equation} which, along with (\ref{eq1}) and (\ref{eq2}), then gives \begin{eqnarray*}
\delta|\vec{H}|^2&=&\vec{H}\cdot\delta\nabla^j\nabla_j\vec{\Phi}\;\;=\;\;\vec{H}\cdot\big[(\delta g^{ij})\partial_i\nabla_j\vec{\Phi}+\nabla^j\delta\nabla_j\vec{\Phi}\big]\\[1ex] &=&\vec{H}\cdot\big[(\delta g^{ij})\vec{h}_{ij}+\pi_{\vec{n}}\nabla^j\delta\nabla_j\vec{\Phi}\big]\\[1ex] &=&\vec{H}\cdot\big[\Delta_\perp\vec{B}+(\vec{B}\cdot\vec{h}^i_j)\vec{h}^j_i+2A^j\nabla_j\vec{H} \big]\:. \end{eqnarray*} Finally, since \begin{equation*}
\delta|g|^{1/2}\;=\;|g|^{1/2}\big[\nabla_jA^j-2\vec{B}\cdot\vec{H} \big]\:, \end{equation*} we obtain \begin{eqnarray*}
\delta\big(|\vec{H}|^2|g|^{1/2}\big)&=&|g|^{1/2}\Big[\vec{H}\cdot\Delta_\perp\vec{B}+(\vec{B}\cdot\vec{h}^i_j)\vec{h}^j_i-2(\vec{B}\cdot\vec{H})|H|^2+\nabla_j\big(|\vec{H}|^2A^j\big) \Big]\\[1ex]
&=&|g|^{1/2}\Big[\vec{B}\cdot\vec{\mathcal{W}}+\nabla_j\big(\vec{H}\cdot\nabla^j\vec{B}-\vec{B}\cdot\nabla^j\vec{H}+|\vec{H}|^2A^j\big) \Big]\:, \end{eqnarray*} where \begin{equation*}
\vec{\mathcal{W}}\;:=\;\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}^i_j)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\:. \end{equation*} Therefore, \begin{equation}\label{diffen}
\delta\int_{\Sigma_0}|\vec{H}|^2\;=\;\int_{\Sigma_0}\Big[\vec{B}\cdot\vec{\mathcal{W}}+\nabla_j\big(\vec{H}\cdot\nabla^j\vec{B}-\vec{B}\cdot\nabla^j\vec{H}+|\vec{H}|^2A^j\big) \Big]\:. \end{equation} This identity holds for every piece of surface $\Sigma_0\subset\Sigma$. We will now consider specific deformations which are known to preserve the Willmore energy (namely translations, rotations, and dilations \cite{BYC3}), and thus for which the right-hand side of (\ref{diffen}) vanishes.
\paragraph{Translations.}
We consider a deformation of the form \begin{equation*} \vec{\Phi}_t\;=\;\vec{\Phi}+t\vec{a}\qquad\text{for some fixed $\vec{a}\in\mathbb{R}^m$}\:. \end{equation*} Hence \begin{equation*} \vec{B}\;=\;\pi_{\vec{n}}\vec{a}\qquad\text{and}\qquad A^j\;=\;\vec{a}\cdot\nabla^j\vec{\Phi}\:. \end{equation*} This gives \begin{eqnarray*}
&&\vec{H}\cdot\nabla^j\vec{B}-\vec{B}\cdot\nabla^j\vec{H}+ |\vec{H}|^2A^j\\[1ex]
&=&\vec{a}\cdot\Big[(\vec{H}\cdot\nabla^j\vec{n}_\alpha-\vec{n}_\alpha\cdot\nabla^j\vec{H})\vec{n}_\alpha+H^\alpha\nabla^j\vec{n}_\alpha+|\vec{H}|^2\nabla^j\vec{\Phi}\Big]\\[1ex]
&=&\vec{a}\cdot\Big[\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi}\Big]\:, \end{eqnarray*} so that (\ref{diffen}) yields \begin{equation*}
\vec{a}\cdot\int_{\Sigma_0}\vec{\mathcal{W}}+\nabla_j\Big[\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi} \Big]\;=\;0\:. \end{equation*} As this holds for all $\vec{a}$ and all $\Sigma_0$, letting \begin{equation}\label{defT}
\vec{T}^j\;:=\;\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi} \end{equation} gives \begin{equation}\label{trans} \nabla_j\vec{T}^j\;=\;-\,\vec{\mathcal{W}}\:. \end{equation} This is equivalent to the conservation law derived in \cite{Riv2} in the case when $\vec{\mathcal{W}}=\vec{0}$ and when the induced metric is conformal with respect to the identity. At the equilibrium, i.e. when $\vec{\mathcal{W}}$ identically vanishes, $\vec{T}^j$ plays in the problem the role of stress-energy tensor. \\
For future convenience, we formally introduce the Hodge decomposition\footnote{Naturally, this is only permitted when working locally, or on a domain whose boundary is contractible to a point.} \begin{equation}\label{defLL}
\vec{T}^j\;=\;\nabla^j\vec{V}+|g|^{-1/2}\epsilon^{kj}\nabla_k\vec{L}\:, \end{equation} for some $\vec{L}$ and some $\vec{V}$ satisfying \begin{equation}\label{defV} -\,\Delta_g\vec{V}\;=\;\vec{\mathcal{W}}\:. \end{equation}
\paragraph{Rotations.}
We consider a deformation of the form \begin{equation*} \vec{\Phi}_t\;=\;\vec{\Phi}+t\star(\vec{b}\wedge\vec{\Phi})\qquad\text{for some fixed $\vec{b}\in\Lambda^{m-2}(\mathbb{R}^m)$}\:. \end{equation*} In this case, we have \begin{equation*} B^\alpha\;=\;-\,\vec{b}\cdot\star(\vec{n}_\alpha\wedge\vec{\Phi})\qquad\text{and}\qquad A^j\;=\;-\,\vec{b}\,\cdot\star\big(\nabla^j\vec{\Phi}\wedge\vec{\Phi}\big)\:. \end{equation*} Hence \begin{eqnarray*}
&&\vec{H}\cdot\nabla^j\vec{B}-\vec{B}\cdot\nabla^j\vec{H}+ |\vec{H}|^2A^j\\[1ex]
&&\hspace{-1cm}=\;\;-\,\vec{b}\cdot\star\Big[(\vec{H}\cdot\nabla^j\vec{n}_\alpha-\vec{n}_\alpha\cdot\nabla^j\vec{H})(\vec{n}_\alpha\wedge\vec{\Phi})+H^\alpha\nabla^j(\vec{n}_\alpha\wedge\vec{\Phi})+|\vec{H}|^2\nabla^j\vec{\Phi}\wedge\vec{\Phi}\Big]\\ &&\hspace{-1cm}=\;\;-\,\vec{b}\cdot\star\big(\vec{T}^j\wedge\vec{\Phi}+\vec{H}\wedge\nabla^j\vec{\Phi}\big)\:, \end{eqnarray*} where we have used the tensor $\vec{T}^j$ defined in (\ref{defT}). Putting this last expression in (\ref{diffen}) and proceeding as in the previous paragraph yields the pointwise equalities \begin{eqnarray}\label{ach1} \vec{\mathcal{W}}\wedge\vec{\Phi}&=&-\,\nabla_j\big(\vec{T}^j\wedge\vec{\Phi}+\vec{H}\wedge\nabla^j\vec{\Phi}\big)\nonumber\\
&\stackrel{\text{(\ref{defLL})}}{\equiv}&-\,\nabla_j\big(\nabla^j\vec{V}\wedge\vec{\Phi}+|g|^{-1/2}\epsilon^{kj}\nabla_k\vec{L}\wedge\vec{\Phi}+\vec{H}\wedge\nabla^j\vec{\Phi}\big)\nonumber\\
&=&-\,\nabla_j\big(\nabla^j\vec{V}\wedge\vec{\Phi}-|g|^{-1/2}\epsilon^{kj}\vec{L}\wedge\nabla_k\vec{\Phi}+\vec{H}\wedge\nabla^j\vec{\Phi}\big)\nonumber\\
&=&-\Delta_g\vec{V}\wedge\vec{\Phi}-\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}+\nabla_j\big(|g|^{-1/2}\epsilon^{kj}\vec{L}\wedge\nabla_k\vec{\Phi}-\vec{H}\wedge\nabla^j\vec{\Phi}\big)\:. \end{eqnarray} Owing to (\ref{defV}), we thus find \begin{equation*}
\nabla_j\big(|g|^{-1/2}\epsilon^{kj}\vec{L}\wedge\nabla_k\vec{\Phi}-\vec{H}\wedge\nabla^j\vec{\Phi}\big)\;=\;\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\:. \end{equation*} It will be convenient to define two 2-vectors $\vec{X}$ and $\vec{R}$ satisfying the Hodge decomposition \begin{equation}\label{defR}
|g|^{-1/2}\epsilon^{kj}\vec{L}\wedge\nabla_k\vec{\Phi}-\vec{H}\wedge\nabla^j\vec{\Phi}\;=\;\nabla^j\vec{X}+|g|^{-1/2}\epsilon^{kj}\nabla_k\vec{R}\:, \end{equation} with thus \begin{equation}\label{defX} \Delta_g\vec{X}\;=\;\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\:. \end{equation}
\paragraph{Dilations.}
We consider a deformation of the form \begin{equation*} \vec{\Phi}_t\;=\;\vec{\Phi}+t\lambda\vec{\Phi}\qquad\text{for some fixed $\lambda\in\mathbb{R}$}\:, \end{equation*} from which we obtain \begin{equation*} B^\alpha\;=\;\lambda\,\vec{n}_\alpha\cdot\vec{\Phi}\qquad\text{and}\qquad A^j\;=\;\lambda\,\nabla^j\vec{\Phi}\cdot\vec{\Phi}\:. \end{equation*} Hence \begin{eqnarray*}
&&\vec{H}\cdot\nabla^j\vec{B}-\vec{B}\cdot\nabla^j\vec{H}+ |\vec{H}|^2A^j\\[1ex]
&&\hspace{-1cm}=\;\;\lambda\Big[(\vec{H}\cdot\nabla^j\vec{n}_\alpha-\vec{n}_\alpha\cdot\nabla^j\vec{H})(\vec{n}_\alpha\cdot\vec{\Phi})+H^\alpha\nabla^j(\vec{n}_\alpha\cdot\vec{\Phi})+|\vec{H}|^2\nabla^j\vec{\Phi}\cdot\vec{\Phi} \Big]\\ &&\hspace{-1cm}=\;\;\lambda\,\vec{T}^j\cdot\vec{\Phi}\:, \end{eqnarray*} where we have used that $\vec{H}\cdot\nabla^j\vec{\Phi}=0$, and where $\vec{T}^j$ is as in (\ref{defT}). \\ Putting this last expression in (\ref{diffen}) and proceeding as before gives the pointwise equalities \begin{eqnarray}\label{ach2} \vec{\mathcal{W}}\cdot\vec{\Phi}&=&-\,\nabla_j\big(\vec{T}^j\cdot\vec{\Phi}\big)\\
&\equiv&-\,\nabla_j\big(\nabla^j\vec{V}\cdot\vec{\Phi}+|g|^{-1/2}\epsilon^{kj}\nabla_k\vec{L}\cdot\vec{\Phi} \big)\nonumber\\
&=&-\,\Delta_g\vec{V}\cdot\vec{\Phi}-\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}+\nabla_j\big(|g|^{-1/2}\epsilon^{kj}\vec{L}\cdot\nabla_j\vec{\Phi}\big)\nonumber\:. \end{eqnarray} Hence, from (\ref{defV}), we find \begin{equation*}
\nabla_j\big(|g|^{-1/2}\epsilon^{kj}\vec{L}\cdot\nabla_k\vec{\Phi}\big)\;=\;\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\:. \end{equation*} We again use a Hodge decomposition to write \begin{equation}\label{defS}
|g|^{-1/2}\epsilon^{kj}\vec{L}\cdot\nabla_k\vec{\Phi}\;=\;\nabla^jY+|g|^{-1/2}\epsilon^{kj}\nabla_k S\:, \end{equation} where \begin{equation}\label{defY} \Delta_g Y\;=\;\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\:. \end{equation}
Our next task consists in relating to each other the ``potentials" $\vec{R}$ and $S$ defined above. Although this is the fruit of a rather elementary computation, the result it yields has far-reaching consequences and which, as far as the author knows, has no direct empirical justification. Recall (\ref{defR}) and (\ref{defS}), namely: \begin{equation}\label{defRS} \left\{\begin{array}{lcl}
\nabla_k\vec{R}&=&\vec{L}\wedge\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\epsilon_{kj}\big(\vec{H}\wedge\nabla^j\vec{\Phi}+\nabla^j\vec{X}\big)\\[1ex]
\nabla_k S&=&\vec{L}\cdot\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\epsilon_{kj}\nabla^jY\:. \end{array}\right. \end{equation} Define the Gauss map \begin{equation*}
\star\,\vec{n}\;:=\;\dfrac{1}{2}\,|g|^{-1/2}\epsilon^{ab}\nabla_a\vec{\Phi}\wedge\nabla_b\vec{\Phi}\:. \end{equation*} We have \begin{eqnarray*}
(\star\,\vec{n})\cdot\nabla_k\vec{R}&=&\dfrac{1}{2}\,|g|^{-1/2}\epsilon^{ab}\big(\nabla_a\vec{\Phi}\wedge\nabla_b\vec{\Phi}\big)\cdot\Big[\vec{L}\wedge\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\epsilon_{kj}\big(\vec{H}\wedge\nabla^j\vec{\Phi}+\nabla^j\vec{X}\big) \Big]\\[1ex]
&=&|g|^{-1/2}\epsilon^{ab}g_{bk}\vec{L}\cdot\nabla_a\vec{\Phi}\,-\,|g|^{1/2}\epsilon_{kj}(\star\,\vec{n})\cdot\nabla^j\vec{X}\\[1ex]
&=&|g|^{-1/2}\epsilon^{ab}g_{bk}\big(\nabla_aS+|g|^{1/2}\epsilon_{aj}\nabla^jY \big)\,-\,|g|^{1/2}\epsilon_{kj}(\star\,\vec{n})\cdot\nabla^j\vec{X}\\[1ex]
&=&|g|^{1/2}\epsilon_{bk}\nabla^bS+\nabla_kY\,-\,|g|^{1/2}\epsilon_{kj}(\star\,\vec{n})\cdot\nabla^j\vec{X}\:, \end{eqnarray*} where we have used that $\vec{H}$ is a normal vector, along with the elementary identities \begin{equation*}
|g|^{-1/2}\epsilon^{ab}g_{bk}\;=\;|g|^{1/2}\epsilon_{bk}g^{ab}\qquad\text{and}\qquad\epsilon^{ab}\epsilon_{aj}\;=\;\delta^{b}_{j}\:. \end{equation*} The latter implies \begin{equation}\label{nablaS}
\nabla^jS\;=\;|g|^{-1/2}\epsilon^{jk}\big((\star\,\vec{n})\cdot\nabla_k\vec{R}-\nabla_kY\big)\,+\,(\star\,\vec{n})\cdot\nabla^j\vec{X}\:. \end{equation} Analogously, we find\footnote{$ (\omega_1\wedge\omega_2)\bullet(\omega_3\wedge\omega_4)\;=\;(\omega_2\cdot\omega_4)\omega_1\wedge\omega_3-(\omega_2\cdot\omega_3)\omega_1\wedge\omega_4-(\omega_1\cdot\omega_4)\omega_2\wedge\omega_3+(\omega_1\cdot\omega_3)\omega_2\wedge\omega_4\:. $} \begin{eqnarray*}
(\star\,\vec{n})\bullet\nabla_k\vec{R}&=&\dfrac{1}{2}\,|g|^{-1/2}\epsilon^{ab}\big(\nabla_a\vec{\Phi}\wedge\nabla_b\vec{\Phi}\big)\bullet\Big[\vec{L}\wedge\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\epsilon_{kj}\big(\vec{H}\wedge\nabla^j\vec{\Phi}+\nabla^j\vec{X}\big) \Big]\\[1ex]
&=&|g|^{-1/2}\epsilon^{ab}g_{bk}\nabla_a\vec{\Phi}\wedge\vec{L}\,+\,|g|^{-1/2}\epsilon^{ab}(\vec{L}\cdot\nabla_a\vec{\Phi})(\nabla_b\vec{\Phi}\wedge\nabla_k\vec{\Phi})\\
&&-\:\epsilon^{ab}\epsilon_{kb}\nabla_a\vec{\Phi}\wedge\vec{H}\,-\,|g|^{1/2}\epsilon_{kj}(\star\,\vec{n})\bullet\nabla^j\vec{X}\\[1ex]
&=&|g|^{1/2}\epsilon_{kb}\vec{L}\wedge\nabla^b\vec{\Phi}\,-\,(\star\,\vec{n})(\vec{L}\cdot\nabla_k\vec{\Phi})\,-\,\nabla_k\vec{\Phi}\wedge\vec{H}\,-\,|g|^{1/2}\epsilon_{kj}(\star\,\vec{n})\bullet\nabla^j\vec{X}\\[1ex]
&=&|g|^{1/2}\epsilon_{kj}\nabla^j\vec{R}+\nabla_k\vec{X}-(\star\,\vec{n})\big(\nabla_kS+|g|^{1/2}\epsilon_{kj}\nabla^jY \big)\,-\,|g|^{1/2}\epsilon_{kj}(\star\,\vec{n})\bullet\nabla^j\vec{X}\:. \end{eqnarray*} It hence follows that there holds \begin{equation}\label{nablaR}
\nabla^j\vec{R}\;=\;|g|^{-1/2}\epsilon^{kj}\big((\star\,\vec{n})\nabla_kS+(\star\,\vec{n})\bullet\nabla_k\vec{R}-\nabla_k\vec{X}\big)\,+\,(\star\,\vec{n})\nabla^jY\,+\,(\star\,\vec{n})\bullet\nabla^j\vec{X}\:. \end{equation} Applying divergence to each of (\ref{nablaS}) and (\ref{nablaR}) gives rise to the {\it conservative Willmore system}: \begin{equation}\label{conswillsys} \left\{\begin{array}{lcl}
|g|^{1/2}\Delta_g S&=&\epsilon^{jk}\partial_j(\star\,\vec{n})\cdot\partial_k\vec{R}
\,+\,|g|^{1/2}\nabla_j\big((\star\,\vec{n})\cdot\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\epsilon^{jk}\Big[\partial_j(\star\,\vec{n})\partial_kS+\partial_j(\star\,\vec{n})\bullet\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\big((\star\,\vec{n})\nabla^jY+(\star\,\vec{n})\bullet\nabla^j\vec{X}\big)\:, \end{array}\right. \end{equation} where $\partial_j$ denotes the derivative in flat local coordinates. \\
\noindent This system is to be supplemented with (\ref{defX}) and (\ref{defY}), which, in turn, are solely determined by the value of the Willmore operator $\vec{\mathcal{W}}$ via equation (\ref{defV}). There is furthermore another useful equation to add to the conservative Willmore system, namely one relating the potentials $S$ and $\vec{R}$ back to the immersion $\vec{\Phi}$. We now derive this identity. Using (\ref{defRS}), it easily follows that \begin{eqnarray*}
&&\hspace{-2cm}\epsilon^{km}\big(\nabla_k\vec{R}+|g|^{1/2}\epsilon_{kj}\nabla^j\vec{X}\big)\bullet\nabla_m\vec{\Phi}\\[1ex]
&=&\epsilon^{km}\big(\vec{L}\wedge\nabla_k\vec{\Phi}-|g|^{1/2}\epsilon_{kj}\vec{H}\wedge\nabla^j\vec{\Phi}\big)\bullet\nabla_m\vec{\Phi}\\[1ex]
&=&\epsilon^{km}\Big[\big(\vec{L}\cdot\nabla_m\vec{\Phi}\big)\nabla_k\vec{\Phi}\,-\,g_{mk}\vec{L}\,-\,|g|^{1/2}\epsilon_{km}\vec{H}\Big]\\[1ex]
&=&\epsilon^{km}\big(\nabla_mS+|g|^{1/2}\epsilon_{mj}\nabla^jY\big)\nabla_k\vec{\Phi}\,-\,2\,|g|^{1/2}\vec{H}\:. \end{eqnarray*} Since $\Delta_g\vec{\Phi}=2\vec{H}$, we thus find \begin{equation}\label{law4}
\partial_j\Big[\epsilon^{jk}\big(S\partial_k\vec{\Phi}+\vec{R}\bullet\partial_k\vec{\Phi} \big)+|g|^{1/2}\nabla^j\vec{\Phi}\Big]\;=\;|g|^{1/2}\big(\nabla^jY\nabla_j\vec{\Phi}+\nabla^j\vec{X}\bullet\nabla_j\vec{\Phi} \big)\:.
\end{equation} At the equilibrium (i.e. when $\vec{\mathcal{W}}=\vec{0}$), the right-hand side of the latter identically vanishes and we recover a conservation law. With the help of a somewhat tedious computation, one verifies that this conservation law follows from the invariance of the Willmore energy by inversion and Noether's theorem. To do so, one may for example consider an infinitesimal variation of the type $\delta\vec{\Phi}\;=\;|\vec{\Phi}|^2\vec{a}-2(\vec{\Phi}\cdot\vec{a})\vec{\Phi}$, for some fixed constant vector $\vec{a}\in\mathbb{R}^m$.\\
We summarize our results in the following pair of systems. \begin{equation}\label{side}
\left\{\begin{array}{rcl}\vec{\mathcal{W}}&=&\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\\[1ex] \Delta_g\vec{V}&=&-\,\vec{\mathcal{W}}\\[1ex] \Delta_g\vec{X}&=&\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\\[1ex] \Delta_g{Y}&=&\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi} \end{array}\right. \end{equation} and \begin{equation}\label{thesys} \left\{\begin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\epsilon^{jk}\partial_j(\star\,\vec{n})\cdot\partial_k\vec{R}
\,+\,|g|^{1/2}\nabla_j\big((\star\,\vec{n})\cdot\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\epsilon^{jk}\Big[\partial_j(\star\,\vec{n})\partial_kS+\partial_j(\star\,\vec{n})\bullet\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\big((\star\,\vec{n})\nabla^jY+(\star\,\vec{n})\bullet\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{\Phi}&=&-\,\epsilon^{jk}\Big[\partial_jS\partial_k\vec{\Phi}+\partial_j\vec{R}\bullet\partial_k\vec{\Phi}\Big]+|g|^{1/2}\big(\nabla^jY\nabla_j\vec{\Phi}+\nabla^j\vec{X}\bullet\nabla_j\vec{\Phi} \big)\:. \end{array}\right. \end{equation}
\noindent In the next section, we will examine more precisely the structure of this system through several examples.
\begin{Rm} Owing to the identities \begin{equation*} \vec{u}\bullet\vec{v}\;=\;(\star\vec{u})\times\vec{v}\quad\text{and}\quad \vec{u}\bullet\vec{w}\;=\;\star\big[(\star\vec{u})\times(\star\vec{w})\big]\:\quad\text{for}\:\:\:\vec{u}\in\Lambda^2(\mathbb{R}^3), \vec{v}\in\Lambda^1(\mathbb{R}^3), \vec{w}\in\Lambda^2(\mathbb{R}^3)\:, \end{equation*} we can in $\mathbb{R}^3$ recast the above systems as \begin{equation}\label{side1}
\left\{\begin{array}{rcl}\vec{\mathcal{W}}&=&\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\\[1ex] \Delta_g\vec{V}&=&-\,\vec{\mathcal{W}}\\[1ex] \Delta_g\vec{X}&=&\nabla^j\vec{V}\times\nabla_j\vec{\Phi}\\[1ex] \Delta_g{Y}&=&\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi} \end{array}\right. \end{equation} and \begin{equation}\label{thesys1} \left\{\begin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\epsilon^{jk}\partial_j\vec{n}\cdot\partial_k\vec{R}
\,+\,|g|^{1/2}\nabla_j\big(\vec{n}\cdot\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\epsilon^{jk}\Big[\partial_j\vec{n}\,\partial_kS+\partial_j\vec{n}\times\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\big(\vec{n}\,\nabla^jY+\vec{n}\times\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{\Phi}&=&-\,\epsilon^{jk}\Big[\partial_jS\,\partial_k\vec{\Phi}+\partial_j\vec{R}\times\partial_k\vec{\Phi}\Big]+|g|^{1/2}\big(\nabla^jY\nabla_j\vec{\Phi}+\nabla^j\vec{X}\times\nabla_j\vec{\Phi} \big)\:. \end{array}\right. \end{equation} In this setting, $\vec{X}$ and $\vec{R}$ are no longer 2-vectors, but rather simply vectors of $\mathbb{R}^3$. \end{Rm}
\section{Examples}
\subsection{Willmore Immersions}
A smooth immersion $\vec{\Phi}:\Sigma\rightarrow\mathbb{R}^{m\ge3}$ of an oriented surface $\Sigma$ with induced metric $g=\vec{\Phi}^*g_{\mathbb{R}^m}$ and corresponding mean curvature vector $\vec{H}$, is said to be Willmore if it is a critical point of the Willmore energy $\int_\Sigma|\vec{H}|^2d\text{vol}_g$. They are known \cite{Wil2,Wei} to satisfy the Euler-Lagrange equation $$
\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\;=\;\vec{0}\:. $$ In the notation of the previous section, this corresponds to the case $\vec{\mathcal{W}}=\vec{0}$. According to (\ref{side}), we have the freedom to set $\vec{V}$, $\vec{X}$, and $Y$ to be identically zero. The Willmore equation then yields the second-order system in divergence form \begin{equation}\label{thesyswillmore} \left\{\begin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\epsilon^{jk}\partial_j(\star\,\vec{n})\cdot\partial_k\vec{R} \\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\epsilon^{jk}\Big[\partial_j(\star\,\vec{n})\partial_kS+\partial_j(\star\,\vec{n})\bullet\partial_k\vec{R}\Big]\\[1ex]
|g|^{1/2}\Delta_g\vec{\Phi}&=&-\,\epsilon^{jk}\Big[\partial_jS\partial_k\vec{\Phi}+\partial_j\vec{R}\bullet\partial_k\vec{\Phi}\Big]\:. \end{array}\right.
\end{equation} This system was originally derived by Rivi\`ere in \cite{Riv2}. For notational reasons, the detailed computations were carried out only in local conformal coordinates, that is when $g_{ij}=\text{e}^{2\lambda}\delta_{ij}$, for some conformal parameter $\lambda$. The analytical advantages of the Willmore system (\ref{thesyswillmore}) have been exploited in numerous works \cite{BR1, BR2, BR3, Riv2, Riv3, Riv4}. The flat divergence form of the operator $|g|^{1/2}\Delta_g$ and the Jacobian-type structure of the right-hand side enable using fine Wente-type estimates in order to produce non-trivial local information about Willmore immersions (see aforementioned works).
\begin{Rm} Any Willmore immersion will satisfy the system (\ref{thesyswillmore}). The converse is however not true. Indeed, in order to derive (\ref{thesyswillmore}), we first obtained the existence of some ``potential" $\vec{L}$ satisfying the first-order equation \begin{equation}\label{ceteq}
|g|^{-1/2}\epsilon^{kj}\nabla_k\vec{L}\;=\;\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi}\:. \end{equation} In doing so, we have gone from the Willmore equation, which is second-order for $\vec{H}$, to the above equation, which is only first-order in $\vec{H}$, thereby introducing in the problem an extraneous degree of freedom. As we shall see in the next section, (\ref{ceteq}) is in fact equivalent to the conformally-constrained Willmore equation, which, as one might suspect, is the Willmore equation supplemented with an additional degree of freedom appearing in the guise of a Lagrange multiplier. \end{Rm}
\subsection{Conformally-Constrained Willmore Immersions}
Varying the Willmore energy $\int_\Sigma|\vec{H}|^2d\text{vol}_g$ in a fixed conformal class (i.e. with infinitesimal, smooth, compactly supported, conformal variations) gives rise to a more general class of surfaces called {\it conformally-constrained Willmore surfaces} whose corresponding Euler-Lagrange equation \cite{BPP, KL, Sch} is expressed as follows. Let $\vec{h}_0$ denote the trace-free part of the second fundamental form, namely \begin{equation*} \vec{h}_0\;:=\;\vec{h}\,-\,\vec{H} g\:. \end{equation*} A conformally-constrained Willmore immersion $\vec{\Phi}$ satisfies \begin{equation}\label{cwe}
\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\;=\;\big(\vec{h}_0\big)_{ij}q^{ij}\:, \end{equation} where $q$ is a transverse\footnote{i.e. $q$ is divergence-free: $\nabla^j q_{ji}=0\:\:\forall i$.} traceless symmetric 2-form. This tensor $q$ plays the role of Lagrange multiplier in the constrained variational problem. \\
In \cite{KS3}, it is shown that under a suitable ``small enough energy" assumption, a minimizer of the Willmore energy in a fixed conformal class exists and is smooth. The existence of a minimizer without any restriction on the energy is also obtained in \cite{Riv3} where it is shown that the minimizer is either smooth (with the possible exclusion of finitely many branch points if the energy is large enough to grant their formation) or else isothermic\footnote{the reader will find in \cite {Riv5} an interesting discussion on isothermic immersions.}. One learns in \cite{Sch} that non-degenerate critical points of the Willmore energy constrained to a fixed conformal class are solutions of the conformally constrained Willmore equation. Continuing along the lines of \cite{Riv3}, further developments are given in \cite{Riv6}, where the author shows that if either the genus of the surface satisfies $g\le2$, or else if the Teichm\"uller class of the immersion is not hyperelliptic\footnote{A class in the Teichm\"uller space is said to be {\it hyperelliptic} if the tensor products of holomorphic 1-forms do not generate the vector space of holomorphic quadratic forms.}, then any critical point $\vec{\Phi}$ of the Willmore energy for $C^1$ perturbations included in a submanifold of the Teichm\"uller space is in fact analytic and (\ref{cwe}) is satisfied for some transverse traceless symmetric 2-tensor $q$. \\
The notion of conformally constrained Willmore surfaces clearly generalizes that of Willmore surfaces, obtained via all smooth compactly supported infinitesimal variations (setting $q\equiv0$ in (\ref{cwe})). In \cite{KL}, it is shown that CMC Clifford tori are conformally constrained Willmore surfaces. In \cite{BR1}, the conformally constrained Willmore equation (\ref{cwe}) arises as that satisfied by the limit of a Palais-Smale sequence for the Willmore functional. \\
Minimal surfaces are examples of Willmore surfaces, while parallel mean curvature surfaces\footnote{parallel mean curvature surfaces satisfy $\pi_{\vec{n}} d\vec{H}\equiv\vec{0}$. They generalize to higher codimension the notion of constant mean curvature surfaces defined in $\mathbb{R}^3$. See [YBer].} are examples of conformally-constrained Willmore surfaces\footnote{{\it a non}-minimal parallel mean curvature surface is however {\it not} Willmore (unless of course it is the conformal transform of a Willmore surface ; e.g. the round sphere).}. Not only is the Willmore energy invariant under reparametrization of the domain, but more remarkably, it is invariant under conformal transformations of $\mathbb{R}^m\cup\{\infty\}$. Hence, the image of a [conformally-constrained] Willmore immersion through a conformal transformation is again a [conformally-constrained] Willmore immersion. It comes thus as no surprise that the class of Willmore immersions [resp. conformally-constrained Willmore immersions] is considerably larger than that of immersions whose mean curvature vanishes [resp. is parallel], which is {\it not} preserved through conformal diffeomorphisms. \\
Comparing (\ref{cwe}) to the first equation in (\ref{side}), we see that $\vec{\mathcal{W}}=-\,\big(\vec{h}_0\big)_{ij}q^{ij}$. Because $q$ is traceless and transverse, we have $$ \big(\vec{h}_0\big)_{ij}q^{ij}\;\equiv\;\vec{h}_{ij}q^{ij}-\vec{H}g_{ij}q^{ij}\;=\;\vec{h}_{ij}q^{ij}\;=\;\nabla_j(q^{ij}\nabla_i\vec{\Phi})\:. $$ Accordingly, choosing $\nabla^j\vec{V}=-\,q^{ij}\nabla_i\vec{\Phi}$ will indeed solve the second equation (\ref{side}). Observe next that $$ \nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\;=\;-\,q^{ij}g_{ij}\;=\;0\:, $$ since $q$ is traceless. Putting this into the fourth equation of (\ref{side}) shows that we may choose $Y\equiv0$. Furthermore, as $q$ is symmetric, it holds $$ \nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\;=\;-\,q^{ij}\nabla_i\vec{\Phi}\wedge\nabla_j\vec{\Phi}\;=\;\vec{0}\:, $$ so that the third equation in (\ref{side}) enables us to choose $\vec{X}\equiv\vec{0}$. \\ Altogether, we see that a conformally-constrained Willmore immersion, just like a ``plain" Willmore immersion (i.e. with $q\equiv0$) satisfies the system (\ref{thesyswillmore}). In fact, it was shown in \cite{BR1} that to any smooth solution $\vec{\Phi}$ of (\ref{thesyswillmore}), there corresponds a transverse, traceless, symmetric 2-form $q$ satisfying (\ref{cwe}).
\subsection{Bilayer Models} Erythrocytes (also called red blood cells) are the body's principal mean of transporting vital oxygen to the organs and tissues. The cytoplasm (i.e. the ``inside") of an erythrocyte is rich in hemoglobin, which chemically tends to bind to oxygen molecules and retain them. To maximize the possible intake of oxygen by each cell, erythrocytes -- unlike all the other types of cells of the human body -- have no nuclei\footnote{this is true for all mammals, not just for humans. However, the red blood cells of birds, fish, and reptiles do contain a nucleus.}. The membrane of an erythrocyte is a bilayer made of amphiphilic molecules. Each molecule is made of a ``head" (rather large) with a proclivity for water, and of a ``tail" (rather thin) with a tendency to avoid water molecules. When such amphiphilic molecules congregate, they naturally arrange themselves in a bilayer, whereby the tails are isolated from water by being sandwiched between two rows of heads. The membrane can then close over itself to form a vesicle. Despite the great biochemical complexity of erythrocytes, some phenomena may be described and explained with the sole help of physical mechanisms\footnote{The most celebrated such phenomenon was first observed by the Polish pathologist Tadeusz Browicz in the late 19$^{\text{th}}$ century \cite{Bro}. Using a microscope, he noted that the luminous intensity reflected by a red blood cell varies erratically along its surface, thereby giving the impression of flickering. In the 1930s, the Dutch physicist Frits Zernicke invented the phase-constrast microscope which revealed that the erratic flickering found by Browicz is in fact the result of very minute and very rapid movements of the cell's membrane. These movements were finally explained in 1975 by French physicists Fran\c{c}oise Brochard and Jean-Fran\c{c}ois Lennon: they are the result of a spontaneous thermic agitation of the membrane, which occurs independently of any particular biological activity \cite{BL}.}. For example, to understand the various shapes an erythrocyte might assume, it is sensible to model the red blood cell by a drop of hemoglobin (``vesicle") whose membrane is made of a lipid bilayer \cite{Lip}. Such ``simple" objects, called {\it liposomes}, do exist in Nature, and they can be engineered artificially\footnote{The adjective ``simple" is to be understood with care: stable vesicles with non-trivial topology can be engineered and observed. See \cite{MB} and \cite{Sei}. }. The membrane of a liposome may be seen as a viscous fluid separated from water by two layers of molecules. Unlike a solid, it can undergo shear stress. Experimental results have however shown that vesicles are very resistant and the stress required to deform a liposome to its breaking-point is so great that in practice, vesicles evolving freely in water are never submitted to such destructive forces. One may thus assume that the membrane of a liposome is incompressible: its area remains constant. The volume it encloses also stays constant, for the inside and the outside of the vesicle are assumed to be isolated. As no shearing and no stretching are possible, one may wonder which forces dictate the shape of a liposome. To understand the morphology of liposomes, Canham \cite{Can}, Evans \cite{Eva}, and Helfrich \cite{Hef} made the postulate that, just as any other elastic material does, the membrane of a liposome tends to minimize its elastic energy. As we recalled in the introduction, the elastic energy of a surface is directly proportional to the Willmore energy. Accordingly, to understand the shape of a liposome, one would seek minimizers of the Willmore energy subject to constraints on the area and on the enclosed volume. A third constraint could be taken into account. As the area of the inner layer of the membrane is slightly smaller than the area of its outer layer, it takes fewer molecules to cover the former as it does to cover the latter. This difference, called {\it asymmetric area difference}, is fixed once and for all when the liposome forms. Indeed, no molecule can move from one layer to the other, for that would require its hydrophilic head to be, for some time, facing away from water. From a theoretical point of view, the relevant energy to study is thus: \begin{equation}\label{canham} E\;:=\;\int_{\Sigma}H^2d\text{vol}_g+\alpha A(\Sigma)+\beta V(\Sigma)+\gamma M(\Sigma)\:, \end{equation} where $H$ denotes the mean curvature scalar\footnote{in this section, we will content ourselves with working in codimension 1.}, $A(\Sigma)$, $V(\Sigma)$, $M(\Sigma)$ denote respectively the area, volume, and asymmetric area difference of the membrane $\Sigma$, and where $\alpha$, $\beta$, and $\gamma$ are three Lagrange multipliers. Depending on the authors' background, the energy (\ref{canham}) bears the names Canham, Helfrich, Canham-Helfrich, bilayer coupling model, spontaneous curvature model, among others.\\ This energy -- in the above form or analogous ones -- appears prominently in the applied sciences literature. It would be impossible to list here a comprehensive account of relevant references. The interested reader will find many more details in \cite{Sei} and \cite{Voi} and the references therein. In the more ``analytical" literature, the energy (\ref{canham}) is seldom found (except, of course, in the case when all three Lagrange multipliers vanish). We will, in time, recall precise instances in which it has been studied. But prior to doing so, it is interesting to pause for a moment and better understand a term which might be confusing to the mathematician reader, namely the asymmetric area difference $M(\Sigma)$. In geometric terms, it is simply the total curvature of $\Sigma$: \begin{equation}\label{defM} M(\Sigma)\;:=\;\int_{\Sigma}H\,d\text{vol}_g\:. \end{equation} This follows from the fact that the infinitesimal variation of the area is the mean curvature, and thus the area difference between two nearby surfaces is the first moment of curvature. Hence, we find the equivalent expression \begin{equation}\label{canam} E\;=\;\int_{\Sigma}\bigg(H+\dfrac{\gamma}{2}\bigg)^{\!2}d\text{vol}_g+\bigg(\alpha-\dfrac{\gamma^2}{4}\bigg) A(\Sigma)+\beta V(\Sigma)\:. \end{equation} This form of the bilayer energy is used, inter alia, in \cite{BWW2, Whe1}, where the constant $-\gamma/2$ is called {\it spontaneous curvature}. \\
From a purely mathematical point of view, one may study the energy (\ref{canham}) not just for embedded surfaces, but more generally for immersions. An appropriate definition for the volume $V$ must be assigned to such an immersion $\vec{\Phi}$. As is shown in \cite{MW}, letting as usual $\vec{n}$ denote the outward unit-normal vector to the surface, one defines \begin{equation*} V(\Sigma)\;:=\;\int_\Sigma\vec{\Phi}^*(d\mathcal{H}^3)\;=\;\int_{\Sigma}\vec{\Phi}\cdot\vec{n}\,d\text{vol}_g\:, \end{equation*} where $d\mathcal{H}^3$ is the Hausdorff measure in $\mathbb{R}^3$. Introducing the latter and (\ref{defM}) into (\ref{canham}) yields \begin{equation}\label{veneer} E\;=\;\int_{\Sigma}\big(H^2+\gamma H+\beta\,\vec{\Phi}\cdot\vec{n}+\alpha\big)\,d\text{vol}_g\:. \end{equation}
We next vary the energy $E$ along a normal variation of the form $\delta\vec{\Phi}=\vec{B}\equiv B\vec{n}$. Using the computations from the previous section, it is not difficult to see that \begin{equation}\label{varr1} \delta\int_\Sigma d\text{vol}_g\;=\;-\,2\int_{\Sigma}\vec{B}\cdot\vec{H} \,d\text{vol}_g\qquad\text{and}\qquad \delta\int_\Sigma H\,d\text{vol}_g\;=\;\int_{\Sigma} (\vec{B}\cdot\vec{n})\bigg(\dfrac{1}{2}h^i_jh^j_i-2H^2\bigg)\,d\text{vol}_g\:. \end{equation} With a bit more effort, in \cite{MW}, it is shown that \begin{equation}\label{varr2} \delta\int_\Sigma \vec{\Phi}\cdot\vec{n}\, d\text{vol}_g\;=\;-\int_{\Sigma}\vec{B}\cdot\vec{n}\,d\text{vol}_g\:. \end{equation} Putting (\ref{diffen}), (\ref{varr1}), and (\ref{varr2}) into (\ref{veneer}) yields the corresponding Euler-Lagrange equation \begin{equation}\label{ELhelf}
\vec{\mathcal{W}}\;:=\;\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\;=\;2\,\alpha\vec{H}+\beta\,\vec{n}-\gamma\bigg(\dfrac{1}{2}h^i_jh^j_i-2H^2 \bigg)\,\vec{n}\:. \end{equation} We now seek a solution to the second equation in (\ref{side1}), namely a vector $\vec{V}$ satisfying $\Delta_g\vec{V}=-\vec{\mathcal{W}}$. To do so, it suffices to observe that \begin{equation*} 2\vec{H}\;=\;\Delta_g\vec{\Phi}\qquad\text{and}\qquad \big(h^i_jh^j_i-4H^2\big)\vec{n}\;=\;\nabla_j\Big[(h^{ij}-2Hg^{ij})\nabla_i\vec{\Phi}\Big]\:, \end{equation*} where we have used the Codazzi-Mainardi identity. Furthermore, it holds \begin{equation*}
2\,\vec{n}\;=\;|g|^{-1/2}\epsilon^{ij}\nabla_i\vec{\Phi}\times\nabla_j\vec{\Phi}\;=\;\nabla_j\big[-|g|^{-1/2}\epsilon^{ij}\vec{\Phi}\times\nabla_i\vec{\Phi} \big]\:. \end{equation*} Accordingly, we may choose \begin{equation*}
\nabla^j\vec{V}\;=\;-\,\alpha\,\nabla^j\vec{\Phi}\,+\,\dfrac{\beta}{2}\,|g|^{-1/2}\epsilon^{ij}\,\vec{\Phi}\times\nabla_i\vec{\Phi}\,+\,\dfrac{\gamma}{2}\big(h^{ij}-2Hg^{ij}\big)\nabla_i\vec{\Phi}\:. \end{equation*} Introduced into the third equation of (\ref{side1}), the latter yields \begin{eqnarray}
\Delta_g\vec{X}&=&\nabla^j\vec{V}\times\nabla_j\vec{\Phi}\;\;=\;\;\dfrac{\beta}{2}\,|g|^{-1/2}\epsilon^{ij}\big(\vec{\Phi}\times\nabla_i\vec{\Phi}\big)\times\nabla_j\vec{\Phi}\nonumber\\[0ex]
&=&\dfrac{\beta}{2}\,|g|^{-1/2}\epsilon^{ij}\big[(\vec{\Phi}\cdot\nabla_j\vec{\Phi})\nabla_i\vec{\Phi}-g_{ij}\vec{\Phi}\big]\;\;=\;\;\dfrac{\beta}{4}\,|g|^{-1/2}\epsilon^{ij}\,\nabla_j|\vec{\Phi}|^2\,\nabla_i\vec{\Phi}\:,\nonumber \end{eqnarray} so that we may choose \begin{equation*}\label{XHel}
\nabla^j\vec{X}\;=\;\dfrac{\beta}{4}\,|g|^{-1/2}\epsilon^{ij}|\vec{\Phi}|^2\,\nabla_i\vec{\Phi}\:. \end{equation*} Then we find \begin{equation}\label{Xprop}
\vec{n}\times\nabla^j\vec{X}\;=\;\dfrac{\beta}{4}\,|\vec{\Phi}|^2\nabla^j\vec{\Phi}\qquad\text{and}\qquad\nabla^j\vec{X}\times\nabla^j\vec{\Phi}\;=\;\dfrac{\beta}{2}\,|\vec{\Phi}|^2\,\vec{n}\:. \end{equation} Analogously, the fourth equation of (\ref{side1}) gives \begin{eqnarray}\label{YHel}
\Delta_gY&=&\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\;\;=\;\;-\,2\alpha\,-\,\gamma H\,+\,\dfrac{\beta}{2}\,|g|^{-1/2}\epsilon^{ij}\big(\vec{\Phi}\times\nabla_i\vec{\Phi}\big)\cdot\nabla_j\vec{\Phi}\nonumber\\[0ex] &=&-\,2\alpha\,-\,\gamma H\,+\,\beta\,\vec{\Phi}\cdot\vec{n}\:. \end{eqnarray}
With (\ref{Xprop}) and (\ref{YHel}), the system (\ref{thesys1}) becomes \begin{equation}\label{thesyshel} \left\{\begin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\epsilon^{jk}\partial_j\vec{n}\cdot\partial_k\vec{R}\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\epsilon^{jk}\Big[\partial_j\vec{n}\,\partial_kS+\partial_j\vec{n}\times\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\bigg(\vec{n}\,\nabla^jY+\dfrac{\beta}{4}\,|\vec{\Phi}|^2\nabla^j\vec{\Phi} \bigg)\\[1.5ex]
|g|^{1/2}\Delta_g\vec{\Phi}&=&-\,\epsilon^{jk}\Big[\partial_jS\,\partial_k\vec{\Phi}+\partial_j\vec{R}\times\partial_k\vec{\Phi}\Big]+|g|^{1/2}\bigg(\nabla_j\vec{\Phi}\nabla^jY+\dfrac{\beta}{2}\,|\vec{\Phi}|^2\,\vec{n} \bigg)\:. \end{array}\right. \end{equation}
Under suitable regularity hypotheses (e.g. that the immersion be locally Lipschitz and lie in the Sobolev space $W^{2,2}$), one can show that the non-Jacobian term in the second equation, namely $$
|g|^{1/2}\nabla_j\bigg(\vec{n}\,\nabla^jY+\dfrac{\beta}{4}\,|\vec{\Phi}|^2\nabla^j\vec{\Phi} \bigg)\:, $$ is a subcritical perturbation of the Jacobian term. Analyzing (\ref{thesyshel}) becomes then very similar to analyzing the Willmore system (\ref{thesyswillmore}). Details may be found in \cite{BWW1}. \\
In some cases, our so-far local considerations can yield global information. If the vesicle we consider has the topology of a sphere, every loop on it is contractible to a point. The Hodge decompositions which we have performed in Section II to deduce the existence of $\vec{V}$, and subsequently that of $\vec{X}$ and $Y$ hold globally. Integrating (\ref{YHel}) over the whole surface $\Sigma$ then gives the {\it balancing condition}: $$ 2\alpha A(\Sigma)+\gamma M(\Sigma)\;=\;\beta V(\Sigma)\:. $$ This condition is well-known in the physics literature \cite{CG, Sei}.
\begin{Rm} Another instance in which minimizing the energy (\ref{veneer}) arises is the isoperimetric problem \cite{KMR, Scy}, which consists in minimizing the Willmore energy under the constraint that the dimensionless isoperimetric ratio $\sigma:=A^3/V^2$ be a given constant in $(0,1]$. As both the Willmore energy and the constraint are invariant under dilation, one might fix the volume $V=1$, forcing the area to satisfy $A=\sigma^{1/3}$. This problem is thus equivalent to minimizing the energy (\ref{veneer}) with $\gamma=0$ (no constraint imposed on the total curvature, but the volume and area are prescribed separately). One is again led to the system (\ref{thesyshel}) and local analytical information may be inferred. \end{Rm}
\subsection{Chen's Problem}
An isometric immersion $\vec{\Phi}:N^{n}\rightarrow\mathbb{R}^{m>n}$ of an $n$-dimensional Riemannian manifold $N^n$ into Euclidean space is called {\it biharmonic} if the corresponding mean-curvature vector $\vec{H}$ satisfies \begin{equation}\label{chen} \Delta_g\vec{H}\;=\;\vec{0}\:. \end{equation} The study of biharmonic submanifolds was initiated by B.-Y. Chen \cite{BYC1} in the mid 1980s as he was seeking a classification of the finite-type submanifolds in Euclidean spaces. Independently, G.Y. Jiang \cite{Jia} also studied (\ref{chen}) in the context of the variational analysis of the biharmonic energy in the sense of Eells and Lemaire. Chen conjectures that a biharmonic immersion is necessarily minimal\footnote{The conjecture as originally stated is rather analytically vague: no particular hypotheses on the regularity of the immersion are {\it a priori} imposed. Many authors consider only smooth immersions.}. Smooth solutions of (\ref{chen}) are known to be minimal for $n=1$ \cite{Dim1}, for $(n,m)=(2,3)$ \cite{Dim2}, and for $(n,m)=(3,4)$ \cite{HV}. A growth condition allows a simple PDE argument to work in great generality \cite{Whe2}. Chen's conjecture has also been solved under a variety of hypotheses (see the recent survey paper \cite{BYC2}). The statement remains nevertheless open in general, and in particular for immersed surfaces in $\mathbb{R}^m$. In this section, we show how our reformulation of the Willmore equation may be used to recast the fourth-order equation (\ref{chen}) in a second-order system with interesting analytical features. \\
Let us begin by inspecting the tangential part of (\ref{chen}), namely, \begin{eqnarray}\label{tgtchen} \vec{0}&=&\pi_T\Delta_g\vec{H}\;=\;\big(\nabla_k\vec{\Phi}\cdot\nabla_j\nabla^j\vec{H})\nabla^k\vec{\Phi}\nonumber\\[1ex] &=&\Big[\nabla_j\big(\nabla_k\vec{\Phi}\cdot\nabla^j\vec{H} \big)-\vec{h}_{jk}\cdot\nabla^j\vec{H} \Big]\nabla^k\vec{\Phi}\;\;=\;\;-\,\Big[\vec{H}\cdot\nabla_j\vec{h}^{j}_{k}+2\,\vec{h}_{jk}\cdot\nabla^j\vec{H} \Big]\nabla^k\vec{\Phi}\nonumber\\[1ex]
&=&-\,\Big[\nabla_k|\vec{H}|^2+2\,\vec{h}_{jk}\cdot\nabla^j\vec{H} \Big]\nabla^k\vec{\Phi}\:, \end{eqnarray} where we have used that $\vec{H}$ is a normal vector, as well as the Codazzi-Mainardi identity. With the help of this equation, one obtains a decisive identity, which we now derive, and which too makes use of the Codazzi-Mainardi equation. \begin{eqnarray}\label{deci}
\nabla_j\Big[|\vec{H}|^2\nabla^j\vec{\Phi}-2(\vec{H}\cdot\vec{h}^{jk})\nabla_k\vec{\Phi}\Big]&=&-\,\nabla_j|\vec{H}|^2\nabla^j\vec{\Phi}+2|\vec{H}|^2\vec{H}-2(\vec{h}^{jk}\cdot\nabla_j\vec{H})\nabla_k\vec{\Phi}-2(\vec{H}\cdot\vec{h}^{jk})\vec{h}_{jk}\nonumber\\[1ex]
&\stackrel{\text{(\ref{tgtchen})}}{=}&2|\vec{H}|^2\vec{H}-2(\vec{H}\cdot\vec{h}^{jk})\vec{h}_{jk}\:. \end{eqnarray} Note that, in general, there holds \begin{eqnarray}\label{bidule} \pi_T\nabla^j\vec{H}&=&\big[\nabla_k\vec{\Phi}\cdot\nabla^j\vec{H}\big]\nabla^k\vec{\Phi}\;\;=\;\;-\,\big[\vec{H}\cdot\vec{h}^{j}_{k}\big]\nabla^k\vec{\Phi}\:. \end{eqnarray} An immersion whose mean curvature satisfies (\ref{chen}) has thus the property that \begin{eqnarray*} \Delta_\perp\vec{H}&:=&\pi_{\vec{n}}\nabla_j\pi_{\vec{n}}\nabla^j\vec{H}\;\;=\;\;\pi_{\vec{n}}\Delta_g\vec{H}\,-\,\pi_{\vec{n}}\nabla_j\pi_T\nabla^j\vec{H}\;\;=\;\;\pi_{\vec{n}}\nabla_j\Big[\big(\vec{H}\cdot\vec{h}^{j}_{k}\big)\nabla^k\vec{\Phi} \Big]\nonumber\\[0ex] &=&\big(\vec{H}\cdot\vec{h}^{j}_{k}\big)\vec{h}^{k}_{j}\:. \end{eqnarray*} Putting the latter into the first equation of (\ref{side}) yields \begin{equation}\label{ELchen}
\vec{\mathcal{W}}\;:=\;\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\;=\;2(\vec{H}\cdot\vec{h}^{jk})\vec{h}_{jk}-2|\vec{H}|^2\vec{H}\:. \end{equation} We now seek a solution to the second equation in (\ref{side1}), namely a vector $\vec{V}$ satisfying $\Delta_g\vec{V}=-\vec{\mathcal{W}}$. To do so, it suffices to compare (\ref{ELchen}) and (\ref{deci}) to see that we may choose \begin{equation*}
\nabla^j\vec{V}\;=\;|\vec{H}|^2\nabla^j\vec{\Phi}-2(\vec{H}\cdot\vec{h}^{jk})\nabla_k\vec{\Phi}\:. \end{equation*} Introduced into the third equation of (\ref{side1}), the latter yields immediately \begin{equation*} \Delta_g\vec{X}\;=\;\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\;=\;\vec{0}\:, \end{equation*} thereby prompting us to choosing $\vec{X}\equiv\vec{0}$. The fourth equation of (\ref{side1}) gives \begin{equation*}
\Delta_gY\;=\;\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\;=\;-\,2|\vec{H}|^2\:. \end{equation*} On the other hand, owing to (\ref{chen}) and (\ref{bidule}), one finds \begin{equation*}
\Delta_g(\vec{\Phi}\cdot\vec{H})\;=\;-\,2|\vec{H}|^2\:, \end{equation*} so that we may choose $Y=\vec{\Phi}\cdot\vec{H}$. Introducing these newly found facts for $\vec{X}$ and $Y$ into the system (\ref{thesys1}) finally gives \begin{equation}\label{thesyschen} \left\{\begin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\epsilon^{jk}\partial_j(\star\,\vec{n})\cdot\partial_k\vec{R}\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\epsilon^{jk}\Big[\partial_j(\star\,\vec{n})\partial_kS+\partial_j(\star\,\vec{n})\bullet\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\big((\star\,\vec{n})\nabla^j(\vec{\Phi}\cdot\vec{H})\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{\Phi}&=&-\,\epsilon^{jk}\Big[\partial_jS\partial_k\vec{\Phi}+\partial_j\vec{R}\bullet\partial_k\vec{\Phi}\Big]+|g|^{1/2}\big(\nabla^j(\vec{\Phi}\cdot\vec{H})\nabla_j\vec{\Phi} \big)\:. \end{array}\right. \end{equation}
As previously noted, the reformulation of (\ref{trans}) in the form (\ref{thesys1}) is mostly useful to reduce the nonlinearities present in (\ref{trans}). Moreover, a local analysis of (\ref{thesys1}) is possible when the non-Jacobian terms on the right-hand side are subcritical perturbations of the Jacobian terms. This is not {\it a priori} the case for (\ref{thesyschen}), and the equation (\ref{chen}) is linear to begin with. Nevertheless, the system (\ref{thesyschen}) has enough suppleness\footnote{owing mostly to the fact that the function $Y:=\vec{\Phi}\cdot\vec{H}$ satisfies $\Delta_gY\le 0$ and $\Delta_g^2Y\le0$.} and enough structural features to deduce interesting analytical facts about solutions of (\ref{chen}), under mild regularity requirements. This is discussed in detail in a work to appear \cite{BWW1}\footnote{see also \cite{Whe3} which contains interesting estimates for equation (\ref{chen}).}.
\subsection{Point-Singularities}
As was shown in \cite{YBer, BR2, Riv2}, the Jacobian-type system (\ref{side}) is particularly suited to the local analysis of point-singularities. The goal of this section is not to present a detailed account of the local analysis of point-singularities -- this is one of the topics of \cite{BWW1} -- but rather to give the reader a few pertinent key arguments on how this could be done. \\
Let $\vec{\Phi}:D^2\setminus\{0\}\rightarrow\mathbb{R}^m$ be a smooth immersion of the unit-disk, continuous at the origin (the origin will be the point-singularity in question). In order to make sense of the Willmore energy of the immersion $\vec{\Phi}$, we suppose that $\int_{D^2}|\vec{H}|^2d\text{vol}_g<\infty$. Our immersion is assumed to satisfy the problem \begin{equation}\label{ptprob}
\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\;=\;\vec{\mathcal{W}}\qquad\text{on}\:\:D^2\setminus\{0\}\:, \end{equation} where the vector $\vec{\mathcal{W}}$ is given. It may depend only on geometric quantities (as is the case in the Willmore problem or in Chen's problem), but it may also involve ``exterior" quantities (as is the case in the conformally-constrained Willmore problem). To simplify the presentation, we will not in this paper discuss the integrability assumptions that must be imposed on $\vec{\mathcal{W}}$ to carry out the procedure that will be outlined. The interested reader is invited to consult \cite{BWW1} for more details on this topic. \\
\noindent As we have shown in (\ref{defT}), equation (\ref{ptprob}) may be rephrased as \begin{equation*}
\partial_j\big(|g|^{1/2}\vec{T}^j\big)\;=\;-\,\vec{\mathcal{W}}\qquad\text{on}\:\:D^2\setminus\{0\}\:, \end{equation*} for some suitable tensor $\vec{T}^j$ defined solely in geometric terms. Consider next the problem \begin{equation*} \Delta_g\vec{V}\;=\;-\,\vec{\mathcal{W}}\qquad\text{on}\:\:D^2\:. \end{equation*} As long as $\vec{\mathcal{W}}$ is not too wildly behaved, this equation will have at least one solution. Let next $\mathcal{L}_g$ satisfy \begin{equation*}
\partial_j\big(|g|^{1/2}\nabla^j\mathcal{L}_g\big)\;=\;\delta_0\qquad\text{on}\:\:D^2\:. \end{equation*} If the immersion is correctly chosen (e.g. $\vec{\Phi}\in W^{2,2}\cap W^{1,\infty}$), the solution $\mathcal{L}_g$ exists and has suitable analytical properties (see \cite{BWW1} for details). \\ We have \begin{equation}\label{poinc1}
\partial_j\Big[|g|^{1/2}\big(\vec{T}^j-\nabla^j\vec{V}-\vec{\beta}\,\nabla^j\mathcal{L}_g\big)\Big]\;=\;\vec{0}\qquad\text{on}\:\:D^2\setminus\{0\}\:, \end{equation} for any constant $\vec{\beta}\in\mathbb{R}^m$, and in particular for the unique $\vec{\beta}$ fulfilling the circulation condition that \begin{equation}\label{poinc2} \int_{\partial D^2}\vec{\nu}\cdot\big(\vec{T}^j-\nabla^j\vec{V}-\vec{\beta}\,\nabla^j\mathcal{L}_g\big)\;=\;0\:, \end{equation} where $\vec{\nu}\in\mathbb{R}^2$ denotes the outer unit normal vector to the boundary of the unit-disk. This vector $\vec{\beta}$ will be called {\it residue}.\\ Bringing together (\ref{poinc1}) and (\ref{poinc2}) and calling upon the Poincar\'e lemma, one infers the existence of an element $\vec{L}$ satisfying \begin{equation}
\vec{T}^j-\nabla^j\vec{V}-\vec{\beta}\,\nabla^j\mathcal{L}_g\;=\;|g|^{-1/2}\epsilon^{kj}\nabla_k\vec{L}\:, \end{equation} with the same notation as before. We are now in the position of repeating {\it mutatis mutandis} the computations derived in the previous section, taking into account the presence of the residue. We define $\vec{X}$ and $Y$ via: \begin{equation}\label{ptside} \left\{\begin{array}{rcl}
\Delta_g\vec{X}&=&\nabla^j\big(\vec{V}+\vec{\beta}\mathcal{L}_g\big)\wedge\nabla_j\vec{\Phi}\\[1ex] \Delta_g{Y}&=&\nabla^j\big(\vec{V}+\vec{\beta}\mathcal{L}_g\big)\cdot\nabla_j\vec{\Phi} \end{array}\right.\qquad\text{on}\:\:D^2\:. \end{equation} One verifies that the following equations hold \begin{equation*} \left\{\begin{array}{rcl}
\nabla^k\Big[\vec{L}\wedge\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\epsilon_{kj}\big(\vec{H}\wedge\nabla^j\vec{\Phi}+\nabla^j\vec{X}\big)\Big]&=&\vec{0}\\[2ex]
\nabla^k\Big[\vec{L}\cdot\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\epsilon_{kj}\nabla^jY\Big]&=&0 \end{array}\right.\qquad\text{on}\:\:D^2\setminus\{0\}\:. \end{equation*} Imposing suitable hypotheses on the integrability of $\vec{\mathcal{W}}$ yields that the bracketed quantities in the latter are square-integrable. With the help of a classical result of Laurent Schwartz \cite{Scw}, the equations may be extended without modification to the whole unit-disk. As before, this grants the existence of two potential functions $S$ and $\vec{R}$ which satisfy (\ref{defRS}) and the system (\ref{thesys}) on $D^2$. The Jacobian-type/divergence-type structure of the system sets the stage for a local analysis argument, which eventually yields a local expansion of the immersion $\vec{\Phi}$ around the point-singularity. This expansion involves the residue $\vec{\beta}$. The procedure was carried out in details for Willmore immersions in \cite{BR2}\footnote{An equivalent notion of residue was also identified in \cite{KS2}.}, and for conformally constrained Willmore immersions in \cite{YBer}. Further considerations can be found in \cite{BWW1}.
\end{document}
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\begin{document}
\title{The pleasures and pains of studying the two-type Richardson model}
\author{Maria Deijfen \thanks{Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden. E-mail: [email protected]} \and Olle H\"{a}ggstr\"{o}m \thanks{Department of Mathematical Sciences, Chalmers University of Technology, 412 96 G\"oteborg, Sweden. E-mail: [email protected]}}
\date{July 2007}
\maketitle
\thispagestyle{empty}
\begin{abstract}
\noindent This paper provides a survey of known results and open problems for the two-type Richardson model, which is a stochastic model for competition on $\mathbb{Z}^d$. In its simplest formulation, the Richardson model describes the evolution of a single infectious entity on $\mathbb{Z}^d$, but more recently the dynamics have been extended to comprise two competing growing entities. For this version of the model, the main question is whether there is a positive probability for both entities to simultaneously grow to occupy infinite parts of the lattice, the conjecture being that the answer is yes if and only if the entities have the same intensity. In this paper attention focuses on the two-type model, but the most important results for the one-type version are also described.
\noindent \emph{Keywords:} Richardson model, first-passage percolation, asymptotic shape, competing growth, coexistence.
\noindent AMS 2000 Subject Classification: 60K35, 82B43. \end{abstract}
\section{Introduction}
Consider an interacting particle system in which, at any time $t$, each site $x\in\mathbb{Z}^d$ is in either of two states, denoted by 0 and 1. A site in state 0 flips to a 1 at rate proportional to the number of nearest neighbors in state 1, while a site in state 1 remains a 1 forever. We may think of sites in state 1 as being occupied by some kind of infectious entity, and the model then describes the propagation of an infection where each infected site tries to infect each of its nearest neighbors on $\mathbb{Z}^d$ at some constant rate $\lambda>0$. More precisely, if at time $t$ a vertex $x$ is infected and a neighboring vertex $y$ is uninfected, then, conditional on the dynamics up to time $t$, the probability that $x$ infects $y$ during a short time window $(t, t+h)$ is $\lambda h + o(h)$. Here and in what follows, sites in state 0 and 1 are referred to as uninfected and infected respectively. This is the intuitive description of the model; a formal definition is given in Section \ref{sect:one-type}.
The model is a special case of a class of models introduced by Richardson (1973), and is commonly referred to as the Richardson model. It has several cousins among processes from mathematical biology, see e.g.\ Eden (1961), Williams and Bjerknes (1972) and Bramson and Griffeath (1981). The model is also a special case of so called first-passage percolation, which was introduced in Hammersley and Welsh (1965) as a model for describing the passage of a fluid through a porous medium. In first-passage percolation, each edge of the $\mathbb{Z}^d$-lattice is equipped with a random variable representing the time it takes for the fluid to traverse the edge, and the Richardson model is obtained by letting these passage times be i.i.d.\ exponential.
Since an infected site stays infected forever, the set of infected sites in the Richardson model increases to cover all of $\mathbb{Z}^d$ as $t\to\infty$, and attention focuses on \emph{how} this set grows. The main result is roughly that the infection grows linearly in time in each fixed direction and that, scaled by a factor $1/t$, the set of infected points converges to a non-random asymptotic shape as $t\to\infty$. To prove that the growth is linear in a fixed direction involves Kingman's subadditive ergodic theorem -- in fact, the study of first-passage percolation was one of the main motivations for the development of subadditive ergodic theory. That the linear growth is preserved when all directions are considered simultaneously is stated in the celebrated shape theorem (Theorem \ref{thm:shape} in Section \ref{sect:two-type_bounded}) which originates from Richardson (1973).
Now consider the following extension of the Richardson model, known as the two-type Richardson model and introduced in H\"{a}ggstr\"{o}m and Pemantle (1998). Instead of two possible states for the sites there are three states, which we denote by 0, 1 and 2. The process then evolves in such a way that, for $i=1,2$, a site in state 0 flips to state $i$ at rate $\lambda_i$ times the number of nearest neighbors in state $i$ and once in state 1 or 2, a site remains in that state forever. Interpreting states 1 and 2 as two different types of infection and state 0 as absence of infection, this gives rise to a model describing the simultaneous spread of two infections on $\mathbb{Z}^d$. To rigorously define the model requires a bit more work; see Section \ref{sect:two-type_bounded}. In what follows we will always assume that $d\geq 2$; the model makes sense also for $d=1$ but the questions considered here become trivial.
A number of similar extensions of (one-type) growth models to (two-type) competition models appear in the literature; see for instance Neuhauser (1992), Durrett and Neuhauser (1997), Kordzakhia and Lalley (2005) and Ferrari et al.\ (2006). These tend to require somewhat different techniques, and results tend not to be easily translated from these other models to the two-type Richardson model (and vice versa). Closer to the latter are (non-Markovian) competition models based on first-passage percolation models with non-exponential passage time variables -- Garet and Marchand (2005), Hoffman (2005:1), Hoffman (2005:2), Garet and Marchand (2006), Gou\'er\'e (2007), Pimentel (2007) -- and a certain continuum model -- Deijfen et al.\ (2004), Deijfen and H\"aggstr\"om (2004), Gou\'er\'e (2007). For ease of exposition, we shall not consider these variations even in cases where results generalize.
The behavior of the two-type Richardson model depends on the initial configuration of the infection and on the ratio between the intensities $\lambda_1$ and $\lambda_2$ of the infection types. Assume first, for simplicity, that the model is started at time 0 from two single sites, the origin being type 1 infected and the site $(1,0,\ldots,0)$ next to the origin being type 2 infected. Three different scenarios for the development of the infection are conceivable:
\begin{itemize} \item[(a)] The type 1 infection at some point completely surrounds type 2, thereby preventing type 2 from growing any further.
\item[(b)] Type 2 similarly strangles type 1.
\item[(c)] Both infections grow to occupy infinitely many sites. \end{itemize}
\noindent It is not hard to see that, regardless of the intensities of the infections, outcomes (a) and (b) where one of the infection types at some point encloses the other have positive probability regardless of $\lambda_1$ and $\lambda_2$. This is because each of (a) and (b) can be guaranteed through some finite initial sequence of infections. In contrast, scenario (c) -- referred to as infinite coexistence -- can never be guaranteed from any finite sequence of infections, and is therefore harder to deal with: the main challenge is to decide whether, for given values of the parameters $\lambda_1$ and $\lambda_2$, this event (c) has positive probability or not. Intuitively, infinite coexistence represents some kind of power balance between the infections, and it seems reasonable to suspect that such a balance is possible if and only if the infections are equally powerful, that is, when $\lambda_1=\lambda_2$. This is Conjecture \ref{samex_conj} in Section \ref{sect:two-type_bounded}, which goes back to H\"{a}ggstr\"{o}m and Pemantle (1998), and, although a lot of progress have been made, it is not yet fully proved. We describe the state of the art in Sections \ref{sect:symmetric} and \ref{sect:nonsymmetric}.
As mentioned above, apart from the intensities, the development of the infections in the two-type model also depends on the initial state of the model. However, if we are only interested in deciding whether the event of infinite coexistence has positive probability or not, it turns out that, as long as the initial configuration is bounded and one of the sets does not completely surround the other, the precise configuration does not matter, that is, whether infinite coexistence is possible or not is determined only by the relation between the intensities. This is proved in Deijfen and H\"{a}ggstr\"{o}m (2006:1); see Theorem \ref{th:startomr} in Section \ref{sect:two-type_bounded} for a precise formulation. Of course one may also consider unbounded initial configurations. Starting with both infection types occupying infinitely many sites means -- apart from in very labored cases -- that they will both infect infinitely many sites. A more interesting case is when one of the infection types starts from an infinite set and the other one from a finite set. We may then ask if outcomes where the finite type infects infinitely many sites have positive probability or not. This question is dealt with in Deijfen and H\"{a}ggstr\"{o}m (2007), and we describe the results in Section \ref{sect:unbounded}.
The dynamics of the two-type Richardson model is deceptively simple, and yet gives rise to intriguing phenomena on a global scale. In this lies a large part of the pleasure indicated in the title. Furthermore, proofs tend to involve elegant probabilistic techniques such as coupling, subadditivity and stochastic comparisons, adding more pleasure. The pain alluded to (which by the way is not so severe that it should dissuade readers from entering this field) comes from the stubborn resistance that some of the central problems have so far put up against attempts to solve them. A case in point is the ``only if'' direction of the aforementioned Conjecture \ref{samex_conj}, saying that infinite coexistence starting from a bounded initial configuration does not occur when $\lambda_1 \neq \lambda_2$.
\section{The one-type model} \label{sect:one-type}
As mentioned in the introduction, the one-type Richardson model is equivalent to first-passage percolation with i.i.d.\ exponential passage times. To make the construction of the model more precise, first define $E_{\mathbb{Z}^d}$ as the edge set for the $\mathbb{Z}^d$ lattice (i.e., each pair of vertices $x,y \in \mathbb{Z}^d$ at Euclidean distance $1$ from each other have an edge $e \in E_{\mathbb{Z}^d}$ connecting them). Then attach i.i.d.\ non-negative random variables $\{\tau(e)\}_{e \in E_{\mathbb{Z}^d}}$ to the edges. We take each $\tau(e)$ to be exponentially distributed with parameter $\lambda>0$, meaning that \[ P(\tau(e)>t) = \exp(-\lambda t) \] for all $t\geq 0$. For $x,y \in \mathbb{Z}^d$, define \begin{equation} \label{eq:path_time} T(x,y) = \inf_\Gamma \sum_{e \in \Gamma} \tau(e) \end{equation} where the infimum is over all paths $\Gamma$ from $x$ to $y$. The Richardson model with a given set $S_0 \subset \mathbb{Z}^d$ of initially infected sites is now defined by taking the set $S_t$ of sites infected at time $t$ to be \begin{equation} \label{eq:infected_at_time_t} S_t = \{ x \in \mathbb{Z}^d : T(y,x) \leq t\mbox{ for some } y \in S_0\} \, . \end{equation} It turns out that the infimum in (\ref{eq:path_time}) is a.s.\ a minimum and attained by a unique path. That $S_t$ grows in the way described in the introduction is a consequence of the memoryless property of the exponential distribution: for any $s,t >0$ we have that
$P(\tau(e)>s+t \, | \tau(e)>s) = \exp(-\lambda t)$.
Note that for any $x,y,z \in\mathbb{Z}^d$ we have $T(x,y)\leq T(x,z)+T(z,y)$. This subadditivity property opens up for the use of subadditive ergodic theory in analyzing the model. To formulate the basic result, let $T(x)$ be the time when the point $x\in\mathbb{Z}^d$ is infected when starting from a single infected site at the origin and write $\mathbf{n}=(n,0,\ldots,0)$. It then follows from the subadditive ergodic theorem -- see e.g.\ Kingman (1968) -- that there is a constant $\mu_\lambda$ such that $T(\mathbf{n})/n\to\mu_\lambda$ almost surely and in $L_1$ as $n\to\infty$. Furthermore, a simple time scaling argument implies that $\mu_\lambda=\lambda\mu_1$ and hence, writing $\mu_1=\mu$, we have that
\begin{equation}\label{eq:time_constant} \lim_{n\to\infty}\frac{T(\mathbf{n})}{n}=\lambda\mu\quad\textrm{a.s. and in }L_1. \end{equation}
\noindent The constant $\mu$ indicates the inverse asymptotic speed of the growth along the axes in a unit rate process and is commonly referred to as the time constant. It turns out that $\mu>0$, so that indeed the growth is linear in time. Similarly, an analog of (\ref{eq:time_constant}) holds in any direction, that is, for any $x\in\mathbb{Z}^d$, there is a constant $\mu(x)>0$ such that $T(nx)/n\to\lambda\mu(x)$. The infection hence grows linearly in time in each fixed direction and the asymptotic speed of the growth in a given direction is an almost sure constant.
We now turn to the shape theorem, which asserts roughly that the linear growth of the infection is preserved also when all directions are considered simultaneously. More precisely, when scaled down by a factor $1/t$ the set $S_t$ converges to a non-random shape $A$. To formalize this, let $\tilde{S}_t\subset \mathbb{R}^d$ be a continuum version of $S_t$ obtained by replacing each $x\in S_t$ by a unit cube centered at $x$.
\begin{theorem}[Shape Theorem] \label{thm:shape} There is a compact convex set $A$ such that, for any $\varepsilon>0$, almost surely $$ (1-\varepsilon)\lambda A\subset\frac{\tilde{S}_t}{t} \subset (1+\varepsilon)\lambda A $$ for large $t$. \end{theorem}
\noindent In the above form, the shape theorem was proved in Kesten (1973) as an improvement on the original ``in probability" version, which appears already in Richardson (1973). See also Cox and Durrett (1988) and Boivin (1991) for generalizations to first-passage percolation processes with more general passage times. Results concerning fluctuations around the asymptotic shape can be found, e.g., in Kesten (1993), Alexander (1993) and Newman and Piza (1995), and, for certain other passage time distributions, in Benjamini et al.\ (2003).
Working out exactly, or even approximately, what the asymptotic shape $A$ is has turned out to be difficult. Obviously the asymptotic shape inherits all symmetries of the $\mathbb{Z}^d$ lattice -- invarince under reflection and permutation of coordiante hyperplanes -- and it is known to be compact and convex, but, apart from this, not much is known about its qualitative features. These difficulties with characterizing the shape revolve around the fact that $\mathbb{Z}^d$ is not rotationally invariant, which causes the growth to behave differently in different directions. For instance, simulations on $\mathbb{Z}^2$ indicate that the asymptotic growth is slightly faster along the axes as compared to the diagonals. There is however no formal proof of this.
Before proceeding with the two-type model, we mention some work concerning properties of the time-minimizing paths in (\ref{eq:path_time}), also known as geodesics. Starting at time $0$ with a single infection at the origin ${\bf 0}$, we denote by $\Gamma(x)$ the (unique) path $\Gamma$ for which the infimum $T({\bf 0}, x)$ in (\ref{eq:path_time}) is attained. Define $\Psi=\cup_{x\in\mathbb{Z}^d} \Gamma(x)$, making $\Psi$ a graph specifying which paths the infection actually takes. It is not hard to see that $\Psi$ is a tree spanning all of $\mathbb{Z}^d$ and hence there must be at least one semi-infinite self-avoiding path from the origin (called an end) in $\Psi$. The issue of whether $\Psi$ has more than one end was noted by H\"aggstr\"om and Pemantle (1998) to be closely related to the issue of infinite coexistence in the two-type Richardson model with $\lambda_1=\lambda_2$: such infinite coexistence happens with positive probability starting from a finite initial configuration if and only if $\Psi$ has at least two ends with positive probability.
We say that an infinite path $x_1,x_2,\ldots$ has asymptotic direction $\hat{x}$ if $x_k/|x_k|\to\hat{x}$ as $k\to\infty$. In $d=2$, it has been conjectured that every end in $\Psi$ has an asymptotic direction and that, for every $x\in\mathbb{R}^2$, there is at least one end (but never more than two) in $\Psi$ with asymptotic direction $\hat{x}$. In particular, this would mean that $\Psi$ has uncountably many ends. For results supporting this conjecture, see Newman (1995) and Newman and Licea (1996). In the former of these papers, the conjecture is shown to be true provided an unproven but highly plausible assumption on the asymptotic shape $A$, saying roughly that the boundary is sufficiently smooth. See also Lalley (2003) for related work.
Results not involving unproven assumptions are comparatively weak: The coexistence result of H\"aggstr\"om and Pemantle (1998) shows for $d=2$ that $\Psi$ has at least two ends with positive probability. This was later improved to $\Psi$ having almost surely at least $2d$ ends, by Hoffman (2005:2) for $d=2$ and by Gou\'er\'e (2007) for higher dimensions.
\section{Introducing two types} \label{sect:two-type_bounded}
The definition of the two-type Richardson model turns out to be simplest in the symmetric case $\lambda_1=\lambda_2$, where the same passage time variables $\{\tau(e)\}_{e \in E_{\mathbb{Z}^d}}$ as in the one-type model can be used, with $\lambda= \lambda_1=\lambda_2$. Suppose we start with an initial configuration $(S^1_0, S^2_0)$ of infected sites, where $S^1_0 \subset \mathbb{Z}^d$ are those initially containing type $1$ infection, and $S^2_0 \subset \mathbb{Z}^d$ are those initially containing type $2$ infection. We wish to define the sets $S_t^1$ and $S_t^2$ of type 1 and type $2$ infected sites for all $t>0$. To this end, set $S_0=S^1_0 \cup S^2_0$, and take the set $S_t=S_t^1 \cup S_t^2$ of infected sites at time $t$ to be given by precisely the same formula (\ref{eq:infected_at_time_t}) as in the one-type model; a vertex $x\in S_t$ is then assigned infection $1$ or $2$ depending on whether the $y \in S_0$ for which \[ \inf\{T(y,x):y\in S_0\} \] is attained is in $S_0^1$ or $S_0^2$.
As in the one-type model, it is a straightforward exercise involving the memoryless property of the exponential distribution to verify that $(S_t^1, S_t^2)_{t \geq 0}$ behaves in terms of infection intensities as described in the introduction.
This construction demonstrates an intimate link between the one-type and the symmetric two-type Richardson model: if we watch the two-type model wearing a pair of of glasses preventing us from distinguishing the two types of infection, what we see behaves exactly as the one-type model. The link between infinite coexistence in the two-type model and the number of ends in the tree of infection $\Psi$ of the one-type model claimed in the previous section is also a consequence of the construction.
In the asymmetric case $\lambda_1 \neq \lambda_2$, the two-type model is somewhat less trivial to define due to the fact that the time it takes for infection to spread along a path depends on the type of infection. There are various ways to deal with this, one being to assign, independently to each $e \in E_{{\mathbb{Z}^d}}$, two independent random variables $\tau_1(e)$ and $\tau_2(e)$, exponentially distributed with respective parameters $\lambda_1$ and $\lambda_2$, representing the time it takes for infections $1$ resp.\ $2$ to traverse $e$. Starting from an intial configuration $(S^1_0, S^2_0)$, we may picture the infections as spreading along the edges, taking time $\tau_1(e)$ or $\tau_2(e)$ to cross $e$ depending on the type of infection, with the extra condition that once a vertex becomes hit by one type of infection it becomes inaccessible for the other type. This is intuitively clear, but readers with a taste for detail may require a more rigorous definition, which however we refrain from here; see H\"aggstr\"om and Pemantle (2000) and Deijfen and H\"aggstr\"om (2006:1).
We now move on to describing conjectures and results. Write $G_i$ for the event that type $i$ infects infinitely many sites on $\mathbb{Z}^d$ and define $G=G_1\cap G_2$. The question at issue is:
\begin{equation}\label{eq:coex?} \textrm{Does $G$ have positive probability?} \end{equation}
\noindent A priori, the answer to this question may depend both on the initial configuration -- that is, on the choice of the sets $S_0^1$ and $S_0^2$ -- and on the ratio between the infection intensities $\lambda_1$ and $\lambda_2$. However, it turns out that, if we are not interested in the actual value of the probability of $G$, but only in whether it is positive or not, then the initial configuration is basically irrelevant, as long as neither of the initial sets completely surrounds the other. This motivates the following definition.
\begin{defn} Let $\xi_1$ and $\xi_2$ be two disjoint finite subsets of $\mathbb{Z}^d$. We say that one of the sets ($\xi_i$) \emph{strangles} the other ($\xi_j$) if there exists no infinite self-avoiding path in $\mathbb{Z}^d$ that starts at a vertex in $\xi_j$ and that does not intersect $\xi_i$. The pair $(\xi_1,\xi_2)$ is said to be \emph{fertile} if neither of the sets strangles the other. \end{defn}
Now write $P^{\lambda_1,\lambda_2}_{\xi_1,\xi_2}$ for the distribution of a two-type process started from $S_0^1=\xi_1$ and $S_0^2=\xi_2$. We then have the following result.
\begin{theorem}\label{th:startomr} Let $(\xi_1,\xi_2)$ and $(\xi_1',\xi_2')$ be two fertile pairs of disjoint finite subsets of $\mathbb{Z}^d$, where $d\geq 2$. For all choices of $(\lambda_1,\lambda_2)$, we have $$ P^{\lambda_1,\lambda_2}_{\xi_1,\xi_2}(A)>0\Leftrightarrow P^{\lambda_1,\lambda_2}_{\xi_1',\xi_2'}(A)>0. $$ \end{theorem}
For connected initial sets $\xi_1$ and $\xi_2$ and $d=2$, this result is proved in H\"{a}ggstr\"{o}m and Pemantle (1998). The idea of the proof in that case is that, by controlling the passage times of only finitely many edges, two processes started from $(\xi_1,\xi_2)$ and $(\xi'_1,\xi'_2)$ respectively can be made to evolve to the same total infected set after some finite time, with the same configuration of the infection types on the boundary. Coupling the processes from this time on and observing that the development of the infections depends only on the boundary configuration yields the result. This argument however breaks down when the initial sets are not connected (since it is then not sure that the same boundary configuration can be obtained in the two processes) and it is unclear whether it applies for $d\geq 3$. Theorem \ref{th:startomr} is proved in full generality in Deijfen and H\"{a}ggstr\"{o}m (2006:1), using a more involved coupling construction.
It follows from Theorem \ref{th:startomr} that the answer to (\ref{eq:coex?}) depends only on the value of the intensities $\lambda_1$ and $\lambda_2$. Hence it is sufficient to consider a process started from $S_0^1=\mathbf{0}$ and $S_0^2=\mathbf{1}$ (recall that $\mathbf{n}=(n,0,\ldots,0)$), and in this case we drop subscripts and write $P^{\lambda_1,\lambda_2}$ for $P^{\lambda_1,\lambda_2}_{{\bf 0}, {\bf 1}}$. Also, by time-scaling, we may assume that $\lambda_1=1$. The following conjecture, where we write $\lambda_2=\lambda$, goes back to H\"{a}ggstr\"{o}m and Pemantle (1998).
\begin{conj}\label{samex_conj} In any dimension $d\geq 2$, we have that $P^{1,\lambda}(G)>0$ if and only if $\lambda=1$. \end{conj}
\noindent The conjecture is no doubt true, although proving it has turned out to be a difficult task. In fact, the ``only if'' direction is not yet fully established. In the following two sections we describe the existing results for $\lambda=1$ and $\lambda\neq 1$ respectively.
\section{The case $\lambda=1$} \label{sect:symmetric}
When $\lambda=1$, we are dealing with two equally powerful infections and Conjecture \ref{samex_conj} predicts a positive probability for infinite coexistence. This part of the conjecture has been proved:
\begin{theorem}\label{th:lambda=1} If $\lambda=1$, we have, for any $d\geq 2$, that $P^{1,\lambda}(G)>0$. \end{theorem}
\noindent This was first proved in the special case $d=2$ by H\"{a}ggstr\"{o}m and Pemantle (1998). That proof has a very ad hoc flavor, and heavily exploits not only the two-dimensionality but also other specific properties of the square lattice, including a lower bound on the time constant $\mu$ in (\ref{eq:time_constant}) that just happens to be good enough. When eventually the result was generalized to higher dimensions, which was done simultaneously and independently by Garet and Marchand (2005) and Hoffman (2005:1), much more appealing proofs were obtained. Yet another distinct proof of Theorem \ref{th:lambda=1} was given by Deijfen and H\"{a}ggstr\"{o}m (2007). All four proofs are different, though if you inspect them for a smallest common denominator you find that they all make critical use of the fact that the time constant $\mu$ is strictly positive. We will give the Garet--Marchand proof below. In Hoffman's proof ergodic theory is applied to the tree of infection $\Psi$ and a so-called Busemann function which is shown to exhibit contradictory behavior under the assumption that infinite coexistence has probability zero. The Deijfen--H\"{a}ggstr\"{o}m proof proceeds via the two-type Richardson model with certain infinite initial configurations (cf.\ Section \ref{sect:unbounded}).
\noindent {\bf Proof of Theorem \ref{th:lambda=1}:} The following argument is due to Garet and Marchand (2005), though our presentation follows more closely the proof of an analogous result in a continuum setting in Deijfen and H\"{a}ggstr\"{o}m (2004) -- a paper that, despite the publication dates, was preceded by and also heavily influenced by Garet and Marchand (2005).
Fix a small $\varepsilon>0$. By Theorem \ref{th:startomr}, we are free to choose any finite starting configuration we want, and here it turns out convenient to begin with a single type $1$ infection at the origin ${\bf 0}$, and a single type $2$ infection at a vertex ${\bf n}=(n,0,\ldots, 0)$, where $n$ is large enough so that \begin{description} \item{(i)} $E[T({\bf 0}, {\bf n})] \leq (1 + \varepsilon) n \mu$, and \item{(ii)} $P(T({\bf 0}, {\bf n}) < (1 - \varepsilon) n \mu) < \varepsilon$; \end{description} note that both (i) and (ii) hold for $n$ large enough due to the asymptotic speed result (\ref{eq:time_constant}). The reader may easily check, for later reference, that (i) and (ii) together with the nonnegativity of $T({\bf 0}, {\bf n})$ imply for any event $B$ with $P(B)=\alpha$ that \begin{equation} \label{eq:key_estimate_GM}
E[T({\bf 0}, {\bf n})\, | \, \neg B] \, \leq \, \left(1 + \frac{3\varepsilon}{1-\alpha}\right) n \mu \, . \end{equation} Next comes an important telescoping idea: for any positive integer $k$ we have \begin{eqnarray*} E[T({\bf 0}, k{\bf n})] & = & E[T({\bf 0}, {\bf n})] + E[T({\bf 0}, 2{\bf n}) - T({\bf 0}, {\bf n})] + E[T({\bf 0}, 3{\bf n}) - T({\bf 0}, 2{\bf n})] \\ & & + \ldots + E[T({\bf 0}, k{\bf n}) - T({\bf 0}, (k-1){\bf n})] \, . \end{eqnarray*} Since $\lim_{k \rightarrow \infty}k^{-1}E[T({\bf 0}, k{\bf n})] = n \mu$, there must exist arbitrarily large $k$ such that \[ E[T({\bf 0}, (k+1){\bf n}) - T({\bf 0}, k{\bf n})] \geq (1 - \varepsilon)n \mu \, . \] By taking ${\bf m}= k{\bf n}$, and by translation and reflection invariance, we may deduce that \begin{equation} \label{eq:for_arbitrarily_large_m} E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m})] \geq (1 - \varepsilon)n \mu \end{equation} for some arbitrarily large $m$. We will pick such an $m$; how large will soon be specified.
The goal is to show that $P(G)>0$, so we may assume for contradiction that $P(G)=0$. By symmetry of the initial configuration, we then have that $P(G_1)=P(G_2)=\frac{1}{2}$. This implies that \[ \lim_{m \rightarrow \infty} P({\bf -m} \mbox{ gets infected by type 2})= \lim_{m \rightarrow \infty} P(T({\bf n}, -{\bf m}) < T({\bf 0}, -{\bf m})) =\frac{1}{2} \] so let us pick $m$ in such a way that \begin{equation} \label{eq:our_chosen_m} P(T({\bf n}, -{\bf m}) < T({\bf 0}, -{\bf m})) \geq \frac{1}{4} \end{equation} while also (\ref{eq:for_arbitrarily_large_m}) holds. Write $B$ for the event in (\ref{eq:our_chosen_m}). The expectation $E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m})]$ may be decomposed as \begin{eqnarray*} E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m})] & = &
E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m}) \, | B]P(B) \\
& & +E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m}) \, | \neg B]P(\neg B) \\
& \leq & E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m}) \, | \neg B]P(\neg B) \\ & \leq & \frac{3}{4}
E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m}) \, | \neg B] \\
& \leq & \frac{3}{4} E[T({\bf n}, {\bf 0}) | \neg B] \\ & \leq & \frac{3}{4} (1 + 4 \varepsilon) n \mu \end{eqnarray*} where the second-to-last inequality is due to the triangle inequality $T({\bf n}, -{\bf m}) \leq T({\bf n}, {\bf 0}) + T({\bf 0}, - {\bf m})$, and the last one uses (\ref{eq:key_estimate_GM}). For small $\varepsilon$, this contradicts (\ref{eq:for_arbitrarily_large_m}), so the proof is complete. ${
\Box}$
\section{The case $\lambda\neq 1$} \label{sect:nonsymmetric}
Let us move on to the case when $\lambda\neq 1$, that is, when the type 2 infection has a different intensity than type 1. It then seems unlikely that the kind of equilibrium which is necessary for infinite coexistence to occur would persist in the long run. However, this part of Conjecture \ref{samex_conj} is not proved. The best result to date is the following theorem from H\"{a}ggstr\"{o}m and Pemantle (2000).
\begin{theorem}\label{th:pain} For any $d\geq 2$, we have $P^{1,\lambda}(G)=0$ for all but at most countably many values of $\lambda$. \end{theorem}
\noindent We leave it to the reader to decide whether this is a very strong or a very weak result: it is very strong in the sense of showing that infinite coexistence has probability $0$ for (Lebesgue)-almost all $\lambda$, but very weak in the sense that infinite coexistence is not ruled out for any given $\lambda$.
The result may seem a bit peculiar at first sight and we will spend some time explaining where it comes from and where the difficulties arise when one tries to strengthen it. Indeed, as formulated in Conjecture \ref{samex_conj}, the belief is that the set $\{\lambda:P^{1,\lambda}(G)>0\}$ in fact consists of the single point $\lambda=1$, but Theorem \ref{th:pain} only asserts that the set is countable.
First note that, by time-scaling and symmetry, we have $P^{1,\lambda}(G)=P^{1,1/\lambda}(G)$ and hence it is enough to consider $\lambda\leq 1$. An essential ingredient in the proof of Theorem \ref{th:pain} is a coupling of the two-type processes $\{P^{1,\lambda}\}_{\lambda\in(0,1]}$ obtained by associating two independent exponential mean 1 variables $\tau_1(e)$ and $\tau_2'(e)$ to each edge $e\in\mathbb{Z}^d$ and then letting the type 2 passage time at parameter value $\lambda$ be given by $\tau_2(e)=\lambda^{-1}\tau_2'(e)$ and the type 1 time (for any $\lambda$) by $\tau_1(e)$. Write $Q$ for the probability measure underlying this coupling and let $G^\lambda$ be the event that infinite coexistence occurs at parameter value $\lambda$. Theorem \ref{th:pain} is obtained by showing that \begin{equation} \label{eq:in_the_coupling} \begin{array}{l} \mbox{with $Q$-probability 1 the event $G^\lambda$ occurs} \\ \mbox{for at most one value of $\lambda\in(0,1]$}. \end{array} \end{equation} Hence, $Q(G^\lambda)$ can be positive for at most countably many $\lambda$, and Theorem \ref{th:pain} then follows by noting that $P^{1,\lambda}(G)=Q(G^\lambda)$.
But why is (\ref{eq:in_the_coupling}) true? Let $G^\lambda_i$ be the event that the type $i$ infection grows unboundedly at parameter value $\lambda$. Then the coupling defining $Q$ can be shown to be monotone in the sense that $G^\lambda_1$ is decreasing in $\lambda$ -- that is, if $G^\lambda_1$ occurs then $G^{\lambda'}_1$ occurs for all $\lambda'<\lambda$ as well -- and $G^\lambda_2$ is increasing in $\lambda$. This kind of monotonicity of the coupling is crucial for proving (\ref{eq:in_the_coupling}), as is the following result, which asserts that, on the event that the type 2 infection survives, the total infected set in a two-type process with distribution $P^{1,\lambda}$, where $\lambda<1$, grows to a first approximation like a one-type process with intensity $\lambda$. More precisely, the speed of the growth in the two-type process is determined by the weaker type 2 infection type. We take $\tilde{S}_t^i$ to denote the union of all unit cubes centered at points in $S_t^i$ and $A$ is the limiting shape for a one-type process with rate 1.
\begin{theorem}\label{th:svag_bestammer} Consider a two-type process with distribution $P^{1,\lambda}$ for some $\lambda\leq 1$. On the event $G_2$ we have, for any $\varepsilon>0$, that almost surely $$ (1-\varepsilon)\lambda A\subset\frac{\tilde{S}_t^1\cup \tilde{S}_t^2}{t}\subset (1+\varepsilon)\lambda A $$ for large $t$. \end{theorem}
\noindent Theorem \ref{th:pain} follows readily from this result and the monotonicity properties of the coupling $Q$. Indeed, fix $\varepsilon>0$ and suppose $G^\lambda$ occurs. Then Theorem \ref{th:svag_bestammer} guarantees that on level $\lambda$ the type 1 infection is eventually contained in $(1+ \varepsilon)\lambda tA$, a conclusion that extends to all $\lambda'>\lambda$, because increasing the type 2 infection rate does not help type 1. On the other hand, for any $\lambda'>\lambda$ we get on level $\lambda'$ that the union of the two infections will -- again by Theorem \ref{th:svag_bestammer} -- eventually contain $(1- \varepsilon)\lambda' tA$, so by taking $\varepsilon$ sufficiently small we see that the type 1 infection is strangled on level $\lambda'$, implying (\ref{eq:in_the_coupling}), and Theorem \ref{th:pain} follows.
We will not prove Theorem \ref{th:svag_bestammer}, but mention that the hard work in proving it lies in establishing a certain key result (Proposition 2.2 in H\"{a}ggstr\"{o}m and Pemantle (2000)) that asserts that if the strong infection type reaches outside $(1 + \varepsilon) \lambda tA$ infinitely often, then the weak type is doomed. The proof of this uses geometrical arguments, the most important ingredient being a certain spiral construction, emanating from the part of the strong of infection reaching beyond $(1 + \varepsilon) \lambda tA$, and designed to allow the strong type to completely surround the weak type before the weak type catches up from inside.
How would one go about to strengthen Theorem \ref{th:pain} and rule out infinite coexistence for all $\lambda \neq 1$? One possibility would be to try to derive a contradiction with Theorem \ref{th:svag_bestammer} from the assumption that the strong infection type grows unboundedly. For instance, intuitively it seems likely that the strong type occupying a positive fraction of the boundary of the infected set would cause the speed of the growth to exceed the speed prescribed by the weak infection type. This type of argument is indeed used in Garet and Marchand (2007) to show, for $d=2$, that on the event of infinite coexistence the fraction of infected sites occupied by the strong infection will tend to $0$ as $t\rightarrow \infty$. This feels like a strong indication that infinite coexistence does not happen.
Another approach to strengthening Theorem \ref{th:pain} in order to obtain the only-if direction of Conjecture \ref{samex_conj} is based on the observation that, since coexistence represents a power balance between the infections, it is reasonable to expect that $P^{1,\lambda}(G)$ decreases as $\lambda$ moves away from 1. We may formulate that intuition as a conjecture:
\begin{conj}\label{conj:monotonitet} For the two-type Richardson model on $\mathbb{Z}^d$ with $d\geq 2$, we have, for $\lambda<\lambda'\in(0,1]$, that $P^{1,\lambda}(G)\leq P^{1,\lambda'}(G)$. \end{conj}
\noindent A confirmation of this conjecture would, in combination with Theorem \ref{th:pain}, clearly establish the only-if direction of Conjecture \ref{samex_conj}: If $P^{1,\lambda}(G)>0$ for some $\lambda<1$, then, according to Conjecture \ref{conj:monotonitet}, we would have $P^{1,\lambda'}(G)>0$ for all $\lambda'\in(\lambda,1]$ as well. But the interval $(\lambda,1]$ is uncountable, yielding a contradiction to Theorem \ref{th:pain}.
Although Conjecture \ref{conj:monotonitet} might seem close to obvious, it has turned out to be very difficult to prove. A natural first attempt would be to use coupling. Consider for instance the coupling $Q$ described above. As pointed out, the events $G^\lambda_1$ and $G^\lambda_2$ that the individual infections grow unboundedly at parameter value $\lambda$ are then monotone in $\lambda$, but one of them is increasing and the other is decreasing, so monotonicity of their intersection $G^\lambda$ does not follow. Hence more sophisticated arguments are needed.
Observing how our colleagues react during seminars and corridor chat, we have noted that it is very tempting to go about trying to prove Conjecture \ref{conj:monotonitet} by abstract and ``easy'' arguments, here meaning arguments that do not involve any specifics about the geometry or graph structure of $\mathbb{Z}^d$. To warn against such attempts, Deijfen and H\"aggstr\"om (2006:2) constructed graphs on which the two-type Richardson model fails to exhibit the monotonicity behavior predicted in Conjecture \ref{conj:monotonitet}. Let us briefly explain the results.
The dynamics of the two-type Richardson model can of course be defined on graphs other than the $\mathbb{Z}^d$ lattice. For a graph $\mathcal{G}$, write Coex$(\mathcal{G})$ for the set of all $\lambda\geq 1$ such that there exists a finite initial configuration $(\xi_1, \xi_2)$ for which the two-type Richardson model with infection intensities $1$ and $\lambda$ started from $(\xi_1, \xi_2)$ yields infinite coexistence with positive probability. Note that, by time-scaling and interchange of the infections, coexistence is possible at parameter value $\lambda$ if and only if it is possible at $\lambda^{-1}$, so no information is lost by restricting to $\lambda\geq 1$. In Deijfen and H\"aggstr\"om (2006:2) examples of graphs $\mathcal{G}$ are given that demonstrate that, among others, the following kinds of coexistence sets Coex$(\mathcal{G})$ are possible:
\begin{itemize} \item[(i)] Coex$(\mathcal{G})$ may be an interval $(a,b)$ with $1<a<b$.
\item[(ii)] For any positive integer $k$ the set Coex$(\mathcal{G})$ may consist of exactly $k$ points.
\item[(iii)] Coex$(\mathcal{G})$ may be countably infinite.
\end{itemize}
\noindent All these phenomena show that the monotonicity suggested in Conjecture \ref{conj:monotonitet} fails for general graphs. However, a reasonable guess is that Conjecture \ref{conj:monotonitet} is true on transitive graphs. Indeed, all counterexamples provided by Deijfen and H\"aggstr\"om are highly non-symmetric (one might even say ugly) with certain parts of the graph being designed specifically with propagation of type 1 in mind, while other parts are meant for type 2. We omit the details.
\section{Unbounded initial configurations} \label{sect:unbounded}
Let us now go back to the $\mathbb{Z}^d$ setting and describe some results from our most recent paper, Deijfen and H\"aggstr\"om (2007), concerning the two-type model with unbounded initial configurations. Roughly, the model will be started from configurations where one of the infections occupies a single site in an infinite ``sea" of the other type. The dynamics is as before and also the question at issue is the same: can both infection types simultaneously infect infinitely many sites? With both types initially occupying infinitely many sites the answer is (apart from in particularly silly cases) obviously yes, so we will focus on configurations where type 1 starts with infinitely many sites and type 2 with finitely many -- for simplicity only one. The question then becomes whether type 2 is able to survive.
To describe the configurations in more detail, write $(x_1,\ldots,x_d)$ for the coordinates of a point $x\in\mathbb{Z}^d$, and define $\mathcal{H}=\{x:x_1=0\}$ and $\mathcal{L}=\{x:x_1 \leq 0\textrm{ and }x_i=0\textrm{ for }i =2,\ldots, d\}$. We will consider the following starting configurations.
\begin{equation}\label{initial_configurations} \begin{array}{rl} I(\mathcal{H}): & \mbox{all points in $\mathcal{H}\backslash\{{\bf 0}\}$ are type 1 infected and}\\ & {\bf 0} \mbox{ is type 2 infected, and}\\ I(\mathcal{L}): & \mbox{all points in $\mathcal{L}\backslash\{{\bf 0}\}$ are type 1 infected and}\\ & {\bf 0} \mbox{ is type 2 infected.} \end{array} \end{equation}
\noindent Interestingly, it turns out that the set of parameter values for which type 2 is able to grow indefinitely is slightly different for these two configuration. First note that, as before, we may restrict to the case $\lambda_1=1$. Write $P^{1,\lambda}_{\mathcal{H},\mathbf{0}}$ and $P^{1,\lambda}_{\mathcal{L},\mathbf{0}}$ for the distribution of the process started from $I(\mathcal{H})$ and $I(\mathcal{L})$ respectively and with type 2 intensity $\lambda$. The following result, where $G_2$ denotes the event that type 2 grows unboundedly, is proved in Deijfen and H\"aggstr\"om (2007).
\begin{theorem}\label{th:unbounded} For the two-type Richardson model in $d\geq 2$ dimensions, we have
\begin{itemize} \item[\rm{(a)}] $P^{1,\lambda}_{\mathcal{H},{\bf 0}}(G_2)>0$ if and only if $\lambda>1$; \item[\rm{\rm{(b)}}]$P^{1,\lambda}_{\mathcal{L},{\bf 0}}(G_2)>0$ if and only if $\lambda\geq 1$. \end{itemize} \end{theorem}
\noindent In words, a strictly stronger type 2 infection will be able to survive in both configurations, but, when the infections have the same intensity, type 2 can survive only in the configuration $I(\mathcal{L})$.
The proof of the if-direction of Theorem \ref{th:unbounded} (a) is based on a lemma stating roughly that the speed of a hampered one-type process, living only inside a tube which is bounded in all directions except one, is close to the speed of an unhampered process when the tube is large. For a two-type process started from $I(\mathcal{H})$, this lemma can be used to show that, if the strong type 2 infection at the origin is successful in the beginning of the time course, it will take off along the $x_1$-axis and grow faster than the surrounding type 1 infection inside a tube around the $x_1$-axis, thereby escaping eradication. The same scenario -- that the type 2 infection rushes away along the $x_1$-axis -- can, by different means, be proved to have positive probability in a process with $\lambda=1$ started from $I(\mathcal{L})$. Infinite growth for type 2 when $\lambda<1$ is ruled out by the key proposition from H\"aggstr\"om and Pemantle (2000) mentioned in Section 3. Proving that type 2 cannot survive in a process with $\lambda=1$ started from $I(\mathcal{H})$ is the most tricky part. The idea is basically to divide $\mathbb{Z}^d$ in different levels, the $l$-th level being all sites with $x_1$-coordinate $l$, and then show that the expected number of type 2 infected sites at level $l$ is constant and equal to 1. It then follows from a certain comparison with a one-type process on each level combined with an application of Levy's 0-1 law that the number of type 2 infected sites at the $l$-th level converges almost surely to 0 as $l\to\infty$.
Finally we mention a question formulated by Itai Benjamini as well as by an anonymous referee of Deijfen and H\"aggstr\"om (2007). We have seen that, when $\lambda=1$, the type 2 infection at the origin can grow unboundedly from $I(\mathcal{L})$ but not from $I(\mathcal{H})$. It is then natural to ask what happens if we interpolate between these two configurations. More precisely, instead of letting type 1 occupy only the negative $x_1$-axis (as in $I(\mathcal{L})$), we let it occupy a cone of constant slope around the same axis. The question then is what the critical slope is for this cone such that there is a positive probability for type 2 to grow unboundedly. That type 2 cannot survive when the cone occupies the whole left half-space follows from Theorem \ref{th:unbounded}, as this situation is equivalent to starting the process from $I(\mathcal{H})$. It seems likely, as suggested by Itai Benjamini, that this is actually also the critical case, that is, infinite growth for type 2 most likely have positive probability for any smaller type 1 cone. This however remains to be proved.
\section*{References}
\noindent Alexander, K.\ (1993): A note on some rates of convergence in first-passage percolation, \emph{Ann. Appl. Probab.} \textbf{3}, 81-90.
\noindent Benjamini, I., Kalai, G. and Schramm, O.\ (2003): First passage percolation has sublinear distance variation, \emph{Ann. Probab.} \textbf{31}, 1970-1978.
\noindent Bramson, M. and Griffeath, D. (1981): On the Williams-Bjerknes tumour growth model I, \emph{Ann. Probab.} \textbf{9}, 173-185.
\noindent Cox, J.T. and Durrett, R. (1981): Some limit theorems for percolation processes with necessary and sufficient conditions, \emph{Ann. Probab.} \textbf{9}, 583-603.
\noindent Deijfen, M.\ and H\"aggstr\"om, O.\ (2004): Coexistence in a two-type continuum growth model, \emph{Adv. Appl. Probab.} \textbf{36}, 973-980.
\noindent Deijfen, M.\ and H\"aggstr\"om, O.\ (2006:1): The initial configuration is irrelevant for the possibility of mutual unbounded growth in the two-type Richardson model, {\em Comb. Probab. Computing} \textbf{15}, 345-353.
\noindent Deijfen, M.\ and H\"{a}ggstr\"{o}m, O.\ (2006:2): Nonmonotonic coexistence regions for the two-type Richardson model on graphs, \emph{Electr. J. Probab.} \textbf{11}, 331-344.
\noindent Deijfen, M.\ and H\"aggstr\"om, O.\ (2007): The two-type Richardson model with unbounded initial configurations, \emph{Ann. Appl. Probab.}, to appear.
\noindent Deijfen, M., H\"aggstr\"om, O.\ and Bagley, J.\ (2004): A stochastic model for competing growth on $R^d$, \emph{Markov Proc. Relat. Fields} \textbf{10}, 217-248.
\noindent Durrett, R. (1988): \emph{Lecture Notes on Particle Systems and Percolation}, Wadsworth $\&$ Brooks/Cole.
\noindent Durrett, R.\ and Neuhauser, C. (1997): Coexistence results for some competition models \emph{Ann. Appl. Probab.} \textbf{7}, 10-45.
\noindent Eden, M. (1961): A two-dimensional growth process, \emph{Proceedings of the 4th Berkeley symposium on mathematical statistics and probability} vol. \textbf{IV}, 223-239, University of California Press.
\noindent Ferrari, P., Martin, J.\ and Pimentel, L. (2006), Roughening and inclination of competition interfaces, \emph{Phys Rev E} \textbf{73}, 031602 (4 p).
\noindent Garet, O.\ and Marchand, R.\ (2005): Coexistence in two-type first-passage percolation models, {\em Ann. Appl. Probab.} {\bf 15}, 298-330.
\noindent Garet, O.\ and Marchand, R.\ (2006): Competition between growths governed by Bernoulli percolation, \emph{Markov Proc. Relat. Fields} \textbf{12}, 695-734.
\noindent Garet, O.\ and Marchand, R.\ (2007): First-passage competition with different speeds: positive density for both species is impossible, preprint, ArXiV math.PR/0608667.
\noindent Gou\'er\'e, J.-B. (2007) Shape of territories in some competing growth models, \emph{Ann. Appl. Probab.}, to appear.
\noindent H\"{a}ggstr\"{o}m, O.\ and Pemantle, R.\ (1998): First passage percolation and a model for competing spatial growth, \emph{J. Appl. Probab.} \textbf{35}, 683-692.
\noindent H\"{a}ggstr\"{o}m, O.\ and Pemantle, R.\ (2000): Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model, \emph{Stoch. Proc. Appl.} \textbf{90}, 207-222.
\noindent Hammersley, J.\ and Welsh D.\ (1965): First passage percolation, subadditive processes, stochastic networks and generalized renewal theory, \emph{1965 Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley}, 61-110, Springer.
\noindent Hoffman, C.\ (2005:1): Coexistence for Richardson type competing spatial growth models, {\em Ann. Appl. Probab.} {\bf 15}, 739-747.
\noindent Hoffman, C.\ (2005:2): Geodesics in first passage percolation, preprint, ArXiV math.PR/0508114.
\noindent Kesten, H.\ (1973): Discussion contribution, \emph{Ann. Probab.} {\bf 1}, 903.
\noindent Kesten, H.\ (1993): On the speed of convergence in first-passage percolation, \emph{Ann. Appl. Probab.} \textbf{3}, 296-338.
\noindent Kingman, J.F.C.\ (1968): The ergodic theory of subadditive stochastic processes, \emph{J. Roy. Statist. Soc. Ser. B} \textbf{30}, 499-510.
\noindent Kordzakhia, G. and Lalley, S. (2005): A two-species competition model on $Z^d$, \emph{Stoch. Proc. Appl.} \textbf{115}, 781-796.
\noindent Lalley, S. (2003): Strict convexity of the limit shape in first-passage percolation, {\em Electr. Comm. Probab.} {\bf 8}, 135--141.
\noindent Licea, C.\ and Newman, C.\ (1996): Geodesics in two-dimensional first-passage percolation, \emph{Ann. Probab.} \textbf{24}, 399-410.
\noindent Neuhauser, C.\ (1992): Ergodic theorems for the multitype contact process, \emph{Probab. Theory Relat. Fields} \textbf{91}, 467-506.
\noindent Newman, C. (1995): A surface view of first passage percolation, \emph{Proc. Int. Congr. Mathematicians} \textbf{1,2} (Zurich 1994), 1017-1023.
\noindent Newman, C.\ and Piza, M.\ (1995): Divergence of shape fluctuations in two dimensions, \emph{Ann. Probab.} \textbf{23}, 977-1005.
\noindent Pimentel, L. (2007): Multitype shape theorems for first passage percolation models, \emph{Adv. Appl. Probab.} \textbf{39}, 53-76.
\noindent Richardson, D.\ (1973): Random growth in a tessellation, \emph{Proc. Cambridge Phil. Soc.} \textbf{74}, 515-528.
\noindent Williams, T. and Bjerknes R. (1972): Stochastic model for abnormal clone spread through epithelial basal layer, \emph{Nature} \textbf{236}, 19-21.
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\begin{document}
\begin{sloppypar}
\title{The $g$-extra connectivity of the strong product of paths and cycles\footnote{The research is supported by National Natural Science Foundation of China (11861066).}}
\author{Qinze Zhu, Yingzhi Tian\footnote{Corresponding author. E-mail: [email protected] (Q. Zhu); [email protected] (Y. Tian).} \\ {\small College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, 830046, PR China}}
\date{} \maketitle
\noindent{\bf Abstract } Let $G$ be a connected graph and $g$ be a non-negative integer. The $g$-extra connectivity of $G$ is the minimum cardinality of a set of vertices in $G$, if it exists, whose removal disconnects $G$ and leaves every component with more than $g$ vertices. The strong product $G_1 \boxtimes G_2$ of graphs $G_1=(V_{1}, E_{1})$ and $G_2=(V_{2}, E_{2})$ is the graph with vertex set $V(G_1 \boxtimes G_2)=V_{1} \times V_{2}$, where two distinct vertices $(x_{1}, x_{2}), (y_{1}, y_{2}) \in V_{1} \times V_{2}$ are adjacent in $G_1 \boxtimes G_2$ if and only if $x_{i}=y_{i}$ or $x_{i} y_{i} \in E_{i}$ for $i=1, 2$. In this paper, we obtain the $g$-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles.
\noindent{\bf Keywords:} Conditional connectivity; $g$-extra connectivity; Strong product; Paths; Cycles
\section{Introduction}
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The $minimum$ $degree$ of $G$ is denoted by $\delta(G)$. A $vertex$ $cut$ in $G$ is a set of vertices whose deletion makes $G$ disconnected. The $connectivity$ $\kappa(G)$ of the graph $G$ is the minimum order of a vertex cut in $G$ if $G$ is not a complete graph; otherwise $\kappa(G)=|V(G)|-1$. Usually, the topology structure of an interconnection network can be modeled by a graph $G$, where $V(G)$ represents the set of nodes and $E(G)$ represents the set of links connecting nodes in the network. Connectivity is used to measure the reliability the network, while it always underestimates the resilience of large networks.
To overcome this deficiency, Harary \cite{Harary} proposed the concept of conditional connectivity. For a graph-theoretic property $\mathcal{P}$, the $conditional$ $connectivity$ $\kappa(G; \mathcal{P})$ is the minimum cardinality of a set of vertices whose deletion disconnects $G$ and every remaining component has property $\mathcal{P}$. Later, F{\`{a}}brega and Fiol \cite{Fabrega} introduced the concept of $g$-extra connectivity, which is a kind of conditional connectivity. Let $g$ be a non-negative integer. A subset $S\subseteq V(G)$ is called a $g$-$extra$ $cut$ if $G-S$ is disconnected and each component of $G-S$ has at least $g+1$ vertices. The $g$-$extra$ $connectivity$ of $G$, denoted by $\kappa_{g}(G)$, is the minimum order of a $g$-extra cut if $G$ has at least one $g$-extra cut; otherwise define $\kappa_{g}(G)=\infty$. If $S$ is a $g$-extra cut in $G$ with order $\kappa_g(G)$, then we call $S$ a $\kappa_g$-$cut$. Since $\kappa_0(G)=\kappa(G)$ for any connected graph $G$ that is not a complete graph, the $g$-extra connectivity can be seen as a generalization of the traditional connectivity. The authors in \cite{Chang} pointed out that there is no polynomial-time algorithm for computing $\kappa_g$ for a general graph. Consequently, much of the work has been focused on the computing of the $g$-extra connectivity of some given graphs, see [1,4,6,8,10-11,16-21] for examples.
The most studied four standard graph products are the Cartesian product, the direct product, the strong product and the lexicographic product. The $Cartesian$ $product$ of two graphs $G_1$ and $G_2$, denoted by $G_1 \square G_2$, is defined on the vertex sets $V(G_1) \times V(G_2)$, and $(x_1, y_1)(x_2, y_2)$ is an edge in $G_1 \square G_2$ if and only if one of the following is true: ($i$) $x_1=x_2$ and $y_1 y_2 \in E(G_2)$; ($ii$) $y_1=y_2$ and $x_1 x_2 \in E(G_1)$.
The $strong$ $product$ $G_1 \boxtimes G_2$ of $G_1$ and $G_2$ is the graph with the vertex set $V(G_1 \boxtimes G_2)=V(G_1) \times V(G_2)$, where two vertices $(x_{1}, y_{1})$, $(x_{2}, y_{2}) \in V(G_1) \times V(G_2)$ are adjacent in $G_1 \boxtimes G_2$ if and only if one of the following holds: ($i$) $x_1=x_2$ and $y_1 y_2 \in E(G_2)$; ($ii$) $y_1=y_2$ and $x_1 x_2 \in E(G_1)$; ($iii$) $x_1 x_2 \in E(G_1)$ and $y_1 y_2 \in E(G_2)$.
{\v{S}}pacapan \cite{Spacapan1} proved that for any nontrivial graphs $G_1$ and $G_2$, $\kappa(G_1 \square G_2)=\min \{\kappa(G_1)|V(G_2)|, \kappa(G_2)|V(G_1)|, \delta(G_1 \square G_2)\}$. L{\"{u}}, Wu, Chen and Lv \cite{Lu} provided bounds for the 1-extra connectivity of the Cartesian product of two connected graphs. Tian and Meng \cite{Tian} determined the exact values of the 1-extra connectivity of the Cartesian product for some class of graphs. In \cite{Chen}, Chen, Meng, Tian and Liu further studied the 2-extra connectivity and the 3-extra connectivity of the Cartesian product of graphs.
Bre{\v{s}}ar and {\v{S}}pacapan \cite{Bresar} determined the edge-connectivity of the strong products of two connected graphs. For the connectivity of the strong product graphs, {\v{S}}pacapan \cite{Spacapan2} obtained $Theorem\ \ref{2}$ in the following. Let $S_i$ be a vertex cut in $G_i$ for $i=1,2$, and let $A_i$ be a component of $ G_i-S_i$ for $i=1,2$. Following the definitions in \cite{Spacapan2}, $I=S_1\times V_2$ or $I=V_1\times S_2$ is called an $I$-set in $G_1\boxtimes G_2$, and $L=(S_1\times A_2)\cup(S_1\times S_2)\cup(A_1\times S_2)$ is called an $L$-set in $G_1\boxtimes G_2$.
\begin{theorem}\label{2} (\cite{Spacapan2}) Let $G_1$ and $G_2$ be two connected graphs. Then every minimum vertex cut in $G_1\boxtimes G_2$ is either an $I$-set or an $L$-set in $G_1\boxtimes G_2$. \end{theorem}
Motivated by the results above, we will study the $g$-extra connectivity of the strong product graphs. In the next section, we introduce some definitions and lemmas. In Section 3, we will give the $g$-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles. Conclusion will be given in Section 4.
\section{Preliminary}
For graph-theoretical terminology and notations not defined here, we follow \cite{Bondy}. Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The $neighborhood$ of a vertex $u$ in $G$ is $N_G(u)=\{v\in V(G)\ |\ v\;\text{is}\; \text{adjacent}\;\text{to} \;\text{the}\; \text{vertex}\; u\}$. Let $A$ be a subset of $V(G)$, the neighborhood of $A$ in $G$ is $N_{G}(A)=\{v \in V(G) \backslash A \ |\ v\;\text{is}\; \text{adjacent}\;\text{to}\; \text{a}\; \text{vertex}\; \text{in}\; A\}$. The subgraph induced by $A$ in $G$ is denoted by $G[A]$. We use $P_n$ to denote the path with order $n$ and $C_n$ to denote the cycle with order $n$.
Let $G_1$ and $G_2$ be two graphs. Define two natural projections $p_1$ and $p_2$ on $V(G_1)\times V(G_2)$ as follows: $p_1(x,y)=x$ and $p_2(x,y)=y$ for any $(x,y)\in V(G_1)\times V(G_2)$. The subgraph induced by $\{(u, y)|u\in V(G_1)\}$ in $G_1\boxtimes G_2$, denoted by $G_{1y}$, is called a $G_1$-$layer$ in $G_1\boxtimes G_2$ for each vertex $y\in V(G_2)$. Analogously, the subgraph induced by $\{(x, v)|v\in V(G_2)\}$ in $G_1\boxtimes G_2$, denoted by ${}_{x}G_2$, is called a $G_2$-$layer$ in $G_1\boxtimes G_2$ for each vertex $x\in V(G_1)$. Clearly, a $G_1$-layer in $G_1\boxtimes G_2$ is isomorphic to $G_1$, and a $G_2$-layer in $G_1\boxtimes G_2$ is isomorphic to $G_2$.
Let $S\subseteq V(G_1\boxtimes G_2)$. For any $x\in V(G_1)$, denote $S\cap V({}_{x}G_2)$ by ${}_{x}S$, and analogously, for any $y\in V(G_2)$, denote $S\cap V(G_{1y})$ by $S_{y}$. Furthermore, we use $\overline{{}_{x}S}=V({}_{x}G_{2})\setminus {}_{x}S$ and $\overline{S_y}=V(G_{1y})\setminus S_y$. By a similar argument as the proof of the second paragraph of $Theorem$ 3.2 in \cite{Spacapan2}, we can obtain the following lemma.
\begin{lemma}\label{1}
Let $G$ be the strong product $G_1\boxtimes G_2$ of two connected graphs $G_1$ and $G_2$, and let $g$ be a non-negative integer. Assume $G$ has $g$-extra cuts and $S$ is a $\kappa_g$-cut of $G$.
(i) If ${}_{x}S\neq\emptyset$ for some $x\in V(G_1)$, then $|{}_{x}S|\geq \kappa(G_2)$.
(ii) If $S_{y}\neq \emptyset$ for some $y\in V(G_2)$, then $|S_{y}|\geq \kappa(G_1)$.
\end{lemma}
\noindent{\bf Proof.} ($i$) Suppose ${ }_{x} S \neq \emptyset$ for some $x\in V(G_1)$. Note that this is obviously true if ${ }_{x} S=V({ }_{x} G_{2})$. If $\overline{{}_{x}S}$ is not contained in one component of $G-S$, then clearly the induced subgraph $G[\overline{{}_{x}S}]$ is not connected, and hence $|{ }_{x} S| \geq \kappa(G_{2})$. If $\overline{{}_{x}S}$ is contained in one component of $G-S$, then choose an arbitrary fixed vertex $(x, y)$ from ${}_{x}S$. Let $H_{1}$ be the component of $G-S$ such that $\overline{{}_{x}S} \subseteq V(H_{1})$ and let $H_{2}=G-S\cup V(H_{1})$. Since $S$ is a $\kappa_{g}$-cut, we find that the vertex $(x, y) \in{ }_{x} S$ has a neighbor $(x_{1}, y_{1}) \in V(H_{2})$. Since $(x_{1}, y_{1}) \in V(H_{2})$, we find that $(x, y_{1}) \in {}_{x} S$, moreover, for any $(x, u) \in \overline{{}_{x}S}$, we find that $(x, u)$ is not adjacent to $(x, y_{1})$, otherwise, $(x, u)$ would be adjacent to $(x_{1}, y_{1})$, which is not true since those two vertices are in different components of $G-S$. Thus if $R={ }_{x} S\setminus\{(x, y_{1})\}$, then $p_{2}(R)$ is a vertex cut in $G_{2}$ and one component of $G_{2}-p_{2}(R)$ is $\{y_{1}\}$. Thus $|{ }_{x} S|=|R|+1 \geq \kappa(G_{2})+1$. Analogously, we can get $|S_{y}| \geq \kappa(G_{1})$ if ($ii$) holds. $\Box$
\section{Main results}
Let $H$ be a subgraph of $G_1\boxtimes G_2$. For the sake of simplicity, we use ${}_{x}H$ instead of ${}_{x}V(H)$ to represent $V(H)\cap V({}_{x}G_2)$ for any $x\in V(G_1)$ and $H_{y}$ to represent $V(H)\cap V(G_{1y})$ for any $y\in V(G_2)$. Since $\kappa_g(P_1\boxtimes P_n)=1$ for $g\leq \lfloor\frac{n-1}{2}\rfloor-1$ and $\kappa_g(P_2\boxtimes P_n)=2$ for $g\leq 2\lfloor\frac{n-1}{2}\rfloor-1$, we assume $m,n\geq3$ in the following theorem.
\begin{theorem}\label{5}
Let $g$ be a non-negative integer and $G=P_m\boxtimes P_n$, where $m,n\geq 3$. If $g\leq min\{n\lfloor\frac{m-1}{2}\rfloor-1, m\lfloor\frac{n-1}{2}\rfloor-1 \}$, then $\kappa_g(G)=min\{m, n, \lceil 2\sqrt{g+1}\ \rceil+1\}$.
\end{theorem}
\noindent{\bf Proof. }Denote $P_m=x_1x_2\dots x_m$ and $P_n=y_1y_2\dots y_n$. Let $S_1=V(P_m)\times \{y_{\lfloor\frac{n-1}{2}\rfloor+1}\}$ and $S_2=\{x_{\lfloor\frac{m-1}{2}\rfloor+1}\}\times V(P_n)$. Since $g\leq min\{n\lfloor\frac{m-1}{2}\rfloor-1, m\lfloor\frac{n-1}{2}\rfloor-1 \}$, we verify that $S_1$ and $S_2$ are two $g$-extra cuts of $G$. Thus $\kappa_g(G)\leq$ min$\{m, n\}$. If $\lceil 2\sqrt{g+1}\ \rceil+1\geq$ min$\{m, n\}$, then $\kappa_g(G)\leq$ min$\{m, n, \lceil 2\sqrt{g+1}\ \rceil+1\}$. If $\lceil 2\sqrt{g+1}\ \rceil+1<$ min$\{m, n\}$, then let $S_3=(J_1\times K_2)\cup (J_1\times J_2)\cup (K_1\times J_2)$, where
$J_1=\{x_{\lceil \sqrt{g+1}\ \rceil+1}\}$, $K_1=\{x_1, x_2, \dots, x_{\lceil \sqrt{g+1}\ \rceil}\}$, $J_2=\{y_{\lceil\frac{g+1}{\lceil \sqrt{g+1}\ \rceil}\rceil+1}\}$ and $K_2=\{y_1, y_2, \dots, y_{\lceil\frac{g+1}{\lceil \sqrt{g+1}\ \rceil}\rceil}\}$. It is routine to verify that $S_3$ is a $g$-extra cut of $G$. By $|S_3|=\lceil \sqrt{g+1}\ \rceil+\lceil\frac{g+1}{\lceil \sqrt{g+1}\ \rceil}\rceil+1=\lceil 2\sqrt{g+1}\ \rceil+1$, we have $\kappa_g(G)\leq \lceil 2\sqrt{g+1}\ \rceil+1$. Therefore, $\kappa_g(G)\leq$ min$\{m, n, \lceil 2\sqrt{g+1}\ \rceil+1\}$ holds.
Now, it is sufficient to prove $\kappa_g(G)\geq$ min$\{m, n, \lceil 2\sqrt{g+1}\ \rceil+1\}$. Assume $S$ is a $\kappa_g$-cut of $G$. We consider two cases in the following.
\noindent{\bf Case 1. } ${}_{x}S\neq\emptyset$ for all $x\in V(P_m)$, or $S_y\neq \emptyset$ for all $y\in V(P_n)$.
Assume ${}_{x}S\neq\emptyset$ for all $x\in V(P_m)$. By $Lemma$ 2.1, $|S|= \sum_{x\in V(P_m)}|{}_{x}S|\geq \kappa(P_n)|V(P_m)|=m$. Analogously, if $S_y\neq \emptyset$ for all $y\in V(P_n)$, then $|S|=\sum_{y\in V(P_n)}|S_y|\geq \kappa(P_m)|V(P_n)|=n$.
\noindent{\bf Case 2. } There exist a vertex $x_a\in V(P_m)$ and a vertex $y_b\in V(P_n)$ such that ${}_{x_a}S=S_{y_b}=\emptyset$.
By the assumption ${}_{x_a}S=S_{y_b}=\emptyset$, we know $V({}_{x_a}G_2)$ and $V(G_{1y_b})$ are contained in a component $H'$ of $G-S$. Let $H$ be another component of $G-S$. Let $p_1(V(H))=\{x_{s+1},x_{s+2},\cdots,x_{s+k}\}$ and $p_2(V(H))=\{y_{t+1},y_{t+2},\cdots,y_{t+h}\}$. Without loss of generality, assume $s+k<a$ and $t+h< b$. Clearly, $|V(H)|\leq kh$. Since $S$ is a $\kappa_g$-cut, we have $N_G(V(H))=S$ and $|V(H)|\geq g+1$. If we can prove $|N_G(V(H))|\geq k+h+1$, then $\kappa_g(G)=|S|=|N_G(V(H))|\geq k+h+1\geq2\sqrt{kh}+1\geq2\sqrt{g+1}+1$ and the theorem holds. Thus, we only need to show that $|N_G(V(H))|\geq k+h+1$ in the remaining proof.
\begin{figure}
\caption{An illustration for the proof of Theorem 3.1.}
\label{1}
\end{figure}
Let $(x_{s+i}, y_{d_i})$ be the vertex in ${}_{x_{s+i}}H$ such that $d_i$ is maximum for $i=1,\cdots, k$, and let $(x_{r_j}, y_{t+j})$ be the vertex in $H_{y_{t+j}}$ such that $r_j$ is maximum for $j=1,\cdots, h$. Denote $D=\{(x_{s+1}, y_{d_1}),\cdots,(x_{s+k}, y_{d_k})\}$ and $R=\{(x_{r_1}, y_{t+1}),\cdots,(x_{r_h}, y_{t+h})\}$. For the convenience of counting, we will construct an injective mapping $f$ from $D\cup R$ to $N_G(V(H)\setminus\{(x_{s+k+1}, y_{d_k+1})\}$. Although $D$ and $R$ may have common elements, we consider the elements in $D$ and $R$ to be different in defining the mapping $f$ below.
First, the mapping $f$ on $D$ is defined as follows. \begin{center} $f((x_{s+i}, y_{d_i}))=(x_{s+i}, y_{d_i+1})$ for $i=1,\cdots, k$. \end{center} Denote $F_1=\{(x_{s+1}, y_{d_1+1}),\cdots,(x_{s+k}, y_{d_k+1})\}$.
Second, for each vertex $(x_{r_j}, y_{t+j})$ satisfying $(x_{r_j+1}, y_{t+j})\notin F_1$, define $f((x_{r_j}, y_{t+j}))=(x_{r_j+1}, y_{t+j})$.
If $(x_{r_j}, y_{t+j})$ satisfies $(x_{r_j+1}, y_{t+j})\notin F_1$ for any $j\in\{1,\cdots, h\}$, then we are done. Otherwise, for each $(x_{r_{j'}}, y_{t+j'})$ satisfying $(x_{r_{j'}+1}, y_{t+j'})\in F_1$, we give the definition as follows. By the definitions of $D$ and $R$, we have $(x_{r_{j'}+1+i}, y_{t+j'+j})\notin V(H)$ for all $i, j\geq 0$, and $\{(x_{r_{j'}},y_{t+j'}),\cdots,(x_{r_{j'}}, y_{d_{r_{j'}-s}})\}\subseteq R$ (see Figure 1 for an illustration). Now, we define $f((x_{r_{j'}}, y_{t+j'}))=(x_{r_{j'}+1}, y_{t+j'+1})$ and change the images of $(x_{r_{j'}}, y_{t+j'+1}),\cdots,(x_{r_{j'}}, y_{d_{r_{j'}-s}})$ to $(x_{r_{j'}+1}, y_{t+j'+2}),\cdots,(x_{r_{j'}+1}, y_{d_{r_{j'}-s}+1})$, respectively. The images of $f$ on $R$ are well defined.
Finally, we have an injective mapping $f$ from $D\cup R$ to $N_G(V(H)\setminus\{(x_{s+k+1}, y_{d_k+1})\}$. Then $\kappa_g(G)=|S|=|N_G(V(H))|\geq |D|+|R|+1\geq k+h+1\geq2\sqrt{kh}+1\geq2\sqrt{g+1}+1$. The proof is thus complete. $\Box$
Since $\kappa_g(C_3\boxtimes P_n)=3$ for $g\leq 3\lfloor\frac{n-1}{2}\rfloor-1$, we assume $m\geq4$ in the following theorem.
\begin{theorem}\label{6} Let $g$ be a non-negative integer and $G=C_m\boxtimes P_n$, where $m\geq4$, $n\geq 3$. If $g\leq min\{n\lfloor\frac{m-2}{2}\rfloor-1, m\lfloor\frac{n-1}{2}\rfloor-1 \}$, then $\kappa_g(G)= min\{m, 2n, \lceil2\sqrt{2(g+1)}\rceil+2\}$. \end{theorem}
\noindent{\bf Proof.} Denote $C_m=x_0x_1\dots x_{m-1} x_{m}$ (where $x_0=x_{m}$) and $P_n=y_1y_2\dots y_n$. The addition of the subscripts of $x$ in the proof is modular $m$ arithmetic. Let $S_1=V(C_m)\times\{y_{\lfloor\frac{n-1}{2}\rfloor+1}\}$ and $S_2=\{x_{0}, x_{\lfloor\frac{m-2}{2}\rfloor+1}\}\times V(P_n)$. Since $g\leq min\{n\lfloor\frac{m-2}{2}\rfloor-1, m\lfloor\frac{n-1}{2}\rfloor-1 \}$, it is routine to check that $S_1$ and $S_2$ are two $g$-extra cuts of $G$. Thus $\kappa_g(G)\leq$ min$\{m, 2n\}$. If $\lceil2\sqrt{2(g+1)}\rceil+2\geq$ min$\{m, 2n\}$, then $\kappa_g(G)\leq$ min$\{m, 2n, \lceil2\sqrt{2(g+1)}\rceil+2\}$. If $\lceil2\sqrt{2(g+1)}\rceil+2<$ min$\{m, 2n\}$, then let $S_3=(J_1\times K_2)\cup (J_1\times J_2)\cup (K_1\times J_2)$, where
$J_1=\{x_{0}, x_{\lceil\sqrt{2(g+1)}\rceil+1} \}$, $K_1=\{x_1, x_2, \dots, x_{\lceil\sqrt{2(g+1)}\rceil}\}$, $J_2=\{y_{\lceil\frac{2(g+1)}{\lceil \sqrt{2(g+1)}\rceil}\rceil+1}\}$ and $K_2=\{y_1, y_2, \dots, y_{\lceil\frac{2(g+1)}{\lceil \sqrt{2(g+1)}\rceil}\rceil}\}$. It is routine to verify that $S_3$ is a $g$-extra cut of $G$. By $|S_3|=\lceil \sqrt{2(g+1)}\rceil+\lceil\frac{2(g+1)}{\lceil \sqrt{2(g+1)}\rceil}\rceil+2=\lceil 2\sqrt{2(g+1)}\rceil+2$, we have $\kappa_g(G)\leq \lceil2\sqrt{2(g+1)}\rceil+2$. Therefore, $\kappa_g(G)\leq$ min$\{m, 2n, \lceil2\sqrt{2(g+1)}\rceil+2\}$.
Now, it is sufficient to prove $\kappa_g(G)\geq$ min$\{m, 2n, \lceil2\sqrt{2(g+1)}\rceil+2\}$. Assume $S$ is a $\kappa_g$-cut of $G$. We consider two cases in the following.
\noindent{\bf Case 1. } ${}_{x}S\neq\emptyset$ for all $x\in V(C_m)$, or $S_y\neq \emptyset$ for all $y\in V(P_n)$.
Assume ${}_{x}S\neq\emptyset$ for all $x\in V(C_m)$. By $Lemma$ 2.1, $|S|= \sum_{x\in V(C_m)}|{}_{x}S|\geq \kappa(P_n)|V(C_m)|=m$. Analogously, if $S_y\neq \emptyset$ for all $y\in V(P_n)$, then $|S|=\sum_{y\in V(P_n)}|S_y|\geq \kappa(C_m)|V(P_n)|=2n$.
\noindent{\bf Case 2. } There exist a vertex $x_a\in V(C_m)$ and a vertex $y_b\in V(P_n)$ such that ${}_{x_a}S=S_{y_b}=\emptyset$.
By the assumption ${}_{x_a}S=S_{y_b}=\emptyset$, we know $V({}_{x_a}G_2)$ and $V(G_{1y_b})$ are contained in a component $H'$ of $G-S$. Let $H$ be another component of $G-S$. Let $p_1(V(H))=\{x_{s+1},x_{s+2},\cdots,x_{s+k}\}$ and $p_2(V(H))=\{y_{t+1},y_{t+2},\cdots,y_{t+h}\}$. Without loss of generality, assume $s+k<a$ and $t+h< b$. Clearly, $|V(H)|\leq kh$. Since $S$ is a $\kappa_g$-cut, we have $N_G(V(H))=S$ and $|V(H)|\geq g+1$. If we can prove $|N_G(V(H))|\geq k+2h+2$, then $\kappa_g(G)=|S|=|N_G(V(H))|\geq k+2h+2\geq2\sqrt{2kh}+2\geq2\sqrt{2(g+1)}+2$ and the theorem holds. Thus, we only need to show that $|N_G(V(H))|\geq k+2h+2$ in the remaining proof.
Let $(x_{s+i}, y_{d_i})$ be the vertex in ${}_{x_{s+i}}H$ such that $d_i$ is maximum for $i=1,\cdots, k$, and let $(x_{l_j}, y_{t+j})$ and $(x_{r_j}, y_{t+j})$ be the vertices in $H_{y_{t+j}}$ such that $l_j$ and $r_j$ are listed in the foremost and in the last along the sequence $(a+1,\cdots,m-1,0,1,\cdots,a-1)$, respectively, for $j=1,\cdots, h$. Denote $D=\{(x_{s+1}, y_{d_1}),\cdots,(x_{s+k}, y_{d_k})\}$, $L=\{(x_{l_1}, y_{t+1}),\cdots,(x_{l_h}, y_{t+h})\}$ and $R=\{(x_{r_1}, y_{t+1}),\cdots,(x_{r_h}, y_{t+h})\}$. For the convenience of counting, we will construct an injective mapping $f$ from $D\cup L\cup R$ to $N_G(V(H)\setminus\{(x_{s}, y_{d_1+1}),(x_{s+k+1}, y_{d_k+1})\}$. Although $D$, $L$ and $R$ may have common elements, we consider the elements in $D$, $L$ and $R$ to be different in defining the mapping $f$ below.
First, the mapping $f$ on $D$ is defined as follows. \begin{center} $f((x_{s+i}, y_{d_i}))=(x_{s+i}, y_{d_i+1})$ for $i=1,\cdots, k$. \end{center} Denote $F_1=\{(x_{s+1}, y_{d_1+1}),\cdots,(x_{s+k}, y_{d_k+1})\}$.
Second, for each vertex $(x_{r_j}, y_{t+j})$ satisfying $(x_{r_j+1}, y_{t+j})\notin F_1$, define $f((x_{r_j}, y_{t+j}))=(x_{r_j+1}, y_{t+j})$.
If $(x_{r_j}, y_{t+j})$ satisfies $(x_{r_j+1}, y_{t+j})\notin F_1$ for any $j\in\{1,\cdots, h\}$, then we are done. Otherwise, for each $(x_{r_{j'}}, y_{t+j'})$ satisfying $(x_{r_{j'}+1}, y_{t+j'})\in F_1$, we give the definition as follows. By the definitions of $D$ and $R$, we have $\{(x_{r_{j'}},y_{t+j'}),\cdots,(x_{r_{j'}}, y_{d_{r_{j'}-s}})\}\subseteq R$. Now, we define $f((x_{r_{j'}}, y_{t+j'}))=(x_{r_{j'}+1}, y_{t+j'+1})$ and change the images of $(x_{r_{j'}}, y_{t+j'+1}),\cdots,(x_{r_{j'}}, y_{d_{r_{j'}-s}})$ to $(x_{r_{j'}+1}, y_{t+j'+2}),\cdots,(x_{r_{j'}+1}, y_{d_{r_{j'}-s}+1})$, respectively. The mapping $f$ on $R$ is defined well .
Third, for each vertex $(x_{l_j}, y_{t+j})$ satisfying $(x_{l_j-1}, y_{t+j})\notin F_1$, define $f((x_{l_j}, y_{t+j}))=(x_{l_j-1}, y_{t+j})$.
If $(x_{l_j}, y_{t+j})$ satisfies $(x_{l_j-1}, y_{t+j})\notin F_1$ for any $j\in\{1,\cdots, h\}$, then we are done. Otherwise, for each $(x_{l_{j'}}, y_{t+j'})$ satisfying $(x_{l_{j'}-1}, y_{t+j'})\in F_1$. By the definitions of $D$ and $L$, we have $\{(x_{l_{j'}},y_{t+j'}),\cdots,(x_{l_{j'}}, y_{d_{l_{j'}-s}})\}\subseteq L$. Now, we define $f((x_{l_{j'}}, y_{t+j'}))=(x_{l_{j'}-1}, y_{t+j'+1})$ and change the images of $(x_{l_{j'}}, y_{t+j'+1}),\cdots,(x_{l_{j'}}, y_{d_{l_{j'}-s}})$ to $(x_{r_{j'}-1}, y_{t+j'+2}),\cdots,(x_{l_{j'}-1}, y_{d_{r_{j'}-s}+1})$, respectively. The definition of $f$ on $L$ is complete.
Finally, we construct an injective mapping $f$ from $D\cup L\cup R$ to $N_G(V(H)\setminus\{(x_{s}, y_{d_1+1}),(x_{s+k+1}, y_{d_k+1})\}$. Then $\kappa_g(G)=|S|=|N_G(V(H))|\geq |D|+|L|+|R|+2\geq k+2h+1\geq2\sqrt{2kh}+2\geq2\sqrt{2(g+1)}+2$. The proof is thus complete. $\Box$
Since $\kappa_g(C_3\boxtimes C_n)=6$ for $g\leq 3\lfloor\frac{n-2}{2}\rfloor-1$, we assume $m, n\geq4$ in the following theorem.
\begin{theorem} Let $g$ be a non-negative integer and $G=C_m\boxtimes C_n$, where $m, n\geq 4$. If $g\leq min\{n\lfloor\frac{m-2}{2}\rfloor-1, m\lfloor\frac{n-2}{2}\rfloor-1 \}$, then $\kappa_g(G)=min\{2m, 2n, \lceil 4\sqrt{g+1}\ \rceil+4\}$. \end{theorem}
\noindent{\bf Proof.} Denote $C_m=x_0x_1\dots x_{m-1} x_{m}$ (where $x_0=x_{m}$) and $C_n=y_0y_1\dots y_n$ (where $y_0=y_{n}$). The addition of the subscripts of $x$ in the proof is modular $m$ arithmetic, and the addition of the subscripts of $y$ in the proof is modular $n$ arithmetic. Let $S_1=V(C_m)\times\{y_{0}, y_{\lfloor\frac{n-2}{2}\rfloor+1}\}$ and $S_2=\{x_{0}, x_{\lfloor\frac{m-2}{2}\rfloor+1}\}\times V(C_n)$. Since $g\leq min\{n\lfloor\frac{m-2}{2}\rfloor-1, m\lfloor\frac{n-2}{2}\rfloor-1 \}$, we can check that $S_1$ and $S_2$ are two $g$-extra cuts of $G$. Thus $\kappa_g(G)\leq$ min$\{2m, 2n\}$. If $\lceil 4\sqrt{g+1}\ \rceil+4\geq$ min$\{2m, 2n\}$, then $\kappa_g(G)\leq$ min$\{2m, 2n, \lceil 4\sqrt{g+1}\ \rceil+4\}$. If $\lceil 4\sqrt{g+1}\ \rceil+4<$ min$\{2m, 2n\}$, then let $S_3=(J_1\times K_2)\cup (J_1\times J_2)\cup (K_1\times J_2)$, where
$J_1=\{x_{0}, x_{\lceil \sqrt{g+1}\ \rceil+1}\}$, $K_1=\{x_1, x_2, \dots, x_{\lceil \sqrt{g+1}\ \rceil}\}$, $J_2=\{y_0,y_{\lceil\frac{g+1}{\lceil \sqrt{g+1}\ \rceil} \rceil+1}\}$ and $K_2=\{y_1, y_2, \dots, y_{\lceil\frac{g+1}{\lceil \sqrt{g+1}\ \rceil}\rceil}\}$. It is routine to verify that $S_3$ is a $g$-extra cut of $G$. By $|S_3|=2\lceil \sqrt{g+1}\rceil+2\lceil\frac{g+1}{\lceil \sqrt{g+1}\ \rceil}\rceil+4=\lceil 4\sqrt{g+1}\rceil+4$, we have $\kappa_g(G)\leq \lceil 4\sqrt{g+1}\ \rceil+4$. Therefore, $\kappa_g(G)\leq$ min$\{2m, 2n, \lceil 4\sqrt{g+1}\ \rceil+4\}$.
Now, it is sufficient to prove $\kappa_g(G)\geq$ min$\{2m, 2n, \lceil 4\sqrt{g+1}\ \rceil+4\}$. Assume $S$ is a $\kappa_g$-cut of $G$. We consider two cases in the following.
\noindent{\bf Case 1. } ${}_{x}S\neq\emptyset$ for all $x\in V(C_m)$, or $S_y\neq \emptyset$ for all $y\in V(C_n)$.
Assume ${}_{x}S\neq\emptyset$ for all $x\in V(C_m)$. By $Lemma$ 2.1, $|S|= \sum_{x\in V(C_m)}|{}_{x}S|\geq \kappa(C_n)|V(C_m)|=2m$. Analogously, if $S_y\neq \emptyset$ for all $y\in V(C_n)$, then $|S|=\sum_{y\in V(C_n)}|S_y|\geq \kappa(C_m)|V(C_n)|=2n$.
\noindent{\bf Case 2. } There exist a vertex $x_a\in V(C_m)$ and a vertex $y_b\in V(C_n)$ such that ${}_{x_a}S=S_{y_b}=\emptyset$.
By the assumption ${}_{x_a}S=S_{y_b}=\emptyset$, we know $V({}_{x_a}G_2)$ and $V(G_{1y_b})$ are contained in a component $H'$ of $G-S$. Let $H$ be another component of $G-S$. Let $p_1(V(H))=\{x_{s+1},x_{s+2},\cdots,x_{s+k}\}$ and $p_2(V(H))=\{y_{t+1},y_{t+2},\cdots,y_{t+h}\}$. Without loss of generality, assume $s+k<a$ and $t+h< b$. Clearly, $|V(H)|\leq kh$. Since $S$ is a $\kappa_g$-cut, we have $N_G(V(H))=S$ and $|V(H)|\geq g+1$. If we can prove $|N_G(V(H))|\geq 2k+2h+4$, then $\kappa_g(G)=|S|=|N_G(V(H))|\geq 2k+2h+4\geq4\sqrt{kh}+4\geq4\sqrt{g+1}+4$ and the theorem holds. Thus, we only need to show that $|N_G(V(H))|\geq 2k+2h+4$ in the remaining proof.
Let $(x_{s+i}, y_{t_i})$ and $(x_{s+i}, y_{d_i})$ be the vertices in ${}_{x_{s+i}}H$ such that $t_i$ and $d_i$ are listed in the foremost and in the last along the sequence $(b+1,\cdots,n-1,0,1,\cdots,b-1)$, respectively, for $i=1,\cdots, k$, and let $(x_{l_j}, y_{t+j})$ and $(x_{r_j}, y_{t+j})$ be the vertices in $H_{y_{t+j}}$ such that $l_j$ and $r_j$ are listed in the foremost and in the last along the sequence $(a+1,\cdots,m-1,0,1,\cdots,a-1)$, respectively, for $j=1,\cdots, h$. Denote $D=\{(x_{s+1}, y_{d_1}),\cdots,(x_{s+k}, y_{d_k})\}$, $T=\{(x_{s+1}, y_{t_1}),\cdots,(x_{s+k}, y_{t_k})\}$, $L=\{(x_{l_1}, y_{t+1}),\cdots,(x_{l_h}, y_{t+h})\}$ and $R=\{(x_{r_1}, y_{t+1}),\cdots,(x_{r_h}, y_{t+h})\}$. For the convenience of counting, we will construct an injective mapping $f$ from $D\cup T\cup L\cup R$ to $N_G(V(H)\setminus\{(x_{s}, y_{d_1+1}), (x_{s}, y_{t_1-1}), (x_{s+k+1}, y_{d_k+1}), (x_{s+k+1}, y_{t_k-1})\}$. Although $D$, $T$, $L$ and $R$ may have common elements, we consider the elements in $D$, $T$, $L$ and $R$ to be different in defining the mapping $f$ below.
First, the mapping $f$ on $D$ is defined as follows. \begin{center} $f((x_{s+i}, y_{d_i}))=(x_{s+i}, y_{d_i+1})$ for $i=1,\cdots, k$. \end{center} Denote $F_1=\{(x_{s+1}, y_{d_1+1}),\cdots,(x_{s+k}, y_{d_k+1})\}$.
Second, the mapping $f$ on $T$ is defined as follows. \begin{center} $f((x_{s+i}, y_{t_i}))=(x_{s+i}, y_{t_i-1})$ for $i=1,\cdots, k$. \end{center} Denote $F_2=\{(x_{s+1}, y_{t_1-1}),\cdots,(x_{s+k}, y_{t_k-1})\}$.
Third, for each vertex $(x_{r_j}, y_{t+j})$ satisfying $(x_{r_j+1}, y_{t+j})\notin F_1$, define $f((x_{r_j}, y_{t+j}))=(x_{r_j+1}, y_{t+j})$.
If $(x_{r_j}, y_{t+j})$ satisfies $(x_{r_j+1}, y_{t+j})\notin F_1$ for any $j\in\{1,\cdots, h\}$, then we are done. Otherwise, for each $(x_{r_{j'}}, y_{t+j'})$ satisfying $(x_{r_{j'}+1}, y_{t+j'})\in F_1$, we define as follows. By the definitions of $D$ and $R$, we have $\{(x_{r_{j'}},y_{t+j'}),\cdots,(x_{r_{j'}}, y_{d_{r_{j'}-s}})\}\subseteq R$. Now, we define $f((x_{r_{j'}}, y_{t+j'}))=(x_{r_{j'}+1}, y_{t+j'+1})$ and change the images of $(x_{r_{j'}}, y_{t+j'+1}),\cdots,(x_{r_{j'}}, y_{d_{r_{j'}-s}})$ to $(x_{r_{j'}+1}, y_{t+j'+2}),\cdots,(x_{r_{j'}+1}, y_{d_{r_{j'}-s}+1})$, respectively.
Fourth, for each vertex $(x_{r_j}, y_{t+j})$ satisfying $(x_{r_j+1}, y_{t+j})\notin F_2$, define $f((x_{r_j}, y_{t+j}))=(x_{r_j+1}, y_{t+j})$.
If $(x_{r_j}, y_{t+j})$ satisfies $(x_{r_j+1}, y_{t+j})\notin F_2$ for any $j\in\{1,\cdots, h\}$, then we are done. Otherwise, for each $(x_{r_{j'}}, y_{t+j'})$ satisfying $(x_{r_{j'}+1}, y_{t+j'})\in F_2$, we define as follows. By the definitions of $T$ and $R$, we have $\{(x_{r_{j'}},y_{t+j'}),\cdots,(x_{r_{j'}}, y_{d_{t_{j'}-s}})\}\subseteq R$. Now, we define $f((x_{r_{j'}}, y_{t+j'}))=(x_{r_{j'}+1}, y_{t+j'-1})$ and change the images of $(x_{r_{j'}}, y_{t+j'-1}),\cdots,(x_{r_{j'}}, y_{d_{t_{j'}-s}})$ to $(x_{r_{j'}+1}, y_{t+j'-2}),\cdots,(x_{r_{j'}+1}, y_{d_{r_{j'}-s}-1})$, respectively.
Note that the proof of four paragraphs above gives the definition of the mapping $f$ on $R$. In the following proof, we will give the definition of the mapping $f$ on $L$.
Fifth, for each vertex $(x_{l_j}, y_{t+j})$ satisfying $(x_{l_j-1}, y_{t+j})\notin F_1$, define $f((x_{l_j}, y_{t+j}))=(x_{l_j-1}, y_{t+j})$.
If $(x_{l_j}, y_{t+j})$ satisfies $(x_{l_j-1}, y_{t+j})\notin F_1$ for any $j\in\{1,\cdots, h\}$, then we are done. Otherwise, for each $(x_{l_{j'}}, y_{t+j'})$ satisfying $(x_{l_{j'}-1}, y_{t+j'})\in F_1$, we define as follows. By the definitions of $D$ and $L$, we have $\{(x_{l_{j'}},y_{t+j'}),\cdots,(x_{l_{j'}}, y_{d_{l_{j'}-s}})\}\subseteq L$. Now, we define $f((x_{l_{j'}}, y_{t+j'}))=(x_{l_{j'}-1}, y_{t+j'+1})$ and change the images of $(x_{l_{j'}}, y_{t+j'+1}),\cdots,(x_{l_{j'}}, y_{d_{l_{j'}-s}})$ to $(x_{r_{j'}-1}, y_{t+j'+2}),\cdots,(x_{l_{j'}-1}, y_{d_{r_{j'}-s}+1})$, respectively.
Sixth, for each vertex $(x_{l_j}, y_{t+j})$ satisfying $(x_{l_j-1}, y_{t+j})\notin F_2$, define $f((x_{l_j}, y_{t+j}))=(x_{l_j-1}, y_{t+j})$.
If $(x_{l_j}, y_{t+j})$ satisfies $(x_{l_j-1}, y_{t+j})\notin F_2$ for any $j\in\{1,\cdots, h\}$, then we are done. Otherwise, for each $(x_{l_{j'}}, y_{t+j'})$ satisfying any $(x_{t_{j'}-1}, y_{t+j'})\in F_2$, we define as follows. By the definitions of $L$ and $T$, we have $\{(x_{l_{j'}},y_{t+j'}),\cdots,(x_{l_{j'}}, y_{d_{t_{j'}-s}})\}\subseteq L$. Now, we define $f((x_{l_{j'}}, y_{t+j'}))=(x_{l_{j'}-1}, y_{t+j'-1})$ and change the images of $(x_{l_{j'}}, y_{t+j'-1}),\cdots,(x_{l_{j'}}, y_{d_{t_{j'}-s}})$ to $(x_{l_{j'}-1}, y_{t+j'-2}),\cdots,(x_{l_{j'}-1}, y_{d_{t_{j'}-s}-1})$, respectively.
Finally, we construct an injective mapping $f$ from $D\cup T\cup L\cup R$ to $N_G(V(H)\setminus\{(x_{s}, y_{d_1+1}), (x_{s}, y_{t_1-1}), (x_{s+k+1}, y_{d_k+1}), (x_{s+k+1}, y_{t_k-1})\}$. Then $\kappa_g(G)=|S|=|N_G(V(H))|\geq |D|+|T|+|L|+|R|+4\geq 2k+2h+4\geq4\sqrt{kh}+4\geq4\sqrt{g+1}+4$. The proof is thus complete. $\Box$
\section{Conclusion}
Graph products are used to construct large graphs from small ones. Strong product is one of the most studied four graph products. As a generalization of traditional connectivity, $g$-extra connectivity can be seen as a refined parameter to measure the reliability of interconnection networks. There is no polynomial-time algorithm to compute the $g\ (\geq1)$-extra connectivity for a general graph. In this paper, we determined the $g$-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles. In the future work, we would like to investigate the $g$-extra connectivity of the strong product of two general graphs.
\end{sloppypar}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{The variety generated by all the ordinal sums of perfect MV-chains} \author{Matteo Bianchi\\{\small Dipartimento di Informatica e Comunicazione}\\{\small Università degli Studi di Milano}\\{\footnotesize \texttt{\href{mailto:[email protected]}{[email protected]}}}} \date{} \maketitle \begin{quote} \small \it ``To my friend Erika, and to her invaluable talent in finding surprisingly deep connections among poetry, art, philosophy and logic.'' \end{quote} \begin{abstract} We present the logic BL$_\text{Chang}$, an axiomatic extension of BL (see \cite{haj}) whose corresponding algebras form the smallest variety containing all the ordinal sums of perfect MV-chains. We will analyze this logic and the corresponding algebraic semantics in the propositional and in the first-order case. As we will see, moreover, the variety of BL$_\text{Chang}$-algebras will be strictly connected to the one generated by Chang's MV-algebra (that is, the variety generated by all the perfect MV-algebras): we will also give some new results concerning these last structures and their logic. \end{abstract} \section{Introduction} MV-algebras were introduced in \cite{chang} as the algebraic counterpart of \L ukasie\-wicz (infinite-valued) logic. During the years these structures have been intensively studied (for a hystorical overview, see \cite{cig}): the book \cite{mun} is a reference monograph on this topic.
Perfect MV-algebras were firstly studied in \cite{dnl2} as a refinement of the notion of local MV-algebras: this analysis was expanded in \cite{dnl1}, where it was also shown that the class of perfect MV-algebras $Perf(MV)$ does not form a variety, and the variety generated by $Perf(MV)$ is also generated by Chang's MV-algebra (see \Cref{sec:mvcha} for the definition). Further studies, about this variety and the associated logic have been done in \cite{bdg, bdg1}.
On the other side, Basic Logic BL and its correspondent variety, BL-algebras, were introduced in \cite{haj}: \L ukasiewicz logic results to be one of the axiomatic extensions of BL and MV-algebras can also be defined as a subclass of BL-algebras. Moreover, the connection between MV-algebras and BL-algebras is even stronger: in fact, as shown in \cite{am}, every ordinal sum of MV-chains is a BL-chain.
For these reasons one can ask if there is a variety of BL-algebras whose chains are (isomorphic to) ordinal sums of perfect MV-chains: even if the answer to this question is negative, we will present the smallest variety (whose correspondent logic is called BL$_\text{Chang}$) containing this class of BL-chains.
As we have anticipated in the abstract, there is a connection between the variety of BL$_\text{Chang}$-algebras and the one generated by Chang's MV-algebra. In fact the first-one is axiomatized (over the variety of BL-algebras) with an equation that, over MV-algebras, is equivalent to the one that axiomatize the variety generated by Chang MV-algebras: however, the two equations are \emph{not} equivalent, over BL. \paragraph*{} The paper is structured as follows: in \Cref{sec:prelim} we introduce the necessary logical and algebraic background: moreover some basic results about perfect MV-algebras and other structures will be listed. In \Cref{sec:blcha} we introduce the main theme of the article: the variety of BL$_\text{Chang}$ and the associated logic. The analysis will be done in the propositional case: completeness results, algebraic and logical properties and also some results about the variety generated by Chang's MV-algebra. We conclude with \Cref{sec:first}, where we will analyze the first-order versions of BL$_\text{Chang}$ and \L$_\text{Chang}$: for the first-one the completeness results will be much more negative. \paragraph{} To conclude, we list the main results. \begin{itemize} \item BL$_\text{Chang}$ enjoys the finite strong completeness (but not the strong one) w.r.t. $\omega\mathcal{V}$, where $\omega\mathcal{V}$ represents the ordinal sum of $\omega$ copies of the disconnected rotation of the standard cancellative hoop. \item \L$_\text{Chang}$ (the logic associated to the variety generated by Chang's MV-algebra) enjoys the finite strong completeness (but not the strong one) w.r.t. $\mathcal{V}$, $\mathcal{V}$ being the disconnected rotation of the standard cancellative hoop. \item There are two BL-chains $\mathcal{A}, \mathcal{B}$ that are strongly complete w.r.t., respectively \L$_\text{Chang}$ and BL$_\text{Chang}$. \item Every \L$_\text{Chang}$-chain that is strongly complete w.r.t. \L$_\text{Chang}$ is also stron\-gly complete w.r.t \L$_\text{Chang}\forall$. \item There is no BL$_\text{Chang}$-chain to be complete w.r.t. BL$_\text{Chang}\forall$. \end{itemize} \section{Preliminaries}\label{sec:prelim} \subsection{Basic concepts} Basic Logic BL was introduced by P. H\'{a}jek in \cite{haj}. It is based over the connectives $\{\&,\to,\bot\}$ and a denumerable set of variables $VAR$. The formulas are defined inductively, as usual (see \cite{haj} for details).
Other derived connectives are the following.
{\em negation}: $\neg \varphi \mathrel{\mathop:}= \varphi\to\bot$; {\em verum} or {\em top}: $\top \mathrel{\mathop:}=\neg\bot$; {\em meet}: $\varphi\land\psi \mathrel{\mathop:}= \varphi\&(\varphi\to\psi)$; {\em join}: $\varphi\vee\psi \mathrel{\mathop:}= ((\varphi\to\psi)\to\psi)\land((\psi\to\varphi)\to\varphi)$. \paragraph*{} \noindent BL is axiomatized as follows. \begin{align} \tag{A1}&(\varphi \rightarrow \psi)\rightarrow ((\psi\rightarrow \chi)\rightarrow(\varphi\rightarrow \chi))\\ \tag{A2}&(\varphi\&\psi)\rightarrow \varphi\\ \tag{A3}&(\varphi\&\psi)\rightarrow(\psi\&\varphi)\\ \tag{A4}&(\varphi\&(\varphi\to\psi))\to(\psi\& (\psi\to\varphi))\\ \tag{A5a}&(\varphi\rightarrow(\psi\rightarrow\chi))\rightarrow((\varphi\&\psi)\rightarrow \chi)\\ \tag{A5b}&((\varphi\&\psi)\rightarrow \chi)\rightarrow(\varphi\rightarrow(\psi\rightarrow\chi))\\ \tag{A6}&((\varphi\rightarrow\psi)\rightarrow\chi)\rightarrow(((\psi\rightarrow\varphi)\rightarrow\chi)\rightarrow\chi)\\ \tag{A7}&\bot\rightarrow\varphi \end{align} {\em Modus ponens} is the only inference rule: \begin{equation} \tag{MP}\frac{\varphi\quad\varphi\to\psi}{\psi}. \end{equation} Among the extensions of BL (logics obtained from it by adding other axioms) there is the well known \L ukasiewicz (infinitely-valued) logic \L, that is, BL plus \begin{equation} \tag*{(INV)}\neg\neg\varphi\to\varphi. \end{equation} On \L ukasiewicz logic we can also define a strong disjunction connective (in the following sections, we will introduce a strong disjunction connective, for BL, that will be proved to be equivalent to the following, over \L) \begin{equation*} \varphi\curlyvee\psi\mathrel{\mathop:}=\neg(\neg\varphi\&\neg\psi). \end{equation*} The notations $\varphi^n$ and $n\varphi$ will indicate $\underbrace{\varphi\&\dots\&\varphi}_{n\text{ times}}$ and $\underbrace{\varphi\curlyvee\dots\curlyvee\varphi}_{n\text{ times}}$.
Given an axiomatic extension L of BL, a formula $\varphi$ and a theory $T$ (a set of formulas), the notation $T\vdash_L\varphi$ indicates that there is a proof of $\varphi$ from the axioms of L and the ones of $T$. The notion of proof is defined like in classical case (see \cite{haj}). \paragraph*{} We now move to the semantics: for all the unexplained notions of universal algebra, we refer to \cite{bs, gra}. \begin{definition} A BL-algebra is an algebraic structure of the form $\mathcal{A}=\left \langle A,*,\Rightarrow,\sqcap,\sqcup,0,1\right \rangle$ such that \begin{itemize} \item $\left \langle A,\sqcap,\sqcup,0,1 \right \rangle$ is a bounded lattice, where $0$ is the bottom and $1$ the top element. \item $\left \langle A,*,1\right \rangle$ is a commutative monoid. \item $\left \langle *,\Rightarrow\right \rangle$ forms a residuated pair,\index{residuated pair} i.e. \begin{equation} \tag*{(res)}z*x\leq y\quad\text{iff}\quad z\leq x\Rightarrow y, \label{eq:res} \end{equation} it can be shown that the only operation that satisfies \ref{eq:res} is $x\Rightarrow y=\max\{z:\ z*x\leq y\}$. \item $\mathcal{A}$ satisfies the following equations \begin{align} \tag*{(pl)}&(x\Rightarrow y)\sqcup(y\Rightarrow x)=1\\ \tag*{(div)}&x\sqcap y=x*(x\Rightarrow y). \end{align} \end{itemize} Two important types of BL-algebras are the followings. \begin{itemize} \item A BL-chain is a totally ordered BL-algebra. \item A standard BL-algebra is a BL-algebra whose support is $[0,1]$. \end{itemize} \end{definition} Notation: in the following, with $\sim x$ we will indicate $x\Rightarrow 0$. \begin{definition} An MV-algebra is a BL-algebra satisfying \begin{equation} \tag*{(inv)}x=\sim\sim x. \end{equation} A well known example of MV-algebra is the standard MV-algebra $[0,1]_\text{\L}=\\\left \langle [0,1], *,\Rightarrow, \min, \max, 0, 1\right \rangle$, where $x*y=\max(0,x+y-1)$ and $x\Rightarrow y=\min(1, 1-x+y)$. \end{definition} In every MV-algebra we define the algebraic equivalent of $\curlyvee$, that is \begin{equation*} x\oplus y\mathrel{\mathop:}=\sim(\sim x * \sim y). \end{equation*} The notations (where $x$ is an element of some BL-algebra) $x^n$ and $nx$ will indicate $\underbrace{x*\dots *x}_{n\text{ times}}$ and $\underbrace{x\oplus\dots\oplus x}_{n\text{ times}}$.
Given a BL-algebra $\mathcal{A}$, the notion of $\mathcal{A}$-evaluation is defined in a truth-functional way (starting from a map $v:\, VAR\to A$, and extending it to formulas), for details see \cite{haj}.
Consider a BL-algebra $\mathcal{A}$, a theory $T$ and a formula $\varphi$. With $\mathcal{A}\models \varphi$ ($\mathcal{A}$ is a model of $\varphi$) we indicate that $v(\varphi)=1$, for every $\mathcal{A}$-evaluation $v$; $\mathcal{A}\models T$ denotes that $\mathcal{A}\models \psi$, for every $\psi\in T$. Finally, the notation $T\models_\mathcal{A}\varphi$ means that if $\mathcal{A}\models T$, then $\mathcal{A}\models \varphi$.
A BL-algebra $\mathcal{A}$ is called L-algebra, where L is an axiomatic extension of BL, whenever $\mathcal{A}$ is a model for all the axioms of L. \begin{definition} Let L be an axiomatic extension of BL and $K$ a class of L-algebras. We say that L is strongly complete (respectively: finitely strongly complete, complete) with respect to $K$ if for every set $T$ of formulas (respectively, for every finite set $T$ of formulas, for $T=\emptyset$) and for every formula $\varphi$ we have \begin{equation*} T\vdash_L\varphi\quad \text{iff}\quad T\models_K\varphi. \end{equation*} \end{definition} \subsection{Perfect MV-algebras, hoops and disconnected rotations}\label{sec:mvcha} We recall that Chang's \emph{MV}-algebra (\cite{chang}) is a BL-algebra of the form \begin{equation*} C=\left \langle \{a_n:\ n\in\mathbb{N}\}\cup\{b_n:\ n\in\mathbb{N}\}, *, \Rightarrow, \sqcap,\sqcup, b_0, a_0\right \rangle. \end{equation*} Where for each $n,m\in \mathbb{N}$, it holds that $b_n<a_m$, and, if $n<m$, then $a_m<a_n,\ b_n<b_m$; moreover $a_0=1,\ b_0=0$ (the top and the bottom element).
The operation $*$ is defined as follows, for each $n,m\in \mathbb{N}$: \begin{equation*} b_n*b_m=b_0,\ b_n*a_m=b_{\max(0,n-m)},\ a_n*a_m=a_{n+m}. \end{equation*} \begin{definition}[\cite{dnl2}] Let $\mathcal{A}$ be an MV-algebra and let $x\in\mathcal{A}$: with $ord(x)$ we mean the least (positive) natural $n$ such that $x^n=0$. If there is no such $n$, then we set $ord(x)=\infty$. \begin{itemize} \item An MV-algebra is called \emph{local}\footnote{Usually, the local MV-algebras are defined as MV-algebras having a unique (proper) maximal ideal. In \cite{dnl2}, however, it is shown that the two definitions are equivalent. We have preferred the other definition since it shows in a more transparent way that perfect MV-algebras are particular cases of local MV-algebras.} if for every element $x$ it holds that \\$ord(x)<\infty$ or $ord(\sim x)<\infty$. \item An MV-algebra is called \emph{perfect} if for every element $x$ it holds that $ord(x)<\infty$ iff $ord(\sim x)=\infty$. \end{itemize} \end{definition} An easy consequence of this definition is that every perfect MV-algebra cannot have a negation fixpoint.
With $Perfect(MV)$ and $Local(MV)$ we will indicate the class of perfect and local MV-algebras. Moreover, given a BL-algebra $\mathcal{A}$, with $\mathbf{V}(\mathcal{A})$ we will denote the variety generated by $\mathcal{A}$. \begin{theorem}[\cite{dnl2}] Every MV-chain is local. \end{theorem} Clearly there are local MV-algebras that are not perfect: $[0,1]_\text{\L}$ is an example.
Now, in \cite{dnl1} it is shown that \begin{theorem}\label{teo:perfcha} \begin{itemize} \item[] \item $\mathbf{V}(C)=\mathbf{V}(Perfect(MV))$, \item $Perfect(MV)=Local(MV)\cap\mathbf{V}(C)$. \end{itemize} \end{theorem} It follows that the class of chains in $\mathbf{V}(C)$ coincides with the one of perfect MV-chains. Moreover \begin{theorem}[\cite{dnl1}]\label{teo:chaeq} An MV-algebra is in the variety $\mathbf{V}(C)$ iff it satisfies the equation $(2x)^2=2(x^2)$. \end{theorem} As shown in \cite{bdg}, the logic correspondent to this variety is axiomatized as {\L} plus $(2\varphi)^2\leftrightarrow 2(\varphi^2)$: we will call it {\L}$_\text{Chang}$. \paragraph*{} We now recall some results about hoops \begin{definition}[\cite{f, bf}] A \emph{hoop} is a structure $\mathcal{A}=\langle A, * ,\Rightarrow ,1\rangle $ such that $\langle A, * ,1\rangle $ is a commutative monoid, and $\Rightarrow $ is a binary operation such that \[ x\Rightarrow x=1,\hspace{0.5cm}x\Rightarrow (y\Rightarrow z)=(x * y)\Rightarrow z\hspace{0.5cm}\mathrm{and}\hspace{0.5cm}x * (x\Rightarrow y)=y * (y\Rightarrow x). \] \end{definition}
In any hoop, the operation $\Rightarrow $ induces a partial order $\le $ defined by $x\le y$ iff $x\Rightarrow y=1$. Moreover, hoops are precisely the partially ordered commutative integral residuated monoids (pocrims) in which the meet operation $\sqcap$ is definable by $x\sqcap y=x * (x\Rightarrow y)$. Finally, hoops satisfy the following divisibility condition: \begin{equation} \tag*{(div)}\text{If } x \le y, \text{ then there is an element } z\text{ such that }z * y=x. \end{equation} We recall a useful result. \begin{definition} Let $\mathcal{A}$ and $\mathcal{B}$ be two algebras of the same language. Then we say that \begin{itemize} \item $\mathcal{A}$ is a partial subalgebra of $\mathcal{B}$ if $A\subseteq B$ and the operations of $\mathcal{A}$ are the ones of $\mathcal{A}$ restricted to $A$. Note that $A$ could not be closed under these operations (in this case these last ones will be undefined over some elements of $A$): in this sense $\mathcal{A}$ is a partial subalgebra. \item $\mathcal{A}$ is partially embeddable into $\mathcal{B}$ when every finite partial subalgebra of $\mathcal{A}$ is embeddable into $\mathcal{B}$. Generalizing this notion to classes of algebras, we say that a class $K$ of algebras is partially embeddable into a class $M$ if every finite partial subalgebra of a member of $K$ is embeddable into a member of $M$. \end{itemize} \end{definition} \begin{definition}\label{def:bound} A \emph{bounded} hoop is a hoop with a minimum element; conversely, an \emph{unbounded} hoop is a hoop without minimum.
Let $\mathcal{A}$ be a bounded hoop with minimum $a$: with $\mathcal{A}^+$ we mean the (partial) subalgebra of $\mathcal{A}$ defined over the universe $A^+=\{x\in A:\, x>x\Rightarrow a\}$.
A hoop is Wajsberg iff it satisfies the equation $(x\Rightarrow y)\Rightarrow y=(y\Rightarrow x)\Rightarrow x$.
A hoop is cancellative iff it satisfies the equation $x=y\Rightarrow(x*y)$. \end{definition}
\begin{proposition}[\cite{f, bf, afm}]\label{prop:canc} Every cancellative hoop is Wajsberg. Totally ordered cancellative hoops coincide with unbounded totally ordered Wajsberg hoops, whereas bounded Wajsberg hoops coincide with (the $0$-free reducts of) MV-algebras. \end{proposition} We now recall a construction introduced in \cite{jen} (and also used in \cite{eghm, neg}), called \emph{disconnected rotation}.
\begin{definition} Let $\mathcal{A}$ be a cancellative hoop. We define an algebra, $\mathcal{A}^*$, called the \emph{disconnected rotation} of $\mathcal{A}$, as follows. Let $\mathcal{A}\times\{0\}$ be a disjoint copy of A. For every $a\in A$ we write $a'$ instead of $\langle a, 0\rangle$. Consider $\left \langle A'= \{a' : a \in A\}, \leq\right \rangle$ with the inverse order and let $A^*\mathrel{\mathop:}= A\cup A'$. We extend these orderings to an order in $A^*$ by putting $a' < b$ for every $a,b\in A$. Finally, we take the following operations in $A^*$: $1\mathrel{\mathop:}= 1_\mathcal{A}$, $0\mathrel{\mathop:}= 1'$, $\sqcap_{\mathcal{A}^*}, \sqcup_{\mathcal{A}^*}$ as the meet and the join with respect to the order over $A^*$. Moreover, \begin{align*} \sim_{\mathcal{A}^*} a\mathrel{\mathop:}=&\begin{cases} a'&\text{if }a\in A\\ b &\text{if }a=b'\in A'
\end{cases}\\ a*_{\mathcal{A}^*}b\mathrel{\mathop:}=&\begin{cases} a*_\mathcal{A} b&\text{if }a, b\in A\\ \sim_{\mathcal{A}^*}(a\Rightarrow_{\mathcal{A}} \sim_{\mathcal{A}^*}b)&\text{if }a\in A, b\in A'\\ \sim_{\mathcal{A}^*}(b\Rightarrow_{\mathcal{A}} \sim_{\mathcal{A}^*}a)&\text{if }a\in A', b\in A\\ 0&\text{if }a, b\in A'
\end{cases}\\ a\Rightarrow_{\mathcal{A}^*}b\mathrel{\mathop:}=&\begin{cases} a\Rightarrow_\mathcal{A} b&\text{if }a, b\in A\\ \sim_{\mathcal{A}^*}(a *_{\mathcal{A}^*} \sim_{\mathcal{A}^*}b)&\text{if }a\in A, b\in A'\\ 1&\text{if }a\in A', b\in A\\ (\sim_{\mathcal{A}^*}b) \Rightarrow_{\mathcal{A}} (\sim_{\mathcal{A}^*}a)&\text{if }a, b\in A'.
\end{cases} \end{align*} \end{definition}
\begin{theorem}[{\cite[theorem 9]{neg}}]\label{teo:perfrot} Let $\mathcal{A}$ be an MV-algebra. The followings are equivalent: \begin{itemize}
\item A is a perfect MV-algebra.
\item A is isomorphic to the disconnected rotation of a cancellative hoop. \end{itemize} \end{theorem} To conclude the section, we present the definition of ordinal sum. \begin{definition}[\cite{am}] Let $\langle I,\le \rangle $ be a totally ordered set with minimum $0$. For all $i\in I$, let $\mathcal{A}_{i}$ be a hoop such that for $i\ne j$, $ A_{i}\cap A_{j}=\{1\}$, and assume that $\mathcal{A}_0$ is bounded. Then $\bigoplus_{i\in I}{\mathcal{A}}_{i}$ (the \emph{ordinal sum} of the family $({\mathcal{A}}_{i})_{i\in I}$) is the structure whose base set is $ \bigcup_{i\in I}A_{i}$, whose bottom is the minimum of $\mathcal{A}_0$, whose top is $1$, and whose operations are \begin{align*} x\Rightarrow y&=\begin{cases} x\Rightarrow ^{{\mathcal{A}}_{i}}y & \mathrm{if} \,\,x,y\in A_{i} \\ y & \mathrm{if}\,\, \exists i>j(x\in A_{i}\,\,\mathrm{and}\,\,y\in A_{j}) \\ 1 & \mathrm{if}\,\,\exists i<j(x\in A_{i}\setminus \{1\}\,\,\mathrm{and }\,\,y\in A_{j}) \end{cases}\\ x* y&=\begin{cases} x * ^{{\mathcal{A}}_{i}}y & \mathrm{if}\,\,x,y\in A_{i} \\ x & \mathrm{if}\,\,\exists i<j(x\in A_{i}\setminus\{1\},\,\,y\in A_{j})\\ y & \mathrm{if}\,\,\exists i<j(y\in A_{i}\setminus\{1\},\,x\in A_{j}) \end{cases} \end{align*} When defining the ordinal sum $\bigoplus_{i\in I}{\mathcal{A}}_{i}$ we will tacitly assume that whenever the condition $A_{i}\cap A_{j}=\left\{ 1\right\} $ is not satisfied for all $i,j\in I$ with $i\neq j$, we will replace the $\mathcal{A}_{i}$ by isomorphic copies satisfying such condition. Moreover if all $\mathcal{A}_i$'s are isomorphic to some $\mathcal{A}$, then we will write $I\mathcal{A}$, instead of $\bigoplus_{i \in I}\mathcal{A}_{i}$. Finally, the ordinal sum of two hoops $\mathcal{A}$ and $\mathcal{B}$ will be denoted by $\mathcal{A}\oplus\mathcal{B}$. \end{definition} Note that, since every bounded Wajsberg hoop is the $0$-free reduct of an MV-algebra, then the previous definition also works with these structures. \begin{theorem}[{\cite[theorem 3.7]{am}}]\label{teo:am} Every BL-chain is isomorphic to an ordinal sum whose first component is an MV-chain and the others are totally ordered Wajsberg hoops. \end{theorem} Note that in \cite{bus} it is presented an alternative and simpler proof of this result. \section{The variety of BL$_\text{Chang}$-algebras}\label{sec:blcha} Consider the following connective \begin{equation*} \varphi\veebar\psi\mathrel{\mathop:}=((\varphi \rightarrow (\varphi \& \psi ))\rightarrow\psi )\land ((\psi \rightarrow (\varphi \& \psi ))\rightarrow \varphi ) \end{equation*} Call $\uplus$ the algebraic operation, over a BL-algebra, corresponding to $\veebar$; we have that \begin{lemma}\label{lem:disgeq} In every MV-algebra the following equation holds \begin{equation*} x\uplus y=x\oplus y. \end{equation*} \end{lemma} \begin{proof} It is easy to check that $x\uplus y=x\oplus y$, over $[0,1]_{MV}$, for every $x,y\in [0,1]$. \end{proof} We now analyze this connective in the context of Wajsberg hoops. \begin{proposition}\label{prop:disj} Let $\mathcal{A}$ be a linearly ordered Wajsberg hoop. Then \begin{itemize} \item If $\mathcal{A}$ is unbounded (i.e. a cancellative hoop), then $x\uplus y=1$, for every $x,y\in\mathcal{A}$. \item If $\mathcal{A}$ is bounded, let $a$ be its minimum. Then, by defining $\sim x\mathrel{\mathop:}= x\Rightarrow a$ and $x\oplus y=\sim(\sim x*\sim y)$ we have that $x\oplus y=x\uplus y$, for every $x,y\in\mathcal{A}$ \end{itemize} \end{proposition} \begin{proof} An easy check. \end{proof} Now, since the variety of cancellative hoops is generated by its linearly ordered members (see \cite{eghm}), then we have that \begin{corollary}\label{cor:disjcanc} The equation $x\uplus y=1$ holds in every cancellative hoop. \end{corollary} We now characterize the behavior of $\uplus$ for the case of BL-chains. \begin{proposition}\label{prop:disg} Let $\mathcal{A}=\bigoplus_{i\in I}\mathcal{A}_i$ be a BL-chain. Then \begin{equation*} x\uplus y=\begin{cases} x\oplus y,&\text{if }x,y\in \mathcal{A}_i\text{ and }\mathcal{A}_i\text{ is bounded }\\ 1,&\text{if }x,y\in \mathcal{A}_i\text{ and }\mathcal{A}_i\text{ is unbounded }\\ \max(x,y),&\text{otherwise}. \end{cases} \end{equation*} for every $x,y\in\mathcal{A}$. \end{proposition} \begin{proof} If $x,y$ belong to the same component of $\mathcal{A}$, then the result follows from \Cref{lem:disgeq} and \Cref{prop:disj}. For the case in which $x$ and $y$ belong to different components of $\mathcal{A}$, this is a direct computation. \end{proof} \begin{remark} From the previous proposition we can argue that $\uplus$ is a good approximation, for BL, of what that $\oplus$ represents for MV-algebras. Note that a similar operation was introduced in \cite{abm}: the main difference with respect to $\uplus$ is that, when $x$ and $y$ belong to different components of a BL-chain, then the operation introduced in \cite{abm} holds $1$. \end{remark} In the following, for every element $x$ of a BL-algebra, with the notation $\overline{n}x$ we will denote $\underbrace{x\uplus\dots\uplus x}_{n\ times}$; analogously $\overline{n}\varphi$ means $\underbrace{\varphi\veebar\dots\veebar\varphi}_{n\ times}$. \begin{definition} We define BL$_\text{Chang}$ as the axiomatic extension of BL, obtained by adding \begin{equation} \tag{cha}(\overline{2}\varphi)^2\leftrightarrow \overline{2}(\varphi^2). \end{equation} That is, writing it in extended form \begin{equation*} (\varphi^2\to(\varphi^2\&\varphi^2)\to\varphi^2)\leftrightarrow((\varphi\to\varphi^2)\to\varphi)^2. \end{equation*} \end{definition} Clearly the variety corresponding to BL$_\text{Chang}$ is given by the class of BL-algebras satisfying the equation $(\overline{2}x)^2=\overline{2}(x^2)$.
Moreover, \begin{definition}\label{def:pseudoperf} We will call pseudo-perfect Wajsberg hoops those Wajsberg hoops satisfying the equation $(\overline{2}x)^2=\overline{2}(x^2)$. \end{definition} \begin{remark}\label{rem:2} Thanks to \Cref{lem:disgeq} we have that \begin{equation*} \vdash_\text{\L}((\overline{2}\varphi)^2\leftrightarrow \overline{2}(\varphi^2))\leftrightarrow ((2\varphi)^2\leftrightarrow 2(\varphi^2)), \end{equation*} that is, if we add $(\overline{2}\varphi)^2\leftrightarrow \overline{2}(\varphi^2)$ or $(2\varphi)^2\leftrightarrow 2(\varphi^2)$ to {\L}, then we obtain the same logic \L$_\text{Chang}$.
These formulas, however are not equivalent over BL: see \Cref{rem:p0} for details. \end{remark} \begin{theorem}\label{teo:chainpswh} Every totally ordered pseudo-perfect Wajsberg hoop is a totally ordered cancellative hoop or (the $0$-free reduct of) a perfect MV-chain.
More in general, the variety of pseudo-perfect Wajsberg hoops coincides with the class of the $0$-free subreducts of members of $\mathbf{V}(C)$. \end{theorem} \begin{proof} In \cite{eghm} it is shown that the variety of Wajsberg hoops coincides with the class of the $0$-free subreducts of MV-algebras. The results easily follow from this fact and from \Cref{prop:canc}, \Cref{teo:chaeq} and \Cref{def:pseudoperf}. \end{proof} As a consequence, we have \begin{theorem}\label{teo:hoopincl} Let $\mathbb{WH}, \mathbb{CH}, ps\mathbb{WH}$ be, respectively, the varieties of Wajsberg hoops, cancellative hoops, pseudo-perfect Wajsberg hoops. Then we have that \begin{equation*} \mathbb{CH}\subset ps\mathbb{WH} \subset \mathbb{WH}. \end{equation*} \end{theorem} \begin{proof} An easy consequence of \Cref{teo:chainpswh}.
The first inclusion follows from the fact that $ps\mathbb{WH}$ contains all the totally ordered cancellative hoops and hence the variety generated by them. For the second inclusion note that, for example, the $0$-free reduct of $[0,1]_\text{\L}$ belongs to $\mathbb{WH}\setminus ps\mathbb{WH}$. \end{proof} We now describe the structure of BL$_\text{Chang}$-chains, with an analogous of the \Cref{teo:am} for BL-chains. \begin{theorem}\label{teo:chainstruct} Every BL$_\text{Chang}$-chain is isomorphic to an ordinal sum whose first component is a perfect MV-chain and the others are totally ordered pseudo-perfect Wajsberg hoops.
It follows that every ordinal sum of perfect MV-chains is a BL$_\text{Chang}$-chain. \end{theorem} \begin{proof} Thanks to \Cref{teo:perfcha,teo:chaeq}, \Cref{rem:2} and \Cref{def:pseudoperf}, we have that every MV-chain (Wajsberg hoop) satisfying the equation $(\overline{2}x)^2=\overline{2}(x^2)$ is perfect (pseudo-perfect): using these facts and \Cref{prop:disg} we have that a BL-chain satisfies the equation $(\overline{2}x)^2=\overline{2}(x^2)$ iff it holds true in all the components of its ordinal sum. From these facts and \Cref{teo:am} we get the result. \end{proof} As a consequence, we obtain the following corollaries. \begin{corollary}\label{cor:blchacont} The variety of BL$_\text{Chang}$-algebras contains the ones of \\product-algebras and G\"{o}del-algebras: however it does not contains the variety of MV-algebras. \end{corollary} \begin{proof} From the previous theorem it is easy to see that the variety of BL$_\text{Chang}$-algebras contains $[0,1]_\Pi$ and $[0,1]_G$, but not $[0,1]_\text{\L}$. \end{proof} \begin{corollary} Every finite BL$_\text{Chang}$-chain is an ordinal sum of a finite number of copies of the two elements boolean algebra. Hence the class of finite BL$_\text{Chang}$-chains coincides with the one of finite G\"{o}del chains. \end{corollary} For this reason it is immediate to see that the finite model property does not hold for BL$_\text{Chang}$.
We conclude with the following remark. \begin{remark}\label{rem:p0} \begin{itemize} \item One can ask if it is possible to axiomatize the class BL$_\text{perf}$ of BL-algebras, whose chains are the BL-algebras that are ordinal sum of perfect MV-chains: the answer, however, is negative. In fact, the class of bounded Wajsberg hoops does not form a variety: for example, it is easy to check that for every bounded pseudo-perfect Wajsberg hoop $\mathcal{A}$, its subalgebra $\mathcal{A}^+$ (see \Cref{def:bound} ) forms a cancellative hoop. Hence BL$_\text{perf}$ cannot be a variety.
However, as we will see in \Cref{sec:propcompl}, the variety of BL$_\text{Chang}$-algebras is the ``best approximation'' of BL$_\text{perf}$, in the sense that it is the smallest variety to contain BL$_\text{perf}$. \item In \cite{dnl4} (see also \cite{ct}) it is studied the variety, called $P_0$, generated by all the perfect BL-algebras (a BL-algebra $\mathcal{A}$ is perfect if, by calling $MV(\mathcal{A})$ the biggest subalgebra of $\mathcal{A}$ to be an MV-algebra, then $MV(\mathcal{A})$ is a perfect MV-algebra). $P_0$ is axiomatized with the equation
\begin{equation}
\tag*{($p_0$)}\sim((\sim(x^2))^2)=(\sim ((\sim x)^2))^2.\label{eq:p0}
\end{equation}
One can ask which is the relation between $P_0$ and the variety of BL$_\text{Chang}$-algebras. The answer is that the variety of BL$_\text{Chang}$-algebras is strictly contained in $P_0$. In fact, an easy check shows that a BL-chain is perfect if and only if the first component of its ordinal sum is a perfect MV-chain. Hence we have:
\begin{itemize}
\item Every BL$_\text{Chang}$-chain is a perfect BL-chain.
\item There are perfect BL-chains that are not BL$_\text{Chang}$-chains: an example is given by $C\oplus [0,1]_\text{\L}$.
\end{itemize}
Now, since the variety of BL$_\text{Chang}$-algebras is generated by its chains (like any variety of BL-algebras, see \cite{haj}), then we get the result.
Finally note that \ref{eq:p0} is equivalent to $2(x^2)=(2x)^2$: hence, differently to what happens over {\L} (see \Cref{rem:2} ), the equations $2(x^2)=(2x)^2$ and $\overline{2}(x^2)=(\overline{2}x)^2$ are not equivalent, over BL. \end{itemize} \end{remark} \subsection{Subdirectly irreducible and simple algebras} We begin with a general result about Wajsberg hoops. \begin{theorem}[{\cite[Corollary 3.11]{f}}] Every subdirectly irreducible Wajsberg hoop is totally ordered. \end{theorem} As a consequence, we have: \begin{corollary}\label{cor:pssubir} Every subdirectly irreducible pseudo-perfect Wajsberg hoop is totally ordered. \end{corollary} We now move to simple algebras.
It is shown in \cite[Theorem 1]{tur} that the simple BL-algebras coincide with the simple MV-algebras, that is, with the subalgebras of $[0,1]_\text{\L}$ (see \cite[Theorem 3.5.1]{mun}). Therefore we have: \begin{theorem} The only simple BL$_\text{Chang}$-algebra is the two elements boolean algebra $\mathbf{2}$. \end{theorem} An easy consequence of this fact is that the only simple {\L}$_\text{Chang}$-algebra is $\mathbf{2}$. \subsection{Completeness}\label{sec:propcompl} We begin with a result about pseudo-perfect Wajsberg hoops. \begin{theorem} The class $pMV$ of $0$-free reducts of perfect MV-chains generates $ps\mathbb{WH}$. \end{theorem} \begin{proof} From \Cref{teo:perfrot,teo:chainpswh} it is easy to check that the variety generated by $pMV$ contains all the totally ordered pseudo-perfect Wajsberg hoops.
From these facts and \Cref{cor:pssubir}, we have that $pMV$ must be generic for $ps\mathbb{WH}$. \end{proof} \begin{theorem}[\cite{dist}]\label{teo:dist} Let L be an axiomatic extension of BL, then L enjoys the finite strong completeness w.r.t a class $K$ of L-algebras iff every countable L-chain is partially embeddable into $K$. \end{theorem} As shown in \cite{haj} product logic enjoys the finite strong completeness w.r.t $[0,1]_\Pi$ and hence every countable product chain is partially embeddable into $[0,1]_\Pi\simeq\mathbf{2}\oplus (0,1]_C$, with $(0,1]_C$ being the standard cancellative hoop (i.e. the $0$-free reduct of $[0,1]_\Pi\setminus\{0\}$). Since every totally ordered product chain is of the form $\mathbf{2}\oplus\mathcal{A}$, where $\mathcal{A}$ is a cancellative hoop (see \cite{eghm}), it follows that: \begin{proposition}\label{prop:cancemb} Every countable totally ordered cancellative hoop partially embeds into $(0,1]_C$. \end{proposition} \begin{theorem}\label{teo:perfho} Every countable perfect MV-chain partially embeds into $\mathcal{V}=(0,1]^*_C$ (i.e. the disconnected rotation of $(0,1]_C$). \end{theorem} \begin{proof} Immediate from \Cref{prop:cancemb} and \Cref{teo:perfrot}. \end{proof} \begin{corollary}\label{cor:comp} The logic {\L}$_\text{Chang}$ is finitely strongly complete w.r.t. $\mathcal{V}$. \end{corollary} \begin{theorem}\label{teo:blccomp} BL$_\text{Chang}$ enjoys the finite strong completeness w.r.t. $\omega\mathcal{V}$. As a consequence, the variety of BL$_\text{Chang}$-algebras is generated by the class of all ordinal sums of perfect MV-chains and hence is the smallest variety to contain this class of algebras. \end{theorem} \begin{proof} Thanks to \Cref{teo:dist} it is enough to show that every countable BL$_\text{Chang}$-chain partially embeds into $\omega\mathcal{V}$ (i.e. the ordinal sum of ``$\omega$ copies'' of $\mathcal{V}$). This fact, however, follows immediately from \Cref{prop:cancemb} and \Cref{teo:chainstruct,teo:perfho}. \end{proof} But we cannot obtain a stronger result: in fact \begin{theorem} BL$_\text{Chang}$ is not strongly complete w.r.t. $\omega\mathcal{V}$. \end{theorem} \begin{proof} Suppose not: from the results of \cite[Theorem 3.5]{dist} this is equivalent to claim that every countable BL$_\text{Chang}$-chain embeds into $\omega\mathcal{V}$. But, this would imply that every countable totally ordered cancellative hoop embeds into $(0,1]_C$: this means that every countable product-chain embeds into $[0,1]_\Pi$, that is product logic is strongly complete w.r.t $[0,1]_\Pi$. As it is well known (see \cite[Corollary 4.1.18]{haj}), this is false. \end{proof} With an analogous proof we obtain \begin{theorem} {\L}$_\text{Chang}$ is not strongly complete w.r.t. $\mathcal{V}$ \end{theorem} However, thanks to \cite[Theorem 3]{monchain} we can claim \begin{theorem}\label{teo:strcompl} There exist a {\L}$_\text{Chang}$-chain $\mathcal{A}$ and a BL$_\text{Chang}$-chain $\mathcal{B}$ such that {\L}$_\text{Chang}$ is strongly complete w.r.t. $\mathcal{A}$ and BL$_\text{Chang}$ is strongly complete w.r.t. $\mathcal{B}$. \end{theorem} \begin{problem} Which can be some concrete examples of such $\mathcal{A}$ and $\mathcal{B}$ ? \end{problem}
\section{First-order logics}\label{sec:first} We assume that the reader is acquainted with the formalization of first-order logics, as developed in \cite{haj, ch}.
Briefly, we work with (first-order) languages without equality, containing only predicate and constant symbols: as quantifiers we have $\forall$ and $\exists$. The notions of terms and formulas are defined inductively like in classical case.
As regards to semantics, given an axiomatic extension L of BL we restrict to L-chains: the first-order version of L is called L$\forall$ (see \cite{haj, ch} for an axiomatization). A first-order $\mathcal{A}$-interpretation ($\mathcal{A}$ being an L-chain) is a structure $\mathbf{M}=\left \langle M, \{r_P\}_{p\in \mathbf{P}}, \{m_c\}_{c\in \mathbf{C}}\right \rangle$, where $M$ is a non-empty set, every $r_P$ is a fuzzy $ariety(P)$-ary relation, over $M$, in which we interpretate the predicate $P$, and every $m_c$ is an element of $M$, in which we map the constant $c$.
Given a map $v:\, VAR\to M$, the interpretation of $\lVert\varphi\rVert_{\mathbf{M}, v}^\mathcal{A}$ in this semantics is defined in a Tarskian way: in particular the universally quantified formulas are defined as the infimum (over $\mathcal{A}$) of truth values, whereas those existentially quantified are evaluated as the supremum. Note that these $\inf$ and $\sup$ could not exist in $\mathcal{A}$: an $\mathcal{A}$-model $\mathbf{M}$ is called \emph{safe} if $\lVert\varphi\rVert^\mathcal{A}_{\mathbf{M}, v}$ is defined for every $\varphi$ and $v$.
A model is called \emph{witnessed} if the universally (existentially) quantified formulas are evaluated by taking the minimum (maximum) of truth values in place of the infimum (supremum): see \cite{witn, ch1, ch} for details.
The notions of soundness and completeness are defined by restricting to safe models (even if in some cases it is possible to enlarge the class of models: see \cite{bm}): see \cite{haj, ch, ch1} for details. \paragraph*{} We begin with a positive result about \L$_\text{Chang}\forall$. \begin{definition} Let L be an axiomatic extension of BL. With L$\forall^w$ we define the extension of L$\forall$ with the following axioms \begin{align} \tag*{(C$\forall$)}&(\exists y)(\varphi(y)\to (\forall x)\varphi(x))\label{cforall}\\ \tag*{(C$\exists$)}&(\exists y)((\exists x)\varphi(x)\to\varphi(y))\label{cexist}. \end{align} \end{definition}
\begin{theorem}[{\cite[Proposition 6]{ch1}}] \L$\forall$ coincides with \L$\forall^w$, that is \\\L$\forall\vdash$\emph{\ref{cforall},\ref{cexist}}. \end{theorem} An immediate consequence is: \begin{corollary}\label{cor:witnluk} Let L be an axiomatic extension of \L. Then L$\forall$ coincides with L$\forall^w$. \end{corollary} \begin{theorem}[{\cite[Theorem 8]{ch1}}]\label{teo:witncompl} Let L be an axiomatic extension of BL. Then L$\forall^w$ enjoys the strong witnessed completeness with respect to the class $K$ of L-chains, i.e. \begin{equation*} T\vdash_{L\forall^w}\varphi\quad\text{iff}\quad \lVert\varphi\rVert_\mathbf{M}^\mathcal{A}=1, \end{equation*} for every theory $T$, formula $\varphi$, algebra $\mathcal{A}\in K$ and witnessed $\mathcal{A}$-model $\mathbf{M}$ such that $\lVert\psi\rVert_\mathbf{M}^\mathcal{A}=1$ for every $\psi\in T$. \end{theorem} \begin{lemma}[{\cite[Lemma 1]{monchain}}]\label{lem:witn} Let L be an axiomatic extension of BL, let $\mathcal{A}$ be an L-chain, let $\mathcal{B}$ be an L-chain such that $A\subseteq B$ and let $\mathbf{M}$ be a witnessed $\mathcal{A}$-structure. Then for every formula $\varphi$ and evaluation $v$, we have $\lVert\varphi\rVert^\mathcal{A}_{\mathbf{M},v}=\lVert\varphi\rVert^\mathcal{B}_{\mathbf{M},v}$. \end{lemma} \begin{theorem} There is a \L$_\text{Chang}$-chain such that \L$_\text{Chang}\forall$ is strongly complete w.r.t. it. More in general, every \L$_\text{Chang}$-chain that is strongly complete w.r.t \L$_\text{Chang}$ is also strongly complete w.r.t. \L$_\text{Chang}\forall$. \end{theorem} \begin{proof} An adaptation of the proof for the analogous result, given in \cite[Theorem 16]{monchain}, for \L$\forall$.
From \Cref{teo:strcompl} we know that there is a \L$_\text{Chang}$-chain $\mathcal{A}$ strongly complete w.r.t. \L$_\text{Chang}$: from \cite[Theorem 3.5]{dist} this is equivalent to claim that every countable \L$_\text{Chang}$-chain embeds into $\mathcal{A}$. We show that $\mathcal{A}$ is also strongly complete w.r.t. \L$_\text{Chang}\forall$.
Suppose that $T\not\vdash_{\text{\L}_\text{Chang}\forall}\varphi$. Thanks to \Cref{cor:witnluk} and \Cref{teo:witncompl} there is a countable \L$_\text{Chang}$-chain $\mathcal{C}$ and a witnessed $\mathcal{C}$-model $\mathbf{M}$ such that $\lVert\psi\rVert_\mathbf{M}^\mathcal{C}=1$, for every $\psi\in T$, but $\lVert\varphi\rVert_\mathbf{M}^\mathcal{C}<1$. Finally, from \Cref{lem:witn} we have that $\lVert\psi\rVert_\mathbf{M}^\mathcal{A}=1$, for every $\psi\in T$ and $\lVert\varphi\rVert_\mathbf{M}^\mathcal{A}=\lVert\varphi\rVert_\mathbf{M}^\mathcal{C}<1$: this completes the proof. \end{proof} For BL$_\text{Chang}\forall$, however, the situation is not so good. \begin{theorem} BL$_\text{Chang}\forall$ cannot enjoy the completeness w.r.t. a single BL$_\text{Chang}$-chain. \end{theorem} \begin{proof} The proof is an adaptation of the analogous result given in \cite[Theorem 17]{monchain} for BL$\forall$.
Let $\mathcal{A}$ be a BL$_\text{Chang}$-chain: call $\mathcal{A}_0$ its first component. We have three cases \begin{itemize} \item $\mathcal{A}_0$ is finite: from \Cref{teo:chainstruct} we have that $\mathcal{A}_0=\mathbf{2}$ and hence $\mathcal{A}\models (\neg\neg x)\rightarrow (\neg\neg x)^2$. However $\mathcal{V}\not\models (\neg\neg x)\rightarrow (\neg\neg x)^2$, where $\mathcal{V}$ is the chain introduced in \Cref{sec:propcompl}, and hence $\mathcal{A}$ cannot be complete w.r.t. BL$_\text{Chang}\forall$. \item $\mathcal{A}_0$ is infinite and dense. As shown in \cite[Theorem 17]{monchain} the formula \\$(\forall x)\neg\neg P(x)\to \neg\neg (\forall x) P(x)$ is a tautology in every BL-chain whose first component is infinite and densely ordered: hence we have that $\mathcal{A}\models (\forall x)\neg\neg P(x)\to \neg\neg (\forall x) P(x)$. However it is easy to check that this formula fails in $[0,1]_G$: take a $[0,1]_G$-model $\mathbf{M}$ with $M=(0,1]$ and such that $r_P(m)=m$. Hence, from \Cref{cor:blchacont}, it follows that BL$_\text{Chang}\forall\not\vdash (\forall x)\neg\neg P(x)\to \neg\neg (\forall x) P(x)$. \item $\mathcal{A}_0$ is infinite and not dense. As shown in \cite[Theorem 17]{monchain} the formula $(\forall x)\neg\neg P(x)\to \neg\neg (\forall x) P(x)\vee \neg (\forall x) P(x)\to ((\forall x) P(x))^2$ is a tautology in every BL-chain whose first component is infinite and not densely ordered: hence we have that $\mathcal{A}\models (\forall x)\neg\neg P(x)\to \neg\neg (\forall x) P(x)\vee \neg (\forall x) P(x)\to ((\forall x) P(x))^2$. Also in this case, however, this formula fails in $[0,1]_G$, using the same model $\mathbf{M}$ of the previous case. \end{itemize} \end{proof}
\end{document}
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arXiv
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\begin{document}
\preprint{APS/123-QED}
\title{Fault-tolerant quantum speedup from constant depth quantum circuits}
\author{Rawad Mezher $^1$$^{,2,3}$}
\email{[email protected]}
\author{Joe Ghalbouni $^2$} \author{Joseph Dgheim $^2$} \author{Damian Markham $^{1,4}$} \email{[email protected]}
\affiliation{(1) Laboratoire d'Informatique de Paris 6, CNRS, Sorbonne Universit\'e, 4 place Jussieu, 75252 Paris Cedex 05, France}
\affiliation{(2) Laboratoire de Physique Appliquée, Faculty of Sciences 2, Lebanese University, 90656 Fanar, Lebanon}
\affiliation{(3) School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh, EH8 9AB}
\affiliation {(4) JFLI, National Institute for Informatics, and the University of Tokyo,
Tokyo, Japan.}
\date{\today}
\begin{abstract}
A defining feature in the field of quantum computing is the potential of a quantum device to outperform its classical counterpart for a specific computational task. By now, several proposals exist showing that certain sampling problems can be done efficiently quantumly, but are not possible efficiently classically, assuming strongly held conjectures in complexity theory. A feature dubbed \emph{quantum speedup}. However, the effect of noise on these proposals is not well understood in general, and in certain cases it is known that simple noise can destroy the quantum speedup.
Here we develop a fault-tolerant version of one family of these sampling problems, which we show can be implemented using quantum circuits of constant depth. We present two constructions, each taking $poly(n)$ physical qubits, some of which are prepared in noisy magic states. The first of our constructions is a constant depth quantum circuit composed of single and two-qubit nearest neighbour Clifford gates in four dimensions. This circuit has one layer of interaction with a classical computer before final measurements. Our second construction is a constant depth quantum circuit with single and two-qubit nearest neighbour Clifford gates in three dimensions, but with two layers of interaction with a classical computer before the final measurements.
For each of these constructions, we show that there is no classical algorithm which can sample according to its output distribution in $poly(n)$ time, assuming two standard complexity theoretic conjectures hold. The noise model we assume is the so-called \emph{local stochastic quantum noise}.
Along the way, we introduce various new concepts such as constant depth magic state distillation (MSD), and constant depth output routing, which arise naturally in measurement based quantum computation (MBQC), but have no constant-depth analogue in the circuit model.
\end{abstract} \maketitle
\textbf{\emph{Introduction }}- Quantum computers promise incredible benefits over their classical counterparts in various areas, from breaking RSA encryption \cite{schor94}, to machine learning \cite{biamonte2017quantum}, and improvements to generic search \cite{grover96}, among others \cite{montanaro2016quantum,olson2017quantum}.
Although these and other examples of quantum algorithms do outperform classical ones, on the practical level, they in general require quantum computers with a high level of fault-tolerance and scalability, the likes of which appear to be out of the reach of current technological developments \cite{Preskill18}.
An interesting question is thus, what can be done with so-called \emph{sub-universal} quantum devices which are not universal, in the sense that they cannot perform any quantum computation, but are realizable in principle by our current technologies.
Several examples of such practically motivated sub-universal models which nevertheless capture a sense of quantum advantage have been discovered in recent years \cite{BJS10,AA11,BMS16PRL,HM18,gao17,BHS+17,HBS+17,MB17,MGDM19,BIS+18,NRK+18,AAB+19,NBG+19,HHB+19,BGK+19,bravyi2018quantum}.
In most of these works, sampling from the output probability distribution of these sub-universal devices has been shown to be classically impossible to do efficiently, provided widely believed complexity theoretic conjectures hold \cite{BJS10,AA11}.
Thus, these devices demonstrate what is known as an exponential $quantum$ $speedup$.
The first experimental demonstration of quantum speedup is a major milestone in quantum information. Recent audacious experimental efforts \cite{AAB+19} and subsequent proposals of their classical simulation \cite{IBM} bring to light the challenges and subtleties of achieving this goal. Statements of quantum speedup are complexity theoretic in nature, making it difficult to pin down when a problem can in practice be simulated or not classically, even if we know in the limit of `infinite size' experiments that efficient classical simulation is impossible. At the same time, the role of noise in simplifying the simulation is ever more important, as systems grow, noise becomes more difficult to control, and it is a subtle question as to when it dominates; and even simple noise can very easily lead to breakdown of quantum speedup. Indeed, in \cite{BMS16,OB18,shchesnovich2019noise,TTT+20,KLF20,yung2017can,gao2018efficient} it was shown that noise generally renders the output probabilities of these devices (which in the noiseless case demonstrate quantum speedup) classically simulable efficiently. There is clearly a great need to understand better the effect of noise, and develop methods of mitigation.
Applying the standard approach to deal with noise in computation, fault-tolerance, is non-trivial in this setting for at least two reasons \cite{fowler2012proof,NC2000,LAR+11,DKL+02}. Firstly, the resources it consumes can be huge.
Secondly, it typically involves operations that step outside of the simplified computational model that makes it attractive in the first place. For example, in \cite{BJS10} the sub-universal model IQP was defined, as essentially the family of circuits where all gates are diagonal in the $X$-basis, and shown to provide sampling problems demonstrating quantum speedup in the noiseless case. However in \cite{BMS16} it was shown that a simple noise model - each output bit undergoes a bit flip with probability $\varepsilon$ - renders the output probabilities of sufficiently anti-concentrated IQP circuits efficiently simulable classically. Interestingly, for this special type of noise, they also show that quanutm speedup can be recovered using classical fault-tolerance and larger encodings of the problem quantumly, still within the IQP framework \cite{BMS16}. However, for more general noise (for example Pauli noise in all the Pauli bases), this does not appear to work, and it is not obvious if it is possible to do so within the constrained computational mode. In this case that would mean maintaining all gates be diagonal in $X$, which is not obvious as typical encoding and syndrome measurements involve more diverse gates.
In this work, we study how quantum speedup can be demonstrated in the presence of noise for a family of sampling problems. We take the \emph{local stochastic quantum noise} (we will also refer to this noise as local stochastic noise) model, commonly studied in the quantum error correction and fault-tolerance literature \cite{BGK+19,FGL18,gottesman2013fault,aliferis2005quantum,aliferis2007accuracy}. Our sampling problems are built on a family of schemes essentially based on local measurements on regular graph states, which correspond to constant depth 2D nearest neighbor quantum circuits showing quantum speedup \cite{gao17,BHS+17,HBS+17,MGDM18,MGDM19,HHB+19}. We show that these can be made fault-tolerant in a way which maintains constant depth of the quantum circuits, albeit with large (but polynomial) overhead in the number of ancilla systems used, and at most two rounds of (efficient) classical computation during the running of the circuit.
We present two different constructions based on two different techniques of fault-tolerance, the first of which involves the use of transversal gates and topological codes each encoding a single logical qubit \cite{BGK+19,fowler2012proof,wang2011surface}. This construction results in a constant depth quantum circuit demonstrating a quantum speedup, but, because of the need for long range transversal gates, can only be viewed as a quantum circuit with single qubit Clifford gates and nearest neighbor two-qubit Clifford gates in 4D (we will henceforth refer to this as our 4D nearest neighbor (NN) architecture). Our second construction avoids using transversal gates by exploiting topological defect-based quantum computation \cite{RHG07}, thereby resulting in a constant depth quantum circuit which is a 3D NN architecture. The tradeoff, unfortunately, is that our 3D NN architecture requires polynomially more ancillas than our 4D NN architecture, and has two layers of interaction with a classical computer, as compared to one such layer in our 4D NN architecture.
Our first construction in 4D uses several techniques from \cite{BGK+19}, in particular regarding the propagation of noise through Clifford circuits. For the second construction, we also develop techniques from \cite{KT19}. In \cite{KT19}, a construction for fault-tolerant quantum speedup was presented which consisted of a constant depth quantum circuit obtained by using defect-based topological quantum computing \cite{RHG07}. This construction is non-adaptive (no interaction with classical computer during running of circuit), and can be viewed as a 3D NN architecture. The main disadvantage of the construction in \cite{KT19} was the magic state distillation (MSD) procedure employed, which makes the scheme impractical in the sense that one should repeat the experiment an exponential number of times in order to observe an instance which is hard for the classical computer to simulate. In both our 3D and 4D NN constructions, we overcome this problem by optimizing our MSD procedure, thereby making the appearance of a hard instance very likely in only a few repetitions of the experiment, a feature called single-instance hardness \cite{gao17}. This, however, comes at the cost of adding adaptive interactions with the classical computer while running the quantum circuit.
This paper is organised as follows. First, we introduce the family of sampling problems using graph states, on which our constructions are based. After briefly defining the noise model, we describe in detail the encoding procedure for our 4D NN architecture. We then describe the effects of noise on our construction, step by step, starting from the Clifford part of the circuit and ending with the MSD, while introducing our optimized MSD techniques based on MBQC, namely constant depth non-adaptive MSD, and MBQC routing. Finally, we explain how to modify, using our optimized MSD techniques, the 3D NN architecture in \cite{KT19} in order to give rise to the single-instance hardness feature \cite{gao17}. Note that in our 3D NN architecture, we use different (fixed) measurement angles to those in \cite{KT19} to construct a different sampling problem having an anti-concentration property \cite{MGDM18,MB17,HBS+17}.
\textbf{\emph{Graph state sampling}} - Our approach is to construct a fault-tolerant version of the architectures based on measurement based quantum computation (MBQC) \cite{RB01}, which have recently been shown to demonstrate a quantum speedup \cite{gao17,BHS+17,HBS+17,MGDM18,MGDM19,HHB+19}.
In these constructions, the sampling is generated by performing local measurements on a large entangled state, known as a graph state. Given a graph $G$, with vertices $V$ and edges $E$, the associated graph state $|G\rangle$, of $|V|$ qubits is defined as
\begin{equation}
\label{eq1PRL}
|G\rangle:=\prod_{\{i,j\} \in E}CZ_{ij}\bigotimes_ {a \in V}|+\rangle_a, \end{equation}
where $|+\rangle:=\dfrac{|0\rangle+|1\rangle}{\sqrt{2}}$ and $CZ_{ij}$ is the controlled-Z gate ($CZ$) acting on qubits $i$ and $j$ connected by an edge. For certain graphs of regular structure, such as the cluster \cite{RB01} or brickwork \cite{BFK+09} states, applying single qubit measurements, of particular choices of angles on the $XY$-plane, effectively samples distributions, in a way that is impossible to do efficiently classically, up to the standard assumptions \cite{BHS+17,HBS+17,gao17,MGDM18,KT19}.
Although our techniques can be applied to $any$ such architecture where the measurement angles in the $XY$-plane of the Bloch sphere are chosen from the set $\left\{0,\dfrac{\pi}{2},\dfrac{\pi}{4}\right\}$ \cite{gao17,BHS+17,HBS+17,MGDM18}; for concreteness we will focus on the architecture of \cite{MGDM18}.
\begin{figure*}
\caption{ Graph state $|G\rangle$ of \cite{MGDM18} together with the pre-specified measurements in the $XY$ plane. This graph state is composed of $n$ rows and $k$ columns as seen in the main text (lower part of figure), and made up of two-qubit gadgets $G_B$ (green rectangles) zoomed in at the upper part of the figure (orange circle and arrow). Blue circles are qubits, blue vertical and horizontal lines are $CZ$ gates, the symbols inside each circle correspond to the angle in the $XY$ plane at which this qubit is measured. The $\pi/4$ symbol is a measurement at an angle $\pi/4$ in the $XY$ plane, similarly for $\pi/2$ and $0$.
In the original construction of \cite{MGDM18}, the red horizontal line is a long range $CZ$, these are used periodically in $|G\rangle$ to connect two consecutive $G_B$ gadgets acting on qubits of either the first row or the last row of $|G\rangle.$ Here, this red horizontal line is a linear cluster of twelve qubits measured at an $XY$ angle of 0, this is in order to make the construction nearest neighbor. Note that this only adds single qubit random Pauli gates to the random gates of \cite{MGDM18}, and therefore does not affect their universality capacity in implementing a $t-$design.}
\label{FIG G original}
\end{figure*}
Following \cite{MGDM18} we start with a regular graph state, closely related to the brickwork state \cite{BFK+09}, composed of $n$ rows and $k$ columns.
Then we (non-adaptively) measure qubits of all but the last column at pre-specified fixed $XY$-angles from the set $\left\{0,\dfrac{\pi}{2},\dfrac{\pi}{4}\right\}$ effectively applying a unitary, on the $n$ unmeasured qubits. This is illustrated in Figure \ref{FIG G original}.
Let $V_1 \subset V$ be the set of qubits which are measured at angle $\dfrac{\pi}{4}$ and $V_2 \subset V$ is the set of qubits which are measured at an $XY$ angle $\dfrac{\pi}{2}$. One can equivalently perform local rotations to the graph state and measure all systems in the $Z$ basis. In this way, if we define \begin{eqnarray}
|G'\rangle:= \left(\bigotimes_{a \in V}H_a \bigotimes_{b\in V_1} Z_b\left(\pi/4\right)\bigotimes_{c\in V_2}Z_c\left(\pi/2\right) \right)|G\rangle, \nonumber \end{eqnarray} where $H$ is the Hadamard unitary and $Z(\theta):=e^{-i{\dfrac{\theta}{2} Z}}$ is a rotation by $\theta$ around Pauli Z, then one can represent the outcome by a measurement result bit string $s\in \{0,1\}^{n.(k-1)}$, with associated resultant state \begin{eqnarray}
\label{eq2PRL}
\langle s|G'\rangle = \dfrac{1}{\sqrt{2^{n.(k-1)}}} U_s|0\rangle^{\otimes n}. \end{eqnarray}
This procedure effectively samples from the ensemble of unitaries $\left\{\dfrac{1}{2^{n.(k-1)}},U_s\right\}.$ It was shown in \cite{MGDM18} that setting $k=O(t^{9}(nt+log(\dfrac{1}{\varepsilon})))$, this ensemble has the property of being an $\varepsilon$-approximate unitary $t$-design \cite{DCE+09} - that is, it approximates sampling on the Haar measure up to the $t$-th moments. This property allows us to reduce the requirements for the proof of quantum speedup since it implies anti-concentration for $t=2$ from \cite{HBS+17}.
Measuring qubits of the last column in the computational ($Z$) basis and denoting the outcome by a bit string $x \in \{0,1\}^n$, our construction samples the bit strings $s,x$ with probability given by \begin{equation}
\label{eq3PRL}
D(s,x)= \dfrac{1}{2^{n.(k-1)}}|\langle x|U_s|0\rangle^{\otimes n}|^2. \end{equation} Fixing $t=2$ and $\varepsilon$ to an appropriate value, in this case the value of $k$ becomes $k=O(n)$, we will use this value of $k$ throughout this work. The results of \cite{HBS+17,MB17} directly imply (see also \cite{MGDM19}), that the distribution \begin{equation} \label{eq4PRL}
D:=\{D(s,x)\} \end{equation} satisfies the following anti-concentration property \cite{MB17,HBS+17} \begin{equation} \label{eq5PRL}
Pr_{s,x}\left(D(s,x) \geq \dfrac{\alpha}{2^{k.n}}\right) \geq \beta, \end{equation} where $\alpha$ is a positive constant, $0<\beta \leq 1$, and $Pr_{s,x}(.)$ is the probability over the uniform choice of bit strings $s$ and $x$.
By using the same techniques as \cite{BMS16PRL,HBS+17, MGDM19}, the following proposition can be shown.
\begin{proposition} \label{prop1PRL} Given that the polynomial Hierarchy (PH) does not collapse to its 3rd level, and that the worst-case hardness of approximating the probabilities of $D$ (Equations (\ref{eq3PRL}) and (\ref{eq4PRL})) extends to average-case; there exists a positive constant $\mu$ such that no $poly(n)$-time classical algorithm $C$ exists that can sample from a probability distribution $D_C$ such that \begin{equation} \label{eq6PRL}
\sum_{s,x}|D_C(s,x)-D(s,x)| \leq \mu. \end{equation}
\end{proposition}
Indeed, as shown in \cite{MGDM18}, (\ref{eq2PRL}) can be viewed as implementing a 1D random circuit, as those in \cite{BHH16}.
In this picture the circuits have depth $O(n)$ (for fixed $t$ and $\varepsilon$) and are composed of 2-qubit gates which are universal on $U(4)$.
These circuits are therefore universal under post-selection implying that there exist probabilities $D(s,x)$ which are hard ($\sharp$ P) to approximate up to relative error 1/4+O(1) \cite{FT17} (this property is referred to as \emph{worst-case hardness} of approximating the probabilities of $D$, or for simplicity worst-case hardness). Worst-case hardness together with the anti-concentration property of Equation (\ref{eq5PRL}) mean that the techniques of \cite{BMS16PRL} directly prove Proposition \ref{prop1PRL}.
Note that Proposition \ref{prop1PRL} is a conditional statement, meaning that it is true up to some conjectures being true.
The first is that the $PH$ does not collapse to its 3rd level, a generalization of $P \neq NP$, which is widely held to be true \cite{gasarch2012guest}. The second conjecture is that the worst-case hardness of the problem extends to average-case, meaning roughly that $most$ outputs are hard to approximate up to relative error 1/4 + O(1). Although this conjecture is less-widely accepted, there exists evidence to support it mainly in the case of random circuits sampling unitaries from the Haar measure \cite{bouland2018quantum,movassagh2019cayley}. Particularly relevant to our case are arguments in \cite{BHS+17,MGDM19} which give convincing evidence that worst-case hardness should extend to average-case for distributions of the form $D(s,x)$ (Equation (\ref{eq3PRL})), where the uniform distribution over bit-strings $s$ effectively makes $D(s,x)$ \emph{more flat} as compared to, say, the outputs of random quantum circuits \cite{bouland2018quantum,movassagh2019cayley} or standard IQP circuits \cite{BMS16PRL}. Also, in \cite{gao17} an average-case hardness conjecture was stated involving an MBQC construction with fixed $XY$ angles, as is the case here. Furthermore, we note that a worst-to-average-case conjecture is effectively always required in all known proofs of hardness of approximate classical sampling up to a constant error in the $l_1$-norm \cite{HM17}.
The circuit implementing this construction is constant depth. To see this, notice that the regularly structured graph states of \cite{BHS+17,gao17,HBS+17,HHB+19,MGDM18,MGDM19} can be constructed from constant depth quantum circuits composed of Hadamard ($H$) and $CZ$ gates \cite{HDE+06}. The measurements, being non-adaptive, may be performed simultaneously (depth one). The explicit form of the circuit can be seen by re-writing the state $|G'\rangle$ as follows \begin{eqnarray} \label{EQN: G' as circuit on Ts and 0s}
|G'\rangle = \bigotimes_{a \in V}H_a \bigotimes_{b\in V_2} && Z_b \left(\pi/2\right) \prod_{\{i,j\} \in E} CZ_{ij}\nonumber\\
&&\bigotimes_{c\in V_1} |T\rangle_c \bigotimes_{d \in V/V_1} H_d |0\rangle_d, \end{eqnarray}
where $|T\rangle = Z(\pi/4)H|0\rangle =(|0\rangle + e^{i\pi/4}|1\rangle)/\sqrt{2}$ is referred to as the $T$-state or magic state. Taking out the $T$-state explicitly as here will be useful for applying fault-tolerant techniques. In this way, these architectures can be viewed as constant depth 2D circuits with NN two-qubit gates \footnote{A recent paper \cite{NLD+20} shows that the outputs of 2D constant depth circuits are generally efficiently simulable classically. However, we note that the circuits discussed here \cite{BHS+17,gao17,HBS+17,HHB+19,MGDM18,MGDM19} correspond to worst-case instances of the circuits in \cite{NLD+20}, where their efficient classical algorithm fails. Indeed, the $XY$ measurement angles performed effectively induce a 1D dynamics which is purely unitary, and which for the choice of $XY$ angles made in \cite{MGDM18,MGDM19} and here in our case typically evolves an input state onto a \emph{volume law entangled} state. The classical algorithm in \cite{NLD+20} is generally inefficient in simulating such volume law entangled states.}.
We will show that this constant depth property prevails in our fault-tolerant version of these architectures as well, in our case using 4D and 3D circuits with NN two-qubit gates. As a final remark, note that the 2D NN circuit presented here has the single-instance hardness property, because the choice of measurement angles is fixed \cite{gao17}.
\textbf{\emph{Noise model}} - Before going into details of the fault-tolerant techniques, we present the noise model which we adopt. We will consider the \emph{local stochastic quantum noise} model, following \cite{FGL18,BGK+19}. Local stochastic noise can be thought of as a type of noise where the probability of the error $E$ occuring decays exponentially with the size of its support. This noise model encompasses errors that can occur in qubit preparations, gate applications, as well as measurements. It also allows for the errors between successive time steps of the circuit to be correlated \cite{FGL18,BGK+19}. More precisely, following the notation in \cite{FGL18,BGK+19}, a local stochastic noise with rate $p$, where $p$ is constant satisfying $0<p<1$, is an $m$-qubit Pauli operator $$E=\otimes_{i=1,...,m}P_{i},$$ where $P_{i} \in \{1,X,Y,Z\}$ are the single qubit Pauli operators, such that
$$Pr\big( F \subseteq Supp(E)\big) \leq p^{|F|},$$ for all $F \subseteq \{1,...,m\}$, where $Supp(E) \subseteq \{1,..,m\}$ is the subset of qubits for which $P_{i} \neq 1$. Also following notation in \cite{FGL18,BGK+19}, we will denote a local stochastic noise with rate $0<p<1$ as $E \sim \mathcal{N}(p)$.
We will use the following property of local stochastic noise, shown in \cite{BGK+19}, which says that all errors for constant depth Clifford circuits can be pushed to the end. Consider a constant depth-$d$ noiseless quantum circuit \begin{equation*} U=U_{d}...U_{1}, \end{equation*} which acts on a prepared input state and is followed by measurements, where each $U_{i}$ for $i=\{1,...,d\}$ is a depth-one circuit composed of single and two-qubit Clifford gates. It was shown in \cite{BGK+19} that a noisy version of this circuit satisfies \begin{eqnarray} \label{eq7PRL} U_{noisy}&=& E_{out}.E_{d}U_{d}....E_{1}U_{1}E_{prep} \nonumber \\ \nonumber &=& E(U_d...U_1) \\ &=& EU, \end{eqnarray} where $E_i \sim \mathcal{N}(p_i)$ for $i \in \{1,...,d\}$, with constant $0<p_{i}<1$ is the noisy implementation of depth-one circuit $U_i$, $E_{prep} \sim \mathcal{N}(p_{prep})$ and $E_{out} \sim \mathcal{N}(p_{out})$ with constants $0<p_{prep},p_{out}<1$ are the errors in the preparation and measurement respectively \footnote{Note that by choosing different values of $p_{prep}$, $p_{out}$, and $p_i$ one can differentiate between the noises of preparation, gate application, and measurement. One can also account for scenarios where some operations could be more faulty than others, as is commonly done for example when assuming two-qubit gates are faultier than single qubit gates \cite{Li15}. }.
For constant depth $d$, $E \sim \mathcal{N}(q)$ where $0<q<1$ is a constant which is a function of $p_1,...,p_d,p_{prep},p_{out}$ \cite{BGK+19} \footnote{For example, when $p_{prep}=p_{out}=p_{1}=...=p_{d}=p$, then $q \leq p^{4^{-d-1}}$ \cite{BGK+19}. Note that $p^{4^{-d-1}}$ is a constant when $d$ is a constant, meaning that $q$ is upper-bounded by a non-zero constant. For a suitable choice of $p$ we can therefore tune $q$ to be below the threshold of fault-tolerant computing with the surface code, where the classical decoding fails with a probability decaying exponentially with the code distance \cite{BGK+19}. }. Equation (\ref{eq7PRL}) shows that the errors accumulating in a constant depth quantum circuit composed of single and two qubit Clifford gates can be treated as a single error $E$. Furthermore, for small enough $q$ (i.e small enough $p_1,...,p_d,p_{prep},p_{out}$ - typically, these should be smaller than the threshold of fault-tolerant computing with the surface code \cite{BGK+19,DKL+02} or of the 3D cluster state \cite{RHG07} in our case), $E$ can be corrected with high probability by using standard techniques in quantum error correction (QEC) \cite{BGK+19,wang2011surface}. Also, $E$ can be propagated until after the measurements, where the error correction procedure is completely classical.
\section*{4D NN architecture}
In this part of the paper, we will describe the construction of our 4D NN architecture demonstrating a quantum speedup. Our approach takes three ingredients, the sampling based on regular graph states mentioned above \cite{gao17,BHS+17,HBS+17,MGDM18,MGDM19,HHB+19}, fault-tolerant single shot preparations of logical qubit states \cite{BGK+19}, and magic state distillation (MSD) \cite{BK05,HHP+17,Li15}. A large part of fault-tolerant techniques follow the work of \cite{BGK+19}, where they present a family of constant depth circuits which give statistics that cannot be reproduced by any classical computer of constant depth. To do so they introduce error correcting codes where it is possible to prepare logical states fault-tolerantly with constant depth, and Clifford gates are transversal. Then, they also show that for local stochastic quantum noise, all errors for Clifford circuits can be traced through to effectively be treated as a final error, meaning that errors do not have to be corrected during the circuit. Together these allow for constant depth fault-tolerant versions of constant depth Clifford circuits. Compared to \cite{BGK+19}, the big difference in our work is the need for non-Clifford operations (for the choice of local measurement angle). To address this, we use so called magic states which can be distilled fault-tolerantly \cite{BK05}. Generally their distillation circuits are not constant depth however, and here we adapt the distillation circuits of \cite{HHP+17} to be constant depth using ideas from MBQC. In particular we do not use feed-forward in the distillation procedure, and instead translate depth of circuits for cost of having to do many copies of constant depth circuits (each being an MSD circuit with no feed-forward) many times in parrallel. We show that, for specific MSD techniques \cite{HHP+17,HH18,Jones13,HH182}, a balance can be reached which gives sufficiently many magic states of high enough fidelity to demonstrate quantum speedup in constant depth with polynomial overhead in number of ancillas. We then use MBQC notions to route in the high fidelity magic states into our sampling circuit. This is also done in constant depth. At this point, interaction with a classical computer is required. This is mainly in order to identify which copies of MSD circuits (which are done in parallel) were successful in distilling magic states of sufficiently high fidelity. After, these high fidelity magic states are taken, together with more ancillas, to make a logical version of the graph state, which is then measured. Effectively we then have two constant depth quantum circuits with an efficient (polynomial) classical computation in between.
The constant depth MBQC distillation, together with the constant depth MBQC routing will ensure that enough magic states with adequately high fidelity are always injected into our sampling problem, thereby enabling us to observe quantum speedup \emph{deterministically} at each run of the experiment, since we would determinstically recover an encoded version of the 2D NN architecture with the single-instance hardness property described in earlier sections \cite{MGDM18}. This is contrary to what happens in \cite{KT19}, where an encoded version of this 2D NN architecture is constructed \emph{probabilistically}, albeit with exponentially low probability of success.
\textbf{\emph{Logical encoding}} -
Following \cite{BGK+19}, we use the folded surface code \cite{bravyi1998quantum,Moussa16,BGK+19}. A single logical qubit is encoded into $l$ physical qubits. We denote the logical versions of states and fault-tolerant gates using a bar, that is, a state $|\psi\rangle$ of $m$ qubits would be encoded onto its logical version $|\overline{\psi}\rangle$ on $m.l$ qubits and operator $U$ would be replaced by logical operator $\overline{U}$. The choice of encoding onto the folded surface code has two main advantages, firstly, Clifford gates have transversal fault-tolerant versions, meaning the fault-tolerant versions of a constant depth Clifford circuit are also constant depth and composed of single and two-qubit Clifford gates acting on physical qubits of the code \cite{BGK+19}. For example $$\overline{X}=\bigotimes_{i \in V_{diag}} X_i,$$
where $V_{diag}$ is the set of physical qubits lying on the main diagonal of the surface code, $X_i$ is a Pauli $X$ operator acting on physical qubit $i$. Similarly for the logical version of the Pauli $Z$ operator
$$\overline{Z}=\bigotimes_{i \in V_{diag}} Z_i.$$
Secondly, the preparation of the logical $|\overline{0}\rangle$ and $|\overline{T}\rangle$ states can be done fault-tolerantly in constant depth \cite{Li15,BGK+19}.
The preparation of the logical $|\overline{0}\rangle$ state can be done fault-tolerantly using the single-shot preparation procedure of \cite{BGK+19}. This requires a constant depth 3D quantum circuit, together with polynomial time classical post-processing, which can be pushed until after measurements of logical qubits of our circuit (see Figure \ref{FIG Overview circuit}).
This constant depth quantum circuit consists of non-adaptive measurements on a 3D cluster state composed of $O(l^{3/2})$ (physical) qubits \cite{BGK+19}.
The 3D cluster state being of regular structure can be prepared in constant depth. The non-adaptive measurements create a two-logical qubit Bell state up to a Pauli operator.
The classical post-processing is in order to trace these Paulis through the Clifford circuits (Figure \ref{FIG Overview circuit}) and correct the measurement results accordingly.
In \cite{BGK+19} it is shown that this preparation process is fault-tolerant, by showing that, in the presence of local stochastic quantum noise the overall noise induced from the preparation, measurements, and Pauli correction is a local stochastic noise with constant rate \cite{BGK+19}.
For our purposes, we will only use one logical qubit of the Bell state
\footnote{One way to do this would be measuring the other logical qubit of the Bell state non-adaptively in $\overline{Z}$, then decoding the result and applying an $\overline{X}$ to the unmeasured logical qubit dependant on the decoded measurement result. This should be done after the recovery Pauli operator of \cite{BGK+19} has been applied. The noise acting on the unmeasured qubit after completion would still be local stochastic with constant rate. Indeed, after applying the recovery operator of \cite{BGK+19}, we are left with a Bell state with some local stochastic noise $E$ \cite{BGK+19}, then after measuring one logical qubit and decoding (which succeeds with high probability if error rates are small), we apply a conditional $\overline{X}$ operator to the unmeasured logical qubit. In the case this $\overline{X}$ is applied, it introduces also a local stochastic noise $E^{'}$, but because $\overline{X}$ is a constant depth Clifford gate with only single qubit gates, $E^{'}$ can be merged with $E$ to give a single local stochastic noise $E^{''}$ which is still local stochastic with constant rate, by the likes of arguments of Equation (\ref{eq7PRL}). In what remains, we incorporate this operation into the classical post-processing needed to apply the recovery operator of the single-shot preparation procedure of \cite{BGK+19}. We will therefore mean by recovery operator hereafter, the Pauli recovery operator of \cite{BGK+19} together with the conditional $\overline{X}$ which is applied to the unmeasured logical qubit. Note also that, as mentioned in the main text, we will often push applying this recovery operator until later parts of the circuit (for example after measuring the non-outputs of all copies of $zMSD$ as well as after the final measurements of $\overline{C_2}$), in that case the arguments for the overall noise being local stochastic still hold and follow similar reasoning as above. }.
The preparation of the logical $T$-state $|\overline{T}\rangle$ can also be done in constant depth by using a technique similar to \cite{Li15}.
Indeed, in the absence of noise, a perfect logical $T$-state can be prepared by the initialization of $l$ physical qubits (over a constant number of rounds), as well as three rounds of full syndrome measurements; as detailed in \cite{Li15}
\footnote{ The constant depth procedure of \cite{Li15} also requires some post-selection (in the presence of noise).
However, this post-selection is usually over measurement results of a small (constant) number of qubits, and the success probability is also a constant \cite{Li15}. We can therefore implement in paralell $O(1)$ runs of this constant depth procedure, and we are guaranteed with high probability that at least one run corresponds to the desired post-selection. }.
Each of the syndrome measurement rounds, because of the locality of the stabilizers in the surface code, can be scheduled in such a way as to be implemented by a constant depth quantum circuit composed of Controlled Nots and ancilla qubit measurements \cite{LAR+11,Li15}. In the presence of noise, this procedure prepares a noisy logical $T$-state (Equation (\ref{eqnoisyy})), starting from a noisy physical qubit $T$-state, and noisy preparations, gates and measurements \cite{Li15}
\footnote{ Although the noise model used in \cite{Li15} is not the same as the one we use here, where in \cite{Li15} they use independent depolarizing noise for preparations and gate application, and with different rates for single and two-qubit gates, we believe their results hold in our case as well. Indeed, viewing a local stochastic noise with rate $p$ on a single qubit, this qubit could experience an error (after preparation, measurement or gate application) with probability $pr \leq p$ (from the definition of local stochastic noise with $|F|=1$, see main text), this is in line with the noise model of \cite{Li15} where the probability of error is exactly $p$. Furthermore, choosing different error rates for local stochastic noise applied after single and two-qubit gates allows mimicking what happens in the noise model of \cite{Li15}. }. However, distillation is required to get sufficiently high quality $T$-states, which will be dealt with separately later. For simplicity, for now we will assume perfect $T$-states.
Starting with the prepared logical $|\overline{0}\rangle$ and $|\overline{T}\rangle$, the logical version of Equation (\ref{EQN: G' as circuit on Ts and 0s}) is written in terms of the constant depth circuit $\overline{C_2}$, \begin{eqnarray}
|\overline{G'}\rangle = \overline{C_2} \bigotimes_{c\in V_1} |\overline{T}\rangle_c \bigotimes_{d \in V/V_1} |\overline{0}\rangle_d \label{EQN: logical G'} \end{eqnarray} where \begin{equation}
\overline{C_2} := \bigotimes_{a \in V}\overline{H_a} \bigotimes_{b\in V_2} \overline{Z_b} \left(\pi/2\right) \prod_{\{i,j\} \in E} \overline{CZ_{ij}} \bigotimes_{d \in V/V_1} \overline{H_d}.
\label{EQN: C_2} \end{equation} Since all gates are Clifford, the physical circuit implementing $\overline{C_2}$ is constant depth. This circuit is the last circuit element in Figure~\ref{FIG Overview circuit} which combines the elements of our construction.
The logical $\overline{Z}$ measurements are carried out by physical $Z$ measurements on the physical qubits of the surface code, and
several classical decoding algorithms have been established \cite{BGK+19,wang2011surface,LAR+11,DKL+02}.
In the noiseless case, the decoding algorithm consists of calculating the sum (modulo 2) of the measurement result from measuring $Z$ on the physical qubits of the main diagonal of the surface code.
In the presence of noise, the decoding algorithm takes as input the (noisy) measurement results of all the $l$ physical qubits of the surface code which are measured in the same basis as the qubits on the main diagonal, and performs a minimal weight perfect matching to correct for the error induced by the noise \cite{BGK+19,fowler2012proof,DKL+02}.
For small enough error rates (below the threshold of fault-tolerant computing with the surface code), the probability that these decoding algorithms fail, that is, the probability that the noise changes the parity of the $\overline{Z}$ measurement result after decoding, decreases exponentially with the code distance $cd$, which for surface codes scales as $cd=O(\sqrt{l})$ \cite{bravyi1998quantum,BGK+19,DKL+02,fowler2012proof}.
Let $$\overline{s}=\{\overline{s}_1,...,\overline{s}_{n.(k-1)}\},$$ denote the measurement results of the logical qubits of all but the last column of $|\overline{G'}\rangle$. Similarly, let
$$\overline{x}=\{ \overline{x}_1,...,\overline{x}_n\},$$ denote the measurement results of the logical qubits of the last column of $|\overline{G'}\rangle$.
If we call $\overline{D}(\overline{s},\overline{x})$ the probability of getting $(\overline{s},\overline{x})$ in the absence of noise, it follows straightforwardly from the logical encoding that
\begin{equation}
\label{eq9PRL}
\overline{D}(\overline{s},\overline{x})=D(s,x),
\end{equation} for all $\overline{s} \in \{0,1\}^{n.(k-1)}$, and $\overline{x} \in \{0,1\}^{n}$, where $D(s,x)$ is as defined in Equations (\ref{eq3PRL}) and (\ref{eq4PRL}).
That is, in the absence of noise, measuring non-adaptively the logical qubits of $|\overline{G'}\rangle$ in $\overline{Z}$ defines a sampling problem with probability distribution $\overline{D}$ demonstrating a quantum speedup, by Proposition \ref{prop1PRL}.
We will now see that this sampling remains robust under local stochastic noise. Noise must be addressed at each part of the construction.
The first being that each depth-one step of the circuit preparing $|\overline{G'}\rangle$ is now followed by a local stochastic noise, as in the example of Equation (\ref{eq7PRL}).
Also, the single-shot preparation procedure of \cite{BGK+19} becomes noisy, however as shown in \cite{BGK+19} this noise is local stochastic with constant error rate and therefore can be treated as a preparation noise in preparing $|\overline{G'}\rangle$, analogous to $E_{prep}$ in Equation (\ref{eq7PRL}). As seen earlier, the circuit preparing $|\overline{G'}\rangle$ is constant depth and composed of single and two-qubit Clifford gates acting on physical qubits. Therefore, we can use the result of \cite{BGK+19}, which is shown in Equation (\ref{eq7PRL}), and treat all the noise accumulating through different steps of the circuit as a single local stochastic noise $E \sim \mathcal{N}(q)$ with a constant rate $0<q<1$, acting on the (classical) measurement outcomes \cite{BGK+19}. Therefore, when $q$ is low enough \cite{BGK+19}, $E$ can be corrected with high probability using the classical decoding algorithms described earlier \cite{BGK+19}. In appendix \ref{APPC}, we show that when the number of physical qubits per logical qubit $l$ scales as \begin{equation} \label{eq10PRL}
l \geq O(log^2(n)), \end{equation}
where $n$ is the number of rows of $|G\rangle$ \footnote{ this is usually the input part of an MBQC \cite{RB01}, which is the basis of our construction }, this suffices for our needs.
More precisely, we denote $\tilde{\overline{D}}_1(\overline{s},\overline{x})$ the probability of getting outcomes $(\overline{s},\overline{x})$ in the presence of stochastic noise, after performing a classical decoding of the measurement results \cite{BGK+19}, but where logical $T$-states are assumed perfect (noisless). Then, if $l$ satisfies Equation (\ref{eq10PRL}), and for small enough error rates (below the threshold of fault-tolerant computing with the surface code \cite{DKL+02}) of preparations, single and two-qubit gates, as well as measurements, $\tilde{\overline{D}}_1(\overline{s},\overline{x})$ can be made $1/poly(n)$ close in $l_1$-norm to the noiseless version (Equation (\ref{eq9PRL})). That is, \begin{equation}
\label{eq11PRL}
\sum_{\overline{s},\overline{x}}|\tilde{\overline{D}}_1(\overline{s},\overline{x})-\overline{D}(\overline{s},\overline{x})| \leq \dfrac{1}{poly(n)}. \end{equation}
This means that for a given constant $\mu_1$, there exists a large enough constant $n_0$, such that for all $n \geq n_0$ classically sampling from $\tilde{\overline{D}}_1$ up to $l_1$-norm error $\mu-\mu_1$ implies, by a triangle inequality, sampling from $\overline{D}$ up to $l_1$-norm error $\mu$, which presents a quantum speedup by Proposition \ref{prop1PRL} \cite{BMS16PRL}. Therefore, we have recovered quantum speedup in the presence of local stochastic noise, assuming perfect $T$-states.
\textbf{\emph{Distillation of $T$-states}}- The final ingredient is the distillation of the $T$-states. The analysis we have done so far assumes we can still prepare perfect logical $T$-states. In reality, however, this is not the case. Indeed, in the presence of noise, the constant depth preparation procedure of \cite{Li15} can only prepare a logical $T$-state with error rate $0<\varepsilon<1$ \begin{equation} \label{eqnoisyy}
\overline{\rho_T}_{noisy}:=(1-\varepsilon)|\overline{T}\rangle \langle \overline{T}| + \varepsilon \eta, \end{equation} with $\eta$ an arbitrary $l$-qubit state. In order to get high purity logical $T$-states, one must employ a technique called magic state distillation (MSD) \cite{BK05}. An MSD circuit is a Clifford circuit which usually takes as input multiple copies of noisy $T$-states $\overline{\rho_T}_{noisy}$, together with some ancillas, and involves measurements and post-selection in order to purify these noisy input states \cite{BK05}. The output of an MSD circuit is a logical $T$-state $\overline{\rho_T}_{out}$ with higher purity than the input one. That is, \begin{equation}
\label{eqPRL12}
\overline{\rho_T}_{out}:=(1-\varepsilon_{out})|\overline{T}\rangle\langle\overline{T}| + \varepsilon _{out}\eta^{'}, \end{equation} with $0<\varepsilon_{out}<\varepsilon<1$, and $\eta^{'}$ an arbitrary $l$-qubit state. For small enough $\varepsilon$ \cite{reichardt2005quantum} \footnote{ This is guaranteed by using the technique of \cite{Li15} if the error rate of preparations, single and two-qubit gates is low enough. Since $\varepsilon$ in \cite{Li15} is generally a function of these error rates. }, $\varepsilon_{out}$ could be made arbitrarily small by repeating the MSD circuit an appropriate number of times \cite{BK05}.
MSD circuits need not in general be constant depth. Our approach to depth is, again, via a translation to the measurement based quantum computing (MBQC) paradigm \cite{RB01}. In MBQC one starts off with graph state, for example the 2D grid cluster state, and computation is carried out through consecutive measurements on individual qubits. In order to preserve determinism these measurements must be corrected for. For a general computation this must be done round by round (the number of rounds typically scales with the depth of the corresponding circuit, though there can be some separation thereof \cite{browne2007generalized}). If we forgo these corrections, we end up applying different unitaries, depending on the outcome of the measurement results - indeed, this is effectively what happens in Equation (\ref{eq2PRL}). Thinking of MBQC now as a circuit, if one could do all measurements at the same time, one could think of it as a constant depth circuit, since all that is needed is to construct the 2D cluster state followed by one round of measurements and corrections, which can be done in constant depth. This is possible for circuits constructed fully of Clifford operations, but not generally, and not for the MSD circuits we use here because of the $T$ gates (or feedforward), so we are forced to sacrifice determinism.
Now, in order to get constant depth MSD, we translate the MSD circuits in \cite{HHP+17} to MBQC. The choice of this MSD construction is argued in appendix \ref{APP subsec zMSD}. Since we want to maintain constant depth, we want to perform all measurements at the same time, however the cost is that it will only succeed if we get the measurement outcomes corresponding to the original circuit of \cite{HHP+17} with successful syndromes. In order to produce enough $T$ states, the trick is simply to do it many times in parallel. That is, we will effectively implement many copies of the MBQC computation, so that we get enough successes. Effectively we trade depth of the corresponding circuit for number of copies and ancillas. Fortunately, for our specifically chosen MSD protocols \cite{HHP+17,HH182}, we will see that this cost is not too high.
Furthermore, this is all done in the logical encoding of the folded surface code. Our construction for this, which we denote $zMSD$, is designed to take copies of the noisy encoded $T$-states (\cite{Li15}) and ancilla in the encoded $|\overline{0}\rangle$ state, and affect $z$ iterations of the fault-tolerant version of MSD protocol in \cite{HHP+17}. As discussed above, this happens only when the correct results occur in the MBQC. In this case we say the $zMSD$ was \emph{successful}. We denote the circuit version of this as $\overline{C_1}$ (see Figure \ref{FIG Overview circuit}). In appendix \ref{app zMSD}, we show that when $zMSD$ is successful, $\varepsilon_{out}$ satisfies \begin{equation} \label{eq13PRL}
\varepsilon_{out} \leq O(\dfrac{1}{n^4}). \end{equation}
We also show that performing $O(n^3log(n))$ copies of $zMSD$ circuits (which can be done in parallel), each of which is composed of $O(log(n))$ logical qubits as seen in appendix \ref{APP subsec zMSD}, guarantees with high probability \begin{equation} \label{eqpsuccprl}
p_{succ} \geq 1-\dfrac{1}{e^{poly(n)}}, \end{equation}
that at least $O(n^2)$ copies of $zMSD$ will be successful (we will refer to these often as successful instances of $zMSD$). Note that $O(n^2)=O(k.n)$ is the number of perfect logical $T$-states needed to create $|\overline{G'}\rangle$ \cite{MGDM18}. Furthermore, because $zMSD$ is constant depth and composed of single and two-qubit Clifford gates, errors can be treated as a single local stochastic noise after the measurements with constant rate (see Equation (\ref{eq7PRL})) which can be corrected classically with high probability when the error rates are low enough using the standard decoding algorithms described previously \cite{fowler2012proof,BGK+19}.
The remaining task is to route these good states into the inputs of the circuit $\overline{C_2}$ (Equation (\ref{EQN: C_2}))
depending on the measurement outcomes - i.e. make sure that only the good outputs go to make $|\overline{G'}\rangle$. The most obvious approach, using control SWAP gates, results in a circuit whose depth scales with $n$. Here, once more, we use MBQC techniques in order to bypass additional circuit depth. The idea is to feed the outputs through a $2D$ cluster graph state, and dependent on the measurement results of the $zMSD$, the routing can be etched out by Pauli Z measurements. Since the graph is regular, and, since the measurements can be made at the same time, this can be done in constant depth, up to Pauli corrections (which can be efficiently traced and dealt with by the classical computation at the end). We denote the fault-tolerant circuit implementing this as $\overline{C_R}$, see Figure \ref{FIG Overview circuit}. Details of the construction can be found in appendix \ref{APP routing}, where we also show that errors remain manageable.
Finally, we denote $\tilde{\overline{D}}_2(\overline{s},\overline{x})$ to mean the probability of observing the outcome $(\overline{s},\overline{x})$ after measuring all logical qubits after $\overline{C_2}$ (Equation (\ref{EQN: C_2})), in the presence of local stochastic noise, and where each $T$-state fed into $\overline{C_2}$ is replaced by $\overline{\rho_T}_{out}$, and performing a classical decoding of these measurement results \cite{BGK+19}. Then, we show, in appendix \ref{app zMSD}, that when $\varepsilon_{out}$ satisfies Equation (\ref{eq13PRL}), \begin{equation}
\label{eq14PRL}
\sum_{\overline{s},\overline{x}}|\tilde{\overline{D}}_2(\overline{s},\overline{x})-\overline{D}(\overline{s},\overline{x})| \leq \dfrac{1}{poly(n)}. \end{equation} Therefore, by the same reasoning as that for $\tilde{\overline{D}}_1$, for small enough error rates, for large enough $n$, and with very high probability $p_{succ}$, we can prepare a constant depth quantum circuit sampling from a noisy distribution $\tilde{\overline{D}}_2$ under local stochastic noise, presenting a quantum speedup.
Our main result can therefore be summarized in the following Theorem, whose proof follows directly from showing that Equation (\ref{eq14PRL}) holds and using Proposition \ref{prop1PRL}.
\begin{theorem} \label{TH1} Assuming that the $PH$ does not collapse to its third level, and that worst-case hardness of the sampling problem (\ref{eq4PRL}) extends to average-case. There exists a positive constant $0<p<1$, and a positive integer $n_o$, such that for all $n \geq n_o$, if the error rates of local stochastic noise in all preparations, gate applications, and measurements in $\overline{C_1}$, $\overline{C_R}$, and $\overline{C_2}$ are upper-bounded by $p$, then with high probability $p_{succ}$ (Equation (\ref{eqpsuccprl})), the sampling problem $\tilde{\overline{D}}_2$ defined by (\ref{eq14PRL}) can be constructed, and no $poly(n)$-time classical algorithm exists which can sample from $\tilde{\overline{D}}_2$ up to a constant $\mu^{'}$ in $l_1-$norm . \end{theorem}
\textbf{\emph{Overview of the 4D NN architecture}} - The overall construction is presented in Figure \ref{FIG Overview circuit} as a combination of the three circuits mentioned above, $\overline{C_1}$ implementing the MSD, the routing of successful $T$-states in $\overline{C_R}$, and the circuit for the construction of the state $|\overline{G'}\rangle$ in $\overline{C_2}$. Overall it takes the noisy logical $|\overline{0}\rangle$ and $\overline{\rho_T}_{noisy}$ states as inputs and the final measurements are fed back to a classical computer ($CC$) to output the error corrected results $\overline{s},\overline{x}$, according to distribution $\tilde{\overline{D}}_2$ (Equation (\ref{eq14PRL})). The preparation of the logical input states is done in constant depth \cite{BGK+19,Li15} and each of these three composite circuits are constant depth, using at most three dimensions. Furthermore, assuming that classical computation is instantaneous, our entire construction can be viewed as a constant depth quantum circuit. Indeed, as already seen $\overline{C_2}$ is constant depth, what remains is to show the same for $\overline{C_1}$ and $\overline{C_R}$. We show this in appendix \ref{APP subsec zMSD} and \ref{APP routing}.
\begin{figure*}\label{FIG Overview circuit}
\end{figure*}
During the circuit, we require some side classical computation, which inputs back into the circuit at one point. Classical information to and from the classical computer are indicated by dotted orange and black lines in Figure \ref{FIG Overview circuit}. First, the measurements of the non-outputs for the $zMSD$ in $\overline{C_1}$, along with measurement results (not illustrated in figure) of (physical) qubits used in preparing $|\overline{0}\rangle$ \cite{BGK+19} states making up the copies of $zMSD$, are fed into the classical computer in order to determine the choice of measurements after the routing circuit $\overline{C_R}$, as indicated by the orange dotted lines in Figure \ref{FIG Overview circuit}. This part simply identifies the successful $zMSD$ outcomes, followed by calculating the routing path.
This is the only point that classical results are fed back into the circuit, all other classical computations can be done after the final measurements. After these final measurements, the remaining measurements are fed back into the computer, indicated by black dotted lines in Figure \ref{FIG Overview circuit}. Together with the measurement results from the state preparations \cite{BGK+19} (not illustrated in the figure) these are incorporated into the classical error correction \cite{BGK+19,edmonds1973operation} giving the outputs $\overline{s},\overline{x}$ with probabilities $\tilde{\overline{D}}_2$. The classical computation can be done in $poly(n)$-time \cite{RBB03,wang2011surface,edmonds1973operation}.
The total number of physical qubits required scales as $O(n^5poly(log(n)))$ (where $n$ scales the size of the original sampling problem (Proposition \ref{prop1PRL})). This breaks down as follows.
$\overline{C_1}$ takes as input $O(n^3log^2(n))$ noisy logical $T$-states $\overline{\rho_T}_{noisy}$ and $O(n^3log^2(n))$ ancillas prepared in $|\overline{0}\rangle$.
$\overline{C_R}$ takes the outputs of $\overline{C_1}$, and additional $O(n^5log(n))$ logical ancillas prepared in $|\overline{0}\rangle$. This dominates the scaling.
$\overline{C_R}$ sends $O(n^2)$ distilled $T$-states to $\overline{C_2}$, which also takes in $O(n^2)$ copies of $|\overline{0}\rangle$. This means that in total we would need $O(n^5log(n))$ logical qubits. Now, each logical qubit is composed of $l \geq O(log^2(n))$ physical qubits (Equation (\ref{eq10PRL})), and some of these logical qubits ,which need to be prepared in $|\overline{0}\rangle$, require an additional overhead of $O(l^{\frac{3}{2}}) \geq O(log^3(n)))$ physical qubits, as seen previously (see also \cite{BGK+19}). Therefore, the total number of physical qubits needed is $\sim O(n^5log^4(n)))=O(n^5poly(log(n))).$
A crucial question relevant to experimental implementations would be calculating the exact values of the error rates of measurements, preparations, and gates needed to achieve fault-tolerant quantum speedup in our construction. Because the quantum depth of our construction is constant and composed of single and two-qubit Clifford gates (as seen previously), we know from \cite{BGK+19} and the likes of Equation (\ref{eq7PRL}) that these error rates are non-zero constants independent of $n$. However, their values may be pessimistically low. A crude estimate of this error rate is $p \sim e^{-4.6 \times 4^{-d-1}}$. This is assuming preparations (including preparation of noisy logical $T$-states for distillation), measurements, and gates all have the same error rate $p$. $d$ is a constant which is the total quantum depth of our construction, which is the sum of the depths of all preparations, gate applications and measurements involved in constructing $zMSD$, routing the outputs of succesful instances of $zMSD$, and constructing $|\overline{G'}\rangle$. This expression is obtained by using the same techniques as \cite{BGK+19}, where the error rate $q$ of $E$ in Equation (\ref{eq7PRL}) is chosen such that it satisfies $q \leq 0.01$. This is in order for classical decoding to fail with probability decaying exponentially with the code distance of the surface code \cite{BGK+19,fowler2012proof}.
This construction is a constant depth quantum circuit implementable on a 4D NN architecture (or a 3D architecture with long range gates). The reason for this is that our original (non fault-tolerant) construction is a 2D NN architecture \cite{MGDM18} as seen previously, and the process of making this architecture fault-tolerant requires adding an additional two dimensions \cite{BGK+19}, albeit while keeping the quantum depth constant, as explained earlier. If we do not want to use long range transversal $CZ$ gates in 3D, and want all the $CZ$ gates to be NN, the only way to do this is to work in 4D. Note that this was not a problem in \cite{BGK+19}, as there the original (non fault-tolerant) circuit was a 1D circuit, and introducing fault-tolerance added two additional dimensions, making their construction constant depth with NN gates in 3D \cite{BGK+19}. Nevertheless, we will show in the next section how to make our construction constant depth in 3D with NN two-qubit gates. We will do this by avoiding the use of transversal gates to implement encoded versions of two-qubit gates; a feature which is naturally found in defect-based topological quantum computing \cite{RHG07}. Armed with the ideas of constant depth MSD and MBQC routing, we shall present in this next section a constant quantum depth fault-tolerant construction demonstrating a quantum speedup with only nearest neighbor $CZ$ gates in 3D.
\section*{3D NN architecture} In this part of the paper, we will explain how the construction for fault-tolerant quantum speedup described earlier can be achieved using a 3D NN architecture, based on the construction of Raussendorf, Harrington, and Goyal (RHG) \cite{RHG07}. Note that in this construction (henceforth referred to as RHG construction), two types of magic states need to be distilled, the $T$-states seen previously, as well as the $Y$-states. A perfect (noiseless) $Y$-state is given by \begin{equation*}
|Y\rangle:=\dfrac{1}{\sqrt{2}}(|0\rangle+ e^{i\frac{\pi}{2}}|1\rangle). \end{equation*} This state is a resource for the phase gate $Z(\pi/2)$. The noisy $Y$-state $\rho_{Y_{noisy}}$ is defined analogously to a noisy $T$-state seen earlier \begin{equation*}
\rho_{Y_{noisy}}:=(1-\varepsilon)|Y\rangle \langle Y| + \varepsilon \eta, \end{equation*} with $0<\varepsilon<1$ representing the noise, and $\eta$ an arbitrary single qubit state.
As already mentioned, the RHG construction was also used in \cite{KT19} to achieve fault-tolerant quantum speedup. However, our construction will differ from \cite{KT19} in mainly two ways. The first, as already mentioned, is that our construction deterministically produces a hard instance, whereas that in \cite{KT19} produces such an instance with exponentially low probability. Secondly, our sampling problem verifies the anti-concentration property by construction \cite{MGDM18}, as explained previously, whereas in \cite{KT19}, this anti-concentration was conjectured. Therefore, in our proofs we assume one less complexity theoretic conjecture ( we use two conjectures in total, see Theorem \ref{TH1} and Proposition \ref{prop1PRL}) as compared to \cite{KT19}. Note that we assume the minimal number of complexity-theoretic conjectures needed to prove quantum speedup, using all currently known techniques \cite{HM17}.
\begin{figure*}\label{fig3DNN}
\end{figure*}
We now very briefly outline the key points in the RHG construction. More detailed explanations can be found in \cite{RHG07,Fujii15,FG08}. In this construction, one starts out with preparing a 3D regular lattice of qubits (call it RHG lattice). This preparation can be done in constant depth by using nearest neighbor $CZ$ gates \cite{RHG07}. This lattice is composed of elementary cells, which can be thought of as smaller 3D lattices building it up. Elementary cells are of two types, primal and dual, and the RHG lattice is composed of a number of interlocked primal and dual cells \cite{RHG07,FG08} . Each elementary cell can be pictured as a cube, with qubits (usually initialized in $|+\rangle$ state) living on the edges and faces of this cube. The RHG lattice is a graph state, and is thus characterized by a set of (local) stabilizer relations \cite{HDE+06}. Errors can be identified by looking at the parity of these stabilizers. Usually, this is done by entangling extra qubits with the systems qubits, these extra qubits are called \emph{syndrome} qubits. However, in the RHG construction this is accounted for by including these syndrome qubits \emph{a priori} when constructing the RHG lattice, this region of syndrome qubits is usually called the \emph{vacuum} region $V$ \cite{RHG07}. Logical qubits in this construction are identified with \emph{defects}. These defects are hole-like regions of the RHG lattice inside of which qubits are measured in the $Z$ basis, effectively eliminating these qubits. Eliminating these qubits (and some of their associated stabilizers) results in extra degrees of freedom which define the logical qubits \cite{RHG07}. Defects can also be primal or dual, depending on whether they are defined on primal or dual lattices. Two defects of the same type (either primal or dual) define a logical qubit. The logical operators $\overline{X}$ and $\overline{Z}$ are products of $X$ operators and $Z$ operators respectively. These products of operators act non-trivially on qubits either encircling each of the two defects, or forming a chain joining the two defects, depending on whether the logical qubit is primal or dual \cite{RHG07,FG08}. By measuring single qubits of the RHG lattice at angles $X$, $Y$, $Z$ and $\dfrac{X+Y}{\sqrt{2}}$, one can perform (primal or dual) logical qubit preparation and measurement in $\overline{X}$ and $\overline{Z}$ bases, preparation of (primal or dual) logical $T$-states and $Y$-states, and logical controlled not ($\overline{CNOT}$) gates between two defects of the same type (this however can only be accomplished by an intermediate step of \emph{braiding} two defects of different types \cite{RHG07}, which is one of the main reasons for the need for two types of defects). If performed perfectly (noiseless case), these operations are universal for quantum computation \cite{RB01}. Note that in our case, as in \cite{KT19}, we will replace measuring qubits in $Y$ and $\dfrac{X+Y}{\sqrt{2}}$ by (equivalently) initializing qubits in $|Y\rangle$ and $|T\rangle$, then measuring these qubits in the $X$ basis. In this way, we will only perform single qubit $X$ and $Z$ measurements. One of the spatial dimensions of the 3D RHG lattice is chosen as \emph{simulated time}, allowing one to perform a logical version of MBQC via single qubit measurements \cite{RHG07}.
The preparation and measurement of logical qubits in the $\overline{X}$ and $\overline{Z}$ bases, as well $\overline{CNOT}$ can all be performed by measuring qubits of the RHG lattice in $X$ and $Z$ \cite{RHG07,FG08}. All these operations can be performed fault-tolerantly, and non-adaptively (up to Pauli corrections, which can be pushed until after measurements, and accounted for, since all our circuits are Clifford \cite{RBB03}.), by choosing the defects to have a large enough perimeter, and a large enough separation \cite{FG08,RHG07}. Indeed, in appendix \ref{appRHG1}, we show that when $L_{m}=O(log(n))$, where $L_{m}$ is the minimum (measured in units of length of an elementary cell) between the perimeter of a defect and the separation between two defects in any direction, we would recover the same fault-tolerance results as our 4D NN architecture under local stochastic noise, albeit with different error rates which we will also calculate in appendix \ref{appRHG1}. The noisy logical $Y$-state and $T$-state preparations can also be prepared non-adaptively up to Pauli corrections by performing $X$ and $Z$ measurements on qubits of the RHG lattice, some of which are intialized in $|Y\rangle$ (for logical $Y$-state preparation) or $|T\rangle$ (for logical $T$-state preparation) \cite{FG08}. However, these preparations are unfortunately non-fault-tolerant (introduce logical errors), and therefore these states must be distilled \cite{RHG07}.
If we could somehow obtain perfect logical $Y$-states, then our constant-depth fault-tolerant 3D NN construction under local stochastic noise would follow a similar analysis as our 4D NN case, and have a circuit exactly the same as that in Figure \ref{FIG Overview circuit} (up to using $\overline{X}$ measurements in place of $\overline{H}$ gates followed by $\overline{Z}$ measurements), with one difference being that instead of using concatenated versions the of MSD circuits of \cite{HHP+17} to construct $\overline{C_1}$, we will use concatenated versions of the MSD circuits of \cite{HH182}. This is in order to preserve the transversality of logical $T$-gates, which allows preparation of logical $T$-states in the RHG construction by using only local measurements \cite{RHG07} \footnote{This replacement of MSD circuits does not change anything in our proofs, because both of these families of MSD circuits satisfy a specific condition regarding the number of noisy $T$-states needed to distill a $T$-state of arbitrary accuracy, and our proof hinges only on this condition being verified (see appendix \ref{APP subsec zMSD}). Furthermore, the probability of distillation succeeding for (non-concatenated) MSD circuits in \cite{HH182} approaches one for low enough $\varepsilon$, as in the MSD circuits of \cite{HHP+17}.}. Unfortunately, distilling logical $Y$-states in the RHG construction is essential. What makes matters worse is that using techniques of the likes of those used in the construction of $\overline{C_1}$, on MSD circuits capable distilling logical $Y$-states up to fidelity $1-\varepsilon_{out}$ (Equation (\ref{eq13PRL})), namely circuits based on the Steane code \cite{RHG07}, leads to circuits with a quasi-polynomial number of ancillas. This is much worse that the polynomial number of ancillas used in circuits $\overline{C_1}$ needed to distill logical $T$-states of the same fidelity $1-\varepsilon_{out}$, and based on the MSD circuits of \cite{HHP+17,HH182} (see appendices \ref{APP subsec zMSD} and \ref{appRHG2}).
Happily, we manage to overcome this limitation by observing two facts about our construction. The first is that the $Z(\pi/2)$ rotations (and thus $Y$-states) are not needed in order to construct our sampling problem. Indeed, in Figure \ref{FIG G original} every qubit measured at an $XY$ angle $\pi/2$ in $G_B$ could be replaced by a linear cluster of three qubits measured respectively at $XY$ angles $\pi/4$, $0$, and $\pi/4$ (these measurements can be implemented by only using logical $T$-states in the fault-tolerant version). To make a graph state of regular shape, we should also replace all qubits at the same vertical level as the $\pi/2$-measured qubits in $G_B$ (see Figure \ref{FIG G original}), and which are always measured at an $XY$ angle $0$, with a linear cluster of three qubits measured at an $XY$ angle $0$. By doing this replacement, the new graph gadget $G^{'}_B$ which is an extension of $G_B$ now defines a so-called partially invertible universal set \cite{MGDM19}. Therefore, by results in \cite{MGDM19}, using $G^{'}_B$ instead of $G_B$ in our construction (Figure \ref{FIG G original}) also results in a sampling problem with distribution $D^{'}=\{D^{'}(s,x)\}$ (where $s$ and $x$ are bit strings defined analogously to those in (\ref{eq3PRL}) and (\ref{eq4PRL})) satisfying both worst-case hardness and the anti-concentration property \cite{MGDM19,HBS+17}. Thus, the distribution $D^{'}$, although different than $D$ ( Equations (\ref{eq3PRL}) and (\ref{eq4PRL})), can be used in the same way as $D$ to demonstrate a quantum speedup (see Proposition \ref{prop1PRL}). Furthermore, all previous results established for $D$ also hold when $D$ is replaced by $D^{'}$.
To see why $G^{'}_B$ defines a partially invertible universal set, call $\mathcal{U}_1 \subset U(4)$ ($\mathcal{U}_2 \subset U(4)$) the set of all random unitaries which can be sampled by measuring the qubits of $G_B$ ($G^{'}_B$) non-adaptively at their perscribed angles. Straightforward calculation shows that $ \mathcal{U}_1 \subset \mathcal{U}_2$. Furthermore, both $ \mathcal{U}_1$ and its complement in $\mathcal{U}_2$ (denoted $\mathcal{U}_2-\mathcal{U}_1$) are (approximately) universal in $U(4)$ since they are composed of unitaries from the gate set of Clifford + T \cite{NC2000,MGDM18}. The set $\mathcal{U}_1$ being both universal in $U(4)$ and inverse containing \cite{MGDM18}, implies that $\mathcal{U}_2$ satisfies all the properties of a partially invertible universal set \cite{MGDM19}.
However, note that in using partially invertible universal sets, for technical reasons \cite{MGDM19}, the number of columns of $|G\rangle$ should now satisfy $k=O(n^3)$, resulting in an increase of overhead of ancilla qubits.
One could keep $k=O(n)$ (as in the original construction with $G_B$) while using only $\pi/4$ and $0$ measurements, by using one of the constructions of \cite{BHS+17}. However, the construction of \cite{BHS+17} does not have a provable anti-concentration, although extensive numerical evidence was provided to support the claim that this family of circuits does indeed anti-concentrate \cite{BHS+17}.
Although $Y$-states are not needed in the construction of our sampling problem, they are still needed to construct MSD circuits for distilling logical $T$-states of fidelity $1-\varepsilon_{out}$ (Equation (\ref{eq13PRL})) \cite{HH182}; which brings us to our second observation. In order to distill logical $T$-states of fidelity $1-\varepsilon_{out}$ (Equation (\ref{eq13PRL})), we only need logical $Y$-states of fidelity $1-\varepsilon^{'}_{out}$ with
\begin{equation}
\label{eqepsilonprimee}
\varepsilon^{'}_{out}=\dfrac{1}{O(poly(log(n)))}.
\end{equation}
In other words, the required output fidelity of the logical $Y$-states need not be as high as that of the logical $T$-states. In appendix \ref{appRHG2}, we show that this leads to a construction of a (constant-depth) non-adaptive MSD (analogous to how $\overline{C_1}$ is constructed) which takes as input a \emph{polynomial} number of logical ancillas, initialized in either noisy logical $Y$-states, $|\overline{+}\rangle$, or $|\overline{0}\rangle$, and which outputs enough logical $Y$- states of fidelity $1-\varepsilon^{'}_{out}$ needed in the subsequent distillation of logical $T$-states. This circuit, which we call $\overline{C^{'}_1}$ and which is based on concatenations of the Steane code \cite{RHG07}, is a constant depth Clifford quantum circuit composed of $\overline{CNOT}$ gates, and followed by non-adaptive $\overline{X}$ and $\overline{Z}$ measurements. $\overline{C^{'}_1}$, as $\overline{C_1}$, prepares the graph states needed for non-adaptive MSD via MBQC (as seen previously). Note that here we will use $\overline{CNOT}$ gates instead of $\overline{CZ}$ gates in order to prepare logical graph states, since these gates are more natural in the RHG construction \cite{RHG07}. The preparation procedure is essentially the same as that with $\overline{CZ}$ modulo some $\overline{H}$ gates, but these logical Hadamards can be absorbed into the initialization procedure (where some qubits become initialized in $|\overline{0}\rangle$ instead of $|\overline{+}\rangle$) and the measurements (where some $\overline{X}$ measurements after $\overline{C^{'}_1}$ are changed to $\overline{Z}$ measurements, and vise versa.). The same holds for all other circuits based on graph states in this construction.
With the distillation of logical $Y$-states taken care of, we now summarize our constant depth construction based on a 3D NN architecture. The circuit of this construction is found in Figure \ref{fig3DNN}. It takes as input logical qubits initialized in the states $|\overline{+}\rangle$, $|\overline{0}\rangle$, $\overline{\rho_T}_{noisy}$, and $\overline{\rho_Y}_{noisy}$, and outputs a bit string $(\overline{s},\overline{x})$ sampled from the distribution $\tilde{\overline{D^{'}}}_2$ demonstrating a quantum speedup (see Theorem \ref{TH1} and Proposition \ref{prop1PRL}). Note that $\tilde{\overline{D^{'}}}_2$ is the fault-tolerant version of the distribution $D^{'}$ defined earlier. $\tilde{\overline{D^{'}}}_2$ is defined analogously to $\tilde{\overline{D}}_2$ in Equation (\ref{eq14PRL}), which is the fault-tolerant version of the distribution $D$ (Equation (\ref{eq4PRL})). Our 3D NN architecture is composed of five constant depth circuits acting on logical qubits, $\overline{C^{'}_1}$, $\overline{C^{'}_R}$, $\overline{C_1}$, $\overline{C_R}$, and $\overline{C_2}$. $\overline{C^{'}_1}$, $\overline{C_1}$, $\overline{C_R}$, and $\overline{C_2}$ are as defined previously, and $\overline{C^{'}_R}$ is a routing circuit, analogous to $\overline{C_R}$, which routes succesfully distilled logical $Y$-states to be used in $\overline{C_1}$. Furthermore, all of these circuits, as well as the preparation of logical qubits, can be constructed by non-adaptive single-qubit $X$ and $Z$ measurements on physical qubits arranged in a 3D RHG lattice, whose preparation is constant depth and involves only nearest neighbor $CZ$ gates. These physical qubits are initialized in the (noisy) states $|+\rangle$, $|Y\rangle$, and $|T\rangle$ \cite{RHG07}. Our construction has two layers of interaction with a classical computer, needed to identify succesfully distilled logical $Y$ and $T$-states respectively. The number of physical qubits needed is $O(n^{11}poly(log(n))$, this calculation is performed in appendix \ref{appRHG2}. The additional overhead as compared to our 4D NN construction comes from mainly two sources, the partially invertible universal set condition \cite{MGDM19}, and the circuits $\overline{C^{'}_R}$ and $\overline{C^{'}_1}$ which arise as a result of needing to distill logical $Y$-states in the 3D RHG construction \cite{RHG07}.
As in our 4D NN architecture, the noise model we use here is the local stochastic quantum noise defined earlier \cite{BGK+19,fawzi2018constant}. Since the circuit needed to construct the 3D RHG lattice is composed of single and two-qubit Clifford gates acting on prepared qubits \cite{RHG07}, all errors of preparations and gate applications can be pushed, together with the measurement errors, until after the measurements; as seen previously. Because the circuit preparing the RHG lattice is constant depth, the overall local stochastic noise has a constant rate (see Equation (\ref{eq7PRL})), and therefore could be corrected with high probability for low enough (constant) error rates of preparation, gate application, and measurements \cite{BGK+19} (see appendix \ref{appRHG1} where we calculate an estimate of these error rates). The error correction, as in our 4D NN architecture, is completely classical and involves minimal weight matching \cite{edmonds1973operation}. This error correction is $poly(n)$-time and is performed at each of the two layers of interaction with the classical computer, as well as after the final measurements. Also, as in the 4D NN case, other $poly(n)$-time classical algorithms are included in the classical post processing; these are in order to identify succesful MSD instances, and identify the measurement patterns of the routing circuits. The classical computer at each layer of interaction as well as after the final measurements takes as input measurement results of qubits involved in the computation, as well as measured qubits in the vacuum region $V$. These vacuum qubits give the error syndrome at multiple steps in the computation, and are therefore needed for the minimal weight matching \cite{RHG07}.
$\textbf{\emph{Discussion}}-$ In summary, we have presented a construction sampling from a distribution demonstrating a quantum speedup, which is robust to noise. Our construction has constant depth in its quantum circuit, and can be thought of as a fault-tolerant version of the (noise free) constant depth quantum speedup based on generating and measuring graph states \cite{gao17,BHS+17,HBS+17,MGDM18,MGDM19,HHB+19}. We have shown how to implement this construction both by using a 4D architecture with nearest neighbor two-qubit gates, or by using a 3D architecture with nearest neighbor two-qubit gates. The circuits of each of these architectures interact at most twice with an (efficient) classical device while running, and have different requirements in terms of overhead of physical ancilla qubits, owing to the fact that they are based on two different constructions for fault-tolerance \cite{BGK+19,RHG07}.
The overheads are large in terms of the number of (physical) qubits, however these may be improved. In any case, our construction is considerably simpler than fault-tolerant full blown quantum computation where circuits are scaling in depth and many adaptive layers are required. Therefore our architectures demonstrate potential for interim demonstration of quantum computational advantage, which may be much more practical. Indeed, if one considers classical computation temporally free, our construction represents a constant time implementation of a sampling problem with fault-tolerance.
We note that although we have presented here a fault-tolerant construction for a specific graph state architecture \cite{MGDM18}, the same techniques can be applied to any of the sampling schemes based on making local $XY$ measurements from the set $\{0,\pi/2,\pi/4\}$ on regular graph states \cite{gao17,BHS+17,HBS+17,MGDM18,MGDM19}.
In particular it can be easily adapted to cases where the measurements are not fixed but chosen at random before the running of the circuit \cite{BHS+17,HBS+17,gao17}. This would essentially just fix the locations of the distilled $T$-states, but it could be done before hand, and would not effect the efficiency of the routing circuits. This has the potential of relating the average-case hardness conjecture to that of other more familiar problems \cite{gao17,BHS+17,BMS16PRL}.
Our work also has potentially another interest, as it can alternatively be viewed as a constant depth quantum circuit which samples from an approximate unitary $t$-design \cite{DCE+09} fault-tolerantly. Indeed, our techniques can be used to directly implement a logical version of Equation (\ref{eq2PRL}), which samples from an approximate $t$-design. These $t$-designs have many useful applications across quantum information theory \cite{DCE+09,emerson2003pseudo,hayden2004randomizing,matthews2015testing,muller2015thermalization,hayden2007black}.
Several interesting approaches for optimization may be considered. One could think of using different quantum error correcting codes, such as those of \cite{LAR+11,fawzi2018constant}, to decrease the overhead of physical qubits. One could also aim to optimize the overhead of both gates and physical qubits of the MSD by using techniques similar to those of \cite{CC19,CN20}.
The ability to efficiently verify quantum speedup is also an important goal. Although this question has already been pursued in the regime of fault-tolerance in \cite{KT19}, and the techniques developped there are directly applicable to our 3D NN architecture; it would be interesting to develop verification techniques more naturally tailored to the graph state approach \cite{HDE+06,RB01} and MBQC \cite{RB01,RBB03}, which we use heavily here. In this direction, the work of \cite{MK18,TMM+19} can be used for this purpose when the measurements (both Clifford and non-Clifford) as well as the $CZ$ and Hadamard gates (needed for the preparation of the graph states \cite{HDE+06}) are assumed $perfect$ (noiseless). Indeed, in this case the verification amounts to verifying that the graph state was correctly prepared, for which \cite{MK18,TMM+19} provide a natural path to do so, by giving good lower bounds (with high confidence) on the fidelity (with respect to the ideal graph state corresponding to the sampling problem) of the prepared graph state in the case where a sufficient amount of stabilizer tests pass \cite{MK18,TMM+19}. These lower bounds on the fidelity, tending asymptotically to one \cite{MK18,TMM+19}, allow one to verify that quantum speedup is being observed, as long as one trusts the local measurement devices (which, being small, can be checked by other means efficiently). This verification of quantum speedup can be done by using the standard relation between the fidelities of two quantum states (which in our case are the ideal state and the state accepted by the verification protocol) and the $l_1$-norm of the two output probability distributions corresponding to measuring the qubits of these two states \cite{NC2000}.
These techniques, however, do not easily extend to the case where the measurements and gates needed for preparation are noisy; since for graph states of size $m$, even for an arbitrarily small (but $constant$, for example below the threshold for fault-tolerant computing) noise strength, the verification protocol might fail (not accept a good state) in the asymptotic ($m \to \infty$) limit (see for example \cite{TMM+19} where the verification accepts with probability one asymptotically only if the noise strenght scales as $1/poly(m)$). We leave this problem for future investigation.
\begin{acknowledgments} We thank David Gosset, Elham Kashefi, Theodoros Kapourniotis, Anthony Leverrier, Ashley Montanaro, Micha\l{} Oszmaniec, and Peter Turner for fruitful discussions and comments. The Authors would like to acknowledge the National Council for Scientific Research of Lebanon (CNRS-L) and the Lebanese University (LU) for granting a doctoral fellowship to R. Mezher. We acknowledge support of the ANR through the ANR-17-CE24-0035 VanQute project. \end{acknowledgments} \nocite{*}
\onecolumngrid
\appendix
\section{Size of encoding and intermediate case hardness of sampling.} \label{APPC}
Here we prove our statements regarding the sufficiency of the size of logical encoding $l$ (Equation (\ref{eq10PRL})) and the proof of hardness in the intermediate case where we have noise in the circuit, but assume perfect $T$-states (Equation (\ref{eq11PRL})). As mentioned in the main text, the probability $p_f$ that the classical decoding fails to correct an error $E \sim \mathcal{N}(p)$ affecting a surface code composed of $l$ physical qubits is given by \cite{DKL+02,BGK+19,fowler2012proof} \begin{equation}
\label{eqappC1}
p_f=e^{-O(cd)}=e^{-O(\sqrt{l})}, \end{equation} when the error rate $p$ is below the threshold for fault-tolerant computing with the surface code \cite{DKL+02}. We will assume, as mentioned in the main text, that the error rates of preparation, single and two qubit gates, and measurements in our construction are small enough, that is, below the threshold of fault-tolerant computing with the surface code, and classical postprocessing is instantaneous. We will also assume that the probabilities of failure of the classical decoding algorithms in each logical qubit are independent (see \footnote{This assumption may seem strong, but it is actually very mild and has no effect on our end result. To see this, suppose we drop this assumption, then the probability of success of all $k.n$ decodings should now be calcuated by a union bound. From the properties of local stochastic noise (namely that local stochastic noise on a subset of qubits of the system is still local stochastic with the same rate \cite{BGK+19}) a decoding of a logical qubit succeeds (is able to identify and correct for the error) with probability $p_{single}=1-p_f=1-e^{-O(\sqrt{l})}$ (when the error rates of all local stochastic noise in our construction are adequately low, i.e below the threshold of fault-tolerant computing with the surface code), therefore the probability that all $k.n$ decodings succeed is given by $P=1-k.n+k.n.p_{single}=1-k.n.e^{-O(\sqrt{l})}$, by a standard bound on the intersection of $k.n$ events derived from a union bound. The assumption we make in the main text results in a good approximation of $P$, and is simpler to state (which is why we used it in the main text). Finally, note that this does not mean that errors between physical qubits of two entangled logical qubits are uncorrelated. Indeed, the correlation between these qubits is accounted for in the propagation rules of local stochastic noise \cite{BGK+19}, since forward propagating local stochastic noise in Clifford circuits composed of single and two-qubit gates generally results in local stochastic noise with higher error rate \cite{BGK+19}.}). Our construction involves classical decoding of the measurement results of $O(k.n)$ logical qubits \footnote{Actually, it is something like $2.k.n$ if we include decoding of measured logical qubits of the Bell states obtained at the end of the single shot procedure of \cite{BGK+19} (see $[57]$). This changes nothing in the analysis we have done, so we chose to omit it in the main text for simplicity.}. After decoding, the probability of observing outcome $(\overline{s},\overline{x})$ is given by \begin{equation}
\label{eqappC2}
\tilde{\overline{D}_1}(\overline{s},\overline{x})=(1-e^{-O(\sqrt{l})})^{O(k.n)}\overline{D}(\overline{s},\overline{x})+\sum_{i}p_{i}\overline{D}_i^{e}(\overline{s},\overline{x}) \end{equation}
where $\overline{D}_i^{e}$ is a distribution corresponding to sampling from the outputs $(\overline{s},\overline{x})$ of $|\overline{G'}\rangle$ in the presence of local stochastic noise, and where the decoding algorithm has failed in at least one logical qubit. $\sum_{i}p_i\overline{D}_i^{e}(\overline{s},\overline{x})$ enumerates all possible ways in which decoding on the $k.n$ logical qubits of $|\overline{G'}\rangle$ can fail. Note that \begin{equation*}
\sum_{i}p_i= 1-(1-e^{-O(\sqrt{l})})^{O(k.n)}. \end{equation*} Now, \begin{equation} \label{eqappCf}
\sum_{\overline{s},\overline{x}}|\tilde{\overline{D}_1}(\overline{s},\overline{x})-\overline{D}(\overline{s},\overline{x})| = \sum_{\overline{s},\overline{x}}|(1-e^{-O(\sqrt{l})})^{O(k.n)}\overline{D}(\overline{s},\overline{x})+\sum_{i}p_{i}\overline{D}_i^{e}(\overline{s},\overline{x})-\overline{D}(\overline{s},\overline{x})| \leq 2(1-(1-e^{-O(\sqrt{l})})^{O(k.n)}). \end{equation} The bound on the right hand side is obtained from a triangle inequality and by noting that $\sum_{\overline{s},\overline{x}}\overline{D}(\overline{s},\overline{x})=\sum_{\overline{s},\overline{x}}\overline{D}_i^{e}(\overline{s},\overline{x})=1$. Choosing \begin{equation}
\label{eqblocksize}
l=r.log^{2}(n)=O(log^2(n)), \end{equation} where $r$ is a positive constant chosen large enough so that the following inequality holds \begin{equation}
\label{eqcondblocksize}
deg(e^{O(\sqrt{l})}) > deg(k.n), \end{equation} where $deg(.)$ represents the highest power of $n$ in the expressions of $e^{O(\sqrt{l})}$ and \\ $O(k.n)$. We can now use (for large enough $n$) the approximation \begin{equation} \label{eqappCapprox} 2\big(1-(1-e^{-O(\sqrt{l})})^{k.n}) \sim 2e^{-O(\sqrt{l})}.k.n=O(\dfrac{1}{n^{\beta}}), \end{equation} with $\beta=deg(e^{O(\sqrt{l})})-deg(k.n)$. Plugging Equation (\ref{eqappCapprox}) in Equation (\ref{eqappCf}) we get \begin{equation*}
\sum_{\overline{s},\overline{x}}|\tilde{\overline{D}_1}(\overline{s},\overline{x})-\overline{D}(\overline{s},\overline{x})| \leq O(\dfrac{1}{n^{\beta}})=\dfrac{1}{poly(n)}. \end{equation*} This completes the proof of Equations (\ref{eq10PRL}) and (\ref{eq11PRL}).
\section{Bounding $\tilde{\overline{D}}_2(\overline{s},\overline{x})$ and Properties of $zMSD$} \label{app zMSD}
\subsection{Bounding $\tilde{\overline{D}}_2(\overline{s},\overline{x})$ (proof of Equation (\ref{eq14PRL}))} Let \begin{equation}
\label{eqappD1}
\tilde{\rho}_{|\overline{G'}\rangle}=\bigotimes_{a\in V} \overline{H}_a \bigotimes_{b \in V_2}\overline{Z(\pi/2)}_b \prod_{\{i,j\}\in E}\overline{CZ}_{ij} \bigotimes_{c\in V/V_1}\overline{H}_c\overline{|0}\rangle_c \langle \overline{0}|_c\overline{H}_c \bigotimes_{d\in V_1}\overline{\rho_T}_{out} \prod_{\{i,j\}\in E}\overline{CZ}_{ij}\bigotimes_{b \in V_2}\overline{Z(-\pi/2)}_b \bigotimes_{a\in V} \overline{H}^{\dagger}_a. \end{equation}
$\tilde{\rho}_{|\overline{G'}\rangle}$ is exactly the same as $|\overline{G'}\rangle$, but with each single logical qubit state $|\overline{T}\rangle$ replaced with $\overline{\rho_T}_{out}$, the output of a succesful instance of $zMSD$ (Equations (\ref{eqPRL12}) and (\ref{eq13PRL})). The probability $\tilde{\overline{D}}_2(\overline{s},\overline{x})$ can be calculated by using the following simple observation \begin{equation}
\label{eqappD2}
\tilde{\overline{D}}_2(\overline{s},\overline{x})=p(\{\overline{s},\overline{x}\}\cap ne)+p(\{\overline{s},\overline{x}\}\cap e), \end{equation}
where $p(\{\overline{s},\overline{x}\}\cap ne)$ is the probability of observing outcome $\{\overline{s},\overline{x}\}$ when no logical error (ne) has occured (that is, that classical decoding did not fail in any logical qubit) , neither in the distillation process, nor in the routing, nor in constructing and measuring $\tilde{\rho}_{|\overline{G'}\rangle}$. $p(\{\overline{s},\overline{x}\}\cap e)$ is the probability of observing $\{\overline{s},\overline{x}\}$ when the decoding algorithm has failed (e) at least on one logical qubit. We will assume that in the case where no logical error has occured, for large enough $n$, the probability $p_{succ}$ (Equation (\ref{eqpsuccprl})) of distilling enough ($O(n^2))$ states $\overline{\rho_T}_{out}$ to construct $\tilde{\rho}_{|\overline{G'}\rangle}$ is equal to one. This is a reasonable assumption since the exponential term in $p_{succ}$ varies much more rapidly than the polynomial terms in our bounds, for large enough $n$. Now, \begin{equation*}
p(\{\overline{s},\overline{x}\}\cap ne)=p(ne).p( \{\overline{s},\overline{x}\}| ne).
\end{equation*}
\begin{equation}
\label{eqpne}
p(ne)=(1-e^{-O(\sqrt{l})})^{O(n^5log^2(n))},
\end{equation} is the probability that that the decoding does not fail on all our $O(n^5log^2(n))$ logical qubits (logical qubits of all copies of $zMSD$, the routing circuit, as well as
$\tilde{\rho}_{|\overline{G'}\rangle}$).
Now, \begin{equation} \label{eqpsxne}
p(\{\overline{s},\overline{x}\}|ne)=\sum_{i_1,...,i_{k.n.l}}\langle i_1...i_{k.n.l} |\tilde{\rho}_{|\overline{G'}\rangle}| i_1...i_{k.n.l}\rangle, \end{equation}
where $|i_1...i_{k.n.l}\rangle$ is a state of $k.l.n$ physical qubits, corresponding to the measurement of the $k.n$ logical qubits of $\tilde{\rho}_{|\overline{G'}\rangle}$, which when decoded gives rise to the bit string $(\overline{s},\overline{x})$.
\begin{equation} \label{eqpsxe}
p(\{\overline{s},\overline{x}\}\cap e)=\sum_{j}p_{e_j}p(\{\overline{s},\overline{x}\}|e_j), \end{equation} where the right hand of Equation (\ref{eqpsxe}) enumerates all possible ways in which decoding on the $O(n^5log^2(n))$ logical qubits could fail. Note that \begin{equation}
\label{eqboundpsxe}
\sum_{\overline{s},\overline{x}}p(\{\overline{s},\overline{x}\}\cap e) \leq \sum_jp_{e_j} \leq 1-p(ne) \leq 1-(1-e^{-O(\sqrt{l})})^{O(n^5log^2(n))}. \end{equation} Replacing Equations (\ref{eqpne})-(\ref{eqpsxe}) in Equation (\ref{eqappD2}) we get
\begin{equation}
\label{eqappD7}
\tilde{\overline{D}}_2(\overline{s},\overline{x})=(1-e^{-O(\sqrt{l})})^{O(n^5log^2(n)))}p(\{\overline{s},\overline{x}\}|ne)+\sum_{j}p_{e_j}p(\{\overline{s},\overline{x}\}|e_j). \end{equation} By using Equations (\ref{eqboundpsxe}) and (\ref{eqappD7}) as well as a triangle inequality. We get that \begin{equation}
\label{eqappD8}
\sum_{\overline{s},\overline{x}}|\tilde{\overline{D}}_2(\overline{s},\overline{x})-p(\{\overline{s},\overline{x}\}|ne)| \leq 2(1-(1-e^{-O(\sqrt{l})})^{O(n^5log^2(n))}). \end{equation} As in appendix \ref{APPC}, chosing $$l =r.log^2(n),$$ but with $r$ chosen so that $$deg(e^{O(\sqrt{l})}) > deg(O(n^5log^2(n))),$$ we get that \begin{equation} \label{eqappD9}
\sum_{\overline{s},\overline{x}}|\tilde{\overline{D}}_2(\overline{s},\overline{x})-p(\{\overline{s},\overline{x}\}|ne)| \leq \dfrac{1}{poly(n)} \end{equation}
by using the same approximations as in appendix \ref{APPC} to bound $2(1-(1-e^{-O(\sqrt{l})})^{O(n^5log^2(n))})$. Now, remark that the fidelity between $\tilde{\rho}_{|\overline{G'}\rangle}$ and $|\overline{G'}\rangle$, denoted as $F$, satisfies (from Equations (\ref{eqappD1}) and (\ref{EQN: logical G'})) \begin{equation} \label{eqappD10}
F \geq (1-\varepsilon_{out})^{O(n^2)}, \end{equation} with $\varepsilon_{out}$ given by Equation (\ref{eq13PRL}). Furthermore, the probabilities $\overline{D}(\overline{s},\overline{x})$
and $p(\{\overline{s},\overline{x}\}|ne)$ satisfy \cite{NC2000} \footnote{Actually, this relation holds for the $l_1$-norm distance between the probability distributions over physical qubits. However, as the absolute value of the sum is less than the sum of absolute values, this relation also holds for the probabilities in Equation (\ref{eqappD11}).} \begin{equation} \label{eqappD11}
\sum_{\overline{s},\overline{x}}|\overline{D}(\overline{s},\overline{x})-p(\{\overline{s},\overline{x}\}|ne)| \leq 2\sqrt{1-F^2}. \end{equation} when $\varepsilon_{out}$ satisfies Equation (\ref{eq13PRL}), $$2\sqrt{1-F^2} \leq 2\sqrt{1-(1-\varepsilon_{out})^{O(n^2)}} \sim 2 \sqrt{O(n^2)\varepsilon_{out}} \leq \dfrac{1}{poly(n)}.$$ Plugging this into Equation (\ref{eqappD11}), then using Equations (\ref{eqappD9}) and (\ref{eqappD11}) and a triangle inequality, we obtain \begin{equation*}
\sum_{\overline{s},\overline{x}}|\tilde{\overline{D}_2}(\overline{s},\overline{x})-\overline{D}(\overline{s},\overline{x})| \leq \dfrac{1}{poly(n)}. \end{equation*} This completes the proof of Equation (\ref{eq14PRL}).
\subsection{Properties of $zMSD$} \label{APP subsec zMSD}
$zMSD$ implements non-adaptively $z$ iterations of the MSD protocol of Theorem 4.1 in \cite{HHP+17}. Note that in the protocol of \cite{HHP+17}, the MSD circuit was for magic states of the form $|H\rangle=cos(\pi/8)|0\rangle+sin(\pi/8)|1\rangle$ whereas in our case we need distillation circuits for $T$-states $|T\rangle $ defined in the main text. However, since $HZ(-\pi/2)|H\rangle=e^{-i\pi/8}|T\rangle$, the circuits in \cite{HHP+17} can be adapted to our case by adding a constant depth layer of $H$ and $Z(-\pi/2)$ gates, whose logical versions can be done fault-tolerantly and also in constant depth in our construction . We call $1MSD$ a circuit which implements non-adaptively one iteration of the protocol of Theorem 4.1 in \cite{HHP+17}. Note that both $zMSD$ and $1MSD$ will be based on non-adaptive MBQC. We will begin by calculating the number of qubits of $1MSD$.
In Theorem 4.1 in \cite{HHP+17}, the MSD circuit takes as input $O(d)$ qubits, where $d$ is a positive integer, uses $O(d^2)$ noisy input $T$-states with fidelity 1-$\varepsilon$ with respect to an ideal (noiseless) $T$-state, and outputs $O(d)$ distilled $T$-states with fidelity 1-$O(\varepsilon^{d})$ with respect to an ideal $T$-state (note that the ratio of the number of noisy input $T$-states to the number of distilled output $T$-states is $\sim d$ for large enough constant $d$ \cite{HHP+17}.). Each time a noisy $T$-state is inserted it affects a noisy $T$-gate, inducing a so-called $T$-gate depth \cite{HHP+17}. The depth of the entire circuit is $O(d^2.log(d))$, where $O(d)$ is the $T$-gate depth, and $O(d.log(d))$ is the depth of the Clifford part of the circuit, which is composed of long-range Cliffords \cite{HHP+17}. Therefore, the MSD circuit is an $O(d)$-qubit circuit of depth $O(d^2.log(d))$. In order to implement this circuit on a regular graph state (for example, the cluster state \cite{RB01}), one must transform the Clifford circuit composed of long range gates, to that composed of nearest neighbor and single qubit Clifford gates, since these single qubit and nearest neighbor two-qubit gates can be implemented by measuring $O(1)$ qubits of a cluster state in the $X$ and $Y$ bases \cite{RB01,RBB03}. An $m$-qubit Clifford gate can be implemented by an $O(m^2)$-depth circuit composed only of gates from the set $\{CZ_{ij},H,Z(\pi/2)\}$ \cite{gottesman1997stabilizer}. Furthermore, $CZ_{ij}$ could be implemented by a circuit of depth $O(i-j)$ composed of nearest neighbor CZ gates \cite{mantri2017universality}. The same arguments hold in the logical picture by replacing $H$, $CZ$, $Z(\pi/2)$, and noisy input $T$-states with their logical versions $\overline{H}$, $\overline{CZ}$, $\overline{Z(\pi/2)}$, and $\overline{\rho_T}_{noisy}$. $m=O(d)$ in our case, thus the number of columns of the cluster state needed to implement $1MSD$ is \begin{equation}
\label{eqnumberofcolumns}
n_{c}=O(d^2log(d)).O(d^2).O(d)=O(d^5log(d)), \end{equation} where the $O(d^2log(d))$ comes from the depth of the MSD circuit with long range Cliffords, $O(d^2)$ is the depth needed to implement an arbitrary Clifford using $\overline{H}$ gates, $\overline{Z(\pi/2)}$ gates, and long range $\overline{CZ}$'s, and the $O(d)$ is an overestimate and represents the number of nearest neighbor $\overline{CZ}$'s needed to give a long range $\overline{CZ}$. The total number of qubits of the cluster state implementing $1MSD$ is then \begin{equation}
\label{eqtotalqubits}
n_{T}=O(d).n_{c}=O(d^{6}.log(d)). \end{equation} $zMSD$ can be thought of as a concatenation of $z$ layers of $1MSD$, where the output of layer $j$ is the input of layer $j+1$. Because the noisy input $T$-states in the protocol of \cite{HHP+17} are injected at different parts of the circuit, this means that the output qubits of layer $j$ should be connected to layer $j+1$ at different positions by means of long range $\overline{CZ}$ gates. Therefore, the graph state implementing $zMSD$ can be seen as cluster states composed of logical qubits, and connected by long range $\overline{CZ}$ gates, as shown in Figure (\ref{fig1}). One could equivalently replace these long range $\overline{CZ}$ gates with a series of $\overline{SWAP}$ gates, which can be implemented (up to Pauli correction by means of non-adaptive $\overline{X}$ and $\overline{Z}$ measurements) on a 2D cluster state with only nearest neighbor $\overline{CZ}$ gates \cite{RB01,RBB03}. Because these long range $\overline{CZ}$ gates act on qubits separated by a distance $poly(d)$, the introduction of $\overline{SWAP}$ gates introduces an additional (constant) overhead of $O(poly(d))$ qubits to $n_T$, but makes the construction of $1MSD$ implementable on a 2D cluster state with only nearest neighbor $CZ$ gates.
\begin{figure}
\caption{Part of the graph state implementing the circuit $zMSD$. Blue filled circles represent logical qubits in the $|\overline{+}\rangle$ state, which when measured implement the Clifford part of the MSD protocol of Theorem 4.1 in \cite{HHP+17}. The green filled circles are noisy input $T$-states $\overline{\rho_T}_{noisy}$. Purple filled circles are the output qubits of the first layer of $zMSD$. When $zMSD$ is successful, these qubits are in a state with fidelity $1-O(\varepsilon^d)$ with respect to the ideal $T$-state $|\overline{T}\rangle$. The orange lines are $\overline{CZ}$ gates. Note that the output qubits of the first layer (purple circles) are connected to the second layer at different positions by means of long range $\overline{CZ}$ gates. These long range $\overline{CZ}$ gates can be implemented in constant depth, since they act each on distinct pairs of qubits. Also, as mentioned in the main text in this appendix, these long range $\overline{CZ}$ gates can be replaced by a series of $\overline{SWAP}$ gates making this construction a constant-depth 2D construction with only nearest-neighbor $\overline{CZ}$ gates. Measurements consist of non-adaptive $\overline{X}$ measurements, $\overline{Z}$ measurements, as well as $\overline{Y}$ measurements. As described in the main text, we could equivalently perform all measurements in $\overline{Z}$, by introducing additional constant depth layers of $\overline{H}$ and $\overline{Z(\pi/2)}$ gates.}
\label{fig1}
\end{figure}
The first layer consists of $N$ copies of cluster states implementing $1MSD$ (see Figure \ref{fig1}), and outputs, when succesful, $N.O(d)=\dfrac{N}{d}.O(d^2)$ $T$-states with fidelity $1-C.\varepsilon^d$ with respect to $|\overline{T}\rangle$, $C$ being a positive constant \cite{HHP+17}. These $T$-states are the input of the second layer which consists of $\dfrac{N}{d}$ copies of cluster states implementing $1MSD$, and outputs, when succesful, $\dfrac{N}{d}.O(d)=\dfrac{N}{d^2}.O(d^2)$ $T$-states with fidelity $C.(C.\varepsilon^{d})^{d}=C^{d+1}.\varepsilon^{d^2}$ with respect to $|\overline{T}\rangle$. Similarly, the $z$th layer will consist of $\dfrac{N}{d^{z-1}}$ copies, and will output, when successful, $\dfrac{N}{d^{z-1}}.O(d)$ $T$-states with fidelity \begin{equation} \label{eqepsilonout} \varepsilon_{out} \sim C^{d^{z-1}}.\varepsilon^{d^z}, \end{equation}
with respect to $|\overline{T}\rangle$. The total number of qubits of the graph state implementing $zMSD$ is then given by \begin{equation}
\label{eqtotalqubitsnmsdl}
n_{NMSD}=(N+\dfrac{N}{d}+\dfrac{N}{d^2}+...).n_T=O(N). \end{equation} $z$ is the last layer, therefore $\dfrac{N}{d^{z-1}} = 1$ and thus \begin{equation} \label{eqN1}
N = d^{z-1}. \end{equation} For a succesful instance of $zMSD$, in order to arrive at Equation (\ref{eq13PRL}), choose $$d^z \geq O(log(n)),$$ this implies that each copy of $zMSD$ is composed of $$n_{NMSD}=O(N)=O(d^{z-1}) \geq O(log(n)),$$ logical qubits, as mentioned in the main text. Indeed, replacing $d^z = a.log(n),$ with $a$ a positive constant in Equation (\ref{eqepsilonout}) yields $$\varepsilon_{out} = \dfrac{1}{n^{a.\alpha}},$$ by a direct calculation, where $\alpha=\dfrac{log(\dfrac{1}{C.\varepsilon^d})}{d}$ while noting that $C.\varepsilon^d < 1$ \cite{HHP+17}. Equation (\ref{eq13PRL}) is therefore obtained for an appropriate choice of $a$ or $\varepsilon$.
Now, we will calculate the probability $p_{szMSD}$ of a single successful instance of $zMSD$. We will assume, rather pessimistically, that only one string of non-adaptive measurement results of $zMSD$ corresponds to a successful instance. This string we will take, by convention, to be the one where all the measurement binaries (after decoding ) are zero. In this case, \begin{equation}
\label{eqpszMSD}
p_{szMSD} \geq \dfrac{1}{2^{n_{NMSD}}}. \end{equation} Note that the lower bound is actually higher than that Equation (\ref{eqpszMSD}) for two reasons. The first is that not all qubits of the graph state implementing $zMSD$ are measured. Indeed, the output qubits of the last layer of $zMSD$ are unmeasured and, in the case when $zMSD$ is successful, are in the state $\overline{\rho_T}_{out}$. The second reason is that some of the measurements correspond, in the successful case, to post-selections which in the protocol of \cite{HHP+17} occur with probability greater than $1/2$ . Indeed, for small enough $\varepsilon$ , the acceptance rate of the protocol of \cite{HHP+17} is approximately 1. Now, $\varepsilon_{out} = \dfrac{1}{n^{\beta}}$, with $\beta \geq 4$, and $n_{NMSD}=\gamma.d^z$ (Equations (\ref{eqtotalqubitsnmsdl}) and (\ref{eqN1})), with $\gamma$ a positive constant. By choosing $$\varepsilon=\dfrac{e^{-\gamma.\beta.log(2)}}{C^{1/d}},$$ and performing a direct calculation using Equation (\ref{eqepsilonout}), we get that $n_{NMSD}=log_2(n)$. Therefore, \begin{equation}
\label{eqpszmsd2}
p_{szMSD} \geq \dfrac{1}{n}. \end{equation}
One might ask, why do other MSD protocols like those of \cite{BK05,BH12}, for example, not work (using our techniques)? The answer to this question has to do with the number of noisy input $T$ states $n_{noisy}$ with fidelity $1-\varepsilon$ with respect to an ideal $T$-state, needed to distill a single $T$-state of sufficiently high fidelity $1-\varepsilon_{out}$ with respect to an ideal $T$-state. $n_{noisy}$ is usually given by \cite{BK05} \begin{equation} \label{eqnnoisy} n_{noisy}=O \Big (log^{\gamma}(\dfrac{1}{\varepsilon_{out}}) \Big ). \end{equation} $\gamma$ is a constant which depends on the error correcting code from which the MSD protocol is derived \cite{HH18}. In the protocol of \cite{HHP+17} (as well as those in \cite{HH182}), $\gamma \sim 1$. Whereas for the Steane code for example \cite{RHG07}, which we used to distill $Y$-states in our 3D NN architecture, $\gamma >1$. $\gamma \sim 1$ in the protocol of \cite{HHP+17} is what allowed us to get a $p_{szMSD}$ of the form of Equation (\ref{eqpszmsd2}). On the other hand, the protocols of \cite{BK05,BH12,RHG07} have a $\gamma >1$, which leads to a lower bound of $p_{szMSD}$ which looks like $1/qp(n)$- by using similar arguments for calculating $n_{NMSD}$- where $qp(n)$ is $quasi$-$polynomial$ in $n$ (if one requires $\varepsilon_{out}=1/poly(n)$). Indeed, $N$ is proportional to $\alpha.n_{noisy}$, where $\alpha$ is the number of output $T$-states with error $\varepsilon_{out}$. Therefore, it follows that $n_{NMSD}=O(N)=O(n_{noisy})$, and that $2^{n_{NMSD}}=2^{O(n_{noisy})}$, which is a quasi-polynomial when $\gamma > 1$. This would mean, using our proof techniques, that we would need a quasi-polynomial in $n$ (which is greater than polynomial in $n$) number of $zMSD$ copies to get a succesful instance, thereby taking us out of the scope of what is considered quantum speedup \footnote{Since quantum speedup is usually defined with respect to quantum devices using polynomial quantum resources \cite{NC2000}.}. Other protocols which we could have used and could have worked are those of \cite{Jones13,HH182} which gives $\gamma \sim 1$, or that of \cite{HH18} which gives $\gamma < 1$, albeit with a huge constant overhead of $2^{58}$ qubits \cite{HH18}.
\subsection{Proof of Equation (\ref{eqpsuccprl})} \label{sec exp fail} We begin by calculating $p_{fail}=1-p_{succ}.$ Suppose we have constructed $M$ copies of $zMSD$, the probability $p_{fail}$ of not getting at least $O(n^2)$ succesful instances of $zMSD$ is given by \begin{equation}
\label{eqappD31}
p_{fail}=\sum_{m=0,...,O(n^2)}{M \choose O(n^2)-m } p^{O(n^2)-m}_{szMSD}(1-p_{szMSD})^{M-O(n^2)+m}. \end{equation} $p_{szMSD} \leq 1-p_{szMSD}$ (Equation (\ref{eqpszmsd2})), \begin{equation}
\label{eqappD32}
p_{fail} \leq \sum_{m=0,...,O(n^2)}{M \choose O(n^2)-m } (1-p_{szMSD})^{M} \end{equation} Taking $M > 2O(n^2),$ \begin{equation}
\label{eqappD33}
\sum_{m=0,...,O(n^2)}{M \choose O(n^2)-m } \leq O(n^2) {M \choose O(n^2)}. \end{equation} Replacing Equation (\ref{eqappD33}) in Equation (\ref{eqappD32}), and using Equation (\ref{eqpszmsd2}), we get \begin{equation}
\label{eqappD34}
p_{fail} \leq O(n^2) {M \choose O(n^2)}(1-\dfrac{1}{n})^{M}. \end{equation} Also, $${M \choose O(n^2)}<M^{O(n^2)}.$$ Replacing this in Equation (\ref{eqappD34}) we get \begin{equation}
p_{fail} \leq O(n^2) M^{ O(n^2)}(1-\dfrac{1}{n})^{M}. \end{equation} Noting that for large enough $n$ $$(1-\dfrac{1}{n})^{n} \sim \dfrac{1}{e},$$ and taking $\dfrac{M}{n}=p(n).O(n^2)$ \begin{equation}
p_{fail} \leq O(n^2)\big(\dfrac{M}{e^{p(n)}}\big)^{O(n^2)}. \end{equation} Choosing $p(n) \geq log(M)=O(log(n))$, we get that $\dfrac{M}{e^{p(n)}} \leq c$, with $c<1$ a constant. In this case, $$p_{fail} \leq O(n^2)c^{O(n^2)} \leq O(n^2)\dfrac{1}{e^{O(n^2)}} \sim \dfrac{1}{e^{O(n^2)}},$$ for large enough $n$. Thus, $$p_{succ} \geq 1-\dfrac{1}{e^{O(n^2)}}.$$ Note that for our choice of $p(n) \geq O(log(n))$, we get that $M=O(n^3)p(n) \geq O(n^3log(n)).$ This completes the proof of Equation (\ref{eqpsuccprl}).
\section{The routing circuit $\overline{C_R}$} \label{APP routing}
The main idea of the MBQC based routing is to use the fact that in a graph state, measurements allow us to etch out desired paths. In particular performing a $\overline{Z}$ measurement removes a vertex and its edges \cite{HDE+06}, as illustrated in Figure \ref{FIG Z measurements on GS}. Once a path is etched out, $\overline{X}$ measurements teleport the state along it. Given $m$ systems to route out of a possible $p$, a grid of size $2pm$ is sufficient for unique paths to be etched out. An example of how this works for a grid is illustrated in Figure \ref{FIG routing} for $m=2$ and $p=7$ \footnote{Again, we can measure all qubits only in $\overline{Z}$ if we add a constant-depth layer of $\overline{H}$ gates to the graph state.}. In our case, we have a total of $O(n^3log(n))$ outputs of all the $zMSD$, of which $O(n^2)$ will be successful, hence the number of ancilla we require scales as $O(n^5log(n))$.
\begin{figure}
\caption{Performing a $Z$ measurement on a vertex of a graph state removes it, up to local Pauli corrections.}
\label{FIG Z measurements on GS}
\end{figure}
\begin{figure}
\caption{Routing via etching out from a grid. The purple vertices on the left represent outputs of the $zMSD$. a) The filled purple vertices are identified as the successful distilled $T$ states from previous measurement results, and the paths to the outputs are identified. b) All other qubits are measured out in $Z$ and the succesful outputs are teleported via $X$ measurements.}
\label{FIG routing}
\end{figure}
It is worth explaining why the overall noise on the routed $\overline{\rho_T}_{out}$ will still be local stochastic with constant rate. Firstly, note that $\overline{C_R}$ is a constant depth Clifford circuit composed of single and two-qubit Clifford gates acting on outputs of $zMSD$ circuits, and therefore all local stochastic noise after each depth one step of this circuit can be treated as a single local stochastic noise $E_{d} \sim \mathcal{N}(m)$ with constant rate $m$ at the end of this circuit, as in Equation (\ref{eq7PRL}) \cite{BGK+19}. The outputs of $zMSD$ circuits are acted upon by local stochastic noise with constant rate (as seen earlier overall noise on $zMSD$ is local stochastic with constant rate, therefore noise acting on a subset of qubits of $zMSD$ (the outputs) is also local stochastic with the same rate \cite{BGK+19}), and therefore can be incorporated as preparation noise (analogous to $E_{prep}$ in Equation (\ref{eq7PRL})) with $E_{d}$ to give a net local stochastic noise $E \sim \mathcal{N}(c)$ with constant rate $c$ acting on qubits of $\overline{C_R}$. After measurements, the unmeasured outputs of $\overline{C_R}$ will also be acted upon by $E^{'} \sim \mathcal{N}(c)$ which is local stochastic with same rate as $E$, but with smaller support, from the properties of local stochastic noise \cite{BGK+19}.
\section{Error correction in our 3D NN architecture} \label{appRHG1} In this section we will show how the probability of failure $p_{fail}$ of decoding in our 3D NN architecture can be made $polynomially$ low. $p_{fail}$ here is equivalent to $1-p(ne)=1/poly(n)$ in appendix \ref{app zMSD}. Thus; obtaining $p_{fail}=1/poly(n)$ allows us to recover the same results for error correction as the 4D NN architecture. We will assume that classical postprocessing is instantaneous. We will work with local stochastic noise and, as discussed in the main text, deal with a single local stochastic noise $E \sim \mathcal{N}(q)$ which is pushed forward until after the measurements \cite{BGK+19} (see Equation (\ref{eq7PRL})). The (constant) rate $q$ satisfies $q \leq 0.0075$ \cite{RHG07}, that is, it is below the threshold of fault-tolerant computing in the RHG construction. As argued in \cite{DKL+02}, the probability $p_{fail}$ can be calculated by calculating the number of ways in which the minimal weight matching results in a non-trivial error, that is, an error stretching across at least $L_{m}$ qubits, where $L_{m}$ is the minimum between the perimeter of the defect and the (minimal) distance between two defects \cite{RHG07,DKL+02}. $p_{fail}$ can be calculated by using the following relation \cite{DKL+02}
\begin{equation}
\label{eqSAP1}
p_{fail} \leq P(n) \sum_{L \geq L_{m}} n(L).prob(L). \end{equation} This relation simply counts the number of ways in which a relevant non-trivial error can occur, this type of error is restricted to errors induced by self-avoiding walks (SAWs) on the lattice, as argued in \cite{DKL+02}. $n(L)=6.5^{L-1}$ calculates all possible SAWs of total lenght $L$ originating from a fixed point in the lattice \cite{DKL+02}, $P(n)=poly(n)$ is the the total number of fixed points (i.e physical qubits) on the lattice, since SAWs can originate at any fixed point, and $prob(L) \leq (4q)^{\frac{L}{2}}$ is the probability that the minimal matching induces an error chain (SAW) of length $L$, this probability is calculated using the techniques in \cite{DKL+02}, but adapted to local stochastic noise (whereas independent depolarizing noise acting on each qubit was considered in \cite{DKL+02}). The sum is over all non-trivial errors of lenght $L_m \leq L \leq poly(n)$. Noting that \begin{equation} \label{eqSAP2}
P(n)\sum_{L \geq L_{m}} n(L).prob(L) \leq poly(n)(poly(n)-L_m)\dfrac{6}{5}(100q)^{\frac{L_{m}}{2}} \leq poly^{'}(n).(0.75)^{\dfrac{L_m}{2}} \sim \dfrac{poly^{'}(n)}{e^{0.06L_m}}, \end{equation} where $poly^{'}(n)$ is some polynomial in $n$. Choosing $L_m=\alpha.log(n)$ with $\alpha$ a positive constant, and replacing Equation (\ref{eqSAP2}) in (\ref{eqSAP1}) we get \begin{equation}
\label{eqSAP3}
p_{fail} \leq \dfrac{poly^{'}(n)}{n^{0.06\alpha}}. \end{equation} Finally, choosing $$0.06\alpha > deg(poly^{'}(n)),$$ and replacing this in Equation (\ref{eqSAP3}) we obtain our desired polynomially low bound for $p_{fail}$ \begin{equation}
\label{eqSAP4}
p_{fail} \leq \dfrac{1}{poly(n)}.
\end{equation}
Now, we want to find an estimate of the individual rates of preparation, gate application and measurement in our 3D NN architecture. Assuming at each layer of the circuit, qubits are acted upon by a local stochastic noise $E \sim N(p)$ with $0<p<1$ a constant, we get that $q \leq 4p^{4^{-D-1}}$ \cite{BGK+19}, where $D$ is the total quantum depth of the RHG construction. $D=6$, one step for preparation, one for (non-adaptive) measurements (assuming instantaneous classical computing as mentioned earlier), and four steps for preparing the RHG lattice \cite{RHG05}. Setting $q \leq 0.0075$ \cite{RHG07}, we get that the errors in preparation, gate application, and measurement should satisfy $p \leq \sim e^{-40000}$. Note that, for completeness, the threshold error rate for the distillation $\varepsilon$ should also be taken into account. Usually, $\varepsilon$ should be lower than some constant \cite{reichardt2005quantum} in order for distilation to be possible, but this is accounted for in the chosen value of $q$ \cite{RHG07}.
\section{ Distillation and overhead in our 3D NN architecture} \label{appRHG2} \subsection{Distillation} In this subsection, we will discuss distillation of logical $Y$-states in our 3D NN construction. The distillation of $T$-states in this construction is exactly the same as in appendix \ref{app zMSD}, but instead of using the protocol of Theorem 4.1 in \cite{HHP+17}, we use the protocol with $\gamma \sim 1$ (see appendix \ref{app zMSD}) in \cite{HH182} which allows transversal implementation of logical $T$-gates and is thus compatible with the RHG construction \cite{RHG07,RHG05}.
The distillation of $Y$-states is done with the $[7, 1, 3]$ Steane code \cite{RHG07}. This code has a $\gamma \sim log(7)/log(3) \sim 1.77$ \cite{HH18}. Therefore, the total number of logical ancilla qubits (including qubits prepared in initial noisy logical $Y$-states $\overline{\rho_{Y}}_{noisy}$) needed to distill a logical $Y$-state of fidelity $1-\varepsilon^{'}_{out}$ is given by \cite{BH12} (see appendix \ref{app zMSD}) \begin{equation}
\label{eqdistillY1}
N_Y=O(log^{1.77}(\dfrac{1}{\varepsilon^{'}_{out}})). \end{equation} Choosing $\varepsilon^{'}_{out}=1/O(poly(log(n)))$ as in the main text, we get that \begin{equation}
\label{eqdistillY2}
N_Y=O(log^{1.77}(poly(log(n))) \sim O(log^{1.77}(log(n))). \end{equation} It is straightforward to see that, for high enough $n$, \begin{equation}
\label{eqdistillY3}
N_Y < log(n). \end{equation} $N_Y$ can be though of as the number of logical qubits of a 2D logical cluster state needed to distill a logical $Y$-state of fidelity $1-\varepsilon^{'}_{out}$. As in appendix \ref{app zMSD}, if we do this MBQC non-adaptively, we only succeed with probability \begin{equation}
\label{eqdistillY4}
P_s \geq \dfrac{1}{2^{N_Y}} \geq \dfrac{1}{n}. \end{equation}
In our case, we need $O(n^5log^2(n))$ logical $Y$-states of fidelity $1-\varepsilon^{'}_{out}$ in order to distill $O(k.n)=O(n^4)$ $T$-states to be used in the construction of $\overline{C_2}$. $O(n^5log^2(n))$ is the number of qubits of $\overline{C_1}$ when $k=O(n^3)$ (number of columns of $|G\rangle$ ). Therefore, by results in appendix \ref{sec exp fail}, we would need $\overline{C^{'}_1}$ to be composed of $O(n^6log^3(n))$ logical qubits in order to distill, with exponentially high probability of success, enough ($O(n^5log^2(n))$) logical $Y$-states with fidelity $1-\varepsilon^{'}_{out}$.
Now, we will see why logical $Y$-states of fidelity $1-\varepsilon^{'}_{out}=1-1/O(poly(log(n)))$ suffice to disill $O(n^5log^2(n))$ $T$-states with fidelity $1-\varepsilon_{out}=1-1/O(poly(n))$. In the construction of $\overline{C_1}$ in appendix \ref{APP subsec zMSD}, replacing a perfect logical $Y$-state with a logical $Y$-state of fidelity $1-\varepsilon^{'}_{out}$, then measuring this state, results in applying the gate $\overline{H}\overline{Z(\pi/2)}$ with probability $1/2(1-\varepsilon^{'}_{out})$ instead of $1/2$ in the perfect logical $Y$-state case. Therefore, the success probability of $zMSD$ becomes in this case \begin{equation}
\label{eqdistillY5}
p_{zMSD} \geq \dfrac{1}{n}(1-\varepsilon^{'}_{out})^{O(log(n))}, \end{equation} as compared with Equation (\ref{eqpszmsd2}) in the perfect logical $Y$ case. By choosing, as we did, $\varepsilon^{'}_{out}=1/poly(log(n))$, the above equation can be rewritten, for large enough $n$, as \begin{equation}
\label{eqdistillY6}
p_{zMSD} \geq \dfrac{1}{n}(1-\dfrac{1}{O(poly(log(n)))})^{O(log(n))} \sim \dfrac{1}{n}(1-\dfrac{1}{O(poly^{'}(log(n)))}) \sim \dfrac{1}{n}. \end{equation} Thus, we have recovered Equation (\ref{eqpszmsd2}), and therefore can now use the same analysis as in appendix \ref{app zMSD} to distill logical $T$-states of fidelity $1-\varepsilon_{out}$ in our 3D NN construction. This will allow us to construct the sampling problem Equation (\ref{eq14PRL}) showing a quantum speedup.
\subsection{Overhead}
In this subsection, we will estimate the overhead (number of physical qubits in the 3D RHG lattice) of our 3D NN construction. As in \cite{RHG07}, we will make use of the concept of a logical elementary cell. Each logical elementary cell is a 3D cluster state composed of $\lambda \times \lambda \times \lambda$ elementary cells (each of which has eighteen qubits). Logical elementary cells can be either primal or dual. Each logical elementary cell contains a single defect. A defect inside a logical elementary cell has a cross section of $d \times d$ (perimeter $4d$) on any plane perpendicular to the direction of simulated time. For our purposes, we will choose $\lambda=O(d)$, and $d=O(log(n))$. This will ensure that the perimeter of the defect ($4d$) and the distance between two defects ($\lambda-d$) satisfy the conditions in appendix \ref{appRHG1}. In this picture, every logical qubit (composed of two defects of the same type) needs $2\times 18 \times \lambda^{3}=O(log^3(n))$ physical qubits. In order to not talk about primal or dual logical qubits (recall that computation is always carried out on logical qubits of same type, but we need braiding between two defects of different type in order to implement some gates such as $\overline{CNOT}$), we will assume each logical qubit needs four cells (two primal, two dual) to be defined, and therefore the number of physical qubits per logical qubit is $4 \times 18 \times \lambda^{3}=O(log^3(n))$. Now, all we need to do is calculate the number of logical qubits we need in total. Preparations of logical qubits in states $|\overline{+}\rangle$, $\overline{\rho_T}_{noisy}$, and $\overline{\rho_Y}_{noisy}$, and applying $\overline{CNOT}$ gates can be done using a constant number of intermediate elementary logical cells \cite{RHG07}. Therefore, we will only need to count the total number of logical qubit inputs for circuits $\overline{C^{'}_1}$, $\overline{C^{'}_R}$, $\overline{C_1}$, $\overline{C_R}$, and $\overline{C_2}$, then multiply this by a constant in order to get the total number of needed logical qubits including preparations and logical CNOT applications. As already calculated in the previous subsection, the total number of logical qubits of $\overline{C^{'}_1}$ is $O(n^6log^3(n))$. The total overhead of circuits $\overline{C_1}$, $\overline{C_R}$, and $\overline{C_2}$ is $O(n^9poly(log(n))$ logical qubits, this is obtained by the same calculations as done in our 4D NN architecture, but with replacing $k=O(n)$ with $k=O(n^3)$, in order for the partially invertible universal set condition to be satisfied \cite{MGDM19}. Finally, the routing circuit $\overline{C^{'}_R}$ (see appendix \ref{APP routing}) needs $O(n^6log^3(n).n^5log^2(n))=O(n^{11}log^5(n))$, this term dominates the scaling. Multiplying $O(n^{11}log^5(n))$ by a constant (to account for preparation and logical CNOT gates overhead), then by $O(log^3(n))$ (to get the number of physical qubits), we get that the overall number of physical qubits needed is $O(n^{11}poly(log(n))$.
\end{document}
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\begin{document}
\title{Two cores of a nonnegative matrix} \thanks{This research was supported by EPSRC grant RRAH15735. Serge\u{\i} Sergeev also acknowledges the support of RFBR-CNRS grant 11-0193106 and RFBR grant 12-01-00886. Bit-Shun Tam acknowledges the support of National Science Council of the Republic of China (Project No. NSC 101-2115-M-032-007)}
\author{Peter Butkovi{\v{c}}} \address{Peter Butkovi{\v{c}}, University of Birmingham, School of Mathematics, Watson Building, Edgbaston B15 2TT, UK} \email{[email protected]}
\author{Hans Schneider} \address{Hans Schneider, University of Wisconsin-Madison, Department of Mathematics, 480 Lincoln Drive, Madison WI 53706-1313, USA} \email{[email protected]}
\author{Serge\u{\i} Sergeev} \address{Serge\u{\i} Sergeev, University of Birmingham, School of Mathematics, Watson Building, Edgbaston B15 2TT, UK} \email{[email protected]}
\author{Bit-Shun Tam} \address{Bit-Shun Tam, Tamkang University, Department of Mathematics, Tamsui, Taiwan 25137, R.O.C. } \email{[email protected]}
\begin{abstract} We prove that the sequence of eigencones (i.e., cones of nonnegative eigenvectors) of positive powers $A^k$ of a nonnegative square matrix $A$ is periodic both in max algebra and in nonnegative linear algebra. Using an argument of Pullman, we also show that the Minkowski sum of the eigencones of powers of $A$ is equal to the core of $A$ defined as the intersection of nonnegative column spans of matrix powers, also in max algebra. Based on this, we describe the set of extremal rays of the core.
The spectral theory of matrix powers and the theory of matrix core is developed in max algebra and in nonnegative linear algebra simultaneously wherever possible, in order to unify and compare both versions of the same theory.
{\it{Keywords:}} Max algebra, nonnegative matrix theory, Perron-Frobenius theory, matrix power, eigenspace, core. \vskip0.1cm {\it{AMS Classification:}} 15A80, 15A18, 15A03,15B48
\end{abstract}
\maketitle
\section{Introduction}
The nonnegative reals $\R_+$ under the usual multiplication give rise to two semirings with addition defined in two ways: first with the usual addition, and second where the role of addition is played by maximum.
Thus we consider the properties of nonnegative matrices with entries in two semirings, the semiring of nonnegative numbers with usual addition and multiplication called ``{\bf nonnegative algebra}'', and the semiring called ``{\bf max(-times) algebra}''.
Our chief object of study is the {\bf core} of a nonnegative matrix $A$. This concept was introduced by Pullman in ~\cite{Pul-71}, and is defined as the intersection of the cones generated by the columns of matrix powers $A^k$. Pullman provided a geometric approach to the Perron-Frobenius theory of nonnegative matrices based on the properties of the core. He investigated the action of a matrix on its core showing that it is bijective and that the extremal rays of the core can be partitioned into periodic orbits. In other words, extremal rays of the core of $A$ are nonnegative eigenvectors of the powers of $A$ (associated with positive eigenvalues).
One of the main purposes of the present paper is to extend Pullman's core to max algebra, thereby investigating the periodic sequence of eigencones of max-algebraic matrix powers. However, following the line of~\cite{BSS,BSS-zeq,KSS}, we develop the theory in max algebra and nonnegative algebra simultaneously, in order to emphasize common features as well as differences, to provide general (simultaneous) proofs where this is possible. We do not aim to obtain new results, relative to~\cite{Pul-71,TS-94}, on the usual core of a nonnegative matrix. However, our unifying approach leads in some cases (e.g., Theorem~\ref{t:tam-schneider} (iii)) to new and more elementary proofs than those given previously. Our motivation is closely related to the Litvinov-Maslov correspondence principle~\cite{LM-98}, viewing the idempotent mathematics (in particular, max algebra) as a ``shadow'' of the ``traditional'' mathematics over real and complex fields.
\if{ Pullman's core can be also seen as closely related to the limits of powers of nonnegative matrices. However it is a different concept. Consider the simple example $$ \begin{pmatrix} 1 & 0 \\ 0 & 0.5 \end{pmatrix}. $$ Then, for any nonnegative $x$, $A^kx$ will tend to a multiple of $( 1,\ 0)^T$ while the core of $A$ is the entire nonnegative orthant ${\mathbb R}^2_+$. }\fi
To the authors' knowledge, the core of a nonnegative matrix has not received much attention in linear algebra. However, a more detailed study has been carried out by Tam and Schneider~\cite{TS-94}, who extended the concept of core to linear mappings preserving a proper cone. The case when the core is a polyhedral (i.e., finitely generated) cone was examined in detail in~\cite[Section 3]{TS-94}, and the results were applied to study the case of nonnegative matrix in~\cite[Section 4]{TS-94}. This work has found further applications in the theory of dynamic systems acting on the path space of a stationary Bratteli diagram. In particular, Bezuglyi~et~al.~\cite{BKMS} describe and exploit a natural correspondence between ergodic measures and extremals of the core of the incidence matrix of such a diagram.
On the other hand, there is much more literature on the related but distinct question of the limiting sets of homogeneous and non-homogeneous Markov chains in nonnegative algebra; see the books by Hartfiel~\cite{Har:02} and Seneta~\cite{Sen:81} and, e.g., the works of Chi~\cite{Chi-96} and Sierksma~\cite{Sie-99}. In max algebra, see the results on the ultimate column span of matrix powers for irreducible matrices (\cite[Theorem 8.3.11]{But:10}, \cite{Ser-09}), and by Merlet~\cite{Mer-10} on the invariant max cone of non-homogeneous matrix products.
The theory of the core relies on the behaviour of matrix powers. In the nonnegative algebra, recall the works of Friedland-Schneider~\cite{FS-80} and Rothblum-Whittle~\cite{RW-82} (on the role of distinguished classes which we call ``spectral classes'', algebraic and geometric growth rates, and various applications). The theory of max-algebraic matrix powers is similar. However, the max-algebraic powers have a well-defined periodic ultimate behaviour starting after sufficiently large time. This ultimate behaviour has been known since the work of Cuninghame-Green~\cite[Theorem 27-9]{CG:79}, Cohen~et~al.~\cite{CDQV-83} (irreducible case), and is described in greater generality and detail, e.g., by Akian, Gaubert and Walsh~\cite{AGW-05}, Gavalec~\cite{Gav:04}, De Schutter~\cite{BdS}, and the authors~\cite{But:10,Ser-11<attr>, SS-11} of the present paper. In particular, the Cyclicity Theorem of Cohen~et~al.~\cite{BCOQ,But:10,CDQV-83,HOW:05}) implies that extremals of the core split into periodic orbits for any irreducible matrix (see Subsection 4.2 below)\footnote{ In fact, many of the cited works and monographs like~\cite{BCOQ,But:10,Gav:04,HOW:05} are written in the setting of {\bf max-plus algebra}. However, this algebra is isomorphic to the max algebra considered here, so the results can be readily translated to the present (max-times) setting.}.
Some results on the eigenvectors of max-algebraic matrix powers have been obtained by Butkovi\v{c} and Cuninghame-Green~\cite{But:10,CGB-07}. The present paper also aims to extend and complete the research initiated in that work.
This paper is organized as follows. In Section~\ref{s:prel} we introduce the basics of irreducible and reducible Perron-Frobenius theory in max algebra and in nonnegative linear algebra.
In Section~\ref{s:key} we formulate the two key results of this paper. The first key result is Main Theorem~\ref{t:core} stating that the matrix core equals to the Minkowski sum of the eigencones of matrix powers (that is, for each positive integer $k$, we take the sum of the eigencones associated with $A^k$, and then we sum over all $k$). The second key result is Main Theorem~\ref{t:periodicity} stating that the sequence of eigencones of matrix powers is periodic and defining the period. This section also contains a table of notations used throughout the paper. Section~\ref{s:core} is devoted to the proof of Main Theorem~\ref{t:core}, taking in ``credit'' the result of Main Theorem~\ref{t:periodicity} (whose proof is deferred to the end of the paper).
In Section~\ref{s:sameaccess} we explain the relation between spectral classes of different matrix powers, and how the eigencones associated with general eigenvalues can be reduced to the case of the greatest eigenvalue, see in particular Theorems~\ref{t:samespectrum} and~\ref{t:reduction}.
In Section~\ref{s:extremals} we describe extremals of the core in both algebras extending~\cite[Theorem~4.7]{TS-94}, see Theorem~\ref{t:tam-schneider}. Prior to this result we formulate the Frobenius-Victory Theorems~\ref{t:FVnonneg} and~\ref{t:FVmaxalg} giving a parallel description of extremals of eigencones in both algebras. In Section~\ref{s:eigencones}, our first goal is to show that the sequence of eigencones of matrix powers in max algebra is periodic, comparing this result with the case of nonnegative matrix algebra, see Theorem~\ref{t:girls}.
Then we study the inclusion relation on eigencones and deduce Main Theorem~\ref{t:periodicity}. The key results are illustrated by a pair of examples in Section~\ref{s:examples}.
\section{Preliminaries} \label{s:prel}
\subsection{Nonnegative matrices and associated graphs} \label{ss:nonneg}
In this paper we are concerned only with nonnegative eigenvalues and nonnegative eigenvectors of a nonnegative matrix. In order to bring our terminology into line with the corresponding theory for max algebra we use the terms eigenvalue and eigenvector in a restrictive fashion appropriate to our semiring point of view. Thus we shall call $\rho$ an {\em eigenvalue} of a nonnegative matrix $A$ (only) if there is a nonnegative eigenvector $x$ of $A$ for $\rho$. Further $x$ will be called an {\em eigenvector} (only) if it is nonnegative. (In the literature $\rho$ is called a distinguished eigenvalue and $x$ a distinguished eigenvector of $A$.) For $x\in\Rp^n$, the {\em support} of $x$, denoted by $\operatorname{supp}(x)$, is the set of indices where $x_i>0$.
In this paper we are led to state the familiar Perron-Frobenius theorem in slightly unusual terms: An irreducible nonnegative matrix $A$ has a unique eigenvalue denoted by $\rho^+(A)$, which is positive (unless $A$ is the $1\times 1$ matrix $0$). Further, the eigenvector $x$ associated with $\rho^+(A)$ is essentially unique, that is all eigenvectors are multiples of $x$. The nonnegative multiples of $x$ constitute the cone of eigenvectors (in the above sense) $V_+(A,\rho^+(A))$ associated with $\rho^+(A)$.
A general (reducible) matrix $A\in\Rp^{n\times n}$ may have several nonnegative eigenvalues with associated cones of nonnegative eigenvectors ({\em eigencones}), and $\rho^+(A)$ will denote the biggest such eigenvalue, in general. Eigenvalue $\rho^+(A)$ is also called the {\em principal eigenvalue}, and $V_+(A,\rho^+(A))$ is called the {\em principal eigencone}.
Recall that a subset $V\subseteq\Rp^n$ is called a (convex) cone if 1) $\alpha v\in V$ for all $v\in V$ and $\alpha\in\R_+$, 2) $u+v\in V$ for $u,v\in V$. Note that cones in the nonnegative orthant can be considered as ``subspaces'', with respect to the semiring of nonnegative numbers (with usual addition and multiplication). In this vein, a cone $V$ is said to be {\em generated} by $S\subseteq\Rp^n$ if each $v\in V$ can be represented as a nonnegative combination $v=\sum_{x\in S} \alpha_x x$ where only finitely many $\alpha_x\in\R_+$ are different from zero. When $V$ is generated (we also say ``spanned'') by $S$, this is denoted $V=\spann_+(S)$. A vector $z$ in a cone $V$ is called an {\em extremal}, if $z=u+v$ and $u,v\in V$ imply $z=\alpha_u u=\alpha_v v$ for some scalars $\alpha_u$ and $\alpha_v$. Any closed cone in $\Rp^n$ is generated by its extremals; in particular, this holds for any finitely generated cone.
Let us recall some basic notions related to (ir)reducibility, which we use also in max algebra. With a matrix $A=(a_{ij})\in\R_+^{n\times n}$ we associate a weighted (di)graph ${\mathcal G}(A)$ with the set of nodes $N=\{1,\dots,n\}$ and set of edges~$E\subseteq N\times N$ containing a pair~$(i,j)$ if and only if~$a_{ij}\neq 0$; the weight of an edge~$(i,j)\in E$ is defined to be~$w(i,j):=a_{ij}$. A graph with just one node and no edges will be called {\em trivial}. A graph with at least one node and at least one edge will be called {\em nontrivial}.
A path $P$ in ${\mathcal G}(A)$ consisting\footnote{In our terminology, a path can visit some nodes more than once.} of the edges $(i_0,i_1),(i_1,i_2),\ldots,(i_{t-1},i_t)$ has {\em length} $l(P):=t$ and {\em weight} $w(P):=w(i_0,i_1) \cdot w(i_1,i_2) \cdots w(i_{t-1},i_t)$, and is called an $i\;-\;j$ path if $i_0=i$ and $i_t=j$. $P$ is called a {\em cycle} if $i_0=i_t$. $P$ is an {\em elementary cycle}, if, further, $i_k\neq i_l$ for all $k,l\in\{1,\ldots,t-1\}$.
Recall that $A=\left( a_{ij}\right) \in \R_+^{n\times n}$ is irreducible if ${\mathcal G}(A)$ is trivial or for any $i,j\in \{1,\ldots, n\}$ there is an $i\;-\;j$ path. Otherwise $A$ is reducible.
Notation $A^{\times k}$ will stand for the usual $k$th power of a nonnegative matrix.
\subsection{Max algebra} \label{ss:max}
By max algebra we understand the set of nonnegative numbers $\R_+$ where the role of addition is played by taking maximum of two numbers: $a\oplus b:=\max(a,b)$, and the multiplication is as in the usual arithmetics. This is carried over to matrices and vectors like in the usual linear algebra so that for two matrices $A=(a_{ij})$ and $B=(b_{ij})$ of appropriate sizes, $(A\oplus B)_{ij}=a_{ij}\oplus b_{ij}$ and $(A\otimes B)_{ij}=\bigoplus_k a_{ik}b_{kj}$.
Notation $A^{\otimes k}$ will stand for the $k$th max-algebraic power.
In max algebra, we have the following analogue of a convex cone. A set $V\subseteq\Rp^n$ is called a {\em max cone} if 1) $\alpha v\in V$ for all $v\in V$ and $\alpha\in\R_+$, 2) $u\oplus v\in V$ for $u,v\in V$. Max cones are also known as idempotent semimodules~\cite{KM:97,LM-98}. A max cone $V$ is said to be {\em generated} by $S\subseteq\Rp^n$ if each $v\in V$ can be represented as a max combination $v=\bigoplus_{x\in S} \alpha_x x$ where only finitely many (nonnegative) $\alpha_x$ are different from zero. When $V$ is generated (we also say ``spanned'') by $S$, this is denoted $V=\spann_{\oplus}(S)$. When $V$ is generated by the columns of a matrix $A$, this is denoted $V=\spann_{\oplus}(A)$. This cone is closed with respect to the usual Euclidean topology~\cite{BSS}.
A vector $z$ in a max cone $V\subseteq\Rp^n$ is called an {\em extremal} if $z=u\oplus v$ and $u,v\in V$ imply $z=u$ or $z=v$. Any finitely generated max cone is generated by its extremals, see Wagneur~\cite{Wag-91} and~\cite{BSS,GK-07} for recent extensions.
The {\em maximum cycle geometric mean} of~$A$ is defined by \begin{equation}\label{mcgm} \lambda(A)=\max\{ w(C)^{1/l(C)}\colon C \text{ is a cycle in }{\mathcal G}(A)\} \enspace. \end{equation}
The cycles with the cycle geometric mean equal to $\lambda(A)$ are called {\em critical}, and the nodes and the edges of ${\mathcal G}(A)$ that belong to critical cycles are called {\em critical}. The set of critical nodes is denoted by $N_c(A)$, the set of critical edges by $E_c(A)$, and these nodes and edges give rise to the {\em critical graph} of $A$, denoted by ${\mathcal C}(A) = (N_c(A), E_c(A))$.
A maximal strongly connected subgraph of ${\mathcal C}(A)$ is called a strongly connected component of ${\mathcal C}(A)$. Observe that ${\mathcal C}(A)$, in general, consists of several nontrivial strongly connected components, and that it never has any edges connecting different strongly connected components.
\if{ The {\em critical graph} of $A$, denoted by ${\mathcal C}(A)$, consists of all nodes and edges belonging to the cycles which attain the maximum in~\eqref{mcgm}. The set of such nodes will be called {\em critical} and denoted $N_c$; the set of such edges will be called {\em critical} and denoted $E_c$. Observe that the critical graph consists of several strongly connected subgraphs of ${\mathcal G}(A)$. Maximal such subgraphs are the {\em strongly connected components} of ${\mathcal C}(A)$, and there are no critical edges connecting different strongly connected components of ${\mathcal C}(A)$. }\fi
If for $A\in\Rp^{n\times n}$ we have $A\otimes x=\rho x$ with $\rho\in\R_+$ and a nonzero $x\in\Rp^n$, then $\rho$ is a {\em max(-algebraic) eigenvalue} and $x$ is a {\em max(-algebraic) eigenvector} associated with $\rho$. The set of max eigenvectors $x$ associated with~$\rho$, with the zero vector adjoined to it, is a max cone. It is denoted by $V_{\oplus}(A,\rho)$.
An irreducible $A\in\Rp^{n\times n}$ has a unique max-algebraic eigenvalue equal to $\lambda(A)$~\cite{BCOQ,But:10,CG:79,HOW:05}. In general $A$ may have several max eigenvalues, and the greatest of them equals $\lambda(A)$. The greatest max eigenvalue will also be denoted by $\rho^{\oplus}(A)$ (thus $\rho^{\oplus}(A)=\lambda(A)$), and called the {\em principal max eigenvalue} of $A$. In the irreducible case, the unique max eigenvalue $\rho^{\oplus}(A)=\lambda(A)$ is also called the {\em max(-algebraic) Perron root}. When max algebra and nonnegative algebra are considered simultaneously (e.g., Section~\ref{s:key}), the principal eigenvalue is denoted by $\rho(A)$.
Unlike in nonnegative algebra, there is an explicit description of $V_{\oplus}(A,\rho^{\oplus}(A))$, see Theorem~\ref{t:FVmaxalg}. This description uses the {\em Kleene star} \begin{equation} \label{def:kleenestar} A^*=I\oplus A\oplus A^{\otimes 2}\oplus A^{\otimes 3}\oplus\ldots. \end{equation}
Series~\eqref{def:kleenestar} converges if and only if $\rho^{\oplus}(A)\leq 1$, in which case $A^*=I\oplus A\oplus\ldots\oplus A^{\otimes(n-1)}$~\cite{BCOQ,But:10,HOW:05}. Note that if $\rho^{\oplus}(A)\neq 0$, then $\rho^{\oplus}(A/\rho^{\oplus}(A))=1$, hence $(A/\rho^{\oplus}(A))^*$ always converges.
The {\em path interpretation} of max-algebraic matrix powers $A^{\otimes l}$ is that each entry $a^{\otimes l}_{ij}$ is equal to the greatest weight of $i\;-\;j$ path with length $l$. Consequently, for $i\neq j$, the entry $a^*_{ij}$ of $A^*$ is equal to the greatest weight of an $i\;-\;j$ path (with no length restrictions).
\subsection{Cyclicity and periodicity} \label{ss:cycl} Consider a nontrivial strongly connected graph ${\mathcal G}$ (that is, a strongly connected graph with at least one node and one edge). Define its {\em cyclicity} $\sigma$ as the gcd of the lengths of all elementary cycles. It is known that for any vertices $i,j$ there exists a number $l$ such that $l(P)\equiv l(\text{mod}\ \sigma)$
for all $i\;-\;j$ paths $P$.
When the length of an $i\;-\;j$ path is a multiple of $\sigma$ (and hence we have the same for all $j\;-\;i$ paths), $i$ and $j$ are said to belong to the same {\em cyclic class}. When the length of this path is $1$ modulo $\sigma$ (in other words, if $l(P)-1$ is a multiple of $\sigma$), the cyclic class of $i$ (resp., of $j$) is {\em previous} (resp., {\em next}) with respect to the class of $j$ (resp., of $i$).
See~\cite[Chapter 8]{But:10} and~\cite{BR,Ser-09,Ser-11<attr>} for more information. Cyclic classes are also known as {\em components of imprimitivity}~\cite{BR}.
The cyclicity of a trivial graph is defined to be $1$, and the unique node of a trivial graph is defined to be its only cyclic class.
We define the cyclicity of a (general) graph containing several strongly connected components to be the lcm of the cyclicities of the components.
For a graph ${\mathcal G}=(N,E)$ with $N=\{1,\ldots,n\}$, define the {\em associated matrix} $A=(a_{ij})\in\{0,1\}^{n\times n}$ by $ a_{ij}=1\Leftrightarrow (i,j)\in E. $ This is a matrix over the Boolean semiring $\mathbb{B}:=\{0,1\}$, where addition is the disjunction and multiplication is the conjunction operation. This semiring is a subsemiring of max algebra, so that it is possible to consider the associated matrix as a matrix in max algebra whose entries are either $0$ or $1$.
For a graph ${\mathcal G}$ and any $k\geq 1$, define ${\mathcal G}^k$ as a graph that has the same vertex set as ${\mathcal G}$ and $(i,j)$ is an edge of ${\mathcal G}^k$ if and only if there is a path of length $k$ on ${\mathcal G}$ connecting $i$ to $j$. Thus, if a Boolean matrix $A$ is associated with ${\mathcal G}$, then the Boolean matrix power $A^{\otimes k}$ is associated with ${\mathcal G}^k$. Powers of Boolean matrices (over the Boolean semiring) are a topic of independent interest, see~Brualdi-Ryser~\cite{BR}, Kim~\cite{Kim}. We will need the following observation.
\begin{theorem}[cf. {\cite[Theorem 3.4.5]{BR}}] \label{t:brualdi} Let ${\mathcal G}$ be a strongly connected graph with cyclicity $\sigma$. \begin{itemize} \item[{\rm (i)}] ${\mathcal G}^k$ consists of gcd $(k,\sigma)$ nontrivial strongly connected components not accessing each other. If ${\mathcal G}$ is nontrivial, then so are all the components of ${\mathcal G}^k$.
\item[{\rm (ii)}] The node set of each component of ${\mathcal G}^k$ consists of $\sigma/$(gcd$(k,\sigma))$ cyclic classes of ${\mathcal G}$. \end{itemize} \end{theorem}
\begin{corollary} \label{c:div-boolean} Let ${\mathcal G}$ be a strongly connected graph with cyclicity $\sigma$, and let $k,l\geq 1$. Then gcd$(k,\sigma)$ divides gcd$(l,\sigma)$ if and only if ${\mathcal G}^k$ and ${\mathcal G}^l$ are such that the node set of every component of ${\mathcal G}^l$ is contained in the node set of a component of ${\mathcal G}^k$. \end{corollary} \begin{proof} Assume that ${\mathcal G}$ is nontrivial.\\
``If''. Since the node set of each component of ${\mathcal G}^k$ consists of $\sigma/$gcd$(k,\sigma)$ cyclic classes of ${\mathcal G}$ and is the disjoint union of the node sets of certain components of ${\mathcal G}^l$, and the node set of each component of ${\mathcal G}^l$ consists of $\sigma/$gcd$(l,\sigma)$ cyclic classes of ${\mathcal G}$, it follows that the node set of each component of ${\mathcal G}^k$ consists of $\frac{\sigma}{\text{gcd}(k,\sigma)}/\frac{\sigma}{\text{gcd}(l,\sigma)}= \frac{\text{gcd}(l,\sigma)}{\text{gcd}(k,\sigma)}$ components of ${\mathcal G}^l$. Therefore, gcd$(k,\sigma)$ divides gcd$(l,\sigma)$.
``Only if.'' Observe that the node sets of the compopnents ${\mathcal G}^k$ and ${\mathcal G}^{\text{gcd}(k,\sigma)}$ (or ${\mathcal G}^l$ and ${\mathcal G}^{\text{gcd}(l,\sigma)}$) are the same: since gcd$(k,\sigma)$ divides $k$, each component of ${\mathcal G}^{\text{gcd}(k,\sigma)}$ splits into several components of ${\mathcal G}^k$, but the total number of components is the same (as gcd$($gcd$(k,\sigma),\sigma)=$gcd$(k,\sigma)$), hence their node sets are the same. The claim follows since the node set of each component of ${\mathcal G}^{\text{gcd}(k,\sigma)}$ splits into several components of ${\mathcal G}^{\text{gcd}(l,\sigma)}$. \end{proof}
Let us formally introduce the definitions related to periodicity and ultimate periodicity of sequences (whose elements are of arbitrary nature). A sequence $\{\Omega_k\}_{k\geq 1}$ is called {\em periodic} if there exists an integer $p$ such that $\Omega_{k+p}$ is identical with $\Omega_k$ for all $k$. The least such $p$ is called the {\em period} of $\{\Omega_k\}_{k\geq 1}$. A sequence $\{\Omega_k\}_{k\geq 1}$ is called {\em ultimately periodic} if the sequence $\{\Omega_k\}_{k\geq T}$ is periodic for some $T\geq 1$. The least such $T$ is called the {\em periodicity threshold} of $\{\Omega_k\}_{k\geq 1}$.
The following observation is crucial in the theory of Boolean matrix powers.
\begin{theorem}[Boolean Cyclicity~\cite{Kim}] \label{t:BoolCycl} Let ${\mathcal G}$ be a strongly connected graph on $n$ nodes, with cyclicity $\sigma$. \begin{itemize} \item [{\rm (i)}] The sequence $\{{\mathcal G}^k\}_{k\geq 1}$ is ultimately periodic with the period $\sigma$. The periodicity threshold, denoted by $T({\mathcal G})$, does not exceed $(n-1)^2+1$. \item[{\rm (ii)}] If ${\mathcal G}$ is nontrivial, then for $k\geq T({\mathcal G})$ and a multiple of $\sigma$, ${\mathcal G}^k$ consists of $\sigma$ complete subgraphs not accessing each other. \end{itemize} \end{theorem} For brevity, we will refer to $T({\mathcal G})$ as the periodicity threshold of ${\mathcal G}$.
We have the following two max-algebraic extensions of Theorem~\ref{t:BoolCycl}.
\begin{theorem}[Cyclicity Theorem, Cohen~et~al.~\cite{CDQV-83}] \label{t:Cycl1} Let $A\in\Rp^{n\times n}$ be irreducible, let $\sigma$ be the cyclicity of ${\mathcal C}(A)$ and $\rho:=\rho^{\oplus}(A)$. Then the sequence $\{(A/\rho)^{\otimes k}\}_{k\geq 1}$ is ultimately periodic with period $\sigma$. \end{theorem}
\begin{theorem}[Cyclicity of Critical Part, Nachtigall~\cite{Nacht}] \label{t:nacht} Let $A\in\Rp^{n\times n}$, $\sigma$ be the cyclicity of ${\mathcal C}(A)$ and $\rho:=\rho^{\oplus}(A)$. Then the sequences $\{(A/\rho)^{\otimes k}_{i\cdot}\}_{k\geq 1}$ and $\{(A/\rho)^{\otimes k}_{\cdot i}\}_{k\geq 1}$, for $i\in N_c(A)$, are ultimately periodic with period $\sigma$. The greatest of their periodicity thresholds, denoted by $T_c(A)$, does not exceed $n^2$. \end{theorem}
\if{ The least number $T$ (resp. $T_c$) satisfying the condition of Theorem~\ref{t:Cycl1} (resp. Theorem~\ref{t:nacht}) is denoted by $T(A)$ (resp. $T_c(A)$), and called the {\em cyclicity threshold} (resp. the {\em critical cyclicity threshold}) of $A$. }\fi
Theorem~\ref{t:Cycl1} is standard~\cite{BCOQ,But:10,HOW:05}, and Theorem~\ref{t:nacht} can also be found as~\cite[Theorem 8.3.6]{But:10}. Here $A_{i\cdot}$ (resp. $A_{\cdot i}$) denote the $i$th row (resp. the $i$th column) of $A$.
When the sequence $\{(A/\rho)^{\otimes k}\}_{k\geq 1}$ (resp. the sequences $\{(A/\rho)^{\otimes k}_{i\cdot}\}_{k\geq 1}$,\\ $\{(A/\rho)^{\otimes k}_{\cdot i}\}_{k\geq 1}$) are ultimately periodic, we also say that the sequence $\{A^{\otimes k}\}_{k\geq 1}$ (resp. $\{A^{\otimes k}_{i\cdot}\}_{k\geq 1}$, $\{A^{\otimes k}_{\cdot i}\}_{k\geq 1}$) is {\em ultimately periodic with growth rate $\rho$}.
Let us conclude with a well-known number-theoretic result concerning the coin problem of Frobenius, which we see as basic for both Boolean and max-algebraic cyclicity.
\begin{lemma}[e.g.,{\cite[Lemma 3.4.2]{BR}}] \label{l:schur} Let $n_1,\ldots, n_m$ be integers such that\\ gcd$(n_1,\ldots,n_m)=k$. Then there exists a number $T$ such that for all integers $l$ with $kl\geq T$, we have $kl=t_1n_1+\ldots +t_mn_m$ for some $t_1,\ldots,t_m\geq 0$. \end{lemma}
\subsection{Diagonal similarity and visualization} \label{ss:viz}
For any $x\in\Rp^n$, we can define $X=\operatorname{diag}(x)$ as the {\em diagonal matrix} whose diagonal entries are equal to the corresponding entries of $x$, and whose off-diagonal entries are zero. If $x$ does not have zero components, the diagonal similarity scaling $A\mapsto X^{-1}AX$ does not change the weights of cycles and eigenvalues (both nonnegative and max); if $z$ is an eigenvector of $X^{-1}AX$ then $Xz$ is an eigenvector of $A$ with the same eigenvalue. This scaling does not change the critical graph ${\mathcal C}(A)=(N_c(A),E_c(A))$. Observe that $(X^{-1}AX)^{\otimes k}=X^{-1}A^{\otimes k}X$, also showing that the periodicity thresholds of max-algebraic matrix powers (Theorems~\ref{t:Cycl1} and~\ref{t:nacht})
do not change after scaling. Of course, we also have $(X^{-1}AX)^{\times k}=X^{-1}A^{\times k}X$ in nonnegative algebra. The technique of nonnegative scaling can be traced back to the works of Fiedler-Pt\'ak~\cite{FP-67}.
When working with the max-algebraic matrix powers, it is often convenient to ``visualize'' the powers of the critical graph. Let $A$ have $\lambda(A)=1$. A diagonal similarity scalling $A\mapsto X^{-1}AX$ is called a {\em strict visualization scaling}~\cite{But:10,SSB} if the matrix $B=X^{-1}AX$ has $b_{ij}\leq 1$, and moreover, $b_{ij}=1$ if and only if $(i,j)\in E_c(A)(=E_c(B))$. Any matrix $B$ satisfying these properties is called {\em strictly visualized}.
\begin{theorem}[Strict Visualization~\cite{But:10,SSB}] \label{t:strictvis} For each $A\in\Rp^{n\times n}$ with $\rho^{\oplus}(A)=1$ {\rm (}that is, $\lambda(A)=1${\rm )}, there exists a strict visualization scaling. \end{theorem}
If $A=(a_{ij})$ has all entries $a_{ij}\leq 1$, then we define the Boolean matrix $A^{[1]}$ with entries \begin{equation} \label{def:A1} a_{ij}^{[1]}= \begin{cases} 1, &\text{if $a_{ij}=1$},\\ 0, &\text{if $a_{ij}<1$}. \end{cases} \end{equation} If $A$ has all entries $a_{ij}\leq 1$ then \begin{equation} \label{e:unimatrix} (A^{\otimes k})^{[1]}=(A^{[1]})^{\otimes k}. \end{equation} Similarly if a vector $x\in\Rp^n$ has $x_i\leq 1$, we define $x^{[1]}$ having $x^{[1]}_i=1$ if $x_i=1$ and $x^{[1]}_i=0$ otherwise. Obviously if $A$ and $x$ have all entries not exceeding $1$ then $(A\otimes x)^{[1]}=A^{[1]}\otimes x^{[1]}$.
If $A$ is strictly visualized, then $a_{ij}^{[1]}=1$ if and only if $(i,j)$ is a critical edge of ${\mathcal G}(A)$. Thus $A^{[1]}$ can be treated as the associated matrix of ${\mathcal C}(A)$ (disregarding the formal difference in dimension). We now show that ${\mathcal C}(A^{\otimes k})={\mathcal C}(A)^k$ and that any power of a strictly visualized matrix is strictly visualized.
\begin{lemma}[cf.~\cite{CGB-07}, {\cite[Prop.~3.3]{Ser-09}}] \label{l:CAk} Let $A\in\Rp^{n\times n}$ and $k\geq 1$. \begin{itemize} \item[{\rm (i)}] ${\mathcal C}(A)^k={\mathcal C}(A^{\otimes k})$. \item[{\rm (ii)}] If $A$ is strictly visualized, then so is $A^{\otimes k}$. \end{itemize} \end{lemma} \begin{proof} Using Theorem~\ref{t:strictvis}, we can assume without loss of generality that $A$ is strictly visualized. Also note that both in ${\mathcal C}(A^{\otimes k})$ and in ${\mathcal C}(A)^k$, each node has ingoing and outgoing edges, hence for part (i) it suffices to prove that the two graphs have the same set of edges.
Applying Theorem~\ref{t:brualdi} (i) to every component of ${\mathcal C}(A)$, we obtain that ${\mathcal C}(A)^k$ also consists of several isolated nontrivial strongly connected graphs. In particular, each edge of ${\mathcal C}(A)^k$ lies on a cycle, so ${\mathcal C}(A)^k$ contains cycles. Observe that ${\mathcal G}(A^{\otimes k})$ does not have edges with weight greater than $1$, while all edges of ${\mathcal C}(A)^k$ have weight $1$, hence all cycles of ${\mathcal C}(A)^k$ have weight $1$. As ${\mathcal C}(A)^k$ is a subgraph of ${\mathcal G}(A^{\otimes k})$, this shows that $\rho^{\oplus}(A^{\otimes k})=\lambda(A^{\otimes k})=1$ and that all cycles of ${\mathcal C}(A)^k$ are critical cycles of ${\mathcal G}(A^{\otimes k})$. Since each edge of ${\mathcal C}(A)^k$ lies on a critical cycle, all edges of ${\mathcal C}(A)^k$ are critical edges of ${\mathcal G}(A^{\otimes k})$.
${\mathcal G}(A^{\otimes k})$ does not have edges with weight greater than $1$, hence every edge of ${\mathcal C}(A^{\otimes k})$ has weight $1$. Equation~\eqref{e:unimatrix} implies that if $a_{ij}^{\otimes k}=1$ then there is a path from $i$ to $j$ composed of the edges with weight $1$. Since $A$ is strictly visualized, such edges are critical. This shows that if $a_{ij}^{\otimes k}=1$ and in particular if $(i,j)$ is an edge of ${\mathcal C}(A^{\otimes k})$, then $(i,j)$ is an edge of ${\mathcal C}(A)^k$. Hence $A^{\otimes k}$ is strictly visualized, and all edges of ${\mathcal C}(A^{\otimes k})$ are edges of ${\mathcal C}(A)^k$.
Thus ${\mathcal C}(A^{\otimes k})$ and ${\mathcal C}(A)^k$ have the same set of edges, so ${\mathcal C}(A^{\otimes k})={\mathcal C}(A)^k$ (and we also showed that $A^{\otimes k}$ is strictly visualized).
\end{proof}
Let $T({\mathcal C}(A))$ be the greatest periodicity threshold of the strongly connected components of ${\mathcal C}(A)$. The following corollary of Lemma~\ref{l:CAk} will be required in Section~\ref{s:eigencones}.
\begin{corollary} \label{TcaTcrit} Let $A\in\Rp^{n\times n}$. Then $T_c(A)\geq T({\mathcal C}(A))$.
\end{corollary} \if{ \begin{proof} Let $\sigma$ be the cyclicity of ${\mathcal C}(A)$. Assume that $A$ is strictly visualized. We are going to show that if $(A^k)_{\cdot i}=A^{k+\sigma}_{\cdot i}$ for all $i\in N_c(A)$ then $({\mathcal C}(A))^k=({\mathcal C}(A))^{k+\sigma}$. Indeed, Lemma~\ref{l:CAk} implies that $({\mathcal C}(A))^k=({\mathcal C}(A))^{k+\sigma}$ is equivalent to $(A^[1])^{\otimes k}= (A^[1])^{\otimes (k+\sigma)}$ and that all nonzero entries of these matrices are in the critical columns and rows. As $(A^k)_{\cdot i}=A^{k+\sigma}_{\cdot i}$ for all $i\in N_c(A)$ is sufficient for $(A^[1])^{\otimes k}= (A^[1])^{\otimes (k+\sigma)}$, the claim follows. \end{proof} }\fi
\subsection{Frobenius normal form}
Every matrix $A=(a_{ij})\in \R_+^{n\times n}$ can be transformed by simultaneous permutations of the rows and columns in almost linear time to a {\em Frobenius normal form}~\cite{BP,BR} \begin{equation} \left( \begin{array}{cccc} A_{11} & 0 & ... & 0 \\ A_{21} & A_{22} & ... & 0 \\ ... & ... & A_{\mu\mu} & ... \\ A_{r1} & A_{r2} & ... & A_{rr} \end{array} \right) , \label{fnf} \end{equation} where $A_{11},...,A_{rr}$ are irreducible square submatrices of $A$. They correspond to the sets of nodes $N_1,\ldots,N_r$ of the strongly connected components of ${\mathcal G}(A)$. Note that in~\eqref{fnf} an edge from a node of $N_{\mu}$ to a node of $N_{\nu}$ in ${\mathcal G}(A)$ may exist only if $\mu\geq \nu.$
Generally, $A_{KL}$ denotes the submatrix of $A$ extracted from the rows with indices in $K\subseteq \{1,\ldots,n\}$ and columns with indices in $L\subseteq \{1,\ldots,n\}$, and $A_{\mu\nu}$ is a shorthand for $A_{N_{\mu}N_{\nu}}$. Accordingly, the subvector $x_{N_{\mu}}$ of $x$ with indices in $N_{\mu}$ will be written as $x_{\mu}$.
If $A$ is in the Frobenius Normal Form \eqref{fnf} then the {\em reduced graph}, denoted by $R(A)$, is the (di)graph whose nodes correspond to $N_{\mu}$, for $\mu=1,\ldots,r$, and the set of arcs is $ \{(\mu,\nu);(\exists k\in N_{\mu})(\exists \ell \in N_{\nu})a_{k\ell }>0\}. $ In max algebra and in nonnegative algebra, the nodes of $R(A)$ are {\em marked} by the corresponding eigenvalues (Perron roots), denoted by $\rho^{\oplus}_{\mu}:=\rho^{\oplus}(A_{\mu\mu})$ (max algebra), $\rho^+_{\mu}:=\rho^+(A_{\mu\mu})$ (nonnegative algebra), and by $\rho_{\mu}$ when both algebras are considered simultaneously.
By a {\em class} of $A$ we mean a node $\mu$ of the reduced graph $R(A)$. It will be convenient to attribute to class $\mu$ the node set and the edge set of ${\mathcal G}(A_{\mu\mu})$, as well as the cyclicity and other parameters, that is, we will say ``nodes of $\mu$'', ``edges of $\mu$'', ``cyclicity of $\mu$'', etc.\footnote{The sets $N_{\mu}$ are also called classes, in the literature. To avoid the confusion, we do not follow this in the present paper.}
A class $\mu$ is trivial if $A_{\mu\mu}$ is the $1\times 1$ zero matrix. Class $\mu$ {\em accesses} class $\nu$, denoted $\mu\to\nu$, if $\mu=\nu$ or if there exists a $\mu\;-\;\nu$ path in $R(A)$. A class is called {\em initial}, resp. {\em final}, if it is not accessed by, resp. if it does not access, any other class. Node $i$ of ${\mathcal G}(A)$ accesses class $\nu$, denoted by $i\to\nu$, if $i$ belongs to a class $\mu$ such that $\mu\to\nu$.
Note that simultaneous permutations of the rows and columns of $A$ are equivalent to calculating $P^{-1}AP,$ where $P$ is a permutation matrix. Such transformations do not change the eigenvalues, and the eigenvectors before and after such a transformation may only differ by the order of their components. Hence we will assume without loss of generality that $A$ is in Frobenius normal form~(\ref{fnf}). Note that a permutation bringing matrix to this form is (relatively) easy to find~\cite{BR}. We will refer to the transformation $A\mapsto P^{-1}AP$ as {\em permutational similarity}.
\subsection{Elements of the Perron-Frobenius theory} \label{ss:pfelts}
In this section we recall the spectral theory of reducible matrices in max algebra and in nonnegative linear algebra. All results are standard: the nonnegative part goes back to Frobenius~\cite{Fro-1912}, Sect.~11, and the max-algebraic counterpart is due to Gaubert~\cite{Gau:92}, Ch.~IV (also see~\cite{But:10} for other references).
A class $\nu$ of $A$ is called a {\em spectral class} of $A$ associated with eigenvalue $\rho\neq 0$, or sometimes $(A,\rho)$-spectral class for short, if \begin{equation} \label{e:speclass} \begin{split} &\rho^{\oplus}_{\nu}=\rho,\ \text{and}\ \mu\to\nu\ \text{implies}\ \rho^{\oplus}_{\mu}\leq\rho^{\oplus}_{\nu}\ \text{(max algebra)},\\ &\rho^+_{\nu}=\rho,\ \text{and}\ \mu\to\nu,\mu\neq\nu\ \text{implies}\ \rho^{+}_{\mu}<\rho^{+}_{\nu}\ \text{(nonnegative algebra)}. \end{split} \end{equation} In both algebras, note that there may be several spectral classes associated with the same eigenvalue.
In nonnegative algebra, spectral classes are called distinguished classes~\cite{Sch-86}, and there are also semi-distinguished classes associated with distinguished generalized eigenvectors of order two or more~\cite{TS-00}. However, these vectors are not contained in the core\footnote{For a polyhedral cone, the core of the cone-preserving map does not contain generalized eigenvectors of order two or more~{\cite[Corollary 4.3]{TS-94}}.}. Also, no suitable max-algebraic analogue of generalized eigenvectors is known to us.
If all classes of $A$ consist of just one element, then the nonnegative and max-algebraic Perron roots are the same. In this case, the spectral classes in nonnegative algebra are also spectral in max algebra. However, in general this is not so. In particular, for a nonnegative matrix $A$, a cycle of ${\mathcal G}(A)$ attaining the maximum cycle geometric mean $\rho^{\oplus}(A)=\lambda(A)$ need not lie in a strongly connected component corresponding to a class with spectral radius $\rho^+(A)$. This is because, if $A_1$, $A_2$ are irreducible nonnegative matrices such that $\rho^+(A_1)<\rho^+(A_2)$, then we need not have $\rho^{\oplus}(A_1)<\rho^{\oplus}(A_2)$. For example, let $A$ be the $3\times 3$ matrix of all $1$'s, and let $B(\epsilon)=(3/2,\epsilon,\epsilon)^T(3/2,\epsilon,\epsilon)$. Then $\rho^+(A)=3$, $\rho^+(B(\epsilon))=9/4+2\epsilon^2$, so $\rho^+(A)>\rho^+(B(\epsilon))$ for sufficiently small $\epsilon>0$, but $\rho^{\oplus}(B(\epsilon))=9/4>1=\rho^{\oplus}(A)$.
Denote by $\Lambda_+(A)$, resp. $\Lambda_{\oplus}(A)$, the set of {\bf nonzero} eigenvalues of $A\in\Rp^{n\times n}$ in nonnegative linear algebra, resp. in max algebra. It will be denoted by $\Lambda(A)$ when both algebras are considered simultaneously, as in the following standard description.
\begin{theorem}[{\cite[Th.~4.5.4]{But:10}}, {\cite[Th.~3.7]{Sch-86}}] \label{t:spectrum} Let $A\in\Rp^{n\times n}$. Then $\Lambda(A)=\{\rho_{\nu}\neq 0\colon \nu\ \text{is spectral}\}$.
\end{theorem}
Theorem~\ref{t:spectrum} encodes the following {\bf two} statements: \begin{equation} \label{e:spectrum2} \Lambda_{\oplus}(A)=\{\rho^{\oplus}_{\nu}\neq 0\colon \nu\ \text{is spectral}\},\quad \Lambda_+(A)=\{\rho^+_{\nu}\neq 0\colon \nu\ \text{is spectral}\}, \end{equation} where the notion of spectral class is defined in two different ways by~\eqref{e:speclass}, in two algebras.
In both algebras, for each $\rho\in\Lambda(A)$ define \begin{equation} \label{amrho} \begin{split} &A_{\rho}:=\rho^{-1} \begin{pmatrix} 0 & 0\\ 0 & A_{\mrho\mrho} \end{pmatrix},\ \text{where}\\ & M_{\rho}:=\{i\colon i\to\nu,\; \nu\ \text{is $(A,\rho)$-spectral}\}\,\enspace . \end{split} \end{equation}
By ``$\nu$ is $(A,\rho)$-spectral'' we mean that $\nu$ is a spectral class of $A$ with $\rho_{\nu}=\rho$. The next proposition, holding both in max algebra and in nonnegative algebra, allows us to reduce the case of arbitrary eigencone to the case of principal eigencone. Here we assume that $A$ is in Frobenius normal form.
\begin{proposition}[\cite{But:10,Gau:92,Sch-86}] \label{p:vamrho} For $A\in\Rp^{n\times n}$ and each $\rho\in\Lambda(A)$, we have $V(A,\rho)=V(A_{\rho},1)$, where $1$ is the principal eigenvalue of $A_{\rho}$. \end{proposition}
For a parallel description of extremals of eigencones\footnote{In nonnegative algebra, \cite[Th. 3.7]{Sch-86} immediately describes both spectral classes and eigencones associated with any eigenvalue.
However, we prefer to split the formulation, following the exposition of~\cite{But:10}. An alternative simultaneous exposition of spectral theory in both algebras can be found in~\cite{KSS}.} in both algebras see Section~\ref{ss:FV}.
In max algebra, using Proposition~\ref{p:vamrho}, we define the {\em critical graph associated with} $\rho\in\Lambda_{\oplus}(A)$ as the critical graph of $A_{\rho}$. By a {\em critical component of $A$} we mean a strongly connected component of the critical graph associated with some $\rho\in\Lambda_{\oplus}(A)$.
In max algebra, the role of spectral classes of $A$ is rather played by these critical components, which will be (in analogy with classes of Frobenius normal form) denoted by $\Tilde{\mu}$, with the node set $N_{\Tilde{\mu}}$. See Section~\ref{ss:critcomp}.
\section{Notation table and key results} \label{s:key}
The following notation table shows how various objects are denoted in nonnegative algebra, max algebra and when both algebras are considered simultaneously.
\begin{equation*} \begin{array}{cccc} & \text{Nonnegative} & \text{Max} & \text{Both}\\ \text{Sum} & \sum & \bigoplus & \sum\\ \text{Matrix power} & A^{\times t} & A^{\otimes t} & A^t\\ \text{Column span} & \spann_+(A) & \spann_{\oplus}(A) & \operatorname{span}(A)\\ \text{Perron root} & \rho^+_{\mu} & \rho^{\oplus}_{\nu} & \rho_{\mu}\\ \text{Spectrum (excl. $0$)} & \Lambda_+(A) & \Lambda_{\oplus}(A) & \Lambda(A)\\ \text{Eigencone} & V_+(A,\rho^+) & V_{\oplus}(A,\rho^{\oplus}) & V(A,\rho)\\ \text{Sum of eigencones} & V_+^{\Sigma}(A) & V_{\oplus}^{\Sigma}(A) & V^{\Sigma}(A)\\ \text{Core} & \core_+(A) & \core_{\oplus}(A) & \operatorname{core}(A)
\end{array} \end{equation*} In the case of max algebra, we also have the critical graph ${\mathcal C}(A)$ (with related concepts and notation), not used in nonnegative algebra.
The core and the sum of eigencones appearing in the table have not been formally introduced. These are the two central notions of this paper, and we now introduce them for both algebras simultaneously.
The {\em core} of a nonnegative matrix $A$ is defined as the intersection of the column spans (in other words, images) of its powers: \begin{equation} \label{def:core} \operatorname{core}(A):=\cap_{i=1}^{\infty} \operatorname{span}(A^i). \end{equation}
The ({\em Minkowski}) {\em sum of eigencones} of a nonnegative matrix $A$ is the cone consisting of all sums of vectors in all $V(A,\rho)$: \begin{equation} \label{va-def} V^{\Sigma}(A):=\sum_{\rho\in\Lambda(A)} V(A,\rho) \end{equation} If $\Lambda(A)=\emptyset$, which happens when $\rho(A)=0$, then we assume that the sum on the right-hand side is $\{0\}$.
The following notations can be seen as the ``global'' definition of cyclicity in nonnegative algebra and in max algebra.
\begin{itemize} \item[1.] Let $\sigma_{\rho}$ be the the lcm of all cyclicities of spectral classes associated with $\rho\in\Lambda_+(A)$ ({\bf nonnegative algebra}), or the cyclicity of critical graph associated with $\rho\in\Lambda_{\oplus}(A)$ ({\bf max algebra}). \item[2.] Let $\sigma_{\Lambda}$ be the lcm of all $\sigma_{\rho}$ where $\rho\in\Lambda(A)$. \end{itemize} The difference between the definitions of $\sigma_{\rho}$ in max algebra and in nonnegative algebra comes from the corresponding versions of Perron-Frobenius theory. In particular, let $A\in\Rp^{n\times n}$ be an irreducible matrix. While in nonnegative algebra the eigencone associated with the Perron root of $A$ is always reduced to a single ray, the number of (appropriately normalized) extremals of the eigencone of $A$ in max algebra is equal to the number of critical components, so that there may be up to $n$ such extremals.
One of the key results of the present paper relates the core with the sum of eigencones. The nonnegative part of this result can be found in Tam-Schneider~\cite[Th.~4.2, part~(iii)]{TS-94}. \begin{maintheorem} \label{t:core} Let $A\in\Rp^{n\times n}$. Then $$\operatorname{core}(A)=\sum_{k\geq 1,\rho\in\Lambda(A)} V(A^k,\rho^k) =V^{\Sigma}(A^{\sigma_{\Lambda}}).$$ \end{maintheorem}
The main part of the proof is given in Section~\ref{s:core}, for both algebras simultaneously. However, this proof takes in ``credit'' some facts, which we will have to show. First of all, we need the equality \begin{equation} \label{e:samespectrum} \Lambda(A^k)=\{\rho^k\colon \rho\in\Lambda(A)\}. \end{equation} This simple relation between $\Lambda(A^k)$ and $\Lambda(A)$, which can be seen as a special case of~\cite[Theorem 3.6(ii)]{KSS}, will be also proved below as Corollary~\ref{c:samespectrum}.
To complete the proof of Main Theorem~\ref{t:core} we also have to study the periodic sequence of eigencones of matrix powers and their sums. On this way we obtain the following key result, both in max and nonnegative algebra.
\begin{maintheorem} \label{t:periodicity} Let $A\in\Rp^{n\times n}$. Then \begin{itemize} \item[{\rm (i)}] $\sigma_{\rho}$, for $\rho\in\Lambda(A)$, is the period of the sequence $\{V(A^k,\rho^k)\}_{k\geq 1}$, and $V(A^k,\rho^k)\subseteq V(A^{\sigma_{\rho}},\rho^{\sigma_{\rho}})$ for all $k\geq 1$; \item[{\rm (ii)}] $\sigma_{\Lambda}$ is the period of the sequence $\{V^{\Sigma}(A^k)\}_{k\geq 1}$, and $V^{\Sigma}(A^k)\subseteq V^{\Sigma}(A^{\sigma_{\Lambda}})$ for all $k\geq 1$. \end{itemize} \end{maintheorem}
Main Theorem~\ref{t:periodicity} is proved in Section~\ref{s:eigencones} as a corollary of Theorems~\ref{t:girls-major} and~\ref{t:girls-minor}, where the inclusion relations between eigencones of matrix powers are studied in detail.
Theorem~\ref{t:tam-schneider}, which gives a detailed description of extremals of both cores, can be also seen as a key result of this paper. However, it is too long to be formulated in advance.
\section{Two cores} \label{s:core}
\subsection{Basics} \label{ss:basics} In this section we investigate the core of a nonnegative matrix defined by~\eqref{def:core}. In the main argument, we consider the cases of max algebra and nonnegative algebra simultaneously.
One of the most elementary and useful properties of the intersection in~\eqref{def:core} is that, actually, \begin{equation} \label{e:monotonic} \operatorname{span}(A)\supseteq\operatorname{span}(A^2)\supseteq\operatorname{span}(A^3)\supseteq\ldots \end{equation}
Generalizing an argument of Pullman~\cite{Pul-71} we will prove that \begin{equation} \label{e:maingoal} \operatorname{core}(A)=\sum_{k\geq 1} V^{\Sigma}(A^k)=\sum_{k\geq 1,\rho\in\Lambda(A)} V(A^k,\rho^k) \end{equation} also in max algebra.
Note that the following inclusion is almost immediate.
\begin{lemma} \label{l:natural} $\sum_{k\geq 1} V^{\Sigma}(A^k)\subseteq\operatorname{core} (A)$. \end{lemma} \begin{proof} $x\in V(A^k,\rho)$ implies that $A^k x=\rho x$ and hence $x=\rho^{-t} A^{kt}x$ for all $t\geq 1$ (using the invertibility of multiplication). Hence $x\in\bigcap_{t\geq 1} \operatorname{span} A^{kt}=\bigcap_{t\geq 1} \operatorname{span}(A^t)$. \end{proof}
So it remains to show the opposite inclusion \begin{equation} \label{e:maingoal2} \operatorname{core}(A)\subseteq\sum_{k\geq 1} V^{\Sigma}(A^k). \end{equation}
Let us first treat the trivial case $\rho(A)=0$, i.e., $\Lambda(A)=\emptyset$. There are only trivial classes in the Frobenius normal form, and ${\mathcal G}(A)$ is acyclic. This implies $A^k=0$ for some $k\geq 1$. In this case $\operatorname{core}(A)=\{0\}$, the sum on the right-hand side is $\{0\}$ by convention,
so~\eqref{e:maingoal} is the trivial "draw" $\{0\}=\{0\}$.
\subsection{Max algebra: cases of ultimate periodicity} \label{ss:maxalg-easy}
In max algebra, unlike the nonnegative algebra, there are wide classes of matrices
for which~\eqref{e:maingoal2} and~\eqref{e:maingoal} follow almost immediately. We list some of them below.\\
${\mathcal S}_1:$ {\em Irreducible matrices}.\\ ${\mathcal S}_2:$ {\em Ultimately periodic matrices.} This is when the sequence $\{A^{\otimes k}\}_{k\geq 1}$ is ultimately periodic with a growth rate $\rho$ (in other words, when the sequence $\{(A/\rho)^{\otimes k}\}_{k\geq 1}$ is ultimately periodic). As shown by Moln\'arov\'a-Pribi\v{s}~\cite{MP-00}, this happens if and only if the Perron roots of all nontrivial classes of $A$ equal $\rho^{\oplus}(A)=\rho$. \\ ${\mathcal S}_3:$ {\em Robust matrices.} For any vector $x\in\Rp^n$ the orbit $\{A^{\otimes k}\otimes x\}_{k\geq 1}$ hits an eigenvector of $A$, meaning that $A^{\otimes T}\otimes x$ is an eigenvector of $A$ for some $T$. This implies that the whole remaining part $\{A^{\otimes k}\otimes x\}_{k\geq T}$ of the orbit (the ``tail'' of the orbit) consists of multiples of that eigenvector $A^{\otimes T}\otimes x$. The notion of robustness was introduced and studied in~\cite{BCG-09}.\\ ${\mathcal S}_4:$ {\em Orbit periodic matrices:} For any vector $x\in\Rp^n$ the orbit $\{A^{\otimes k}\otimes x\}_{k\geq 1}$ hits an eigenvector of $A^{\otimes \sigma_x}$ for some $\sigma_x$, implying that the remaining ``tail'' of the orbit $\{(A^{\otimes k}\otimes x\}_{k\geq 1}$ is periodic with some growth rate. See~\cite[Section 7]{SS-11} for characterization.\\ ${\mathcal S}_5:$ {\em Column periodic matrices.} This is when for all $i$ we have $(A^{\otimes (t+\sigma_i)})_{\cdot i}=\rho_i^{\sigma_i} A^{\otimes t}_{\cdot i}$ for all large enough $t$ and some $\rho_i$ and $\sigma_i$.
Observe that ${\mathcal S}_1\subseteq{\mathcal S}_2\subseteq{\mathcal S}_4\subseteq{\mathcal S}_5$ and ${\mathcal S}_3\subseteq{\mathcal S}_4$. Indeed, ${\mathcal S}_1\subseteq{\mathcal S}_2$ is the Cyclicity Theorem~\ref{t:Cycl1}. For the inclusion ${\mathcal S}_2\subseteq{\mathcal S}_4$ observe that, if $A$ is ultimately periodic then $A^{\otimes(t+\sigma)}=\rho^{\sigma}A^{\otimes t}$ and hence $A^{\otimes (t+\sigma)}\otimes x=\rho^{\sigma} A^{\otimes t}\otimes x$ holds for all $x\in\Rp^n$ and all big enough $t$. Observe that ${\mathcal S}_3$ is a special case of ${\mathcal S}_4$, which is a special case of ${\mathcal S}_5$ since the columns of matrix powers can be considered as orbits of the unit vectors.
To see that~\eqref{e:maingoal2} holds in all these cases, note that in the column periodic case all column sequences $\{A_{\cdot i}^t\}_{t\geq 1}$ end up with periodically repeating eigenvectors of $A^{\otimes \sigma_i}$ or the zero vector, which implies that $\spann_{\oplus}(A^{\otimes t})\subseteq\bigoplus_{k\geq 1} V_{\oplus}^{\Sigma}(A^{\otimes k})\subseteq\core_{\oplus}(A)$ and hence $\spann_{\oplus}(A^{\otimes t})=\core_{\oplus}(A)$ for all large enough $t$. Thus, {\em finite stabilization of the core} occurs in all these classes. A necessary and sufficient condition for this finite stabilization is described in~\cite{BSS-inprep}.
\subsection{Core: a general argument} \label{ss:pullman} The original argument of Pullman~\cite[Section 2]{Pul-71} used the separation of a point from a closed convex cone by an open homogeneous halfspace (that contains the cone and does not contain the point).
In the case of max algebra, Nitica and Singer~\cite{NS-07I} showed that at each point $x\in\Rp^n$ there are at most $n$ maximal max-cones not containing this point. These {\em conical semispaces}, used to separate $x$ from any max cone not containing $x$, turn out to be open. Hence they can be used in the max version of Pullman's argument.
However, for the sake of a simultaneous proof we will exploit the following analytic argument instead of separation. By $B(x,\epsilon)$ we denote the intersection of the open ball centered at $x\in\Rp^n$ of radius $\epsilon$ with $\Rp^n$. In the remaining part of Section~\ref{s:core} we consider {\bf both algebras simultaneously.}
\begin{lemma} \label{l:analytic} Let $x^1,\ldots,x^m\in\Rp^n$ be nonzero and let $z\notin\operatorname{span}(x^1,\ldots,x^m)$. Then there exists $\epsilon>0$ such that $z\notin\operatorname{span}(B(x^1,\epsilon),\ldots,B(x^m,\epsilon))$. \end{lemma} \begin{proof} By contradiction assume that for each $\epsilon$ there exist points $y^i(\epsilon)\in B(x^i,\epsilon)$ and nonnegative scalars $\mu_i(\epsilon)$ such that \begin{equation} \label{e:epscomb} z=\sum_{i=1}^m \mu_i(\epsilon) y^i(\epsilon). \end{equation} Since $y^i(\epsilon)\to x^i$ as $\epsilon\to 0$ and $x^i$ are nonzero, we can assume that $y^i(\epsilon)$ are bounded from below by nonzero vectors $v^i$, and then
$z\geq \sum_{i=1}^m \mu_i(\epsilon) v^i$ for all $\epsilon$,
implying that $\mu_i(\epsilon)$ are uniformly bounded from above. By compactness we can assume that $\mu_i(\epsilon)$ converge to some $\mu_i\in\R_+$, and then~\eqref{e:epscomb} implies by continuity that $z=\sum_{i=1}^m \mu_i x^i$, a contradiction. \end{proof}
\begin{theorem}[{\cite[Theorem~2.1]{Pul-71}}] \label{t:pullman} Assume that $\{K_l\}$ for $l\geq 1$, is a sequence of cones in $\Rp^n$ such that $K_{l+1}\subseteq K_l$ for all $l$, and each of them generated by no more than $k$ nonzero vectors. Then the intersection $K=\cap_{l=1}^{\infty}K_l$ is also generated by no more than $k$ vectors. \end{theorem} \begin{proof}
Let $K_l=\operatorname{span}(y^{l1},\ldots,y^{lk})$ (where some of the vectors
$y^{l1},\ldots,y^{lk}$ may be repeated when $K_l$ is generated by less than $k$ nonzero vectors), and consider the sequences of normalized vectors $\{y^{li}/||y^{li}||\}_{l\geq 1}$ for
$i=1,\ldots,k$, where $||u||:=\max u_i$ (or any other norm). As the set $\{u\colon ||u||=1\}$ is compact, we can find a subsequence $\{l_t\}_{t\geq 1}$ such that for $i=1,\ldots,k$, the sequence
$\{y^{l_ti}/||y^{l_ti}||\}_{t\geq 1}$ converges to a finite vector
$u^i$, which is nonzero since $||u^i||=1$. We will assume that
$||y^{l_t i}||=1$ for all $i$ and $t$.
We now show that $u^1,\ldots,u^k\in K$. Consider any $i=1,\ldots,k$. For each $s$, $y^{l_t i}\in K_s$ for all sufficiently large $t$. As $\{y^{l_ti}\}_{t\geq 1}$ converges to $u^i$ and $K_s$ is closed, we have $u^i\in K_s$. Since this is true for each $s$, we have $u^i\in\cap_{s=1}^{\infty} K_s=K$.
Thus $u^1,\ldots,u^k\in K$, and $\operatorname{span}(u^1,\ldots,u^k)\subseteq K$. We claim that also $K\subseteq \operatorname{span}(u^1,\ldots, u^k)$. Assume to the contrary that there is $z\in K$ that is not in $\operatorname{span}(u^1,\ldots, u^k)$. Then by Lemma~\ref{l:analytic} there exists $\epsilon>0$ such that $z\notin\operatorname{span}(B(u^1,\epsilon),\ldots,B(u^k,\epsilon))$. Since the sequence $\{y^{l_ti}\}_{t\geq 1}$ converges to $u^i$, we have $y^{l_ti}\in B(u^i,\epsilon)$ for $t$ large enough, and $$ \operatorname{span}(y^{l_t1},\ldots,y^{l_tk})\subseteq \operatorname{span}(B(u^1,\epsilon),\ldots,B(u^k,\epsilon)) $$ But $z$ belongs to $K_{l_t}=\operatorname{span}(y^{l_t1},\ldots,y^{l_tk})$ since it belongs to the intersection of all these cones, a contradiction. \end{proof}
Theorem~\ref{t:pullman} applies to the sequence $\{\operatorname{span}(A^t)\}_{t\geq 1}$,
so $\operatorname{core}(A)$ is generated by no more than $n$ vectors.
\begin{proposition}[{\cite[Lemma~2.3]{Pul-71}}] \label{p:surj} The mapping induced by $A$ on its core is a surjection. \end{proposition} \begin{proof} First note that $A$ does induce a mapping on its core. If $z\in\operatorname{core}(A)$ then for each $t$ there exists $x^t$ such that $z=A^tx^t$. Hence $Az=A^{t+1}x^t$, so $Az\in\cap_{t\geq 2}\operatorname{span} A^t=\operatorname{core}(A)$.
Next, let $m$ be such that $A^m$ has the greatest number of zero columns (we assume that $A$ is not nilpotent; recall that a zero column in $A^k$ remains zero in all subsequent powers). If $z=A^tx^t$ for $t\geq m+1$, we also can represent it as $A^{m+1}u^t$, where $u^t:=A^{t-m-1}x^t$. The components of $u^t$ corresponding to the nonzero columns of $A^{m+1}$ are bounded since $A^{m+1}u^t=z$. So we can assume that the sequence of subvectors of $u^{t}$ with these components converges. Then the sequence $y^t:=A^m u^t$ also converges, since the indices of nonzero columns of $A^m$ coincide with those of $A^{m+1}$, which are the indices of the converging subvectors of $u^t$. Let $y$ be the limit of $y^t$. Since $y^s=A^{s-1} x^s$ are in $\operatorname{span}(A^t)$ for all $s>t$, and since $\operatorname{span}(A^t)$ are closed, we obtain $y\in\operatorname{span}(A^t)$ for all $t$. Thus we found $y\in\operatorname{core}(A)$ satisfying $Ay=z$. \end{proof}
Theorem~\ref{t:pullman} and Proposition~\ref{p:surj} show that the core is generated by finitely many vectors in $\Rp^n$ and that the mapping induced by $A$ on its core is ``onto''.
Now we use the fact that a finitely generated cone in the nonnegative orthant (and more generally, closed cone) is generated by its extremals both in nonnegative algebra and in max algebra, see~\cite{BSS,Wag-91}.
\begin{proposition}[{\cite[Theorem 2.2]{Pul-71}}] \label{p:permute} The mapping induced by $A$ on the extremal generators of its core is a permutation (i.e., a bijection). \end{proposition} \begin{proof}
Let $\operatorname{core}(A)=\operatorname{span}(u^1,\ldots,u^k)$ where $u^1,\ldots,u^k$ are extremals of the core. Suppose that $x^j$ is a preimage of $u^j$ in the core, that is, $Ax^j=u^j$ for some $x^j\in\operatorname{core}(A)$, $j=1,\ldots,k$. Then $x^j=\sum_{i=1}^k \alpha_i u^i$ for some nonnegative coefficients $\alpha_1,\ldots,\alpha_k$, and $u^j=\sum_{i=1}^k\alpha_i A u^i$. Since $u^j$ is extremal, it follows that $u^j$ is proportional to $Au^i$ for some $i$. Thus for each $j\in\{1,\ldots,k\}$ there exists an $i\in\{1,\ldots,k\}$ such that $Au^i$ is a positive multiple of $u^j$. But since for each $i\in\{1,\ldots,k\}$ there is at most one $j$ such that $Au^i$ is a positive multiple of $u^j$, it follows that $A$ induces a bijection on the set of extremal generators of its core. \end{proof}
We are now ready to prove~\eqref{e:maingoal} and Main Theorem~\ref{t:core} taking the periodicity of the eigencone sequence (Main Theorem~\ref{t:periodicity}) in ``credit''.
\begin{proof}[Proof of Main Theorem~\ref{t:core}] Proposition~\ref{p:permute} implies that all extremals of $\operatorname{core}(A)$ are eigenvectors of $A^q$, where $q$ denotes the order of the permutation induced by $A$ on the extremals of $\operatorname{core}(A)$.
Hence $\operatorname{core}(A)$ is a subcone of the sum of all eigencones of all powers of $A$, which is the inclusion relation~\eqref{e:maingoal2}. Combining this with the reverse inclusion of Lemma~\ref{l:natural} we obtain that $\operatorname{core}(A)$ is precisely the sum of all eigencones of all powers of $A$, and using ~\eqref{e:samespectrum} (proved in Section~\ref{s:sameaccess} below), we obtain the first part of the equality of Main Theorem~\ref{t:core}. The last part of the equality of Main Theorem~\ref{t:core} now follows from the periodicity of eigencones formulated in Main Theorem~\ref{t:periodicity}, or more precisely, from the weaker result of Theorem~\ref{t:girls} proved in Section~\ref{s:eigencones}. \end{proof}
\section{Spectral classes and critical components of matrix powers} \label{s:sameaccess}
This section is rather of technical importance. It shows that the union of node sets of all spectral classes is invariant under matrix powering, and that access relations between spectral classes in all matrix powers are essentially the same. Further, the case of an arbitrary eigenvalue can be reduced to the case of the principal eigenvalue for all powers simultaneously (in both algebras). At the end of the section we consider the critical components of max-algebraic powers.
\subsection{Classes and access relations}
As in Section~\ref{s:core}, the arguments are presented in both algebras simultaneously. This is due to the fact that the edge sets of ${\mathcal G}(A^{\otimes k})$ and ${\mathcal G}(A^{\times k})$ are the same for any $k$ and that the definitions of spectral classes in both algebras are alike. Results of this section can be traced back, for the case of nonnegative algebra, to the classical work of Frobenius~\cite{Fro-1912}, see remarks on the very first page of \cite{Fro-1912} concerning the powers of an irreducible nonnegative matrix\footnote{Frobenius defines (what we could call) the cyclicity or index of imprimitivity $k$ of an irreducible $S$ as the number of eigenvalues that lie on the spectral circle. He then remarks ``If $A$ is primitive, then every power of $A$ is again primitive and a certain power and all subsequent powers are positive''. This is followed by ``If $A$ is imprimitive, then $A^m$ consists of $d$ irreducible parts where $d$ is the greatest common divisor of $m$ and $k$. Further, $A^m$ is completely reducible. The characteristic functions of the components differ only in the powers of the variable'' (which provides a converse to the preceding assertion). And then ``The matrix $A^k$ is the lowest power of $A$ whose components are all primitive''. The three quotations cover Lemma~\ref{l:sameperron} in the case of nonnegative algebra.}.
The reader is also referred to the monographs of Minc~\cite{Minc}, Ber\-man-Plem\-mons~\cite{BP}, Brua\-ldi-Ry\-ser~\cite{BR}, and we will often cite the work of Tam-Sch\-nei\-der~\cite[Section 4]{TS-94} containing all of our results in this section, in nonnegative algebra.
\begin{lemma}[cf. {\cite[Ch.~5, Ex.~6.9]{BP}}, {\cite[Lemma~4.5]{TS-94}} ] \label{l:sameperron} Let $A$ be irreducible with the unique eigenvalue $\rho$, let ${\mathcal G}(A)$ have cyclicity $\sigma$ and $k$ be a positive integer. \begin{itemize} \item[{\rm (i)}] $A^k$ is permutationally similar to a direct sum of gcd$(k,\sigma)$ irreducible blocks with eigenvalues $\rho^k$, and $A^k$ does not have eigenvalues other than $\rho^k$. \item[{\rm (ii)}] If $k$ is a multiple of $\sigma$, then the sets of indices in these blocks coincide with the cyclic classes of ${\mathcal G}(A)$. \item[{\rm (iii)}] If $\operatorname{supp}(x)$ is a cyclic class of ${\mathcal G}(A)$, then $\operatorname{supp}(Ax)$ is the previous cyclic class.
\end{itemize} \end{lemma} \begin{proof}
(i): Assuming without loss of generality $\rho=1$, let $X=\operatorname{diag}(x)$ for a positive eigenvector $x\in V(A,\rho)$ and consider $B:=X^{-1}AX$ which is stochastic (nonnegative algebra), or max-stochastic, i.e., such that $\bigoplus_{j=1}^n b_{ij}=1$ holds for all $i$ (max algebra). By Theorem~\ref{t:brualdi}, $B^k$ is permutationally similar to a direct sum of gcd$(k,\sigma)$ irreducible isolated blocks. These blocks are stochastic (or max-stochastic), hence they all have an eigenvector $(1,\ldots,1)$ associated with the unique eigenvalue $1$. If $x\in V(A^k,\Tilde{\rho})$ for some $\Tilde{\rho}$, then its subvectors corresponding to the irreducible blocks of $A^k$ are also eigenvectors of those blocks, or zero vectors. Hence $\Tilde{\rho}=1$, which is the only eigenvalue of $A^k$.
(ii): By Theorem~\ref{t:brualdi}, ${\mathcal G}(A)$ splits into gcd$(k,\sigma)=\sigma$ components, and each of them contains exactly one cyclic class of ${\mathcal G}(A)$.
(iii): Use the definition of cyclic classes and that each node has an ingoing edge. \end{proof}
\begin{lemma} \label{l:samenonzero} Both in max algebra and in nonnegative linear algebra, the trivial classes of $A^k$ are the same for all $k$.
\end{lemma} \begin{proof} In both algebras, an index belongs to a class with nonzero Perron root if and only if the associated graph contains a cycle with a nonzero weight traversing the node with that index. This property is invariant under taking matrix powers, hence the claim. \end{proof}
In both algebras, each class $\mu$ of $A$ with cyclicity $\sigma$ corresponds to an irreducible submatrix $A_{\mu\mu}$. It is easy to see that $(A^k)_{\mu\mu}=(A_{\mu\mu})^k$. Applying Lemma~\ref{l:sameperron} to $A_{\mu\mu}$ we see that $\mu$ gives rise to gcd$(k,\sigma)$ classes in $A^k$, which are said to be {\em derived} from their common {\em ancestor} $\mu$. If $\mu$ is trivial, then it gives rise to a unique trivial derived class of $A^k$, and if $\mu$ is non-trivial then all the derived classes are nontrivial as well. The classes of $A^k$ and $A^l$ derived from the common ancestor will be called {\em related}. Note that this is an equivalence relation on the set of classes of all powers of $A$. Evidently, a class of $A^k$ is derived from a class of $A$ if and only if its index set is contained in the index set of the latter class. It is also clear that each class of $A^k$ has an ancestor in $A$.
We now observe that access relations in matrix powers are ``essentially the same''. This has identical proof in max algebra and nonnegative algebra.
\begin{lemma} \label{l:sameaccess} Let $A\in\Rp^{n\times n}$. For all $k,l\geq 1$ and $\rho>0$, if an index $i\in\{1,\ldots,n\}$ accesses (resp. is accessed by) a class with Perron root $\rho^k$ in $A^k$ then $i$ accesses (resp. is accessed by) a related class with Perron root $\rho^l$ in $A^l$. \end{lemma} \begin{proof}
We deduce from Lemma~\ref{l:sameperron} and Lemma~\ref{l:samenonzero} that the index set of each class of $A^k$ with Perron root $\rho^k$ is contained in the ancestor class of $A$ with Perron root $\rho$. Then, $i$ accessing (resp. being accessed by) a class in $A^k$ implies $i$ accessing (resp. being accessed by) its ancestor in $A$. Since $\rho>0$, this ancestor class is nontrivial, so the access path can be extended to have a length divisible by $l$, by means of a path contained in the ancestor class. By Lemma~\ref{l:sameperron}, the ancestor decomposes in $A^l$ into several classes with the common Perron root $\rho^l$, and $i$ accesses (resp. is accessed by) one of them.
\end{proof}
\begin{theorem}[{\cite[Corollary 4.6]{TS-94}}] \label{t:samespectrum} Let $A\in\Rp^{n\times n}$. \begin{itemize} \item[{\rm (i)}] If a class $\mu$ is spectral in $A$, then so are the classes derived from it in $A^k$. Conversely, each spectral class of $A^k$ is derived from a spectral class of $A$. \item[{\rm (ii)}] For each class $\mu$ of $A$ with cyclicity $\sigma$, there are gcd$(k,\sigma)$ classes of $A^k$ derived from it. If $k$ is a multiple of $\sigma$ then the index sets of derived classes are the cyclic classes of $\mu$. \end{itemize} \end{theorem} \begin{proof} (i): We will prove the following equivalent statement: For each pair $\mu,\nu$ where $\mu$ is a class in $A$ and $\nu$ is a class derived from $\mu$ in $A^k$, we have that $\mu$ is non-spectral if and only if $\nu$ is non-spectral.
Observe that by Lemma~\ref{l:samenonzero}, the Perron root of $\mu$ is $0$ if and only if the Perron root of $\nu$ is $0$. In this case, both $\mu$ and $\nu$ are non-spectral (by definition). Further, let $\rho>0$ be the Perron root of $\mu$. Then, by Lemma~\ref{l:sameperron}, the Perron root of $\nu$ is $\rho^k$. Let $i$ be an index in $\nu$. It also belongs to $\mu$.
If $\mu$ is non-spectral, then $i$ is accessed in $A$ by a class with Perron root $\rho'$ such that $\rho'>\rho$ in max algebra, resp. $\rho'\geq\rho$ in nonnegative algebra. By Lemma~\ref{l:sameaccess}, there is a class of $A^k$, which accesses $i$ in $A^k$ and has Perron root $(\rho')^k$. Since we have $(\rho')^k>\rho^k$ in max algebra or resp. $(\rho')^k\geq \rho^k$ in nonnegative algebra, we obtain that $\nu$, being the class to which $i$ belongs in $A^k$, is also non-spectral.
Conversely, if $\nu$ is non-spectral, then $i$ is accessed in $A^k$ by a class $\theta$ with Perron root equal to $\Tilde{\rho}^k$ for some $\Tilde{\rho}$, and such that $\Tilde{\rho}^k>\rho^k$ in max algebra, resp. $\Tilde{\rho}^k\geq\rho^k$ in nonnegative algebra. The ancestor of $\theta$ in $A$ accesses\footnote{This can be observed immediately, or obtained by applying Lemma~\ref{l:sameaccess}.} $i$ in $A$ and has Perron root $\Tilde{\rho}$. Since we have $\Tilde{\rho}>\rho$ in max algebra or resp. $\Tilde{\rho}\geq \rho$ in nonnegative algebra, we obtain that $\mu$, being the class to which $i$ belongs in $A$, is also non-spectral. Part (i) is proved.
(ii): This part follows directly from Lemma~\ref{l:sameperron} parts (i) and (ii). \end{proof}
\begin{corollary} \label{c:samespectrum} Let $A\in\Rp^{n\times n}$ and $k\geq 1$. Then $\Lambda(A^k)=\{\rho^k\colon\rho\in\Lambda(A)\}.$ \end{corollary} \begin{proof} By Theorem~\ref{t:spectrum}, the nonzero eigenvalues of $A$ (resp. $A^k$) are precisely the Perron roots of the spectral classes of $A$ (resp. $A^k$). By Theorem~\ref{t:samespectrum}(i), if a class of $A$ is spectral, then so is any class derived from it in $A^k$. This implies
that $\Lambda(A^k)\subseteq \{\rho^k\colon \rho\in\Lambda(A)\}.$ The converse inclusion follows from the converse part of Theorem~\ref{t:samespectrum}(i). \end{proof}
Let us note yet another corollary of Theorem~\ref{t:samespectrum}. For $A\in\Rp^{n\times n}$ and $\rho\geq 0$,
let $N(A,\rho)$ be the union of index sets of all classes of $A$ with Perron root $\rho$, and $N^s(A,\rho)$ be the union of index sets of all {\bf spectral} classes of $A$ with Perron root $\rho$. Obviously, $N^s(A,\rho)\subseteq N(A,\rho)$, and both sets (as defined for arbitrary $\rho\geq 0$) are possibly empty.
\begin{corollary} \label{c:specindex} Let $A\in\Rp^{n\times n}$, $\rho\in\R_+$ and $k\geq 1$. Then \begin{itemize} \item[{\rm (i)}] $N(A^k,\rho^k)=N(A,\rho)$, \item[{\rm (ii)}] $N^s(A^k,\rho^k)=N^s(A,\rho)$. \end{itemize} \end{corollary} \begin{proof} (i): This part follows from Lemmas~\ref{l:sameperron} and~\ref{l:samenonzero}.
(ii): Inclusion $N^s(A,\rho)\subseteq N^s(A^k,\rho^k)$ follows from the direct part of Theorem~\ref{t:samespectrum}(i), and inclusion\\ $N^s(A^k,\rho^k)\subseteq N^s(A,\rho)$ follows from the converse part of Theorem~\ref{t:samespectrum}(i).
\end{proof}
For the eigencones of $A\in\Rp^{n\times n}$, the case of an arbitrary $\rho\in\Lambda(A)$ can be reduced to the case of the principal eigenvalue: $V(A,\rho)=V(A_{\rho},1)$ (Proposition~\ref{p:vamrho}). Now we extend this reduction to the case of $V(A^k,\rho^k)$, for any $k\geq 1$. As in the case of Propositon~\ref{p:vamrho}, we assume that $A$ is in Frobenius normal form.
\begin{theorem} \label{t:reduction} Let $k\geq 1$ and $\rho\in \Lambda(A)$. \begin{itemize} \item[{\rm (i)}] The set of all indices having access to the spectral classes of $A^k$ with the eigenvalue $\rho^k$ equals $M_{\rho}$, for each $k$. \item[{\rm (ii)}] $(A^k)_{M_{\rho}M_{\rho}}=\rho^k (A_{\rho})^k_{M_{\rho}M_{\rho}}$. \item[{\rm (iii)}] $V(A^k,\rho^k)=V((A_{\rho})^k,1)$.
\end{itemize} \end{theorem} \begin{proof} (i): Apply Corollary~\ref{c:specindex}~part~(ii) and Lemma~\ref{l:sameaccess}. (ii): Use that $M_{\rho}$ is initial in ${\mathcal G}(A)$. (iii): By Proposition~\ref{p:vamrho} we have (assuming that $A^k$ is in Frobenius normal form) that $V(A^k,\rho^k)=V((A_{\rho^k})^k,1)$ where, instead of~\eqref{amrho}, \begin{equation} \label{amrhok} \begin{split} & A^k_{\rho^k}:=\rho^{-k} \begin{pmatrix} 0 & 0\\ 0 & A^k_{M_{\rho}^kM_{\rho}^k} \end{pmatrix},\ \text{and}\\ & M_{\rho}^k:=\{i\colon i\to\nu,\; \nu\ \text{is $(A^k,\rho^k)$-spectral}\} \end{split} \end{equation} By part (i) $M_{\rho}^k=M_{\rho}$, hence $A_{\rho^k}^k=(A_{\rho})^k$ and the claim follows. \end{proof}
\subsection{Critical components}
\label{ss:critcomp}
In max algebra, when $A$ is assumed to be strictly visualized, each component $\Tilde{\mu}$ of ${\mathcal C}(A)$ with cyclicity $\sigma$ corresponds to an irreducible submatrix $A^{[1]}_{\Tilde{\mu}\Tilde{\mu}}$ (as in the case of classes, $A_{\Tilde{\mu}\Tilde{\mu}}$ is a shorthand for $A_{N_{\Tilde{\mu}}N_{\Tilde{\mu}}}$). Using the strict visualization and Lemma~\ref{l:CAk} we see that $(A^{\otimes k})^{[1]}_{\Tilde{\mu}\Tilde{\mu}}=(A^{[1]}_{\Tilde{\mu}\Tilde{\mu}})^{\otimes k}$. Applying Lemma~\ref{l:sameperron}(i) to $A^{[1]}_{\Tilde{\mu}\Tilde{\mu}}$ we see that $\Tilde{\mu}$ gives rise to gcd$(k,\sigma)$ critical components in $A^{\otimes k}$. As in the case of classes, these components are said to be derived from their common ancestor $\Tilde{\mu}$.
Evidently a component of ${\mathcal C}(A^{\otimes k})$ is derived from a component of ${\mathcal C}(A)$ if and only if its index set is contained in the index set of the latter component. Following this line we now formulate an analogue of Theorem~\ref{t:samespectrum} (and some other results).
\begin{theorem}[cf. {\cite[Theorem 8.2.6]{But:10}}, {\cite[Theorem 2.3]{CGB-07}}] \label{t:samecritical} Let $A\in\Rp^{n\times n}$. \begin{itemize} \item[{\rm (i)}] For each component $\Tilde{\mu}$ of ${\mathcal C}(A)$ with cyclicity $\sigma$, there are gcd$(k,\sigma)$ components of ${\mathcal C}(A^{\otimes k})$ derived from it. Conversely, each component of ${\mathcal C}(A^{\otimes k})$ is derived from a component of ${\mathcal C}(A)$. If $k$ is a multiple of $\sigma$, then index sets in the derived components are the cyclic classes of $\Tilde{\mu}$. \item[{\rm (ii)}] The sets of critical indices of $A^{\otimes k}$ for $k=1,2,\ldots$ are identical. \item[{\rm (iii)}] If $A$ is strictly visualized, $x_i\leq 1$ for all $i$ and $\operatorname{supp}(x^{[1]})$ is a cyclic class of $\Tilde{\mu}$, then $\operatorname{supp}((A\otimes x)^{[1]})$ is the previous cyclic class of $\Tilde{\mu}$. \end{itemize} \end{theorem} \begin{proof} (i),(ii): Both statements are based on the fact that ${\mathcal C}(A^{\otimes k})=({\mathcal C}(A))^k$, shown in Lemma~\ref{l:CAk}. To obtain (i), also apply Theorem~\ref{t:brualdi} to a component $\Tilde{\mu}$ of ${\mathcal C}(A)$. (iii): Use $(A\otimes x)^{[1]}=A^{[1]}\otimes x^{[1]}$, the definition of cyclic classes and the fact that each node in $\Tilde{\mu}$ has an ingoing edge. \end{proof}
\section{Describing extremals} \label{s:extremals}
The aim of this section is to describe the extremals of the core, in both algebras. To this end, we first give a parallel description of extremals of eigencones (the Frobenius-Victory theorems).
\subsection{Extremals of the eigencones} \label{ss:FV}
We now describe the principal eigencones in nonnegative linear algebra and then in max algebra. By means of Proposition~\ref{p:vamrho}, this description can be obviously extended to the general case. As in Section~\ref{ss:pfelts}, both descriptions are essentially known: see~\cite{But:10,Fro-1912,Gau:92, Sch-86}.
We emphasize that the vectors $x^{(\mu)}$ and $x^{(\Tilde{\mu})}$ appearing below are {\bf full-size}.
\begin{theorem}[Frobenius-Victory{\cite[Th.~3.7]{Sch-86}}] \label{t:FVnonneg} Let $A\in\Rp^{n\times n}$ have $\rho^+(A)=1$. \begin{itemize} \item[{\rm (i)}] Each spectral class $\mu$ with $\rho^+_{\mu}=1$ corresponds to an eigenvector $x^{(\mu)}$, whose support consists of all indices in the classes that have access to $\mu$, and all vectors $x$ of $V_+(A,1)$ with $\operatorname{supp} x=\operatorname{supp} x^{(\mu)}$ are multiples of $x^{(\mu)}$. \item[{\rm (ii)}] $V_+(A,1)$ is generated by $x^{(\mu)}$ of {\rm (i)}, for $\mu$ ranging over all spectral classes with $\rho^+_{\mu}=1$. \item[{\rm (iii)}] $x^{(\mu)}$ of {\rm (i)} are extremals of $V_+(A,1)$. {\rm (}Moreover, $x^{(\mu)}$ are linearly independent.{\rm )} \end{itemize} \end{theorem}
Note that the extremality and the {\bf usual} linear independence of $x^{(\mu)}$ (involving linear combinations with possibly negative coefficients) can be deduced from the description of supports in part (i), and from the fact that in nonnegative algebra, spectral classes associated with the same $\rho$ do not access each other. This linear independence also means that $V_+(A,1)$ is a simplicial cone. See also~\cite[Th.~4.1]{Sch-86}.
\begin{theorem}[{\cite[Th.~4.3.5]{But:10}}, {\cite[Th.~2.8]{SSB}}] \label{t:FVmaxalg} Let $A\in\Rp^{n\times n}$ have $\rho^{\oplus}(A)=1$. \begin{itemize} \item[{\rm (i)}] Each component $\Tilde{\mu}$ of ${\mathcal C}(A)$ corresponds to an eigenvector $x^{(\Tilde{\mu})}$ defined as one of the columns $A^*_{\cdot i}$ with $i\in N_{\Tilde{\mu}}$, all columns with $i\in N_{\Tilde{\mu}}$ being multiples of each other. \item[{\rm (i')}] Each component $\Tilde{\mu}$ of ${\mathcal C}(A)$ is contained in a (spectral) class $\mu$ with $\rho^{\oplus}_{\mu}=1$, and the support of each $x^{(\Tilde{\mu})}$ of {\rm (i)} consists of all indices in the classes that have access to $\mu$.
\item[{\rm (ii)}] $V_{\oplus}(A,1)$ is generated by $x^{(\Tilde{\mu})}$ of {\rm (i)}, for $\Tilde{\mu}$ ranging over all components of ${\mathcal C}(A)$. \item[{\rm (iii)}] $x^{(\Tilde{\mu})}$ of {\rm (i)} are extremals in $V_{\oplus}(A,1)$. {\rm (}Moreover, $x^{(\Tilde{\mu})}$ are strongly linearly independent in the sense of~\cite{But-03}.{\rm )} \end{itemize} \end{theorem}
To verify (i'), not explicitly stated in the references, use (i) and the path interpretation of~$A^*$.
Vectors $x^{(\Tilde{\mu})}$ of Theorem~\ref{t:FVmaxalg} are also called the {\em fundamental eigenvectors} of $A$, in max algebra. Applying a strict visualization scaling (Theorem~\ref{t:strictvis}) allows us to get further details on these fundamental eigenvectors.
\begin{proposition}[{\cite[Prop.~4.1]{SSB}}] \label{p:xmuvis} Let $A\in\Rp^{n\times n}$ be strictly visualized (in particular, $\rho^{\oplus}(A)=1$). Then \begin{itemize} \item[{\rm (i)}] For each component $\Tilde{\mu}$ of ${\mathcal C}(A)$, $x^{(\Tilde{\mu})}$ of Theorem~\ref{t:FVmaxalg} can be canonically chosen as $A^*_{\cdot i}$ for any $i\in N_{\Tilde{\mu}}$, all columns with $i\in N_{\Tilde{\mu}}$ being {\em equal} to each other. \item[{\rm (ii)}] $x^{(\Tilde{\mu})}_i\leq 1$ for all $i$. Moreover, $\operatorname{supp}(x^{(\Tilde{\mu})[1]})=N_{\Tilde{\mu}}$.
\end{itemize} \end{proposition}
\subsection{Extremals of the core} \label{ss:extremals}
Let us start with the following observation in both algebras.
\begin{proposition} \label{p:extremals} For each $k\geq 1$, the set of extremals of $V^{\Sigma}(A^k)$ is the union of the sets of extremals of $V(A^k,\rho^k)$ for $\rho\in\Lambda(A)$. \end{proposition} \begin{proof} Due to the fact that $\Lambda(A^k)=\{\rho^k\colon\rho\in\Lambda(A)\}$, we can assume without loss of generality that $k=1$.
1. As $V^{\Sigma}(A)$ is the sum of $V(A,\rho)$ for $\rho\in\Lambda(A)$, it is generated by the extremals of $V(A,\rho)$ for $\rho\in\Lambda(A)$. Hence each extremal of $V^{\Sigma}(A)$ is an extremal of $V(A,\rho)$ for some $\rho\in\Lambda(A)$.
2. Let $x\in V(A,\rho_{\mu})$, for some spectral class $\mu$, be extremal. Assume without loss of generality that $\rho_{\mu}=1$, and by contradiction that there exist vectors $y^{\kappa}$, all of them extremals of $V^{\Sigma}(A)$, such that $x=\sum_{\kappa} y^{\kappa}$. By above, all vectors $y^{\kappa}$ are eigenvectors of $A$. If there is $y^{\kappa}$ associated with an eigenvalue $\rho_{\nu}>1$, then applying $A^t$ we obtain $x= (\rho_{\nu})^t y^{\kappa}+\ldots$, which is impossible at large enough $t$. So $\rho_{\nu}\leq 1$. With this in mind, if there is $y^{\kappa}$ associated with $\rho_{\nu}<1$, then 1) in nonnegative algebra we obtain $Ax>A\sum_{\kappa} y^{\kappa}$, a contradiction; 2) in max algebra, all nonzero entries of $A\otimes y^{\kappa}$ go below the corresponding entries of $x$ meaning that $y^{\kappa}$ is redundant. Thus we are left only with $y^{\kappa}$ associated with $\rho^{\oplus}_{\nu}=1$, which is a contradiction: an extremal $x\in V_{\oplus}(A,1)$ appears as a ``sum'' of other extremals of $V_{\oplus}(A,1)$ not proportional to $x$. \end{proof}
A vector $x\in\Rp^n$ is called {\em normalized} if $\max x_i=1$. Recall the notation $\sigma_{\rho}$ introduced in Section~\ref{s:key}.
\begin{theorem}[cf.~{\cite[Theorem~4.7]{TS-94}}] \label{t:tam-schneider} Let $A\in\Rp^{n\times n}$. \begin{itemize} \item[{\rm (i)}] The set of extremals of $\operatorname{core}(A)$ is the union of the sets of extremals of $V(A^{\sigma_{\rho}},\rho^{\sigma_{\rho}})$ for all $\rho\in\Lambda(A)$. \item[{\rm (ii)}] {\bf In nonnegative algebra}, each spectral class $\mu$ with cyclicity $\sigma_{\mu}$ corresponds to a set of distinct $\sigma_{\mu}$ normalized extremals of $\core_+(A)$, such that there exists an index in their support that belongs to $\mu$, and each index in their support has access to $\mu$.\\ {\bf In max algebra}, each critical component $\Tilde{\mu}$ with cyclicity $\sigma_{\Tilde{\mu}}$ associated with some $\rho\in\Lambda_{\oplus}(A)$ corresponds to a set of distinct $\sigma_{\Tilde{\mu}}$ normalized extremals $x$ of $\core_{\oplus}(A)$, which are (normalized) columns of $(A_{\rho}^{\sigma_{\rho}})^*$ with indices in $N_{\Tilde{\mu}}$. \item[{\rm (iii)}] Each set of extremals described in {\rm (ii)} forms a simple cycle under the action of $A$. \item[{\rm (iv)}] There are no normalized extremals other than those described in {\rm (ii)}. {\bf In nonnegative algebra,} the total number of normalized extremals equals the sum of cyclicities of all spectral classes of $A$. {\bf In max algebra,} the total number of normalized extremals equals the sum of cyclicities of all critical components of $A$.
\end{itemize} \end{theorem}
\begin{proof} (i) follows from Main Theorem~\ref{t:core} and Proposition~\ref{p:extremals}.
For the proof of (ii) and (iii) we can fix $\rho=\rho_{\mu}\in\Lambda(A)$, assume $A=A_{\rho}$ (using Theorem~\ref{t:reduction}) and $\sigma:=\sigma_{\rho}$. In max algebra, we also assume that $A$ is strictly visualized.
(ii) {\bf In nonnegative algebra,} observe that by Theorem~\ref{t:samespectrum}, each spectral class $\mu$ of $A$ gives rise to $\sigma_{\mu}$ spectral classes in $A^{\times\sigma}$, whose node sets are cyclic classes of $\mu$ (note that $\sigma_{\mu}$ divides $\sigma$). According to Frobenius-Victory Theorem~\ref{t:FVnonneg}, these classes give rise to normalized extremals of $V_+(A^{\times\sigma},1)$, and the conditions on support follow from Theorem~\ref{t:FVnonneg} and Lemma~\ref{l:sameaccess}.
(iii):
Let $x$ be an extremal described above. Then $\operatorname{supp}(x)\cap N_{\mu}$ is a cyclic class of $\mu$ and $\operatorname{supp}(Ax)\cap N_{\mu}$ is the previous cyclic class of $\mu$, by Lemma~\ref{l:sameperron} part (iii). It can be checked that all indices in $\operatorname{supp}(Ax)$ also have access to $\mu$. By Proposition~\ref{p:permute}, $Ax$ is an extremal of $\core_+(A)$, and hence an extremal of $V_+(A^{\times\sigma},1)$. Theorem~\ref{t:FVnonneg} identifies $Ax$ with the extremal associated with the previous cyclic class of $\mu$.
Vectors $x$, $Ax,\ldots,A^{\times\sigma_{\mu}-1}x$ are distinct since the intersections of their supports with $N_{\mu}$ are disjoint, so they are exactly the set of extremals associated with $\mu$. Note that $A^{\times\sigma_{\mu}}x=x$, as $\operatorname{supp}(A^{\times\sigma_{\mu}}x)\cap N_{\mu}=\operatorname{supp}(x)\cap N_{\mu}$, and both vectors are extremals of $V_+(A^{\times\sigma},1)$.
(ii): {\bf In max algebra,} observe that by Theorem~\ref{t:samecritical}(i) each component $\Tilde{\mu}$ of ${\mathcal C}(A)$ gives rise to $\sigma_{\Tilde{\mu}}$ components of ${\mathcal C}(A^{\otimes\sigma})$, whose node sets are the cyclic classes of $\Tilde{\mu}$ (note that $\sigma_{\Tilde{\mu}}$ divides $\sigma$). These components correspond to $\sigma_{\Tilde{\mu}}$ columns of $(A^{\otimes\sigma})^*$ with indices in different cyclic classes of $\Tilde{\mu}$, which are by Theorem~\ref{t:samecritical}(i) the node sets of components of ${\mathcal C}(A^{\otimes\sigma})$. By Theorem~\ref{t:FVmaxalg} these columns of $(A^{\otimes\sigma})^*$ are extremals of $V_{\oplus}(A^{\otimes\sigma},1)$, and Proposition~\ref{p:xmuvis}(ii) implies that they are normalized.
(iii):
Let $x$ be an extremal described above. By Proposition~\ref{p:xmuvis} and Theorem~\ref{t:samecritical}(i) $\operatorname{supp}(x^{[1]})$ is a cyclic class of $\Tilde{\mu}$, and by Theorem~\ref{t:samecritical}(iii) $\operatorname{supp}((A\otimes x)^{[1]})$ is the previous cyclic class of $\Tilde{\mu}$. By Proposition~\ref{p:permute}, $A\otimes x$ is an extremal of $\core_{\oplus}(A)$, and hence an extremal of $V_{\oplus}(A^{\otimes\sigma},1)$. Proposition~\ref{p:xmuvis} identifies $A\otimes x$ with the extremal associated with the previous cyclic class of $\Tilde{\mu}$.
Vectors $x$, $A\otimes x,\ldots,A^{\otimes\sigma_{\Tilde{\mu}}-1}x$ are distinct since their booleanizations $x^{[1]}$, $(A\otimes x)^{[1]},\ldots,(A^{\otimes\sigma_{\Tilde{\mu}}-1}\otimes x)^{[1]}$ are distinct,
so they are exactly the set of extremals associated with $\Tilde{\mu}$. Note that $A^{\otimes\sigma_{\Tilde{\mu}}}\otimes x=x$, as $(A^{\otimes\sigma_{\Tilde{\mu}}}\otimes x)^{[1]}=x^{[1]}$ and both vectors are extremals of $V_{\oplus}(A^{\otimes\sigma},1)$.
(iv): {\bf In both algebras}, the converse part of Theorem~\ref{t:samespectrum} (i) shows that there are no spectral classes of $A^{\sigma}$ other than the ones derived from the spectral classes of $A$. {\bf In nonnegative algebra,} this shows that there are no extremals other than described in (ii). {\bf In max algebra}, on top of that, the converse part of Theorem~\ref{t:samecritical} (i) shows that there are no components of ${\mathcal C}(A_{\rho}^{\otimes\sigma})$ other than the ones derived from the components ${\mathcal C}(A_{\rho})$, for $\rho\in\Lambda_{\oplus}(A)$, hence there are no extremals other than described in (ii). {\bf In both algebras}, it remains to count the extremals described in (ii). \end{proof}
\section{Sequence of eigencones}
\label{s:eigencones}
The main aim of this section is to investigate the periodicity of eigencones and to prove Main Theorem~\ref{t:periodicity}. Unlike in Section~\ref{s:core}, the proof of periodicity will be different for the cases of max algebra and nonnegative algebra. The periods of eigencone sequences in max algebra and in nonnegative linear algebra are also in general different, for the same nonnegative matrix (see Section~\ref{s:examples} for an example). To this end, recall the definitions of $\sigma_{\rho}$ and $\sigma_{\Lambda}$ given in Section~\ref{s:key}, which will be used below.
\subsection{Periodicity of the sequence}
\label{ss:period}
We first observe that in both algebras \begin{equation} \label{e:inclusion-eig} \begin{split} k\;\makebox{divides}\; l &\Rightarrow V(A^k,\rho^k)\subseteq V(A^l,\rho^l)\quad\forall\rho\in\Lambda(A),\\ k\;\makebox{divides}\; l &\Rightarrow V^{\Sigma}(A^k)\subseteq V^{\Sigma}(A^l). \end{split} \end{equation}
We now prove that the sequence of eigencones is periodic.
\begin{theorem} \label{t:girls} Let $A\in\Rp^{n\times n}$ and $\rho\in\Lambda(A)$. \begin{itemize} \item[{\rm (i)}] $V(A^k,\rho^k)=V(A^{k+\sigma_{\rho}},\rho^{k+\sigma_{\rho}})$, and $V(A^k,\rho^k)\subseteq V(A^{\sigma_{\rho}},\rho^{\sigma_{\rho}})$ for all $k\geq 1$. \item[{\rm (ii)}] $V^{\Sigma}(A^k)=V^{\Sigma}(A^{k+\sigma_{\Lambda}})$ and $V^{\Sigma}(A^k)\subseteq V^{\Sigma}(A^{\sigma_{\Lambda}})$ for all $k\geq 1$. \end{itemize} \end{theorem} \begin{proof} We will give two separate proofs of part (i), for the case of max algebra and the case of nonnegative algebra. Part (ii) follows from part (i).
In both algebras, we can assume without loss of generality that $\rho=1$, and using Theorem~\ref{t:reduction}, that this is the greatest eigenvalue of $A$.
{\bf In max algebra}, by Theorem~\ref{t:nacht}, columns of $A^{\otimes r}$ with indices in $N_c(A)$ are periodic for $r\geqT_c(A)$. Recall that by Corollary~\ref{TcaTcrit}, $T_c(A)$ is not less than $T({\mathcal C}(A))$, which is the greatest periodicity threshold of the strongly connected components of ${\mathcal C}(A)$. By Theorem~\ref{t:BoolCycl} part (ii), $({\mathcal C}(A))^{t\sigma}$ consists of complete graphs for $t\sigma\geq T({\mathcal C}(A))$, in particular, it contains loops $(i,i)$ for all $i\inN_c(A)$. Hence $$ a_{ii}^{\otimes(t\sigma)}=1\quad\forall i\inN_c(A),\ t\geq\lceil\frac{T_c(A)}{\sigma}\rceil, $$ and $$ a_{ki}^{\otimes (l+t\sigma)}\geq a^{\otimes l}_{ki} a_{ii}^{\otimes (t\sigma)}=a_{ki}^{\otimes l} \quad\forall i\inN_c(A),\ \forall k,l,\ \forall t\geq\lceil\frac{T_c(A)}{\sigma}\rceil, $$ or, in terms of columns of matrix powers, $$ A_{\cdot i}^{\otimes (l+t\sigma)}\geq A_{\cdot i}^{\otimes l}\quad \forall i\inN_c(A),\ \forall l,\ \forall t\geq\lceil\frac{T_c(A)}{\sigma}\rceil. $$ Multiplying this inequality repeatedly by $A^{\otimes l}$ we obtain $A_{\cdot i}^{\otimes (kl+t\sigma)}\geq A_{\cdot i}^{\otimes(kl)}$ for all $k\geq 1$, and $A_{\cdot i}^{\otimes (k(l+t\sigma))}\geq A_{\cdot i}^{\otimes(kl)}$ for all $k\geq 1$.
Hence we obtain \begin{equation} \label{e:*ineq} (A^{\otimes (l+t\sigma)})^*_{\cdot i}\geq (A^{\otimes l})^*_{\cdot i}\quad\forall i\inN_c(A),\ \forall l,\ \forall t\geq\lceil\frac{T_c(A)}{\sigma}\rceil. \end{equation} On the other hand, using the ultimate periodicity of critical columns we have \begin{equation*}
(A^{\otimes (l+t\sigma)})^*_{\cdot i}=\bigoplus\{A_{\cdot i}^{\otimes s}\colon s\equiv kl\text{(mod} \sigma),\ k\geq 1,\ s\geqT_c(A) \} \end{equation*} for all $l$ and all $t\sigma\geqT_c(A)$, while generally \begin{equation*} \label{e:sweepineq} (A^{\otimes l})^*_{\cdot i}\geq \bigoplus\{A_{\cdot i}^{\otimes s}\colon s\equiv kl\text{(mod} \sigma),\ k\geq 1,\ s\geqT_c(A) \}\ \forall l, \end{equation*} implying the reverse inequality with respect to~\eqref{e:*ineq}. It follows that \begin{equation} \label{e:*eq} (A^{\otimes (l+t\sigma)})^*_{\cdot i}= (A^{\otimes l})^*_{\cdot i}\quad\forall i\inN_c(A),\ \forall l,\ \forall t\geq\lceil\frac{T_c(A)}{\sigma}\rceil,
\end{equation} therefore $(A^{\otimes (l+\sigma)})^*_{\cdot i}=(A^{\otimes (l+t\sigma+\sigma)})^*_{\cdot i}=(A^{\otimes (l+t\sigma)})^*_{\cdot i} =(A^{\otimes l})^*_{\cdot i}$ for all critical indices $i$ and all $l$. Since $V(A^{\otimes l},1)$ is generated by the critical columns of $(A^{\otimes l})^*$, and the critical indices of $A^{\otimes l}$ are $N_c(A)$ by Theorem~\ref{t:samecritical}(ii), the periodicity $V_{\oplus}(A^{\otimes l},\rho^l)=V_{\oplus}(A^{\otimes (l+\sigma)},\rho^{l+\sigma})$ follows. Using this and~\eqref{e:inclusion-eig} we obtain $V_{\oplus}(A^{\otimes l},\rho^l)\subseteqV_{\oplus}(A^{\otimes(l\sigma)},\rho^{l\sigma})=V_{\oplus}(A^{\otimes\sigma},\rho^{\sigma})$ for each $l$ and $\rho\in\Lambda_{\oplus}(A)$.
{\bf In nonnegative algebra,} also assume that all final classes (and hence only them) have Perron root $\rho=1$. Final classes of $A^{\times l}$ are derived from the final classes of $A$; they (and no other classes) have Perron root $\rho^l$. By Theorem~\ref{t:samespectrum}(i) and~Corollary~\ref{c:div-boolean}, for any $t\geq 0$, the spectral classes of $A^{\times l}$ and $A^{\times (l+t\sigma)}$ with Perron root $1$ have the same sets of nodes, which we denote by $N_1,\ldots,N_m$ (assuming that their number is $m\geq 1$).
By the Frobenius-Victory Theorem~\ref{t:FVnonneg}, the cone $V_+(A^{\times l},1)$ is generated by $m$ extremals $x^{(1)},\ldots,x^{(m)}$ with the support condition of Theorem~\ref{t:FVnonneg}(i) from which we infer that the subvectors $x^{(\mu)}_{\mu}$ (i.e., $x^{(\mu)}_{N_{\mu}}$) are positive, while $x^{(\mu)}_{\nu}$ (i.e., $x^{(\mu)}_{N_{\nu}}$) are zero for all $\mu\neq \nu$ from $1$ to $m$, since the different spectral classes by~\eqref{e:speclass} do not access each other, in the nonnegative linear algebra. Analogously the cone $V_+(A^{\times(l+t\sigma)},1)$ is generated by $m$ eigenvectors $y^{(1)},\ldots,y^{(m)}$ such that the subvectors $y^{(\mu)}_{\mu}$ are positive, while $y^{(\mu)}_{\nu}=0$ for all $\mu\neq \nu$ from $1$ to $m$.
Assume first that $l=\sigma$. As $V_+(A^{\times\sigma},1)\subseteq V_+(A^{\times (t\sigma)},1)$, each $x^{(\mu)}$ is a nonnegative
linear combination of $y^{(1)},\ldots,y^{(m)}$, and this implies $x^{(\mu)}=y^{(\mu)}$ for all $\mu=1,\ldots,m$. Hence $V_+(A^{\times (t\sigma)},1)=V_+(A^{\times\sigma},1)$ for all $t\geq 0$.
We also obtain $V_+(A^{\times l},1)\subseteq V_+(A^{\times (\sigma l)},1)=V_+(A^{\times\sigma},1)$ for all $l$. Thus $V_+(A^{\times l},1)\subseteqV_+(A^{\times (t\sigma)},1)$, and therefore $V_+(A^{\times l},1)\subseteq V_+(A^{\times (l+t\sigma)},1)$. Now if $V_+(A^{\times l},1)$, resp. $V_+(A^{\times (l+t\sigma)},1)$ are generated by $x^{(1)},\ldots,x^{(m)}$, resp. $y^{(1)},\ldots, y^{(m)}$ described above and each $x^{(\mu)}$ is a nonnegative
linear combination of $y^{(1)},\ldots,y^{(m)}$, this again implies $x^{(\mu)}=y^{(\mu)}$ for all $\mu=1,\ldots,m$, and $V_+(A^{\times (l+t\sigma)},1)=V_+(A^{\times l},1)$ for all $t\geq 0$ and all $l$.
Using this and~\eqref{e:inclusion-eig} we obtain $V_+(A^{\times l},\rho^l)\subseteqV_+(A^{\times(l\sigma)},\rho^{l\sigma})=V_+(A^{\times\sigma},\rho^{\sigma})$ for each $l$ and $\rho\in\Lambda_+(A)$. \end{proof}
\if{ for the sum of all eigencones $V^{\Sigma}(A^k)=\sum_{\rho\in\Lambda(A)} V(A^k,\rho^k)$. Summarizing Theorems~\ref{t:girls-max} and~\ref{t:girls-nonneg} and using $\sigma_{\rho}$ and $\sigma_{\Lambda}$
we obtain the following formulation (now uniting both algebras).
\begin{theorem} \label{t:girls} Let $A\in\Rp^{n\times n}$. For all $l\geq 1$ and $\rho\in\Lambda(A)$, \begin{itemize} \item[(i)] $V(A^l,\rho^l)=V(A^{l+\sigma_{\rho}},\rho^{l+\sigma_{\rho}})$ and $V(A^l,\rho^l)\subseteq V(A^{\sigma_{\rho}},\rho^{\sigma_{\rho}})$, \item[(ii)] $V^{\Sigma}(A^l)=V^{\Sigma}(A^{l+\sigma_{\Lambda}})$ and $V^{\Sigma}(A^l)\subseteq V^{\Sigma}(A^{\sigma_{\Lambda}})$. \end{itemize} \end{theorem} }\fi
\subsection{Inclusion and divisibility} \label{ss:incl}
We now show that the inclusion relations between the eigencones of different powers of a matrix, in both algebras, strictly follow divisibility of exponents of matrix powers with respect to $\sigma_{\rho}$ and $\sigma_{\Lambda}$.
We start with a corollary of Theorem~\ref{t:girls}.
\begin{lemma} \label{l:gcd-if} Let $k,l\geq 1$ and $\rho\in\Lambda(A)$. \begin{itemize} \item[{\rm (i)}] $V(A^k,\rho^k)=V(A^{\text{gcd}(\sigma_{\rho},k)},\rho^{\text{gcd}(\sigma_{\rho},k)})$ and $V^{\Sigma}(A^k)=V^{\Sigma}(A^{\text{gcd}(\sigma_{\Lambda},k)})$. \item[{\rm (ii)}] gcd$(k,\sigma_{\rho})=$ gcd$(l,\sigma_{\rho})$ implies $V(A^k,\rho^k)=V(A^l,\rho^l)$, and gcd$(k,\sigma_{\Lambda})=$ gcd$(l,\sigma_{\Lambda})$ implies $V^{\Sigma}(A^k)=V^{\Sigma}(A^l)$. \end{itemize} \end{lemma} \begin{proof} (i): Let $\sigma:=\sigma_{\rho}$, and $s:=$gcd$(k,\sigma)$. If $s=\sigma$ then $k$ is a multiple of $\sigma$ and $V(A^k,\rho^k)=V(A^s,\rho^s)$ by Theorems~\ref{t:girls}(i). Otherwise, since $s$ divides $k$, we have $V(A^s,\rho^s)\subseteq V(A^k,\rho^k)$. In view of the periodicity (Theorem~\ref{t:girls}(i)), it suffices to find $t$ such that $V(A^k,\rho^k)\subseteq V(A^{s+t\sigma},\rho^{s+t\sigma})$. For this, observe that $s+t\sigma$ is a multiple of $s=$ gcd$(k,\sigma)$. By Lemma~\ref{l:schur} (the Frobenius coin problem), for big enough $t$ it can be expressed as $t_1 k + t_2\sigma$ where $t_1,t_2\geq 0$. Moreover $t_1\neq 0$, for otherwise we have $s=\sigma$. Then we obtain \begin{equation*} \begin{split} V(A^k,\rho^k)&\subseteq V(A^{t_1k},\rho^{t_1k})=V(A^{t_1k+t_2\sigma},\rho^{t_1 k+t_2\sigma})\\ &=V(A^{s+t\sigma},\rho^{s+t\sigma})=V(A^s,\rho^s), \end{split} \end{equation*} and the first part of the claim follows. The second part is obtained similarly, using Theorem~\ref{t:girls}(ii) instead of Theorems~\ref{t:girls}(i).
(ii) follows from (i). \end{proof}
\begin{theorem} \label{t:girls-major} Let $A\in\Rp^{n\times n}$ and $\sigma$ be either the cyclicity of a spectral class of $A$ {\bf (nonnegative algebra)} or the cyclicity of a critical component of $A$ {\bf (max algebra)}.The following are equivalent for all positive $k,l$: \begin{itemize} \item[{\rm (i)}] gcd$(k,\sigma)$ divides gcd$(l,\sigma)$ for all cyclicities $\sigma$; \item[{\rm (ii)}] gcd$(k,\sigma_{\rho})$ divides gcd$(l,\sigma_{\rho})$ for all $\rho\in\Lambda(A)$; \item[{\rm (iii)}] gcd$(k,\sigma_{\Lambda})$ divides gcd$(l,\sigma_{\Lambda})$; \item[{\rm (iv)}] $V(A^k,\rho^k)\subseteq V(A^l,\rho^l)$ for all $\rho\in\Lambda(A)$ and \item[{\rm (v)}] $V^{\Sigma}(A^k)\subseteq V^{\Sigma}(A^l)$. \end{itemize} \end{theorem} \begin{proof} (i)$\Rightarrow$(ii)$\Rightarrow$(iii) follow from elementary number theory. (ii)$\Rightarrow$ (iv) and (iii)$\Rightarrow$(v) follow from~\eqref{e:inclusion-eig} and Lemma~\ref{l:gcd-if} part (i) (which is essentially based on Theorem~\ref{t:girls}). (iv)$\Rightarrow$(v) is trivial. It only remains to show that (v)$\Rightarrow$ (i).
(v)$\Rightarrow$ (i): {\bf In both algebras}, take an extremal $x\in V(A^k,\rho^k)$. As $V^{\Sigma}(A^k)\subseteq V^{\Sigma}(A^l)$, this vector can be represented as $x=\sum_i y^i$, where $y^i$ are extremals of $V^{\Sigma}(A^l)$. Each $y^i$ is an extremal of $V(A^l,\Tilde{\rho}^l)$ for some $\Tilde{\rho}\in\Lambda(A)$ (as we will see, only the extremals with $\Tilde{\rho}=\rho$ are important). By Frobenius-Victory Theorems~\ref{t:FVnonneg} and~\ref{t:FVmaxalg} and Theorem~\ref{t:samespectrum}(i), there is a unique spectral class $\mu$ of $A$ to which all indices in $\operatorname{supp}(x)$ have access. Since $\operatorname{supp}(y^i)\subseteq\operatorname{supp}(x)$, we are restricted to the submatrix $A_{JJ}$ where $J$ is the set of all indices accessing $\mu$ in $A$. In other words, we can assume without loss of generality that $\mu$ is the only final class in $A$, hence $\rho$ is the greatest eigenvalue, and $\rho=1$. Note that $\operatorname{supp}(x)\cap N_{\mu}\neq\emptyset$.
{\bf In nonnegative algebra,} restricting the equality $x=\sum_i y^i$ to $N_{\mu}$ we obtain \begin{equation} \label{e:eqsupps1} \operatorname{supp}(x_{\mu})=\bigcup_i \operatorname{supp}(y^i_{\mu}). \end{equation} If $\operatorname{supp}(y^i_{\mu})$ is non-empty, then $y^i$ is associated with a spectral class of $A^{\times l}$ whose nodes are in $N_{\mu}$.
Theorem~\ref{t:FVnonneg}(i) implies that $\operatorname{supp}(y^i_{\mu})$ consists of all indices in a class of $A_{\mu\mu}^{\times l}$. As $x$ can be any extremal eigenvector of $A^{\times k}$ with $\operatorname{supp} x\cap N_{\mu}\neq\emptyset$, \eqref{e:eqsupps1} shows that each class of $A_{\mu\mu}^{\times k}$ (corresponding to $x$) splits into several classes of $A_{\mu\mu}^{\times l}$ (corresponding to $y^i$). By Corollary~\ref{c:div-boolean} this is only possible when gcd$(k,\sigma)$ divides gcd$(l,\sigma)$, where $\sigma$ is the cyclicity of the spectral class $\mu$.
{\bf In max algebra,} since $\rho=1$, assume without loss of generality that $A$ is strictly visualized. In this case $A$ and $x$ have all coordinates not exceeding $1$. Recall that $x^{[1]}$ is the Boolean vector defined by $x^{[1]}_i=1$ $\Leftrightarrow$ $x_i=1$. Vector $x$ corresponds to a unique critical component $\Tilde{\mu}$ of ${\mathcal C}(A)$ with the node set $N_{\Tilde{\mu}}$. Then instead of~\eqref{e:eqsupps1} we obtain \begin{equation} \label{e:eqsupps2} x^{[1]}=\bigoplus_i y^{i[1]}\quad \Rightarrow\quad \operatorname{supp}(x^{[1]}_{\Tilde{\mu}})=\bigcup_i \operatorname{supp}(y^{i[1]}_{\Tilde{\mu}}), \end{equation} where $\operatorname{supp}(x^{[1]})=\operatorname{supp}(x^{[1]}_{\Tilde{\mu}})$ by Proposition~\ref{p:xmuvis}(ii) and Theorem~\ref{t:samecritical}(i), and hence also $\operatorname{supp}(y^{i[1]})=\operatorname{supp}(y^{i[1]}_{\Tilde{\mu}})$. If $\operatorname{supp}(y^{i[1]}_{\Tilde{\mu}})$ is non-empty then also $\operatorname{supp}(y^i_{N_{\mu}})$ is non-empty so that $y^i$ is associated with the eigenvalue $1$.
\if{Lemma~\ref{l:sameperron} implies that $\operatorname{supp}(y^{i[1]}_{N_{\Tilde{\mu}}})$ comprises the set of indices in several classes of $(A^{[1]}_{N_{\Tilde{\mu}}N_{\Tilde{\mu}}})^{\otimes l}$ (as it can be argued, in only one class if $y^{i}$ is extremal). }\fi As $y^i$ is extremal, Proposition~\ref{p:xmuvis}(ii) implies that $\operatorname{supp}(y^{i[1]}_{\Tilde{\mu}})$ consists of all indices in a class of $(A^{[1]}_{\Tilde{\mu}\Tilde{\mu}})^{\otimes l}$. As $x$ can be any extremal eigenvector of $A^{\otimes k}$ with $\operatorname{supp} (x^{[1]})\cap N_{\Tilde{\mu}}\neq\emptyset$, \eqref{e:eqsupps2} shows that each class of $(A^{[1]}_{\Tilde{\mu}\Tilde{\mu}})^{\otimes k}$ splits into several classes of $(A^{[1]}_{\Tilde{\mu}\Tilde{\mu}})^{\otimes l}$. By Corollary~\ref{c:div-boolean} this is only possible when gcd$(k,\sigma)$ divides gcd$(l,\sigma)$, where $\sigma$ is the cyclicity of the critical component $\Tilde{\mu}$. \end{proof}
Let us also formulate the following version restricted to some $\rho\in\Lambda(A)$.
\begin{theorem} \label{t:girls-minor} Let $A\in\Rp^{n\times n}$, and let $\sigma$ be either the cyclicity of a spectral class {\bf (nonnegative algebra)} or the cyclicity of a critical component {\bf (max algebra)} associated with some $\rho\in\Lambda(A)$. The following are equivalent for all positive $k,l$: \begin{itemize} \item[{\rm (i)}] gcd$(k,\sigma)$ divides gcd$(l,\sigma)$ for all cyclicities $\sigma$; \item[{\rm (ii)}] gcd$(k,\sigma_{\rho})$ divides gcd$(l,\sigma_{\rho})$; \item[{\rm (iii)}] $V(A^k,\rho^k)\subseteq V(A^l,\rho^l)$. \end{itemize} \end{theorem} \begin{proof} (i)$\Rightarrow$(ii) follows from the elementary number theory, and (ii)$\Rightarrow$(iii) follows from~\eqref{e:inclusion-eig} and Lemma~\ref{l:gcd-if}(i). The proof of (iii)$\Rightarrow$(i) follows the lines of the proof of Theorem~\ref{t:girls-major} (v)$\Rightarrow$ (i), with a slight simplification that $\Tilde{\rho}=\rho$ and further, $x$ and all $y^i$ in $x=\sum_i y^i$ are associated with the same eigenvalue. \end{proof}
We are now ready to deduce Main Theorem~\ref{t:periodicity}
\begin{proof}[Proof of Main Theorem~\ref{t:periodicity}]
We prove the first part. The inclusion $V(A^k,\rho^k)\subseteq V(A^{\sigma},\rho^{\sigma})$ was proved in Theorem~\ref{t:girls} (i), and we are left to show that $\sigma_{\rho}$ is the least such $p$ that $V(A^{k+p},\rho^{k+p})=V(A^k,\rho^k)$ for all $k\geq 1$. But taking $k=\sigma_{\rho}$ and using Theorem~\ref{t:girls-minor} (ii)$\Leftrightarrow$(iii), we obtain gcd$(\sigma_{\rho}+p,\sigma_{\rho})=\sigma_{\rho}$, implying that $\sigma_{\rho}$ divides $\sigma_{\rho}+p$, so $\sigma_{\rho}$ divides $p$. Since Theorem~\ref{t:girls} (i) also shows that $V(A^{k+\sigma_{\rho}},\rho^{k+\sigma_{\rho}})=V(A^k,\rho^k)$ for all $k\geq 1$, the result follows.
The second part can be proved similarly, using Theorem~\ref{t:girls}(ii) and Theorem~\ref{t:girls-major} (iii)$\Leftrightarrow$(v). \end{proof}
\section{Examples} \label{s:examples} We consider two examples of reducible nonnegative matrices, examining their core in max algebra and in nonnegative linear algebra.
\noindent {\em Example 1.} Take \begin{equation} A= \begin{pmatrix}
0.1206 & 0 & 0 & 0 & 0\\
0.5895 & 0.2904 & 1 & 0.8797 & 0.4253\\
0.2262 & 0.6171 & 0.3439 & 1 & 0.3127\\
0.3846 & 0.2653 & 0.5841 & 0.2607 & 1\\
0.5830 & 1 & 0.1078 & 0.5944 & 0.1788 \end{pmatrix}\,\enspace . \end{equation} $A$ has two classes with node sets $\{1\}$ and $\{2,3,4,5\}$. Both in max algebra and in nonnegative linear algebra, the only spectral class arises from $M:=\{2,3,4,5\}$. The max-algebraic Perron root of this class is $\rho^{\oplus}(A)=1$, and the critical graph consists of just one cycle $2\to 3\to 4\to 5\to 2$.
The eigencones $V_{\oplus}(A,1)$, $V_{\oplus}(A^{\otimes 2},1)$, $V_{\oplus}(A^{\otimes 3},1)$ and $V_{\oplus}(A^{\otimes 4},1)$ are generated by the last four columns of the Kleene stars $A^*$, $(A^{\otimes 2})^*$, $(A^{\otimes 3})^*$, $(A^{\otimes 4})^*$. Namely, \begin{equation*} \begin{split} V_{\oplus}(A,1)&=V_{\oplus}(A^{\otimes 3},1)=\spann_{\oplus}\{(0\ 1\ 1\ 1\ 1)\},\\ V_{\oplus}(A^{\otimes 2},1)=\spann_{\oplus} &\{(0,1,0.8797,1,0.8797),\ (0,0.8797,1,0.8797,1)\},\\ V_{\oplus}(A^{\otimes 4},1)=\spann_{\oplus} &\{(0,1,0.6807,0.7738,0.8797),\ (0,0.8797,1,0.6807,0.7738),\\
&(0,0.7738,0.8797,1,0.6807),\ (0,0.6807,0.7738,0.8797,1)\} \end{split} \end{equation*}
By Main Theorem~\ref{t:core}, $\core_{\oplus}(A)$ is equal to $V_{\oplus}(A^{\otimes 4},1)$. Computing the max-algebraic powers of $A$ we see that the sequence of submatrices $A^{\otimes t}_{MM}$ becomes periodic after $t=10$, with period $4$. In particular, \begin{equation} A^{\otimes 10}= \begin{pmatrix}
\alpha & 0 & 0 & 0 & 0\\
0.4511 & 0.7738 & 0.6807 & 1 & 0.8797\\
0.5128 & 0.8797 & 0.7738 & 0.6807 & 1\\
0.5830 & 1 & 0.8797 & 0.7738 & 0.6807\\
0.5895 & 0.6807 & 1 & 0.8797 & 0.7738 \end{pmatrix}, \end{equation} where $0<\alpha<0.0001$. Observe that the last four columns are precisely the ones that generate $V_{\oplus}(A^{\otimes 4},1)$. Moreover, if $\alpha$ was $0$ then the first column would be the following max-combination of the last four columns: $$ a^{\otimes 10}_{41}A^{\otimes 10}_{\cdot 2}\oplus a^{\otimes 10}_{51}A^{\otimes 10}_{\cdot 3}\oplus a^{\otimes 10}_{21}A^{\otimes 10}_{\cdot 4}\oplus a^{\otimes 10}_{31}A^{\otimes 10}_{\cdot 5}. $$ On the one hand, the first column of $A^{\otimes t}$ cannot be a max-combination of the last four columns for any $t>0$ since $a_{11}^{\otimes t}>0$. On the other hand, $a_{11}^{\otimes t}\to 0$ as $t\to\infty$ ensuring that the first column belongs to the core ``in the limit''.
Figure~\ref{f:petersfigure1} gives a symbolic illustration of what is going on in this example.
\begin{figure}
\caption{The spans of matrix powers (upper curve) and the periodic sequence of their eigencones (lower graph) in Example 1 (max algebra).}
\label{f:petersfigure1}
\end{figure}
\if{ \begin{figure}
\caption{The spans of matrix powers (upper curve) and the periodic sequence of their eigencones (lowergraph) in Example 1 (max algebra).}
\label{f:petersfigure1}
\end{figure} }\fi
In {\bf nonnegative algebra}, the block $A_{MM}$ with $M=\{2,3,4,5\}$ is also the only spectral block . Its Perron root is approximately $\rho^+(A)=2.2101$, and the corresponding eigencone is $$ V_+(A,\rho^+(A))=\spann_+\{(0,\ 0.5750,\ 0.5107,\ 0.4593,\ 0.4445)\}. $$ Taking the usual powers of $(A/\rho^+(A))$ we see that $$ \left(A/\rho^+(A)\right)^{\times 12}= \begin{pmatrix}
\alpha & 0 & 0 & 0 & 0\\
0.2457 & 0.2752 & 0.2711 & 0.3453 & 0.2693\\
0.2182 & 0.2444 & 0.2408 & 0.3067 & 0.2392\\
0.1963 & 0.2198 & 0.2165 & 0.2759 & 0.2151\\
0.1899 & 0.2127 & 0.2096 & 0.2670 & 0.2082 \end{pmatrix}, $$ where $0<\alpha<0.0001$, and that the first four digits of all entries in all higher powers are the same. It can be verified that the submatrix $(A/\rho^+(A))^{\times 12}_{MM}$ is, approximately, the outer product of the Perron eigenvector with itself, while the first column is also almost proportional to it.
\noindent {\em Example 2.} Take \begin{equation} A= \begin{pmatrix}
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0\\
0.6718 & 0.2240 & 0.5805 & 0.1868\\
0.6951 & 0.6678 & 0.4753 & 0.3735 \end{pmatrix}\,\enspace . \end{equation}
This matrix has two classes $\mu$ and $\nu$ with index sets $\{1,2\}$ and $\{3,4\}$, and both classes are spectral, in both algebras. In max algebra $\rho^{\oplus}_{\mu}=1$ and $\rho^{\oplus}_{\nu}=a_{33}<1$. The eigencones of matrix powers associated with $\rho^{\oplus}_{\mu}=1$ are \begin{equation*} \begin{split} V_{\oplus}(A,1)&=\spann_{\oplus}\{(1,1,0.6718,0.6951)\},\\ V_{\oplus}(A^{\otimes 2},1)&= \spann_{\oplus}\{(1,0,0.3900,0.6678),\ (0,1,0.6718,0.6951)\},\\ \end{split} \end{equation*} and the eigencone associated with $\rho^{\oplus}_{\nu}$ is generated by the third column of the matrix: $$ V_{\oplus}(A,\rho^{\oplus}_{\nu})=\spann_{\oplus}\{(0,\ 0,\ 0.5805,\ 0.4753)\}. $$
By Main Theorem~\ref{t:core}, $\core_{\oplus}(A)$ is equal to the (max-algebraic) sum of $V_{\oplus}(A^{\otimes 2},1)$ and $V_{\oplus}(A,\rho^{\oplus}_{\nu})$. To this end, observe that already in the second max-algebraic power \begin{equation} A^{\otimes 2}= \begin{pmatrix} 1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0.3900 & 0.6718 & 0.3370 & 0.1084\\
0.6678 & 0.6951 & 0.2759 & 0.1395 \end{pmatrix} \end{equation} the first two columns are the generators of $V_{\oplus}(A^{\otimes 2},1)$. However, the last column is still not proportional to the third one which shows that $\spann_{\oplus}(A^{\otimes 2})\neq \core_{\oplus}(A)$. However, it can be checked that this happens in $\spann_{\oplus}(A^{\otimes 4})$, with the first two columns still equal to the generators of $V_{\oplus}(A^{\otimes 2},1)$, which shows that $\spann_{\oplus}(A^{\otimes 4})$ is the sum of above mentioned max cones, and hence $\spann_{\oplus}(A^{\otimes 4})=\spann_{\oplus}(A^{\otimes 5})=\ldots=\core_{\oplus}(A)$. Hence we see that $A$ is column periodic (${\mathcal S}_5$) and the core finitely stabilizes. See Figure~\ref{f:petersfigure2} for a symbolic illustration.
\begin{figure}
\caption{The spans of matrix powers (upper graph) and the periodic sequence of their eigencones (lower graph) in Example 2 (max algebra)}
\label{f:petersfigure2}
\end{figure}
\if{ \begin{figure}
\caption{The spans of matrix powers (upper curve) and the periodic sequence of their eigencones (lower graph) in Example 2 (max algebra)}
\label{f:petersfigure2}
\end{figure} }\fi
In {\bf nonnegative algebra}, $\rho^+_{\mu}=1$ and $\rho^+_{\nu}=0.7924$. Computing the eigenvectors of $A$ and $A^{\times 2}$ yields \begin{equation*} \begin{split} V_+(A,1)&=\spann_+\{(0.1326,0.1326,0.6218,0.7604)\},\\ V_+(A^{\times 2},1)&= \spann_+\{(0.2646,0,0.5815,0.7693),\ (0,0.2566,0.6391,0.7251)\},\\ \end{split} \end{equation*} and $$ V_+(A,\rho^+_{\nu})=\spann_+\{(0,\ 0,\ 0.6612,\ 0.7502)\}. $$
Here $\core_+(A)$ is equal to the ordinary (Minkowski) sum of $V_+(A^{\times 2},1)$ and $V_+(A,\rho^+_{\nu})$. To this end, it can be observed that, within the first $4$ digits, the first two columns of $A^{\times t}$ become approximately periodic after $t=50$, and the columns of powers of the normalized submatrix $A_{\nu\nu}/\rho^+_{\nu}$ approximately stabilize after $t=7$. Of course, there is no finite stabilization of the core in this case. However, the structure of the nonnegative core is similar to the max-algebraic counterpart described above.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{frontmatter}
\title{Capacity of the range in dimension $5$} \runtitle{Capacity of the range in dimension $5$}
\author{\fnms{Bruno} \snm{Schapira}\ead[label=e1]{[email protected]}} \address{Aix-Marseille Universit\'e, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France\\ \printead{e1}}
\runauthor{Bruno Schapira}
\begin{abstract} We prove a Central limit theorem for the capacity of the range of a symmetric random walk on $\mathbb Z^5$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\ge 5$, for the probability that the ranges of two independent random walks intersect. The latter are then used for computing covariances of some intersection events, at the leading order. \end{abstract}
\begin{keyword}[class=MSC]
\kwd{60F05; 60G50; 60J45}
\end{keyword}
\begin{keyword} \kwd{Random Walk} \kwd{Range} \kwd{Capacity} \kwd{Central Limit Theorem} \kwd{Intersection of random walk ranges} \end{keyword}
\end{frontmatter}
\section{Introduction}\label{sec:intro} Consider a random walk $(S_n)_{n\ge 0}$ on $\mathbb{Z}^d$, that is a process of the form $S_n=S_0+X_1+\dots + X_n$, where the $(X_i)_{i\ge 1}$ are independent and identically distributed. A general question is to understand the geometric properties of its range, that is the random set $\mathcal{R}_n:=\{S_0,\dots,S_n\}$, and more specifically to analyze its large scale limiting behavior as the time $n$ is growing. In their pioneering work, Dvoretzky and Erd\'os \cite{DE51} proved a strong law of large numbers for the number of distinct sites in $\mathcal{R}_n$, in any dimension $d\ge 1$. Later a central limit theorem was obtained first by Jain and Orey \cite{JO69} in dimensions $d\ge 5$, then by Jain and Pruitt \cite{JP71} in dimension $3$ and higher, and finally by Le Gall \cite{LG86} in dimension $2$, under fairly general hypotheses on the common law of the $(X_i)_{i\ge 1}$. Furthermore, a lot of activity has been focused on analyzing the large and moderate deviations, which we will not discuss here.
More recently some papers were concerned with other functionals of the range, including its entropy \cite{BKYY}, and its boundary \cite{AS17, BKYY, BY, DGK, Ok16}. Here we will be interested in another natural way to measure the size of the range, which also captures some properties of its shape. Namely we will consider its Newtonian capacity, defined for a finite subset $A\subset \mathbb{Z}^d$, as \begin{equation}\label{cap.def} \mathrm{Cap}(A) :=\sum_{x\in A} \mathbb{P}_x[H_A^+ = \infty], \end{equation} where $\mathbb{P}_x$ is the law of the walk starting from $x$, and $H_A^+$ denotes the first return time to $A$ (see \eqref{HAHA+} below). Actually the first study of the capacity of the range goes back to the earlier work by Jain and Orey \cite{JO69}, who proved a law of large numbers in any dimension $d\ge 3$; and more precisely that almost surely, as $n\to \infty$, \begin{equation}\label{LLN.cap} \frac 1n \mathrm{Cap}(\mathcal{R}_n)\to \gamma_d, \end{equation} for some constant $\gamma_d$, which is nonzero if and only if $d\ge 5$ -- the latter observation being actually directly related to the fact that it is only in dimension $5$ and higher that two independent ranges have a positive probability not to intersect each other. However, until very recently to our knowledge there were no other work on the capacity of the range, even though the results of Lawler on the intersection of random walks incidentally gave a sharp asymptotic behavior of the mean in dimension four, see \cite{Law91}.
In a series of recent papers \cite{Chang, ASS18, ASS19}, the central limit theorem has been established for the simple random walk in any dimension $d\ge 3$, except for the case of dimension $5$, which remained unsolved so far. The main goal of this paper is to fill this gap, but in the mean time we obtain general results on the probability that the ranges of two independent walks intersect, which might be of independent interest. We furthermore obtain estimates for the covariances between such events, which is arguably one of the main novelty of our work; but we shall come back on this point a bit later.
Our hypotheses on the random walk are quite general: we only require that the distribution of the $(X_i)_{i\ge 1}$ is a symmetric and irreducible probability measure on $\mathbb{Z}^d$, which has a finite $d$-th moment. Under these hypotheses our first result is the following. \begin{theoremA}\label{theoremA} Assume $d=5$. There exists a constant $\sigma>0$, such that as $n\to \infty$, $$\operatorname{Var}(\mathrm{Cap}(\mathcal{R}_n)) \sim \sigma^2 \, n\log n.$$ \end{theoremA} We then deduce a central limit theorem. \begin{theoremB}\label{theoremB} Assume $d=5$. Then, $$\frac{\mathrm{Cap}(\mathcal{R}_n) - \gamma_5 n}{\sigma \sqrt{n\log n}}\quad \stackrel{(\mathcal L)}{\underset{n\to \infty}{\Longrightarrow}} \quad \mathcal N(0,1).$$ \end{theoremB} As already mentioned, along the proof we also obtain a precise asymptotic estimate for the probability that the ranges of two independent walks starting from far away intersect. Previously to our knowledge only the order of magnitude up to multiplicative constants had been established, see \cite{Law91}. Since our proof works the same in any dimension $d\ge 5$, we state our result in this general setting. Recall that to each random walk one can associate a norm (see below for a formal definition), which we denote here by $\mathcal J(\cdot)$ (in particular in the case of the simple random walk it coincides with the Euclidean norm). \begin{theoremC}\label{theoremC} Assume $d\ge 5$. Let $S$ and $\widetilde S$ be two independent random walks starting from the origin (with the same distribution).
There exists a constant $c>0$, such that as $\|x\|\to \infty$, $$\mathbb{P}\left[\mathcal{R}_\infty \cap (x+\widetilde \mathcal{R}_\infty)\neq \varnothing\right]\sim \frac{c}{\mathcal J(x)^{d-4}}.$$ \end{theoremC}
In fact we obtain a stronger and more general result. Indeed, first we get some control on the second order term, and show that it is $\mathcal{O} (\|x\|^{4-d-\nu})$, for some constant $\nu>0$. Moreover, we also consider some functionals of the position of one of the two walks at its hitting time of the other range. More precisely, we obtain asymptotic estimates for quantities of the form $\mathbb{E}[F(S_\tau){\text{\Large $\mathfrak 1$}}\{\tau<\infty\}]$, with $\tau$ the hitting time of the range $x+\widetilde \mathcal{R}_\infty$, for functions $F$ satisfying some regularity property, see \eqref{cond.F}. In particular, it applies to functions of the form $F(x)=1/\mathcal J(x)^\alpha$, for any $\alpha\in [0,1]$, for which we obtain that for some constants $\nu>0$, and $c>0$,
$$ \mathbb{E}\left[\frac{ {\text{\Large $\mathfrak 1$}}\{\tau<\infty\} }{1+\mathcal J(S_\tau)^\alpha}\right] = \frac{c}{\mathcal J(x)^{d-4+\alpha}} + \mathcal{O}\left(\|x\|^{4-\alpha-d- \nu}\right).$$ Moreover, the same kind of estimates is obtained when one considers rather $\tau$ as the hitting time of $x+\widetilde \mathcal{R}[0,\ell]$, with $\ell$ a finite integer. These results are then used to derive asymptotic estimates for covariances of hitting events in the following four situations: let $S$, $S^1$, $S^2$, and $S^3$, be four independent random walks on $\mathbb{Z}^5$, all starting from the origin and consider either \begin{itemize} \item[$(i)$] $A=\{\mathcal{R}_\infty^1 \cap \mathcal{R}[k,\infty)\neq \varnothing\}, \ \text{and}\ B= \{\mathcal{R}_\infty^2 \cap (S_k + \mathcal{R}_\infty^3 )\neq \varnothing\}$, \item[$(ii)$] $A=\{\mathcal{R}_\infty^1 \cap \mathcal{R}[k,\infty)\neq \varnothing\}, \text{ and } B= \{(S_k+\mathcal{R}_\infty^2) \cap \mathcal{R}[k+1,\infty)\neq \varnothing\}$, \item[$(iii)$] $A=\{\mathcal{R}_\infty^1 \cap \mathcal{R}[k,\infty)\neq \varnothing\}, \ \text{and}\ B= \{(S_k+\mathcal{R}_\infty^2) \cap \mathcal{R}[0,k-1] \neq \varnothing\}$, \item[$(iv)$] $A=\{\mathcal{R}_\infty^1 \cap \mathcal{R}[1,k] \neq \varnothing\}, \ \text{and}\ B= \{(S_k+\mathcal{R}_\infty^2) \cap \mathcal{R}[0,k-1] \neq \varnothing\}$. \end{itemize} In all these cases, we show that for some constant $c>0$, as $k\to \infty$, $$\operatorname{Cov}(A,B) \sim \frac{c}{k}. $$ Case $(i)$ is the easiest, and follows directly from Theorem C, since actually one can see that in this case both $\mathbb{P}[A\cap B]$ and $\mathbb{P}[A]\cdot \mathbb{P}[B]$ are asymptotically equivalent to a constant times the inverse of $k$. However, the other cases are more intricate, partly due to some cancellations that occur between the two terms, which, if estimated separately, are both of order $1/\sqrt{k}$ in cases $(ii)$ and $(iii)$, or even of order $1$ in case $(iv)$. In these cases, we rely on the extensions of Theorem C, that we just mentioned above. More precisely in case $(ii)$ we rely on the general result applied
with the functions $F(x)=1/\|x\|$, and its convolution with the distribution of $S_k$, while in cases $(iii)$ and $(iv)$ we use the extension to hitting times of finite windows of the range. We stress also that showing the positivity of the constants $c$ here is a delicate part of the proof, especially in case $(iv)$, where it relies on the following inequality:
$$\int_{0\le s \le t\le 1}\left( \mathbb{E}\left[\frac{1}{\|\beta_s-\beta_1\|^3\cdot \|\beta_t\|^3}\right] -\mathbb{E}\left[\frac{1}{\|\beta_s-\beta_1\|^3}\right] \mathbb{E}\left[\frac{1}{\|\beta_t\|^3}\right]\right) \, ds\, dt>0,$$ with $(\beta_u)_{u\ge 0}$ a standard Brownian motion in $\mathbb{R}^5$.
The paper is organized as follows. The next section is devoted to preliminaries, in particular we fix the main notation, recall known results on the transition kernel and the Green's function, and derive some basic estimates. In Section 3 we give the plan of the proof of Theorem A, which is cut into a number of intermediate results: Propositions \ref{prop.error}--\ref{prop.phipsi.2}. Propositions \ref{prop.error}--\ref{prop.phi0} are then proved in Sections $4$--$6$. The last one, which is also the most delicate one, requires Theorem C and its extensions. Its proof is therefore postponed to Section 8, while we first prove our general results on the intersection of two independent ranges in Section $7$, which is written in the general setting of random walks on $\mathbb{Z}^d$, for any $d\ge 5$, and can be read independently of the rest of the paper. Finally Section 9 is devoted to the proof of Theorem B, which is done by following a relatively well-established general scheme, based on the Lindeberg-Feller theorem for triangular arrays.
\section{Preliminaries} \subsection{Notation} We recall that we assume the law of the $(X_i)_{i\ge 1}$ to be a symmetric and irreducible probability measure\footnote{symmetric means that for all $x\in \mathbb{Z}^d$, $\mathbb{P}[X_1=x]=\mathbb{P}[X_1=-x]$, and irreducible means that for all $x$,
$\mathbb{P}[S_n=x]>0$, for some $n\ge 1$.} on $\mathbb{Z}^d$, $d\ge 5$, with a finite $d$-th moment\footnote{this means that $\mathbb{E}[\|X_1\|^d]<\infty$, with $\|\cdot \|$ the Euclidean norm.}. The walk is called aperiodic if the probability to be at the origin at time $n$ is nonzero for all $n$ large enough, and it is called bipartite if this probability is nonzero only when $n$ is even. Note that only these two cases may appear for a symmetric random walk.
Recall also that for $x\in \mathbb{Z}^d$, we denote by $\mathbb{P}_x$ the law of the walk starting from $S_0=x$. When $x=0$, we simply write it as $\mathbb{P}$. We denote its total range as $\mathcal{R}_\infty :=\{S_k\}_{k\ge 0}$, and for $0\le k\le n\le +\infty$, set $\mathcal{R}[k,n]:=\{S_k,\dots,S_n\}$.
For an integer $k\ge 2$, the law of $k$ independent random walks (with the same step distribution) starting from some $x_1,\dots, x_k\in \mathbb{Z}^5$, is denoted by $\mathbb{P}_{x_1,\dots,x_k}$, or simply by $\mathbb{P}$ when they all start from the origin.
We define \begin{equation}\label{HAHA+} H_A:=\inf\{n\ge 0\ : \ S_n\in A\},\quad \text{and} \quad H_A^+ :=\inf\{n\ge 1\ :\ S_n\in A\}, \end{equation} respectively for the hitting time and first return time to a subset $A\subset \mathbb{Z}^d$, that we abbreviate respectively as $H_x$ and $H_x^+$ when $A$ is a singleton $\{x\}$.
We let $\|x\|$ be the Euclidean norm of $x\in \mathbb{Z}^d$. If $X_1$ has covariance matrix $\Gamma= \Lambda \Lambda^t$, we define its associated norm as
$$\mathcal J^*(x) := |x\cdot \Gamma^{-1} x|^{1/2} = \|\Lambda^{-1} x\|,$$ and set $\mathcal J(x)= d^{-1/2}\mathcal J^*(x)$ (see \cite{LL} p.4 for more details).
For $a$ and $b$ some nonnegative reals, we let $a\wedge b:=\min(a,b)$ and $a\vee b:= \max(a,b)$.
We use the letters $c$ and $C$ to denote constants (which could depend on the covariance matrix of the walk), whose values might change from line to line. We also use standard notation for the comparison of functions: we write $f=\mathcal{O}(g)$, or sometimes $f\lesssim g$, if there exists a constant $C>0$, such that $f(x) \le Cg(x)$, for all $x$. Likewise, $f=o(g)$ means that $f/g \to 0$, and $f\sim g$ means that $f$ and $g$ are equivalent, that is if $|f-g| = o(f)$. Finally we write $f\asymp g$, when both $f=\mathcal{O}(g)$, and $g=\mathcal{O}(f)$.
\subsection{Transition kernel and Green's function} We denote by $p_n(x)$ the probability that a random walk starting from the origin ends up at position $x\in \mathbb{Z}^d$ after $n$ steps, that is $p_n(x):=\mathbb{P}[S_n=x]$, and note that for any $x,y\in \mathbb{Z}^d$, one has $\mathbb{P}_x[S_n=y] = p_n(y-x)$. Recall the definitions of $\Gamma$ and $\mathcal J^*$ from the previous subsection, and define \begin{equation}\label{pnbar} \overline p_n(x) := \frac {1}{(2\pi n)^{d/2} \sqrt{\det \Gamma}} \cdot e^{-\frac{\mathcal J^*(x)^2}{2n}}. \end{equation} The first tool we shall need is a local central limit theorem, roughly saying that $p_n(x)$ is well approximated by $\overline p_n(x)$, under appropriate hypotheses. Such result has a long history, see in particular the standard books by Feller \cite{Feller} and Spitzer \cite{Spitzer}. We refer here to the more recent book of Lawler and Limic \cite{LL}, and more precisely to their Theorem 2.3.5 in the case of an
aperiodic random walk, and to (the proof of) their Theorem 2.1.3 in the case of bipartite walks, which provide the result we need under minimal hypotheses (in particular it only requires a finite fourth-moment for $\|X_1\|$). \begin{theorem}[{\bf Local Central Limit Theorem}]\label{LCLT} There exists a constant $C>0$, such that for all $n\ge 1$, and all $x\in \mathbb{Z}^d$, \begin{eqnarray*}
|p_n(x)-\overline p_n(x)| \le \frac{C}{n^{(d+2)/2}}, \end{eqnarray*} in the case of an aperiodic walk, and for bipartite walks,
$$|p_n(x)+p_{n+1}(x)-2\overline p_n(x)| \le \frac{C}{n^{(d+2)/2}}.$$ \end{theorem}
In addition, under our hypotheses (in particular assuming $\mathbb{E}[\|X_1\|^d]<\infty$), there exists a constant $C>0$, such that for any $n\ge 1$ and any $x\in \mathbb{Z}^d$ (see Proposition 2.4.6 in \cite{LL}), \begin{equation}\label{pn.largex} p_n(x)\le C\cdot \left\{ \begin{array}{ll}
n^{-d/2} & \text{if }\|x\|\le \sqrt n,\\
\|x\|^{-d} & \text{if }\|x\|>\sqrt n. \end{array} \right. \end{equation} It is also known (see the proof of Proposition 2.4.6 in \cite{LL}) that \begin{equation}\label{norm.Sn}
\mathbb{E}[\|S_n\|^d] =\mathcal{O}(n^{d/2}). \end{equation} Together with the reflection principle (see Proposition 1.6.2 in \cite{LL}), and Markov's inequality, this gives that for any $n\ge 1$ and $r\ge 1$, \begin{equation}\label{Sn.large}
\mathbb{P}\left[\max_{0\le k\le n} \|S_k\|\ge r\right] \le C \cdot \left(\frac{\sqrt n}{r}\right)^{d}. \end{equation} Now we define for $\ell \ge 0$, $G_\ell(x) := \sum_{n\ge \ell} p_n(x)$. The {\bf Green's function} is the function $G:=G_0$. A union bound gives \begin{equation}\label{Green.hit}
\mathbb{P}[x\in \mathcal{R}[\ell,\infty)] \le G_\ell(x).
\end{equation} By \eqref{pn.largex} there exists a constant $C>0$, such that for any $x\in \mathbb{Z}^d$, and $\ell \ge 0$, \begin{equation}\label{Green}
G_\ell(x) \le \frac{C}{\|x\|^{d-2} + \ell^{\frac{d-2}{2}} + 1}. \end{equation}
It follows from this bound (together with the corresponding lower bound $G(x)\ge c\|x\|^{2-d}$, which can be deduced from Theorem \ref{LCLT}), and the fact that $G$ is harmonic on $\mathbb{Z}^d\setminus\{0\}$, that the hitting probability of a ball is bounded as follows (see the proof of \cite[Proposition 6.4.2]{LL}): \begin{equation}\label{hit.ball}
\mathbb{P}_x\left[\eta_r<\infty \right] =\mathcal{O}\left(\frac{r^{d-2}}{1+\|x\|^{d-2}}\right), \quad \text{with}\quad \eta_r:=\inf\{n\ge 0\ :\ \|S_n\|\le r\}. \end{equation} We shall need as well some control on the overshoot. We state the result we need as a lemma and provide a short proof for the sake of completeness. \begin{lemma}[{\bf Overshoot Lemma}]\label{hit.ball.overshoot}
There exists a constant $C>0$, such that for all $r\ge 1$, and all $x\in \mathbb{Z}^d$, with $\|x\|\ge r$, \begin{equation*}
\mathbb{P}_x[\eta_r<\infty,\, \|S_{\eta_r}\| \le r/2] \le \frac{C}{1+\|x\|^{d-2}}. \end{equation*} \end{lemma} \begin{proof} We closely follow the proof of Lemma 5.1.9 in \cite{LL}. Note first that one can alway assume that $r$ is large enough, for otherwise the result follows from \eqref{hit.ball}. Then define for $k\ge 0$,
$$Y_k:= \sum_{n= 0}^{\eta_r} {\text{\Large $\mathfrak 1$}}\{r+k \le \|S_n\|< r+(k+1)\}.$$ Let
$$g(x,k) = \mathbb{E}_x[Y_k] = \sum_{n=0}^\infty \mathbb{P}_x[r+k \le \|S_n\|\le r+k+1,\, n< \eta_r].$$ One has \begin{align*}
& \mathbb{P}_x[\eta_r<\infty, \, \|S_{\eta_r}\| \le r/2] = \sum_{n=0}^\infty \mathbb{P}_x[\eta_r=n+1, \, \|S_{\eta_r}\| \le r/2] \\ & = \sum_{n=0}^\infty \sum_{k=0}^\infty
\mathbb{P}_x[\eta_r=n+1, \, \|S_{\eta_r}\| \le r/2,\, r+k \le \|S_n\|< r+k+1]\\
& \le \sum_{k=0}^\infty \sum_{n=0}^\infty \mathbb{P}_x\left[\eta_r>n,\, r+k \le \|S_n\|\le r+k+1,\, \|S_{n+1}-S_n\| \ge \frac r2 + k\right]\\
& = \sum_{k=0}^\infty g(x,k) \mathbb{P}\left[\|X_1\| \ge \frac r2 + k \right] = \sum_{k=0}^\infty g(x,k) \sum_{\ell = k}^\infty \mathbb{P}\left[\frac r2 +\ell \le \|X_1\|< \frac r2 + \ell+1 \right]\\
& = \sum_{\ell = 0}^\infty \mathbb{P}\left[\frac r2 +\ell \le \|X_1\|< \frac r2 + \ell+1 \right]\sum_{k=0}^\ell g(x,k). \end{align*}
Now Theorem \ref{LCLT} shows that one has $\mathbb{P}_z[\|S_{\ell^2}\| \le r]\ge \rho$, for some constant $\rho>0$, uniformly in $r$ (large enough), $\ell\ge 1$, and $r\le \|z\|\le r+\ell$. It follows, exactly as in the proof of Lemma 5.1.9 from \cite{LL}, that for any $\ell\ge 1$,
$$\max_{\|z\|\le r+\ell} \sum_{0\le k< \ell} g(z,k) \le \frac{\ell^2}{\rho}.$$ Using in addition \eqref{hit.ball}, we get with the Markov property,
$$\sum_{0\le k< \ell} g(x,k) \le C \frac{(r+\ell)^{d-2}}{1+\|x\|^{d-2}} \cdot \ell^2,$$ for some constant $C>0$. As a consequence one has \begin{align*}
& \mathbb{P}_x[\eta_r<\infty, \, \|S_{\eta_r}\| \le r/2] \\
& \le \frac C{1+\|x\|^{d-2}} \sum_{\ell = 0}^\infty \mathbb{P}\left[\frac r2+ \ell \le \|X_1\|< \frac r2 +\ell + 1\right](r+\ell)^{d-2}(\ell+1) ^2\\
& \le \frac C{1+\|x\|^{d-2}} \mathbb{E}\left[\|X_1\|^{d-2}(\|X_1\| - r/2)^2{\text{\Large $\mathfrak 1$}}\{\|X_1\|\ge r/2\}\right] \le \frac{C}{1+\|x\|^{d-2}},
\end{align*}
since by hypothesis, the $d$-th moment of $X_1$ is finite. \end{proof}
\subsection{Basic tools} We prove here some elementary facts, which will be needed throughout the paper, and which are immediate consequences of the results from the previous subsection.
\begin{lemma}\label{lem.upconvolG} There exists $C>0$, such that for all $x\in \mathbb{Z}^d$, and $\ell \ge 0$,
$$ \sum_{z\in \mathbb{Z}^d} G_\ell(z) G(z-x) \le \frac{C}{\|x\|^{d-4}+\ell^{\frac{d-4}{2}} + 1}.$$ \end{lemma} \begin{proof} Assume first that $\ell =0$. Then by \eqref{Green}, \begin{align*}
\sum_{z\in \mathbb{Z}^d} G(z) G(z-x) & \lesssim \frac{1}{1+\|x\|^{d-2}} \left(\sum_{\|z\|\le 2\|x\|} \frac{1}{1+\|z\|^{d-2}} +\sum_{\|z-x\|\le \frac{\|x\|}{2}} \frac{1}{1+\|z-x\|^{d-2}} \right)\\
& \quad + \sum_{\|z\|\ge 2\|x\|}\frac{1}{1+\|z\|^{2(d-2)}} \lesssim \frac{1}{1+\|x\|^{d-4}}. \end{align*}
Assume next that $\ell\ge 1$. We distinguish two cases: if $\|x\|\le \sqrt \ell$, then by using \eqref{Green} again we deduce,
$$\sum_{z\in \mathbb{Z}^d} G_\ell(z) G(z-x) \lesssim \frac{1}{\ell^{d/2}}\cdot \sum_{\|z\|\le 2\sqrt \ell} \frac{1}{1+ \|z-x\|^{d-2}} + \sum_{\|z\|\ge 2\sqrt \ell} \frac 1{\|z\|^{2(d-2)}} \lesssim \frac{1}{ \ell^{\frac{d-4}2}}.$$
When $\|x\| >\sqrt \ell$, the result follows from case $\ell =0$, since $G_\ell(z) \le G(z)$. \end{proof}
\begin{lemma}\label{lem.sumG} One has, \begin{equation}\label{exp.Green} \sup_{x\in \mathbb{Z}^d} \, \mathbb{E}[G(S_n-x)] = \mathcal{O}\left(\frac 1{n^{\frac{d-2}{2}}}\right), \end{equation} and for any $\alpha \in [0,d)$, \begin{equation}\label{exp.Green.x}
\sup_{n\ge 0} \, \mathbb{E}\left[\frac 1{1+\|S_n-x\|^\alpha } \right] = \mathcal{O}\left(\frac 1{1+\|x\|^\alpha}\right). \end{equation} Moreover, when $d=5$, \begin{equation} \label{sumG2} \mathbb{E}\left[\Big(\sum_{n\ge k} G(S_n)\Big)^2\right] = \mathcal{O}\left(\frac 1k\right). \end{equation} \end{lemma} \begin{proof}
For \eqref{exp.Green}, we proceed similarly as in the proof of Lemma \ref{lem.upconvolG}. If $\|x\| \le \sqrt{n}$, one has using \eqref{pn.largex} and \eqref{Green}, \begin{align*}
& \mathbb{E}[G(S_n-x)] = \sum_{z\in \mathbb{Z}^d}p_n(z) G(z-x) \\
& \lesssim \frac 1{n^{d/2}} \sum_{\|z\|\le 2\sqrt n} \frac{1}{1+\|z-x\|^{d-2}} + \sum_{\|z\|>2\sqrt n} \frac {1}{\|z\|^{2d-2}} \lesssim n^{\frac{2-d}{2}}, \end{align*}
while if $\|x\|>\sqrt{n}$, we get as well
$$\mathbb{E}[G(S_n-x)]\lesssim \frac 1{n^{d/2}} \sum_{\|z\|\le \sqrt n/2} \frac{1}{\|x\|^{d-2}} + \sum_{\|z\|>\sqrt n/2} \frac {1}{\|z\|^d(1+\|z-x\|)^{d-2}} \lesssim n^{\frac{2-d}{2}}.$$ Considering now \eqref{exp.Green.x}, we write \begin{align*}
& \mathbb{E}\left[\frac 1{1+\|S_n-x\|^\alpha} \right] \le \frac{C}{1+\|x\|^\alpha} + \sum_{\|z-x\|\le \|x\|/2} \frac{p_n(z)}{1+\|z-x\|^\alpha} \\
& \stackrel{\eqref{pn.largex}}{\lesssim} \frac{1}{1+\|x\|^\alpha} + \frac{1}{1+\|x\|^d} \sum_{\|z-x\|\le \|x\|/2} \frac{1}{1+\|z-x\|^\alpha} \lesssim \frac{1}{1+\|x\|^\alpha}. \end{align*} Finally for \eqref{sumG2}, one has using the Markov property at the second line, \begin{align*} &\mathbb{E}\left[\Big(\sum_{n\ge k} G(S_n)\Big)^2\right] = \sum_{x,y} G(x)G(y)\mathbb{E}\left[\sum_{n,m\ge k}{\text{\Large $\mathfrak 1$}}\{S_n=x,S_m=y\}\right]\\ &\le 2\sum_{x,y} G(x)G(y)\sum_{n\ge k}\sum_{\ell \ge 0} p_n(x)p_\ell(y-x)= 2\sum_{x,y} G(x)G(y)G_k(x)G(y-x)\\
& \stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} \sum_x \frac{1}{\|x\|^4}G_k(x) \stackrel{\eqref{Green}}{\lesssim} \frac 1k. \end{align*} \end{proof}
The next result deals with the probability that two independent ranges intersect. Despite its proof is a rather straightforward consequence of the previous results, it already provides upper bounds of the right order (only off by a multiplicative constant). \begin{lemma}\label{lem.simplehit} Let $S$ and $\widetilde S$ be two independent walks starting respectively from the origin and some $x\in \mathbb{Z}^d$. Let also $\ell$ and $m$ be two given nonnegative integers (possibly infinite for $m$). Define $$\tau:=\inf\{n\ge 0\ :\ \widetilde S_n\in \mathcal{R}[\ell,\ell+m]\}.$$ Then, for any function $F:\mathbb{Z}^d\to \mathbb{R}_+$, \begin{equation}\label{lem.hit.1} \mathbb{E}_{0,x}[{\text{\Large $\mathfrak 1$}}\{\tau <\infty\}F(\widetilde S_\tau)] \le \sum_{i=\ell}^{\ell + m} \mathbb{E}[G(S_i-x)F(S_i)]. \end{equation} In particular, uniformly in $\ell$ and $m$, \begin{equation} \label{lem.hit.2}
\mathbb{P}_{0,x}[\tau<\infty] = \mathcal{O}\left(\frac{1}{1+\|x\|^{d-4} }\right). \end{equation} Moreover, uniformly in $x\in \mathbb{Z}^d$, \begin{equation}\label{lem.hit.3} \mathbb{P}_{0,x}[\tau<\infty] = \left\{ \begin{array}{ll}
\mathcal{O}\left(m\cdot \ell^{\frac{2-d}2}\right) & \text{if }m<\infty \\ \mathcal{O}\left(\ell^{\frac{4-d}2} \right) & \text{if }m=\infty. \end{array} \right. \end{equation} \end{lemma} \begin{proof} The first statement follows from \eqref{Green.hit}. Indeed using this, and the independence between $S$ and $\widetilde S$, we deduce that \begin{align*} \mathbb{E}_{0,x}[{\text{\Large $\mathfrak 1$}}\{\tau <\infty\}F(\widetilde S_\tau)] & \le \sum_{i=\ell}^{\ell + m} \mathbb{E}_{0,x}[{\text{\Large $\mathfrak 1$}}\{S_i\in \widetilde \mathcal{R}_\infty \} F(S_i)] \stackrel{\eqref{Green.hit}}{\le} \sum_{i=\ell}^{\ell +m} \mathbb{E}[G(S_i-x)F(S_i)]. \end{align*} For \eqref{lem.hit.2}, note first that it suffices to consider the case when $\ell=0$ and $m=\infty$, as otherwise the probability is just smaller. Taking now $F\equiv 1$ in \eqref{lem.hit.1}, and using Lemma \ref{lem.upconvolG} gives the result. Similarly \eqref{lem.hit.3} directly follows from \eqref{lem.hit.1} and \eqref{exp.Green}. \end{proof}
\section{Scheme of proof of Theorem A} \subsection{A last passage decomposition for the capacity of the range} We provide here a last passage decomposition for the capacity of the range, in the same fashion as the well-known decomposition for the size of the range, which goes back to the seminal paper by Dvoretzky and Erd\'os \cite{DE51}, and which was also used by Jain and Pruitt \cite{JP71} for their proof of the central limit theorem. We note that Jain and Orey \cite{JO69} used as well a similar decomposition in their analysis of the capacity of the range (in fact they used instead a first passage decomposition).
So let $(S_n)_{n\ge 0}$ be some random walk starting from the origin, and set $$\varphi_k^n:= \mathbb{P}_{S_k}[H_{\mathcal{R}_n}^+=\infty\mid \mathcal{R}_n], \text{ and } Z_k^n := {\text{\Large $\mathfrak 1$}}\{S_\ell \neq S_k,\ \text{for all } \ell = k+1,\dots, n\},$$ for all $0\le k\le n$, By definition of the capacity \eqref{cap.def}, one can write by recording the sites of $\mathcal{R}_n$ according to their last visit, $$\mathrm{Cap}(\mathcal{R}_n)=\sum_{k=0}^n Z_k^n\cdot \varphi_k^n.$$ A first simplification is to remove the dependance in $n$ in each of the terms in the sum. To do this, we need some additional notation: we consider $(S_n)_{n\in \mathbb{Z}}$ a two-sided random walk starting from the origin (that is $(S_n)_{n\ge 0}$ and $(S_{-n})_{n\ge 0}$ are two independent walks starting from the origin), and denote its total range by $\overline \mathcal{R}_\infty :=\{S_n\}_{n\in \mathbb{Z}}$. Then for $k\ge 0$, let $$\varphi(k):=\mathbb{P}_{S_k}[H_{\overline \mathcal{R}_\infty}^+ = \infty \mid (S_n)_{n\in \mathbb{Z}}], \text{ and } Z(k):= {\text{\Large $\mathfrak 1$}}\{S_\ell \neq S_k,\text{ for all }\ell \ge k+1 \}.$$ We note that $\varphi(k)$ can be zero with nonzero probability, but that $\mathbb{E}[\varphi(k)]\neq 0$ (see the proof of Theorem 6.5.10 in \cite{LL}). We then define $$\mathcal C_n : = \sum_{k=0}^n Z(k)\varphi(k),\quad \text{ and }\quad W_n:=\mathrm{Cap}(\mathcal{R}_n) - \mathcal C_n.$$ We will prove in a moment the following estimate. \begin{lemma} \label{lem.Wn} One has $$\mathbb{E}[W_n^2] = \mathcal O(n).$$ \end{lemma} Given this result, Theorem A reduces to an estimate of the variance of $\mathcal C_n$. To this end, we first observe that $$\operatorname{Var}(\mathcal C_n) = 2 \sum_{0\le \ell < k\le n} \operatorname{Cov}( Z(\ell)\varphi(\ell), Z(k)\varphi(k)) + \mathcal{O}(n).$$ Furthermore, by translation invariance, for any $\ell < k$, $$\operatorname{Cov}( Z(\ell)\varphi(\ell), Z(k)\varphi(k)) = \operatorname{Cov}(Z(0)\varphi(0), Z(k-\ell)\varphi(k-\ell)),$$ so that in fact $$\operatorname{Var}(\mathcal C_n) = 2\sum_{\ell = 1}^n \sum_{k=1}^{\ell} \operatorname{Cov}( Z(0)\varphi(0), Z(k)\varphi(k)) + \mathcal{O}(n).$$ Thus Theorem A is a direct consequence of the following theorem. \begin{theorem}\label{prop.cov} There exists a constant $\sigma>0$, such that \begin{equation*} \operatorname{Cov}( Z(0)\varphi(0), Z(k)\varphi(k)) \sim \frac{\sigma^2}{2k}. \end{equation*} \end{theorem} This result is the core of the paper, and uses in particular Theorem C (in fact some more general statement, see Theorem \ref{thm.asymptotic}). More details about its proof will be given in the next subsection, but first we show that $W_n$ is negligible by giving the proof of Lemma \ref{lem.Wn}.
\begin{proof}[Proof of Lemma \ref{lem.Wn}] Note that $W_n=W_{n,1} + W_{n,2}$, with $$W_{n,1} = \sum_{k=0}^n (Z_k^n - Z(k))\varphi_k^n, \quad \text{and}\quad W_{n,2} = \sum_{k=0}^n(\varphi_k^n- \varphi(k)) Z(k).$$ Consider first the term $W_{n,1}$ which is easier. Observe that $Z_k^n-Z(k)$ is nonnegative and bounded by the indicator function of the event $\{S_k\in \mathcal{R}[n+1,\infty)\}$. Bounding also $\varphi_k^n$ by one, we get \begin{align*} \mathbb{E}[W_{n,1}^2] & \le \sum_{\ell = 0}^n \sum_{k=0}^n \mathbb{E}[(Z_\ell^n-Z(\ell))(Z_k^n -Z(k))] \\ &\le \sum_{\ell = 0}^n \sum_{k=0}^n \mathbb{P}\left[S_\ell \in \mathcal{R}[n+1,\infty), \, S_k\in \mathcal{R}[n+1,\infty)\right]. \end{align*} Then noting that $(S_{n+1-k}-S_{n+1})_{k\ge 0}$ and $(S_{n+1+k}-S_{n+1})_{k\ge 0}$ are two independent random walks starting from the origin, we obtain \begin{align*}
\mathbb{E}[W_{n,1}^2] & \le \sum_{\ell =1}^{n+1} \sum_{k=1}^{n+1} \mathbb{P}[H_{S_\ell} <\infty, \, H_{S_k}<\infty]\le 2 \sum_{\ell = 1}^{n+1} \sum_{k=1}^{n+1} \mathbb{P}[H_{S_\ell} \le H_{S_k} <\infty] \\ &\le 2 \sum_{1\le \ell \le k\le n+1} \mathbb{P}[H_{S_\ell} \le H_{S_k} <\infty] + \mathbb{P}[H_{S_k} \le H_{S_\ell} <\infty]. \end{align*} Using next the Markov property and \eqref{Green.hit}, we get with $S$ and $\widetilde S$ two independent random walks starting from the origin, \begin{align*}
\mathbb{E}[W_{n,1}^2] & \le 2 \sum_{1\le \ell \le k\le n+1} \mathbb{E}[G(S_\ell) G(S_k-S_\ell)] + \mathbb{E}[G(S_k) G(S_k- S_\ell)]\\ &\le 2 \sum_{\ell = 1}^{n+1} \sum_{k=0}^n \mathbb{E}[G(S_\ell)] \cdot \mathbb{E}[G(S_k)] + \mathbb{E}[G(S_\ell + \widetilde S_k) G(\widetilde S_k)]\\ &\le 4 \left(\sup_{x\in \mathbb{Z}^5} \sum_{\ell \ge 0} \mathbb{E}[G(x+S_\ell)]\right)^2 \stackrel{\eqref{exp.Green}}{=} \mathcal O(1). \end{align*} We proceed similarly with $W_{n,2}$. Observe first that for any $k\ge 0$, $$0 \le \varphi_k^n - \varphi(k) \le \mathbb{P}_{S_k}[H_{\mathcal{R}(-\infty,0]}<\infty\mid S] + \mathbb{P}_{S_k}[H_{\mathcal{R}[n,\infty)}<\infty\mid S].$$ Furthermore, for any $0\le \ell \le k\le n$, the two terms $\mathbb{P}_{S_\ell}[H_{\mathcal{R}(-\infty,0]}<\infty\mid S]$ and $\mathbb{P}_{S_k}[H_{\mathcal{R}[n,\infty)}<\infty\mid S]$ are independent. Therefore, \begin{align}\label{Wn2}
\nonumber \mathbb{E}[W_{n,2}^2] \le & \sum_{\ell = 0}^n \sum_{k=0}^n \mathbb{E}[(\varphi_\ell^n-\varphi(\ell))(\varphi_k^n-\varphi(k))] \le 2\left(\sum_{\ell = 0}^n \mathbb{P}\left[ H_{\mathcal{R}[\ell, \infty)}<\infty\right]\right)^2 \\ & + 4 \sum_{0\le \ell \le k\le n} \mathbb{P}\left[\mathcal{R}^3_\infty \cap (S_\ell + \mathcal{R}^1_\infty) \neq \varnothing, \, \mathcal{R}^3_\infty \cap (S_k+\mathcal{R}_\infty^2)\neq \varnothing \right], \end{align} where in the last term $\mathcal{R}^1_\infty$, $\mathcal{R}^2_\infty$ and $\mathcal{R}^3_\infty$ are the ranges of three (one-sided) independent walks, independent of $(S_n)_{n\ge 0}$, starting from the origin (denoting here $(S_{-n})_{n\ge 0}$ as another walk $(S^3_n)_{n\ge 0}$). Now \eqref{lem.hit.3} already shows that the first term on the right hand side of \eqref{Wn2} is $\mathcal{O}(n)$. For the second one, note that for any $0\le \ell \le k\le n$, one has \begin{align*} &\mathbb{P}\left[\mathcal{R}^3_\infty \cap (S_\ell + \mathcal{R}^1_\infty) \neq \varnothing, \, \mathcal{R}^3_\infty \cap (S_k+\mathcal{R}_\infty^2)\neq \varnothing \right]\\
\le & \
\mathbb{E}\left[|\mathcal{R}^3_\infty \cap (S_\ell + \mathcal{R}^1_\infty)| \cdot | \mathcal{R}^3_\infty \cap (S_k+\mathcal{R}_\infty^2)| \right]\\
= &\ \mathbb{E}\left[\mathbb{E}[|\mathcal{R}^3_\infty \cap (S_\ell + \mathcal{R}^1_\infty)|\mid S,\, S^3] \cdot \mathbb{E}[| \mathcal{R}^3_\infty \cap (S_k+\mathcal{R}_\infty^2)|\mid S,\, S^3] \right] \\
\stackrel{\eqref{Green.hit}}{\le} & \mathbb{E}\left[ \Big(\sum_{m\ge 0} G(S^3_m - S_\ell) \Big) \Big(\sum_{m\ge 0} G(S^3_m - S_k) \Big) \right] = \mathbb{E}\left[ \Big(\sum_{m\ge k} G(S_m - S_{k-\ell}) \Big) \Big(\sum_{m\ge k} G(S_m) \Big) \right]\\ \le & \ \mathbb{E}\left[ \Big(\sum_{m\ge \ell} G(S_m) \Big)^2\right]^{1/2} \cdot \mathbb{E}\left[ \Big(\sum_{m\ge k} G(S_m)\Big)^2\right]^{1/2} \stackrel{\eqref{sumG2}}{=} \mathcal{O}\left(\frac{1}{1+\sqrt{k\ell}}\right), \end{align*} using invariance by time reversal at the penultimate line, and Cauchy-Schwarz at the last one. This concludes the proof of the lemma. \end{proof}
\subsection{Scheme of proof of Theorem \ref{prop.cov}} We provide here some decomposition of $\varphi(0)$ and $\varphi(k)$ into a sum of terms involving intersection and non-intersection probabilities of different parts of the path $(S_n)_{n\in \mathbb{Z}}$. For this, we consider some sequence of integers $(\varepsilon_k)_{k\ge 1}$ satisfying $k>2\varepsilon_k$, for all $k\ge 3$, and whose value will be fixed later. A first step in our analysis is to reduce the influence of the random variables $Z(0)$ and $Z(k)$, which play a very minor role in the whole proof. Thus we define $$Z_0:={\text{\Large $\mathfrak 1$}}\{S_\ell\neq 0,\, \forall \ell=1,\dots,\varepsilon_k\}, \text{ and } Z_k:={\text{\Large $\mathfrak 1$}}\{S_\ell\neq S_k, \, \forall \ell =k+1,\dots,k+\varepsilon_k\}.$$ Note that these notation are slightly misleading (as in fact $Z_0$ and $Z_k$ depend on $\varepsilon_k$, but this shall hopefully not cause any confusion). One has
$$\mathbb{E}[|Z(0)- Z_0|]= \mathbb{P}[0\in \mathcal{R}[\varepsilon_k+1,\infty)] \stackrel{\eqref{Green.hit}}{\le} G_{\varepsilon_k}(0) \stackrel{\eqref{Green}}{=} \mathcal{O}(\varepsilon_k^{-3/2}),$$
and the same estimate holds for $\mathbb{E}[|Z(k)-Z_k|]$, by the Markov property. Therefore, $$\operatorname{Cov}(Z(0)\varphi(0),Z(k)\varphi(k)) = \operatorname{Cov}(Z_0\varphi(0),Z_k\varphi(k)) + \mathcal{O}(\varepsilon_k^{-3/2}).$$ Then recall that we consider a two-sided walk $(S_n)_{n\in \mathbb{Z}}$, and that $\varphi(0) = \mathbb{P}[H_{\mathcal{R}(-\infty, \infty)}^+=\infty\mid S]$. Thus one can decompose $\varphi(0)$ as follows: $$\varphi(0) =\varphi_0- \varphi_1- \varphi_2-\varphi_3 + \varphi_{1,2} +\varphi_{1,3} + \varphi_{2,3} - \varphi_{1,2,3},$$ with $$\varphi_0:=\mathbb{P}[H_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}^+=\infty\mid S],\quad \varphi_1: = \mathbb{P}[H^+_{\mathcal{R}(-\infty,-\varepsilon_k-1]}<\infty, \, H_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}^+=\infty \mid S], $$ $$\varphi_2 := \mathbb{P}[H^+_{\mathcal{R}[\varepsilon_k+1,k]}<\infty, H_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}^+=\infty \mid S], \, \varphi_3 := \mathbb{P}[H^+_{\mathcal{R}[k+1,\infty)}<\infty , H_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}^+=\infty\mid S],$$ $$\varphi_{1,2}: = \mathbb{P}[H^+_{\mathcal{R}(-\infty,-\varepsilon_k-1]}<\infty , \, H^+_{\mathcal{R}[\varepsilon_k+1,k]}<\infty , \, H_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}^+=\infty \mid S], $$ $$\varphi_{1,3} := \mathbb{P}[H^+_{\mathcal{R}(-\infty,-\varepsilon_k-1]}<\infty , \, H^+_{\mathcal{R}[k+1,\infty)}<\infty, \, H_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}^+=\infty \mid S],$$ $$\varphi_{2,3} := \mathbb{P}[H^+_{\mathcal{R}[\varepsilon_k+1,k]}<\infty,\, H^+_{\mathcal{R}[k+1,\infty)}<\infty, \, H_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}^+=\infty \mid S],$$ $$\varphi_{1,2,3} := \mathbb{P}[H^+_{\mathcal{R}(-\infty, -\varepsilon_k-1]}<\infty, H^+_{\mathcal{R}[\varepsilon_k+1,k]}<\infty, H^+_{\mathcal{R}[k+1,\infty)}<\infty, H_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}^+=\infty \mid S].$$ We decompose similarly $$\varphi(k)=\psi_0 - \psi_1 - \psi_2 - \psi_3 + \psi_{1,2} + \psi_{1,3} + \psi_{2,3} - \psi_{1,2,3},$$ where index $0$ refers to the event of avoiding $\mathcal{R}[k-\varepsilon_k,k+\varepsilon_k]$, index $1$ to the event of hitting $\mathcal{R}(-\infty,-1]$, index $2$ to the event of hitting $\mathcal{R}[0,k-\varepsilon_k-1]$ and index $3$ to the event of hitting $\mathcal{R}[k+\varepsilon_k+1,\infty)$ (for a walk starting from $S_k$ this time). Note that $\varphi_0$ and $\psi_0$ are independent. Then write \begin{align}\label{main.dec}
& \operatorname{Cov}(Z_0\varphi(0),Z_k\varphi(k)) = -\sum_{i=1}^3 \left(\operatorname{Cov}(Z_0\varphi_i,Z_k\psi_0)+ \operatorname{Cov}(Z_0\varphi_0,Z_k\psi_i)\right) \\ \nonumber & + \sum_{i,j=1}^3 \operatorname{Cov}(Z_0\varphi_i , Z_k\psi_j) +\sum_{1\le i <j\le 3} \left(\operatorname{Cov}(Z_0\varphi_{i,j},Z_k\psi_0) + \operatorname{Cov}(Z_0\varphi_0,Z_k\psi_{i,j})\right)+ R_{0,k}, \end{align} where $R_{0,k}$ is an error term. Our first task will be to show that it is negligible. \begin{proposition}\label{prop.error}
One has $|R_{0,k}| = \mathcal{O}\left(\varepsilon_k^{-3/2}\right)$. \end{proposition} The second step is the following. \begin{proposition}\label{prop.ij0} One has \begin{itemize} \item[(i)]
$|\operatorname{Cov}(Z_0\varphi_{1,2},Z_k\psi_0)|+|\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_{2,3})| = \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right)$,
\item[(ii)] $|\operatorname{Cov}(Z_0\varphi_{1,3},Z_k\psi_0)| + |\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_{1,3})| = \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k}) + \frac{1}{\varepsilon_k^{3/4} \sqrt k}\right)$, \item[(iii)]
$|\operatorname{Cov}(Z_0\varphi_{2,3},Z_k\psi_0)| +|\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_{1,2})|= \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k})+ \frac{1}{\varepsilon_k^{3/4} \sqrt k}\right)$. \end{itemize} \end{proposition} In the same fashion as Part (i) of the previous proposition, we show: \begin{proposition}\label{prop.phipsi.1} For any $1\le i<j\le 3$,
$$|\operatorname{Cov}(Z_0\varphi_i,Z_k\psi_j)| =\mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right), \quad |\operatorname{Cov}(Z_0\varphi_j,Z_k\psi_i)| =\mathcal{O}\left(\frac{1}{\varepsilon_k}\right).$$ \end{proposition} The next step deals with the first sum in the right-hand side of \eqref{main.dec}. \begin{proposition}\label{prop.phi0} There exists a constant $\alpha\in (0,1)$, such that $$\operatorname{Cov}(Z_0\varphi_1, Z_k\psi_0) = \operatorname{Cov}(Z_0\varphi_0,Z_k\psi_3) = 0,$$
$$|\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_0)|+ |\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_2)| = \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}} \right),$$
$$|\operatorname{Cov}(Z_0\varphi_3,Z_k\psi_0)|+ |\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_1)| = \mathcal{O}\left(\frac{ \varepsilon_k^\alpha}{k^{1+\alpha}} \right).$$ \end{proposition} At this point one can already deduce the bound $\operatorname{Var}(\mathrm{Cap}(\mathcal{R}_n))= \mathcal{O}(n \log n)$, just applying the previous propositions with say $\varepsilon_k:=\lfloor k/4\rfloor$.
In order to obtain the finer asymptotic result stated in Theorem \ref{prop.cov}, it remains to identify the leading terms in \eqref{main.dec}, which is the most delicate part. The result reads as follows. \begin{proposition}\label{prop.phipsi.2} There exists $\delta>0$, such that if $\varepsilon_k\ge k^{1-\delta}$ and $\varepsilon_k = o(k)$, then for some positive constants $(\sigma_{i,j})_{1\le i\le j\le 3}$, $$\operatorname{Cov}(Z_0\varphi_j,Z_k\psi_i)\sim \operatorname{Cov}(Z_0\varphi_{4-i},Z_k\psi_{4-j}) \sim \frac{\sigma_{i,j}}{k}.$$ \end{proposition} Note that Theorem \ref{prop.cov} is a direct consequence of \eqref{main.dec} and Propositions \ref{prop.error}--\ref{prop.phipsi.2}, which we prove now in the following sections.
\section{Proof of Proposition \ref{prop.error}} We divide the proof into two lemmas. \begin{lemma}\label{lem.123} One has $$\mathbb{E}[\varphi_{1,2,3}] = \mathcal{O}\left(\frac{1}{\varepsilon_k \sqrt k}\right),\quad \text{and}\quad \mathbb{E}[\psi_{1,2,3}] = \mathcal{O}\left(\frac{1}{\varepsilon_k\sqrt k}\right).$$ \end{lemma}
\begin{lemma}\label{lem.ijl} For any $1\le i<j\le 3$, and any $1\le \ell\le 3$, $$ \mathbb{E}[\varphi_{i,j}\psi_\ell] =\mathcal{O}\left(\varepsilon_k^{-3/2} \right), \quad \text{and} \quad \mathbb{E}[\varphi_{i,j}] \cdot \mathbb{E}[\psi_\ell] =\mathcal{O}\left(\varepsilon_k^{-3/2} \right) .$$ \end{lemma} Observe that the $(\varphi_{i,j})_{i,j}$ and $(\psi_{i,j})_{i,j}$ have the same law (up to reordering), and similarly for the $(\varphi_i)_i$ and $(\psi_i)_{i}$. Furthermore, $\varphi_{i,j}\le \varphi_i$ for any $i,j$. Therefore by definition of $R_{0,k}$ the proof of Proposition \ref{prop.error} readily follows from these two lemmas. For their proofs, we will use the following fact. \begin{lemma}\label{lem.prep.123} There exists $C>0$, such that for any $x,y\in \mathbb{Z}^5$, $0\le \ell \le m$, \begin{equation*} \sum_{i=\ell}^m \sum_{z\in \mathbb{Z}^5} p_i(z) G(z-y) p_{m-i}(z-x) \le
\frac{C}{(1+\|x\|+\sqrt m)^5} \left(\frac 1{1+\|y-x\|} + \frac{1}{1+\sqrt{\ell}+\|y\|}\right). \end{equation*} \end{lemma} \begin{proof}
Consider first the case $\|x\|\le \sqrt m$. By \eqref{pn.largex} and Lemma \ref{lem.upconvolG}, $$ \sum_{i=\ell}^{\lfloor m/2\rfloor } \sum_{z\in \mathbb{Z}^5} p_i(z) G(z-y) p_{m-i}(z-x) \lesssim \frac{1}{1+m^{5/2}} \sum_{z\in \mathbb{Z}^5} G_{\ell}(z) G(z-y)
\lesssim \frac{(1+ m)^{-5/2}}{1+\sqrt{\ell}+\|y\|}, $$ with the convention that the first sum is zero when $m<2\ell$, and $$\sum_{i=\lfloor m/2\rfloor }^m \sum_{z\in \mathbb{Z}^5} p_i(z) G(z-y) p_{m-i}(z-x) \lesssim \frac{1}{1+m^{5/2}} \sum_{z\in \mathbb{Z}^5} G(z-y) G(z-x)
\lesssim \frac{(1+m)^{-5/2}} {1+\|y-x\|}. $$
Likewise, when $\|x\|>\sqrt m$, applying again \eqref{pn.largex} and Lemma \ref{lem.upconvolG}, we get \begin{align*}
& \sum_{i=\ell}^{m} \sum_{\|z-x\| \ge \frac{\|x\|}{2}} p_i(z) G(z-y) p_{m-i}(z-x) \lesssim \frac{1}{\|x\|^5} \sum_{z\in \mathbb{Z}^5} G_{\ell}(z)G(z-y) \lesssim \frac{\|x\|^{-5}}{ 1+\sqrt{\ell}+\|y\|},\\
& \sum_{i=\ell}^{m} \sum_{\|z-x\| \le \frac{\|x\|}{2}} p_i(z) G(z-y) p_{m-i}(z-x) \lesssim \frac{1}{\|x\|^5} \sum_{z\in \mathbb{Z}^5} G(z-y)G(z-x) \lesssim \frac{\|x\|^{-5}}{1+\|y-x\|}, \end{align*} which concludes the proof of the lemma. \end{proof}
One can now give the proof of Lemma \ref{lem.123}. \begin{proof}[Proof of Lemma \ref{lem.123}] Since $\varphi_{1,2,3}$ and $\psi_{1,2,3}$ have the same law, it suffices to prove the result for $\varphi_{1,2,3}$. Let $(S_n)_{n\in \mathbb{Z}}$ and $(\widetilde S_n)_{n\ge 0}$ be two independent random walks starting from the origin. Define $$\tau_1:=\inf\{n\ge 1\, :\, \widetilde S_n \in \mathcal{R}(-\infty,-\varepsilon_k-1]\},\ \tau_2:=\inf\{n\ge 1\, :\, \widetilde S_n \in \mathcal{R}[\varepsilon_k+1,k]\},$$ and $$\tau_3:= \inf\{n\ge 1\, :\, \widetilde S_n\in \mathcal{R}[k+1,\infty)\}.$$ One has \begin{equation}\label{phi123.tauij} \mathbb{E}[\varphi_{1,2,3}] \le \sum_{i_1\neq i_2 \neq i_3} \mathbb{P}[\tau_{i_1}\le \tau_{i_2}\le \tau_{i_3}]. \end{equation} We first consider the term corresponding to $i_1=1$, $i_2=2$, and $i_3=3$. One has by the Markov property, \begin{align*}
\mathbb{P}[\tau_1\le \tau_2\le \tau_3<\infty] \stackrel{\eqref{lem.hit.2}}{\lesssim} \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_2<\infty\}}{1+\|\widetilde S_{\tau_2} - S_k\|}\right]\stackrel{\eqref{lem.hit.1}}{\lesssim} \sum_{i=\varepsilon_k}^k \mathbb{E}\left[ \frac{G(S_i-\widetilde S_{\tau_1}){\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}}{1+\| S_i - S_k\|}\right]. \end{align*} Now define $\mathcal{G}_i:=\sigma((S_j)_{j\le i})\vee \sigma((\widetilde S_n)_{n\ge 0})$, and note that $\tau_1$ is $\mathcal{G}_i$-measurable for any $i\ge 0$. Moreover, the Markov property and \eqref{pn.largex} show that
$$\mathbb{E}\left[\frac{1}{1+\|S_i - S_k\|}\mid \mathcal{G}_i\right] \lesssim \frac{1}{\sqrt{k-i}}.$$ Therefore, \begin{align*} & \mathbb{P}[\tau_1\le \tau_2\le \tau_3<\infty] \lesssim \sum_{i=\varepsilon_k}^k \mathbb{E}\left[{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}\cdot \frac{G(S_i-\widetilde S_{\tau_1})}{1+\sqrt{k-i}}\right]\\ & \lesssim \sum_{z\in \mathbb{Z}^5}\mathbb{P}[\tau_1<\infty, \, \widetilde S_{\tau_1}=z] \cdot\left( \sum_{i=\varepsilon_k}^{k/2} \frac{\mathbb{E}[G(S_i-z)]}{\sqrt k} + \sum_{i=k/2}^k \frac{\mathbb{E}[G(S_i-z)]}{1+\sqrt{k-i}}\right)\\ & \stackrel{\eqref{exp.Green}}{\lesssim} \frac{1}{\sqrt{k\varepsilon_k}}\cdot \mathbb{P}[\tau_1<\infty] \stackrel{\eqref{lem.hit.2}}{\lesssim}\frac 1{\varepsilon_k\sqrt k}. \end{align*} We consider next the term corresponding to $i_1=1$, $i_2=3$ and $i_3=2$, whose analysis slightly differs from the previous one. First Lemma \ref{lem.prep.123} gives \begin{align}\label{tau132}
& \mathbb{P}[\tau_1\le \tau_3\le \tau_2<\infty] =\sum_{x,y\in \mathbb{Z}^5} \mathbb{E}\left[{\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_3<\infty, \widetilde S_{\tau_3}=y, S_k=x\} \sum_{i=\varepsilon_k}^k G(S_i-y) \right]\\ \nonumber = & \sum_{x,y\in \mathbb{Z}^5} \left(\sum_{i=\varepsilon_k}^k \sum_{z\in \mathbb{Z}^5} p_i(z)G(z-y) p_{k-i}(x-z)\right) \mathbb{P}\left[\tau_1\le \tau_3<\infty, \widetilde S_{\tau_3}=y\mid S_k=x\right]\\
\nonumber \lesssim & \sum_{x\in \mathbb{Z}^5} \frac{1}{(\|x\| + \sqrt k)^5}\left(\frac{\mathbb{P}[\tau_1\le \tau_3 <\infty\mid S_k=x]}{\sqrt{\varepsilon_k}} +
\mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big| \ S_k=x\right] \right). \end{align} We then have \begin{align*} & \mathbb{P}[\tau_1\le \tau_3<\infty\mid S_k=x] \stackrel{\eqref{lem.hit.2}}{\lesssim}
\mathbb{E}\left[ \frac { {\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}}{1+\|\widetilde S_{\tau_1}-x\|}\right] \\
&\stackrel{\eqref{lem.hit.1}}{\lesssim} \sum_{y\in \mathbb{Z}^5} \frac{G_{\varepsilon_k}(y) G(y)}{1+\|y-x\|} \stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} \frac{1}{(1+\|x\|) \sqrt \varepsilon_k} + \sum_{\|y-x\|\le \frac{\|x\|}{2}}\frac{G_{\varepsilon_k}(y) G(y)}{1+\|y-x\|} . \end{align*}
Moreover, when $\|x\|\ge \sqrt {\varepsilon_k}$, one has \begin{align*}
\sum_{\|y-x\|\le \frac{\|x\|}{2}} \frac{G_{\varepsilon_k}(y) G(y)}{1+\|y-x\|} \stackrel{\eqref{Green}}{\lesssim} \frac{1}{\|x\|^6} \sum_{\|y-x\|\le \frac{\|x\|}{2}} \frac{1}{1+\|y-x\|}
\lesssim \frac{1}{\|x\|^2}, \end{align*}
while, when $\|x\|\le \sqrt {\varepsilon_k}$,
$$\sum_{\|y-x\|\le \frac{\|x\|}{2}} \frac{G_{\varepsilon_k}(y) G(y)}{1+\|y-x\|} \stackrel{\eqref{Green}}{\lesssim} (1+\|x\|) \varepsilon_k^{-3/2} \lesssim \frac {1}{\varepsilon_k}.$$ Therefore, it holds for any $x$, \begin{equation}\label{tau132.1}
\mathbb{P}[\tau_1\le \tau_3<\infty\mid S_k=x] \lesssim \frac{1}{(1+\|x\|)\sqrt \varepsilon_k}. \end{equation} Similarly, one has \begin{align}\label{tau132.2}
\nonumber & \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big| \ S_k=x\right] \le \mathbb{E}\left[ \sum_{y\in \mathbb{Z}^5} \frac{G(y-\widetilde S_{\tau_1}) G(y-x)}{1+\|y-x\|}{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}\right]\\
& \le \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}}{1+\|\widetilde S_{\tau_1} - x\|^2}\right] \le \sum_{y\in \mathbb{Z}^5} \frac{G_{\varepsilon_k}(y) G(y)}{1+\|y-x\|^2}
\lesssim \frac{1}{(1+\|x\|^2)\sqrt{\varepsilon_k}}. \end{align} Injecting \eqref{tau132.1} and \eqref{tau132.2} into \eqref{tau132} finally gives $$\mathbb{P}[\tau_1\le \tau_2\le \tau_3<\infty] \lesssim \frac{1}{\varepsilon_k\sqrt k}.$$ The other terms in \eqref{phi123.tauij} are entirely similar, so this concludes the proof of the lemma. \end{proof}
For the proof of Lemma \ref{lem.ijl}, one needs some additional estimates that we state as two separate lemmas. \begin{lemma}\label{lem.prep.ijl} There exists a constant $C>0$, such that for any $x,y\in \mathbb{Z}^5$, \begin{align*}
\sum_{i=\varepsilon_k}^{k-\varepsilon_k}\sum_{z\in \mathbb{Z}^5} & \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}\left(\frac{1}{1+\|z-x\|} + \frac 1{\sqrt{k-i}}\right) \\ & \le C\cdot \left\{ \begin{array}{ll}
\frac{1}{k^{5/2}}\left( \frac 1{1+\|x\|^2} + \frac{1}{\varepsilon_k}\right) + \frac{1}{k^{3/2}\varepsilon_k^{3/2}(1+\|y-x\|)}&\quad \text{if }\|x\|\le \sqrt k\\
\frac{1}{\|x\|^5\varepsilon_k}\left(1+\frac{k}{\sqrt{\varepsilon_k}(1+\|y-x\|)} \right) &\quad \text{if }\|x\|>\sqrt k. \end{array} \right. \end{align*} \end{lemma} \begin{proof}
We proceed similarly as for the proof of Lemma \ref{lem.prep.123}. Assume first that $\|x\|\le \sqrt k$. On one hand, using Lemma \ref{lem.upconvolG}, we get
$$\sum_{i=\varepsilon_k}^{k/2} \frac{1}{\sqrt{k-i}} \sum_{z\in \mathbb{Z}^5} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}\lesssim \frac{1}{k^3} \sum_{z\in \mathbb{Z}^5} G_{\varepsilon_k}(z) G(z-y) \lesssim \frac{1}{k^{5/2}\sqrt{k\varepsilon_k}},$$ and, \begin{align*}
&\sum_{i=\varepsilon_k}^{k/2} \sum_{z\in \mathbb{Z}^5} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5(1+\|z-x\|)} \lesssim \frac{1}{k^{5/2}}\sum_{z\in \mathbb{Z}^5} \frac{G_{\varepsilon_k}(z)G(z-y)}{1+\|z-x\|}\\
&\lesssim \frac{1}{k^{5/2}} \left(\sum_{\|z-x\|\ge \frac{\|x\|}{2}} \frac{G_{\varepsilon_k}(z)G(z-y)}{1+\|z-x\|} + \sum_{\|z-x\|\le \frac{\|x\|}{2}} \frac{G_{\varepsilon_k}(z)G(z-y)}{1+\|z-x\|}\right) \\
& \lesssim \frac{1}{k^{5/2}} \left(\frac{1}{(1+\|x\|)\sqrt{\varepsilon_k}} + \frac{1}{1+\|x\|^2}\right) \lesssim \frac{1}{k^{5/2}} \left(\frac{1}{1+\|x\|^2}+ \frac 1{\varepsilon_k}\right). \end{align*} On the other hand, by \eqref{pn.largex} \begin{align*}
\sum_{i=k/2}^{k-\varepsilon_k} \sum_{\|z\|>2\sqrt k} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}\left(\frac{1}{1+\|z-x\|}+ \frac{1}{\sqrt{k-i}} \right)
\lesssim \frac{1}{k^2}\sum_{\|z\|>2\sqrt k} \frac{G(z-y)}{\|z\|^5} \lesssim k^{-\frac 72}. \end{align*} Furthermore, \begin{align*}
\sum_{i=k/2}^{k-\varepsilon_k} \frac 1{\sqrt{k-i}} \sum_{\|z\|\le 2\sqrt k} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}
\lesssim \frac{1}{k^2\varepsilon_k}\sum_{\|z\|\le 2\sqrt k} \frac{G(z-y)}{1+\|z-x\|^3} \lesssim \frac{(k^2\varepsilon_k)^{-1}}{1+\|y-x\|}, \end{align*} and \begin{align*}
\sum_{i=\frac k2}^{k-\varepsilon_k} \sum_{\|z\|\le 2\sqrt k} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}\frac{1}{1+\|z-x\|} & \lesssim \frac{1}{k^{3/2}\varepsilon_k^{3/2}}\sum_{\|z\|\le 2\sqrt k} \frac{G(z-y)}{1+\|z-x\|^3} \\
& \lesssim \frac{1}{k^{3/2}\varepsilon_k^{3/2}} \frac{1}{1+\|y-x\|}. \end{align*}
Assume now that $\|x\|>\sqrt k$. One has on one hand, using Lemma \ref{lem.upconvolG}, \begin{align*}
\sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{\|z-x\|\ge \frac{\|x\|}2} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}\left(\frac{1}{1+\|z-x\|}+ \frac{1}{\sqrt{k-i}} \right)
\lesssim \frac{1}{\|x\|^5 \varepsilon_k}. \end{align*} On the other hand, \begin{align*}
\sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{\|z-x\|\le \frac{\|x\|}2} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5} \frac{1}{1+\|z-x\|} & \lesssim \frac{k}{\|x\|^5 \varepsilon_k^{3/2}} \sum_{z\in \mathbb{Z}^5} \frac{G(z-y)}{1+\|z-x\|^3} \\
& \lesssim \frac{k}{\|x\|^5 \varepsilon_k^{3/2}(1+\|y-x\|)}, \end{align*} and \begin{align*}
\sum_{i=\varepsilon_k}^{k-\varepsilon_k}\frac 1{\sqrt {k-i}} \sum_{\|z-x\|\le \frac{\|x\|}2} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5} & \lesssim \frac{\sqrt k}{\|x\|^5 \varepsilon_k} \sum_{z\in \mathbb{Z}^5} \frac{G(z-y)}{1+\|y-x\|^3} \\
& \lesssim \frac{\sqrt k}{\|x\|^5 \varepsilon_k(1+\|y-x\|)}, \end{align*} concluding the proof of the lemma. \end{proof}
\begin{lemma}\label{lem.prep.ijl2} There exists a constant $C>0$, such that for any $x,y\in \mathbb{Z}^5$, \begin{align*}
& \sum_{v\in \mathbb{Z}^5} \frac{1}{(\|v\|+\sqrt k)^5} \left(\frac{1}{1+\|x-v\|} + \frac{1}{ 1+\|x\|}\right)\frac 1{(\|x-v\|+ \sqrt{\varepsilon_k} )^5} \left(\frac{1}{1+\|y-x\|}+\frac 1{1+\|y-v\|}\right)\\ & \le C\cdot \left\{ \begin{array}{ll}
\frac{1}{k^2\varepsilon_k} \left( \frac 1{\sqrt {\varepsilon_k}} + \frac{1}{1+\|x\|} +\frac 1{1+\|y-x\|}+\frac{\sqrt{\varepsilon_k}}{(1+\|x\|) (1+\|y-x\|)}\right) & \quad \text{if }\|x\|\le \sqrt k\\
\frac{\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})}{\|x\|^5\sqrt{\varepsilon_k}} \left(\frac{1}{1+\|y-x\|} + \frac 1{\sqrt k} \right) & \quad \text{if }\|x\|>\sqrt k. \end{array} \right. \end{align*} \end{lemma} \begin{proof}
Assume first that $\|x\| \le \sqrt k$. In this case it suffices to notice that on one hand, for any $\alpha\in \{3,4\}$, one has
$$\sum_{\|v\|\le 2 \sqrt k} \frac{1}{(1+\|x-v\|^\alpha)(1+\|y-v\|^{4-\alpha})} = \mathcal{O}(\sqrt k),$$ and on the other hand, for any $\alpha, \beta \in \{0,1\}$,
$$\sum_{\|v\|>2\sqrt k} \frac{1}{\|v\|^{10+\alpha} (1+\|y-v\|)^\beta} = \mathcal{O}(k^{-5/2 - \alpha - \beta}). $$
Assume next that $\|x\|>\sqrt k$. In this case it is enough to observe that
$$\sum_{\|v\|\le \frac{\sqrt k}{2}} \left(\frac{1}{1+\|x-v\|} + \frac{1}{\|x\|}\right) \left(\frac{1}{1+\|y-x\|}+\frac 1{1+\|y-v\|}\right) \lesssim \frac {k^2}{(1+\|y-x\|)},$$
$$\sum_{\|v\|\ge \frac{\sqrt k}{2}} \frac{1}{\|v\|^5 (\sqrt{\varepsilon_k}+\|x-v\|)^5} \lesssim \frac{\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})}{\|x\|^5} ,$$
$$\sum_{\|v\|\ge \frac{\sqrt k}{2}} \frac{1}{\|v\|^5 (\sqrt{\varepsilon_k}+\|x-v\|)^5(1+\|y-v\|)} \lesssim \frac{\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})}{\|x\|^5}\left(\frac 1{\sqrt k} + \frac 1{1+\|y-x\|}\right). $$ \end{proof}
\begin{proof}[Proof of Lemma \ref{lem.ijl}] First note that for any $\ell$, one has $\mathbb{E}[\psi_\ell] = \mathcal{O}(\varepsilon_k^{-1/2})$, by \eqref{lem.hit.3}. Using also similar arguments as in the proof of Lemma \ref{lem.123}, that we will not reproduce here, one can see that $\mathbb{E}[\varphi_{i,j}] = \mathcal{O}(\varepsilon_k^{-1})$, for any $i\neq j$. Thus only the terms of the form $\mathbb{E}[\varphi_{i,j}\psi_\ell]$ are at stake.
Let $(S_n)_{n\in \mathbb{Z}}$, $(\widetilde S_n)_{n\ge 0}$ and $(\widehat S_n)_{n\ge 0}$ be three independent walks starting from the origin. Recall the definition of $\tau_1$, $\tau_2$ and $\tau_3$ from the proof of Lemma \ref{lem.123}, and define analogously $$\widehat \tau_1:=\inf\{n\ge 1 : S_k+\widehat S_n \in \mathcal{R}(-\infty,-1]\}, \, \widehat \tau_2:=\inf\{n\ge 1 : S_k+\widehat S_n \in \mathcal{R}[0,k-\varepsilon_k-1]\},$$ and $$\widehat \tau_3:=\inf\{n\ge 1 : S_k+\widehat S_n \in \mathcal{R}[k+\varepsilon_k+1,\infty)\}.$$ When $\ell\neq i,j$, one can take advantage of the independence between the different parts of the range of $S$, at least once we condition on the value of $S_k$. This allows for instance to write $$\mathbb{E}[\varphi_{1,2}\psi_3] \le \mathbb{P}[\tau_1<\infty,\, \tau_2<\infty,\, \widehat \tau_3<\infty] =\mathbb{P}[\tau_1<\infty,\, \tau_2<\infty] \mathbb{P}[\widehat \tau_3<\infty] \lesssim \varepsilon_k^{-3/2},$$ using independence for the second equality and our previous estimates for the last one. Similarly, \begin{align*} & \mathbb{E}[\varphi_{1,3} \psi_2] \le \sum_{x\in \mathbb{Z}} \mathbb{P}[\tau_1<\infty, \, \tau_3<\infty \mid S_k=x] \times \mathbb{P}[\widehat \tau_2<\infty,\, S_k=x] \\
&\lesssim \sum_{x\in \mathbb{Z}^5} \frac{1}{(1+\|x\|)\sqrt \varepsilon_k}\cdot \frac 1{(1+\|x\| + \sqrt k)^5}\left(\frac{1}{1+\|x\|}+\frac 1{\sqrt{\varepsilon_k}}\right) \lesssim \frac{1}{\varepsilon_k \sqrt k}, \end{align*} using \eqref{tau132.1} and Lemma \ref{lem.prep.123} for the second inequality. The term $\mathbb{E}[\varphi_{2,3} \psi_1]$ is handled similarly. We consider now the other cases. One has \begin{equation}\label{phi233} \mathbb{E}[\varphi_{2,3} \psi_3] \le \mathbb{P}[\tau_2\le \tau_3<\infty,\, \widehat \tau_3<\infty] + \mathbb{P}[\tau_3\le \tau_2<\infty,\, \widehat \tau_3<\infty]. \end{equation} By using the Markov property at time $\tau_2$, one can write \begin{align*} & \mathbb{P}[\tau_2\le \tau_3<\infty,\, \widehat \tau_3<\infty] \\ & \le \sum_{x,y\in \mathbb{Z}^5} \mathbb{E}\left[\left(\sum_{i=0}^\infty G(S_i-y+x)\right)\left(\sum_{j=\varepsilon_k}^\infty G(S_j) \right) \right] \mathbb{P}[\tau_2<\infty, \widetilde S_{\tau_2}=y, S_k=x]. \end{align*} Then applying Lemmas \ref{lem.upconvolG} and \ref{lem.prep.123}, we get \begin{align}\label{tau33} \nonumber & \mathbb{E}\left[ \left(\sum_{i=0}^{\varepsilon_k} G(S_i-y+x)\right) \left(\sum_{j=\varepsilon_k}^\infty G(S_j) \right)\right]\\ & \nonumber = \sum_{v\in \mathbb{Z}^5} \mathbb{E}\left[\left(\sum_{i=0}^{\varepsilon_k} G(S_i-y+x) \right){\text{\Large $\mathfrak 1$}}\{S_{\varepsilon_k} = v\} \right] \mathbb{E}\left[\left(\sum_{j=0}^\infty G(S_j+v) \right)\right]\\
\nonumber & \lesssim \sum_{v\in \mathbb{Z}^5} \frac{1}{1+\|v\|}\cdot \left(\sum_{i=0}^{\varepsilon_k} p_i(z) G(z-y+x) p_{\varepsilon_k-i}(v-z) \right)\\
& \lesssim \sum_{v\in \mathbb{Z}^5} \frac{1}{1+\|v\|} \frac{1}{(\|v\|+ \sqrt{\varepsilon_k})^5} \left(\frac{1}{1+\|v-y+x\|} + \frac{1}{1+\|y-x\|}\right) \lesssim \frac{\varepsilon_k^{-1/2}}{ 1+\|y-x\|}. \end{align} Likewise, \begin{align}\label{tau33bis}
\nonumber \mathbb{E}\left[ \left(\sum_{i=\varepsilon_k}^\infty G(S_i-y+x)\right)\left(\sum_{j=\varepsilon_k}^\infty G(S_j) \right)\right] &\le \sum_{z\in \mathbb{Z}^5} G_{\varepsilon_k}(z) \left(\frac{G(z-y+x)}{1+\|z\|} + \frac{G(z)}{1+\|z-y+x\|}\right)\\
&\lesssim \frac{1}{\sqrt{\varepsilon_k} (1+\|y-x\|)}. \end{align} Recall now that by \eqref{lem.hit.3}, one has $\mathbb{P}[\tau_2<\infty] \lesssim \varepsilon_k^{-1/2}$. Moreover, from the proof of Lemma \ref{lem.123}, one can deduce that
$$\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}}{\|\widetilde S_{\tau_2}-S_k\|}\right] \lesssim \frac{1}{\sqrt {k \varepsilon_k}}.$$ Combining all these estimates we conclude that $$\mathbb{P}[\tau_2\le \tau_3<\infty,\, \widehat \tau_3<\infty] \lesssim \frac 1{\varepsilon_k\sqrt k}.$$ We deal next with the second term in the right-hand side of \eqref{phi233}. Applying the Markov property at time $\tau_3$, and then Lemma \ref{lem.prep.123}, we obtain \begin{align}\label{tau323.start} \nonumber & \mathbb{P}[\tau_3\le \tau_2<\infty,\, \widehat \tau_3<\infty] \\ &\nonumber \le \sum_{x,y\in \mathbb{Z}^5}\left(\sum_{i=\varepsilon_k}^k \mathbb{E}[G(S_i-y){\text{\Large $\mathfrak 1$}}\{S_k=x\}]\right) \mathbb{P}[\tau_3<\infty, \widehat \tau_3<\infty, \widetilde S_{\tau_3} = y\mid S_k=x]\\
\nonumber & \lesssim \sum_{x,y\in \mathbb{Z}^5} \frac{1}{(\|x\|+\sqrt k)^5} \left(\frac{1}{1+\|y-x\|}+\frac{1}{\sqrt{\varepsilon_k}}\right) \mathbb{P}[\tau_3<\infty, \widehat \tau_3<\infty, \widetilde S_{\tau_3} = y\mid S_k=x]\\
\nonumber &\lesssim \sum_{x\in \mathbb{Z}^5} \frac{1}{(\|x\|+\sqrt k)^5}\left( \frac{\mathbb{P}[\tau_3<\infty, \widehat \tau_3<\infty\mid S_k=x]}{\sqrt{\varepsilon_k}} +
\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty, \widehat \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big| \ S_k=x\right]\right)\\
& \lesssim \sum_{x\in \mathbb{Z}^5} \frac{1}{(\|x\|+\sqrt k)^5}\left( \frac{1}{\varepsilon_k(1+\|x\|)} + \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty, \widehat \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big| \ S_k=x\right]\right), \end{align} using also \eqref{tau33} and \eqref{tau33bis} (with $y=0$) for the last inequality. We use now \eqref{hit.ball} and Lemma \ref{hit.ball.overshoot} to remove the denominator in the last expectation above. Define for $r\ge 0$, and $x\in \mathbb{Z}^5$,
$$\eta_r(x):=\inf\{n\ge 0\ :\ \|\widetilde S_n -x\|\le r\}.$$
On the event when $r/2\le \|\widetilde S_{\eta_r(x)} -x\| \le r$, one applies the Markov property at time $\eta_r(x)$, and we deduce from \eqref{hit.ball} and Lemma \ref{hit.ball.overshoot} that \begin{align*}
&\mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty, \, \widehat \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big|\ S_k=x\right] \le \frac{\mathbb{P}[\tau_3<\infty, \, \widehat \tau_3<\infty\mid S_k=x]}{1+\|x\| } \\
&\qquad + \sum_{i=0}^{\log_2\|x\|}
\frac{\mathbb{P}\left[\tau_3<\infty, \, \widehat \tau_3<\infty,\, 2^i \le \|\widetilde S_{\tau_3} -x\| \le 2^{i+1}\mid S_k=x\right]}{2^i} \\
& \lesssim \frac{1}{\sqrt{\varepsilon_k}(1+\|x\|^2)} + \sum_{i=0}^{\log_2\|x\|} \frac{\mathbb{P}\left[\eta_{2^{i+1}}(x)\le \tau_3<\infty, \, \widehat \tau_3<\infty\mid S_k=x\right]}{2^i} \\
& \lesssim \frac{\varepsilon_k^{-1/2}}{1+\|x\|^2} + \frac{\mathbb{P}[\widehat \tau_3<\infty]}{1+\|x\|^3} + \sum_{i=0}^{\log_2\|x\|} \frac{2^{2i}}{1+\|x\|^3} \max_{\|z\|\ge 2^i} \mathbb{P}_{0,0,z}\left[H_{\mathcal{R}[\varepsilon_k,\infty)}<\infty, \widetilde H_{\mathcal{R}_\infty}<\infty \right], \end{align*} where in the last probability, $H$ and $\widetilde H$ refer to hitting times by two independent walks, independent of $S$, starting respectively from the origin and from $z$. Then it follows from \eqref{tau33} and \eqref{tau33bis} that \begin{equation}\label{remove.denominator}
\mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty, \widehat \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big|\ S_k=x\right] \lesssim \frac{1}{\sqrt{\varepsilon_k}(1+\|x\|^2)}. \end{equation} Combining this with \eqref{tau323.start}, it yields that \begin{align*} \mathbb{P}[\tau_2\le \tau_3<\infty, \widehat \tau_3 <\infty] \lesssim \frac 1{\varepsilon_k\sqrt k}. \end{align*} The terms $\mathbb{E}[\varphi_{1,3} \psi_3]$ and $\mathbb{E}[\varphi_{1,3}\psi_1]$ are entirely similar, and we omit repeating the proof. Thus it only remains to consider the terms $\mathbb{E}[\varphi_{2,3} \psi_2]$ and $\mathbb{E}[\varphi_{1,2} \psi_2]$. Since they are also similar we only give the details for the former. We start again by writing \begin{equation}\label{tau232} \mathbb{E}[\varphi_{2,3} \psi_2]\le \mathbb{P}[\tau_2\le \tau_3<\infty, \, \widehat \tau_2 <\infty] + \mathbb{P}[\tau_3\le \tau_2<\infty, \, \widehat \tau_2<\infty]. \end{equation} Then one has \begin{align}\label{Sigmai} &\mathbb{P}[\tau_3\le \tau_2<\infty, \, \widehat \tau_2 <\infty] \\ \nonumber \le & \sum_{x,y\in \mathbb{Z}^5}\mathbb{E}\left[\left(\sum_{i=\varepsilon_k}^k G(S_i-y)\right) \left( \sum_{j=0}^{k-\varepsilon_k} G(S_j-x)\right) {\text{\Large $\mathfrak 1$}}\{S_k=x\} \right]
\mathbb{P}[\tau_3<\infty, \widetilde S_{\tau_3}=y\mid S_k=x]\\ \nonumber \le & \sum_{x,y\in \mathbb{Z}^5} \left(\sum_{i=\varepsilon_k}^k \sum_{j=0}^{k-\varepsilon_k}\sum_{z,w\in \mathbb{Z}^5} \mathbb{P}[S_i= z, S_j=w, S_k=x] G(z-y)G(w-x)\right) \\ \nonumber & \qquad \times \mathbb{P}[\tau_3<\infty, \widetilde S_{\tau_3}=y\mid S_k=x]. \end{align} Now for any $x,y\in \mathbb{Z}^5$, \begin{align*} & \Sigma_1(x,y):= \sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{j=\varepsilon_k}^{k-\varepsilon_k} \sum_{z,w\in \mathbb{Z}^5} \mathbb{P}[S_i= z,\, S_j=w,\, S_k=x] G(z-y)G(w-x)\\
& \le 2\sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{z\in \mathbb{Z}^5}p_i(z) G(z-y) \left(\sum_{j=i}^{k-\varepsilon_k} \sum_{w\in \mathbb{Z}^5} p_{j-i}(w-z)G(w-x) p_{k-j}(x-w) \right)\\ & = 2\sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{z\in \mathbb{Z}^5}p_i(z) G(z-y) \left(\sum_{j=\varepsilon_k}^k \sum_{w\in \mathbb{Z}^5} p_j(w) G(w) p_{k-i-j}(w+x-z) \right)\\
& \stackrel{\text{Lemma } \ref{lem.prep.123}}{\lesssim} \sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{z\in \mathbb{Z}^5} \frac{p_i(z)G(z-y)}{(\|z-x\| + \sqrt{k-i})^5} \left(\frac 1{1+\|z-x\|} + \frac{1}{\sqrt{k-i}}\right)\\
& \stackrel{\text{Lemma } \ref{lem.prep.ijl}}{\lesssim} \left\{ \begin{array}{ll}
\frac{1}{k^{5/2}}\left( \frac 1{1+\|x\|^2} + \frac{1}{\varepsilon_k}\right) + \frac{1}{k^{3/2}\varepsilon_k^{3/2}(1+\|y-x\|)} & \text{if }\|x\|\le \sqrt k\\
\frac{1}{\|x\|^5\varepsilon_k}\left(1+ \frac{k}{\sqrt{\varepsilon_k}(1+\|y-x\|)}\right) &\text{if }\|x\|>\sqrt k. \end{array} \right. \end{align*} We also have \begin{align*} & \Sigma_2(x,y) := \sum_{i=k-\varepsilon_k}^k \sum_{j=0}^{k-\varepsilon_k} \sum_{z,w\in \mathbb{Z}^5} \mathbb{P}[S_i= z, S_j=w, S_k=x] G(z-y)G(w-x)\\
=& \sum_{i=k-\varepsilon_k}^k \sum_{j=0}^{k-\varepsilon_k} \sum_{z,v,w\in \mathbb{Z}^5} \mathbb{P}[S_j= w, S_{k-\varepsilon_k} = v, S_i=z, S_k=x] G(z-y)G(w-x)\\
=& \sum_{v\in \mathbb{Z}^5} \left(\sum_{j=0}^{k-\varepsilon_k} \sum_{w\in \mathbb{Z}^5} p_j(w) p_{k-\varepsilon_k-j}(v-w) G(w-x)\right) \left(\sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} p_i(z-v) p_{\varepsilon_k-i}(x-z)G(z-y)\right), \end{align*} and applying then Lemmas \ref{lem.prep.123} and \ref{lem.prep.ijl2}, gives \begin{align*} & \Sigma_2(x,y) \\
\lesssim & \sum_{v\in \mathbb{Z}^5} \frac{1}{(\|v\|+\sqrt k)^5}\left(\frac{1}{1+\|x-v\|} + \frac{1}{ 1+\|x\|}\right)\frac 1{(\|x-v\|+ \sqrt{\varepsilon_k} )^5} \left(\frac{1}{1+\|y-x\|}+\frac 1{1+\|y-v\|}\right)\\
\lesssim & \left\{ \begin{array}{ll}
\frac{1}{k^2\varepsilon_k} \left( \frac 1{\sqrt {\varepsilon_k}} + \frac{1}{1+\|x\|} +\frac 1{1+\|y-x\|}+\frac{\sqrt{\varepsilon_k}}{(1+\|x\|) (1+\|y-x\|)}\right) & \quad \text{if }\|x\|\le \sqrt k\\
\frac{\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})}{\|x\|^5\sqrt{\varepsilon_k}} \left(\frac{1}{1+\|y-x\|} + \frac 1{\sqrt k} \right) &\quad \text{if }\|x\|>\sqrt k. \end{array} \right. \end{align*} Likewise, by reversing time, one has \begin{align*} & \Sigma_3(x,y):= \sum_{i=\varepsilon_k}^k \sum_{j=0}^{\varepsilon_k} \sum_{z,w\in \mathbb{Z}^5} \mathbb{P}[S_i= z, S_j=w, S_k=x] G(z-y)G(w-x)\\
=& \sum_{i=0}^{k-\varepsilon_k} \sum_{j=k-\varepsilon_k}^k \sum_{z,v,w\in \mathbb{Z}^5} \mathbb{P}[S_i = z-x, S_{k-\varepsilon_k} = v-x, S_j= w-x, S_k=-x] G(z-y)G(w-x)\\
=& \sum_{v\in \mathbb{Z}^5} \left(\sum_{i=0}^{k-\varepsilon_k} \sum_{z\in \mathbb{Z}^5} p_i(z-x) p_{k-\varepsilon_k-i}(v-z) G(z-y)\right) \left(\sum_{j=0}^{\varepsilon_k} \sum_{w\in \mathbb{Z}^5} p_j(w-v) p_{\varepsilon_k-j}(w)G(w-x)\right) \\
\lesssim & \sum_{v\in \mathbb{Z}^5} \frac{1}{(\|v-x\|+\sqrt k)^5}\left(\frac{1}{1+\|y-v\|} + \frac{1}{ 1+\|y-x\|}\right)\frac 1{(\|v\|+ \sqrt{\varepsilon_k} )^5} \left(\frac{1}{1+\|x\|}+\frac 1{1+\|x-v\|}\right), \end{align*} and then a similar argument as in the proof of Lemma \ref{lem.prep.ijl2} gives the same bound for $\Sigma_3(x,y)$ as for $\Sigma_2(x,y)$. Now recall that \eqref{Sigmai} yields $$\mathbb{P}[\tau_3\le \tau_2<\infty, \widehat \tau_2 <\infty] \le \sum_{x,y\in \mathbb{Z}^5} \left(\Sigma_1(x,y) + \Sigma_2(x,y) + \Sigma_3(x,y)\right) \mathbb{P}[\tau_3<\infty, \widetilde S_{\tau_3}=y\mid S_k=x]. $$ Recall also that by \eqref{lem.hit.2},
$$\mathbb{P}[\tau_3<\infty\mid S_k=x] \lesssim \frac{1}{1+\|x\|},$$ and \begin{align*}
\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} -x \| }\, \Big|\, S_k=x\right]
\le \sum_{y\in \mathbb{Z}^5} \frac{G(y) G(y-x)}{1+\|y-x\|}
\lesssim \frac{1}{1+\|x\|^2}. \end{align*}
Furthermore, for any $\alpha\in\{1,2,3\}$, and any $\beta\ge 6$,
$$\sum_{\|x\|\le \sqrt k} \frac{1}{1+\|x\|^\alpha} \lesssim k^{\frac{5-\alpha}{2}}, \quad \sum_{\|x\|\ge \sqrt k} \frac {\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})} {\|x\|^{\beta}} \le \sum_{\|x\|\ge \sqrt{\varepsilon_k}} \frac {\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})} {\|x\|^{\beta}}\lesssim \varepsilon_k^{\frac{5-\beta}{2}}.$$
Putting all these pieces together we conclude that
$$\mathbb{P}[\tau_3\le \tau_2<\infty, \, \widehat \tau_2 <\infty] \lesssim \varepsilon_k^{-3/2}. $$ We deal now with the other term in \eqref{tau232}. As previously, we first write using the Markov property, and then using \eqref{lem.hit.1} and Lemma \ref{lem.upconvolG}, \begin{align*}
\mathbb{P}[\tau_2\le \tau_3<\infty, \, \widehat \tau_2 <\infty] \le \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_2<\infty, \, \widehat \tau_2<\infty\}}{1+\|\widetilde S_{\tau_2} - S_k\|}\right]. \end{align*} Then using \eqref{hit.ball} and Lemma \ref{hit.ball.overshoot} one can handle the denominator in the last expectation, the same way as for \eqref{remove.denominator}, and we conclude similarly that \begin{align*} \mathbb{P}[\tau_2\le \tau_3<\infty, \, \widehat \tau_2 <\infty] \lesssim \varepsilon_k^{-3/2}. \end{align*} This finishes the proof of Lemma \ref{lem.ijl}. \end{proof}
\section{Proof of Propositions \ref{prop.ij0} and \ref{prop.phipsi.1}} For the proof of these propositions we shall need the following estimate. \begin{lemma}\label{lem.prep.0ij} One has for all $x,y\in \mathbb{Z}^5$, \begin{align*} \sum_{i=k-\varepsilon_k}^k & \mathbb{E}\left[G(S_i-y) {\text{\Large $\mathfrak 1$}}\{S_k=x\}\right] \\
& \lesssim \varepsilon_k \left(\frac{\log (2+\frac{\|y-x\|}{\sqrt{\varepsilon_k}})}{(\|x\|+\sqrt k)^5(\|y-x\| + \sqrt{\varepsilon_k})^3}
+ \frac{\log (2+\frac{\|y\|}{\sqrt{k}})}{(\|x\|+\sqrt{\varepsilon_k})^5(\|y\| + \sqrt{k})^3}\right). \end{align*} \end{lemma} \begin{proof} One has using \eqref{pn.largex} and \eqref{Green}, \begin{align*} &\sum_{i=k-\varepsilon_k}^k \mathbb{E}\left[G(S_i-y){\text{\Large $\mathfrak 1$}}\{S_k=x\}\right] = \sum_{i=k-\varepsilon_k}^k \sum_{z\in \mathbb{Z}^5} p_i(z) G(z-y) p_{k-i}(x-z)\\
& \lesssim \sum_{z\in \mathbb{Z}^5} \frac{\varepsilon_k}{(\|z\| + \sqrt k)^5(1+\|z-y\|^3) (\|x-z\| +\sqrt{\varepsilon_k})^5} \\
& \lesssim \frac{1}{\varepsilon_k^{3/2}(\|x\|+\sqrt k)^5} \sum_{\|z-x\|\le \sqrt{\varepsilon_k}}\frac 1{1+\|z-y\|^3} \\
& \quad + \frac{\varepsilon_k}{(\|x\|+\sqrt k)^5} \sum_{\sqrt{\varepsilon_k}\le \|z-x\|\le \frac{\|x\|}{2}} \frac{1}{(1+\|z-y\|^3)(1+\|z-x\|^5)} \\
&\quad + \frac{\varepsilon_k}{(\|x\|+\sqrt {\varepsilon_k})^5} \sum_{\|z-x\|\ge \frac{\|x\|}{2}}\frac{1}{(\|z\|+\sqrt k)^5(1+\|z-y\|^3)}. \end{align*} Then it suffices to observe that
$$ \sum_{\|z-x\|\le \sqrt{\varepsilon_k}}\frac 1{1+\|z-y\|^3} \lesssim \frac{\varepsilon_k^{5/2}}{(\|y-x\| + \sqrt{\varepsilon_k})^3},$$
$$ \sum_{\sqrt{\varepsilon_k}\le \|z-x\|\le \frac{\|x\|}{2}} \frac{1}{(1+\|z-y\|^3)(1+\|z-x\|^5)} \lesssim
\frac{\log(2+\frac{\|y-x\|}{\sqrt{\varepsilon_k}})}{(\|y-x\| + \sqrt{\varepsilon_k})^3}, $$
$$ \sum_{z\in \mathbb{Z}^5}\frac{1}{(\|z\|+\sqrt k)^5(1+\|z-y\|^3)} \lesssim \frac{\log (2+\frac{\|y\|}{\sqrt{k}})}{(\|y\| + \sqrt{k})^3}. $$ \end{proof}
\begin{proof}[Proof of Proposition \ref{prop.ij0} (i)] This part is the easiest: it suffices to observe that $\varphi_{1,2}$ is a sum of one term which is independent of $Z_k\psi_0$ and another one, whose expectation is negligible. To be more precise, define $$\varphi_{1,2}^0 := \mathbb{P}\left[H_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}^+ = \infty,\, H^+_{\mathcal{R}(-\infty,-\varepsilon_k-1]}<\infty, \, H^+_{\mathcal{R}[\varepsilon_k+1,k-\varepsilon_k-1]}<\infty \mid S\right],$$ and note that $Z_0\varphi_{1,2}^0$ is independent of $Z_k\psi_0$. It follows that
$$|\operatorname{Cov}(Z_0\varphi_{1,2},Z_k\psi_0)| =|\operatorname{Cov}(Z_0(\varphi_{1,2}-\varphi_{1,2}^0),Z_k\psi_0)|\le \mathbb{P}\left[\tau_1<\infty, \, \tau_*<\infty\right],$$ with $\tau_1$ and $\tau_*$ the hitting times respectively of $\mathcal{R}(-\infty,-\varepsilon_k]$ and $\mathcal{R}[k-\varepsilon_k,k]$ by another walk $\widetilde S$ starting from the origin, independent of $S$. Now, using \eqref{pn.largex}, we get \begin{align*} \mathbb{P}[\tau_1\le \tau_*<\infty] & \le \mathbb{E}\left[{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty \} \left(\sum_{i=k-\varepsilon_k}^k G(S_i-\widetilde S_{\tau_1})\right)\right] \\ & \le \sum_{y\in \mathbb{Z}^5} \left(\sum_{z\in \mathbb{Z}^5} \sum_{i=k-\varepsilon_k}^k p_i(z) G(z-y) \right)\mathbb{P}[\tau_1<\infty, \, \widetilde S_{\tau_1} = y]\\ & \lesssim \frac{\varepsilon_k}{k^{3/2}} \, \mathbb{P}[\tau_1<\infty] \stackrel{\eqref{lem.hit.3}}{\lesssim} \frac{\sqrt{\varepsilon_k}}{k^{3/2}}. \end{align*} Likewise, using now Lemma \ref{lem.upconvolG}, \begin{align*} \mathbb{P}[\tau_*\le \tau_1<\infty] & \le \mathbb{E}\left[{\text{\Large $\mathfrak 1$}}\{\tau_*<\infty \} \left(\sum_{i=\varepsilon_k}^\infty G(S_{-i}-\widetilde S_{\tau_*})\right)\right] \\ & \le \sum_{y\in \mathbb{Z}^5} \left(\sum_{z\in \mathbb{Z}^5} G_{\varepsilon_k}(z) G(z-y) \right)\mathbb{P}[\tau_*<\infty, \, \widetilde S_{\tau_*} = y]\\ & \lesssim \frac{1}{\sqrt{\varepsilon_k}}\, \mathbb{P}[\tau_*<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}, \end{align*} and the first part of (i) follows. But since $Z_0$ and $Z_k$ have played no role here, the same computation gives the result for the covariance between $Z_0\varphi_0$ and $Z_k \psi_{2,3}$ as well. \end{proof}
\begin{proof}[Proof of Proposition \ref{prop.ij0} (ii)-(iii)] These parts are more involved. Since they are entirely similar, we only prove (iii), and as for (i) we only give the details for the covariance between $Z_0 \varphi_{2,3}$ and $Z_k\psi_0$, since $Z_0$ and $Z_k$ will not play any role here. We define similarly as in the proof of (i), $$\varphi_{2,3}^0 := \mathbb{P}\left[H_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}^+ = \infty,\, H^+_{\mathcal{R}[\varepsilon_k,k-\varepsilon_k]}<\infty, \, H^+_{\mathcal{R}[k+\varepsilon_k,\infty)}<\infty \mid S\right],$$ but observe that this time, the term $\varphi_{2,3}^0$ is no more independent of $\psi_0$. This entails some additional difficulty, on which we shall come back later, but first we show that one can indeed replace $\varphi_{2,3}$ by $\varphi_{2,3}^0$ in the computation of the covariance. For this, denote respectively by $\tau_2$, $\tau_3$, $\tau_*$ and $\tau_{**}$ the hitting times of $\mathcal{R}[\varepsilon_k,k]$, $\mathcal{R}[k,\infty)$, $\mathcal{R}[k-\varepsilon_k,k]$, and $\mathcal{R}[k,k+\varepsilon_k]$ by $\widetilde S$. One has \begin{align*}
\mathbb{E}[|\varphi_{2,3} - \varphi_{2,3}^0|]& \le \mathbb{P}[\tau_2<\infty, \, \tau_{**}<\infty] +\mathbb{P}[\tau_3<\infty, \, \tau_*<\infty]. \end{align*} Using \eqref{pn.largex}, \eqref{lem.hit.1} and Lemma \ref{lem.upconvolG}, we get \begin{align*}
\mathbb{P}[\tau_*\le \tau_3<\infty] & \le \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_*<\infty\}}{1+\|\widetilde S_{\tau_*} - S_k\|}\right] \le \sum_{i=k-\varepsilon_k}^k \mathbb{E}\left[ \frac{G(S_i)}{1+\|S_i-S_k\|}\right] \\ &\lesssim \sum_{i=k-\varepsilon_k}^k \mathbb{E}\left[ \frac{G(S_i)}{1+\sqrt{k-i} }\right] \lesssim \sum_{z\in \mathbb{Z}^5} \sum_{i=k-\varepsilon_k}^k \frac{p_i(z) G(z)}{1+\sqrt{k-i}}\\
& \lesssim \sqrt{\varepsilon_k} \sum_{z\in \mathbb{Z}^5}\frac 1{(\|z\| + \sqrt k)^5} G(z) \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}. \end{align*} Next, applying Lemma \ref{lem.prep.0ij}, we get \begin{align*} & \mathbb{P}[\tau_3\le \tau_*<\infty] \\ & \le \sum_{x,y\in \mathbb{Z}^5} \mathbb{E}\left[\left(\sum_{i=k-\varepsilon_k}^k G(S_i-y)\right){\text{\Large $\mathfrak 1$}}\{S_k=x\}\right] \mathbb{P}[\tau_3<\infty, \widetilde S_{\tau_3} = y\mid S_k=x]\\
& \lesssim \varepsilon_k \sum_{x\in \mathbb{Z}^5} \left( \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty\}\log (2+\frac{\|\widetilde S_{\tau_3}-x\|}{\sqrt{\varepsilon_k}})}{(\|x\|+\sqrt k)^5(\sqrt{\varepsilon_k}+\|\widetilde S_{\tau_3} - x\|)^3}\, \Big|\, S_k=x \right] \right. \\
& \qquad \left. + \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty\}\log(2+\frac{\|\widetilde S_{\tau_3}\|}{\sqrt k})}{(\|x\|+\sqrt{\varepsilon_k})^5(\sqrt{k}+\|\widetilde S_{\tau_3}\|)^3}\, \Big|\, S_k=x \right] \right). \end{align*} Moreover, \begin{align*}
\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty\}\log (2+\frac{\|\widetilde S_{\tau_3}-x\|}{\sqrt{\varepsilon_k}})}{(\sqrt{\varepsilon_k}+\|\widetilde S_{\tau_3} - x\|)^3}\, \Big|\, S_k=x \right] & \stackrel{\eqref{lem.hit.1}}{\le} \sum_{y\in \mathbb{Z}^5} \frac{G(y) G(y-x)
\log (2+\frac{\|y-x\|}{\sqrt{\varepsilon_k}})}{ (\sqrt{\varepsilon_k}+\|y - x\|)^3} \\
& \lesssim \frac{1}{\sqrt{\varepsilon_k}(1+\|x\|)^3}, \end{align*} and \begin{align*}
\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty\}\log(2+\frac{\|\widetilde S_{\tau_3}\|}{\sqrt k})}{(\sqrt{k}+\|\widetilde S_{\tau_3}\|)^3}\, \Big|\, S_k=x \right] & \stackrel{\eqref{lem.hit.1}}{\le} \sum_{y\in \mathbb{Z}^5} \frac{G(y) G(y-x)\log(2+\frac{\|y\|}{\sqrt k})}{ (\sqrt{k}+\|y \|)^3} \\
& \lesssim \frac{1}{\sqrt{k}(1+\|x\|)(\sqrt k + \|x\|)^2}. \end{align*} Furthermore, it holds
$$\sum_{x\in \mathbb{Z}^5} \frac{1}{(\|x\|+\sqrt k)^5(1+\|x\|)^3} \lesssim \frac{1}{k^{3/2}},$$
$$\sum_{x\in \mathbb{Z}^5} \frac{1}{(\|x\|+\sqrt{\varepsilon_k})^5(1+\|x\|)(\sqrt k + \|x\|)^2} \lesssim \frac{1}{\sqrt{k\varepsilon_k}},$$ which altogether proves that $$ \mathbb{P}[\tau_3\le \tau_*<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.$$ Likewise, \begin{align*}
\mathbb{P}[\tau_2\le \tau_{**}<\infty] \le \sum_{x,y\in \mathbb{Z}^5} \mathbb{E}\left[\sum_{i=0}^{\varepsilon_k} G(S_i-y+x) \right] \mathbb{P}[\tau_2<\infty, \widetilde S_{\tau_2} = y, S_k=x], \end{align*} and using \eqref{Green}, we get \begin{align*} & \mathbb{E}\left[\sum_{i=0}^{\varepsilon_k} G(S_i-y+x) \right] = \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} p_i(z) G(z-y+x)\\
& \lesssim \sum_{\|z\|\le \sqrt{\varepsilon_k}} G(z) G(z-y+x) + \varepsilon_k\, \sum_{\|z\|\ge \sqrt{\varepsilon_k}} \frac{G(z-y+x)}{\|z\|^5} \\
&\lesssim \frac{\varepsilon_k}{(\|y-x\| +\sqrt{\varepsilon_k})^2(1+\|y-x\|)} + \varepsilon_k \, \frac{\log\left(2+\frac{\|y-x\|}{\sqrt{\varepsilon_k}}\right) }{(\|y-x\| + \sqrt{\varepsilon_k})^3}\\
& \lesssim \varepsilon_k \, \frac{\log\left(2+\frac{\|y-x\|}{\sqrt{\varepsilon_k}}\right) }{(\|y-x\| + \sqrt{\varepsilon_k})^2(1+\|y-x\|)}. \end{align*} Therefore, using the Markov property, \begin{align*}
&\mathbb{P}[\tau_2\le \tau_{**}<\infty] \lesssim \varepsilon_k \cdot \mathbb{E}\left[ \frac{\log\left(2+\frac{\|\widetilde S_{\tau_2}-S_k\|}{\sqrt{\varepsilon_k}}\right) \cdot {\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}}{(\|\widetilde S_{\tau_2}-S_k\| + \sqrt{\varepsilon_k})^2(1+\|\widetilde S_{\tau_2} - S_k\|)} \right]\\
&\lesssim \varepsilon_k \sum_{i=\varepsilon_k}^k \mathbb{E}[G(S_i)] \cdot \mathbb{E}\left[\frac{\log\left(2+\frac{\|S_{k-i}\|}{\sqrt{\varepsilon_k}}\right) }{(\|S_{k-i}\| + \sqrt{\varepsilon_k})^2(1+\| S_{k-i}\|)} \right]. \end{align*} Furthermore, using \eqref{pn.largex} we obtain after straightforward computations,
$$\mathbb{E}\left[\frac{\log\left(2+\frac{\|S_{k-i}\|}{\sqrt{\varepsilon_k}}\right) }{(\|S_{k-i}\| + \sqrt{\varepsilon_k})^2(1+\| S_{k-i}\|)} \right] \lesssim \frac{\log\left(2+\frac{k-i}{\varepsilon_k}\right) }{\sqrt{k-i}(\varepsilon_k + k-i)},$$ and using in addition \eqref{exp.Green}, we conclude that $$\mathbb{P}[\tau_2\le \tau_{**}<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k}).$$ Similarly, using Lemma \ref{lem.prep.123} we get \begin{align*} &\mathbb{P}[\tau_{**} \le \tau_2<\infty] \\
& = \sum_{x,y\in \mathbb{Z}^5} \mathbb{P}[\tau_{**}<\infty,\, \widetilde S_{\tau_{**}} = y \mid S_k=x] \cdot \mathbb{E}\left[\sum_{i=\varepsilon_k}^k G(S_i-y) {\text{\Large $\mathfrak 1$}}\{S_k=x\}\right] \\
& \lesssim \sum_{x\in\mathbb{Z}^5} \frac{1}{(\|x\| + \sqrt{k})^5} \left(\mathbb{E}\left[\frac{1\{\tau_{**}<\infty\}}{ 1+\|\widetilde S_{\tau_{**}}-x\|}\, \Big|\, S_k=x\right] + \frac{\mathbb{P}[\tau_{**}<\infty\mid S_k=x]}{\sqrt{\varepsilon_k}}\right). \end{align*} Moreover, one has \begin{align*}
\mathbb{P}[\tau_{**}<\infty\mid S_k=x] &\le \sum_{i=0}^{\varepsilon_k} \mathbb{E}[G(S_i+x)]\lesssim \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} \frac{1}{(1+\|z\| + \sqrt i)^5(1+\|z+x\|^3)} \\
&\lesssim \sum_{\|z\| \le \sqrt{\varepsilon_k}} \frac{1}{(1+\|z\|^3)(1+\|z+x\|^3)} + \sum_{\|z\|\ge \sqrt{\varepsilon_k}} \frac{\varepsilon_k}{\|z\|^5(1+\|z+x\|^3)}\\
&\lesssim \frac{\varepsilon_k \log(2+\frac{\|x\|}{\sqrt{\varepsilon_k}})}{(\sqrt{\varepsilon_k}+\|x\|)^2(1+\|x\|)}, \end{align*} and likewise \begin{align*}
\mathbb{E}\left[\frac{1\{\tau_{**}<\infty\}}{ 1+\|\widetilde S_{\tau_{**}}-x\|}\, \Big|\, S_k=x\right] &\le \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} \frac{1}{(1+\|z\| + \sqrt i)^5(1+\|z-x\|^3)(1+\|z\|)} \\
&\lesssim \sum_{\|z\| \le \sqrt{\varepsilon_k}} \frac{1}{(1+\|z\|^4)(1+\|z-x\|^3)} + \sum_{\|z\|\ge \sqrt{\varepsilon_k}} \frac{\varepsilon_k}{\|z\|^6(1+\|z-x\|^3)}\\
&\lesssim \frac{\sqrt{\varepsilon_k} }{(\|x\|+\sqrt{\varepsilon_k})(1+\|x\|^2)}. \end{align*} Then it follows as above that $$\mathbb{P}[\tau_{**} \le \tau_2<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k}).$$ In other words we have proved that
$$\mathbb{E}[|\varphi_{2,3} - \varphi_{2,3}^0|]\lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k}).$$ We then have to deal with the fact that $Z_0\varphi_{2,3}^0$ is not really independent of $Z_k\psi_0$. Therefore, we introduce the new random variables $$\widetilde Z_k := {\text{\Large $\mathfrak 1$}}\{S_i \neq S_k \ \forall i=k+1,\dots,\varepsilon'_k\}, \ \widetilde \psi_0:=\mathbb{P}_{S_k}\left[H^+_{\mathcal{R}[k-\varepsilon'_k,k+\varepsilon'_k]}=\infty\mid S\right],$$ where $(\varepsilon'_k)_{k\ge 0}$ is another sequence of integers, whose value will be fixed later. For the moment we only assume that it satisfies $\varepsilon'_k\le \varepsilon_k/4$, for all $k$. One has by \eqref{Green} and \eqref{lem.hit.3}, \begin{equation}\label{tildepsi0}
\mathbb{E}[|Z_k\psi_0 - \widetilde Z_k\widetilde \psi_0|] \lesssim \frac{1}{\sqrt{\varepsilon'_k}}. \end{equation} Furthermore, for any $y\in \mathbb{Z}^5$, \begin{align}\label{cov.230} & \mathbb{E}\left[\varphi_{2,3}^0 \mid S_{k+\varepsilon_k} -S_{k-\varepsilon_k}= y\right] = \sum_{x\in \mathbb{Z}^5} \mathbb{E}\left[\varphi_{2,3}^0 {\text{\Large $\mathfrak 1$}}\{S_{k-\varepsilon_k}=x\}\mid S_{k+\varepsilon_k} -S_{k-\varepsilon_k}= y\right]\\
\nonumber & \le \sum_{x\in \mathbb{Z}^5}\mathbb{P}\left[\widetilde \mathcal{R}_\infty \cap \mathcal{R}[\varepsilon_k,k-\varepsilon_k]\neq \varnothing,\, \widetilde \mathcal{R}_\infty \cap (x+y+\widehat \mathcal{R}_\infty)\neq \varnothing,\, S_{k-\varepsilon_k}=x\right], \end{align} where in the last probability, $\widetilde \mathcal{R}_\infty$ and $\widehat \mathcal{R}_\infty$ are the ranges of two independent walks, independent of $S$, starting from the origin. Now $x$ and $y$ being fixed, define $$\tau_1:=\inf\{n\ge 0 : \widetilde S_n\in \mathcal{R}[\varepsilon_k,k-\varepsilon_k]\}, \ \tau_2:= \inf\{n\ge 0 : \widetilde S_n\in (x+y + \widehat \mathcal{R}_\infty)\}.$$ Applying \eqref{lem.hit.1} and the Markov property we get \begin{align*}
&\mathbb{P}[\tau_1\le \tau_2<\infty,\, S_{k-\varepsilon_k} = x] \le \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty,\, S_{k-\varepsilon_k}=x\}}{1+\|\widetilde S_{\tau_1} - (x+y)\|}\right] \\
& \le \sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{z\in \mathbb{Z}^5}\frac{p_i(z) G(z) p_{k-\varepsilon_k-i}(x-z)}{1+\|z-(x+y)\|} \\
&\lesssim \frac{1}{(\|x\|+\sqrt k)^5}\left(\frac{1}{\sqrt{\varepsilon_k} (1+\|x+y\|)} + \frac{1}{1+\|x\|^2}\right), \end{align*} using also similar computations as in the proof of Lemma \ref{lem.prep.123} for the last inequality. It follows that for some constant $C>0$, independent of $y$, $$\sum_{x\in \mathbb{Z}^5} \mathbb{P}[\tau_1\le \tau_2<\infty,\, S_{k-\varepsilon_k} = x] \lesssim \frac{1}{\sqrt{k\varepsilon_k}}.$$ On the other hand, by Lemmas \ref{lem.prep.123} and \ref{lem.simplehit}, \begin{align*}
\mathbb{P}[\tau_2\le \tau_1<\infty,\, S_{k-\varepsilon_k} = x] & \lesssim \frac{1}{( \| x \| +\sqrt k)^5} \left(\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\} }{1+\|\widetilde S_{\tau_2} - x\|}\right] + \frac{\mathbb{P}[\tau_2<\infty]}{\sqrt{\varepsilon_k} }\right) \\
& \lesssim \frac{1}{(\|x\|+\sqrt k)^5}\left(\frac{1}{\sqrt{\varepsilon_k} (1+\|x+y\|)} + \frac{1}{1+\|x\|^2}\right), \end{align*} and it follows as well that $$\sum_{x\in \mathbb{Z}^5} \mathbb{P}[\tau_2\le \tau_1<\infty, S_{k-\varepsilon_k} = x] \lesssim \frac{1}{\sqrt{k\varepsilon_k}}.$$ Coming back to \eqref{cov.230}, we deduce that \begin{equation}\label{phi230.cond}
\mathbb{E}\left[\varphi_{2,3}^0 \mid S_{k+\varepsilon_k} -S_{k-\varepsilon_k}= y\right] \lesssim \frac{1}{\sqrt{k\varepsilon_k}},
\end{equation}
with an implicit constant independent of $y$. Together with \eqref{tildepsi0}, this gives \begin{align*}
& \mathbb{E}\left[\varphi_{2,3}^0|Z_k\psi_0-\widetilde Z_k\widetilde \psi_0|\right] \\
& = \sum_{y\in \mathbb{Z}^5} \mathbb{E}\left[\varphi_{2,3}^0 \mid S_{k+\varepsilon_k} -S_{k-\varepsilon_k}= y\right]\cdot \mathbb{E}\left[|Z_k\psi_0-\widetilde Z_k\widetilde \psi_0|{\text{\Large $\mathfrak 1$}}\{S_{k+\varepsilon_k}-S_{k-\varepsilon_k} = y\}\right] \\ &\lesssim \frac{1}{\sqrt{k\varepsilon_k\varepsilon'_k}}. \end{align*} Thus at this point we have shown that $$ \operatorname{Cov}(Z_0\varphi_{2,3},Z_k\psi_0) = \operatorname{Cov}(Z_0\varphi_{2,3}^0,\widetilde Z_k\widetilde \psi_0) +\mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k}) + \frac{1}{\sqrt{k\varepsilon_k\varepsilon'_k}}\right).$$ Note next that \begin{align*}
\operatorname{Cov}(Z_0\varphi_{2,3}^0,\widetilde Z_k\widetilde \psi_0) & = \sum_{y,z\in \mathbb{Z}^5} \mathbb{E}\left[Z_0\varphi_{2,3}^0 \mid S_{k+\varepsilon_k} - S_{k-\varepsilon'_k} =y\right] \\
& \times \mathbb{E}\left[\widetilde Z_k \widetilde \psi_0{\text{\Large $\mathfrak 1$}}\{S_{k+\varepsilon'_k}-S_{k-\varepsilon'_k}=z\}\right] \left(p_{\varepsilon_k-\varepsilon'_k}(y-z) - p_{\varepsilon_k+\varepsilon'_k}(y)\right). \end{align*} Moreover, one can show exactly as \eqref{phi230.cond} that uniformly in $y$, $$\mathbb{E}\left[\varphi_{2,3}^0 \mid S_{k+\varepsilon_k} - S_{k-\varepsilon'_k} =y\right] \lesssim \frac{1}{\sqrt{k\varepsilon_k}}.$$ Therefore by using also \eqref{Sn.large} and Theorem \ref{LCLT}, we see that \begin{align*}
& |\operatorname{Cov}(Z_0\varphi_{2,3}^0,\widetilde Z_k\widetilde \psi_0) | \\
& \lesssim \frac{1}{\sqrt{k\varepsilon_k}} \sum_{\|y\| \le \varepsilon_k^{\frac 6{10}}} \sum_{\|z\|\le \varepsilon_k^{\frac 1{10}}\cdot \sqrt{\varepsilon'_k} } p_{2\varepsilon'_k}(z)\, |\overline p_{\varepsilon_k-\varepsilon'_k}(y-z) - \overline p_{\varepsilon_k+\varepsilon'_k}(y)| + \frac 1{\varepsilon_k \sqrt{k} }. \end{align*} Now straightforward computations show that for $y$ and $z$ as in the two sums above, one has for some constant $c>0$,
$$|\overline p_{\varepsilon_k-\varepsilon'_k}(y-z) - \overline p_{\varepsilon_k+\varepsilon'_k}(y)| \lesssim \left(\frac{\|z\|}{\sqrt{\varepsilon_k}} + \frac{\varepsilon'_k}{\varepsilon_k}\right)\overline p_{\varepsilon_k-\varepsilon'_k}(cy),$$ at least when $\varepsilon'_k\le \sqrt{\varepsilon_k}$, as will be assumed in a moment.
Using also that $\sum_z \|z\|p_{2\varepsilon'_k}(z) \lesssim \sqrt{\varepsilon'_k}$, we deduce that
$$ |\operatorname{Cov}(Z_0\varphi_{2,3}^0,\widetilde Z_k\widetilde \psi_0) | = \mathcal{O}\left(\frac{\sqrt{\varepsilon'_k}}{\varepsilon_k \sqrt k}\right).$$ This concludes the proof as we choose $\varepsilon'_k = \lfloor \sqrt{\varepsilon_k} \rfloor$. \end{proof}
We can now quickly give the proof of Proposition \ref{prop.phipsi.1}.
\begin{proof}[Proof of Proposition \ref{prop.phipsi.1}] \underline{Case $1\le i<j\le 3$.} First note that $Z_0\varphi_1$ and $Z_k\psi_3$ are independent, so only the cases $i=1$ and $j=2$, or $i=2$ and $j=3$ are at stake. Let us only consider the case $i=2$ and $j=3$, since the other one is entirely similar. Define, in the same fashion as in the proof of Proposition \ref{prop.ij0}, $$\varphi_2^0:= \mathbb{P}\left[H^+_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}=\infty,\, H^+_{\mathcal{R}[\varepsilon_k+1,k-\varepsilon_k]}<\infty\mid S\right]. $$ One has by using independence and translation invariance,
$$\mathbb{E}[|\varphi_2-\varphi_2^0| \psi_3] \le \mathbb{P}[H_{\mathcal{R}[k-\varepsilon_k,k]}<\infty]\cdot \mathbb{P}[H_{\mathcal{R}[\varepsilon_k,\infty)}<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}},$$ which entails $$\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_3) = \operatorname{Cov}(Z_0\varphi_2^0,Z_k\psi_3) + \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right) \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}},$$ since $Z_0\varphi_2^0$ and $Z_k\psi_3$ are independent.
\underline{Case $1\le j\le i\le 3$}. Here one can use entirely similar arguments as those from the proof of Lemma \ref{lem.ijl}, and we therefore omit the details. \end{proof}
\section{Proof of Proposition \ref{prop.phi0}} We need to estimate here the covariances $\operatorname{Cov}(Z_0\varphi_i, Z_k\psi_0)$ and $\operatorname{Cov}(Z_0\varphi_0, Z_k\psi_{4-i})$, for all $1\le i \le 3$.
\underline{Case $i=1$.} It suffices to observe that $Z_0\varphi_1$ and $Z_k\psi_0$ are independent, as are $Z_0\varphi_0$ and $Z_k\psi_3$. Thus their covariances are equal to zero.
\underline{Case $i=2$.} We first consider the covariance between $Z_0\varphi_2$ and $Z_k\psi_0$, which is easier to handle. Define $$\widetilde \varphi_2:=\mathbb{P}\left[H^+_{\mathcal{R}[-\varepsilon_k,k-\varepsilon_k-1]}=\infty,\, H^+_{\mathcal{R}[k-\varepsilon_k,k]}<\infty\mid S\right],$$ and note that $Z_0(\varphi_2 - \widetilde \varphi_2)$ is independent of $Z_k\psi_0$. Therefore $$\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_0) = \operatorname{Cov}(Z_0\widetilde \varphi_2, Z_k\psi_0).$$ Then we decompose $\psi_0$ as $\psi_0=\psi_0^1-\psi_0^2$, where $$\psi_0^1:=\mathbb{P}_{S_k}[H^+_{\mathcal{R}[k,k+\varepsilon_k]}=\infty\mid S],\ \psi_0^2:=\mathbb{P}_{S_k}[H^+_{\mathcal{R}[k,k+\varepsilon_k]}=\infty, H^+_{\mathcal{R}[k-\varepsilon_k,k-1]}<\infty \mid S].$$ Using now that $Z_k\psi_0^1$ is independent of $Z_0\widetilde \varphi_2$ we get $$\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_0) =- \operatorname{Cov}(Z_0\widetilde \varphi_2, Z_k\psi_0^2).$$ Let $(\widetilde S_n)_{n\ge 0}$ and $(\widehat S_n)_{n\ge 0}$ be two independent walks starting from the origin, and define $$\tau_1:=\inf \{n\ge 0 : S_{k-n}\in \widetilde \mathcal{R}[1,\infty)\},\ \tau_2:=\inf\{n\ge 0 : S_{k-n}\in (S_k + \widehat \mathcal{R}[1,\infty))\}.$$ We decompose \begin{align*} & \operatorname{Cov}(Z_0\widetilde \varphi_2, Z_k\psi_0^2)\\ & = \mathbb{E}\left[Z_0\widetilde \varphi_2Z_k\psi_0^2{\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_2\}\right] + \mathbb{E}\left[Z_0\widetilde \varphi_2Z_k\psi_0^2{\text{\Large $\mathfrak 1$}}\{\tau_1> \tau_2\}\right] - \mathbb{E}[Z_0\widetilde \varphi_2] \mathbb{E}[Z_k\psi_0^2]. \end{align*} We bound the first term on the right-hand side simply by the probability of the event $\{\tau_1\le \tau_2\le \varepsilon_k\}$, which we treat later, and for the difference between the last two terms, we use that
$$\left| {\text{\Large $\mathfrak 1$}}\{\tau_2<\tau_1\le \varepsilon_k\} - \sum_{i=0}^{\varepsilon_k} {\text{\Large $\mathfrak 1$}}\left\{\tau_2=i,\, H^+_{\mathcal{R}[k-\varepsilon_k,k-i-1]}<\infty\right\}\right| \le {\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_2\le \varepsilon_k\}.$$ Using also that the event $\{\tau_2=i\}$ is independent of $(S_n)_{n\le k-i}$, we deduce that \begin{align*}
& |\operatorname{Cov}(Z_0\widetilde \varphi_2, Z_k\psi_0^2)| \\
&\le 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \sum_{i=0}^{\varepsilon_k} \mathbb{P}[\tau_2=i] \left|\mathbb{P}\left[H^+_{\mathcal{R}[k-\varepsilon_k,k-i]}<\infty \right] - \mathbb{P}\left[H^+_{\mathcal{R}[k-\varepsilon_k,k]}<\infty \right] \right|\\
& \le 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \sum_{i=0}^{\varepsilon_k} \mathbb{P}[\tau_2=i] \cdot \mathbb{P}\left[H^+_{\mathcal{R}[k-i,k]}<\infty \right] \\
& \stackrel{\eqref{lem.hit.3}}{\le} 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \frac{C}{k^{3/2}}\sum_{i=0}^{\varepsilon_k} i \mathbb{P}[\tau_2=i] \\
& \le 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \frac{C}{k^{3/2}}\sum_{i=0}^{\varepsilon_k} \mathbb{P}[\tau_2\ge i]\\
& \stackrel{\eqref{lem.hit.3}}{\le} 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \frac{C}{k^{3/2}}\sum_{i=0}^{\varepsilon_k} \frac{1}{\sqrt i} \le 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \frac{C\sqrt{\varepsilon_k}}{k^{3/2}}. \end{align*} Then it amounts to bound the probability of $\tau_1$ being smaller than $\tau_2$: \begin{align*} &\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] =\sum_{x,y\in \mathbb{Z}^5} \sum_{i=0}^{\varepsilon_k} \mathbb{P}\left[\tau_1=i, i\le \tau_2\le \varepsilon_k, S_k=x,\, S_{k-i} = x+y\right]\\
\le & \sum_{x,y\in \mathbb{Z}^5} \sum_{i=0}^{\varepsilon_k} \mathbb{P}\left[\tau_1=i, S_{k-i}=x+y, (x+\widehat \mathcal{R}_\infty) \cap \mathcal{R}[k-\varepsilon_k,k-i]\neq \varnothing, S_k=x\right]\\ \le & \sum_{x,y\in \mathbb{Z}^5} \sum_{i=0}^{\varepsilon_k} \mathbb{P}\left[\widetilde \mathcal{R}_\infty \cap (x+\mathcal{R}[0,i-1])=\varnothing, S_i=y,\, x+y\in \widetilde \mathcal{R}_\infty\right]\\ & \qquad \times \mathbb{P}\left[ \widehat \mathcal{R}_\infty \cap (y+ \mathcal{R}[0,\varepsilon_k-i])\neq \varnothing, S_{k-i}=-x-y\right], \end{align*} using invariance by time reversal of $S$, and where we stress the fact that in the first probability in the last line, $\mathcal{R}$ and $\widetilde \mathcal{R}$ are two independent ranges starting from the origin. Now the last probability can be bounded using \eqref{Green.hit} and Lemma \ref{lem.prep.123}, which give \begin{align*}
&\mathbb{P}\left[ \widehat \mathcal{R}_\infty \cap (y+ \mathcal{R}[0,\varepsilon_k-i])\neq \varnothing, S_{k-i}=-x-y\right] \le \sum_{j=0}^{\varepsilon_k-i} \mathbb{E}\left[G(S_j+y){\text{\Large $\mathfrak 1$}}\{S_{k-i} = -x-y\}\right]\\ =& \sum_{j=0}^{\varepsilon_k-i} \sum_{z\in \mathbb{Z}^5} p_j(z) G(z+y)p_{k-i-j}(z+x+y) =\sum_{j=k-\varepsilon_k}^{k-i} \sum_{z\in \mathbb{Z}^5} p_j(z) G(z-x)p_{k- i-j}(z-x-y) \\
\lesssim & \frac{1}{(\|x+y\|+\sqrt k)^5}\left(\frac 1{1+\|y\|} + \frac 1{\sqrt k + \|x\|}\right). \end{align*} It follows that \begin{align*}
\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] \lesssim \sum_{x,y\in \mathbb{Z}^5} \sum_{i=0}^{\varepsilon_k} \frac{G(x+y) p_i(y)}{(\|x+y\| + \sqrt k)^5}\left(\frac 1{1+\|y\|} + \frac 1{\sqrt k+\|x\|}\right), \end{align*} and then standard computations show that \begin{equation}\label{tau12eps} \mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}. \end{equation} Taking all these estimates together proves that $$ \operatorname{Cov}(Z_0\varphi_2,Z_k\psi_0) \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.$$ We consider now the covariance between $Z_0\varphi_0$ and $Z_k\psi_2$. Here a new problem arises due to the random variable $Z_0$, which does not play the same role as $Z_k$, but one can use similar arguments. In particular the previous proof gives $$\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_2) = -\operatorname{Cov}((1-Z_0)\varphi_0,Z_k\psi_2) + \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right).$$ Then we decompose as well $\varphi_0=\varphi_0^1 - \varphi_0^2$, with $$\varphi_0^1:=\mathbb{P}[H^+_{\mathcal{R}[k-\varepsilon_k,k]}=\infty\mid S],\ \varphi_0^2:=\mathbb{P}[H^+_{\mathcal{R}[k-\varepsilon_k,k]}=\infty, H^+_{\mathcal{R}[k+1,k+\varepsilon_k]}<\infty \mid S].$$ Using independence we get $$\operatorname{Cov}((1-Z_0)\varphi_0^1,Z_k\psi_2) = \mathbb{E}[\varphi_0^1]\cdot \operatorname{Cov}((1-Z_0),Z_k\psi_2).$$ Then we define in the same fashion as above, $$\widetilde \tau_0:= \inf\{n\ge 1 : S_n=0\}, \ \widetilde \tau_2:= \inf\{n\ge 0 : S_n\in (S_k+\widehat \mathcal{R}[1,\infty))\},$$ with $\widehat \mathcal{R}$ the range of an independent walk starting from the origin. Recall that by definition $1-Z_0 = {\text{\Large $\mathfrak 1$}}\{\widetilde \tau_0 \le \varepsilon_k\}$. Thus one can write $$\operatorname{Cov}((1-Z_0),Z_k\psi_2) = \mathbb{E}[Z_k\psi_2 {\text{\Large $\mathfrak 1$}}\{\widetilde \tau_2 \le \widetilde \tau_0\le \varepsilon_k\}] + \mathbb{E}[Z_k\psi_2 {\text{\Large $\mathfrak 1$}}\{\widetilde \tau_0 < \widetilde \tau_2\}] - \mathbb{P}[\widetilde \tau_0\le \varepsilon_k] \mathbb{E}[Z_k\psi_2].$$ On one hand, using \eqref{Green.hit}, the Markov property, and \eqref{exp.Green}, \begin{align*} & \mathbb{E}[Z_k\psi_2 {\text{\Large $\mathfrak 1$}}\{\widetilde \tau_2 \le \widetilde \tau_0\le \varepsilon_k\}] \le \mathbb{P}[\widetilde \tau_2 \le \widetilde \tau_0 \le \varepsilon_k]\le \sum_{y\in \mathbb{Z}^5} \mathbb{P}[\widetilde \tau_2\le \varepsilon_k,\, S_{\widetilde \tau_2} =y] \cdot G(y)\\ & \le \sum_{i=0}^{\varepsilon_k} \mathbb{E}\left[G(S_i-S_k)G(S_i)\right] \le \sum_{i=0}^{\varepsilon_k} \mathbb{E}[G(S_{k-i})]\cdot \mathbb{E}[G(S_i)] \lesssim \frac{1}{k^{3/2}} \sum_{i=0}^{\varepsilon_k} \frac{1}{1+i^{3/2}} \lesssim \frac{1}{k^{3/2}}. \end{align*} On the other hand, similarly as above, \begin{align}\label{tau20eps} \nonumber & \mathbb{E}[Z_k\psi_2 {\text{\Large $\mathfrak 1$}}\{\widetilde \tau_0 < \widetilde \tau_2\}] - \mathbb{P}[\widetilde \tau_0\le \varepsilon_k]\cdot \mathbb{E}[Z_k\psi_2] \\ \nonumber & \le \mathbb{P}[\widetilde \tau_2 \le \widetilde \tau_0 \le \varepsilon_k] + \sum_{i=1}^{\varepsilon_k}\mathbb{P}[\widetilde \tau_0=i] \left(\mathbb{P}\left[(S_k+\widehat \mathcal{R}[1,\infty))\cap \mathcal{R}[i+1,\varepsilon_k]\neq \varnothing\right] - \mathbb{P}[\widetilde \tau_2\le \varepsilon_k]\right)\\ \nonumber &\lesssim \frac{1}{k^{3/2}} + \sum_{i=1}^{\varepsilon_k}\mathbb{P}[\widetilde \tau_0=i] \mathbb{P}[\widetilde \tau_2\le i] \stackrel{\eqref{lem.hit.3}}{\lesssim} \frac{1}{k^{3/2}} + \frac{1}{k^{3/2}}\sum_{i=1}^{\varepsilon_k} i \mathbb{P}[\widetilde \tau_0=i] \\ & \lesssim \frac{1}{k^{3/2}} + \frac{1}{k^{3/2}}\sum_{i=1}^{\varepsilon_k} \mathbb{P}[\widetilde \tau_0\ge i] \stackrel{\eqref{Green.hit}, \eqref{Green}}{\lesssim} \frac{1}{k^{3/2}}+ \frac{1}{k^{3/2}}\sum_{i=1}^{\varepsilon_k} \frac{1}{1+i^{3/2}} \lesssim \frac{1}{k^{3/2}}. \end{align} In other terms, we have already shown that
$$|\operatorname{Cov}((1-Z_0)\varphi_0^1,Z_k\psi_2) | \lesssim \frac{1}{k^{3/2}}.$$ The case when $\varphi_0^1$ is replaced by $\varphi_0^2$ is entirely similar. Indeed, we define $$\widetilde \tau_1:=\inf\{n\ge 0 : S_n\in \widetilde \mathcal{R}[1,\infty)\},$$ with $\widetilde \mathcal{R}$ the range of a random walk starting from the origin, independent of $S$ and $\widehat \mathcal{R}$. Then we set $\widetilde \tau_{0,1} := \max(\widetilde \tau_0,\widetilde \tau_1)$, and exactly as for \eqref{tau12eps} and \eqref{tau20eps}, one has \begin{equation*} \mathbb{P}[\widetilde \tau_2\le \widetilde \tau_{0,1} \le \varepsilon_k] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}, \end{equation*} and \begin{align*} & \mathbb{E} \left[(1-Z_0) \varphi_0^2 Z_k\psi_2 {\text{\Large $\mathfrak 1$}}\{ \widetilde \tau_{0,1}<\widetilde \tau_2 \} \right] - \mathbb{E}[(1-Z_0) \varphi_0^2]\cdot \mathbb{E}[ Z_k\psi_2] \\ & \le \mathbb{P}[\widetilde \tau_2\le \widetilde \tau_{0,1} \le \varepsilon_k] +\sum_{i=0}^{\varepsilon_k} \mathbb{P}[\widetilde \tau_{0,1}= i] \cdot \mathbb{P}[\widetilde \tau_2\le i] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}. \end{align*} Altogether, this gives
$$|\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_2)|\lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.$$
\underline{Case $i=3$.} We only need to treat the case of the covariance between $Z_0\varphi_3$ and $Z_k\psi_0$, as the other one is entirely similar here. Define $$\widetilde \varphi_3:=\mathbb{P}\left[H^+_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]\cup \mathcal{R}[k+\varepsilon_k+1,\infty)}=\infty,\, H^+_{\mathcal{R}[k,k+\varepsilon_k]}<\infty\mid S\right].$$ The proof of the case $i=2$, already shows that
$$|\operatorname{Cov}(Z_0\widetilde \varphi_3,Z_k\psi_0)|\lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.$$ Define next $$h_3:=\varphi_3-\widetilde \varphi_3=\mathbb{P}\left[H^+_{\mathcal{R}[-\varepsilon_k,\varepsilon_k]}=\infty,\, H^+_{\mathcal{R}[k+\varepsilon_k+1,\infty)}<\infty\mid S\right].$$ Assume for a moment that $\varepsilon_k\ge k^{\frac{9}{20}}$. We will see later another argument when this condition is not satisfied. Then define $\varepsilon_k':= \lfloor \varepsilon_k^{10/9}/k^{1/9}\rfloor$, and note that one has $\varepsilon'_k\le \varepsilon_k$. Write $\psi_0=\psi'_0+h_0$, with $$\psi_0':= \mathbb{P}\left[H^+_{\mathcal{R}[k-\varepsilon'_k+1,k+\varepsilon'_k-1]}=\infty \mid S\right], $$ and $$h_0:= \mathbb{P}\left[H^+_{\mathcal{R}[k-\varepsilon'_k+1,k+\varepsilon'_k-1]}=\infty, \, H^+_{\mathcal{R}[k-\varepsilon_k,k-\varepsilon'_k]\cup\mathcal{R}[k+\varepsilon'_k,k+\varepsilon_k] } <\infty \mid S\right]. $$ Define also $$Z'_k:={\text{\Large $\mathfrak 1$}}\{S_\ell\neq S_k,\text{ for all }\ell=k+1,\dots,k+\varepsilon'_k-1\}.$$ One has $$\operatorname{Cov}(Z_0h_3,Z_k\psi_0) = \operatorname{Cov}(Z_0h_3,Z'_k\psi'_0) + \operatorname{Cov}(Z_0h_3,Z'_kh_0) +\operatorname{Cov}(Z_0h_3,(Z_k-Z'_k)\psi_0) .$$ For the last of the three terms, one can simply notice that, using the Markov property at the first return time to $S_k$ (for the walk $S$), and then \eqref{Green.hit}, \eqref{Green}, and \eqref{lem.hit.3}, we get \begin{align*} \mathbb{E}[h_3(Z_k-Z'_k)] & \le \mathbb{E}[Z_k-Z'_k] \times \mathbb{P}[\widetilde \mathcal{R}_\infty \cap \mathcal{R}\left[k,\infty)\neq \varnothing\right] \\ & \lesssim \frac{1}{(\varepsilon'_k)^{3/2}\sqrt k} \lesssim \frac 1{\varepsilon_k^{5/3}k^{1/3} }\lesssim \frac{1}{k^{\frac{13}{12}}}, \end{align*} using our hypothesis on $\varepsilon_k$ for the last equality. As a consequence, it also holds
$$|\operatorname{Cov}(Z_0h_3,(Z_k-Z'_k)\psi_0)| \lesssim k^{-\frac{13}{12}}. $$ Next we write \begin{equation}\label{cov.i3} \operatorname{Cov}(Z_0h_3,Z'_kh_0) = \sum_{x,y\in \mathbb{Z}^5} (p_{k-2\varepsilon_k}(x-y) - p_k(x)) H_1(y)H_2(x), \end{equation} where $$H_1(y) := \mathbb{E}\left[Z'_kh_0 {\text{\Large $\mathfrak 1$}}\{S_{k+\varepsilon_k} - S_{k-\varepsilon_k} = y\}\right], \ H_2(x) := \mathbb{E}\left[Z_0h_3 \mid S_{k+\varepsilon_k} - S_{\varepsilon_k} = x\right].$$ Define $r_k:=(k/\varepsilon'_k)^{1/8}$. By using symmetry and translation invariance, \begin{align*}
&\sum_{\|y\|\ge \sqrt{\varepsilon_k} r_k} H_1(y) \le \mathbb{P}\left[H_{\mathcal{R}[-\varepsilon_k,-\varepsilon'_k]\cup\mathcal{R}[\varepsilon'_k,\varepsilon_k]}<\infty, \, \|S_{\varepsilon_k} - S_{-\varepsilon_k}\|\ge \sqrt{\varepsilon_k} r_k\right]\\
&\le 2 \mathbb{P}\left[H_{\mathcal{R}[\varepsilon'_k,\varepsilon_k]}<\infty, \, \|S_{\varepsilon_k} \|\ge \sqrt{\varepsilon_k} \frac{r_k}{2}\right] + 2\mathbb{P}\left[H_{\mathcal{R}[\varepsilon'_k,\varepsilon_k]}<\infty, \, \|S_{-\varepsilon_k} \|\ge \sqrt{\varepsilon_k} \frac{r_k}{2} \right] \\
& \stackrel{\eqref{lem.hit.3}, \, \eqref{Sn.large}}{\le} 2 \mathbb{P}\left[H_{\mathcal{R}[\varepsilon'_k,\varepsilon_k]}<\infty, \, \|S_{\varepsilon_k} \|\ge \sqrt{\varepsilon_k} \frac{r_k}2\right] + \frac{C}{\sqrt{\varepsilon'_k} r_k^5}. \end{align*} Considering the first probability on the right-hand side, define $\tau$ as the first hitting time (for $S$), after time $\varepsilon'_k$, of another independent walk $\widetilde S$ (starting from the origin). One has \begin{align*}
& \mathbb{P}\left[H_{\mathcal{R}[\varepsilon'_k,\varepsilon_k]}<\infty, \, \|S_{\varepsilon_k} \|\ge \sqrt{\varepsilon_k} \frac{r_k}2\right] \\
& \le \mathbb{P}[\|S_\tau\| \ge \sqrt{\varepsilon_k} \frac{r_k}4,\, \tau\le \varepsilon_k] + \mathbb{P}[\|S_{\varepsilon_k} - S_\tau\| \ge \sqrt{\varepsilon_k} \frac{r_k}4,\, \tau\le \varepsilon_k] . \end{align*} Using then the Markov property at time $\tau$, we deduce with \eqref{lem.hit.3} and \eqref{Sn.large},
$$\mathbb{P}[\|S_{\varepsilon_k} - S_\tau\| \ge \sqrt{\varepsilon_k} \frac{r_k}4,\, \tau\le \varepsilon_k] \lesssim \frac{1}{\sqrt{\varepsilon'_k} r_k^5}.$$ Likewise, using the Markov property at the first time when the walk exit the ball of radius $\sqrt{\varepsilon_k} r_k/4$, and applying then \eqref{Sn.large} and \eqref{lem.hit.2}, we get as well
$$ \mathbb{P}[\|S_\tau\| \ge \sqrt{\varepsilon_k} \frac{r_k}4,\, \tau\le \varepsilon_k] \lesssim \frac{1}{\sqrt{\varepsilon_k} r_k^6}.$$ Furthermore, for any $y$, one has
$$\sum_{x\in \mathbb{Z}^5} p_{k-2\varepsilon_k}(x-y) H_2(x) \stackrel{\eqref{pn.largex}, \eqref{lem.hit.2}}{\lesssim} \sum_{x\in \mathbb{Z}^5} \frac{1}{(1+\|x+y\|)(\|x\|+\sqrt{k})^5} \lesssim \frac{1}{\sqrt k},$$ with an implicit constant, which is uniform in $y$ (and the same holds with $p_k(x)$ instead of $p_{k-2\varepsilon_k}(x-y)$).
Similarly, define $r'_k:=(k/\varepsilon'_k)^{\frac 1{10}}$. One has for any $y$, with $\|y\|\le \sqrt{\varepsilon_k}r_k$,
$$\sum_{\|x\|\ge \sqrt{k}r_k'} p_{k-2\varepsilon_k}(x-y) H_2(x) \stackrel{\eqref{Sn.large}, \eqref{lem.hit.2}}{\lesssim} \frac{1}{\sqrt{k}(r'_k)^6}.$$ Therefore coming back to \eqref{cov.i3}, and using that by \eqref{lem.hit.2}, $\sum_y H_1(y)\lesssim 1/\sqrt{\varepsilon'_k}$, we get \begin{align*} & \operatorname{Cov}(Z_0h_3,Z'_kh_0) \\
& = \sum_{\|x\|\le \sqrt{k}r'_k} \sum_{\|y\|\le \sqrt{\varepsilon_k}r_k} (p_{k-2\varepsilon_k}(x-y) - p_k(x)) H_1(y)H_2(x) + \mathcal{O}\left(\frac{1}{\sqrt{k\varepsilon'_k}(r'_k)^6} + \frac{1}{\sqrt{k\varepsilon'_k} r_k^5}\right)\\
& = \sum_{\|x\|\le \sqrt{k}r'_k} \sum_{\|y\|\le \sqrt{\varepsilon_k}r_k} (p_{k-2\varepsilon_k}(x-y) - p_k(x)) H_1(y)H_2(x) + \mathcal{O}\left(\frac{(\varepsilon'_k)^{\frac{1}{10} } }{k^{\frac{11}{10}}}\right). \end{align*} Now we use the fact $H_1(y) = H_1(-y)$. Thus the last sum is equal to half of the following: \begin{align*}
&\sum_{\|x\|\le \sqrt{k}r'_k} \sum_{\|y\|\le \sqrt{\varepsilon_k}r_k} (p_{k-2\varepsilon_k}(x-y) + p_{k-2\varepsilon_k}(x+y) - 2p_k(x)) H_1(y)H_2(x) \\
&\stackrel{\text{Theorem }\ref{LCLT},\eqref{lem.hit.2}}{\le} \sum_{\|x\|\le \sqrt{k} r'_k} \sum_{\|y\|\le \sqrt{\varepsilon_k}r_k} (\overline p_{k-2\varepsilon_k}(x-y) + \overline p_{k-2\varepsilon_k}(x+y) - 2\overline p_k(x)) H_1(y)H_2(x) \\ & \qquad + \mathcal{O}\left(\frac{(r'_k)^4}{k^{3/2}\sqrt{\varepsilon'_k}} \right), \end{align*} (with an additional factor $2$ in front in case of a bipartite walk). Note that the error term above is $\mathcal{O}(k^{-11/10})$, by definition of $r'_k$. Moreover, straightforward computations show that for any $x$ and $y$ as in the sum above,
$$|\overline p_{k-2\varepsilon_k}(x-y) + \overline p_{k-2\varepsilon_k}(x+y) - 2\overline p_k(x)| \lesssim \left(\frac{\|y\|^2+\varepsilon_k}{k} \right) \overline p_k(cx). $$ In addition one has (with the notation as above for $\tau$), \begin{align*}
&\sum_{y\in \mathbb{Z}^5} \|y\|^2\, H_1(y) \le 2 \mathbb{E}\left[\|S_{\varepsilon_k} - S_{-\varepsilon_k}\|^2 {\text{\Large $\mathfrak 1$}}\{\tau\le \varepsilon_k\}\right] \\
& \le 4 \mathbb{E}[\|S_{\varepsilon_k}\|^2] \mathbb{P}[\tau\le \varepsilon_k] + 4 \mathbb{E}\left[\|S_{\varepsilon_k}\|^2 {\text{\Large $\mathfrak 1$}}\{\tau\le \varepsilon_k\}\right] \\
& \stackrel{\eqref{Sn.large}, \eqref{lem.hit.3}}{\lesssim} \frac{ \varepsilon_k}{\sqrt{\varepsilon'_k}} + \mathbb{E}\left[\|S_{\tau}\|^2 {\text{\Large $\mathfrak 1$}}\{\tau\le \varepsilon_k\}\right] + \mathbb{E}\left[\|S_{\varepsilon_k}-S_{\tau}\|^2 {\text{\Large $\mathfrak 1$}}\{\tau\le \varepsilon_k\}\right] \\
& \stackrel{\eqref{Sn.large}, \eqref{lem.hit.3}}{\lesssim} \frac{ \varepsilon_k}{\sqrt{\varepsilon'_k}} + \sum_{r\ge \sqrt{\varepsilon_k}} r \mathbb{P}\left[\|S_{\tau}\|\ge r, \, \tau\le \varepsilon_k\right] \stackrel{\eqref{Sn.large}, \eqref{lem.hit.2}}{\lesssim} \frac{ \varepsilon_k}{\sqrt{\varepsilon'_k}}, \end{align*} using also the Markov property in the last two inequalities (at time $\tau$ for the first one, and at the exit time of the ball of radius $r$ for the second one). Altogether, this gives
$$|\operatorname{Cov}(Z_0h_3,Z'_kh_0)| \lesssim \frac{\varepsilon_k }{k^{3/2}\sqrt{\varepsilon'_k}} + \frac{(\varepsilon'_k)^{\frac{1}{10} } }{k^{\frac{11}{10}}} \lesssim \frac{(\varepsilon_k)^{\frac{1}{9} } }{k^{ \frac{10}{9} }}. $$ In other words, for any sequence $(\varepsilon_k)_{k\ge 1}$, such that $\varepsilon_k\ge k^{9/20}$, one has $$\operatorname{Cov}(Z_0h_3,Z_k\psi_0) = \operatorname{Cov}(Z_0h_3,Z'_k\psi'_0) + \mathcal{O}\left(\frac{(\varepsilon_k)^{\frac{1}{9} } }{k^{ \frac{10}{9} }} + \frac{1}{k^{\frac{13}{12}} }\right).$$ One can then iterate the argument with the sequence $(\varepsilon'_k)$ in place of $(\varepsilon_k)$, and (after at most a logarithmic number of steps), we are left to consider a sequence $(\varepsilon_k)$, satisfying $\varepsilon_k\le k^{9/20}$. In this case, we use similar arguments as above. Define $\widetilde H_1(y)$ as $H_1(y)$, but with $Z_k\psi_0$ instead of $Z'_kh_0$ in the expectation, and choose $r_k:= \sqrt{k/\varepsilon_k}$, and $r'_k=k^{\frac 1{10}}$. Then we obtain exactly as above, \begin{align*} &\operatorname{Cov}(Z_0h_3,Z_k\psi_0) \\
& = \sum_{\|x\|\le \sqrt k r'_k} \sum_{\|y\|\le \sqrt{k}} ( p_{k-2\varepsilon_k}(x-y) - p_k(x)) \widetilde H_1(y) H_2(x) + \mathcal{O}\left(\frac{1}{r_k^5\sqrt k} + \frac 1{(r'_k)^6 \sqrt k}\right)\\
& = \sum_{\|x\|\le \sqrt k r'_k} \sum_{\|y\|\le \sqrt{k}} ( \overline p_{k-2\varepsilon_k}(x-y) - \overline p_k(x)) \widetilde H_1(y) H_2(x) + \mathcal{O}\left(\frac{1}{k^{\frac {11}{10}}} \right)\\ & \lesssim \frac{\varepsilon_k}{k^{3/2}} + \frac{1}{k^{\frac {11}{10}}} \lesssim \frac{1}{k^{\frac {21}{20}}}, \end{align*} which concludes the proof of the proposition.
\section{Intersection of two random walks and proof of Theorem C} \label{sec.thmC} In this section we prove a general result, which will be needed for proving Proposition \ref{prop.phipsi.2}, and which also gives Theorem C as a corollary. First we introduce some general condition for a function $F:\mathbb{Z}^d\to \mathbb{R}$, namely: \begin{equation}\label{cond.F} \begin{array}{c} \text{there exists a constant $C_F>0$, such that }\\
|F(y) - F(x)|\le C_F\, \frac{\|y-x\|}{1+\|y\|} \cdot |F(x)|, \quad \text{for all }x,y\in \mathbb{Z}^d. \end{array} \end{equation} Note that any function satisfying \eqref{cond.F} is automatically bounded. Observe also that this condition is satisfied by functions which are equivalent to $c/\mathcal J(x)^\alpha$, for some constants $\alpha\in [0,1]$, and $c>0$.
On the other hand, it is not satisfied by functions which are $o(1/\| x\|)$, as $\|x\|\to \infty$.
However, this is fine, since the only two cases that will be of interest for us here are when either $F$ is constant, or when $F(x)$ is of order $1/\|x\|$. Now for a general function $F:\mathbb{Z}^d\to \mathbb{R}$, we define for $r>0$,
$$\overline F(r) := \sup_{r\le \|x\|\le r+1} |F(x)|.$$ Then, set $$I_F(r):= \frac {\log (2+r)}{r^{d-2}}\int_0^r s\cdot \overline F(s)\, ds + \int_r^\infty \frac{\overline F(s)\log(2+s)}{s^{d-3} }\, ds,$$ and, with $\chi_d(r):= 1+(\log (2+r)){\text{\Large $\mathfrak 1$}}_{\{d=5\}}$, $$ J_F(r):= \frac{\chi_d(r)}{r^{d-2}} \int_0^r \overline F(s)\, ds + \int_r^\infty \frac{\overline F(s)\chi_d(s)}{s^{d-2}}\, ds.$$
\begin{theorem}\label{thm.asymptotic} Let $(S_n)_{n\ge 0}$ and $(\widetilde S_n)_{n\ge 0}$ be two independent random walks on $\mathbb{Z}^d$, $d\ge 5$, starting respectively from the origin and some $x\in \mathbb{Z}^d$. Let $\ell \in \mathbb{N}\cup \{\infty\}$, and define $$\tau:=\inf\{n\ge 0\, : \, \widetilde S_n \in \mathcal{R}[0,\ell] \}.$$ There exists $\nu\in (0,1)$, such that for any $F:\mathbb{Z}^d\to \mathbb{R}$, satisfying \eqref{cond.F}, \begin{align}\label{thm.asymp.formula} \mathbb{E}_{0,x}\left[F(\widetilde S_\tau) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\}\right] = &\ \frac {\gamma_d}{\kappa}\cdot \mathbb{E}\left[\sum_{i=0}^\ell G(S_i-x)F(S_i)\right] \\
\nonumber & + \mathcal{O}\left(\frac{I_F(\|x\|)}{(\ell\wedge \|x\|)^\nu} + (\ell\wedge \|x\|)^\nu J_F(\|x\|)\right), \end{align} where $\gamma_d$ is as in \eqref{LLN.cap}, and $\kappa$ is some positive constant given by $$\kappa:=\mathbb{E}\left[\Big(\sum_{n\in \mathbb{Z}} G(S_n)\Big)\cdot \mathbb{P}\left[H^+_{\overline \mathcal{R}_\infty}=+\infty \mid \overline \mathcal{R}_\infty \right]\cdot {\text{\Large $\mathfrak 1$}}\{S_n\neq 0,\, \forall n\ge 1\}\right],$$ with $(S_n)_{n\in\mathbb{Z}}$ a two-sided walk starting from the origin and $\overline \mathcal{R}_\infty := \{S_n\}_{n\in \mathbb{Z}}$. \end{theorem}
\begin{remark}\emph{Note that when $F(x) \sim c/\mathcal J(x)^{\alpha}$, for some constants $\alpha \in [0,1]$ and $c>0$, then $I_F(r)$ and $J_F(r)$ are respectively of order $1/r^{d-4+ \alpha}$, and $1/r^{d-3+\alpha}$ (up to logarithmic factors), while one could show that
$$\mathbb{E}\left[\sum_{i=0}^\ell G(S_i-x)F(S_i)\right] \sim \frac{c'}{\mathcal J(x)^{d-4+\alpha}}, \quad \text{as }\|x\|\to \infty\text{ and }\ell/\|x\|^2\to \infty,$$ for some other constant $c'>0$ (see below for a proof at least when $\ell = \infty$ and $\alpha=0$). Therefore in these cases Theorem \ref{thm.asymptotic} provides a true equivalent for the term on the left-hand side of \eqref{thm.asymp.formula}. } \end{remark}
\begin{remark} \emph{This theorem strengthens Theorem C in two aspects: on one hand it allows to consider functionals of the position of one of the two walks at its hitting time of the other path, and on the other hand it also allows to consider only a finite time horizon for one of the two walks (not mentioning the fact that it gives additionally some bound on the error term). Both these aspects will be needed later (the first one in the proof of Lemma \ref{lem.var.2}, and the second one in the proofs of Lemmas \ref{lem.var.3} and \ref{lem.var.4}). } \end{remark}
Given this result one obtains Theorem C as a corollary. To see this, we first recall an asymptotic result on the Green's function:
in any dimension $d\ge 5$, under our hypotheses on $\mu$, there exists a constant $c_d>0$, such that as $\|x\|\to \infty$, \begin{equation}\label{Green.asymp}
G(x)= \frac{c_d}{\mathcal J(x)^{d-2}} + \mathcal{O}(\|x\|^{1-d}). \end{equation} This result is proved in \cite{Uchiyama98} under only the hypothesis that $X_1$ has a finite $(d-1)$-th moment (we refer also to Theorem 4.3.5 in \cite{LL}, for a proof under the stronger hypothesis that $X_1$ has a finite $(d+1)$-th moment). One also needs the following elementary fact: \begin{lemma}\label{Green.convolution}
There exists a positive constant $c$, such that as $\|x\|\to \infty$,
$$\sum_{y\in \mathbb{Z}^d\setminus\{0,x\}} \frac{1}{\mathcal J(y)^{d-2}\cdot \mathcal J(y-x)^{d-2}} = \frac{c}{\mathcal J(x)^{d-4}} + \mathcal{O}\left(\frac 1{\|x\|^{d-3}}\right).$$ \end{lemma} \begin{proof} The proof follows by first an approximation by an integral, and then a change of variables. More precisely, letting $u:=x/\mathcal J(x)$, one has \begin{align*} & \sum_{y\in \mathbb{Z}^d\setminus\{0,x\} } \frac{1}{\mathcal J(y)^{d-2} \mathcal J(y-x)^{d-2}} = \ \int_{\mathbb{R}^d} \frac{1}{\mathcal J(y)^{d-2} \mathcal J(y-x)^{d-2}} \, dy +
\mathcal{O}(\|x\|^{3-d}) \\
& = \frac 1{\mathcal J(x)^{d-4}} \int_{\mathbb{R}^5}\frac{1}{\mathcal J(y)^{d-2} \mathcal J(y-u)^{d-2} }\, dy + \mathcal{O}(\|x\|^{3-d}), \end{align*} and it suffices to observe that by rotational invariance, the last integral is independent of $x$. \end{proof}
\begin{proof}[Proof of Theorem C] The result follows from Theorem \ref{thm.asymptotic}, by taking $F\equiv 1$ and $\ell = \infty$, and then by using \eqref{Green.asymp} together with Lemma \ref{Green.convolution}. \end{proof}
It amounts now to prove Theorem \ref{thm.asymptotic}. For this, we need some technical estimates that we gather in Lemma \ref{lem.thm.asymptotic} below. Since we believe this is not the most interesting part, we defer its proof to the end of this section.
\begin{lemma}\label{lem.thm.asymptotic} Assume that $F$ satisfies \eqref{cond.F}. Then \begin{enumerate} \item There exists a constant $C>0$, such that for any $x\in \mathbb{Z}^d$, \begin{equation}\label{lem.thm.asymp.1}
\sum_{i=0}^\infty \mathbb{E}\left[\left(\sum_{j=0}^\infty G(S_j-S_i)\frac{\|S_j-S_i\|}{1+\|S_j\|}\right) \cdot |F(S_i)| G(S_i-x)\right] \le C J_F(\|x\|). \end{equation} \item There exists $C>0$, such that for any $R>0$, and any $x\in \mathbb{Z}^d$, \begin{equation}\label{lem.thm.asymp.2}
\sum_{i=0}^\infty \mathbb{E}\left[\left(\sum_{|j-i|\ge R}G(S_j-S_i)\right) |F(S_i)| G(S_i-x)\right] \le \frac{C}{R^{\frac{d-4}2}}\cdot I_F(\|x\|), \end{equation} \begin{equation}\label{lem.thm.asymp.2bis}
\sum_{i=0}^\infty \mathbb{E}\left[\left(\sum_{|j-i|\ge R}G(S_j-S_i) |F(S_j)|\right) G(S_i-x)\right] \le \frac{C}{R^{\frac{d-4}2} }\cdot I_F(\|x\|) . \end{equation} \end{enumerate} \end{lemma}
One also need some standard results from (discrete) potential theory. If $\Lambda$ is a nonempty finite subset of $\mathbb{Z}^d$, containing the origin, we define
$$\text{rad}(\Lambda):=1+\sup_{x\in \Lambda} \|x\|,$$ and also consider for $x\in \Lambda$, $$e_\Lambda(x):=\mathbb{P}_x[H_\Lambda^+=\infty], \quad \text{and} \quad \overline e_\Lambda(x):=\frac{e_\Lambda(x)}{\mathrm{Cap}(\Lambda)}.$$ The measure $\overline e_\Lambda$ is sometimes called the harmonic measure of $\Lambda$ from infinity, due to the next result. \begin{lemma}\label{lem.potential}
There exists a constant $C>0$, such that for any finite subset $\Lambda\subseteq \mathbb{Z}^d$ containing the origin, and any $y\in \mathbb{Z}^d$, with $\| y\|>2\text{rad}(\Lambda)$, \begin{eqnarray}\label{cap.hitting}
\mathbb{P}_y[H_\Lambda<\infty] \le C\cdot \frac{\mathrm{Cap}(\Lambda)}{1+\|y\|^{d-2}}. \end{eqnarray} Furthermore, for any $x\in \Lambda$, and any $y\in \mathbb{Z}^d$, \begin{eqnarray}\label{harm.hit}
\Big| \mathbb{P}_y[S_{H_\Lambda}=x\mid H_\Lambda<\infty] - \overline e_\Lambda(x)\Big| \le C\cdot \frac{\text{rad}(\Lambda)}{1+\|y\|}. \end{eqnarray} \end{lemma} This lemma is proved in \cite{LL} for finite range random walks. The proof extends to our setting, but some little care is needed, so we shall give some details at the end of this section. Assuming this, one can now give the proof of our main result.
\begin{proof}[Proof of Theorem \ref{thm.asymptotic}] The proof consists in computing the quantity \begin{equation}\label{eq.A} A:= \mathbb{E}_{0,x}\left[\sum_{i=0}^\ell \sum_{j=0}^\infty {\text{\Large $\mathfrak 1$}}\{S_i = \widetilde S_j\}F(S_i)\right], \end{equation} in two different ways\footnote{This idea goes back to the seminal paper of Erd\'os and Taylor \cite{ET60}, even though it was not used properly there and was corrected only a few years later by Lawler, see \cite{Law91}.}. On one hand, by integrating with respect to the law of $\widetilde S$ first, we obtain \begin{equation}\label{A.first} A= \mathbb{E}\left[\sum_{i=0}^\ell G(S_i-x)F(S_i)\right]. \end{equation} On the other hand, the double sum in \eqref{eq.A} is nonzero only when $\tau$ is finite. Therefore, using also the Markov property at time $\tau$, we get \begin{align*} A &= \mathbb{E}_{0,x}\left[\left(\sum_{i=0}^\ell \sum_{j=0}^\infty {\text{\Large $\mathfrak 1$}}\{S_i = \widetilde S_j\}F(S_i)\right) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\} \right]\\ & = \sum_{i=0}^\ell \mathbb{E}_{0,x}\left[ \left( \sum_{j=0}^\ell G(S_j-S_i) F(S_j)\right)Z_i^\ell \cdot {\text{\Large $\mathfrak 1$}}\{\tau<\infty, \widetilde S_\tau = S_i\} \right], \end{align*} where we recall that $Z_i^\ell = {\text{\Large $\mathfrak 1$}}\{S_j \neq S_i,\, \forall j=i+1,\dots,\ell\}$. The computation of this last expression is divided in a few steps.
\underline{Step 1.} Set $$B:= \sum_{i=0}^\ell \mathbb{E}_{0,x}\left[ \left( \sum_{j=0}^\ell G(S_j-S_i) \right)F(S_i)Z_i^\ell \cdot {\text{\Large $\mathfrak 1$}}\{\tau<\infty, \widetilde S_\tau = S_i\} \right],$$ and note that, \begin{align*}
& |A-B| \stackrel{\eqref{cond.F}}{\le} C_F\, \sum_{i=0}^\ell \mathbb{E}_{0,x}\left[ \left( \sum_{j=0}^\ell G(S_j-S_i)\frac{\|S_j-S_i\|}{(1+\|S_j\|)} \right)|F(S_i)| {\text{\Large $\mathfrak 1$}}\{S_i\in \widetilde \mathcal{R}_\infty\} \right]\\
&\stackrel{\eqref{Green.hit}}{\le} C_F\, \sum_{i=0}^\ell \mathbb{E}\left[ \left( \sum_{j=0}^\ell G(S_j-S_i)\frac{\|S_j-S_i\|}{(1+\|S_j\|)} \right)|F(S_i)| G(S_i-x)\right] \stackrel{\eqref{lem.thm.asymp.1}}{=} \mathcal{O}\left(J_F(\|x\|) \right). \end{align*}
\underline{Step 2.} Consider now some positive integer $R$, and define $$D_R:= \sum_{i=0}^{\ell} \mathbb{E}_{0,x}\left[ \mathcal G_{i,R,\ell} F(S_i)Z_i^\ell \cdot {\text{\Large $\mathfrak 1$}}\{\tau<\infty, \widetilde S_\tau = S_i\} \right],$$ with $\mathcal G_{i,R,\ell}:= \sum_{j=(i-R)\vee 0}^{(i+R)\wedge \ell} G(S_j-S_i)$. One has
$$|B-D_R| \stackrel{\eqref{Green.hit}}{\le} \sum_{i=0}^{\ell} \mathbb{E}\left[ \left(\sum_{|j-i|>R} G(S_j-S_i)\right) |F(S_i)|G(S_i-x)\right]
\stackrel{\eqref{lem.thm.asymp.2}}{\lesssim} \frac{ I_F(\|x\|)}{R^{\frac{d-4}2}}.$$
\underline{Step 3.} Let $R$ be an integer larger than $2$, and such that $\ell\wedge \|x\|^2 \ge R^6$. Let $M:=\lfloor \ell / R^5\rfloor -1$, and define for $0\le m\le M$, $$I_m:=\{mR^5+R^3,\dots, (m+1)R^5-R^3\}, \text{ and } J_m:=\{mR^5,\dots, (m+1)R^5-1\}.$$ Define further $$E_R := \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}_{0,x}\left[ \mathcal G_{i,R} F(S_i)Z_i^\ell \cdot {\text{\Large $\mathfrak 1$}}\{\tau<\infty, \widetilde S_\tau = S_i\} \right],$$ with $\mathcal G_{i,R} := \sum_{j=i-R}^{i+R} G(S_j-S_i)$. One has, bounding $\mathcal G_{i,R}$ by $(2R+1)G(0)$, \begin{align*}
|D_R - E_R| & \le (2R+1) G(0)\\
& \times \left\{ \sum_{m=0}^M \sum_{i\in J_m\setminus I_m} \mathbb{E} \left[|F(S_i)| G(S_i-x) \right] + \sum_{i=(M+1) R^5}^\ell \mathbb{E} \left[|F(S_i)| G(S_i-x) \right] \right\}, \end{align*} with the convention that the last sum is zero when $\ell$ is infinite. Using $\ell \ge R^6$, we get \begin{align*}
&\sum_{i=(M+1) R^5}^\ell \mathbb{E} \left[|F(S_i)| G(S_i-x) \right] \le \sum_{z\in \mathbb{Z}^d} |F(z)| G(z-x) \sum_{i=(M+1)R^5}^{(M+2)R^5} p_i(z) \\
&\stackrel{\eqref{pn.largex},\, \eqref{Green}}{\lesssim} \frac{R^5}{\ell} \sum_{z\in \mathbb{Z}^d} \frac{|F(z)|}{(1+ \|z-x\|^{d-2})(1+\|z\|^{d-2})} \lesssim \frac{R^5}{ \ell} \cdot I_F(\|x\|). \end{align*}
Likewise, since $\|x\|^2\ge R^6$, \begin{align}\label{final.step3}
\nonumber & \sum_{m=0}^M \sum_{i\in J_m\setminus I_m} \mathbb{E} \left[|F(S_i)| G(S_i-x) \right] \le \sum_{z\in \mathbb{Z}^d} \frac{|F(z)|}{1+\|z-x\|^{d-2}} \sum_{m=0}^M \sum_{i\in J_m\setminus I_m} p_i(z)\\
\nonumber &\stackrel{\eqref{Green}}{\lesssim} \frac{1}{1+\|x\|^{d-2}} \sum_{\|z\|^2\le R^5} \frac{1}{1+\|z\|^{d-2}} \\
\nonumber & \qquad + \sum_{\|z\|^2 \ge R^5} \frac{|F(z)|}{1+\|z-x\|^{d-2}} \sum_{m=0}^M \sum_{i\in J_m\setminus I_m} \left(\frac{{\text{\Large $\mathfrak 1$}}\{i\le \|z\|^2\}}{1+\|z\|^d} + \frac{{\text{\Large $\mathfrak 1$}}\{i\ge \|z\|^2\}}{i^{d/2}}\right) \\
& \lesssim \frac{R^5}{1+\|x\|^{d-2}} + \frac{1}{R^2} \cdot I_F(\|x\|), \end{align} using for the last inequality that the proportion of indices $i$ which are not in one of the $I_m$'s, is of order $1/R^2$.
\underline{Step 4.} For $0\le m \le M+1$, set $$\mathcal{R}^{(m)}:=\mathcal{R}[mR^5,(m+1)R^5-1], \quad \text{and}\quad \tau_m:= \inf\{ n \ge 0 \, :\, \widetilde S_n \in \mathcal{R}^{(m)}\}.$$ Then let $$F_R := \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}_{0,x}\left[ \mathcal G_{i,R} F(S_i)Z_i^\ell \cdot {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty, \widetilde S_{\tau_m} = S_i\} \right].$$ Since by definition $\tau\le \tau_m$, for any $m$, one has for any $i\in I_m$, \begin{align*}
& |\mathbb{P}_{0,x}[\tau<\infty, \widetilde S_{\tau} = S_i\mid S] - \mathbb{P}_{0,x}[\tau_m<\infty, \widetilde S_{\tau_m} = S_i\mid S] | \\
& \le \mathbb{P}_{0,x}[\tau<\tau_m<\infty, \widetilde S_{\tau_m}=S_i\mid S]
\le \sum_{j\notin J_m} \mathbb{P}_{0,x}[\tau<\tau_m<\infty, \widetilde S_{\tau} = S_j, \widetilde S_{\tau_m} = S_i\mid S]\\ &\stackrel{\eqref{Green.hit}}{\le} \sum_{j\notin J_m} G(S_j-x) G(S_i-S_j). \end{align*} Therefore, bounding again $\mathcal G_{i,R}$ by $(2R+1)G(0)$, we get \begin{align*}
|E_R-F_R| & \lesssim R \, \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}\left[\left(\sum_{j\notin J_m} G(S_i-S_j) G(S_j-x)\right)\cdot |F(S_i)| \right]\\
&\lesssim R \, \sum_{i=0}^\infty \mathbb{E}\left[\left(\sum_{j\, :\, |j-i|\ge R^3} G(S_i-S_j) G(S_j-x)\right)\cdot |F(S_i)| \right] \\
& \stackrel{\eqref{lem.thm.asymp.2bis}}{\lesssim} \frac{1}{ R^{3\frac{d-4}{2}-1}} \cdot I_F(\|x\|)\lesssim \frac{1}{\sqrt R} \cdot I_F(\|x\|). \end{align*}
\underline{Step 5.} For $m\ge 0$ and $i\in I_m$, define $$e_i^m := \mathbb{P}_{S_i} \left[H_{\mathcal{R}^{(m)}}^+=\infty\mid S\right], \quad \text{and}\quad \overline e_i^m:= \frac{e_i^m}{\mathrm{Cap}(\mathcal{R}^{(m)}) }.$$ Then let $$H_R: = \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}_{0,x}\left[ \mathcal G_{i,R}F(S_i)Z_i^\ell \overline e_i^m \cdot {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right].$$ Applying \eqref{harm.hit} to the sets $\Lambda_m:=\mathcal{R}^{(m)}-S_{i_m}$, we get for any $m\ge 0$, and any $i\in I_m$, \begin{eqnarray}\label{harm.application}
\left| \mathbb{P}_{0,x}[\widetilde S_{\tau_m} = S_i\mid \tau_m<\infty, S] - \overline e_i^m \right| \le C\, \frac{\text{rad}(\Lambda_m)}{1+\|x-S_{i_m}\|}. \end{eqnarray} By \eqref{cap.hitting}, it also holds \begin{align}\label{hit.cap.application}
\nonumber \mathbb{P}_{0,x}[\tau_m<\infty \mid S ] & \le \frac{ CR^5}{1+\|x-S_{i_m}\|^{d-2}} + {\text{\Large $\mathfrak 1$}}\{\|x-S_{i_m} \| \le 2\text{rad}(\Lambda_m)\}\\
& \lesssim \frac{ R^5+\text{rad}(\Lambda_m)^{d-2}}{1+\|x-S_{i_m}\|^{d-2}}, \end{align}
using that $\mathrm{Cap}(\Lambda_m)\le |\Lambda_m| \le R^5$. Note also that by \eqref{norm.Sn} and Doob's $L^p$-inequality (see Theorem 4.3.3 in \cite{Dur}), one has for any $1< p\le d$, \begin{equation}\label{Doob} \mathbb{E}[\text{rad}(\Lambda_m)^p] = \mathcal{O}(R^{\frac{5p}{2}}). \end{equation} Therefore, \begin{align*}
& |F_R - H_R| \stackrel{\eqref{harm.application}}{\lesssim} R
\sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}_{0,x}\left[ \frac{|F(S_i)|\cdot \text{rad}(\Lambda_m)}{1+\|x-S_{i_m}\|} {\text{\Large $\mathfrak 1$}}\{ \tau_m<\infty\}\right] \\ &\stackrel{\eqref{cond.F}}{\lesssim} R^6
\sum_{m=0}^M \mathbb{E}_{0,x}\left[ \frac{|F(S_{i_m})|\cdot \text{rad}(\Lambda_m)^2}{1+\|x-S_{i_m}\|} {\text{\Large $\mathfrak 1$}}\{ \tau_m<\infty\}\right] \\
& \stackrel{\eqref{hit.cap.application}, \eqref{Doob}}{\lesssim} R^{6+\frac{5d}{2}} \sum_{m=0}^M \mathbb{E}\left[ \frac{|F(S_{i_m})|}{1+\|x-S_{i_m}\|^{d-1}}\right] \lesssim R^{6+\frac{5d}{2}}\, \sum_{z\in \mathbb{Z}^d} \frac{|F(z)|G(z)}{1+\|x-z\|^{d-1}} \\
& \stackrel{\eqref{Green}}{\lesssim} \frac{R^{6+\frac{5d}{2}}}{1+\|x\|} \cdot I_F(\|x\|). \end{align*}
\underline{Step 6.} Let $$K_R:= \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell \overline e_i^m\right] \cdot \mathbb{E}\left[F(S_{i_m}) {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right].$$ One has, using the Markov property and a similar argument as in the previous step, \begin{align*}
& |K_R-H_R| \stackrel{\eqref{cond.F}}{\lesssim}
R \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}_{0,x}\left[\frac{|F(S_{i_m})|\cdot (1+\|S_i-S_{i_m}\|^2)}{1+\|S_{i_m}\|}\cdot {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right] \\
& \stackrel{\eqref{hit.cap.application}, \eqref{Sn.large}}{\lesssim} R^{6+\frac{5d}{2}} \sum_{m=0}^M \mathbb{E}\left[ \frac{|F(S_{i_m})|}{(1+\|S_{i_m}\|)(1+\|x-S_{i_m}\|^{d-2})}\right] \lesssim R^{6+\frac{5d}{2}}\cdot J_F(\|x\|). \end{align*}
\underline{Step 7.} Finally we define $$\widetilde A:= \frac{\kappa}{\gamma_d} \cdot \mathbb{E}_{0,x}\left[F(\widetilde S_\tau) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\} \right].$$ We recall that one has (see Lemmas 2.1 and 2.2 in \cite{AS19}), \begin{eqnarray}\label{easy.variance} \mathbb{E}\left[\left(\mathrm{Cap}(\mathcal{R}_n) - \gamma_d n\right)^2\right] = \mathcal{O}(n(\log n)^2). \end{eqnarray} It also holds for any nonempty subset $\Lambda\subseteq \mathbb{Z}^d$, \begin{eqnarray}\label{min.cap}
\mathrm{Cap}(\Lambda) \ge c|\Lambda|^{1-\frac 2d}\ge c|\Lambda|^3, \end{eqnarray}
using $d\ge 5$ for the second inequality (while the first inequality follows from \cite[Proposition 6.5.5]{LL} applied to the constant function equal to $c/|\Lambda|^{2/d}$, with $c>0$ small enough). As a consequence, for any $m\ge 0$ and any $i\in I_m$, \begin{align*}
& \left|\mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell \overline e_i^m\right] - \frac{\mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell e_i^m\right]}{\gamma_dR^5} \right| \lesssim
\frac{1}{R^4} \mathbb{E}\left[ \frac {|\mathrm{Cap}(\mathcal{R}^{(m)}) - \gamma_dR^5|}{\mathrm{Cap}(\mathcal{R}^{(m)})}\right] \\
\stackrel{\eqref{easy.variance}}{\lesssim} & \frac{\log R}{R^{3/2}} \mathbb{E}\left[\frac 1{\mathrm{Cap}(\mathcal{R}^{(m)})^2}\right]^{1/2}
\stackrel{\eqref{min.cap}}{\lesssim} \frac{\log R}{R^{3/2}} \left(\frac{\mathbb{P}[\mathrm{Cap}(\mathcal{R}^{(m)}) \le \gamma_d R^5/2]}{R^6} + \frac{1}{R^{10}}\right)^{1/2} \\
\stackrel{\eqref{easy.variance}}{\lesssim} & \frac{\log R}{R^{3/2}} \left(\frac{(\log R)^2}{R^{11}} + \frac{1}{R^{10}}\right)^{1/2} \lesssim \frac{1}{R^6}. \end{align*} Next, recall that $Z(i)={\text{\Large $\mathfrak 1$}}\{S_j\neq S_i,\, \forall j>i\}$, and note that
$$|\mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell e_i^m\right] - \mathbb{E} \left[ \mathcal G_{i,R} Z(i) e_i^m\right]| \stackrel{\eqref{Green.hit},\, \eqref{Green}}{\lesssim} \frac{1}{R^{7/2}}.$$ Moreover, letting $e_i:=\mathbb{P}_{S_i}[H^+_{\overline \mathcal{R}_\infty} = \infty\mid \overline \mathcal{R}_\infty]$ (where we recall $\overline \mathcal{R}_\infty$ is the range of a two-sided random walk), one has
$$ |\mathbb{E} \left[ \mathcal G_{i,R} Z_i e_i^m\right] - \mathbb{E} \left[ \mathcal G_{i,R} Z_i e_i \right]| \stackrel{\eqref{lem.hit.3}}{\lesssim} \frac{1}{\sqrt R}, $$
$$| \mathbb{E} \left[ \mathcal G_{i,R} Z_i e_i \right] - \kappa| \le 2\, \mathbb{E}\left[\sum_{j>R} G(S_j)\right] \stackrel{\eqref{exp.Green}}{\lesssim}\frac 1{\sqrt R}.$$ Altogether this gives for any $m\ge 0$ and any $i\in I_m$,
$$\left|\mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell \overline e_i^m\right] - \frac{\kappa}{\gamma_dR^5}\right|\lesssim \frac{1}{R^{5+\frac 12}},$$ and thus for any $m\ge 0$,
$$\left| \left(\sum_{i\in I_m} \mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell \overline e_i^m\right] \right) - \frac{\kappa}{\gamma_d}\right|\lesssim \frac{1}{\sqrt R}.$$ Now, a similar argument as in Step 6 shows that
$$\sum_{m=0}^M \left|\mathbb{E}_{0,x}\left[F(S_{i_m}) {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right] - \mathbb{E}_{0,x}\left[F(\widetilde S_{\tau_m}) {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right]\right| \lesssim R^{\frac{5d}2} J_F(\|x\|). $$ Furthermore, using that \begin{align*} F(\widetilde S_\tau){\text{\Large $\mathfrak 1$}}\{\tau<\infty\} & = \sum_{m=0}^{M+1} F(\widetilde S_{\tau_m}){\text{\Large $\mathfrak 1$}}\{\tau=\tau_m<\infty\}\\ & =\sum_{m=0}^{M+1} F(\widetilde S_{\tau_m})({\text{\Large $\mathfrak 1$}} \{\tau_m<\infty\} - {\text{\Large $\mathfrak 1$}}\{\tau<\tau_m<\infty\}), \end{align*} (with the convention that the term corresponding to index $M+1$ is zero when $\ell =\infty$) we get, \begin{align*}
& \left| \sum_{m=0}^M \mathbb{E}_{0,x}\left[F(\widetilde S_{\tau_m}) {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right] - \mathbb{E}_{0,x}\left[F(\widetilde S_{\tau}) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\} \right]\right| \\
& \lesssim \mathbb{P}_{0,x}[\tau_{M+1}<\infty] + \sum_{m=0}^M \mathbb{E}_{0,x}\left[|F(\widetilde S_{\tau_m})| {\text{\Large $\mathfrak 1$}}\{\tau<\tau_m<\infty\}\right]. \end{align*} Using \eqref{hit.cap.application}, \eqref{Doob} and \eqref{exp.Green.x}, we get
$$\mathbb{P}_{0,x}[\tau_{M+1}<\infty] \lesssim \frac{R^{\frac{5(d-2)}2}}{1+\|x\|^{d-2} }.$$ On the other hand, for any $m\ge 0$, \begin{align*}
&\mathbb{E}\left[|F(\widetilde S_{\tau_m})| {\text{\Large $\mathfrak 1$}}\{\tau<\tau_m<\infty\}\right] \le \sum_{j\in J_m} \sum_{i\notin J_m} \mathbb{E}\left[|F(S_j)| G(S_i-S_j)G(S_i-x)\right]\\
& \le \sum_{j\in I_m} \sum_{|j-i|>R^3} \mathbb{E}\left[|F(S_j)| G(S_i-S_j)G(S_i-x)\right] \\
& \qquad + \sum_{j\in J_m\setminus I_m} \sum_{i\notin J_m} \mathbb{E}\left[|F(S_j)|G(S_i-S_j)G(S_i-x)\right]. \end{align*} The first sum is handled as in Step 4. Namely, \begin{align*}
& \sum_{m=0}^M \sum_{j\in I_m} \sum_{|j-i|>R^3} \mathbb{E}\left[|F(S_j)| G(S_i-S_j)G(S_i-x)\right] \\
& \le \sum_{j\ge 0} \sum_{|j-i|>R^3} \mathbb{E}\left[|F(S_j)| G(S_i-S_j)G(S_i-x)\right] \stackrel{\eqref{lem.thm.asymp.2bis}}{\lesssim} \frac{I_F(\|x\|)}{R^{3/2}} . \end{align*} Similarly, defining $\widetilde J_m:=\{mR^5,\dots, mR^5+R\} \cup \{(m+1)R^5-R,\dots,(m+1)R^5-1\}$, one has, \begin{align*}
&\sum_{m=0}^M \sum_{j\in J_m\setminus I_m} \sum_{i\notin J_m} \mathbb{E}\left[|F(S_j)|G(S_i-S_j)G(S_i-x)\right] \\
& \le \sum_{m=0}^M \sum_{j\in J_m\setminus I_m} \sum_{|i-j|>R} \mathbb{E}\left[|F(S_j)|G(S_i-S_j)G(S_i-x)\right] \\
& \quad +\sum_{m=0}^M \sum_{j\in J_m\setminus I_m} \sum_{i\notin J_m,\, |i-j|\le R} \mathbb{E}\left[|F(S_j)|G(S_i-S_j)G(S_i-x)\right] \\
& \stackrel{\eqref{lem.thm.asymp.2bis}, \eqref{cond.F}}{\lesssim} \frac{I_F(\|x\|)}{\sqrt R} + \sum_{m=0}^M \sum_{j\in J_m\setminus I_m} \sum_{i\notin J_m,\, |i-j|\le R} \mathbb{E}\left[|F(S_i)|G(S_i-x)\right]\\
& \lesssim \frac{ I_F(\|x\|)}{\sqrt R} + R \sum_{m= 0}^M \sum_{i \in \widetilde J_m} \mathbb{E}\left[|F(S_i)|G(S_i-x)\right] \lesssim \frac{I_F(\|x\|)}{\sqrt R} + \frac{R^5}{1+\|x\|^{d-2}}, \end{align*} using for the last inequality the same argument as in \eqref{final.step3}. Note also that
$$\mathbb{E}[|F(\widetilde S_\tau)|{\text{\Large $\mathfrak 1$}}\{\tau<\infty\}] \stackrel{\eqref{lem.hit.1}}{\le} \sum_{i\ge 0} \mathbb{E}[|F(S_i)|G(S_i-x)]\lesssim I_F(\|x\|). $$ Therefore, putting all pieces together yields
$$|K_R - \widetilde A| \lesssim \frac{I_F(\|x\|)}{\sqrt{R}} + R^{\frac{5d}2}\cdot J_F(\| x\|) + \frac{R^{\frac{5(d-2)}{2}}}{1+\|x\|^{d-2}}.$$
\underline{Step 8.}
Altogether the previous steps show that for any $R$ large enough, any $\ell \ge 1$, and any $x\in\mathbb{Z}^d$, satisfying $\ell\wedge \|x\|^2 \ge R^6$,
$$|A-\widetilde A| \lesssim \left(\frac{1}{\sqrt{R}} + \frac{R^{6+\frac{5d}{2}}}{1+\|x\|}\right) \cdot I_F(\|x\|) + \frac{R^{\frac{5(d-2)}{2}}}{1+\|x\|^{d-2}} + R^{6+\frac{5d}{2}}\cdot J_F(\|x\|). $$
The proof of the theorem follows by taking for $R$ a sufficiently small power of $\|x\|\wedge \ell$, and observing that
for any function $F$ satisfying \eqref{cond.F}, one has $\liminf_{\|z\|\to \infty} |F(z)|/\|z\|>0$, and thus also $I_F(\|x\|) \ge \frac{c}{1+\|x\|^{d-3}}$. \end{proof}
It amounts now to give the proofs of Lemmas \ref{lem.thm.asymptotic} and \ref{lem.potential}.
\begin{proof}[Proof of Lemma \ref{lem.thm.asymptotic}] We start with the proof of \eqref{lem.thm.asymp.1}. Recall the definition of $\chi_d$ given just above Theorem \ref{thm.asymptotic}. One has for any $i\ge 0$, \begin{align*}
&\mathbb{E}\left[\sum_{j= i+1}^\infty G(S_j-S_i) \frac{\|S_j-S_i\|}{1+\|S_j\|} \mid S_i \right] \stackrel{\eqref{Green}}{\lesssim}
\mathbb{E}\left[\sum_{j= i+1}^\infty \frac{1}{(1+\|S_j - S_i\|^{d-3})(1+\|S_j\|)} \mid S_i \right] \\
& \lesssim \sum_{z\in \mathbb{Z}^d} G(z) \frac{1}{(1+\|z\|^{d-3})(1+\|S_i+z\|)}\stackrel{\eqref{Green}}{\lesssim} \frac{\chi_d(\|S_i\|)}{1+\|S_i\|}, \end{align*} and moreover, \begin{align}\label{lem.FG}
\nonumber &\sum_{i=0}^\infty \mathbb{E}\left[\frac{|F(S_i)|\chi_d(\|S_i\|)}{1+\|S_i\|}G(S_i-x)\right] = \sum_{z\in \mathbb{Z}^d} G(z) \frac{|F(z)|\chi_d(\|z\|)}{1+\|z\|} G(z-x) \\
\nonumber \stackrel{\eqref{Green}}{\lesssim}& \frac{ \chi_d(\|x\|)}{1+\|x\|^{d-2}} \sum_{\|z\|\le \frac{\|x\|}{2}} \frac{|F(z)|}{1+\|z\|^{d-1}} + \sum_{\|z\|\ge \frac{\|x\|}{2}} \frac{|F(z)|\chi_d(\|z\|)}{1+\|z\|^{2d-3}} \\
\nonumber & \qquad + \frac{\chi_d(\|x\|)}{1+\|x\|^{d-1}} \sum_{\|z-x\|\le \frac{\|x\|}{2}} \frac{|F(z)|}{1+\|z-x\|^{d-2}} \\
\stackrel{\eqref{cond.F}}{\lesssim} & J_F(\|x\|/2) + \frac{|F(x)|\chi_d(\|x\|)}{1+\|x\|^{d-3}}\lesssim J_F(\|x\|), \end{align} where the last inequality follows from the fact that by \eqref{cond.F},
$$ \int_{\|x\|/2}^{\|x\|} \frac{\overline F(s)\chi_d(s)}{s^{d-2}} \, ds\, \asymp\, \frac{|F(x)|\chi_d(\|x\|)}{1+\|x\|^{d-3}}\, \asymp\, \frac{\chi_d(\|x\|)}{1+\|x\|^{d-2}} \int_{\|x\|/2}^{\|x\|} \overline F(s)\, ds.$$ Thus
$$\sum_{i=0}^\infty \sum_{j=i+1}^\infty \mathbb{E}\left[G(S_j-S_i) \frac{\|S_j-S_i\|}{1+\|S_j\|} |F(S_i)| G(S_i-x) \right] = \mathcal{O}(J_F(\|x\|)).$$ On the other hand, for any $j\ge 0$, \begin{align}\label{lem.FG.2}
\nonumber &\mathbb{E}\left[\sum_{i= j+1}^\infty G(S_j-S_i) \|S_j-S_i\| \cdot |F(S_i)|G(S_i-x) \mid S_j \right] \\
\nonumber \stackrel{\eqref{Green}}{\lesssim} &
\sum_{i=j+1}^\infty \mathbb{E}\left[ \frac{|F(S_i)|G(S_i-x)}{1+\|S_j - S_i\|^{d-3}} \mid S_j \right]
\stackrel{\eqref{Green}}{\lesssim} \sum_{z\in \mathbb{Z}^d} \frac{|F(S_j+z)| G(S_j + z-x)}{1+\|z\|^{2d-5}}\\ \nonumber \stackrel{\eqref{cond.F}, \eqref{Green}}{\lesssim} & \sum_{z\in \mathbb{Z}^d}
\frac{|F(S_j)|}{(1+\|z\|^{2d-5})(1+\|S_j+z-x\|^{d-2})} + \frac{1}{1+\|S_j\|^{2d-5}}\sum_{\|u\|\le \|S_j\|}\frac{|F(u)| }{1+\|u-x\|^{d-2}} \\
\nonumber \stackrel{\eqref{cond.F}}{\lesssim} & \frac{|F(S_j)|\chi_d(\|S_j-x\|)}{1+\|S_j-x\|^{d-2}} + \frac{{\text{\Large $\mathfrak 1$}}\{\|S_j\|\le \|x\|/2\}\cdot |F(S_j)|}{(1+\|x\|^{d-2})(1+\|S_j\|^{d-5})} \\
\nonumber & \qquad + \frac{{\text{\Large $\mathfrak 1$}}\{\|S_j\|\ge \|x\|/2\}}{1+\|S_j\|^{2d-5}}\left(|F(x)|(1+\| x\|^2) + |F(S_j)|(1+\|S_j\|^2)\right) \\
\lesssim & \frac{|F(S_j)|\chi_d(\|S_j-x\|)}{1+\|S_j-x\|^{d-2}} + \frac{{\text{\Large $\mathfrak 1$}}\{\|S_j\|\le \|x\|\} |F(S_j)|}{1+\|x\|^{d-2}} + \frac{{\text{\Large $\mathfrak 1$}}\{\|S_j\|\ge \|x\| \} |F(S_j)|}{1+\|S_j\|^{d-2}}, \end{align}
where for the last two inequalities we used that by \eqref{cond.F}, if $\|u\|\le \|v\|$, then $|F(u)|\lesssim | F(v)| (1+\|v\|)/(1+\|u\|)$, and also that $d\ge 5$ for the last one. Moreover, for any $r\ge 0$
$$\sum_{j=0}^\infty \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|S_j\|\le r\} \cdot |F(S_j)| }{1+\|S_j\|} \right] = \sum_{\|z\| \le r} \frac{G(z)|F(z)|}{1+\|z\|} \stackrel{\eqref{Green}}{=}\mathcal{O} \left(\int_0^{r} \overline F(s)\, ds\right),$$
$$\sum_{j=0}^\infty \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|S_j\|\ge r\} \cdot |F(S_j)| }{1+\|S_j\|^{d-1}} \right] = \sum_{\|z\| \ge r} \frac{G(z)|F(z)|}{1+\|z\|^{d-1}} \stackrel{\eqref{Green}}{=}\mathcal{O} \left(\int_{r}^\infty \frac{\overline F(s)}{s^{d-2}}\, ds\right).$$ Using also similar computations as in \eqref{lem.FG} to handle the first term in \eqref{lem.FG.2}, we conclude that
$$\sum_{j=0}^\infty \sum_{i=j+1}^\infty \mathbb{E}\left[G(S_j-S_i) \frac{\|S_j-S_i\|}{1+\|S_j\|} |F(S_i)| G(S_i-x) \right] = \mathcal{O}(J_F(\|x\|)),$$ which finishes the proof of \eqref{lem.thm.asymp.1}.
We then move to the proof of \eqref{lem.thm.asymp.2}. First note that for any $i\ge 0$, $$\mathbb{E}\left[\sum_{j\ge i+R} G(S_j-S_i)\mid S_i \right] = \mathbb{E}\left[\sum_{j\ge R} G(S_j) \right] \stackrel{\eqref{exp.Green}}{=} \mathcal{O}\left(R^{\frac{4-d}{2}}\right),$$ and furthermore, \begin{equation}\label{lem.FG.3}
\sum_{i=0}^\infty \mathbb{E}[|F(S_i)|G(S_i-x)] = \sum_{z\in \mathbb{Z}^d} |F(z)| G(z-x)G(z) \stackrel{ \eqref{cond.F},\, \eqref{Green}}{=} \mathcal{O}(I_F(\|x\|)), \end{equation} which together give the desired upper bound for the sum on the set $\{0\le i\le j-R\}$. On the other hand, for any $j\ge 0$, we get as for \eqref{lem.FG.2}, \begin{align*}
&\mathbb{E}\left[\sum_{i\ge j+R} G(S_j-S_i)|F(S_i)|G(S_i-x) \mid S_j \right] = \sum_{z\in \mathbb{Z}^d} G(z)|F(S_j+z)| G(S_j+z-x) G_R(z) \\
& \stackrel{\eqref{Green}}{\lesssim} \frac{1}{R^{\frac{d-4}{2}}} \cdot \sum_{z\in \mathbb{Z}^d} \frac{|F(S_j+z)|}{ (1+\|z\|^d)(1+\|S_j+z-x\|^{d-2})}\\
& \stackrel{ \eqref{cond.F}}{\lesssim} \frac{1}{R^{\frac{d-4}{2}}} \left\{ \sum_{z\in \mathbb{Z}^d} \frac{|F(S_j)|}{ (1+\|z\|^d)(1+\|S_j+z-x\|^{d-2})} + \frac 1{1+\|S_j\|^d} \sum_{\|u\|\le \|S_j\|} \frac{|F(u)|}{1+\|u-x\|^{d-2}}\right\}\\
&\lesssim \frac{1}{R^{\frac{d-4}{2}}} \left\{ \frac{|F(S_j)|\log (2+\|S_j-x\|)}{1+\|S_j-x\|^{d-2}} + \frac{|F(S_j)|}{1+\|x\|^{d-2}+\|S_j\|^{d-2}} \right\}. \end{align*} Then similar computation as above, see e.g. \eqref{lem.FG.3}, give \begin{equation}\label{FSj}
\sum_{j\ge 0} \mathbb{E}\left[\frac{|F(S_j)|\log (2+\|S_j-x\|)}{1+\|S_j-x\|^{d-2}}\right] = \mathcal{O}(I_F(\|x\|)), \end{equation} \begin{equation*}
\sum_{j\ge 0} \mathbb{E}\left[\frac{|F(S_j)|}{1+\|x\|^{d-2} + \|S_j\|^{d-2}}\right] = \mathcal{O}(I_F(\|x\|)), \end{equation*} which altogether proves \eqref{lem.thm.asymp.2}.
The proof of \eqref{lem.thm.asymp.2bis} is entirely similar: on one hand, for any $i\ge 0$, \begin{align*}
& \mathbb{E}\left[\sum_{j= i+R}^\infty G(S_j-S_i) |F(S_j)| \mid S_i \right] \stackrel{\eqref{cond.F}}{\lesssim} \mathbb{E}\left[\sum_{j= i+R}^\infty G(S_j-S_i) \frac{\|S_j-S_i\|}{1+\|S_j\|} \mid S_i \right] |F(S_i)| \\
& \lesssim \sum_{z\in \mathbb{Z}^d} G_R(z) \frac{|F(S_i)|}{(1+\|z\|^{d-3})(1+\|S_i+z\|)} \\
& \lesssim \sum_{z\in \mathbb{Z}^d} \frac{|F(S_i)|}{(R^{\frac{d-2}{2}} + \|z\|^{d-2})(1+\|z\|^{d-3})(1+\|S_i+z\|)} \lesssim \frac{|F(S_i)|}{R^{\frac{d-4}{2}}},
\end{align*} and together with \eqref{FSj}, this yields
$$\sum_{i=0}^\infty \sum_{j=i+R}^\infty \mathbb{E}\left[G(S_j-S_i) |F(S_j)| G(S_i-x) \right]\lesssim \frac{I_F(\|x\|)}{R^{\frac{d-4}{2}} }.$$ On the other hand, for any $j\ge 0$, using \eqref{Green}, \begin{align*}
\mathbb{E}\left[\sum_{i\ge j+R} G(S_j-S_i)G(S_i-x) \mid S_j \right] \lesssim \sum_{z\in \mathbb{Z}^d} \frac{G(S_j + z-x)}{R^{\frac{d-4}{2}}(1+\|z\|^d)}
\lesssim \frac{\log (2+ \|S_j-x\|)}{R^{\frac{d-4}{2}}(1+\|S_j-x\|^{d-2})}, \end{align*} and we conclude the proof of \eqref{lem.thm.asymp.2bis} using \eqref{FSj} again. \end{proof}
\begin{proof}[Proof of Lemma \ref{lem.potential}] The first statement follows directly from \eqref{Green.asymp} and the last-exit decomposition (see Proposition 4.6.4 (c) in \cite{LL}): $$\mathbb{P}_y[H_\Lambda<\infty] = \sum_{x\in \Lambda} G(y-x) e_\Lambda(x).$$
Indeed if $\|y\|>2 \text{rad}(\Lambda)$, using \eqref{Green} we get $G(y-x)\le C\|y\|^{2-d}$, for some constant $C>0$ independent of $x\in \Lambda$, which gives well \eqref{cap.hitting}, since by definition $\sum_{x\in \Lambda} e_\Lambda(x) = \mathrm{Cap}(\Lambda)$.
The second statement is more involved. Note that one can always assume $\mathcal J(y)>C\text{rad}(\Lambda)$, for some constant $C>0$, for otherwise the result is trivial. We use similar notation as in \cite{LL}. In particular $G_A(x,y)$ denotes the Green's function restricted to a subset $A\subseteq \mathbb{Z}^d$, that is the expected number of visits to $y$ before exiting $A$ for a random walk starting from $x$, and $H_A(x,y)=\mathbb{P}_x[H_{A^c}=y]$. We also let $\mathcal{C}_n$ denote the (discrete) ball of radius $n$ for the norm $\mathcal J(\cdot)$. Then exactly as in \cite{LL} (see Lemma 6.3.3 and Proposition 6.3.5 thereof), one can see using \eqref{Green.asymp} that for all $n\ge 1$, \begin{equation}\label{GA}
\left| G_{\mathcal{C}_{n}}(x,w) - G_{\mathcal{C}_{n}}(0,w) \right| \le C \frac{\|x\|}{1+\|w\|} \, G_{\mathcal{C}_{n}}(0,w), \end{equation} for all $x\in \mathcal{C}_{n/4}$, and all $w$ satisfying $2\mathcal J(x) \le \mathcal J(w) \le n/2$. One can then derive an analogous estimate for the (discrete) derivative of $H_{\mathcal{C}_n}$. Define $A_n=\mathcal{C}_n\setminus \mathcal{C}_{n/2}$, and $\rho = H^+_{A_n^c}$. By the last-exit decomposition (see \cite[Lemma 6.3.6]{LL}), one has for $x\in \mathcal{C}_{n/8}$ and $z\notin \mathcal{C}_n$, \begin{align*}
\nonumber & |H_{\mathcal{C}_n}(x,z) - H_{\mathcal{C}_n}(0,z)|\le \sum_{w\in \mathcal{C}_{n/2}}|G_{\mathcal{C}_n}(x,w) - G_{\mathcal{C}_n}(0,w)|\cdot \mathbb{P}_w[S_\rho = z]\\
\nonumber & \stackrel{\eqref{GA}, \eqref{Green}}{\lesssim} \frac{\|x\|}{n}\cdot H_{\mathcal{C}_n}(0,z) + \sum_{2\mathcal J(x)\le \mathcal J(w)\le \frac n4} \frac{\|x\|}{\|w\|^{d-1}} \mathbb{P}_w[S_\rho = z] \\
& \qquad + \sum_{\mathcal J(w) \le 2\mathcal J(x)} \left(\frac 1{1+\|w-x\|^{d-2}} + \frac 1{1+\|w\|^{d-2}}\right)\mathbb{P}_w[S_\rho = z]. \end{align*} Now, observe that for any $y\notin \mathcal{C}_n$, any $w\in \mathcal{C}_{n/4}$, and any $A\subseteq \mathbb{Z}^d$, \begin{align*}
\sum_{z\notin \mathcal{C}_n} G_{A} (y,z) \mathbb{P}_w[S_\rho = z] \lesssim \sum_{z\notin \mathcal{C}_n} \mathbb{P}_w[S_\rho = z] \lesssim \mathbb{P}_w[\mathcal J(S_1)>\frac n2] \lesssim \mathbb{P}[\mathcal J(X_1)> \frac n4] \lesssim n^{-d}, \end{align*} using that by hypothesis $\mathcal J(X_1)$ has a finite $d$-th moment. It follows from the last two displays that \begin{align}\label{potentiel.3}
\sum_{z\notin \mathcal{C}_n} G_{A} (y,z) H_{\mathcal{C}_n}(x,z) = \left(\sum_{z\notin \mathcal{C}_n} G_{A} (y,z) H_{\mathcal{C}_n}(0,z)\right)\left(1+\mathcal{O}\Big(\frac{\|x\|}{n}\Big)\right)+ \mathcal{O}\left( \frac{\|x\|}{n^{d-1}} \right). \end{align}
Now let $\Lambda$ be some finite subset of $\mathbb{Z}^d$ containing the origin, and let $m:=\sup\{\mathcal J(u)\, :\, \|u\|\le 2\text{rad}(\Lambda)\}$. Note that $m=\mathcal{O}(\text{rad}(\Lambda))$, and thus one can assume $\mathcal J(y)>16m$. Set $n:=\mathcal J(y)-1$. Using again the last-exit decomposition and symmetry of the step distribution, we get for any $x\in \Lambda$, \begin{align}\label{potentiel.4} \mathbb{P}_y[S_{H_\Lambda} =x,\, H_\Lambda<\infty] = \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z) \mathbb{P}_x[S_{\tau_n} = z,\, \tau_n<H_\Lambda^+], \end{align} with $\tau_n:= H_{\mathcal{C}_n^c}$. We then write, using the Markov property, \begin{align}\label{potentiel.5} \nonumber \mathbb{P}_x[S_{\tau_n} = z,\, \tau_n<H_\Lambda^+]& =\sum_{x'\in \mathcal{C}_{n/8}\setminus \mathcal{C}_m}\mathbb{P}_x[\tau_m<H_\Lambda^+,\, S_{\tau_m} =x']\cdot \mathbb{P}_{x'}[S_{\tau_n} = z,\, \tau_n<H_\Lambda^+] \\ & \qquad + \mathbb{P}_x\left[\mathcal J(S_{\tau_m}) >\frac n8,\, S_{\tau_n}=z\right], \end{align} with $\tau_m:=H_{\mathcal{C}_m^c}$. Concerning the last term we note that \begin{align}\label{potentiel.5.bis} \nonumber & \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z) \mathbb{P}_x\left[\mathcal J(S_{\tau_m}) >\frac n8,\, S_{\tau_n}=z\right] \\ \nonumber & \stackrel{\eqref{Green.hit}}{\le} \sum_{z\notin \mathcal{C}_n} G(z-y) \left\{\mathbb{P}_x[S_{\tau_m}=z] + \sum_{u\in \mathcal{C}_n\setminus \mathcal{C}_{n/8}} \mathbb{P}_x[S_{\tau_m} =u]G(z-u)\right\}\\
\nonumber & \stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} \sum_{z\notin \mathcal{C}_n} G(z-y) \mathbb{P}_x[S_{\tau_m}=z] + \sum_{u\in \mathcal{C}_n\setminus \mathcal{C}_{n/8}} \frac{\mathbb{P}_x[S_{\tau_m} =u]}{\|y-u\|^{d-4}} \\ \nonumber & \lesssim \mathbb{P}_x[\mathcal J(S_{\tau_m})>n/8] \lesssim \sum_{u\in \mathcal{C}_m} G_{\mathcal{C}_m}(x,u) \mathbb{P}[J(X_1)>\frac n8 - m] \\ & \stackrel{\eqref{Green}}{=}\mathcal{O}\left(\frac{m^2}{n^d}\right) = \mathcal{O}\left(\frac{m}{n^{d-1}}\right), \end{align} applying once more the last-exit decomposition at the penultimate line, and the hypothesis that $\mathcal J(X_1)$ has a finite $d$-th moment at the end. We handle next the sum in the right-hand side of \eqref{potentiel.5}. First note that \eqref{potentiel.3} gives for any $x'\in \mathcal{C}_{n/8}$, \begin{align}\label{potentiel.6} \nonumber \sum_{z\notin \mathcal{C}_n} & G_{\Lambda^c}(y,z)\mathbb{P}_{x'}[S_{\tau_n} = z] = \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(x', z)\\
& = \left(\sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{\|x'\|}n\Big)\right) + \mathcal{O}\left( \frac{\|x'\|}{n^{d-1}} \right). \end{align} Observe then two facts. On one hand, by the last exit-decomposition and symmetry of the step distribution, \begin{equation}\label{pot.1} \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(0, z) \le \sum_{z\notin \mathcal{C}_n} G_{\mathbb{Z}^d\setminus \{0\}}(y,z)H_{\mathcal{C}_n}(0, z) = \mathbb{P}[H_y<\infty] \stackrel{\eqref{Green.hit}, \eqref{Green}}{\lesssim} n^{2-d}, \end{equation} and on the other hand by Proposition 4.6.2 in \cite{LL}, \begin{align}\label{pot.2} & \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z) H_{\mathcal{C}_n}(0, z) \\ \nonumber & = \sum_{z\notin \mathcal{C}_n} G_{\mathbb{Z}^d\setminus \{0\}}(y,z)H_{\mathcal{C}_n}(0, z) + \sum_{z\notin \mathcal{C}_n} \left(G_{\Lambda^c}(y,z) - G_{\mathbb{Z}^d\setminus \{0\}}(y,z)\right) H_{\mathcal{C}_n}(0, z) \\ \nonumber & \ge \mathbb{P}[H_y<\infty] - \mathcal{O}\left(\mathbb{P}_y[H_\Lambda<\infty]\sum_{z\notin \mathcal{C}_n} G(z) H_{\mathcal{C}_n}(0, z)\right) \\ \nonumber & \stackrel{\eqref{hit.ball}}{\ge} \mathbb{P}[H_y<\infty] - \mathcal{O}\left(n^{2-d}\sum_{z\notin \mathcal{C}_n} G(z)^2\right)
\stackrel{\eqref{Green.asymp}}{\ge} \frac{c}{n^{d-2}}. \end{align} This last fact, combined with \eqref{potentiel.6} gives therefore, for any $x'\in \mathcal{C}_{n/8}$, \begin{align}\label{potentiel.6.bis}
\sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)\mathbb{P}_{x'}[S_{\tau_n} = z] = \left(\sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{\|x'\|}n\Big)\right). \end{align} By the Markov property, we get as well \begin{align*}
\sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)\mathbb{P}_{x'}[S_{\tau_n} = z\mid H_\Lambda<\tau_n ] = \left(\sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{m}n\Big)\right), \end{align*} since by definition $\Lambda\subseteq \mathcal{C}_m\subset \mathcal{C}_{n/8}$, and thus \begin{align*}
\sum_{z\notin \mathcal{C}_n} & G_{\Lambda^c}(y,z)\mathbb{P}_{x'}[S_{\tau_n} = z,\, H_\Lambda<\tau_n ] \\
& = \mathbb{P}_{x'}[H_\Lambda<\tau_n]\left( \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{m}n\Big)\right). \end{align*} Subtracting this from \eqref{potentiel.6.bis}, we get for $x'\in \mathcal{C}_{n/8}\setminus \mathcal{C}_m$, \begin{align*}
\sum_{z\notin \mathcal{C}_n} & G_{\Lambda^c}(y,z)\mathbb{P}_{x'}[S_{\tau_n} = z,\, \tau_n<H_\Lambda ] \\
& = \mathbb{P}_{x'}[\tau_n<H_\Lambda]\left( \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{\|x'\|}n\Big)\right), \end{align*} since by \eqref{hit.ball}, one has $\mathbb{P}_{x'}[\tau_n<H_\Lambda]>c$, for some constant $c>0$, for any $x'\notin \mathcal{C}_m$ (note that the stopping time theorem gives in fact
$\mathbb{P}_{x'}[H_\Lambda<\infty] \le G(x') / \inf_{\|u\|\le \text{rad}(\Lambda)} G(u)$, and thus by using \eqref{Green.asymp}, one can ensure $\mathbb{P}_{x'}[H_\Lambda<\infty]\le 1-c$, by taking $\|x'\|$ large enough, which is always possible). Combining this with \eqref{potentiel.4}, \eqref{potentiel.5} and \eqref{potentiel.5.bis}, and using as well \eqref{pot.1} and \eqref{pot.2}, we get \begin{align*} & \mathbb{P}_y[S_{H_\Lambda} =x,\, H_\Lambda<\infty] = \mathbb{P}_x[\tau_n<H_\Lambda]\left( \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(0, z)\right) \\
& \qquad + \mathcal{O}\left(\frac 1{n^{d-1}}\sum_{x'\in\mathcal{C}_{n/8} \setminus \mathcal{C}_m} \mathbb{P}_x[S_{\tau_m}=x'] \cdot \|x'\| \right) + \mathcal{O}\left(\frac{m}{n^{d-1}}\right)\\
\stackrel{\eqref{hit.ball}}{=} & e_\Lambda(x)\left( \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{m}{n}\Big)\right)+ \mathcal{O}\left(\frac 1{n^{d-1}}\sum_{r=2m}^{n/8} \frac{m^2}{r^{d-1}} \right) + \mathcal{O}\left(\frac{m}{n^{d-1}}\right)\\
= & \ e_\Lambda(x)\left( \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{m}{n}\Big)\right), \end{align*} using the same argument as in \eqref{potentiel.5.bis} for bounding $\mathbb{P}_{x'}[\mathcal J(S_{\tau_m})\ge r]$, when $r\ge 2m$. Summing over $x\in \Lambda$ gives \begin{align*} \mathbb{P}_y[H_\Lambda<\infty]= \mathrm{Cap}(\Lambda) \left( \sum_{z\notin \mathcal{C}_n} G_{\Lambda^c}(y,z)H_{\mathcal{C}_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{m}{n}\Big)\right), \end{align*} and the proof of the lemma follows from the last two displays. \end{proof}
\section{Proof of Proposition \ref{prop.phipsi.2}} The proof is divided in four steps, corresponding to the next four lemmas.
\begin{lemma} \label{lem.var.1} Assume that $\varepsilon_k\to \infty$, and $\varepsilon_k/k\to 0$. There exists a constant $\sigma_{1,3}>0$, such that $$\operatorname{Cov}(Z_0\varphi_3, Z_k\psi_1) \sim \frac{\sigma_{1,3}}{k}.$$ \end{lemma}
\begin{lemma} \label{lem.var.2} There exist positive constants $\delta$ and $\sigma_{1,1}$, such that when $\varepsilon_k\ge k^{1-\delta}$, and $\varepsilon_k/k\to 0$, $$ \operatorname{Cov}(Z_0\varphi_1,Z_k\psi_1) \sim \operatorname{Cov}(Z_0\varphi_3, Z_k\psi_3) \sim \frac{\sigma_{1,1}}{k}.$$ \end{lemma}
\begin{lemma}\label{lem.var.3} There exist positive constants $\delta$ and $\sigma_{1,2}$, such that when $\varepsilon_k\ge k^{1-\delta}$, and $\varepsilon_k/k\to 0$, $$ \operatorname{Cov}(Z_0\varphi_2,Z_k\psi_1) \sim \operatorname{Cov}(Z_0\varphi_3, Z_k\psi_2) \sim \frac{\sigma_{1,2}}{k}.$$ \end{lemma}
\begin{lemma}\label{lem.var.4} There exist positive constants $\delta$ and $\sigma_{2,2}$, such that when $\varepsilon_k\ge k^{1-\delta}$, and $\varepsilon_k/k\to 0$, $$ \operatorname{Cov}(Z_0\varphi_2,Z_k\psi_2) \sim \frac{\sigma_{2,2}}{k}.$$ \end{lemma}
\subsection{Proof of Lemma \ref{lem.var.1}} We assume now to simplify notation that the distribution $\mu$ is aperiodic, but it should be clear from the proof that the case of a bipartite walk could be handled similarly.
The first step is to show that \begin{equation}\label{var.1.1}
\operatorname{Cov}(Z_0\varphi_3, Z_k \psi_1) = \rho^2 \left\{\sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x^2 - \left(\sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x\right)^2\right\} + o\left(\frac 1k\right),
\end{equation} where $\rho$ and $\varphi_x$ are defined respectively as \begin{eqnarray}\label{def.rho} \rho:= \mathbb{E}\left[ \mathbb{P}\left[H^+_{\overline \mathcal{R}_\infty } = \infty \mid (S_n)_{n\in \mathbb{Z}} \right] \cdot {\text{\Large $\mathfrak 1$}}\{S_\ell \neq 0,\, \forall \ell \ge 1\}\right] , \end{eqnarray} and $$\varphi_x : = \mathbb{P}_{0,x} [\mathcal{R}_\infty \cap \widetilde \mathcal{R}_\infty \neq \varnothing].$$ To see this, one needs to dissociate $Z_0$ and $Z_k$, as well as the events of avoiding $\mathcal{R}[-\varepsilon_k,\varepsilon_k]$ and $\mathcal{R}[k-\varepsilon_k,k+\varepsilon_k]$ by two independent walks starting respectively from the origin and from $S_k$, which are local events (in the sense that they only concern small parts of the different paths), from the events of hitting $\mathcal{R}[k+1,\infty)$ and $\mathcal{R}(-\infty,-1]$ by these two walks, which involve different parts of the trajectories.
To be more precise, consider $(S_n^1)_{n\ge 0}$ and $(S_n^2)_{n\ge 0}$, two independent random walks starting from the origin, and independent of $(S_n)_{n\in \mathbb{Z}}$. Then define $$\tau_1:= \inf\{n\ge \varepsilon_k : S_n^1 \in \mathcal{R}[k+\varepsilon_k,\infty) \}, \ \tau_2:= \inf \{n\ge \varepsilon_k : S_k + S_n^2 \in \mathcal{R}(-\infty,-\varepsilon_k]\}.$$ We first consider the term $\mathbb{E}[Z_0\varphi_3]$. Let $$\tau_{0,1}:=\inf \left\{n\ge \varepsilon_k \, :\, S_n^1 \in \mathcal{R}[-\varepsilon_k,\varepsilon_k]\right\},$$ and $$\Delta_{0,3}:= \mathbb{E}\left[Z_0\cdot {\text{\Large $\mathfrak 1$}}\{\mathcal{R}^1[1,\varepsilon_k]\cap \mathcal{R}[-\varepsilon_k,\varepsilon_k] =\varnothing\}\cdot {\text{\Large $\mathfrak 1$}}\{ \tau_1 <\infty\}\right].$$ One has, \begin{align*}
\left| \mathbb{E}[Z_0\varphi_3] - \Delta_{0,3}\right| &\le \mathbb{P}\left[\tau_{0,1}<\infty,\, \tau_1<\infty \right] + \mathbb{P}\left[ \mathcal{R}^1[0,\varepsilon_k] \cap \mathcal{R}[k,\infty)\neq \varnothing \right] \\ & \qquad +\mathbb{P}[\mathcal{R}^1_\infty \cap \mathcal{R}[k,k+\varepsilon_k]\neq \varnothing] \\ & \stackrel{\eqref{lem.hit.3}}{\le} \mathbb{P}[\tau_1\le \tau_{0,1}<\infty] + \mathbb{P}[\tau_{0,1}\le \tau_1<\infty ] + \mathcal{O}\left(\frac{\varepsilon_k}{k^{3/2}} \right). \end{align*} Next, conditioning on $\mathcal{R}[-\varepsilon_k,\varepsilon_k]$ and using the Markov property at time $\tau_{0,1}$, we get with $X=S_{\varepsilon_k} - S^1_{\tau_{0,1}}$, \begin{align*} \mathbb{P}[\tau_{0,1}\le \tau_1<\infty ] &\le \mathbb{E}\left[\mathbb{P}_{0,X}[\mathcal{R}[k,\infty)\cap \widetilde \mathcal{R}_\infty \neq \varnothing] \cdot {\text{\Large $\mathfrak 1$}}\{\tau_{0,1}<\infty\}\right] \\ & \stackrel{\eqref{lem.hit.3}}{=} \mathcal{O}\left(\frac{\mathbb{P}[\tau_{0,1}<\infty] }{\sqrt{k}}\right) \stackrel{\eqref{lem.hit.3}}{=} \mathcal{O}\left(\frac 1{\sqrt {k \varepsilon_k}} \right). \end{align*} Likewise, using the Markov property at time $\tau_1$, we get \begin{align*} & \mathbb{P}[\tau_1\le \tau_{0,1}<\infty ] \stackrel{\eqref{lem.hit.1}}{\le} \mathbb{E}\left[\left(\sum_{j=-\varepsilon_k}^{\varepsilon_k} G(S_j- S_{\tau_1}^1)\right) {\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}\right] \\ & \stackrel{\eqref{lem.hit.1}}{\le}\sum_{i=k+\varepsilon_k}^\infty \sum_{j=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[G(S_j- S_i)G(S_i-S^1_{\varepsilon_k})\right]\\ & \le (2\varepsilon_k+1) \sup_{x\in \mathbb{Z}^5} \sum_{i=k}^\infty \mathbb{E}\left[G(S_i)G(S_i-x)\right] \\ & \le (2\varepsilon_k+1) \sup_{x\in \mathbb{Z}^5} \sum_{z\in \mathbb{Z}^5} G(z) G(z-x) G_k(z) \stackrel{\eqref{Green}, \, \text{Lemma }\ref{lem.upconvolG}}{=}\mathcal{O}\left(\frac{\varepsilon_k}{k^{3/2}}\right). \end{align*} Now define for any $y_1,y_2\in \mathbb{Z}^5$, \begin{equation}\label{Hy1y2} H(y_1,y_2):= \mathbb{E}\left[Z_0 {\text{\Large $\mathfrak 1$}}\{\mathcal{R}^1[1,\varepsilon_k]\cap \mathcal{R}[-\varepsilon_k,\varepsilon_k] =\varnothing, S_{\varepsilon_k }= y_1, S^1_{\varepsilon_k} = y_2 \}\right]. \end{equation} One has by the Markov property \begin{align*} \Delta_{0,3} =\sum_{x\in \mathbb{Z}^5} \sum_{y_1,y_2\in \mathbb{Z}^5} H(y_1,y_2) p_k(x+y_2-y_1) \varphi_x. \end{align*}
Observe that typically $\|y_1\|$ and $\|y_2\|$ are much smaller than $\|x\|$, and thus $p_k(x+y_2-y_1)$ should be also typically close to $p_k(x)$. To make this precise, consider $(\chi_k)_{k\ge 1}$ some sequence of positive integers, such that $\varepsilon_k \chi_k^3 \le k$, for all $k\ge 1$, and $\chi_k\to \infty$, as $k\to \infty$. One has using Cauchy-Schwarz at the third line, \begin{align*}
& \sum_{\|x\|^2\le k/\chi_k} \sum_{y_1,y_2\in \mathbb{Z}^5} H(y_1,y_2) p_k(x+y_2-y_1)\varphi_{x} \\
& \le \sum_{\|x\|^2\le k/\chi_k } \sum_{y_2\in \mathbb{Z}^5} p_{\varepsilon_k}(y_2) p_{k+\varepsilon_k}(x) \varphi_{x-y_2}
\stackrel{\eqref{lem.hit.2}}{\lesssim} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|S_{k+\varepsilon_k}\|^2 \le k/\chi_k\}}{1+\|S_{k+\varepsilon_k} - S^1_{\varepsilon_k}\|}\right]\\
& \lesssim \mathbb{E}\left[\frac 1{1+\|S_{k+2\varepsilon_k}\|^2}\right]^{1/2}\cdot \mathbb{P}\left[\|S_{k+\varepsilon_k}\|^2\le k/\chi_k\right]^{1/2}\stackrel{\eqref{pn.largex}}{\lesssim} \frac 1{\sqrt{k}\cdot \chi_k^{5/4}}. \end{align*} Likewise, using just \eqref{Sn.large} at the end instead of \eqref{pn.largex}, we get
$$\sum_{\|x\|^2 \ge k\chi_k} \sum_{y_1,y_2\in \mathbb{Z}^5} H(y_1,y_2) p_k(x+y_2-y_1)\varphi_{x} \lesssim \frac 1{\sqrt{k}\cdot \chi_k^{5/4}},$$
and one can handle the sums on the sets $\{\|y_1\|^2\ge \varepsilon_k\chi_k\}$ and $\{\|y_2\|^2\ge \varepsilon_k\chi_k\}$ similarly. Therefore, it holds
$$\Delta_{0,3} =\sum_{k/\chi_k \le \|x\|^2 \le k \chi_k } \sum_{\|y_1\|^2\le \varepsilon_k \chi_k} \sum_{\|y_2\|^2\le \varepsilon_k \chi_k} H(y_1,y_2) p_k(x+y_2-y_1) \varphi_x + \mathcal{O}\left(\frac 1{\sqrt{k}\cdot \chi_k^{5/4}} \right).$$ Moreover, Theorem \ref{LCLT} shows that for any $x,y_1,y_2$ as in the three sums above, one has
$$|p_k(x+y_2-y_1)-p_k(x)| = \mathcal{O}\left(\frac{\sqrt{\varepsilon_k} \cdot \chi_k}{\sqrt k}\cdot p_k(x) + \frac 1{k^{7/2}}\right).$$ Note also that by \eqref{lem.hit.2}, one has \begin{equation}\label{sum.pkxphix} \sum_{x,y_1,y_2\in \mathbb{Z}^5} H(y_1,y_2)p_k(x) \varphi_x\le \sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x = \mathcal{O}\left(\frac 1{\sqrt{k}}\right). \end{equation}
Using as well that $\sqrt{\varepsilon_k}\chi_k \le \sqrt {k/\chi_k}$, and $\sum_{\|x\|^2 \le k\chi_k} \varphi_x = \mathcal{O}(k^2\chi_k^2)$, we get \begin{align*} \Delta_{0,3} =\rho_k \sum_{x\in \mathbb{Z}^5 } p_k(x) \varphi_x + \mathcal{O}\left(\frac{1}{\sqrt{k\cdot \chi_k}} + \frac{\chi_k^2}{k^{3/2}}\right), \end{align*} with $$\rho_k:= \sum_{y_1,y_2\in \mathbb{Z}^5} H(y_1,y_2) = \mathbb{E}\left[Z_0\cdot {\text{\Large $\mathfrak 1$}}\{\mathcal{R}^1[1,\varepsilon_k] \cap \mathcal{R}[-\varepsilon_k,\varepsilon_k] = \varnothing\}\right].$$ Note furthermore that one can always take $\chi_k$ such that $\chi_k=o( \sqrt k)$, and that by \eqref{Green.hit}, \eqref{Green} and \eqref{lem.hit.3}, one has
$|\rho_k - \rho| \lesssim \varepsilon_k^{-1/2}$. This gives \begin{eqnarray}\label{Z0phi3.final} \mathbb{E}[Z_0\varphi_3] = \rho \sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x + o\left(\frac{1}{\sqrt{k}}\right). \end{eqnarray} By symmetry the same estimate holds for $\mathbb{E}[Z_k\psi_1]$, and thus using again \eqref{sum.pkxphix}, it entails $$\mathbb{E}[Z_0\varphi_3] \cdot \mathbb{E}[Z_k\psi_1] = \rho^2\left( \sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x\right)^2 + o\left(\frac{1}{k}\right).$$ The estimate of $\mathbb{E}[Z_0\varphi_3Z_k\psi_1]$ is done along the same line, but is a bit more involved. Indeed, let \begin{align*} \Delta_{1,3}:= \mathbb{E}\left[\right. &Z_0Z_k {\text{\Large $\mathfrak 1$}}\{\mathcal{R}^1[1,\varepsilon_k]\cap \mathcal{R}[-\varepsilon_k,\varepsilon_k] =\varnothing\} \\ & \left. \times {\text{\Large $\mathfrak 1$}}\{(S_k+\mathcal{R}^2[1,\varepsilon_k])\cap \mathcal{R}[k-\varepsilon_k,k+\varepsilon_k] =\varnothing, \tau_1 <\infty, \tau_2<\infty\}\right]. \end{align*} The difference between $\mathbb{E}[Z_0\varphi_3Z_k\psi_1]$ and $\Delta_{1,3}$ can be controlled roughly as above, but one needs additionally to handle the probability of $\tau_2$ being finite. Namely one has using symmetry, \begin{align}\label{Delta13}
& |\mathbb{E}[Z_0\varphi_3Z_k\psi_1] - \Delta_{1,3} | \le 2\left(\mathbb{P}\left[\tau_{0,1}<\infty,\, \tau_1<\infty, \, \overline \tau_2<\infty \right] \right. \\
\nonumber + & \mathbb{P}\left[ \mathcal{R}^1[0,\varepsilon_k] \cap \mathcal{R}[k,\infty)\neq \varnothing,\, \overline \tau_2<\infty \right] + \left. \mathbb{P}[\mathcal{R}^1_\infty \cap \mathcal{R}[k,k+\varepsilon_k]\neq \varnothing, \, \overline \tau_2<\infty]\right), \end{align} with $$\overline \tau_2:= \inf\{n\ge 0 \, : \, S_k+S^2_n \in \mathcal{R}(-\infty,0]\}.$$ The last term in \eqref{Delta13} is handled as follows: \begin{align*} &\mathbb{P}\left[\mathcal{R}^1_\infty \cap \mathcal{R}[k,k+\varepsilon_k]\neq \varnothing, \overline \tau_2<\infty\right] =\sum_{x\in \mathbb{Z}^5} \mathbb{P}[\mathcal{R}^1_\infty \cap \mathcal{R}\left[k,k+\varepsilon_k]\neq \varnothing, \overline \tau_2<\infty, S_k=x\right] \\ & \stackrel{\eqref{lem.hit.1}}{\le} \sum_{x\in \mathbb{Z}^5} p_k(x)\varphi_x \sum_{i=0}^{\varepsilon_k} \mathbb{E}[G(S_i+x)]
\stackrel{\eqref{Green}, \eqref{lem.hit.2}, \eqref{exp.Green.x}}{\lesssim} \varepsilon_k \sum_{x\in \mathbb{Z}^5} \frac{p_k(x)}{1+\|x\|^4} \stackrel{\eqref{pn.largex}}{\lesssim} \frac{\varepsilon_k}{k^2}. \end{align*} The same arguments give as well $$\mathbb{P}\left[ \mathcal{R}^1[0,\varepsilon_k] \cap \mathcal{R}[k,\infty)\neq \varnothing,\, \overline \tau_2<\infty \right] \lesssim \frac{\varepsilon_k}{k^2},$$ $$\mathbb{P}\left[\tau_{0,1}<\infty,\, \tau_1<\infty, \, \overline \tau_2<\infty \right] = \mathbb{P}\left[\tau_{0,1}<\infty,\, \tau_1<\infty, \, \tau_2<\infty \right] + \mathcal{O}\left(\frac{\varepsilon_k}{k^2}\right).$$ Then we can write, \begin{align*} &\mathbb{P}\left[\tau_{0,1}\le \tau_1<\infty, \tau_2<\infty \right] = \mathbb{E} \left[ \mathbb{P}_{0,S_{k+\varepsilon_k}-S_{\tau_{0,1}}}[\mathcal{R}_\infty \cap \widetilde \mathcal{R}_\infty \neq \varnothing] {\text{\Large $\mathfrak 1$}}\{\tau_{0,1}<\infty, \tau_2<\infty\}\right] \\
& \stackrel{\eqref{lem.hit.1}, \eqref{lem.hit.2}}{\lesssim} \sum_{i=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[\frac {1}{1+\|S_{k+\varepsilon_k}-S_i\|}\cdot \frac{G(S_i-S^1_{\varepsilon_k})}{1+\|S_k-S_{-\varepsilon_k}\|}\right]\\
& \stackrel{\eqref{exp.Green}}{\lesssim}\frac{1}{\varepsilon_k^{3/2}} \sum_{i=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[\frac {1}{1+\|S_k-S_i\|}\cdot \frac{1}{1+\|S_k-S_{-\varepsilon_k}\|}\right] \\
& \lesssim \frac{1}{\sqrt{\varepsilon_k}} \max_{k-\varepsilon_k\le j \le k+\varepsilon_k}\, \sup_{u\in\mathbb{Z}^d}\, \mathbb{E}\left[\frac {1}{1+\|S_j\|}\cdot \frac{1}{1+\|S_j+u\|}\right] \lesssim \frac{1}{k\sqrt{\varepsilon_k}}, \end{align*} where the last equality follows from straightforward computations, using \eqref{pn.largex}. On the other hand, \begin{align*} &\mathbb{P}\left[\tau_1\le \tau_{0,1}<\infty, \tau_2<\infty \right] \stackrel{\eqref{lem.hit.1}, \eqref{lem.hit.2}}{\lesssim} \sum_{i=k+\varepsilon_k}^\infty \sum_{j=-\varepsilon_k}^{\varepsilon_k}
\mathbb{E}\left[\frac{G(S_j-S_i)G(S_i-S_{\varepsilon_k}^1)}{1+\|S_k-S_{-\varepsilon_k}\|}\right] \\ & \stackrel{\eqref{Green}, \eqref{exp.Green.x}}{\lesssim} \sum_{j=-\varepsilon_k}^{\varepsilon_k} \sum_{i=k+\varepsilon_k}^\infty
\mathbb{E}\left[\frac{G(S_j-S_i)}{(1+\|S_i\|^3)(1+\|S_k-S_{-\varepsilon_k}\|)}\right] \\
& \lesssim \sum_{j=-\varepsilon_k}^{\varepsilon_k}\sum_{z\in \mathbb{Z}^d} G_{\varepsilon_k}(z) \mathbb{E}\left[\frac{G(z+ S_k-S_j)}{(1+\|z+S_k\|^3)(1+\|S_k-S_{-\varepsilon_k}\|)}\right]. \end{align*} Note now that for $x,y\in \mathbb{Z}^5$, by \eqref{Green} and Lemma \ref{lem.upconvolG}, \begin{align*}
\sum_{z\in \mathbb{Z}^d} \frac{G_{\varepsilon_k}(z)}{(1+\|z-x\|^3)(1+\|z-y\|^3)} \lesssim \frac{1}{1+\|x\|^3} \left(\frac 1{\sqrt{\varepsilon_k}} + \frac{1}{1+\|y-x\|}\right) . \end{align*} It follows that \begin{align*} &\mathbb{P}\left[\tau_1\le \tau_{0,1}<\infty, \tau_2<\infty \right] \lesssim \sum_{j=-\varepsilon_k}^{\varepsilon_k}
\mathbb{E}\left[\frac{1}{(1+\|S_k\|^3)(1+\|S_k-S_{-\varepsilon_k}\|)}\left(\frac 1{\sqrt{\varepsilon_k}} + \frac 1{1+\|S_j\|}\right) \right]\\
& \stackrel{\eqref{exp.Green.x}}{\lesssim} \mathbb{E}\left[\frac {\sqrt{\varepsilon_k}}{1+\|S_k\|^4}\right] + \sum_{j=-\varepsilon_k}^0 \mathbb{E}\left[\frac 1{(1+\|S_k\|^3)(1+\|S_k-S_j\|)(1+\|S_j\|)}\right] \\
& \qquad + \sum_{j=1}^{\varepsilon_k} \mathbb{E}\left[\frac 1{(1+\|S_k\|^4)(1+\|S_j\|)}\right]\\
& \lesssim \frac{1}{k^2} \left(\sqrt{\varepsilon_k} + \sum_{j=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[\frac 1{1+\|S_j\|}\right] \right)\lesssim \frac{\sqrt{\varepsilon_k}}{k^2}, \end{align*} using for the third inequality that by \eqref{pn.largex}, it holds uniformly in $x\in \mathbb{Z}^5$ and $j\le \varepsilon_k$,
$$\mathbb{E}\left[\frac 1{1+\|S_k-S_j+x\|^4}\right] \lesssim k^{-2}, \quad \mathbb{E}\left[\frac 1{(1+\|S_k\|^3) (1+\|S_k+x\|)}\right] \lesssim k^{-2}.$$ Now we are left with computing $\Delta_{1.3}$. This step is essentially the same as above, so we omit to give all the details. We first define for $y_1,y_2,y_3 \in \mathbb{Z}^5$, $$H(y_1,y_2,y_3):= \mathbb{E}\left [Z_0 {\text{\Large $\mathfrak 1$}}\{\mathcal{R}^1[1,\varepsilon_k]\cap \mathcal{R}[-\varepsilon_k,\varepsilon_k] =\varnothing, S_{\varepsilon_k} =y_1, S^1_{\varepsilon_k} = y_2, S_{-\varepsilon_k} = y_3\}\right],$$ and note that $$\Delta_{1,3}= \sum_{\substack{y_1,y_2,y_3\in \mathbb{Z}^5 \\ z_1,z_2,z_3\in \mathbb{Z}^5 \\ x\in \mathbb{Z}^5}} H(y_1,y_2,y_3) H(z_1,z_2,z_3) p_{k-2\varepsilon_k}(x-y_1+z_3) \varphi_{x+z_1-y_2} \varphi_{x + z_2-y_3}.$$
Observe here that by Theorem C, $\varphi_{x+z_1-y_2}$ is equivalent to $\varphi_x$, when $\|z_1\|$ and $\|y_2\|$ are small when compared to $\|x\|$, and similarly for $\varphi_{x+z_2-y_3}$. Thus using similar arguments as above, and in particular that by \eqref{pn.largex} and \eqref{lem.hit.2}, \begin{equation}\label{sum.pkxphix.2} \sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x^2 = \mathcal{O}\left(\frac 1k\right), \end{equation} we obtain $$\Delta_{1,3} = \rho^2 \sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x^2 + o\left(\frac 1k\right).$$ Putting all pieces together gives \eqref{var.1.1}. Using in addition \eqref{pn.largex}, \eqref{lem.hit.2} and Theorem \ref{LCLT}, we deduce that $$\operatorname{Cov}(Z_0\varphi_3, Z_k \psi_1) = \rho^2 \left\{\sum_{x\in \mathbb{Z}^5} \overline p_k(x) \varphi_x^2 - \left(\sum_{x\in \mathbb{Z}^5} \overline p_k(x) \varphi_x\right)^2\right\} + o\left(\frac 1k\right).$$ Then Theorem C, together with \eqref{sum.pkxphix} and \eqref{sum.pkxphix.2} show that $$\operatorname{Cov}(Z_0\varphi_3, Z_k \psi_1) = \sigma \left\{\sum_{x\in \mathbb{Z}^5} \frac{\overline p_k(x)}{1+\mathcal J(x)^2} - \left(\sum_{x\in \mathbb{Z}^5} \frac{\overline p_k(x)}{1+\mathcal J(x)} \right)^2\right\} +o\left(\frac 1k\right), $$ for some constant $\sigma>0$. Finally an approximation of the series with an integral and a change of variables gives, with $c_0:=(2\pi)^{-5/2} (\det \Gamma)^{-1/2}$, \begin{align*}
\operatorname{Cov}(Z_0\varphi_3, Z_k \psi_1) & = \frac{\sigma c_0}{k} \left\{\int_{\mathbb{R}^5} \frac{e^{- 5 \mathcal J(x)^2/2}}{\mathcal J(x)^2} \, dx - c_0\left(\int_{\mathbb{R}^5} \frac{e^{- 5 \mathcal J(x)^2/2}}{\mathcal J(x)} \, dx \right)^2\right\} +o\left(\frac 1k\right). \end{align*} The last step of the proof is to observe that the difference between the two terms in the curly bracket is well a positive real. This follows simply by Cauchy-Schwarz, once we observe that $c_0\int_{\mathbb{R}^5} e^{-5 \mathcal J(x)^2/2} \, dx = 1$, which itself can be deduced for instance from the fact that $1= \sum_{x\in \mathbb{Z}^5} p_k(x) \sim c_0\int_{\mathbb{R}^5} e^{- 5 \mathcal J(x)^2/2} \, dx$, by the above arguments. This concludes the proof of Lemma \ref{lem.var.1}.
$\square$
\subsection{Proof of Lemma \ref{lem.var.2}} Let us concentrate on the term $\operatorname{Cov}(Z_0\varphi_3,Z_k\psi_3)$, the estimate of $\operatorname{Cov}(Z_0\varphi_1,Z_k\psi_1)$ being entirely similar. We also assume to simplify notation that the walk is aperiodic.
We consider as in the proof of the previous lemma $(S_n^1)_{n\ge 0}$ and $(S_n^2)_{n\ge 0}$ two independent random walks starting from the origin, independent of $(S_n)_{n\in \mathbb{Z}}$, and define this time $$\tau_1:= \inf\{n\ge k+\varepsilon_k : S_n \in \mathcal{R}^1[\varepsilon_k,\infty) \}, \ \tau_2:= \inf \{n\ge k+\varepsilon_k : S_n \in S_k+\mathcal{R}^2[\sqrt \varepsilon_k,\infty)\}.$$ Define as well $$\overline \tau_1:= \inf\{n\ge k+\varepsilon_k : S_n \in \mathcal{R}^1_\infty \}, \ \overline \tau_2:= \inf \{n\ge k+\varepsilon_k : S_n \in S_k+\mathcal{R}^2_\infty\}.$$ \underline{Step $1$.} Our first task is to show that \begin{equation}\label{cov.33.first} \operatorname{Cov}(Z_0\varphi_3,Z_k\psi_3) = \rho^2 \cdot \operatorname{Cov}\left({\text{\Large $\mathfrak 1$}}\{\overline \tau_1<\infty\} ,\, {\text{\Large $\mathfrak 1$}}\{\overline \tau_2<\infty\}\right) + o\left(\frac 1k\right), \end{equation} with $\rho$ as defined in \eqref{def.rho}. This step is essentially the same as in the proof of Lemma \ref{lem.var.1}, but with some additional technical difficulties, so let us give some details. First, the proof of Lemma \ref{lem.var.1} shows that (using the same notation), $$\mathbb{E}[Z_0\varphi_3] = \Delta_{0,3} + \mathcal{O}\left(\frac{1}{\sqrt{k\varepsilon_k}} + \frac{\varepsilon_k}{k^{3/2}}\right),$$ and that for any sequence $(\chi_k)_{k\ge 1}$ going to infinity with $\varepsilon_k \chi_k^{2+\frac 14} \le k$,
$$\Delta_{0,3} = \sum_{k/\chi_k \le \|x\|^2 \le k \chi_k } \sum_{\substack{\|y_1\|^2\le \varepsilon_k \chi_k \\ \|y_2\|^2\le \varepsilon_k \chi_k}} H(y_1,y_2) p_k(x+y_2-y_1) \varphi_x + \mathcal{O}\left(\frac 1{\sqrt{k}\cdot \chi_k^{5/4}} \right).$$ Observe moreover, that by symmetry $H(y_1,y_2) = H(-y_1,-y_2)$, and that by Theorem \ref{LCLT}, for any $x$, $y_1$, and $y_2$ as above, for some constant $c>0$,
$$\left|p_k(x+y_2-y_1) + p_k(x+y_1-y_2) - p_k(x) \right| = \mathcal{O}\left(\frac{\varepsilon_k\chi_k}{k} \overline p_k(cx) + \frac 1{k^{7/2}}\right),$$ It follows that one can improve the bound \eqref{Z0phi3.final} into \begin{align}\label{Z0phi3.bis} \nonumber \mathbb{E}[Z_0\varphi_3] & = \rho \sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x + \mathcal{O}\left(\frac{\varepsilon_k\chi_k}{k^{3/2}} + \frac{\chi_k^2}{k^{3/2}} +\frac 1{\sqrt{k}\cdot \chi_k^{5/4}} + \frac{1}{\sqrt{k\varepsilon_k}} + \frac{\varepsilon_k}{k^{3/2}} \right)\\ & = \rho \, \mathbb{P}[\overline\tau_1<\infty] + \mathcal{O}\left(\frac{\varepsilon_k\chi_k}{k^{3/2}} + \frac{\chi_k^2}{k^{3/2}} +\frac 1{\sqrt{k}\cdot \chi_k^{5/4}} + \frac{1}{\sqrt{k\varepsilon_k}} + \frac{\varepsilon_k}{k^{3/2}} \right). \end{align} Since by \eqref{lem.hit.3} one has $$\mathbb{E}[Z_k\psi_3] \le \mathbb{E}[\psi_3] = \mathcal{O}\left(\frac 1{\sqrt{\varepsilon_k}}\right),$$ this yields by taking $\chi_k^{2+1/4} := k/\varepsilon_k$, and $\varepsilon_k\ge k^{2/3}$ (but still $\varepsilon_k= o(k)$), \begin{align}\label{Z03.1} \mathbb{E}[Z_0\varphi_3] \cdot \mathbb{E}[Z_k\psi_3] = \rho\, \mathbb{P}[\overline\tau_1<\infty]\cdot \mathbb{E}[Z_k\psi_3] + o\left(\frac 1k \right). \end{align} We next seek an analogous estimate for $\mathbb{E}[Z_k\psi_3]$. Define $Z'_k:=1\{S_{k+i}\neq S_k,\, \forall i=1,\dots,\varepsilon_k^{3/4}\}$, and $$\Delta_0:= \mathbb{E}\left[Z'_k\cdot {\text{\Large $\mathfrak 1$}}\left\{\mathcal{R}[k-\varepsilon_k,k+\varepsilon_k^{3/4}] \cap (S_k+\mathcal{R}^2[1,\sqrt{\varepsilon_k}])=\varnothing, \, \tau_2<\infty \right\}\right].$$ Note that (with $\mathcal{R}$ and $\widetilde \mathcal{R}$ two independent walks), \begin{align*}
\left| \mathbb{E}[Z_k\psi_3] - \Delta_0\right| & \le \mathbb{P}\left[0\in \mathcal{R}[\varepsilon_k^{3/4},\varepsilon_k]\right] + \mathbb{P}\left[\widetilde \mathcal{R}[0,\sqrt {\varepsilon_k}]\cap \mathcal{R}[\varepsilon_k,\infty)\neq \varnothing\right] \\ & + \mathbb{P}\left[\widetilde \mathcal{R}_\infty\cap \mathcal{R}[\varepsilon_k^{3/4},\varepsilon_k] \neq \varnothing, \widetilde \mathcal{R}_\infty\cap \mathcal{R}[\varepsilon_k,\infty) \neq \varnothing \right] \\
& + \mathbb{P}\left[\widetilde \mathcal{R}[\sqrt{\varepsilon_k},\infty) \cap \mathcal{R}[-\varepsilon_k,\varepsilon_k]\neq \varnothing, \widetilde \mathcal{R}[\sqrt{\varepsilon_k},\infty) \cap \mathcal{R}[\varepsilon_k,\infty)\neq \varnothing \right]. \end{align*} Moreover, \begin{equation}\label{ZkZ'k} \mathbb{P}\left[0\in \mathcal{R}[\varepsilon_k^{3/4},\varepsilon_k]\right] \stackrel{\eqref{Green.hit}, \eqref{exp.Green}}{\lesssim} \varepsilon_k^{-9/8},\quad \mathbb{P}\left[\widetilde \mathcal{R}[0,\sqrt {\varepsilon_k}]\cap \mathcal{R}[\varepsilon_k,\infty)\neq \varnothing\right] \stackrel{\eqref{lem.hit.3}}{\lesssim} \varepsilon_k^{- 1}. \end{equation} Using also the same computation as in the proof of Lemma \ref{lem.123}, we get \begin{equation*} \mathbb{P}\left[\widetilde \mathcal{R}_\infty\cap \mathcal{R}[\varepsilon_k^{3/4},\varepsilon_k] \neq \varnothing,\, \widetilde \mathcal{R}_\infty\cap \mathcal{R}[\varepsilon_k,\infty) \neq \varnothing \right] \lesssim \varepsilon_k^{-\frac 38 - \frac 12}, \end{equation*} \begin{equation} \label{tau01tau2}
\mathbb{P}\left[\widetilde \mathcal{R}[\sqrt{\varepsilon_k},\infty) \cap \mathcal{R}[-\varepsilon_k,\varepsilon_k]\neq \varnothing, \widetilde \mathcal{R}[\sqrt{\varepsilon_k},\infty) \cap \mathcal{R}[\varepsilon_k,\infty)\neq \varnothing \right] \lesssim \varepsilon_k^{-\frac 14 - \frac 12}. \end{equation} As a consequence \begin{align}\label{Zk3.1} \mathbb{E}[Z_k\psi_3] = \Delta_0 + \mathcal{O}\left(\varepsilon_k^{-3/4}\right). \end{align} Introduce now \begin{align*} \widetilde H(y_1,y_2) := \mathbb{E}\left[Z'_k\cdot \right. & {\text{\Large $\mathfrak 1$}}\{\mathcal{R}[k-\varepsilon_k,k+\varepsilon_k^{3/4}]\cap (S_k+ \mathcal{R}^2[1,\sqrt{\varepsilon_k}])=\varnothing\} \\ & \times \left. {\text{\Large $\mathfrak 1$}}\{S_{k+\varepsilon_k^{3/4}}-S_k = y_1, S^2_{\sqrt{\varepsilon_k}} = y_2\}\right], \end{align*} and note that $$\Delta_0 = \sum_{x\in \mathbb{Z}^d} \sum_{y_1,y_2\in \mathbb{Z}^d} \widetilde H(y_1,y_2) p_{\varepsilon_k-\varepsilon_k^{3/4}}(x+y_2-y_1) \varphi_x.$$ Let $\chi_k:= \varepsilon_k^{1/8}$. As above, we can see that \begin{align*}
\Delta_0 & = \sum_{\substack{\varepsilon_k/\chi_k\le \|x\|^2\le \varepsilon_k\chi_k \\ \|y_1\|^2\le \varepsilon_k^{3/4}\chi_k \\ \|y_2\|^2 \le \sqrt{\varepsilon_k}\chi_k}} \widetilde H(y_1,y_2) p_{\varepsilon_k-\varepsilon_k^{3/4}}(x+y_2-y_1) \varphi_x + \mathcal{O}\left(\frac{1}{\sqrt{\varepsilon_k} \chi_k^{5/4}}\right)\\
&= \left(\sum_{y_1,y_2\in \mathbb{Z}^d} \widetilde H(y_1,y_2)\right) \left(\sum_{x\in \mathbb{Z}^d} p_{\varepsilon_k}(x)\varphi_x\right) + \mathcal{O}\left( \frac{\chi_k}{\varepsilon_k^{3/4}} + \frac{\chi_k^2}{\varepsilon_k^{3/2}} + \frac{1}{\sqrt{\varepsilon_k} \chi_k^{5/4}}\right) \\
& = \rho\cdot \mathbb{P}[\overline \tau_2<\infty] + \mathcal{O}(\varepsilon_k^{-5/8}). \end{align*} Then by taking $\varepsilon_k\ge k^{5/6}$, and recalling \eqref{Z03.1} and \eqref{Zk3.1}, we obtain \begin{align}\label{Z03.2} \mathbb{E}[Z_0\varphi_3] \cdot \mathbb{E}[Z_k\psi_3] = \rho^2\cdot \mathbb{P}[\overline\tau_1<\infty]\cdot \mathbb{P}[\overline \tau_2<\infty]+ o\left(\frac 1k \right). \end{align} Finally, let \begin{align*} \Delta_{3,3}:= & \mathbb{E} [Z_0Z'_k {\text{\Large $\mathfrak 1$}}\{\mathcal{R}^1[1,\varepsilon_k]\cap \mathcal{R}[-\varepsilon_k,\varepsilon_k] =\varnothing\}\\ & \times {\text{\Large $\mathfrak 1$}}\{ (S_k+\mathcal{R}^2[1,\sqrt{\varepsilon_k}])\cap \mathcal{R}[k-\varepsilon_k^{\frac 34},k+\varepsilon_k^{\frac 34}] =\varnothing, \tau_1 <\infty, \tau_2<\infty \}]. \end{align*} It amounts to estimate the difference between $\Delta_{3,3}$ and $\mathbb{E}[Z_0Z_k\varphi_3\psi_3]$. Define $$\widetilde \tau_1:=\inf\{n\ge k+\varepsilon_k : S_n\in \mathcal{R}^1[0,\varepsilon_k]\}, \ \widetilde \tau_2:=\inf\{n\ge k+\varepsilon_k : S_n\in S_k+\mathcal{R}^2[0,\sqrt{\varepsilon_k}]\}.$$ Observe first that \begin{align}\label{tilde1bar2}
\nonumber & \mathbb{P}[\widetilde \tau_1\le \overline \tau_2<\infty] \stackrel{\eqref{lem.hit.2}}{\lesssim} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\widetilde \tau_1<\infty\}}{1+\|S_{\widetilde \tau_1}-S_k\|} \right]
\stackrel{\eqref{lem.hit.1}}{\lesssim} \sum_{i=0}^{\varepsilon_k} \mathbb{E}\left[\frac{G(S_i^1-S_{k+\varepsilon_k})}{1+\|S_i^1-S_k\|} \right]\\
\nonumber & \lesssim \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} p_i(z) \mathbb{E}\left[\frac{G(z-S_{k+\varepsilon_k})}{1+\|z-S_k\|} \right]\stackrel{\eqref{pn.largex}}{\lesssim}
\sum_{z\in \mathbb{Z}^5} \frac {\sqrt{\varepsilon_k}}{1+\|z\|^4}\, \mathbb{E}\left[\frac{G(z-S_{k+\varepsilon_k})}{1+\|z-S_k\|} \right] \\
&\stackrel{\eqref{Green}}{\lesssim} \mathbb{E}\left[\frac{\sqrt{\varepsilon_k}}{(1+\|S_{k+\varepsilon_k}\|^2)(1+\|S_k\|)} \right]
\stackrel{\eqref{exp.Green.x}}{\lesssim} \mathbb{E}\left[\frac{\sqrt{\varepsilon_k}}{1+\|S_k\|^3} \right] \stackrel{\eqref{exp.Green}}{\lesssim} \frac{\sqrt{\varepsilon_k}}{k^{3/2}}, \end{align} and likewise, \begin{align*} & \mathbb{P}[\overline \tau_1\le \widetilde \tau_2<\infty] \stackrel{\eqref{lem.hit.1}}{\le} \sum_{j\ge 0}\sum_{i=0}^{\varepsilon_k}\mathbb{E}\left[G(S_k + S_i^2 - S_j^1) G(S_j^1- S_{k+\varepsilon_k})\right] \\ & = \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5}\mathbb{E}\left[G(z) G(S_k + S_i^2 - z) G(z- S_{k+\varepsilon_k})\right] \\
& \le C\sum_{i=0}^{\varepsilon_k} \mathbb{E}\left[\frac 1{1+\|S_k + S_i^2\|^3}\left(\frac 1{1+\|S_{k+\varepsilon_k}\|} +\frac 1{1+ \|S_{k+\varepsilon_k}-S_k-S_i^2\|}\right)\right]\\
& \stackrel{\eqref{exp.Green},\, \eqref{exp.Green.x}}{\le} C\mathbb{E}\left[\frac {\varepsilon_k}{1+\|S_k\|^4}\right] +C \mathbb{E}\left[\frac {\sqrt{\varepsilon_k}}{1+\|S_k\|^3} \right]= \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right). \end{align*} Additionally, it follows directly from \eqref{lem.hit.3} that $$\mathbb{P}[\overline \tau_2 \le \widetilde \tau_1<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}, \quad \text{and} \quad \mathbb{P}[\widetilde \tau_2 \le \overline \tau_1<\infty] \lesssim \frac{1}{\varepsilon_k \sqrt k},$$ which altogether yields
$$|\mathbb{P}[\overline \tau_1<\infty,\, \overline \tau_2<\infty] - \mathbb{P}[\tau_1<\infty,\, \tau_2<\infty]| \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}} + \frac{1}{\varepsilon_k \sqrt k}.$$ Similar computations give also \begin{equation}\label{tau1tau2} \mathbb{P}[\overline \tau_1<\infty, \, \overline \tau_2<\infty] \lesssim \frac 1{\sqrt{k \varepsilon_k}}. \end{equation} Next, using \eqref{ZkZ'k} and the Markov property, we get
$$\mathbb{E}[|Z_k-Z'_k|{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}] \lesssim \frac 1{\varepsilon_k^{9/8}\sqrt k}.$$ Thus, for $\varepsilon_k \ge k^{5/6}$, \begin{align*}
\left|\mathbb{E}[Z_0Z_k\varphi_3\psi_3] - \Delta_{3,3}\right| & \le \mathbb{P}[\tau_{0,1}<\infty, \tau_1<\infty, \tau_2<\infty] +\mathbb{P}[\tau_{0,2}<\infty, \tau_1<\infty, \tau_2<\infty] \\ & \qquad + \mathbb{P}[\widetilde \tau_{0,2}<\infty, \tau_1<\infty, \tau_2<\infty] + o\left(\frac 1k\right), \end{align*} where $\tau_{0,1}$ is as defined in the proof of Lemma \ref{lem.var.1}, $$\tau_{0,2} : =\inf\{n\ge \sqrt{\varepsilon_k} : S_k + S_n^2 \in \mathcal{R}[k-\varepsilon_k,k+\varepsilon_k]\},$$ and $$\widetilde \tau_{0,2} : =\inf\{n\le \sqrt{\varepsilon_k} : S_k + S_n^2 \in \mathcal{R}[k-\varepsilon_k,k-\varepsilon_k^{3/4}]\cup \mathcal{R}[k+\varepsilon_k^{3/4},k+\varepsilon_k]\}.$$ Applying \eqref{lem.hit.3} twice already shows that $$\mathbb{P}[\widetilde \tau_{0,2}<\infty,\, \tau_1<\infty] \lesssim \frac{1}{\sqrt k} \cdot \mathbb{P}[\widetilde \tau_{0,2}<\infty] \lesssim \frac{1}{\sqrt{k}\varepsilon_k^{5/8}} = o\left(\frac 1k\right). $$ Then, notice that \eqref{tilde1bar2} entails $$\mathbb{P}[\mathcal{R}[k+\varepsilon_k,\infty) \cap \mathcal{R}^1[0,\tau_{0,1}]\neq \varnothing, \, S^1_{\tau_{0,1}} \in \mathcal{R}[-\varepsilon_k,0]] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.$$ On the other hand, \begin{align*} &\mathbb{P}[\mathcal{R}[k+\varepsilon_k,\infty) \cap \mathcal{R}^1[0,\tau_{0,1}]\neq \varnothing, S^1_{\tau_{0,1}} \in \mathcal{R}[0,\varepsilon_k]] \\
\stackrel{\eqref{lem.hit.1}}{\le} & \sum_{i=0}^{\varepsilon_k} \sum_{j=k+\varepsilon_k}^\infty \mathbb{E}[G(S_i-S_{k+j})G(S_{k+j} - S_k)] = \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} \mathbb{E}[G(S_i-S_k + z)G(z)G_{\varepsilon_k}(z)] \\ \stackrel{\eqref{exp.Green}}{\lesssim} & \frac{\varepsilon_k}{k^{3/2}} \sum_{z\in \mathbb{Z}^5} G(z)G_{\varepsilon_k}(z)\stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} \frac{\sqrt{\varepsilon_k}}{k^{3/2}}. \end{align*} By \eqref{lem.hit.1} and \eqref{exp.Green}, one has with $\widetilde \mathcal{R}_\infty$ an independent copy of $\mathcal{R}_\infty$, \begin{align*} & \mathbb{P}[\tau_{0,1}<\infty, \tau_2<\infty, \mathcal{R}[k+\varepsilon_k,\infty) \cap \mathcal{R}^1[\tau_{0,1},\infty)\neq \varnothing ] \\ \lesssim & \frac{1}{\sqrt{\varepsilon_k}} \max_{-\varepsilon_k\le i\le \varepsilon_k}\mathbb{P}[\tau_2<\infty,\, \mathcal{R}[k+\varepsilon_k,\infty) \cap (S_i+ \widetilde \mathcal{R}_\infty) \neq \varnothing ] \lesssim \frac{1}{\varepsilon_k\sqrt{k}}, \end{align*} where the last equality follows from \eqref{tau1tau2}. Thus $$\mathbb{P}[\tau_{0,1}<\infty, \tau_1<\infty, \tau_2<\infty] =o\left(\frac 1k\right).$$ In a similar fashion, one has $$\mathbb{P}[\tau_{0,2}<\infty, \tau_2\le \tau_1<\infty] \stackrel{\eqref{lem.hit.3}}{\lesssim} \frac{1}{\sqrt k} \mathbb{P}[\tau_{0,2}<\infty, \tau_2<\infty] \stackrel{\eqref{tau01tau2}}{\lesssim} \frac{1}{\varepsilon_k^{3/4}\sqrt{k}},$$ as well as, \begin{align*} & \mathbb{P}\left[\tau_{0,2}<\infty, \, \tau_1\le \tau_2<\infty,\, S_{\tau_2}\in (S_k+\mathcal{R}^2[0, \tau_{0,2}])\right] \\
\stackrel{\eqref{lem.hit.1}}{\le} & \sum_{i=k-\varepsilon_k}^{k+\varepsilon_k} \sum_{j\ge 0} \sum_{\ell \ge 0} \mathbb{E}[G(S_i-\widetilde S_j - S^1_\ell) G(\widetilde S_j + S^1_\ell-S_k) G(S^1_\ell - S_{k+\varepsilon_k})] \\ \le & \sum_{i=k-\varepsilon_k}^{k+\varepsilon_k} \sum_{\ell \ge 0} \sum_{z\in\mathbb{Z}^5} \mathbb{E}[G(z) G(S_i- S^1_\ell - z) G(z + S^1_\ell-S_k) G(S^1_\ell - S_{k+\varepsilon_k})]\\
\stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} & \sum_{i=k-\varepsilon_k}^{k+\varepsilon_k} \sum_{\ell \ge 0}
\mathbb{E}\left[\frac{G(S^1_\ell - S_{k+\varepsilon_k})}{1+\|S^1_\ell - S_k\|^3} \left(\frac 1{1+\|S_\ell^1- S_i\|} + \frac 1{1+\|S_i-S_k\|}\right)\right]\\
\stackrel{\eqref{exp.Green}, \eqref{exp.Green.x}}{\lesssim} & \sum_{i=0}^{\varepsilon_k} \sum_{\ell \ge 0}
\left\{\mathbb{E}\left[\frac{\varepsilon_k^{-3/2}}{1+\|S^1_\ell - S_k\|^3} \left(\frac 1{1+\|S_\ell^1- S_{k-i}\|} + \frac 1{1+\|S_{k-i}-S_k\|}\right)\right] \right. \\
+& \left. \mathbb{E}\left[\frac{1}{(1+\|S^1_\ell - S_{k+i}\|^3)(1+\|S^1_\ell - S_k\|^3)} \left(\frac 1{1+\|S_\ell^1- S_{k+i}\|} + \frac 1{1+\|S_{k+i}-S_k\|}\right)\right] \right\}\\
\stackrel{\eqref{pn.largex}, \eqref{exp.Green.x}}{\lesssim} & \sum_{i=0}^{\varepsilon_k} \sum_{\ell \ge 0}
\left\{\mathbb{E}\left[\frac{\varepsilon_k^{-3/2}}{1+\|S^1_\ell - S_{k-i}\|^3} \left(\frac 1{1+\|S_\ell^1- S_{k-i}\|} + \frac 1{1+\sqrt{i}}\right)\right] \right. \\
& \qquad + \left. \mathbb{E}\left[\frac{(1+i)^{-1/2}}{1+\|S^1_\ell - S_k\|^6} \right] \right\} \\
\lesssim &\ \frac {\sqrt{\varepsilon_k}}{k^{3/2}} , \end{align*} and \begin{align*} & \mathbb{P}\left[\tau_{0,2}<\infty, \, \tau_1\le \tau_2<\infty,\, S_{\tau_2}\in (S_k+\mathcal{R}^2[\tau_{0,2},\infty))\right] \\
\stackrel{\eqref{Green.hit}}{\le} & \sum_{i=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[G(S_{k+i}-S_k - S^2_{\sqrt{\varepsilon_k}}) {\text{\Large $\mathfrak 1$}}\{\tau_1<\infty,\, \mathcal{R}[\tau_1,\infty) \cap (S_{k+i}+ \widetilde \mathcal{R}_\infty) \neq \varnothing\}\right]\\\
\stackrel{\eqref{lem.hit.2}}{\lesssim} & \sum_{i=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[G(S_{k+i}-S_k - S^2_{\sqrt{\varepsilon_k}})
\frac{{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}}{1+ \|S_{\tau_1} - S_{k+i}\|} \right] \\
\stackrel{\eqref{lem.hit.1}}{\lesssim} & \sum_{i=-\varepsilon_k}^{\varepsilon_k}\sum_{j\ge k+\varepsilon_k} \mathbb{E}\left[\frac{G(S_{k+i}-S_k - S^2_{\sqrt{\varepsilon_k}}) G(S_j)}{1+ \|S_j - S_{k+i}\|} \right] \\
\lesssim & \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} \left\{\mathbb{E}\left[\frac{G(S_{k-i}-S_k - S^2_{\sqrt{\varepsilon_k}}) G(S_k+z)G(z)}{1+ \|z + S_k - S_{k-i}\|} \right] \right. \\
& \qquad \left. + \mathbb{E}\left[\frac{G(S_{k+i}-S_k - S^2_{\sqrt{\varepsilon_k}}) G(S_{k+i}+z)G(z)}{1+ \|z\|} \right] \right\}\\
\lesssim & \sum_{i=0}^{\varepsilon_k} \left\{\mathbb{E}\left[\frac{G(S_{k-i}-S_k - S^2_{\sqrt{\varepsilon_k}}) }{1+ \|S_k\|^2} \right]
+ \mathbb{E}\left[\frac{G(S_{k+i}-S_k - S^2_{\sqrt{\varepsilon_k}}) }{1+ \|S_{k+i}\|^2} \right] \right\}\\
\stackrel{\eqref{exp.Green}, \eqref{exp.Green.x}}{\lesssim} & \frac{1}{\varepsilon_k^{3/4}}\sum_{i=0}^{\sqrt{\varepsilon_k}} \mathbb{E}\left[\frac{1}{1+ \|S_k\|^2} +\frac{1}{1+ \|S_{k+i}\|^2}\right] + \sum_{i = \sqrt{\varepsilon_k}}^{\varepsilon_k} \mathbb{E}\left[\frac{G(S_{k-i}-S_k)}{1+ \|S_k\|^2} +\frac{G(S_{k+i}-S_k)}{1+ \|S_{k+i}\|^2}\right]\\
\stackrel{\eqref{pn.largex}, \eqref{exp.Green}}{\lesssim} & \frac{1}{\varepsilon_k^{1/4}k} + \sum_{i = \sqrt{\varepsilon_k}}^{\varepsilon_k} \frac 1{i^{3/2}}\cdot \mathbb{E}\left[\frac{1}{1+ \|S_{k-i}\|^2} +\frac{1}{1+ \|S_k\|^2}\right] \lesssim \frac 1{\varepsilon_k^{1/4} k}. \end{align*} Thus at this point we have shown that \begin{eqnarray}\label{approx.Delta3}
\left|\mathbb{E}[Z_0Z_k\varphi_3\psi_3] - \Delta_{3,3}\right| = o\left(\frac 1k\right).
\end{eqnarray} Now define \begin{align*} \widetilde H(z_1,z_2,z_3) := \mathbb{P}\left[0\notin \mathcal{R}[1,\varepsilon_k^{3/4}], \right. & \widetilde \mathcal{R}[1,\sqrt{\varepsilon_k}]\cap \mathcal{R}[-\varepsilon_k^{ 3/4},\varepsilon_k^{ 3/4}]=\varnothing, \\
& \left. S_{\varepsilon_k^{3/4}}=z_1, S_{-\varepsilon_k^{3/4}}=z_3, \widetilde S_{\sqrt{\varepsilon_k}} = z_3\right],
\end{align*} and recall also the definition of $H(y_1,y_2)$ given in \eqref{Hy1y2}. One has $$\Delta_{3,3} = \sum H(y_1,y_2) \widetilde H(z_1,z_2,z_3) p_{k-\varepsilon_k - \varepsilon_k^{3/4}}(x - y_1+y_2+z_3-z_2) p_{\varepsilon_k- \varepsilon_k^{3/4}}(u-z_1+z_2) \varphi_{x,u},$$ where the sum runs over all $x,u,y_1,y_2,z_1,z_2,z_3\in \mathbb{Z}^5$, and \begin{align*} \varphi_{x,u} := \mathbb{P}[\overline \tau_1<\infty,\, \overline \tau_2<\infty\mid S_k = x,\, S_{k+\varepsilon_k} = x+u]. \end{align*} Note that the same argument as for \eqref{tau1tau2} gives also \begin{eqnarray}\label{phixu}
\varphi_{x,u} \lesssim \frac 1{1+\|u\|}\left(\frac 1{1+\|x+u\|} +\frac 1{1+\|x\|}\right). \end{eqnarray} Using this it is possible to see that in the expression of $\Delta_{3,3}$ given just above, one can restrict the sum to typical values of the parameters. Indeed, consider for instance the sum on atypically large values of $x$. More precisely, take $\chi_k$, such that $\varepsilon_k \chi_k^{2+1/4} =k$, and note that by \eqref{phixu}, \begin{align*}
& \sum_{\substack{\|x\|^2\ge k\chi_k \\ u,y_1,y_2,z_1,z_2,z_3}} H(y_1,y_2) \widetilde H(z_1,z_2,z_3) p_{k-\varepsilon_k - \varepsilon_k^{3/4}}(x - y_1+y_2+z_3-z_2) p_{\varepsilon_k- \varepsilon_k^{3/4}}(u-z_1+z_2) \varphi_{x,u}\\
& \le \mathbb{P}\left[ \|S_k-S_{\varepsilon_k}^1\|\ge \sqrt{k\chi_k}, \tau_1<\infty, \tau_2<\infty\right]\le \mathbb{P}\left[ \|S_k-S_{\varepsilon_k}^1\|\ge \sqrt{k\chi_k}, \tau_1<\infty, \overline \tau_2<\infty\right] \\
& \lesssim \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|S_k-S_{\varepsilon_k}^1\|\ge \sqrt{k\chi_k}\}}{1+\|S_{k+\varepsilon_k}-S_k\|} \left(\frac {1}{1+\|S_k-S^1_{\varepsilon_k}\|} + \frac 1{1+\|S_{k+\varepsilon_k}-S^1_{\varepsilon_k}\|}\right)\right] \\ & \lesssim \frac{1}{\chi_k^{5/4}\sqrt{k\varepsilon_k} }, \end{align*} where the last equality follows by applying Cauchy-Schwarz inequality and \eqref{Sn.large}. The other cases are entirely similar. Thus $\Delta_{3,3}$ is well approximated by the sums on typical values of the parameters (similarly as for $\Delta_0$ for instance), and then we can deduce with Theorem \ref{LCLT} and \eqref{phixu} that $$\Delta_{3,3} = \rho^2\cdot \mathbb{P}[\overline \tau_1<\infty,\, \overline \tau_2<\infty]+ o\left(\frac 1k\right).$$ Together with \eqref{approx.Delta3} and \eqref{Z03.2} this proves \eqref{cov.33.first}.
\underline{Step $2$.} For a (possibly random) time $T$, set $$\overline \tau_1\circ T := \inf \{n\ge T\vee \varepsilon_k : S_n \in \mathcal{R}^1_\infty\},\ \overline \tau_2\circ T := \inf \{n\ge T\vee \varepsilon_k : S_n \in (S_k+\mathcal{R}^2_\infty)\}. $$ Observe that \begin{equation}\label{main.tau.1} \mathbb{P}[\overline \tau_1\le \overline \tau_2<\infty] = \mathbb{P}[\overline \tau_1\le \overline \tau_2\circ \overline \tau_1<\infty] - \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2\le \overline \tau_2\circ \overline \tau_1\circ \overline \tau_2<\infty], \end{equation} and symmetrically, \begin{equation}\label{main.tau.2} \mathbb{P}[\overline \tau_2\le \overline \tau_1<\infty] = \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2<\infty] - \mathbb{P}[\overline \tau_1\le\overline \tau_2\circ \overline \tau_1\le \overline \tau_1\circ \overline \tau_2\circ \overline \tau_1<\infty]. \end{equation} Our aim here is to show that the two error terms appearing in \eqref{main.tau.1} and \eqref{main.tau.2} are negligible. Applying repeatedly \eqref{lem.hit.1} gives \begin{align*} E_1& := \mathbb{P}[\overline \tau_1\le\overline \tau_2\circ \overline \tau_1\le \overline \tau_1\circ \overline \tau_2\circ \overline \tau_1<\infty] \\ &\lesssim \sum_{j\ge 0} \sum_{\ell \ge 0} \sum_{m\ge 0} \mathbb{E}\left[ G(S_j^1 - S_k - S_\ell^2) G(S_k + S_\ell^2 - S_m^1) G(S_m^1 - S_{k+\varepsilon_k})\right]\\ & \stackrel{\eqref{exp.Green.x}}{\lesssim} \sum_{j\ge 0} \sum_{\ell \ge 0} \sum_{m\ge 0} \mathbb{E}\left[ G(S_j^1 - S_k - S_\ell^2) G(S_k + S_\ell^2 - S_m^1) G(S_m^1 - S_k)\right]\\ & \lesssim \sum_{j\ge 0} \sum_{m\ge 0} G(z) \mathbb{E}\left[ G(S_j^1 - S_k - z) G(S_k + z - S_m^1) G(S_m^1 - S_k)\right]. \end{align*} Note also that by using Lemma \ref{lem.upconvolG} and \eqref{Green}, we get
$$\sum_{z\in \mathbb{Z}^5}G(z-x) G(z-y) G(z) \lesssim \frac 1{1+\|x\|^3} \left(\frac 1{1+\|y\|} + \frac {1}{1+ \|y-x\|} \right).$$ Thus, distinguishing also the two cases $j\le m$ and $m\le j$, we obtain \begin{align*}
E_1& \lesssim \sum_{j\ge 0} \sum_{m\ge 0}\mathbb{E}\left[\frac {G(S_m^1 - S_k)}{1+\|S_j^1-S_k\|^3} \left( \frac 1{1+\|S_m^1-S_k\|} + \frac 1{1+\|S_m^1- S_j^1\|}\right) \right] \\
& \lesssim \sum_{j\ge 0} \sum_{z\in \mathbb{Z}^5} G(z)\left\{ \mathbb{E}\left[\frac {G(z+S_j^1 - S_k)}{1+\|S_j^1-S_k\|^3} \left( \frac 1{1+\|z+S_j^1-S_k\|} + \frac 1{1+\|z\|}\right) \right] \right. \\
& \qquad \left. + \mathbb{E}\left[\frac {G(S_j^1 - S_k)}{1+\|z+S_j^1-S_k\|^3} \left( \frac 1{1+\|S_j^1-S_k\|} + \frac 1{1+\|z\|}\right) \right]\right\}\\
& \lesssim \sum_{j\ge 0} \mathbb{E}\left[\frac {1}{1+\|S_j^1-S_k\|^5} \right] \lesssim \mathbb{E}\left[\frac {\log (1+\|S_k\|)}{1+\|S_k\|^3}\right] \lesssim \frac{\log k}{k^{3/2}}. \end{align*} Similarly, \begin{align*} & \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2\le \overline \tau_2\circ \overline \tau_1\circ \overline \tau_2<\infty] \\ &\lesssim \sum_{j\ge 0} \sum_{\ell \ge 0} \sum_{m\ge 0} \mathbb{E}\left[ G(S_j^2 + S_k - S_\ell^1) G(S_\ell^1 - S_k- S_m^2) G(S_m^2 + S_k - S_{k+\varepsilon_k})\right]\\
& \stackrel{\eqref{exp.Green}, \eqref{exp.Green.x}}{\lesssim} \frac {1}{\sqrt{\varepsilon_k}} \sum_{j\ge 0} \sum_{\ell \ge 0} \sum_{m\ge 0}\mathbb{E}\left[ \frac{G(S_j^2 + S_k - S_\ell^1) G(S_\ell^1 - S_k- S_m^2) }{1+\|S_m^2\|^2} \right]\\
&\lesssim \frac{1}{\sqrt{\varepsilon_k}} \sum_{j\ge 0} \sum_{m\ge 0}\mathbb{E}\left[\frac 1{(1+\|S_m^2\|^2)(1+\|S_j^2+S_k\|^3)} \left( \frac 1{1+\|S_m^2+S_k\|} + \frac 1{1+\|S_m^2- S_j^2\|}\right) \right] \\
&\lesssim \frac{1}{\sqrt{\varepsilon_k}} \sum_{j\ge 0}\mathbb{E}\left[\frac 1{(1+\|S_j^2\|)(1+\|S_j^2+S_k\|^3)} +\frac 1{(1+\|S_j^2\|^2)(1+\|S_j^2+S_k\|^2)} \right] \\
&\lesssim \frac{1}{\sqrt{\varepsilon_k}}\cdot \mathbb{E}\left[\frac {\log (1+\|S_k\|)}{1+\|S_k\|^2}\right] \lesssim \frac{\log k}{k \sqrt{\varepsilon_k}}. \end{align*}
\underline{Step $3$.} We now come to the estimate of the two main terms in \eqref{main.tau.1} and \eqref{main.tau.2}. In fact it will be convenient to replace $\overline \tau_1$ in the first one by $$\widehat \tau_1:= \inf \{n\ge k : S_n\in \mathcal{R}_\infty^1\}.$$ The error made by doing this is bounded as follows: by shifting the origin to $S_k$, and using symmetry of the step distribution, we can write \begin{align*}
& \left| \mathbb{P}[\overline \tau_1\le \overline \tau_2\circ \overline \tau_1<\infty] - \mathbb{P}[\widehat \tau_1\le \overline \tau_2\circ \widehat \tau_1<\infty]\right| \le \mathbb{P}\left[\mathcal{R}_\infty^1\cap \mathcal{R}[k,k+\varepsilon_k] \neq \varnothing, \overline \tau_2<\infty\right] \\ & \stackrel{\eqref{Green.hit}}{\le} \mathbb{E}\left[\left(\sum_{i=0}^{\varepsilon_k} G(S_i- \widetilde S_k)\right) \left(\sum_{j=\varepsilon_k}^\infty G(S_j)\right)\right] \\ & = \mathbb{E}\left[\left(\sum_{i=0}^{\varepsilon_k} G(S_i- \widetilde S_k)\right) \left(\sum_{z\in \mathbb{Z}^5} G(z) G(z+S_{\varepsilon_k})\right)\right] \\
&\stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} \sum_{i=0}^{\varepsilon_k} \mathbb{E}\left[ \frac{G(S_i- \widetilde S_k)}{1+\|S_{\varepsilon_k}\|}\right] \stackrel{\eqref{exp.Green}}{\lesssim} \frac {\varepsilon_k}{k^{3/2}}\cdot \mathbb{E}\left[ \frac 1{1+\|S_{\varepsilon_k}\|}\right] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}. \end{align*} Moreover, using Theorem C, the Markov property and symmetry of the step distribution, we get for some constant $c>0$, \begin{align*} & \mathbb{P}[\widehat \tau_1\le \overline \tau_2\circ \widehat \tau_1<\infty]
= c \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\widehat \tau_1<\infty\}}{1+\mathcal{J}(S_{\widehat \tau_1} - S_k) }\right] + o\left(\frac 1k\right) \\
& = c \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\widehat \tau_1<\infty\}}{1+\mathcal{J}(S_{\widehat \tau_1}) }\right] + o\left(\frac 1k\right)
= c \sum_{x\in \mathbb{Z}^5}p_k(x) \, \mathbb{E}_{0,x} \left[F(S_\tau) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\}\right] + o\left(\frac 1k\right), \end{align*} with $\tau$ the hitting time of two independent walks starting respectively from the origin and from $x$, and $F(z) := 1/(1+ \mathcal{J}(z))$. Note that the bound $o(1/k)$ on the error term in the last display comes from the fact that
$$\mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\widehat \tau_1<\infty\}}{1+\mathcal{J}(S_{\widehat \tau_1})} \right] \stackrel{\eqref{lem.hit.1}}{\lesssim} \sum_{j\ge 0} \mathbb{E}\left[\frac{G(\widetilde S_j-S_k)}{1+\|\widetilde S_j\|}\right] \lesssim \sum_{z\in \mathbb{Z}^5}
\mathbb{E}\left[\frac{G(z)G(z-S_k)}{1+\| z\|}\right] \lesssim \frac 1k. $$ Then by applying Theorem \ref{thm.asymptotic}, we get \begin{equation}\label{main.tau.1.2}
\mathbb{P}[\widehat \tau_1\le \overline \tau_2\circ \widehat \tau_1<\infty] = c_0 \sum_{x\in \mathbb{Z}^5} p_k(x) \sum_{z\in \mathbb{Z}^5} \frac{G(z)G(z-x)}{1+\mathcal J(z)} + o\left(\frac 1k\right), \end{equation} for some constant $c_0>0$. Likewise, by Theorem \ref{thm.asymptotic} one has for some constant $\nu\in (0,1)$, \begin{align*} \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2<\infty] & = c\, \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\overline \tau_2<\infty\} }{1+ \mathcal{J}(S_{\overline \tau_2})}\right] + \mathcal{O}\left(\mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\overline \tau_2<\infty\} }{1+ \mathcal{J}(S_{\overline \tau_2})^{1+\nu}} \right] \right). \end{align*} Furthermore, \begin{align*}
& \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\overline \tau_2<\infty\} }{1+ \mathcal{J}(S_{\overline \tau_2})^{1+\nu}} \right] \lesssim \sum_{j\ge 0} \mathbb{E}\left[ \frac {G(S_j^2+S_k-S_{k+\varepsilon_k}) }{1+ \| S_j^2+S_k\|^{1+\nu}} \right] \\
& \stackrel{\eqref{exp.Green}, \eqref{exp.Green.x}}{\lesssim} \frac{1}{\sqrt{\varepsilon_k}}\sum_{j\ge 0} \mathbb{E}\left[ \frac {1 }{(1+\|S_j^2\|^2)(1+ \| S_j^2+S_k\|^{1+\nu})} \right] \\
& \lesssim \frac{1}{\sqrt{\varepsilon_k}} \mathbb{E}\left[\frac{\log (1+\|S_k\|)}{1+\|S_k\|^{1+\nu}}\right] \lesssim \frac {\log k}{ k^{(1+ \nu)/2}\sqrt{\varepsilon_k}}.
\end{align*} Therefore, taking $\varepsilon_k\ge k^{1-\nu/2}$, we get \begin{align}\label{main.tau.2.1}
\nonumber \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2<\infty] & = c\, \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\overline \tau_2<\infty\} }{1+ \mathcal{J}(S_{\overline \tau_2})}\right] + o\left(\frac 1k\right) \\ \nonumber & = c \sum_{u\in \mathbb{Z}^5} p_{\varepsilon_k}(u) \mathbb{E}_{0,u} \left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau<\infty\} }{1+ \mathcal{J}(S_\tau - S_k)}\right] + o\left(\frac 1k\right) \\
& = c \sum_{u\in \mathbb{Z}^5} p_{\varepsilon_k}(u) \mathbb{E}_{0,u} \left[\widetilde F(S_\tau) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\} \right] + o\left(\frac 1k\right), \end{align} with $\tau$ the hitting time of two independent walks starting respectively from the origin and from $u$, and $$\widetilde F(z):= \mathbb{E}\left[\frac 1{1+ \mathcal{J}(z-S_k)} \right]. $$ We claim that this function $\widetilde F$ satisfies \eqref{cond.F}, for some constant $C_{\widetilde F}$ which is independent of $k$. Indeed, first notice that
$$\widetilde F(z) \asymp \frac 1{1+\|z\| + \sqrt{k}},\quad \text{and}\quad \mathbb{E}\left[ \frac 1{1+\mathcal{J}(z-S_k)^2}\right] \asymp \frac 1{1+\|z\|^2 + k},$$ which can be seen by using Theorem \ref{LCLT}. Moreover, by triangle inequality, and Cauchy-Schwarz, \begin{align*}
|\widetilde F(y) - \widetilde F(z) | & \lesssim \mathbb{E}\left[\frac {\|y-z\|}{(1+\|y-S_k\|)(1+\|z-S_k\|)}\right] \\
& \lesssim \|y-z\| \, \mathbb{E}\left[\frac 1{1+\|y-S_k\|^2}\right]^{\frac 12} \mathbb{E}\left[\frac 1{1+\|z-S_k\|^2}\right]^{\frac 12} \\
& \lesssim \frac{\|y-z\|}{(1+\|y\|+\sqrt k)(1+\|z\| +\sqrt{k})} \lesssim \frac{\|y-z\|}{1+\|y\|}\cdot \widetilde F(z), \end{align*} which is the desired condition \eqref{cond.F}. Therefore, coming back to \eqref{main.tau.2.1} and applying Theorem \ref{thm.asymptotic} once more gives, \begin{align}\label{main.tau.2.1.bis} \nonumber \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2<\infty] & = c_0\sum_{u\in \mathbb{Z}^5} p_{\varepsilon_k}(u) \sum_{z\in \mathbb{Z}^5} G(z)G(z-u)\widetilde F(z) + o\left(\frac 1k\right) \\
& = c_0\sum_{u\in \mathbb{Z}^5} \sum_{x\in \mathbb{Z}^5}p_{\varepsilon_k}(u) p_k(x) \sum_{z\in \mathbb{Z}^5} \frac{G(z)G(z-u)}{1+\mathcal J(z-x)} + o\left(\frac 1k\right). \end{align} Similarly, one has \begin{align}\label{main.tau.product} \nonumber &\mathbb{P}[\overline \tau_1<\infty] \cdot \mathbb{P}[\overline \tau_2<\infty] =\mathbb{P}[\widehat \tau_1<\infty] \cdot \mathbb{P}[\overline \tau_2<\infty] + \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right) \\ & = c_0 \sum_{u\in \mathbb{Z}^5} \sum_{x\in \mathbb{Z}^5} p_{\varepsilon_k}(u)p_k(x) \sum_{z\in \mathbb{Z}^5} \frac{G(z)G(z-u)}{1+\mathcal J(x)} + o\left(\frac 1k\right). \end{align} Note in particular that the constant $c_0$ that appears here is the same as in \eqref{main.tau.1.2} and \eqref{main.tau.2.1.bis}.
\underline{Step $4$.} We claim now that when one takes the difference between the two expressions in \eqref{main.tau.2.1.bis} and \eqref{main.tau.product}, one can remove the parameter $u$ from the factor $G(z-u)$ (and then absorb the sum over $u$). Indeed, note that for any $z$ with $\mathcal{J}(z)\le \mathcal{J}(x)/2$, one has
$$\left| \frac 1{1+\mathcal{J}(z+x)} + \frac 1{1+\mathcal{J}(z-x)} - \frac 2{1+\mathcal{J}(x)}\right| \lesssim \frac{\|z\|^2}{1+\|x\|^3}.$$ It follows that, for any $\chi_k\ge 2$, \begin{align*}
& \sum_{\substack{u,x\in \mathbb{Z}^5 \\ \mathcal{J}(z) \le \frac{\mathcal{J}(x)}{\chi_k}}} p_{\varepsilon_k}(u) p_k(x) G(z) G(z-u) \left|\frac 1{1+\mathcal{J}(z-x)}+\frac 1{1+\mathcal{J}(z+x)} - \frac 2{1+\mathcal{J}(x)}\right| \\
&\lesssim \sum_{x\in \mathbb{Z}^5} \frac{p_k(x)}{1+\|x\|^3} \sum_{\mathcal{J}(z) \le \mathcal{J}(x)/\chi_k } \frac{\mathbb{E}[G(z-S_{\varepsilon_k})]}{1+\|z\|} \stackrel{\eqref{exp.Green.x}}{\lesssim} \frac{1}{k\chi_k}. \end{align*} In the same way, for any $z$ with $\mathcal{J}(z) \ge 2\mathcal{J}(u)$, one has
$$|G(z-u) -G(z)| \lesssim \frac{\|u\|}{1+\|z\|^4},$$
$$\left|\frac 1{1+\mathcal{J}(z-x)} - \frac 1{1+\mathcal{J}(x)}\right| \lesssim \frac{\|z\|}{(1+\|x\|)(1+\|z-x\|)}.$$ Therefore, for any $\chi_k\ge 2$, \begin{align*}
& \sum_{\substack{u,x\in \mathbb{Z}^5 \\ \mathcal{J}(z) \ge (\mathcal{J}(u)\chi_k)\vee \frac{\mathcal{J}(x)}{\chi_k}}} p_{\varepsilon_k}(u) p_k(x) G(z) |G(z-u)-G(z)| \left|\frac 1{1+\mathcal{J}(z-x)} - \frac 1{1+\mathcal{J}(x)}\right| \\
&\lesssim \sqrt{\varepsilon_k} \sum_{x\in \mathbb{Z}^5} \frac{p_k(x)}{1+\|x\|} \sum_{\mathcal{J}(z) \ge \mathcal{J}(x)/\chi_k } \frac{1}{\|z\|^6(1+\|z-x\|)}
\stackrel{\eqref{exp.Green.x}}{\lesssim} \frac{\chi_k^2\sqrt{\varepsilon_k}}{k^{3/2}}. \end{align*} On the other hand by taking $\chi_k = (k/\varepsilon_k)^{1/6}$, we get using \eqref{pn.largex} and \eqref{Sn.large}, \begin{align*}
\sum_{\substack{x,z\in \mathbb{Z}^5 \\ \mathcal{J}(u)\ge \sqrt{\varepsilon_k}\chi_k }} p_{\varepsilon_k}(u) p_k(x) G(z) G(z-u) \left(\frac 1{1+\mathcal{J}(z-x)} + \frac 1{1+\mathcal{J}(x)}\right) \lesssim \frac{1}{\chi_k^5\sqrt{k\varepsilon_k}} = o \left(\frac 1k\right), \end{align*} \begin{align*} & \sum_{\substack{u,z\in \mathbb{Z}^5 \\ \mathcal{J}(x)\le \sqrt{k}/\chi_k }} p_{\varepsilon_k}(u) p_k(x) G(z) G(z-u) \left(\frac 1{1+\mathcal{J}(z-x)} + \frac 1{1+\mathcal{J}(x)}\right) = o \left(\frac 1k\right). \end{align*} As a consequence, since $\mathcal{J}(u)\le \sqrt{\varepsilon_k} \chi_k$ and $\mathcal{J}(x)\ge \sqrt{k}/\chi_k$, implies $\mathcal{J}(u)\le \mathcal{J}(x)/\chi_k$, with our choice of $\chi_k$, we get as wanted (using also symmetry of the step distribution) that \begin{align}\label{main.tau.combined.1} & \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2<\infty] - \mathbb{P}[\overline \tau_1<\infty] \cdot \mathbb{P}[\overline \tau_2<\infty] \\ \nonumber & = c_0 \sum_{x,z\in \mathbb{Z}^5} p_k(x) G(z)^2 \left(\frac 1{1+\mathcal{J}(z-x)} - \frac{1}{1+\mathcal J(x)}\right) + o \left(\frac 1k\right)\\ \nonumber &= \frac{c_0}{2} \sum_{x,z\in \mathbb{Z}^5} p_k(x) G(z)^2 \left(\frac 1{1+\mathcal{J}(z-x)} +\frac 1{1+\mathcal{J}(z+x)}- \frac{2}{1+\mathcal J(x)}\right) + o \left(\frac 1k\right). \end{align}
\underline{Step 5.} The previous steps show that $$\operatorname{Cov}\left(\{\overline \tau_1<\infty\} , \{\overline \tau_2<\infty\}\right) = c_0 \sum_{x,z\in \mathbb{Z}^5} p_k(x) \left(\frac{G(z)G(z-x)}{1+\mathcal{J}(z)} + \frac{G(z)^2}{1+\mathcal{J}(z-x)} - \frac{G(z)^2}{1+\mathcal{J}(x)}\right).$$ Now by approximating the series with an integral (recall \eqref{Green.asymp}), and doing a change of variables, we get with $u:=x/\mathcal{J}(x)$ and $v:=\Lambda^{-1} u$, and for some constant $c>0$ (that might change from line to line), \begin{align}\label{last.lem.integral} \nonumber & \sum_{z\in \mathbb{Z}^5}\left(\frac{G(z)G(z-x)}{1+\mathcal{J}(z)} + \frac{G(z)^2}{1+\mathcal{J}(z-x)} - \frac{G(z)^2}{1+\mathcal{J}(x)}\right) \\ \nonumber & \sim c \int_{\mathbb{R}^5} \left\{\frac{1}{\mathcal{J}(z)^4\cdot \mathcal{J}(z-x)^3} + \frac{1}{\mathcal{J}(z)^6} \left(\frac 1{\mathcal{J}(z-x)} -\frac 1{\mathcal{J}(x)}\right)\right\}\, dz \\ \nonumber & = \frac{c}{\mathcal{J}(x)^2} \int_{\mathbb{R}^5}\left\{\frac{1}{\mathcal{J}(z)^4\cdot \mathcal{J}(z-u)^3} + \frac{1}{\mathcal{J}(z)^6} \left(\frac 1{\mathcal{J}(z-u)} -1\right)\right\}\, dz \\
& = \frac{c}{\mathcal{J}(x)^2} \int_{\mathbb{R}^5} \left\{\frac{1}{\|z\|^4\cdot \|z-v\|^3} + \frac{1}{\|z\|^6}\left(\frac 1{\|z-v\|} -1\right)\right\} \, dz. \end{align} Note that the last integral is convergent and independent of $v$ (and thus of $x$ as well) by rotational invariance. Therefore, since $\sum_{x\in \mathbb{Z}^5} p_k(x) / \mathcal{J}(x)^2\sim \sigma/k$, for some constant $\sigma>0$ (for instance by applying Theorem \ref{LCLT}), it only remains to show that the integral above is positive.
To see this, we use that the map $z\mapsto \|z\|^{-3}$ is harmonic outside the origin, and thus satisfies the mean value property on $\mathbb{R}^5\setminus \{0\}$. In particular, using also the rotational invariance, this shows (with $\mathcal{B}_1$ the unit Euclidean ball and $\partial \mathcal{B}_1$ the unit sphere), \begin{align}\label{last.lem2.a}
\int_{\mathcal{B}_1^c} \frac{1}{\|z\|^4\cdot \|z-v\|^3}\, dz &= \frac 1{|\partial \mathcal{B}_1|} \int_{\partial \mathcal{B}_1} \, dv \int_{\mathcal{B}_1^c} \frac{1}{\|z\|^4\cdot \|z-v\|^3}\, dz\\
\nonumber & = \int_{\mathcal{B}_1^c} \frac 1{\|z\|^7} \, dz = c_1 \int_1^\infty \frac 1{r^3}\, dr = \frac {c_1}2, \end{align} for some constant $c_1>0$. Likewise, \begin{equation}\label{last.lem2.b}
\int_{\mathcal{B}_1} \frac{1}{\|z\|^4\cdot \|z-v\|^3}\, dz = \frac{c_1}{|\partial \mathcal{B}_1|} \int_0^1 \, dr \int_{\partial \mathcal{B}_1} \frac{du}{\|ru - v\|^3} = c_1, \end{equation} with the same constant $c_1$ as in the previous display. On the other hand \begin{equation}\label{last.lem2.c}
\int_{\mathcal{B}_1^c} \frac 1{\|z\|^6}\, dz = c_1 \int_1^\infty \frac 1{r^2} \, dr = c_1. \end{equation} Furthermore, using again the rotational invariance, \begin{align}\label{last.lem.2}
& \int_{\mathcal{B}_1} \frac{1}{\|z\|^6}\left(\frac 1{\|z-v\|} -1\right) \, dz = \int_{\mathcal{B}_1} \frac{1}{\|z\|^6}\left(\frac 1{2\|z-v\|} + \frac 1{2\|z+v\|} -1\right) \, dz \\
\nonumber & = \frac{c_1}{|\partial \mathcal{B}_1|} \int_0^1 \frac {dr}{r^2} \int_{\partial \mathcal{B}_1}\left(\frac 1{2\|v-ru\|} + \frac 1{2\|v+ru\|} -1\right)\, du. \end{align} Now we claim that for any $u,v\in \partial \mathcal{B}_1$, and any $r\in (0,1)$, \begin{equation}\label{claim.geom}
\frac 12\left(\frac 1{\|v-ru\|} + \frac 1{\|v+ru\|}\right) \ge \frac{1}{\sqrt{1+r^2}}. \end{equation} Before we prove this claim, let us see how we can conclude the proof. It suffices to notice that, if $f(s) = (1+s^2)^{-1/2}$, then $f'(s) \ge - s$, for all $s\in (0,1)$, and thus \begin{equation}\label{lower.sqrt} \frac 1{\sqrt{1+r^2}} - 1 = f(r) - f(0) \ge - \int_0^r s\, ds \ge -r^2/2. \end{equation} Inserting this and \eqref{claim.geom} in \eqref{last.lem.2} gives
$$ \int_{\mathcal{B}_1} \frac{1}{\|z\|^6}\left(\frac 1{\|z-v\|} -1\right) \, dz \ge -\frac {c_1}{2}. $$ Together with \eqref{last.lem2.a}, \eqref{last.lem2.b}, \eqref{last.lem2.c}, this shows that the integral in \eqref{last.lem.integral} is well positive. Thus all that remains to do is proving the claim \eqref{claim.geom}. Since the origin, $v$, $v+ru$, and $v-ru$ all lie in a common two-dimensional plane, one can always work in the complex plane, and assume for simplicity that $v= 1$, and $u=e^{i\theta}$, for some $\theta\in [0,\pi/2]$. In this case, the claim is equivalent to showing that $$\frac 12 \left( \frac 1{\sqrt{1+ r^2 + 2r\cos \theta} }+ \frac 1{\sqrt{1+ r^2 - 2r\cos \theta}}\right) \ge \frac 1{\sqrt{1+r^2}},$$ which is easily obtained using that the left hand side is a decreasing function of $\theta$. This concludes the proof of Lemma \ref{lem.var.2}.
$\square$
\begin{remark}\emph{Note that the estimate of the covariance mentioned in the introduction, in case $(ii)$, can now be done as well. Indeed, denoting by $$\widehat \tau_2:= \inf\{n\ge k+1\, :\, S_n\in S_k + \mathcal{R}_\infty^2\},$$ it only remains to show that
$$\left|\mathbb{P}[\widehat \tau_2\le k+\varepsilon_k, \, \overline \tau_1<\infty ] - \mathbb{P}[\widehat \tau_2\le k+ \varepsilon_k] \cdot \mathbb{P}[\overline \tau_1<\infty]\right| = o\left(\frac 1k\right).$$ Using similar estimates as above we get, with $\chi_k = (k/\varepsilon_k)^{4/5}$, \begin{align*}
& \left|\mathbb{P}[\widehat \tau_2\le k+ \varepsilon_k, \overline \tau_1<\infty ] - \mathbb{P}[\widehat \tau_2\le k+ \varepsilon_k] \cdot \mathbb{P}[\overline \tau_1<\infty]\right| \\
\stackrel{\eqref{Sn.large}}{=} & \left|\mathbb{P}[\widehat \tau_2\le k+ \varepsilon_k, \|S_{\widehat \tau_2} - S_k\|\le \sqrt{\varepsilon_k\chi_k}, \overline \tau_1<\infty ] - \mathbb{P}[\widehat \tau_2\le k+ \varepsilon_k] \mathbb{P}[\overline \tau_1<\infty]\right| + \mathcal{O}\left(\frac 1{\sqrt k \chi_k^{\frac 52}}\right)\\
=& \sum_{\substack{x\in \mathbb{Z}^5 \\ \|y\|\le \sqrt{\varepsilon_k \chi_k}} } \left|\frac{p_k(x+y) + p_k(x-y)}{2}-p_k(x)\right| \mathbb{P}[\widehat \tau_2\le k+\varepsilon_k, S_{\widehat \tau_2} -S_k= y] \varphi_x + \mathcal{O}\left(\frac 1{\sqrt k \chi_k^{\frac 52}}\right)\\
\lesssim & \frac 1{k^{\frac 32}} \mathbb{E}\left[\|S_{\widehat \tau_2}-S_k\|^2 {\text{\Large $\mathfrak 1$}}\{\|S_{\widehat \tau_2}-S_k\|\le \sqrt{\varepsilon_k\chi_k} \}\right] + \frac 1{\sqrt k \chi_k^{\frac 52}}\lesssim \frac 1{\sqrt k \chi_k^{\frac 52}}+ \frac {\sqrt{\varepsilon_k\chi_k}}{k^{\frac 32}}, \end{align*}
using that by \eqref{lem.hit.2} and the Markov property, one has $\mathbb{P}[\|S_{\widehat \tau_2}-S_k\|\ge t] \lesssim \frac 1t $.
} \end{remark}
\subsection{Proof of Lemma \ref{lem.var.3}} We consider only the case of $\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_1)$, the other one being entirely similar. Define $$\tau_1:= \inf\{n\ge 0 : S_n^1\in \mathcal{R}[\varepsilon_k,k]\},\ \tau_2:=\inf\{n\ge 0 : S_k+S_n^2\in \mathcal{R}(-\infty, 0] \},$$ with $S^1$ and $S^2$ two independent walks, independent of $S$. The first step is to see that $$\operatorname{Cov}(Z_0\varphi_3,Z_k\psi_2)= \rho^2\cdot \operatorname{Cov}({\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\},{\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}) +o\left(\frac 1k\right),$$ with $\rho$ as in \eqref{def.rho}. Since the proof of this fact has exactly the same flavor as in the two previous lemmas, we omit the details and directly move to the next step.
Let $\eta\in (0,1/2)$ be some fixed constant (which will be sent to zero later). Notice first that \begin{align}\label{eta.tau1}
& \nonumber \mathbb{P}\left[S^1_{\tau_1} \in \mathcal{R}[(1-\eta) k,k],\, \tau_2<\infty\right] \stackrel{\eqref{Green.hit}, \eqref{lem.hit.2}}{\lesssim} \sum_{i=\lfloor (1-\eta)k\rfloor }^k
\mathbb{E}\left[\frac{G(S_i)}{1+\|S_k\|}\right] \\ & \stackrel{\eqref{exp.Green}}{\lesssim} \sum_{i=\lfloor (1-\eta)k\rfloor }^k \frac{\mathbb{E}\left[G(S_i)\right] }{1+\sqrt{k-i}}
\stackrel{\eqref{exp.Green}}{\lesssim}\frac{\sqrt{\eta}}{k}. \end{align} Next, fix another constant $\delta\in (0,1/4)$ (which will be soon chosen small enough). Then let $N: = \lfloor (1-\eta)k/\varepsilon_k^{1-\delta}\rfloor$, and for $i=1,\dots,N$, define $$\tau_1^i:= \inf\{n\ge 0 \, :\, S_n^1 \in \mathcal{R}[k_i,k_{i+1}]\},\quad \text{with}\quad k_i:= \varepsilon_k + i\lfloor \varepsilon_k^{1-\delta}\rfloor .$$ We claim that with sufficiently high probability, at most one of these hitting times is finite. Indeed, for $i\le N$, set $I_i := \{k_i,\dots,k_{i+1}\}$, and notice that \begin{align*} &\sum_{1\le i< j\le N} \mathbb{P}[\tau_1^i <\infty, \, \tau_1^j<\infty,\, \tau_2<\infty] \\
\le & \sum_{1\le i< j\le N} \left(\mathbb{P}[\tau_1^i \le \tau_1^j<\infty,\, \tau_2<\infty] + \mathbb{P}[\tau_1^j \le \tau_1^i<\infty,\, \tau_2<\infty]\right) \\
\stackrel{\eqref{lem.hit.1}, \eqref{lem.hit.2}}{\lesssim} & \sum_{\substack{i=1,\dots,N, j\neq i \\ \ell \in I_i, m\in I_j}} \mathbb{E}\left[\frac{G(S_\ell - S_m) G(S_m)}{1+\|S_k\|} \right] \lesssim \frac{1}{\sqrt{k}} \sum_{\substack{i=1,\dots,N, j\neq i \\ \ell \in I_i, m\in I_j}} \mathbb{E}\left[G(S_\ell - S_m) G(S_m)\right]\\
\stackrel{\eqref{exp.Green}, \eqref{exp.Green.x}}{\lesssim} & \frac{1}{\sqrt{k}} \sum_{\substack{i=1,\dots, N, j\neq i \\ \ell \in I_i, m\in I_j}} \frac{1}{(1+ |m-\ell |^{3/2}) (m\wedge \ell)^{3/2}} \lesssim \frac {N\varepsilon_k^{(1-\delta)/2}}{\varepsilon_k^{3/2}\sqrt k} = o\left(\frac 1k\right), \end{align*} where the last equality follows by assuming $\varepsilon_k\ge k^{1-c}$, with $c>0$ small enough. Therefore, as claimed $$\mathbb{P}[\tau_1<\infty,\, \tau_2<\infty ] = \sum_{i=1}^{N} \mathbb{P}[\tau_1^i <\infty, \, \tau_2<\infty] + o\left(\frac 1k\right), $$ and one can show as well that, $$\mathbb{P}[\tau_1<\infty]\cdot \mathbb{P}[ \tau_2<\infty ] = \sum_{i=1}^{N-2} \mathbb{P}[\tau_1^i <\infty] \cdot \mathbb{P}[\tau_2<\infty] +o\left(\frac 1k\right).$$ Next, observe that for any $i\le N$, using H\"older's inequality at the third line, \begin{align*}
&\mathbb{P}\left[\tau_1^i <\infty, \tau_2<\infty, \|S_{k_{i+1}} - S_{k_i}\|^2\ge \varepsilon_k^{1-\delta/2}\right] \stackrel{\eqref{Green.hit}, \eqref{lem.hit.2}}{\lesssim} \sum_{j=k_i}^{k_{i+1}} \mathbb{E}\left[ \frac{G(S_j){\text{\Large $\mathfrak 1$}}\{\|S_{k_{i+1}} - S_{k_i}\|^2\ge \varepsilon_k^{1-\delta/2}\} }{1+\|S_k\|}\right]\\
& \stackrel{\eqref{exp.Green}}{\lesssim} \frac {1}{\sqrt k} \sum_{j=k_i}^{k_{i+1}} \mathbb{E}\left[ G(S_j){\text{\Large $\mathfrak 1$}}\{\|S_{k_{i+1}} - S_{k_i}\|^2\ge \varepsilon_k^{1-\delta/2}\} \right]\\
& \lesssim \frac {1}{\sqrt k}\left( \sum_{j=k_i}^{k_{i+1}} \mathbb{E}\left[\frac 1{1+\|S_j\|^4}\right]^{3/4}\right) \cdot \mathbb{P}\left[\|S_{k_{i+1}} - S_{k_i}\|^2\ge \varepsilon_k^{1-\delta/2}\right]^{1/4} \\ & \stackrel{\eqref{Sn.large}}{\lesssim} \frac {\varepsilon_k^{1-\delta}}{ k_i^{3/2}\sqrt{k}} \cdot \frac 1{\varepsilon_k^{5\delta/16}}= o\left(\frac 1{Nk}\right), \end{align*} by choosing again $\varepsilon_k\ge k^{1-c}$, with $c$ small enough. Similarly, one has using Cauchy-Schwarz, \begin{align*}
& \mathbb{P}\left[\tau_1^i <\infty, \tau_2<\infty, \|S_k - S_{k_{i+1}}\|^2\ge k \varepsilon_k^{\delta/2}\right] \lesssim \sum_{j=k_i}^{k_{i+1}} \mathbb{E}\left[ \frac{G(S_j){\text{\Large $\mathfrak 1$}}\{\|S_k - S_{k_{i+1}}\|^2 \ge k \varepsilon_k^{\delta/2}\} }{1+\|S_k\|}\right]\\
& \lesssim \frac{1}{\varepsilon_k^{5\delta/8}}\sum_{j=k_i}^{k_{i+1}} \mathbb{E}\left[ G(S_j) \mathbb{E}\left[\frac 1{1+\|S_k\|^2} \mid S_j \right]^{1/2}\right]\lesssim \frac {\varepsilon_k^{1-\delta}}{ k_i^{3/2}\sqrt{k}} \cdot \frac 1{\varepsilon_k^{5\delta/8}}= o\left(\frac 1{Nk}\right). \end{align*} As a consequence, using also Theorem \ref{LCLT}, one has for $i\le N$, and with $\ell := k_{i+1}-k_i$,
\begin{align*} & \mathbb{P}[\tau_1^i <\infty, \, \tau_2<\infty] \\
& =\sum_{x\in \mathbb{Z}^5} \sum_{\substack{ \|z\|^2 \le k \varepsilon_k^{\delta/2} \\ \|y\|^2 \le \varepsilon_k^{1-\delta/2} }} p_{k_i}(x)\mathbb{P}_{0,x}\left[ \mathcal{R}_\infty \cap \widetilde \mathcal{R}[0,\ell] \neq\varnothing, \widetilde S_{\ell} = y\right] p_{k-k_{i+1}}(z-y) \varphi_{x+z} + o\left(\frac 1{Nk}\right)\\
& = \sum_{x\in \mathbb{Z}^5} \sum_{\underset{\|z\|^2 \le k \varepsilon_k^{\delta/2} }{\|y\|^2 \le \varepsilon_k^{1-\delta/2}} } p_{k_i}(x) \mathbb{P}_{0,x} \left[ \mathcal{R}_\infty \cap \widetilde \mathcal{R}[0,\ell] \neq\varnothing, \widetilde S_\ell = y\right] p_{k-k_i}(z) \varphi_{x+z} + o\left(\frac 1{Nk}\right)\\ & = \sum_{x,z\in \mathbb{Z}^5} p_{k_i}(x) \mathbb{P}_{0,x} \left[ \mathcal{R}_\infty \cap \widetilde \mathcal{R}[0,\ell] \neq\varnothing \right] p_{k-k_i}(z) \varphi_{x+z} + o\left(\frac 1{Nk}\right).
\end{align*} Moreover, Theorem \ref{thm.asymptotic} yields for any nonzero $x\in \mathbb{Z}^5$, and some $\nu>0$,
\begin{align}\label{RtildeRell}
\mathbb{P}_{0,x} \left[ \mathcal{R}_\infty \cap \widetilde \mathcal{R}[0,\ell] \neq\varnothing \right] = \frac{\gamma_5}{\kappa}\cdot \mathbb{E}\left[ \sum_{j=0}^{\ell} G(x+\widetilde S_j) \right] + \mathcal{O}\left(\frac {\log(1+ \|x\|)}{\|x\| (\|x\|\wedge \ell)^{\nu}}\right). \end{align} Note also that for any $\varepsilon \in [0,1]$,
\begin{align*}
\sum_{x,z\in \mathbb{Z}^5} \frac{p_{k_i}(x)}{1+\|x\|^{1+\varepsilon}} p_{k-k_i}(z) \varphi_{x+z} = \mathbb{E}\left[\frac 1{(1+\|S_{k_i}\|^{1+\varepsilon})(1+\|S_k\|)}\right]
\lesssim \frac 1{\sqrt{k_i}^{1+\varepsilon} \sqrt k},
\end{align*}
and thus
$$\sum_{i=1}^N \sum_{x,z\in \mathbb{Z}^5} \frac{p_{k_i}(x)}{1+\|x\|^{1+\varepsilon}} p_{k-k_i}(z) \varphi_{x+z} = \mathcal{O}\left(\frac 1{\ell k^{\varepsilon}}\right).$$
In particular, the error term in \eqref{RtildeRell} can be neglected, as we take for instance $\delta=\nu/2$, and $\varepsilon_k\ge k^{1-c}$, with $c$ small enough. It amounts now to estimate the other term in \eqref{RtildeRell}. By \eqref{pn.largex}, for any $x\in \mathbb{Z}^5$ and $j\ge 0$, \begin{equation*}
\mathbb{E}[G(x+S_j)] = G_j(x) = G(x)-\mathcal O(\frac{j}{1+ \|x\|^d}). \end{equation*} As will become clear the error term can be neglected here. Furthermore, similar computations as above show that for any $j\in \{k_i,\dots,k_{i+1}\}$, $$\sum_{x,z\in \mathbb{Z}^5} p_{k_i}(x) G(x) p_{k-k_i}(z) \varphi_{x+z} = \sum_{x,z\in \mathbb{Z}^5} p_j(x) G(x) p_{k-j}(z) \varphi_{x+z}+ o\left(\frac 1{Nk}\right),$$ Altogether, and applying once more Theorem \ref{thm.asymptotic}, this gives for some $c_0>0$,
\begin{equation}\label{cov.3.sum}
\sum_{i=1}^N \mathbb{P}[\tau_1^i <\infty, \tau_2<\infty] = \sum_{j=\varepsilon_k}^{(1-\eta) k} \mathbb{E}[G(S_j)\varphi_{S_k}]+ o\left(\frac 1k\right) =c_0 \sum_{j=\varepsilon_k}^{\lfloor (1-\eta) k\rfloor} \mathbb{E}\left[\frac{G(S_j)}{1+\mathcal{J}(S_k)}\right]+ o\left(\frac 1k\right). \end{equation} We treat the first terms of the sum separately. Concerning the other ones notice that by \eqref{Green.asymp} and Donsker's invariance principle, one has \begin{align*}
\sum_{j=\lfloor \eta k\rfloor}^{\lfloor (1-\eta)k\rfloor} \mathbb{E}\left[\frac{G(S_j)}{1+\mathcal{J}(S_k)}\right] & = \frac 1k \int_\eta^{1-\eta} \mathbb{E}\left[\frac{G(\Lambda \beta_s)}{\mathcal{J}(\Lambda \beta_1)}\right] \, ds+ o\left(\frac 1k\right) \\
& = \frac {c_5}k \int_\eta^{1-\eta} \mathbb{E}\left[\frac{1}{\|\beta_s\|^3 \cdot \|\beta_1\|}\right] \, ds+ o\left(\frac 1k\right), \end{align*} with $(\beta_s)_{s\ge 0}$ a standard Brownian motion, and $c_5>0$ the constant that appears in \eqref{Green.asymp}. In the same way, one has \begin{align*} \sum_{i=1}^N \mathbb{P}[\tau_1^i <\infty] \cdot \mathbb{P}[\tau_2<\infty] & = c_0 \sum_{j=\varepsilon_k}^{\lfloor \eta k\rfloor} \mathbb{E}[G(S_j)]\mathbb{E}\left[\frac{1}{1+\mathcal{J}(S_k)}\right] \\
& \quad + \frac{c_0c_5}{k} \int_\eta^{1-\eta}\mathbb{E}\left[\frac{1}{\|\beta_s\|^3}\right] \mathbb{E}\left[ \frac 1{\|\beta_1\|}\right] \, ds+ o\left(\frac 1k\right), \end{align*} with the same constant $c_0$, as in \eqref{cov.3.sum}. We next handle the sum of the first terms in \eqref{cov.3.sum} and show that its difference with the sum from the previous display is negligible. Indeed, observe already that with $\chi_k := k/(\eta\varepsilon_k)$,
$$\sum_{j=\varepsilon_k}^{\lfloor \eta k\rfloor} \mathbb{E}\left[\frac{G(S_j){\text{\Large $\mathfrak 1$}}\{\|S_j\|\ge \eta^{1/4} \sqrt k\}}{1+\mathcal{J}(S_k)}\right]+ \mathbb{E}\left[\frac{G(S_j){\text{\Large $\mathfrak 1$}}\{\|S_k\|\ge \sqrt{k\chi_k}\}}{1+\mathcal{J}(S_k)}\right] \lesssim \frac{\eta^{1/4}}{k}. $$ Thus one has, using Theorem \ref{LCLT}, \begin{align}\label{eta.tau1.bis}
&\sum_{j=\varepsilon_k}^{\lfloor \eta k\rfloor}\left| \mathbb{E}\left[\frac{G(S_j)}{1+\mathcal{J}(S_k)}\right] - \mathbb{E}[G(S_j)]\cdot \mathbb{E}\left[\frac{1}{1+\mathcal{J}(S_k)}\right]\right| \\
\nonumber & \lesssim \sum_{j=\varepsilon_k}^{\lfloor \eta k\rfloor} \sum_{\underset{\|x\|\le \eta^{1/4}\sqrt{k}}{\|z\|\le \sqrt{k\chi_k}} } \frac{p_j(x)G(x)}{1+\|z\|} \left|\overline p_{k-j}(z-x) +\overline p_{k-j}(z+x) -2 \overline p_k(z)\right| + \frac {\eta^{1/4}}k \lesssim \frac {\eta^{1/4}}k. \end{align} Define now for $s\in (0,1]$,
$$ H_s: = \mathbb{E}\left[\frac 1{\|\beta_s\|^3 \|\beta_1\|}\right] - \mathbb{E}\left[\frac 1{\|\beta_s\|^3}\right] \cdot \mathbb{E}\left[\frac 1{\|\beta_1\|}\right].$$ Let $f_s(\cdot)$ be the density of $\beta_s$ and notice that as $s\to 0$, \begin{align*}
& H_s=\int_{\mathbb{R}^5} \int_{\mathbb{R}^5} \frac { f_s(x) f_{1-s}(y)}{\|x\|^3 \|x+y\|} \, dx\, dy - \int_{\mathbb{R}^5} \int_{\mathbb{R}^5} \frac { f_s(x) f_1(y)}{\|x\|^3 \|y\|} \, dx\, dy \\
& = \frac {1}{s^{3/2}} \int_{\mathbb{R}^5} \int_{\mathbb{R}^5} \frac{f_1(x)f_1(y)}{\|x\|^3} \left(
\frac {1}{\|y\sqrt{1-s} +x \sqrt s \| } -\frac {1}{\|y\|}\right) \, dx\, dy\\
& = \frac {1}{s^{3/2}} \int_{\mathbb{R}^5} \int_{\mathbb{R}^5} \frac{f_1(x)f_1(y)}{\|x\|^3\|y\|}
\left\{\left(\frac 12 + \frac{\|x\|^2}{2\|y\|^2} + \frac{\langle x,y\rangle^2}{\|y\|^4} \right)s + \mathcal{O}(s^{3/2}) \right\}\, dx\, dy =\frac c{\sqrt{s}} + \mathcal{O}(1), \end{align*} with $c>0$. Thus the map $s\mapsto H_s$ is integrable at $0$, and since it is also continuous on $(0,1]$, its integral on this interval is well defined. Since $\eta$ can be taken arbitrarily small in \eqref{eta.tau1} and \eqref{eta.tau1.bis}, in order to finish the proof it just remains to show that the integral of $H_s$ on $(0,1]$ is positive.
To this end, note first that $\widetilde \beta_{1-s}:= \beta_1 - \beta_s$ is independent of $\beta_s$. We use then \eqref{claim.geom}, which implies,
with $q=\mathbb{E}[1/\|\beta_1\|^3]$, \begin{align*}
& \mathbb{E}\left[\frac 1{\|\beta_s\|^3 \|\beta_1\|}\right] = \mathbb{E}\left[\frac 1{\|\beta_s\|^3 \|\beta_s + \widetilde \beta_{1-s}\|}\right]
\ge \mathbb{E}\left[\frac 1{\|\beta_s\|^3 \sqrt{\|\beta_s\|^2 + \|\widetilde \beta_{1-s}\|^2}}\right]\\
& = \frac{(5q)^2}{s^{3/2}} \int_0^\infty \int_0^\infty \frac{r e^{-\frac 52 r^2} u^4 e^{-\frac 52u^2}}{\sqrt{sr^2 + (1-s)u^2}} \, dr\, du\\
& = \frac{q^2}{5s^{3/2}} \int_0^\infty \int_0^\infty \frac{r e^{-\frac{r^2}2} u^4 e^{-\frac{u^2}2}}{\sqrt{sr^2 + (1-s)u^2}} \, dr\, du.
\end{align*}
We split the double integral in two parts, one on the set $\{sr^2\le (1-s)u^2\}$, and the other one on the complementary set $\{sr^2\ge (1-s)u^2\}$.
Call respectively $I_s^1$ and $I_s^2$ the integrals on these two sets. For $I_s^1$, \eqref{lower.sqrt} gives
\begin{align*}
I_s^1 & \ge \frac 1{\sqrt{1-s}} \int_0^\infty u^3 e^{-\frac{u^2}2} \int_0^{\sqrt{\frac{1-s}{s}}u} r e^{-\frac{r^2}2}\, dr \, du\\ & \qquad - \frac{s}{2(1-s)^{3/2}} \int_0^\infty u e^{-\frac{u^2}2} \int_0^{\sqrt{\frac{1-s}{s}}u} r^3 e^{-\frac{r^2}2}\, dr \, du\\ & = \frac{2(1-s^2)}{\sqrt{1-s}} + \frac{s^2}{\sqrt{1-s}} - \frac{s}{\sqrt{1-s}}= \frac{2-s - s^2}{\sqrt{1-s}}.
\end{align*} For $I_s^2$ we simply use the rough bound: $$I_s^2 \ge \frac 1{\sqrt{2s}} \int_0^\infty \int_0^\infty e^{-\frac{r^2}2} u^4 e^{-\frac{u^2}2}{\text{\Large $\mathfrak 1$}}\{sr^2\ge (1-s)u^2\} \, dr \, du, $$ which entails \begin{align*} & \int_0^1 \frac{I_s^2}{s^{3/2}}\, ds \ge \frac 1{\sqrt 2} \int_0^\infty \int_0^\infty e^{-\frac{r^2}2} u^4 e^{-\frac{u^2}2} \left(\int_{\frac{u^2}{u^2+r^2}}^1 \frac 1{s^2}\, ds\right)\, dr \, du \\ & = \frac 1{\sqrt 2} \left(\int_0^\infty r^2 e^{-\frac{r^2}2} \, dr\right)^2= \frac 1{\sqrt 2} \left(\int_0^\infty e^{-\frac{r^2}2}\, dr\right)^2 =\frac{\pi}{2\sqrt 2}>1, \end{align*} where for the last inequality we use $\sqrt{2}<3/2$. Note now that
$$\mathbb{E}\left[\frac 1{\|\beta_s\|^3}\right] \cdot \mathbb{E}\left[\frac 1 {\|\beta_1\|}\right] = \frac {2q^2}{5s^{3/2}} , $$
and \begin{align*} & \int_0^1 \frac{I_s^1 - 2}{s^{3/2}}\, ds \ge \int_0^1 s^{-3/2} \left\{(2- s- s^2)(1+\frac s2 + \frac{3s^2}{8}) - 2\right\} \, ds \\ & = - \int_0^1 (\frac 34 \sqrt s + \frac 78 s^{3/2} +\frac {3}{8} s^{5/2} ) \, ds = - (\frac 12 + \frac 7{20} + \frac{3}{28}) = - \frac{134}{140} > -1. \end{align*} Altogether this shows that the integral of $H_s$ on $(0,1]$ is well positive as wanted. This concludes the proof of the lemma.
$\square$
\begin{remark}\emph{The value of $H_1$ can be computed explicitely and one can check that it is positive. Similarly, by computing the leading order term in $I_s^2$, we could show that $H_s$ is also positive in a neighborhood of the origin, but it would be interesting to know whether $H_s$ is positive for all $s\in (0,1)$. } \end{remark}
\subsection{Proof of Lemma \ref{lem.var.4}} We define here $$\tau_1:= \inf\{n\ge 0 : S_n^1\in \mathcal{R}[\varepsilon_k,k-\varepsilon_k]\},\ \tau_2:=\inf\{n\ge 0 : S_k+S_n^2\in \mathcal{R}[\varepsilon_k,k-\varepsilon_k]\},$$ with $S^1$ and $S^2$ two independent walks, independent of $S$. As in the previous lemma, we omit the details of the fact that $$\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_2)= \rho^2\cdot \operatorname{Cov}({\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\},{\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}) +o\left(\frac 1k\right).$$ Then we define $N:=\lfloor (k-3\varepsilon_k)/\varepsilon_k\rfloor$ and let $(\tau_1^i)_{i=1,\dots,N}$ be as in the proof of Lemma \ref{lem.var.3}. Define also $(\tau_2^i)_{i=1,\dots,N}$ analogously. Similarly as before one can see that \begin{eqnarray}\label{tau1i.tau2j} \mathbb{P}[\tau_1<\infty, \tau_2<\infty ]= \sum_{i=1}^{N} \sum_{j=1}^{N} \mathbb{P}[\tau_1^i <\infty, \tau_2^j <\infty ] + o\left(\frac 1k\right). \end{eqnarray}
Note also that for any $i$ and $j$, with $|i-j| \le 1$, by \eqref{Green.hit} and \eqref{exp.Green}, $$\mathbb{P}[\tau_1^i<\infty, \, \tau_2^j<\infty] = \mathcal{O}\left(\frac {\varepsilon_k^{2(1-\delta)}}{k_i^{3/2} (k-k_i)^{3/2} }\right), $$
so that in \eqref{tau1i.tau2j}, one can consider only the sum on the indices $i$ and $j$ satisfying $|i-j|\ge 2$. Furthermore, when $i<j$, the events $\{\tau_1^i<\infty\}$ and $\{\tau_2^j<\infty\}$ are independent. Thus altogether this gives \begin{align*}
\operatorname{Cov}( &{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}, {\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}) \\
& = \sum_{i = 1}^{N-2} \sum_{j=i+2}^N \left( \mathbb{P}[ \tau_1^j <\infty, \tau_2^i<\infty] - \mathbb{P}[\tau_1^j<\infty] \mathbb{P}[\tau_2^i<\infty] \right) + o\left(\frac 1k\right). \end{align*} Then by following carefully the same steps as in the proof of the previous lemma we arrive at $$\operatorname{Cov}({\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}, \, {\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}) = \frac {c}{k} \int_0^1 \widetilde H_t\, dt + o\left(\frac 1k\right), $$ with $c>0$ some positive constant and,
$$\widetilde H_t := \int_0^t \left(\mathbb{E}\left[\frac 1{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right] - \mathbb{E}\left[\frac 1{\|\beta_s-\beta_1\|^3}\right] \cdot\mathbb{E}\left[\frac 1{ \|\beta_t\|^3} \right] \right) \, ds,$$ at least provided we show first that $\widetilde H_t$ it is well defined and that its integral over $[0,1]$ is convergent.
However, observe that for any $t\in (0,1)$, one has with $q=\mathbb{E}[\|\beta_1\|^{-3}]$,
$$\int_0^t \mathbb{E}\left[\frac 1{\|\beta_s-\beta_1\|^3}\right] \cdot\mathbb{E}\left[\frac 1{ \|\beta_t\|^3} \right] =\frac{q^2}{t^{3/2}} \int_0^t \frac 1{(1-s)^{3/2}}\, ds = \frac{2q^2(1-\sqrt{1-t})}{t^{3/2}\sqrt{1-t}},$$ and therefore this part is integrable on $[0,1]$. This implies in fact that the other part in the definition of $\widetilde H_t$ is also well defined and integrable, since we already know that $\operatorname{Cov}({\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}, \, {\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\})=\mathcal{O}(1/k)$. Thus it only remains to show that the integral of $\widetilde H_t$ on $[0,1]$ is positive. To this end, we write $\beta_t = \beta_s + \gamma_{t-s}$, and $\beta_1 = \beta_s + \gamma_{t-s} + \delta_{1-t}$, with $(\gamma_u)_{u\ge 0}$ and $(\delta_u)_{u\ge 0}$
two independent Brownian motions, independent of $\beta$. Furthermore, using that the map $z\mapsto 1/\|z\|^3$ is harmonic outside the origin, we can compute: \begin{align*}
& I_1:= \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|\beta_s\|\ge \|\gamma_{t-s}\|\ge \|\delta_{1-t}\| \} }{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right] = \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|\beta_s\|\ge \|\gamma_{t-s}\|\ge \|\delta_{1-t}\| \} }{\|\gamma_{t-s} + \delta_{1-t}\|^3 \cdot \|\beta_s\|^3} \right] \\
= & \frac{5q}{s^{3/2}} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|\gamma_{t-s}\|\ge \|\delta_{1-t}\| \} }{\|\gamma_{t-s} + \delta_{1-t}\|^3} \int_{\frac{\|\gamma_{t-s}\|}{\sqrt s}}^\infty re^{-\frac 52 r^2} \, dr\right] = \frac{q}{s^{3/2}} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|\gamma_{t-s}\|\ge \|\delta_{1-t}\| \}}{\|\gamma_{t-s} + \delta_{1-t}\|^3} e^{-\frac 5{2s} \|\gamma_{t-s}\|^2} \right] \\
=& \frac{q}{s^{3/2}} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|\gamma_{t-s}\|\ge \|\delta_{1-t}\| \}}{\|\gamma_{t-s}\|^3} e^{-\frac 5{2s} \|\gamma_{t-s}\|^2} \right] = \frac{5q^2}{s^{3/2}(t-s)^{3/2}} \mathbb{E}\left[\int_{\frac {\|\delta_{1-t}\|}{\sqrt{t-s}}}^\infty r e^{-\frac 52 r^2 (1+ \frac {t-s}{s})}\, dr \right] \\
=& \frac{q^2}{\sqrt{s}(t-s)^{3/2}t}\mathbb{E}\left[ e^{- \frac{\|\delta_{1-t}\|^2t}{s(t-s)}}\right] = \frac{5q^3}{\sqrt{s}(t-s)^{3/2}t} \int_0^\infty r^4 e^{-\frac 52r^2(1+ \frac{t(1-t)}{s(t-s)})}\, dr = \frac{q^2s^2(t-s)}{t\, \Delta^{5/2}}, \end{align*} with $$\Delta := t(1-t) + s(t-s) = (1-t)(t-s) + s(1-s).$$ Likewise, \begin{align*}
I_2&:= \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|\beta_s\|\ge\|\gamma_{t-s}\|,\, \|\delta_{1-t}\| \ge \|\gamma_{t-s}\| \} }{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right] = \frac{q}{s^{3/2}} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|\gamma_{t-s}\|\le \|\delta_{1-t}\| \}}{\|\gamma_{t-s} + \delta_{1-t}\|^3} e^{-\frac 5{2s} \|\gamma_{t-s}\|^2} \right] \\
=& \frac{q}{s^{3/2}} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|\gamma_{t-s}\|\le \|\delta_{1-t}\| \}}{\|\delta_{1-t}\|^3} e^{-\frac 5{2s} \|\gamma_{t-s}\|^2} \right]
= \frac{5q^2}{s^{3/2}(1-t)^{3/2}} \mathbb{E}\left[ e^{-\frac 5{2s} \|\gamma_{t-s}\|^2} \int_{\frac {\|\gamma_{t-s}\|}{\sqrt{1-t}}}^\infty r e^{-\frac 52 r^2}\, dr \right] \\
=& \frac{q^2}{s^{3/2}(1-t)^{3/2}} \mathbb{E}\left[ e^{-\frac 5{2} \|\gamma_{t-s}\|^2(\frac 1s + \frac {1}{1-t})}\right] = \frac{q^2s(1-t)}{\Delta^{5/2}}. \end{align*} Define as well \begin{align*}
I_3 & := \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|\beta_s\|\le \|\gamma_{t-s}\|\le \|\delta_{1-t}\| \} }{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right], \end{align*} \begin{align*}
I_4:=\mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{ \|\delta_{1-t}\|\le\|\beta_s\|\le \|\gamma_{t-s}\| \} }{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right], \ I_5:= \mathbb{E}\left[\frac { {\text{\Large $\mathfrak 1$}}\{\|\beta_s\|\le \|\delta_{1-t}\|\le \|\gamma_{t-s}\| \} }{ \|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right]. \end{align*} Note that by symmetry one has $$\int_{0\le s\le t \le 1} I_1 \, ds \, dt = \int_{0\le s\le t \le 1} I_3 \, ds \, dt,\text{ and } \int_{0\le s\le t \le 1} I_4 \, ds \, dt = \int_{0\le s\le t \le 1} I_5 \, ds \, dt. $$ Observe also that, $$I_1+I_2 = \frac{q^2s}{t \Delta^{3/2}}. $$ Moreover, using symmetry again, we can see that $$\int_0^t \frac {s-t/2}{ \Delta^{3/2}} \, ds = 0,$$ and thus $$\int_0^t (I_1+I_2)\, ds = \frac {q^2}{2} \int_0^t \frac{1}{ \Delta^{3/2}}\, ds. $$ Likewise, \begin{align*} & \int_{0\le s\le t\le 1} I_1\, ds \, dt = \int_{0\le s\le t\le 1} \frac{q^2s(t-s)^2}{t\Delta^{5/2}}\, ds \, dt = \frac 12\int_{0\le s\le t\le 1} \frac{q^2s(t-s)}{\Delta^{5/2}}\, ds \, dt \\
=& \int_{0\le s\le t\le 1} \frac{q^2(1-t)(t-s)}{2\Delta^{5/2}}\, ds \, dt = \int_{0\le s\le t\le 1} \frac {q^2t(1-t)}{4\Delta^{5/2}}\, ds\, dt = \int_{0\le s\le t\le 1} \frac {q^2}{6\Delta^{3/2}}\, ds\, dt. \end{align*} It follows that $$\int_{0\le s\le t\le 1} (I_1+I_2+I_3) \, ds\, dt = \frac {2q^2}3 \int_{0\le s\le t\le 1} \Delta^{-3/2}\, ds\, dt.$$ We consider now the term $I_4$, which is a bit more complicated to compute, thus we only give a lower bound on a suitable interval. To be more precise, we first define for $r\ge 0$ and $\lambda\ge 0$, $$F(r):=\int_0^r s^4e^{-5s^2/2}\, ds,\quad \text{and}\quad F_2(\lambda, r):=\int_0^r F(\lambda s) s^4e^{-5s^2/2}\, ds,$$ and then we write, \begin{align*}
& I_4= \mathbb{E}\left[ \frac { {\text{\Large $\mathfrak 1$}}\{ \|\delta_{1-t}\|\le\|\beta_s\|\le \|\gamma_{t-s}\| \} }{ \|\gamma_{t-s}\|^6} \right]
= 5q\cdot \mathbb{E}\left[\frac { {\text{\Large $\mathfrak 1$}}\{ \|\beta_s\|\le \|\gamma_{t-s}\| \} }{ \|\gamma_{t-s}\|^6} F\left(\frac{\|\beta_s\|}{\sqrt{1-t}}\right)\right]\\
=& \mathbb{E}\left[\frac {(5q)^2}{ \|\gamma_{t-s}\|^6} F_2\left(\frac{\sqrt s}{\sqrt{1-t}},\frac{\|\gamma_{t-s}\|}{\sqrt s}\right)\right] = \frac{(5q)^3}{(t-s)^3} \int_0^\infty \frac{e^{-\frac{5r^2}{2}}}{r^2} F_2\left(\frac{\sqrt s}{\sqrt{1-t}},r\frac{\sqrt{t-s}}{\sqrt s}\right)\, dr \\
=& \frac{(5q)^3}{(t-s)^3} \left\{ \frac{\sqrt{t-s}}{\sqrt s} \int_0^\infty F\left(r\frac{\sqrt{t-s}}{\sqrt{1-t}}\right) r^3e^{-\frac{5r^2}{2}}\, dr -5\int_0^\infty F_2\left(\frac{\sqrt s}{\sqrt{1-t}},r\frac{\sqrt{t-s}}{\sqrt s}\right) e^{-\frac{5r^2}{2}}\, dr \right\}\\
\ge & \frac{(5q)^3}{(t-s)^3} \left\{ \frac{(t-s)^{\frac 32}}{s^{3/2}} \int_0^\infty F\left( r\frac{\sqrt{t-s}}{\sqrt {1-t}}\right) r^3 e^{-\frac {5r^2t}{2s}}\, dr +\frac{(2s-t)\sqrt{t-s}}{s^{3/2}} \int_0^\infty F\left(r\frac{\sqrt{t-s}}{\sqrt{1-t}}\right) r^3e^{-\frac {5r^2}{2}}\, dr \right\}, \end{align*} using that $$F_2(\lambda,r)\le \frac 15 r^3F(\lambda r)(1-e^{-5r^2/2}).$$ Therefore, if $t/2\le s\le t$, \begin{align*} I_4& \ge \frac{(5q)^3}{[s(t-s)]^{3/2}} \int_0^\infty r^3 F\left( r\frac{\sqrt{t-s}}{\sqrt {1-t}}\right) e^{-\frac {5r^2t}{2s}}\, dr \\ & = \frac{(5q)^3\sqrt s}{t^2(t-s)^{3/2}} \int_0^\infty r^3 F\left( r\frac{\sqrt{s(t-s)}}{\sqrt {t(1-t)}}\right) e^{-5r^2/2}\, dr\\ & \ge \frac{2\cdot 5^2q^3\sqrt s}{t^2(t-s)^{3/2}}\int_0^\infty F\left( r\frac{\sqrt{s(t-s)}}{\sqrt {t(1-t)}}\right) re^{-\frac{5r^2}{2}}\, dr =\frac{2\cdot 5q^3s^3(t-s)}{t^2[t(1-t)]^{5/2}}\int_0^\infty r^4 e^{-\frac{5r^2\Delta}{2t(1-t)}}\, dr\\ & = \frac{2 q^2s^3(t-s)}{t^2\Delta^{5/2}}\ge \frac{q^2s(t-s)}{2\Delta^{5/2}}, \end{align*} and as a consequence, \begin{align*} \int_{0\le s\le t\le 1} I_4\, ds\, dt & \ge \int_{t/2\le s\le t\le 1}I_4\, ds\, dt \ge \frac {q^2}2 \int_{t/2\le s\le t\le 1}\, \frac{s(t-s)}{\Delta^{5/2}}\, ds\, dt \\ &=\frac {q^2}4 \int_{0\le s\le t\le 1}\frac{s(t-s)}{\Delta^{5/2}}\, ds\, dt = \frac {q^2}{12} \int_{0\le s\le t\le 1}\Delta^{-3/2}\, ds\, dt. \end{align*} Putting all these estimates together yields
$$\int_{0\le s\le t\le 1} \mathbb{E}\left[\frac 1{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right] \, ds\, dt = \sum_{k=1}^5 \int_{0\le s\le t\le 1} I_k\, ds\, dt \ge \frac 56 \int_{0\le s\le t\le 1} \Delta^{-3/2}\, ds\,dt.$$ Thus it just remains to show that \begin{eqnarray}\label{finalgoal} \int_{0\le s\le t\le 1} \Delta^{-3/2}\, ds\,dt \ge \frac 65 \int_{0\le s\le t\le 1} \widetilde \Delta^{-3/2}\, ds\,dt, \end{eqnarray} where $\widetilde \Delta := t(1-s)$. Note that $\Delta = \widetilde \Delta +(t-s)^2$. Recall also that for any $\alpha \in \mathbb{R}$, and any $x\in (-1,1)$, \begin{eqnarray}\label{DL1+x} (1+x)^\alpha = 1+ \sum_{i\ge 1}\frac{\alpha (\alpha-1)\dots(\alpha-i+1)}{i!} x^i. \end{eqnarray} Thus $$\frac 1{\Delta^{3/2}} = \frac 1{\widetilde \Delta^{3/2}} \left(1+\sum_{k\ge 1} \frac{(3/2)(5/2)\dots (k+1/2)}{k!} \cdot \frac{(t-s)^{2k}}{\widetilde \Delta^k}\right).$$ One needs now to compute the coefficients $C_k$ defined by $$C_k := \frac{(3/2)(5/2)\dots (k+1/2)}{k!} \int_{0\le s\le t\le 1} \frac{(t-s)^{2k}}{\widetilde \Delta^{k+3/2}}\, ds\,dt.$$ We claim that one has for any $k\ge 0$, \begin{eqnarray}\label{Ckformula} C_k= \frac{2^{2k+2}}{2k+1}(-1)^k \Sigma_k, \end{eqnarray} with $\Sigma_0=1$, and for $k\ge 1$, $$\Sigma_k = 1+ \sum_{i=1}^{2k}(-1)^i \frac{(k+1/2)(k-1/2)\dots(k-i +3/2)}{i!}.$$ We will prove this formula in a moment, but let us conclude the proof of the lemma first, assuming it is true. Straightforward computations show by \eqref{Ckformula} that $$C_0 = 4,\quad C_1= \frac 23, \quad \text{and}\quad C_2= \frac {3}{10},$$ and $C_0+C_1+C_2\ge 6C_0/5$, gives \eqref{finalgoal} as wanted.
So let us prove \eqref{Ckformula} now. Note that one can assume $k\ge 1$, as the result for $k=0$ is immediate. By \eqref{DL1+x}, one has $$(1-s)^{-k-3/2}= 1+ \sum_{i\ge 1} \frac{(k+3/2)(k+5/2)\dots (k+ i +1/2)}{i!} s^i.$$ Thus by integrating by parts, we get $$\int_0^t \frac{(t-s)^{2k}}{(1-s)^{k+3/2}} \, ds = (2k)! \sum_{i\ge 0} \frac{(k+3/2)\dots(k+i+1/2)}{(2k + i +1)!}\cdot t^{2k+i+1},$$ and then $$\int_0^1 \int_0^t \frac{(t-s)^{2k}}{t^{k+3/2}(1-s)^{k+3/2}} \, ds\, dt = (2k)! \sum_{i\ge 0} \frac{(k+3/2)\dots(k+i-1/2)}{(2k + i +1)!}.$$ As a consequence, \begin{align*} C_k& = \frac{(2k)!}{k!} \sum_{i\ge 0} \frac{(3/2)(5/2)\dots(k+i-1/2)}{(2k+i+1)!} \\
& = \frac{(2k)!}{(k+1/2)(k-1/2)\dots(3/2)(1/2)^2\cdot k!} \sum_{i\ge 0} \frac{|(k+1/2)(k-1/2)\dots(-k-i+1/2)|}{(2k + i +1)!} \\
& = \frac{2^{2k+2}}{2k+1} \sum_{i\ge 0} \frac{|(k+1/2)(k-1/2)\dots(-k-i+1/2)|}{(2k + i +1)!}, \end{align*} and it just remains to observe that the last sum is well equal to $\Sigma_k$. The latter is obtained by taking the limit as $t$ goes to $1$ in the formula \eqref{DL1+x} for $(1-t)^{k+1/2}$. This concludes the proof of Lemma \ref{lem.var.4}.
$\square$
\begin{remark}
\emph{It would be interesting to show that the covariance between $1/\|\beta_s-\beta_1\|^3$ and $1/\|\beta_t\|^3$ itself is positive for all $0\le s\le t\le 1$, and not just its integral, as we have just shown. } \end{remark}
\section{Proof of Theorem B} The proof of Theorem B is based on the Lindeberg-Feller theorem for triangular arrays, that we recall for convenience (see Theorem 3.4.5 in \cite{Dur}): \begin{theorem}[Lindeberg-Feller]\label{thm:lind} For each $n$ let $(X_{n,i}: \, 1\leq i\leq n)$ be a collection of independent random variables with zero mean. Suppose that the following two conditions are satisfied \newline {\rm{(i)}} $\sum_{i=1}^{n}\mathbb{E}[X_{n,i}^2] \to \sigma^2>0$ as $n\to \infty$, and \newline
{\rm{(ii)}} $\sum_{i=1}^{n}\mathbb{E}\left[(X_{n,i})^2{\text{\Large $\mathfrak 1$}}\{|X_{n,i}|>\varepsilon\}\right] \to 0$, as $n\to \infty$, for all $\varepsilon>0$. \newline Then, $S_n=X_{n,1}+\ldots + X_{n,n} \Longrightarrow \mathcal{N}(0,\sigma^2)$, as $n\to \infty$. \end{theorem} In order to apply this result, one needs three ingredients. The first one is an asymptotic estimate for the variance of the capacity of the range, which is given by our Theorem A. The second ingredient is a decomposition of the capacity of two sets as a sum of the capacities of the two sets minus some error term, in the spirit of the inclusion-exclusion formula for the cardinality of a set, which allows to decompose the capacity of the range up to time $n$ into a sum of independent pieces having the law of the capacity of the range up to a smaller time index, and finally the last ingredient is a sufficiently good bound on the centered fourth moment.
This strategy has been already employed successfully for the capacity of the range in dimension six and more in \cite{ASS18} (and for the size of the range as well, see \cite{JO69, JP71}). In this case the asymptotic of the variance followed simply from a sub-additivity argument, but the last two ingredients are entirely similar in dimension $5$ and in higher dimension. In particular one has the following decomposition (see Proposition 1.6 in \cite{ASS19}): for any two subsets $A,B\subset \mathbb{Z}^d$, $d\ge 3$, \begin{eqnarray}\label{cap.decomp} \mathrm{Cap}(A\cup B) = \mathrm{Cap}(A) + \mathrm{Cap}(B) - \chi(A,B), \end{eqnarray} where $\chi(A,B)$ is some error term. Its precise expression is not so important here. All one needs to know is that
$$|\chi(A,B)| \le 3\sum_{x\in A}\sum_{y\in B} G(x,y),$$ so that by \cite[Lemma 3.2]{ASS18}, if $\mathcal{R}_n$ and $\widetilde \mathcal{R}_n$ are the ranges of two independent walks in $\mathbb{Z}^5$, then \begin{eqnarray}\label{bound.chin} \mathbb{E}[\chi(\mathcal{R}_n,\widetilde \mathcal{R}_n)^4] = \mathcal{O}(n^2). \end{eqnarray} We note that the result is shown for the simple random walk only in \cite{ASS18}, but the proof applies as well to our setting (in particular Lemma 3.1 thereof also follows from \eqref{exp.Green}). Now as noticed already by Le Gall in his paper \cite{LG86} (see his remark (iii) p.503), a good bound on the centered fourth moment follows from \eqref{cap.decomp} and \eqref{bound.chin}, and the triangle inequality in $L^4$. More precisely in dimension $5$, one obtains (see for instance the proof of Lemma 4.2 in \cite{ASS18} for some more details): \begin{eqnarray}\label{cap.fourth} \mathbb{E}\left[\left(\mathrm{Cap}(\mathcal{R}_n)-\mathbb{E}[\mathrm{Cap}(\mathcal{R}_n)]\right)^4\right] = \mathcal{O}(n^2(\log n)^4). \end{eqnarray} Actually we would even obtain the slightly better bound $\mathcal{O}(n^2(\log n)^2)$, using our new bound on the variance $\operatorname{Var}(\mathrm{Cap}(\mathcal{R}_n))=\mathcal{O}(n\log n)$, but this is not needed here. Using next a dyadic decomposition of $n$, one can write with $T:=\lfloor n/(\log n)^4\rfloor$, \begin{eqnarray}\label{cpRn}
\mathrm{Cap}(\mathcal{R}_n) = \sum_{i=0}^{\lfloor n/T\rfloor} \mathrm{Cap}(\mathcal{R}^{(i)}_T) - R_n,
\end{eqnarray} where the $(\mathcal{R}^{(i)}_T)_{i=0,\dots,n/T}$ are independent pieces of the range of length either $T$ or $T+1$, and $$ R_n= \sum_{\ell =1}^L \sum_{i=0}^{2^{\ell-1}} \chi(\mathcal{R}^{(2i)}_{n/2^\ell},\mathcal{R}^{(2i+1)}_{n/2^\ell}), $$ is a triangular array of error terms (with $L=\log_2(\log n)^4$). Then it follows from \eqref{bound.chin}, that \begin{align*} \operatorname{Var}(R_n) &\le L \sum_{\ell=1}^L \operatorname{Var}\left(\sum_{i=1}^{2^{\ell-1}} \chi(\mathcal{R}^{(2i)}_{n/2^\ell},\mathcal{R}^{(2i+1)}_{n/2^\ell})\right)\le L\sum_{\ell=1}^L \sum_{i=1}^{2^{\ell-1}} \operatorname{Var}\left(\chi(\mathcal{R}^{(2i)}_{n/2^\ell},\mathcal{R}^{(2i+1)}_{n/2^\ell})\right)\\ & = \mathcal{O}(L^2 n)=\mathcal{O}(n(\log \log n)^2). \end{align*} In particular $(R_n-\mathbb{E}[R_n])/\sqrt{n\log n}$ converges in probability to $0$. Thus one is just led to show the convergence in law of the remaining sum in \eqref{cpRn}. For this, one can apply Theorem \ref{thm:lind}, with $$X_{n,i}:=\frac{\mathrm{Cap}(\mathcal{R}^{(i)}_T)-\mathbb{E}\left[\mathrm{Cap}(\mathcal{R}^{(i)}_T)\right]}{\sqrt{n\log n}}.$$ Indeed, Condition (i) of the theorem follows from Theorem A, and Condition (ii) follows from \eqref{cap.fourth} and Markov's inequality (more details can be found in \cite{ASS18}). This concludes the proof of Theorem B.
$\square$
\section*{Acknowledgments} We thank Fran{\c c}oise P\`ene for enlightening discussions at an early stage of this project, and Pierre Tarrago for Reference \cite{Uchiyama98}. We also warmly thank Amine Asselah and Perla Sousi for our many discussions related to the subject of this work, which grew out of it. The author was supported by the ANR SWIWS (ANR-17-CE40-0032) and MALIN (ANR-16-CE93-0003).
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Homomorphisms of planar $(m, n)$-colored-mixed graphs to planar targets}
\begin{abstract} An $(m, n)$-colored-mixed graph $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an $(m,n)$-colored-mixed graph $G$ to another $(m, n)$-colored-mixed graph $H$ is a morphism $\varphi:V(G)\rightarrow V(H)$ such that each edge (resp. arc) of $G$ is mapped to an edge (resp. arc) of $H$ of the same color (and orientation). An $(m,n)$-colored-mixed graph $T$ is said to be $P_g^{(m, n)}$-universal if every graph in $P_g^{(m, n)}$ (the planar $(m, n)$-colored-mixed graphs with girth at least $g$) admits a homomorphism to $T$.
We show that planar $P_g^{(m, n)}$-universal graphs do not exist for $2m+n\ge3$ (and any value of $g$) and find a minimal (in the number vertices) planar $P_g^{(m, n)}$-universal graphs in the other cases. \end{abstract}
\section{Introduction}
The concept of homomorphisms of $(m, n)$-colored-mixed graph was introduced by J. Nes\v{e}t\v{r}il and A. Raspaud~\cite{MNCM} in order to generalize homomorphisms of $k$-edge-colored graphs and oriented graphs.
An \emph{$(m, n)$-colored-mixed graph} $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not allow two arcs or edges to have the same endpoints. The case $m=0$ and $n=1$ corresponds to simple graphs, $m=1$ and $n=0$ to oriented graphs and $m=0$ and $n=k$ to $k$-edge-colored graphs. For the case $m=0$ and $n = 2$ ($2$-edge-colored graphs) we refer to the two types of edges as \emph{blue} and \emph{red} edges.
A \emph{homomorphism} from an $(m, n)$-colored-mixed graph $G$ to another $(m, n)$-colored-mixed graph $H$ is a mapping $\varphi:V(G) \rightarrow V(H)$ such that every edge (resp. arc) of $G$ is mapped to an edge (resp. arc) of $H$ of the same color (and orientation). If $G$ admits a homomorphism to $H$, we say that $G$ is \emph{$H$-colorable} since this homomorphism can be seen as a coloring of the vertices of $G$ using the vertices of $H$ as colors. The edges and arcs of $H$ (and their colors) give us the rules that this coloring must follow. Given a class of graphs $\mathcal{C}$, a graph is \emph{$\mathcal{C}$-universal} if for every graph $G \in \mathcal{C}$ is $H$-colorable. The class $P_g^{(m, n)}$ contains every planar $(m, n)$-colored-mixed graph with girth at least $g$.
In this paper, we consider some planar $P_g^{(m, n)}$-universal graphs with $k$ vertices. They are depicted in Figures~\ref{fig:t_oriented} and~\ref{fig:t_2edge}. The known results about this topic are as follows.
\begin{theorem}\label{thm:known}{\ } \begin{enumerate} \item $K_4$ is a planar $P^{(0,1)}_3$-universal graph. This is the four color theorem. \item $K_3$ is a planar $P^{(0,1)}_4$-universal graph. This is Grötzsch's Theorem \cite{grotzsch}. \item $\overrightarrow{C_6^2}$ is a planar $P_{16}^{(1,0)}$-universal graph~\cite{P10}. \end{enumerate} \end{theorem}
Our first result shows that, in addition to the case of $(0,1)$-graphs covered by Theorems~\ref{thm:known}.1 and~\ref{thm:known}.2, our topic is actually restricted to the cases of oriented graphs (i.e., $(m,n)=(1,0)$) and 2-edge-colored graphs (i.e., $(m,n)=(0,2)$).
\begin{theorem}\label{thm:Pmn} For every $g\ge3$, there exists no planar $P_g^{(m,n)}$-universal graph if $2m+n\ge3$. \end{theorem}
As Theorems~\ref{thm:known}.1 and~\ref{thm:known}.2 show for $(0,1)$-graphs, there might exist a trade-off between minimizing the girth $g$ and the number of vertices of the universal graph, for a fixed pair $(m,n)$. For oriented graphs, Theorem~\ref{thm:known}.3 tries to minimize the girth. For oriented graphs and 2-edge-colored graphs, we choose instead to minimize the number of vertices of the universal graph.
\begin{theorem}\label{thm:positive}{\ } \begin{enumerate} \item $\overrightarrow{T_5}$ is a planar $P_{28}^{(1,0)}$-universal graph on 5 vertices. \item $T_6$ is a planar $P_{22}^{(0, 2)}$-universal graph on 6 vertices. \end{enumerate} \end{theorem}
The following results shows that Theorem~\ref{thm:positive} is optimal in terms of the number of vertices of the universal graph.
\begin{theorem}\label{thm:negative}{\ } \begin{enumerate} \item For every $g\ge3$, there exists an oriented bipartite cactus graph (i.e., $K_4^-$ minor-free graph) with girth at least $g$ and oriented chromatic number at least 5. \item For every $g\ge3$, there exists a 2-edge-colored bipartite outerplanar graph (i.e., $(K_4^-,K_{2,3})$ minor-free graph) with girth at least $g$ that does not map to a planar graph with at most 5 vertices. \end{enumerate} \end{theorem}
Most probably, Theorem~\ref{thm:positive} is not optimal in terms of girth. The following constructions give lower bounds on the girth.
\begin{theorem}\label{thm:ce}{\ } \begin{enumerate} \item There exists an oriented bipartite 2-outerplanar graph with girth $14$ that does not map to $\overrightarrow{T_5}$. \item There exists a 2-edge-colored planar graph with girth $11$ that does not map to $T_6$. \item There exists a 2-edge-colored bipartite planar graph with girth $10$ that does not map to $T_6$. \end{enumerate} \end{theorem}
\begin{figure}
\caption{The $P_{28}^{(1,0)}$-universal graph overrightarrow$(T_5)$.}
\label{fig:t_oriented}
\caption{The $P_{22}^{(0,2)}$-universal graph $T_6$.}
\label{fig:t_2edge}
\end{figure}
Next, we obtain the following complexity dichotomies:
\begin{theorem}\label{thm:NPC}{\ } \begin{enumerate} \item For any fixed girth $g\geqslant 3$, either every graph in $P_g^{(1,0)}$ maps to $\overrightarrow{T_5}$ or it is NP-complete to decide whether a graph in $P_g^{(1,0)}$ maps to $\overrightarrow{T_5}$. Either every bipartite graph in $P_g^{(1,0)}$ maps to $\overrightarrow{T_5}$ or it is NP-complete to decide whether a bipartite graph in $P_g^{(1,0)}$ maps to $\overrightarrow{T_5}$. \item Either every graph in $P_g^{(0,2)}$ maps to $T_6$ or it is NP-complete to decide whether a graph in $P_g^{(1,0)}$ maps to $T_6$. Either every bipartite graph in $P_g^{(0,2)}$ maps to $T_6$ or it is NP-complete to decide whether a bipartite graph in $P_g^{(1,0)}$ maps to $T_6$. \end{enumerate} \end{theorem}
Finally, we can use Theorem~\ref{thm:NPC} with the non-colorable graphs in Theorem~\ref{thm:ce}.
\begin{corollary}\label{cor:cor}{\ } \begin{enumerate} \item Deciding whether a bipartite graph in $P_{14}^{(1,0)}$ maps to $\overrightarrow{T_5}$ is NP-complete. \item Deciding whether a graph in $P_{11}^{(0,2)}$ maps to $T_6$ is NP-complete. \item Deciding whether a bipartite graph in $P_{10}^{(0,2)}$ maps to $T_6$ is NP-complete. \end{enumerate} \end{corollary}
A 2-edge-colored path or cycle is said to be \emph{alternating} if any two adjacent edges have distinct colors.
\begin{proposition}[folklore]\label{prop:3n-6}{\ } \begin{itemize} \item Every planar simple graph on $n$ vertices has at most $3n-6$ edges. \item Every planar simple graph satisfies $(\mad(G)-2)\cdot(g(G)-2)<4$. \end{itemize} \end{proposition}
\section{Proof of Theorem~\ref{thm:positive}} We use the discharging method for both results in Theorem~\ref{thm:positive}. The following lemma will handle the discharging part. We call a vertex of degree $n$ an $n$-vertex and a vertex of degree at least $n$ an $n^+$-vertex. If there is a path made only of $2$-vertices linking two vertices $u$ and $v$, we say that $v$ is a weak-neighbor of $u$. If $v$ is a neighbor of $u$, we also say that $v$ is a weak-neighbor of $u$. We call a (weak-)neighbor of degree $n$ an $n$-(weak-)neighbor.
\begin{lemma}\label{lem:discharge} Let $k$ be a non-negative integer. Let $G$ be a graph with minimum degree 2 such that every 3-vertex has at most $k$ 2-weak-neighbors and every path contains at most $\tfrac{k+1}2$ consecutive 2-vertices. Then $\mad(G)\ge2+\tfrac2{k+2}$. In particular, $G$ cannot be a planar graph with girth at least $2k+6$. \end{lemma}
\begin{proof} Let $G$ be as stated. Every vertex has an initial charge equal to its degree. Every $3^+$-vertex gives $\tfrac1{k+2}$ to each of its 2-weak-neighbors. Let us check that the final charge $ch(v)$ of every vertex $v$ is at least $2+\tfrac2{k+2}$. \begin{itemize}
\item If $d(v)=2$, then $v$ receives $\tfrac1{k+2}$ from both of its 3-weak-neighbors. Thus $ch(v)=2+\tfrac2{k+2}$.
\item If $d(v)=3$, then $v$ gives $\tfrac1{k+2}$ to each of its 2-weak-neighbors. Thus $ch(v)\ge3-\tfrac{k}{k+2}=2+\tfrac2{k+2}$.
\item If $d(v)=d\ge4$, then $v$ has at most $\tfrac{k+1}2$ 2-weak-neighbors in each of the $d$ incident paths.
Thus $ch(v)\geqslant d-d\paren{\tfrac{k+1}2}\paren{\tfrac1{k+2}}=\tfrac d2\paren{1+\tfrac1{k+2}}\ge2+\tfrac2{k+2}$. \end{itemize} This implies that $mad(G)\ge2+\frac2{k+2}$. Finally, if $G$ is planar, then the girth of $G$ cannot be at least $2k+6$, since otherwise $(\mad(G)-2)\cdot(g(G)-2)\geqslant\paren{2+\tfrac2{k+2}-2}\paren{2k+6-2}=\paren{\tfrac2{k+2}}\paren{2k+4}=4$, which contradicts Proposition~\ref{prop:3n-6}. \end{proof}
\subsection{Proof of Theorem~\ref{thm:positive}.1} We prove that the oriented planar graph $\overrightarrow{T_5}$ on 5 vertices from Figure~\ref{fig:t_oriented} is $P_{28}^{(1,0)}$-universal by contradiction. Assume that $G$ is an oriented planar graphs with girth at least $28$ that does not admit a homomorphism to $\overrightarrow{T_5}$ and is minimal with respect to the number of vertices. By minimality, $G$ cannot contain a vertex $v$ with degree at most one since a $\overrightarrow{T_5}$-coloring of $G-v$ can be extended to $G$. Similarly, $G$ does not contain the following configurations.
\begin{itemize} \item A path with 6 consecutive 2-vertices. \item A $3$-vertex with at least 12 2-weak-neighbors. \end{itemize}
Suppose that $G$ contains a path $u_0u_1u_2u_3u_4u_5u_6u_7$ such that the degree of $u_i$ is two for $1\leqslant i\le6$. By minimality of $G$, $G-{u_1,u_2,u_3,u_4,u_5,u_6}$ admits a $\overrightarrow{T_5}$-coloring $\varphi$. We checked on a computer that for any $\varphi(v_0)$ and $\varphi(v_6)$ in $V\paren{\overrightarrow{T_5}}$ and every possible orientation of the 7 arcs $u_iu_{i+1}$, we can always extend $\varphi$ into a $\overrightarrow{T_5}$-coloring of $G$, a contradiction.
Suppose that $G$ contains a 3-vertex $v$ with at least 12 2-weak-neighbors. Let $u_1$, $u_2$, $u_3$ be the $3^+$-weak-neighbors of $v$ and let $l_i$ be the number of common 2-weak-neighbors of $v$ and $u_i$, i.e., $2$-vertices on the path between $v$ and $l_i$. Without loss of generality and by the previous discussion, we have $5\geqslant l_1\geqslant l_2\geqslant l_3$ and $l_1+l_2+l_3\ge12$. So we have to consider the following cases: \begin{itemize} \item\textbf{Case 1:} $l_1=5$, $l_2=5$, $l_3=2$. \item\textbf{Case 2:} $l_1=5$, $l_2=4$, $l_3=3$. \item\textbf{Case 3:} $l_1=4$, $l_2=4$, $l_3=4$. \end{itemize}
By minimality, the graph $G'$ obtained from $G$ by removing $v$ and its 2-weak-neighbors admits a $\overrightarrow{T_5}$-coloring $\varphi$. Let us show that in all three cases, we can extend $\varphi$ into a $\overrightarrow{T_5}$-coloring of $G$ to get a contradiction.
With an extensive search on a computer we found that if a vertex $v$ is connected to a vertex $u$ colored in $\varphi(u)$ by a path made of $l$ 2-vertices ($0\leqslant l\le5$) then $v$ can be colored in:
\begin{itemize} \item at least 1 color if $l=0$, \item at least 2 colors if $l=1$, \item at least 2 colors if $l=2$ (the sets $\acc{c, d, e}$ and $\acc{b, c, d}$ are the only sets of size 3 that can be forbidden from $v$), \item at least 3 colors if $l=3$, \item at least 4 colors if $l=4$ and \item at least 4 colors if $l=5$ (only the sets $\acc{b}$, $\acc{c}$, and $\acc{e}$ can be forbidden from $v$). \end{itemize}
In Case 1, $u_3$ forbids at most 3 colors from $v$ since $l_3=2$. If it forbids less than $3$ colors, we will be able to find a color for $v$ since $u_1$ and $u_2$ forbid at most 1 color from $v$. The only sets of 3 colors that $u_3$ can forbid are $\acc{b,c,d}$ and $\acc{c, d, e}$. Since $u_1$ and $u_2$ can each only forbid $b$, $c$ or $e$, we can always find a color for $v$.
In Case 2, $u_1$ and $u_2$ each forbid at most one color and $u_3$ forbids at most $2$ colors so there remains at least one color for $v$.
In Case 3, $u_1$, $u_2$, and $u_3$ each forbid at most one color, so there remains at least two colors for $v$.
We can always extend $\varphi$ into a $\overrightarrow{T_5}$-coloring of $G$, a contradiction.
So $G$ contains at most 5 consecutive 2-vertices and every 3-vertex has at most 11 2-weak-neighbors. Using Lemma~\ref{lem:discharge} with $k=11$ contradicts the fact that the girth of $G$ is at least 28.
\subsection{Proof of Theorem~\ref{thm:positive}.2} We prove that the 2-edge-colored planar graph $T_6$ on 6 vertices from Figure~\ref{fig:t_2edge} is $P_{22}^{(0,2)}$-universal by contradiction. Assume that $G$ is a 2-edge-colored planar graphs with girth at least $22$ that does not admit a homomorphism to $T_6$ and is minimal with respect to the number of vertices. By minimality, $G$ cannot contain a vertex $v$ with degree at most one since a $T_6$-coloring of $G-v$ can be extended to $G$. Similarly, $G$ does not contain the following configurations.
\begin{itemize} \item A path with 5 consecutive 2-vertices. \item A $3$-vertex with at least 9 2-weak-neighbors. \end{itemize}
Suppose that $G$ contains a path $u_0u_1u_2u_3u_4u_5u_6$ such that the degree of $u_i$ is two for $1\leqslant i\le5$. By minimality of $G$, $G-{u_1, u_2, u_3, u_4, u_5}$ admits a $T_6$-coloring $\varphi$. We checked on a computer that for any $\varphi(v_0)$ and $\varphi(v_6)$ in $V(T)$ and every possible colors of the 6 edges $u_iu_{i+1}$, we can always extend $\varphi$ into a $T_6$-coloring of $G$, a contradiction.
Suppose that $G$ contains a 3-vertex $v$ with at least 9 2-weak-neighbors. Let $u_1$, $u_2$, $u_3$ be the $3^+$-weak-neighbors of $v$ and let $l_i$ be the number of common 2-weak-neighbors of $v$ and $u_i$, i.e., $2$-vertices on the path between $v$ and $l_i$. Without loss of generality and by the previous discussion, we have $4\geqslant l_1\geqslant l_2\geqslant l_3$ and $l_1+l_2+l_3\geqslant 9$. So we have to consider the following cases:
\begin{itemize} \item\textbf{Case 1:} $l_1=3$, $l_2=3$, $l_3=3$. \item\textbf{Case 2:} $l_1=4$, $l_2=3$, $l_3=2$. \item\textbf{Case 3:} $l_1=4$, $l_2=4$, $l_3=1$. \end{itemize}
By minimality of $G$, the graph $G'$ obtained from $G$ by removing $v$ and its 2-weak-neighbors admits a $T_6$-coloring $\varphi$. Let us show that in all three cases, we can extend $\varphi$ into a $T_6$-coloring of $G$ to get a contradiction.
With an extensive search on a computer we found that if a vertex $v$ is connected to a vertex $u$ colored in $\varphi(u)$ by a path $P$ made of $l$ 2-vertices ($0\leqslant l\leqslant 4$) then $v$ can be colored in:
\begin{itemize} \item at least 1 color if $l=0$ (the sets ${a, c, d, e, f}$ and ${b, c, d, e, f}$ of colors are the only sets of size 5 that can be forbidden from $v$ for some $\varphi(u)\in T$ and edge-colors on $P$), \item at least 2 colors if $l=1$ (the sets ${a, b, c, f}$ and ${b, c, e, f}$ are the only sets of size 4 that can be forbidden from $v$), \item at least 3 colors if $l=2$ (the sets ${b, c, f}$, ${c, e, f}$ and ${d, e, f}$ are the only sets of size 3 that can be forbidden from $v$), \item at least 4 colors if $l=3$ (the set ${c, b}$ is the only set of size 2 that can be forbidden from $v$), and \item at least 5 colors if $l=4$ (the sets ${c}$ and ${f}$ are the only sets of size 1 that can be forbidden from $v$). \end{itemize}
Suppose that we are in Case 1. Vertices $u_1$, $u_2$, and $u_3$ each forbid at most 2 colors from $v$ since $l_1=l_2=l_3=3$. Suppose that $u_1$ forbids 2 colors. It has to forbid colors $c$ and $f$ (since it is the only pair of colors that can be forbidden by a path made of 3 2-vertices). If $u_2$ or $u_3$ also forbids 2 colors, they will forbid the exact same pair of colors. We can therefore assume that they each forbid 1 color from $v$. There are 6 available colors in $T_6$, so we can always find a color for $v$ and extend $\varphi$ to a $T_6$-coloring of $G$, a contradiction. We proceed similarly for the other two cases.
So $G$ contains at most 4 consecutive 2-vertices and every 3-vertex has at most 8 2-weak-neighbors. Then Lemma~\ref{lem:discharge} with $k=8$ contradicts the fact that the girth of $G$ is at least 22.
\section{Proof of Theorem~\ref{thm:negative}.1} We construct an oriented bipartite cactus graph with girth at least $g$ and oriented chromatic number at least 5. Let $g'$ be such that $g'\geqslant g$ and $g'\equiv4\pmod{6}$. Consider a circuit $v_1,\cdots,v_{g'}$. Clearly, the oriented chromatic number of this circuit is 4 and the only tournament on 4 vertices it can map to is the tournament $\overrightarrow{T_4}$ induced by the vertices $a$, $b$, $c$, and $d$ in $\overrightarrow{T_5}$. Now we consider the cycle $C=w_1,\cdots,w_{g'}$ containing the arcs $w_{2i-1}w_{2i}$ with $1\leqslant i\leqslant g'/2$, $w_{2i+1}w_{2i}$ with $1\leqslant i\leqslant g'/2-1$, and $w_{g'}w_1$.
Suppose for contradiction that $C$ admits a homomorphism $\varphi$ such that $\varphi(w_1)=d$. This implies that $\varphi(w_2)=a$, $\varphi(w_3)=d$, $\varphi(w_4)=a$, and so on until $\varphi(w_{g'})=a$. Since $\varphi(w_{g'})=a$ and $\varphi(w_1)=d$, $w_{g'}w_1$ should map to $ad$, which is not an arc of $\overrightarrow{T_4}$, a contradiction.
Our cactus graph is then obtain from the circuit $v_1,\cdots,v_{g'}$ and $g'$ copies of $C$ by identifying every vertex $v_i$ with the vertex $w_1$ of a copy of $C$. This cactus graph does not map to $\overrightarrow{T_4}$ since one of the $v_i$ would have to map to $d$ and then the copy of $C$ attached to $v_i$ would not be $\overrightarrow{T_4}$-colorable.
\section{Proof of Theorem~\ref{thm:negative}.2} We construct a 2-edge-colored bipartite outerplanar graph with girth at least $g$ that does not map to a 2-edge-colored planar graph with at most 5 vertices. Let $g'$ be such that $g'\geqslant g$ and $g'\equiv2\pmod{4}$. Consider an alternating cycle $C=v_0,\cdots,v_{g'-1}$. For every $0\leqslant i\leqslant g'-3$, we add $g'-2$ 2-vertices $w_{i,1},\cdots,w_{i,g'-2}$ that form the path $P_i=v_iw_{i,1}\cdots w_{i,g'-2}v_{i+1}$ such that the edges of $P_i$ get the color distinct from the color of the edge $v_iv_{i+1}$. Let $G$ be the obtained graph. The 2-edge-colored chromatic number of $C$ is 5. So without loss of generality, we assume for contradiction that $G$ admits a homomorphism $\varphi$ to a 2-edge-colored planar graph $H$ on 5 vertices. Let us define $\mathcal{E}=\bigcup_{i\texttt{ even}}\varphi(v_i)$ and $\mathcal{O}=\bigcup_{i\texttt{ odd}}\varphi(v_i)$. Since $C$ is alternating, $\varphi(v_i)\ne\varphi(v_{i+2})$ (indices are modulo $g'$). Since $g'\equiv2\pmod{4}$, there is an odd number of $v_i$ with an even (resp. odd) index. Thus, $\abs{\mathcal{E}}\ge3$ and $\abs{\mathcal{O}}\ge3$. Therefore we must have $\mathcal{E}\cap\mathcal{O}\ne\emptyset$.
Notice that every two vertices $v_i$ and $v_j$ in $G$ are joined by a blue path and a red path such that the lengths of these paths have the same parity as $i-j$. Thus, the blue (resp. red) edges of $H$ must induce a connected spanning subgraph of $H$. Since $|V(H)|=5$, $H$ contains at least 4 blue (resp. red) edges. Since red and blue edges play symmetric roles in $G$ and since $|E(H)|\le9$ by Proposition~\ref{prop:3n-6}, we assume without loss of generality that $H$ contains exactly 4 blue edges. Moreover, these 4 blue edges induce a tree. In particular, the blue edges induce a bipartite graph which partitions $V(H)$ into 2 parts. Thus, every $v_i$ with even index is mapped into one part of $V(H)$ and every $v_i$ with odd index is mapped into the other part of $V(H)$. So $\mathcal{E}\cap\mathcal{O}=\emptyset$, which is a contradiction.
\section{Proof of Theorem~\ref{thm:Pmn}} Let $T$ be a $P_g^{(m, n)}$-universal planar graph for some $g$ that is minimal with respect to the subgraph order.
By minimality of $T$, there exists a graph $G \in P_g^{(m, n)}$ such that every color in $T$ has to be used at least once to color $G$. Without loss of generality, $G$ is connected, since otherwise we can replace $G$ by the connected graph obtained from $G$ by choosing a vertex in each component of $G$ and identifying them. We create a graph $G'$ from $G$ as follows:
For each edge or arc $uv$ we create $4m+n$ paths starting at $u$ and ending at $v$ made of vertices of degree 2:
\begin{itemize} \item For each type of edge, we create a path made of $g-1$ edges of this type.
\item For each type of arc, we create two paths made of $g-1$ arcs of this type such that the paths alternate between forward and backward arcs. We make the paths such that $u$ is the tail of the first arc of one path and the head of the first arc of the other path.
\item Similarly, for each type of arc we create two paths made of $g$ arcs of this type such that the paths alternate between forward and backward arcs. We make the paths such that $u$ is the tail of the first arc of one path and the head of the first arc of the other path. \end{itemize}
Notice that $G'$ is in $P_g^{(m, n)}$ and thus admits a homomorphism $\varphi$ to $T$. Since $G$ is connected and every color in $T$ has to be used at least once to color $G$, we can find for each pair of vertices and $(c_1, c_2)$ in $T$ and each type of edge a path $(v_1, v_2,\cdots, v_l)$ in $G'$ made only of edges of this type such that $\varphi(v_1)=c_1$ and $\varphi(v_l)=c_2$. \newline
This implies that for every pair of vertices $(c_1, c_2)$ in $T$ and each type of edge, there exists a walk from $c_1$ to $c_2$ made of edges of this type. Therefore, for $1\leqslant j\leqslant n$, the subgraph induced by $E_j(T)$ is connected and contains all the vertices of $T$.
So $E_j(T)$ contains a spanning tree of $T$. Thus $T$ contains at least $|V(T)|-1$ edges of each type.\newline
Similarly, we can find for each pair of vertices $(c_1, c_2)$ in $T$ and each type of arc a path of even length $(v_1, v_2,\cdots, v_{2l-1})$ in $G'$ made only of arcs of this type, starting with a forward arc and alternating between forward and backward arcs such that $\varphi(v_1)=c_i$ and $\varphi(v_l)=c_2$. We can also find a path of the same kind with odd length.\newline
This implies that for every pair of vertices $(c_1, c_2)$ in $T$ and each type of arc there exist a walk of odd length and a walk of even length from $c_1$ to $c_2$ made of arcs of this type, starting with a forward arc and alternating between forward and backward arcs. Let $p$ be the maximum of the length of all these paths. Given one of these walks of length $l$, we can also find a walk of length $l+2$ that satisfies the same constraints by going through the last arc of the walk twice more. Therefore, for every $l\geqslant p$, every pair of vertices $(c_1, c_2)$ in $T$, and every type of arc, it is possible to find a homomorphism from the path $P$ of length $l$ made of arcs of this type, starting with a forward arc and alternating between forward and backward arcs to $T$ such that the first vertex is colored in $c_1$ and the last vertex is colored in $c_2$.\newline
We now show that this implies that $|A_j(T)|\ge2|V(T)|-1$ for $1\leqslant j\leqslant m$. Let $P$ be a path $(v_1, v_2,\cdots, v_p, v_{p+1})$ of length $p$ starting with a forward arc and alternating between forward and backward arcs of the same type. We color $v_1$ in some vertex $c$ of $T$. Let $C_i$ be the set of colors in which vertex $v_i$ could be colored. We know that $C_1=c$ and $C_2$ is the set of direct successors of $c$. Set $C_3$ is the set of direct predecessors of vertices in $C_2$ so $C_1\subseteq C_3$ and, more generally, $C_i \subseteq C_i+2$. Let $uv$ be an arc in $T$. If $u\in C_i$ with $i$ odd, then $v\in C_{i+1}$. If $v\in C_i$ with $i$ even then $u\in C_{i+1}$. We can see that $uv$ is capable of adding at most one vertex to a $C_i$ (and every $C_j$ with $j\equiv i\mod 2$ and $i\leqslant j$). We know that $C_{p+1}=V(T)$ hence $T$ contains at least $2|V(T)|-1$ arcs of each type.\newline
Therefore, the underlying graph of $T$ contains at least $m\paren{2|V(T)|-1}+n\paren{|V(T)|-1}=\paren{2m+n}|V(T)|-m-n$ edges, which contradicts Proposition~\ref{prop:3n-6} for $2m+n\ge3$.
\section{Proof of Theorem~\ref{thm:ce}.1} We construct an oriented bipartite 2-outerplanar graph with girth $14$ that does not map to $\overrightarrow{T_5}$.
The oriented graph $X$ is a cycle on 14 vertices $v_0,\cdots,v_{13}$ such that the tail of every arc is the vertex with even index, except for the arc $\overrightarrow{v_{13}v_0}$. Suppose for contradiction that $X$ has a $\overrightarrow{T_5}$-coloring $h$ such that no vertex with even index maps to $b$. The directed path $v_{12}v_{13}v_0$ implies that $h(v_{12})\ne h(v_0)$. If $h(v_0)=a$, then $h(v_1)\in\acc{b,c}$ and $h(v_2)=a$ since $h(v_2)\ne b$. By contagion, $h(v_0)=h(v_2)=\cdots=h(v_{12})=a$, which is a contradiction. Thus $h(v_0)\ne a$. If $h(v_0)=c$, then $h(v_1)=d$ and $h(v_2)=c$ since $h(v_2)\ne b$. By contagion, $h(v_0)=h(v_2)=\cdots=h(v_{12})=c$, which is a contradiction. Thus $h(v_0)\ne c$. So $h(v_0)\not\in\acc{a,b,c}$, that is, $h(v_0)\in\acc{d,e}$. Similarly, $h(v_{12})\in\acc{d,e}$. Notice that $\overrightarrow{T_5}$ does not contain a directed path $xyz$ such that $x$ and $z$ belong to $\acc{d,e}$. So the path $v_{12}v_{13}v_0$ cannot be mapped to $\overrightarrow{T_5}$. Thus $X$ does not have a $\overrightarrow{T_5}$-coloring $h$ such that no vertex with even index maps to $b$.
Consider now the path $P$ on 7 vertices $p_0,\cdots,p_6$ with the arcs $\overrightarrow{p_1p_0}$, $\overrightarrow{p_1p_2}$, $\overrightarrow{p_3p_2}$, $\overrightarrow{p_4p_3}$, $\overrightarrow{p_5p_4}$, $\overrightarrow{p_5p_6}$. It is easy to check that there exists no $\overrightarrow{T_5}$-coloring $h$ of $P$ such that $h(p_0)=h(p_6)=b$.
We construct the graph $Y$ as follows: we take 8 copies of $X$ called $X_{\texttt{main}}$, $X_0$, $X_2$, $X_4$, $\cdots$, $X_{12}$. For every couple $(i,j)\in\acc{0,2,4,6,8,10,12}^2$, we take a copy $P_{i,j}$ of $P$, we identify the vertex $p_0$ of $P_{i,j}$ with the vertex $v_i$ of $X_{\texttt{main}}$ and we identify the vertex $p_6$ of $P_{i,j}$ with the vertex $v_j$ of $H_i$.
So $Y$ is our oriented bipartite 2-outerplanar graph with girth $14$. Suppose for contradiction that $Y$ has a $\overrightarrow{T_5}$-coloring $h$. By previous discussion, there exists $i\in\acc{0,2,4,6,8,10,12}$ such that the vertex $v_i$ of $X_{\texttt{main}}$ maps to $b$. Also, there exists $j\in\acc{0,2,4,6,8,10,12}$ such that the vertex $v_j$ of $X_i$ maps to $b$. So the corresponding path $P_{i,j}$ is such that $h(p_0)=h(p_6)=b$, a contradiction. Thus $Y$ does not map to $\overrightarrow{T_5}$.
\section{Proof of Theorem~\ref{thm:ce}.2} We construct a 2-edge-colored 2-outerplanar graph with girth $11$ that does not map to $T_6$. We take 12 copies $X_0,\cdots,X_{11}$ of a cycle of length $11$ such that every edge is red. Let $v_{i,j}$ denote the $j^{\text{\tiny th}}$ vertex of $X_i$. For every $0\leqslant i\leqslant 10$ and $0\leqslant j\leqslant 10$, we add a path consisting of 5 blue edges between $v_{i,11}$ and $v_{j,i}$.
Notice that in any $T_6$-coloring of a red odd cycle, one vertex must map to $c$. So we suppose without loss of generality that $v_{0,11}$ maps to $c$. We also suppose without loss of generality that $v_{0,0}$ maps to $c$. The blue path between $v_{0,11}$ and $v_{0,0}$ should map to a blue walk of length 5 from $c$ to $c$ in $T_6$. Since $T_6$ contains no such walk, our graph does not map to $T_6$.
\section{Proof of Theorem~\ref{thm:ce}.3} We construct a 2-edge-colored bipartite 2-outerplanar graph with girth $10$ that does not map to $T_6$. By Theorem~\ref{thm:negative}.2, there exists a bipartite outerplanar graph $M$ with girth at least $10$ such that for every $T_6$-coloring $h$ of $M$, there exists a vertex $v$ in $M$ such that $h(v)=c$.
Let $X$ be the graph obtained as follows. Take a main copy $Y$ of $M$. For every vertex $v$ of $Y$, take a copy $Y_v$ of $M$. Since $Y_v$ is bipartite, let $A$ and $B$ the two independent sets of $Y_v$. For every vertex $w$ of $A$, we add a path consisting of 5 blue edges between $v$ and $w$. For every vertex $w$ of $B$, we add a path consisting of 4 edges colored (blue, blue, red, blue) between $v$ and $w$.
Notice that $X$ is indeed a bipartite 2-outerplanar graph with girth $10$. We have seen in the previous proof that $T_6$ contains no blue walk of length 5 from $c$ to $c$. We also check that $T_6$ contains no walk of length 4 colored (blue, blue, red, blue) from $c$ to $c$. By the property of $M$, for every $T_6$-coloring $h$ of $X$, there exist a vertex $v$ in $Y$ and a vertex $w$ in $Y_v$ such that $h(v)=h(w)=c$. Then $h$ cannot be extended to the path of length 4 or 5 between $v$ and $w$. So $X$ does not map to $T_6$.
\section{Proof of Theorem~\ref{thm:NPC}.1} Let $g$ be the largest integer such that there exists a graph in $P_g^{(1,0)}$ that does not map to $\overrightarrow{T_5}$. Let $G\in P_g^{(1,0)}$ be a graph that does not map to $\overrightarrow{T_5}$ and such that the underlying graph of $G$ is minimal with respect to the homomorphism order.
Let $G'$ be obtained from $G$ by removing an arbitrary arc $v_0v_3$ and adding two vertices $v_1$ and $v_2$ and the arcs $v_0v_1$, $v_2v_1$, $v_2v_3$. By minimality, $G'$ admits a homomorphism $\varphi$ to $\overrightarrow{T_5}$. Suppose for contradiction that $\varphi(v_2)=c$. This implies that $\varphi(v_1)=\varphi(v_3)=d$. Thus $\varphi$ provides a $\overrightarrow{T_5}$-coloring of $G$, a contradiction. So $\varphi(v_2)\ne c$ and, similarly, $\varphi(v_2)\ne e$.
Given a set $S$ of vertices of $\overrightarrow{T_5}$, we say that we force $S$ if we specify a graph $H$ and a vertex $v\in V(H)$ such that for every vertex $x\in V\paren{\overrightarrow{T_5}}$, we have $x\in S$ if and only if there exists a $\overrightarrow{T_5}$-coloring $\varphi$ of $H$ such that $\varphi(v)=x$. Thus, with the graph $G'$ and the vertex $v_2$, we force a non-empty set $\mathcal{S}\subset V\paren{\overrightarrow{T_5}}\setminus\acc{c,e}=\acc{a,b,d}$.
We use a series of constructions in order to eventually force the set $\acc{a,b,c,d}$ starting from $\mathcal{S}$. Recall that $\acc{a,b,c,d}$ induces the tournament $\overrightarrow{T_4}$. We thus reduce $\overrightarrow{T_5}$-coloring to $\overrightarrow{T_4}$-coloring, which is NP-complete for subcubic bipartite planar graphs with any given girth~\cite{GO15}.
These constructions are summarized in the tree depicted in Figure~\ref{fig:oriented}. The vertices of this forest contain the non-empty subsets of $\acc{a,b,d}$ and a few other sets. In this tree, an arc from $S_1$ to $S_2$ means that if we can force $S_1$, then we can force $S_2$. Every arc has a label indicating the construction that is performed. In every case, we suppose that $S_1$ is forced on the vertex $v$ of a graph $H_1$ and we construct a graph $H_2$ that forces $S_2$ on the vertex $w$.
\begin{figure}
\caption{Forcing the set $\acc{a,b,c,d}$.}
\label{fig:oriented}
\end{figure}
\begin{itemize} \item Arcs labelled "out": The set $S_2$ is the out-neighborhood of $S_1$ in $\overrightarrow{T_5}$. We construct $H_2$ from $H_1$ by adding a vertex $w$ and the arc $vw$. Thus, $S_2$ is indeed forced on the vertex $w$ of $H_2$. \item Arcs labelled "in": The set $S_2$ is the in-neighborhood of $S_1$ in $\overrightarrow{T_5}$. We construct $H_2$ from $H_1$ by adding a vertex $w$ and the arc $wv$. Thus, $S_2$ is indeed forced on the vertex $w$ of $H_2$. \item Arc labelled "Z": Let $g'$ be the smallest integer such that $g'\geqslant g$ and $g'\equiv4\pmod{6}$. We consider a circuit $v_1,\cdots,v_{g'}$. For $2\leqslant i\leqslant g'$, we take a copy of $H_1$ and we identify its vertex $v$ with $v_i$. We thus obtain the graph $H_2$ and we set $w=v_2$. Let $\varphi$ be any $T_6$-coloring of $H_2$. By construction, $\acc{\varphi(v_2),\cdots,\varphi(v_{g'})}\subset S_1=\acc{a,b,d}$. A circuit of length $\not\equiv0\pmod{3}$ cannot map to the 3-circuit induced by $\acc{a,b,d}$, so $\varphi(v_1)\in\acc{c,e}$. If $\varphi(v_1)=c$ then $\varphi(v_2)=d$ and if $\varphi(v_1)=e$ then $\varphi(v_2)=a$. Thus $S_2=\acc{ad}$. \end{itemize}
\section{Proof of Theorem~\ref{thm:NPC}.2} Let $g$ be the largest integer such that there exists a graph in $P_g^{(0,2)}$ that does not map to $T_6$. Let $G\in P_g^{(0,2)}$ be a graph that does not map to $T_6$ and such that the underlying graph of $G$ is minimal with respect to the homomorphism order.
Let $G'$ be obtained from $G$ by subdividing an arbitrary edge $v_0v_3$ twice to create the path $v_0v_1v_2v_3$ such that the edges $v_0v_1$ and $v_1v_2$ are red and the edge $v_2v_3$ gets the color of the original edge $v_0v_3$. By minimality, $G'$ admits a homomorphism $\varphi$ to $T_6$. Suppose for contradiction that $\varphi(v_1)=f$. This implies that $\varphi(v_0)=\varphi(v_2)=b$. Thus $\varphi$ provides a $T_6$-coloring of $G$, a contradiction.
Given a set $S$ of vertices of $T_6$, we say that we force $S$ if we specify a graph $H$ and a vertex $v\in V(H)$ such that for every vertex $x\in V(T_6)$, we have $x\in S$ if and only if there exists $T_6$-coloring $\varphi$ of $H$ such that $\varphi(v)=x$. Thus, with the graph $G'$ and the vertex $v_1$, we force a non-empty set $\mathcal{S}\subset V(T_6)\setminus\acc{f}=\acc{a,b,c,d,e}$.
Recall that the core of a graph is the smallest subgraph which is also a homomorphic image. We say that a subset $S$ of $V(T_6)$ is \emph{good} if the core of the subgraph induced by $S$ is isomorphic to the graph $T_4$ which is a a clique on 4 vertices such that both the red and the blue edges induce a path of length $3$. We use a series of constructions in order to eventually force a good set starting from $\mathcal{S}$. We thus reduce $T_6$-coloring to $T_4$-coloring, which is NP-complete for subcubic bipartite planar graphs with any given girth~\cite{MO17}.
These constructions are summarized in the forest depicted in Figure~\ref{fig:2edge}. The vertices of this forest are the non-empty subsets of $\acc{a,b,c,d,e}$ together with a few auxiliary sets of vertices containing $f$. In this forest, an arc from $S_1$ to $S_2$ means that if we can force $S_1$, then we can force $S_2$. Every set with no outgoing arc is good. We detail below the construction that is performed for each arc. In every case, we suppose that $S_1$ is forced on the vertex $v$ of a graph $H_1$ and we construct a graph $H_2$ that forces $S_2$ on the vertex $w$.
\begin{figure}
\caption{Forcing a good set.}
\label{fig:2edge}
\end{figure}
\begin{itemize} \item Blue arcs: The set $S_2$ is the blue neighborhood of $S_1$ in $T_6$. We construct $H_2$ from $H_1$ by adding a vertex $w$ adjacent to $v$ such that $vw$ is blue. Thus, $S_2$ is indeed forced on the vertex $w$ of $H_2$. \item Red arcs: The set $S_2$ is the red neighborhood of $S_1$ in $T_6$. The construction is as above except that the edge $vw$ is red. \item Dashed blue arcs: The set $S_2$ is the set of vertices incident to a blue edge contained in the subgraph induced by $S_1$ in $T_6$. We construct $H_2$ from two copies of $H_1$ by adding a blue edge between the vertex $v$ of one copy and the vertex $v$ of the other copy. Then $w$ is one of the vertices $v$. \item Dashed red arcs: The set $S_2$ is the set of vertices incident to a red edge contained in the subgraph induced by $S_1$ in $T_6$. The construction is as above except that the added edge is red. \item Arc labelled "X": Let $g'=2\ceil{g/2}$. We consider an even cycle $v_1,\cdots,v_{g'}$ such that $v_1v_{g'}$ is red and the other edges are blue. For every vertex $v_i$, we take a copy of $H_1$ and we identify its vertex $v$ with $v_i$. We thus obtain the graph $H_2$ and we set $w=v_1$. Let $\varphi$ be any $T_6$-coloring of $H_2$. In any $T_6$-coloring of $H_2$, the cycle $v_1,\cdots,v_{g'}$ maps to a 4-cycle with exactly one red edge contained in the subgraph of $T_6$ induced by $S_1=\acc{a,b,c,d,e}$. These 4-cycles are $aedb$ with red edge $ae$ and $cdba$ with red edge $cd$. Since $w$ is incident to the red edge in the cycle $v_1,\cdots,v_{g'}$, $w$ can be mapped to $a$, $e$, $c$, or $d$ but not to $b$. Thus $S_2=\acc{a,c,d,e}$. \item Arc labelled "Y": We consider an alternating cycle $v_0,\cdots,v_{8g-1}$. For every vertex $v_i$, we take a copy of $H_1$ and we identify its vertex $v$ with $v_i$. We obtain the graph $H_2$ by adding the vertex $x$ adjacent to $v_0$ and $v_{4g+2}$ such that $xv_0$ and $xv_{4g+2}$ are blue. We set $w=v_0$. In any $T_6$-coloring $\varphi$ of $H_2$, the cycle $v_1,\cdots,v_{g'}$ maps to the alternating $4$-cycle $acde$ contained in $S_1=\acc{a,c,d,e}$ such that $\varphi(v_i)=\varphi(v_{i+4\pmod{8g}})$. So, a priori, either $\acc{\varphi(v_0),\varphi(v_{4g+2})}=\acc{a,d}$ or $\acc{\varphi(v_0),\varphi(v_{4g+2})}=\acc{c,e}$. In the former case, we can extend $\varphi$ to $H_2$ by setting $\varphi(x)=b$. In the latter case, we cannot color $x$ since $c$ and $e$ have no common blue neighbor in $T_6$. Thus, $\acc{\varphi(v_0),\varphi(v_{4g+2})}=\acc{a,d}$ and $S_2=\acc{a,d}$. \end{itemize}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Zero-sum $K_m$ over $\Z$ and the story of $K_4$}
\begin{center}
\begin{multicols}{2}
Yair Caro\\[1ex] {\small Dept. of Mathematics and Physics\\ University of Haifa-Oranim\\ Tivon 36006, Israel\\ [email protected]}
\columnbreak
Adriana Hansberg\\[1ex] {\small Instituto de Matem\'aticas\\ UNAM Juriquilla\\ Quer\'etaro, Mexico\\ [email protected]}\\[2ex]
\end{multicols}
Amanda Montejano\\[1ex] {\small UMDI, Facultad de Ciencias\\ UNAM Juriquilla\\ Quer\'etaro, Mexico\\ [email protected]}\\[4ex]
\end{center}
\begin{abstract}
We prove the following results solving a problem raised in Caro-Yuster \cite{CY3}. For a positive integer $m\geq 2$, $m\neq 4$, there are infinitely many values of $n$ such that the following holds: There is a weighting function $f:E(K_n)\to \{-1,1\}$ (and hence a weighting function $f: E(K_n)\to \{-1,0,1\}$), such that $\sum_{e\in E(K_n)}f(e)=0$ but, for every copy $H$ of $K_m$ in $K_n$, $\sum_{e\in E(H)}f(e)\neq 0$. On the other hand, for every integer $n\geq 5$ and every weighting function $f:E(K_n)\to \{-1,1\}$ such that $|\sum_{e\in E(K_n)}f(e)|\leq \binom{n}{2}-h(n)$, where $h(n)=2(n+1)$ if $n \equiv 0$ (mod $4$) and $h(n)=2n$ if $n \not\equiv 0$ (mod $4$), there is always a copy $H$ of $K_4$ in $K_n$ for which $\sum_{e\in E(H)}f(e)=0$, and the value of $h(n)$ is sharp.
\end{abstract}
\section{Introduction}
Our main source of motivation is a recent paper of Caro and Yuster \cite{CY3}, extending classical zero-sum Ramsey theory to weighting functions $f:E(K_n)\to \{-r,-r+1, \cdots ,0, \cdots , r-1,r\}$ seeking zero-sum copies of a given graph $H$ subject to the obviously necessary condition that $|\sum_{e\in E(K_n)}f(e)|$ is bounded away from $ r\binom{n}{2}$, or even in the extreme case where $|\sum_{e\in E(K_n)}f(e)|=0$.
In zero-sum Ramsey theory, one studies functions $f:E(K_n)\to X$, where $X$ is usually the cyclic group $\mbox{${\mathbb Z}$}_k$ or (less often) an arbitrary finite abelian group. The goal is to show that, under some necessary divisibility conditions imposed on the number of the edges $e(H)$ of a graph $H$ and for sufficiently large $n$, there is always a zero-sum copy of $H$, where by a zero-sum copy of $H$ we mean a copy of $H$ in $K_n$ for which $\sum_{e\in E(H)}f(e)=0$ (where $0$ is the neutral element of $X$). For several results concerning zero-sum Ramsey theory for graphs see \cite{AC,BD1,BD2,C1,C2,CY1,FK,SS}, for zero-sum Ramsey problems concerning matrices and linear algebraic techniques see \cite{BCRY,CY2,WW,W}.
The following notation was introduced in \cite{CY3} and the following zero-sum problems over $\mbox{${\mathbb Z}$}$
were considered. For positive integers $r$ and $q$, an \emph{$(r,q)$-weighting} of the edges of the complete graph $K_n$ is a function $f:E(K_n)\to \{-r, \cdots ,r\}$ such that $|\sum_{e\in E(K_n)}f(e)|<q$. The general problem considered in \cite{CY3} is to find nontrivial conditions on the $(r,q)$-weightings that guarantee the existence of certain bounded-weight subgraphs and even zero-weighted subgraphs (also called \emph{zero-sum} subgraphs). So, given a subgraph $H$ of $K_n$, and a weighting $f:E(K_n)\to \{-r,\cdots ,r\}$, the \emph{weight} of $H$ is defined as $w(H)=\sum_{e\in E(H)}f(e)$, and we say that $H$ is a \emph{zero-sum graph} if $w(H)=0$. Finally, we say that a weighting function $f:E(K_n) \to \{-1,1\}$ is \emph{zero-sum-$H$ free} if it contains no zero-sum copy of $H$.
Among the many results proved in \cite{CY3}, the following theorem and open problem are the main motivation of this paper.
\begin{theorem}[Caro and Yuster, \cite{CY3}]\label{thm:CY}
For a real $\epsilon >0$ the following holds. For $n$ sufficiently large, any weighting $f:E(K_n)\to \{-1,0,1\}$ with $|\sum_{e\in E(K_n)}f(e)|\leq (1-\epsilon)n^2/6$ contains a zero-sum copy of $K_4$. On the other hand, for any positive integer $m$ which is not of the form $m = 4d^2$, there are infinitely many integers $n$ for which there is a weighting $f:E(K_n)\to \{-1,0,1\}$ with $\sum_{e\in E(K_n)}f(e)=0$ without a zero-sum copy of $K_m$. \end{theorem}
The authors posed the following complementary problem:
\begin{problem}[Caro and Yuster, \cite{CY3}]\label{probl} For an integer $m = 4d^2$, is it true that, for $n$ sufficiently large, any weighting $f:E(K_n)\to \{-1,0,1\}$ with $\sum_{e\in E(K_n)}f(e)=0$ contains a zero-sum copy of $K_m$? \end{problem}
The main result in this paper is a negative answer to the above problem for any $m \geq 2$ except for $m=4$ already for weightings of the form $f:E(K_n)\to \{-1,1\}$ with $\sum_{e\in E(K_n)}f(e)=0$. On the other hand, concerning the study of the existence of zero-sum copies of $K_4$, we prove a result analogous to Theorem \ref{thm:CY} where the range $\{-1,1\}$ instead of $\{-1,0,1\}$ is considered. Finally, we show that Theorem \ref{thm:CY} can neither be extended to wider ranges. To be more precise, we gather our results in the following theorem.
\begin{theorem}\label{thm:main} \hspace{1cm} \begin{enumerate}
\item For any positive integer $m \geq2$, $m\neq 4$, there are infinitely many values of $n$ such that the following holds: There is a weighting function $f:E(K_n)\to \{-1,1\}$ with $\sum_{e\in E(K_n)}f(e)=0$ which is zero-sum-$K_m$ free.
\item Let $n$ be an integer such that $n\geq 5$. Define $g(n)=2(n+1)$ if $n \equiv 0$ (mod $4$) and $g(n)=2n$ if $n \not\equiv 0$ (mod $4$). Then, for any weighting $f:E(K_n)\to \{-1,1\}$ such that $|\sum_{e\in E(K_n)}f(e)|\leq \binom{n}{2}-g(n)$, there is a zero-sum copy of $K_4$.
\item There are infinitely many values of $n$ such that the following holds: There is a weighting function $f:E(K_n)\to \{-2,-1,0,1,2\}$ with $\sum_{e\in E(K_n)}f(e)=0$ which is zero-sum-$K_4$ free.
\end{enumerate} \end{theorem}
Theorem \ref{thm:main} together with the above Theorem \ref{thm:CY} do not only solve Problem \ref{probl}, they also supply a good understanding of the situation concerning $K_4$ as the value of $g(n)$ is sharp and the upper bound $(1-\epsilon)n^2/6$ in Theorem \ref{thm:CY} is nearly sharp, as already observed in \cite{CY3}.
We will use the following notation. Given a weighting $f:E(K_n)\to \{-r, \cdots, r\}$ and $i\in \ \{-r, \cdots, r\}$, denote by $E(i)$ the set of the $i$-weighted edges, that is, $E(i)=f^{-1}(i)$ and define $e(i)=|E(i)|$. Given a vertex $x\in V(K_n)$ we use $deg_{i}(x)$ to denote the number of $i$-weighted edges incident to $x$, that is, $deg_{i}(x)=|\{u:f(xu)=i\}|$.
In Section \ref{sec:K4} we will prove instances 2 and 3 of Theorem \ref{thm:main}, corresponding to the study of the existence of zero-sum copies of $K_4$. In order to prove instance 2, we will use an equivalent formulation consequence of the following remark.
\begin{remark}\label{rem:eq}
A weighting $f:E(K_n)\to \{-1,1\}$ satisfies $\left|\sum_{e\in E(K_n)}f(e)\right|\leq \binom{n}{2}-g(n)$ if and only if $\min\{e(-1),e(1)\}\geq \frac{1}{2}g(n)$. \end{remark}
The remark follows from the fact that $e(1)+e(-1)=\binom{n}{2}$, which implies $|\sum_{e\in E(K_n)}f(e)|=|e(1)-e(-1)|=\max\{e(-1),e(1)\}-\min\{e(-1),e(1)\}=\binom{n}{2}-2\min\{e(-1),e(1)\}$.
In Section \ref{sec:K4} we will also prove that instance 2 of Theorem \ref{thm:main} is best posible by exhibiting, for each $n\geq5$, a weighting function $f:E(K_n)\to \{-1,1\}$ with $\min\{e(-1),e(1)\}= \frac{1}{2}g(n)-1$ and no zero-sum copies of $K_4$. Moreover, we will characterize the extremal functions.
Finally, relying heavily on Pell equations and some classical biquadratic Diophantine equations,
in Section \ref{sec:Kk} we will prove instance 1 of Theorem \ref{thm:main}, corresponding to the study of the existence of zero-sum copies of $K_m$ in $0$-weighted weightings, where $m \neq 4$.
\section{The case of $K_4$}\label{sec:K4}
We will use standard graph theoretical notation to denote particular graphs. Having said this, $K_{1,3}$ will stand for the star with three leaves, $K_3 \cup K_1$ for the disjoint union of a triangle and a vertex, $P_k$ for a path with $k$ edges, and $C_k$ for a cycle with $k$ edges.
A weighting function $f:E(K_n) \to \{-1,1\}$ is \emph{zero-sum-$K_4$ free} if and only if the graph induced by $E(-1)$ (or equivalently $E(1)$) is $\{K_{1,3}, K_3 \cup K_1, P_3\}$-free (in the induced sense). The following lemma, which characterizes the $K_3$-free subclass of the family of $\{K_{1,3}, K_3 \cup K_1, P_3\}$-free graphs, will be useful in proving the forthcoming results. We define \[h(n)= \left\{ \begin{array}{rl}
n+1, & \mbox{ if } n\equiv 0 \mbox{ (mod $4$), and} \\
n, & \mbox{ otherwise. } \\ \end{array}\right.\]
\begin{lemma}\label{lem:triangle-free} Let $G$ be a $\{K_{1,3}, K_3, P_3\}$-free graph on $n$ vertices. Then each component of $G$ is isomorphic to one of $C_4$, $K_1$, $K_2$ or $P_2$. Moreover, $e(G) \le h(n)-1$, and equality holds if and only if $G \cong J \cup \bigcup_{i=1}^{q} C_4$, where $J \in \{\emptyset, K_1, K_2, P_2\}$ and $q = \lfloor \frac{n}{4} \rfloor$. \end{lemma}
\begin{proof}
Let $J$ be a connected component of $G$. If $J$ has at most $3$ vertices, then it is easy to see that $J \in \{K_1, K_2, P_2\}$. So assume that $J$ has at least $4$ vertices. Then, since $J$ is $\{K_3, P_3\}$-free, we infer that $J$ has no vertex of degree larger than $2$ and so we can deduce that $J \cong C_4$. Further, we note that $e(J) = |J|$ if $J \cong C_4$, and $e(J) = |J|-1$ otherwise. This implies that, among all $\{K_{1,3}, K_3, P_3\}$-free graphs on $n$ vertices, $G$ has maximum number of edges if and only if $G \cong J \cup \bigcup_{i=1}^{q} C_4$, where $J \in \{\emptyset, K_1, K_2, P_2\}$ and $q = \lfloor \frac{n}{4} \rfloor$. Since, clearly, $e(J \cup \bigcup_{i=1}^{q} C_4) = h(n) -1$, the proof is complete. \end{proof}
\begin{lemma}\label{lem:one_is_K3-free} Let $n\geq 5$ and $f:E(K_n)\to \{-1,1\}$ be a zero-sum-$K_4$ free coloring. Let $G_{-1}$ and $G_1$ be the graphs induced by $E(-1)$ and $E(1)$, respectively. Then at least one of $G_{-1}$ or $G_1$ is triangle-free. \end{lemma}
\begin{proof} Suppose for contradiction that both $G_{-1}$ and $G_1$ have a triangle. Let $abc$ be a triangle in $G_{-1}$ and $uvw$ a triangle in $G_1$. Suppose first that $abc$ and $uvw$ have a vertex in common, say $a=u$. Consider the graph $J$ induced by the $(-1)$-edges among vertices in $\{a,b,c,v,w\}$. If a vertex $x \in \{b,c\}$ is neighbor of both $v$ and $w$, then $\{x,v,w,a\}$ would induce a $K_{1,3}$ in $G_{-1}$, which is not possible. If no vertex $x \in \{b,c\}$ is adjacent to some $y \in \{v,w\}$, then $\{a,b,c,y\}$ would induce a $K_3\cup K_1$ in $G_{-1}$, which again is not possible. Hence, $\{b,c,v,w\}$ induces two independent edges. But then $\{b,c,v,w\}$ induces a $P_3$ in $G_{-1}$, a contradiction. Hence, we can assume that any pair of triangles such that one has only $(-1)$-edges and the other only $1$-edges are vertex disjoint. This implies that from any vertex in $\{u,v,w\}$ there is at most one $(-1)$-edge to vertices from $\{a,b,c\}$. Analogously, from any vertex in $\{a,b,c\}$ there is at most one $1$-edge to vertices from $\{u,v,w\}$. But this implies that there are at most $6$ edges between $\{a,b,c\}$ and $\{u,v,w\}$, which is false. Since in all cases we obtain a contradiction, we conclude that at least one of $G_{-1}$ or $G_1$ is triangle-free. \end{proof}
By Remark \ref{rem:eq}, the next result is equivalent to instance 2 of Theorem \ref{thm:main}.
\begin{theorem}\label{thm:k4} Let $n\geq 5$ and $f:E(K_n)\to \{-1,1\}$ such that $\min\{e(-1),e(1)\}\geq h(n)$. Then there is a zero-sum $K_4$. \end{theorem}
\begin{proof}
Let $f:E(K_n)\to \{-1,1\}$ be such that $\min\{e(-1),e(1)\}\geq h(n)$ and suppose for contradiction that it has no zero-sum $K_4$. Let $G_{-1}$ and $G_1$ be the graphs induced by $E(-1)$ and $E(1)$, respectively. Then both $G_{-1}$ and $G_1$ are $\{K_{1,3}, K_3 \cup K_1, P_3\}$-free graphs. By Lemma \ref{lem:one_is_K3-free}, $G_{-1}$ or $G_1$ is $K_3$-free. So we may assume, without loss of generality, that $G_{-1}$ is triangle-free. It follows by Lemma \ref{lem:triangle-free} that $e(-1) = |E(G_{-1})| \le h(n)-1$, which is a contradiction to the hypothesis. \end{proof}
The following theorem shows that Theorem \ref{thm:k4} is best possible and characterizes the extremal zero-sum-$K_4$ free weightings. We will use Mantel's Theorem, that any graph on $n$ vertices and at least $\frac{n^2}{4}+1$ edges contains a copy of $K_3$.
\begin{theorem}\label{thm:k4_sharp} Let $n\geq 5$ and $f:E(K_n)\to \{-1,1\}$ such that $e(1) = h(n)-1$. Then $f$ is zero-sum-$K_4$ free if and only if the graph induced by $E(1)$ is isomorphic to $J \cup \bigcup_{i=1}^{q} C_4$, where $J \in \{\emptyset, K_1, K_2, P_2\}$ and $q = \lfloor \frac{n}{4} \rfloor$. \end{theorem}
\begin{proof} If the graph induced by $E(1)$ is isomorphic to $J \cup \bigcup_{i=1}^{q} C_4$, where $J \in \{\emptyset, K_1, K_2, P_2\}$ and $q = \lfloor \frac{n}{4} \rfloor$, it is easy to check that $f$ is zero-sum-$K_4$ free. Conversely, let $f$ be zero-sum-$K_4$ free. Then the graphs $G_{-1}$ and $G_1$ induced by $E(-1)$ and $E(1)$, respectively, are both $\{K_{1,3}, K_3 \cup K_1, P_3\}$-free. If $n = 5$, it is easy to check that the only $\{K_{1,3}, K_3 \cup K_1, P_3\}$-free graph with $h(5) - 1 = 4$ edges is isomorphic to $C_4 \cup K_1$, and so we are done. Hence, we may assume that $n \ge 6$. Observe that \[e(-1) = \frac{n(n-1)}{2} - h(n)+1 \ge \frac{n(n-1)}{2} - n = \frac{n(n-3)}{2},\] whose right-hand side is at least $\frac{n^2}{4}$ for $n \ge 6$. Hence, by Mantel's Theorem, $G_{-1}$ has a triangle, and, by Lemma \ref{lem:one_is_K3-free}, this implies that $G_1$ is triangle-free. It follows that $G_1$ is a $\{K_{1,3}, K_3, P_3\}$-free graph on $n$ vertices and with $h(n) - 1$ edges. Thus, with Lemma \ref{lem:triangle-free}, we obtain that $G_1$ is isomorphic to $J \cup \bigcup_{i=1}^{q} C_4$, where $J \in \{\emptyset, K_1, K_2, P_2\}$ and $q = \lfloor \frac{n}{4} \rfloor$, and we are done. \end{proof}
The following theorem is instance 2 from Theorem \ref{thm:main}. It shows that, whenever we take a wider range for the weighting function $f$, we cannot hope for a result as in Theorem \ref{thm:k4} anymore.
\begin{theorem}\label{thm:larger_range} There are infinitely many values of $n$ such that the following holds: There is a weighting function $f:E(K_n)\to \{-2,-1,0,1,2\}$ with $\sum_{e\in E(K_n)}f(e)=0$ which is zero-sum-$K_4$ free. \end{theorem}
\begin{proof} Let $X \cup Y$ be a partition of the vertex set of $K_n$ and consider the weighting function $f:E(K_n)\to \{-2,-1,0,1,2\}$ such that \[ f(uv) = \left\{\begin{array}{ll} -2, & \mbox{if } u,v \in X\\ 1, & \mbox{if } u,v \in Y\\ 0, & \mbox{otherwise}. \end{array} \right. \] Clearly, $f$ is zero-sum-$K_4$ free. On the other hand, $\sum_{e\in E(K_n)}f(e)=0$ if and only if \[
-2\frac{|X|(|X|-1)}{2} + \frac{|Y|(|Y|-1)}{2} = 0, \]
which is equivalent to $(2|Y|-1)^2-2(2|X|-1)^2 = -1$. Hence, solving the latter equation is equivalent to solve the following Pell's equation \begin{equation}\label{eq:pell-pythago} y^2-2x^2 = -1, \end{equation}
for (odd) integers $x = 2|X|-1$ and $y= 2|Y|-1$. It is well-known that the Diophantine equation $y^2-2x^2=\pm 1$ has infinitely many solutions given by \[x_{k}=\frac{a^k-b^k}{a-b}=\frac{a^k-b^k}{2\sqrt{2}}, \hspace{2ex} y_k=\frac{a^k+b^k}{2},\]
where $a=1+\sqrt{2}$, $b=1-\sqrt{2}$ and $k \in \mathbb{N}$. Moreover, since $y_k^2 - 2x_k^2 = (-1)^k$, the solutions for equation (\ref{eq:pell-pythago}) are the pairs $(x_k,y_k)$ where $k$ is odd. Observe also that, for odd $k$, $x_k$ and $y_k$ are odd, too. Hence, each odd $k$ gives us a solution $(\frac{x_k+1}{2}, \frac{y_k+1}{2})$ for $(|X|,|Y|)$ and thus for $n = \frac{x_k+y_k}{2}+1$ and we are done. \end{proof}
For the sake of comprehension, let us compute small values of $n=\frac{x_k+y_k}{2}+1$ and exhibit how the partition $(|X|,|Y|)=(\frac{x_k+1}{2}, \frac{y_k+1}{2})$ gives a zero-sum weighting function $f$ as described in the theorem. Recall that we only want to consider solutions for odd values of $k$.
So we have $(x_1,x_3,x_5,\dots)=(1,5,29,\dots)$ and $(y_1,y_3,y_5,\dots)=(1,7,41,\dots)$, and the corresponding sequence of $n$'s is $(2,7,36,\dots )$. The case of $n=2$ is not interesting for vaquity reasons. For $n=7$, the partition is $(|X|,|Y|)=(3,4)$, thus there will be $ \binom{3}{2}$ edges weighted with $-2$, $ \binom{4}{2}$ edges weighted with $1$ and the rest of edges weighted with $0$, adding up to zero. For $n=36$, the partition is $(|X|,|Y|)=(15,21)$, and the sum of weighted edges is $-2\binom{15}{2}+1\binom{21}{2}=(-2) \cdot 105+1 \cdot 210=0$.
\section{The case of $K_m$, $m\neq 4$}\label{sec:Kk}
A \emph{balanced} $\{-1,1\}$-weighting function $f:E(K_n)\to \{-1,1\}$ is a function for which $e(-1)=e(1)$. In Section \ref{sec:K4}, we prove that, for $n\geq 5$, any function $f:E(K_n)\to \{-1,1\}$ with sufficiently many edges assigned to each type contains a zero-sum $K_4$. In this section, we prove that this is not true for $K_m$ with $m \in \mathbb{N} \setminus \{1, 4\}$. In other words, we exhibit, for infinitely many values of $n$, the existence of a balanced weighting function $f:E(K_n)\to \{-1,1\}$ without a zero-sum copies of $K_m$, where $m \neq 1,4$. In order to define those functions, consider first the following Pell equation: \begin{equation}\label{eq:pell} 8x^2-8x+1=y^2. \end{equation} It is well known that such a Diophantine equation has infinitely many solutions given by the recursion \[(x_1,y_1)=(1,1),\] \[(x_2,y_2)=(3,7),\] \[y_k=6y_{k-1}-y_{k-2}, \hspace{.5cm} x_{k}=\frac{y_k+x_{k-1}+1}{3}.\]
\begin{lemma}\label{lem:bal1} Let $n$ be a positive integer and consider the complete graph $K_n$ and a partition $V(K_n) = A \cup B$ of its vertex set. Then the function $f:E(K_n)\to \{-1,1\}$ defined as \[f(e)= \left\{ \begin{array}{rl}
-1, & \mbox{ if } e\subset A \\
1, & \mbox{ otherwise, } \\ \end{array}\right.\]
is balanced if and only if $n = \frac{1+y_k}{2}$ and $|A| = x_k$ for some $k \in\mathbb{N}$. \end{lemma}
\begin{proof}
Suppose first that $f$ is balanced and let $|A| = x$. Then $$e(-1)=\frac{x(x-1)}{2} = \frac{1}{2} \binom{n}{2},$$ which yields \[ n^2-n-(2x^2-2x)=0, \]
and therefore $n = \frac{1+\sqrt{8x^2-8x+1}}{2}$. But this is only an integer if $8x^2-8x+1 = y^2$ for some integer $y$, and we obtain equation (\ref{eq:pell}). Hence, $|A| = x = x_k$ and $n = \frac{1+y_k}{2}$ for some $k \in \mathbb{N}$.\\
Conversely, suppose that $n = \frac{1+y_k}{2}$ and $|A| = x_k$ for some $k \in\mathbb{N}$. Then \[ n = \frac{1+y_k}{2} = \frac{1+\sqrt{y_k^2}}{2} = \frac{1+\sqrt{8x_k^2-8x_k+1}}{2}. \] Thus $n$ is the positive root of \begin{equation}\label{eq:n_k} n^2-n-(2x_k^2-2x_k)=0, \end{equation} which is equivalent to \[ \frac{x_k(x_k-1)}{2}=\frac{1}{2}\binom{n}{2}. \] Since the left hand side of this equation is precisely $e(-1)$ and the right hand side is half the number of the edges of $K_n$, it follows that $f$ is balanced. \end{proof}
\begin{lemma}\label{lem:bal2} Let $n$ be a positive integer and consider the complete graph $K_n$ and a partition $V(K_n) = A \cup B$ of its vertex set. Then the function \[f(e)= \left\{ \begin{array}{rl}
-1, & \mbox{ if } e\subset A \mbox{ or } e\subset B \\
1, & \mbox{ otherwise, } \\ \end{array}\right.\]
is balanced if and only if $n = k^2$ and $\{|A|, |B| \} = \{\frac{1}{2}k(k+1), \frac{1}{2}k(k-1)\}$ for some $k \in\mathbb{N}$. \end{lemma}
\begin{proof}
Suppose first that $f$ is balanced and let $|A| = w$. Then \[ e(1) = w(n-w)=\frac{1}{2} \binom{n}{2}, \] which is equivalent to \[ w^2 - nw + \frac{1}{4}n(n-1)=0. \] Hence, \begin{equation}\label{eq:n(w)} w = \frac{n \pm \sqrt{n}}{2}, \end{equation}
which is an integer if and only if $n = k^2$ for some $k \in \mathbb{N}$. So we obtain $n = k^2$ and $w \in \{\frac{1}{2}k(k+1), \frac{1}{2}k(k-1)\}$. Since $|B| = n - |A| = k^2 - w$, it follows easily that $\{|A|, |B| \} = \{\frac{1}{2}k(k+1), \frac{1}{2}k(k-1)\}$ and we are done.\\
Conversely, suppose that $n = k^2$ and $\{|A|, |B| \} = \{\frac{1}{2}k(k+1), \frac{1}{2}k(k-1)\}$ for some $k \in\mathbb{N}$. Without loss of generality, assume that $|A| = \frac{1}{2}k(k+1)$. Then \[
e(1) = |A| (n - |A|) = \frac{1}{2}k(k+1) \left( k^2 - \frac{1}{2}k(k+1)\right) = \frac{1}{4} k^2(k^2-1) = \frac{1}{2} \binom{n}{2},\] implying that $f$ is balanced. \end{proof}
We define the set $S_1$ as the set of all integers $n_k = \frac{1+y_k}{2}$, $k \in \mathbb{N}$, where $(x_k,y_k)$ is the k-th solution of (\ref{eq:pell}), that is,
$$S_1 = \left\{\frac{1+y_k}{2} \; | \; k \in \mathbb{N} \right\}.$$ Further, let $S_2$ be the set of all integer squares, that is,
$$S_2 = \left\{k^2 \; | \; k \in \mathbb{N} \right\}.$$
Lemmas \ref{lem:bal1} and \ref{lem:bal2} yield the following corollary.
\begin{corollary}\label{cor:Km_Si} For any integer $m \in \mathbb{N} \setminus (S_1 \cap S_2)$, there are infinitely many positive integers $n$ such that there exists a balanced function $f:E(K_n)\to \{-1,1\}$ without zero-sum $K_m$. \end{corollary}
\begin{proof} Let $m \in \mathbb{N} \setminus (S_1 \cap S_2)$. By Lemmas \ref{lem:bal1} and \ref{lem:bal2}, there is a balanced function $f:E(K_n)\to \{-1,1\}$ for each $n \in S_1 \cup S_2$. Suppose that there is a zero-sum $K_m$ in such a weighting $f$ for a given $n \in S_1 \cup S_2$. Then, the function $f$ restricted to the edges of $K_m$ is a balanced function on $E(K_m)$, which is not possible by Lemmas \ref{lem:bal1} and \ref{lem:bal2} since $m \in \mathbb{N} \setminus S_1 \cap S_2$. Since $S_1 \cup S_2$ has infinitely many elements, it follows that there are infinitely many positive integers $n$ such that there exists a balanced function $f:E(K_n)\to \{-1,1\}$ without zero-sum $K_m$. \end{proof}
Now we can state the main result of this section, which is equivalent to instance 3 of Theorem \ref{thm:main}.
\begin{theorem} For any integer $m \in \mathbb{N} \setminus \{1, 4\}$, there are infinitely many positive integers $n$ such that there exists a balanced weighting $f:E(K_n)\to \{-1,1\}$ which is zero-sum-$K_m$ free. \end{theorem}
\begin{proof} By Corollary \ref{cor:Km_Si}, for any $m \in \mathbb{N} \setminus (S_1\cap S_2)$, there are infinitely many positive integers $n$, such that there exists a balanced weighting function $f:E(K_n)\to \{-1,1\}$ without a zero-sum $K_m$. We will show that $S_1 \cap S_2=\{1,4\}$. Let $q$ be an integer such that $q^2\in S_1$ (and thus $q^2\in S_1\cap S_2$). Then $q^2$ must be the positive root of equation (\ref{eq:n_k}) for some $x_k$. Thus we need to know for which positive integers $q$ and $x$ the following is possible: \begin{equation}\label{eq:qx} q^4-q^2-(2x^2-2x)=0. \end{equation} Note that equation (\ref{eq:qx}) can be written as: \begin{equation}\label{eq:QX} Q^2-2X^2=-1. \end{equation} where $Q=2q^2-1$ and $X=2x-1$. Again (as in the proof of Theorem \ref{thm:larger_range}), we have to deal with the Diophantine equation $Q^2-2X^2=\pm 1$, which has infinitely many solutions given by \[Q_k=\frac{a^k+b^k}{2}, \hspace{.5cm} X_{k}=\frac{a^k-b^k}{a-b}=\frac{a^k-b^k}{2\sqrt{2}},\] where $a=1+\sqrt{2}$ and $b=1-\sqrt{2}$. Since we need to solve equation (\ref{eq:QX}) (that is, with $-1$ on the right side), we know that $k$ must be odd. Therefore, according to the definition of $Q$, we need to determine all odd $k$'s such that $$Q_k+1=2q^2,$$ or equivalently, $$2Q_k+2=4q^2,$$ and so, \begin{equation}\label{eq:ab} a^k+b^k+a+b=(2q)^2. \end{equation} We consider two cases:\\
\noindent \emph{Case 1.} If $k\equiv 1$ (mod $4$), we will prove that the left side of equation (\ref{eq:ab}) is $4Q_{\frac{k-1}{2}}Q_{\frac{k+1}{2}}$. Note that $ab=-1$ and, since in this case $\frac{k-1}{2}$ is even, we have $(ab)^{\frac{k-1}{2}}=(-1)^{\frac{k-1}{2}}=1$. Hence, \begin{align*} a^k+b^k+a+b&=a^k+b^k+(ab)^{\frac{k-1}{2}}(a+b)\\
&=a^{\frac{k-1}{2}}a^{\frac{k+1}{2}}+b^{\frac{k-1}{2}}b^{\frac{k+1}{2}}+a^{\frac{k-1}{2}}b^{\frac{k-1}{2}}(a+b)\\
&=a^{\frac{k-1}{2}}a^{\frac{k+1}{2}}+b^{\frac{k-1}{2}}b^{\frac{k+1}{2}}+a^{\frac{k+1}{2}}b^{\frac{k-1}{2}}+a^{\frac{k-1}{2}}b^{\frac{k+1}{2}}\\
&=(a^{\frac{k-1}{2}}+b^{\frac{k-1}{2}})(a^{\frac{k+1}{2}}+b^{\frac{k+1}{2}})\\
&=4Q_{\frac{k-1}{2}}Q_{\frac{k+1}{2}}. \end{align*} Thus, by (\ref{eq:ab}), we conclude that $Q_{\frac{k-1}{2}}Q_{\frac{k+1}{2}}$ is a perfect square. We know that, for all $i$, $Q_i$ and $Q_{i+1}$ are coprimes. Thus, it follows that both $Q_{\frac{k-1}{2}}$ and $Q_{\frac{k+1}{2}}$ are perfect squares. Coming back to equation (\ref{eq:QX}), the following must be satisfied \begin{equation}\label{eq:YX} Y^4-2X^2=-1 \end{equation} where $Q_{\frac{k+1}{2}}=Y^2$. But, the only possible solution for the Diophantine equation (\ref{eq:YX}) is $(Y,X)=(\pm 1,1)$. Hence, $Q_{\frac{k+1}{2}}= 1$, which means that $k=1$, and so $Q = Q_1 = 1$. Since $Q=2q^2-1$ and $q>0$, we conclude that $q=1$.\\
\noindent \emph{Case 2.} If $k\equiv 3$ (mod $4$), then we will prove that the left side of equation (\ref{eq:ab}) is $8X_{\frac{k-1}{2}}X_{\frac{k+1}{2}}$. Recall that $ab=-1$ and, since in this case $\frac{k+1}{2}$ is even, we have $(ab)^{\frac{k+1}{2}}=(-1)^{\frac{k+1}{2}}=1$. Hence, \begin{align*} a^k+b^k+a+b&=a^k+b^k+(ab)^{\frac{k+1}{2}}(a+b)\\ &=a^k+b^k-(ab)^{\frac{k-1}{2}}(a+b)\\
&=a^{\frac{k-1}{2}}a^{\frac{k+1}{2}}+b^{\frac{k-1}{2}}b^{\frac{k+1}{2}}-a^{\frac{k-1}{2}}b^{\frac{k-1}{2}}(a+b)\\
&=a^{\frac{k-1}{2}}a^{\frac{k+1}{2}}+b^{\frac{k-1}{2}}b^{\frac{k+1}{2}}-a^{\frac{k+1}{2}}b^{\frac{k-1}{2}}-a^{\frac{k-1}{2}}b^{\frac{k+1}{2}}\\
&=(a^{\frac{k-1}{2}}-b^{\frac{k-1}{2}})(a^{\frac{k+1}{2}}-b^{\frac{k+1}{2}})\\
&=8X_{\frac{k-1}{2}}X_{\frac{k+1}{2}}. \end{align*} Thus, by (\ref{eq:ab}), we conclude that $2X_{\frac{k-1}{2}}X_{\frac{k+1}{2}}$ is a perfect square. We know that $X_{\frac{k-1}{2}}$and $X_{\frac{k+1}{2}}$ have different parity. Observe that, for $k\equiv 3$ (mod $4$), $X_{\frac{k-1}{2}}$ is odd and $X_{\frac{k+1}{2}}$ is even. Since for all $i$, $X_i$ and $X_{i+1}$ are coprimes, also $X_{\frac{k-1}{2}}$ and $2X_{\frac{k+1}{2}}$ are coprimes, from which it follows that both $X_{\frac{k-1}{2}}$ and $2X_{\frac{k+1}{2}}$ are perfect squares. Particularly, coming back to equation (\ref{eq:QX}), we obtain \begin{equation}\label{eq:QW} Q^2-2W^4=-1 \end{equation} where $X_{\frac{k-1}{2}}=W^2$. Note that equation (\ref{eq:QW}) is the well known Ljunggren Equation $1+Q^2=2W^4$. Such a Diophantine equation has solutions only for $W=1$ and $W=13$, which correspond respectively to $X_1$ and $X_7$ (because $X_1=1=1^2$ and $X_7=169=13^2$). Therefore, we have two possibilities, either $k=3$ (that is $X_{\frac{3-1}{2}}=X_1$), or $k=15$ (that is $X_{\frac{15-1}{2}}=X_7$). The second case is disclaimed since $X_{\frac{15+1}{2}}=X_8=408=2 \cdot (204)$ and $204$ is not a perfect square. The first case, corresponding to $k=3$, leads to $X_{\frac{3-1}{2}}=X_1=1$ and $X_{\frac{3+1}{2}}=X_2=2$. The solution $(Q_1,X_1)=(1,1)$ gives $q=1$ as we saw in Case 1. The solution $(Q_2,X_2)=(3,2)$ gives $q=2$ (since $Q=2q^2-1$ an $q>0$).
From both cases we conclude that, if $q^2\in S_1\cap S_2$ then either $q=1$ or $q=2$. Hence, $S_1\cap S_2=\{1,4\}$ which concludes the proof. \end{proof}
\section{Conclusions}
While the situation about zero-sum copies of $K_m$ over $\mbox{${\mathbb Z}$}$-weightings is fairly clear now, a lot of interesting results can be proved when the graphs in question are not complete graphs. Several examples are given in \cite{CY3} (for example, certain complete bipartite graphs and many more), and in a forthcoming paper \cite{CHM} which is under preparation.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Concurrence of Two Identical Atoms in a Rectangular Waveguide: Linear Approximation with Single Excitation}
\author{Lijuan \surname{Hu}} \affiliation{Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center of Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China} \author{Guiyuan \surname{Lu}} \affiliation{Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center of Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China} \author{Jing \surname{Lu}} \affiliation{Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center of Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China} \author{Lan \surname{Zhou}} \thanks{Corresponding author} \email{[email protected]} \affiliation{Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center of Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China}
\begin{abstract} We study two two-level systems (TLSs) interacting with a reservoir of guided modes confined in a rectangular waveguide. For the energy separation of the identical TLSs far away from the cutoff frequencies of transverse modes, the delay-differential equations are obtained with single excitation initial in the TLSs. The effects of the inter-TLS distance on the time evolution of the concurrence of the TLSs are examined. \end{abstract}
\pacs{03.65.Yz, 03.65.-w}
\maketitle
\section{Introduction}
Quantum entanglement is a nonlocal correlation of multipartite quantum systems, which distinguishes the quantum world from the classical world. Due to its important role in quantum computation and communication, it is a physical resource which quantum technologies are based on. However, the inevitable interaction of quantum systems with their surrounding environments induces decoherence of quantum systems, which degrades the entanglement of quantum systems. Understanding the dynamics of the entanglement is desirable to be able to manipulate entanglement states in a practical way as well as the question of emergent classicality from quantum theory. Entanglement dynamics is studied under local decoherence (two particles in an entangled state are coupled to its own environment individually), a peculiar dynamical feature of entangled state is that complete disentanglement is achieved in finite time although complete decoherence takes an infinite time, which is termed ``entanglement sudden death''~\cite{ESD1,ESD2}. The assumption of local decoherence requires that two two-level systems (TLSs), e.g. atoms, are sufficiently separated. It is well known that the radiation field emitted by an atom may influence the dynamics of its closely spaced atoms~\cite {Dicke,Lehmberg70,Feng41,Ordon70,BermanPRA76}. The entanglement can be generated in a two-atom system after a finite time by their cooperative spontaneous emission, or the destroyed entanglement may reappear suddenly after its death, which is known as sudden birth of entanglement~\cite{ESB}.
In quantum network, stationary qubits generate, store, and process quantum information at quantum nodes, and flying qubit transmit quantum information between the nodes through quantum channels. A distributed quantum network requires coherently transferring quantum information among stationary qubits, flying qubits, and between stationary qubits and flying qubits. With the development of techniques in quantum information, an alternative waveguide-based quantum electrodynamics (QED) system has emerged as a promising candidate for achieving quantum network~\cite{Tien,HamRMP82,Caruso} . In this system, atoms are located at quantum nodes and photons propagating along the network are confined in a waveguide. Inside a one-dimensional (1D) waveguide, the electromagnetic field is confined spatially in two dimensions and propagates along the remaining one, which is called guided modes. The spectrum of the guided modes is continuous. The coupling of the electromagnetic field to a TLS can be increased by reducing the transverse size of the guided modes. Therefore the study of entanglement dynamics in systems embedded in waveguides is of importance. A waveguide with a cross section has many guided modes~\cite{TETMmode}, e.g. transverse-magnetic (TM) modes or transverse-electric (TE) ones. However, most work only consider one guided mode of the waveguide~\cite {Fans,ZLPRL08,ZLQrouter,Zheng,LawPRA78,TShiSun,PRA14Red,Ordonez}. In this paper, we consider the dynamic behavior of bipartite entanglement involving two identical TLSs which is implanted into the 1D rectangular hollow metallic waveguide. Since local addressing is difficult, we assume that there is no direct interaction between the TLSs, the TLSs and the field share initially a single excitation. By considering the energy separation of the TLSs is far away from the cutoff frequencies of the transverse modes, the delay differential equations are obtained for two TLSs' amplitudes with the field initially in vacuum, where multiple guided modes are included. The spatial separation of the two TLSs introduces the position-dependent phase factor and the time delay (finite time required for light to travel from one TLS to the other) in each transverse mode. The phase factors and the time delays are different in different transverse modes. The effect of the phase factors and the time delays on the entanglement dynamics of the TLSs are studied in details by considering the TLSs interacting with single transverse mode and double transverse modes.
This paper is organized as follows. In Sec.~\ref{Sec:2}, we introduce the model and establish the notation. In Sec.~\ref{Sec:3}, we derive the relevant equations describing the dynamics of the system for the TLSs being initially excited and the waveguide mode in the vacuum state, and investigate the effect of spatial separation on the dynamics of entanglement between two identical TLSs, which is characterized by concurrence. We make a conclusion in Sec.~\ref{Sec:4}.
\section{\label{Sec:2}Two TLSs in a rectangular waveguide}
We consider a rectangular hollow metallic waveguide with the area $A=ab$ ($ a=2b$) of its cross section, as shown in Fig.~\ref{Fig1.eps}. The axes of the waveguide parallel to the $z$ axis, and the waveguide is infinite along the $z$ axis. Since the translation invariance is maintained along the $z$ axis, all the components of the electromagnetic field describing the guided mode depend on the coordinate z as $e^{ikz}$. The guided mode can be characterized by three wave numbers $\{k_{x},k_{y},k_{z}\}$. The spatial confinement of the electromagnetic field along the $xy$ plane makes the appearance of the two non-negative integers $m$ and $n$, which related to wave numbers along the $x$ and $y$ directions by $k_{x}=m\pi /a$ and $ k_{y}=n\pi /b$. In this waveguide, there are two types of guiding modes~\cite{TETMmode,HuangPRA,Shahmoon,liqongPRA,zhouCTP69}:the transverse-magnetic modes $TM_{mn}$ ($H_{z}=0$) and the transverse-electric modes $TE_{mn}$ ($ E_{z}=0$). Two idential TLSs, named TLS $1$ and TLS $2$, are separately located inside the waveguide at positions $\vec{r}_{1}=(a/2,b/2,z_{1})$ and $ \vec{r}_{2}=(a/2,b/2,z_{2})$, the distance between the TLSs is denoted by $ d=z_{2}-z_{1}$. The free Hamiltonian of the TLSs read \begin{equation} H_{a}=\sum\limits_{l=1}^{2}\hbar \omega _{A}\sigma _{l}^{+}\sigma _{l}^{-} \label{2-A1} \end{equation} where $\omega _{A}$ are the energy difference between the excited state $
|e\rangle $ and the ground state $|g\rangle $, and $\sigma _{l}^{+}\equiv \left\vert e_{l}\right\rangle \left\langle g_{l}\right\vert $ ($\sigma _{l}^{-}\equiv \left\vert g_{l}\right\rangle \left\langle e_{l}\right\vert $ ) is the rising (lowing) atomic operator of the $l$-$th$ TLS. We assume the dipoles of TLSs are along the $z$ axis. In this case, only the $TM_{mn}$ guided modes are interacted with the TLSs. The free Hamiltonian of the field reads \begin{equation} H_{f}=\sum_{j}\int dk\hbar \omega _{jk}\hat{a}_{jk}^{\dagger }\hat{a}_{jk} \label{2-A2} \end{equation} where $\hat{a}_{jk}^{\dagger }$ ($\hat{a}_{jk}$) is the creation (annihilation) operator of the $TM_{mn}$ modes. Here,we have replaced $(m,n)$ with the sequence number\ $j$, i.e., $j=1,2,3...$ denoting $ TM_{11},TM_{31},TM_{51}\cdots$, respectively. For each guided mode, the dispersion relation is given by $\omega _{jk}=\sqrt{\Omega _{j}^{2}+c^{2}k^{2}} $, where $\Omega _{mn}=c\sqrt{(m\pi /a)^{2}+(n\pi /b)^{2}}$ is the cutoff frequency. No electromagnetic field can be guided if their frequency is smaller than the cutoff frequency $\Omega _{1}$. The interaction between the TLSs and the the electromagnetic field is written as \begin{equation} H_{int}=\sum_{l=1}^{2}\sum_{j}\int dk\hbar \frac{g_{jl}}{\sqrt{\omega _{jk}}} e^{ikz_{l}}S_{l}^{-}\hat{a}_{k}^{\dagger }+h.c. \label{2-A3} \end{equation} in the electric dipole and rotating wave approximations, where $ g_{jl}=\Omega _{j}\mu _{l}/\sqrt{A\pi \epsilon _{0}}$ and $\mu _{l}$ the magnitude of the dipole of the $l$-th TLS. We assume that $\mu _{1}=\mu _{2}=\mu $ is real. Then the parameter $g_{jl}$ becomes \begin{equation} g_{j}=\frac{\Omega _{j}\mu \sin \left( \frac{m\pi }{2}\right) \sin \left( \frac{n\pi }{2}\right) }{\sqrt{\hbar A\pi \epsilon _{0}}}, \label{2-A4} \end{equation} where $\epsilon _{0}$ is the permittivity of free space. The TLS's position is presented in the exponential function in Eq.(\ref{2-A3}). \begin{figure}
\caption{(color online) Schematic illustration for an infinite waveguide of rectangular cross section $A=ab$ (a) coupling to two TLSs (b) located at $ \vec{r}_{1}=(a/2,b/2,z)$ and $\vec{r}_{2}=(a/2,b/2,z_2)$.}
\label{Fig1.eps}
\end{figure} The total system, the two TLSs and the photons in quantum electromagnetic field, is described by the Hamiltonian \begin{equation} H=H_{f}+H_{a}+H_{int} \label{2-A5} \end{equation} The total system is a closed system. However, each subsystem is an open system. When we are only interested in the dynamics of TLSs, the quantum electromagnetic field can be regarded as an environment.
\section{\label{Sec:3} Entanglement Dynamics}
Any state of the two TLSs are linear superposition of the basis of the separable product states $\left\vert 1\right\rangle =\left\vert g_{1}g_{2}\right\rangle $, $\left\vert 2\right\rangle =\left\vert e_{1}g_{2}\right\rangle $, $\left\vert 3\right\rangle =\left\vert g_{1}e_{2}\right\rangle $, and $\left\vert 4\right\rangle =\left\vert e_{1}e_{2}\right\rangle $. Since the number of quanta is conserved in this system, the wavefunction of the total system can be written as: \begin{equation} \left\vert \psi (t)\right\rangle =b_{1}\left\vert 20\right\rangle +b_{2}\left\vert 30\right\rangle +\sum_{j}\int dkb_{jk}a_{jk}^{\dagger }\left\vert 10\right\rangle \label{2-A6} \end{equation} in single excitation subspace, where $\left\vert 0\right\rangle $ is the vacuum state of the quantum field. The first term in Eq.(\ref{2-A6})presents TLS $1$ in the excited state with no excitations in the field, $b_{1}\left( t\right) $ is the corresponding amplitude, the second term in Eq.(\ref{2-A6} ) presents TLS $2$ in the excited state with photons in the vacuum, whereas the third term in Eq.(\ref{2-A6}) describes all TLSs in the ground state with a photon emitted at a mode $k$ of the TM$_{j}$ guided mode, $ b_{jk}\left( t\right) $ is the corresponding amplitude. The initial state of the system is denoted by the amplitudes $b_{1}\left( 0\right) ,b_{2}\left( 0\right) $, $b_{jk}\left( 0\right) =0$. The Schr\"{o}dinger equation results in the following coupled equation of the amplitudes \begin{subequations} \label{2-A7} \begin{eqnarray} \dot{b}_{1} &=&-i\omega _{A}b_{1}-\sum_{j}\int dk\frac{b_{jk}g_{j}}{\sqrt{ \omega _{jk}}}e^{-ikz_{1}} \\ \dot{b}_{2} &=&-i\omega _{A}b_{2}-\sum_{j}\int dk\frac{b_{jk}g_{j}}{\sqrt{ \omega _{jk}}}e^{-ikz_{2}} \\ \dot{b}_{jk} &=&-i\omega _{jk}b_{jk}+\frac{g_{j}e^{ikz_{1}}}{\sqrt{\omega _{jk}}}\left( b_{1}+b_{2}e^{ikd}\right) \end{eqnarray} We introduce three new variables to remove the high-frequency effect \end{subequations} \begin{subequations} \label{2-A8} \begin{eqnarray} b_{1}(t) &=&B_{1}(t)e^{-i\omega _{A}t}, \\ b_{2}(t) &=&B_{2}(t)e^{-i\omega _{A}t}, \\ b_{jk}(t) &=&B_{jk}\left( t\right) e^{-i\omega _{jk}t}, \end{eqnarray} then, formally integrate equation of $B_{jk}\left( t\right) $, which is later inserted into the equations for $B_{1}\left( t\right) $ and $ B_{2}\left( t\right) $. The probability amplitude for one TLS being excited is determined by two coupled integro-differential equations. Assuming that the frequency $\omega _{A}$ is far away from the cutoff frequencies $\Omega _{j}$, we can expand $\omega _{jk}$ around $\omega _{A}$ up to the linear term \end{subequations} \begin{equation} \omega _{jk}=\omega _{A}+v_{j}\left( k-k_{j0}\right) , \label{2-A10} \end{equation} where the wavelength of the emitted radiation $k_{j0}=\sqrt{\omega _{A}^{2}-\Omega _{j}^{2}}/c$ is determined by $\omega _{jk_{0}}=\omega _{A}$ , and the group velocity \begin{equation}
v_{j}\equiv \frac{d\omega _{jk}}{dk}|_{k=k_{j0}}=\frac{c\sqrt{\omega _{A}^{2}-\Omega _{j}^{2}}}{\omega _{A}} \label{2-A11} \end{equation} is different for different TM$_{j}$ guided modes. Integrating over all wave vectors $k$ gives rise to a linear combination of $\delta \left( t-\tau -\tau _{j}\right) $ and $\delta \left( t-\tau_j \right) $, where $\tau _{j}=d/v_{j}$ is the time delay taking by a photon traveling from one TLS to the other TLS in the given transverse mode $j$. The differential equations governing the dynamics of two TLSs read \begin{subequations} \label{2-A12} \begin{eqnarray} \left( \partial _{t}+\gamma \right) B_{1}(t) &=&-\sum_{j}\gamma _{j}e^{i\varphi _{j}}B_{2}\left( t-\tau_{j}\right) \Theta \left( t-\tau_{j}\right) \\ \left( \partial _{t}+\gamma \right) B_{2}(t) &=&-\sum_{j}\gamma _{j}e^{i\varphi _{j}}B_{1}\left( t-\tau_{j}\right) \Theta \left( t-\tau_{j}\right) \end{eqnarray} where we have defined the phase $\varphi _{j}=k_{j0}d$ due to the distance between the TLSs, the decay rate $\gamma _{j}=\pi \left\vert g_{j}\right\vert ^{2}/(v_{j}\omega _{A})$ caused by the interaction between the TLSs and the vacuum field in a give transverse mode $j$, $\Theta(x)$ is the Heaviside unit step function, i.e., $\Theta(x)=1$ for $x>0$, and $ \Theta(x)=0$ for $x<0$. The decay to all $TM_{j}$ modes is denoted by $ \gamma =\sum_{j}\gamma _{j}$, the retard effect~\cite {Milonni74,cookPRA53,DungPRA59,DornPRA66,RistPRA78,GulfPRA12,JingPLA377} has been implied by the symbol $\tau _{j}$. At times less than minimum $\tau _{j}$, two TLSs decay as if they are isolated in rectangular waveguide. After the time $\min\tau _{j}$ the TLS recognizes the other TLS due to its absorption of photons. As time goes on, reemissions and reabsorptions of photons by two TLSs might produce interference, which leads to the change of atomic upper state population. It is convenient to write Eq.(~\ref{2-A12})
in the Dicke symmetric state $|s\rangle =(|2\rangle +|3\rangle )/\sqrt{2}$
and antisymmetric state $|a\rangle =(|2\rangle -|3\rangle )/\sqrt{2}$ \end{subequations} \begin{subequations} \label{2-A13} \begin{eqnarray} \left( \partial _{t}+\gamma \right) C_{s}(t) &=&-\sum_{j}\gamma _{j}e^{i\varphi _{j}}C_{s}\left( t-\tau_{j}\right) \Theta \left( t-\tau_{j}\right) \\ \left( \partial _{t}+\gamma \right) C_{a}(t) &=&\sum_{j}\gamma _{j}e^{i\varphi _{j}}C_{a}\left( t-\tau_{j}\right) \Theta \left( t-\tau_{j}\right) \end{eqnarray} which allow either TLS 1 or TLS 2 to be excited with equal probability. They are degenerate eigenstates of Hamiltonian $H_a$. The equations for the amplitudes of the Dicke states are not coupled.
\begin{figure}
\caption{(Color online) The concurrence between the TLSs as functions of the dimensionless time $t/\protect\tau_{1}$ with initial condition $C_{s}(0)=1$ for different phase $\protect\varphi _{1}=2n\protect\pi$ (black solid curve), $\protect\varphi _{1}=2n\protect\pi+\protect\pi$ (blue dotted curve), $\protect\varphi _{1}=2n\protect\pi+\protect\pi/2$ (green dot-dashed curve), $\protect\varphi _{1}=2n\protect\pi+\protect\pi/4$ (red dashed curve) in (a)$n=2$, (b)$n=20$, (c)$n=150$. We have set the following parameters: $a=2b$, $\protect\omega_{A}=(\Omega_{11}+\Omega_{31})/2$, $ \protect\gamma _{1}\protect\lambda _{1}/v_{1}=0.05 $.}
\label{Fig2.eps}
\end{figure}
To measure the amount of the entanglement, we use concurrence as the quantifier~\cite{Wootters}. By taking a partial trace over the waveguide degrees of freedom, the initial density matrix of the two TLSs is of an X-form in the two-qubit standard basis $\{\left\vert 1\right\rangle ,\left\vert 2\right\rangle ,\left\vert 3\right\rangle ,\left\vert 4\right\rangle \}$. The concurrence for this type of state can be calculated easily as \end{subequations} \begin{equation} \label{2-A14} C(t)=\max (0,2\left\vert B_{1}(t)B_{2}^{\ast }(t)\right\vert ) \end{equation} which can also expressed as the function of the amplitudes of the Dicke states by the relation \begin{subequations} \label{2-A15} \begin{eqnarray} C_{s}(t) &=&\frac{B_{1}\left( t\right) +B_{2}\left( t\right) }{\sqrt{2}}, \\ C_{a}(t) &=&\frac{B_{1}\left( t\right) -B_{2}\left( t\right) }{\sqrt{2}}. \end{eqnarray} We expect that the position-dependent phase factors $e^{i\varphi _{j}}$ and the delay times $\tau _{j}$ will lead to a modification of the entanglement among the TLSs.
\subsection{Single transverse mode}
In the frequency band between $\Omega _{11}$ and $\Omega _{31}$, the waveguide is said to be single-moded. The TLSs with the transition frequency $\omega _{A}\in \left( \Omega _{11},\Omega _{31}\right) $ only emit photons into the TM$_{11}$ ($j=1$) guided mode. In this case, the time behavior of Dicke states reads \end{subequations} \begin{subequations} \label{2-B1} \begin{eqnarray} C_{s}(t) &=&C_{s0}\sum_{n=0}^{\infty }\frac{\left( -\gamma _{1}e^{i\varphi _{1}}\right) ^{n}}{n!}t_{n}^{n}e^{-\gamma _{1}t_{n}}, \\ C_{a}(t) &=&C_{a0}\sum_{n=0}^{\infty }\frac{\left( \gamma _{1}e^{i\varphi _{1}}\right) ^{n}}{n!}t_{n}^{n}e^{-\gamma _{1}t_{n}}. \end{eqnarray} where $t_{n}=t-n\tau _{1}$, and $C_{s0}$ and $C_{a0}$ are the initial amplitude. The time axis is divided into intervals of length $\tau _{1}$. A step character is presented in Eqs.(\ref{2-B1}). For $t\in \left[ 0,\tau _{1} \right] $, both amplitudes, $C_{s}$ and $C_{a}$, decay exponentially with decay rate $\gamma _{1}$. The underlying physics is that one TLS requires at least the time $\tau _{1}$ to recognize the other TLS. For $t\in \left[ \tau _{1},2\tau _{1}\right] $, the absorption and reemission of light by each TLS produce the interference, which results in a energy change between two TLSs.
\begin{figure}
\caption{(Color online) The concurrence between the TLSs as a function of the dimensionless time $t/\protect\tau _{1}$ with initial condition $ C_{a}(0)=1$ for distance $d=0$ (black solid line), $d=10\protect\lambda _{1}$ (green dot-dashed line), $d=200\protect\lambda _{1}$ (red dashed line). Other parameters are the same as in Fig.~\ref{Fig2.eps}.}
\label{Fig3.eps}
\end{figure}
From Eq.(\ref{2-B1}), one can observe that there is a $\pi $ phase difference between the amplitudes $C_{s}(t)$ and $C_{a}(t)$. Hence, we assume that the two TLSs are initially prepared in the symmetric state $
C_{s0}=1$ to study the effect of the inter-TLS distance on the the dynamics of entanglement between the TLSs. In this case, the concurrence takes the maximum value between $0$ and $|C_{s}(t)|^{2}$. In Fig.~\ref{Fig2.eps}, we have numerically plotted the concurrence as a function of time $t$ in units of $\tau _{1}$ with $\gamma _{1}\lambda _{1}/\tau _{1}=0.05$, where the wavelength $\lambda _{1}k_{10}=2\pi $. In the time interval $t\in \left[ 0,\tau _{1}\right] $, two TLSs radiate spontaneously, so the concurrence decays exponentially with time. As time goes on, the inter-TLS distance have influenced the entanglement dynamics via phase $\varphi _{1}$ and delay time $\tau _{1}$. When the two TLSs are close together, two TLSs act collectively, the system dynamics is independent of the finite propagating time of the light, which has been shown in Fig.~\ref{Fig2.eps}(a) with $ \gamma _{1}\tau _{1}\ll 1$. There is stationary two-TLS entanglement when the inter-TLS distance equals an odd integer number of $\lambda _{1}/2$, (i.e., $\varphi _{1}=2n\pi +\pi $). And a small deviation of the special position leads to the entanglement decaying asymptotically to zero. The entanglement loses fast when the inter-TLS distance equals an integer number of $\lambda _{1}$ corresponding to $\varphi _{1}=2n\pi $. For $\gamma _{1}\tau _{1}\ll 1$, The dependence of the entanglement on phase in Fig.~\ref {Fig2.eps}(a) can be understood by letting $\tau _{1}\rightarrow 0$. In this case, the amplitude of state $\left\vert s\right\rangle $ becomes \end{subequations} \begin{equation} C_{s}(t)=C_{s0}\exp \left[ -t\gamma _{1}(1+\cos \varphi _{1})-it\gamma _{1}\sin \varphi _{1}\right] \label{2-B2} \end{equation}
It can be observed from Eq.~(\ref{2-B2}) that $|C_{s}(t)|$ exponentially decays with time, it decays fast when $\varphi _{1}=2n\pi $ and keeps its initial value when $\varphi _{1}=2n\pi +\pi $. Although one can explain the relation of entanglement with phase by Eq.~(\ref{2-B2}), the probability of finding the TLSs in the initial state is less than unity in Fig.~\ref {Fig2.eps}(a) when $\varphi _{1}=2n\pi +\pi $. Hence, the stationary two-TLS entanglement indicates a superposition of the symmetry state in the absence of photons and the ground TLS state in the presence of a photon. As the inter-TLS separation increases a little bit to meet $\gamma _{1}\tau _{1}\sim 1$ in Fig.~\ref{Fig2.eps}(b), both the decay time and the phase play important roles due to the interference. The interference produced by multiple reemissions and reabsorptions of photon results in an oscillatory entanglement. Panel (c) of Fig.~\ref{Fig2.eps} illustrates the dynamics of entanglement for a larger inter-TLS distance with $\gamma _{1}\tau _{1}\gg 1$ . It can be observed that the phase does not make any sense. At early time, each initially excited TLS emits light to the waveguide, the entanglement begins to decrease abruptly from one. Then the concurrence keeps small and approximates to zero until the time $t=\tau _{1}$, at which the excitation get absorbed by each TLS. As soon as the emitted photon returns to the TLSs, the entanglement is created. Then, the decrease of entanglement begins. The entanglement of the TLSs exhibits peaks due to the iteration of the process where a photon emitted by one of the atoms is reabsorbed by another atom, but its periodic maxima are reduced in magnitude as $t$ increases because the energy is carried away from TLSs by the forward-going waves emitted by TLS 1 and the backward-going waves emitted by TLS 2. In this case, the amplitudes are approximately described by \begin{equation} C_{s}(t)\propto \frac{\left( -\gamma _{1}e^{i\varphi _{1}}\right) ^{n}}{n!} t_{n}^{n}e^{-\gamma _{1}t_{n}} \label{2-BA} \end{equation} in each time interval $\left[ n\tau _{1},\left( n+1\right) \tau _{1}\right]$ . \begin{figure*}
\caption{(Color online) The time evolution of the concurrence between the TLSs as a function of the dimensionless $t/\protect\tau _{1}$ with TLSs initial in the antisymmetry state for phases $\protect\varphi _{1}=2n\protect \pi $ in (a) $n=4$, (b) $n=10$, (c) $n=30$, (d) $n=3000$. Here, the concurrence for $d=0$ is given in the black dot-dashed lines. The concurrence that only TM$_{11}$ mode is considered is presented by blue dashed lines. The concurrence that both TM$_{11}$ and TM$_{31}$ modes are considered is presented by the red solid lines. We have set the following parameters: $a=2b$, $\protect\omega _{A}=(\Omega _{31}+\Omega _{51})/2$, $ \protect\gamma _{1}\protect\lambda _{1}/v_{1}=0.0086$.}
\label{Fig4.eps}
\end{figure*}
For TLSs initially in the antisymmetry state, the concurrence exhibits the similar behavior to the symmetry state with a $\pi$ phase difference. However, when the inter-TLS spacing $d=0$, the antisymmetry state is a dark state which does not interact with the electromagnetic field, it means the the probability of finding the TLSs in the initial state is unity at any time, so the concurrence is unchanged and remains its initial value one (see the solid black line in Fig.~\ref{Fig3.eps}). We also plot the concurrence as a function of the dimensionless time $t/\tau_{1}$ for phase $\varphi _{1}=2n\pi $ with $n=10$ (the green dot-dashed line) and $n=200$ (red dashed line) in Fig.~\ref{Fig3.eps}, where all the position-dependent phase factors $e^{i\varphi_1}$ are equal. It can be seen that the cooperative effect become lower and lower as the inter-TLS distance increases, so does the maximum of concurrence.
\subsection{two transverse modes}
A TLS in its upper state radiates waves into the continua resonant with itself. As the transition frequency of the TLSs increases, there are additional guided modes taking part in the interaction with the TLSs. Due to their different wave numbers, the inter-TLS distance introduces different flight time $\tau _{j}$ of light between the TLSs as well as different phases $\varphi _{j}$. From the definition of $\tau _{j}$ and $\varphi _{j}$ , we know that $\tau _{j}<\tau _{j+1}$ and $\varphi _{j}<\varphi _{j+1}$ for the given distance $d$. In this section, we assume that the transition frequency of the TLSs is smaller than the cutoff frequency $\Omega _{51}$ and larger than the cutoff frequency $\Omega _{31}$, this means that the emit photons will propagate in guided modes TM$_{11}$ and TM$_{51}$. The delay-differential equation Eq.(~\ref{2-A13}) reduces to \begin{subequations} \label{2-B3} \begin{eqnarray} \left( \partial _{t}+\gamma \right) C_{s}(t) &=&-\alpha _{1}C_{s}\left( t-\tau _{1}\right) \Theta \left( t-\tau _{1}\right) \notag \\ &&-\alpha _{2}C_{s}\left( t-\tau _{2}\right) \Theta \left( t-\tau _{2}\right) \\ \left( \partial _{t}+\gamma \right) C_{a}(t) &=&\alpha _{1}C_{a}\left( t-\tau _{1}\right) \Theta \left( t-\tau _{1}\right) \notag \\ &&+\alpha _{2}C_{a}\left( t-\tau _{2}\right) \Theta \left( t-\tau _{2}\right) \end{eqnarray}
where $\gamma =\gamma _{1}+\gamma _{2}$, and $\alpha _{j}=\gamma _{j}e^{i\varphi _{j}}$ ($j=1,2$). We first discuss the case with $d=0$. It is clear by inspection of Eq.~(\ref{2-B3}) that the initial entanglement determined by the state $|s\rangle $ decreases more quickly in time, on the contrary, the initial entanglement determined by the state $|a\rangle $ does not change in time as shown by the black lines in Fig.~\ref{Fig4.eps}.
To analyze the influence of the number of the transverse modes on the entanglement which interact with TLSs, we fix the phase $\varphi _{1}=2n\pi $ with $n=4$ in Fig.~\ref{Fig4.eps}(a), $n=10$ in Fig.~\ref{Fig4.eps}(b), $ n=30 $ in Fig.~\ref{Fig4.eps}(c), $n=3000$ in Fig.~\ref{Fig4.eps}(d). We plotted the time behavior of the concurrence between TLS by only considering the TM$_{11}$ mode in Eq.(\ref{2-B3}), which is shown in blue dashed lines in Fig.~\ref{Fig4.eps}. The red solid lines in Fig.~\ref{Fig4.eps} present the time behavior of the concurrence by considering both TM$_{11}$ and TM$ _{31}$ modes in Eq.~(\ref{2-B3}). It can be found that increasing the number of the transverse modes which interact with TLSs leads to exponentially decay with a rate $\gamma _{1}+\gamma _{2}$ up to time $t=\tau _{1}$. After this, the phase $\varphi _{1}$ begins to have an effect until time $t=\tau _{2}$. After time $\tau _{2}$, the dynamics can be dramatically affected by the phases $\varphi _{j}$ and delay times $\tau _{j}$ induced by the inter-TLS distance. The blue lines of panels (a) and (b) in Fig.~\ref {Fig4.eps} show that the concurrence remains constant in time after the time $t=\tau _{1}$ in TM$_{11}$ mode, which indicate that the delay time $\tau _{1}$ is negligibly small. As the phase factor $e^{i\varphi _{1}}$ is fixed, the behavior of the red solid lines is completely determined by the phase $ \varphi _{2}$ and delay time $\tau _{2}$ in TM$_{31}$ mode. We see from panel (a) that the entanglement decays almost exponentially in time after delay time $\tau _{2}$, which means that a part of the emitted energy from one TLS is transferred directly to another TLS, so there is no delay in the absorption of the energy by another TLS in both TM$_{11}$ and TM$_{31}$ modes. In this case, we can solve Eq.(\ref{2-B3}) by letting $\tau _{1},\tau _{2}\rightarrow 0$ \end{subequations} \begin{subequations} \label{2-B4} \begin{eqnarray} C_{s}(t) &=&C_{s0}\exp \left[ -(\gamma _{1}+\alpha _{1})t-(\gamma _{2}+\alpha _{2})t\right] , \\ C_{a}(t) &=&C_{a0}\exp \left[ -(\gamma _{1}-\alpha _{1})t-(\gamma _{2}-\alpha _{2})t\right] . \end{eqnarray} The norm of amplitudes are completely determined by phases leading to exponential decay of the entanglement. The red solid line in panel (b) exhibits behavior different from that in panel (a), indicating that the $ \varphi _{2}$ and delay time $\tau _{2}$ play an equal role. The part of excitation emitted into TM$_{11}$ mode is immediately reabsorbed by the other one, but the part of excitation emitted into TM$_{31}$ mode undergoes delay, however, wave interference still produced at each exchanges of the excitation between the TLSs in TM$_{31}$ mode. In this case, we can solve Eq.(\ref{2-B3}) by letting $\tau _{1}\rightarrow 0$ \end{subequations} \begin{subequations} \label{2-B5} \begin{eqnarray} &&C_{s}=C_{s0}\sum_{n=0}^{\infty }\frac{\left( -\alpha _{2}\right) ^{n}}{n!} e^{-\left( \gamma +\alpha _{1}\right) \left( t-n\tau _{2}\right) }\left( t-n\tau _{2}\right) ^{n} \\ &&C_{a}=C_{a0}\sum_{n=0}^{\infty }\frac{\alpha _{2}^{n}}{n!}e^{-\left( \gamma -\alpha _{1}\right) \left( t-n\tau _{2}\right) }\left( t-n\tau _{2}\right) ^{n} \end{eqnarray} Panel (c) shows that as $d$ increased, the delay time should be taken into account in TM$_{11}$ mode besides the wave interference. In this case, we can use Laplace transformation and geometric series expansion to solve Eq.( \ref{2-B3}), the solutions \end{subequations} \begin{subequations} \label{2-B6} \begin{eqnarray} C_{s} &=&C_{s0}\sum_{n=0}^{\infty }\sum_{k=0}^{n}C_n^k\alpha _{1}^{k}\alpha _{2}^{n-k}\frac{\left(\tau _{nk}-t\right) ^{n}}{n!}e^{-\gamma \left( t-\tau _{nk}\right) }, \\ C_{a} &=&C_{a0}\sum_{n=0}^{\infty }\sum_{k=0}^{n}C_n^k\alpha _{1}^{k}\alpha _{2}^{n-k}\frac{\left( t-\tau _{nk}\right) ^{n}}{n!}e^{-\gamma \left( t-\tau _{nk}\right) }, \end{eqnarray} are coherent sums over contributions starting at different instants of time $ \tau _{nk}=k\tau _{1}+\left( n-k\right) \tau _{2}$, where $C_n^k=\frac{k!}{
n!(n-k)!}$. Each term of the sum has a well-defined phase, and are damped by an exponential function at rate $\gamma$. Interference is possible if the amplitudes do not decay appreciably over the time $\tau_{nk}$.As $d$ is large enough so that $\min_{p,q}{\gamma|p\tau_2-q\tau_1|}\gg 1$ with non-negative integers $p$ and $q$ which are not zero at the same time,the phase factor plays no role as shown in panel (d). The wave packets of the emitted excitation is bouncing back and forth between two TLSs until its intensity is damped to zero. So there are collapses and revivals of the concurrence until the amplitude of the revivals damped to zero.
\section{\label{Sec:4} conclusion}
We have studied the effects of the inter-TLS distance on the entanglement properties of two identical TLSs located inside a rectangular hollow metallic waveguide of transverse dimensions $a$ and $b$. When the energy separation of the TLS is far away from the cutoff frequencies of the transverse modes and there is single excitation in the system, the Schrodinger equation for the wave function with single excitation initial in the TLSs is reduced to the delay differential equations for the amplitudes of two TLSs, where phase factors and delay times are induced by the finite distance between the TLSs. The delay differential equations are solved exactly for the TLSs interacting with either single transverse mode or double transverse modes of the waveguide, which directly reveals the retarded character of multiple reemissions and reabsorptions of photons between the TLSs. For the inter-TLS distance $d=0$, there exists an anti-symmetry state decoupled with the field modes, so the entanglement can be generated if the TLSs are initial in a separate state, later trapped in the anti-symmetry state. As the TLSs are close together such that the time delay $\max \{\tau _{j}\}$ is much smaller than the TLS decay time $\gamma
^{-1}$, the excitation emitted by one TLS into the field is absorbed immediately by the other. The dynamic of the entanglement are dramatically affected by phases, leading to an enhanced and inhibited exponential decay of the concurrence when only one transverse mode are considered, and an exponential decay when more transverse modes are involved. As the inter-TLS distance increases, both phases and delay times affect the concurrence. There is a proper delay of reabsorption after reemission of photons but interference is possible if the amplitudes of TLSs do not decay appreciably over time $\tau_{nk}$. As $d$ is large enough so that $\min_{p,q}{\gamma|p\tau_2-q\tau_1|}\gg 1$ with non-negative integers $p$ and $q$ which are not zero at the same time,the phase factor plays no role. There appear collapses and revivals of the entanglement of the TLSs. We note that our studies focus on the the dependence of the concurrence on the inter-TLS distance but it is easy to study the dependence of the concurrence on the initial state of the system with the exact solution.
\begin{acknowledgments} This work was supported by NSFC Grants No. 11434011, No. 11575058. \end{acknowledgments}
\end{subequations}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Greenberger-Horne-Zeilinger theorem for $N$ qudits}
\author{Junghee Ryu}
\affiliation{Institute of Theoretical Physics and Astrophysics, University of Gda\'{n}sk, 80-952 Gda\'{n}sk, Poland} \affiliation{Department of Physics, Hanyang University, Seoul 133-791, Korea}
\author{Changhyoup Lee}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543} \affiliation{Department of Physics, Hanyang University, Seoul 133-791, Korea}
\author{Marek \.{Z}ukowski}
\affiliation{Institute of Theoretical Physics and Astrophysics, University of Gda\'{n}sk, 80-952 Gda\'{n}sk, Poland}
\author{Jinhyoung Lee}
\affiliation{Department of Physics, Hanyang University, Seoul 133-791, Korea} \affiliation{Center for Macroscopic Quantum Control, Seoul National University, Seoul, 151-742, Korea}
\begin{abstract} We generalize Greenberger-Horne-Zeilinger (GHZ) theorem to an arbitrary number of $D$-dimensional systems. Contrary to conventional approaches using compatible composite observables, we employ incompatible and concurrent observables, whose common eigenstate is still a generalized GHZ state. It is these concurrent observables which enable to prove a genuinely $N$-partite and $D$-dimensional GHZ theorem. Our principal idea is illustrated for a four-partite system with $D$ which is an arbitrary multiple of $3$. By extending to $N$ qudits, we show that GHZ theorem holds as long as $N$ is not divisible by all nonunit divisors of $D$, smaller than $N$. \end{abstract} \pacs{} \maketitle
\newcommand{\bra}[1]{\langle #1\vert} \newcommand{\ket}[1]{\vert #1\rangle} \newcommand{\abs}[1]{\vert#1\vert} \newcommand{\avg}[1]{\langle#1\rangle}
\newcommand{\braket}[2]{\langle{#1}|{#2}\rangle} \newcommand{\commute}[2]{\left[{#1},{#2}\right]}
\newtheorem{theorem}{Theorem}
\section{Introduction}
The inconsistency of (local) hidden variable theories with quantum mechanics fascinates many researchers. It has been discussed in many theoretical~\cite{Bell64,*Clauser69,*Specker60,*Kochen67,*Leggett03} and experimental works~\cite{Aspect82,*Lapkiewicz11,*Groeblacher07}. Bell's theorem, one of the most profound discoveries concerning the foundations of quantum mechanics, states that any local realistic theory is incompatible with quantitative predictions of quantum mechanics. Even though Bell's theorem was studied mostly in terms of statistical inequalities, a more striking conflict, without inequalities, was also shown for a multiqubit system by Greenberger, Horne, and Zeilinger (GHZ)~\cite{GHZ89,*Pan12}. They derived an {\it all-versus-nothing} contradiction based on perfect correlations for so-called GHZ states. This leads to a direct refutation of Einstein-Podolsky-Rosen (EPR) ideas on the relation between locality and elements of reality with quantum mechanics~\cite{EPR35}.
This is a striking blow right into the very basic ideas linked with local hidden variables. After all, EPR used the concept of (local) elements of reality to support their claim that quantum mechanics is incomplete. All this can be best explained using the three particle GHZ paradox. Take a state $\ket{GHZ}=\frac{1}{\sqrt{2}}(\ket{+++}-\ket{---})$, where $\ket{\pm}$ denotes states associated with the eigenvalues $\pm1$ of the local Pauli $\sigma_z$ operator. The
operators $\sigma_x\otimes\sigma_x\otimes\sigma_x$, $\sigma_x\otimes\sigma_y\otimes\sigma_y$, $\sigma_y\otimes\sigma_x\otimes\sigma_y$, and $\sigma_y\otimes\sigma_y\otimes\sigma_x$ all commute, and their eigenstate is $|GHZ\rangle$. The eigenvalues are $-1$, $1$, $1$, and $1$, respectively, which signify perfect (GHZ)-EPR correlations. This would please any local realist. Assume that the particles are far away from each other, and three distant independent observers can perform experiments on them, choosing at will the observables. For example, the first and the second one may choose $\sigma_x$ and their measurement results are $1$ and $-1$, respectively. In such a case, they can together predict with certainty what would have been the result of the third observer had he or she chosen to measure also $\sigma_x$. Simply the local results must multiply to the eigenvalue of the joint observable $\sigma_x\otimes\sigma_x\otimes\sigma_x$, and this is $-1$. Thus, the third observer, if the hypothetical case of him or her choosing to measure $\sigma_x$ really happens, must for sure get $1$. Thus, as EPR would say, this value is an {\em element of reality}, because in no way the distant choices, and obtained results can influence anything that happens at the location of the third observer (especially if measurements actions, are spatially separated events in the relativistic sense of this term, and the measurement choices are made some time after the particles are emitted from a common, say central, source). For such counterfactual reasonings one can use any of the four perfect correlation cases for the joint measurements given above, and apply to each observer. Thus, it seems that one can ascribe elements of reality of to all local situations, no matter whether the local observable is $\sigma_x$ or $\sigma_y$. Note that these are incommensurable. Let us denote such elements of reality, related with a single emission act of three particles, by $r_{w}^k$, where $w=x,y$ denotes the observable, and $k=1,2,3$ denotes the observer. Obviously $r_{w}^k=\pm 1. $ For the four cases of GHZ-EPR perfect correlations one therefore must have $r_{x}^1r_{x}^2r_{x}^3=-1$, and $r_{x}^1r_{y}^2r_{y}^3=1$, $r_{y}^1r_{x}^2r_{y}^3=1$, $r_{y}^1r_{y}^2r_{x}^3=1$. If one multiplies these four relations side by side, one gets $1=-1$. Thus an attempt of introducing EPR elements of reality leads to a nonsense. {\em Ergo}, elements of reality are a nonsense. No other argument against local realism could be more striking.
Extending Bell's theorem to more complex systems such as multipartite and/or high-dimensional systems ~\cite{Mermin90c,*Werner01,*Zukowski02,*Collins02,*Laskowski04,*Son06,*James10} is important not only for a deeper understanding of foundations of quantum mechanics. It is associated with developing new applications in quantum information processing, such as quantum cryptography, secret sharing, quantum teleportation, reduction of communication complexity, quantum key distribution, and random numbers generation~\cite{Ekert91,Horodecki96,Cleve97, *Brukner04,Zukowski98,*Hillery99,*Kempe99,*Scarani01,*Barrett05,*Acin07,Pironio10}. Similarly to Bell's theorem, also all-versus-nothing tests, which we call GHZ theorem, have been generalized to higher dimensional systems. For the sake of convenience, we shall use the tuple $(N, M, D)$ to denote $N$ parties, $M$ measurements for each party, and $D$ distinct outcomes for each measurement. In Ref.~\cite{Zukowski99}, the GHZ theorem was derived for a $(D+1, 2, D)$ problem. A probabilistic but conclusive GHZ-like test was shown for $(D,2,D)$ in Ref.~\cite{Kaszlikowski02}. The $(N,2,D)$ problem for odd $N>D$ and even $D$ was studied by Cerf {\it et al.}~\cite{Cerf02a}. Lee {\it et al.} showed the GHZ theorem for more general cases, $(\mbox{odd}~N, 2, \mbox{even}~D)$, by an unconventional approach using incompatible observables~\cite{Lee06}. Recently, Tang {\it et al.} generalized GHZ theorem to the $N (\geq4)$-partite case and even-$D$ dimensional systems with the help of GHZ graphs~\cite{Tang13}. Despite such an intensive progress in extending GHZ theorem, many cases of $N$-partite and $D$-dimensional systems remain still as open problems.
We generalize the GHZ theorem to three or higher $D$-dimensional systems. To this end, we employ concurrent composite observables which, in contrast with the standard approach, are mutually incompatible but still have a common eigenstate, here a generalized GHZ state. They can be realized by multiport beam splitters and phase shifters, as it is shown in Refs.~\cite{Zukowski99,Lee06}. We first illustrate our principal idea with four $3d$-dimensional systems and then provide a systematic method, so as to extend it to three or higher $D$-dimensional systems. Finally, we show a GHZ-type contradiction, as long as $N$ is not divided by all nonunit divisors of $D$, smaller than $N$. Our generalization is genuinely $N$-partite {\it and} $D$-dimensional and can reproduce the previous results~\cite{GHZ89,Cerf02a,Zukowski99,Lee06}. This approach can lead to a general GHZ theorem for $N$ qudits.
\section{Concurrent observables}{\label{sec_concurrent}}
Some sets of observables have a common eigenstate. If a system is prepared in the eigenstate, the measurement results for such observables are concurrently appearing with certainty. Such observables are called ``concurrent''~\cite{Lee06}. For a quantum system of dimension $D (>2)$, consider two Hermitian operators $\hat{A}$ and $\hat{B}$ such that $\hat{A}=a\ket{\psi}\bra{\psi}+\hat{A}_{\psi}^{\perp}$ and $\hat{B}=b\ket{\psi}\bra{\psi}+\hat{B}_{\psi}^{\perp}$ with $\hat{A}_{\psi}^{\perp}(\hat{B}_{\psi}^{\perp}) \ket{\psi}=0$. The state $\ket{\psi}$ is then a common eigenstate of both observables as $\hat{A}(\hat{B})\ket{\psi}=a(b)\ket{\psi}$, even if $[\hat{A},\hat{B}]=[\hat{A}_{\psi}^{\perp},\hat{B}_{\psi}^{\perp} ]\neq0$~\footnote{Note that compatible observables are clearly concurrent.}.
Such concurrent observables can be constructed by the method introduced in Ref.~\cite{Lee06}. Consider a unitary operator $\hat{U}$, which is of the form of $\hat{U}=e^{i \phi} \ket{\psi}\bra{\psi}+\hat{U}_{\psi}^{\perp}$ with $\hat{U}_{\psi}^{\perp} \ket{\psi}=0$. Here $\hat{U}_{\psi}^{\perp}$ is a unitary operator on a space $\mathcal{H}_{\psi}^{\perp}$ which is defined by the requirement $\mathcal{H} =\mathcal{H}_{\psi} \oplus \mathcal{H}_{\psi}^{\perp}$, where $\mathcal{H}_{\psi}$ is the one-dimensional space containing $\ket{\psi}$. Every such unitary operator leaves the state $\ket{\psi}$ unchanged, up to a global phase: If the state $\ket{\psi}$ satisfies $\hat{A}\ket{\psi}=\lambda \ket{\psi}$, then all transformed operators $\hat{B}_{U}=\hat{U} \hat{A} \hat{U}^{\dagger}$ are concurrent with $\hat{A}$.
Consider $N$ qudits prepared in a generalized GHZ state $\ket{\psi}=\frac{1}{\sqrt{D}}\sum_{n=0}^{D-1} \bigotimes_{k=1}^{N} \ket{n}_{k}$, where $\ket{n}$ denotes a basis state for a qudit. This GHZ state is a common eigenstate with the unity eigenvalue of any composite observable $\hat{X}^{\otimes N} \equiv \hat{X}\otimes \hat{X} \otimes \cdots \otimes \hat{X}$ as $\hat{X}^{\otimes N} \ket{\psi}=\ket{\psi}$, where the local observable $\hat{X}$ is defined by applying quantum Fourier transformation $\hat{F}$ on a reference unitary observable $\hat{Z}=\sum_{n=0}^{D-1} \omega^n \ket{n} \bra{n}$ with $\omega=\exp({2 \pi i/D})$, that is $\hat{X}=\hat{F}\hat{Z}\hat{F}^{\dagger}$~\footnote{Here, we shall use local unitary observables $\hat{V}=\sum_{n=0}^{D-1}\omega^n \left|n\right>_{\mathrm{v}}\left<n\right|$, where $\omega=\exp(2 \pi i/D)$, in $D$-dimensional Hilbert space. The observable $\hat{V}$ is unitary, however, one can uniquely relate it with a Hermitian observable $\hat{H}$ by requiring $\hat{V}=\exp(i \hat{H})$. Therefore the complex eigenvalues of $\hat{V}$ can be associated with the measurement results, denoted by real eigenvalues of $\hat{H}$. Such a unitary representation leads to a simplification of mathematics without changing any physical results~\cite{Cerf02a, Lee06}.}. An eigenvector of $\hat{X}$ associated with the eigenvalue $\omega^n$ is given by $ \ket{n}_\mathrm{x} =\hat{F}\ket{n} = \frac{1}{\sqrt{D}} \sum_{m=0}^{D-1} \omega^{nm} \ket{m}$. With the standard basis set $\{\ket{n} \}$, the observable $\hat{X}$ is written as $\hat{X} = \sum_{n=0}^{D-1} \ket{n}\bra{n+1}$, where $\ket{n} \equiv \ket{n \mod D}$.
To construct a set of concurrent composite observables, we employ a unitary operation in the form of $\hat{U}= \bigotimes_{k=1}^{N} \hat{P}_{k} (f_k)$ with a phase shifter $\hat{P}_k=\sum_{n=0}^{D-1} \omega^{f_k(n)} \ket{n}\bra{n}$. If ``phases" $f_{k}(n)$ satisfy a condition, \begin{equation} \sum_{k=1}^{N} f_{k} (n) \equiv 0 \mod D, \label{invariant_condition} \end{equation} for each $n$, then the unitary operator $\hat{U}$ leaves the GHZ state $\ket{\psi}$ invariant. This simple invariance condition enables one to construct a large number of concurrent observables which have a common eigenstate of the generalized GHZ state.
Let us apply the unitary operation $\hat{U}$ with the phases $f_{k}(n)=\alpha_{k} n$ with rational numbers $\alpha_{k}$ to $\hat{X}^{\otimes N}$. If the phases $f_{k} (n)$ satisfy the invariance condition~(\ref{invariant_condition}), the transformed observable $\hat{U} \hat{X}^{\otimes N} \hat{U}^{\dagger}=\hat{X}(\alpha_1)\otimes \hat{X}(\alpha_2)\otimes \cdots \otimes\hat{X}(\alpha_N)$ is concurrent with $\hat{X}^{\otimes N}$, i.e., $\hat{U} \hat{X}^{\otimes N} \hat{U}^{\dagger} \ket{\psi}=\ket{\psi}$. For each eigenvalue $\omega^{n}$, the eigenvector of the local observable $\hat{X}({\alpha})$ is given by applying the phase shifter $\hat{P}$ on $\ket{n}_{\mathrm{x}}$ as $\ket{n}_{\alpha} = \hat{P}\ket{n}_\mathrm{x}= \frac{1}{\sqrt{D}}\sum_{m=0}^{D-1} \omega^{(n+\alpha)m} \ket{m}$. The observable $\hat{X}({\alpha})$ can be written in the standard basis set $\{\ket{n} \}$ as \begin{equation} \hat{X}({\alpha})=\omega^{-\alpha} \left( \sum_{n=0}^{D-2} \ket{n}\bra{n+1} + \omega^{\alpha D} \ket{D-1}\bra{0} \right). \label{obs_y} \end{equation} Note that if $\alpha$ is an integer, the measurement basis set $\{ \ket{n}_{\alpha}\}$ of $\hat{X}(\alpha)$ will be the same as $\{\ket{n}_\mathrm{x} \}$ of $\hat{X}$ except the ordering, i.e., $\ket{n}_{\alpha} = \ket{n + \alpha}_\mathrm{x} $. Thus, $\hat{X}(\alpha) = \omega^{-\alpha} \hat{X}$. That is, the observable $\hat{X}(\alpha)$ is equivalent to $\hat{X}$, up to a phase factor $\omega^{-\alpha}$. Let $\ket{n}_{\alpha}$ be the eigenstate of $\hat{X}(\alpha)$ associated with eigenvalue $\omega^n$, and $\ket{m}_{\beta}$ the eigenstate of $\hat{X}(\beta)$ associated with $\omega^m$. If and only if $\alpha$ differs from $\beta$ by an integer, then two measurement bases satisfy $\abs{{}_{\alpha}\bra{n} m \rangle_{\beta}}^2 = \delta_{D} (\gamma)$, where $\gamma=m-n+\beta-\alpha$. Here $\delta_{D}(\gamma)=1$ if $\gamma$ is congruent to zero modulo $D$ and otherwise $\delta_{D}(\gamma)=0$. That is, if $\beta - \alpha$ is not an integer, two local observables $\hat{X}(\alpha)$ and $\hat{X}(\beta)$ are inequivalent.
\section{Generalized GHZ theorem}{\label{sec_4partite}}
\subsection{Four-qudit system} We first illustrate our idea by considering a four-qudit system. Already this case goes significantly beyond the previous studies~\cite{Zukowski99, Cerf02a, Lee06, Tang13}. Take a four-qudit GHZ state $\ket{\psi}=\frac{1}{\sqrt{D}}\sum_{n=0}^{D-1} \ket{n,n,n,n}$ (here $D$ is assumed to be an integral multiple of 3).
The qudits are distributed to four sufficiently separated parties. Each party performs one of two nondegenerate local measurements on his or her qudit, each of which produces distinguishable $D$ outcomes. We represent the measurement for the $k$-th party by $\hat{M}_k$, and the eigenvalues of the observables are of the form $\omega^{m_k}$, where $m_k$ is an integer. One can denote a joint probability that each party obtains the result $\omega^{m_k}$ by $\mathcal{P}(m_1,m_2,m_3,m_4)$, and define a correlation function $E_{\mathrm{QM}} (M_1,M_2,M_3,M_4)=\bra{\psi}\hat{M}_1\otimes\hat{M}_2\otimes\hat{M}_3\otimes\hat{M}_4\ket{\psi}$, equivalently the quantum average of products of the measurement results: $\sum_{m_{1}=0}^{D-1}\cdots\sum_{m_{4}=0}^{D-1} \omega^{\sum_{i=1}^{4} m_i} \mathcal{P} (m_1,m_2,m_3,m_4)$.
When all measurements are $\hat{X}$, that is, each $\hat{M}_i=\hat{X}$, since $\hat{X} \otimes \hat{X} \otimes \hat{X} \otimes \hat{X} \ket{\psi} = \ket{\psi}$, one has $ E_{\mathrm{QM}} (X,X,X,X)=1$. This implies that we have a perfect GHZ correlation. If arbitrary three parties know their own outcomes $\omega^{x_i}$ of measurements $\hat{X}$, then they can predict with certainty the remaining party's outcome. We will denote such a perfect correlation by \begin{equation} C_{\mathrm{QM}} (x_1 + x_2 + x_3 + x_4 \equiv 0 ). \label{eq:perfect_correlation} \end{equation} The sum is taken modulo $D$; such a convention is used below in all formulas.
Let us construct concurrent composite observables from the observable $\hat{v}_0 = \hat{X}^{\otimes 4}$, by applying a unitary operator $\hat{U}_{1}=\hat{P}_{1}\otimes\hat{P}_{2}^{\otimes3}$, with the phase shifters $\hat{P}_k$ of phases $f_{1}(n)=(D-1)n$ and $f_{2}(n)=n/3$. One of the new observables is $\hat{v}_{1} \equiv \hat{U}_{1} \hat{v}_{0} \hat{U}_{1}^{\dagger} = \hat{X}(D-1) \otimes \hat{X}(1/3)^{\otimes 3}$. The phases $f_k (n)$ satisfy the invariance condition~(\ref{invariant_condition}), $f_1 (n) + 3 f_2 (n) \equiv 0 \mod D$. Thus, the observable $\hat{v}_1$ has $|\psi\rangle $ as its eigenstate with eigenvalue $1$. By Eq.~(\ref{obs_y}) one has $\hat{X}(D-1) =\omega \hat{X}(0)$, and $\hat{Y} \equiv \hat{X}(1/3)$ is given by \begin{equation} \hat{Y}=\omega^{-1/3} \left( \sum_{n=0}^{D-2} \ket{n}\bra{n+1} + \omega^{D/3} \ket{D-1}\bra{0} \right). \label{4_obs_y} \end{equation} The observable $\hat{Y}=\hat{X}(1/3)$ is not equivalent to $\hat{X}=\hat{X}(0)$ [see the explanation below Eq.~(\ref{obs_y}]. The other three concurrent observables are obtained by $\hat{v}_l \equiv \hat{U}_{l} \hat{v}_0 \hat{U}_l^{\dagger},~l\in \{2,3,4\}$, where the unitary operators $\hat{U}_l$ are composed of the cyclic permutations of the local phase shifters: $\hat{U}_{2}=\hat{P}_{2}\otimes\hat{P}_{1}\otimes\hat{P}_{2}\otimes\hat{P}_{2}, \hat{U}_{3}=\hat{P}_{2}\otimes\hat{P}_{2}\otimes\hat{P}_{1}\otimes\hat{P}_{2}$, and $\hat{U}_{4}=\hat{P}_{2}\otimes\hat{P}_{2}\otimes\hat{P}_{2}\otimes\hat{P}_{1}$. The perfect correlations for $\hat{v}_{i}$ $(i=1,\dots,4)$ are, respectively, given by \begin{eqnarray} &C_{\mathrm{QM}} (x_1 + y_2 + y_3 + y_4 \equiv -1) &, \nonumber \\ &C_{\mathrm{QM}} (y_1 + x_2 + y_3 + y_4 \equiv -1) &, \nonumber \\ &C_{\mathrm{QM}} (y_1 + y_2 + x_3 + y_4 \equiv -1) &, \nonumber \\ &C_{\mathrm{QM}} (y_1 + y_2 + y_3 + x_4 \equiv -1) &, \label{prob_QM} \end{eqnarray} where $\omega^{y_i}$ is an outcome of measurement $\hat{Y}$ for the $i$-th party.
Local realistic theories assume that the outcomes of the measurements are predetermined, before the actual measurements. This implies that the values of local realistic predictions for the correlations (\ref{prob_QM}), for each experimental run, must satisfy: \begin{eqnarray} & \omega^{x_1}\omega^{y_2}\omega^{ y_3}\omega^{y_4} = \omega^{-1},& \nonumber \\ & \omega^{y_1}\omega^{x_2}\omega^{ y_3}\omega^{y_4} = \omega^{-1},& \nonumber \\ & \omega^{y_1}\omega^{y_2}\omega^{ x_3}\omega^{y_4} = \omega^{-1},& \nonumber \\ & \omega^{y_1}\omega^{y_2}\omega^{ y_3}\omega^{x_4} = \omega^{-1}.& \label{eq_LHV2} \end{eqnarray} If local realism is also to reproduce quantum perfect correlations for the case of $\hat{v}_0 = \hat{X} ^{\otimes4}$, one must have
for each run, \begin{eqnarray} & \omega^{x_1}\omega^{x_2}\omega^{ x_3}\omega^{x_4} =1.& \label{eq_LHV2B} \end{eqnarray} However, one sees that this is possible only provided $\omega^{ -3 \sum_{i=1}^{4} y_{i}-4} =1 $ if one multiplies side-by-side all equations (\ref{eq_LHV2}). As $\omega=\exp(2\pi i/D) $, if $D=3d$ where $d$ is an integer, one can use the fact that an elementary algebra shows that there is no integer solution of the equation $3{\bf y} + 4 \equiv 0 \mod D$. Thus local realistic correlation (\ref{eq_LHV2B}) is impossible.
It is worth noting that the approach with concurrent observables relaxes the restrictions of early studies requiring {\it compatible} observables. This enables one to generalize GHZ contradictions beyond the case $N>D$ studied in Refs.~\cite{Zukowski99,Cerf02a}.
Note, that to prove the four-partite GHZ contradiction, we chose the local dimension $D$ and the number of the observables $\hat{Y}$'s in the considered correlation functions, $N_2$, such that the greatest common divisor (gcd) of $D$ and $N_2$, here $\mathrm{gcd}(D,N_2)=3$, does not divide the number of parties $N=4$, or equivalently the number $N_1$ of the observables $\hat{X}$ (here equal to $1$). This mathematical property plays a central role in the generalization of the GHZ contradiction to an arbitrary number of parties.
\subsection{Extending to $N$ qudits system}{\label{sec_extend}}
We extend our approach into a general case of $N$ qudits, $N\geq3$, such that $N$ is nondivisible by any nonunit divisor of $D$, smaller than $N$. To this end, we use a set of $(N+1)$ concurrent observables given by $\hat{v}_{0}=\hat{X}^{\otimes N}$ and $N$ observables of the following forms:
$\hat{v}_{1}=\omega \hat{X}^{\otimes N_1} \otimes \hat{Y}^{\otimes N_2}$ and
$\hat{v}_{k}= \hat{Y}^{\otimes k-1}\otimes \omega \hat{X}^{\otimes N_1} \otimes \hat{Y}^{\otimes N_2-k+1}$ for $k=2, \dots, N_2 +1$ and finally
$\hat{v}_{k}= \hat{X}^{\otimes k-N_2 -1} \otimes \hat{Y}^{\otimes N_2}\otimes \omega \hat{X}^{\otimes N-k+1} $ for $k=N_2+2,\dots, N$.
The composite observable $\hat{v}_1$ is obtained by a unitary transformation, $\hat{U}_{1}=\hat{P}_{1}\otimes \openone^{\otimes N_1 -1} \otimes \hat{P}_{2}^{\otimes N_2}$, of the observable $\hat{v}_{0}$, i.e., $\hat{v}_{1}=\hat{U}_{1}\hat{v}_{0}\hat{U}_{1}^{\dagger}$, with the phase shifters $\hat{P}_1$ and $\hat{P}_2$ of phases $f_{1} (n)=(D-1)n$ and $f_{2}(n)=n/N_2$, respectively. The local observables $\hat{X}=\hat{X}(0)$ and $\hat{Y}=\hat{X}(1/N_{2})$ are given by Eq.~(\ref{obs_y}). Likewise, we obtain the other concurrent observables $\hat{v}_{l}~(2 \leq l \leq N)$ by cyclic permutations in the unitary operator $\hat{U}_{1}$, as it was done for the four-partite case. The phases satisfy the invariance condition~(\ref{invariant_condition}) as $f_1 (n) + N_2 f_2 (n) \equiv 0 \mod D$. Thus, the $N$-partite generalized GHZ state $\ket{\psi}=\frac{1}{\sqrt{D}}\sum_{n=0}^{D-1} \bigotimes_{k=1}^{N} \ket{n}_{k}$ is a common eigenstate of all the $(N+1)$ concurrent observables $\hat{v}_l$ $(l=0,\dots,N)$, with the same eigenvalue $1$. This leads to the following values of correlation functions (for later convenience we use party indices $i=1, \dots, N$, which will later on allow us to get a more concise notation in formulas). If all local observables are $\hat{X}$, that is, for global $\hat{v}_0$, one has $E_{\mathrm{QM}} (X,X,\dots,X) = 1$. Thus we have a perfect correlation which can be denoted, in the way introduced earlier, as $C_{\mathrm{QM}} (\sum_{i=1}^{N}x_i \equiv 0)$. For $\hat{v}_k$, where $k=1,2,\dots,N$, one has perfect correlations of the following forms: for $k=1$, $C_{\mathrm{QM}} ( \sum_{i=1}^{N_1}x_i + \sum_{i=N_1+1}^{N}y_{i} \equiv -1)$,
and for $k=2,\dots,N_2 +1$, $C_{\mathrm{QM}} ( \sum_{i=1}^{k-1}y_i + \sum_{i=k}^{N_1+k-1}x_{i} + \sum_{i=N_1+k}^{N}y_{i} \equiv -1)$,
and finally for $k=N_2+2,\dots, N$, $C_{\mathrm{QM}} ( \sum_{i=1}^{k-N_2 -1}x_i + \sum_{i=k-N_2}^{k-1}y_{i} + \sum_{i=k}^{N}x_{i} \equiv -1)$.
Following similar arguments as in the case of the four-partite GHZ contradiction, we obtain the following condition for the local realistic correlation function for the composite observable $\hat{v}_{0}=\hat{X}^{\otimes N}$, to have value equal to the quantum prediction, that is, 1. It reads (modulo $D$) \begin{equation} N_1 \sum_{i=1}^{N} x_{i} \equiv -N_2 \sum_{i=1}^{N} y_{i} - N\equiv 0 . \label{general_LR_condition} \end{equation} However, if $N_2$ is an integral multiple of $g$ but $N$ cannot be divided by $g$, then there are no solutions of ${\bf y}=\sum_{i} y_{i}$ to the equation $N_2 {\bf y} + N \equiv 0 \mod D$. The greatest common divisor of $N_2$ and $D$ is an integral multiple of $g$, i.e., gcd($N_2, D$)=$kg$ for some positive integer $k$ but $kg$ cannot divide $N$ as $N$ is not an integer multiple of $g$. Thus we have a contradiction with the quantum prediction.
In order to show a GHZ contradiction for $N$-partite and $D$-dimensional system, we choose that (a) $D=dg$, (b) $N_2 = \eta g$, where $d$ and $\eta$ are positive integers, and (c) $N$ cannot be divided by $g$. Choosing the integer $g$, a nonunit divisor $D$, plays a crucial role. For example, consider {\em four} six-dimensional systems. The nonunit divisors $g$, smaller than $N=4$, are $2$ and $3$. If we choose $g=2$, then we are unable to see any four-partite GHZ contradiction as the greatest common divisor (gcd) of $N_2$ and $D$, $\mathrm{gcd}(N_2 = 2, D=6)=2$, divides $N=4$. On the other hand, if we choose $g=3$, the four-partite GHZ contradiction can be proved as $\mathrm{gcd}(N_2 =3 , D=6)=3$ and $g=3$ does not divide $N=4$. This is a specific example of a GHZ contradiction for a $(4,2,3d)$ problem. As a consequence, we conclude that one is always able to prove the GHZ contradiction for the $N(\geq 3)$-partite and $D(\geq 2)$-dimensional systems as long as $N$ cannot be divided by all nonunit divisors of $D$.
For appropriate values of $N$ and $D$, our approach reproduces the previous works~\cite{Cerf02a,Zukowski99,Lee06}. A GHZ contradiction for $(D+1)$ qudits shown in Ref.~\cite{Zukowski99} is reproduced by choosing $N_1=1$ and $N_2=D$ in our method, and noticing the fact that $N_1=1$ is indivisible by $D=\mathrm{gcd} (N_2 =D, D)$. The case of $(\mathrm{odd}~N, 2, \mathrm{even}~D)$ studied in Refs.~\cite{Cerf02a,Lee06} can be also proved by choosing a nonunit divisor $g=2$ of $D$ and an arbitrary odd integer $N_1$. One can also easily check that if $N_2 = D=2$ and $N_1 =1$, then our contradiction is reduced to the original GHZ theorem~\cite{GHZ89}.
\subsection{Genuinely $N$-partite $D$-dimensional case}
The GHZ theorem for the two-dimensional systems seems to be fully understood. However, for more complex systems this is not so. Cerf {\it et al.} suggested a criterion for a genuinely $N$-partite ($D$-dimensional) GHZ contradiction~\cite{Cerf02a}. It arises only for the given full $N$-partite ($D$-dimensional) system, but not for any $n(<N)$-partite subset, or for an effectively lower dimensionality of the involved observables. For example, the three-qubit classic GHZ theorem can be put as the theorem for three qutrits, and specific entangled GHZ states involving only two-dimensional subspaces for each qutrit.
The GHZ contradiction we show here is a genuinely $N$-partite one, as it is constructed using a set of composite observables composed of cyclic permutations. Let us explain this with the four-partite GHZ contradiction, where we used the five concurrent composite observables: $\hat{X} \otimes \hat{X} \otimes \hat{X} \otimes \hat{X}$, $\hat{X} \otimes \hat{Y} \otimes \hat{Y} \otimes \hat{Y}$, $\hat{Y} \otimes \hat{X} \otimes \hat{Y} \otimes \hat{Y}$, $\hat{Y} \otimes \hat{Y} \otimes \hat{X} \otimes \hat{Y}$, and $\hat{Y} \otimes \hat{Y} \otimes \hat{Y} \otimes \hat{X}$. In such circumstances, if we eliminate one of the parties, we are unable to show a GHZ contradiction with the remaining observables. The four-partite GHZ state is no longer their common eigenstate. Similar argument can be put forward in the case of our $N$-partite theorem.
The genuine $D$ dimensionality is reflected by the fact that the operators are undecomposable to a direct sum of any subdimensional observables~\cite{Lee06}. In other words, if two local observables $\hat{X}$ and $\hat{Y}$ can be simultaneously block diagonalized by some similarity transformation $\hat{S}$ such that $\hat{S}\hat{X}\hat{S}^{\dagger}=\hat{X}_1 \oplus \cdots \oplus \hat{X}_K$ and $\hat{S}\hat{Y}\hat{S}^{\dagger}=\hat{Y}_1 \oplus \cdots \oplus \hat{Y}_K$, then there exist some eigenstates $\ket{n}_{\alpha}$ of $\hat{X}$ and $\ket{m}_{\beta}$ of $\hat{Y}$ such that ${}_{\alpha}\bra{n} \hat{S}^{\dagger} \hat{S} \ket{m}_{\beta}=0$ and one can find a sub-dimensional GHZ contradiction. However, there are no such eigenstates in our method because for every $n$ and $m$, $\abs{{}_{\alpha}\langle n \ket{m}_{\beta}}^2 = \frac{\sin^2 (\pi \xi)}{D^2 \sin^2 \left[(\pi/D) \xi \right]} > 0$,
where $\xi=m-n+\beta-\alpha$. As, the local observables $\hat{X}=\hat{X}(\alpha)$ and $\hat{Y}=\hat{X}(\beta)$ are such that $\beta - \alpha$ is not an integer, $\xi$ is a nonintegral rational number. Thus, our GHZ contradiction is genuinely $D$ dimensional.
\section{Summary}
We construct a generalized GHZ contradiction for {\it multipartite} and {\it high}-dimensional systems. The GHZ theorem holds as long as $N$ is not divisible by all nonunit divisors of $D$, smaller than $N$. We also demonstrate that our formulation of a generalized GHZ contradiction is genuinely $N$ partite {\it and} $D$ dimensional. For this purpose, we employ concurrent composite observables, which have a generalized GHZ state as a common eigenstate (even though these observables are {\it incompatible}). Our approach, by using concurrent observables, enables us to find a broader class of GHZ contradictions. There remain still more possibilities for constructing concurrent observables, which may help in further extension of the GHZ theorem. We hope that our approach of concurrent observables would be useful in the search of other kinds of quantum correlations, which are impossible classically.
\acknowledgements We thank \v{C}. Brukner for discussions. The work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (Grants No. 2010-0015059 and No. 2010-0018295). J.R. and M.\.{Z}. are supported by the Foundation for Polish Science TEAM project cofinanced by the EU European Regional Development Fund and a NCBiR-CHIST-ERA Project QUASAR. C.L. is supported by the National Research Foundation and Ministry of Education, Singapore.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Global Optimization with Parametric Function Approximation}
\begin{abstract} We consider the problem of global optimization with noisy zeroth order oracles — a well-motivated problem useful for various applications ranging from hyper-parameter tuning for deep learning to new material design. Existing work relies on Gaussian processes or other non-parametric family, which suffers from the curse of dimensionality. In this paper, we propose a new algorithm GO-UCB that leverages a parametric family of functions (e.g., neural networks) instead. Under a realizable assumption and a few other mild geometric conditions, we show that GO-UCB achieves a cumulative regret of $\tilde{O}(\sqrt{T})$ where $T$ is the time horizon. At the core of GO-UCB is a carefully designed uncertainty set over parameters based on gradients that allows optimistic exploration. Synthetic and real-world experiments illustrate GO-UCB works better than Bayesian optimization approaches in high dimensional cases, even if the model is misspecified. \end{abstract}
\section{Introduction}\label{sec:intro} We consider the problem of finding a global optimal solution to the following optimization problem \begin{align*} \max_{x \in \mathcal{X}} f(x), \end{align*} where $f: \mathcal{X} \rightarrow \mathbb{R}$ is an unknown non-convex function that is not necessarily differentiable in $x$.
This problem is well-motivated by many real-world applications. For example, the accuracy of a trained neural network on a validation set is complex non-convex function of a set of hyper-parameters (e.g., learning rate, momentum, weight decay, dropout, depth, width, choice of activation functions ...) that one needs to maximize \citep{kandasamy2020tuning}. Also in material design, researchers want to synthesize ceramic materials, e.g., titanium dioxide ($\mathrm{TiO}_2$) thin films, using microwave radiation \citep{nakamura2017design} where the film property is a non-convex function of parameters including temperature, solution concentration, pressure, and processing time. Efficiently solving such non-convex optimization problems could significantly reduce energy cost.
We assume having access to only noisy function evaluations, i.e., at round $t$, we select a point $x_t$ and receive a noisy function value $y_t$, \begin{align} y_t = f(x_t) + \eta_t,\label{eq:y} \end{align} where $\eta_t$ for $t=1,...,T$ are \emph{independent}, \emph{zero-mean}, $\sigma$-\emph{sub-Gaussian} noise. This is known as the \emph{noisy zeroth-order oracle} setting in optimization literature. Let $f^*$ be the optimal function value, following the tradition of Bayesian optimization (see e.g., \citet{frazier2018tutorial} for a review), throughout this paper, we use \emph{cumulative regret} as the evaluation criterion, defined as \begin{align*} R_T = \sum_{t=1}^T r_t = \sum_{t=1}^T f^* - f(x_t), \end{align*} where $r_t$ is called instantaneous regret at round $t$. An algorithm $\mathcal{A}$ is said to be a no-regret algorithm if $\lim_{T \rightarrow \infty} R_T(\mathcal{A})/T=0$.
Generally speaking, solving a global non-convex optimization is NP-hard \citep{jain2017non} and we need additional assumptions to efficiently proceed. Bayesian optimization assumes the objective function $f$ is drawn from a Gaussian process prior. \citet{srinivas2010gaussian} proposed the GP-UCB approach, which iteratively queries the argmax of an upper confidence bound of the current posterior belief, before updating the posterior belief using the new data point. However, Gaussian process relies on kernels, e.g., squared error kernel or Mat\'ern kernel, which suffer from the curse of dimensionality. A folklore rule-of-thumb is that GP-UCB becomes unwieldy when the dimension is larger than $10$.
A naive approach is to passively query $T$ data points uniformly at random, estimate $f$ by $\hat{f}$ using supervised learning, then return the maximizer of the plug-in estimator $\hat{x} =\mathop{\mathrm{argmax}}_{x \in \mathcal{X}}\hat{f}(x)$. This may side-step the curse-of-dimensionality depending on which supervised learning model we use. The drawback of this passive query model is that it does not consider the structure of the function nor does it quickly ``zoom-in'' to the region of the space that is nearly optimal. In contrast, an active query model allows the algorithm to iteratively interact with the function. At round $t$, the model collects information from all previous rounds $1,...,t-1$ and decides where to query next.
\textbf{GO-UCB Algorithm.} In this paper, we develop an algorithm that allows Bayesian optimization-style active queries to work for general supervised learning-based function approximation. We assume that the supervised learning model $f_w:\mathcal{X} \rightarrow \mathbb{R}$ is differentiable w.r.t. its $d_w$-dimensional parameter vector $w\in \mathcal{W}\subset \mathbb{R}^{d_w}$ and that the function class $\mathcal{F}=\{f_w | w\in\mathcal{W}\}$ is flexible enough such that the true objective function $f = f_{w^*}$ for some $w^*\in\mathcal{W}$, i.e., $\mathcal{F}$ is \emph{realizable}. Our algorithm --- \emph{Global Optimization via Upper Confidence Bound} (GO-UCB) --- has two phases:
\fbox{\parbox{0.9\columnwidth}{The \textit{GO-UCB} Framework: \small \noindent \begin{itemize}
\item Phase I: Uniformly explore $n$ data points.
\item Phase II: Optimistically explore $T$ data points. \end{itemize}}}
The goal of Phase I to sufficiently explore the function and make sure the estimated parameter $\hat{w}_0$ is close enough to true parameter $w^*$ such that exploration in Phase II are efficient. To solve the estimation problem, we rely on a regression oracle that is able to return an estimated $\hat{w}_0$ after $n$ observations. In details, after Phase I we have a dataset $\{(x_j,y_j)\}_{j=1}^{n}$, then \begin{align} \hat{w}_0 \leftarrow \mathop{\mathrm{argmin}}_{w\in\mathcal{W}} \sum_{j=1}^n (f_w(x_j) - y_j)^2.\label{eq:regression_oracle} \end{align} This problem is known as a \emph{non-linear least square} problem. It is computationally hard in the worst-case, but many algorithms are known (e.g., SGD, Gauss-Newton, Levenberg-Marquardt) to effectively solve this problem in practice. Our theoretical analysis of $\hat{w}_0$ uses techniques from \citet{nowak2007complexity}. See Section \ref{sec:mle} for details.
In Phase II, exploration is conducted following the principle of ``Optimism in the Face of Uncertainty'', i.e., the parameter is optimized within an uncertainty region that always contains the true parameter $w^*$. Existing work in bandit algorithms provide techniques that work when $f_w$ is a linear function \citep{abbasi2011improved} or a generalized linear function \citep{li2017provably}, but no solution to general differentiable function is known. At the core of our GO-UCB is a carefully designed uncertainty ball $\mathrm{Ball}_t$ over parameters based on gradients, which allows techniques from the linear bandit \citep{abbasi2011improved} to be adapted for the non-linear case. In detail, the ball is defined to be centered at $\hat{w}_t$ --- the solution to a regularized online regression problem after $t-1$ rounds of observations. And the radius of the ball is measured by the covariance matrix of the gradient vectors of all previous rounds. We prove that $w^*$ is always trapped within the ball with high probability.
\textbf{Contributions.} In summary, our main contributions are: \begin{enumerate} \item We initiate the study of global optimization problem with parametric function approximation and proposed a new optimistic exploration algorithm --- GO-UCB. \item Under realizable assumption and geometric conditions, we prove GO-UCB converges to the global optima with cumulative regret at the order of $\tilde{O}(\sqrt{T})$ where $T$ is the time horizon. \item GO-UCB does not suffer from the curse of dimensionality like Gaussian processes-based Bayesian optimization methods. The unknown objective function $f$ can be high-dimensional, non-convex, non-differentiable, and even discontinuous in its input domain. \item Synthetic test function and real-world hyperparameter tuning experiments show that GO-UCB works better than all compared Bayesian optimization methods in both realizable and misspecified settings. \end{enumerate}
\textbf{Technical novelties.} The design of GO-UCB algorithm builds upon the work of \citet{abbasi2011improved} and \citet{agarwal2021rl}, but requires substantial technical novelties as we handle a generic nonlinear parametric function approximation. Specifically: \begin{enumerate}
\item LinUCB analysis (e.g., self-normalized Martingale concentration, elliptical potential lemmas \citep{abbasi2011improved,agarwal2021rl}) is not applicable for nonlinear function approximation, but we showed that they can be adapted for this purpose if we can \emph{localize} the learner to a neighborhood of $w^*$.
\item We identify a new set of structural assumptions under which we can localize the learner sufficiently with only $O(\sqrt{T})$ rounds of pure exploration.
\item Showing that $w^*$ remains inside the parameter uncertainty ball $\mathrm{Ball}_t, \forall t \in [T]$ is challenging. We solve this problem by setting regularization centered at the initialization parameter $\hat{w}_0$ and presenting novel inductive proof of a lemma showing $\forall t\in [T], \hat{w}_t$ converges to $w^*$ in $\ell_2$-distance at the same rate. \end{enumerate} These new techniques could be of independent interest.
\section{Related Work}\label{sec:rw} Global non-convex optimization is an important problem that can be found in a lot of research communities and real-world applications, e.g., optimization \citep{rinnooy1987stochastic1,rinnooy1987stochastic2}, machine learning \citep{bubeck2011x,malherbe2017global}, hyperparameter tuning \citep{hazan2018hyperparameter}, neural architecture search \citep{kandasamy2018neural,wang2020learning}, and material discovery \citep{frazier2016bayesian}.
One of the most prominent approaches to this problem is Bayesian Optimization (BO) \citep{shahriari2015taking}, in which the objective function is modeled by a Gaussian Process (GP) \citep{williams2006gaussian}, so that the uncertainty can be updated under the Bayesian formalism. Among the many notable algorithms in GP-based BO \citep{srinivas2010gaussian,jones1998efficient,bull2011convergence,frazier2009knowledge,agrawal2013thompson,cai2021on}, GP-UCB \citep{srinivas2010gaussian} is the closest to our paper because our algorithm also selects data points in a UCB (upper confidence bound) style but the construction of the UCB in our paper is different since we are not working with GPs. \citet{scarlett2017lower} proves lower bounds on regret for noisy Gaussian process bandit optimization. GPs are highly flexible and can approximate any smooth functions, but such flexibility comes at a price to play --- curse of dimensionality. Most BO algorithms do not work well when $d>10$. Notable exceptions include the work of \citet{shekhar2018gaussian,calandriello2019gaussian,eriksson2019scalable,salgia2021domain,rando2022ada} who designed more specialized BO algorithms for high-dimensional tasks.
Besides GP, other nonparametric families were considered for global optimization tasks, but they, too, suffer from the curse of dimensionality. We refer readers to \citet{wang2018optimization} and the references therein.
Our problem is also connected to the bandits literature \citep{li2019nearly,foster2020beyond,russo2013eluder,filippi2010parametric}. The global optimization problem can be written as a nonlinear bandits problem in which queries points as actions and the function evaluations are rewards. However, no bandits algorithms can simultaneously handle an infinite action space and a generic nonlinear reward function. Here ``generic'' means the reward function is much more general than a linear or generalized linear function \citep{filippi2010parametric}. To the best of our knowledge, we are the first to address the infinite-armed bandit problems with a general differentiable value function (albeit with some additional assumptions).
A recent line of work studied bandits and global optimization with neural function approximation \citep{zhou2020neural,zhang2020neural,dai2022sample}. The main difference from us is that these results still rely on Gaussian processes with a Neural Tangent Kernel in their analysis, thus intrinsically linear. Their regret bounds also require the width of the neural network to be much larger than the number of samples to be sublinear. In contrast, our results apply to general nonlinear function approximations and do not require overparameterization.
\section{Preliminaries}\label{sec:pre}
\subsection{Notations}
We use $[n]$ to denote the set $\{1,2,...,n\}$. The algorithm queries $n$ points in Phase I and $T$ points in Phase II. Let $\mathcal{X} \subset \mathbb{R}^{d_x}$ and $\mathcal{Y} \subset \mathbb{R}$ denote the domain and range of $f$, and $\mathcal{W} \subset [0, 1]^{d_w}$ denote the parameter space of a family of functions $\mathcal{F} := \{f_w: \mathcal{X}\rightarrow \mathcal{Y} | w\in \mathcal{W}\}$. For convenience, we denote the bivariate function $f_w(x)$ by $f_x(w)$ when $w$ is the variable of interest. $\nabla f_x(w)$ and $\nabla^2 f_x(w)$ denote the gradient and Hessian of function $f$ w.r.t. $w$. $L(w) := \mathbb{E}_{x \sim \mathcal{U}} (f_x(w)-f_x(w^*))^2$ denotes the (expected) risk function where $\mathcal{U}$ is uniform distribution. For a vector $x$, its $\ell_p$ norm is denoted by $\|x\|_p = (\sum_{i=1}^d |x_i|^p)^{1/p}$ for $1\leq p < \infty$.
For a matrix $A$, its operator norm is denoted by $\|A\|_\mathrm{op}$.
For a vector $x$ and a square matrix $A$, define $\|x\|^2_A = x^\top A x$. Throughout this paper, we use standard big $O$ notations; and to improve the readability, we use $\tilde{O}$ to hide poly-logarithmic factors. For reader's easy reference, we list all symbols and notations in Appendix \ref{sec:table}.
\subsection{Assumptions} Here we list main assumptions that we will work with throughout this paper. The first assumption says that we have access to a differentiable function family that contains the unknown objective function.
\begin{assumption}[Realizability]\label{ass:parameter_class} There exists $w^*\in \mathcal{W}$ such that the unknown objective function $f= f_{w^*}$. Also, assume $\mathcal{W}\subset [0,1]^{d_w}$. This is w.l.o.g. for any compact $\mathcal{W}$. \end{assumption} Realizable parameter class is a common assumption in literature \citep{chu2011contextual,foster2018practical,foster2020beyond}, usually the starting point of a line of research for a new problem because one doesn't need to worry about extra regret incurred by misspecified parameter. Although in this paper we only theoretically study the realizable parameter class, our GO-UCB algorithm empirically works well in misspecified tasks too. The second assumption is on properties of function approximation. \begin{assumption}[Bounded, differentiable and smooth function approximation]\label{ass:objective} There exist constants $F, C_g, C_h > 0$ such that
$\forall x \in \mathcal{X}, \forall w \in \mathcal{W}$, it holds that $|f_x(w)|\leq F,$ \begin{align*}
\|\nabla f_x(w) \|_2 \leq C_g, \quad\text{ and }\quad
\|\nabla^2 f_x(w)\|_\mathrm{op} \leq C_h. \end{align*} \end{assumption}
The third assumption is on the expected loss function over uniform distribution in Phase I of GO-UCB.
\begin{assumption}[Geometric conditions on the loss function]\label{ass:loss} $L(w)=\mathbb{E}_{x\sim \mathcal{U}}(f_x(w)-f_x(w^*))^2$ satisfies $(\tau, \gamma)$-growth condition or local $\mu$-strong convexity at $w^*$, i.e., $\forall w \in \mathcal{W}$, \begin{align*}
\min\left\{\frac{\mu}{2}\|w-w^*\|_2^2,\frac{\tau}{2}\|w-w^*\|_2^\gamma \right\} \leq L(w)-L(w^*). \end{align*} Also, $L(w)$ is a self-concordant function at $w^*$. \end{assumption}
\begin{figure}
\caption{Example of a highly non-convex $L(w)$ satisfying Assumption \ref{ass:loss}. Solid lines denote the actual lower bound by taking $\min$ over strong convexity and growth condition. $L(w)$ is strongly convex near $w^*$ but can be highly non-convex away from $w^*$. }
\label{fig:loss}
\end{figure}
\begin{remark}
This assumption is strictly weaker than global strongly convex because it does not require convexity except in a local neighborhood near the global optimal $w^*$, i.e., $\{w|\|w - w^*\|_2 \leq (\tau/\mu)^\frac{1}{2-\gamma}\}$ and it does not limit the number of spurious local minima, as the global $\gamma$ growth condition only gives a mild lower bound as $w$ moves away from $w^*$. See Figure \ref{fig:loss} for an example. Our results work even if $\gamma$ is a small constant $<1$. \end{remark}
Note the assumption is only made on the expected loss function w.r.t. uniform distribution $\mathcal{U}$ as a function of $w$, rather than objective function $f_{w^*}(x)$. The problem to be optimized can still be arbitrarily complex in terms of $\mathcal{X}$, e.g., high-dimensional and non-continuous functions. As an example, in Gaussian process based Bayesian optimization approaches, $f_{w^*}(x)$ belongs to a reproducing kernel Hilbert space, but its loss function is globally convex in its ``infinite dimensional'' parameter $w$. Also, we no longer need this assumption in Phase II.
Self-concordance is needed for technical reasons, but again it is only required near $w^*$. It means that for a given point $w$ such that $\|w - w^*\|_{\nabla^2 L(w^*)} \leq 1$, then \begin{align*}
(1-\|w - w^*\|_{\nabla^2 L(w^*)})^2 \nabla^2 L(w^*) \preceq \nabla^2 L(w) \preceq \frac{\nabla^2 L(w^*)}{(1-\|w-w^*\|_{\nabla^2 L(w^*)})^2}. \end{align*} See Example 4 of \citet{zhang2017improved} for some self-concordance function examples.
\textbf{Additional notations.} For convenience, we define $\zeta >0$ such that $\|\nabla^2 L(w^*)\|_\mathrm{op} \leq \zeta$. The existence of a finite $\zeta$ is implied by Assumption~\ref{ass:objective} and it suffices to take $\zeta = 2C_g^2$ because $\nabla^2 L(w^*) = \mathbb{E}_{x\sim \mathcal{U}}[ 2\nabla f_x(w^*) \nabla f_x(w^*)^\top]$.
\section{Main Results}\label{sec:main} In Section \ref{sec:alg}, we state our Global Optimization with Upper Confidence Bound (GO-UCB) algorithm and explain key design points of it. Then in Section \ref{sec:regret_bound}, we prove that its cumulative regret bound is at the rate of $\tilde{O}(\sqrt{T})$.
\subsection{Algorithm}\label{sec:alg}
Our GO-UCB algorithm, shown in Algorithm \ref{alg:go_ucb}, has two phases. Phase I does uniform exploration in $n$ rounds and Phase II does optimistic exploration in $T$ rounds. In Step 1 of Phase I, $n$ is chosen to be large enough such that the objective function can be sufficiently explored. Step 2-3 are doing uniform sampling. In Step 5, we call regression oracle to estimate $\hat{w}_0$ given all observations in Phase I as in eq. \eqref{eq:regression_oracle}. Adapted from \citet{nowak2007complexity}, we prove the convergence rate of $\|\hat{w}_0 - w^*\|_2$ is at the rate of $\tilde{O}(1/\sqrt{n})$. See Theorem \ref{thm:mle_guarantee} for details.
\begin{algorithm}[!b] \caption{GO-UCB}
\label{alg:go_ucb}
{\bf Input:}
Time horizon $T$, uniform exploration phase length $n$, uniform distribution $\mathcal{U}$, regression oracle $\mathrm{Oracle}$, regularization weight $\lambda$,
confidence sequence $\beta_t$ for $t=1,2,...,T$.\\
{\bf Phase I} (Uniform exploration)
\begin{algorithmic}[1]
\FOR{$j = 1,...,n$}
\STATE Sample $x_j \sim \mathcal{U}(\mathcal{X})$.
\STATE Observe $y_j = f(x_j) + \eta_j$.
\ENDFOR
\STATE Estimate $\hat{w}_0 \leftarrow \mathrm{Oracle}(x_1, y_1, ..., x_n, y_n)$.
\end{algorithmic}
{\bf Phase II} (Optimistic exploration)
\begin{algorithmic}[1]
\FOR{$t = 1,...,T$}
\STATE Update $\Sigma_t$ by eq. \eqref{eq:sigma_t} with the input $\lambda$.
\STATE Update $\hat{w}_t$ by eq. \eqref{eq:inner} with the input $\lambda$.
\STATE Update $\mathrm{Ball}_t$ by eq. \eqref{eq:ball} with the input $\beta_t$.
\STATE Select $x_t=\mathop{\mathrm{argmax}}_{x \in \mathcal{X}} \max_{w \in \mathrm{Ball}_t} f_x(w)$.
\STATE Observe $y_t = f(x_t) + \eta_t$.
\ENDFOR
\end{algorithmic}
{\bf Output:}
$\hat{x} \sim \mathcal{U} (\{x_1, ..., x_T\})$. \end{algorithm}
The key challenge of Phase II of GO-UCB is to design an acquisition function to select $x_t, \forall t \in [T]$. Since we are using parametric function to approximate the objective function, we heavily rely on a feasible parameter uncertainty region $\mathrm{Ball}_t, \forall t \in [T]$, which should always contain the true parameter $w^*$ throughout the process. The shape of $\mathrm{Ball}_t$ is measured by the covariance matrix $\Sigma_t$, defined as \begin{align} \Sigma_t = \lambda I + \sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top.\label{eq:sigma_t} \end{align} Note $i$ is indexing over both $x$ and $w$, which means that as time $t$ goes from $0$ to $T$, the update to $\Sigma_t$ is always rank one. It allows us to bound the change of $\Sigma_t$ from $t=0$ to $T$.
$\mathrm{Ball}_t$ is centered at $\hat{w}_t$, the newly estimated parameter at round $t$. In Step 2, we update the estimated $\hat{w}_t$ by solving the following optimization problem: \begin{align}
\hat{w}_t &= \mathop{\mathrm{argmin}}_{w}\frac{\lambda}{2} \|w - \hat{w}_0\|_2^2 + \frac{1}{2} \sum_{i=0}^{t-1} ((w-\hat{w}_i)^\top \nabla f_{x_i}(\hat{w}_i) + f_{x_i}(\hat{w}_i) - y_i)^2.\label{eq:opt_inner} \end{align} The optimization problem is an online regularized least square problem involving gradients from all previous rounds, i.e., $\nabla f_{x_i}(\hat{w}_i), \forall i \in [T]$. The intuition behind it is that we use gradients to approximate the function since we are dealing with generic objective function. We set the regularization w.r.t. $\hat{w}_0$ rather than $0$ because from regression oracle we know how close is $\hat{w}_0$ to $w^*$. By setting the gradient of objective function in eq. \eqref{eq:opt_inner} to be $0$, the closed form solution of $\hat{w}_t$ is \begin{align} \hat{w}_t &= \Sigma^{-1}_t \left(\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) (\nabla f_{x_i}(\hat{w}_i)^\top \hat{w}_i +y_i - f_{x_i}(\hat{w}_i)) \right) + \lambda \Sigma^{-1}_t \hat{w}_0.\label{eq:inner} \end{align}
Now we move to our definition of $\mathrm{Ball}_t$, shown as \begin{align}
\mathrm{Ball}_t = \{w: \|w-\hat{w}_t\|^2_{\Sigma_t} \leq \beta_t\},\label{eq:ball} \end{align} where $\beta_t$ is a pre-defined monotonically increasing sequence that we will specify later. Following the ``optimism in the face of uncertainty'' idea, our ball is centered at $\hat{w}_t$ with $\beta_t$ being the radius and $\Sigma_t$ measuring the shape. $\beta_t$ ensures that the true parameter $w^*$ is always contained in $\mathrm{Ball}_t$ w.h.p. In Section \ref{sec:ball}, we will show that it suffices to choose \begin{align} \beta_t &= \tilde{\Theta}\bigg(d_w \sigma^2 + \frac{d_w F^2}{\mu} + \frac{d^3_w F^4t}{\mu^2 T}\bigg),\label{eq:beta_t} \end{align} where $\tilde{\Theta}$ hides logarithmic terms in $t, T$ and $1/\delta$ (w.p. $1-\delta$).
Then in Step 5 of Phase II, $x_t$ is selected by joint optimization over $x \in \mathcal{X}$ and $w \in \mathrm{Ball}_t$. Finally, we collect all observations in $T$ rounds and output $\hat{x}$ by uniformly sampling over $\{x_1,...,x_T\}$.
\subsection{Regret Upper Bound}\label{sec:regret_bound} Now we present the cumulative regret upper bound of GO-UCB algorithm.
\begin{theorem}[Cumulative regret of GO-UCB]\label{thm:cr} Suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold. Let $C$ denote a universal constant and $C_\lambda$ denote a constant that is independent to $T$. Assume \begin{equation}\label{eq:n_lower_main}
T > C \max \left\{ \frac{2^{\gamma-2} d_w F^2\zeta^\frac{\gamma}{2} \iota}{\tau}, \frac{d_w F^2 \zeta \iota}{\mu} \right\}^2, \end{equation} where $\iota$ is a logarithmic term depending on $n, C_h, 2/\delta$. Algorithm~\ref{alg:go_ucb} with parameters $n = \sqrt{T}$, $\lambda = C_\lambda \sqrt{T}$ and $\beta_{1:T}$ as in eq. \eqref{eq:beta_t} obeys that with probability $> 1-\delta$, \begin{align} R_{\sqrt{T}+T} &= \tilde{O}\left(\sqrt{T} F + \sqrt{T\beta_T d_w + T\beta^2_T}\right) = \tilde{O}\left(\frac{d^3_w F^4 \sqrt{T}}{\mu^2}\right),\label{eq:final_R_T} \end{align} where $d_w, F$ are constants. \end{theorem}
Let us highlight a few interesting aspects of the result. \begin{remark} Without Gaussian process assumption, we propose the first algorithm to solve global optimization problem with $\tilde{O}(\sqrt{T})$ cumulative regret, which is dimension free in terms of its input domain $\mathcal{X}$. GO-UCB is a no-regret algorithm since $\lim_{T \rightarrow \infty} R_T/T =0$, and the output $\hat{x}$ satisfies that $f^* - \mathbb{E}[ f(\hat{x})] \leq \tilde{O}(1/\sqrt{T})$. The dependence in $T$ is optimal up to logarithmic factors, as it matches the lower bound for linear bandits \citep[Theorem 3]{dani2008stochastic}. \end{remark} \begin{remark}[Choice of $\lambda$] One important deviation from the classical linear bandit analysis is that we require a regularization that centers around $\hat{w}_0$ and the regularization weight $\lambda$ to be $C_\lambda \sqrt{T}$, comparing to $\lambda = O(1)$ in the linear case. The choice is to ensure that $\hat{w}_t$ stays within the local neighborhood of $\hat{w}_0$, and to delicately balance different terms that appear in the regret analysis to ensure that the overall regret bound is $\tilde{O}(\sqrt{T})$. \end{remark} \begin{remark}[Choice of $n$] We choose $n =\sqrt{T}$, therefore, it puts sample complexity requirement on $T$ shown in eq. \eqref{eq:n_lower_main}. The choice of $n$ plays two roles here. First, it guarantees that the regression result $\hat{w}_0$ lies in the neighboring region of $w^*$ of the loss function $L(w)$ with high probability. The neighboring region of $w^*$ has nice properties, e.g., local strong convexity, which allow us to build the upper bound of $\ell_2$-distance between $\hat{w}_0$ and $w^*$. Second, in Phase I, we are doing uniform sampling over the function so the cumulative regret in Phase I is bounded by $2Fn=2F\sqrt{T}$ which is at the same $\tilde{O}(\sqrt{T})$ rate as that in Phase II. \end{remark}
\section{Proof Overview}\label{sec:proof} In this section, we give a proof sketch of all theoretical results. A key insight of our analysis is that there is more mileage that seminal techniques developed by \citet{abbasi2011improved} for analyzing linearly parameterized bandits problems in analyzing non-linear bandits, though we need to localize to a nearly optimal region and carefully handle the non-linear components via more aggressive regularization. Other assumptions that give rise to a similarly good initialization may work too and our new proof can be of independent interest in analyzing other extensions of LinUCB, e.g., to contextual bandits, reinforcement learning and other problems.
In detail, first we prove the estimation error bound of $\hat{w}_0$ for Phase I of GO-UCB algorithm, then prove the feasibility of $\mathrm{Ball}_t$. Finally by putting everything together we prove the cumulative regret bound of GO-UCB algorithm. We list all auxiliary lemmas in Appendix \ref{sec:auxiliary} and show complete proofs in Appendix \ref{sec:miss}.
\subsection{Regression Oracle Guarantee}\label{sec:mle}
The goal of Phase I of GO-UCB is to sufficiently explore the unknown objective function with $n$ uniform queries and obtain an estimated parameter $\hat{w}_0$. By assuming access to a regression oracle, we prove the convergence bound of $\hat{w}_0$ w.r.t. $w^*$, i.e., $\|\hat{w}_0-w^*\|^2_2$. To get started, we need the following regression oracle lemma.
\begin{lemma}[Adapted from \citet{nowak2007complexity})]\label{lem:mle_oracle} Suppose Assumption \ref{ass:parameter_class} \& \ref{ass:objective} hold. There is an absolute constant $C'$, such that after round $n$ in Phase I of Algorithm \ref{alg:go_ucb}, with probability $>1 - \delta/2$, regression oracle estimated $\hat{w}_0$ satisfies \begin{align*} \mathbb{E}_{x \sim \mathcal{U}} [(f_x(\hat{w}_0) - f_x(w^*))^2] \leq \frac{C' d_w F^2 \iota}{n}, \end{align*} where $\iota$ is the logarithmic term depending on $n, C_h, 2/\delta$. \end{lemma}
\citet{nowak2007complexity} proves that expected square error of Empirical Risk Minimization (ERM) estimator can be bounded at the rate of $\tilde{O}(1/n)$ with high probability, rather than $\tilde{O}(1/\sqrt{n})$ rate achieved by Chernoff/Hoeffding bounds. It works with realizable and misspecified settings. Proof of Lemma \ref{lem:mle_oracle} includes simplifying it with regression oracle, Assumption \ref{ass:parameter_class}, and $\varepsilon$-covering number argument over parameter class. Basically Lemma \ref{lem:mle_oracle} says that expected square error of $f_x(\hat{w}_0)$ converges to $f_x(w^*)$ at the rate of $\tilde{O}(1/n)$ with high probability. Based on it, we prove the following regression oracle guarantee.
\begin{theorem}[Regression oracle guarantee]\label{thm:mle_guarantee} Suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold. There is an absolute constant $C$ such that after round $n$ in Phase I of Algorithm \ref{alg:go_ucb} where $n$ satisfies \begin{align*} n \geq C \max \left\{ \frac{2^{\gamma-2}d_w F^2 \zeta^\frac{\gamma}{2} \iota}{\tau}, \frac{d_w F^2 \zeta \iota}{\mu} \right\}, \end{align*} with probability $> 1-\delta/2$, regression oracle estimated $\hat{w}_0$ satisfies \begin{align*}
\|\hat{w}_0 - w^*\|^2_2 \leq \frac{C d_w F^2 \iota}{\mu n}, \end{align*} where $\iota$ is the logarithmic term depending on $n, C_h, 2/\delta$. \end{theorem}
Compared with Lemma \ref{lem:mle_oracle}, there is an extra sample complexity requirement on $n$ because we need $n$ to be sufficiently large such that the function can be sufficiently explored and more importantly $\hat{w}_0$ falls into the neighboring region (strongly convex region) of $w^*$. See Figure \ref{fig:loss} for illustration. It is also the reason why strong convexity parameter $\mu$ appears in the denominator of the upper bound.
\subsection{Feasibility of $\mathrm{Ball}_t$}\label{sec:ball} The following lemma is the key part of algorithm design of GO-UCB. It says that our definition of $\mathrm{Ball}_t$ is appropriate, i.e., throughout all rounds in Phase II, $w^*$ is contained in $\mathrm{Ball}_t$ with high probability.
\begin{lemma}[Feasibility of $\mathrm{Ball}_t$]\label{lem:feasible_ball} Set $\Sigma_t, \hat{w}_t$ as in eq. \eqref{eq:sigma_t}, \eqref{eq:inner}. Set $\beta_t$ as \begin{align} \beta_t = \tilde{\Theta} \left(d_w \sigma^2 + \frac{d_w F^2}{\mu} + \frac{d^3_w F^4 t }{\mu^2 T}\right).\label{eq:beta_2} \end{align} Suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold and choose $n= \sqrt{T}, \lambda = C_\lambda \sqrt{T}$. Then $\forall t \in [T]$ in Phase II of Algorithm \ref{alg:go_ucb}, w.p. $>1-\delta$, \begin{align*}
\|\hat{w}_t - w^*\|^2_{\Sigma_t} &\leq \beta_t. \end{align*} \end{lemma}
For reader's easy reference, we write our choice of $\beta_t$ again in eq. \eqref{eq:beta_2}. Note this lemma requires careful choices of $\lambda$ and $n$ because $\beta_t$ appears later in the cumulative regret bound and $\beta_t$ is required to be at the rate of $\tilde{O}(1)$. The proof has three steps. First we obtain the closed form solution of $\hat{w}_t$ as in eq. \eqref{eq:inner}. Next we use induction to prove that $\forall t \in [T], \|\hat{w}_t - w^*\|_2 \leq \tilde{O}(\tilde{C}/\sqrt{n})$ for some universal constant $\tilde{C}$. Finally we prove $\|\hat{w}_t - w^*\|^2_{\Sigma_t} \leq \beta_t$.
\subsection{Regret Analysis}\label{sec:reg} To prove cumulative regrets bound of GO-UCB algorithm, we need following two lemmas of instantaneous regrets in Phase II of GO-UCB.
\begin{lemma}[Instantaneous regret bound]\label{lem:instant_regret} Set $\Sigma_t, \hat{w}_t, \beta_t$ as in eq. \eqref{eq:sigma_t}, \eqref{eq:inner}, \& \eqref{eq:beta_2} and suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold, then with probability $> 1- \delta$,
$w^*$ is contained in $\mathrm{Ball}_t$. Define $u_t = \|\nabla f_{x_t}(\hat{w}_t)\|_{\Sigma^{-1}_t}$, then $\forall t \in [T]$ in Phase II of Algorithm \ref{alg:go_ucb}, \begin{align*} r_t \leq 2\sqrt{\beta_t}u_t + \frac{2\beta_t C_h}{\lambda}. \end{align*} \end{lemma} The first term of the upper bound is pretty standard, seen also in LinUCB \citep{abbasi2011improved} and GP-UCB \citep{srinivas2010gaussian}. After we apply first order gradient approximation of the objective function, the second term is the upper bound of the high order residual term, which introduces extra challenge to derive the upper bound.
Technically, proof of Lemma \ref{lem:instant_regret} requires $w^*$ is contained in our parameter uncertainty ball $\mathrm{Ball}_t$ with high probability throughout Phase II of GO-UCB, which has been proven in Lemma \ref{lem:feasible_ball}. Later, the proof utilizes Taylor's theorem and uses the convexity of $\mathrm{Ball}_t$ twice. See Appendix \ref{sec:regret}. The next lemma is an extension of Lemma \ref{lem:instant_regret}, where the proof uses monotonically increasing property of $\beta_t$ in $t$.
\begin{lemma}[Sum of squared instantaneous regret bound]\label{lem:sos_instant_regret} Set $\Sigma_t, \hat{w}_t, \beta_t$ as in eq. \eqref{eq:sigma_t}, \eqref{eq:inner}, \& \eqref{eq:beta_2} and suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold, then with probability $> 1- \delta$, $w^*$ is contained in $\mathrm{Ball}_t$ and $\forall t \in [T]$ in Phase II of Algorithm \ref{alg:go_ucb}, \begin{align*} \sum_{t=1}^T r^2_t \leq 16\beta_T d_w \log \left(1 + \frac{TC_g^2}{d_w \lambda}\right) + \frac{8\beta^2_T C^2_h T}{\lambda^2}. \end{align*} \end{lemma}
Proof of Theorem \ref{thm:cr} is by putting everything together.
\section{Experiments}\label{sec:experiment} We compare our GO-UCB algorithm with four Bayesian Optimization (BO) algorithms: GP-EI \citep{jones1998efficient}, GP-PI \citep{kushner1964new}, GP-UCB \citep{srinivas2010gaussian}, and Trust Region BO (TuRBO) \citep{eriksson2019scalable}, where the first three are classical methods and TuRBO is a more advanced algorithm designed for high-dimensional cases.
To run GO-UCB, we choose our model $\hat{f}$ to be a two linear layer neural network with sigmoid function being the activation function: \begin{align*} \hat{f}(x) = \textrm{linear2}(\textrm{sigmoid}(\textrm{linear1}(x))), \end{align*} where $w_1, b_1$ denote the weight and bias of $\textrm{linear1}$ layer and $w_2, b_2$ denote those of $\textrm{linear2}$ layer. Specifically, we set $w_1 \in \mathbb{R}^{25\times d_x}, b_1 \in \mathbb{R}^{25}, w_2 \in \mathbb{R}^{25}, b_2 \in \mathbb{R}$, meaning the dimension of activation function is $25$. All implementations are based on BoTorch framework \citep{balandat2020botorch} and sklearn package \citep{head2021skopt} with default parameter settings. To help readers reproduce our results, implementation details are shown in Appendix \ref{sec:imp_goucb} and we will release code once our paper gets accepted.
\subsection{Synthetic Experiments}\label{sec:syn} First, we test all algorithms on three high dimensional synthetic functions defined on $[-5, 5]^{d_x}$ where $d_x = 20$, including both realizable and misspecified cases. The first test function $f_1$ is created by setting all elements in $w_1, b_1, w_2, b_2$ in $\hat{f}$ to be $1$, so $f_1$ is a realizable function given $\hat{f}$. The second and third test functions $f_2, f_3$ are Styblinski-Tang function and Rastrigin function, defined as follows: \begin{align*} f_2 &= -\frac{1}{2} \sum_{i=1}^{20} x_i^4 - 16 x^2_i + 5 x_i,\\ f_3 &= -200 + \sum_{i=1}^{20} 10 \cos(2 \pi x_i) - x^2_i, \end{align*} where $x_i$ denotes the $i$-th element in its $20$ dimensions, so $f_2, f_3$ are misspecified functions given $\hat{f}$.
We set $n=5, T=25$ for $f_1$ and $n=8,T=64$ for $f_2, f_3$. To reduce the effect of randomness in all algorithms, we repeat the whole optimization process for $5$ times for all algorithms and report mean and error bar of cumulative regrets. The error bar is measured by Wald's test with $95\%$ confidence, i.e., $1.96 \nu/\sqrt{5}$ where $\nu$ is standard deviation of cumulative regrets and $5$ is the number of repetitions.
From Figure \ref{fig:simulation}, we learn that in all tasks our GO-UCB algorithm performs better than all other four BO approaches. Among BO approches, TuRBO performs the best since it is specifically designed for high-dimensional tasks. In Figure \ref{fig:simulation}(a), mean of cumulative regrets of GO-UCB and TuRBO stays the same when $t \geq 22$, which means that both of them have found the global optima, but GO-UCB algorithm is able to find the optimal point shortly after Phase I and enjoys the least error bar. It is well expected since $f_1$ is a realizable function for $\hat{f}$. Unfortunately, GP-UCB, GP-EI, and GP-PI incur almost linear regrets, showing the bad performances of classical BO algorithms in high-dimensional cases.
In Figure \ref{fig:simulation}(b) and \ref{fig:simulation}(c), all methods are suffering from linear regret because $f_2, f_3$ are misspecified functions. The gap between GO-UCB and other methods is smaller in Figure \ref{fig:simulation}(c) than in \ref{fig:simulation}(b) because optimizing $f_3$ is more challenging than $f_2$ since $f_3$ has much more local optimal points.
\begin{figure*}
\caption{Cumulative regrets (the lower the better) of all algorithms on $20$-dimensional $f_1, f_2, f_3$ synthetic functions.
}
\label{fig:simulation}
\end{figure*}
\begin{figure*}
\caption{Cumulative regrets (the lower the better) of all algorithms in real-world hyperparameter tuning task on Breast-cancer dataset. }
\label{fig:real}
\end{figure*}
\subsection{Real-World Experiments}\label{sec:real} To illustrate the GO-UCB algorithm works in real-world tasks, we do hyperparameter tuning experiments on three tasks using three classifiers. Three UCI datasets \citep{Dua:2019} are Breat-cancer, Australian, and Diabetes, and three classifiers are random forest, multi-layer perceptron, and gradient boosting where each of them has $7,8,11$ hyperparameters. For each classifier on each dataset, the function mapping from hyperparameters to classification accuracy is the black-box function that we are maximizing, so the input space dimension $d_x=7,8,11$ for each classifier. We use cumulative regret to evaluate hyperparameter tuning performances, however, best accuracy $f^*$ is unknown ahead of time so we set it to be the best empirical accuracy of each task. To reduce the effect of randomness, we divide each dataset into 5 folds and every time use 4 folds for training and remaining 1 fold for testing. We report mean and error bar of cumulative regrets where error bar is measured by Wald's test, the same as synthetic experiments.
Figure \ref{fig:real} shows results on Breast-cancer dataset. In Figure \ref{fig:real}(b)(c) GO-UCB performs statistically much better that all other BO algorithms since there is almost no error bar gap between TuRBO and GO-UCB. It shows that GO-UCB can be deployed in real-world applications to replace BO methods. Also, in Figure \ref{fig:real}(b) Phase I of GO-UCB is not good but GO-UCB is able to perform better than others in Phase II, which shows the effectiveness of Phase II of GO-UCB. In Figure \ref{fig:real}(a) all algorithms have similar performances. In Figure \ref{fig:real}(b), TuRBO performs similarly as GP-UCB, GP-EI, and GP-PI when $t \leq 23$, but after $t=23$ it performs better and shows a curved regret line by finding optimal points. Results on Australian and Diabetes datasets are shown in Appendix \ref{sec:real_detail} where similar algorithm performances can be seen.
\subsection{Discussion on Practical Usage} Note in experiments, we choose parametric model $\hat{f}$ to be a two linear layer neural network. In more real-world experiments, one can choose the model $\hat{f}$ in GO-UCB to be simpler functions or much more complex functions, e.g., deep neural networks, depending on task requirements.
\section{Conclusion}\label{sec:conclusion} Global non-convex optimization is an important problem that widely exists in many real-world applications, e.g., deep learning hyper-parameter tuning and new material design. However, solving this optimization problem in general is NP-hard. Existing work relies on Gaussian process assumption, e.g., Bayesian optimization, or other non-parametric family which suffers from the curse of dimensionality.
We propose the first algorithm to solve such global optimization with parametric function approximation, which shows a new way of global optimization. GO-UCB first uniformly explores the function and collects a set of observation points and then uses the optimistic exploration to actively select points. At the core of GO-UCB is a carefully designed uncertainty set over parameters based on gradients that allow optimistic exploration. Under realizable parameter class assumption and a few mild geometric conditions, our theoretical analysis shows that cumulative regret of GO-UCB is at the rate of $\tilde{O}(\sqrt{T})$, which is dimension-free in terms of function domain $\mathcal{X}$.
Our high-dimensional synthetic test shows that GO-UCB works better than BO methods even in misspecified setting. Moreover, GO-UCB performs better than BO algorithms in real-world hyperparameter tuning tasks, which may be of independent interest beyond this paper.
There is $\mu$, the strongly convexity parameter, in the denominator of upper bound in Theorem \ref{thm:cr}. $\mu$ can be small in practice, thus the upper bound can be large. Developing the cumulative regret bound containing a term depending on $\mu$ but being independent to $T$ remains a future problem.
\subsection*{Acknowledgments} The work is supported by NSF Award \#1934641 and \#2134214.
\onecolumn \appendix
\section{Notation Table}\label{sec:table} \begin{table}[!htbp] \centering \caption{Symbols and notations.}\label{tab:notations} \begin{tabular}{ccl} \noalign{
} \hline \textbf{Symbol} & \textbf{Definition} & \textbf{Description} \\ \hline
$\|A\|_\mathrm{op}$ & & operator norm\\ \hline $\mathrm{Ball}_t$ & eq. \eqref{eq:ball} & parameter uncertainty region at round $t$\\ \hline $\beta_t$ & eq. \eqref{eq:beta_t} & parameter uncertainty region radius at round $t$\\ \hline $C, \zeta$ & & constants\\ \hline $d_x$ & & domain dimension \\ \hline $d_w$ & & parameter dimension \\ \hline $\delta$ & & failure probability \\ \hline $\varepsilon$ & & covering number discretization distance \\ \hline $\eta$ & $\sigma$-sub-Gaussian & observation noise \\ \hline $f_w(x)$ & & objective function at $x$ parameterized by $w$ \\ \hline $f_x(w)$ & & objective function at $w$ parameterized by $x$ \\ \hline $\nabla f_x(w)$ & & 1st order derivative w.r.t. $w$ parameterized by $x$ \\ \hline $\nabla^2 f_x(w)$ & & 2nd order derivative w.r.t. $w$ parameterized by $x$ \\ \hline $F$ & & function range constant bound \\ \hline $\gamma, \tau$ & & growth condition parameters \\ \hline $\iota, \iota', \iota{''}$ & & logarithmic terms \\ \hline $L(w)$ & $\mathbb{E}[(f_x(w) - f_x(w^*))^2]$ & expected loss function \\ \hline $\lambda$ & & regularization parameter \\ \hline $\mu$ & & strong convexity parameter \\ \hline $n$ & & time horizon in Phase I \\ \hline $[n]$ & $\{1,2,...,n\}$ & integer set of size $n$\\ \hline $\mathrm{Oracle}$ & & regression oracle \\ \hline $r_t$ & $f_{w^*}(x^*) - f_{w^*}(x_t)$ & instantaneous regret at round $t$ \\ \hline $R_T$ & $\sum_{t=1}^T r_t$ & cumulative regret after round $T$\\ \hline $\Sigma_t$ & eq. \eqref{eq:sigma_t} & covariance matrix at round $t$ \\ \hline $T$ & & time horizon in Phase II \\ \hline $\mathcal{U}$ & & uniform distribution \\ \hline $w$ & $w \in \mathcal{W}$ & function parameter \\ \hline $w^*$ & $w^* \in \mathcal{W}$ & true parameter \\ \hline $\hat{w}_0$ & & oracle-estimated parameter after Phase I \\ \hline $\hat{w}_t$ & eq. \eqref{eq:inner} & updated parameter at round $t$\\ \hline $\mathcal{W}$ & $\mathcal{W} \subseteq [0,1]^{d_w}$ & parameter space \\ \hline $x$ & $x \in \mathcal{X}$ & data point \\\hline $x^*$ & & optimal data point \\\hline
$\|x\|_p$ &$(\sum_{i=1}^d |x_i|^p)^{1/p}$ & $\ell_p$ norm \\\hline
$\|x\|_A$ &$\sqrt{x^\top A x}$ & distance defined by square matrix $A$ \\\hline $\mathcal{X}$ & $\mathcal{X} \subseteq \mathbb{R}^{d_x}$ & function domain \\ \hline $\mathcal{Y}$ & $\mathcal{Y} = [-F, F]$ & function range \\ \hline \end{tabular} \end{table}
\section{Auxiliary Technical Lemmas}\label{sec:auxiliary} In this section, we list auxiliary lemmas that are used in proofs.
\begin{lemma}[Adapted from eq. (5) (6) of \citet{nowak2007complexity}]\label{lem:regression} Given a dataset $\{x_i,y_i\}_{j=1}^n$ where $y_j$ is generated from eq. \eqref{eq:y} and $f_0$ is the underlying true function. Let $\hat{f}$ be an ERM estimator taking values in $\mathcal{F}$ where $\mathcal{F}$ is a finite set and $\mathcal{F} \subset \{f: [0,1]^d \rightarrow [-F,F]\}$ for some $F \geq 1$. Then with probability $> 1- \delta$, $\hat{f}$ satisfies that \begin{align}
\mathbb{E}[(\hat{f} - f_0)^2] \leq \left(\frac{1+\alpha}{1-\alpha}\right) \left( \inf_{f\in \mathcal{F} } \mathbb{E}[(f - f_0)^2] + \frac{F^2 \log(|\mathcal{F}|) \log(2)}{n\alpha}\right) + \frac{2\log(2/\delta)}{n\alpha}, \end{align} for all $\alpha \in (0, 1]$. \end{lemma}
\begin{lemma}[Sherman-Morrison lemma \citep{sherman1950adjustment}]\label{lem:sherman} Let $A$ denote a matrix and $b,c$ denote two vectors. Then \begin{align} (A + bc^\top)^{-1} = A^{-1} - \frac{A^{-1}bc^\top A^{-1}}{1+ c^\top A^{-1} b}. \end{align} \end{lemma}
\begin{lemma}[Self-normalized bound for vector-valued martingales \citep{abbasi2011improved,agarwal2021rl}]\label{lem:self_norm}
Let $\{\eta_i\}_{i=1}^\infty$ be a real-valued stochastic process with corresponding filtration $\{\mathcal{F}_i\}_{i=1}^\infty$ such that $\eta_i$ is $\mathcal{F}_i$ measurable, $\mathbb{E}[\eta_i | \mathcal{F}_{i-1} ] = 0$, and $\eta_i$ is conditionally $\sigma$-sub-Gaussian with $\sigma \in \mathbb{R}^+$. Let $\{X_i\}_{i=1}^\infty$ be a stochastic process with $X_i \in \mathcal{H}$ (some Hilbert space) and $X_i$ being $F_t$ measurable. Assume that a linear operator $\Sigma:\mathcal{H} \rightarrow \mathcal{H}$ is positive definite, i.e., $x^\top \Sigma x > 0$ for any $x \in \mathcal{H}$. For any $t$, define the linear operator $\Sigma_t = \Sigma_0 + \sum_{i=1}^t X_i X_i^\top$ (here $xx^\top$ denotes outer-product in $\mathcal{H}$). With probability at least $1-\delta$, we have for all $t\geq 1$: \begin{align}
\left\|\sum_{i=1}^t X_i \eta_i \right\|^2_{\Sigma_t^{-1}} \leq \sigma^2 \log \left(\frac{\det(\Sigma_t) \det(\Sigma_0)^{-1}}{\delta^2} \right). \end{align} \end{lemma}
\section{Missing Proofs}\label{sec:miss} In this section, we show complete proofs of all technical results in the main paper. For reader's easy reference, we define $\iota$ as a logarithmic term depending on $n, C_h, 2/\delta$ (w.p. $>1-\delta/2$), $\iota'$ as a logarithmic term depending on $t, d_w, C_g, 1/\lambda, 2/\delta$ (w.p. $>1-\delta/2$), and $\iota{''}$ as a logarithmic term depending on $t, d_w, C_g, 1/\lambda$.
\subsection{Regression Oracle Guarantee}
\begin{lemma}[Restatement of Lemma \ref{lem:mle_oracle}] Suppose Assumption \ref{ass:parameter_class} \& \ref{ass:objective} hold. There is an absolute constant $C'$, such that after round $n$ in Phase I of Algorithm \ref{alg:go_ucb}, with probability $>1 - \delta/2$, regression oracle estimated $\hat{w}_0$ satisfies \begin{align*} \mathbb{E}_{x \sim \mathcal{U}} [(f_x(\hat{w}_0) - f_x(w^*))^2] \leq \frac{C' d_w F^2 \iota}{n}, \end{align*} where $\iota$ is the logarithmic term depending on $n, C_h, 2/\delta$. \end{lemma} \begin{proof} The regression oracle lemma establishes on Lemma \ref{lem:regression} which works only for finite function class. In order to work with our continuous parameter class $\mathcal{W}$, we need $\varepsilon$-covering number argument.
First, let $\tilde{w}, \widetilde{\mathcal{W}}$ denote the ERM parameter and finite parameter class after applying covering number argument on $\mathcal{W}$. By Lemma \ref{lem:regression}, we find that with probability $>1-\delta/2$, \begin{align}
\mathbb{E}_{x \sim \mathcal{U}}[(f_x(\tilde{w}) - f_x(w^*))^2] &\leq \left(\frac{1+\alpha}{1-\alpha}\right) \left( \inf_{w \in \widetilde{\mathcal{W}} \cup \{w^*\}} \mathbb{E}_{x\sim \mathcal{U}}[(f_x(w) - f_x(w^*))^2] + \frac{F^2 \log(|\widetilde{\mathcal{W}}|) \log(2)}{n\alpha}\right) \nonumber \\ &\qquad + \frac{2\log(4/\delta)}{n\alpha}\\
&\leq \left(\frac{1+\alpha}{1-\alpha}\right) \left( \frac{F^2 \log(|\widetilde{\mathcal{W}}|) \log(2)}{n\alpha}\right) + \frac{2\log(4/\delta)}{n\alpha}, \end{align}
where the second inequality is by realizable assumption (Assumption \ref{ass:parameter_class}). Our parameter class $\mathcal{W} \subseteq [0, 1]^{d_w}$, so $\log(|\widetilde{\mathcal{W}}|) = \log(1/\varepsilon^{d_w})= d_w\log(1/\varepsilon)$ and the new upper bound is that with probability $> 1-\delta/2$, \begin{align} \mathbb{E}_{x\sim \mathcal{U}}[(f_x(\tilde{w}) - f_x(w^*))^2] \leq C^{''}\left(\frac{d_w F^2 \log(1/\varepsilon)}{n} + \frac{\log(2/\delta)}{n}\right), \end{align} where $C^{''}$ is a universal constant obtained by choosing $\alpha=1/2$. Note $\tilde{w}$ is the ERM parameter in $\widetilde{\mathcal{W}}$ after discretization, not our target parameter $\hat{w}_0 \in \mathcal{W}$. By $(a+b)^2\leq 2a^2 + 2b^2$, \begin{align} \mathbb{E}_{x\sim \mathcal{U}}[(f_x(\hat{w}_0) - f_x(w^*))^2] &\leq 2\mathbb{E}_{x\sim \mathcal{U}}[(f_x(\Hat{w}_0) - f_x(\tilde{w}))^2] + 2\mathbb{E}_{x\sim \mathcal{U}}[(f_x(\tilde{w}) - f_x(w^*))^2]\\ &\leq 2\varepsilon^2 C_h^2 + 2C^{''}\left(\frac{d_w F^2 \log(1/\varepsilon)}{n} + \frac{\log(2/\delta)}{n}\right)\label{eq:interm1} \end{align} where the second line applies Lemma~\ref{lem:regression}, discretization error $\varepsilon$, and Assumption \ref{ass:objective}. By choosing $\varepsilon = 1/\sqrt{n C_h^2}$, we get $$ \eqref{eq:interm1} =\frac{2}{n} + \frac{C^{''}d_w F^2\log(n C_h^2)}{n} + \frac{2C^{''}\log(2/\delta)}{n} \leq C' \frac{d_w F^2\log(n C_h^2) +\log(2/\delta)}{n}$$ where we can take $C' = 2C^{''}$ (assuming $2<C^{''}d_w F^2\log(n C_h^2)$). The proof completes by defining $\iota$ as the logarithmic term depending on $n, C_h, 2/\delta$. \end{proof}
\begin{theorem}[Restatement of Theorem \ref{thm:mle_guarantee}] Suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold. There is an absolute constant $C$ such that after round $n$ in Phase I of Algorithm \ref{alg:go_ucb} where $n$ satisfies \begin{align*} n \geq C \max \left\{ \frac{2^{\gamma-2}d_w F^2 \zeta^\frac{\gamma}{2} \iota}{\tau}, \frac{d_w F^2 \zeta \iota}{\mu} \right\}, \end{align*} with probability $> 1-\delta/2$, regression oracle estimated $\hat{w}_0$ satisfies \begin{align*}
\|\hat{w}_0 - w^*\|^2_2 \leq \frac{C d_w F^2 \iota}{\mu n}, \end{align*} where $\iota$ is the logarithmic term depending on $n, C_h, 2/\delta$. \end{theorem} \begin{proof} Recall the definition of expected loss function $L(w) = \mathbb{E}_{x \sim \mathcal{U}}(f_x(w)-f_x(w^*))^2$ and the second order Taylor's theorem, $L(\hat{w}_0)$ at $w^*$ can be written as \begin{align}
L(\hat{w}_0) &= L(w^*) + (\hat{w}_0 - w^*) \nabla L(w^*) + \frac{1}{2} \|\hat{w}_0 - w^*\|^2_{\nabla^2 L(\tilde{w})}, \end{align} where $\tilde{w}$ lies between $\hat{w}_0$ and $w^*$. Also, because $\nabla L(w^*) = \nabla E_{x\sim \mathcal{U}}(f_x(w^*) - f_x(w^*))^2=0$, then with probability $> 1-\delta/2$, \begin{align}
\frac{1}{2} \|\hat{w}_0 - w^*\|^2_{\nabla^2 L(\tilde{w})} = L(\hat{w}_0) - L(w^*) \leq \frac{C' d_w F^2 \iota}{n},\label{eq:ll} \end{align} where the inequality is due to Lemma \ref{lem:mle_oracle}.
Next, we prove the following lemma stating after certain number of $n$ samples, $\|\hat{w}_0 - w^*\|_{\nabla^2 L(w^*)}$ can be bounded by a constant, e.g., $\frac{1}{2}$. \begin{lemma}\label{lem:n_hessian} Suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold. There is an absolute constant $C'$ such that after round $n$ in Phase I of Algorithm \ref{alg:go_ucb} where $n$ satisfies \begin{align*} n \geq \max \left\{ \frac{2^{\gamma+1}C' d_w F^2\zeta^\frac{\gamma}{2} \iota}{\tau}, \frac{8 C'd_w F^2 \zeta \iota }{\mu} \right\}, \end{align*} then with probability $> 1-\delta/2$, \begin{align*}
\|\hat{w}_0 - w^*\|_{\nabla^2 L(w^*)} \leq \frac{1}{2}. \end{align*} \end{lemma} \begin{proof} Consider two cases of relationships between $\hat{w}_0$ and $w^*$.
\textbf{Case I.} If $\hat{w}_0$ is far away from $w^*$, i.e., $\|\hat{w}_0 - w^*\|_2 \geq (\tau/\mu)^{1/(2-\gamma)}$, by growth condition (Assumption \ref{ass:loss}), \begin{align}
\frac{\tau}{2}\|\hat{w}_0-w^*\|^\gamma_2 &\leq L(\hat{w}_0)-L(w^*) \leq \frac{C' d_w F^2 \iota}{n}, \end{align} where the second inequality is due to eq. \eqref{eq:ll}. Then, \begin{align}
\|\hat{w}_0-w^*\|_{\nabla^2 L(w^*)} &\leq \sqrt{\zeta} \|\hat{w}_0-w^*\|_2 \leq \sqrt{\zeta}\left(\frac{2C' d_w F^2 \iota}{\tau n}\right)^\frac{1}{\gamma}, \end{align} where the first inequality is due to smoothness of loss function in Assumption \ref{ass:loss}. Therefore, setting $\sqrt{\zeta}(\frac{2 C' d_w F^2 \iota}{\tau n})^\frac{1}{\gamma}\leq \frac{1}{2}$ will result in a sample complexity bound on $n$: \begin{align} n \geq \frac{2^{\gamma+1}C' d_w F^2\zeta^\frac{\gamma}{2} \iota}{\tau}.\label{eq:n_1} \end{align}
\textbf{Case II.} If $\hat{w}_0$ is close to $w^*$, i.e., $\|\hat{w}_0 - w^*\|_2 \leq (\tau/\mu)^{1/(2-\gamma)}$, by local strong convexity (Assumption \ref{ass:loss}), \begin{align}
\frac{\mu}{2} \|\hat{w}_0-w^*\|^2_2 &\leq L(\hat{w}_0)-L(w^*) \leq \frac{C' d_w F^2 \iota}{n }, \end{align} where the second inequality is due to eq. \eqref{eq:ll}. Then, \begin{align}
\|\hat{w}_0-w^*\|_{\nabla^2 L(w^*)}&\leq \sqrt{\zeta} \|\hat{w}_0-w^*\|_2 \leq \sqrt{\frac{2\zeta C' d_w F^2 \iota}{\mu n}}. \end{align} where the first inequality is again due to smoothness of loss function in Assumption \ref{ass:loss}. Therefore, setting $\sqrt{\frac{2\zeta C' d_w F^2 \iota}{\mu n}}\leq \frac{1}{2}$ will lead to another sample complexity bound on $n$: \begin{align} n\geq \frac{8 C' d_w F^2 \zeta \iota}{\mu}.\label{eq:n_2} \end{align}
The proof completes by combining \textbf{Case I} and \textbf{Case II} discussion. \end{proof}
Now we continue the proof of Theorem \ref{thm:mle_guarantee}. By self-concordance assumption (Assumption \ref{ass:loss}), \begin{align}
(1-\|\tilde{w}- w^*\|_{\nabla^2 L(w^*)})^2 \|\hat{w}_0- w^*\|^2_{\nabla^2 L(w^*)}\leq \|\hat{w}_0- w^*\|^2_{\nabla^2 L(\tilde{w})}.\label{eq:self} \end{align} The LHS can be further lower bounded by \begin{align}
\frac{1}{4}\|\hat{w}_0- w^*\|^2_{\nabla^2 L(w^*)} &\leq (1-\|\hat{w}_0- w^*\|_{\nabla^2 L(w^*)})^2 \|\hat{w}_0- w^*\|^2_{\nabla^2 L(w^*)}\\
&\leq (1-\|\tilde{w}- w^*\|_{\nabla^2 L(w^*)})^2 \|\hat{w}_0- w^*\|^2_{\nabla^2 L(w^*)},\label{eq:self_con} \end{align} where the first inequality is because of Lemma \ref{lem:n_hessian} and the second inequality is due to the fact that $\hat{w}$ lies between $\hat{w}_0$ and $w^*$. Therefore, \begin{align}
\|\hat{w}_0 - w^*\|^2_{\nabla^2 L(w^*)} \leq \frac{8C' d_w F^2 \iota }{n}. \end{align}
The proof completes by inequality $\|\hat{w}_0 - w^*\|^2_2 \leq \|\hat{w}_0 - w^*\|^2_{\nabla^2 L(w^*)}/\mu$ due to $\mu$-strongly convexity of $L(w)$ at $w^*$ (Assumption \ref{ass:loss}) and defining $C=8C'$. \end{proof}
\subsection{Properties of Covariance Matrix $\Sigma_t$}\label{sec:sigma} In eq. \eqref{eq:sigma_t}, $\Sigma_t$ is defined as $\lambda I + \sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top$. In this section, we prove three lemmas saying the change of $\Sigma_t$ as $t \in 1,...,T$ is bounded in Phase II of GO-UCB. The key observation is that at each round $i$, the change made to $\Sigma_t$ is $\nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top$, which is only rank one.
\begin{lemma}[Adapted from \citet{agarwal2021rl}]\label{lem:det}
Set $\Sigma_t, \hat{w}_t$ as in eq. \eqref{eq:sigma_t} \& \eqref{eq:inner}, suppose Assumption \ref{ass:parameter_class} \& \ref{ass:loss} hold, and define $u_t = \|\nabla f_{x_t}(\hat{w}_t)\|_{\Sigma^{-1}_t}$. Then \begin{align*} \det \Sigma_t = \det \Sigma_0 \prod_{i=0}^{t-1} (1 + u^2_i). \end{align*} \end{lemma} \begin{proof} Recall the definition of $\Sigma_t = \lambda I + \sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top$ and we can show that \begin{align} \det \Sigma_{t+1} &= \det (\Sigma_t + \nabla f_{x_t}(w_t) \nabla f_{x_t}(w_t)^\top)\\ &= \det (\Sigma^\frac{1}{2}_t(I + \Sigma^{-\frac{1}{2}}_t \nabla f_{x_t}(w_t) \nabla f_{x_t}(w_t)^\top \Sigma^{-\frac{1}{2}}_t)\Sigma^\frac{1}{2}_t)\\ &= \det (\Sigma_t) \det(I + \Sigma^{-\frac{1}{2}}_t \nabla f_{x_t}(w_t) (\Sigma^{-\frac{1}{2}}_t \nabla f_{x_t}(w_t))^\top)\\ &= \det (\Sigma_t) \det(I + v_t v_t^\top), \end{align}
where $v_t = \Sigma^{-\frac{1}{2}}_t \nabla f_{x_t}(w_t)$. Recall $u_t$ is defined as $\|\nabla f_{x_t}(\hat{w}_t)\|_{\Sigma_t^{-1}}$. Because $v_t v^\top_t$ is a rank one matrix, $\det(I + v_t v^\top_t) = 1 + u^2_t$. The proof completes by induction. \end{proof}
\begin{lemma}[Adapted from \citet{agarwal2021rl}]\label{lem:log_det} Set $\Sigma_t$ as in eq. \eqref{eq:sigma_t} and suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold. Then \begin{align*} \log \left(\frac{\det \Sigma_{t-1}}{\det \Sigma_0}\right) \leq d_w \log \left(1 + \frac{t C_g^2}{d_w \lambda}\right). \end{align*} \end{lemma} Proof of Lemma \ref{lem:det} directly follows definition of $\Sigma_t$ and proof of Lemma \ref{lem:log_det} involves Lemma \ref{lem:det} and inequality of arithmetic and geometric means. Note $C_g$ is a constant coming from Assumption \ref{ass:objective}. We do not claim any novelty in proofs of these two lemmas which replace feature vector in linear bandit \citep{agarwal2021rl} with gradient vectors. \begin{proof} Let $\xi_1,...,\xi_{d_w}$ denote eigenvalues of $\sum_{i=0}^{t-1} \nabla f_{x_i}(w_i) \nabla f_{x_i}(w_i)^\top$, then \begin{align}
\sum_{k=1}^{d_w} \xi_k = \mathrm{tr} \left(\sum_{i=0}^{t-1} \nabla f_{x_i}(w_i) \nabla f_{x_i}(w_i)^\top \right) = \sum_{i=0}^{t-1} \|\nabla f_{x_i}(w_i)\|^2_2 \leq t C_g^2,\label{eq:t_c_g} \end{align} where the inequality is by Assumption \ref{ass:objective}. By Lemma \ref{lem:det}, \begin{align} \log \left(\frac{\det \Sigma_{t-1}}{\det \Sigma_0}\right) &\leq \log \det \left(I + \frac{1}{\lambda} \sum_{i=0}^{t-1} \nabla f_{x_i}(w_i) \nabla f_{x_i}(w_i)^\top \right)\\ &= \log \left(\prod_{k=1}^{d_w} (1 + \xi_k/\lambda)\right)\\ &= d_w \log \left(\prod_{k=1}^{d_w} (1 + \xi_k/\lambda)\right)^{1/{d_w}}\\ &\leq d_w \log \left(\frac{1}{d_w} \sum_{k=1}^{d_w} (1 + \xi_k/\lambda)\right)\\ &\leq d_w \log \left(1+ \frac{t C_g^2}{d_w \lambda}\right), \end{align} where the second inequality is by inequality of arithmetic and geometric means and the last inequality is due to eq. \eqref{eq:t_c_g}. \end{proof}
\begin{lemma}\label{lem:sum_of_square} Set $\Sigma_t, \hat{w}_t$ as in eq. \eqref{eq:sigma_t} \& \eqref{eq:inner} and suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold. Then \begin{align*} \sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i)^\top \Sigma^{-1}_t \nabla f_{x_i}(\hat{w}_i) \leq 2d_w \log\left(1 + \frac{tC^2_g}{d_w \lambda}\right). \end{align*} \end{lemma}
A trivial bound of LHS in Lemma \ref{lem:sum_of_square} could be simply $O(t C^2_g/\lambda)$. Lemma \ref{lem:sum_of_square} is important because it saves the upper bound to be $O(\log(t C^2_g/\lambda))$, which allows us to build a feasible parameter uncertainty ball, shown in the next section.
\begin{proof} First, we prove $\forall i \in \{0, 1,..., t-1\}, 0 < \nabla f_{x_i}(\hat{w}_i)^\top \Sigma^{-1}_t \nabla f_{x_i}(\hat{w}_i) < 1$. Recall the definition of $\Sigma_t$, it's easy to see that $\Sigma_t$ is a positive definite matrix and thus $0 < \nabla f_{x_i}(\hat{w}_i)^\top \Sigma^{-1}_t \nabla f_{x_i}(\hat{w}_i)$. To prove it's smaller than $1$, we need to decompose $\Sigma_t$ and write \begin{align} &\quad\ \nabla f_{x_i}(\hat{w}_i)^\top \Sigma^{-1}_t \nabla f_{x_i}(\hat{w}_i)\\ &= \nabla f_{x_i}(\hat{w}_i)^\top \left(\lambda I + \sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top \right)^{-1} \nabla f_{x_i}(\hat{w}_i)\\ &= \nabla f_{x_i}(\hat{w}_i)^\top \left(\nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top - \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top + \lambda I + \sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top \right)^{-1} \nabla f_{x_i}(\hat{w}_i). \end{align} Let $A = - \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top + \lambda I + \sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top$, and it becomes \begin{align} \nabla f_{x_i}(\hat{w}_i)^\top \Sigma^{-1}_t \nabla f_{x_i}(\hat{w}_i) = \nabla f_{x_i}(\hat{w}_i)^\top (\nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top + A)^{-1} \nabla f_{x_i}(\hat{w}_i). \end{align} By applying Sherman-Morrison lemma (Lemma \ref{lem:sherman}), we have \begin{align} &\quad \ \nabla f_{x_i}(\hat{w}_i)^\top \Sigma^{-1}_t \nabla f_{x_i}(\hat{w}_i)\\ &= \nabla f_{x_i}(\hat{w}_i)^\top \left(A^{-1} - \frac{A^{-1} \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top A^{-1}}{1+ \nabla f_{x_i}(\hat{w}_i)^\top A^{-1} \nabla f_{x_i}(\hat{w}_i)} \right)\nabla f_{x_i}(\hat{w}_i)\\ &= \nabla f_{x_i}(\hat{w}_i)^\top A^{-1} \nabla f_{x_i}(\hat{w}_i) - \frac{\nabla f_{x_i}(\hat{w}_i)^\top A^{-1} \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top A^{-1} \nabla f_{x_i}(\hat{w}_i)}{1+ \nabla f_{x_i}(\hat{w}_i)^\top A^{-1} \nabla f_{x_i}(\hat{w}_i)}\\ &= \frac{\nabla f_{x_i}(\hat{w}_i)^\top A^{-1} \nabla f_{x_i}(\hat{w}_i)}{1+ \nabla f_{x_i}(\hat{w}_i)^\top A^{-1} \nabla f_{x_i}(\hat{w}_i)} < 1. \end{align} Next, we use the fact that $\forall x \in (0,1), x \leq 2\log(1+ x)$, and we have \begin{align} \sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i)^\top \Sigma^{-1}_t \nabla f_{x_i}(\hat{w}_i) &\leq \sum_{i=0}^{t-1} 2\log\left( 1+ \nabla f_{x_i}(\hat{w}_i)^\top \Sigma^{-1}_t \nabla f_{x_i}(\hat{w}_i)\right)\\ &\leq 2\log\left(\frac{\det \Sigma_{t-1}}{\det \Sigma_0} \right)\\ &\leq 2d_w \log\left( 1 + \frac{tC^2_g}{d_w\lambda}\right), \end{align} where the last two inequalities are due to Lemma \ref{lem:det} and \ref{lem:log_det}. \end{proof}
\subsection{Feasibility of $\mathrm{Ball}_t$}
\begin{lemma}[Restatement of Lemma \ref{lem:feasible_ball}] Set $\Sigma_t, \hat{w}_t$ as in eq. \eqref{eq:sigma_t}, \eqref{eq:inner}. Set $\beta_t$ as \begin{align*} \beta_t = \tilde{\Theta} \left(d_w \sigma^2 + \frac{d_w F^2}{\mu} + \frac{d^3_w F^4 t }{\mu^2 T}\right). \end{align*} Suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold and choose $n= \sqrt{T}, \lambda = C_\lambda \sqrt{T}$. Then $\forall t \in [T]$ in Phase II of Algorithm \ref{alg:go_ucb}, w.p. $>1-\delta$, \begin{align*}
\|\hat{w}_t - w^*\|^2_{\Sigma_t} &\leq \beta_t. \end{align*} \end{lemma} \begin{proof}
The proof has three steps. First we obtain the closed form solution of $\hat{w}_t$. Next we derive the upper bound of $\|\hat{w}_i - w^*\|^2_2$. Finally we use it to prove that the upper bound of $\|\hat{w}_t - w^*\|^2_{\Sigma_t}$ matches our choice of $\beta_t$.
\textbf{Step 1: Closed form solution of $\hat{w}_t$.} The optimal criterion for the objective function in eq. \eqref{eq:opt_inner} is \begin{align} 0= \lambda (\hat{w}_t - \hat{w}_0) + \sum_{i=0}^{t-1} ((\hat{w}_t - \hat{w}_i)^\top \nabla f_{x_i}(\hat{w}_i) + f_{x_i}(\hat{w}_i) - y_i) \nabla f_{x_i}(\hat{w}_i). \end{align} Rearrange the equation and we have \begin{align} \lambda (\hat{w}_t- \hat{w}_0) + \sum_{i=0}^{t-1} (\hat{w}_t - \hat{w}_i)^\top \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i) &= \sum_{i=0}^{t-1} (y_i - f_{x_i}(\hat{w}_i) ) \nabla f_{x_i}(\hat{w}_i),\\ \lambda (\hat{w}_t- \hat{w}_0) + \sum_{i=0}^{t-1} (\hat{w}_t - \hat{w}_i)^\top \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i) &= \sum_{i=0}^{t-1} (y_i - f_{x_i}(w^*) + f_{x_i}(w^*) - f_{x_i}(\hat{w}_i) ) \nabla f_{x_i}(\hat{w}_i),\\ \lambda (\hat{w}_t - \hat{w}_0) + \sum_{i=0}^{t-1} \hat{w}^\top_t \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i) &= \sum_{i=0}^{t-1} (\hat{w}^\top_i \nabla f_{x_i}(\hat{w}_i) +\eta_i + f_{x_i}(w^*) - f_{x_i}(\hat{w}_i) ) \nabla f_{x_i}(\hat{w}_i),\\ \hat{w}_t\left(\lambda I + \sum_{i=1}^{t-1} \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top \right) - \lambda \hat{w}_0 &= \sum_{i=0}^{t-1} (\hat{w}^\top_i \nabla f_{x_i}(\hat{w}_i) + \eta_i + f_{x_i}(w^*) - f_{x_i}(\hat{w}_i))\nabla f_{x_i}(\hat{w}_i),\\ \hat{w}_t \Sigma_t &= \lambda \hat{w}_0 + \sum_{i=0}^{t-1} (\hat{w}^\top_i \nabla f_{x_i}(\hat{w}_i) + \eta_i + f_{x_i}(w^*) - f_{x_i}(\hat{w}_i))\nabla f_{x_i}(\hat{w}_i), \end{align} where the second line is by removing and adding back $f_{x_i}(w^*)$, the third line is due to definition of observation noise $\eta$ and the last line is by our choice of $\Sigma_t$ (eq. \eqref{eq:sigma_t}). Now we have the closed form solution of $\hat{w}_t$. Further, $\hat{w}_t - w^*$ can be written as \begin{align} \hat{w}_t - w^* &= \Sigma^{-1}_t \left(\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) (\nabla f_{x_i}(\hat{w}_i)^\top \hat{w}_i +\eta_i + f_{x_i}(w^*) - f_{x_i}(\hat{w}_i)) \right) + \lambda \Sigma^{-1}_t \hat{w}_0 - \Sigma^{-1}_t \Sigma_t w^*\\ &= \Sigma^{-1}_t \left(\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) (\nabla f_{x_i}(\hat{w}_i)^\top \hat{w}_i +\eta_i + f_{x_i}(w^*) - f_{x_i}(\hat{w}_i)) \right) + \lambda \Sigma^{-1}_t (\hat{w}_0 - w^*)\nonumber\\ &\qquad - \Sigma^{-1}_t \left(\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \nabla f_{x_i}(\hat{w}_i)^\top\right) w^* \\ &= \Sigma^{-1}_t \left(\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) (\nabla f_{x_i}(\hat{w}_i)^\top (\hat{w}_i - w^*) +\eta_i + f_{x_i}(w^*) - f_{x_i}(\hat{w}_i)) \right) + \lambda \Sigma^{-1}_t (\hat{w}_0- w^*)\\
&= \Sigma^{-1}_t \left(\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \frac{1}{2} \|w^* - \hat{w}_i\|^2_{\nabla^2 f_{x_i}(\tilde{w})}\right) + \Sigma^{-1}_t \left(\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \eta_i \right) +\lambda \Sigma^{-1}_t (\hat{w}_0 - w^*),\label{eq:w_t_w_star} \end{align} where the second line is again by our choice of $\Sigma_t$ and the last equation is by the second order Taylor's theorem of $f_{x_i}(w^*)$ at $\hat{w}_i$ where $\tilde{w}$ lies between $w^*$ and $\hat{w}_i$.
\textbf{Step 2: Upper bound of $\|\hat{w}_i - w^*\|^2_2$.} Note eq. \eqref{eq:w_t_w_star} holds $\forall i \in [T]$ because all $\hat{w}_i$ are obtained through the same optimization problem, which means \begin{align}
\hat{w}_i - w^* = \Sigma^{-1}_i \left(\sum_{\rho=0}^{i-1} \nabla f_{x_\rho}(\hat{w}_\rho) \frac{1}{2} \|w^* - \hat{w}_\rho\|^2_{\nabla^2 f_{x_\rho}(\tilde{w})}\right) + \Sigma^{-1}_i \left(\sum_{\rho=0}^{i-1} \nabla f_{x_\rho}(\hat{w}_\rho) \eta_\rho \right) +\lambda \Sigma^{-1}_i (\hat{w}_0 - w^*). \end{align} By inequality $(a+b+c)^2 \leq 4a^2 + 4b^2 + 4c^2$ and definition of $\Sigma_i$, we take the square of both sides and get \begin{align}
\|\hat{w}_i - w^*\|^2_2 &\leq \frac{4}{\lambda}\left\|\sum_{\rho=0}^{i-1} \nabla f_{x_\rho}(\hat{w}_\rho) \eta_\rho \right\|^2_{\Sigma^{-1}_i} + 4\|\hat{w}_0 - w^*\|^2_2 + \frac{1}{\lambda}\left\|\sum_{\rho=0}^{i-1} \nabla f_{x_\rho}(\hat{w}_\rho) \|w^* - \hat{w}_\rho\|^2_{\nabla^2 f_{x_\rho}(\tilde{w}_\rho)} \right\|^2_{\Sigma^{-1}_i}.\label{eq:w_i} \end{align}
Now we use induction to prove the convergence rate of $\|\hat{w}_i - w^*\|^2_2, \forall i \in [T]$. Recall at the very beginning of Phase II, by Theorem \ref{thm:mle_guarantee}, with probability $> 1- \delta/2$, \begin{align}
\|\hat{w}_0 - w^*\|^2_2 \leq \frac{C d_w F^2 \iota}{\mu n}. \end{align}
To derive a claim based on induction. Formally, we suppose at round $i$, there exists some universal constant $\tilde{C}$ such that with probability $> 1- \delta/2$, \begin{align}
\|\hat{w}_i - w^*\|^2_2 &\leq \frac{\tilde{C} d_w F^2 \iota}{\mu n}. \end{align} Our task is to prove that at round $i+1$ with probability $> 1- \delta/2$, \begin{align}
\|\hat{w}_{i+1} - w^*\|^2_2 &\leq \frac{\tilde{C} d_w F^2 \iota}{\mu n}. \end{align} Note $\tilde{C}$ is for induction purpose, which can be different from $C$.
From eq. \eqref{eq:w_i}, at round $i+1$ we can write \begin{align}
\|\hat{w}_{i+1} - w^*\|^2_2 &\leq \frac{4\sigma^2}{\lambda} \log\left(\frac{\det(\Sigma_i)\det(\Sigma_0)^{-1}}{\delta^2_i} \right) + \frac{4Cd_w F^2 \iota}{\mu n } + \frac{1}{\lambda}\left\|\sum_{\rho=0}^{i} \nabla f_{x_\rho}(\hat{w}_\rho) \|w^* - \hat{w}_\rho\|^2_{\nabla^2 f_{x_\rho}(\tilde{w}_\rho)} \right\|^2_{\Sigma^{-1}_{i+1}}\\ &\leq \frac{4\sigma^2}{\lambda} \left(d_w \log \left(1+\frac{i C_g^2}{d_w \lambda} \right) + \log\left(\frac{\pi^2 i^2}{3\delta}\right) \right) + \frac{4Cd_w F^2 \iota}{\mu n } \nonumber\\
&\qquad + \frac{1}{\lambda}\left\|\sum_{\rho=0}^{i} \nabla f_{x_\rho}(\hat{w}_\rho) \|w^* - \hat{w}_\rho\|^2_{\nabla^2 f_{x_\rho}(\tilde{w}_\rho)} \right\|^2_{\Sigma^{-1}_{i+1}}\\
&\leq \frac{4d_w \sigma^2 \iota'}{\lambda} + \frac{4Cd_w F^2 \iota}{\mu n } + \frac{1}{\lambda}\left\|\sum_{\rho=0}^{i} \nabla f_{x_\rho}(\hat{w}_\rho) \|w^* - \hat{w}_\rho\|^2_{\nabla^2 f_{x_\rho}(\tilde{w}_\rho)} \right\|^2_{\Sigma^{-1}_{i+1}}, \end{align}
where the first inequality is due to self-normalized bound for vector-valued martingales (Lemma \ref{lem:self_norm} in Appendix \ref{sec:auxiliary}) and Theorem \ref{thm:mle_guarantee}, the second inequality is by Lemma \ref{lem:log_det} and our choice of $\delta_i= 3\delta/(\pi^2 i^2)$, and the last inequality is by defining $\iota'$ as the logarithmic term depending on $i, d_w, C_g, 1/\lambda, 2/\delta$ (with probability $> 1- \delta/2$). The choice of $\delta_i$ guarantees the total failure probability over $t$ rounds is no larger than $\delta/2$. Now we use our assumption $\|\hat{w}_i - w^*\|^2_2 \leq \frac{\tilde{C} d_w F^2 \iota}{\mu n}$ to bound the last term. \begin{align}
\|\hat{w}_{i+1} - w^*\|^2_2 &\leq \frac{4d_w \sigma^2 \iota'}{\lambda} + \frac{4Cd_w F^2 \iota}{\mu n } + \frac{1}{\lambda}\left\|\frac{\tilde{C} C_h d_w F^2 \iota}{\mu n} \sum_{\rho=0}^i \nabla f_{x_\rho}(\hat{w}_\rho) \right\|^2_{\Sigma^{-1}_{i+1}}\\ &\leq \frac{4d_w \sigma^2 \iota'}{\lambda} + \frac{4Cd_w F^2 \iota}{\mu n } + \frac{\tilde{C}^2 C^2_h d_w^2 F^4 \iota^2}{\mu^2 \lambda n^2}\left(\sum_{\rho=0}^i \sqrt{\nabla f_{x_\rho}(\hat{w}_\rho)^\top \Sigma^{-1}_{i+1} \nabla f_{x_\rho}(\hat{w}_\rho)} \right)^2\\ &\leq \frac{4d_w \sigma^2 \iota'}{\lambda} + \frac{4Cd_w F^2 \iota}{\mu n } + \frac{\tilde{C}^2 C^2_h d_w^2 F^4 \iota^2}{\mu^2 \lambda n^2}\left(\sum_{\rho=0}^i 1\right)\left(\sum_{\rho=0}^i \nabla f_{x_\rho}(\hat{w}_\rho)^\top \Sigma^{-1}_{i+1} \nabla f_{x_\rho}(\hat{w}_\rho) \right)\\ &\leq \frac{4d_w \sigma^2 \iota'}{\lambda} + \frac{4Cd_w F^2 \iota}{\mu n } + \frac{\tilde{C}^2 C^2_h d^3_w F^4 i \iota{''} \iota^2}{\mu^2 \lambda n^2}, \end{align} where the first inequality is due to smoothness of loss function in Assumption \ref{ass:loss}, the third inequality is by Cauchy-Schwarz inequality, and the last inequality is because of Lemma \ref{lem:sum_of_square} and defining $\iota{''}$ as logarithmic term depending on $i, d_w, C_g, 1/\lambda$.
What we need is that there exists some universal constant $\tilde{C}$ such that \begin{align} \frac{4d_w \sigma^2 \iota'}{\lambda} + \frac{4C d_w F^2 \iota}{\mu n} + \frac{\tilde{C}^2 C^2_h d^3_w F^4 i \iota^2 \iota{''}}{\lambda \mu^2 n^2} \leq \frac{\tilde{C}d_w F^2 \iota}{\mu n}. \end{align} Note the LHS is monotonically increasing w.r.t $i$ so the inequality must hold when $i=T$, i.e., \begin{align} \frac{4d_w \sigma^2 \iota'}{\lambda} + \frac{4C d_w F^2 \iota}{\mu n} + \frac{\tilde{C}^2 C^2_h d^3_w F^4 T \iota^2 \iota{''}}{\lambda \mu^2 n^2} \leq \frac{\tilde{C}d_w F^2 \iota}{\mu n}. \end{align} Recall the range of our function is $[-F, F]$, given any distribution, the variance $\sigma^2$ can always be upper bounded by $ F^2/4$, so we just need to show that \begin{align} \frac{d_w F^2 \iota'}{\lambda} + \frac{4C d_w F^2 \iota}{\mu n} + \frac{\tilde{C}^2 C^2_h d^3_w F^4 T \iota^2 \iota{''}}{\lambda \mu^2 n^2} &\leq \frac{\tilde{C}d_w F^2 \iota}{\mu n},\\ \mu^2 n^2 \iota' + 4 \lambda \mu n C \iota + \tilde{C}^2 C^2_h d^2_w F^2 T \iota^2 \iota{''} &\leq \lambda \mu n \tilde{C} \iota,\\ \tilde{C}^2 C^2_h d^2_w F^2 T \iota^2 \iota{''} - \tilde{C}\lambda \mu n \iota + \mu^2 n^2 \iota' + 4\lambda \mu n C \iota &\leq 0, \end{align} where the second and third lines are by rearrangement. A feasible solution on $\tilde{C}$ requires \begin{align} \lambda^2 \mu^2 n^2 \iota^2 - 4C^2_h d^2_w F^2 T \iota^2 \iota{''} (\mu^2 n^2 \iota' + 4\lambda \mu n C \iota) &\geq 0,\\ \lambda^2 \mu^2 n - 4C^2_h d^2_w F^2 T\iota{''} (\mu^2 n \iota' + 4\lambda \mu C \iota) &\geq 0,\label{eq:b^2-4ac} \end{align} where the second line is by rearrangement. Our choices of $\lambda = C_\lambda \sqrt{T}, n=\sqrt{T}$ where $C_\lambda$ is a constant independent to $T$ ensure eq. \eqref{eq:b^2-4ac} holds, thus such a $\tilde{C}$ exists. Therefore, by induction, we prove that $\forall i \in [T]$ there exists a universal constant $\tilde{C}$ such that with probability $> 1- \delta/2$, \begin{align}
\|\hat{w}_i - w^*\|^2_2 \leq \frac{\tilde{C}d_w F^2\iota}{n}. \end{align} With this result, now we are ready to move to \textbf{Step 3}.
\textbf{Step 3: Upper bound of $\|\hat{w}_t - w^*\|^2_{\Sigma_t}$.} Multiply both sides of eq. \eqref{eq:w_t_w_star} by $\Sigma^\frac{1}{2}_t$ and we have \begin{align}
\Sigma^{\frac{1}{2}}_t(\hat{w}_t - w^*) &\leq \frac{1}{2} \Sigma^{-\frac{1}{2}}_t \left(\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i)\|w^* - \hat{w}_i\|^2_{\nabla^2 f_{x_i}(\tilde{w})}\right) + \Sigma^{-\frac{1}{2}}_t \left(\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \eta_i \right) + \lambda \Sigma^{-\frac{1}{2}}_t (\hat{w}_0 - w^*). \end{align} Take square of both sides and by inequality $(a+b+c)^2\leq 4a^2 + 4b^2 + 4c^2$ we obtain \begin{align}
\|\hat{w}_t - w^*\|^2_{\Sigma_t} &\leq 4\left\| \sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \eta_i\right\|^2_{\Sigma_t^{-1}} + 4\lambda^2 \|\hat{w}_0 - w^*\|^2_{\Sigma^{-1}_t} + \left\|\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i)\|w^* - \hat{w}_i\|^2_{\nabla^2 f_{x_i}(\tilde{w})}\right\|^2_{\Sigma^{-1}_t}. \end{align} The remaining proof closely follows \textbf{Step 2}, i.e., \begin{align}
\|\hat{w}_t - w^*\|^2_{\Sigma_t} &\leq 4d_w \sigma^2 \iota' + \frac{4\lambda C d_w F^2 \iota}{\mu n} + \left\|\frac{\tilde{C}C_h d_w F^2 \iota}{\mu n} \sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i) \right\|^2_{\Sigma^{-1}_{t}}\\ &\leq 4d_w \sigma^2 \iota' + \frac{4\lambda C d_w F^2 \iota}{\mu n} + \frac{\tilde{C}^2 C^2_h d_w^2 F^4 \iota^2}{\mu^2 n^2}\left(\sum_{i=0}^{t-1} \sqrt{\nabla f_{x_i}(\hat{w}_i)^\top \Sigma^{-1}_{t} \nabla f_{x_i}(\hat{w}_i)} \right)^2\\ &\leq 4d_w \sigma^2 \iota' + \frac{4\lambda C d_w F^2 \iota}{\mu n } + \frac{\tilde{C}^2 C^2_h d_w^2 F^4 \iota^2}{\mu^2 n^2}\left(\sum_{i=0}^{t-1} 1\right)\left(\sum_{i=0}^{t-1} \nabla f_{x_i}(\hat{w}_i)^\top \Sigma^{-1}_t \nabla f_{x_i}(\hat{w}_i) \right)\\ &\leq 4d_w \sigma^2 \iota' + \frac{4\lambda Cd_w F^2 \iota}{\mu n } + \frac{\tilde{C}^2 C^2_h d_w^3 F^4 t \iota{''} \iota^2}{\mu^2 n^2}\\ &\leq O\left(d_w \sigma^2 \iota' + \frac{d_w F^2 \iota}{\mu} + \frac{d^3_w F^4 t \iota{''} \iota^2 }{\mu^2 T}\right), \end{align} where the last inequality is by our choices of $\lambda=C_\lambda \sqrt{T}, n = \sqrt{T}$. Therefore, our choice of \begin{align} \beta_t &= \tilde{\Theta}\left(d_w \sigma^2 + \frac{d_w F^2 }{\mu} + \frac{d^3_w F^4 t }{\mu^2 T}\right) \end{align} guarantees that $w^*$ is always contained in $\mathrm{Ball}_t$ with probability $1- \delta$. \end{proof}
\subsection{Regret Analysis}\label{sec:regret}
\begin{lemma}[Restatement of Lemma \ref{lem:instant_regret}] Set $\Sigma_t, \hat{w}_t, \beta_t$ as in eq. \eqref{eq:sigma_t}, \eqref{eq:inner}, \& \eqref{eq:beta_2} and suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold, then with probability $> 1- \delta$,
$w^*$ is contained in $\mathrm{Ball}_t$. Define $u_t = \|\nabla f_{x_t}(\hat{w}_t)\|_{\Sigma^{-1}_t}$, then $\forall t \in [T]$ in Phase II of Algorithm \ref{alg:go_ucb}, \begin{align*} r_t \leq 2\sqrt{\beta_t}u_t + \frac{2\beta_t C_h}{\lambda}. \end{align*} \end{lemma} \begin{proof} By definition of instantaneous regret $r_t$, \begin{align} r_t &= f_{x^*}(w^*) - f_{x_t}(w^*). \end{align} Recall the selection process of $x_t$ and define $\tilde{w} = \mathop{\mathrm{argmax}}_{w \in \mathrm{Ball}_t} f_{x_t}(w)$, \begin{align} r_t \leq f_{x_t}(\tilde{w}) - f_{x_t}(w^*)= (\tilde{w} - w^*)^\top \nabla f_{x_t}(\dot{w}), \end{align} where the equation is by first order Taylor's theorem and $\dot{w}$ lies between $\tilde{w}$ and $w^*$ which means $\dot{w}$ is guaranteed to be in $\mathrm{Ball}_t$ since $\mathrm{Ball}_t$ is convex. Then, by adding and removing terms, \begin{align} r_t &= (\tilde{w} - \hat{w}_t + \hat{w}_t - w^*)^\top (\nabla f_{x_t}(\hat{w}_t) - \nabla f_{x_t}(\hat{w}_t) + \nabla f_{x_t}(\dot{w}))\\
&\leq \|\tilde{w}-\hat{w}_t\|_{\Sigma_t} \|\nabla f_{x_t}(\hat{w}_t)\|_{\Sigma_t^{-1}} + \|\hat{w}_t - w^*\|_{\Sigma_t} \|\nabla f_{x_t}(\hat{w}_t)\|_{\Sigma_t^{-1}} + (\tilde{w} - \hat{w}_t)^\top (\nabla f_{x_t}(\dot{w}_t) - \nabla f_{x_t}(\hat{w}_t))\nonumber\\ &\qquad + (\hat{w}_t - w^*)^\top (\nabla f_{x_t}(\dot{w}) - \nabla f_{x_t}(\hat{w}_t)), \end{align}
where the last inequality is due to Holder's inequality. By definitions of $\beta_t$ in $\mathrm{Ball}_t$ and $u_t = \|\nabla f_{x_t}(\hat{w}_t)\|_{\Sigma^{-1}_t}$, \begin{align} r_t &\leq 2\sqrt{\beta_t} u_t + (\tilde{w} - \hat{w}_t)^\top (\nabla f_{x_t}(\dot{w}) - \nabla f_{x_t}(\hat{w}_t)) + (\hat{w}_t - w^*)^\top (\nabla f_{x_t}(\dot{w}) - \nabla f_{x_t}(\hat{w}_t)). \end{align} Again by first order Taylor's theorem where $\ddot{w}$ lies between $\dot{w}$ and $\hat{w}$ and thus $\ddot{w}$ lies in $\mathrm{Ball}_t$, \begin{align} r_t &\leq 2\sqrt{\beta_t} u_t + (\tilde{w}-\hat{w}_t)^\top \Sigma^\frac{1}{2}_t \Sigma^{-\frac{1}{2}}_t \nabla^2 f_{x_t}(\ddot{w}) \Sigma^{-\frac{1}{2}}_t \Sigma^\frac{1}{2}_t (\dot{w}-\hat{w}_t)\nonumber\\ &\qquad + (\hat{w}_t- w^*)^\top \Sigma^\frac{1}{2}_t \Sigma^{-\frac{1}{2}}_t \nabla^2 f_{x_t}(\ddot{w}) \Sigma^{-\frac{1}{2}}_t \Sigma^\frac{1}{2}_t (\dot{w}-\hat{w}_t)\\
&\leq 2\sqrt{\beta_t} u_t + \|(\tilde{w}-\hat{w}_t)^\top \Sigma^\frac{1}{2}_t\|_2 \|\Sigma^{-\frac{1}{2}}_t \nabla^2 f_{x_t}(\ddot{w}) \Sigma^{-\frac{1}{2}}_t\|_\mathrm{op} \|\Sigma^\frac{1}{2}_t (\dot{w}-\hat{w}_t)\|_2 \nonumber\\
&\qquad + \|(\hat{w}_t - w^*)^\top \Sigma^\frac{1}{2}_t\|_2 \|\Sigma^{-\frac{1}{2}}_t \nabla^2 f_{x_t}(\ddot{w}) \Sigma^{-\frac{1}{2}}_t\|_\mathrm{op} \|\Sigma^\frac{1}{2}_t (\dot{w}-\hat{w}_t)\|_2\\ &\leq 2\sqrt{\beta_t}u_t + \frac{2\beta_t C_h}{\lambda}, \end{align} where the second inequality is by Holder's inequality and the last inequality is due to definition of $\beta_t$ in $\mathrm{Ball}_t$, Assumption \ref{ass:objective}, and our choice of $\Sigma_t$. \end{proof}
\begin{lemma}[Restatement of Lemma \ref{lem:sos_instant_regret}] Set $\Sigma_t, \hat{w}_t, \beta_t$ as in eq. \eqref{eq:sigma_t}, \eqref{eq:inner}, \& \eqref{eq:beta_2} and suppose Assumption \ref{ass:parameter_class}, \ref{ass:objective}, \& \ref{ass:loss} hold, then with probability $> 1- \delta$, $w^*$ is contained in $\mathrm{Ball}_t$ and $\forall t \in [T]$ in Phase II of Algorithm \ref{alg:go_ucb}, \begin{align*} \sum_{t=1}^T r^2_t \leq 16\beta_T d_w \log \left(1 + \frac{TC_g^2}{d_w \lambda}\right) + \frac{8\beta^2_T C^2_h T}{\lambda^2}. \end{align*} \end{lemma} \begin{proof} By Lemma \ref{lem:instant_regret} and inequality $(a+b)^2 \leq 2a^2 + 2b^2$, \begin{align} \sum_{t=1}^T r^2_t &\leq \sum_{t=1}^T 8\beta_t u^2_t + \frac{8 \beta^2_t C^2_h}{\lambda^2}\\ &\leq 8\beta_T \sum_{i=1}^T u^2_t + \frac{8\beta^2_T C^2_h T}{\lambda^2}\\ &\leq 16\beta_T d_w \log \left(1 + \frac{TC_g^2}{d_w \lambda}\right) + \frac{8\beta^2_T C^2_h T}{\lambda^2}, \end{align} where the second inequality is due to $\beta_t$ is increasing in $t$ and the last inequality is by Lemma \ref{lem:sum_of_square}. \end{proof}
By putting everything together, we are ready to prove the main cumulative regret theorem.
\begin{proof}[Proof of Theorem \ref{thm:cr}] By definition of cumulative regret including both Phase I and II, \begin{align*} R_{\sqrt{T}+T} &= \sum_{j=1}^{\sqrt{T}} r_j + \sum_{t=1}^T r_t\\ &\leq 2\sqrt{T}F + \sqrt{T\sum_{t=1}^T r^2_t}\\ &\leq 2\sqrt{T}F + \sqrt{16T\beta_T d_w \log \left(1 + \frac{T C_g^2}{d_w \lambda}\right) + \frac{8T^2\beta^2_T C^2_h}{\lambda^2}}\\ &\leq \tilde{O}\left(\sqrt{T} F + \sqrt{T\beta_T d_w + T \beta^2_T }\right), \end{align*} where the first inequality is due to function range and Cauchy-Schwarz inequality, the second inequality is by Lemma \ref{lem:sos_instant_regret} and the last inequality is obtained by setting $\lambda=C_\lambda \sqrt{T}, n=\sqrt{T}$ as required by Lemma \ref{lem:feasible_ball}.
Recall that $\beta_t$ is defined in eq. \eqref{eq:beta_2}, so \begin{align*} \beta_T &= \tilde{\Theta}\left(d_w \sigma^2 + \frac{d_w F^2}{\mu} + \frac{d^3_w F^4}{\mu^2} \right) \leq \tilde{O}\left(\frac{d^3_w F^4}{\mu^2} \right). \end{align*} The proof completes by plugging in upper bound of $\beta_T$. \end{proof}
\section{Additional Experimental Details} In addition to Experiments section in main paper, in this section, we show details of algorithm implementation and and real-world experiments.
\subsection{Implementation of GO-UCB}\label{sec:imp_goucb} Noise parameter $\sigma=0.01$. Regression oracle in GO-UCB is approximated by stochastic gradient descent algorithm on our two linear layer neural network model with mean squared error loss, $2,000$ iterations and $10^{-11}$ learning rate. Exactly solving optimization problem in Step 5 of Phase II may not be computationally tractable, so we use iterative gradient ascent algorithm over $x$ and $w$ with $2,000$ iterations and $10^{-4}$ learning rate. $\beta_t$ is set as $d^3_w F^4 t/T$. $\lambda$ is set as $\sqrt{T}\log^2 T$.
\subsection{Real-world Experiments}\label{sec:real_detail}
Hyperparameters can be continuous or categorical, however, in order to fairly compare GO-UCB with Bayesian optimization methods, in all hyperparameter tuning tasks, we set function domain to be $[0, 10]^{d_x}$, a continuous domain. If a hyperparameter is categorical, we allocate equal length domain for each hyperparameter. For example, the seventh hyperparameter of random forest is a bool value, True or False and we define $[0, 5)$ as True and $[5,10]$ as False. If a hyperparameter is continuous, we set linear mapping from the hyperparameter domain to $[0,10]$. For example, the sixth hyperparameter of multi-layer perceptron is a float value in $(0,1)$ thus we multiply it by $10$ and map it to $(0, 10)$.
\textbf{Hyperparameters in hyperparameter tuning tasks.} We list hyperparameters in all three tasks as follows.
\begin{itemize} \item Classification with Random Forest. \begin{enumerate}
\item Number of trees in the forest, (integer, [20, 200]).
\item Criterion, (string, ``gini'', ``entropy'', or ``logloss'').
\item Maximum depth of the tree, (integer, [1, 10]).
\item Minimum number of samples required to split an internal node, (integer, [2, 10]).
\item Minimum number of samples required to be at a leaf node, (integer, [1, 10]).
\item Maximum number of features to consider when looking for the best split, (string, ``sqrt'' or ``log2''). \item Bootstrap, (bool, True or False). \end{enumerate} \item Classification with Multi-layer Perceptron. \begin{enumerate}
\item Activation function (string, ``identity'', ``logistic'', ``tanh'', or ``relu'').
\item Strength of the L2 regularization term, (float, [$10^{-6}, 10^{-2}$]).
\item Initial learning rate used, (float, [$10^{-6}, 10^{-2}$]).
\item Maximum number of iterations, (integer, [100, 300]).
\item Whether to shuffle samples in each iteration, (bool, True or False).
\item Exponential decay rate for estimates of first moment vector, (float, (0, 1)).
\item Exponential decay rate for estimates of second moment vector (float, (0, 1)).
\item Maximum number of epochs to not meet tolerance improvement, (integer, [1, 10]). \end{enumerate}
\item Classification with Gradient Boosting. \begin{enumerate}
\item Loss, (string, ``logloss'' or ``exponential''). \item Learning rate, (float, (0, 1)). \item Number of estimators, (integer, [20, 200]). \item Fraction of samples to be used for fitting the individual base learners, (float, (0, 1)). \item Function to measure the quality of a split, (string, ``friedman mse'' or ``squared error''). \item Minimum number of samples required to split an internal node, (integer, [2, 10]). \item Minimum number of samples required to be at a leaf node, (integer, [1, 10]). \item Minimum weighted fraction of the sum total of weights, (float, (0, 0.5)). \item Maximum depth of the individual regression estimators, (integer, [1, 10]). \item Number of features to consider when looking for the best split, (float, ``sqrt'' or ``log2''). \item Maximum number of leaf nodes in best-first fashion, (integer, [2, 10]). \end{enumerate} \end{itemize}
\textbf{Results on Australian and Diabetes datasets.} We show experimental results of hyperparameter tuning tasks on Australian and Diabetes datasets in Figure \ref{fig:aus} and Figure \ref{fig:dia}. Our proposed GO-UCB algorithm performs consistently better than all other algorithms, which is the same as on Breast-cancer dataset in main paper.
\begin{figure*}
\caption{Cumulative regrets (the lower the better) of all algorithms in real-world hyperparameter tuning task on Australian dataset.
}
\label{fig:aus}
\end{figure*}
\begin{figure*}
\caption{Cumulative regrets (the lower the better) of all algorithms in real-world hyperparameter tuning task on Diabetes dataset.
}
\label{fig:dia}
\end{figure*}
\end{document}
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\begin{document}
\title{Hermitian
structures on the derived category of coherent sheaves} \author{ \\ \\
Jos\'e I. Burgos Gil\footnote{Partially supported by grant
MTM2009-14163-C02-01 and CSIC research project 2009501001.}\\
\small{Instituto de Ciencias Matem\'aticas (ICMAT-CSIC-UAM-UCM-UC3)}\\
\small{Consejo Superior de Investigaciones Cient\'ificas (CSIC)}\\
\small{Spain}\\
\small{\texttt{[email protected]}}\\
\and
Gerard Freixas i Montplet\footnote{Partially supported by
grant MTM2009-14163-C02-01}\\
\small{Institut de Math\'ematiques de Jussieu (IMJ)}\\
\small{Centre National de la Recherche Scientifique (CNRS)}\\
\small{France}\\
\small{\texttt{[email protected]}}\\
\and
R\u azvan Li\c tcanu\footnote{Supported by CNCSIS -UEFISCSU,
project number PNII - IDEI 2228/2008 .}\\
\small{Faculty of Mathematics}\\
\small{University Al. I. Cuza Iasi}\\
\small{Romania}\\
\small{\texttt{[email protected]}} }
\maketitle \begin{abstract}
The main objective of the present paper is to set up the theoretical
basis and the language needed to deal with the problem of direct
images of hermitian vector bundles for projective non-necessarily
smooth morphisms. To this end, we first define hermitian structures on the
objects of the bounded derived category of coherent sheaves on a
smooth complex variety. Secondly we extend the theory of Bott-Chern
classes to these hermitian structures. Finally we introduce the
category $\oSm_{\ast/{\mathbb C}}$ whose morphisms are projective morphisms
with a hermitian structure on the relative tangent complex. \end{abstract}
\section{Introduction} \label{sec:introduction}
Derived categories were introduced in the 60's of the last century by Grothendieck and Verdier in order to study and generalize duality phenomenons in Algebraic Geometry (see \cite{Hartshorne:rd}, \cite{Verdier:MR1453167}). Since then, derived categories had become a standard tool in Algebra and Geometry and the right framework to define derived functors and to study homological properties. A paradigmatic example is the definition of direct image of sheaves. Given a map $\pi\colon X\to Y$ between varieties and a sheaf $\mathcal{F}$ on $X$, there is a notion of direct image $\pi _{\ast}\mathcal{F}$. We are not specifying what kind of variety or sheaf we are talking about because the same circle of ideas can be used in many different settings. This direct image is not exact in the sense that if $f\colon \mathcal{F}\to \mathcal{G}$ is a surjective map of sheaves, the induced morphism $\pi _{\ast}f\colon \pi _{\ast}\mathcal{F}\to \pi _{\ast}\mathcal{G}$ is not necessarily surjective. One then can define a \emph{derived} functor $R\pi _{\ast}$ that takes values in the derived category of sheaves on $Y$ and that is exact in an appropriate sense. This functor encodes a lot of information about the topology of the fibres of the map $\pi $.
The interest for the derived category of coherent sheaves on a variety exploded with the celebrated 1994 lecture by Kontsevich \cite{Kontsevich:MR1403918}, interpreting mirror symmetry as an equivalence between the derived category of the Fukaya category of certain symplectic manifold and the derived category of coherent sheaves of a dual complex manifold. In the last decades, many interesting results about the derived category of coherent sheaves have been obtained, like Bondal-Orlov Theorem \cite{BondalOrlov:MR1818984} that shows that a projective variety with ample canonical or anti-canonical bundle can be recovered from its derived category of coherent sheaves. Moreover, new tools for studying algebraic varieties have been developed in the context of derived categories like the Fourier-Mukai transform \cite{Mukai:FT}. The interested reader is referred to books like \cite{Huybrechts:FM} and \cite{Bartoccietal:MR2511017} for a thorough exposition of recent developments in this area.
Hermitian vector bundles are ubiquitous in Mathematics. An interesting problem is to define the direct image of hermitian vector bundles. More concretely, let $\pi \colon X\to Y$ be a proper holomorphic map of complex manifolds and let $\overline E=(E,h)$ be a hermitian holomorphic vector bundle on $X$. We would like to define the direct image $\pi_{\ast}\overline E$ as something as close as possible to a hermitian vector bundle on $Y$. The information that would be easier to extract from such a direct image is encoded in the determinant of the cohomology \cite{Deligne:dc}, that can be defined directly. Assume that $\pi $ is a submersion and that we have chosen a hermitian metric on the relative tangent bundle $T_{\pi }$ of $\pi $ satisfying certain technical conditions. Then the determinant line bundle $\lambda (E)=\det(R\pi _{\ast}E)$ can be equipped with the Quillen metric (\cite{Quillen:dCRo}, \cite{BismutFreed:EFI}, \cite{BismutFreed:EFII}), that depends on the metrics on $E$ and $T_{\pi }$ and is constructed using the analytic torsion \cite{RaySinger:ATCM}. The Quillen metric has applications in Arithmetic Geometry (\cite{Faltings:cas}, \cite{Deligne:dc}, \cite{GilletSoule:aRRt}) and also in String Theory (\cite{Yau:MR915812}, \cite{AlvarezGaumeetals:MR908551}). Assume furthermore that the higher direct image sheaves $R^{i}\pi _{\ast}E$ are locally free. In general it is not possible to define an analogue of the Quillen metric as a hermitian metric on each vector bundle $R^{i}\pi _{\ast}E$. But following Bismut and K\"ohler \cite{Bismut-Kohler}, one can do something almost as good. We can define the $L^{2}$-metric on $R^{i}\pi _{\ast}E$ and \emph{correct} it using the higher analytic torsion form. Although this \emph{corrected metric} is not properly a hermitian metric, it is enough for constructing characteristic forms and it appears in the Arithmetic Grothendieck-Riemann-Roch Theorem in higher degrees~\cite{GilletRoesslerSoule:_arith_rieman_roch_theor_in_higher_degrees_p}.
The main objective of the present paper is to set up the theoretical basis and the language needed to deal with the problem of direct images of hermitian vector bundles for projective non-necessarily smooth morphisms. This program will be continued in the subsequent paper \cite{BurgosFreixasLitcanu:GenAnTor} where we give an axiomatic characterization of analytic torsion forms and we generalize them to projective morphisms. The ultimate goal of this program is to state and prove an Arithmetic Grothendieck-Riemann-Roch Theorem for general projective morphisms. This last result will be the topic of a forthcoming paper.
When dealing with direct images of hermitian vector bundles for non smooth morphisms, one is naturally led to consider hermitian structures on objects of the bounded derived category of coherent sheaves $\Db$. One reason for this is that, for a non-smooth projective morphism $\pi $, instead of the relative tangent bundle one should consider the relative tangent complex, that defines an object of $\Db(X)$. Another reason is that, in general, the higher direct images $R^{i}\pi _{\ast}E$ are coherent sheaves and the derived direct image $R\pi _{\ast}E$ is an object of $\Db(Y)$.
Thus the first goal of this paper is to define hermitian structures. A possible starting point is to define a hermitian metric on an object $\mathcal{F}$ of $\Db(X)$ as an isomorphism $E\dashrightarrow \mathcal{F}$ in $\Db(X)$, with $E$ a bounded complex of vector bundles, together with a choice of a hermitian metric on each constituent vector bundle of $E$. Here we find a problem, because even being $X$ smooth, in the bounded derived category of coherent sheaves of $X$, not every object can be represented by a bounded complex of locally free sheaves (see \cite{Voisin:counterHcK} and Remark \ref{rem:3}). Thus the previous idea does not work for general complex manifolds. To avoid this problem we will restrict ourselves to the algebraic category. Thus, from now on the letters $X,\ Y,\dots$ will denote smooth algebraic varieties over ${\mathbb C}$, and all sheaves will be algebraic.
With the previous definition of hermitian metric, for each object of $\Db(X)$ we obtain a class of metrics that is too wide. Different constructions that ought to produce the same metric produce in fact different metrics. This indicates that we may define a hermitian structure as an equivalence class of hermitian metrics.
Let us be more precise. Being $\Db(X)$ a triangulated category, to every morphism $\mathcal{F}\overset{f}{\dashrightarrow}\mathcal{G}$ in $\Db(X)$ we can associate its cone, that is defined up to a (not unique) isomorphism by the fact that \begin{displaymath}
\mathcal{F}\dashrightarrow \mathcal{G}\dashrightarrow \cone(f)\dashrightarrow \mathcal{F}[1] \end{displaymath} is a distinguished triangle. If now $\mathcal{F}$ and $\mathcal{G}$ are provided with hermitian metrics, we want that $\cone(f)$ has an induced hermitian structure that is well defined up to \emph{isometry}. By choosing a representative of the map $f$ by means of morphisms of complexes of vector bundles, we can induce a hermitian metric on $\cone(f)$, but this hermitian metric depends on the choices. The idea behind the definition of hermitian structures is to introduce the finest equivalence relation between metrics such that all possible induced hermitian metrics on $\cone(f)$ are equivalent.
Once we have defined hermitian structures a new invariant of $X$ can be naturally defined. Namely, the set of hermitian structures on a zero object of $\Db(X)$ is an abelian group that we denote $\KA(X)$ (Definition \ref{def:KA}). In the same way that $K_{0}(X)$ is the universal abelian group for additive characteristic classes of vector bundles, $\KA(X)$ is the universal abelian group for secondary characteristic classes of acyclic complexes of hermitian vector bundles (Theorem~\ref{thm:8}).
Secondary characteristic classes constitute other of the central topics of this paper. Recall that to each vector bundle we can associate its Chern character, that is an additive characteristic class. If the vector bundle is provided with a hermitian metric, we can use Chern-Weil theory to construct a concrete representative of the Chern character, that is a differential form. This characteristic form is additive only for orthogonally split short exact sequences and not for general short exact sequences. Bott-Chern classes were introduced in \cite{BottChern:hvb} and are secondary classes that measure the lack of additivity of the characteristic forms.
The Bott-Chern classes have been extensively used in Arakelov Geometry (\cite{GilletSoule:vbhm}, \cite{BismutGilletSoule:at}) and they can be used to construct characteristic classes in higher $K$-theory (\cite{Burgos-Wang}). The second goal of this paper is to extend the definition of additive Bott-Chern classes to the derived category. This is the most general definition of additive Bott-Chern classes and encompasses both, the Bott-Chern classes defined in \cite{BismutGilletSoule:at} and the ones defined in \cite{Ma:MR1765553} (Example \ref{exm:1}).
Finally, recall that the hermitian structure on the direct image of a hermitian vector bundle should also depend on a hermitian structure on the relative tangent complex. Thus the last goal of this paper is to introduce the category $\overline{\Sm}_{\ast/{\mathbb C}}$ (Definition \ref{def:16}, Theorem \ref{thm:17}). The objects of this category are smooth algebraic varieties over ${\mathbb C}$ and the morphisms are pairs $\overline{f}=(f,\overline{T}_{f})$ formed by a projective morphism of smooth complex varieties $f$, together with a hermitian structure on the relative tangent complex $T_{f}$. The main difficulty here is to define the composition of two such morphisms.
The remarkable fact is that the hermitian cone construction enables us to define a composition rule for these morphisms.
We describe with more detail the contents of each section.
In Section \ref{sec:meager-complexes} we define and characterize the notion of \textit{meager complex} (Definition \ref{def:9} and Theorem \ref{thm:3}). Roughly speaking, meager complexes are bounded acyclic complexes of hermitian vector bundles whose Bott-Chern classes vanish for structural reasons. We then introduce the concept of tight morphism (Definition \ref{def:tight_morphism}) and tight equivalence relation (Definition \ref{def:17}) between bounded complexes of hermitian vector bundles. We explain a series of useful computational rules on the monoid of hermitian vector bundles modulo tight equivalence relation, that we call \textit{acyclic calculus} (Theorem \ref{thm:7}). We prove that the submonoid of acyclic complexes modulo meager complexes has a structure of abelian group, this is the group $\KA(X)$ mentioned previously.
With these tools at hand, in Section \ref{sec:oDb} we define hermitian structures on objects of $\Db(X)$ and we introduce the category $\oDb(X)$. The objects of the category $\oDb(X)$ are objects of $\Db(X)$ together with a hermitian structure, and the morphisms are just morphisms in $\Db(X)$. Theorem \ref{thm:13} is devoted to describe the structure of the forgetful functor $\oDb(X)\to\Db(X)$. In particular, we show that the group $\KA(X)$ acts on the fibers of this functor, freely and transitively.
An important example of use of hermitian structures is the construction of the \textit{hermitian cone} of a morphism in $\oDb(X)$ (Definition \ref{def:her_cone}), which is well defined only up to tight isomorphism. We also study several elementary constructions in $\oDb(X)$. Here we mention the classes of isomorphisms and distinguished triangles in $\oDb(X)$. These classes lie in the group $\KA(X)$ and their properties are listed in Theorem \ref{thm:10}. As an application we show that $\KA(X)$ receives classes from $K_{1}(X)$ (Proposition \ref{prop:K1_to_KA}).
Section \ref{sec:bott-chern-classes} is devoted to the extension of Bott-Chern classes to the derived category. For every additive genus, we associate to each isomorphism or distinguished triangle in $\oDb(X)$ a Bott-Chern class satisfying properties analogous to the classical ones.
We conclude the paper with Section \ref{sec:multiplicative-genera}, where we extend the definition of Bott-Chern classes to multiplicative genera and in particular to the Todd genus. In this section we also define the category $\overline{\Sm}_{\ast/{\mathbb C}}$.
\textbf{Acknowledgements:} We would like to thank the following institutions where part of the research conducting to this paper was done. The CRM in Bellaterra (Spain), the CIRM in Luminy (France), the Morningside Institute of Beijing (China), the University of Barcelona and the IMUB, the Alexandru Ioan Cuza University of Iasi, the Institut de Math\'ematiques de Jussieu and the ICMAT (Madrid).
We would also like to thank D. Eriksson and D. R\"ossler for several discussions on the subject of this paper.
\section{Meager complexes and acyclic calculus} \label{sec:meager-complexes}
The aim of this section is to construct a universal group for additive Bott-Chern classes of acyclic complexes of hermitian vector bundles. To this end we first introduce and study the class of meager complexes. Any Bott-Chern class that is additive for certain short exact sequences of acyclic complexes (see \ref{thm:8}) and that vanishes on orthogonally split complexes, necessarily vanishes on meager complexes. Then we develop an acyclic calculus that will ease the task to check if a particular complex is meager. Finally we introduce the group $\KA$, which is the universal group for additive Bott-Chern classes.
Let $X$ be a complex algebraic variety over ${\mathbb C}$, namely a reduced and separated scheme of finite type over ${\mathbb C}$. We denote by $\Vb(X)$ the exact category of bounded complexes of algebraic vector bundles on $X$. Assume in addition that $X$ is smooth over ${\mathbb C}$. Then $\oV(X)$ is defined as the category of pairs $\overline E=(E,h)$, where $E\in \Ob \Vb(X)$ and $h$ is a smooth hermitian metric on the complex of analytic vector bundle $E^{\text{{\rm an}}}$. From now on we shall make no distinction between $E$ and $E^{\text{{\rm an}}}$. The complex $E$ will be called \emph{the underlying complex of } $\overline E$. We will denote by the symbol $\sim$ the quasi-isomorphisms in any of the above categories.
A basic construction in $\Vb(X)$ is the cone of a morphism of complexes. Recall that, if $f\colon E\to F$ is such a morphism, then, as a graded vector bundle $\cone(f)=E[1]\oplus F$ and the differential is given by $\dd(x,y)=(-\dd x,f(x)+\dd y).$ We can extend the cone construction easily to $\oV(X)$ as follows.
\begin{definition} \label{def:14} If $f\colon \overline E\to \overline F$ is a morphism in $\oV(X)$, \emph{the hermitian cone} of $f$, denoted by $\ocone(f)$, is defined as the cone of $f$ provided with the orthogonal sum hermitian metric.
When the morphism is clear from the context we will sometimes denote $\ocone(f)$ by $\ocone(\overline E,\overline F)$. \end{definition}
\begin{remark} \label{rem:2} Let $f\colon \overline E\to \overline F$ be a
morphism in $\oV(X)$. Then
there is an exact sequence of complexes
\begin{displaymath}
0\longrightarrow \overline F\longrightarrow \ocone(f)
\longrightarrow \overline E[1]\longrightarrow 0,
\end{displaymath}
whose constituent short exact sequences are orthogonally
split. Conversely, if
\begin{displaymath}
0\longrightarrow \overline F\longrightarrow \overline G
\longrightarrow \overline E[1]\longrightarrow 0
\end{displaymath}
is a short exact sequence all whose constituent exact sequences are
orthogonally split, then there is a natural section $s\colon E[1]\to
G$. The image of $\dd s-s\dd$ belongs to
$F$ and, in fact, determines a morphism
of complexes
\begin{displaymath}
f_{s}:=\dd s-s\dd\colon \overline E
\longrightarrow \overline F.
\end{displaymath}
Moreover, there is a natural isometry $\overline G \cong \ocone
(f_{s})$. \end{remark}
The hermitian cone has the following useful property.
\begin{lemma}\label{lemm:13}
Consider a diagram in $\oV(X)$
\begin{displaymath}
\xymatrix{
\overline{E}'\ar[r]^{f'}\ar[d]_{g'} &\overline{F}'\ar[d]^{g}\\
\overline{E}\ar[r]^{f} &\overline{F}.
}
\end{displaymath}
Assume that the diagram is commutative up to homotopy and fix a
homotopy $h$. The homotopy $h$ induces
morphisms of complexes
\begin{align*}
&\psi \colon\ocone(f')\longrightarrow\ocone(f)\\
&\phi \colon\ocone(-g')\longrightarrow\ocone(g)
\end{align*}
and there is a natural isometry of complexes
\begin{displaymath}
\ocone(\phi )\overset{\sim}{\longrightarrow}\ocone(\psi ).
\end{displaymath}
Morever, let $h'$ be a second homotopy between $g\circ f'$ and
$f\circ g'$ and let $\psi '$ be the induced morphism. If there
exists a higher homotopy between $h$ and $h'$, then $\psi
$ and $\psi '$ are homotopically equivalent.
\end{lemma}
\begin{proof}
Since $h\colon E'\to F[-1]$ is a homotopy between $gf'$ and $fg'$,
we have
\begin{equation}\label{eq:homotopy_square_1}
gf'-fg'=\dd h+h\dd.
\end{equation}
First of all, define the arrow $\psi \colon\ocone(f')\to\ocone(f)$ by
the following rule:
\begin{displaymath}
\psi (x',y')=(g'(x'),g(y')+h(x')).
\end{displaymath}
From the definition of the differential of a cone and the homotopy
relation (\ref{eq:homotopy_square_1}), one easily checks that $\psi $
is a morphism of complexes. Now apply the same construction to the
diagram
\begin{equation}\label{eq:homotopy_square_2}
\xymatrix{
\overline{E}'\ar[r]^{-g'}\ar[d]_{-f'} &\overline{E}\ar[d]^{f}\\
\overline{F'}\ar[r]^{g} &\overline{F}.
}
\end{equation}
The diagram (\ref{eq:homotopy_square_2}) is still commutative up
to homotopy and $h$ provides such a homotopy. We obtain a
morphism of complexes $\phi :\ocone(-g')\to\ocone(g)$, defined by the
rule
\begin{displaymath}
\phi (x',x)=(-f'(x'),f(x)+h(x')).
\end{displaymath}
One easily checks that a suitable reordering of factors sets an
isometry of complexes between $\ocone(\phi )$ and $\ocone(\psi )$.
Assume now that $h'$ is a second homotopy and that there is a
higher homotopy $s\colon \overline E' \to \overline F[-2]$ such that
\begin{displaymath}
h'-h=\dd s- s \dd.
\end{displaymath}
Let $H\colon \ocone(f')\to \ocone(f)[-1]$ be given by
$H(x',y')=(0,s(x'))$. Then
\begin{displaymath}
\psi '-\psi =\dd H + H \dd.
\end{displaymath}
Hence $\psi $ and $\psi' $ are homotopically equivalent.
\end{proof}
Recall that, given a morphism of complexes $f\colon \overline E \to \overline
F$, we use the abuse of notation $\ocone(f)=\ocone(\overline E, \overline
F)$. As seen in the previous lemma, sometimes it is natural to consider
$\ocone(-f)$. With the notation above it will be denoted also by
$\ocone(\overline E,\overline F)$. Note that this ambiguity is harmless because
there is a natural isometry between $\ocone(f)$ and $\ocone(-f)$. Of
course, when more than one morphism between $\overline E$ and $\overline F$ is
considered, the above notation should be avoided.
With this convention, Lemma \ref{lemm:13} can
be written as
\begin{equation}
\label{eq:53}
\ocone(\ocone(\overline E',\overline E),\ocone(\overline F',\overline F))\cong
\ocone(\ocone(\overline E',\overline F'),\ocone(\overline E,\overline F)).
\end{equation}
\begin{definition}\label{def:11}
We will denote by $\mathscr{M}_{0}=\mathscr{M}_{0}(X)$ the subclass of
$\oV(X)$ consisting of
\begin{enumerate}
\item the orthogonally split complexes;
\item all objects $\overline E$ such that there
is an acyclic complex $\overline F$ of $\oV(X)$, and an isometry $\overline E
\to \overline F\oplus \overline F[1]$.
\end{enumerate} \end{definition}
We want to stabilize $\mathscr{M}_{0}$ with respect to hermitian cones.
\begin{definition}\label{def:9}
We will denote by $\mathscr{M}=\mathscr{M}(X)$ the smallest subclass of
$\oV(X)$ that satisfies the following
properties:
\begin{enumerate}
\item \label{item:12} it contains $\mathscr{M}_{0}$;
\item \label{item:13} if $f\colon \overline
E\to \overline F$ is a morphism and two
of $\overline E$, $\overline F$ and $\ocone(f)$ belong
to $\mathscr{M}$, then so does the third.
\end{enumerate}
The elements of $\mathscr{M}(X)$ will be called \emph{meager
complexes}. \end{definition}
We next give a characterization of meager complexes. For this, we introduce two auxiliary classes. \begin{definition} \label{def:12}
\begin{enumerate}
\item \label{item:29} Let $\mathscr{M}_{F}$ be the subclass of
$\oV(X)$ that contains all complexes $\overline E$ that have a
finite filtration $\Fil$ such that
\begin{enumerate}
\item[({\bf A})] \label{item:14} for every $p,n\in {\mathbb Z}$, the exact
sequences
\begin{displaymath}
0\to \Fil^{p+1}\overline E^{n}\to \Fil^{p}\overline E^{n}\to
\Gr^{p}_{\Fil}\overline E^{n}\to 0,
\end{displaymath}
with the induced metrics, are orthogonally split short exact
sequences of vector bundles;
\item[({\bf B})] \label{item:15} the complexes
$\Gr^{\bullet}_{\Fil}\overline E$ belong to $\mathscr{M}_{0}$.
\end{enumerate}
\item \label{item:32} Let $\mathscr{M}_{S}$ be the subclass of
$\oV(X)$ that contains all complexes $\overline E$ such that there
is a morphism of complexes $f\colon \overline E\to \overline F$
and both $\overline F$ and $\ocone (f)$ belong to
$\mathscr{M}_{F}$.
\end{enumerate} \end{definition}
\begin{lemma} \label{lemm:10} Let $0\to \overline E\to \overline F\to \overline G\to
0$ be an exact sequence in $\oV(X)$ whose
constituent rows are orthogonally split. Assume $\overline E$ and $\overline
G$ are in $\mathscr{M}_{F}$ . Then $\overline F\in\mathscr{M}_{F}$. In
particular, $\mathscr{M}_{F}$ is closed under cone formation.
\end{lemma}
\begin{proof}
For the first claim, notice that the filtrations of $\overline E$ and $\overline G$
induce a filtration on $\overline F$ satisfying conditions
\ref{def:12}~({\bf A}) and \ref{def:12}~({\bf B}). The second
claim then follows by Remark \ref{rem:2}.
\end{proof}
\begin{example} \label{exm:2}
Given any complex $\overline E\in \Ob \oV(X)$, the complex
$\ocone(\Id_{\overline E})$ belongs to
$\mathscr{M}_{F}$. This can be seen by induction on the length of
$\overline E$ using Lemma \ref{lemm:10} and the b\^ete filtration of $\overline
E$. For the starting point of the induction one takes into account
that, if $\overline E$ has only one non zero
degree, then $\ocone(\Id_{\overline E})$ is orthogonally split. In fact,
this argument shows something slightly stronger. Namely, the complex
$\ocone(\Id_{\overline E})$ admits a finite filtration $\Fil$ satisfying
\ref{def:12}~({\bf A}) and such that the complexes
$\Gr^{\bullet}_{\Fil} \ocone(\Id_{\overline E})$ are orthogonally
split. \end{example}
\begin{theorem}\label{thm:3} The equality
\begin{math}
\mathscr{M}=\mathscr{M}_{S}
\end{math}
holds. \end{theorem} \begin{proof}
We start by proving that $\mathscr{M}_{F}\subset \mathscr{M}$.
Let $\overline E\in \mathscr{M}_{F}$ and let $\Fil$ be any
filtration that satisfies conditions \ref{def:12}~({\bf A}) and
\ref{def:12}~({\bf B}). We show that $
\overline E\in \mathscr{M}$ by induction
on the length of $\Fil$. If
$\Fil $ has length one, then $\overline E$ belongs to
$\mathscr{M}_{0}\subset \mathscr{M}$. If the length of $\Fil$ is
$k>1$, let $p$ be
such that $\Fil^{p}\overline E=\overline E$ and
$\Fil^{p+1}\overline E\not = \overline E$. On the one hand, $\Gr
^{p}_{\Fil}\overline E [-1]\in
\mathscr{M}_{0}\subset \mathscr{M}$ and, on the other hand, the
filtration $\Fil$
induces a filtration on $\Fil^{p+1}\overline E$ fulfilling
conditions \ref{def:12}~({\bf A}) and \ref{def:12}~({\bf B}) and has
length $k-1$. Thus, by induction hypothesis, $\Fil^{p+1}\overline
E\in \mathscr{M}$. Then, by
Lemma \ref{lemm:10}, we deduce that $\overline E\in
\mathscr{M}$.
Clearly, the fact that $\mathscr{M}_{F}\subset \mathscr{M}$
implies that $\mathscr{M}_{S}\subset \mathscr{M}$. Thus, to
prove the theorem, it only remains to show that $\mathscr{M}_{S}$
satisfies the condition \ref{def:9}~\ref{item:13}.
The content of the next result is that
the apparent asymmetry in the definition of
$\mathscr{M}_{S}$ is not real.
\begin{lemma}\label{lemm:9}
Let $\overline E\in \Ob \oV(X)$. Then there is a morphism $f\colon
\overline
E\to \overline F$ with $\overline F$ and
$\ocone(f)$ in $\mathscr{M}_{F}$ if and only if there is a
morphism $g\colon \overline G\to \overline E$ with
$\overline G$ and
$\ocone(g)$ in $\mathscr{M}_{F}$.
\end{lemma}
\begin{proof}
Assume that there is a morphism $f\colon \overline
E\to \overline F$ with $\overline F$ and
$\ocone(f)$ in $\mathscr{M}_{F}$. Then, write $\overline
G= \ocone(f)[-1]$ and let $g\colon \overline G\to
\overline E$ be the natural map. By hypothesis, $\overline
G\in \mathscr{M}_{F}$. Moreover, since there is a natural isometry
\begin{displaymath}
\ocone(\ocone(\overline E,\overline
F)[-1],\overline E)\cong
\ocone(\ocone(\Id_{\overline E})[-1],\overline
F),
\end{displaymath}
by Example \ref{exm:2} and Lemma \ref{lemm:10} we obtain that
$\ocone(g)\in \mathscr{M}_{F}$. Thus we have proved one
implication. The proof of the other implication is analogous.
\end{proof}
Let now $f\colon \overline E\to \overline F$ be a morphism of
complexes with $\overline E, \overline F\in
\mathscr{M}_{S}$. We want to show that $\ocone(f)\in
\mathscr{M}_{S}$. By Lemma \ref{lemm:9}, there are morphisms of
complexes
$g\colon \overline G\to \overline E$ and
$h\colon \overline H\to \overline F$ with $\overline
G,\ \overline H,\ \ocone(g),\ \ocone(h)\in
\mathscr{M}_{F}$. We consider the map $\overline G\to
\ocone(h)$ induced by $f\circ g$. Then we write
\begin{displaymath}
\overline{G'}=\ocone(\overline G, \ocone(h) )[-1].
\end{displaymath}
By Lemma \ref{lemm:10}, we have that $\overline{G'}\in
\mathscr{M}_{F}$. We denote by $g'\colon G'\to E$ and $k\colon G'\to H$ the
maps $g'(a,b,c)=g(a)$ and $k(a,b,c)=-b$.
There is an exact sequence
\begin{displaymath}
0\to \ocone(h)\to \ocone(g')\to \ocone(g)\to 0
\end{displaymath}
whose constituent short exact sequences are orthogonally split.
Since $\ocone(h)$ and $\ocone(g)$ belong to $\mathscr{M}_{F}$,
Lemma \ref{lemm:10} insures that $\ocone(g')$ belongs to
$\mathscr{M}_{F}$ as well.
There is a diagram
\begin{equation}\label{eq:51}
\xymatrix{
\overline {G'} \ar[d]_{g'} \ar[r]^{k} & \overline H\ar[d]^{h} \\
\overline E \ar[r]^{f} & \overline F
}
\end{equation}
that commutes up to homotopy. We fix the homotopy $s\colon \overline G'\to
F$ given by $s(a,b,c)=c$. By Lemma \ref{lemm:13} there is
a natural isometry
\begin{displaymath}
\ocone(\ocone(g'),\ocone(h))\cong \ocone(\ocone(-k),\ocone(f)).
\end{displaymath}
Applying Lemma \ref{lemm:10} again, we have that $\ocone(-k)$ and
$\ocone(\ocone(g'),\ocone(h))$ belong to
$\mathscr{M}_{F}$. Therefore $\ocone(f)$ belongs to
$\mathscr{M}_{S}$.
\begin{lemma} \label{lemm:11}
Let $f\colon \overline E\to \overline F$ be a morphism in $\oV(X)$.
\begin{enumerate}
\item \label{item:18} If $\overline E\in \mathscr{M}_{S}$ and $\ocone(f)\in
\mathscr{M}_{F}$ then $\overline F\in \mathscr{M}_{S}$.
\item \label{item:19} If $\overline F\in \mathscr{M}_{S}$ and $\ocone(f)\in
\mathscr{M}_{F}$ then $\overline E\in \mathscr{M}_{S}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume that $\overline E\in \mathscr{M}_{S}$ and $\ocone(f)\in
\mathscr{M}_{F}$. Let $g\colon \overline G\to \overline E$ with $\overline G\in
\mathscr{M}_{F}$ and $\ocone(g)\in \mathscr{M}_{F}$. By Lemma
\ref{lemm:10} and Example \ref{exm:2}, $\ocone(\ocone(\Id_{\overline
G}),\ocone(f))\in \mathscr{M}_{F}$. But
there is a natural isometry of complexes
\begin{displaymath}
\ocone(\ocone(\Id_{\overline G}),\ocone(f))\cong
\ocone(\ocone(\ocone(g)[-1],\overline G),\overline F).
\end{displaymath}
Since, by Lemma \ref{lemm:10}, $\ocone(\ocone(g)[-1],\overline G)\in
\mathscr{M}_{F}$, then $\overline F\in \mathscr{M}_{S}$.
The second statement of the lemma is proved using the dual
argument.
\end{proof}
\begin{lemma} \label{lemm:12}
Let $f\colon \overline E\to \overline F$ be a morphism in $\oV(X)$.
\begin{enumerate}
\item \label{item:20} If $\overline E\in \mathscr{M}_{F}$ and $\ocone(f)\in
\mathscr{M}_{S}$ then $\overline F\in \mathscr{M}_{S}$.
\item \label{item:21} If $\overline F\in \mathscr{M}_{F}$ and $\ocone(f)\in
\mathscr{M}_{S}$ then $\overline E\in \mathscr{M}_{S}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume that $\overline E\in \mathscr{M}_{F}$ and $\ocone(f)\in
\mathscr{M}_{S}$. Let $g\colon \overline G\to \ocone(f)$ with $\overline G$
and $\ocone(\overline G, \ocone(f))$ in $\mathscr{M}_{F}$. There is a
natural isometry of complexes
\begin{displaymath}
\ocone(\overline G,\ocone(f)))\cong \ocone(\ocone(\overline G[-1],\overline
E),\overline F)
\end{displaymath}
that shows $\overline F\in \mathscr{M}_{S}$.
The second statement of the lemma is proved by a dual argument.
\end{proof}
Assume now that $f\colon \overline E\to \overline F$ is a morphism in $\oV(X)$
and $\overline E,\ \ocone(f)\in
\mathscr{M}_{S}$. Let $g\colon \overline G \to \overline E$ with $\overline G,\
\ocone(g)\in \mathscr{M}_{F}$. There is a natural isometry
\begin{displaymath}
\ocone(\ocone(\overline G,\overline E),\ocone(\Id_{\overline F}))
\cong
\ocone(\ocone(\overline G,\overline F),\ocone(\overline E, \overline F)),
\end{displaymath}
that implies $\ocone(\ocone(\overline G,\overline F),\ocone(\overline E, \overline
F))\in \mathscr{M}_{F}$. By Lemma \ref{lemm:11}, we deduce that
$\ocone(\overline G,\overline F)\in \mathscr{M}_{S}$. By Lemma \ref{lemm:12},
$\overline F\in \mathscr{M}_{S}$.
With $f$ as above, the fact that, if $\overline F$ and $\ocone(f)$ belong
to $\mathscr{M}_{S}$ so does $\overline E$, is proved by a similar
argument. In conclusion, $\mathscr{M}_{S}$ satisfies the condition
\ref{def:9}~\ref{item:13}, hence $\mathscr{M}\subset
\mathscr{M}_{S}$, which completes the proof of the
theorem. \end{proof}
The class of meager complexes satisfies the next list of properties, that follow almost directly from Theorem \ref{thm:3}. \begin{theorem} \label{thm:4}
\begin{enumerate}
\item \label{item:22} If $\overline E$ is a meager complex and $\overline
F$ is a hermitian vector bundle, then the complexes
$\overline F \otimes \overline E$, $\Hom(\overline F,\overline E)$ and $\Hom(\overline E,\overline
F)$, with the induced metrics, are meager.
\item \label{item:23} If $\overline E^{\ast,\ast}$ is a bounded
double complex of hermitian vector bundles and all rows (or
columns) are meager complexes,
then the complex $\Tot(\overline E^{\ast,\ast})$ is meager.
\item \label{item:24} If $\overline E$ is a meager complex and $\overline F$ is another
complex of hermitian vector bundles, then the complexes
\begin{align*}
\overline E\otimes \overline F&=\Tot((\overline F^{i}\otimes \overline E^{j})_{i,j}),\\
\uHom (\overline E,\overline F)&= \Tot(\Hom ((\overline E^{-i},\overline F^{j})_{i,j}))\text{
and }\\
\uHom (\overline F,\overline E)&= \Tot(\Hom ((\overline F^{-i},\overline E^{j})_{i,j})),
\end{align*}
are meager.
\item \label{item:25} If $f\colon X\to Y$ is a morphism of smooth complex
varieties and $\overline E$ is a meager complex on $Y$, then $f^{\ast}
\overline E$ is a meager complex on $X$.
\end{enumerate} \end{theorem}
We now introduce the notion of tight morphism.
\begin{definition}\label{def:tight_morphism} A morphism $f\colon\overline E\to \overline F$ in $\oV(X)$ is said to be \emph{tight} if $\ocone(f)$ is a meager complex. \end{definition}
\begin{proposition} \label{prop:8}
\begin{enumerate}
\item Every meager complex is acyclic.
\item Every tight morphism is a quasi-isomorphism.
\end{enumerate} \end{proposition} \begin{proof}
Let $\overline E\in \mathscr{M}_{F}(X)$. Let $\Fil$ be any
filtration that satisfies conditions
\ref{def:12}~({\bf A}) and \ref{def:12}~({\bf B}).
By definition, the
complexes $\Gr_{\Fil}^{p}\overline E$ belong to $\mathscr{M}_{0}$,
so they are acyclic. Hence
$\overline E$ is acyclic.
If $\overline E\in
\mathscr{M}_{S}(X)$, let $\overline F$ and $\ocone(f)$
be as in Definition \ref{def:12}~\ref{item:32}. Then, $\overline F$
and
$\ocone(f)$ are acyclic, hence $\overline E$ is also
acyclic. Thus we have proved the first statement.
The second statement is a direct consequence of the first one. \end{proof}
Many arguments used for proving that a certain complex is meager or a certain morphism is tight involve cumbersome diagrams. In order to ease these arguments we will develop a calculus of acyclic complexes.
Before starting we need some preliminary lemmas.
\begin{lemma} \label{lemm:16}
Let $\overline E$, $\overline F$ be objects of $\oV(X)$. Then the following
conditions are equivalent.
\begin{enumerate}
\item \label{item:37} There exists an object $\overline G$ and a diagram
\begin{displaymath}
\xymatrix{
& \overline G \ar[dl]^{\sim}_{f} \ar[dr]^{g}&\\
\overline E && \overline F,}
\end{displaymath}
such that $\ocone(g)\oplus \ocone(f)[1]$ is meager.
\item \label{item:38}
There exists an object $\overline G$ and a diagram
\begin{displaymath}
\xymatrix{
& \overline G \ar[dl]^{\sim}_{f} \ar[dr]^{g}&\\
\overline E && \overline F,}
\end{displaymath}
such that $f$ and $g$ are tight morphisms.
\end{enumerate} \end{lemma} \begin{proof}
Clearly, \ref{item:38} implies \ref{item:37}. To prove the converse
implication, if $\overline G$ satisfies the conditions of \ref{item:37},
we put $G'=G\oplus \ocone(f)$ and consider the morphisms $f'\colon \overline G'\to
E$ and $g'\colon G'\to F$ induced by the first projection $G'\to
G$. Then
\begin{displaymath}
\ocone(f')=\ocone(f)\oplus \ocone(f)[1],
\end{displaymath}
that is meager because $\ocone(f)$ is acyclic, and
\begin{displaymath}
\ocone(g')=\ocone(g)\oplus \ocone(f)[1],
\end{displaymath}
that is meager by hypothesis. \end{proof}
\begin{lemma}\label{lemm:15} Any diagram of tight morphisms, of the following types:
\begin{equation}\label{diag:1}
\begin{array}{ccc}
\xymatrix{
\overline E\ar[rd]_{f} & &\overline G\ar[ld]^{g}\\
&\overline F
} &
\quad \quad &
\xymatrix{
&\overline H\ar[rd]^{g'}\ar[ld]_{f'} &\\
\overline E & &\overline G
}\\
(i) && (ii)
\end{array}
\end{equation}
can be completed into a diagram of tight morphisms
\begin{equation}\label{eq:meager_1}
\xymatrix{
&\overline H\ar[ld]_{f'}\ar[rd]^{g'} &\\
\overline E \ar[rd]_{f}& &\overline G\ar[ld]^{g}\\
&\overline F, &
}
\end{equation} which commutes up to homotopy. \end{lemma} \begin{proof}
We prove the statement only for the case (i), the other one being analogous.
Note that there is a natural arrow $\overline
G\to\ocone(f)$. Define
\begin{displaymath}
\overline H=\ocone(\overline G,\ocone(f))[-1].
\end{displaymath}
With this choice, diagram (\ref{eq:meager_1} i) becomes commutative up
to homotopy, taking the projection $H\to F[-1]$ as homotopy. We
first show that $\ocone(\overline H, \overline G)$ is meager. Indeed, there is a
natural isometry
\begin{displaymath}
\ocone(\overline H,\overline G)
\cong\ocone(\ocone(\Id_{\overline G}), \ocone(\overline E,\overline F)[-1])
\end{displaymath}
and the right hand side complex is meager. Now for $\ocone(\overline H,
\overline E)$. By Lemma \ref{lemm:13}, there is an isometry
\begin{equation}
\ocone(\ocone(\overline H, \overline E),\ocone(\overline G, \overline F))
\cong
\ocone(\ocone(\overline H, \overline G),\ocone(\overline E, \overline F)).
\end{equation}
The right hand side complex is meager, hence the left hand side is
meager as well. Since, by hypothesis, $\ocone(\overline G, \overline F)$ is
meager, the same is true for $\ocone(\overline H, \overline E)$. \end{proof}
\begin{definition} \label{def:17}
We will say that two complexes $\overline E$ and $\overline F$ are \emph{tightly
related} if any of the equivalent conditions of Lemma
\ref{lemm:16} holds. \end{definition}
It is easy to see, using Lemma \ref{lemm:15}, that to be tightly related is an equivalence relation.
\begin{definition} \label{def:10}
We denote by $\oV(X)/\mathscr{M}$ the set of classes of
tightly related complexes. The class of a complex $\overline E$ will be
denoted $[\overline E]$. \end{definition}
\begin{theorem}[Acyclic calculus] \label{thm:7}
\begin{enumerate}
\item \label{item:34} For a complex $\overline E\in \Ob\oV(X)$, the class
$[\overline E]=0$ if and only if $\overline E\in \mathscr{M}$.
\item \label{item:33} The operation $\oplus$ induces an operation,
that we denote $+$, in $\oV(X)/\mathscr{M}$. With this operation
$\oV(X)/\mathscr{M}$ is an associative abelian semigroup.
\item \label{item:39} For a complex $\overline E$, there exists a complex
$\overline F$ such that $[\overline F]+[\overline E]=0$, if and only if $\overline E$ is acyclic. In
this case $[\overline E[1]]=-[\overline E]$.
\item \label{item:35} For every morphism $f\colon \overline E \to \overline F$,
if $E$ is acyclic, then the equality
\begin{displaymath}
[\ocone(\overline E,\overline F)]=[\overline F]-[\overline E]
\end{displaymath}
holds.
\item \label{item:40} For every morphism $f\colon \overline E \to \overline F$,
if $F$ is acyclic, then the equality
\begin{displaymath}
[\ocone(\overline E,\overline F)]=[\overline F]+[\overline E[1]]
\end{displaymath}
holds.
\item \label{item:36} Given a diagram
\begin{displaymath}
\xymatrix{
\overline{E}'\ar[r]^{f'}\ar[d]_{g'} &\overline{F}'\ar[d]^{g}\\
\overline{E}\ar[r]^{f} &\overline{F}
}
\end{displaymath}
in $\oV(X)$, that commutes up to homotopy, then for every choice
of homotopy we have
\begin{displaymath}
[\ocone(\ocone(f'),\ocone(f))]=[\ocone(\ocone(-g'),\ocone(g))].
\end{displaymath}
\item \label{item:41} Let $f\colon \overline E\to \overline F$, $g\colon \overline
F\to \overline G$ be morphisms of complexes. Then
\begin{align*}
[\ocone(\ocone(g\circ f),\ocone(g))]&=[\ocone(f)[1]],\\
[\ocone(\ocone(f),\ocone(g\circ f))]&=[\ocone(g)].
\end{align*}
If one of $f$ or $g$ are quasi-isomorphisms, then
\begin{displaymath}
[\ocone(g\circ f)]=[\ocone(g)]+[\ocone(f)].
\end{displaymath}
If $g\circ f$ is a quasi-isomorphism, then
\begin{displaymath}
[\ocone(g)]=[\ocone(f)[1]]+[\ocone(g\circ f)].
\end{displaymath}
\end{enumerate} \end{theorem} \begin{proof}
The statements \ref{item:34} and \ref{item:33} are immediate.
For assertion \ref{item:39}, observe that, if $\overline E$ is
acyclic, then $\overline E\oplus \overline E[1]$ is meager. Thus
\begin{displaymath}
[\overline E]+[\overline E[1]]=[\overline E\oplus \overline E[1]]=0.
\end{displaymath}
Conversely, if $[\overline F]+[\overline E]=0$, then $\overline F\oplus \overline E$ is
meager, hence acyclic. Thus $\overline E$ is acyclic.
For property \ref{item:35} we consider the map $\overline F\oplus \overline E[1]\to
\ocone(f)$ defined by the map $\overline F\to \ocone(f)$. There is a
natural isometry
\begin{displaymath}
\ocone(\overline F\oplus \overline E[1],\ocone(f))\cong
\ocone(\overline E\oplus \overline E[1],\ocone(\Id_{F})).
\end{displaymath}
Since the right hand complex is meager, so is the first. In consequence
\begin{displaymath}
[\ocone(f)]=[\overline F\oplus \overline E[1]]=[\overline F]+[\overline E[1]]=[\overline F]-[\overline E].
\end{displaymath}
Statement \ref{item:40} is proved analogously.
Statement \ref{item:36} is a direct consequence of Lemma
\ref{lemm:13}.
Statement \ref{item:41} is an easy consequence of the previous
properties. \end{proof}
\begin{remark}\label{rem:9}
In $f\colon \overline E \to \overline F$ is a morphism and neither $\overline E$ nor
$\overline F$ are acyclic, then $[\ocone(f)]$ depends on the homotopy
class of $f$ and not only on $\overline E$ and $\overline F$. For instance, let
$\overline E$ be a non-acyclic complex of hermitian bundles. Consider the
zero map and the identity map $0,\Id\colon \overline E\to \overline E$. Since,
by Example \ref{exm:2}, we know that $\ocone (\Id)$ is
meager, then $[\ocone (\Id)]=0$. By contrast,
\begin{displaymath}
[\ocone(0)]=[\overline E]+[\overline E[-1]]\not = 0
\end{displaymath}
because $\overline E$ is not acyclic. This implies that we can not extend
Theorem \ref{thm:7}~\ref{item:35} or \ref{item:40} to the case
when none of the complexes are acyclic. \end{remark}
\begin{corollary}\label{cor:3}
\begin{enumerate}
\item \label{item:42}
Let
\begin{displaymath}
0\longrightarrow \overline E\longrightarrow \overline F\longrightarrow \overline G
\longrightarrow 0
\end{displaymath}
be a short exact sequence in $\oV(X)$ all whose constituent short
exact sequences are orthogonally split. If either $\overline E$ or
$\overline G$ is acyclic, then
\begin{displaymath}
[\overline F]=[\overline E]+[\overline G].
\end{displaymath}
\item \label{item:58} Let $\overline E^{\ast,\ast}$ be a bounded double complex of
hermitian vector bundles. If the columns of $\overline E^{\ast,\ast}$
are acyclic, then
\begin{displaymath}
[\Tot(\overline E^{\ast,\ast})]=\sum_{k}(-1)^{k} [\overline E^{k,\ast}].
\end{displaymath}
If the rows are acyclic, then
\begin{displaymath}
[\Tot(\overline E^{\ast,\ast})]=\sum_{k}(-1)^{k} [\overline E^{\ast,k}].
\end{displaymath}
In particular, if rows and columns are acyclic
\begin{displaymath}
\sum_{k}(-1)^{k} [\overline E^{k,\ast}]=\sum_{k}(-1)^{k} [\overline E^{\ast,k}].
\end{displaymath}
\end{enumerate} \end{corollary} \begin{proof}
The first item follows from Theorem \ref{thm:7}~\ref{item:35}
and \ref{item:40}, by using Remark \ref{rem:2}. The second assertion
follows from the first by induction on the size of the complex, by
using the usual filtration of $\Tot(E^{\ast,\ast})$. \end{proof}
As an example of the use of the acyclic calculus we prove
\begin{proposition}\label{prop:6}
Let $f\colon \overline E\to\overline F$ and $g\colon \overline F\to\overline G$ be morphisms
of complexes. If two of $f,\ g,\ g\circ f$ are tight, then so is
the third. \end{proposition}
\begin{proof}
Since tight morphisms are quasi-isomorphisms, by Theorem
\ref{thm:7}~\ref{item:41}
\begin{displaymath}
[\ocone(g\circ f)]=[\ocone(f)]+[\ocone(g)].
\end{displaymath}
Hence the result follows from \ref{thm:7}~\ref{item:34}. \end{proof}
\begin{definition}\label{def:KA}
We will denote by $\KA(X)$ the set of invertible elements of
$\oV(X)/\mathscr{M}$. This is an abelian subgroup. By Theorem
\ref{thm:7}~\ref{item:39} the group $\KA(X)$
agrees with the image in $\oV(X)/\mathscr{M}$ of the class of
acyclic complexes. \end{definition}
The group $\KA(X)$ is a universal abelian group for additive Bott-Chern classes. More precisely, let us denote by $\oVo(X)$ the full subcategory of $\oV(X)$ of acyclic complexes.
\begin{theorem} \label{thm:8}
Let $\mathscr{G}$ be an abelian group and let $\varphi\colon \Ob \oVo(X) \to
\mathscr{G}$ be an assignment such that
\begin{enumerate}
\item (Normalization) Every complex of the form
\begin{displaymath}
\overline E\colon\quad 0\longrightarrow \overline A
\overset{\Id}{\longrightarrow}
\overline A \longrightarrow 0
\end{displaymath} satisfies $\varphi(\overline E)=0$.
\item (Additivity for exact sequences) For every short exact
sequence in $\oVo(X)$
\begin{displaymath}
0\longrightarrow \overline E\longrightarrow \overline F\longrightarrow \overline G
\longrightarrow 0,
\end{displaymath}
all whose constituent short
exact sequences are orthogonally split, we have
\begin{displaymath}
\varphi(\overline F)=\varphi(\overline E)+\varphi(\overline G).
\end{displaymath}
\end{enumerate} Then $\varphi$ factorizes through a group homomorphism $\widetilde \varphi\colon \KA(X)\to \mathscr{G}$. \end{theorem} \begin{proof}
The second condition tells us that $\varphi$ is a morphism of
semigroups. Thus we only need to show that it vanishes on meager
complexes.
Again by the second condition, it is enough to prove that $\varphi$
vanishes on the class $\mathscr{M}_{0}$. Both conditions together
imply that $\varphi$ vanishes on orthogonally split
complexes. Therefore, by Example
\ref{exm:2}, it vanishes on complexes of the form
$\ocone(\Id_{E})$. Once more by the second condition, if $E$ is acyclic,
\begin{displaymath}
\varphi(E)+\varphi(E[1])=\varphi(\ocone(\Id_{E}))=0.
\end{displaymath}
Thus $\varphi$ vanishes also on the complexes described in Definition \ref{def:11} (ii).
Hence $\varphi$ vanishes on the class $\mathscr{M}$. \end{proof}
\begin{remark}
The considerations of this section carry over to the category of
complex analytic varieties. If $M$ is a complex analytic variety, one thus obtains for
instance a group $\KA^{\text{{\rm an}}}(M)$. Observe that, by GAGA principle, whenever $X$ is a
proper smooth algebraic variety over ${\mathbb C}$, the group $\KA^{\text{{\rm an}}}(X^{\text{{\rm an}}})$ is
canonically isomorphic to $\KA(X)$. \end{remark}
As an example, we consider the simplest case $\Spec {\mathbb C}$ and we compute the group $\KA(\Spec {\mathbb C})$. Given an acyclic complex $E$ of ${\mathbb C}$-vector spaces, there is a canonical isomorphism \begin{displaymath}
\alpha :\det E\longrightarrow {\mathbb C}. \end{displaymath}
If we have an acyclic complex of hermitian vector bundles $\overline E$, there is an induced metric on $\det E$. If we put on ${\mathbb C}$ the trivial hermitian metric, then there is a well defined positive real number $\|\alpha \|$, namely the norm of the isomorphism $\alpha$.
\begin{theorem}
The assignment $\overline E\mapsto \log\|\alpha \|$ induces an isomorphism
\begin{displaymath}
\widetilde{\tau }\colon \KA(\Spec
{\mathbb C})\overset{\simeq}{\longrightarrow}{\mathbb R}.
\end{displaymath} \end{theorem} \begin{proof}
First, we observe that the assignment in the theorem
satisfies the hypothesis of Theorem \ref{thm:8}. Thus,
$\widetilde{\tau }$ exists and is a group morphism.
Second, for every $a\in {\mathbb R}$ we consider the acyclic complex
\begin{displaymath}
e^{a}:= 0\longrightarrow
\overline{{\mathbb C}}\overset{e^{a}}{\longrightarrow}
\overline{{\mathbb C}}\longrightarrow 0,
\end{displaymath}
where $\overline {{\mathbb C}}$ has the standard metric and the left copy of
$\overline {{\mathbb C}}$ sits in degree $0$.
Since $\widetilde {\tau }([e^{a}])=a$ we deduce that $\widetilde{\tau }$ is
surjective.
Next we prove that the complexes of the form $[e^{a}]$ form a set of
generators of $\KA(\Spec {\mathbb C})$.
Let
$\overline E=(\overline E^{\ast},f^{\ast})$ be an acyclic complex. Let $r=\sum_{i}
\rk(E^{i})$. We will show by induction on $r$ that
$[\overline E]=\sum_{k}(-1)^{i_{k}}[e^{a_{k}}]$ for certain integers
$i_{k}$ and real numbers $a_{k}$. Let $n$ be the
smallest integer such that $f^{n}\colon E^{n}\to E^{n+1}$ is
non-zero. Let $v\in E^{n}\setminus \{0\}$. By acyclicity, $f^{n}$ is
injective, hence $\|f^{n}(v)\|\not=0.$ Set $i_{1}=n$ and
$a_{1}=\log(\|f^{n}(v)\|/\|v\|)$ and consider the diagram
\begin{displaymath}
\xymatrix{
& &0\ar[d] &0\ar[d] & &\\
&0\ar[r] &\overline{{\mathbb C}}\ar[r]^{e^{a}}\ar[d]^{\gamma ^{n}}
&\overline{{\mathbb C}}\ar[r]\ar[d]^{\gamma ^{n+1}}
&0 \ar[r]\ar[d]
& \dots
\\
&0\ar[r] &\overline E^{n}\ar[r]\ar[d] &\overline E^{n+1}\ar[r]\ar[d] &\overline E^{n+2}\ar[r]\ar[d] &\dots
\\
&0\ar[r] &\overline F^{n}\ar[r]\ar[d] &\overline F^{n+1}\ar[r]\ar[d] &\overline F^{n+2}\ar[r]\ar[d] &\dots
\\
& &0 &0 &0 & &
}
\end{displaymath}
where $\gamma ^{n}(1)=v$, $\gamma ^{n+1}(1)=f^{n}(v)$ and all the
columns are orthogonally split short exact sequences. By Corollary
\ref{cor:3}~\ref{item:42} and Theorem \ref{thm:7}~\ref{item:39}, we have
\begin{displaymath}
[\overline E]=(-1)^{i_{1}}[e^{a_{1}}]+[\overline F].
\end{displaymath}
Thus we deduce the claim.
Considering now the diagram
\begin{displaymath}
\xymatrix{ \overline {\mathbb C} \ar[r]^{e^{a}}\ar[d]_{\Id}& \overline {\mathbb C}\ar[d]^{e^{b}}\\
\overline {\mathbb C} \ar[r]_{e^{a+b}}& \overline {\mathbb C}
}
\end{displaymath}
and using Corollary \ref{cor:3}~\ref{item:58} we deduce that
$[e^{a}]+[e^{b}]=[e^{a+b}]$ and $[e^{-a}]=-[e^{a}]$. Therefore
every element of $\KA(\Spec {\mathbb C})$ is of the form $[e^{a}]$. Hence
$\widetilde{\tau }$ is also injective. \end{proof}
\section{Definition of $\oDb(X)$ and basic
constructions}\label{sec:oDb}
Let $X$ be a smooth algebraic variety over ${\mathbb C}$. We denote by $\Coh(X)$ the abelian category of coherent sheaves on $X$ and by $\Db(X)$ its bounded derived category. The objects of $\Db(X)$ are complexes of quasi-coherent sheaves with bounded coherent cohomology. The reader is referred to \cite{GelfandManin:MR1950475} for an introduction to derived categories. For notational convenience, we also introduce $\Cb(X)$, the abelian category of bounded cochain complexes of coherent sheaves on $X$. Arrows in $\Db(X)$ will be written as $\dashrightarrow$, while arrows in $\Cb(X)$ will be denoted by $\rightarrow$. The symbol $\sim$ will mean either quasi-isomorphism in $\Cb(X)$ or isomorphism in $\Db(X)$. Every functor from $\Db(X)$ to another category will tacitly be assumed to be the derived functor. Therefore we will denote just by $\Rd f_{\ast}$, $\Ld f^{\ast}$, $\otimes^{\Ld}$ and $\Rd\uHom$ the derived direct image, inverse image, tensor product and internal Hom. Finally, we will refer to (complexes of) locally free sheaves by normal upper case letters (such as $F$) whereas we reserve script upper case letters for (complexes of) quasi-coherent sheaves in general (for instance $\mathcal{F}$).
\begin{remark}\label{rem:3}
Because $X$ is in particular a smooth noetherian scheme over ${\mathbb C}$,
every object $\mathcal{F}$ of $\Cb(X)$ admits a quasi-isomorphism
$F\to \mathcal{F}$, with $F$ an object of $\Vb(X)$. Hence, if
$\mathcal{F}$ is an object in $\Db(X)$, then there is an
isomorphism $F\dashrightarrow\mathcal{F}'$ in $\Db(X)$, for some
object $F\in\Vb(X)$. In general, the analogous
statement is no longer true if we work with complex manifolds, as
shown by the counterexample \cite[Appendix,
Cor. A.5]{Voisin:counterHcK}.
\end{remark}
For the sake of completeness, we recall how morphisms in $\Db(X)$ between bounded complexes of vector bundles can be represented.
\begin{lemma}\label{lemma:morphisms_Db}
\begin{enumerate}
\item \label{item:26} Let $F, G$ be bounded complexes of vector
bundles on $X$. Every morphism $F\dashrightarrow G$ in $\Db(X)$ may be
represented by a diagram in $\Cb(X)$
\begin{displaymath}
\xymatrix{
&E\ar[ld]_{f}\ar[rd]^{g} &\\
F & &G,
}
\end{displaymath}
where $E\in \Ob \Vb(X)$ and $f$ is a quasi-iso\-morphism.
\item \label{item:27} Let $E$, $E'$, $F$, $G$ be bounded
complexes of
vector bundles on $X$. Let $f$, $f'$, $g$, $g'$
be morphisms in $\Cb(X)$ as in the diagram below, with $f$, $f'$
quasi-isomorphisms. These data define the same morphism $F \dashrightarrow
G$ in $\Db(X)$ if, and only if, there exists a bounded complex of
vector bundles $E''$ and a diagram
\begin{displaymath}
\xymatrix{
& & E''\ar[ld]_{\alpha}\ar[rd]^{\beta} & &\\
&E\ar[ld]_{f}\ar[rrrd] &{}_g\hspace{2cm} {}_{f'}
&E'\ar[llld]\ar[rd]^{g'} &\\
F & & & &G,
}
\end{displaymath}
whose squares are commutative up to homotopy and where $\alpha$
and $\beta$ are quasi-isomorphisms.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows from the equivalence of $\Db(X)$ with the localization
of the homotopy category of $\Cb(X)$ with respect to the class of
quasi-isomorphisms and Remark \ref{rem:3}.
\end{proof}
\begin{proposition} \label{prop:7} Let $f\colon \overline E\to \overline E$ be an
endomorphism in $\oV(X)$ that represents $\Id_{E}$ in $\Db(X)$. Then
$\ocone(f)$ is meager. \end{proposition} \begin{proof}
By Lemma \ref{lemma:morphisms_Db}~\ref{item:27}, the fact that $f$
represents the identity in $\Db(X)$ means that there are diagrams
\begin{displaymath}
\xymatrix{
\overline E' \ar[r]^{\alpha }_{\sim} \ar[d]_{\beta }^{\sim}& \overline E \ar[d]^{\Id_{E}} &&
\overline E' \ar[r]^{\alpha }_{\sim} \ar[d]_{\beta }^{\sim}& \overline E\ar[d]^{f}\\
\overline E \ar[r]_{\Id_{E}}& \overline E, && \overline E \ar[r]_{\Id_{E}}& \overline E,
}
\end{displaymath}
that commute up to homotopy. By Theorem \ref{thm:7}~\ref{item:35}
and \ref{item:36} the equalities
\begin{align*}
[\ocone(\alpha )]-[\ocone(\Id_{E})]&=
[\ocone(\beta )]-[\ocone(\Id_{E})]\\
[\ocone(\alpha )]-[\ocone(\Id_{E})]&= [\ocone(\beta )]-[\ocone(f)]
\end{align*}
hold in the group $\KA(X)$ (observe that these relations do not
depend on the choice of homotopies).Therefore
\begin{displaymath}
[\ocone(f)]=[\ocone(\Id_{E})]=0.
\end{displaymath}
Hence $\ocone(f)$ is meager. \end{proof}
\begin{definition}\label{definition:hermitian_structure}
Let $\mathcal{F}$ be an object of $\Db(X)$. A \textit{hermitian
metric on} $\mathcal{F}$ consists of the following data:
\begin{enumerate}
\item[--] an isomorphism
$E\overset{\sim}{\dashrightarrow}\mathcal{F}$ in $\Db(X)$, where
$E\in \Ob \Vb(X)$;
\item[--] an object $\overline E\in \Ob \oV(X)$, whose underlying complex
is $E$.
\end{enumerate}
We write $\overline{E}\dashrightarrow\mathcal{F}$ to refer to the
data above and we call it a \textit{metrized object of} $\Db(X)$.
\end{definition}
Our next task is to define the category $\oDb(X)$, whose objects are objects of $\Db(X)$ provided with equivalence classes of metrics. We will show that in this category there is a hermitian cone well defined up to isometries.
\begin{lemma}\label{lemm:14}
Let $\overline E, \overline E'\in \Ob(\oV(X))$ and consider an arrow $E\dashrightarrow E'$ in
$\Db(X)$.
Then the following statements are equivalent:
\begin{enumerate}
\item \label{item:30} for any diagram
\begin{equation}\label{diag:2}
\xymatrix{
& E'' \ar[dl]_{\sim} \ar[dr]&\\
E && E',}
\end{equation}
that represents $E \dashrightarrow E'$, and any choice of hermitian metric on
$E''$, the complex
\begin{equation}\label{eq:54}
\ocone(\overline E'',\overline E)[1]\oplus \ocone(\overline E'',\overline E')
\end{equation}
is meager;
\item \label{item:31} there is a diagram (\ref{diag:2})
that represents $E \dashrightarrow E'$, and a choice of hermitian metric on
$E''$, such that the complex (\ref{eq:54}) is meager;
\item \label{item:31bis}there is a diagram (\ref{diag:2})
that represents $E \dashrightarrow E'$, and a choice of hermitian metric on
$E''$, such that the arrows $\overline E''\to \overline E$ and $\overline E'\to \overline
E'$ are tight morphisms.
\end{enumerate} \end{lemma} \begin{proof}
Clearly \ref{item:30} implies \ref{item:31}. To prove the converse
we assume the existence of a $\overline E''$ such that the complex
\eqref{eq:54} is meager, and let $\overline E'''$ be any complex that satisfies the
hypothesis of \ref{item:30}. Then there is a diagram
\begin{displaymath}
\xymatrix{
& & E^{''''}\ar[ld]_{\alpha}\ar[rd]^{\beta} & &\\
&E''\ar[ld]_{f}\ar[rrrd] &{}_g\hspace{2cm} {}_{f'}
&E'''\ar[llld]\ar[rd]^{g'} &\\
E & & & &E'
}
\end{displaymath}
whose squares commute up to homotopy. Using acyclic calculus we have
\begin{multline*}
[\ocone(g')]-[\ocone(f')]=\\
[\ocone(\beta )]+[\ocone(g)]-[\ocone(\alpha )]
-[\ocone(\beta )]-[\ocone(f)]+[\ocone(\alpha )]=\\
[\ocone(g)]-[\ocone(f)]=0.
\end{multline*}
Now repeat the argument of Lemma \ref{lemm:16} to prove that
\ref{item:31} and \ref{item:31bis} are equivalent. The only point is
to observe that the diagram constructed in Lemma \ref{lemm:16}
represents the same morphism in the derived category as the
original diagram. \end{proof}
\begin{definition}\label{def:13}
Let $\mathcal{F}\in \Ob\Db(X)$ and let $\overline E\dashrightarrow \mathcal{F}$ and
$\overline E'\dashrightarrow \mathcal{F}$ be two hermitian metrics on
$\mathcal{F}$. We say that they \emph{fit tight} if the induced
arrow $\overline E\dashrightarrow\overline E'$ satisfies any of the equivalent conditions
of Lemma \ref{lemm:14} \end{definition}
\begin{theorem} \label{thm:5} The relation ``to fit tight'' is an
equivalence relation. \end{theorem} \begin{proof}
The reflexivity and the symmetry are obvious. To prove the
transitivity, consider a diagram
\begin{displaymath}
\xymatrix{
& \overline F \ar[dl]_{f} \ar[dr]^{g}& & \overline F' \ar[dl]_{f'} \ar[dr]^{g'} &\\
\overline E & & \overline E' & & \overline E'' ,
}
\end{displaymath}
where all the arrows are tight morphisms and $f$, $f'$ are quasi-isomorphisms. By Lemma \ref{lemm:15},
this diagram can be completed into a diagram
\begin{displaymath}
\xymatrix{
& & \overline F'' \ar[dl]_{\alpha } \ar[dr]^{\beta }& & \\
& \overline F \ar[dl]_{f} \ar[dr]^{g}& & \overline F' \ar[dl]_{f'} \ar[dr]^{g'} &\\
\overline E & & \overline E' & & \overline E'' ,
}
\end{displaymath}
where all the arrows are tight morphisms and the square commutes up
to homotopy. Now observe that $f\circ\alpha$ and $g'\circ\beta$
represent the morphism $E\dashrightarrow E''$ in $\Db(X)$ and are both tight
morphisms by Proposition \ref{prop:6}. This finishes the proof.
\end{proof}
\begin{definition}\label{def:category_oDb}
We denote by $\oDb(X)$ the category whose objects are pairs
$\ocF=(\mathcal{F},h)$ where $\mathcal{F}$ is an object of $\Db(X)$
and $h$ is an equivalence class of metrics that fit tight, and with
morphisms
\begin{displaymath}
\Hom_{\oDb(X)}(\overline{\mathcal{F}},\overline{\mathcal{G}})
=
\Hom_{\Db(X)}(\mathcal{F},\mathcal{G}).
\end{displaymath}
A class $h$ of metrics will be called \emph{a hermitian structure},
and may be referenced by any representative $\overline E\dashrightarrow \mathcal{F}$
or, if the arrow is clear, by the complex $\overline E$. We will denote by
$\overline 0\in \Ob \oDb(X)$ a zero object of $\Db(X)$ provided with a
trivial hermitian structure given by any meager complex.
If the underlying complex to an object $\overline{\mathcal{F}}$ is
acyclic, then its hermitian structure has a well defined class in
$\KA(X)$. We will use the notation $[\overline{\mathcal{F}}]$ for this
class. \end{definition}
\begin{definition}
A morphism in $\oDb(X)$, $f\colon (\overline E\dashrightarrow\mathcal{F})
\dashrightarrow(\overline F\dashrightarrow \mathcal{G})$, is called \emph{a tight
isomorphism} whenever the underlying morphism $f\colon
\mathcal{F}\dashrightarrow \mathcal{G}$ is an isomorphism and the metric on
$\mathcal{G}$ induced by $f$ and $\overline E$ fits tight with $\overline F$. An
object of $\oDb(X)$ will be called \emph{meager} if it is tightly
isomorphic to the zero object with the trivial metric. \end{definition}
\begin{remark} \label{rem:5} A word of warning should be said about
the use of acyclic calculus to show that a particular map is a tight
isomorphism. There is an assignment $\Ob\oDb(X)\to
\oV(X)/\mathscr{M}$ that sends $\overline E\dashrightarrow \mathcal{F}$ to $[\overline
E]$. This assignment is not injective. For instance, let $r>0$ be a
real number and consider the trivial bundle $\mathcal{O}_{X}$ with
the trivial metric $\|1\|=1$ and with the metric $\|1\|'=1/r$. Then
the product by $r$ induces an isometry between both bundles. Hence,
if $\overline E$ and $\overline E'$ are the complexes that have
$\mathcal{O}_{X}$ in degree $0$ with the above hermitian metrics,
then $[\overline E]=[\overline E']$, but they define different hermitian
structures on $\mathcal{O}_{X}$ because the product by $r$ does not
represent $\Id_{\mathcal{O}_{X}}$.
Thus the right procedure to show that a morphism $f\colon (\overline E\dashrightarrow
\mathcal{F})\dashrightarrow (\overline F\dashrightarrow \mathcal{G})$ is a tight isomorphism, is
to construct a diagram
\begin{displaymath}
\xymatrix{
& \overline G \ar[dl]^{\sim}_{\alpha } \ar[dr]^{\beta }&\\
\overline E && \overline F}
\end{displaymath}
that represents $f$ and use the acyclic calculus to show that
$[\ocone(\beta )]-[\ocone(\alpha )]=0$. \end{remark}
By definition, the forgetful functor $\mathfrak{F}\colon \oDb(X)\to \Db(X)$ is fully faithful. The structure of this functor will be given in the next result that we suggestively summarize by saying that $\oDb(X)$ is a principal fibered category over $\Db(X)$ with structural group $\KA(X)$ provided with a flat connection.
\begin{theorem}\label{thm:13}
The functor $\mathfrak{F}\colon \oDb(X)\to \Db(X)$ defines a
structure of category fibered in grupoids. Moreover
\begin{enumerate}
\item \label{item:43} The fiber $\mathfrak{F}^{-1}(0)$ is the
grupoid associated to the abelian group
$\KA(X)$. The object $\overline 0$ is the neutral element of $\KA(X)$.
\item \label{item:44} For any object $\mathcal{F}$ of $\Db(X)$, the
fiber $\mathfrak{F}^{-1}(\mathcal{F})$ is the grupoid associated
to a torsor over $\KA(X)$. The action of $\KA(X)$ over
$\mathfrak{F}^{-1}(\mathcal{F})$ is given by orthogonal direct
sum. We will denote this action by $+$.
\item \label{item:45} Every isomorphism $f\colon \mathcal{F}\dashrightarrow
\mathcal{G}$ in $\Db(X)$ determines an isomorphism of
$\KA(X)$-torsors
\begin{displaymath}
\mathfrak{t}_{f}\colon \mathfrak{F}^{-1}(\mathcal{F})\longrightarrow
\mathfrak{F}^{-1}(\mathcal{G}),
\end{displaymath}
that sends the hermitian structure $\overline E\overset{\epsilon }{\dashrightarrow}
\mathcal{F}$ to the hermitian structure $\overline E\overset{f\circ
\epsilon }{\dashrightarrow} \mathcal{G}$. This isomorphism will be called
the parallel transport along $f$.
\item \label{item:46} Given two isomorphisms $f\colon
\mathcal{F}\dashrightarrow \mathcal{G}$ and $g\colon \mathcal{G}\dashrightarrow
\mathcal{H}$, the equality
$$\mathfrak{t}_{g\circ f}=\mathfrak{t}_{g}\circ \mathfrak{t}_{f}$$
holds. \end{enumerate}
\end{theorem} \begin{proof} Recall that $\mathfrak{F}^{-1}(\mathcal{F})$ is the
subcategory of $\oDb(X)$ whose objects satisfy
$\mathfrak{F}(A)=\mathcal{F}$ and whose morphisms satisfy
$\mathfrak{F}(f)=\Id_{\mathcal{F}}$. The first assertion is
trivial. To prove that $\mathfrak{F}^{-1}(\mathcal{F})$ is a torsor
under $\KA(X)$, we need to show that $\KA(X)$ acts freely and
transitively on this fiber. For the freeness, it is enough to
observe that if for $\overline E\in\oV(X)$ and $\overline M\in\oVo(X)$, the
complexes $\overline E$ and $\overline E\oplus \overline M$ represent the same
hermitian structure, then the inclusion $\overline E\hookrightarrow \overline
E\oplus\overline M$ is tight. Hence $\ocone(\overline E, \overline E\oplus \overline M)$ is
meager. Since
\begin{displaymath}
\ocone(\overline E, \overline E\oplus \overline M)=\ocone(\overline E, \overline E)\oplus \overline M
\end{displaymath}
and $\ocone(\overline E, \overline E)$ is meager, we deduce that $\overline M$ is
meager. For the transitivity, any two hermitian structures on
$\mathcal{F}$ are related by a diagram
\begin{displaymath}
\xymatrix{
&\overline E''\ar[ld]_{\sim}^{f}\ar[rd]^{\sim}_{g} &\\
\overline E & &\overline E'.
}
\end{displaymath}
After possibly replacing $\overline E''$ by $\overline E''\oplus\ocone(f)$, we
may assume that $f$ is tight. We consider the natural arrow $\overline
E''\to \overline E'\oplus\ocone(g)[1]$ induced by $g$. Observe that
$\ocone(g)[1]$ is acyclic. Finally, we find
\begin{displaymath}
\ocone(\overline E'', \overline E'\oplus\ocone(g)[1])=\ocone(g)\oplus\ocone(g)[1],
\end{displaymath}
that is meager. Thus the hermitian structure represented by $\overline
E''$ agrees with the hermitian structure represented by $\overline
E'\oplus\ocone(g)[1]$.
The remaining properties are straightforward. \end{proof}
Our next objective is to define the cone of a morphism in $\oDb(X)$. This will be an object of $\oDb(X)$ uniquely defined up to tight isomorphism. Let $f\colon (\overline{E}\dashrightarrow\mathcal{F})\dashrightarrow (\overline{E}'\dashrightarrow\mathcal{G})$ be a morphism in $\oDb(X)$, where $\overline E$ and $\overline E'$ are representatives of the hermitian structures.
\begin{definition}\label{def:her_cone}
A \textit{hermitian cone} of $f$, denoted $\ocone(f)$, is an
object $(\cone(f),h_{f})$ of $\oDb(X)$ where:
\begin{enumerate}
\item[--] $\cone(f)\in\Ob\Db(X)$ is a choice of cone of $f$. Namely
an object of $\Db(X)$ completing $f$ into a distinguished
triangle;
\item[--] $h_{f}$ is a hermitian structure on $\cone(f)$
constructed as follows. The morphism $f$ induces an arrow $E\dashrightarrow
E'$. Choose any bounded complex $E''$ of vector bundles with a
diagram
\begin{displaymath}
\xymatrix{
&E''\ar[ld]^{\sim}\ar[rd] &\\
E & &E'
}
\end{displaymath}
that represents $E\dashrightarrow E'$, and an arbitrary hermitian metric on
$E''$. Put
\begin{equation}\label{eq:her_cone}
\overline{C}(f)=\ocone(\overline{E}'',\overline{E})[1]\oplus\ocone(\overline{E}'',\overline{E}').
\end{equation}
There are morphisms defined as compositions
\begin{displaymath}
\overline{E}'\longrightarrow \ocone(\overline{E}'',\overline{E}')
\longrightarrow \overline{C}(f),
\end{displaymath}
where the second arrow is the natural inclusion, and
\begin{displaymath}
\overline{C}(f) \longrightarrow \ocone(\overline{E}'',\overline{E}')
\longrightarrow \overline{E}''[1] \longrightarrow
\overline{E}[1],
\end{displaymath}
where the first arrow is the natural projection.
These morphisms fit into a natural distinguished triangle completing
$\overline{E}\dashrightarrow\overline{E}'$. By the axioms of triangulated category,
there is a quasi-isomorphism $\overline{C}(f)\dashrightarrow\cone(f)$ such that
the following diagram (where the rows are distinguished triangles)
\begin{displaymath}
\xymatrix{
\overline{E}\ar@{-->}[r]\ar@{-->}[d] &\overline{E}'\ar@{-->}[r]\ar@{-->}[d]
&\overline{C}(f)\ar@{-->}[d]\ar@{-->}[r] &\overline{E}[1]\ar@{-->}[d]\\
\mathcal{F}\ar@{-->}[r] &\mathcal{G}\ar@{-->}[r]
&\cone(f)\ar@{-->}[r] &\mathcal{F}[1]
}
\end{displaymath}
commutes. We take the hermitian structure that
$\overline{C}(f)\dashrightarrow\cone(f)$ defines on $\cone(f)$. By Theorem
\ref{thm:6bis} below, this hermitian structure does not depend on
the particular choice of arrow $\overline{C}(f)\dashrightarrow\cone(f)$. Moreover,
by Theorem \ref{thm:6}, the hermitian structure will not depend
on the choices of representatives of hermitian structures nor on
the choice of $\overline{E}''$.
\end{enumerate}
\end{definition}
\begin{remark}\label{rem:8}
The factor $\ocone(\overline{E}'',\overline{E})[1]$ has to be seen as a
correction term to take into account the difference of metrics from
$\overline E $ and $\overline E''$. We would have obtained an equivalent
definition using the factor $\ocone(\overline{E}'',\overline{E})[-1]$.
\end{remark}
\begin{theorem}\label{thm:6bis}
Let
\begin{displaymath}
\xymatrix{
\mathcal{F}\ar@{-->}[r]\ar[d]^{\Id}
&\mathcal{G}\ar@{-->}[r]\ar[d]^{\Id}
&\mathcal{H}\ar@{-->}[r]\ar@{-->}[d]^{\alpha }
& \mathcal{F}[1]\ar@{-->}[r] \ar[d]^{\Id}
&\dots\\
\mathcal{F}\ar@{-->}[r]
&\mathcal{G}\ar@{-->}[r]
&\mathcal{H}\ar@{-->}[r]
& \mathcal{F}[1]\ar@{-->}[r]
&\dots
}
\end{displaymath}
be a commutative diagram in $\Db(X)$, where the rows are the same
distinguished triangle. Let $\overline H\dashrightarrow \mathcal{H}$ be any
hermitian structure. Then $\alpha\colon (\overline H\dashrightarrow
\mathcal{H})\dashrightarrow (\overline H\dashrightarrow \mathcal{H}) $ is a tight
isomorphism.
\end{theorem}
\begin{proof}
First of all, we claim that if
$\gamma:\overline{\mathcal{B}}\dashrightarrow\overline{\mathcal{H}}$ is any isomorphism,
then $\gamma^{-1}\circ\alpha\circ\gamma$ is tight if, and only if,
$\alpha$ is tight. Indeed, denote by $\overline G\dashrightarrow\mathcal{B}$ a
representative of the hermitian structure on
$\overline{\mathcal{B}}$. Then there is a diagram
\begin{displaymath}
\xymatrix{
& & &\overline
R\ar[ld]^{t_{1}}_{\sim}\ar[rd]_{t_{2}}^{\sim} & &\\
& &\overline P\ar[ld]^{w_{1}}_{\sim}\ar[rd]_{w_{2}}^{\sim}
& &\overline Q\ar[ld]^{w_{3}}_{\sim}\ar[rd]_{w_{4}}
^{\sim}&\\
&\overline{G}'\ar[ld]_{\sim}^{u}\ar[rd]^{\sim}_{v} &
&\overline{H}'\ar[rd]^{\sim}_{f}\ar[ld]_{\sim}^{g} &
&\overline{G}'\ar[ld]_{\sim}^{v}\ar[rd]^{\sim}_{u}&\\
\overline G\ar@{-->}[rr] & &\overline H\ar@{-->}[rr] &
&\overline H\ar@{-->}[rr] & &\overline G
}
\end{displaymath}
for the liftings of $\gamma^{-1}$, $\alpha$, $\gamma$ to
representatives, as well as for their composites, all whose squares
are commutative up to homotopy. By acyclic calculus, we have the
following chain of equalities
\begin{multline*}
[\ocone(u\circ w_{1}\circ t_{1})[1]]+[\ocone(u\circ w_{4}\circ
t_{2})]=\\
[\ocone(u)[1]]+[\ocone(v)]+[\ocone(g)[1]]+
[\ocone(f)]+[\ocone(v)[1]]+[\ocone(u)]=\\
[\ocone(g)[1]]+[\ocone(f)].
\end{multline*}
Thus, the right hand side vanishes if and only if the left hand
side vanishes, proving the claim. This observation allows to reduce
the proof of the lemma to the following situation: consider a
diagram of complexes of hermitian vector bundles
\begin{displaymath}
\xymatrix{
\overline{E}\ar[d]^{\Id}\ar[r]^{f}
&\overline{F}\ar[d]^{\Id}\ar[r]^{\iota}
&\overline{\cone}(f)\ar[r]^{\pi}\ar@{-->}[d]^{\phi}_{\sim} &
\overline{E}[1]\ar[d]^{\Id}\ar[r] &\dots\\
\overline{E}\ar[r]^{f} &\overline{F}\ar[r]^{\iota}
&\overline{\cone}(f)\ar[r]^{\pi} &\overline{E}[1]\ar[r]
&\dots,
}
\end{displaymath}
which commutes in $\Db(X)$. We need to show that $\phi$ is a tight
isomorphism. The commutativity of the diagram translates into the
existence of bounded complexes of hermitian vector bundles
$\overline{P}$ and $\overline{Q}$ and a diagram
\begin{displaymath}
\xymatrix{
& &
&\overline{\cone}(f)\ar[rd]^{\pi}\ar@{-->}[dd]^{\phi}_{\sim}
&\\
\overline{F}\ar@/^/[rrru]^{\iota}\ar@/_/[rrrd]_{\iota}
&\overline{P}\ar[l]_{\hspace{0.2cm} j}^{\sim}\ar[r]^{g}
&\overline{Q}
\ar[ru]_{u}^{\sim}\ar[rd]^{v}_{\sim} & &\overline{E}[1]\\
& &
&\overline{\cone}(f)\ar[ru]_{\pi} &
}
\end{displaymath}
fulfilling the following properties: (a) $j$, $u$, $v$ are
quasi-isomorphisms; (b) the squares formed by $\iota, j, g, u$ and
$\iota, j, g, v$ are commutative up to homotopy; (c) the morphisms
$u$, $v$ induce $\phi$ in the derived category. We deduce a
commutative up to homotopy square
\begin{displaymath}
\xymatrix{
\ocone(g)\ar[d]_{\tilde{v}}^{\sim}\ar[r]^{\tilde{u}}_{\sim}
&\ocone(\iota)\ar[d]^{\tilde{\pi}}_{\sim}\\
\ocone(\iota)\ar[r]^{\tilde{\pi}}_{\sim} &\overline{E}[1].
}
\end{displaymath}
The arrows $\tilde{u}$, $\tilde{v}$ are induced by $j,u$ and $j, v$
respectively. Observe they are quasi-isomorphisms. Also the natural
projection $\tilde{\pi}$ is a quasi-isomorphism. By acyclic
calculus, we have
\begin{displaymath}
[\ocone(\tilde{\pi})]+[\ocone(\tilde{u})]=[\ocone(\tilde{\pi})]+[\ocone(\tilde{v})].
\end{displaymath}
Therefore we find
\begin{equation}\label{eq:59}
[\ocone(\tilde{u})]=[\ocone(\tilde{v})].
\end{equation}
Finally, notice there is an exact sequence
\begin{displaymath}
0\longrightarrow\ocone(u)\longrightarrow
\ocone(\tilde{u})\longrightarrow \ocone(j[1]) \longrightarrow 0,
\end{displaymath}
whose rows are orthogonally split. Therefore,
\begin{equation}\label{eq:60}
[\ocone(\tilde{u})]=[\ocone(u)]+[\ocone(j[1])].
\end{equation}
Similarly we prove
\begin{equation}\label{eq:61}
[\ocone(\tilde{v})]=[\ocone(v)]+[\ocone(j[1])].
\end{equation}
From equations (\ref{eq:59})--(\ref{eq:61}) we infer
\begin{displaymath}
[\ocone(u)[1]]+[\ocone(v)]=0,
\end{displaymath}
as was to be shown.
\end{proof}
\begin{theorem} \label{thm:6} The object $\overline{C}(f)$ of equation
\eqref{eq:her_cone} is well defined up to tight isomorphism.
\end{theorem}
\begin{proof}
We first show the independence on the choice of $\overline E''$, up to
tight isomorphism. To this end, it is enough to assume that there
is a diagram
\begin{displaymath}
\xymatrix{
&& \overline E''' \ar[dl]_{\sim} \ar[rdd]&\\
& \overline E''\ar[dl]_{\sim}\ar[rrd]&&\\
\overline E &&&\overline E'
}
\end{displaymath}
such that the triangle commutes up to homotopy. Fix such a homotopy. Then
\begin{align*}
[\ocone(\ocone(\overline E''',\overline E'),\ocone(\overline E'',\overline E'))]&=
-[\ocone(E''', E'')],\\
[\ocone(\ocone(\overline E''',\overline E),\ocone(\overline E'',\overline E))]&=
-[\ocone(E''', E'')].
\end{align*}
By Lemma \ref{lemm:13}, the left hand sides of these relations
agree and hence this implies that the hermitian structure does not
depend on the choice of $\overline E''$.
We now prove the independence on the choice of the representative
$\overline E$. Let $\overline F\to \overline E$ be a tight morphism. Then we can
construct a diagram
\begin{displaymath}
\xymatrix{
& \overline E''' \ar[ddl]_{\sim} \ar[rd]^{\sim}&&\\
&& \overline E''\ar[dl]_{\sim}\ar[rd]&\\
\overline F \ar[r]^{\sim} &\overline E &&\overline E',
}
\end{displaymath}
where the square commutes up to homotopy. Choose one
homotopy. Taking into account Lemma \ref{lemm:13}, we find
\begin{align*}
[\ocone(\ocone(\overline E''',\overline E'),\ocone(\overline E'',\overline E'))]&=
-[\ocone(E''', E'')],\\
[\ocone(\ocone(\overline E''',\overline F),\ocone(\overline E'',\overline E))]&=
-[\ocone(E''', E'')]+[\ocone(\overline F,\overline
E)]\\
&=-[\ocone(E''', E'')].
\end{align*}
Hence the definitions of $\overline{C}(f)$ using $\overline E$ or $\overline F$ agree
up to tight isomorphism. The remaining possible choices of
representatives are treated analogously.
\end{proof}
\begin{remark}
The construction of $\ocone(f)$ involves the choice of $\cone(f)$,
which is unique up to isomorphism. Since the construction of
$\overline{C}(f)$ \eqref{eq:her_cone} does not depend on the choice of
$\cone(f)$, by Theorem \ref{thm:6bis}, we see that different
choices of $\cone(f)$ give rise to tightly isomorphic hermitian
cones. Therefore $\ocone(f)$ is well defined up to tight isomorphism
and we will usually call it \emph{the}
hermitian cone of $f$. When the morphism is clear, we will also write
$\ocone(\overline{\mathcal{F}},\overline{\mathcal{G}})$ to refer to it. \end{remark}
The hermitian cone satisfies the same relations than the usual cone.
\begin{proposition} \label{prop:10} Let $f\colon \overline {\mathcal{F}}\dashrightarrow
\overline {\mathcal{G}}$ be a morphism in $\oDb(X)$. Then, the natural
morphisms
\begin{gather*}
\ocone(\overline{\mathcal{G}},\ocone(f))\dashrightarrow
\overline{\mathcal{F}}[1],\\
\overline{\mathcal{G}}\dashrightarrow \ocone(\ocone(f)[-1],\overline{\mathcal{F}})
\end{gather*}
are tight isomorphisms. \end{proposition} \begin{proof}
After choosing representatives, there are isometries
\begin{align*}
\ocone(\ocone(\overline{\mathcal{G}},\ocone(f)),
\overline{\mathcal{F}}[1])\cong &
\ocone(\ocone(\Id_{\mathcal{F}}),\ocone(\Id_{\mathcal{G}}))\cong\\
&\ocone(\overline{\mathcal{G}}, \ocone(\ocone(f)[-1],\overline{\mathcal{F}})).
\end{align*}
Since the middle term is meager, the same is true for the other two. \end{proof}
We next extend some basic constructions in $\Db(X)$ to $\oDb(X)$.
\noindent\textbf{Derived tensor product.} Let $\overline{\mathcal{F}}_{i}=( \overline{E}_{i}\dashrightarrow\mathcal{F}_{i})$, $i=1,2$, be objects of $\oDb(X)$. The derived tensor product $\mathcal{F}_{1}\otimes^{\Ld} \mathcal{F}_{2}$ is endowed with a natural hermitian structure \begin{equation}
\label{eq:29}
\overline{E}_{1}\otimes\overline{E}_{2}\dashrightarrow
\mathcal{F}_{1}\otimes^{\Ld} \mathcal{F}_{2}, \end{equation} that is well defined by Theorem \ref{thm:4}~\ref{item:24}. We write $\overline{\mathcal{F}}_{1}\otimes^{\Ld} \overline{\mathcal{F}}_{2}$ for the resulting object in $\oDb(X)$.
\noindent\textbf{Derived internal $\Hom$ and dual objects.} Let $\overline{\mathcal{F}}_{i}=( \overline{E}_{i}\dashrightarrow\mathcal{F}_{i})$, $i=1,2$, be objects of $\oDb(X)$. The derived internal $\Hom$, $ \Rd\uHom(\mathcal{F}_{1},\mathcal{F}_{2})$ is endowed with a natural hermitian structure \begin{equation}
\label{eq:22}
\uHom(\overline{E}_{1},\overline{E}_{2})\dashrightarrow
\Rd\uHom(\mathcal{F}_{1},\mathcal{F}_{2}), \end{equation} that is well defined by Theorem \ref{thm:4}~\ref{item:24}. We write $\Rd\uHom(\overline{\mathcal{F}}_{1},\overline{\mathcal{F}}_{2})$ for the resulting object in $\oDb(X)$.
In particular, denote by $\overline {\mathcal{O}}_{X}$ the structural sheaf with the metric $\|1\|=1$. Then, for every object $\overline{\mathcal{F}} \in \oDb(X)$, the \emph{dual object} is defined to be \begin{equation}
\label{eq:30}
\overline{\mathcal{F}}^{\vee}=\Rd\uHom(\overline
{\mathcal{F}},\overline{\mathcal{O}}_{X}). \end{equation}
\noindent\textbf{Left derived inverse image.} Let
$g\colon X^{\prime}\rightarrow X$ be a morphism of smooth algebraic
varieties over ${\mathbb C}$ and $\overline{\mathcal{F}}=(\overline E\dashrightarrow \mathcal{F})\in
\Ob \oDb(X)$. Then the left derived inverse image $\Ld
g^{\ast}(\mathcal{F})$ is equipped with the hermitian structure
$g^{\ast}(\overline{E})\dashrightarrow\Ld g^{\ast}(\mathcal{F})$,
that is well defined up to tight isomorphism by Theorem
\ref{thm:4}~\ref{item:25}. As it is customary, we will pretend that
$\Ld g^{\ast}$ is a functor. The notation for the corresponding
object in $\oDb(X^{\prime})$ is $\Ld
g^{\ast}(\overline{\mathcal{F}})$. If $f\colon
\overline{\mathcal{F}}_{1}\dashrightarrow \overline{\mathcal{F}}_{2}$
is a morphism in $\oDb(X)$, we denote by $\Ld g^{\ast}(f)\colon \Ld
g^{\ast}(\overline{\mathcal{F}}_{1})\dashrightarrow\Ld g^{\ast}
(\overline{\mathcal{F}}_{2})$ its left derived inverse image by $g$.
The functor $\Ld g^{\ast}$ preserves the structure of principal
fibered category with flat connection and the formation of hermitian
cones. Namely we have the following result that is easily proved.
\begin{theorem} \label{thm:9} Let $g\colon X^{\prime}\rightarrow X$
be a morphism of smooth algebraic varieties over ${\mathbb C}$ and let
$f\colon \overline {\mathcal{F}}_{1}\dashrightarrow \overline {\mathcal{F}}_{2}$ be a
morphism in $\oDb(X)$.
\begin{enumerate}
\item The functor $\Ld g^{\ast}$ preserves the forgetful functor:
\begin{displaymath}
\mathfrak{F}\circ \Ld g^{\ast}=\Ld g^{\ast}\circ \mathfrak{F}
\end{displaymath}
\item The restriction $\Ld g^{\ast}\colon\KA(X)\to \KA(X')$ is a
group homomorphism.
\item The functor $\Ld g^{\ast}$ is equivariant with respect to the
actions of $\KA(X)$ and $\KA(X')$.
\item The functor $\Ld g^{\ast}$ preserves parallel transport: if
$f$ is an isomorphism, then
\begin{displaymath}
\Ld g^{\ast}\circ \mathfrak{t}_{f}=\mathfrak{t}_{\Ld
g^{\ast}(f)}\circ \Ld g^{\ast}.
\end{displaymath}
\item The functor $\Ld g^{\ast}$ preserves hermitian cones:
\begin{displaymath}
\Ld g^{\ast}(\ocone(f))=\ocone(\Ld g^{\ast}(f)).
\end{displaymath}
\end{enumerate}
\end{theorem}
$\square$
\noindent\textbf{Classes of isomorphisms and distinguished
triangles.}
Let $f\colon \overline {\mathcal{F}}\overset{\sim}{\dashrightarrow}
\overline{\mathcal{G}}$ be an isomorphism in $\oDb(X)$. To it, we attach a
class $[f]\in \KA(X)$ that measures the default of being a tight
isomorphism. This class is defined using the hermitian cone.
\begin{equation}
\label{eq:57}
[f]=[\ocone(f)].
\end{equation}
Observe the abuse of notation: we wrote $[\ocone(f)]$ for the class
in $\KA(X)$ of the hermitian structure of a hermitian cone of
$f$. This is well defined, since the hermitian cone is unique up to
tight isomorphism. Alternatively, we can construct $[f]$ using
parallel transport as follows. There is a unique element $\overline
A\in\KA(X)$ such that
\begin{displaymath}
\overline {\mathcal{G}}=\mathfrak{t}_{f}\overline {\mathcal{F}}+\overline A.
\end{displaymath}
We denote this element by $\overline {\mathcal{G}}-\mathfrak{t}_{f}\overline
{\mathcal{F}}$. Then
\begin{displaymath}
[f]=\overline {\mathcal{G}}-\mathfrak{t}_{f}\overline
{\mathcal{F}}.
\end{displaymath}
By the very definition of parallel transport, both definitions
clearly agree.
\begin{definition}\label{def:6}
A \textit{distinguished triangle in} $\oDb(X)$, consists in a
diagram
\begin{equation}\label{eq:64}
\overline{\tau}=(u,v,w):
\overline{\mathcal{F}}\overset{u}{\dashrightarrow}
\overline{\mathcal{G}}
\overset{v}{\dashrightarrow}\overline{\mathcal{H}}
\overset{w}{\dashrightarrow}\overline{\mathcal{F}}[1]
\overset{u}{\dashrightarrow}\dots
\end{equation}
in $\oDb(X)$, whose underlying morphisms in $\Db(X)$ form a
distinguished triangle. We will say that it is \emph{tightly
distinguished} if there is a commutative diagram
\begin{equation}\label{eq:63}
\xymatrix{
\overline{\mathcal{F}}\ar@{-->}[r]\ar[d]^{\Id}
&\overline{\mathcal{G}}\ar@{-->}[r]\ar[d]^{\Id}
&\ocone(\overline {\mathcal{F}},\overline{\mathcal{G}})
\ar@{-->}[r]\ar@{-->}[d]^{\alpha }
& \overline{\mathcal{F}}[1]\ar@{-->}[r] \ar[d]^{\Id}
&\dots\\
\overline{\mathcal{F}}\ar@{-->}[r]
&\overline{\mathcal{G}}\ar@{-->}[r]
&\overline{\mathcal{H}}\ar@{-->}[r]
& \overline{\mathcal{F}}[1]\ar@{-->}[r]
&\dots,
}
\end{equation}
with $\alpha $ a tight isomorphism.
\end{definition}
To every distinguished triangle in $\oDb(X)$ we can associate a class
in $\KA(X)$ that measures the default of being tightly distinguished.
Let $\overline {\tau }$ be a distinguished triangle as in
\eqref{eq:64}. Then there is a diagram as \eqref{eq:63}, but with
$\alpha $ an isomorphism non-necessarily tight. Then we define
\begin{equation}
\label{eq:65}
[\overline{\tau }]=[\alpha ].
\end{equation}
By Theorem \ref{thm:6bis}, the class $[\alpha]$ does not depend on
the particular choice of morphism $\alpha$ in $\oDb(X)$ for which
\eqref{eq:63} commutes. Hence \eqref{eq:65} only depends on
$\overline{\tau}$.
\begin{theorem}\label{thm:10}\
\begin{enumerate}
\item \label{item:51} Let $f$ be an isomorphism in $\oDb(X)$
(respectively $\overline \tau $ a distinguished triangle). Then $[f]=0$
(respectively $[\overline \tau ]=0$) if and only if $f$ is a tight isomorphism
(respectively $\overline \tau $ is tightly distinguished).
\item \label{item:50} Let $g\colon X'\to X$ be a morphism of smooth
complex varieties, let $f$ be an isomorphism in $\oDb(X)$ and
$\overline \tau $ a distinguished triangle in $\oDb(X)$. Then
\begin{displaymath}
\Ld g^{\ast}[f]=[\Ld g^{\ast}f],\qquad \Ld g^{\ast}[\overline \tau
]=[\Ld g^{\ast}\overline \tau ].
\end{displaymath}
In particular, tight isomorphisms and tightly distinguished
triangles are preserved under left derived inverse images.
\item \label{item:16}
Let $f\colon
\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}$ and
$h\colon \overline{\mathcal{G}}\dashrightarrow
\overline{\mathcal{H}}$ be two isomorphisms in $\oDb(X)$. Then:
\begin{displaymath}
[h\circ f]=[h]+[f].
\end{displaymath}
In particular, $[f^{-1}]=-[f]$.
\item \label{item:17} For any distinguished triangle $\overline \tau $ in
$\oDb(X)$ as in Definition \ref{def:6}, the rotated triangle
\begin{displaymath}
\overline{\tau}'\colon\
\overline{\mathcal{G}}\overset{v}{\dashrightarrow}\overline{\mathcal{H}}
\overset{w}{\dashrightarrow}\overline{\mathcal{F}}[1]
\overset{-u[1]}{\dashrightarrow}\overline{\mathcal{G}}[1]
\overset{v[1]}{\dashrightarrow}\dots
\end{displaymath}
satisfies
\begin{math}
[\overline \tau ']=-[\overline \tau ].
\end{math}
In particular, rotating preserves tightly distinguished
triangles.
\item \label{item:28} For any acyclic complex $\overline{\mathcal{F}}$,
we have
\begin{displaymath}
[\overline{\mathcal{F}}\to 0\to
0\to\dots]=
[\overline{\mathcal{F}}].
\end{displaymath}
\item \label{item:47} If
$f\colon\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}$ is an isomorphism
in $\oDb(X)$, then
\begin{displaymath}
[0\to\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}
\to\dots]=[f].
\end{displaymath}
\item \label{item:48} For a commutative diagram of distinguished
triangles
\begin{displaymath}
\xymatrix{
\overline \tau \ar@{-->}[d]
&\overline{\mathcal{F}}\ar@{-->}[r]\ar@{-->}[d]^{f}_{\sim}
&\overline{\mathcal{G}}\ar@{-->}[r]\ar@{-->}[d]^{g}_{\sim}
&\overline{\mathcal{H}}
\ar@{-->}[r]\ar@{-->}[d]^{h} _{\sim}
& \overline{\mathcal{F}}[1]\ar@{-->}[r] \ar@{-->}[d]^{f[1]}_{\sim}
&\dots\\
\overline \tau ' &\overline{\mathcal{F}}'\ar@{-->}[r]
&\overline{\mathcal{G}}'\ar@{-->}[r]
&\overline{\mathcal{H}}'\ar@{-->}[r]
& \overline{\mathcal{F}}'[1]\ar@{-->}[r]
&\dots,
}
\end{displaymath}
the following relation holds:
\begin{displaymath}
[\overline \tau ']
-[\overline \tau ]=
[f]-[g]+[h].
\end{displaymath}
\item \label{item:49} For a commutative diagram of distinguished
triangles
\begin{equation}\label{eq:66}
\xymatrix{
\overline \tau \ar@{-->}[d]
&\overline{\mathcal{F}}\ar@{-->}[r]\ar@{-->}[d]
&\overline{\mathcal{G}}\ar@{-->}[r]\ar@{-->}[d]
&\overline{\mathcal{H}}
\ar@{-->}[r]\ar@{-->}[d]
& \overline{\mathcal{F}}[1]\ar@{-->}[r] \ar@{-->}[d]
&\dots\\
\overline \tau' \ar@{-->}[d]
&\overline{\mathcal{F}}'\ar@{-->}[r]\ar@{-->}[d]
&\overline{\mathcal{G}}'\ar@{-->}[r]\ar@{-->}[d]
&\overline{\mathcal{H}}'
\ar@{-->}[r]\ar@{-->}[d]
& \overline{\mathcal{F}}'[1]\ar@{-->}[r] \ar@{-->}[d]
&\dots\\
\overline \tau '' &\overline{\mathcal{F}}''\ar@{-->}[r]\ar@{-->}[d]
&\overline{\mathcal{G}}''\ar@{-->}[r] \ar@{-->}[d]
&\overline{\mathcal{H}}''\ar@{-->}[r] \ar@{-->}[d]
& \overline{\mathcal{F}}''[1]\ar@{-->}[r] \ar@{-->}[d]
&\dots\\
&\overline{\mathcal{F}}[1]\ar@{-->}[r]\ar@{-->}[d]
&\overline{\mathcal{G}}[1]\ar@{-->}[r]\ar@{-->}[d]
&\overline{\mathcal{H}}[1]
\ar@{-->}[r]\ar@{-->}[d] &\overline{\mathcal{F}}[2]\ar@{-->}[r]
\ar@{-->}[d]
&\dots&\\
&\vdots &\vdots &\vdots & \vdots &&\\
& \overline \eta \ar@{-->}[r] & \overline \eta' \ar@{-->}[r] & \overline \eta''
&&&
}
\end{equation}
the following relation holds:
\begin{displaymath}
[\overline \tau ]-[\overline \tau' ]
+[\overline \tau'' ]=
[\overline \eta ]-[\overline \eta' ]
+[\overline \eta'' ].
\end{displaymath}
\end{enumerate}
\end{theorem}
\begin{proof}
The first two statements are clear. For the third, we may assume
that $f$ and $g$ are realized by quasi-isomorphisms
\begin{displaymath}
f\colon \overline F\longrightarrow \overline G,
\quad g\colon \overline G\longrightarrow \overline H.
\end{displaymath}
Then the result follows from Theorem \ref{thm:7}~\ref{item:41}.
The fourth assertion is a consequence of Proposition
\ref{prop:10}. Then \ref{item:28}, \ref{item:47} and \ref{item:48}
follow from equation \eqref{eq:65} and the fourth statement. The
last property is derived from \ref{item:48} by comparing the
diagram to a diagram of tightly distinguished triangles.
\end{proof}
As an application of the class in $\KA(X)$ attached to a
distinguished triangle, we exhibit a natural morphism
$K_{1}(X)\to\KA(X)$. This is included for the sake of completeness,
but won't be needed in the sequel.
\begin{proposition}\label{prop:K1_to_KA}
There is a natural morphism of groups $K_{1}(X)\to\KA(X)$. \end{proposition} \begin{proof}
We follow the definitions and notations of \cite{Burgos-Wang}. From
\emph{loc. cit.} we know it is enough to construct a morphism of
groups
\begin{equation}\label{eq:K1-KA}
H_{1}(\widetilde{\mathbb{Z}}C(X))\to\KA(X).
\end{equation}
By definition, the piece of degree $n$ of the homological complex
$\widetilde{\mathbb{Z}}C(X)$ is
\begin{displaymath}
\widetilde{\mathbb{Z}}C_{n}(X)=\mathbb{Z}C_{n}(X)/D_{n}.
\end{displaymath}
Here $\mathbb{Z}C_{n}(X)$ stands for the free abelian group on
metrized exact $n$-cubes and $D_{n}$ is the subgroup of degenerate
elements. A metrized exact $1$-cube is a short exact sequence of
hermitian vector bundles. Hence, for such a $1$-cube
$\overline{\varepsilon}$, there is a well defined class in
$\KA(X)$. Observe that this class coincides with the class of
$\overline{\varepsilon}$ thought as a distinguished triangle in
$\oDb(X)$. Because $\KA(X)$ is an abelian group, it follows the
existence of a morphism of groups
\begin{displaymath}
\mathbb{Z}C_{1}(X)\longrightarrow\KA(X).
\end{displaymath}
From the definition of degenerate cube \cite[Def. 3.3]{Burgos-Wang}
and the construction of $\KA(X)$, this morphism clearly factors
through $\widetilde{\mathbb{Z}}C_{1}(X)$. The definition of the
differential $d$ of the complex $\widetilde{\mathbb{Z}}C(X)$
\cite[(3.2)]{Burgos-Wang} and Theorem \ref{thm:10} \ref{item:49}
ensure that $d\mathbb{Z}C_{2}(X)$ is in the kernel of the
morphism. Hence we derive the existence of a morphism
\eqref{eq:K1-KA}. \end{proof}
\noindent\textbf{Classes of complexes and of direct images of
complexes.} In \cite[Section 2]{BurgosLitcanu:SingularBC} the notion of homological exact sequences of metrized coherent sheaves is treated. In the present article, this situation will arise in later considerations. Therefore we provide the link between the point of view of \emph{loc. cit.} and the formalism adopted here. The reader will find no difficulty to translate it to cohomological complexes.
Consider a homological complex \begin{displaymath}
\overline{\varepsilon }:\quad
0\to \overline{\mathcal{F}}_{m}\to \dots \to
\overline{\mathcal{F}}_{l}\to 0 \end{displaymath} of metrized coherent sheaves, namely coherent sheaves provided with hermitian structures $\overline {\mathcal{F}}_{i}=(\mathcal{F}_{i},\overline F_{i}\dashrightarrow \mathcal{F}_{i})$. We may equivalently see $\overline{\varepsilon}$ as a cohomological complex, by the usual relabeling $\overline{\mathcal{F}}^{-i}=\overline{\mathcal{F}}_{i}$. This will be freely used in the sequel, especially in cone constructions. \begin{definition}\label{def:1}
The complex $\overline \varepsilon $ defines an object $[\overline \varepsilon
]\in \Ob \oDb(X)$ that is determined inductively by the condition
\begin{displaymath}
[\overline \varepsilon ]=\ocone(\overline{\mathcal{F }}_{m}[m],[\sigma _{<m}\overline
\varepsilon ]).
\end{displaymath}
Here $\sigma _{<m}$ is the homological b\^ete filtration and
$\overline{\mathcal{F
}}_{m}$ denotes a cohomological complex concentrated in degree
zero. Hence, $\overline{\mathcal{F }}_{m}[m]$ is a cohomological complex
concentrated in degree $-m$. \end{definition}
If $\overline{E}$ is a hermitian vector bundle on $X$, then \begin{math}
[\overline{\varepsilon}\otimes \overline{E}]=[\overline{\varepsilon}]\otimes \overline{E} \end{math}. According to Definition \ref{def:category_oDb}, if $\varepsilon $ is an acyclic complex, then we also have the corresponding class $[[\overline{\varepsilon}]]$ in $\KA(X)$. We will employ the lighter notation $[\overline{\varepsilon}]$ for this class.
Given a morphism $\varphi\colon \overline{\varepsilon}\to\overline{\mu}$ of bounded complexes of metrized coherent sheaves, the pieces of the complex $\cone(\varepsilon,\mu)$ are naturally endowed with hermitian metrics. We thus get a complex of metrized coherent sheaves $\overline{\cone(\varepsilon,\mu)}$. Hence Definition \ref{def:1} provides an object $[\overline{\cone(\varepsilon,\mu)}]$ in $\oDb(X)$. On the other hand, Definition \ref{def:her_cone} attaches to $\varphi$ the hermitian cone $\ocone([\overline{\varepsilon}],[\overline{\mu}])$, which is well defined up to tight isomorphism. Both constructions actually agree. \begin{lemma}\label{lemm:3}
Let $\overline{\varepsilon}\to\overline{\mu}$ be a morphism of bounded complexes
of metrized coherent sheaves on $X$. Then there is a tight
isomorphism
\begin{displaymath}
\ocone([\overline{\varepsilon}],[\overline{\mu}])\cong [\overline{\cone(\varepsilon,\mu)}],
\end{displaymath} \end{lemma} \begin{proof}
The case when $\varepsilon$ and $\mu$ are both concentrated in a
single degree $d$ is clear. The general case follows by induction
taking into account Definition \ref{def:1}. \end{proof}
Assume now that $f\colon X\to Y$ is a morphism of smooth complex varieties and, for each complex $\Rd f_{\ast}\mathcal{F}_{i}$, we have chosen a hermitian structure $\overline{\Rd f_{\ast} \mathcal{F}_{i}}=(\overline E_{i}\dashrightarrow \Rd f_{\ast} \mathcal{F}_{i})$. Denote by $\overline {\Rd
f_{\ast}\varepsilon }$ this choice of metrics. \begin{definition} \label{def:5} The family of hermitian structures
$\overline {\Rd f_{\ast}\varepsilon }$ defines an object $[\overline{\Rd
f_{\ast}\varepsilon }]\in \Ob \oDb(Y)$ that is determined
inductively by the condition
\begin{displaymath}
[\overline {\Rd f_{\ast}\varepsilon }]=\ocone (\overline {\Rd f_{\ast}
\mathcal{F}_{m}}[m],[\overline {\Rd f_{\ast}\sigma _{< m} \varepsilon} ]).
\end{displaymath} \end{definition}
We remark that the notation $\overline {\Rd
f_{\ast}\varepsilon }$ means that the hermitian structure is chosen after taking the direct image and it is not determined by the hermitian structure on $\overline \varepsilon $.
If $\overline{F}$ is a hermitian vector bundle on $Y$, then the object $[\overline{\Rd f_{\ast}(\varepsilon\otimes f^{\ast} F)}]$ (whose definition is obvious) satisfies \begin{displaymath}
[\overline{\Rd f_{\ast}(\varepsilon\otimes f^{\ast} F)}]
=[\overline{\Rd f_{\ast}\varepsilon}]\otimes\overline{F}. \end{displaymath} Notice also that if $\varepsilon$ is an acyclic complex on $X$, we have the class $[\overline {\Rd f_{\ast}\varepsilon}]\in \KA(Y)$.
Let $\varepsilon\to\mu$ be a morphism of bounded complexes of coherent sheaves on $X$ and $f\colon X\to Y$ a morphism of smooth complex varieties. Fix choices of metrics $\overline{\Rd f_{\ast}\varepsilon}$ and $\overline{\Rd f_{\ast}\mu}$. Then there is an obvious choice of metrics on $\Rd f_{\ast}\cone(\varepsilon,\mu)$, that we denote $\overline{\Rd
f_{\ast}\cone(\varepsilon,\mu)}$, and hence an object $[\overline{\Rd
f_{\ast}\cone(\varepsilon,\mu)}]$ in $\oDb(Y)$. On the other hand, we also have the hermitian cone $\ocone([\overline{\Rd
f_{\ast}\varepsilon}],[\overline{\Rd f_{\ast}\mu}])$. Again both definitions agree.
\begin{lemma}\label{lemm:3bis}
Let $\varepsilon\to\mu$ be a morphism of bounded complexes of
coherent sheaves on $X$ and $f\colon X\to Y$ a morphism of smooth
complex varieties. Assume that families of metrics $\overline{\Rd
f_{\ast}\varepsilon}$ and $\overline{\Rd f_{\ast}\mu}$ are chosen. Then there is a
tight isomorphism
\begin{displaymath}
\ocone([\overline{\Rd f_{\ast}\varepsilon}],[\overline{\Rd f_{\ast}\mu}])\cong [\overline{\Rd f_{\ast}\cone(\varepsilon,\mu)}].
\end{displaymath} \end{lemma} \begin{proof}
If $\varepsilon$ and $\mu$ are concentrated in a single degree $d$,
then the statement is obvious. The proof follows by induction and
Definition \ref{def:5}. \end{proof}
The objects we have defined are compatible with short exact sequences, in the sense of the following statement. \begin{proposition} \label{prop:4} Consider a commutative diagram of
exact sequences of coherent sheaves on $X$
\begin{displaymath}
\xymatrix{
& & &0\ar[d] & &0\ar[d] & \\
&\mu' &0\ar[r] &\mathcal{F}_{m}'\ar[r]\ar[d] &\dots\ar[r] &\mathcal{F}_{l}'\ar[r]\ar[d] &0
\\
&\mu &0\ar[r] &\mathcal{F}_{m}\ar[r]\ar[d] &\dots\ar[r] &\mathcal{F}_{l}\ar[r]\ar[d] &0
\\
&\mu'' &0\ar[r] &\mathcal{F}_{m}''\ar[r]\ar[d] &\dots\ar[r] &\mathcal{F}_{l}''\ar[r]\ar[d] &0
\\
& & &0 & &0 & \\
& & &\xi_{m} &\dots &\xi_{l}. &
}
\end{displaymath}
Let $f\colon X\to Y$ be a morphism of smooth complex varieties and
choose hermitian structures on the sheaves $\mathcal{F}_{j}'$,
$\mathcal{F}_{j}$, $\mathcal{F}_{j}''$ and on the objects $\Rd
f_{\ast}\mathcal{F}_{j}'$, $\Rd f_{\ast}\mathcal{F}_{j}$ and $\Rd
f_{\ast}\mathcal{F}_{j}''$, $j=l,\dots, m$. Then the following
equalities hold in $\KA(X)$ and $\KA(Y)$, respectively:
\begin{align*}
&\sum_{j}(-1)^{j}[\overline{\xi}_{j}]=[\overline{\mu}']-[\overline{\mu}]+[\overline{\mu}''],\\
&\sum_{j}(-1)^{j}[\overline{\Rd f_{\ast}\xi}_{j}]=[\overline{\Rd
f_{\ast}\mu}']-[\overline{\Rd f_{\ast} \mu}]+[\overline{\Rd f_{\ast}\mu}''].
\end{align*} \end{proposition} \begin{proof}
The lemma follows inductively taking into account definitions
\ref{def:1} and \ref{def:5} and Theorem \ref{thm:10}~\ref{item:49}. \end{proof} \begin{corollary}\label{cor:5}
Let $\overline{\varepsilon}\to\overline{\mu} $ be a morphism of exact sequences
of metrized coherent sheaves. Let $f\colon X\to Y$ be a morphism of
smooth complex varieties and fix families of metrics $\overline{\Rd
f_{\ast}\varepsilon}$ and $\overline{\Rd f_{\ast}\mu}$. Then the following
equalities in $\KA(X)$ and $\KA(Y)$, respectively, hold
\begin{align}
&[\overline{\cone(\varepsilon,\mu)}]=[\overline{\mu}]-[\overline{\varepsilon}],\\
&[\overline{\Rd f_{\ast}\cone(\varepsilon,\mu})]=[\overline{\Rd
f_{\ast}\mu}]-[\overline{\Rd f_{\ast}\varepsilon}].
\end{align} \end{corollary} \begin{proof}
The result readily follows from lemmas \ref{lemm:3}, \ref{lemm:3bis}
and Proposition \ref{prop:4}. \end{proof}
\noindent\textbf{Hermitian structures on cohomology.} Let $\mathcal{F}$ be an object of $\Db(X)$ and denote by $\mathcal{H}$ its cohomology complex. Observe that $\mathcal{H}$ is a bounded complex with 0 differentials. By the preceding discussion and because $\KA(X)$ acts transitively on hermitian structures, giving a hermitian structure on $\mathcal{H}$ amounts to give hermitian structures on the individual pieces $\mathcal{H}^{i}$. We show that to these data there is attached a natural hermitian structure on the complex $\mathcal{F}$. This situation will arise when considering cohomology sheaves endowed with $L^2$ metric structures. The construction is recursive. If the cohomology complex is trivial, then we endow $\mathcal{F}$ with the trivial hermitian structure. Otherwise, let $\mathcal{H}^{m}$ be the highest non-zero cohomology sheaf. The canonical filtration $\tau ^{\leq m}$ is given by \begin{displaymath}
\tau^{\leq
m}\mathcal{F}\colon\quad\dots\to\mathcal{F}^{m-2}\to\mathcal{F}^{m-1}
\to\ker(\dd^{m})\to 0. \end{displaymath} By the condition on the highest non vanishing cohomology sheaf, the natural inclusion is a quasi-isomorphism: \begin{equation}\label{eq:her_coh_1}
\tau^{\leq m}\mathcal{F}\overset{\sim}{\longrightarrow}\mathcal{F}. \end{equation} We also introduce the subcomplex \begin{displaymath}
\widetilde{\mathcal{F}}\colon\quad\dots\to\mathcal{F}^{m-2}\to
\mathcal{F}^{m-1}\to{\im}(\dd^{m-1})\to 0. \end{displaymath} Observe that the cohomology complex of $\widetilde{\mathcal{F}}$ is the b\^ete truncation $\mathcal{H}/\sigma^{\ge m}\mathcal{H}$. By induction, $\widetilde{\mathcal{F}}$ carries an induced hermitian structure. We also have an exact sequence \begin{equation}\label{eq:her_cor_2}
0\to\widetilde{\mathcal{F}}\to\tau^{\leq m}\mathcal{F}\to\mathcal{H}^{m}[-m]\to 0. \end{equation} Taking into account the quasi-isomorphism \eqref{eq:her_coh_1} and the exact sequence \eqref{eq:her_cor_2}, we construct a natural commutative diagram of distinguished triangles in $\Db(X)$ \begin{displaymath}
\xymatrix{
\mathcal{H}^{m}[-m-1]\ar@{-->}[r]^{\hspace{0.6cm} 0}\ar[d]^{\Id} &\widetilde{\mathcal{F}}\ar@{-->}[r]\ar[d]^{\Id}
&\mathcal{F}\ar@{-->}[r]\ar@{-->}[d]^{\sim} &\mathcal{H}^{m}[m]\ar[d]^{\Id}\\
\mathcal{H}^{m}[-m-1]\ar@{-->}[r]^{\hspace{0.6cm} 0} &\widetilde{\mathcal{F}}\ar@{-->}[r]
&\cone(\mathcal{H}^{m}[-m-1],\widetilde{\mathcal{F}})\ar@{-->}[r] &\mathcal{H}^{m}[m].
} \end{displaymath} By the hermitian cone construction and Theorem \ref{thm:6bis}, we see that hermitian structures on $\widetilde{\mathcal{F}}$ and $\mathcal{H}^{m}$ induce a well defined hermitian structure on $\mathcal{F}$.
\begin{definition}\label{def:her_coh}
Let $\mathcal{F}$ be an object of $\Db(X)$ with cohomology complex
$\mathcal{H}$. Assume the pieces $\mathcal{H}^{i}$ are endowed with
hermitian structures. The hermitian structure on $\mathcal{F}$
constructed above will be called the \emph{hermitian structure induced by
the hermitian structure on the cohomology complex} and will be denoted $(\mathcal{F},\overline{\mathcal{H}})$. \end{definition}
The following proposition is a direct consequence of the definitions.
\begin{proposition}\label{prop:her_coh}
Let $\varphi\colon\mathcal{F}_{1}\dashrightarrow\mathcal{F}_{2}$ be an
isomorphism in $\Db(X)$. Assume the pieces of the cohomology
complexes $\mathcal{H}_{1}$, $\mathcal{H}_{2}$ of $\mathcal{F}_{1}$,
$\mathcal{F}_{2}$ are endowed with hermitian structures. If the
induced isomorphism in cohomology
$\varphi_{\ast}\colon\mathcal{H}_{1}\to\mathcal{H}_{2}$ is tight,
then $\varphi$ is tight for the induced hermitian structures on
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$. \end{proposition}
$\square$
\section{Bott-Chern classes for isomorphisms and distinguished triangles in $\oDb(X)$} \label{sec:bott-chern-classes}
In this section we will define Bott-Chern classes for isomorphisms and distinguished triangles in $\oDb(X)$. The natural context where one can define the Bott-Chern classes is that of Deligne complexes. For details about Deligne complexes the reader is referred to \cite{Burgos:CDB} and \cite{BurgosKramerKuehn:cacg}. In this section we will use the same notations as in \cite{BurgosLitcanu:SingularBC} \S1. In particular, the \emph{Deligne algebra of differential forms} on $X$ is denoted by \begin{math}
\mathcal{D}^{\ast}(X,\ast). \end{math} and we use the notation \begin{displaymath}
\widetilde
{\mathcal{D}}^{n}(X,p)=\left. \mathcal{D}^{n}(X,p)\right/
\dd_{\mathcal{D}}\mathcal{D}^{n-1}(X,p). \end{displaymath}
When characterizing axiomatically Bott-Chern classes, the basic tool to exploit the functoriality axiom is to use a deformation parametrized by ${\mathbb P}^{1}$. This argument leads to the following lemma that will be used to prove the uniqueness of the Bott-Chern classes introduced in this section.
\begin{lemma} \label{lemm:1}
Let $X$ be a smooth complex variety. Let
$\widetilde \varphi$ be an assignment that, to each smooth morphism
of complex
varieties $g\colon X'\to X$ and each acyclic complex $\overline A$ of
hermitian vector bundles on $X'$
assigns a class
\begin{displaymath}
\widetilde \varphi(\overline A)\in \bigoplus
_{n,p}\widetilde{\mathcal{D}}^{n}(X',p)
\end{displaymath}
fulfilling the following properties:
\begin{enumerate}
\item (Differential equation) the equality
\begin{math}
\dd_{\mathcal{D}}\widetilde \varphi(\overline A)=0
\end{math}
holds;
\item (Functoriality) for each morphism of smooth complex varieties
$h\colon X''\to X'$ with $g\circ h $ smooth, we have
\begin{math}
h^{\ast} \widetilde \varphi(\overline A)=\widetilde \varphi(h^{\ast}\overline A)
\end{math};
\item (Normalization) if $\overline A$ is orthogonally split, then $\widetilde
\varphi(\overline A)=0$.
\end{enumerate}
Then $\widetilde \varphi=0$. \end{lemma} \begin{proof}
The argument of the proof of \cite[Thm.
2.3]{BurgosLitcanu:SingularBC} applies \emph{mutatis mutandis} to
the present situation. \end{proof}
\begin{definition} \label{def:24}
An \emph{additive genus in Deligne cohomology} is a characteristic class $\varphi$ for vector bundles of any rank in the sense of \cite[Def. 1.5]{BurgosLitcanu:SingularBC} that satisfies the equation \begin{equation}
\label{eq:75}
\varphi(E_{1}\oplus E_{2})=\varphi(E_{1})+\varphi(E_{2}). \end{equation}
\end{definition} Let ${\mathbb D}$ denote the base ring for Deligne cohomology (see \cite{BurgosLitcanu:SingularBC} before Definition 1.5). A consequence of \cite[Thm. 1.8]{BurgosLitcanu:SingularBC} is that there is a bijection between the set of additive genera in Deligne cohomology and the set of power series in one variable ${\mathbb D}[[x]]$. To each power series $\varphi \in {\mathbb D}[[x]]$ it corresponds the unique additive genus such that \begin{displaymath}
\varphi(L)=\varphi (c_{1}(L)) \end{displaymath} for every line bundle $L$.
\begin{definition} \label{def:25}
A \emph{real additive genus} is an additive genus such that the
corresponding power series belong to ${\mathbb R}[[x]]$. \end{definition}
\begin{remark}\label{rem:7}
It is clear that, if $\varphi$ is a real additive genus, then for each
vector bundle $E$ we have
\begin{displaymath}
\varphi(E)\in \bigoplus_{p}H_{\mathcal{D}}^{2p}(X,{\mathbb R}(p))
\end{displaymath}
\end{remark}
We now focus on additive genera, for instance the Chern character is a real additive genus. Let $\varphi $ be such a genus. Using Chern-Weil theory, to each hermitian vector bundle $\overline E$ on $X$ we can attach a closed characteristic form \begin{displaymath}
\varphi (\overline E)\in \bigoplus_{n,p}\mathcal{D}^{n}(X,p). \end{displaymath} If $\overline E$ is an object of $\oV(X)$, then we define $$\varphi (\overline E)=\sum_{i} (-1)^{i} \varphi (\overline E^{i}).$$ If $\overline E$ is acyclic, following \cite[Sec. 2]{BurgosLitcanu:SingularBC}, we associate to it a Bott-Chern characteristic class \begin{displaymath}
\widetilde \varphi (\overline E)\in
\bigoplus_{n,p}\widetilde {\mathcal{D}}^{n-1}(X,p) \end{displaymath} that satisfies the differential equation \begin{displaymath}
\dd_{\mathcal{D}}\widetilde \varphi (\overline E)=\varphi
(\overline E). \end{displaymath}
In fact, \cite[Thm. 2.3]{BurgosLitcanu:SingularBC} for additive genera can be restated as follows.
\begin{proposition} \label{prop:9}
Let $\varphi $ be an additive genus. Then there
is a unique group homomorphism
\begin{displaymath}
\widetilde \varphi \colon \KA(X)\to \bigoplus_{n,p}\widetilde
{\mathcal{D}}^{n-1}(X,p)
\end{displaymath}
satisfying the properties:
\begin{enumerate}
\item (Differential equation) $\dd_{\mathcal{D}}
\widetilde\varphi(\overline E)= \varphi(\overline E).$
\item (Functoriality) If $f\colon X\to Y$ is a morphism of smooth
complex varieties, then
$
\widetilde{\varphi}(\Ld f^{\ast}(\overline{E}))=f^{\ast}(\widetilde{\varphi}(\overline{E})).
$
\end{enumerate} \end{proposition} \begin{proof}
For the uniqueness, we observe that, if $\widetilde \varphi$ is a group
homomorphism then $\widetilde\varphi (\overline 0)=0$. Hence, if $\overline E$
is a orthogonally split complex, then it is meager and therefore
$\widetilde\varphi (\overline E)=0$. Thus, the assignment that, to each acyclic
complex bounded $\overline E$, associates the class $\widetilde
\varphi([\overline E])$ satisfies the
conditions of \cite[Thm. 2.3]{BurgosLitcanu:SingularBC}, hence is
unique. For the existence, we note that
Bott-Chern classes for additive genera satisfy the
hypothesis of Theorem \ref{thm:8}. Hence the result follows. \end{proof}
\begin{remark} \label{rem:6} If \begin{displaymath}
\overline{\varepsilon }:\quad
0\to \overline{\mathcal{F}}_{m}\to \dots \to
\overline{\mathcal{F}}_{l}\to 0 \end{displaymath} is an acyclic complex of coherent sheaves on $X$ provided with hermitian structures $\overline {\mathcal{F}}_{i}=(\mathcal{F}_{i},\overline F_{i}\dashrightarrow \mathcal{F}_{i})$, by Definition \ref{def:1} we have an object $[\overline \varepsilon ]\in \KA(X)$, hence a class $\widetilde \varphi([\overline \varepsilon ])$. In the case of the Chern character, in \cite[Thm. 2.24]{BurgosLitcanu:SingularBC}, a class $\widetilde {\ch}(\overline \varepsilon )$ is defined. It follows from \cite[Thm. 2.24]{BurgosLitcanu:SingularBC} that both classes agree. That is, $\widetilde{\ch}([\overline \varepsilon ])=\widetilde{\ch}(\overline \varepsilon )$. For this reason we will denote $\widetilde \varphi([\overline \varepsilon ])$ by $\widetilde \varphi(\overline \varepsilon)$. \end{remark}
\begin{definition}\label{definition:forms_complexes}
Let
$\overline{\mathcal{F}}=(\overline E\overset{\sim}{\dashrightarrow}\mathcal{F})$
be an object of $\oDb(X)$. Let $\varphi$ denote
an additive genus. We denote the form
\begin{equation*}
\varphi(\overline{\mathcal{F}})=\varphi(\overline E)\in \bigoplus_{n,p}\mathcal{D}^{n}(X,p)
\end{equation*}
and the class
\begin{equation*}
\varphi(\mathcal{F})=[\varphi(\overline E)]\in
\bigoplus_{n,p}H_{\mathcal{D}}^{n}(X,{\mathbb R}(p)).
\end{equation*}
Note that the form $\varphi(\overline{\mathcal{F}})$ only depends on the hermitian structure
and not on a particular representative thanks to Proposition
\ref{prop:7} and Proposition \ref{prop:9}. The class
$\varphi(\mathcal{F})$ only depends on the object $\mathcal{F}$ and
not on the hermitian structure.
\end{definition} \begin{remark}
The reason to restrict to additive genera when working with the
derived category is now clear: there is no
canonical way to attach a rank to $\oplus_{i\even}\mathcal{F}^{i}$
(respectively $\oplus_{i\odd} \mathcal{F}^{i}$). The naive choice
$\rk(\oplus_{i\even} E^{i})$ (respectively $\rk(\oplus_{i\odd} E^{i})$)
does depend on $E\dashrightarrow\mathcal{F}$. Thus we can not define
Bott-Chern classes by the general rule from
\cite{BurgosLitcanu:SingularBC}. The case of a multiplicative genus
such as the Todd genus will be considered later. \end{remark}
Next we will construct Bott-Chern classes for isomorphisms in $\oDb(X)$.
\begin{definition}
Let $f\colon \overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}$
be a morphism in $\oDb(X)$ and $\varphi$ an additive
genus. We define the differential form
\begin{equation*}
\varphi(f)=\varphi(\overline{\mathcal{G}})-
\varphi(\overline{\mathcal{F}}).
\end{equation*} \end{definition}
\begin{theorem}\label{theorem:ch_tilde_qiso}
Let $\varphi$ be an additive genus. There is a unique way to attach to
every isomorphism in $\oDb(X)$
\begin{math}
f\colon (\overline{F}\dashrightarrow\mathcal{F})
\overset{\sim}{\dashrightarrow}
(\overline{G}\dashrightarrow\mathcal{G})
\end{math}
a Bott-Chern class
\begin{displaymath}
\widetilde{\varphi}(f)\in\bigoplus_{n,p}\widetilde{\mathcal{D}}^{n-1}(X,p)
\end{displaymath}
such that the following axioms are satisfied:
\begin{enumerate}
\item (Differential equation)
\begin{math}
\dd_{\mathcal{D}}\widetilde{\varphi}(f)=\varphi(f).
\end{math}
\item (Functoriality) If $g\colon X'\rightarrow X$ is a morphism of smooth
noetherian schemes over ${\mathbb C}$, then
\begin{displaymath}
\widetilde{\varphi}(\Ld g^{\ast}(f))=g^{\ast}(\widetilde{\varphi}(f)).
\end{displaymath}
\item (Normalization) If $f$ is a tight isomorphism, then
\begin{math}
\widetilde{\varphi}(f)=0.
\end{math}
\end{enumerate} \end{theorem} \begin{proof} For the existence we define \begin{equation}
\label{eq:56}
\widetilde \varphi(f)=\widetilde \varphi([f]), \end{equation} where $[f]\in\KA(X)$ is the class of $f$ given by equation \eqref{eq:57}. That $\widetilde \varphi$ satisfies the axioms follows from Proposition \ref{prop:9} and Theorem \ref{thm:9}.
We now focus on the uniqueness. Assume such a theory $f\mapsto\widetilde{\varphi}_{0}(f)$ exists. Fix $f$ as in the statement. Since $\widetilde \varphi_{0}$ is well defined, by replacing $\overline F$ by one that is tightly related, we may assume that $f$ is realized by a morphism of complexes \begin{displaymath}
f\colon \overline F\longrightarrow \overline G. \end{displaymath} We factorize $f$ as \begin{displaymath}
\overline F\overset{\alpha }{\longrightarrow }
\overline G\oplus \ocone(\overline F,\overline G)[-1]
\overset{\beta }{\longrightarrow}
\overline G, \end{displaymath} where both arrows are zero on the second factor of the middle complex. Since $\alpha $ is a tight morphism and $\ocone(\overline F,\overline G)[-1]$ is acyclic, we are reduced to the case when $\overline F=\overline G\oplus \overline A$, with $\overline A$ an acyclic complex and $f$ is the projection onto the first factor.
For each smooth morphism $g\colon X'\to X$ and each acyclic complex of vector bundles $\overline E$ on $X'$, we denote \begin{displaymath}
\widetilde \varphi_{1}(\overline E)=\widetilde \varphi_{0} (g^{\ast}\overline G\oplus \overline E\to g^{\ast}\overline G)+\widetilde \varphi(\overline E), \end{displaymath} where $\widetilde \varphi$ is the usual Bott-Chern form for acyclic complexes of hermitian vector bundles associated to $\varphi$. Then $\widetilde \varphi_{1}$ satisfies the hypothesis of Lemma \ref{lemm:1}, so $\widetilde \varphi_{1}=0$. Therefore \begin{math}
\widetilde \varphi(f)=-\widetilde \varphi(\overline A). \end{math} \end{proof}
\begin{proposition}\label{proposition:ch_tilde_comp} Let $f\colon \overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}$ and $g\colon \overline{\mathcal{G}}\dashrightarrow \overline{\mathcal{H}}$ be two isomorphisms in $\oDb(X)$. Then: \begin{displaymath}
\widetilde{\varphi}(g\circ f)=
\widetilde{\varphi}(g)+\widetilde{\varphi}(f). \end{displaymath} In particular, $\widetilde{\varphi}(f^{-1})=-\widetilde{\varphi}(f)$. \end{proposition} \begin{proof}
The statement follows from Theorem \ref{thm:10}~\ref{item:16}. \end{proof}
The Bott-Chern classes behave well under shift.
\begin{proposition}
Let $f\colon \overline{\mathcal{F}}\dashrightarrow \overline{\mathcal{G}}$ be an
isomorphism in $\oDb(X)$. Let $f[i]\colon \overline{\mathcal{F}}[i]\dashrightarrow
\overline{\mathcal{G}}[i]$ be the shifted isomorphism. Then
\begin{displaymath}
(-1)^{i}\widetilde{\varphi}(f[i])=\widetilde{\varphi}(f).
\end{displaymath} \end{proposition} \begin{proof}
The assignment $f\mapsto (-1)^{i}\widetilde{\varphi}(f[i])$
satisfies the characterizing properties of Theorem
\ref{theorem:ch_tilde_qiso}. Hence it agrees with $\widetilde
\varphi$. \end{proof}
The following notation will be sometimes used. \begin{notation} Let $\mathcal{F}$ be an object of $\Db(X)$ and consider two choices of hermitian structures $\overline{\mathcal{F}}$ and $\overline{\mathcal{F}}'$. Then we write \begin{displaymath}
\widetilde{\varphi}(\overline{\mathcal{F}},\overline{\mathcal{F}}')=
\widetilde{\varphi}(\overline{\mathcal{F}}
\overset{\Id}{\dashrightarrow}\overline{\mathcal{F}}'). \end{displaymath} Thus $\dd_{\mathcal{D}}\widetilde{\varphi} (\overline{\mathcal{F}},\overline{\mathcal{F}}') = \varphi(\overline{\mathcal{F}}')-\varphi(\overline{\mathcal{F}}). $ \end{notation}
\begin{example}\label{exm:1}
Let $\overline
{\mathcal{F}}=(\mathcal{F},\mathcal{F}\dashrightarrow\overline E)$
be an object of $\oDb(X)$. Let $\mathcal{H}^{i}$ denote the
cohomology sheaves of $\mathcal{F}$ and assume that we have chosen
hermitian structures $\overline{\mathcal{H}}^{i} $ of each
$\mathcal{H}^{i}$. In the case when the sheaves $\mathcal{H}^{i}$
are vector bundles and the hermitian structures are hermitian
metrics, X. Ma, in the paper \cite{Ma:MR1765553}, has associated to
these data a Bott-Chern class, that we
denote $M(\overline {\mathcal{F}},\overline {\mathcal{H}})$. By
the characterization given by Ma of $M(\overline {\mathcal{F}},\overline
{\mathcal{H}})$, it is immediate that
\begin{displaymath}
M(\overline {\mathcal{F}},\overline {\mathcal{H}})=
\cht(\overline {\mathcal{F}},(\mathcal{F},\overline {\mathcal{H}})),
\end{displaymath}
where $(\mathcal{F},\overline {\mathcal{H}})$ is as in Definition
\ref{def:her_coh}. \end{example}
Our next aim is to construct Bott-Chern classes for distinguished triangles.
\begin{definition}
Let $\overline \tau $ be a distinguished triangle in $\oDb(X)$, \begin{displaymath}
\overline{\tau}\colon\
\overline{\mathcal{F}}\overset{u}{\dashrightarrow}\overline{\mathcal{G}}
\overset{v}{\dashrightarrow}\overline{\mathcal{H}}
\overset{w}{\dashrightarrow}\overline{\mathcal{F}}[1]
\overset{u}{\dashrightarrow}\dots
\end{displaymath} For an additive genus $\varphi$, we attach the differential form
\begin{displaymath}
\varphi(\overline{\tau})=\varphi(\overline{\mathcal{F}})-
\varphi(\overline{\mathcal{G}})+\varphi(\overline{\mathcal{H}}).
\end{displaymath} \end{definition}
Notice that if $\overline \tau$ is tightly distinguished, then $\varphi(\overline \tau )=0$. Moreover, for any distinguished triangle $\overline \tau $ as above, the rotated triangle \begin{displaymath}
\overline{\tau}'\colon\
\overline{\mathcal{G}}\overset{v}{\dashrightarrow}\overline{\mathcal{H}}
\overset{w}{\dashrightarrow}\overline{\mathcal{F}}[1]
\overset{-u[1]}{\dashrightarrow}\overline{\mathcal{G}}[1]
\overset{v[1]}{\dashrightarrow}\dots \end{displaymath} satisfies \begin{math}
\varphi(\overline \tau ')=-\varphi(\overline \tau). \end{math}
\begin{theorem}\label{thm:16}
Let $\varphi$ be an additive genus. There is a unique way to attach to
every distinguished triangle in $\oDb(X)$
\begin{displaymath}
\overline{\tau}:\quad
\overline{\mathcal{F}}\overset{u}{\dashrightarrow}\overline{\mathcal{G}}
\overset{v}{\dashrightarrow}\overline{\mathcal{H}}
\overset{w}{\dashrightarrow}\overline{\mathcal{F}}[1]
\overset{u[1]}{\dashrightarrow}\dots
\end{displaymath}
a Bott-Chern class
\begin{displaymath}
\widetilde{\varphi}(\overline{\tau})
\in\bigoplus_{n,p}\widetilde{\mathcal{D}}^{n-1}(X,p)
\end{displaymath}
such that the following axioms are satisfied:
\begin{enumerate}
\item (Differential equation)
\begin{math}
\dd_{\mathcal{D}}\widetilde{\varphi}(\overline{\tau})=
\varphi(\overline{\tau}).
\end{math}
\item (Functoriality) If $g\colon X^{\prime}\rightarrow X$ is a morphism of smooth noetherian schemes over ${\mathbb C}$, then
\begin{displaymath}
\widetilde{\varphi}(\Ld g^{\ast}(\overline{\tau}))=g^{\ast}\widetilde{\varphi}(\overline{\tau}).
\end{displaymath}
\item (Normalization) If $\overline{\tau}$ is tightly distinguished, then
\begin{math}
\widetilde{\varphi}(\overline{\tau})=0.
\end{math}
\end{enumerate} \end{theorem}
\begin{proof} To show the existence we write \begin{equation}
\label{eq:58}
\widetilde \varphi(\overline \tau)=\widetilde \varphi([\overline \tau ]). \end{equation} Theorem \ref{thm:10} implies that it satisfies the axioms.
To prove the uniqueness, observe that, by replacing representatives of the hermitian structures by tightly related ones, we may assume that the distinguished triangle is represented by \begin{displaymath}
\overline F \longrightarrow \overline G \longrightarrow \ocone(\overline F,\overline
G)\oplus \overline K \longrightarrow \overline F[1], \end{displaymath} with $\overline K$ acyclic. Then Lemma \ref{lemm:1} shows that the axioms imply \begin{math}
\widetilde \varphi(\overline \tau )=\widetilde \varphi(\overline K). \end{math} \end{proof}
\begin{remark}\label{rem:1} The normalization axiom can be replaced by
the apparently weaker condition that $\widetilde \varphi(\overline \tau
)=0$ for all distinguished triangles of the form
\begin{displaymath}
\overline {\mathcal{F}}\dashrightarrow \overline {\mathcal{F}}\overset{\perp}{\oplus}
\overline {\mathcal{G}}\dashrightarrow \overline {\mathcal{G}} \dashrightarrow
\end{displaymath}
where the maps are the natural inclusion and projection. \end{remark}
Theorem \ref{thm:10}~\ref{item:17}-\ref{item:49} can be easily translated to Bott-Chern classes.
\section{Multiplicative genera, the Todd genus and the category
$\oSm_{\ast/{\mathbb C}}$} \label{sec:multiplicative-genera} Let $\psi $ be a multiplicative genus, such that the piece of degree zero is $\psi ^{0}=1$, and \begin{displaymath}
\varphi=\log(\psi ). \end{displaymath} It is a well defined additive genus, because, by the condition above, the power series $\log (\psi )$ contains only finitely many terms in each degree.
If $\overline \theta $ is either a hermitian vector bundle, a complex of hermitian vector bundles, a morphism in $\oDb(X)$ or a distinguished triangle in $\oDb(X)$ we can write \begin{displaymath}
\psi (\overline \theta )=\exp(\varphi(\overline \theta )). \end{displaymath}
All the results of the previous sections can be translated to the multiplicative genus $\psi $. In particular, if $\overline \theta $ is an acyclic complex of hermitian vector bundles, an isomorphism in $\oDb(X)$ or a distinguished triangle in $\oDb(X)$, we define a Bott-Chern class \begin{displaymath}
\widetilde \psi_{m} (\overline \theta)=
\frac{\exp(\varphi(\overline \theta))-1}{\varphi(\overline \theta)}
\widetilde\varphi(\overline \theta). \end{displaymath}
\begin{theorem}\label{thm:11} The characteristic class $\widetilde
\psi_{m} (\overline \theta)$ satisfies:
\begin{enumerate}
\item (Differential equation)
\begin{math}
\dd_{\mathcal{D}}\widetilde{\psi}_{m}(\overline{\theta
})=\psi(\overline{\theta })-1.
\end{math}
\item (Functoriality) If $g\colon X^{\prime}\rightarrow X$ is a morphism
of smooth noetherian schemes over ${\mathbb C}$, then
\begin{displaymath}
\widetilde{\psi}_{m}(\Ld g^{\ast}(\overline{\theta }))
=g^{\ast}\widetilde{\psi}_{m}(\overline{\theta }).
\end{displaymath}
\item (Normalization) If $\overline{\theta }$ is either a meager
complex, a tight isomorphism or a tightly distinguished triangle,
then
\begin{math}
\widetilde{\psi}_{m}(\overline{\theta })=0.
\end{math}
\end{enumerate}
Moreover $\widetilde \psi_{m} $ is uniquely characterized by these
properties. \end{theorem}
\begin{remark}
For an acyclic complex of vector bundles $\overline E$, using the general
procedure for arbitrary symmetric power series, we can associate a
Bott-Chern class $\widetilde \psi (\overline E)$ (see for instance
\cite[Thm. 2.3]{BurgosLitcanu:SingularBC}) that satisfies the
differential equation
\begin{displaymath}
\dd_{\mathcal{D}} \widetilde \psi (\overline E)= \prod _{k \text{ even }}\psi (\overline
E^{k})-\prod _{k \text{ odd }}\psi (\overline
E^{k}),
\end{displaymath}
whereas $\widetilde \psi _{m}$ satisfies the differential equation
\begin{equation} \label{eq:62}
\dd_{\mathcal{D}} \widetilde \psi _{m}(\overline E)= \prod _{k}\psi (\overline
E^{k})^{(-1)^{k}}-1.
\end{equation}
In fact both Bott-Chern classes are related by
\begin{equation}\label{eq:76}
\widetilde \psi _{m}(\overline E)=
\widetilde \psi (\overline E)\prod_{k \text{ odd}}\psi (\overline E^{k})^{-1}.
\end{equation} \end{remark}
The main example of a multiplicative genus with the above properties is the Todd genus $\Td$. From now on we will treat only this case. Following the above procedure, to the Todd genus we can associate two Bott-Chern classes for acyclic complexes of vector bundles: the one given by the general theory, denoted by $\widetilde {\Td}$, and the one given by the theory of multiplicative genera, denoted $\widetilde {\Td}_{m}$. Both are related by the equation \eqref{eq:76}. Note however that, for isomorphisms and distinguished triangles in $\oDb(X)$, we only have the multiplicative version.
We now consider morphisms between smooth complex varieties and relative hermitian structures.
\begin{definition} Let $f\colon X\rightarrow Y$ be a morphism of smooth complex varieties. The \textit{tangent complex} of $f$ is the complex
\begin{displaymath}
T_{f}\colon \quad 0\longrightarrow T_{X}
\overset{df}{\longrightarrow} f^{\ast}T_{Y}\longrightarrow 0
\end{displaymath}
where $T_{X}$ is placed in degree 0 and $f^{\ast}T_{Y}$ is placed in degree 1. It defines an object $T_{f}\in \Ob \Db(X)$. A \emph{relative hermitian structure on} $f$ is the choice of an object $\overline T_{f}\in \oDb(X)$ over $T_{f}$.
\end{definition}
The following particular situations are of special interest:
\begin{enumerate} \item[--] suppose $f\colon X\hookrightarrow Y$ is a closed immersion. Let
$N_{X/Y}[-1]$ be the normal bundle to $X$ in $Y$, considered as a
complex concentrated in degree 1. By definition, there is a natural
quasi-isomorphism $p\colon T_{f}\overset{\sim}{\rightarrow} N_{X/Y}[-1]$ in
$\Cb(X)$, and hence an isomorphism $p^{-1}\colon N_{X/Y}[-1]\overset{\sim}
{\dashrightarrow} T_{f}$ in $\Db(X)$. Therefore, a hermitian metric $h$
on the vector bundle $N_{X/Y}$ naturally induces a hermitian
structure $p^{-1}\colon (N_{X/Y}[-1],h) \dashrightarrow T_{f}$ on $T_{f}$. Let
$\overline{T}_{f}$ be the corresponding object in $\oDb(X)$. Then we have
\begin{displaymath}
\Td(\overline{T}_{f})=\Td(N_{X/Y}[-1],h)=\Td(N_{X/Y},h)^{-1};
\end{displaymath}
\item[--] suppose $f\colon X\rightarrow Y$ is a smooth morphism. Let
$T_{X/Y}$ be the relative tangent bundle on $X$, considered as a
complex concentrated in degree $0$. By definition, there is a
natural quasi-isomorphism $\iota\colon T_{X/Y}\overset{\sim}{\rightarrow}
T_{f}$ in $\Cb(X)$. Any choice of hermitian metric $h$ on $T_{X/Y}$
naturally induces a hermitian structure
$\iota\colon (T_{X/Y},h)\dashrightarrow T_{f}$. If $\overline{T}_{f}$ denotes
the corresponding object in $\oDb(X)$, then we find
\begin{displaymath}
\Td(\overline{T}_{f})=\Td(T_{X/Y},h).
\end{displaymath}
\end{enumerate}
Let now $g\colon Y\rightarrow Z$ be another morphism of smooth varieties over ${\mathbb C}$. The tangent complexes $T_{f}$, $T_{g}$ and $T_{g\circ f}$ fit into a distinguished triangle in $\Db(X)$
\begin{displaymath}
\mathcal{T}\colon T_f\dashrightarrow T_{g\circ f}\dashrightarrow
\Ld f^{\ast}T_g\dashrightarrow T_f[1].
\end{displaymath}
\begin{definition}\label{def:16} We denote $\oSm_{\ast/{\mathbb C}}$ the following data:
(i) The class $\rm{Ob}\,\oSm_{\ast/{\mathbb C}}$ of smooth complex varieties.
(ii) For each $X,Y\in \rm{Ob}\,\oSm_{\ast/{\mathbb C}}$, a set of morphisms $\oSm_{\ast/{\mathbb C}}(X,Y)$ whose elements are pairs $\overline f=(f,\overline T_{f})$, where $f\colon X\to Y$ is a projective morphism and $\overline T_{f}$ is a hermitian structure on $T_{f}$. When $\overline f$ is given we will denote the hermitian structure by $T_{\overline f}$. A hermitian structure on $T_{f}$ will also be called a hermitian structure on $f$.
(iii) For each pair of morphisms $\overline f\colon X\to Y$ and $\overline g\colon Y\to Z$, the composition defined as \begin{displaymath}
\overline g\circ \overline f=(g\circ f,\ocone(\Ld f^{\ast} T_{\overline g}[-1], T_{\overline f})). \end{displaymath} \end{definition}
We shall prove (Theorem \ref{thm:17}) that $\oSm_{\ast/{\mathbb C}}$ is a category. Before this, we proceed with some examples emphasizing some properties of the composition rule.
\begin{example}\label{exm:3}
Let $f\colon X\to Y$
and $g\colon Y\to Z$, be projective morphisms of smooth complex
varieties. Assume
that we have chosen hermitian metrics on the tangent vector bundles
$T_{X}$, $T_{Y}$ and $T_{Z}$. Denote by $\overline f$, $\overline g$ and
$\overline{g\circ f}$ the morphism of $\oSm_{\ast/{\mathbb C}}$ determined by
these metrics. Then
\begin{displaymath}
\overline g\circ \overline f=\overline{g\circ f}.
\end{displaymath}
This is seen as follows. By the choice of metrics, there is a tight
isomorphism
\begin{displaymath}
\ocone(T_{\overline f},T_{\overline{g\circ f}})\to \Ld
f^{\ast}T_{\overline g}.
\end{displaymath}
Then the natural maps
\begin{displaymath}
T_{\overline g\circ \overline f}\to \ocone(\Ld
f^{\ast}T_{\overline g}[-1],T_{\overline f})\to
\ocone(\ocone(T_{\overline f},T_{\overline{g\circ f}})[-1],T_{\overline f})\to
T_{\overline{g\circ f}}
\end{displaymath}
are tight isomorphisms. \end{example}
\begin{example} \label{exm:4}
Let $f\colon X\to Y$
and $g\colon Y\to Z$, be smooth projective morphisms of smooth complex
varieties. Choose hermitian metrics on the relative tangent
vector bundles
$T_{f}$, $T_{g}$ and $T_{g\circ f}$. Denote by $\overline f$, $\overline g$ and
$\overline{g\circ f}$ the morphism of $\oSm_{\ast/{\mathbb C}}$ determined by
these metrics. There is a short exact sequence of hermitian vector
bundles
\begin{displaymath}
\overline \varepsilon \colon
0\longrightarrow \overline T_{f}
\longrightarrow \overline T_{g\circ f}
\longrightarrow f^{\ast} \overline T_{g}
\longrightarrow 0,
\end{displaymath}
that we consider as an acyclic complex declaring $f^{\ast} \overline T_{g}$
of degree $0$. The morphism $f^{\ast}T_{\overline g}[-1]\dashrightarrow T_{\overline f}$ is
represented by the diagram
\begin{displaymath}
\xymatrix{
& \ocone(T_{\overline f},T_{\overline{g\circ f}})[-1]
\ar[dl]_{\sim} \ar[rd] &\\
f^{\ast}T_{\overline g}[-1] && T_{\overline f}.
}
\end{displaymath} Thus, by the definition of a composition we have \begin{displaymath}
T_{\overline g \circ \overline f}=
\ocone(\ocone(T_{\overline f},T_{\overline{g\circ f}})[-1],f^{\ast}T_{\overline
g}[-1])[1]\oplus
\ocone(\ocone(T_{\overline f},T_{\overline{g\circ f}})[-1],T_{\overline f}). \end{displaymath} In general this hermitian structure is different to $T_{\overline{g\circ
f}}$.
\emph{Claim.} The equality of hermitian structures \begin{equation}
\label{eq:84}
T_{\overline g \circ \overline f}=
T_{\overline{g\circ f}}+ [\overline \varepsilon ] \end{equation} holds. \begin{proof}[Proof of the claim]
We have a commutative diagram of distinguished triangles
\begin{displaymath}
\xymatrix{
\overline{\varepsilon} &T_{\overline{f}}\ar[r]\ar[d]_{\Id} &T_{\overline{g\circ f}}\ar[r]\ar[d]
&f^{\ast}T_{\overline{g}}\ar[d]_{\Id}\ar@{-->}[r] &T_{\overline{f}}[1]\ar[d]_{\Id}\\
\overline{\tau} &T_{\overline{f}}\ar[r] &T_{\overline{g}\circ \overline{f}}\ar[r]
&f^{\ast}T_{\overline{g}}\ar@{-->}[r] &T_{\overline{f}}[1].
}
\end{displaymath} By construction the triangle $\overline{\tau}$ is tightly distinguished, hence $[\overline{\tau}]=0$. Therefore, according to Theorem \ref{thm:10} \ref{item:48}, we have \begin{displaymath}
[T_{\overline{g\circ f}}\rightarrow T_{\overline{g}\circ\overline{f}}]=[\overline{\varepsilon}]. \end{displaymath} The claim follows. \end{proof} \end{example}
\begin{theorem}\label{thm:17} $\oSm_{\ast/{\mathbb C}}$ is a category. \end{theorem}
\noindent\emph{Proof} The only non-trivial fact to prove is the associativity of the composition, given by the following lemma: \begin{lemma}\label{lem:Sm_cat}
Let $\overline{f}:X\to Y$, $\overline{g}:Y\to Z$ and $\overline{h}:Z\to W$ be
projective morphisms together with hermitian structures. Then
$\overline{h}\circ(\overline{g}\circ\overline{f})=(\overline{h}\circ\overline{g})\circ\overline{f}$. \end{lemma} \begin{proof}
First of all we observe that if the hermitian structures on
$\overline{f}$, $\overline{g}$ and $\overline{h}$ come from fixed hermitian metrics on
$T_{X}$, $T_{Y}$, $T_{Z}$ and $T_{W}$, Example \ref{exm:3} ensures
that the proposition holds. For the general case, it is enough to
see that if the proposition holds for a fixed choice of hermitian
structures $\overline{f}$, $\overline{g}$, $\overline{h}$, and we change the metric on
$f$, $g$ or $h$, then the proposition holds for the new choice of
metrics. We treat, for instance, the case when we change the
hermitian structure on $g$, the proof of the other cases being analogous.
Denote by $\overline{g}'$ the new hermitian structure on
$g$. Then there exists a unique class $\varepsilon\in\KA(Y)$ such
that $T_{\overline{g}'}=T_{\overline{g}}+\varepsilon$. According to the
definitions, we have
\begin{displaymath}
T_{\overline{h}\circ(\overline{g}'\circ\overline{f})}=
\ocone((g\circ
f)^{\ast}T_{\overline{h}}[-1],\ocone(f^{\ast}(T_{\overline{g}}+\varepsilon)[-1],T_{\overline{f}}))
=T_{\overline{h}\circ(\overline{g}\circ\overline{f})}+f^{\ast}\varepsilon.
\end{displaymath}
Similarly, we find
\begin{displaymath}
T_{(\overline{h}\circ\overline{g}')\circ\overline{f}}=
\ocone(f^{\ast}\ocone(g^{\ast}T_{\overline{h}}[-1],T_{\overline{g}})[-1]+
f^{\ast}(-\varepsilon),
T_{\overline{f}})
=T_{(\overline{h}\circ\overline{g})\circ\overline{f}}+f^{\ast}\varepsilon.
\end{displaymath}
By assumption,
$T_{\overline{h}\circ(\overline{g}\circ\overline{f})}=T_{(\overline{h}\circ\overline{g})\circ\overline{f}}$. Hence
the relations above show
\begin{displaymath}
T_{\overline{h}\circ(\overline{g}'\circ\overline{f})}=T_{(\overline{h}\circ\overline{g}')\circ\overline{f}}.
\end{displaymath}
This concludes the proofs of Lemma \ref{lem:Sm_cat} and of Theorem \ref{thm:17}. \end{proof}
Let $f\colon X\to Y$ and $g\colon Y\to Z$ be projective morphisms of smooth complex varieties. By the definition of composition, hermitian structures on $f$ and $g$ determine a hermitian structure on $g\circ f$. Conversely we have the following result.
\begin{lemma}\label{lemm:4}
Let $\overline {g}$ and $\overline {g\circ f}$ be hermitian structures on $g$
and $g\circ f$. Then there is a unique hermitian structure $\overline f$ on
$f$ such that
\begin{equation}
\label{eq:85}
\overline{g\circ f}=\overline g\circ \overline f.
\end{equation} \end{lemma} \begin{proof}
We have the distinguished triangle
\begin{displaymath}
T_{f}\dashrightarrow T_{g\circ f}\dashrightarrow f^{\ast}T_{g}\dashrightarrow T_{f}[1].
\end{displaymath}
The unique hermitian structure that satisfies equation
\eqref{eq:85} is $\ocone(T_{\overline{g\circ
f}},f^{\ast}T_{\overline{g}})[-1]$. \end{proof}
\begin{remark}
By contrast with the preceding result, it is not true in general that
hermitian structures $\overline f$ and $\overline{g\circ f}$ determine a
unique hermitian structure $\overline g$ that satisfies equation
\eqref{eq:85}. For instance, if $X=\emptyset$, then any hermitian
structure on $g$ will satisfy this equation. \end{remark}
If $\Sm_{\ast/{\mathbb C}}$ denotes the category of smooth complex varieties and projective morphisms and $\mathfrak{F}\colon \oSm_{\ast/{\mathbb C}}\to \Sm_{\ast/{\mathbb C}}$ is the forgetful functor, for any object $X$ we have that \begin{align*}
\Ob \mathfrak{F}^{-1}(X)&=\{X\},\\
\Hom_{\mathfrak{F}^{-1}(X)}(X,X)&=\KA(X). \end{align*}
To any arrow $\overline f\colon X\to Y$ in $\oSm_{\ast/{\mathbb C}}$ we associate a Todd form \begin{equation}\label{eq:1}
\Td(\overline f):=\Td(T_{\overline f})\in \bigoplus_{p}\mathcal{D}^{2p}(X,p). \end{equation}
The following simple properties of $\Td(\overline f)$ follow directly from the definitions.
\begin{proposition}
\begin{enumerate}
\item Let $\overline f\colon X\to Y$ and $\overline{g}\colon Y\to Z$ be
morphisms in $\oSm_{\ast/{\mathbb C}}$. Then
\begin{displaymath}
\Td(\overline g\circ\overline f)=f^{\ast}\Td(\overline{g})\bullet\Td(\overline{f}).
\end{displaymath}
\item Let $f,f'\colon X\to Y$ be two morphisms in $\oSm_{\ast/{\mathbb C}}$
with the same underlying algebraic morphism. There is an
isomorphism $\overline \theta \colon T_{\overline f}\to T_{\overline f'}$ whose
Bott-Chern class $\widetilde{\Td}_{m}(\overline\theta)$ satisfies
\begin{displaymath}
\dd_{\mathcal{D}}\widetilde {\Td}_{m}(\overline \theta )=
\Td(T_{\overline f'}) \Td(T_{\overline f})^{-1}-1.
\end{displaymath}
\end{enumerate} \end{proposition}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title[Higher order elliptic systems] {The conormal derivative problem for higher order elliptic systems with irregular coefficients}
\author[H. Dong]{Hongjie Dong} \address[H. Dong]{Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA} \email{Hongjie\[email protected]} \thanks{H. Dong was partially supported by the NSF under agreement DMS-0800129 and DMS-1056737.}
\author[D. Kim]{Doyoon Kim} \address[D. Kim]{Department of Applied Mathematics, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 446-701, Republic of Korea} \email{[email protected]} \thanks{D. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0013960).}
\subjclass[2000]{35K52, 35J58,35R05}
\date{}
\begin{abstract} We prove $L_p$ estimates of solutions to a conormal derivative problem for divergence form complex-valued higher-order elliptic systems on a half space and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one direction and have small mean oscillations in the orthogonal directions on each small ball. Our results are new even in the second-order case. The corresponding results for the Dirichlet problem were obtained recently in \cite{DK10}. \end{abstract}
\maketitle
\section{Introduction}
This paper is concerned with $L_p$ theory for higher-order elliptic systems in divergence form with {\em conormal} derivative boundary conditions. Our focus is to seek minimal regularity assumptions on the leading coefficients of elliptic systems defined on regular and irregular domains. The paper is a continuation of \cite{DK09_01,DK10}, where the authors considered higher-order systems in the whole space and on domains with {\em Dirichlet} boundary conditions.
There is a vast literature on $L_p$ theory for second-order and higher-order elliptic and parabolic equations/systems with {\em constant} or {\em uniformly continuous} coefficients. We refer the reader to the classical work \cite{A65,ADN64,Solo,LSU,Fried}. Concerning possibly discontinuous coefficients, a notable class is the set of bounded functions with vanishing mean oscillations (VMO). This class of coefficients was firstly introduced in \cite{CFL1,CFL2} in the case of second-order non-divergence form elliptic equations, and further considered by a number of authors in various contexts, including higher-order equations and systems; see, for instance, \cite{CFF,HHH,Mi06,PS3}.
Recently, in \cite{DK09_01,DK10} the authors studied the {\em Dirichlet} problem for higher-order elliptic and parabolic systems with possibly measurable coefficients. In \cite{DK09_01}, we established the $L_p$-solvability of both divergence and non-divergence form systems with coefficients (called $\text{VMO}_x$ coefficients in \cite{Krylov_2005}) having locally small mean oscillations with respect to the spatial variables, and measurable in the time variable in the parabolic case. While in \cite{DK10}, divergence form elliptic and parabolic systems of arbitrary order are considered in the whole space, on a half space, and on Reifenberg flat domains, with {\em variably partially BMO coefficients}. This class of coefficients was introduced in \cite{Krylov08} in the context of second-order non-divergence form elliptic equations in the whole space, and naturally appears in the homogenization of layered materials; see, for instance, \cite{CKV}. It was later considered by the authors of the present article in \cite{DK09_02,DK10} and by Byun and Wang in \cite{BW10}. Loosely speaking, on each cylinder (or ball in the elliptic case), the coefficients are allowed to be merely measurable in one spatial direction called the {\em measurable direction}, which may vary for different cylinders. It is also assumed that the coefficients have small mean oscillations in the orthogonal directions, and near the boundary the measurable direction is sufficiently close to the ``normal'' direction of the boundary. Note that the boundary of a Reifenberg flat domain is locally trapped in thin discs, which allows the boundary to have a fractal structure; cf. \eqref{eq3.49}. Thus the normal direction of the boundary may not be well defined for Reifenberg flat domains, so instead we take the normal direction of the top surface of these thin discs.
The proofs in \cite{DK09_01,DK10} are in the spirit of \cite{Krylov_2005} by N. V. Krylov, in which the author gave a unified approach of $L_p$ estimates for both divergence and non-divergence second-order elliptic and parabolic equations in the whole space with $\text{VMO}_x$ coefficients. One of the crucial steps in \cite{Krylov_2005} is to establish certain interior {\em mean oscillation estimates}\footnote{Also see relevant early work \cite{Iw83,DM93}.} of solutions to equations with ``simple'' coefficients, which are measurable functions of the time variable only. Then the estimates for equations with $\text{VMO}_x$ coefficients follow from the mean oscillation estimates combined with a perturbation argument. In this connection, we point out that in \cite{DK09_01,DK10} a great deal of efforts were made to derive boundary and interior mean oscillation estimates for solutions to higher-order systems. For systems in Reifenberg flat domains, we also used an idea in \cite{CaPe98}.
In this paper, we study a {\em conormal} derivative problem for elliptic operators in divergence form of order $2m$: \begin{equation} \label{eq0617_02}
\mathcal{L} \textit{\textbf{u}} :=\sum_{|\alpha|\le m,|\beta|\le m}D^\alpha(a_{\alpha\beta}D^\beta\textit{\textbf{u}}), \end{equation} where $\alpha$ and $\beta$ are $d$-dimensional multi-indices, $a_{\alpha\beta}=[a_{\alpha\beta}^{ij}(x)]_{i,j=1}^n$ are $n\times n$ complex matrix-valued functions, and $\textit{\textbf{u}}$ is a complex vector-valued function. For $\alpha=(\alpha_1,\ldots,\alpha_d)$, we use the notation $D^\alpha \textit{\textbf{u}} =D_1^{\alpha_1}\ldots D_d^{\alpha_d} \textit{\textbf{u}}$. All the coefficients are assumed to be bounded and measurable, and $\mathcal{L}$ is uniformly elliptic; cf. \eqref{eq11.28}. Consider the following elliptic system \begin{equation}
\label{eq9.15}
(-1)^m\mathcal{L} \textit{\textbf{u}}+\lambda \textit{\textbf{u}}=\sum_{|\alpha|\le m}D^\alpha\textit{\textbf{f}}_\alpha \end{equation} on a domain $\Omega$ in $\mathbb{R}^d$, where $\textit{\textbf{f}}_\alpha \in L_p(\Omega)$, $p\in (1,\infty)$, and $\lambda\ge 0$ is a constant. A function $\textit{\textbf{u}}\in W^m_p$ is said to be a weak solution to \eqref{eq9.15} on $\Omega$ with the conormal derivative boundary condition associated with $\textit{\textbf{f}}_\alpha$ (on $\partial \Omega$) if \begin{equation}
\label{eq3.02}
\int_{\Omega}\sum_{|\alpha|\le m,|\beta|\le m}(-1)^{m+|\alpha|} D^\alpha\phi \cdot a_{\alpha\beta}D^\beta\textit{\textbf{u}} +\lambda \phi\cdot \textit{\textbf{u}}\,dx
=\sum_{|\alpha|\le m}\int_{\Omega}(-1)^{|\alpha|}D^\alpha\phi\cdot \textit{\textbf{f}}_\alpha\,dx \end{equation} for any test function $\phi=(\phi^1,\phi^2,\ldots,\phi^n)\in W^m_q(\Omega)$, where $q=p/(p-1)$. We emphasize that the phrase ``associated with $\textit{\textbf{f}}_\alpha$'' is appended after ``the conormal derivative boundary condition'' because for different representations of the right-hand side of \eqref{eq9.15}, even if they are pointwise equal, the weak formulation \eqref{eq3.02} could still be different. In the sequel, we omit this phrase when there is no confusion. We note that the equation above can also be understood as $$
\int_{\Omega}\sum_{|\alpha|\le m,|\beta|\le m}(-1)^{m+|\alpha|} D^\alpha\phi \cdot a_{\alpha\beta}D^\beta\textit{\textbf{u}} +\lambda \phi\cdot \textit{\textbf{u}}\,dx=\textit{\textbf{F}}(\phi)\quad \forall \phi\in W^m_{q}(\Omega), $$ where $\textit{\textbf{F}}$ is a given vector-valued bounded linear functional on $W^m_{q}(\Omega)$. The main objective of the paper is to show the unique $W^m_p(\Omega)$-solvability of \eqref{eq9.15} on a half space or on a possibly unbounded Reifenberg domain with the same regularity conditions on the leading coefficients, that is, {\em variably partially BMO coefficients}, as those in \cite{DK10}. See Section \ref{sec082001} for the precise statements of the assumptions and main results.
Notably, our results are new even for second-order scalar equations. In the literature, an $L_p$ estimate for the conormal derivative problem can be found in \cite{BW05}, where the authors consider second-order divergence elliptic equations without lower-order terms and with coefficients small BMO with respect to all variables on bounded Reifenberg domains. The proof in \cite{BW05} contains a compactness argument, which does not apply to equations with coefficients measurable in some direction discussed in the current paper. For other results about the conormal derivative problem, we refer the reader to \cite{Lieb1} and \cite{Lieb2}.
We prove the main theorems by following the strategy in \cite{DK10}. First, for systems with homogeneous right-hand side and coefficients measurable in one direction, we estimate the H\"older norm of certain linear combinations of $D^m\textit{\textbf{u}}$ in the interior of the domain, as well as near the boundary if the boundary is flat and perpendicular to the measurable direction. Then by using the H\"{o}lder estimates, we proceed to establish mean oscillation estimates of solutions to elliptic systems. As is expected, the obstruction is in the boundary mean oscillation estimates, to which we give a more detailed account. Note that when obtaining mean oscillation estimates of solutions, even in the half space case we do not require the measurable direction to be exactly perpendicular to the boundary, but allow it to be sufficiently close to the normal direction. For the Dirichlet problem in \cite{DK10}, we used a delicate cut-off argument together with a generalized Hardy's inequality. However, this method no longer works for the conormal derivative problem as solutions do not vanish on the boundary. The key observation in this paper is Lemma \ref{lem4.2} which shows that if one modifies the right-hand side a little bit, then the function $\textit{\textbf{u}}$ itself still satisfies the system with the conormal derivative boundary condition on a subdomain with a flat boundary perpendicular to the measurable direction. This argument is also readily adapted to elliptic systems on Reifenberg flat domains with variably partially BMO coefficients.
The corresponding parabolic problem, however, seems to be still out of reach by the argument mentioned above. In fact, in the modified equation in Lemma \ref{lem4.2} there would be an extra term involving $\textit{\textbf{u}}_t$ on the right-hand side. At the time of this writing, it is not clear to us how to estimate this term.
The remaining part of the paper is organized as follows. We state the main theorems in the next section. Section \ref{sec_aux} contains some auxiliary results including $L_2$-estimates, interior and boundary H\"older estimates, and approximations of Reifenberg domains. In Section \ref{sec4} we establish the interior and boundary mean oscillation estimates and then prove the solvability of systems on a half space. Finally we deal with elliptic systems on a Reifenberg flat domain in Section \ref{Reifenberg}.
We finish the introduction by fixing some notation. By $\mathbb{R}^d$ we mean a $d$-dimensional Euclidean space, a point in $\mathbb{R}^d$ is denoted by $x=(x_1,\ldots,x_d)=(x_1,x')$, and $\{e_j\}_{j=1}^d$ is the standard basis of $\mathbb{R}^d$. Throughout the paper, $\Omega$ indicates an open set in $\mathbb{R}^d$. For vectors $\xi,\eta\in \mathbb{C}^n$, we denote $$ (\xi,\eta)=\sum_{i=1}^n \xi^i\overline{\eta^i}. $$ For a function $f$ defined on a subset $\mathcal{D}$ in $\mathbb{R}^{d}$, we set \begin{equation*} (f)_{\mathcal{D}} = \dashint_{\mathcal{D}}
f(x) \, dx= \frac{1}{|\mathcal{D}|} \int_{\mathcal{D}} f(x) \, dx, \end{equation*}
where $|\mathcal{D}|$ is the $d$-dimensional Lebesgue measure of $\mathcal{D}$. Denote \begin{align*} \mathbb{R}^d_+ &= \{(x_1,x') \in \mathbb{R}^d: x_1 > 0\},\\
B_r(x) &= \{ y \in \mathbb{R}^d: |x-y| < r\},\quad B'_r(x') = \{ y' \in
\mathbb{R}^{d-1}: |x'-y'| < r\},\\ B_r^+(x)&=B_r(x)\cap \mathbb{R}^d_+,\quad \Gamma_r(x)=B_r(x)\cap \partial\mathbb{R}^d_+,\quad \Omega_r(x)=B_r(x)\cap \Omega. \end{align*} For a domain $\Omega$ in $\mathbb{R}^d$, we define the solution spaces $W_p^m(\Omega)$ as follows: \begin{equation*} W_p^m(\Omega) =\{u\in L_p(\Omega): D^{\alpha}u \in L_p(\Omega), 1
\le |\alpha| \le m \}, \end{equation*} $$
\|u\|_{W_p^m(\Omega)} = \sum_{|\alpha|\le m}
\|D^{\alpha}u\|_{L_p(\Omega)}. $$ We denote $C_{\text{loc}}^{\infty}(\mathcal{D})$ to be the set of all infinitely differentiable functions on $\mathcal{D}$, and $C_0^{\infty}(\mathcal{D})$ the set of infinitely differentiable functions with compact support $\Subset \mathcal{D}$.
\section{Main results} \label{sec082001}
Throughout the paper, we assume that the $n \times n$ complex-valued coefficient matrices $a_{\alpha\beta}$ are measurable and bounded, and the leading coefficients
$a_{\alpha\beta}$, $|\alpha|=|\beta|=m$, satisfy an ellipticity condition. More precisely, we assume: \begin{enumerate}
\item There exists a constant $\delta \in (0,1)$ such that the leading coefficients $a_{\alpha\beta}$, $|\alpha|=|\beta|=m$, satisfy \begin{equation}
\label{eq11.28}
\delta |\xi|^2 \le
\sum_{|\alpha|=|\beta|=m}\Re(a_{\alpha\beta}(x) \xi_{\beta}, \xi_{\alpha}), \quad
|a_{\alpha\beta}| \le \delta^{-1} \end{equation} for any $ x \in \mathbb{R}^{d}$ and $\xi =
(\xi_{\alpha})_{|\alpha|=m}$, $\xi_{\alpha} \in \mathbb{C}^n$. Here we use $\Re(f)$ to denote the real part of $f$.
\item All the lower-order coefficients $a_{\alpha\beta}$,
$|\alpha| \ne m$ or $|\beta| \ne m$, are bounded by a constant $K\ge 1$. \end{enumerate}
We note that the ellipticity condition \eqref{eq11.28} can be relaxed. For instance, the operator $\mathcal{L}=D_1^4+D_2^4$ is allowed when $d=m=2$. See Remark 2.5 of \cite{DK10}.
Throughout the paper we write $\{\bar{a}_{\alpha\beta}\}_{|\alpha|=|\beta|=m} \in \mathbb{A}$ whenever the $n \times n$ complex-valued matrices $\bar{a}_{\alpha\beta}=\bar{a}_{\alpha\beta}(y_1)$ are measurable functions satisfying the condition \eqref{eq11.28}. For a linear map $\mathcal{T}$ from $\mathbb{R}^d$ to $\mathbb{R}^d$, we write $\mathcal{T} \in \mathbb{O}$ if $\mathcal{T}$ is of the form $$ \mathcal{T}(x) = \rho x + \xi, $$ where $\rho$ is a $d \times d$ orthogonal matrix and $\xi \in \mathbb{R}^d$.
Let $\mathcal{L}$ be the elliptic operator defined in \eqref{eq0617_02}. Our first result is about the conormal derivative problem on a half space. The following mild regularity assumption is imposed on the leading coefficients, with a parameter $\gamma \in (0,1/4)$ to be determined later. \begin{assumption}[$\gamma$] \label{assumption20100901} There is a constant $R_0\in (0,1]$ such that the following hold with $B:=B_r(x_0)$.
(i) For any $x_0\in \mathbb{R}^d_+$ and any $r\in \left(0,R_0\right]$ so that
$B\subset \mathbb{R}^{d}_+$, one can find $\mathcal{T}_B \in \mathbb{O}$ and coefficient matrices $\{\bar a_{\alpha\beta}\}_{|\alpha|=|\beta|=m} \in \mathbb{A}$ satisfying \begin{equation}
\label{eq10_23}
\sup_{|\alpha|=|\beta|=m}\int_B |a_{\alpha\beta}(x) -
\bar{a}_{\alpha\beta}(y_1)| \, dx \le \gamma |B|, \end{equation} where $y = \mathcal{T}_B (x)$.
(ii) For any $x_0\in \partial \mathbb{R}^d_+$ and any $r\in (0,R_0]$, one can find $\mathcal{T}_B \in \mathbb{O}$ satisfying $\rho_{11}\ge \cos (\gamma/2)$
and coefficient matrices $\{\bar a_{\alpha\beta}\}_{|\alpha|=|\beta|=m} \in \mathbb{A}$ satisfying \eqref{eq10_23}. \end{assumption}
The condition $\rho_{11}\ge \cos (\gamma/2)$ with a sufficiently small $\gamma$ means that at any boundary point the $y_1$-direction is sufficiently close to the $x_1$-direction, i.e., the normal direction of the boundary.
\begin{theorem}[Systems on a half space]
\label{thm3} Let $\Omega=\mathbb{R}^d_+$, $p \in (1,\infty)$, and $$
\textit{\textbf{f}}_\alpha= (f_\alpha^1, \ldots, f_\alpha^n)^{\text{tr}} \in L_p(\Omega), \quad |\alpha|\le m. $$ Then there exists a constant $\gamma=\gamma(d,n,m,p,\delta)$ such that, under Assumption \ref{assumption20100901} ($\gamma$), the following hold true.
\noindent (i) For any $\textit{\textbf{u}} \in W^m_p(\Omega)$ satisfying \begin{equation}
\label{eq1.55}
(-1)^m\mathcal{L} \textit{\textbf{u}} +\lambda \textit{\textbf{u}} = \sum_{|\alpha|\le m}D^\alpha \textit{\textbf{f}}_\alpha \end{equation} in $\Omega$ and the conormal derivative condition on $\partial\Omega$, we have \begin{equation*}
\sum_{|\alpha|\le m}\lambda^{1-\frac {|\alpha|} {2m}} \|D^\alpha
\textit{\textbf{u}} \|_{L_p(\Omega)} \le N \sum_{|\alpha|\le m}\lambda^{\frac
{|\alpha|} {2m}} \| \textit{\textbf{f}}_\alpha \|_{L_p(\Omega)}, \end{equation*} provided that $\lambda \ge \lambda_0$, where $N$ and $\lambda_0 \ge 0$ depend only on $d$, $n$, $m$, $p$, $\delta$, $K$ and $R_0$.
\noindent (ii) For any $\lambda > \lambda_0$, there exists a unique solution $\textit{\textbf{u}} \in W_p^m(\Omega)$ to \eqref{eq1.55} with the conormal derivative boundary condition.
\noindent (iii) If all the lower-order coefficients of $\mathcal{L}$ are zero and the leading coefficients are measurable functions of $x_1\in \mathbb{R}$ only, then one can take $\lambda_0=0$. \end{theorem}
For elliptic systems on a Reifenberg flat domain which is possibly unbounded, we impose a similar regularity assumption on $a_{\alpha\beta}$ as in Assumption \ref{assumption20100901}. Near the boundary, we require that in each small scale the direction in which the coefficients are only measurable coincides with the ``normal'' direction of a certain thin disc, which contains a portion of $\partial\Omega$. More precisely, we assume the following, where the parameter $\gamma\in (0,1/50)$ will be determined later. \begin{assumption}[$\gamma$]
\label{assump1} There is a constant $R_0\in (0,1]$ such that the following hold.
(i) For any $x\in \Omega$ and any $r\in (0,R_0]$ such that $B_r(x)\subset \Omega$, there is an orthogonal coordinate system depending on $x$ and $r$ such that in this new coordinate system, we have \begin{equation}
\label{eq13.07}
\dashint_{B_r(x)}\Big| a_{\alpha\beta}(y_1, y') -
\dashint_{B'_r(x')} a_{\alpha\beta}(y_1,z') \, dz' \Big| \, dy\le \gamma. \end{equation}
(ii) The domain $\Omega$ is Reifenberg flat: for any $x\in \partial\Omega$ and $r\in (0,R_0]$, there is an orthogonal coordinate system depending on $x$ and $r$ such that in this new coordinate system, we have \eqref{eq13.07} and \begin{equation}
\label{eq3.49}
\{(y_1,y'):x_1+\gamma r<y_1\}\cap B_r(x)
\subset\Omega_r(x)
\subset \{(y_1,y'):x_1-\gamma r<y_1\}\cap B_r(x). \end{equation} \end{assumption}
In particular, if the boundary $\partial \Omega$ is locally the graph of a Lipschitz continuous function with a small Lipschitz constant, then $\Omega$ is Reifenberg flat. Thus all $C^1$ domains are Reifenberg flat for any $\gamma>0$.
The next theorem is about the conormal derivative problem on a Reifenberg flat domain.
\begin{theorem}[Systems on a Reifenberg flat domain]
\label{thm5} Let $\Omega$ be a domain in $\mathbb{R}^d$ and $p \in (1,\infty)$. Then there exists a constant $\gamma=\gamma(d,n,m,p,\delta)$ such that, under Assumption \ref{assump1} ($\gamma$), the following hold true.
\noindent (i) Let $\textit{\textbf{f}}_\alpha= (f_\alpha^1, \ldots, f_\alpha^n)^{\text{tr}} \in L_p(\Omega)$, $|\alpha|\le m$. For any $\textit{\textbf{u}} \in W^m_p(\Omega)$ satisfying \begin{equation}
\label{eq11_01}
(-1)^m \mathcal{L} \textit{\textbf{u}} +\lambda \textit{\textbf{u}} = \sum_{|\alpha|\le m}D^\alpha \textit{\textbf{f}}_\alpha \quad \text{in}\quad \Omega \end{equation} with the conormal derivative condition on $\partial\Omega$, we have \begin{equation*}
\sum_{|\alpha|\le m}\lambda^{1-\frac {|\alpha|} {2m}} \|D^\alpha
\textit{\textbf{u}} \|_{L_p(\Omega)} \le N \sum_{|\alpha|\le m}\lambda^{\frac
{|\alpha|} {2m}} \| \textit{\textbf{f}}_\alpha \|_{L_p(\Omega)}, \end{equation*} provided that $\lambda \ge \lambda_0$, where $N$ and $\lambda_0 \ge 0$ depend only on $d$, $n$, $m$, $p$, $\delta$, $K$, and $R_0$.
\noindent (ii) For any $\lambda > \lambda_0$ and $\textit{\textbf{f}}_\alpha \in L_p(\Omega)$, $|\alpha|\le m$, there exists a unique solution $\textit{\textbf{u}} \in W_p^m(\Omega)$ to \eqref{eq11_01} with the conormal derivative boundary condition. \end{theorem}
For $\lambda=0$, we have the following solvability result for systems without lower-order terms on bounded domains.
\begin{corollary}
\label{cor7}
Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, and $p \in (1,\infty)$. Assume that $a_{\alpha\beta}\equiv 0$ for any $\alpha,\beta$ satisfying $|\alpha|+|\beta|<2m$. Then there exists a constant $\gamma=\gamma(d,n,m,p,\delta)$ such that, under Assumption
\ref{assump1} ($\gamma$), for any $\textit{\textbf{f}}_\alpha \in L_p(\Omega)$, $|\alpha|= m$, there exists a solution $\textit{\textbf{u}} \in W_p^m(\Omega)$ to \begin{equation}
\label{eq22.34}
(-1)^m \mathcal{L} \textit{\textbf{u}} = \sum_{|\alpha|= m}D^\alpha \textit{\textbf{f}}_\alpha \quad \text{in}\quad \Omega \end{equation} with the conormal derivative boundary condition, and $\textit{\textbf{u}}$ satisfies \begin{equation}
\label{eq23.08}
\|D^m \textit{\textbf{u}} \|_{L_p(\Omega)} \le N \sum_{|\alpha|= m}\|\textit{\textbf{f}}_\alpha
\|_{L_p(\Omega)}, \end{equation} where $N$ depends only on $d$, $n$, $m$, $p$, $\delta$, $K$,
$R_0$, and $|\Omega|$. Such a solution is unique up to a polynomial of order at most $m-1$. \end{corollary}
Finally, we present a result for second-order scalar elliptic equations in the form \begin{equation}
\label{eq1005_2} D_i(a_{ij}D_j u)+D_i(a_i u)+b_iD_i u+cu=\operatorname{div} g+f\quad \text{in}\,\,\Omega \end{equation} with the conormal derivative boundary condition. The result generalizes Theorem 5 of \cite{DongKim08a}, in which bounded Lipschitz domains with small Lipschitz constants are considered. It also extends the main result of \cite{BW05} to equations with lower-order terms and with leading coefficients in a more general class.
In the theorem below we assume that all the coefficients are bounded and measurable, and $a_{ij}$ satisfies \eqref{eq11.28} with $m=1$. As usual, we say that $D_i a_i+c\le 0$ in $\Omega$ holds in the weak sense if $$ \int_\Omega(-a_i D_i\phi+c\phi)\,dx\le 0 $$ for any nonnegative $\phi\in C_0^\infty(\Omega)$. By Assumption ($\text{H}$) we mean that $$ \int_\Omega(-a_i D_i\phi+c\phi)\,dx=0\quad \forall \phi\in C^\infty(\overline{\Omega}). $$ Similarly, Assumption ($\text{H}^*$) is satisfied if $$ \int_\Omega(b_i D_i\phi+c\phi)\,dx=0\quad \forall \phi\in C^\infty(\overline{\Omega}). $$
\begin{theorem}[Scalar equations on a bounded domain]
\label{thmB} Let $p\in (1,\infty)$ and $\Omega$ be a bounded domain. Assume $D_ia_i+c\le 0$ in $\Omega$ in the weak sense. Then there exists a constant $\gamma=\gamma(d,p,\delta)$ such that, under Assumption \ref{assump1} ($\gamma$), the following hold true.
\noindent (i) If Assumption ($\text{H}$) is satisfied, then for any $f$, $g = (g_1, \cdots, g_d) \in L_{p}(\Omega)$, the equation \eqref{eq1005_2} has a unique up to a constant solution $u\in W^1_p(\Omega)$ provided that Assumption ($\text{H}^*$) is also satisfied. Moreover, we have \begin{equation*}
\|Du\|_{L_p(\Omega)}\le N\|f\|_{L_p(\Omega)}+N\|g\|_{L_p(\Omega)}. \end{equation*}
\noindent (ii) If Assumption ($\text{H}$) is not satisfied, the solution is unique and we have \begin{equation*}
\|u\|_{W^1_p(\Omega)}\le N\|f\|_{L_p(\Omega)}+N\|g\|_{L_p(\Omega)}. \end{equation*} The constants $N$ are independent of $f$, $g$, and $u$. \end{theorem}
\section{Some auxiliary estimates} \label{sec_aux}
In this section we consider operators without lower-order terms. Denote $$
\mathcal{L}_0 \textit{\textbf{u}} = \sum_{|\alpha|=|\beta|=m}D^\alpha( a_{\alpha\beta} D^\beta \textit{\textbf{u}}). $$
\subsection{$L_2$-estimates} \label{sec3.1} The following $L_2$-estimate for elliptic operators in divergence form with measurable coefficients is classical. We give a sketched proof for the sake of completeness.
\begin{theorem} \label{theorem08061901} Let $\Omega = \mathbb{R}^d$ or $\mathbb{R}^d_+$. There exists $N = N(d,m,n, \delta)$ such that, for any $\lambda \ge 0$, \begin{equation}
\label{eq2010_01}
\sum_{|\alpha|\le m}\lambda^{1-\frac {|\alpha|} {2m}} \|D^\alpha \textit{\textbf{u}} \|_{L_2(\Omega)}
\le N \sum_{|\alpha|\le m}\lambda^{\frac {|\alpha|} {2m}} \| \textit{\textbf{f}}_\alpha \|_{L_2(\Omega)}, \end{equation}
provided that $\textit{\textbf{u}} \in W_2^m(\Omega)$ and $\textit{\textbf{f}}_\alpha \in L_2(\Omega)$, $|\alpha|\le m$, satisfy \begin{equation} \label{eq080501}
(-1)^m\mathcal{L}_0 \textit{\textbf{u}} + \lambda \textit{\textbf{u}} = \sum_{|\alpha|\le m}D^\alpha \textit{\textbf{f}}_\alpha \end{equation}
in $\Omega$ with the conormal derivative condition on $\partial\Omega$. Furthermore, for any $\lambda > 0$ and $\textit{\textbf{f}}_\alpha \in L_2(\Omega),|\alpha|\le m$, there exists a unique solution $\textit{\textbf{u}}\in W_2^m(\Omega)$ to the equation \eqref{eq080501} in $\Omega$ with the conormal derivative boundary condition. \end{theorem} \begin{proof} By the method of continuity and a standard density argument, it suffices to prove the estimate \eqref{eq2010_01} for $\textit{\textbf{u}} \in C^{\infty}(\overline{\Omega})\cap W_2^m(\Omega)$. From the equation, it follows that \begin{equation*}
\int_{\Omega} \left[ (D^{\alpha}\textit{\textbf{u}}, a_{\alpha\beta}D^{\beta}\textit{\textbf{u}})
+ \lambda |\textit{\textbf{u}}|^2 \right]\,dx
= \sum_{|\alpha|\le m} (-1)^{|\alpha|} \int_{\Omega} (D^{\alpha}\textit{\textbf{u}}, \textit{\textbf{f}}_{\alpha}) \, dx. \end{equation*} By the uniform ellipticity \eqref{eq11.28}, we get $$
\delta \int_{\Omega} |D^m\textit{\textbf{u}}|^2 \, dx \le \int_{\Omega} \Re(a_{\alpha\beta} D^\beta \textit{\textbf{u}}, D^\alpha \textit{\textbf{u}}) \, dx. $$ Hence, for any $\varepsilon>0$, \begin{align*}
&\delta \int_{\Omega} |D^m\textit{\textbf{u}}|^2 \, dx+ \lambda \int_{\Omega} |\textit{\textbf{u}}|^2 \, dx
\le \sum_{|\alpha|\le m}(-1)^{|\alpha|} \int_{\Omega}\Re(D^\alpha \textit{\textbf{u}}, \textit{\textbf{f}}_\alpha) \, dx\\
&\le \varepsilon \sum_{|\alpha|\le m}\lambda^{\frac {m-|\alpha|} m} \int_{\Omega} |D^\alpha \textit{\textbf{u}}|^2 \, dx + N\varepsilon^{-1} \sum_{|\alpha|\le m}\lambda^{-\frac {m-|\alpha|} m}\int_{\Omega} |\textit{\textbf{f}}_\alpha|^2 \, dx. \end{align*} To finish the proof, it suffices to use interpolation inequalities
and choose $\varepsilon$ sufficiently small depending on $\delta$, $d$, $m$, and $n$. \end{proof}
We say that a function $\textit{\textbf{u}}\in W_p(\Omega)$ satisfies \eqref{eq9.15} with the conormal derivative condition on $\Gamma\subset \partial\Omega$ if $u$ satisfies \eqref{eq3.02} for any $\phi\in W^m_q(\Omega)$ which is supported on $\Omega\cup \Gamma$.
By Theorem \ref{theorem08061901} and adapting the proofs of Lemmas 3.2 and 7.2 in \cite{DK10} to the conormal case, we have the following local $L_2$-estimate.
\begin{lemma}
\label{lem6.2} Let $0<r<R<\infty$. Assume $\textit{\textbf{u}}\in C_{\text{loc}}^\infty(\overline{\mathbb{R}^{d}_+})$ satisfies \begin{equation}
\label{eq2.54} \mathcal{L}_0 \textit{\textbf{u}}=0 \end{equation} in $B_{R}^+$ with the conormal derivative boundary condition on $\Gamma_R$. Then there exists a constant $N=N(d,m,n,\delta)$ such that for $j=1,\ldots,m$, $$
\|D^j\textit{\textbf{u}}\|_{L_2(B_r^+)}\leq N(R-r)^{-j}\|\textit{\textbf{u}}\|_{L_2(B_R^+)}. $$ \end{lemma}
\begin{corollary}
\label{cor6.3}
Let $0<r<R<\infty$ and $a_{\alpha\beta}=a_{\alpha\beta}(x_1)$, $|\alpha|=|\beta|=m$. Assume that $\textit{\textbf{u}}\in C_{\text{loc}}^\infty(\overline{\mathbb{R}^{d}_+})$ satisfies \eqref{eq2.54}
in $B_R^+$ with the conormal derivative boundary condition on $\Gamma_R$. Then for any multi-index $\theta$ satisfying $\theta_1\le m$ and $|\theta|\ge m$, we have \begin{equation*}
\|D^\theta\textit{\textbf{u}}\|_{L_2(B_r^+)}\le N\|D^m\textit{\textbf{u}}\|_{L_2(B_R^+)}, \end{equation*} where $N=N(d,m,n,\delta, R, r, \theta)$. \end{corollary} \begin{proof} It is easily seen that $D_{x'}^{mk} \textit{\textbf{u}},k=1,2,\ldots,$ also satisfies \eqref{eq2.54} with the conormal derivative boundary condition on $\Gamma_R$. Then by applying Lemma \ref{lem6.2} repeatedly, we obtain \begin{equation*}
\|D^mD^{mk}_{x'}\textit{\textbf{u}}\|_{L_2(B_{R'}^+)}\le N\|D^m \textit{\textbf{u}}\|_{L_2(B_{R}^+)}, \end{equation*} where $R'=(r+R)/2$. From this inequality and the interpolation inequality, we get the desired estimate. \end{proof}
By using a Sobolev-type inequality, we shall obtain from Corollary \ref{cor6.3} a H\"older estimate of all the $m$-th derivatives of $\textit{\textbf{u}}$ except $D^{\bar\alpha} \textit{\textbf{u}}$, where $\bar\alpha=me_1=(m,0,\ldots,0)$. To compensate this lack of regularity of $D^{\bar\alpha} \textit{\textbf{u}}$, we consider the quantity $$
\Theta:=\sum_{|\beta|=m}a_{\bar\alpha\beta}D^\beta \textit{\textbf{u}}. $$
We recall the following useful estimate proved in \cite[Corollary 4.4]{DK10}.
\begin{lemma}
\label{corA.2} Let $k\ge 1$ be an integer, $r\in (0,\infty)$, $p\in [1,\infty]$, $\mathcal{D}=[0,r]^{d}$, and $u(x)\in L_p(\mathcal{D})$. Assume that $D_1^k u=f_0+D_1f_1+\ldots+D_1^{k-1}f_{k-1}$ in $\mathcal{D}$, where $f_j\in L_p(\mathcal{D}),j=0,\ldots,k-1$. Then $D_1 u \in L_p(\mathcal{D})$ and \begin{equation*}
\|D_1 u\|_{L_p(\mathcal{D})}\le N\|u\|_{L_p(\mathcal{D})}+N\sum_{j=0}^{k-1}\|f_j\|_{L_p(\mathcal{D})}, \end{equation*} where $N=N(d,k, r)>0$. \end{lemma}
\begin{corollary}
\label{lem6.6} Let $0<r<R<\infty$ and $a_{\alpha\beta}=a_{\alpha\beta}(x_1)$. Assume $\textit{\textbf{u}}\in C_{\text{loc}}^\infty(\overline{\mathbb{R}^{d}_+})$ satisfies \eqref{eq2.54} in $B_R^+$ with the conormal derivative boundary condition on $\Gamma_R$. Then, for any nonnegative integer $j$, \begin{equation*}
\|D^j_{x'} \Theta\|_{L_2(B_r^+)}
+ \|D^j_{x'} D_1 \Theta \|_{L_2(B_r^+)}
\le N \|D^m\textit{\textbf{u}}\|_{L_2(B_R^+)}, \end{equation*} where $N=N(d, m,n,r,R,\delta, j)>0$. \end{corollary}
\begin{proof} Due to Corollary \ref{cor6.3} and the fact that $D_{x'}^j\textit{\textbf{u}}$ satisfies \eqref{eq2.54} with the conormal derivative boundary condition, it suffices to prove the desired inequality when $j = 0$ and $R$ is replaced by another $R'$ such that $r < R' < R$. Obviously, we have \begin{equation*}
\|\Theta \|_{L_2(B_r^+)}
\le N \|D^m\textit{\textbf{u}}\|_{L_2(B_r^+)}. \end{equation*} Thus we prove that, for $R'=(r+R)/2$, \begin{equation}
\label{eq0924}
\|D_1 \Theta \|_{L_2(B_r^+)}
\le N \|D^m\textit{\textbf{u}}\|_{L_2(B_{R'}^+)}. \end{equation} From \eqref{eq2.54}, in $B_R^+$ we have $$
D_1^m\Theta=-\sum_{\substack{|\alpha|=|\beta|=m\\\alpha_1<m}} D^\alpha(a_{\alpha\beta} D^\beta \textit{\textbf{u}})
=-\sum_{\substack{|\alpha|=|\beta|=m\\\alpha_1<m}}D_1^{\alpha_1} (a_{\alpha\beta}D_{x'}^{\alpha'}D^\beta \textit{\textbf{u}}). $$ Then the estimate \eqref{eq0924} follows from Lemma \ref{corA.2} with a covering argument and Corollary \ref{cor6.3}. The corollary is proved. \end{proof}
\subsection{H\"older estimates}
By using the $L_2$ estimates obtained in Section \ref{sec3.1}, in this section we shall derive several H\"older estimates of derivatives of $\textit{\textbf{u}}$. As usual, for $\mu\in (0,1)$ and a function $u$ defined on $\mathcal{D} \subset \mathbb{R}^{d} $, we denote $$ [u]_{C^{\mu}(\mathcal{D})}
= \sup_{\substack{x,y\in\mathcal{D}\\x\ne y}}\frac{|u(x)-u(y)|}
{|x-y|^{\mu}}, $$ $$
\|u\|_{C^{\mu}(\mathcal{D})}=[u]_{C^{\mu}(\mathcal{D})}+\|u\|_{L_{\infty}(\mathcal{D})}. $$
\begin{lemma}
\label{lem6.4} Let $a_{\alpha\beta}=a_{\alpha\beta}(x_1)$. Assume that $\textit{\textbf{u}}\in C_{\text{loc}}^\infty(\overline{\mathbb{R}^{d}_+})$ satisfies
\eqref{eq2.54} in $B_2^+$ with the conormal derivative boundary condition on $\Gamma_2$. Then for any $\alpha$ satisfying $|\alpha|=m$ and $\alpha_1<m$, we have \begin{equation*}
\|\Theta\|_{C^{1/2}(B_1^+)}+\|D^\alpha\textit{\textbf{u}}\|_{C^{1/2}(B_1^+)}
\le N \|D^m\textit{\textbf{u}}\|_{L_2(B_2^+)}, \end{equation*} where $N=N(d,m,n,\delta)>0$. \end{lemma} \begin{proof} The lemma follows from the proof of Lemma 4.1 in \cite{DK10} by using Corollaries \ref{cor6.3} and \ref{lem6.6}. \end{proof}
For $\lambda\ge0$, let \begin{equation*}
U=\sum_{|\alpha| \le m}\lambda^{\frac 1 2-\frac {|\alpha|}{2m}} |D^\alpha \textit{\textbf{u}}|,
\quad U'=\sum_{|\alpha|\le m,\alpha_1<m} \lambda^{\frac 1 2-\frac {|\alpha|} {2m}} |D^\alpha \textit{\textbf{u}}|. \end{equation*} Notably, since the matrix $[a_{\bar\alpha\bar\alpha}^{ij}]_{i,j=1}^n$ is positive definite, we have \begin{equation}
\label{eq12.01}
N^{-1}U\le U'+|\Theta|\le NU, \end{equation} where $N=N(d,m,n,\delta)$.
\begin{lemma}
\label{lem6.7} Let $a_{\alpha\beta}=a_{\alpha\beta}(x_1)$ and $\lambda\ge 0$. Assume that $\textit{\textbf{u}}\in C_{\text{loc}}^\infty(\overline{\mathbb{R}^{d}_+})$ satisfies \begin{equation*}
(-1)^m\mathcal{L}_0 \textit{\textbf{u}}+\lambda \textit{\textbf{u}}=0 \end{equation*} in $B_2^+$ with the conormal derivative condition on $\Gamma_2$. Then we have \begin{align}
\label{eq3.11}
\|\Theta\|_{C^{1/2}(B_1^+)}+\|U'\|_{C^{1/2}(B_1^+)}
\le N \|U\|_{L_2(B_2^+)},\\
\label{eq14.51}
\|U\|_{L_\infty(B_1^+)}\le N\|U\|_{L_2(B_2^+)}, \end{align} where $N=N(d,m,n,\delta)>0$. \end{lemma} \begin{proof} First we prove \eqref{eq3.11}. The case when $\lambda=0$ follows from Lemma \ref{lem6.4}. To deal with the case $\lambda>0$, we follow an idea by S. Agmon, which was originally used in a quite different situation. Let $ \eta(y)=\cos(\lambda^{1/(2m)}y)+\sin(\lambda^{1/(2m)}y) $ so that $\eta$ satisfies $$ D^{2m}\eta=(-1)^m\lambda \eta,\quad
\eta(0)=1,\quad |D^j\eta(0)|=\lambda^{j/(2m)},\,\,\,j=1,2,\ldots. $$ Let $z = (x,y)$ be a point in $\mathbb{R}^{d+1}$, where $x \in \mathbb{R}^{d}$, $y \in \mathbb{R}$, and $\hat{\textit{\textbf{u}}}(z)$ and $\hat{B}_r^+$ be given by $$ \hat{\textit{\textbf{u}}}(z) = \hat{\textit{\textbf{u}}}(x,y) = \textit{\textbf{u}}(x)\eta(y), \quad
\hat{B}_r^+ = \{ |z| < r: z \in \mathbb{R}^{d+1},x_1>0 \}. $$ Also define $$
\hat\Theta=\sum_{|\beta|=m}a_{\bar\alpha\beta}D^{(\beta,0)} \hat\textit{\textbf{u}}. $$ It is easily seen that $\hat{\textit{\textbf{u}}}$ satisfies $$ (-1)^m\mathcal{L}_0\hat{\textit{\textbf{u}}}+(-1)^mD^{2m}_y \hat{\textit{\textbf{u}}} = 0 $$ in $\hat{B}_2^+$ with the conormal derivative condition on $\hat{B}_2\cap \partial \mathbb{R}^{d+1}_+$. By Lemma \ref{lem6.4} applied to $\hat{\textit{\textbf{u}}}$ we have \begin{equation} \label{eq0804}
\|\hat\Theta\|_{C^{1/2}(\hat{B}_1^+)}+\big\| D_z^\beta\hat{\textit{\textbf{u}}}\big\|_{C^{1/2}(\hat{B}_1^+)}
\le N(d,m,n,\delta) \|D^m_z\hat{\textit{\textbf{u}}}\|_{L_2(\hat{B}_2^+)} \end{equation}
for any $\beta=(\beta_1,\ldots,\beta_{d+1})$ satisfying $|\beta|=m$ and $\beta_1<m$. Notice that for any $\alpha=(\alpha_1,\ldots,\alpha_d)$ satisfying $|\alpha|\le m$ and $\alpha_1<m$, $$
\lambda^{\frac 1 2-\frac {|\alpha|} {2m}} \big\|D^\alpha\textit{\textbf{u}}\big\|_{C^{1/2}(B_1^+)}
\le N\big\| D^\beta_z\hat{\textit{\textbf{u}}} \big\|_{C^{1/2}(\hat{B}_1^+)},\quad \beta=(\alpha_1,\ldots,\alpha_d,m-|\alpha|), $$ $$
\|\Theta\|_{C^{1/2}(B_1^+)}\le \|\hat\Theta\|_{C^{1/2}(\hat{B}_1^+)}, $$ and $D_z^m \hat{\textit{\textbf{u}}}$ is a linear combination of $$ \lambda^{\frac 1 2-\frac k {2m}}\cos( \lambda^{\frac 1 {2m}} y) D^k_x\textit{\textbf{u}}, \quad \lambda^{\frac 1 2-\frac k {2m}}\sin( \lambda^{\frac 1 {2m}} y) D^k_x\textit{\textbf{u}},\quad k=0,1,\ldots,m. $$ Thus the right-hand side of \eqref{eq0804} is less than the right-hand side of \eqref{eq3.11}. This completes the proof of \eqref{eq3.11}. Finally, we get \eqref{eq14.51} from \eqref{eq3.11} and \eqref{eq12.01}. \end{proof}
Similarly, we have the following interior estimate.
\begin{lemma}
\label{cor3.5} Let $a_{\alpha\beta}=a_{\alpha\beta}(x_1)$ and $\lambda\ge0$. Assume that $\textit{\textbf{u}}\in C_{\text{loc}}^\infty(\mathbb{R}^{d})$ satisfies \begin{equation*}
(-1)^m\mathcal{L}_0 \textit{\textbf{u}}+\lambda \textit{\textbf{u}}=0 \end{equation*} in $B_2$. Then we have \begin{align*}
\|\Theta\|_{C^{1/2}(B_1)}+\left\|U'\right\|_{C^{1/2}(B_1)}
\le N \|U\|_{L_2(B_2)},\\
\|U\|_{L_\infty(B_1)}\le N\|U\|_{L_2(B_2)}, \end{align*} where $N=N(d,m,n,\delta)>0$. \end{lemma}
\subsection{The maximal function theorem and a generalized Fefferman-Stein theorem} We recall the maximal function theorem and a generalized Fefferman-Stein theorem. Let $$ \mathcal{Q}=\{B_r(x): x \in \mathbb{R}^{d}, r \in (0,\infty)\}. $$ For a function $g$ defined in $\mathbb{R}^{d}$, the maximal function of $g$ is given by $$
\mathcal{M} g (x) = \sup_{B \in \mathcal{Q}, x \in B} \dashint_{B} |g(y)| \, dy. $$ By the Hardy--Littlewood maximal function theorem, $$
\| \mathcal{M} g \|_{L_p(\mathbb{R}^{d})} \le N \| g\|_{L_p(\mathbb{R}^{d})}, $$ if $g \in L_p(\mathbb{R}^{d})$, where $1 < p < \infty$ and $N = N(d,p)$.
Theorem \ref{th081201} below is from \cite{Krylov08} and can be considered as a generalized version of the Fefferman-Stein Theorem. To state the theorem, let $$ \mathbb{C}_l = \{ C_l(i_1, \ldots, i_d), i_1, \ldots, i_d \in \mathbb{Z}, i_1\ge 0 \}, \quad l \in \mathbb{Z} $$ be the collection of partitions given by dyadic cubes in $\mathbb{R}^{d}_+$ \begin{equation*}
[ i_1 2^{-l}, (i_1+1)2^{-l} ) \times \ldots \times [ i_d 2^{-l}, (i_d+1)2^{-l} ). \end{equation*}
\begin{theorem} \label{th081201}
Let $p \in (1, \infty)$, and $U,V,F\in L_{1,\text{loc}}(\mathbb{R}^{d}_+)$. Assume that we have $|U| \le V$ and, for each $l \in \mathbb{Z}$ and $C \in \mathbb{C}_l$, there exists a measurable function $U^C$ on $C$
such that $|U| \le U^C \le V$ on $C$ and \begin{equation*}
\int_C |U^C - \left(U^C\right)_C| \,dx \le \int_C F(x) \,dx. \end{equation*} Then $$
\| U \|_{L_p(\mathbb{R}^{d}_+)}^p
\le N(d,p) \|F\|_{L_p(\mathbb{R}^{d}_+)}\| V \|_{L_p(\mathbb{R}^{d}_+)}^{p-1}. $$ \end{theorem}
\subsection{Approximations of Reifenberg domains}
Let $\Omega$ be a domain in $\mathbb{R}^d$. Throughout this subsection, we assume that, for any $x\in \partial\Omega$ and $r\in (0,1]$, $\Omega$ satisfies \eqref{eq3.49} in an appropriate coordinate system. That is, $\Omega$ satisfies the following assumption with $\gamma<1/50$.
\begin{assumption}[$\gamma$]
\label{assump11} There is a constant $R_0\in (0,1]$ such that the following holds. For any $x\in \partial\Omega$ and $r\in (0,R_0]$, there is a coordinate system depending on $x$ and $r$ such that in this new coordinate system, we have \begin{equation}
\label{eq1005_1}
\{(y_1,y'):x_1+\gamma r<y_1\}\cap B_r(x)
\subset\Omega_r(x)
\subset \{(y_1,y'):x_1-\gamma r<y_1\}\cap B_r(x). \end{equation} \end{assumption}
For any $\varepsilon\in (0,1)$, we define \begin{equation}
\label{eq2.22}
\Omega^\varepsilon=\{x\in \Omega\,|\,\text{dist}(x,\partial\Omega)>\varepsilon\}. \end{equation}
We say that a domain is a Lipschitz domain if locally the boundary is the graph of a Lipschitz function in some coordinate system. More precisely,
\begin{assumption}[$\theta$]
\label{assump3} There is a constant $R_1\in (0,1]$ such that, for any $x\in \partial\Omega$ and $r\in(0,R_1]$, there exists a Lipschitz function $\phi$: $\mathbb{R}^{d-1}\to \mathbb{R}$ such that $$ \Omega\cap B_r(x_0) = \{x \in B_r(x_0)\, :\, x_1 >\phi(x')\} $$ and $$
\sup_{x',y'\in B_r'(x_0'),x' \neq y'}\frac {|\phi(y')-\phi(x')|}{|y'-x'|}\le \theta $$ in some coordinate system. \end{assumption}
We note that if $\Omega$ satisfies Assumption \ref{assump3} ($\theta$) with a constant $R_1$, then $\Omega$ satisfies Assumption \ref{assump11} with $R_1$ and $\theta$ in place of $R_0$ and $\gamma$, respectively.
Next we show that $\Omega^\varepsilon$ is a Lipschitz domain and Reifenberg flat with uniform parameters if $\Omega$ is Reifenberg flat. A related result was proved in \cite{BW07} which, in our opinion, contains a flaw.
\begin{lemma}
\label{lem3.11} Let $\Omega$ satisfy Assumption \ref{assump11} ($\gamma$). Then for any $\varepsilon\in (0,R_0/4)$, $\Omega^\varepsilon$ satisfies Assumption \ref{assump11} ($N_0\gamma^{1/2}$) with $R_0/2$ in place of $R_0$, and satisfies Assumption \ref{assump3} ($N_0\gamma^{1/2}$) with $R_1=\varepsilon$. Here $N_0$ is a universal constant. \end{lemma}
\begin{proof} We first prove that $\Omega^\varepsilon$ satisfies Assumption \ref{assump3} ($N_0\gamma^{1/2}$) with $R_1 = \varepsilon >0$. In particular, we show that, for each $x_0 \in \partial\Omega^{\varepsilon}$, there exists a function $\phi:\mathbb{R}^{d-1} \to \mathbb{R}$ such that \begin{equation}
\label{eq0930} \Omega^{\varepsilon} \cap B_{\varepsilon}(x_0) = \{ x \in B_{\varepsilon}(x_0) : x_1 > \phi(x') \}, \quad
\frac{|\phi(y') - \phi(x')|}{|x'-y'|} \le N_0 \gamma^{1/2} \end{equation} for all $x',y' \in B'_{\varepsilon}(x_0')$, $x' \ne y'$. Indeed, this implies Assumption \ref{assump3} ($N_0\gamma^{1/2}$) since for a fixed $x_0 \in \partial \Omega^\varepsilon$ we can use the same $\phi$ for all $r \in (0,\varepsilon)$.
Let $0$ be a point on $\partial \Omega$ such that $|x_0 - 0| = \varepsilon$. That is, we have a coordinate system and $r_0 := 4 \varepsilon < R_0$ such that $\partial \Omega \cap B_{r_0}(0)$ is trapped between $\{x_1 = \gamma r_0\}$ and $\{x_1 = - \gamma r_0\}$. See Figure \ref{fg3}. Note that $B_{\varepsilon}(x_0) \subset B_{r_0}(0)$ since, for $x \in B_{\varepsilon}(x_0)$, $$
|x| \le |x-x_0| + |x_0| < 2\varepsilon < r_0 = 4\varepsilon. $$ We show that for any $y, z \in \partial \Omega^\varepsilon \cap B_{\varepsilon}(x_0)$ \begin{equation}
\label{eq1003_1}
|y_1 - z_1| \le N_0 \gamma^{1/2} |y'-z'|, \end{equation} which implies \eqref{eq0930}. For $y, z \in \partial \Omega^\varepsilon \cap B_{\varepsilon}(x_0)$, we see that \begin{equation}
\label{eq1003_2} \varepsilon - \gamma r_0 < y_1 < \varepsilon + \gamma r_0, \quad \varepsilon - \gamma r_0 < z_1 < \varepsilon + \gamma r_0. \end{equation} Without loss of generality we assume that $y_1 \ge z_1$. \begin{figure}\label{fg3}
\end{figure}
To prove \eqref{eq1003_1}, let us consider two cases. First, let $\varepsilon \gamma^{1/2} \le |y'-z'|$. In this case, due to the inequalities \eqref{eq1003_2}, we have $$
\frac{|y_1-z_1|}{|y'-z'|} \le \frac{2\gamma r_0}{\varepsilon \gamma^{1/2}} = 8 \gamma^{1/2}, $$ which proves \eqref{eq1003_1}.
Now let $|y'-z'| \le \varepsilon \gamma^{1/2}$. In this case, find $w \in \partial \Omega$ such that $|y-w| = \varepsilon$. Note that $B_\varepsilon(w) \subset B_{r_0}(0)$ since $$
|w| \le |w-y| + |y-x_0| + |x_0| < 3 \varepsilon < r_0 = 4\varepsilon. $$
We estimate $|w'-z'|$ as follows. Using the fact that $-\gamma r_0 < w_1 < \gamma r_0$ and the first inequality in \eqref{eq1003_2}, we have $$
|y_1-w_1| \ge \varepsilon - 2 \gamma r_0 >0. $$ Thus using the equality $$
|w'-y'|^2 + |w_1-y_1|^2 = \varepsilon^2, $$ we see that $$
|w'-y'|^2 \le \varepsilon^2 - (\varepsilon - 2\gamma r_0)^2 \le 4 \varepsilon \gamma r_0 = 4^2 \varepsilon^2 \gamma. $$ Hence $$
|w'-z'| \le |w'-y'| + |y'-z'| < 5\varepsilon \gamma^{1/2}. $$
Since $y_1 \ge z_1$, $|w'-z'| \le 5\varepsilon \gamma^{1/2}$, and $z$ is above the ball $B_\varepsilon(\omega)$ (recall that $\gamma<1/50$), it follows that $$
\frac{y_1 - z_1}{|y'-z'|}
\le \frac{d}{dx} \left(-\sqrt{\varepsilon^2 - x^2}\right) \bigg|_{x=5 \varepsilon \gamma^{1/2}} \le N_0 \gamma^{1/2}. $$ Thus \eqref{eq1003_1} is proved. Therefore, we have proved that $\Omega^\varepsilon$ satisfies Assumption \ref{assump3} ($N_0\gamma^{1/2}$) with $R_1 = \varepsilon$. As pointed out earlier, this shows that $\Omega^\varepsilon$ satisfies \eqref{eq1005_1} for all $0 < r < \varepsilon$. Thus in order to completely prove that $\Omega^\varepsilon$ satisfies Assumption \ref{assump11} ($N_0\gamma^{1/2}$) with $R_0/2$, we need to prove that $\Omega^\varepsilon$ satisfies \eqref{eq1005_1} for $\varepsilon \le r < R_0/2$.
Let $\varepsilon \le r < R_0/2$ and $x_0 \in \partial\Omega^{\varepsilon}$. Find $0 \in \partial\Omega$
such that $|x_0 - 0| = \varepsilon$. Then $$ B_r(x_0) \subset B_R(0), $$ where $R = \varepsilon+r < R_0$. Then the first coordinate $x_1$ of the point $x \in \partial \Omega^{\varepsilon} \cap B_r(x_0)$ is trapped by $$ \varepsilon - \gamma R < x_1 < \varepsilon + \gamma R, $$ which is the same as $$ \varepsilon - \gamma ( \varepsilon+r) < x_1 < \varepsilon + \gamma ( \varepsilon+r). $$ Note that $$ \gamma (\varepsilon + r) \le 2 \gamma r \le 2 \gamma^{1/2}r. $$ Thus each $x_1$ of $x \in \partial\Omega^\varepsilon \cap B_r(x_0)$ satisfies \eqref{eq1005_1} with $2\gamma^{1/2}$ in place of $\gamma$. The lemma is proved. \end{proof}
The next approximation result is well known. See, for instance, \cite{Lie85}.
\begin{lemma}
\label{lem3.19} Let $\Omega$ be a domain in $\mathbb{R}^d$ and satisfy Assumption \ref{assump3} ($\theta$) with some $\theta>0$ and $R_1\in (0,1]$. Then there exists a sequence of expanding smooth subdomains $\Omega^k,k=1,2,\ldots$, such that $\Omega^k\to \Omega$ as $k\to \infty$ and each $\Omega^k$ satisfies Assumption \ref{assump3} ($N_0\theta$) with $R_1/2$ in place of $R_1$. Here $N_0$ is a universal constant. \end{lemma}
\section{Systems on a half space}
\label{sec4}
\subsection{Estimates of mean oscillations} Now we prove the following estimate of mean oscillations. As in Section \ref{sec_aux}, we assume that all the lower-order coefficients of $\mathcal{L}$ are zero. For $\textit{\textbf{f}}_\alpha= (f_\alpha^1, \ldots, f_\alpha^n)^{\text{tr}}$, we denote $$
F=\sum_{|\alpha|\le m}\lambda^{\frac {|\alpha|} {2m}-\frac 1 2}|\textit{\textbf{f}}_\alpha|. $$
\begin{proposition}
\label{prop7.9} Let $x_0\in \overline{\mathbb{R}^d_+}$, $\gamma \in (0,1/4)$, $r\in (0,\infty)$, $\kappa\in [64,\infty)$, $\lambda\ge 0$, $\nu \in (2,\infty)$, $\nu'=2\nu/(\nu-2)$, and $\textit{\textbf{f}}_\alpha= (f_\alpha^1, \ldots, f_\alpha^n)^{\text{tr}} \in L_{2,\text{loc}}(\overline{\mathbb{R}^{d}_+})$. Assume that $\kappa r\le R_0$ and $\textit{\textbf{u}}\in W_{\nu,\text{loc}}^m(\overline{\mathbb{R}^d_+})$ satisfies \begin{equation}
\label{eq11.13}
(-1)^m\mathcal{L} \textit{\textbf{u}}+\lambda \textit{\textbf{u}}=\sum_{|\alpha|\le m}D^\alpha \textit{\textbf{f}}_\alpha \end{equation} in $B^+_{\kappa r}(x_0)$ with the conormal derivative condition on $\Gamma_{\kappa r}(x_0)$. Then under Assumption \ref{assumption20100901} ($\gamma$), there exists a function $U^B$ depending on $B^+:=B^+_{\kappa r}(x_0)$ such that $N^{-1}U\le U^B\le NU$ and $$
\big(|U^B-(U^B)_{B_r^+(x_0)}|\big)_{B_r^+(x_0)} \le N(\kappa^{-1/2}+(\kappa \gamma)^{1/2} \kappa^{d/2}) \big(U^2\big)_{B^+_{\kappa r}(x_0)}^{1/2} $$ \begin{equation}
\label{eq5.42} +N\kappa^{d/2}\left[(F^2)_{B^+_{\kappa r}(x_0)}^{1/2}+\gamma^{1/\nu'} (U^\nu)_{B^+_{\kappa r}(x_0)}^{1/\nu}\right], \end{equation} where $N=N(d,m,n,\delta,\nu)>0$. \end{proposition}
The proof of the proposition is split into two cases.
{\em Case 1: the first coordinate of $x_0$ $\ge \kappa r/16$.} In this case, we have $$ B_r^+(x_0)=B_r(x_0)\subset B_{\kappa r/16}(x_0)\subset \mathbb{R}^{d}_+. $$ With $B_{\kappa r/16}$ in place of $B^+_{\kappa r}$ in the right-hand side of \eqref{eq5.42}, the problem is reduced to an interior mean oscillation estimate. Thus the proof can be done in the same way as in Proposition 7.10 in \cite{DK10} using Theorem \ref{theorem08061901} and Lemma \ref{cor3.5}.
{\em Case 2: $0\le$ the first coordinate of $x_0$ $< \kappa r/16$.} Notice that in this case, \begin{equation}
\label{eq22.15} B_r^+(x_0)\subset B^+_{\kappa r/8}(\hat x_0) \subset B^+_{\kappa r/4}(\hat x_0)\subset B^+_{\kappa r/2}(\hat x_0) \subset B^+_{\kappa r}(x_0), \end{equation} where $\hat x_0:=(0,x_0')$. Denote $R=\kappa r/2(< R_0)$. Because of Assumption \ref{assumption20100901}, after a linear transformation, which is an orthogonal transformation determined by $B=B_R(\hat x_0)$ followed by a translation downward, we may assume \begin{equation}
\label{eq17.29} B_R^+(\hat{y}_0)
\subset\Omega_{R}(\hat{y}_0)
\subset \{(y_1,y'):-2\gamma R< y_1\}\cap B_{R}(\hat{y}_0) \end{equation} and \begin{equation}
\label{eq17_50}
\sup_{|\alpha|=|\beta|=m}\int_{B_R(\hat{y}_0)} |a_{\alpha\beta}(x) - \bar{a}_{\alpha\beta}(y_1)| \, dy \le \gamma |B_R|. \end{equation} \begin{figure}
\caption{$\hat y_0 $ (or $y_0$) is the new coordinates of $\hat x_0$ (or $x_0$).}
\label{fg1}
\end{figure} Here $\Omega$ is the image of $\mathbb{R}^d_+$ under the linear transformation and $\hat y_0 $ (or $y_0$) is the new coordinates of $\hat x_0$ (or $x_0$). See Figure \ref{fg1}. Then \eqref{eq22.15} becomes \begin{equation}
\label{eq22.16} \Omega_r(y_0)\subset \Omega_{R/4}(\hat y_0) \subset \Omega_{R/2}(\hat y_0) \subset \Omega_{R}(\hat y_0) \subset \Omega_{\kappa r}(y_0). \end{equation}
For convenience of notation, in the new coordinate system we still denote the corresponding unknown function, the coefficients, and the data by $\textit{\textbf{u}}$, $a_{\alpha\beta}$, $\bar{a}_{\alpha\beta}$, and $\textit{\textbf{f}}_\alpha$, respectively. Note that, without loss of generality, we may assume that the coefficients $\bar{a}_{\alpha\beta}(y_1)$ in \eqref{eq17_50} are infinitely differentiable.
Below we present a few lemmas, which should be read as parts of the proof of the second case.
Let us introduce the following well-known extension operator. Let $\{c_1, \cdots, c_{m}\}$ be the solution to the system: \begin{equation} \label{eq0514-02} \sum_{k=1}^{m} \left(-\frac{1}{k}\right)^j c_k = 1, \quad j=0,\cdots,m-1. \end{equation} For a function $w$ defined on $\mathbb{R}^d_+$, set \begin{equation*}
\mathcal{E}_{m} w = \left\{ \begin{aligned} &w(y_1,y') \quad \text{if} \quad y_1 > 0\\ &\sum_{k=1}^{m} c_k w(-\frac{1}{k}y_1,y') \quad \text{otherwise} \end{aligned} \right.. \end{equation*} Note that $\mathcal{E}_{m} w \in W^{m}_{2,\text{loc}}(\mathbb{R}^d)$ if $w \in W^{m}_{2,\text{loc}}(\overline{\mathbb{R}^d_+})$. Indeed, by \eqref{eq0514-02} $$
D_1^j \left(\sum_{k=1}^{m} c_k w(-\frac{1}{k}y_1,y')\right)\bigg|_{y_1=0} = \sum_{k=1}^{m} \left(-\frac1k\right)^j c_k D_1^jw(0,y') = D_1^jw(0,y') $$ for $j = 0, \cdots, m-1$.
Denote $\Omega^*=\mathbb{R}^d_-\cap\Omega\cap B_R(\hat y_0)$. Recall that in the new coordinate system we still denote the corresponding unknown function, the coefficients, and the data by $\textit{\textbf{u}}$, $a_{\alpha\beta}$, $\bar{a}_{\alpha\beta}$, and $\textit{\textbf{f}}_\alpha$, respectively. Throughout the end of this subsection, the derivatives are taken with respect to the $y$-coordinates. The following lemma contains the key observation in the proof of Proposition \ref{prop7.9}.
\begin{lemma}
\label{lem4.2} The function $\textit{\textbf{u}}$ satisfies \begin{align}
\label{eq17.23b}
(-1)^m \mathcal{L}_0 \textit{\textbf{u}}+\lambda \textit{\textbf{u}}&=(-1)^m \sum_{|\alpha|=|\beta|=m}D^\alpha\left( (\bar a_{\alpha\beta}- a_{\alpha\beta})D^\beta \textit{\textbf{u}}\right)\\
&\quad + \sum_{|\alpha|\le m} D^\alpha \tilde \textit{\textbf{f}}_\alpha+\sum_{|\alpha|=m}D^\alpha \textit{\textbf{g}}_\alpha-\lambda \textit{\textbf{h}}\nonumber \end{align} in $B_R^+(\hat y_0)$ with the conormal derivative boundary condition on $\Gamma_R(\hat y_0)$. In the above, $\mathcal{L}_0$ is the differential operator with the coefficients $\bar a_{\alpha\beta}$ from \eqref{eq17_50}, and \begin{align*} \tilde \textit{\textbf{f}}_\alpha&=\textit{\textbf{f}}_\alpha+c_{\alpha,k} \textit{\textbf{f}}_\alpha(-ky_1,y')\,1_{(-ky_1,y')\in \Omega^*},\\
\textit{\textbf{g}}_\alpha&=c_{\alpha,k}(-1)^{m+1}\sum_{|\beta|=m}\sum_{k=1}^m a_{\alpha\beta}(-ky_1,y') (D^{\beta}\textit{\textbf{u}})(-ky_1,y')\,1_{(-ky_1,y')\in \Omega^*},\\ \textit{\textbf{h}}&=\sum_{k=1}^m kc_k\textit{\textbf{u}}(-ky_1,y')\, 1_{(-ky_1,y')\in \Omega^*}, \end{align*} where $c_{\alpha,k}=(-1)^{\alpha_1}c_kk^{-\alpha_1+1}$ are constants. \end{lemma} \begin{proof} Take a test function $\phi=(\phi^1,\phi^2,\ldots,\phi^n)\in W^m_2(B_R^+(\hat y_0))$ which vanishes near $\mathbb{R}^d_+\cap \partial B_R(\hat y_0)$. Due to \eqref{eq17.29}, it is easily seen that $\mathcal{E}_m \phi\in W^m_2(\Omega_R(\hat y_0))$ and vanishes near $\Omega\cap \partial B_R(\hat y_0)$. Since $\textit{\textbf{u}}$ satisfies \eqref{eq11.13} with the conormal derivative condition on $\partial\Omega\cap B_R(\hat y_0)$, we have $$
\int_{\Omega_R(\hat y_0)}\sum_{|\alpha|=|\beta|=m}D^\alpha\mathcal{E}_m\phi \cdot a_{\alpha\beta}D^\beta\textit{\textbf{u}} +\lambda \mathcal{E}_m\phi\cdot \textit{\textbf{u}}\,dy $$ $$
=\sum_{|\alpha|\le m}\int_{\Omega_R(\hat y_0)}(-1)^{|\alpha|}D^\alpha\mathcal{E}_m\phi\cdot \textit{\textbf{f}}_\alpha\, dy. $$ From this identity and the definition of the extension operator $\mathcal{E}_m$, a straightforward calculation gives $$
\int_{B_R^+(\hat y_0)}\sum_{|\alpha|=|\beta|=m}D^\alpha\phi \cdot \bar a_{\alpha\beta}D^\beta\textit{\textbf{u}} +\lambda \phi\cdot \textit{\textbf{u}}\,dy $$ $$
=\sum_{|\alpha|=|\beta|=m} \int_{B_R^+(\hat y_0)}D^\alpha\phi \cdot (\bar a_{\alpha\beta}-a_{\alpha\beta})D^\beta\textit{\textbf{u}} \, dy
+\sum_{|\alpha|\le m}\int_{B_R^+(\hat y_0)}(-1)^{|\alpha|}D^\alpha\phi\cdot \tilde\textit{\textbf{f}}_\alpha\,dy $$ $$
=\sum_{|\alpha|=m} \int_{B_R^+(\hat y_0)}D^\alpha\phi \cdot
(-1)^{|\alpha|}\textit{\textbf{g}}_\alpha \,dy -\lambda \int_{B_R^+(\hat y_0)}\phi\cdot\textit{\textbf{h}}\,dy. $$ The lemma is proved. \end{proof}
Set $$
G_\alpha = (-1)^m \sum_{|\beta|=m} \left(\bar a_{\alpha\beta} - a_{\alpha\beta}\right)D^\beta u + \tilde{\textit{\textbf{f}}}_\alpha + \textit{\textbf{g}}_\alpha \quad \text{for} \quad
|\alpha|=m, $$ $$ G_\alpha = \tilde{\textit{\textbf{f}}}_\alpha \quad \text{for}
\quad 0\le |\alpha| < m. $$ We see that $G_\alpha \in L_2(B_R^+(\hat{y}_0))$, and by \eqref{eq17.23b} $$
(-1)^m \mathcal{L}_0 \textit{\textbf{u}} + \lambda \textit{\textbf{u}} = \sum_{|\alpha| \le m} D^{\alpha} G_\alpha - \lambda \textit{\textbf{h}}. $$ Take $\varphi$ to be an infinitely differentiable function such that $$ 0 \le \varphi \le 1, \quad \varphi = 1 \,\, \text{on} \,\, B_{R/2}(\hat y_0), \quad \varphi = 0 \,\, \text{outside} \,\, B_R(\hat y_0). $$ Then we find a unique solution $\textit{\textbf{w}} \in W_2^m(\mathbb{R}^d_+)$ satisfying \begin{equation}
\label{eq08_01}
(-1)^m \mathcal{L}_0 \textit{\textbf{w}} + \lambda \textit{\textbf{w}} = \sum_{|\alpha| \le m} D^{\alpha} (\varphi G_\alpha) - \lambda \varphi\textit{\textbf{h}} \end{equation} with the conormal derivative condition on $\partial \mathbb{R}^d_+$. By Theorem \ref{theorem08061901} we have \begin{equation}
\label{eq08_02}
\sum_{|\alpha|\le m}\lambda^{\frac12-\frac {|\alpha|} {2m}} \|D^\alpha \textit{\textbf{w}} \|_{L_2(\mathbb{R}^d_+)}
\le N \sum_{|\alpha|\le m}\lambda^{\frac {|\alpha|} {2m}-\frac12} \|\varphi G_\alpha\|_{L_2(\mathbb{R}^d_+)}
+ N \lambda \|\varphi \textit{\textbf{h}}\|_{L_2(\mathbb{R}^d_+)}. \end{equation}
Now we set $\textit{\textbf{v}} := \textit{\textbf{u}}-\textit{\textbf{w}}$ in $B_R(\hat y_0) \cap \overline{\mathbb{R}^d_+}$. Then $\textit{\textbf{v}}$ satisfies \begin{equation}
\label{eq08_03} (-1)^m \mathcal{L}_0 \textit{\textbf{v}} + \lambda \textit{\textbf{v}} = 0 \end{equation} in $B_{R/2}^+(\hat y_0)$ with the conormal derivative condition on $\Gamma_{R/2}(\hat y_0)$. Since the coefficients of $\mathcal{L}_0$ are infinitely differentiable, by the classical theory $\textit{\textbf{v}}$ is infinitely differentiable in $B_{R/2}(\hat{y}_0) \cap \overline{\mathbb{R}^d_+}$.
Recall that $$
U=\sum_{|\alpha| \le m}\lambda^{\frac 1 2-\frac {|\alpha|}{2m}} |D^\alpha \textit{\textbf{u}}|,
\quad F=\sum_{|\alpha|\le m}\lambda^{\frac {|\alpha|} {2m}-\frac 1 2}|\textit{\textbf{f}}_\alpha|. $$
\begin{lemma} We have \begin{equation}
\label{eq21.52h}
\sum_{k=0}^m\lambda^{\frac 1 2-\frac k {2m}}(|D^k \textit{\textbf{w}}|^2)_{B_R^+(\hat y_0)}^{1/2} \le N\gamma^{1/ {\nu'}} (U^\nu)_{\Omega_R(\hat y_0)}^{1/ \nu}+ N(F^2)_{\Omega_R(\hat y_0)}^{1/2}, \end{equation} where $\nu$ and $\nu'$ are from Proposition \ref{prop7.9}. \end{lemma}
\begin{proof} By \eqref{eq08_02} and the definition of $G_\alpha$, we have \begin{multline*}
\sum_{|\alpha|\le m}\lambda^{\frac 1 2-\frac {|\alpha|} {2m}}\|D^{\alpha} \textit{\textbf{w}}\|_{L_2(\mathbb{R}^d_+)}
\le N \sum_{|\alpha|=|\beta|=m}\|\varphi(\bar a_{\alpha\beta}- a_{\alpha\beta})D^\beta \textit{\textbf{u}}\|_{L_2(\mathbb{R}^d_+)} \\
+N\sum_{|\alpha|\le m}\lambda^{\frac{|\alpha|}{2m}-\frac 1 2} \|\varphi\tilde \textit{\textbf{f}}_\alpha\|_{L_2(\mathbb{R}^d_+)}
+N \sum_{|\beta|=m}\|\varphi \textit{\textbf{g}}_\alpha\|_{L_2(\mathbb{R}^d_+)}
+N \lambda \|\varphi \textit{\textbf{h}}\|_{L_2(\mathbb{R}^d_+)}. \end{multline*}
Note that $\Omega^*$ lies in the strip $B_R(\hat y_0)\cap \{y: -2\gamma R<y_1<0\}$. Thus, by the definitions of $\tilde \textit{\textbf{f}}_\alpha$, $\textit{\textbf{g}}_\alpha$, and $\textit{\textbf{h}}$, it follows that the left-hand side of \eqref{eq21.52h} is less than a constant times \begin{align*}
\sum_{|\alpha|=|\beta|=m}&\left(|(\bar a_{\alpha\beta}- a_{\alpha\beta})D^\beta \textit{\textbf{u}}|^2\right)^{1/2}_{B_R^+(\hat y_0)}
+ \sum_{|\alpha|\le m}\lambda^{\frac{|\alpha|}{2m}-\frac 1 2} \left(|\textit{\textbf{f}}_\alpha|^2\right)^{1/2}_{\Omega_R(\hat y_0)}\\
&+ \left(I_{\{-2\gamma R<y_1<0\}}|D^{m}\textit{\textbf{u}}|^2\right)^{1/2}_{\Omega_R(\hat y_0)}+\lambda
\left(I_{\{-2\gamma R<y_1<0\}}|\textit{\textbf{u}}|^2\right)^{1/2}_{\Omega_R(\hat y_0)}\\ &:=I_1+I_2+I_3+I_4. \end{align*} By using \eqref{eq17_50} and H\"older's inequality, we see that $$ I_1\le N\gamma^{1/ {\nu'}} (U^\nu)_{\Omega_R(\hat y_0)}^{1/ \nu}. $$ It is clear that $I_2$ is bounded by $N(F^2)_{\Omega_R(\hat y_0)}^{1/2}$. Observe that by H\"older's inequality we have \begin{align*} \left(I_{\{-2\gamma R < y_1 < 0\}}
|D^{m}\textit{\textbf{u}}|^2\right)^{1/2}_{\Omega_R(\hat y_0)} &\le \left(I_{\{-2\gamma R < y_1 < 0\}}\right)_{\Omega_R(\hat y_0)}^{1/\nu'}
\left(|D^m\textit{\textbf{u}}|^\nu\right)^{1/\nu}_{\Omega_R(\hat y_0)} \\
&\le N\gamma^{1/\nu'}(|D^m\textit{\textbf{u}}|^{\nu})_{\Omega_R(\hat y_0)}^{1/\nu}. \end{align*} Thus $I_3$ is also bounded by $N\gamma^{1/ {\nu'}} (U^\nu)_{\Omega_R(\hat y_0)}^{1/ \nu}$. In a similar way, $I_4$ is bounded by $N\gamma^{1/ {\nu'}} (U^\nu)_{\Omega_R(\hat y_0)}^{1/ \nu}$. Therefore, we conclude \eqref{eq21.52h}. \end{proof}
Now we are ready to complete the proof of Proposition \ref{prop7.9}. We shall show that $U^B:=U'+|\Theta|$ satisfies the inequalities in the proposition. First, we consider the case when $\kappa \gamma \le 1/10$. By \eqref{eq21.52h}, it follows that \begin{equation}
\label{eq18.34h} (W^2)_{B_{R}^+(\hat y_0)}^{1/2} \le N\gamma^{1/ {\nu'}} (U^\nu)_{\Omega_{R}(\hat y_0)}^{1/ \nu}+ N(F^2)_{\Omega_{R}(\hat y_0)}^{1/2}. \end{equation}
Noting that $B_r^+(y_0) \subset B_R^+(\hat y_0)$ and, since $\kappa \gamma\le 1/10$, $|B_R^+(\hat y_0)|/|B_r^+(y_0)| \le N(d) \kappa^{d}$, we obtain from \eqref{eq18.34h} \begin{equation}
\label{eq28_01} (W^2)_{B_r^+(y_0)}^{1/2} \le N\kappa^{\frac d 2} \left(\gamma^{1/ {\nu'}} (U^\nu)_{\Omega_{R}(\hat y_0)}^{1/ \nu}+ (F^2)_{\Omega_{R}(\hat y_0)}^{1/2}\right). \end{equation} Next we denote $$ \mathcal{D}_1=\Omega_{r}(y_0)\cap \{y_1<0\}=\Omega^*, \quad \mathcal{D}_2=B_{r}^+(y_0), \quad \mathcal{D}_3=B_{R/4}^+(\hat y_0). $$ Since $\textit{\textbf{v}}$ in \eqref{eq08_03} is infinitely differentiable, by applying Lemma \ref{lem6.7} to the system \eqref{eq08_03} with a scaling argument, we compute \begin{multline}
\label{eq23.10}
\big(|V'-(V')_{\mathcal{D}_2}|\big)_{\mathcal{D}_2}
+\big(|\hat \Theta-(\hat \Theta)_{\mathcal{D}_2}|\big)_{\mathcal{D}_2} \le N r^{1/2}\big([V']_{C^{1/2}(\mathcal{D}_2)} +[\hat \Theta]_{C^{1/2}( \mathcal{D}_2)}\big)\\ \le N r^{1/2}\big([V']_{C^{1/2}(\mathcal{D}_3)} +[\hat \Theta]_{C^{1/2}( \mathcal{D}_3)}\big) \le N\kappa^{-1/2}(V^2)_{B^+_{R/2}(\hat y_0)}^{1/2}. \end{multline} Thanks to the fact that $\kappa\gamma\le 1/10$, we have \begin{equation}
\label{eq21.31}
|\mathcal{D}_1|\le N\kappa\gamma |\mathcal{D}_2|,\quad
|\Omega_R(\hat y_0)|\le N\kappa^d |\mathcal{D}_2|. \end{equation} By combining \eqref{eq18.34h}, \eqref{eq28_01}, and \eqref{eq23.10}, we get \begin{align*}
\big(|&U^{B}-(U^B)_{\mathcal{D}_2}|\big)_{\mathcal{D}_2}\nonumber\\
&\le N\big(|V'-(V')_{\mathcal{D}_2}|\big)_{\mathcal{D}_2}
+ N \big(|\hat \Theta-(\hat \Theta)_{\mathcal{D}_2}|\big)_{{\mathcal{D}_2}} + N \big(W\big)_{\mathcal{D}_2}\nonumber\\ &\le N\kappa^{-1/2}(U^2)_{B^+_{R/2}(\hat y_0)}^{1/2} +N\kappa^{\frac d 2} \left(\gamma^{1/ {\nu'}} (U^\nu)_{\Omega_{R}(\hat y_0)}^{1/ \nu}+ (F^2)_{\Omega_{R}(\hat y_0)}^{1/2}\right). \end{align*} By using the triangle inequality and the assumption $\kappa \gamma\le 1/10$, $$
\big(|U^B-(U^B)_{\Omega_r(y_0)}|\big)_{\Omega_r(y_0)}
\le N\big(|U^B-(U^B)_{\mathcal{D}_2}|\big)_{\mathcal{D}_2} $$ $$ +N\kappa\gamma (U^B)_{\mathcal{D}_2} +N(1_{\mathcal{D}_1} U^B )_{\Omega_r(y_0)}. $$ We use \eqref{eq12.01}, \eqref{eq21.31}, and H\"older's inequality to bound the last two terms on the right-hand side above as follows: $$
\kappa \gamma (U^B)_{\mathcal{D}_2}\le N\kappa \gamma (|U|^2)^{1/2}_{\mathcal{D}_2}
\le N\kappa \gamma \kappa^{d/2}(|U|^2)^{1/2}_{\Omega_R(\hat y_0)}, $$ \begin{equation*}
(1_{\mathcal{D}_1} U^B )_{\Omega_r(y_0)}\le (1_{\mathcal{D}_1})^{1/2}_{\Omega_r(y_0)}(|U|^2)^{1/2}_{\Omega_r(y_0)}
\le N(\kappa \gamma)^{1/2} \kappa^{d/2}(|U|^2)_{\Omega_R(\hat y_0)}^{1/2}. \end{equation*} Therefore, \begin{multline}
\label{eq21.12}
\big(|U^B-(U^B)_{\Omega_r(y_0)}|\big)_{\Omega_r(y_0)}
\le N\big(\kappa^{-1/2}+(\kappa \gamma)^{1/2} \kappa^{d/2}\big) (|U|^2)_{\Omega_R(\hat y_0)}^{1/2}\\ +N\kappa^{\frac d 2} \left(\gamma^{1/ {\nu'}} (U^\nu)_{\Omega_{R}(\hat y_0)}^{1/ \nu}+(F^2)_{\Omega_{R}(\hat y_0)}^{1/2}\right). \end{multline}
In the remaining case when $\kappa\gamma>1/10$, by \eqref{eq12.01} and \eqref{eq22.16}, \begin{align*}
\big(|U^B-(U^B)_{\Omega_r(y_0)}|\big)_{\Omega_r(y_0)}&\le N(U^B)_{\Omega_r(y_0)}\le N(U)_{\Omega_r(y_0)}\\
&\le N(|U|^2)_{\Omega_r(y_0)}^{1/2}
\le N\kappa^{d/2}(|U|^2)_{\Omega_R(\hat y_0)}^{1/2}, \end{align*}
where in the last inequality, we used the obvious inequality $|\Omega_R(\hat y_0)|\le N\kappa^d|\Omega_r(y_0)|$. Therefore, in this case, \eqref{eq21.12} still holds. Finally, we transform the obtained inequality back to the original coordinates to get the inequality \eqref{eq5.42}. This completes the proof of Proposition \ref{prop7.9}.
\subsection{Proof of Theorem \ref{thm3}} We finish the proof of Theorem \ref{thm3} in this subsection. First we observe that by taking a sufficiently large $\lambda_0$ and using interpolation inequalities, we can move all the lower-order terms of $\mathcal{L} \textit{\textbf{u}}$ to the right-hand side. Thus in the sequel we assume that all the lower-order coefficients of $\mathcal{L}$ are zero.
Recall the definition of $\mathbb{C}_l,l\in \mathbb{Z}$ above Theorem \ref{th081201}. Notice that if $x\in C\in \mathbb{C}_l$, then for the smallest $r>0$ such that $C\subset B_r(x)$ we have $$
\dashint_{C} \dashint_{C}|g(y)-g(z)| \,dy\,dz\leq N(d)
\dashint_{B^+_r(x)} \dashint_{B^+_r(x)}|g(y)-g(z)| \,dy\,dz. $$ We use this inequality in the proof of the following corollary.
\begin{corollary} \label{cor001b} Let $\gamma \in (0,1/4)$, $\lambda > 0$, $\nu\in (2,\infty)$, $\nu'=2\nu/(\nu-2)$, and $z_0 \in \overline{\mathbb{R}^d_+}$. Assume that $\textit{\textbf{u}}\in W_{\nu,\text{loc}}^m(\overline{\mathbb{R}^d_+})$ vanishes outside $B_{\gamma R_0}(z_0)$ and satisfies $$
(-1)^m\mathcal{L} \textit{\textbf{u}}+\lambda \textit{\textbf{u}}=\sum_{|\alpha|\le m}D^\alpha \textit{\textbf{f}}_\alpha $$ locally in $\mathbb{R}^d_+$ with the conormal derivative condition on $\partial \mathbb{R}^d_+$, where $\textit{\textbf{f}}_\alpha\in L_{2,\text{loc}}(\overline{\mathbb{R}^d_+})$. Then under Assumption \ref{assumption20100901} ($\gamma$), for each $l \in \mathbb{Z}$, $C \in \mathbb{C}_l$, and $\kappa \ge 64$, there exists a function $U^C$ depending on $C$ such that $N^{-1}U\le U^C\le NU$ and \begin{equation}
\label{eq08_04}
\left(|U^C-(U^C)_{C}|\right)_{C} \le N \left(F_{\kappa}\right)_C, \end{equation} where $N=N(d,\delta,m,n,\tau)$ and \begin{align*} F_{\kappa}&= \big(\kappa^{-1/2}+(\kappa \gamma)^{1/2}\kappa^{d/2}\big)\big(\mathcal{M} (1_{\mathbb{R}^d_+}U^2)\big)^{1/2}\\ &\quad +\kappa^{d/2}\big[\big(\mathcal{M}(1_{\mathbb{R}^d_+}F^2)\big)^{1/2}+\gamma^{1/\nu'} (\mathcal{M}(1_{\mathbb{R}^d_+}U^{\nu}))^{1/\nu}\big]. \end{align*} \end{corollary}
\begin{proof} For each $\kappa \ge 64$ and $C \in \mathbb{C}_l$, let $B_r(x_0)$ be the smallest ball containing $C$. Clearly, $x_0 \in \overline{\mathbb{R}^d_+}$.
If $\kappa r > R_0$, then we take $U^C = U$. Note that the volumes of $C$, $B_r(x_0)$, and $B^+_r(x_0)$ are comparable, and $C\subset B_r^+(x_0)$. Then by the triangle inequality and H\"{o}lder's inequality, the left-hand side of \eqref{eq08_04} is less than \begin{multline*} N \left(U^2\right)^{1/2}_{B_r^+(x_0)} \le N \kappa^{d/2} \big( 1_{B^+_{\gamma R_0}(z_0)}U^2 \big)^{1/2}_{B_{\kappa r}(x_0)}\\ \le N \kappa^{d/2} \big( 1_{B_{\gamma R_0(z_0)}} \big)_{B_{\kappa r}(x_0)}^{1/\nu'} \big( 1_{\mathbb{R}^d_+ }U^\nu \big)^{1/\nu}_{B_{\kappa r}(x_0)} \le N \kappa^{d/2} \gamma^{1/\nu'} \big( 1_{\mathbb{R}^d_+ }U^\nu \big)^{1/\nu}_{B_{\kappa r}(x_0)}. \end{multline*}
Here the first inequality is because $|B_{\kappa r}(x_0)|\le 2\kappa^d |B_r^+(x_0)|$ and $U$ vanishes outside $B_{\gamma R_0}(z_0)$. The second inequality follows from H\"older's inequality. In the last inequality we used $\kappa r>R_0$ and $\gamma^d \le \gamma$. Note that \begin{equation}
\label{eq08_05} \big( 1_{\mathbb{R}^d_+ }U^\nu \big)^{1/\nu}_{B_{\kappa r}(x_0)} \le \mathcal{M}^{1/\nu} \big(1_{\mathbb{R}^d_+ }U^\nu \big)(x) \end{equation} for all $x \in C$. Hence the inequality \eqref{eq08_04} follows.
If $\kappa r \le R_0$, from Proposition \ref{prop7.9}, we find $U^B$ with $B^+ = B^+_{\kappa r}(x_0)$. Take $U^C = U^B$. Then by Proposition \ref{prop7.9} we have \begin{equation}
\label{eq08_06}
\left( |U^C - \left(U^C\right)_C| \right)_C \le N(d) I, \end{equation} where $I$ is the right-hand side of the inequality \eqref{eq5.42}. Note that, for example, $$ \left( U^2 \right)_{B_{\kappa r}^+(x_0)} \le N(d) \big( 1_{\mathbb{R}^d_+}U^2 \big)_{B_{\kappa r}(x_0)}. $$ Using this and inequalities like \eqref{eq08_05}, we see that \eqref{eq08_06} implies the desired inequality \eqref{eq08_04}. \end{proof}
\begin{theorem} \label{theorem001b} Let $p \in (2,\infty)$, $\lambda > 0$, $z_0\in \overline{\mathbb{R}^d_+}$, and $\textit{\textbf{f}}_{\alpha} \in L_p(\mathbb{R}^d_+)$. There exist positive constants $\gamma\in (0,1/4)$ and $N$, depending only on $d$, $\delta$, $m$, $n$, $p$, such that under Assumption \ref{assumption20100901} ($\gamma$), for $\textit{\textbf{u}} \in W_p^m(\mathbb{R}^d_+)$ vanishing outside $B_{\gamma R_0}(z_0)$ and satisfying $$
(-1)^m\mathcal{L} \textit{\textbf{u}}+\lambda \textit{\textbf{u}}=\sum_{|\alpha|\le m}D^\alpha \textit{\textbf{f}}_\alpha $$ in $\mathbb{R}^d_+$ with the conormal derivative condition on $\partial \mathbb{R}^d_+$, we have $$
\|U\|_{L_p(\mathbb{R}^d_+)}
\le N \| F\|_{L_p(\mathbb{R}^d_+)}, $$ where $N = N(d,\delta,m,n,p)$. \end{theorem}
\begin{proof} Let $\gamma > 0$ and $\kappa \ge 64$ be constants to be specified below. Take a constant $\nu$ such that $p > \nu > 2$. Then we see that $\textit{\textbf{u}} \in W_{\nu,\text{loc}}^m(\overline{\mathbb{R}^d_+})$ and all the conditions in Corollary \ref{cor001b} are satisfied.
For each $l \in \mathbb{Z}$ and $C \in \mathbb{C}_l$, let $U^C$ be the function from Corollary \ref{cor001b}. Then by Corollary \ref{cor001b} and Theorem \ref{th081201} we have $$
\| U \|_{L_p(\mathbb{R}^d_+)}^p \le N \|F_\kappa\|_{L_p(\mathbb{R}^d_+)} \|U\|_{L_p(\mathbb{R}^d_+)}^{p-1}. $$ The implies that $$
\|U \|_{L_p(\mathbb{R}^d_+)} \le N \|F_\kappa\|_{L_p(\mathbb{R}^d_+)}. $$ Now we observe that by the Hardy--Littlewood maximal function theorem \begin{multline*}
\|F_\kappa\|_{L_p(\mathbb{R}^d_+)}
\le \|F_\kappa\|_{L_p(\mathbb{R}^d)}
\le N \big(\kappa^{-1/2} + (\kappa \gamma)^{1/2} \kappa^{d/2}\big)\|1_{\mathbb{R}^d_+} U\|_{L_p(\mathbb{R}^d)}\\
+ N \kappa^{d/2} \|1_{\mathbb{R}^d_+} F\|_{L_p(\mathbb{R}^d)} + N \kappa^{d/2}\gamma^{1/\nu'} \|1_{\mathbb{R}^d_+} U\|_{L_p(\mathbb{R}^d)}. \end{multline*} To complete the proof, it remains to choose a sufficiently large $\kappa$, and then a sufficiently small $\gamma$ so that $$ N \big(\kappa^{-1/2} + (\kappa \gamma)^{1/2} \kappa^{d/2}\big) + N \kappa^{d/2}\gamma^{1/\nu'} < 1/2. $$ \end{proof}
\begin{proof}[Proof of Theorem \ref{thm3}] We treat the following three cases separately.
{\em Case 1: $p=2$.} In this case, the theorem follows from Theorem \ref{theorem08061901}.
{\em Case 2: $p\in (2,\infty)$.} Assertion (i) follows from Theorem \ref{theorem001b} and the standard partition of unity argument. Then Assertion (ii) is derived from Assertion (i) by using the method of continuity. Finally, Assertion (iii) is due to a standard scaling argument.
{\em Case 3: $p\in (1,2)$.} In this case, Assertion (i) is a consequence of the duality argument and the $W^m_q$-solvability obtained above for $q=p/(p-1)\in (2,\infty)$. With the a priori estimate, the remaining part of the theorem is proved in the same way as in Case 2. The theorem is proved. \end{proof}
\section{Systems on a Reifenberg flat domain}
\label{Reifenberg}
In this section, we consider elliptic systems on a Reifenberg flat domain. The crucial ingredients of the proofs below are the interior and the boundary estimates established in Sections \ref{sec_aux}, a result in \cite{Sa80,KS80} on the ``crawling of ink drops'', and an idea in \cite{CaPe98}.
By a scaling, in the sequel we may assume $R_0=1$ in Assumption \ref{assump1}. Recall the definitions of $U$ and $F$ in Sections \ref{sec_aux} and \ref{sec4}.
\begin{lemma}
\label{lem7.3}
Let $\gamma \in (0,1/50)$, $R\in (0,1]$, $\lambda\in (0,\infty)$, $\nu\in (2,\infty)$, $\nu'=2\nu/(\nu-2)$, $\textit{\textbf{f}}_\alpha= (f_\alpha^1, \ldots, f_\alpha^n)^{\text{tr}} \in L_{2,\text{loc}}(\overline{\Omega})$, $|\alpha|\le m$. Assume that $a_{\alpha\beta}\equiv 0$ for any $\alpha,\beta$ satisfying $|\alpha|+|\beta|<2m$ and that $\textit{\textbf{u}}\in W_{\nu, \text{loc}}^m(\overline{\Omega})$ satisfies \eqref{eq11_01} locally in $\Omega$ with the conormal derivative condition on $\partial \Omega$. Then the following hold true.
\noindent(i) Suppose $0\in \Omega$, $\text{dist}(0,\partial\Omega)\ge R$, and Assumption \ref{assump1} ($\gamma$) (i) holds at the origin. Then there exists nonnegative functions $V$ and $W$ in $B_R$ such that $U \le V+W$ in $B_R$, and $V$ and $W$ satisfy \begin{equation*}
(W^2)_{B_R}^{1/2} \le N\gamma^{1/\nu'} (U^\nu)_{B_R}^{1/\nu}+ N(F^2)_{B_R}^{1/2} \end{equation*} and \begin{equation*}
\|V\|_{L_\infty(B_{R/4})} \le N\gamma^{1/\nu'} (U^\nu)_{B_{R}}^{1/ \nu}+ N(F^2)_{B_{R}}^{1/2}+ N(U^2)_{B_{R}}^{1/2}, \end{equation*} where $N=N(d,n,m,\delta,\nu)>0$ is a constant.
\noindent(ii) Suppose $0\in \partial\Omega$ and Assumption \ref{assump1} ($\gamma$) (ii) holds at the origin. Then there exists nonnegative functions $V$ and $W$ in $\Omega_R$ such that $U \le V+W$ in $\Omega_R$, and $W$ and $V$ satisfy \begin{equation}
\label{eq17.10} (W^2)_{\Omega_R}^{1/2} \le N\gamma^{1 /\nu'} (U^\nu)_{\Omega_R}^{1/\nu}+ N(F^2)_{\Omega_R}^{1/2} \end{equation} and \begin{equation} \label{eq17.11}
\|V\|_{L_\infty(\Omega_{R/4})} \le N\gamma^{1/\nu'} (U^\nu)_{\Omega_R}^{1/\nu} +N(F^2)_{\Omega_R}^{1/2}+ N(U^2)_{\Omega_R}^{1/2}, \end{equation} where $N=N(d,n,m,\delta,\nu)>0$ is a constant. \end{lemma}
\begin{proof} The proof is similar to that of Proposition \ref{prop7.9} with some modifications. We assume that Assumption \ref{assump1} holds in the original coordinates. Without loss of generality, we may further assume that the coefficients $\bar a_{\alpha\beta}$ are infinitely differentiable.
Assertion (i) is basically an interior estimate which does not involve boundary conditions, so the proof is exactly the same as that of Assertion (i) in \cite[Lemma 8.3]{DK10}.
Next, we prove Assertion (ii). Due to Assumption \ref{assump1}, by shifting the origin upward, we can assume that $$ B_R^+(x_0) \subset \Omega_R(x_0) \subset \{(x_1,x') : -2\gamma R < x_1 \} \cap B_R(x_0) $$ where $x_0 \in \partial \Omega$ (see Figure \ref{fg2}). Define $\bar a_{\alpha\beta}$ as in Section \ref{sec4}. \begin{figure}\label{fg2}
\end{figure} Then $\textit{\textbf{u}}$ satisfies \eqref{eq17.23b} in $B_R^+(x_0)$ with the conormal derivative condition on $\Gamma_R(x_0)$. By following the argument in the proof of Proposition \ref{prop7.9}, we can find $\textit{\textbf{w}} \in W_2^m(\mathbb{R}^d_+)$ and $\textit{\textbf{v}} \in W_{2}^m(B_R^+(x_0))$ such that $\textit{\textbf{u}} = \textit{\textbf{w}} + \textit{\textbf{v}}$, the function $\textit{\textbf{w}}$ satisfies \begin{equation}
\label{eq08_07}
\sum_{k=0}^m\lambda^{\frac 1 2-\frac k {2m}}(|D^k \textit{\textbf{w}}|^2)_{B_R^+(x_0)}^{1/2} \le N\gamma^{1/ {\nu'}} (U^\nu)_{\Omega_R(x_0)}^{1/ \nu}+ N(F^2)_{\Omega_R(x_0)}^{1/2}, \end{equation} and the function $\textit{\textbf{v}}$ satisfies $$ (-1)^m \mathcal{L} \textit{\textbf{v}} + \lambda \textit{\textbf{v}} = 0 $$ in $B_{R/2}^+(x_0)$ with the conormal derivative condition on $\Gamma_{R/2}(x_0)$. We define $V$ and $W$ in $B_R^+(x_0)$ as in Section \ref{sec4}. As noted in the proof of Proposition \ref{prop7.9}, we can assume that $\textit{\textbf{v}}$ is infinitely differentiable. Applying Lemma \ref{lem6.7}, we get \begin{equation}
\label{eq08_08}
\|V\|_{L_\infty(B^+_{R/4}(x_0))} \le N (V^2)_{B^+_{R/2}(x_0)}^{1/2}. \end{equation} Now we extend $W$ and $V$ on $\Omega^* = \mathbb{R}^d_- \cap \Omega_R(x_0)$ by setting $W = U$ and $V = 0$, respectively. Then we see that by H\"{o}lder's inequality and \eqref{eq08_07} \begin{align*} \left(W^2\right)^{1/2}_{\Omega_R(x_0)}
&= \left[ \frac{1}{|\Omega_R(x_0)|}\int_{B_R^+(x_0)} W^2 \, dx
+ \frac{1}{|\Omega_R(x_0)|} \int_{\Omega^*} U^2 \, dx \right]^{1/2}\\ &\le N \left( W^2 \right)_{B_R^+(x_0)}^{1/2} + N \left( 1_{\Omega^*} \right)_{\Omega_R(x_0)}^{1/\nu'} \left( U^\nu \right)^{1/\nu}_{\Omega_R(x_0)}\\ &\le N\gamma^{1/ {\nu'}} (U^\nu)_{\Omega_R(x_0)}^{1/ \nu}+ N(F^2)_{\Omega_R(x_0)}^{1/2}. \end{align*} Upon recalling that the origin was shifted from $x_0$, we arrive at \eqref{eq17.10}. To prove \eqref{eq17.11} we observe that by \eqref{eq08_08} and the fact that $V\le U+W$ $$
\|V\|_{L_\infty(\Omega_{R/4}(x_0))}
= \|V\|_{L_\infty(B_{R/4}^+(x_0))} \le N \left(V^2\right)^{1/2}_{B_{R/2}^+(x_0)} $$ $$ \le N \left(V^2\right)^{1/2}_{\Omega_R(x_0)} \le N \left(W^2\right)^{1/2}_{\Omega_R(x_0)} + N \left(U^2\right)^{1/2}_{\Omega_R(x_0)}. $$ This together with \eqref{eq17.10} gives \eqref{eq17.11}. This completes the proof of the lemma. \end{proof}
For a function $f$ on a set $\mathcal{D}\subset\mathbb{R}^{d}$, we define its maximal function $\mathcal{M} f$ by $\mathcal{M} f=\mathcal{M} (I_{\mathcal{D}}f)$. For any $s>0$, we introduce two level sets $$ \mathcal{A}(s)=\{x\in \Omega:U> s\}, $$ $$ \mathcal{B}(s)=\Big\{x\in \Omega:\gamma^{-1/\nu'}\big(\mathcal{M}(F^2)\big)^{1/2}+ \big(\mathcal{M}(U^\nu)\big)^{1/\nu}>s\Big\}. $$ With Lemma \ref{lem7.3} in hand, we get the following corollary.
\begin{corollary}
\label{cor7.5} Under the assumptions of Lemma \ref{lem7.3}, suppose $0\in \overline{\Omega}$ and Assumption \ref{assump1} ($\gamma$) holds. Let $s\in (0,\infty)$ be a constant. Then there exists a constant $\kappa\in (1,\infty)$, depending only on $d$, $n$, $m$, $\delta$, and $\nu$, such that the following holds. If \begin{equation}
\label{eq15.59}
\big|\Omega_{R/32}\cap \mathcal{A}(\kappa s)\big|> \gamma^{2/\nu'} |\Omega_{R/32}|, \end{equation} then we have $\Omega_{R/32}\subset \mathcal{B}(s)$. \end{corollary} \begin{proof} By dividing $\textit{\textbf{u}}$ and $\textit{\textbf{f}}$ by $s$, we may assume $s=1$. We prove by contradiction. Suppose at a point $x\in \Omega_{R/32}$, we have \begin{equation}
\label{eq15.38} \gamma^{-1/\nu'}\big(\mathcal{M}(F^2)(x)\big)^{1/2}+ \big(\mathcal{M}(U^\nu)(x)\big)^{1/\nu}\le 1. \end{equation} Let us consider two cases.
First we consider the case when $\text{dist}(0,\partial \Omega)\ge R/8$. Notice that $$ x\in\Omega_{R/32}=B_{R/32}\subset B_{R/8}\subset \Omega. $$ Due to Lemma \ref{lem7.3} (i), we have $U \le V + W$ and, by \eqref{eq15.38}, \begin{equation}
\label{eq15.47}
\|V\|_{L_\infty(B_{R/32})} \le N_1, \quad (W^2)_{B_{R/8}}^{1/2} \le N_1\gamma^{1/\nu'}, \end{equation} where $N_1$ and constants $N_i$ below depend only on $d$, $n$, $m$, $\delta$, and $\nu$. By \eqref{eq15.47}, the triangle inequality and Chebyshev's inequality, we get \begin{multline}
\label{eq5.20}
\big|\Omega_{R/32}\cap \mathcal{A}(\kappa)\big|=
\big|\{x\in \Omega_{R/32}: U>\kappa\}\big|\\
\le \big|\{x\in \Omega_{R/32}: W>\kappa-N_1\}\big|
\le (\kappa-N_1)^{-2} N_1^2 \gamma^{2/\nu'}|B_{R/8}|, \end{multline} which contradicts with \eqref{eq15.59} if we choose $\kappa$ sufficiently large.
Next we consider the case when $\text{dist}(0,\partial \Omega)< R/8$. We take $y\in \partial\Omega$ such that $|y|=\text{dist}(0,\partial \Omega)$. Notice that in this case we have $$ x\in\Omega_{R/32}\subset \Omega_{R/4}(y)\subset \Omega_{R}(y). $$ Due to Lemma \ref{lem7.3} (ii), we have $U\le V+ W$ in $\Omega_R(y)$ and, by \eqref{eq15.38}, \begin{equation}
\label{eq17.23c}
\|V\|_{L_\infty(\Omega_{R/32})}\le \|V\|_{L_\infty(\Omega_{R/4}(y))} \le N_2, \quad (W^2)_{\Omega_{R}(y)}^{1/2} \le N_2\gamma^{1/\nu'}. \end{equation}
By \eqref{eq17.23c}, the triangle inequality and Chebyshev's inequality, we still get \eqref{eq5.20} with $N_2$ in place of $N_1$, which contradicts with \eqref{eq15.59} if we choose $\kappa$ sufficiently large. \end{proof}
\begin{theorem} \label{theorem101} Let $p \in (2,\infty)$, $\lambda > 0$, $x_0\in \mathbb{R}^{d}$
and $\textit{\textbf{f}}_{\alpha} \in L_p(\Omega)$. Suppose that $a_{\alpha\beta}\equiv 0$ for any $\alpha,\beta$ satisfying $|\alpha|+|\beta|<2m$, and $\textit{\textbf{u}} \in W_p^m(\Omega)$ is supported on $B_{\gamma}(x_0)\cap \overline{\Omega}$ and satisfies \eqref{eq11_01} in $\Omega$ with the conormal derivative boundary condition. There exist positive constants $\gamma \in (0,1/50)$ and $N$, depending only on $d,\delta,m,n, p$, such that, under Assumption \ref{assump1} ($\gamma$) we have \begin{equation}
\label{eq16.00}
\|U \|_{L_p(\Omega)}
\le N \|F\|_{L_p(\Omega)}, \end{equation} where $N = N(d,\delta,m,n,p)$. \end{theorem}
\begin{proof} We fix $\nu=p/2+1$ and let $\nu'=2\nu/(\nu-2)$. Then we see that $\textit{\textbf{u}} \in W_{\nu,\text{loc}}^m(\overline{\Omega})$. Let $\kappa$ be the constant in Corollary \ref{cor7.5}. Recall the elementary identity: $$
\|f\|_{L_p(\mathcal{D})}^p=p\int_0^\infty \big|\{x\in\mathcal{D}:|f(x)|> s\}\big|s^{p-1}\,ds, $$ which implies that \begin{equation}
\label{eq5.07}
\|U\|_{L_p(\Omega)}^p=p\kappa^p \int_0^\infty|\mathcal{A}(\kappa s)|s^{p-1}\,ds. \end{equation} Thus, to obtain \eqref{eq16.00} we need to estimate $\mathcal{A}(\kappa s)$. First, we note that by Chebyshev's inequality \begin{equation}
\label{eq17.57}
|\mathcal{A}(\kappa s)|\le (\kappa s)^{-2}\|U\|_{L_2(\Omega)}^2. \end{equation}
When $\kappa s \ge \gamma^{-1/\nu'} \|U\|_{L_2(\Omega)}$, this indicates that $$
|\mathcal{A}(\kappa s)|\le \gamma^{2/\nu'}. $$ With the above inequality and Corollary 5.2 in hand, we see that all the conditions of the ``crawling of ink drops'' lemma are satisfied; see \cite{Sa80}, \cite[Sect. 2]{KS80}, or \cite[Lemma 3]{BW05} for the lemma. Hence we have \begin{equation}
\label{eq18.34}
|\mathcal{A}(\kappa s)|\le N_4\gamma^{2/\nu'}|\mathcal{B}(s)|. \end{equation}
Now we estimate $\|U\|_{L_p(\Omega)}^p$ in \eqref{eq5.07} by splitting the integral into two depending on the range of $s$. If $\kappa s \ge \gamma^{-1/\nu'}\|U\|_{L_2(\Omega)}$, we use \eqref{eq18.34}. Otherwise, we use \eqref{eq17.57}. Then it follows that \begin{align*}
\|U\|_{L_p(\Omega)}^p &\le N_5\gamma^{(2-p)/\nu'}\big(\|U\|_{L_2(\Omega)}^p+
\big\|\big(\mathcal{M}(F^2)\big)^{1/2}\big\|_{L_p(\Omega)}^p\big)\\
&\quad +N_5\gamma^{2/\nu'}\big\|\big(\mathcal{M}(U^\nu)\big)^{1/\nu}
\big\|_{L_p(\Omega)}^p\\
&\le N_5\gamma^{(2-p)/\nu'}\|U\|_{L_2(\Omega)}^p+N_6\gamma^{(2-p)/\nu'}
\|F\|_{L_p(\Omega)}^p+N_6\gamma^{2/\nu'}\|U\|_{L_p(\Omega)}^p, \end{align*} where we used the Hardy--Littlewood maximal function theorem in the last inequality. By H\"older's inequality, \begin{equation}
\label{eq20.31bb}
\|U\|_{L_2(\Omega)}=\|U\|_{L_2(B_\gamma(x_0)\cap\Omega)}
\le N\|U\|_{L_p(\Omega)}\gamma^{d(1/2-1/p)}. \end{equation} By the choice of $\nu$, $d(p/2-1)+(2-p)/\nu'>2/\nu'$. Thus, we get $$
\|U\|_{L_p(\Omega)}^p \le N_6\gamma^{(2-p)/\nu'}
\|F\|_{L_p(\Omega)}^p
+N_6\gamma^{2/\nu'}\|U\|_{L_p(\Omega)}^p. $$ To get the estimate \eqref{eq16.00}, it suffices to take $\gamma=\gamma(d,n,m,\delta,p)\in (0,1/50]$ sufficiently small such that $N_6\gamma^{2/\nu'}\le 1/2$. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm5}] We again consider the following three cases separately.
{\em Case 1: $p=2$.} In this case, the theorem follows directly from the well-known Lax--Milgram lemma.
{\em Case 2: $p\in (2,\infty)$.} Assertion (i) follows from Theorem \ref{theorem101} and the standard partition of unity argument. By the method of continuity, for Assertion (ii) it suffices to prove the solvability for the operator $\mathcal{L}_1:=\delta_{ij}\Delta^m$, which is not immediate because the domain $\Omega$ is irregular. We approximate $\Omega$ by regular domains. Recall the definition of $\Omega^\varepsilon$ in \eqref{eq2.22}. By Lemma \ref{lem3.11}, for any $\varepsilon\in (0,1/4)$, $\Omega^\varepsilon$ satisfies Assumption \ref{assump3} ($N_0 \gamma^{1/2}$) with a constant $R_1(\varepsilon)>0$. Thanks to Lemma \ref{lem3.19}, there is a sequence of expanding smooth domains $\Omega^{\varepsilon,k}$ which converges to $\Omega^\varepsilon$ as $k\to \infty$. Moreover, $\Omega^{\varepsilon,k}$ satisfies Assumption \ref{assump3} ($N_0 \gamma^{1/2}$) with the constant $R_1(\varepsilon)/2$ which is independent of $k$. In particular, $\Omega^{\varepsilon,k}$ satisfies Assumption \ref{assump1} ($N_0\gamma^{1/2}$) with the constant $R_1(\varepsilon)/2$. By the classical result, there is a constant $\lambda_\varepsilon=\lambda_\varepsilon(d,n,m,p,\delta)\ge \lambda_0$ such that, for any $\lambda>\lambda_\varepsilon$, the equation $$
(-1)^m\mathcal{L}_1\textit{\textbf{u}}+\lambda \textit{\textbf{u}}=\sum_{|\alpha|\le m}D^\alpha \textit{\textbf{f}}_\alpha\quad \text{in}\,\,\Omega^{\varepsilon,k} $$ with the conformal derivative boundary condition has a unique solution $\textit{\textbf{u}}^{\varepsilon,k}\in W^{m}_p(\Omega^{\varepsilon,k})$. The a priori estimate above gives \begin{equation}
\label{eq3.07}
\|\textit{\textbf{u}}^{\varepsilon,k}\|_{W^m_p(\Omega^{\varepsilon,k})}\le N_\varepsilon, \end{equation}
where $N_\varepsilon>0$ is a constant independent of $k$. By the weak compactness, there is a subsequence, which is still denoted by $\textit{\textbf{u}}^{\varepsilon,k}$, and functions $\textit{\textbf{v}}^\varepsilon,\textit{\textbf{v}}^\varepsilon_\alpha\in L_p(\Omega^{\varepsilon}),1\le |\alpha|\le m$, such that weakly in $L_p(\Omega^{\varepsilon})$, $$
\textit{\textbf{u}}^{\varepsilon,k}I_{\Omega^{\varepsilon,k}}\rightharpoonup \textit{\textbf{v}}^\varepsilon,\quad D^\alpha \textit{\textbf{u}}^{\varepsilon,k}I_{\Omega^{\varepsilon,k}}\rightharpoonup \textit{\textbf{v}}^\varepsilon_\alpha\quad \forall\, \alpha,\,1\le |\alpha|\le m. $$ It is easily seen that $\textit{\textbf{v}}^\varepsilon_\alpha=D^\alpha \textit{\textbf{v}}^\varepsilon$ in the sense of distributions. Thus, by \eqref{eq3.07} and the weak convergence,
$\textit{\textbf{v}}^{\varepsilon}\in W^m_p(\Omega^\varepsilon)$ is a solution to \begin{equation}
\label{eq4.58}
(-1)^m\mathcal{L}_1\textit{\textbf{u}}+\lambda \textit{\textbf{u}}=\sum_{|\alpha|\le m}D^\alpha \textit{\textbf{f}}_\alpha\quad \text{in}\,\,\Omega^{\varepsilon} \end{equation} with the conormal derivative boundary condition. We have proved the solvability for any $\lambda>\lambda_\varepsilon$. Recall that, by Lemma \ref{lem3.11}, $\Omega^\varepsilon$ satisfies Assumption \ref{assump1} ($N_0\gamma^{1/2}$) with $R_0=1/2$. By the a priori estimate in Assertion (i) and the method of continuity, for any $\lambda>\lambda_0$ there is a unique solution $\textit{\textbf{u}}^\varepsilon\in W^m_p(\Omega^\varepsilon)$ to \eqref{eq4.58} with the conormal derivative boundary condition. Moreover, we have \begin{equation}
\label{eq3.07b}
\|\textit{\textbf{u}}^{\varepsilon}\|_{W^m_p(\Omega^{\varepsilon})}\le N, \end{equation}
where $N$ is a constant independent of $\varepsilon$. Again by the weak compactness, there is a subsequence $\textit{\textbf{u}}^{\varepsilon_j}$, and functions $\textit{\textbf{u}},\textit{\textbf{u}}_\alpha\in L_p(\Omega),1\le |\alpha|\le m$, such that weakly in $L_p(\Omega)$, $$
\textit{\textbf{u}}^{\varepsilon_j}I_{\Omega^{\varepsilon_j}}\rightharpoonup \textit{\textbf{u}},\quad D^\alpha \textit{\textbf{u}}^{\varepsilon_j}I_{\Omega^{\varepsilon_j}}\rightharpoonup \textit{\textbf{u}}_\alpha\quad \forall\, \alpha,\,1\le |\alpha|\le m. $$ It is easily seen that $\textit{\textbf{u}}_\alpha=D^\alpha \textit{\textbf{u}}$ in the sense of distributions. Thus, by \eqref{eq3.07b} and the weak convergence, $\textit{\textbf{u}}\in W^m_p(\Omega)$ is a solution to \begin{equation*}
(-1)^m\mathcal{L}_1\textit{\textbf{u}}+\lambda \textit{\textbf{u}}=\sum_{|\alpha|\le m}D^\alpha \textit{\textbf{f}}_\alpha\quad \text{in}\,\,\Omega \end{equation*} with the conormal derivative boundary condition. The uniqueness then follows from the a priori estimate. This completes the proof of Assertion (ii).
{\em Case 3: $p\in (1,2)$.} The a priori estimate in Assertion (i) is a directly consequence of the solvability when $p\in (2,\infty)$ and the duality argument. Then the solvability in Assertion (ii) follows from the a priori estimate by using the same argument as in Case 2.
The theorem is proved. \end{proof}
We now give the proofs of Corollary \ref{cor7} and Theorem \ref{thmB}.
\begin{proof}[Proof of Corollary \ref{cor7}] {\em Case 1: $p=2$.} We define a Hilbert space $$
H:=\{\textit{\textbf{u}}\in W^m_2(\Omega)\,|\,(\textit{\textbf{u}})_\Omega=(D\textit{\textbf{u}})_\Omega=\ldots=(D^{m-1}\textit{\textbf{u}})_\Omega=0\}. $$ By the Lax--Milgram lemma, there is a unique $\textit{\textbf{u}}\in H$ such that for any $\textit{\textbf{v}}\in H$, \begin{equation}
\label{eq22.44} \int_\Omega a_{\alpha\beta}D^\beta \textit{\textbf{u}} D^\alpha \textit{\textbf{v}}\,dx=
\sum_{|\alpha|=m}\int_\Omega (-1)^{|\alpha|}\textit{\textbf{f}}_\alpha D^\alpha \textit{\textbf{v}} \end{equation} and $$
\|D^m \textit{\textbf{u}}\|_{L_2(\Omega)}\le N\sum_{|\alpha|=m}\|\textit{\textbf{f}}_\alpha\|_{L_2(\Omega)}. $$ Note that any function $\textit{\textbf{v}}\in W^m_2(\Omega)$ can be decomposed as a sum of a function in $H$ and a polynomial of degree at most $m-1$. Therefore, \eqref{eq22.44} also holds for any $\textit{\textbf{v}}\in W^m_2(\Omega)$. This implies that $\textit{\textbf{u}}\in W_2^m(\Omega)$ is a solution to the original equation. On the other hand, by the uniqueness of the solution in $H$, any solution $\textit{\textbf{w}}\in W_2^m(\Omega)$ can only differ from $\textit{\textbf{u}}$ by a polynomial of order at most $m-1$.
{\em Case 2: $p\in (2,\infty)$.} First we suppose that $p$ satisfies $1/p>1/2-1/d$. Since $\Omega$ is bounded, $\textit{\textbf{f}}\in L_2(\Omega)$. Let $\textit{\textbf{u}}$ be the unique $H$-solution to the equation. By Theorem \ref{thm5}, there is a unique solution $\textit{\textbf{v}}\in W^m_p(\Omega)$ to the equation \begin{equation}
\label{eq6.01}
(-1)^m\mathcal{L} \textit{\textbf{v}}+(\lambda_0+1)\textit{\textbf{v}}=\sum_{|\alpha|= m}D^\alpha \textit{\textbf{f}}_\alpha+(\lambda_0+1)\textit{\textbf{u}}\quad \text{in}\,\,\Omega \end{equation} with the conormal derivative boundary condition. Moreover, we have \begin{equation}
\label{eq23.17}
\|\textit{\textbf{v}} \|_{W^{m}_p(\Omega)}
\le N \sum_{|\alpha|= m}\|\textit{\textbf{f}}_\alpha \|_{L_p(\Omega)}+N\|\textit{\textbf{u}} \|_{L_p(\Omega)}. \end{equation} By the Sobolev imbedding theorem and the $W^m_2$ estimate, we have $$
\|\textit{\textbf{u}}\|_{L_p(\Omega)}\le N\|\textit{\textbf{u}}\|_{W^1_2(\Omega)}\le N\sum_{|\alpha|= m}\|\textit{\textbf{f}}_\alpha\|_{L_2(\Omega)}\le N\sum_{|\alpha|= m}\|\textit{\textbf{f}}_\alpha\|_{L_{p}(\Omega)}, $$ which together with \eqref{eq23.17} gives $$
\|\textit{\textbf{v}} \|_{W^{m}_p(\Omega)}\le N \sum_{|\alpha|= m}\|\textit{\textbf{f}}_\alpha \|_{L_p(\Omega)}. $$ Since both $\textit{\textbf{v}}$ and $\textit{\textbf{u}}$ are $W^m_2(\Omega)$-solutions to \eqref{eq6.01}, by Theorem \ref{thm5} we have $\textit{\textbf{u}}=\textit{\textbf{v}}$. Therefore, the solvability is proved under the assumption $1/p>1/2-1/d$. The general case follows by using a bootstrap argument. The uniqueness is due to the uniqueness of $W^m_2$-solutions.
{\em Case 3: $p\in (1,2)$.} By the duality argument and Case 2, we have the a priori estimate \eqref{eq23.08} for any $\textit{\textbf{u}}\in W^m_p(\Omega)$ satisfying \eqref{eq22.34} with the conormal derivative boundary condition. For the solvability, we take a sequence $$ \textit{\textbf{f}}^{\,k}_\alpha=\min\{\max\{\textit{\textbf{f}}_\alpha,-k\},k\}\in L_2(\Omega) $$ which converges to $\textit{\textbf{f}}_\alpha$ in $L_p(\Omega)$. Let $\textit{\textbf{u}}^k$ be the $H$-solution to the equation with the right-hand side $\textit{\textbf{f}}^{\,k}_\alpha$. Since $\Omega$ is bounded, we have $\textit{\textbf{u}}^k\in W^m_p(\Omega)$. By the a priori estimate, $\textit{\textbf{u}}^k$ is a Cauchy sequence in $W^m_p(\Omega)$. Then it is easily seen that the limit $\textit{\textbf{u}}$ is the $W^m_p(\Omega)$-solution to the original equation. Next we show the uniqueness. Let $\textit{\textbf{u}}_1$ be another $W^m_p(\Omega)$-solution to the equation. Then $\textit{\textbf{v}}:=\textit{\textbf{u}}-\textit{\textbf{u}}_1\in W^m_p(\Omega)$ satisfies the equation with the zero right-hand side. Following the bootstrap argument in Case 2, we infer that $\textit{\textbf{v}}\in W^m_2(\Omega)$. Therefore, by Case 1, $\textit{\textbf{v}}$ must be a polynomial of degree at most $m-1$.
The corollary is proved. \end{proof}
\begin{proof}[Proof of Theorem \ref{thmB}] The theorem is proved in the same way as Corollary \ref{cor7} in Cases 2 and 3 by using the classical $W^1_2$-estimate of the conormal derivative problem on a domain with a finite measure; see Theorem 13 of \cite{DongKim08a}. We remark that although in Theorem 13 (i) of \cite{DongKim08a} it is assumed that $b_i=c=0$, the same proof works under the relaxed condition $-D_ib_i+c=0$ in $\Omega$ and $b_in_i=0$ on $\partial\Omega$ in the weak sense, i.e., Assumption ($\text{H}^*$). \end{proof}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\begin{abstract}
This article deals with the uniqueness and stability issues in the inverse problem of determining the unbounded potential of the Schr\"odinger operator in a bounded domain of $\mathbb{R}^n$, $n \geq 3$, endowed with Robin boundary condition, from knowledge of its boundary spectral data. These data are defined by the pairs formed by the eigenvalues and either partial or full Dirichlet measurement of the eigenfunctions on the boundary of the domain. \end{abstract}
\maketitle
\section{Introduction}
In the present article $\Omega$ is a $C^{1,1}$ bounded domain of $\mathbb{R}^n$, $n\ge 3$, with boundary $\Gamma$, and we equip the two spaces $H:=L^2(\Omega)$ and $V:=H^1(\Omega)$ with their usual scalar product. Put $p:=2n/(n+2)$ and let $p^\ast:=2n/(n-2)$ be its conjugate number, in such a way that $V$ is continuously embedded in $L^{p^\ast}(\Omega)$.
\subsection{The Robin Laplacian} \label{sec-RL} For $\alpha \in L^\infty (\Gamma,\mathbb{R})$ and $q\in L^{n/2} (\Omega,\mathbb{R})$, we introduce the following continuous sesquilinear form $\mathfrak{a} : V\times V\rightarrow \mathbb{C}$ \[ \mathfrak{a}(u,v)=\int_\Omega \nabla u\cdot \nabla \overline{v}dx+\int_\Omega qu\overline{v}dx+\int_\Gamma \alpha u\overline{v}ds(x),\quad u,v\in V. \]
Throughout the entire text, we assume that $\alpha \ge -\mathfrak{c}$ for some constant $\mathfrak{c} \in (0, \mathfrak{n}^{-2})$ almost everywhere on $\Gamma$, where $\mathfrak{n}$ denotes the norm of the (bounded) trace operator $u\in V\mapsto u_{|\Gamma}\in L^2(\Gamma)$. Set \[
\mathrm{Q}(\rho,\aleph):=\{q\in L^\rho(\Omega,\mathbb{R});\; \|q\|_{L^\rho(\Omega)}\le \aleph\},\quad \rho \ge n/2,\; \aleph >0. \] Then, arguing as in the derivation of \cite[Lemma A2]{Po}, we obtain that \begin{equation}\label{ii1}
\|qu^2\|_{L^1(\Omega)}\le \epsilon \|u\|_V^2+C_\epsilon\|u\|_H^2,\quad q\in \mathrm{Q}(n/2,\aleph),\; u\in V,\; \epsilon >0, \end{equation} for some constant $C_\epsilon>0$ depending only of $n$, $\Omega$, $\aleph$ and $\epsilon$. Further, applying \eqref{ii1} with $\epsilon =\kappa:=(1-\mathfrak{c}\mathfrak{n}^2)/2$ yields \begin{equation}\label{co}
\mathfrak{a}(u,u)+ \lambda ^\ast\|u\|_H^2\ge \kappa \|u\|_V^2,\quad u\in V, \end{equation} where $\lambda^\ast>0$ is a constant which depends only on $n$, $\Omega$, $\mathfrak{c}$ and $\aleph$. Let us consider the bounded operator $A:V\rightarrow V^\ast$ defined by \[ \langle Au,v\rangle=\mathfrak{a}(u,v),\quad u,v\in V, \] where $\langle \cdot,\cdot \rangle$ denotes the duality pairing between an arbitrary Banach space and its dual. Notice that $A$ is self-adjoint and coercive according to \eqref{co}.
\subsection{Boundary spectral data} \label{sec-BSD} With reference to \cite[Theorem 2.37]{Mc}, the spectrum of $A$ consists of its eigenvalues $\lambda_k$, $k \in \mathbb{N}:=\{1,2,\ldots \}$, arranged in non-decreasing order and repeated with the (finite) multiplicity, \[ -\infty <\lambda_1\le \lambda_2\le \ldots \le \lambda_k\le \ldots, \quad \mbox{and\ such\ that}\quad \lim_{k \to \infty}\lambda_k \rightarrow \infty. \] Moreover, there exists an orthonormal basis $\{ \phi_k,\ k \in \mathbb{N} \}$ of $H$, made of eigenfunctions $\phi_k\in V$ of $A$, satisfying \[ \mathfrak{a}(\phi_k,v)=\lambda_k(\phi_k,v),\quad v\in V,\quad k \in \mathbb{N}, \] where $(\cdot,\cdot)$ is the usual scalar product in $H$. For the sake of shortness, we write \[
\psi_k:=\phi_k{_{|\Gamma}},\quad k \in \mathbb{N}. \]
Recall that for $u\in V$, we have $\Delta u\in H^{-1}(\Omega)$, the space dual to $H_0^1(\Omega)$, but that it is not guaranteed that $\Delta u$ lie in $V^\ast$ (which is strictly embedded in $H^{-1}(\Omega)$). Thus, we introduce \[ W:=\{u\in V; \Delta u\in V^\ast\}. \] Endowed with its natural norm \[
\|u\|_{W}=\|u\|_V+\|\Delta u\|_{V^\ast},\quad u\in W, \] is a Banach space. Next, for $\varphi \in H^{1/2}(\Gamma)$, we set \[
\dot{\varphi}:=\{v\in V;\; v_{|\Gamma}=\varphi\}, \] and we equip the space $H^{1/2}(\Gamma)$ with its graph norm \[
\|\varphi\|_{H^{1/2}(\Gamma)}=\min\{\|v\|_V;\; v\in \dot{\varphi}\}. \] Now, for $u\in W$ fixed, we put \[ \Phi_u (v):=\langle \Delta u , v\rangle+(\nabla u,\nabla v),\quad v\in V, \] apply the Cauchy-Schwarz inequality, and get that \begin{equation}\label{0.1}
|\Phi_u (v)|\le \|\Delta u\|_{V^\ast}\|v\|_V+\|u\|_V\|v\|_V \leq \|u\|_W\|v\|_V. \end{equation}
Moreover, since $C_0^\infty (\Omega)$ is dense in $H_0^1(\Omega)$, it is easy to see that $H_0^1(\Omega)\subset \ker \Phi_u$ and consequently that $\Phi_u (v)$ depends only on $v_{|\Gamma}$. This enables us to define the normal derivative of $u$, denoted by $\partial_\nu u$, as the unique vector in $H^{-1/2}(\Gamma)$ satisfying \[ \langle \partial_\nu u , \varphi\rangle =\Phi_u (v),\quad v \in \dot{\varphi}\hskip.2cm \mbox{is arbitrary}. \] As a consequence we have \[
\|\partial_\nu u\|_{H^{-1/2}(\Gamma)}\le \|u\|_W, \] by \eqref{0.1}, and the following generalized Green formula: \begin{equation} \label{ggf}
\langle\Delta u , v\rangle+(\nabla u,\nabla v)=\langle \partial_\nu u , v_{|\Gamma}\rangle,\quad u\in W,\; v\in V. \end{equation}
Pick $f\in V^\ast$ and $\mu \in \mathbb{C}$, and let $u\in V$ satisfy \begin{equation}\label{vf} \mathfrak{a}(u,v)+\mu(u,v)=\langle f , v\rangle ,\quad v\in V. \end{equation} Using that $C_0^\infty(\Omega) \subset V$, we obtain that \[
\int_\Omega \nabla u\cdot \nabla \overline{v}dx +\int_\Omega qu\overline{v}dx+\mu\int_\Omega u\overline{v}dx=\langle f| v \rangle,\quad v\in C_0^\infty (\Omega), \] which yields $-\Delta u+qu+\mu u=f$ in $\mathscr{D}'(\Omega)$. Thus, bearing in mind that $qu \in V^\ast$, we have $u\in W$, and the generalized Green formula \eqref{ggf} provides \[
\langle \partial_\nu u+\alpha u_{|\Gamma} , v_{|\Gamma}\rangle =0,\quad v\in V. \]
Since $v\in V\mapsto v_{|\Gamma}\in H^{1/2}(\Gamma)$ is surjective, the above line reads $\partial_\nu u+\alpha u_{|\Gamma}=0$, showing that \eqref{vf} is the variational formulation of the following boundary value problem (BVP): \[
(-\Delta +q+\mu)u=f\; \mathrm{in}\; \Omega,\quad \partial_\nu u+\alpha u_{|\Gamma}=0\; \mathrm{on}\; \Gamma. \] Thus, taking $\mu=\lambda_k$ for all $k \in \mathbb{N}$, we find that $\phi_k\in W$ satisfies \begin{equation}\label{ee}
(-\Delta +q-\lambda_k)\phi_k=0\; \mathrm{in}\; \Omega,\quad \partial_\nu \phi_k+\alpha \phi_k{_{|\Gamma}}=0\; \mathrm{on}\; \Gamma. \end{equation}
\subsection{Statement of the results} We stick to the notations of the previous sections, that is to say that we write $\tilde{\lambda}_k$ (resp., $\tilde{\phi}_k$, $\tilde{\psi}_k$), $k \in \mathbb{N}$, instead of $\lambda_k$ (resp., $\phi_k$, $\psi_k$) when the potential $\tilde{q}$ is substituted for $q$. Our first result is as follows.
\begin{theorem}\label{theorem1} Let $q$ and $\tilde{q}$ be in $L^{r}(\Omega,\mathbb{R})$, where $r=n/2$ when $n \ge 4$ and $r >n/2$ when $n=3$, and let $\ell \in \mathbb{N}$. Then, the conditions \[ \lambda_k=\tilde{\lambda}_k\ \mbox{for\ all}\ k \ge \ell\quad \mbox{and}\quad \psi_k=\tilde{\psi}_k\ \mbox{on}\ \Gamma\ \mbox{for\ all}\ k\ge 1, \] yield that $q=\tilde{q}$ in $\Omega$. \end{theorem} The claim of Theorem \ref{theorem1} was first established for smooth bounded potentials, in the peculiar case where $\ell=1$, by Nachman, Sylvester and Uhlmann in \cite{NSU}. In the same context (of smooth bounded potentials), their result was extended to $\ell \geq 1$ through a heuristic approach in \cite{Sm}.
In view of stating our stability results, we denote by $\ell^\infty$ (resp. $\ell^2$) the Banach (resp., Hilbert) space of bounded (resp. squared summable) sequences of complex numbers $(z_k)$ , equipped with the norm \[
\|(z_k)\|_{\ell^\infty}:=\sup_{k\ge 1}|z_k|\ \left({\rm resp.,}\ \|(z_k)\|_{\ell^2}:=\left(\sum_{k\ge 1}|z_k|^2\right)^{1/2}\right), \] and let \[
\ell^2(L^2(\Gamma)):=\left\{ (w_k) \in L^2(\Gamma)^{\mathbb N}\; \mbox{such that}\; (\|w_k\|_{L^2(\Gamma)})\in \ell^2 \right\} \] be endowed with its natural norm \[
\|(w_k)\|_{\ell^2(L^2(\Gamma))}:=\|(\|w_k\|_{L^2(\Gamma)})\|_{\ell^2}. \]
\begin{theorem}\label{theorem2} Fix $\aleph \in (0,\infty)$ and let
$(q,\tilde{q})\in \mathrm{Q}(r,\aleph)^2$, where $r=n/2$ when $n \ge 4$ and $r >n/2$ when $n = 3$, satisfy $q-\tilde{q} \in L^2(\Omega)$. Assume that $(\lambda_k-\tilde{\lambda}_k)\in \ell^\infty$ fulfills $\|(\lambda_k-\tilde{\lambda}_k)\|_{\ell^\infty}\le \aleph$ and that $(\psi_k-\tilde{\psi}_k)\in \ell^2(L^2(\Gamma))$. Then, we have \[
\|q-\tilde{q}\|_{H^{-1}(\Omega)}\le C\left( \|(\lambda_k-\tilde{\lambda}_k)\|_{\ell^\infty}+ \|(\psi_k-\tilde{\psi}_k)\|_{\ell^2(L^2(\Gamma))} \right)^{2(1-2\beta)/(3(n+2))}, \] where $\beta:=\max \left( 0,n(2-r)/(2r) \right)$ and $C$ is a positive constant depending only on $n$, $\Omega$, $\aleph$ and $\mathfrak{c}$. \end{theorem}
\begin{remark}\label{remark1} {\rm (i) It is worth noticing that we have $\beta=0$ when $n \ge 4$, whereas $\beta \in [0,1/2)$ when $n=3$. Moreover, in the latter case we see that $\beta$ converges to $1/2$ (resp., $0$) as $r$ approaches $3/2$ (resp., $2$) from above (resp., below). \\ (ii) We have $q-\tilde{q} \in L^2(\Omega)$ for all $(q,\tilde{q}) \in \mathrm{Q}(n/2,\aleph)^2$, provided that $n \ge 4$. Nevertheless, this is no longer true when $n=3$, even if $(q,\tilde{q})$ is taken in $\mathrm{Q}(r,\aleph)^2$ with $r \in (n/2,2)$. Hence the additional requirement of Theorem \ref{theorem2} that $q-\tilde{q} \in L^2(\Omega)$ in the three-dimensional case.\\
(iii) When $q-\tilde{q} \in L^\infty(\Omega)$, we have $(\lambda_k-\tilde{\lambda}_k)\in \ell^\infty$ and $\|(\lambda_k-\tilde{\lambda}_k)\|_{\ell^\infty}\le \|q-\tilde{q}\|_{L^\infty(\Omega)}$, by the min-max principle. Thus, Theorem \ref{theorem2} remains valid by replacing the condition $\|(\lambda_k-\tilde{\lambda}_k)\|_{\ell^\infty}\le \aleph$ by the stronger assumption $\|q-\tilde{q}\|_{L^\infty(\Omega)}\le \aleph$. } \end{remark}
To the best of our knowledge, there is no comparable stability result available in the mathematical literature for Robin boundary conditions, even when the potentials are assumed to be bounded. Nevertheless, it should be pointed out that the variable coefficients case was recently addressed by \cite{BCKPS} in the framework of Dirichlet boundary conditions.
Further downsizing the data needed for retrieving the unknown potential, we seek a stability inequality requesting a local Dirichlet boundary measurement of the eigenfunctions only, i.e. boundary observation of the $\psi_k$'s and $\tilde{\psi}_k$'s that is performed on astrict subset of $\Gamma$. For this purpose we consider a subdomain $\Omega_0$ of $\Omega$ such that $\overline{\Omega}_0$ is a neighborhood of $\Gamma$ in $\overline{\Omega}$, a fixed nonempty open subset $\Gamma_{\ast}$ of $\Gamma$, and for all $\vartheta \in (0,\infty)$ we introduce the function $\Psi_\vartheta : [0,\infty) \to \mathbb{R}$ as \begin{equation} \label{def-Phi} \Phi_\vartheta(t):= \left\{ \begin{array}{cl} 0 & \mbox{if}\ t=0 \\
|\ln t|^{-\vartheta} & \mbox{if}\ t \in (0,1/e) \\ t & \mbox{if}\ t \in [1/e,\infty). \end{array} \right. \end{equation} The corresponding local stability estimate can be stated as follows.
\begin{theorem} \label{theorem3} For $\aleph \in (0,\infty)$ fixed, let $(q,\tilde{q}) \in \mathrm{Q}(n,\aleph)^2$ satisfy $q=\tilde{q}$ on $\Omega_0$. Assume that $\alpha \in C^{0,1}(\Gamma)$, and suppose that $(\lambda_k-\tilde{\lambda}_k)\in \ell^\infty$ and that $(k^{\mathfrak{t}}(\psi_k-\tilde{\psi}_k))\in \ell^2(L^2(\Gamma))$ for some $\mathfrak{t}>4/n+1$, with \[
\|(\lambda_k-\tilde{\lambda}_k)\|_{\ell^\infty}\le \aleph,\quad \|(k^\mathfrak{t}(\psi_k-\tilde{\psi}_k))\|_{\ell^2(L^2(\Gamma))}\le \aleph. \]
Then there exist two constants $C>0$ and $\vartheta>0$, both of them depending only on $n$, $\Omega$, $\Omega_0$, $\Gamma_{\ast}$, $\aleph$, $\mathfrak{c}$ and $\|\alpha\|_{C^{0,1}(\Gamma)}$, such that we have: \begin{equation} \label{thm3}
\|q-\tilde{q}\|_{H^{-1}(\Omega)}\le C\Phi_\vartheta\left( \|(\lambda_k-\tilde{\lambda}_k)\|_{\ell^\infty}+ \|(k^{-\mathfrak{t}+2/n}(\psi_k-\tilde{\psi}_k))\|_{\ell^2(H^1(\Gamma_{\ast}))} \right). \end{equation} \end{theorem}
\begin{remark}\label{remark2} {\rm
Bearing in mind that the $k$-th eigenvalue, $k \ge 1$, of the unperturbed Dirichlet Laplacian (i.e. the operator $A$ associated with $q=0$ in $\Omega$ and $\alpha=0$ on $\Gamma$) scales like $k^{2/n}$ when $k$ becomes large, see e.g. \cite[Theorem III.36 and Remark III.37]{Be}, we obtain by combining the min-max principle with \eqref{ii1}, that for all $q\in \mathrm{Q}(n,\aleph)$, \begin{equation}\label{waf}
C^{-1}k^{2/n}\le 1+|\lambda_k|\le Ck^{2/n},\ k\ge 1, \end{equation}
where $C \in (1,\infty)$ is a constant depending only on $n$, $\Omega$, $\mathfrak{c}$ and $\aleph$. In light of Lemma \ref{lemma5.0} below, establishing the $H^2$-regularity of the eigenfunctions $\phi_k$, $k \ge 1$, of $A$, and the energy estimate \eqref{5.1}, it follows from \eqref{waf} that $(k^{-\mathfrak{t}+2/n}\psi_k)\in \ell^2(H^1(\Gamma))$. Therefore, we have $\|(k^{-\mathfrak{t}+2/n}(\psi_k-\tilde{\psi}_k))\|_{\ell^2(H^1(\Gamma_{\ast}))}<\infty$ on the right hand side of \eqref{thm3}. } \end{remark}
\subsection{A short bibliography of the existing literature} \label{sec-literature} The first published uniqueness result for the multidimensional Borg-Levinson problem can be found in \cite{NSU}. The breakthrough idea of the authors of this article was to relate the inverse spectral problem under analysis to the one of determining the bounded potential by the corresponding elliptic Dirichlet-to-Neumann map. This can be understood from the fact that, the Schwartz kernel of the elliptic Dirichlet-to-Neumann operator can be, at least heuristically, fully expressed in terms of the eigenvalues and the normal derivatives of the eigenfunctions. Later on, \cite{Is} proved that the result of \cite{NSU}, which assumes complete knowledge of the boundary spectral data, remains valid when finitely many of them remain unknown.
The stability issue for multidimensional Borg-Levinson type problems was first examined in \cite{AS}. The authors proceed by relating the spectral data to the corresponding hyperbolic Dirichlet-to-Neumann operator, which stably determines the bounded electric potential. We refer the reader to \cite{BCY1,BCY2,BD} for alternative inverse stability results based on this approach.
In all the aforementioned results, the number of unknown spectral data is at most finite (that is to say that the data are either complete or incomplete). Nevertheless, it was proved in \cite{CS} that asymptotic knowledge of the boundary spectral data is enough to H\"older stably retrieve the bounded potential. This result was improved in \cite{KKS,So} by removing all quantitative information on the eigenfunctions of the stability inequality, at the expense of an additional summability condition on their boundary measurements.
In all the articles cited above in this section, the unknown potential is supposed to be bounded. The unique determination of unbounded potentials by either complete or incomplete boundary spectral data is discussed in \cite{PS, Po}, whereas the stability issue for the same problem, but in the variable coefficients case, is examined in \cite{BCKPS}. As for the treatment of the inverse problem of determining the unbounded potential from asymptotic knowledge of the spectral data, we refer the reader to \cite{BKMS} for the uniqueness issue, and to \cite{KS} for the stability issue.
All the above mentioned results were obtained for multidimensional Laplace operators endowed with Dirichlet boundary conditions, except for \cite{NSU} which proved that full knowledge of the boundary spectral data of the Robin Laplacian uniquely determines the unknown electric potential. But, apart from the claim, based on a heuristic approach, of \cite{Sm}, that incomplete knowledge of the spectral data of the multidimensional Robin Laplacian uniquely determines the unknown bounded potential, it seems that, even for a bounded unknown potential $q$, there is no reconstruction result of $q$ by incomplete spectral data, available in the mathematical literature for such operators. In the present article we prove not only unique identification by incomplete spectral data, but also stable determination by either full or local boundary spectral data, of the singular potential of the multidimensional Robin Laplacian.
\subsection{Outline} The remaining part of this paper is structured as follows. In Section \ref{sec-pre} we gather several technical results which are needed by the proof of the three main results of this article. Then we proceed with the proof of Theorems \ref{theorem1}, \ref{theorem2} and \ref{theorem3} in Section \ref{sec-proof}.
\section{Preliminaries} \label{sec-pre}
In this section we collect several preliminary results that are needed by the proof of the main results of this article. We start by noticing, upon applying \eqref{co} with $u=\phi_k$, $k\ge 1$, that \begin{equation}\label{lb} \lambda_k>-\lambda^\ast,\quad k\ge 1. \end{equation}
\subsection{Resolvent estimates}
By \cite[Corollary 2.39]{Mc}, the operator $A-\lambda : V \to V^\ast$ has a bounded inverse whenever $\lambda \in \rho (A):=\mathbb{C}\setminus \sigma(A)$, the resolvent set of $A$. Furthermore, for all $f \in V^\ast$ we have \begin{equation}\label{rf} (A-\lambda)^{-1}f=\sum_{k\ge 1} \frac{\langle f , \phi_k \rangle}{\lambda_k-\lambda}\phi_k, \end{equation} where the series converges in $V$.
For further use, we now establish that the resolvent $(A-\lambda)^{-1}$ may be regarded as a bounded operator from $H$ into the space $K:=\{u\in H;\; Au \in H\}$ endowed with the norm \[
\|u\|_K:=\|u\|_H+\|Au\|_H,\quad u\in K. \]
\begin{lemma} \label{lemma1} For all $\lambda \in \rho(A)$, the operator $(A-\lambda)^{-1}$ is bounded from $H$ into $K$. \end{lemma} \begin{proof} Put $u:=(A-\lambda)^{-1}f$ where $f\in H$ is fixed. Then, we have $(u , \phi_k )=(f,\phi_k)/(\lambda_k-\lambda)$ for all $k \ge 1$, from \eqref{rf}, whence
\begin{equation} \label{es8} Au=\sum_{k\ge1} \frac{\lambda_k}{\lambda_k-\lambda} (f,\phi_k) \phi_k, \end{equation} according to \cite[Theorem 2.37]{Mc}, the series being convergent in $V^\ast$. Moreover, since
$$\sum_{k\ge 1} \frac{\lambda_k^2}{|\lambda_k-\lambda|^2} | (f,\phi_k) |^2 \le \| (\lambda_k / (\lambda_k-\lambda)) \|_{\ell^\infty}^2 \| f \|_H^2<\infty, $$
by the Parseval theorem, the right hand side on \eqref{es8} lies in $H$. Therefore, we have $Au\in H$ and $\| A u \|_H \le \| (\lambda_k / (\lambda_k-\lambda)) \|_{\ell^\infty} \| f \|_H$, and consequently
$u \in K$ and $\| u \|_K \le \| ((1+\lambda_k) / (\lambda_k-\lambda)) \|_{\ell^\infty} \| f \|_H$. \end{proof}
\begin{proposition}\label{proposition1} Let $q\in \mathrm{Q}(n/2,\aleph)$ and let $\lambda \in \rho(A)$. Then, for all $f \in V^\ast$, the following estimate \begin{equation} \label{re2}
\|(A-\lambda)^{-1}f\|_V \le C \|((\lambda_k+\lambda^\ast)/(\lambda_k-\lambda))\|_{\ell^\infty} \|f\|_{V^\ast} \end{equation}
holds with $C=\kappa^{-1/2} \| (A+\lambda_*)^{-1}\|_{\mathcal{B}(V^\ast,V)}$, where $\mathcal{B}(V^\ast,V)$ denotes the space of linear bounded operators from $V^\ast$ to $V$. Moreover, in the special case where $f \in H$, we have \begin{equation}
\label{re1}
\|(A-\lambda)^{-1}f\|_H\le \|(1/(\lambda_k-\lambda))\|_{\ell^\infty}\|f\|_H.
\end{equation}
\end{proposition}
\begin{proof} Since \eqref{re1} follows directly from \eqref{rf} and the Parseval formula, it is enough to prove \eqref{re2}. To this purpose we set $u:=(A-\lambda)^{-1} f$ and notice from the obvious identity $\Delta u = (q-\lambda) u - f \in V^\ast$ that $u \in W$. Therefore, by applying \eqref{ggf} with $v=u$, we infer from the coercivity estimate \eqref{co} that \begin{equation} \label{es7}
\kappa \|u\|_V^2\le \langle (A+\lambda^\ast) u , u \rangle_{V^\ast,V}. \end{equation} Let us assume for a while that $f \in H$. Then, with reference to \eqref{es8}, we have $$ (A+\lambda^\ast) u = \sum_{k \ge 1} \frac{\lambda_k+\lambda^\ast}{\lambda_k-\lambda} (f,\phi_k) \phi_k, $$ where the series converges in $H$. It follows from this, \eqref{rf} and \eqref{es7} that \begin{eqnarray} \label{es9}
\kappa \|u\|_V^2 & \le & \sum_{k \ge 1} \frac{\lambda_k+\lambda^\ast}{|\lambda_k-\lambda|^2} |(f,\phi_k)|^2 \\
& \le & \| ( (\lambda_k+\lambda^\ast) / (\lambda_k-\lambda) ) \|_{\ell^\infty}^2
\sum_{k \ge 1} \frac{|(f,\phi_k)|^2}{\lambda_k+\lambda^\ast}. \nonumber \end{eqnarray} Further, taking into account that
$$ \sum_{k \ge 1} \frac{|(f,\phi_k)|^2}{\lambda_k+\lambda^\ast} = \| (A+\lambda^\ast)^{-1} f \|_H^2$$
according to \eqref{rf} and the Parseval formula, and then using that $\| (A+\lambda^\ast)^{-1} f \|_H \le \| (A+\lambda_*)^{-1}\|_{\mathcal{B}(V^\ast,V)} \| f \|_{V^\ast}$, we infer from \eqref{es9} that \begin{equation} \label{es9b}
\|u\|_V \leq \kappa^{-1/2} \| (A+\lambda_*)^{-1}\|_{\mathcal{B}(V^\ast,V)} \| ( (\lambda_k+\lambda^\ast) / (\lambda_k-\lambda) ) \|_{\ell^\infty}\| f \|_{V^\ast}. \end{equation} Finally, keeping in mind that $u=(A-\lambda)^{-1} f$ and that $(A-\lambda)^{-1} \in \mathcal{B}(V^\ast,V)$, \eqref{re2} follows readily from \eqref{es9b} by density of $H$ in $V^\ast$. \end{proof}
As a byproduct of Proposition \ref{proposition1}, we have the following:
\begin{corollary}\label{corollary1} Let $q\in \mathrm{Q}(n/2,\aleph)$. Then, for all $\tau \in [1,+\infty)$ we have \begin{equation} \label{re4}
\|(A-(\tau+i)^2)^{-1}f\|_H\le (2\tau)^{-1}\|f\|_H,\ f \in H. \end{equation} Moreover, for all $\tau \ge \tau_\ast=:1+(\max(0,2-\lambda^\ast))^{1/2}$, we have \begin{equation} \label{re5}
\|(A-(\tau+i)^2)^{-1}f\|_V \le C (\tau+\lambda^\ast) \|f\|_{V^\ast},\ f \in V^{\ast}, \end{equation} where $C$ is the same constant as in \eqref{re2}. \end{corollary} \begin{proof} As \eqref{re4} is a straightforward consequence of \eqref{re1}, we shall only prove \eqref{re5}. To do that, we refer to \eqref{re2} and notice that \begin{equation} \label{es10}
\frac{\lambda_k+\lambda^\ast}{|\lambda_k-(\tau+i)^2|}= \frac{\lambda_k+\lambda^\ast}{\left( (\lambda_k-(\tau^2-1))^2+4\tau^2 \right)^{1/2}} \leq 2 \Theta(\lambda_k),\ k \geq 1, \end{equation} where we have set
$\Theta(t):=(t+\lambda^\ast)/ (|t-(\tau^2-1)|+ 2\tau)$ for all $t \in [-\lambda^\ast,\infty)$. Further, taking into account that $\Theta$ is a decreasing function on $[\tau^2-1,\infty)$, provided that $\tau \geq \tau_\ast$, we easily get that $$ \sup_{t \in [-\lambda^\ast,+\infty)} \Theta(t) \le \frac{\tau^2-1+\lambda^\ast}{2 \tau} \le \frac{\tau+\lambda^\ast}{2}, $$ which along with \eqref{re2} and \eqref{es10}, yields \eqref{re5}. \end{proof}
\begin{proposition}\label{proposition2} Let $q \in \mathrm{Q}(n/2,\aleph)$. Then, there exists a constant $C>0$, depending only on $n$, $\Omega$, $\mathfrak{c}$ and $\aleph$, such that for all $\sigma \in [0,1]$ and all $f \in L^{p_\sigma}(\Omega)$, we have \begin{equation}\label{re7}
\|(A-(\tau+i)^2)^{-1}f\|_{L^{p_\sigma^\ast}(\Omega)}\le C\tau ^{-1+2\sigma}\|f\|_{L^{p_\sigma}(\Omega)},\ \tau \in [\tau_\ast,\infty), \end{equation} where $p_\sigma:=2n / (n+2 \sigma)$ and $p_\sigma^\ast:=2n / (n-2 \sigma)$ is the conjugate integer to $p_\sigma$.
\end{proposition}
\begin{proof} In light of \eqref{re5}, we have for all $f \in L^p(\Omega)$, $$
\| (A-(\tau+i)^2)^{-1} f \|_{L^{p^\ast}(\Omega)} \le C \tau \| f \|_{L^p(\Omega)},\ \tau \in [\tau_\ast,\infty), $$ by the Sobolev embedding theorem, where $C$ is a positive constant depending only on $n$, $\Omega$, $\mathfrak{c}$ and $\aleph$. Thus, \eqref{re7} follows from this and \eqref{re4} by interpolating between $H=L^{p_0}(\Omega)$ and $L^p(\Omega)=L^{p_1}(\Omega)$ with the aid of the Riesz-Thorin theorem (see, e.g. \cite[Theorem IX.17]{RS2} \end{proof}
\subsection{Asymptotic spectral analysis}
Set $\mathfrak{H}:=H^2(\Omega)$ if $n\ne 4$ and put $\mathfrak{H}:=H^{2+\epsilon}(\Omega)$ for some arbitrary $\epsilon >0$, if $n=4$. We notice that $\mathfrak{H} \subset L^\infty(\Omega)$ and that the embedding is continuous, provided that $n=3$ or $n=4$, while $\mathfrak{H}$ is continuously embedded in $L^{2n/(n-4)}(\Omega)$ when $n>4$. The main purpose for bringing $\mathfrak{H}$ into the analysis here is the following useful property: $fu\in H$ whenever $f\in L^{\max(2,n/2)}(\Omega)$ and $u\in \mathfrak{H}$.
Next we introduce the subspace \[
\mathfrak{h}:=\{ g=\partial_\nu G +\alpha G_{|\Gamma};\; G\in \mathfrak{H}\} \] of $L^2(\Gamma)$, equipped with its natural quotient norm \[
\|g\|_{\mathfrak{h}}:=\min\{ \|G\|_{\mathfrak{H}};\; G\in \dot{g}\},\quad g\in \mathfrak{h}, \] where \[
\dot{g}:=\{ G\in \mathfrak{H};\; \partial_\nu G+\alpha G_{|\Gamma}=g\},\quad g\in \mathfrak{h}, \] and we consider the non homogenous BVP: \begin{equation}\label{bvp1}
(-\Delta +q-\lambda )u=0\; \mathrm{in}\; \Omega ,\quad \partial_\nu u+\alpha u_{|\Gamma} =g\; \mathrm{on}\; \Gamma . \end{equation}
We first examine the well-posedness of \eqref{bvp1}.
\begin{lemma}\label{lemma2} Let $\lambda\in \rho(A)$ and let $g\in \mathfrak{h}$. Then, the function \begin{equation}\label{sol1} u_\lambda (g):=(A-\lambda )^{-1}(\Delta -q+\lambda)G+G \end{equation} is independent of $G\in \dot{g}$. Moreover, $u_\lambda (g)\in W$ is the unique solution to \eqref{bvp1} and is expressed as \begin{equation}\label{rep1} u_\lambda (g)=\sum_{k\ge 1}\frac{(g,\psi_k)}{\lambda_k-\lambda}\phi_k \end{equation} in $H$, where $(\cdot,\cdot)$ denotes the usual scalar product in $L^2(\Gamma)$. \end{lemma}
\begin{proof} Since $G\in \mathfrak{H}$, it is clear that $(\Delta -q+\lambda)G\in H$. Thus, the right hand side of \eqref{sol1} lies in $W$ and it is obviously a solution to the BVP \eqref{bvp1}. Moreover, $\lambda$ being taken in the resolvent set of $A$, this solution is unique.
Further, for all $G_1$ and $G_2$ in $\dot{g}$, it is easy to check that $\partial_\nu (G_1-G_2) + \alpha (G_1-G_2)=0$ on $\Gamma$ and that $(A-\lambda )^{-1}(\Delta -q+\lambda)(G_1-G_2)=-(G_1-G_2)$ in $\Omega$. Therefore, the function $u_\lambda (g)$ given by \eqref{sol1}, is independent of $G\in \dot{g}$.
We turn now to showing \eqref{rep1}. To do that we apply the generalized Green formula \eqref{ggf} with $u=u_\lambda(g)$ and $v=\phi_k$, $k \geq 1$. We obtain \[
\langle \Delta u_\lambda(g) , \phi_k\rangle+(\nabla u_\lambda(g)|\nabla \phi_k)=\langle \partial_\nu u_\lambda(g) , \psi_k\rangle, \] which may be equivalently rewritten as \begin{equation}\label{2.1}
((q-\lambda) u_\lambda(g),\phi_k)+(\nabla u_\lambda(g),\nabla \phi_k)=\langle g-\alpha u_\lambda(g)_{|\Gamma} ,\psi_k\rangle. \end{equation} Doing the same with $u=\phi_k$ and $v=u_\lambda(g)$, and taking the conjugate of both sides of the obtained equality, we find that
$$( u_\lambda(g), (q-\lambda_k) \phi_k)+( \nabla u_\lambda(g), \nabla \phi_k)=-\langle u_\lambda(g)_{|\Gamma} , \alpha \psi_k \rangle. $$ Bearing in mind that $q$ and $\alpha$ are real-valued, and that $\lambda_k \in \mathbb{R}$, this entails that \begin{equation}\label{2.2}
((q-\lambda_k)u_\lambda(g),\phi_k)+(\nabla u_\lambda(g),\nabla \phi_k)=-\langle \alpha u_\lambda(g)_{|\Gamma} , \psi_k\rangle. \end{equation} Now, taking the difference of \eqref{2.1} with \eqref{2.2}, we end up getting that \[ (\lambda_k-\lambda )(u_\lambda(g),\phi_k)=\langle g , \psi_k \rangle=(g,\psi_k). \] This and the basic identity \[ u_\lambda(g)=\sum_{k\ge 1}(u,\phi_k)\phi_k \] yield \eqref{rep1}. \end{proof}
The series on the right hand side of \eqref{rep1} converges only in $H$ and thus we cannot deduce an expression of the trace $u_\lambda(g)_{| \Gamma}$ in terms of $\lambda_k$ and $\psi_k$, $k \geq 1$, directly from \eqref{rep1}. To circumvent this difficulty we establish the following lemma:
\begin{lemma}\label{lemma4} Let $g\in \mathfrak{h}$. Then, for all $\lambda$ and $\mu$ in $\rho(A)$, we have \begin{equation}\label{s2}
u_\lambda(g){_{|\Gamma}} -u_\mu(g){_{|\Gamma}}=(\lambda-\mu)\sum_{k\ge 1}\frac{(g,\psi_k)}{(\lambda_k-\lambda)(\lambda_k-\mu)}\psi_k, \end{equation} and the series converges in $H^{1/2}(\Gamma)$. \end{lemma}
\begin{proof} Notice that \[ (-\Delta +q-\lambda )(u_\lambda -u_\mu)=(\lambda-\mu)u_\mu \]
in $\Omega$ and that $\partial_\nu(u_\lambda -u_\mu)+\alpha(u_\lambda -u_\mu)_{|\Gamma}=0$ on $\Gamma$, where, for shortness sake, we write $u_\lambda =u_\lambda (g)$ and $u_\mu=u_\mu(g)$. Thus, we have \[ u_\lambda -u_\mu=(\lambda-\mu)(A-\lambda)^{-1}u_\mu =(\lambda-\mu)\sum_{k\ge 1}\frac{(u_\mu,\phi_k)}{\lambda_k-\lambda}\phi_k. \] On the other hand, since \[ (u_\mu,\phi_k)=\frac{(g,\psi_k)}{\lambda_k-\mu},\ k \geq 1, \] from \eqref{rep1}, we obtain that \begin{equation}\label{s1} u_\lambda -u_\mu=(\lambda-\mu)\sum_{k\ge 1}\frac{(g,\psi_k)}{(\lambda_k-\lambda)(\lambda_k-\mu)}\phi_k. \end{equation} Moreover, we have \[ \sum_{k\ge 1}\frac{(g,\psi_k)}{(\lambda_k-\lambda)(\lambda_k-\mu)}(A-\lambda)\phi_k=\sum_{k\ge 1}\frac{(g,\psi_k)}{\lambda_k-\mu}\phi_k, \] the series being convergent in $H$. It follows from this and \eqref{s1} that \[ u_\lambda -u_\mu=(\lambda-\mu) (A-\lambda)^{-1}\sum_{k\ge 1}\frac{(g,\psi_k)}{\lambda_k-\mu}\phi_k, \] where the series on the right hand side of \eqref{s1} converges in $V$. As a consequence we have \begin{equation}
u_\lambda{_{|\Gamma}} -u_\mu{_{|\Gamma}}=(\lambda-\mu)\sum_{k\ge 1}\frac{(g,\psi_k)}{(\lambda_k-\lambda)(\lambda_k-\mu)}\psi_k, \end{equation} the series being convergent in $H^{1/2}(\Gamma)$. \end{proof}
Next, we establish the following {\it a priori} estimate for the solution to \eqref{bvp1}.
\begin{lemma}\label{lemma3} Let $q\in \mathrm{Q}(n/2,\aleph)$. Then, there exist two constants $\lambda_+>0$ and $C>0$, depending only on $n$, $\Omega$, $\aleph$ and $\mathfrak{c}$, such that for all $\lambda \in (-\infty,-\lambda_+]$ and all $g\in \mathfrak{h}$, the solution $u_\lambda (g)$ to \eqref{bvp1} satisfies the estimate \begin{equation}\label{lim1}
|\lambda|^{1/2} \|u_\lambda(g)\|_H+\|u_\lambda (g)\|_V\le C\|g\|_{L^2(\Gamma)}. \end{equation} \end{lemma}
\begin{proof} Fix $\lambda \in \rho(A)\cap (-\infty ,0)$. We apply the generalized Green formula \eqref{ggf} with $u=v:=u_\lambda$, where we write $u_\lambda$ instead of $u_\lambda(g)$. We get that \begin{equation}\label{ae1}
|\lambda| \|u_\lambda\|_H^2+\|\nabla u_\lambda\|_H^2 \leq \|qu_\lambda ^2\|_{L^1(\Omega)}-(\alpha u_\lambda,u_\lambda)+(g,u_\lambda). \end{equation} Next, $\epsilon$ being fixed in $(0,+\infty)$, we combine \eqref{ii1} with \eqref{ae1} and obtain \begin{equation}\label{ae2}
|\lambda |\|u_\lambda\|_H^2+\|\nabla u_\lambda\|_H^2\le \epsilon\|u_\lambda\|_V^2+C_\epsilon \|u\|_H^2+\mathfrak{c}\mathfrak{n} ^2\|u\|_V^2+\mathfrak{n} \|g\|_{L^2(\Gamma)}\|u_\lambda\|_V, \end{equation} where $C_\epsilon$ is a positive constant depending only on $n$, $\Omega$, $\aleph$ and $\epsilon$. Taking $\epsilon = \kappa =(1- \mathfrak{c}\mathfrak{n}^2)/2$ in \eqref{ae2} then yields \[
(|\lambda |-1-C_\kappa)\|u_\lambda\|_H^2+\kappa\|u_\lambda\|_V^2\le \mathfrak{n} \|g\|_{L^2(\Gamma)}\|u_\lambda\|_V. \] As a consequence we have
$$ | \lambda | \| u_\lambda \|_H^2 + \| u_\lambda \|_V^2 \leq \frac{2\mathfrak{n}^2}{\kappa^2} \|g\|_{L^2(\Gamma)}^2, $$
whenever $|\lambda| \geq (1+C_\kappa) \slash (1 - \kappa \slash 4)$, and \eqref{lim1} follows readily from this. \end{proof}
Armed with Lemma \ref{lemma3} we can examine the dependence of (the trace of) the solution to the BVP \eqref{bvp1} with respect to $q$. More precisely, we shall establish that the influence of the potential on $u_\lambda(g)$ is, in some sense, dimmed as the spectral parameter $\lambda$ goes to $-\infty$. \begin{lemma}\label{lemma5.1} Let $q$ and $\tilde{q}$ be in $\mathrm{Q}(n/2,\aleph)$.
Then, for all $g\in \mathfrak{h}$, we have \begin{equation}\label{lim2}
\lim_{\lambda=\Re \lambda \rightarrow -\infty}\|u_\lambda(g)_{|\Gamma}-\tilde{u}_\lambda(g)_{|\Gamma}\|_{H^{1/2}(\Gamma)}=0. \end{equation} \end{lemma}
\begin{proof} Let $\lambda \in (-\infty,-\lambda_+]$, where $\lambda_+$ is the same as in Lemma \ref{lemma3}. We use the same notation as in the proof of Lemma \ref{lemma3} and write $u_\lambda$ (resp., $\tilde{u}_\lambda$) instead of $u_\lambda(g)$ (resp., $\tilde{u}_\lambda(g)$). Since \[ (-\Delta +q-\lambda )(u_\lambda - \tilde{u}_\lambda)=(\tilde{q}-q)\tilde{u}_\lambda\quad \mathrm{in}\; \Omega \] and \[
\partial_\nu(u_\lambda - \tilde{u}_\lambda)+\alpha (u_\lambda - \tilde{u}_\lambda)_{|\Gamma}=0\quad \mbox{on}\; \Gamma, \] we have $$ u_\lambda - \tilde{u}_\lambda=(A-\lambda)^{-1} ((\tilde{q}-q)\tilde{u}_\lambda), $$ whence \begin{equation} \label{a1}
\| u_\lambda - \tilde{u}_\lambda \|_V \le C \|((\lambda_k+\lambda^\ast)/(\lambda_k-\lambda))\|_{\ell^\infty} \| (\tilde{q}-q)\tilde{u}_\lambda\|_{V^\ast}, \end{equation} by \eqref{re2}, where $C$ is a positive constant which is independent of $\lambda$.
We are left with the task of estimating $\| (\tilde{q}-q)\tilde{u}_\lambda\|_{V^\ast}$. For this purpose, we notice from $\tilde{q}-q \in L^{n/2}(\Omega)$ and from $\tilde{u}_\lambda \in L^{p^\ast}(\Omega)$ that $(\tilde{q}-q)\tilde{u}_\lambda \in L^p(\Omega)$. Thus, bearing in mind that the embedding $V \subset L^{p^\ast}(\Omega)$ is continuous, we infer from H\"older's inequality that \begin{eqnarray*}
\| (\tilde{q}-q)\tilde{u}_\lambda\|_{V^\ast}
& \le & \| \tilde{q}-q \|_{L^{n/2}(\Omega)} \| \tilde{u}_\lambda \|_{L^{p^\ast}(\Omega)} \\
& \le & 2 \aleph \| \tilde{u}_\lambda \|_{V}. \end{eqnarray*} In light of \eqref{lim1}, this entails that
$$\| (\tilde{q}-q)\tilde{u}_\lambda\|_{V^\ast} \leq C \|g\|_{L^2(\Gamma)},$$
for some constant $C$ depending only on $n$, $\Omega$, $\aleph$ and $\mathfrak{c}$. From this, \eqref{a1} and the continuity of the trace operator $w\in V\mapsto w_{|\Gamma}\in H^{1/2}(\Gamma)$, we obtain that \[
\|(u_\lambda)_{| \Gamma}-(\tilde{u}_\lambda)_{|\Gamma}\|_{H^{1/2}(\Gamma)}\le C \|((\lambda_k+\lambda^\ast)/(\lambda_k-\lambda))\|_{\ell^\infty} \|g\|_{L^2(\Gamma)}, \] where $C$ is independent of $\lambda$. Now \eqref{lim2} follows immediately from this upon sending $\lambda$ to $-\infty$ on both sides of the above inequality. \end{proof}
\subsection{$H^2$-regularity of the eigenfunctions} For all $q \in L^{n/2}(\Omega)$, we have $\phi_k \in V$, $k \ge 1$, but it is no guaranteed in general that $\phi_k \in H^2(\Omega)$. Nevertheless, we shall establish that the regularity of the eigenfunctions of $A$ can be upgraded to $H^2$, provided that the potential $q$ is taken in $L^n(\Omega)$.
\begin{lemma}\label{lemma5.0} Let $q\in \mathrm{Q}(n,\aleph)$ and assume that $\alpha\in C^{0,1}(\Gamma)$. Then, for all $k\in \mathbb{N}$, we have $\phi_k\in H^2(\Omega)$ and the estimate \begin{equation}\label{5.1}
\|\phi_k\|_{H^2(\Omega)}\le C(1+|\lambda_k|), \end{equation}
where $C$ is a positive constant depending on $n$, $\Omega$ and $\aleph$ and $\|\alpha\|_{C^{0,1}(\Gamma)}$. \end{lemma}
\begin{proof} Let us start by noticing from \eqref{co} that \begin{equation}\label{5.0}
\|\phi_k\|_V\le \kappa^{-1/2}(\lambda_k+\lambda ^\ast)^{1/2},\ k \geq 1. \end{equation} On the other hand we have $q\phi_k\in H$ for all $k\in \mathbb{N}$, and the estimate
\begin{equation}\label{5.0.1}
\|q\phi_k\|_H\le \|q\|_{L^n(\Omega)}\|\phi_k\|_{L^{p^\ast}(\Omega)}\le C_0 \|\phi_k\|_V,
\end{equation} where $C_0$ is a positive constant depending only on $n$, $\Omega$, $\mathfrak{c}$ and $\aleph$.
Next, bearing in mind that $\alpha \phi_k{_{|\Gamma}} \in H^{1/2}(\Gamma)$, we pick
$\phi_k^0\in H^2(\Omega)$ such that $\partial_\nu \phi_k^0=\alpha \phi_k{_{|\Gamma}}$. Evidently, we have
\[
-\Delta (\phi_k+\phi_k^0)=(\lambda_k-q)\phi_k-\Delta\phi_k^0\; \mbox{in}\; \Omega \quad \mbox{and} \quad \partial_\nu(\phi_k+\phi_k^0)=0\; \mbox{on}\; \Gamma.
\] Since $(\lambda_k-q)\phi_k-\Delta\phi_k^0\in H$, \cite[Theorem 3.17]{Tr} then yields that $\phi_k+\phi_k^0\in H^2(\Omega)$. As a consequence we have $\phi_k=(\phi_k+\phi_k^0) -\phi_k^0 \in H^2(\Omega)$ and \[
\|\phi_k\|_{H^2(\Omega)}\le C_1(\|(\lambda_k-q)\phi_k\|_H+\|\phi_k\|_V) \]
for some constant $C_1>0$ which depends only on $n$, $\Omega$ and $\|\alpha\|_{C^{0,1}(\Gamma)}$, by \cite[Lemma 3.181]{Tr} (see also \cite[Theorem 2.3.3.6]{Gr}). Putting this together with \eqref{5.0}-\eqref{5.0.1}, we obtain \eqref{5.1}. \end{proof}
\section{Proof of Theorems \ref{theorem1}, \ref{theorem2} and \ref{theorem3}} \label{sec-proof}
\subsection{Proof of Theorem \ref{theorem1}} \label{sec-prthm1}
We use the same notations as in the previous sections. Namely, we denote by $\tilde{A}$ is the operator generated in $H$ by $\mathfrak{a}$ where $\tilde{q}$ is substituted for $q$, and we write $u_\lambda$ (resp., $\tilde{u}_\lambda$) instead of $u_\lambda (g)$ (resp., $\tilde{u}_\lambda(g)$). Let $\lambda \in \mathbb{C}\setminus \mathbb{R}$ and pick $\mu$ in $\rho(A)\cap \rho(\tilde{A})$. Depending on whether $\ell=1$ or $\ell \ge 2$, we have either \[
(u_\lambda)_{|\Gamma} -(u_\mu)_{|\Gamma}=(\tilde{u}_\lambda)_{|\Gamma} - (\tilde{u}_\mu)_{|\Gamma} \] or \begin{eqnarray*}
& & (u_\lambda)_{|\Gamma} -(u_\mu)_{|\Gamma}-(\lambda-\mu) \sum_{k= 1}^{\ell-1}\frac{(g,\psi_k)}{(\lambda_k-\lambda)(\lambda_k-\mu)}\psi_k \\
&=& (\tilde{u}_\lambda)_{|\Gamma} -(\tilde{u}_\mu)_{|\Gamma}-(\lambda-\mu)\sum_{k= 1}^{\ell-1}\frac{(g,\psi_k)}{(\tilde{\lambda}_k-\lambda)(\tilde{\lambda}_k-\mu)}\psi_k, \end{eqnarray*} by virtue of \eqref{s2}. Sending $\Re \mu$ to $-\infty$ in these two identities, where $\Re \mu$ denotes the real part of $\mu$, we get with the help of \eqref{lim2} that \begin{equation}\label{RtoD}
(u_\lambda)_{|\Gamma}- (\tilde{u}_\lambda)_{|\Gamma}=R_\lambda^\ell, \end{equation} where \[ R_\lambda^\ell=R_\lambda^\ell (g) := \left\{ \begin{array}{ll} 0 & \mbox{if}\ \ell=1 \\ \sum_{k= 1}^{\ell-1}\frac{(\tilde{\lambda}_k -\lambda_k)(g,\psi_k)}{(\lambda_k-\lambda)(\tilde{\lambda}_k-\lambda)}\psi_k & \mbox{if}\ \ell \ge 2. \end{array} \right. \] Notice for further use that there exists $\lambda_*>0$ such that the estimate \begin{equation}\label{Re}
| \langle R_\lambda ^\ell ,h\rangle|\le \frac{C_\ell}{|\lambda|^2}\|g\|_{L^2(\Gamma)}\|h\|_{L^2(\Gamma)},\quad |\lambda |\ge \lambda_*,\quad g,h\in \mathfrak{h}, \end{equation} holds for some constant $C_\ell=C_\ell(q,\tilde{q})$ which is independent of $\lambda$.
Let us now consider two functions $G \in \mathfrak{H}$ and $H \in \mathfrak{H}$, that will be made precise below, and put $u:=(A-\lambda)^{-1}(\Delta -q+\lambda)G+G$, $g:=\partial_\nu G+\alpha G_{|\Gamma}$ and $h:=\partial_\nu H +\alpha H_{|\Gamma}$. Then, bearing in mind that $\partial_\nu u + u_{| \Gamma}=g$, the Green formula yields that \begin{equation} \label{eq-G} \int_\Gamma u\overline{h}ds(x)=\int_\Gamma g\overline{H}ds(x)+\int_\Omega (u\Delta\overline{H}-\Delta u\overline{H})dx. \end{equation} Further, taking into account that $\Delta u=(q-\lambda )u$ in $\Omega$, we see that \begin{eqnarray*} u\Delta\overline{H}-\Delta u\overline{H} & =& u (\Delta -q+\lambda )\overline{H} \\ & = & \left( (A-\lambda)^{-1}(\Delta -q+\lambda )G+G \right) (\Delta -q+\lambda )\overline{H}. \end{eqnarray*} Thus, assuming that $(\Delta +\lambda)G=(\Delta +\lambda)H=0$, the above identity reduces to \[ u\Delta\overline{H}-\Delta u\overline{H}= -\left( -(A-\lambda)^{-1}qG+G \right) q\overline{H}, \] and \eqref{eq-G} then reads \begin{equation}\label{id1} \int_\Gamma u\overline{h}ds(x)=\int_\Gamma g\overline{H}ds(x)-\int_\Omega \left( -(A-\lambda)^{-1}qG+G \right) q\overline{H} dx. \end{equation}
This being said, we set $\lambda_\tau:=(\tau+i)^2$ for some fixed $\tau \in [1,+\infty)$, pick two vectors $\omega$ and $\theta$ in $\mathbb{S}^{n-1}$, and we consider the special case where \[ G(x)=\mathfrak{e}_{\lambda_\tau,\omega}(x):=e^{i\sqrt{\lambda_\tau}\omega \cdot x},\quad \overline{H}(x)=\mathfrak{e}_{\lambda_\tau,-\theta}(x):= e^{-i\sqrt{\lambda_\tau}\theta \cdot x}. \] Next, we put $$ S(\lambda_\tau,\omega ,\theta) :=\int_\Gamma u_\lambda (g)\overline{h}ds(x),\ \quad \tilde{S}(\lambda_\tau,\omega ,\theta) :=\int_\Gamma \tilde{u}_\lambda (g)\overline{h}ds(x), $$ in such a way that \begin{equation} \label{eq-S} S(\lambda_\tau,\omega ,\theta)-\tilde{S}(\lambda,\omega ,\theta)=\langle R_{\lambda_\tau} ^\ell(g) , h\rangle. \end{equation} Then, taking into account that \[ g(x)=(i\sqrt{\lambda_\tau}\omega \cdot \nu+\alpha)e^{i\sqrt{\lambda_\tau}\omega \cdot x},\quad \overline{h}(x)= (-i\sqrt{\lambda_\tau}\theta \cdot \nu+\alpha)e^{-i\sqrt{\lambda_\tau}\theta \cdot x}, \]
we have $\|g\|_{L^2(\Gamma)}\|h\|_{L^2(\Gamma)}\le C\tau^2$ for some positive constant $C$ which is independent of $\omega$, $\theta$ and $\tau$, and we infer from \eqref{Re} and \eqref{eq-S} that \begin{equation} \label{S1} \lim_{\tau \rightarrow \infty}\sup_{\omega,\theta \in \mathbb{S}^{n-1}} \left( S(\lambda_\tau,\omega ,\theta)-\tilde{S}(\lambda_\tau,\omega ,\theta) \right)=0. \end{equation} On the other hand, \eqref{id1} reads \begin{equation} \label{S1b} S(\lambda_\tau,\omega ,\theta) =S_0(\lambda_\tau,\omega ,\theta) +\int_\Gamma(i\sqrt{\lambda_\tau}\omega \cdot\nu +\alpha)e^{-i\sqrt{\lambda_\tau}(\theta-\omega)\cdot x}ds(x), \end{equation} where \begin{equation} \label{id3} S_0(\lambda_\tau,\omega ,\theta):=\int_\Omega (A-\lambda_\tau)^{-1}(q\mathfrak{e}_{\lambda_\tau,\omega})q\mathfrak{e}_{\lambda_\tau,-\theta}dx-\int_\Omega qe^{-i\sqrt{\lambda_\tau}(\theta-\omega)\cdot x}dx. \end{equation}
Now, we fix $\xi$ in $\mathbb{R}^n$, pick $\eta \in \mathbb{S}^{n-1}$ such that $\xi \cdot \eta =0$, and for all $\tau \in \left( |\xi|/2,+\infty \right)$ we set \begin{equation} \label{es4}
\omega_\tau :=\left(1-|\xi|^2/(4\tau ^2)\right)^{1/2}\eta -\xi/(2\tau),\quad \theta_\tau :=\left(1-|\xi|^2/(4\tau ^2)\right)^{1/2}\eta +\xi/(2\tau) \end{equation} in such a way that \begin{equation} \label{id3b} \lim_{\tau \rightarrow +\infty} \sqrt{\lambda_\tau}(\theta_\tau-\omega_\tau) = \xi. \end{equation} Evidently, we have \begin{equation} \label{gs0}
\|\mathfrak{e}_{\lambda_\tau,\omega_\tau}\|_{L^\infty (\Omega)}\le \|e^{|x|} \|_{L^\infty (\Omega)},\quad \|\mathfrak{e}_{\lambda_\tau,-\theta_\tau}\|_{L^\infty (\Omega)}\le \|e^{|x|} \|_{L^\infty (\Omega)}. \end{equation} Next, with reference to the notations $\beta=\max \left( 0,n(2-r)/(2r) \right)$ and $p_\sigma=2n / (n+2\sigma)$, $\sigma \in [0,1]$, of Theorem \ref{theorem2} and Proposition \ref{proposition2}, respectively, we see that $\beta=0$ and hence that $p_\beta=p_0=2$, when $n \ge 4$, whereas $p_\beta=r \in (3/2,2)$, when $n=3$. Thus, we have $p_\beta\le r$ whenever $n \ge 3$, and consequently $q \in L^{p_\beta}(\Omega)$. It follows from this and \eqref{gs0} that $q\mathfrak{e}_{\lambda_\tau,\omega_\tau}$ and $q\mathfrak{e}_{\lambda_\tau,-\theta_\tau}$ lie in $L^{p_\beta}(\Omega)$ and satisfy the estimate \begin{equation} \label{gs1}
\|q\mathfrak{e}_{\lambda_\tau,\omega_\tau}\|_{L^{p_\beta}(\Omega)}+\|q\mathfrak{e}_{\lambda_\tau,-\theta_\tau} \|_{L^{p_\beta}(\Omega)}\le C \| q \|_{L^r(\Omega)},\quad \tau \in (|\xi|/2,\infty), \end{equation}
for some positive constant $C=C(n,\Omega)$ depending only on $n$ and $\Omega$. Moreover, for all $\tau \ge \max(|\xi|/2,\tau_\ast)$, we have \begin{eqnarray} \label{gm0}
& & \left| \int_\Omega (A-\lambda_\tau)^{-1}(q\mathfrak{e}_{\lambda_\tau,\omega_\tau})q\mathfrak{e}_{\lambda_\tau,-\theta_\tau}dx \right| \\
& \leq & \| (A-\lambda_\tau)^{-1}(q\mathfrak{e}_{\lambda_\tau,\omega_\tau}) \|_{L^{p_\beta^\ast}(\Omega)} \|q\mathfrak{e}_{\lambda_\tau,-\theta_\tau} \|_{L^{p_\beta}(\Omega)} \nonumber\\
& \le & C \tau^{-1+2\beta} \|q\mathfrak{e}_{\lambda_\tau,\omega_\tau}\|_{L^{p_\beta}(\Omega)} \|q\mathfrak{e}_{\lambda_\tau,-\theta_\tau} \|_{L^{p_\beta}(\Omega)},\ \nonumber \end{eqnarray} by \eqref{re7}, where $C>0$ is independent of $\tau$. Since $\beta \in [0,1/2)$ from its very definition, we infer from \eqref{gs1}-\eqref{gm0} that \begin{equation} \label{gs2}
\lim_{\tau \rightarrow \infty}\left| \int_\Omega (A-\lambda_\tau)^{-1}(q\mathfrak{e}_{\lambda_\tau,\omega_\tau})q\mathfrak{e}_{\lambda_\tau,-\theta_\tau}dx \right|=0, \end{equation} which together with \eqref{id3}-\eqref{id3b} yields that \[ \lim_{\tau \rightarrow \infty} S_0(\lambda_\tau ,\omega_\tau ,\theta_\tau) =-\int_\Omega qe^{-i\xi \cdot x},\quad \xi \in \mathbb{R}^n. \] From this and the identity \[ \lim_{\tau \rightarrow \infty}\left( S_0(\lambda_\tau ,\omega_\tau ,\theta_\tau)-\tilde{S}_0(\lambda_\tau ,\omega_\tau ,\theta_\tau) \right)=\lim_{\tau \rightarrow \infty} \left( S(\lambda_\tau,\omega_\tau ,\theta_\tau)-\tilde{S}(\lambda_\tau,\omega_\tau ,\theta_\tau) \right)=0, \] arising from \eqref{S1}-\eqref{S1b}, it then follows that \[ \int_\Omega (q-\tilde{q})e^{-i\xi \cdot x}dx=0,\quad \xi \in \mathbb{R}^n. \] Otherwise stated, the Fourier transform of $(q-\tilde{q})\chi_\Omega$, where $\chi_\Omega$ is the characteristic function of $\Omega$, is identically zero in $\mathscr{S}'(\mathbb{R}^n)$. By the injectivity of the Fourier transformation, this entails that $q=\tilde{q}$ in $\Omega$.
\subsection{Proof of Theorem \ref{theorem2}} \label{sec-prthm2} Pick $\omega$ and $\theta$ be in $\mathbb{S}^{n-1}$, and let $\lambda\in \mathbb{C}\setminus\mathbb{R}$. We use the same notations as in the proof of Theorem \ref{theorem1}. Namely, for all $x \in \Gamma$, we write $$ g(x)=g_\lambda(x)=(i\sqrt{\lambda}\omega \cdot \nu+\alpha)e^{i\sqrt{\lambda}\omega \cdot x},\ \overline{h}(x)=\overline{h}_\lambda(x)= (-i\sqrt{\lambda}\theta \cdot \nu+\alpha)e^{-i\sqrt{\lambda}\theta \cdot x}$$ and we recall that $S(\lambda,\omega ,\theta)=\int_\Gamma u_\lambda (g)\overline{h}ds(x)$. Next, for all $\mu \in \rho(A)\cap \rho(\tilde{A})$ we set \begin{equation} \label{4.0} T(\lambda ,\mu)=T(\lambda ,\mu,\omega ,\theta):=S(\lambda,\omega ,\theta)-S(\mu,\omega ,\theta)=\int_\Gamma \left(u_\lambda (g)-u_\mu (g) \right)\overline{h}ds(x). \end{equation} By Lemma \ref{lemma4}, we have
\[ T(\lambda ,\mu)= (\lambda-\mu)\sum_{k\ge 1}\frac{d_k}{(\lambda_k-\lambda)(\lambda_k-\mu)},\ d_k:=(g,\psi_k)(\psi_k,h), \]
and hence \begin{equation} \label{dt} T(\lambda ,\mu)-\tilde{T}(\lambda ,\mu)=U(\lambda ,\mu)+V(\lambda ,\mu), \end{equation} where \begin{eqnarray} U(\lambda ,\mu)&:=&\sum_{k\ge 1} \frac{\lambda-\mu}{\lambda_k-\mu} \frac{d_k-\tilde{d}_k}{\lambda_k-\lambda}, \label{du} \\ V(\lambda ,\mu)&:=&\sum_{k\ge 1}\left(\frac{\lambda-\mu}{(\lambda_k-\lambda)(\lambda_k-\mu)}-\frac{\lambda-\mu}{(\tilde{\lambda}_k-\lambda)(\tilde{\lambda}_k-\mu)}\right)\tilde{d}_k. \label{dv} \end{eqnarray}
Notice that for all $k \in \mathbb{N}$, we have $d_k-\tilde{d}_k=(g,\psi_k-\tilde{\psi}_k) (\psi_k,h)+ (g,\tilde{\psi}_k)(\psi_k-\tilde{\psi}_k,h)$, which immediately entails that
\begin{equation} \label{es0}
\frac{|d_k-\tilde{d}_k|}{|\lambda_k-\lambda|}\le \left(\frac{|(g|\psi_k)|}{|\lambda_k-\lambda|}\|h\|_{L^2(\Gamma)}+\rho_k(\lambda)\frac{|(\tilde{\psi}_k|h)|}{|\tilde{\lambda}_k-\lambda|}\|g\|_{L^2(\Gamma)}\right)\|\psi_k-\tilde{\psi}_k\|_{L^2(\Gamma)}, \end{equation}
where $\rho_k(\lambda):=|\tilde{\lambda}_k-\lambda|/|\lambda_k-\lambda|$. Further, since $0 \le \rho_k(\lambda) \le 1+|\lambda_k-\tilde{\lambda}_k| / |\lambda_k-\lambda|$ and $(\lambda_k-\tilde{\lambda}_k)\in \ell^\infty$ by assumption, with $\|(\lambda_k-\tilde{\lambda}_k)\|_{\ell^\infty}\le \aleph$, it is apparent that $(\rho_k(\lambda ))\in \ell ^\infty$ and that $$
\|(\rho_k(\lambda ))\|_{\ell ^\infty}\le \zeta(\lambda):=1+\frac{\aleph}{|\Im \lambda|}, $$ where $\Im \lambda$ denotes the imaginary part of $\lambda$. Thus, applying the Cauchy-Schwarz inequality in \eqref{es0} and Parseval's theorem to the representation formula \eqref{rep1} in Lemma \ref{lemma2}, we get that \begin{equation} \label{es1}
\sum_{k=1}^N \frac{|d_k-\tilde{d}_k|}{|\lambda_k-\lambda|}\le M(\lambda)\|(\psi_k-\tilde{\psi}_k)\|_{\ell^2(L^2(\Gamma))},\ N \in \mathbb{N}, \end{equation} where \begin{equation} \label{def-M}
M(\lambda):=\|h\|_{L^2(\Gamma)}\|u_\lambda (g)\|_H +\zeta(\lambda)\|g\|_{L^2(\Gamma)}\|\tilde{u}_\lambda (h)\|_H. \end{equation}
As a consequence we have $\sum_{k\ge 1} |d_k-\tilde{d}_k| / |\lambda_k-\lambda|<\infty$. Furthermore, taking into account that
$$ \frac{|\lambda-\mu|}{|\lambda_k-\mu|} \le 1 + \frac{|\lambda|}{\lambda_1},\ \mu \in (-\infty,-\lambda_1], $$ we apply the dominated convergence theorem to \eqref{du} and find that \begin{equation}\label{4.1} \lim_{\mu=\Re \mu \rightarrow -\infty}U(\lambda ,\mu)=\sum_{k\ge 1}\frac{d_k-\tilde{d}_k}{\lambda_k-\lambda}=:\mathcal{U}(\lambda). \end{equation} Moreover, we have \begin{equation}\label{4.4}
|\mathcal{U}(\lambda)|
\le M(\lambda)\|(\psi_k-\tilde{\psi}_k)\|_{\ell^2(L^2(\Gamma))}, \end{equation} according to \eqref{es1}.
Arguing as before with $V$ defined by \eqref{dv} instead of $U$, we obtain in a similar fashion that \begin{equation}\label{4.3} \lim_{\mu=\Re \mu \rightarrow -\infty}V(\lambda ,\mu)=\sum_{k\ge 1} \frac{\tilde{\lambda}_k-\lambda_k}{(\lambda_k-\lambda)(\tilde{\lambda}_k-\lambda)} \tilde{d}_k=:\mathcal{V}(\lambda)
\end{equation} and that
\begin{equation}\label{4.7}
|\mathcal{V}(\lambda)|\le \zeta(\lambda)\|(\tilde{\lambda}_k-\lambda_k)\|_{\ell^\infty}\| \tilde{u}_\lambda(g)\|_H\|\tilde{u}_\lambda(h)\|_H. \end{equation}
Having seen this, we refer to \eqref{4.0}-\eqref{dt} and deduce from Lemma \ref{lemma5.1}, \eqref{4.1} and \eqref{4.3} that \begin{equation} \label{es3} \int_\Gamma \left( u_\lambda (g)-\tilde{u}_\lambda (g) \right) \overline{h}ds(x)=\mathcal{U}(\lambda)+\mathcal{V}(\lambda). \end{equation}
Now, taking $\lambda=\lambda_\tau=(\tau+i)^2$ for some fixed $\tau \in \left( |\xi|/2, \infty \right)$ and $(\omega,\theta)=(\omega_\tau,\theta_\tau)$, where $\omega_\tau$ and $\theta_\tau$ are the same as in \eqref{es4}, we combine \eqref{S1b}-\eqref{id3} with \eqref{es3}. We obtain that the Fourier transform $\hat{b}$ of $b:=(\tilde{q}-q)\chi_\Omega$, reads \begin{equation}\label{4.6} \hat{b}((1+i/\tau)\xi)= \mathcal{U}(\lambda_\tau )+\mathcal{V}(\lambda_\tau)+\mathfrak{R}(\lambda_\tau), \end{equation} where \[ \mathfrak{R}(\lambda_\tau):=\int_\Omega (\tilde{A}-\lambda_\tau)^{-1}(\tilde{q}\mathfrak{e}_{\lambda_\tau,\omega_\tau})\tilde{q}\mathfrak{e}_{\lambda_\tau,-\theta_\tau}dx-\int_\Omega (A-\lambda_\tau)^{-1}(q\mathfrak{e}_{\lambda_\tau,\omega_\tau})q\mathfrak{e}_{\lambda_\tau,-\theta_\tau}dx. \]
Moreover, for all $\tau \ge \max(|\xi|/2,\tau_\ast)$, we have \begin{equation}\label{4.8.1}
|\mathfrak{R}(\lambda_\tau)|\le C \tau^{-1+2\beta}, \end{equation} by \eqref{gs1}-\eqref{gm0}, where $\beta \in [0,1/2)$ is defined in Theorem \ref{theorem2} and $\tau_\ast$ is the same as in Corollary \ref{corollary1}. Here and in the remaining part of this proof, $C$ denotes a positive constant depending only on $n$, $\Omega$, $\aleph$ and $\mathfrak{c}$, which may change from line to line.
On the other hand, using that \begin{eqnarray*}
\left| \hat{b}((1+i/\tau)\xi) -\hat{b}(\xi) \right| & = & \left| \int_{\mathbb{R}^n} e^{-i \xi \cdot x} \left( e^{\frac{\xi}{\tau} \cdot x} - 1 \right) b(x) dx \right| \\
& \le & \frac{| \xi |}{\tau} \left( \sup_{x \in \Omega} e^{(| \xi | / \tau) |x|} \right) \| b \|_{L^1(\mathbb{R}^n)}, \end{eqnarray*}
we get in a similar way to \cite[Eq. (5.1)]{CS} that $$
|\hat{b}(\xi)|\le |\hat{b}((1+i/\tau)\xi)|+\frac{c|\xi|}{\tau}e^{c|\xi|/\tau}\aleph,\ \tau \in (|\xi|/2, \infty), $$ for some positive constant $c$ depending only on $\Omega$. Putting this together with
\eqref{4.6}-\eqref{4.8.1} we find that for all $\tau \ge \max(|\xi|/2,\tau_\ast)$, \begin{equation}\label{4.9}
|\hat{b}(\xi)|\le \frac{C}{\tau^{1-2\beta}}+\frac{c|\xi|}{\tau}e^{c|\xi|/\tau}\aleph+|\mathcal{U}(\lambda_\tau )|+|\mathcal{V}(\lambda_\tau)|. \end{equation}
To upper bound $|\mathcal{U}(\lambda_\tau )|+|\mathcal{V}(\lambda_\tau)|$ on the right hand side of \eqref{4.9}, we recall from \eqref{sol1} that $u_{\lambda_\tau}(g)=-(A-\lambda_\tau)^{-1} (q \mathfrak{e}_{\lambda_\tau,\omega_\tau}) + \mathfrak{e}_{\lambda_\tau,\omega_\tau}$ and that $\tilde{u}_{\lambda_\tau}(h)=-(\tilde{A}-\lambda_\tau)^{-1} (\tilde{q} \mathfrak{e}_{\lambda_\tau,-\theta_\tau})+\mathfrak{e}_{\lambda_\tau,-\theta_\tau}$, and we combine \eqref{re7} with
\eqref{gs0} and \eqref{gs1}: We get for all $\tau \ge \tau_{\xi}:=\max(1,|\xi|/2,\tau_\ast)$, that
$$\| u_{\lambda_\tau}(g) \|_H + \| \tilde{u}_{\lambda_\tau}(h) \|_H \leq C. $$
This together with the basic estimate $\| g \|_{L^2(\Gamma)} + \| h \|_{L^2(\Gamma)} \le C \tau$, \eqref{def-M}, \eqref{4.4} and \eqref{4.7}, yield that $$
|\mathcal{U}(\lambda_\tau )|+|\mathcal{V}(\lambda_\tau)|\le C\left(\tau\|(\psi_k-\tilde{\psi}_k)\|_{\ell^2(L^2(\Gamma))}+\|(\tilde{\lambda}_k-\lambda_k)\|_{\ell^\infty}\right),\ \tau \in [ \tau_{\xi},\infty ). $$ Inserting this into \eqref{4.9}, we find that \begin{equation}\label{4.12}
|\hat{b}(\xi)|\le \frac{C}{\tau^{1-2\beta}}+\frac{c|\xi|}{\tau}e^{c|\xi|/\tau}\aleph+C \tau \delta,\ \tau \in [ \tau_{\xi},\infty ), \end{equation} where we have set \begin{equation} \label{def-delta}
\delta:= \| (\psi_k-\tilde{\psi}_k)\|_{\ell^2(L^2(\Gamma))}+\|(\tilde{\lambda}_k-\lambda_k)\|_{\ell^\infty}. \end{equation}
Let $\varrho \in (0,1)$ to be made precise further. For all $\tau \in [\tau_\ast,\infty)$, where $\tau_\ast$ is defined in Corollary \ref{corollary1}, it is apparent that the condition $\tau \ge \tau_{\xi}$ is automatically satisfied whenever $\xi \in B(0,\tau^\varrho):= \{ \xi \in \mathbb{R}^n,\ |\xi|< \tau^\varrho \}$. Thus, squaring both sides of \eqref{4.12} and integrating the obtained inequality over $B(0,\tau^\varrho)$, we get that $$
\|\hat{b}\|_{L^2(B(0,\tau ^\varrho))}^2\le C\left( \tau^{-2(1-2 \beta) + \varrho n}+ e^{2c\tau^{-(1-\varrho)}} \tau^{\varrho (n+2)-2}+\tau^{2+\varrho n}\delta ^2 \right),\ \tau \in [\tau_\ast,\infty). $$ Then, taking $\varrho=(1-2\beta)/(n+2)$ in the above line, we obtain that \begin{equation} \label{sta1}
\|\hat{b}\|_{L^2(B(0,\tau ^{(1-2\beta)/(n+2)}))}^2\le C \left( \tau^{-(1-2\beta)}+\tau^{(3n+4)/(n+2)}\delta ^2 \right), \tau \in [\tau_\ast,\infty). \end{equation} On the other hand, using that the Fourier transform is an isometry from $L^2(\mathbb{R}^n)$ to itself, we have for all $\tau \in [\tau_\ast,\infty)$, \begin{eqnarray*}
\int_{\mathbb{R}^n \setminus B(0,\tau^{(1-2\beta)/(n+2)})} (1+|\xi|^2|)^{-1}|\hat{b}(\xi)|^2d\xi & \le &
\tau^{-2(1-2\beta)/(n+2)}\|b\|_{L^2(\mathbb{R}^n)}^2 \\ & \le & C \tau^{-2(1-2\beta)/(n+2)}, \end{eqnarray*} which together with \eqref{sta1} yields that \[
\|b\|_{H^{-1}(\mathbb{R}^n)}^2\le \tau^{-2(1-2\beta)/(n+2)}+\tau^{(3n+4)/(n+2)}\delta ^2,\ \tau \in [\tau_\ast,\infty). \] Assuming that $\delta < \left( 2(1-2\beta)/(3n+4) \right)^{1 /2}=:\delta_0$, we get by minimizing the right hand side of the above estimate with respect to $\tau \in [\tau_\ast,\infty)$, that $$
\|b\|_{H^{-1}(\mathbb{R}^n)}\le C\delta^{2(1-2\beta)/(3(n+2))}, $$
and the desired stability inequality follows from this upon recalling that $\|q-\tilde{q}\|_{H^{-1}(\Omega)}\le \|b\|_{H^{-1}(\mathbb{R}^n)}$. Finally, we complete the proof by noticing that for all $\delta \ge \delta_0$, we have
$$ \|q-\tilde{q}\|_{H^{-1}(\Omega)}\le \|q-\tilde{q}\|_{L^2(\Omega)} \leq \left( 2 \aleph \delta_0^{-2(1-2\beta)/(3(n+2))} \right) \delta^{2(1-2\beta)/(3(n+2))}. $$
\subsection{Proof of Theorem \ref{theorem3}} Upon possibly substituting $q+\lambda^\ast+1$ (resp., $\tilde{q}+\lambda^\ast+1$) for $q$ (resp., $\tilde{q}$), we shall assume without loss of generality in the sequel, that $\lambda_k\ge 1$ (resp., $\tilde{\lambda}_k \ge 1$) for all $k \ge 1$.
Next, taking into account that $q=\tilde{q}$ in $\Omega_0$, we notice that the function $u_k:=\phi_k-\tilde{\phi}_k$, $k \ge 1$, satisfies \begin{equation} \label{es20} (-\Delta +q-\lambda_k)u_k=(\lambda_k-\tilde{\lambda}_k)\tilde{\phi}_k\; \mbox{in}\; \Omega_0,\quad \partial_\nu u_k+\alpha u_k=0\; \mbox{on}\; \Gamma. \end{equation} Now, let us recall from \cite[Theorem 2.2]{BCKPS} that for all $s \in (0,1/2)$ fixed, there exist three constants $C=C(n,\Omega_0,\Gamma_{\ast})>0$, $\mathfrak{b}=\mathfrak{b}(n,\Omega_0,\Gamma_{\ast},s)>0$ and $\gamma=\gamma(n,\Omega_0)>0$, such that for all $r \in (0,1)$ and all $\lambda \in [0,+\infty)$, we have \begin{equation}\label{UC1}
C\left(\|u\|_{H^1(\Gamma_0)}+ \|\partial_\nu u\|_{L^2(\Gamma_0)}\right)\le r^{s/4}\|u\|_{H^2(\Omega_0)}+e^{\mathfrak{b}r^{-\gamma}}\mathfrak{C}_\lambda (u),\ u\in H^2(\Omega_0), \end{equation}
where we have set $\Gamma_0:=\partial \Omega_0$ and \[
\mathfrak{C}_\lambda (u) :=(1+\lambda)\left(\|u\|_{H^1(\Gamma_{\ast})}+\| \partial_\nu u\|_{L^2(\Gamma_{\ast})}\right)+\|(\Delta -q+\lambda)u\|_{L^2(\Omega_0)}. \]
Thus, in light of \eqref{5.1} and the embedding $\Gamma \subset \Gamma_0$, we deduce from \eqref{es20} upon applying \eqref{UC1} with $(\lambda,u)=(\lambda_k,(u_k)_{| \Omega_0})$, $k \geq 1$, that for all $r \in (0,1)$, we have \begin{eqnarray*}
& & C\|\psi_k-\tilde{\psi}_k\|_{L^2(\Gamma)} \\
& \le & r^{s/4}( \lambda_k +\tilde{\lambda}_k )+e^{\mathfrak{b} r^{-\gamma}}\left((1+\|\alpha\|_{C^{0,1}(\Gamma)})\lambda_k\|\psi_k-\tilde{\psi}_k\|_{H^1(\Gamma_{\ast})}+|\lambda_k-\tilde{\lambda}_k|\right), \end{eqnarray*} for some constant $C>0$ depending only on $n$, $\Omega$, $\Omega_0$, $\Gamma_\ast$, $\aleph$ and $s$.
From this and Weyl's asymptotic formula \eqref{waf}, it then follows for all $k \ge 1$ and all $r \in (0,1)$, that \begin{equation} \label{es21}
C\|\psi_k-\tilde{\psi}_k\|_{L^2(\Gamma)}^2\le r^{s/2}k^{4/n}+e^{2\mathfrak{b}r^{-\gamma}}\left(k^{4/n}\|\psi_k-\tilde{\psi}_k\|_{H^1(\Gamma_{\ast})}^2 + |\lambda_k-\tilde{\lambda}_k|^2 \right). \end{equation} Here and in the remaining part of this proof, $C$ denotes a generic positive constant depending only on $n$, $\Omega$, $\Omega_0$, $\Gamma_\ast$, $\aleph$ and $\alpha$, which may change from one line to another. Since the constant $C$ is independent of $k \geq 1$ and since $\sum_{k\ge 1}k^{-2\mathfrak{t}+4/n}<\infty$ as we have $2\mathfrak{t}>1+4/n$, we find upon multiplying both sides of \eqref{es21} by $k^{-2\mathfrak{t}}$ and then summing up the result over $k \geq 1$, that \begin{eqnarray} \label{es22}
& & C \|(k^{-\mathfrak{t}} (\psi_k-\tilde{\psi}_k))\|_{\ell ^2(L^2(\Gamma))}^2 \\
& \le & r^{s/2}
+e^{2\mathfrak{b}r^{-\gamma}} \left( \| (k^{-\mathfrak{t}+2/n} (\psi_k-\tilde{\psi}_k))\|_{\ell^2(H^1(\Gamma_{\ast}))}^2 + \| (k^{-\mathfrak{t}} (\lambda_k-\tilde{\lambda}_k)) \|_{\ell^2}^2 \right), \nonumber
\end{eqnarray} uniformly in $r \in (0,1)$.
Further, taking into account that $(k^{\mathfrak{t}}(\psi_k-\tilde{\phi}_k)) \in \ell^2(L^2(\Gamma))$ and $\|(k^{\mathfrak{t}}(\psi_k-\tilde{\phi}_k))\|_{\ell ^2(L^2(\Gamma))}\le \aleph$, we have \begin{eqnarray*}
\|(\psi_k-\tilde{\psi}_k)\|_{\ell ^2(L^2(\Gamma))}^2
&\le &\|(k^{\mathfrak{t}}(\psi_k-\tilde{\phi}_k))\|_{\ell ^2(L^2(\Gamma))}\|(k^{-\mathfrak{t}}(\psi_k-\tilde{\psi}_k))\|_{\ell ^2(L^2(\Gamma))} \\
& \le & \aleph\|(k^{-\mathfrak{t}}(\psi_k-\tilde{\psi}_k))\|_{\ell ^2(L^2(\Gamma))}, \end{eqnarray*} by the Cauchy-Schwarz inequality, and hence \begin{eqnarray} \label{es23}
& & C \|(\psi_k-\tilde{\psi}_k)\|_{\ell ^2(L^2(\Gamma))}^2 \\ & \le & r^{s/4}
+e^{\mathfrak{b}r^{-\gamma}} \left( \| (k^{-\mathfrak{t}} (\lambda_k-\tilde{\lambda}_k)) \|_{\ell^2} + \| (k^{-\mathfrak{t}+2/n} (\psi_k-\tilde{\psi}_k))\|_{\ell^2(H^1(\Gamma_{\ast}))} \right), \nonumber \end{eqnarray} whenever $r \in (0,1)$, by \eqref{es22}. Moreover, since
$$\| (k^{-\mathfrak{t}} (\lambda_k-\tilde{\lambda}_k)) \|_{\ell^2} \le \left( \sum_{k\ge 1} k^{-2\mathfrak{t}}\right)^{1/2}
\| (\lambda_k-\tilde{\lambda}_k)) \|_{\ell^\infty}$$ and $\sum_{k\ge 1} k^{-2\mathfrak{t}}<\infty$ as we assumed that $2\mathfrak{t}>1+n/2$, \eqref{es23} then provides \begin{equation} \label{es24}
\|(\psi_k-\tilde{\psi}_k)\|_{\ell ^2(L^2(\Gamma))}^2 \le C \left( r^{s/4} +e^{\mathfrak{b}r^{-\gamma}} \delta_\ast \right),\ r \in (0,1), \end{equation} where we have set $$ \delta_\ast :=
\| (\lambda_k-\tilde{\lambda}_k) \|_{\ell^\infty} + \| (k^{-\mathfrak{t}+2/n} (\psi_k-\tilde{\psi}_k))\|_{\ell^2(H^1(\Gamma_{\ast}))}. $$ Next, with reference to \eqref{def-delta} we have \begin{eqnarray*}
\delta^2 & \le & 2 \left( \|(\lambda_k-\tilde{\lambda}_k)\|_{\ell^\infty}^2 + \|(\psi_k-\tilde{\psi}_k)\|_{\ell ^2(L^2(\Gamma))}^2 \right) \\
& \le & 2 \left( \aleph \|(\lambda_k-\tilde{\lambda}_k)\|_{\ell^\infty} + \|(\psi_k-\tilde{\psi}_k)\|_{\ell ^2(L^2(\Gamma))}^2 \right). \nonumber \end{eqnarray*}
Moreover, since $\|(\lambda_k-\tilde{\lambda}_k)\|_{\ell^\infty} \le e^{\mathfrak{b}r^{-\gamma}} \delta_\ast$ whenever $r \in (0,1)$, the above inequality combined with \eqref{es24} yield that \begin{equation} \label{e1} \delta^2 \leq C \left( r^{s/4}+e^{\mathfrak{b}r^{-\gamma}} \delta_\ast \right),\ r \in (0,1). \end{equation}
On the other hand, we have $$
\|q-\tilde{q}\|_{H^{-1}(\Omega)}\le C \delta^{2 (1-2\beta) / (3(n+2))}, $$ from Theorem \ref{theorem2}. Putting this together with \eqref{e1}, we obtain that \begin{equation}\label{e3}
\|q-\tilde{q}\|_{H^{-1}(\Omega)}\le C \left( r^{s/4}+e^{\mathfrak{b}r^{-\gamma}}\delta_\ast \right)^{(1-2\beta)/(3(n+2))},\ r \in (0,1). \end{equation}
Let us now examine the two cases $\delta_\ast \in (0,1/e)$ and $\delta_\ast \in [1/e,\infty)$ separately. We start with $\delta_\ast \in (0,1/e)$ and take $r=| \ln \delta_\ast |^{-1/\gamma} \in (0,1)$ in \eqref{e3}, getting that \begin{eqnarray*}
\|q-\tilde{q}\|_{H^{-1}(\Omega)} & \le & C \left( | \ln \delta_\ast |^{-s/(4\gamma)}+\delta_\ast^{(\mathfrak{b}+1)} \right)^{(1-2\beta)/(3(n+2))} \\
& \le & C \left( | \ln \delta_\ast |^{-s/(4\gamma)}+ e^{-(\mathfrak{b}+1)} | \ln \delta_\ast |^{-(\mathfrak{b}+1)} \right)^{(1-2\beta)/(3(n+2))}, \end{eqnarray*}
where we used in the last line that $\delta_\ast \le 1/(e | \ln \delta_\ast |)$. This immediately yields \begin{equation}\label{e4}
\|q-\tilde{q}\|_{H^{-1}(\Omega)} \le C | \ln \delta_\ast |^{-\vartheta},\ \delta_\ast \in (0,1/e), \end{equation} where $$\vartheta:=\min \left( s/(4\gamma) , \mathfrak{b}+1 \right)(1-2\beta)/(3(n+2)).$$ Next, for $\delta_\ast \in [1/e,\infty)$, we get upon choosing, say, $r=1/2$ in \eqref{e3}, and then taking into account that $r < 1 \le e \delta_\ast$ and $(1-2\beta)/(3(n+2)) \ge 0$, that \begin{eqnarray*}
\|q-\tilde{q}\|_{H^{-1}(\Omega)} & \le & C \left( (e\delta_\ast)^{s/4}+e^{2^\gamma \mathfrak{b}-1} e \delta_\ast \right)^{(1-2\beta)/(3(n+2))} \\ & \le & C (e\delta_\ast)^{(1-2\beta)/(3(n+2))} \\ & \le & C \delta_\ast. \end{eqnarray*} Now, with reference to \eqref{def-Phi}, the stability estimate \eqref{thm3} follows readily from this and \eqref{e4}.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Triple correlations of Fourier coefficients of cusp forms}
\author[Y. Lin]{Yongxiao Lin} \address{Department of Mathematics, The Ohio State University\\ 231 W 18th Avenue\\ Columbus, Ohio 43210-1174} \email{[email protected]}
\begin{abstract} We treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms. As a consequence, we obtain an upper bound for correlation of three Hecke eigenvalues of holomorphic cusp forms $\sum_{H\leq h\leq 2H}W\big(\frac{h}{H}\big)\sum_{X\leq n\leq 2X}\lambda_{1}(n-h)\lambda_{2}(n)\lambda_{3}(n+h)$, which is nontrivial provided that $H\geq X^{2/3+\varepsilon}$. The result can be viewed as a cuspidal analogue of a recent result of Blomer \cite{blo} on triple correlations of divisor functions. \end{abstract}
\subjclass[2010]{Primary 11F30; Secondary 11F72}
\keywords{Triple correlation, cusp forms, circle method, Kuznetsov's trace formula, large sieve inequalities}
\maketitle
\section{Introduction}
Recently Blomer \cite{blo} established an asymptotic formula with power saving error term for certain types of triple correlations of divisor functions. Motivated by his work, we are going to prove a cuspidal analogue. While shifted convolutions of two Fourier coefficients have been an extensively studied topic, there seem to be few results available on the correlation of three Fourier coefficients, with power saving error term. For instance, one of the highly interesting and open problems in analytic number theory is to find an asymptotic formula for \begin{equation} D_h(X)=\sum_{n\leq X}\tau(n-h)\tau(n)\tau(n+h), \end{equation} where $\tau(n)$ denotes the divisor function. It is conjectured that \begin{equation}\label{D_h} D_h(X)\sim c_h X(\log X)^3, \end{equation} as $X\rightarrow \infty$, for some positive constant $c_h$. Browning \cite{bro} suggests that one should take
\begin{equation}
c_h=\frac{11}{8}f(h)\prod_{p}\left(1-\frac{1}{p}\right)^2\left(1+\frac{2}{p}\right),
\end{equation}
for an explicit multiplicative function $f(h)$, and is able to prove that \eqref{D_h} is true on average, namely
\begin{equation}
\sum_{h\leq H}\left(D_h(X)-c_h X(\log X)^3\right)=o(HX(\log X)^3)
\end{equation}
for $H\geq X^{3/4+\varepsilon}$.
Using spectral theory of automorphic forms, Blomer \cite{blo} improved the range of $H$ substantially to $H\geq X^{1/3+\varepsilon}$ and produced a power saving error term. Furthermore, Blomer's approach seems to be flexible enough to be adapted to the study of more general correlation sums.
\begin{Theorem}[Blomer \cite{blo}\label{blomer's}] Let $W$ be a smooth function with compact support in $[1,2]$ and Mellin transform $\widehat{W}$. Let $1\leq H\leq X/3$ and $r_d(n)$ be the Ramanujan sum. Let $a(n)$, for $X\leq n\leq 2X$, be any sequence of complex numbers. Then \begin{equation} \begin{split} \sum_{h}W\left(\frac{h}{H}\right)\sum_{X\leq n\leq 2X}a(n)\tau(n+h)\tau(n-h)&=H\widehat{W}(1)\sum_{X\leq n\leq 2X}a(n)\sum_{d}\frac{r_d(2n)}{d^2}(\log n+2\gamma-2\log d)^2
\\&+O\left(X^{\varepsilon}\left(\frac{H^2}{X^{1/2}}+HX^{1/4}+(XH)^{1/2}+\frac{X}{H^{1/2}}\right)\|a\|_2\right). \end{split} \end{equation} \end{Theorem}
Here $\|a\|_2=(\sum|a_n|^2)^{\frac{1}{2}}$ is the $\ell^2$-norm. The first term in the $O$-term above comes from the smooth approximation of the sum on the left. The second one comes from the treatment of the ``minus-case'' after application of Voronoi summation of the divisor function. The last two terms come from spectral methods.
As a corollary, Blomer obtains
\begin{Corollary}[Blomer \cite{blo}] Let $W$ be a smooth function with compact support in $[1,2]$ and Mellin transform $\widehat{W}$. Let $1\leq H\leq X/3$. Then \begin{equation} \begin{split} \sum_{h}W\left(\frac{h}{H}\right)\sum_{X\leq n\leq 2X}\tau_k(n)\tau(n+h)\tau(n-h)&=\widehat{W}(1)XHQ_{k+1}(\log X) \\&+O\left(X^{\varepsilon}(H^2+HX^{1-\frac{1}{k+2}}+XH^{1/2}+X^{3/2}H^{-1/2})\right), \end{split} \end{equation} where $\tau_k$ is the $k$-th fold divisor function, and $Q_{k+1}$ is a degree $k+1$ polynomial. \end{Corollary}
The result is non-trivial for $X^{\frac{1}{3}+\varepsilon}\leq H\leq X^{1-\varepsilon}$, which in the case of $k=2$, substantially improves Browning's result.
Note that since the divisor function can be viewed as the Fourier coefficient of Eisenstein series, one naturally would ask what will be the case if the divisor functions are replaced by Fourier coefficients of cusp forms. Blomer \cite{blo} remarked that if one uses Jutila's circle method and argues as in \cite{bl-mi}, then one might obtain an analogous result. The purpose of this note is to carry this out in detail. It turns out that new difficulties arise, making it difficult to obtain a range for $H$ as good as the divisor function case. Namely, we are not able to `open' the Fourier coefficient as Blomer does with the divisor function. We obtain the following result.
\begin{Theorem}\label{maintheorem} Let $1\leq H\leq X/3$. Let $W$ be a smooth function with compact support in $[1,2]$, and $a(n)$, $X\leq n\leq 2X$, be any sequence of complex numbers. Let $\lambda_1(n), \lambda_2(n)$ be Hecke eigenvalues of holomorphic Hecke eigencuspforms of weight $\kappa_1$, $\kappa_2$ for $\mathrm{SL}_2(\mathbb{Z})$ respectively. Then \begin{equation}\label{finalbdd}
\sum_{h}W\left(\frac{h}{H}\right)\sum_{X\leq n\leq 2X}a(n)\lambda_{1}(n+h)\lambda_{2}(n-h)\ll X^{\varepsilon}\frac{X}{H}\left((XH)^{1/2}+\frac{X}{H^{1/2}}\right)\|a\|_2. \end{equation} \end{Theorem}
One should compare our result with the third and fourth terms of the $O$-term in Blomer's theorem, as both of them naturally come from the spectral theory of automorphic forms. The result is nontrivial as long as $H\geq X^{\frac{2}{3}+\varepsilon}$.
As an immediate consequence, we have \begin{Corollary} Let $1\leq H\leq X/3$. Let $W$ be a smooth function with compact support in $[1,2]$. Let $\lambda_1(n), \lambda_2(n), \lambda_3(n)$ be Hecke eigenvalues of holomorphic Hecke eigencuspforms of weight $\kappa_1$, $\kappa_2$ and $\kappa_3$ for $\mathrm{SL}_2(\mathbb{Z})$ respectively. Then \begin{equation} \sum_{h}W\left(\frac{h}{H}\right)\sum_{X\leq n\leq 2X}\lambda_{1}(n-h)\lambda_{2}(n)\lambda_{3}(n+h)\ll X^{\varepsilon}\min\left(XH,\frac{X^2}{H^{1/2}}\right). \end{equation} \end{Corollary}
The result is nontrivial only for $H\geq X^{\frac{2}{3}+\varepsilon}$. One can remove the smooth function $W$ in the $h$-sum, as in \cite{blo}.
If on the other hand, one allows one of the Fourier coefficients to be non-cuspidal, then the advantage to open the divisor function enables us to enlarge the range of $H$ to $H\geq X^{\frac{1}{3}+\varepsilon}$. This feature of decomposable functions was pointed out by Meurman in \cite{Meu}. For instance one has the following result which follows from the same line of proof of Blomer \cite{blo}, although it is not explicitly stated as such in that work. \begin{Corollary}[Blomer \cite{blo}] Let $1\leq H\leq X/3$. Let $W$ be a smooth function with compact support in $[1,2]$, and $a(n)$, $X\leq n\leq 2X$, be any sequence of complex numbers. Let $\lambda(n)$ be Hecke eigenvalues of a holomorphic Hecke eigencuspform of weight $\kappa$ for $\mathrm{SL}_2(\mathbb{Z})$. Then \begin{equation}
\sum_{h}W\left(\frac{h}{H}\right)\sum_{X\leq n\leq 2X}a(n)\tau(n+h)\lambda(n-h)\ll X^{\varepsilon}\left(\frac{H^2}{X^{1/2}}+HX^{1/4}+(XH)^{1/2}+\frac{X}{H^{1/2}}\right)\|a\|_2. \end{equation} \end{Corollary}
\textbf{Notation.} $x\asymp X$ means $X\leq x\leq 2X$. We will use the $\varepsilon$-convention: $\varepsilon>0$ is arbitrarily small but not necessarily the same at each occurrence.
\section{Preliminaries} In this section we collect some lemmas that will be used in our proof. First let us recall the Voronoi summation formula for holomorphic Hecke eigenvalues. \begin{Lemma}[{\cite[Theorem A.4]{KMV1}}]\label{voronoi} Assume $(b,c)=1$. Let $V$ be a smooth compactly supported function. Let $N>0$ and let $\lambda(n)$ denote Hecke eigenvalues of a holomorphic Hecke eigencuspform of weight $\kappa$ for $\mathrm{SL}_2(\mathbb{Z})$. Then \begin{equation} \sum_{n}\lambda(n)e\left(\frac{bn}{c}\right)V\left(\frac{n}{N}\right)=\frac{N}{c}\sum_{n}\lambda(n)e\left(-\frac{\bar{b}n}{c}\right)\cdot 2\pi i^\kappa\int_{0}^{\infty}V(x)J_{k-1}\left(\frac{4\pi\sqrt{nNx}}{c}\right)\mathrm{d}x \end{equation} \end{Lemma}
We will use the following variant of Jutila's circle method \cite{jut1}, \cite{jut}.
\begin{Lemma}\label{jutila}Let $Q\geq 1$ and $Q^{-2}\leq \delta\leq Q^{-1}$ be two parameters. Let $w$ be a nonnegative function supported in $[Q,2Q]$ and satisfies $\|w\|_\infty\leq1$ and $\sum_{c}w(c)>0$. Let $\mathbb{I}_S$ be the characteristic function of the set $S$. Define\label{jutila} \begin{equation} \tilde{I}(\alpha)=\frac{1}{2\delta \Lambda}\sum_{c}w(c)\sumstar_{d(\mathrm{mod} c)}\mathbb{I}_{[\frac{d}{c}-\delta,\frac{d}{c}+\delta]}(\alpha), \end{equation} where $\Lambda=\sum_{c}w(c)\phi(c)$. Then $\tilde{I}(\alpha)$ is a good approximation of $\mathbb{I}_{[0,1]}$ in the sense that \begin{equation}
\int_{0}^{1}|1-\tilde{I}(\alpha)|^2\mathrm{d}\alpha\ll_{\varepsilon} \frac{Q^{2+\varepsilon}}{\delta\Lambda^2}. \end{equation} \end{Lemma}
The feature of the circle method in our application, as in \cite{bl-mi}, is that the parameter $Q$ turns out to be just a ``catalyst'', not entering into the final bound. This will become clear at the final stage of our argument.
In order to state Kuznetsov's trace formula and the large sieve inequalities, let us define the following integral transforms for a smooth function $\phi: [0,\infty)\rightarrow \mathbb{C}$ satisfying $\phi(0)=\phi'(0)=0$, $\phi^{(j)}(x)\ll (1+x)^{-3}$ for $0\leq j\leq 3$: $$\dot{\phi}(k)=4i^k\int_{0}^{\infty}\phi(x)J_{k-1}(x)\frac{\mathrm{d}x}{x},\ \ \tilde{\phi}(t)=2\pi i\int_{0}^{\infty}\phi(x)\frac{J_{2it}(x)-J_{-2it}(x)}{\sinh(\pi t)}\frac{\mathrm{d}x}{x}.$$
We use the notations in \cite{BHM} and \cite{blo}. Let $\mathcal{B}_k$ be an orthonormal basis of the space of holomorphic cusp forms of level $1$ and weight $k$. Let $\mathcal{B}$ be a fixed orthonormal basis of Hecke-Maass eigenforms of level 1. For $f\in \mathcal{B}_k$, we write its Fourier expansion at $\infty$ as $$f(z)=\sum_{n\geq 1}\rho_f(n)(4\pi n)^{k/2}e(nz).$$
For $f\in \mathcal{B}$ with spectral parameter $t$, we write its Fourier expansion as
$$f(z)=\sum_{n\neq 0}\rho_f(n)W_{0,it}(4\pi|n|y)e(nx),$$
where $W_{0,it}(4\pi|n|y)=(y/\pi)^{1/2}K_{it}(y/2)$ is a Whittaker function.
Finally for the Eisenstein series $E(z,s)$, we write its Fourier expansion at $s=\frac{1}{2}+it$ as \begin{equation*}
E\left(z,\frac{1}{2}+it\right)=y^{\frac{1}{2}+it}+\varphi\left(1/2+it\right)y^{\frac{1}{2}-it}+\sum_{n\neq0}\rho(n,t)W_{0,it}(4\pi|n|y)e(nx). \end{equation*}
Then we have the following Kuznetsov's trace formula in the notation of \cite{BHM}. \begin{Lemma}\label{kuznetsov} Let $a, b>0$ be integers, then
\begin{equation} \begin{split} \sum_{c\geq 1}\frac{1}{c}S(a,b;c)\phi\left(\frac{4\pi\sqrt{ab}}{c}\right)&=\sum_{\substack{k\geq 2\\ k\ \mathrm{even}}}\sum_{f\in\mathcal{B}_k}\dot{\phi}(k)\Gamma(k)\sqrt{ab}\,\overline{\rho_f(a)}\rho_f(b) \\&+\sum_{f\in\mathcal{B}}\tilde{\phi}(t_f)\frac{\sqrt{ab}}{\cosh (\pi t_f)}\overline{\rho_f(a)}\rho_f(b)+\frac{1}{4\pi}\int_{-\infty}^{\infty}\tilde{\phi}(t)\frac{\sqrt{ab}}{\cosh (\pi t)}\overline{\rho(a,t)}\rho(b,t)\mathrm{d}t. \end{split} \end{equation} \end{Lemma}
We will use the following spectral large sieve inequalities of Deshouillers and Iwaniec \cite{de-iw}. \begin{Lemma}\label{largesieve} Let $T, M\geq 1$. Let $(a_m)$, $M\leq m\leq 2M$, be a sequence of complex numbers, then all of the three quantities\\
$$\sum_{\substack{2\leq k\leq T\\ k\ \mathrm{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k}\big|\sum_{m}a_m\sqrt{m}\rho_f(m)\big|^2, \sum_{\substack{f\in\mathcal{B}\\ |t_f|\leq T}}\frac{1}{\cosh (\pi t_f)}\big|\sum_{m}a_m\sqrt{m}\rho_f(\pm m)\big|^2,$$
$$\int_{-T}^{T}\frac{1}{\cosh (\pi t)}\big|\sum_{m}a_m\sqrt{m}\rho(\pm m,t)\big|^2 \mathrm{d}t$$ are bounded by
$$M^{\varepsilon}(T^2+M)\sum_{m}|a_m|^2.$$ \end{Lemma}
We need the following lemma of Blomer-Mili{\'c}evi{\'c} \cite{bl-mi} for a certain type of Bessel transform, which provides the asymptotic behavior of the weight function. \begin{Lemma}\label{wstar} Let $W$ be a fixed smooth function with support in $[1,2]$ satisfying $W^{(j)}(x)\ll_j 1$ for all $j$. Let $\nu\in \mathbb{C}$ be a fixed number with $\Re \nu\geq 0$. Define\label{boundofw} \begin{equation} W^{\star}(z,w)=\int_{0}^{\infty}W(y)J_{\nu}(4\pi\sqrt{yw+z})\mathrm{d}y. \end{equation}
Fix $C\geq 1$ and $A,\varepsilon>0$. Then for $z\geq 4|w|>0$ we have \begin{equation} W^{\star}(z,w)=W_+(z,w)z^{-1/4}e(2\sqrt{z})+W_-(z,w)z^{-1/4}e(-2\sqrt{z})+O_A(C^{-A}) \end{equation} for suitable functions $W_\pm$ satisfying \begin{equation}\label{wplusminus} z^i w^j\frac{\partial^i}{\partial z^i}\frac{\partial^j}{\partial w^j}W_\pm(z,w)\begin{cases}=0, & \quad \sqrt{z}/w\leq C^{-\varepsilon}, \\ \ll C^{\varepsilon(i+j)}, & \quad \mathrm{otherwise},\\ \end{cases} \end{equation} for any $i,j\in \mathbb{N}_0$. The implied constants depend on $i, j$ and $\nu$. \end{Lemma}
The lemma holds true for both positive and negative $w$, as long as $z\geq 4|w|$. For a proof, see \cite[Lemma 17]{bl-mi} and the remark after that.
From Lemma \ref{wstar}, one has the following easy consequence. \begin{Corollary}[{\cite[Corollary 18]{bl-mi}}]\label{realpart} Let $\omega$ be a smooth function supported in a rectangle $[c_1,c_2]\times[c_1,c_2]$ for two constants $c_2>c_1>0$, and let $Z\gg 1$, $W>1$ be two parameters such that $c_1Z\geq 4c_2W$, then the double Mellin transform \begin{equation*} \widehat{W}_{\pm}(s,t)=\int_{0}^{\infty}\int_{0}^{\infty}W_{\pm}(z,w)\omega\left(\frac{z}{Z},\frac{w}{W}\right)z^sw^t\frac{\mathrm{d}z}{z}\frac{\mathrm{d}w}{w} \end{equation*} is holomorphic on $\mathbb{C}^2$, and satisfies \begin{equation*}
\widehat{W}_{\pm}(s,t)\ll_{A,B,\varepsilon,\Re s, \Re t} C^{\varepsilon}(1+|s|)^{-A}(1+|t|)^{-B} \end{equation*} on vertical lines.
\end{Corollary} We will use this corollary when we try to separate variables, after the application of Kuznetsov's trace formula. Then, the following lemma of \cite{blo} will help us to truncate the lengths of summations. See also \cite[Lemma 3]{jut}. \begin{Lemma}[{\cite[Lemma 5]{blo}}] \label{lemmaofphi} Let $\mathcal{Z}\gg1$,$\tau\in\mathbb{R}$, $\alpha\in[-4/5,4/5]$ and $w$ be a smooth compactly supported function. For \begin{equation} \phi(z)=e^{\pm iz\alpha}w\left(\frac{z}{\mathcal{Z}}\right)\left(\frac{z}{\mathcal{Z}}\right)^{i\tau}, \end{equation} we have \begin{equation}
\dot{\phi}(k)\ll_A\frac{1+|\tau|}{\mathcal{Z}}\left(1+\frac{k}{\mathcal{Z}}\right)^{-A}, \tilde{\phi}(t)\ll_A\left(1+\frac{|t|+\mathcal{Z}}{1+|\tau|}\right)^{-A} \end{equation} for $t\in\mathbb{R}$, $k\in\mathbb{N}$ and any $A\geq 0$. \end{Lemma}
\begin{Remark} From Lemma \ref{lemmaofphi} we see that $\dot{\phi}(k)$ is negligibly small as long as $|k|>\mathcal{Z}$, so that later one can truncate summation over $k$ at $\mathcal{Z}$, up to a negligible error. Moreover, in application, $\mathcal{Z}$ will usually be relatively larger than $\tau$, so that the contribution from $\tilde{\phi}(t)$ is always negligible. In particular, after the application of Kuznetsov's trace formula later, we only need to treat the contribution from the holomorphic spectrum, since the Maass forms and Eisenstein series parts will be negligible due to the rapid decay of the weight function $\tilde{\phi}(t)$. \end{Remark}
\section{Proof of the theorem} Let $\lambda_1(n), \lambda_2(n)$ be the Hecke eigenvalues of holomorphic Hecke eigencuspforms of weight $\kappa_1$, $\kappa_2$ for $\mathrm{SL}_2(\mathbb{Z})$. We change the order of summation of $n$ and $h$, fix $n\asymp X$, and first deal with the sum over $h$ \begin{equation}\label{originalsum} E(n):=\sum_{h\asymp H}\lambda_{1}\left(n+h\right)\lambda_{2}\left(n-h\right)W\left(\frac{h}{H}\right). \end{equation}
Let $n+h=m_1$ and $n-h=m_2$, then $m_1+m_2=2n$, and we can rewrite the summation above as
\begin{equation}\label{shiftedsum} \sum_{m_1+m_2=2n}\lambda_{1}(m_1)\lambda_{2}(m_2)W\left(\frac{m_1-n}{H}\right)V\left(\frac{n-m_2}{H'}\right). \end{equation} Here $V$ is a redundant smooth function used to keep track of the support of $m_2$, and $H'$ is a parameter to be determined later, satisfying $H\leq H'\leq X/3$. Later we will see that in our case, $H'=X/3$ will give the best result. Here one can take, say, $V=W$.
\begin{Remark} Let us make a comment on the difference between the shifted convolution sum in \eqref{shiftedsum} with the one treated by Blomer and Mili{\'c}evi{\'c}, \cite[(3.7)]{bl-mi}. In our case, we have localized both the variables $m_1$ and $m_2$ to vary in intervals \textbf{around} $n$. In Blomer and Mili{\'c}evi{\'c}'s case, one of the variables varies in an interval of length $2H$ around $n$, while the other variable varies, say, in $[H/2,2H]$. One should also note that for $f$ a holomorphic cusp form, the sum $\sum_{m_1+m_2=X}\lambda_{f}(m_1)\lambda_{f}(m_2)$ is just the $X$-th Fourier coefficient of the cusp form $f^2$ and thus one can apply Deligne's estimate. \end{Remark}
Now we follow the approach of Jutila \cite{jut} and Blomer and Mili{\'c}evi{\'c} \cite[sections 7 $\&$ 8]{bl-mi}. Let $C=X^{1000}$ be a large parameter. Apply Lemma \ref{jutila} with $Q=C$ and $\delta=C^{-1}$. Let $w_0$ be a fixed smooth function supported in $[1,2]$. Let $w(c)=w_0(c/C)$. In particular we have $\Lambda=\sum_{c}w_0(c/C)\phi(c)\gg C^{2-\varepsilon}$. Detecting the condition $m_1+m_2=2n$ by $\int_{0}^{1}e(\alpha(m_1+m_2-2n))\mathrm{d} \alpha$ and applying Jutila's circle method, we have \begin{equation*} E(n)=\widetilde{E}(n)+\mathrm{Error}, \end{equation*} where \begin{equation} \begin{split} \widetilde{E}(n)=\frac{1}{2\delta}\int_{-\delta}^{\delta}&\frac{1}{\Lambda}\sum_{c}w_0\left(\frac{c}{C}\right)\sumstar_{d(c)}e\left(\frac{-2nd}{c}\right)\sum_{m_1}\lambda_{1}(m_1)e\left(\frac{dm_1}{c}\right)W\left(\frac{m_1-n}{H}\right)e(\eta(m_1-n)) \\&\cdot \sum_{m_2}\lambda_{2}(m_2)e\left(\frac{dm_2}{c}\right)V\left(\frac{n-m_2}{H'}\right)e(-\eta(n-m_2))\mathrm{d}\eta, \end{split} \end{equation} and \begin{equation*} \begin{split} \mathrm{Error}&=\int_{0}^{1}\sum_{m_1}\sum_{m_2}\lambda_{1}(m_1)\lambda_{2}(m_2)W\left(\frac{m_1-n}{H}\right)V\left(\frac{n-m_2}{H'}\right)e(\alpha(m_1+m_2-2n))(1-\tilde{I}(\alpha))\mathrm{d} \alpha\\
&\ll\left(\sum_{m_1}\left|\lambda_{1}(m_1)W\big(\frac{m_1-n}{H}\big)\right|\right)\left(\sum_{m_2}\left|\lambda_{2}(m_2)V\big(\frac{n-m_2}{H'}\big)\right|\right)\cdot \frac{C^{1+\varepsilon}}{\delta^{1/2}\Lambda}\\ &\ll\frac{X^{2}C^{1+\varepsilon}}{\delta^{1/2}\Lambda}\ll C^{-2/5} \end{split} \end{equation*}
by the Cauchy-Schwarz inequality, Lemma \ref{jutila} and the bound $\sum_{n\leq x}|\lambda_f(n)|^2\ll_f x$.
Denote $W_\eta(x)=W(x)e(\eta x)$ and $V_{-\eta}(x)=V(x)e(-\eta x)$. Then \begin{equation} \begin{split} \widetilde{E}(n)=\frac{1}{2\delta}\int_{-\delta}^{\delta}&\frac{1}{\Lambda}\sum_{c}w_0\left(\frac{c}{C}\right)\sumstar_{d(c)}e\left(\frac{-2nd}{c}\right)\sum_{m_1}\lambda_{1}(m_1)e\left(\frac{dm_1}{c}\right)W_{\eta H}\left(\frac{m_1-n}{H}\right)\\ &\cdot \sum_{m_2}\lambda_{2}(m_2)e\left(\frac{dm_2}{c}\right)V_{-\eta H'}\left(\frac{n-m_2}{H'}\right)\mathrm{d}\eta. \end{split} \end{equation}
Note that since $|\eta|\leq C^{-1}=X^{-1000}$ is very small, the functions $W_{\eta H}$ and $V_{-\eta H'}$ have nice properties inherited from $W$. In particular, one has $W^{(j)}_{\eta H},\ V^{(j)}_{-\eta H'}\ll 1$ for any $j\geq 0$, supported in $[1,2]$, uniformly for $|\eta|\leq C^{-1}$.
Applying Voronoi summation formula to the $m_1$-sum, we have \begin{equation} \sum_{m_1}\lambda_{1}(m_1)e\left(\frac{dm_1}{c}\right)W_{\eta H}\left(\frac{m_1-n}{H}\right)=\frac{H}{c}\sum_{m_1}\lambda_{1}(m_1)e\left(-\frac{\bar{d}m_1}{c}\right)W^{\star}_{\eta H}\left(\frac{m_1n}{c^2},\frac{m_1H}{c^2}\right), \end{equation} where \begin{equation} W^{\star}_{\eta H}(z,w)=2\pi i^{\kappa_1}\int_{0}^{\infty}W_{\eta H}(y)J_{\kappa_1-1}(4\pi\sqrt{yw+z})\mathrm{d}y. \end{equation}
Similarly, \begin{equation} \sum_{m_2}\lambda_{2}(m_2)e\left(\frac{dm_2}{c}\right)V_{-\eta H'}\left(\frac{n-m_2}{H'}\right)=\frac{H'}{c}\sum_{m_2}\lambda_{2}(m_2)e\left(-\frac{\bar{d}m_2}{c}\right)V^{\star}_{-\eta H'}\left(\frac{m_2n}{c^2},\frac{m_2H'}{c^2}\right), \end{equation} with \begin{equation} V^{\star}_{-\eta H'}(z,w)=2\pi i^{\kappa_2}\int_{0}^{\infty}V_{-\eta H'}(y)J_{\kappa_2-1}(4\pi\sqrt{-yw+z})\mathrm{d}y. \end{equation}
Substituting these back into $\widetilde{E}(n)$, we get
\begin{equation} \widetilde{E}(n)=\frac{1}{2\delta}\int_{-\delta}^{\delta} \widetilde{E}_\eta(n)\mathrm{d}\eta, \end{equation} with \begin{equation} \begin{split} \widetilde{E}_\eta(n)=&\frac{HH'}{\Lambda}\sum_{c}\frac{w_0(c/C)}{c^2}\sum_{m_1}\sum_{m_2}\lambda_{1}(m_1)\lambda_{2}(m_2)S(m_1+m_2,2n;c)\\ &\cdot W^{\star}_{\eta H}\left(\frac{m_1n}{c^2},\frac{m_1H}{c^2}\right)V^{\star}_{-\eta H'}\left(\frac{m_2n}{c^2},\frac{m_2H'}{c^2}\right). \end{split} \end{equation}
If one can establish a bound for $\sum_{n\asymp X}a(n)\widetilde{E}_\eta(n)$, uniformly in $\eta$, then the same bound will hold true for $\sum_{n\asymp X}a(n)E(n)$, and we are done.
By Lemma \ref{boundofw}, we can write \begin{equation*} W^{\star}_{\eta H}\left(\frac{m_1n}{c^2},\frac{m_1H}{c^2}\right)=\sum_{\pm}W_{\pm}\left(\frac{m_1n}{c^2},\frac{m_1H}{c^2}\right)\left(\frac{m_1n}{c^2}\right)^{-\frac{1}{4}}e\left(\pm2\frac{\sqrt{m_1n}}{c}\right)+O\left(C^{-A}\right), \end{equation*} for some $W_{\pm}$ satisfying \eqref{wplusminus}.
Similarly we have \begin{equation*}V^{\star}_{-\eta H'}\left(\frac{m_2n}{c^2},\frac{m_2H'}{c^2}\right)=\sum^{\pm}V^{\pm}\left(\frac{m_2n}{c^2},\frac{m_2H'}{c^2}\right)\left(\frac{m_2n}{c^2}\right)^{-\frac{1}{4}}e\left(\pm2\frac{\sqrt{m_2n}}{c}\right)+O\left(C^{-A}\right), \end{equation*} for some $V^{\pm}$ satisfying similar conditions. Note that from now on we will not display dependence on the parameter $\eta$ anymore, since all of the estimations we obtain will be uniform in $\eta$, by our previous remark on the nice properties of $W_{\eta H}$ and $V_{-\eta H'}$.
By the bound \eqref{wplusminus}, we can restrict the lengths of the new sums over $m_1$, $m_2$ to \begin{equation*} m_1\leq T_1:=\frac{C^{2+\varepsilon}X}{H^2},\ m_2\leq T_2:=\frac{C^{2+\varepsilon}X}{H'^2}, \end{equation*} up to a negligible error.
Now we further restrict the lengths of summations to dyadic segments. That is, we assume
$$m_1\asymp \mathcal{M}_1,\ m_2\asymp \mathcal{M}_2,$$ where $\mathcal{M}_1\ll T_1,\ \mathcal{M}_2\ll T_2$.
Denote $b=m_1+m_2$. Then $$b\asymp \mathcal{M}_1+\mathcal{M}_2.$$
Then $\widetilde{E}(n)$ will be a sum of at most $O((\log C)^3)$ terms, of the following form. \begin{equation} \begin{split} \widetilde{E}(n,\mathcal{M}_1,\mathcal{M}_2)&=\frac{HH'n^{-\frac{1}{2}}}{\Lambda}\sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\sum_{\substack{m_1+m_2=b\\ m_1\asymp\mathcal{M}_1,\ m_2\asymp\mathcal{M}_2}}\lambda_{1}(m_1)m_1^{-\frac{1}{4}}\lambda_{2}(m_2)m_2^{-\frac{1}{4}}\sum_{c}\frac{S(b,2n;c)}{c}w_0\left(\frac{c}{C}\right) \\& \cdot\sum_{\pm}\sum^{\pm}W_{\pm}\left(\frac{m_1n}{c^2},\frac{m_1H}{c^2}\right)V^{\pm}\left(\frac{m_2n}{c^2},\frac{m_2H'}{c^2}\right)e\left(\pm2\frac{\sqrt{m_1n}}{c}\right)e\left(\pm2\frac{\sqrt{m_2n}}{c}\right)+O(C^{-10}). \end{split} \end{equation}
To prepare for the application of Kuznetsov's trace formula, we combine the weight functions above and write \begin{equation} \widetilde{E}(n,\mathcal{M}_1,\mathcal{M}_2)=\frac{HH'n^{-\frac{1}{2}}}{\Lambda}\sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\sum_{\substack{m_1+m_2=b\\ m_1\asymp\mathcal{M}_1,\ m_2\asymp\mathcal{M}_2}}\lambda_{1}(m_1)m_1^{-\frac{1}{4}}\lambda_{2}(m_2)m_2^{-\frac{1}{4}}\sum_{c}\frac{S(b,2n;c)}{c}\Phi\left(\frac{4\pi\sqrt{2nb}}{c}\right), \end{equation} where \begin{equation} \Phi(z)=\sum_{\pm}\sum^{\pm}w_0\left(\frac{4\pi\sqrt{2nb}}{zC}\right)W_{\pm}\left(\frac{m_1z^2}{32\pi^2b},\frac{m_1Hz^2}{32\pi^2nb}\right) V^{\pm}\left(\frac{m_2z^2}{32\pi^2b},\frac{m_2H'z^2}{32\pi^2nb}\right)e^{\pm i\sqrt{\frac{m_1}{2b}}z}e^{\pm i\sqrt{\frac{m_2}{2b}}z}. \end{equation}
Note that the support of $w_0$ implies that we can restrict $z$ to $$z\asymp \mathcal{Z}:=\frac{\sqrt{X(\mathcal{M}_1+\mathcal{M}_2)}}{C}.$$
We can attach a redundant smooth weight function $w_2(\frac{z}{\mathcal{Z}})$ of compact support $[1/100\mathcal{Z},100\mathcal{Z}]$ that is constantly $1$ on $[1/20\mathcal{Z},20\mathcal{Z}]$, to $\Phi(z)$.
Now we separate the variables. We do this by Mellin inversion to the functions $w_0$, $W_{\pm}$ and $V^{\pm}$ (using Corollary \ref{realpart}). This can be done with almost no loss since these functions are non-oscillatory, similar to \cite{bl-mi} and \cite{bfkmm}. Thus we have \begin{equation} \begin{split} \Phi(z)=&\sum_{\pm}\sum^{\pm}w_0\left(\frac{4\pi\sqrt{2nb}}{zC}\right)W_{\pm}\left(\frac{m_1z^2}{32\pi^2b},\frac{m_1Hz^2}{32\pi^2nb}\right)V^{\pm}\left(\frac{m_2z^2}{32\pi^2b},\frac{m_2H'z^2}{32\pi^2nb}\right)\\ &\cdot e^{\pm iz(\sqrt{\frac{m_1}{2b}}\pm\sqrt{\frac{m_2}{2b}})}w_2\left(\frac{z}{\mathcal{Z}}\right)\\ =&\sum_{\pm}\sum^{\pm}\int_{\mathcal{C}}\widehat{w}_0(s_1)\widehat{W}_{\pm}(s_2,s_3)\widehat{V}^{\pm}(s_4, s_5) \bigg(\frac{4\pi\sqrt{2nb}}{zC}\bigg)^{-s_1}\bigg(\frac{m_1z^2}{32\pi^2b}\bigg)^{-s_2}\\ &\cdot\bigg(\frac{m_1Hz^2}{32\pi^2nb}\bigg)^{-s_3}\bigg(\frac{m_2z^2}{32\pi^2b}\bigg)^{-s_4}\bigg(\frac{m_2H'z^2}{32\pi^2nb}\bigg)^{-s_5}\mathrm{d} s\cdot e^{\pm iz(\sqrt{\frac{m_1}{2b}}\pm\sqrt{\frac{m_2}{2b}})}w_2\left(\frac{z}{\mathcal{Z}}\right)\\ =&\sum_{\pm}\sum^{\pm}\int_{\mathcal{C}}\widehat{w}_0(s_1)\widehat{W}_{\pm}(s_2,s_3)\widehat{V}^{\pm}(s_4, s_5)(4\pi\sqrt{2})^{-s_1}(32\pi^2)^{s_2+s_3+s_4+s_5}C^{s_1}\mathcal{Z}^{s_1-2s_2-2s_3-2s_4-2s_5}\\ &\cdot (\mathcal{M}_1+\mathcal{M}_2)^{-\frac{s_1}{2}+s_2+s_3+s_4+s_5}\mathcal{M}_1^{-s_2-s_3}\mathcal{M}_2^{-s_4-s_5}H^{-s_3}H'^{-s_5}n^{-\frac{s_1}{2}+s_3+s_5}\left(\frac{b}{\mathcal{M}_1+\mathcal{M}_2}\right)^{-\frac{s_1}{2}+s_2+s_3+s_4+s_5}\\ &\cdot\bigg(\frac{m_1}{\mathcal{M}_1}\bigg)^{-s_2-s_3}\bigg(\frac{m_2}{\mathcal{M}_2}\bigg)^{-s_4-s_5} \bigg(\frac{z}{\mathcal{Z}}\bigg)^{s_1-2s_2-2s_3-2s_4-2s_5}e^{\pm iz(\sqrt{\frac{m_1}{2b}}\pm\sqrt{\frac{m_2}{2b}})}w_2\left(\frac{z}{\mathcal{Z}}\right)\mathrm{d}s, \end{split} \end{equation} where $\mathcal{C}$ is the fivefold contour taken over the vertical lines $\Re s_1=\Re s_2=\Re s_3=\Re s_4=\Re s_5=0$.
Here we denote $\mathrm{d}s=\frac{1}{(2\pi i)^5}\prod_{j=1}^{5}\mathrm{d} s_j$. Note that due to the rapid decay of $\widehat{w}_1(s_1)\widehat{W}_{\pm}(s_2,s_3)\widehat{V}^{\pm}(s_4, s_5)$ along vertical lines, we can truncate the integrals above at $|\Im s_i|\leq C^\varepsilon, 1\leq i\leq 5$, at the cost of a negligible error. We denote the truncated contour by $\widetilde{\mathcal{C}}$.
We arrive at \begin{equation} \begin{split} \widetilde{E}(n,\mathcal{M}_1,\mathcal{M}_2)&=\frac{HH'}{\Lambda}(\mathcal{M}_1\mathcal{M}_2)^{-\frac{1}{4}}\sum_{\pm}\sum^{\pm}\int_{\widetilde{\mathcal{C}}}\widehat{w}_0(s_1)\widehat{W}_{\pm}(s_2,s_3)\widehat{V}^{\pm}(s_4, s_5)(4\pi\sqrt{2})^{-s_1}(32\pi^2)^{s_2+s_3+s_4+s_5} \\&\cdot C^{s_1}\mathcal{Z}^{s_1-2s_2-2s_3-2s_4-2s_5}(\mathcal{M}_1+\mathcal{M}_2)^{-\frac{s_1}{2}+s_2+s_3+s_4+s_5}\mathcal{M}_1^{-s_2-s_3}\mathcal{M}_2^{-s_4-s_5}H^{-s_3}H'^{-s_5}\\ &\cdot n^{-\frac{1}{2}-\frac{s_1}{2}+s_3+s_5}\sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\left(\frac{b}{\mathcal{M}_1+\mathcal{M}_2}\right)^{-\frac{s_1}{2}+s_2+s_3+s_4+s_5}\sum_{\substack{m_1+m_2=b\\ m_1\asymp\mathcal{M}_1,\ m_2\asymp\mathcal{M}_2}}\lambda_{1}(m_1)\lambda_{2}(m_2) \\&\cdot\left(\frac{m_1}{\mathcal{M}_1}\right)^{-\frac{1}{4}-s_2-s_3}\left(\frac{m_2}{\mathcal{M}_2}\right)^{-\frac{1}{4}-s_4-s_5}\sum_{c}\frac{S(b,2n;c)}{c}\Theta\left(\frac{4\pi\sqrt{2nb}}{c}\right)\mathrm{d}s+O(C^{-10}), \end{split} \end{equation} where \begin{equation}\label{function} \Theta(z)=e^{\pm iz(\sqrt{\frac{m_1}{2b}}\pm\sqrt{\frac{m_2}{2b}})}w_2\left(\frac{z}{\mathcal{Z}}\right)\left(\frac{z}{\mathcal{Z}}\right)^{s_1-2s_2-2s_3-2s_4-2s_5}. \end{equation}
Now we apply Kuznetsov's trace formula to the $c$-sum. By Lemma \ref{lemmaofphi}, the spectral sums can be truncated at $C^{\varepsilon}\mathcal{Z}$. Hence we obtain \begin{equation} \sum_{c}\frac{S(b,2n;c)}{c}\Theta\left(\frac{4\pi\sqrt{2nb}}{c}\right)=\mathcal{H}(n)+\mathcal{M}(n)+\mathcal{E}(n)+\text{(negligible error)}, \end{equation} where \begin{equation}\label{ksum} \begin{split} & \mathcal{H}(n)=\sum_{\substack{2\leq k\leq C^{\varepsilon}\mathcal{Z}\\ k\ \mathrm{even}}}\sum_{f\in\mathcal{B}_k}\Gamma(k)\cdot4i^k\int_{0}^{\infty}\Theta(z)J_{k-1}(z)\frac{\mathrm{d}z}{z}\cdot \sqrt{2n}\rho_f(2n)\sqrt{b}\,\overline{\rho_f(b)}, \\
& \mathcal{M}(n)=\sum_{\substack{f\in\mathcal{B}\\|t_f|\leq C^{\varepsilon}\mathcal{Z}}}2\pi i\int_{0}^{\infty}\Theta(z)\frac{J_{2it_f}(z)-J_{-2it_f}(z)}{\sinh(\pi t_f)}\frac{\mathrm{d}z}{z}\cdot \frac{\sqrt{2nb}}{\cosh (\pi t_f)}\rho_f(2n)\overline{\rho_f(b)}, \\
& \mathcal{E}(n)=\frac{1}{4\pi}\int_{|t|\leq C^{\varepsilon}\mathcal{Z}}2\pi i\int_{0}^{\infty}\Theta(z)\frac{J_{2it}(z)-J_{-2it}(z)}{\sinh(\pi t)}\frac{\mathrm{d}z}{z}\cdot \frac{\sqrt{2nb}}{\cosh (\pi t)}\rho(2n,t)\overline{\rho(b,t)}\mathrm{d}t \end{split} \end{equation} are contributions from the holomorphic modular forms, Maass forms, and Eisenstein series respectively.
Now we are ready to sum over $a(n)$, $X\leq n\leq 2X$. Summing over $n$, we get
\begin{equation} \begin{split} &\sum_{n\asymp X}a(n)\widetilde{E}(n,\mathcal{M}_1,\mathcal{M}_2)\\ &=\frac{HH'}{\Lambda}(\mathcal{M}_1\mathcal{M}_2)^{-\frac{1}{4}}\sum_{\pm}\sum^{\pm}\int_{\widetilde{\mathcal{C}}}\widehat{w}_0(s_1)\widehat{W}_{\pm}(s_2,s_3)\widehat{V}^{\pm}(s_4, s_5)(4\pi\sqrt{2})^{-s_1}(32\pi^2)^{s_2+s_3+s_4+s_5}\\ &\cdot C^{s_1}\mathcal{Z}^{s_1-2s_2-2s_3-2s_4-2s_5}(\mathcal{M}_1+\mathcal{M}_2)^{-\frac{s_1}{2}+s_2+s_3+s_4+s_5}\mathcal{M}_1^{-s_2-s_3}\mathcal{M}_2^{-s_4-s_5}H^{-s_3}H'^{-s_5}\\ &\cdot\sum_{n\asymp X}a(n)n^{-\frac{1}{2}-\frac{s_1}{2}+s_3+s_5}\sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\left(\frac{b}{\mathcal{M}_1+\mathcal{M}_2}\right)^{-\frac{s_1}{2}+s_2+s_3+s_4+s_5}\sum_{\substack{m_1+m_2=b\\ m_1\asymp\mathcal{M}_1,\ m_2\asymp\mathcal{M}_2}}\lambda_{1}(m_1)\lambda_{2}(m_2)\\ &\cdot\bigg(\frac{m_1}{\mathcal{M}_1}\bigg)^{-\frac{1}{4}-s_2-s_3}\bigg(\frac{m_2}{\mathcal{M}_2}\bigg)^{-\frac{1}{4}-s_4-s_5}\bigg(\mathcal{H}(n)+\mathcal{M}(n)+\mathcal{E}(n)\bigg)\mathrm{d}s+O(C^{-10})\\ &:=\mathcal{HH}+\mathcal{MM}+\mathcal{EE}+O(C^{-10}), \end{split} \end{equation} with $\mathcal{HH}$, $\mathcal{MM}$, $\mathcal{EE}$ being contributions from holomorphic forms, Maass forms and Eisenstein series respectively.
Note that for the function $\Theta(z)$ in \eqref{function}, the imaginary part $\Im (s_1-2s_2-2s_3-2s_4-2s_5)\ll C^{\varepsilon}$, which is relatively small compared to $\mathcal{Z}$. By Lemma \ref{lemmaofphi} and the remark after it, it suffices to deal with the contribution from the holomorphic spectrum, since the contributions from the other two parts will be similar or even smaller. Also note that since we are considering the full modular group case, we do not have exceptional eigenvalues contribution in the Maass spectrum.
Now we focus on the holomorphic contribution, which is \begin{equation} \begin{split} \mathcal{HH}&=\frac{C^{\varepsilon}HH'}{\Lambda}(\mathcal{M}_1\mathcal{M}_2)^{-\frac{1}{4}}\sum_{\pm}\sum^{\pm}\int_{\widetilde{\mathcal{C}}}\widehat{w}_0(s_1)\widehat{W}_{\pm}(s_2,s_3)\widehat{V}^{\pm}(s_4, s_5)(4\pi\sqrt{2})^{-s_1}(32\pi^2)^{s_2+s_3+s_4+s_5} \\&\cdot C^{s_1}\mathcal{Z}^{s_1-2s_2-2s_3-2s_4-2s_5}(\mathcal{M}_1+\mathcal{M}_2)^{-\frac{s_1}{2}+s_2+s_3+s_4+s_5}\mathcal{M}_1^{-s_2-s_3}\mathcal{M}_2^{-s_4-s_5}H^{-s_3}H'^{-s_5}\\ &\cdot\int_{0}^{\infty}\sum_{\substack{2\leq k\leq C^{\varepsilon}\mathcal{Z}\\ k\ \mathrm{even}}}4i^k\Gamma(k)\sum_{f\in\mathcal{B}_k}\sum_{n\asymp X}a(n)n^{-\frac{1}{2}-\frac{s_1}{2}+s_3+s_5}\sqrt{2n}\rho_f(2n)\cdot \sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\sqrt{b}\,\overline{\rho_f(b)}\\ &\cdot\left(\frac{b}{\mathcal{M}_1+\mathcal{M}_2}\right)^{-\frac{s_1}{2}+s_2+s_3+s_4+s_5} \sum_{\substack{m_1+m_2=b\\ m_1\asymp\mathcal{M}_1,\ m_2\asymp\mathcal{M}_2}}\lambda_{1}(m_1)\lambda_{2}(m_2)\left(\frac{m_1}{\mathcal{M}_1}\right)^{-\frac{1}{4}-s_2-s_3} \left(\frac{m_2}{\mathcal{M}_2}\right)^{-\frac{1}{4}-s_4-s_5}\\ &\cdot e^{\pm iz(\sqrt{m_1}\pm\sqrt{m_2})}w_2\left(\frac{\sqrt{2b}z}{\mathcal{Z}}\right)\left(\frac{\sqrt{2b}z}{\mathcal{Z}}\right)^{s_1-2s_2-2s_3-2s_4-2s_5}J_{k-1}(\sqrt{2b}z)\frac{\mathrm{d}z}{z}\mathrm{d}s, \end{split} \end{equation} after making the change of variable $z\mapsto \sqrt{2b}z$. The support of the smooth function $w_2$ implies that $z\asymp \frac{\sqrt{X}}{C}$ in the inner integral.
Bounding the $z$-integral and $s_i$-integrals trivially, we have \begin{equation} \begin{split} \mathcal{HH}&\ll\frac{C^{\varepsilon}HH'}{\Lambda}(\mathcal{M}_1\mathcal{M}_2)^{-\frac{1}{4}}\cdot
\sup_{\substack{|u_1|, |u_2|,|u_3|,|u_4|\leq C^{\varepsilon}\\z\asymp \frac{\sqrt{X}}{C}}}\bigg|\sum_{\substack{2\leq k\leq C^{\varepsilon}\mathcal{Z}\\ k\ \mathrm{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k}\sum_{n\asymp X}a(n)n^{-\frac{1}{2}+iu_4}\sqrt{2n}\rho_f(2n)\\ &\cdot \sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\sqrt{b}\,\overline{\rho_f(b)}J_{k-1}(\sqrt{2b}z)\cdot w_2\left(\frac{\sqrt{2b}z}{\mathcal{Z}}\right)\left(\frac{\sqrt{2b}z}{\mathcal{Z}}\right)^{-2iu_3}\left(\frac{b}{\mathcal{M}_1+\mathcal{M}_2}\right)^{iu_3}\\
&\cdot\sum_{\substack{m_1+m_2=b\\ m_1\asymp\mathcal{M}_1,\ m_2\asymp\mathcal{M}_2}}\lambda_{1}(m_1)\lambda_{2}(m_2)\left(\frac{m_1}{\mathcal{M}_1}\right)^{-\frac{1}{4}+iu_1}\left(\frac{m_2}{\mathcal{M}_2}\right)^{-\frac{1}{4}+iu_2}e^{\pm iz(\sqrt{m_1}\pm\sqrt{m_2})}\bigg|. \end{split} \end{equation}
The Cauchy-Schwarz inequality further implies that \begin{equation}\label{twosup} \begin{split} \mathcal{HH}&\ll\frac{C^{\varepsilon}HH'}{\Lambda}(\mathcal{M}_1\mathcal{M}_2)^{-\frac{1}{4}}\cdot
\bigg(\sup_{|u_4|\leq C^{\varepsilon}}\sum_{\substack{2\leq k\leq C^{\varepsilon}\mathcal{Z}\\ k\ \mathrm{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k}\bigg |\sum_{n\asymp X}a(n)n^{-\frac{1}{2}+iu_4}\sqrt{2n}\rho_f(2n)\bigg|^2\bigg)^{\frac{1}{2}}
\\&\cdot\bigg(\sup_{\substack{|u_1|, |u_2|,|u_3|\leq C^{\varepsilon}\\z\asymp \frac{\sqrt{X}}{C}}}\sum_{\substack{2\leq k\leq C^{\varepsilon}\mathcal{Z}\\ k\ \mathrm{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k}\bigg|\sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\sqrt{b}\,\overline{\rho_f(b)}J_{k-1}(\sqrt{2b}z)\gamma^{\star}(b,z)\bigg|^2\bigg)^{\frac{1}{2}}, \end{split} \end{equation} where \begin{equation}\label{gammastar} \begin{split} \gamma^{\star}(b,z)&:=w_2\left(\frac{\sqrt{2b}z}{\mathcal{Z}}\right)\left(\frac{\sqrt{2b}z}{\mathcal{Z}}\right)^{-2iu_3}\left(\frac{b}{\mathcal{M}_1+\mathcal{M}_2}\right)^{iu_3}\\ &\cdot\sum_{\substack{m_1+m_2=b\\ m_1\asymp\mathcal{M}_1,\ m_2\asymp\mathcal{M}_2}}\lambda_{1}(m_1)\lambda_{2}(m_2)\left(\frac{m_1}{\mathcal{M}_1}\right)^{-\frac{1}{4}+iu_1}e^{\pm iz\sqrt{m_1}}\left(\frac{m_2}{\mathcal{M}_2}\right)^{-\frac{1}{4}+iu_2}e^{\pm iz\sqrt{m_2}}. \end{split} \end{equation}
The large sieve inequalities yield that \begin{equation*}
\bigg(\sup_{|u_4|\leq C^{\varepsilon}}\sum_{\substack{2\leq k\leq C^{\varepsilon}\mathcal{Z}\\ k\ \mathrm{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k}\bigg |\sum_{n\asymp X}a(n)n^{-\frac{1}{2}+iu_4}\sqrt{2n}\rho_f(2n)\bigg|^2\bigg)^{\frac{1}{2}}
\ll \mathcal{C}^{\varepsilon}\left(\mathcal{Z}^2+X\right)^{\frac{1}{2}}X^{-\frac{1}{2}}\|a\|_2. \end{equation*}
Now it remains to deal with the second line of \eqref{twosup}. We denote \begin{equation*}
(\star\star):=\sup_{\substack{|u_1|, |u_2|,|u_3|\leq C^{\varepsilon}\\z\asymp \frac{\sqrt{X}}{C}}}\sum_{\substack{2\leq k\leq C^{\varepsilon}\mathcal{Z}\\ k\ \mathrm{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k}\bigg|\sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\sqrt{b}\,\overline{\rho_f(b)}J_{k-1}(\sqrt{2b}z)\gamma^{\star}(b,z)\bigg|^2. \end{equation*}
We want to separate the $k$-variable from the argument of the $J$-Bessel function, so that one can apply the large sieve inequalities. By the integral representation $J_{k-1}(x)=\frac{1}{\pi}\int_0^{\pi}\cos((k-1)\xi-x\sin\xi)\mathrm{d}\xi$, \cite[(8.411.1)]{GR07}, we have \begin{equation*} J_{k-1}(\sqrt{2b}z)=\frac{1}{2\pi}\int_0^{\pi}\left(e^{i(k-1)\xi}e^{-i\sqrt{2b}z\sin\xi}+e^{-i(k-1)\xi}e^{i\sqrt{2b}z\sin\xi}\right)\mathrm{d}\xi. \end{equation*}
Hence \begin{equation*} \begin{split}
(\star\star)&\ll\sup_{\substack{|u_1|, |u_2|,|u_3|\leq C^{\varepsilon}\\z\asymp \frac{\sqrt{X}}{C}}}\sum_{\substack{2\leq k\leq C^{\varepsilon}\mathcal{Z}\\ k\ \mathrm{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k}\bigg|\int_0^{\pi}e^{i(k-1)\xi}\sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\sqrt{b}\,\overline{\rho_f(b)}\cdot e^{-i\sqrt{2b}z\sin\xi}\gamma^{\star}(b,z)\mathrm{d}\xi\bigg|^2\\
&\ll\int_0^{\pi}\sup_{\substack{|u_1|, |u_2|,|u_3|\leq C^{\varepsilon}\\z\asymp \frac{\sqrt{X}}{C}}}\sum_{\substack{2\leq k\leq C^{\varepsilon}\mathcal{Z}\\ k\ \mathrm{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k}\bigg|\sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\sqrt{b}\,\overline{\rho_f(b)}\cdot e^{-i\sqrt{2b}z\sin\xi}\gamma^{\star}(b,z)\bigg|^2\mathrm{d}\xi\\
&\ll \mathcal{C}^{\varepsilon}\left(\mathcal{Z}^2+\mathcal{M}_1+\mathcal{M}_2\right)\sup_{z\asymp \frac{\sqrt{X}}{C}}\sum_{b}\big|\gamma^{\star}(b,z)\big|^2 \end{split} \end{equation*} by Lemma \ref{largesieve}.
In summary, we have arrived at \begin{equation*}
\mathcal{HH}\ll\frac{C^{\varepsilon}HH'}{\Lambda}X^{-\frac{1}{2}}(\mathcal{M}_1\mathcal{M}_2)^{-\frac{1}{4}}\cdot \left(\mathcal{Z}^2+X\right)^{\frac{1}{2}}\|a\|_2\cdot \left(\mathcal{Z}^2+\mathcal{M}_1+\mathcal{M}_2\right)^{\frac{1}{2}}\sup_{z\asymp \frac{\sqrt{X}}{C}}\|\gamma^{\star}\|_2. \end{equation*}
Now for our purpose it remains to deal with the $\ell^2$-norm $\|\gamma^{\star}\|_2=\left(\sum_{b}\big|\gamma^{\star}(b,z)\big|^2\right)^{\frac{1}{2}}$, where $\gamma^{\star}(b,z)$ is defined in \eqref{gammastar}. By Parseval,
\begin{equation} \begin{split}
\sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\big|\gamma^{\star}(b,z)\big|^2\ll&
\int_{0}^{1}\left|\sum_{m_1\asymp\mathcal{M}_1}\lambda_{1}(m_1)\left(\frac{m_1}{\mathcal{M}_1}\right)^{-\frac{1}{4}+iu_1}e^{\pm iz\sqrt{m_1}}e(m_1\alpha)\right|^2\\
&\cdot\left|\sum_{m_2\asymp\mathcal{M}_2}\lambda_{2}(m_2)\left(\frac{m_2}{\mathcal{M}_2}\right)^{-\frac{1}{4}+iu_2}e^{\pm iz\sqrt{m_2}}e(m_2\alpha)\right|^2\mathrm{d}\alpha. \end{split} \end{equation}
Note that both $u_1$ and $u_2$ are of negligible size here. The sup-norm of the $m_2$-sum is bounded by $C^{\varepsilon}(\mathcal{M}_2^{1/2}+z\mathcal{M}_2)$, by Wilton's bound $\sum_{n\leq x}\lambda_f(n)e(\alpha n)\ll_f x^{1/2+\varepsilon}$ and partial summation. Next we open the square, getting \begin{equation*}
\sum_{b}\big|\gamma^{\star}(b,z)\big|^2\ll C^{\varepsilon}(\mathcal{M}_2^{1/2}+z\mathcal{M}_2)^2\sum_{m_1\asymp\mathcal{M}_1}|\lambda_{1}(m_1)|^2\ll C^{\varepsilon}(\mathcal{M}_2^{1/2}+z\mathcal{M}_2)^2\mathcal{M}_1. \end{equation*}
Similar sums without the twisted weight functions have appeared in the work of Jutila \cite{jut}. See also Blomer and Mili{\'c}evi{\'c} \cite[(8.5)]{bl-mi} for a similar sum.
Recall $\ \mathcal{M}_1\ll T_1=C^{\varepsilon}\frac{C^{2}X}{H^2}$, $\mathcal{M}_2\ll T_2=C^{\varepsilon}\frac{C^{2}X}{H'^2}$ $\mathcal{Z}=\frac{\sqrt{X(\mathcal{M}_1+\mathcal{M}_2)}}{C}\ll C^{\varepsilon}\frac{X}{H}$, $\Lambda\gg C^{2-\varepsilon}$, and $C=X^{1000}$ is a large parameter. In particular, $\sup_{z\asymp \frac{\sqrt{X}}{C}}\sum_{b}\big|\gamma^{\star}(b,z)\big|^2\ll C^{\varepsilon}(\frac{X}{H'})^2\mathcal{M}_1\mathcal{M}_2$.
Then \begin{equation*} \begin{split}
\mathcal{HH}&\ll\frac{C^{\varepsilon}HH'}{C^2}X^{-\frac{1}{2}}(\mathcal{M}_1\mathcal{M}_2)^{-\frac{1}{4}}\cdot \left(\mathcal{Z}^2+X\right)^{\frac{1}{2}}\|a\|_2\cdot \left(\mathcal{Z}^2+\mathcal{M}_1+\mathcal{M}_2\right)^{\frac{1}{2}}\cdot \frac{X}{H'}\left(\mathcal{M}_1\mathcal{M}_2\right)^{\frac{1}{2}}\\
&\ll\frac{C^{\varepsilon}HH'}{C^2} X^{-\frac{1}{2}}\left(\frac{C^{2}X}{H^2}\cdot\frac{C^{2}X}{H'^2}\right)^{\frac{1}{4}}\left(\frac{X^2}{H^2}+X\right)^{\frac{1}{2}}\cdot \left(\frac{X^2}{H^2}+\frac{C^{2}X}{H^2}\right)^{\frac{1}{2}}\cdot\frac{X}{H'}\cdot \|a\|_2\\
&\ll C^{\varepsilon}(HH')^{\frac{1}{2}}\left(\frac{X^2}{H^2}+X\right)^{\frac{1}{2}}\cdot\frac{X^{1/2}}{H}\cdot\frac{X}{H'}\|a\|_2\\
&\ll C^{\varepsilon}\frac{X^{3/2}}{(HH')^{\frac{1}{2}}}\left(\frac{X}{H}+X^{\frac{1}{2}}\right)\|a\|_2. \end{split} \end{equation*}
By taking $H'=X/3$, we have $\mathcal{HH}\ll C^{\varepsilon}\frac{X}{H}\left(\frac{X}{H^{1/2}}+(XH)^{\frac{1}{2}}\right)\|a\|_2$, and hence
\begin{equation*}\sum_{X\leq n\leq 2X}a(n)E(n)\ll C^{\varepsilon}\frac{X}{H}\left(\frac{X}{H^{1/2}}+(XH)^{\frac{1}{2}}\right)\|a\|_2. \end{equation*}
\subsection*{Acknowledgements.} The author is grateful to Valentin Blomer for several helpful comments and suggestions during the preparation of this article. He also thanks Sheng-Chi Liu for drawing his attention to \cite{blo} and Roman Holowinsky for his encouragement and comments.
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title[The Bochner-Riesz means for Fourier-Bessel expansions] {The Bochner-Riesz means for Fourier-Bessel expansions: norm inequalities for the maximal operator and almost everywhere convergence}
\author[\'O. Ciaurri and L. Roncal]{\'Oscar Ciaurri and Luz Roncal} \address{Departamento de Matem\'aticas y Computaci\'on\\ Universidad de La Rioja\\ 26004 Logro\~no, Spain} \email{[email protected], [email protected]}
\thanks{Research supported by the grant MTM2009-12740-C03-03 from Spanish Government.}
\keywords{Fourier-Bessel expansions, Bochner-Riesz means, almost everywhere convergence, maximal operators, weighted inequalities}
\subjclass[2010]{Primary: 42C10, Secondary: 42C20, 42A45}
\begin{abstract} In this paper, we develop a thorough analysis of the boundedness properties of the maximal operator for the Bochner-Riesz means related to the Fourier-Bessel expansions. For this operator, we study weighted and unweighted inequalities in the spaces $L^p((0,1),x^{2\nu+1}\, dx)$. Moreover, weak and restricted weak type inequalities are obtained for the critical values of~$p$. As a consequence, we deduce the almost everywhere pointwise convergence of these means. \end{abstract}
\maketitle
\section{Introduction and main results}
Let $J_{\nu}$ be the Bessel function of order $\nu$. For $\nu>-1$ we have that \[ \int_0^1 J_{\nu}(s_jx)J_{\nu}(s_kx)x\,dx= \frac12 (J_{\nu+1}(s_j))^2\delta_{j,k},\quad j,k=1,2,\dots \] where $\{s_j\}_{j\ge 1}$ denotes the sequence of successive positive zeros of $J_{\nu}$. From the previous identity we can check that the system of functions \begin{equation} \label{eq:FBesselSystemI}
\psi_j(x)=\frac{\sqrt{2}}{|J_{\nu+1}(s_j)|}x^{-\nu}J_{\nu}(s_jx),\quad j=1,2,\dots \end{equation} is orthonormal and complete in $L^2((0,1),d\mu_\nu)$, with $d\mu_\nu(x)=x^{2\nu+1}\, dx$ (for the completeness, see~\cite{Hochstadt}). Given a function $f$ on $(0,1)$, its Fourier series associated with this system, named as Fourier-Bessel series, is defined by \begin{equation}\label{SeriesCoeficientes} f\sim \sum_{j=1}^\infty a_j(f)\psi_j,\qquad\text{with}\qquad a_j(f)=\int_0^1 f(y)\psi_j(y)\,d\mu_{\nu}(y), \end{equation} provided the integral exists. When $\nu=n/2-1$, for $n\in \mathbb{N}$ and $n\ge 2$, the functions $\psi_j$ are the eigenfunctions of the radial Laplacian in the multidimensional ball $B^n$. The eigenvalues are the elements of the sequence $\{s_j^2\}_{j\ge 1}$. The Fourier-Bessel series corresponds with the radial case of the multidimensional Fourier-Bessel expansions analyzed in~\cite{Bal-Cor}.
For each $\delta>0$, we define the Bochner-Riesz means for Fourier-Bessel series as \begin{equation*}
\mathcal{B}_R^{\delta}(f,x)=\sum_{j\ge 1} \left(1-\frac{s_j^2}{R^2}\right)_+^{\delta}a_j(f)\psi_j(x), \end{equation*} where $R>0$ and $(1-s^2)_+=\max\{1-s^2,0\}$. Bochner-Riesz means are a regular summation method used oftenly in harmonic analysis. It is very common to analyze regular summation methods for Fourier series when the convergence of the partial sum fails. Ces\`{a}ro means are other of the most usual summation methods. B. Muckenhoupt and D. W. Webb \cite{Mu-We} give inequalities for Ces\`{a}ro means of Laguerre polynomial series and for the supremum of these means with certain parameters and $1<p\leq \infty$. For $p=1$, they prove a weak type result. They also obtain similar estimates for Ces\`{a}ro means of Hermite polynomial series and for the supremum of those means in \cite{Mu-We-Her}. An almost everywhere convergence result is obtained as a corollary in \cite{Mu-We} and \cite{Mu-We-Her}. The result about Laguerre polynomials is an extension of a previous result in \cite{Stem}. This kind of matters has been also studied by the first author and J. L. Varona in \cite{Ciau-Var} for the Ces\`{a}ro means of generalized Hermite expansions. The Ces\`{a}ro means for Jacobi polynomials were analyzed by S. Chanillo and B. Muckenhoupt in \cite{Ch-Muc}. The Bochner-Riesz means themselves have been analyzed for the Fourier transform and their boundedness properties in $L^p(\mathbb{R}^n)$ is an important unsolved problem for $n>2$ (the case $n=2$ is well understood, see \cite{Car-Sjo}).
The target of this paper is twofold. First we will analyze the almost everywhere (a. e.) convergence, for functions in $L^p((0,1),d\mu_\nu)$, of the Bochner-Riesz means for Fourier-Bessel expansions. By the general theory \cite[Ch. 2]{Duoa}, to obtain this result we need to estimate the maximal operator \[
\mathcal{B}^{\delta}(f,x)=\sup_{R>0}\left|\mathcal{B}_R^{\delta}(f,x)\right|, \] in the $L^p((0,1),d\mu_\nu)$ spaces. A deep analysis of the boundedness properties of this operator will be the second goal of our paper. This part of our work is strongly inspired by the results given in \cite{Ch-Muc} for the Fourier-Jacobi expansions.
Before giving our results we introduce some notation. Being $p_0=\frac{4(\nu+1)}{2\nu+3+2\delta}$ and $p_1=\frac{4(\nu+1)}{2\nu+1-2\delta}$, we define \begin{align} \label{eq:endpoints0} p_0(\delta)&=\begin{cases} 1,& \delta> \nu+1/2\text{ or } -1<\nu\le -1/2,\\ p_0,& \delta\le \nu+1/2\text{ and } \nu>-1/2, \end{cases}\\
\notag p_1(\delta)&=\begin{cases} \infty,& \delta> \nu+1/2\text{ or } -1<\nu\le -1/2,\\ p_1,& \delta\le \nu+1/2\text{ and } \nu>-1/2. \end{cases} \end{align}
Concerning to the a. e. convergence of the Bochner-Riesz means, our result reads as follows \begin{Theo} \label{th:fin} Let $\nu>-1$, $\delta>0$, and $1\le p<\infty$. Then, \begin{equation*}
\mathcal{B}_R^\delta(f,x)\to f(x)\quad \text{a. e., for $f\in L^p((0,1),d\mu_\nu)$} \end{equation*} if and only if $p_0(\delta)\le p$, where $p_0(\delta)$ is as in \eqref{eq:endpoints0}. \end{Theo} Proof of Theorem \ref{th:fin} is contained in Section \ref{sec:Proof Th1} and is based on the following arguments. On one hand, to prove the necessity part, we will show the existence of functions in $L^p((0,1),d\mu_{\nu})$ for $p<p_0(\delta)$ such that $\mathcal{B}^\delta_R$ diverges for them. In order to do this, we will use a reasoning similar to the one given by C. Meaney in \cite{Meaney} that we describe in Section \ref{sec:Proof Th1}. On the other hand, for the sufficiency, observe that the convergence result follows from the study of the maximal operator $\mathcal{B}^\delta f$. Indeed, it is sufficient to get $(p_0(\delta),p_0(\delta))$-weak type estimates for this operator and this will be the content of Theorem \ref{th:AcDebilMaxRedonda}.
Regarding the boundedness properties of $\mathcal{B}^\delta f$ we have the following facts. First, a result containing the $(p,p)$-strong type inequality. \begin{Theo} \label{th:max} Let $\nu>-1$, $\delta>0$, and $1< p\le\infty$. Then, \begin{equation*}
\left\|\mathcal{B}^{\delta}f\right\|_{L^p((0,1),d\mu_{\nu})}\le C
\|f\|_{L^p((0,1),d\mu_{\nu})} \end{equation*} if and only if \[ \begin{cases} 1<p\le \infty, &\text{for $-1<\nu\le -1/2$ or $\delta>\nu+1/2$},\\ p_0<p<p_1, &\text{for $\delta\le \nu+1/2$ and $\nu>-1/2$.} \end{cases} \] \end{Theo} In the lower critical value of $p_0(\delta)$ we can prove a $(p_0(\delta),p_0(\delta))$-weak type estimate. \begin{Theo} \label{th:AcDebilMaxRedonda} Let $\nu>-1$, $\delta>0$, and $p_0(\delta)$ be the number in \eqref{eq:endpoints0}. Then, \[
\left\|\mathcal{B}^{\delta}f\right\|_{L^{p_0(\delta),\infty}((0,1),d\mu_{\nu})}\le C \|f\|_{L^{p_0(\delta)}((0,1),d\mu_{\nu})}, \] with $C$ independent of $f$. \end{Theo} Finally, for the upper critical value, when $0<\delta<\nu+1/2$ and $\nu>-1/2$, it is possible to obtain a $(p_1,p_1)$-restricted weak type estimate. \begin{Theo} \label{th:AcDebilRestMaxRedonda} Let $\nu>-1/2$ and $0<\delta<\nu+1/2$. Then, \[
\left\|\mathcal{B}^{\delta}\chi_E\right\|_{L^{p_1,\infty}((0,1),d\mu_{\nu})}\le C \|\chi_E\|_{L^{p_1}((0,1),d\mu_{\nu})}, \] for all measurable subsets $E$ of $(0,1)$ and $C$ independent of $E$. \end{Theo} The previous results about norm inequalities are summarized in Figure 1 (case $-1<\nu \le -1/2$) and Figure 2 (case $\nu>-1/2$).
\begin{center} \begin{tikzpicture}[scale=2.45] \fill[black!10!white] (0.1,0.1) -- (2.1,0.1) -- (2.1,2.1) -- (0.1,2.1) -- cycle; \draw[thick, dashed] (2.1,0.1) -- (2.1,2.1);
\draw[very thick] (0.1,2.1) -- (0.1,0.1); \draw[very thin] (0.1,0.1) -- (2.15,0.1); \node at (0.1,0) {$0$}; \node at (2.1,0) {$1$}; \node at (2.2,0.1) {$\frac{1}{p}$}; \node at (0.1,2.175) {$\delta$}; \draw (0,1.4) node [rotate=90] {\tiny{$(p,p)$-strong}}; \draw (2.2,1.4) node [rotate=90] {\tiny{$(p,p)$-weak}}; \draw (1.1,-0.2) node {Figure 1: case $-1<\nu\le-\tfrac{1}{2}$.};
\fill[black!10!white] (2.7,0.75) -- (3.35,0.1) -- (4.05,0.1) -- (4.7,0.75) -- (4.7,2.1) -- (2.7,2.1) -- cycle; \draw[thick, dashed] (4.7,0.75) -- (4.7,2.1);
\draw[very thick] (2.7,2.1) -- (2.7,0.78);
\filldraw[fill=white] (2.7,0.75) circle (0.8pt); \draw[thick, dotted] (2.72,0.73) -- (3.35,0.1); \draw[thick, dashed] (4.05,0.1) -- (4.7,0.75); \draw[very thin] (2.7,0.1) -- (4.75,0.1); \draw[very thin] (2.7,0.1) -- (2.7,0.72); \draw[very thin] (4.7,0.1) -- (4.7,0.13); \node at (2.7,0) {$0$}; \node at (4.7,0) {$1$}; \node at (4.8,0.1) {$\frac{1}{p}$}; \node at (2.7,2.175) {$\delta$}; \draw (2.43,0.75) node {\tiny{$\delta=\nu+\tfrac{1}{2}$}}; \draw (3.35,0) node {\tiny{$\tfrac{2\nu+1}{4(\nu+1)}$}}; \draw (4.05,0) node {\tiny{$\tfrac{2\nu+3}{4(\nu+1)}$}}; \draw (2.6,1.4) node [rotate=90] {\tiny{$(p,p)$-strong}}; \draw (3.1,0.45) node [rotate=-45] {\tiny{$(p,p)$-restric. weak}}; \draw (4.3,0.45) node [rotate=45] {\tiny{$(p,p)$-weak}}; \draw (4.8,1.4) node [rotate=90] {\tiny{$(p,p)$-weak}}; \draw (3.7,-0.2) node {Figure 2: case $\nu>-\tfrac{1}{2}$.}; \end{tikzpicture} \end{center}
At this point, a comment is in order. Note that J. E. Gilbert \cite{Gi} also proves weak type norm inequalities for maximal operators associated with orthogonal expansions. The method used cannot be applied in our case, and the reason is the same as can be read in \cite{Ch-Muc}, at the end of Sections 15 and 16 therein. Following the technique in \cite{Gi} we have to analyze some weak type inequalities for Hardy operator and its adjoint with weights and these inequalities do not hold for $p=p_0$ and $p=p_1$.
The proof of the sufficiency in Theorem \ref{th:max} will be deduced from a more general result in which we analyze the boundedness of the operator $\mathcal{B}^\delta f$ with potential weights. Before stating it, we need a previous definition. We say that the parameters $(b,B,\nu,\delta)$ satisfy the $C_p$ conditions if \begin{align}
b& > \frac{-2(\nu+1)}{p} \,\,\, (\ge \text{ if }p=\infty), \label{ec:con1B}\\
B& < 2(\nu+1)\left(1-\frac1p\right) \,\,\, (\le \text{ if }
p=1), \label{ec:con2B}\\
b& > 2(\nu+1)\left(\frac12-\frac1p\right)-\delta-\frac12\,\,\,
(\ge \text{ if }p=\infty), \label{ec:con3B}\\
B& \le 2(\nu+1)\left(\frac12-\frac1p\right)+\delta+\frac12, \label{ec:con4B}\\
B &\le b \label{ec:con5B}, \end{align} and in at least one of each of the following pairs the inequality is strict: \eqref{ec:con2B} and \eqref{ec:con5B}, \eqref{ec:con3B} and \eqref{ec:con5B}, and \eqref{ec:con4B} and \eqref{ec:con5B} except for $p=\infty$. The result concerning inequalities with potential weights is the following. \begin{Theo} \label{th:AcFuerteMaxRedonda} Let $\nu>-1$, $\delta>0$, and $1< p\le\infty$. If $(b,B,\nu,\delta)$ satisfy the $C_p$ conditions, then \begin{equation*}
\left\|x^b\mathcal{B}^{\delta}f\right\|_{L^p((0,1),d\mu_{\nu})}\le C \|x^Bf\|_{L^p((0,1),d\mu_{\nu})}, \end{equation*} with $C$ independent of $f$. \end{Theo}
A result similar to Theorem \ref{th:AcFuerteMaxRedonda} for the partial sum operator was proved in \cite [Theorem 1]{GuPeRuVa}. It followed from a weighted version of a general Gilbert's maximal transference theorem, see \cite[Theorem 1]{Gi}. The weighted extension of Gilbert's result given in \cite{GuPeRuVa} depended heavily on the $A_p$ theory and it can not be used in our case because it did not capture all the information relative to the weights. On the other hand, it is also remarkable the paper by K. Stempak \cite{Stem2} in which maximal inequalities for the partial sum operator of Fourier-Bessel expansions and divergence and convergence results are discussed.
The necessity in Theorem \ref{th:max} will follow by showing that the operator $\mathcal{B}^\delta f$ is neither $(p_1,p_1)$-weak nor $(p_0,p_0)$-strong for $\nu>-1/2$ and $0<\delta\le \nu+1/2$. This is the content of the next theorems. \begin{Theo} \label{th:noweak} Let $\nu>-1/2$. Then \[
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}
\|\mathcal{B}^{\delta}_{R}f\|_{L^{p_1,\infty}((0,1),d\mu_\nu)}\ge C (\log R)^{1/p_0}, \] if $0<\delta<\nu+1/2$; and \[
\sup_{\|f\|_{L^\infty((0,1),d\mu_\nu)}=1}
\|\mathcal{B}^{\delta}_{R}f\|_{L^{\infty}((0,1),d\mu_\nu)}\ge C \log R, \] if $\delta= \nu+1/2$. \end{Theo}
\begin{Theo} \label{th:nostrong} Let $\nu>-1/2$. Then \[ \sup_{E\subset
(0,1)}\frac{\|\mathcal{B}^{\delta}_{R}\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}{\|\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}\ge C (\log R)^{1/p_0}, \] if $0<\delta<\nu+1/2$; and \[
\sup_{\|f\|_{L^1((0,1),d\mu_\nu)}=1}
\|\mathcal{B}^{\delta}_{R}f\|_{L^{1}((0,1),d\mu_\nu)}\ge C \log R, \] if $\delta= \nu+1/2$. \end{Theo}
The paper is organized as follows. In the next section, we give the proof of Theorem \ref{th:fin}. In Section \ref{sec:proofAcFuerte} we first relate the Bochner-Riesz means $\mathcal{B}_R^{\delta}$ to the Bochner-Riesz means operator associated with the Fourier-Bessel system in the Lebesgue measure setting. Then, we prove weighted inequalities for the supremum of this new operator. With the connection between these means and the operator $\mathcal{B}_R^{\delta}$, we obtain Theorem \ref{th:AcFuerteMaxRedonda} and, as a consequence, the sufficiency of Theorem \ref{th:max}. Sections \ref{sec:ProofThAcDebilMaxRedonda} and \ref{sec:acdelrest} will be devoted to the proofs of Theorems \ref{th:AcDebilMaxRedonda} and \ref{th:AcDebilRestMaxRedonda}, respectively. The proofs of Theorems \ref{th:noweak} and \ref{th:nostrong} are contained in Section \ref{sec:negativeths}. One of the main ingredients in the proofs of Theorems \ref{th:noweak} and \ref{th:nostrong} will be Lemma \ref{lem:pol}, this lemma is rather technical and it will be proved in the Section \ref{sec:techlemma}.
Throughout the paper, we will use the following notation: for each $p\in[1,\infty]$, we will denote by $p'$ the conjugate of $p$, that is, $\tfrac{1}{p}+\tfrac{1}{p'}=1$. We shall write $X\simeq Y$ when simultaneously $X\le C Y$ and $Y \le C X$.
\section{Proof of Theorem \ref{th:fin}}\label{sec:Proof Th1}
The proof of the sufficiency follows from Theorem \ref{th:AcDebilMaxRedonda} and standard arguments.
In order to prove the necessity, let us see that, for $0<\delta<\nu+1/2$ and $\nu>-1/2$, there exists a function $f\in L^{p}((0,1),d\mu_{\nu})$, $p\in [1,p_0)$, for which $\mathcal{B}_R^{\delta}(f,x)$ diverges. We follow some ideas contained in \cite{Meaney} and \cite{Stem2}.
First, we need a few more ingredients. Recall the well-known asymptotics for the Bessel functions (see \cite[Chapter 7]{Wat}) \begin{equation}\label{zero} J_\nu(z) = \frac{z^\nu}{2^\nu \Gamma(\nu+1)} + O(z^{\nu+2}),
\quad |z|<1,\quad |\arg(z)|\leq\pi, \end{equation} and \begin{equation} \label{infty} J_\nu(z)=\sqrt{\frac{2}{\pi z}}\left[ \cos\left(z-\frac{\nu\pi}2
- \frac\pi4 \right) + O(e^{\mathop{\rm Im}(z)}z^{-1}) \right], \quad |z|
\ge 1,\quad |\arg(z)|\leq\pi-\theta, \end{equation} where $D_{\nu}=-(\nu\pi/2+\pi/4)$. It will also be useful the fact that (cf. \cite[(2.6)]{OScon}) \begin{equation} \label{eq:zerosCons} s_{j}=O(j). \end{equation} For our purposes, we need estimates for the $L^p$ norms of the functions $\psi_j$. These estimates are contained in the following lemma, whose proof can be read in \cite[Lemma 2.1]{Ci-RoWave}. \begin{Lem} \label{Lem:NormaFunc} Let $1\le p\le\infty$ and $\nu>-1$. Then, for $\nu>-1/2$, \begin{equation*}
\|\psi_j\|_{L^p((0,1),d\mu_{\nu})}\simeq \begin{cases} j^{(\nu+1/2)-\frac{2(\nu+1)}{p}}, & \text{if $p>\frac{2(\nu+1)}{\nu+1/2}$},\\ (\log j)^{1/p}, & \text{if $p=\frac{2(\nu+1)}{\nu+1/2}$},\\ 1, & \text{if $p<\frac{2(\nu+1)}{\nu+1/2}$}, \end{cases} \end{equation*} and, for $-1<\nu\le-1/2$, \begin{equation*}
\|\psi_j\|_{L^p((0,1),d\mu_{\nu})}\simeq \begin{cases} 1, & \text{if $p<\infty$},\\ j^{\nu+1/2},& \text{if $p=\infty$}. \end{cases} \end{equation*} \end{Lem} We will also use a slight modification of a result by G. H. Hardy and M. Riesz for the Riesz means of order $\delta$, that is contained in \cite[Theorem 21]{HaRi}. We present here this
result, adapted to the Bochner-Riesz means. We denote by $S_R(f,x)$ the partial sum associated to the Fourier-Bessel expansion, namely \[ S_R(f,x)=\sum_{0<s_j\le R} a_j(f)\psi_j(x). \] The result reads as follows. \begin{Lem} \label{lem:HardyRiesz} Suppose that $f$ can be expressed as a Fourier-Bessel expansion and for some $\delta>0$ and $x\in(0,1)$ its Bochner-Riesz means $\mathcal{B}_R^\delta(f,x)$ converges to $c$ as $R\rightarrow \infty$. Then, for $s_n\le R < s_{n+1}$, \[
|S_R(f,x)-c|\le A_{\delta}n^{\delta}\sup_{0<t\le s_{n+1}}|\mathcal{B}_{t}^{\delta}(f,x)|. \] \end{Lem} By using this lemma, we can write \begin{equation} \label{ConsecuenciaLemaHardyRiesz}
|a_j(f)\psi_j(x)|=|(S_{s_{j}}(f,x)-c)-(S_{s_{j-1}}(f,x)-c)|\le A_{\delta}j^\delta\sup_{0<t\le s_{j+1}}|\mathcal{B}_{t}^{\delta}(f,x)|. \end{equation} Let us proceed with the proof of the necessity. Let $1\le p<p_0$. Note that $p_0'=p_1$. Therefore, $p'>p_0'>\tfrac{2(\nu+1)}{\nu+1/2}$, and $\delta<\nu+1/2-\frac{2(\nu+1)}{p'}:=\lambda$. By Lemma
\ref{Lem:NormaFunc}, $\|\psi_j\|_{L^{p'}((0,1),d\mu_{\nu})}\ge C j^{\lambda}$. Then, we have that the mapping $f\mapsto a_j(f)$, where $a_j(f)$ was given in \eqref{SeriesCoeficientes}, is a bounded linear functional on $L^{p}((0,1),d\mu_{\nu})$ with norm bounded below by a constant multiple of $j^\lambda$. By uniform boundedness principle, for $p$ conjugate to $p'$ and each $0\le \varepsilon<\lambda$, there is a function $f_0\in L^p((0,1), d\mu_{\nu})$ so that $a_j(f_0)j^{-\varepsilon}\rightarrow \infty$ as $j\rightarrow \infty$. By taking $\varepsilon=\delta$, we have that \begin{equation}\label{coeficienteInfinito} a_j(f_0)j^{-\delta}\rightarrow \infty \quad \textrm{ as } \quad j\rightarrow \infty. \end{equation}
Suppose now that $B_R^\delta(f_0,x)$ converges. Then, by Egoroff's theorem, it converges on a subset $E$ of positive measure in $(0,1)$ and, clearly, we can think that $E\subset (\eta, 1)$ for some fixed $\eta>0$. For each $x\in E$, we can consider $j$ such that $s_j x\ge 1$ and, by \eqref{infty}, \begin{align*}
|a_j(f_0)\psi_j(x)|&=\big|a_j(f_0)
\Big(\frac{\sqrt2}{|J_{\nu+1}(s_j)|}
x^{-\nu}J_{\nu}(s_jx)\\
&-\frac{\sqrt2}{|J_{\nu+1}(s_j)|}x^{-\nu}
\Big(\frac{2}{\pi s_jx}\Big)^{1/2}
\cos(s_jx+D_{\nu})\Big)\\
&+a_j(f_0) \frac{\sqrt2}{|J_{\nu+1}(s_j)|}x^{-\nu}
\Big(\frac{2}{\pi s_jx}\Big)^{1/2}
\cos(s_jx+D_{\nu})\big|\\
&=Cs_j^{-1/2}\frac{\sqrt 2}{|J_{\nu+1}(s_j)|}|a_j(f_0)x^{-\nu-1/2}
\big(O((s_jx)^{-1})+\cos(s_jx+D_{\nu})\big)|\\
&\simeq |a_j(f_0)x^{-\nu-1/2}(\cos(s_jx+D_{\nu})+O((s_jx)^{-1}))|. \end{align*} By \eqref{ConsecuenciaLemaHardyRiesz} on this set $E$, \[
|a_j(f_0)x^{-\nu-1/2}(\cos(s_jx+D_{\nu})+O((j)^{-1}))|\le A_{\delta}j^{\delta}\sup_{0<t\le s_{j+1}}|\mathcal{B}_{t}^{\delta}(f_0,x)|\le K_Ej^{\delta}, \] uniformly on $x\in E$. We also used \eqref{eq:zerosCons} in the latter. The inequality above is equivalent to $$
|a_j(f_0)(\cos(s_jx+D_{\nu})+O(j^{-1}))|\le K_E x^{\nu+1/2}j^{\delta}\le K_{E}j^{\delta}. $$ Therefore, \begin{equation} \label{ec:boundFj}
|a_j(f_0)j^{-\delta}(\cos(s_jx+D_{\nu})+O((j)^{-1}))|\le K_E. \end{equation} Now, taking the functions \[ F_j(x)=a_j(f_0)j^{-\delta}(\cos(s_jx+D_{\nu})+O(j^{-1})), \qquad x\in E, \] and using an argument based on the Cantor-Lebesgue and Riemann-Lebesgue theorems, see \cite[Section 1.5]{Meaney} and \cite[Section IX.1]{Zyg}, we obtain that \[
\int_E |F_j(x)|^2\, dx\ge C |a_j(f_0)j^{-\delta}|^2|E|, \]
where, as usual, $|E|$ denotes the Lebesgue measure of the set $E$. On the other hand, by \eqref{ec:boundFj}, \[
\int_E |F_j(x)|^2\, dx\le K_E^2 |E|. \] Then, from the previous estimates, it follows that
$|a_j(f_0)j^{-\delta}|\le C$, which contradicts \eqref{coeficienteInfinito}.
\section{Bochner-Riesz means for Fourier-Bessel expansions in the Lebesgue measure setting. Proof of Theorem \ref{th:AcFuerteMaxRedonda}}\label{sec:proofAcFuerte}
For our convenience, we are going to introduce a new orthonormal system. We will take the functions \[
\phi_j(x)=\frac{\sqrt{2x}J_{\nu}(s_jx)}{|J_{\nu+1}(s_j)|},\quad j=1,2,\dots. \] These functions are a slight modification of the functions \eqref{eq:FBesselSystemI}; in fact, \begin{equation} \label{eq:Relation} \phi_j(x)=x^{\nu+1/2}\psi_j(x). \end{equation} The system $\{\phi_j(x)\}_{j\ge1}$ is a complete orthonormal basis of $L^2((0,1),dx)$.
In this case, the corresponding Fourier-Bessel expansion of a function $f$ is \[ f\sim\sum_{j=1}^{\infty}b_j(f) \phi_j(x), \qquad \text{with} \qquad b_j(f)=\left(\int_0^1 f(y)\phi_j(y)\, dy\right) \] provided the integral exists, and for $\delta>0$ the Bochner-Riesz means of this expansion are \[ B_R^{\delta}(f,x)=\sum_{j\ge 1} \left(1-\frac{s_j^2}{R^2}\right)_+^{\delta}b_j(f)\phi_j(x), \] where $R>0$ and $(1-s^2)_+=\max\{1-s^2,0\}$. It follows that \[ B_R^{\delta}(f,x)=\int_0^1 f(y)K_R^\delta(x,y)\, dy \] where \begin{equation} \label{ec:kern} K_R^\delta(x,y)=\sum_{j\ge 1}\left(1-\frac{s_j^2}{R^2}\right)_+^{\delta}\phi_j(x)\phi_j(y). \end{equation}
Our next target is the proof of Theorem \ref{th:AcFuerteMaxRedonda}. Taking into account that \[ \mathcal{B}_R^\delta f(x)=\int_0^1 f(y)\mathcal{K}_R^\delta(x,y) \, d\mu_\nu(y), \] where \[ \mathcal{K}_R^\delta(x,y)=\sum_{j\ge 1}\left(1-\frac{s_j^2}{R^2}\right)_+^{\delta}\psi_j(x)\psi_j(y), \] it is clear, from \eqref{eq:Relation}, that $\mathcal{K}_R^\delta(x,y)=(xy)^{-(\nu+1/2)}K_R^{\delta}(x,y)$. Then, it is verified that the inequality \[
\|x^b\mathcal{B}^{\delta}(f,x)\|_{L^p((0,1),d\mu_{\nu})}\le C\|x^Bf(x)\|_{L^p((0,1),d\mu_{\nu})} \] is equivalent to \begin{equation*}
\|x^{b+(\nu+1/2)(2/p-1)}B^{\delta}(f,x)\|_{L^p((0,1),dx)}\le C\|x^{B+(\nu+1/2)(2/p-1)}f(x)\|_{L^p((0,1),dx)}, \end{equation*} that is, we can focus on the study of a weighted inequality for the operator $B_R^{\delta}(f,x)$. The first results about convergence of this operator can be found in \cite{Ci-Ro}.
We are going to prove an inequality of the form \begin{equation*}
\|x^{a}B^{\delta}(f,x)\|_{L^p((0,1),dx)}\le C
\|x^{A}f(x)\|_{L^p((0,1),dx)} \end{equation*} for $\delta>0$, $1< p\leq \infty$, under certain conditions for $a, A,\nu$ and $\delta$. Besides, a weighted weak type result for
$\sup_{R>0}|B_R^{\delta}(f,x)|$ will be proved for $p=1$. The abovementioned conditions are the following. Let $\nu>-1$, $\delta>0$ and $1\leq p\leq \infty$; parameters $(a,A,\nu,\delta)$ will be said to satisfy the $c_p$ conditions provided \begin{align}
a & > -1/p-(\nu+1/2) \,\,\, (\ge \text{ if } p=\infty), \label{ec:con1}\\
A & < 1-1/p+(\nu+1/2)\,\,\, (\le \text{ if } p=1),\label{ec:con2}\\
a &> -\delta-1/p\,\,\, (\ge \text{ if }p=\infty),\label{ec:con3}\\
A &\le 1+\delta-1/p, \label{ec:con4}\\
A &\le a\label{ec:con5} \end{align} and in at least one of each of the following pairs the inequality is strict: \eqref{ec:con2} and \eqref{ec:con5}, \eqref{ec:con3} and \eqref{ec:con5}, and \eqref{ec:con4} and \eqref{ec:con5} except for $p=\infty$.
The main results in this section are the following: \begin{Theo} \label{th:main1} Let $\nu>-1$, $\delta>0$ and $1< p\le \infty$. If $(a, A, \nu, \delta)$ satisfy the $c_p$ conditions, then \[
\|x^{a}B^{\delta}(f,x)\|_{L^p((0,1),dx)}\le C
\|x^{A}f(x)\|_{L^p((0,1),dx)}, \] with $C$ independent of $f$. \end{Theo}
\begin{Theo} \label{th:main2} Let $\nu>-1$ and $\delta>0$. If $(a, A, \nu, \delta)$ satisfy the $c_1$ conditions and \[ E_{\lambda}=\left\{x\in (0,1)\colon x^{a}
\sup_{R>0}\left(|B_R^{\delta}(f,x)|\right)>\lambda \right\}, \]
then \[
|E_{\lambda}|\leq C \frac{\|x^{A}
f(x)\|_{L^1((0,1),dx)}}{\lambda}, \] with $C$ independent of $f$ and $\lambda$. \end{Theo}
Note that, taking $a=b+(\nu+1/2)(2/p-1)$ and $A=B+(\nu+1/2)(2/p-1)$, Theorem \ref{th:AcFuerteMaxRedonda} follows from Theorem \ref{th:main1}.
The proofs of Theorem \ref{th:main1} and Theorem \ref{th:main2} will be achieved by decomposing the square $(0,1)\times (0,1)$ into five regions and obtaining the estimates therein. The regions will be: \begin{align} \label{regions} \notag A_1&=\{(x,y):0 < x, y\leq 4/R\},\\
\notag A_2&=\{(x,y):4/R<\max\{x,y\}<1,\, |x-y|\le 2/R \},\\
A_3&=\{(x,y): 4/R \leq x < 1,\, 0 < y\leq x/2\},\\ \notag A_4&=\{(x,y):0 < x \leq y/2,\, 4/R \leq y< 1\}, \\ \notag A_5&=\{(x,y): 4/R < x < 1, \, x/2 < y< x- 2/R\}\\ \notag &\kern25pt\cup \{(x,y): y/2 < x \leq y-2/R,\, 4/R \leq y<1\}. \end{align}
Theorem~\ref{th:main1} and Theorem~\ref{th:main2} will follow by showing that, if $1\leq p\leq \infty$, then \begin{equation}
\label{eq:des_1} \left\|\sup_{R> 0}\int_0^1 y^{-A}x^a|K_R^\delta(x,y)||f(y)|\chi_{A_j}\,dy\right\|_{L^p((0,1),dx)}
\leq C\|f(x)\|_{L^p((0,1),dx)} \end{equation} holds for $j=1,3,4$ and that \begin{equation}
\label{eq:des_2} \int_0^1 y^{-A}x^a|K_R^\delta(x,y)||f(y)|\chi_{A_j}\,dy \leq C M (f,x), \end{equation} for $j=2,5$, where $M$ is the Hardy-Littlewood maximal function of $f$, and $C$ is independent of $R, x$ and $f$. These results and the fact that $M$ is $(1,1)$-weak and $(p,p)$-strong if $1<p\leq \infty$ complete the proofs.
To get \eqref{eq:des_1} and \eqref{eq:des_2} we will use a very precise pointwise estimate for the kernel $K_R^\delta(x,y)$, obtained in \cite{Ci-Ro}; there, it was shown that \begin{equation} \label{ec:kernel}
|K_R^\delta(x,y)|\le C \begin{cases} (xy)^{\nu+1/2}R^{2(\nu+1)}, & (x,y) \in A_1,\\ R, & (x,y) \in A_2\\
\frac{\Phi_\nu(Rx)\Phi_{\nu}(Ry)}{R^{\delta}|x-y|^{\delta+1}}, & (x,y) \in A_3\cup A_4 \cup A_5, \end{cases} \end{equation} with \begin{equation} \label{ec:aux} \Phi_\nu(t)=\begin{cases}t^{\nu+1/2}, & \text{ if $0<t<2$},\\ 1,& \text{ if $t\ge 2$}.\end{cases} \end{equation}
The proof of \eqref{eq:des_2} follows from the given estimate for the kernel $K_R^\delta(x,y)$ and $y^{-A}x^a\simeq C$ in $A_2\cup A_5$ because $A\le a$. In the case of $A_2$, from
$|K_R^\delta(x,y)|\le C R$ we deduce easily the required inequality. For $A_5$ the result is a consequence of
$\Phi_{\nu}(Rx)\Phi_\nu(Ry)\le C$ and of a decomposition of the region in strips such that $R|x-y|\simeq 2^{k}$, with $k=0,\dots, [\log_2 R]-1$; this can be seen in \cite[p. 109]{Ci-Ro}
In this manner, to complete the proofs of Theorem~\ref{th:main1} and Theorem~\ref{th:main2} we only have to show \eqref{eq:des_1} for $j=1,3,4$ in the conditions $c_p$ for $1\le p \le \infty$, and this is the content of Corollary~\ref{cor:corolario2} in Subsection~\ref{subsec:reg1}. In its turn, Corollary~\ref{cor:corolario2} follows from Lemmas~\ref{lem:lema7} and~\ref{lem:lema8} in the same subsection. Previously, Subsection \ref{subsec:lemmas} contains some technical lemmas that will be used in the proofs of Lemmas~\ref{lem:lema7} and~\ref{lem:lema8}.
\subsection{Technical Lemmas} \label{subsec:lemmas}
To prove \eqref{eq:des_1} for $j=1,3,4$ we will use an interpolation argument based on six lemmas. These are stated below. They are small modifications of the six lemmas contained in Section 3 of \cite{Mu-We} where a sketch of their proofs can be found.
\begin{Lem} \label{lem:lema1} Let $\xi_0>0$, if $r<-1$, $r+t\leq-1$ and $r+s+t\leq-1$, then for $p=1$ \[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{\xi_0\leq\xi\leq x}\xi^s
\int_\xi^x y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq C\|f(x)\|_{L^p((0,\infty),dx)} \] with $C$ independent of $f$. If $r\leq0$, $r+t\leq-1$ and $r+s+t\leq-1$ with equality holding in at most one of the first two inequalities, then this holds for $p=\infty$. \end{Lem}
\begin{Lem} \label{lem:lema2} Let $\xi_0>0$, if $t\leq0$, $r+t\leq-1$ and $r+s+t\leq-1$, with strict inequality in the last two in case of equality in the first, then for $p=1$ \[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{\xi_0\leq\xi\leq x}\xi^s
\int_x^\infty y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq C\|f(x)\|_{L^p((0,\infty),dx)} \] with $C$ independent of $f$. If $t<-1$, $r+t\leq-1$ and $r+s+t\leq-1$, then this holds for $p=\infty$. \end{Lem}
\begin{Lem} \label{lem:lema3} If $s<0$, $s+t\leq0$ and $r+s+t\leq-1$,with equality holding in at most one of the last two inequalities, then for $p=1$ \[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{\xi\geq x}\xi^s \int_x^\xi y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq C\|f(x)\|_{L^p((0,\infty),dx)} \] with $C$ independent of $f$. If $s<0$, $s+t\leq-1$ and $r+s+t\leq-1$ this holds for $p=\infty$. \end{Lem}
\begin{Lem} \label{lem:lema4} If $t\leq0$, $s+t\leq0$ and $r+s+t\leq-1$,with strict inequality holding in the first two in case the third is an equality, then for $p=1$ \[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{\xi\geq x}\xi^s
\int_\xi^\infty y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq C\|f(x)\|_{L^p((0,\infty),dx)} \] with $C$ independent of $f$. If $t<-1$, $s+t\leq-1$ and $r+s+t\leq-1$ then this holds for $p=\infty$. \end{Lem}
\begin{Lem} \label{lem:lema5} If $s<0$, $r+s<-1$ and $r+s+t\leq-1$, then for $p=1$ \[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{\xi\geq x}\xi^s \int_1^x y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq C\|f(x)\|_{L^p((0,\infty),dx)} \] with $C$ independent of $f$. If $s<0$, $r+s\leq 0$ and $r+s+t\leq-1$, with equality holding in at most one of the last two inequalities, this holds for $p=\infty$. \end{Lem}
\begin{Lem} \label{lem:lema6} If $r<-1$, $r+s<-1$ and $r+s+t\leq-1$, then for $p=1$ \[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{1\leq\xi\leq x}\xi^s
\int_1^{\xi} y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq C\|f(x)\|_{L^p((0,\infty),dx)} \] with $C$ independent of $f$. If $r\leq0$, $r+s\leq0$ and $r+s+t\leq-1$, with equality in at most one of the last two inequalities, this holds for $p=\infty$. \end{Lem}
\subsection{Proofs of Theorem~\ref{th:main1} and Theorem~\ref{th:main2} for regions $A_1$, $A_3$ and $A_4$}
\label{subsec:reg1}
This section contains the proofs of the inequality \eqref{eq:des_1} for regions $A_1$, $A_3$ and $A_4$. The results we will prove are included in the following \begin{Lem} \label{lem:lema7} If $\nu>-1$, $\delta>0$, $R>0$, $j=1, 3, 4$ and $(a, A, \nu, \delta)$ satisfy the $c_1$ conditions, then \eqref{eq:des_1} holds for $p=1$ with $C$ independent of $f$. \end{Lem}
\begin{Lem} \label{lem:lema8} If $\nu>-1$, $\delta>0$, $R>0$, $j=1, 3, 4$ and $(a, A, \nu, \delta)$ satisfy the $c_{\infty}$ conditions, then \eqref{eq:des_1} holds for $p=\infty$ with $C$ independent of $f$. \end{Lem}
\begin{Cor} \label{cor:corolario2} If $1\leq p\leq\infty$, $\nu>-1$, $\delta>0$, $R>0$, $(a, A, \nu, \delta)$ satisfy the $c_p$ conditions and $j=1,3,4$, then \eqref{eq:des_1} holds with $C$ independent of $f$. \end{Cor}
\textbf{Proof of Corollary~\ref{cor:corolario2}}. It is enough to observe that if $1< p<\infty$ and $(a, A, \nu, \delta)$ satisfy the $c_p$ conditions, then $(a-1+1/p, A-1+1/p, \nu, \delta)$ satisfy the $c_1$ conditions. So, by Lemma~\ref{lem:lema7} \begin{multline*}
\left\|\sup_{R\geq 0}\int_0^1 y^{-A+1-1/p}x^{a-1+1/p}|K_R^{\delta}(x,y)|\chi_{A_j}(x,y)|f(y)|\, dy \right\|_{L^1((0,1),dx)}\\\leq C\|f(x)\|_{L^1((0,1),dx)}, \end{multline*} and this is equivalent to \[ \int_0^1 x^{a+1/p}\left(\sup_{R\geq 0}\int_0^1
|K_R^{\delta}(x,y)|\chi_{A_j}(x,y)|f(y)|\, dy\right)
\frac{dx}{x}\leq C\int_0^1 x^{A+1/p}|f(x)|\frac{dx}{x}, \] where $j=1,3,4$. Similarly, if $(a,A,\nu,\delta)$ verify the $c_p$ conditions, then $(a+1/p, A+1/p, \nu, \delta )$ satisfy the $c_{\infty}$ conditions. Hence, by Lemma~\ref{lem:lema8} \begin{multline*}
\left\|x^{a+1/p}\sup_{R\geq 0}\int_0^1
|K_R^{\delta}(x,y)|\chi_{A_j}(x,y)|f(y)|\, dy\right\|_{L^{\infty}((0,1),dx)}
\\\leq C\|x^{A+1/p}f(x)\|_{L^{\infty}((0,1),dx)}. \end{multline*} Now, we can use the Marcinkiewicz interpolation theorem to obtain the inequality \begin{multline*}
\int_0^1 \left(x^{a+1/p} \left(\sup_{R\geq 0}\int_0^1
|K_R^{\delta}(x,y)|\chi_{A_j}(x,y)|f(y)|\, dy\right)\right)^p \frac{dx}{x}\\
\leq C\int_0^1 \left(x^{A+1/p}|f(x)|\right)^p\frac{dx}{x}, \end{multline*} for $1<p<\infty$ and the proof is finished.
Finally, we will prove Lemmas~\ref{lem:lema7} and~\ref{lem:lema8} for $A_j$, $j=1, 3$ and $4$, separately.
\textbf{Proof of Lemma~\ref{lem:lema7} and Lemma~\ref{lem:lema8} for $A_1$}. First of all, we have to note that $B_R^\delta (f,x)=0$ when $0<R<s_1$, being $s_1$ the first positive zero of $J_\nu$. Using the estimate~\eqref{ec:kernel}, the left side of \eqref{eq:des_1} in this case is bounded by \[
C\left\|x^{a+\nu+1/2}\chi_{[0,1]}(x)\sup_{s_1<R\leq 4/x}R^{2(\nu+1)} \int_0^{4/R}y^{-A+\nu+1/2}|f(y)|\, dy\right\|_{L^p((0,1),dx)}. \] Making the change of variables $x=4/u$ and $y=4/v$, we have \[
C\left\|u^{-a-\nu-\frac12-\frac2p}\chi_{[4,\infty)}(u)
\sup_{s_1\leq R \leq u}R^{2(\nu+1)}\int_R^\infty v^{A-(\nu+\frac12)-2+\frac2p}g(v)\,dv\right\|_{L^p((0,\infty),du)}, \]
where $\|\cdot\|_{L^p((0,\infty),du)}$ denotes the $L^p$ norm in the variable $u$, and \[
g(v)=v^{-2/p}|f(4v^{-1})|. \] Note that function $g(v)$ is supported in $(1,\infty)$ and
$\|g\|_{L^p((0,\infty),du)}=\|f\|_{L^p((0,1),dx)}$. The function $g$ will be used through the subsection, but the value $4$ may be changed by another one, at some points, without comment. Now, splitting the inner integral at $u$, we obtain the sum of \begin{equation} \label{ec:pa11}
C\left\|u^{-a-\nu-\frac12-\frac2p}\chi_{[4,\infty)}(u)
\sup_{s_1\leq R \leq u}R^{2(\nu+1)}\int_R^u v^{A-(\nu+\frac12)-2+\frac2p}g(v)\,dv\right\|_{L^p((0,\infty),du)} \end{equation} and \begin{equation} \label{ec:pa12}
C\left\|u^{-a-\nu-\frac12-\frac2p}\chi_{[4,\infty)}(u)
\sup_{s_1\leq R \leq u}R^{2(\nu+1)}\int_u^\infty v^{A-(\nu+\frac12)-2+\frac2p}g(v)\,dv\right\|_{L^p((0,\infty),du)}. \end{equation} From Lemma~\ref{lem:lema1} we get the required estimate for \eqref{ec:pa11}, using conditions \eqref{ec:con1} and \eqref{ec:con5}; Lemma~\ref{lem:lema2} is applied to inequality \eqref{ec:pa12}, there we need conditions \eqref{ec:con2} and \eqref{ec:con5} and the restriction on them. This completes the proof of Lemmas~\ref{lem:lema7} and~\ref{lem:lema8} for $j=1$.
\textbf{Proof of Lemma~\ref{lem:lema7} and Lemma~\ref{lem:lema8} for $A_3$}. Clearly, the left side of \eqref{eq:des_1} is bounded by \[
C\left\|x^a \chi_{[4/R,1]}(x)\sup_{4/x\leq R}\int_0^{x/2}y^{-A}
|K_R^{\delta}(x,y)||f(y)|\, dy\right\|_{L^p((0,1),dx)}. \] Splitting the inner integral at $2/R$, using the bound for the kernel given in \eqref{ec:kernel} and the definition of $\Phi_\nu$, we have this expression majorized by the sum of \begin{equation}
\label{ec:pa21} \left\|x^a \chi_{[0,1]}(x)\sup_{4/x\leq R}\int_0^{2/R}|f(y)|\frac{(Ry)^{\nu+1/2}y^{-A}}{R^{\delta}|x-y|^{\delta+1}}\, dy\right\|_{L^p((0,1),dx)} \end{equation} and \begin{equation}
\label{ec:pa22} \left\|x^a \chi_{[0,1]}(x)\sup_{4/x\leq R}\int_{2/R}^{x/2}\frac{|f(y)|y^{-A}}{R^{\delta}|x-y|^{\delta+1}}\, dy
\right\|_{L^p((0,1),dx)}. \end{equation}
For \eqref{ec:pa21}, taking into account that $|x-y|\simeq x$ in $A_3$, the changes of variables $x=4/u$, $y=2/v$ give us \[
\left\|u^{-a+(\delta+1)-\frac 2p}\chi_{[4,\infty)}(u)\sup_{u\leq R} R^{-\delta+(\nu+1/2)}\int_R^{\infty}v^{-(\nu+1/2)+A+\frac 2p-2}g(v)\, dv\right\|_{L^p((0,\infty),du)}. \] Lemma~\ref{lem:lema4} can be used here. The required conditions for $p=1$ are \eqref{ec:con2}, \eqref{ec:con4} and \eqref{ec:con5} with the restriction in the pairs therein. For $p=\infty$ the same inequalities are needed.
On the other hand, in \eqref{ec:pa22}, using again that $|x-y|\simeq x$, by changing of variables $x=4/u$ and $y=2/v$ we have \begin{multline*}
C\left\|u^{-a+(\delta+1)-\frac
2p}\chi_{[4,\infty)}(u)\sup_{u\leq R}R^{-\delta}\int_{2u}^R
v^{A+\frac 2p-2}g(v)\, dv\right\|_{L^p((0,\infty),du)}\\
\leq C\left\|u^{-a+(\delta+1)-\frac
2p}\chi_{[4,\infty)}(u)\sup_{u\leq R}R^{-\delta}\int_u^R
v^{A+\frac 2p-2}g(v)\, dv\right\|_{L^p((0,\infty),du)}. \end{multline*} Lemma~\ref{lem:lema3} can then be applied. For $p=1$, we need $\delta>0$, which is an hypothesis, and \eqref{ec:con4} and \eqref{ec:con5} with its corresponding restriction. For $p=\infty$ the inequalities are the same, with the requirement that \eqref{ec:con4} is strict. This completes the proof of Lemmas~\ref{lem:lema7} and~\ref{lem:lema8} for $j=3$.
\textbf{Proof of Lemma~\ref{lem:lema7} and Lemma~\ref{lem:lema8} for $A_4$}. In this case, the left hand side of \eqref{eq:des_1} is estimated by \[
C\left\|x^a \chi_{[0,1/2]}(x)\sup_{R>4}\int_{\max(4/R,2x)}^1 y^{-A}|K_R^{\delta}(x,y)||f(y)|\, dy\right\|_{L^p((0,1),dx)}. \] To majorize this, we decompose the $R$-range in two regions: $4<R\leq 2/x$ and $R\geq 2/x$. In this manner, with the bound for the kernel given in \eqref{ec:kernel} and the definition of $\Phi_\nu$, the previous norm is controlled by the sum of \[
C\left\|x^a \chi_{[0,1/2]}(x)\sup_{4<R\leq 2/x}
\int_{4/R}^1 |f(y)|\frac{(Rx)^{\nu+1/2}y^{-A}}{R^{\delta}|x-y|^{\delta+1}}\, dy
\right\|_{L^p((0,1),dx)} \] and \[
C\left\|x^a \chi_{[0,1/2]}(x)\sup_{R\geq 2/x}\int_{2x}^1
\frac{|f(y)|y^{-A}}{R^{\delta}|x-y|^{\delta+1}}\, dy\right\|_{L^p((0,1),dx)}. \]
Next, using that $|x-y|\simeq y$ in $A_4$, with the changes of variables $x=2/u$ and $y=1/v$ the previous norms are controlled by \begin{equation}
\label{ec:pa31} C\left\|u^{-a-\frac 2p-(\nu+\frac 12)}\chi_{[4,\infty)}(u)\sup_{4<R\leq u}R^{-\delta+(\nu +\frac 12)} \int_1^{R/4}v^{A+\frac 2p
-2+(\delta+1)}g(v)\,dv\right\|_{L^p((0,\infty),du)} \end{equation} and \begin{equation}
\label{ec:pa32} C\left\|u^{-a-\frac 2p}\chi_{[4,\infty)}(u)\sup_{R\geq u}R^{-\delta}\int_1^{u/4}
v^{A+\frac 2p -2+(\delta+1)}g(v)\,dv\right\|_{L^p((0,\infty),du)}. \end{equation} In \eqref{ec:pa31}, we use Lemma~\ref{lem:lema6}; for $p=1$, conditions \eqref{ec:con1}, \eqref{ec:con3} and \eqref{ec:con5} are needed; we need the same for $p=\infty$. For \eqref{ec:pa32}, Lemma~\ref{lem:lema5} requires the hypothesis $\delta>0$ and conditions \eqref{ec:con3} and \eqref{ec:con5} for $p=1$ and the same for $p=\infty$ with the restrictions in the pairs therein. This proves Lemmas~\ref{lem:lema7} and~\ref{lem:lema8} for $j=4$.
\section{Proof of Theorem \ref{th:AcDebilMaxRedonda}} \label{sec:ProofThAcDebilMaxRedonda}
Now we shall prove Theorem \ref{th:AcDebilMaxRedonda}. First note that, by \eqref{eq:Relation}, we can write \begin{equation*} \mathcal{B}_R^{\delta}(f,x) =\int_0^1f(y)\left(\frac{y}{x}\right)^{\nu+1/2}K_R^{\delta}(x,y)\,dy, \end{equation*} where $K_R^\delta$ is the kernel in \eqref{ec:kern}. By taking $g(y)=f(y)y^{\nu+1/2}$, to prove the result it is enough to check that \[
\int_{E}\,d\mu_\nu(x)\le \frac{C}{\lambda^{p}}
\int_0^1|g(x)|^{p}x^{(\nu+1/2)(2-p)}\,dx, \] where $E=\left\{x\in(0,1):
\sup_{R>0}x^{-(\nu+1/2)}\int_0^1|g(y)||K_R^{\delta}(x,y)|\,dy>\lambda\right\}$ and $p=p_0(\delta)$. We decompose $E$ into four regions, such that $E=\bigcup_{i=1}^{4}J_i$, where \begin{equation*}
J_i=\left\{x\in(0,1): \sup_{R>0}x^{-(\nu+1/2)}\int_0^{1}|g(y)|
\chi_{B_i}(x,y)|K_R^{\delta}(x,y)|\,dy>\lambda\right\} \end{equation*} for $i=1,\dots,4$, with $B_1=A_1$, $B_2=A_2\cup A_5$, $B_3=A_3$, and $B_4=A_4$ where the sets $A_i$ were defined in \eqref{regions}. Note also that $\int_{E}\,d\mu_\nu(x)\le\sum_{i=1}^4\int_{J_i}\,d\mu_\nu(x)$, then we need to prove that \begin{equation} \label{ec:boundweak}
\int_{J_i}\,d\mu_\nu(x)\le \frac{C}{\lambda^{p}}
\int_0^1|g(x)|^{p}x^{(\nu+1/2)(2-p)}\,dx, \end{equation} for $i=1,\dots,4$ and $p=p_0(\delta)$. At some points along the proof we will use the notation \begin{equation} \label{eq:integral}
I_p:=\int_0^{1}|g(y)|^{p}y^{(\nu+1/2)(2-p)}\,dy. \end{equation}
In $J_1$, by applying \eqref{ec:kernel} and H\"older inequality with $p=p_0$, we have \begin{multline*}
x^{-(\nu+1/2)}\int_0^{1}|g(y)|\chi_{B_1}(x,y)
|K_R^{\delta}(x,y)|\,dy\\
\begin{aligned}
&\le Cx^{-(\nu+1/2)}\int_0^{4/R}
|g(y)|(xy)^{\nu+1/2}R^{2(\nu+1)}\,dy\\
&\le C R^{2(\nu+1)}\left(\int_0^{4/R}
|g(y)|^{p_0}y^{(\nu+1/2)(2-p_0)}\,dy\right)^{1/p_0}
\left(\int_0^{4/R}y^{(2\nu+1)}\,dy\right)^{1/p'_0}\\
&=C R^{\frac{2(\nu+1)}{p_0}}
\left(\int_0^{4/R}|g(y)|^{p_0}y^{(\nu+1/2)(2-p_0)}\,dy\right)^{1/p_0}
\le C R^{\frac{2(\nu+1)}{p_0}}I_{p_0}^{1/p_0}.
\end{aligned} \end{multline*} Therefore, \begin{align*}
\sup_{R>0}x^{-(\nu+1/2)}
\int_0^{1}|g(y)|\chi_{B_1}(x,y)|K_R^{\delta}(x,y)|\,dy
&\le C\sup_{R>0}\chi_{[0,4/R]}(x)R^{\frac{2(\nu+1)}{p_0}}
I_{p_0}^{1/p_0}\\&\le C
x^{-\frac{2(\nu+1)}{p_0}}I_{p_0}^{1/p_0}. \end{align*} In the case $p=1$, it is clear that \[
x^{-(\nu+1/2)}\int_0^{1}|g(y)|\chi_{B_1}(x,y)|K_R^{\delta}(x,y)|\,dy\le C R^{2(\nu+1)}I_1 \] and \[ \sup_{R>0}x^{-(\nu+1/2)}
\int_0^{1}|g(y)|\chi_{B_1}(x,y)|K_R^{\delta}(x,y)|\,dy\le C x^{-2(\nu+1)}I_1. \] Hence, for $p=p_0(\delta)$, \[
J_1\subseteq \{x\in(0,1): C x^{-\frac{2(\nu+1)}{p}}I_{p}^{1/p} >\lambda\}, \] and this gives \eqref{ec:boundweak} for $i=1$.
In $J_3$, note first that \begin{multline*}
\sup_{R>0}x^{-(\nu+1/2)}
\int_0^{1}|g(y)|\chi_{B_3}(x,y)|K_R^{\delta}(x,y)|\,dy\\
=\sup_{R>0}x^{-(\nu+1/2)}\chi_{[4/R,1]}(x)
\left(\int_0^{2/R}|g(y)||K_R^{\delta}(x,y)|\,dy+
\int_{2/R}^{x/2}|g(y)||K_R^{\delta}(x,y)|\,dy\right)\\:=R_1+R_2. \end{multline*} For $R_1$, using \eqref{ec:kernel}, the inequality $x/2<x-y$, which holds in $B_3$, and H\"older inequality with $p=p_0$, \begin{align*}
R_1
&\le \sup_{R>0}x^{-(\nu+3/2+\delta)}\chi_{[4/R,1]}(x)
\int_0^{2/R}R^{\nu+1/2-\delta}y^{\nu+1/2}|g(y)|\,dy\\
&\le\sup_{R>0}x^{-(\nu+3/2+\delta)}\chi_{[4/R,1]}(x)
R^{\nu+1/2-\delta}R^{-\frac{2(\nu+1)}{p'_0}}I_{p_0}^{1/p_0}
\le C x^{-\frac{2(\nu+1)}{p_0}}I_{p_0}^{1/p_0}, \end{align*} where $I_{p_0}$ is the same as in \eqref{eq:integral}. In the case $p=1$, the estimate $R_1\le C x^{-2(\nu+1)}I_{1}$ can be obtained easily.
On the other hand, for $R_2$, by using \eqref{ec:kernel} and H\"older inequality with $p=p_0$ again, \begin{align*}
R_2
&\le \sup_{R>0}x^{-(\nu+3/2+\delta)}
\chi_{[4/R,1]}(x)I_{p_0}^{1/p_0}R^{-\delta}
\left(\int_{2/R}^{x/2}y^{-(\nu+1/2)\frac{(2-p_0)p'_0}{p_0}}\,dy\right)^{1/p'_0}\\
&\le \sup_{R>0}x^{-(\nu+3/2+\delta)}
\chi_{[4/R,1]}(x)I_{p_0}^{1/p_0}R^{-\delta}
\left(\int_{2/R}^{x/2}y^{(\nu+1/2)\frac{2-p_0}{1-p_0}}\,dy\right)^{1/p'_0}. \end{align*} Using that $(\nu+1/2)\frac{2-p_0}{1-p_0}<-1$ and $4/R<x<1$, we have that \begin{align*}
R^{-\delta}\left(\int_{2/R}^{x/2}
y^{(\nu+1/2)\frac{2-p_0}{1-p_0}}\,dy\right)^{1/p'_0}
\le C\left(R^{-(\nu+1/2)\frac{2-p_0}{1-p_0}-1}\right)^{1/p'_0}
R^{-\delta}= C \end{align*} and the last inequality is true because the exponent of $R$ is zero. Then \[ R_2\le C x^{\frac{-2(\nu+1)}{p_0}}I_{p_0}^{1/p_0}. \] In the case $p=1$ applying H\"older inequality, then \begin{equation*}
R_2\le \sup_{R>0}x^{-(\nu+3/2+\delta)}\chi_{[4/R,1]}(x)I_1
\,R^{-\delta}\sup_{y\in[2/R,x/2]}y^{-(\nu+1/2)}. \end{equation*} Now, if $\nu+1/2>0$ and $\nu+1/2<\delta$, \begin{multline*} \sup_{R>0}\chi_{[4/R,1]}(x)R^{-\delta}\sup_{y\in[2/R,x/2]}y^{-(\nu+1/2)}\\ =C\sup_{R>0}\chi_{[4/R,1]}(x)R^{\nu+1/2-\delta}\le Cx^{-\nu-1/2+\delta}; \end{multline*} and if $\nu+1/2\le0$, \begin{multline*} \sup_{R>0}\chi_{[4/R,1]}(x)R^{-\delta}\sup_{y\in[2/R,x/2]}y^{-(\nu+1/2)}\\ =C\sup_{R>0}\chi_{[4/R,1]}(x)R^{-\delta}x^{-(\nu+1/2)}\le Cx^{-\nu-1/2+\delta}. \end{multline*} In this manner \[ R_2\le C x^{-2(\nu+1)}I_{1}. \] Therefore, collecting the estimates for $R_1$ and $R_2$ for $p=p_0$ and $p=1$, we have shown that \[
J_3\subseteq \{x\in(0,1): C x^{\frac{-2(\nu+1)}{p}}(x)I^{1/p}
>\lambda\}, \] hence we can deduce \eqref{ec:boundweak} for $i=3$.
For the region $J_4$, we proceed as follows \begin{align*}
\sup_{R>0}x^{-(\nu+1/2)}&
\int_{0}^1|g(y)|\chi_{B_4}(x,y)|K_R^\delta(x,y)|\,dy\\
&\le \sup_{R>0}x^{-(\nu+1/2)}\chi_{[0,2/R]}(x)
\int_{4/R}^1|g(y)||K_R^\delta(x,y)|\,dy\\
&\kern20pt+\sup_{R>0}x^{-(\nu+1/2)}\chi_{[2/R,1]}(x)
\int_{2x}^1|g(y)||K_R^\delta(x,y)|\,dy\\
&\le C\sup_{R>0}x^{-(\nu+1/2)}\chi_{[0,2/R]}(x)(Rx)^{\nu+1/2}
\int_{4/R}^1\frac{|g(y)|}{R^{\delta}|x-y|^{\delta+1}}\,dy\\
&\kern20pt+ C\sup_{R>0}x^{-(\nu+1/2)}\chi_{[2/R,1]}(x)
\int_{2x}^1\frac{|g(y)|}{R^{\delta}|x-y|^{\delta+1}}\,dy:=S_1+S_2. \end{align*} We first deal with $S_1$, we use that $y-x>y/2$, then \begin{align*}
S_1\le &C\sup_{R>0}\chi_{[0,2/R]}(x) R^{\nu+1/2-\delta}
\int_{4/R}^1\frac{|g(y)|}{y^{\delta+1}}\,dy\\
&\le C\sup_{R>0}\chi_{[0,2/R]}(x) R^{\nu+1}\int_{4/R}^1\frac{|g(y)|}{\sqrt{y}}\,dy
\le C x^{-(\nu+1)}\int_x^1\frac{|g(y)|}{\sqrt{y}}\,dy. \end{align*} Now for $p=p_0$ or $p=1$, we have that $2\nu+1-p(\nu+1)>-1$ and Hardy's inequality \cite[Lemma 3.14, p. 196]{SteinWeiss} is applied in the following estimate \begin{align*}
\int_0^1|S_1(x)|^{p}x^{2\nu+1}\,dx& \le C
\int_0^1\left(\int_x^1\frac{|g(y)|}{\sqrt{y}}\,dy\right)^{p}
x^{2\nu+1-p(\nu+1)}\,dx\\
&\le C \int_0^1\left|\frac{g(y)}{\sqrt y}\right|^{p}y^{2\nu+1-p\nu}\,dy
=C\int_0^1|g(y)|^{p}y^{(\nu+1/2)(2-p)}\,dy. \end{align*} Concerning $S_2$, observe that $\sup_{R>0}\chi_{[2/R,1]}(x)R^{-\delta}\le Cx^{\delta}$, thus \[
S_2\le C x^{-\nu-1/2+\delta}\int_x^1\frac{|g(y)|}{y^{\delta+1}}\,dy. \] Since for $p=p_0$ or $p=1$ we have that $2\nu+1-p(\nu+1/2-\delta)>-1$, we can use again Hardy's inequality to complete the required estimate. Indeed, \begin{align*}
\int_0^1|S_2(x)|^{p}x^{2\nu+1}\,dx&
\le C\int_0^1\left(\int_x^1\frac{|g(y)|}{y^{\delta+1}}\,dy\right)^{p}
x^{2\nu+1-p(\nu+1/2-\delta)}\,dx\\
&\le C\int_0^1\left|\frac{g(y)}{y^{\delta+1}}\right|^{p}
y^{2\nu+1-p(\nu+1/2-\delta)+p}\,dy\\&
=C\int_0^1|g(y)|^{p}y^{(\nu+1/2)(2-p)}\,dy. \end{align*} With the inequalities for $S_1$ and $S_2$, we can conclude \eqref{ec:boundweak} for $i=4$.
To prove \eqref{ec:boundweak} for $i=2$ we define, for $k$ a nonnegative integer, the intervals \[ I_k=[2^{-k-1},2^{-k}], \qquad N_k=[2^{-k-3},2^{-k+2}] \]
and the function $g_k(y)=|g(y)|\chi_{I_k}(y)$. By using \eqref{ec:kernel} for $x/2<y<2x$, with $x\in (0,1)$, we have the bound \[
|K_R^\delta (x,y)|\le \frac{C}{R^{\delta}(|x-y|+2/R)^{\delta+1}}. \] Then \[ J_{2}\subset \left\{x\in (0,1): \sup_{R>0}\sum_{k=0}^\infty \int_{x/2}^{\min{\{2x,1\}}}
\frac{g_k(t)}{R^{\delta}(|x-y|+2/R)^{\delta+1}}\, dy> C \lambda x^{\nu+1/2}\right\}. \] Since at most three of these integrals are not zero for each $x\in (0,1)$ \begin{align*} J_2&\subset \bigcup_{k=0}^\infty \left\{x\in (0,1): 3\sup_{R>0}\int_{x/2}^{\min{\{2x,1\}}}
\frac{g_k(t)}{R^{\delta}(|x-y|+2/R)^{\delta+1}}\, dy> C \lambda x^{\nu+1/2}\right\}\\ &\subset \bigcup_{k=0}^\infty \left\{x\in N_k : M(g_k,x)> C \lambda x^{\nu+1/2}\right\} \end{align*} where in the las step we have used that \[ \sup_{R>0}\int_{x/2}^{\min{\{2x,1\}}}
\frac{g_k(t)}{R^{\delta}(|x-y|+2/R)^{\delta+1}}\, dy\le C M(g_k,x). \] By using the estimate $x\simeq 2^{-k}$ for $x\in N_k$, we can check easily that \[ J_2\subset \bigcup_{k=1}^\infty \left\{x \in N_k : M(g_k,x)> C \lambda 2^{-k(\nu+1/2)}\right\}. \] Finally by using again that $x\simeq 2^{-k}$ for $x\in I_k, N_k$ and the weak type norm inequality for the Hardy-Littlewood maximal function we have \begin{align*} \int_{J_2}x^{2\nu+1}\, dx &\le C \sum_{k=0}^\infty 2^{-k(2\nu+1)} \int_{\left\{x\in N_k : M(g_k,x)> C \lambda 2^{-k(\nu+1/2)}\right\}}\, dx\\&\le C \sum_{k=0}^\infty
\frac{2^{pk(\nu+1/2)-k(2\nu+1)}}{\lambda^p}\int_{I_k}|g(y)|^p\, dy\\&\le \frac{C}{\lambda^p}\int_0^1 |g(y)|^p y^{(\nu+1/2)(2-p)}\, dy \end{align*} and the proof is complete.
\section{Proof of Theorem \ref{th:AcDebilRestMaxRedonda}} \label{sec:acdelrest}
To conclude the result we have to prove \eqref{ec:boundweak} with $g(x)=\chi_E(x)$ and $p=p_1$. For $J_1$ and $J_2$ the result follows by using the steps given in the proof of Theorem \ref{th:AcDebilMaxRedonda} for the same intervals. To analyze $J_3$ we proceed as we did for $J_4$ in the proof of Theorem \ref{th:AcDebilMaxRedonda}. In this case we obtain that \begin{multline*} \sup_{R>0}x^{-(\nu+1/2)}\int_0^1
|g(y)|\chi_{B_3}(x,y)|K_r^\delta(x,y)|\\\le C\left(x^{-(\nu+1)}\int_0^x \frac{|g(y)|}{\sqrt{y}}\, dy+
x^{-(\nu+3/2+\delta)}\int_0^x |g(y)| y^\delta\, dy\right). \end{multline*} Now taking into account that for $p=p_1$ we have $2\nu+1-p(\nu+1)<-1$ and $2\nu+1-p(\nu+3/2+\delta)<-1$ we can apply Hardy's inequalities to obtain that \[
\int_0^1\left(x^{-(\nu+1)}\int_0^x \frac{|g(y)|}{\sqrt{y}}\, dy\right)^{p}x^{2\nu+1}\, dx\le C \int_0^1 |g(y)|^p y^{(\nu+1/2)(2-p)}\, dy \] and \[
\int_0^1\left(x^{-(\nu+3/2+\delta)}\int_0^x |g(y)| y^\delta\, dy\right)^{p}x^{2\nu+1}\, dx\le C \int_0^1 |g(y)|^p y^{(\nu+1/2)(2-p)}\, dy, \] with these two inequalities we can deduce that \eqref{ec:boundweak} holds for $J_3$ with $p=p_1$ in this case.
The main difference with the previous proof appears in the analysis of $J_4$. To deal with this case, we have to use the following lemma \cite[Lemma 16.5]{Ch-Muc} \begin{Lem} \label{lem:Muck} If $1<p<\infty$, $a>-1$, and $E\subset [0,\infty)$, then \[ \left(\int_{E}x^a\, dx\right)^p\le 2^p(a+1)^{1-p}\int_{E}x^{(a+1)p-1}\, dx. \] \end{Lem} In this case, it is enough to prove that \[ \int_{\mathcal{J}}\, d\mu_\nu(x)\le \frac{C}{\lambda^p}\int_{0}^1 \chi_E(y)\,d\mu_\nu(y), \] where \begin{equation*}
\mathcal{J}=\left\{x\in(0,1):
\sup_{R>0}x^{-(\nu+1/2)}\int_0^{1}\chi_E(y)
\chi_{B_4}(x,y)y^{\nu+1/2}|K_R^{\delta}(x,y)|\,dy>\lambda\right\}, \end{equation*} and this can be deduced immediately by using the inclusion \begin{equation} \label{ec:final} \mathcal{J}\subseteq [0,\min\{1,H\}] \end{equation} with \[ H^{2(\nu+1)}=\frac{C}{\lambda^p}\int_{0}^1 \chi_E(y)\,d\mu_\nu(y). \] Let's prove \eqref{ec:final}. By using \eqref{ec:kern} and the estimate $y-x>y/2$, we have \begin{multline*} \sup_{R>0}x^{-(\nu+1/2)}\int_0^{1}\chi_E(y)
\chi_{B_4}(x,y)y^{\nu+1/2}|K_R^{\delta}(x,y)|\,dy \\ \le C\sup_{R>0} R^{-\delta+\nu+1/2}\chi_{[0,2/R]}(x)\int_{4/R}^1\chi_E(y) y^{-\delta+\nu-1/2}\, dy\\ + C\sup_{R>0} R^{-\delta}x^{-(\nu+1/2)}\chi_{[2/R,1]}(x)\int_{2x}^1 \chi_E(y) y^{-\delta+\nu-1/2}\, dy. \end{multline*} In the first summand we can use that $R^{-\delta+\nu+1/2}\le C x^{\delta-\nu-1/2}$ and in the second one that $R^{-\delta}\le x^{\delta}$. Moreover observing that with $p=p_1$ it holds $-\delta+\nu+1/2=2(\nu+1)/p$ we obtain that \begin{align*}
\sup_{R>0}x^{-(\nu+1/2)}\int_0^{1}\chi_E(y)\chi_{B_4}y^{\nu+1/2}|K_R^{\delta}(x,y)|\,dy&\le C x^{-2(\nu+1)/p}\int_{E}y^{-1+2(\nu+1)/p}\, dy\\ &\le C x^{-2(\nu+1)/p}\int_{E}\,d\mu_\nu(y), \end{align*} where in the last step we have used Lemma \ref{lem:Muck}, and this is enough to deduce the inclusion in \eqref{ec:final}.
\section{Proofs of Theorem \ref{th:noweak} and Theorem \ref{th:nostrong}} \label{sec:negativeths}
This section will be devoted to the proofs of Theorem \ref{th:noweak} and Theorem \ref{th:nostrong}. To this end we need a suitable identity for the kernel and in order to do that we have to introduce some notation. $H_{\nu}^{(1)}$ will denote the Hankel function of the first kind, and it is defined as follows \[ H_{\nu}^{(1)}(z)=J_{\nu}(z)+iY_{\nu}(z), \] where $Y_{\nu}$ denotes the Weber's function, given by \begin{equation*} Y_{\nu}(z)=\frac{J_{\nu}(z)\cos \nu \pi-J_{-\nu}(z)}{\sin \nu \pi},\,\, \nu\notin \mathbb{Z}, \text{ and } Y_n(z)=\lim_{\nu\to n}\frac{J_{\nu}(z)\cos \nu \pi-J_{-\nu}(z)}{\sin \nu \pi}. \end{equation*} From these definitions, we have \begin{equation*} H_{\nu}^{(1)}(z)=\frac{J_{-\nu}(z)-e^{-\nu \pi i}J_{\nu}(z)}{i\sin \nu \pi}, \,\, \nu\notin \mathbb{Z},\\ \text{ and } H_n^{(1)}(z)=\lim_{\nu\to n}\frac{J_{-\nu}(z)-e^{-\nu \pi i}J_{\nu}(z)}{i\sin \nu \pi}. \end{equation*} For the function $H_\nu^{(1)}$, the asymptotic \begin{equation}
\label{inftyH} H_{\nu}^{(1)}(z)=\sqrt{\frac{2}{\pi z}}e^{i(z-\nu\pi/2-\pi/4)}[A+O(z^{-1})], \quad |z|>1,\quad -\pi < \arg(z)<2\pi, \end{equation} holds for some constant $A$.
In \cite[Lemma 1]{Ci-Ro} the following lemma was proved \begin{Lem} \label{lem:expresnucleo} For $R>0$ the following holds: \[K_R^\delta(x,y)=I_{R,1}^\delta(x,y)+I_{R,2}^\delta(x,y)\] with \[ I_{R,1}^{\delta}(x,y)=(xy)^{1/2}\int_0^{R}z\multiJ_{\nu}(zx)J_{\nu}(zy)\, dz \] and \[ I_{R,2}^{\delta}(x,y)=\lim_{\varepsilon\to 0} \frac{(xy)^{1/2}}{2}\int_{\mathbf{S_\varepsilon}} \multi \frac{z H^{(1)}_{\nu}(z)J_{\nu}(zx)J_{\nu}(zy)}{J_{\nu}(z)}\,dz, \] where, for each $\varepsilon>0$, $\mathbf{S_\varepsilon}$ is the path of integration given by the interval $R+i[\varepsilon,\infty)$ in the direction of increasing imaginary part and the interval $-R+i[\varepsilon,\infty)$ in the opposite direction. \end{Lem}
Then, by Lemma \ref{lem:expresnucleo} we have \[ \mathcal{K}_R^\delta(x,y)=\mathcal{I}_{R,1}^\delta(x,y)+\mathcal{I}_{R,2}^\delta(x,y) \] where $\mathcal{I}_{R,j}^\delta(x,y)=(xy)^{-(\nu+1/2)}I_{R,j}^{\delta}(x,y)$ for $j=1,2$. The main tool to deduce our negative results will be the following lemma
\begin{Lem} \label{lem:zero} For $\nu>-1/2$, $\delta>0$, and $R>0$ it is verified that \[ \mathcal{K}_R^\delta(0,y)=\frac{2^{\delta-\nu}\Gamma(\delta+1)}{\Gamma(\nu+1)}R^{2(\nu+1)} \frac{J_{\nu+\delta+1}(yR)}{(yR)^{\nu+\delta+1}}+\mathcal{I}_{R,2}^\delta(0,y), \] where \begin{equation}
\label{ec:boundI} \left|\mathcal{I}_{R,2}^\delta(0,y)\right|\le C\begin{cases} R^{2\nu-\delta+1}, & yR\le 1,\\ R^{\nu-\delta+1/2}y^{-(\nu+1/2)}, & yR>1. \end{cases} \end{equation} \end{Lem} \begin{proof} From \eqref{zero}, it is clear that \[ \mathcal{I}_{R,1}^\delta(0,y)=\frac{y^{-\nu}}{2^\nu\Gamma(\nu+1)}\int_0^R z^{\nu+1}\multiJ_{\nu}(zy)\, dz. \] Now, by using Sonine's identity \cite[Ch. 12, 12.11, p. 373]{Wat} \[ \int_0^1 s^{\nu+1}\left(1-s^2\right)^\deltaJ_{\nu}(sy)\, ds=2^{\delta}\Gamma(\delta+1)\frac{J_{\nu+\delta+1}(y)}{y^{\delta+1}}, \qquad \nu,\delta>-1, \] we deduce the leading term of the expression for $\mathcal{K}_{R}^\delta(0,y)$.
To control the term \[ \mathcal{I}_{R,2}^\delta(0,y)=\lim_{\varepsilon\to 0}\frac{y^{-(\nu+1/2)}}{2}\int_{\mathbf{S_\varepsilon}} \multi \frac{z^{\nu+1/2} H^{(1)}_{\nu}(z)(zy)^{1/2}J_{\nu}(zy)}{J_{\nu}(z)}\,dz, \] we start by using the asymptotic expansions given in \eqref{inftyH} and \eqref{infty} for $H_{\nu}^{(1)}(z)$ and $J_{\nu}(z)$. We see that on $\mathbf{S_\varepsilon}$, the path of integration described in Lemma \ref{lem:expresnucleo}, for $t=\mathop{\rm Im}(z)$ the estimate \[
\left|\frac{H_{\nu}(z)}{J_{\nu}(z)}\right|\leq C e^{-2t}, \] holds for $t>0$. Now, from \eqref{zero} and \eqref{infty}, it is clear that for $z=\pm R+it$ \[
|\sqrt{zy}J_{\nu}(zy)|\le Ce^{yt}\Phi_\nu((R+t)y) \] where $\Phi_\nu$ is the function in \eqref{ec:aux}. Then \[
|\mathcal{I}_{R,2}^\delta(0,y)|\le C y^{-(\nu+1/2)}R^{-2\delta}\int_0^\infty t^{\delta}(R+t)^{\nu+\delta+1/2}\Phi_\nu((R+t)y)e^{-(2-y)t}\, dt. \] If $y>1/R$ we have the inequality $\Phi_\nu((R+t)y)\le C$, then \begin{align*}
|\mathcal{I}_{R,2}^\delta(0,y)| &\le C y^{-(\nu+1/2)} R^{-2\delta} \int_0^\infty t^{\delta}(R+t)^{\nu+\delta+1/2} e^{-(2-y)t}\, dt\\ &\le C y^{-(\nu+1/2)}R^{-\delta}(R^{\nu+1/2}+R^{-\delta})\le C R^{\nu-\delta+1/2}y^{-(\nu+1/2)} \end{align*} and \eqref{ec:boundI} follows in this case. If $y\le 1/R$ we obtain the bound in \eqref{ec:boundI} with the estimate $\Phi_\nu((R+t)y)\le C (\Phi_\nu(yR)+(yt)^{\nu+1/2})$. Indeed, \begin{multline*}
|\mathcal{I}_{R,2}^\delta(0,y)| \le C y^{-(\nu+1/2)} R^{-2\delta} \Phi_\nu(yR) \int_0^\infty t^{\delta}(R+t)^{\nu+\delta+1/2} e^{-(2-y)t}\, dt\\+ C R^{-2\delta} \int_0^\infty t^{\nu+\delta+1/2}(R+t)^{\nu+\delta+1/2} e^{-(2-y)t}\, dt\\\le C (R^{2\nu-\delta+1}+R^{\nu-2\delta+1/2}+R^{\nu-\delta+1/2}+R^{-2\delta})\le R^{2\nu-\delta+1}. \end{multline*} \end{proof}
\begin{Lem} \label{lem:cota0} For $\nu>-1/2$ and $0<\delta\le \nu+1/2$, the estimate \[
\|\mathcal{K}_R^\delta(0,y)\|_{L^{p_0}((0,1),d\mu_\nu)}\ge C R^{\nu-\delta+1/2}(\log R)^{1/p_0} \] holds. \end{Lem} \begin{proof} We will use the decomposition in Lemma \ref{lem:zero}. By using \eqref{zero} and \eqref{infty} as was done in \cite[Lemma 2.1]{Ci-RoWave} we obtain that \[
\left\|R^{2(\nu+1)}
\frac{J_{\nu+\delta+1}(yR)}{(yR)^{\nu+\delta+1}}\right\|_{L^{p_0}((0,1),d\mu_\nu)}\ge C R^{\nu-\delta+1/2}(\log R)^{1/p_0}. \] With the bound \eqref{ec:boundI} it can be deduced that \[
\left\|\mathcal{I}_{R,2}^\delta(0,y)\right\|_{L^{p_0}((0,1),d\mu_\nu)}\le C R^{\nu-\delta+1/2}. \] With the previous estimates the proof is completed. \end{proof}
Finally, the last element that we need to prove Theorems \ref{th:noweak} and \ref{th:nostrong} is the norm inequality for
finite linear combinations of the functions $\{\psi_j\}_{j\ge 1}$ contained in the next lemma. Its proof is long and technical and it will be done in the last section.
\begin{Lem} \label{lem:pol} For $\nu>-1/2$, $R>0$, $1<p<\infty$ and $f$ a linear combination of the functions $\{\psi_j\}_{1\le j\le N(R)}$ with $N(R)$ a positive integer such that $N(R)\simeq R$, the inequality \[
\|f\|_{L^\infty ((0,1),d\mu_\nu)}\le C R^{2(\nu+1)/p}\|f\|_{L^{p,\infty} ((0,1),d\mu_\nu)} \] holds. \end{Lem}
\begin{proof}[Proof of Theorem \ref{th:noweak}] With the bound in Lemma \ref{lem:cota0} we have \begin{align*} (\log R)^{1/p_0}&\le C R^{-2(\nu+1)/p_1}
\left\|\mathcal{K}_{R}^\delta(0,y)\right\|_{L^{p_0}((0,1),d\mu_\nu)}\\ & = C R^{-2(\nu+1)/p_1}
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}\left|\int_0^1
\mathcal{K}_{R}^\delta(0,y) f(y)\, d\mu_\nu\right|\\ & = C R^{-2(\nu+1)/p_1}
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}\left|\mathcal{B}_{R}^\delta f(0)\right|. \end{align*} From the previous estimate the result for $\delta=\nu+1/2$ follows. In the case $\delta<\nu+1/2$ it is obtained by using Lemma \ref{lem:pol} because \begin{multline*} R^{-2(\nu+1)/p_1}
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}\left|\mathcal{B}_{R}^\delta f(0)\right|\\\le C
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}\left\|\mathcal{B}_{R}^\delta f(x)\right\|_{L^{p_1,\infty}((0,1),d\mu_\nu)} \end{multline*} since $\mathcal{B}_{R}^\delta f(x)$ is a linear combination of the functions $\{\psi_j\}_{1\le j\le N(R)}$ with $N(R)\simeq R$. \end{proof}
\begin{proof}[Proof of Theorem \ref{th:nostrong}] In the case $\delta <\nu+1/2$, the result follows from Theorem \ref{th:noweak} by using a duality argument. Indeed, it is clear that \begin{align} \sup_{E\subset
(0,1)}\frac{\|\mathcal{B}^{\delta}_{R}\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}
{\|\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}} &=\sup_{E\subset (0,1)}
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}
\frac{\left|\int_0^1f(y)\mathcal{B}^{\delta}_{R}\chi_E(y)\, d\mu_\nu\right|}
{\|\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}\notag\\
&= \sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1} \sup_{E\subset (0,1)}
\frac{\left|\int_0^1\chi_E(y)\mathcal{B}^{\delta}_{R}f(y)\, d\mu_\nu\right|}
{\|\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}\label{ec:lambdacero}. \end{align} By Theorem \ref{th:noweak} it is possible to choose a function $g$
such that $\|g\|_{L^{p_1}((0,1),d\mu_\nu)}=1$ and \[
\|\mathcal{B}_{R}^\delta g(x)\|_{L^{p_1,\infty}((0,1),d\mu_\nu)}\ge C (\log R)^{1/p_0}. \] Then, with the notation \[ \mu_\nu(E)=\int_{E}\, d\mu_\nu, \] we have \begin{equation} \label{ec:lambda} \lambda^{p_1}\mu_\nu(A)\ge C (\log R)^{p_1/p_0}, \end{equation}
for some positive $\lambda$ and $A=\{x\in (0,1): |B_{R}^\delta g(x)|>\lambda\}$. Now, we consider the subsets of $A$ \[ A_1=\{x\in (0,1): B_{R}^\delta g(x)>\lambda\} \qquad\text{ and } \qquad A_2=\{x\in (0,1): B_{R}^\delta g(x)<-\lambda\} \] and we define $D=A_1$ if $\mu_\nu(A_1)\ge \mu_\nu(A)/2$ and $D=A_2$ otherwise. Then, by \eqref{ec:lambda}, we deduce that \begin{equation} \label{ec:lambda2} \lambda \ge C \frac{(\log R)^{1/p_0}}{\mu_\nu(D)^{1/p_1}}. \end{equation} Taking $f=g$ and $E=D$ in \eqref{ec:lambdacero} and using \eqref{ec:lambda2}, we see that \[ \sup_{E\subset
(0,1)}\frac{\|\mathcal{B}^{\delta}_{R}\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}
{\|\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}} \ge C
\lambda\frac{\mu_\nu(D)}{\|\chi_D\|_{L^{p_0}((0,1),d\mu_\nu)}} \ge C (\log R)^{1/p_0} \] and the proof is complete in this case. For $\delta=\nu+1/2$ the result follows from Theorem \ref{th:noweak} with a standard duality argument. \end{proof}
\section{Proof of Lemma \ref{lem:pol}} \label{sec:techlemma}
To proceed with the proof of Lemma \ref{lem:pol} we need some auxiliary results that are included in this section.
We start by defining a new operator. For each non-negative integer $r$, we consider the vector of coefficients $\alpha=(\alpha_1,\dots,\alpha_{r+1})$ and we define \[ T_{r,R,\alpha}f(x)=\sum_{\ell=1}^{r+1}\alpha_\ell \mathcal{B}_{\ell R}^{r}f(x). \] This new operator is an analogous of the \textit{generalized delayed means} considered in \cite{SteinDuke}. In \cite{SteinDuke} the operator is defined in terms of the Ces\`{a}ro means instead of the Bochner-Riesz means. The properties of $T_{r,R,\alpha}$ that we need are summarized in the next lemma
\begin{Lem} \label{lem:delay} For each non-negative integer $r$ and $\nu\ge -1/2$, the following statements hold \begin{enumerate} \item[a)] $T_{r,R,\alpha}f$ is a linear combination of the functions $\{\psi_j\}_{1\le j\le N((r+1)R)}$, where $N((r+1)R)$ is a non-negative integer such that $N((r+1)R)\simeq (r+1)R$;
\item[b)] there exists a vector of coefficients $\alpha$, verifying that $|\alpha_\ell|\le A$, for $\ell=1,\dots, r+1$, with $A$ independent of $R$ and such that $T_{r,R,\alpha}f(x)=f(x)$ for each linear combination of the functions $\{\psi_j\}_{1\le j\le N(R)}$ where $N(R)$ is a positive integer. Moreover, in this case, for $r>\nu+1/2$, \[
\|Tf_{r,R,\alpha}\|_{L^1 ((0,1),d\mu_\nu)}\le C\|f\|_{L^1((0,1),d\mu_\nu)}\] and
\[\|T_{r,R,\alpha}f\|_{L^\infty ((0,1),d\mu_\nu)}\le C
\|f\|_{L^\infty((0,1),d\mu_\nu)}, \] with $C$ independent of $R$ and $f$. \end{enumerate} \end{Lem} \begin{proof} Part a) is a consequence of the definition of $T_{r,R,\alpha}$ and the fact that the $m$-th zero of the Bessel function $J_\nu$, with $\nu \ge-1/2$, is contained in the interval $(m\pi+\nu\pi/2+\pi/2,m\pi+\nu\pi/2+3\pi/4)$.
To prove b) we consider $f(x)=\sum_{j=1}^{N(R)} a_j \psi_j(x)$. In order to obtain the vector of coefficients such that $T_{r,R,\alpha}f(x)=f(x)$ the equations \[ \sum_{\ell=1}^{r+1}\alpha_\ell \left(1-\frac{s_{k}^2}{(\ell R)^2}\right)^r=1, \] for all $k=1,\dots,N(R)$, should be verified. After some elementary manipulations each one of the previous equations can be written as \[ \sum_{j=0}^r s_k^{2j}\binom{r}{j}\frac{(-1)^j}{R^{2j}}\sum_{\ell=1}^{r+1} \frac{\alpha_\ell}{\ell^{2j}}=1 \] and this can be considered as a polynomial in $s_k^2$ which must be equal $1$, therefore we have the system of equations \[ \sum_{\ell=1}^{r+1}\frac{\alpha_\ell}{\ell^{2j}}=\delta_{j,0}, \qquad j=0,\dots,r. \] This system has an unique solution because the determinant of the matrix of coefficients is a Vandermonde's one. Of course for each
$\ell=1,\dots,r+1$, it is verified that $|\alpha_\ell|\le A$, with $A$ a constant depending on $r$ but not on $N(R)$.
The norm estimates are consequence of the uniform boundedness \[
\|\mathcal{B}_R^\delta f\|_{L^p((0,1),d\mu_\nu)}\le C
\|f\|_{L^p((0,1),d\mu_\nu)}, \] for $p=1$ and $p=\infty$ when $\delta > \nu+1/2$ (see \cite{Ci-Ro}). \end{proof}
In the next lemma we will control the $L^\infty$-norm of a finite linear combination of the functions $\{\psi_j\}_{j\ge 1}$ by its $L^1$-norm.
\begin{Lem} \label{lem:infty1} If $\nu>-1/2$ and $f(x)$ is a linear combination of the functions $\{\psi_j\}_{1\le j\le N(R)}$ with $N(R)$ a positive integer such that $N(R)\simeq R$, the inequality \[
\|f\|_{L^\infty ((0,1),d\mu_\nu)}\le C R^{2(\nu+1)}\|f\|_{L^1((0,1),d\mu_\nu)} \] holds. \end{Lem} \begin{proof} It is clear that \[ f(x)=\sum_{j=1}^{N(R)} \psi_j(x)\int_0^1 f(y) \psi_j(y)\, d\mu_\nu(y). \] Now, using H\"{o}lder inequality and Lemma \ref{Lem:NormaFunc} we have \begin{align*}
\|f\|_{L^\infty ((0,1),d\mu_\nu)}&\le C\sum_{j=1}^{N(R)}
\|\psi_j\|_{L^\infty
((0,1),d\mu_\nu)}^2\|f\|_{L^{1}((0,1),d\mu_\nu)}\\&\le C
\|f\|_{L^{1}((0,1),d\mu_\nu)} \sum_{j=1}^{N(R)}j^{2\nu+1}\le C R^{2(\nu+1)}\|f\|_{L^1((0,1),d\mu_\nu)}. \end{align*} \end{proof}
The following lemma is a version in the space $((0,1),d\mu_\nu)$ of Lemma 19.1 in \cite{Ch-Muc}. The proof can be done in the same way, with the appropriate changes, so we omit it.
\begin{Lem} \label{lem:fuerdeb} Let $\nu>-1$, $1<p<\infty$ and $T$ be a linear operator defined for functions in $L^1((0,1),d\mu_\nu)$ and such that \[
\|Tf\|_{L^\infty ((0,1),d\mu_\nu)}\le A
\|f\|_{L^1((0,1),d\mu_\nu)} \,\text{ and }\,\|Tf\|_{L^\infty
((0,1),d\mu_\nu)}\le B \|f\|_{L^\infty((0,1),d\mu_\nu)}, \] then \[
\|Tf\|_{L^\infty ((0,1),d\mu_\nu)}\le C A^{1/p}B^{1/p'}\|f\|_{L^{p,\infty}((0,1),d\mu_\nu)}. \] \end{Lem}
Now, we are prepared to conclude the proof of Lemma \ref{lem:pol}.
\begin{proof}[Proof of Lemma \ref{lem:pol}] We consider the operator $T_{r,R,\alpha}f$ given in Lemma \ref{lem:delay} b) with $r>\nu+1/2$. By Lemma \ref{lem:delay} and Lemma \ref{lem:infty1} we have \begin{align*}
\|T_{r,R,\alpha}f\|_{L^\infty((0,1),d\mu_\nu)}&\le C ((r+1)R)^{2(\nu+1)}
\|T_{r,R,\alpha}f\|_{L^1((0,1),d\mu_\nu)}\\&\le C R^{2(\nu+1)}\|f\|_{L^1((0,1),d\mu_\nu)}. \end{align*} From b) in Lemma \ref{lem:delay} we obtain the estimate \[
\|T_{r,R,\alpha}f\|_{L^\infty((0,1),d\mu_\nu)}\le C
\|f\|_{L^\infty((0,1),d\mu_\nu)}. \] So, by using Lemma \ref{lem:fuerdeb}, we obtain the inequality \[
\|T_{r,R,\alpha}f\|_{L^\infty((0,1),d\mu_\nu)}\le C
R^{2(\nu+1)/p}\|f\|_{L^{p,\infty}((0,1),d\mu_\nu)} \] for any $f\in L^1((0,1),d\mu_{\nu})$. Now, since $T_{r,R,\alpha}f(x)=f(x)$ for a linear combination of the functions $\{\psi_j\}_{1\le j\le N(R)}$, the proof is complete. \end{proof}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\baselineskip=26pt
\address[F\'abio~P.~Machado] {Institute of Mathematics and Statistics \\ University of S\~ao Paulo \\ Rua do Mat\~ao 1010, CEP 05508-090, S\~ao Paulo, SP, Brazil.}
\address[Valdivino~V.~Junior] {Federal University of Goias \\ Campus Samambaia, CEP 74001-970, Goi\^ania, GO, Brazil.}
\address[Alejandro Roldan-Correa] {Instituto de Matem\'aticas, Universidad de Antioquia, Calle 67, no 53-108, Medellin, Colombia}
\title[Colonization and collapse on Homogeneous Trees]{Colonization and collapse on Homogeneous Trees} \author{Valdivino~V.~Junior} \author{F\'abio~P.~Machado} \author{Alejandro Rold\'an-Correa}
\noindent \email{[email protected], [email protected], [email protected]}
\thanks{Research supported by CNPq (306927/2007-1), FAPESP (2010/50884-4) and Universidad de Antioquia (SUI XXXXX).}
\keywords{Branching processes, Coupling, Catastrophes, Population dynamics.}
\subjclass[2010]{60J80, 60J85, 92D25}
\date{\today}
\begin{abstract}
We investigate a basic immigration process where co\-lo\-nies grow, during a random time, according to a general counting process until collapse. Upon collapse a random amount of indivi\-duals survive. These survivors try independently establishing new colonies at neighbour sites. Here we consider this general process subject to two schemes, Poisson growth with geometric catastrophe and Yule growth with binomial catastrophe. Independent of everything else colonies growth, during an exponential time, as a Poisson (or Yule) process and right after that exponential time their size is reduced according to geometric (or binomial) law. Each survivor tries independently, to start a new colony at a neighbour site of a homogeneous tree. That colony will thrive until its collapse, and so on. We study conditions on the set of parameters for these processes to survive, present relevant bounds for the probability of survival, for the number of vertices that were colonized and for the reach of the colonies compared to the starting point. \end{abstract}
\singlespacing
\maketitle \section{Introduction} \label{S: Introduction} Biological populations are subject to disasters that can cause from a partial elimination of the individuals until their total extinction. When a disaster occurs surviving individuals may react in different ways. A strategy adopted by some populations is the dispersion. In this case, individuals migrate, trying to create new colonies in other locations, there may be competition or collaboration between individuals of the same colony. Once they settle down a new colony in a new spot, again another disaster can strike, which causes a new collapse.
In this type of population dynamics there are some issues to consider, such as: What is the duration of colonization until the moment of the disaster? How much the population grows until be hit? How many individuals will survive? How survivors react when facing a disaster?
In recent articles, the main variables considered in population modeling are (i) the spatial structure where the colonies are located and individuals can move, (ii) the lifetime of a colony until the moment of collapse, (iii) the evolution of the number of individuals in the colony (random or deterministic growth, possible deaths or migration), (iv) the way the cathastrophes affects the size of the colony allowing or not the survival of some individuals and (v) whether the individuals that survive to the catastrophe are able to spread out.
Brockwell \textit{et al.}~\cite{BGR1982} and later Artalejo \textit{et al.}~\cite{AEL2007} considered a model for the growth of a population subject to collapse. In their model, two types of effects when a disaster strikes are analyzed separately, \textit{binomial effect} and \textit{geometric effect}. After the collapse, the survivors remain together in the same colony (there is no dispersion). They carried out an extensive analysis including first extinction time, number of individuals removed, survival time of a tagged individual, and maximum population size reached between two consecutive extinctions.
More recently, Schinazi~\cite{S2014} and Machado \textit{et al.}~\cite{MRS2015} proposed stochastic models for this kind of population dynamics. For these models they concluded that dispersion is a good survival strategy. Latter Junior \textit{et al.}~\cite{VAF2016} showed nice combinations of a type of catastrophe, spatial restriction and individual survival probability when facing the catastrophe where dispersion may not be a good strategy for group survival. For a comprehensive literature overview and motivation see Kapodistria \textit{et al.}~\cite{KPR2016}.
The paper is divided into four sections. In Section 2 we present a general model for the growth of populations subject to collapses, introduce the variables of interest, notation and two particular schemes: Poisson growth with geometric catastrophe and Yule growth with binomial catastrophe. In Section 3 we present the main results of the paper while their proofs are in Section 4.
\section{Colonization and Collapse models} \label{S: CCM}
In the beginning all vertices of $\mathbb{G}$, a infinite conected graph, are empty except for the origin where there is one individual. Besides that, at any time each colony is started by a single individual. The number of individuals in each colony behaves as $\mathcal{C}$, a Counting Process. To each colony is associated a non-negative random variable $T$ which defines its lifetime. After a period of time $T$, that colony collapses and the vertex where it is placed becomes empty. At the time of collapse, with a random effect $\mathcal{E}$, some individuals in the colony are able to survive while others die. By simplicity we represent this quantity by $N$. Note that this random quantity depends on the Counting Process which defines the growth of the colony, on the distribution of $T$ and on how the collapse afects the group of individuals present in the colony at time $T$. Each one individual that survives ($N$ individuals) tries to found a new colony on one of the nearest neighbour vertices by first picking one of them at random. If the chosen vertex is occupied, that individual dies, otherwise the individual founds there a new colony. We denote the Colonization and Collapse model generally described here either by ${\{{\mathbb G}; N\}}$ or ${\{\mathbb{G}; \mathcal{C}, \mathcal{E},T \}}$, a stochastic process whose state space is $\mathbb{N}^{\mathbb{T}^d}$. Along this paper we concentrate our attention on ${\mathbb T}^d$, a homogeneous tree where every vertex has $d+1$ nearest neighbours and on ${\mathbb T}^d_+$, a tree whose only difference from ${\mathbb T}^d$ is that its origin has degree $d$.
\begin{defn} Let us consider the following random variables \begin{itemize} \item $I_d:$ the number of colonies created from the beginning to the end of the process; \item $M_d:$ the distance from the origin to the furthest vertex where a colony is created; \item $\{X_t\}_{0 \le t \le T}$ growth process for the amount of individuals in a colony. \end{itemize} \end{defn}
We work in details some specific cases.
\begin{itemize} \item $T:$ Lifetime of a colony \begin{itemize} \item $T \sim {\mathcal Exp}(1)$, Exponential with mean 1 \[ P[T<t] = 1- e^{- t}, \ t > 0 \] \end{itemize} \item $X_t:$ Growth of the number of individuals \begin{itemize} \item $X_t \sim {\mathcal Poisson}(\lambda t)$, a Poisson point process with rate $\lambda$ \[P[X_t = k] = \frac{e^{-\lambda} \lambda^{k-1}}{(k-1)!} , \ k \in \{1,2,...\} \] \item $X_t \sim {\mathcal Geom}(e^{-\lambda t})$, a Yule process with rate $\lambda$ \[P[X_t = k] = e^{-\lambda t} (1-e^{-\lambda t})^{k-1} , \ k \in \{1,2,...\} \] \end{itemize} \item $N:$ Number of individuals able to survive \begin{itemize}
\item $N | X_T \sim {\mathcal B}(X_T,p),$ Binomial catastrophe
\[P[N = m | X_T = k] = {k \choose m} p^m(1-p)^{k-m} , \ m \in \{0, 1,...,k\} \]
\item $N | X_T = X_T- \min\{{\mathcal Geom}(p)-1;X_T\} \sim {\mathcal G}_{X_T}(p),$ Geometric catastrophe
\[ P[N = m | X_T] = \left\{ \begin{array}{ll} p (1-p)^{X_T - m} & \mbox{if $m \in \{1,...,X_T$\}} \\ \\ (1-p)^{X_T} & \mbox{if $m = 0$} \end{array} \right. \] \end{itemize} \end{itemize}
In general it is true that \begin{align*}
\mathbb{P}(N = n) &= \int_0^{\infty} \mathbb{P}(N=n|T=t)f_T(t)dt \\
\mathbb{P}(N = n|T=t) &= \sum_{x=n}^{\infty} \mathbb{P}(X_T=x|T=t)\mathbb{P}(N=n|X_T=x ; T=t). \end{align*}
Suppose that individuals are born following a Poisson process at rate $\lambda$, that the collapse time follows an exponential random variable with average 1 ($T \sim {\mathcal Exp}(1)$) and the individuals are exposed to the collapse effects, one by one, until the first individual survive, if any, then the collapse effects stop. If the collapse effects reach a fixed individual, it survives with probability $p$, meaning that $N_T \sim {\mathcal G}_{X_T}(p)$ (Geometric catastrophe) or ${\mathcal G}(p)$ for short. Let us consider the distribution of the number of survivals at collapse times \begin{align*} \mathbb{P}(N = 0) = \int_{0}^{\infty}e^{-t}\sum_{j=0}^{\infty}\frac {e^{-\lambda t}(\lambda t)^j}{j!}(1-p)^{j+1}dt = \frac{1-p}{1 +\lambda p } \end{align*} and for $n \geq 1$: \begin{align*} \mathbb{P}(N = n) = \int_{0}^{\infty}e^{-t}\sum_{j=n-1}^ {\infty}\frac{e^{-\lambda t}(\lambda t)^j}{j!} p(1-p)^{j+1-n}dt = \left(\frac{\lambda }{\lambda + 1}\right )^{n-1} \frac{p}{\lambda p +1}. \end{align*}
\noindent In this case the probability generating function is of $N$ is \begin{align} \label{eq: fgpP} \mathbb{E}(s^N) =& \frac{1-p}{1+\lambda p}+\sum_{n=1}^{\infty} s^n \left(\frac{\lambda }{1+\lambda }\right)^{n-1} \left(\frac{p}{\lambda p+1}\right) \\ =& \frac{1}{\lambda p +1}\left[1-p+\frac{(\lambda+1)ps}{1+\lambda-\lambda s}\right]. \end{align} while its average is $\displaystyle \mathbb{E}(N) = \frac{p (\lambda +1)^2}{(\lambda p + 1)}.$
Suppose now that individuals are born following a Yule process at rate $\lambda$, that $T \sim {\mathcal Exp}(1)$ and that the disaster reach the individuals simultaneously and independently of everything else. Assuming that each individual survives with probability $p$, we have that $N_T \sim {\mathcal B}(X_T, p)$ (Binomial catastrophe) or ${\mathcal B}(p)$ for short. Let us consider the distribution of the number of survivals at collapse times. \begin{align*} \mathbb{P}(N = 0) =&\int_0^\infty e^{-t}\sum_{j= 1}^\infty e^{-\lambda t}(1-e^{-\lambda t})^{j-1}(1-p)^{j}dt \\ =&\frac{1-p}{\lambda+1}\ {_2F_1}\left(1,1;2+\frac{1}{\lambda};{1-p}\right). \end{align*} \noindent and for $n \geq 1$ \begin{align*} \mathbb{P}(N = n) =& \int_0^\infty e^{-t}\sum_{j= n}^\infty e^{-\lambda t}(1-e^{-\lambda t})^{j-1}{j \choose n}p^n(1-p)^{j-n}dt \\ =&\frac{p^k}{\lambda}B\left(k,1+\frac{1}{\lambda}\right){_2F_1}\left(k+1,k;k+1+\frac{1}{\lambda};{1-p}\right). \end{align*} In this setup the probability generating function of $N$ is \begin{eqnarray}\label{eq: fgpY} \mathbb{E}(s^N)&=&\sum_{n=0}^{\infty} s^n \int_{0}^{\infty} e^{-t} \sum_{k=n \vee 1}^{\infty} e^{-\lambda t}(1-e^{-\lambda t})^{k-1} {k \choose n} p^n(1-p)^{k-n} \ dt\nonumber \\ &=&\frac{ps+1-p}{\lambda +1}\ {_2F_1}\left(1,1;2+\frac{1}{\lambda}; p(s-1)+1\right) \end{eqnarray} and its average is \begin{eqnarray}\label{eq: averageN} \mathbb{E}(N)= \left\{\begin{array}{cl} \displaystyle\frac{p}{1-\lambda} & ,\text{ se } \lambda<1 \\ \\ \infty &, \text{ se } \lambda\geq 1. \end{array}\right.\nonumber \end{eqnarray}
\section{Main Results} \label{S: Homogeneous Trees}
$\{ {\mathbb T}^d, \mathcal{C}, \mathcal{E}, T \}$ is a stochastic process whose state space is $\mathbb{N}^{\mathbb{T}^d}$ and whose evolution (status at time $t$) is denoted by $\eta_t$. For a vertex $x \in \mathbb{T}^d$, $\{\eta_t(x)=i\}$ means that at the time $t$ there are $i$
individuals at the vertex $x$. We consider $|\eta_t| = \sum_{x \in \mathbb{T}^d} \eta_t(x)$.
\subsection{Phase Transition} \label{SS: PT}
\begin{defn} Let $\eta_t$ be the process ${\{\bbT^d; \mathcal{C}, \mathcal{E},T \}}$. Let us define the event
\[ V_d = \{ |\eta_t| > 0, \hbox{ for all } t \ge 0 \}. \]
If $\mathbb{P}(V_d) > 0$ we say that the process ${\{\bbT^d; \mathcal{C}, \mathcal{E},T \}}$ {\it survives}. Otherwise, we say that the process ${\{\bbT^d; \mathcal{C}, \mathcal{E},T \}}$ {\it dies out }. \end{defn}
\begin{teo} \label{T: MCC1H} Consider the process ${\{\bbT^d; N\}}$. Then $\mathbb{P}(V_d) = 0 $ if \begin{displaymath} \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] \geq \frac{d}{d+1} \end{displaymath} and $\mathbb{P}(V_d) > 0 $ if \begin{displaymath} \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] < \frac{d-1}{d}. \end{displaymath} \end{teo}
\begin{cor} \label{C: MCC1HP} Consider the process ${\{\bbT^d; \mathcal{P}(\lambda), \mathcal{G}(p) \}}$.
\begin{itemize} \item[$(i)$] $\mathbb{P}(V_d) = 0$ if \begin{equation}\label{C:tfp1} (\lambda^2d + \lambda d + \lambda +d + 1)p \leq \lambda + d + 1. \end{equation}
\item[$(ii)$] $\mathbb{P}(V_d) > 0$ if \begin{equation}\label{C:tfp2} (\lambda^2d - \lambda^2+ \lambda d - \lambda +d )p > \lambda + d + 1. \end{equation}
\end{itemize} \end{cor}
\begin{cor} \label{C: MCC1HPY} Consider the process ${\{\bbT^d; \mathcal{Y}(\lambda), \mathcal{B}(p) \}}$.
\begin{itemize} \item[$(i)$] $\mathbb{P}(V_d) = 0 $ if \begin{equation}\label{C:tfy1} _2 F_1 \left(1,1; 2 +\frac{1}{\lambda}; \frac{d(1-p) +1}{d+1} \right) \geq \frac{d(\lambda +1)}{d +1 - p}. \end{equation}
\item[$(ii)$] $\mathbb{P}(V_d) > 0 $ if \begin{equation}\label{C:tfy2} _2 F_1 \left(1,1; 2 +\frac{1}{\lambda}; \frac{d(1-p) +1}{d+1} \right) < \frac{(d^2-1)(\lambda +1)}{d(d+1-p)}. \end{equation} \end{itemize} \end{cor}
Observe that for the process ${\{\mathbb{G}; \mathcal{C}, \mathcal{E},T \}},\hbox{ when } \mathcal{C} \in \{\mathcal{Y}(\lambda), \mathcal{P}(\lambda)\} \hbox{ and } \mathcal{E} \in \{\mathcal{G}(p), \mathcal{B}(p)\},$ by a coupling argument one can see that $\mathbb{P}(V_d)$ is a non-decreasing function of $\lambda$ and also of $p$. Moreover, the function $\lambda_c(p)$, defined by $$\lambda_c(p):=\inf\{\lambda: \mathbb{P}(V_d) >0 \},$$ is a non-increasing function of $p$, with $\lambda_c(1)=0$ and $\lambda_c(0)=\infty$.
\begin{defn} Let $\eta_t$ be a ${\{\mathbb{G}; \mathcal{C}, \mathcal{E},T \}} \hbox{ for } \mathcal{C} \in \{\mathcal{Y}(\lambda), \mathcal{P}(\lambda)\} \hbox{ and } \mathcal{E} \in \{\mathcal{G}(p), \mathcal{B}(p)\},$ with $0<p<1$. We say that $\eta_t$ exhibits \textit{phase transition} on $\lambda$ if $0<\lambda_c(p)<\infty.$ \end{defn}
Machado \textit{et al.}(2016) proved phase transition on $\lambda $ for the process ${\{\bbT^d; \mathcal{Y}(\lambda), \mathcal{B}(p) \}}$. So, there exists a function $\lambda_c(\cdot):(0,1)\rightarrow \mathbb{R}^+$ whose graphic splits the parametric space $\lambda \times p$ into two regions. For those values of $(\lambda,p)$ above the curve $\lambda_c(p)$, there is survival in ${\{\bbT^d; \mathcal{Y}(\lambda), \mathcal{B}(p) \}}$ with positive probability. Moreover, for those values of $(\lambda,p)$ below the curve $\lambda_c(p)$ extinction occurs in ${\{\bbT^d; \mathcal{Y}(\lambda), \mathcal{B}(p) \}}$ with probability 1.
However, it is not known anything about the continuity and strict monotonicity (in $p$) of the function $ \lambda_c(p) $. If there is continuity and strict monotonicity, then the process also has phase transition in $p$ for each $\lambda \in (0, \infty)$ fixed.
In order to answer the question about phase transition on $p$ for the process ${\{\mathbb{G}; \mathcal{C}, \mathcal{E},T \}}, \\ \hbox{ when } \mathcal{C} \in \{\mathcal{Y}(\lambda), \mathcal{P}(\lambda)\} \hbox{ and } \mathcal{E} \in \{\mathcal{G}(p), \mathcal{B}(p)\},$ we start with the following definition \[p_c(\lambda):=\inf\{p: \mathbb{P}(V_d) > 0 \}.\]
\begin{defn} Let $\eta_t$ be a ${\{\mathbb{G}; \mathcal{C}, \mathcal{E},T \}} \hbox{ for } \mathcal{C} \in \{\mathcal{Y}(\lambda), \mathcal{P}(\lambda)\} \hbox{ and } \mathcal{E} \in \{\mathcal{G}(p), \mathcal{B}(p)\},$ with $\lambda \in (0, \infty)$ fixed. We say that $\eta_t$ exhibits \textit{phase transition} on $p$ if $0<p_c(\lambda)<1.$ \end{defn}
The item $(i)$ of Corollary~\ref{C: MCC1HPY} coincides with item $(iii)$ of Theorem 3.1 from Machado {\it et al.}~\cite{MRS2015}. The novelty of Corollary~\ref{C: MCC1HPY} is its item $(ii)$ which provides a suficient condition for survival. Corollary~\ref{C: MCC1HPY} guarantees phase transition in $p$ for ${\{\bbT^d; \mathcal{Y}(\lambda), \mathcal{B}(p) \}}$ for $\lambda$ large enough, and gives lower and upper bounds for $\lambda_c(p)$.
\begin{exa} Consider ${\{\bbT^4; \mathcal{Y}(\lambda), \mathcal{B}(p) \}}$. The equalities in (\ref{C:tfy1}) and (\ref{C:tfy2}) provide lower and upper bounds, respectively, for $\lambda_c(p)$. See Figure \ref{F: LimitanteTransicaoAnt}. These bounds guarantees phase transition in $p$ for $\lambda>\lambda_4^*.$ Where $\lambda_d^*$ is an upper bound for $\displaystyle\lim_{p\rightarrow 1^-}\lambda_c(p)$, where the former is the solution for
\begin{displaymath}
_2 F_1 \left(1,1; 2 +\frac{1}{\lambda}; \frac{1}{d+1} \right) = \frac{(d^2-1)(\lambda +1)}{d^2},
\end{displaymath} see Corollary \ref{C: MCC1HPY} $(ii)$. The following table shows computations for $\lambda_d^*$ for some values of $d$
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}\hline
$d$ & 2 &3 & 4 & 5 & 6 & 10 \\ \hline
$\lambda_d^*$ & 0.4555826 & 0.1613016& 0.08212601 & 0.04961835 & 0.03315455 & 0.01110147 \\ \hline
\end{tabular}
\end{center}
\begin{figure}\label{F: LimitanteTransicaoAnt}
\end{figure}
\end{exa}
\begin{exa} Consider $\{{\mathbb T}^{4}; \mathcal{P}(\lambda), \mathcal{G}(p) \}$. The equalities in (\ref{C:tfp1}) and (\ref{C:tfp2}) provide lower and upper bounds, respectively, for $\lambda_c(p)$. See Figure \ref{F:Poison-Gem-d4}. These bounds guarantees phase transition in $p$ for $\lambda>\lambda_4^*.$ Where $\lambda_d^*$ is an upper bound for $\displaystyle\lim_{p\rightarrow 1^-}\lambda_c(p)$, where the former is the solution for
\begin{displaymath}
(\lambda^2d - \lambda^2+ \lambda d - \lambda +d )p = \lambda + d + 1,
\end{displaymath}
when $p=1,$ see Corollary \ref{C: MCC1HP} $(ii)$. Thus, $$\lambda_d^*=\frac{1}{d-1}.$$
\begin{figure}\label{F:Poison-Gem-d4}
\end{figure}
\end{exa}
\subsection{Probability of Survival} \label{SS: SP}
We denote by $T(n,k)$ the number of surjective functions $f:A \to B $, where $|A| = n$ and $|B| = k$, whose value is given, by the inclusion-exclusion principle (see Tucker~\cite{Tucker} p. 319), by \begin{displaymath} T(n,k) = \sum_{i=0}^{k} \left [ (-1)^i \binom{k}{i}(k-i)^n \right], n \geq k. \end{displaymath}
\begin{teo} \label{T: MCCHPE1} Consider the process ${\{\bbT^d; N\}}$. We have that \begin{displaymath}
\sum_{r=1}^{d+1} \left [(1 - \rho^r)\binom{d+1}{r}\sum_{n=r}^{\infty}\frac{T(n,r)}{(d+1)^n}\mathbb{P}(N=n) \right] \leq \mathbb{P}(V_d) \leq 1-\psi \end{displaymath} where $\psi$ and $\rho$ are, respectively, the smallest non-negative solutions of \begin{align*} &\sum_{y=0}^{d+1} \left[ s^y\binom{d+1}{y}\sum_{n=y}^{\infty}\frac{T(n,y)}{(d+1)^n}\mathbb{P}(N=n) \right] = s,\\ &\sum_{y=1}^{d} \left [ s^y\binom{d}{y}\sum_{n=y}^{\infty}\frac{T(n,y)+T(n,y+1)}{(d+1)^n}\mathbb{P}(N=n)\right ] = s - \sum_{n=0}^{\infty}\frac{\mathbb{P}(N=n)}{(d+1)^n} . \end{align*} \end{teo}
\begin{teo} \label{T: MCCHPE1L} Consider the process ${\{\bbT^d; N\}}$. We have that \[ \lim_{d \to \infty} \mathbb{P}(V_d) = 1 - \nu \] where $\nu$ is the smallest non-negative solution of $ \mathbb{E}(s^N) = s.$ \end{teo}
\begin{cor} \label{C: MCC1HPS} Consider the process ${\{\bbT^d; \mathcal{P}(\lambda), \mathcal{G}(p) \}}$. Then \[ \lim_{d \to \infty} \mathbb{P}(V_d) = \max \left \{ 0, \frac{p(\lambda^2 + \lambda +1)}{\lambda (1+ \lambda p)} \right \}. \] \end{cor}
\begin{exa} Consider the process ${\{\bbT^{d}; \mathcal{P}(5), \mathcal{G}(0.6) \}}$. If $d=10$ then \begin{displaymath} \mathbb{P}(N = n ) = \left\{
\begin{array}{ll}
\frac{9}{50}\left(\frac{5}{6} \right )^n, & \hbox{$n \geq 1$;} \\
\frac{1}{10}, & \hbox{$n=0$.}
\end{array}
\right. \end{displaymath} By using Theorem~\ref{T: MCCHPE1} we have that $\psi = 0.12226$ and $\rho = 0.143256$. Then \[ 0.8733 \leq \mathbb{P}(V_{10}) \leq 0.8778.\] Besides \[ \displaystyle \lim_{d \to \infty} \mathbb{P}(V_d) = 0.93. \] \end{exa}
\begin{cor} \label{C: MCC1HPS2} Consider the process ${\{\bbT^d; \mathcal{Y}(\lambda), \mathcal{B}(p) \}}$. Then \[ \lim_{d \to \infty} \mathbb{P}(V_d) =1 - \nu \] where $\nu$ is the smallest non-negative solution of \[ _2 F_1 \left ( 1,1;2 + \frac{1}{\lambda}; p(s-1) + 1 \right ) = \frac{ s(\lambda +1)}{p(s + 1)}. \] \end{cor} \begin{exa} Consider the process ${\{\bbT^d; \mathcal{Y}(\lambda), \mathcal{B}(p) \}}$. If $\lambda = 2$ and $p = 0.5$ then, by using Corolary~\ref{C: MCC1HPS2} \[ \displaystyle \lim_{d \to \infty} \mathbb{P}(V_d) = 0.680977.\] \end{exa}
\subsection{The reach of the process} \label{SS: RP}
In order to show results for the reach of the process, meaning the distance from the origin to the furthest vertex where a colony is created, let us define a few technical quantities
\begin{defn} \begin{align*} \alpha&= d \left [ 1 - \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] \right ] \\ \beta &= (d+1) \left [ 1 - \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] \right ] = \alpha + 1- \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right]\\ D &= \max \left \{ 2; \frac{\beta}{\beta - \mathbb{P}(N \neq 0)} \right\} \\ B& =d(d-1)\left[ 1 - 2\mathbb{E} \left ( \left(\frac{d }{d+1} \right)^N \right) + \mathbb{E} \left ( \left(\frac{d-1 }{d+1} \right)^N \right) \right] \end{align*} \end{defn}
\begin{teo}\label{T: LEIT} Consider the process ${\{\bbT^d; N\}}$. Assume that \begin{displaymath} \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] > \frac{d-1}{d} \end{displaymath} We have that \begin{displaymath} \frac{[1+D(1-\beta)][1-\beta^{m+1}]}{1+ D(1-\beta)-\beta^{m+1}} \leq \mathbb{P}(M_d \leq m) \leq \frac{[1 + \frac{\alpha(1-\alpha)}{B}](1-\alpha^{m+1})}{ 1 + \frac{\alpha(1-\alpha)}{B} - \alpha^{m+1}} \end{displaymath} and \begin{displaymath}
\frac{\alpha^2}{2(B+ \alpha)} + \alpha(1-\alpha)\frac{\ln \left[1 - \frac{\alpha B}{B + \alpha(1-\alpha)}\right]}{B \ln \alpha} \leq \mathbb{E}(M_d) \leq \frac{D \beta}{D+1} + D(1-\beta)\frac{\ln \left[1 - \frac{\beta}{1 + D(1-\beta)}\right]}{\ln \beta}. \end{displaymath} \end{teo}
\begin{cor} \label{C: CTM1} Consider the process ${\{\bbT^d; \mathcal{P}(\lambda), \mathcal{G}(p) \}}$. If \[ (\lambda^2d + \lambda d + \lambda +d + 1)p < \lambda + d + 1\] then Theorem~\ref{T: LEIT} holds under the values \begin{align*} \alpha &= \frac{dp(\lambda + 1)^2}{(d + \lambda + 1)(\lambda p + 1)},\ \beta = \frac{(d+1)p(\lambda + 1)^2}{(d + \lambda + 1)(\lambda p + 1)},\ D = \max \left \{ 2; \frac{(d+1)(\lambda + 1)}{d \lambda} \right\},\\ B& = 2d(d-1) \left [ \frac{(\lambda +1)^2(2(\lambda p + 1) - 1) + (\lambda +1)(p-1)d - (\lambda p+1)d^2}{(d + 2\lambda + 1)(d + 2 \lambda + 1)(\lambda p + 1)} \right ]. \end{align*} \end{cor}
\begin{teo}\label{T: LEITL} Consider the process ${\{\bbT^d; N\}}$. We have that \begin{displaymath} M_d \overset{D}{\to} M, \end{displaymath} where $\mathbb{P}( M \leq m) = g_{m+1}(0)$, being $g(s) = \mathbb{E}(s^N) $ and $g_{m+1}(s) = \overset{ m+1 \textrm { times }} {g(g(\cdots g(s)) \cdots )}$. \end{teo}
\begin{cor} \label{C: LEITL} Consider the process ${\{\bbT^d; \mathcal{P}(\lambda), \mathcal{G}(p) \}}$. \begin{itemize} \item[$(i)$] If $ p \neq {(\lambda^2 + \lambda + 1)}^{-1}$ then \[ \mathbb{P}(M \leq m) = \frac{1 - \left (\frac{(\lambda + 1)^2 p}{\lambda p + 1} \right )^{m+1}}{1 - \frac{\lambda(\lambda p +1)}{(1-p)(\lambda p + 1)}\left (\frac{(\lambda +1)^2 p}{\lambda p + 1} \right )^{m+1}}, \ m \geq 0 \] and \[ \mathbb{E}(M)= \frac{(1- p(\lambda^2 + \lambda + 1))}{\lambda (\lambda p + 1))} \lim_{ s \to \infty} \left [ \psi_{\gamma} \left( 1 - \frac{\ln{\frac{(\lambda +1)(1-p)}{\lambda ( \lambda p + 1)}}}{ \ln {\gamma}} \right) - \psi_{\gamma} \left( s - \frac{\ln{\frac{(\lambda +1)(1-p)}{\lambda ( \lambda p + 1)}}}{ \ln {\gamma}} + 1 \right) \right ] \] where $\displaystyle \gamma = \frac{(\lambda +1)^2 p}{\lambda p + 1}$ and $\displaystyle \psi_a(z) = -\ln(1-a) + \ln(a) \sum_{n=0}^{\infty} \frac{a^{n+z}}{1-a^{n+z}},$ being $\psi_a(z)$ known as the $a$-digama function.\\ \item[$(ii)$] If $ p = {(\lambda^2 + \lambda + 1)}^{-1}$ then \[ \mathbb{P}(M \leq m) = \frac{(m+1)\lambda}{(m+1)\lambda +1}\] and \[ \mathbb{E}(M)= \infty.\] \end{itemize} \end{cor}
\subsection{Number of collonies in the process} \label{SS: NC}
\begin{teo} \label{T: NC} Consider the process ${\{\bbT^d; N\}}$. If \begin{displaymath} \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] > \frac{d-1}{d} \end{displaymath} then \begin{displaymath} \mathbb{E}(I_d) \leq (1 - \beta)^{-1} \textrm { and } \end{displaymath} \begin{displaymath} \mathbb{E}(I_d) \geq \sum_{r=1}^{d+1} \left [[1 + r\theta] \dbinom{d+1}{r} \sum_{n=r}^{\infty}\frac{T(n,r)}{(d+1)^n}\mathbb{P}(N=n) \right] + \mathbb{P}(N=0) \end{displaymath} \noindent where $\theta = (1 - \alpha)^{-1}.$ Besides that, if $\mathbb{E}(N) < 1$ (the subcritical case) \[ \lim_{d \to \infty} \mathbb{E}(I_d) = \frac{1}{1 - \mathbb{E}(N)}. \] \end{teo}
\section{Proofs} \label{S: Proofs}
In order to prove the main results we define auxiliary processes whose understanding will provide bounds for the processes defined at introduction. In the first two auxiliary process, denoted by ${U\{\bbT^d; N\}}$ and ${U\{\bbT^d_+; N\}}$, every time a colony collapses the survival individuals are only allowed to choose neighbour vertices which are further (compared to the origin) that the vertex where their colony was placed. In other words an individual is not allowed to choose the neighbour vertex which has been already colonized. We refer to this process as \textit{Self Avoiding}. The last two auxiliary process, denoted by ${L\{\bbT^d; N\}}$ and ${L\{\bbT^d_+; N\}}$, while the survival individuals are allowed to choose the neighbour vertex which has been already colonized, those who does that are not able to colonize it as this place is considered hostile or infertile. We refer to this process as \textit{Move Forward or Die}. In both processes, $Y$, the number of new colonies at collapse times in a vertex $x$ equals the number of diferent neighbours chosen which are located further from the origin than $x$ is. Besides that, every new colony starts with only one individual.
\begin{prop} \label{P: CFGP} Consider a sequence of random variables $\{Y_d\}_{ d \in \mathbb{N}}$ whose sequence of probability generating functions is $\{g_{Y_d}(s)\}_{ d \in \mathbb{N}}$ and a random variable $Y$ such that $Y_d \overset{D}{\to} Y$. Then $g_{Y_d, m}(s)$, the $m-th$ composition of $g_{Y_d}(s)$, converges to $ g_{Y,m}(s)$, where $ g_{Y,m}(s)$ is the $m-th$ composition of $g_Y(s)$, the probability generating function of $Y$. \end{prop} \begin{proof}[Proof of Proposition~\ref{P: CFGP}] From the fact that $Y_d \overset{D}{\to} Y$ it follows that $\displaystyle g_Y(s) = \lim_{d \to \infty} g_{Y_d}(s)$.
\[ \lim_{d \to \infty} g_{Y_d, 2}(s) = \lim_{d \to \infty} g_{Y_d}(g_{Y_d}(s)) = \lim_{d \to \infty} \mathbb{E} \left [ \left ( \mathbb{E}(s^{Y_d}) \right)^{Y_d} \right ] \]
From the Dominated Convergence Theorem~\cite[Theorem 9.1 page 26]{Thorisson} (observe that $[\mathbb{E}(s^{Y_d})]^{Y_d} \in [0,1]$)
\[ \lim_{d \to \infty} \mathbb{E} \left [ [\mathbb{E}(s^{Y_d})]^{Y_d} \right ] = \mathbb{E} \left [ \lim_{d \to \infty} [\mathbb{E}(s^{Y_d})]^{Y_d} \right].
\]
Again, from the Dominated Convergence Theorem~\cite[Theorem 9.1 page 26]{Thorisson} (observe that $s^{Y_d} \in [0,1]$ and that $Y_d$ converges to $Y$ in distribution) $Y_d \ln \mathbb{E}(s^{Y_d})$ converges in distribution to $Y \ln \mathbb{E}(s^Y).$
So we conclude that \[ \mathbb{E} \left [ \lim_{d \to \infty} \left ( \mathbb{E}(s^{Y_d}) \right)^{Y_d} \right ] = \mathbb{E} \left [ [ \mathbb{E}(s^Y) ]^Y \right ]
\] and then
\[ \lim_{d \to \infty} g_{Y_d, 2}(s)= \lim_{d \to \infty} \mathbb{E} \left [ \left ( \mathbb{E}(s^{Y_d}) \right)^{Y_d} \right ] = \mathbb{E} \left [ [ \mathbb{E}(s^Y) ]^Y \right ] = g_{Y,2}(s).
\]
By induction one can prove that $\displaystyle\lim_{d \to \infty} g_{Y_d, m}(s) = g_{Y,m}(s).$ \end{proof}
\begin{prop} \label{P: ConvBranching} Let $\{Z_n \}_{n \geq 0}, \{Z_{1,n} \}_{n \geq 0}, \{Z_{2,n} \}_{n \geq 0}, \cdots$ be a branching processes and $Y, Y_1, Y_2, \cdots $, respectively, their offspring distributions. Supose that \begin{enumerate} \item[(i)] $Y_d \overset{D}{\to} Y$; \item[(ii)] $\mathbb{P}(Y_d \geq k) \leq \mathbb{P}(Y_{d+1} \geq k)$, for all $k$ and for all $d$; \end{enumerate} Then, if $\nu_d$ is the pro\-ba\-bi\-li\-ty of the extinction of the process $\{Z_{d,n} \}_{n \geq 0}$ and $\nu$ is the pro\-ba\-bi\-li\-ty of the extinction of the process $\{Z_{n} \}_{n \geq 0}$ we have that \[ \lim_{d \to \infty}\nu_d = \nu. \] \end{prop}
\begin{proof}[Proof of Proposition~\ref{P: ConvBranching}] From \textit{(i)}, \textit{(ii)} and by using a coupling argument we have that
\begin{equation}\label{1}
\nu_d \ge \nu_{d+1} \ge \lim_{d \to \infty} \nu_d =: \nu_L \ge \nu. \end{equation}
From the fact that $Y_d \overset{D}{\to} Y$ and \cite[Theorem 25.8, page 335]{Billingsley} we have that
\begin{equation}\label{A} \phi_d(s):=\mathbb{E}[s^{Y_d}]\underset{d\to\infty}\longrightarrow\mathbb{E}[s^Y] := \phi(s). \end{equation}
Let $s\in[0,1]$ fixed and $f(y):=s^y, \ y\in\mathbb{N}$. Clearly, $f$ is non-increasing and therefore from \textit{(ii)} and \cite[equation (3.3), page 6]{Thorisson} we have that \begin{equation}\label{B} \phi_{d+1}(s)\leq \phi_{d}(s). \end{equation}
\noindent From (\ref{A}), (\ref{B}) and Dini's Theorem, we have that \begin{equation}\label{C} \phi_{d} (\cdot) \longrightarrow \phi(\cdot) \text{ uniformly.} \end{equation}
\noindent From (\ref{1}), (\ref{C}) and \cite[Exercise 9 - Chapter 7]{Rudin}: \begin{equation}\label{D} \lim_{d\to \infty}\phi_d(\nu_d)=\phi(\nu_L) \end{equation}
\noindent Finally, given that $\phi_d(\nu_d)=\nu_d$, from (\ref{D}) we obtain that \begin{equation}\label{E} \phi(\nu_L)=\nu_L. \end{equation}
\noindent From the convexity of $\phi(s)$ it follows that $\phi(s) = s$ (the fixed points of $\phi(\cdot)$) for at most two points in $[0,1]$. It is known that [see~\cite{Harris2002}, Theorem 6.1 and its proof] if $\nu<1$, the fixed points of $\phi(\cdot)$ are $s=\nu$ and $s=1$. If $\nu=1$, the unique solution is 1. So there are two cases to be considered.
\textbf{1.} If $\nu_d < 1$ for some $d\geq 1$, then from (\ref{1}) it follows that $\nu_L < 1$. If $\nu_L < 1$, it follows from (\ref{E}) that $\nu_L = \nu$.
\textbf{2.} If $\nu_d = 1$ for all $d\geq1$, then \[ \mathbb{E}(Y_d) \leq 1 \textrm { for all } d\geq1. \] Then, \begin{equation} \label{lim}
\lim_{d \to \infty} \mathbb{E}(Y_d) \leq 1. \end{equation}
From \textit{(ii)} we have that $\mathbb{P}(Y_d \geq k) \leq \mathbb{P}(Y \geq k)$, for all $k$ and all $d$. From \textit{(i)}, \textit{(ii)} and a non standart version of Fatou Lemma~\cite[page 230]{Ash} (applied to the sequence $a_{d,j} = j {\mathbb P}(Y_d=j)$), it follows that
\begin{equation} \label{TCD} \liminf_{d \to \infty} \mathbb{E}(Y_d) \ge \mathbb{E}(Y). \end{equation}
From (\ref{lim}) and (\ref{TCD}), it follows that $\nu=1.$ Then, from (\ref{E}) we have that $\nu_L=\nu.$ \end{proof}
\subsection{${U\{\bbT^d; N\}}$: The Self Avoiding model}
\begin{prop} \label{P: CCSRSR3} Consider the process ${U\{\bbT^d; N\}}$. $\mathbb{P}(V_d) > 0 $ if and only if \begin{displaymath} \mathbb{E} \left [ \left(\frac{d - 1}{d} \right)^N \right] < \frac {d-1}{d} \end{displaymath} \end{prop}
\begin{proof}[Proof of Proposition~\ref{P: CCSRSR3}] First of all observe that for a fixed distribution for $N$, the processes ${U\{\bbT^d; N\}}$ and ${U\{\bbT^d_+; N\}}$ either both survives or both die. Next observe that the process ${U\{\bbT^d_+; N\}}$ behaves as a homogeneous branching process. Every vertex $x$ which is colonized produces $Y_d$ new colonies (whose distribution depends only on $N$) on the $d$ neighbour vertices which located are further from the origin than $x$ is. By conditioning one can see that \begin{equation} \label{E: MediaY}
\mathbb{E}(Y_d) = d \sum_{n=0}^{\infty} \left [\left(1 - \left(\frac{d-1}{d}\right)^n\right)\mathbb {P}(N = n) \right] = d \left[ 1- \mathbb{E}\left [\left(\frac{d-1}{d} \right)^N \right] \right]. \end{equation}
From the theory of homogeneous branching processes we see that ${U\{\bbT^d_+; N\}}$ (and also ${U\{\bbT^d; N\}}$) survives if and only if $\mathbb{E}[ (\frac{d - 1}{d} )^N ] < \frac {d-1}{d}.$ \end{proof}
\begin{prop} \label{P: CCSRSR1V} Consider the process ${U\{\bbT^d; N\}}$. Then \begin{displaymath} \mathbb{P}(V_d) = \sum_{r=1}^{d+1}\left [(1 - \psi^r)\binom{d+1} {r}\sum_{n=r}^{\infty}\frac{T(n,r)}{(d+1)^n}\mathbb{P}(N=n) \right] \end{displaymath} where $\psi$, the extinction probability for the process ${U\{\bbT^d_+; N\}}$, is the smallest non-negative solution of \begin{displaymath} \sum_{y=0}^{d} \left[ s^y\binom{d}{y}\sum_{n=y}^{\infty}\frac{T(n,y)} {d^n}\mathbb{P}(N=n) \right] = s. \end{displaymath} On the sub critical regime, which means \begin{displaymath} \mathbb{E} \left [ \left(\frac{d - 1}{d} \right)^N \right] > \frac {d-1}{d}, \end{displaymath} it holds that \begin{displaymath} \mathbb{E}(I_d) = \sum_{r=1}^{d} \left [[1 + r\theta_u]\dbinom{d+1}{r} \sum_ {n=r}^{\infty}\frac{T(n,r)}{(d+1)^n}\mathbb{P}(N=n) \right] + \mathbb{P}(N=0) \end{displaymath} where \begin{displaymath} \theta_u = \left \{ 1- d \left [ 1 - \mathbb{E} \left ( \left(\frac{d - 1}{d} \right)^N \right) \right ]\right\}^{-1}. \end{displaymath} \end{prop} \begin{proof}[Proof of Proposition~\ref{P: CCSRSR1V}] Let $Y_{d,R}$ be the number of colonies created at the neighbour vertices of the origin from its colony at the collapse time. Then \begin{displaymath}
\mathbb{P}(V_d) = \sum_{r=0}^{d+1} \mathbb{P}(V_d| Y_{d,R} = r) \mathbb{P} (Y_{d,R} = r) \end{displaymath} where \begin{displaymath} \mathbb{P}(Y_{d,R}=r) =\sum_{n=r}^{\infty} \left[ \mathbb{P}(N=n) \frac {\dbinom{d+1}{r}T(n,r)}{(d+1)^n} \right] \textrm { for } r =0,1,2, \cdots, d+1. \end{displaymath} because \begin{displaymath}
\mathbb{P}(Y_{d,R}=r | N = n) =\frac {\dbinom{d+1}{r}T(n,r)}{(d+1)^n}. \end{displaymath}
Given that $Y_{d,R} = r$ one have $r$ independent ${U\{\bbT^d_+; N\}}$ processes living on $r$ independent rooted trees. Every vertex $x$ which is colonized, on some of these trees, right after the collapse will have $N$ survival individuals. These individuals will produce $Y_d$ new colonies (whose distribution depends only on $N$) on the $d$ neighbour vertices which are located further from the origin than $x$ is. So we have that \begin{displaymath}
\mathbb{P}(Y_d=y | N = n) = \frac{\dbinom{d}{y}T(n,y)}{d^n}. \end{displaymath} From this, \begin{displaymath} \mathbb{P}(Y_d=y) =\sum_{n=y}^{\infty} \left[ \mathbb{P}(N=n) \frac {\dbinom{d}{y}T(n,y)}{d^n} \right] \textrm { for } y =0,1,2, \cdots, d, \end{displaymath} and \begin{displaymath} \mathbb{E}(s^{Y_d}) = \sum_{y=0}^{d} \left[ s^y\binom{d}{y}\sum_{n=y}^ {\infty}\frac{T(n,y)}{d^n}\mathbb{P}(N=n) \right]. \end{displaymath}
Then $\mathbb{P}(V_d^C | Y_{d,R} = r) = \psi^r \hbox{ for } r =0,1,2, \cdots, d+1$ and \begin{displaymath} \mathbb{P}(V_d) = \sum_{r=1}^{d+1}\left [(1 - \psi^r)\binom{d+1} {r}\sum_{n=r}^{\infty}\frac{T(n,r)}{(d+1)^n}\mathbb{P}(N=n) \right] \end{displaymath} As for the second part of the proposition \begin{displaymath}
\mathbb{E}(I_d) = \sum_{r=0}^{d+1} \mathbb{E}(I_d| Y_{d,R} = r) \mathbb{P} (Y_{d,R}= r). \end{displaymath}
Besides that, $\mathbb{E}(I_d| Y_{d,R} = r) = r\theta_u + 1$ (see Stirzaker~\cite[Exercise 2b, page 280]{Stirzaker}). \end{proof}
\begin{prop} \label{P: CCSRSR1VL} Consider the process ${U\{\bbT^d; N\}}$. Then \begin{equation} \label{limPVD} \lim_{d \to \infty} \mathbb{P}(V_d) = 1 - \nu \end{equation} where $\nu$ is the smallest non-negative solution of $ \mathbb{E}(s^N) = s$. Besides that, if $\mathbb{E}(N) < 1$ (the subcritical case) then \begin{equation} \label{limEID} \lim_{d \to \infty} \mathbb{E}(I_d) = \frac{1}{1 - \mathbb{E}(N)}. \end{equation} \end{prop}
\begin{proof}[Proof of Proposition~\ref{P: CCSRSR1VL}] In order to prove~(\ref{limPVD}) one has to apply Proposition~\ref{P: ConvBranching}, observing that $Y_d \overset{D}{\to} N$ and $Y_{d,R} \overset{D}{\to} N.$ Moreover to prove~(\ref{limEID}) observe that \[
\lim_{d \to \infty} \mathbb{E}(I_d) = \lim_{d \to \infty} \sum_{r=0}^{d+1} \mathbb{E}(I_d|Y_{d,R}=r){\mathbb P}(Y_{d,R}=r).\] As $Y_d \overset{D}{\to} N$ and $Y_{d,R} \overset{D}{\to} N $ then
\[ \lim_{d \to \infty} \mathbb{E}(I_d|Y_{d,R}=r) = \lim_{d \to \infty} r\theta_u +1 = \frac{r}{1-\mathbb{E}(N)}+1 \] and the result follows from the Dominated Convergence Theorem~\cite[Theorem 9.1 page 26]{Thorisson}. \end{proof}
\begin{prop} \label{P: LIMTSRSR1} Consider the process ${U\{\bbT^d_+; N\}}$. Assuming \begin{displaymath} \mathbb{E} \left [ \left(\frac{d - 1}{d} \right)^N \right] > \frac {d-1}{d} \end{displaymath} we have that \begin{displaymath} \frac{[1+D(1-\mu)][1-\mu^{m+1}]}{1+ D(1-\mu)-\mu^{m+1}} \leq \mathbb{P}(M_d \leq m) \leq \frac{[1 + \frac{\mu(1-\mu)}{B}](1-\mu^{m+1})}{ 1 + \frac{\mu(1-\mu)}{B} - \mu^{m+1}} \end{displaymath} and \begin{displaymath}
\frac{\mu^2}{2(B+ \mu)} + \mu(1-\mu)\frac{\ln \left[1 - \frac{\mu B}{B + \mu(1-\mu)}\right]}{B \ln \mu} \leq \mathbb{E}(M_d) \leq \frac{D \mu}{D+1} + D(1-\mu)\frac{\ln \left[1 - \frac{\mu}{1 + D(1-\mu)}\right]}{\ln \mu} \end{displaymath} where \begin{align*} \mu &= d \left [ 1 - \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] \right ] \\ D &= \max \left \{ 2; \frac{g^{\prime}(1)}{g^{\prime}(1) - \mathbb{P}(N \neq 0)} \right\} \\ B & = d(d-1)\left[ 1 - 2\mathbb{E} \left ( \left(\frac{d-1 }{d} \right)^N \right) + \mathbb{E} \left ( \left(\frac{d-2 }{d} \right)^N \right) \right]. \end{align*} Moreover, \begin{displaymath} M_d \overset{D}{\to} M, \end{displaymath} where $\mathbb{P}( M \leq m) = g_{m+1}(0)$, being $g(s) = \mathbb{E}(s^N) $ and $g_{m+1}(s) = \overset{ m+1 \textrm { times }} {g(g(\cdots g(s)) \cdots )}$. \end{prop} \begin{proof}[Proof of Proposition~\ref{P: LIMTSRSR1}]
Every vertex $x$ which is colonized produces $Y_d$ new colonies (whose distribution depends only on $N$) on the $d$ neighbour vertices which are located further from the origin than $x$ is. The random variable $Y_d$ can be seen as $Y_d = \sum_{i=1}^{d}I_i$ where for $i=1, \dots, d$ \begin{displaymath} I_i = \left\{ \begin{array}{ll} 1, & \hbox{the $i-th$ neighbour of $x$ is colonized} \\ 0, & \hbox{else.} \\ \end{array} \right. \end{displaymath} Defining $g_{Y_d}(s)$ as the generating function of $Y_d$ observe that equation~(\ref{E: MediaY}) gives $g_{Y_d}^{\prime}(1)$. Moreover \begin{displaymath} {Y_d}^2 = \left (\sum_{i=1}^{d}I_i \right )^2 = \sum_{i=1}^{d}I_i^2 + 2\sum_{1 \leq i < j \leq d}I_iI_j \end{displaymath} and \begin{displaymath}
\mathbb{E} \left({Y_d}^2 \right ) = d \mathbb{E} \left (I_1^2 \right ) + d(d-1) \mathbb{E}(I_1I_2) \end{displaymath} and finally \begin{displaymath}
\mathbb{E} \left({Y_d}^2 \right ) = d \left[ 1 - \mathbb{E} \left [\left( \frac{d-1}{d} \right)^N \right] \right] + d(d-1) \left [ 1 -2 \mathbb{E} \left [\left ( \frac{d-1}{d} \right )^N \right] + \mathbb{E} \left [\left ( \frac{d-2}{d} \right )^N \right] \right ]. \end{displaymath} Then \begin{align*} g_{Y_d}^{\prime \prime}(1) = \mathbb{E} \left(Y_d(Y_d-1) \right ) = d(d-1) \left [ 1 -2 \mathbb{E} \left [\left ( \frac{d-1}{d} \right )^N \right] + \mathbb{E} \left [\left ( \frac{d-2}{d} \right )^N \right] \right ]. \end{align*} Then the result follows from Theorem 1 page 331 in~\cite{AA}, where $ m = g_{Y_d}^{\prime}(1)$.
The convergence $M_d \overset{D}{\to} M$ follows from the fact that $Y_d \overset{D}{\to} N$ when $ d \to \infty$ and from Proposition \ref{P: CFGP}. \end{proof}
\subsection{{${L\{\bbT^d; N\}}$: Move Forward or Die}}
\begin{prop} \label{P: CCSRCR3} Consider the process ${L\{\bbT^d; N\}}$. $\mathbb{P}(V_d) > 0 $ if and only if \begin{displaymath} \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] < \frac{d- 1}{d} \end{displaymath} \end{prop}
\begin{proof}[Proof of Proposition~\ref{P: CCSRCR3}] First of all observe that for a fixed distribution for $N$, the processes ${L\{\bbT^d; N\}}$ and ${L\{\bbT^d_+; N\}}$ either both survives or both die. Next observe that the process ${L\{\bbT^d_+; N\}}$ behaves as a homogeneous branching process. Every vertex $x$ which is colonized produces a bunch of survival individuals right after the collapse which are willing to jump to one of the $d+1$ nearest neighbours vertices of $x$. All those which jump towards the origin get killed. So, $Y_d$ new colonies will be found on the $d$ neighbour vertices which are located further from the origin than $x$ is. By conditioning one can see that \begin{equation} \label{E: MediaYL}
\mathbb{E}(Y_d) = d\sum_{n=0}^{\infty} \left [\left(1 - \left(\frac{d}{d+1}\right)^n\right)\mathbb {P}(N = n) \right] = d \left[ 1- \mathbb{E} \left [\left(\frac{d}{d+1} \right)^N \right ]\right] \end{equation}
From the theory of homogeneous branching processes we see that ${L\{\bbT^d_+; N\}}$ (and also ${L\{\bbT^d; N\}}$) survives if and only if $\mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] < \frac{d-1}{d}.$ \end{proof}
\begin{prop} \label{P: CCSRCR1V} Consider the process ${L\{\bbT^d; N\}}$. Then \begin{displaymath} \mathbb{P}(V_d) = \sum_{r=1}^{d+1}\left [(1 - \rho^r)\binom{d+1} {r}\sum_{n=r}^{\infty}\frac{T(n,r)}{(d+1)^n}\mathbb{P}(N=n) \right] \end{displaymath} where $\rho$, the extinction probability for the process ${L\{\bbT^d_+; N\}}$, is the smallest non-negative solution of \begin{displaymath} \sum_{y=0}^{d} \left[ s^y\binom{d}{y}\sum_{n=y}^{\infty}\frac{T(n,y)+ T(n,y+1)}{(d+1)^n}\mathbb{P}(N=n) \right] = s. \end{displaymath} On the subcritical regime, which means \begin{displaymath} \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] > \frac{d- 1}{d}, \end{displaymath} it holds that \begin{displaymath} \mathbb{E}(I_d) = \sum_{r=1}^{d+1} \left [[1 + r\theta_l] \dbinom{d+1}{r} \sum_{n=r}^{\infty}\frac{T(n,r)}{(d+1)^n}\mathbb{P}(N=n) \right] + \mathbb{P}(N=0) \end{displaymath} where \begin{displaymath} \theta_l = \left \{ 1- d \left [ 1 - \mathbb{E} \left ( \left(\frac{d }{d +1} \right)^N \right) \right ]\right\}^{-1}. \end{displaymath} \end{prop}
\begin{proof}[Proof of Proposition~\ref{P: CCSRCR1V}] Let $Y_{d,R}$ be the number of colonies created at the neighbour vertices of the origin from its colony at the collapse time. Then \begin{displaymath}
\mathbb{P}(V_d) = \sum_{r=0}^{d+1} \mathbb{P}(V_d| Y_{d,R} = r) \mathbb{P} (Y_{d,R} = r) \end{displaymath} where \begin{displaymath} \mathbb{P}(Y_{d,R}=r) =\sum_{n=r}^{\infty} \left[ \mathbb{P}(N=n) \frac {\dbinom{d+1}{r}T(n,r)}{(d+1)^n} \right] \textrm { for } r =0,1,2, \cdots, d+1. \end{displaymath} because \begin{displaymath}
\mathbb{P}(Y_{d,R}=r | N = n) = \frac {\dbinom{d+1}{r}T(n,r)}{(d+1)^n}. \end{displaymath}
Given that $Y_{d,R} = r$ one have $r$ independent ${L\{\bbT^d_+; N\}}$ processes living on $r$ independent rooted trees. Every vertex $x$ which is colonized, on some of these trees, right after the collapse will have $N$ survival individuals. These individuals will produce $Y_d$ new colonies (whose distribution depends only on $N$) on the $d$ neighbour vertices which are located further from the origin than $x$ is. So we have that
\begin{displaymath}
\mathbb{P}(Y_d=y | N = n) = \frac{\dbinom{d}{y}[T(n,y)+ T(n,y+1)]}{(d +1)^n} \end{displaymath} From this, \begin{displaymath} \mathbb{P}(Y_d=y) =\sum_{n=y}^{\infty} \left[ \mathbb{P}(N=n) \frac {\dbinom{d}{y}[T(n,y)+ T(n,y+1)]}{(d+1)^n} \right] \textrm { for } y =0,1,2, \cdots, d. \end{displaymath} and \begin{displaymath} \mathbb{E}(s^{Y_d}) = \sum_{y=0}^{d}s^y \sum_{n=y}^{\infty} \left[ \mathbb{P}(N=n) \frac{\dbinom{d}{y}[T(n,y)+ T(n,y+1)]}{(d+1)^n} \right]. \end{displaymath}
Then $\mathbb{P}({V_d}^C | Y_{d,R} = r) = \rho^r,\ r =0,1,2, \cdots, d+1$ and \begin{displaymath} \mathbb{P}(V_d) = \sum_{r=1}^{d+1}\left [(1 - \rho^r)\binom{d+1} {r}\sum_{n=r}^{\infty}\frac{T(n,r)}{(d+1)^n}\mathbb{P}(N=n) \right] \end{displaymath} As for the second part of the proposition \begin{displaymath}
\mathbb{E}(I_d) = \sum_{r=0}^{d+1} \mathbb{E}(I_d| Y_{d,R} = r) \mathbb{P} (Y_{d,R} = r). \end{displaymath}
Besides that, $\mathbb{E}(I_d| Y_{d,R} = r) = r\theta_l + 1$ (see Stirzaker~\cite[Exercise 2b, page 280]{Stirzaker}). \end{proof}
\begin{prop} \label{P: RCR1VL2} Consider the process ${L\{\bbT^d; N\}}$. Then, \begin{equation} \label{limPVD2} \lim_{d \to \infty} \mathbb{P}(V_d) = 1 - \nu \end{equation} where $\nu$ is the smallest non-negative solution of $ \mathbb{E}(s^N) = s$. Besides that, if $\mathbb{E}(N) < 1$ (the subcritical case) then \begin{equation} \label{limEID2} \lim_{d \to \infty} \mathbb{E}(I_d) = \frac{1}{1 - \mathbb{E}(N)}. \end{equation} \end{prop}
\begin{proof}[Proof of Proposition~\ref{P: RCR1VL2}] In order to prove~(\ref{limPVD2}) one has to aply Proposition~\ref{P: ConvBranching}, observing that $Y_d \overset{D}{\to} N$ and $Y_{d,R} \overset{D}{\to} N.$ For the proof of~(\ref{limEID2}) observe that \[
\lim_{d \to \infty} \mathbb{E}(I_d) = \lim_{d \to \infty} \sum_{r=0}^{d+1} \mathbb{E}(I_d|Y_{d,R}=r){\mathbb P}(Y_{d,R}=r).\] As $Y_d \overset{D}{\to} N$ and $Y_{d,R} \overset{D}{\to} N $ then
\[ \lim_{d \to \infty} \mathbb{E}(I_d|Y_{d,R}=r) = \lim_{d \to \infty} r \theta_l +1 = \frac{r}{1-\mathbb{E}(N)}+1. \] The result follows from the Dominated Convergence Theorem~\cite[Theorem 9.1 page 26]{Thorisson}. \end{proof}
\begin{prop} \label{P: LIMTSRCR1} Consider the process ${L\{\bbT^d_+; N\}}$. Assuming \begin{displaymath} \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] > \frac{d- 1}{d} \end{displaymath} We have that \begin{displaymath} \frac{[1+D(1-\mu)][1-\mu^{m+1}]}{1+ D(1-\mu)-\mu^{m+1}} \leq \mathbb{P}(M_d \leq m) \leq \frac{[1 + \frac{\mu(1-\mu)}{B}](1-\mu^{m+1})}{ 1 + \frac{\mu(1-\mu)}{B} - \mu^{m+1}} \end{displaymath} and \begin{displaymath}
\frac{\mu^2}{2(B+ \mu)} + \mu(1-\mu)\frac{\ln \left[1 - \frac{\mu B}{B + \mu(1-\mu)}\right]}{B \ln \mu} \leq \mathbb{E}(M_d) \leq \frac{D \mu}{D+1} + D(1-\mu)\frac{\ln \left[1 - \frac{\mu}{1 + D(1-\mu)}\right]}{\ln \mu} \end{displaymath} where \begin{align*} \mu &= d \left [ 1 - \mathbb{E} \left [ \left(\frac{d}{d+1} \right)^N \right] \right ] \\ D &= \max \left \{2; \frac{\mu}{\mu - \mathbb{P}(N \neq 0)} \right\} \\ B& = d(d-1)\left[ 1 - 2\mathbb{E} \left ( \left(\frac{d }{d+1} \right)^N \right) + \mathbb{E} \left ( \left(\frac{d-1 }{d+1} \right)^N \right) \right]. \end{align*} Besides that, \begin{displaymath} M_d \overset{D}{\to} M, \end{displaymath} where $\mathbb{P}( M \leq m) = g_{m+1}(0)$, being $g(s) = \mathbb{E}(s^N) $ and $g_{m+1}(s) = \overset{ m+1 \textrm { times }} {g(g(\cdots g(s)) \cdots )}$. \end{prop}
\begin{proof}[Proof of Proposition~\ref{P: LIMTSRCR1}] Every vertex $x$ which is colonized produces $Y_d$ new colonies (whose distribution depends only on $N$) on the $d$ neighbour vertices which are located further from the origin than $x$ is. The random variable $Y_d$ can be seen as $Y_d = \sum_{i=1}^{d}I_i$ where for $i=1, \dots, d$ \begin{displaymath} I_i = \left\{ \begin{array}{ll} 1, & \hbox{the $i-th$ neighbour of $x$ is colonized} \\ 0, & \hbox{else.} \\ \end{array} \right. \end{displaymath} Defining $g_{Y_d}(s)$ as the generating function of $Y_d$ observe that equation~(\ref{E: MediaYL}) gives $g_{Y_d}^{\prime}(1)$. Moreover \begin{displaymath}
{Y_d}^2 = \left (\sum_{i=1}^{d}I_i \right )^2 = \sum_{i=1}^{d}I_i^2 + 2\sum_{1 \leq i < j \leq d}I_iI_j \end{displaymath} and \begin{displaymath}
\mathbb{E} \left({Y_d}^2 \right ) = d \mathbb{E} \left (I_1^2 \right ) + d(d-1) \mathbb{E}(I_1I_2) \end{displaymath} and finally \begin{displaymath}
\mathbb{E} \left({Y_d}^2 \right ) = d \left[ 1 - \mathbb{E} \left [\left( \frac{d}{d+1} \right)^N \right] \right] + d(d-1) \left [ 1 -2 \mathbb{E} \left [\left ( \frac{d}{d+1} \right )^N \right] + \mathbb{E} \left [\left ( \frac{d-1}{d+1} \right )^N \right] \right ] \end{displaymath} Then \begin{align*} g_{Y_d}^{\prime \prime}(1) = \mathbb{E} \left(Y_d(Y_d-1) \right ) = d(d-1) \left [ 1 -2 \mathbb{E} \left [\left ( \frac{d}{d+1} \right )^N \right] + \mathbb{E} \left [\left ( \frac{d-1}{d+1} \right )^N \right] \right ] \end{align*} Then the result follows from Theorem 1 page 331 in~\cite{AA}, where $ m = g_{Y_d}^{\prime}(1)$.
The convergence $M_d \overset{D}{\to} M$ follows from the fact that $Y_d \overset{D}{\to} N$ when $ d \to \infty$ and from Proposition \ref{P: CFGP}. \end{proof}
\subsection{Proofs of the main results}
First we define a coupling between the processes ${\{\bbT^d; N\}}$ and ${L\{\bbT^d_+; N\}}$ in such a way that the former is dominated by the earlier. Every colony in ${L\{\bbT^d_+; N\}}$ is associated to a colony in ${\{\bbT^d; N\}}$. As a consequence, if the process ${\{\bbT^d; N\}}$ dies out, the same happens to ${L\{\bbT^d_+; N\}}$.
At every collapse time at a vertex $x$ in the original model, a non-empty group of individuals that tries to colonize the neighbour vertex to $x$ which is closer to the origin than $x$ will create there a new colony provided that that vertex is empty. In the model ${L\{\bbT^d_+; N\}}$ the same non-empty group of individuals that tries to colonize the same vertex, imediately dies.
Next we define a coupling between the processes ${\{\bbT^d; N\}}$ and ${U\{\bbT^{d+1}_+; N\}}$ in such a way that the former dominates the earlier. Every colony in ${\{\bbT^d; N\}}$ can be associated to a colony in ${U\{\bbT^{d+1}_+; N\}}$. As a consequence if the process ${U\{\bbT^{d+1}_+; N\}}$ dies out, the same happens to ${\{\bbT^d; N\}}$.
At every collapse time at a vertex $x$ we associate the neighbour vertex to $x$ which is closer to the origin than $x$ to the extra vertex on the model ${U\{\bbT^{d+1}_+; N\}}$. In the original model, a non-empty group of individuals that tries to colonize the neighbour vertex to $x$ which is closer to the origin than $x$ will create there a new colony provided that that vertex is empty. In the model ${U\{\bbT^{d+1}_+; N\}}$ the same non-empty group of individuals that tries to colonize the extra vertex, founds a new colonony there.
\begin{proof}[Proof of Theorem~\ref{T: MCC1H}]
The result follows from the fact that the process ${\{\bbT^d; N\}}$ dominates the process ${L\{\bbT^d_+; N\}}$ and by its turn, is dominated by the process ${U\{\bbT^{d+1}_+; N\}}$, together with Propositions~\ref{P: CCSRSR3} and~\ref{P: CCSRCR3}.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{C: MCC1HP}] Assuming $s=\frac{d}{d+1}$ in (\ref{eq: fgpP}) and applying Theorem~\ref{T: MCC1H} the result follows. \end{proof}
\begin{proof}[Proof of Corollary~\ref{C: MCC1HPY}] Assuming $s=\frac{d}{d+1}$ in (\ref{eq: fgpY}) and applying Theorem~\ref{T: MCC1H} the result follows. \end{proof}
\begin{proof}[Proof of Theorem~\ref{T: MCCHPE1}] The result follows from the fact that the process ${\{\bbT^d; N\}}$ dominates the process ${L\{\bbT^d_+; N\}}$ and by its turn, is dominated by the process ${U\{\bbT^{d+1}_+; N\}}$, together with Propositions~\ref{P: CCSRSR1V} and~\ref{P: CCSRCR1V}. \end{proof}
\begin{proof}[Proof of Theorem~\ref{T: MCCHPE1L}] The result follows from the fact that the process ${\{\bbT^d; N\}}$ dominates the process ${L\{\bbT^d_+; N\}}$ and by its turn, is dominated by the process ${U\{\bbT^{d+1}_+; N\}}$, together with Propositions~\ref{P: CCSRSR1VL} and~\ref{P: RCR1VL2}. \end{proof}
\begin{proof}[Proof of Corollary~\ref{C: MCC1HPS}] The proof is just a matter of computing the smallest positive fixed point for the generating function of $N$ (the smallest positive $s$ such that $\mathbb{E}(s^N) = s$) for $\mathbb{E}(s^N)$ given in~(\ref{eq: fgpP}). \end{proof}
\begin{proof}[Proof of Corollary~\ref{C: MCC1HPS2}] The proof is just a matter of computing the smallest positive fixed point for the generating function of $N$ (the smallest positive $s$ such that $\mathbb{E}(s^N) = s$) for $\mathbb{E}(s^N)$ given in~(\ref{eq: fgpY}).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{T: LEIT}] The result follows from the fact that the process ${\{\bbT^d; N\}}$ dominates the process ${L\{\bbT^d_+; N\}}$ and by its turn, is dominated by the process ${U\{\bbT^{d+1}_+; N\}}$, together with Propositions~\ref{P: LIMTSRSR1} and~\ref{P: LIMTSRCR1}. \end{proof}
\begin{proof}[Proof of Corollary~\ref{C: CTM1}] The proof is just a matter of computing the generating function of $N$ (see Equation~(\ref{eq: fgpP})) on both values $s = \frac{d}{d+1}$ and $s = \frac{d-1}{d+1}$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{T: LEITL}] The result follows from the fact that the process ${\{\bbT^d; N\}}$ dominates the process ${L\{\bbT^d_+; N\}}$ and by its turn, is dominated by the process ${U\{\bbT^{d+1}_+; N\}}$, together with Propositions~\ref{P: LIMTSRSR1} and~\ref{P: LIMTSRCR1}. \end{proof}
\begin{defn} \label{fgpfl} A fractional linear generating function is a probability generating function of the form \begin{equation*} f(b,c;s) = 1 - \frac{b}{1-c} + \frac{bs}{1-cs}, \ 0 \leq s \leq 1. \end{equation*} where $0 \leq b \leq 1$, $0 \leq c \leq 1$, and $b+c \leq 1$. \end{defn}
\begin{proof}[Proof of Corollary~\ref{C: LEITL}] Observe that the generating function of $N$ given in (\ref{eq: fgpP}) is a fractional linear generating function. The results follow from equations (3.1) and (3.2) in \cite{AA} page 330 and from Theorem \ref{T: LEITL}. \end{proof}
\begin{proof}[Proof of Theorem~\ref{T: NC}] The result follows from the fact that the process ${\{\bbT^d; N\}}$ dominates the process ${L\{\bbT^d_+; N\}}$ and by its turn, is dominated by the process ${U\{\bbT^{d+1}_+; N\}}$, together with Propositions~\ref{P: CCSRSR1V} and~\ref{P: CCSRCR1V}. \end{proof}
\end{document}
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\begin{document}
\title{On Minimizing Tardy Processing Time, Max-Min Skewed Convolution, and Triangular Structured ILPs hanks{Part of this research was done during the Discrete Optimization trimester program at Hausdorff Research Institute for Mathematics (HIM) in Bonn, Germany.}
\begin{abstract} The starting point of this paper is the problem of scheduling $n$ jobs with processing times and due dates on a single machine so as to minimize the total processing time of tardy jobs, i.e., $1\,|\,|\,\sum p_j U_j$\xspace. This problem was identified by Bringmann et al.~(Algorithmica 2022) as a natural subquadratic-time special case of the classic $1\,|\,|\,\sum w_j U_j$\xspace problem, which likely requires time quadratic in the total processing time $P$, because of a fine-grained lower bound. Bringmann et al.~obtain their $\widetilde{O}(P^{7/4})$ time scheduling algorithm through a new variant of convolution, dubbed Max-Min Skewed Convolution, which they solve in $\widetilde{O}(n^{7/4})$ time. Our main technical contribution is a faster and simpler convolution algorithm running in $\widetilde{O}(n^{5/3})$ time. It implies an $\widetilde{O}(P^{5/3})$ time algorithm for $1\,|\,|\,\sum p_j U_j$\xspace, but may also be of independent interest.
Inspired by recent developments for the Subset Sum and Knapsack problems, we study $1\,|\,|\,\sum p_j U_j$\xspace parameterized by the maximum job processing time $p_{\max}$. With proximity techniques borrowed from integer linear programming (ILP), we show structural properties of the problem that, coupled with a new dynamic programming formulation, lead to an $\widetilde{O}(n+p_{\max}^3)$ time algorithm. Moreover, in the setting with multiple machines, we use similar techniques to get an $n \cdot p_{\max}^{O(m)}$ time algorithm for $Pm\,|\,|\,\sum p_j U_j$\xspace.
Finally, we point out that the considered problems exhibit a particular triangular block structure in the constraint matrices of their ILP formulations. In light of recent ILP research, a question that arises is whether one can devise a generic algorithm for such a class of ILPs. We give a negative answer to this question: we show that already a slight generalization of the structure of the scheduling ILP leads to a strongly NP-hard problem. \end{abstract} \pagebreak \section{Introduction} We consider the scheduling problem $1\,|\,|\,\sum p_j U_j$\xspace of minimizing the total sum of processing times of \emph{tardy} jobs, where we are given a set of $n$ jobs, numbered from $1$ to $n$, and each job $j$ has a processing time $p_j$ and a due date $d_j$. A schedule is defined by a permutation $\sigma: \{1, \ldots ,n \} \to \{1, \ldots ,n \}$ of the jobs, and, based on this schedule $\sigma$, the \emph{completion time} of jobs is defined. The completion time $C_j$ of a job $j$ is $C_j = \sum_{i \, : \, \sigma(i) \leq \sigma(j)} p_i$. The objective of the problem is to find a schedule $\sigma$ that minimizes the sum of processing times of jobs that miss their due date $d_j$ (called \emph{tardy}), i.e., \[
\min_{\sigma} \ \sum_{\mathclap{j \, : \, C_j > d_j}} p_j. \] Note that for tardy jobs we pay their penalty regardless of the actual completion time. Therefore, in the remainder of the paper, we will use the equivalent problem formulation that we have to select a subset of jobs $S\subseteq \{1,\ldots,n\}$ such that all selected job can be completed by their due dates. In this case it can be assumed that the jobs from $S$ are scheduled by the earliest-due-date-first order~\cite{lawler1969functional_deadline}. One could also think of the problem as a scenario where the jobs that cannot be scheduled before their due dates on the available machine have to be outsourced somewhere else, and the cost of doing this is proportional to the total size of these outsourced jobs. These two properties are typical for hybrid cloud computing platforms.
For the case where all due dates are identical, the scheduling problem is equivalent to the classic Subset Sum problem. In the latter problem we are given a (multi-)set of numbers $\{a_1, \ldots , a_n\}$ and a target value $t$, and the objective is to find a subset of the numbers that sums up exactly to $t$, which is equivalent to the problem of finding the maximum subset sum that is at most $t$. With arbitrary due dates, $1\,|\,|\,\sum p_j U_j$\xspace behaves like a multi-level generalization of the Subset Sum problem, where upper bounds are imposed not only on the whole sum, but also on prefix sums.
In recent years there has been considerable attention on developing fast pseudopolynomial time algorithms solving the Subset Sum problem. Most prominent is the algorithm by Bringmann~\cite{bringmann2017subset_sum_near_linear} solving the problem in time $\widetilde{O}(t)$.\footnote{The $\widetilde O$ notation hides polylogarithmic factors.} The algorithm relies, among other techniques, on the use of Boolean convolution, which can be computed very efficiently in time $O(n \log n)$ by Fast Fourier Transform. Pseudopolynomial time algorithms have also been studied for a parameter $a_{\max} = \max_i a_i$, which is stronger, i.e., $a_{\max} \le t$. Eisenbrand and Weismantel~\cite{eisenbrand2018proximity} developed an algorithm for integer linear programming (ILP) that, when applied to Subset Sum, gives a running time of $O(n + a_{\max}^3)$, the first algorithm with a running time of the form $O(n + \mathrm{poly}(a_{\max}))$. Based on the Steinitz Lemma they developed proximity results, which can be used to reduce the size of $t$ and therefore solve the problem within a running time independent of the size of the target value $t$. The currently fastest algorithm in this regard is by Polak, Rohwedder and Węgrzycki~\cite{polak_subsetsum_an} with running time $\widetilde O(n+ a_{\max}^{5/3})$.
Given the recent attention on pseudopolynomial time algorithms for Subset Sum, it is not surprising that also~$1\,|\,|\,\sum p_j U_j$\xspace has been considered in this direction. Already in the late '60s, Lawler and Moore~\cite{lawler1969functional_deadline} considered this problem or, more precisely, the general form of $1\,|\,|\,\sum w_j U_j$\xspace, where the penalty for each job being tardy is not necessarily $p_j$, but can be an independent weight $w_j$. They solved this problem in time $O(nP)$, where $P$ is the sum of processing times over all jobs. Their algorithm follows from a (by now) standard dynamic programming approach. This general form of the problem is unlikely to admit better algorithms: under a common hardness assumption the running time of this algorithm is essentially the best possible. Indeed, assuming that $(\min,+)$-convolution cannot be solved in subquadratic time (a common hardness assumption in fine-grained complexity), there is no algorithm that solves $1\,|\,|\,\sum w_j U_j$\xspace in time $O(P^{2-\epsilon})$, for any $\epsilon>0$. This follows from the fact that $1\,|\,|\,\sum w_j U_j$\xspace generalizes Knapsack and the hardness already holds for Knapsack~\cite{cygan2019_minconv,KunnemannPS17_finegrainedDP}. Since the hardness does not hold for Subset Sum (the case of Knapsack where profits equal weights), one could hope that it also does not hold either for the special case of $1\,|\,|\,\sum w_j U_j$\xspace, where penalties equal processing times, which is precisely~$1\,|\,|\,\sum p_j U_j$\xspace. Indeed, Bringmann, Fischer, Hermelin, Shabtay, and Wellnitz~\cite{bringmann_deadline_scheduling} recently obtained a subquadratic algorithm with running time $\widetilde O(P^{7/4})$ for the problem. Along with this main result, they also consider other parameters: the sum of distinct due dates and the number of distinct due dates. See also Hermelin et al.~\cite{hermelin_deadline_scheduling} for more related results.
\paragraph*{Max-min skewed convolution.} Typically, a convolution has two input vectors $a$ and $b$ and outputs a vector $c$, where $c[k] = \oplus_{i + j = k} (a[i] \otimes a[j])$. Different kinds of operators $\oplus$ and $\otimes$ have been studied, and, if the operators require only constant time, then a quadratic time algorithm is trivial. However, if, for example, ``$\oplus$'' $=$ ``$+$'' (standard addition) and ``$\otimes$'' $=$ ``$\cdot$'' (standard multiplication), then Fast Fourier Transform can solve convolution very efficiently in time $O(n\log n)$. If, on the other hand, ``$\oplus$'' $=$ ``$\max$'' and ``$\otimes$'' $=$ ``$+$'', it is generally believed that no algorithm can solve the problem in time $O(n^{2 - \epsilon})$ for some fixed $\epsilon > 0$~\cite{cygan2019_minconv}. Convolution problems have been studied extensively in fine-grained complexity, partly because they serve as good subroutines for other problems, see also~\cite{BringmannKW19, Lincoln0W20} for other recent examples. The scheduling algorithm by Bringmann et al.~\cite{bringmann_deadline_scheduling} works by first reducing the problem to a new variant of convolution, called max-min skewed convolution, and then solving this problem in subquadratic time. Max-min skewed convolution is defined as the problem where we are given vectors $(a[0],a[1],\dotsc,a[n-1])$ and $(b[0],b[1],\dotsc,b[n-1])$ as well as a third vector $(d[0],d[1],\dotsc,d[2n-1])$ and our goal is to compute, for each $k = 0,1,\dotsc,2n-1$, the value \begin{equation*}
c[k] = \max_{i + j = k}\{\min\{a[i], b[j] + d[k]\} . \end{equation*} This extends the standard max-min convolution, where $d[k] = 0$ for all $k$. Max-min convolution can be solved non-trivially in time $\widetilde O(n^{3/2})$~\cite{maxminconv}. Bringmann et al.~develop an algorithm with running time $\widetilde O(n^{7/4})$ for max-min skewed convolution, when $d[k] = k$ for all $k$ (which is the relevant case for $1\,|\,|\,\sum p_j U_j$\xspace). By their reduction this implies an $\widetilde O(P^{7/4})$ time algorithm for~$1\,|\,|\,\sum p_j U_j$\xspace.
Our first result and our main technical contribution is a faster, simpler and more general algorithm for max-min skewed convolution. \begin{theorem} Max-min skewed convolution can be computed in time $O(n^{5/3} \log n)$. \end{theorem} As a direct consequence, we obtain an improved algorithm for~$1\,|\,|\,\sum p_j U_j$\xspace. \begin{corollary} The problem $1\,|\,|\,\sum p_j U_j$\xspace can be solved in $\widetilde O(P^{5/3})$ time, where $P$ is the sum of processing times. \end{corollary} Since convolution algorithms are often used as building blocks for other problems, we believe that this result is of independent interest, in particular, since our algorithm also works for arbitrary $d[k]$. From a technical point of view, the algorithm can be seen as a two-dimensional generalization of the approach used in the $\widetilde O(n^{3/2})$ time algorithm for max-min convolution~\cite{maxminconv}. This is quite different and more compact than the algorithm by Bringmann et al.~\cite{bringmann_deadline_scheduling}, which only relies on the ideas in~\cite{maxminconv} indirectly by invoking max-min convolution as a black box.
\paragraph*{Parameterization by the maximum processing time.} As mentioned before, Subset Sum has been extensively studied with respect to running times in the maximum value $a_{\max}$. Our second contribution is that we show that running time in similar spirit can also be achieved for~$1\,|\,|\,\sum p_j U_j$\xspace. Based on techniques from integer programming, namely, Steinitz Lemma type of arguments that have also been crucial for Subset Sum, we show new structural properties of solutions for the problem. Based on this structural understanding, we develop an algorithm that leads to the following result. \begin{theorem} The problem~$1\,|\,|\,\sum p_j U_j$\xspace can be solved in time $\widetilde O(n + p^3_{\max})$. \end{theorem} For the generalized problem $Pm\,|\,|\,\sum p_j U_j$\xspace with multiple machines, we present an algorithm relying on a different structural property. \begin{theorem} The problem~$Pm\,|\,|\,\sum p_j U_j$\xspace can be solved in time $O(n \cdot p_{\max}^{O(m)})$. \end{theorem} Similar algorithmic results with running times depending on the parameter $p_{\max}$ have been developed for makespan scheduling and minimizing weighted completion time (without due dates) \cite{knop2017scheduling}.
\paragraph*{Integer programming generalization and lower bounds.}
The problem $1\,|\,|\,\sum p_j U_j$\xspace can be formulated as an integer linear program (ILP) with binary variables as follows:
\begin{equation} \text{maximize} \quad \sum_j p_jx_j \quad \text{subject to} \quad \begin{pmatrix} p_1 & 0 & \cdots & 0\\ p_1 & p_2 & \ddots & \\ \vdots & & \ddots & 0 \\ p_1 & p_2 & \cdots & p_n \end{pmatrix} \cdot x \le \begin{pmatrix}d_1\\ d_2 \\ \vdots \\ d_n\end{pmatrix}, \quad x \in \{0, 1\}^n. \label{ILP_problem} \end{equation} Here, variable $x_j$ indicates whether job $j$ is selected in the solution (or, in the alternative formulation, finished within its due date). The objective is to maximize the processing time of the selected jobs. The necessary and sufficient conditions are that the total volume of selected jobs with due date at most some time $t$ does not exceed $t$. It suffices to consider only constraints for $t$ equal to one of the jobs' due dates.
Clearly, the shape of the non-zeros in the constraint matrix of the ILP exhibits a very special structure, a certain triangular shape. In recent years there has been significant attention in the area of structured integer programming on identifying parameters and structures for which integer programming is tractable~\cite{DBLP:conf/soda/CslovjecsekEHRW21, CslovjecsekEPVW21, JansenKL21, klein_multistage, KouteckyLO18, EisenbrandHK18}. The most prominent example in this line of work are so-called $n$-fold integer programs (see~\cite{DBLP:conf/soda/CslovjecsekEHRW21} and references therein), which are of the form \[ \text{maximize} \quad c^T x \quad \text{subject to} \quad \begin{pmatrix} A_1 & A_2 & \cdots & A_n \\ B_1 & 0 & & 0\\ 0 & B_2 & \ddots & \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & B_n \end{pmatrix} \cdot x = b \quad \text{and} \quad \forall_i \ x_i \in \mathbb{Z}_{\geqslant 0}. \]
Here $A_i$ and $B_i$ are ``small'' matrices in the sense that it is considered acceptable for the running time to depend superpolynomially (e.g., exponentially) on their parameters. It is known that these types of integer programs can be solved in FPT time $\max_i f(A_i, B_i) \cdot \mathrm{poly}(|I|)$, where $f(A_i, B_i)$ is a function that depends only on the matrices $A_i$ and $B_i$ (potentially superpolynomially), but not on $n$ or any other parameters, and $\mathrm{poly}(|I|)$ is some polynomial in the encoding length of the input~\cite{DBLP:conf/soda/CslovjecsekEHRW21}. There exist generalizations of this result and also other tractable structures, but none of them captures the triangular shape of~(\ref{ILP_problem}).
We now consider a natural generalization of $n$-fold ILPs that also contains the triangle structure~(\ref{ILP_problem}), in the remainder referred to as \emph{triangle-fold} ILPs. \[ \text{maximize} \quad c^T x \quad \text{subject to} \quad \begin{pmatrix} A_1 & 0 & \cdots & 0\\ A_1 & A_2 & \ddots & \\ \vdots & & \ddots & 0 \\ A_1 & A_2 & \cdots & A_n \\ B_1 & 0 & & 0\\ 0 & B_2 & \ddots & \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & B_n \end{pmatrix} \cdot x \le b \quad \text{and} \quad \forall_i \ x_i \in \mathbb{Z}_{\geqslant 0}. \]
Let us elaborate on some of the design choices and why they come naturally. First, it is obvious that this structure generalizes $n$-fold, because constraints can be ``disabled'' by selecting a right-hand side of $\infty$ (or some sufficiently large number). After disabling a prefix of constraints, we end up with exactly the $n$-fold structure except that we have inequality ($\le$) constraints instead of equality constraints. Clearly, equality constraints can easily be emulated by duplicating and negating the constraints. The reason we chose inequality is that with equality constraints this ILP would directly decompose into independent small subproblems, which would be uninteresting from an algorithmic point of view and also not general enough to capture~(\ref{ILP_problem}). The matrices $B_1, \dotsc, B_n$ can, for example, be used to express lower and upper bounds on variables, such as the constraints $x_i \in \{0, 1\}$ in~(\ref{ILP_problem}).
With this formulation it is also notable that the scheduling problem on multiple machines, $Pm\,|\,|\,\sum p_j U_j$\xspace, can be modelled easily: instead of one decision variable $x_j \in \{0, 1\}$ for each job $j$, we introduce variables $x_{j,1},x_{j,2},\dotsc,x_{j,m} \in \{0, 1\}$, one for each machine. Then we use the constraints $p_1 x_{1,i} + p_2 x_{2,i} + \cdots + p_j x_{j,i} \le d_j$ for each machine $i$ and job $j$, which ensure that on each machine the selected jobs can be finished within their respective due date. Finally, we use constraints of the form $x_{j,1} + x_{j,2} + \cdots + x_{j,m} \le 1$ to guarantee that each job is scheduled on at most one machine. It can easily be verified that these constraints have the form of a triangle-fold where the small submatrices have dimension only dependent on $m$.
Given the positive results for $1\,|\,|\,\sum p_j U_j$\xspace and $Pm\,|\,|\,\sum p_j U_j$\xspace, one may hope to develop a general theory for triangle-folds. Instead of specific techniques for these (and potentially other) problems, in this way one could create general techniques that apply to many problems. For example, having an algorithm with the running time of the form $\max_i f(A_i, B_i) \cdot \mathrm{poly}(|I|)$ (like we have for $n$-folds), one would directly get an FPT algorithm for $Pm\,|\,|\,\sum p_j U_j$\xspace with parameters $p_{\max}$ and $m$. However, we show strong hardness results, which indicate that such a generalization is not possible. \begin{theorem}\label{th:trianglefold}
There exist fixed matrices $A_i, B_i$ of constant dimensions for which
testing feasibility of triangle-fold ILPs is NP-hard.
This holds even in the case when $A_1 = A_2 = \cdots = A_n$ and
$B_1 = B_2 = \cdots = B_n$. \end{theorem} This hardness result has a surprising quality: not only are there no FPT algorithms for triangle-folds, but also no XP algorithms, which is in stark contrast to other similarly shaped cases.
\section{Max-min skewed convolution} Recall that max-min skewed convolution is defined as the problem where, given vectors $a$, $b$ and $d$, we want to compute, for each $k = 0,1,\dotsc,2n-1$, the value \begin{equation*}
c[k] = \max_{i + j = k}\{\min\{a[i], b[j] + d[k]\} . \end{equation*} In this section we give an algorithm that solves this problem in time $O(n^{5/3}\log n)$. \begin{figure}
\caption{Matrix $M_k$ for $k = 6$ and vectors $a$ and $b$, which it is derived from.
The entry in cell for row $a[i]$ and column $b[j]$ describes whether there exist $i', j'$ with $i' + j' = k$, $a[i'] \ge a[i]$ and $b[j'] \ge b[j]$. For example, the entry for $a[2]$ and $b[2]$ is $1$ because $a[3] \geq a[2]$ and $b[3] \geq b[2]$.}
\label{fig:conv-preprocessing}
\end{figure} Assume without loss of generality that the values in $a$ and $b$ are all different. This can easily be achieved by multiplying all values (including those in $d$) by $4 n$ and adding a different number from $0,1,\dotsc,2n-1$ to each entry in $a$ and $b$. After computing the convolution we then divide the output numbers by $4n$ and round down to reverse the effect of these changes.
The algorithm consists of two phases. In the first phase we build a data structure to aid the computation of each $c[k]$. Then in the second phase we compute each element $c[k]$ separately with a binary search. The goal of the data structure is thus to efficiently answer queries of the form ``is $c[k] > v$?'' for some given $k$ and $v$.
First we introduce a quite excessive data structure. Building it explicitly is impossible within our running time goals, but it will help gain intuition. Suppose for every $k$ we have a matrix $M_k$ (see Figure~\ref{fig:conv-preprocessing}), where the rows correspond to the values $a[0],a[1],\dotsc,a[n-1]$, the columns correspond to $b[0],b[1],\dotsc,b[n-1]$, and the entries tell us whether, for some $a[i], b[j]$, there exist $i'$ and $j'$ with $i' + j' = k$, $a[i'] \ge a[i]$ and $b[j'] \ge b[j]$. This matrix contains enough information to answer the queries above: To determine whether $c[k] > v$, we need to check if there are $i', j'$ with $i' + j' = k$ and $a[i'] > v$ and $b[j'] > v - d[k]$. Choose the smallest $a[i]$ such that $a[i] > v$ and the smallest $b[j]$ such that $b[j] > v - d[k]$. If such $a[i]$ or $b[j]$ does not exist, then we already know that $c[k] \le v$. Otherwise, if both elements exist, the entry $M_k[a[i], b[j]]$ directly gives us the answer: If it is $0$, then for all $i' + j' = k$ either (1) $a[i'] < a[i]$ and hence (since $a[i]$ is the smallest element greater than $v$) $a[i'] \le v$, or (2) $b[j'] < b[j]$ and hence $b[j'] \le v - d[k]$; thus $c[k] \le v$. The converse is also true: Suppose that $c[k] \le v$. Then for all $i'$, $j'$ with $i' + j' = k$ we have that either $a[i'] \le v < a[i]$ or $b[j'] \le v - d[k] < b[j]$, and thus the matrix entry at $(a[i], b[j])$ is $0$.
It is obvious that we cannot afford to explicitly construct the matrices described above; their space alone would be cubic in $n$. Instead, we are only going to compute a few carefully chosen values of these matrices, that allow us to recover any other value in sublinear time. First, reorder the columns and rows so that the corresponding values ($a[i]$ or $b[j]$) are increasing. We call the resulting matrix $M^{\mathrm{sort}}_k$, see also Figure~\ref{fig:conv-preprocessing2}. Clearly, this does not change the information stored in it, but one can observe that now the rows and columns are each nonincreasing. We will compute only the values at intersections of every $\lfloor n/p \rfloor$-th row and every $\lfloor n / p \rfloor$-th column, for a parameter $p \in \mathbb N$, which is going to be specified later. This means we are computing a total of $O(n p^2)$ many values. Although computing a single entry of one of the matrices $M^{\mathrm{sort}}_k$ would require linear time, we will show that computing the same entry for all matrices (all $k$) can be done much more efficiently than in $O(n^2)$ time.
Indeed, for fixed $u, w \in \mathbb Z$, it takes only $O(n \log n)$ time to compute, for all $k$, whether there exists $i'$ and $j'$ with $i' + j' = k$, $a[i'] \ge u$ and $b[j'] \ge v$. This fact follows from a standard application of Fast Fourier Transformation (FFT): we construct two vectors $a', b'$ where \begin{equation*}
a'[i] = \begin{cases}
1 &\text{ if } a[i] \ge u, \\
0 &\text{ otherwise,}
\end{cases}
\quad
b'[i] = \begin{cases}
1 &\text{ if } b[j] \ge w, \\
0 &\text{ otherwise,}
\end{cases} \end{equation*} and compute their $(+, \cdot)$-convolution with FFT. For non-zero output entries there exist $i', j'$ as above, and for zero entries they do not. It follows that computing the selected $O(n p^2)$ values of $M^{\mathrm{sort}}_0, M^{\mathrm{sort}}_1, \ldots, M^{\mathrm{sort}}_{2n-2}$ can be done in time $O(p^2 \cdot n \log n)$. \begin{figure}\label{fig:conv-preprocessing2}
\end{figure}
We now consider the second phase, where the algorithm computes $c[k]$, for each $k$ separately, using binary search. To this end, we design a procedure (see Algorithm~\ref{alg:bs}) to determine, given $k \in \{0,1,2,\ldots,2n-2\}$ and $v \in \mathbb{Z}$, whether $c[k] > v$. The procedure will run in $O(n/p)$ time. As described in the beginning of the proof, this corresponds to computing the value of a specific cell $(a[i], b[j])$ in the matrix $M^{\mathrm{sort}}_k$. If this cell happens to be among the precomputed values, we are done. Otherwise, consider the $\lfloor n/p \rfloor \times \lfloor n/p \rfloor$ submatrix that encloses $(a[i], b[j])$ and whose corners are among the precomputed values. If the lower right corner $(a[i''], b[j''])$ is equal to one, then entry $(a[i], b[j])$ must also be one by monotonicity. Hence, assume otherwise. The entry $(a[i], b[j])$ could still be one, but this happens only if there is a \emph{witness} $(i', j')$ that satisfies $i' + j' = k$ and \begin{enumerate}
\item $a[i''] > a[i'] \ge a[i]$ and $b[j'] \ge b[j]$, or
\item $b[j''] > b[j'] \ge b[j]$ and $a[i'] \ge a[i]$. \end{enumerate} The number of possible witnesses for the first case is bounded by $n / p$, since there are only $\lfloor n/p \rfloor$ many values $a[i']$ between $a[i]$ and $a[i'']$ (since they are in the same $\lfloor n/p \rfloor \times \lfloor n/p \rfloor$ submatrix) and the corresponding $j'$ is fully determined by $i'$. Likewise, there are at most $n / p$ many possible witnesses for the second case. Hence, we can compute the value of the cell $(a[i], b[j])$ by exhaustively checking all these candidates for a witness, i.e., \[\{ (i', k-i') \mid a[i'] \in [a[i], a[i'']) \} \cup \{ (k-j', j') \mid b[j'] \in [b[j], b[j'']) \}.\]
\begin{algorithm} \caption{Procedure used in binary search to check whether $c[k] > v$.} \label{alg:bs} $\mathrlap{i}\hphantom{j} \longleftarrow \argmin \{a[i] \mid a[i] > v\}$\; $j \longleftarrow \argmin \{b[j] \mid b[j] > v - d[k]\}$\; \lIf{$i$ \rm{or} $j$ \rm{does not exist}}{\KwRet{\textsc{no}}} $(a[i''], b[j'']) \longleftarrow$ the closest precomputed cell below and to the right of $M^{\mathrm{sort}}_k[a[i], b[j]]$\; \lIf{$M^{\mathrm{sort}}_k[a[i''],b[j'']] = 1$}{\KwRet{\textsc{yes}}} \ForEach{$i' \in \{i' \mid a[i] \le a[i'] < a[i'']\}$}{
\lIf{$b[k - i'] \ge b[j]$}{\KwRet{\textsc{yes}}}} \ForEach{$j' \in \{j' \mid b[j] \le b[j'] < b[j'']\}$}{
\lIf{$a[k - j'] \ge a[i]$}{\KwRet{\textsc{yes}}}} \KwRet{\textsc{no}}\; \end{algorithm}
Finally, let us note that, for a fixed $k$, the value of $c[k]$ has to be among the following $2n$ values: $a[0], a[1], \ldots, a[n-1], b[0] + d[k], b[1] + d[k], \ldots, b[n-1] + d[k]$. A careful binary search only over these values makes the number of iterations logarithmic in $n$, and not in the maximum possible output value. Indeed, after the preprocessing, we already have access to sorted versions of the lists $a[0],a[1],\dotsc,a[n-1]$ and $b[0],b[1],\dotsc,b[n-1]$ and, in particular, to a sorted version of $b[0] + d[k], b[1] + d[k], \ldots, b[n-1] + d[k]$ (since this is just a constant offset of the latter list). We then first binary search for the lowest upper bound of $c[k]$ in $a[0],a[1],\dotsc,a[n-1]$ and then in $b[0] + d[k], b[1] + d[k], \ldots, b[n-1] + d[k]$ in order to determine the exact value of $c[k]$.
The total running time of both phases is $O(p^2 \cdot n \log(n) + n \cdot \log(n) \cdot n/p)$. We set $p = n^{1/3}$ in order to balance the two terms, and this gives the desired $O(n^{5/3} \log n)$ running time.
\section{Parameterizing by the maximum processing time} In this section we study algorithms for~$1\,|\,|\,\sum p_j U_j$\xspace with running time optimized for $p_{\max}$ and $n$ instead of $P$. We present an algorithm with a running time of $\widetilde O(n + p_{\max}^3)$. Such a running time is particularly appealing when $n$ is much larger than $p_{\max}$. This complements Lawler and Moore's algorithm with complexity $O(nP) \le O(n^2 p_{\max})$, which is fast when $n$ is small. Interestingly, in both cases we have roughly cubic dependence on $n + p_{\max}$.
Our result is based on a central structural property that we prove for an optimal solution and a sophisticated dynamic programming algorithm that recovers solutions of this form. The structural property uses exchange arguments that are similar to an approach used by Eisenbrand and Weismantel~\cite{eisenbrand2018proximity} to prove proximity results in integer programming via the Steinitz Lemma.
In the following, a solution is characterized by the subset of jobs that are finished within their due date. Once such a subset is selected, jobs in this subset can be scheduled in non-decreasing order sorted by their due date. In the remainder we assume that $d_1 \le d_2 \le \cdots \le d_n$. This sorting can be done efficiently: we may assume without loss of generality that all due dates are at most $np_{\max}$. Then radix sort requires only time $O(n \log_n (n p_{\max})) \le O(n + p_{\max})$. \begin{lemma}\label{lem:pmax-structur}
There exists an optimal solution $S\subseteq \{1,2,\dotsc,n\}$
such that for all $i=1,2,\dotsc,n$ it holds that either
\begin{enumerate}
\item $|\{1,2,\dotsc,i\} \cap S| < 2 p_{\max}$, or
\item $|\{i + 1, i + 2,\dotsc, n\} \setminus S| < 2 p_{\max}$.
\end{enumerate} \end{lemma} \begin{proof} Let $S$ be an optimal solution that does not satisfy this property for some $i$. It is easy to see that if we find some $A \subseteq S \cap \{1,2,\dotsc,i\}$ and $B \subseteq \{i+1,i+2,\dotsc,n\} \setminus S$ with the same volumes (that is, $\sum_{j\in A} p_j = \sum_{j\in B} p_j$), then $(S \setminus A) \cup B$ would form an optimal solution that is closer to satisfying the property. Note that the solution $(S \setminus A) \cup B$ is feasible since the due dates of the jobs in $B$ are strictly larger than the due dates of the jobs in $A$.
Since both 1.~and 2.~are false for $i$, we have that $A' = S \cap \{1,2,\dotsc,i\}$ and $B' = \{i+1, i+2,\dotsc,n\} \setminus S$ both have cardinality at least $2 p_{\max}$. We construct $A$ and $B$ algorithmically as follows. Starting with $A_1 = B_1 = \emptyset$ we iterate over $k = 1,2,\dotsc, 2 p_{max}$. In each iteration we check whether $\sum_{j\in A_k} p_j - \sum_{j\in B_k} p_j$ is positive or not. If it is positive, we set $B_{k+1} = B_k \cup \{j\}$ for some $j\in B'\setminus B_k$ and $A_{k+1} = A_k$; otherwise we set $A_{k+1} = A_k \cup \{j\}$ for some $j \in A'\setminus A_k$. The difference $\sum_{j\in A_k} p_j - \sum_{j\in B_k} p_j$ is always between $-p_{\max}$ and $p_{\max} - 1$. Hence, by pidgeon-hole principle there are two indices $k < h$ such that $\sum_{j\in A_k} p_j - \sum_{j\in B_k} p_j = \sum_{j\in A_h} p_j - \sum_{j\in B_h} p_j$. Since $A_k \subseteq A_h$ and $B_k \subseteq B_h$ by construction, we can simply set $A = A_h \setminus A_k$ and $B = B_h \setminus B_k$ and it follows that $\sum_{j\in A} p_j = \sum_{j\in B} p_j$. \end{proof}
\begin{corollary}\label{cor:pmax-structur}
There exists an optimal solution $S\subseteq \{1,2,\dotsc,n\}$ and an index $i\in\{1,2,\dotsc,n\}$
such that
\begin{enumerate}
\item $|\{1,2,\dotsc,i\} \cap S| \le 2 p_{\max}$, and
\item $|\{i + 1, i + 2,\dotsc, n\} \setminus S| < 2 p_{\max}$.
\end{enumerate} \end{corollary} \begin{proof}
Consider the solution $S$ as it Lemma~\ref{lem:pmax-structur} and let $i$ be the maximum index such that $|S \cap \{1,2,\dotsc,i\}| < 2 p_{\max}$. If $i = n$ the corollary's statement follows. Otherwise, we set $i' = i+1$. Then
$|S \cap \{1,2,\dotsc,i'\}| = 2 p_{\max}$ and by virtue of Lemma~\ref{lem:pmax-structur} it must hold that
$|S \setminus \{i' + 1, i' + 2,\dotsc, n\}| < 2 p_{\max}$. \end{proof}
Although we do not know the index $i$ from Corollary~\ref{cor:pmax-structur}, we can compute efficiently an index, which is equally good for our purposes. \begin{lemma}\label{lem:pmax-idx} In time $O(n)$ we can compute an index $\ell$ such that there exists an optimal solution $S$ with \begin{enumerate}
\item $p(\{1,2,\dotsc,\ell\} \cap S) \le O(p^2_{\max})$, and
\item $p(\{\ell+1,\ell+2,\dotsc,n\} \setminus S) \le O(p^2_{\max})$, \end{enumerate} where $p(X) = \sum_{i \in X} p_i$, for a subset $X \subseteq \{1, \ldots , n \}$. \end{lemma} \begin{proof} For $j=1,2,\dotsc,n$ let $t_j$ denote the (possibly negative) maximum time such that we can schedule all jobs $j,j+1,\dotsc,n$ after $t_j$ and before their respective due dates. It is easy to see that \begin{equation*}
t_j = \min_{j' \ge j} \bigg( d_{j'} - \sum_{j \le k \le j'} p_k\bigg) \ . \end{equation*} It follows that $t_{j} = \min\{t_{j+1}, d_j\} - p_{j}$ and thus we can compute all values $t_j$ in time $O(n)$. Now let $h$ be the biggest index such that $t_h < 0$. If such an index does not exist, it is trivial to compute the optimal solution by scheduling all jobs. Further, let $k < h$ be the biggest index such that $\sum_{j=k}^h p_j > 2 p_{\max}^2$ or $k = 1$ if no such index exists. Similarly, let $\ell > h$ be the smallest index such that $\sum_{j=h}^{\ell} p_j > 4 p_{\max}^2$ or $\ell = n$ if this does not exist. Let $i$ be the index as in Corollary~\ref{cor:pmax-structur}. We will now argue that either $k \le i \le \ell$ or $\ell$ satisfies the claim trivially. This finishes the proof, since $p(\{i, i+1, \dotsc, \ell\}) \le p(\{k, k+1, \dotsc, \ell\}) \le O(p_{\max}^2)$ and thus the properties in this lemma, which $i$ satisfies due to Corollary~\ref{cor:pmax-structur}, transfer to $\ell$.
As for $k\le i$, we may assume that $k > 1$ and therefore $\sum_{j=k}^{\ell} p_j > 2 p^2_{\max}$. Notice that there must jobs in $\{k,k+1,\dotsc,n\}$ of total volume more than $2 p_{\max}^2$, which are not in $S$. This is because $t_k < t_h - 2 p^2_{\max} < - 2 p^2_{\max}$. On the other hand, from Property~1 of Corollary~\ref{cor:pmax-structur} it follows that $p(\{i+1,i+2,\dotsc,n\}\setminus S) < 2 p_{\max}^2$. This implies that $k\le i$.
We will now show that $i \le \ell$ or $\ell$ satisfies the lemma's statement trivially. Assume w.l.o.g.\ that $\ell < n$ and thus $\sum_{j=h}^{\ell} p_j > 4 p^2_{\max}$. Notice that $t_\ell \ge t_h + 4 p_{\max}^2 \ge 2 p_{\max}^2 + p_{\max}$. If $S \cap \{1,2,\dotsc,\ell\}$ contains jobs of a total processing time more than $2 p_{\max}^2$, then $i\le \ell$ follows directly because of Corollary~\ref{cor:pmax-structur}. Conversely, if the total processing time is at most $2 p_{\max}^2$, then we can schedule all these jobs and also all jobs in $\{\ell + 1, \ell + 2, \dotsc, n\}$ (since $t_{\ell} \ge 2 p_{\max}^2$), which must be optimal. Hence, $\ell$ satisfies the properties of the lemma. \end{proof} An immediate consequence of the previous lemma is that we can estimate the optimum up to an additive error of $O(p_{\max}^2)$. We use this to perform a binary search with $O(\log(p_{\max}))$ iterations. In each iteration we need to check if there is a solution of at least a given value $v$. For this it suffices to devise an algorithm that decides whether there exists a solution that runs a subset of jobs without any idle time (between $0$ and $d_n$) or not. To reduce to this problem, we add dummy jobs with due date $d_n$ and total processing time $d_n - v$; more precisely, $\min \{p_{\max}, d_n - v\}$ many jobs with processing time $1$ and $\max\{0, \lceil (d_n - v - p_{\max}) / p_{\max} \rceil\}$ many jobs with processing time $p_{\max}$. Using that w.l.o.g.\ $d_n \le n p_{\max}$, we can bound the total number of added jobs by $O(n + p_{\max})$, which is insignificant for our target running time. If there exists a schedule without any idle time, we remove the dummy jobs and obtain a schedule where jobs with total processing time at least $d_n - (d_n - v) = v$ finish within their due dates. If on the other hand there is a schedule for a total processing time $v' \ge v$, we can add dummy jobs of size exactly $d_n - v'$ to arrive at a schedule without idle time. For the remainder of the section we consider this decision problem. We will create two data structures that allow us to efficiently query information about the two partial solutions from Lemma~\ref{lem:pmax-idx}, that is, the solution for jobs $1,2,\dotsc,\ell$ and that for $\ell+1,\ell+2,\dotsc,n$.
\begin{lemma}\label{lem:pmax-dyn1}
In time $O(n + p_{\max}^3 \log(p_{\max}))$ we can compute
for each $T = 1,2,\dotsc, O(p_{\max}^2)$ whether there is a subset
of $\{1,2,\dotsc,\ell\}$ which can be run exactly during $[0, T]$. \end{lemma} \begin{proof} We will efficiently compute for each $T = 0,1,\dotsc,O(p_{\max}^2)$ the smallest index $i_T$ such that there exists some $A_T\subseteq \{1,2,\dotsc,i_T\}$, which runs on the machine exactly for the time interval $[0,T]$. Then it only remains to check if $i_T \le \ell$ for each $T$.
Consider $i_T$ for some fixed $T$. Then $i_T\in A_T$, since otherwise $i_T$ would not be minimal. Further, we can assume that job $i_T$ is processed exactly during $[T - p_{i_T}, T]$, since $i_T$ has the highest due date among jobs in $A_T$. It follows that $i_{T'} < i_T$ for $T' = T - p_i$. Using this idea we can calculate each $i_T$ using dynamic programming. Assume that we have already computed $i_{T'}$ for all $T' < T$. Then to compute $i_T$ we iterate over all processing times $k=1,2,\dotsc,p_{\max}$. We find the smallest index $i$ such that $p_i = k$, $i_{T - k} < i$, and $d_i \ge T$. Among the indices we get for each $k$ we set $i_T$ as the smallest, that is, \[i_T = \min_{k \in [p_{\max}]} \min \, \{ i \mid p_i = k, i > i_{T-k}, d_i \ge T\} . \] The inner minimum can be computed using a binary search over the $O(p_{\max}^2)$ jobs with $p_i = k$ and smallest due date, since the values of $i_T$ do not change when restricting to the $O(p_{\max}^2) \ge T$ jobs with smallest due date. It follows that computing $i_T$ can be done in $O(p_{\max} \cdot \log(p_{\max}))$. Computing $i_T$ for all $T$ requires time $O(p_{\max}^3 \cdot \log(p_{\max}))$. \end{proof}
\begin{lemma}\label{lem:pmax-dyn2}
In time $O(n + p_{\max}^3 \log(p_{\max}))$ we can compute
for each $T = 1,2,\dotsc, O(p_{\max}^2)$ whether there is some
$A\subseteq\{\ell+1,\ell+2,\dotsc,n\}$,
such that $\{\ell+1,\ell+2,\dotsc,n\} \setminus A$
can be run exactly during $[d_n + T - \sum_{j=\ell+1}^n p_j, d_n]$. \end{lemma} \begin{proof} For every $T = 1,\dotsc,O(p_{\max}^2)$ determine $i_T$, which is the largest value such that there exists a set $A_T \subseteq \{i_T,i_T + 1,\dotsc,n\}$ with $\sum_{j\in A_T} p_j = T$, such that $\{i+1,i+2,\dotsc,n\} \setminus A_T$ can be run exactly during the interval $[d_n + T - \sum_{j=i+1}^n p_j, d_n]$. Consider one such $i_T$ and corresponding $A_T$. Let $j\in A_T$ has the minimal due date. Then for $T' = T - p_j$ we have $i_{T'} > i_T$. To compute $i_T$ we can proceed as follows. Guess the processing time $k$ of the job in $A_T$ with the smallest due date. Then find the maximum $j$ with $p_j = k$ and $j < i_{T - k}$. Finally, verify that there is indeed a schedule. For this consider the schedule of all jobs $\{\ell+1,\ell+2,\dotsc,n\}$, where we run them as late as possible (a schedule that is not idle in $[d_n - \sum_{i=\ell+1}^n p_i, d_n]$. Here, we look at the ``earliness'' of each job in $\{\ell+1,\dotsc,j-1\}$. This needs to be at least $T$ for each of the jobs.
We can check the earliness efficiently by precomputing the mentioned schedule and building a Cartesian tree with the earliness values. This data structure can be built in $O(n)$ and allows us to query for the minimum value of an interval in constant time. \end{proof} We can now combine Lemmas~\ref{lem:pmax-idx}-\ref{lem:pmax-dyn2} to conclude the algorithm's description. Suppose that $S$ is a set of jobs corresponding to a solution, where the machine is busy for all of $[0, d_n]$. By Lemma~\ref{lem:pmax-idx} there exist some $T, T' \le O(p_{\max}^2)$ such that $p(\{1,2,\dotsc,\ell\} \cap S) = T$ and $p(\{\ell+1,\ell+2,\dotsc,n\} \setminus S) = T'$. In particular, the machine will be busy with jobs of $\{1,2,\dotsc,\ell\} \cap S$ during $[0, T]$ and with jobs of $\{\ell + 1, \ell + 2,\dotsc,n\} \cap S$ during $[T, d_n]$. The choice of $T$ and $T'$ implies that \begin{equation*}
d_n = p(S) = p(\{1,\dotsc,\ell\} \cap S) + p(\{\ell + 1,\dotsc,n\} \cap S) = T + p(\{\ell + 1, \dotsc, n\}) - T' . \end{equation*} Hence, $T = d_n + T' - p(\{\ell + 1, \dotsc, n\})$. In order to find a schedule where the machine is busy between $[0, d_n]$ we proceed as follows. We iterate over every potential $T = 0,1,\dotsc,O(p_{\max}^2)$. Then we check using Lemma~\ref{lem:pmax-dyn1} whether there exists a schedule of jobs in $\{1,2,\dotsc,\ell\}$ where the machine is busy during $[0, T]$. For the correct choice of $T$ this is satisfied. Finally, using Lemma~\ref{lem:pmax-dyn2} we check whether there is a subset of jobs in $\{\ell+1,\ell+2, \dotsc, n\}$ that can be run during $[T, d_n] = [d_n + T' - p(\{\ell + 1, \dotsc, n\}), d_n]$. The preprocessing of Lemmas~\ref{lem:pmax-idx}-\ref{lem:pmax-dyn2} requires time $O(n + p^3_{\max} \log(p_{\max}))$. Then determining the solution requires only $O(p_{\max}^2)$, which is dominated by the preprocessing. Finally, notice that we lose another factor of $\log(p_{\max})$ due to the binary search.
\section{Multiple machines} In this section we present an algorithm that solves~$Pm\,|\,|\,\sum p_j U_j$\xspace in $n \cdot p^{O(m)}_{\max}$ time. We assume that jobs are ordered non-decreasingly by due dates (using radix sort as in the previous section), and that all considered schedules use earliest-due-date-first on each machine. We will now prove a structural lemma that enables our result. \begin{lemma}\label{lem:Pm}
There is an optimal schedule such that
for every job $j$ and every pair of machines $i, i'$ we have
\begin{equation}\label{eq:machines-balance}
|\ell_i(j) - \ell_{i'}(j)| \le O(p_{\max}^2) ,
\end{equation}
where $\ell_i(j)$ denotes the total volume of jobs $1,2,\dotsc,j$ scheduled on machine $i$.
Furthermore, the schedule satisfies, for every $j$ and $i$, that
\begin{equation}\label{eq:machines-lb}
\ell_i(j) \le \min_{j' > j}\left\{ d_{j'} - \frac{1}{m} \sum_{j''=j+1}^{j'} p_{j''} \right\} + O(p_{\max}^2) .
\end{equation} \end{lemma} \begin{proof}
The proof relies on swapping arguments.
As a first step we make sure that between $\ell_1(n),\dotsc,\ell_m(n)$
the imbalance is at most $p_{\max}$. If it is bigger, we simply move the
last job from the higher loaded machine to the lower loaded machine.
Now we augment this solution to obtain that, for every two machines $i$, $i'$ and every~$j$, we satisfy~\eqref{eq:machines-balance}.
Let $j$, $i$, and $i'$ be such that $\ell_i(j) > \ell_{i'}(j) + 3 p_{\max}^2$.
This implies that on $i'$ there is a set of jobs $A$
with due dates greater than $d_j$ that is processed between $\ell_{i'}(d_j)$
and at least $\ell_i(d_j) - p_{\max}$ (the latter because of the balance of total machine loads). Thus, $p(A) \ge 3 p^2_{\max} - p_{\max}$ and consequently $|A| \ge 2 p_{\max}$.
On $i$ let $B$ be the set of jobs with due dates at most
$d_j$ that are fully processed
between $\ell_{i'}(d_j)$ and $\ell_i(d_j)$. Then also
$p(B) \ge 3 p^2_{\max} - p_{\max}$ and $|B| \ge 2 p_{\max}$.
By the same arguments
as in Lemma~\ref{lem:pmax-structur}, there are non-empty
subsets $A'\subseteq A$ and $B' \subseteq B$ with equal processing time.
We swap $A'$ and $B'$, which improves the balance between $\ell_i(j)$
and $\ell_{i'}(j)$.
We now need to carefully apply these swaps so that after a finite number of them we have no large imbalances anymore.
We start with low values of $j$, balance all $\ell_i(j)$, and then increase $j$.
However, it might be that balancing
$\ell_i(j)$ and $\ell_{i'}(j)$ affects the values $\ell_i(j')$ for $j' < j$. To avoid this, we will use some buffer space. Notice that because
\begin{equation*}
\sum_i \ell_i(j) = \sum_{j' : \text{$j'$ is scheduled and } j' \le j} p_{j'} ,
\end{equation*}
it follows that a swap does not change the average $\ell_i(j)$ over all $i$.
If we already balanced for all $j' \le j$,
then we will skip some values $j'' \ge j$
and only establish (\ref{eq:machines-balance})
again for the first $j''$ such that
\begin{equation*}
\frac 1 m \sum_i \ell_i(j'') > \frac 1 m \sum_i[\ell_i(j)] + 6 p^2_{\max} .
\end{equation*}
It is not hard to see that the balance for job prefixes between $j$ and $j''$ then also holds (with a slightly worse constant).
To balance the values for $j''$, a careful look at the swapping procedure reveals that it suffices to move jobs, which are scheduled after
$\frac 1 m \sum_i[\ell_i(j'')] - 3 p^2_{\max}$.
Such jobs cannot have a due date lower than $d_j$,
since $\max_i \ell_i(j) \le \frac 1 m \sum_i [\ell_i(j)] + 3 p^2_{\max} \le \frac 1 m \sum_i \ell_i(j'') - 3 p^2_{\max}$. Hence, the swaps do not affect the values $\ell_i(j')$ for $j' \le j$.
Continuing these swaps, we can establish~\eqref{eq:machines-balance}.
Let us now consider~\eqref{eq:machines-lb}. Suppose that for some job $j$ and machine $i$ we have
\begin{equation*}
\ell_i(j) > \min_{j' > j}\left\{ d_{j'} - \frac{1}{m} \sum_{j''=j+1}^{j'} p_{j''} \right\} + \Omega(p_{\max}^2) \ .
\end{equation*}
With a sufficiently large hidden constant (compared to~\eqref{eq:machines-balance}) it follows that
there is a volume of at least $3p_{\max}^2$ of jobs $j'$ with $d_{j'} > d_j$ that are not scheduled
by this optimal solution. This follows simply from
considering the available space on the machines.
Also with a sufficiently large constant it holds that $\ell_i(j) > 3 p^2_{\max}$.
In the same way as earlier in the proof, we can find two non-empty sets of jobs $A'$ and $B'$ where
$A'$ consists only of jobs $j'$ with $d_{j'} > d_j$ that are not scheduled and $B'$ consists only of
jobs $j'$ with $d_{j'} \le d_j$, which are scheduled on machine $i$. We can swap these two sets and
retain an optimal solution. Indeed, this may lead to a new violation of~\eqref{eq:machines-balance}.
In an alternating manner we establish~\eqref{eq:machines-balance} and then perform a swap for~\eqref{eq:machines-lb}.
Since every time the latter swap is performed the average due dates of the scheduled jobs increases, the process
must terminate eventually. \end{proof} Having this structural property, we solve the problem by dynamic programming: for every $j = 1,2,\dotsc,n$ and every potential machine-load pattern $(\ell_1(j),\ell_2(j),\dotsc,\ell_m(j))$ we store the highest volume of jobs in $1,2,\dotsc,j$ that achieves these machine loads or lower machine loads on all machines. By Lemma~\ref{lem:Pm}, we may restrict ourselves to patterns where machine loads differ pairwise by only $O(p_{\max}^2)$, but this is not sufficient to obtain our desired running time. For each job $j$ we will only look at patterns where all machines $i$ satisfy \begin{equation*}
\ell_i(j) = \min_{j'>j} \left\{d_{j'} - \frac{1}{m}\sum_{j'>j} p_{j'} \right\} + k, \qquad \text{for } k \in \{-O(p_{\max}^2),\dotsc,O(p_{\max}^2)\}. \end{equation*} By the second part of Lemma~\ref{lem:Pm} it is clear that we can ignore larger values of $k$, but it needs to be clarified why we can ignore smaller values of $k$ as well. In the case that $\ell_i(j) < \min_{j'>j} \{d_{j'} - \frac{1}{m}\sum_{j'>j} p_{j'} \} - \Omega(p^2_{\max})$ for some $i$ and $j$ (with sufficiently large hidden constants), we may assume with~\eqref{eq:machines-balance} that all machines $i'$ satisfy $\ell_i(j) \le \min_{j'>j} \{d_{j'} - \frac{1}{m}\sum_{j''=j+1}^{j'} p_{j''} \} - p_{\max}$. It is not hard to see that starting with such a solution we can greedily add all remaining jobs $j'>j$ to the schedule without violating any due date. This is still true if we increase the machine loads until we reach $k = -O(p_{\max}^2)$. It is therefore not necessary to remember any lower load.
For each $j = 1,2,\dotsc,n$, the number of patterns $(\ell_1(j), \ell_2(j),\dotsc,\ell_m(j))$ can now be bounded by $p_{\max}^{O(m)}$, so there are in total $n \cdot p_{\max}^{O(m)}$ states in the dynamic program. Calculations for each state take $O(m)$ time, because job $j$ is either scheduled as the last job on one of the $m$ machines, or not scheduled at all. Hence, we obtain an $n \cdot p_{\max}^{O(m)}$ running time.
\section{Hardness of ILP with triangular block structure}
In this section we will show that it is NP-hard to decide if there is a feasible solution to an integer linear program of the form
\[ \begin{pmatrix} A & 0 & \cdots & 0\\ A & A & \ddots & \vdots \\ \vdots & & \ddots & 0 \\ A & A & \cdots & A \end{pmatrix} \cdot \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} \leqslant \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix}, \quad \forall_i \ 0 \leq x_i \leq u_i, \quad \forall_i \ x_i \in \mathbb{Z}, \] even when $A$ is a matrix of constant size with integers entries of at most a constant absolute values, $b_i \in \{0, 1, +\infty\}$ for $i = 1,2,\dotsc,m$, and $u_i \in \{0, +\infty\}$ for $i=1,2,\dotsc,n$. This implies Theorem~\ref{th:trianglefold} by taking $B_1, B_2, \dotsc, B_n$ each as an identity matrix and a negated identity matrix on top of each other, which allows us to easily implement the bounds on the variables.
We will give a reduction from the Subset Sum problem, which asks, given $n$ nonnegative integers $a_1, a_2, \ldots, a_n \in \mathbb{Z}_{\geqslant 0}$, and a target value $t \in \mathbb{Z}_{\geqslant 0}$, whether there exists a subset of $\{a_1, a_2, \ldots, a_n\}$ that sums up to $t$. The Subset Sum problem is weakly NP-hard~\cite{Karp72}, so the reduction will have to to deal with $n$-bit input numbers.
Before we describe the actual reduction, let us start with two general observations that we are going to use multiple times later in the proof.
First, even though the theorem statement speaks, for clarity, about an ILP structure with only ``$\leqslant$'' constraints, we can actually have all three types of constraints, i.e., ``$\leqslant$'', ``$=$'', and ``$\geqslant$''. Indeed, it suffices to append to the matrix $A$ a negated copy of each row, in order to be able to specify for each constraint not only an upper bound but also a lower bound (and use $+\infty$ when only one bound is needed). An equality constraint can then be expressed as an upper bound and a lower bound with the same value.
Second, even though the constraint matrix has a very rigid repetitive structure, we can selectively cancel each individual row, by setting the corresponding constraint to ``$\leq +\infty$'', or each individual column -- by setting the upper bound of the corresponding variable to $0$.
For now, let us consider matrix $A$ composed of four submatrices $B$, $C$, $D$, $E$, arranged in a $2 \times 2$ grid, as follows: \[ A = \begin{pmatrix} B & C \\ D & E \end{pmatrix}; \quad \begin{pmatrix} A & & & & \\ A & A & & & \\ A & A & A & & \\ \vdots & & &\ddots \end{pmatrix} = \begin{pmatrix} B & C & & & & & \\ D & E & & & & & \\ B & C & B & C & & & \\ D & E & D & E & & & \\ B & C & B & C & B & C & \\ D & E & D & E & D & E & \\ \vdots & & & & & & \ddots \\ \end{pmatrix}. \]
In every odd row of $A$'s we will cancel the bottom $(D, E)$ row, and in every even row -- the upper $(B, C)$ row, and similarly for columns, so that we obtain the following structure of the constraint matrix:
\[ \begin{pmatrix} B & \tikzmark{ta} C & \tikzmark{tb} & & & \tikzmark{tc} & \tikzmark{td} & & \\ \tikzmark{la} D & E & & & & & & & \tikzmark{ra} \\ \tikzmark{lb} B & C & B & C & & & & & \tikzmark{rb} \\ D & E & D & E & & & & & \\ B & C & B & C & B & C & & & \\ \tikzmark{lc}D & E & D & E & D & E & & & \tikzmark{rc} \\ \tikzmark{ld}B & C & B & C & B & C & B & C & \tikzmark{rd} \\ D & E & D & E & D & E & D & E & \\ \vdots & \tikzmark{ba} & \tikzmark{bb} & & & \tikzmark{bc} & \tikzmark{bd} & & \ddots \\ \end{pmatrix} \cong \begin{pmatrix} B & & & & \\ D & E & & & \\ B & C & B & & \\ D & E & D & E & \\ \vdots & & & & \ddots \end{pmatrix} \]
\begin{tikzpicture}[overlay,remember picture] \draw ($(pic cs:la)+(0pt,5pt)$) -- ($(pic cs:ra)+(0pt,5pt)$); \draw ($(pic cs:lb)+(0pt,5pt)$) -- ($(pic cs:rb)+(0pt,5pt)$); \draw ($(pic cs:lc)+(0pt,5pt)$) -- ($(pic cs:rc)+(0pt,5pt)$); \draw ($(pic cs:ld)+(0pt,5pt)$) -- ($(pic cs:rd)+(0pt,5pt)$); \draw ($(pic cs:ta)+(4pt,11pt)$) -- ($(pic cs:ba)+(0pt,0pt)$); \draw ($(pic cs:tb)+(0pt,11pt)$) -- ($(pic cs:bb)+(0pt,0pt)$); \draw ($(pic cs:tc)+(0pt,11pt)$) -- ($(pic cs:bc)+(0pt,0pt)$); \draw ($(pic cs:td)+(0pt,11pt)$) -- ($(pic cs:bd)+(0pt,0pt)$); \end{tikzpicture}
Now, let us set the four submatrices of $A$ to be $1 \times 1$ matrices with the following values. \[ B = \begin{pmatrix} 1 \end{pmatrix}, \quad C = \begin{pmatrix} -2 \end{pmatrix}, \quad D = \begin{pmatrix} 1 \end{pmatrix}, \quad E = \begin{pmatrix} -1 \end{pmatrix}. \]
Let us consider a constraint matrix composed of $2n$ block-rows and $2n$ block-columns. We set the first constraint to ``$\leq 1$'', and all the remaining constraints to ``$=0$''. Let us denote the variables corresponding to the first column of $A$ by $y_1, y_2, \ldots, y_n$, and those to the second column by $z_1, z_2, \ldots, z_n$. We have the following ILP:
\[ \begin{pmatrix} 1 & & & & & & \\ 1 & -1 & & & & & \\ 1 & -2 & 1 & & & & \\ 1 & -1 & 1 & -1 & & & \\ 1 & -2 & 1 & -2 & 1 & & \\ 1 & -1 & 1 & -1 & 1 & -1 & \\ \vdots & & & & & & \ddots \end{pmatrix} \cdot \begin{pmatrix} y_1 \\ z_1 \\ y_2 \\ z_2 \\ y_3 \\ z_3 \\ \vdots \end{pmatrix}\quad\begin{matrix} \leqslant \\ = \\ = \\ = \\ = \\ = \\ \vdots \end{matrix}\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \vdots \end{pmatrix} \]
Observe that $z_i = y_i$ and $y_{i+1} = y_1 + \cdots + y_i$ for every $i$. Since $y_1 \in \{0, 1\}$, it is easy to verify that there are exactly two solutions to this ILP. Indeed, either $y_i = z_i = 0$ for every $i$, or $y_1 = z_1 = 1$ and $y_{i+1}=z_{i+1} = 2^i$ for every $i$. In other words, either $z = (0,0,0,\ldots)$, or $z = (1, 1, 2, 4, 8, \ldots)$. We will call these two solutions \emph{all-zeros} and \emph{powers-of-two}, respectively.
Now, let us add one more column, namely $(1, 0)$, to matrix $A$, which therefore looks now as follows: \[ A = \begin{pmatrix} 1 & -2 & 1 \\ 1 & -1 & 0 \end{pmatrix}. \] The newly added column (and the corresponding variable) shall be cancelled (by setting the corresponding upper bound to $0$) in all but the last copy of $A$, which in turn shall have the other two columns cancelled. Let us call $w$ the variable corresponding to the only non-cancelled copy of the $(1, 0)$ column. Both solutions to the previous ILP extend to the current one, with $w = \sum_i z_i$. Note that, in both solutions, both $\sum_i (y_i - 2 z_i) + w = 0$, and $\sum_i (y_i - z_i) = 0$. Therefore, if we append another copy of the ILP to itself, as follows, \[ \begin{pmatrix} \tikzmark{startofmatrix} 1 & & & & & & & & & \\ 1 & -1 & & & & & & & & \\ 1 & -2 & 1 & & & & & & &\\ 1 & -1 & 1 & -1 & & & & & &\\ \vdots & & & & \ddots & & & & & \\ 1 & -2 & 1 & -2 & \cdots & 1 & & &\\ 1 & -1 & 1 & -1 & \cdots & 0 \tikzmark{endoffirst} & & &\\ 1 & -2 & 1 & -2 & \cdots & 1 & \tikzmark{startofsecond} 1 & &\\ 1 & -1 & 1 & -1 & \cdots & 0 & 1 & -1 &\\ \vdots & & & & & & & & \ddots \tikzmark{endofmatrix} \end{pmatrix} \cdot \begin{pmatrix} \tikzmark{startofvars} y_1 \\ z_1 \\ y_2 \\ z_2 \\ \vdots \\ w \tikzmark{endoffirstvars} \\ \tikzmark{startofsecondvars} y'_1 \\ z'_1 \\ \vdots \tikzmark{endofvars} \end{pmatrix}\quad\begin{matrix} \tikzmark{startofcons} \leqslant \\ = \\ = \\ = \\ \vdots \\ = \\ = \\ \tikzmark{startofsecondcons} \leqslant \\ = \\ \vdots \end{matrix}\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ \vdots \\ 0 \\ 0 \tikzmark{endoffirstcons} \\ 1 \\ 0 \\ \vdots \tikzmark{endofcons} \end{pmatrix}, \] the two copies are independent from each other, and we get an ILP that has exactly four feasible solutions: both $z$ and $z'$ can be either all-zeros or powers-of-two, independently, giving four choices in total. \begin{tikzpicture}[overlay,remember picture] \draw[dashed] ($(pic cs:startofmatrix)+(-2pt,.9em)$) rectangle ($(pic cs:endoffirst)+(2pt,-2pt)$); \draw[dashed] ($(pic cs:startofsecond)+(-2pt,.9em)$) rectangle ($(pic cs:endofmatrix)+(2pt,-2pt)$);
\draw[dashed] ($(pic cs:startofvars)+(-10pt,.85em)$) rectangle ($(pic cs:endoffirstvars)+(10pt,-1pt)$); \draw[dashed] ($(pic cs:startofsecondvars)+(-10pt,.85em)$) rectangle ($(pic cs:endofvars)+(13pt,-1pt)$);
\draw[dashed] ($(pic cs:startofcons)+(-3pt,.85em)$) rectangle ($(pic cs:endoffirstcons)+(10pt,-1pt)$); \draw[dashed] ($(pic cs:startofsecondcons)+(-3pt,.85em)$) rectangle ($(pic cs:endofcons)+(13pt,-1pt)$); \end{tikzpicture}
Let $n$ be the number of elements in the Subset Sum instance we reduce from. We copy the above construction $n+1$ times, and we will call each copy a \emph{super-block}. In the last super-block we change the first constraint from ``$\leqslant 1$'' to ``$=1$'', effectively forcing the powers-of-two solution. Therefore, the resulting ILP has exactly $2^n$ feasible solutions -- two choices for each of the first $n$ super-blocks, one choice for the last super-block. We will denote by $z_{i,j}$ the $j$-th $z$-variable in the $i$-th super-block.
Now, we replace the $z$-column of $A$ with three identical copies of it, and each variable $z_{i,j}$ with three variables $p_{i,j}$, $q_{i,j}$, $r_{i,j}$.
For each $i$, $j$ we will set to $0$ exactly two out of the three upper bounds of $p_{i,j}$, $q_{i,j}$, $r_{i,j}$. Therefore, the solutions of the ILP after the replacement map one-to-one to the solutions of the ILP before the replacement, with $z_{i,j} = p_{i,j} + q_{i,j} + r_{i,j}$. Let $a_1, a_2, \ldots, a_n \in \mathbb{Z}_{\geqslant 0}$ be the elements in the Subset Sum instance we reduce from, and let $t \in \mathbb{Z}_{\geqslant 0}$ be the target value. The upper bounds are set as follows. For every $i$ and for $j=1$, we set $p_{i,1} \leqslant +\infty$ and $q_{i,1}, r_{i,1} \leqslant 0$. For $i = 1, 2, \ldots, n$, we set \begin{align*} &p_{i,j} \leqslant + \infty &&\text{and} &&q_{i,j} \leqslant 0 &&\text{if the $(j-1)$-th bit of $a_i$ is zero, and}\\ &p_{i,j} \leqslant 0 &&\text{and} &&q_{i,j} \leqslant +\infty &&\text{if the $(j-1)$-th bit of $a_i$ is one};\end{align*} in both cases $r_{i,j} \leqslant 0$. For $i = n + 1$ we look at $t$ instead of $a_i$, and we swap the roles of the $q$-variables and $r$-variables, i.e., we set \begin{align*} &p_{n+1,j} \leqslant + \infty &&\text{and} &&r_{n+1,j} \leqslant 0 &&\text{if the $(j-1)$-th bit of $t$ is zero, and}\\ &p_{n+1,j} \leqslant 0 &&\text{and} &&r_{n+1,j} \leqslant +\infty &&\text{if the $(j-1)$-th bit of $t$ is one};\end{align*} and in both cases $q_{n+1,j} \leqslant 0$.
Note that, for $i = 1, 2, \ldots, n$, depending on whether the part of the solution corresponding to the $i$-th super-block is the all-zeros or powers-of-two, $\sum_j q_{i,j}$ equals either $0$ or $a_i$. Hence, the set of the sums of all $q$-variables over all feasible solutions to the ILP is exactly the set of Subset Sums of $\{a_1, a_2, \ldots, a_n\}$. Moreover, $\sum_j r_{n+1,j} = t$.
We need one last step to finish the description of the ILP. We add to matrix A row $(0, 0, 1, -1, 0)$, so it looks as follows: \[ A = \begin{pmatrix} 1 & -2 & -2 & -2 & 1 \\ 1 & -1 & -1 & -1 & 0 \\ 0 & 0 & 1 & -1 & 0 \end{pmatrix}. \]
The newly added row (and the corresponding constraint) shall be cancelled (by setting the constraint to ``$\leqslant +\infty$'') in all but the last row of the whole constraint matrix. That last constraint in turn shall be set to ``$= 0$'', so that we will have $\sum_i \sum_j q_{i,j} - \sum_i \sum_j r_{i,j} = 0$, i.e., $\sum_i \sum_j q_{i,j} = t$. Hence, the final constructed ILP has a feasible solution if and only if there is a choice of all-zeros and powers-of-two solutions for each super-block that corresponds to a choice of a subset of $\{a_1, a_2, \ldots, a_n\}$ that sums up to $t$. In other words, the ILP has a feasible solution if and only if the Subset Sum instance has a feasible solution.
Finally, let us note that the ILP has $O(n^2)$ variables, and the desired structure.
\end{document}
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arXiv
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\begin{document}
\preprint{APS/123-QED}
\title{State mapping and discontinuous entanglement transfer in a multipartite open system} \author{Matteo Bina}
\affiliation{Dipartimento di Fisica, Universit\`{a} di Milano, I-20133 Milano, Italy}
\affiliation{CNISM, UdR Milano, I-20133, Milano, Italy} \author{Federico Casagrande} \affiliation{Dipartimento di Fisica, Universit\`{a} di Milano, I-20133 Milano, Italy} \affiliation{CNISM, UdR Milano, I-20133, Milano, Italy} \email{[email protected]} \author{Marco G. Genoni} \affiliation{Dipartimento di Fisica, Universit\`{a} di Milano, I-20133 Milano, Italy} \affiliation{CNISM, UdR Milano, I-20133, Milano, Italy} \author{Alfredo Lulli} \affiliation{Dipartimento di Fisica, Universit\`{a} di Milano, I-20133 Milano, Italy} \affiliation{CNISM, UdR Milano, I-20133, Milano, Italy} \author{Matteo G. A. Paris} \affiliation{Dipartimento di Fisica, Universit\`{a} di Milano, I-20133 Milano, Italy} \affiliation{CNISM, UdR Milano, I-20133, Milano, Italy} \affiliation{ISI Foundation, I-10133, Torino, Italy}
\date{\today}
\begin{abstract} We describe the transfer of quantum information and correlations from an entangled tripartite bosonic system to three separate qubits through their local environments also in the presence of various dissipative effects. Optimal state mapping and entanglement transfer are shown in the framework of optical cavity quantum electrodynamics involving qubit-like radiation states and two-level atoms via the mediation of cavity modes. For an input GHZ state mixed with white noise we show the occurrence of sudden death and birth of entanglement, that is discontinuously exchanged among the tripartite subsystems. \end{abstract} \pacs{03.67.Mn,42.50.Pq} \maketitle
\section{Introduction} As early as in 1935 Einstein, Podolski, and Rosen \cite{EPR} as well as Schr\"{o}dinger \cite{Sch} drew the attention on the correlations in quantum composite systems and the problems raised by their properties. Much later, theoretical \cite{Bell} and experimental \cite{Aspect} cornerstones elucidated the issue of nonlocality. Entanglement is currently viewed as the key resource for quantum information (QI) processing \cite{Nielsen}, where it allowed a number of achievements such as teleportation \cite{Bennett}, cryptography \cite{Gisin} and enhanced measurements \cite{entame}. The deep meaning of multipartite entanglement, its quantification and detection \cite{Guhne1}, the possible applications, are the object of massive investigation. As a matter of fact, optical systems have been a privileged framework for encoding and manipulating quantum information, since bipartite and multipartite entanglement may be effectively generated either in the discrete or continuous variable regime. On the other hand, the development of QI also requires localized registers, e.g. for the storage of entanglement in quantum memories. \\\indent Cavity quantum electrodynamics (CQED) \cite{Haroche_Raimond} is a relevant scenario for this kind of investigations and has been addressed for the analysis of entanglement transfer of bipartite \cite{P04,P204,P304,mem,Hald00,Son02,Zou06,Cas07,CLP2008} and multipartite \cite{Paternostro,Casagrande,Cirac,Ser06} entanglement. In this framework we present a complete study on the entanglement dynamics of a nine parties open system whose implementation could be feasible in the optical regime of CQED \cite{Nussmann,ions}. In particular we describe a system where three radiation modes, prepared in qubit-like entangled states, are coupled by optical fibers to three separate optical cavities each of them containing a trapped two-level atom. This paradigmatic example allows us investigating multipartite entanglement transfer and swapping in a more realistic way than in \cite{Paternostro,Casagrande}, shedding light on fundamental processes related to quantum interfaces and memories in quantum networks \cite{Cirac,Ser06}. \\ \indent We demonstrate that a complete mapping of pure entangled states onto the tripartite atomic subsystem occurs when the external field has fed the cavities. If this field is then switched off, the quantum correlations can be periodically mapped onto the tripartite atomic and cavity mode subsystems according to a triple Jaynes-Cummings (JC) dynamics \cite{JC}. In the case of external radiation prepared in a mixed Werner state we deal with the recently observed phenomenon of entanglement sudden death (ESD) (and birth (ESB)) \cite{YuPRL,Almeida}, that in our case involves the abrupt vanishing (and raising) of quantum correlations in tripartite systems. Though this is in general a still open problem, nevertheless we can show the occurrence of discontinuous exchange of tripartite entanglement via ESD effects. We also describe the dissipative effects introduced by the presence of external environments, such as the decay of cavity modes, atomic excitations, and fiber modes. Some results are then reported in the case that the coupling between external and cavity mode fields is generalized from monomode to multimode.\\ \indent In Sec. II we introduce the model of the physical system. In Sec. III we derive all main results concerning state mapping, entanglement transfer, and entanglement sudden death, also adding the presence of external environments. The case of multimode external-cavity field coupling is addressed in Sec. IV. Some conclusive remarks are reported in Sec. V.
\section{Model of the physical system} \indent We consider an entangled tripartite bosonic system (f), prepared in general in a mixed state, interacting with three qubits (a) through their local environments (c) also in the presence of dissipative effects. In the interaction picture the system Hamiltonian has the form: \begin{eqnarray} \label{H_int} \hat{\mathcal{H}}^I=\hbar \big\{\sum_{J=A,B,C}\big[ g_J(\hat{c}_J\hat{\sigma}^{\dag}_J+\hat{c}^{\dag}_J\hat{\sigma}_J)\big]+\,\nonumber\\ +\sum_{J,K=A,B,C}\big[\nu_{J,K}(t)(\hat{c}_J\hat{f}^{\dag}_K+\hat{c}^{\dag}_J\hat{f}_K)\big]\big\} \end{eqnarray} The operators $\hat{c}_J,\hat{c}^{\dag}_J$ ($\hat{f}_J,\hat{f}_J^{\dag}$) are the annihilation and creation operators for the cavity (external radiation) modes, while $\hat{\sigma}_J,\hat{\sigma}_J^{\dag}$ are the raising and lowering operators for the atomic qubits in each subsystem ($J=$A,B,C). We consider real coupling constant $g_J$ for the atom-cavity mode interaction and $\nu_{J,K}(t)$ for the interaction of each cavity mode with the three modes of driving external radiation. We take time dependent constants in order to simulate the interaction switching-off at a suitable time $\tau_{off}$ for the external field. Now we take into account three processes of dissipation: the cavity losses at rate $\kappa_c$ due to interaction with a thermal bath with a mean photon number $\bar{n}$, the atomic spontaneous emission with a decay rate $\gamma_a$ for the upper level, and a loss of photons inside the fibers at a rate $\kappa_f$. All dissipative effects can be described under the Markovian limit by standard Liouville superoperators so that the time evolution of the whole system density operator $\hat{\rho}(t)$ can be described by the following ME written in the Lindblad form: \begin{eqnarray}\label{ME} \dot{\hat{\rho}}&=&-\frac{i}{\hbar}\left [\hat{\mathcal{H}}_e,\hat{\rho}\right ]+\sum_{J=A,B,C} \big[\hat{C}_{f,J}\hat{\rho}\hat{C}_{f,J}^{\dag}+\hat{C}^{(g)}_{c,J}\hat{\rho}\hat{C}_{c,J}^{(g)\dag}+\,\nonumber\\ &+&\hat{C}^{(l)}_{c,J}\hat{\rho}\hat{C}_{c,J}^{(l)\dag}+\hat{C}_{a,J}\rho\hat{C}_{a,J}^{\dag}\big] \end{eqnarray} where the non-Hermitian effective Hamiltonian is \begin{eqnarray}\label{He} \hat{\mathcal{H}}_e&=&\hat{\mathcal{H}}^I-\frac{i\hbar}{2}\sum_{J}\big[\hat{C}^{\dag}_{f,J}\hat{C}_{f,J}+\hat{C}^{(g)\dag}_{c,J}\hat{C}^{(g)}_{c,J}+\,\nonumber\\ &+&\hat{C}^{(l)\dag}_{c,J}\hat{C}^{(l)}_{c,J}+\hat{C}^{\dag}_{a,J}\hat{C}_{a,J}\big]. \end{eqnarray} The jump operators for the atoms are $\hat{C}_{a,J}=\sqrt{\gamma_a}\hat{\sigma}_J$, for the fiber losses $\hat{C}_{f,J}=\sqrt{\kappa_f}\hat{f}_J$, and for the cavity modes $\hat{C}_{c,J}^{(l)}=\sqrt{\kappa_c(\bar{n}+1)}\hat{c}_J$ (loss of a photon) and $\hat{C}_{c,J}^{(g)}=\sqrt{\kappa_c\bar{n}}\hat{c}^{\dag}_J$ (gain of a photon). The above ME can be solved numerically by the Monte Carlo Wave Function method \cite{MCWF}. From now on we consider dimensionless parameters, all scaled to the coupling constant $g_A$, and times $\tau=g_At$.\\ \indent As a significative example, the implementation of our scheme may be realized in the optical regime of CQED by choosing a continuous variable (CV) entangled field for the subsystem (f) and two-level atoms as qubits (a). Each qubit is trapped in a one-sided optical cavity (c), where the radiation modes can be coupled to the cavity modes via optical fibers as in \cite{SMB}. In optical cavities thermal noise is negligible ($\bar{n}\cong 0$), spontaneous emission can be effectively suppressed, and single atoms can remain trapped even for several seconds \cite{Nussmann}.\\ \indent Here we focus on external field prepared in a qubit-like entangled state $\hat{\rho}_f(0)$ because this is the condition for high entanglement transfer for CV field \cite{Casagrande}. The generation of photon number multimode entangled radiation was recently demonstrated \cite{Papp}. Under qubit-like behavior we can describe the entanglement of all three-qubit subsystems (a, c, f) by combining the information from tripartite negativity \cite{Sabin}, entanglement witnesses \cite{Acin} for the two inequivalent classes GHZ and W \cite{Dur}, and recently proposed criteria for separability \cite{Ghune2}. In fact, the tripartite negativity $E^{(\alpha)} (\tau)$($\alpha$=a, c, f), defined as the geometric mean of the three bipartite negativities \cite{Vidal}, is an entanglement measure providing only a sufficient condition for entanglement detection, though its positivity guarantees the GHZ-distillability that is an important feature in QI.
\section{State mapping and tripartite entanglement transfer for single mode coupling} \subsection{Hamiltonian regime for external field in qubit-like pure states} \indent We first illustrate the Hamiltonian dynamics ($\{\tilde{\kappa}_f,\tilde{\kappa}_c,\tilde{\gamma}_a\}\ll 1$) for the external field prepared in a qubit-like entangled pure state $\ket{\Psi(0)}_f$, atoms prepared in the lower state $\ket{ggg}_a$, and cavities in the vacuum state $\ket{000}_c$ . By choosing $\nu_{JK}(\tau)=0$ if $J\neq K$ and $\nu_{J,J}=g_A$ we describe single-mode fibers each one propagating a mode of the entangled radiation. We show that under optimal conditions for mode matching it is possible to map $\ket{\Psi(0)}_f$ onto atomic and cavity mode states for suitable interaction times. Overall we are dealing with an interacting 9-qubit system, though the input field will be switched off at a time $\tau_{off}$ such that the atomic probability of excited state $p_e(\tau)$ reaches the maximum. Injected field switch-off can be obtained, e.g., by rotating fiber polarization.\\ \indent In Fig.~\ref{fig:fig1} we show numerical results for the external field prepared in the GHZ state $\ket{\Psi(0)}_f=(\ket{000}_f+\ket{111}_f)/\sqrt{2}$. In the time interval $0<\tau \leq\tau_{off}$ (transient regime) each flying qubit transfers its excitation to the cavity which in turn passes it onto the atom (see Fig.~\ref{fig:fig1}a). The cavity mode, simultaneously coupled to the external field and to the atom, exchanges energy according to a Tavis-Cummings dynamics at an effective frequency $g\sqrt{2}$ \cite{Haroche_Raimond,TC}. During the transient up to time $\tau_{off}=\pi/\sqrt{2}$ the mean photon number $N^{(c)}(\tau)\equiv\langle \hat{c}^\dag \hat{c}\rangle(\tau)$ in each cavity completes a cycle. In the same period the atomic excitation probability $p_e(\tau)$ reaches its maximum value, while the input field has completely fed the cavity, i.e., its mean photon number $N^{(f)}(\tau)\equiv\langle \hat{f}^\dag \hat{f}\rangle(\tau)$ vanishes. \begin{figure}
\caption{Dynamics for the external field in a GHZ state: (a) $N^{(c)}$(dashed), $N^{(f)}$(dotted) and $p_e$(solid); (b) $E^{(\alpha)}$ for atoms (solid), cavity modes (dashed) and external field (dotted); (c) $\mu^{(a)}$ (solid) and $F_{\phi}^{(a)}$ with $\phi=0$ (dashed), $\phi=\pi$ (dotted); (d) $\mu^{(c)}$ (solid) and $F_{\phi}^{(c)}$ with $\phi=-\pi/2$ (dashed), $\phi=+\pi/2$ (dotted).}
\label{fig:fig1}
\end{figure} In Fig.~\ref{fig:fig1}b we show that in the transient the atomic tripartite negativity is always positive and $E^{(a)}(\tau_{off})=1$, that is the value of the injected GHZ state. Until $\tau_{off}$ the dynamics maps the whole initial state $\ket{\Psi(0)}_f\otimes\ket{000}_c\otimes\ket{ggg}_a$ onto the pure state $\ket{000}_f\otimes\ket{000}_c\otimes \ket{\Psi(0)}_a$, where $\ket{\Psi(0)}_a$ is obtained from $\ket{\Psi(0)}_f$ by the correspondence $\ket{0}_f\rightarrow\ket{g}_a$ and $\ket{1}_f\rightarrow\ket{e}_a$. This is confirmed in Fig.~\ref{fig:fig1}c by the time evolution of the purity $\mu^{(a)}(\tau)=\hbox{Tr}_{a}[\hat{\rho}^2_{\alpha}(\tau)]$ and the fidelity $F^{(a)}(\tau)={}_{a} \meanvalue{\Psi(0)}{\hat{\rho}_{a}(\tau)}{\Psi(0)}_{a}$, where $\hat{\rho}_{a}(\tau)$ is the atomic reduced density operator. As for the cavity mode dynamics we note that (see Fig.~\ref{fig:fig1}b,d) the local maximum of $E^{(c)}(\tau_{off}/2)$ does not correspond to a pure state, i.e. the initial state $\ket{\Psi(0)}_f$ cannot be mapped onto the cavity modes during the transient regime. The entanglement is only partially transferred to the cavity modes nevertheless allowing the building up of full atomic entanglement later on. This dynamics is quite different than in \cite{Casagrande} where the entangled field was mapped onto the cavity modes before the interaction with the atoms.\\ \indent At the end of the transient regime the external radiation is turned off and the subsequent dynamics is described by a triple JC ruled by oscillations at the vacuum Rabi frequency $2g$, hence with a dimensionless period $\pi$ as shown by cavity mean photon number and atomic probability in Fig.~\ref{fig:fig1}a. The purities $\mu^{(a,c)}(\tau)$ in Figs.~\ref{fig:fig1}c,d oscillate at a double frequency between pure entangled (maximum negativity) and separable (zero negativity) states. In particular, at times $\tau_m=\tau_{off}+m\pi$ ($m=0,1,2...)$ the atoms are in the entangled states $\hat{U}^{(a)}_{\phi}\ket{\Psi(0)}_a$, where $\hat{U}^{(a)}_{\phi}=\bigotimes_{J}e^{-i\phi\hat{\sigma}_J^{\dag}\hat{\sigma}_{J}}$ is a local phase operator where $\phi=0$ ($\phi=\pi$) applies for even (odd) values of $m$, that are the peaks of $E^{(a)}(\tau)$ in Fig.~\ref{fig:fig1}b. At times $\tau_n=\tau_{off}+(n+\frac{1}{2})\pi$ ($n=0,1,2...$) the cavity mode states are obtained by applying $\hat{U}^{(c)}_{\phi}=\bigotimes_{J}e^{-i\phi\hat{c}_J^{\dag}\hat{c}_{J}}$, where $\phi=-\frac{\pi}{2}$ $(+\frac{\pi}{2})$ for even (odd) values of $n$, to the state $\ket{\Psi(0)}_c$ derived from $\ket{\Psi(0)}_f$ by the correspondence $\ket{0}_f\leftrightarrow\ket{0}_c$ and $\ket{1}_f\leftrightarrow\ket{1}_c$. By choosing to turn off the external field at times shorter than $\tau_{off}$ we find a progressive reduction in the entanglement transfer to the atomic and cavity subsystems, simulating the effect of non perfect cavity mirror transmittance. Up to 10\% changes in the value of $\tau_{off}$, the fidelity $F^{(a)}(\tau_{off})$ remains above 99.9\%. We remark that the state mapping process can be obtained for any $\ket{\Psi(0)}_f$ written in a generalized Schmidt decomposition \cite{Dur}, as well as for mixed states as described below.\\ \subsection{Tripartite entanglement sudden death for external field in a Werner state} \indent For the injected field we consider the Werner state $\hat{\rho}_f(0)=(1-p)\ketbra{GHZ}{GHZ}+\frac{p}{8} \hat{I}$, $(0\leq p\leq 1)$, because it is relevant in QI and it is possible to fully classify its entanglement as a function of parameter $p$. In fact, for $0\leq p<\frac{2}{7}$ the state belongs to the GHZ class and to the W class up to $p=\frac{4}{7}$. The tripartite negativity is positive up to $4/5$, and due to the structure of $\hat{\rho}_f(0)$, the state is clearly inseparable under all bipartitions (INS), i.e. it cannot be written as convex combination of biseparable states. For $4/5\leq p\leq1$ it is known that the state is fully separable \cite{Pittenger,Ghune2}.\\ \indent The system dynamics can be divided into a transient and an oscillatory regime, and the state mapping of $\hat{\rho}_f(0)$ onto atoms (cavity modes) still occurs at times $\tau_m$ ($\tau_n$). Out of these times the density matrices of all subsystems lose the form of a GHZ state mixed with white noise but still preserve invariance under all permutations of the three qubits and present only one non vanishing coherence as in $\hat{\rho}_f(0)$. This greatly helps us in the entanglement classifications in the plane $(\tau,p)$ shown in Fig.~\ref{fig:fig2}. In fact, in the regions where $E^{(\alpha)}(\tau)>0$ but out of W class we can exclude the biseparability. The fully separability criteria in \cite{Ghune2} are violated only where $E^{(a)}(\tau)>0$ so that if $E^{(a)}(\tau)=0$ the state may be fully separable or biseparable. Nevertheless, in the latter case the state should be symmetric and biseparable under all bipartitions and hence it is fully separable \cite{Kraus}. For any fixed value of $p$ in the range $0<p<4/5$ we thus show the occurrence of entanglement sudden death and birth at the boundaries between fully separable and INS states. In particular, for $0<p<4/7$ we find genuine tripartite ESD and ESB phenomena. Note that, for a fixed value of $p$, the passage of atomic state during time evolution from W-class to GHZ-class (or viceversa) entangled states is permitted by the non-unitarity of the partial trace over non-atomic degrees of freedom (so that the overall operation on the initial three qubits is not SLOCC). We also notice that for times $\tau\geq\tau_{off}$ we can solve exactly the triple JC dynamics, confirming our numerical results and generalizing \cite{ESDChina} to mixed states.\\ \indent In Fig.~\ref{fig:fig2}b we see that, for increasing values of $p$, there is an increase of both the slope of $E^{(a)}(\tau)$ and the time interval of fully separability. \begin{figure}
\caption{ ESD/ESB for external field in a GHZ state mixed with white noise. a) Regions in the plain ($\tau,p$) for atomic entanglement of type GHZ, W, INS, and fully separable (black). b) Sections $E^{(a)}(\tau)$ for selected values of $p$. c) Zoom on $E^{(\alpha)}(\tau_{off}/2)$ for field (dotted), cavity modes (dashed), and atoms (solid) with $p=0$ (black), $p=0.2$ (blue), $p=0.4$ (green), $p=0.6$ (red). d) Classification for cavity mode entanglement.}
\label{fig:fig2}
\end{figure} In Fig.~\ref{fig:fig2}c we show in detail the transient dynamics of the tripartite negativities $E^{(\alpha)}(\tau,p)$ ($\alpha=a,c,f$) in the crucial region around $\tau_{off}/2$. We consider some values of $p$ where the atoms exhibit in times different classes of entanglement. We see that for $p=0.2$, where the input state has GHZ class entanglement, the ESB of subsystems (c),(a) anticipates the ESD of (f),(c), and there is an interval around $\tau_{off}/2$ where all three subsystems are entangled (of INS-type). As $p$ grows, hence the initial state becomes more noisy, the effects of ESD occur earlier and those of ESB later. For $p=0.4$, involving W-class entanglement, only at most two subsystems are simultaneously entangled (first (f),(c) and then (c),(a)). For $p=0.6$, involving only entanglement of INS-type, the cavity modes do not entangle at all (see Fig.~\ref{fig:fig2}d). They physically mediate the discontinuous entanglement transfer from (f) to (a), where for $p\rightarrow 4/5$ the time interval without any entanglement increases while the entanglement level vanishes.\\ \subsection{Effect of dissipation on state mapping} \indent In the perspective of experimental implementation for QI purposes an important issue is the effect of dissipation on both state mapping and entanglement transfer. For external field prepared in a GHZ pure state we first evaluated the effect of cavity decay rates in the range $0<\tilde{\kappa}_c\leq0.5$ for negligible values of all other decay rates. We consider as function of $\tilde{\kappa}_c$ the behavior of the fidelities $F^{(\alpha)}(\tau_{m,n})$ and the tripartite negativities $E^{(\alpha)}(\tau_{m,n})$ ($\alpha=c,f$) at the first peaks $(m=n=0)$. \begin{figure}
\caption{ Effect of cavity mode dissipation. At the first peaks $\tau_{m,n}$ with $m=n=0$ we consider the tripartite negativities as function of $\tilde{\kappa}_c\equiv\frac{\kappa_c}{g_A}$: $F^{(a)}(\tau_0)$ (1), $E^{(a)}(\tau_0)$ (2), $F^{(c)}(\tau_0)$ (3), $E^{(c)}(\tau_0)$ (4).}
\label{fig:fig3}
\end{figure} In Fig.~\ref{fig:fig3} we see that the above functions of $\tilde{\kappa}_c$ can be well fitted by exponential functions, whose decay rates for the atomic subsystem are $\beta^{(a)}_{F}=0.75$, $\beta^{(a)}_{E}=1.09$, and for the cavity modes $\beta^{(c)}_{F}=1.80$, $\beta^{(c)}_{E}=2.94$. As expected, quantum state mapping and entanglement transfer are by far more efficient onto atomic than cavity qubits. For instance, if $\tilde{\kappa}_c = 0.1$ we obtain a state mapping onto the atoms (cavity modes) with a fidelity of $\cong0.93$ ($\cong0.83$).\\ \indent We can now add the further dissipative effect of atomic decay. For instance we find that, for an atomic decay rate $\tilde{\gamma}_a=0.03$ and in the presence of cavity decay with a rate $\tilde{\kappa}_c = 0.1$, the fidelity of the atomic (cavity mode) subsystem reduces by $4.4\%$ ($8.9\%$).\\ \indent Finally, we evaluate the effect of losses in the fibers used to inject the external field into each cavity. Clearly, this effect is relevant only up to the time $\tau_{off}=2.22$. We evaluated the effects of fiber decay rates $\tilde{\kappa}_f$ up to 1.0 for negligible values of atomic and cavity decay rates ($\tilde{\kappa}_c<<1$, $\tilde{\gamma}_a<<1$) (see Fig.~\ref{fig:fig4}). We show the effect of parameter $\tilde{\kappa}_f$ on cavity field mean photon number $N^{(c)}(\tau_{off}/2)$ and atomic excitation probability $p_e(\tau_{off})$ and we see that the amount of energy transferred to the atoms and to the cavity modes decreases exponentially for increasing values of $\tilde{\kappa}_f$; the decay rates are $\cong 0.42$ and $\cong 0.82$, respectively. Also the behavior of the tripartite negativity $E^{(a)}(\tau_{0})$ and fidelity $F^{(a)}(\tau_{0})$ at the first peak can be described versus $\tilde{\kappa}_f$ by exponential functions whose decay rates are $\cong1.51$ and $\cong 1.95$, respectively. \begin{figure}
\caption{ Effect of fiber mode decay rate $\tilde{\kappa}_f\equiv\kappa_f/g_A$ for $\tilde{\kappa}_c<<1$, $\tilde{\gamma}_a<<1$. We evaluate at time $\tau_{off}$ the atomic tripartite negativity $E^{a}$(solid), the fidelity $F^{a}$(dash-dot), and the atomic probability $p_e$(dot), and at time $\tau_{off}/2$ the cavity mean photon number $N^{(c)}$(dash).}
\label{fig:fig4}
\end{figure} \section{State mapping and tripartite entanglement transfer for multi-mode coupling} \indent Finally, we consider multi-mode coupling of the external field to each cavity mode. For simplicity we choose equal coupling constants $\tilde{\nu}_{J,K}\equiv\nu_{J,K}/g_A\neq 0$ if $K\neq J$ and we consider values in the range $0-1.4$. In the transient regime the dynamics is sharply modified with respect to the case of single mode fiber shown in Fig.~\ref{fig:fig1}. By increasing the values of $\tilde{\nu}_{J,K}$ the period of energy exchange decreases from $2\pi/\sqrt{2}$ to $\cong2.6$. The maximum of cavity mode mean photon number grows up to $N^{(c)}\cong 0.41$ whereas the maximum of atomic excitation probability decreases to $p_{e}\cong 0.24$. The external field mean photon number does not vanish but it reaches a minimum, that can be always found between the two maxima of $N^{(c)}(\tau)$ and $p_{e}(\tau)$, such that $ 0.002<N^{(f)}<0.02$ changing $\tilde{\nu}_{J,K}$ from 0.1 to 1.4. We investigate the differences in the entanglement transfer for three selections of switching-off time $\tau_{off}$ corresponding to the maximum of $p_{e}(\tau)$, the minimum of $N^{(f)}(\tau)$, and the maximum of $N^{(c)}(\tau)$. In Fig.~\ref{fig:fig5}a we show the dependence of $\tau_{off}$ on $\tilde{\nu}_{J,K\neq J}$. Switching off the external field at times $\tau_{off}$ corresponding to the maxima of $p_{e}(\tau)$, as in the previous case with single-mode fibers, we find (Fig.~\ref{fig:fig5}b,c) that the maxima of tripartite negativities $E^{(\alpha)}(\tau)$ after the transient regime reduce for increasing values of $\tilde{\nu}_{J,K}$ for both atomic and cavity mode subsystems. \begin{figure}
\caption{ Effect of multimode coupling. a) Dependence of $\tau_{off}$ on the coupling constants $\tilde{\nu}_{J,K}$ for different choices of switching-off the external field: maximum of $p_{e}(\tau)$ (o), minimum of $N^{(f)}(\tau)$ (x), and maximum of $N^{(c)}(\tau)$ (+). Tripartite negativities $E^{(\alpha)}$ ($\alpha=a,c$) for $\tilde{\nu}_{J,K}=0$ (solid gray), 0.3 (dashed), 0.6 (dotted), 1.0 (dashed-dotted), and 1.4 (solid black): b,c) $\tau_{off}$ in the maximum of $p_{e}(\tau)$; d,e) $\tau_{off}$ in the minimum of $N^{(f)}(\tau)$; f,g) $\tau_{off}$ in the maximum of $N^{(c)}(\tau)$.}
\label{fig:fig5}
\end{figure} If we consider $\tau_{off}$ corresponding to the minimum of $N^{(f)}(\tau)$ (Fig.~\ref{fig:fig5}d,e) we observe a small reduction of the peak values of $E^{(\alpha)}(\tau)$. Finally, if we turn off the external field at the first maximum of the cavity field mean photon number we note that by increasing the values of $\tilde{\nu}_{J,K}$ it is possible to improve the entanglement transfer (Fig.~\ref{fig:fig5}f,g). The peak value of tripartite negativity grows up to $\cong0.93$ for $\tilde{\nu}_{J,K}=1.4$ and the fidelity up to $\cong0.95$ for both subsystems (a) and (c). We remark that these values cannot be significantly increased for larger values of $\tilde{\nu}_{J,K}$. In conclusion, for all the above choices of switching-off time $\tau_{off}$ we observe that, by increasing the values of $\tilde{\nu}_{J,K}$, the amount of entanglement that can be transferred to the cavity modes in the transient regime also increases. This is due to the fact that the amount of energy transferred to each cavity mode increases: in fact, the peak value of $N^{(c)}(\tau)$ progressively grows up from $\cong 0.25$ to $\cong 0.41$. Nevertheless, multimode coupling for larger values of $\tau_{off}$ results in a less favorite condition for entanglement transfer.\\ \section{conclusions}
\indent In this paper we have addressed the transfer of quantum information and entanglement from a tripartite bosonic system to three localized qubits through their environments, also in the presence of external environments. We considered an implementation in the optical regime of CQED based on CV photon number entangled fields and atomic qubits trapped in one-sided optical cavities, where the radiation modes can couple to cavity modes by optical fibers.\\ \indent In the nine-qubit transient regime the quantum state is mapped from tripartite entangled radiation to tripartite atomic system via the cavity modes. After the transient we switch off the external field. The subsequent triple JC dynamics, that we solved analytically (numerically) for pure (mixed) input states, shows how the effect of mapping can further affect the atom-cavity six qubits system. Its relevance is in the possible manipulation for QI purposes of entanglement stored in separate qubits of atomic or bosonic nature. Hence the interest to put quantitative limits dictated by cavity, atomic and fiber mode decays, that we evaluated at the times where the transfer protocol is optimal.\\ \indent In the case of a GHZ input state mixed with white noise, we provide a full characterization of the separability properties of the tripartite subsystems. We can then show the occurrence of entanglement sudden death effects at the tripartite level, deriving the conditions for the repeated occurrence of discontinuous exchange of quantum correlations among the tripartite subsystems. This is an issue of fundamental interest as well as worth investigating for all applications in quantum information processing, remarkably computing and error correction, where disentanglement, which may be faster than decoherence, has to be carefully controlled.\\ \indent An extension and comparison to other types of entangled qubit-like input fields and experimentally available CV fields \cite{CV} will be presented elsewhere \cite{Bina}.
\acknowledgements This work has been partially supported by the CNR-CNISM convention.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title[Fubini-Tonelli type theorem]{Fubini-Tonelli type theorem for non product measures in a product space} \author{ Jorge Salazar}
\address{DMAT, Universidade de \'Evora, \'Evora - Portugal} \email{[email protected]}
\subjclass{ } \keywords{}
\date{}
\begin{abstract} I prove a theorem about iterated integrals for non-product measures in a product space. The first task is to show the existence of a family of measures on the second space, indexed by the points on of the first space (outside a negligible set), such that integrating the measures on the index against the first marginal gives back the original measure (see Theorem \ref{teo}). At the end, I give a simple application in Optimal Transport. \end{abstract} \maketitle
\section{Introduction} \noindent The Fubini-Tonelli theorem states that the integral of a function defined on a product space, against a measure which is a product of measures on the factor spaces, can be obtained by iterated integration, i.e. integrating one variable (against its marginal measure) at the time. \vskip 2mm \noindent
If a measure on a product space is not a product measure, is it still possible to decompose the measure and evaluate the integral using iterated integration? To better understand the problem, imaging we are dealing with a measure $ \zeta $ which is absolutely continuous with respect to the product measure $ \mu\otimes\nu $, i.e. there is a function $ \delta: X\times Y\rightarrow \left[ 0, \infty\right] $, such that for all measurable set $C\subseteq X\times Y$,
\[ \zeta\left( C\right) =\int_{C} \delta(x,y)\,\mu\otimes\nu\left( dx, dy \right) . \]
\vskip 2mm
\noindent
Then, using the classical Fubini-Tonelli theorem, we can decompose this integral into
\[ \zeta\left( C\right) =\int_X\left( \int_{C_x} \delta(x,y)\, \nu\left( dy \right) \right) \mu\left( dx \right) ,\]
where $C_x :=\left\lbrace y\in Y;\, \left( x,y\right)\in C \right\rbrace $
is the slice of $C$ at $X$.
Accordingly, $\nu$ decomposes into the measures
\[ \nu_x(dy):=\delta(x,y)\, \nu\left( dy \right) , \]
which integrates against $\mu$ to give $\zeta$. Symbolically,
\[ \zeta\left( dx, dy \right) =\nu_x(dy)\, \mu\left( dx \right) . \]
With this decomposition the order of integration is not interchangeable, since the first measure depends on the second variable.
To interchange the order of integration, we must decompose $\mu$ in a similar way,
\[ \mu_y(dx):=\delta(x,y)\, \mu\left( dx \right) , \]
which integrates against $\nu$ to give $\zeta$. Symbolically,
\[ \zeta\left( dx, dy \right) =\mu_y(dx)\, \nu\left( dy \right) . \]
\noindent
In this paper, the existence of this kind of decomposition is established for arbitrary Borel probability measures on
the product of two complete, separable, locally compact,
metric spaces (see Theorem \ref{teo}). I restricted myself to the case of probability measures to simplify the discourse, although the results stay valid for $\sigma-$finite measures. \vskip 2mm \noindent
In the best of my knowledge, there is nothing of the kind in the literature on foundations of measure theory that describes similar results. Nonetheless, this question is natural and I believe it may provide a useful calculation and/or analytical tool as much as the classical Fubini-Tonelli theorem does.
\vskip 2mm
\noindent
Optimal Transport, for example, deals with fixed marginal probability measures and a minimal cost is seek among all the couplings of the given marginal probabilities, i.e. among all the probabilities on the product space, such that the marginal measures are the ones given.
It would be a nice research project to look for a new characterization of the optimal transport plans in terms of the measures along the ``fibers'', obtained from the decomposition described in Theorem \ref{teo}. In section \ref{OT}, I give a simple application of Theorem \ref{teo}, showing that a pair of competitive price functions, whose integral with respect to some transference plan matches the transport cost, are conjugate to each other almost surely. This complements the Kantorovich duality theorem on the nature, regarding convexity/concavity, of pair of competitive prices maximizing the profit. See Villani's book \cite{vill}, page 70, for a very detailed discussion of the Kantorovich theorem.
\vskip 2mm
\noindent Another interesting project is the application of Theorem \ref{teo} to the study of measures on the Tangent bundle of Riemannian manifolds. Indeed, the local charts of the tangent bundle are Cartesian products of Euclidean open sets. Using local charts, we can transport the measure to this product to be decomposed and then sent back the family of measures fiber-wise. In the literature, the measures on tangent bundles are a kind of product measures, as is the volume obtained from the Sasaki \cite{sasaki} metric, or the measure on the unit sphere on the tangent space, integrated against the volume element of the base manifold. I believe Theorem \ref{teo} is a tool that could help exploring general integration on tangent bundles.
\section{Main theorem}
{\thm\label{teo} Let $X\times Y$ be the product of two complete, separable, locally compact,
metric spaces. We equip $X$, $Y$, and $X\times Y$ with their Borel $\sigma-$algebras, denoted by $\mathfrak{B}_X$, $\mathfrak{B}_Y$, and $\mathfrak{B}_{X\times Y}$ respectively.
\vskip 3pt
\noindent
Let $\zeta$ be a probability measure on $\mathfrak{B}_{X\times Y}$, and denote $\mu$ and $\nu$ the marginal probabilities on $\mathfrak{B}_X$, $\mathfrak{B}_Y$ respectively. i.e.
\[ \forall A\in \mathfrak{B}_X, \ \mu(A)= \zeta(A\times Y) \]
and
\[ \forall B\in \mathfrak{B}_Y, \ \nu(B)= \zeta(X\times B) .\] \vskip 3pt \noindent Then, outside an exceptional $\mu-$negligible set $E_1 \in \mathfrak{B}_X$ ($\mu\left( E_1\right) = 0$), for all
$ x \in X\setminus E_1 $, there is a measure $\nu_x$ defined on $\mathfrak{B}_Y$, such that for all $C\in \mathfrak{B}_{X\times Y}$, the function \begin{equation}\label{meas0}
x \in X\setminus E_1 \longrightarrow \nu_x\left( C_x \right) ,
\end{equation} where $ C_x = C\cap\left( \left\lbrace x \right\rbrace\times Y \right) $, is $\mathfrak{B}_X-$measurable and \begin{equation}\label{int0}
\zeta(C)= \int_X\nu_x\left(C_x \right) \mu(dx).
\end{equation}
In particular,
\[ \forall B\in \mathfrak{B}_Y, \ \nu(B)= \int_{X} \nu_x\left(B \right) \mu(dx) ,\] Moreover, for all positive $\mathfrak{B}_{X\times Y}-$measurable function, $f: X\times Y\rightarrow \mathbb{R}$, \begin{equation}\label{meas0f} x \in X\setminus E_1 \longrightarrow \int_Y f(x,y)\, \nu_x\left(dy\right) \end{equation}
is $\mathfrak{B}_X-$measurable and
\begin{equation}\label{int0f}
\int_{X\times Y} f(x,y)\, \zeta\left(dx,dy\right)= \int_X \left( \int_Y f(x,y)\, \nu_x\left(dy\right) \right) \mu(dx).
\end{equation}
\vskip 3mm \noindent Likewise, there is a $\nu-$negligible set $E_2 \in \mathfrak{B}_Y$, such that for every $\, y\in Y\setminus E_2 $
there is a measure $\mu_y$ on $\mathfrak{B}_X$, such that for all $C\in \mathfrak{B}_{X\times Y}$, the function \[ y\in Y\setminus E_2 \longrightarrow \mu_y\left(C_y \right) ,\] where $ C_y = C\cap\left( X \times\left\lbrace y \right\rbrace \right) $, is $\mathfrak{B}_{ Y}-$measurable and \[ \zeta(C)= \int_{X} \mu_y\left(C_y \right) \mu(dx). \] In particular, \[ \forall A\in \mathfrak{B}_X, \ \mu(A)=\int_{Y} \mu_y\left(A \right) \nu(dx) .\] Moreover, for all positive $\mathfrak{B}_{X\times Y}-$measurable function, $f: X\times Y\rightarrow \mathbb{R}$, \[ y\in Y\setminus E_2 \rightarrow \int_{X} f(x,y)\, \mu_y\left(dx\right) \] is $\mathfrak{B}_{Y}-$measurable and \[ \int_{X\times Y} f(x,y)\, \zeta \left(dx, dy\right) = \int_{Y} \left( \int_{X} f(x,y)\, \mu_y \left(dx\right) \right) \nu(dy). \]
\vskip 3mm \noindent As a consequence, given a $\mathfrak{B}_{X\times Y}-$measurable function, $f: X\times Y\rightarrow \mathbb{R}$, the following affirmations are equivalent
\vskip 2mm
\begin{enumerate}
\item $f: X\times Y\rightarrow \mathbb{R}$ is $\zeta-$integrable. \vskip 3mm
\item $ x \in X\setminus E_1 \rightarrow \int_Y \left| f(x,y)\right| \, \nu_x\left( dy\right) $ is $\mu-$integrable. \vskip 3mm
\item $ y\in Y\setminus E_2 \rightarrow \int_{X}\left| f(x,y)\right| \, \mu_y\left( dx\right) $ is $\nu-$integrable. \end{enumerate} \vskip 2mm\noindent And \[ \int_{X\times Y} f(x,y)\, \zeta\left( dx, dy\right)= \int_X \left( \int_Y f(x,y)\, \nu_x\left( dy\right) \right) \mu\left( dx\right) . \] \[ \qquad \qquad \qquad \qquad \qquad = \int_{Y} \left( \int_{X} f(x,y)\, \mu_y \left( dx\right) \right) \nu\left( dy\right) . \] }
\section{Proof of Theorem \ref{teo}}
\noindent \emph{Note about the notation}: We will use $x$ and $y$ to denote generic points in $X$ and $Y$ respectively. In this way, $B_r\left( x\right) $ automatically refers to a ball in $X$, of center $x$ and and radius $r$, while $B_r\left( y\right)$ represents a ball in $Y$ (different space, different metric). \vskip 2mm \noindent The proof of Theorem \ref{teo} will be given in several steps.
\subsection{Definition of $ \mathit{l}_x $}
Let $\mathcal{Y} $ be a dense subset of $Y$. Denote by $ \mathcal{B} $ the set of open balls $B_r\left( y\right) $ with center $y\in \mathcal{Y} $ and radius $ r\in\mathbb{Q} $. i.e. \begin{equation}\label{balls} \mathcal{B}:= \left\lbrace B_r \left( y\right) ;\, y\in\mathcal{Y},\ r\in \mathbb{Q} \right\rbrace . \end{equation}
\noindent Consider also the complement of the closed balls, \[ \mathcal{B}_{\mathrm{c}} :=\left\lbrace Y\setminus \overline{B}_r\left( y\right) ; {B}_r\left( y\right) \in\mathcal{B} \right\rbrace . \]
\noindent Finally, let $ \mathcal{L} $ be the set of finite unions of finite intersections of elements of $ \mathcal{B}\cup \mathcal{B}_{\mathrm{c}} $ (note that $\emptyset\in \mathcal{L}$). \vskip 2mm \noindent For each $O\in \mathcal{L}$, define the measure \[ A\in \mathfrak{B}_X\rightarrow \mu_O \left( A\right) =\zeta\left( A\times O\right) .\]
\noindent Since $\mu_O \left( A\right) \le \mu\left( A\right) $, $\mu_O $ is absolutely continuous with respect to $ \mu $. By Radon-Nikodym's Theorem, there is a density function $ \frac{d\mu_O}{d\mu} $, defined $\mu-$almost surely, such that $\mu_O $ is represented as an integral of this density against $\mu$. \vskip 2mm \noindent To obtain a common exceptional $\mu-$negligible set outside which $ \frac{d\mu_O}{d\mu} $ is well defined (by a formula) for all $ O\in \mathcal{L} $, we choose the version of $ \frac{d\mu_O}{ d\mu} $ given by the limit of the quotient of balls. To avoid talking about measurability issues, we fix once and for all a sequence $\rho_k$ decreasing to $0$. Given $ O\in \mathcal{L} $, define \[ \overline{\mathit{l}}_x\left( O\right) :=\limsup_{k\rightarrow \infty} \frac{\mu_O \left( B_{\rho_k}(x)\right) }{\mu \left( B_{\rho_k}(x)\right) },\] and \[ \underline{\mathit{l}}_x\left( O\right) :=\liminf_{k\rightarrow \infty} \frac{\mu_O \left( B_{\rho_k}(x)\right) }{\mu \left( B_{\rho_k}(x)\right) },\] \vskip 2mm \noindent It is well known, by a generalization of Lebesgue differentiation theorem (see for example Federer \cite{fed}, section 2.9),
that $ \overline{\mathit{l}}_x $ and $ \underline{\mathit{l}}_x $ are versions of
$ \frac{d\mu_O}{d\mu} $. i.e. For all $ A\in \mathfrak{B}_X$, \begin{equation}\label{int} \mu_O \left( A\right) =\int_A \overline{\mathit{l}}_x\left( O\right) \mu\left( dx\right) =\int_A \underline{\mathit{l}}_x \left( O\right) \mu\left( dx\right) . \end{equation} In particular, $$ \overline{\mathit{l}}_x\left( O\right) = \underline{\mathit{l}}_x\left( O\right) ,\ \mu-\mathrm{a.s.}.$$ \noindent Let $E_O$ be the exceptional set where the limit does not exist. i.e. $$E_O:=\left\lbrace x\in X;\ \overline{\mathit{l}}_x\left( O\right) -\underline{\mathit{l}}_x \left( O\right) >0\right\rbrace \in \mathfrak{B}_X$$
\noindent Put \[ E:=\bigcup_{O\in \mathcal{L} } E_O. \]
\noindent Since $\mu\left( E_O\right) =0$ for all $ O\in \mathcal{L} $, and $\mathcal{L}$ is numerable, we have $$\mu\left( E\right) =0 .$$
For $O\in \mathcal{L}$, and $x\in X\setminus E$, put
$$ \mathit{l}_x\left( O\right) = \overline{\mathit{l}}_x \left( O\right) =\underline{\mathit{l}}_x \left( O\right) .$$
\vskip 2pt \noindent Recapitulating, for all $O\in \mathcal{L}$ and every $ A\in \mathfrak{B}_Y $, by (\ref{int}) we have \begin{equation}\label{int1}
\zeta\left( A\times O\right) =\int_A {\mathit{l}}_x\left( O\right) \mu\left( dx\right) . \end{equation}
\subsection{Outer measure $ \nu^*_x $}
Changing the standpoint, we fix $x\in X\setminus E$ and consider the set function $$ O\in \mathcal{L}\rightarrow \mathit{l}_x\left( O\right) .$$ For future reference, observe that $ \mathit{l}_x $ has the following properties: For all $ O$ and $ \tilde{O}\in \mathcal{L} $, \vskip 3pt \noindent \emph{Finite additivity} \begin{equation}\label{add}
\mathit{l}_x\left( O\right) +\mathit{l}_x\left( \tilde{O}\right) = \mathit{l}_x\left( O\cup \tilde{O}\right) + \mathit{l}_x\left( O\cap \tilde{O}\right) , \end{equation} \vskip 2pt \noindent \emph{Finite subadditivity}
\begin{equation}\label{subadd} \mathit{l}_x\left( O\cup \tilde{O}\right)\le \mathit{l}_x\left( O\right) +\mathit{l}_x\left( \tilde{O}\right)
\end{equation} \vskip 2pt \noindent
\emph{Monotonicity} \begin{equation}\label{mono}
O\subseteq \tilde{O} \, \Rightarrow \, \mathit{l}_x\left( O\right) \le \mathit{l}_x\left( \tilde{O}\right) \end{equation} \vskip 3pt \noindent We need a measure on $\mathfrak{B}_Y$, capable of fulfilling the role of $\mathit{l}_x$ in equation (\ref{int1}). Let us start by defining the outer measure \begin{equation} \label{outer} C\subseteq Y\rightarrow \nu^*_x\left( C\right) :=\inf\sum_{i=1}^{\infty}\mathit{l}_x\left( O_i\right) , \end{equation} where the infimum is taken over all the covers $ \left\lbrace O_i \right\rbrace _{i\in \mathbb{N}} \subseteq \mathcal{L} $ of $C$. i.e. \[ C\subseteq \bigcup_{i=1}^\infty O_i \, ,\ \mathrm{and}\ \forall i\in \mathbb{N}, \, O_i\in \mathcal{L}\]
\noindent It is well known that $\nu^*_x$, restricted to the set of $\nu^*_x-$measurable sets, is a measure (denoted $\nu_x$). What we need to prove are: Firstly, that every Borel subset of $Y$ (i.e. in $\mathfrak{B}_Y$) is $\nu^*_x-$measurable and secondly that the integration property (\ref{int1}) is preserved when $\mathit{l}_x$ is replaced by $\nu_x$ (and therefore valid for any set in $\mathfrak{B}_Y$). \vskip 2mm \noindent
Unfortunately, $\mathit{l}_x$ is not countably subadditive, as a result, $\nu^*_x$ is not the extension of $\mathit{l}_x$, hardening our task a little bit. Indeed, for all $ O\in \mathcal{L} $, we clearly have $ \nu^*_x(O)\le \mathit{l}_x(O)$, but the inverse inequality may fail, as the following example shows. \vskip 3mm \noindent \emph{Example}: Let $X=Y=\left[ 0,1\right] $. Let $\mu=\nu$ be the Lebesgue measure on $ \left[ 0,1\right] $ and $\zeta$ the normalized length on the diagonal $\left\lbrace \left(x,x \right) ;\, x\in \left[ 0,1\right] \right\rbrace $. \vskip 3mm \noindent Observe that for every $ x\in \left] 0,1\right[ \, $ and $\rho_k$ small enough, \[ \frac{\mu_{\left[ 0,x\right[} \left( B_{\rho_k}(x)\right) }{\mu \left( B_{\rho_k}(x)\right) }=\frac{1}{2}.\]
\noindent So, $ \mathit{l}_x \left( \left[ 0,x\right[\right) = 1/2 $, while $ \nu_x^* \left( \left[ 0,x\right[\right) = 0 $. In fact, we can cover $ \left[ 0,x\right[ $ with a sequence of intervals $ \left[ 0,x_n\right[ $, where $x_n \in \mathbb{Q}$ increases to $x$. Each one of the intervals $ \left[ 0,x_n\right[ $ verify \[ \frac{\mu_{\left[ 0,x_n\right[} \left( B_{\rho_k}(x)\right) }{\mu \left( B_{\rho_k}(x)\right) }=0, \] for all $\rho_k$ small enough. Therefore, for all $n\in \mathbb{N}$, $ \mathit{l}_x \left( \left[ 0,x_n\right[\right) =0 $ and \[ \nu_x^* \left( \left[ 0,x\right[\right) \le \sum_{i=1}^\infty \mathit{l}_x \left( \left[ 0,x_n\right[\right) = 0 .\]
\vskip 3mm \noindent \subsection{Borel subsets of $Y$ are $\nu^*_x-$measurable} Let's prove first that any open ball $B_r(y)\in\mathcal{L}$ is $\nu^*_x-$measurable. To this end, fix $C\subseteq Y$ and a cover $ \left\lbrace O_i\right\rbrace_{i\in\mathbb{N}} \subseteq \mathcal{L}$ of $C$. We must show that \begin{equation}\label{measurable} \nu^*_x \left( C \cap B_r(y) \right) + \nu^*_x \left( C\setminus B_r(y)\right) \le \sum_{i=1}^\infty \mathit{l}_x\left( O_i\right). \end{equation} \noindent Since $ O_i\cap B_r(y) \in \mathcal{L}$ and $ \left\lbrace O_i\cap B_r(y) \right\rbrace_{i\in\mathbb{N}} $ is a covering of $ C\cap B_r(y) $, \begin{equation}\label{uno}
\nu^*_x\left( C\cap B_r(y)\right) \le \sum_{i=1}^\infty \mathit{l}_x\left( O_i\cap B_r(y) \right) . \end{equation} \noindent Now, let $\alpha_{i}<1$, $\alpha_{i}\in\mathbb{Q}$. Then, $O_i\setminus \overline{B}_{ \alpha_{i} r}(y)\in \mathcal{L}$ and $ \left\lbrace O_i\setminus \overline{B}_{ \alpha_{k_i} r}(y) \right\rbrace_{i\in\mathbb{N}} $ is a covering of $C\setminus B_r(y) $. So,
\begin{equation}\label{dos}
\nu^*_x\left( C\setminus B_r(y)\right) \le \sum_{i=1}^\infty \mathit{l}_x\left( O_i\setminus \overline{B}_{ \alpha_i r}(y) \right) .
\end{equation}
By (\ref{add}) and (\ref{mono}), we have
\begin{equation}\label{tres}
\mathit{l}_x\left( O_i\cap B_r(y) \right) + \mathit{l}_x\left( O_i\setminus \overline{B}_{ \alpha r}(y) \right) \le \mathit{l}_x\left( O_i \right) + \mathit{l}_x \left( B_r(y) \setminus \overline{B}_{ \alpha_i r}(y) \right).
\end{equation} \vskip 2mm \noindent Adding (\ref{uno}) and (\ref{dos}), and using (\ref{tres}), we obtain \[\nu^*_x\left( C\cap B_r(y)\right) +\nu^*_x\left( C\setminus B_r(y)\right) \] \[\le \sum_{i=1}^\infty \mathit{l}_x\left( O_i \right) +\sum_{i=1}^\infty \mathit{l}_x\left( B_r(y)\setminus \overline{B}_{ \alpha_i r}(y) \right) .\] \vskip 2mm \noindent The result follows if we can make the second sum as small as we want. Unfortunately, for a fixed $x\in E$, we might fail to do so, even though \begin{equation}\label{empty} \bigcap_{i=1}^\infty B_r(y) \setminus \overline{B}_{ \alpha_i r}(y) =\emptyset , \end{equation} for any sequence $\alpha_i\rightarrow 1$. In fact, we can not switch the limits in \[ \lim_{i\rightarrow \infty}\mathit{l}_x\left( B_r(y)\setminus \overline{B}_{ \alpha_i r}(y) \right) = \lim_{i\rightarrow \infty}\lim_{k\rightarrow \infty} \frac{\zeta \left( B_{\rho_k}\left( x\right) \times \left( B_r(y)\setminus \overline{B}_{ \alpha_i r}(y) \right) \right) }{\mu\left( B_{\rho_k}\left( x\right) \right)} .\] So, we need to look back at what happens for $x$ variable and check whether we can solve the problem by throwing away a few more points (meaning to enlarge $E$). \vskip 2mm \noindent
By (\ref{int1}) and ($\ref{empty}$), and $\gamma < 1$, \[ \int_X \mathit{l}_x\left( B_r(y) \setminus \overline{B}_{ \gamma r}(y)\right) \mu\left( dx\right) =\nu\left( B_r(y) \setminus \overline{B}_{ \gamma r}(y)\right) \longrightarrow_{\gamma \nearrow 1} 0. \] Therefore, \begin{equation}\label{null-a.e.} \mathit{l}_x\left( B_r(y) \setminus \overline{B}_{ \gamma r}(y)\right) \longrightarrow_{\gamma \nearrow 1} 0,\ \mu-\mathrm{a.s.} \end{equation} \vskip 2mm \noindent Now, fix an increasing sequence $ \left\lbrace \gamma_j\right\rbrace_{j\in\mathbb{N}} \subseteq \mathbb{Q}$, $ \gamma_j \rightarrow 1$. Define, for all $y\in\mathcal{Y}$ and $r\in\mathbb{Q}$, \[ {E}_{r,y}:=\left\lbrace x\in X\setminus E ;\ \liminf_{j\rightarrow \infty}\mathit{l}_x\left( B_r(y) \setminus \overline{B}_{ \gamma_j r}(y)\right) > 0 \right\rbrace .\] By (\ref{null-a.e.}), $\mu\left( {E}_{r,y}\right) =0 $. Since \[ {E}_1:= E\cup\bigcup_{r\in\mathbb{Q} ,\, y\in\mathcal{Y} }{E}_{r,y} \] is a countable union of sets of $\mu-$measure 0, we have $$\mu\left( {E}_1\right) =0 . $$ For all $x\in X\setminus E_1 $, and every $\epsilon>0$, we can choose a subsequence $ \left\lbrace \alpha_i\right\rbrace_{i\in\mathbb{N}} $ of $ \left\lbrace \gamma_j\right\rbrace_{k\in\mathbb{N}} $, such that \[ \sum_{i=1}^\infty \mathit{l}_x\left( B_r(y)\setminus \overline{B}_{ \alpha_i r}(y) \right) < \epsilon .\] This completes the proof of (\ref{measurable}). \vskip 2mm \noindent Consequently, for all $x \in X\setminus E_1$, $\nu_x$ is a measure defined at least in the $\sigma-$field generated by $\mathcal{B}$, i.e. $\mathfrak{B}_Y$.
\subsection{Measurability and integrability for compact sets} Our task now is to prove that given $B\in \mathfrak{B}_Y$, the function \begin{equation}\label{meas}
x\in X\setminus E_1 \longrightarrow \nu_x\left( B\right) \end{equation} is measurable and, for all $A\in \mathfrak{B}_X$ \begin{equation}\label{int2} \zeta\left( A\times B\right) =\int_A \nu_x\left( B\right) \mu\left( dx\right) . \end{equation} \vskip 2pt \noindent Let's consider first a finite intersection of closed, compact balls $$ \overline{B}=\overline{B}_{r_1}(y_1) \cap\cdots\cap \overline{B}_{r_n}(y_n) ,$$ with $y_1,\cdots,y_n\in\mathcal{Y}$ and $ r_1,\cdots,r_n\in\mathbb{Q} $.
By the measurability of $\mathit{l}_x$ and (\ref{int1}), the properties (\ref{meas}) and (\ref{int2}) are proven at once if we show \begin{equation}\label{l_x} \nu_x \left( \overline{B} \right) =1-\mathit{l}_x\left( Y\setminus \overline{B}\right) ,\ \mu-\mathrm{a.s.} \end{equation} (We use $\mathit{l}_x\left( Y\setminus \overline{B}\right) $ just because $\mathit{l}_x $ is not defined for $ \overline{B}$.) \vskip 2pt \noindent Let $O_1,\, O_2, \cdots, O_m\subseteq \mathcal{L}$ be a covering of $ \overline{B} $. We can assume the covering is finite, since $\overline{B}$ is compact and the sets in $\mathcal{L}$ are open. \vskip 3pt \noindent Clearly, for all $ x\in X\setminus E $, \[ \lim_{k\rightarrow \infty} \frac{\zeta\left( {B}_{\rho_k}(x)\times \overline{B}\right) }{\mu\left( {B}_{\rho_k}(x)\right) }=\lim_{k\rightarrow \infty} \left( 1-\frac{\zeta\left( {B}_{\rho_k}(x)\times \left( Y\setminus \overline{B} \right)\right) }{\mu\left( {B}_{\rho_k} (x)\right) }\right) =1-\mathit{l}_x\left( Y\setminus \overline{B}\right) . \] Since the covering is finite, by (\ref{subadd}),
\[ \lim_{k\rightarrow \infty} \frac{\zeta\left( {B}_{\rho_k}(x)\times \overline{B}\right) }{\mu\left( {B}_{\rho_k}(x)\right) }\le \mathit{l}_x\left( \bigcup_{i=1}^{n} O_i\right) \le \mathit{l}_x\left( O_1\right) + \cdots + \mathit{l}_x\left( O_m\right). \] Then, \[ 1-\mathit{l}_x\left( Y\setminus \overline{B}\right) \le \nu_x \left( \overline{B} \right) .\] \vskip 2pt \noindent On the other hand, given $ \eta >1 $, $ \eta\in\mathbb{Q} $, and denoting \[ B_\eta={B}_{\eta r_1}(y_1) \cap\cdots\cap {B}_{\eta r_n}(y_n) , \] we have \[ \nu_x \left( \overline{B}\right) \le \mathit{l}_x\left( {B}_\eta \right) .\] Consequently, taking any sequence $\eta_k\searrow 1$, $ \eta_k \in\mathbb{Q} $, we have \[ 1-\mathit{l}_x\left( Y\setminus \overline{B}\right) \le \nu_x \left( \overline{B}\right) \le \lim_{k\rightarrow\infty}\mathit{l}_x\left( {B}_{\eta_k}\right) .\] \vskip 2pt \noindent Since the functions at the left and at the right of the above inequalities are $\mathfrak{B}_{X}-$measurable, and equal between them $\mu-$almost-surely, using (\ref{int1}), we have proved (\ref{l_x}) and, a fortiori, (\ref{meas}) and (\ref{int2}), at least for a finite intersection of compact balls.
\subsection{Measurability and integrability for Borel sets}
Let $\mathcal{M}$ be the collection of sets $B\in \mathfrak{B}_Y$ actually verifying (\ref{meas}) and (\ref{int2}).
\vskip 3pt
\noindent
Clearly, $\emptyset\in \mathcal{M} $ and $ Y\setminus B\in \mathcal{M} $, whenever $B \in \mathcal{M} $. \vskip 3pt \noindent Now, take a disjoint sequence $B_1,B_2,\, \cdots \, \in \mathcal{M}$ ($B_i\cap B_j= \emptyset $, $ i\neq j $). Since, for all $x\in X\setminus E_1 $, $\nu_x$ is a measure on $\mathfrak{B}_Y$, \begin{equation}\label{suma}
\nu_x\left( \bigcup_{i=1}^{\infty} B_i\right) = \sum_{i=1}^{\infty}\nu_x\left( B_i\right) . \end{equation} By (\ref{meas}), $ x\rightarrow \nu_x\left( \bigcup_{i=1}^{\infty} B_i\right) $ is $\mathfrak{B}_{X}-$measurable, being a countable sum of $\mathfrak{B}_{X}-$measurable functions.
\vskip 2pt \noindent By (\ref{suma}), the monotone convergence theorem, and (\ref{int2}) (remember $B_i \in \mathcal{M}$, for all $i\in \mathbb{N}$), given $ A\in\mathfrak{B}_X $, \[ \int_A \nu_x\left( \bigcup_{i=1}^{\infty} B_i\right) \mu\left( dx\right) = \int_A \sum_{i=1}^{\infty}\nu_x\left( B_i\right) \mu\left( dx\right) \] \[ \ \quad \qquad \qquad \qquad \qquad = \sum_{i=1}^{\infty} \int_A \nu_x\left( B_i\right) \mu\left( dx\right) \] \[ \qquad \quad \qquad \qquad = \sum_{i=1}^{\infty} \zeta\left( A\times B_i\right) \] \[ \qquad \qquad \qquad \qquad = \zeta \left( \bigcup_{i=1}^{\infty} A\times B_i\right) . \] \[ \qquad \qquad \qquad \qquad = \zeta \left( A\times\bigcup_{i=1}^{\infty} B_i\right) . \]
Then,
\[ \bigcup_{i=1}^{\infty} B_i\in \mathcal{M}. \] Since $\mathcal{M}$ contains all the finite intersections of compact balls with center in $\mathcal{Y}$ and rational radius, by the $\pi-\lambda$ theorem (see \cite{Bi}, page 36),
$$ \mathcal{M}=\mathfrak{B}_Y. $$
\subsection{Proof of the theorem} We have proven so far (\ref{meas0}) and (\ref{int0}) for sets of the form $C=A\times B$, with $ A\in \mathfrak{B}_X $ and $ B\in \mathfrak{B}_Y $. \vskip 2mm \noindent Using a similar argument as before, let $\tilde{\mathcal{M}}$ denote the collection of sets $C\in \mathfrak{B}_{X\times Y} $ verifying (\ref{meas0}) and (\ref{int0}). We readily see that $\emptyset\in \mathcal{M} $.
\vskip 2mm \noindent Now, let $C\in\tilde{\mathcal{M}} $. Since $\left( \left( X\times Y\right) \setminus C\right)_x = Y \setminus C_x $, the function \[ x\rightarrow \nu_x\left(\left( \left( X\times Y\right) \setminus C\right) _x \right) = 1- \nu_x\left( C_x\right) \]
is $ \mathfrak{B}_{X}-$measurable, and \[ \int_X \nu_x\left(\left( \left( X\times Y\right) \setminus C\right)_x \right) \mu\left( dx\right) = 1- \int_X \nu_x\left(C_x \right) \mu\left( dx \right) \] \[ \ \qquad \qquad \qquad \qquad = 1-\zeta\left( C\right)\] \[\ \qquad \qquad \qquad \qquad \qquad \ \quad = \zeta\left( \left( X\times Y\right) \setminus C\right). \]
\noindent Then, for all $C\in\tilde{\mathcal{M}} $, we have $\left( X\times Y\right) \setminus C \in\tilde{\mathcal{M}} $. \vskip 3pt \noindent Finally, let $C_1,C_2,\, \cdots \, \in \tilde{\mathcal{M}} $ a sequence of disjoint sets ($C_i\cap C_j= \emptyset $, for all $ i\neq j $). Since $\nu_x$ is a measure on $\mathfrak{B}_Y$, \begin{equation}\label{suma1} \nu_x\left( \left( \bigcup_{i=1}^{\infty} C_i\right)_x\,\right) = \nu_x\left( \bigcup_{i=1}^{\infty} \left( C_i\right) _x\right) = \sum_{i=1}^{\infty}\nu_x\left( \left( C_i\right)_x \right) . \end{equation} Then, $ x\rightarrow \nu_x\left( \left( \bigcup_{i=1}^{\infty} C_i\right)_x \,\right) $ is $\mathfrak{B}_{X}-$measurable, being a sum of $\mathfrak{B}_{X}-$measurable functions, since $C_i \in \tilde{\mathcal{M}}$, for all $i\in \mathbb{N}$.
\vskip 3pt\noindent By (\ref{suma1}), the monotone convergence theorem, and (\ref{int0}) ($C_i \in \tilde{\mathcal{M}}$), \[ \int_X \nu_x\left( \left( \bigcup_{i=1}^{\infty} C_i\right)_x\,\right) \mu\left( dx \right) = \int_X \sum_{i=1}^{\infty}\nu_x\left( \left( C_i\right) _x\right) \mu\left( dx \right) \ \quad \quad \] \[ \ \quad \qquad \qquad \qquad \qquad = \sum_{i=1}^{\infty} \int_X \nu_x\left( \left( C_i\right) _x\right) \mu\left( dx \right) \] \[ \qquad \qquad = \sum_{i=1}^{\infty} \zeta\left( C_i\right) \] \[ \ \qquad \qquad = \zeta \left( \bigcup_{i=1}^{\infty} C_i\right) \] Then, \[ \bigcup_{i=1}^{\infty} C_i\in \tilde{\mathcal{M}}. \] \vskip 2mm \noindent Since \[ \left\lbrace A\times B;\, A\in \mathfrak{B}_X ,\, B\in \mathfrak{B}_Y \right\rbrace \subseteq \tilde{\mathcal{M}} ,\]
by the $\pi-\lambda$ theorem, \[ \tilde{\mathcal{M}} = {\mathfrak{B}_{X\times Y}} .\] \vskip 3mm \noindent The remaining of the proof is standard. Assume $f :{X\times Y}\rightarrow \mathbb{R}$ is a positive, $\mathfrak{B}_{X\times Y}-$measurable functions. Then, $f$ can be approached by an increasing sequence of simple, $\mathfrak{B}_{X\times Y}-$measurable functions (linear combinations of characteristic functions) \[ f_k(x,y)=\sum_{i=1}^{n_k} \lambda_{k,i}C_{k,i},\] where $\lambda_{k,i}\in\mathbb{R}$ and $C_{k,i} \in \mathfrak{B}_{X \times Y}$ ($i=1,\cdots, n_k $, $ k\in \mathbb{N}$). \vskip 3mm \noindent By linearity, for each $ f_k$, properties (\ref{meas0f}) and (\ref{int0f}) are a direct consequence of (\ref{meas0}) and (\ref{int0}). Passing to the limit as $k\rightarrow \infty$, equations (\ref{meas0f}) and (\ref{int0f}) are conserved, therefore valid for any positive function like $f$. \vskip 3mm \noindent To establish the integrability equivalence, we apply the preceding result to the positive and negative parts of the given function. In this way, the proof of Theorem \ref{teo} is complete.
\section{An aplication to Optimal Transport} \label{OT}
\noindent In this section we use Theorem \ref{teo} to show that a pair of competitive price functions, whose integral with respect to some transference plan equals the transport cost, are conjugate to each other almost surely, complementing Kantorovich's duality theorem on the nature of a pairs of competitive prices maximizing the profit. See Villani's book \cite{vill}, page 70, for a very detailed discussion on Kantorovich's theorem, in particular
Theorem 5.1, part \textit{(ii)}, item \textit{(d)}. For this result, we do not need lower semicontinuity of the cost and other assumptions used to prove Kantorovich's theorem, so we state the following lemma in its simplest form, using the notations in Theorem \ref{teo}, by the way.
{\lem\label{ot} Let $ c : X \times Y \rightarrow \mathbb{R} \cup \left\lbrace +\infty\right\rbrace $ be a ${\mathfrak{B}_{X\times Y}}-$measurable,
cost function, and
$ \psi :X\rightarrow \mathbb{R}\cup \left\lbrace +\infty\right\rbrace \ \mathrm{and}\ \phi :Y\rightarrow \mathbb{R}\cup \left\lbrace -\infty\right\rbrace $
be a pair of competitive prices, i.e.
\[ \forall \, \left( x,y\right) \in X\times Y, \ \phi \left( y\right) -\psi\left( x\right) \le c\left( x,y\right) . \] Let $\pi$ be a probability measure on ${\mathfrak{B}_{X\times Y}}$, with marginal measures $\mu$ on ${\mathfrak{B}_X}$ and $\nu$ on ${\mathfrak{B}_Y}$. (i.e. $\pi$ is a transference plan between $\mu$ and $\nu$.) Assume that $ \phi -\psi$ and $ c $ are $\pi-$integrable and \[ \phi \left( y\right) -\psi\left( x\right) = c\left( x,y\right) , \ \pi-\mathrm{a.s.} . \] Then \begin{equation}\label{intes01} \psi\left( x\right) = \sup_{y\in Y}\left(\phi \left( y\right) - c\left( x,y\right) \right) , \ \mu-\mathrm{a.s.} \end{equation} and \begin{equation}\label{intes02}
\phi \left( y\right) = \inf_{x\in X}\left( \psi \left( y\right) + c\left( x,y\right)\right) , \ \nu-\mathrm{a.s.}
\end{equation} } \vskip 2mm \noindent \emph{Proof}: Since $ \psi+ c-\phi = 0 $, $\pi-$a.s., \[\int_{X\times Y} \left( \psi(x)+ c(x,y)-\phi(y) \right) \pi\left(dx,dy\right)= 0. \] Using the decomposition given by Theorem \ref{teo}, equation (\ref{int0f}), \begin{equation}\label{nullint} \int_X \left( \int_Y \left( \psi(x)+ c(x,y)-\phi(y) \right) \nu_x\left(dy\right) \right) \mu(dx)=0. \end{equation} Since $ \psi+ c-\phi \ge 0 $, \begin{equation}\label{sup} \psi(x)\ge \sup_{y\in Y} \left(\phi(y) - c(x,y) \right) \end{equation} and, for all $x$ where it is defined, the function \[ x\longrightarrow \int_Y \left( \psi(x)+ c(x,y)-\phi(y) \right) \nu_x\left(dy\right)
\] is nonnegative. By (\ref{nullint}), \[ \int_Y \left( \psi(x)+ c(x,y)-\phi(y) \right) \nu_x\left(dy\right) = 0,\ \mu-\mathrm{a.s.} \] Then,
\begin{equation}\label{intas} \psi(x)= \int_Y \left(\phi(y) - c(x,y)\right) \nu_x\left(dy\right)\le \sup_{y\in Y} \left(\phi(y) - c(x,y) \right) ,\ \mu-\mathrm{a.s.} \end{equation} Combining (\ref{sup}) and (\ref{intas}), we obtain (\ref{intes01}). Equation (\ref{intes02}) is validated in a similar way.
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\setcounter{page}{271} \title[Fuzzy functions and $L$-Top]{Fuzzy functions and an extension of the category $L$-Top of Chang-Goguen $L$-topological spaces} \author{Alexander P. \v{S}ostak} \address{Department of Mathematics\\ University of Latvia\\ Riga\\ Latvia} \email{[email protected]} \thanks{Partly supported by grant 01.0530 of Latvijas Zin\=atnes Padome} \subjclass[2000]{03E72, 18A05, 54A40} \keywords{Fuzzy category} \thanks{This article will be revised and submitted for publication elsewhere.} \thanks{Alexander P. \v{S}ostak, {\em Fuzzy functions and an extension of the category $L$-Top of Chang-Goguen $L$-topological spaces}, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp.~271--294, Topology Atlas, Toronto, 2002} \begin{abstract} We study $\mathcal{F}TOP(L)$, a fuzzy category with fuzzy functions in the role of morphisms. This category has the same objects as the category L-TOP of Chang-Goguen L-topological spaces, but an essentially wider class of morphisms---so called fuzzy functions introduced earlier in our joint work with U. H\"ohle and H. Porst. \end{abstract} \maketitle
\section*{Introduction}
In research works where fuzzy sets are involved, in particular, in Fuzzy Topology, mostly certain usual functions are taken as morphisms: they can be certain mappings between corresponding sets, or between the fuzzy powersets of these sets, etc. On the other hand, in our joint works with U.~H\"ohle and H.E.~Porst \cite{HPS1}, \cite{HPS2} a certain class of $L$-relations (i.e.\ mappings $F: X\times Y \to L$) was distinguished which we view as ($L$-){\it fuzzy functions} from a set $X$ to a set $Y$; these fuzzy functions play the role of morphisms in an {\it $L$-fuzzy category} of sets {$\mathcal{F}SET(L)$}, introduced in \cite{HPS2}.
Later on we constructed a fuzzy category {$\mathcal{F}TOP(L)$}\ related to topology with fuzzy functions in the role of morphisms, see \cite{So2000}. Further, in \cite{So2001} a certain uniform counterpart of {$\mathcal{F}TOP(L)$}\ was introduced. Our aim here is to continue the study of {$\mathcal{F}TOP(L)$}. In particular, we show that the top frame {$\mathcal{F}TOP(L)$}$^\top$ of the fuzzy category {$\mathcal{F}TOP(L)$}\ is a topological category in H. Herrlich's sense \cite{AHS}) over the top frame {$\mathcal{F}SET(L)$}$^\top$ of the fuzzy category {$\mathcal{F}SET(L)$}.
In order to make exposition self-contained, we start with Section 1 Prerequisities, where we briefly recall the three basic concepts which are essentially used in this work: they are the concepts of a $GL$-monoid (see e.g.\ \cite{Ho91}, \cite{Ho94}, etc.); of an $L$-valued set (see e.g.\ \cite{Ho92}, etc.), and of an $L$-fuzzy category (see e.g.\ \cite{So91}, \cite{So92}, \cite{So97}, etc.). In Section 2 we consider basic facts about fuzzy functions and introduce the $L$-fuzzy category {$\mathcal{F}SET(L)$}\ \cite {HPS1}, \cite{HPS2}. The properties of this fuzzy category and some related categories are the subject of Section 3. {$\mathcal{F}SET(L)$}\ is used as the ground category for the $L$-fuzzy category {$\mathcal{F}TOP(L)$}\ whose objects are Chang-Goguen $L$-topological spaces \cite{Ch}, \cite{Go73}, and whose morphisms are certain fuzzy functions, i.e.\ morphisms from {$\mathcal{F}SET(L)$}. Fuzzy category {$\mathcal{F}TOP(L)$}\ is considered in Section 4. Its crisp top frame {$\mathcal{F}TOP(L)^\top$}\ is studied in Section 5. In particular, it is shown that {$\mathcal{F}TOP(L)^\top$}\ is a topological category over {$\mathcal{F}SET(L)$}$^\top$. Finally, in Section 6 we consider the behaviour of the topological property of compactness with respect to fuzzy functions --- in other words in the context of the fuzzy category {$\mathcal{F}TOP(L)$}\ and, specifically, in the context of the category {$\mathcal{F}TOP(L)^\top$}.
\section{Prerequisities}
\subsection{$GL$-monoids}
Let $(L, \leq)$ be a complete infinitely distributive lattice, i.e.\ $(L, \leq)$ is a partially ordered set such that for every subset $A \subset L$ the join $\bigvee A$ and the meet $\bigwedge A$ are defined and $(\bigvee A) \wedge \alpha = \bigvee \{ a\wedge \alpha) \mid a \in A \}$ and $(\bigwedge A) \vee \alpha = \bigwedge \{a\vee \alpha) \mid a \in A \}$ for every $\alpha \in L$. In particular, $\bigvee L =: \top$ and $\bigwedge L =: \bot$ are respectively the universal upper and the universal lower bounds in $L$. We assume that $\bot \ne \top$, i.e.\ $L$ has at least two elements.
A $GL-$monoid (see \cite{Ho91}, \cite{Ho92}, \cite{Ho94}) is a complete lattice enriched with a further binary operation $*$, i.e.\ a triple $(L, \leq, *)$ such that: \begin{enumerate} \item[(1)] $*$ is monotone, i.e.\ $\alpha \leq \beta$ implies $\alpha * \gamma \leq \beta * \gamma$, $\forall \alpha, \beta, \gamma \in {\it L}$; \item[(2)] $*$ is commutative, i.e.\ $\alpha * \beta = \beta * \alpha$, $\forall \alpha, \beta \in {\it L}$; \item[(3)] $*$ is associative, i.e.\ $\alpha * (\beta * \gamma) = (\alpha * \beta) * \gamma$, $\forall \alpha, \beta, \gamma \in L$; \item[(4)] $(L,\leq,*)$ is integral, i.e.\ $\top$ acts as the unity: $\alpha * \top = \alpha$, $\forall \alpha \in {\it L}$; \item[(5)] $\bot$ acts as the zero element in $(L, \leq, *)$, i.e.\ $\alpha * \bot = \bot$, $\forall \alpha \in {\it L}$; \item[(6)] $*$ is distributive over arbitrary joins, i.e.\ $\alpha * (\bigvee_j \beta_j) = \bigvee_j (\alpha * \beta_j)$, $\forall \alpha \in {\it L}, \forall \{ \beta_j : j \in J \} \subset {\it L}$; \item[(7)] $(L, \leq, *)$ is divisible, i.e.\ $\alpha \leq \beta$ implies existence of $\gamma \in L$ such that $\alpha = \beta * \gamma$. \end{enumerate}
It is known that every $GL-$monoid is residuated, i.e.\ there exists a further binary operation ``$\longmapsto$'' (implication) on $L$ satisfying the following condition: $$\alpha * \beta \leq \gamma \Longleftrightarrow \alpha \leq (\beta \longmapsto \gamma) \qquad \forall \alpha, \beta, \gamma \in L.$$ Explicitly implication is given by $$\alpha \longmapsto \beta = \bigvee \{ \lambda \in L \mid \alpha * \lambda \leq \beta \}.$$
Below we list some useful properties of $GL-$monoids (see e.g.\ \cite{Ho91}, \cite{Ho92}, \cite{Ho94}): \begin{enumerate} \item[(i)] $\alpha \longmapsto \beta = \top \Longleftrightarrow \alpha \leq \beta$; \item[(ii)] $\alpha \longmapsto (\bigwedge_i \beta_i) = \bigwedge_i (\alpha \longmapsto \beta_i)$; \item[(iii)] $(\bigvee_i \alpha_i) \longmapsto \beta = \bigwedge_i (\alpha_i \longmapsto \beta)$; \item[(v)] $\alpha * (\bigwedge_i \beta_i) = \bigwedge_i (\alpha * \beta_i)$; \item[(vi)] $(\alpha \longmapsto \gamma ) * (\gamma \longmapsto \beta) \leq \alpha \longmapsto \beta$; \item[(vii)] $\alpha * \beta \leq (\alpha * \alpha) \vee (\beta * \beta)$. \end{enumerate}
Important examples of $GL$-monoids are Heyting algebras and $MV$-alg\-ebras. Namely, a {\it Heyting algebra} is $GL$-monoid of the type $(L,\leq,\wedge,\vee,\wedge)$ (i.e.\ in case of a Heyting algebra $\wedge = *$), cf.\ e.g.\ \cite{Jhst}. A $GL$-monoid is called an {\it $MV$-algebra} if $(\alpha \longmapsto \bot) \longmapsto \bot = \alpha \quad \forall \alpha \in L$, \cite{Ch58}, \cite{Ch59}, see also \cite[Lemma 2.14]{Ho94}. Thus in an $MV$-algebra an order reversing involution $^c: L \to L$ can be naturally defined by setting $\alpha^c := \alpha \longmapsto \bot \quad \forall \alpha \in L$.
If $X$ is a set and $L$ is a $GL$-monoid, then the fuzzy powerset $L^X$ in an obvious way can be pointwise endowed with a structure of a $GL$-monoid. In particular the $L$-sets $1_X$ and $0_X$ defined by $1_X (x):= \top$ and $0_X (x) := \bot$ $\forall x \in X$ are respectively the universal upper and lower bounds in $L^X$.
In the sequel $L$ denotes an arbitrary $GL$-monoid.
\subsection{$L$-valued sets}
Following U.~H\"ohle (cf.\ e.g.\ \cite{Ho92}) by a (global) {\it $L$-valued set} we call a pair $(X,E)$ where $X$ is a set and $E$ is an {\it $L$-valued equality}, i.e.\ a mapping $E: X \times X \to L$ such that \begin{enumerate} \item[(1eq)] $E(x,x) = \top$ \item[(2eq)] $E(x,y) = E(y,x) \quad \forall x, y \in X$; \item[(3eq)] $E(x,y)*E(y,z) \leq E(x,z) \quad \forall x, y, z \in X$. \end{enumerate}
A mapping $f: (X,E_X) \to (Y,E_Y)$ is called {\it extensional} if $$E_X(x,x') \leq E_Y(f(x),f(x'))\ \forall x,x' \in X.$$ Let $SET(L)$ denote the category whose objects are $L$-valued sets and whose morphisms are extensional mappings between the corresponding $L$-valued sets. Further, recall that an $L$-set, or more precisely, an $L$-subset of a set $X$ is just a mapping $A: X \to L$. In case $(X,E)$ is an $L$-valued set, its $L$-subset $A$ is called {\it extensional} if $$\bigvee_{x\in X} A(x) * E(x,x') \leq A(x') \quad \forall x' \in X.$$
\subsection{{\it L}-fuzzy categories}
\begin{definition}[\cite{So91}, \cite{So92}, \cite{So97}, \cite{So99}] An {\it L}-fuzzy category is a quintuple $\mathcal{C} = (\OBC, \omega, \MC, \mu, \circ)$ where $\mathcal{C}_\bot = (\mathcal{O}b(\mathcal{C}), \mathcal{M}(\mathcal{C}), \circ)$ is a usual (classical) category called {\em the bottom frame} of the fuzzy category $\mathcal{C}$; $\omega: \mathcal{O}b(\mathcal{C}) \longrightarrow {\it L} $ is an $L$-subclass of the class of objects $\mathcal{O}b(\mathcal{C})$ of $\mathcal{C}_\bot$ and $\mu : \mathcal{M}(\mathcal{C}) \longrightarrow {\it L} $ is an $L$-subclass of the class of morphisms $\mathcal{M}(\mathcal{C})$ of $\mathcal{C}_\bot$. Besides $\omega$ and $\mu$ must satisfy the following conditions: \begin{enumerate} \item[(1)] if $f: X \to Y$, then $ \mu (f) \leq \omega (X) \wedge \omega (Y)$; \item[(2)] $\mu (g \circ f) \geq \mu (g) * \mu (f)$ whenever composition $g \circ f$ is defined; \item[(3)] if $e_X: X \to X$ is the identity morphism, then $\mu(e_X) = \omega(X)$. \end{enumerate} \end{definition}
Given an {\it L}-fuzzy category $ \mathcal{C} = (\OBC, \omega, \MC, \mu, \circ)$ and $X \in \mathcal{O}b(\mathcal{C})$, the intuitive meaning of the value $\omega (X)$ is the {\it degree} to which a potential object $X$ of the {\it L}-fuzzy category $\mathcal{C}$ is indeed its object; similarly, for $f \in \mathcal{M}(\mathcal{C})$ the intuitive meaning of $\mu (f)$ is the degree to which a potential morphism $f$ of $\mathcal{C}$ is indeed its morphism.
\begin{definition} Let $\mathcal{C} = (\OBC, \omega, \MC, \mu, \circ)$ be an {\it L}-fuzzy category. By an ({\it L}-fuzzy) subcategory of $\mathcal{C}$ we call an {\it L}-fuzzy category $$\mathcal{C}' = (\OBC, \omega', \MC, \mu', \circ)$$ where $\omega' \leq \omega$ and $\mu' \leq \mu$. A subcategory $\mathcal{C}'$ of the category $\mathcal{C}$ is called full if $\mu'(f) = \mu(f) \wedge \omega'(X) \wedge \omega'(Y)$ for every $f \in \mathcal{M}_\mathcal{C} (X,Y)$, and all $X$,$Y \in \mathcal{O}b(\mathcal{C})$. \end{definition}
Thus an {\it L}-fuzzy category and its subcategory have the same classes of potential objects and morphisms. The only difference of a subcategory from the whole category is in {\it L}-fuzzy classes of objects and morphisms, i.e.\ in the belongness degrees of potential objects and morphisms.
Let $\mathcal{C} = (\mathcal{O}b(\mathcal{C}), \mathcal{M}(\mathcal{C}), \circ)$ be a crisp category and $\mathcal{D} = (\mathcal{O}b(\mathcal{D}), \mathcal{M}(\mathcal{D}), \circ)$ be its subcategory. Then for every $GL$-monoid {\it L}\ the category $\mathcal{D}$ can be identified with the {\it L}-fuzzy subcategory $$\tilde{\mathcal{D}} = (\mathcal{O}b(\mathcal{C}), \omega', \mathcal{M}(\mathcal{C}), \mu', \circ)$$ of $\mathcal{C}$ such that $\omega'(X) = \top$ if $X \in \mathcal{O}b(\mathcal{D})$ and $\omega'(X) = \bot$ otherwise; $\mu' (f) = \top$ if $f \in \mathcal{M}(\mathcal{D})$ and $\mu' (f) = \bot$ otherwise. In particular, ${\tilde{\mathcal{D}}}_\top = \mathcal{D}$.
On the other hand sometimes it is convenient to identify a fuzzy subcategory $$\mathcal{C}' = (\OBC, \omega', \MC, \mu', \circ)$$ of the fuzzy category $$\mathcal{C} = (\OBC, \omega, \MC, \mu, \circ)$$ with the fuzzy category $$\mathcal{D} = (\OBD, \omega_\mathcal{D}, \MD, \mu_\mathcal{D},\circ)$$ where $$\mathcal{O}b(\mathcal{D}) := \{X \in \mathcal{O}b(\mathcal{C}) \mid \omega'(X) \ne \bot \}$$ and $$\mathcal{M}(\mathcal{D}) := \{f \in \mathcal{M}(\mathcal{C}) \mid \mu'(f) \ne \bot \}$$ and $\omega_\mathcal{D}$ and $\mu_\mathcal{D}$ are restrictions of $\omega'$ and $\mu'$ to $\mathcal{O}b(\mathcal{D})$ and $\mathcal{M}(\mathcal{D})$ respectively.
\section{Fuzzy functions and fuzzy category {$\mathcal{F}SET(L)$}}
As it was already mentioned above, the concept of a fuzzy function and the corresponding fuzzy category {$\mathcal{F}SET(L)$}\ were introduced in \cite{HPS1}, \cite{HPS2}. There were studied also basic properties of fuzzy functions. In this section we recall those definitions and results from \cite{HPS2} which will be needed in the sequel\footnote{Actually, the subject of \cite{HPS1}, \cite{HPS2} was a more general fuzzy category $L$-{$\mathcal{F}SET(L)$}\ containing {$\mathcal{F}SET(L)$}\ as a full subcategory. However, since for the merits of this work the category {$\mathcal{F}SET(L)$}\ is of importance, when discussing results from \cite{HPS1}, \cite{HPS2} we reformulate (simplify) them for the case of {$\mathcal{F}SET(L)$}\ without mentioning this every time explicitly}. Besides some new needed facts about fuzzy functions will be established here, too.
\subsection{Fuzzy functions and category {$\mathcal{F}SET(L)$}}
\begin{definition}[cf.\ {\cite[2.1]{HPS1}}] A fuzzy function\footnote{Probably, the name {\it an $L$-fuzzy function} would be more adequate here. However, since the $GL$-monoid $L$ is considered to be fixed, and since the prefix ``$L$'' appears in the text very often, we prefer to say just {\it a fuzzy function}} $F$ from an $L$-valued set $(X,E_X)$ to $(Y,E_Y)$ (in symbols $F: (X,E_X) \rightarrowtail (Y,E_Y))$) is a mapping $F: X \times Y \to L$ such that \begin{enumerate} \item[(1ff)] $F(x,y) * E_Y(y,y') \leq F(x,y') \quad \forall x \in X, \forall y,y' \in Y$; \item[(2ff)] $E_X(x,x') * F(x,y) \leq F(x',y) \quad \forall x,x' \in X, \forall y \in Y$; \item[(3ff)] $F(x,y) * F(x,y') \leq E_Y(y,y') \quad \forall x \in X, \forall y,y' \in Y$. \end{enumerate} \end{definition}
{\small Notice that conditions (1ff)--(2ff) say that $F$ is a certain $L-$relation, while axiom (3ff) together with evaluation $\mu(F)$ (see Subsection \ref{fsetl}) specify that the $L$-relation $F$ is a fuzzy {\it function}}.
\begin{remark} Let $F: (X,E_X) \rightarrowtail (Y,E_Y)$ be a fuzzy function, $X' \subset X$, $Y' \subset Y$, and let the $L$-valued equalities $E_{X'}$ and $E_{Y'}$ on $X'$ and $Y'$ be defined as the restrictions of the equalities $E_X$ and $E_Y$ respectively. Then defining a mapping $F': X'\times Y' \to L$ by the equality $F'(x,y) = F(x,y)\ \forall x \in X', \forall y \in Y'$ a fuzzy function $F': (X',E_{X'}) \rightarrowtail (Y',E_{Y'})$ is obtained. We refer to it as the {\it restriction of $F$} to the subspaces $(X',E_{X'})$ $(Y',E_{Y'})$ \end{remark}
Given two fuzzy functions $F: (X,E_X) \rightarrowtail (Y,E_Y)$ and $G: (Y,E_Y) \rightarrowtail (Z,E_Z)$ we define their {\it composition} $G \circ F: (X,E_X) \rightarrowtail (Z,E_Z)$ by the formula $$(G\circ F)(x,z) = \bigvee_{y\in Y} \bigl( F(x,y) * G(y,z))\bigr).$$
In \cite{HPS2} it was shown that the composition $G \circ F$ is indeed a fuzzy function and that the operation of composition is associative. Further, if we define the identity morphism by the corresponding $L$-valued equality: $E_X: (X,E_X) \rightarrowtail (X,E_X),$ we come to a category {$\mathcal{F}SET(L)$}\ whose objects are $L$-valued sets and whose morphisms are fuzzy functions $F: (X,E_X) \rightarrowtail (Y,E_Y)$.
\subsection{Fuzzy category {$\mathcal{F}SET(L)$}}\label{fsetl}
Given a fuzzy function $F: (X,E_X) \rightarrowtail (Y,E_Y)$ let $$\mu(F) = \inf_x \sup_y F(x,y).$$ Thus we define an $L$-subclass $\mu$ of the class of all morphisms of {$\mathcal{F}SET(L)$}. In case $\mu(F) \geq \alpha$ we refer to $F$ as a {\it fuzzy $\alpha$-function}.
If $F: (X,E_X) \rightarrowtail (Y,E_Y) \mbox{ and } G: (Y,E_Y) \rightarrowtail (Z,E_Z)$ are fuzzy functions, then $\mu (G \circ F) \geq \mu(G) * \mu(F)$ \cite{HPS2}. Further, given an $L$-valued set $(X,E)$ let $\omega(X,E) := \mu(E) = \inf_x E(x,x) = \top$. Thus a {\it fuzzy category} {$\mathcal{F}SET(L)$} = $(FSET(L), \omega, \mu)$ is obtained.
\begin{example}\label{ex-crisp} Let $* = \wedge$ and $E_Y$ be a crisp equality on $Y$, i.e.\ $E_Y(y,y') = \top \mbox{ iff } y = y'$, and $E_Y(y,y') = \bot$ otherwise. Then every fuzzy function $F: (X,E_X) \rightarrowtail (Y,E_Y)$ such that $\mu(F) = \top$ is uniquely determined by a usual function $f: X \to Y$. Indeed, let $f(x) = y \mbox{ iff } F(x,y) = \top$. Then condition (3ff) implies that there cannot be $f(x)=y$, $f(x) = y'$ for two different $y$, $y' \in Y$ and condition $\mu(F) = \top$ guarantees that for every $x \in X$ one can find $y \in Y$ such that $f(x)=y$. If besides $E_X$ is crisp, then, vice versa, every mapping $f: X \to Y$ can be viewed as a fuzzy mapping $F: (X,E_X) \rightarrowtail (Y,E_Y)$ (since the conditions of extensionality (2ff) and (3ff) are automatically fulfilled in this case) \end{example}
\begin{remark} If $F'\colon (X',E_{X'}) \rightarrowtail (Y,E_{Y})$ is the restriction of $F\colon (X,E_X) \rightarrowtail (Y,E_Y)$ (see Remark above) and $\mu(F) \geq \alpha$, then $\mu(F') \geq \alpha$. However, generally the restriction $F'\colon (X',E_{X'}) \rightarrowtail (Y',E_{Y'})$ of $F\colon (X,E_X) \rightarrowtail (Y,E_Y)$ may fail to satisfy condition $\mu(F') \geq \alpha$. \end{remark}
\subsection{Images and preimages of $L$-sets under fuzzy functions}\label{impref}
Given a fuzzy function $F: (X, E_X) \rightarrowtail (Y, E_Y)$ and $L$-subsets $A: X \to L$ and $B: Y \to L$ of $X$ and $Y$ respectively, we define the fuzzy set $F^{\rightarrow}(A): Y \to L$ (the image of $A$ under $F$) by the equality $F^{\rightarrow}(A)(y) = \bigvee_x F(x,y) * A(x)$ and the fuzzy set $F^{\leftarrow}(B): X \to L$ (the preimage of $B$ under $F$) by the equality $F^{\leftarrow}(B)(x) = \bigvee_y F(x,y) * B(y)$.
Note that if $A \in L^X$ is extensional, then $F^{\rightarrow}(A) \in L^Y$ is extensional (by (2ff)) and if $B \in L^Y$ is extensional, then $F^{\leftarrow}(B) \in L^X$ is extensional (by (3ff)).
\begin{proposition}[Basic properties of images and preimages of $L$-sets under fuzzy functions] \label{im-pr} \mbox{} \begin{enumerate} \item $F^{\rightarrow}(\bigvee_{i \in \mathcal{I}} (A_i) = \bigvee_{i \in \mathcal{I}} F^{\rightarrow}(A_i) \qquad \forall \{A_i: i \in {\mathcal{I}} \} \subset L^X$; \item $F^{\rightarrow}(A_1\bigwedge A_2) \leq F^{\rightarrow}(A_1) \bigwedge F^{\rightarrow}(A_2) \qquad \forall A_1, A_2 \in L^X$; \item If $L$-sets $B_i$ are {\em extensional}, then $$ \bigwedge_{i \in {\mathcal{I}}} F^{\leftarrow}(B_i)*\mu(F)^2 \leq F^{\leftarrow}(\bigwedge_{i \in \mathcal{I}} B_i) \leq \bigwedge_{i \in \mathcal{I}} F^{\leftarrow}(B_i) \qquad \forall \{B_i: i \in \mathcal{I} \} \subset L^Y. $$ In particular, if $\mu(F) = \top$, then $F^{\leftarrow}(\bigwedge_{i\in \mathcal{I}} B_i) = \bigwedge_{i \in \mathcal{I}} F^{\leftarrow}(B_i)$ for every family of extensinal $L$-sets $\{B_i: i \in \mathcal{I} \} \subset L^Y$. \item $F^{\leftarrow}(\bigvee_{i \in \mathcal{I}} B_i) = \bigvee_{i \in \mathcal{I}} F^{\leftarrow}(B_i) \qquad \forall \{B_i: i \in {\mathcal{I}} \} \subset L^Y$. \item $A*\mu(F)^2 \leq F^{\leftarrow}(F^{\rightarrow}(A))$ for every $A \in L^X$. \item $F^{\to}\bigl(F^{\gets}(B)\bigr) \leq B$ for every {\em extensional} $L$-set $B\in L^Y$. \item $F^{\leftarrow}(c_Y) \geq \mu(F) * c$ where $c_Y: Y \to L$ is the constant function taking value $c \in L$. In particular, $F^{\leftarrow}(c_Y) = c$ if $\mu(F) = \top$. \end{enumerate} \end{proposition}
\begin{proof} (1). $$ \begin{array}{lll} \bigl(\bigvee_i F^{\rightarrow}(A_i)\bigr)(y)& =& \bigvee_i \bigvee_x \bigl(F(x,y) * A_i(x) \bigr)\\ & =& \bigvee_x \bigvee_i \bigl(F(x,y) * A_i (x)\bigr)\\ & =& \bigvee_x (F(x,y) * (\bigvee_i A_i)(x))\\ & =& F^{\rightarrow}(\bigvee_i A_i)(y). \end{array} $$
(2). The validity of (2) follows from the monotonicity of $F^{\to}$.
(3). To prove property 3 we first establish the following inequality \begin{equation}\label{basic1} \bigvee_{y\in Y}\bigl(F(x,y)\bigr)^2 \geq \bigl(\bigvee_{y\in Y} F(x,y)\bigr)^2. \end{equation} Indeed, by a property (vii) of a GL-monoid $$ \begin{array}{lll} \Bigl(\bigvee_{y\in Y} F(x,y)\Bigr)^2& =& \Bigl(\bigvee_{y\in Y} F(x,y)\Bigr) * \Bigl(\bigvee_{y'\in Y} F(x,y')\Bigr)\\ & =& \bigvee_{y,y'\in Y} \Bigl(F(x,y) * F(x,y') \Bigr)\\ & \leq& \bigvee_{y,y'\in Y} \Bigl(F(x,y)^2 \vee F(x,y')^2\Bigr)\\ & =& \bigvee_{y\in Y} \Bigl(F(x,y)\Bigr)^2. \end{array}$$ In particular, it follows from (\ref{basic1}) that \begin{equation}\label{basic2} \forall x \in X\ \bigvee_{y\in Y} \bigl(F(x,y)\bigr)^2 \geq \mu(F)^2. \end{equation} Now, applying (\ref{basic2}) and taking into account extensionality of $L$-sets $B_i$, we proceed as follows: $$ \begin{array}{lll} \multicolumn{3}{l}{ \Bigl(\bigwedge_i F^{\gets}(B_i)\Bigr)(x) * \bigl(\mu(F)\bigr)^2 } \\ & \leq& \Bigl(\bigwedge_i \bigl(\bigvee_{y_i \in Y} (F(x,y_i)*B_i(y_i)\bigr)\Bigr) * \bigvee_{y\in Y}\bigl(F(x,y)\bigr)^2\\ & =& \bigvee_{y\in Y} \Bigl(\bigl(F(x,y)\bigr)^2 * \bigwedge_i\bigl(\bigvee_{y_i \in Y} F(x,y_i) * B_i(y_i)\bigr)\\ & =& \bigvee_{y\in Y}\Bigl(F(x,y) * \bigl(\bigwedge_i \bigl(F(x,y) * \bigl(\bigvee_{y_i \in Y} F(x,y_i) * B_i(y_i)\bigr)\bigr)\Bigr)\\ & =& \bigvee_{y\in Y}\Bigl(F(x,y) * \bigl(\bigwedge_i \bigl(\bigvee_{y_i\in Y} (F(x,y) * F(x,y_i)\bigr) \bigr) * B_i(y_i)\bigr)\Bigr)\\ & \leq& \bigvee_{y\in Y}\Bigl(F(x,y) * \bigl(\bigwedge_i \bigl(\bigvee_{y_i \in Y} E(y,y_i) * B_i(y_i)\bigr)\bigr)\Bigr)\\ & \leq& \bigvee_{y\in Y}\bigl(F(x,y) * \bigl(\bigwedge_i B_i(y)\bigr)\bigr)\\ & =& F^{\gets}\bigl(\bigwedge_i B_i)\bigr)(x),$$ \end{array} $$ and hence $$\Bigl(\bigwedge_i F^{\gets}(B_i)\Bigr)* \bigl(\mu(F)\bigr)^2 \leq F^{\gets}\bigl(\bigwedge_i B_i)\bigr).$$ To complete the proof notice that the inequality $$F^{\leftarrow}(\bigwedge_{i \in \mathcal{I}} B_i) \leq \bigwedge_{i \in \mathcal{I}} F^{\leftarrow}(B_i)$$ is obvious.
(4). The proof of (4) is similar to the proof of (1) and is therefore omitted.
(5). Let $A \in L^X$, then $$ \begin{array}{lll} F^{\leftarrow}(F^{\rightarrow}(A))(x)& =& \bigvee_y (F(x,y)*F^{\rightarrow}(A)(y))\\ & =& \bigvee_y \bigl(F(x,y)*\bigl(\bigvee_{x'} F(x',y) * A(x') \bigr)\bigr)\\ & \geq& \bigvee_y (F(x,y)^2 * A(x))\\ & \geq& (\mu(F))^2 * A(x) \end{array} $$ for every $x\in X$, and hence 5 holds.
(6). To show property 6 assume that $B\in L^Y$ is extensional. Then $$ \begin{array}{lll} F^{\rightarrow}(F^{\leftarrow}(B))(y)& =& \bigvee_x\bigl(F(x,y) * F^{\gets}(B)(x)\bigr)\\ & =& \bigvee_x F(x,y) * \bigl(\bigvee_{y'} F(x,y') * B(y') \bigr)\\ & =& \bigvee_{x \in X, y\in Y} \bigl(F(x,y) * F(x,y') * B(y')\bigr)\\ & \leq& E_Y(y,y')*B(y')\\ & \leq& B(y), \end{array} $$ and hence $F^{\to}\bigl(F^{\gets}(B)\bigr) \leq B$.
(7). The proof of property 7 is straightforward and therefore omitted. \end{proof}
\begin{comments} \mbox{} \begin{enumerate} \item Properties 1,2 and 4 were proved in \cite[Proposition 3.2]{HPS2}. Here we reproduce these proofs in order to make the article self-contained. \item The inequality in item 2 of the previous proposition obviously cannot be improved even in the crisp case. \item One can show that the condition of extensionality cannot be omitted in items 3 and 6. \item The idea of the proof of Property 3 was communicated to the author by U.~H\"ohle in Prague at TOPOSYM in August 2001. \item In \cite{HPS2} there was established the following version of Property 3 without the assumption of extensionality of $L$-sets $B_i$ in case $L$ is completely distributive: $$ {(\bigwedge_{i \in \mathcal{I}} F^{\leftarrow}(B_i))}^5 \leq F^{\leftarrow}(\bigwedge_{i \in \mathcal{I}} B_i) \leq \bigwedge_{i \in \mathcal{I}} F^{\leftarrow}(B_i) \qquad \forall \{B_i: i \in \mathcal{I} \} \subset L^Y$$ and $$ \bigwedge_{i \in \mathcal{I}} F^{\leftarrow}(B_i) = F^{\leftarrow}(\bigwedge_{i \in \mathcal{I}} B_i) \qquad \forall \{B_i: i \in \mathcal{I} \} \subset L^Y, \mbox{ in case } * = \wedge$$ \end{enumerate} \end{comments}
\subsection{Injectivity, surjectivity and bijectivity of fuzzy functions}
\begin{definition}\label{inj} A fuzzy function $F: (X, E_X) \rightarrowtail (Y, E_Y)$ is called {\it injective}, if \begin{itemize} \item[(inj)] $F(x,y) * F(x',y) \leq E_X(x,x') \quad \forall x,x' \in X, \forall y \in Y$. \end{itemize} \end{definition}
\begin{definition}\label{sur} Given a fuzzy function $F: (X, E_X) \rightarrowtail (Y, E_Y)$, we define its degree of surjectivity by the equality: $$\sigma(F) := \inf_y \sup_x F(x,y)$$ In particular, a fuzzy function $F$ is called $\alpha$-surjective, if $\sigma(F) \geq \alpha$.
In case $F$ is injective and $\alpha$-surjective, it is called {\it $\alpha$-bijective}. \end{definition}
\begin{remark} Let $(X,E_X)$, $(Y,E_Y)$ be $L$-valued sets and $(X',E_{X'})$, $(Y',E_{Y'})$ be their subspaces. Obviously, the restriction $F'\colon (X',E_{X'}) \rightarrowtail (Y',E_{Y'})$ of an injection $F\colon (X,E_X) \rightarrowtail (Y,E_Y)$ is an injection. The restriction $F'\colon (X,E_{X}) \rightarrowtail (Y',E_{Y'})$ of an $\alpha$-surjection $F\colon (X,E_X) \rightarrowtail (Y,E_Y)$ is an $\alpha$-surjection. On the other hand, generally the restriction $F'\colon (X',E_{X'}) \rightarrowtail (Y',E_{Y'})$ of an $\alpha$-surjection $F\colon (X,E_X) \rightarrowtail (Y,E_Y)$ may fail to be an $\alpha$-surjection. \end{remark}
A fuzzy function $F: (X, E_X) \rightarrowtail (Y, E_Y)$ determines a fuzzy {\it relation} $F^{-1}: X\times Y \to L$ by setting $F^{-1}(y,x) = F(x,y)$\ $\forall x \in X$, $\forall y \in Y$.
\begin{proposition}[Basic properties of injections, $\alpha$-surjections and $\alpha$-bi\-jections] \label{in-sur} \mbox{} \begin{enumerate} \item $F^{-1} :(Y,E_Y) \rightarrowtail (X,E_X)$ is a fuzzy function iff $F$ is injective (actually $F^{-1}$ satisfies (3ff) iff F satisfies (inj)) \item $F$ is $\alpha$-bijective iff $F^{-1}$ is $\alpha$-bijective. \item If $F$ is injective, and L-sets $A_i$ are extensional, then $$ (\bigwedge_i F^{\rightarrow}(A_i))*(\sigma(F))^2 \leq F^{\rightarrow}(\bigwedge_i A_i) \leq \bigwedge_i F^{\rightarrow}(A_i) \qquad \forall \{A_i : i \in {\mathcal{I}}\} \subset L^X. $$ In particular, if $F$ is $\top$-bijective, then $$ (\bigwedge_i F^{\rightarrow}(A_i) = F^{\rightarrow}(\bigwedge_i A_i) \qquad \forall \{A_i : i \in {\mathcal{I}}\} \subset L^X.$$ \item $F^{\rightarrow}(F^{\leftarrow}(B)) \geq \sigma(F)^2 * B$. In particular, if $F$ is $\top$-surjective and $B$ is extensional, then $F^{\rightarrow}(F^{\leftarrow}(B)) = B$. \item $F^{\rightarrow}(c_X) \geq \sigma(F) * c$ where $c_X: X \to L$ is the constant function with value $c$. In particular, $F^{\rightarrow}(c_X) = c$ if $\sigma(F) = \top$. \end{enumerate} \end{proposition}
\begin{proof} Properties 1 and 2 follow directly from the definitions.
(2). The proof of Property 3 is analogous to the proof of item 3 of Proposition \ref{im-pr}:
First, reasoning as in the proof of $(\ref{basic1})$ we establish the following inequality \begin{equation}\label{basic3} \bigvee_{x\in X}\bigl(F(x,y)\bigr)^2 \geq \bigl(\bigvee_{x\in X} F(x,y)\bigr)^2. \end{equation} In particular, from here it follows that \begin{equation}\label{basic4} \forall y \in Y\ \bigvee_{x\in X} \bigl(F(x,y)^2\bigr) \geq \sigma(F)^2. \end{equation} Now, applying $(\ref{basic4})$ and taking into account extensionality of $L$-sets $A_i$, we proceed as follows: $$ \begin{array}{lll} \multicolumn{3}{l}{ \Bigl(\bigwedge_i F^{\to}(A_i)\Bigr)(x) * \bigl(\sigma(F)\bigr)^2 } \\ & \leq& \Bigl(\bigwedge_i \bigl(\bigvee_{x_i \in X} (F(x_i,y)*A_i(x_i)\bigr)\Bigr) * \bigvee_{x\in X}\bigl(F(x,y)\bigr)^2 \\ & =& \bigvee_{x\in X} \Bigl(\bigl(F(x,y)\bigr)^2 * \bigwedge_i\bigl(\bigvee_{x_i \in X} F(x_i,y) * A_i(x_i)\bigr) \\ & =& \bigvee_{x\in X}\Bigl(F(x,y) * \bigl(\bigwedge_i \bigl(F(x,y) * \bigl(\bigvee_{x_i \in X} F(x_i,y) * A_i(x_i)\bigr)\bigr)\Bigr) \\ & = & \bigvee_{x\in X}\Bigl(F(x,y) * \bigl(\bigwedge_i \bigl(\bigvee_{x_i\in X} (F(x,y) * F(x_i,y)\bigr) \bigr) * A_i(x_i)\bigr)\Bigr) \\ & \leq& \bigvee_{x\in X}\Bigl(F(x,y) * \bigl(\bigwedge_i \bigl(\bigvee_{x_i \in X} E_X(x,x_i) * A_i(x_i)\bigr)\bigr)\Bigr) \\ & \leq& \bigvee_{x\in X}\bigl(F(x,y) * \bigl(\bigwedge_i A_i(x)\bigr)\bigr) \\ & =& F^{\to}\bigl(\bigwedge_i A_i)\bigr)(y), \end{array} $$ and hence $$\Bigl(\bigwedge_i F^{\to}(A_i)\Bigr)* \bigl(\sigma(F)\bigr)^2 \leq F^{\to}\bigl(\bigwedge_i A_i)\bigr).$$ To complete the proof notice that the inequality $$F^{\rightarrow}(\bigwedge_{i \in \mathcal{I}} A_i) \leq \bigwedge_{i \in \mathcal{I}} F^{\rightarrow}(A_i)$$ is obvious.
(4). Let $B \in L^Y$, then $$ \begin{array}{lll} F^{\rightarrow}(F^{\leftarrow}(B))(y)& =& \bigvee_x \bigl(F(x,y)*F^{\gets}(B)(x)\bigr)\\ & =& \bigvee_x F(x,y) * \bigl(\bigvee_{y'} F(x,y') * B(y) \bigr)\\ & \geq& \bigvee_x F(x,y) * F(x,y) * B(y)\\ & \geq& \sigma (F)^2 * B(y), \end{array} $$ and hence the first inequality in item 4 is proved. From here and Proposition \ref{im-pr} (6) the second statement of item 4 follows.
(5) The proof of the last property is straightforward and therefore omitted. \end{proof}
\begin{question} We do not know whether inequalities in items 3, 4 and 5 can be improved. \end{question}
\begin{comments} \mbox{} \begin{enumerate} \item Properties 1 and 2 were first established in \cite{HPS2}. \item In \cite{HPS2} the following version of Property 3 was proved:\\ If $L$ is completely distributive and $F$ is injective, then $$ (\bigwedge_i F^{\rightarrow}(A_i))^5 \leq F^{\rightarrow}(\bigwedge_i A_i) \leq \bigwedge_i F^{\rightarrow}(A_i) \qquad \forall \{A_i : i \in {\mathcal{I}}\} \subset L^X $$ and $$F^{\rightarrow}(\bigwedge_i A_i) = \bigwedge_i F^{\rightarrow}(A_i) \quad \mbox{ in case } \wedge = *$$ No extensionality is assumed in these cases. \item In case of an ordinary function $f:X \to Y$ the equality $$f^{\to}\bigl(\bigwedge_{i\in \mathcal{I}} (A_i)\bigr) = \bigwedge_{i\in\mathcal{I}} f^{\to}(A_i)$$ holds just under assumption that $f$ is injective. On the other hand, in case of a fuzzy function $F$ to get a reasonable counterpart of this property we need to assume that $F$ is bijective. The reason for this, as we see it, is that in case of an ordinary function $f$, when proving the equality, we actually deal only with points belonging to the image $f(X)$, while the rest of $Y \setminus f(X)$ does not play any role. On the other hand, in case of a fuzzy function $F: X \rightarrowtail Y$ the whole $Y$ is ``an image of $X$ to a certain extent'', and therefore, when operating with images of $L$-sets, we need to take into account, to what extent a point $y$ is in the ``image'' of $X$. \end{enumerate} \end{comments}
\section{Further properties of fuzzy category {$\mathcal{F}SET(L)$}}
In this section we continue to study properties of the fuzzy category {$\mathcal{F}SET(L)$}. As different from the previous section, were our principal interest was in the ``set-theoretic'' aspect of fuzzy functions, here shall be mainly interested in their properties of ``categorical nature''.
First we shall specify the two (crisp) categories related to {$\mathcal{F}SET(L)$}: namely, its bottom frame {$\mathcal{F}SET(L)$}$^\bot$ (={$\mathcal{F}SET(L)$}\ (this category was introduced already in Section 2) and its top frame {$\mathcal{F}SET(L)$}$^\top$. The last one will be of special importance for us. By definition its morphisms $F$ satisfy condition $\mu(F) = \top$, and as we have seen in the previous section, fuzzy functions satisfying this condition ``behave themselves much more like ordinary functions'' than general fuzzy functions. Respectively, the results which we are able to establish about {$\mathcal{F}SET(L)$}$^\top$ and about topological category {$\mathcal{F}TOP(L)^\top$} based on it, are more complete and nice, then their more general counterparts.
Second, note that the ``classical'' category $SET(L)$ of $L$-valued sets can be naturally viewed as a subcategory of {$\mathcal{F}SET(L)$}$^\top$. In case $L=\{0,1\}$ obviously the two categories collapse into the category SET of sets. On the other hand, starting with the category SET (=SET$(\{0,1\})$) (i.e.\ $L=\{0,1\}$) of sets and enriching it with respective fuzzy functions, we obtain again the category SET as {$\mathcal{F}SET(L)$}$^\top$ and obtain the category of sets and partial functions as {$\mathcal{F}SET(L)$}$^\bot$.
\subsection{Preimages of $L$-valued equalities under fuzzy functions}
Let an $L$-valued set $(Y,E_Y)$, a set $X$ and a mapping $F: X \times Y \to L$ be given. We are interested to find the largest $L$-valued equality $E_X$ on $X$ for which $F: (X,E_X) \to (Y,E_Y)$ is a fuzzy function. This $L$-valued equality will be called {\it the preimage of $E_Y$ under $F$} and will be denoted $F^{\gets}(E_Y)$.
Note first that the axioms \begin{enumerate} \item[(1ff)] $F(x,y)*E_Y(y,y') \leq F(x,y')$, and \item[(3ff)] $F(x,y)*F(x,y') \leq E(y,y')$ \end{enumerate} do not depend on the $L$-valued equality on $X$ and hence we have to demand that the mapping $F$ originally satisfies them. To satisfy the last axiom \begin{enumerate} \item[(2ff)] $E_X(x,x') * F(x,y) \leq F(x',y)$ \end{enumerate} in an ``optimal way'' we define $$E_X(x,x') := \bigwedge_y \Bigl(\bigl( F(x,y) \longmapsto F(x',y)\bigr) \wedge \bigl(F(x',y) \longmapsto F(x,y) \bigr)\Bigr).$$ Then $E_X: X\times X \to L$ is an $L$-valued equality on $X$. Indeed, the validity of properties $E_X(x,x) = \top$ and $E_X(x,x') = E_X(x',x)$ is obvious. To establish the last property, i.e.\ $E_X(x,x') * E_X(x',x'') \leq E_X(x,x'')$, we proceed as follows: $$ \begin{array}{lll} \multicolumn{3}{l}{ E_X(x,x') * E_X(x',x'') } \\ & =& \bigwedge_y \Bigl( \bigl(F(x,y) \longmapsto F(x',y)\bigr) \wedge \bigl(F(x',y) \longmapsto F(x,y)\bigr) \Bigr) \\ & & * \bigwedge_y \Bigl( \bigl(F(x',y) \longmapsto F(x'',y)\bigr) \wedge \bigl(F(x'',y) \longmapsto F(x',y)\bigr) \Bigr) \\ & \leq& \bigwedge_y \Bigl( \bigl(F(x,y) \longmapsto F(x',y)\bigr) \wedge \bigl(F(x',y) \longmapsto F(x,y) \bigr) \\ & & \ * \bigl(F(x',y) \longmapsto F(x'',y)\bigr) \wedge \bigl(F(x'',y) \longmapsto F(x',y)\bigr) \Bigr) \\ & \leq& \bigwedge_y \Bigl( \bigl(F(x,y) \longmapsto F(x',y)\bigr) * \bigl(F(x',y) \longmapsto F(x,y) \bigr) \\ & & \ \wedge (\bigl(F(x',y) \longmapsto F(x'',y)\bigr) * \bigl(F(x'',y) \longmapsto F(x',y) \bigr) \Bigr) \\ & \leq& \bigwedge_y \Bigl( \bigl(F(x,y) \longmapsto F(x'',y)\bigr) \wedge \bigl(F(x'',y) \longmapsto F(x,y)\bigr) \Bigr) \\ & =& E_X(x,x''). \end{array} $$ Further, just from the definition of $E_X$ it is clear that $F$ satisfies the axiom {\it (2ff)} and hence it is indeed a fuzzy function $F: (X,E_X) \rightarrowtail (Y,E_Y).$ Moreover, from the definition of $E_X$ it is easy to note that it is really the largest $L$-valued equality on $X$ for which $F$ satisfies axiom {\it (2ff)}.
Finally, note that the value $\mu(F)$ is an inner property of the mapping $F: X\times Y \to L$ and does not depend on $L$-valued equalities on these sets.
\begin{question} We do not know whether the preimage $F^{\gets}(E_Y)$ is the initial structure for the source $F: X \rightarrowtail (Y,E_Y)$ in {$\mathcal{F}SET(L)$}. Namely, given an $L$-valued set $(Z,E_Z)$ and a ``fuzzy quasi-function'' $G: (Z,E_Z) \rightarrowtail X$ is it true that composition $F \circ G: (Z,E_Z) \rightarrowtail (Y,E_Y)$ is a fuzzy function if and only if $G: (Z,E_Z) \rightarrowtail (X,E_X)$ is a fuzzy function? By a fuzzy quasi-function we mean that $G$ satisfies properties {\it (1ff)} and {\it (3ff)} which do not depend on the equality on $X$. \end{question}
\subsection{Images of $L$-valued equalities under fuzzy functions}
Let an $L$-valued set $(X,E_X)$, a set $Y$ and a mapping $F: X \times Y \to L$ be given. We are interested to find the smallest $L$-valued equality $E_Y$ on $Y$ for which $F: (X,E_X) \to (Y,E_Y)$ is a fuzzy function. This $L$-valued equality will be called {\it the image of $E_X$ under $F$} and will be denoted $F^{\to}(E_X)$.
Note first that the axiom \begin{enumerate} \item[(2ff)] $E_X(x,x') * F(x,y) \leq F(x',y)$ \end{enumerate} does not depend on the $L$-valued equality on $Y$ and hence we have to demand that the mapping $F$ originally satisfies it. Therefore we have to bother that $F$ satisfies the remaining two axioms: \begin{enumerate} \item[(1ff)] $F(x,y)*E_Y(y,y') \leq F(x,y')$, and \item[(3ff)] $F(x,y)*F(x,y') \leq E(y,y')$ \end{enumerate} These conditions can be rewritten in the form of the double inequality: $$ \begin{array}{lll} F(x,y)*F(x,y')& \leq& E_Y(y,y')\\ & \leq& \bigl(F(x,y')\longmapsto F(x,y)\bigr) \wedge \bigl(F(x,y) \longmapsto F(x,y')\bigr). \end{array} $$ Defining $E_Y$ by the equality $$E_Y(y,y') = \bigvee_x \bigl(F(x,y) * F(x,y')\bigr),$$ we shall obviously satisfy both of them. Moreover, it is clear that $E_Y$ satisfies property (3ff) and besides $E_Y$ cannot be diminished without loosing this property. Hence we have to show only that $E_Y$ is indeed an $L$-valued equality. However, to prove this we need the assumption that $\sigma(F) = \top$, that is $F$ is $\top$-surjective. Note that $$E_Y(y,y) = \bigvee_x \bigl(F(x,y)*F(x,y)\bigr) \geq (\sigma(F))^2,$$ and hence the first axiom is justified in case $\sigma(F) = \top$.
The equality $E_Y(y,y') = E_Y(y',y)$ is obvious.
Finally, to establish the last property, we proceed as follows. Let $y,y',y'' \in Y$. Then $$ \begin{array}{lll} \multicolumn{3}{l}{ E_Y(y,y')*E_Y(y',y'') } \\ & =& \bigvee_x\bigl(F(x,y)*F(x,y')\bigr) \, * \, \bigvee_x\bigl(F(x,y')*F(x,y'')\bigr) \\ & =& \bigvee_{x,x'} \bigl(F(x,y)*F(x,y')* F(x',y')*F(x',y'')\bigr)\\ & \leq& \bigvee_{x,x'} \bigl(F(x,y)* E_X(x,x')*F(x',y'')\bigr)\\ & \leq& \bigvee_x \bigl(F(x,y) * F(x,y'')\bigr) \\ & =& E(y,y''). \end{array} $$
\begin{question} We do not know whether the image $F^{\to}(E_X)$ is the final structure for the sink $F: (X,E_X) \rightarrowtail Y$ in {$\mathcal{F}SET(L)$}\ in case $\sigma(F) = \top$. Namely, given an $L$-valued set $(Z,E_Z)$ and a ``fuzzy almost-function'' $G: Y \rightarrowtail (Z,E_Z) $ is it true that composition $F \circ G: (Z,E_Z) \rightarrowtail (Y,E_Y)$ is a fuzzy function if and only if $G: (Y,E_Y) \rightarrowtail (Z,E_Z)$ is a fuzzy function? By a fuzzy almost-function we mean that $G$ satisfies property (2ff) which does not depend on the equality on $Y$. \end{question}
\subsection{Products in {$\mathcal{F}SET(L)^\top$}}
Let $\mathcal{Y} = \{(Y_i, E_i): i \in \mathcal{I} \}$ be a family of $L$-valued sets and let $Y = \prod_i Y_i $ be the product of the corresponding sets. We introduce the $L$-valued equality $E: Y\times Y \to L$ on $Y$ by setting $E_Y(y,y') = \bigwedge_{i\in\mathcal{I}} E_i(y_i,{y'}_i)$ where $y=(y_i)_{i\in\mathcal{I}}$, $y'=({y'}_i)_{i\in\mathcal{I}}$. Further, let $p_i :Y \to Y_i$ be the projection. Then the pair $(Y,E)$ thus defined with the family of projections $p_i :Y \to Y_i$, $i \in \mathcal{I}$, is the product of the family $\mathcal{Y}$ in the category {$\mathcal{F}SET(L)^\top$}.
To show this notice first that, since the morphisms in this category are fuzzy functions, a projection $p_{i_0}: Y \to Y_{i_0}$ must be realized as the fuzzy function $p_{i_0}: Y \times Y_{i_0} \to L$ such that $p_{i_0}(y,y^0_{i_0}) = \top$ if and only if the $i_0$-coordinate of $y$ is $y^0_{i_0}$ and $p_{i_0}(y,y^0_{i_0}) = \bot$ otherwise.
Next, let $F_i: (X,E_X) \rightarrowtail (Y_i,{E_i})$, $i \in \mathcal{I}$ be a family of fuzzy functions. We define the fuzzy function $F: (X,E_X) \rightarrowtail (Y,E_Y)$ by the equality: $$F(x,y) = \bigwedge_{i\in\mathcal{I}} F_i(x,y_i).$$ It is obvious that $\mu(F) = \top$ and hence $F$ is in {$\mathcal{F}SET(L)^\top$}. Finally, notice that the composition $$(X,E_X) \ \stackrel{F}{\longrightarrow} \ (Y,E_Y) \ \stackrel{p_{i_0}}{\longrightarrow} \ (Y_{i_0},E_{i_0})$$ is the fuzzy function $$F_{i_0}: (X,E_X) \rightarrowtail (Y_{i_0},E_{i_0}).$$ Indeed, let $x^0 \in X$ and $y^0_{i_0} \in Y_{i_0}$. Then, taking into account that $\mu(F_i) = \top$ for all $i \in \mathcal{I}$, we get $$ \begin{array}{lll} (p_{i_0} \circ F)(x^o,y^0_{i_0})& =& \bigvee_{y\in Y} \bigl(p_{i_0}(y,y^0_{i_0}) \wedge F(x^0,y)\bigr)\\ & =& \bigvee_{y\in Y} \bigl(p_{i_0}(y,y^0_{i_0}) \wedge \bigwedge_{i\in\mathcal{I}} F_i(x^0,y_i)\bigr)\\ & =& F_{i_0}(x^0,y_{i_0}). \end{array}$$
\begin{question} We do not know whether products in {$\mathcal{F}SET(L)$}\ can be defined in a reasonable way. \end{question}
\subsection{Coproducts in {$\mathcal{F}SET(L)$}}
Let $\mathcal{X}$ = $\{(X_i,E_i): i \in \mathcal{I} \}$ be a family of $L$-valued sets, let $X = \bigcup X_i$ be the disjoint sum of sets $X_i$. Further, let $q_i: X_i \to X$ be the inclusion map. We introduce the $L$-equality on $X_0$ by setting $E(x,x') = E_i(x,x')$ if $(x,x') \in X_i \times X_i$ for some $i \in \mathcal{I}$ and $E(x,x') = \bot$ otherwise (cf.\ \cite{Ho92}). An easy verification shows that $(X,E)$ is the coproduct of $\mathcal{X}$ in {$\mathcal{F}SET(L)$}\ and hence, in particular, in {$\mathcal{F}SET(L)^\top$}.
Indeed, given a family of fuzzy functions $F_i: (X_i,E_i) \to (Y,E_Y)$, let the fuzzy function $$\oplus_{i\in\mathcal{I}} F_i: (X,E) \to (Y,E_Y)$$ be defined by $$\oplus_{i\in\mathcal{I}} F_i(x,y) = F_{i_0}(x,y) \mbox{ whenever } x \in X_{i_0}.$$ Then for $x=x_{i_0} \in X_{i_0}$ we have $$ \begin{array}{lll} \bigl(\oplus_{i\in\mathcal{I}} F_i \circ q_{i_0} \bigr)(x,y)& =& \bigvee_{x'\in X} \Bigl(q_{i_0}(x,x') \wedge \bigl(\oplus_{i\in\mathcal{I}} F_i (x',y)\bigr)\Bigr)\\ & =& F_{i_0}(x,y). \end{array} $$
\subsection{Subobjects in {$\mathcal{F}SET(L)$}}
Let $(X,E)$ be an $L$-valued set, let $Y \subset X$ and let $e: Y \to X$ be the natural embedding. Further, let $E_Y := e^{\gets}(E)$ be the preimage of the $L$-valued equality $E$. Explicitly, in this case this means that $E_Y(y,y') = E(y,y')$ for all $y,y' \in Y$. One can easily see that $(Y,E_Y)$ is a subobject of $(X,E)$ in the fuzzy category {$\mathcal{F}SET(L)$}.
\section{Fuzzy category {$\mathcal{F}TOP(L)$}}
\subsection{Basic concepts}
\begin{definition}[see \cite{So2000}, cf.\ also \cite{Ch}, \cite{Go73}, \cite{HoSo99}]\label{LFTop} A family $\tau_X \subset L^X$ of {\it extensional} $L$-sets\footnote{Since $L$-topology is defined on an {\it $L$-valued set} $X$ the condition of extensionality of elements of $L$-topology seems natural. Besides the assumption of extensionality is already implicitly included in the definition of a fuzzy function.} is called an $L$-topology on an $L$-valued set $(X,E_X)$ if it is closed under finite meets, arbitrary joins and contains $0_X$ and $1_X$. Corresponding triple $(X,E_X,\tau_X)$ will be called an $L$-valued $L$-topological space or just an $L$-topological space for short. A fuzzy function $F: (X,E_X,\tau_X) \rightarrowtail (Y,E_Y,\tau_Y)$ is called {\it continuous} if $F^{\leftarrow}(V) \in \tau_X$ for all $V \in \tau_Y$. \end{definition}
$L$-topological spaces and continuous fuzzy mappings between them form the fuzzy category which will be denoted {$\mathcal{F}TOP(L)$}. Indeed, let $$F\colon (X,E_X,\tau_X) \rightarrowtail (Y,E_Y,\tau_Y)\ \mbox{and}\ G\colon (Y,E_Y,\tau_Y) \rightarrowtail (Z,E_Z,\tau_Z)$$ be continuous fuzzy functions and let $W \in \tau_Z$. Then $$ \begin{array}{lll} (G \circ F)^{\gets}(W)(x)& =& \bigvee_z\Bigl(\bigl(G \circ F)(x,z) * W(z)\Bigr)\\ & =& \bigvee_{z,y} \Bigl(F(x,y)*G(y,z)*W(z)\Bigr). \end{array} $$ On the other hand, $G^{\gets}(W)(y) = \bigvee_z G(y,z) * W(z)$ and $$ \begin{array}{lll} F^{\gets}\bigl(G^{\gets}(W)\bigr)(x)& =& \bigvee_y F(x,y) *\bigl(\bigvee_z (G(y,z) * W(z))\bigr)\\ & =& \bigvee_{z,y} \Bigl(F(x,y)*G(y,z)*W(z)\Bigr). \end{array} $$ Thus $(G \circ F)^{\gets}(W) = G^{\gets}(F^{\gets}(W))$ for every $W$, and hence composition of continuous fuzzy functions is continuous. Besides, we have seen already before that $\mu(G \circ F) \geq \mu(G) * \mu(F)$. Finally, $E_X^{\leftarrow}(B) = B$ for every {\it extensional} $B \in L^X$ and hence the identity mapping $E_X: (X,E_X,\tau_X) \rightarrowtail (X,E_X,\tau_X)$ is continuous.
\begin{remark} In case when $L$-valued equality $E_X$ is crisp, i.e.\ when $X$ is an ordinary set, the above definition of an L-topology on $X$ reduces to the ``classical'' definition of an $L$-topology in the sense of Chang and Goguen, \cite{Ch}, \cite{Go73}. \end{remark}
\begin{remark} Some (ordinary) subcategories of the fuzzy category {$\mathcal{F}TOP(L)$}\ will be of special interest for us. Namely, let {$\mathcal{F}TOP(L)$}$^\bot$ =: FTOP(L) denote the bottom frame of {$\mathcal{F}TOP(L)$}, let {$\mathcal{F}TOP(L)^\top$}\ be the top frame of {$\mathcal{F}TOP(L)$}, and finally let L-TOP(L) denote the subcategory of {$\mathcal{F}TOP(L)$}\ whose morphisms are ordinary functions. Obviously the ``classical'' category L-TOP of Chang-Goguen $L$-topological spaces can be obtained as a full subcategory L-TOP(L) whose objects carry crisp equalities. Another way to obtain L-TOP is to consider fuzzy subcategory of {$\mathcal{F}TOP(L)$} whose objects carry crisp equalities and whose morphisms satisfy condition $\mu(F) > \bot.$ \end{remark}
In case when $L$ is an $MV$-algebra and involution $^c: L \to L$ on $L$ is defined in the standard way, i.e.\ $\alpha^c := \alpha \longmapsto \bot$ we can reasonably introduce the notion of a closed $L$-set in an $L$-topological space:
\begin{definition} An $L$-set $A$ in an $L$-topological space $(X,E_X,\tau_X)$ is called closed if $A^c \in \tau_X$ where $A^c \in L^X$ is defined by the equality $$A^c(x) := A(x) \longmapsto \bot \quad \forall x \in X.$$ \end{definition}
Let $\mathcal{C}_X$ denote the family of all closed $L$-sets in $(X,E_X,\tau_X)$. In case when $L$ is an $MV$-algebra the families of sets $\tau_X$ and $\mathcal{C}_X$ mutually determine each other: $$A \in \tau_X \Longleftrightarrow A^c \in \mathcal{C}_X.$$
\subsection{Analysis of continuity}
Since the operation of taking preimages $F^{\leftarrow}$ commutes with joins, and in case when $\mu(F)=\top$ also with meets (see Proposition \ref{im-pr}), one can easily verify the following
\begin{theorem}\label{cont} Let $(X,E_X,\tau_X)$ and $(Y,E_Y,\tau_Y)$ be $L$-topological spaces, $\beta_Y$ be a base of $\tau_Y$, $\xi_Y$ its subbase and let $F: X \rightarrowtail Y$ be a fuzzy function. Then the following are equivalent: \begin{enumerate} \item[(1con)] $F$ is continuous; \item[(2con)] for every $V \in \beta_Y$ it holds $F^{\leftarrow}(V) \in \tau_X$; \item[(3con)] $F^{\leftarrow}(Int_Y(B)) \leq Int_X (F^{\leftarrow}(B))$, for every $B \in L^Y$ where $Int_X$ and $Int_Y$ are the corresponding $L$-interior operators on $X$ and $Y$ respectively. \end{enumerate} In case when $\mu(F) = \top$ these conditions are equivalent also to the following \begin{enumerate} \item[(4con)] for every $V \in \xi_Y$ it holds $F^{\leftarrow}(V) \in \tau_X$. \end{enumerate} \end{theorem}
In case when $L$ is an MV-algebra one can characterize continuity of a fuzzy function by means of closed $L$-sets and $L$-closure operators:
\begin{theorem}\label{cont-cl} Let $(L,\leq,\vee,\wedge,*)$ be an MV-algebra, $(X,E_X,\tau_X)$ and $(Y,E_Y,\tau_Y)$ be $L$-topological spaces and $F: X \rightarrowtail Y$ be a fuzzy function. Further, let $\mathcal{C}_X$, $\mathcal{C}_Y$ denote the families of closed $L$-sets and $cl_X$, $cl_Y$ denote the closure operators in $(X,E_X,\tau_X)$ and $(Y,E_Y,\tau_Y)$ respectively. Then the following two conditions are equivalent: \begin{enumerate} \item[(1con)] $F$ is continuous; \item[(5con)] For every $B \in \mathcal{C}_Y$ it follows $F^{\leftarrow}(B) \in \mathcal{C}_X$. \end{enumerate} In case when $\mu(F) = \top$, the previous conditions are equivalent to the following: \begin{enumerate} \item[(6con)] For every $A \in L^X$ it holds $F^{\rightarrow}(cl_X(A)) \leq cl_Y(F^{\rightarrow}(A)).$ \end{enumerate} \end{theorem}
\begin{proof} In case when $L$ is equipped with an order reversing involution, as it is in our situation, families of closed and open $L$-sets mutually determine each other. Therefore, to verify the equivalence of (1con) and (5con) it is sufficient to notice that for every $B \in L^Y$ and every $x \in X$ it holds $$ \begin{array}{lll} F^{\leftarrow}(B^c)(x)& =& \bigvee_y \bigl(F(x,y) * (B(y) \longmapsto \bot)\bigr)\\ & =& \bigvee_y \bigl(F(x,y) * B(y) \longmapsto \bot \bigr)\\ & =& \bigl(\bigvee_y (F(x,y) * B(y))\bigr)^c\\ & =& (F^{\leftarrow}(B))^c(x), \end{array} $$ and hence $$ F^{\leftarrow}(B^c) = ( F^{\leftarrow}(B))^c \quad \forall B \in L^Y, $$ i.e.\ operation of taking preimages preserves involution.
To show implication (5con) $\Longrightarrow$ (6con) under assumption $\mu(F)=\top$ let $A \in L^X$. Then, according to Proposition \ref{im-pr} (5), $$A \leq F^{\leftarrow}(F^{\rightarrow}(A)) \leq F^{\leftarrow}(cl_Y(F^{\rightarrow}(A))),$$ and hence, by (5con), also $$cl_X (A) \leq F^{\leftarrow}(cl_Y (F^{\rightarrow}(A))).$$ Now, by monotonicity of the image operator and by Proposition \ref{im-pr} (6) (taking into account that $cl_X A$ is extensional as a closed $L$-set), we get: $$ F^{\rightarrow}(cl_X (A)) \leq F^{\rightarrow}\bigl(F^{\leftarrow}(cl_Y (F^{\rightarrow}(A)))\bigr) \leq cl_Y (F^{\rightarrow}(A)).$$
Conversely, to show implication (6con) $\Longrightarrow$ (5con) let $B \in \mathcal{C}_Y$ and let $F^{\gets}(B) := A$. Then, by (6con), $$F^{\to}(cl_X (A)) \leq cl_Y (F^{\to} (A)) \leq cl_Y (B) = B.$$ In virtue of Proposition \ref{im-pr} (5) and taking into account that $\mu(F) = \top$, it follows from here that $cl_X (A) \leq F^{\gets}(B) = A$, and hence $cl_X (A) = A$. \end{proof}
\subsection{Fuzzy $\alpha$-homeomorphisms and fuzzy $\alpha$-homeomorphic spaces}
The following definition naturally stems from Definitions \ref{inj}, \ref{sur} and \ref{LFTop} and item 2 of Proposition \ref{im-pr}:
\begin{definition} Given $L$-topological spaces $(X,E_X,\tau_X)$ and $(Y,E_Y,\tau_Y)$, a fuzzy function $F: X \rightarrowtail Y$ is called a fuzzy $\alpha$-homeomorphism if $\mu(F) \geq \alpha$, $\sigma(F) \geq \alpha$, it is injective, continuous, and the inverse fuzzy function $F^{-1}: Y \rightarrowtail X$ is also continuous. Spaces $(X,E_X,\tau_X)$ and $(Y,E_Y,\tau_Y)$ are called fuzzy $\alpha$-homeomorphic if there exists a fuzzy $\alpha$-homeomorphism $F: (X,E_X,\tau_X) \rightarrowtail (Y,E_Y,\tau_Y)$. \end{definition}
One can easily verify that composition of two fuzzy $\alpha$-homeomorphisms is a fuzzy $\alpha^2$-home\-omorphism; in particular, composition of fuzzy $\top$-homeo\-morphisms is a fuzzy $\top$-homeo\-morphism, and hence fuzzy $\top$-homeo\-morph\-isms determine the equivalence relation $\stackrel{\top}{\approx}$ on the class of all $L$-topological spaces. Besides, since every (usual) homeomorphism is obviously a fuzzy $\top$-homeo\-morphism, homeomorphic spaces are also fuzzy $\top$-homeo\-morphic: $$(X,E_X,\tau_X) \approx (Y,E_Y,\tau_Y) \Longrightarrow (X,E_X,\tau_X) \stackrel{\top}{\approx} (Y,E_Y,\tau_Y).$$ The converse generally does not hold:
\begin{example} Let $L$ be the unit interval $[0,1]$ viewed as an $MV$-algebra (i.e.\ $\alpha * \beta = max\{\alpha+\beta-1, 0\}$, let $(X,\varrho)$ be an uncountable separable metric space such that $\varrho(x,x') \leq 1$ $\forall x,x' \in X$, and let $Y$ be its countable dense subset. Further, let the $L$-valued equality on $X$ be defined by $E_X(x,x') := 1 - \varrho(x,x')$ and let $E_Y$ be its restriction to $Y$. Let $\tau_X$ be any $L$-topology on an $L$-valued set $(X,E_X)$ (in particular, one can take $\tau_X := \{c_X \mid c \in [0,1] \}$). Finally, let a fuzzy function $F: (X,E_X) \rightarrowtail (Y,E_Y)$ be defined by $F(x,y) := 1 - \varrho(x,y)$. It is easy to see that $F$ is a $\top$-homeomorphism and hence $(X,E_X,\tau_X) \stackrel{\top}{\approx} (Y,E_Y,\tau_Y)$. On the other hand $(X,E_X,\tau_X) \not{\approx} (Y,E_Y,\tau_Y)$ just for set-theoretical reasons. \end{example}
\section{Category {$\mathcal{F}TOP(L)^\top$}}
Let {$\mathcal{F}TOP(L)^\top$}\ be the top-frame of {$\mathcal{F}TOP(L)$}, i.e.\ {$\mathcal{F}TOP(L)^\top$}\ is a category whose objects are the same as in {$\mathcal{F}TOP(L)$}, that is L-topological spaces, and morphisms are continuous fuzzy functions $F: (X,E_X,\tau_X) \rightarrowtail (Y,E_Y,\tau_Y)$ such that $\mu(F) = \top.$
Note, that as different from the fuzzy category {$\mathcal{F}TOP(L)$}, {$\mathcal{F}TOP(L)^\top$}\ is a usual category. Applying Theorem \ref{cont}, we come to the following result:
\begin{theorem}\label{cont-top} Let $(X,E_X,\tau_X)$ and $(Y,E_Y,\tau_Y)$ be $L$-topological spaces, $\beta_Y$ be a base of $\tau_Y$, $\xi_Y$ its subbase and let $F: X \rightarrowtail Y$ be a fuzzy function. Then the following conditions are equivalent: \begin{enumerate} \item[(1con)] $F$ is continuous; \item[(2con)] for every $V \in \beta_Y$ it holds $F^{\leftarrow}(V) \in \tau_X$; \item[(3con)] $F^{\leftarrow}(Int_Y(B)) \leq Int_X (F^{\leftarrow}(B))$, where $Int_X$ and $Int_Y$ are the corresponding $L$-interior operators on $X$ and $Y$ respectively; \item[(4con)] for every $V \in \xi_Y$ it holds $F^{\leftarrow}(V) \in \tau_X$. \end{enumerate} \end{theorem}
In case when $L$ is an MV-algebra, we get from \ref{cont-cl}
\begin{theorem}\label{cont-cltop} Let $(L,\leq,\vee,\wedge,*)$ be an MV-algebra, $(X,E_X,\tau_X)$ and $(Y,E_Y,\tau_Y)$ be $L$-topological spaces and $F: X \rightarrowtail Y$ be a a morphism in {$\mathcal{F}TOP(L)^\top$}. Further, let $\mathcal{C}_X$, $\mathcal{C}_Y$ denote the families of closed $L$-sets and $cl_X$, $cl_Y$ denote the closure operators on $(X,E_X,\tau_X)$ and $(Y,E_Y,\tau_Y)$ respectively. Then the following two conditions are equivalent: \begin{enumerate} \item[(1con)] $F$ is continuous; \item[(5con)] For every $B \in \mathcal{C}_Y$ it follows $F^{\leftarrow}(B) \in \mathcal{C}_X$; \item[(6con)] For every $A \in L^X$ it holds $F^{\rightarrow}(cl_X(A)) \leq cl_Y(F^{\rightarrow}(A))$. \end{enumerate} \end{theorem}
\begin{theorem}\label{topcat} $$\mbox{ {$\mathcal{F}TOP(L)^\top$} is a topological category over the category {$\mathcal{F}SET(L)^\top$}. }$$ \end{theorem}
\begin{proof} Since intersection of any family of L-topologies is an L-topology, {$\mathcal{F}TOP(L)^\top$}\ is fiber complete. Therefore we have to show only that any structured source in {$\mathcal{F}SET(L)^\top$}\ $F: (X,E_X) \rightarrowtail (Y,E_Y,\tau_Y)$ has a unique initial lift. Let $$\tau_X := F^{\gets}(\tau_Y):= \{F^{\gets}(V) \mid V \in \tau_Y \}.$$ Then from theorem \ref{im-pr} it follows that $\tau_X$ is closed under taking finite meets and arbitrary joins. Furthermore, obviously $F^{\gets}(0_Y) = 0_X$ and taking into account condition $\mu(F)=\top$ one easily establishes that $F^{\gets}(1_Y) = 1_X$. Therefore, taking into account that preimages of extensional $L$-sets are extensional, (see Subsection \ref{impref}) we conclude that the family $\tau_X$ is an $L$-topology on $X$.
Further, just from the construction of $\tau_X$ it is clear that $F: (X,E_X,\tau_X) \rightarrowtail (Y,E_Y,\tau_Y)$ is continuous and, moreover, $\tau_X$ is the weakest L-topology on $X$ with this property.
Let now $(Z,E_Z,\tau_Z)$ be an $L$-topological space and $H: (Z,E_Z) \rightarrowtail (X,E_X)$ a fuzzy function such that the composition $G := H \circ F: (Z,E_Z,\tau_Z) \rightarrowtail (Y,E_Y,\tau_Y)$ is continuous. To complete the proof we have to show that $H$ is continuous.
Indeed, let $U \in \tau_X$. Then there exists $V \in \tau_Y$ such that $U = F^{\gets}(V)$. Therefore $$H^{\gets}(U) = H^{\gets}\bigl(F^{\gets}(V)\bigr) = G^{\gets}(V) \in \tau_Z$$ and hence $H$ is continuous. \end{proof}
\subsection{Products in {$\mathcal{F}TOP(L)^\top$}}
Our next aim is to give an explicite description of the product in {$\mathcal{F}TOP(L)^\top$}.
Given a family $\mathcal{Y} = \{(Y_i, E_i, \tau_i): i \in \mathcal{I} \}$ of $L$-topological spaces, let $(Y,E)$ be the product of the corresponding $L$-valued sets $\{(Y_i, E_i): i \in \mathcal{I} \}$ in {$\mathcal{F}SET(L)^\top$} and let $p_{i}: Y \to Y_{i}$ be the projections.
Further, for each $U_i \in \tau_i$ let $\hat U_i := p_i^{-1}(U_i)$. Then the family $\xi := \{\hat U_i: U_i \in \tau_i, i \in \mathcal{I} \}$ is a subbase of an $L$-topology $\tau$ on the product $L$-valued set $(X,E_X)$ which is known to be the product $L$-topology for L-topological spaces $\{(Y_i,\tau_i) \mid i \in \mathcal{I} \}$ in the category L-TOP. In its turn the triple $(Y,E,\tau)$ is the product of $L$-topological spaces $\{(Y_i, E_i, \tau_i): i \in \mathcal{I} \}$ in the category {$\mathcal{F}TOP(L)^\top$}. Indeed, let $(Z,E_Z,\tau_Z)$ be an $L$-topological space and $\{ F_i: Z \rightarrowtail Y_i \mid i \in \mathcal{I} \}$ be a family of continuous fuzzy mappings. Then, defining a mapping $F: Z\times Y \to L$ by $F(z,y) = \wedge_{i\in\mathcal{I}} F_i(z,y_i)$ we obtain a fuzzy function $F: Z \rightarrowtail Y$ such that $\mu(F) = \bigwedge_{i\in\mathcal{I}}\mu(F_i) = \top$ and besides it can be easily seen that for every $i_0 \in \mathcal{I}$, every $z \in Z$, and for every $U_{i_0} \in \tau_{i_0}$ it holds: $$\begin{array}{lll} F^{\leftarrow}(\hat U_{i_0})(z)& =& \bigvee_{y \in Y} \Bigl(\bigl(\bigwedge_{i\in\mathcal{I}}F_i(z,y_i)\bigr) * U_{i}(y_i)\Bigr) \\ & =& \bigvee_{y_i \in Y_i} \Bigl(\bigwedge_{i\in\mathcal{I}} \bigl(F_i(z,y_i)\bigr) * U_{i_0}(y_{i_0})\Bigr)\\ & =& \bigwedge_{i\in \mathcal{I}} \Bigl(\bigvee_{y_i \in Y_i} F_i(z,y_i)\Bigr) \wedge \bigvee_{y_{i_0} \in X_{i_0}} \bigl(F_{i_0} (z,y_{i_0}) * U_{i_0}(y_{i_0})\bigr) \\ & =& \top \wedge F^{\gets}_{i_0}(U_{i_0})(z)\\ & =& F^{\gets}_{i_0}(U_{i_0})(z). \end{array}$$ Hence continuity of all $F_i$ guarantees the continuity of $F: (Z,E_Z,\tau_Z) \rightarrowtail (Y,E,\tau)$. Thus $(Y,E,\tau)$ is indeed the product of $\mathcal{Y} = \{(Y_i, E_i, \tau_i): i \in \mathcal{I} \}$ in {$\mathcal{F}SET(L)$}$^\top$.
\begin{question} We do not know whether products in the fuzzy category {$\mathcal{F}TOP(L)$}\ exist. \end{question}
\subsection{Subspaces in {$\mathcal{F}TOP(L)^\top$}}
Let $(X,E,\tau)$ be an $L$-valued $L$-topological space, $Y \subset X$ and let $(Y,E_Y)$ be the subobject of the L-valued set $(X,E)$ in the category {$\mathcal{F}SET(L)$}. Further, let $\tau_Y$ be the subspace L-topology, that is $(Y,\tau_Y)$ is a subspace of the $L$-topological space $(X,\tau)$ in the category L-TOP. Then it is clear that the triple $(Y,E_Y,\tau_Y)$ is the subobject of $(X,E, \tau)$ in the category {$\mathcal{F}TOP(L)^\top$} (and in the fuzzy category {$\mathcal{F}TOP(L)$}\ as well).
\subsection{Coproducts in {$\mathcal{F}TOP(L)^\top$}}
Given a family $\mathcal{X} := \{(X_i,E_i,\tau_i) \}$ of $L$-topological spaces let $(X,E)$ be the direct sum of the corresponding $L$-valued sets $(X_i,E_i)$ in {$\mathcal{F}SET(L)$}. Further, let $\tau$ be the $L$-topology on $X$ determined by the subbase $\bigcup_{i\in\mathcal{I}} \tau_i \subset 2^X$. In other words $(X,\tau)$ is the coproduct of $L$-topological spaces $(X_i,\tau_i)$ in the category L-TOP. Then the triple $(X,E,\tau)$ is the coproduct of the family $\mathcal{X} := \{(X_i,E_i,\tau_i) \}$ in the category {$\mathcal{F}TOP(L)^\top$} (and in the fuzzy category {$\mathcal{F}TOP(L)$} as well). Indeed, let $q_i: (X_i,E_i,\tau_i) \to (X,E,\tau), i\in\mathcal{I}$ denote the canonical embeddings. Further, consider an $L$-topological space $(Y,E_Y,\tau_Y)$ and a family of continuous fuzzy functions $F_i: (X_i,E_i,\tau_i) \rightarrowtail (Y,E_Y,\tau_Y)$. Then, by setting $F(x,y) := F_i(x_i,y)$ whenever $x = x_i \in X_i$, we obtain a continuous fuzzy function $F: (X,E,\tau) \rightarrowtail (Y,E_Y,\tau_Y)$ (i.e.\ a mapping $F: X\times Y \to L$) such that $F_i = q_i \circ F$ for every $i \in \mathcal{I}$.
\subsection{Quotients in {$\mathcal{F}TOP(L)^\top$}}
Let $(X,E_X,\tau_X)$ be an $L$-topological space, let $q: X \to Y$ be a surjective mapping. Further, let $q^{\to}(E_X) =: E_Y$ be the image of the $L$-valued equality $E_X$ and let $\tau_Y = \{V \in L^Y \mid q^{-1}(V) \in \tau_X \}$, that is $\tau_Y$ is the quotient $L$-topology determined by the mapping $q: (X,\tau) \to Y$ in the category L-TOP. Then $(Y,E_Y,\tau_Y)$ is the quotient object in the category {$\mathcal{F}TOP(L)^\top$}. Indeed, consider a fuzzy function $F: (X,E_X,\tau_X) \rightarrowtail (Z,E_Z,\tau_Z)$ and let $G: (Y,E_Y) \rightarrowtail (Z,E_Z)$ be a morphism in {$\mathcal{F}SET(L)$}\ such that $q \circ G = F$. Then an easy verification shows that the fuzzy function $G: (Y,E_Y,\tau_Y) \rightarrowtail (Z,E_Z,\tau_Z)$ is continuous (i.e.\ a morphism in {$\mathcal{F}TOP(L)^\top$}) if and only if $F: (X,E_X,\tau_X) \to (Z,E_Z,\tau_Z)$ is continuous (i.e.\ a morphism in {$\mathcal{F}TOP(L)^\top$}).
Our next aim is to consider the behaviour of some topological properties of L-valued L-topological spaces in respect of fuzzy function. In this work we restrict our interest to the property of compactness. Some other topological properties, in particular, connectedness and separation properties will be studied in a subsequent work.
\section{Compactness}
\subsection{Preservation of compactness by fuzzy functions}
One of the basic facts of general topology --- both classic and ``fuzzy'', is preservation of compactness type properties by continuous mappings. Here we present a counterpart of this fact in {$\mathcal{F}TOP(L)$}. However, since in literature on fuzzy topology different definitions of compactness can be found, first we must specify which one of compactness notions will be used.
\begin{definition}\label{comp} An $L$-topological space $(X,E,\tau)$ will be called $(\alpha, \beta)$-compact where $\alpha, \beta \in L$, if for every family $\mathcal{U} \subset \tau$ such that $\bigvee \mathcal{U} \geq \alpha$ there exists a finite subfamily $\mathcal{U}_0 \subset \mathcal{U}$ such that $\bigvee\mathcal{U}_0 \geq \beta$. An $(\alpha,\alpha)$-compact space will be called just $\alpha$-compact.\footnote{Note that Chang's definition of compactness \cite{Ch} for a $[0,1]$-topological space is equivalent to our $1$-compactness. An $[0,1]$-topological space is compact in Lowen's sense \cite{Lo76} if it is $(\alpha,\beta)$-compact for all $\alpha \in [0,1]$ and all $\beta < \alpha$} \end{definition}
\begin{theorem}\label{pr-comp} Let $(X,E_X, \tau_X)$, $(Y, E_Y, \tau_Y)$ be $L$-topological spaces, $F:X \rightarrowtail Y $ be a continuous fuzzy function such that $\mu(F) \geq \beta$, and $\sigma(F) \geq \gamma$. If $X$ is $\alpha * \beta$-compact, then $Y$ is $(\alpha, \alpha*\beta*\gamma)$-compact. \end{theorem}
\begin{proof} Let $\mathcal{V} \subset \tau_Y$ be such that $\bigvee \mathcal{V} \geq \alpha$. Then, applying Proposition \ref{im-pr} (4), (7) and taking in view monotonicity of $F^{\gets}$, we get $$\bigvee_{V \in \mathcal{V}} F^{\leftarrow}(V) = F^{\leftarrow}(\bigvee_{V \in \mathcal{V}}V) \geq F^{\leftarrow}(\alpha) \geq \alpha * \beta.$$ Now, since $(X,E_X,\tau_X)$ is $\alpha * \beta$-compact, it follows that there exists a finite subfamily $\mathcal{V}_0 \subset \mathcal{V}$ such that $$\bigvee_{V\in \mathcal{V}_0} F^{\leftarrow}(V) \geq \alpha*\beta.$$ Applying Propositions \ref{im-pr} (6),(4) and \ref{in-sur} (5) we obtain: $$\bigvee_{V\in\mathcal{V}_0} V \geq F^{\rightarrow}\Bigl(F^{\leftarrow}\bigl(\bigvee_{V\in\mathcal{V}_0} V\bigr)\Bigr) = F^{\rightarrow}\Bigl(\bigvee_{V\in\mathcal{V}_0} \bigl(F^{\leftarrow}(V)\bigr)\Bigr) \geq F^{\rightarrow}\bigl(\alpha * \beta\bigr) \geq \alpha*\beta*\gamma.$$ \end{proof}
\begin{corollary} Let $(X,E_X, \tau_X)$, $(Y,E_Y,\tau_Y)$ be $L$-topological spaces, $F:X \rightarrowtail Y$ be a fuzzy function such that $\mu(F) = \top $ and $\sigma(F) = \top$. If $X$ is $\alpha$-compact, then $Y$ is also $\alpha$-compact. \end{corollary}
\subsection{Compactness in case of an $MV$-algebra}
In case $L$ is an MV-algebra one can characterize compactness by systems of closed $L$-sets:
\begin{proposition} Let $(X,E_X,\tau_X)$ be an $L$-topological space and let $\mathcal{C}_X$ be the family of its closed $L$-sets. Then the space $(X,E_X,\tau_X)$ is $(\alpha,\beta)$-compact if and only if for every $\mathcal{A} \subset \mathcal{C}_X$ the following implication follows: $$ \mbox{if } \bigwedge_{A\in \mathcal{A}_0} A \not\leq \beta^c \mbox{ for every finite family } \mathcal{A}_0 \subset \mathcal{A}, \mbox{ then } \bigwedge_{A\in \mathcal{A}} A \not\leq \alpha^c. $$ \end{proposition}
\begin{proof} One has just to take involutions ``$\longmapsto \bot$'' in the definition of $(\alpha,\beta)$-compactenss and apply De Morgan law. \end{proof}
\subsection{Perfect mappings: case of an MV-algebra $L$}
In order to study preservation of compactness by preimages of fuzzy functions we introduce the property of $(\alpha,\beta)$-perfectness of a fuzzy function. Since we shall operate with closed $L$-sets, from the beginning it will be assumed that $L$ is an $MV$-algebra.
First we shall extend the notion of compactness for $L$-subsets of $L$-topolog\-ical spaces. We shall say that an $L$-set $S: X \to L$ is $(\alpha,\beta)$-compact if for every family $\mathcal{A}$ of closed $L$-sets of $X$ the following implication holds: $$ \mbox{ if } S\wedge\bigl(\bigwedge_{A\in \mathcal{A}_0} A\bigr) \not\leq \beta^c \mbox{ for every finite } \mathcal{A}_0 \in \mathcal{A} \mbox{ then } S\wedge\bigl(\bigwedge_{A\in \mathcal{A}} A \bigr) \not\leq \alpha^c. $$
Further, since the preimage $F^{\gets}(y_0): X \to L$ of a point $y_0 \in Y$ under a fuzzy function $F: X \rightarrowtail Y$ is obviously determined by the equality $$F^{\gets}(y_0)(x) = \bigvee_{y\in Y} F(x,y)*y_0(y) = F(x,y_0),$$ the general definition of $(\alpha,\beta)$-compactness of an $L$-set in this case means the following:
The preimage $F^{\gets}(y_0)$ of a point $y_0$ under a fuzzy function $F$ is $(\alpha,\beta)$-compact if for every family $\mathcal{A}$ of closed sets of $X$ the following implication holds: $$ \bigvee_x \Bigl(F(x,y_0)\wedge(\bigwedge_{A\in \mathcal{A}_0} A(x))\Bigr) \not\leq \beta^c\ \
\forall \mathcal{A}_0 \subset \mathcal{A}, |\mathcal{A}_0| < \aleph_0 $$ implies $$\bigvee_x \bigl(F(x,y_0) \wedge (\bigwedge_{A\in \mathcal{A}} A(x))\bigr) \not\leq \alpha^c.$$
Now we can introduce the following
\begin{definition} A continuous fuzzy mapping $F\colon (X,E_X,\tau_X) \rightarrowtail (Y,E_Y,\tau_Y)$ is called $(\alpha,\beta)$-perfect if \begin{itemize} \item $F$ is closed, i.e.\ $F^{\to}(A) \in \mathcal{C}_Y$ for every $A \in \mathcal{C}_X;$ \item the preimage $F^{\gets}(y)$ of every point $y \in Y$ is $(\alpha,\beta)$-compact. \end{itemize} \end{definition}
\begin{theorem} Let $F: (X,E_X,\tau_X) \rightarrowtail (Y,E_Y,\tau_Y)$ be an $(\alpha,\gamma)$-perfect fuzzy function such that $\mu(F) = \top$ and $\sigma(F) = \top$. If the space $(Y,E_Y,\tau_Y)$ is $(\gamma,\beta)$-compact, then the space $(X,E_X,\tau_X)$ is $(\alpha,\beta)$-compact. \end{theorem}
\begin{proof} Let $\mathcal{A}$ be a family of closed $L$-sets in $X$ such that $\bigwedge_{A\in\mathcal{A}_0} A \not \leq \beta^c$. Without loss of generality we may assume that $\mathcal{A}$ is closed under taking finite meets. Let $B := B_A := F^{\to}(A)$ and let $\mathcal{B} := \{B_A: A \in \mathcal{A} \}$. Then, since $\mu(F) = \top$, by \ref{im-pr} (7) it follows that $B \not \leq \beta^c\ \forall B \in \mathcal{B}$, and moreover, since $\mathcal{A}$ is assumed to be closed under finite meets, $$ \begin{array}{lll} B_{A_1}\wedge\ldots\wedge B_{A_n}& =& F^{\to}(A_1)\wedge\ldots\wedge F^{\to}(A_n)\\ & \geq& F^{\to}(A_1 \wedge\ldots\wedge A_n)\\ & =& F^{\to}(A), \end{array} $$ for some $A \in \mathcal{A}$, and hence $\bigwedge_{B\in\mathcal{B}_0}(B) \not \leq \beta^c$ for every finite subfamily $\mathcal{B}_0 \subset \mathcal{B}$. Hence, by $(\gamma,\beta)$-compactness of the space $(Y,E_Y,\tau_Y)$ we conclude that $\bigwedge_{B \in \mathcal{B}}(B) \not \leq \gamma^c$, and therefore there exists a point $y_0 \in Y$ such that $F^{\to}(A)(y_0) = B_A(y_0) \not \leq \gamma^c$ for all $A \in \mathcal{A}$. Now, applying $(\alpha,\gamma)$-compactness of the preimage $F^{\gets}(y_0)$ and recalling that $\mathcal{A}$ was assumed to be closed under taking finite meets, we conclude that $$\bigvee_x \bigl(F(x,y_0) \wedge (\bigwedge_{A\in \mathcal{A}} A(x))\bigr) \not\leq \alpha^c.$$ and hence, furthermore, $$\bigwedge_{A\in\mathcal{A}} A \not\leq \alpha^c.$$ \end{proof}
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\begin{document}
\title{On thermalization of two-level quantum systems}
\section{Introduction}
The study of evolution of open systems towards equilibrium has always been a challenging problem in Statistical Mechanics. The difficulty lies in prescribing a form of interaction between the system and the environment at the microscopic level that will give rise to equilibration. It has been evaded by proposing the so called {\it H-theorem} which states that a system attains equilibrium when the entropy function is maximized over the accessible states of the system.
Although this has proved to be a very efficient way to calculate and work with equilibrium states, the heart of the problem remains unsolved. We look at this thermodynamic problem from a quantum mechanical perspective. Quantum Thermodynamics have received a lot attention in the recent past \cite{rt1,rt2}. The concepts and laws of thermodynamics are presumably valid only in macroscopic regime. To see how the laws and definitions of thermodynamic quantities viz heat, work, etc behave in microscopic regime is one of the main objectives of Quantum Thermodynamics.
There has been a number of works \cite{pop1,pop2,short1,short2,gogolin} where the problem of equilibration is looked at from a quantum mechanical perspective. For example, Linden et al. \cite{linden} looked into the problem of smallest possible quantum refrigerator. In the process, they considered a two-qubit system as a refrigerator in which one qubit acts as the system to be cooled while the other works as the coil of the refrigerator by extracting heat from the body (to be cooled), and releasing it to the environment. The two-qubit refrigerator is derived from the equilibrium (steady) state solution of a three-qubit master equation which the authors provided phenomenologically. This motivated us to see if, instead of following this phenomenological approach, a microscopic description for the thermalization process (equilibration to a thermal state) is possible through a thermalizing Hamiltonian. Such a simulation of the thermalization process can serve at least two purposes: (i) simulating a natural thermalization process in lab, and (ii) comparing different time scales (e.g., time scales for thermalization versus interaction time scales of different constituents of the system) without assuming a priori their ordering.
To completely characterize the joint Hamiltonian of the system and environment that results in equilibration of the system, is a formidable task. So, instead we ask the following question: whether for a given thermalization process of a system, there exists an ancilla in a specific state and a joint Hamiltonian of system-ancilla that gives rise to the exact process of equilibration on the system. In this paper, we provide an affirmative answer to this question in the case of quantum-optical master equation.
We work out a {\it thermalizing Hamiltonian} $H_{th}$ for the quantum-optical master equation \cite{text1} which gives rise to thermal equilibration of a qubit. We find that a single-qubit ancilla initialized in a thermal state is sufficient for such a dynamics to be mimicked.
Our next aim is to look for such simulations of thermalization process which evolves under the action of non-Markovian dynamics. We analyse such situations further by considering a general form of thermalizing Hamiltonian of which the quantum-optical master equation dynamics is a special case. We work out the necessary and sufficient conditions for Markovianity of the system dynamics given a form of the simulating interaction Hamiltonian. Note that not every non-Markovian dynamics gives rise to equilibration of the system, and thereby, thermalization. Our approach here provides one possible way of generating a thermalizing non-Markovian dynamics through the prescription of a simulating Hamiltonian. It is worth mentioning here that, as there are a number of definitions of Markovianity in quantum mechanical scenario \cite{markov1,markov2,markov3,markov4} we stick to the definition of completely positive (CP) divisibility \cite{markov3,markov4} and use the characterization of Wolf et. al. \cite{cirac} for finding out the aforementioned conditions.
An interesting model of thermalization was proposed by V. Scarani et. al. \cite{scarani}. Another model of thermalization (for spin-$\frac{1}{2}$ systems) has been developed by Kleinbolting and Klesse \cite{klesse}. In these works, they used the swap operation between system and bath to give rise to thermalization. But a drawback of these methods is that the system is fully thermalized after a \textit{finite} time interval, which would imply that the thermalizing map is a function of only the temperature to which the system will thermalize and the time interval taken to reach it. This proposition seems to be unrealistic as this does not take into account the intricacies of the system, environment or the correlations shared between them, that might affect the process of thermalization.
In \cite{oliveira1,oliveira2,oliveira3}, M. J. de Oliveira has shown another novel approach to thermalization for systems in contact with an environment (typically, heat reservoirs). In \cite{oliveira1}, a quantum Focker-Planck-Kramers (FPK) equation is derived via canonical quantization of the classical FPK equation to account for quantum dissipation of systems interacting with environment. The dissipation term is chosen such that the system equilibriates to the Gibbs thermal state i.e. system thermalizes. In \cite{oliveira2,oliveira3}, the quantum FPK equation is further exploited to study heat transport properties in harmonic oscillator chains and bosonic systems. Although our approach to thermalization also begins with solving a master equation, it differs from de Oliveira's in that our aim is to derive simulating Hamiltonians for thermalization and thereby study generic features of thermalization in open quantum systems.
First, we describe the thermalizing process of a qubit as a pin map. We then look at the quantum optical master equation for a qubit to find out its time dependent solutions. The affine transformation relating the initial state and the time-evolved state is then described. This affine transformation is then parametrized to find out the thermalizing Hamiltonian $H_{th}$ with a single-qubit ancilla simulating the heat bath. In the following section, we consider the thermalizing Hamiltonian in a more general form and derive the conditions on the time dependence for thermalization to occur (in the infinite time limit). We also derive the necessary and sufficient conditions for such a Hamiltonian to lead to Markovian dynamics for the system evolution. Further, we derive the Lindblad type master equation for a system dynamics arising out of our general form of Hamiltonian. And finally, we draw our conclusions.
\section{Form of thermalizing Hamiltonian}
The starting point of our work is realising that thermalization can be achieved through several ways, one of which being Markovian master equations with a thermal bath. Therefore, we take a Markovian master equation, the quantum optical master equation, where a qubit (two levels of an atom) is in contact with a bath (a system of non-interacting radiation field). Given the fact that all Markovian master equations with thermal baths give rise to equilibration to {\it thermal states} $(\rho_{th} = e^{-H/k_B T})$, with $H$ being the system Hamiltonian, we try to figure out a joint Hamiltonian between the system and an ancilla, which will do the same. This system-ancilla Hamiltonian, which will henceforth be called as thermalizing Hamiltonian $H_{th}$, will give rise to a unitary process where the system (two levels of the atom) will equilibriate to a (constant) thermal state.
To calculate the thermalizing Hamiltonian $H_{th}$, we find the affine transformation on the Bloch vector of the system qubit that will give rise to the same evolution as the quantum optical master equation. In doing so, we realize that the affine transformation is a special case of the generalized amplitude damping channel \cite{text2}. We then refer to a result by Narang and Arvind \cite{narang} where it is shown that it is enough for certain qubit channels to have a single-qubit mixed state ancilla to simulate the action of the channel as a sub-system dynamics of a system-ancilla unitary evolution. It may be noted here that Terhal et. al. \cite{smolin} have shown that certain single-qubit channels can only be simulated through qutrit mixed state environments. Incidentally, our affine transformation fits into the criterion for single-qubit ancilla as in Narang and Arvind \cite{narang}, and we find a two-qubit Hamiltonian that simulates the evolution of the system qubit via the quantum optical master equation.
Given below are the details of the aforesaid process. We will be working in the computational basis unless mentioned otherwise.
\subsection{Thermalizing maps for a qubit: Pin Map} Before we introduce the optical master equation for a two-level quantum system, we look for the most general way a qubit can lead to thermalization -- a qubit channel -- a completely positive trace preserving map $\mathcal{N}:\mathcal{L}(\mathbb{C}^2)\rightarrow \mathcal{L}(\mathbb{C}^{2})$ such that $\mathcal{N}(\rho)=\rho_{th}=\text{diag}(p,1-p)$ for all single-qubit states $\rho$ with $0\leq p\leq 1$. Such a map is called a {\it pin} map. Here, $\mathcal{L}(\mathbb{C}^2)$ is the set of all bounded linear operators $A: \mathbb{C}^2\rightarrow\mathbb{C}^2$. Thus, we have: \begin{equation} \label{N} \mathcal{N} = \begin{bmatrix} p & 0 & 0 & p\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 1-p & 0 & 0 & 1-p \end{bmatrix}. \end{equation} The Kraus operator for $\mathcal{N}$ are: \begin{gather*} \label{k1} K_{00} = \begin{bmatrix} \sqrt{p} & 0\\ 0 & 0 \end{bmatrix}, ~~~ K_{01} = \begin{bmatrix} 0 & \sqrt{p} \\ 0 & 0 \end{bmatrix}, \end{gather*}
\begin{gather} \label{k2} K_{10} = \begin{bmatrix} 0 & 0\\ \sqrt{1-p} & 0 \end{bmatrix}, ~~~ K_{11} = \begin{bmatrix} 0 & 0 \\ 0 & \sqrt{1-p} \end{bmatrix}. \end{gather}
\subsection{Optical master equation} It would have been useful to have a dynamical version of the pin map, whose Kraus operators are given in equation (\ref{k2}). This would then give rise to a master equation corresponding to pin map, and thereby, for thermalization. In the absence of such a dynamical version in general, we now look at the optical master equation to come up with one possible dynamical version of the Kraus operators in equation (\ref{k2}).
We choose the following Markovian master equation (quantum optical master equation) which corresponds to a qubit interacting with a bosonic thermal bath under Markovian conditions.
\begin{equation} \label{lind} \begin{split} \frac{d\rho (t)}{dt} &= \gamma_0 (N +1) \Big(\sigma_-\rho(t)\sigma_+ - \frac{1}{2} \{ \sigma_+\sigma_-,\rho(t) \}\Big)\\ &~~ + \gamma_0 N \Big(\sigma_+\rho(t)\sigma_- - \frac{1}{2}\{\sigma_-\sigma_+,\rho(t)\}\Big) \end{split} \end{equation}
Here, $N=(\exp\frac{E(\omega)}{k_B T}-1)^{-1}$ is the Planck distribution, $k_B$ is the Boltzmann constant, $T$ is temperature of the heat bath and $E(\omega)=\hbar\omega$ is the energy of the system at frequency $\omega$. $\gamma_0$ is the spontaneous emission rate of the bath, and $\gamma = \gamma_0 (2N+1)$ is the total emission rate (including thermally induced emission and absorption processes). Here we have neglected the free evolution part. For more details, refer to \cite{text1}.
If the initial system qubit state is given by $\rho(0)=\frac{1}{2}(\mathbb{1}+\bar{r}(0).\bar{\sigma})$, where $\bar{r}(0) = (r_1(0),r_2(0),r_3(0))$, the master equation can be readily solved by choosing the time-evolved state to be $\rho(t)=\frac{1}{2}(\mathbb{1}+\bar{r}(t).\bar{\sigma})$ where $\bar{r}(t) = (r_1(t),r_2(t),r_3(t))$. We find, $r_1(t) = r_1(0) e^{-\gamma t/2}, r_2(t) = r_2(0) e^{-\gamma t/2}, r_3(t) = (r_3(0)+g) e^{-\gamma t} - g.$
Here $g =\frac{\gamma_0}{\gamma}=(2N+1)^{-1}$ and so, $g\in [0,1]$. $g$ gives us a measure of the temperature $T$. It can be easily seen that, higher the value of $g$, lower the temperature and vice versa. Specifically, $g=0$ for $T=\infty$ and $g=1$ for $T=0$. The steady state solution for the system is a thermal state as expected, and corresponds to the Bloch vector $(0,0,-g)$. Explicitly, \begin{equation} \label{gval} \rho_{th} = \text{diag}(\frac{1-g}{2},\frac{1+g}{2}). \end{equation}
\subsection{Affine Transformation} Any single-qubit channel can be written as an affine transformation of the form $r_i(t) = \sum_{j=0}^{3}M_{ij} r_j(0) + C_i$ \cite{text2,narang}. Thus, we can express the corresponding affine transformation for our solution (given in equation (\textcolor{red}{4})) as a $3\times3$ matrix $M$ and a column matrix $C$:
\begin{equation} \label{M1} M = \begin{bmatrix} e^{-\gamma t/2} & 0 & 0\\ 0 & e^{-\gamma t/2} & 0\\ 0 & 0 & e^{-\gamma t} \end{bmatrix}, C = \begin{bmatrix} 0\\ 0\\ g(e^{-\gamma t}-1) \end{bmatrix}. \end{equation}
Here, we notice that this affine transformation is a special kind of generalized amplitude damping channel. Amplitude damping channels describe the effect of energy dissipation to environment at finite temperature. The affine transformation for a generalized amplitude damping channel has two positive parameters $B$, $p \in [0,1]$. It is given by: \begin{equation} M_{GAD} = \begin{bmatrix} \sqrt{1-B} & 0 & 0\\ 0 & \sqrt{1-B} & 0\\ 0 & 0 & 1-B \end{bmatrix}, \end{equation} \begin{equation} C_{GAD} = \begin{bmatrix} 0\\ 0\\ B(2p-1) \end{bmatrix}. \end{equation}
We can see that our thermalization process is a generalized amplitude damping channel with the parameter $p<\frac{1}{2}$.
\subsection{Parametrizing the transformation}
In \cite{narang}, Narang and Arvind used a single-qubit mixed state ancilla to parametrize the affine transformation of a single-qubit channel. We follow their technique to simulate our dynamical process for thermalization. To do so, we consider a single-qubit mixed state ancilla of the form $\rho_e=(1-\lambda)\frac{\mathbb{1}}{2}+\lambda|\phi\rangle\langle\phi|.$ where $\frac{\mathbb{1}}{2}$ is the maximally mixed state and $|\phi\rangle$ is a general pure state given by, $|\phi\rangle= \cos\Big( \frac{\xi}{2}\Big)|0\rangle+ e^{-i\eta}\sin\Big(\frac{\xi}{2}\Big)|1\rangle.$
If $\rho_e$ plays the role of a bath state of a single-qubit system then evolution through the most general two-qubit unitary $U$ (upto a freedom of local unitary actions), given in equation (\ref{ud}) below, will result in the following affine transformation for the system qubit, as given in equations (\ref{M}) and (\ref{C}) below. Apart from $\eta,\xi,\lambda$, three more parameters $\alpha,\beta,\delta$ are needed to completely identify the channel. Thus, the class of single-qubit channels which can be simulated by a single-qubit mixed state ancilla is a six parameter family ($\alpha,\beta,\delta,\eta,\xi,\lambda$) of affine transformations:
\begin{equation} \label{ud} U = \begin{bmatrix} \cos\frac{\alpha+\delta}{2} & 0 & 0 & i\sin\frac{\alpha+\delta}{2}\\ 0 & e^{-i\beta}\cos\frac{\alpha-\delta}{2} & ie^{-i\beta}\sin\frac{\alpha-\delta}{2} & 0\\ 0 & ie^{-i\beta}\sin\frac{\alpha-\delta}{2} & e^{-i\beta}\cos\frac{\alpha-\delta}{2} & 0\\ i\sin\frac{\alpha+\delta}{2} & 0 & 0 & \cos\frac{\alpha+\delta}{2}\\ \end{bmatrix} \end{equation}
\begin{equation} \label{C}
C = \begin{bmatrix} -\lambda\sin\delta\sin\beta\sin\xi\cos\eta\\ -\lambda\sin\alpha\sin\beta\sin\xi\sin\eta\\ -\lambda\sin\alpha\sin\delta\cos\xi \end{bmatrix}. \end{equation} \begin{widetext} \begin{equation} \label{M} M = \begin{bmatrix} \cos\delta\cos\beta & \lambda\cos{\delta}\sin{\beta}\cos\xi & -\lambda\sin\delta\cos\beta\sin\eta\sin\xi\\ -\lambda\cos\alpha\sin\beta\cos\xi & \cos\alpha\cos\beta & \lambda\sin\alpha\cos\beta\cos\eta\sin\xi\\ -\lambda\cos\alpha\sin\delta\sin\eta\sin\xi & -\lambda\sin\alpha\cos\delta\sin\xi\cos\eta & \cos\alpha\cos\delta \end{bmatrix}. \end{equation} \end{widetext}
Refer to Appendix A for details of our calculation.
It is also important to note here that by using the ancilla qubit, we are only simulating the dynamics of the system qubit leading to the infinite time thermalization. More specifically, we do not have the ancilla state remaining static, as is the case for the bosonic bath. The ancilla state does in fact change.
Now, we compare the parametrized forms of C and M in equations (\ref{C}) and (\ref{M}) respectively with the affine transformation of quantum optical master equation in (\ref{M1}). Thus, we get a joint unitary giving rise to thermalization. One can check that the unitary does indeed lead to thermalization in the infinite time limit. Equivalently, this can also be seen by calculating the Kraus operators for the system qubit from the joint unitary operator and then applying infinite time limit, $\lim_{t\to\infty} \rho_s(t) = \rho_{th}.$
Now, the thermalizing Hamiltonian $(H_{th})$ is found and details of the derivation are given in Appendix B.
$H_{th}$ is of the following form, \begin{equation} \label{hth}
H_{th}(t) = f(t) \Big(|\phi^+\rangle\langle\phi^+ | - |\phi^-\rangle\langle\phi^- |\Big), \end{equation}
where, \begin{align} \label{ft} &f(t)= \frac{\gamma e^{-\gamma t/2}}{2\sqrt{1- e^{-\gamma t}}},\\ &\ket{\phi^\pm} = \frac{1}{\sqrt{2}}(\ket{00} \pm \ket{11}). \end{align}
The most general two-qubit time-dependent Hamiltonian which gives rise to the affine transformation (\ref{M1}), by acting on tensor product of arbitrary intitial state of the system qubit and the initial state of the ancilla qubit being $\rho_{th}$ (given in equation (\ref{gval})), is of the form given in equation (\ref{hth}) above.
\section{On Markovianity of Dynamics for Thermalization}
Given a 2-qubit Hamiltonian of the form, \begin{equation} \label{Hft} H(t) = f(t) (\ket{\phi^+}\bra{\phi^+}-\ket{\phi^-}\bra{\phi^-}) \end{equation} where $\ket{\phi^\pm}=(\ket{00}\pm\ket{11})/\sqrt{2}$, we can ask what are the conditions on $f(t)$ such that the system will thermalize in the asymptotic time limit. Moreover, we can ask when the evolution of the system follows Markovian dynamics. The main reason behind the search for generic properties of $f(t)$ in the above equation is to look for a generic Hamiltonian (involving ancilla) method for thermalization which does not necssarily follow from the optical master equation - in the latter case the system is known to thermalize in the infinite time limit. We can rewrite eq. (\ref{Hft}) in the Pauli basis as $f(t)(\sigma_x\otimes\sigma_x - \sigma_y\otimes\sigma_y)$. This represents a kind of spin exchange interaction similar to the double-quantum Hamiltonian used in NMR experiments \cite{dqh}.In particular, $f(t)$ can be interpreted as a time-dependent coupling strength between the spins. Such Hamiltonians can in principle be realized in lab.
\subsection{Thermalization} Given an arbitrary initial state for the system (say, $\rho_s^i$) and an initial thermal state for the ancilla (say, $\rho_e^i=\frac{1}{2}\text{diag}(1+g,1-g)$), we can derive the condition on a generic $f(t)$ such that the system will thermalize in the infinite time limit i.e. by imposing the following constraint,
\begin{equation*} \lim_{t\to\infty}\text{Tr}_e \Big[U(t,0) (\rho_s^i\otimes\rho_e^i)U(t,0)^{\dagger}\Big]=\text{diag}(\frac{1-g}{2},\frac{1+g}{2}) \end{equation*} where $U(t,0)=\exp{\left(-i\int_0^t H(\tau)d\tau \right)}$ with $H(\tau)$ defined above in (\ref{Hft}) and the RHS is as we saw in (\ref{gval}).
This condition for thermalization is finally found to be,
\begin{equation} \lim_{t\to\infty} F(t) = (2n+1)\frac{\pi}{2} \end{equation} where, $F(t)=\int_0^t f(\tau)d\tau$ and $n$ is any integer. \subsection{Markovianity of System Evolution} Another interesting question we can raise is about the nature of the system evolution under such a Hamiltonian - will it be Markovian always? To answer this we refer to \cite{cirac} in which the authors have produced necessary and sufficient conditions for a given master equation $\dot{\rho}=L_t[\rho]$ to be Markovian (CP divisible) in nature. These conditions are:
\begin{itemize} \item $L_t$ must be hermiticity preserving. \item $L_t^*(\mathbb{1})=0$, \text{and} \item $\omega_c L_t^\Gamma\omega_c \geqslant 0$, \end{itemize} for all times $t$, where $L_t^*$ and $L_t^\Gamma$ are the adjoint map and Choi map of $L_t$ respectively. $\omega_c=\mathbb{I}-\ket{\omega}\bra{\omega}$ is the projector onto the orthogonal complement of the maximally entangled state $\ket{\omega}=\sum_i \frac{1}{\sqrt{2}}\ket{i,i}$.
It can be seen that the hermiticity preserving condition will always be satisfied for our particular case. Imposing the other conditions, we obtain the following necessary and sufficient constraints on the time dependence of the Hamiltonian for ensuring Markovianity of the dynamical map, \begin{align}
0\leqslant F(t) &\leqslant \frac{\pi}{2},~\forall t\\
\frac{d}{dt}F(t)&\geqslant 0,~\forall t \end{align}
Note that alternatively, we can have a monotonically decreasing $F(t)$ bounded between $[-\frac{\pi}{2},0]$ if we choose $-f(t)$ in our Hamiltonian (\ref{Hft}).
We may now think of a functional form of $f(t)$ which satisfies the thermalization condition but violates the markovianity conditions - namely that $F(t)$ be monotonic and bounded. A simple example for such a non-Markovian thermalizing form is,
\begin{equation} \label{eg} F(t)=\frac{\sin(20 t)}{1+10 t}+(1-e^{-t})\frac{\pi}{2} \end{equation}
FIG. 1 plots $F(t)=\int_0^t f(\tau)d\tau$ for $f(\tau)$ given by equation (\ref{ft}) and also $F(t)$ given by equation (\ref{eg}).
\begin{figure}
\caption{(colour online) Red solid line is the $F(t)$ corresponding to non-Markovian thermalizing Hamiltonian while the black corresponds to that of our Markovian thermalizing form. Note that both converge to $\frac{\pi}{2}$ asymptotically and hence signify thermalization.}
\end{figure}
In the preceding sections, we have derived a specific form of thermalizing Hamiltonian from the quantum optical master equation and then we generalized it by identifying conditions for the dynamics to be Markovian. We now derive the master equation that refers to the system dynamics for thermalization under our specific form of Hamiltonian given by equation (\ref{Hft}). It is found to be of the following form,
\begin{equation} \label{lindtype} \begin{split} \frac{d \rho (t)}{dt} &= \gamma_1(t) \Big(\sigma_-\rho(t)\sigma_+ - \frac{1}{2} \{ \sigma_+\sigma_-,\rho(t) \}\Big)\\ &~~ + \gamma_2(t) \Big(\sigma_+\rho(t)\sigma_-\frac{1}{2}\{\sigma_-\sigma_+,\rho(t)\}\Big) \end{split} \end{equation}
where, \begin{align*} \gamma_1(t) &= (1+g) f(t) \tan [F(t)] \\ \gamma_2(t) &= (1-g) f(t) \tan [F(t)] \end{align*}
Here, $F(t)=\int_0^t f(\tau)d\tau$ and $g$ is the parameter referring to the bath temperature used in defining the initial ancilla state as $\sigma_e(0)=\frac{1}{2}(\mathbb{1}+g \sigma_3)$. For more details regarding the derivation of master equation, refer to Appendix C.
The above form of master equation is immediately reminiscent of the Lindblad (Markovian) form that we have used at the beginning in equation (\ref{lind}), hence we have a master equation that is of the Lindblad type, but with time-dependent coefficients $\gamma_1(t)$ and $\gamma_2(t)$. It has been shown that the negativity of decoherence rates represent non-Markovianity \cite{hall}. Simply put, if the decoherence rates remain non-negative for all time, then the master equation represents a Markovian evolution. On the other hand, if for some time interval, it becomes negative, the dynamics is necessarily non-Markovian.
Thus, we have derived a class of master equations that can describe both Markovian as well as non-Markovian thermalization depending on the choice of $f(t)$ in the Hamiltonian.
For example, consider the non-Markovian $F(t)$ we have defined in equation (\ref{eg}). The corresponding $f(t)$ is calculated by taking the derivative of $F(t)$ and
thus, we can derive the master equation governing such a dynamics. It can be checked that the coefficients $\gamma_1(t)$ and $\gamma_2(t)$ will not be non-negative for all time. Thus, it is seen to signify the non-Markovian nature of the dynamics.
It can also be seen that when we consider the $f(t)$ we originally derived given by equation (\ref{ft}), we recover the quantum optical master equation (\ref{lind}) with $\gamma_1(t)$ and $\gamma_2(t)$ reducing to the appropriate time-independent, positive coefficients.
\section{Conclusion} In this paper, we look at a Markovian master equation of a qubit that leads to thermalization and simulate it through a unitary process by replacing the thermal bath with a single-qubit mixed state ancilla. Thus, we derive a thermalizing Hamiltonian for a single qubit corresponding to the quantum optical master equation.
Although a Markovian model of thermalization has been used here, there exist non-Markovian models as well. Those models need not necessarily be simulatable through a single-qubit ancilla (mixed or pure). For example, we considered the case of post-Markovian master equation as in \cite{manis,lidar} and find that a single-qubit ancilla is not sufficient to simulate the thermalization process described therein.
We derive necessary and sufficient conditions for thermalization and Markovianity of the state evolution under a specific form (\ref{Hft}) of system-ancilla Hamiltonian. We find that it is indeed possible for us to have non-Markovian thermalization processes even for this specific kind of Hamiltonian we have described in this work.
We also derive a Lindblad type master equation for system dynamics arising out of the Hamiltonian described in our work. We see that it is possible to find signature of non-Markovian dynamics based on the negativity of decoherence rates in the master equation.
In principle, the method we have employed can be used for finding simulating Hamiltonians in higher (finite) dimensions as well. It is non-trivial because the parametrization of unitary operators for higher dimensions aren't readily available as was the case for 2-qubit unitaries. Nevertheless, in the case of infinite dimensional systems (eg: quantum harmonic oscillators), covariance matrices can be employed to proceed in this direction. As a future project, it would be interesting to study simulating thermalizing Hamiltonians for single mode harmonic oscillator.
We expect that our result will stimulate further interest in finding out the fundamental dynamics that leads to thermalization (for example, studying adiabaticity in open quantum systems). As an extension to this work, we hope to look into more general thermalization models (including non-Markovian) which will require two-qubit ancillae. Also, finding similar thermalizing Hamiltonian models for leaking cavity modes of radiation fields is an intriguing future project.
\textbf{Acknowledgements:} We would like to thank Daniel Alonso and Ramandeep Johal for insightful comments. We would also like to acknowledge productive discussions with Sandip Goyal, Manik Banik, Arindam Mallick, and George Thomas.
\section{Appendix A}
Here, we discuss the explicit calculations involved in parametrizing $C$ and $M$ matrices (appearing in equations (\ref{C}) and (\ref{M})) of the single-qubit channels simulatable through a single-qubit mixed state ancilla.
The form of $U$, given in equation (\ref{ud}), can be re-written after a simple basis change in the following way, \begin{equation} \begin{split} U &= K_0(\mathbb{1}^{(s)}\otimes \mathbb{1}^{(e)}) + K_1(\sigma_1^{(s)} \otimes \sigma_1^{(e)})\\
&+ K_2(\sigma_2^{(s)} \otimes \sigma_2^{(e)})+ K_3(\sigma_3^{(s)} \otimes \sigma_3^{(e)}) \end{split}\tag{A.1} \end{equation} where, \begin{equation}
\begin{split}
&K_0 = \frac{1}{2}\left(\cos\frac{\alpha+\delta}{2} + e^{-i\beta}\cos\frac{\alpha-\delta}{2}\right),\\
&K_1 = \frac{i}{2}\left(\sin\frac{\alpha+\delta}{2} + e^{-i\beta}\sin\frac{\alpha-\delta}{2}\right),\\
&K_2 = \frac{-i}{2}\left(\sin\frac{\alpha+\delta}{2} - e^{-i\beta}\sin\frac{\alpha-\delta}{2}\right),\\
&K_3 = \frac{1}{2}\left(\cos\frac{\alpha+\delta}{2} - e^{-i\beta}\cos\frac{\alpha-\delta}{2}\right).
\end{split}\tag{A.2} \end{equation}
Now recalling the form of the mixed state ancilla $\rho_e$ and using an arbitrary initial state for the system qubit $\rho_s = \frac{1}{2}(\mathbb{1}+\bar{r}.\bar{\sigma})$, we can define the composite initial state, $\rho_{se}^{initial}=\rho_s\otimes\rho_e.$
The final time-evolved state of the system qubit can be found as, $\rho_s^{final}= \text{Tr}_e \Big[U \rho_{se}^{initial} (U)^{\dagger}\Big]$.
Now we can find out the components of $\rho_s^{final}$ in the basis $\{\sigma_1^{(s)},\sigma_2^{(s)},\sigma_3^{(s)}\}$ by computing Tr$[\sigma_i^{(s)}\rho_s^{final}]$. Thereby, we can read out the elements of $M$ and $C$. For example, consider $i=3$, we get: \begin{equation} \text{Tr}[\sigma_3^{(s)}\rho_s^{final}] = M_{31}n_1+M_{32}n_2+M_{33}n_3+C_3.\tag{A.3} \end{equation} Finally, we get the parametrized matrices $M$ and $C$ produced earlier.
\section{Appendix B}
To find the thermalizing Hamiltonian, we first need to find the values of the parameters that match with our particular case. For this, we compare the affine transformation for the quantum optical case in equations (\ref{M1}) with the parametrized matrices in equations (\ref{C}) and (\ref{M}). It can be easily seen that there exists a set of parameters as given below: \begin{equation} \label{set1} \lambda=g,\cos\alpha=\cos\delta= e^{\frac{-\gamma t}{2}},\cos\beta=\pm 1 =\cos\xi,\tag{B.1} \end{equation} where $\eta$ can be arbitrary. So finally, we get the mixed state ancilla as the following thermal state, \begin{equation*}
\rho_e = \frac{1+g}{2}|0\rangle\langle 0| + \frac{1-g}{2}|1\rangle\langle 1|. \end{equation*}
Putting the values from equation (\ref{set1}) in the form of unitary given in equation (\ref{ud}), we get the unitary for the thermalization process. Note that we now have a time dependent unitary, \begin{equation} U(t,0) = \begin{bmatrix} e^{\frac{-\gamma t}{2}} & 0 & 0 & i\sqrt{1- e^{-\gamma t}}\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ i\sqrt{1- e^{-\gamma t}} & 0 & 0 & e^{\frac{-\gamma t}{2}} \end{bmatrix}.\tag{B.2} \end{equation} From here, we can calculate $H_{th}$ as follows. We know, $$U(t_2,t_1) = \exp{\bigg(-i\int_{t_1}^{t_2} H(s) ds\bigg)}, \text{and}$$ \begin{equation*} \begin{split} U(t+\Delta t,t) &= \exp{\left(-i\int_{t}^{t+\Delta t} H(s) ds\right)}\\ &\approx \mathbb{1} -i\Delta t H(t) \end{split} \end{equation*} Using the semi-group property of $U(t)$ (which holds good for small time interval $\Delta t$ even if $H$ is time-dependent) we get, \begin{equation*} \begin{split} U(t+\Delta t,0) &= U(t+\Delta t,t)U(t,0)\\ \Rightarrow U(t+\Delta t,t) &= U(t+\Delta t,0)U^{\dagger} (t,0)\\ &= \Big(U(t,0) + \Delta t\frac{dU(t,0)}{dt}+\cdots\Big)U^{\dagger}(t,0)\\ &\approx \mathbb{1} + \Delta t \frac{dU(t,0)}{dt}U^{\dagger}(t,0) \end{split} \end{equation*} Comparing with the RHS of the previous equation, we get: \begin{equation*} H_{th}(t) = i\left(\frac{dU(t,0)}{dt}\right)U^{\dagger}(t,0) \end{equation*} Thus, we get: \begin{equation} \begin{split}
H_{th} &= \frac{\pm \gamma e^{\frac{-\gamma t}{2}}}{2\sqrt{1-e^{-\gamma t}}} \Big(|00\rangle\langle11| + |11\rangle\langle 00|\Big)\\
&= f(t) \big(|\phi^+\rangle\langle\phi^+| - |\phi^-\rangle\langle\phi^-|\big) \end{split} \tag{B.3} \end{equation} Without loss of generality, we choose the positive sign for the $f(t)$ in this paper.
\section{Appendix C}
We consider a Hamiltonian of the form (\ref{Hft}), with fixed initial state of ancilla qubit as $\sigma_e(0)=\frac{1}{2}(\mathbb{1}+g \sigma_3)$ (i.e. a thermal state with temperature defined through $g$ as previously explained) and an arbitrary initial state of system qubit $\rho_s(0)=\frac{1}{2}(\mathbb{1}+\bar{r}.\bar{\sigma})$ with $\bar{r}=(x,y,z)$. The time evolved state of the system under the action of such a Hamiltonian can be calculated as, \begin{equation} \rho_s(t)= \text{Tr}_e \Big[U(t,0) \rho_{s}(0)\otimes\sigma_e(0) (U(t,0))^{\dagger}\Big].\tag{C.1} \end{equation} where, $U(t,0)=\exp{\left(-i\int_0^t H(\tau)d\tau \right)}$.
Now, we use the mathematical prescription described in the Appendix of \cite{jordan} to derive the master equation for such a dynamics. First, we express $\rho_s(0)$ and $\rho_s(t)$ as vectors in the operator space of the system which has basis $\{\mathbb{1},\sigma_1,\sigma_2,\sigma_3\}$.The density matrix of the system can be represented by a $4\times1$ vector, and a superoperator on the system can be represented by a $4\times4$ matrix. In this representation, $\boldsymbol{v}_0=\frac{1}{2}[1,x,y,z]^{T}$ is the vector form of the initial arbitrary density matrix of the system qubit and the vector form of the system qubit at time $t$ is, \begin{equation} \boldsymbol{v}_t=\frac{1}{2}[1,C_t x,C_t y,C_t^2 z + g S_t^2]^{T}=Q_t \boldsymbol{v}_0\tag{C.2} \end{equation} where, $C_t\equiv \cos(F(t)),S_t\equiv \sin(F(t))$ and $Q_t$ is the matrix representation of the system qubit evolution from the initial time to the time $t$, \begin{equation} Q_t=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & C_t & 0 & 0\\ 0 & 0 & C_t & 0\\ g S_t^2 & 0 & 0 & C_t^2 \end{array}\right).\tag{C.3} \end{equation}
It can be seen that $Q_t$ is invertible for finite $t$. Thus we can find that,$\partial_t\boldsymbol{v}_t=\dot{Q}_t \boldsymbol{v}_0=\dot{Q}_t Q^{-1}_t\boldsymbol{v}_t.$
Thus, $\dot{Q}_t Q^{-1}_t$ is the matrix representation of the linear transformation corresponding to the time derivative of the system density matrix, and \begin{equation} \dot{Q}_t Q^{-1}_t=\left(\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & \alpha_t & 0 & 0\\ 0 & 0 & \alpha_t & 0\\ \beta_t & 0 & 0 & 2\alpha_t \end{array}\right).\tag{C.4} \end{equation} where, $\alpha_t =-f(t)\tan(F(t)),\beta_t = 2gf(t)\tan(F(t))$.
Now we can find the superoperator corresponding to $\dot{Q}_t Q^{-1}_t$. In order to do this, we need to know the matrix representations $s_{ij}$ for the basis of the superoperator $\sigma_i[\cdot]\sigma_j$. These representations are easy to find and are given in equation (S15) in \cite{hall}. Decomposing $\dot{Q}_t Q^{-1}_t$ into the matrix representation, we get, $\dot{Q}_t Q^{-1}_t=\sum_{i,j=0}^3 a_{ij} s_{ij}.$
In our particular case, the non-zero components $a_{ij}$ turns out to be $a_{00}=4\alpha_t,a_{03}=a_{30}=\beta_t,a_{11}=a_{22}=-2\alpha_t$ and $a_{21}=-a{12}=i\beta_t$. Now by de-vectorizing, the master equation can be written as,
\begin{equation} \begin{split} \partial_t \rho(t) &= 4\alpha_t\rho -2\alpha_t(\sigma_1\rho\sigma_1+\sigma_2\rho\sigma_2)\\ &~~ +i\beta_t(\sigma_2\rho\sigma_1-\sigma_1\rho\sigma_2)+\beta_t\{\rho,\sigma_3\} \end{split}\tag{C.5} \end{equation}
Using the fact that $\sigma_\pm=\sigma_1\pm i \sigma_2$, the above equation can easily be recast into the Lindblad type master equation as given in equation (\ref{lindtype}).
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Bound States for Magic State Distillation in Fault-Tolerant Quantum Computation}
\author{Earl T. Campbell} \affiliation{Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, UK.} \author{Dan E. Browne} \affiliation{Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, UK.}
\begin{abstract} Magic state distillation is an important primitive in fault-tolerant quantum computation. The magic states are pure non-stabilizer states which can be distilled from certain mixed non-stabilizer states via Clifford group operations alone. Because of the Gottesman-Knill theorem, mixtures of Pauli eigenstates are not expected to be magic state distillable, but it has been an open question whether all mixed states outside this set may be distilled. In this Letter we show that, when resources are finitely limited, non-distillable states exist outside the stabilizer octahedron. In analogy with the bound entangled states, which arise in entanglement theory, we call such states bound states for magic state distillation. \pacs{03.67.Pp} \end{abstract}
\maketitle
The significant noise and decoherence in quantum systems means that harnessing these systems for computational tasks must be performed fault tolerantly~\cite{STEANE95,G01a}. In a wide variety of setups only a limited set of gates, known as the \textit{Clifford group}, are implemented in a manifestly fault tolerant manner. Examples include some anyonic topological quantum computers~\cite{Moore90,Lloyd02,Doucot02}, post-selected quantum computers~\cite{Knill05,Rei02a} and measurement based topological quantum computers~\cite{RHG01a}. This motivates the problem of when such devices, with practically error free Clifford gates, may be promoted to a full quantum computer. The celebrated Gottesman-Knill theorem shows that a Clifford circuit acting on stabilizer states --- simultaneous eigenstates of several Pauli operators --- can be efficiently simulated by a classical computer~\cite{G02a}. However, given a resource of pure non-stabilizer states, we can implement gates outside the Clifford group. For example, a qubit in an eigenstate of the Hadamard enables one to implement a $\pi/8$ phase gate that when supplementing the Clifford group gives a dense covering of all unitary operations~\cite{BraKit05}, and so enables universal quantum computation.
Preparation of non-stabilizer states would usually require a non-Clifford operation, so in this context, one would require that even noisy copies of these states enable high fidelity quantum computation. Bravyi and Kitaev~\cite{BraKit05} showed that this can be achieved. Coining the term \textit{magic state distillation}, they showed that most mixed non-stabilizer states can be \textit{distilled} via Clifford group circuits to fewer copies of a lower entropy state, reaching in {the limit of infinite iterations a pure non-stabilizer \textit{magic state}. However, the protocols they presented do not succeed for all mixed non-stabilizer states. Bravyi and Kitaev were not satisfied by the ambiguous status of these states and concluded that ``\textit{The most exciting open problem is to understand the computational power of the model in [this] region of parameters.}''. Either all non-stabilizer states are efficiently distillable by an undiscovered protocol, or there exist non-stabilizer states that are impossible to distill. Such undistillable states we call \textit{bound states} for magic state distillation, in analogy with bound states in entanglement distillation~\cite{Horodecki99activate}. Here we make progress by showing that bound states exist for a very broad class of protocols. By showing that a single round of a finite sized protocol will not improve these states, it follows that repeating such a protocol, even with an infinite number of iterations, will also have no benefit. Hence, we explain why all known protocols fail to distill some states.
The single-qubit stabilizer states, for which the Gottesman-Knill theorem applies, are the six pure stabilizer states (the eigenstates of $\pm X, \pm Y$ and $\pm Z$) and \earl{any incoherent mixture of these}. In the Bloch sphere, this convex set with 6 vertices forms the \textit{stabilizer octahedron} partially shown in figure~\ref{fig:Outline}a. Single-qubit states have density matrices: \begin{equation} \label{eqn:NONedge}
\rho(f, \vec{a}) = \left( \mbox{\small 1} \!\! \mbox{1} + (2f-1)( a_{X} X + a_{Y} Y + a_{Z} Z ) \right) / 2, \end{equation}
\earl{where $\vec{a}=(a_{X}, a_{Y}, a_{Z})$ is a unit vector}, and $f$ is the fidelity w.r.t the pure state $|\psi_{\vec{a}}\rangle\langle\psi_\vec{a}|=(\mbox{\small 1} \!\! \mbox{1} + a_{X}X+a_{Y}Y+a_{Z}Z)/2$. Stabilizer states satisfy: \begin{equation} \label{eqn:OctSurface}
|2f-1|(|a_{X}|+|a_{Y}|+|a_{Z}|) \leq 1 \end{equation} where the equality holds for states on the surface of the octahedron, and we denote the fidelity of such surface states as ${f^S_{\vec{a}}}$, which is unique assuming $f \geq 1/2$.
\begin{figure}
\caption{One octant of the Bloch sphere with various regions and directions shown. (a) The blue region shows the stabilizer states in one octant. Each octant is identical, with all stabilizer states forming an octahedron. (b) The yellow plane is the distillation threshold for the 5 qubit code, with the direction of $\vec{a}_{T}$ shown. The yellow plane is parallel to the underlying blue face of the stabilizer octahedron, but is displaced by a small gap. (c) The three green planes are the thresholds for the Steane code, with each plane differing by local Clifford gates. These planes meet the stabilizer octahedron at its edges. The three vectors are axes of H-like gates, e.g. $\vec{a}_{H}$; (d) The combined region of states distilled by either the 5 qubit code or the Steane code. This region only touches the stabilizer octahedron at its edges, and no other known protocol is tight in any other direction.}
\label{fig:Outline}
\end{figure}
Prior protocols for magic state distillation~\cite{BraKit05,Rei01a,Rei02a,Rei03a} increase fidelity towards eigenstates of Clifford gates, such as the Hadamard $H$ and the $T$ gate\footnote{The T gate performs, $TXT^{\dagger}=Y$, $TYT^{\dagger}=Z$.}. These eigenstates have $\vec{a}_{H}=(1,0,1)/\sqrt{2}$ and $\vec{a}_{T}=(1,1,1)/\sqrt{3}$, with $f=1$ for ideal magic states. Given the ability to prepare a mixed non-stabilizer state, $\rho$, we can perform an operation called \textit{polarization}, or \textit{twirling}, that brings $\rho$ onto a symmetry axis of the octahedron. For example, by randomly applying $\mbox{\small 1} \!\! \mbox{1}$, $T$ or $T^{\dagger}$, we \earl{map} $\rho \rightarrow \rho(f, \vec{a}_{T})$.
Bravyi and Kitaev proposed the following protocol~\cite{BraKit05} for $\ket{T}$ state distillation: (1) Prepare 5 copies of $\rho(f,\vec{a}_{T})$; (2) Measure the 4 stabilizers of the five-qubit error correcting code; (3) If all measurements give $+1$, the protocol succeeds and the encoded state is decoded into a single qubit state, and otherwise restart. Upon a successful implementation of this protocol the output qubit has a fidelity $F(f)$ plotted in figure~\ref{fig:fidelities}b. Provided the initial fidelity is greater than some threshold, a successful implementation yields a higher fidelity. This protocol has a non-tight threshold, and exhibits a gap between the threshold and the set of stabilizer states. Because the initial state was twirled onto the T axis, the threshold forms a plane in the Bloch sphere (see figure~\ref{fig:Outline}). In contrast, Reichardt has proposed a protocol that does have a tight threshold for distillation of $\rho(f, \vec{a}_{H})$ states in a $H$-like direction~\cite{Rei01a}. His protocol is similar to above, but uses 7 qubits each attempt and measures the 6 stabilizers of the STEANE code~\cite{STEANE95}. In figure~\ref{fig:fidelities}a we show the performance of this protocol, where there is no threshold gap. When the initial mixture is not of the form $\rho(f, \vec{a}_{H})$, we twirl the initial mixture onto the $H$ axis. Hence, the threshold forms a plane for each $H$-like direction (see figure~\ref{fig:Outline}). Although the protocol is tight in directions crossing an octahedron edge, the protocol fails to distill some mixed states just above the octahedron faces, and so is not tight in all directions. Even the combined region of states distilled by all known protocols still leaves a set of states above the octahedron faces, whose distillability properties are unknown.
Here we show that for all size $n$ protocols there is a region of bound states above the octahedron faces. More formally, we considering all states $\rho(f, \vec{a}_{P})$ where $\vec{a}_{P}$ has all positive (non-zero) components. Having all components as non-zero excludes states above octahedron edges. Considering only states in the positive octant is completely general as Clifford gates enable movement between octants. Many copies of bound states cannot be used to improve on a single copy, and below we formalize the idea of \textit{not improved} and state our main result.
\begin{defin} We say $\rho'$ is not an improvement on $\rho(f, \vec{a}_{P})$, when $\rho'$ is a convex mixture of $C_{i}\rho(f, \vec{a}_{P})C_{i}^{\dagger}$ and stabilizer states, where $C_{i}$ are Clifford group gates. \end{defin}
\begin{theorem} \label{THMgeneralcase} Consider a device capable of ideal Clifford gates, preparation of stabilizer states, classical feedforward and Pauli measurements. For any protocol on this device that takes $\rho(f, \vec{a}_{P})^{\otimes n}$ and outputs a single qubit, $\rho'$, there exists an $\epsilon > 0$ such that $\rho'$ is not an improvement on $\rho(f, \vec{a}_{P}) $ for $f \leq {f^S_{\vec{a}_{P}}}+\epsilon$. \end{theorem}
Theorem~\ref{THMgeneralcase} covers \earl{a wide} class of protocols, \earl{which} attain a fidelity that is upper-bounded by a narrower class of protocols~\cite{Camp09c}, such that theorem~\ref{THMgeneralcase} follows from:
\begin{theorem} \label{THMstabilizercase} Consider all protocols that follow these steps: (i) prepare $\rho(f, \vec{a}_{P})^{\otimes n}$; (ii) measure the $n-1$ generators of an $n$ qubit stabilizer code $\mathcal{S}_{n-1}$ with one logical qubit; (iii) postselect on all ``+1" measurement outcomes; (iv) decode the stabilizer code and output the logical qubit as the single qubit state $\rho'$. For all such protocols there exists an $\epsilon > 0$ such that $\rho'$ is not an improvement on $\rho(f, \vec{a}_{P}) $ for $f \leq {f^S_{\vec{a}_{P}}}+\epsilon$. \end{theorem}
\begin{figure}\label{fig:fidelities}
\end{figure}
Prior protocols, such as those based on the STEANE code and 5 qubit code, are covered explicitly by theorem~\ref{THMstabilizercase}. Here we use the structure of stabilizer codes to prove theorem~\ref{THMstabilizercase}, with theorem~\ref{THMgeneralcase} following directly from the results of~\cite{Camp09c}, where such distillation protocols are shown to have equal efficacy with more general Clifford protocols. It is crucial to consider the implication of these theorems when an $n$-qubit protocol is iterated $m$ times. When a single round provides no improvement on the initial resource, the input into the second round will only differ by Clifford group operations, and hence our theorem applies to the second, and all subsequent, rounds. Hence, repeated iteration cannot be used to circumvent our theorem. Before proving these theorems, we derive a pair of powerful lemmas that identify bound states. \begin{lem} \label{LEM1} Consider n copies of an octahedron surface state $\rho({f^S_{\vec{a}_{P}}}, \vec{a}_{P})$ projected onto the codespace of $\mathcal{S}_{n-1}$ and then decoded. If the output qubit is in the octahedron interior, then there exists an $\epsilon >0$ such that for $f\leq {f^S_{\vec{a}_{P}}}+\epsilon$ the same projection on $\rho(f, \vec{a}_{P})^{\otimes n}$ also projects onto a mixed stabilizer state. \end{lem} This lemma follows directly from the dependence of the output on $f$, which for finite $n$ is always continuous. We can observe this lemma at work in figure~\ref{fig:fidelities}b. Our next lemma identifies when octahedron surface states are projected into the octahedron interior. Before stating this we must establish some notation. An initial state, $\rho({f^S_{\vec{a}_{P}}}, \vec{a}_{P})^{\otimes n}$, is an ensemble of pure stabilizer states: \begin{equation} \label{eqn:initial_ensemble}
\rho({f^S_{\vec{a}_{P}}}, \vec{a}_{P})^{\otimes n}=\sum_{\vec{g} \in \{X, Y, Z\}^{n}} q_{\vec{g}} \kb{\Psi_{\vec{g}}}{\Psi_{\vec{g}}}, \end{equation} where $\ket{\Psi_{\vec{g}}}$ is stabilized, $g\ket{\Psi_{\vec{g}}}=\ket{\Psi_{\vec{g}}}$, by the group $\mathcal{G}_{\vec{g}}$ generated by $\vec{g}=$($g_{1}$, $g_{2}$,... $g_{n}$). The operator $g_{i}$ is $X_{i}$, $Y_{i}$ or $Z_{i}$, with $i$ labeling the qubit on which it acts. Each contribution has a weighting $q_{\vec{g}} = \prod_{i} ( a_{g_{i}} /(a_{X}+a_{Y}+a_{Z} ) )$. Measuring the generators of $\mathcal{S}_{n-1}$ and post-selecting on ``+1'' outcomes, projects onto the codespace of $\mathcal{S}_{n-1}$ with projector $P=\sum_{s \in \mathcal{S}_{n-1}}s/2^{n-1} $, producing: \begin{equation} \label{eqn:final_state}
\frac{P \rho({f^S_{\vec{a}_{P}}}, \vec{a}_{P})^{\otimes n} P}{\ensuremath{\mathrm{tr}} [ P \rho({f^S_{\vec{a}_{P}}}, \vec{a}_{P})^{\otimes n} P ] } = \sum_{\vec{g} \in \{ X, Y, Z \}^{n} } q'_{\vec{g}} \kb{\Psi'_{\vec{g}}}{\Psi'_{\vec{g}}},
\end{equation} with projected terms, $\ket{\Psi'_{\vec{g}}}$, of new weighting $ q'_{\vec{g}}$. Each $\ket{\Psi'_{\vec{g}}}$ has its stabilizer generated by ($G_{\vec{g}}, s_{1}, s_{2},.... s_{n-1}$), where $G_{\vec{g}}$ is an independent generator that: (a) was present in the initial group $G_{\vec{g}} \in \mathcal{G}_{\vec{g}}$; and (b) commutes with the measurement stabilizers $G_{\vec{g}} \mathcal{S}_{n-1}= \mathcal{S}_{n-1}G_{\vec{g}}$. In other words, it must be equivalent to one of six logical Pauli operators of the codespace. We denote the set of logical operators as $\mathcal{L}$, and its elements $\pm X_{L}, \pm Y_{L}$ and $\pm Z_{L}$, and so $G_{\vec{g}} \in \mathcal{L} . \mathcal{S}_{n-1}$. This defines a decoding via the Clifford map, $X_{L}\rightarrow X_{1}$ and $Z_{L} \rightarrow Z_{1}$. Since there are only six distinct logical states, we can combine many terms in equation~\ref{eqn:final_state}: \begin{equation}
\frac{P \rho({f^S_{\vec{a}_{P}}}, \vec{a}_{P})^{\otimes n} P}{\ensuremath{\mathrm{tr}} [ P \rho({f^S_{\vec{a}_{P}}}, \vec{a}_{P})^{\otimes n} P ] } = \sum_{L \in \mathcal{L} } q_{L} \kb{\Psi_{L}}{\Psi_{L}} , \end{equation} where $\ket{\Psi_{L}}$ has stabilizer generators ($L, s_{1}, s_{2},.... s_{n-1} $). The new weighting is $q_ {L} = \sum q'_{\vec{g}}$ with the sum taken over all $\vec{g}$ that generate $\mathcal{G}_{\vec{g}}$ containing an element $G_{\vec{g}} \in L.\mathcal{S}_{n-1}$. We can now state the next lemma:
\begin{lem} \label{LEM2} Given $n$ copies of $\rho({f^S_{\vec{a}_{P}}}, \vec{a}_{P})$ projected into the codespace of $\mathcal{S}_{n-1}$ and decoded, the output qubit is in the octahedron interior if there exist any two pure states in the initial ensemble, $\ket{\Psi_\vec{g}}$ and $\ket{\Psi_\vec{g'}}$ (defined in equation~\ref{eqn:initial_ensemble}), such that both: \begin{enumerate}
\item[(i)] the projected pure states are orthogonal, so that $L \in \mathcal{G}_{\vec{g}}$ and $-s L \in \mathcal{G}_{\vec{g}'}$ where $L \in \mathcal{L}$ and $s \in \mathcal{S}_{n-1}$; \\
\textit{and}
\item[(ii)] upon projection $\ket{\Psi_\vec{g}}$ and $\ket{\Psi_\vec{g'}}$ do not vanish, so $q'_{\vec{g}} \neq 0$ and $q'_{\vec{g}'} \neq 0$. \end{enumerate} \end{lem} \noindent We prove this lemma by contradiction. From equation~\ref{eqn:OctSurface}, and $(2f-1)a_{L}=(q_{L}-q_{-L})$, surface states satisfy:
\begin{equation}
|q_{X_{L}}-q_{-X_{L}}|+ |q_{Y_{L}}-q_{-Y_{L}}|+ |q_{Z_{L}}-q_{-Z_{L}}|=1, \end{equation}
and we assume to the contrary that the projected state has this form. Since $q_{\pm L}$ are non-negative reals, we have $|q_{L}-q_{-L}|=q_{L}+q_{-L}-2\mathrm{Min}(q_{L}, q_{-L})$, where $\mathrm{Min}(q_{L}, q_{-L})$ is the minimum of $q_{L}$ and $q_{-L}$. Along with the normalization condition, $\sum_{L} q_{L}=1$, this entails: \begin{equation} \mathrm{Min}(q_{X_{L}}, q_{-X_{L}})+\mathrm{Min}(q_{Y_{L}}, q_{-Y_{L}})+\mathrm{Min}(q_{Z_{L}}, q_{-Z_{L}}) =0. \nonumber \end{equation} Since all terms are positive, no cancellations can occur and so every term must vanish, hence $\mathrm{Min}(q_{L}, q_{-L}) = 0, \forall L$. However, conditions (\textit{i}) and (\textit{ii}) of the lemma entail that there exists a non-vanishing $\mathrm{Min}(q_{L}, q_{-L})$, as $q_{L}\geq q'_{\vec{g}} \neq 0$ and $q_{-L}\geq q'_{\vec{g}'} \neq 0$. Having arrived at this contradiction, we conclude the falsity of the assumption that the projected state remains on the octahedron surface, and so must be in the octahedron interior. This proves lemma~\ref{LEM2}, and we now show that lemma~\ref{LEM2} applies to all stabilizer reductions that do not trivially take $\rho(f, \vec{a}_{P})^{\otimes n} \rightarrow C_{i} \rho(f, \vec{a}_{P}) C_{i}^{\dagger}$.
Our proof continues by finding canonical generators for the code $\mathcal{S}_{n-1}$. A related method has been used to prove that all stabilizer states are local Clifford equivalent to a graph state~\cite{VdNDDM01a}, and we review this first. All stabilizer states have a stabilizer $\mathcal{S}_{n}$ with $n$ generators. Each generator is a tensor product of $n$ single-qubit Pauli operators. This can be visualized as an $n$ by $n$ matrix with elements that are Pauli operators, each row a generator and each column a qubit. Different, yet equivalent, generators are produced by row multiplication, via which we can produce a canonical form. In this form column $i$ has a non-trivial Pauli operator $A_{i}$ that appears on the diagonal, and all other operators in that column are either the identity or another operator $B_{i}$. Note that $A_{i}$ and $B_{i}$ compose a third non-trivial Pauli $A_{i}B_{i}=i(-1)^{\gamma_{i}}C_{i}$ with $\gamma_{i}=0,1$. Hence, all stabilizer states differ from some graph state by only local Cliffords that map $(A_{i}, B_{i}) \rightarrow (X_{i}, Z_{i})$.
A code, $\mathcal{S}_{n-1}$, has one less generator than the number of qubits, and so more columns than rows. We can apply the diagonalisation procedure on an $n-1$ by $n-1$ submatrix, to bring this submatrix into canonical form. Hence, we can find generators of $\mathcal{S}_{n-1}$ such that: \begin{equation}
s_{j} = (-1)^{\alpha_{j}} A_{j} ( \prod_{k \neq j, n} B_{k}^{\beta_{k,j}} ) T_{j, n} , \end{equation} where the variables $\beta_{k,j}=0,1$ denote whether $B_{k}$ or $\mbox{\small 1} \!\! \mbox{1}_{k}$ is present, and $\alpha_{j}=0,1$ defines the phase. With the $n^{\mathrm{th}}$ column out of canonical form, this leaves the $n^{\mathrm{th}}$ qubit operator $T_{j,n}$ unspecified. However, if all these generators have $T_{j,n}=\mbox{\small 1} \!\! \mbox{1}_{n}$, then the protocol is trivial and projects $n-1$ qubits into a known stabilizer state and the last qubit untouched, and so no improvement is made for any $f$. Hence, herein we assume the non-trivial case; in particular we assume stabilizer $T_{n-1, n} \neq \mbox{\small 1} \!\! \mbox{1}_{n}$. Since, we can always relabel qubits this is completely general. Furthermore, we can define $T_{n-1, n}=A_{n}$. \earl{Now} we can define a logical operator in the codespace of $\mathcal{S}_{n-1}$: \begin{eqnarray}
Z_{L} & = & \left( \prod_{1 \leq j \leq n-2} B^{\zeta_{j}}_{j} \right) B_{n-1} B_{n}, \end{eqnarray} where the variables $\zeta_{j}=0,1$ are uniquely fixed by commutation relations $Z_{L} s_{j}=s_{j} Z_{L}$. Note that $Z_{L}$ has some inbuilt freedom as $B_{n}$ is not fixed other than that $B_{n}\neq A_{n}, \mbox{\small 1} \!\! \mbox{1}_{n}$, which is equivalent to free choice of $\gamma_{n}$ in the expression $A_{n}B_{n}=i (-1)^{\gamma_{n}}C_{n}$. Now we enquire whether the final state contains two terms stabilized by $Z_{L}$ and $-sZ_{L}$ respectively, hence satisfying the conditions for lemma~\ref{LEM2}. If we consider the product of $Z_{L}$ and $s_{n-1}$, and choose $\gamma_{n}=\alpha_{n-1} + \gamma_{n-1}$ mod 2, we have: \begin{equation} - s_{n-1} Z_{L} = \left( \prod_{1 \leq k \leq n-2} B_{k}^{\beta_{k, n-1}+\zeta_{k}} \right) C_{n-1} C_{n} . \end{equation} Our choice of $\gamma_{n}$ ensures a minus sign on the left hand side, which aids in finding $\ket{\Psi_{\vec{g}}}$ and $\ket{\Psi_{\vec{g}'}}$ that satisfy our lemma by being stabilized by $G_{\vec{g}} = Z_{L}$ and $G_{\vec{g}'} =-s_{n-1}Z_{L}$ respectively. This criterion is fulfilled when: \begin{eqnarray*} \vec{g} & = & ( B_{1}, B_{2}, ....., B_{n-2}, B_{n-1}, B_{n}) ,\\ \nonumber \vec{g'} & = & ( B_{1}, B_{2}, ....., B_{n-2}, C_{n-1}, C_{n}) . \end{eqnarray*} These states only vanish under projection, $q'_{\vec{g}}, q'_{\vec{g'}}=0$, if they are stabilized by the negative of some element of the code $\mathcal{S}_{n-1}$. To prove they don't vanish, we first observe that every element of $\mathcal{G}_{\vec{g}}$ and $\mathcal{G}_{\vec{g}'}$ has either $\mbox{\small 1} \!\! \mbox{1}_{j}$ or $B_{j}$ acting on qubit $j$, for all $j=1,2,...n-2$. The only elements of $\mathcal{S}_{n-1}$ for which this is true are $\mbox{\small 1} \!\! \mbox{1}$ and $s_{n-1}$, but $s_{n-1}$ has $A_{n-1}A_{n}$ acting on the last two qubits and neither $\mathcal{G}_{\vec{g}}$ or $\mathcal{G}_{\vec{g}'}$ contain any such element.
Using a canonical form of the generators of $\mathcal{S}_{n-1}$, we have shown that non-trivial codes always satisfy the conditions of lemma~\ref{LEM2}. That is, all non-trivial codespace projections take many surface states into the octahedron interior. From the continuity expressed by lemma~\ref{LEM1}, this entails the existance of a finite region of non-stabilizer states that are also projected into the octahedron. Hence, all $n$-copy protocols do no improve on a single copy for some region of bound states above the octahedron faces, completing the proof. This does not contradict known tight thresholds in edge directions, as these directions have $\vec{a}$ with one zero component.
Although our proof holds for protocols using fixed and finite $n$ copies of $\rho(f, \vec{a}_{P})$, we could conceive of a protocol that varies $n$. If this varying-$n$ protocol has an $n$-dependent threshold, $f^{T}_{\vec{a}_{P}}(n)$, and $f^{T}_{\vec{a}_{P}}(n) \rightarrow f^{S}_{\vec{a}_{P}}$ as $n \rightarrow \infty$, then its threshold would be arbitrarily suppressible. \earl{Repeated iterations of a protocol, or equivalently employing concatenation of a single-qubit code, will not change the threshold. However, one could consider a broader class of protocols consisting of iterates that act on p qubits and output q qubits (for $p > q > 1$) followed by a final round outputting a single qubit. Such protocols map $n$ qubits to $1$ qubit, with $n$ growing each iterate, but with only $p$ qubits involved in each iterate. This implies that multi-qubit output iterates may suppress the threshold effectively, and are worth further study. Currently, no such protocol is known.} As such, in the asymptotic regime, bound magic states may not exist. However, numerical evidence so far indicates that smaller codes tend to produce better thresholds than larger codes. Nevertheless, the theorem does not rule out infinite cases from attaining a tight threshold. In the regime of finite resources, bound states do exist, and it is interesting ask what computational power Clifford circuits acting on such states possess. Can we find methods of efficiently classically simulating bound states; or can bound states be exploited in algorithms that offer a speedup over classical computation?
Furthermore, our proof assumes a protocol acting on identical copies, which invites study into whether our results extend to non-identical copies. In particular, following the analogy with entanglement distillation, we speculate that bound magic states may be distillable via ``catalysis'', where some non-consumed distillable resource activates the distillation~\cite{Horodecki99activate}. Finally we note that noisy Clifford gates can also enable quantum computation~\cite{Plenio08,vanDam09}, and we conjecture that a similar theorem will apply to a class of noisy Clifford gates analogous to states just above the octahedron faces.
The authors would like to thank Shashank Virmani, Matthew Hoban, Tobias Osborne, Ben Reichardt and Steve Flammia for interesting discussions. We acknowledge support from the Royal Commission for the Exhibition of 1851, the QIP IRC, QNET and the National Research Foundation and Ministry of Education, Singapore
\end{document}
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arXiv
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\begin{document}
\title{Growth of solutions for QG and 2D Euler equations}
\author{Diego Cordoba \\ {\small Department of Mathematics} \\ {\small University of Chicago} \\ {\small 5734 University Av, Il 60637} \\ {\small Telephone: 773 702-9787, e-mail: [email protected]} \\ {\small and} \\ Charles Fefferman\thanks{Partially supported by NSF grant DMS 0070692.}\\ {\small Princeton University} \\ {\small Fine Hall, Washington Road, NJ 08544} \\ {\small Phone: 609-258 4205, e-mail: [email protected]} \\ }
\date{January 17 2001}
\maketitle
\markboth{QG and 2D Euler equations}{D.Cordoba and C.Fefferman}
\newtheorem {Thm}{Theorem} \newtheorem {Def}{Definition} \newtheorem {Lm}{Lemma} \newtheorem {prop}{Proposition} \newtheorem {Rem}{Remark} \newtheorem {Cor}{Corollary} \def\mathcal{\mathcal} \newtheorem {Ack*}{Acknowledgments}
\section{Abstract}
We study the rate of growth of sharp fronts of the Quasi-geostrophic equation and 2D incompressible Euler equations.. The development of sharp fronts are due to a mechanism that piles up level sets very fast. Under a semi-uniform collapse, we obtain a lower bound on the minimum distance between the level sets.
\section{Introduction}
The work of Constantin-Majda-Tabak [1] developed an analogy between the Quasi-geostrophic and 3D Euler equations. Constantin, Majda and Tabak proposed a candidate for a singularity for the Quasi-geostrophic equation. Their numerics showed evidence of a blow-up for a particular initial data, where the level sets of the temperature contain a hyperbolic saddle. The arms of the saddle tend to close in finite time, producing a a sharp front. Numerics studies done later by Ohikitani-Yamada [8] and Constantin-Nie-Schorgofer [2], with the same initial data, suggested that instead of a singularity the derivatives of the temperature where increasing as double exponential in time.
The study of collapse on a curve was first studied in [1] for the Quasi-geostrophic equation where they considered a simplified ansatz for classical frontogenesis with trivial topology. At the time of collapse, the scalar $\theta$ is discontinues across the curve $x_2 = f(x_1)$ with different limiting values for the temperature on each side of the front. They show that under this topology the directional field remains smooth up to the collapse, which contradicts the following theorem proven in [1]: \begin{eqnarray*}
\text{If locally the direction field remains smooth as t}\ \ \ \ \\ \text{ approaches $T_*$, then no finite singularity is possible}\ \ \ \ \\
\text{ as t approaches $T_*$.}\ \ \ \ \quad \quad \quad \quad \quad \quad \quad \quad \end{eqnarray*} The simplified ansatz with trivial topology studied in [1] does not describe a hyperbolic saddle.
Under the definition of a simple hyperbolic saddle, in [3], it was shown that the angle of the saddle can not decrease faster than a double exponential in time.
The criterion obtained in [5] for a sharp front formation for a general two dimensional incompressible flow is : \begin{eqnarray*} \text{A necessary condition to have a sharp front at time T is}\\
\int_{0}^{T}|u|_{L^{\infty}}(s) ds = \infty\ \quad \quad \quad \quad \ \ \ \ \quad \end{eqnarray*}
For the Quasi-geostrophic equation it is not known if the quantity $\int_{0}^{T}|u|_{L^{\infty}}(s) ds$ diverges or not. And the criterion does not say how fast the arms of a saddle can close.
In this paper we do not assume anything on the velocity field, and we show that under a semi-uniform collapse the distance between two level curves cannot decrease faster than a double exponential in time. The semi-uniform collapse assumption greatly weakens the assumptions made in [1] for an ansatz for classical frontogenesis, and the simple hyperbolic saddle in [3].
In the case of 2D incompressible Euler equation we are interested in the large time behavior of solutions.
The two equations we discus in this paper, have in common the property that a scalar function is convected by the flow, which implies that the level curves are transported by the flow. The possible singular scenario is due to level curves approaching each other very fast which will lead to a fast growth on the gradient of the scalar function. Below we study the semi-uniform collapse of two level sets on a curve. By semi-uniform collapse we mean that the distance of the two curves in any point are comparable.
The equations we study are as follows:
\bc {\underline{The Quasi-geostrophic (QG) Equation}} \ec
Here the unknowns are a scalar $\theta(x,t)$ and a velocity field $u(x,t) = (u_1(x,t), u_2(x,t)) \in R^2$, defined for $t\in[0,T^*)$ with $T^*\leq \infty$, and for $x \in \Omega$ where $\Omega = R^2$ or $R^2/Z^2$. The equations for $\theta$, u are as follows \begin{eqnarray} \left (\partial_t + u\cdot\nabla_x \right ) \theta = 0 \\ u = \nabla_{x}^{\perp}\psi\ \ and \ \ \psi = (-\triangle_x)^{-\frac{1}{2}}\theta, \nonumber \end{eqnarray} where $\nabla_{x}^{\perp} f = (-\frac{\partial f}{\partial x_2}, \frac{\partial f}{\partial x_1})$ for scalar functions f. The initial condition is $\theta(x,0) = \theta_0(x)$ for a smooth initial datum $\theta_0$.
\bc {\underline{The Two-Dimensional Euler Equation}} \ec
The unknown is an incompressible velocity field u(x,t) as above with vorticity denoted by $\omega$. The 2D Euler equation may be written in the form \begin{eqnarray} \left (\partial_t + u\cdot\nabla_x \right ) \omega = 0 \\
u = \nabla_{x}^{\perp}\psi\ \ and \ \ \psi = (-\triangle_x)^{-1}\omega, \nonumber \end{eqnarray} with u(x,0) equal to a given smooth divergence free $u_0(x)$.
\section{Results}
Asssume that q = q(x,t) is a solution to (1) or (2), and that a level curve of q can be parameterized by
\begin{eqnarray} x_2=\phi_{\rho}(x_1,t)\ \ for\ \ x_1\in[a,b] \label{eq:1} \end{eqnarray} with $\phi_{\rho}\in C^1([a,b]\cap [0,T^*))$, in the sense that \begin{eqnarray} q(x_1,\phi_{\rho}(x_1,t), t) = G(\rho)\ \ for\ \ x_1\in[a,b],\label{eq:2} \end{eqnarray} and for certain $\rho$ to be specified below.
The stream function $\psi$ satisfies \begin{eqnarray} \nabla^{\perp}\psi=u. \end{eqnarray}
From (3) and (4), we have \begin{eqnarray} \frac{\partial q}{\partial x_1} + \frac{\partial q}{\partial x_2} \frac{\partial\phi_{\rho}}{\partial x_1} = 0 \label{eq:3} \end{eqnarray} \begin{eqnarray} \frac{\partial q}{\partial t} + \frac{\partial q}{\partial x_2} \frac{\partial\phi_{\rho}}{\partial t} = 0 \label{eq:4} \end{eqnarray}
By (1), (2), (5), (6) and (7) we obtain \begin{eqnarray*} \frac{\partial\phi_{\rho}}{\partial t} & = & -\frac{\frac{\partial q}{\partial t}}{\frac{\partial q}{\partial x_2}} = \frac{<-\frac{\partial\psi}{\partial x_2},\frac{\partial\psi}{\partial x_1}>\cdot <\frac{\partial q}{\partial x_1},\frac{\partial q}{\partial x_2} >}{\frac{\partial q}{\partial x_2}} \\ & = & <-\frac{\partial\psi}{\partial x_2},\frac{\partial\psi}{\partial x_1}>\cdot <\frac{\frac{\partial q}{\partial x_1}}{\frac{\partial q}{\partial x_2}}, 1> \\ & = & <-\frac{\partial\psi}{\partial x_2},\frac{\partial\psi}{\partial x_1}>\cdot <-\frac{\partial\phi_{\rho}}{\partial x_1}, 1> \end{eqnarray*} Next \begin{eqnarray*} \frac{\partial}{\partial x_1}\left (\psi(x_1,\phi_{\rho}(x_1,t), t)\right ) & = & \frac{\partial\psi}{\partial x_1} + \frac{\partial\psi}{\partial x_2} \frac{\partial\phi_{\rho}}{\partial x_1} \\ & = & <-\frac{\partial\psi}{\partial x_2},\frac{\partial\psi}{\partial x_1}>\cdot <-\frac{\partial\phi_{\rho}}{\partial x_1}, 1> \end{eqnarray*}
Therefore \begin{eqnarray} \frac{\partial\phi_{\rho}}{\partial t} = \frac{\partial}{\partial x_1}\left (\psi(x_1,\phi_{\rho}(x_1,t), t)\right ) \label{eq:5} \end{eqnarray}
With this formula we can write a explicit equation for the change of time of the area between two fixed points a, b and two level curves $(\phi_{\rho_1}, \phi_{\rho_2})$; \begin{eqnarray} \frac{d }{d t}\left ( \int_{a}^{b} [\phi_{\rho_2}(x_1,t) - \phi_{\rho_1}(x_1,t)] dx_1 \right ) \nonumber \\ = \psi(b,\phi_{\rho_2}(b,t), t) - \psi(a,\phi_{\rho_2}(a,t), t) \nonumber \\
+ \psi (a,\phi_{\rho_1}(a,t), t) - \psi(b,\phi_{\rho_1}(b,t), t) \label{eq:6} \end{eqnarray}
Assume that two level curves $\phi_{\rho_1}$ and $\phi_{\rho_2}$ collapse when t tends to $T^*$ uniformly in $a\leq x_1\leq b$ i.e. $$ \phi_{\rho_2}(x_1,t) - \phi_{\rho_1}(x_1,t) \sim \frac{1}{b - a} \int_{a}^{b} [\phi_{\rho_2}(x_1,t) - \phi_{\rho_1}(x_1,t)] dx_1 $$ In other words; the distance between two level sets are comparable for $a \leq x_1 \leq b$.
Let $$
\delta(x_1,t) = |\phi_{\rho_2}(x_1,t) - \phi_{\rho_1}(x_1,t)| $$ be the thickness of the front.
We define semi-uniform collapse on a curve if (3) and (4) holds and there exists a constant $c$, independent of t, such that $$ min \delta (x_1,t) \geq c\cdot max \delta (x_1,t) $$ for $a\leq x_1 \leq b$, and for all $t\in [0,T^*)$.
We call the length b-a of the interval [a,b] the length of the front.
Now we can state the following theorem
\begin{Thm} For a QG solution with a semi-uniform front, the thickness $\delta(t)$ satisfies \begin{eqnarray*} \delta(t) > e^{-e^{At + B}} \ \ for\ \ all\ \ t\in[0,T^*). \end{eqnarray*} Here, the constants A and B may be taken to depend only on the length of the front, the semi-uniformity constant, the initial thickness $\delta(0)$, and the norm of the initial datum $\theta_0(x)$ in $L^1\cap L^{\infty}$. \end{Thm}
Proof: From (9) we have
\begin{eqnarray}
|\frac{d}{d t} A(t)| < \frac{C}{b - a} sup_{a\leq x_1\leq b} |\psi(x_1,\phi_{\rho_2}(x_1,t), t) - \psi(x_1,\phi_{\rho_2}(x_1,t), t)| \end{eqnarray} where \begin{eqnarray*} A(t) = \frac{1}{b - a} \int_{a}^{b} [\phi_{\rho_2}(x_1,t) & - &\phi_{\rho_1}(x_1,t)] dx_1, \end{eqnarray*} and C is determined by the semi-uniformity constant c.
The estimate of the difference of the value of the stream function at two different points that are close to each other is obtained by writing the stream function as follows; \begin{eqnarray*}
\psi(x,t) = - \int_{\Omega}\frac{\theta(x + y,t)}{|y|} dy, \end{eqnarray*} and this is because $\psi = (-\triangle_x)^{-\frac{1}{2}}\theta$.
Therefore \begin{eqnarray*}
\psi(z_1, t) - \psi(z_2,t) & = & \int_{\Omega}\theta(y)(\frac{1}{|y - z_1|} - \frac{1}{|y - z_2|}) dy \\ & = & \int_{|y - z_1| \leq 2\tau} + \int_{2\tau < |y - z_2| \leq k} + \int_{k < |y - z_1| } \\ & \equiv & I_{1} + I_{2} + I_{3}.\end{eqnarray*}
where $\tau = |z_1 - z_2| $.
Furthermore \begin{eqnarray*}
|I_{1}| & \leq & ||\theta||_{L^{\infty}} \cdot\int_{|y - z_1| \leq 2\tau}(\frac{1}{|y - z_1|} + \frac{1}{|y - z_2|}) dy \\ & \leq & C\tau \end{eqnarray*}
We define s to be a point in the line between $z_1$ and $z_2$, then $|y - z_1|
\leq 2|y - s|$ and $I_{2}$ can be estimated by \begin{eqnarray*}
|I_{2}| &\leq& C\tau \cdot \int_{2\tau < |y - z_1| \leq k}max_{s}|\nabla(\frac{1}{|y - s|})| dy \\ &\leq& C\tau \cdot
\int_{2\tau < |y - z_1| \leq k}max_{s}\frac{1}{|y - s|^{2}} dy \\ &\leq&
C\tau \cdot |\log \tau| \end{eqnarray*}
We use the conservation of energy to estimate $I_{3}$ by \begin{eqnarray*}
|I_{3}| \leq C \cdot \tau \end{eqnarray*}
Finally, by choosing $\tau = |z_1 - z_2|$ we obtain \begin{eqnarray}
|\psi(z_1, t) - \psi(z_2,t)| \leq M|z_1 - z_2||log |z_1 - z_2|| \end{eqnarray} where M is a constant that depend on the initial data $\theta_0$. (See details in [3].)
Then we have \begin{eqnarray*}
|\frac{d}{d t} A(t)| & \leq & \frac{M}{b - a} sup_{a\leq x_1\leq b}|\phi_{\rho_2}(x_1,t) - \phi_{\rho_1}(x_1,t)||log |\phi_{\rho_2}(x_1,t) - \phi_{\rho_1}(x_1,t)|| \\
& \leq & \frac{C\cdot M}{\cdot(b - a)} |A(t)||log A(t)| \end{eqnarray*} and therefore \begin{eqnarray*} A(t) >> A(0)e^{-e^{\frac{C\cdot M}{\cdot(b - a)} t}} \end{eqnarray*}
\begin{Thm} For a 2D Euler solution with a semi-uniform front, the thickness $\delta(t)$ satisfies \begin{eqnarray*} \delta(t) > e^{-[At + B]} \ \ for\ \ all\ \ t\in[0,T^*). \end{eqnarray*} \end{Thm}
Here, the constants A and B may be taken to depend only on the length of the front, the semi-uniformity constant, the initial thickness $\delta(0)$, and the norm of the initial vorticity in $L^1\cap L^{\infty}$.
The proof theorem 2 is similar to theorem 1 with the difference that instead of the estimate (11), we have \begin{eqnarray*}
|\psi(z_1, t) - \psi(z_2,t)|\leq M|z_1 - z_2| \end{eqnarray*} where M is a constant that depend on the initial data $u_0$. (See details in [3].)
Similar estimates can be obtain for 2D ideal Magneto-hydrodynamics (MHD) Equation, with the extra assumption that $\int_{0}^{T^*}|u|_{L^{\infty}}(s) ds$ is bounded up to the time of the blow-up. This estimates are consequence of applying the Mean value theorem in (10). Nevertheless in the case of MHD these estimates improve the results obtain in [6].
\begin{Ack*}
This work was initially supported by the American Institute of Mathematics. \end{Ack*}
\end{document}
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\begin{document}
\subject{Dissertation} \title{On Time-Reversal Equivariant\\ Hermitian Operator Valued Maps\\
from a 2D-Torus Phase Space} \author{Moritz Schulte} \date{\today} \maketitle
\cleardoublepage \tableofcontents
\addchap{Acknowledgements}
First of all, I would like to express my sincerest gratitude to my supervisor Professor Huckleberry. His help and guidance have been invaluable to me throughout the project. The mathematical discussions we shared were crucial to my progress, in particular when he helped with several of the proofs by suggesting key ideas. The work enabled me to grasp what mathematical research is like.
I would also like to thank my parents, my family and my friends for supporting me throughout my studies and research.
And finally, there is somebody whose role throughout the critical phase of my PhD project I would like to underline: Thank you, Christina, for your unwavering support, especially when some of the proofs had given me a hard time. Your part in the completion of this paper can hardly be overestimated.
This research was conducted as part of the the Sonderforschungsbereich TR12 (SFB TR12) ``Symmetries and Universality in Mesoscopic Systems'' and was partially financed by the Deutsche Forschungsgemeinschaft (DFG) for which I am very thankful.
\chapter{Introduction}
In this paper we are primarily concerned with the equivariant homotopy classification of maps from a torus phase space $X$ to $\mathcal{H}_n^*$, the space of $n \times n$, non-singular hermitian operators. The equivariance is with respect to a time-reversal involution on $X$ and an involution on $\mathcal{H}_n^*$ defining a certain symmetry class. This is in contrast to related works in this area (e\,g. \cite{Kennedy}), which deal with the classification of equivariant maps defined on \emph{momentum space}, where the time-reversal involution on momentum space is given by $k \mapsto -k$. Here we consider the model problem where the phase space is a $2$-dimensional symplectic torus $(X,\omega)$ and the time-reversal involution $T$ is antisymplectic, i.\,e. \begin{align*}
T^*(\omega) = -\omega. \end{align*} In this case it is known (see e.\,g. \cite{Seppala}, \cite{Tehrani}) that up to equivariant symplectic diffeomorphisms there are exactly three classes of antisymplectic involutions on $X$. Representatives for these three classes can be given by \begin{align*}
[z] \mapsto [\overline{z}] \;\;\text{,}\;\; [z] \mapsto [i\overline{z}] \;\;\text{and}\;\; [z] \mapsto [\overline{z} + \text{\textonehalf}], \end{align*} where $X$ is regarded as the square torus $\mathbb{C}/\Lambda$ defined by the lattice $\Lambda = \LatGen{1}{i}$. In this paper we only deal with the first two involutions, as these are the only ones, which have fixed points. In particular, in local coordinates they are of the form \begin{align*}
(q,p) \mapsto (q,-p). \end{align*} We call the first one \emph{type I} involution and the second one \emph{type II} involution. On $\mathcal{H}_n^*$ resp. $\mathcal{H}_n$ we define the involution, which we also denote by $T$, to be \begin{align*}
H \mapsto \overline{H} = \ltrans{H}. \end{align*} We believe, though, that the methods described in this paper, are also applicable, when $X$ is equipped with a time-reversal involution coming from the third class of (free) antisymplectic involutions on $X$ and when physically relevant image spaces other than $(\mathcal{H}_n,T)$ are considered.
Letting $G$ be the cyclic group $C_2$ of order two and $X$ and $\mathcal{H}_n$ be equipped as above with $G$-actions, our main results in chapter~\ref{ChapterClassification} give a complete description of the spaces \begin{align*}
[X,\mathcal{H}_n^*]_G \end{align*} of $G$-equivariant homotopy classes of maps $X \to \mathcal{H}_n^*$, where $X$ is equipped with the type I or the type II involution. These descriptions are in terms of numerical topological invariants. It turns out that the topological invariant given by the \emph{total
degree} together with certain \emph{fixed point degrees} defines a \emph{complete} topological invariant of equivariant homotopy. Recall that an invariant is called \emph{complete} if the equality of the invariants associated to two objects (equivariant maps in our case) implies the equivalence of the objects (the maps are equivariantly homotopic). It is important to emphasize that not all combinations of total degree and fixed point degrees are realizable; a certain elementary number theoretic condition is necessary and sufficient for the existence of a map with given invariants.
In chapter~\ref{ChapterJumps} we study the situation where a certain controlled degeneration is allowed. In formal terms this is a curve \begin{align*}
f\colon [-1,1] \times X \to \mathcal{H}_n \end{align*} where the image $\Im(f_t)$ is contained in $\mathcal{H}_n^*$ for $t \not= 0$ and where the degeneration $f_0(x)$ being singular is only allowed at isolated points in $X$. The interesting case is where $f_{-1}$ and $f_{+1}$ are in distinct $G$-homotopy classes, i.\,e. where we have a sort of \emph{jump curve}. In this case we show exactly which jumps are possible and produce maps with given jumps.
This research was carried out in the context of the interdisciplinary project SFB TR12. Thus we have attempted to present a work that is mostly self-contained; for the convenience of the reader, the appendix (Chapter~\ref{ChapterAppendix}) includes some well-known background material.
\section{Outline}
The equivariant homotopy classification of maps $X \to \mathcal{H}_n^*$ is contained in chapter~\ref{ChapterClassification}. We begin by identifying the connected components of $\mathcal{H}_n^*$. These are the subspaces $\mathcal{H}_{(p,q)}$ of the matrices with $p$ positive and $q$ negative eigenvalues ($p + q = n$). It follows that this \emph{eigenvalue signature} $(p,q)$ is a homotopy invariant of maps $X \to \mathcal{H}_n^*$ and the classification of maps to $\mathcal{H}_n^*$ reduces to the classification of maps to $\mathcal{H}_{(p,q)}$, where $(p,q)$ takes on all possible eigenvalue signatures. We then begin by examining the case where $X$ is equipped with the type I involution and $n=2$. An ad-hoc computation shows that $\mathcal{H}_{(2,0)}$ resp. $\mathcal{H}_{(0,2)}$ are equivariantly contractible to plus resp. minus the identity matrix, whereas $\mathcal{H}_{(1,1)}$ contains a $U(2)$-orbit diffeomorphic to $S^2$ as equivariant strong deformation retract. The induced $T$-action on the sphere is given by a reflection along the $x,y$-plane. This motivates the need to classify equivariant maps $X \to S^2$. Also note that the $2$-sphere can be equivariantly identified with the projective space $\mathbb{P}_1$, on which the involution is the standard real structure given by complex conjugation. Since $X$ and $S^2$ are both $2$-dimensional manifolds, we obtain the (Brouwer) degree, which we call \emph{total degree}, as a homotopy invariant of maps $X \to S^2$. The equivariance requires that fixed point sets are mapped to fixed point sets. The fixed point set in the torus -- for the type I involution -- consists of two disjoint circles -- $C_0$ and $C_1$ -- while the fixed point set in the sphere consists of only the equator $E$. This defines two additional invariants of equivariant homotopy: The \emph{fixed point degrees}, which are the degrees of the restrictions of the maps to the respective fixed point circles in $X$, regarded as maps to the equator. This gives rise the \emph{degree triple map} $\mathcal{T}$, which sends an equivariant map $f\colon X \to S^2$ to the degree triple $\Triple{d_0}{d}{d_1}$, where $d$ is the total degree of $f$ and $d_0$, $d_1$ denote the fixed point degrees of $f$. A convenient property of the type I involution is the fact that a fundamental region for this involution is given by a \emph{cylinder}, which we call $Z$. The boundary circles of the cylinder are the fixed point circles in $X$. Maps $(Z,C_0 \cup C_1) \to (S^2,E)$ can be uniquely equivariantly extended to the full torus $X$. It follows that the equivariant homotopy classification of maps $X \to S^2$ reduces to the (non-equivariant) homotopy classification of maps $(Z, C_0 \cup C_1) \to (S^2,E)$. We then regard maps from the cylinder $Z = I \times S^1$ to $S^2$ as \emph{curves} in the free loop space $\mathcal{L}S^2$. After this translation of the problem and a certain normalization procedure on the boundary circles, we are dealing with the problem of computing homotopy classes of curves in $\mathcal{L}S^2$ (with fixed endpoints). The key point now is to define the \emph{degree map}, which associates to curves in $\mathcal{L}S^2$ the degree of its equivariant extension to the full torus. The degree map factors through the space of homotopy class of curves. Furthermore it satisfies a certain compatibility condition with respect to the concatenation of curves. In particular, the degree map becomes a group homomorphism, when it is restricted to a based fundamental group. This can be used for proving that the restriction of the degree map to based fundamental groups is injective. The injectivity of this map is then the basis for the proof of the statement that maps with the same degree triple are equivariantly homotopic. To summarize the above: The degree triple map $\mathcal{T}$ is well-defined on the set of equivariant homotopy classes of maps $X \to S^2$ and defines a bijection to a subset of $\mathbb{Z}^3$. We call a degree triple, which is contained in the image of the degree triple map, \emph{realizable}. As a preperation for describing the image of the degree triple map we express the concatenation of curves in $\mathcal{L}S^2$ in terms of a binary operation (``concatenation'') on the space of degree triples: \begin{align*}
\Triple{d_0}{d}{d_1} \bullet \Triple{d_1}{d'}{d_2} = \Triple{d_0}{d+d'}{d_2}. \end{align*} This concatenation operation is only defined for degree triples which are compatible in the sense that the fixed point degrees in the middle coincide. Using this formalism we observe that the image of the degree triple map $\Im(\mathcal{T})$ is closed under the concatenation operation. Furthermore we show that certain basic triples, e.\,g. $\Triple{0}{1}{0}$, are not contained in the image of the degree triple map. These are the key tools for completely describing the image $\Im(\mathcal{T})$ using the formalism of degree triple concatenations. It turns out that the number theoretic condition \begin{align}
\label{DegreeTripleCondition}
d \equiv d_0 + d_1 \mod 2 \end{align} is sufficient and necessary for a degree triple $\Triple{d_0}{d}{d_1}$ to be in the image of the degree triple map. This completely solves the problem for the type I involution in the case $n=2$.
In the next step we reduce the classification problem for the type II involution to the type I classification. The key observation here is that maps $X \to S^2$, where $X$ is equipped with the type II involution, can be normalized on a subspace $A \subset X$ such that they push down to the quotient $X/A$, which happens to be equivariantly diffeomorphic to $S^2$. The action on the $2$-sphere is again given by a reflection along the $x,y$-plane. After another normalization procedure, concerning the images of the two poles of $S^2$, we prove a one-to-one correspondence between equivariant maps $S^2 \to S^2$ and equivariant maps from a torus equipped with the type I involution to $S^2$ having the one fixed point degree be zero. This allows us to use the results for the type I involution. In particular it implies that we are dealing with degree \emph{pairs} instead of degree triples. The degree pair map associates to an equivariant map $f\colon X \to S^2$ its degree pair $\Pair{d_C}{d}$ where $d_C$ is the fixed point degree and $d$ the total degree of $f$. Because of this correspondence between maps defined on a type II torus and maps defined on a type I torus, the condition (\ref{DegreeTripleCondition}) for degree triples to be in the image of the degree triple map induces a corresponding condition for a pair $\Pair{d_C}{d}$ to be in the image of the degree pair map: \begin{align}
\label{DegreePairCondition}
d \equiv d_C \mod 2. \end{align} This completes the case $n=2$ for both involutions.
For the general case $n > 2$ we begin with an examination of the connected components $\mathcal{H}_{(p,q)}$ of $\mathcal{H}_n^*$. As in the case $n=2$ we prove that the components $\mathcal{H}_{(n,0)}$ and $\mathcal{H}_{(0,n)}$ are equivariantly contractible to a point. Fundamental for the following considerations is the result that the components $\mathcal{H}_{(p,q)}$ with $0 < p,q < n$ have a $U(n)$-orbit as equivariant strong deformation retract, where $U(n)$ acts on $\mathcal{H}_{(p,q)}$ by conjugation. This orbit is equivariantly diffeomorphic to the complex Grassmann manifold $\text{Gr}_p(\mathbb{C}^n)$ on which the involution $T$ acts as \begin{align*}
V \mapsto \overline{V}. \end{align*} Thus, the problem of classifying equivariant maps $X \to \mathcal{H}_{(p,q)}$ is equivalent to classifying equivariant maps $X \to \text{Gr}_p(\mathbb{C}^n)$. Now we fix the standard flag in $\mathbb{C}^n$ and consider the -- with respect to this flag -- unique (complex) one-dimensional Schubert variety $\mathcal{S}$ in $\text{Gr}_p(\mathbb{C}^n)$. In this situation we prove, using basic Hausdorff dimension theoretical results, that it is possible to iteratively remove certain parts of the Grassmann manifold $\text{Gr}_p(\mathbb{C}^n)$ so that in the end we obtain an equivariant strong deformation retract to the Schubert variety $\mathcal{S}$. The latter is equivariantly biholomorphic to the complex projective space $\mathbb{P}_1$ equipped with the standard real structure. This allows us to use the results for the $n=2$ case. In terms of the invariants, the only difference between the case $n = 2$ and $n > 2$ is that in the former case the fixed point degrees are integers from the set $\mathbb{Z}$ while in the latter case they are only from the set $\{0,1\}$. This stems from the fact that in the case $n>2$ the fundamental group of the $T$-fixed points in the Grassmann manifold is not infinite cyclic anymore, but cyclic of order two. In this setup, the conditions for degree triples resp. pairs to be realizable are exactly (\ref{DegreeTripleCondition}) and (\ref{DegreePairCondition}). This completes the classification of equivariant maps $X \to \mathcal{H}_n^*$.
In chapter~\ref{ChapterJumps} we construct curves \begin{align*}
H\colon [-1,1] \times X \to \mathcal{H}_n \end{align*} of equivariant maps with the property that the image $\Im(H_t)$ is contained in $\mathcal{H}_n^*$ for $t\not= 0$ and $H_{-1}$ and $H_{+1}$ are not equivariantly homotopic as maps to $\mathcal{H}_n^*$. This is only possible if we allow a certain degeneration at $t=0$, which means that there exists a non-empty \emph{singular set} $S(H_0)$ consisting of the points $x \in X$ such that the matrix $H_0(x)$ is singular. To make the construction non-trivial we require the singular set $S(H_0)$ to be discrete. As in the previous chapter we first consider the type I involution in the special case $n=2$. As a first result we obtain that jump curves from $H_{-1}$ to $H_{+1}$ with discrete singular set can only exist if the eigenvalue signatures of $H_{-1}$ and $H_{+1}$ coincide. Hence we fix an eigenvalue signature $(p,q)$. The construction of \emph{jump
curves} is based on the decomposition of degree triples as a concatenation of simpler triples. First we show how to construct jumps for certain ``model maps'' for basic degree triples. Subsequently we extend this method such that we can construct jumps from any equivariant homotopy class to any other -- as long as the eigenvalue signature remains unchanged.
For the type II involution in the case $n=2$, we can employ the same correspondence that has been used during the homotopy classification to turn jump curves for the type I involution into corresponding jump curves for the type II involution. This completes the construction of jump curves in the case $n=2$.
For the general case $n > 2$ we construct jumps curves using the embedding of the Schubert variety $\mathcal{S} \cong \mathbb{P}_1$ into $\text{Gr}_p(\mathbb{C}^n)$. In other words: The jump curves we construct in the higher dimensional situation really come from jump curves in the case $n=2$.
\chapter{Equivariant Homotopy Classification} \label{ChapterClassification}
As indicated in the introduction, we may regard the torus $X$ as being defined by the standard square lattice $\Lambda = \LatGen{1}{i}$. As a complex manifold it carries a canonical orientation. Furthermore, we regard $X$ as being equipped with the real structure $T([z]) = [\overline{z}]$ (type I) or with the real structure $T([z]) = [i\overline{z}]$ (type II). Recall that $G$ denotes the cyclic group $C_2$ of order two. The $G$-action on $\mathcal{H}_n$ is assumed to be given by matrix transposition or -- equivalently -- complex conjugation: \begin{align*}
T\colon \mathcal{H}_n &\to \mathcal{H}_n\\
H &\mapsto \ltrans{H} = \overline{H}. \end{align*} The goal in this chapter is to completely describe the sets $[X,\mathcal{H}_n^*]_G$ of $G$-\parbox{0pt}{}equivariant homotopy classes of maps $X \to \mathcal{H}_n^*$ ($n \geq 2$). Recall the basic definition of equivariant homotopy: \begin{definition}
Let $X$ and $Y$ be two $G$-spaces and $f_0, f_1\colon X \to Y$ two
$G$-maps. Note that $I \times X$ ($I$ is the unit interval $[0,1]$)
can be regarded as a $G$-space by defining $T(t,x) = (t,T(X))$. Then
$f_0$ and $f_1$ are said to be \emph{$G$-homotopic} (or
\emph{equivariantly homotopic}\footnote{We use the terms
``$G$-homotopy'' and ``equivariant homotopy'' interchangeably.})
if there exists a $G$-map $f\colon I \times X \to Y$ such that
$f(0,\cdot) = f_0$ and $f(1,\cdot) = f_1$. \end{definition}
The first invariant of maps $X \to \mathcal{H}_n^*$ we consider is the \emph{eigenvalue signature}. We begin with the following \begin{definition}
Let $H$ be a non-singular $n \times n$ matrix and
\begin{align*}
\lambda^+_1 \geq \ldots \geq \lambda^+_p > 0 > \lambda^-_1 \ldots \geq \lambda^-_q
\end{align*}
the eigenvalues of $H$ (repetitions allowed). Then we call $(p,q)$
the \emph{eigenvalue signature} (or simply \emph{signature}) of $H$
and set $\text{sig}\, H = (p,q)$. \end{definition} The connected components of the space $\mathcal{H}_n^*$ are the $n+1$ open subspaces $\mathcal{H}_{(p,q)}$, where \begin{align*}
\mathcal{H}_{(p,q)} = \left\{H \in \mathcal{H}_n^*\colon \text{sig}\, H = (p,q)\right\}. \end{align*} Thus we can write \begin{align*}
[X,\mathcal{H}_n^*]_G = \dot{\bigcup_{p+q=n}}\;[X,\mathcal{H}_{(p,q)}]_G. \end{align*} In particular this means that the signature $\text{sig}\,(H)$ for maps $H\colon X \to \mathcal{H}_n^*$ is well-defined and a topological invariant. It turns out that the components $\mathcal{H}_{(n,0)}$ and $\mathcal{H}_{(0,n)}$ are equivariantly contractible but the components $\mathcal{H}_{(p,q)}$ with $0<p,q<n$ are topologically interesting as they have Grassmann manifolds as equivariant strong deformation retract. The classification of maps to a component $\mathcal{H}_{(p,q)}$ for a fixed (non-definite) eigenvalue signature $(p,q)$ is in this way reduced to the problem of classifying equivariant maps to complex Grassmann manifolds $\text{Gr}_p(\mathbb{C}^n)$ on which the $G$-action is given by standard complex conjugation.
As a preparation for proving a general statement about the set $[X,\mathcal{H}_n^*]_G$, we first consider the special case $n=2$ (theorem~\ref{HamiltonianClassificationRank2}, p.~\pageref{HamiltonianClassificationRank2}). As mentioned earlier, the only non-trivial cases occur for maps whose images are contained in the components $\mathcal{H}_{(p,q)}$ of mixed signature. In the case $n=2$ this boils down to $\mathcal{H}_{(1,1)}$. This space has the $2$-sphere, equipped with the involution given by a reflection along the $x,y$-plane, as equivariant strong deformation retract.\footnote{The 2-sphere is regarded as being embedded as the
unit sphere in $\mathbb{R}^3$} Thus, the problem of describing $[X,\mathcal{H}_{(1,1)}]_G$ is reduced to classifying equivariant maps $X \to S^2$. We do this for the type~I and the type~II involution seperately, but it turns out that the type II case can be reduced to the type I case (see theorem~\ref{Classification1} and theorem~\ref{Classification2}). The result is that for both involutions the complete invariants of maps $X \to S^2$ consist of two pieces of data: First the \emph{total degree} $d$ (which is also an invariant of non-equivariant homotopy) and second the so-called \emph{fixed point degrees} -- two integers (named $d_0, d_1$) for the type I involution and one integer (named $d_C$) for the type~II involution. Note that the notion of fixed point degree only makes sense for \emph{equivariant} maps, as they are required to map the fixed point set in $X$ into the fixed point set in $S^2$. Furthermore we obtain the statement that not all combinations of total degree and fixed point degrees are realizable by $G$-maps. As mentioned in the introduction, the conditions which are sufficient and necessary are $d \equiv d_0 + d_1 \mod 2$ for the type I involution and $d \equiv d_C \mod 2$ for the type II involution.
After having proven a classification statement for $n=2$, we observe that this case is really the fundamental case to which the general case ($n$ arbitrary) can be reduced to by means of a retraction argument. In each Grassmannian $\text{Gr}_p(\mathbb{C}^n)$ there exists a Schubert variety $\mathcal{S} \cong \mathbb{P}_1$; after fixing a full flag in $\mathbb{C}^n$, this the unique one-dimensional (over $\mathbb{C}$) Schubert variety which generates the second homology groups $H_2(\text{Gr}_p(\mathbb{C}^n),\mathbb{Z})$. After iteratively cutting out certain parts of a general Grassmannian $\text{Gr}_p(\mathbb{C}^n)$ we obtain an equivariant retraction to this embedded Schubert variety. It can be equivariantly identified with $\mathbb{P}_1$ on which the involution is given by the standard real structure: \begin{align*}
[z_0:z_1] \mapsto \left[\overline{z_0}:\overline{z_1}\right] \end{align*} This space can be equivariantly identified with $S^2$. Hence, we can use the previous results for the case $n=2$.
\section{Maps to $\mathcal{H}_2$} \label{SectionN=2}
In this section we provide a complete description of the set $[X,\mathcal{H}_2^*]_G$, where $X$ is equipped with a real structure $T$, which is either the type I or the type II involution. This case is of particular importance, since all the other cases (i.\,e. $n > 2$) can be reduced to this one. Note that a general equivariant map $H\colon X \to \mathcal{H}_2$ is of the form \begin{align}
\label{GeneralRank2Map}
H =
\begin{pmatrix}
a_1 & b \\
\overline{b} & a_2
\end{pmatrix}, \end{align} where $a_1$, $a_2$ are functions $X \to \mathbb{R}$ and $b$ is a function $X \to \mathbb{C}$ satisfying \begin{align}
\label{N=2MatrixEquivarianceConditionsA}
&a_j \circ T = a_j \;\text{for $j=1,2$}\\
\label{N=2MatrixEquivarianceConditionsB}
&b \circ T = \overline{b} \end{align} A map into $\mathcal{H}_n^*$ has the additional property that
$\det H = a_1 a_2 - |b|^2$ is nowhere zero. As we have mentioned previously, the space $\mathcal{H}_2^*$ decomposes into the disjoint union \begin{align*}
\mathcal{H}_{(2,0)} \,\dot{\cup}\, \mathcal{H}_{(1,1)} \,\dot{\cup}\, \mathcal{H}_{(0,2)}, \end{align*} where $(2,0)$, $(1,1)$ and $(0,2)$ are the possible eigenvalue signatures for maps $X \to \mathcal{H}_n^*$. First we note: \begin{remark}
\label{Rank2DefiniteComponentsContractible}
The components $\mathcal{H}_{(2,0)}$ and $\mathcal{H}_{(0,2)}$ have
$\{\pm \I{2}\}$ as equivariant strong deformation
retract.\footnote{Here, $\I{k}$ is the $k \times k$ identity
matrix.} \end{remark} \begin{proof}
It suffices to consider the component $\mathcal{H}_{(2,0)}$, as the
proof for the other is exactly the same. We define the following
strong deformation retract:
\begin{align*}
\rho\colon I \times \mathcal{H}_{(2,0)} &\to \mathcal{H}_{(2,0)}\\
(t,H) &\mapsto (1-t)H + t\I{2}.
\end{align*}
We have to check that $\rho$ really maps into
$\mathcal{H}_{(2,0)}$. For this, let $H$ be any positive-definite
hermitian matrix. Then $(1-t)H$ is also positive-definite for $t \in
(0,1)$. If $\det((1-t)H + t\I{2}) = 0$, this means that the negative
number $-t$ is an eigenvalue of $(1-t)H$, which is a
contradiction. Equivariance of $\rho_t$ follows from the fact that
the equivariance conditions (\ref{N=2MatrixEquivarianceConditionsA})
and (\ref{N=2MatrixEquivarianceConditionsB}) are compatible with
scaling by real numbers. \end{proof}
At this point we begin using the space $i\mathfrak{su}_2$, which is the vector space of traceless hermitian operators: \begin{align*}
i\mathfrak{su}_2 = \left\{
\begin{pmatrix}
a & \phantom{-}b \\
\overline{b} & -a
\end{pmatrix}\colon a \in \mathbb{R} \;\text{and}\; b \in \mathbb{C}\right\}. \end{align*} It can be identified with $\mathbb{R}^3$ via the linear isomorphism \begin{align}
\label{EmbeddingU2OrbitAsUnitSphere}
\Psi\colon \begin{pmatrix}
a & b \\
\overline{b} & -a
\end{pmatrix} \mapsto
\begin{pmatrix}
a \\
\Re(b) \\
\Im(b)
\end{pmatrix}. \end{align} We regard $\mathbb{R}^3$ as being equipped with the involution, which reflects along the $x,y$-plane: \begin{align*}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix} \mapsto
\begin{pmatrix}
\phantom{-}x \\
\phantom{-}y \\
-z
\end{pmatrix}. \end{align*} With respect to this action, the diffeomorphism $i\mathfrak{su}_2 \xrightarrow{\;\sim\;} \mathbb{R}^3$ is equivariant. Regarding the topology of the component $\mathcal{H}_{(1,1)}$ we state: \begin{remark}
\label{Rank2TraceZeroReduction}
The component $\mathcal{H}_{(1,1)}$ has
$i\mathfrak{su}_2\smallsetminus\{0\}$ as equivariant, strong deformation
retract. \end{remark} \begin{proof}
A general matrix in $i\mathfrak{su}_2$ is of the form
\begin{align*}
\begin{pmatrix}
a & \phantom{-}b \\
\overline{b} & -a
\end{pmatrix}
\end{align*}
with $a$ real and $b$ complex. Its eigenvalues are
\begin{align*}
\pm\sqrt{a^2 + |b|^2}.
\end{align*}
This implies that $i\mathfrak{su}_2\smallsetminus\{0\}$ is contained
in $\mathcal{H}_{(1,1)}$. Now we prove that $\mathcal{H}_{(1,1)}$
has the space $i\mathfrak{su}_2\smallsetminus\{0\}$ as equivariant,
strong deformation retract. A computation shows that the two
eigenvalue functions $\lambda_1,\lambda_2$ of the general map $H$
(\ref{GeneralRank2Map}) are
\begin{align*}
\lambda_{1,2} = \frac{a_1+a_2}{2} \pm \sqrt{\left(\frac{a_1 + a_2}{2}\right)^2 + |b|^2 - a_1 a_2}.
\end{align*}
It is a straightforward computation to show that $\det H > 0$
implies that $H$ is positive or negative definite. From this we can
conclude that the eigenvalue signature being $(1,1)$ implies $\det H
< 0$, that is $|b|^2 - a_1 a_2 > 0$. Now define
\begin{align*}
c = \frac{a_1 + a_2}{2}
\end{align*}
and consider the strong deformation retract
\begin{align*}
\rho\colon I \times \mathcal{H}_{(1,1)} &\to \mathcal{H}_{(1,1)}\\
(t,H) &\mapsto (1-t)H - tc\I{2}.
\end{align*}
For this to be well-defined we need $\det(\rho(t,H)) <
0$ for all $t$ and $H$:
\begin{align*}
\det(\rho(t,H)) &= (a_1 - tc)(a_2 - tc) - |b|^2\\
&= a_1 a_2 - |b|^2 - tc(a_1 + a_2) + (tc)^2\\
&< - tc(a_1 + a_2) + (tc)^2\\
&= -\frac{(a_1 + a_2)^2}{2} + t\frac{(a_1 + a_2)^2}{4}\\
&= \left(\frac{a_1 + a_2}{2}\right)^2(t-2) < 0.
\end{align*}
Furthermore, $\rho$ stabilizes $i\mathfrak{su}_2\smallsetminus\{0\}$ and
$\rho(1,\mathcal{H}_{(1,1)}) = i\mathfrak{su}_2\smallsetminus\{0\}$. As
in the previous remark, equivariance of $\rho_t$ follows since the
equivariance conditions (\ref{N=2MatrixEquivarianceConditionsA}) and
(\ref{N=2MatrixEquivarianceConditionsB}) are compatible with scaling
by real numbers. \end{proof}
Clearly, the isomorphism $\Psi$ maps the zero matrix to the origin. The space $\mathbb{R}^3\smallsetminus\{0\}$ has the unit $2$-sphere as equivariant strong deformation retract. Under $\Psi$ this corresponds to the $U(2)$ orbit of $\I{1}{1} = \text{Diag}\,(1,-1)$ in $i\mathfrak{su}_2$. This orbit consists of the matrices \begin{align*}
\left\{
\begin{pmatrix}
|a|^2 - |b|^2 & 2a\overline{c} \\
2\overline{a}c & |b|^2 - |a|^2
\end{pmatrix}\colon \;\text{where}\;
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\;\text{is unitary.}\right\} \end{align*}
This proves the following \begin{proposition}
\label{Rank2MixedSignatureReduction}
The space $i\mathfrak{su}_2\smallsetminus\{0\}$ has the $U(2)$-orbit of
$\I{1}{1}$ as equivariant strong deformation retract. The above
isomorphispm $\Psi$ equivariantly identifies the $U(2)$-orbit with
the unit sphere in $\mathbb{R}^3$ on which the involution acts as a
reflection along the $x,y$-plane. \qed \end{proposition}
This reduces the classification problem to the classification of equivariant maps $X \to S^2$. In order to discuss mapping degrees of maps to $S^2$ we need to fix an orientation on the sphere and on its equator $E$. We choose the orientation on $S^2$ to be defined by an outer normal vector field on the sphere. Furthermore we define the orientation on the equator $E$ to be such that the loop \begin{align}
\label{DegreeOneMapOnEquator}
S^1 &\to E \subset S^2 \subset \mathbb{R}^3\\
z &\mapsto
\begin{pmatrix}
\cos(\arg z)\\
\sin(\arg z)\\
0
\end{pmatrix}\nonumber \end{align} has positive degree one\footnote{These choices are arbitrary. Choosing
the opposite orientation on the sphere or on the equator only
changes the signs appearing in concrete examples, but not the
general discussion.}. Later, when discussing the case $n > 2$, it will become important to equivariantly identify $S^2$ with the projective space $\mathbb{P}_1$. The action on $\mathbb{P}_1$ is given by standard complex conjugation \begin{align*}
[z_0:z_1] \mapsto \left[\overline{z_0}:\overline{z_1}\right]. \end{align*} \label{P1S2OrientationDiscussion}Note that $\mathbb{P}_1$ is canonically oriented as a complex manifold. Let $\psi$ be an equivariant, orientation-preserving diffeomorphism $S^2 \to \mathbb{P}_1$. This identifies the compactified real line $\smash{\widehat{\mathbb{R}}}$ with the equator $E$. We can assume that this identification of $S^2$ with $\mathbb{P}_1$ is such that a loop of degree $+1$ into the compactified real line $\smash{\widehat{\mathbb{R}}} \subset \mathbb{P}_1$ is given by \begin{align*}
S^1 &\to \mathbb{RP}_1\\
z &\mapsto
\begin{cases}
\infty & \;\text{if $\arg(z) \in 2\pi\mathbb{Z}$}\\
\tan\left(\frac{\arg z - \pi}{2}\right) & \;\text{else}\\
\end{cases}. \end{align*} That is, the loop starts at $\infty$, then traverses the negative real numbers until it reaches zero, then it traverses the positive real numbers until it reaches $\infty$ again.
As indicated in the introduction, the relevant topological invariants of maps $X \to S^2$ consist of two pieces of data: \begin{enumerate} \item The \emph{total degree}, which is the usual Brouwer degree, and \item The \emph{fixed point degrees}, which are the Brouwer degrees of
the restrictions of the maps to the components of the fixed point sets
of the respective involution on $X$. \end{enumerate} For the type I involution this fixed point set consists of two disjoint circles. Hence we obtain \emph{degree triples} of the form $\Triple{d_0}{d}{d_1}$ as invariants, where the $d_j$ are the fixed point degrees and $d$ is the total degree. For the type II involution the fixed point set is a single circle, which is why the invariants we work with are \emph{degree pairs} of the form $\Pair{d_C}{d}$ ($d$ being the total degree and $d_C$ the fixed point degree). We call a degree triple resp. a degree pair \emph{realizable} if it occurs as invariant of a $G$-maps. The exact definitions will be given in definition~\ref{DefinitionTriple} and definition~\ref{DefinitionDegreePair}.
The main result in this section is the following \begin{restatable*}{theorem}{HamiltonianClassificationRankTwo}
\label{HamiltonianClassificationRank2}
Let $X$ be a torus equipped with either the type I or the type II
involution. Then:
\begin{enumerate}[(i)]
\item The sets $[X,\mathcal{H}_{(2,0)}]_G$ and
$[X,\mathcal{H}_{(0,2)}]_G$ are trivial (i.\,e. one-point sets).
\item Two $G$-maps $X \to \mathcal{H}_{(1,1)}$ are $G$-homotopic iff
their degree triples (type I) resp. their degree pairs (type II)
agree.
\item The realizable degree triples $\Triple{d_0}{d}{d_1}$ (type I)
resp. degree pairs $\Pair{d_C}{d}$ (type II) are exactly those
which satisfy
\begin{align*}
d \equiv d_0 + d_1 \mod 2 \;\;\text{resp.}\;\;d \equiv d_C \mod 2.
\end{align*}
\end{enumerate} \end{restatable*} The proof will be given in section~\ref{SectionHamiltonianClassificationRank2} (p.~\pageref{HamiltonianClassificationRankTwoProof}).
\subsection{Maps to $S^2$} \label{SectionMapsTo2Sphere}
After the reduction described above, we can regard $G$-maps $X \to \mathcal{H}_{(1,1)}$ as having their image contained in the oriented $2$-sphere $S^2 \subset \mathbb{R}^3$. Throughout this section we use a fixed orientation-preserving identification of the equator $E$ with $S^1$. Furthermore, we let $p_0 \in E$ be the fixed base point in the equator, which corresponds -- under this identification -- to $1 \in S^1$. This point will be used later for bringing the maps into a convenient “normal form”. We will make frequent use of the following theorem: \begin{theorem}
\label{HopfTheorem}
Let $M$ be a closed, $n$-dimensional manifold. Then maps $M \to S^n$
are homotopic if and only if their degrees agree. \end{theorem} \begin{proof}
See e.\,g. \cite[p.~50]{Milnor}. \end{proof}
An important tool for the classification will be the study of homotopies on the fixed point sets. For this we start with a general definition: \begin{definition}
\label{DefinitionFixpointDegree} (Fixed point degree) Let $f\colon M
\to Y$ be a $G$-map between $G$-manifolds. Assume that the fixed point
set $Y^G$ is a closed, oriented manifold $K$ of dimension $m$ and
that the fixed point set $M^G$ is the disjoint union
\begin{align*}
M^G = \bigcup_{j=0,\ldots,k} K_j
\end{align*}
of closed, oriented (connected) manifolds $K_j$ each of dimension
$m$. Then we define the \emph{fixed point degrees} to be the
$k$-tuple $\left(d_0,\ldots,d_k\right)$ where $d_j$ is the degree of
the map $\restr{f}{K_j}\colon K_j \to K$. \end{definition}
\begin{remark}
\label{FixpointDegreeInvariant}
The fixed point degrees are invariants of $G$-homotopies. \end{remark} \begin{proof}
Let $M$ and $Y$ be two $G$-manifolds as in
definition~\ref{DefinitionFixpointDegree} and let $f,f'$ be two
$G$-homotopic maps $M \to Y$. Denote by $(d_0,\ldots,d_k)$
resp. $(d'_0,\ldots,d'_k)$ their fixed point degrees. Assume that
$d_j \neq d'_j$ for some $j$. By assumption, there exists a
$G$-homotopy $H\colon I \times X \to Y$ from $f$ to $f'$. This
homotopy can be restricted to $I \times K_j$, which can be regarded
as a homotopy $h\colon I \times K_j \to K$. The degree $d_j$ is the
degree of the map $h_0$ and $d'_j$ is the degree of the map
$h_1$. But the degree is a homotopy invariant, hence $d_j \neq d'_j$
yields a contradiction. \end{proof}
From now on we handle the involution types I and II seperately.
\subsubsection{Type I} \label{SectionMapsToSphereClassI}
In this section the real structure $T$ on $X$ is the type I involution: \begin{align*}
T\colon X &\to X\\
[z] &\mapsto [\overline{z}]. \end{align*} Furthermore, we regard the torus $X$ as being equipped with the structure of a $G$-CW complex (see e.g. \cite[p.~16]{May} or \cite[p.~1]{ShahArticle}). Such a $G$-CW structure is depicted in figure~\ref{FigureClassICWDecomposition}. We continue with the \emph{cylinder reduction} method.
\paragraph{Cylinder Reduction}
In order to describe the cylinder reduction we need the notion of a fundamental region: \begin{definition}
\label{DefinitionFundamentalRegion}
A \emph{fundamental region} for the $G$-action on $X$ is a connected
subset $R \subset X$ such that each $G$-orbit in $X$ intersects $R$
in a single point. \end{definition}
\begin{figure}
\caption{Type I $G$-CW setup of the torus $X$.}
\label{FigureClassICWDecomposition}
\label{FigureClassIFundamentalRegion}
\label{3figs}
\end{figure}
The type I involution is particularly convenient to work with, since for this action there exists a sub-$G$-CW complex that is a fundamental region for the $T$-action (see figure~\ref{FigureClassIFundamentalRegion}). Geometrically this fundamental region is a closed cylinder (including its boundary circles), which we denote by $Z$. The fixed point set $X^G$ is the disjoint union of two circles $C_0$ and $C_1$. The cylinder $Z$ is bounded by these circles. We identify $Z$ with $I \times S^1$. In this setup, the boundary circles are $C_0 = \{0\} \times S^1$ and $C_1 = \{1\} \times S^1$. For convenience we also set $C = C_0 \cup C_1$. In this situation we note the following general lemma: \begin{lemma}
\label{Class1EquivariantExtension}
Let $Y$ be a $G$-space, $Y^G$ the $T$-fixed point set in $Y$ and
$$
f\colon (Z,C) \to (Y,Y^G)
$$
a map. Then there exists a unique equivariant extension of $f$ to a
$G$-map $X \to Y$ which we denote by $\smash{\hat{f}}$. The same applies
to homotopies of such maps. \end{lemma} \begin{proof}
We only prove the statement for a given homotopy
\begin{align*}
H\colon I \times (Z,C) \to (Y,Y^G).
\end{align*}
Denote the complementary cylinder by $Z'$ (note that $Z \cap Z' = C
= C_0 \cup C_1$). Then we can construct the equivariant extension
$\smash{\widehat{H}}$ of $H$ by defining
\begin{align*}
\widehat{H}\colon I \times X &\to Y\\
(t,x) &\mapsto
\begin{cases}
H(t,x) & \;\text{if $x \in Z$}\\
T(H(t,T(x))) & \;\text{if $x \in Z'$}\\
\end{cases}
\end{align*}
Since $H(t,\cdot)$ is a map $(Z,C) \to (Y,Y^G)$, this definition of
$\smash{\widehat{H}}$ is well-defined on the boundary circles $C_0$ and
$C_1$ and globally continuous. \end{proof}
For the concrete situation $(Y,Y^G) = (S^2,E)$ this implies: \begin{remark}
\label{CylinderReduction} (Cylinder reduction) Every $G$-map
$f\colon X \to S^2$ is completely determined by its restriction
$\restr{f}{Z}$ to the cylinder $Z$. Discussing $G$-homotopies of
maps $X \to S^2$ is equivalent to discussing homotopies of maps
$(Z, C) \to (S^2,E)$. \end{remark}
\label{CjS1Identification} For the following we need to fix an identification of $C_0$ and $C_1$ with $S^1$.\footnote{Here, $S^1$ is regarded as being equipped
with the standard orientation.} Let us assume that the identification of the circles $C_j$ with $S^1$ is such that the loops \begin{align*}
\gamma_j\colon I &\to X\\
t &\mapsto \left[t + \frac{j}{2}i\right] \;\text{ for $j=0,1$} \end{align*} both define loops of degree one. Recall that loops in $S^1$ are up to homotopy determined by their degree. Based on this observation we define a normal form for equivariant maps $X \to S^2$: \begin{definition}
\label{DefinitionBoundaryNormalization}
(Type I normalization) Let $f\colon X \to S^2$ be a $G$-map (resp. a
map $(Z,C) \to (S^2,E)$). Then $f$ is \emph{type I normalized} if
the restrictions
\begin{align*}
\Restr{f}{C_j}\colon C_j \to E
\end{align*}
are, using the fixed identifications of the circles $C_j$ and $E$
with $S^1$, of the form $z \mapsto z^{k_j}$ for some integers $k_j$. \end{definition}
\begin{remark}
\label{BoundaryNormalization}
Let $f\colon (Z,C) \to (S^2,E)$ be a map (or let $f$ be a $G$-map $X
\to S^2$). Via the homotopy extension property (resp. its
equivariant version, see corollary~\ref{G-HEP}) of the pair $(Z, C)$
(resp. $(X,C)$), there exists a $G$-homotopy from $f$ to a map $f'$,
which is type I normalized. This means that, using the respective
identifications of the $C_j$ and $E$ with $S^1$, the map
$\restr{f'}{C_j}$ is of the form $z \mapsto z^{k_j}$ where $k_j$
denotes the respective fixed point degree on the circle $C_j$. \qed \end{remark}
Given a $G$-map $f\colon X \to S^2$ we have already defined its fixed point degree $\Bideg{d_0}{d_1}$ (see definition~\ref{DefinitionFixpointDegree}). Given a map $f\colon (Z, C) \to (S^2,E)$ we define its fixed point degree to be the fixed point degree of its equivariant extension $X \to S^2$.
\paragraph{The Free Loop Space}
By the cylinder reduction (remark~\ref{CylinderReduction}) it suffices to classify maps $(Z,C) \to (S^2,E)$ up to homotopy. The cylinder $Z$ is just $S^1$ times some closed interval (e.\,g. the closed unit interval $I$). Recall that $\mathcal{L}S^2$, the free loop space of the space $S^2$, is the space of all maps $S^1 \to S^2$: \begin{align*}
\mathcal{L}S^2 = \mathcal{M}(S^1,S^2) = \{f\colon S^1 \to S^2\}. \end{align*} First we make a general remark about the topology on the free loop space: \begin{remark}
The space $S^1$ is Hausdorff and locally compact. Hence, in the
terminology of e.\,g. \cite{ArticleEscardoHeckmann}, it is
\emph{exponentiable}, which means that for any other spaces $Y$ and
$A$, the natural bijection
\begin{align*}
\mathcal{M}(A,\mathcal{M}(S^1,Y)) = \left(\Maps{S^1}{Y}\right)^A
\cong \Maps{A \times S^1}{Y} = \mathcal{M}(A \times S^1,Y)
\end{align*}
is compatible with the topology in the sense that it preserves
continuity. See \cite{ArticleEscardoHeckmann}. The mapping spaces
are regarded as being equipped with the compact-open topology. \qed \end{remark}
As an immediate consequence of the above we note: \begin{remark}
A map $f\colon (Z,C) \to (S^2,E)$ can be regarded as a curve in
$\mathcal{L}S^2$ starting and ending at loops whose image are
contained in $E$. After type I normalization the start and end curve
are distinguished and indexed by $\mathbb{Z}$. \qed \end{remark}
We do not make any distinction in the notation; we silently identify maps $(Z,C) \to (S^2,E)$ with curves in $\mathcal{L}S^2$.
\begin{definition}
Let $\alpha$ be a curve in $\mathcal{L}S^2$ from $p_1$ to $p_2$ and
$\beta$ a curve from $p_2$ to $p_3$. Then we denote by
$\beta \ast \alpha$ the concatenation of the curves $\alpha$ and
$\beta$, which is a curve from $p_1$ to $p_3$. Given a curve
$\alpha$, we denote by $\alpha^{-1}$ the curve with time reversed. \end{definition}
\begin{figure}
\caption{Concatenation of maps from the cylinder.}
\label{fig:ConcatCylinderMaps}
\end{figure}
Using the isomorphism \begin{align}
\label{FLSIso}
\mathcal{M}(I,\mathcal{L}S^2) \cong \mathcal{M}(Z,S^2), \end{align} concatenation of curves induces a corresponding operation on the space of (type I normalized) maps $(Z,C) \to (S^2,E)$. This is depicted in figure~\ref{fig:ConcatCylinderMaps}. We need to be careful to distinguish the inverse of a map from its inverse regarded as a curve. But this should always be clear from the context.
Of course, the homotopy classification of curves in $\mathcal{L}S^2$ depends on the topology of $\mathcal{L}S^2$. Our first remark towards understanding the topology of the latter is: \begin{remark}
The free loop space $\mathcal{L}S^2$ is path-connected. \end{remark} \begin{proof}
Let $\alpha, \beta \in \mathcal{L}S^2$ be two loops in $S^2$. Since
$\pi_1(S^2) = 0$, any loop can be deformed to the constant loop at
some chosen point in $S^2$. By transitivity, this implies that
$\alpha$ and $\beta$ can be deformed into each other. In other
words, there is a path from $\alpha$ to $\beta$, regarded as
elements of the free loop space. \end{proof}
Now we consider the set $\pi(\mathcal{L}S^2;\gamma_0,\gamma_1)$ of homotopy classes of curves in $\mathcal{L}S^2$ with fixed endpoints $\gamma_0$ and $\gamma_1$. Using lemma~\ref{LemmaCurvesNullHomotopic} we can conclude that two curves $f, g\colon I \to \mathcal{L}S^2$ from $\gamma_0$ to $\gamma_1$ are homotopic if and only if $g^{-1} \ast f$ is null-homotopic in $\pi(\mathcal{L}S^2,\gamma_0)$. This underlines that we need to understand the fundamental group $\pi_1(\mathcal{L}S^2)$. In the next step we compute the fundamental group $\pi_1(\mathcal{L}S^2)$. For this we make use of the following fibration (see e.\,g. \cite{AgaudeArticle}) \begin{align}
\label{LoopFibration}
\Omega S^2 \hookrightarrow \mathcal{L}S^2 \to S^2 \end{align} where $\Omega S^2$ denotes the space of \emph{based} loops in $S^2$ at some fixed basepoint, e.\,g. $p_0 \in E$. The map $\mathcal{L}S^2 \to S^2$ is the evaluation map, which takes a loop in $\mathcal{L}S^2$, i.\,e. a map $S^1 \to S^2$, and evaluates it at $1$. Now we can state: \begin{lemma}
\label{FundamentalGroupOfFLS}
The fundamental group $\pi_1(\mathcal{L}S^2)$ of the free loop space
of $S^2$ is infinite cyclic. \end{lemma} \begin{proof}
We consider the fibration (\ref{LoopFibration}) and develop it into
a long exact homotopy sequence. The relevant part of this sequence
is:
\begin{align}
\ldots \rightarrow \mathbb{Z} \cong \pi_2(S^2) \rightarrow \pi_1(\Omega S^2) \rightarrow \pi_1(\mathcal{L}S^2) \rightarrow \pi_1(S^2) \cong 0
\end{align}
It is known that
\begin{align*}
\pi_1(\Omega S^2) \cong \pi_2(S^2) \cong \mathbb{Z}.
\end{align*}
Therefore, by exactness of the above sequence, it follows that
$\pi_1(\mathcal{L}S^2)$ must be a quotient of $\mathbb{Z}$,
i.\,e. either the trivial group, $\mathbb{Z}$ or one of the finite
groups $\mathbb{Z}_n$. We prove that it is $\mathbb{Z}$ with the
help of a geometric argument: Let $\gamma_1$ be the curve in $S^2$
which parametrizes the equator $E$ with degree one and choose
$\gamma_1$ as the basepoint for the fundamental group and then
construct an infinite family of maps $X \to S^2$ with the same fixed
point degree $(1,1)$ but with different global degrees -- the latter
implies by Hopf's theorem that they cannot be homotopic. In
particular they cannot be $G$-homotopic and hence the curves
associated to their restrictions to $Z$ cannot be homotopic.
The construction goes as follows: We construct a map $\varphi$ from
the cylinder $Z = I \times S^1$ to $S^2$ such that
$\varphi_0 \equiv \varphi_1$ and $\varphi_0$ and $\varphi_1$ are
degree 1 maps $S^1 \to E$ as follows: Assume that for $t=0$ we have
an embedding of $S^1$ as the equator $E \subset S^2$. Then we fix
two antipodal points on the equator and begin rotating the equator
on the Riemann sphere while keeping the two antipodal points fixed
for all $t$. We can rotate this $S^1$-copy on the sphere as fast as
we want, we just need to make sure that at $t=1$ we again match up
with the equator. By making the correct number (i.\,e. an even
number) of rotations we obtain a map $\varphi$ from $Z$ to $S^2$
whose equivariant extension to the full torus $X$ has the desired
fixed point degree $(1,1)$. By increasing the number of rotations,
we can generate homotopically distinct $G$-maps from the torus into
the sphere. Thus we obtain an infinite family of maps with fixed
point degree $\Bideg{1}{1}$ and with distinct total
degrees. Therefore, the fundamental group cannot be finite, but must
be infinite cyclic. \qedhere \end{proof}
\paragraph{The Degree Map}
In the following we introduce the \emph{total degree} into the discussion. The fixed point degree together with this total degree will turn out to be a complete invariant of equivariant homotopy. The degree map will be the fundamental tool for the proof. We would like to associate to a curve in $\mathcal{L}S^2$ the degree of its equivariant extension. This only makes sense for a certain subset of all curves. Thus we define: \begin{definition}
For each $j \in \mathbb{Z}$ we let $\mathscr{D}_j \in
\mathcal{L}S^2$ be the normalized loop in
$E \subset S^2$ of degree $j$. We set
\begin{align*}
\mathcal{L}_{\text{std}} = \left\{\mathscr{D}_j\colon j \in \mathbb{Z}\right\} \subset \mathcal{L}S^2.
\end{align*}
Let $\mathscr{P}$ be the set of curves in $\mathcal{L}S^2$ starting
and ending at points in $\mathcal{L}_{\text{std}}$:
\begin{align*}
\mathscr{P} = \left\{\alpha\colon I \to \mathcal{L}S^2\mid \alpha(0), \alpha(1) \in \mathcal{L}_{\text{std}}\right\}.
\end{align*} \end{definition} The set $\mathscr{P}$ is a suitable domain of definition for the degree map, which we now define: \begin{definition}
\label{DefDegreeMap}
The \emph{degree map} on $\mathscr{P}$ is the map
\begin{align*}
\text{deg}\colon \mathscr{P} &\to \mathbb{Z}\\
\alpha &\mapsto \deg \widehat{\alpha},
\end{align*}
where $\smash{\widehat{\alpha}}$ denotes the equivariant extension
of $\alpha$. That is, $\alpha$, which is a curve in the free loop
space $\mathcal{L}S^2$, is to be regarded as a map
$(Z,C) \to (S^2,E)$ and $\smash{\widehat{\alpha}}\colon X \to S^2$
is its equivariant extension to the full torus (see
lemma~\ref{Class1EquivariantExtension}). \end{definition} An immediate property of the degree map is: \begin{remark}
\label{DegOfConstantCurveZero}
Let $\alpha \in \mathscr{P}$ be the constant curve at some $\gamma
\in \mathcal{L}_{\text{std}}$. Then $\deg \alpha = 0$. \end{remark} \begin{proof}
The equivariant extension $\smash{\widehat{\alpha}}$ has its image contained
in $E \subset S^2$. Hence it is not surjective as a map to $S^2$,
which implies $\deg \smash{\widehat{\alpha}} = 0$. \end{proof} For curves $\alpha$, $\beta$ we introduce the symbols $\alpha \simeq \beta$ to express that $\alpha$ and $\beta$ are homotopic as curves (with fixed endpoints). \begin{remark}
\label{DegreeMapInvariantUnderHomotopies}
When $\alpha, \beta \in \mathscr{P}$ are two curves with common
endpoints, then $\alpha \simeq \beta$ implies $\deg \alpha = \deg
\beta$. In other words: the degree map is also well-defined
on $\mathscr{P}/{\simeq}$, that is $\mathscr{P}$ modulo homotopy. \end{remark} \begin{proof}
Homotopies between two curves can be regarded as homotopies between
maps $Z \to S^2$ rel $C$. Such homotopies can also be equivariantly
extended, giving homotopies between the respective equivariant
extensions of the two maps $X \to S^2$. Therefore, their total
degree must agree. \end{proof} We have just seen that the degree map factors through $\mathscr{P}/{\simeq}$. The next goal is to prove an injectivity property of the degree map. Crucial for its proof is that the degree map is compatible with the aforementioned concatenation operation of loops: \begin{lemma}
\label{DegreeCompatibleWithConcatenation}
Let $\gamma_1, \gamma_2, \gamma_3$ be in $\mathcal{L}_{\text{std}}$,
furthermore let $\alpha \in \mathscr{P}$ be a path in
$\mathcal{L}S^2$ from $\gamma_1$ to $\gamma_2$ and $\beta$ be a path
in $\mathcal{L}S^2$ from $\gamma_2$ to $\gamma_3$. Then
$\beta \ast \alpha$ is a path from $\gamma_1$ to $\gamma_3$ and
\begin{align*}
\deg (\beta \ast \alpha) = \deg \beta + \deg \alpha.
\end{align*} \end{lemma}
\begin{proof}
Let $\alpha$ and $\beta$ be paths in $\mathcal{L}S^2$ with the
properties mentioned above. To prove the lemma, we have to show
that
\begin{align*}
\deg \widehat{\beta \ast \alpha} = \deg \widehat{\beta} + \deg
\widehat{\alpha}.
\end{align*}
The idea we employ here is that -- after smooth approximation of the
maps -- we can count preimage points of a regular fiber to compute
the total degrees. We regard the curves $\alpha$ and $\beta$ as maps
from the cylinder $Z = I \times S^1$ to $S^2$. We can assume that
$\alpha$ and $\beta$ are smooth on $Z$; the concatenation
$\beta \ast \alpha$ does not have to be smooth at the glueing
curves. But after a smooth approximation of $\beta \ast \alpha$ near
this glueing curve, we can assume that the equivariant extensions
$\smash{\widehat{\alpha}}$, $\smash{\widehat{\beta}}$ and
$\smash{\widehat{\beta \ast \alpha}}$ are smooth maps from the torus
into $S^2$.
Now let $q \in S^2$ be a regular point of
$\smash{\widehat{\beta \ast \alpha}}$, away from the neighborhood of
the boundary circles where we have smoothed the map
$\beta \ast \alpha$. We now consider its fiber over $q$. This
consists of all points $p_1,\ldots,p_k$ with
$\smash{\widehat{\alpha}}(p_j) = q$ and all points
$p'_1,\ldots,p'_\ell$ with $\smash{\widehat{\beta}}(p'_j) =
q$.
Summing the points $p_j$ (resp. $p'_j$) respecting the orientation
signs yields $\deg\smash{\widehat{\alpha}}$
(resp. $\deg\smash{\widehat{\beta}}$). Therefore, the points
$p_1,\ldots,p_k,p'_1,\ldots,p'_\ell$ with their respective
orientation signs attached add up to
$\deg\smash{\widehat{\alpha}} + \deg\smash{\widehat{\beta}}$. \end{proof}
Note that lemma~\ref{DegreeCompatibleWithConcatenation} also implies that \begin{align*}
\deg\left(\alpha^{-1}\right) = -\deg\left(\alpha\right), \end{align*} where $\alpha^{-1}$ is the curve $\alpha\colon I \to \mathcal{L}S^2$ with reversed timed. The space $\mathscr{P}$ does not carry a group structure in the usual sense, but of course the based fundamental groups $\pi_1(\mathcal{L}S^2,\gamma)$ do carry a group structure. When $\gamma$ is in $\mathcal{L}_{\text{std}}$, then $\pi_1(\mathcal{L}S^2,\gamma) \subset \mathscr{P}/{\simeq}$ and we can note the following \begin{remark}
The degree map induces a group homomorphism on fundamental groups:
\begin{align*}
\text{deg}\colon \pi_1(\mathcal{L}S^2, \mathscr{D}_j) \to \mathbb{Z}
\end{align*}
for every $j \in \mathbb{Z}$. \end{remark} \begin{proof}
Let $\mathscr{D}_j$ be in $\mathcal{L}_{\text{std}}$. By
remark~\ref{DegreeMapInvariantUnderHomotopies} we know that the
degree map is well-defined as a map on the fundamental group
$\pi_1(\mathcal{L}S^2,\mathscr{D}_j)$. By
remark~\ref{DegOfConstantCurveZero} the neutral element in
$\pi_1(\mathcal{L}S^2,\mathscr{D}_j)$, that is the homotopy class of
the constant curve at $\mathscr{D}_j$, gets mapped to $0 \in
\mathbb{Z}$. The group structure in the fundamental group is given
be the concatenation of loops at the
basepoint. Lemma~\ref{DegreeCompatibleWithConcatenation} shows that
the degree map is compatible with the group structures. Hence, the
degree map is a group homomorphism on fundamental groups. \end{proof}
Now that we know that restrictions of the degree map to fundamental groups are group homomorphisms it is useful to compute its kernel. For this we need a lemma, which deals with the special case of a fixed point degree of the form $\Bideg{d_0}{d_0}$. To formulate the statement, we introduce an equivalence relation $\sim$ on the cylinder $Z$, which identifies the circles $C_0, C_1$: $(0,z) \sim (1,z)$. The resulting space $\smash{\widetilde{Z}} = Z/{\sim}$ is a 2-torus.\label{CylinderBoundaryCirclesIdentified} Now we can state:
\begin{lemma}
\label{MapExtensionDegreeEquality}
A map $f\colon (Z,C) \to (S^2,E)$ with $\restr{f}{C_0} \equiv
\restr{f}{C_1}$ has even degree. More concretely: The map $f$
induces a map $\smash{\tilde{f}}\colon \smash{\widetilde{Z}} \to S^2$ and
\begin{align*}
\deg \hat{f} = 2 \deg \tilde{f}.
\end{align*}
In particular, the equivariant extension $\smash{\hat{f}}$ has degree
zero iff the induced map $\smash{\tilde{f}}$ has degree zero. \end{lemma}
\begin{proof}
Note that on the LHS we have the degree of the equivariant extension
$\smash{\hat{f}}\colon X \to S^2$ of $f$ while on the RHS we have (two
times) the degree of the induced map $\smash{\tilde{f}}\colon
\smash{\widetilde{Z}} \to S^2$.
We prove the statement by counting preimage points. The torus
$\smash{\widetilde{Z}}$ carries a differentiable structure. The
induced map $\smash{\tilde{f}}\colon \smash{\widetilde{Z}} \to S^2$
does not have to be smooth near $C \subset
\smash{\widetilde{Z}}$.
But after a smooth approximation near $C$ we can assume that it is
globally smooth. By the same reason the equivariant extension
$\smash{\hat{f}}$ defined on $X$ is globally smooth.
Let $q$ be a regular value of the smoothed map $f$, away from the
circle $C$. This implies that $q$ is also a regular value of
$\smash{\tilde{f}}$ and $\smash{\hat{f}}$. Let $d$ be the degree of
$\smash{\tilde{f}}$. The fiber of $q$ under $\smash{\tilde{f}}$ can thus be
written as
\begin{align*}
\tilde{f}^{-1}(q) = \{p_1,\ldots,p_k\}.
\end{align*}
Denoting the respective orientation signs at each preimage point
$p_j$ with $\sigma_j$ we can express the degree $d$ as
\begin{align*}
d = \sum_{1\leq j \leq k} \sigma_j.
\end{align*}
To simplify the proof we assume that $q$ has been chosen such that
$T(q)$ is also a regular value of $\smash{\hat{f}}$ and that $q$
(and therefore also $T(q)$) has no preimage points on the boundary
circles.
Recall that when regarding the cylinder $Z$ as being embedded in $X$
with boundary circles $C_0$ and $C_1$, the equivariant extension
$\smash{\hat{f}}$ is the map
\begin{align*}
\hat{f}\colon X &\to S^2\\
x &\mapsto
\begin{cases}
f(x) & \;\text{if $x \in Z$}\\
T \circ f \circ T(x) & \;\text{if $x \in Z'$}\\
\end{cases}
\end{align*}
where $Z'$ denotes the complementary cylinder in $X$ whose
intersection with $Z$ consists of the circles $C_0 \cup C_1$. It
remains to compare the fibers $\smash{\hat{f}^{-1}}(q)$ and
$\smash{\tilde{f}^{-1}}(q)$. Since $q$ is assumed to have no preimage
points in the boundary circles, we can write the first fiber as
\begin{align*}
\hat{f}^{-1}(q) = \Restr{\hat{f}}{Z}^{-1}(q) \cup \Restr{\hat{f}}{Z'}^{-1}(q).
\end{align*}
By definition of $\smash{\hat{f}}$, the preimage points in the cylinder
$Z$ are exactly those of $\smash{\tilde{f}}$, with the same orientation
signs. We now prove
\begin{align*}
\Restr{\hat{f}}{Z'}^{-1}(q) = T\left(\Restr{\hat{f}}{Z}^{-1}(T(q))\right).
\end{align*}
For this, let $p$ be in $Z'$ with
$\restr{\smash{\hat{f}}}{Z'}(p) = q$. This implies
$T \circ \restr{\smash{\hat{f}}}{Z'}(p) = T(q)$, which in turn
implies by equivariance
\begin{align*}
\Restr{\hat{f}}{Z}(T(p)) = T(q).
\end{align*}
In other words, after defining $\smash{\tilde{p}} = T(p) \in Z$,
we have: $p = T(\smash{\tilde{p}})$ with $\smash{\tilde{p}} \in Z$
and
\begin{align*}
\Restr{\smash{\tilde{f}}}{Z}(\smash{\tilde{p}}) =
\Restr{\smash{\tilde{f}}}{Z}(T(p)) =
T \circ \Restr{\smash{\tilde{f}}}{Z'}(p) = T(q).
\end{align*}
On the other hand, let $\smash{\tilde{p}}$ be in
$T\left(\restr{\smash{\hat{f}}}{Z}^{-1}(T(q))\right)$. This
means that $\smash{\tilde{p}} = p$ with
$\restr{\smash{\hat{f}}}{Z}(p) = T(q)$. Set
$p = T(\smash{\tilde{p}})$. Then:
\begin{align*}
\Restr{\hat{f}}{Z'}(\hat{p}) = \Restr{\hat{f}}{Z'}(T(p)) = T \circ \Restr{\hat{f}}{Z}(p) = q,
\end{align*}
which means $\smash{\tilde{p}} \in \restr{\smash{\hat{f}}}{Z'}^{-1}(q)$.
Since $T$ bijectively sends $Z$ to $Z'$ it follows that the number
of preimage points on $Z$ and $Z'$ coincide. Also, since
$\smash{\hat{f}}$ restricted to (the interior of) $Z'$ is the
composition of $f$ with two orientation-reversing diffeomorphisms,
the orientation sign at each preimage point $p$ in $Z$ is the same
as the orientation sign of the corresponding preimage point $T(p)$
in $Z'$. This means that summing (respecting the orientation) over
all the preimage points in the fiber $\smash{\hat{f}}^{-1}(q)$
yields exactly $2 \deg \smash{\tilde{f}}$. \end{proof}
Now we can prove the desired injectivity property of the degree map: \begin{proposition}
\label{DegreeMapInjective}
For any $\gamma \in \mathcal{L}_{\text{std}}$, the degree homomorphism
\begin{align*}
\deg\colon \pi_1(\mathcal{L}S^2,\gamma) \to \mathbb{Z}
\end{align*}
is injective. \end{proposition} \begin{proof}
We prove that the degree homomorphism has trivial kernel. By
path-connected- ness of $\mathcal{L}S^2$ all the based fundamental
groups of $\mathcal{L}S^2$ are isomorphic. Hence it suffices to
prove the statement for one fixed loop $\gamma$. Therefore we reduce
the problem to a simpler case and take $\gamma$ to be the constant
loop at the base point $p_0$ in $E$.
Let $\alpha$ be a type I normalized map of the cylinder $Z$ into
$S^2$ with fixed point degree $\Bideg{0}{0}$ and $\deg \alpha = 0$. As before,
$\alpha$ can be regarded as curve in $\mathcal{L}S^2$ and $[\alpha]$
defines an element in the fundamental group
$\pi_1(\mathcal{L}S^2,\gamma)$. Let $\widetilde{Z} = Z/{\sim}$ be the
torus which results from the cylinder $Z$ by identifying the circles
$C_0$ and $C_1$ (see
p.\pageref{CylinderBoundaryCirclesIdentified}). Since, by
construction, $\restr{\alpha}{C_0} \equiv \restr{\alpha}{C_1} \equiv
\gamma$, $\alpha$ induces a map $\widetilde{Z} \to S^2$ which, by
lemma~\ref{MapExtensionDegreeEquality}, also has degree zero.
Denote the generator of the fundamental group of the torus
$\widetilde{Z}$ which corresponds to $C_0$ resp. $C_1$ by $C$.
Let $C'$ denote the other generator of the fundamental group such
that the intersection number of $C$ with $C'$ is $+1$. The
restriction of $f\colon \widetilde{Z} \to S^2$ to $C$ is already
constant. Using the homotopy extension property together with the
simply-connectedness of $S^2$ we can assume that $f$ is also
constant along $C'$. Therefore we can collapse these two generators
and $f$ induces a map
$\widetilde{Z}/(C \cup C') \cong S^2 \to S^2 \cong S^2$. By
remark~\ref{RemarkMapOnQuotientDegreeUnchanged} the map
$f\colon \widetilde{Z}/(C \cup C') \to S^2$ also has degree zero. By
Hopf's theorem, this map is null-homotopic (even as a map of pointed
spaces, as can be shown by another application of the homotopy
extension property). This corresponds to a null-homotopy of the
curve $\alpha$ with fixed basis curve $\gamma$, hence
$[\alpha] = 0$, which finishes the proof. \end{proof}
\paragraph{The Degree Triple}
In the following we combine the total degree and the fixed point degrees. For this we make the following \begin{definition}
\label{DefinitionTriple}
(Degree triple map) The \emph{degree triple map} is the map
\begin{align*}
\mathcal{T}\colon \mathcal{M}_G(X,S^2) &\to \mathbb{Z}^3\\
f &\mapsto \Triple{d_0}{d}{d_1},
\end{align*}
where $\Bideg{d_0}{d_1}$ is the fixed point degree of $f$ and $d$ is
the total degree of $f$. For a map $f\colon X \to S^2$ we define its
\emph{degree triple} (or simply \emph{triple}) to be
$\mathcal{T}(f)$. We call a given triple \emph{realizable} if it is
contained in the image $\Im(\mathcal{T})$. For a map $f\colon (Z,C)
\to (S^2,E)$ we define its \emph{degree triple} to be the degree
triple of the equivariant extension $\smash{\hat{f}}\colon X \to S^2$. \end{definition}
As an immediate consequence we obtain: \begin{remark}
\label{DegreeTripleInvariant}
The degree triple of a map is a $G$-homotopy invariant. \qed \end{remark} Thus, the degree triple map factors through $[X,S^2]_G$. In the following we analyze the image $\Im(\mathcal{T})$.
\begin{remark}
\label{RemarkTripleGlobalDegree}
Let $f$ be a $G$-map $X \to S^2$ with fixed point degrees
$\Bideg{d_0}{d_0}$. Then the total degree $\deg f$ must be even. In
other words, the degree triples $\Triple{d_0}{2k+1}{d_0}$ are not
contained in the image $\Im(\mathcal{T})$.\end{remark} \begin{proof}
Assume that the map $f$ is type I normalized. The restriction
$\restr{f}{Z}$ to the cylinder then satisfies the assumptions of
lemma~\ref{MapExtensionDegreeEquality}, hence we can conclude that
its equivariant extension $\smash{\hat{f}}$ has even degree. But the
equivariant extension of $\restr{f}{Z}$ is $f$ itself, hence $f$
must have even degree. \end{proof} In particular, the triple $\Triple{0}{1}{0}$ is not contained in $\Im(\mathcal{T})$. This gives rise to the $\mod 2$ condition \begin{align}
\label{DiscussionParityCondition}
d \equiv d_0 + d_1 \mod 2, \end{align} which was already mentioned in the outline and must hold for any realizable degree triple. The details will be explained in paragraph~\ref{ParagraphClassificationType1MainResult} (p.~\pageref{ParagraphClassificationType1MainResult}). In order to deduce the above ``parity condition'' (\ref{DiscussionParityCondition}) for general degree triples we introduce an algebraic structure on the set of realizable triples. This is the following binary operation for triples: \begin{definition}
Two triples $\Triple{d_0}{d}{d_1}$ and $\Triple{d_0'}{d'}{d_1'}$ are
called \emph{compatible} if $d_1 = d_0'$. Given two compatible
triples $\Triple{d_0}{d}{d_1}$ and $\Triple{d_1}{d'}{d_2}$, then we
define $\Triple{d_0}{d}{d_1} \bullet \Triple{d_1}{d'}{d_2}$ to be
the triple
\begin{align*}
\Triple{d_0}{d + d'}{d_2}.
\end{align*} \end{definition} We call this binary operation \emph{concatenation} of triples. Its usefulness is illustrated by the following remark: \begin{remark}
\label{ImageOfTripleMapClosedUnderConcatenation}
When the triples $\Triple{d_0}{d}{d_1}$ and $\Triple{d_1}{d'}{d_2}$
are in $\Im(\mathcal{T})$, then so is
\begin{align*}
\Triple{d_0}{d}{d_1} \bullet \Triple{d_1}{d'}{d_2}.
\end{align*} \end{remark} \begin{proof}
Assume that the triples $\Triple{d_0}{d}{d_1}$
resp. $\Triple{d_1}{d'}{d_2}$ are realized by the paths $\alpha$
resp. $\beta$ in $\mathscr{P}$ satisfying $\alpha(1) = \beta(0)$. By
definition,
\begin{align*}
\Triple{d_0}{d}{d_1} \bullet \Triple{d_1}{d'}{d_2} = \Triple{d_0}{d+d'}{d_2}\;.
\end{align*}
We need to show the existence of a path in the loop space
$\mathcal{L}S^2$ from $\alpha(0)$ to $\beta(1)$ with total degree
$d + d'$ of its associated equivariant extension. Recall that $d$
(resp. $d'$) is the degree of the equivariant extension
$\widehat{\alpha}$ (resp. $\widehat{\beta}$). The concatenation
$\alpha \ast \beta$ is a path in loop space from $d_0$ to $d_2$.
Its total degree is defined to be the degree of the equivariant
extension $\widehat{\alpha \ast \beta}$. By
lemma~\ref{DegreeCompatibleWithConcatenation} this is the same as
$\deg\widehat{\alpha} + \deg\widehat{\beta} = d + d'$. \end{proof}
In other words: the image $\Im(\mathcal{T})$ is closed under the concatenation operation of degree triples. In the following lemma we encapsulate several fundamental properties of the image $\Im(\mathcal{T})$. \begin{proposition}
\label{TripleBuildingBlocks}
Let $d_0, d_1$ be integers. Then:
\begin{enumerate}[(i)]
\item If the triple $\Triple{d_0}{d}{d_1}$ is in $\Im(\mathcal{T})$, then so is
$\Triple{d_0}{-d}{d_1}$.
\item For any fixed point degree $\Bideg{d_0}{d_1}$, the triple
$\Triple{d_0}{d_0 - d_1}{d_1}$ is in $\Im(\mathcal{T})$. In
particular, the triple $\Triple{1}{1}{0}$ is in
$\Im(\mathcal{T})$.
\item For any $k$, the triple $\Triple{0}{2k}{0}$ is in $\Im(\mathcal{T})$.
\item If $\Triple{d_0}{d}{d_1}$ is in $\Im(\mathcal{T})$, then so is
$\Triple{d_1}{d}{d_0}$.
\end{enumerate} \end{proposition} \begin{proof}
Regarding (i): Assume that $f\colon X \to S^2$ is a map with
$\mathcal{T}(f) = \Triple{d_0}{d}{d_1}$. Note that $T\colon S^2 \to
S^2$ is an orientation-reversing diffeomorphism of $S^2$ which keeps
the equator $E$ fixed. Therefore:
\begin{align*}
\mathcal{T}(T \circ f) = \Triple{d_0}{-d}{d_1}.
\end{align*}
In other words: the triple $\Triple{d_0}{-d}{d_1}$ is realizable.
Regarding (ii): We need to construct a map $X \to S^2$ with fixed
point degrees $\Bideg{d_0}{d_1}$ and total degree $d_0 - d_1$. For
this, let $D$ be the closed unit disk in the complex plane:
\begin{align*}
D = \{z \in \mathbb{C}\colon |z| \leq 1\}.
\end{align*}
Let $\iota_D$ be the embedding of $D$ onto one of the two hemispheres.:
\begin{align}
\label{IotaHemisphereEmbedding}
\iota\colon D &\hookrightarrow S^2\\
r e^{i\varphi} &\mapsto
\begin{pmatrix}
r\cos \varphi \\
r\sin \varphi \\
\pm\sqrt{1-r^2}
\end{pmatrix}.
\end{align}
We pick the sign defining either the lower or the upper hemisphere
such that $\iota_D$ is orientation preserving. Now we define the
map
\begin{align*}
f\colon Z &\to D\\
(t, \varphi) &\mapsto \begin{cases}
(1-2t) e^{i d_0 \varphi} & \text{ for $0 \leq t \leq \frac{1}{2}$}\\
(2t-1) e^{i d_1 \varphi} & \text{ for $\frac{1}{2} \leq t \leq 1$}.
\end{cases}
\end{align*}
To ease the notation, set
\begin{align*}
Z_1 = \left(0,\frac{1}{2}\right) \times S^1 \;\text{ and }\; Z_2 &= \left(\frac{1}{2},1\right) \times S^1.
\end{align*}
Then we consider the composition map
\begin{align*}
F = \iota_D \circ f\colon Z \to S^2.
\end{align*}
The restriction $\restr{F}{C_j}$ to the boundary circles is of the
form:
\begin{align*}
\Restr{F}{C_j}\colon S^1 &\to E \subset S^2\\
\varphi &\mapsto
\begin{pmatrix}
\cos(d_j \varphi) \\
\sin(d_j \varphi) \\
0
\end{pmatrix}.
\end{align*}
Hence, by construction, $F$ has fixed point degrees
$\Bideg{d_0}{d_1}$ (compare with (\ref{DegreeOneMapOnEquator}),
p.~\pageref{DegreeOneMapOnEquator}). It remains to show that the
equivariant extension $\smash{\hat{F}}$ has total degree
$d_0 - d_1$. At least away from the boundary circles,
$\smash{\hat{F}}$ is a smooth map. Thus, let $q \in S^2$ be a
regular value of $\smash{\hat{F}}$, say, on the hemisphere
$\iota_D(D)$, but not one of the poles (they are not regular values
for $f$). The point $q$ corresponds to some point
$\smash{\tilde{q}} = \smash{\tilde{t}e^{i\tilde{\varphi}}}$ in the
closed unit disk $D$ with $\smash{\tilde{t}} \in (0,1)$. By
construction, the fiber $\smash{\hat{F}^{-1}}(q)$ will be the same as
the fiber $F^{-1}(\tilde{q})$, which contains exactly $d_0 + d_1$
points in the cylinder $Z$:
\begin{align*}
q_{j,k} = \left(\frac{1+(-1)^{j+1}\tilde{t}}{2}, d_j\left(\tilde{\varphi} + \frac{2k}{d_j}\pi\right)\right)\; \text{for $j=0,1$ and $k=0,1,\ldots,d_j$}.
\end{align*}
The points $q_{0,k}$, $k=0,\ldots,d_1-1$, are of the form
\begin{align*}
q_{0,k} = \left(\frac{1-\tilde{t}}{2}, d_1\left(\tilde{\varphi} + \frac{2k}{d_1}\pi\right)\right)
\end{align*}
which means that they are contained in the cylinder half $Z_1$. The
points $q_{1,k}$, $k=0,\ldots,d_2-1$ are of the form
\begin{align*}
q_{2,k} = \left(\frac{1+\tilde{t}}{2}, d_2\left(\tilde{\varphi} + \frac{2k}{d_2}\pi\right)\right)
\end{align*}
and therefore they are contained in $Z_2$. A computation shows
that
the orientations signs at the points $q_{1,k} \in Z_1$ are the
opposite of those at the points $q_{2,k} \in Z_2$. Therefore,
summing over the points in the generic fiber
$\smash{\hat{F}^{-1}}(q)$ yields $\pm (d_0 - d_1)$ as total degree
of $\smash{\hat{F}}$. Since $\iota_D$ has been chosen
orientation-preserving, the total degree is $d_0 - d_1$.
Regarding (iii): First note that by (i) and (ii) the triple
$\Triple{0}{1}{1}$ is in $\Im(\mathcal{T})$. But then the triple
$\Triple{0}{1}{1} \bullet \Triple{1}{1}{0} = \Triple{0}{2}{0}$ is
also in $\Im(\mathcal{T})$. Concatenation of this triple with itself
$k$ times yields the desired triple $\Triple{0}{2k}{0}$.
Regarding (iv): Let $\tau\colon X \to X$ be the translation inside
the torus, which swaps the boundary circles (e.\,g. $[z] \mapsto [z
+ \text{\textonehalf}i]$ in our model). Then, composing a map of type
$\Triple{d_0}{d}{d_1}$ with $\tau$ yields a map of type
$\Triple{d_1}{d}{d_0}$.
This finishes the proof. \end{proof}
\label{RemarkAboutNormalForms} The proof of proposition~\ref{TripleBuildingBlocks} (ii) is of particular importance -- it contains a constructive method for producing equivariant maps $X \to S^2$ with triple $\Triple{d_0}{d_0-d_1}{d_1}$. These triples can be regarded as basic building blocks for equivariant maps $X \to S^2$, since all other triples can be built from triples of this form by concatenation.
\begin{definition}
We call the maps constructed in proposition~\ref{TripleBuildingBlocks}
(ii) \emph{maps in normal form} for the triple
$\Triple{d_0}{d_0-d_1}{d_1}$. \end{definition}
Note that the normal form map for a triple $\Triple{d_0}{d_1-d_0}{d_1}$ is by definition the normal form map for $\Triple{d_0}{d_0-d_1}{d_1}$ composed with $T\colon S^2 \to S^2$.
\paragraph{Main Result} \label{ParagraphClassificationType1MainResult}
In this paragraph we state and prove the main classification result for the type I involution. The statement is: \begin{theorem}
\label{Classification1}
The $G$-homotopy class of a map $f \in \mathcal{M}_G(X,S^2)$ is
uniquely determined by its degree triple $\mathcal{T}(f)$. The image
$\Im(\mathcal{T})$ of the degree triple map
$\mathcal{T}\colon \mathcal{M}_G(X,S^2) \to \mathbb{Z}^3$ consists of those
triples $\Triple{d_0}{d}{d_1}$ satisfying
\begin{align*}
d \equiv d_0 + d_1 \mod 2.
\end{align*} \end{theorem}
We need one last lemma in order to prove this theorem: \begin{lemma}
\label{HomotopyOnCylinder}
Let $f$ and $g$ be two type I normalized maps $(Z, C) \to (S^2,E)$
with the same triple $\Triple{d_0}{d}{d_1}$. Then $f$ and $g$ are
homotopic rel $C$. \end{lemma} \begin{proof}
Both maps can be regarded as curves $f,g\colon I \to
\mathcal{L}S^2$. Since they are assumed to be type I normalized,
they start at the same curve $\gamma_0 \in \mathcal{L}_{\text{std}}$
and end at the same curve $\gamma_1 \in
\mathcal{L}_{\text{std}}$. We need to prove the existence of a
homotopy between the curves $f$ and $g$ in $\mathcal{L}S^2$.
By lemma~\ref{LemmaCurvesNullHomotopic} it suffices to show that the
loop $g^{-1} \ast f$ based at $\gamma_0$ is null-homotopic. To
prove this we use the degree map restricted to the fundamental group
based at $\gamma_0$:
\begin{align*}
\deg\colon \pi_1(\mathcal{L}S^2,\gamma_0) &\to \mathbb{Z}\\
[\gamma] &\mapsto \deg \hat{\gamma}.
\end{align*}
From lemma~\ref{DegreeCompatibleWithConcatenation} it follows that
\begin{align*}
\deg (g^{-1} \ast f) = \deg f - \deg g = \deg \hat{f} - \deg \hat{g} = \deg f - \deg g = d - d = 0.
\end{align*}
Because of remark~\ref{DegreeMapInvariantUnderHomotopies} this can
also be stated on the level of homotopy classes:
\begin{align*}
\deg [g^{-1} \ast f] = \deg [f] - \deg [g] = 0.
\end{align*}
Therefore, using the injectivity of the degree map
(proposition~\ref{DegreeMapInjective}) we can conclude that $[g^{-1}
\ast f] = 0$ in $\pi_1(\mathcal{L}S^2,\gamma_1)$. In other
words:
\begin{align*}
g^{-1} \ast f \simeq c_{\gamma_0},
\end{align*}
where $c_{\gamma_0}$ denotes the constant curve in $\mathcal{L}S^2$ at
$\gamma_0$. Now, lemma~\ref{LemmaCurvesNullHomotopic} implies that
$f$ and $g$ are homotopic by means of a homotopy $h\colon I
\times I \to \mathcal{L}S^2$. This homotopy induces a homotopy
\begin{align*}
H\colon I \times (Z,C) &\to (S^2,E)\\
(t,(s,z)) &\mapsto h(t,s,z)
\end{align*}
between the maps $f$ and $g$. \end{proof}
Finally, we can prove theorem~\ref{Classification1}: \begin{proof}
It is clear by remark~\ref{DegreeTripleInvariant} that the degree
triple $\Triple{d_0}{d}{d_1}$ is a $G$-homotopy invariant. Now we
show that it is a \emph{complete} invariant in the sense that
\begin{align*}
\mathcal{T}(f) = \mathcal{T}(g) \;\Rightarrow\; f \simeq_G g,
\end{align*}
where $f \simeq_G g$ means that the maps $f$ and $g$ are
$G$-homotopic. For this, let $f$ and $g$ be two $G$-maps with the
same triple $\Triple{d_0}{d}{d_1}$.
By remark~\ref{BoundaryNormalization} we can assume that $f$ and $g$
are type~I normalized. Now we use remark~\ref{CylinderReduction} and
reduce the construction of a $G$-homotopy from $f$ to $g$ to the
construction of a homotopy from $\restr{f}{Z}$ to $\restr{g}{Z}$ rel
$C$. The assumptions of lemma~\ref{HomotopyOnCylinder} are
satisfied, hence this lemma provides us with a homotopy $H\colon I
\times (Z,C) \to (S^2,E)$ rel $C$. Such a homotopy can be
equivariantly extended to all of $X$, establishing a $G$-homotopy
between $f$ and $g$ as maps $X \to S^2$.
It remains to compute the image $\Im(\mathcal{T})$. To simplify the
notation, set
\begin{align*}
p =
\begin{cases}
0 & \;\text{if $d_0 + d_1$ is even}\\
1 & \;\text{if $d_0 + d_1$ is odd}.
\end{cases}
\end{align*}
First we show that for any $d$ of the form $2k + p$ the triple
$\Triple{d_0}{d}{d_1}$ is contained in the image $\Im(\mathcal{T})$,
then we show that the assumption of the triple $\Triple{d_0}{2k + 1
+ p}{d_1}$ being contained in $\Im(\mathcal{T})$ leads to a
contradiction. To ease the notation, let $\sigma_j$ be the sign of
$d_j$, i.\,e. $d_j = \sigma_j|d_j|$, for $j=0,1$. By definition of
$p$ we know that $2k + p - (d_0 + d_1)$ is an even number. Hence the
triple
\begin{align*}
t = \Triple{0}{2k+p-(d_0+d_1)}{0}
\end{align*}
is contained in $\Im(\mathcal{T})$ by
proposition~\ref{TripleBuildingBlocks} (ii). We define the following
triples:
\begin{align*}
t_1 =& \Triple{\sigma_0}{\sigma_0}{0}\\
t_2 =&\Triple{2\sigma_0}{\sigma_0}{\sigma_0}\\
& \ldots\\
t_{|d_0|} =&\Triple{|d_0|\sigma_0}{\sigma_0}{|d_0|\sigma_0 - \sigma_0}\\
u_1 =& \Triple{0}{\sigma_1}{\sigma_1}\\
u_2 =& \Triple{\sigma_1}{\sigma_1}{2\sigma_1}\\
& \ldots\\
u_{|d_1|} =& \Triple{|d_1|\sigma_1 - \sigma_1}{\sigma_1}{|d_1|\sigma_1},
\end{align*}
which are contained in $\Im(\mathcal{T})$ by
proposition~\ref{TripleBuildingBlocks} (ii). From this we can form the
triple
\begin{align*}
\tilde{t} &= t_{|d_0|} \bullet \ldots \bullet t_2 \bullet t_1 \bullet t \bullet u_1 \bullet u_2 \bullet \ldots \bullet u_{|d_1|}\\
&= \Triple{d_0}{2k+p}{d_1}
\end{align*}
Therefore, the triple $\tilde{t}$ is contained in
$\Im(\mathcal{T})$. On the other hand, assume that the triple $t =
\Triple{d_0}{2k + 1 + p}{d_1}$ is contained in $\Im(\mathcal{T})$
for some $k$. As above, we define the following realizable triples
\begin{align*}
t_1 =& \Triple{0}{-\sigma_1}{\sigma_1}\\
t_2 =&\Triple{\sigma_1}{-\sigma_1}{2\sigma_1}\\
& \ldots\\
t_{|d_0|} =&\Triple{|d_0|\sigma_1 - \sigma_1}{-\sigma_1}{|d_0|\sigma_1}\\
u_1 =& \Triple{\sigma_2}{-\sigma_2}{0}\\
u_2 =& \Triple{2\sigma_2}{-\sigma_2}{\sigma_2}\\
& \ldots\\
u_{|d_1|} =& \Triple{|d_1|\sigma_2}{-\sigma_2}{|d_1|\sigma_2 - \sigma_2}.
\end{align*}
This allows us to form the following triple
\begin{align*}
\tilde{t} &= t_1 \bullet t_2 \bullet \ldots \bullet t_{|d_0|} \bullet t \bullet u_{|d_1|} \bullet \ldots \bullet u_2 \bullet u_1 \\
&= \Triple{0}{2k + 1 + p - (d_0 + d_1)}{0},
\end{align*}
which then also has to be in $\Im(\mathcal{T})$. But, by definition,
$p - (d_0 + d_1)$ is an even number, hence $2k + 1 + p - (d_0 +
d_1)$ is odd, contrary to
remark~\ref{RemarkTripleGlobalDegree}. Hence, the assumption that
$\Triple{d_0}{2k+1+p}{d_1}$ is in the image $\Im(\mathcal{T})$
cannot hold. \end{proof}
A consequence of the above is that we can identify equivariant homotopy classes with their associated degree triples. At this point we underline that the proof of theorem~\ref{Classification1} and lemma~\ref{HomotopyOnCylinder} shows that when two type I normalized $G$-maps $f, g\colon X \to S^2$ have the same degree triple, then they are not only equivariantly homotopic, but they are equivariantly homotopic rel $C = C_0 \cup C_1$. This will be of particular importance in the next section.
\subsubsection{Type II}
In this section, the torus $X$ -- still defined in terms of the standard square lattice -- is equipped with the type II involution \begin{align*}
T\colon X &\to X\\
[z] &\mapsto [i\overline{z}]. \end{align*} As before, we can regard the torus $X$ as a $G$-CW complex. The $G$-CW structure we use for the type II involution is depicted in figure~\ref{MSInvolutionFigureCW}. \label{ParagraphGeometryOfClass2} The first major difference when compared with type I is the fact that the fixed point set in the torus consists of a single circle $C$, not two. In the universal cover $\mathbb{C}$ this circle can be described as the diagonal line $x + ix$. Furthermore, the complement of the circle $C$ in the torus is still connected. The choice of ``fundamental region'' in this case is not completely obvious. Indeed, we need a new definition -- the notion of a fundamental region in the sense of definition~\ref{DefinitionFundamentalRegion} is not suitable for the type II involution: \begin{definition}
\label{PseudofundamentalRegion}
A connected subset $R \subset X$ is called a \emph{pseudofundamental
region} for the $G$-action on $X$ if there exists a subset $R'
\subset \partial R$ in the boundary of $R$ such that $R\smallsetminus R'$
is a fundamental region. \end{definition}
The idea behind the proof of the classification for type II is to identify a pseudofundamental region $R \subset X$, then bring the maps into a certain normal form such that we can collapse certain parts of $\partial R$ and the maps push down to this quotient. In the quotient, the geometry of the $T$-action is easier to understand and the image of $R$ in the quotient will be fundamental region for the induced $T$-action on the quotient.
\begin{figure}
\caption{Type II $G$-CW Setup of the Torus $X$.}
\label{MSInvolutionFigureCW}
\label{MSInvolutionFigureCWDelta}
\end{figure}
The pseudofundamental region we use for the type II $T$-action on $X$ is \begin{align*}
R = \left\{[x+iy]\colon 0 \leq x \leq 1 \text{ and } 0 \leq y \leq x\right\} \end{align*}
See figure~\ref{MSInvolutionFigureCWDelta} for a depiction of this fundamental region. Note that $T$-equivariance of a map requires the values of this map along the circles $A_1$ and $A_2$ to be compatible, because $T$ maps $A_1$ to $A_2$ and vice versa. To make this precise (and to ease the notation) we make the following definition. \begin{definition}
\label{Class2CompatibilityCondition}
For $f\colon X \to S^2$ resp. $f\colon R \to S^2$ we let
$f_{A_j}\colon I \to S^2$ ($j=1,2$) be the map
\begin{align*}
f_{A_j}\colon t \mapsto
\begin{cases}
f\left([t]\right) & \;\text{for $j=1$}\\
f\left([1 + it]\right) & \;\text{for $j=2$}.
\end{cases}
\end{align*}
For convenience we also set $A = A_1 \cup A_2$. \end{definition}
Then we can reformulate what it means for a map $f\colon X \to S^2$ to be equivariant: \begin{remark}
\label{RemarkClassIIEquivariance}
Let $F\colon X \to S^2$ be a $G$-map. Then its restriction $f =
\restr{F}{R}$ to $R$ satisfies
\begin{enumerate}[(i)]
\item $f(C) \subset E$ and
\item $f_{A_1} = T \circ f_{A_2}$.
\end{enumerate} \end{remark}
\begin{lemma}
\label{Class2EquivariantExtension}
Maps $R \to S^2$ resp. homotopies $I \times X \to S^2$ with the
above two properties extend uniquely to equivariant maps $X \to S^2$
resp. equivariant homotopies $I \times X \to S^2$. \end{lemma} \begin{proof}
This is basically the same proof as for
lemma~\ref{Class1EquivariantExtension}. We let $R'$ be the opposite
pseudofundamental region, which has the intersection $A_1 \cup A_2
\cup C$ with $R$. Then we define the extension of the map on $R'$ to
be the $T$-conjugate of the map on $R$. The compatibility condition
along $A_1 \cup A_2$ (from
definition~\ref{Class2CompatibilityCondition}) guarantees that the
resulting extension is globally well-defined on $X$. \end{proof}
\paragraph{Homotopy Invariant}
In the type I classification we have identified degree \emph{triples} as the desired (complete) homotopy invariant; in the type II case we need something slightly different. In order to obtain well-defined notion of fixed point degree we must choose an orientation on the fixed point circle $C \subset X$ (compare with p.~\pageref{CjS1Identification}). \label{CircleOrientationRemark} Although we assume for this paragraph that an orientation has already been chosen, we will wait until paragraph~\ref{RedcutionToClassI} (p.~\pageref{RedcutionToClassI}) before we fix this orientation on $C$.
\begin{definition}
\label{DefinitionDegreePair} (Degree pair map)
The \emph{degree pair map} is the map
\begin{align*}
\mathcal{P}\colon \mathcal{M}_G(X,S^2) &\to \mathbb{Z}^2\\
f &\mapsto \Pair{d_C}{d},
\end{align*}
where $d$ is the total degree of the map $f$ and $d_C$ is the
fixed point degree of $f$. For a map $f\colon X \to S^2$ we define its
\emph{degree pair} (or simply \emph{pair}) to be
$\mathcal{P}(f)$. We call a given pair \emph{realizable} if it is
contained in the image $\Im(\mathcal{P})$. \end{definition} We need to be careful to distinguish the \emph{degree pair} $\Pair{d_C}{d}$ of an equivariant map $X \to S^2$ (for the type II involution) from the \emph{fixed point degrees} $\Bideg{d_0}{d_1}$ of an equivariant map $X \to S^2$ (for the type I involution), but this should always be clear from the context. After having defined the degree pair, we immediately obtain: \begin{remark}
\label{DegreePairInvariant}
The degree pair is a $G$-homotopy invariant. \qed \end{remark}
\paragraph{Reduction to Type I} \label{RedcutionToClassI}
Although, at first glance, the type II involution appears to be quite different from the type I involution, it is nevertheless possible to use the results from the type I classification (section~\ref{SectionMapsToSphereClassI}).
\begin{definition}
\label{DefinitionClass2Normalized} (Type II normalization)
Let $f\colon X \to S^2$ be a $G$-map. We say $f$ is \emph{type II
normalized} if
\begin{enumerate}[(i)]
\item $f_{A_1} = f_{A_2} = c_{p_0}$, where
$c_{p_0}\colon X \to S^2$ denotes the constant map whose image is
$\{p_0\} \subset E$ and
\item $f$ is normalized on the circle $C$ in the sense that, using
the identifications of $C$ and $E$ with $S^1$, the map
$\restr{f}{C}$ corresponds to $z \mapsto z^k$ for some
$k$.
\end{enumerate} \end{definition}
\begin{proposition}
\label{Class2Normalization}
Any $G$-map $f\colon X \to S^2$ is $G$-homotopic to a map $f'$ which
is type II normalized. \end{proposition}
\begin{proof}
The statement is proved in several steps. Consider $X$ with the
$G$-CW-decomposition as in figure~\ref{MSInvolutionFigureCW}
consisting of the six cells $e^0$, $e^1_{1,2,3}$, and
$e^2_{1,2}$. Restrict $f$ to the pseudo-fundamental region $R$. Then
$A = A_1 \cup A_2$ is a subcomplex of $R$. We now successively apply
two homotopies to $f$. Observe that the maps $f_{A_j}$ ($j=1,2$) are
loops in $S^2$ at some point $p \in E$. Using the
simply-connectedness of $S^2$ there exists a null-homotopy
$\rho\colon I \times I \to S^2$ of $f_{A_1}$ to the constant curve
$c_p$. For the first homotopy, we define $H$ on $\{0\} \times R$ to
be the original map $f$ and on $I \times A_1$ resp. $I \times A_2$
we set
\begin{align*}
(t,s) \mapsto \rho(t,s) \;\;\text{resp.}\; (t,1+is) \mapsto T \circ \rho(t,s).
\end{align*}
By the homotopy extension property (HEP, see proposition~\ref{HEP})
this can be extended to a homotopy $H\colon I \times R \to S^2$ and
the resulting map $H(1,\cdot)$, which we denote by
$\smash{\tilde{f}}$, has the property that
$\smash{\tilde{f}}(A_j) = \{p\}$ ($j=1,2$).
Define the second homotopy as follows: On $\{0\} \times R$ we define
$H$ to be the map $\smash{\tilde{f}}$. On $I \times C$ we make a
homotopy $H_C\colon I \times C \to E$ to the normalized form $z
\mapsto z^k$. In particular, $e_0$ will then be mapped to $p_0 \in
E$. But when we change the map on $C$ we only need to adjust it
accordingly on $A$, hence we define the homotopy on $I \times A_1$
resp. $I \times A_2$ to be
\begin{align*}
(t,s) \mapsto H_C(t,0) \;\text{resp.}\; (t,1+is) \mapsto H_C(t,0).
\end{align*}
The map $H$ is then well-defined on $I \times e_0$ and extends by
the HEP to a homotopy $I \times R \to S^2$. By
lemma~\ref{Class2EquivariantExtension}, we can equivariantly extend
this homotopy to a homotopy $I \times X \to S^2$ and the resulting
map has the desired properties. \end{proof}
For the reduction to type I it will be very useful to replace the torus $X$ with the quotient $X/A$. This will be formalized next. With the above remark we can assume without loss of generality that any map in $\mathcal{M}_G(X,S^2)$ is type II normalized. Type II normalized maps $f\colon X \to S^2$ push down to the quotient $X/A \to S^2$. Let us denote the topological quotient $X \to X/A$ by $\pi_A$ and the image of $A$ under $\pi_A$ with $A/A$. In the next lemma we study the geometry of $X/A$. \begin{lemma}
\label{ClassIIQuotientSphereIdentification}
We state:
\begin{enumerate}[(i)]
\item The $T$-action on $X$ pushes down to the quotient.
\item The projection map $\pi_A\colon X \to X/A$ is equivariant.
\item The space $(X/A,T)$ can then be equivariantly identified with
$(S^2, T)$, where $T$ acts on $S^2$ as the standard reflection
along the equator.
\end{enumerate} \end{lemma}
\begin{proof}
Regarding (i): First we prove that the $T$-action on $X$ pushes down
to $X/A$: The $T$-action stabilizes $A \subset X$. Hence, $T$ is
compatible with the relation on $X$, which identifies the points in
$A$. By theorem~\ref{TopologyMapInducedOnQuotient} $T$ therefore
induces a map $X/A \to X/A$, which we, by abuse of notation, again
denote by $T$. Since $T\colon X \to X$ is an involutive
homeomorphism on $X$, the same applies to $T\colon X/A \to X/A$ (see
e.\,g. theorem~\ref{TopologyMapInducedOnQuotient}).
Regarding (ii): This is true by definition of the $T$-action on the
quotient: $T([x]) = [T(x)]$.
Regarding (iii): The space $X/A$ consists of two $2$-cells, glued
together along their boundary, which is stabilized by $T$. We can
define an equivariant homeomorphism to $S^2$ by mapping one of the
$2$-cells to e.\,g. the lower hemisphere of the sphere, mapping the
boundary of the $2$-cell to the equator. The map on the second cell
can then be defined by equivariant extension, thus mapping it to the
upper hemisphere. \end{proof}
Thus we can from now on identify $X/A$ with $S^2$ as topological manifolds and by this identification equip $X/A$ with the smooth structure from $S^2$. We denote the point in $S^2$ corresponding to $A/A \in X/A$ by $P_0$. The projection map $\pi_A$ induces an equivariant map \begin{align}
\label{TypeIIDiscussionS2}
\pi_{S^2}\colon X \to S^2, \end{align} whose restriction $X\smallsetminus A \to S^2\smallsetminus\{P_0\}$ is smooth. The degree of $\pi_{S^2}$ is either $+1$ or $-1$ but we can assume that it is $+1$ (if it were $-1$, we could compose with a reflection along the equator in $S^2$). We must be careful to not confuse the 2-sphere introduced in (\ref{TypeIIDiscussionS2}) with the 2-sphere which has been identified as a deformation retract of $\mathcal{H}_{(1,1)}$. In particular their equators must be considered as distinct objects. Thus we denote the equator of the 2-sphere which appears as a quotient of the torus $X$ by $E'$.
Now, let $(X',T)$ be a type I torus, that is, the torus $\mathbb{C}/\Lambda$ equipped with the type I involution $T$ and denote the north resp. the south pole of $S^2$ by $O_\pm$. Observe that also the sets $X'\smallsetminus C_0$ and $S^2\smallsetminus\{O_\pm\}$ are $G$-spaces. The former is an open cylinder and the latter is a doubly punctured 2-sphere. We note: \begin{remark}
\label{Class1TorusWithSphereIdentification}
There exists a smooth and orientation preserving $G$-diffeomorphism
\begin{align*}
\Psi\colon X'\smallsetminus C_0 \xrightarrow{\;\sim\;} S^2\smallsetminus \{O_\pm\}.
\end{align*} \end{remark}
\begin{proof}
Regard $S^2$ as being embedded as the unit-sphere in $\mathbb{R}^3$. Then we
define the map $\Psi$ as follows:
\begin{align*}
\Psi\colon X'\smallsetminus C_0 &\xrightarrow{\;\sim\;} S^2\smallsetminus \{O_\pm\}\\
[x+iy] &\mapsto
\begin{pmatrix}
\sqrt{1-(2y-1)^2} \cos(2\pi x) \\
\sqrt{1-(2y-1)^2} \sin(2\pi x) \\
2y-1
\end{pmatrix}
\end{align*}
This defines an equivariant and orientation preserving
diffeomorphism. \end{proof}
On p.~\pageref{CircleOrientationRemark} we mentioned that we still need to decide for an orientation on the circle $C \subset X$, which is equivalent to chosing an identification of $C$ with $S^1$ -- we will do this now. Observe that the restriction of the above map $\Psi$ to $C_1 \subset X'$ identifies the circle $C_1$ with the equator $E'$ in $S^2$, which the map $\pi_{S^2}$ identifies with with $C \subset X$. We define the identifications of $C \subset X$ and $E' \subset S^2$ with $S^1$ in terms of this identification with $C_1 \subset X'$. Analogously to definition~\ref{DefinitionClass2Normalized} we make the following \begin{definition}
We call a $G$-map $f\colon S^2 \to S^2$ \emph{type II
normalized}\footnote{as a map defined on the 2-sphere, not on the
type II torus $X$}, if the north pole $O_+$ and the south pole
$O_-$ both are mapped to $p_0 \in E \subset S^2$ and the map is
normalized on the equator $E'$ according to the above identification
of $E'$ with $S^1$. \end{definition}
\begin{remark}
Any $G$-map $S^2 \to S^2$ is equivariantly homotopic to a type II
normalized $G$-map. \end{remark}
\begin{proof}
This follows with the equivariant homotopy extension
(corollary~\ref{G-HEP}): There exists a $G$-CW decomposition of
$S^2$ such that the set $B = \{O_\pm\} \cup E'$ is a $G$-CW
subcomplex of $S^2$. \end{proof}
To summarize the above discussion: Above we have constructed a map from the set of type II normalized $G$-maps $X \to S^2$ to the set of type II normalized $G$-maps $S^2 \to S^2$. We denote this map by $\psi$ and state: \begin{proposition}
\label{Class2CorrespondenceXS2}
The map $\psi$ is a bijection. Furthermore, the degree pairs of $f$
and $\psi(f)$ agree. \end{proposition}
\begin{proof}
Let us begin by reviewing how the map $\psi$ is defined: If $f\colon
X \to S^2$ is a type II normalized map, it is by definition constant
along $A$. Hence it pushes down to a $G$-map $f\colon X/A \to
S^2$. Now $X/A$ can be equivariantly identified with $S^2$, hence,
by means of this identification, $f$ defines a $G$-map $S^2 \to
S^2$. By construction these two maps are related by the following
diagram:
\[
\begin{xy}
\xymatrix{
X \ar[d]_{\pi_{S^2}} \ar[dr]^f \\
S^2 \ar[r]_{f'} & S^2
}
\end{xy}
\]
Injectivity of this map is clear: If $\psi(f) = \psi(g)$, then also
\begin{align*}
f = \psi(f) \circ \pi_{S^2} = \psi(g) \circ \pi_{S^2} = g.
\end{align*}
For the surjectivity, let $f'\colon S^2 \to S^2$ be a type II
normalized $G$-map, then $f = f' \circ \pi_{S^2}\colon X \to S^2$ is a
type II normalized $G$-map such that $\psi(f) = f'$.
Since $\pi_{S^2}$ has degree $+1$ and using the functorial property
of homology we see that the total degrees of $f$ and $f'$ must
agree. Regarding the fixed point degree: $\pi_{S^2}$ homeomorphically
maps the circle $C \subset X$ to the equator $E \subset S^2$. Now
the only possibility would be that the fixed point degrees differ by a
sign. But, by definition, the identifications of $C \subset X$ and
$E \subset S^2$ with $S^1$ differ only by $\pi_{S^2}$ and above the
orientation of $C \subset X$ has been chosen such that
$\restr{\pi_{S^2}}{C}$ preserves the orientation. \end{proof}
This correspondence between type II normalized $G$-maps $X \to S^2$ and type II normalized $G$-maps $S^2 \to S^2$ also implies a statement on the level of equivariant homotopy: \begin{remark}
\label{Class2HomotopiesLiftFromSphere}
Given two type II normalized $G$-maps $f,g\colon X \to S^2$ and an
equivariant homotopy $H\colon S^2 \to S^2$ between the induced maps
$f',g'\colon S^2 \to S^2$, then $f$ and $g$ are also equivariantly
homotopic. \end{remark}
\begin{proof}
Compose the homotopy with the map $\mathrm{id} \times \pi_{S^2}$. \end{proof}
We now describe a basic correspondence between type II normalized $G$-maps $S^2 \to S^2$ and type I normalized $G$-maps $X' \to S^2$. Assume $f$ is a type II normalized $G$-map $S^2 \to S^2$. This induces a a map \begin{align*}
\phi(f)\colon X' &\to S^2\\
p &\mapsto
\begin{cases}
f \circ \Psi (p) & \;\text{if $p \not\in C_0$}\\
p_0 & \;\text{if $p \in C_0$}
\end{cases} \end{align*} Since $f$ is type II normalized, i.\,e. $f(O_\pm) = p_0$, the map $\phi(f)$ is continuous everywhere and type I normalized as a map $X' \to S^2$. By construction $f$ and $\phi(f)$ are related by \begin{align}
\label{PsiCorrespondence}
\Restr{f}{S^2\smallsetminus\{O_\pm\}} \circ \Psi = \Restr{\phi(f)}{X'\smallsetminus
C_0}, \end{align} where $\Psi$ is the orientation preserving diffeomorphism from remark~\ref{Class1TorusWithSphereIdentification}. The reduction to type I is now summarized in the following two statements: \begin{proposition}
\label{LemmaClass2ReductionToClass1}
The above map $\psi$ defines a bijection between type II normalized
$G$-maps $S^2 \to S^2$ and type I normalized $G$-maps $X' \to S^2$ whose
fixed point degree along the circle $C_0$ is zero. For a type II
normalized $G$-map $f$, the degree pair of $f$ is $\Pair{d_C}{d}$
iff the degree triple of $\phi(f)$ is $\Triple{0}{d}{d_C}$. \end{proposition}
\begin{proof}
Regarding the injectivity: Let $f$ and $g$ be two type II normalized
$G$-maps $S^2 \to S^2$ such that $\phi(f) = \phi(g)$. By
(\ref{PsiCorrespondence}) it follows that
$\restr{f}{S^2\smallsetminus\{O_\pm\}} =
\restr{g}{S^2\smallsetminus\{O_\pm\}}$. Since $f$ and $g$ are assumed to
be type II normalized, hence in particular continuous, this implies
that $f = g$. Regarding the surjectivity: Let $f'$ be a type I
normalized $G$-map $X' \to S^2$ with fixed point degree
$\Bideg{0}{d_1}$. Then (\ref{PsiCorrespondence}) defines a map
$f\colon S^2\smallsetminus\{O_\pm\} \to S^2$ which, by defining $f(O_\pm) =
p_0$, uniquely extends to a type II normalized $G$-map $S^2 \to
S^2$. By construction we obtain $\phi(f) = f'$.
Regarding the correspondence between the degree invariants: Assume
we are given a $G$-map $f\colon S^2 \to S^2$ with degree pair
$\Pair{d_C}{d}$. By corollary~\ref{SmoothApproximation2}, $f$ can be
equivariantly smoothed. Note that the map $\Psi$ in
(\ref{PsiCorrespondence}) is an orientation preserving
diffeomorphism. This implies that $\phi(f)$ is smooth, at least away
from the boundary circle $C_0$. Using
theorem~\ref{SmoothApproximation1} we can smooth $\phi(f)$ near the
circle $C_0$ while keeping it unchanged away from $C_0$. Now, let
$q$ be a regular value of $f$ such that its preimage points
$p_1,\ldots,p_m$ are contained in $S^2\smallsetminus\{O_\pm\}$. Then $q$
is also a regular of $\phi(f)$ and its preimage points are
$\Psi^{-1}(p_j)$, $j=1,\ldots,m$. Since $\Psi$ preserves the
orientation, $\deg f = \deg \phi(f)$. By definition of the
identification of $E \subset S^2$ with $S^1$ (through $\Psi$), the
fixed point degree $d_C$ remains unchanged as well. Finally, by
construction, the fixed point degree of $\phi(f)$ along $C_0$ is zero. \end{proof}
Now we can show that equivariant homotopies for maps $X' \to S^2$ induce equivariant homotopies for maps $S^2 \to S^2$: \begin{proposition}
\label{LemmaClass2ReductionToClass1Homotopy}
Given two type II normalized $G$-maps $f,g\colon S^2 \to S^2$ with
the same degree pair, then they are $G$-homotopic rel $\{O_\pm\}$. \end{proposition}
\begin{proof}
Without loss of generality we can assume that $f$ and $g$ are both
type II normalized $G$-maps $S^2 \to S^2$ with the same degree pair.
Then, by proposition~\ref{LemmaClass2ReductionToClass1}, the induced
maps $\phi(f),\phi(g)\colon X' \to S^2$ share the same degree
triple. Hence there exists a $G$-homotopy $H'$ from $\phi(f)$ to
$\phi(g)$ which is in particular relative with respect to the
boundary circle $C_0$. In other words, throughout the homotopy, $H'$
maps the circle
$C_0$ to $p_0 \in E \subset S^2$. This homotopy then induces a an
equivariant homotopy $H$ between $f$ and $g$:
\begin{align*}
H\colon I \times S^2 &\to S^2\\
(t,p) &\to
\begin{cases}
H'(t,\Psi^{-1}(p)) & \;\text{if $p \not\in \{O_\pm\}$}\\
p_0 & \;\text{if $p \in \{O_\pm\}$}
\end{cases}
\end{align*}
Hence, $f$ and $g$ are equivariantly homotopic rel $\{O_\pm\}$. \end{proof}
\paragraph{Main Result}
The main result in this section is the following \begin{theorem}
\label{Classification2}
The $G$-homotopy class of a map $f \in \mathcal{M}_G(X,S^2)$ is
uniquely determined by its degree pair $\mathcal{P}(f)$. The image
$\Im(\mathcal{P})$ of the degree pair map
$\mathcal{P}\colon \mathcal{M}_G(X,S^2) \to \mathbb{Z}^2$ consists of those
pairs $\Pair{d_C}{d}$ satisfying
\begin{align*}
d \equiv d_C \mod 2.
\end{align*} \end{theorem}
\begin{proof}
First we prove that the degree pairs $\Pair{d_C}{d}$ are a complete
invariant of equivariant homotopy for maps $X \to S^2$. For this,
let $f$ and $g$ be two $G$-maps $X \to S^2$ with the same degree
pair $\Pair{d_C}{d}$. Without loss of generality we can assume that
$f$ and $g$ are type II normalized
(Proposition~\ref{Class2Normalization}). In this case, by
proposition~\ref{Class2CorrespondenceXS2}, $f$ and $g$ push down to
type II normalized maps $f',g'\colon S^2 \to S^2$ with the same
degree pair. With
proposition~\ref{LemmaClass2ReductionToClass1Homotopy} it now
follows that $f'$ and $g'$ are equivariantly homotopic. Then
remark~\ref{Class2HomotopiesLiftFromSphere} establishes the
existence of an equivariant homotopy from $f$ to $g$.
Regarding the image of $\Im(\mathcal{P})$: Let $d_C$ be an
integer. We now have to compute all possible integers $d$ such that
the degree pair $\Pair{d_C}{d}$ is contained in the
$\Im(\mathcal{P})$. Let us first deduce a necessery condition for
the given degree pair to be in the image $\Im(\mathcal{P})$. For
this, let $f\colon X \to S^2$ be a $G$-map with the aforementioned
degree pair. By proposition~\ref{Class2CorrespondenceXS2} we obtain an
induced $G$-map $f'\colon S^2 \to S^2$ with the same degree pair. Now
proposition~\ref{LemmaClass2ReductionToClass1} implies the existence
of an induced map $\phi(f)\colon X' \to S^2$ with the degree triple
$\Triple{0}{d}{d_C}$. By theorem~\ref{Classification1}, we must have
$d \equiv d_C \mod 2$. This shows that the condition is necessary
for a pair to be contained in $\Im(\mathcal{P})$; what remains to
show is that this condition is also sufficient. Let $f''\colon X'
\to S^2$ be a $G$-map with degree triple $\Triple{0}{d}{d_C}$. By
proposition~\ref{LemmaClass2ReductionToClass1} there exists a
$G$-map $f'\colon S^2 \to S^2$ with the degree pair $\Pair{d_C}{d}$
and by proposition~\ref{Class2CorrespondenceXS2} this induces a
$G$-map $f\colon X \to S^2$ with the same degree pair. This proves the
statement. \end{proof}
\subsection{Classification of Maps to $\mathcal{H}_2^*$} \label{SectionHamiltonianClassificationRank2}
Note that so far the degree triple map (resp. the degree pair map) has only been defined on $\mathcal{M}_G(X,S^2)$. But since $S^2$ has been shown to be an equivariant strong deformation retract, we can also define the degree triple map (resp. degree pair map) on $\mathcal{M}_G(X,\mathcal{H}_{(1,1)})$. In particular this allows us to speak of degree triples (resp. of degree pairs) of $G$-maps $X \to \mathcal{H}_{(1,1)}$. Now we state and prove the main result of this section:
\HamiltonianClassificationRankTwo \begin{proof}
\label{HamiltonianClassificationRankTwoProof}
The fact that the cases $\text{sig}\, = (2,0)$ and $\text{sig}\, = (0,2)$ constitute
$G$-homotopy classes on their own has already been shown at the
beginning of this chapter
(remark~\ref{Rank2DefiniteComponentsContractible}). Regarding the
case $\text{sig}\, = (1,1)$: By remark~\ref{Rank2MixedSignatureReduction},
$\mathcal{H}_{(1,1)}$ has $S^2$ as equivariant, strong deformation
retract. Therefore,
\begin{align*}
[X,\mathcal{H}_{(1,1)}]_G \cong [X,S^2]_G.
\end{align*}
Then theorem~\ref{Classification1} (for the type I involution) and
theorem~\ref{Classification2} (for the type II involution) complete
the proof. \end{proof}
\subsection{An Example from Complex Analysis}
The Weierstrass $\wp$-function is a meromorphic and doubly-periodic function on the complex plane associated to a lattice $\Lambda$. It can be defined as follows: \begin{align*}
\wp(z) = \frac{1}{z^2} + \sum_{\lambda \in \Lambda\smallsetminus\{0\}} \left(\frac{1}{(z-\lambda)^2} - \frac{1}{\lambda^2}\right). \end{align*} This function can be regarded as a holomorphic map to $\mathbb{P}_1$. Since the lattice $\Lambda$ is stable under complex conjugation, we have the identity \begin{align*}
\wp(\overline{z}) = \overline{\wp(z)}. \end{align*} By the orientation-preserving idenfication $\mathbb{P}_1 \cong S^2$ (see p.~\ref{P1S2OrientationDiscussion})we can regard the map $\wp$ as an equivariant map $X \to S^2$. Since its total degree as a map to $\mathbb{P}_1$ is two, its degree as a map to $S^2$ is also two. Analogously, the derivative $\wp'$ defines an equivariant map $X \to \mathbb{P}_1 \cong S^2$ of degree three. Furthermore we state: \begin{remark}
\label{BidegreeOfWP}
The Weierstrass $\wp$-function (resp. $\wp'$), regarded as a map
\begin{align*}
X \to \mathbb{P}_1 \cong S^2 \hookrightarrow \mathcal{H}_{(1,1)}
\end{align*}
has fixed point degrees $\Bideg{0}{0}$ (resp. $\Bideg{1}{0}$). \end{remark} \begin{proof}
Regarding $\wp$: The restriction $\restr{\wp}{C_0}$ to be boundary
circle $C_0$ is not surjective to the compactified real line
$\smash{\widehat{\mathbb{R}}}$, since in the square lattice case, the only zero of
$\wp(z)$ is at the point $z=\text{\textonehalf}(1+i)$ (see
e.\,g. \cite{EichlerZagier}). Similarly, the restriction of $\wp$
to the boundary circle is not surjective to $\mathbb{RP}_1$, because the
only pole of $\wp$ is at $z = 0$ (of order 2). This proves that the
fixed point degrees are both zero.
Regarding $\wp'$: Since $\wp'(z)$ has its only pole at $z=0$, its
restriction to the circle $C_1 = I + \text{\textonehalf}i$ cannot be
surjective to $\smash{\widehat{\mathbb{R}}}$, therefore its fixed point
degree $d_1$ must be zero. But its fixed point degree $d_0$ is
$\pm 1$: It is known that the only pole of $\wp'(z_0)$ is at $z=0$
and its only zero in $C_0$ is at $z=\text{\textonehalf}$. A
computation shows that $\wp'$ is negative along
$(0,\text{\textonehalf})$. In other words, along the curve segment
$(0,\text{\textonehalf})$, the image under $\wp'$ moves from
$\infty$ to the negative real numbers and finally to $0$. By the
identity
\begin{align*}
\wp'(-z) = -\wp'(z),
\end{align*}
it follows that on the curve segment $(\text{\textonehalf})$ the
image under $\wp'$ moves from $0$ to the positive real numbers until
it finally reaches $\infty$. By definition of the orientation on
$\smash{\widehat{\mathbb{R}}}$ (see the discussion on
p.~\ref{P1S2OrientationDiscussion}), this is a loop of degree $+1$. \end{proof}
For the type II involution we consider the scaled Weierstrass functions \begin{align*}
&F = i\wp\\
\text{and}\; &G = e^{i\frac{3\pi}{4}} \wp'. \end{align*} As above, they can be considered as equivariant maps $X \to S^2$. \begin{remark}
The fixed point degree of $F$ is $0$ and that of $G$ is $1$. \end{remark} \begin{proof}
Regarding $F$: The map $F(z)$ has its only pole at $z=0$ and its
only zero at $\text{\textonehalf}(1+i)$. Thus, when $z$ linearly
moves from the origin to the point $\text{\textonehalf}(1+i)$, then
its image under $F$ defines a curve from $\infty$ to $0$. Since $F$
is an even function it follows that on the second segment of the
diagonal circle, $F$ reverses the previous curve, going back from
$0$ to $\infty$. Thus the fixed point degree for $F$ is $0$.
Regarding $G$: Along the curve segment from $0$ to
$\text{\textonehalf}(1+i)$ it defines a curve from $\infty$ to
$0$. A computation shows that this curve is along the negative real
numbers. Since $G$ is not an odd map, its restriction to the segment
from $\text{\textonehalf}(1+i)$ to $1+i$ defines a curve from $0$ to
$\infty$, but with the opposite sign, i.\,e. along the positive real
numbers. This defines a degree $+1$ loop. \end{proof} To summarize the above: \begin{remark}
With respect to the identification $\mathbb{P}_1 \cong S^2$ described on
p.~\ref{P1S2OrientationDiscussion}, we have:
\begin{align*}
&\mathcal{T}(\wp) = \Triple{0}{2}{0}\\
&\mathcal{T}(\wp') = \Triple{1}{3}{0}\\
&\mathcal{P}(i\wp) = \Pair{0}{2}\\
&\mathcal{P}(e^{i\frac{3\pi}{4}} \wp') = \Pair{1}{3}.
\end{align*} \end{remark}
By choosing a different equivariant identification $\mathbb{P}_1 \cong S^2$, we might introduce signs for the total degrees or for the fixed point degrees or for both. For instance, we can always compose the $G$-diffeomorphism $\mathbb{P}_1 \to S^2$ with reflections on $S^2$. Thus, in some sense, the numbers in the above remark are only well-defined up to sign.
\section{Maps to $\mathcal{H}_n$}
In this chapter we generalize the results of the previous section to arbitrary $n > 2$. For this we begin by noting that the unitary group $U(n)$ acts on $\mathcal{H}_{(p,q)}$ by conjugation. Since every matrix in $\mathcal{H}_{(p,q)}$ can be diagonalized by unitary matrices, the set of $U(n)$-orbits in $\mathcal{H}_{(p,q)}$ is parametrized by the set of (unordered) real eigenvalues $\lambda_1^+, \ldots, \lambda_p^+, \lambda_1^-,\ldots, \lambda^-_q$. The involution $T$ stabilizes the components $\mathcal{H}_{(p,q)}$, since the matrices $H$ and $T(H)$ clearly have the same spectrum. The $T$-action is also compatible with the $U(n)$-orbit structure of $\mathcal{H}_{(p,q)}$, as shown in the next remark: \begin{remark}
\label{RemarkRealStructureRestrictsToEachOrbit}
The $T$-action on $\mathcal{H}_n$ stabilizes each $U(n)$-orbit in
each $\mathcal{H}_{(p,q)}$. \end{remark} \begin{proof}
Let $D = \text{Diag}\,(\lambda_1,\ldots,\lambda_n)$ be a diagonal matrix in
$\mathcal{H}_{(p,q)}$. In particular all $\lambda_j$ are non-zero
real numbers. Let $\mathcal{O}$ be the $U(n)$-orbit of $D$. We have
to show that $\mathcal{O}$ is $T$-stable. For this, let $M$ be a
matrix in $\mathcal{O}$. By definition it is of the form $M = UDU^*$
for some $U \in U(n)$. Then we have
\begin{align*}
T(UDU^*) = \overline{UDU^*} = \overline{U}D\overline{U}^*.
\end{align*}
But since $U \in U(n)$ implies $\overline{U} \in U(n)$, we can
conclude that $T(UDU^*)$ is again contained in the orbit
$\mathcal{O}$. \end{proof}
In each component $\mathcal{H}_{(p,q)}$ there is a $U(n)$-orbit which is particularly convenient to work with: \begin{remark}
\label{MinimalOrbitInComponent}
The $U(n)$-orbit of the block diagonal matrix
\begin{align*}
\I{p}{q} = \begin{pmatrix}
\I{p} & 0 \\
0 & -\I{q}
\end{pmatrix}
\end{align*}
is diffeomorphic to the Grassmannian $\text{Gr}_p(\mathbb{C}^n)$. \end{remark}
\begin{proof}
The $U(n)$-isotropy of $\I{p}{q}$ is $U(p) \times U(q)$. Hence, for
the $U(n)$-orbit we have the following smooth identification:
\begin{equation*}
U(n).\I{p}{q} \cong \frac{U(n)}{U(p) \times U(q)} \cong \text{Gr}_p(\mathbb{C}^n)\qedhere
\end{equation*} \end{proof}
In order to reduce the classification of maps to $\mathcal{H}_{(p,q)}$ to the classification of maps to this Grassmann manifold, it is important to understand the $T$-action on the latter. For this we make the following remark: \begin{remark}
\label{InducedActionOnGrassmannian}
The induced $T$-action on $\text{Gr}_p(\mathbb{C}^n)$ is given by
\begin{align*}
V \mapsto \overline{V},
\end{align*}
for each $p$-dimensional subvector space $V \subset \mathbb{C}^n$. \end{remark}
\begin{proof}
As a first step we compute the induced $T$-action under the
identification
\begin{align*}
\frac{U(n)}{U(p) \times U(q)} &\xrightarrow{\sim} U(\I{p}{q}) \subset \mathcal{H}_{(p,q)}\\
[U] &\mapsto U\I{p}{q}U^*.
\end{align*}
Now, let $[U]$ be a point in the quotient. The above isomorphism
sends $[U]$ to $U \I{p}{q} U^*$. The $T$-action on $\mathcal{H}_n$
maps this to
$\smash{\overline{U \I{p}{q} U^*} = \overline{U} \I{p}{q}
\overline{U}^*}$,
which under the above isomorphism, corresponds to the point
$\smash{\left[\overline{U}\right]}$ in the quotient. Thus, the
induced $T$-action on the quotient is via
$\smash{T\left([U]\right) = \left[\overline{U}\right]}$. For the
next step we need to use the isomorphism
\begin{align*}
\frac{U(n)}{U(p) \times U(q)} &\xrightarrow{\;\sim\;} \text{Gr}_p(\mathbb{C}^n)\\
[U] &\mapsto U(V_0),
\end{align*}
where $V_0$ denotes a fixed base point in the Grassmannian,
e.\,g. the subvector space spanned by $e_1,\ldots,e_p$. Now, let $V
\in \text{Gr}_p(\mathbb{C}^n)$ be a subvector space of $\mathbb{C}^n$
generated by the vectors $v_1,\ldots,v_p$. This subspace uniquely
corresponds to a coset $[U]$ such that $U(V_0) = V$. Recalling that
$V_0$ is by definition generated by the standard basis
$e_1,\ldots,e_p$, we see that the first $p$ coloumns
$u_1,\ldots,u_p$ of $U$ constitute a unitary basis of the
subvector space $V$. The $T$-action on the quotient maps $[U]$ to
$\smash{\left[\overline{U}\right]}$, which then corresponds to the
subvector space $\smash{\overline{U}}(V_0)$. This subvector space can be
described in terms of the basis
\begin{align*}
\overline{U}(e_1),\ldots,\overline{U}(e_p),
\end{align*}
which is just $\overline{u_1},\ldots,\overline{u_p}$. Therefore,
the induced $T$-action on $\text{Gr}_p(\mathbb{C}^n)$ is
\begin{align*}
V \mapsto \overline{V},
\end{align*}
which finishes the proof. \end{proof}
As a special case we also note: \begin{remark}
\label{InducedActionOnProjectiveSpace}
The induced action on $\mathbb{P}_n = \text{Gr}_1(\mathbb{C}^{n+1})$ in terms
of homogeneous coordinates is given by
\begin{align*}
[z_0:\ldots:z_n] \mapsto \left[\overline{z_0}:\ldots:\overline{z_n}\right].
\end{align*} \end{remark}
\begin{proof}
Follows from remark~\ref{InducedActionOnGrassmannian} together with
the fact that the homogeneous coordinates for a given line $L =
\mathbb{C}(\ell_1,\ldots,\ell_{n+1})$ in $\mathbb{C}^{n+1}$ are
$[\ell_1:\ldots:\ell_{n+1}]$. \end{proof}
\begin{remark}
Regard the Grassmannian $\text{Gr}_k(\mathbb{C}^n)$ as being equipped with the
standard real structure given by complex conjugation. If $T$ is a
real structure on the $n$-dimensional vectorspace $W$, then there is
a biholomorphic $G$-map
\begin{align*}
\text{Gr}_k(W) \xrightarrow{\;\sim\;} \text{Gr}_k(\mathbb{C}^n).
\end{align*} \end{remark}
\begin{proof}
The real structure $T$ on $W$ induces a decomposition of $W$ as $W =
W_\mathbb{R} \oplus iW_\mathbb{R}$, where $W_\mathbb{R}$ is an $n$-dimensional, real
subvector space of $W$. The vectorspace $W$ can be identified with
$\mathbb{C}^n$ by sending an orthonormal basis $w_1,\ldots,w_n$ of $W_\mathbb{R}$ to
the standard basis $e_1,\ldots,e_n$. This is equivariant by
construction and induces an equivariant biholomorphism $\text{Gr}_k(W) \to
\text{Gr}_k(\mathbb{C}^n)$. \end{proof}
Let us now look at the topology of the two connected components $\mathcal{H}_{(n,0)}$ and $\mathcal{H}_{(0,n)}$ of $\mathcal{H}_n^*$: \begin{remark}
\label{RemarkDefiniteComponentsRetractable}
The component $\mathcal{H}_{(n,0)}$ (resp. $\mathcal{H}_{(0,n)}$)
has $\{\I{n}\}$ (resp. $\{-\I{n}\}$) as a strong
equivariant deformation retract. \end{remark}
\begin{proof}
We only prove the statement for the component $\mathcal{H}_{(n,0)}$.
For this we define:
\begin{align*}
\rho\colon I \times \mathcal{H}_{(n,0)} &\to \mathcal{H}_{(n,0)}\\
(t,A) &\mapsto (1-t)A + t\I{n}
\end{align*}
Clearly, for $t=0$ this is the identity on $\mathcal{H}_{(n,0)}$ and
for $t=1$ it is constant at the identity $\I{n}$. Hence, in
order to show that this is a well-defined deformation retract, we
have to prove that there exists no matrix $A \in
\mathcal{H}_{(n,0)}$ such that $\rho_t(A)$ is singular for some $t
\in (0,1)$.
Let $A$ be a positive definite matrix in $\mathcal{H}_n$. By
assumption we know that $\det A$, which coincides with the product
of all eigenvalues of $A$, is positive. Let us assume
that $\det \rho_t(A) = 0$. By definition this means
\begin{align}
\label{PositiveDefiniteRetraction}
\det ((1-t)A + t\I{n}) = 0.
\end{align}
This means that $-t$ is an eigenvalue of $A' := (1-t)A$. But $A'$ is
just $A$, scaled by a positive real number. Hence $A'$ is also
positive definite and therefore cannot have a negative eigenvalue
$-t$. Thus we obtain a contradiction, which proves that $\rho_t$
really has its image contained in $\mathcal{H}_{(n,0)}$. The proof
for the positive negative case works completely analogously.
What is now left to show is equivariance of the above deformation
retract. But this follows from the fact that the involution on
$\mathcal{H}_n$ acts by conjugation and this is compatible with
scaling by real numbers as done in
(\ref{PositiveDefiniteRetraction}). \end{proof}
This shows that the components $\mathcal{H}_{(n,0)}$ and $\mathcal{H}_{(0,n)}$ are equivariantly contractible to a point, thus the spaces $[X,\mathcal{H}_{(n,0)}]_G$ and $[X,\mathcal{H}_{(0,n)}]_G$ are trivial. On the other hand, as we will see in the following, interesting topology will occur in the case of mixed signatures: For $0 < p,q < n$ the components $\mathcal{H}_{(p,q)}$ have the $U(n)$-orbit of $\I{p}{q}$ as equivariant strong deformation retract\footnote{Here $\I{p}{q}$ denotes the block diagonal matrix
with $\I{p}$ in the upper left corner and $-\I{q}$ in the lower
right corner.}. Proving this requires some preperations; we begin by introducing the following surjection: \begin{align*}
\pi\colon \mathcal{H}_{(p,q)} &\to \text{Gr}_p(\mathbb{C}^n)\\
H &\mapsto E^+(H), \end{align*} where $E^+(H)$ denotes the direct sum of the positive eigenspaces of $H$. This is a $p$-dimensional subspace of $\mathbb{C}^n$. Fixing a basepoint $E_0 = \left<e_1,\ldots,e_p\right>$, this defines a fiber bundle with neutral fiber $F = \pi^{-1}(\{E_0\})$. The fiber consists of those matrices in $\mathcal{H}_{(p,q)}$ which have $E_0$ as the direct sum of their positive eigenspaces. Note that $K := U(n)$ acts transitively on $\text{Gr}_p(\mathbb{C}^n)$. Thus, denoting the stabilizer of $E_0$ in $K$ with $L$, we can identify $\text{Gr}_p(\mathbb{C}^n)$ with $K/L$. We equip the product $K \times F$ with a $K$-action given by multiplication on the first factor: \begin{align*}
k(k',H) \mapsto (kk', H). \end{align*} The $L$-action on $K \times F$ is given by \begin{align*}
\ell(k,H) = (k\ell,\ell^*H\ell). \end{align*} It induces the quotient $K \times F \to K \times_L F$. The $K$-action on $K \times_L F$ is then given by \begin{align*}
k[(k',H)] = [k(k',H)] = [(kk',H)]. \end{align*} The quotient $K \times_L F \to K/L$ is $K$-equivariant by definition of the respective $K$-actions. We obtain the following diagram of $K$-spaces: \[ \xymatrix{
K \times F \ar[r] \ar[d] & K \times_L F \ar[d]\\
K \ar[r] & K/L } \]
After these remarks we state: \begin{proposition}
\label{GrassmannianStrongDeformationRetract}
The component $\mathcal{H}_{(p,q)}$ has the orbit $U(n).\I{p}{q}$ as
$T$-equivariant, strong deformation retract. \end{proposition} \begin{proof}
The idea of this proof is to use the above diagram and define a
strong deformation retract of the fiber $F$ which we then globalize
to a strong deformation retraction on $K \times_L F$. Using the
identifications $K \times_L F \cong \mathcal{H}_{(p,q)}$ and $K/L
\cong \text{Gr}_p(\mathbb{C}^n)$ this defines a $T$-equivariant strong deformation
retract from $\mathcal{H}_{(p,q)}$ to $U(n).\I{p}{q} \subset
\mathcal{H}_{(p,q)}$ (see figure~\ref{fig:UnOrbitsInHpq}).
Note that we have the following isomorphism
\begin{align}
\label{EquivariantFiberBundleIsoPsi}
\Psi\colon K \times_L F &\xrightarrow{\;\sim\;} \mathcal{H}_{(p,q)}\\
\left[(k,H)\right] &\mapsto kHk^*.\nonumber
\end{align}
Set $\Sigma = K.\I{p}{q} \subset \mathcal{H}_{(p,q)}$. This defines
(the image of) a global section of the bundle $K \times_L F \to
K/L$. Observe that a matrix $H \in F$ is, by definition,
positive-definite on $E_0 = \left<e_1,\ldots,e_p\right>$ and
therefore negative-definite on $E_0^\perp =
\smash{\left<e_{p+1},\ldots,e_{p+q}\right>}$\footnote{On $\mathbb{C}^n$ we use the
standard unitary structure.}. This implies that $H$ is of the form
\begin{align*}
H =
\begin{pmatrix}
H_p & 0 \\
0 & H_q
\end{pmatrix},
\end{align*}
where $H_p$ and $H_q$ are both hermitian, $H_p$ is positive-definite
on $E_0$ and $H_q$ is negative-definite on $E_0^\perp$. Now we can
define a retraction $\rho\colon I \times F \to F$ of the neutral
fiber:
\begin{align}
\label{EquationRetractRho}
\rho_t(H) =
\begin{pmatrix}
(1-t)H_p + t\I{p} & 0 \\
0 & (1-t)H_q - t\I{q}
\end{pmatrix}.
\end{align}
This is a well-defined map with image in the fiber $F$: The proof of
remark~\ref{RemarkDefiniteComponentsRetractable} shows that the
intermediate matrices during a homotopy of a definite matrix to
$+\I_p$ resp. $-\I_q$ stay definite. This generalizes to the
situation at hand: Given a matrix $H$ of signature $(p,q)$, then
$\rho_t(H)$ will be of the same signature for every $t$. Therefore,
$\rho_t(F) \subset F$ for all $t$. Next, extend $\rho$ to a map
\begin{align*}
\hat{\rho}\colon I \times K \times F &\to K \times F\\
(t,(k,H)) &\mapsto (k,\rho_t(H))
\end{align*}
Now we prove that $\smash{\hat{\rho}}$ is $L$-equivariant and
therefore it pushes down to a map
\begin{align*}
I \times K \times_L F \to K \times_L F
\end{align*}
For this, let $\ell$ be in $L$. Then
\begin{align*}
\ell(\hat{\rho}(t,k,H)) = \ell(k,\rho_t(H)) = (k\ell,\ell^* \rho_t(H) \ell)
\end{align*}
On the other hand:
\begin{align*}
\hat{\rho}(\ell(t,k,H)) = \hat{\rho}(t,k\ell,\ell^* H \ell) = (k\ell, \rho_t(\ell^* H \ell))
\end{align*}
Now, $L$-equivariance follows from the fact that matrix conjugation
commutes with addition and scalar multiplication of matrices and
therefore
\begin{align*}
\ell^* \rho_t(H) \ell = \rho_t(\ell^* H \ell).
\end{align*}
By the isomorphism $K \times_L F \cong \mathcal{H}_{(p,q)}$,
$\smash{\hat{\rho}}$ induces a map
$\smash{\tilde{\rho}}\colon I \times \mathcal{H}_{(p,q)} \to
\mathcal{H}_{(p,q)}$.
Note that $\rho_0$ and therefore also $\smash{\hat{\rho_0}}$ as well as
the push-down to the quotient is the identity. On the other hand,
$\rho_1$ retracts the fiber $F$ to $\I{p}{q}$. Thus,
$\smash{\hat{\rho_1}}$ retracts $K \times F$ to $K \times
\{\I{p}{q}\}$.
Using the isomorphism $K \times_L F \cong \mathcal{H}_{(p,q)}$, it
follows that $\smash{\tilde{\rho}}$, retracts $\mathcal{H}_{(p,q)}$ to
$\Sigma = K.\I{p}{q}$.
It remains to prove that $\smash{\tilde{\rho}}_t$ is $T$-equivariant for
each $t$:
\begin{align}
\label{EquationRetractionEquivariance}
\tilde{\rho}_t(\overline{H}) = \overline{\tilde{\rho}_t(H)}.
\end{align}
A short computation shows that the isomorphism $\Psi$
(\ref{EquivariantFiberBundleIsoPsi}) is $T$-equivariant with respect
to the $T$-action on $K \times_L F$ given by
\begin{align*}
T\left(\left[(k,H)\right]\right) = \left[\left(\overline{k},\overline{H}\right)\right].
\end{align*}
Therefore, in order to prove (\ref{EquationRetractionEquivariance})
we need to show that the induced map $I \times K \times_L F \to K
\times_L F$ is $T$-equivariant. By definition this boils down to
showing that
\begin{align*}
\left[\left(\overline{k},\rho_t\left(\overline{H}\right)\right)\right] = \left[\left(\overline{k},\overline{\rho_t(H)}\right)\right],
\end{align*}
which is a direct consequence of the definition of $\rho_t$ as in
(\ref{EquationRetractRho}). Summarizing the above we have seen that
there exists a strong deformation retract $\smash{\tilde{\rho}}$ from
$\mathcal{H}_{(p,q)}$ to the orbit $K.\I{p}{q}$ which is
$T$-equviariant. \end{proof} \begin{figure}
\caption{The $U(n)$-orbits in $\mathcal{H}_{(p,q)}$.}
\label{fig:UnOrbitsInHpq}
\end{figure}
The $U(n)$-orbit of $\I{p}{q}$ can be equivariantly and diffeomorphically identified with the complex Grassmann manifold $\text{Gr}_p(\mathbb{C}^n)$. This reduces the problem of describing the sets $\smash{[X,\mathcal{H}_{(p,q)}]_G}$ to the problem of describing the sets $\smash{[X,\text{Gr}_p(\mathbb{C}^n)]_G}$. As a first step towards the study of equivariant maps to $\text{Gr}_p(\mathbb{C}^n)$ we describe $(\text{Gr}_p(\mathbb{C}^n))_{\mathbb{R}}$, the space of real points with respect to the real structure $T$ on $\text{Gr}_p(\mathbb{C}^n)$: \begin{remark}
The space $(\text{Gr}_p(\mathbb{C}^n))_{\mathbb{R}}$ of real points is
diffeomorphic to the real Grassmannian $\text{Gr}_p(\mathbb{R}^n)$. \end{remark}
\begin{proof}
Given any $T$-stable subspace $V$ in $\mathbb{C}^n$, then $T$ defines a
decomposition $V = V_\mathbb{R} \oplus i V_\mathbb{R}$ where $V_\mathbb{R} = \text{Fix}\, T$ is a
subspace of $\mathbb{R}^n$. Note that the group $O(n,\mathbb{R})$ acts transitively
on $\text{Gr}_p(\mathbb{R}^n)$. Therefore, for any two $T$-stable $p$-dimensional
subvector spaces $V, V' \subset \mathbb{C}^n$, there exists an orthogonal
transformation $g$ mapping the $V_\mathbb{R}$ to $V'_\mathbb{R}$. Such a
transformation extends uniquely to a unitary transformation of
$\mathbb{C}^n$ which sends $V$ to $V'$. Now, a computation shows that the
stabilizer of the base point
$V_0 = \smash{\left<e_1,\ldots,e_p\right>}$ in $O(n,R)$ consists of
the matrices of the form
\begin{align*}
g =
\begin{pmatrix}
g_1 & 0 \\
0 & g_2
\end{pmatrix},
\end{align*}
where $g_1 \in O(p,\mathbb{R})$ and $g_2 \in O(q,\mathbb{R})$. It follows that
\begin{equation*}
(\text{Gr}_p(\mathbb{C}^n))_\mathbb{R} \cong \frac{O(n,\mathbb{R})}{O(p,\mathbb{R}) \times O(q,\mathbb{R})} \cong \text{Gr}_p(\mathbb{R}^n).\qedhere
\end{equation*} \end{proof}
We now proceed with the classification of equivariant maps to the space of non-singular $n \times n$ hermitian matrices: \begin{align*}
H\colon X \to \mathcal{H}^*_n. \end{align*} As shown earlier, such a map has a well-defined signature $(p,q)$ (with $p+q = n$) specifying that $H(x)$ has exactly $p$ positive eigenvalues and $q$ negative eigenvalues for every $x \in X$. The classification for the image space $\mathcal{H}_n$ will be achieved in the following way: \begin{enumerate} \item First we handle the classification of maps $X \to \mathcal{H}_3$
seperately. In this situation the relevant image space can be
reduced to the Grassmannian $\text{Gr}_1(\mathbb{C}^3) \cong \mathbb{P}_2$. The
classification with respect to this image space works by making a
reduction to the case of maps to $\mathbb{P}_1$. \item Then, for a general Grassmannian $\text{Gr}_p(\mathbb{C}^n)$ we apply
a reduction method iteratively until we have reduced the situation
to a copy of $\mathbb{P}_1$ embedded in $\text{Gr}_p(\mathbb{C}^n)$ as a Schubert
variety. The seperate discussion of $\mathbb{P}_2$ will act as a guiding
example and provide useful statements for the final step of the
iterative reduction. \end{enumerate}
For the following we need a notion of \emph{total degree}, as the standard notion of degree (i.\,e. the Brouwer degree) is not applicable here, since $X$ and a general Grassmann manifold $\text{Gr}_p(\mathbb{C}^n)$ have different dimensions. But because of the convenient geometrical properties of Grassmannians (see proposition~\ref{TopologyGrassmannians}) we in fact do not have to change much: The second homology group is still infinite cyclic, which allows us to generalize the notion of total degree. For this, let $f\colon X \to \text{Gr}_p(\mathbb{C}^n)$ be a $G$-map. The homology group $H_2(X,\mathbb{Z})$ is generated by the fundamental class $[X]$. The homology group $H_2(\text{Gr}_p(\mathbb{C}^n),\mathbb{Z})$ is generated by $[\mathcal{S}]$, where $\mathcal{S}$ denotes the Schubert variety \begin{align}
\label{DefOfSchubertVarietyC}
\mathcal{S} = \left\{E \in \text{Gr}_p(\mathbb{C}^n)\colon \mathbb{C}^{p-1} \subset E \subset \mathbb{C}^{p+1}\right\}. \end{align} Here, $\mathbb{C}^\ell$ is regarded as being embedded in $\mathbb{C}^n$, for $\ell < n$, via \begin{align*}
(z_1,\ldots,z_\ell) \mapsto (z_1,\ldots,z_\ell,\underbrace{0,\ldots,0}_{n-\ell}). \end{align*} In the terminology of e.\,g. \cite{GriffithsHarris}, $\mathcal{S}$ is the Schubert variety\label{SchubertVarietyDiscussion} \begin{align*}
\mathcal{S} = \left\{E \in \text{Gr}_p(\mathbb{C}^k)\colon \dim(E \cap V_{k-p+i-a_i}) \geq i \;\text{for all $i=1,\ldots,p$}\right\} \end{align*} defined by the sequence \begin{align*}
a_i =
\begin{cases}
n - p & \;\text{for $1 \leq i < p-1$} \\
n - p - 1 & \;\text{for $p - 1 \leq i \leq p$}.
\end{cases} \end{align*} After having fixed the standard flag $V_1 \subset V_2 \subset \ldots \subset V_n$, where $V_j$ is the vector space generated by the standard basis vectors $e_1,\ldots,e_j$, $\mathcal{S}$ is the unique one-dimensional Schubert variety in $\text{Gr}_p(\mathbb{C}^n)$. It is biholomorphic to $\mathbb{P}_1$ and comes equippd with its canonical orientation as a complex manifold. As it has been discussed on p.~\pageref{P1S2OrientationDiscussion}, this space can be equivariantly identified with the $2$-sphere. Now we can make the following \begin{definition}
\label{TotalDegreeGr}
Let $f\colon X \to \text{Gr}_p(\mathbb{C}^n)$ be a $G$-map. Then its induced map
on the second homology is of the form
\begin{align*}
f_*\colon H_2(X,\mathbb{Z}) &\to H_2(\text{Gr}_p(\mathbb{C}^n),\mathbb{Z})\\
[X] &\mapsto d[\mathcal{S}].
\end{align*}
We define the \emph{total degree} of $f$ to be this integer $d$ and
write $\deg f$ for the total degree of $f$. \end{definition} Clearly, the total degree defined this way is a homotopy invariant of maps $X \to \text{Gr}_p(\mathbb{C}^n)$. We also need a slight generalization of definition~\ref{DefinitionFixpointDegree} to account for the fundamental group of $(\text{Gr}_p(\mathbb{C}^n))_{\mathbb{R}}$ not being infinite cyclic unless $p = 1$ and $n=2$: \begin{definition}
\label{DefinitionFixpointSignature} (Fixed point signature) Let
$f\colon M \to Y$ be a $G$-map between $G$-manifolds. Assume that
the fixed point set $Y^G \subset Y$ is path-connected, $\pi_1(Y^G) =
C_2$ and that the fixed point set $M^G$ is the disjoint union
\begin{align*}
M^G = \bigcup_{j=0,\ldots,k} K_j
\end{align*}
of circles $K_j$. Then we define the \emph{fixed point signature} of
$f$ to be the $k$-tuple $(m_0,\ldots,m_k)$ where
\begin{align*}
m_j =
\begin{cases}
0 & \;\text{if the loop $\Restr{f}{K_j}\colon K_j \to Y^G$ is contractible in $Y^G$}\\
1 & \;\text{else}.
\end{cases}
\end{align*} \end{definition}
In the following let $n > 2$, i.\,e. we exclude the case $\text{Gr}_p(\mathbb{C}^n) = \mathbb{P}_1$, which has already been dealt with in section~\ref{SectionN=2}. For the type I involution on $X$ we make the following definition: \begin{definition} (Degree triple map)
For $n > 2$, we define the \emph{degree triple map} to be the map
\begin{align*}
\mathcal{T}\colon \mathcal{M}_G(X,\text{Gr}_p(\mathbb{C}^n)) &\to \{0,1\} \times \mathbb{Z} \times \{0,1\}\\
f &\mapsto \Triple{m_0}{d}{m_1},
\end{align*}
where $\Bideg{m_0}{m_1}$ is the fixed point signature of $f$ with
respect to the circles $C_0, C_1$ and $d$ is the total degree of
$f$. We call a given triple \emph{realizable} if it is contained in
the image $\Im(\mathcal{T})$. For a map
\begin{align*}
f\colon (Z,C) \to \left(\text{Gr}_p(\mathbb{C}^n),\text{Gr}_p(\mathbb{C}^n)_\mathbb{R}\right)
\end{align*}
we define its \emph{degree triple} to be the degree triple of the
equivariant extension $\smash{\hat{f}}\colon X \to \text{Gr}_p(\mathbb{C}^n)$. \end{definition}
For the type II involution on $X$ we define: \begin{definition} (Degree pair map)
For $n > 2$, we define the \emph{degree pair map} to be the map
\begin{align*}
\mathcal{P}\colon \mathcal{M}_G(X,\text{Gr}_p(\mathbb{C}^n)) &\to \mathbb{Z} \times \{0,1\}\\
f &\mapsto \Pair{m}{d},
\end{align*}
where $(m)$ is the fixed point signature of $f$ and $d$ is the total
degree of $f$. We call a given pair \emph{realizable} if it is
contained in the image $\Im(\mathcal{P})$. For a map
\begin{align*}
f\colon (Z,C) \to \left(\text{Gr}_p(\mathbb{C}^n),\text{Gr}_p(\mathbb{C}^n)_\mathbb{R}\right)
\end{align*}
we define its \emph{degree pair} to be the degree pair of the
equivariant extension $\smash{\hat{f}}\colon X \to \text{Gr}_p(\mathbb{C}^n)$. \end{definition}
The degree triple resp. the degree pair is clearly an invariant of equivariant homotopy.
\subsection{Maps to $\mathcal{H}_3$}
In this section we study the equivariant homotopy of maps $X \to \mathcal{H}_3^*$. The connected components of $\mathcal{H}_3^*$ are parameterized by the eigenvalue signatures $(3,0)$, $(2,1)$, $(1,2)$ and $(0,3)$. The group actions are exactly the same as in the first chapter: we can regard $X$ as being defined by the standard square torus, equipped with either the type I or the type II real structure and the involution on $\mathcal{H}_3$ is given by conjugation. It turns out that the classification of maps in the case where the image space is $\mathcal{H}_3^*$ is not fundamentally different from the classification where it is $\mathcal{H}_2^*$. As before we start with a reduction procedure.
Let $H\colon X \to \mathcal{H}_3^*$ be a map with the signature $(p,q)$. Hence, $H$ can be regarded as a map $X \to \mathcal{H}_{(p,q)}$. In the concrete case of maps to $\mathcal{H}_3$ with mixed signature we can now conclude with proposition~\ref{GrassmannianStrongDeformationRetract} that $\mathcal{H}_{(2,1)}$ (resp. $\mathcal{H}_{(1,2)}$) can be equivariantly retracted to the orbit $U(3).\I{2}{1}$ (resp. $U(3).\I{1}{2}$). These orbits are equivariantly and diffeomorphically identifiable with $\text{Gr}_2(\mathbb{C}^3)$ (resp. $\text{Gr}_1(\mathbb{C}^3)$). Since these two Grassmann manifolds are by remark~\ref{GrassmannianIdentificationEquivariant} equivalent as $G$-spaces, we can assume without loss of generality that we are dealing with $G$-maps of the form \begin{align*}
X \to \text{Gr}_1(\mathbb{C}^3) \cong \mathbb{P}_2, \end{align*} where the involution $T$ on $\mathbb{P}_2$ is given by standard complex conjugation: \begin{align*}
T\colon [z_0:z_1:z_2] \mapsto \left[\overline{z_0}:\overline{z_1}:\overline{z_2}\right]. \end{align*}
\subsubsection{Maps to $\mathbb{P}_2$} \label{SectionClassificationP2}
For the following, let $p_0$ be the base point $[0:0:1] \in \mathbb{P}_2$ and regard $\mathbb{P}_1$ as being embedded in $\mathbb{P}_2$ as \begin{align*}
[z_0:z_1] \mapsto [z_0:z_1:0]. \end{align*}
\begin{lemma}
\label{RetractFromP2toP1}
There exists an equivariant strong deformation retract $\rho$ from
$\mathbb{P}_2\smallsetminus\{p_0\}$ to $\mathbb{P}_1 \subset
\mathbb{P}_2$. \end{lemma}
\begin{proof}
Define $\rho$ as follows:
\begin{align*}
\rho\colon I \times \mathbb{P}_2\smallsetminus\{p_0\} &\to \mathbb{P}_2\smallsetminus\{p_0\}\\
\left(t,[z_0:z_1:z_2]\right) &\mapsto [z_0:z_1:(1-t) z_2].
\end{align*}
For $t=0$ this is the identity of $\mathbb{P}_2\smallsetminus\{p_0\}$,
for $t=1$, this is the map
\begin{align*}
[z_0:z_1:z_2] \mapsto [z_0:z_1:0].
\end{align*}
This is well-defined, since $\rho$ is only defined on the complement
of the point $p_0$. This construction is equivariant, because the
multiplication with real numbers commutes with complex conjugation. \end{proof}
\begin{figure}
\caption{$\mathbb{P}_2\smallsetminus\{p_0\}$ retracts to $\mathbb{P}_1 \subset \mathbb{P}_2$.}
\label{P2VectorBundleOverP1}
\end{figure}
The strong deformation retract $\rho$ (see figure~\ref{P2VectorBundleOverP1} for a depiction) constructed in the previous lemma also defines a strong deformation retract of the fixed point set $(\mathbb{P}_2)_\mathbb{R}$ to the fixed point set $(\mathbb{P}_1)_\mathbb{R}$. For understanding what happens to fixed point signatures of maps $X \to \mathbb{P}_2$ during such a deformation retract we make the following remark: \begin{remark}
\label{NonTrivialLoopInRPn}
Assume $n > 1$. Let $\gamma\colon I \to \mathbb{RP}_1$ be the loop which
wraps around $\mathbb{RP}_1$ once. Let $\iota_n$ be the following embedding:
\begin{align*}
\iota_n\colon \mathbb{RP}_1 &\to \mathbb{RP}_n\\
[x_0:x_1] &\mapsto [x_0:x_1:0:\ldots:0].
\end{align*}
Then $\iota_n \circ \gamma$ defines the (up to homotopy) unique
non-trivial loop in $\mathbb{RP}_n$. \end{remark}
\begin{proof}
To simplify the notation set $\iota = \iota_n$.
We have the following commutative diagram
\[
\begin{xy}
\xymatrix{
S^1 \ar@{^{(}->}[r]^{\hat{\iota}} \ar[d] & S^n \ar[d]\\
\mathbb{RP}_1 \ar@{^{(}->}[r]_\iota & \mathbb{RP}_n
}
\end{xy}
\]
Here, $\smash{\hat{\iota}}$ can be regarded as the restriction of
the embedding $\mathbb{R}^2 \hookrightarrow \mathbb{R}^{n+1}$ to
$S^1 \subset \mathbb{R}^2$. Denote the loop which wraps around
$\mathbb{RP}_1$ once by $\gamma\colon I \to \mathbb{RP}_1$. It can
be lifted to a path $\smash{\hat{\gamma}}\colon I \to S^1$, which
makes a one-half loop in $S^1$. Then, by commutativity of the
diagram, it follows that
$\smash{\widehat{\iota \circ \gamma}} := \smash{\hat{\iota}} \circ
\smash{\hat{\gamma}}$
is a lift of $\iota \circ \gamma$, the latter being a representant
of the homotopy class $\iota_*[\gamma] \in \pi_1(\mathbb{RP}_n)$. It is
well known (see e.\,g. \cite[p.~74]{Hatcher}) that $\pi_1(\mathbb{RP}_n)$
(for $n > 1$) is generated by the projection to $\mathbb{RP}_n$ of a
curve in the universal covering space $S^n$ which connects two
antipodal points. Thus it is enough to show that
$\widehat{\iota \circ \gamma}$ is not a closed loop in $S^n$. But
this is clear, since $\smash{\hat{\gamma}}$ is not a closed loop and
$S^1 \hookrightarrow S^n$ is just an embedding. \end{proof}
As a consequence of the previous remark we can formulate: \begin{remark}
\label{RPnFundamentalGroupIsomorphism}
Assume $n > 1$ and let $\iota$ be the embedding
\begin{align*}
\iota\colon \mathbb{RP}_n &\to \mathbb{RP}_{n+1}\\
[x_0:\ldots:x_n] &\mapsto [x_0:\ldots:x_n:0].
\end{align*}
Then the induced map $\iota_*\colon \pi_1(\mathbb{RP}_n) \to
\pi_1(\mathbb{RP}_{n+1})$ is an isomorphism. \end{remark}
\begin{proof}
Since both fundamental groups are cyclic of order 2 it suffices to
show that $\iota_*$ maps the non-trivial homotopy class $[\gamma]
\in \pi_1(\mathbb{RP}_n)$ to the non-trivial homotopy class in
$\pi_1(\mathbb{RP}_{n+1})$. Let $\alpha$ be the non-trivial loop in
$\mathbb{RP}_n$ which is defined in terms of the embedding of $\mathbb{RP}_1
\hookrightarrow \mathbb{RP}_n$ (see
remark~\ref{NonTrivialLoopInRPn}). Then we have $[\gamma] =
[\alpha]$. The homotopy class $\iota_*([\gamma])$ can thus be
regarded as the homotopy class of the loop $\iota \circ \alpha$. But
this is just the non-trivial loop constructed in the previous remark
for $\mathbb{RP}_{n+1}$. Therefore, $\iota \circ \alpha$ is non-trivial
and the statement is proven. \end{proof}
Note that there is a canonical embedding of $S^1 \cong \mathbb{RP}_1 \hookrightarrow \mathbb{RP}_2$ as the line at infinity: \begin{align*}
[x_1:x_2] \mapsto [x_1:x_2:0] \end{align*} Fix a point $p_0$ in $\mathbb{RP}_1$. We now define two loops: Let $\gamma_0$ be the loop which is constant at $p_0$ and let $\gamma_1$ be a fixed loop which wraps around $\mathbb{RP}_1 \subset \mathbb{RP}_2$ once. By remark~\ref{NonTrivialLoopInRPn} this defines the non-trivial element in $\pi_1(\mathbb{RP}_2)$.
\begin{definition}
\label{DefinitionFixpointNormalizationP2}
A $G$-map $f\colon X \to \mathbb{P}_2$ is \emph{fixed point normalized} if
$\restr{f}{C_j}$ is either $\gamma_0$ or $\gamma_1$ for all fixed point
circles $C_j$ in $X$. \end{definition}
\begin{remark}
\label{FixpointNormalizationP2}
Let $X$ be equipped with the type I or the type II involution. Then
every $G$-map $f\colon X \to \mathbb{P}_2$ can be fixed point normalized. \end{remark} \begin{proof}
Using the equivariant version of the homotopy extension property of
the pair $(X,C)$ (see e.\,g. corollary~\ref{G-HEP}) we can prescribe
a homotopy on each of the the fixed point circles of the torus which
deforms the corresponding loop to either $\gamma_0$ or $\gamma_1$,
depending on the fixed point signature of the map. \end{proof}
Thus, after this normalization we can assume that maps with the same fixed point signature agree along the boundary circles.
\begin{lemma}
\label{ReductionP2toP1}
Let $f\colon X \to \mathbb{P}_2$ be a fixed point normalized $G$-map
for the type I or the type II involution. Then:
\begin{enumerate}[(i)]
\item The map $f$ is equivariantly homotopic to a map $f'$ which has
its image contained in $\mathbb{P}_1 \subset \mathbb{P}_2$.
\item The degree of $f$ (definition~\ref{TotalDegreeGr}) agrees with
$\deg f'$, where $f'$ is regarded as a map $X \to \mathbb{P}_1$.
\end{enumerate} \end{lemma}
\begin{proof}
Regarding (i): Let $\smash{\widehat{C}}$ be the union of the
(finitely many) fixed point circles. Because of dimension reasons
there exists a point $p \in \mathbb{RP}_2$ which is not contained in
the image of $\smash{\restr{f}{\widehat{C}}}$. Note that $\SOR{3}$ acts
transitively on $\mathbb{RP}_2$. Thus, let $g_1 \in \SOR{3}$ be the
transformation in $\SOR{3}$ which sends $p$ to $p_0 :=
[0:0:1]$.
This transformation $g_1$ can also be regarded as a transformation
of $\mathbb{P}_2$, since $\SOR{3} \subset \mathrm{SU}(3)$. Also, since
$\SOR{3}$ is path-connected we can find a path $g_t$ from the
identity in $\SOR{3}$ to $g_1$. The composition $g_t \circ f$ is
then a homotopy of $f$ to a map $\smash{\tilde{f}}$ such that
$p_0 \not\in \Im \smash{\tilde{f}}$.
This homotopy is also equivariant. Thus, without loss of generality
we can assume that, at least after a $G$-homotopy, a given $G$-map
$f\colon X \to \mathbb{P}_2$ does not have $p_0$ in its image. Let
us assume that the given map $f$ has this property. Then we can
compose $f$ with the equivariant strong deformation retract from
remark~\ref{RetractFromP2toP1} and therefore obtain an equivariant
homotopy from $f$ to a map $f'$ such that $\Im f' \subset
\mathbb{P}_1 \subset \mathbb{P}_2$.
Regarding (ii): The degree $\deg f$ is the number $d$ such that
\begin{align*}
f_*\colon H_2(X,\mathbb{Z}) \to H_2(\mathbb{P}_2,\mathbb{Z})
\end{align*}
is the map
\begin{align*}
[X] \mapsto d[\mathbb{P}_1].
\end{align*}
while the degree $\deg f'$ is the number $d'$ such that
\begin{align*}
f'_*\colon H_2(X,\mathbb{Z}) \to H_2(\mathbb{P}_1,\mathbb{Z})
\end{align*}
is the map
\begin{align*}
[X] \mapsto d'[\mathbb{P}_1].
\end{align*}
The inclusion $\iota\colon \mathbb{P}_1 \hookrightarrow
\mathbb{P}_2$ induces an isomorphism in the second homology
group. Thus we can also regard $f_*$ as taking on values in
$H_2(\mathbb{P}_2,\mathbb{Z})$. But since $f$ and $\iota \circ f'$
are homotopic, their induced maps in homology must agree. This means
$d = d'$. \end{proof}
\paragraph{Type I}
In this paragraph we are assuming the involution on $X$ to be of type I.
\begin{remark}
\label{ReductionP2toP1Class1}
Let $f\colon X \to \mathbb{P}_2$ be a fixed point normalized
$G$-map. Then the map $f$ has fixed point signature $\Pair{m_0}{m_1}$
iff $f'$ (from lemma~\ref{ReductionP2toP1}) has fixed point degree
$\Bideg{m_0}{m_1}$. \end{remark}
\begin{proof}
Let us first assume that $\restr{f}{C_j}\colon C_j \to
\mathbb{RP}_2$ is trivial, i.\,e. $m_j = 0$. Since $f$ is fixed point
normalized, we know that $\restr{f}{C_j}$ is the constant
map. Therefore, after making the deformation retract to $\mathbb{RP}_1
\subset \mathbb{P}_1$ it is still constant, hence has degree
zero. On the other hand, if a map $S^1 \to \mathbb{RP}_1 \subset
\mathbb{P}_1$ has degree zero, then it is already null-homotopic as
a map to $\mathbb{RP}_1$. Therefore it will also be null-homotopic as a
map to $\mathbb{RP}_2 \supset \mathbb{RP}_1$.
Let us now assume that $\restr{f}{C_j}\colon C_j \to \mathbb{RP}_2$
is non-trivial, i.\,e. $m_j = 1$. In this case $\restr{f}{C_j}$ is
the map which wraps around $\mathbb{RP}_1$ once. In particular, this
loop will be fixed when composing with the strong deformation
retract to $\mathbb{P}_1$. Therefore, after the retraction, the
resulting map will still have degree one along the boundary circle
$C_j$. The reverse direction follows from
remark~\ref{NonTrivialLoopInRPn}, where we discussed the loop which
wraps around $\mathbb{RP}_1$ once; using the canonical embedding
$\mathbb{RP}_1 \hookrightarrow \mathbb{RP}_2$, this loops defines a
non-trivial element in $\pi_1(\mathbb{RP}_2)$. \end{proof}
\begin{theorem}
\label{TheoremClassificationMapsToP2Class1}
The $G$-homotopy class of a map $f \in \mathcal{M}_G(X,\mathbb{P}_2)$ is
uniquely determined by its degree triple $\mathcal{T}(f)$. The image
$\Im(\mathcal{T})$ of the degree triple map consists of those
triples $\Triple{m_0}{d}{m_1}$ (with $m_0, m_1 \in \{0,1\}$)
satisfying
\begin{align*}
d \equiv m_0 + m_1 \mod 2.
\end{align*} \end{theorem}
\begin{proof}
First we show that two maps with the same degree triple are
equivariantly homotopic. Thus, let $f$ and $g$ be two $G$-maps $X
\to \mathbb{P}_2$, both having the triple $\Triple{m_0}{d}{m_1}$.
Using lemma~\ref{ReductionP2toP1} (i) we can assume that $f$ and $g$
have their images contained in $\mathbb{P}_1 \subset
\mathbb{P}_2$. Applying lemma~\ref{ReductionP2toP1} (ii) and
lemma~\ref{ReductionP2toP1Class1} it then follows that the degree
triple of $f$ and $g$, regarded as a map $X \to \mathbb{P}_1$ is
again $\Triple{m_0}{d}{m_1}$. Hence, by
theorem~\ref{Classification1}, these maps are equivariantly
homotopic.
Regarding the image $\Im(\mathcal{T})$: Let us first assume that
$\Triple{m_0}{d}{m_1}$ is in the image. Then, by definition, there
exists a map $f\colon X \to \mathbb{P}_2$ with this
triple. Lemma~\ref{ReductionP2toP1} then implies the existence of a
map $f'\colon X \to \mathbb{P}_1$ with the same triple. Hence, by
theorem~\ref{Classification1}, $d \equiv m_0 + m_1 \mod 2$. On the
other hand, given a triple $\Triple{m_0}{d}{m_1}$ with $m_0,m_1 \in
\{0,1\}$ and $d \equiv m_0 + m_1 \mod 2$, there exists a map
$f\colon X \to S^2 \cong \mathbb{P}_1$ with this triple. Composing
with the embedding $\mathbb{P}_1 \hookrightarrow \mathbb{P}_2$ we
have produced a map $X \to \mathbb{P}_2$ which, again by
lemma~\ref{ReductionP2toP1}, has the same triple
$\Triple{m_0}{d}{m_1}$. \end{proof}
\paragraph{Type II}
In this paragraph we are assuming the involution on $X$ to be of type II.
\begin{remark}
\label{ReductionP2toP1Class2}
Let $f\colon X \to \mathbb{P}_2$ be a fixed point normalized
$G$-map. Then the map $f$ has the fixed point signature $m$ iff $f'$
(from lemma~\ref{ReductionP2toP1}) has the fixed point degree $m$. \end{remark}
\begin{proof}
The proof is exactly the same as in
lemma~\ref{ReductionP2toP1Class1}. One only has consider the single
circle $C$ in $X$ instead of two circles $C_0$ and $C_1$. \end{proof}
\begin{theorem}
\label{TheoremClassificationMapsToP2Class2}
The $G$-homotopy class of a map $f \in \mathcal{M}_G(X,\mathbb{P}_2)$ is
uniquely determined by its degree pair $\mathcal{P}(f)$. The image
$\Im(\mathcal{P})$ of the degree pair map consists of those pairs
$\Pair{m}{d}$ (with $m \in \{0,1\}$) satisfying
\begin{align*}
d \equiv m \mod 2.
\end{align*} \end{theorem}
\begin{proof}
The proof works completely analogously to the proof of
theorem~\ref{TheoremClassificationMapsToP2Class1}. For the first
statement it suffices to show that two maps with the same degree
pair are equivariantly homotopic. Thus, let $f$ and $g$ be two
$G$-maps $X \to \mathbb{P}_2$, both having the pair
$\Pair{m}{d}$. Using lemma~\ref{ReductionP2toP1} (i) we can assume
that $f$ and $g$ have their images contained in $\mathbb{P}_1 \subset
\mathbb{P}_2$. Applying lemma~\ref{ReductionP2toP1} (ii) and
lemma~\ref{ReductionP2toP1Class2} it then follows that the degree
pair of $f$ and $g$, regarded as a map $X \to \mathbb{P}_1$ is again
$\Pair{m}{d}$. Hence, by theorem~\ref{Classification2}, these maps
are equivariantly homotopic, which proves the statement.
Regarding the image $\Im(\mathcal{P})$: Let us assume that
$\Pair{m}{d}$ is in the image. By definition there exists a map
$f\colon X \to \mathbb{P}_2$ with this
pair. Lemma~\ref{ReductionP2toP1} then implies the existence of a
map $f'\colon X \to \mathbb{P}_1$ with the same pair. Hence, by
theorem~\ref{Classification2}, $d \equiv m \mod 2$ is a necessary
condition for a pair to be contained in $\Im(\mathcal{P})$. On the
other hand, given a pair $\Pair{m}{d}$ with $m \in \{0,1\}$ and $d
\equiv m \mod 2$, there exists a map $f\colon X \to S^2 \cong
\mathbb{P}_1$ with this pair. Composing with the embedding
$\mathbb{P}_1 \hookrightarrow \mathbb{P}_2$ we have produced a map
$X \to \mathbb{P}_2$ which, again by lemma~\ref{ReductionP2toP1},
has the same pair $\Pair{m}{d}$. Therefore, $\Pair{m}{d}$ is
contained in the image $\Im(\mathcal{P})$. \end{proof}
\subsubsection{Classification of Maps to $\mathcal{H}_3^*$}
So far the degree triple map (resp. the degree pair map) has only been defined for $G$-maps $X \to \mathbb{P}_2$. But after fixing an identification of the orbit $U(3).\I{p}{q} \subset \mathcal{H}_{(p,q)}$, which has been shown to be an equivariant strong deformation retract, with $\mathbb{P}_2$, the degree triple map (resp. the degree pair map) is also defined on all mapping spaces $\mathcal{M}_G(X,\mathcal{H}_{(p,q)})$ with $p+q=3$. This remark then allows us to state and prove the main result of this section:
\begin{theorem}
\label{HamiltonianClassificationRank3}
Let $X$ be a torus equipped with either the type I or the type II
involution. Then:
\begin{enumerate}[(i)]
\item The sets $[X,\mathcal{H}_{(3,0)}]_G$ and
$[X,\mathcal{H}_{(0,3)}]_G$ are trivial.
\item Two $G$-maps $X \to \mathcal{H}_{(p,q)}$ (with $0<p,q<3$) are
$G$-homotopic iff their degree triples (type I) resp. their degree
pairs (type II) agree.
\item The realizable degree triples $\Triple{m_0}{d}{m_1}$ (type I)
resp. degree pairs $\Pair{m}{d}$ (type II) are exactly those
which satisfy
\begin{align*}
d \equiv m_0 + m_1 \mod 2 \;\text{resp.}\;d \equiv m \mod 2.
\end{align*}
\end{enumerate} \end{theorem}
\begin{proof}
The $\text{sig}\, = (3,0)$ resp. the $\text{sig}\, = (0,3)$ case is handled by
remark~\ref{RemarkDefiniteComponentsRetractable}, which proves that
maps $X \to \mathcal{H}_{(2,0)}$ (resp. $X \to \mathcal{H}_{(0,2)}$)
are equivariantly retractable to the map which is constant the
identity (resp. minus identity). In the case of the signature being
$(2,1)$ (resp. $(1,2)$) the image space has the projective space
$\mathbb{P}_2$ as an equivariant strong deformation retract. In this case
theorem~\ref{TheoremClassificationMapsToP2Class1} (for type I) and
theorem~\ref{TheoremClassificationMapsToP2Class2} (for type II)
complete the proof. \end{proof}
\begin{remark}
\label{HamiltonianMapRealizationRank3}
Clearly, the case $\text{sig}\, = (3,0)$ (resp. $\text{sig}\, = (0,3)$) is trivially
realized by the map, which is constant the identity (resp. minus
identity). Representants for the homotopy classes for $\text{sig}\, = (2,1)$
and $\text{sig}\, = (1,2)$ can be constructed as follows: Let
$f\colon X \to S^2$ be a $G$-map. Using the orientation-preserving
diffeomorphism $S^2 \xrightarrow{\;\sim\;} \mathbb{P}_1$, we can regard
$f$ as a map to $\mathbb{P}_1$. Composing this map with the embedding
$\mathbb{P}_1 \hookrightarrow \mathbb{P}_2$ yields a $G$-map
$\smash{\tilde{f}}\colon X \to \mathbb{P}_2$. Finally, we can compose
$\smash{\tilde{f}}$ with one of the two embeddings
$\iota_{(2,1)},\iota_{(1,2)}\colon \mathbb{P}_2 \hookrightarrow
\mathcal{H}_3^*$,
which embed $\mathbb{P}_2$ in the respective component of
$\mathcal{H}_3^*$ associated to the signature $(2,1)$ or $(1,2)$. \end{remark}
\subsection{Iterative Retraction Method}
In this section we describe a reduction procedure which can be used iteratively until we arrive in a known situation, i.\,e. $\text{Gr}_1(\mathbb{C}^2) \cong \mathbb{P}_1$. We call this procedure \emph{iterative
retraction} of Grassmann manifolds. Recall section~\ref{SectionClassificationP2}, in particular lemma~\ref{ReductionP2toP1}: There we start with a $G$-map $f\colon X \to \mathbb{P}_2$ and remove a certain subset $S$ (in this case, $S$ is a point) from $\mathbb{P}_2$ which -- at least after a $G$-homotopy -- is not contained in the image $\Im(f)$. The resulting space $\mathbb{P}_2\smallsetminus S$ has the submanifold $\mathbb{P}_1 \subset \mathbb{P}_2$ as equivariant, strong deformation retract. In this section we show that this method can be generalized to general Grassmann manifolds $\text{Gr}_p(\mathbb{C}^n)$. Fundamental for this reduction is the fact that the Schubert variety $\mathcal{S} \subset \text{Gr}_p(\mathbb{C}^n)$ is equivariantly identifiable with $\mathbb{P}_1 \cong S^2$ (see the definition of discussion of $\mathcal{S}$ on p.~\pageref{DefOfSchubertVarietyC}).
The strategy in this chapter is to identify a higher-dimensional analog of the subset $S$ in $\text{Gr}_p(\mathbb{C}^\ell)$ such that $\text{Gr}_p(\mathbb{C}^\ell)\smallsetminus S$ has $\text{Gr}_p(\mathbb{C}^{\ell-1}) \subset \text{Gr}_p(\mathbb{C}^n)$ as equivariant strong deformation retract. This reduction step is the basic building block for the reducing the classification of maps to $\text{Gr}_p(\mathbb{C}^n)$ to the classification of maps to the Schubert variety $\mathcal{S}$. It works as follows:
Starting with a $G$-map $f\colon X \to \text{Gr}_p(\mathbb{C}^n)$, we identify a subset $S$ with the aforementioned property and such that (after a $G$-homotopy) the image $\Im(f)$ does not intersect $S$. Then, using the equivariant deformation retract to $\text{Gr}_p(\mathbb{C}^{n-1})$, we can regard $f$ as having its image contained in $\text{Gr}_p(\mathbb{C}^{n-1})$. This step can be repeated until it can be assumed that the map $f$ has its image contained in $\text{Gr}_p(\mathbb{C}^{p+1})$. The latter can be equivariantly identified with $\text{Gr}_1(\mathbb{C}^{p+1})$. It follows that the reduction procedure just outlined can be repeated again until it can be assumed that the map $f$ as its image contained in $\text{Gr}_1(\mathbb{C}^2)$. This subspace $\text{Gr}_1(\mathbb{C}^2)$ identified using the procedure just described is exactly the Schubert variety $\mathcal{S}$. We begin with the following remark in this direction: \begin{remark}
\label{GrassmannianIdentificationEquivariant}
Using the standard unitary structure on $\mathbb{C}^n$, the canonical map
\begin{align*}
\Psi\colon \text{Gr}_k(\mathbb{C}^n) \to \text{Gr}_{n-k}(\mathbb{C}^n),
\end{align*}
where both Grassmannians are equipped with the real structure $T$
sending a space $V$ to $\overline{V}$, is equivariant with respect
to $T$. \end{remark}
\begin{proof}
It must be shown that, for $V \in \text{Gr}_k(\mathbb{C}^n)$, $T(V)^\perp =
T(V^\perp)$. This is equivalent to showing that $T(V)$ and
$T(V^\perp)$ are orthogonal with respect to the standard hermitian
form on $\mathbb{C}^n$. Thus, let $T(v)$ be in $T(V)$ (with $v \in
V$) and $T(w)$ be in $T(V^\perp)$ (with $w \in V^\perp$). We have to
show that $\left<T(v),T(w)\right> = 0$. Equivalently we can show
that $\smash{\overline{\left<T(v),T(w)\right>}} = 0$. A short computation
yields:
\begin{align*}
\overline{\left<T(v),T(w)\right>} = \overline{T(v)^*T(w)} = \overline{\overline{v}^*\overline{w}} = v^*w = 0.
\end{align*}
This proves that $T(V)$ is orthogonal to $T(V^\perp)$ and therefore
$T(V)^\perp = T(V^\perp)$. \end{proof}
For the technical arguments in this section we require that the smooth manifolds $X$ and $\text{Gr}_p(\mathbb{C}^n)$ are both smoothly embedded in some euclidean space (using e.\,g. the Whitney embedding theorem). By $\dim_H(\cdot)$ we denote the Hausdorff dimension of a topological space. See section~\ref{AppendixHausdorffDimension} for some introductory statements about Hausdorff dimensions. The set $S$, being the higher dimensional analog of a point in the $\mathbb{P}_2$ situation, will be denoted by $\mathcal{L}_L$, where $L$ is a line\footnote{By \emph{line} we mean a $1$-dimensional complex
subspace.} in $\mathbb{C}^n$, and is defined as follows: \begin{align*}
\mathcal{L}_L = \left\{E \in \text{Gr}_p(\mathbb{C}^n)\colon L \subset E\}\right. \end{align*}
\begin{remark}
\label{LLManifold}
The set $\mathcal{L}_L$ is a submanifold of $\text{Gr}_p(\mathbb{C}^n)$ of
complex dimension $(p-1)(n-p)$. \end{remark} \begin{proof}
Consider the map
\begin{align*}
f\colon \text{Gr}_{p-1}(L^\perp) &\to \text{Gr}_p(\mathbb{C}^n)\\
E &\mapsto \left<L,E\right>,
\end{align*}
where $\left<L,E\right>$ denotes the $p$-plane that is spanned by
the line $L$ together with the $p-1$-dimensional plane $E \subset
L^\perp$. Clearly, $\Im f = \mathcal{L}_L$. One can write down $f$
in local coordinates for the Grassmannians, which shows that $f$ is
a holomorphic immersion. Furthermore, it is injective. Together with
the compactness of $\text{Gr}_p(L^\perp)$ this shows that $f$ is an
holomorphic embedding (see
e.\,g. \cite[p.~214]{FritzscheGrauert}). In other words,
$\mathcal{L}_L$ is biholomorphically equivalent to
$\text{Gr}_{p-1}(\mathbb{C}^{n-1})$, a complex manifold of dimension $(p-1)(n-p)$. \end{proof}
Let $\mathbb{C}^n = L \oplus W$ be a decomposition of $\mathbb{C}^n$ as direct sum of a $T$-stable line $L$ and a 1-codi\-men\-sio\-nal, $T$-stable subvector space $W$. In this setup, we can regard the Grassmann manifold $\text{Gr}_p(W)$ as a submanifold of $\text{Gr}_p(\mathbb{C}^n)$. The usefulness of the submanifold $\mathcal{L}_L$ is illustrated by the following lemma: \begin{lemma}
\label{GrassmannianDeformationRetract}
The space $\text{Gr}_p(\mathbb{C}^n)\smallsetminus \mathcal{L}_L$ has $\text{Gr}_p(W)$ as
equivariant strong deformation retract. \end{lemma} \begin{proof}
To ease the notation set $\Omega = \text{Gr}_p(\mathbb{C}^n)\smallsetminus
\mathcal{L}_L$. The idea of this proof is that
\begin{align*}
\pi\colon \Omega \to \text{Gr}_p(W)
\end{align*}
can be regarded as a complex rank-$k$ vector bundle, where $\pi$ is
defined in terms of the projection $\mathbb{C}^n \to W$. Here we use the
fact that a $p$-dimensional plane $E$ in $\mathbb{C}^n$ which does not
contain the line $L$ projects down to a $p$-dimensional plane in
$W$.
The local trivializations of this bundle are defined using local
trivializations of the tautological bundle $\mathcal{T} \to
\text{Gr}_p(W)$: Let $U_\alpha$ be a trivializing neighborhood in $\text{Gr}_p(W)$ of
the tautological bundle and
\begin{align*}
\psi_\alpha\colon \Restr{\mathcal{T}}{U_\alpha} \to U_\alpha \times \mathbb{C}^k
\end{align*}
a local trivialization. In other words, the $k$ maps
\begin{align*}
f_{\alpha,j}\colon U_\alpha &\to \Restr{\mathcal{T}}{U_\alpha}\\
E &\mapsto \psi^{-1}_\alpha(E,e_j)\;\text{ for $j=1,\ldots,k$}
\end{align*}
define a local frame over $U_\alpha$. Let $\ell$ be non-zero vector
in the line $L$. Notice that the fiber of $\Omega \to \text{Gr}_p(W)$ over
a plane $E \in U_\alpha$ consists of those planes $\smash{\widehat{E}}$
which are spanned by bases of the form
\begin{align*}
\lambda_1 \ell + f_{\alpha,1}(E), \ldots, \lambda_k \ell + f_{\alpha,k}(E),
\end{align*}
for a unique vector $\lambda = (\lambda_1,\ldots,\lambda_k)$. The
uniqueness follows from the fact that every vector in $\mathbb{C} = L \oplus
W$ uniquely decomposes into $v_L + v_W$. Now we can define a local
trivialization for $\Omega \to \text{Gr}_p(W)$ over $U_\alpha$:
\begin{align*}
\varphi_\alpha\colon U_\alpha &\to \mathbb{C}^k\\
\widehat{E} &\mapsto \left(\pi\left(\widehat{E}\right), \lambda\right).
\end{align*}
This also defines the vectorspace structure on the fibers: For a given $k$-vector $\lambda$, define
\begin{align*}
\widehat{E}_\lambda = \left<\lambda_1 \ell + f_{\alpha,1}(E),\ldots,\lambda_k \ell + f_{\alpha,k}(E)\right>.
\end{align*}
Using this notation we obtain
\begin{align*}
\widehat{E}_\lambda + \widehat{E}_\mu = \widehat{E}_{\lambda + \mu} \;\text{for all $\lambda,\mu \in \mathbb{C}^k$} \;\text{and}\; a \widehat{E}_{\lambda} = \widehat{E}_{\alpha\lambda} \;\text{for all $a \in \mathbb{C}$}.
\end{align*}
Thus, $\pi\colon \Omega \to \text{Gr}_p(W)$ defines a rank-$k$ vector
bundle. Note that the zero section of this bundle can be naturally
identified with $\text{Gr}_p(W)$. Thus it remains to show that there is a
$G$-equivariant retraction of $\Omega$ to its zero section. We
define the retraction $\varphi_t\colon \Omega \to \Omega$ in local
trivializations for a covering $\{U_\alpha\}$ via
\begin{align*}
\varphi_{\alpha}\colon I \times \mathbb{C}^k &\to \mathbb{C}^k\\
\lambda &\mapsto t\lambda.
\end{align*}
We now show that this gives a globally well-defined map
$\varphi\colon I \times \Omega \to \Omega$: Let $E$ be a plane in
$U_\alpha \cap U_\beta$. Over $U_\alpha$, the map $\varphi_\alpha$
is defined using the frame $f_{\alpha,1},\ldots,f_{\alpha,k}$, while
$\varphi_\beta$ is defined using the frame
$f_{\beta,1},\ldots,f_{\beta,k}$. Let $\smash{\widehat{E}}$ be a
plane in $\restr{\Omega}{U_\alpha \cap U_\beta}$ such that
\begin{align*}
\widehat{E} = \left<\ell^\alpha_1 + w^\alpha_1,\ldots,\ell_k^\alpha + w_k^\alpha\right> = \left<\ell^\beta_1 + w^\beta_1,\ldots,\ell^\beta_k + w^\beta_k\right>,
\end{align*}
where
\begin{align*}
w^\alpha_j = f_{\alpha,j}\left(\pi\left(\widehat{E}\right)\right) \;\text{resp.}\; w^\beta_j = f_{\beta,j}\left(\pi\left(\widehat{E}\right)\right).
\end{align*}
Since these two bases generate the same plane, there exists a $k
\times k$ matrix $A = (a_{ij})$ such that
\begin{align*}
\ell^\alpha_j + w^\alpha_j = \sum a_{ij}\left(\ell^\beta_i + w^\beta_i\right).
\end{align*}
Using the fact that the intersection $L \cap W$ is trivial we obtain
\begin{align*}
\ell^\alpha_j = \sum a_{ij} \ell^\beta_i \;\text{ and }\; w^\alpha_j = \sum a_{ij} w^\beta_i.
\end{align*}
But then the bases
\begin{align*}
t\ell^\alpha_1 + w^\alpha_1,\ldots,t\ell_k^\alpha + w_k^\alpha \;\text{ and }\; t\ell^\beta_1 + w^\beta_1,\ldots,t\ell^\beta_k + w^\beta_k
\end{align*}
are related by the same matrix $A$, hence they define the same plane
$\smash{\widehat{E}}_t$. In other words, the deformation retract $\varphi_t$
is globally well-defined on $\Omega$. In particular the above shows
that for any plane $E = \left<\ell_1 + w_1, \ldots, \ell_k +
w_k\right>$, where the $W$-vectors are not necessarily the ones
defined by one of the trivializing frames of the tautological bundle
we have
\begin{align*}
\varphi_t(E) = \left<(1-t)\ell_1 + w_1, \ldots, (1-t)\ell_k + w_k\right>.
\end{align*}
Now, using the fact that $L$ and $W$ are both $T$-stable,
equivariance follows by a computation:
\begin{align*}
\varphi_t \circ T(E) &= \varphi_t\left(T\left(\left<\ell_1 + w_1, \ldots, \ell_k + w_k\right>\right)\right) \\
&= \varphi_t\left(\left<\overline{\ell_1 + w_1}, \ldots, \overline{\ell_k + w_k}\right>\right) \\
&= \left<(1-t)\overline{\ell_1} + \overline{w_1}, \ldots, (1-t)\overline{\ell_k} + \overline{w_k}\right> \\
&= \left<\overline{(1-t)\ell_1 + w_1}, \ldots, \overline{(1-t)\ell_k + w_k}\right> \\
&= T\left(\left<(1-t)\ell_1 + w_1, \ldots, (1-t)\ell_k + w_k\right>\right) \\
&= T\left(\varphi_t\left(\left<\ell_1 + w_1, \ldots, \ell_k + w_k\right>\right)\right)\\
&= T \circ \varphi_t(E).
\end{align*}
This proves the statement. \end{proof}
The next intermediate result will be the statement that, given a map $H$ from $X$ to $\text{Gr}_p(\mathbb{C}^n)$ where $n - 1 > p$, there always exists a line $L$ in $\mathbb{C}^n$ such that -- at least after a $G$-homotopy -- the image of $H$ does not intersect $\mathcal{L}_L$. Then, lemma~\ref{GrassmannianDeformationRetract} is applicable and we obtain an equivariant homotopy from $H$ to a map $H'$ having its image contained in the lower-dimensional Grassmannian embedded in $\text{Gr}_p(\mathbb{C}^n)$. We begin by outlining some technical preparations and introducing a convenient notation. For a plane $p$-plane $E$ in $\mathbb{C}^n$ we set $E_T = E \cap T(E)$. Note that $0 \leq \dim E_T \leq p$. For $k=0,\ldots,p$ define $\mathcal{E}_k = \{E \in \text{Gr}_p(\mathbb{C}^n)\colon \dim E_T = k\}$. This induces a stratification of $\text{Gr}_p(\mathbb{C}^n)$: \begin{align*}
\text{Gr}_p(\mathbb{C}^n) = \bigcup_{k=0,\ldots,p} \mathcal{E}_k. \end{align*} Note that in particular we have \begin{align*}
\mathcal{E}_p = \left\{E \in \text{Gr}_p(\mathbb{C}^n)\colon T(E) = E\right\} = (\text{Gr}_p(\mathbb{C}^n))_{\mathbb{R}} \end{align*} Define the following incidence set: \begin{align*}
\mathcal{I} = \left\{(L,E) \in \mathbb{P}(\mathbb{C}^n) \times \text{Gr}_p(\mathbb{C}^n)\colon L \subset E\right\}. \end{align*} The set $\mathcal{I}$ is a manifold, which can be seen by considering the diagonal action of $U(n)$ on the product manifold $\mathbb{P}(\mathbb{C}^n) \times \text{Gr}_p(\mathbb{C}^n)$: $U(n)$ acts transitively on $\mathcal{I}$. Furthermore we introduce \begin{align*}
\mathcal{I}_{\mathbb{R}} = \left\{(L,E) \in (\mathbb{P}(\mathbb{C}^n))_{\mathbb{R}} \times \text{Gr}_p(\mathbb{C}^n)\colon L \subset E\right\} = \left\{(L,E) \in \mathcal{I}\colon T(L) = L\right\}. \end{align*} We obtain two projections, namely \begin{align*}
& \pi_1\colon \mathcal{I} \to \mathbb{P}(\mathbb{C}^n)\\
\;\text{and}\; & \pi_2\colon \mathcal{I} \to \text{Gr}_p(\mathbb{C}^n), \end{align*} together with their ``real'' counterparts \begin{align*}
& \pi_{1,\mathbb{R}}\colon \mathcal{I}_{\mathbb{R}} \to (\mathbb{P}(\mathbb{C}^n))_\mathbb{R}\\
\;\text{and}\; & \pi_{2,\mathbb{R}}\colon \mathcal{I}_{\mathbb{R}} \to \text{Gr}_p(\mathbb{C}^n). \end{align*} We regard the incidence manifold $\mathcal{I}$ as being smoothly embedded in some $\mathbb{R}^N$ (e.\,g. with the Whitney embedding theorem).
Defining $M_k := \smash{\pi_{2,\mathbb{R}}^{-1}}(\mathcal{E}_k)$ we obtain -- for each $k$ -- a fiber bundle \begin{align*}
M_k \xrightarrow{\;\pi_{2,\mathbb{R}}\;} \mathcal{E}_k \end{align*} with the fiber, over a plane $E \in \mathcal{E}_k$, being \begin{align*}
F_k &= \left\{(L,E)\colon \text{$L \in \mathbb{P}(\mathbb{C}^n)$, $T(L) = L$ and $L \subset E$}\right\}\\
&= \left\{(L,E)\colon \text{$L \in \mathbb{P}(\mathbb{C}^n)$ and $L \subset E_T$}\right\}\\
&\cong \mathbb{P}(E_T)\\
&\cong \mathbb{P}_{k-1} \end{align*} for all $k=1,\ldots,p$. In particular we obtain the estimate \begin{align}
\label{IterativeRetractionDiscussionFirstEstimate}
\dim_{\mathbb{R}} F_k = \dim_{\mathbb{C}}(E_T) - 1 = k - 1 \leq p - 1. \end{align}
Let us now focus on the image $\Im H$ of a map $H\colon X \to \text{Gr}_p(\mathbb{C}^n)$. Without loss of generality we can assume that $H$ is equivariantly smoothed (theorem~\ref{SmoothApproximation1}). For each $k=0,\ldots,p$ we set $H(X)_k = H(X) \cap \mathcal{E}_k$ and $M = \smash{\pi_{2,\mathbb{R}}^{-1}}(H(X)) \subset \mathcal{I}_\mathbb{R}$. Note that \begin{align*}
M = \left\{(L,E) \in (\mathbb{P}(\mathbb{C}^n))_\mathbb{R} \times \text{Gr}_p(\mathbb{C}^n)\colon \text{$E \in \Im H$ and $L \subset E$}\right\}. \end{align*} Having the above formalism in place, we can now focus on the fundamental problem: Understanding the image of $M$ under the projection $\smash{\pi_{1,\mathbb{R}}}$. The following remark highlights what kind of dimension estimate we need in order to conclude the existence of a line $L$ such that the image $\Im H$ does not intersect $\mathcal{L}_L$. \begin{remark}
If the inequality
\begin{align*}
\text{dim}_H(\pi_{1,\mathbb{R}}(M)) < \text{dim}_H(\mathbb{P}(\mathbb{C}^n)_\mathbb{R}) = n-1
\end{align*}
is satisfied, then there exists a line $L$ in $\mathbb{C}^n$ such that
$\mathcal{L}_L \cap H(X) = \emptyset$. \end{remark}
\begin{proof}
The inequality implies that
$\pi_{1,\mathbb{R}}(M) \subsetneq \mathbb{P}(\mathbb{C}^n)_\mathbb{R}$, since otherwise both sets
would have the same Hausdorff dimension. Recall that
$M = \smash{\pi_{2,\mathbb{R}}^{-1}}(H(X))$. Thus, the non-surjectivity of
$\smash{\restr{\pi_{1,\mathbb{R}}}{M}}$ means that there exists a $T$-fixed
line $L$ in $\mathbb{C}^n$ which has no $\smash{\pi_{1,\mathbb{R}}}$-preimage in the
set
\begin{align*}
\pi_{2,\mathbb{R}}^{-1}(H(X)) = \left\{(L,E) \in \mathbb{P}(\mathbb{C}^n)_\mathbb{R} \times \text{Gr}_p(\mathbb{C}^n)\colon \text{$E \in H(X)$ and $L \subset E$}\right\}.
\end{align*}
If there was a plane $E \in H(X)$ with $L \subset E$, then $(L,E)$
would be in ${\pi_{2,\mathbb{R}}}^{-1}(H(X))$, which is a contradiction. Hence,
there cannot exist such a plane $E$ in the image of $H$. \end{proof}
By the above remark together with theorem~\ref{HDimProperties} (5) it suffices to prove that $\text{dim}_H(M)$ is smaller than $n - 1$ in order to conclude the existence of a $T$-fixed line $L$ such that $\Im H$ has empty intersection with $\mathcal{L}_L$. This is the next goal. The preimage $M$ can be written as a finite union \begin{equation*}
M = \bigcup_{k=0,\ldots,p} {\pi_{2,\mathbb{R}}}^{-1}(H(X)_k). \end{equation*} With corollary~\ref{HDimViaSetDecomposition} it follows that \begin{align}
\label{HDimOfMAsMax}
\text{dim}_H M = \max_k \left\{\text{dim}_H(\pi_{2,\mathbb{R}}^{-1}(H(X)_k))\right\}. \end{align}
But each set $\smash{\pi_{2,\mathbb{R}}^{-1}}(H(X)_k)$ is contained in the total space $M_k$ of the fiber bundle $M_k \to \mathcal{E}_k$. Applying corollary~\ref{HDimOfTotalSpace} yields \begin{align*}
\text{dim}_H\left(\pi_{2,\mathbb{R}}^{-1}(H(X)_k)\right) = \text{dim}_H(H(X)_k) + \dim(F_k). \end{align*} Substituting the dimension of the fiber computed in (\ref{IterativeRetractionDiscussionFirstEstimate}) yields \begin{align*}
\text{dim}_H\left(\pi_{2,\mathbb{R}}^{-1}(H(X)_k)\right) = \text{dim}_H(H(X)_k) + k - 1. \end{align*} Therefore, (\ref{HDimOfMAsMax}) implies: \begin{align}
\label{HDimOfMAsMaxConcrete}
\text{dim}_H(M) = \max_k \left\{\text{dim}_H(H(X)_k) + k - 1\right\}. \end{align} Thus, in order to control $\text{dim}_H(M)$ it suffices to control each $\text{dim}_H(H(X)_k)$. But since $H(X)$ cannot have Hausdorff dimension bigger than two ($H$ is assumed to be smooth) and under the assumption that $p \leq n - 2$, we directly obtain the estimate \begin{align*}
\text{dim}_H(M) &\leq \max\{p, \text{dim}_H(H(X)_p) + p - 1\}\\
&\leq \max\{n-2,\text{dim}_H(H(X)_p) + p - 1\}. \end{align*} Thus we only have to control the dimension of $H(X)_p$. In order to obtain the desired estimate $\text{dim}_H(M) < n-1$ it suffices to have $\text{dim}_H(H(X)_p) \leq 1$. In the following we show that -- up to $G$-homotopy -- this can be assumed. Although, for proving the main result of this section, it suffices to reduce a general Grassmannian $\text{Gr}_p(\mathbb{C}^n)$ until we arrive at $\text{Gr}_1(\mathbb{C}^3) \cong \mathbb{P}_2$, we state the following in its general form, i.\,e. including the statement for the reduction from $\mathbb{P}_2$ to $\mathbb{P}_1$: \begin{proposition}
\label{PropIterativeRetractionDecomposition}
Given an equviariant map $f\colon X \to \text{Gr}_p(\mathbb{C}^n)$ with
$n \geq 3$ and $1 \leq p \leq n-2$, there exists a decomposition of
$\mathbb{C}^n$ into the direct sum of a line
$L \subset \mathbb{C}^n$ and a 1-codimensional subvector space
$W \subset \mathbb{C}^n$, both $T$-stable, such that
$\mathcal{L}_L \cap \Im f = \emptyset$. \end{proposition}
\begin{proof}
We prove the statement in two parts. First we prove that the
statement is true for $0 < p \leq n - 3$ and then we seperately deal
with the case $p = n - 2$. Both proofs work by analyzing the image
$\Im f$ and then using dimension estimates to deduce the existence
of a line $L \subset \mathbb{C}^n$ with $\mathcal{L}_L \cap \Im f$
being the empty set. Without loss of generality we can assume that
$f$ is a smooth $G$-map (see
theorem~\ref{SmoothApproximation1}). Recall that smooth functions
are, in particular, Lipschitz continuous.
First assume $p \leq n - 3$. Since $\text{dim}_H(X) = 2$, it follows that
$\text{dim}_H(H(X)) \leq 2$ by theorem~\ref{HDimProperties} (5). In
particular, $\text{dim}_H(H(X)_k) \leq 2$ for all $k=0,\ldots,p$. Thus,
using (\ref{HDimOfMAsMaxConcrete}) we obtain
\begin{align*}
\text{dim}_H(M) &= \max_{k=0,\ldots,p} \left\{\text{dim}_H(H(X)_k) + k - 1\right\} \\
&\leq p + 1 \\
&\leq n - 2 \\
&< n - 1.
\end{align*}
Hence, the non-surjectivity of $\smash{\restr{\pi_{1,\mathbb{R}}}{M}}$ is
established in the case $p \leq n - 3$. For the remaining case $p =
n - 2$ we can apply lemma~\ref{LemmaSmallIntersectionWithStrata}
(see below) which guarantees that, at least after a $G$-homotopy:
\begin{align*}
\text{dim}_H(H(X)_p) \leq 1.
\end{align*}
In this case we can write
\begin{align*}
\text{dim}_H(M) = \max_{k=0,\ldots,p} \{\text{dim}_H(H(X)_k) + k - 1\} \leq p = n - 2 < n - 1.
\end{align*}
Thus, in both cases we have constructed a homotopy from the map $H$
to a map $H'$ such that $\mathcal{L}_L \cap \Im(H') = \emptyset$ for
some $T$-fixed line $L \subset \mathbb{C}^n$. Take $W$ to be the orthogonal
complement $L^\perp \subset \mathbb{C}^n$. In particular $W$ is also
$T$-stable.
This finishes the proof. \end{proof}
The following lemma completes the proof of the previous proposition~\ref{PropIterativeRetractionDecomposition}: \begin{lemma}
\label{LemmaSmallIntersectionWithStrata}
Every $G$-map $H\colon X \to \text{Gr}_p(\mathbb{C}^n)$ is
equivariantly homotopic to a $G$-map $H'$ such that $\text{dim}_H(H'(X)_p)
\leq 1$. \end{lemma}
\begin{proof}
Recall that $H(X)_p = H(X) \cap \mathcal{E}_p$ and $\mathcal{E}_p =
(\text{Gr}_p(\mathbb{C}^n))_{\mathbb{R}}$. Thus in order to minimize the dimension of
$H(X)_p$ we need to modify $H$ by moving its image away from the
real points in $\text{Gr}_p(\mathbb{C}^n)$. To ease the notation we set $Y =
\text{Gr}_p(\mathbb{C}^n)$. Now we let
\begin{align*}
N \xrightarrow{\;\pi_{Y,\mathbb{R}}\;} Y_\mathbb{R}
\end{align*}
be the normal vector bundle of $Y_{\mathbb{R}}$ in $Y$.
The bundle $N$ comes equipped with a bundle norm $\|\cdot\|$.
Now we can use the standard method
of diffeomorphically identifying an open tubular neighborhood $U$ of
the 0-section of $N$ (which can be identified with $Y_\mathbb{R}$ itself)
with an open neighborhood $V$ of $Y_\mathbb{R}$ in $Y$. We denote this
diffeomorphism $U \to V$ by $\Psi$.
In the following we construct an equivariant homotopy $H_t$ such
that $H_0 = H$ and $H_1$ has the desired property of $\text{dim}_H(H_1(X)
\cap Y_\mathbb{R})$ being at most one.
For this, let $s$ be a generic section in $\Gamma(Y_\mathbb{R},N)$. In
particular this means that there are only finitely many points over
which $s$ vanishes. Furthermore, let $\chi$ be a smooth cut-off
function which is constantly one in a small neighborhood $U'
\subset\subset U$ of the zero section in $N$ and which vanishes
outside of $U$. Now define
\begin{align*}
g^s_t(v) = v - t\chi(v)s(\pi_{Y,\mathbb{R}}(v)).
\end{align*}
After scaling $s$ by a constant such that $\|s(\cdot)\|$ is
sufficiently small, $g^s_t$ defines a diffeomorphism $U \to U$.
Then, via the diffeomorphism $\Psi$, we obtain an induced
diffeomorphism $\smash{\tilde{g}^s_t}\colon V \to V$. Since the
cut-off function $\chi$ vanishes near the boundary of $U$,
$\smash{\tilde{g}^s_t}$ extends to a diffeomorphism $Y \to Y$ which
is the identity outside of $V$. Define
\begin{align*}
H^s_t := \tilde{g}^s_t \circ H\colon X \to Y.
\end{align*}
Note that $H^s_t$ is not necessarily equivariant anymore for
positive $t$. This will be corrected later in the proof by
restricting the maps $H^s_t$ to a (pseudo-)fundamental region $R$ of
$X$ and then equivariantly extending to all of $X$. As a first step
towards equivariance of $H^s_t$, we need to make sure that $H^s_t$
maps $\text{Fix}\, T \subset X$ into $Y_\mathbb{R}$ for all $t$. To guarantee this,
let $f$ be a $C^\infty$ function on $Y_\mathbb{R}$ such that
\begin{align}
\label{IterativeRetractionProofHelperFunction}
\{f = 0 \} = \Psi^{-1}(H(\text{Fix}\, T)) \subset U.
\end{align}
By multiplying the section $s$ with this function $f$ (and, by abuse
of notation, denoting the resulting section again $s$) it is
guaranteed that $\smash{\tilde{g}^s_t}$ is the identity along
$\text{Fix}\, T \subset X$, hence $H^s_t$ still maps the fixed point set
$X^T$ into $Y^T = Y_\mathbb{R}$ for all $t$. In the next step we need to
estimate of the ``critical set''
\begin{align*}
C_s = \{x \in X\colon H^s_1(x) \in Y_\mathbb{R}\}.
\end{align*}
By construction, a matrix $H^s_1(x)$ is contained in the real part
$Y_\mathbb{R}$ iff $\Psi^{-1} \circ H^s_1(x)$ is contained in the
zero-section in the normal bundle $N$. By definition this means
\begin{align*}
g^s_1 \circ \Psi^{-1} \circ H(x) = 0,
\end{align*}
which, by definition of $g^s_t$, is equivalent to saying that
\begin{align*}
\Psi^{-1}(H(x)) = s\left(\pi_{Y,\mathbb{R}}\left(\Psi^{-1}(H(x))\right)\right).
\end{align*}
Using the fact that $C_s$ must be contained in $H^{-1}(V)$ we can
conclude that
\begin{align*}
C_s = \left\{x \in H^{-1}(V)\colon \Psi^{-1} \circ H(x) = s \circ \pi_{Y,\mathbb{R}} \circ \Psi^{-1} \circ H(x)\right\}.
\end{align*}
We have to prove that, for some choice of $s$, $\text{dim}_H(C_s) \leq 1$.
If $H(x) \in Y_\mathbb{R}$ for some $x$, then $H^s_1(x) \in Y_\mathbb{R}$ is almost
never satisfied ($s$ is almost never zero). Hence, it suffices to
estimate the dimension of the set
\begin{align*}
C_s \smallsetminus H^{-1}(Y_\mathbb{R}) = \{x \in \Omega\colon \Psi^{-1} \circ H(x) = s \circ \pi_{Y,\mathbb{R}} \circ \Psi^{-1} \circ H(x)\},
\end{align*}
where
\begin{align*}
\Omega = H^{-1}(V)\smallsetminus H^{-1}(Y_\mathbb{R}).
\end{align*}
We show that by scaling the section $s$ appropriately, we obtain the
desired estimate $\text{dim}_H(C_s) \leq 1$. Observe that $\Omega$ is an
open set in $X$, hence in particular a real manifold of dimension
two, not neccesarily connected. But it has at most countably many
connected components which we denote by $\{\Omega_j\}_{j \in J}$.
Define the following function
\begin{align*}
h_s\colon \Omega &\to \mathbb{R}\\
x &\mapsto \frac{\|s(\pi_{Y,\mathbb{R}}(\Psi^{-1}(H(x))))\|}{\|\Psi^{-1}(H(x))\|}.
\end{align*}
This quotient is well-defined on $\Omega$, since $\Omega$ does not
include the $H$-preimage of $Y_\mathbb{R}$, which corresponds to the
zero-section in $U$. Therefore $h_s$ defines a smooth function on
$\Omega$. It now follows that
\begin{align*}
C_s \smallsetminus H^{-1}(Y_\mathbb{R}) = h_s^{-1}(\{1\}).
\end{align*}
Now we introduce the scaling of the section $s$ such that
$h_s^{-1}(\{1\})$ is at most one-dimensional. Since the number of
connected components $\Omega_j$ ist at most countable, there exists
a number $\varepsilon$ arbitrary close to 1 such that there exists
no component $\Omega_j$ on which $h_s \equiv \varepsilon$ and
furthermore such that $\varepsilon$ is a regular value for $h_s$ on
all the components $\Omega_j$ on which $h_s$ is not constant. It
then follows that $h_s^{-1}(\{\varepsilon\})$ is one-dimensional and
\begin{align*}
h_s^{-1}(\{\varepsilon\}) = h_{\varepsilon s}^{-1}(\{1\}) = C_{\varepsilon s}\smallsetminus H^{-1}(Y_\mathbb{R}).
\end{align*}
Thus, for the section $\varepsilon s$ we have the desired dimension
estimate of the critical set.
Finally we correct the missing equivariance of $H_1$ as follows: In
the type I case we let $Z$ be the cylinder fundamental region and
restrict the homotopy $H_t$ just constructed to $Z$. By
remark~\ref{Class1EquivariantExtension}, the homotopy $\restr{H}{Z}$
extends uniquely to an equivariant homotopy
$\smash{\widetilde{H}}\colon I \times X \to Y$. Define
$H' = \widetilde{H}_1$.
It remains to check that the above estimate of $h_s^{-1}(\{1\})$
remains valid. But this is follows with
corollary~\ref{HDimViaSetDecomposition}, which proves the statement
for the type I involution.
For the type II case we let $R$ be the pseudofundamental region
introduced in the beginning of section
(p.~\pageref{ParagraphGeometryOfClass2}). In this case we need to
make sure that we do not destroy the equivariance propery on the set
$A = A_1 \cup A_2$ (see p.~\pageref{ParagraphGeometryOfClass2}). In
order to be able to use the method of restriction (to $R$) followed
by equivariant extension, we need to make sure that the homotopy
$\restr{H}{I \times R}$ behaves well on the boundary of $R$ (see
remark~\ref{RemarkClassIIEquivariance}). For this we make two small
adjustments to the above construction. First, using the homotopy
extension property together with the simply-connectedness of
$Y$
we make a homotopy to the original map $H$ such that it is constant
along $A_1, A_2 \subset \partial R$ (compare with the type II
normalization, in particular proposition~\ref{Class2Normalization},
p.~\ref{Class2Normalization}). It follows that the images $H(A_1)$
and $H(A_2)$ are one-point sets contained in $Y_\mathbb{R}$. Second, we
modify the function $f$ introduced in
(\ref{IterativeRetractionProofHelperFunction}): Instead of letting
this helper function $f$ vanish exactly over $\Psi^{-1}(H(\text{Fix}\, T))$,
we let it vanish over the bigger set $\Psi^{-1}(H(\partial R))$
(notice that $\text{Fix}\, T \subset \partial R$). It then follows that the
homotopy $H_t$ does not change $H$ along the image $H(\partial R)
\subset V$. In particular, it preserves the compatibility condition
on $A \subset \partial R$ which is required for the
equivariance. Now we can construct the equivariant extension to all
of $X$ as in the type I case above. The desired dimension estimate
remains satisfied because of
corollary~\ref{HDimViaSetDecomposition}. \end{proof}
Having proposition~\ref{PropIterativeRetractionDecomposition} in place, we formulate the main result of this section: \begin{restatable*}{proposition}{PropositionReductionOfGrToCurve}
\label{LemmaReductionOfGrToP1}
Assume $n \geq 3$ and $1 \leq p \leq n-1$. Let $f\colon X \to
\text{Gr}_p(\mathbb{C}^n)$ be a $G$-map. Then $f$ is equivariantly homotopic to
the map $\iota \circ f'$ where $\Im(f')$ is contained in the
Schubert variety $\mathcal{S}$ and $\iota$ is the above embedding of
$\mathcal{S}$ into $\text{Gr}_p(\mathbb{C}^n)$. By identifying $\mathcal{S} \cong
\mathbb{P}_1$, the degree triples (resp. degree pairs) of $f\colon X \to
\text{Gr}_p(\mathbb{C}^n)$ and $f'\colon X \to \mathcal{S} \cong \mathbb{P}_1$ agree. \end{restatable*} Its proof will be given on p.~\pageref{LemmaReductionOfGrToP1}. By proposition~\ref{PropIterativeRetractionDecomposition} we know there exists a line $L$ such that $\mathcal{L}_L$ is not contained in the image of a given map $H$. For using this statement as the building block for the iterative retraction procedure it is convenient to be able to normalize this line $L$. This is made precise in the following remark: \begin{remark}
\label{SOnRNormalizingInGrassmannian}
Let $L$ be a $T$-stable line in $\mathbb{C}^n$. Then there exists a curve
$g(t)$ in $\SOR{n}$ such that $g(0) = \mathrm{Id}$ and $g(1)$ maps
$\mathcal{L}_L$ to $\mathcal{L}_{L_0}$ where $L_0$ is the $n$-th
standard line $L_0 = \mathbb{C}.e_n$. \end{remark}
\begin{proof}
By assumption the line $L$ is $T$-stable. This implies that $L$ is
generated by a vector $v_n$ of unit length such that $T(v_n) =
v_n$. In other words, $v_n$ is in $(\mathbb{C}^n)_\mathbb{R} = \mathbb{R}^n$. Furthermore,
the orthogonal complement $L^\perp$ of $L$ is also
$T$-invariant. Let $(v_1,\ldots,v_{n-1})$ be an orthonormal basis of
$(L^\perp)_\mathbb{R}$. Then, $(v_1,\ldots,v_n)$ is an orthonormal basis of
$\mathbb{C}^n$ consisting solely of real vectors. Define $g$ as the element
of $\SOR{n}$ which maps $v_j$ to $e_j$ for all
$j=1,\ldots,n$. Using the path-connectedness of $\SOR{n}$ we can
find a path $g(t)$ such that $g(0) = \mathrm{Id}$ and $g(1) = g$. Hence, by
construction, $g(1)$ maps the line $L$ to $L_0$. It remains to show
that
\begin{align*}
g(\mathcal{L}_L) = \mathcal{L}_{L_0}.
\end{align*}
For this, let $E$ be a $p$-plane in $\mathcal{L}_L$. Then, by
definition, $L \subset E$. But then also $g(L) = L_0 \subset
g(E)$. On the other hand, given a plane $E$ in with $L_0 \subset E$,
then define $E' = g^{-1}(E)$. It follows that $L \subset E'$, on
other words $E' \in \mathcal{L}_L$. Now we see that $E = g(E') \in
\mathcal{L}_L$. \end{proof}
Combining the above we can make a reduction from $\text{Gr}_p(\mathbb{C}^n)$ to the smaller manifold $\text{Gr}_p(\mathbb{C}^{n-1})$: \begin{proposition}
\label{PropositionIterativeRetraction}
Given an equivariant map $f\colon X \to \text{Gr}_p(\mathbb{C}^n)$ where
$3 \leq n$ and $1 \leq p \leq n - 2$, then there exists a
$G$-homotopy from $f$ to a map $f'$ whose image is contained in
$\text{Gr}_p(\mathbb{C}^{n-1})$, where $\mathbb{C}^{n-1}$ is regarded as being
embedded in $\mathbb{C}^n$ as $(z_1,\ldots,z_n) \mapsto (z_1,\ldots,z_n,0)$. \end{proposition}
\begin{proof}
Due to the conditions on $p$ and $n$,
proposition~\ref{PropIterativeRetractionDecomposition} is
applicable. Hence there exists a $T$-stable decomposition $\mathbb{C}^n = L
\oplus W$ such that $\mathcal{L}_L \cap \Im f = \emptyset$. Using
remark~\ref{SOnRNormalizingInGrassmannian} it follows that there
exists a curve $g(t)$ of $\SOR{n}$ transformations such that
$g(1)$ maps $\mathcal{L}_L$ to $\mathcal{L}_{L_0}$ where $L_0$ is
the standard line $\mathbb{C}.e_n$. Now define a homotopy $F_t =
g(t)f$. Then, by construction, $F_0 = f$ and $\Im F_1 \cap
\mathcal{L}_{L_0} = \emptyset$. In this situation
lemma~\ref{GrassmannianDeformationRetract} is applicable and we
obtain an equivariant homotopy from $f$ to a map $f'$ such that
$\Im(f')$ is contained in the lower dimensional Grassmannian
$\text{Gr}_p(\mathbb{C}^{n-1})$. \end{proof}
Taking the degree invariants (triples and pairs) into account we state the following addition to the previous proposition: \begin{lemma}
\label{DegreeInvariantsConstantDuringRetraction}
As before, assume $n \geq 4$ and $1 \leq p \leq n - 2$. Let $f$ be a
$G$-map $X \to \text{Gr}_p(C^n)$ and denote the canonical embedding
$\text{Gr}_p(\mathbb{C}^{n-1}) \hookrightarrow \text{Gr}_p(\mathbb{C}^n)$ by $\iota$. Assume
there exists a $G$-map $f'\colon X \to \text{Gr}_p(\mathbb{C}^{n-1})$ such that
$f$ and $\iota \circ f'$ are equivariantly homotopic. Then,
depending on the involution type, the degree triples (type I)
resp. the degree pairs (type II) of $f$ and $f'$ agree for any two
such maps. \end{lemma}
\begin{proof}
The existence of the map $f'$ is the statement of
proposition~\ref{PropositionIterativeRetraction}: we obtain a map $f'$,
equivariantly homotopic to $f$, whose image is contained in
$\text{Gr}_p(\mathbb{C}^{n-1})$. Thus, we can regard $f'$ as a map to
$\text{Gr}_p(\mathbb{C}^{n-1})$ and clearly we then obtain $f = \iota \circ f'$.
Regarding the degree invariants: By
remark~\ref{GrassmannianEmbeddingIsoInHomology}, the embedding
$\iota$ induces an isomorphism
\begin{align*}
\iota_*\colon H_2(\text{Gr}_p(\mathbb{C}^{n-1}),\mathbb{Z}) \xrightarrow{\;\sim\;} H_2(\text{Gr}_p(\mathbb{C}^n),\mathbb{Z}).
\end{align*}
Note that $\iota_*$ maps a generating cycle $[\mathcal{C}]$ of
$H_2(\text{Gr}_p(\mathbb{C}^{k-1}),\mathbb{Z})$ to the generating cycle $\iota_*([C]) =
[\iota(\mathcal{C})]$ of $H_2(\text{Gr}_p(\mathbb{C}^k),\mathbb{Z})$. We regard
$\mathcal{C}$ as being canonically oriented as a complex manifold.
Since $f_* = \iota_* \circ f'_*$, it follows that $f_*$ and $f'_*$
are defined in terms of the same multiplication factor; in other
words: The total degree of the map does not change when we regard it
as a map to the lower dimensional Grassmann manifold.
It remains to check that the fixed point signatures do not change
during the retraction. For this let $C \subset X$ be one of the
fixed point circles (any of the two circles in type I or the unique
circle in type II). The restrictions
\begin{align*}
\Restr{f}{C}\colon C \to \text{Gr}_p(\mathbb{R}^n) \;\text{ and }\; \Restr{\iota \circ f'}{C}\colon C \to \text{Gr}_p(\mathbb{R}^n)
\end{align*}
are homotopic, thus they define the same class $[\gamma]$ in the
fundamental group of $\text{Gr}_p(\mathbb{R}^n)$. By construction $f'$ has its
image contained in $\text{Gr}_p(\mathbb{R}^{n-1}) \subset \text{Gr}_p(\mathbb{R}^n)$, thus its
restriction to the circle $C$ can be regarded as a map $C \to
\text{Gr}_p(\mathbb{R}^{n-1})$, defining a homotopy class $[\gamma']$ in
$\pi_1(\text{Gr}_p(\mathbb{R}^{n-1}))$. By
remark~\ref{RealGrassmannianEmbeddingIsoInHomotopy}, the inclusion
$\text{Gr}_p(\mathbb{R}^{n-1}) \hookrightarrow \text{Gr}_p(\mathbb{R}^n)$ induces an isomorphism
of their fundamental groups if $n \geq 4$. This implies that $[\gamma]$
and $[\gamma']$ are either both trivial or both non-trivial, which
proves the statement. \end{proof} To complete the previous lemma we need the following two remarks: \begin{remark}
\label{GrassmannianEmbeddingIsoInHomology}
Assume $0 < p < k - 1$. The embedding
\begin{align*}
\iota\colon \text{Gr}_p(\mathbb{C}^{k-1}) \hookrightarrow \text{Gr}_p(\mathbb{C}^k)
\end{align*}
of Grassmann manifold induces an isomorphism on their second homology
groups. More precisely, let $\mathcal{S}$ be the Schubert variety
generating $H_2(\text{Gr}_p(\mathbb{C}^{k-1}),\mathbb{Z})$ (see
p.~\pageref{DefOfSchubertVarietyC}), then
\begin{align*}
\iota_*([\mathcal{S}]) = [\iota(\mathcal{S})].
\end{align*} \end{remark}
\begin{proof}
Note that the second homology groups of complex Grassmann manifolds
are infinite cyclic. Fix the standard flag in $\mathbb{C}^k$ and let
$\mathcal{C}$ be the Schubert variety with respect to this flag
which generates $H_2(\text{Gr}_p(\mathbb{C}^{k-1}),\mathbb{Z})$. By means of the embedding
$\iota$ it can also be regarded as being contained in the bigger
Grassmannian $\text{Gr}_p(\mathbb{C}^k)$, where it also generates
$H_2(\text{Gr}_p(\mathbb{C}^k),\mathbb{Z})$. In other words: the embedding $\iota$ maps
the generator of $H_2(\text{Gr}_p(\mathbb{C}^{k-1}),\mathbb{Z})$ to the generator of
$H_2(\text{Gr}_p(\mathbb{C}^k),\mathbb{Z})$. It follows that the induced map
\begin{align*}
\iota_*\colon H_2\left(\text{Gr}_p\left(\mathbb{C}^{k-1}\right),\mathbb{Z}\right) \to H_2\left(\text{Gr}_p\left(\mathbb{C}^k\right),\mathbb{Z}\right)
\end{align*}
is an isomorphism. \end{proof}
\begin{remark}
\label{RealGrassmannianEmbeddingIsoInHomotopy}
Let $k \geq 4$. The embedding
\begin{align*}
\iota\colon \text{Gr}_p(\mathbb{R}^{k-1}) \hookrightarrow \text{Gr}_p(\mathbb{R}^k)
\end{align*}
of real Grassmannians induces an isomorphism of their fundamental
groups. \end{remark}
\begin{proof}
We begin the proof with a general remark: For every Grassmannian
$\text{Gr}_p(\mathbb{R}^m)$ we have the following double cover
\begin{align}
\label{GrassmannianCovering}
\widetilde{\text{Gr}}_p(\mathbb{R}^m) \to \text{Gr}_p(\mathbb{R}^m),
\end{align}
where $\smash{\widetilde{\text{Gr}}_p}(\mathbb{R}^k)$ denotes the \emph{oriented}
Grassmannian. The covering map is given by forgetting the
orientation of each subspace. It is known that for $m > 2$ the
oriented Grassmannian $\text{Gr}_p(\mathbb{R}^m)$ is simply connected. Thus, by
assumption about $k$, the oriented Grassmannians
$\smash{\widetilde{\text{Gr}}_p}(\mathbb{R}^{k-1})$ and
$\smash{\widetilde{\text{Gr}}_p}(\mathbb{R}^k)$ are
simply-connected.
Thus, (\ref{GrassmannianCovering}) defines the universal cover of
$\text{Gr}_p(\mathbb{R}^m)$, and this implies that $\pi_1(\text{Gr}_p(\mathbb{R}^m))$ is
isomorphic to the Deck transformation group
$\text{Deck}\,(\smash{\widetilde{\text{Gr}}_p}(\mathbb{R}^m)/\text{Gr}_p(\mathbb{R}^m))$ (see
\cite[p.~71]{Hatcher}). The Deck transformation group in this case
consists of the single homeomorphism $\sigma$, which flips the
orientation on each subspace, thus it is $C_2$ and we obtain
$\pi_1(\text{Gr}_p(\mathbb{R}^k)) \cong C_2$.
To show that the induced map
\begin{align*}
\iota_*\colon \pi_1(\text{Gr}_p(\mathbb{R}^{k-1})) \to \pi_1(\text{Gr}_p(\mathbb{R}^k))
\end{align*}
is an isomorphism it suffices to show that a non-trivial loop
$\gamma$ in $\text{Gr}_p(\mathbb{R}^{k-1})$ will still be non-trivial when it is,
using the embedding $\iota$, regarded as a loop in $\text{Gr}_p(\mathbb{R}^k)$. We
have the following diagram:
\[
\xymatrix{
\widetilde{\text{Gr}}_p(\mathbb{R}^{k-1}) \ar[d] \ar@{^{(}->}[r]^{\hat{\iota}} & \widetilde{\text{Gr}}_p(\mathbb{R}^k) \ar[d]\\
\text{Gr}_p(\mathbb{R}^{k-1}) \ar@{^{(}->}[r]_{\iota} & \text{Gr}_p(\mathbb{R}^k)
}
\]
Let $\gamma$ be a non-trivial loop in $\text{Gr}_p(\mathbb{R}^{k-1})$, say
$\gamma(0) = \gamma(1) = E$. Its lift to the universal cover is a
non-closed curve $\smash{\hat{\gamma}}$ with
$\smash{\hat{\gamma}}(0) = E^+$ and $\smash{\hat{\gamma}}(1) = E^-$,
where $E^+$ and $E^-$ denote the same plane $E$ but equipped with
different orientations, i.\,e. $\sigma(E^+) = \sigma(E^-)$. It
follows that $\smash{\hat{\iota}} \circ \smash{\hat{\gamma}}$ is a lift of
$\iota \circ \gamma$. The curve
$\smash{\hat{\iota}} \circ \smash{\hat{\gamma}}$ is not closed, as it is
still a curve whose endpoints are related by the
orientation-flipping map $\sigma$. Under the isomorphism from the
Deck transformation group to the fundamental group of the base (see
e.\,g. on p.~34 the proof of theorem~5.6 in
\cite{ForsterRiemannSurfaces}), $\sigma$ corresponds to the curve
$\iota \circ \gamma$. Since $\sigma$ is non-trivial, so is
$\iota \circ \gamma$. This proves that the embedding $\iota$ induces
an isomorphism on the fundamental groups. \end{proof}
Now we can finally prove our main reduction statement: \PropositionReductionOfGrToCurve \begin{proof}
Note that in the case $n=3$, this is just the statement of
lemma~\ref{ReductionP2toP1} together with
remark~\ref{ReductionP2toP1Class1}
resp. remark~\ref{ReductionP2toP1Class2}.
If $n \geq 4$, apply proposition~\ref{PropositionIterativeRetraction}
iteratively until we arrive at $n = p + 1$, producing a $G$-homotopy
from $f$ to a map $f'$ whose image is contained in
$\text{Gr}_p(\mathbb{C}^{p+1})$. Note that by assumption $p + 1 > 3$. This space
can be equivariantly identified with $\text{Gr}_1(\mathbb{C}^{p+1})$ by
remark~\ref{GrassmannianIdentificationEquivariant}. Now
proposition~\ref{PropositionIterativeRetraction} can be applied again
iteratively to the map
\begin{align*}
\tilde{f}\colon X \to \text{Gr}_p(\mathbb{C}^{p+1}) \cong \text{Gr}_1(\mathbb{C}^{p+1})
\end{align*}
until we arrive at $p = 2$, yielding a map
\begin{align*}
f'\colon X \to \text{Gr}_1(\mathbb{C}^3) \cong \mathbb{P}_2.
\end{align*}
By lemma~\ref{DegreeInvariantsConstantDuringRetraction}, up to this
point, the degree triple (type I) resp. the degree pair (type II) of
the map is unchanged. For the last reduction step to $\text{Gr}_1(\mathbb{C}^2)$
we first apply a $G$-homotopy in order to make sure that both maps
are fixed point normalized (see
definition~\ref{DefinitionFixpointNormalizationP2}). Then
lemma~\ref{ReductionP2toP1} and remark~\ref{ReductionP2toP1Class1}
(type I) resp. remark~\ref{ReductionP2toP1Class2} (type II) together
with remark~\ref{GrassmannianEmbeddingIsoInHomology} imply that the
final reduction step to $\mathcal{S} \cong \mathbb{P}_1$ also keeps the
degree triple (resp. the degree pair) unchanged. Thus, in the end we
have an equivariant homotopy from $f$ to a map whose image is
contained in $\mathcal{S}$ and whose degree triples (resp. pairs) as
a map $X \to \mathcal{S} \cong \mathbb{P}_1$ are those of $f$. \end{proof}
We can now prove the main result for the equivariant homotopy classification of maps $X \to \text{Gr}_p(\mathbb{C}^n)$: \begin{theorem}
\label{ClassificationMapsToGrassmannians}
Let the torus $X$ be equipped with the type I involution (resp. the
type II involution). Assume $n > 3$ and $1 < p < n$. Then the
homotopy class of a map $f$ in $\mathcal{M}_G(X,\text{Gr}_p(\mathbb{C}^n))$ is
completely determined by its degree triple (type I) resp. its degree
pair (type II). Furthermore, the image $\Im(\mathcal{T})$
(resp. $\Im(\mathcal{P})$) consists of those degree triples
$\Triple{m_0}{d}{m_1}$ (resp. degree pairs $\Pair{m}{d}$) satisfying
\begin{align*}
d \equiv m_0 + m_1 \mod 2 \;\text{(resp. $d \equiv m \mod 2$)}.
\end{align*} \end{theorem}
\begin{proof}
We only prove the statement for the type I involution case, as the
other case works analogously. Let $f$ and $g$ be two $G$-maps with
the same degree triple. By proposition~\ref{LemmaReductionOfGrToP1}
these maps are equivariantly homotopic to maps $\iota \circ f'$ and
$\iota \circ g'$, where $f'$ and $g'$ are $G$-maps $X \to
\mathcal{S}$ and $\iota$ is the embedding of the Schubert variety
$\mathcal{S}$ into $\text{Gr}_p(\mathbb{C}^n)$. Furthermore, by the same
statement, the degree invariants remain unchanged.
By transitivity, this proves the first statement.
Regarding the image $\Im(\mathcal{T})$ (resp. $\Im(\mathcal{P})$):
As before, we can use the embedding
\begin{align*}
\mathcal{S} \hookrightarrow \text{Gr}_p(\mathbb{C}^n)
\end{align*}
together with the iterative retraction method to show that the
conditions
\begin{align*}
d \equiv m_0 + m_1 \mod 2\;\; \text{(type I)}\;\;\text{resp.}\;\; d \equiv m \mod 2\;\; \text{(type II)}
\end{align*}
are both sufficient and necessary for the degree triples
(resp. degree pairs) to be in the image of $\mathcal{T}$ (type I)
resp. $\mathcal{P}$ (type II). \end{proof}
\subsection{Classification of Maps to $\mathcal{H}_n^*$}
As in the previous cases we note that the degree triple map (resp. the degree pair) is so far only defined on the mapping spaces $\mathcal{M}_G(X,\text{Gr}_p(\mathbb{C}^n))$. But after fixing an identification of each orbit $U(n).\I{p}{q} \subset \mathcal{H}_{(p,q)}$ ($0 < p,q < n$) with $\text{Gr}_p(\mathbb{C}^n)$, the degree triple map (resp. the degree pair map) is also defined on the mapping spaces $\mathcal{M}_G(X,\mathcal{H}_{(p,q)})$ for each non-definite signature $(p,q)$. This allows us to state and prove the main result of this section:
\begin{theorem}
\label{HamiltonianClassificationRankN}
Let $X$ be a torus equipped with either the type I or the type II
involution. Assume $n \geq 3$ and $0 < p,q < 1$. Then:
\begin{enumerate}[(i)]
\item The sets $[X,\mathcal{H}_{(n,0)}]_G$ and
$[X,\mathcal{H}_{(0,n)}]_G$ are trivial.
\item Two $G$-maps $X \to \mathcal{H}_{(p,q)}$ are $G$-homotopic iff
their degree triples (type I) resp. their degree pairs (type II)
agree.
\item The realizable degree triples $\Triple{m_0}{d}{m_1}$ (type I)
resp. degree pairs $\Pair{m}{d}$ (type II) are exactly those
which satisfy
\begin{align*}
d \equiv m_0 + m_1 \mod 2 \;\text{ resp. }\;d \equiv m \mod 2.
\end{align*}
\end{enumerate} \end{theorem}
\begin{proof}
The case $n=3$ has already been dealt with in
theorem~\ref{HamiltonianClassificationRank3}. Therefore it suffices
to consider the case $n>3$. The topological triviality of the
definite signature cases is handled in
remark~\ref{RemarkDefiniteComponentsRetractable}. Let $f$ and $g$ be
two equivariant maps $X \to \smash{\mathcal{H}_{(p,q)}^*}$. By
proposition~\ref{GrassmannianStrongDeformationRetract} the image
space has $\text{Gr}_p(\mathbb{C}^n)$ as equivariant deformation retract. Thus,
$f$ and $g$ can be regarded as $G$-maps $X \to \text{Gr}_p(\mathbb{C}^n)$ with the
same degree triple resp. the same degree pair. Now
theorem~\ref{ClassificationMapsToGrassmannians} can be applied. Part
(i) implies that $f$ and $g$ are $G$-homotopic while part (ii)
contains the statement about realizable degree triples resp. degree
invariants. \end{proof}
Remark~\ref{HamiltonianMapRealizationRank3} explains how to concretely realize maps to $\mathcal{H}_3$ of a given degree invariant. This works equally well in the general situation. Let $(p,q)$ be a fixed signature ($p+q=n$) such that $0 < p,q < n$. If $n$ is smaller then $4$ we end up in one projective situations already handled (i.\,e. maps to $\mathbb{P}_1$ or maps to $\mathbb{P}_2$). Thus, assume $n \geq 4$. In section~\ref{SectionN=2} we have seen how to construct maps of a (realizable) degree triple resp. degree pair to $S^2 \cong \mathbb{P}_1$. We then identify $\mathbb{P}_1$ with the Schubert variety $\mathcal{S}$ and compose this map with the embedding $\mathcal{C} \hookrightarrow \text{Gr}_p(\mathbb{C}^n)$ to obtain a map into the $\text{Gr}_p(\mathbb{C}^n)$. The latter Grassmannian needs to be embedded as the $U(n)$-orbit of $\I{p}{q}$ into $\mathcal{H}_{(p,q)}$ (see proposition~\ref{GrassmannianStrongDeformationRetract}).
\chapter{Topological Jumps} \label{ChapterJumps}
In this chapter we construct curves \begin{align*}
H\colon [-1,1] \times X \to \mathcal{H}_n \end{align*} of equivariant maps $X \to \mathcal{H}_n$ such that the maps $H_{-1}$ and $H_{+1}$, whose images are assumed to be contained in $\mathcal{H}_n^*$, represent distinct $G$-homotopy classes. In order to make this precise, we make the following definitions: \begin{definition}
Let $H\colon X \to \mathcal{H}_n$ be a $G$-map. Then we define its
\emph{singular set} to be the set
\begin{align*}
S(H) = \{x \in X\colon \det H(x) = 0\} \subset X.
\end{align*}
A $G$-map $X \to \mathcal{H}_n$ is called \emph{singular}
(resp. non-singular) if its singular set is non-empty (resp. empty). \end{definition} \begin{definition}
\label{DefinitionJumpCurve}
A \emph{jump curve} from $H_-$ to $H_+$ (both in
$\mathcal{M}_G(X,\mathcal{H}_n)$) is a $G$-map\footnote{$G$ is
assumed to act trivially on the interval.}
$H\colon [-1,1] \times X \to \mathcal{H}_n$ such that
\begin{enumerate}[(i)]
\item $H_{\pm 1} = H_\pm$ and
\item $H_t = H(t,\cdot)$ is non-singular for $t \neq 0$.
\end{enumerate} \end{definition} Given two $G$-maps $H_\pm\colon X \to \mathcal{H}_n$ belonging to distinct $G$-homotopy classes and a jump curve $H_t$ from $H_-$ to $H_+$, then $H_0$ must be singular. Otherwise $H_t$ would induce a $G$-homotopy $I \times X \to \mathcal{H}_n^*$ from $H_-$ to $H_+$, which would imply that $H_-$ and $H_+$ are equivariantly homotopic.
Of course, given a two $G$-maps $H_\pm\colon X \to \mathcal{H}_n$, we can always consider the affine curve \begin{align*}
(1-t)H_- + tH_+ \end{align*} of $G$-maps connecting $H_-$ and $H_+$ in the vectorspace $\mathcal{H}_n$. But in this case we have no control over the singular set; neither is it guaranteed that the degeneration only occurs at $t=0$, nor that the singularity set $S(H_0)$ is in some sense ``small''. In this chapter we construct jump curves obeying the restriction that the singular set $S(H_0)$ is \emph{discrete}. The main result of this chapter is the description of a procedure for constructing jump curves for $G$-maps $X \to \mathcal{H}_{(p,q)} \subset \mathcal{H}_n^*$ ($n \geq 2$) from any $G$-homotopy class to any other $G$-homotopy class with a finite singular set; the only requirement is that the the signature $(p,q)$ remains unchanged. Note that jumps from one signature $(p,q)$ to a different signature $(p',q')$ are not possible with a finite singular set as shown in the following remark: \begin{remark}
If a curve $H_t\colon X \to \mathcal{H}^*_n$ of $G$-maps whose only
degeneration occurs at $t=0$ jumps from one signature $(p,q)$ to a
different signature $(p',q')$, then the singular set
$S(H_0)$ is the whole space $X$. \end{remark} \begin{proof}
Under the assumption that $(p,q) \not= (p',q')$, let $x$ be an
arbitray point in $X$. We have to show that $H_0(x)$ is
singular. Assume the opposite, i.\,e. that $H_0(x)$ is
non-singular. Then $c(t) = H_t(x)$ is a continuous curve in
$\mathcal{H}_n^*$ with $c(-1) \in \mathcal{H}_{(p,q)}$ and $c(1) \in
\mathcal{H}_{(p',q')}$. But for $(p,q) \neq (p',q')$,
$\mathcal{H}_{(p,q)}$ and $\mathcal{H}_{(p',q')}$ denote two
distinct connected components of $\mathcal{H}^*_n$, this yields a
contradiction. Therefore, $S(H_0) = X$. \end{proof} Note that in order to be able to construct jump curves, it does not suffice to consider maps of the type $X \to Y$ where $Y$ is a Grassmannians $\text{Gr}_p(\mathbb{C}^n)$, as these spaces are the deformation retracts of the components $\mathcal{H}_{(p,q)}$, which consist entirely of non-singular matrices. Instead we need to let the image space $Y$ be a subspace of the closure of $\mathcal{H}_{(p,q)}$ having non-empty intersection with its boundary: \begin{align*}
& Y \subset \text{cl}\,(\mathcal{H}_{(p,q)})\\
& Y \cap \partial\mathcal{H}_{(p,q)} \not= \emptyset. \end{align*} For the curve $H_t$ to be a jump curve it must satisfy \begin{align*}
\Im(H_0) \cap \partial\mathcal{H}_{(p,q)} \not= \emptyset, \end{align*} as otherwise it would not be singular at $t=0$ and the possibility of jumps would be excluded. See figure~\ref{fig:jumps} for a depiction of a jump curve.
As in the previous chapter, we handle the case $n=2$ first and then use the methods for $n=2$ for proving a general statement. For the $n=2$ case we will use as image space a certain subspace of $\mathrm{cl}(\mathcal{H}_{(1,1)})$, namely the vector space $i\mathfrak{su}_2 \subset \mathrm{cl}(\mathcal{H}_{(p,q)})$ consisting of the hermitian operators of trace zero in $\mathcal{H}_2$. The space $i\mathfrak{su}_2\smallsetminus\{0\}$ contains the $U(2)$-orbit of $\I{1}{1}$, which we have already identified as an equivariant strong deformation retract of $\mathcal{H}_{(1,1)}$ (see proposition~\ref{Rank2MixedSignatureReduction}), and the origin in $i\mathfrak{su}_2$, which is the unique singular matrix among the hermitian matrices of trace zero, is contained in the boundary $\partial\mathcal{H}_{(1,1)}$. For the general case we embed $i\mathfrak{su}_2$ into $\mathcal{H}_{(p,q)}$ such that it contains the -- with respect to the standard flag for $\mathbb{C}^n$ -- unique one-dimensional Schubert variety $\mathcal{S} \subset \text{Gr}_p(\mathbb{C}^n)$ (see the discussion on p.~\pageref{SchubertVarietyDiscussion}). This setup allows us to use the same methods as for the $n=2$ case.
\begin{figure}
\caption{A jump curve.}
\label{fig:jumps}
\end{figure}
\section{Maps into $\mathrm{cl}(\mathcal{H}_2)$}
Let us first understand how jumps can occur for maps into $\mathcal{H}_2$. Since the components $\mathcal{H}_{(2,0)}$ and $\mathcal{H}_{(0,2)}$ are topologically trivial, the only relevant component to consider is $\mathcal{H}_{(1,1)}$. Recall the definition of the vector space $i\mathfrak{su}_2$: \begin{align*}
i\mathfrak{su}_2 = \left\{
\begin{pmatrix}
a & \phantom{-}b \\
\overline{b} & -a
\end{pmatrix}\colon a \in \mathbb{R}, b \in \mathbb{C}\right\}. \end{align*} The space $i\mathfrak{su}_2\smallsetminus\{0\}$ is contained in $\mathcal{H}_{(1,1)}$. It is important that this space is not \emph{entirely} contained in $\mathcal{H}_{(1,1)}$, as we have \begin{align*}
i\mathfrak{su}_2 \cap \partial\mathcal{H}_{(1,1)} = \left\{
\begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix}\right\} \end{align*} As discussed earlier, $i\mathfrak{su}_2$ can be linearly identified with $\mathbb{R}^3$ via \begin{align}
\label{JumpDiscussionIsomToR3}
\begin{pmatrix}
a & \phantom{-}b \\
\overline{b} & -a
\end{pmatrix} \mapsto
\begin{pmatrix}
a \\
\Re(b) \\
\Im(b)
\end{pmatrix}. \end{align} Recall that the $U(2)$-orbit of the diagonal matrix $\I{1}{1}$ is contained in $i\mathfrak{su}_2\smallsetminus\{0\}$ and is identified by the isomorphism (\ref{JumpDiscussionIsomToR3}) with the unit sphere in $\mathbb{R}^3$.
In the following we construct jump curves as follows: Given two $G$-maps $H_\pm\colon X \to S^2$, we regard them as maps into $S^2 \subset \mathbb{R}^3$ and then construct a map of the form $H\colon [-1,1] \times X \to \mathbb{R}^3$ such that $H_{-1} = H_-$ and $H_{+1} = H_+$. First we introduce some new definitions, which are slightly more suitable for this concrete approach than definition~\ref{DefinitionJumpCurve}: \begin{definition}
Let $H\colon X \to \mathbb{R}^3$ be a $G$-map. We define the \emph{singular
set} $S(H)$ of $H$ to be the fiber $H^{-1}(\{0\})$. The singular
set of a map $(Z,C) \to (\mathbb{R}^3,\{z = 0\})$ is the singular set of its
equivariant extension. \end{definition}
\begin{definition}
\label{DefinitionJumpCurveR3}
A \emph{jump curve} from $H_-$ to $H_+$ (both in
$\mathcal{M}_G(X,\mathbb{R}^3\smallsetminus\{0\})$) is a $G$-map
$H\colon [-1,1] \times X \to \mathbb{R}^3$ such that
\begin{enumerate}[(i)]
\item $H_{\pm 1} = H_\pm$ and
\item $0 \not\in \Im(H_t)$ for $t \neq 0$.
\end{enumerate}
Futhermore, a \emph{jump curve} from
\begin{align*}
H_- \in \mathcal{M}\left((Z,C),\left(\mathbb{R}^3\smallsetminus\{0\},\left(\mathbb{R}^3\smallsetminus\{0\}\right)\cap\{z=0\}\right)\right)
\end{align*}
to
\begin{align*}
H_+ \in \mathcal{M}\left((Z,C),\left(\mathbb{R}^3\smallsetminus\{0\},\left(\mathbb{R}^3\smallsetminus\{0\}\right)\cap\{z=0\}\right)\right)
\end{align*}
is a $G$-map $H\colon [-1,1] \times (Z,C) \to (\mathbb{R}^3,\{z = 0\})$
such that the equivariant extension of $H$ to $[-1,1] \times X$ is a
jump curve from the equivariant extension of $H_-$ to the
equivariant extension of $H_+$. \end{definition} By means of the isomorphism (\ref{JumpDiscussionIsomToR3}), a jump curve between maps $X \to \mathbb{R}^3\smallsetminus\{0\}$ induces a jump curve between maps $X \to \mathcal{H}_n^*$. For the concrete construction of the curve $H_t$ we often require that the maps $H_\pm$ are in \emph{normal form} (see p.~\pageref{RemarkAboutNormalForms}) or in \emph{modified normal form} (see p.~\pageref{DefModifiedNormalForm}, definition~\ref{DefModifiedNormalForm}). We use the terms \emph{curves} and \emph{homotopies} (of $G$-maps) interchangeably.
\subsection{Type I}
In this section, $X$ always denotes the torus equipped with the type I involution. For illustrative purposes, we begin with a very naive jumping method, which does not satisfy our requirement of the singular set being small: \begin{lemma}
Let $X$ be the type I torus and let $d_0^\pm, d_1^\pm, d_0^\pm,
d_1^\pm$ be integers. Denote by $H_\pm\colon X \to
\mathcal{H}_{(1,1)}$ the $G$-maps in normal form for the triples
\begin{align*}
\Triple{d_0^\pm}{d_0^\pm-d_1^\pm}{d_1^\pm}.
\end{align*}
Then there exists a jump curve
\begin{align*}
H\colon [-1,1] \times X \to \text{cl}\,(\mathcal{H}_{(1,1)})
\end{align*}
from $H_-$ to $H_+$ such that the singular set $S(H_0)$ consists of
the two circles $C_0 \cup C_1$. \end{lemma}
\begin{proof}
Let $H_\pm\colon X \to S^2 \subset \mathbb{R}^3$ be the maps in normal form
for the triples
$\smash{\Triple{d_0^\pm}{d_0^\pm - d_1^\pm}{d_1^\pm}}$ as it has
been constructed as part of the proof of
proposition~\ref{TripleBuildingBlocks} (ii). In the following we
construct a jump curve
\begin{align*}
H\colon [-1,1] \times X \to \mathbb{R}^3
\end{align*}
from $H_-$ to $H_+$. The maps $H_\pm$ are of the form
\begin{align*}
H_{\pm} =
\begin{pmatrix}
x_\pm \\
y_\pm \\
z_\pm
\end{pmatrix}.
\end{align*}
Since both maps are in normal form (see proof of
proposition~\ref{TripleBuildingBlocks}), they have the convenient
property that $z_- = z_+$. The functions $x_\pm$ and $y_\pm$ define
the rotation defined by the respective degree triples. Note that a
matrix $H_\pm(p)$ is singular iff $x_\pm(p) = y_\pm(p) = z(p) = 0$.
The functions $x_-$, $y_-$ and $z$ (resp. $x_+$, $y_+$ and $z$) do
not simultaneously vanish, because the maps $H_\pm$ are non-singular
by assumption. Now we define the jump curve as:
\begin{align*}
H_t = \begin{pmatrix}
x_t \\
y_t \\
z
\end{pmatrix}
\end{align*}
where
\begin{align*}
x_t =
\begin{cases}
|t| x_- & \;\text{for $t < 0$}\\
0 & \;\text{for $t = 0$}\\
|t| x_+ & \;\text{for $t > 0$}\\
\end{cases}
\;\;\text{and}\;\;
y_t =
\begin{cases}
|t| y_- & \;\text{for $t < 0$}\\
0 & \;\text{for $t = 0$}\\
|t| y_+ & \;\text{for $t > 0$}\\
\end{cases}
\end{align*}
Then $H_t$ defines a non-singular map for $t \neq 0$. The only
points of degeneracy which were introduced by the $t$-scaling are
those points in $X$ where together with $z$ also $t$ vanishes. Since
the maps $H_\pm$ are in normal form, this means
\begin{equation*}
S(H_0) = \{p \in X\colon z(p) = 0 \} = C_0 \cup C_1,
\end{equation*}
which finishes the proof. \end{proof}
For the construction of more interesting topological jumps we employ a new normal form for maps for triples $\Triple{d_0}{d}{d_1}$. This is described in the following definition \begin{definition}
\label{DefModifiedNormalForm}
For triples $\Triple{d_0}{\pm(d_0 - d_1)}{d_0}$ we define $G$-maps
$F\colon X \to S^2 \subset \mathbb{R}^3$ in \emph{modified normal form} as
follows: Let $D$ be the closed two-disk, $D = \{z \in
\mathbb{C}\colon |z| \leq 1\}$ and let $\iota_D\colon (D,\partial D)
\hookrightarrow (S^2,E)$ be the embedding
(\ref{IotaHemisphereEmbedding} from
p.~\pageref{IotaHemisphereEmbedding} of $D$ onto one of the
hemispheres of $S^2$. By $Z$ we denote the fundamental region
cylinder of the torus $X$. Now define the map
\begin{align*}
f\colon (Z,C) &\to (D,\partial D)\\
(t, \varphi) &\mapsto \begin{cases}
(1-2t) e^{i d_1 \varphi} + 2t & \text{ for $0 \leq t \leq \frac{1}{2}$}\\
(2t-1) e^{i d_2 \varphi} + 2(1-t) & \text{ for $\frac{1}{2} \leq t \leq 1$}
\end{cases}
\end{align*}
and let $F\colon X \to S^2$ be the equivariant extension of the
composition map $\iota_D \circ f$ (see
lemma~\ref{Class1EquivariantExtension}). The resulting map has the
triple $\Triple{d_0}{d_0-d_1}{d_1}$. \end{definition} The fact that the map $F$ constructed above has the triple $\Triple{d_0}{d_0-d_1}{d_0}$ can be shown as in the proof for proposition~\ref{TripleBuildingBlocks}~(ii). The crucial point of this modified normal form is that the fibers over the north-
resp. south pole of the sphere contains only a finite number of points, precisely $|d_0| + |d_1|$ for each of the poles. In the original normal form the fibers over the poles were copies of $S^1$ in $X$ (see figure~\ref{FigureModifiedNormalForm}).
\begin{figure}
\caption{Homotopy Visualization on the closed $2$-Disk.}
\label{FigureModifiedNormalForm}
\end{figure}
Now we construct a jump curve with only a discrete set of singular points. This jump curve ``flips'' the total degree of maps in modified normal form for basic triples. \begin{lemma}
\label{JumpsRank2Class1TotalDegree}
Let $d_0$ and $d_1$ be two integers and assume that the maps
\begin{align*}
H_\pm\colon (Z,C) \to \left(S^2,E\right) \subset \left(\mathbb{R}^3,\{z=0\}\right)
\end{align*}
have the triples $\Triple{d_0}{\pm(d_1-d_0)}{d_1}$. Then there
exists a jump curve
\begin{align*}
H\colon [-1,1] \times (Z,C) \to \left(\mathbb{R}^3,\{z=0\}\right)
\end{align*}
from $H_-$ to $H_+$ such that the singular set of the equivariant
extension of $H_0$ consists of $2(|d_0| + |d_1|)$ isolated points in
$X\smallsetminus (C_0 \cup C_1)$. If the maps are already normalized
along the boundary circles $C_0$ and $C_1$, then this homotopy $H_t$
can be chosen to be relative to $C_0 \cup C_1$. \end{lemma}
\begin{proof}
After a $G$-homotopy we can assume that the maps $H_\pm$ are in
modified normal form for the triples
$\Triple{d_0}{\pm(d_1-d_0)}{d_1}$. This homotopy can be chosen to be
relative to $C_0 \cup C_1$ if the original maps $H_\pm$ are already
type I normalized. It now suffices to prove the statement under the
assumption that $H_\pm$ are in modified normal form. We can regard
$H_\pm$ as maps
\begin{align*}
H_\pm\colon (Z,C) &\to (\mathbb{R}^3,\{z=0\})\\
p &\mapsto
\begin{pmatrix}
x_\pm(p) \\
y_\pm(p) \\
z_\pm(p)
\end{pmatrix}
\end{align*}
where the functions $x_\pm$, $y_\pm$ and $z_\pm$ satisfy:
\begin{align*}
& x_\pm \circ T = x_\pm\\
& y_\pm \circ T = y_\pm\\
& z_\pm \circ T = -z_\pm.
\end{align*}
Note that the functions $x_-$, $y_-$ and $z_-$ (resp. $x_+$, $y_+$
and $z_+$) do not simultaneously vanish. Also, since the maps in
modified normal form for triples of the form $\Triple{d_0}{\pm(d_1 -
d_0)}{d_1}$ only differ by a reflection along the $x,y$-plane, we
can conclude that $x_- = x_+$ , $y_- = y_+$ and $z_- = -z_+$. We set
$x = x_\pm$ and $y = y_\pm$ and define the jump curve $H_t$ as
follows:
\begin{align*}
H\colon [-1,1] \times (Z,C) &\to (\mathbb{R}^3,\{z=0\})\\
(t,p) &\mapsto
\begin{pmatrix}
x(p)\\
y(p)\\
z_t(p)
\end{pmatrix}
\end{align*}
where
\begin{align*}
z_t(p) =
\begin{cases}
|t| z_-(p) & \;\text{for $t < 0$} \\
0 & \;\text{for $t = 0$} \\
|t| z_+(p) & \;\text{for $t > 0$}
\end{cases}.
\end{align*}
As long as $t \neq 0$, the map $H_t$ is non-singular. For $t=0$, the
singular set $S(H_0)$ consists of those points $p \in X$ which are
mapped to the south resp. the north pole (this is the set $\{x = y =
0\}$). In the cylinder $Z \subset X$ we obtain $|d_0|$ singular
points during the shrinking process of the degree $d_0$ loop and
$|d_1|$ points during the enlarging process of the degree $d_1$
loop. By equivariant extension
(lemma~\ref{Class1EquivariantExtension}), the same applies to the
complementary fundamental cylinder $Z'$, yielding a total number of
$2(|d_0|+|d_1|)$ singular points in $X$.
Equivariance of the curve is guaranteed because the equivariance
conditions on the functions $x_\pm$, $y_\pm$ and $z$ are compatible
with scaling by real numbers. \end{proof}
Given integers $d_0$ and $d_1$, let \begin{align*}
H^{\Triple{d_0}{\pm(d_1-d_0)}{d_1}}\colon [-1,1] \times (Z,C) \to (\mathbb{R}^3,\{z=0\}). \end{align*} denote the jump curve from $H_-$ to $H_+$ (with $\mathcal{T}(H_\pm) = \Triple{d_0}{\pm(d_1-d_0)}{d_1}$) whose existence is guaranteed by lemma~\ref{JumpsRank2Class1TotalDegree}. In the following we consider jump curves, which modify the fixed point degrees. For the proof of the next lemma it is useful to introduce a new notation for maps $(Z,C) \to (S^2,E)$ having triples of the form $\Triple{d_0}{d_0-d_1}{d_1}$. Let us recall how the normal form maps $(Z,C) \to (S^2,E)$ for such triples were constructed in the proof of proposition~\ref{TripleBuildingBlocks} (ii). The cylinder is to be regarded as $I \times S^1$ where we use $s$ as the interval coordinate. Denote the closed unit disk in $\mathbb{C}$ by $D$. Let $\gamma_d\colon S^1 \to S^1$ be the normalized loop of degree $d$. Let $\iota_D$ be the embedding (\ref{IotaHemisphereEmbedding}) of the disk $D$ as one of the hemispheres in $S^2$ and let $f$ be the following map: \begin{align*}
f\colon [0,1] \times S^1 &\to D\\
(s,z) &\mapsto
\begin{cases}
(1-2s)\gamma_{d_0}(z) & \;\text{for $0 \leq t \leq \onehalf$}\\
(2s-1)\gamma_{d_1}(z) & \;\text{for $\onehalf \leq t \leq 1$}\\
\end{cases}. \end{align*} In particular, $f_0$ and $f_1$ both map $S^1$ to the boundary of $D$. Therefore, $\iota_D \circ f$ can be regarded as a map \begin{align*}
(Z,C) \to \left(S^2,E\right) \subset \left(\mathbb{R}^3,\{z=0\}\right). \end{align*} We can then define a map for the above triple as the equivariant extension of the composition $\iota_D \circ f$ (see lemma~\ref{Class1EquivariantExtension}): \begin{align*}
F = \widehat{\iota_D \circ f}. \end{align*} In other words, $F$ is the equivariant extension of the map \begin{align}
\label{TopJumpsDefBigF}
I \times S^1 &\to S^2\\
(s,z) &\mapsto
\begin{cases}
\begin{pmatrix}
(1-2s)\Re(\gamma_{d_0}(z))\\
(1-2s)\Im(\gamma_{d_0}(z)\\
-2\sqrt{s-s^2}
\end{pmatrix} & \;\text{for $s \in \left[0,\onehalf\right]$}\\
\begin{pmatrix}
(1-2s)\Re(\gamma_{d_1}(z))\\
(1-2s)\Im(\gamma_{d_1}(z))\\
-2\sqrt{s-s^2}
\end{pmatrix} & \;\text{for $s \in \left[\onehalf,1\right]$}
\end{cases}.\nonumber \end{align} What this shows is that normal form maps for a triple $\Triple{d_0}{\pm(d_0-d_1)}{d_1}$ can be defined in terms of two loops $S^1 \to S^1$ which are of degree $d_0$ resp. of degree $d_1$. This motivates the following \begin{definition}
By $F_{\alpha,\beta}$ we denote the map $(Z,C) \to
\left(S^2,E\right)$ which is defined in terms of the loops $\alpha$
and $\beta$ as described above. In particular, $F_{\alpha,\beta}$
has the triple $\Triple{\deg \alpha}{\deg \alpha - \deg \beta}{\deg
\beta}$. \end{definition}
Recall that for loops $\gamma_{d_0}, \ldots,\gamma_{d_n}$ we can form the concatenation loop \begin{align*}
\gamma_{d_n} \ast \ldots \ast \gamma_{d_0}\colon S^1 &\to S^1\\
z &\mapsto \gamma_{d_j}(z_{n+1}) \text{, where } \arg z \in\left[\frac{2\pi j}{n+1},\frac{2\pi(j+1)}{n+1}\right] \end{align*}
\begin{lemma}
\label{JumpsRank2Class1FixpointDegree}
Let $d^\pm$ be two integers and $H_\pm\colon (Z,C) \to
\left(S^2,E\right) \subset (\mathbb{R}^3,\{z=0\})$ be maps with the degree
triples
\begin{align*}
\Triple{d^\pm}{d^\pm}{0}\;\text{(resp. $\Triple{0}{d^\pm}{d^\pm}$)}.
\end{align*}
Then there exists a jump curve
$H\colon [-1,1] \times (Z,C) \to \left(\mathbb{R}^3,\{z=0\}\right)$ from
$H_-$ to $H_+$ such that the singular set $S(H_0)$ consists of
$|d^+ - d^-|$ isolated points in $C_0$ (resp. in $C_1$). If the
original maps $H_\pm$ are already normalized along the boundary
circle $C_1$ (resp. $C_0$), then $H_t$ can be chosen relative to
$C_1$ (resp. $C_0$). \end{lemma}
\begin{proof}
We prove the statement only for the triples
$\Triple{d^\pm}{d^\pm}{0}$; the other case works completely
analogously. After a $G$-homotopy we can assume that $H_-$ is
defined in terms of the loops $\gamma_k^{-1} \ast \gamma_k \ast
\gamma_{d^-}$ and $\gamma_0$:
\begin{align}
H_- = F_{\gamma_k^{-1} \ast \gamma_k \ast \gamma_{d^-},\gamma_0}.
\end{align}
After defining $k = d^+ - d^-$ and after another $G$-homotopy we can
assume that $H_+$ is defined in terms of the loops $\gamma_0 \ast
\gamma_k \ast \gamma_{d^-}$ and $\gamma_0$:
\begin{align}
H_+ = F_{\gamma_0 \ast \gamma_k \ast \gamma_{d^-},\gamma_0}.
\end{align}
In order to prove the statement it suffices to construct a jump
curve $H_t$ from $H_-$ to $H_+$. We define the curve $H_t$ as:
\begin{align*}
H_t = F_{\iota_t \ast \gamma_k \ast \gamma_{d^-},\gamma_0},
\end{align*}
where $\iota_t$ (``jump'') is the following null-homotopy of the
loop $\gamma_k^{-1}$:
\begin{align*}
\iota_t\colon [-1,1] \times S^1 &\to \mathbb{C}\\
(t,z) &\mapsto \frac{(1-t)z^{-k} + t + 1}{2}.
\end{align*}
By construction we have $H_{\pm 1} = H_\pm$. Let us compute the
singular set of $H_t$. From the definition of the map
$F_{\alpha,\beta}$ in (\ref{TopJumpsDefBigF}) we can conclude that
the singular set consists of the common zeros of
\begin{align*}
& (1-2s)\iota_t \ast \gamma_k \ast \gamma_{d^-}(z)\\
\text{and}\; & -2\sqrt{s-s^2}.
\end{align*}
Since the square root only vanishes for $s \in \{0,1\}$ and $(1-2s)$
does not vanish for these values of $s$, the only possibility for
both functions to vanish is
\begin{align*}
s = 0 \;\text{and}\;\iota_t \ast \gamma_k \ast \gamma_{d^-}(z) = 0.
\end{align*}
By construction, $\iota_t$ crosses the origin once and that happens
at $t=0$. Therefore, $H_t$ is only singular for $t = 0$ and the
singular set consists of $|k| = |d^+ - d^-|$ points in $C_0 \subset
Z$. \end{proof}
Note that this method also changes the total degree of the map. That is to be expected, since we have all the freedom possible for changing the fixed point degrees, but as we have seen earlier, not all combinations of fixed point degrees and total degree are allowed. The previous lemma also covers the special case $d^- = d^+$, in which there are no singular points introduced. For two integers $d^\pm$ we denote the jump curve from $H_-$ to $H_+$ (with $\mathcal{T}(H_\pm) = \Triple{d^\pm}{d^\pm}{0}$ resp. $\mathcal{T}(H_\pm) = \Triple{0}{d^\pm}{d^\pm}$) whose existence is guaranteed by lemma~\ref{JumpsRank2Class1FixpointDegree} by \begin{align*}
& H^{\Triple{d^\pm}{d^\pm}{0}}\colon [-1,1] \times (Z,C) \to \left(\mathbb{R}^3,\{z=0\}\right)\\
\text{resp. }\; & H^{\Triple{0}{d^\pm}{d^\pm}}\colon [-1,1] \times (Z,C) \to \left(\mathbb{R}^3,\{z=0\}\right). \end{align*}
Now we combine the results of lemma~\ref{JumpsRank2Class1TotalDegree} and~\ref{JumpsRank2Class1FixpointDegree} into one theorem: \begin{theorem}
\label{JumpsRank2Class1}
Let the torus $X$ be eqipped with the type I involution and let
$H_\pm\colon X \to S^2 \subset \mathbb{R}^3$ be two $G$-maps with triples
$\Triple{d_0^\pm}{d^\pm}{d_1^\pm}$. Then there exists a jump curve
$H\colon [-1,1] \times X \to \mathbb{R}^3$ of $G$-maps from $H_-$ to $H_+$
such that the singular set $S(H_0) \subset X$ consists of
\begin{align*}
|d_0^+ - d_0^-| + |d_1^+ - d_1^-| + d^+ - (d_0^+ + d_1^+) - \left[d^- - (d_0^- + d_1^-)\right]
\end{align*}
isolated points.
If the original maps $H_\pm$ are type I normalized along the
boundary circles $C_0$ and $C_1$ and $d_0^- = d_0^+$ or $d_1^- =
d_1^+$, then the curve $H_t$ can be chosen such that it does not
modify the maps on the respective boundary circles where they agree. \end{theorem}
\begin{proof}
Assume we are given two maps $H_\pm\colon X \to S^2 \subset \mathbb{R}^3$
having the degree triples $\Triple{d_0^\pm}{d^\pm}{d_1^\pm}$. Define
\begin{align*}
k = \frac{d^+ - (d_0^+ + d_1^+) - \left[d^- - (d_0^- + d_1^-)\right]}{2}.
\end{align*}
Since $d^\pm - (d_0^\pm + d_1^\pm) \equiv 0 \mod 2$, $k$ is an integer.
Note that we then have the identity
\begin{align}
\label{JumpsRank2Class1ProofDiscussionD}
d^+ = d_0^+ + 2k + d^- - d_0^- - d_1^- + d_1^+.
\end{align}
The maps $H_\pm$ are equivariantly homotopic (as non-singular maps
$X \to S^2 \subset \mathbb{R}^3$) to maps $H'_\pm$ which are assumed to be
the concatenations of simpler maps, corresponding to the triple
decomposition
\begin{align}
\label{JumpCurveTripleDecomposition}
\Triple{d_0^\pm}{d_0^\pm}{0} &\bullet \Triple{0}{k}{k}\\
&\bullet \Triple{k}{\pm k}{0}\nonumber \\
&\bullet \Triple{0}{d^--(d_0^- + d_1^-)}{0}\nonumber\\
&\bullet \Triple{0}{d_1^\pm}{d_1^\pm}\nonumber.
\end{align}
If the original maps $H_\pm$ are already normalized along any of the
fixed point circles and if they have the same degree there, then the
homotopy to the maps described by the aforementioned triples can be
chosen relatively to those boundary circles. After this reduction to
the maps $H'_\pm$, it suffices to prove the existence of a jump
curve from $H'_-$ to $H'_+$. The restrictions of the maps for the
triple (\ref{JumpCurveTripleDecomposition}) to the cylinder $Z$ are
defined as the concatenation of five maps $H^j$, each defined on one
fifth $Z_j$ of the cylinder $Z$, $j=0,\ldots,4$. We can assume that
each map $H^j$ is normalized on the boundary circles of its
subcylinder $Z_j$. Note that $H'_-$ and $H'_+$ agree on
$Z_1 \cup Z_3$ and differ only on $Z_0 \cup Z_2 \cup Z_4$. This
allows us to define a curve
\begin{align*}
H'\colon [-1,1] \times (Z,C) &\to (\mathbb{R}^3,\{z=0\})\\
(t,p) &\mapsto
\begin{cases}
H^{\Triple{d_0^\pm}{d_0^\pm}{0}}(t,p) & \;\text{if $p \in Z_0$}\\
H'_-(t,p) & \;\text{if $p \in Z_1$}\\
H^{\Triple{k}{\pm k}{0}}(t,p) & \;\text{if $p \in Z_2$}\\
H'_-(t,p) & \;\text{if $p \in Z_3$}\\
H^{\Triple{0}{d_1^\pm}{d_1^\pm}}(t,p) & \;\text{if $p \in Z_4$}
\end{cases}
\end{align*}
Note that if $d_0^+ = d_0^-$ or $d_1^+ = d_1^-$, then this homotopy
$H_t$ is relative to the respective boundary circles on which the
fixed point degrees agree.
Equivariant extension of $H'$ yields a $G$-map
\begin{align*}
\widehat{H'}\colon [-1,1] \times X \to \mathbb{R}^3
\end{align*}
satisfying the desired properties (see
lemma~\ref{Class1EquivariantExtension}). By
lemma~\ref{JumpsRank2Class1FixpointDegree} we obtain $|d_0^+ -
d_0^-|$ singular points from $H^{\Triple{d_0^\pm}{d_0^\pm}{0}}$,
$|d_1^+ - d_1^-|$ singular points from
$H^{\Triple{0}{d_1^\pm}{d_1^\pm}}$. To this we have to add the
singular set added by $H^{\Triple{k}{\pm k}{0}}$, which (by
lemma~\ref{JumpsRank2Class1TotalDegree}) and by
(\ref{JumpsRank2Class1ProofDiscussionD}) consists of
\begin{align*}
2k = d^+ - (d_0^+ + d_1^+) - \left[d^- - (d_0^- + d_1^-)\right]
\end{align*}
points. In total these are
\begin{align*}
|d_0^+ - d_0^-| + |d_1^+ - d_1^-| + d^+ - (d_0^+ + d_1^+) -
\left[d^- - (d_0^- + d_1^-)\right]
\end{align*}
singular points on the torus. \end{proof}
The following corollary is a reformulation of theorem~\ref{JumpsRank2Class1Corollary} using the isomorphism $\mathbb{R}^3 \cong i\mathfrak{su}_2$: \begin{corollary}
\label{JumpsRank2Class1Corollary}
Let the torus $X$ be equipped with the type I involution and let
\begin{align*}
H_\pm\colon X \to i\mathfrak{su}_2\smallsetminus\{0\} \subset \mathcal{H}_{(1,1)}
\end{align*}
be two $G$-maps with the triples
$\Triple{d_0^\pm}{d^\pm}{d_1^\pm}$. Then there exists a jump curve
\begin{align*}
H\colon [-1,1] \times X \to i\mathfrak{su}_2 \subset \mathcal{H}_{(1,1)}
\end{align*}
from $H_-$ to $H_+$ such that the singular set $S(H_0) \subset X$
consists of
\begin{align*}
|d_0^+ - d_0^-| + |d_1^+ - d_1^-| + d^+ - (d_0^+ + d_1^+) - \left[d^- - (d_0^- + d_1^-)\right]
\end{align*}
isolated points. \end{corollary}
\subsection{Type II}
As in the classification chapter, it is possible to deduce results for the type II involution from the respective results of the type I involution. Let $(X,T)$ be the torus equipped with the type II involution and $(X',T')$ be the torus equipped with the type I involution. As an immediate corollary to theorem~\ref{JumpsRank2Class1} note: \begin{corollary}
\label{JumpsRank2Class2}
Let $H_\pm\colon X' \to S^2 \subset \mathbb{R}^3$ be two $G$-maps with
triples $\Triple{0}{d^\pm}{d_1^\pm}$. Then there exists a jump curve
$H\colon [-1,1] \times X' \to \mathbb{R}^3$ from $H_-$ to $H_+$ such that
the singular set $S(H_0) \subset X'$ consists of
\begin{align*}
|d_1^+ - d_1^-| + d^+ - d^- + d_1^- - d_1^+
\end{align*}
isolated points. In particular, if the maps $H_-$ and $H_+$ are
normalized along the boundary circle $C_0$ (meaning that
$H_\pm(C_0) = \{p_0\} \in E \subset S^2$), then the curve $H_t$ can
be chosen such that $H_t(C_0) = \{p_0\}$ for all $t$. \end{corollary}
\begin{proof}
This is just the special case $d_0^\pm = 0$ of
theorem~\ref{JumpsRank2Class1}. \end{proof}
For reducing jump curves for the type II involution to jump curves for the type I involution, recall the equivariant and orientation preserving diffeomorphism \begin{align*}
\Psi\colon X'\smallsetminus C_0 \xrightarrow{\;\sim\;} S^2\smallsetminus \{O_\pm\}. \end{align*} from remark~\ref{Class1TorusWithSphereIdentification} which we have used for identifying type I normalized $G$-maps $X' \to S^2$ with type II normalized $G$-maps $S^2 \to S^2$. We can use the same method for jump curves: \begin{proposition}
\label{JumpsRank2Class2CompleteSphere}
Let $H_\pm$ be two type II normalized $G$-maps
$S^2 \to S^2 \subset \mathbb{R}^3$ with degree pairs
$\smash{\Pair{d_E^\pm}{d^\pm}}$. Then there exists a jump curve
$H\colon [-1,1] \times S^2 \to \mathbb{R}^3$ from $H_-$ to $H_+$ such that
the singular set $S(H_0)$ consists of
\begin{align*}
|d_E^+ - d_E^-| + d^+ - d^- + d_E^- - d_E^+
\end{align*}
isolated points in $X$. \end{proposition}
\begin{proof}
By lemma~\ref{LemmaClass2ReductionToClass1}, the maps $H_\pm$ induce
type I normalized equivariant maps
$H'_\pm\colon X' \to S^2 \subset \mathbb{R}^3$ with the degree triples
$\Triple{0}{d^\pm}{d_E^\pm}$ such that
\begin{align*}
\Restr{H'_\pm}{X'\smallsetminus C_0} = \Restr{H_\pm}{S^2\smallsetminus\{O_\pm\}} \circ \Psi.
\end{align*}
Using corollary~\ref{JumpsRank2Class2} we obtain a jump curve
$H'\colon [-1,1] \times X' \to \mathbb{R}^3$ from $H'_-$ to $H'_+$ such that
the singular set $S(H'_0)$ consists of
\begin{align}
\label{SingularPointsJumpsRank2Class2CompleteSphere}
|d_E^+ - d_E^-| + d^+ - d^- + d_E^- - d_E^+
\end{align}
points. In particular this curve has the property that
$H'_t(C_0) = \{p_0\}$ for all times $t$. The last property allows us
to define the curve
\begin{align*}
H\colon [-1,1] \times S^2 &\to \mathbb{R}^3\\
(t,p) &\mapsto
\begin{cases}
H'\left(t,\Psi^{-1}(p)\right) & \;\text{if $p \not\in \{O_\pm\}$}\\
p_0 & \;\text{if $p \in \{O_\pm\}$}\\
\end{cases}
\end{align*}
Since $H'_t$ is non-singular for $t \not= 0$, the same applies to
$H$. By construction
$H_t$ is a jump curve from $H_-$ to $H_+$. The diffeomorphism $\Psi$
maps the singular set $S(H'_0)$ onto the singular set $S(H_0)$, thus
the number of singular points of $S(H_0)$ is also given by
(\ref{SingularPointsJumpsRank2Class2CompleteSphere}). This finishes
the proof. \end{proof}
Now we can deduce the following result for type II jump curves: \begin{theorem}
\label{JumpsRank2Class2Complete}
Let $X$ be the torus equipped with the type II involution and let
$H_\pm$ be two $G$-maps $X \to S^2 \subset \mathbb{R}^3$ with degree pairs
$\Pair{d_C^\pm}{d^\pm}$. Then there exists a jump curve
$H\colon [-1,1] \times X \to \mathbb{R}^3$ from $H_-$ to $H_+$ such that the
singular set $S(H_0)$ consists of
\begin{align*}
|d_C^+ - d_C^-| + d^+ - d^- + d_C^- - d_C^+
\end{align*}
isolated points in $X$. \end{theorem}
\begin{proof}
After a $G$-homotopy we can assume that both maps $H_\pm$ are type
II normalized. Therefore they push down to maps $H'_\pm$ on the
quotient $X/A$, which can be equivariantly identified with
$S^2$. Thus $H'_\pm$ can be regarded as type II normalized maps
$S^2 \to S^2$ and we have the identity
$H_\pm = H'_\pm \circ \smash{\pi_{S^2}}$. By
remark~\ref{Class2CorrespondenceXS2} the degree pairs of the maps
$H'_\pm$ are the same as those of the original maps $H_\pm$. Now
proposition~\ref{JumpsRank2Class2CompleteSphere} guarantees the
existence of a jump curve $H'\colon I \times S^2 \to \mathbb{R}^3$ from
$H'_-$ to $H'_+$ with
\begin{align*}
|d_C^+ - d_C^-| + d^+ - d^- + d_C^- - d_C^+
\end{align*}
singular points on $S^2$. Then the composition
$H'_t \circ \smash{\pi_{S^2}}$ defines a jump curve from $H_-$ to
$H_+$ with the same number of singular points. \end{proof}
As in the case of the type I involution we use the isomorphism $\mathbb{R}^3 \cong i\mathfrak{su}_2$ to note the following reformulation: \begin{corollary}
\label{JumpsRank2Class2CompleteCorollary}
Let $X$ be the torus equipped with the type II involution and let
$H_\pm$ be two $G$-maps $X \to i\mathfrak{su}_2 \subset
\mathcal{H}_{(1,1)}$ with degree pairs $\Pair{d_C^\pm}{d^\pm}$. Then
there exists a jump curve
\begin{align*}
H\colon [-1,1] \times X \to i\mathfrak{su}_2 \subset \mathcal{H}_{(1,1)}
\end{align*}
from $H_-$ to $H_+$ such that the singular set $S(H_0)$ consists of
\begin{align*}
|d_C^+ - d_C^-| + d^+ - d^- + d_C^- - d_C^+
\end{align*}
isolated points in $X$. \end{corollary}
\section{Maps into $\mathrm{cl}(\mathcal{H}_n)$}
In this section we assume $n > 2$. Recall that $\mathcal{H}_{(p,q)}$ contains the orbit $U(n).\I{p}{q}$ as an equivariant strong deformation retract. This orbit can be equivariantly identified with the Grassmann manifold $\text{Gr}_p(\mathbb{C}^n)$. In order to employ the method described in the previous section in the higher dimensional setting, we embed the space $i\mathfrak{su}_2$ into $\mathrm{cl}(\mathcal{H}_{(p,q)})$ such that $i\mathfrak{su}_2\smallsetminus\{0\} \subset U(n).\I{p}{q}$ and the origin in $i\mathfrak{su}_2$ defines a singular matrix in the boundary of this orbit. As we have seen earlier, the topology on the mapping space is only non-trivial in the case of a mixed-signature. Thus we can assume that $0 < p,q < n$. The space $i\mathfrak{su}_2$ can be equivariantly embedded into $\mathrm{cl}(\mathcal{H}_{(p,q)})$ as an affine subspace via \begin{align}
\label{isu2Embedding}
\iota\colon i\mathfrak{su}_2 &\hookrightarrow \mathrm{cl}(\mathcal{H}_{(p,q)})\\
\xi &\mapsto \begin{pmatrix}
\I{p-1} & 0 & 0\\
0 & \xi & 0 \\
0 & 0 & -\I{q-1}
\end{pmatrix}\nonumber \end{align} Similarly, the group $U(2)$ can be embedded into $U(n)$ via \begin{align*}
U(2) &\hookrightarrow U(n)\\
U &\mapsto \begin{pmatrix}
\I{p-1} & 0 & 0\\
0 & U & 0 \\
0 & 0 & -\I{q-1}
\end{pmatrix}. \end{align*} Since $U(n)$ acts on $\mathcal{H}_{(p,q)}$ via conjugation, so does $U(2)$, by regarding each element of $U(2)$ as being embedded in $U(n)$. The $U(2)$-orbit of $\I{p}{q}$ is the same as $\iota(U(2).\I{1}{1})$. It is contained as a subspace in the orbit $U(n).\I{p}{q}$ and corresponds -- under the isomorphism $\text{Gr}_p(\mathbb{C}^n) \cong U(n).\I{p}{q}$ -- to the Schubert variety $\mathcal{S} \cong \mathbb{P}_1$ (see the remarks about the Schubert variety $\mathcal{S}$ on p.~\pageref{SchubertVarietyDiscussion}). This follows from the fact that $U(2)$ acts transitively on \begin{align*}
\mathcal{S} = \left\{E \in \text{Gr}_p(\mathbb{C}^n)\colon \mathbb{C}^{p-1} \subset E \subset \mathbb{C}^{p+1}\right\}. \end{align*} Under the above embedding $i\mathfrak{su}_2 \hookrightarrow \mathrm{cl}(\mathcal{H}_{(p,q)})$, the origin of $i\mathfrak{su}_2$ corresponds to a singular matrix, which is contained in the boundary $\partial\mathcal{H}_{(p,q)}$.
Since $\mathcal{H}_{(p,q)}$ has $\text{Gr}_p(\mathbb{C}^n)$ as equivariant deformation retract, these two spaces and also their real parts have the same fundamental group: \begin{align*}
\pi_1\left(\left(\mathcal{H}_{(p,q)}\right)_\mathbb{R}\right) \cong \pi_1\left(\left(\text{Gr}_p\left(\mathbb{C}^n\right)\right)_\mathbb{R}\right) \cong C_2. \end{align*} We write $C_2$ additively as the group $\mathbb{Z}_2$ and identify elements of $\pi_1((\mathcal{H}_{(p,q)})_\mathbb{R})$ with $\mathbb{Z}_2$. Thus we can regard the restrictions of maps $X \to \mathcal{H}_{(p,q)}$ to the boundary circle(s) in $X$ as defining an element in $\pi_1((\mathcal{H}_{(p,q)})_\mathbb{R}) \cong \{0,1\}$.
\begin{theorem}
\label{JumpsRankN}
Assume $n = p + q > 2$. Let $X$ be the torus equipped with the type
I (resp. type II) involution and let $H_\pm$ be two $G$-maps $X \to
\mathcal{H}_{(p,q)}$ with the degree triples
$\smash{\Triple{m_0^\pm}{d^\pm}{m_1^\pm}}$ resp. the degree pairs
$\smash{\Pair{m_C^\pm}{d^\pm}}$. Then there exists a jump curve
\begin{align*}
H\colon [-1,1] \times X \to \mathrm{cl}(\mathcal{H}_{(p,q)})
\end{align*}
from $H_-$ to $H_+$ such that the singular set $S(H_0)$ consists of
\begin{align*}
& |m_0^+ - m_0^-| + |m_1^+ - m_1^-| + d^+ - (m_0^+ + m_1^+) - \left[d^- - (m_0^- + m_1^-)\right]\\
\text{resp.}\; & |m_C^+ - m_C^-| + d^+ - d^- + m_C^- - m_C^+
\end{align*}
isolated points. \end{theorem}
\begin{proof}
Let $H_\pm$ be $G$-maps $X \to \mathcal{H}_{(p,q)}$ with degree
triples $\Triple{m_0^\pm}{d^\pm}{m_1^\pm}$ resp. degree pairs
$\Pair{m_C^\pm}{d^\pm}$. By applying
proposition~\ref{LemmaReductionOfGrToP1} we can assume that the maps
$H_\pm$ have their images contained in the Schubert variety
$\mathcal{S} \subset \text{Gr}_p(\mathbb{C}^n)$. The Schubert variety is the
$U(2)$-orbit of $\I{p}{q}$, which is $\iota(U(2).\I{1}{1})$. That
is, it is contained in the image of the embedding
\begin{align}
\label{TopJumpsGrIotaEmbedding}
\iota\colon i\mathfrak{su}_2 \hookrightarrow \mathrm{cl}(\mathcal{H}_{(p,q)})
\end{align}
from (\ref{isu2Embedding}). This means that we can assume that the
maps $H_\pm$ are of the form
\begin{align*}
H_\pm = \iota \circ H'_\pm\colon X \to \mathrm{cl}(\mathcal{H}_{(p,q)})
\end{align*}
where the $H'_\pm$ are $G$-maps
\begin{align*}
H'_\pm\colon X \to U(2).\I{1}{1} \subset i\mathfrak{su}_2
\end{align*}
By construction, their degree triples resp. their degree pairs of
$H'_\pm$ agree with those of $H_\pm$.
Now corollary~\ref{JumpsRank2Class1Corollary} (type I)
resp. corollary~\ref{JumpsRank2Class2CompleteCorollary} (type II)
implies the existence of a jump curve
\begin{align*}
H'\colon [-1,1] \times X \to i\mathfrak{su}_2
\end{align*}
from $H'_-$ to $H'_+$ such that the singular set $S(H_0)$ consists
of
\begin{align}
\label{TopJumpsGrSingularPoints}
& |m_0^+ - m_0^-| + |m_1^+ - m_1^-| + d^+ - (m_0^+ + m_1^+) - \left[d^- - (m_0^- + m_1^-)\right]\\
\text{resp.}\; & |m_C^+ - m_C^-| + d^+ - d^- + m_C^- - m_C^+\nonumber
\end{align}
isolated points in $X$. Composing $H'_t$ with $\iota$ yields a jump
curve
\begin{align*}
H = \iota \circ H'\colon X \to \mathrm{cl}(\mathcal{H}_{(p,q)})
\end{align*}
from $H_-$ to $H_+$ whose number of isolated points is the same as
that of $H'_t$, namely that given in
(\ref{TopJumpsGrSingularPoints}). \end{proof}
\section{An Example from Physics}
In this section we present an example jump curve coming from physics (see e.\,g. \cite[p.~11]{PrimerTopIns}). For this we regard the torus $X$ as $\mathbb{R}^2/\Lambda$, where \begin{align*}
\Lambda = \LatGen{
\begin{pmatrix}
2\pi \\
0
\end{pmatrix}}{
\begin{pmatrix}
0 \\
2\pi
\end{pmatrix}}. \end{align*} The orientation on $X$ is the orientation inherited from $\mathbb{R}^2$. Employing physics notation for the coordinates on the torus phase space, we define the jump curve as \begin{align}
H_t(q,p) = \begin{pmatrix}
t - \cos q - \cos p & \sin q - i \sin p \\
\sin q + i \sin p & -(t - \cos q - \cos p)
\end{pmatrix}. \end{align} Each map $H_t$ is equivariant with respect to the type I involution. This curve differs from the jump curves we have constructed earlier in that it depends on a continuous parameter $t \in \mathbb{R}$ with multiple jumps taking place as $t$ varies. These occur precisely at $t=-2$, $t=0$ and at $t=2$. The associated curve of scaled maps into $S^2 \subset \mathbb{R}^3$ is \begin{align*}
\widetilde{H}_t(q,p) =
c_t(q,p)
\begin{pmatrix}
t - \cos q - \cos p\\
\phantom{-}\sin q\\
-\sin p
\end{pmatrix}, \end{align*} where \begin{align*}
c_t(q,p) = \frac{1}{\sqrt{(t - \cos q \cos p)^2 + (\sin q)^2 + (\sin p)^2}}. \end{align*} In the following we compute the degree triples $\mathcal{T}(H_t)$. For this it is convenient to decompose $\smash{\widetilde{H}}_t$ as \begin{align*}
\widetilde{H}_t(q,p) =
c_t(q,p)\left[
\begin{pmatrix}
t - \cos p\\
0\\
-\sin p
\end{pmatrix} + \begin{pmatrix}
- \cos q\\
\sin q\\
0
\end{pmatrix}\right]. \end{align*} Now we can regard $\smash{\widetilde{H}}_t$ as a family of circles in the $x,y$-plane whose centers in $\mathbb{R}^3$ varies with $t$ and $p$. It follows that $\widetilde{H}_t$ is not surjective (as a map to $S^2$)
for $|t| \in (2,\infty)$, therefore we have $\smash{\deg \widetilde{H}_t} = 0$ for these $t$. Furthermore, when $t$ is in a neighborhood of the origin and transitions from a negative real number to a positive real number, then the total degree of $\smash{\widetilde{H}}_t$ flips its sign. Hence it suffices to compute the total degree of e.\,g. $\smash{\widetilde{H}}_1$. This can be done by counting preimage points, for example in the fiber $\smash{\widetilde{H}}_1^{-1}(y_0)$ where $y_0 = \ltrans{(0,1,0)}$. A computation shows that the $y_0$-fiber consists only of the point $(q_0,p_0) = (\text{\textonehalf}\pi,0)$. Using the orientation of $S^2$ induced by an outer normal vector field one can find that the orientation sign of the map $\smash{\widetilde{H}}_1$ at $(q_0,p_0)$ is positive. This then implies that $\deg \smash{\widetilde{H}}_t = 1$ for $t \in (0,2)$ and $\deg \smash{\widetilde{H}}_t = -1$ for $t \in (-2,0)$.
Regarding the fixed point degrees: For $p=0$ resp. $p=\pi$ the map $\smash{\widetilde{H}}_t(p,\cdot)$ can be written as \begin{align}
\label{ExampleFixpointMapP=0}
& q \mapsto
c_t(q,0)\left[
\begin{pmatrix}
t - 1\\
0 \\
0
\end{pmatrix} +
\begin{pmatrix}
-\cos q \\
\phantom{-}\sin q \\
0
\end{pmatrix}\right]\\ \text{resp.}\;\; & q \mapsto
c_t(q,\pi)\left[
\begin{pmatrix}
t + 1\\
0 \\
0
\end{pmatrix} +
\begin{pmatrix}
-\cos q \\
\phantom{-}\sin q \\
0
\end{pmatrix}\right]\nonumber \end{align} Using the orientation on the equator $E \subset S^2$ as defined on p.~\pageref{DegreeOneMapOnEquator} it follows that the fixed point degree for $p=0$ is $0$ for $t \in \mathbb{R}\smallsetminus[0,2]$, and $-1$ for $t \in (0,2)$. The fixed point degree for $p=\pi$ is $0$ for $t \in \mathbb{R}\smallsetminus[-2,0]$ and $-1$ for $t \in (-2,0)$. To summarize the above: \begin{remark}
The following table lists the degree triples for $H_t$ depending on
$t$, where $t$ is assumed to be in one of the $t$-intervals such
that $H_t$ is non-singular:
\begin{center}
\footnotesize
\begin{tabular}[h]{ll}
\toprule
\textbf{$t$-interval} & \textbf{degree triple $\mathcal{T}(H_t)$} \\
\midrule
$(-\infty,-2)$ & $\Triple{0}{0}{0}$ \\
$(-2,0)$ & $\Triple{0}{-1}{-1}$ \\
$(0,2)$ & $\Triple{-1}{1}{0}$ \\
$(2,\infty)$ & $\Triple{0}{0}{0}$ \\
\bottomrule
\end{tabular}
\end{center} \end{remark}
Now we consider a generalization of the above map $H_t$. We define \begin{align}
H_t^m(q,p) = c_t(q,mp) \begin{pmatrix}
t - \cos q - \cos(mp) & \sin q - i \sin(mp) \\
\sin q + i \sin (mp) & -(t - \cos q - \cos(mp))
\end{pmatrix}, \end{align} with $m$ a positive integer. The maps $H_t^m$ are still type I equivariant. They can be expressed as the composition $H_t \circ p_m(q,p)$, where $p_m$ is the following $m:1$-cover of $X$: \begin{align*}
p_m\colon X &\to X\\
(q,p) &\mapsto (q,mp). \end{align*} In particular, $p_m$ has degree $m$, which implies that $\deg H_t^m = m\deg H_t$. Let us now compute the fixed point degrees of $H_t^m$. These depend on the parity of $m$. If $m$ is even, then the $2\pi$-periodicity of the trigonometric functions implies that the fixed point degrees for $p=0$ and $p=\pi$ must be the same. They are given by the map (\ref{ExampleFixpointMapP=0}). Therefore, the fixed point degrees for even $m$ must be $(0,0)$ for $t \in \mathbb{R}\smallsetminus[0,2]$ and $(-1,-1)$ otherwise. The periodicity of the trigonometric functions implies that fixed point degrees for odd $m$ are the same as those for $m=1$. We obtain the following result: \begin{remark}
The following table lists the degree triples for $H_t^m$ for even
$m$ varying with $t$ such that $H_t^m$ is non-singular:
\begin{center}
\footnotesize
\begin{tabular}[h]{ll}
\toprule
\textbf{$t$-interval} & \textbf{degree triple $\mathcal{T}(H^m_t)$} \\
\midrule
$(-\infty,-2)$ & $\Triple{0}{0}{0}$ \\
$(-2,0)$ & $\Triple{0}{-m}{0}$ \\
$(0,2)$ & $\Triple{-1}{m}{-1}$ \\
$(2,\infty)$ & $\Triple{0}{0}{0}$ \\
\bottomrule
\end{tabular}
\end{center} \end{remark} In particular, this example shows how the degree triples vary with $t$ in such a way that the condition $d \equiv d_0 + d_1 \mod 2$ is always satisfied.
\appendix \chapter{Appendix} \label{ChapterAppendix}
This chapter lists some standard material, which is included for the convenience of the reader.
\section{Topology}
Taken from \cite{Hatcher}: \begin{proposition}
\label{HEP} (Homotopy Extension Property) Let $X$ be a CW complex
and $A \subset X$ a CW subcomplex. Then the CW pair $(X,A)$ has the
homotopy extension property (HEP) for all spaces. \end{proposition} \begin{proof}
See \cite[p.\,15]{Hatcher}. \end{proof}
\begin{lemma}
\label{LemmaCurvesNullHomotopic}
Let $X$ be a topological space and $f,g\colon I \to X$ be two curves
in $X$ with $f(0) = g(0) = p$ and $f(1) = g(1) = q$. Then $g^{-1}
\ast f$ is null-homotopic if and only if $f$ and $g$ are homotopic
(with fixed endpoints). \end{lemma} \begin{proof}
The one direction is trivial: When $f$ and $g$ are homotopic, then
we can define a null-homotopy $H\colon I \times I \to X$ as follows:
During $0 \leq t \leq \text{\textonehalf}$ use a homotopy from $f$
to $g$ to build a homotopy from $g^{-1} \ast f$ to $g^{-1} \ast
g$. Then, during $\text{\textonehalf} \leq t \leq 1$ we shrink $g^{-1} \ast
g$ the constant curve at $p$.
Now assume that we are given a homotopy from $\gamma_1^{-1} \ast
\gamma_0$ to the constant curve at $p$. This is a map $H\colon I
\times I \to X$. On the left, top and right boundary of the square
$I$ the map $H$ takes on the value $p$, on the bottom boundary,
i.\,e. $H_0$, this is the curve $\gamma^{-1}_1 \ast \gamma_0$. We use
this map to construct a new map $\gamma\colon I \times I \to X$
which is a homotopy from $\gamma_0$ to $\gamma_1$ as follows:
We define $\gamma_t\colon I \to X$ to be the map $I
\xhookrightarrow{\iota_t} I \times I \xrightarrow{H} X$ where the
embedding $\iota_t$ is defined as follows:
\begin{align*}
\iota_t\colon I &\to I \times I\\
s &\mapsto \frac{1}{\max \{|\cos \pi(1-t)|,|\sin \pi(1-t)|\}} \begin{pmatrix}
\frac{\cos \pi(1-t)}{2}\\
i \sin \pi(1-t)
\end{pmatrix} +
\begin{pmatrix}
1\\
2
\end{pmatrix}
\end{align*}
With this definition, $\iota_0$ is the embedding $s \mapsto
\text{\textonehalf}(1-s)$ and $\iota_1$ is the embeding $s \mapsto
\text{\textonehalf}(1+s)$. Hence, $\gamma_t$ is indeed a homotopy
from $\gamma_0$ to $\gamma_1$. Furthermore, the construction makes
sure that for every $t$, $\iota_t(0) = \text{\textonehalf}$ and
$\iota_t(1)$ is always contained in the left, top or right boundary
of $I \times I$. This translates to the fact that $\gamma_t(0) = p$
and $\gamma_t(1) = q$ for all $t$. \end{proof}
\begin{theorem}
\label{WhitneyApproximationTheorem}
(Taken from John Lee, Whitney Approximation Theorem) Suppose $N$ is
a smooth manifold with or without boundary, $M$ is a smooth manifold
(without boundary), and $F\colon N \to M$ is a continuous map. Then
$F$ is homotopic to a smooth map. If $F$ is already smooth on a
closed subset $A \subseteq N$, then the homotopy can be taken to be
relative to $A$. \end{theorem} \begin{proof}
See \cite[p.\,141]{JohnLee}. \end{proof}
Here is a theorem by Whitney, taken from \cite{JohnLee}: \begin{theorem}
\label{SmoothApproximation0}
Let $N$ and $M$ be smooth manifolds and let $F\colon N \to M$ be a
map. Then $F$ is homotopic to a smooth map
$\smash{\widetilde{F}}\colon N \to M$. If $F$ is smooth on a closed
subset $A \subset N$, then the homotopy can be taken to be relative
to $A$. \end{theorem} \begin{proof}
See \cite[p.~142]{JohnLee} \end{proof}
The following two statements can be found in \cite{Bredon}: \begin{theorem}
\label{SmoothApproximation1}
Let $G$ be a compact Lie group acting smoothly on the manifolds $M$
and $N$. Let $\varphi\colon M \to N$ be an equivariant map. Then
$\varphi$ can be approximated by a smooth equivariant map
$\psi\colon M \to N$ which is equivariantly homotopic to $\varphi$
by a homotopy approximating the constant homotopy. Moreover, if
$\varphi$ is already smooth on the closed invariant set
$A \subset M$, then $\psi$ can be chosen to coincide with $\varphi$
on $A$, and the homotopy between $\varphi$ and $\psi$ to be constant
there. \end{theorem} \begin{proof}
See \cite[p.\,317]{Bredon}. \end{proof}
\begin{corollary}
\label{SmoothApproximation2}
Let $G, M, N$ be as in [the previous theorem]. Then any equivariant
map $M \to N$ is equivariantly homotopic to a smooth equivariant
map. Moreover, if two smooth equivariant maps $M \to N$ are
equivariantly homotopic, then they are so by a smooth equivariant
homotopy. \end{corollary} \begin{proof}
\cite[p.\,317]{Bredon}. \end{proof}
The following theorem is taken (and translated) from \cite{StoeckerZieschang}: \begin{theorem}
\label{TopologyMapInducedOnQuotient}
Let $f\colon X \to Y$ be a continuous map, which is compatible with
given equivalence relations $R$ resp. $S$ on $X$ resp. $Y$ (which
means: $x \sim_R x'$ implies $f(x) \sim_S f(x')$). Then
$f'([x]_R) = [f(x)]_S$\footnote{Here $[x]_R$ denotes the equivalence
class of $x$ under the relation $R$ (likewise for $y$ and $S$).}
defines a continuous map $f'\colon X/R \to Y/S$; it is called the
map induced by $f$. If $f$ is a homeomorphism and $f^{-1}$ is also
compatible with the relations $R$ resp. $S$, then the induced map
$f'$ is also a homeomorphism. \end{theorem} \begin{proof}
See \cite[p.~9]{StoeckerZieschang}. \end{proof}
The same applies to the following \begin{theorem}
\label{QuotientOfCWIsCW}
Let $(X,A)$ be a CW pair and $A \neq \emptyset$, then $X/A$ is a
CW complex with the zero cell $[A] \in X/A$ and the cells of the
form $p(e)$, where $p\colon X \to X/A$ denotes the identifying map
and $e$ is a cell in $X\smallsetminus A$. \end{theorem} \begin{proof}
See \cite[p.~93]{StoeckerZieschang}. \end{proof}
Taken from \cite{May}: \begin{theorem}
\label{G-HELP} (G-HELP) Let $A$ be a subcomplex of a $G$-CW
complex $X$ of dimension $\nu$ and let $e\colon Y \to Y$ be a
$\nu$-equivalence. Suppose given maps $g\colon A \to Y$, $h\colon A
\times I \to Z$, and $f\colon X \to Z$ such that $e \circ g = h
\circ i_1$ and $f \circ i = h \circ i_0$ in the following diagram:
\[
\label{HELP-Diagram}
\begin{tikzcd}
A \arrow{rr}{i_0} \arrow{dd}{i} & & A \times I \arrow{dd} \arrow{dl}{h} & & A \arrow{ll}{i_1} \arrow{dl}{g} \arrow{dd}{i} \\
& Z & & Y \arrow[crossing over,near start]{ll}{e} \\
X \arrow{ru}{f} \arrow{rr}{i_0} & & X \times I \arrow[dashed]{ul}{\tilde{h}} & & X \arrow{ll}{i_1} \arrow[dashed]{lu}{\tilde{g}} \\
\end{tikzcd}
\]
Then there exist maps $\smash{\tilde{g}}$ and $\smash{\tilde{h}}$
that make the diagram commute. \end{theorem} \begin{proof}
See \cite[p.~17]{May}. \end{proof}
From this we can deduce a statement about equivariant homotopy extensions for $G$-CW complexes: \begin{corollary}
\label{G-HEP}
(Equivariant HEP) Let $G$ be a topological group. Let $X$ and $Y$
be $G$-CW complexes and $A$ a $G$-CW subcomplex of $X$. Then the
$G$-CW pair $(X,A)$ has the equivariant homotopy extension
property. That is, given a $G$-map $f\colon X \to Y$ and a
$G$-homotopy $h\colon I \times A \to Y$. Then $h$ extends to a
homotopy $H\colon I \times X \to Y$ such that $H_0 \equiv f$ and
$\restr{H}{I \times A} \equiv h$. \end{corollary} \begin{proof}
Set $Z = Y$ and let $f\colon Y \to Z$ be the identity. In particular
this makes $f$ be a $\nu$-equivalence where we set $\nu(H) =
\dim(X)$ for all subgroups $H \subset G$. Then, the equivariant
homotopy extension property follows from the LHS square of the
diagram (\ref{HELP-Diagram}). \end{proof}
\begin{remark}
\label{RemarkQuotientDegreeOne}
Let $M$ be an $n$-dimensional, connected, closed, oriented
CW manifold and $A \subset M$ such that
\begin{enumerate}[(i)]
\item $A$ is contained in the $n-1$ skeleton $M^{n-1}$ and
\item $M/A$ is topologically an $n$-dimensional, connected, closed,
orientable manifold.
\end{enumerate}
Then the quotient $M/A$ can be equipped with an orientation such
that the projection map $M \to M/A$ has degree $+1$. \end{remark}
\begin{proof}
The manifold $M$ is assumed to be oriented, which corresponds to a
choice of fundamental class $[M]$ in $H_n(M,\mathbb{Z}) \cong \mathbb{Z}$. Denote
the quotient map with
\begin{align*}
\pi\colon M \to M/A.
\end{align*}
Since $A$ is contained in $M^{n-1}$, $\pi$ maps the $n$-cells in $M$
homeomorphically to the $n$-cells in the quotient $M/A$. It follows
that there exists an $n$-ball in one of the $n$-cells in $M/A$ such
that $\pi^{-1}(B)$ is a single $n$-ball in $M$ which gets mapped
homeomorphically to $B$. Fix a fundamental class $[M/A]$ of
$H_n(M/A,\mathbb{Z})$. Using exercise~8 from \cite[p.~258]{Hatcher} we can
conclude that the degree of $\pi$ is $\pm 1$, depending on wether it
is orientation preserving or orientation reversing. In case it is
orientation reversing we can chose the opposite orientation on $M/A$
such that $\deg \pi = +1$. \end{proof}
A direct consequence of the previous remark is: \begin{remark}
\label{RemarkMapOnQuotientDegreeUnchanged}
Let $M$ and $N$ be $n$-dimensional, connected, closed, oriented
manifolds and $f\colon M \to N$ a map of degree $d$. Furthermore,
assume that $f$ is constant on $A \subset M$ such that it induces a
map $f'\colon M/A \to N$. If the pair $(M,A)$ satisfies the
assumptions of remark~\ref{RemarkQuotientDegreeOne}, then $f'$ also
has degree $d$. \end{remark}
\begin{proof}
Denote the quotient map with $\pi\colon M \to M/A$. We then have the
following commutative diagram:
\[
\xymatrix{
M \ar[d]_\pi \ar[dr]^f\\
M/A \ar[r]_{f'} & N
}
\]
In homology we obtain:
\[
\xymatrix{
H_n(M,\mathbb{Z}) \ar[d]_{\pi_*} \ar[dr]^{f_*}\\
H_n(M/A,\mathbb{Z}) \ar[r]_{f'_*} & H_n(N,\mathbb{Z})
}
\]
By remark~\ref{RemarkQuotientDegreeOne} the quotient $M/A$ can be
equipped with an orientation such that, after identifying all
homology groups with $\mathbb{Z}$, $\pi_*$ is multiplication by $+1$ and
$f'_*$ is multiplication by $d'$. Thus we have:
\[
\xymatrix{
\mathbb{Z} \ar[d]_{\cdot +1} \ar[dr]^{\cdot d}\\
\mathbb{Z} \ar[r]_{\cdot d'} & \mathbb{Z}
}
\]
By commutativity of the diagram it follows that the degree of
$f'$ is $'d = d$. \end{proof}
\begin{proposition}
\label{TopologyGrassmannians}
Assume $0 < p < n$. Then:
\begin{enumerate}[(i)]
\item The second homology group $H_2(\text{Gr}_p(\mathbb{C}^n),\mathbb{Z})$ is infinite
cyclic for all $p$ and $n$. Fixing a full flag of $\mathbb{C}^n$ induces a
decomposition of $\text{Gr}_p(\mathbb{C}^n)$ into Schubert cells. With respect
to a fixed flag there exists a unique (complex) 1-dimensional
Schubert variety, which can be regarded as the generator of
$H_2(\text{Gr}_p(\mathbb{C}^n),\mathbb{Z})$.
\item The fundamental group $\pi_1(\text{Gr}_p(\mathbb{R}^n))$ is cyclic of order
two unless $p=1$ and $n=2$, in which case it is infinite cyclic.
\end{enumerate} \qed \end{proposition}
\section{Hausdorff Dimension} \label{AppendixHausdorffDimension}
For completeness we quote a theorem taken from \cite[p.~515]{SchleicherArticle}: \begin{theorem}
\label{HDimProperties}
Hausdorff dimension has the following properties:
\begin{enumerate}[(1)]
\item if $X \subset Y$, then $\text{dim}_H X \leq \text{dim}_H Y$;
\item if $X_i$ is a countable collection of sets with $\text{dim}_H X_i
\leq d$, then
\begin{align*}
\text{dim}_H \bigcup_i X_i \leq d
\end{align*}
\item if $X$ is countable, then $\text{dim}_H X = 0$;
\item if $X \subset \mathbb{R}^d$, then $\text{dim}_H X \leq d$;
\item if $f\colon X \to f(X)$ is a Lipschitz map, then $\text{dim}_H(f(X))
\leq \text{dim}_H(X)$.
\item if $\text{dim}_H X = d$ and $\text{dim}_H Y = d'$, then $\text{dim}_H(X \times Y) \geq d + d'$;
\item if $X$ is connected and contains more than one point, then
$\text{dim}_H X \geq 1$; more generally, the Hausdorff dimension of any
set is no smaller than its topological dimension;
\item if a subset $X$ of $\mathbb{R}^n$ has finite positive $d$-dimensional
Lebesgue measure, then
\begin{align*}
\text{dim}_H X = d.
\end{align*}
\end{enumerate} \end{theorem}
Although the Hausdorff dimension is not invariant under homeomorphisms, it is invariant under diffeomorphisms: \begin{corollary}
\label{HdimDiffeomorphismInvariant}
If $f\colon X \to Y$ is a diffeomorphism (between metric spaces),
then $\text{dim}_H X = \text{dim}_H Y$. \end{corollary} \begin{proof}
Let $f\colon X \to Y$ be a diffeomorphism. In particular, $f$ and
$f^{-1}$ are both Lipschitz continuous. Thus, by
theorem~\ref{HDimProperties} (5) we obtain
\begin{align*}
& \text{dim}_H(Y) = \text{dim}_H(f(X)) \leq \text{dim}_H(X)\\
\;\text{and}\; & \text{dim}_H(X) = \text{dim}_H(f^{-1}(Y)) \leq \text{dim}_H Y
\end{align*}
and the statement follows. \end{proof}
\begin{corollary}
\label{HDimViaSetDecomposition}
If $X$ is the finite union of sets $X_i$, then
\begin{align*}
\text{dim}_H X = \max_i \text{dim}_H X_i.
\end{align*} \end{corollary}
\begin{proof}
Each $X_i$ is contained in $X$, hence by
theorem~\ref{HDimProperties} (1) we obtain
\begin{align*}
\text{dim}_H X_i \leq \text{dim}_H X \;\text{for all $i$}.
\end{align*}
In other words:
\begin{align*}
\max_i \text{dim}_H X_i \leq \text{dim}_H X
\end{align*}
On the other hand, noting the trivial fact
\begin{align*}
\text{dim}_H X_i \leq \max_i \text{dim}_H X_i
\end{align*}
and using (2) of the same theorem we can conclude
\begin{align*}
\text{dim}_H X = \text{dim}_H\left(\bigcup_i X_i\right) \leq \max_i(\text{dim}_H X_i).
\end{align*}
Thus we get the desired equation:
\begin{align*}
\text{dim}_H X = \max_i(\text{dim}_H X_i).
\end{align*} \end{proof}
\begin{proposition}
\label{PropositionHausdorffPackingDimension}
For sets $A, B \subset \mathbb{R}^n$ we have
\begin{align*}
\text{dim}_H(A) + \text{dim}_H(B) \leq \text{dim}_H(A \times B) \leq \text{dim}_H(A) + \text{dim}_P(B)
\end{align*} \end{proposition}
\begin{proof}
See \cite[p.~1]{XiaoArticle}. \end{proof}
For a submanifold $B \subset \mathbb{R}^n$ the packing dimension $\text{dim}_P(B)$, the Hausdorff dimension $\text{dim}_H(B)$ and the usual manifold dimension $\dim(B)$ agree\footnote{See e.\,g. \cite[p.~48]{Falconer}}. Thus we obtain for submanifolds $A,B \subset \mathbb{R}^n$: \begin{align}
\label{HDimOfProduct}
\text{dim}_H(A \times B) = \text{dim}_H(A) + \dim(B). \end{align} This allows us to prove the following: \begin{corollary}
\label{HDimOfTotalSpace}
Let $E \xrightarrow{\;\pi\;} X$ be a fiber bundle over the smooth
manifold $X$ where $E$ is a smooth submanifold of $\mathbb{R}^N$ such that
the fiber $F$ is also a smooth manifold.
Let $A$ be a subset in $X$. Then $\text{dim}_H(\pi^{-1}(A)) = \text{dim}_H(A) +
\dim(F)$. \end{corollary}
\begin{proof}
The statement follows by using using a trivializing covering of $X$
together with corollary~\ref{HdimDiffeomorphismInvariant},
corollary~\ref{HDimViaSetDecomposition} and (\ref{HDimOfProduct}). \end{proof}
\addchap{Notation}
{\footnotesize \begin{longtable}[ht]{lp{11.6cm}}
$\emptyset$ & The empty set\\
$I$ & The closed unit interval $[0,1]$\\
$\widehat{\mathbb{R}}$ & Compactified real line, $\widehat{\mathbb{R}} = \mathbb{R} \cup \{\infty\}$\\
$\LatGen{\omega_1}{\omega_2}$ & The lattice in $\mathbb{C}$ generated by $\omega_1$ and $\omega_2$\\
$C_n$ & The cyclic group of order $n$\\
$\mathcal{M}(X,Y)$ & The space of maps $X \to Y$\\
$\mathcal{M}((X,A),(Y,B))$ & The space of maps $(X,A) \to (Y,B)$\\
$\mathcal{M}_G(X,Y)$ & The space of continuous $G$-maps $X \to Y$ (both $G$-spaces)\\
$f \simeq g$ & The maps $f$ and $g$ are homotopic\\
$f \simeq_G g$ & The maps $f$ and $g$ are $G$-equivariantly homotopic\\
$[X,Y]_G$ & The $G$-homotopy classes of maps $X \to Y$\\
$\Triple{d_0}{d}{d_1}$ & Degree triple for equivariant homotopy classes (see p.~\pageref{DefinitionTriple})\\
$\mathcal{L}X$ & The free loop space of $X$\\
$\Omega X$ & The space of based loops in $X$\\
$\Omega (X,x_0)$ & The space of based loops in $X$ with basepoint $x_0$\\
$\mathbb{H}^+$ & The upper half plane in $\mathbb{C}$\\
$\pi(X,p)$ & The fundamental group of $X$ with basepoint $p$\\
$\pi(X;p,q)$ & The homotopy classes of curves in $X$ with fixed endpoints $p$ and $q$\\
$\mathcal{H}_n$ & The set of complex, hermitian $n \times n$ matrices\\
$\mathcal{H}^*_n$ & The set of complex, hermitian, non-singular $n \times n$ matrices\\
$\mathcal{H}_{(p,q)}$ & The subset of $\mathcal{H}_{p+q}^*$ consisting of matrices with eigenvalue signature $(p,q)$\\
$c_p\colon X \to Y$ & The constant map, which sends every $x \in X$ to $p \in Y$\\
iff & if and only if\\
LHS, RHS & Left-hand side, right-hand side\\
$X^G$ & For a $G$-space $X$, $X^G = \{x \in X\colon G(x) = \{x\}\}$\\
$\ltrans{M}$ & Matrix transpose of the matrix $M$\\
$M^*$ & For a matrix $M$, $M^*$ denotes its conjugate-transpose, i.\,e. $M^* = \ltrans{\overline{M}}$\\
$e^n$ & An open $n$-cell\\
$\gamma_2 \ast \gamma_1$ & Concatenation of curves $\gamma_1$ and $\gamma_2$ where $\gamma_1(1) = \gamma_2(0)$\\
$\gamma^{-1}$ & For a curve $\gamma$, $\gamma^{-1}$ denotes the same curve with reversed time\\
$\mathbb{P}_n$ & The $n$-dimensional complex projective space\\
$\mathbb{RP}_n$ & The $n$-dimensional real projective space\\
$\I_n$ & The $n \times n$ identity matrix\\
$\I{p}{q}$ & The block diagonal matrix $\text{Diag}\,(\I{p},-\I{q})$\\
$\text{dim}_H X$ & The Hausdorff dimension of the topological space $X$\\
$\text{dim}_P X$ & The packing dimension of the topological space $X$\\
$\text{Gr}_k(\mathbb{C}^n)$ & Grassmannian of $k$-dimensional complex subvectorspaces in $\mathbb{C}^n$\\
$\text{Gr}_k(\mathbb{R}^n)$ & Grassmannian of $k$-dimensional real subvectorspaces in $\mathbb{R}^n$\\
$\mathrm{cl}(X)$ & Closure of the topological space $X$\\
$\partial X$ & Boundary of the topological space $X$ \end{longtable}}
All maps are assumed to be continuous, unless otherwise stated. When discussing the dimension of complex geometric objects we refer to its \emph{complex} dimension unless we explicitely use the term \emph{real
dimension}.
\listoffigures
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{ Randomized Block Coordinate Descent \
for Online and Stochastic Optimization}
\begin{abstract} Two types of low cost-per-iteration gradient descent methods have been extensively studied in parallel. One is online or stochastic gradient descent ( OGD/SGD), and the other is randomzied coordinate descent (RBCD). In this paper, we combine the two types of methods together and propose online randomized block coordinate descent (ORBCD). At each iteration, ORBCD only computes the partial gradient of one block coordinate of one mini-batch samples. ORBCD is well suited for the composite minimization problem where one function is the average of the losses of a large number of samples and the other is a simple regularizer defined on high dimensional variables. We show that the iteration complexity of ORBCD has the same order as OGD or SGD. For strongly convex functions, by reducing the variance of stochastic gradients, we show that ORBCD can converge at a geometric rate in expectation, matching the convergence rate of SGD with variance reduction and RBCD. \end{abstract}
\section{Introduction}
In recent years, considerable efforts in machine learning have been devoted to solving the following composite objective minimization problem: \begin{align}\label{eq:compositeobj} \min_{\mathbf{x}}~f(\mathbf{x}) + g(\mathbf{x}) = \frac{1}{I}\sum_{i=1}^{I}f_i(\mathbf{x}) + \sum_{j=1}^{J}g_j(\mathbf{x}_j)~, \end{align} where $\mathbf{x}\in\R^{n\times 1}$ and $\mathbf{x}_j$ is a block coordinate of $\mathbf{x}$. $f(\mathbf{x})$ is the average of some smooth functions, and $g(\mathbf{x})$ is a \emph{simple} function which may be non-smooth. In particular, $g(\mathbf{x})$ is block separable and blocks are non-overlapping. A variety of machine learning and statistics problems can be cast into the problem~\myref{eq:compositeobj}. In regularized risk minimization problems~\cite{hastie09:statlearn}, $f$ is the average of losses of a large number of samples and $g$ is a simple regularizer on high dimensional features to induce structural sparsity~\cite{bach11:sparse}. While $f$ is separable among samples, $g$ is separable among features.
For example, in lasso~\cite{tibs96:lasso}, $f_i$ is a square loss or logistic loss function and $g(\mathbf{x}) = \lambda \| \mathbf{x} \|_1$ where $\lambda$ is the tuning parameter. In group lasso~\cite{yuan07:glasso}, $g_j(\mathbf{x}_j) = \lambda\| \mathbf{x}_j \|_2$, which enforces group sparsity among variables. To induce both group sparsity and sparsity, sparse group lasso~\cite{friedman:sglasso} uses composite regularizers $g_j(\mathbf{x}_j) = \lambda_1\| \mathbf{x}_j \|_2 + \lambda_2 \|\mathbf{x}_j\|_1$ where $\lambda_1$ and $\lambda_2$ are the tuning parameters.
Due to the simplicity, gradient descent (GD) type methods have been widely used to solve problem~\myref{eq:compositeobj}. If $g_j$ is nonsmooth but simple enough for \emph{proximal mapping}, it is better to just use the gradient of $f_i$ but keep $g_j$ untouched in GD. This variant of GD is often called proximal splitting~\cite{comb09:prox} or proximal gradient descent (PGD)~\cite{tseng08:apgm,beck09:pgm} or forward/backward splitting method (FOBOS)~\cite{duchi09}. Without loss of generality, we simply use GD to represent GD and its variants in the rest of this paper. Let $m$ be the number of samples and $n$ be dimension of features. $m$ samples are divided into $I$ blocks (mini-batch), and $n$ features are divided into $J$ non-overlapping blocks. If both $m$ and $n$ are large, solving~\myref{eq:compositeobj} using batch methods like gradient descent (GD) type methods is computationally expensive. To address the computational bottleneck, two types of low cost-per-iteration methods, online/stochastic gradient descent (OGD/SGD)~\cite{Robi51:SP,Judi09:SP,celu06,Zinkevich03,haak06:logregret,Duchi10_comid,duchi09,xiao10} and randomized block coordinate descent (RBCD)~\cite{nesterov10:rbcd,bkbg11:pbcd,rita13:pbcd,rita12:rbcd}, have been rigorously studied in both theory and applications.
Instead of computing gradients of all samples in GD at each iteration, OGD/SGD only computes the gradient of one block samples, and thus the cost-per-iteration is just one $I$-th of GD. For large scale problems, it has been shown that OGD/SGD is faster than GD~\cite{tari13:pdsvm,shsisr07:pegasos,shte09:sgd}. OGD and SGD have been generalized to handle composite objective functions~\cite{nest07:composite,comb09:prox,tseng08:apgm,beck09:pgm,Duchi10_comid,duchi09,xiao10}. OGD and SGD use a decreasing step size and converge at a slower rate than GD. In stochastic optimization, the slow convergence speed is caused by the variance of stochastic gradients due to random samples, and considerable efforts have thus been devoted to reducing the variance to accelerate SGD~\cite{bach12:sgdlinear,bach13:sgdaverage,xiao14:psgdvd,zhang13:sgdvd,jin13:sgdmix,jin13:sgdlinear}. Stochastic average gradient (SVG)~\cite{bach12:sgdlinear} is the first SGD algorithm achieving the linear convergence rate for stronly convex functions, catching up with the convergence speed of GD~\cite{nesterov04:convex}. However, SVG needs to store all gradients, which becomes an issue for large scale datasets. It is also difficult to understand the intuition behind the proof of SVG. To address the issue of storage and better explain the faster convergence,~\cite{zhang13:sgdvd} proposed an explicit variance reduction scheme into SGD. The two scheme SGD is refered as stochastic variance reduction gradient (SVRG). SVRG computes the full gradient periodically and progressively mitigates the variance of stochastic gradient by removing the difference between the full gradient and stochastic gradient. For smooth and strongly convex functions, SVRG converges at a geometric rate in expectation. Compared to SVG, SVRG is free from the storage of full gradients and has a much simpler proof. The similar idea was also proposed independently by~\cite{jin13:sgdmix}. The results of SVRG is then improved in~\cite{kori13:ssgd}. In~\cite{xiao14:psgdvd}, SVRG is generalized to solve composite minimization problem by incorporating the variance reduction technique into proximal gradient method.
On the other hand, RBCD~\cite{nesterov10:rbcd,rita12:rbcd,luxiao13:rbcd,shte09:sgd,chang08:bcdsvm,hsieh08:dcdsvm,osher09:cdcs} has become increasingly popular due to high dimensional problem with structural regularizers. RBCD randomly chooses a block coordinate to update at each iteration. The iteration complexity of RBCD was established in~\cite{nesterov10:rbcd}, improved and generalized to composite minimization problem by~\cite{rita12:rbcd,luxiao13:rbcd}. RBCD can choose a constant step size and converge at the same rate as GD, although the constant is usually $J$ times worse~\cite{nesterov10:rbcd,rita12:rbcd,luxiao13:rbcd}. Compared to GD, the cost-per-iteration of RBCD is much cheaper. Block coordinate descent (BCD) methods have also been studied under a deterministic cyclic order~\cite{sate13:cbcd,tseng01:ds,luo02:cbcd}. Although the convergence of cyclic BCD has been established~\cite{tseng01:ds,luo02:cbcd}, the iteration of complexity is still unknown except for special cases~\cite{sate13:cbcd}.
While OGD/SGD is well suitable for problems with a large number of samples, RBCD is suitable for high dimension problems with non-overlapping composite regularizers. For large scale high dimensional problems with non-overlapping composite regularizers, it is not economic enough to use one of them. Either method alone may not suitable for problems when data is distributed across space and time or partially available at the moment~\cite{nesterov10:rbcd}. In addition, SVRG is not suitable for problems when the computation of full gradient at one time is expensive. In this paper, we propose a new method named online randomized block coordinate descent (ORBCD) which combines the well-known OGD/SGD and RBCD together. ORBCD first randomly picks up one block samples and one block coordinates, then performs the block coordinate gradient descent on the randomly chosen samples at each iteration. Essentially, ORBCD performs RBCD in the online and stochastic setting. If $f_i$ is a linear function, the cost-per-iteration of ORBCD is $O(1)$ and thus is far smaller than $O(n)$ in OGD/SGD and $O(m)$ in RBCD. We show that the iteration complexity for ORBCD has the same order as OGD/SGD. In the stochastic setting, ORBCD is still suffered from the variance of stochastic gradient. To accelerate the convergence speed of ORBCD, we adopt the varaince reduction technique~\cite{zhang13:sgdvd} to alleviate the effect of randomness. As expected, the linear convergence rate for ORBCD with variance reduction (ORBCDVD) is established for strongly convex functions for stochastic optimization. Moreover, ORBCDVD does not necessarily require to compute the full gradient at once which is necessary in SVRG and prox-SVRG. Instead, a block coordinate of full gradient is computed at each iteration and then stored for the next retrieval in ORBCDVD.
The rest of the paper is organized as follows. In Section~\ref{sec:relate}, we review the SGD and RBCD. ORBCD and ORBCD with variance reduction are proposed in Section~\ref{sec:orbcd}. The convergence results are given in Section~\ref{sec:theory}. The paper is concluded in Section~\ref{sec:conclusion}.
\section{Related Work}\label{sec:relate} In this section, we briefly review the two types of low cost-per-iteration gradient descent (GD) methods, i.e., OGD/SGD and RBCD. Applying GD on~\myref{eq:compositeobj}, we have the following iterate: \begin{align}\label{eq:fobos}
\mathbf{x}^{t+1} = \argmin_{\mathbf{x}}~\langle \nabla f(\mathbf{x}^t), \mathbf{x} \rangle + g(\mathbf{x}) + \frac{\eta_t}{2} \| \mathbf{x} - \mathbf{x}^t \|_2^2~. \end{align} In some cases, e.g. $g(\mathbf{x})$ is $\ell_1$ norm,~\myref{eq:fobos} can have a closed-form solution. \subsection{Online and Stochastic Gradient Descent} In~\myref{eq:fobos}, it requires to compute the full gradient of $m$ samples at each iteration, which could be computationally expensive if $m$ is too large. Instead, OGD/SGD simply computes the gradient of one block samples.
In the online setting, at time $t+1$, OGD first presents a solution $\mathbf{x}^{t+1}$ by solving \begin{align}
\mathbf{x}^{t+1} = \argmin_{\mathbf{x}}~\langle \nabla f_t(\mathbf{x}^t), \mathbf{x} \rangle + g(\mathbf{x}) + \frac{\eta_t}{2} \| \mathbf{x} - \mathbf{x}^t \|_2^2~. \end{align} where $f_t$ is given and assumed to be convex. Then a function $f_{t+1}$ is revealed which incurs the loss $f_t(\mathbf{x}^t)$. The performance of OGD is measured by the regret bound, which is the discrepancy between the cumulative loss over $T$ rounds and the best decision in hindsight, \begin{align} R(T) = \sum_{t=1}^{T} { [f_t(\mathbf{x}^t) + g(\mathbf{x}^t)] - [f_t(\mathbf{x}^*)+g(\mathbf{x}^*)]}~, \end{align} where $\mathbf{x}^*$ is the best result in hindsight. The regret bound of OGD is $O(\sqrt{T})$ when using decreasing step size $\eta_t = O(\frac{1}{\sqrt{t}})$. For strongly convex functions, the regret bound of OGD is $O(\log T)$ when using the step size $\eta_t = O(\frac{1}{t})$. Since $f_t$ can be any convex function, OGD considers the worst case and thus the mentioned regret bounds are optimal.
In the stochastic setting, SGD first randomly picks up $i_t$-th block samples and then computes the gradient of the selected samples as follows: \begin{align}\label{eq:sgd}
\mathbf{x}^{t+1} = \argmin_{\mathbf{x}}~\langle \nabla f_{i_t}(\mathbf{x}^t), \mathbf{x} \rangle + g(\mathbf{x}) + \frac{\eta_t}{2} \| \mathbf{x} - \mathbf{x}^t \|_2^2~. \end{align} $\mathbf{x}^t$ depends on the observed realization of the random variable $\xi = \{ i_1, \cdots, i_{t-1}\}$ or generally $\{ \mathbf{x}^1, \cdots, \mathbf{x}^{t-1} \}$. Due to the effect of variance of stochastic gradient, SGD has to choose decreasing step size, i.e., $\eta_t = O(\frac{1}{\sqrt{t}})$, leading to slow convergence speed. For general convex functions, SGD converges at a rate of $O(\frac{1}{\sqrt{t}})$. For strongly convex functions, SGD converges at a rate of $O(\frac{1}{t})$. In contrast, GD converges linearly if functions are strongly convex.
To accelerate the SGD by reducing the variance of stochastic gradient, stochastic variance reduced gradient (SVRG) was proposed by~\cite{zhang13:sgdvd}.~\cite{xiao14:psgdvd} extends SVRG to composite functions~\myref{eq:compositeobj}, called prox-SVRG. SVRGs have two stages, i.e., outer stage and inner stage. The outer stage maintains an estimate $\tilde{\mathbf{x}}$ of the optimal point $x^*$ and computes the full gradient of $\tilde{\mathbf{x}}$ \begin{align} \tilde{\mu} &= \frac{1}{n} \sum_{i=1}^{n} \nabla f_i(\tilde{\mathbf{x}}) = \nabla f(\tilde{\mathbf{x}})~. \end{align} After the inner stage is completed, the outer stage updates $\tilde{\mathbf{x}}$. At the inner stage, SVRG first randomly picks $i_t$-th sample, then modifies stochastis gradient by subtracting the difference between the full gradient and stochastic gradient at $\tilde{\mathbf{x}}$, \begin{align} \mathbf{v}_{t} &= \nabla f_{i_t}(\mathbf{x}^t) - \nabla f_{i_t}(\tilde{\mathbf{x}}) + \tilde{\mu}~. \end{align} It can be shown that the expectation of $\mathbf{v}_{t}$ given $\mathbf{x}^{t-1}$ is the full gradient at $\mathbf{x}^t$, i.e., $\mathbb{E}\mathbf{v}_{t} = \nabla f(\mathbf{x}^t)$. Although $\mathbf{v}_t$ is also a stochastic gradient, the variance of stochastic gradient progressively decreases. Replacing $\nabla f_{i_t}(\mathbf{x}^t)$ by $\mathbf{v}_t$ in SGD step~\myref{eq:sgd}, \begin{align}
\mathbf{x}^{t+1} & = \argmin_{\mathbf{x}}~\langle \mathbf{v}_{t}, \mathbf{x} \rangle + g(\mathbf{x}) + \frac{\eta}{2} \| \mathbf{x} - \mathbf{x}^t \|_2^2~. \end{align} By reduding the variance of stochastic gradient, $\mathbf{x}^t$ can converge to $\mathbf{x}^*$ at the same rate as GD, which has been proved in~\cite{zhang13:sgdvd,xiao14:psgdvd}.
For strongly convex functions, prox-SVRG~\cite{xiao14:psgdvd} can converge linearly in expectation if $\eta > 4L$ and $m$ satisfy the following condition: \begin{align}\label{eq:svrg_rho} \rho = \frac{\eta^2}{\gamma(\eta-4L)m} + \frac{4L(m+1)}{(\eta-4L)m} < 1~. \end{align} where $L$ is the constant of Lipschitz continuous gradient. Note the step size is $1/\eta$ here.
\subsection{Randomized Block Coordinate Descent} Assume $\mathbf{x}_{j} (1\leq j \leq J)$ are non-overlapping blocks. At iteration $t$, RBCD~\cite{nesterov10:rbcd,rita12:rbcd,luxiao13:rbcd} randomly picks $j_t$-th coordinate and solves the following problem: \begin{align}\label{eq:rbcd}
\mathbf{x}_{j_t}^{t+1} = \argmin_{\mathbf{x}_{j_t}}~\langle \nabla_{j_t} f(\mathbf{x}^t), \mathbf{x}_{j_t} \rangle + g_{j_t}(\mathbf{x}_{j_t}) + \frac{\eta_t}{2} \| \mathbf{x}_{j_t} - \mathbf{x}_{j_t}^t \|_2^2~. \end{align} Therefore, $\mathbf{x}^{t+1} = (\mathbf{x}_{j_t}^{t+1}, \mathbf{x}_{k\neq j_t}^t)$. $\mathbf{x}^t$ depends on the observed realization of the random variable \begin{align} \xi = \{ j_1, \cdots, j_{t-1}\}~. \end{align} Setting the step size $\eta_t = L_{j_t}$ where $L_{j_t}$ is the Lipshitz constant of $j_t$-th coordinate of the gradient $\nabla f(\mathbf{x}^t)$, the iteration complexity of RBCD is
$O(\frac{1}{t})$. For strongly convex function, RBCD has a linear convergence rate. Therefore, RBCD converges at the same rate as GD, although the constant is $J$ times larger~\cite{nesterov10:rbcd,rita12:rbcd,luxiao13:rbcd}.
\iffalse Here we briefly review and simplify the proof of the iteration complexity of RBCD in~\cite{nesterov10:rbcd,rita12:rbcd,luxiao13:rbcd}, which paves the way for the proof of ORBCD.
\begin{thm} RBCD has the following iteration complexity: \begin{align}
\mathbb{E}_{\xi_{T-1}}f(\mathbf{x}^T) - f(\mathbf{x}) & \leq \frac{J \left[ \mathbb{E} [f(\mathbf{x}^1)] - f(\mathbf{x}^*) + \frac{L}{2} \mathbb{E}\| \mathbf{x} - \mathbf{x}^1 \|_2^2 \right ]}{T} ~. \end{align} \end{thm} Denoting $g'_{j_t}(\mathbf{x}_{j_t}^{t+1}) \in \partial g_{j_t}(\mathbf{x}_{j_t}^{t+1})$, the optimality condition of~\myref{eq:rbcd} is \begin{align} \langle \nabla_{j_t} f(\mathbf{x}^t) + g'_{j_t}(\mathbf{x}_{j_t}^{t+1}) + \eta_t (\mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t) , \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t} \rangle \leq 0~. \end{align} Rearranging the terms yields \begin{align} & \langle \nabla_{j_t} f(\mathbf{x}^t) + g'_{j_t}(\mathbf{x}_{j_t}^{t+1}) , \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t} \rangle \leq - \eta_t \langle \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t , \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t} \rangle \nonumber \\
& \leq \frac{\eta_t}{2} ( \| \mathbf{x}_{j_t} - \mathbf{x}_{j_t}^t \|_2^2 - \| \mathbf{x}_{j_t} - \mathbf{x}_{j_t}^{t+1} \|_2^2 - \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 ) \nonumber \\
& \leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2 - \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 ) ~. \end{align} Using the smoothness of $f$ and $\mathbf{x}^{t+1} = (\mathbf{x}_{j_t}^{t+1}, \mathbf{x}_{k\neq j_t}^t)$, we have \begin{align} & f(\mathbf{x}^{t+1}) + g(\mathbf{x}^{t+1})- [ f(\mathbf{x}^t) + g(\mathbf{x}^t) ] \nonumber \\
& \leq \langle \nabla_{j_t} f(\mathbf{x}^t), \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \rangle + \frac{L_{j_t}}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 + g_{j_t}(\mathbf{x}_{j_t}^{t+1}) - g_{j_t}(\mathbf{x}_{j_t}) - [g_{j_t}(\mathbf{x}_{j_t}^{t}) - g_{j_t}(\mathbf{x}_{j_t}) ] \nonumber \\
& = \langle \nabla_{j_t} f(\mathbf{x}^t) + g'_{j_t}(\mathbf{x}_{j_t}^{t+1}) , \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t} \rangle + \frac{L_{j_t}}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 - \langle \nabla_{j_t} f(\mathbf{x}^t), \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle - [g_{j_t}(\mathbf{x}_{j_t}^{t}) - g_{j_t}(\mathbf{x}_{j_t}) ]\nonumber \\
& \leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{L_{j_t} - \eta_t}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 - \langle \nabla_{j_t} f(\mathbf{x}^t), \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle - [g_{j_t}(\mathbf{x}_{j_t}^{t}) - g_{j_t}(\mathbf{x}_{j_t}) ] \nonumber \\
& \leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) - \langle \nabla_{j_t} f(\mathbf{x}^t), \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle - [g_{j_t}(\mathbf{x}_{j_t}^{t}) - g_{j_t}(\mathbf{x}_{j_t}) ]~. \end{align} where the last inequality is obtained by setting $\eta_t = L_{j_t}$. Conditioned on $\mathbf{x}^t$, take expectation over $j_t$ gives \begin{align}\label{eq1}
& \mathbb{E}_{j_t}[f(\mathbf{x}^{t+1}) + g(\mathbf{x}^{t+1})|\mathbf{x}^t] - [f(\mathbf{x}^t) +g(\mathbf{x}^t)] \nonumber \\
& \leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \mathbb{E}_{j_t}\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) - \frac{1}{J} \sum_{j=1}^J \langle \nabla_{j_t} f(\mathbf{x}^t), \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle - \frac{1}{J}[g(\mathbf{x}^t) - g(\mathbf{x}) ]\nonumber \\
& = \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \mathbb{E}_{j_t}\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) - \frac{1}{J} \langle \nabla f(\mathbf{x}^t), \mathbf{x}^{t} - \mathbf{x} \rangle - \frac{1}{J}[g(\mathbf{x}^t) - g(\mathbf{x}) ] \nonumber \\
& \leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \mathbb{E}_{j_t}\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) - \frac{1}{J} [ f(\mathbf{x}^t) + g(\mathbf{x}^t) - (f(\mathbf{x}) + g(\mathbf{x})) ]~. \end{align} Taking expectation over $\xi_t$, we have \begin{align} & \mathbb{E}_{\xi_t}[f(\mathbf{x}^{t+1}) + g(\mathbf{x}^{t+1})] -\mathbb{E}_{\xi_{t-1}} [ f(\mathbf{x}^t) + g(\mathbf{x}^t)] \nonumber \\
& \leq \frac{\eta_t}{2} ( \mathbb{E}_{\xi_{t-1}}\| \mathbf{x} - \mathbf{x}^t \|_2^2 - \mathbb{E}_{\xi_t}\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) - \frac{1}{J} \{ \mathbb{E}_{\xi_{t-1}} [ f(\mathbf{x}^t) +g(\mathbf{x}^t) ] - \mathbb{E}_{\xi_{t-1}} [ f(\mathbf{x}) + g(\mathbf{x})] \} ~. \end{align} Setting $\mathbf{x} = \mathbf{x}^t$ gives \begin{align}
& \mathbb{E}_{\xi_{t}}[f(\mathbf{x}^{t+1}) + g(\mathbf{x}^{t+1})] - \mathbb{E}_{\xi_{t-1}} [ f(\mathbf{x}^t) + g(\mathbf{x}^{t})] \leq - \frac{\eta_t}{2} \mathbb{E}_{\xi_t}[\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2]~. \end{align} Thus, $\mathbb{E}_{\xi_{t-1}}[f(\mathbf{x}^t) + g(\mathbf{x}^{t})]$ decreases monotonically.
Let $\mathbf{x} = \mathbf{x}^*$, which is an optimal solution. Rearranging the temrs of~\myref{eq1} yields \begin{align} &\mathbb{E}_{\xi_{t}}[f(\mathbf{x}^t) + g(\mathbf{x}^t)] - [ f(\mathbf{x}^*) + g(\mathbf{x}^*) ] \nonumber \\
&\leq J \left[ \mathbb{E}_{\xi_{t-1}} [f(\mathbf{x}^t) + g(\mathbf{x}^{t}) ] - \mathbb{E}_{\xi_{t}}[f(\mathbf{x}^{t+1}) + g(\mathbf{x}^{t+1})] + \frac{\eta_t}{2} ( \mathbb{E}_{\xi_{t-1}}\| \mathbf{x}^* - \mathbf{x}^t \|_2^2 - \mathbb{E}_{\xi_{t}}[\| \mathbf{x}^* - \mathbf{x}^{t+1} \|_2^2]) \right]~. \end{align} Let $\eta_t = L = \max_j {L_{j}} $. Summing over $t$, we have \begin{align} & \sum_{t=1}^T\mathbb{E}_{\xi_{t-1}}[f(\mathbf{x}^t) + g(\mathbf{x}^{t})] - [ f(\mathbf{x}^*) + g(\mathbf{x}^{*})] \nonumber \\
& \leq J \left\{ f(\mathbf{x}^1) + g(\mathbf{x}^{1}) -\mathbb{E}_{\xi_{T}} [ f(\mathbf{x}^{T+1})+ g(\mathbf{x}^{T+1})] + \frac{L}{2}\| \mathbf{x}^* - \mathbf{x}^1 \|_2^2 \right \} \nonumber \\
& \leq J \left\{ [f(\mathbf{x}^1) + g(\mathbf{x}^{1})] - [ f(\mathbf{x}^*) +g(\mathbf{x}^*)] + \frac{L}{2} \| \mathbf{x}^* - \mathbf{x}^1 \|_2^2 \right \} ~. \end{align} Using the monotonicity of $\mathbb{E}_{\xi_{t-1}}f(\mathbf{x}^t)$ and dividing $T$ on both sides complete the proof. \qed \fi
\section{Online Randomized Block Coordinate Descent}\label{sec:orbcd} In this section, our goal is to combine OGD/SGD and RBCD together to solve problem~\myref{eq:compositeobj}. We call the algorithm online randomized block coordinate descent (ORBCD), which computes one block coordinate of the gradient of one block of samples at each iteration. ORBCD essentially performs RBCD in online and stochastic setting.
Let $\{ \mathbf{x}_1, \cdots, \mathbf{x}_J \}, \mathbf{x}_j\in \R^{n_j\times 1}$ be J non-overlapping blocks of $\mathbf{x}$. Let $U_j \in \R^{n\times n_j}$ be $n_j$ columns of an $n\times n$ permutation matrix $\mathbf{U}$, corresponding to $j$ block coordinates in $\mathbf{x}$. For any partition of $\mathbf{x}$ and $\mathbf{U}$, \begin{align} \mathbf{x} = \sum_{j=1}^{J}U_j\mathbf{x}_j~, \mathbf{x}_j = U_j^T\mathbf{x}~. \end{align} The $j$-th coordinate of gradient of $f$ can be denoted as \begin{align} \nabla_j f(\mathbf{x}) = U_j^T \nabla f(\mathbf{x})~. \end{align} Throughout the paper, we assume that the minimum of problem~\myref{eq:compositeobj} is attained. In addition, ORBCD needs the following assumption :
\begin{asm}\label{asm:orbcd1} $f_t$ or $f_i$ has block-wise Lipschitz continuous gradient with constant $L_j$, e.g., \begin{align}
\| \nabla_j f_t(\mathbf{x} + U_j h_j ) - \nabla_j f_t(\mathbf{x}) \|_2 \leq L_j \| h_j \|_2 \leq L \| h_j \|_2~, \end{align} where $L = \max_j L_j$. \end{asm} \begin{asm}\label{asm:orbcd2}
1. $\| \nabla f_t (\mathbf{x}^t) \|_2 \leq R_f $, or $\| \nabla f (\mathbf{x}^t) \|_2 \leq R_f $;
2. $\mathbf{x}^t$ is assumed in a bounded set ${\cal X}$, i.e., $\sup_{\mathbf{x},\mathbf{y} \in {\cal X}} \| \mathbf{x} - \mathbf{y} \|_2 = D$. \end{asm} While the Assumption~\ref{asm:orbcd1} is used in RBCD, the Assumption~\ref{asm:orbcd2} is used in OGD/SGD. We may assume the sum of two functions is strongly convex. \begin{asm}\label{asm:orbcd3} $f_t(\mathbf{x}) + g(\mathbf{x})$ or $f(\mathbf{x}) + g(\mathbf{x})$ is $\gamma$-strongly convex, e.g., we have \begin{align}\label{eq:stronggcov}
f_t(\mathbf{x}) + g(\mathbf{x}) \geq f_t(\mathbf{y}) + g(\mathbf{y}) + \langle \nabla f_t(\mathbf{y}) + g'(\mathbf{y}), \mathbf{x} - \mathbf{x}^{t} \rangle + \frac{\gamma}{2} \| \mathbf{x} - \mathbf{y} \|_2^2~. \end{align} where $\gamma > 0$ and $g'(\mathbf{y})$ denotes the subgradient of $g$ at $\mathbf{y}$. \end{asm}
\subsection{ORBCD for Online Learning} In online setting, ORBCD considers the worst case and runs at rounds. At time $t$, given any function $f_t$ which may be agnostic, ORBCD randomly chooses $j_t$-th block coordinate and presents the solution by solving the following problem: \begin{align}\label{eq:orbcdo}
\mathbf{x}_{j_t}^{t+1} &= \argmin_{\mathbf{x}_{j_t}}~\langle \nabla_{j_t} f_t(\mathbf{x}^t), \mathbf{x}_{j_t} \rangle + g_{j_t}(\mathbf{x}_{j_t}) + \frac{\eta_t}{2} \| \mathbf{x}_{j_t} - \mathbf{x}_{j_t}^t \|_2^2 \nonumber \\ & = \text{Prox}_{g_{j_t}}(\mathbf{x}_{j_t} -\frac{1}{\eta_t}\nabla_{j_t} f_t(\mathbf{x}^t) )~, \end{align} where $\text{Prox}$ denotes the proximal mapping. If $f_t$ is a linear function, e.g., $f_t = l_t\mathbf{x}^t$, then $\nabla_{j_t} f_t(\mathbf{x}^t) = l_{j_t}$, so solving~\myref{eq:orbcdo} is $J$ times cheaper than OGD. Thus, $\mathbf{x}^{t+1} = ( \mathbf{x}_{j_t}^{t+1}, \mathbf{x}_{k\neq j_t}^t)$, or \begin{align} \mathbf{x}^{t+1} = \mathbf{x}^t + U_{j_t}(\mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t)~. \end{align} Then, ORBCD receives a loss function $f_{t+1}(\mathbf{x})$ which incurs the loss $f_{t+1}(\mathbf{x}^{t+1})$. The algorithm is summarized in Algorithm~\ref{alg:orbcd_online}.
$\mathbf{x}^t$ is independent of $j_t$ but depends on the sequence of observed realization of the random variable \begin{align} \xi = \{ j_1, \cdots, j_{t-1} \}. \end{align} Let $\mathbf{x}^*$ be the best solution in hindsight. The regret bound of ORBCD is defined as \begin{align} R(T) = \sum_{t=1}^T\left \{ \mathbb{E}_{\xi}[ f_t(\mathbf{x}^t) + g(\mathbf{x}^t) ] - [f_t(\mathbf{x}^*) +g(\mathbf{x}^*)] \right \}~. \end{align} By setting $\eta_t = \sqrt{t} + L$ where $L=\max_jL_j$, the regret bound of ORBCD is $O(\sqrt{T})$. For strongly convex functions, the regret bound of ORBCD is $O(\log T)$ by setting $\eta_t = \frac{\gamma t}{J} + L$.
\begin{algorithm*}[tb] \caption{Online Randomized Block Coordinate Descent for Online Learning} \label{alg:orbcd_online} \begin{algorithmic}[1]
\STATE {\bfseries Initialization:} $\mathbf{x}^1 = \mathbf{0}$ \FOR{$t=1 \text{ to } T$} \STATE randomly pick up $j_t$ block coordinates
\STATE $\mathbf{x}_{j_t}^{t+1} = \argmin_{\mathbf{x}_{j_t} \in {\cal X}_j}~\langle \nabla_{j_t} f_t(\mathbf{x}^t), \mathbf{x}_{j_t} \rangle + g_{j_t}(\mathbf{x}_{j_t}) + \frac{\eta_t}{2} \| \mathbf{x}_{j_t} - \mathbf{x}_{j_t}^t \|_2^2$~ \STATE $\mathbf{x}^{t+1} = \mathbf{x}^t + U_{j_t}(\mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t)$ \STATE receives the function $f_{t+1}(\mathbf{x}) + g(\mathbf{x})$ and incurs the loss $f_{t+1}(\mathbf{x}^{t+1}) + g(\mathbf{x}^{t+1})$ \ENDFOR \end{algorithmic} \end{algorithm*}
\subsection{ORBCD for Stochastic Optimization} In the stochastic setting, ORBCD first randomly picks up $i_t$-th block sample and then randomly chooses $j_t$-th block coordinate. The algorithm has the following iterate: \begin{align}\label{eq:orbcds}
\mathbf{x}_{j_t}^{t+1} & = \argmin_{\mathbf{x}_{j_t}}~\langle \nabla_{j_t} f_{i_t}(\mathbf{x}^t), \mathbf{x}_{j_t} \rangle + g_{j_t}(\mathbf{x}_{j_t}) + \frac{\eta_t}{2} \| \mathbf{x}_{j_t} - \mathbf{x}_{j_t}^t \|_2^2 \nonumber \\ & = \text{Prox}_{g_{j_t}}(\mathbf{x}_{j_t} -\nabla_{j_t} f_{i_t}(\mathbf{x}^t) )~. \end{align} For high dimensional problem with non-overlapping composite regularizers, solving~\myref{eq:orbcds} is computationally cheaper than solving~\myref{eq:sgd} in SGD. The algorithm of ORBCD in both settings is summarized in Algorithm~\ref{alg:orbcd_stochastic}.
$\mathbf{x}^{t+1}$ depends on $(i_t, j_t)$, but $j_{t}$ and $i_{t}$ are independent. $\mathbf{x}^t$ is independent of $(i_t, j_t)$ but depends on the observed realization of the random variables \begin{align} \xi = \{ ( i_1, j_1), \cdots, (i_{t-1}, j_{t-1}) \}~. \end{align} The online-stochastic conversion rule~\cite{Duchi10_comid,duchi09,xiao10} still holds here. The iteration complexity of ORBCD can be obtained by dividing the regret bounds in the online setting by $T$. Setting $\eta_t = \sqrt{t} + L$ where $L=\max_jL_j$, the iteration complexity of ORBCD is \begin{align} \mathbb{E}_{\xi} [ f(\bar{\mathbf{x}}^t) + g(\bar{\mathbf{x}}^t) ] - [f(\mathbf{x}) +g(\mathbf{x})] \leq O(\frac{1}{\sqrt{T}})~. \end{align} For strongly convex functions, setting $\eta_t = \frac{\gamma t}{J} + L$, \begin{align} \mathbb{E}_{\xi} [ f(\bar{\mathbf{x}}^t) + g(\bar{\mathbf{x}}^t) ] - [f(\mathbf{x}) +g(\mathbf{x})] \leq O(\frac{\log T}{T})~. \end{align} The iteration complexity of ORBCD match that of SGD. Simiarlar as SGD, the convergence speed of ORBCD is also slowed down by the variance of stochastic gradient.
\begin{algorithm*}[tb] \caption{Online Randomized Block Coordinate Descent for Stochastic Optimization} \label{alg:orbcd_stochastic} \begin{algorithmic}[1]
\STATE {\bfseries Initialization:} $\mathbf{x}^1 = \mathbf{0}$ \FOR{$t=1 \text{ to } T$} \STATE randomly pick up $i_t$ block samples and $j_t$ block coordinates
\STATE $\mathbf{x}_{j_t}^{t+1} = \argmin_{\mathbf{x}_{j_t} \in {\cal X}_j}~\langle \nabla_{j_t} f_{i_t}(\mathbf{x}^t), \mathbf{x}_{j_t} \rangle + g_{j_t}(\mathbf{x}_{j_t}) + \frac{\eta_t}{2} \| \mathbf{x}_{j_t} - \mathbf{x}_{j_t}^t \|_2^2$~ \STATE $\mathbf{x}^{t+1} = \mathbf{x}^t + U_{j_t}(\mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t)$ \ENDFOR \end{algorithmic} \end{algorithm*}
\begin{algorithm*}[tb] \caption{Online Randomized Block Coordinate Descent with Variance Reduction} \label{alg:orbcdvd} \begin{algorithmic}[1]
\STATE {\bfseries Initialization:} $\mathbf{x}^1 = \mathbf{0}$ \FOR{$t=2 \text{ to } T$} \STATE $\mathbf{x}_0 = \tilde{\mathbf{x}} = \mathbf{x}^t$. \FOR{$k = 0\textbf{ to } m-1$} \STATE randomly pick up $i_k$ block samples \STATE randomly pick up $j_k$ block coordinates \STATE $\mathbf{v}_{j_k}^{i_k} = \nabla_{j_k} f_{i_k}(\mathbf{x}^{k}) - \nabla_{j_k} f_{i_k}(\tilde{\mathbf{x}}) + \tilde{\mu}_{j_k}$ where $\tilde{\mu}_{j_k} = \nabla_{j_k} f(\tilde{\mathbf{x}})$
\STATE $\mathbf{x}_{j_k}^{k} = \argmin_{\mathbf{x}_{j_k} }~\langle \mathbf{v}_{j_k}^{i_k}, \mathbf{x}_{j_k} \rangle + g_{j_k}(\mathbf{x}_{j_k}) + \frac{\eta_k}{2} \| \mathbf{x}_{j_k} - \mathbf{x}_{j_k}^{k}\|_2^2$~ \STATE $\mathbf{x}^{k+1} = \mathbf{x}^k + U_{j_k}(\mathbf{x}_{j_j}^{k+1} - \mathbf{x}_{j_k}^k)$ \ENDFOR \STATE $\mathbf{x}^{t+1} = \mathbf{x}^m$ or $\frac{1}{m}\sum_{k=1}^{m}\mathbf{x}^k$ \ENDFOR \end{algorithmic} \end{algorithm*} \subsection{ORBCD with variance reduction} In the stochastic setting, we apply the variance reduction technique~\cite{xiao14:psgdvd,zhang13:sgdvd} to accelerate the rate of convergence of ORBCD, abbreviated as ORBCDVD. As SVRG and prox-SVRG, ORBCDVD consists of two stages. At time $t+1$, the outer stage maintains an estimate $\tilde{\mathbf{x}} = \mathbf{x}^t$ of the optimal $\mathbf{x}^*$ and updates $\tilde{\mathbf{x}}$ every $m+1$ iterations. The inner stage takes $m$ iterations which is indexed by $k = 0,\cdots, m-1$. At the $k$-th iteration, ORBCDVD randomly picks $i_k$-th sample and $j_k$-th coordinate and compute \begin{align} \mathbf{v}_{j_k}^{i_k} &= \nabla_{j_k} f_{i_k}(\mathbf{x}^k) - \nabla_{j_k} f_{i_k}(\tilde{\mathbf{x}}) + \tilde{\mu}_{j_k}~, \label{eq:orbcdvd_vij} \end{align} where \begin{align}\label{eq:orbcdvd_mu} \tilde{\mu}_{j_k} = \frac{1}{n} \sum_{i=1}^{n} \nabla_{j_k} f_i(\tilde{\mathbf{x}}) = \nabla_{j_k} f(\tilde{\mathbf{x}})~. \end{align} $\mathbf{v}_{j_t}^{i_t}$ depends on $(i_t, j_t)$, and $i_t$ and $j_t$ are independent. Conditioned on $\mathbf{x}^k$, taking expectation over $i_k, j_k$ gives
\begin{align} \mathbb{E}\mathbf{v}_{j_k}^{i_k} &= \mathbb{E}_{i_k} \mathbb{E}_{j_k}[\nabla_{j_k} f_{i_k}(\mathbf{x}^k) - \nabla_{j_k} f_{i_k}(\tilde{\mathbf{x}}) + \tilde{\mu}_{j_k}] \nonumber \\ &= \frac{1}{J}\mathbb{E}_{i_k} [\nabla f_{i_k}(\mathbf{x}^k) - \nabla f_{i_k}(\tilde{\mathbf{x}}) + \tilde{\mu} ] \nonumber \\ & = \frac{1}{J}\nabla f(\mathbf{x}^k)~. \end{align}
Although $\mathbf{v}_{j_k}^{i_k}$ is stochastic gradient, the variance $\mathbb{E} \| \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{x}^k) \|_2^2$ decreases progressively and is smaller than $\mathbb{E} \| \nabla f_{i_t}(\mathbf{x}^t) - \nabla f(\mathbf{x}^t) \|_2^2$. Using the variance reduced gradient $\mathbf{v}_{j_k}^{i_k}$, ORBCD then performs RBCD as follows: \begin{align}
\mathbf{x}_{j_k}^{k+1} &= \argmin_{\mathbf{x}_{j_k}}~ \langle \mathbf{v}_{j_k}^{i_k}, \mathbf{x}_{j_k} \rangle + g_{j_k}(\mathbf{x}_{j_k}) + \frac{\eta}{2} \| \mathbf{x}_{j_k} - \mathbf{x}_{j_k}^k \|_2^2 \label{eq:orbcdvd_xj}~. \end{align} After $m$ iterations, the outer stage updates $\mathbf{x}^{t+1}$ which is either $\mathbf{x}^m$ or $\frac{1}{m}\sum_{k=1}^{m}\mathbf{x}^k$. The algorithm is summarized in Algorithm~\ref{alg:orbcdvd}. At the outer stage, ORBCDVD does not necessarily require to compute the full gradient at once. If the computation of full gradient requires substantial computational efforts, SVRG has to stop and complete the full gradient step before making progress. In contrast, $\tilde{\mu}$ can be partially computed at each iteration and then stored for the next retrieval in ORBCDVD.
Assume $\eta > 2L$ and $m$ satisfy the following condition: \begin{align}\label{eq:orbcdvd_rho} \rho = \frac{L(m+1)}{(\eta-2L)m} + \frac{(\eta-L)J}{(\eta-2L)m} - \frac{1}{m}+ \frac{\eta (\eta-L)J}{(\eta-2L)m\gamma} < 1~, \end{align} Then $h(\mathbf{x})$ converges linearly in expectation, i.e., \begin{align}
\mathbb{E}_{\xi} [f(\mathbf{x}^t) + g(\mathbf{x}^t) - (f(\mathbf{x}^*) + g(\mathbf{x}^*) ] \leq O(\rho^t)~. \end{align}
Setting $\eta = 4L$ in~\myref{eq:orbcdvd_rho} yields \begin{align} \rho = \frac{m+1}{2m} + \frac{3J}{2m} - \frac{1}{m}+ \frac{6JL}{m\gamma} \leq \frac{1}{2} + \frac{3 J}{2m}(1+\frac{4 L}{\gamma})~. \end{align} Setting $m = 18JL/\gamma$, then \begin{align} \rho \leq \frac{1}{2} + \frac{1}{12}(\frac{\gamma}{L}+4) \approx \frac{11}{12}~. \end{align} where we assume $\gamma/L \approx 1$ for simplicity.
\section{The Rate of Convergence}\label{sec:theory} The following lemma is a key building block of the proof of the convergence of ORBCD in both online and stochastic setting. \begin{lem} Let the Assumption~\ref{asm:orbcd1} and \ref{asm:orbcd2} hold. Let $\mathbf{x}^t$ be the sequences generated by ORBCD. $j_t$ is sampled randomly and uniformly from $\{1,\cdots, J \}$. We have \begin{align}\label{eq:orbcd_key_lem}
& \langle \nabla_{j_t} f_t(\mathbf{x}^t) + g'_{j_t}(\mathbf{x}_{j_t}^t), \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle \leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{R_f^2}{2(\eta_t - L)} + g(\mathbf{x}^t) - g(\mathbf{x}^{t+1}) ~. \end{align} where $L = \max_j L_j$. \end{lem} \noindent{\itshape Proof:}\hspace*{1em} The optimality condition is \begin{align} \langle \nabla_{j_t} f_t(\mathbf{x}^t) + \eta_t (\mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t) + g'_{j_t}(\mathbf{x}_{j_t}^{t+1}), \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t} \rangle \leq 0~. \end{align}
Rearranging the terms yields \begin{align} & \langle \nabla_{j_t} f_t(\mathbf{x}^t) + g'_{j_t}(\mathbf{x}_{j_t}^{t+1}) , \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t} \rangle \leq - \eta_t \langle \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t , \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t} \rangle \nonumber \\
& \leq \frac{\eta_t}{2} ( \| \mathbf{x}_{j_t} - \mathbf{x}_{j_t}^t \|_2^2 - \| \mathbf{x}_{j_t} - \mathbf{x}_{j_t}^{t+1} \|_2^2 - \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 ) \nonumber \\
& = \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2 - \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 ) ~, \end{align} where the last equality uses $\mathbf{x}^{t+1} = (\mathbf{x}_{j_t}^{t+1}, \mathbf{x}_{k\neq {j_t}}^t)$. By the smoothness of $f_t$, we have \begin{align}
f_t(\mathbf{x}^{t+1}) \leq f_t(\mathbf{x}^t) + \langle \nabla_j f_t(\mathbf{x}^t), \mathbf{x}_j^{t+1} - \mathbf{x}_j^t \rangle + \frac{L_j}{2} \| \mathbf{x}_j^{t+1} - \mathbf{x}_j^t \|_2^2~. \end{align} Since $\mathbf{x}^{t+1} - \mathbf{x}^t = U_{j_t}(\mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^{t})$, \iffalse the convexity of $g$ gives \begin{align} & g(\mathbf{x}^{t+1}) - g(\mathbf{x}^t) \leq \langle g'(\mathbf{x}^{t+1}), \mathbf{x}^{t+1} - \mathbf{x}^t \rangle \leq \langle g'_{j_t}(\mathbf{x}_{j_t}^{t+1}) + \sum_{\mathbb{I}_{j_t} \cap \mathbb{I}_k \neq \emptyset} g'_{k}(\mathbf{x}_{k}^{t+1}), \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \rangle \nonumber \\
& \leq \langle g'_{j_t}(\mathbf{x}_{j_t}^{t+1}) , \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \rangle + \frac{1}{2\alpha}\| \sum_{\mathbb{I}_{j_t} \cap \mathbb{I}_k \neq \emptyset} g'_{k}(\mathbf{x}_{k}^{t+1}) \|_2^2 + \frac{\alpha}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 \nonumber \\
& \leq \langle g'_{j_t}(\mathbf{x}_{j_t}^{t+1}) , \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \rangle + \frac{(J-1)R_g^2}{2\alpha} + \frac{\alpha}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2~. \end{align} \begin{align} g(\mathbf{x}^{t+1}) - g(\mathbf{x}^t) \leq \langle g'_{j_t}(\mathbf{x}_{j_t}^{t+1}) , \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \rangle~. \end{align} \begin{align}
\| \mathbf{x}_{k}^{t+1} - \mathbf{x}_{k}^t \|_2^2 = \| \mathbf{x}_{\mathbb{I}_{j_t} \cap \mathbb{I}_k \neq \emptyset}^{t+1} - \mathbf{x}_{\mathbb{I}_{j_t} \cap \mathbb{I}_k \neq \emptyset}^t \|_2^2 \leq \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2~. \end{align} \begin{align}
\langle \sum_{\mathbb{I}_{j_t} \cap \mathbb{I}_k \neq \emptyset} g'_{k}(\mathbf{x}_{k}^{t+1}), \mathbf{x}_{k}^{t+1} - \mathbf{x}_{k}^t \rangle \leq \frac{(J-1)R_g^2}{2\alpha} + \frac{\alpha}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2~. \end{align} \fi we have \begin{align} & f_t(\mathbf{x}^{t+1}) + g(\mathbf{x}^{t+1}) - [f_t(\mathbf{x}^t) + g(\mathbf{x}^t)] \nonumber \\
& \leq \langle \nabla_{j_t} f_t(\mathbf{x}^t), \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \rangle + \frac{L_{j_t}}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 + g_{j_t}(\mathbf{x}_{j_t}^{t+1}) - g_{j_t}(\mathbf{x}_{j_t}) + g_{j_t}(\mathbf{x}_{j_t}^{t}) - g_{j_t}(\mathbf{x}_{j_t}) \nonumber \\
& \leq \langle \nabla_{j_t} f_t(\mathbf{x}^t) + g'_{j_t}(\mathbf{x}_{j_t}^{t+1}), \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t} \rangle + \frac{L_{j_t}}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 - \langle \nabla_{j_t} f_t(\mathbf{x}^t) + g'_{j_t}(\mathbf{x}_{j_t}^{t}), \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle \nonumber \\
& \leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{L_{j_t} - \eta_t}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 - \langle \nabla_{j_t} f_t(\mathbf{x}^t) + g'_{j_t}(\mathbf{x}_{j_t}^{t}), \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle ~. \end{align} Rearranging the terms yields \begin{align}\label{eq:lem1}
\langle \nabla_{j_t} f_t(\mathbf{x}^t) + g_{j_t}'(\mathbf{x}^t) , \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle &\leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{L_{j_t} - \eta_t}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 \nonumber \\ &+ f_t(\mathbf{x}^t) + g(\mathbf{x}^t)- [ f_t(\mathbf{x}^{t+1}) + g(\mathbf{x}^{t+1}) ]~. \end{align} The convexity of $f_t$ gives \begin{align}
f_t(\mathbf{x}^t) - f_t(\mathbf{x}^{t+1}) \leq \langle \nabla f_t(\mathbf{x}^t) , \mathbf{x}^t - \mathbf{x}^{t+1} \rangle = \langle \nabla_{j_t} f_t(\mathbf{x}^t) , \mathbf{x}_{j_t}^t - \mathbf{x}_{j_t}^{t+1} \rangle \leq \frac{1}{2\alpha} \| \nabla_{j_t} f_t(\mathbf{x}^t) \|_2^2 + \frac{\alpha}{2} \| \mathbf{x}_{j_t}^t - \mathbf{x}_{j_t}^{t+1} \|_2^2~. \end{align} where the equality uses $\mathbf{x}^{t+1} = (\mathbf{x}_{j_t}^{t+1}, \mathbf{x}_{k\neq {j_t}}^t)$. Plugging into~\myref{eq:lem1}, we have \begin{align} & \langle \nabla_{j_t} f_t(\mathbf{x}^t) + g'_{j_t}(\mathbf{x}_{j_t}^t), \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle \nonumber \\
& \leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{L_{j_t} - \eta_t}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 + \langle \nabla_{j_t} f_t(\mathbf{x}^t) , \mathbf{x}_{j_t}^t - \mathbf{x}_{j_t}^{t+1} \rangle + g(\mathbf{x}^t) - g(\mathbf{x}^{t+1}) \nonumber \\
& \leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{L_{j_t} - \eta_t}{2} \| \mathbf{x}_{j_t}^{t+1} - \mathbf{x}_{j_t}^t \|_2^2 + \frac{\alpha}{2}\| \mathbf{x}_{j_t}^t - \mathbf{x}_{j_t}^{t+1} \|_2^2 + \frac{1}{2\alpha} \| \nabla_{j_t} f_t(\mathbf{x}^t) \|_2^2 ~. \end{align} Let $ L = \max_{j} L_j$. Setting $\alpha = \eta_t - L$ where $\eta_t > L$ completes the proof. \qed
This lemma is also a key building block in the proof of iteration complexity of GD, OGD/SGD and RBCD. In GD, by setting $\eta_t = L$, the iteration complexity of GD can be established. In RBCD, by simply setting $\eta_t = L_{j_t}$, the iteration complexity of RBCD can be established.
\subsection{Online Optimization} Note $\mathbf{x}^t$ depends on the sequence of observed realization of the random variable $\xi = \{ j_1, \cdots, j_{t-1} \}$. The following theorem establishes the regret bound of ORBCD. \begin{thm} Let $\eta_t = \sqrt{t} + L$ in the ORBCD and the Assumption~\ref{asm:orbcd1} and \ref{asm:orbcd2} hold. $j_t$ is sampled randomly and uniformly from $\{1,\cdots, J \}$. The regret bound $R(T)$ of ORBCD is \begin{align} R(T) \leq J ( \frac{\sqrt{T} + L}{2}D^2 + \sqrt{T} R^2 + g(\mathbf{x}^1) - g(\mathbf{x}^*) )~. \end{align} \end{thm} \noindent{\itshape Proof:}\hspace*{1em} In~\myref{eq:orbcd_key_lem}, conditioned on $\mathbf{x}^t$, take expectation over $j_t$, we have \begin{align}\label{eq:a}
\frac{1}{J} \langle \nabla f_t(\mathbf{x}^t) + g'(\mathbf{x}^t), \mathbf{x}^{t} - \mathbf{x} \rangle &\leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \mathbb{E}\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{R^2}{2(\eta_t - L)} + g(\mathbf{x}^t) - \mathbb{E}g(\mathbf{x}^{t+1}) ~. \end{align} Using the convexity, we have \begin{align} f_t(\mathbf{x}^t) + g(\mathbf{x}^t) - [f_t(\mathbf{x}) + g(\mathbf{x})] \leq \langle \nabla f_t(\mathbf{x}^t) + g'(\mathbf{x}^t), \mathbf{x}^{t} - \mathbf{x} \rangle~. \end{align} Together with~\myref{eq:a}, we have \begin{align}
f_t(\mathbf{x}^t) + g(\mathbf{x}^t) - [f_t(\mathbf{x}) + g(\mathbf{x}) ] &\leq J \left \{ \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \mathbb{E}\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{R^2}{2(\eta_t - L)} + g(\mathbf{x}^t) - \mathbb{E}g(\mathbf{x}^{t+1}) \right \}~. \end{align} Taking expectation over $\xi$ on both sides, we have \begin{align}
\mathbb{E}_{\xi} \left [ f_t(\mathbf{x}^t) + g(\mathbf{x}^t) - [f_t(\mathbf{x}) + g(\mathbf{x}) ] \right ] &\leq J \left \{ \frac{\eta_t}{2} ( \mathbb{E}_{\xi}\| \mathbf{x} - \mathbf{x}^t \|_2^2 - \mathbb{E}_{\xi}\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) \right .\nonumber \\ & + \left. \frac{R^2}{2(\eta_t - L)} + \mathbb{E}_{\xi} g(\mathbf{x}^t) - \mathbb{E}_{\xi}g(\mathbf{x}^{t+1}) \right \}~. \end{align} Summing over $t$ and setting $\eta_t = \sqrt{t} + L$, we obtain the regret bound \begin{align}\label{eq:orbcd_rgt0} & R(T) = \sum_{t=1}^T\left \{ \mathbb{E}_{\xi} [ f_t(\mathbf{x}^t) + g(\mathbf{x}^t) ] - [f_t(\mathbf{x}) +g(\mathbf{x})] \right \} \nonumber \\
&\leq J \left \{ - \frac{\eta_{T}}{2} \mathbb{E}_{\xi}\| \mathbf{x} - \mathbf{x}^{T+1} \|_2^2 + \sum_{t=1}^{T}(\eta_{t} - \eta_{t-1}) \mathbb{E}_{\xi}\| \mathbf{x} - \mathbf{x}^{t} \|_2^2 + \sum_{t=1}^{T}\frac{R^2}{2(\eta_t - L)} + g(\mathbf{x}^1) - \mathbb{E}_{\xi}g(\mathbf{x}^{T+1}) \right \} \nonumber \\ & \leq J \left \{ \frac{\eta_T}{2} D^2 + \sum_{t=1}^{T}\frac{R^2}{2(\eta_t - L)} + g(\mathbf{x}^1) - g(\mathbf{x}^*) \right \} \nonumber \\ & \leq J \left \{ \frac{\sqrt{T} + L}{2} D^2 + \sum_{t=1}^{T}\frac{R^2}{2\sqrt{t} } + g(\mathbf{x}^1) - g(\mathbf{x}^*) \right \} \nonumber \\ & \leq J ( \frac{\sqrt{T} + L}{2} D^2+ \sqrt{T} R^2 + g(\mathbf{x}^1) - g(\mathbf{x}^*) )~, \end{align} which completes the proof. \qed
If one of the functions is strongly convex, ORBCD can achieve a $\log(T)$ regret bound, which is established in the following theorem. \begin{thm}\label{thm:orbcd_rgt_strong} Let the Assumption~\ref{asm:orbcd1}-\ref{asm:orbcd3} hold and $\eta_t = \frac{\gamma t}{J} + L$ in ORBCD. $j_t$ is sampled randomly and uniformly from $\{1,\cdots, J \}$. The regret bound $R(T)$ of ORBCD is \begin{align} R(T) \leq J^2R^2 \log(T) + J(g(\mathbf{x}^1) - g(\mathbf{x}^*) ) ~. \end{align} \end{thm} \noindent{\itshape Proof:}\hspace*{1em} Using the strong convexity of $f_t + g$ in~\myref{eq:stronggcov}, we have \begin{align}
f_t(\mathbf{x}^t) + g(\mathbf{x}^t) - [f_t(\mathbf{x}) + g(\mathbf{x})] \leq \langle \nabla f_t(\mathbf{x}^t) + g'(\mathbf{x}^t), \mathbf{x}^{t} - \mathbf{x} \rangle - \frac{\gamma}{2} \| \mathbf{x} - \mathbf{x}^t \|_2^2~. \end{align} Together with~\myref{eq:a}, we have \begin{align}
f_t(\mathbf{x}^t) + g(\mathbf{x}^t) - [f_t(\mathbf{x}) + g(\mathbf{x}) ] &\leq \frac{J\eta_t - \gamma }{2} \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \frac{J\eta_t}{2} \mathbb{E}\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) \nonumber \\ & + \frac{JR^2}{2(\eta_t - L)} + J [ g(\mathbf{x}^t) - \mathbb{E}g(\mathbf{x}^{t+1}) ] ~. \end{align} Taking expectation over $\xi$ on both sides, we have \begin{align}
\mathbb{E}_{\xi} \left [ f_t(\mathbf{x}^t) + g(\mathbf{x}^t) - [f_t(\mathbf{x}) + g(\mathbf{x}) ] \right ] &\leq \frac{J\eta_t - \gamma}{2} \mathbb{E}_{\xi}\| \mathbf{x} - \mathbf{x}^t \|_2^2 - \frac{J\eta_t}{2}\mathbb{E}_{\xi}[\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2]) \nonumber \\ & + \frac{JR^2}{2(\eta_t - L)} + J [ \mathbb{E}_{\xi} g(\mathbf{x}^t) - \mathbb{E}_{\xi}g(\mathbf{x}^{t+1}) ]~. \end{align} Summing over $t$ and setting $\eta_t = \frac{\gamma t}{J} + L$, we obtain the regret bound \begin{align} & R(T) = \sum_{t=1}^T\left \{ \mathbb{E}_{\xi} [ f_t(\mathbf{x}^t) + g(\mathbf{x}^t) ] - [f_t(\mathbf{x}) +g(\mathbf{x})] \right \} \nonumber \\
&\leq - \frac{J\eta_{T}}{2} \mathbb{E}_{\xi}\| \mathbf{x} - \mathbf{x}^{T+1} \|_2^2 + \sum_{t=1}^{T}\frac{J\eta_{t} -\gamma - J\eta_{t-1}}{2}\mathbb{E}_{\xi}\| \mathbf{x} - \mathbf{x}^{t} \|_2^2 + \sum_{t=1}^{T}\frac{JR^2}{2(\eta_t - L)} + J ( g(\mathbf{x}^1) - \mathbb{E}_{\xi}g(\mathbf{x}^{T+1}) ) \nonumber \\ & \leq \sum_{t=1}^{T}\frac{J^2R^2}{2\gamma t } + J(g(\mathbf{x}^1) - g(\mathbf{x}^*) ) \nonumber \\ & \leq J^2R^2 \log(T) + J(g(\mathbf{x}^1) - g(\mathbf{x}^*) ) ~, \end{align} which completes the proof. \qed
In general, ORBCD can achieve the same order of regret bound as OGD and other first-order online optimization methods, although the constant could be $J$ times larger.
\iffalse If setting $f_t = f$, ORBCD turns to batch optimization or randomized overlapping block coordinate descent (ROLBCD). By dividing the regret bound by $T$ and denoting $\bar{\mathbf{x}}^T = \frac{1}{T}\sum_{t=1}^{T} \mathbf{x}^t$, we obtain the iteration complexity of ROLBCD, i.e., \begin{align} \mathbb{E}_{\xi_{T-1}^j}[ f(\bar{\mathbf{x}}^T) + g(\bar{\mathbf{x}}^T) ] - [f(\mathbf{x}) +g(\mathbf{x})] = \frac{R(T)}{T} ~. \end{align} The iteration complexity of ROLBCD is $O(\frac{1}{\sqrt{T}})$, which is worse than RBCD. \fi \subsection{Stochastic Optimization}
In the stochastic setting, ORBCD first randomly chooses the $i_t$-th block sample and the $j_t$-th block coordinate. $j_{t}$ and $i_{t}$ are independent. $\mathbf{x}^t$ depends on the observed realization of the random variables $\xi = \{ ( i_1, j_1), \cdots, (i_{t-1}, j_{t-1}) \}$. The following theorem establishes the iteration complexity of ORBCD for general convex functions. \begin{thm}\label{thm:orbcd_stc_ic} Let $\eta_t = \sqrt{t} + L$ and $\bar{\mathbf{x}}^T = \frac{1}{T} \sum_{t=1}^{T}\mathbf{x}^t $ in the ORBCD. $i_t, j_t$ are sampled randomly and uniformly from $\{1,\cdots, I \}$ and $\{1,\cdots, J \}$ respectively. The iteration complexity of ORBCD is \begin{align} \mathbb{E}_{\xi} [ f(\bar{\mathbf{x}}^t) + g(\bar{\mathbf{x}}^t) ] - [f(\mathbf{x}) +g(\mathbf{x})] \leq \frac{J ( \frac{\sqrt{T} + L}{2} D^2+ \sqrt{T} R^2 + g(\mathbf{x}^1) - g(\mathbf{x}^*) )}{T}~. \end{align} \end{thm} \noindent{\itshape Proof:}\hspace*{1em} In the stochastic setting, let $f_t$ be $f_{i_t}$ in~\myref{eq:orbcd_key_lem}, we have \begin{align}\label{eq:orbcd_key_stoc}
\langle \nabla_{j_t} f_{i_t}(\mathbf{x}^t) + g_{j_t}'(\mathbf{x}^t) , \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle \leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{R^2}{2(\eta_t - L)} + g(\mathbf{x}^t) - g(\mathbf{x}^{t+1}) ~. \end{align} Note $i_t, j_t$ are independent of $\mathbf{x}^t$. Conditioned on $\mathbf{x}^t$, taking expectation over $i_t$ and $j_t$, the RHS is \begin{align} & \mathbb{E}\langle \nabla_{j_t} f_{i_t}(\mathbf{x}^t) + g_{j_t}'(\mathbf{x}^t) , \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle = \mathbb{E}_{i_t} [ \mathbb{E}_{j_t} [ \langle \nabla_{j_t} f_{i_t}(\mathbf{x}^t) + g_{j_t}'(\mathbf{x}^t), \mathbf{x}_{j_t}^{t} - \mathbf{x}_{j_t} \rangle ] ] \nonumber \\ & = \frac{1}{J} \mathbb{E}_{i_t} [ \langle \nabla f_{i_t}(\mathbf{x}^t), \mathbf{x}^{t} - \mathbf{x} \rangle + \langle g'(\mathbf{x}^t) , \mathbf{x}^{t} - \mathbf{x} \rangle ] \nonumber \\ & = \frac{1}{J}\langle \nabla f(\mathbf{x}^t) + g'(\mathbf{x}^t) , \mathbf{x}^{t} - \mathbf{x} \rangle ~. \end{align} Plugging back into~\myref{eq:orbcd_key_stoc}, we have \begin{align}\label{eq:orbcd_stc_0} & \frac{1}{J}\langle \nabla f(\mathbf{x}^t) + g'(\mathbf{x}^t) , \mathbf{x}^{t} - \mathbf{x} \rangle \nonumber \\
&\leq \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \mathbb{E}\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{R^2}{2(\eta_t - L)} + g(\mathbf{x}^t) - \mathbb{E} g(\mathbf{x}^{t+1}) ~. \end{align} Using the convexity of $f + g$, we have \begin{align} f(\mathbf{x}^t) + g(\mathbf{x}^t) - [f(\mathbf{x}) + g(\mathbf{x})] \leq \langle \nabla f(\mathbf{x}^t) + g'(\mathbf{x}^t), \mathbf{x}^{t} - \mathbf{x} \rangle~. \end{align} Together with~\myref{eq:orbcd_stc_0}, we have \begin{align}
f(\mathbf{x}^t) + g(\mathbf{x}^t) - [f(\mathbf{x}) + g(\mathbf{x}) ] &\leq J \left \{ \frac{\eta_t}{2} ( \| \mathbf{x} - \mathbf{x}^t \|_2^2 - \mathbb{E}\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2) + \frac{R^2}{2(\eta_t - L)} + g(\mathbf{x}^t) - \mathbb{E}g(\mathbf{x}^{t+1}) \right \}~. \end{align} Taking expectation over $\xi$ on both sides, we have \begin{align}
\mathbb{E}_{\xi} \left [ f(\mathbf{x}^t) + g(\mathbf{x}^t) \right ] - [f(\mathbf{x}) + g(\mathbf{x}) ] &\leq J \left \{ \frac{\eta_t}{2} ( \mathbb{E}_{\xi}\| \mathbf{x} - \mathbf{x}^t \|_2^2 - \mathbb{E}_{\xi}[\| \mathbf{x} - \mathbf{x}^{t+1} \|_2^2]) \right .\nonumber \\ & + \left. \frac{R^2}{2(\eta_t - L)} + \mathbb{E}_{\xi} g(\mathbf{x}^t) - \mathbb{E}_{\xi}g(\mathbf{x}^{t+1}) \right \}~. \end{align} Summing over $t$ and setting $\eta_t = \sqrt{t} + L$, following similar derivation in~\myref{eq:orbcd_rgt0}, we have \begin{align} \sum_{t=1}^T\left \{ \mathbb{E}_{\xi} [ f(\mathbf{x}^t) + g(\mathbf{x}^t) ] - [f(\mathbf{x}) +g(\mathbf{x})] \right \} \leq J ( \frac{\sqrt{T} + L}{2} D^2+ \sqrt{T} R^2 + g(\mathbf{x}^1) - g(\mathbf{x}^*) )~. \end{align} Dividing both sides by $T$, using the Jensen's inequality and denoting $\bar{\mathbf{x}}^T = \frac{1}{T}\sum_{t=1}^{T}\mathbf{x}^t$ complete the proof. \qed
For strongly convex functions, we have the following results. \begin{thm} For strongly convex function, setting $\eta_t = \frac{\gamma t}{J} + L$ in the ORBCD. $i_t, j_t$ are sampled randomly and uniformly from $\{1,\cdots, I \}$ and $\{1,\cdots, J \}$ respectively. Let $\bar{\mathbf{x}}^T = \frac{1}{T} \sum_{t=1}^{T}\mathbf{x}^t $. The iteration complexity of ORBCD is \begin{align} \mathbb{E}_{\xi} [ f(\bar{\mathbf{x}}^T) + g(\bar{\mathbf{x}}^T) ] - [f(\mathbf{x}) +g(\mathbf{x})] \leq \frac{J^2R^2 \log(T) + J(g(\mathbf{x}^1) - g(\mathbf{x}^*) ) }{T}~. \end{align} \end{thm} \noindent{\itshape Proof:}\hspace*{1em} If $f+g$ is strongly convex, we have \begin{align}
f(\mathbf{x}^t) + g(\mathbf{x}^t) - [f(\mathbf{x}) + g(\mathbf{x})] \leq \langle \nabla f(\mathbf{x}^t) + g'(\mathbf{x}^t), \mathbf{x}^{t} - \mathbf{x} \rangle - \frac{\gamma}{2} \| \mathbf{x} - \mathbf{x}^t \|_2^2~. \end{align} Plugging back into~\myref{eq:orbcd_stc_0}, following similar derivation in Theorem~\ref{thm:orbcd_rgt_strong} and Theorem~\ref{thm:orbcd_stc_ic} complete the proof. \qed
\subsection{ORBCD with Variance Reduction} According to the Theorem 2.1.5 in~\cite{nesterov04:convex}, the block-wise Lipschitz gradient in Assumption~\ref{asm:orbcd1} can also be rewritten as follows: \begin{align}
& f_i(\mathbf{x}) \leq f_i(\mathbf{y}) + \langle \nabla_j f_i(\mathbf{x}) - \nabla_j f_i(\mathbf{y}), \mathbf{x}_j - \mathbf{y}_j\rangle + \frac{L}{2} \| \mathbf{x}_j - \mathbf{y}_j\|_2^2~, \label{eq:blk_lip1} \\
&\| \nabla_j f_i(\mathbf{x}) - \nabla_j f_i(\mathbf{y}) \|_2^2 \leq L \langle \nabla_j f_i(\mathbf{x}) - \nabla_j f_i(\mathbf{y}), \mathbf{x}_j - \mathbf{y}_j\rangle~.\label{eq:blk_lip2} \end{align}
Let $\mathbf{x}^*$ be an optimal solution. Define an upper bound of $f(\mathbf{x}) + g(\mathbf{x}) -(f(\mathbf{x}^*)+g(\mathbf{x}^*))$ as \begin{align}\label{eq:def_h} h(\mathbf{x},\mathbf{x}^*)= \langle \nabla f(\mathbf{x}), \mathbf{x} - \mathbf{x}^*\rangle + g(\mathbf{x}) - g(\mathbf{x}^*)~. \end{align}
If $f(\mathbf{x}) + g(\mathbf{x})$ is strongly convex, we have \begin{align}\label{eq:orbcd_strong_h}
h(\mathbf{x},\mathbf{x}^*) \geq f(\mathbf{x}) - f(\mathbf{x}^*) + g(\mathbf{x}) - g(\mathbf{x}^*) \geq \frac{\gamma}{2} \| \mathbf{x} - \mathbf{x}^* \|_2^2 ~. \end{align}
\begin{lem}\label{lem:orbcdvd_lem1} Let $\mathbf{x}^*$ be an optimal solution and the Assumption~\ref{asm:orbcd1}, we have \begin{align}
\frac{1}{I} \sum_{i=1}^{I} \| \nabla f_i(\mathbf{x}) - \nabla f_i(\mathbf{x}^*) \|_2^2 \leq L h(\mathbf{x},\mathbf{x}^*)~. \end{align} where $h$ is defined in~\myref{eq:def_h}. \end{lem} \noindent{\itshape Proof:}\hspace*{1em} Since the Assumption~\ref{asm:orbcd1} hold, we have using \begin{align}\label{eq:orbcd_bd_h}
&\frac{1}{I} \sum_{i=1}^{I} \| \nabla f_i(\mathbf{x}) - \nabla f_i(\mathbf{x}^*) \|_2^2 = \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^J \| \nabla_j f_i(\mathbf{x}) - \nabla_j f_i(\mathbf{x}^*) \|_2^2 \nonumber \\ &\leq \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^J L \langle \nabla_j f_i(\mathbf{x}) - \nabla_j f_i(\mathbf{x}^*), \mathbf{x}_j - \mathbf{x}_j^*\rangle \nonumber \\ & = L [ \langle \nabla f(\mathbf{x}), \mathbf{x} - \mathbf{x}^*\rangle + \langle \nabla f(\mathbf{x}^*), \mathbf{x}^* - \mathbf{x} \rangle]~, \end{align} where the inequality uses~\myref{eq:blk_lip2}. For an optimal solution $\mathbf{x}^*$, $g'(\mathbf{x}^*) + \nabla f(\mathbf{x}^*) = 0$ where $g'(\mathbf{x}^*)$ is the subgradient of $g$ at $\mathbf{x}^*$. The second term in~\myref{eq:orbcd_bd_h} can be rewritten as \begin{align} & \langle \nabla f(\mathbf{x}^*), \mathbf{x}^* - \mathbf{x} \rangle = - \langle g'(\mathbf{x}^*), \mathbf{x}^* - \mathbf{x} \rangle = g(\mathbf{x}) - g(\mathbf{x}^*) ~. \end{align} Plugging into~\myref{eq:orbcd_bd_h} and using~\myref{eq:def_h} complete the proof. \qed
\begin{lem}\label{lem:orbcdvd_lem2} Let $\mathbf{v}_{j_k}^{i_k} $ and $\mathbf{x}_{j_k}^{k+1}$ be generated by~\myref{eq:orbcdvd_vij}-\myref{eq:orbcdvd_xj}. Conditioned on $\mathbf{x}^k$, we have \begin{align}
\mathbb{E} \| \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{x}^k) \|_2^2 \leq \frac{2L}{J} [h(\mathbf{x}^{k},\mathbf{x}^*) + h(\tilde{\mathbf{x}},\mathbf{x}^*)]~. \end{align} \end{lem} \noindent{\itshape Proof:}\hspace*{1em} Conditioned on $\mathbf{x}^k$, we have \begin{align} \mathbb{E}_{i_k}[ \nabla f_{i_k}(\mathbf{x}^k) - \nabla f_{i_k}(\tilde{\mathbf{x}}) + \tilde{\mu} ] = \frac{1}{I} \sum_{i=1}^{I} [ \nabla f_{i}(\mathbf{x}^k) - \nabla f_{i}(\tilde{\mathbf{x}}) + \tilde{\mu} ] = \nabla f(\mathbf{x}^k)~. \end{align} Note $\mathbf{x}^k$ is independent of $i_k, j_k$. $i_k$ and $j_k$ are independent. Conditioned on $\mathbf{x}^k$, taking expectation over $i_k, j_k$ and using~\myref{eq:orbcdvd_vij} give \begin{align}
&\mathbb{E}\| \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{x}^k) \|_2^2 = \mathbb{E}_{i_k} [ \mathbb{E}_{j_k}\| \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{x}^k) \|_2^2] \nonumber \\
&= \mathbb{E}_{i_k}[ \mathbb{E}_{j_k}\| \nabla_{j_k} f_{i_k}(\mathbf{x}^k) - \nabla_{j_k} f_{i_k}(\tilde{\mathbf{x}}) + \tilde{\mu}_{j_k} - \nabla_{j_k} f(\mathbf{x}^k) \|_2^2] \nonumber \\
&= \frac{1}{J}\mathbb{E}_{i_k}\| \nabla f_{i_k}(\mathbf{x}^k) - \nabla f_{i_k}(\tilde{\mathbf{x}}) + \tilde{\mu} - \nabla f(\mathbf{x}^k) \|_2^2 \nonumber \\
& \leq \frac{1}{J} \mathbb{E}_{i_k}\| \nabla f_{i_k}(\mathbf{x}^k) - \nabla f_{i_k}(\tilde{\mathbf{x}}) \|_2^2 \nonumber \\
& \leq \frac{2}{J} \mathbb{E}_{i_k}\| \nabla f_{i_k}(\mathbf{x}^k) - \nabla f_{i_k}(\mathbf{x}^*) \|_2^2 + \frac{2}{J} \mathbb{E}_{i_k}\| \nabla f_{i_k}(\tilde{\mathbf{x}}) - \nabla f_{i_k}(\mathbf{x}^*) \|_2^2 \nonumber \\
& = \frac{2}{IJ} \sum_{i=1}^{I}\| \nabla f_i(\mathbf{x}^k) - \nabla f_i(\mathbf{x}^*) \|_2^2 + \frac{2}{IJ}\sum_{i=1}^{I} \| \nabla f_i(\tilde{\mathbf{x}}) - \nabla f_i(\mathbf{x}^*) \|_2^2 \nonumber \\ & \leq \frac{2L}{J} [ h(\mathbf{x}^{k}, \mathbf{x}^*) + h(\tilde{\mathbf{x}}, \mathbf{x}^*)]~. \end{align}
The first inequality uses the fact $\mathbb{E} \| \zeta - \mathbb{E}\zeta \|_2^2 \leq \mathbb{E} \| \zeta \|_2^2$ given a random variable $\zeta$, the second inequality uses $\| \mathbf{a} + \mathbf{b} \|_2^2 \leq 2 \| \mathbf{a} \|_2^2 + 2\|\mathbf{b}\|_2^2$, and the last inequality uses Lemma~\ref{lem:orbcdvd_lem1}. \qed
\begin{lem}\label{lem:orbcdvd_lem3} Under Assumption~\ref{asm:orbcd1}, $f(\mathbf{x}) = \frac{1}{I} \sum_{i=1}^{I}f_i(\mathbf{x})$ has block-wise Lipschitz continuous gradient with constant $L$, i.e., \begin{align}
\| \nabla_j f(\mathbf{x} + U_j h_j ) - \nabla_j f(\mathbf{x}) \|_2 \leq L \| h_j \|_2~. \end{align} \end{lem} \noindent{\itshape Proof:}\hspace*{1em} Using the fact that $f(\mathbf{x}) = \frac{1}{I} \sum_{i=1}^{I}f_i(\mathbf{x})$, we have \begin{align}
&\| \nabla_j f(\mathbf{x} + U_j h_j ) - \nabla_j f(\mathbf{x}) \|_2 = \| \frac{1}{I} \sum_{i=1}^{I} [\nabla_j f_i(\mathbf{x} + U_j h_j ) - \nabla_j f_i(\mathbf{x}) ] \|_2 \nonumber \\
& \leq \frac{1}{I} \sum_{i=1}^{I} \| \nabla_j f_i(\mathbf{x} + U_j h_j ) - \nabla_j f_i(\mathbf{x}) \|_2 \nonumber \\
& \leq L \| h_j \|_2~, \end{align} where the first inequality uses the Jensen's inequality and the second inequality uses the Assumption~\ref{asm:orbcd1}. \qed
Now, we are ready to establish the linear convergence rate of ORBCD with variance reduction for strongly convex functions. \begin{thm}\label{thm:orbcdvd} Let $\mathbf{x}^t$ be generated by ORBCD with variance reduction~\myref{eq:orbcdvd_mu}-\myref{eq:orbcdvd_xj}. $j_k$ is sampled randomly and uniformly from $\{1,\cdots, J \}$. Assume $\eta > 2L$ and $m$ satisfy the following condition: \begin{align} \rho = \frac{L(m+1)}{(\eta-2L)m} + \frac{(\eta-L)J}{(\eta-2L)m} - \frac{1}{m}+ \frac{\eta (\eta-L)J}{(\eta-2L)m\gamma} < 1~, \end{align} Then ORBCDVD converges linearly in expectation, i.e., \begin{align}
\mathbb{E}_{\xi} [ f(\mathbf{x}^t) + g(\mathbf{x}^t) - (f(\mathbf{x}^*)+g(\mathbf{x}^*) ] \leq \rho^t [ \mathbb{E}_{\xi} h(\mathbf{x}^1, \mathbf{x}^*)]~. \end{align} where $h$ is defined in~\myref{eq:def_h}. \end{thm} \noindent{\itshape Proof:}\hspace*{1em} The optimality condition of~\myref{eq:orbcdvd_xj} is \begin{align} \langle \mathbf{v}_{j_k}^{i_k} + \eta (\mathbf{x}_{j_k}^{k+1} - \mathbf{x}_{j_k}^k) + g'_{j_k}(\mathbf{x}_{j_k}^{k+1}), \mathbf{x}_{j_k}^{k+1} - \mathbf{x}_{j_k} \rangle \leq 0~. \end{align}
Rearranging the terms yields \begin{align} & \langle \mathbf{v}_{j_k}^{i_k} + g'_{j_k}(\mathbf{x}_{j_k}^{k+1}) , \mathbf{x}_{j_k}^{k+1} - \mathbf{x}_{j_k} \rangle \leq - \eta \langle \mathbf{x}_{j_k}^{k+1} - \mathbf{x}_{j_k}^k , \mathbf{x}_{j_k}^{k+1} - \mathbf{x}_{j_k} \rangle \nonumber \\
& \leq \frac{\eta}{2} ( \| \mathbf{x}_{j_k} - \mathbf{x}_{j_k}^k \|_2^2 - \| \mathbf{x}_{j_k} - \mathbf{x}_{j_k}^{k+1} \|_2^2 - \| \mathbf{x}_{j_k}^{k+1} - \mathbf{x}_{j_k}^k \|_2^2 ) \nonumber \\
& = \frac{\eta}{2} ( \| \mathbf{x} - \mathbf{x}^k \|_2^2 - \| \mathbf{x} - \mathbf{x}^{k+1} \|_2^2 - \| \mathbf{x}_{j_k}^{k+1} - \mathbf{x}_{j_k}^k \|_2^2 ) ~, \end{align} where the last equality uses $\mathbf{x}^{k+1} = (\mathbf{x}_{j_k}^{k+1}, \mathbf{x}_{k\neq {j_k}}^t)$. Using the convecxity of $g_j$ and the fact that $g(\mathbf{x}^k) - g(\mathbf{x}^{k+1}) = g_{j_k}(\mathbf{x}^k) - g_{j_k}(\mathbf{x}^{k+1})$, we have \begin{align} & \langle \mathbf{v}_{j_k}^{i_k} , \mathbf{x}_{j_k}^{k} - \mathbf{x}_{j_k} \rangle + g_{j_k}(\mathbf{x}^k) - g_{j_k}(\mathbf{x}) \leq \langle \mathbf{v}_{j_k}^{i_k} , \mathbf{x}_{j_k}^{k} - \mathbf{x}_{j_k}^{k+1} \rangle + g(\mathbf{x}^k) - g(\mathbf{x}^{k+1}) \nonumber \\
& + \frac{\eta}{2} ( \| \mathbf{x} - \mathbf{x}^k \|_2^2 - \| \mathbf{x} - \mathbf{x}^{k+1} \|_2^2 - \| \mathbf{x}_{j_k}^{k+1} - \mathbf{x}_{j_k}^k \|_2^2 ) ~. \end{align} According to Lemma~\ref{lem:orbcdvd_lem3} and using~\myref{eq:blk_lip1}, we have \begin{align}
\langle \nabla_{j_k} f(\mathbf{x}^k), \mathbf{x}_{j_k}^{k} - \mathbf{x}_{j_k}^{k+1} \rangle \leq f(\mathbf{x}^k) - f(\mathbf{x}^{k+1}) + \frac{L}{2} \| \mathbf{x}_{j_k}^{k} - \mathbf{x}_{j_k}^{k+1} \|_2^2~. \end{align} Letting $\mathbf{x} = \mathbf{x}^*$ and using the smoothness of $f$, we have \begin{align} & \langle \mathbf{v}_{j_k}^{i_k} , \mathbf{x}_{j_k}^{k} - \mathbf{x}_{j_k} \rangle + g_{j_k}(\mathbf{x}^k) - g_{j_k}(\mathbf{x}^*) \leq \langle \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{x}^k), \mathbf{x}_{j_k}^{k} - \mathbf{x}_{j_k}^{k+1} \rangle + f(\mathbf{x}^k) + g(\mathbf{x}^k) - [f(\mathbf{x}^{k+1})+g(\mathbf{x}^{k+1})] \nonumber \\
& + \frac{\eta}{2} ( \| \mathbf{x}^* - \mathbf{x}^k \|_2^2 - \| \mathbf{x}^* - \mathbf{x}^{k+1} \|_2^2 - \| \mathbf{x}_{j_k}^{k+1} - \mathbf{x}_{j_k}^k \|_2^2 ) + \frac{L}{2} \| \mathbf{x}_{j_k}^{k} - \mathbf{x}_{j_k}^{k+1} \|_2^2\nonumber \\
& \leq \frac{1}{2(\eta-L)} \| \mathbf{v}_{j_k}^{i_k} - \nabla_{j_k} f(\mathbf{x}^k) \|_2^2 + f(\mathbf{x}^k) + g(\mathbf{x}^k) - [f(\mathbf{x}^{k+1})+g(\mathbf{x}^{k+1})] + \frac{\eta}{2} ( \| \mathbf{x}^* - \mathbf{x}^k \|_2^2 - \| \mathbf{x}^* - \mathbf{x}^{k+1} \|_2^2 ) ~. \end{align} Taking expectation over $i_k, j_k$ on both sides and using Lemma~\ref{lem:orbcdvd_lem2}, we have \begin{align}\label{eq:orbcdvd_expbd} & \mathbb{E} [ \langle \mathbf{v}_{j_k}^{i_k} , \mathbf{x}_{j_k}^{k} - \mathbf{x}_{j_k}^* \rangle + g_{j_k}(\mathbf{x}^k) - g_{j_k}(\mathbf{x}^*)] \nonumber \\ &\leq \frac{L}{J(\eta-L)} [h(\mathbf{x}^k,\mathbf{x}^*) + h(\tilde{\mathbf{x}},\mathbf{x}^*)] + f(\mathbf{x}^k) + g(\mathbf{x}^k) - \mathbb{E}[f(\mathbf{x}^{k+1})+g(\mathbf{x}^{k+1})] \nonumber \\
& + \frac{\eta}{2} ( \| \mathbf{x}^* - \mathbf{x}^k \|_2^2 - \mathbb{E}\| \mathbf{x}^* - \mathbf{x}^{k+1} \|_2^2 )~. \end{align} The left hand side can be rewritten as \begin{align} & \mathbb{E} [\langle \mathbf{v}_{j_k}^{i_k} , \mathbf{x}_{j_k}^{k} - \mathbf{x}_{j_k}^* \rangle + g_{j_k}(\mathbf{x}^k) - g_{j_k}(\mathbf{x}^*)] = \frac{1}{J} [ \mathbb{E}_{i_k}\langle \mathbf{v}^{i_k} , \mathbf{x}^k - \mathbf{x}^* \rangle + g(\mathbf{x}^k) -g(\mathbf{x}^*) ] \nonumber \\ & = \frac{1}{J} [ \langle \nabla f(\mathbf{x}^k) , \mathbf{x}^k - \mathbf{x}^* \rangle + g(\mathbf{x}^k) -g(\mathbf{x}^*) ] = \frac{1}{J} h(\mathbf{x}^k,\mathbf{x}^*)~. \end{align} Plugging into~\myref{eq:orbcdvd_expbd} gives \begin{align} \frac{1}{J} [ h(\mathbf{x}^k,\mathbf{x}^*) ] & \leq \frac{L}{J(\eta-L)} [h(\mathbf{x}^k,\mathbf{x}^*) + h(\tilde{\mathbf{x}},\mathbf{x}^*)] + f(\mathbf{x}^k) + g(\mathbf{x}^k) - \mathbb{E}[f(\mathbf{x}^{k+1})+g(\mathbf{x}^{k+1})] \nonumber \\
&+ \frac{\eta}{2} ( \| \mathbf{x}^* - \mathbf{x}^k \|_2^2 - \mathbb{E}\| \mathbf{x}^* - \mathbf{x}^{k+1} \|_2^2 ) \nonumber \\ & \leq \frac{L}{J(\eta-L)} [h(\mathbf{x}^k,\mathbf{x}^*) + h(\tilde{\mathbf{x}},\mathbf{x}^*)] + f(\mathbf{x}^k) + g(\mathbf{x}^k) - \mathbb{E}[f(\mathbf{x}^{k+1})+g(\mathbf{x}^{k+1})] \nonumber \\
&+ \frac{\eta}{2} ( \| \mathbf{x}^* - \mathbf{x}^k \|_2^2 - \mathbb{E}\| \mathbf{x}^* - \mathbf{x}^{k+1} \|_2^2 ) ~, \end{align} Rearranging the terms yields \begin{align} \frac{\eta - 2L}{J(\eta-L)} h(\mathbf{x}^k,\mathbf{x}^*) &\leq \frac{L}{J(\eta-L)}[ h(\tilde{\mathbf{x}},\mathbf{x}^*) ] + f(\mathbf{x}^k) + g(\mathbf{x}^k) - \mathbb{E}[f(\mathbf{x}^{k+1})+g(\mathbf{x}^{k+1})] \nonumber \\
& + \frac{\eta}{2} ( \| \mathbf{x}^* - \mathbf{x}^k \|_2^2 - \mathbb{E}\| \mathbf{x}^* - \mathbf{x}^{k+1} \|_2^2 )~. \end{align} At time $t+1$, we have $\mathbf{x}_0 = \tilde{\mathbf{x}} = \mathbf{x}^t$. Summing over $k = 0,\cdots, m$ and taking expectation with respect to the history of random variable $\xi$, we have \begin{align} \frac{\eta - 2L}{J(\eta-L)} \sum_{k=0}^{m} \mathbb{E}_{\xi}h(\mathbf{x}_k,\mathbf{x}^*) &\leq \frac{L(m+1)}{J(\eta-L)} \mathbb{E}_{\xi}h(\tilde{\mathbf{x}},\mathbf{x}^*) + \mathbb{E}_{\xi}[ f(\mathbf{x}_0) + g(\mathbf{x}_0) ] - \mathbb{E}_{\xi} [ f(\mathbf{x}_{m+1}) + g(\mathbf{x}_{m+1})] \nonumber \\
&+ \frac{\eta}{2} ( \mathbb{E}_{\xi}\| \mathbf{x}^* - \mathbf{x}_0 \|_2^2 - \mathbb{E}_{\xi}\| \mathbf{x}^* - \mathbf{x}_{m+1} \|_2^2 ) \nonumber \\
&\leq \frac{Lm}{J(\eta-L)} \mathbb{E}_{\xi}h(\tilde{\mathbf{x}},\mathbf{x}^*) + \mathbb{E}_{\xi}h(\mathbf{x}_0,\mathbf{x}^*) + \frac{\eta}{2} \mathbb{E}_{\xi}\| \mathbf{x}^* - \mathbf{x}_0 \|_2^2 \nonumber ~, \end{align} where the last inequality uses \begin{align} f(\mathbf{x}_0) + g(\mathbf{x}_0) - [ f(\mathbf{x}_{m+1}) + g(\mathbf{x}_{m+1})] & \leq f(\mathbf{x}_0) + g(\mathbf{x}_0) - [ f(\mathbf{x}^*) + g(\mathbf{x}^*)] \nonumber \\ & \leq \langle\nabla f(\mathbf{x}_0), \mathbf{x}_0 - \mathbf{x}^* \rangle + g(\mathbf{x}_0) - g(\mathbf{x}^*) \nonumber \\ & = h(\mathbf{x}_0,\mathbf{x}^*)~. \end{align} Rearranging the terms gives \begin{align}
\frac{\eta -2L}{J(\eta-L)} \sum_{k=1}^{m} \mathbb{E}_{\xi}h(\mathbf{x}^k,\mathbf{x}^*) \leq \frac{L(m+1)}{J(\eta-L)} \mathbb{E}_{\xi}h(\tilde{\mathbf{x}},\mathbf{x}^*) + (1- \frac{\eta -2 L}{J(\eta-L)} ) \mathbb{E}_{\xi}h(\mathbf{x}_0,\mathbf{x}^*) + \frac{\eta}{2} \mathbb{E}_{\xi}\| \mathbf{x}^* - \mathbf{x}_0 \|_2^2~. \end{align} Pick $x^{t+1}$ so that $h(\mathbf{x}^{t+1}) \leq h(\mathbf{x}_k), 1\leq k \leq m$, we have \begin{align}\label{eq:orbcdvd_lineareq0}
\frac{\eta - 2L}{J(\eta-L)} m \mathbb{E}_{\xi} h(\mathbf{x}^{t+1},\mathbf{x}^*) \leq [ \frac{L(m+1)}{J(\eta-L)} + 1- \frac{\eta -2 L}{J(\eta-L)} ] \mathbb{E}_{\xi}h(\mathbf{x}^t,\mathbf{x}^*) + \frac{\eta}{2} \mathbb{E}_{\xi}\| \mathbf{x}^* - \mathbf{x}^t \|_2^2~, \end{align} where ther right hand side uses $\mathbf{x}^t = \mathbf{x}_0 = \tilde{\mathbf{x}}$. Using~\myref{eq:orbcd_strong_h}, we have \begin{align} \frac{\eta - 2L}{J(\eta-L)} m \mathbb{E}_{\xi} h(\mathbf{x}^{t+1} ,\mathbf{x}^*) \leq [ \frac{L(m+1)}{J(\eta-L)} + 1- \frac{\eta -2 L}{J(\eta-L)} +\frac{\eta}{\gamma} ] \mathbb{E}_{\xi}h(\mathbf{x}^t,\mathbf{x}^*)~. \end{align} Dividing both sides by $\frac{\eta - 2L}{J(\eta-L)} m$, we have \begin{align}
\mathbb{E}_{\xi} h(\mathbf{x}^{t+1},\mathbf{x}^*) \leq \rho \mathbb{E}_{\xi}h(\mathbf{x}^t,\mathbf{x}^*)~, \end{align} where \begin{align} \rho = \frac{L(m+1)}{(\eta-2L)m} + \frac{(\eta-L)J}{(\eta-2L)m} - \frac{1}{m}+ \frac{\eta (\eta-L)J}{(\eta-2L)m\gamma} < 1~, \end{align} which completes the proof. \qed
\section{Conclusions}\label{sec:conclusion} We proposed online randomized block coordinate descent (ORBCD) which combines online/stochastic gradient descent and randomized block coordinate descent. ORBCD is well suitable for large scale high dimensional problems with non-overlapping composite regularizers. We established the rate of convergence for ORBCD, which has the same order as OGD/SGD. For stochastic optimization with strongly convex functions, ORBCD can converge at a geometric rate in expectation by reducing the variance of stochastic gradient.
\section*{Acknowledgment} H.W. and A.B. acknowledge the support of NSF via IIS-0953274, IIS-1029711, IIS- 0916750, IIS-0812183, NASA grant NNX12AQ39A, and the technical support from the University of Minnesota Supercomputing Institute. A.B. acknowledges support from IBM and Yahoo. H.W. acknowledges the support of DDF (2013-2014) from the University of Minnesota. H.W. also thanks Renqiang Min and Mehrdad Mahdavi for mentioning the papers about variance reduction when the author was in the NEC Research Lab, America.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title[\tiny Approximation with respect to polynomials with constant coefficients]{The best m-term approximation with respect to polynomials with constant coefficients} \author{Pablo M. Bern\'a} \address{Pablo M. Bern\'a \\ Instituto Universitario de Matem\'atica Pura y Aplicada \\ Universitat Polit\`ecnica de Val\`encia \\ 46022 Valencia, Spain} \email{[email protected]} \author{\'Oscar Blasco} \address{\'Oscar Blasco \\ Departamento de An\'alisis Matem\'atico \\ Universidad de Valencia, Campus de Burjassot \\ 46100 Valencia, Spain} \email{[email protected]}
\thanks{The first author is partially supported by GVA PROMETEOII/2013/013 and 19368/PI/14 (\textit{Fundaci\'on S\'eneca}, Regi\'on de Murcia, Spain). The second author is partially supported by MTM2014-53009-P (MINECO, Spain).} \subjclass{41A65, 41A46, 46B15.}
\keywords{thresholding greedy algorithm; m-term approximation; weight-greedy basis. }
\begin{abstract} In this paper we show that that greedy bases can be defined as those where the error term using $m$-greedy approximant is uniformly bounded by the best $m$-term approximation with respect to polynomials with constant coefficients in the context of the weak greedy algorithm and weights. \end{abstract}
\maketitle
\section{Introduction }
Let $(\SX,\Vert \cdot \Vert)$ be an infinite-dimensional real Banach space and let $\mathscr B = (e_n)_{n=1}^\infty$ be a normalized Schauder basis of $\SX$ with biorthogonal functionals $(e_n^*)_{n=1}^\infty$. Throughout the paper, for each finite set $A\subset \SN$ we write $|A|$ for the cardinal of the set $A$, $1_A=\sum_{j\in A} e_j$ and $P_A(x)=\sum_{n\in A}e_n^*(x) e_n$. Given a collection of signs $(\eta_j)_{j\in A}\in\lbrace\pm 1\rbrace$ with $|A|<\infty$, we write $1_{\eta A} = \sum_{n\in A}\eta_j e_j\in \SX$ and we use the notation $[1_{\eta A}]$ and $[e_n, n\in A]$ for the one-dimensional subspace and the $|A|-$dimensional subspace generated by generated by $1_{\eta A}$ and by $\lbrace e_n, n\in A\rbrace$ respectively. For each $x\in\SX$ and $m\in \SN$, S.V. Konyagin and V.N. Temlyakov defined in \cite{VT} the \textbf{$m$-th greedy approximant} of $x$ by $$\mathcal{G}_m(x) = \sum_{j=1}^m e_{\rho(j)}^*(x)e_{\rho(j)},$$ where $\rho$ is a greedy ordering, that is $\rho : \SN \longrightarrow \SN$ is a permutation such that $supp(x) = \lbrace n: e_n^*(x)\neq 0\rbrace \subseteq \rho(\SN)$ and $\vert e_{\rho(j)}^*(x)\vert \geq \vert e_{\rho(i)}^*(x)\vert$ for $j\leq i$. The collection $(\mathcal{G}_m)_{m=1}^\infty$ is called the \textbf{Thresholding Greedy Algorithm} (TGA).
This algorithm is usually a good candidate to obtain the \textbf{best m-term approximation} with regard to $\mathscr B$, defined by $$ \sigma_m(x,\mathscr B)_\SX =\sigma_m(x) := \inf\lbrace d(x,[e_n, n\in A]) : A\subset \SN, \vert A\vert = m\rbrace.$$
The bases satisfying \begin{equation} \label{old}\Vert x-\mathcal{G}_m(x)\Vert \leq C\sigma_m(x),\;\; \forall x\in\SX, \forall m\in\SN,\end{equation} where $C$ is an absolute constant are called \textbf{greedy bases} (see \cite{VT}).
The first characterization of greedy bases was given by S.V. Konyagin and V. N. Temlyakov in \cite{VT} who established that a basis is greedy if and only if it is unconditional and democratic (where a basis is said to be democratic if there exists $C>0$ so that $\|1_A\|\le C \|1_B\|$ for any pair of finite sets $A$ and $B$ with $|A|=|B|$).
Let us also recall two possible extensions of the greedy algorithm and the greedy basis. The first one consists in taking the $m$ terms with near-biggest coefficients and generating the Weak Greedy Algorithm (WGA) introduced by V.N. Temlyakov in \cite{T}.
For each $t\in (0,1]$,
a finite set $\Ga\subset\SN$ is called a $t$-greedy set for $x\in\SX$, for short $\Ga\in\mathscr G(x,t)$, if \[
\min_{n\in \Ga}|{\be^*_n}(x)|\,\geq\,t\max_{n\notin\Ga}|{\be^*_n}(x)|, \]
and write $\Ga\in\mathscr G(x,t,N)$ if in addition $|\Ga|=N$. A \textbf{$t$-greedy operator of order $N$} is a mapping $G^t:\SX\to\SX$ such that \[ G^t(x)=\sum_{n\in \Ga_x}{\be^*_n}(x){\mathbf e}_n, \quad \mbox{for some }\Ga_x\in \mathscr G(x,t,N). \]
A basis is called \textbf{$t$-greedy} if there exists $C(t)>0$ such that \begin{eqnarray} \Vert x-G^t(x)\Vert \leq C(t)\sigma_m(x)\; \forall x\in\SX, \forall m\in\SN, \forall G^t\in \mathscr G(x,t,m). \end{eqnarray}
It was shown that a basis is $t$-greedy for some $0<t\le 1$ if and only if it is $t$-greedy for all $0<t\le 1$. From the proof it follows that greedy basis are also $t$-greedy basis with constant $C(t)= O(1/t)$ as $t\to 0$.
The second one consists in replacing $|A|$ by $w(A)=\sum_{n\in A} w_n$ and it was considered by G. Kerkyacharian, D. Picard and V.N. Temlyakov in \cite{KPT} (see also \cite[Definition 16]{Tem}). Given a weight sequence $\omega = \lbrace \omega_n\rbrace_{n=1}^\infty, \omega_n >0$ and a positive real number $\delta>0$, they defined $$\sigma_\delta ^\omega (x) = \inf \lbrace d(x,[e_n, n\in A]) : A\subset \SN, \omega(A)\leq \delta\rbrace$$ where $\omega(A) := \sum_{n\in A}\omega_n$, with $A\subset\mathbb{N}$.
They called \textbf{weight-greedy bases} ($\omega$- greedy bases) to those bases satisfying \begin{eqnarray}\label{wt} \Vert x-\mathcal{G}_m(x)\Vert \leq C \sigma_{\omega(A_m)}^\omega (x),\; \forall x\in\SX, \forall m\in\SN, \end{eqnarray}
where $C>0$ is an absolute constant and $A_m = supp(\mathcal{G}_m(x))$. Moreover, they proved in \cite{KPT} that $\mathscr B$ is a $\omega$- greedy basis if and only if it is unconditional and $w$-democratic (where a basis is $w$-democratic whenever there exists $C>0$ so that $\|1_A\|\le C \|1_B\|$ for any pair of finite sets $A$ and $B$ with $w(A)\le w(B)$).
This generalization was motivated by the work of A. Cohen, R.A. DeVore and R. Hochmuth in \cite{CDH} where the basis was indexed by dyadic intervals and $w_\alpha (\Lambda)=\sum_{I\in \Lambda}|I|^\alpha$. Later in 2013, similar considerations were considered by E. Hern\'andez and D. Vera to prove some inclusions of approximation spaces (see \cite{HV}).
Let us summarize and use the following combined definition. \begin{defi} Let $\mathscr B$ be a normalized Schauder basis in $\mathbb{X}$, $0<t\le 1$ and weight sequence $\omega = \lbrace \omega_n\rbrace_{n=1}^\infty$ with $\omega_n >0$. We say that $\mathscr B$ is \textbf{$(t,\omega)$-greedy} if there exists $C(t)>0$ such that \begin{equation}\label{g} \Vert x-G^t(x)\Vert \leq C(t)\sigma^w_{m(t)}(x)\; \forall x\in\SX, \forall m\in\SN, \forall G^t\in \mathscr G(x,t,m) \end{equation} where $A_m(t)=supp (G^t(x))$ and $m(t)=w(A_m(t))$. \end{defi}
The authors introduced (see \cite{BB}) the best $m$-term approximation with respect to polynomials with constant coefficients as follows: $$\mathcal{D}^*_m(x) := \inf \lbrace d(x,[1_{\eta A}]) : A\subset \SN, (\eta_n)\in \{\pm 1\}, \vert A\vert = m\rbrace.$$
Obviously, $\sigma_m(x)\leq \mathcal{D}^*_m(x)$ but, while $\sigma_m(x)\to 0$ as $m\to\infty$ it was shown that for orthonormal bases in Hilbert spaces we have $\mathcal{D}^*_m(x)\to \|x\|$ as $m\to\infty$. The following result establishes a new description of greedy bases using the best $m$-term approximation with respect to polynomials with constant coefficients. \begin{theorem} (\cite[Theorem 3.6]{BB}) Let $\SX$ be a Banach space and $\mathscr B$ a Schauder basis of $\SX$.
(i) If there exists $C>0$ such that
\begin{equation}\label{new}\Vert x-\mathcal{G}_m(x)\Vert \leq C\mathcal{D}^*_m(x),\; \forall x\in \SX,\; \forall m\in \mathbb{N},\end{equation}
then $\mathscr B$ is $C$-suppression unconditional and $C$-symmetric for largest coefficients.
(ii) If $\mathscr B$ is $K_s$-suppression unconditional and $C_s$-symmetric for largest coefficients then
$$\Vert x-\mathcal{G}_m(x)\Vert \leq (K_s C_s)\sigma_m(x),\; \forall x\in \SX,\; \forall m\in \mathbb{N}.$$
\end{theorem}
The concepts of suppression unconditional and symmetric for largest coefficients bases can be found in \cite{BB,AA2,AW,DKOSS,VT}. We recall here that a basis is \textbf{$K_s$-suppression unconditional} if the projection operator is uniformly bounded, that is to say $$\Vert P_A(x)\Vert \leq K_s\Vert x\Vert,\; \forall x\in\SX,\forall A\subset \SN$$
and $\mathscr B$ is \textbf{$C_s$-symmetric for largest coefficients} if $$\Vert x+t1_{\varepsilon A}\Vert \leq C_s\Vert x+t1_{\varepsilon' B}\Vert,$$ for any $\vert A\vert = \vert B\vert$, $A\cap B=\emptyset$, $supp(x) \cap (A\cup B) = \emptyset$, $(\varepsilon_j), (\varepsilon'_j) \in \lbrace \pm 1\rbrace$ and $t = \max\lbrace \vert e_n^*(x)\vert : n\in supp(x)\rbrace$.
In this note we shall give a direct proof of the equivalence between condition (\ref{old}) and (\ref{new}) even in the setting of $(t,w)$-greedy basis.
Let us now introduce our best $m$-term approximation with respect to polynomials with constant coefficients associated to a weight sequence and the basic property to be considered in the paper.
\begin{defi} Let $\mathscr B$ be a normalized Schauder basis in $\mathbb{X}$, $0<t\le 1$ and a weight sequence $\omega = \lbrace \omega_n\rbrace_{n=1}^\infty$ with $\omega_n >0$. We denote by $$\mathcal{D}_{\delta}^\omega (x) := \inf \lbrace d(x,[1_{\eta A}]) : A\subset \SN, (\eta_n)\in \{\pm 1\}, \omega(A)\leq \delta\rbrace.$$
The basis $\mathscr B$ is said to $(t,w)$-greedy for polynomials with constant coefficients, denoted to have {\bf $(t,w)$-PCCG property}, if there exists $D(t)>0$ such that \begin{equation}\label{ng}\Vert x-G^t(x)\Vert \leq D(t)\mathcal{D}_{m(t)}^\omega (x), \forall x\in\SX, \forall m\in\SN, \forall G^t\in \mathscr G(x,t,m)\end{equation} where $A_m(t)=supp(G^t(x))$ and $m(t)=\omega(A_m(t))$.
In the case $t=1$ and $w(A)=|A|$ we simply call it the {\bf PCCG property}. \end{defi} Of course $\sigma_\delta ^\omega(x) \leq \mathcal{D}_\delta^\omega(x)$ for all $\delta>0$, hence if the basis is $(t,\omega)$-greedy then (\ref{ng}) holds with the $D(t)=C(t)$. We now formulate our main result which produces a direct proof of the result in \cite{BB} and give the extension to $t$-greedy and weighted greedy versions. \begin{theorem} Let $\mathscr B$ be a normalized Schauder basis in $\mathbb{X}$ and let $\omega = \lbrace \omega_n\rbrace_{n=1}^\infty$ be a weight sequence with $\omega_n >0$ for all $n\in \mathbb N$. The following are equivalent:
(i) There exist $0<s\le 1$ such that $\mathscr B$ has the $(s,w)$-PCCG property.
(ii) $\mathscr B$ is $(t,\omega)$-greedy for all $0<t\le 1$. \end{theorem}
\begin{proof} Only the implication (i) $\Longrightarrow$ (ii) needs a proof. Let us assume that (\ref{ng}) holds for some $0<s\le 1$. Let $0<t\le 1$, $x \in\SX$, $m\in \mathbb N$ and $G^t\in \mathscr G(x,t,m)$. We write $G^t(x) = P_{A_m(t)}(x)$ with $A_m(t)\in \mathcal G(x,t,m)$. For each $\varepsilon >0$ we choose $z = \sum_{n\in B}e_n^*(x)e_n$ with $\omega(B)\leq \omega(A_m(t))$ and $\Vert x-z\Vert \leq \sigma_{\omega(A_m)}^\omega (x) + \varepsilon$.
We write $$x- P_{A_m(t)}(x)= x- P_{A_m(t)\cup B} (x) +P_{B\setminus A_m(t)}(x).$$ Taking into account that $P_{B\setminus A_m(t)}(x)\in co(\lbrace S 1_{\eta (B\setminus A_m(t))} : \vert \eta_j\vert = 1\rbrace)$ for any $S\ge \underset{j\in B\setminus A_m(t)}{\max}\vert e_j^*(x)\vert$, it suffices to show that there exists $R\ge 1$ and $C(t)>0$ such that \begin{equation}\label{final}
\|x- P_{A_m(t)\cup B} (x) + R\gamma 1_{\eta(B\setminus A_m(t))}\|\le C(t) \|x-z\| \end{equation} for any choice of signs $(\eta_j)_{j\in B\setminus A_m(t)}$ where $\gamma = \underset{j\in B\setminus A_m(t)}{\max}\vert e_j^*(x)\vert$.
Let us assume first that $t\geq s$. We shall show that \begin{equation} \label{two}\Vert x- P_{(A_m(t)\cup B)}(x)+\frac{t}{s}\gamma 1_{\eta B\setminus A_m(t)}\Vert \le D(s)\Vert x- P_B(x)\Vert\end{equation} for any choice of signs $(\eta_j)_{j\in B\setminus A_m(t)}$.
Given $(\eta_j)_{j\in B\setminus A_m(t)}$ we consider $$y_{\eta}= x- P_B(x)+ \frac{t}{s}\gamma 1_{\eta(B\setminus A_m(t))}=\sum_{n\notin B} e_n^*(x)e_n+ \sum_{n\in B\setminus A_m(t)}\frac{t}{s}\gamma\eta_n e_n .$$
Note that $$\min_{n\in A_m(t)\setminus B}|e_n^*(y_\eta)|= \min_{n\in A_m(t)\setminus B}|e_n^*(x)|\ge \min_{n\in A_m(t)}|e_n^*(x)|$$ and $$s\max_{n\in (A_m(t)\setminus B)^c}|e_n^*(y_\eta)|=\max\{ s\max_{n\notin A_m(t)}|e_n^*(x)|, t\gamma\} .$$ Therefore, since $t\ge s$, we conclude that
$$\min_{n\in A_m(t)\setminus B}|e_n^*(y_\eta)|\ge s \max_{n\in (A_m(t)\setminus B)^c}|e_n^*(y_\eta)|.$$
Hence $A_m(t)\setminus B \in \mathcal G(y_\eta, s, N)$ with $N = \vert A_m(t)\setminus B\vert$. We write $G^s(y_\eta) = P_{A_m(t)\setminus B}(x)$ and notice that $$ y_\eta- G^s(y_\eta)= x- P_{A_m(t)\cup B}(x)+ \frac{t}{s}\gamma 1_{\eta(B\setminus A_m(t))}. $$
Since $\omega(B)\leq \omega(A_m(t))$ we have also that $\omega(B\setminus A_m(t))\leq \omega(A_m(t)\setminus B)$. Hence for $N(s)=\omega(A_m(t)\setminus B)$ we conclude \begin{eqnarray*} \Vert x- P_{(A_m(t)\cup B)}(x)+\frac{t}{s}\gamma 1_{\eta B\setminus A_m(t)}\Vert &\leq& D(s)\mathcal{D}_{N(s)}^\omega (y_\eta)\\\nonumber
&\leq& D(s) \Vert y_\eta-\frac{t}{s}\gamma1_{\eta B\setminus A_m(t)}\Vert\\\nonumber &=&D(s)\Vert x- P_B(x)\Vert. \end{eqnarray*}
Now, let $y=x-z+ \mu 1_B$ for $\mu = s \, \underset{j\notin B}{\max}\vert e_j^*(x-z)\vert +\underset{j\in B}{\max}\vert e_j^*(x-z)\vert.$\newline
Then $$\min_{j\in B} |\mu + e_n^*(x-z)|\ge s\max_{j\notin B} |e_n^*(x-z)|,$$ which gives that $B\in \mathcal G(y,s, |B|)$ and we obtain $G^s(y) = P_B(x-z)+\mu 1_{B}$. Hence \begin{equation}\label{three} \Vert x-P_B(x)\Vert = \Vert y-G^s(y)\Vert \le D(s)\Vert y - \mu 1_B\Vert= D(s)\Vert x-z\Vert. \end{equation}
Therefore, by $\eqref{two}$ and $\eqref{three}$ we obtain
$$\Vert x- P_{(A_m(t)\cup B)}(x)+\frac{t}{s}\gamma 1_{\eta B\setminus A_m(t)}\Vert\le D(s)^2\|x-z\|.$$ Then, for $s\le t$ we obtain that $\mathscr B$ is $(t,w)$-greedy with constant $C(t)\le D(s)^2$.
We now consider the case $s>t$. We use the following estimates: $$\Vert x- P_{(A_m(t)\cup B)}(x)+\gamma 1_{\eta B\setminus A_m(t)}\Vert\le \Vert x- P_{ B}(x)\Vert+\Vert P_{A_m(t)\setminus B}(x)\Vert+\gamma \Vert1_{\eta B\setminus A_m(t)}\Vert.$$ Arguing as above, using now $$\tilde y_{\eta}= P_{A_m(t)\setminus B}(x)+ \frac{t}{s}\gamma 1_{\eta(B\setminus A_m(t))},$$
we conclude that $\frac{t}{s}\gamma \Vert1_{\eta B\setminus A_m(t)}\Vert\le D(s)\|P_{A_m(t)\setminus B}(x)\|$.
The argument used to show (\ref{three}) gives $ \|z- P_C z\|\le D(s) \|z\|$ for all $z\in \mathbb{X}$ and finite set $C$. Therefore
$$\Vert P_{A_m(t)\setminus B}(x)\Vert= \Vert P_{A_m(t)}(x-P_{B}x)\Vert\le (1+ D(s))\|x-P_Bx\|.$$ Putting all together we have
$$\Vert x- P_{(A_m(t)\cup B)}(x)+\gamma 1_{\eta B\setminus A_m(t)}\Vert\le (2+\frac{t+s}{t}D(s))\|x-P_Bx\|,$$ and therefore $\mathscr B$ is $(t,w)$-greedy with constant $C(t)\le (2+\frac{t+s}{t}D(s))D(s).$ \end{proof}
\begin{corollary} If $t=1$ and $\omega(A) = \vert A\vert$, then $\mathscr B$ has the PCCG property if and only if $\mathscr B$ is greedy. \end{corollary}
\begin{corollary} If $\omega(A) = \vert A\vert$, then $\mathscr B$ has the $t$-PCCG property if and only if $\mathscr B$ is $t$-greedy. \end{corollary}
\section{A remark on the Haar system}
Throughout this section $|E|$ stands for the Lebesgue measure of a set in $[0,1]$, $\mathcal D$ for the family of dyadic intervals in $[0,1]$ and $card(\Lambda)$ for the number of dyadic elements in $\Lambda$. We denote by $\mathcal{H}:=\lbrace H_I\rbrace$ the Haar basis in $[0,1]$, that is to say $$H_{[0,1]} (x) = 1\; \text{for}\; x\in [0,1),$$ and for $I\in \mathcal D$ of the form $I = [(j-1)2^{-n}, j2^{-n})$, $j=1,..,2^n$, $n= 0,1,...$ we have \begin{displaymath} H_{I}(x) = \left\{ \begin{array}{ll} 2^{n/2} & \mbox{if $x\in [(j-1)2^{-n}, (j-\frac{1}{2})2^{-n})$,} \\ -2^{n/2} & \mbox{if $x\in [(j-\frac{1}{2})2^{-n}, j2^{-n})$,} \\ 0 & \mbox{otherwise.} \end{array} \right. \end{displaymath}
We write $$c_I(f) := \langle f,H_I\rangle = \int_0^1 f(x)H_I(x)dx \hbox{ and } c_I(f,p):= \Vert c_I(f)H_I\Vert_p, \quad 1\le p<\infty.$$
It is well known that $\mathcal H$ is an orthonormal basis in $L^2([0,1])$ and for $1<p<\infty$ we can use the Littlewood-Paley's Theorem which gives \begin{equation}\label{lp} c_p \left\Vert \left( \sum_I \vert c_I(f,p)\frac{H_I}{\Vert H_I\Vert_p}\vert^2\right)^{1/2}\right\Vert_p \leq \Vert f\Vert_p \leq C_p\left\Vert \left( \sum_I \vert c_I(f,p)\frac{H_I}{\Vert H_I\Vert_p}\vert^2\right)^{1/2}\right\Vert_p \end{equation}
to conclude that $(\frac{H_I}{\Vert H_I\Vert_p})_I$ is an unconditional basis in $L^p([0,1])$. Denoting $f<<_p g$ whenever $c_I(f,p)\le c_I(g,p)$ for all dyadic intervals $I$ we obtain from (\ref{lp}) the existence of a constant $K_p$ such that
\begin{equation}\label{uncon}
\|f\|_p\le K_p \|g\|_p \quad \forall f, g\in L^p([0,1]) \hbox{ with } f<<_p g,
\end{equation}
and also
\begin{equation}\label{h}
\|P_\Lambda g\|\le K_p \|g\|_p \quad \forall g\in L^p \quad \forall \Lambda\subset \mathcal D. \end{equation} Regarding the greedyness of the Haar basis it was V. N. Temlyakov the first one who proved (see \cite{T}) that the every wavelet basis $L_p$-equivalent to the Haar basis is $t$-greedy in $L_p([0,1])$ with $1<p<\infty$ for any $0<t\le 1$.
Let $\omega:[0,1]\to \mathbb R^+$ be a measurable weight and, as usual, we denote $\omega(I)=\int_I \omega(x)dx$ and $m_I(\omega)=\frac{\omega(I)}{|I|}$ for any $I\in \mathcal D$. In the space $L^p(\omega)=L^p([0,1],\omega)$ we denote $\|f\|_{p, \omega}=(\int_0^1 |f(x)|^p\omega(x) dx)^{1/p}$ and $$ c_I(f,p,\omega):= \Vert c_I(f)H_I\Vert_{p,\omega}= |c_I(f)|\frac{\omega(I)^{1/p}}{|I|^{1/2}}. $$ Recall that $\omega$ is said to be a dyadic $A_p$-weight (denoted $\omega \in A^{d}_p$) if \begin{equation} A^{d}_p(\omega)= \sup_{I\in \mathcal D} m_I(\omega) \Big( m_I(\omega^{-1/(p-1)})\Big)^{p-1}<\infty. \end{equation}
As one may expect, Littlewood-Paley theory holds for weights in the dyadic $A_p$-class. \begin{theorem} (see \cite{ABM, I} for the multidimensional case) If $\omega \in A^{d}_p$ then \begin{eqnarray}\label{uncon} \Vert f\Vert_{p,\omega} \approx\left\Vert \left( \sum_I \vert c_I(f,p,\omega)\frac{H_I}{\Vert H_I\Vert_{p,\omega}}\vert^2\right)^{1/2}\right\Vert_{p,\omega}. \end{eqnarray} In particular $(\frac{H_I}{\Vert H_I\Vert_{p,\omega}})_I$ is an unconditional basis in $L^p(\omega)$ for $1<p<\infty$. \end{theorem}
The greedyness of the Haar basis in $L^p(\omega)$ goes back to M. Izuki (see \cite{I, IS}) who showed that this holds for weights in the class $A_p^d$.
We shall use the ideas in these papers to show that the Haar basis satisfies the PCCG property for certain spaces defined using the Littlewood-Paley theory.
\begin{defi} Let $\omega:[0,1]\to \mathbb R^+$ be a measurable weight and $1\le p<\infty$. For each finite set of dyadic intervals $\Lambda$ we define
$f_\Lambda=\sum_{I\in \Lambda} c_I(f)H_I=\sum_{I\in \Lambda} c_I(f,p,\omega)\frac{H_I}{\Vert H_I\Vert_{p,\omega}}$ and write $$\|f\|_{X^p(\omega)}= \left\Vert\left( \sum_{I\in \Lambda} \vert c_I(f,p,\omega)\frac{H_I}{\Vert H_I\Vert_{p,\omega}}\vert^2\right)^{1/2}\right\Vert_{p,\omega}.$$
The closure of $span(f_\Lambda: card(\Lambda)<\infty)$ under this norm will be denoted $X^p(\omega)$.
\end{defi}
From the definition $(\frac{H_I}{\Vert H_I\Vert_{p,\omega}})_I$ is an unconditional basis with constant 1 in $X^p(\omega)$ and due to (\ref{uncon}) $X^p(\omega)=L^p(\omega)$ whenever $\omega\in A_p^d$. Our aim is to analyze conditions on the weight $\omega$ for the basis to be greedy. For such a purpose we do not need the weight to belong to $A_p^d$. In fact analyzing the proof in \cite{I, IS} one notices that only the dyadic reverse doubling condition (see \cite[p. 141]{GCRF}) was used. Recall that a weight $\omega$ is said to satisfies {\bf the dyadic reverse doubling condition} if there exists $\delta<1$ such that
\begin{equation}\label{dc}\omega(I')\le \delta \omega (I), \forall I,I'\in \mathcal D \hbox{ with } I'\subsetneq I.
\end{equation}
Let us introduce certain weaker conditions.
\begin{defi} Let $\alpha>0$ and $\omega$ be a measurable weight. We shall say that $\omega$ satisfies {\bf the dyadic reverse Carleson condition} of order $\alpha$ with constant $C>0$ whenever
\begin{equation}\label{cc}\sum_{I\in \mathcal D, J\subseteq I}\omega(I)^{-\alpha}\le C \omega(J)^{-\alpha}, \forall J\in \mathcal D .
\end{equation}
\end{defi}
\begin{defi} Let $\alpha>0$ and two sequences $(w_I)_{I\in \mathcal D}$ and $(v_I)_{I\in \mathcal D}$ of positive real numbers. We say that the pair $\Big((w_I)_{I\in \mathcal D}, (v_I)_{I\in \mathcal D}\Big)$ satisfies $\alpha-{\bf DRCC }$ with constant $C>0$ whenever
\begin{equation}\label{cc1}\sum_{I\in \mathcal D, J\subseteq I}w_I^{-\alpha}\le C v_J^{-\alpha}, \forall J\in \mathcal D .
\end{equation}
\end{defi}
\begin{Remark} \label{n} (i) If $\omega\in \cup_{p> 1}A_p^w$ then $\omega$ satisfies the dyadic reverse doubling condition (see \cite[p 141]{GCRF}).
(ii) If $\omega$ satisfies the dyadic reverse doubling condition then $\omega$ satisfies the dyadic reverse Carleson condition of order $\alpha$ with constant $\frac{1}{1-\delta^\alpha}$ for any $\alpha>0$.
Indeed, $$\sum_{J\subseteq I}\omega(I)^{-\alpha}\le \omega(J)^{-\alpha}+ \omega(J)^{-\alpha}\sum_{m=1}^{\infty} \delta^{m\alpha}\le \frac{1}{1-\delta^\alpha}\omega(J)^{-\alpha}.$$
(iii) If $\omega$ satisfies the dyadic reverse Carleson condition of order $\alpha$ and $w_I=\omega(I)$ for each $I\in \mathcal D$ then $\Big((w_I)_{I\in \mathcal D},(w_I)_{I\in \mathcal D}\Big)$ satisfies $\alpha$-{\bf DRCC }. \end{Remark}
We need the following lemmas, whose proofs are essentially included in \cite{CDH, I, IS}.
\begin{lemma} \label{1c} Let $\omega$ be a weight and $(v_I)_{I\in \mathcal D}$ be a sequence of positive real numbers such that
$\Big((v_I)_{I\in \mathcal D}, (\omega(I))_{I\in \mathcal D}\Big)$ satisfies $ 1$-{\bf DRCC } with constant $C$. Then \begin{equation}\label{dem}
\left(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}\right)^{1/p}\le C\left\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}, \forall 1\le p<\infty. \end{equation} \end{lemma} \begin{proof} We first write
\begin{equation}\label{main} \left\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}= \left(\int_0^1 (\sum_{I\in \Lambda} \omega(I)^{-2/p}\chi_I)^{p/2}\omega(x) dx\right)^{1/p}. \end{equation} Let $I(x)$ denote the minimal dyadic interval in $\Lambda$ with regard to the inclusion relation that contains $x$. Now we use that $$\sum_{I\in \mathcal D, I(x)\subseteq I}v_I^{-1}\le C \omega(I(x))^{-1}$$ to conclude that \begin{eqnarray*} (\sum_{I\in \Lambda} \frac{\omega(I)}{v_I})^{1/p}&=& \Big(\sum_{I\in\Lambda} \int_{I}v_I^{-1}\omega(x)dx\Big)^{1/p}= \Big(\int_0^1(\sum_{I\in \Lambda}v_I^{-1}\chi_I(x))\omega(x)dx\Big)^{1/p}\\ &\le & C\Big(\int_0^1 \omega(I(x))^{-1}\omega(x)dx\Big)^{1/p}\le C\Big(\int_0^1(\sum_{I\in \Lambda}\omega(I)^{-2/p}\chi_I(x))^{p/2}\omega(x)dx\Big)^{1/p}\\
&=& C\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_p}\|_{X^p(\omega)}. \end{eqnarray*} The proof is complete. \end{proof} \begin{lemma} \label{2p}Let $1<p<\infty$, $\omega$ be a weight and $(v_I)_{I\in \mathcal D}$ of positive real numbers. If $\Big((\omega(I))_{I\in \mathcal D}, (v_I)_{I\in \mathcal D}\Big)$ satisfies $2/p$-{\bf DRCC } with constant $C>0$ then \begin{equation}
\left\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}\le C \left(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}\right)^{1/p} \end{equation} for all finite family $\Lambda$ of dyadic intervals. \end{lemma} \begin{proof}
Let $E = \cup_{I\in\Lambda}I$. As above
$I(x)$ stands for the minimal dyadic interval in $\Lambda$ with regard to the inclusion relation that contains $x$. From (\ref{cc1}) we have that
\begin{equation}\label{1}
\sum_{I\in \Lambda}\omega(I)^{-2/p}\chi_I(x)\le Cv_{I(x)}^{-2/p}, \quad x\in E.\end{equation} Now denote for each $I\in \Lambda$, $\tilde I=\{x\in E: I(x)=I\}$. Clearly $\tilde I\subseteq I$ and $E=\cup_{I\in \Lambda}\tilde I $. Hence applying (\ref{main}) and (\ref{1}) we obtain \begin{eqnarray*}
\left\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_p}\right\|_{X^p(\omega)}&\le& C\left(\int_E
v_{I(x)}^{-1}\omega(x) dx\right)^{1/p} = C\left(\int_{\cup_{I\in \Lambda} \tilde I}
v_{I(x)}^{-1}\omega(x) dx\right)^{1/p} \\ &\le& C \Big(\sum_{I\in \Lambda} \int_{\tilde I}v_{I}^{-1}\omega(x)dx\Big)^{1/p}\le C \Big(\sum_{I\in \Lambda} v_I^{-1}\int_{I}\omega(x)dx\Big)^{1/p}\\ &=& C(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I})^{1/p}. \end{eqnarray*} The proof is now complete. \end{proof}
Combining Remark \ref{n} and Lemmas \ref{1c} and \ref{2p} we obtain the following corollary. \begin{corollary} Let $1<p<\infty$, $\omega$ be a weight satisfying the dyadic reverse doubling condition then \begin{equation} \label{basic}
\left\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}\approx card(\Lambda)^{1/p} \end{equation} for all finite family $\Lambda$ of dyadic intervals. \end{corollary}
\begin{corollary} Let $1<p<\infty$, $\omega$ be a weight and $(v_I)_{I\in \mathcal D}$ of positive real numbers. If $\Big((\omega(I))_{I\in \mathcal D}, (v_I)_{I\in \mathcal D}\Big)$ satisfies $2/p'$-{\bf DRCC } with constant $C>0$ then \begin{equation}\label{dem0}
\left(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}\right)^{1/p}\le C \Big(\max_{I\in \Lambda}\frac{\omega(I)}{v_I}\Big)\left\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)} \end{equation} for all finite family $\Lambda$ of dyadic intervals. \end{corollary} \begin{proof} Note that, using Lemma \ref{2p}, we have \begin{eqnarray*} \sum_{I\in \Lambda} \frac{\omega(I)}{v_I}&=& \int_0^1 (\sum_{I\in\Lambda} v_I^{-1}\chi_I(x))\omega(x)dx\\ &\le & \int_0^1 (\sum_{I\in\Lambda} \omega(I)^{-2/p}\chi_I)^{1/2}(\sum_{I\in\Lambda} v_I^{-2}\omega(I)^{2/p}\chi_I(x))^{1/2}\omega(x)dx\\ &\le & \Big(\int_0^1 (\sum_{I\in\Lambda} \omega(I)^{-2/p}\chi_I)^{p/2}\omega(x)dx\Big)^{1/p}\Big(\int_0^1(\sum_{I\in\Lambda} v_I^{-2}\omega(I)^{2/p}\chi_I(x))^{p'/2}\omega(x)dx\Big)^{1/p'}\\
&\le& \left\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)} \left\|\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}\frac{H_I}{\Vert H_I\Vert_{p',\omega}}\right\|_{X^{p'}(\omega)}\\
&\le& \left(\max_{I\in \Lambda}\frac{\omega(I)}{v_I}\right)\left\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}\left\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p',\omega}}\right\|_{X^{p'}(\omega)}\\
&\le& C\left(\max_{I\in \Lambda}\frac{\omega(I)}{v_I}\right)\left\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}\left(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}\right)^{1/p'}. \end{eqnarray*} The result now follows. \end{proof} Taking into account that dyadic reverse Carleson condition of order $\alpha$ implies dyadic reverse Carleson condition of order $\beta$ for $\beta>\alpha$ we obtain the following fact. \begin{corollary} Let $1<p<\infty$, $\omega$ be a weight satisfying the dyadic reverse Carleson condition of order $\min\{2/p', 2/p\}$ then \begin{equation} \label{basic}
\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\|_{X^p(\omega)}\approx card(\Lambda)^{1/p} \end{equation} for all finite family $\Lambda$ of dyadic intervals. \end{corollary}
\begin{theorem} Let $1<p<\infty$, $0<t\le 1$, $(w_I)_{I\in \mathcal D}$ be a sequence of real numbers such that $$0<m_0=\inf_{I\in \mathcal D}w_I\le \sup_{I\in \mathcal D}w_I=M_0<\infty$$ and let $\omega$ be a weight satisfying the dyadic reverse Carleson condition of order $\min\{1, 2/p\}$ with constant $C>0$. Then the Haar basis has the $(t, w_I)$-PCCG property in $X^p(\omega)$.
\end{theorem}
\begin{proof} Let $f\in X^p(\omega)$ and let $\Lambda^t_m$ be a set of $m$ dyadic intervals where $$\min_{I\in \Lambda^t_m}c_I(f,p, \omega)\ge t \max_{I'\notin \Lambda^t_m}c_{I'}(f,p, \omega) .$$
For each $\alpha\in \mathbb R$, $(\varepsilon_n)\in \{\pm 1\}$ and $\Lambda'$ with $\sum_{J\in \Lambda'}w_J\le \sum_{I\in \Lambda_m^t}w_I$ we need to show that $\|f-P_{\Lambda^t_m}(f)\|_{X^p(\omega)}\le C(t) \|f-\alpha 1_{\varepsilon \Lambda'}\|_{X^p(w)}$ for some constant $C(t)>0$. From triangular inequality $$\Vert f-P_{\Lambda^t_m}(f)\Vert_{X^p(\omega)} \leq \Vert P_{(\Lambda_m \cup \Lambda')^c}(f-\alpha 1_{\varepsilon \Lambda'})\Vert_{X^p(\omega)} + \Vert P_{\Lambda'\setminus \Lambda^t_m}(f)\Vert_{X^p(\omega)}$$ and the fact $\Vert P_{\Lambda}(f-\alpha 1_{\varepsilon B})\Vert_{X^p(\omega)}\le \Vert f-\alpha 1_{\varepsilon B}\Vert_{X^p(\omega)}$ for any $\Lambda$
we only need to show that there exists $C>0$ such that
$$\Vert P_{\Lambda'\setminus \Lambda^t_m}(f)\Vert_{X^p(\omega)}\le C \|f-\alpha 1_{\varepsilon \Lambda'}\|_{X^p(\omega)}.$$
Set $v_I=\frac{\omega(I)}{w_I}$ and observe that $\Big((\omega(I))_{I\in \mathcal D}, (v_I)_{I\in \mathcal D}\Big)$ satisfies $2/p$-{DRCC } with constant $M_0C$ and $\Big( (v_I)_{I\in \mathcal D}, \omega(I)_{I\in \mathcal D}\Big)$ satisfies $1$-{DRCC } with constant $C/m_0$. Note that $\sum_{J\in \Lambda'}w_J\le \sum_{I\in \Lambda_m^t}w_I$ implies that $$ \sum_{J\in \Lambda'\setminus \Lambda_m^t}\frac{\omega(J)}{v_{J}}\le \sum_{I\in \Lambda_m^t\setminus \Lambda'}\frac{\omega(I)}{v_{I}} $$ and then, invoking Lemma \ref{2p} and Lemma \ref{1c}, we get the estimates \begin{eqnarray*} \Vert P_{\Lambda'\setminus \Lambda_m^t}(f)\Vert_{X^p(\omega)} &\leq& \Vert \underset{I\in \Lambda'\setminus A_m}{\max} c_I(f,p,\omega)1_{\Lambda'\setminus A_m}\Vert_{X^p(\omega)}\\
&\le& C M_0 \underset{I\in \Lambda'\setminus \Lambda_m^t}{\max} c_I(f,p,\omega) (\sum_{J\in \Lambda'\setminus \Lambda_m^t}\frac{\omega(J)}{v_{J}})^{1/p}\\ &\leq&t^{-1} C M_0\underset{I\in \Lambda_m^t\setminus \Lambda'}{\min}c_I(f,p,\omega)(\sum_{I\in \Lambda_m^t\setminus \Lambda'}\frac{\omega(I)}{v_I})^{1/p} \\
&\leq& \frac{C^2 M_0}{tm_0}\Vert \underset{I\in \Lambda_m^t\setminus \Lambda'}{\min} c_I(f,p,\omega)1_{\Lambda_m^t\setminus \Lambda'}\Vert_{X^p(\omega)}\\ &\leq& \frac{C^2 M_0}{tm_0}\Vert P_{\Lambda_m^t\setminus \Lambda'}(f)\Vert_{X^p(\omega)} \\ &=& \frac{C^2 M_0}{tm_0}\Vert P_{\Lambda_m^t\setminus \Lambda'}(f-\alpha 1_{\varepsilon B})\Vert_{X^p(\omega)} \\
&\leq &\frac{C^2 M_0}{tm_0} \|f-\alpha 1_{\varepsilon \Lambda'}\|_{X^p(\omega)}. \end{eqnarray*}
This completes the proof with $C(t)= 1+\frac{C^2 M_0}{tm_0}$. \end{proof}
\begin{corollary} (i) If $\omega\in A_p^d$ then
the Haar basis has the $t$-PCCG property (and hence is $t$-greedy) in $L^p(\omega)$ with $1<p<\infty$.
(ii) The Haar basis has the $(t,w_I)$-PCCG property (and hence is $(t, w_I)$-greedy) in $L^p([0,1])$ for any sequence $(w_I)_{I\in\mathcal{D}}$ with $0<\inf w_I\le \sup w_I <\infty.$
\end{corollary} \noindent{\it \bf Acknowledgment:} The authors would like to thank to G. Garrig\'os and E. Hern\'andez for useful conversations during the elaboration of this paper.
\vspace*{-1.5cm}
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\begin{document}
\title{A multiplicatively symmetrized version of the Chung-Diaconis-Graham random process} \author{Martin Hildebrand \footnote{Department of Mathematics and Statistics, University at Albany, State University of New York, Albany, NY 12222. {\tt [email protected]}}} \maketitle \begin{abstract} This paper considers random processes of the form $X_{n+1}=a_nX_n+b_n\pmod p$ where $p$ is odd, $X_0=0$, $(a_0,b_0), (a_1,b_1), (a_2,b_2),...$ are i.i.d., and $a_n$ and $b_n$ are independent with $P(a_n=2)=P(a_n=(p+1)/2)=1/2$ and $P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$. This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order $(\log p)^2$ steps suffice for $X_n$ to be close to uniformly distributed on the integers mod $p$ for all odd $p$ while order $(\log p)^2$ steps are necessary for $X_n$ to be close to uniformly distributed on the intgers mod $p$. \end{abstract}
\section{Introduction}
Chung, Diaconis, and Graham~\cite{cdg} comsidered random processes of the form $X_{n+1}=2X_n+b_n\pmod p$ where $p$ is odd, $X_0=0$, and $b_0, b_1, b_2,...$ are i.i.d. with $P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$. They showed that order $(\log p)\log(\log p)$ steps suffice to make $X_n$ close to uniformly distributed on the integers mod $p$. Diaconis~\cite{diaconis} asked about random processes of the form $X_{n+1}=a_nX_n+b_n \pmod p$ where $p$ is odd, $X_0=0$, and $(a_0,b_0), (a_1,b_1), (a_2,b_2),...$ are i.i.d. with $a_n$ and $b_n$ being independent, $P(a_n=2)=P(a_n=(p+1)/2)=1/2$ and $P(b_n=1)=P(b_n=-1)=1/2$. In his Ph.D. thesis, the author~\cite{mvhphd} showed that order $(\log p)^2$ steps suffice to make $X_n$ close to uniformly distributed on the integers mod $p$ and that order $(\log p)^2$ steps
are necessary to make $X_n$ close to uniformly distributed on the integers mod $p$. The techniques used there can be readily adapted if the distribution is changed so that $P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$; in this case, these techniques show that order $((\log p)(\log(\log p)))^2$ steps suffice to make $X_n$ close to uniformly distributed on the integers mod $p$ for all odd integers $p$ and order $(\log p)^2$ steps suffice for almost all odd integers $p$
while order $(\log p)^2$ steps are necessary to make $X_n$ close to uniformly distributed in the integrs mod $p$. This paper shows that this result can be improved to show that order $(\log p)^2$ steps suffice to make $X_n$ close to uniformly distributed on the integers mod $p$ for all odd integers $p$.
\section{Some Background, Notation, and Main Result}
We let the integers mod $p$ be denoted by ${\mathbb Z}/p{\mathbb Z}$. We may denote elements of this group by $0, 1,..., p-1$ instead of $0+p{\mathbb Z}, 1+p{\mathbb Z},...,(p-1)+{\mathbb Z}$.
A probability $P$ on the integers mod $p$ satifies $P(s)\ge 0$ for $s\in{\mathbb Z}/p{\mathbb Z}$ and $\sum_{s\in{\mathbb Z}/p{\mathbb Z}}P(s)=1$.
We use the variation distance to measure how far a probability $P$ on ${\mathbb Z}/p{\mathbb Z}$ is from the uniform distribution on ${\mathbb Z}/p{\mathbb Z}$. This distance is given by \[
\|P-U\|=\frac{1}{2}\sum_{s\in{\mathbb Z}/p{\mathbb Z}}\left|P(s)-\frac{1}{p}\right|
=\max_{A\subset{\mathbb Z}/p{\mathbb Z}}|P(A)-U(A)| \]
where $P(A)=\sum_{s\in A}P(s)$ and $U(A)=\sum_{s\in A}1/p=|A|/p$. Note that $\|P-U\|\le 1$ for all probabilities $P$ on ${\mathbb Z}/p{\mathbb Z}$.
\begin{proposition} \label{probmixture} If $P=p_1P_1+p_2P_2+...+p_mP_m$ where $p_1, p_2,..., p_m$ are positive real numbers summing to $1$, then \[
\|P-U\|\le\sum_{i=1}^mp_i\|P_i-U\|. \] \end{proposition} This proposition can be readily shown using the triangle inequality.
If $P$ is a probability on ${\mathbb Z}/p{\mathbb Z}$, define the Fourier tranform \[\hat P(k)=\sum_{j=0}^{p-1}P(j)e^{2\pi ijk/p}\]
for $k=0, 1,..., p-1$. The Upper Bound Lemma of Diaconis and Shahshahani (see, for example, Diaconis~\cite{diaconis}, p. 24) implies \[
\|P-U\|^2\le\frac{1}{4}\sum_{k=1}^{p-1}|\hat P(k)|^2. \]
The main theorem is \begin{theorem} \label{mainthm} Suppose $X_0=0$ and $p$ is an odd integer greater than $1$. Let $X_{n+1}=a_nX_n+b_n \pmod p$ where $(a_0,b_0), (a_1,b_1), (a_2,b_2),...$ are i.i.d. such that $a_n$ and $b_n$ are independent,
$P(a_n=2)=P(a_n=(p+1)/2)=1/2$, and $P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$. Let $P_n(j)=P(X_n=j)$ for $j\in{\mathbb Z}/p{\mathbb Z}$. Let $\epsilon>0$ be given. For some $c>0$, if $n>c(\log p)^2$, then $\|P_n-U\|<\epsilon$. \end{theorem}
\section{Beginnings of the argument}
Observe that \begin{eqnarray*} X_0&=&0\\ X_1&=&b_0\\ X_2&=&a_1b_0+b_1\\ X_3&=&a_2a_1b_0+a_2b_1+b_2\\ &&...\\ X_n&=&a_{n-1}...a_2a_1b_0+a_{n-1}...a_2b_1+...+a_{n-1}b_{n-2}+b_{n-1} \end{eqnarray*}
We shall focus on the distribution of $X_n$ given values for $a_1, a_2,..., a_{n-1}$. In the case where $a_{n-1}=2$, $a_{n-2}=(p+1)/2$, $a_{n-3}=2$, $a_{n-4}=(p+1)/2$, etc., then \[ X_n=2(b_{n-2}+b_{n-4}+...)+(b_{n-1}+b_{n-3}+...) \pmod p . \] If $n=c(\log p)^2$, then $X_n$ lies between $-(3/2)c(\log p)^2$ and $(3/2)c(\log p)^2$ and, for large enough $p$, will not be close to uniformly distributed on the integers mod $p$. In the case where $a_{n-1}=2$, $a_{n-2}=2$, $a_{n-3}=2$, ..., $a_0=2$, then results of Chung, Diaconis, and Graham~\cite{cdg} show that order $(\log p)\log(\log p)$ steps suffice to make $X_n$ close to uniformly distributed on the integers mod $p$, and so order $(\log p)^2$ steps suffice as well.
Let $P_n(a_{n-1}, a_{n-2},...,a_1)(s)=P(a_{n-1}...a_1b_0+a_{n-1}...a_2b_1+...+a_{n-1}b_{n-2}+b_{n-1}=s \pmod p )$ where $b_0, b_1,...,b_{n-1}$ are i.i.d. uniform on $\{1,0,-1\}$.
We shall show \begin{theorem} \label{indiviudalcases} Let $\epsilon>0$ be given. There exists a constant $c>0$ such that if $n>c(\log p)^2$, then \[
\|P_n(a_{n-1}, a_{n-2},..., a_1)-U\|<\epsilon/2 \] except for a set $A$ of values $(a_{n-1}, a_{n-2},..., a_1)$ in
$\{2,(p+1)/2\}^{n-1}$ where $|A|<(\epsilon/2)2^{n-1}$. ($\{2,(p+1)/2\}^{n-1}$ is the set of $(n-1)$-tuples with entries in $\{2,(p+1)/2\}$.) \end{theorem}
By Proposition~\ref{probmixture}, Theorem~\ref{indiviudalcases} implies Theorem~\ref{mainthm}.
\section{Random Walk on the Exponent} \label{rwexp}
Suppose $a_0, a_1, a_2,...$ are i.i.d. with $P(a_1=2)=P(a_1=(p+1)/2)=1/2$. In the integers mod $p$, one can view $(p+1)/2$ as $2^{-1}$, the multiplicative inverse of $2$. So $1, a_{n-1}, a_{n-1}a_{n-2}, a_{n-1}a_{n-2}a_{n-3},...$ can be viewed as $2^{w_0}, 2^{w_1}, 2^{w_2}, 2^{w_3},...$ where $w_0=0$ and $w_{j+1}-w_j$ are i.i.d. for $j=0, 1, 2,...$ with $P(w_{j+1}-w_j=1)=P(w_{j+1}-w_j=-1)=1/2$.
Let $M_j=\max\{w_0, w_1,..., w_j\}$ and $m_j=\min\{w_0, w_1,..., w_j\}$.
By Theorem 1 of Section III.7 of Feller~\cite{feller}, $P(M_j=\ell)=p_{j,\ell}+p_{j,\ell+1}$ where $p_{j,\ell}={j \choose (j+\ell)/2}2^{-j}$ where the binomial coefficient is $0$ unless $(j+\ell)/2$ is an integer between $0$ and $j$, inclusive. Thus by Central Limit Theorem considerations, for some constant $c_1>0$, if $\epsilon_1>0$ and $j=\lceil c_1(\log p)^2\rceil$, then $P(M_j\le 0.5\log_2p)<\epsilon_1/4$ for sufficiently large $p$, and, by symmetry, $P(-m_j\le 0.5\log_2p)<\epsilon_1/4$ for sufficiently large $p$. Also by Central Limit Theorem considerations, for some constant $c_2>0$, $P(M_j\ge (c_2/2)\log_2p)<\epsilon_1/4$ and $P(-m_j\ge (c_2/2)\log_2p)<\epsilon_1/4$ for sufficiently large $p$. So if
$j=\lceil c_1(\log p)^2\rceil$, $P(\log_2p<M_j-m_j<c_2\log_2p)>1-\epsilon_1$ for sufficiently large $p$. If this event does not hold, then $(a_{n-1},a_{n-2},...,a_1)$ might be in the set $A$.
Exercise III.10 of Feller~\cite{feller} gives \[ z_{r,2n}=\frac{1}{2^{2n-r}}{2n-r\choose n} \] where $z_{r,2n}$ is the probability of exactly $r$ returns to the origin in the first $2n$ steps of the symmetric nearest neighbor random walk on the integers. Observe \[ z_{0,2n}=\frac{1}{2^{2n}}{2n\choose n}\sim \frac{1}{\sqrt{\pi n}}, \] which is approximately a multiple of $1/\log p$ if $n$ is approximately a multiple of $(\log p)^2$.
Observe that if $r\ge 0$, then \begin{eqnarray*} \frac{z_{r+1,2n}}{z_{r,2n}}&=&\frac{1/2^{2n-r-1}}{1/2^{2n-r}} \frac{{2n-r-1\choose n}}{{2n-r\choose n}}\\ &=&2\frac{n-r}{2n-r}\\ &\le &1. \end{eqnarray*} Thus $z_{r+1,2n}\le z_{r,2n}$.
For $k\in[m_j,M_j]$ with $j\lceil c_1(\log p)^2\rceil$,
let $R(k)$ be the number of $i$ such that $w_i=k$ where $0<i-\min_i\{w_i=k\}\le(\log p)^2$. Observe that $P(R(k)\le f(p))\le c_3(f(p)+1)/\log p$ for some positive constant $c_3$.
For some positive constant $c_4$, observe that
$E(|\{k:R(k)\le f(p), m_j\le k\le M_j\}|\ | \log_2p<M_j-m_j<c_2(\log_2p)) \le c_4(f(p)+1)$. Thus by Markov's inequality,
$P(|\{k:R(k)\le f(p), m_j\le k\le M_j\}|\ge c_5(f(p)+1)| \log_2p<M_j-m_j< c_2(\log_2p))\le c_4/c_5$.
\section{Fourier transform argument}
Let $\tilde P_n(a_{n-1}, a_{n-2},..., a_1)(s)=P(2^n(a_{n-1}a_{n-2}...a_1b_0+ a_{n-1}a_{n-2}...a_2b_1+...+a_{n-1}b_{n-2}+b_{n-1})=s \pmod p )$
where $b_0, b_1,..., b_{n-1}$ are i.i.d. uniform on $\{1, 0, -1\}$. Observe $\|\tilde P_n(a_{n-1}, a_{n-2},..., a_1)-U\|=
\|P_n(a_{n-1}, a_{n-2},..., a_1)-U\|$ since $p$ is odd. Note that all powers of $2$ in $2^na_{n-1}a_{n-2}...a_1$, $2^na_{n-1}a_{n-2}...a_2$, ..., $2^na_{n-1}$, $2^n$ are nonnegative.
The Upper Bound Lemma implies \begin{eqnarray*}
\|\tilde P_n(a_{n-1}, a_{n-2},..., a_1)-U\|&\le&\frac{1}{4}\sum_{m=1}^{p-1} \prod_{\ell=n+m_j}^{n+M_j}\left(\frac{1}{3}+\frac{2}{3}\cos(2\pi 2^{\ell}m/p) \right)^{2R(\ell-n)}\\ &\times &\prod_{r=j+1}^{n-1}\left(\frac{1}{3}+\frac{2}{3}\cos(2\pi 2^{n+w_r}m/p) \right)^2. \end{eqnarray*} Note that the first product term is for times up to $j$ and the second product term is for times after $j$. Recall $j=\lceil c_1(\log p)^2\rceil$.
Note that \[ \left(\frac{1}{3}+\frac{2}{3}\cos(2\pi 2^{\ell}m/p)\right)^{2R(\ell-n)} \le \cases{9^{-R(\ell-n)}&if $1/4\le \{2^{\ell}m/p\}<3/4$\cr 1&otherwise} \] and \[ \left(\frac{1}{3}+\frac{2}{3}\cos(2\pi 2^{n+w_r}m/p)\right)^2 \le \cases{1/9&if $1/4\le \{2^{n+w_r}m/p\}<3/4$\cr 1&otherwise} \] where $\{x\}$ is the fractional part of $x$.
Assume
$|\{k:R(k)\le {c_6} \log(\log p), m_j\le k\le M_j\}|<c_5(\log(\log p)+1)$ where $c_5$ is such that $c_4c_6/c_5<\epsilon_2$ where $\epsilon_2>0$ is given and $j=\lceil c_1(\log p)^2\rceil$ and
$|\{k:R(k)<(\log(\log p))^{2.1}, m_j\le k\le M_j\}|<(\log(\log p))^{2.5}$. Also assume $\log_2p<M_j-m_j<c_2(\log_2p)$, If these assumptions don't hold, then
$(a_{n-1},a_{n-2},...,a_1)$ might be in the set $A$. We shall consider various cases for $m$.
{\underbar {Case 1}}: $m$ is such that for some $\ell\in[n+m_j,n+M_j]$, $1/4\le\{2^{\ell}m/p\}<3/4$ and $R(\ell-n)>(\log(\log p))^{2.1}$. Let $S_1$ be the set of such $m$ in $1, 2,..., p-1$. Then, by arguments similar to those in Chung, Diaconis, and Graham~\cite{cdg} \[ \sum_{m\in S_1}\prod_{\ell=n+m_j}^{n+M_j}\left(\frac{1}{3}+\frac{2}{3} \cos(2\pi 2^{\ell}m/p)\right)^{2R(\ell-n)}<\epsilon. \] Details appear in Section~\ref{fouriervalues}.
{\underbar {Case 2}}: $m\notin S_1$ and for $b$ values of $\ell\in[n+m_j,n+M_j]$, $1/4\le\{2^{\ell}m/p\}<3/4$ and
${c_6} \log(\log p)<R(\ell-n)\le(\log(\log p))^{2.1}.$ Let $S_{2,b}$ be the set of such $m$ in $1, 2,..., p-1$.
Let's consider the binary expansion of $m/p$; in particular, consider the positions $n+m_j+1$ through $n+M_j+1$. If $1/4\le\{2^{\ell}m/p\}<3/4$, then there is an ``alternation'' between positions $(\ell+1)$ and $(\ell+2)$, i.e. there is a $1$ followed by a $0$ or a $0$ followed by a $1$. We say an alternation follows position $\ell$ if there is an alternation between positions $\ell+1$ and $\ell+2$. Alternations will start following $b$ of no more than $(\log(\log p))^{2.5}$ positions $\ell$ where ${c_6} \log(\log p)<R(\ell-n)<(\log(\log p))^{2.1}$, and alternations may or may not start following each of no more than $c_5(\log(\log p)+1)$ positions $\ell$ with $R(\ell-n)\le {c_6} \log(\log p)$. No other alternations may occur. Place $n+m_j+1$ may be either $0$ or $1$.
Places $n+m_j+1$ through $n+M_j+1$ of the binary expansion of $m/p$ are unique for each $m$ in $\{1,2,...,p-1\}$ since $M_j-m_j>\log_2p$ by an observation similar to the blocks in the argument of Chung, Diaconis, and Graham~\cite{cdg} being unique. So \begin{eqnarray*}
|S_{2,b}|&\le &2\cdot 2^{c_5(\log(\log p)+1)}{\lfloor(\log(\log p))^{2.5}\rfloor \choose b}\\ &\le &2\cdot 2^{c_5(\log(\log p)+1)}(\log(\log p))^{2.5b} \end{eqnarray*}
If $m\in S_{2,b}$, then \[ \prod_{\ell=n+m_j}^{n+M_j}\left(\frac{1}{3}+\frac{2}{3}\cos(2\pi 2^{\ell}m/p) \right)^{2R(\ell-n)} \le (1/9)^{b{c_6} \log(\log p)}. \] So \begin{eqnarray*} &&\sum_{m\in S_{2,b}}\prod_{\ell=n+m_j}^{n+M_j}\left(\frac{1}{3}+ \frac{2}{3}\cos(2\pi 2^{\ell}m/p)\right)^{2R(\ell-n)} \\ &\le & 2\cdot 2^{c_5(\log (\log p)+1)}((\log(\log p))^{2.5} (1/9)^{{c_6} \log(\log p)})^b \end{eqnarray*}
Note that for large enough $p$, $(\log(\log p))^{2.5}(1/9)^{{c_6} \log(\log p)}<1/2$. Also observe for $b\ge b_{\min}$ where $b_{\min}$ is a value depending on $c_5$ and ${c_6}$, \[ 2^{c_5(\log (\log p)+1)}((\log(\log p))^{2.5}(1/9)^{{c_6} \log(\log p)})^b\rightarrow 0 \] as $p\rightarrow\infty$. Thus \[ \sum_{b=b_{\min}}^{\infty} 2^{c_5(\log(\log p)+1)}((\log(\log p))^{2.5} (1/9)^{{c_6} \log(\log p)})^b\rightarrow 0 \] and \[ \sum_{b=b_{\min}}^{\infty}\sum_{m\in S_{2,b}}\prod_{\ell=n+m_j}^{n+M_j} \left(\frac{1}{3}+\frac{2}{3}\cos(2\pi 2^{\ell}m/p)\right)^{2R(\ell-n)} \rightarrow 0. \] So all we need to consider are $m\in S_{2,b}$ where $b<b_{\min}$.
To consider such $m$, we shall look at further steps in the Fourier transform. We shall use the following lemma. \begin{lemma} Let $\epsilon^{\prime}>0$ be given. Let $d$ be a positive number. For some constant $c_7>0$, except with probability no more than $\epsilon^{\prime}$, \[ \max_{\ell=d+1}^{d+\lfloor c_7(\log p)^2\rfloor}w_{\ell}-\min_{\ell=d+1}^{d+\lfloor c_7(\log p)^2\rfloor} w_{\ell}>2\log_2p. \] If this inequality holds, then, given $m\in\{1, 2,..., p-1\}$, $1/4\le \{2^{\ell}m/p\}<3/4$ for some $\ell\in\{d+1, d+2,..., d+\lfloor c_7(\log p)^2 \rfloor\}$. With probability at least $1-(\log(\log p))^{2.5}/\log p$, \[
|\{h:\ell+1\le h\le\ell+(\log p)^2, w_{\ell}=w_h\}|>(\log(\log p))^{2.1}. \] \end{lemma}
{\it Proof:} Similar to reasoning in section~\ref{rwexp}, the existence of $c_7$ follows by Central Limit Theorem considerations and Theorem 1 of Section III.7 of Feller~\cite{feller}. The existence of such $\ell$ follows since for each positive integer $k$, at least one of $\{2^km/p\}$, $\{2^{k+1}m/p\}$,...,$\{2^{k+\lfloor 2\log_2p\rfloor-1}m/p\}$ lies in $[1/4,3/4)$. The result
on $|\{h:\ell+1\le h\le\ell+(\log p)^2, w_{\ell}=w_h\}|$ follows similarly to the earlier argument that $P(R(k)\le f(p))\le c_3(f(p)+1)/\log p$.
$\Box$
Suppose $n_{before}$ is the number of $m$ being considered, i.e. need further Fourier transform terms before going an additional $\lfloor c_7(\log p)^2\rfloor +\lfloor(\log p)^2\rfloor$ terms. Afterwards, we will need to continue to consider only $m$ such that $\ell$ in the lemma exists and
$|\{h:\ell+1\le h\le \ell+(\log_2p)^2: w_{\ell}=w_jh\}|<(\log(\log p))^{2.1}$; otherwise we have sufficient additional terms in the Fourier transform; see Section~\ref{fouriervalues}. Except for at most $(\epsilon^{\prime}+o(1))2^{n-1}$ $(n-1)$-tuples in $A$, $n_{after}\le n_{before}(\log(\log p))^{2.5}/\log p$ where $n_{after}$ is the number of $m$ still being considered after going the additional $\lfloor c_7(\log p)^2\rfloor+\lfloor(\log p)^2\rfloor$ steps. Repeating this a fixed number $f$ times will give $n_{after}<1$, i.e. $n_{after}=0$ except for at most $f(\epsilon^{\prime}+o(1))2^{n-1}$ $(n-1)$-tuples in $A$.
\section{Bounding the Fourier transform sums} \label{fouriervalues}
Some of the ideas in this section, for example ``alternations'', come from Chung, Diaconis, and Graham~\cite{cdg}.
Suppose $m\in S_1$. If \[ g(x)=\cases{1/9&if $1/4\le\{x\}<3/4$ \cr 1&otherwise,} \] then \begin{eqnarray*} \prod_{\ell=n+m_j}^{n+M_j}\left(\frac{1}{3}+\frac{2}{3}\cos(2\pi 2^{\ell}m/p) \right)^{2R(\ell-n)} &\le& \prod_{\ell=n+m_j}^{n+M+j}(g(2^{\ell}m/p))^{R(\ell-n)}\\ &\le&(1/9)^{{c_6} \log(\log p)A(B_m)} \end{eqnarray*} where $A(B_m)$ is the number of ``alternations'' in the first $M_j-m_j$ positions of the binary expansion of $\{2^{n+m_j}m/p\}$. An alternation in the binary expansion $.\alpha_1\alpha_2\alpha_3...$ occurs when $\alpha_i\ne\alpha_{i+1}$. There will be an alternation in the first $\lceil\log_2p\rceil$ positions of the binary expansion of $\{2^{n+m_j}m/p\}$ if $m\in\{1, 2,..., p-1\}$, and for different $m\in\{1, 2,..., p-1\}$, the first $\lceil \log_2p\rceil$ positions of the binary expansion of $\{2^{n+m_j}m/p\}$ will differ. The inequality ending $<(1/9)^{{c_6} \log(\log p)A(B_m)}$ occurs since for some $\ell\in[n+m_j,n+M_j]$ with $1/4\le\{2^{\ell}m/p\}<3/4$, $R(\ell-n)\ge(\log(\log p))^{2.1}$ and the $R(\ell-n)$ powers of $1/9$ also cover all $c_5(\log(\log p)+1)$ terms of the from $(1/9)^{R(\ell-n)}$ with $\ell$ such that
$R(\ell-n)\le{c_6}\log(\log p)$ if $p$ is large enough.
Observe \begin{eqnarray*} \sum_{m\in S_1}(1/9)^{{c_6} \log(\log p)A(B_m)}&\le& \sum_{m=1}^{p-1}(1/9)^{{c_6} \log(\log p)A(B_m)}\\ &\le& 2\sum_{s=1}^{M_j-m_j}{M_j-m_j\choose s}(1/9)^{{c_6} \log(\log p)s}\\ &\le& 2\sum_{s=1}^{M_j-m_j}(M_j-m_j)^s(1/9)^{{c_6} \log(\log p)s}\\ &\rightarrow&0 \end{eqnarray*} as $p\rightarrow\infty$ if $\log_2p<M_j-m_j<c_2(\log p)$ and ${c_6}$ is large enough.
Now suppose $m\in S_{2,0}$ and for some $\ell$ with $1/4\le\{2^{\ell}m/p\}<3/4$ where $\ell<n-(\log p)^2$ and
$|\{h:\ell+1\le h\le(\log p)^2,w_{\ell}=w_h\}|\ge(\log(\log p))^{2.1}$, then \begin{eqnarray*} && \prod_{\ell=n+m_j}^{n+M_j}\left(\frac{1}{3}+\frac{2}{3}\cos(2\pi 2^{\ell}m/p) \right)^{2R(\ell-n)} \\ && \times \prod_{r=j+1}^{n-1}\left(\frac{1}{3}+\frac{2}{3}\cos(2\pi 2^{n+w_r}m/p)\right)^2 \\ &\le& (1/9)^{{c_6} \log(\log p) A(B_m)}. \end{eqnarray*} In other words, the powers of $1/9$ for these values of $h$ cover all $c_5(\log(\log p)+1)$ terms of the form $(1/9)^{R(\ell-n)}$ with $\ell$ such that
$R(\ell-n)\le {c_6}\log(\log p)$ if $p$ is large enough. By reasoning similar to the sum involving $m\in S_1$, \[ \sum_{m\in S_{2,0}}(1/9)^{{c_6} \log(\log p) A(B_m)} \rightarrow 0 \] as $p\rightarrow\infty$.
\section{Lower Bound}
The argument for the lower bound is more straightforward and is based upon \cite{mvhphd}.
\begin{theorem} \label{lowerbound} Suppose $X_n$, $a_n$, $b_n$, and $p$ are as in Theorem~\ref{mainthm}. Let
$\epsilon>0$ be given. For some $c>0$, if $n<c(\log p)^2$ for large enough $p$, then $\|P_n-U\|>1-\epsilon$. \end{theorem}
{\it Proof:} Let $m_j$ and $M_j$ be as in Section~\ref{rwexp}. For some $c>0$, if $n=\lfloor c(\log p)^2\rfloor$, then $P(m_j\le -0.25\log_2p)<\epsilon/3$ and $P(M_j\ge 0.25\log_2p)<\epsilon/3$. If $m_j>-0.25\log_2p$ and $M_j<0.25\log_2p$, then $2^{\lceil -0.25\log_2p\rceil}X_n$ lies in the interval $[-{\sqrt p}c(\log p)^2, {\sqrt p}c(\log p)^2]$, and so
$\|P_n-U\|\ge(1-2\epsilon/3)-(2{\sqrt p}c(\log p)^2+1)/p>1-\epsilon$ for sufficiently large $p$.
\section{Discussion of Generalizations for $a_n$}
One can ask if the results generalize to the case where $a$ is a fixed integer greater than $1$, $(a,p)=1$, and $P(a_n=a)=P(a_n=a^{-1})=1/2$. The results indeed should generalize. Chapter 3 of Hildebrand~\cite{mvhphd} gives a result if $P(a_n=a)=1$. This result gives an upper bound similar to the original Chung-Diaconis-Graham result with $P(a_n=2)=1$ and involves an $a$-ary expansion along with a generalization of alternations in a Fourier transform argument. The random walk on the exponent should work with powers of $a$ instead of powers of $2$. The Fourier transform argument may consider the interval $[1/a^2,1-1/a^2)$ instead of $[1/4,3/4)$. The constant $1/9$ may be replaced by another constant less than $1$. One needs to be careful with the size of the analogue of $S_{2,b}$.
Also Breuillard and Varj\'u~\cite{bv} consider the Chung-Diaconis-Graham process with $P(a_n=a)=1$ where $a$ is not fixed. One might explore cases where $P(a_n=a)=P(a_n=a^{-1})=1/2$ where $a$ is not fixed but does have a multiplicative inverse in the integers mod $p$.
\section{Questions for Further Study}
Eberhard and Varj\'u~\cite{ev} prove and locate a cut-off phenomonon for most odd integers $p$ in the original Chung-Diaconis-Graham random process. However, the diffusive nature of the random walk on the exponent suggests that a cut-off phenomenon might not appear in the multiplicatively symmetrized version. Exploring this question more rigorously is a problem for further study.
The Chung-Diaconis-Graham random process can be extended to multiple dimensions. Klyachko~\cite{klyachko} considers random processes of the form $X_{N+1}=A_NX_N+B_N \pmod p$ where $X_N$ is a random vector in $({\mathbb Z}/p{\mathbb Z})\times({\mathbb Z}/p{\mathbb Z})$ and $A_N$ is a fixed $2\times 2$ matrix with some conditions. Perhaps techniques in this paper could be combined with Klyachko's result to get a result for the case where $A_N$ is a fixed $2\times 2$ matrix or its inverse with probability $1/2$ each.
\section{Acknowledgment}
The author would like to thank the referee for some suggestions.
This is a preprint of an article published in {\it Journal of Theoretical Probability}. The final authenticated version is available online at
{\tt https://doi.org/10.1007/s10959-021-01088-3}.
\end{document}
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\begin{document}
\title{Sharp asymptotic behavior of solutions of the $3d$ Vlasov-Maxwell system with small data}
\begin{abstract} We study the asymptotic properties of the small data solutions of the Vlasov-Maxwell system in dimension three. No neutral hypothesis nor compact support assumptions are made on the data. In particular, the initial decay in the velocity variable is optimal. We use vector field methods to obtain sharp pointwise decay estimates in null directions on the electromagnetic field and its derivatives. For the Vlasov field and its derivatives, we obtain, as in \cite{FJS3}, optimal pointwise decay estimates by a vector field method where the commutators are modification of those of the free relativistic transport equation. In order to control high velocities and to deal with non integrable source terms, we make fundamental use of the null structure of the system and of several hierarchies in the commuted equations. \end{abstract}
\tableofcontents \section{Introduction}
This article is concerned with the asymptotic behavior of small data solutions to the three-dimensional Vlasov-Maxwell system. These equations, used to model collisionless plasma, describe, for one species of particles\footnote{Our results can be extended without any additional difficulty to several species of particles.}, a distribution function $f$ and an electromagnetic field which will be reprensented by a two form $F_{\mu \nu}$. The equations are given by\footnote{We will use all along this paper the Einstein summation convention so that, for instance, $v^i \partial_i f = \sum_{i=1}^3 v^i \partial_i f$ and $\nabla^{\mu} F_{\mu \nu} = \sum_{\mu=0}^3 \nabla^{\mu} F_{\mu \nu}$. The latin indices goes from $1$ to $3$ and the greek indices from $0$ to $3$.} \begin{eqnarray}\label{VM1} v^0\partial_t f+v^i \partial_i f +ev^{\mu}{ F_{\mu}}^{ j} \partial_{v^j} f & = & 0, \\ \label{VM2} \nabla^{\mu} F_{\mu \nu} & = & e J(f)_{\nu} \hspace{2mm} := \hspace{2mm} e\int_{v \in \mathbb{R}^3} \frac{v_{\nu}}{v^0} f dv, \\ \label{VM3} \nabla^{\mu} {}^* \! F_{\mu \nu} & = & 0, \end{eqnarray}
where $v^0=\sqrt{m^2+|v|^2}$, $m>0$ is the mass of the particles and $e \in \mathbb{R}^*$ their charge. For convenience, we will take $m=1$ and $e=1$ for the remainder of this paper. The particle density $f$ is a non-negative\footnote{In this article, the sign of $f$ does not play any role.} function of $(t,x,v) \in \mathbb{R}_+ \times \mathbb{R}^3 \times \mathbb{R}^3$, while the electromagnetic field $F$ and its Hodge dual ${}^* \! F $ are $2$-forms depending on $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^3$. We can recover the more common form of the Vlasov-Maxwell system using the relations $$E^i=F_{0i} \hspace{8mm} \text{and} \hspace{8mm} B^i=-{}^* \! F_{0i},$$ so that the equations can be rewritten as \begin{flalign*}
& \hspace{3cm} \sqrt{1+|v|^2} \partial_t f+v^i \partial_i f + (\sqrt{1+|v|^2} E+v \times B) \cdot \nabla_v f = 0, & \\
& \hspace{3cm} \nabla \cdot E = \int_{v \in \mathbb{R}^3}fdv, \hspace{1.1cm} \partial_t E^j = (\nabla \times B)^j -\int_{v \in \mathbb{R}^3} \frac{v^j}{\sqrt{1+|v|^2}}fdv, & \\ & \hspace{3cm} \nabla \cdot B = 0, \hspace{2,2cm} \partial_t B = - \nabla \times E. & \end{flalign*} We refer to \cite{Glassey} for a detailed introduction to this system. \subsection{Small data results for the Vlasov-Maxwell system}
The first result on global existence with small data for the Vlasov-Maxwell system in $3d$ was obtained by Glassey-Strauss in \cite{GSt} and then extended to the nearly neutral case in \cite{Sc}. This result required compactly supported data (in $x$ and in $v$) and shows that $\int_v f dv \lesssim \frac{\epsilon}{(1+t)^3}$, which coincides with the linear decay. They also obtain estimates for the electromagnetic field and its derivatives of first order, but they do not control higher order derivatives of the solutions. The result established by Schaeffer in \cite{Sc} allows particles with high velocity but still requires the data to be compactly supported in space\footnote{Note also that when the Vlasov field is not compactly supported (in $v$), the decay estimate obtained in \cite{Sc} on its velocity average contains a loss.}.
In \cite{dim4}, using vector field methods, we proved optimal decay estimates on small data solutions and their derivatives of the Vlasov-Maxwell system in high dimensions $d \geq 4$ without any compact support assumption on the initial data. We also obtained that similar results hold when the particles are massless ($m=0$) under the additional assumption that $f$ vanishes for small velocities\footnote{Note that there exists initial data violating this condition and such that the system does not admit a local classical solution (see Section $8$ of \cite{dim4}).}.
A better understanding of the null condition of the system led us in our recent work \cite{massless} to an extension of these results to the massless 3d case. In \cite{ext} we study the asymptotic properties of solutions to the massive Vlasov-Maxwell in the exterior of a light cone for mildly decaying initial data. Due to the strong decay satisfied by the particle density in such a region we will be able to lower the initial decay hypothesis on the electromagnetic field and then avoid any difficulty related to the presence of a non-zero total charge.
The results of this paper establish sharp decay estimates on the small data solutions to the three-dimensional Vlasov-Maxwell system. The hypotheses on the particle density in the variable $v$ are optimal in the sense that we merely suppose $f$ (as well as its derivatives) to be initially integrable in $v$, which is a necessarily condition for the source term of the Maxwell equations to be well defined.
Recently, Wang proved independently in \cite{Wang} a similar result for the $3d$ massive Vlasov-Maxwell system. Using both vector field methods and Fourier analysis, he does not require compact support assumptions on the initial data but strong polynomial decay hypotheses in $(x,v)$ on $f$ and obtained optimal pointwise decay estimates on $\int_v f dv$ and its derivatives.
\subsection{Vector fields and modified vector fields for the Vlasov equations}
The vector field method of Klainerman was first introduced in \cite{Kl85} for the study of nonlinear wave equations. It relies on energy estimates, the algebra $\mathbb{P}$ of the Killing vector fields of the Minkowski space and conformal Killing vector fields, which are used as commutators and multipliers, and weighted functional inequalities now known as Klainerman-Sobolev inequalities.
In \cite{FJS}, the vector field method was adapted to relativistic transport equations and applied to the small data solutions of the Vlasov-Nordstr\"om system in dimensions $d \geq 4$. It provided sharp asymptotics on the solutions and their derivatives. Key to the extension of the method is the fact that even if $Z \in \mathbb{P}$ does not commute with the free transport operator $T:= v^{\mu} \partial_{\mu}$, its complete lift\footnote{The expression of the complete lift of a vector field of the Minkowski space is presented in Definition \ref{defliftcomplete}.} $\widehat{Z}$ does. The case of the dimension $3$, studied in \cite{FJS2}, required to consider modifications of the commutation vector fields of the form $Y=\widehat{Z}+\Phi^{\nu} \partial_{\nu}$, where $\widehat{Z}$ is a complete lift of a Killing field (and thus commute with the free transport operator) while the coefficients $\Phi$ are constructed by solving a transport equation depending on the solution itself. In \cite{Poisson} (see also \cite{Xianglong}), similar results were proved for the Vlasov-Poisson equations and, again, the three-dimensionsal case required to modify the set of commutation vector fields in order to compensate the worst source terms in the commuted transport equations. Let us also mention \cite{rVP}, where the asymptotic behavior of the spherically symmetric small data solutions of the massless relativistic Vlasov-Poisson system are studied\footnote{Note that the Lorentz boosts cannot be used as commutation vector fields for this system since the Vlasov equation and the Poisson equation have different speed of propagation.}. Vector field methods led to a proof of the stability of the Minkowski spacetime for the Einstein-Vlasov system, obtained independently by \cite{FJS3} and \cite{Lindblad}.
Note that vector field methods can also be used to derive integrated decay for solutions to the the massless Vlasov equation on curved background such as slowly rotating Kerr spacetime (see \cite{ABJ}).
\subsection{Charged electromagnetic field}
In order to present our main result, we introduce in this subsection the pure charge part and the chargeless part of a $2$-form. \begin{Def} Let $G$ be a sufficiently regular $2$-form defined on $[0,T[ \times \mathbb{R}^3$. The total charge $Q_G(t)$ of $G$ is defined as $$ Q_G(t) \hspace{2mm} = \hspace{2mm} \lim_{r \rightarrow + \infty} \int_{\mathbb{S}_{t,r}} \frac{x^i}{r}G_{0i} d \mathbb{S}_{t,r},$$ where $\mathbb{S}_{t,r}$ is the sphere of radius $r$ of the hypersurface $\{t \} \times \mathbb{R}^3$ which is centered at the origin $x=0$. \end{Def} If $(f,F)$ is a sufficiently regular solution to the Vlasov-Maxwell system, $Q_F$ is a conserved quantity. More precisely, $$ \forall \hspace{0.5mm} t \in [0,T[, \hspace{2cm} Q_F(t)=Q_F(0)= \int_{x \in \mathbb{R}^3} \int_{v \in \mathbb{R}^3} f(0,x,v) dv dx.$$
Note that the derivatives of $F$ are automatically chargeless (see Appendix $C$ of \cite{massless}). The presence of a non-zero charge implies $\int_{\mathbb{R}^3} r|F|^2 dx = +\infty$ and prevents us from propagating strong weighted $L^2$ norms on the electromagnetic field. This leads us to decompose $2$-forms into two parts. For this, let $\chi : \mathbb{R} \rightarrow [0,1]$ be a cut-off function such that $$ \forall \hspace{0.5mm} s \leq -2, \hspace{3mm} \chi(s) =1 \hspace{1cm} \text{and} \hspace{1cm} \forall \hspace{0.5mm} s \geq -1, \hspace{3mm} \chi(s) =0.$$ \begin{Def}\label{defpure1} Let $G$ be a sufficiently regular $2$-form with total charge $Q_G$. We define the pure charge part $\overline{G}$ and the chargeless part $\widetilde{G}$ of $G$ as $$\overline{G}(t,x) := \chi(t-r) \frac{Q_G(t)}{4 \pi r^2} \frac{x_i}{r} dt \wedge dx^i \hspace{1cm} \text{and} \hspace{1cm} \widetilde{G} := G-\overline{G}.$$ \end{Def}
One can then verify that $Q_{\overline{G}}=Q_G$ and $Q_{\widetilde{G}}=0$, so that the hypothesis $\int_{\mathbb{R}^3} r|\widetilde{G}|^2 dx = +\infty$ is consistent. Notice moreover that $G=\widetilde{G}$ in the interior of the light cone.
The study of non linear systems with a presence of charge was initiated by \cite{Shu} in the context of the Maxwell-Klein Gordon equations. The first complete proof of such a result was given by Lindblad and Sterbenz in \cite{LS} and improved later by Yang (see \cite{Yang}). Let us also mention the work of \cite{Bieri}.
\subsection{Statement of the main result}
\begin{Def} We say that $(f_0,F_0)$ is an initial data set for the Vlasov-Maxwell system if $f_0 : \mathbb{R}^3_x \times \mathbb{R}^3_v \rightarrow \mathbb{R}$ and the $2$-form $F_0$ are both sufficiently regular and satisfy the constraint equations $$\nabla^i (F_{0})_{i0} =- \int_{v \in \mathbb{R}^3} f_0 dv \hspace{10mm} \text{and} \hspace{10mm} \nabla^i {}^* \! (F_0)_{i0} =0.$$ \end{Def} The main result of this article is the following theorem.
\begin{Th}\label{theorem} Let $N \geq 11$, $\epsilon >0$, $(f_0,F_0)$ an initial data set for the Vlasov-Maxwell equations \eqref{VM1}-\eqref{VM3}and $(f,F)$ be the unique classical solution to the system arising from $(f_0,F_0)$. If
$$ \sum_{ |\beta|+|\kappa| \leq N+3} \int_{x \in \mathbb{R}^3} \int_{v \in \mathbb{R}^3} (1+|x|)^{2N+3}(1+|v|)^{|\kappa|} \left| \partial_{x}^{\beta} \partial_v^{\kappa} f_0 \right| dv dx + \sum_{ |\gamma| \leq N+2} \int_{x \in \mathbb{R}^3} (1+|x|)^{2 |\gamma|+1} \left| \nabla_x^{\gamma} \widetilde{F}_0 \right|^2 dx \leq \epsilon ,$$ then there exists $C>0$, $M \in \mathbb{N}$ and $\epsilon_0>0$ such that, if $\epsilon \leq \epsilon_0$, $(f,F)$ is a global solution to the Vlasov-Maxwell system and verifies the following estimates. \begin{itemize} \item Energy bounds for the electromagnetic field $F$ and its chargeless part: $\forall$ $t \in \mathbb{R}_+$,
$$ \sum_{\begin{subarray}{} Z^{\gamma} \in \mathbb{K}^{|\gamma|} \\ \hspace{1mm} |\gamma| \leq N \end{subarray}} \int_{|x| \geq t} \tau_+ \left( | \alpha ( \mathcal{L}_{ Z^{\gamma}}(\widetilde{F}) ) |^2 + | \rho ( \mathcal{L}_{ Z^{\gamma}}(\widetilde{F}) )|^2 + |\sigma ( \mathcal{L}_{ Z^{\gamma}}(\widetilde{F}) ) |^2 \right)+\tau_- |\underline{\alpha} ( \mathcal{L}_{ Z^{\gamma}}(\widetilde{F}) ) |^2 dx \leq C\epsilon ,$$
$$ \hspace{-0.3cm} \sum_{\begin{subarray}{} Z^{\gamma} \in \mathbb{K}^{|\gamma|} \\ \hspace{1mm} |\gamma| \leq N \end{subarray}} \int_{|x| \leq t} \hspace{-0.2mm} \tau_+ \left( \left| \alpha \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) \right|^2 \hspace{-0.2mm} + \hspace{-0.2mm} \left| \rho \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) \right|^2 \hspace{-0.2mm} + \hspace{-0.2mm} \left|\sigma \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) \right|^2 \right)+\tau_- \left|\underline{\alpha} \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) \right|^2 dx \leq C\epsilon \log^{2M}(3+t) .$$
\item Pointwise decay estimates for the null components of\footnote{If $|x| \geq t+1$, the logarithmical growth can be removed for the components $\alpha$ and $\underline{\alpha}$.} $\mathcal{L}_{Z^{\gamma}}(F)$: $\forall$ $|\gamma| \leq N-6$, $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^3$, \begin{flalign*}
& \hspace{0.5cm} |\alpha(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \sqrt{\epsilon}\frac{\log(3+t)}{\tau_+^2} , \hspace{30mm} |\underline{\alpha}(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \sqrt{\epsilon}\frac{\log(3+t)}{\tau_+\tau_-} ,& \\
& \hspace{0.5cm} |\rho(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \sqrt{\epsilon} \frac{\log^2(3+t)}{\tau_+^2}, \hspace{29mm} |\sigma(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \sqrt{\epsilon}\frac{\log^2(3+t)}{\tau_+^2}.& \end{flalign*} \item Energy bounds for the Vlasov field: $\forall$ $t \in \mathbb{R}_+$,
$$\sum_{\begin{subarray}{} \hspace{0.5mm} Y^{\beta} \in \mathbb{Y}^{|\beta|} \\ \hspace{1mm} |\beta| \leq N \end{subarray}} \int_{x \in \mathbb{R}^3} \int_{v \in \mathbb{R}^3} \left|Y^{\beta} f \right| dv dx \leq C\epsilon.$$
\item Pointwise decay estimates for the velocity averages of $Y^{\beta} f$: $\forall$ $|\beta| \leq N-3$, $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^3$,
$$\int_{ v \in \mathbb{R}^3} \left| Y^{\beta} f \right| dv \lesssim \frac{\epsilon}{\tau_+^2 \tau_-} \hspace{5mm} \text{and} \hspace{5mm} \int_{ v \in \mathbb{R}^3} \left| Y^{\beta} f \right| \frac{dv}{(v^0)^2} \lesssim \epsilon \frac{1}{\tau_+^3} \mathds{1}_{t \geq |x|}+ \epsilon\frac{\log^2(3+t)}{\tau_+^3\tau_-} \mathds{1}_{|x| \geq t}$$. \end{itemize} \end{Th} \begin{Rq}
For the highest derivatives of $f_0$, those of order at least $N-2$, we could save four powers of $|x|$ in the condition on the initial norm and even more for those of order at least $N+1$. We could also avoid any hypothesis on the derivatives of order $N+1$ and $N+2$ of $F_0$ (see Remark \ref{rqgainH}). \end{Rq} \begin{Rq} Assuming more decay on $\widetilde{F}$ and its derivatives at $t=0$, we could use the Morawetz vector field as a multiplier, propagate a stronger energy norm and obtain better decay estimates on its null components in the exterior of the light cone. We could recover the decay rates of the free Maxwell equations (see \cite{CK}) on $\alpha (F)$, $\underline{\alpha} (F)$ and $\sigma (F)$, but not on $\rho(F)$. We cannot obtain a better decay rate than $\tau_+^{-2}$ on $\rho (F)$ because of the presence of the charge. With our approach, we cannot recover the sourceless behavior in the interior region because of the slow decay of $\int_v f dv$. \end{Rq}
\subsection{Key elements of the proof} \subsubsection{Modified vector fields} In \cite{dim4}, we observed that commuting \eqref{VM1} with the complete lift of a Killing vector field gives problematic source terms. More precisely, if $Z \in \mathbb{P}$, \begin{equation}\label{sourcetermintro} [T_F, \widehat{Z} ] f= -v^{\mu} {\mathcal{L}_Z(F)_{\mu}}^{ j} \partial_{v^j} f, \hspace{10mm} \text{with} \hspace{3mm} T_F = v^{\mu}\partial_{\mu}+v^{\mu} {F_{\mu}}^{ j} \partial_{v^j}. \end{equation} The difficulty comes from the presence of $\partial_v$, which is not part of the commutation vector fields, since in the linear case ($F=0$) $\partial_v f$ essentially behaves as $t\partial_{t,x} f$. However, one can see that the source term has the same form as the non-linearity $v^{\mu} {F_{\mu}}^{ j} \partial_{v^j} f$. In \cite{dim4}, we controlled the error terms by taking advantage of their null structure and the strong decay rates given by high dimensions. Unfortunately, our method does not apply in dimension $3$ since even assuming a full understanding of the null structure of the system, we would face logarithmic divergences. The same problem arises for other Vlasov systems and were solved using modified vector fields in order to cancel the worst source terms in the commutation formula. Let us mention again the works of \cite{FJS2} for the Vlasov-Nordstr\"om system, \cite{Poisson} for the Vlasov-Poisson equations, \cite{FJS3} and \cite{Lindblad} for the Einstein-Vlasov system. We will thus consider vector fields of the form $Y=\widehat{Z}+\Phi^{\nu}\partial_{\nu}$, where the coefficients $\Phi^{\nu}$ are themselves solutions to transport equations, growing logarithmically. As a consequence, we will need to adapt the Klainerman-Sobolev inequalities for velocity averages and the result of Theorem $1.1$ of \cite{dim4} in order to replace the original vector fields by the modified ones. \subsubsection{The electromagnetic field and the non-zero total charge} Because of the presence of a non-zero total charge, i.e. $ \lim_{r \rightarrow + \infty} \int_{ \mathbb{S}_{0,r} } \frac{x^i}{r} (F_0)_{0i} d \mathbb{S}_{0,r} \neq 0$, we have, at $t=0$,
$$\int_{\mathbb{R}^3} (1+r) \left| \frac{x^i}{r} F_{0i} \right|^2 dx = \int_{\mathbb{R}^3} (1+r) |\rho(F)|^2 dx= + \infty$$
and we cannot propagate $L^2$ bounds on $\int_{\mathbb{R}^3} (1+t+r) |\rho(F)(t,x)|^2 dx$. However, provided that we can control the flux of the electromagnetic field on the light cone $t=r$, we can propagate weighted $L^2$ norms of $F$ in the interior region. To deal with the exterior of the light cone, recall from Definition \ref{defpure1} the decomposition \begin{equation}\label{explicit} F = \widetilde{F}+\overline{F}, \hspace{1cm} \text{with} \hspace{1cm} \overline{F}(t,x) := \chi(t-r) \frac{Q_F}{4 \pi r^2} dr \wedge dt . \end{equation}
The hypothesis $\int_{\mathbb{R}^3} (1+|x|) | \widetilde{F} (0,.)| dx < + \infty$ is consistent with the chargelessness of $\widetilde{F}$ and we can then propagate weighted energy norms of $\widetilde{F}$ and bound the flux of $F$ on the light cone. On the other hand, we have at our disposal pointwise estimates on $\overline{F}$ and its derivatives through the explicit formula \eqref{explicit}. These informations will allow us to deduce pointwise decay estimates on the null components of $F$ in both the exterior and the interior regions.
Another problem arises from the source terms of the commuted Maxwell equations, which need to be written with our modified vector fields. This leads us, as \cite{FJS2} and \cite{FJS3}, to rather consider them of the form $Y=\widehat{Z}+\Phi^{i}X_i$, where $X_i=\partial_i+\frac{v^i}{v^0}\partial_t$. The $X_i$ vector fields enjoy a kind of null condition\footnote{Note that they were also used in \cite{dim4} to improve the decay estimate on $\partial \int_v f ds$.} and allow us to avoid a small growth on the electromagnetic field norms which would prevent us to close our energy estimates\footnote{We make similar manipulations to recover the standard decay rate on the modified Klainerman-Sobolev inequalities.}. However, at the top order, a loss of derivative does not allow us to take advantage of them and creates a $t^{\eta}$-loss, with $\eta >0$ a small constant. A key step is to make sure that $\| \left| Y^{\kappa} \Phi \right|^2 Y f \|_{L^1_{x,v}}$, for $|\kappa|=N-1$, does not grow faster than $t^{\eta}$. \subsubsection{High velocities and null structure of the system}
After commuting the transport equation satisfied by the coefficients $\Phi^i$ and in order to prove energy estimates, we are led to control integrals such as
$$\int_0^t \int_{\mathbb{R}^3} \int_{v \in \mathbb{R}^3}(s+|x|) \left| \mathcal{L}_Z(F) f \right| dv dx ds.$$
If $f$ vanishes for high velocities, the characteristics of the transport equations have velocities bounded away from $1$. If $f$ is moreover initially compactly supported in space, its spatial support is ultimately disjoint from the light cone and, assuming enough decay on the Maxwell field, one can prove $$|\mathcal{L}_Z(F) f| \lesssim (1+t+r)^{-1}(1+|t-r|)^{-1}| f | \lesssim (1+t+r)^{-2}| f |,$$ so that \begin{equation}\label{eq:pb1}
\int_0^t \int_{\mathbb{R}^3} \int_{v \in \mathbb{R}^3}(s+|x|) \left| \mathcal{L}_Z(F) f \right| dv dx ds \lesssim \int_0^t (1+s)^{-1} ds,
\end{equation}
which is almost uniformly bounded in time\footnote{Dealing with these small growth is the next problem addressed.}. As we do not make any compact support assumption on the initial data, we cannot expect $f$ to vanish for high velocities and certain characteristics of the transport operator ultimately approach those of the Maxwell equations. We circumvent this difficulty by taking advantage of the null structure of the error term given in \eqref{sourcetermintro}, which, in some sense, allows us to transform decay in $|t-r|$ into decay in $t+r$. The key is that certain null components of $v$, $\mathcal{L}_Z(F)$ and $\nabla_v f :=(0,\partial_{v^1} f,\partial_{v^2}f,\partial_{v^3}f)$ behave better than others and we will see in Lemma \ref{nullG} that no product of three bad components appears. More precisely, noting $c \prec d$ if $d$ is expected to behave better than $c$, we have, $$v^L \prec v^A, \hspace{1mm} v^{\underline{L}}, \hspace{10mm} \underline{\alpha}(\mathcal{L}_Z(F)) \prec \rho (\mathcal{L}_Z(F)) \sim \sigma( \mathcal{L}_Z(F) ) \prec \alpha( \mathcal{L}_Z(F) ) \hspace{10mm} \text{and} \hspace{6mm} \left( \nabla_v f \right)^A \prec \left( \nabla_v f \right)^{r}.$$
In the exterior of the light cone (and for the massless relativistic transport operator), we have $v^A \prec v^{\underline{L}}$ since $v^{\underline{L}}$ permits to integrate along outgoing null cones\footnote{The angular component $v^A$ can, in some sense, merely do half of it since $|v^A| \lesssim \sqrt{v^0 v^{\underline{L}}}$.} and they are both bounded by $(1+t+r)^{-1}v^0\sum_{z \in \mathbf{k}_1} |z|$, where $\mathbf{k}_1$ is a set of weigths preserved by the free transport operator. In the interior region, the angular components still satisfies the same properties whereas $v^{\underline{L}}$ merely satisfies the inequality \begin{equation}\label{eq:intro1}
v^{\underline{L}} \lesssim \frac{|t-r|}{1+t+r}v^0+\frac{v^0}{1+t+r} \sum_{z \in \mathbf{k}} |z| \hspace{10mm} \text{( see Lemma \ref{weights1})}. \end{equation}
This inequality is crucial for us to close the energy estimates on the electromagnetic field without assuming more initial decay in $v$ on $f$. It gives a decay rate of $(1+t+r)^{-3}$ on $\int_v \frac{v^{\underline{L}}}{v^0} |f| dv$ by only using a Klainerman-Sobolev inequality (Theorem \ref{decayopti} and Proposition \ref{decayopti2} would cost us two powers of $v^0$). As $1 \lesssim v^0 v^{\underline{L}}$ for massive particles, we obtain, combining \eqref{eq:intro1} and Theorem \ref{decayopti}, for $g$ a solution to $v^{\mu} \partial_{\mu} g =0$,
$$ \forall \hspace{0.5mm} t \geq |x|, \hspace{10mm} \int_{v \in \mathbb{R}^3} |g|(t,x,v)dv \lesssim \frac{(1+|t-r|)^k}{(1+t+r)^{3+k}} \sum_{|\beta| \leq 3} \left\| (v^0)^{2k+2}(1+r)^k \widehat{Z}^{\beta}g \right\|_{L^1_{x,v}}(t=0).$$
In the exterior region, the estimate can be improved by removing the factor $(1+|t-r|)^k$ (however one looses one power of $r$ in the initial norm). This remarkable behavior reflects that the particles do not reach the speed of light so that $\int_{v \in \mathbb{R}^3} |g| dv$ enjoys much better decay properties along null rays than along time-like directions and should be compared with solutions to the Klein-Gordon equation (see \cite{Kl93}).
\subsubsection{Hierarchy in the equations}
Because of certains source terms of the commuted transport equation, we cannot avoid a small growth on certain $L^1$ norms as it is suggested by \eqref{eq:pb1}. In order to close the energy estimates, we then consider several hierarchies in the energy norms of the particle density, in the spirit of \cite{LR} for the Einstein equations or \cite{FJS3} for the Einstein-Vlasov system. Let us show how a hierarchy related to the weights $z \in \mathbf{k}_1$ preserved by the free massive transport operator (which are defined in Subsection \ref{sectionweights}) naturally appears. \begin{itemize} \item The worst source terms of the transport equation satisfied by $Yf$ are of the form $(t+r)X_i(F_{\mu \nu})\partial_{t,x} f$. \item Using the improved decay properties given by $X_i$ (see \eqref{eq:X}), we have
$$ \left| (t+r)X_i(F_{\mu \nu})\partial_{t,x} f \right| \lesssim \sum_{Z \in \mathbb{K}} |\nabla_Z F| \sum_{z \in \mathbf{k}_1} |z\partial_{t,x} f|.$$
\item Then, we can obtain a good bound on $\| Yf \|_{L^1_{x,v}}$ provided we have a satisfying one on $\| z \partial_{t,x} f \|_{L^1_{x,v}}$. We will then work with energy norms controlling $\| z^{N_0-\beta_P} Y^{\beta} f \|_{L^1_{x,v}}$, where $\beta_P$ is the number of non-translations composing $Y^{\beta}$.
\item At the top order, we will have to deal with terms such as $(t+r)z^{N_0}\partial_{t,x}^{\gamma}(F_{\mu \nu})\partial_{t,x}^{\beta} f$ and we will this time use the extra decay $(1+|t-r|)^{-1}$ given by the translations $\partial_{t,x}^{\gamma}$. \end{itemize}
\subsection{Structure of the paper}
In Section \ref{sec2} we introduce the notations used in this article. Basic results on the electromagnetic field as well as fundamental relations between the null components of the velocity vector $v$ and the weights preserved by the free transport operator are also presented. Section \ref{sec3} is devoted to the commutation vector fields. The construction and basic properties of the modified vector fields are in particular presented. Section \ref{sec4} contains the energy estimates and the pointwise decay estimates used to control both fields. Section \ref{secpurecharge} is devoted to properties satisfied by the pure charge part of the electromagnetic field. In Section \ref{sec6} we describe the main steps of the proof of Theorem \ref{theorem} and present the bootstrap assumptions. In Section \ref{sec7}, we derive pointwise decay estimates on the solutions and the $\Phi$ coefficients of the modified vector fields using only the bootstrap assumptions. Section \ref{sec8} (respectively Section \ref{sec12}) concerns the improvement of the bootstrap assumptions on the norms of the particle density (respectively the electromagnetic field). A key step consists in improving the estimates on the velocity averages near the light cone (cf. Proposition \ref{Xdecay}). In Section \ref{sec11}, we prove $L^2$ estimates for $\int_v|Y^{\beta}f|dv$ in order to improve the energy estimates on the Maxwell field.
\section{Notations and preliminaries}\label{sec2}
\subsection{Basic notations}
In this paper we work on the $3+1$ dimensional Minkowski spacetime $(\mathbb{R}^{3+1},\eta)$. We will use two sets of coordinates, the Cartesian $(t,x^1,x^2,x^3)$, in which $\eta=diag(-1,1,1,1)$, and null coordinates $(\underline{u},u,\omega_1,\omega_2)$, where $$\underline{u}=t+r, \hspace{5mm} u=t-r$$ and $(\omega_1,\omega_2)$ are spherical variables, which are spherical coordinates on the spheres $(t,r)=constant$. These coordinates are defined globally on $\mathbb{R}^{3+1}$ apart from the usual degeneration of spherical coordinates and at $r=0$. We will also use the following classical weights, $$\tau_+:= \sqrt{1+\underline{u}^2} \hspace{8mm} \text{and} \hspace{8mm} \tau_-:= \sqrt{1+u^2}.$$ We denote by $(e_1,e_2)$ an orthonormal basis on the spheres and by $\slashed{\nabla}$ the intrinsic covariant differentiation on the spheres $(t,r)=constant$. Capital Latin indices (such as $A$ or $B$) will always correspond to spherical variables. The null derivatives are defined by $$L=\partial_t+\partial_r \hspace{3mm} \text{and} \hspace{3mm} \underline{L}=\partial_t-\partial_r, \hspace{3mm} \text{so that} \hspace{3mm} L(\underline{u})=2, \hspace{2mm} L(u)=0, \hspace{2mm} \underline{L}( \underline{u})=0 \hspace{2mm} \text{and} \hspace{2mm} \underline{L}(u)=2.$$
The velocity vector $(v^{\mu})_{0 \leq \mu \leq 3}$ is parametrized by $(v^i)_{1 \leq i \leq 3}$ and $v^0=\sqrt{1+|v|^2}$ since we take the mass to be $1$. We introduce the operator $$T : f \mapsto v^{\mu} \partial_{\mu} f,$$ defined for all sufficiently regular functions $f : [0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v$, and we denote $(0,\partial_{v^1}g, \partial_{v^2}g,\partial_{v^3}g)$ by $\nabla_v g$ so that \eqref{VM1} can be rewritten $$T_F(f) := v^{\mu} \partial_{\mu} f +F \left( v, \nabla_v f \right) =0.$$ We will use the notation $D_1 \lesssim D_2$ for an inequality such as $ D_1 \leq C D_2$, where $C>0$ is a positive constant independent of the solutions but which could depend on $N \in \mathbb{N}$, the maximal order of commutation. Finally we will raise and lower indices using the Minkowski metric $\eta$. For instance, $\nabla^{\mu} = \eta^{\nu \mu} \nabla_{\nu}$ so that $\nabla^{\partial_t}=-\nabla_{\partial_t}$ and $\nabla^{\partial_i}=\nabla_{\partial_i}$ for all $1 \leq i \leq 3$.
\subsection{Basic tools for the study of the electromagnetic field}\label{basicelec}
As we describe the electromagnetic field in geometric form, it will be represented, throughout this article, by a $2$-form. Let $F$ be a $2$-form defined on $[0,T[ \times \mathbb{R}^3_x$. Its Hodge dual ${}^* \! F$ is the $2$-form given by $${}^* \! F_{\mu \nu} = \frac{1}{2} F^{\lambda \sigma} \varepsilon_{ \lambda \sigma \mu \nu},$$ where $\varepsilon_{ \lambda \sigma \mu \nu}$ are the components of the Levi-Civita symbol. The null decomposition of $F$, introduced by \cite{CK}, is denoted by $(\alpha(F), \underline{\alpha}(F), \rho(F), \sigma (F))$, where $$\alpha_A(F) = F_{AL}, \hspace{5mm} \underline{\alpha}_A(F)= F_{A \underline{L}}, \hspace{5mm} \rho(F)= \frac{1}{2} F_{L \underline{L} } \hspace{5mm} \text{and} \hspace{5mm} \sigma(F) =F_{12}.$$ Finally, the energy-momentum tensor of $F$ is $$T[F]_{\mu \nu} := F_{\mu \beta} {F_{\nu}}^{\beta}- \frac{1}{4}\eta_{\mu \nu} F_{\rho \sigma} F^{\rho \sigma}.$$ Note that $T[F]_{\mu \nu}$ is symmetric and traceless, i.e. $T[F]_{\mu \nu}=T[F]_{\nu \mu}$ and ${T[F]_{\mu}}^{\mu}=0$. This last point is specific to the dimension $3$ and engenders additional difficulties in the analysis of the Maxwell equations in high dimension (see Section $3.3.2$ in \cite{dim4} for more details).
We have the following alternative form of the Maxwell equations (for a proof, see \cite{CK} or Lemmas $2.2$ and $D.3$ of \cite{massless}).
\begin{Lem}\label{maxwellbis} Let $G$ be a $2$-form and $J$ be a $1$-form both sufficiently regular and such that \begin{eqnarray} \nonumber \nabla^{\mu} G_{\mu \nu} & = & J_{\nu} \\ \nonumber \nabla^{\mu} {}^* \! G_{\mu \nu} & = & 0. \end{eqnarray} Then, $$\nabla_{[ \lambda} G_{\mu \nu ]} = 0 \hspace{4mm} \text{and} \hspace{4mm} \nabla_{[ \lambda} {}^* \! G_{\mu \nu ]} = \varepsilon_{\lambda \mu \nu \kappa} J^{\kappa}.$$ We also have, if $(\alpha, \underline{\alpha}, \rho, \sigma)$ is the null decomposition of $G$, \begin{eqnarray} \nabla_{\underline{L}} \hspace{0.5mm} \rho-\frac{2}{r} \rho+ \slashed{\nabla}^A \underline{\alpha}_A & = & J_{\underline{L}}, \label{eq:nullmax1} \\ \nabla_{\underline{L}} \hspace{0.5mm} \sigma-\frac{2}{r} \sigma+ \varepsilon^{AB} \slashed{\nabla}_A \underline{\alpha}_B & = & 0, \label{eq:nullmax2} \\ \nabla_{\underline{L}} \hspace{0.5mm} \alpha_A-\frac{\alpha_A}{r}+\slashed{\nabla}_{e_A} \rho+\varepsilon_{BA} \slashed{\nabla}_{e_B} \sigma &=& J_A. \label{eq:nullmax4} \end{eqnarray} \end{Lem}
We can then compute the divergence of the energy momentum tensor of a $2$-form.
\begin{Cor}\label{tensorderiv} Let $G$ and $J$ be as in the previous lemma. Then, $\nabla^{\mu} T[G]_{\mu \nu}=G_{\nu \lambda} J^{\lambda}$. \end{Cor}
\begin{proof} Using the previous lemma, we have \begin{eqnarray} \nonumber G_{\mu \rho} \nabla^{\mu} {G_{\nu}}^{\rho}& = & G^{\mu \rho} \nabla_{\mu} G_{\nu \rho} \\ \nonumber & = & \frac{1}{2} G^{\mu \rho} (\nabla_{\mu} G_{\nu \rho}-\nabla_{\rho} G_{\nu \mu}) \\ \nonumber & = & \frac{1}{2} G^{\mu \rho} \nabla_{\nu} G_{\mu \rho} \\ \nonumber & = & \frac{1}{4} \nabla_{\nu} (G^{\mu \rho} G_{\mu \rho}). \end{eqnarray}
Hence, $$\nabla^{\mu} T[G]_{\mu \nu} = \nabla^{\mu} (G_{\mu \rho}){G_{\nu}}^{\rho}+\frac{1}{4} \nabla_{\nu} (G^{\mu \rho} G_{\mu \rho})-\frac{1}{4}\eta_{\mu \nu} \nabla^{\mu} (G^{\sigma \rho} G_{\sigma \rho})=G_{\nu \rho} J^{\rho}.$$ \end{proof}
Finally, we recall the values of the null components of the energy-momentum tensor of a $2$-form. \begin{Lem}\label{tensorcompo} Let $G$ be $2$-form. We have
$$T[G]_{L L}=|\alpha(G)|^2, \hspace{8mm} T[G]_{\underline{L} \underline{L} }=|\underline{\alpha}(G)|^2 \hspace{8mm} \text{and} \hspace{8mm} T[G]_{L \underline{L}}=|\rho(G)|^2+|\sigma(G)|^2.$$ \end{Lem}
\subsection{Weights preserved by the flow and null components of the velocity vector}\label{sectionweights}
Let $(v^L,v^{\underline{L}},v^A,v^B)$ be the null components of the velocity vector, so that $$v=v^L L+ v^{\underline{L}} \underline{L}+v^Ae_A, \hspace{8mm} v^L=\frac{v^0+\frac{x_i}{r}v^i}{2} \hspace{8mm} \text{and} \hspace{8mm} v^{\underline{L}}=\frac{v^0-\frac{x_i}{r}v^i}{2}.$$ As in \cite{FJS}, we introduce the following set of weights, $$ \mathbf{k}_1 := \left\{\frac{v^{\mu}}{v^0} \hspace{1mm} / \hspace{1mm} 0 \leq \mu \leq 3 \right\} \cup \left\{ z_{\mu \nu} \hspace{1mm} / \hspace{1mm} \mu \neq \nu \right\}, \hspace{1cm} \text{with} \hspace{1cm } z_{\mu \nu} := x^{\mu}\frac{v^{\nu}}{v^0}-x^{\nu}\frac{v^{\mu}}{v^0}.$$ Note that \begin{equation}\label{weightpreserv} \forall \hspace{0.5mm} z \in \mathbf{k}_1, \hspace{8mm} T(z)=0. \end{equation}
Recall that if $\mathbf{k}_0 := \mathbf{k}_1 \cup \{ x^{\mu} v_{\mu} \}$, then $v^{\underline{L}} \lesssim \tau_+^{-1} \sum_{w \in \mathbf{k}_0} |w|$. Unfortunately, $x^{\mu} v_{\mu}$ is not preserved by\footnote{Note however that $x^{\mu} v_{\mu}$ is preserved by $|v| \partial_t+x^i \partial_i$, the massless relativistic transport operator.} $T$ so we will not be able to take advantage of this inequality in this paper. In the following lemma, we try to recover (part of) this extra decay. We also recall inequalities involving other null components of $v$, which will be used all along this paper. \begin{Lem}\label{weights1} The following estimates hold,
$$ 1 \leq 4v^0v^{\underline{L}}, \hspace{8mm} |v^A| \lesssim \sqrt{v^Lv^{\underline{L}}}, \hspace{8mm} |v^A| \lesssim \frac{v^0}{\tau_+} \sum_{z \in \mathbf{k}_1} |z|, \hspace{8mm} \text{and} \hspace{8mm} v^{\underline{L}} \lesssim \frac{\tau_-}{\tau_+} v^0+\frac{v^0}{\tau_+}\sum_{z \in \mathbf{k}_1}|z|.$$ \end{Lem} \begin{proof}
Note first that, as $v^0= \sqrt{1+|v|^2}$,
$$ 4r^2v^Lv^{\underline{L}} \hspace{2mm} = \hspace{2mm} r^2+r^2 |v|^2-|x^i|^2|v_i|^2-2\sum_{1 \leq k < l \leq n}x^kx^lv^kv^l \hspace{2mm} = \hspace{2mm} r^2+\sum_{1 \leq k < l \leq n} |z_{kl}|^2.$$
It gives us the first inequality since $v^L \leq v^0$. For the second one, use also that $rv^A=v^0C_A^{i,j} z_{ij}$, where $C_A^{i,j}$ are bounded functions on the sphere such that $re_A = C_A^{i,j} (x^i \partial_j-x^j \partial_i)$. The third one follows from $|v^A| \leq v^0$ and
$$|v^A| \lesssim \frac{v^0}{r} \sum_{1 \leq i < j \leq 3} |z_{ij}| = \frac{v^0}{tr} \sum_{1 \leq i < j \leq 3} \left| x^i\left( \frac{v^j}{v^0}t-x^j+x^j \right)-x^j\left( \frac{v^i}{v^0}t-x^i+x^i \right) \right| \lesssim \frac{v^0}{t} \sum_{q=1}^3 |z_{0q}|.$$
For the last inequality, note first that $v^{\underline{L}} \leq v^0$, which treats the case $t+|x| \leq 2$. Otherwise, use $$2tv^{\underline{L}}=tv^0-\frac{x^i}{r}tv_i = tv^0-v^0\frac{x^iz_{0i}}{r}-v^0r=(t-r)v^0-\frac{x^i}{r}z_{0i}v^0 \hspace{5mm} \text{and} \hspace{5mm} r v^{\underline{L}} =(r-t) v^{\underline{L}}+tv^{\underline{L}}.$$ \end{proof} \begin{Rq}\label{rqweights1}
Note that $v^{\underline{L}} \lesssim \frac{v^0}{\tau_+} \sum_{z \in \mathbf{k}_1} |z|$ holds in the exterior region. Indeed, if $r \geq t$,
$$v^0(r-t) \leq v^0|x|-|v|t \leq |v^0 x-tv| \leq \sum_{i=1}^3 |v^0x^i-tv^i|= v^0 \sum_{i=1}^3 |z_{0i}|.$$ We also point out that $1 \lesssim v^0 v^{\underline{L}}$ is specific to massive particles. \end{Rq}
Finally, we consider an ordering on $\mathbf{k}_1$ such that $\mathbf{k}_1 = \{ z_i \hspace{1mm} / \hspace{1mm} 1 \leq i \leq |\mathbf{k}_1| \}$. \begin{Def}\label{orderk1}
If $ \kappa \in \llbracket 1, |\mathbf{k}_1| \rrbracket^r$, we define $z^{\kappa} := z_{\kappa_1}...z_{\kappa_r}$. \end{Def} \subsection{Various subsets of the Minkowski spacetime}\label{secsubsets}
We now introduce several subsets of $\mathbb{R}_+ \times \mathbb{R}^3$ depending on $t \in \mathbb{R}_+$, $r \in \mathbb{R}_+$ or $u \in \mathbb{R}$. Let $\Sigma_t$, $\mathbb{S}_{t,r}$, $C_u(t)$ and $V_u(t)$ be defined as
\begin{flalign*}
& \hspace{0.5cm} \Sigma_t := \{t\} \times \mathbb{R}^n, \hspace{5.4cm} C_u(t):= \{(s,y) \in \mathbb{R}_+ \times \mathbb{R}^3 / \hspace{1mm} s \leq t, \hspace{1mm} s-|y|=u \}, & \\
& \hspace{5mm} \mathbb{S}_{t,r}:= \{ (s,y) \in \mathbb{R}_+ \times \mathbb{R}^3 \hspace{1mm} / \hspace{1mm} (s,|y|)=(t,r) \} \hspace{4mm} \text{and} \hspace{4mm} V_u(t) := \{ (s,y) \in \mathbb{R}_+ \times \mathbb{R}^3 / \hspace{1mm} s \leq t, \hspace{1mm} s-|y| \leq u \}. & \end{flalign*} The volume form on $C_u(t)$ is given by $dC_u(t)=\sqrt{2}^{-1}r^{2}d\underline{u}d \mathbb{S}^{2}$, where $ d \mathbb{S}^{2}$ is the standard metric on the $2$ dimensional unit sphere.
\begin{tikzpicture} \draw [-{Straight Barb[angle'=60,scale=3.5]}] (0,-0.3)--(0,5); \fill[color=gray!35] (2,0)--(5,3)--(9.8,3)--(9.8,0)--(1,0); \node[align=center,font=\bfseries, yshift=-2em] (title)
at (current bounding box.south)
{The sets $\Sigma_t$, $C_u(t)$ and $V_u(t)$}; \draw (0,3)--(9.8,3) node[scale=1.5,right]{$\Sigma_t$}; \draw (2,0.2)--(2,-0.2); \draw [-{Straight Barb[angle'=60,scale=3.5]}] (0,0)--(9.8,0) node[scale=1.5,right]{$\Sigma_0$}; \draw[densely dashed] (2,0)--(5,3) node[scale=1.5,left, midway] {$C_u(t)$}; \draw (6,1.5) node[ color=black!100, scale=1.5] {$V_u(t)$}; \draw (0,-0.5) node[scale=1.5]{$r=0$}; \draw (2,-0.5) node[scale=1.5]{$-u$}; \draw (-0.5,4.7) node[scale=1.5]{$t$}; \draw (9.5,-0.5) node[scale=1.5]{$r$}; \end{tikzpicture}
We will use the following subsets, given for $ \underline{u} \in \mathbb{R}_+$, specifically in the proof of Proposition \ref{Phi1},
$$ \underline{V}_{\underline{u}}(t) := \{ (s,y) \in \mathbb{R}_+ \times \mathbb{R}^3 / \hspace{1mm} s \leq t, \hspace{1mm} s+|y| \leq \underline{u} \}.$$ For $b \geq 0$ and $t \in \mathbb{R}_+$, define $\Sigma^b_t$ and $\overline{\Sigma}^b_t$ as
$$\Sigma^b_t:= \{ t \} \times \{ x \in \mathbb{R}^3 \hspace{1mm} / \hspace{1mm} |x| \leq t-b \} \hspace{6mm} \text{and} \hspace{6mm} \overline{\Sigma}^b_t:= \{ t \} \times \{ x \in \mathbb{R}^3 \hspace{1mm} / \hspace{1mm} |x| \geq t-b \}.$$ We also introduce a dyadic partition of $\mathbb{R}_+$ by considering the sequence $(t_i)_{i \in \mathbb{N}}$ and the functions $(T_i(t))_{i \in \mathbb{N}}$ defined by $$t_0=0, \hspace{5mm} t_i = 2^i \hspace{5mm} \text{if} \hspace{5mm} i \geq 1, \hspace{5mm} \text{and} \hspace{5mm} T_{i}(t)= t \mathds{1}_{t \leq t_i}(t)+t_i \mathds{1}_{t > t_i}(t).$$ We then define the truncated cones $C^i_u(t)$ adapted to this partition by
$$C_u^i(t) := \left\{ (s,y) \in \mathbb{R}_+ \times \mathbb{R}^3 \hspace{2mm} / \hspace{2mm} t_i \leq s \leq T_{i+1}(t), \hspace{2mm} s-|y| =u \right\}= \left\{ (s,y) \in C_u(t) \hspace{2mm} / \hspace{2mm} t_i \leq s \leq T_{i+1}(t) \right\}.$$ The following lemma will be used several times during this paper. It depicts that we can foliate $[0,t] \times \mathbb{R}^3$ by $(\Sigma_s)_{0 \leq s \leq t}$, $(C_u(t))_{u \leq t}$ or $(C^i_u(t))_{u \leq t, i \in \mathbb{N}}$. \begin{Lem}\label{foliationexpli} Let $t>0$ and $g \in L^1([0,t] \times \mathbb{R}^3)$. Then $$ \int_0^t \int_{\Sigma_s} g dx ds \hspace{2mm} = \hspace{2mm} \int_{u=-\infty}^t \int_{C_u(t)} g dC_u(t) \frac{du}{\sqrt{2}} \hspace{2mm} = \hspace{2mm} \sum_{i=0}^{+ \infty} \int_{u=-\infty}^t \int_{C^i_u(t)} g dC^i_u(t) \frac{du}{\sqrt{2}}.$$ \end{Lem}
Note that the sum over $i$ is in fact finite. The second foliation will allow us to exploit $t-r$ decay since $\| \tau_-^{-1} \|_{L^{\infty}(C_u(t)}=\tau_-^{-1}$ whereas $\|\tau_-^{-1}\|_{L^{\infty}(\Sigma_s)}=1$. The last foliation will be used to take advantage of time decay on $C_u(t)$ (the problem comes from $\|\tau_+^{-1}\|_{L^{\infty}(C_u(t))} = \tau_-^{-1}$). More precisely, let $0 < \delta < a$ and suppose for instance that, $$\forall \hspace{0.5mm} t \in [0,T[, \hspace{6mm} \int_{C_u(t)} g dC_u(t) \leq C (1+t)^{\delta}, \hspace{5mm} \text{so that} \hspace{5mm} \int_{C_u^i(t)} g dC^i_u(t) \leq C (1+T_{i+1}(t))^{\delta} \leq C (1+t_{i+1})^{\delta} .$$ Then, $$ \int_{C_u(t)} \tau_+^{-a}g dC_u(t) \leq \sum_{i=0}^{+ \infty} \int_{C^i_u(t)} (1+s)^{-a} g dC^i_u(t) \leq \sum_{i=0}^{+ \infty} (1+t_{i})^{-a} \int_{C^i_u(t)} g dC^i_u(t) \leq 3^aC \sum_{i=0}^{+ \infty} (1+2^{i+1})^{\delta-a}.$$ As $\delta-a <0$, we obtain a bound independent of $T$. \subsection{An integral estimate}
A proof of the following inequality can be found in the appendix $B$ of \cite{FJS}.
\begin{Lem}\label{intesti}
Let $m \in \mathbb{N}^*$ and let $a$, $b \in \mathbb{R}$, such that $a+b >m$ and $b \neq 1$. Then $$\exists \hspace{0.5mm} C_{a,b,m} >0, \hspace{0.5mm} \forall \hspace{0.5mm} t \in \mathbb{R}_+, \hspace{1.5cm} \int_0^{+ \infty} \frac{r^{m-1}}{\tau_+^a \tau_-^b}dr \leq C_{a,b,m} \frac{1+t^{b-1}}{1+t^{a+b-m}} .$$ \end{Lem} \section{Vector fields and modified vector fields}\label{sec3}
For all this section, we consider $F$ a suffciently regular $2$-form.
\subsection{The vector fields of the Poincaré group and their complete lift}
We present in this section the commutation vector fields of the Maxwell equations and those of the relativistic transport operator (we will modified them to study the Vlasov equation). Let $\mathbb{P}$ be the generators of Poincaré group of the Minkowski spacetime, i.e. the set containing \begin{flalign*} & \hspace{1cm} \bullet \text{the translations\footnotemark} \hspace{18mm} \partial_{\mu}, \hspace{2mm} 0 \leq \mu \leq 3, & \\ & \hspace{1cm} \bullet \text{the rotations} \hspace{25mm} \Omega_{ij}=x^i\partial_{j}-x^j \partial_i, \hspace{2mm} 1 \leq i < j \leq 3, & \\ & \hspace{1cm} \bullet \text{the hyperbolic rotations} \hspace{8mm} \Omega_{0k}=t\partial_{k}+x^k \partial_t, \hspace{2mm} 1 \leq k \leq 3. \end{flalign*} \footnotetext{In this article, we will denote $\partial_{x^i}$, for $1 \leq i \leq 3$, by $\partial_{i}$ and sometimes $\partial_t$ by $\partial_0$.} We also consider $\mathbb{T}:= \{ \partial_{t}, \hspace{1mm} \partial_1, \hspace{1mm} \partial_2, \hspace{1mm} \partial_3\}$ and $\mathbb{O} := \{ \Omega_{12}, \hspace{1mm} \Omega_{13}, \hspace{1mm} \Omega_{23} \}$, the subsets of $\mathbb{P}$ containing respectively the translations and the rotational vector fields as well as $\mathbb{K}:= \mathbb{P} \cup \{ S \}$, where $S=x^{\mu} \partial_{\mu}$ is the scaling vector field. The set $\mathbb{K}$ is well known for commuting with the wave and the Maxwell equations (see Subsection \ref{subseccomuMax}). However, to commute the operator $T=v^{\mu} \partial_{\mu}$, one should consider the complete lifts of the elements of $\mathbb{P}$. \begin{Def}\label{defliftcomplete}
Let $W=W^{\beta} \partial_{\beta}$ be a vector field. Then, the complete lift $\widehat{W}$ of $W$ is defined by $$\widehat{W}=W^{\beta} \partial_{\beta}+v^{\gamma} \frac{\partial W^i}{\partial x^{\gamma}} \partial_{v^i}.$$ We then have $\widehat{\partial}_{\mu}=\partial_{\mu}$ for all $0 \leq \mu \leq 3$ and $$\widehat{\Omega}_{ij}=x^i \partial_j-x^j \partial_i+v^i \partial_{v^j}-v^j \partial_{v^i}, \hspace{2mm} \text{for} \hspace{2mm} 1 \leq i < j \leq 3, \hspace{6mm} \text{and} \hspace{6mm} \widehat{\Omega}_{0k} = t\partial_k+x^k \partial_t+v^0 \partial_{v^k}, \hspace{2mm} \text{for} \hspace{2mm} 1 \leq k \leq 3.$$ \end{Def} One can check that $[T,\widehat{Z}]=0$ for all $Z \in \mathbb{P}$. Since $[T,S]=T$, we consider $$\widehat{\mathbb{P}}_0 := \{ \widehat{Z} \hspace{1mm} / \hspace{1mm} Z \in \mathbb{P} \} \cup \{ S \}$$ and we will, for simplicity, denote by $\widehat{Z}$ an arbitrary vector field of $\widehat{\mathbb{P}}_0$, even if $S$ is not a complete lift. The weights introduced in Subsection \ref{sectionweights} are, in a certain sense, preserved by the action of $\widehat{\mathbb{P}}_0$.
\begin{Lem}\label{weights} Let $z \in \mathbf{k}_1$, $\widehat{Z} \in \widehat{\mathbb{P}}_0$ and $j \in \mathbb{N}$. Then
$$\widehat{Z}(v^0z) \in v^0 \mathbf{k}_1 \cup \{ 0 \} \hspace{8mm} \text{and} \hspace{8mm} \left| \widehat{Z} (z^j) \right| \leq 3j \sum_{w \in \mathbf{k}_1} |w|^j.$$ \end{Lem}
\begin{proof} Let us consider for instance $tv^1-x^1v^0$, $x^1v^2-x^2v^1$, $\widehat{\Omega}_{01}$ and $\widehat{\Omega}_{02}$. We have \begin{eqnarray} \nonumber \widehat{\Omega}_{01}(x^1v^2-x^2v^1 ) & = & tv^2-x^2v^0, \hspace{2cm} \widehat{\Omega}_{01}(tv^1-x^1v^0) \hspace{2mm} = \hspace{2mm} 0, \\ \nonumber \widehat{\Omega}_{02}(x^1v^2-x^2v^1 ) & = & x^1v^0-tv^1 \hspace{8mm} \text{and} \hspace{8mm} \widehat{\Omega}_{02}(tv^1-x^1v^0) \hspace{2mm} = \hspace{2mm} x^2v^1-x^1v^2. \end{eqnarray} The other cases are similar. Consequently,
$$\left| \widehat{Z} (z^j) \right| = \left| \widehat{Z} \left(\frac{1}{(v^0)^j}(v^0z)^j \right) \right| \leq j|z|^j+\frac{j}{(v^0)^j}\left| \widehat{Z} \left(v^0z \right) \right| |v^0z|^{j-1} \leq j|z|^j +j\frac{|\widehat{Z}(v^0z)|^j}{(v^0)^j}+j|z|^j,$$
since $|w||z|^{a-1} \leq |w|^a+|z|^a$ when $a \geq 1$. \end{proof} The vector fields introduced in this section and the averaging in $v$ almost commute in the following sense (we refer to \cite{FJS} or to Lemma \ref{lift2} below for a proof). \begin{Lem}\label{lift} Let $f : [0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v \rightarrow \mathbb{R} $ be a sufficiently regular function. We have, almost everywhere,
$$\forall \hspace{0.5mm} Z \in \mathbb{K}, \hspace{8mm} \left|Z\left( \int_{v \in \mathbb{R}^3 } |f| dv \right) \right| \lesssim \sum_{ \begin{subarray}{} \widehat{Z}^{\beta} \in \widehat{\mathbb{P}}_0^{|\beta|} \\ \hspace{1mm} |\beta| \leq 1 \end{subarray}} \int_{v \in \mathbb{R}^3 } |\widehat{Z}^{\beta} f | dv .$$
Similar estimates hold for $\int_{v \in \mathbb{R}^3} (v^0)^k |f| dv$. For instance,
$$\left| S\left( \int_{v \in \mathbb{R}^3 } (v^0)^{-2}|f| dv \right) \right| \lesssim \int_{v \in \mathbb{R}^3 } (v^0)^{-2}|Sf| dv.$$ \end{Lem} The vector spaces engendered by each of the sets defined in this section are actually algebras. \begin{Lem} Let $\mathbb{L}$ be either $\mathbb{K}$, $\mathbb{P}$, $\mathbb{O}$, $\mathbb{T}$ or $\widehat{\mathbb{P}}_0$. Then for all $(Z_1,Z_2) \in \mathbb{L}^2$, $[Z_1,Z_2]$ is a linear combinations of vector fields of $\mathbb{L}$. Note also that if $Z_2=\partial \in \mathbb{T}$, then $[Z_1,\partial]$ can be written as a linear combination of translations. \end{Lem}
We consider an ordering on each of the sets $\mathbb{O}$, $\mathbb{P}$, $\mathbb{K}$ and $\widehat{\mathbb{P}}_0$. We take orderings such that, if $\mathbb{P}= \{ Z^i / \hspace{2mm} 1 \leq i \leq |\mathbb{P}| \}$, then $\mathbb{K}= \{ Z^i / \hspace{2mm} 1 \leq i \leq |\mathbb{K}| \}$, with $Z^{|\mathbb{K}|}=S$, and
$$ \widehat{\mathbb{P}}_0= \left\{ \widehat{Z}^i / \hspace{2mm} 1 \leq i \leq |\widehat{\mathbb{P}}_0| \right\}, \hspace{2mm} \text{with} \hspace{2mm} \left( \widehat{Z}^i \right)_{ 1 \leq i \leq |\mathbb{P}|}=\left( \widehat{Z^i} \right)_{ 1 \leq i \leq |\mathbb{P}|} \hspace{2mm} \text{and} \hspace{2mm} \widehat{Z}^{|\widehat{\mathbb{P}}_0|}=S .$$
If $\mathbb{L}$ denotes $\mathbb{O}$, $\mathbb{P}$, $\mathbb{K}$ or $\widehat{\mathbb{P}}_0$, and $\beta \in \{1, ..., |\mathbb{L}| \}^r$, with $r \in \mathbb{N}^*$, we will denote the differential operator $\Gamma^{\beta_1}...\Gamma^{\beta_r} \in \mathbb{L}^{|\beta|}$ by $\Gamma^{\beta}$. For a vector field $W$, we denote the Lie derivative with respect to $W$ by $\mathcal{L}_W$ and if $Z^{\gamma} \in \mathbb{K}^{r}$, we will write $\mathcal{L}_{Z^{\gamma}}$ for $\mathcal{L}_{Z^{\gamma_1}}... \mathcal{L}_{Z^{\gamma_r}}$. The following definition will be useful to lighten the notations in the presentation of commutation formulas.
\begin{Def}\label{goodcoeff} We call good coefficient $c(t,x,v)$ any function $c$ of $(t,x,v)$ such that
$$ \forall \hspace{0.5mm} Q \in \mathbb{N}, \hspace{1mm} \exists \hspace{0.5mm} C_Q >0, \hspace{2mm} \forall \hspace{0.5mm} |\beta| \leq Q, \hspace{2mm} (t,x,v) \in \mathbb{R}_+ \times \mathbb{R}_x^3 \times \mathbb{R}_v^3 \setminus \{ 0 \} \times \{ 0 \} \times \mathbb{R}_v^3, \hspace{8mm} \left| \widehat{Z}^{\beta} \left( c(t,x,v) \right) \right| \leq C_Q.$$ Similarly, we call good coefficient $c(v)$ any function $c$ such that
$$ \forall \hspace{0.5mm} Q \in \mathbb{N}, \hspace{2mm} \exists \hspace{0.5mm} C_Q >0, \hspace{2mm} \forall \hspace{0.5mm} |\beta| \leq Q, \hspace{2mm} v \in \mathbb{R}^3, \hspace{8mm} \left| \widehat{Z}^{\beta} \left( c(v) \right) \right| \leq C_Q.$$ Finally, we will say that $B$ is a linear combination, with good coefficients $c(v)$, of $(B^i)_{1 \leq i \leq M}$ if there exists good coefficients $(c_i(v))_{1 \leq i \leq M}$ such that $B=c_i B^i$. We define similarly a linear combination with good coefficients $c(t,x,v)$. \end{Def}
These good coefficients introduced here are to be thought of bounded functions which remain bounded when they are differentiated (by $\widehat{\mathbb{P}}_0$ derivatives) or multiplied between them. In the remainder of this paper, we will denote by $c(t,x,v)$ (or $c_Z(t,x,v)$, $c_i(t,x,v)$) any such functions. Note that $\widehat{Z}^{\beta} \left( c(t,x,v) \right)$ is not necessarily defined on $\{ 0 \} \times \{ 0 \} \times \mathbb{R}_v^3$ as, for instance, $c(t,x,v)=\frac{x^1}{t+r} \frac{v^2}{v^0}$ satisfies these conditions. Typically, the good coefficients $c(v)$ will be of the form $\widehat{Z}^{\gamma} \left( \frac{v^i}{v^0} \right)$.
Let us recall, by the following classical result, that the derivatives tangential to the cone behave better than others.
\begin{Lem}\label{goodderiv} The following relations hold, $$(t-r)\underline{L}=S-\frac{x^i}{r}\Omega_{0i}, \hspace{3mm} (t+r)L=S+\frac{x^i}{r}\Omega_{0i} \hspace{3mm} \text{and} \hspace{3mm} re_A=\sum_{1 \leq i < j \leq 3} C^{i,j}_A \Omega_{ij},$$ where the $C^{i,j}_A$ are uniformly bounded and depends only on spherical variables. In the same spirit, we have $$(t-r)\partial_t =\frac{t}{t+r}S-\frac{x^i}{t+r}\Omega_{0i} \hspace{3mm} \text{and} \hspace{3mm} (t-r) \partial_i = \frac{t}{t+r} \Omega_{0i}- \frac{x^i}{t+r}S- \frac{x^j}{t+r} \Omega_{ij}.$$ \end{Lem}
As mentioned in the introduction, we will crucially use the vector fields $(X_i)_{1 \leq i \leq 3}$, defined by \begin{equation}\label{eq:defXi} X_i := \partial_i+\frac{v^i}{v^0}\partial_t. \end{equation} They provide extra decay in particular cases since \begin{equation}\label{eq:Xi} X_i= \frac{1}{t} \left( \Omega_{0i}+z_{0i} \partial_t \right). \end{equation} We also have, using Lemma \ref{goodderiv} and $(1+t+r)X_i=X_i+2tX_i+(r-t)X_i$, that there exists good coefficients $c_Z(t,x,v)$ such that \begin{equation}\label{eq:X} (1+t+r)X_i=2z_{0i} \partial_t +\sum_{Z \in \mathbb{K}} c_Z(t,x,v) Z. \end{equation} By a slight abuse of notation, we will write $\mathcal{L}_{X_i}(F)$ for $\mathcal{L}_{\partial_i}(F)+\frac{v^i}{v^0} \mathcal{L}_{\partial_t}(F)$. We are now interested in the compatibility of these extra decay with the Lie derivative of a $2$-form and its null decomposition. \begin{Pro}\label{ExtradecayLie} Let $G$ be a sufficiently regular $2$-form. Then, with $z=t\frac{v^i}{v^0}-x^i$ if $X=X_i$ and $\zeta \in \{ \alpha, \underline{\alpha}, \rho, \sigma \}$, we have \begin{eqnarray}
\left| \mathcal{L}_{\partial}(G) \right| & \lesssim & \frac{1}{\tau_-} \sum_{Z \in \mathbb{K} } \left|\nabla_Z G \right| \hspace{2mm} \lesssim \hspace{2mm} \frac{1}{\tau_-} \sum_{ |\gamma| \leq 1 } \left| \mathcal{L}_{Z^{\gamma}}(G) \right| , \label{eq:goodlie} \\
\left| \mathcal{L}_{X}(G) \right| & \lesssim & \frac{1}{\tau_+} \left( |z| | \nabla_{\partial_t} G|+ \sum_{Z \in \mathbb{K}} \left|\nabla_Z G \right| \right), \label{eq:Xdecay} \\
\tau_-\left| \nabla_{\underline{L}} \zeta \right|+\tau_+\left| \nabla_L \zeta \right|+(1+r) \left| \slashed{\nabla} \zeta \right| & \lesssim & \sum_{|\gamma| \leq 1 } \left| \zeta \left( \mathcal{L}_{Z^{\gamma}}(G) \right) \right|, \label{eq:zeta} \\
\left| \zeta \left( \mathcal{L}_{\partial} (G) \right) \right| & \lesssim & \sum_{|\gamma| \leq 1 } \frac{1}{\tau_-} \left| \zeta \left( \mathcal{L}_{Z^{\gamma}}(G) \right) \right|+\frac{1}{\tau_+} \left| \mathcal{L}_{Z^{\gamma}}(G) \right|. \label{eq:zeta2} \end{eqnarray} \end{Pro} \begin{proof} To obtain the first two identities, use Lemma \ref{goodderiv} as well as \eqref{eq:X} and then remark that if $\Gamma$ is a translation or an homogeneous vector field,
$$ |\nabla_{\Gamma}(G)| \lesssim \left| \mathcal{L}_{\Gamma}(G) \right|+|G|.$$ For \eqref{eq:zeta}, we refer to Lemma $D.2$ of \cite{massless}. Finally, the last inequality comes from \eqref{eq:goodlie} if $2t \leq \max(r,1)$ and from $$\partial_i=\frac{\Omega_{0i}}{t}-\frac{x^i}{2t} L-\frac{x^i}{2t} \underline{L} \hspace{1cm} \text{and} \hspace{1cm} \eqref{eq:zeta} \hspace{6mm} \text{if} \hspace{3mm} 2t \geq \max(r,1).$$ \end{proof}
\begin{Rq}
We do not have, for instance, $\left| \rho \left( \mathcal{L}_{\partial_k} (G) \right) \right| \lesssim \sum_{|\gamma| \leq 1} \tau_-^{-1} \left| \rho \left( \mathcal{L}_{Z^{\gamma}} (G) \right) \right|$, for $1 \leq k \leq 3$. \end{Rq} \begin{Rq}
If $G$ solves the Maxwell equations $\nabla^{\mu} G_{\mu \nu} = J_{\nu}$ and $\nabla^{\mu} {}^* \! G_{\mu \nu} =0$, a better estimate can be obtained on $\alpha( \mathcal{L}_{\partial} (G) )$. Indeed, as $|\nabla_{\partial} \alpha | \leq | \nabla_{L} \alpha |+ |\underline{L} \alpha |+ | \slashed{\nabla} \alpha|$, \eqref{eq:zeta} and Lemma \ref{maxwellbis} gives us,
$$\forall \hspace{0.5mm} |x| \geq 1+\frac{t}{2}, \hspace{0.6cm} |\alpha( \mathcal{L}_{\partial} (G) ) |(t,x) \lesssim |J_A|+ \frac{1}{\tau_+} \sum_{|\gamma| \leq 1} \Big( |\alpha ( \mathcal{L}_{Z^{\gamma}} (G) ) |(t,x)+|\sigma ( \mathcal{L}_{Z^{\gamma}} (G) ) |(t,x)+|\rho ( \mathcal{L}_{Z^{\gamma}} (G) ) |(t,x) \Big).$$ We make the choice to work with \eqref{eq:zeta2} since it does not directly require a bound on the source term of the Maxwell equation, which lightens the proof of Theorem \ref{theorem} (otherwise we would have, among others, to consider more bootstrap assumptions). \end{Rq} \subsection{Modified vector fields and the first order commutation formula}
We start this section with the following commutation formula and we refer to Lemma $2.8$ of \cite{massless} for a proof\footnote{Note that a similar result is proved in Lemma \ref{calculF} below.}.
\begin{Lem}\label{basiccomuf} If $\widehat{Z} \in \widehat{\mathbb{P}}_0 \setminus \{ S \}$, then $$[T_F,\widehat{Z}]( f) = -\mathcal{L}_{Z}(F)(v,\nabla_v f) \hspace{8mm} \text{and} \hspace{8mm} [T_F,S]( f) = F(v,\nabla_v f)-\mathcal{L}_{S}(F)(v,\nabla_v f).$$ \end{Lem} In order to estimate quantities such as $\mathcal{L}_{Z}(F)(v,\nabla_v f)$, we rewrite $\nabla_v f$ in terms of the commutation vector fields (i.e. the elements of $\widehat{\mathbb{P}}_0$). Schematically, if we neglect the null structure of the system, we have, since $v^0\partial_{v^i}= \widehat{\Omega}_{0i}-t\partial_i-x^i\partial_t$, \begin{eqnarray}
\nonumber \left| \mathcal{L}_{Z}(F)(v,\nabla_v f) \right| & \lesssim & v^0\left| \mathcal{L}_{Z}(F) \right| |\partial_{v} f | \\ \nonumber
& \sim & \tau_+ \left| \mathcal{L}_{Z}(F) \right| |\partial_{t,x} f |+\text{l.o.t.}, \end{eqnarray} so that the $v$ derivatives engender a $\tau_+$-loss. The modified vector fields, constructed below, will allow us to absorb the worst terms in the commuted equations. \begin{Def}\label{defphi} Let $\mathbb{Y}_0$ be the set of vector fields defined by $$\mathbb{Y}_0:=\{ \widehat{Z}+\Phi_{\widehat{Z}}^j X_j \hspace{2mm} / \hspace{2mm} \widehat{Z} \in \widehat{\mathbb{P}}_0 \setminus \mathbb{T} \},$$ where $\Phi_{\widehat{Z}}^j : [0,T] \times \mathbb{R}^n_x \times \mathbb{R}^n_v $ are smooth functions which will be specified below and the $X_j$ are defined in \eqref{eq:defXi}. We will denote $\widehat{\Omega}_{0k}+\Phi_{\widehat{\Omega}_{0k}}^j X_j$ by $Y_{0k}$ and, more generally, $\widehat{Z}+\Phi_{\widehat{Z}}^j X_j$ by $Y_{\widehat{Z}}$. We also introduce the sets $$\mathbb{Y} := \mathbb{Y}_0 \cup \mathbb{T} \hspace{8mm} \text{and} \hspace{8mm} \mathbb{Y}_X:= \mathbb{Y} \cup \{ X_1,X_2,X_3 \}.$$
We consider an ordering on $\mathbb{Y}$ and $\mathbb{Y}_X$ compatible with $\widehat{\mathbb{P}}_0$ in the sense that if $\mathbb{Y} = \{ Y^i \hspace{1mm} / \hspace{1mm} 1 \leq i \leq |\mathbb{Y}| \}$, then $Y^i=\widehat{Z}^i+\Phi^k_{\widehat{Z}^i}X_k$ or $Y^i=\partial_{\mu}=\widehat{Z}^i$. We suppose moreover that $X_j$ is the $(|\mathbb{Y}|+j)^{th}$ element of $\mathbb{Y}_X$. Most of the time, for a vector field $Y \in \mathbb{Y}_0$, we will simply write $Y=\widehat{Z}+\Phi X$.
Let $\widehat{Z} \in \widehat{\mathbb{P}}_0 \setminus \{S \}$ and $1 \leq k \leq 3$. $\Phi_{\widehat{Z}}^k$ and $ \Phi^k_S$ are defined such as \begin{equation}\label{defPhicoeff} \hspace{-1.5mm} T_F(\Phi^k_{\widehat{Z}} )=-t\frac{v^{\mu}}{v^0}\mathcal{L}_Z(F)_{\mu k}, \hspace{7.5mm} T_F(\Phi^k_S)=t\frac{v^{\mu}}{v^0}\left(F_{\mu k}-\mathcal{L}_S(F)_{\mu k} \right) \hspace{6mm} \text{and} \hspace{6mm} \Phi_{\widehat{Z}}^k(0,.,.)=\Phi_{S}^k(0,.,.)=0. \end{equation} \end{Def} As explained during the introduction, we consider the $X_i$ vector fields rather than translations in view of \eqref{eq:X}. We are then led to compute $[T_F,X_i]$. \begin{Lem}\label{ComuX} Let $1 \leq i \leq 3$. We have $$[T_F,X_i]=-\mathcal{L}_{X_i}(F)(v,\nabla_v )+\frac{v^{\mu}}{v^0} F_{\mu X_i} \partial_t.$$ \end{Lem} \begin{proof} One juste has to notice that $$[T_F,X_i]=\frac{v^i}{v^0}[T_F,\partial_t]+[T_F,\partial_i]+F\left(v,\nabla_v \left( \frac{v^i}{v^0} \right) \right) \partial_t$$ and $v^{\mu} v^j F_{\mu j} =-v^{\mu} v^0 F_{\mu 0}$, as $F$ is antisymmetric. \end{proof} Finally, we study the commutator between the transport operator and these modified vector fields. The following relation, \begin{equation}\label{eq:vderiv} \partial_{v^i}=\frac{1}{v^0} \left( Y_{0i}-\Phi^j_{\widehat{\Omega}_{0i}} X_j-t X_i+ z_{0i}\partial_t \right), \end{equation} will be useful to express the $v$ derivatives in terms of the commutation vector fields \begin{Pro}\label{Comfirst} Let $Y \in \mathbb{Y}_0 \backslash \{ Y_S \}$. We have, using \eqref{defPhicoeff} \begin{eqnarray} \nonumber [T_F,Y] & = & -\frac{v^{\mu}}{v^0}{\mathcal{L}_Z(F)_{\mu}}^j \left(Y_{0j}-\Phi^k_{\widehat{\Omega}_{0j}} X_k+z_{0j}\partial_t \right) -\Phi^j_{\widehat{Z}}\mathcal{L}_{X_j}(F)(v,\nabla_v )+\Phi^j_{\widehat{Z}}\frac{v^{\mu}}{v^0} F_{\mu X_j} \partial_t, \\ \nonumber [T_F,Y_S] & = & \frac{v^{\mu}}{v^0}\left({F_{\mu}}^j-{\mathcal{L}_S(F)_{\mu}}^j \right) \left(Y_{0j}-\Phi^k_{\widehat{\Omega}_{0j}} X_k+z_{0j}\partial_t \right)-\Phi^j_{S}\mathcal{L}_{X_j}(F)(v,\nabla_v ) +\Phi^j_{S}\frac{v^{\mu}}{v^0} F_{\mu X_j} \partial_t. \end{eqnarray} \end{Pro} \begin{proof} We only treat the case $Y \in \mathbb{Y}_0 \setminus \{ Y_S \}$ (the computations are similar for $Y_S$). Using Lemmas \ref{basiccomuf} and \ref{ComuX} as well as \eqref{eq:vderiv}, we have \begin{eqnarray} \nonumber [T_F,Y] & = & [T_F,\widehat{Z}]+[T_F,\Phi^j_{\widehat{Z}} X_j] \\ \nonumber & = & -\mathcal{L}_Z(F)(v,\nabla_v )+T_F(\Phi^j_{\widehat{Z}} ) X_j +\Phi^j_{\widehat{Z}} [T_F,X_j]. \\ \nonumber & = & -\mathcal{L}_Z(F)(v,\nabla_v )+T_F(\Phi^j_{\widehat{Z}} )X_j -\Phi^j_{\widehat{Z}}\mathcal{L}_{X_j}(F)(v,\nabla_v )+\Phi^j_{\widehat{Z}}\frac{v^{\mu}}{v^0} F_{\mu X_j} \partial_t \\ \nonumber & = & -\frac{v^{\mu}}{v^0}{\mathcal{L}_Z(F)_{\mu}}^j \left(Y_{0j}-\Phi^k_{\widehat{\Omega}_{0j}} X_k+z_{0j}\partial_t \right)+ \left(t\frac{v^{\mu}}{v^0}{\mathcal{L}_Z(F)_{\mu}}^j+T_F(\Phi^j_{\widehat{Z}} ) \right) X_j \\ \nonumber & & -\Phi^j_{\widehat{Z}}\mathcal{L}_{X_j}(F)(v,\nabla_v )+\Phi^j_{\widehat{Z}}\frac{v^{\mu}}{v^0} F_{\mu X_j} \partial_t. \end{eqnarray} To conclude, recall from \eqref{defPhicoeff} that $t\frac{v^{\mu}}{v^0}{\mathcal{L}_Z(F)_{\mu}}^j+T_F(\Phi^j_{\widehat{Z}} )=0$. \end{proof} \begin{Rq}
As we will have $|\Phi| \lesssim \log^2(1+\tau_+)$, a good control on $z_{0j} \partial_t f$ and in view of the improved decay given by $X_j$ (see Proposition \ref{ExtradecayLie}), it holds schematically
$$ \left| [T_F,Y](f) \right| \lesssim \log^2 (1+\tau_+) \left| \mathcal{L}_Z(F)\right| |Y f|,$$
which is much better than $ \left| [T_F,\widehat{Z}](f) \right| \lesssim \tau_+ \left| \mathcal{L}_Z(F) \right| |\partial_{t,x} f|$. \end{Rq} Let us introduce some notations for the presentation of the higher order commutation formula. \begin{Def}
Let $Y^{\beta} \in \mathbb{Y}^{|\beta|}$. We denote by $\beta_T$ the number of translations composing $Y^{\beta}$ and by $\beta_P$ the number of modified vector fields (the elements of $\mathbb{Y}_0$). Note that $\beta_T$ denotes also the number of translations composing $\widehat{Z}^{\beta}$ and $Z^{\beta}$ and $\beta_P$ the number of elements of $\widehat{\mathbb{P}}_0 \setminus \mathbb{T}$ or $\mathbb{K} \setminus \mathbb{T}$. We have
$$|\beta|= \beta_T+\beta_P$$
and, for instance, if $Y^{\beta}=\partial_t Y_1 \partial_3$, $|\beta|=3$, $\beta_T=2$ and $\beta_P=1$. We define similarly $\beta_X$ if $Y^{\beta} \in \mathbb{Y}^{|\beta|}_X$. \end{Def} \begin{Def}\label{Pkp} Let $k=(k_T,k_P) \in \mathbb{N}^2$ and $ p \in \mathbb{N}$. We will denote by $P_{k,p}(\Phi)$ any linear combination of terms such as
$$ \prod_{j=1}^p Y^{\beta_j}(\Phi), \hspace{3mm} \text{with} \hspace{3mm} Y^{\beta_j} \in \mathbb{Y}^{|\beta_j|}, \hspace{3mm} \sum_{j=1}^p |\beta_j| = |k|, \hspace{3mm} \sum_{j=1}^p \left(\beta_j \right)_P = k_P$$
and where $\Phi$ denotes any of the $\Phi$ coefficients. Note that $\sum_{j=1}^p \left(\beta_j \right)_T = k_T$. Finally, if $ \min_{j} |\beta_j| \geq 1$, we will denote $\prod_{j=1}^p Y^{\beta_j}(\Phi)$ by $P_{\beta}(\Phi)$, where $\beta=(\beta_1,...\beta_p)$. \end{Def} \begin{Def} Let $k=(k_T,k_P,k_X) \in \mathbb{N}^3$ and $ p \in \mathbb{N}$. We will denote by $P^X_{k,p}(\Phi)$ any linear combination of terms such as
$$ \prod_{j=1}^p Y^{\beta_j}(\Phi), \hspace{3mm} \text{with} \hspace{3mm} Y^{\beta_j} \in \mathbb{Y}^{|\beta_j|}, \hspace{3mm} \sum_{j=1}^p |\beta_j| = |k|, \hspace{3mm} \sum_{j=1}^p \left(\beta_j \right)_P = k_P, \hspace{3mm} \sum_{j=1}^p \left(\beta_j \right)_X = k_X \hspace{3mm} \text{and} \hspace{3mm} \min_{1 \leq j \leq p} \left( \beta_j \right)_X \geq 1.$$ We will also denote $ \prod_{j=1}^p Y^{\beta_j}(\Phi)$ by $P^X_{\beta}(\Phi)$. \end{Def} \begin{Rq}
For convenience, if $p=0$, we will take $P_{k,p}(\Phi)=1$. Similarly, if $|\beta|=0$, we will take $P_{\beta}(\Phi)=P^X_{\beta}(\Phi)=1$. \end{Rq} In view of presenting the higher order commutation formulas, let us gather the source terms in different categories. \begin{Pro}\label{Comufirst} Let $Y \in \mathbb{Y} \setminus \mathbb{T}$. In what follows, $0 \leq \nu \leq 3$. The commutator $[T_F,Y]$ can be written as a linear combination, with $c(v)$ coefficients, of terms such as \begin{itemize}
\item $ \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \Gamma $, where $|\gamma| \leq 1$ and $\Gamma \in \mathbb{Y}_0$.
\item $\Phi \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \partial_{t,x} $, where $|\gamma| \leq 1$.
\item $z \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \partial_{t,x} $, where $|\gamma| \leq 1$ and $z \in \mathbf{k}_1$. \item $\Phi \mathcal{L}_{X}(F)(v,\nabla_v )$. \end{itemize} \end{Pro} Finally, let us adapt Lemma \ref{lift} to our modified vector fields. \begin{Lem}\label{lift2}
Let $f : [0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v \rightarrow \mathbb{R} $ be a sufficiently regular function and suppose that for all $|\beta| \leq 1$, $|Y^{\beta} \Phi| \lesssim \log^{\frac{7}{2}}(1+\tau_+)$. Then, we have, almost everywhere,
$$\forall \hspace{0.5mm} Z \in \mathbb{K}, \hspace{8mm} \left|Z\left( \int_{v \in \mathbb{R}^3 } |f| dv \right) \right| \lesssim \sum_{ \begin{subarray}{} Y \in \mathbb{Y} \\ z \in \mathbf{k}_1 \end{subarray}} \int_{v \in \mathbb{R}^3 } \left( |Yf|+|f| +|X(\Phi)f| + \frac{ \log^7 (1 + \tau_+)}{\tau_+} \left(|z \partial_t f|+|zf| \right) \right) dv .$$ \end{Lem}
\begin{proof} Consider, for instance, the rotation $\Omega_{12}$. We have by integration by parts, as $\Omega_{12}=\widehat{\Omega}_{12}-v^{1} \partial_{v^2}+v^2 \partial_{v^1}$,
$$ \Omega_{12}\left( \int_{v \in \mathbb{R}^3 } |f| dv \right) = \int_{v \in \mathbb{R}^3} \widehat{\Omega}_{12} (|f|) dv -\int_{v \in \mathbb{R}^3} \left( v^1\partial_{v^2} -v^2 \partial_{v^1} \right)(|f|) dv= \int_{v \in \mathbb{R}^3} \widehat{\Omega}_{12} (|f|) dv.$$
This proves Lemma \ref{lift} for $\Omega_{12}$ since $| \widehat{\Omega}_{12} (|f|) |= | \frac{f}{|f|}\widehat{\Omega}_{12} (f) | \leq |\widehat{\Omega}_{12} (f)|$. On the other hand, \begin{eqnarray}\label{45:eq}
\int_{v \in \mathbb{R}^3} \widehat{\Omega}_{12} (|f|) dv & = & \int_{v \in \mathbb{R}^3} \left( \widehat{\Omega}_{12} +\Phi_{\widehat{\Omega}_{12}}^k X_k -\Phi_{\widehat{\Omega}_{12}}^k X_k \right)(|f|) dv \\
& = & \int_{v \in \mathbb{R}^3} \frac{f}{|f|} Y_{\widehat{\Omega}_{12}} f dv+\int_{v \in \mathbb{R}^3} X_k \left( \Phi^k_{\widehat{\Omega}_{12}} \right) |f| dv -\int_{v \in \mathbb{R}^3} X_k \left( \Phi^k_{\widehat{\Omega}_{12}} |f| \right) dv \label{eq:lif33}. \end{eqnarray} \eqref{45:eq} implies the result if $t+r \leq 1$. Otherwise, if $t \geq r$, note that by \eqref{eq:Xi}, \begin{eqnarray}
\nonumber \int_{v \in \mathbb{R}^3} X_k \left( \Phi^k_{\widehat{\Omega}_{12}} |f| \right) dv & = & \frac{1}{t}\int_{v \in \mathbb{R}^3} \left( \Omega_{0k}+z_{0k} \partial_t \right) \left( \Phi^k_{\widehat{\Omega}_{12}} |f| \right) dv \\ \nonumber
& = & \frac{1}{t}\int_{v \in \mathbb{R}^3} \left( Y_{0k}-v^0\partial_{v^k}-\Phi^q_{\widehat{\Omega}_{0k}}X_q+z_{0k} \partial_t \right) \left( \Phi^k_{\widehat{\Omega}_{12}} |f| \right)dv \\ \nonumber & = & \frac{1}{t}\int_{v \in \mathbb{R}^3} \left( Y_{0k}+\frac{v_k}{v^0}-\Phi^q_{\widehat{\Omega}_{0k}}X_q+z_{0k} \partial_t \right) \left( \Phi^k_{\widehat{\Omega}_{12}} |f| \right)dv . \end{eqnarray}
Consequently, in view of the bounds on $Y^{\beta} \Phi$ for $|\beta| \leq 1$,
$$ \left| \int_{v \in \mathbb{R}^3} X_k \left( \Phi^k_{\widehat{\Omega}_{12}} |f| \right) dv \right| \lesssim \sum_{Y \in \mathbb{Y}} \sum_{z \in \mathbf{k}_1} \int_{v \in \mathbb{R}^3} |Yf|+|f|+\frac{|z| \log^7 (1+t)}{t} \left( |\partial_t f|+|f| \right) dv ,$$ and it remains to combine it with \eqref{eq:lif33}. When $t \leq r$, one can use $rX_k=tX_k+(r-t)X_k$ and Lemma \ref{goodderiv}. \end{proof}
\begin{Rq}\label{lift3}
If moreover $|\Phi| \lesssim \log^2(1+\tau_+)$, one can prove similarly that, for $Z \in \mathbb{K}$, $z \in \mathbf{k}_1$ and $j \in \mathbb{N}^*$,
$$ \left|Z \left( \int_{v } |z^jf| dv \right) \right| \lesssim \hspace{1mm} j \hspace{-1mm} \sum_{ \begin{subarray}{} |\xi|+|\beta| \leq 1 \\ \hspace{2.5mm} w \in \mathbf{k}_1 \end{subarray}} \int_{v } |w^jP^X_{\xi}(\Phi) Y^{\beta} f|+\log^2(3+t)|w^{j-1}f|+\frac{ \log^7(1+\tau_+)|w|^{j+1}}{\tau_+} \left(| \partial_t f|+|f| \right) dv .$$ To prove this inequality, apply Lemma \ref{lift2} to $z^j f$ and use the two following properties,
$$|Y(z^j)| \leq |\widehat{Z}(z^j)|+|\Phi X(z^j)| \lesssim j \left( \sum_{w \in \mathbf{k}_1} |w|^j+ \log^2(1+\tau_+) |z|^{j-1} \right) \hspace{6mm} \text{and} \hspace{6mm} \sum_{w \in \mathbf{k}_1} |w||z|^j \lesssim \sum_{w \in \mathbf{k}_1} |w|^{j+1}.$$ It remains to apply Remark \ref{rqweights1} in order to get
$$\forall \hspace{0.5mm} |x| \geq 1+2t, \hspace{1cm} \log^2(1+\tau_+) |z|^{j-1} \lesssim \frac{\log^2(3+r)}{r} \sum_{w \in \mathbf{k}_1} |w z^{j-1} | \lesssim \sum_{w \in \mathbf{k}_1} |w^j|$$
and to note that $\log (1+\tau_+) \lesssim \log (3+t)$ if $|x| \leq 1+2t$. \end{Rq}
\subsection{Higher order commutation formula}
The following lemma will be useful for upcoming computations. \begin{Lem}\label{calculF} Let $G$ be a sufficiently regular $2$-form and $g$ a sufficiently regular function defined respectively on $[0,T[ \times \mathbb{R}^3$ and $[0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v$. Let also $Y=\widehat{Z}+\Phi X \in \mathbb{Y}_0$ and $\nu \in \llbracket 0,3 \rrbracket$. We have, with $n_Z=0$ is $Z \in \mathbb{P}$ and $n_S=-1$, \begin{eqnarray} \nonumber Y \left( v^{\mu}G_{\mu \nu} \right) \hspace{-1.5mm} & = & \hspace{-1.5mm} v^{\mu}\mathcal{L}_{Z }(G)_{\mu \nu}+n_Zv^{\mu} G_{\mu \nu} +\Phi v^{\mu}\mathcal{L}_{X }(G)_{\mu \nu}+v^{\mu}G_{\mu [Z,\partial_{\nu}]}, \\ \nonumber Y \left( G \left( v , \nabla_v g \right) \right) \hspace{-1.5mm} & = & \hspace{-1.5mm} \mathcal{L}_Z(G) \left( v , \nabla_v g \right)+2n_ZG \left( v ,\nabla_v g \right)+\Phi \mathcal{L}_X(G) \left( v , \nabla_v g \right)+G \left( v , \nabla_v \widehat{Z} g \right)+c(v)\Phi G \left( v , \nabla_v \partial g \right) . \end{eqnarray} For $i \in \llbracket 1,3 \rrbracket$, $ Y \left( v^{\mu} \mathcal{L}_{X_i}(G)_{\mu \nu} \right)$ can be written as a linear combination, with $c(v)$ coefficients, of terms of the form
$$ \Phi^p v^{\mu} \mathcal{L}_{X Z^{\gamma} }(G)_{\mu \theta}, \hspace{3mm} \text{with} \hspace{3mm} 0 \leq \theta \leq 3 \hspace{3mm} \text{and} \hspace{3mm} \max(p,|\gamma|) \leq 1.$$ Finally, $Y \left( \mathcal{L}_{X_i}(G) \left( v , \nabla_v g \right) \right)$ can be written as a linear combination, with $c(v)$ coefficients, of terms of the form
$$ \Phi^p \mathcal{L}_{X Z^{\gamma} }(G) \left( v, \nabla \left( \widehat{Z}^{\kappa} g \right) \right), \hspace{3mm} \text{with} \hspace{3mm} \max(|\gamma|+|\kappa|,p+\kappa_P) \leq 1.$$ \end{Lem}
\begin{proof} Let $Z_v=\widehat{Z}-Z$ so that $Y=Z+Z_v+\Phi X$. We prove the second and the fourth properties (the first and the third ones are easier). We have \begin{eqnarray} \nonumber Y \left( G \left( v , \nabla_v g \right) \right) & = & \mathcal{L}_Z(G) \left( v, \nabla_v g \right)+G \left( [Z,v], \nabla_v g \right)+G \left( v,[Z,\nabla_v g] \right)+G \left( Z_v(v), \nabla_v g\right)+G \left( v, Z_v \left(\nabla_v g \right) \right) \\ \nonumber & &+\Phi \mathcal{L}_{X}(G) \left( v, \nabla_v g \right)+c(v) \Phi G \left( v , \nabla_v \partial g \right). \end{eqnarray} Note now that \begin{itemize} \item $S_v=0$ and $[S,v]=-v$, \item $[Z,v]=-Z_v(v)$ if $Z \in \mathbb{P}$. \end{itemize} The second identity is then implied by \begin{itemize} \item $[\partial, \nabla_v g]=\nabla_v \partial(g)$ and $[S, \nabla_v g ]= \nabla_v S(g)-\nabla_v g$. \item $[Z, \nabla_v g]+Z_v \left( \nabla_v g \right)= \nabla_v \widehat{Z}(g)$ if $Z \in \mathbb{O}$. \item $[\Omega_{0i}, \nabla_v g]+(\Omega_{0i})_v \left( \nabla_v g \right)= \nabla_v \widehat{Z}(g)-\frac{v}{v^0} \partial_{v^i}$ and $G(v,v)=0$ as $G$ is a $2$-form. \end{itemize} We now prove the fourth identity. We treat the case $Y=\widehat{Z}+\Phi X \in \mathbb{Y}_0 \setminus \{ Y_S \}$ as the computations are similar for $Y_S$. On the one hand, since $[\partial,X_i]=0$ and $X_k= \partial_k+\frac{v^k}{v^0}\partial_t$, one can easily check that $\Phi X_k \left( \mathcal{L}_{X_i}(G) \left( v , \nabla_v g \right) \right)$ gives four terms of the expected form. On the other hand, $$\widehat{Z} \left( \mathcal{L}_{X_i}(G) \left( v , \nabla_v g \right) \right)=\widehat{Z} \left( \mathcal{L}_{\partial_i}(G) \left( v , \nabla_v g \right) \right) +\widehat{Z} \left(\frac{v^i}{v^0} \mathcal{L}_{\partial_t}(G) \left( v , \nabla_v g \right) \right).$$ Applying the second equality of this Lemma to $\mathcal{L}_{\partial}(G)$, $g$ and $\widehat{Z}$ (which is equal to $Y$ when $\Phi=0$), we have \begin{eqnarray} \nonumber \widehat{Z} \left( \mathcal{L}_{\partial_i}(G) \left( v , \nabla_v g \right) \right) & = & \mathcal{L}_{Z \partial_i}(G) \left( v , \nabla_v g \right) +\mathcal{L}_{\partial_i}(G) \left( v , \nabla_v \widehat{Z} g \right) \\ \nonumber \widehat{Z} \left(\frac{v^i}{v^0} \mathcal{L}_{\partial_t}(G) \left( v , \nabla_v g \right) \right) & = & \widehat{Z} \left( \frac{v^i}{v^0} \right) \mathcal{L}_{\partial_t}(G) \left( v , \nabla_v g \right)+\frac{v^i}{v^0}\mathcal{L}_{Z \partial_t}(G) \left( v , \nabla_v g \right) +\frac{v^i}{v^0}\mathcal{L}_{\partial_t}(G) \left( v , \nabla_v \widehat{Z} g \right) \end{eqnarray} The sum of the last terms of these two identities is of the expected form. The same holds for the sum of the three other terms since \begin{eqnarray} \nonumber [\Omega_{0j},\partial_i]+\frac{v^i}{v^0}[\Omega_{0j},\partial_t]+v^0 \partial_{v^j}\left( \frac{v^i}{v^0} \right) \partial_t \hspace{-2mm} & = & \hspace{-2mm} -\delta_{j}^{i} \partial_t-\frac{v^i}{v^0}\partial_j-\frac{v^i v^j}{(v^0)^2} \partial_t+\delta_{j}^{i} \partial_t= -\frac{v^i}{v^0} X_j=c(v) X_j, \\ \nonumber [\Omega_{kj},\partial_i]+\frac{v^i}{v^0}[\Omega_{kj},\partial_t]+\left( v^k \partial_{v^j}-v^j \partial_{v^k} \right) \left( \frac{v^i}{v^0} \right) \partial_t \hspace{-2mm} & = & \hspace{-2mm} \delta_{j}^i \partial_k-\delta_k^i \partial_j+\left(\frac{v^k \delta_j^i-v^j \delta_k^i}{v^0} \right)\partial_t= \delta_j^i X_k -\delta_k^i X_j,\\ \nonumber [S,\partial_i]+\frac{v^i}{v^0}[S,\partial_t] \hspace{-2mm} & = & \hspace{-2mm} - \partial_i-\frac{v^i}{v^0}\partial_t=- X_i . \end{eqnarray} \end{proof}
We are now ready to present the higher order commutation formula. To lighten its presentation and facilitate its future usage, we introduce $\mathbb{G}:= \widehat{\mathbb{P}}_0 \cup \mathbb{Y}_0$, on which we consider an ordering. A combination of vector fields of $\mathbb{G}$ will always be denoted by $\Gamma^{\sigma}$ and we will also denote by $\sigma_T$ its number of translations and by $\sigma_P= |\sigma|-\sigma_T$ its number of homogeneous vector fields. In Lemma \ref{GammatoYLem} below, we will express $\Gamma^{\sigma}$ in terms of $\Phi$ coefficients and $\mathbb{Y}$ vector fields.
\begin{Pro}\label{ComuVlasov} Let $\beta$ be a multi-index. In what follows, $\nu \in \llbracket 0 , 3 \rrbracket$. The commutator $[T_F,Y^{\beta}]$ can be written as a linear combination, with $c(v)$ coefficients, of the following terms. \begin{itemize} \item \begin{equation}\label{eq:com1}
z^d P_{k,p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{ \mu \nu} Y^{\sigma}, \tag{type 1-$\beta$}
\end{equation}
where \hspace{2mm} $z \in \mathbf{k}_1$, \hspace{2mm} $d \in \{ 0,1 \}$, \hspace{2mm} $|\sigma| \geq 1$ \hspace{2mm} $\max ( |\gamma|, |k|+|\gamma|, |k|+|\sigma| ) \leq |\beta|$, \hspace{2mm} $|k|+|\gamma|+|\sigma| \leq |\beta|+1$ \hspace{2mm} and \hspace{2mm} $p+k_P+\sigma_P+d \leq \beta_P$. Note also that, as \hspace{2mm} $|\sigma| \geq 1$, \hspace{2mm} $|k| \leq |\beta|- 1$. \item \begin{equation}\label{eq:com2} P_{k,p}(\Phi) \mathcal{L}_{X Z^{\gamma_0}}(F) \left( v, \nabla_v \Gamma^{\sigma} \right), \tag{type 2-$\beta$} \end{equation}
where \hspace{2mm} $|k|+|\gamma_0|+|\sigma| \leq |\beta|-1$, \hspace{2mm} $p+k_P+\sigma_P \leq \beta_P$ \hspace{2mm} and \hspace{2mm} $p \geq 1$. \item \begin{equation}\label{eq:com4}
P_{k,p}(\Phi) \mathcal{L}_{ \partial Z^{\gamma_0}}(F) \left( v,\nabla_v \Gamma^{\sigma} \right), \tag{type 3-$\beta$}
\end{equation}
where \hspace{2mm} $|k|+|\gamma_0|+|\sigma| \leq |\beta|-1$, \hspace{2mm} $p+|\gamma_0| \leq |\beta|-1$ \hspace{2mm} and \hspace{2mm} $p+k_P+\sigma_P \leq \beta_P$.
\end{itemize} \end{Pro}
\begin{proof}
The result follows from an induction on $|\beta|$, Proposition \ref{Comufirst} (which treats the case $|\beta| =1$) and $$[T_F,YY^{\beta_0}]=Y[T_F,Y^{\beta_0}]+[T_F,Y]Y^{\beta_0}.$$
Let $ Q \in \mathbb{N}$ and suppose that the commutation formula holds for all $|\beta_0| \leq Q$. We then fix a multi-index $|\beta_0|=Q$, consider $Y \in Y$ and denote the multi-index corresponding to $YY^{\beta_0}$ by $\beta$. Then, $|\beta|=|\beta_0|+1$.
Suppose first that $Y=\partial$ is a translation so that $\beta_P=(\beta_0)_P$. Then, using Lemma \ref{basiccomuf}, we have $$ [T_F,\partial]Y^{\beta_0} = -\mathcal{L}_{\partial}(F)(v,\nabla_v Y^{\beta_0}), $$
which is a term of \eqref{eq:com4} as $|\beta_0| = |\beta|-1$ and $(\beta_0)_P=\beta_P$. Using the induction hypothesis, $\partial[T_F,Y^{\beta_0}]$ can be written as a linear combination with good coefficients $c(v)$ of terms of the form\footnote{We do not mention the $c(v)$ coefficients here since $\partial \left( c(v) \right) =0$.} \begin{itemize}
\item $ \partial \left( z^d P_{k,p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{ \mu \nu} Y^{\sigma} \right) $, with $z \in \mathbf{k}_1$, $d \in \{0,1 \}$, $|\sigma| \geq 1$, $\max ( |\gamma|, |k|+|\gamma|, |k|+ |\sigma| ) \leq |\beta_0|$, $|k|+|\gamma|+|\sigma| \leq |\beta_0|+1$ and $p+k_P+\sigma_P+d \leq (\beta_0)_P$. This leads to the sum of the following terms. \begin{itemize} \item $\partial(z^d) P_{k,p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{ \mu \nu} Y^{\sigma}$, which is of \eqref{eq:com1} since $\partial(z)=0$ or $\frac{v^{\lambda}}{v^0}$. \item $z^d P_{(k_T+1,k_P),p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{ \mu \nu} Y^{\sigma}+z^dP_{k,p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{\partial Z^{\gamma}}(F)_{ \mu \nu} Y^{\sigma}+z^dP_{k,p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{ \mu \nu} \partial Y^{\sigma},$ which is the sum of terms of \eqref{eq:com1} (as, namely, $k_P$ does not increase and $(\sigma_0)_P=\sigma_P$ if $Y^{\sigma_0}=\partial Y^{\sigma}$). \end{itemize}
\item $\partial \left( P_{k,p}(\Phi) \mathcal{L}_{ \partial Z^{\gamma_0}}(F) \left( v,\nabla_v \Gamma^{\sigma} \right) \right)$, with $|k|+|\gamma_0|+|\sigma| \leq |\beta_0|-1$, $p+|\gamma_0| \leq |\beta_0|-1$ and $p+k_P+\sigma_P \leq (\beta_0)_P$. We then obtain $$ P_{(k_T+1,k_P),p}(\Phi) \mathcal{L}_{ \partial Z^{\gamma_0}}(F) \hspace{-0.2mm} \left( v,\nabla_v \Gamma^{\sigma} \right), \hspace{2.3mm} P_{k,p}(\Phi)\mathcal{L}_{ \partial \partial Z^{\gamma_0}}(F) \hspace{-0.2mm} \left( v,\nabla_v \Gamma^{\sigma} \right) \hspace{2.3mm} \text{and} \hspace{2.3mm} P_{k,p}(\Phi) \mathcal{L}_{ \partial Z^{\gamma_0}}(F) \hspace{-0.2mm} \left( v,\nabla_v \partial \Gamma^{\sigma} \right),$$
which are all of \eqref{eq:com4} since $|k|+|\gamma_0|+|\sigma|+1 \leq |\beta_0|=|\beta|-1$, $p+|\gamma_0|+1 \leq |\beta|-1$ and, if $\Gamma^{\overline{\sigma}} = \partial \Gamma^{\sigma}$, $p+k_P+\overline{\sigma}_P=p+k_P+\sigma_P \leq \left( \beta_0 \right)_P = \beta_P$.
\item $\partial \left( P_{k,p}(\Phi) \mathcal{L}_{ X Z^{\gamma_0}}(F) \left( v,\nabla_v \Gamma^{\sigma} \right) \right)$, with $|k|+|\gamma_0|+|\sigma| \leq |\beta_0|-1$, $p+k_P+\sigma_P \leq (\beta_0)_P$ and $p \geq1$. We then obtain, as $[\partial,X]=0$, $$ P_{(k_T+1,k_P),p}(\Phi) \mathcal{L}_{ X Z^{\gamma_0}}(F) \hspace{-0.2mm} \left( v,\nabla_v \Gamma^{\sigma} \right) \hspace{-0.3mm} , \hspace{1.7mm} P_{k,p}(\Phi)\mathcal{L}_{X \partial Z^{\gamma_0}}(F) \hspace{-0.2mm} \left( v,\nabla_v \Gamma^{\sigma} \right) \hspace{1.7mm} \text{and} \hspace{1.7mm} P_{k,p}(\Phi)\mathcal{L}_{ X Z^{\gamma_0}}(F) \hspace{-0.2mm} \left( v,\nabla_v \partial \Gamma^{\sigma} \right) \hspace{-0.3mm} ,$$
which are all of \eqref{eq:com2} since, for instance, $|k|+|\gamma_0|+|\sigma|+1 \leq |\beta_0| = |\beta|-1$. \end{itemize} We now suppose that $Y \in \mathbb{Y} \setminus \mathbb{T}$, so that $\beta_P = (\beta_0)_P+1$. We will write schematically that $Y=\widehat{Z}+\Phi X$. Using Proposition \ref{Comufirst}, we have that $[T_F,Y]Y^{\beta_0}$ can be written as a linear combination, with $c(v)$ coefficients, of the following terms. \begin{itemize}
\item $ \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \Gamma Y^{\beta_0} $, where $|\gamma| \leq 1$ and $\Gamma \in \mathbb{Y}$, which is of \eqref{eq:com1}.
\item $\Phi^{1-d}z^d \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \partial Y^{\beta_0}$, where $|\gamma| \leq 1$, $d \in \{0,1 \}$ and $z \in \mathbf{k}_1$, which is of \eqref{eq:com1} since, if $\xi$ is the multi-index corresponding to $\partial Y^{\beta_0}$, $\xi_P = (\beta_0)_P < \beta_P$.
\item $ \Phi \mathcal{L}_{X}(F)(v,\nabla_v Y^{\beta_0} )$, which is of \eqref{eq:com2} since $|\beta_0| \leq |\beta|-1$ and $1+(\beta_0)_P \leq \beta_P$. \end{itemize} It then remains to compute $Y[T_F,Y^{\beta_0}]$. Using the induction hypothesis, it can be written as a linear combination of terms of the form \begin{itemize}
\item $ Y \left(c(v) z^d P_{k,p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} Y^{\sigma} \right),$ with $z \in \mathbf{k}_1$, $d \in \{0,1 \}$, $|\sigma| \geq 1$, $\max ( |\gamma|,|k|+|\gamma|, |k|+ |\sigma| ) \leq |\beta_0|$, $|k|+|\gamma|+|\sigma| \leq |\beta_0|+1$ and $p+k_P+\sigma_P+d \leq (\beta_0)_P$. It leads to the following error terms. \begin{itemize} \item $ Y\left( \frac{c(v)}{v^0} \right) z^dP_{k,p}(\Phi) v^{\mu}\mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} Y^{\sigma} $, which is of \eqref{eq:com1} since $Y\left( \frac{c(v)}{v^0} \right) = \widehat{Z} \left( \frac{c(v)}{v^0} \right) = \frac{c_0(v)}{v^0} $. \item $c(v)Y \left( z^d \right) P_{k,p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F) Y^{\sigma}$, which is a linear combination of terms of \eqref{eq:com1} since, by Lemma \ref{weights}, $$Y(z)=\widehat{Z}(z)+\Phi^i_{\widehat{Z}} X_i(z)=c_0(v)z+z'+\Phi^i_{\widehat{Z}}c_i(v), \hspace{2mm} \text{where} \hspace{2mm} z' \in \mathbf{k}_1, \hspace{2mm} \text{and} \hspace{2mm} p+1+k_P+\sigma_P+1 \leq \beta_P.$$ \item $c(v)z^d P_{(k_T,k_P+1),p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{ \mu \nu} Y^{\sigma}+c(v)z^d P_{k,p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{ \mu \nu} YY^{\sigma}$, which is the sum of terms of \eqref{eq:com1}, since $p+k_P+\sigma_P+d+1 \leq (\beta_0)_P+1 = \beta_P$.
\item $c(v)z^dP_{k,p+p_0}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\xi}Z^{\gamma}}(F)_{ \mu \theta} Y^{\sigma}$, with $\max (p_0 ,|\xi| ) \leq 1$, which is given by the first identity of Lemma \ref{calculF}. These terms are of \eqref{eq:com1} since $|k|+|\gamma|+|\xi|+|\sigma| \leq |\beta_0|+2 = |\beta|+1$ and $|\gamma|+|\xi| \leq |\beta|$. \end{itemize} For the remaining terms, we suppose for simplicity that $c(v)=1$, as we have just see that $Y \left(c(v) \right)$ is a good coefficient.
\item $ Y \Big( P_{k,p}(\Phi) \mathcal{L}_{ X Z^{\gamma_0}}(F) \left( v , \nabla_v \Gamma^{\sigma} \right) \Big) $, with $|k|+|\gamma_0|+|\sigma| \leq |\beta_0|-1$, $p+k_P+\sigma_P \leq (\beta_0)_P$ and $p \geq 1$. It gives us $$P_{(k_T,k_P+1),p}(\Phi) \mathcal{L}_{X Z^{\gamma_0}}(F) \left( v , \nabla_v \Gamma^{\sigma} \right), $$ which is of \eqref{eq:com2} since, $p+k_P+1+\sigma_P \leq (\beta_0)_P+1=\beta_P$. We also obtain, using the fourth identity of Lemma \ref{calculF},
$$c(v)P_{k,p+p_0}(\Phi)\mathcal{L}_{X Z^{\delta} Z^{\gamma_0}} (F) \left( v , \nabla_v \widehat{Z}^{\xi}\Gamma^{\sigma} \right), \hspace{3mm} \text{with} \hspace{3mm} \max(|\delta|+|\xi|,p_0+ \xi_P) \leq 1.$$
They are all of \eqref{eq:com2} since $|k|+|\gamma_0|+|\delta|+|\sigma|+|\xi| \leq |\beta_0|=|\beta|-1$, $p+p_0+k_P+\sigma_P+\xi_P \leq (\beta_0)_P+1=\beta_P$ and $p+p_0 \geq p \geq 1$.
\item $ Y \Big(P_{k,p}(\Phi) \mathcal{L}_{\partial Z^{\gamma_0}}(F) \left( v , \nabla_v \Gamma^{\sigma} \right) \Big) $, with $|k|+|\gamma_0|+|\sigma| \leq |\beta_0|-1$, $p+|\gamma_0| \leq |\beta_0|-1$ and $p+k_P+\sigma_P \leq (\beta_0)_P$. We obtain \begin{itemize} \item $P_{(k_T,k_P+1),p}(\Phi) \mathcal{L}_{\partial Z^{\gamma_0}}(F) \left( v , \nabla_v \Gamma^{\sigma} \right) $, clearly of \eqref{eq:com4}, \end{itemize} and, using the second identity of Lemma \ref{calculF}, \begin{itemize} \item $ P_{k,p+1}(\Phi)\mathcal{L}_{X \partial Z^{\gamma_0}} (F) \left( v , \nabla_v \Gamma^{\sigma} \right)$, which is of \eqref{eq:com2}, and
$$c(v)P_{k,p+p_0}(\Phi)\mathcal{L}_{Z^{\delta} \partial Z^{\gamma_0}} (F) \left( v , \nabla_v \widehat{Z}^{\xi}\Gamma^{\sigma} \right), \hspace{3mm} \text{with} \hspace{3mm} |\delta|+|\xi| \leq 1, \hspace{3mm} p_0+|\delta| \leq 1 \hspace{3mm} \text{and} \hspace{3mm} p_0+ \xi_P \leq 1.$$
As $p+p_0+|\gamma_0|+|\delta| \leq p+|\gamma_0|+1 \leq |\beta|-1$, $p+p_0+k_P+\sigma_P+\xi_P \leq (\beta_0)_P+1=\beta_P$ and, if $|\delta|=1$, $[Z^{\delta}, \partial ] \in \mathbb{T} \cup \{ 0 \}$, we can conclude that these terms are of \eqref{eq:com4}. \end{itemize} \end{itemize} \end{proof}
\begin{Rq}\label{rqjustifnorm}
To deal with the weight $\tau_+$ in the terms of \eqref{eq:com2} and \eqref{eq:com4} (hidden by the $v$ derivatives), we will take advantage of the extra decay given by the $X$ vector fields or the translations $\partial_{\mu}$ through Proposition \ref{ExtradecayLie}. To deal with the terms of \eqref{eq:com1}, when $d=1$, we will need to control the $L^1$ norm of $\sum_{w \in \mathbf{k}_1} |w|^{q+1}P_{k,p}(\Phi)Y^{\sigma}f$, with $k_P+\sigma_P < \beta_P$, in order to control $\||z|^q Y^{\beta}f\|_{L^1_{x,v}}$. \end{Rq}
As we will need to bound norms such as $\| P_{\xi}(\Phi) Y^{\beta} f \|_{L^1_{x,v}}$, we will apply Proposition \ref{ComuVlasov} to $\Phi$ and we then need to compute the derivatives of $T_F(\Phi)$. This is the purpose of the next proposition. \begin{Pro}\label{sourcePhi}
Let $Y^{\beta} \in \mathbb{Y}^{|\beta|}$ and $Z^{\gamma_1} \in \mathbb{K}^{|\gamma_1|}$ (we will apply the result for $|\gamma_1| \leq 1$). Then, $$ Y^{\beta} \left( t \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma_1}}(F)_{\mu \zeta} \right)$$ can be written as a linear combination, with $c(v)$ coefficients, of the following terms, with $0 \leq \theta, \nu \leq 3$ and $p \leq |\beta|$.
\begin{equation}\label{equa1}
\hspace{-0.5cm} x^{\theta} \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu}, \hspace{18mm} \text{where} \hspace{6mm} |\gamma| \leq |\beta| \hspace{17mm} \text{and} \hspace{6mm} \gamma_T=\beta_T. \tag{family $\beta-1$}
\end{equation} \begin{equation}\label{equa1bis}
\hspace{-0.3cm} P_{k,p}(\Phi)\frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu}, \hspace{10mm} \text{where} \hspace{5mm} |k|+|\gamma| \leq |\beta|-1 \hspace{6mm} \text{and} \hspace{6mm} k_P \leq \beta_P. \tag{family $\beta-2$}
\end{equation}
\begin{equation}\label{equa2}
x^{\theta}P_{k,p}(\Phi) \frac{v^{\mu}}{v^0} \mathcal{L}_{X Z^{\gamma} Z^{\gamma_1} }(F)_{\mu \nu}, \hspace{6mm} \text{where} \hspace{5mm} |k|+|\gamma| \leq |\beta|-1 \hspace{6mm} \text{and} \hspace{6mm} k_P < \beta_P. \tag{family $\beta-3$}
\end{equation} \end{Pro}
\begin{proof}
Let us prove this by induction on $|\beta|$. The result holds for $|\beta|=0$. We then consider $Y^{\beta}=YY^{\beta_0} \in \mathbb{Y}^{|\beta|}$ and we suppose that the Proposition holds for $\beta_0$. Suppose first that $Y= \partial$, so that $\beta_P=(\beta_0)_P$. Using the induction hypothesis, $\partial Y^{\beta_0} \left( t \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma_1}}(F)_{\mu \nu} \right)$ can be written as a linear combination, with good coefficients $c(v)$, of the following terms. \begin{itemize}
\item $ \partial (x^{\theta}) \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu}$, with $|\gamma| \leq |\beta_0| < |\beta|$, which is part of \eqref{equa1bis}.
\item $x^{\theta} \frac{v^{\mu}}{v^0} \mathcal{L}_{\partial Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu}$, with $1+|\gamma| \leq 1+|\beta_0|=|\beta|$. Denoting $\partial Z^{\gamma}$ by $Z^{\xi}$, we have $\xi_T=1+\gamma_T=1+(\beta_0)_T=\beta_T$ and this term is part of \eqref{equa1}.
\item $ P_{(k_T+1,k_P),p}(\Phi)\frac{v^{\mu}}{v^0} \mathcal{L}_{ Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu}$, with $|k|+1+|\gamma| \leq |\beta|-1+1=|\beta|-1$ and $k_P \leq (\beta_0)_P = \beta_P$, which is part of \eqref{equa1bis}.
\item $ P_{k,p}(\Phi)\frac{v^{\mu}}{v^0} \mathcal{L}_{\partial Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu}$, with $|k|+|\gamma|+1 \leq |\beta_0|-1+1=|\beta|-1$ and $k_P \leq (\beta_0)_P = \beta_P$, which is part of \eqref{equa1bis}.
\item $\partial(x^{\theta}) P_{k,p}(\Phi) \frac{v^{\mu}}{v^0} \mathcal{L}_{X Z^{\gamma} Z^{\gamma_1} }(F)_{\mu \nu}$, with $|k|+|\gamma| \leq |\beta_0|-1 \leq |\beta|-2$ and $k_P < (\beta_0)_P=\beta_P$, which is then equal to $0$ or part of \eqref{equa1bis}.
\item $x^{\theta} P_{(k_T+1,k_P),p}(\Phi) \frac{v^{\mu}}{v^0} \mathcal{L}_{X Z^{\gamma} Z^{\gamma_1} }(F)_{\mu \nu}$, with $|k|+1+|\gamma| \leq |\beta_0|-1+1=|\beta|-1$ and $k_P < (\beta_0)_P=\beta_P$, which is then part of \eqref{equa2}.
\item $x^{\theta}P_{k,p}(\Phi) \frac{v^{\mu}}{v^0} \mathcal{L}_{\partial X Z^{\gamma} Z^{\gamma_1} }(F)_{\mu \nu}$, with $|k|+|\gamma|+1 \leq |\beta|-1$ and $k_P < \beta_P$, which is part of \eqref{equa2}, as $[\partial, X ]=0$. \end{itemize} Suppose now that $Y=\widehat{Z}+\Phi X \in \mathbb{Y}_0$. We then have $\beta_P=(\beta_0)_P+1$ and $(\beta_0)_T=\beta_T$. In the following, we will skip the case where $Y$ hits $c(v)(v^0)^{-1}$ and we suppose for simplicty that $c(v)=1$. Note however that this case is straightforward since $$ Y\left( \frac{c(v)}{v^0} \right)= \widehat{Z} \left( \frac{c(v)}{v^0} \right)= \frac{\widehat{Z}(c(v))}{v^0}+c(v) \widehat{Z} \left( \frac{1}{v^0} \right) = \frac{c_1(v)}{v^0} .$$ Using again the induction hypothesis, $Y Y^{\beta_0} \left( t \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma_1}}(F)_{\mu \zeta} \right)$ can be written as a linear combination of the following terms.
\begin{itemize}
\item $ Y (x^{\theta}) \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu}$, with $|\gamma| \leq |\beta_0| < |\beta|$ and $\gamma_T=(\beta_0)_T=\beta_T$. As, schematically (with $\delta=0$ or $\delta=1$), \begin{equation}\label{eq:53} Y(x^{\theta})=\widehat{Z}(x^{\theta})+\Phi X(x^{\theta})=\delta x^{\kappa}+c(v)\Phi, \end{equation} This leads to terms of \eqref{equa1} and \eqref{equa1bis}.
\item $x^{\theta} \frac{1}{v^0} Y \left( v^{\mu} \mathcal{L}_{ Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu} \right)$, with $|\gamma| \leq |\beta_0|$ and $\gamma_T=(\beta_0)_T=\beta_T$. Using the first identity of Lemma \ref{calculF}, we have that $Y \left( v^{\mu} \mathcal{L}_{ Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \theta} \right)$ is a linear combination of terms such as
$$v^{\mu}\mathcal{L}_{ Z^{\gamma_0} Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \lambda} , \hspace{3mm} \text{with} \hspace{3mm} |\gamma_0| \leq 1, \hspace{3mm} (\gamma_0)_T=0, \hspace{3mm} \text{and} \hspace{3mm} 0 \leq \lambda \leq 3,$$ leading to terms of \eqref{equa1}, and $$\Phi v^{\mu}\mathcal{L}_{ X Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu},$$
giving terms of \eqref{equa2}, as $|\gamma| \leq |\beta_0|=|\beta|-1$.
\item $\frac{1}{v^0} Y \left( P_{k,p}(\Phi) \right) v^{\mu} \mathcal{L}_{ Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu} $, with $|k|+|\gamma| \leq |\beta_0|-1$ and $k_P \leq \beta_P$. We obtain terms of \eqref{equa1bis}, since
$$Y \left( P_{k,p}(\Phi) \right)=P_{(k_T,k_P+1),p}(\Phi), \hspace{3mm} |k|+1+|\gamma| \leq |\beta|-1 \hspace{3mm} \text{and} \hspace{3mm} k_P+1 \leq (\beta_0)_P+1 = \beta_P .$$
\item $\frac{1}{v^0} P_{k,p}(\Phi) Y \left( v^{\mu} \mathcal{L}_{ Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu} \right)$, with $|k|+|\gamma| \leq |\beta_0|-1$ and $k_P \leq (\beta_0)_P$. Using the first identity of Lemma \ref{calculF}, we have that $Y \left( v^{\mu} \mathcal{L}_{ Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \nu} \right)$ is a linear combination of terms of the form
$$c(v) \Phi^r v^{\mu}\mathcal{L}_{ Z^{\gamma_0} Z^{\gamma} Z^{\gamma_1}}(F)_{\mu \lambda} , \hspace{6mm} \text{with} \hspace{6mm} \max(r,|\gamma_0|) \leq 1 \hspace{6mm} \text{and} \hspace{6mm} 0 \leq \lambda \leq 3.$$
We then obtain terms of \eqref{equa1bis}, as $|k|+|\gamma|+|\gamma_0| \leq |\beta_0|=|\beta|-1$ and $k_P \leq \beta_P$.
\item $Y\left(x^{\theta} \right)P_{k,p}(\Phi) \frac{v^{\mu}}{v^0} \mathcal{L}_{XZ^{\gamma} Z^{\gamma_1} }(F)_{\mu \nu}$, with $|k|+|\gamma| \leq |\beta_0|-1$ and $k_P < (\beta_0)_P$, which, using \eqref{eq:53}, gives terms of \eqref{equa1bis} and \eqref{equa2}.
\item $ x^{\theta}P_{(k_T,k_P+1),p}(\Phi) \frac{v^{\mu}}{v^0} \mathcal{L}_{X Z^{\gamma} Z^{\gamma_1} }(F)_{\mu \nu}$, with $|k|+1+|\gamma| \leq |\beta_0|-1+1=|\beta|-1$ and $k_P+1 < (\beta_0)_P+1=\beta_P$, which is part of \eqref{equa2}.
\item $x^{\theta}P_{k,p}(\Phi)\frac{1}{v^0} Y \left( v^{\mu} \mathcal{L}_{ X Z^{\gamma} Z^{\gamma_1} }(F)_{\mu \nu} \right)$, with $|k|+|\gamma| \leq |\beta_0|-1$ and $k_P < (\beta_0)_P$. By the third point of Lemma \ref{calculF}, we can write $Y \left( v^{\mu} \mathcal{L}_{X Z^{\gamma} Z^{\gamma_1} }(F)_{\mu \nu} \right)$ as a linear combination of terms such as
$$c(v) \Phi^r v^{\mu} \mathcal{L}_{ X Z^{\gamma_0} Z^{\gamma} Z^{\gamma_1} }(F)_{\mu \lambda}, \hspace{3mm} \text{with} \hspace{3mm} \max(r,|\gamma_0|) \leq 1 \hspace{3mm} \text{and} \hspace{3mm} 0 \leq \lambda \leq 3.$$
It gives us terms of \eqref{equa2}, as $|k|+|\gamma_0|+|\gamma| \leq |\beta_0|=|\beta|-1$ and $k_P < \beta_P$. \end{itemize} \end{proof}
The worst terms are those of \eqref{equa1}. They do not appear in the source term of $T_F \left( P^X_{\zeta}(\Phi) \right)$, which explains why our estimate on $\| P^X_{\zeta}(\Phi) Y^{\beta} f \|_{L^1_{x,v}}$ will be better than the one on $\| P_{\xi}(\Phi) Y^{\beta} f \|_{L^1_{x,v}}$.
\begin{Pro}\label{sourceXPhi}
Let $Y^{\overline{\beta}} \in \mathbb{Y}_X^{|\overline{\beta}|}$, with $\overline{\beta}_X \geq 1$, $Z^{\gamma_1} \in \mathbb{K}^{|\gamma_1|}$ and $\beta$ be a multi-index associated to $\mathbb{Y}$ such that $\beta_P=\overline{\beta}_P$ and $\beta_T=\overline{\beta}_T+\overline{\beta}_X$. Then, $ Y^{\overline{\beta}} \left( t \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma_1}}(F)_{\mu \zeta} \right)$ can be written as a linear combination of terms of \eqref{equa1bis}, \eqref{equa2} and, \begin{equation}\label{equa2bis}
\text{if} \hspace{6mm} \beta_P=0, \hspace{6mm} x^{\theta} \frac{v^{\mu}}{v^0} \mathcal{L}_{X Z^{\gamma} Z^{\gamma_1} }(F)_{\mu \nu}, \hspace{6mm} \text{where} \hspace{5mm} |\gamma| \leq |\beta|-1. \tag{family $\beta-3-bis$} \end{equation} \end{Pro} \begin{proof} The proof is similar to the previous one. The difference comes from the fact a $X$ vector field necessarily have to hit a term of the first family, giving either a term of the second family or of the third-bis family, where we we do not have the condition $k_P < \beta_P$ since $k_P$ and $\beta_P$ could be both equal to $0$. \end{proof}
\subsection{The null structure of $G(v,\nabla_v g)$} In this subsection, we consider $G$, a $2$-form defined on $[0,T[ \times \mathbb{R}^3$, and $g$, a function defined on $[0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v$, both sufficiently regular. We investigate in this subsection the null structure of $G(v,\nabla_v g)$ in view of studying the error terms obtained in Proposition \ref{ComuVlasov}. Let us denote by $(\alpha, \underline{\alpha}, \rho, \sigma)$ the null decomposition of $G$. Then, expressing $G \left( v, \nabla_v g \right)$ in null coordinates, we obtain a linear combination of the following terms. \begin{itemize} \item The terms with the radial component of $\nabla_v g$ (remark that $\left( \nabla_v g \right)^L =- \left( \nabla_v g \right)^{\underline{L}}=\left( \nabla_v g \right)^r$), \begin{equation}\label{eq:radi} v^L \rho \left( \nabla_v g \right)^{\underline{L}}, \hspace{8mm} v^{\underline{L}} \rho \left( \nabla_v g \right)^{L}, \hspace{8mm} v^A \alpha_A \left( \nabla_v g \right)^{L} \hspace{8mm} \text{and} \hspace{8mm} v^A \underline{\alpha}_A \left( \nabla_v g \right)^{\underline{L}}. \end{equation} \item The terms with an angular component of $\nabla g$, \begin{equation}\label{eq:angu} \varepsilon_{BA} v^B \sigma \left( \nabla_v g \right)^{A}, \hspace{12mm} v^{L} \alpha_A \left( \nabla_v g \right)^{A} \hspace{8mm} \text{and} \hspace{8mm} v^{\underline{L}} \underline{\alpha}_A \left( \nabla_v g \right)^{A}. \end{equation} \end{itemize} We are then led to bound the null components of $\nabla_v g$. A naive estimate, using $v^0\partial_{v^k}= Y_k-\Phi X-t\partial_k-x^k \partial_t$, gives \begin{equation}\label{naive2}
\left| \left( \nabla_v g \right)^{L} \right|, \hspace{1mm} \left| \left( \nabla_v g \right)^{\underline{L}} \right|, \hspace{1mm} \left| \left( \nabla_v g \right)^{A} \right| \leq \left| \nabla_v g \right| \lesssim \frac{\tau_++|\Phi|}{v^0} |\nabla_{t,x} g |+\frac{1}{v^0}\sum_{Y \in \mathbb{Y}} |Y g|. \end{equation} With these inequalities, using our schematic notations $c \prec d$ if $d$ is expected to behave better than $c$, we have $v^L \rho \left( \nabla_v g \right)^{\underline{L}} \prec \varepsilon_{BA} v^B \sigma \left( \nabla_v g \right)^{A}$, since $v^L \prec v^B$ and $\rho \sim \sigma$. The purpose of the following result is to improve \eqref{naive2} for the radial component in order to have a better control on $v^L \rho \left( \nabla_v g \right)^{\underline{L}}$. \begin{Lem}\label{vradial} Let $g$ be a sufficiently regular function, $z \in \mathbf{k}_1$ and $j \in \mathbb{N}^*$. We have
$$ \left| \left( \nabla_v g \right)^{r} \right| \lesssim \frac{\tau_-+|\Phi|}{v^0} |\nabla_{t,x} g |+\frac{1}{v^0}\sum_{Y \in \mathbb{Y}} |Y g| \hspace{10mm} \text{and} \hspace{10mm} \left| \left( \nabla_v z^j \right)^r \right| \lesssim \frac{\tau_-}{v^0}|z|^{j-1}+\frac{1}{v^0} \sum_{w \in \mathbf{k}_1} |w |^j.$$ \end{Lem} \begin{proof} We have $$( \nabla_v g )^r=\frac{x^i}{r}\partial_{v^i} g \hspace{8mm} \text{and} \hspace{8mm} \frac{x^i}{rv^0}(t\partial_i+x_i\partial_t)=\frac{1}{v^0}(t\partial_r+r\partial_t)=\frac{1}{v^0}(S+(r-t)\underline{L}),$$ so that, using $\partial_{v^i}=\frac{1}{v^0}(\widehat{\Omega}_{0i}-t\partial_i-x_i\partial_t)$, \begin{equation}\label{equ:proof} ( \nabla_v g )^r = \frac{x^i }{rv^0}\widehat{\Omega}_{0i} \left(g \right)-\frac{1}{v^0}S \left( g \right) +\frac{t-r}{v^0}\underline{L} \left( g \right). \end{equation}
To prove the first inequality, it only remains to write schematically that $\widehat{\Omega}_{0i}=Y_{0i}-\Phi X$, $S=Y_S-\Phi X$ and to use the triangle inequality. To complete the proof of the second inequality, apply \eqref{equ:proof} to $g=z^j$, recall from Lemma \ref{weights} that $ \left| \widehat{Z} \left( z^j \right) \right| \lesssim \sum_{z \in \mathbf{k}_1} |w|^j$ and use that $\left| \underline{L} \left( z^j \right) \right| \lesssim |z|^{j-1}$. \end{proof} For the terms containing an angular component, note that they are also composed by either $\alpha$, the better null component of the electromagnetic field, $v^A$ or $v^{\underline{L}}$. The following lemma is fundamental for us to estimate the energy norms of the Vlasov field. \begin{Lem}\label{nullG}
We can bound $\left| G(v, \nabla_v g ) \right|$ either by
$$ \left( |\rho|+|\underline{\alpha}| \right) \left( \sum_{ Y \in \mathbb{Y}} |Y(g)| \hspace{-0.2mm}+ \hspace{-0.2mm} \left( \tau_-+|\Phi|+\sum_{w \in \mathbf{k}_1} |w| \right) |\nabla_{t,x} g | \hspace{-0.5mm} \right)+ \left(|\alpha|+\sqrt{\frac{v^{\underline{L}}}{v^0}}|\sigma| \right) \hspace{-1mm} \left( \sum_{ Y \in \mathbb{Y}} |Y(g)| \hspace{-0.2mm} + \hspace{-0.2mm} (\tau_++|\Phi|) |\nabla_{t,x} g | \right)$$ or by
$$ \left( |\alpha|+|\rho|+\sqrt{\frac{v^{\underline{L}}}{v^0} }|\sigma|+\sqrt{\frac{v^{\underline{L}}}{v^0} }|\underline{\alpha}| \right) \left( \sum_{ Y \in \mathbb{Y}} |Y(g)|+ \left( \tau_++|\Phi| \right) |\nabla_{t,x} g | \right)$$ \end{Lem} \begin{proof}
The proof consists in bounding the terms given in \eqref{eq:radi} and \eqref{eq:angu}. By Lemma \ref{vradial} and $|v^A| \lesssim \sqrt{v^0v^{\underline{L}}}$, one has
$$ \left| v^L \rho \left( \nabla_v g \right)^{\underline{L}}-v^{\underline{L}} \rho \left( \nabla_v g \right)^L+v^A \underline{\alpha}_A \left( \nabla_v g \right)^{\underline{L}} \right| \lesssim \left( |\rho|+\sqrt{\frac{v^{\underline{L}}}{v^0}}|\underline{\alpha}| \right)\left( \sum_{ Y \in \mathbb{Y}} |Y(g)|+ \left( \tau_-+|\Phi| \right) |\nabla_{t,x} g | \right) .$$
As $v^0 \partial_{v^i} = Y_i-\Phi X-x^i \partial_t-t \partial_i$ and $|v^B | \lesssim \sqrt{v^0 v^{\underline{L}}}$, we obtain
$$ \left| v^L \alpha_A \left( \nabla_v g \right)^A+v^A \alpha_A \left( \nabla_v g \right)^{L}+v^B \sigma_{BA} \left( \nabla_v g \right)^A \right| \lesssim \left(|\alpha|+\sqrt{\frac{v^{\underline{L}}}{v^0}}|\sigma| \right) \left( \sum_{ Y \in \mathbb{Y}} |Y(g)|+ (\tau_++\Phi|) |\nabla_{t,x} g | \right).$$ Finally, using $v^0 \partial_{v^i} = Y_i-\Phi X-x^i \partial_t-t \partial_i$ and Lemma \ref{weights1} (for the first inequality), we get \begin{eqnarray}
\nonumber \left| v^{\underline{L}} \underline{\alpha}_A \left( \nabla_v g \right)^A \right| & \lesssim & |\underline{\alpha}| \left( \sum_{ Y \in \mathbb{Y}} |Y(g)|+ \left( \tau_-+|\Phi|+\sum_{w \in \mathbf{k}_1} |w| \right) |\nabla_{t,x} g | \right) \\ \nonumber
\left| v^{\underline{L}} \underline{\alpha}_A \left( \nabla_v g \right)^A \right| & \lesssim & \sqrt{\frac{v^{\underline{L}}}{v^0}} |\underline{\alpha}| \left( \sum_{ Y \in \mathbb{Y}} |Y(g)|+ \left( \tau_++|\Phi| \right) |\nabla_{t,x} g | \right). \end{eqnarray} \end{proof} \begin{Rq}
The second inequality will be used in extremal cases of the hierarchies considered, where we will not be able to take advantage of the weights $w \in \mathbf{k}_1$ in front of $|\nabla_{t,x} g|$ and where the terms $\sum_{Y \in \mathbb{Y}_0} |Y g |$ will force us to estimate a weight $z \in \mathbf{k}_1$ by $\tau_+$ (see Proposition \ref{ComuPkp} below). \end{Rq} \subsection{Source term of $T_F(z^jP_{\xi}(\Phi) Y^{\beta}f)$}
In view of Remark \ref{rqjustifnorm}, we will consider hierarchised energy norms controling, for $Q$ a fixed integer, $\| z^{Q-\xi_P-\beta_P} P_{\xi}(\Phi) Y^{\beta} f \|_{L^1_{x,v}}$. In order to estimate them, we compute in this subsection the source term of $T_F(z^jP_{\xi}(\Phi) Y^{\beta}f)$. We start by the following technical result. \begin{Lem}\label{GammatoYLem}
Let $h : [0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v \rightarrow \mathbb{R}$ be a sufficiently regular function and $\Gamma^{\sigma} \in \mathbb{G}^{|\sigma|}$. Then, \begin{eqnarray}
\nonumber \Gamma^{\sigma} h & = & \sum_{\begin{subarray}{} \hspace{1mm} |g|+|\overline{\sigma}| \leq |\sigma| \\ \hspace{2mm} |g| \leq |\sigma|-1 \\ r+g_P+\overline{\sigma}_P \leq \sigma_P \end{subarray} } c^{g,r}_{\overline{\sigma}}(v) P_{g,r}(\Phi) Y^{\overline{\sigma}} h ,\\ \nonumber
\left| \partial_{v^i} \left( \Gamma^{\sigma} h \right) \right| & \lesssim & \sum_{\delta=0}^1 \sum_{\begin{subarray}{} \hspace{3.5mm} |g|+|\overline{\sigma}| \leq |\sigma|+1 \\ \hspace{9mm} |g| \leq |\sigma| \\ r+g_P+\overline{\sigma}_P+\delta \leq \sigma_P+1 \end{subarray} } \tau_+^\delta \left| P_{g,r}(\Phi) Y^{\overline{\sigma}} h \right|. \end{eqnarray} \begin{proof}
The first formula can be proved by induction on $|\sigma|$, using that $\widehat{Z}=Y-\Phi X$ for each $\widehat{Z}$ composing $\Gamma^{\sigma}$. The inequality then follows using $v^0 \partial_{v^i}=Y_i-\Phi X-t \partial_i-x^i \partial_t$. \end{proof} \end{Lem} \begin{Pro}\label{ComuPkp}
Let $N \in \mathbb{N}$ and $N_0 \geq N$. Consider $\zeta^0$ and $\beta$ multi-indices such that $|\zeta^0|+|\beta| \leq N$ and $|\zeta^0| \leq N-1$. Let also $z \in \mathbf{k}_1$ and $j \leq N_0-\zeta^0_P-\beta_P$. Then, $T_F(z^jP_{\zeta^0}(\Phi) Y^{\beta} f)$ can be bounded by a linear combination of the following terms, where $|\gamma|+|\zeta| \leq |\zeta^0|+|\beta|$. \begin{itemize} \item \begin{equation}\label{eq:cat0}
\left| F \left(v, \nabla_v \left( z^j \right) \right) P_{\zeta^0}(\Phi) Y^{\beta} f \right|. \tag{category $0$} \end{equation} \item \begin{equation}\label{eq:cat1}
\left( \left| \nabla_{Z^{\gamma}} F \right|+\frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right| +\frac{\tau_+}{\tau_-} \sqrt{\frac{v^{\underline{L}}}{v^0}} \left| \sigma \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right| \right)\left| \Phi \right|^n \left|w^i P_{\zeta}(\Phi) Y^{\kappa} f \right|, \tag{category $1$} \end{equation}
where \hspace{2mm} $n \leq 2N$, \hspace{2mm} $w \in \mathbf{k}_1$, \hspace{2mm} $|\zeta|+|\gamma|+|\kappa| \leq |\zeta^0|+|\beta|+1$, \hspace{2mm} $i \leq N_0 -\zeta_P-\kappa_P$, \hspace{2mm} $\max( |\gamma|, |\zeta|+|\kappa|) \leq |\zeta^0|+|\beta|$ \hspace{2mm} and \hspace{2mm} $|\zeta| \leq N-1$. \item \begin{equation}\label{eq:cat3}
\hspace{-10mm} \frac{\tau_+}{\tau_-} |\rho \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) | \left|z^{j-1} P_{\zeta}(\Phi) Y^{\sigma} f \right| \hspace{5mm} \text{and} \hspace{5mm} \frac{\tau_+}{\tau_-} \sqrt{\frac{v^{\underline{L}}}{v^0}}\left| \underline{\alpha} \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) \right| \left| z^i P_{\zeta}(\Phi) Y^{\kappa} f \right|, \tag{category $2$} \end{equation}
where \hspace{2mm} $|\zeta|+|\gamma|+|\kappa| \leq |\zeta^0|+|\beta|+1$, \hspace{2mm} $j-1$, $i=N_0-\zeta_P-\kappa_P$, \hspace{2mm} $\max( |\gamma|, |\zeta|+|\kappa|) \leq |\zeta^0|+|\beta|$ \hspace{2mm} and \hspace{2mm} $|\zeta| \leq N-1$. Morevover, we have $i \leq j$. \item \begin{equation}\label{eq:cat4}
\tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \theta} z^j P_{\zeta}(\Phi) Y^{\beta} f \right|, \tag{category $3$} \end{equation}
with \hspace{2mm}$ |\zeta| < |\zeta^0|$, \hspace{2mm} $\zeta_T+\gamma_T = \zeta^0_T$, \hspace{2mm} $\zeta_P \leq \zeta^0_P$, and \hspace{2mm} $|\zeta|+|\gamma| \leq |\zeta^0|+1$. This implies $j \leq N_0-\zeta_P-\beta_P$. \end{itemize}
Note that the terms of \eqref{eq:cat3} only appears when $j=N_0-k_P-\beta_P$ and the ones of \eqref{eq:cat4} when $|\zeta^0| \geq 1$. \end{Pro} \begin{proof} The first thing to remark is that $$T_F(z^jP_{\zeta^0}(\Phi) Y^{\beta} f)=F \left(v, \nabla_v \left( z^j \right) \right) P_{\zeta^0}(\Phi) Y^{\beta} f +z^jT_F(P_{\zeta^0}(\Phi))Y^{\beta} f+z^jP_{\zeta^0}(\Phi) T_F(Y^{\beta} f ).$$
We immediately obtain the terms of \eqref{eq:cat0}. Let us then consider $z^jP_{\zeta^0}(\Phi) T_F(Y^{\beta} f )$. Using Proposition \ref{ComuVlasov}, it can be written as a linear combination of terms of \eqref{eq:com1}, \eqref{eq:com2} or \eqref{eq:com4} (applied to $f$), multiplied by $z^jP_{\zeta^0}(\Phi)$. Consequently, $|z^jP_{\zeta^0}(\Phi) T_F(Y^{\beta} f )|$ can be bounded by a linear combination of \begin{itemize}
\item $|z|^j\left| w^d Z^{\gamma}(F_{\mu \nu}) \right| \left| P_{k,p}(\Phi)P_{\zeta^0}(\Phi) Y^{\kappa} f \right|$, with $d \in \{0,1 \}$, $w \in \mathbf{k}_1$, $|\sigma| \geq 1$, $\max( |\gamma|, |k|+|\gamma|, |k|+|\kappa|,|k|+1 ) \leq |\beta|$, $|k|+|\gamma|+|\kappa| \leq |\beta|+1$ and $p+k_P+\kappa_P+d \leq \beta_P$. Now, note that
$$ \exists \hspace{0.5mm} n, \hspace{0.5mm} \zeta \hspace{3mm} \text{such that} \hspace{3mm} P_{k,p}(\Phi) P_{\zeta^0}(\Phi) = \Phi^n P_{\zeta}(\Phi), \hspace{3mm} n \leq |\beta|, \hspace{3mm} \zeta_T=k_T+\zeta^0_T \hspace{3mm} \text{and} \hspace{3mm} \zeta_P=k_P+\zeta^0_P.$$
Consequently, $|\zeta|=|k|+|\zeta^0| \leq |\zeta^0|+|\beta|-1 \leq N-1$, \hspace{2mm} $|\zeta|+|\gamma| =|k|+|\zeta^0|+|\gamma| \leq |\zeta^0|+|\beta|$, $$|\zeta|+|\kappa|=|k|+|\zeta^0|+|\kappa| \leq |\zeta^0| + |\beta| \hspace{3mm} \text{and} \hspace{3mm} |\zeta|+|\gamma|+|\kappa| \leq |k|+|\zeta^0|+|\gamma|+|\kappa| \leq |\zeta^0|+|\beta|+1.$$ Since $$k_P+\kappa_P+d \leq \beta_P \hspace{3mm} \text{and} \hspace{3mm} \zeta_P=k_P+ \zeta^0 _P, \hspace{3mm} \text{we have} \hspace{3mm} j+d \leq N_0-\zeta_P-\kappa_P.$$ Finally, as $|z^j w^d| \leq |z|^{j+d}+|w|^{j+d}$, we obtain terms of \eqref{eq:cat1}.
\item $|z|^j\left| P_{k,p}(\Phi) \mathcal{L}_{ X Z^{\gamma_0}}(F)\left( v, \nabla_v \left( \Gamma^{\sigma} f \right) \right) P_{\zeta^0}(\Phi) \right|$, with $|k|+|\gamma_0|+|\sigma| \leq |\beta|-1$, $p+k_P+\sigma_P \leq \beta_P$ and $p \geq 1$. Then, apply Lemma \ref{GammatoYLem} in order to get
$$\left| \nabla_v \left( \Gamma^{\sigma} f \right) \right| \lesssim \sum_{\delta=0}^1 \sum_{\begin{subarray}{} \hspace{3mm} |g|+|\overline{\sigma}| \leq |\sigma|+1 \\ \hspace{5mm} |g| \leq |\sigma| \\ r+g_P+\overline{\sigma}_P+\delta \leq \sigma_P+1 \end{subarray} } \tau_+^\delta \left| P_{g,r}(\Phi) Y^{\overline{\sigma}} f \right|.$$
Fix parameters $(\delta, g , r, \overline{\sigma})$ as in the right hand side of the previous inequality and consider first the case $\delta=0$. Then, $|z|^j\left| \mathcal{L}_{ X Z^{\gamma_0}}(F) \right| \left| P_{k,p}(\Phi) P_{g,r}(\Phi)P_{\zeta^0}(\Phi) Y^{\overline{\sigma}} f \right|$ can be bounded by terms such as
$$ |z|^j\left| Z^{\gamma}(F_{\mu \nu}) \right| \left|\Phi^n P_{\zeta}(\Phi) Y^{\overline{\sigma}} f \right| \hspace{-0.3mm} , \hspace{1.9mm} \text{with} \hspace{1.9mm} |\gamma| \leq |\gamma_0|+1, \hspace{1.9mm} n \leq p+r, \hspace{2mm} \zeta_T=k_T+g_T+ \zeta^0_T , \hspace{1.9mm} \zeta_P=k_P+g_P+\zeta^0_P .$$
We then have $n \leq 2|\beta|$, $|\zeta|+|\gamma|+|\overline{\sigma}| \leq |k|+|g|+|\zeta^0|+|\gamma_0|+1+|\overline{\sigma}| \leq |\zeta^0|+|\beta|+1$, $|\zeta|+|\overline{\sigma}| \leq |\zeta^0|+ |\beta|$ and $|\zeta| \leq |\zeta^0|+|\beta|-1$. As $$\zeta_P+\overline{\sigma}_P =k_P+g_P+\zeta^0_P+\overline{\sigma}_P \leq k_P+\sigma_P+1+\zeta^0_P \leq \zeta^0_P+\beta_P,$$ we have $j \leq N_0-\zeta_P-\overline{\sigma}_P$. If $\delta=1$, use the inequality \eqref{eq:Xdecay} of Proposition \ref{ExtradecayLie} to compensate the weight $\tau_+$. The only difference is that it brings a weight $w \in \mathbf{k}_1$. To handle it, use $|z^j w | \leq |z|^{j+1}+|w|^{j+1}$ and $$\zeta_P+\overline{\sigma}_P =k_P+g_P+\zeta^0_P+\overline{\sigma}_P \leq k_P+\sigma_P+1-\delta+\zeta^0_P \leq \zeta^0_P+\beta_P-1,$$ so that $j+1 \leq N_0-\zeta_P-\beta_P$. In both cases, we then have terms of \eqref{eq:cat1}.
\item $|z|^j\left| P_{k,p}(\Phi) \mathcal{L}_{ \partial Z^{\gamma_0}}(F)\left( v, \nabla_v \left( \Gamma^{\sigma^0} f \right) \right) P_{\zeta_0}(\Phi) \right|$, with $|k|+|\gamma_0|+|\sigma^0| \leq |\beta|-1$, $p+|\gamma_0| \leq |\beta|-1$ and $p+k_P+\sigma^0_P \leq \beta_P$, which arises from a term of \eqref{eq:com4}. Applying Lemma \ref{GammatoYLem}, we can schematically suppose that
$$ \Gamma^{\sigma^0} = c(v) \Phi^r P_{\chi}(\Phi) Y^{\kappa} \hspace{3mm} \text{with} \hspace{3mm} |\chi|+|\kappa| \leq |\sigma^0|, \hspace{3mm} |\chi| \leq |\sigma^0|-1 \hspace{3mm} \text{and} \hspace{3mm} r+r_{\chi} +\chi_P+\kappa_P \leq \sigma^0_P,$$ where $r_{\chi}$ is the number of $\Phi$ coefficients in $P_{\chi}(\Phi)$. As $Y \left( c(v) \right)$ is a good coefficient, $c(v)$ does not play any role in what follows and we then suppose for simplicity that $c(v)=1$. We suppose moreover, in order to not have a weight in excess, that \begin{equation}\label{condihyp} j+k_P+\chi_P+\kappa_P < N_0-\zeta^0_P \end{equation} and we will treat the remaining cases below. Using the first inequality of Lemma \ref{nullG} and denoting by $(\alpha, \underline{\alpha}, \rho, \sigma)$ the null decomposition of $\mathcal{L}_{\partial Z^{\gamma_0}}(F)$, we can bound the quantity considered here by the sum of the three following terms \begin{equation}\label{eq:unus}
|z|^j\left| P_{k,p}(\Phi) P_{\zeta_0}(\Phi) \right| \left( |\alpha|+|\rho|+\sqrt{\frac{v^{\underline{L}}}{v^0}}|\sigma|+|\underline{\alpha}| \right) \sum_{ Y \in \mathbb{Y}_0} \left| Y \left( \Phi^r P_{\chi}(\Phi) Y^{\kappa} f \right) \right|, \end{equation} \begin{equation}\label{eq:duo}
|z|^j\left| P_{k,p}(\Phi) P_{\zeta_0}(\Phi) \right| \left( |\rho|+ |\underline{\alpha}| \right) \left( \tau_- +|\Phi|+ \hspace{-1mm} \sum_{w \in \mathbf{k}_1} |w| \right) \left| \nabla_{t,x} \left( \Phi^r P_{\chi}(\Phi) Y^{\kappa} f \right) \right|, \end{equation} \begin{equation}\label{eq:tres}
|z|^j \left| P_{k,p}(\Phi) P_{\zeta_0}(\Phi) \right| \left(\tau_++|\Phi| \right)\left(|\alpha|+\sqrt{\frac{v^{\underline{L}}}{v^0}} |\sigma| \right) \left| \nabla_{t,x} \left( \Phi^r P_{\chi}(\Phi) Y^{\kappa} f \right) \right|. \end{equation} Let us start by \eqref{eq:unus}. We have schematically, for $Y \in \mathbb{Y}_0$, $Y^{\kappa^1}=Y^{\kappa}$ and $Y^{\kappa^2}=Y Y^{\kappa}$,
$$P_{k,p}(\Phi) P_{\zeta^0}(\Phi) Y \left( \Phi^r P_{\chi}(\Phi) Y^{\kappa} f \right) = \Phi^{n_1}P_{\zeta^1}(\Phi) Y^{\kappa^1} f+\Phi^{n_2}P_{\zeta^2}(\Phi) Y^{\kappa^2} f,$$ $$\text{with} \hspace{3mm} |n_i| \leq p+r, \hspace{3mm} |\zeta^i|=|k|+|\zeta^0|+|\chi|+\delta_1^{i} \hspace{3mm} \text{and} \hspace{3mm} \zeta^i_P=k_P+\zeta^0_P+\chi_P+\delta_{1}^{i}.$$ We have, according to \eqref{condihyp}, $$j+\zeta^i_P+\kappa^i_P = \zeta^0_P+j +k_P+\chi_P+\kappa_P+1 \leq N_0.$$ Consequently, as \begin{equation}\label{bound45}
|\alpha|+|\rho|+\sqrt{\frac{v^{\underline{L}}}{v^0}}|\sigma|+ |\underline{\alpha}| \lesssim \left| \mathcal{L}_{ \partial Z^{\gamma}}(F) \right| \lesssim \sum_{|\gamma| \leq |\gamma_0|+1} \left|\nabla_{ Z^{\gamma}} F \right| \hspace{3mm} \text{and} \hspace{3mm} |\zeta^i|+|\gamma|+|\kappa^i| \leq |\beta|+|\zeta^0|+1,
\end{equation} we obtain terms of \eqref{eq:cat1} (the other conditions are easy to check).
Let us focus now on \eqref{eq:duo} and \eqref{eq:tres}. Defining $Y^{\kappa^3}=Y^{\kappa}$ and $Y^{\kappa^4}= \partial Y^{\kappa}$, we have schematically
$$P_{k,p}(\Phi) P_{\zeta^0}(\Phi) \partial \left( \Phi^r P_{\chi}(\Phi) Y^{\kappa} f \right)= \Phi^{n_3}P_{\zeta^3}(\Phi) Y^{\sigma^3} f+\Phi^{n_4}P_{\zeta^4}(\Phi) Y^{\kappa^4} f,$$ $$\text{with} \hspace{3mm} |n_i| \leq p+r \leq 2|\beta|-2, \hspace{3mm} |\zeta^i|=|k|+|\zeta^0|+|\chi|+\delta_{i}^{3} \hspace{3mm} \text{and} \hspace{3mm} \zeta^i_P=k_P+\zeta^0_P+\chi_P.$$ This time, one obtains $j +1 \leq N_0-\zeta^i_P-\kappa^i_P $. As, by inequality \eqref{eq:zeta2} of Proposition \ref{ExtradecayLie},
$$\left( |\rho|+ |\underline{\alpha}| \right) \lesssim \frac{1}{\tau_-}\sum_{|\gamma| \leq |\gamma_0|+1} \left| \nabla_{Z^{\gamma}} F \right|, \hspace{5mm} |\alpha| \lesssim \sum_{|\gamma| \leq |\gamma_0|+1}\frac{1}{\tau_-} |\alpha (\mathcal{L}_{Z^{\gamma}}(F))|+ \frac{1}{\tau_+}\left| \nabla_{Z^{\gamma}} F \right| ,$$
$$ |\sigma| \lesssim \sum_{|\gamma| \leq |\gamma_0|+1}\frac{1}{\tau_-} |\sigma (\mathcal{L}_{Z^{\gamma}}(F))|+ \frac{1}{\tau_+}\left| \nabla_{Z^{\gamma}} F \right| \hspace{5mm} \text{and} \hspace{5mm} |z^j w | \leq |z|^{j+1}+|w|^{j+1},$$ \eqref{eq:duo} and \eqref{eq:tres} also give us terms of \eqref{eq:cat1}. \item We now treat the remaining terms arising from those of \eqref{eq:com4}, for which $$j+k_P+\chi_P+\kappa_P=N_0-\zeta^0_P.$$ This equality can only occur if $j=N_0-\zeta^0_P-\beta_P$ and $k_P+\chi_P+\kappa_P=\beta_P$. It implies $p+r+r_{\chi}=0$ and we then have to study terms of the form
$$|z|^j\left| \mathcal{L}_{ \partial Z^{\gamma_0}}(F)\left( v, \nabla_v \left( Y^{\kappa} f \right) \right) P_{\zeta^0}(\Phi) \right|, \hspace{2mm} \text{with} \hspace{2mm} |\gamma_0|+|\kappa| \leq |\beta|-1.$$ Using the second inequality of Lemma \ref{nullG}, and denoting again the null decomposition of $\mathcal{L}_{\partial Z^{\gamma_0}}(F)$ by $(\alpha, \underline{\alpha}, \rho, \sigma)$, we can bound it by quantities such as
$$\left| \Phi \right| \left| \mathcal{L}_{ \partial Z^{\gamma_0}}(F) \right| \left| z^j P_{\zeta^0}(\Phi) \partial Y^{\kappa} f \right|, \hspace{3mm} \text{leading to terms of} \hspace{3mm} \eqref{eq:cat1}, $$ \begin{equation}\label{3:eq}
|\rho| \left| P_{\zeta^0}(\Phi) \right| \left( \tau_+|z|^{j-1}\left| Y Y^{\sigma} f \right|+\tau_- |z|^j \left| \partial Y^{\kappa} f \right| \right), \hspace{3mm} \text{with} \hspace{3mm} Y \in \mathbb{Y}_0, \hspace{3mm} \text{and}
\end{equation} \begin{equation}\label{2:eq}
\left( |\alpha|+\sqrt{\frac{v^{\underline{L}}}{v^0}}|\sigma|+\sqrt{\frac{v^{\underline{L}}}{v^0}} |\underline{\alpha}| \right) \left| P_{\zeta^0}(\Phi) \right| \left( \tau_+|z|^{j-1}\left| Y Y^{\kappa} f \right|+\tau_+ |z|^j \left| \partial Y^{\kappa} f \right| \right), \hspace{3mm} \text{with} \hspace{3mm} Y \in \mathbb{Y}_0.
\end{equation}
If $Y Y^{\kappa}=Y^{\chi^1}$ and $\partial Y^{\kappa}=Y^{\chi^2}$, we have $$|\zeta^0|+|\chi^i| \leq |k|+|\beta|, \hspace{8mm} j-1 = N_0-\zeta^0_P-\chi^1_P \hspace{8mm} \text{and} \hspace{8mm} j = N_0-\zeta^0_P-\chi_P^2.$$ Thus, \eqref{3:eq} and \eqref{2:eq} give terms of \eqref{eq:cat1} and \eqref{eq:cat3} since we have, according to inequality \eqref{eq:zeta2} of Proposition \ref{ExtradecayLie} and for $\varphi \in \{\alpha, \underline{\alpha}, \rho, \sigma \}$,
$$ |\varphi| \lesssim \sum_{|\gamma| \leq |\gamma_0|+1} \tau_-^{-1} \left| \varphi \left( \mathcal{L}_{Z^{\gamma}} (F) \right) \right|+\tau_+^{-1} \left| \nabla_{Z^{\gamma}} F \right| .$$ \end{itemize}
It then remains to bound $T_F(P_{\zeta^0}(\Phi))z^jY^{\beta}f$. If $|\zeta^0| \geq 1$, there exists $ 1 \leq p \leq |\zeta^0|$ and $\left( \xi^i \right)_{1 \leq i \leq p}$ such that
$$P_{\zeta^0}(\Phi) = \prod_{i=1}^p Y^{\xi^i} \Phi, \hspace{8mm} \min_{1 \leq i \leq p} |\xi^i| \geq 1, \hspace{8mm} \sum_{i=1}^p |\xi^i|=|k| \hspace{8mm} \text{and} \hspace{8mm} \sum_{i=1}^p (\xi^i)_T=k_T.$$ Then, $T_F(P_{\zeta_0}(\Phi))=\sum_{i=1}^p T_F(Y^{\xi^i} \Phi ) \prod_{j \neq i} Y^{\xi^j} \Phi$ and let us, for instance, bound $T_F(Y^{\xi^1} \Phi) Y^{\beta} f \prod_{j =2}^p Y^{\xi^j} \Phi$. To lighten the notation, we define $\chi$ such that $$P_{\chi}(\Phi)=\prod_{j =2}^p Y^{\xi^j} \Phi, \hspace{8mm} \text{so that} \hspace{8mm} (\chi_T,\chi_P)=\left(\zeta^0_T-\xi^1_T,\zeta^0_P-\xi^1_P \right).$$
Using Propositions \ref{ComuVlasov} and \ref{sourcePhi} (with $|\gamma_1| \leq 1$), $T_F(Y^{\xi_1} \Phi) P_{\chi}(\Phi) Y^{\beta} f $ can be written as a linear combination of terms of $(type \hspace{1mm} 1-\xi_1)$, $(type \hspace{1mm} 2-\xi_1)$, $(type \hspace{1mm} 3-\xi_1)$ (applied to $\Phi$), $(family \hspace{1mm} 1-\xi_1)$, $(family \hspace{1mm} 2-\xi_1)$ and $(family \hspace{1mm} 3-\xi_1)$, multiplied by $P_{\chi}(\Phi) Y^{\beta} f$. The treatment of the first three type of terms is similar to those which arise from $z^j P_{\zeta^0}(\Phi)T_F(Y^{\beta} f )$, so we only give details for the first one. We then have to bound \begin{itemize}
\item $|z|^j\left| Z^{\gamma}(F_{\mu \nu}) \right| \left|w^d P_{k,p}(\Phi) Y^{\kappa} \Phi P_{\chi}(\Phi) Y^{\beta} f \right|$, with $d \in \{0,1 \}$, $w \in \mathbf{k}_1$, $|\kappa| \geq 1$ $\max( |\gamma|, |k|+|\gamma|, |k|+|\kappa| ) \leq |\xi^1|$, $|k|+|\gamma|+|\kappa| \leq |\xi^1|+1$ and $p+k_P+\kappa_P+d \leq \xi^1_P$. Note now that
$$P_{k,p}(\Phi) Y^{\kappa} \Phi P_{\chi}(\Phi)= \Phi^n P_{\zeta}(\Phi), \hspace{3mm} \text{with} \hspace{3mm} n \leq p \leq |\xi^1| , \hspace{3mm} \zeta_T=k_T+\kappa_T+\chi_T \hspace{3mm} \text{and} \hspace{3mm} \zeta_P=k_P+\kappa_P+\chi_P.$$ Note moreover that
$$|\zeta|+|\gamma|+|\beta|=|k|+|\gamma|+|\kappa|+|\chi|+|\beta| \leq |\xi^1|+|\chi|+|\beta|+1 = |\zeta^0|+|\beta|+1, \hspace{3mm} |\zeta|+|\beta| \leq |\zeta^0|+|\beta|$$ and $\zeta_P+\beta_P+d=k_P+\kappa_P+d+\chi_P+\beta_P \leq \xi^1_P+\chi_P+\beta_P= \zeta^0_P+\beta_P$, which proves that this is a term of \eqref{eq:cat1}.
\item $\tau_+|z|^j \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \theta} P_{\chi}(\Phi) Y^{\beta} f \right| $, with $|\gamma| \leq |\xi^1|+1$ and $\gamma_T=\xi^1_T$. It is part of \eqref{eq:cat4} as
$$|\chi| < |k|, \hspace{6mm} \chi_T+\gamma_T=\chi_T+\xi^1_T = \zeta^0_T, \hspace{6mm} \chi_P \leq \zeta^0_P \hspace{6mm} \text{and} \hspace{6mm} |\chi|+|\gamma| \leq |\chi|+|\xi^1|+1 =|\zeta^0|+1.$$
\item $\left| Z^{\gamma}(F_{\mu \nu})\right| \left| z^j P_{k,p}(\Phi) P_{\chi} (\Phi)Y^{\beta} f \right| $, with $|k|+ |\gamma| \leq |\xi^1|-1$, $k_P \leq \xi^1_P$ and $p \leq |\xi^1|$, which is part of \eqref{eq:cat1}. Indeed, we can write
$$P_{k,p}(\Phi) P_{\chi} (\Phi) = \Phi^r P_{\zeta}(\Phi), \hspace{3mm} \text{with} \hspace{3mm} r \leq p \leq |\xi^1|, \hspace{3mm} \left( \zeta_T, \zeta_P \right) = \left( k_T+\chi_T,k_P+\chi_P \right)$$
and we then have $|\zeta|+|\gamma| = |k|+|\gamma|+|\chi| \leq |\xi^1|+|\chi| \leq |\zeta^0|$,
$$|\zeta|+|\gamma|+|\beta| \leq |\xi^1|+|\chi| +|\beta| \leq |\zeta^0|+|\beta| \hspace{3mm} \text{and} \hspace{3mm} \zeta_P+\beta_P \leq \xi^1_P+\chi_P+\beta_P = \zeta^0_P+\beta_P$$
\item $\tau_+ \left| \mathcal{L}_{X Z^{\gamma_0}}(F)\right| \left|z^j P_{k,p}(\Phi) P_{\chi} (\Phi) Y^{\beta} f \right|$, with $|k|+ |\gamma_0| \leq |\xi_1|-1$, $k_P < \xi^1_P$ and $p \leq |\xi^1|$. By inequality \eqref{eq:Xdecay} of Proposition \ref{ExtradecayLie}
$$\exists \hspace{1mm} w \in \mathbf{k}_1, \hspace{8mm} \tau_+\left| \mathcal{L}_{X Z^{\gamma_0}}(F)\right| \lesssim (1+|w|) \sum_{|\gamma| \leq |\gamma_0|+1} \left| \nabla_{Z^{\gamma}} F \right|.$$
Note moreover that $k_P+\chi_P+\beta_P \leq \xi^1_P-1+\chi_P+\beta_P < \zeta^0_P+\beta_P$, as\footnote{Note that this term could appear only if $\xi^1_P \geq 1$.} $k_P < \xi^1_P$. We then have $j+1 \leq N_0-k_P-\chi_P-\beta_P$ and we obtain, using $|z^jw| \leq |z|^{j+1}+|w|^{j+1}$ and writting again $P_{k,p}(\Phi) P_{\chi} (\Phi) = \Phi^r P_{\zeta}(\Phi)$, terms which are in \eqref{eq:cat1} (the other conditions can be checked as previously). \end{itemize} \end{proof}
\begin{Rq}\label{hierarchyjustification} There is three types of terms which bring us to consider a hierarchy on the quantities of the form $z^j P_{\xi}(\Phi) Y^{\beta} f$. \begin{itemize} \item Those of \eqref{eq:cat0}, as $\nabla_v \left( z^j \right)$ creates (at least) a $\tau_-$-loss and since $\tau_- F \sim \tau_+^{-1}$.
\item The first ones of \eqref{eq:cat3}. Indeed, we will have $|\rho| \lesssim \tau_+^{- \frac{3}{2}}\tau_-^{-\frac{1}{2}}$, so, using\footnote{We will be able to lose one power of $v^0$ as it is suggested by the energy estimate of Proposition \ref{energyf}.} $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$,
$$\frac{\tau_+}{\tau_-}|\rho| \lesssim \frac{v^0}{\tau_+}+\frac{v^{\underline{L}}}{\tau_-^3}.$$ $v^{\underline{L}} \tau_-^{-3}$ will give an integrable term, as the component $v^{\underline{L}}$ will allow us to use the foliation $(u,C_u(t))$ of $[0,t] \times \mathbb{R}^3_x$. However, $v^0 \tau_+^{-1}$ will create a logarithmical growth. \item The ones of \eqref{eq:cat4}, because of the $\tau_+$ weight and the fact that even the better component of $\mathcal{L}_{Z^{\gamma}}(F)$ will not have a better decay rate than $\tau_+^{-2}$. \end{itemize}
We will then classify them by $|\xi|+|\beta|$ and $j$, as one of these quantities is lowered in each of these terms. \end{Rq} \begin{Rq}\label{deuxblocs}
Let $\beta$ and, for $i \in \{1,2\}$, $\zeta^i$ be multi-indices such that $|\zeta^i|+|\beta| \leq N$, $|\zeta^1| \leq N-1$ and $N_0 \geq 2N-1$. We can adapt the previous proposition to $T_F \left( z^j P_{\zeta_1}(\Phi) P_{\zeta_2}(\Phi) Y^{\beta} f \right)$. One just has \begin{itemize} \item to add the factor $P_{\zeta_2}(\Phi)$ (or $P_{\zeta_1}(\Phi)$) in the terms of each categories and \item to replace conditions such as $j \leq N_0-\zeta_P-\sigma_P$ by $j \leq N_0-\zeta_P - \zeta^2_P-\sigma_P$ (or $j \leq N_0-\zeta_P - \zeta^1_P-\sigma_P$). \end{itemize} \end{Rq}
The worst terms are those of \eqref{eq:cat4} as they are responsible for the stronger growth of the top order energy norms. However, as suggested by the following proposition, we will have better estimates on $\| z^j P_{\xi}^X(\Phi) Y^{\beta} \|_{L^1_{x,v}}$.
\begin{Pro}\label{ComuPkpX}
Let $N \in \mathbb{N}$, $z \in \mathbf{k}_1$, $N_0 \geq N$, $\xi^0$, $\beta$ and $j \in \mathbb{N}$ be such that $|\xi^0| \leq N-1$, $|\xi^0|+|\beta| \leq N$ and $j \leq N_0-\xi^0_P-\beta_P$. Then, $T_F(z^j P^X_{\xi^0}(\Phi) Y^{\beta} f)$ can be bounded by a linear combination of terms of \eqref{eq:cat0}, \eqref{eq:cat1}\hspace{-0.1mm}, \eqref{eq:cat3} and \begin{equation}\label{eq:cat4bis}
\frac{\tau_+}{\tau_-} \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{ Z^{\gamma}}(F)_{\mu \nu} w^{j} P^X_{\xi}(\Phi) Y^{\beta} f \right|, \tag{category $3-X$} \end{equation}
with \hspace{2mm} $\xi_X < \xi^0_X$, \hspace{2mm} $\xi_T \leq \xi^0_T$, \hspace{2mm} $\xi_P \leq \xi^0_P$, \hspace{2mm} $|\xi|+|\gamma|+|\beta| \leq |\xi|+|\beta|+1$, \hspace{2mm} $|\gamma| \leq |\xi|+1$, \hspace{2mm} $w \in \mathbf{k}_1$ \hspace{2mm} and \hspace{2mm} $j = N_0-\zeta_P-\beta_P$.
Note that the terms of \eqref{eq:cat3} only appear when $j=N_0-\xi^0_P-\beta_P$ and those of \eqref{eq:cat4bis} if $j=N_0-\xi^0_P-\beta_P$ and $|\xi^0| \geq 1$. \end{Pro} \begin{proof} Proposition \ref{ComuVlasov} also holds for $Y^{\beta} \in \mathbb{Y}_X$ in view of Lemma \ref{ComuX} and the fact that $X$ can be considered as $c(v) \partial$. Then, one only has to follow the proof of the previous proposition and to apply Proposition \ref{sourceXPhi} where we used Proposition \ref{sourcePhi}. Hence, instead of terms of \eqref{eq:cat4}, we obtain
$$ \tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{X Z^{\gamma}} (F)_{\mu \nu} z^j P_{\chi}^X(\Phi) Y^{\beta} f \right|, \hspace{3mm} \text{with} \hspace{3mm} |\gamma| \leq |\xi^1|, \hspace{3mm} \chi_X < \xi^0_X, \hspace{3mm} \chi_T \leq \xi^0_T \hspace{3mm} \text{and} \hspace{3mm} \chi_P \leq \xi^0_P.$$ Apply now the second and then the first inequality of Proposition \ref{ExtradecayLie} to obtain that
$$\tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{X Z^{\gamma}} (F)_{\mu \theta} z^j P_{\chi}^X(\Phi) Y^{\beta} f \right| \lesssim \left| P_{\chi}^X(\Phi) Y^{\beta} f \right| \sum_{|\delta| \leq |\xi_1|+1} \hspace{-0.6mm} \left( \sum_{w \in \mathbf{k}_1 } \frac{ |w|^{j+1}}{\tau_-}\left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\delta}}(F)_{\mu \theta} \right|+|z|^j\left| \mathcal{L}_{Z^{\delta}}(F)\right| \hspace{-0.6mm} \right)$$ which leads to terms of \eqref{eq:cat4bis} (if $j=N_0-\chi_P-\beta_P$) and \eqref{eq:cat1} (as $P_{\chi}^X(\Phi)$ can be bounded by a linear combination of $P_{\chi^0}(\Phi)$ with $\chi^0_T = \chi_T+\chi_X$ and $\chi^0_P \leq \chi_P$). \end{proof} \begin{Rq}
As we will mostly apply this commutation formula with a lower $N_0$ than for our utilizations of Proposition \ref{ComuPkp} or for $|\xi^0|=0$, we will have to deal with terms of \eqref{eq:cat4bis} only once (for \eqref{Auxenergy}). \end{Rq}
\subsection{Commutation of the Maxwell equations}\label{subseccomuMax}
We recall the following property (see Lemma $2.8$ of \cite{massless} for a proof). \begin{Lem}\label{basiccom} Let $G$ and $M$ be respectively a $2$-form and a $1$-form such that $\nabla^{\mu} G_{\mu \nu}=M_{\nu}$. Then, $$\forall \hspace{0.5mm} Z \in \mathbb{P}, \hspace{5mm} \nabla^{\mu} \mathcal{L}_{Z}(G)_{\mu \nu} = \mathcal{L}_{Z} (M)_{\nu} \hspace{10mm} \text{and} \hspace{10mm} \nabla^{\mu} \mathcal{L}_{S}(G)_{\mu \nu} = \mathcal{L}_{S} (M)_{\nu} +2M_{\nu}.$$ If $g$ is a sufficiently regular function such that $\nabla^{\mu} G_{\mu \nu} = J(g)_{\nu}$, then $$\forall \hspace{0.5mm} Z \in \mathbb{P}, \hspace{5mm} \nabla^{\mu} \mathcal{L}_{Z}(G)_{\mu \nu} = J(\widehat{Z} g)_{\nu} \hspace{10mm} \text{and} \hspace{10mm} \nabla^{\mu} \mathcal{L}_{S}(G)_{\mu \nu} = J(Sg)_{\nu}+3J(g)_{\nu}.$$ \end{Lem}
We need to adapt this formula since we will control $Yf$ and not $\widehat{Z}f$. We cannot close the estimates using only the formula
$$ J(\widehat{Z} f)=J(Y f ) -J(\Phi^k_{\widehat{Z}} X_k f )$$ as we will have $\|\Phi \|_{L^{\infty}_{v}} \lesssim \log^2 (\tau_+ )$ and since this small loss would prevent us to close the energy estimates. \begin{Pro}\label{ComuMax1} Let $Z \in \mathbb{K}$. Then, for $0 \leq \nu \leq 3$, $\nabla^{\mu} \mathcal{L}_Z(F)_{\mu \nu}$ can be written as a linear combination of the following terms. \begin{itemize}
\item $\int_v \frac{v_{\nu}}{v^0}( X \Phi )^j Y^{\kappa} f dv$, with $j+|\kappa| \leq 1$.
\item $\frac{1}{\tau_+} \int_v c(t,x,v) z P_{k,p}(\Phi) Y^{\kappa} f dv$, with $z \in \mathbf{k}_1$, $p+|k|+|\kappa| \leq 3$ and $|k|+|\kappa| \leq 1$. \end{itemize} \end{Pro} \begin{Rq}\label{rq11} We would obtain a similar proposition if $J(f)_{\nu}$ was equal to $\int_v c_{\nu}(v) f dv$, except that we would have to replace $\frac{v_{\nu}}{v^0}$, in the first terms, by certain good coefficients $c(v)$. \end{Rq} \begin{proof} If $Z \in \mathbb{T}$, the result ensues from Lemma \ref{basiccom}. Otherwise, we have, using \eqref{eq:X} \begin{eqnarray} \nonumber J(\widehat{Z} f) & = & J(Yf)-J(\Phi^{k} X_{k} f) \\ \nonumber & = & J(Yf)_{\nu}+J(X_k(\Phi^k) f)_{\nu}-J(X_k(\Phi^k f)) \\ \nonumber & = & J(Yf)+J(X_k(\Phi^k) f)-\frac{1}{1+t+r} \sum_{k=1}^3 J\left( \left(2z_{0k}\partial_t+\sum_{Z \in \mathbb{K}} c_Z(t,x,v) Z \right)(\Phi^k f) \right) . \end{eqnarray}
Now, note that $J(z_{0k} \partial_t ( \Phi^k f ))= J(z_{0k} \Phi \partial_t f+z_{0k} \partial_t(\Phi) f)$ and, for $Z \in \mathbb{K} \setminus \mathbb{T}$ (in the computations below, we consider $Z=\Omega_{0i}$, but the other cases are similar), by integration by parts in $v$, \begin{eqnarray} \nonumber J\left( Z(\Phi^k f) \right) \hspace{-1.4mm} & = & \hspace{-1.4mm} J\left((Y-v^{0}\partial_{v^i}-\Phi^q X_q)(\Phi^k f) \right) \\ \nonumber & = & \hspace{-1.4mm} J \left(Y(\Phi^k)f+\Phi^k Y(f)-\Phi^q X_q ( \Phi^k ) f +\Phi^q \Phi^k X_q (f) \right) \hspace{-0.3mm} + \hspace{-0.3mm} \left(\int_v \Phi^k f dv \right) dx^{i} \hspace{-0.3mm} - \hspace{-0.3mm} \left( \int_v \Phi^k f \frac{v_i}{v^0}dv \right) dx^{0} \hspace{-0.2mm} , \end{eqnarray} where $dx^{\mu}$ is the differential of $x^{\mu}$. \end{proof} We are now ready to establish the higher order commutation formula. \begin{Pro}\label{ComuMaxN} Let $R \in \mathbb{N}$ and $Z^{\beta} \in \mathbb{K}^{R}$. Then, for all $0 \leq \nu \leq 3$, $\nabla^{\mu} \mathcal{L}_{Z^{\beta}}(F)_{\mu \nu}$ can be written as a linear combination of terms such as \begin{equation}\label{eq:comu1}
\int_v \frac{v_{\nu}}{v^0} P^X_{\xi}( \Phi ) Y^{\kappa} f dv, \hspace{3mm} \text{with} \hspace{3mm} |\xi|+|\kappa| \leq R, \tag{type $1-R$} \end{equation} \begin{equation}\label{eq:comu2}
\frac{1}{\tau_+}\int_v c(t,x,v) zP_{k,p}(\Phi) Y^{\kappa} f dv, \hspace{3mm} \text{with} \hspace{3mm} p+|k|+|\kappa| \leq 3R \hspace{3mm} \text{and} \hspace{3mm} k+|\kappa| \leq R.\tag{type $2-R$} \end{equation} \end{Pro}
\begin{proof} We will use during the proof the following properties, arising from Lemma \ref{weights} and the definition of the $X_i$ vector field, \begin{equation}\label{eq:Yz} \forall \hspace{0.5mm} (Y,z) \in \mathbb{Y} \times \mathbf{k}_1, \hspace{3mm} \exists \hspace{0.5mm} z' \in \mathbf{k}_1, \hspace{3mm} Y(z)=c_1(v)z+z'+c_2(v)\Phi, \end{equation} \begin{equation}\label{eq:PX} P^X_{\xi}(\Phi) = \sum_{\begin{subarray}{} \zeta_T = \xi_T+\xi_X \\ \hspace{2mm} \zeta_P \leq \xi_P \end{subarray}} c^\zeta(v) P_{\zeta}(\Phi). \end{equation}
Let us suppose that the formula holds for all $|\beta_0| \leq R-1 $, with $R \geq 2$ (for $R-1=1$, see Proposition \ref{ComuMax1}). Let $(Z,Z^{\beta_0}) \in \mathbb{K} \times \mathbb{K}^{|\beta_0|}$ with $|\beta_0|=R-1$ and consider the multi-index $\beta$ such that $Z^{\beta}=Z Z^{\beta_0}$. We fix $\nu \in \llbracket 0,3 \rrbracket$. By the first order commutation formula, Remark \ref{rq11} and the induction hypothesis, $\nabla^{\mu} \mathcal{L}_{Z^{\beta}}(F)_{\mu \nu}$ can be written as a linear combination of the following terms (to lighten the notations, we drop the good coefficients $c(t,x,v)$ in the integrands of the terms given by Proposition \ref{ComuMax1}). \begin{itemize}
\item $\int_v \frac{v_{\nu}}{v^0} \left( X \Phi \right)^j Y^{\kappa^0} \left( P_{\xi}^X( \Phi ) Y^{\kappa} f \right) dv$, with $j+|\kappa^0| \leq 1$ and $|\xi|+|\kappa| \leq R-1$. It leads to $\int_v \frac{v_{\nu}}{v^0} P_{\xi}^X( \Phi ) Y^{\kappa} f dv$, $$ \int_v \frac{v_{\nu}}{v^0} X(\Phi) P^X_{\xi}( \Phi ) Y^{\kappa} f dv, \hspace{5mm} \int_v \frac{v_{\nu}}{v^0} Y \left(P^X_{\xi}( \Phi ) \right) Y^{\kappa} f dv \hspace{5mm} \text{and} \hspace{5mm} \int_v \frac{v_{\nu}}{v^0} P^X_{\xi}( \Phi ) Y^{\kappa^0} Y^{\kappa} f dv,$$
which are all of \eqref{eq:comu1} since $Y\left(P^X_{\xi}( \Phi ) \right)=P_{\zeta}^X(\Phi)$, with $|\zeta|=|\xi|+1$, and $|\xi|+1+|\kappa| \leq R$.
\item $\int_v c(v) \left( X \Phi \right)^j Y^{\kappa^0} \left( \frac{z}{\tau_+} c(t,x,v) P_{k,p}(\Phi) Y^{\kappa} f \right) dv$, with $j+|\kappa^0| \leq 1$, $z \in \mathbf{k}_1$, $p+|k|+|\kappa| \leq 3R-3$ and $|k|+|\kappa| \leq R-1$. For simplicity, we suppose $c(v)=1$. As $$ Y \left( \frac{1}{\tau_+} c(t,x,v) \right) = \frac{1}{\tau_+} c_1(t,x,v)+ \frac{1}{\tau_+} c_2(t,x,v) \Phi,$$ we obtain, dropping the dependance in $(t,x,v)$ of the good coefficients, the following terms (with the first one corresponding to $j=1$ and the other ones to $j=0$). $$\frac{1}{\tau_+}\int_v c zP_{(k_T+1,k_P),p+1}(\Phi) Y^{\kappa} f dv, \hspace{5mm} \frac{1}{\tau_+}\int_v (c+c_1) zP_{k,p}(\Phi) Y^{\kappa} f dv, \hspace{5mm} \frac{1}{\tau_+}\int_v c_2 zP_{k,p+1}(\Phi) Y^{\kappa} f dv, $$ $$\frac{1}{\tau_+}\int_v c zP_{(k_T+\kappa^0_T,k_P+\kappa^0_P),p}(\Phi) Y^{\kappa} f dv, \hspace{5mm} \frac{1}{\tau_+}\int_v c Y(z) P_{k,p}(\Phi) Y^{\kappa} f dv, \hspace{5mm} \frac{1}{\tau_+}\int_v c z P_{k,p}(\Phi) Y^{\kappa^0} Y^{\kappa} f dv.$$ It is now easy to check that all these terms are of \eqref{eq:comu2} (for the penultimate term, recall in particular \eqref{eq:Yz}). For instance, for the first one, we have
$$(p+1)+(|k|+1)+|\kappa|=(p+|k|+|\kappa|)+2 \leq 3R-1 \leq 3R \hspace{3mm} \text{and} \hspace{3mm} (|k|+1)+|\kappa| \leq (|k|+|\kappa|)+1 \leq R.$$
\item $\frac{1}{\tau_+} \int_v zP_{k^0,p^0}(\Phi) Y^{\kappa^0} \left( P_{\xi}^X( \Phi ) Y^{\kappa} f \right) dv$, with $p^0+|k^0|+|\kappa^0| \leq 3$, $|k^0|+|\kappa^0| \leq 1$ and $|\xi|+|\kappa| \leq R-1$. According to \eqref{eq:PX}, we can suppose without loss of generality that $P_{\xi}^X( \Phi )=c(v) P_{\zeta}( \Phi )$, with $|\zeta| \leq |\xi|$. If $|k^0|=1$, we obtain
$$ \frac{1}{\tau_+} \int_v c(v) zP_{(\zeta_T+k^0_T,\zeta_P+k^0_P),r}( \Phi ) Y^{\kappa} f dv, \hspace{1cm} \text{with} \hspace{1cm} r \leq |\zeta|+p^0,$$
which is of \eqref{eq:comu2} since $$(|\zeta|+p^0)+(|\zeta|+|k^0|)+|\kappa| \leq (p^0+|k^0|)+2(|\xi|+|\kappa|)\leq 2R+1 \leq 3R \hspace{3mm} \text{and} \hspace{3mm} (|\zeta|+|k^0|)+|\kappa| \leq R.$$
If $|k_0|=0$, we obtain, with $r \leq |\zeta|+p^0$ and since $Y^{\kappa_0}(c(v))=c_1(v)$, $$ \frac{1}{\tau_+} \int_v (c+c_1)(v) zP_{(\zeta_T,\zeta_P),r}(\Phi ) Y^{\kappa} f dv, \hspace{5mm} \frac{1}{\tau_+} \int_v c(v) zP_{(\zeta_T,\zeta_P),r}( \Phi ) Y^{\kappa^0} Y^{\kappa} f dv \hspace{5mm} \text{and} \hspace{5mm}$$ $$\frac{1}{\tau_+} \int_v c(v) zP_{(\zeta_T+\kappa^0_T,\zeta_P+\kappa^0_P),r}( \Phi ) Y^{\kappa} f dv, $$ which are of \eqref{eq:comu2} since
$$|\zeta|+1+|\kappa| \leq R \hspace{3mm} \text{and} \hspace{3mm} |\zeta|+p^0+|\zeta|+|\kappa^0|+|\kappa| \leq 3+2R-2 \leq 3R .$$
\item $\frac{1}{\tau_+} \int_v w P_{k^0,p^0}(\Phi) Y^{\kappa^0} \left( \frac{z}{\tau_+} c(t,x,v) P_{k,p}(\Phi) Y^{\kappa} f \right) dv$, with $(w,z) \in \mathbf{k}_1^2$, $p^0+|k^0|+|\kappa^0| \leq 3$, $|k^0|+|\kappa^0| \leq 1$, $p+|k|+|\kappa| \leq 3R-3$ and $|k|+|\kappa| \leq R-1$.
If $|k^0|=1$, we obtain the term $$\frac{1}{\tau_+} \int_v c_0(t,x,v) wP_{k+k^0,p+p^0}(\Phi) Y^{\sigma} f dv, \hspace{3mm} \text{where} \hspace{3mm} c_0(t,x,v):=c(t,x,v)\frac{z}{\tau_+},$$ which is of \eqref{eq:comu2} since
$$|k+k^0|+(p+p^0)+|\kappa| \leq (p+|k|+|\kappa|)+(p^0+|k^0|) \leq 3R \hspace{3mm} \text{and} \hspace{3mm} |k+k^0|+|\kappa|=(|k|+|\kappa|)+1 \leq R.$$
If $|k_0|=0$, using that $$ \frac{z}{\tau_+}c(t,x,v)+Y^{\kappa^0} \left( \frac{z}{\tau_+} c(t,x,v) \right) = c_3(t,x,v)+c_4(t,x,v)\Phi,$$ we obtain the following terms of \eqref{eq:comu2}, $$\frac{1}{\tau_+} \int_v \left( c_3(t,x,v) P_{k,p+p^0}(\Phi)+c_4(t,x,v) P_{k,p+p^0+1}(\Phi) \right) w Y^{\kappa} f dv, $$ $$\frac{1}{\tau_+} \int_v c_0(t,x,v)w P_{k,p+p^0}(\Phi) Y^{\kappa^0}Y^{\kappa} f dv \hspace{6mm} \text{and} \hspace{6mm} \frac{1}{\tau_+} \int_v c_0(t,x,v)w P_{(k_T,\kappa^0_T,k_P+\kappa^0_P),p+p^0}(\Phi) Y^{\kappa} f.$$ \end{itemize} \end{proof}
Recall from the transport equation satisfied by the $\Phi$ coefficients that, in order to estimate $Y^{\gamma} \Phi$, we need to control $\mathcal{L}_{Z^{\beta}}(F)$ with $|\beta|=|\gamma|+1$. Consequently, at the top order, we will rather use the following commutation formula.
\begin{Pro}\label{CommuFsimple}
Let $Z^{\beta} \in \mathbb{K}^{|\beta|}$. Then,
$$\nabla^{\mu} \mathcal{L}_{Z^{\beta}}(F)_{\mu \nu} = \sum_{\begin{subarray}{} |q|+|\kappa| \leq |\beta| \\ \hspace{1.5mm} |q| \leq |\beta|-1 \\ \hspace{1mm} p \leq q_X+\kappa_T \end{subarray}} J \left( c^{k,q}_{\kappa}(v) P_{q,p}(\Phi) Y^{\kappa} f \right),$$ where $P_{q,p}(\Phi)$ can contain $\mathbb{Y}_X$, and not merely $\mathbb{Y}$, derivatives of $\Phi$. We then denote by $q_X$ its number of $X$ derivatives. \end{Pro} \begin{proof} Iterating Lemma \ref{basiccom}, we have \begin{equation}\label{comuiterbasic}
\nabla^{\mu} \mathcal{L}_{Z^{\beta}}(F)_{\mu \nu} = \sum_{|\gamma| \leq |\beta| } C^{\beta}_{\gamma} J \left( \widehat{Z}^{\gamma} f \right). \end{equation}
The result then follows from an induction on $|\gamma|$. Indeed, write $\widehat{Z}^{\gamma}=\widehat{Z} \widehat{Z}^{\gamma_0}$ and suppose that \begin{equation}\label{lifttomodif}
\widehat{Z}^{\gamma_0} f=\sum_{\begin{subarray}{} |q|+|\kappa| \leq |\gamma_0| \\ \hspace{1.5mm} |q| \leq |\gamma_0|-1 \\ \hspace{1mm} p \leq q_X+\kappa_T \end{subarray}} c^{k,q}_{\kappa}(v) P_{q,p}(\Phi) Y^{\kappa} f . \end{equation} If $\widehat{Z}=\partial \in \mathbb{T}$, then
$$\widehat{Z}^{\gamma} f = \sum_{\begin{subarray}{} |q|+|\kappa| \leq |\gamma_0| \\ \hspace{1.5mm} |q| \leq |\gamma_0|-1 \\ \hspace{1mm} p \leq q_X+\kappa_T \end{subarray}} c^{k,q}_{\kappa}(v) P_{(q_T+1,q_P,q_X),p}(\Phi) Y^{\kappa} f+c^{k,q}_{\kappa}(v) P_{q,p}(\Phi) \partial Y^{\kappa} f = \sum_{\begin{subarray}{} |q|+|\kappa| \leq |\gamma| \\ \hspace{1.5mm} |q| \leq |\gamma|-1 \\ \hspace{1mm} p \leq q_X+\kappa_T \end{subarray}} c^{k,q}_{\kappa}(v) P_{q,p}(\Phi) Y^{\kappa} f.$$ Otherwise $\gamma_P=(\gamma_0)_P+1$ and write $\widehat{Z}=Y-\Phi X$ with $Y \in \mathbb{Y}_0$. Hence, using $X Y^{\kappa} f=c(v) \partial Y^{\kappa} f$, \begin{eqnarray}
\nonumber \widehat{Z}^{\gamma} f \hspace{-2mm} & = & \hspace{-2mm} \sum_{\begin{subarray}{} |q|+|\kappa| \leq |\gamma_0| \\ \hspace{1.5mm} |q| \leq |\gamma_0|-1 \\ \hspace{1mm} p \leq q_X+\kappa_T \end{subarray}} \Big( Y \left(c^{k,q}_{\kappa}(v) \right) P_{q,p}(\Phi) Y^{\kappa} f+ c^{k,q}_{\kappa}(v) P_{(q_T,q_P+1,q_X),p}(\Phi) Y^{\kappa} f +c^{k,q}_{\kappa}(v) P_{q,p}(\Phi) YY^{\kappa} f \\ \nonumber
& & \hspace{-4.5mm} +c^{k,q}_{\kappa}(v) P_{(q_T,q_P,q_X+1),p+1}(\Phi) Y^{\kappa} f +c^{k,q}_{\kappa}(v) P_{(q_T,q_P,q_X),p+1}(\Phi) c(v) \partial Y^{\kappa} \Big) \lesssim \hspace{-0.2mm} \sum_{\begin{subarray}{} |q|+|\kappa| \leq |\gamma| \\ \hspace{1.5mm} |q| \leq |\gamma|-1 \\ \hspace{1mm} p \leq q_X+\kappa_T \end{subarray}} c^{k,q}_{\kappa}(v) P_{q,p}(\Phi) Y^{\kappa} f . \end{eqnarray} \end{proof}
\section{Energy and pointwise decay estimates}\label{sec4}
In this section, we recall classical energy estimates for both the electromagnetic field and the Vlasov field and how to obtain pointwise decay estimates from them. For that purpose, we need to prove Klainerman-Sobolev inequalities for velocity averages, similar to Theorem $8$ of \cite{FJS} or Theorem $1.1$ of \cite{dim4}, adapted to modified vector fields.
\subsection{Energy estimates}\label{energy} For the particle density, we will use the following approximate conservation law. \begin{Pro}\label{energyf} Let $H : [0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v \rightarrow \mathbb{R} $ and $g_0 : \mathbb{R}^3_x \times \mathbb{R}^3_v \rightarrow \mathbb{R}$ be two sufficiently regular functions and $F$ a sufficiently regular $2$-form defined on $[0,T[ \times \mathbb{R}^3$. Then, $g$, the unique classical solution of \begin{eqnarray} \nonumber T_F(g)&=&H \\ \nonumber g(0,.,.)&=&g_0, \end{eqnarray} satisfies the following estimate,
$$\forall \hspace{0.5mm} t \in [0,T[, \hspace{1cm} \| g \|_{L^1_{x,v}}(t)+ \sup_{u \in \mathbb{R}} \left\|\frac{v^{\underline{L}}}{v^0} g \right\|_{L^1(C_u(t))L^1_{v}} \leq 2 \| g_0 \|_{L^1_{x,v}}+2 \int_0^t \int_{\Sigma_s} \int_v |H | \frac{dv}{v^0}dxds.$$ \end{Pro}
\begin{proof}
The estimate follows from the divergence theorem, applied to $\int_v \frac{v^{\mu}}{v^0}|f|dv$ in $[0,t] \times \mathbb{R}^3$ and $V_u(t)$, for all $u \leq t$. We refer to Proposition $3.1$ of \cite{massless} for more details. \end{proof}
We consider, for the remainder of this section, a $2$-form $G$ and a $1$-form $J$, both defined on $[0,T[ \times \mathbb{R}^3$ and sufficiently regular, such that \begin{eqnarray} \nonumber \nabla^{\mu} G_{\mu \nu} & = & J_{\nu} \\ \nonumber \nabla^{\mu} {}^* \! G_{\mu \nu} & = & 0. \end{eqnarray}
We denote by $(\alpha,\underline{\alpha},\rho,\sigma)$ the null decomposition of $G$. As $\int_{\Sigma_0} r \rho(G)|(0,x)dx=+\infty$ when the total charge is non-zero, we cannot control norms such as $\left\|\sqrt{\tau_+} \rho \right\|_{L^2(\Sigma_t)}$ and we then separate the study of the electromagnetic field in two parts. \begin{itemize} \item The exterior of the light cone, where we propagate $L^2$ norms on the chargeless part $\widetilde{F}$ of $F$ (introduced, as $\overline{F}$, in Definition \ref{defpure1}), which has a finite initial weighted energy norm. The pure charge part $\overline{F}$ is given by an explicit formula, which describes directly its asymptotic behavior. As $F=\widetilde{F}+\overline{F}$, we are then able to obtain pointwise decay estimates on the null components of $F$. \item The interior of the light cone, where we can propagate $L^2$ weighted norms of $F$ since we control its flux on $C_0(t)$ with the bounds obtained on $\widetilde{F}$ in the exterior region. \end{itemize} We then introduce the following energy norms.
\begin{Def}\label{defMax1} Let $N \in \mathbb{N}$. We define, for $t \in [0,T[$, \begin{eqnarray}
\nonumber \mathcal{E}^0[G](t) & : = & \int_{\Sigma_t}\left( |\alpha|^2+|\underline{\alpha}|^2+2|\rho|^2+2|\sigma|^2 \right)dx +\sup_{u \leq t} \int_{C_u(t)} \left( |\alpha|^2+|\rho|^2+|\sigma|^2 \right) dC_u(t), \\
\nonumber \mathcal{E}_N^0[G](t) & : = & \sum_{\begin{subarray}{l} \hspace{0.5mm} Z^{\gamma} \in \mathbb{K}^{|\gamma|} \\ \hspace{1mm} |\gamma| \leq N \end{subarray}} \mathcal{E}^0_N[\mathcal{L}_{ Z^{\gamma}}(G)](t), \\
\nonumber \mathcal{E}^{S, u \geq 0}[G](t) & := & \int_{\Sigma^0_t} \tau_+ \left( |\alpha|^2+|\rho|^2+|\sigma|^2 \right)+\tau_- |\underline{\alpha}| dx+ \sup_{0 \leq u \leq t} \int_{C_u(t)} \tau_+ |\alpha|^2+\tau_-\left( |\rho|^2+|\sigma|^2 \right) d C_u(t) . \\
\nonumber \mathcal{E}_N[G](t) & := &\sum_{\begin{subarray}{l} \hspace{0.5mm} Z^{\gamma} \in \mathbb{K}^{|\gamma|} \\ \hspace{1mm} |\gamma| \leq N \end{subarray}} \mathcal{E}^{S, u \geq 0}_N[\mathcal{L}_{ Z^{\gamma}}(G)](t) \\
\nonumber \mathcal{E}^{S,u \leq 0}[G](t) & := & \int_{\overline{\Sigma}^{0}_t} \tau_+ \left( |\alpha|^2+|\rho|^2+|\sigma|^2 \right)+\tau_- |\underline{\alpha}| dx+ \sup_{ u \leq 0} \int_{C_u(t)} \tau_+ |\alpha|^2+\tau_-\left( |\rho|^2+|\sigma|^2 \right) d C_u(t) \\
\nonumber \mathcal{E}^{Ext}_N[G](t) & : = & \sum_{\begin{subarray}{l} \hspace{0.5mm} Z^{\gamma} \in \mathbb{K}^{|\gamma|} \\ \hspace{1mm} |\gamma| \leq N \end{subarray}} \mathcal{E}^{S,u \leq 0}_N[\mathcal{L}_{Z^{\gamma}}(G)](t). \end{eqnarray} \end{Def} The following estimates hold. \begin{Pro}\label{energyMax1} Let $\overline{S} := S+ \partial_t \mathds{1}_{u >0}+2 \tau_- \partial_t \mathds{1}_{u \leq 0}$. For all $ t \in [0,T[$, \begin{eqnarray}
\nonumber \mathcal{E}^0[G](t) & \leq & 2\mathcal{E}^0[G](0) + 8\int_0^t \int_{\Sigma_s} |G_{\mu 0} J^{\mu}| dx ds \\ \nonumber
\mathcal{E}^{S,u \leq 0}[G](t) & \leq & 6\mathcal{E}^{S,u \leq 0}[G](0) + 8\int_0^t \int_{\overline{\Sigma}^{0}_s } \left| \overline{S}^{\nu} G_{\nu \mu } J^{\mu} \right| dx ds \\ \nonumber
\mathcal{E}^{S,u \geq 0}[G](t) & \leq & 3\mathcal{E}^{S,u \leq 0}[\widetilde{G}](t) + 8\int_0^t \int_{\Sigma^{0}_s } \left| \overline{S}^{\nu} G_{\nu \mu } J^{\mu} \right| dx ds. \end{eqnarray} \end{Pro} \begin{proof} For the first inequality, apply the divergence theorem to $T_{\mu 0}[G]$ in $[0,t] \times \mathbb{R}^3$ and $V_u(t)$, for all $u \leq t$. Let us give more details for the other ones. Denoting $ T[G]$ by $T$ and using Lemma \ref{tensorcompo}, we have, if $u \leq 0$, \begin{eqnarray} \nonumber \nabla^{\mu} \left( \tau_- T_{\mu 0} \right) & = & \tau_-\nabla^{\mu} T_{\mu 0}-\frac{1}{2}\underline{L} \left( \tau_- \right) T_{L 0} \\ \nonumber
& = & \tau_- \nabla^{\mu} T_{\mu 0} -\frac{u}{2\tau_-} \left( \left| \alpha \right|^2+\left| \rho \right|^2+\left| \sigma \right|^2 \right) \hspace{2mm} \geq \hspace{2mm} \tau_- \nabla^{\mu} T_{\mu 0}. \end{eqnarray} Consequently, applying Corollary \ref{tensorderiv} and the divergence theorem in $V_{u_0}(t)$, for $u_0 \leq 0$, we obtain \begin{equation}\label{eq:1}
\int_{\overline{\Sigma}^{u_0}_t} \tau_- T_{00}dx + \frac{1}{\sqrt{2}} \int_{C_{u_0}(t)} \tau_-T_{L0}dC_{u_0}(t) \leq \int_{\overline{\Sigma}^{u_0}_0} \sqrt{1+r^2} T_{00}dx-\int_0^t \int_{\overline{\Sigma}^{u_0}_s } \tau_- G_{0 \nu} J^{\nu} dx ds.
\end{equation} On the other hand, as $\nabla^{\mu} S^{\nu}+\nabla^{\nu} S^{\mu}=2\eta^{\mu \nu}$ and ${T_{\mu}}^{\mu}=0$, we have \begin{eqnarray} \nonumber \nabla^{\mu} \left( T_{\mu \nu} S^{\nu} \right) & = & \nabla^{\mu} T_{\mu \nu}S^{\nu}+T_{\mu \nu} \nabla^{\mu} S^{\nu} \\ \nonumber & = & G_{\nu \lambda} J^{\lambda} S^{\nu} +\frac{1}{2} T_{\mu \nu} \left( \nabla^{\mu} S^{\nu}+\nabla^{\nu} S^{\mu} \right) \\ \nonumber & = & G_{\nu \lambda} J^{\lambda} S^{\nu}. \end{eqnarray} Applying again the divergence theorem in $V_{u_0}(t)$, for all $u_0 \leq 0$, we get \begin{equation}\label{eq:2}
\int_{\overline{\Sigma}^{u_0}_t} T_{0 \nu} S^{\nu} dx + \frac{1}{\sqrt{2}} \int_{C_{u_0}(t)} T_{L \nu} S^{\nu}dC_{u_0}(t) = \int_{\overline{\Sigma}^{u_0}_0} T_{0 \nu} S^{\nu} dx-\int_0^t \int_{\overline{\Sigma}^{u_0}_s } G_{\mu \nu} J^{\mu} S^{\nu} dx ds.
\end{equation} Using Lemma \ref{tensorcompo} and $2S=(t+r)L+(t-r) \underline{L}$, notice that \begin{flalign*}
& \hspace{1.3cm} 4\tau_-T_{00} = \tau_-\left( |\alpha|^2+|\underline{\alpha}|^2+2|\rho|^2+2|\sigma|^2 \right), \hspace{10mm} 4T_{0 \nu} S^{\nu} = (t+r)|\alpha|+(t-r)|\underline{\alpha}|+2t(|\rho|+|\sigma|), & \\
& \hspace{1.3cm} 2 \tau_- T_{L0}= \tau_- \left( |\alpha|^2+|\rho|^2+|\sigma|^2 \right), \hspace{23mm} 2 T_{L \nu} S^{\nu} = (t+r) |\alpha|^2+(t-r)|\rho|^2+(t-r)|\sigma|^2, & \end{flalign*} and then add twice \eqref{eq:1} to \eqref{eq:2}. The second estimate then follows and we now turn on the last one. Recall that $\nabla^{\mu} T_{\mu \nu} G=G_{\nu \lambda} J^{\lambda}$ and $\nabla^{\mu} \left( T_{\mu \nu} S^{\nu} \right) = G_{\nu \lambda} J^{\lambda} S^{\nu}$. Hence, by the divergence theorem applied in $[0,t] \times \mathbb{R}^3 \setminus V_{0}(t)$, we obtain \begin{equation}\label{eq:1bis}
\int_{\Sigma^0_t} \left( T_{00} +T_{0 \nu} S^{\nu} \right)dx = \frac{1}{\sqrt{2}} \int_{C_{0}(t)} \left( T_{L0} +T_{L \nu} S^{\nu} \right) d C_{0}(t) - \int_0^t \int_{\Sigma^{0}_s } G_{0 \nu} J^{\nu} + S^{\nu} G_{\nu \mu } J^{\mu} dx ds.
\end{equation}
By Lemma \ref{tensorcompo}, we have $4T_{00} = \left( |\alpha|^2+|\underline{\alpha}|^2+2|\rho|^2+2|\sigma|^2 \right) $, so that \begin{equation}\label{eq:1bbis}
4T_{00} +4T_{0 \nu} S^{\nu} \geq \tau_+|\alpha|^2+\tau_-|\underline{\alpha}|^2+\tau_+|\rho|^2+\tau_+|\sigma|^2 \geq 0 \hspace{1cm} \text{on} \hspace{5mm} \Sigma^{0}_t. \end{equation} Consequently, the divergence theorem applied in $ V_{u}(t) \setminus V_0(t)$, for $0 \leq u \leq t$, gives \begin{equation}\label{eq:2bis} \frac{1}{\sqrt{2}} \int_{C_{u}(t)} \left( T_{L0} +T_{L \nu} S^{\nu} \right) dC_u(t) \leq \frac{1}{\sqrt{2}} \int_{C_{0}(t)} \left( T_{L0} +T_{L \nu} S^{\nu} \right) d C_0(t) - \int_{V_u(t) \setminus V_0(t)} \left( G_{0 \nu} J^{\nu} + S^{\nu} G_{\nu \mu } J^{\mu} \right).
\end{equation}
Not now that $ T_{L0} +T_{L \nu} S^{\nu} \geq \tau_+|\alpha|^2+\tau_-|\rho|^2+\tau_-|\sigma|^2$ if $u \geq 0$ since \begin{flalign*}
& \hspace{0.8cm} 2 T_{L0}= |\alpha|^2+|\rho|^2+|\sigma|^2 \hspace{7mm} \text{and} \hspace{7mm} 2 T_{L \nu} S^{\nu} = (t+r) |\alpha|^2+(t-r)|\rho|^2+(t-r)|\sigma|^2. & \end{flalign*} It then remains to take the $\sup$ over all $0 \leq u \leq t$ in \eqref{eq:2bis}, to combine it with \eqref{eq:1bis}, \eqref{eq:1bbis} and to remark that \begin{eqnarray}
\nonumber 2\int_{C_{0}(t)} T_{L0} +T_{L \nu} S^{\nu} d C_0(t) & \leq & \int_{C_{0}(t)} |\rho|^2+|\sigma|^2 d C_{0}(t)+\int_{C_0(t)} \tau_+ |\alpha|^2 d C_0(t) \\ \nonumber & \leq & \mathcal{E}^{S, u \leq 0}[\widetilde{G}](t), \end{eqnarray} since $G=\widetilde{G}$ on $C_0(t)$. \end{proof} \subsection{Pointwise decay estimates} \subsubsection{Decay estimates for velocity averages} As the set of our commutation vector fields is not $\widehat{\mathbb{P}}_0$, we need to modify the following standard Klainerman-Sobolev inequality, which was proved in \cite{FJS} (see Theorem $8$).
\begin{Pro}\label{KSstandard} Let $g$ be a sufficiently regular function defined on $[0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v$. Then, for all $(t,x) \in [0,T[ \times \mathbb{R}^3$,
$$\forall \hspace{0.5mm} (t,x) \in [0,T[ \times \mathbb{R}^3, \hspace{1cm} \int_{v \in \mathbb{R}^3} |g(t,x,v)| dv \lesssim \frac{1}{\tau_+^2 \tau_-} \sum_{\begin{subarray}{l} \widehat{Z}^{\beta} \in \widehat{\mathbb{P}}_0^{|\beta|} \\ \hspace{1mm} |\beta| \leq 3 \end{subarray}}\|\widehat{Z}^{\beta} g \|_{L^1_{x,v}}(t).$$ \end{Pro}
We need to rewrite it using the modified vector fields. For the remainder of this section, $g$ will be a sufficiently regular function defined on $[0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v$. We also consider $F$, a regular $2$-form, so that we can consider the $\Phi$ coefficients introduced in Definition \ref{defphi} and we suppose that they satisfy the following pointwise estimates, with $M_1 \geq 7$ a fixed integer. For all $(t,x,v) \in [0,T[ \times \mathbb{R}^3 \times \mathbb{R}^3$,
$$|Y \Phi|(t,x,v) \lesssim \log^{\frac{7}{2}}(1+\tau_+), \hspace{8mm} |\Phi|(t,x,v) \lesssim \log^2(1+\tau_+) \hspace{8mm} \text{and} \hspace{8mm} \sum_{ |\kappa| \leq 3} |Y^{\kappa} \Phi|(t,x,v) \lesssim \log^{M_1}(1+\tau_+).$$
\begin{Pro}\label{KS1} For all $(t,x) \in [0,T[ \times \mathbb{R}^3$,
$$\tau_+^2 \tau_- \int_{v \in \mathbb{R}^3} |g(t,x,v)| dv \lesssim \sum_{ |\xi|+|\beta| \leq 3 } \left\| P^X_{\xi}(\Phi)Y^{\beta} g \right\|_{L^1_{x,v}} \hspace{-0.8mm} (t)+ \sum_{ |\kappa| \leq \min(2+\kappa_T,3)} \sum_{z \in \mathbf{k}_1}\frac{\log^{6M_1}(3+t)}{1+t} \left\| z Y^{\kappa} g \right\|_{L^1_{x,v}} \hspace{-0.8mm} (t) .$$ \end{Pro} \begin{Rq}
This inequality is suitable for us since we will bound $\left\| P^X_{\xi}(\Phi)Y^{\beta} g \right\|_{L^1_{x,v}} $ without any growth in $t$. Moreover, observe that $Y^{\kappa}$ contains at least a translation if $|\kappa|=3$, which is compatible with our hierarchy on the weights $z \in \mathbf{k}_1$ (see Remark \ref{rqjustifnorm}). \end{Rq} \begin{proof}
Let $(t,x) \in [0,T[ \times \mathbb{R}^n$. Consider first the case $|x| \leq \frac{1+t}{2}$, so that, with $\tau := 1+t$,
$$\forall \hspace{0.5mm} |y| \leq \frac{1}{4}, \hspace{3mm} \tau \leq 10(1+|t-|x+\tau y||).$$ For a sufficiently regular function $h$, we then have, using Lemmas \ref{goodderiv} and then \ref{lift2}, \begin{eqnarray} \nonumber
\left| \partial_{y^i} \left( \int_v |h|(t,x+\tau y,v) dv \right) \right| & = & \left| \tau \partial_i \int_v |h|(t,x+\tau y,v) dv \right| \\ \nonumber
& \lesssim & \left| (1+|t-|x+\tau y||) \partial_i \int_v |h|(t,x+\tau y,v) dv \right| \\ \nonumber & \lesssim & \sum_{Z \in \mathbb{K}} \left| Z \int_v |h|(t,x+\tau y,v) dv \right| \\ \nonumber & \lesssim & \sum_{\begin{subarray}{l} |\xi|+|\beta| \leq 1 \\ \hspace{3mm} p \leq 1 \end{subarray}} \sum_{z \in \mathbf{k}_1} \int_v \hspace{-0.5mm} \left( |P^X_{\xi}(\Phi) Y^{\beta} h|+\frac{\log^7 (1+\tau_+)}{\tau_+} |z \partial^p_t h| \right) (t,x+\tau y,v) dv . \end{eqnarray}
Using a one dimensional Sobolev inequality, we obtain, for $\delta=\frac{1}{4 \sqrt{3}}$ (so that $|y| \leq \frac{1}{4}$ if $|y^i| \leq \delta$ for all $1 \leq i \leq 3$), \begin{eqnarray}
\nonumber \int_v |g|(t,x,v) dv & \lesssim & \sum_{n=0}^1 \int_{|y^1| \leq \delta} \left| \left(\partial_{y^1} \right)^n \int_v |g|(t,x+\tau(y^1,0,0),v) dv \right| dy^1 \\ \nonumber
& \lesssim & \sum_{\begin{subarray}{l} |\xi|+|\beta| \leq 1 \\ \hspace{3mm} p \leq 1 \\ \hspace{2.5mm} z \in \mathbf{k}_1 \end{subarray}} \int_{|y^1| \leq \delta} \int_v \left( |P^X_{\xi}(\Phi) Y^{\beta}g|+\frac{\log^7 ( 3+t)}{1+t} |z \partial^p_t g| \right) (t,x+\tau (y^1,0,0),v) dv dy^1 \hspace{-0.7mm}. \end{eqnarray}
Repeating the argument for $y^2$ and the functions $\int_v P^X_{\xi}(\Phi) Y^{\beta}g dv$ and $ \int_v z \partial^p_t g dv$, we get, as $|z| \leq 2t$ in the region considered and dropping the dependence in $(t,x+\tau(y^1,y^2,0),v)$ of the functions in the integral,
$$\int_v |g|(t,x,v) dv \lesssim \sum_{\begin{subarray}{l} |\xi|+|\beta| \leq 2 \\ \hspace{2.5mm} z \in \mathbf{k}_1 \end{subarray}} \sum_{\begin{subarray}{l} |\zeta|+|\kappa| \leq 2 \\ |\kappa| \leq 1+\kappa_T \end{subarray}} \int_{|y^1| \leq \delta} \int_{|y^2| \leq \delta} \int_v |P^X_{\xi}(\Phi) Y^{\beta} g|+\frac{\log^{14} (3+t)}{1+t} |z P^X_{\zeta}(\Phi) Y^{\kappa} g| dv dy^1 dy^2.$$ Repeating again the argument for the variable $y^3$, we finally obtain
$$\int_v |g|(t,x,v) dv \lesssim \sum_{\begin{subarray}{l} |\xi|+|\beta| \leq 3 \\ \hspace{2.5mm} z \in \mathbf{k}_1 \end{subarray}} \sum_{\begin{subarray}{l} |\zeta|+|\kappa| \leq 3 \\ |\kappa| \leq 2+\kappa_T \end{subarray}} \int_{|y| \leq \frac{1}{4}} \int_v |P^X_{\xi}(\Phi) Y^{\beta} g|+ \frac{\log^{21} (3+t)}{1+t} |z P_{\zeta}^X(\Phi) Y^{\kappa} g|dv(t,x+\tau y)dy.$$
It then remains to remark that $\left| P^X_{\zeta}(\Phi) \right| \lesssim \log^{3M_1}(3+t)$ on the domain of integration and to make the change of variables $z=\tau y$. Note now that one can prove similarly that, for a sufficiently regular function $h$, \begin{equation}\label{eq:sobsphere}
\int_v |h|(t,r,\theta,\phi)dv \lesssim \sum_{\begin{subarray}{l} |\xi|+|\beta| \leq 2 \\ \hspace{1mm} z \in \mathbf{k}_1 \end{subarray}} \sum_{ |\kappa| \leq \min(1+\kappa_T,2)} \int_{\mathbb{S}^2} \int_v |P_{\xi}^X(\Phi)Y^{\beta}h|+\frac{\log^{14+2M_1} (1+\tau_+)}{\tau_+}|zY^{\kappa} h | dv d\mathbb{S}^2 (t,r). \end{equation} Indeed, by a one dimensional Sobolev inequality, we have
$$\int_v |f|(t,r,\theta,\phi,v)dv \lesssim \sum_{r=0}^1 \int_{\omega_1} \left| \left(\partial_{\omega_1} \right)^r \int_v |f|(t,r,\theta+\omega_1,\phi,v)dv \right| d\omega_1.$$
Then, since $\partial_{\omega_1}$ ( and $\partial_{\omega_2}$) can be written as a combination with bounded coefficients of the rotational vector fields $\Omega_{ij}$, we can repeat the previous argument. Finally, let us suppose that $\frac{1+t}{2} \leq |x |$. We have, using again Lemmas \ref{goodderiv} and \ref{lift2},
\begin{eqnarray}
\nonumber |x|^2 \tau_- \int_v |g|(t,x,v)dv & = & -|x|^2\int_{|x|}^{+ \infty} \partial_r \left( \tau_- \int_v |g|(t,r,\theta,\phi,v)dv \right)dr \\ \nonumber
& \lesssim & \int_{|x|}^{+ \infty} \int_v | g|(t,r,\theta,\phi,v)dvr^2dr+\int_{|x|}^{+ \infty}\left| \tau_- \partial_r \int_v |g|(t,r,\theta,\phi,v)dv \right| r^2 dr \\ \nonumber
& \leq & \sum_{\begin{subarray}{l} |\xi|+|\beta| \leq 1 \\ \hspace{3mm} p \leq 1 \end{subarray}} \sum_{ w \in \mathbf{k}_1} \int_{0}^{+ \infty} \int_v \hspace{-0.5mm} \left( \hspace{-0.5mm} |P^X_{\xi}(\Phi) Y^{\beta} g|+\frac{\log^7 (3+t)}{1+t} |w \partial^p_t g| \right)(t,r,\theta,\phi,v) dv r^2 dr.
\end{eqnarray}
It then remains to apply \eqref{eq:sobsphere} to the functions $P^X_{\xi}(\Phi) Y^{\beta} g$ and $z \partial^p_t g$ and to remark that $|z| \leq 2\tau_+$. \end{proof} A similar, but more general, result holds. \begin{Cor}\label{KS2} Let $z \in \mathbf{k}_1$ and $j \in \mathbb{N}$. Then, for all $(t,x) \in [0,T[ \times \mathbb{R}^3$, \begin{eqnarray}
\nonumber \int_{v \in \mathbb{R}^n} |z|^j|g(t,x,v)| dv & \lesssim & \frac{1}{\tau_+^2 \tau_-} \sum_{w \in \mathbf{k}_1} \Bigg( \sum_{d=0}^{\min(3,j)} \sum_{|\xi|+|\beta| \leq 3-d} \log^{2d}(3+t) \left\| w^{j-d} P^X_{\xi}(\Phi) Y^{\beta} g \right\|_{L^1_{x,v}} \hspace{-0.8mm} (t) \\ \nonumber
& & \hspace{4.5cm} +\frac{\log^{6M_1}(3+t)}{1+t} \sum_{ |\kappa| \leq \min(2+\kappa_T,3) } \|w^{j+1} Y^{\kappa} f \|_{L^1_{x,v}} \hspace{-0.8mm} (t) \Bigg) . \end{eqnarray} \end{Cor} \begin{proof} One only has to follow the proof of Proposition \ref{KS1} and to use Remark \eqref{lift3} instead of Lemma \ref{lift2}). \end{proof} A weaker version of this inequality will be used in Subsection \ref{subsecH}. \begin{Cor}\label{KS3} Let $z \in \mathbf{k}_1$ and $j \in \mathbb{N}$. Then, for all $(t,x) \in [0,T[ \times \mathbb{R}^3$, \begin{eqnarray}
\nonumber \int_{v \in \mathbb{R}^n} |z|^j|g(t,x,v)| dv & \lesssim & \frac{1}{\tau_+^2 \tau_-} \sum_{w \in \mathbf{k}_1} \Bigg( \sum_{d=0}^{\min(3,j)} \sum_{|\beta| \leq 3-d} \log^{2d+M_1}(3+t) \left\| w^{j-d} Y^{\beta} g \right\|_{L^1_{x,v}} \hspace{-0.8mm} (t) \\ \nonumber
& & \hspace{4.5cm} +\frac{\log^{6M_1}(3+t)}{1+t} \sum_{ |\kappa| \leq \min(2+\kappa_T,3) } \|w^{j+1} Y^{\kappa} f \|_{L^1_{x,v}} \hspace{-0.8mm} (t) \Bigg) . \end{eqnarray} \end{Cor} \begin{proof} Start by applying Corollary \ref{KS2}. It remains to bound the terms of the form
$$\left\| w^{j-d} P^X_{\xi}(\Phi) Y^{\beta} g \right\|_{L^1_v L^1(\Sigma_t)}, \hspace{1cm} \text{with} \hspace{1cm} d \leq \min(3,j), \hspace{5mm} |\xi|+|\beta| \leq 3-d \hspace{5mm} \text{and} \hspace{5mm} |\xi| \geq 1.$$
For this, we divide $\Sigma_t$ in two regions, the one where $r \leq 1+2t$ and its complement. As $|P^X_{\xi}(\Phi)| \lesssim \log^{M_1}(1+\tau_+)$ and $\tau_+ \lesssim 1+t$ if $r \leq 1+2t$, we have
$$\left\| w^{j-d} P^X_{\xi}(\Phi) Y^{\beta} g \right\|_{L^1_v L^1(|y| \leq 2t)} \lesssim \log^{M_1}(3+t) \left\| w^{j-d} Y^{\beta} g \right\|_{L^1_v L^1(\Sigma_t)}.$$
Now recall from Remark \ref{rqweights1} that $1+r \lesssim \sum_{z_0 \in \mathbf{k}_1} |z_0|$ and $|P^X_{\xi}(\Phi)|(1+r)^{-1} \lesssim \frac{\log^{M_1}(3+t)}{1+t}$ if $r \geq 1+ 2t$, so that
$$\left\| w^{j-d} P^X_{\xi}(\Phi) Y^{\beta} g \right\|_{L^1_v L^1(|y| \geq 2t)} \lesssim \frac{\log^{M_1}(3+t)}{1+t} \sum_{z_0 \in \mathbf{k}_1} \left\| z_0^{j+1} Y^{\beta} g \right\|_{L^1_v L^1(\Sigma_t)}.$$
The result follows from $|\beta| \leq 2-d \leq 2+\beta_T$. \end{proof}
We are now interested in adapting Theorem $1.1$ of \cite{dim4} to the modified vector fields.
\begin{Th}\label{decayopti}
Suppose that $\sum_{|\kappa| \leq 3} \|Y^{\kappa} \Phi\|_{L^{\infty}_{x,v}} (0) \lesssim 1$. Let $H : [0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v \rightarrow \mathbb{R}$ and $h_0 : \mathbb{R}^3_x \times \mathbb{R}^3_v \rightarrow \mathbb{R}$ be two sufficiently regular functions and $h$ the unique classical solution of \begin{eqnarray} \nonumber T_F(h) & = & H \\ \nonumber h(0,.,.) & = & h_0. \end{eqnarray}
Consider also $z \in \mathbf{k}_1$ and $j \in \mathbb{N}$. Then, for all $(t,x) \in [0,T[ \times \mathbb{R}^3$ such that $t \geq |x|$, \begin{eqnarray}
\nonumber \tau_+^3\int_v |z^jh|(t,x,v)\frac{dv}{(v^0)^2} & \lesssim & \sum_{ |\beta| \leq 3 } \| (1+r)^{|\beta|+j} \partial^{\beta}_{t,x} h \|_{L^1_x L^1_v} (0) \\ \nonumber
& &+\sum_{\begin{subarray}{} |\xi|+|\beta| \leq 3 \\ \hspace{1.5mm} w \in \mathbf{k}_1 \end{subarray} } \hspace{1.5mm} \sum_{\begin{subarray}{} 0 \leq d \leq 3 \\ \delta \in \{0,1\} \end{subarray} }\frac{ \log^{2d}(3+t)}{\sqrt{1+t}^{\delta}} \int_0^t \int_{\Sigma_s} \int_v \left| T_F \left(w^{j-d+\delta} P^X_{\xi}(\Phi)Y^{\beta} h \right) \right| \frac{dv}{v^0} dx ds, \end{eqnarray}
where $|\xi|=0$ and $|\beta| \leq \min(2+\beta_T,3)$ if $\delta=1$. \end{Th} \begin{proof}
If $|x| \leq \frac{t}{2}$, the result follows from Corollary \ref{KS2} and the energy estimate of Proposition \ref{energyf}. If $\frac{t}{2} \leq |x| \leq t$, we refer to Section $5$ of \cite{dim4}, where Lemma $5.2$ can be rewritten in the same spirit as we rewrite Proposition \ref{KSstandard} with modified vector fields. \end{proof} To deal with the exterior, we use the following result. \begin{Pro}\label{decayopti2}
For all $(t,x) \in [0,T[ \times \mathbb{R}^3$ such that $|x| \geq t$, we have
$$\int_v |g|(t,x,v) \frac{dv}{(v^0)^2} \lesssim \frac{1}{\tau_+} \sum_{w \in \mathbf{k}_1} \int_v |w|| g|(t,x,v) dv.$$ \end{Pro} \begin{proof}
Let $|x| \geq t$. If $|x| \leq 1$, $\tau_+ \leq 3$ and the estimate holds. Otherwise, $\tau_+ \leq 3|x|$ so, as $\left( x^i-t \frac{v^i}{v^0} \right) \in \mathbf{k}_1$ and
$$\left| x-t\frac{v}{v^0} \right| \geq |x|-t\frac{|v|}{v^0} \geq |x| \frac{(v^0)^2-|v|^2}{v^0(v^0+|v|)} \geq \frac{|x|}{2(v^0)^2}, \hspace{3mm} \text{we have} \hspace{3mm} \int_v |g|(t,x,v) \frac{dv}{(v^0)^2} \lesssim \frac{1}{|x|} \sum_{w \in \mathbf{k}} \int_v |w||g|(t,x,v)dv.$$ \end{proof} \begin{Rq} Using $1 \lesssim v^0 v^{\underline{L}}$ and Lemma \ref{weights1}, we can obtain a similar inequality for the interior of the light cone, at the cost of a $\tau_-$-loss. Note however that because of the presence of the weights $w \in \mathbf{k}_1$, this estimate, combined with Corollary \ref{KS2}, is slightly weaker than Theorem \ref{decayopti}. During the proof, this difference will lead to a slower decay rate insufficient to close the energy estimates. \end{Rq}
\subsubsection{Decay estimates for the electromagnetic field}
We start by presenting weighted Sobolev inequalities for general tensor fields. Then we will use them in order to obtain improved decay estimates for the null components of a $2$-form\footnote{Note however that our improved estimates on the components $\alpha$, $\rho$ and $\sigma$ require the $2$-form $G$ to satisfy $\nabla^{\mu} {}^* \! G_{\mu \nu} =0$.}. In order to treat the interior of the light cone (or rather the domain in which $|x| \leq 1+\frac{1}{2}t$), we will use the following result.
\begin{Lem}\label{decayint} Let $U$ be a smooth tensor field defined on $[0,T[ \times \mathbb{R}^3$. Then,
$$\forall \hspace{0.5mm} t \in [0,T[, \hspace{8mm} \sup_{|x| \leq 1+\frac{t}{2}} |U(t,x)| \lesssim \frac{1}{(1+t)^2} \sum_{|\gamma| \leq 2} \| \sqrt{\tau_-} \mathcal{L}_{Z^{\gamma}}(U)(t,y) \|_{L^2 \left( |y| \leq 2+\frac{3}{4}t \right)}.$$ \end{Lem} \begin{proof}
As $|\mathcal{L}_{Z^{\gamma}}(U)| \lesssim \sum_{|\beta| \leq |\gamma|} \sum_{\mu, \nu} | Z^{\beta} (U_{\mu \nu})|$, we can restrict ourselves to the case of a scalar function. Let $t \in \mathbb{R}_+$ and $|x| \leq 1+ \frac{1}{2}t$. Apply a standard $L^2$ Sobolev inequality to $V: y \mapsto U(t,x+\frac{1+t}{4}y)$ and then make a change of variables to get
$$|U(t,x)|=|V(0)| \lesssim \sum_{|\beta| \leq 2} \| \partial_x^{\beta} V \|_{L^2_y(|y| \leq 1)} \lesssim \left( \frac{1+t}{4} \right)^{-\frac{3}{2}} \sum_{|\beta| \leq 2} \left( \frac{1+t}{4} \right)^{|\beta|} \| \partial_x^{\beta} U(t,.) \|_{L^2_y(|y-x| \leq \frac{1+t}{4})}.$$
Observe now that $|y-x| \leq \frac{1+t}{4}$ implies $|y| \leq 2+\frac{3}{4}t$ and that $1+t \lesssim \tau_-$ on that domain. By Lemma \ref{goodderiv} and since $[Z, \partial] \in \mathbb{T} \cup \{0 \}$, it follows
$$( 1+t )^{|\beta|+\frac{1}{2}} \| \partial_x^{\beta} U(t,.) \|_{L^2_y(|y-x| \leq \frac{1+t}{4})} \hspace{1.5mm} \lesssim \hspace{1.5mm} \| \tau_-^{|\beta|+\frac{1}{2}} \partial_x^{\beta} U(t,.) \|_{L^2_y(|y| \leq 2+\frac{3}{4}t)} \hspace{1.5mm} \lesssim \hspace{1.5mm} \sum_{|\gamma| \leq |\beta|} \| \sqrt{\tau_-} Z^{\gamma} U(t,.) \|_{L^2_y(|y| \leq 2+\frac{3}{4}t)}.$$ \end{proof} For the remaining region, we have the three following inequalities, coming from Lemma $2.3$ (or rather from its proof for the second estimate) of \cite{CK}. We will use, for a smooth tensor field $V$, the pointwise norm
$$ |V|^2_{\mathbb{O},k} := \sum_{p \leq k} \sum_{\Omega^{\gamma} \in \mathbb{O}^{p}} | \mathcal{L}_{\Omega^{\gamma}}(V)|^2.$$ \begin{Lem}\label{Sob} Let $U$ be a sufficiently regular tensor field defined on $\mathbb{R}^3$. Then, for $t \in \mathbb{R}_+$, \begin{eqnarray}
\nonumber \forall \hspace{0.5mm} |x| \geq \frac{t}{2}+1, \hspace{10mm} |U(x)| & \lesssim & \frac{1}{|x|\tau_-^{\frac{1}{2}}} \left( \int_{ |y| \geq \frac{t}{2}+1} |U(y)|^2_{\mathbb{O},2}+\tau_-^2|\nabla_{\partial_r} U(y) |^2_{\mathbb{O},1} dy \right)^{\frac{1}{2}}, \\ \nonumber
\forall \hspace{0.5mm} |x| > t, \hspace{10mm} |U(x)| & \lesssim & \frac{1}{|x|\tau_-^{\frac{1}{2}}} \left( \int_{ |y| \geq t} |U(y)|^2_{\mathbb{O},2}+\tau_-^2|\nabla_{\partial_r} U(y) |^2_{\mathbb{O},1} dy \right)^{\frac{1}{2}}, \\ \nonumber
\forall \hspace{0.5mm} x \neq 0, \hspace{10mm} |U(x)| & \lesssim & \frac{1}{|x|^{\frac{3}{2}}} \left( \int_{|y| \geq |x|} |U(y)|^2_{\mathbb{O},2}+|y|^2|\nabla_{\partial_r} U(y) |^2_{\mathbb{O},1} dy \right)^{\frac{1}{2}}. \end{eqnarray} \end{Lem}
Recall that $G$ and $J$ satisfy \begin{eqnarray} \nonumber \nabla^{\mu} G_{\mu \nu} & =& J_{\nu} \\ \nonumber \nabla^{\mu} {}^* \! G_{ \mu \nu } & = & 0 \end{eqnarray} and that $(\alpha, \underline{\alpha}, \rho, \sigma)$ denotes the null decomposition of $G$. Before proving pointwise decay estimates on the components of $G$, we recall the following classical result and we refer, for instance, to Lemma $D.1$ of \cite{massless} for a proof. Concretely, it means that $\mathcal{L}_{\Omega}$, for $\Omega \in \mathbb{O}$, $\nabla_{\partial_r}$, $\nabla_{\underline{L}}$ and $\nabla_L$ commute with the null decomposition. \begin{Lem}\label{randrotcom} Let $\Omega \in \mathbb{O}$. Then, denoting by $\zeta$ any of the null component $\alpha$, $\underline{\alpha}$, $\rho$ or $\sigma$, $$ [\mathcal{L}_{\Omega}, \nabla_{\partial_r}] G=0, \hspace{1.2cm} \mathcal{L}_{\Omega}(\zeta(G))= \zeta ( \mathcal{L}_{\Omega}(G) ) \hspace{1.2cm} \text{and} \hspace{1.2cm} \nabla_{\partial_r}(\zeta(G))= \zeta ( \nabla_{\partial_r}(G) ) .$$ Similar results hold for $\mathcal{L}_{\Omega}$ and $\nabla_{\partial_t}$, $\nabla_L$ or $\nabla_{\underline{L}}$. For instance, $\nabla_{L}(\zeta(G))= \zeta ( \nabla_{L}(G) )$. \end{Lem} \begin{Pro}\label{decayMaxwell} We have, for all $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^3$, \begin{eqnarray}
\nonumber |\rho|(t,x) , \hspace{2mm} |\sigma|(t,x) & \lesssim & \frac{ \sqrt{\mathcal{E}_2[G](t)+\mathcal{E}_2^{Ext}[G](t)}}{\tau_+^{\frac{3}{2}}\tau_-^{\frac{1}{2}}}, \\ \nonumber
|\alpha|(t,x) & \lesssim & \frac{\sqrt{ \mathcal{E}_2[G](t)+\mathcal{E}_2^{Ext}[G](t)}+\sum_{|\kappa| \leq 1} \|r^{\frac{3}{2}} \mathcal{L}_{Z^{\kappa}}(J)_A\|_{L^2(\Sigma_t)}}{\tau_+^2} \\ \nonumber
|\underline{\alpha}|(t,x) & \lesssim & \min\left( \frac{\sqrt{\mathcal{E}_2[G](t)+\mathcal{E}_2^{Ext}[G](t)}}{\tau_+ \tau_-}, \frac{\sqrt{\mathcal{E}^0_2[G](t)}}{\tau_+ \tau_-^{\frac{1}{2}}} \right). \end{eqnarray}
Moreover, if $|x| \geq \max (t,1)$, the term involving $\mathcal{E}_2[G](t)$ on the right hand side of each of these three estimates can be removed. \end{Pro}
\begin{Rq} As we will have a small loss on $\mathcal{E}_2[F]$ and not on $\mathcal{E}^0_2[F]$, the second estimate on $\underline{\alpha}$ is here for certain situations, where we will need a decay rate of degree at least $1$ in the $t+r$ direction. \end{Rq}
\begin{proof}
Let $(t,x) \in [0,T[ \times \mathbb{R}^3$. If $|x| \leq 1+\frac{1}{2}t$, $\tau_- \leq \tau_+ \leq 2+2t$ so the result immediately follows from Lemma \ref{decayint}. We then focus on the case $|x| \geq 1+\frac{t}{2}$. During this proof, $\Omega^{\beta}$ will always denote a combination of rotational vector fields, i.e. $\Omega^{\beta} \in \mathbb{O}^{|\beta|}$. Let $\zeta$ be either $\alpha$, $ \rho$ or $ \sigma$. As, by Lemma \ref{randrotcom}, $\nabla_{\partial_r}$ and $\mathcal{L}_{\Omega}$ commute with the null decomposition, we have, applying Lemma \ref{Sob},
$$r^3 \tau_- |\zeta|^2 \lesssim \int_{ |y| \geq \frac{t}{2}+1} |\sqrt{r} \zeta |^2_{\mathbb{O},2}+\tau_-^2|\nabla_{\partial_r} (\sqrt{r} \zeta) |_{\mathbb{O},1}^2 dy \lesssim \sum_{\begin{subarray}{} |\gamma| \leq 2 \\ |\beta| \leq 1 \end{subarray}} \int_{ |y| \geq \frac{t}{2}+1} r| \zeta ( \mathcal{L}_{Z^{\gamma}} (G) |^2+r\tau_-^2| \zeta ( \mathcal{L}_{\Omega^{\beta}} (\nabla_{\partial_r} G)) |^2 dy.$$ As $\nabla_{\partial_r}$ commute with $\mathcal{L}_{\Omega}$ and since $\nabla_{\partial_r}$ commute with the null decomposition (see Lemma \ref{randrotcom}), we have, using $2\partial_r= L-\underline{L}$ and \eqref{eq:zeta}, \begin{equation}\label{zetaeq2}
| \zeta ( \mathcal{L}_{\Omega} (\nabla_{\partial_r} G)) |+| \zeta ( \nabla_{\partial_r} G) | \hspace{2mm} \lesssim \hspace{2mm} | \nabla_{\partial_r} \zeta ( \mathcal{L}_{\Omega} (G) |+| \nabla_{\partial_r} \zeta ( G) | \hspace{2mm} \lesssim \hspace{2mm} \frac{1}{\tau_-}\sum_{ |\gamma| \leq 2} | \zeta ( \mathcal{L}_{Z^{\gamma}} (G) |.
\end{equation} As $\tau_+ \lesssim r \leq \tau_+$ in the region considered, it finally comes
$$\tau_+^3 \tau_- |\zeta|^2 \lesssim \sum_{|\gamma| \leq 2} \int_{ |y| \geq \frac{t}{2}+1} \tau_+| \zeta ( \mathcal{L}_{Z^{\gamma}} (G) |^2 dx \lesssim \mathcal{E}_2[G](t)+\mathcal{E}^{Ext}_2[G](t).$$ Let us improve now the estimate on $\alpha$. As, by Lemma \ref{basiccom}, $\nabla^{\mu} \mathcal{L}_{\Omega} (G)_{\mu \nu} = \mathcal{L}_{\Omega}(J)_{\nu}$ and $\nabla^{\mu} {}^* \! \mathcal{L}_{\Omega} (G)_{\mu \nu} = 0$ for all $\Omega \in \mathbb{O}$, we have according to Lemma \ref{maxwellbis} that
$$\forall \hspace{0.5mm} |\beta| \leq 1, \hspace{15mm} \nabla_{\underline{L}} \alpha(\mathcal{L}_{\Omega^{\beta}} (G))_A=\frac{1}{r}\alpha(\mathcal{L}_{\Omega^{\beta}} (G))_A-\slashed{\nabla}_{e_A}\rho (\mathcal{L}_{\Omega^{\beta}} (G)) +\varepsilon_{AB} \slashed{\nabla}_{e_B} \sigma (\mathcal{L}_{\Omega^{\beta}} (G))+\mathcal{L}_{\Omega^{\beta}}(J)_A.$$ Thus, using \eqref{eq:zeta}, we obtain, for all $\Omega \in \mathbb{O}$, \begin{equation}\label{alphaeq2}
| \alpha ( \nabla_{\partial_r} G) |+| \alpha ( \mathcal{L}_{\Omega} (\nabla_{\partial_r} G)) | \lesssim \left|J_A \right| +\left| \mathcal{L}_{\Omega} (J)_A \right|+ \frac{1}{r}\sum_{ |\gamma| \leq 2} \left( | \alpha ( \mathcal{L}_{Z^{\gamma}} (G) |+| \rho ( \mathcal{L}_{Z^{\gamma}} (G) |+| \sigma ( \mathcal{L}_{Z^{\gamma}} (G) | \right).
\end{equation} Hence, utilizing this time the third inequality of Lemma \ref{Sob} and \eqref{alphaeq2} instead of \eqref{zetaeq2}, we get
$$\tau_+^4 |\alpha|^2 \lesssim r^4 |\alpha|^2 \lesssim \int_{ |y| \geq |x|} |\sqrt{r} \alpha|^2_{\mathbb{O},2}+r^2|\nabla_{\partial_r} ( \sqrt{r} \alpha) |_{\mathbb{O},1}^2 dy \lesssim \mathcal{E}_2[G](t)+\mathcal{E}_2^{Ext}[G](t)+\sum_{|\kappa| \leq 1} \|r^{\frac{3}{2}} \mathcal{L}_{Z^{\kappa}}(J)_A\|^2_{L^2(\Sigma_t)}.$$ Using the same arguments as previously, one has \begin{eqnarray}
\nonumber \int_{|y| \geq \frac{t}{2}+1} \left| \underline{\alpha} \right|^2_{\mathbb{O},2} +\tau_-^2 \left| \nabla_{\partial_r} \underline{\alpha} \right|_{\mathbb{O},1}^2 dy & \lesssim & \mathcal{E}^0_2[G](t), \\ \nonumber
\int_{ |y| \geq \frac{t}{2}+1} \left| \sqrt{\tau_-} \underline{\alpha} \right|^2_{\mathbb{O},2} +\tau_-^2 \left| \nabla_{\partial_r} \left( \sqrt{\tau_-} \underline{\alpha} \right) \right|_{\mathbb{O},1}^2 dy & \lesssim & \mathcal{E}_2[G](t)+\mathcal{E}_2^{Ext}[G](t)
\end{eqnarray}
and a last application of Lemma \ref{Sob} gives us the result. The estimates for the region $|x| \geq \max (t,1)$ can be obtained similarly, using the second inequality of Lemma \ref{Sob} instead of the first one. \end{proof} Losing two derivatives more, one can improve the decay rate of $\rho$ and $\sigma$ near the light cone. \begin{Pro}\label{Probetteresti} Let $M \in \mathbb{N}$, $C>0$ and assume that
\begin{equation}\label{eq:imprdecay} \forall \hspace{0.5mm} (t,x) \in [0,T[ \times \mathbb{R}^3, \hspace{1cm} \sum_{|\gamma| \leq 1} |\mathcal{L}_{Z^{\gamma}}(G)|(t,x)+|J_{\underline{L}}|(t,x) \hspace{2mm} \leq \hspace{2mm} C\frac{\log^M(3+t)}{\tau_+ \tau_-}. \end{equation} Then, we have
\begin{equation}\label{eq:imprdecay2} \forall \hspace{0.5mm} (t,x) \in [0,T[ \times \mathbb{R}^3, \hspace{1cm} |\rho|(t,x)+|\sigma|(t,x) \hspace{2mm} \lesssim \hspace{2mm} C\frac{\log^{M+1}(3+t)}{\tau_+^2}. \end{equation} \end{Pro} \begin{proof} Let $(t,x)=(t,r \omega) \in [0,T[ \times \mathbb{R}^3$. If $r \leq \frac{t+1}{2}$ or $t \leq \frac{r+1}{2}$ the inequalities follow from \eqref{eq:imprdecay} since $\tau_+ \lesssim \tau_-$ in these two cases. We then suppose that $\frac{t+1}{2} \leq r \leq 2t-1$, so that $\tau_+ \leq 10\min(r,t)$. Hence, we obtain from equations \eqref{eq:nullmax1}-\eqref{eq:nullmax2} of Lemma \ref{maxwellbis} and \eqref{eq:zeta} that \begin{equation}\label{eq:fortheproof3}
|\nabla_{\underline{L}} \hspace{0.5mm} \rho|(t,x)+|\nabla_{\underline{L}} \hspace{0.5mm} \sigma|(t,x) \hspace{2mm} \lesssim \hspace{2mm} |J_{\underline{L}}|(t,x)+ \frac{1}{\tau_+} \sum_{|\gamma| \leq 1} |\mathcal{L}_{Z^{\gamma}}(G)|(t,x).
\end{equation}
Let $\zeta$ be either $\rho$ or $\sigma$ and $$\varphi ( \underline{u}, u) := \zeta \left( \frac{\underline{u}+u}{2}, \frac{\underline{u}-u}{2} \omega \right), \quad \text{so that, by \eqref{eq:fortheproof3} and \eqref{eq:imprdecay}}, \quad |\nabla_{\underline{L}} \varphi |(\underline{u},u) \lesssim C\frac{\log^M \left(3+\frac{\underline{u}+u}{2} \right)}{(1+\underline{u})^2(1+|u|)}.$$ \begin{itemize} \item If $r \geq t$, we then have \begin{eqnarray}
\nonumber |\zeta|(t,x) & = & |\varphi|(t+r,t-r) \hspace{2mm} \leq \hspace{2mm} \int_{u=-t-r}^{t-r} |\nabla_{\underline{L}} \varphi |(t+r,u) du + |\varphi|(t+r,-t-r) \\ \nonumber
& \lesssim & \int_{u=-t-r}^{t-r} |\nabla_{\underline{L}} \varphi |(t+r,u) du + |\zeta|(0,(t+r)\omega) \\ \nonumber
& \lesssim & C \frac{ \log^M \left(3+t \right)}{(1+t+r)^2} \int_{u=-t-r}^{t-r} \frac{du}{1+|u|} + \frac{C}{(1+t+r)^2} \hspace{2mm} \lesssim \hspace{2mm} C \frac{\log^M \left(3+t \right)}{(1+t+r)^2} \log(1+t+r). \end{eqnarray} It then remains to use that $t+r \lesssim 1+t$ in the region studied. \item If $r \leq t$, we obtain using the previous estimate, \begin{eqnarray}
\nonumber |\zeta|(t,x) & = & |\varphi|(t+r,t-r) \hspace{2mm} \leq \hspace{2mm} \int_{u=0}^{t-r} |\nabla_{\underline{L}} \varphi |(t+r,u) du + |\varphi|(t+r,0) \\ \nonumber
& \lesssim & \int_{u=0}^{t-r} |\nabla_{\underline{L}} \varphi |(t+r,u) du + |\zeta|\left( \frac{t+r}{2}, \frac{t+r}{2} \right) \\ \nonumber
& \lesssim & C \frac{ \log^M \left(3+t \right)}{(1+t+r)^2} \int_{u=0}^{t-r} \frac{du}{1+|u|} + C \frac{\log^{M+1} \left(3+\frac{t+r}{2} \right)}{(1+t+r)^2} \hspace{2mm} \lesssim \hspace{2mm} C \frac{\log^{M+1} \left(3+t \right)}{(1+t+r)^2} . \end{eqnarray} \end{itemize} This concludes the proof. \end{proof}
\begin{Rq}
Assuming enough decay on $|F|(t=0)$ and on the spherical components of the source term $J_A$, one could prove similarly that $|\alpha| \lesssim \log^{M+2}(3+t) \frac{\tau_-}{\tau_+^3}$. \end{Rq}
\section{The pure charge part of the electromagnetic field}\label{secpurecharge}
As we will consider an electromagnetic field with a non-zero total charge, $\int_{\mathbb{R}^3} r|\rho(F)| dx$ will be infinite and we will not be able to apply the results of the previous section to $F$ and its derivatives. As mentioned earlier, we will split $F$ in $\widetilde{F}+\overline{F}$, where $\widetilde{F}$ and $\overline{F}$ are introduced in Definition \ref{defpure1}. We will then apply the results of the previous section to the chargeless field $\widetilde{F}$, which will allow us to derive pointwise estimates on $F$ since the field $\overline{F}$ is completely determined. More precisely, we will use the following properties of the pure charge part $\overline{F}$ of $F$. \begin{Pro}\label{propcharge} Let $F$ be a $2$-form with a constant total charge $Q_F$ and $\overline{F}$ its pure charge part $$\overline{F}(t,x) := \chi(t-r) \frac{Q_F}{4 \pi r^2} \frac{x_i}{r} dt \wedge dx^i.$$
Then, \begin{enumerate} \item $\overline{F}$ is supported in $\cup_{t \geq 0} V_{-1}(t)$ and $\widetilde{F}$ is chargeless. \item $\rho(\overline{F})(t,x)=-\frac{Q_F}{4 \pi r^2} \chi(t-r)$, \hspace{2mm} $\alpha(\overline{F})=0$, \hspace{2mm} $\underline{\alpha}(\overline{F})=0$ \hspace{2mm} and \hspace{2mm} $\sigma(\overline{F})=0$.
\item $\forall \hspace{0.5mm} Z^{\gamma} \in \mathbb{K}^{|\gamma|}$, \hspace{1mm} $\exists \hspace{0.5mm} C_{\gamma} >0$, \hspace{5mm} $|\mathcal{L}_{Z^{\gamma}} (\overline{F}) | \leq C_{\gamma} |Q_F| \tau_+^{-2}$. \item $\overline{F}$ satisfies the Maxwell equations $\nabla^{\mu} \overline{F}_{\mu \nu} = \overline{J}_{\nu}$ and $\nabla^{\mu} {}^* \! \overline{F}_{\mu \nu} =0$, with $\overline{J}$ such that $$\overline{J}_0(t,x)= \frac{Q_F}{4 \pi r^2} \chi'(t-r) \hspace{5mm} \text{and} \hspace{5mm} \overline{J}_i(t,x) =-\frac{Q_F}{4 \pi r^2} \frac{x_i}{r} \chi'(t-r).$$
$\overline{J}$ is then supported in $\{ (s,y) \in \mathbb{R}_+ \times \mathbb{R}^3 \hspace{1mm} / \hspace{1mm} -2 \leq t-|y| \leq -1 \}$ and its derivatives satisfy
$$ \forall \hspace{0.5mm} Z^{\gamma} \in \mathbb{K}^{|\gamma|}, \hspace{1mm} \exists \hspace{0.5mm} \widetilde{C}_{\gamma} >0, \hspace{10mm} |\mathcal{L}_{Z^{\gamma}} (\overline{J})^L |+\tau_+|\mathcal{L}_{Z^{\gamma}} (\overline{J})^A |+\tau_+^2|\mathcal{L}_{Z^{\gamma}} (\overline{J})^{\underline{L}} | \leq \frac{\widetilde{C}_{\gamma} |Q_F|}{ \tau_+^2}.$$ \end{enumerate} \end{Pro} \begin{proof} The first point follows from the definitions of $\overline{F}$, $\chi$ and $$ Q_{\widetilde{F}}(t) \hspace{1mm} = \hspace{1mm} Q_F-Q_{\overline{F}}(t) \hspace{1mm} = \hspace{1mm} Q_F-\lim_{r \rightarrow + \infty} \left( \int_{\mathbb{S}_{t,r}} \frac{x^i}{r} \overline{F}_{0i} d \mathbb{S}_{t,r} \right) \hspace{1mm} = \hspace{1mm} Q_F- \frac{Q_F}{4 \pi r^2} \int_{\mathbb{S}_{t,r}} d \mathbb{S}_{t,r} \hspace{1mm} = \hspace{1mm} 0.$$ The second point is straightforward and depicts that $\overline{F}$ has a vanishing magnetic part and a radial electric part. The third point can be obtained using that, \begin{itemize} \item for a $2$-form $G$ and a vector field $\Gamma$, $\mathcal{L}_{\Gamma}(G)_{\mu \nu} = \Gamma(G_{\mu \nu})+\partial_{\mu} (\Gamma^{\lambda} ) G_{\lambda \nu}+\partial_{\nu} ( \Gamma^{\lambda} ) G_{\mu \lambda}$. \item For all $Z \in \mathbb{K}$, $Z$ is either a translation or a homogeneous vector field. \item For a function $\chi_0 : u \mapsto \chi_0(u)$, we have $\Omega_{ij}(\chi_0(u))=0$, $$ \partial_{t} (\chi_0(u))= \chi_0'(u), \hspace{0.6cm} \partial_{i} (\chi_0(u))= -\frac{x^i}{r}\chi_0'(u), \hspace{0.6cm} S(\chi_0(u))= u \chi_0'(u), \hspace{0.6cm} \Omega_{0i} (\chi(u)) = -\frac{x^i}{r}u \chi_0'(u).$$
\item $1+t \leq \tau_+ \lesssim r$ on the support of $\overline{F}$ and $ |u| \leq \tau_- \leq \sqrt{5}$ on the support of $\chi'$. \end{itemize} Consequently, one has
$$\forall \hspace{0.5mm} Z^{\xi} \in \mathbb{K}^{|\xi|}, \hspace{5mm} Z^{\xi} \left( \frac{x^i}{r^3} \chi(t-r) \right) \leq C_{\xi,\chi} \tau_+^{-2} \hspace{1cm} \text{and} \hspace{1cm} \left| \mathcal{L}_{Z^{\gamma}}(\overline{F}) \right| \lesssim \sum_{|\kappa| \leq |\gamma| } \sum_{\mu=0}^3 \sum_{\nu = 0}^3 \left| Z^{\kappa}(\overline{F}_{\mu \nu}) \right| \lesssim \frac{C_{\gamma}}{\tau_+^2}.$$ The equations $\nabla^{\mu} {}^* \! \overline{F}_{\mu \nu}$, equivalent to $\nabla_{[ \lambda} \overline{F}_{\mu \nu]}=0$ by Proposition \ref{maxwellbis}, follow from $\overline{F}_{ij}=0$ and that the electric part of $\overline{F}$ is radial, so that $\nabla_i \overline{F}_{ 0j}-\nabla_j \overline{F}_{0i} =0$. The other ones ensue from straightforward computations, \begin{eqnarray} \nonumber \nabla^{i} \overline{F}_{i0} \hspace{-1mm} & = & \hspace{-1mm} -\frac{Q_F}{4 \pi} \partial_i \hspace{-0.2mm} \left( \frac{x^i}{r^3} \chi(t-r) \hspace{-0.2mm} \right) \hspace{0.8mm} = \hspace{0.8mm} -\frac{Q_F}{4 \pi} \left( \hspace{-0.2mm} \left( \frac{3}{r^3} -3\frac{x_i x^i}{r^5} \right) \hspace{-0.2mm} \chi(t-r)- \frac{x^i}{r^3} \times \frac{x_i}{r} \chi'(t-r) \hspace{-0.2mm} \right) \hspace{0.8mm} = \hspace{0.8mm} \frac{Q_F}{4 \pi r^2} \chi'(t-r), \\ \nonumber \nabla^{\mu} \overline{F}_{\mu i} \hspace{-1mm} & = & \hspace{-1mm} -\partial_t \overline{F}_{0i} \hspace{1mm} = \hspace{1mm} -\frac{Q_F}{4 \pi} \frac{x^i}{r^3} \chi'(t-r). \end{eqnarray} For the estimates on the derivatives of $\overline{J}$, we refer to \cite{LS} (equations $(3.52a)-(3.52c)$). \end{proof}
\section{Bootstrap assumptions and strategy of the proof}\label{sec6}
Let, for the remainder of this article, $N \in \mathbb{N}$ such that $N \geq 11$ and $M \in \mathbb{N}$ which will be fixed during the proof. Let also $0 < \eta < \frac{1}{16}$ and $(f_0,F_0)$ be an initial data set satisfying the assumptions of Theorem \ref{theorem}. By a standard local well-posedness argument, there exists a unique maximal solution $(f,F)$ of the Vlasov-Maxwell system defined on $[0,T^*[$, with $T^* \in \mathbb{R}_+^* \cup \{+ \infty \}$. Let us now introduce the energy norms used for the analysis of the particle density. \begin{Def}\label{normVlasov} Let $Q \leq N$, $q \in \mathbb{N}$ and $a = M+1$. For $g$ a sufficiently regular function, we define the following energy norms, \begin{eqnarray}
\nonumber \mathbb{E}[g](t) & := & \| g \|_{L^1_{x,v} }(t) +\int_{C_u(t)} \int_v \frac{v^{\underline{L}}}{v^0} \left| g \right| dv dC_u(t), \\ \nonumber
\mathbb{E}^{q}_Q[g](t) & := & \sum_{\begin{subarray}{l} 1 \leq i \leq 2 \\ \hspace{1mm} z \in \mathbf{k}_1 \end{subarray}} \sum_{ \begin{subarray}{} |\xi^i|+|\beta| \leq Q \\ \hspace{1mm} |\xi^i| \leq Q-1 \end{subarray}} \sum_{j=0}^{2N-1+q- \xi^1_P-\xi^2_P-\beta_P} \log^{- (j+ |\xi^1|+|\xi^2|+|\beta|)a}(3+t) \mathbb{E} \left[ z^j P_{\xi^1}(\Phi)P_{\xi^2}(\Phi)Y^{\beta} f \right] \hspace{-1mm} (t), \\ \nonumber
\overline{\mathbb{E}}_N[g](t) & := & \sum_{\begin{subarray}{l} 1 \leq i \leq 2 \\ \hspace{1mm} z \in \mathbf{k}_1 \end{subarray}} \sum_{ \begin{subarray}{} |\xi^i|+|\beta| \leq Q \\ \hspace{1mm} |\xi^i| \leq Q-1 \end{subarray}} \sum_{j=0}^{2N-1- \xi^1_P-\xi^2_P-\beta_P} \hspace{-1mm} \log^{-aj}(3+t) \mathbb{E} \left[ z^j P_{\xi^1}(\Phi)P_{\xi^2}(\Phi)Y^{\beta} f \right] \hspace{-1mm} (t), \\ \nonumber
\mathbb{E}^X_{N-1}[f](t) & := & \sum_{\begin{subarray}{l} 1 \leq i \leq 2 \\ \hspace{1mm} z \in \mathbf{k}_1 \end{subarray}} \sum_{ |\zeta^i|+|\beta| \leq N-1} \sum_{j=0}^{2N-2-\zeta^1_P-\zeta^2_P-\beta_P} \log^{-2j}(3+t) \mathbb{E} \left[ z^j P^X_{\zeta^1}(\Phi)P^X_{\zeta^2}(\Phi)Y^{\beta} f \right]\hspace{-1mm} (t), \\ \nonumber
\mathbb{E}^X_{N}[f](t) & := & \sum_{ z \in \mathbf{k}_1} \sum_{ \begin{subarray}{} |\zeta|+|\beta| \leq N \\ \hspace{1mm} |\zeta| \leq N-1 \end{subarray}} \sum_{j=0}^{2N-2-\zeta_P-\beta_P} \log^{-2j}(3+t) \mathbb{E} \left[ z^j P^X_{\zeta}(\Phi)Y^{\beta} f \right]\hspace{-1mm} (t) . \end{eqnarray} To understand the presence of the logarithmical weights, see Remark \ref{hierarchyjustification}. \end{Def} In order to control the derivatives of the $\Phi$ coefficients and $\overline{\mathbb{E}}_N[f]$ at $t=0$, we prove the following result. \begin{Pro}\label{Phi0}
Let $|\beta| \leq N-1$ a multi index and $Y^{\beta} \in \mathbb{Y}^{|\beta|}$. Then, at $t=0$, \begin{eqnarray}
\nonumber \max \left( |Y^{\beta} \Phi |, | \widehat{Z}^{\beta} \Phi | \right) & \lesssim & \frac{1+r^2}{v^0} \sum_{|\gamma| \leq |\beta| -1} \left| \mathcal{L}_{Z^{\gamma}}(F) \right| \\ \nonumber & \lesssim & \frac{\sqrt{\epsilon}}{v^0 }. \end{eqnarray} \end{Pro} \begin{proof} Note that the second inequality ensues from \begin{equation}\label{decayF00}
\sum_{|\gamma| \leq N-2} \left\| \mathcal{L}_{Z^{\gamma}}(F) \right\|_{L^{\infty}(\Sigma_0)} \lesssim \frac{\sqrt{\epsilon}}{1+r^2}, \end{equation}
which comes from Proposition \ref{decayMaxwell}. Let us now prove the first inequality. Unless the opposite is mentioned explicitly (as in \eqref{equa7}), all functions considered here will be evaluated at $t=0$. As $\Phi(0,.,.)=0$, the result holds for $|\beta|=0$. Let $1 \leq |\beta| \leq N-1$ and suppose that the result holds for all $|\sigma| < |\beta|$. Note that, for instance, $$ Y_2 Y_1 \Phi = \widehat{Z}_2 \widehat{Z}_1 \Phi +\Phi X \widehat{Z}_1 \Phi+Y_2(\Phi) X \Phi+\Phi \widehat{Z}_2 X \Phi + \Phi \Phi X X \Phi.$$ More generally, we have, \begin{equation}\label{equa5}
\left| Y^{\beta} \Phi \right| \lesssim \sum_{\begin{subarray}{} p \leq |k|+|\sigma| \leq |\beta| \\ \hspace{2.5mm} k < |\beta| \end{subarray} } P_{k,p}(\Phi) \widehat{Z}^{\sigma} \Phi. \end{equation}
Consequently, using the induction hypothesis, we only have to prove the result for $\widehat{Z}^{\beta} \Phi$. Indeed, as $|k| < |\beta|$, by \eqref{decayF00}, \begin{equation}\label{equa3}
|P_{k,p}(\Phi) \widehat{Z}^{\sigma}(\Phi) | \lesssim |\widehat{Z}^{\sigma}(\Phi)| \left|\frac{1+r^{2}}{v^0}\right|^p \sum_{|\gamma| \leq N-2 } \left| \mathcal{L}_{Z^{\gamma}}(F) \right|^p \lesssim |\widehat{Z}^{\sigma}(\Phi)|. \end{equation}
Combining \eqref{equa5} and \eqref{equa3}, we would then obtain the inequality on $|Y^{\beta} \Phi|$, if we would have it on $\widehat{Z}^{\sigma} \Phi$ for all $|\sigma| \leq |\beta|$. Let us then prove that the result holds for $\widehat{Z}^{\beta} \Phi$ and suppose, for simplicity, that $\Phi=\Phi^k_{\widehat{Z}}$, with $\widehat{Z} \neq S$. Remark that
$$|\widehat{Z}^{\beta} \Phi | \lesssim \sum_{|\alpha_2|+|\alpha_1|+q \leq |\beta|}(1+|x|)^{|\alpha_1|+q}(v^0)^{|\alpha_2|} |\partial_{v}^{\alpha_2} \partial_x^{\alpha_1} \partial_t^q \Phi|$$ and let us prove by induction on $q$ that \begin{equation}\label{equa6}
\forall \hspace{0.5mm} |\alpha_2|+|\alpha_1|+q \leq |\beta|, \hspace{8mm} (1+|x|)^{|\alpha_1|+q}(v^0)^{|\alpha_2|} |\partial_{v}^{\alpha_2} \partial_x^{\alpha_1} \partial_t^q \Phi| \lesssim \frac{1+r^2}{v^0} \sum_{|\gamma| \leq |\beta| -1} \left| \mathcal{L}_{Z^{\gamma}}(F) \right|. \end{equation} Recall that for $t \in [0,T^*[$, \begin{equation}\label{equa7} T_F(\Phi)=v^{\mu} \partial_{\mu} \Phi + F(v,\nabla_v \Phi)=-t\frac{v^{\mu}}{v^0} \mathcal{L}_Z(F)_{\mu k}. \end{equation}
As $\Phi(0,.,.)=0$ and $v^0\partial_t \Phi=-v^i \partial_i \Phi-F(v,\nabla_v \Phi)$, implying $\partial_t \Phi(0,.,.)=0$, \eqref{equa6} holds for $q \leq 1$. Let $2 \leq q \leq |\beta|$ and suppose that \eqref{equa6} is satisfied for all $q_0 < q$. Let $|\alpha_2|+|\alpha_1| \leq |\beta|-q$. Using the commutation formula given by Lemma \ref{basiccomuf}, we have (at $t=0$),
$$v^0\partial_x^{\alpha_1} \partial_t^q \Phi=-v^i \partial_i\partial_x^{\alpha_1} \partial_t^{q-1} \Phi-\frac{v^{\mu}}{v^0}\mathcal{L}_{\partial_x^{\alpha_1} \partial_t^{q-2} Z}(F)_{\mu k}+\sum_{|\gamma_1|+q_1+|\gamma_2| = |\alpha_1|+q-1} C^1_{\gamma_1, \gamma_2} \mathcal{L}_{\partial^{\gamma_2}}(F)(v,\nabla_v \partial_{x}^{\gamma_1} \partial_t^{q_1} \Phi),$$ Dividing the previous equality by $v^0$, taking the $\partial_v^{\alpha_2}$ derivatives of each side and using Lemma \ref{goodderiv}, we obtain \begin{eqnarray}
\nonumber |\partial_v^{\alpha_2} \partial_x^{\alpha_1} \partial_t^q \Phi| & \lesssim & \sum_{|\alpha_3| \leq |\alpha_2|} (v^0)^{-|\alpha_2|+|\alpha_3|}|\partial^{\alpha_3}_v\partial_x\partial_x^{\alpha_1} \partial_t^{q-1} \Phi|+\sum_{|\gamma| \leq |\alpha_1|+q-2} \frac{1}{(v^0)^{1+|\alpha_2|}(1+r)^{|\alpha_1|+q-2}}\left|\mathcal{L}_{Z^{\gamma} Z}(F)\right|\\ \nonumber & & +\sum_{\begin{subarray}{} |\gamma_1|+q_1+n = |\alpha_1|+q-1 \\ \hspace{5mm} 1 \leq |\alpha_4| \leq |\alpha_2|+1 \end{subarray}} \sum_{|\gamma_2| \leq n} \frac{1}{(v^0)^{|\alpha_2|-|\alpha_4|+1}(1+r)^n}\left| \mathcal{L}_{Z^{\gamma_2}}(F) \right| |\partial_v^{\alpha_4}\partial_{x}^{\gamma_1} \partial_t^{q_1} \Phi|. \end{eqnarray}
It then remains to multiply both sides of the inequality by $(v^0)^{|\alpha_2|}(1+r)^{|\alpha_1|+q}$ and \begin{itemize}
\item To bound $(v^0)^{|\alpha_2|}(1+r)^{|\alpha_1|+q} (v^0)^{-|\alpha_2|+|\alpha_3|}|\partial^{\alpha_3}_v\partial_x\partial_x^{\alpha_1} \partial_t^{q-1} \Phi|$ with the induction hypothesis.
\item To remark that $(v^0)^{|\alpha_2|}(1+r)^{|\alpha_1|+q} \frac{1}{(v^0)^{1+|\alpha_2|}(1+r)^{|\alpha_1|+q-2}}\left|\mathcal{L}_{Z^{\gamma} Z}(F)\right|$ has the desired form.
\item To note that, using $|\gamma_1|+q_1+1 = |\alpha_1|+q-n$ and the induction hypothesis, \begin{eqnarray}
\nonumber \frac{(v^0)^{|\alpha_2|}(1+r)^{|\alpha_1|+q}}{(v^0)^{|\alpha_2|-|\alpha_4|+1}(1+r)^n} |\partial_v^{\alpha_4}\partial_{x}^{\gamma_1} \partial_t^{q_1} \Phi|\left| \mathcal{L}_{Z^{\gamma_2}}(F) \right| \hspace{-2mm} & = & \hspace{-2mm} \frac{1+r}{v^0}(v^0)^{|\alpha_4|}(1+r)^{|\gamma_1|+q_1}|\partial_v^{\alpha_4}\partial_{x}^{\gamma_1} \partial_t^{q_1} \Phi|\left| \mathcal{L}_{Z^{\gamma_2}}(F) \right| \\ \nonumber
& \lesssim & \hspace{-2mm} \frac{1+r}{v^0} \left| \mathcal{L}_{Z^{\gamma_2}}(F) \right| \hspace{-1mm} \sum_{|\zeta| \leq |\alpha_4|+|\gamma_1|+q_1-1} \hspace{-1mm} \frac{(1+r)^2}{v^0} \left| \mathcal{L}_{Z^{\zeta}}(F) \right| \\ \nonumber
& \lesssim & \hspace{-2mm} \sum_{|\zeta| \leq |\alpha_2|+|\alpha_1|+q-1} \frac{(1+r)^2}{v^0} \left| \mathcal{L}_{Z^{\zeta}}(F) \right| , \end{eqnarray}
since $\left| \mathcal{L}_{Z^{\gamma_2}}(F) \right| \lesssim (1+r)^{-2}$, as $|\gamma_2| \leq |\alpha_1|+q-1 \leq |\beta|-1 \leq N-2$. This concludes the proof of the Proposition. \end{itemize} \end{proof} \begin{Cor}\label{coroinit} There exists $\widetilde{C} >0$ a constant depending only on $N$ such that $ \mathbb{E}^{4}_N[f](0) \leq \widetilde{C} \epsilon = \widetilde{\epsilon}$. Without loss of generality and in order to lighten the notations, we suppose that $\mathbb{E}^{4}_N[f](0) \leq \epsilon$. \end{Cor} \begin{proof}
All the functions considered here are evaluated at $t=0$. Consider multi-indices $\xi_1$, $\xi_2$ and $\beta$ such that, for $i \in \{1 , 2 \}$, $\max(|\xi^i|+1,|\xi^i|+|\beta|) \leq N$ and $j \leq 2N+3-\xi^1_P-\xi^2_P-\beta_P$. Then,
$$\left|z^j P_{\xi^1}(\Phi) P_{\xi^2}(\Phi) Y^{\beta} f \right| \leq \left|z^j P_{\xi^1}(\Phi) P_{\xi^2}(\Phi) \widehat{Z}^{\beta} f \right|+ \sum_{\begin{subarray}{} \hspace{1mm} |k|+|\kappa| \leq |\beta| \\ \hspace{1mm} |k| \leq |\beta|-1 \\ p+k_P+\kappa_P < \beta_P \end{subarray} } \left|z^j P_{\xi^1}(\Phi) P_{\xi^2}(\Phi) P_{k,p}(\Phi) \widehat{Z}^{\kappa} f \right|.$$ Using the previous proposition and the assumptions on $f_0$, one gets, with $C_1 >0$ a constant,
$$ \mathbb{E}^{4}_N[f](0) \leq (1+C_1 \sqrt{\epsilon} )\sum_{\begin{subarray}{l} \hspace{0.5mm} \widehat{Z}^{\beta} \in \widehat{\mathbb{P}}_0^{|\beta|} \\ |\beta| \leq N \end{subarray}} \| z^{2N+3-\beta_P} \widehat{Z}^{\beta} f \|_{L^1_{x,v}}(0) .$$ By similar computations than in Appendix $B$ of \cite{massless}, we can bound the right hand side of the last inequality by $\widetilde{C} \epsilon$ using the smallness hypothesis on $(f_0,F_0)$. \end{proof} By a continuity argument and the previous corollary, there exists a largest time $T \in ]0,T^*[$ such that, for all $t \in [0,T[$, \begin{eqnarray}\label{bootf1} \mathbb{E}^4_{N-3}[f](t) & \leq & 4\epsilon, \\ \mathbb{E}^{0}_{N-1}[f](t) & \leq & 4\epsilon, \label{bootf2} \\ \overline{\mathbb{E}}_{N}[f](t) & \leq & 4\epsilon(1+t)^{\eta}, \label{bootf3} \\
\sum_{|\beta| \leq N-2} \left\| r^{\frac{3}{2}} \int_v \frac{v^A}{v^0} \widehat{Z}^{\beta} f dv \right\|_{L^2(\Sigma_t)} & \leq & \sqrt{\epsilon}, \label{bootL2} \\ \mathcal{E}^0_{N}[F](t) & \leq & 4\epsilon, \label{bootF1} \\ \mathcal{E}^{Ext}_N[\widetilde{F}](t) & \leq & 8 \epsilon, \label{bootext} \\ \mathcal{E}_{N-3}[F](t) & \leq & 30\epsilon \log^2(3+t), \label{bootF2} \\ \mathcal{E}_{N-1}[F](t) & \leq & 30\epsilon \log^{2M}(3+t), \label{bootF3} \\ \mathcal{E}_{N}[F](t) & \leq & 30\epsilon(1+t)^{\eta}. \label{bootF4} \end{eqnarray}
The remainder of the proof will then consist in improving our bootstrap assumptions, which will prove that $(f,F)$ is a global solution to the $3d$ massive Vlasov-Maxwell system. The other points of the theorem will be obtained during the proof, which is divided in four main parts.
\begin{enumerate} \item First, we will obtain pointwise decay estimates on the particle density, the electromagnetic field and then on the derivatives of the $\Phi$ coefficients, using the bootstrap assumptions.
\item Then, we will improve the bootstrap assumptions \eqref{bootf1}, \eqref{bootf2} and \eqref{bootf3} by several applications of the energy estimate of Proposition \ref{energyf} and the commutation formula of Proposition \ref{ComuPkp}. The computations will also lead to optimal pointwise decay estimates on $\int_v |Y^{\beta} f | \frac{dv}{(v^0)^2}$.
\item The next step consists in proving enough decay on the $L^2$ norms of $\int_v |zY^{\beta} f | dv$, which will permit us to improve the bootstrap assumption \eqref{bootL2}. \item Finally, we will improve the bootstrap assumptions \eqref{bootF1}-\eqref{bootF4} by using the energy estimates of Proposition \ref{energyMax1}. \end{enumerate} \section{Immediate consequences of the bootstrap assumptions}\label{sec7}
In this section, we prove pointwise estimates on the Maxwell field, the $\Phi$ coefficients and the Vlasov field. We start with the electromagnetic field.
\begin{Pro}\label{decayF}
We have, for all $|\gamma| \leq N-3$ and $(t,x) \in [0,T[ \times \mathbb{R}^3$, \begin{flalign*}
& \hspace{0.5cm} |\alpha(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \sqrt{\epsilon}\frac{\log^M(3+t)}{\tau_+^2}, \hspace{17mm} |\underline{\alpha}(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \sqrt{\epsilon}\min \left( \frac{1}{\tau_+\tau_-^{\frac{1}{2}}}, \frac{\log^M(3+t)}{\tau_+\tau_-} \right) \hspace{-0.5mm} ,& \\
& \hspace{0.5cm} |\sigma(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \sqrt{\epsilon} \frac{\log^M(3+t)}{\tau_+^{\frac{3}{2}}\tau_-^{\frac{1}{2}}}, \hspace{17mm} |\rho(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \sqrt{\epsilon}\frac{\log^M(3+t)}{\tau_+^{\frac{3}{2}}\tau_-^{\frac{1}{2}}}.& \end{flalign*}
Moreover, if $|x| \geq t$, \begin{flalign*}
& \hspace{0.5cm} |\alpha(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \frac{\sqrt{\epsilon}}{\tau_+^2}, \hspace{35mm} |\underline{\alpha}(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \frac{\sqrt{\epsilon}}{\tau_+\tau_-}, & \\
& \hspace{0.5cm} |\sigma(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \frac{\sqrt{\epsilon}}{\tau_+^{\frac{3}{2}}\tau_-^{\frac{1}{2}}}, \hspace{31mm} |\rho(\mathcal{L}_{Z^{\gamma}}(F))|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \frac{\sqrt{\epsilon}}{\tau_+^{\frac{3}{2}}\tau_-^{\frac{1}{2}}}. & \end{flalign*} We also have
$$ \forall \hspace{0.5mm} (t,x) \in [0,T[ \times \mathbb{R}^3, \hspace{1.5cm} \sum_{|\kappa| \leq N} \left| \mathcal{L}_{Z^{\kappa}}(\overline{F}) \right|(t,x) \hspace{1mm} \lesssim \hspace{1mm} \frac{\epsilon}{\tau_+^2}.$$ \end{Pro}
\begin{Rq}\label{lowderiv}
If $|\gamma| \leq N-5$, we can replace the $\log^M(3+t)$-loss in the interior of the lightcone by a $\log(3+t)$-loss (for this, use the bootstrap assumption \eqref{bootF2} instead of \eqref{bootF3} in the proof below). \end{Rq} \begin{Rq}\label{decayoftheo}
Applying Proposition \ref{Probetteresti} and using the estimate \eqref{decayf} proved below, we can also improve the decay rates of the components $\rho$ and $\sigma$ near the light cone. We have, for all $|\gamma| \leq N-6$, \begin{eqnarray}
\forall \hspace{0.5mm} (t,x) \in [0,T[ \times \mathbb{R}^3, \hspace{1cm} |\rho(\mathcal{L}_{Z^{\gamma}}(F))|(t,x)+|\sigma (\mathcal{L}_{Z^{\gamma}}(F))|(t,x) & \lesssim & \sqrt{\epsilon} \frac{\log^2(3+t)}{\tau_+^2} . \label{kevatal} \end{eqnarray} \end{Rq}
\begin{proof}
The last estimate, concerning $\overline{F}$, ensues from Proposition \ref{propcharge} and $|Q_F| \leq \| f_0 \|_{L^1_{x,v}} \leq \epsilon$. The estimate $\tau_+ \sqrt{\tau_-}|\underline{\alpha}| \lesssim \sqrt{\epsilon}$ follows from Proposition \ref{decayMaxwell} and the bootstrap assumption \eqref{bootF1}. Note that the other estimates hold with $F$ replaced by $\widetilde{F}$ since $\mathcal{E}_{N-1}[F]=\mathcal{E}_{N-1}[\widetilde{F}]$ and according to Proposition \ref{decayMaxwell} and the bootstrap assumptions \eqref{bootext}, \eqref{bootF3} and \eqref{bootL2}. It then remains to use $F=\widetilde{F}+\overline{F}$ and the estimates obtained on $\overline{F}$ and $\widetilde{F}$. \end{proof} \begin{Rq}\label{justif} Even if the pointwise decay estimates \eqref{kevatal}, which correspond to the ones written in Theorem \ref{theorem}, are stronger than the ones given by Proposition \ref{decayF} (or Remark \ref{lowderiv}) in the region located near the light cone, we will not work with them for two reasons. \begin{enumerate} \item Using these stronger decay rates do not simplify the proof. We compensate the lack of decay in $t+r$ of the estimates given by Proposition \ref{decayF} for the components $\rho$ and $\sigma$ by taking advantage of the inequality\footnote{We are able to use this inequality in the energy estimates as the degree in $v$ of the source terms of $T_F(Y^{\beta} f)$ is $0$ whereas the one of $v^{\mu} \partial_{\mu}Y^{\beta}f$ is equal to $1$.} $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$ and the good properties of $v^{\underline{L}}$. \item Compared to the estimates given by Remark \ref{lowderiv}, \eqref{kevatal} requires to control one derivative more of the electromagnetic field in $L^2$. Working with them would then force us to take $N \geq 12$. \end{enumerate} \end{Rq} We now turn on the $\Phi$ coefficients and start by the following lemma. \begin{Lem}\label{LemPhi}
Let $G$, $G_1$, $G_2 : [0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v \rightarrow \mathbb{R}$ and $\varphi_0 : \mathbb{R}^3_x \times \mathbb{R}^3_v \rightarrow \mathbb{R}$ be four sufficiently regular functions such that $|G| \leq G_1+G_2$. Let $\varphi$, $\widetilde{\varphi}$, $\varphi_1$ and $\varphi_2$ be such that $$ T_F( \varphi ) =G, \hspace{5mm} \varphi(0,.,.)=\varphi_0, \hspace{12mm} T_F(\widetilde{\varphi})=0, \hspace{5mm} \widetilde{\varphi}(0,.,.)=\varphi_0$$ and, for $i \in \{1,2 \}$, $$ T_F(\varphi_i)=G_i, \hspace{5mm} \varphi_i(0,.,.)=0.$$ Then, on $[0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v$,
$$|\varphi| \leq |\widetilde{\varphi}|+|\varphi_1|+|\varphi_2|.$$ \end{Lem} \begin{proof} Denoting by $X(s,t,x,v)$ and $V(s,t,x,v)$ the characteristics of the transport operator, we have by Duhamel's formula, \begin{eqnarray}
\nonumber |\varphi|(t,x,v) & = & \left| \widetilde{\varphi}(t,x,v)+\int_0^t \frac{G}{v^0} \left( s,X(s,t,x,v),V(s,t,x,v) \right) ds \right| \\ \nonumber
& \leq & |\widetilde{\varphi}|(t,x,v)+\int_0^t \frac{G_1+G_2}{v^0} \left( s,X(s,t,x,v),V(s,t,x,v) \right) ds \\ \nonumber
& = & |\widetilde{\varphi}|(t,x,v)+|\varphi_1|(t,x,v)+|\varphi_2|(t,x,v). \end{eqnarray} \end{proof} \begin{Pro}\label{Phi1} We have, $\forall \hspace{0.5mm} (t,x,v) \in [0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v$
$$ |\Phi|(t,x,v) \lesssim \sqrt{\epsilon} \log^2 (1+\tau_+), \hspace{3mm} |\partial_{t,x} \Phi| (t,x,v) \lesssim \sqrt{\epsilon} \log^{\frac{3}{2}}(1+\tau_+) \hspace{3mm} \text{and} \hspace{3mm} |Y \Phi|(t,x,v) \lesssim \sqrt{\epsilon} \log^{\frac{7}{2}} (1+\tau_+).$$ \end{Pro} \begin{proof}
We will obtain this result through the previous Lemma and by parameterizing the characteristics of the operator $T_F$ by $t$ or by $u$. Let us start by $\Phi$ and recall that, schematically, $T_F(\Phi)=-t\frac{v^{\mu}}{v^0} \mathcal{L}_Z(F)_{\mu k}$. Denoting by $(\alpha, \underline{\alpha}, \rho, \sigma)$ the null decomposition of $\mathcal{L}_Z(F)$ and using $|v^A| \lesssim \sqrt{v^0 v^{\underline{L}}}$ (see Lemma \ref{weights1}), we have \begin{eqnarray}
\nonumber \left| \frac{v^{\mu}}{v^0} \mathcal{L}_Z(F)_{\mu k} \right| & \lesssim & \frac{v^L+|v^A|}{v^0}|\alpha|+\frac{v^L+v^{\underline{L}}}{v^0}|\rho|+\frac{|v^A|}{v^0}|\sigma|+\frac{v^{\underline{L}}+|v^A|}{v^0}|\underline{\alpha}| \\ \nonumber
& \lesssim & |\alpha|+|\rho|+|\sigma|+\sqrt{\frac{v^{\underline{L}}}{v^0}}|\underline{\alpha}|. \end{eqnarray} Using the pointwise estimates given by Remark \ref{lowderiv} as well as the inequalities $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$, which comes from Lemma \ref{weights1}, and $2ab \leq a^2+b^2$, we get \begin{equation}\label{eq:firstphiesti}
\tau_+\left| \frac{v^{\mu}}{v^0} \mathcal{L}_Z(F)_{\mu k} \right| \hspace{2mm} \lesssim \hspace{2mm} \sqrt{\frac{\epsilon v^0 v^{\underline{L}}}{\tau_+ \tau_-}}\log(3+t)+v^{\underline{L}}\frac{\sqrt{\epsilon}}{ \tau_-} \log(3+t) \hspace{2mm} \lesssim \hspace{2mm} \frac{v^0 \sqrt{\epsilon}}{\tau_+}\log(3+t)+\frac{v^{\underline{L}}\sqrt{\epsilon}}{\tau_-} \log(3+t). \end{equation} Consider now the functions $\varphi_1$ and $\varphi_2$ such that $$T_F(\varphi_1) = \frac{v^0 \sqrt{\epsilon}}{\tau_+}\log(3+t), \hspace{8mm} T_F(\varphi_2) = \frac{v^{\underline{L}}\sqrt{\epsilon}}{\tau_-}\log(3+t) \hspace{6mm} \text{and} \hspace{6mm} \varphi_1(0,.,.)=\varphi_2(0,.,.)=0.$$
According to Lemma \ref{LemPhi}, we have $|\Phi| \lesssim |\varphi_1|+|\varphi_2|$. In order to estimate $\varphi_1$, we will parametrize the characteristics of the operator $T_F$ by $t$. More precisely, let $(X_{s,y,v}(t),V_{s,y,v}(t))$ be the value in $t$ of the characteristic which is equal to $(y,v)$ in $t=s$, with $s < T$. Dropping the indices $s$, $y$ and $w$, we have $$ \frac{dX^i}{dt}(t) = \frac{V^i(t)}{V^0(t)} \hspace{1.5cm} \text{and} \hspace{1.5cm} \frac{d V^i}{dt}(t) = \frac{V^{\mu}(t)}{V^0(t)} {F_{ \mu}}^{ i}(t,X(t)).$$ Duhamel's formula gives
$$ |\varphi_1|(s,y,v) \hspace{2mm} \lesssim \hspace{2mm} \sqrt{\epsilon} \int_0^s \frac{\log(3+t) }{\tau_+ \big( t, X_{t,y,v}(t) \big)} ds \hspace{2mm} \leq \hspace{2mm} \sqrt{\epsilon} \int_0^s \frac{\log(3+t)}{1+t} ds \hspace{2mm} \leq \hspace{2mm} \sqrt{\epsilon} \log^2 (3+s).$$ For $\varphi_2$, we parameterize the characteristics of $T_F$ by\footnote{Note that $T_F=2v^{\underline{L}} \partial_u +2v^L \partial_{\underline{u}}+v^A e_A+F(v,\nabla_v)$} $u$. For a point $(s,y) \in [0,T[ \times \mathbb{R}^3$, we will write its coordinates in the null frame as $(z,\underline{z},\omega_1, \omega_2)$. Let $(\underline{U}_{z,\underline{z},\omega_1, \omega_2,v}(u), \Omega^1_{z,\underline{z},\omega_1, \omega_2,v}(u),\Omega^2_{z,\underline{z},\omega_1, \omega_2,v}(u),V_{z,\underline{z},\omega_1, \omega_2,v}(u))$ be the value in $u$ of the characteristic which is equal to $(s,y,v)=(z,\underline{z},\omega_1,\omega_2,v)$ in $u=z$. Dropping the indices $z$, $\underline{z}$, $\omega_1$, $\omega_2$ and $v$, we have $$\frac{d \underline{U}}{du}(u)= \frac{V^L(u)}{V^{\underline{L}}(u)}, \hspace{1.5cm} \frac{d \Omega^A}{du}(u) = \frac{V^A(u)}{2V^{\underline{L}}(u)} \hspace{1.5cm} \text{and} \hspace{1.5cm} \frac{d V^i}{du}(u) = \frac{V^{\mu}(u)}{2V^{\underline{L}}(u)} {F_{\mu}}^{ i}(u,\underline{U}(u),\Omega (u)).$$ Note that $u \mapsto \frac{1}{2}(u+\underline{U}(u))$ vanishes in a unique $z_0$ such that $ -\underline{z} \leq z_0 \leq z$, i.e. the characteristic reaches the hypersurface $\Sigma_0$ once and only once, at $u=z_0$. This can be noticed on the following picture, representing a possible trajectory of $(u,\underline{U}(u))$, which has to be in the backward light cone of $(z,\underline{z})$ by finite time of propagation,
\begin{tikzpicture} \draw [-{Straight Barb[angle'=60,scale=3.5]}] (0,-0.3)--(0,4); \fill[color=gray!35] (1,0)--(4,3)--(7,0)--(1,0); \node[align=center,font=\bfseries, yshift=-2em] (title)
at (current bounding box.south)
{\hspace{1cm} The trajectory of $\left (u,\underline{U} \left(u \right) \right)$ for $u \leq z$.}; \draw [-{Straight Barb[angle'=60,scale=3.5]}] (0,0)--(9,0) node[scale=1.5,right]{$\Sigma_0$}; \fill (4,3) circle[radius=2pt]; \fill (1,0) circle[radius=2pt]; \fill (7,0) circle[radius=2pt]; \draw (4.6,3) node[scale=1]{$(z,\underline{z})$}; \draw (1,-0.3) node[scale=1]{$(0,z)$}; \draw (7,-0.3) node[scale=1]{$(0,\underline{z})$}; \draw (0,-0.5) node[scale=1.5]{$r=0$}; \draw (-0.5,3.7) node[scale=1.5]{$t$}; \draw (8.7,-0.5) node[scale=1.5]{$r$}; \draw[scale=1,domain=sqrt(3)+1:4,smooth,variable=\x,black] plot (\x,{0.5*(\x-1)*(\x-1)-1.5}); \end{tikzpicture}
or by noticing that $$g(u):=u+\underline{U}(u) \hspace{12mm} \text{satisfies} \hspace{12mm} g'(u) \hspace{1mm} \geq \hspace{1mm} 1+ \frac{V^L\left( u \right)}{V^{\underline{L}}\left( u \right)} \hspace{1mm} \geq \hspace{1mm} 1$$ so that $g$ vanishes in $z_0$ such that $-\underline{z}=z-(z+\underline{z}) \leq z_0 \leq z$. Similarly, one can prove (or observe) that $\sup_{z_0 \leq u \leq z } \underline{U}(u) \leq \underline{z}$. It then comes that \begin{equation}\label{eq:varphi2}
|\varphi_2|(s,y,v) \hspace{2mm} \lesssim \hspace{2mm} \sqrt{\epsilon} \int_{z_0}^z \frac{\log \left( 3+\underline{U} \left(u \right) \right)}{ \tau_-(u,\underline{U} \left( u \right))} du \hspace{2mm} \lesssim \hspace{2mm} \sqrt{\epsilon} \log (3+ \underline{z} ) \int_{-\underline{z}}^z \frac{1}{1+|u|} d u \hspace{2mm} \lesssim \hspace{2mm} \sqrt{\epsilon} \log^2(1+\underline{z}),
\end{equation}
which allows us to deduce that $|\Phi|(s,y,v) \lesssim \sqrt{\epsilon} \log^2 (3+s+|y|)$. We prove the other estimates by the continuity method. Let $0 < T_0 < T$ and $ \underline{u} > 0$ be the largest time and null ingoing coordinate such that \begin{equation}\label{bootPhi}
|\nabla_{t,x} \Phi| (t,x,v) \leq C \sqrt{\epsilon} \log^{\frac{3}{2}} (1+\tau_+) \hspace{8mm} \text{and} \hspace{8mm} \sum_{Y \in \mathbb{Y}_0} |Y \Phi| (t,x,v) \leq C \sqrt{\epsilon} \log^{\frac{7}{2}} (1+\tau_+) \end{equation} hold for all $(t,x,v) \in \underline{V}_{\underline{u}}(T_0) \times \mathbb{R}^3_v$ and where the constant $C>0$ will be specified below. The goal now is to improve the estimates of \eqref{bootPhi}. Using the commutation formula of Lemma \ref{basiccomuf} and the definition of $\Phi$, we have (in the case where $\Phi$ is not associated to the scaling vector field), for $\partial \in \mathbb{T}$, $$T_F \left( \partial \Phi \right) = - \mathcal{L}_{\partial}(F)(v,\nabla_v \Phi)-\partial \left( t\frac{v^{\mu}}{v^0}\mathcal{L}_{Z}(F)_{\mu k} \right).$$ With $\delta = \partial (t) \in \{0,1 \}$, one has $$ \partial \left( t\frac{v^{\mu}}{v^0}\mathcal{L}_{Z}(F)_{\mu k} \right) = \delta \frac{v^{\mu}}{v^0}\mathcal{L}_{Z}(F)_{\mu k}+t\frac{v^{\mu}}{v^0}\mathcal{L}_{\partial Z}(F)_{\mu k}.$$
Using successively the inequality \eqref{eq:zeta2}, the pointwise decay estimates\footnote{Note that we use the estimate $|\underline{\alpha}| \lesssim \sqrt{\epsilon} \tau_+^{-1}\tau_-^{-\frac{1}{2}}$ here in order to obtain a decay rate of $\tau_+^{-1}$ in the $t+r$ direction.} given by Remark \ref{lowderiv} and the inequalities $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$, $2ab \leq a^2+b^2$, we get \begin{eqnarray}
\nonumber t\frac{v^{\mu}}{v^0}\mathcal{L}_{\partial Z}(F)_{\mu k} & \lesssim & \tau_+ \left( |\alpha ( \mathcal{L}_{\partial Z}(F) ) |+|\rho (\mathcal{L}_{\partial Z}(F))|+| \sigma ( \mathcal{L}_{\partial Z}(F) ) |+ \sqrt{\frac{v^{\underline{L}}}{v^0}} |\underline{\alpha} ( \mathcal{L}_{\partial Z}(F) ) | \right) \\ \nonumber
& \lesssim & \frac{\tau_+}{\tau_-} \sqrt{v^0 v^{\underline{L}}} \sum_{|\beta|\leq 2} \Big(\frac{\tau_-}{\tau_+} \left| \mathcal{L}_{Z^{\beta}}(F) \right|+ |\alpha ( \mathcal{L}_{Z^{\beta}}(F) ) |+|\rho (\mathcal{L}_{ Z^{\beta} }(F))|+| \sigma ( \mathcal{L}_{Z^{\beta}}(F) ) | \\ \nonumber
& & + \sqrt{\frac{v^{\underline{L}}}{v^0}} |\underline{\alpha} ( \mathcal{L}_{ Z^{\beta}}(F) ) | \Big) \\ & \lesssim & \sqrt{v^0 v^{\underline{L}}} \frac{\sqrt{\epsilon} \log(3+t)}{\tau_+^{\frac{1}{2}} \tau_-^{\frac{3}{2}}}+ v^{\underline{L}} \frac{\tau_+}{\tau_-} \frac{\sqrt{\epsilon} }{\tau_+ \tau_-^{\frac{1}{2}}} \hspace{2mm} \lesssim \hspace{2mm} \sqrt{\epsilon} \frac{ v^0 }{\tau_+} \log^{\frac{1}{2}}(3+t) + \sqrt{\epsilon} \frac{ v^{\underline{L}} }{\tau_-^{\frac{3}{2}}} \log^{\frac{3}{2}}(3+t). \label{bootPhiT1} \end{eqnarray} Similarly, \begin{eqnarray}
\nonumber \frac{v^{\mu}}{v^0}\mathcal{L}_{ Z}(F)_{\mu k} & \lesssim & \left( |\alpha ( \mathcal{L}_{ Z}(F) ) |+|\rho (\mathcal{L}_{ Z}(F))|+| \sigma ( \mathcal{L}_{ Z}(F) ) |+ \sqrt{\frac{v^{\underline{L}}}{v^0}} |\underline{\alpha} ( \mathcal{L}_{ Z}(F) ) | \right) \\ & \lesssim & \frac{\sqrt{\epsilon} \log(3+t)}{\tau_+^{\frac{3}{2}} \tau_-^{\frac{1}{2}}}+ v^{\underline{L}} \frac{\sqrt{\epsilon} }{\tau_+ \tau_-^{\frac{1}{2}}} \hspace{2mm} \lesssim \hspace{2mm} \sqrt{\epsilon} \frac{ v^0 }{\tau_+^{\frac{5}{4}}} + \sqrt{\epsilon} \frac{ v^{\underline{L}} }{\tau_-^{\frac{5}{4}}} . \label{bootPhiT2} \end{eqnarray}
Expressing $\mathcal{L}_{\partial}(F)(v,\nabla_v \Phi)$ in null components, denoting by $(\alpha, \underline{\alpha}, \rho, \sigma)$ the null decomposition of $\mathcal{L}_{\partial}(F)$ and using the inequalities $|v^A| \lesssim \sqrt{v^0v^{\underline{L}}}$, $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$ (see Lemma \ref{weights1}), one has \begin{equation}\label{eq:referlater}
\left|\mathcal{L}_{\partial}(F)(v,\nabla_v \Phi) \right| \lesssim \sqrt{v^0v^{\underline{L}}}|\rho | \left| \left( \nabla_v \Phi \right)^r\right|+\left(\sqrt{v^0v^{\underline{L}}} |\alpha |+v^{\underline{L}}|\underline{\alpha} |+v^{\underline{L}} |\sigma | \right) \left| \nabla_v \Phi \right|.
\end{equation} Using Lemma \ref{vradial}, $v^0 \partial_{v^i}=Y_i-\Phi X-t\partial_i-x^i \partial_t$ and the bootstrap assumption on the $\Phi$ coefficients \eqref{bootPhi}, we obtain \begin{eqnarray}
\nonumber \left| \left( \nabla_v \Phi \right)^r\right| & \lesssim & \sum_{Y \in \mathbb{Y}_0} |Y \Phi |+ |\Phi| |X( \Phi ) |+ \tau_- |\nabla_{t,x} \Phi | \hspace{2mm} \lesssim \hspace{2mm} C \sqrt{\epsilon} \log^{\frac{7}{2}} (1+\tau_+)+C\sqrt{\epsilon} \tau_- \log^{\frac{3}{2}} (1+\tau_+) , \\ \nonumber
\left| \nabla_v \Phi \right| & \lesssim & \sum_{Y \in \mathbb{Y}_0} |Y \Phi |+ |\Phi| |X( \Phi ) |+ \tau_+ |\nabla_{t,x} \Phi | \hspace{2mm} \lesssim \hspace{2mm} C\sqrt{\epsilon} \log^{\frac{7}{2}} (1+\tau_+)+C\sqrt{\epsilon} \tau_+ \log^{\frac{3}{2}} (1+\tau_+).
\end{eqnarray} We then deduce, by \eqref{eq:zeta2} and the pointwise estimates given by Remark \ref{lowderiv}, \begin{eqnarray}
\nonumber \sqrt{v^0v^{\underline{L}}}|\rho | \left| \left( \nabla_v \Phi \right)^r\right|+\sqrt{v^0v^{\underline{L}}} |\alpha |\left| \nabla_v \Phi \right| & \lesssim & C\epsilon \frac{\sqrt{v^0 v^{\underline{L}}}}{\tau_+ \tau_- } \log^{\frac{5}{2}} (1+\tau_+) \hspace{2mm} \lesssim \hspace{2mm} C \epsilon \frac{v^0}{\tau_+^{\frac{3}{2}}}+ C \epsilon \frac{v^{\underline{L}}}{\tau_-^2}, \\ \nonumber
\left( v^{\underline{L}}|\underline{\alpha} |+v^{\underline{L}} |\sigma | \right) \left| \nabla_v \Phi \right| & \lesssim & C \epsilon \frac{v^{\underline{L}}}{\tau_-^{\frac{3}{2}}} \log^{\frac{3}{2}}(1+\tau_+). \end{eqnarray} Combining these two last estimates with \eqref{bootPhiT1} and \eqref{bootPhiT2}, we get
$$ \left| T_F \left( \partial \Phi \right) \right| \lesssim (\sqrt{\epsilon}+C\epsilon) \frac{v^0}{\tau_+} \log^{\frac{1}{2}}(1+\tau_+)+ (\sqrt{\epsilon}+C\epsilon) \frac{v^{\underline{L}}}{\tau_-^{\frac{5}{4}}} \log^{\frac{3}{2}}(1+\tau_+).$$ We then split $\partial \Phi$ in three functions $\widetilde{\psi}+\psi_1+\psi_2$ such that $\psi_1(0,.,.)=\psi_2(0,.,.)=0$, $\widetilde{\psi}(0,.,.)=\partial \Phi (0,.,.)$, $$ T_F(\psi_1)=(\sqrt{\epsilon}+C\epsilon) \frac{v^0}{\tau_+} \log^{\frac{1}{2}}(1+\tau_+), \hspace{10mm} T_F(\psi_2)=(\sqrt{\epsilon}+C\epsilon) \frac{v^{\underline{L}}}{\tau_-^{\frac{5}{4}}} \log^{\frac{3}{2}}(1+\tau_+) \hspace{10mm} \text{and} \hspace{10mm} T_F(\widetilde{\psi})=0 .$$
According to Proposition \ref{Phi0}, we have $\|\widetilde{\psi} \|_{L^{\infty}_{t,x,v}} = \| \partial \Phi (0,.,.) \|_{L^{\infty}_{x,v}} \lesssim \sqrt{\epsilon}$. Fix now $(s,y,v) \in \underline{V}_{\underline{u}}(T_0) \times \mathbb{R}^3_v$ and let $(z,\underline{z}, \omega_1, \omega_2)$ be the coordinates of $(s,y)$ in the null frame. Keeping the notations used previously in this proof, we have \begin{eqnarray}
\nonumber |\psi_1|(s,y,v) & \lesssim & (\sqrt{\epsilon}+C\epsilon) \int_0^s \frac{ \log^{\frac{1}{2}} (1+\tau_+(t,X(t)))}{\tau_+ (t,X(t))} dt \\ & \lesssim & (\sqrt{\epsilon}+C\epsilon) \int_0^{s} \frac{ \log^{\frac{1}{2}} (3+t)}{1+t} dt \hspace{2mm} \lesssim \hspace{2mm} (\sqrt{\epsilon}+C\epsilon) \log^{\frac{3}{2}} (3+t), \label{eq:repeat1} \\ \nonumber
|\psi_2|(s,y,v) & \lesssim & (\sqrt{\epsilon}+C\epsilon) \int_{z_0}^z \frac{\log^{\frac{3}{2}} \left( 1+\tau_+(u,\underline{U} \left(u \right)) \right)}{ \tau_-^{\frac{5}{4}}(u,\underline{U} \left( u \right))} du \\ & \lesssim & (\sqrt{\epsilon}+C\epsilon) \log^{\frac{3}{2}} (3+ \underline{z} ) \int_{-\underline{z}}^z \frac{1}{(1+|u|)^{\frac{5}{4}}} d u \hspace{2mm} \lesssim \hspace{2mm} (\sqrt{\epsilon}+C\epsilon) \log^{\frac{3}{2}}(3+\underline{z}). \label{eq:repeat2} \end{eqnarray} Thus, there exists $C_1 >0$ such that
$$ \forall \hspace{0.5mm} (s,y,v) \in \underline{V}_{\underline{u}}(T_0) \times \mathbb{R}^3_v, \hspace{1cm} |\nabla_{t,x} \Phi |(s,y,v) \leq C_1(\sqrt{\epsilon}+C \epsilon ) \log^{\frac{3}{2}}(1+\tau_+(s,y))$$ and we can then improve the bootstrap assumption on $\nabla_{t,x} \Phi$ if $C$ is choosen large enough and $\epsilon$ small enough. It remains to study $Y \Phi$ with $Y \in \mathbb{Y}_0$. Using Lemma \ref{Comufirst}, $T_F(Y \Phi)$ can be bounded by a linear combination of terms of the form
$$ \left| \frac{v^{\mu}}{v^0}\mathcal{L}_Z(F)_{\mu k} Y \Phi \right|, \hspace{9mm} \tau_+\left| \frac{v^{\mu}}{v^0}\mathcal{L}_Z(F)_{\mu k} \partial_{t,x} \Phi \right|, \hspace{9mm} \left| \Phi \mathcal{L}_{\partial}(F)(v, \nabla_v \Phi ) \right| \hspace{9mm} \text{and} \hspace{9mm} \left| Y \left( t \frac{v^{\mu}}{v^0} \mathcal{L}_{Z}(F)_{\mu k} \right) \right|.$$
Using the bootstrap assumption \eqref{bootPhi} in order to estimate $|Y \Phi |$ and reasoning as for \eqref{bootPhiT2}, one obtains
$$ \left| \frac{v^{\mu}}{v^0}\mathcal{L}_Z(F)_{\mu k} Y \Phi \right| \hspace{2mm} \lesssim \hspace{2mm} C\epsilon \frac{v^0}{\tau_+^{\frac{5}{4}}}+C\epsilon \frac{v^{\underline{L}}}{\tau_-^{\frac{5}{4}}} .$$
Bounding $|\partial_{t,x} \Phi|$ with the bootstrap assumption \eqref{bootPhi} and using the inequality \eqref{eq:firstphiesti}, it follows
$$\tau_+\left| \frac{v^{\mu}}{v^0}\mathcal{L}_Z(F)_{\mu k} \partial \Phi \right| \hspace{2mm} \lesssim \hspace{2mm} C\epsilon \frac{v^0}{\tau_+}\log^{\frac{5}{2}}(1+\tau_+)+C\epsilon \frac{v^{\underline{L}}}{\tau_-}\log^{\frac{5}{2}}(1+\tau_+).$$
As $|\Phi| \lesssim \sqrt{\epsilon} \log^2(1+\tau_+)$, we get, using the bound obtained on the left hand side of \eqref{eq:referlater}, $$\Phi \mathcal{L}_{\partial}(F)(v, \nabla_v \Phi ) \hspace{2mm} \lesssim \hspace{2mm} C\epsilon \frac{v^0}{\tau_+^{\frac{3}{2}}} \log^2(1+\tau_+)+C\epsilon \frac{v^{\underline{L}}}{\tau_-^{\frac{3}{2}}} \log^{\frac{7}{2}}(1+\tau_+).$$ For the remaining term, one has schematically, by the first equality of Lemma \ref{calculF},
$$ \left| Y \left( t \frac{v^{\mu}}{v^0} \mathcal{L}_{Z}(F)_{\mu k} \right) \right| \hspace{2mm} \lesssim \hspace{2mm} \left(\tau_++|\Phi| \right) \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z}(F)_{\mu \theta} \right|+\tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{ZZ}(F)_{\mu k} \right|+\tau_+ |\Phi|\left| \frac{v^{\mu}}{v^0} \mathcal{L}_{\partial Z}(F)_{\mu k} \right|. $$
Using $|\Phi| \lesssim \log^2(1+\tau_+) \leq \tau_+$ and following \eqref{eq:firstphiesti}, we get
$$ \left(\tau_++|\Phi| \right) \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z}(F)_{\mu \theta} \right|+\tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{ZZ}(F)_{\mu k} \right| \hspace{2mm} \lesssim \hspace{2mm} \sqrt{\epsilon} \frac{v^0}{\tau_+} \log (1+\tau_+)+\sqrt{\epsilon} \frac{v^{\underline{L}}}{\tau_-} \log (1+\tau_+).$$
Combining \eqref{bootPhiT1} with $|\Phi| \lesssim \log^2(1+\tau_+)$, we obtain
$$ \tau_+ |\Phi|\left| \frac{v^{\mu}}{v^0} \mathcal{L}_{\partial Z}(F)_{\mu k} \right| \hspace{2mm} \lesssim \hspace{2mm} \sqrt{\epsilon} \frac{v^0}{\tau_+} \log^{\frac{5}{2}} (1+\tau_+)+\sqrt{\epsilon} \frac{v^{\underline{L}}}{\tau_-^{\frac{3}{2}}} \log^{\frac{7}{2}} (1+\tau_+).$$ Consequently, one has
$$\left| T_F ( Y \Phi ) \right| \hspace{2mm} \lesssim \hspace{2mm} (\sqrt{\epsilon}+C \epsilon) \frac{v^0}{\tau_+} \log^{\frac{5}{2}} (1+\tau_+)+(\sqrt{\epsilon}+C \epsilon) \frac{v^{\underline{L}}}{\tau_-^{\frac{5}{4}}} \log^{\frac{7}{2}} (1+\tau_+)+(\sqrt{\epsilon}+C \epsilon) \frac{v^{\underline{L}}}{\tau_-} \log^{\frac{5}{2}} (1+\tau_+).$$
One can then split $Y \Phi$ in three functions $\widetilde{\varsigma}$, $\varsigma_1$ and $\varsigma_2$ defined as $\widetilde{\psi}$, $\psi_1$ and $\psi_2$ previously. We have $\| \widetilde{\varsigma} \|_{L^{\infty}_{t,x,v}} \lesssim \sqrt{\epsilon}$ since $\|Y \Phi \|_{L^{\infty}_{x,v}}(0) \lesssim \sqrt{\epsilon}$ (see Proposition \ref{Phi0}) and we can obtain $|\varsigma_1|+|\varsigma_2| \lesssim ( \sqrt{\epsilon}+C \epsilon ) \log^{\frac{7}{2}} (1+\tau_+)$ by similar computations as those of \eqref{eq:repeat1}, \eqref{eq:repeat2} and \eqref{eq:varphi2}. So, taking $C$ large enough and $\epsilon$ small enough, we can improve the bootstrap assumption on $Y \Phi$ and conclude the proof. \end{proof} For the higher order derivatives, we have the following result. \begin{Pro} For all $(Q_1,Q_2) \in \llbracket 0, N-4 \rrbracket^2$ satisfying $Q_2 \leq Q_1$, there exists $R(Q_1,Q_2) \in \mathbb{N}$ such that
$$\forall \hspace{0.5mm} |\beta| \leq N-4, \hspace{3mm} (t,x) \in [0,T[ \times \mathbb{R}^3, \hspace{12mm} \left| Y^{\beta} \Phi \right|(t,x) \lesssim \sqrt{\epsilon} \log^{R(|\beta|,\beta_P)} (1+\tau_+ ).$$ Note that $R(Q_1,Q_2)$ is independent of $M$ if $Q_1 \leq N-6$. \end{Pro} \begin{proof}
The proof is similar to the previous one and we only sketch it. We process by induction on $Q_1$ and, at $Q_1$ fixed, we make an induction on $Q_2$. Let $|\beta| \leq N-4$ and suppose that the result holds for all $ Q_1 \leq |\beta|$ and $Q_2 \leq \beta_P$ satisfying $Q_1 < |\beta|$ or $Q_2 < \beta_P$. Let $0 < T_0 < T$ and $\underline{u} >0$ be such that
$$ \forall \hspace{0.5mm} (t,x,v) \in \underline{V}_{\underline{u}}(T_0) \times \mathbb{R}^3_v, \hspace{1cm} |Y^{\beta} \Phi|(t,x,v) \leq C \sqrt{\epsilon} \log^{R(|\beta|,\beta_P)}(1+\tau_+),$$ with $C>0$ a constant sufficiently large. We now sketch the improvement of this bootstrap assumption, which will imply the desired result. The source terms of $T_F(Y^{\beta} \Phi)$, given by Propositions \ref{ComuVlasov} and \ref{sourcePhi}, can be gathered in two categories. \begin{itemize}
\item The ones where there is no $\Phi$ coefficient derived more than $|\beta| -1$ times, which can then be bounded by the induction hypothesis and give logarithmical growths, as in the proof of the previous Proposition. We then choose $R(|\beta|,\beta_P)$ sufficiently large to fit with these growths.
\item The ones where a $\Phi$ coefficient is derived $|\beta|$ times. Note then that they all come from Proposition \ref{ComuVlasov}, when $|\sigma| = |\beta|$ for the quantities of \eqref{eq:com1} and when $|\sigma|=|\beta|-1$ for the other ones. We then focus on the most problematic ones (with a $\tau_+$ or $\tau_-$ weight, which can come from a weight $z \in \mathbf{k}_1$ for the terms of \eqref{eq:com1}), leading us to integrate along the characteristics of $T_F$ the following expressions. \end{itemize} \begin{equation}\label{eq:Phi11}
\tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} Y^{\kappa} \Phi \right|, \hspace{5mm} \text{with} \hspace{5mm} |\gamma| \leq N-3, \hspace{5mm} |\kappa| = |\beta| \hspace{5mm} \text{and} \hspace{5mm} \kappa_P < \beta_P, \end{equation} \begin{equation}\label{eq:Phi22}
\left| \Phi^{p} \mathcal{L}_{\partial Z^{\gamma_0}} (F) \left( v, \Gamma^{\kappa} \Phi \right) \right|, \hspace{5mm} \text{with} \hspace{5mm} |\gamma_0| \leq N-4, \hspace{5mm} |\kappa| = |\beta|-1 \hspace{5mm} \text{and} \hspace{5mm} p+\kappa_P \leq \beta_P. \end{equation} To deal with \eqref{eq:Phi11}, use the induction hypothesis, as $\kappa_P < \beta_P$. For the other terms, recall from Lemma \ref{GammatoYLem} that we can schematically suppose that
$$\Gamma^{\kappa} \Phi = P_{q,n}(\Phi) Y^{\zeta} \Phi, \hspace{5mm} \text{with} \hspace{5mm} |q|+|\zeta| \leq |\beta|-1, \hspace{5mm} |q| \leq |\beta|-2 \hspace{5mm} \text{and} \hspace{5mm} n+q_P+\zeta_P = \kappa_P.$$ Expressing \eqref{eq:Phi22} in null coordinates and transforming the $v$ derivatives with Lemma \ref{vradial} or $v^0 \partial_{v^i}=Y_i-\Phi X-x^i \partial_t-t \partial_i$, we obtain the following bad terms,
$$ \left( \tau_- |\rho|+ \tau_+ |\alpha|+\tau_+\sqrt{\frac{v^{\underline{L}}}{v^0}}\left( |\sigma|+|\underline{\alpha}| \right) \right) \Phi^p \partial_{t,x} \left( P_{q,n}(\Phi) Y^{\zeta} \Phi \right).$$
Then, note that there is no derivatives of order $|\beta|$ in $\Phi^p \partial_{t,x} \left( P_{q,n}(\Phi) \right) Y^{\zeta} \Phi$ so that these terms can be handled using the induction hypothesis. It then remains to study the terms related to $ P_{q,n+p}(\Phi) \partial_{t,x} Y^{\zeta} \Phi$. If $\zeta_P < \beta_P$, we can treat them using again the induction hypothesis. Otherwise $p+n=0$ and we can follow the treatment of \eqref{eq:referlater}. Finally, the fact that $R(|\beta|,\beta_P)$ is independent of $M$ if $|\beta| \leq N-6$ follows from Remark \ref{lowderiv} and that we merely need pointwise estimates on the derivatives of $F$ up to order $N-5$ in order to bound $Y^{\xi} \Phi$, with $|\xi| \leq N-6$. \end{proof} \begin{Rq}\label{estiPkp} There exist $(M_1,M_2) \in \mathbb{N}^2$, with $M_1$ independent of $M$, such that, for all $p \leq 3N$ and $(t,x,v) \in [0,T[ \times \mathbb{R}^3 \times \mathbb{R}^3$,
$$ \sum_{|k| \leq N-6} |P_{k,p}(\Phi)|(t,x,v) \lesssim \log^{M_1}(1+\tau_+) \hspace{10mm} \text{and} \hspace{10mm} \sum_{|k| \leq N-4} |P_{k,p}(\Phi)|(t,x,v) \lesssim \log^{M_2}(1+\tau_+).$$ \end{Rq}
We are now able to apply the Klainerman-Sobolev inequalities of Proposition \ref{KS1} and Corollary \ref{KS2}. Combined with the bootstrap assumptions \eqref{bootf1}, \eqref{bootf3} and the estimates on the $\Phi$ coefficients, one immediately obtains that, for any $z \in \mathbf{k}_1$, $\max(|\xi|+|\beta|,|\xi|+1) \leq N-6$, $j \leq 2N-\xi_P-\beta_P$, \begin{equation}\label{decayf}
\forall \hspace{0.5mm} (t,x) \in [0,T[\times \mathbb{R}^3, \qquad \int_v |z^jP_{\xi}(\Phi)Y^{\beta} f|(t,x,v) dv \lesssim \epsilon \frac{\log^{(j+|\xi|+|\beta|+3)a}(3+t)}{\tau_+^2\tau_-}. \end{equation}
\section{Improvement of the bootstrap assumptions \eqref{bootf1}, \eqref{bootf2} and \eqref{bootf3} }\label{sec8}
As the improvement of all the energy bounds concerning $f$ are similar, we unify them as much as possible. Hence, let us consider \begin{itemize} \item $Q \in \{ N-3,N-1,N\}$, $n_{N-3}=4$, $n_{N-1}=0$ and $n_N=0$.
\item Multi-indices $\beta^0$, $\xi^0$ and $\xi^2$ such that $\max (|\xi^0|+|\beta^0|, 1+|\xi^0| ) \leq Q$ and $\max (|\xi^2|+|\beta^0|, 1+|\xi^2| ) \leq Q$. \item A weight $z_0 \in \mathbf{k}_1$ and $q \leq 2N-1+n_Q-\xi^0_P-\xi^2_P-\beta^0_P$. \end{itemize} According to the energy estimate of Propostion \ref{energyf}, Corollary \ref{coroinit} and since $\xi^0$ and $\xi^2$ play a symmetric role, we could improve \eqref{bootf1}-\eqref{bootf3}, for $\epsilon$ small enough, if we prove that \begin{eqnarray}\label{improvebootf}
\int_0^t \int_{\Sigma_s} \int_v \left| T_F \left( z^q_0 P_{\xi^0}(\Phi) Y^{\beta^0} f \right) P_{\xi^2}(\Phi) \right| \frac{dv}{v^0} dx ds & \lesssim & \epsilon^{\frac{3}{2}}(1+t)^{\eta} \log^{aq}(3+t) \hspace{3mm} \text{if} \hspace{3mm} Q =N, \\
& \lesssim & \epsilon^{\frac{3}{2}} \log^{(q+|\xi^0|+|\xi^2|+|\beta^0|)a}(3+t) \hspace{3mm} \text{otherwise}. \label{improvebootf2} \end{eqnarray}
For that purpose, we will bound the spacetime integral of the terms given by Proposition \ref{ComuPkp}, applied to $z^q_0 P_{\xi^0}(\Phi) Y^{\beta^0} f$. We start, in Subsection \ref{sec81}, by covering the term of \eqref{eq:cat0}. Subsection \ref{sec82} (respectively \ref{ref:L2elec}) is devoted to the study of the expressions of the other categories for which the electromagnetic field is derived less than $N-3$ times (respectively more than $N-2$ times). Finally, we treat the more critical terms in Subsection \ref{sec86}. In Subsection \ref{Ximpro}, we bound $\mathbb{E}_N^X[f]$, $ \mathbb{E}_{N-1}^X[f]$ and we improve the decay estimate of $\int_v (v^0)^{-2} |Y^{\beta} f|dv$ near the light cone.
\subsection{The terms of \eqref{eq:cat0}}\label{sec81}
The purpose of this Subsection is to prove the following proposition.
\begin{Pro}\label{M1}
Let $\xi^1$, $\xi^2$ and $\beta$ such that $\max(1+|\xi^i|,|\xi^i|+|\beta|) \leq N$ for $i \in \{1,2 \}$. Consider also $z \in \mathbf{k}_1$, $r \in \mathbb{N}^*$, $0 \leq \kappa \leq \eta$, $0 < j \leq 2N+3-\xi^1-\xi^2_P-\beta_P$ and suppose that, $$\forall \hspace{0.5mm} t \in [0,T[, \hspace{8mm} \mathbb{E} \left[ z^j P_{\xi^1}(\Phi)P_{\xi^2}(\Phi) Y^{\beta} f \right](t)+\log^2(3+t)\mathbb{E} \left[ z^{j-1} P_{\xi^1}(\Phi)P_{\xi^2}(\Phi) Y^{\beta} f \right](t) \lesssim \epsilon (1+t)^{\kappa} \log^r(3+t).$$ Then,
$$ \int_0^t \int_{\Sigma_s} \int_v \left| F\left(v,\nabla_v z^j\right) P_{\xi^1}(\Phi)P_{\xi^2}(\Phi) Y^{\beta} f \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}}(1+t)^{\kappa} \log^{r}(3+t).$$ \end{Pro} \begin{proof} To lighten the notations, we denote $P_{\xi^1}(\Phi)P_{\xi^2}(\Phi) Y^{\beta}f$ by $h$ and, for $d \in \{0,1 \}$, $\mathbb{E} \left[ z^{j-d} h \right]$ by $H_{j-d}$, so that
$$H_{j-d}(t) \hspace{1mm} = \hspace{1mm} \| z^{j-d} h \|_{L^1_{x,v}}(t)+\sup_{u \in \mathbb{R}} \int_{C_u(t)} \int_v \frac{ v^{\underline{L}}}{v^0}|z^{j-d} h| dv dC_u(t) \hspace{1mm} \lesssim \hspace{1mm} \epsilon (1+t)^{\kappa} \log^{r-2d}(3+t).$$ Using Lemmas \ref{weights1} and \ref{vradial}, we have
$$ \left| \left( \nabla_v z^j \right)^L \right|, \hspace{1mm} \left| \left( \nabla_v z^j \right)^{\underline{L}} \right|, \hspace{1mm} \frac{|v^A|+v^{\underline{L}}}{v^0}\left| \left( \nabla_v z^j \right)^A \right| \lesssim \frac{\tau_-}{v^0}|z|^{j-1}+\frac{1}{v^0} \sum_{w \in \mathbf{k}_1} |w|^j.$$
Hence, the decomposition of $F\left(v,\nabla_v |z|^j\right)$ in our null frame brings us to control the integral, over $[0,T] \times \mathbb{R}^3_x \times \mathbb{R}^3_v$, of\footnote{The second term comes from $\alpha(F)_Av^L\left(\nabla_v |z|^j \right)^A$.}
$$\left( \tau_-|w|^{j-1}+|w|^j \right)(|\rho(F)|+|\alpha(F)|+|\sigma(F)|+|\underline{\alpha}(F)|)\frac{|h|}{v^0} \hspace{5mm} \text{and} \hspace{5mm} \left( \tau_+|w|^{j-1}+ |w|^j \right) |\alpha(F)|\frac{|h|}{v^0}.$$ According to Remark \ref{lowderiv} and using $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$ (see Lemma \ref{weights1}), we have
$$ \tau_-(|\rho(F)|+|\sigma(F)|+|\underline{\alpha}(F)|)+\tau_+|\alpha(F)| \lesssim \sqrt{\epsilon} \frac{\log(3+t)}{\tau_+}, \hspace{6mm} |\rho(F)|+|\sigma(F)|+|\underline{\alpha}(F)|+|\alpha(F)| \lesssim \sqrt{\epsilon} \frac{v^0}{\tau_+^{\frac{3}{2}}}+\sqrt{\epsilon}\frac{v^{\underline{L}}}{\tau_-^{\frac{3}{2}}}.$$ The result is then implied by the following two estimates, \begin{eqnarray}
\nonumber \int_0^t \int_{\Sigma_s} \int_v \sqrt{\epsilon} |h|\left( \frac{|w|^{j-1}}{1+s}\log(3+s)+\frac{|w|^j}{(1+s)^{\frac{3}{2}}} \right) dvdxds \hspace{-0.5mm} & \lesssim & \hspace{-0.5mm} \sqrt{\epsilon} \int_0^t \frac{\log(3+s)}{1+s} H_{j-1}(s)ds+\int_0^t \frac{H_{j}(s)}{(1+s)^{\frac{3}{2}}} ds \\ \nonumber & \lesssim & \hspace{-0.5mm} \epsilon^{\frac{3}{2}} \int_0^t \frac{\log^{r-1}(3+t)}{(1+s)^{1-\kappa}} +\frac{\log^{r}(3+t)}{(1+s)^{\frac{5}{4}-\kappa}} ds \\ \nonumber
& \lesssim & \epsilon^{\frac{3}{2}}(1+t)^{\kappa} \log^r(3+t), \\
\nonumber \int_0^t \int_{\Sigma_s} \frac{\sqrt{\epsilon}}{\tau_-^{\frac{3}{2}}} \int_v \frac{v^{\underline{L}}}{v^0} \left| w^j h \right| dvdxds \hspace{-0.5mm} & = & \hspace{-0.5mm} \int_{u=-\infty}^t \frac{\sqrt{\epsilon}}{\tau_-^{\frac{3}{2}}} \int_{C_u(t)} \int_v \frac{v^{\underline{L}}}{v^0} \left| w^j h \right| dv dC_u(t) du \\ \nonumber & \lesssim & \hspace{-0.5mm} \sqrt{\epsilon} H_j(t) \int_{u=-\infty}^{+ \infty} \frac{du}{\tau_-^{\frac{3}{2}}} \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} (1+t)^{\kappa} \log^r (3+t). \end{eqnarray} \end{proof}
\subsection{Bounds on several spacetime integrals}\label{sec82}
We estimate in this subsection the spacetime integral of the source terms of \eqref{eq:cat1}-\eqref{eq:cat4} of $T_F(z_0^q P_{\xi^0}(\Phi) Y^{\beta^0}f )$, multiplied by $(v^0)^{-1} P_{\xi^2}(\Phi)$, where the electromagnetic field is derived less than $N-3$ times. We then fix, for the remainder of the subsection, \begin{itemize}
\item multi-indices $\gamma$, $\beta$ and $\xi^1$ such that $$|\gamma| \leq N-3, \hspace{3mm} |\xi^1|+ |\gamma| + |\beta| \leq Q+1, \hspace{3mm} |\beta| \leq |\beta^0|, \hspace{3mm} |\xi^1|+|\beta| \leq |\xi^0|+|\beta^0| \leq Q \hspace{3mm} \text{and} \hspace{3mm} |\xi^1| \leq Q-1.$$ \item $n \leq 2N$, \hspace{2mm} $z \in \mathbf{k}_1$ \hspace{2mm} and \hspace{2mm} $j \in \mathbb{N}$ \hspace{2mm} such that \hspace{2mm} $j \leq 2N-1+n_Q-\xi^1_P-\xi^2_P-\beta_P$.
\item We will make more restrictive hypotheses for the study of the terms of \eqref{eq:cat3} and \eqref{eq:cat4}. For instance, for the last ones, we will take $|\xi^1| < |\xi^0|$ and $j=q$. This has to do with their properties described in Proposition \ref{ComuPkp}. \end{itemize}
Note that $|\xi^2|+|\beta| \leq Q$. To lighten the notations, we introduce $$h := z^j P_{\xi^1}(\Phi) P_{\xi^2}(\Phi) Y^{\beta} f.$$ We start by treating the terms of \eqref{eq:cat1}. \begin{Pro}\label{M2} Under the bootstrap assumptions \eqref{bootf1}-\eqref{bootf3}, we have,
$$I_1:=\int_0^t \int_{\Sigma_s} \int_v |\Phi|^n \left( \left| \nabla_{Z^{\gamma}} F \right|+\frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(F) \right)\right|+\frac{\tau_+}{\tau_-}\sqrt{\frac{v^{\underline{L}}}{v^0}} \left| \sigma \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right| \right) \left| h \right| \frac{dv}{v^0} dx ds \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}.$$ \end{Pro} \begin{proof} According to Propositions \ref{Phi1}, \ref{decayF} and $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$, we have \begin{eqnarray}
\nonumber \left| \Phi \right|^n \left| \nabla_{Z^{\gamma}} F \right|+\left| \Phi \right|^n\frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(F) \right)\right|+\left| \Phi \right|^n \frac{\tau_+}{\tau_-}\sqrt{\frac{v^{\underline{L}}}{v^0}} \left| \sigma \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right| \hspace{-1.5mm} & \lesssim & \hspace{-1.5mm} \sqrt{\epsilon}\log^{4N+M}(3+t)\left( \frac{\sqrt{v^0 v^{\underline{L}}}}{\tau_+\tau_-}+ \frac{v^{\underline{L}}}{\tau_+^{\frac{1}{2}}\tau_-^{\frac{3}{2}}} \right) \\ \nonumber & \lesssim & \hspace{-1.5mm} \sqrt{\epsilon} \frac{v^0}{\tau_+^{\frac{5}{4}}}+\sqrt{\epsilon} \frac{v^{\underline{L}}}{\tau_+^{\frac{1}{4}}\tau_-^{\frac{3}{2}}}. \end{eqnarray} Then, \begin{eqnarray}
\nonumber I_1 & \lesssim & \int_0^t \int_{\Sigma_s} \frac{\sqrt{\epsilon}}{\tau_+^{\frac{5}{4}}} \int_v |h| dv dx ds + \int_0^t \int_{\Sigma_s} \frac{\sqrt{\epsilon}}{\tau_+^{\frac{1}{4}}\tau_-^{\frac{3}{2}}} \int_v \frac{ v^{\underline{L}}}{v^0} |h| dv dx ds \\ \nonumber
& \lesssim & \sqrt{\epsilon} \int_0^t \frac{\mathbb{E}[h](s)}{(1+s)^{\frac{5}{4}}}ds+\sqrt{\epsilon} \int_{u=-\infty}^t \int_{C_u(t)} \frac{1}{\tau_+^{\frac{1}{4}}\tau_-^{\frac{3}{2}}} \int_v \frac{ v^{\underline{L}}}{v^0} |h| dv d C_u(t) du. \end{eqnarray} Recall now the definition of $(t_i)_{i \in \mathbb{N}}$, $(T_i(t))_{i \in \mathbb{N}}$ and $C_u^i(t)$ from Subsection \ref{secsubsets}. By the bootstrap assumption \eqref{bootf3} and $2\eta < \frac{1}{8}$, we have
$$\mathbb{E}[h](s) \lesssim \epsilon(1+s)^{\frac{1}{8}} \hspace{5mm} \text{and} \hspace{5mm} \sup_{u \in \mathbb{R}} \int_{C_u^i(t)} \int_v v^0 v^{\underline{L}} |h| dv dC_u^i(t) \lesssim \epsilon (1+T_{i+1}(t))^{2 \eta} \lesssim \epsilon (1+t_{i+1})^{ \frac{1}{8}},$$ so that, using also\footnote{Note that the sum over $i$ is actually finite as $C^i_u(t) = \varnothing$ for $i \geq \log_2(1+t)$.} $1+t_{i+1} \leq 2(1+t_i) $ and Lemma \ref{foliationexpli}, \begin{eqnarray} \nonumber \sqrt{\epsilon} \int_0^t \frac{\mathbb{E}[h](s)}{(1+s)^{\frac{5}{4}}} & \lesssim & \epsilon^{\frac{3}{2}} \int_0^{+ \infty} \frac{ds}{(1+s)^{\frac{9}{8}}} \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}, \\
\nonumber \sqrt{\epsilon} \int_{u=-\infty}^t \int_{C_u(t)} \frac{1}{\tau_+^{\frac{1}{4}}\tau_-^{\frac{3}{2}}} \int_v \frac{ v^{\underline{L}}}{v^0} |h| dv d C_u(t) du \hspace{-1mm} & = & \hspace{-1mm} \sqrt{\epsilon} \int_{u=-\infty}^t \sum_{i=0}^{+ \infty} \int_{C^i_u(t)} \frac{1}{\tau_+^{\frac{1}{4}}\tau_-^{\frac{3}{2}}} \int_v \frac{ v^{\underline{L}}}{v^0} |h| dv d C^i_u(t) du \\ \nonumber
& \lesssim & \sqrt{\epsilon} \int_{u=-\infty}^t \frac{1}{\tau_-^{\frac{3}{2}}} \sum_{i=0}^{+ \infty}\frac{1}{(1+t_i)^{\frac{1}{4}}} \int_{C^i_u(t)} \int_v \frac{ v^{\underline{L}}}{v^0} |h| dv d C^i_u(t) du \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}} \int_{u=-\infty}^t \frac{du}{\tau_-^{\frac{3}{2}}} \sum_{i=0}^{+ \infty}\frac{(1+t_{i+1})^{\frac{1}{8}}}{(1+t_{i+1})^{\frac{1}{4}}} \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}} \int_{u = - \infty}^{+\infty} \frac{du}{\tau_-^{\frac{3}{2}}} \sum_{i=0}^{+ \infty} 2^{-\frac{i}{8}} \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}. \end{eqnarray} \end{proof} We now start to bound the problematic terms. \begin{Pro}\label{M3} We study here the terms of \eqref{eq:cat3}. If, for $\kappa \geq 0$ and $r \in \mathbb{N}$, $$
\mathbb{E}[h](t)= \left\| h \right\|_{L^1_{x,v}}(t)+\sup_{u \in \mathbb{R}} \int_{C_u(t)} \int_v \frac{ v^{\underline{L}}}{v^0} |h| dv dC_u(t) \lesssim \epsilon (1+s)^{\kappa} \log^r(3+t), \hspace{3mm} \text{then} $$
\begin{flalign*}
& \hspace{1cm} I^1_3:=\int_{0}^t \int_{\Sigma_s} \frac{\tau_+}{\tau_-} \left| \underline{\alpha} \left( \mathcal{L}_{Z^{\gamma}} ( F) \right) \right| \int_v \sqrt{\frac{v^{\underline{L}}}{v^0}} \left| h \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}}(1+s)^{\kappa} \log^r(3+t) \hspace{3mm} \text{and} & \\
& \hspace{1cm} I^2_3:=\int_{0}^t \int_{\Sigma_s} \frac{\tau_+}{\tau_-} \left| \rho \left( \mathcal{L}_{Z^{\gamma}} ( F) \right) \right|\int_v \left| h \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}}(1+s)^{\kappa} \log^{r+a}(3+t) .& \end{flalign*} \end{Pro} \begin{Rq} The extra $\log^{a}(3+t)$-growth on $I^2_3$, compared to $I^1_3$, will not avoid us to close the energy estimates in view of the hierarchies in the energy norms. Indeed, we have $j=q-1$ (in $I^2_3$) according to the properties of the terms of \eqref{eq:cat3} (in $I_3^1$, we merely have $j \leq q$). \end{Rq} \begin{proof}
Recall first from Lemma \ref{weights1} that $1+|v^A| \lesssim \sqrt{v^0 v^{\underline{L}}}$. Then, using Proposition \ref{decayF} and the inequality $2CD \leq C^2+D^2$, one obtains
$$ \sqrt{\frac{v^{\underline{L}}}{v^0}} \frac{\tau_+}{\tau_-}\left|\underline{\alpha} \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) \right| \lesssim \sqrt{\epsilon} \frac{v^{\underline{L}}}{\tau_-^{\frac{3}{2}}} \hspace{8mm} \text{and} \hspace{8mm} \frac{\tau_+}{\tau_-} \left| \rho \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right| \lesssim \sqrt{\epsilon} \log^M(3+t) \frac{v^0}{\tau_+}+\sqrt{\epsilon} \log^M(3+t) \frac{v^{\underline{L}} }{ \tau_-^3} .$$ We then have, as $a = M+1$, \begin{eqnarray}
\nonumber I^1_3 & \lesssim & \int_{u = -\infty}^t \frac{\sqrt{\epsilon}}{\tau_-^{\frac{3}{2}}} \int_{C_u(t)} \int_v \frac{ v^{\underline{L}}}{v^0} |h| dv dC_u(t) du \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} \mathbb{E}[h](t) \int_{u=-\infty}^{+\infty} \frac{du}{\tau_-^{\frac{3}{2}}} \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} (1+s)^{\kappa} \log^r(3+t) , \\
\nonumber I^2_3 & \lesssim & \sqrt{\epsilon} \int_0^t \int_{\Sigma_s} \frac{\log^M(3+s)}{\tau_+} \int_v |h| dv dx ds +\sqrt{\epsilon} \log^M (3+t) \int_{u = -\infty}^t \frac{\sqrt{\epsilon}}{\tau_-^{\frac{3}{2}}} \int_{C_u(t)} \int_v \frac{ v^{\underline{L}}}{v^0} |h| dv dC_u(t) du \\ \nonumber & \lesssim & \sqrt{\epsilon} \int_0^t \frac{\log^{r+M}(3+s)}{(1+s)^{1- \kappa}} ds+ \epsilon^{\frac{3}{2}} (1+t)^{\kappa} \log^{r+M}(3+t) \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}}(1+t)^{\kappa} \log^{r+M+1}(3+t) \hspace{2mm} = \hspace{2mm} \epsilon^{\frac{3}{2}}(1+t)^{\kappa} \log^{r+a}(3+t) . \end{eqnarray}
\end{proof} We finally end this subsection by the following estimate. \begin{Pro}\label{MM3}
We suppose here that $\max( |\xi^1|+|\beta|, |\xi^1|+1) \leq N-1$. Then, \begin{eqnarray}
\nonumber I_4 \hspace{2mm} := \hspace{2mm} \int_0^t \int_{\Sigma_s} \tau_+ \int_v \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \right| \left| h \right| \frac{dv}{v^0} dx ds & \lesssim & \epsilon^{\frac{3}{2}} \log^{(1+j+|\xi^1|+|\xi^2|+|\beta|)a}(3+t) \hspace{1cm} \text{if} \hspace{3mm} |\xi^2| \leq N-2, \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}} (1+t)^{\frac{3}{4} \eta} \hspace{4cm} \text{otherwise}. \end{eqnarray} \end{Pro} \begin{Rq}\label{rq:i4}
To understand the extra hypothesis made in this proposition, recall from the properties of the terms of \eqref{eq:cat4} that we can assume $|\xi^1| < |\xi^0|$, $\beta=\beta^0$ and $j=q$. We then have
$$1+j+|\xi^1|+|\xi^2|+|\beta| \leq q+|\xi^0|+|\xi^2|+|\beta^0|.$$ \end{Rq} \begin{proof}
Let us denote by $(\alpha, \underline{\alpha}, \rho, \sigma)$ the null decomposition of $\mathcal{L}_{Z^{\gamma}}(F)$. Using $1+|v^A| \leq \sqrt{v^0 v^{\underline{L}}}$ and Proposition \ref{decayF}, we have \begin{eqnarray}
\nonumber \tau_+ \left| \frac{v^{\mu}}{(v^0)^2} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \right| \hspace{-1mm} & \lesssim & \hspace{-1mm} \tau_+ \sqrt{\frac{v^{\underline{L}}}{v^0}} \left( |\alpha|+|\rho|+|\sigma| \right)+\tau_+\frac{v^{\underline{L}}}{v^0}|\underline{\alpha}| \\ \nonumber & \lesssim & \hspace{-1mm} \sqrt{\epsilon} \sqrt{\frac{v^{\underline{L}}}{v^0}} \frac{\log^M (3+t)}{\sqrt{\tau_+ \tau_-}}+\sqrt{\epsilon} \frac{v^{\underline{L}}}{v^0} \frac{\log^M (3+t)}{\tau_-} \hspace{1.2mm} \lesssim \hspace{1.2mm} \sqrt{\epsilon} \frac{\log^M (3+t)}{\tau_+}+\sqrt{\epsilon} \frac{v^{\underline{L}}}{v^0} \frac{\log^M (3+t)}{\tau_-}. \end{eqnarray}
As $\tau_- \sim \tau_+$ away from the light cone (for, say\footnote{If $(s,y)$ is in one of these regions of $[0,t] \times \mathbb{R}^3$, we have $|y| \geq 2s$ or $|y| \leq \frac{s}{2}$.}, $u \leq -t$ and $ u \geq \frac{t}{2}$), we finally obtain that \begin{eqnarray}
\nonumber I_4 \hspace{-1mm} & = & \hspace{-1mm} \sqrt{\epsilon} \int_0^t \frac{\log^M(3+s)}{1+s} \int_{ \Sigma_s } \int_v |h| dv dx ds + \sqrt{\epsilon} \log^M (3+t) \int_{u=-t}^{\frac{t}{2}} \frac{1}{\tau_-} \int_{C_u(t)} \int_v \frac{ v^{\underline{L}}}{v^0} |h| dv dC_u(t) du \\ \nonumber & \lesssim & \hspace{-1mm} \sqrt{\epsilon} \log^M(3+t) \sup_{[0,t]} \mathbb{E}[h] \int_0^t \hspace{-0.5mm} \frac{ds}{1+s}+\sqrt{\epsilon}\log^M(3+t) \mathbb{E}[h](t) \int_{u=-t}^t \hspace{-0.5mm} \frac{du}{\tau_-} \hspace{1.2mm} \lesssim \hspace{1.2mm} \sqrt{\epsilon} \log^{a}(3+t) \sup_{[0,t]} \mathbb{E}[h] . \end{eqnarray}
If $|\xi^2| \leq N-2$, the bootstrap assumption \eqref{bootf1} or \eqref{bootf2} gives
$$ \sup_{[0,t]} \mathbb{E}[h] \leq \epsilon \log^{(j+|\xi^1|+|\xi^2|+|\beta|)a}(3+t)$$
and we can conclude the proof in that case. If $|\xi^2|=N-1$, we have $j \leq 2N-1-\xi^1_P-\xi^2_P-\beta_P$ since this case appears only if $Q=N$. Let $(i_1,i_2) \in \mathbb{N}^2$ be such that $$i_1+i_2=2j, \hspace{1cm} i_1 \leq 2N-1-2 \xi^1_P-\beta_P \hspace{1cm} \text{and} \hspace{1cm} i_2 \leq 2N-1-2 \xi^2_P-\beta_P.$$ Using the bootstrap assumptions \eqref{bootf2} and \eqref{bootf3}, we have \begin{eqnarray}
\nonumber \mathbb{E}[h](t) & = & \int_{\Sigma_t} \int_v \left| z^j P_{\xi^1}(\Phi) P_{\xi^2}(\Phi) Y^{\beta} f \right| dv dx \\ \nonumber
& \lesssim & \left| \int_{\Sigma_t} \int_v \left| z^{i_1} P_{\xi^1}(\Phi)^2 Y^{\beta} f \right| dv dx \int_{\Sigma_t} \int_v \left| z^{i_2} P_{\xi^2}(\Phi)^2 Y^{\beta} f \right| dv dx \right|^{\frac{1}{2}} \\ \nonumber
& \lesssim & \left| \log^{(i_1+2|\xi^1|+|\beta|)a}(3+t) \mathbb{E}_{N-1}^0[f](t) \log^{a i_2}(3+t) \overline{\mathbb{E}}_N[f](t) \right|^{\frac{1}{2}} \hspace{2mm} \lesssim \hspace{2mm} \epsilon (1+t)^{\frac{3}{4} \eta}, \end{eqnarray} which ends the proof. \end{proof} Note now that Propositions \ref{ComuPkp}, \ref{M1}, \ref{M2}, \ref{M3} and \ref{MM3} imply \eqref{improvebootf2} for $Q=N-3$, so that $\mathbb{E}^4_{N-3}[f] \leq 3 \epsilon$ on $[0,T[$. \subsection{Completion of the bounds on the spacetime integrals}\label{ref:L2elec}
In this subsection, we bound the spacetime integrals considered previously when the electromagnetic field is differentiated too many times to be estimated pointwise. For this, we make crucial use of the pointwise decay estimates on the velocity averages of $ \left| z^j P_{\zeta}(\Phi) Y^{\beta} f \right|$ which are given by \eqref{decayf}. The terms studied here appear only if $|\xi^0|+|\beta^0| \geq N-2$ since otherwise the electromagnetic field would be differentiated at most $N-3$ times. We then fix, for the remainder of the subsection, $Q \in \{N-1,N \}$, \begin{itemize}
\item multi-indices $\gamma$, $\beta$ and $\xi^1$ such that \hspace{1mm} $N-2 \leq |\gamma| \leq N$, $$ |\gamma|+|\xi^1| \leq Q, \hspace{3mm} |\xi^1|+ |\gamma| + |\beta| \leq Q+1, \hspace{3mm} |\beta| \leq |\beta^0|, \hspace{3mm} |\xi^1|+|\beta| \leq |\xi^0|+|\beta^0| \leq Q \hspace{3mm} \text{and} \hspace{3mm} |\xi^1| \leq Q-1.$$ \item $n \leq 2N$, \hspace{2mm} $z \in \mathbf{k}_1$ \hspace{2mm} and \hspace{2mm} $j \in \mathbb{N}$ \hspace{2mm} such that \hspace{2mm} $j \leq 2N-1-\xi^1_P-\xi^2_P-\beta_P$. \item Consistently with Proposition \ref{ComuPkp}, we will, in certain cases, make more assumptions on $\xi^1$ or $j$, such as $j \leq q$ for the terms of \eqref{eq:cat3}. \end{itemize}
Note that $|\xi^2|+|\beta| \leq Q$ and that there exists $i_1$ and $i_2$ such as $$i_1+i_2=2j, \hspace{5mm} i_1 \leq 2N-1-2\xi^1_P-\beta_P \hspace{5mm} \text{and} \hspace{5mm} i_2 \leq 2N-1-2\xi^2_P-\beta_P.$$ To lighten the notations, we introduce $$h := z^j P_{\xi^1}(\Phi) P_{\xi^2}(\Phi) Y^{\beta} f, \hspace{10mm} h_1 := z^{i_1} P_{\xi^1}(\Phi)^2 Y^{\beta} f \hspace{10mm} \text{and} \hspace{10mm} h_2 := z^{i_2} P_{\xi^2}(\Phi)^2 Y^{\beta} f,$$
so that $\left| h \right| = \sqrt{| h_1 h_2 |}$. As $|\gamma| \geq N-2$, we have $|\xi^1| \leq 2 \leq N-7$ and $2|\xi^1| + |\beta| \leq 5 \leq N-6$. Thus, by Lemma \ref{weights1} and \eqref{decayf}, we have, for all $(t,x) \in [0,T[ \times \mathbb{R}^3$, \begin{equation}\label{eq:h1}
\tau_+^3 \int_v |h_1| \frac{dv}{(v^0)^2}+\tau_+^2 \tau_- \int_v |h_1| dv \hspace{1mm} \lesssim \hspace{1mm} \int_v \left(\tau_+^3 \frac{v^{\underline{L}}}{v^0}+\tau_+^2 \tau_- \right) |h_1| dv \hspace{1mm} \lesssim \hspace{1mm} \epsilon \log^{(4+i_1+2|\xi^1|+|\beta|)a}(3+t). \end{equation} Using Remark \ref{rqweights1}, we have, \begin{equation}\label{eq:h11}
\forall \hspace{0.5mm} |x| \geq t, \hspace{1cm} \tau_+^3 \tau_- \int_v |h_1| \frac{dv}{(v^0)^2} \hspace{1mm} \lesssim \hspace{1mm} \tau_+^3 \tau_- \int_v \frac{v^{\underline{L}}}{v^0} |h_1| dv \hspace{1mm} \lesssim \hspace{1mm} \epsilon \log^{(4+i_1+2|\xi^1|+|\beta|)a}(3+t). \end{equation} \begin{Pro}\label{M21} The following estimates hold,
$$I^1_1 := \int_0^t \int_{\Sigma_s} \int_v |\Phi|^n\left| \nabla_{Z^{\gamma}} F \right| \left| h \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}}, \hspace{6mm} I^2_1 := \int_0^t \int_{\Sigma_s} \int_v |\Phi|^n \frac{\tau_+}{\tau_-}\sqrt{\frac{v^{\underline{L}}}{v^0}} \left| \sigma \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right| \left| h \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}}$$
$$ \text{and} \hspace{8mm} I^3_1 := \int_0^t \int_{\Sigma_s} \int_v |\Phi|^n \frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right| \left| h \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}}.$$ \end{Pro} \begin{proof}
Using the Cauchy-Schwarz inequality twice (in $x$ and then in $v$), $ \| \nabla_{Z^{\gamma}} F \|^2_{L^2(\Sigma_t)} \lesssim \mathcal{E}_N^0[F](t) \leq 4\epsilon $, $|\Phi| \lesssim \sqrt{\epsilon} \log^2(1+\tau_+)$, $\overline{\mathbb{E}}_N[f](t) \lesssim \epsilon (1+t)^{\eta}$ and \eqref{eq:h1}, we have \begin{eqnarray}
\nonumber I^1_1 & \lesssim & \int_0^t \| \nabla_{Z^{\gamma}} F \|_{L^2(\Sigma_s)} \left\| \int_v |\Phi|^{n} |h| \frac{dv}{v^0} \right\|_{L^2(\Sigma_s)}ds \\ \nonumber
& \lesssim & \sqrt{\epsilon} \int_0^t \left\| \log^{8N}(1+\tau_+) \int_v |h_1| \frac{dv}{(v^0)^2} \int_v |h_2| dv \right\|_{L^1(\Sigma_s)}^{\frac{1}{2}}ds \\ \nonumber
& \lesssim & \sqrt{\epsilon} \int_0^t \left\| \log^{8N}(1+\tau_+) \int_v |h_1| \frac{dv}{(v^0)^2} \right\|_{L^{\infty}(\Sigma_s)}^{\frac{1}{2}} \sqrt{\mathbb{E}[h_2](s)}ds \\ \nonumber & \lesssim & \epsilon \int_0^t \frac{\log^{4N+3Na}(3+s)}{(1+s)^{\frac{3}{2}}} \log^{ai_2}(3+s) \overline{\mathbb{E}}_N[f](s) ds \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}. \end{eqnarray} For the second one, recall from the bootstrap assumptions \eqref{bootF1} and \eqref{bootf3} that for all $t \in [0,T[$ and $i \in \mathbb{N}$,
$$ \int_{C^i_u(t)} \hspace{-0.5mm} |\sigma|^2 dC^i_u(t) \leq \mathcal{E}_N^0[F](t_{i+1}(t)) \lesssim \epsilon \hspace{3.2mm} \text{and} \hspace{3.2mm} \sup_{u \in \mathbb{R}} \int_{C_u^i(t)} \int_v \frac{ v^{\underline{L}}}{v^0} \left| h_2 \right| dv dC^i_u(t) \lesssim \mathbb{E}[h_2](T_{i+1}(t)) \lesssim \epsilon (1+t_{i+1})^{2\eta}.$$ Hence, using this time a null foliation, one has \begin{eqnarray}
\nonumber I_1^2 & \lesssim & \sum_{i=0}^{+ \infty}\int_{u=-\infty}^t \frac{1}{\tau_-} \left| \int_{C^i_u(t)} |\sigma \left( \mathcal{L}_{Z^{\gamma}} ( F) \right)|^2 dC^i_u(t) \int_{C_u^i(t)} \tau_+^{2} \left| \int_v |\Phi|^n\sqrt{\frac{ v^{\underline{L}}}{v^0}} |h| \frac{dv}{v^0} \right|^2 dC_u^i(t) \right|^{\frac{1}{2}} du \\ \nonumber
& \lesssim & \sqrt{\epsilon} \sum_{i=0}^{+ \infty} \int_{u=-\infty}^t \frac{1}{\tau_-} \left| \int_{C_u^i(t)} \tau_+^{2} \log^{8N}(1+\tau_+) \int_v \left| h_1 \right| \frac{dv}{(v^0)^2} \int_v \frac{ v^{\underline{L}}}{v^0} \left| h_2 \right| dv dC_u^i(t) \right|^{\frac{1}{2}} du \\ \nonumber
& \lesssim & \sqrt{\epsilon} \sum_{i=0}^{+ \infty }\int_{u=-\infty}^t \frac{1}{\tau_-} \left| \int_{C_u^i(t)} \frac{1}{\tau_+^{\frac{3}{4}}} \int_v \frac{ v^{\underline{L}}}{v^0} \left| h_2 \right| dv dC_u^i(t) \right|^{\frac{1}{2}} du \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}} \int_{u=-\infty}^{+ \infty} \frac{du}{\tau_-^{\frac{9}{8}}} \sum_{i=0}^{+ \infty} \frac{(1+t_{i+1})^{\eta}}{(1+t_i)^{\frac{1}{4}}} \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}. \end{eqnarray} For the last one, use first that $F=\widetilde{F}+\overline{F}$ to get
$$I^3_1 = I_1^{\widetilde{F}}+I_1^{\overline{F}} := \int_0^t \int_{\Sigma_s} \int_v |\Phi|^n \frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(\widetilde{F}) \right) \right| \left| h \right| \frac{dv}{v^0} dx ds+\int_0^t \int_{\Sigma_s} \int_v |\Phi|^n \frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(\overline{F}) \right) \right| \left| h \right| \frac{dv}{v^0} dx ds.$$
By Proposition \ref{decayF}, we have $|\mathcal{L}_{Z^{\gamma}}(\overline{F})| \lesssim \epsilon \tau_+^{-2}$. Hence, using $|\Phi| \lesssim \log^2(1+\tau_+)$ and $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$, we have
$$ |\Phi|^n \frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(\overline{F}) \right) \right| \lesssim \frac{ \epsilon \sqrt{v^0v^{\underline{L}}} }{\sqrt{v^0}\tau_+^{\frac{3}{4}} \tau_-} \leq \epsilon \frac{v^0}{\tau_+^{\frac{5}{4}}} + \epsilon \frac{ v^{\underline{L}} }{\tau_+^{\frac{1}{4}} \tau_-^2}$$ and we can bound $I_1^{\overline{F}}$ by $\epsilon^{\frac{3}{2}}$ as $I_1$ in Proposition \ref{M2}. For $I_1^{\widetilde{F}}$, remark first that, by the bootstrap assumptions \eqref{bootext}, \eqref{bootF4} and since $F=\widetilde{F}$ in the interior of the light cone,
$$ \int_{C^i_u(t)} \tau_+\left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(\widetilde{F}) \right) \right|^2 dC_u^i(t) \lesssim \mathcal{E}_N[F](T_{i+1}(t))+\mathcal{E}^{Ext}_{N}[\widetilde{F}](T_{i+1}(t)) \lesssim \epsilon (1+t_{i+1})^{\eta}.$$
It then comes, using $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$, $16 \eta < 1$ and $\int_v |\Phi|^n |h_1| dv \lesssim \epsilon \tau_+^{-\frac{3}{2}}\tau_-^{-1}$, that \begin{eqnarray}
\nonumber I_1^{\widetilde{F}} & \lesssim & \sum_{i=0}^{+ \infty}\int_{u=-\infty}^t \frac{1}{\tau_-} \left| \int_{C^i_u(t)} \tau_+ \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right|^2 dC_u^i(t) \int_{C_u^i(t)} \tau_+ \left| \int_v |\Phi|^n \sqrt{\frac{ v^{\underline{L}}}{v^0}} |h| dv \right|^2 dC_u^i(t) \right|^{\frac{1}{2}} du \\ \nonumber
& \lesssim & \sqrt{\epsilon} \sum_{i=0}^{+ \infty} (1+t_{i+1})^{\eta}\int_{u=-\infty}^t \frac{1}{\tau_-} \left| \int_{C_u^i(t)} \tau_+ \int_v |\Phi|^n \left| h_1 \right| dv \int_v \frac{ v^{\underline{L}}}{v^0} \left| h_2 \right| dv dC_u^i(t) \right|^{\frac{1}{2}} du \\ \nonumber & \lesssim & \sqrt{\epsilon} \sum_{i=0}^{+ \infty }\frac{(1+t_{i+1})^{2\eta}}{(1+t_i)^{\frac{1}{4}}} \int_{u=-\infty}^{+\infty} \frac{1}{\tau_-^{\frac{3}{2}}} \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} \sum_{i=0}^{+ \infty} 2^{-\frac{i}{4}(1-8 \eta)} \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}. \end{eqnarray} \end{proof} We now turn on the problematic terms. \begin{Pro}\label{M32}
If $|\xi_2| \leq N-2$, we have \begin{eqnarray}
\nonumber I^1_3 & = & \int_{0}^t \int_{\Sigma_s} \frac{\tau_+}{\tau_-} \left| \underline{\alpha} \left( \mathcal{L}_{Z^{\gamma}} ( F) \right) \right| \int_v \sqrt{\frac{v^{\underline{L}}}{v^0}} |h| \frac{dv}{v^0} dx ds \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} \log^{(3+j+|\xi_1|+|\xi_2|+|\beta|)a} (3+t) \hspace{3mm} \text{and} \\ \nonumber I^2_3 & = & \int_{0}^t \int_{\Sigma_s} \frac{\tau_+}{\tau_-} \left| \rho \left( \mathcal{L}_{Z^{\gamma}} ( F) \right) \right|\int_v |h| \frac{dv}{v^0} dx ds \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} \log^{(2+j+|\xi_1|+|\xi_2|+|\beta|)a} (3+t). \end{eqnarray}
Otherwise, $|\xi^2|=N-1$ and $I^1_3 +I^2_3 \lesssim \epsilon^{\frac{3}{2}} (1+t)^{\frac{3}{4} \eta}$. \end{Pro} \begin{Rq}\label{nopb} Note that these estimates are sufficient to improve the bootstrap assumptions \eqref{bootf2} and \eqref{bootf3}. Indeed, \begin{itemize}
\item the case $|\xi^2|=N-1$ concerns only the study of $\overline{\mathbb{E}}_N[f]$.
\item Even if the bound on $I^2_3+I^1_3$, when $|\xi^2| \leq N-2$ could seem to possess a factor $\log^{3a}(3+t)$ in excess, one has to keep in mind that $|\gamma| \geq N-2$, so $|\xi^1|+|\beta| \leq 3$ and $|\xi^0|+|\beta^0| \geq N-2$. Moreover, by the properties of the terms of \eqref{eq:cat3}, $j \leq q$. We then have, as $N \geq 8$,
$$j+3+|\xi^1|+|\xi^2|+|\beta| \leq q+|\xi^0|+|\xi^2|+|\beta^0|.$$ \end{itemize} \end{Rq} \begin{proof} Throughout this proof, we will use \eqref{eq:h1} and the bootstrap assumption \eqref{bootF1}, which implies
$$ \left\| \underline{\alpha} \left( \mathcal{L}_{Z^{\gamma}} ( F) \right) \right\|_{L^2 \left( \Sigma_t \right) } + \sup_{u \in \mathbb{R}} \left\| \rho \left( \mathcal{L}_{Z^{\gamma}} ( F) \right) \right\|_{L^2 \left( C_u(t) \right) } \lesssim \sqrt{\mathcal{E}_{N}^0[F](t)} \lesssim \epsilon^{\frac{1}{2}}.$$ Applying the Cauchy-Schwarz inequality twice (in $(t,x)$ and then in $v$), we get \begin{eqnarray}
\nonumber I^1_3 & \lesssim & \left| \int_{0}^t \frac{\left\| \underline{\alpha} \left( \mathcal{L}_{Z^{\gamma}} ( F) \right) \right\|_{L^2 \left( \Sigma_s \right) }}{1+s} ds \int_{u=- \infty}^t \int_{C_u(t)} \frac{\tau_+^3}{\tau_-^2} \left| \int_v \sqrt{\frac{v^{\underline{L}}}{v^0}} \left| h\right| \frac{dv}{v^0} \right|^2 dC_u(t) du \right|^{\frac{1}{2}} \\ \nonumber
& \lesssim & \epsilon^{\frac{1}{2}}\log^{\frac{1}{2}}(1+t) \left| \int_{u=- \infty}^t \frac{1}{\tau_-^2} \int_{C_u(t)} \int_v \frac{v^{\underline{L}}}{v^0} \left| h_2 \right| dv dC_u(t) du \right|^{\frac{1}{2}} \sup_{u \in \mathbb{R}} \left\| \tau_+^3 \int_v |h_1| \frac{dv}{(v^0)^2} \right\|_{L^{\infty} \left( C_u(t) \right) }^{\frac{1}{2}} \\ \nonumber
& \lesssim & \epsilon \log^{\frac{1}{2}+\frac{a}{2} \left(4+i_1+ 2|\xi|^1+|\beta| \right)}(3+t) \sqrt{\mathbb{E}[h_2](t)}. \end{eqnarray} Using $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$ and the Cauchy-Schwarz inequality (this time in $(\underline{u},\omega_1,\omega_2)$ and then in $v$), we obtain \begin{eqnarray}
\nonumber I^2_3 & \lesssim & \int_{u=-\infty}^t \left\| \rho \left( \mathcal{L}_{Z^{\gamma}} ( F) \right) \right\|_{L^2 \left( C_u(t) \right) } \left| \int_{C_u(t)} \frac{\tau_+^2}{\tau_-^2} \left| \int_v \sqrt{\frac{v^{\underline{L}}}{v^0}} \left| h \right| dv \right|^2 dC_u(t) \right|^{\frac{1}{2}} du \\ \nonumber
& \lesssim & \epsilon^{\frac{1}{2}} \int_{u = - \infty}^t \frac{1}{\tau_-^{\frac{3}{2}}} \left\| \tau_+^2 \tau_- \int_v |h_1| dv \right\|_{L^{\infty} \left( C_u(t) \right) }^{\frac{1}{2}} \left| \int_{C_u(t)} \int_v \frac{v^{\underline{L}}}{v^0} \left| h_2 \right| dv dC_u(t) \right|^{\frac{1}{2}} du \\ \nonumber
& \lesssim & \epsilon \log^{\frac{a}{2} \left(4+i_1 +2|\xi|^1+|\beta| \right)} \sqrt{\mathbb{E}[h_2](t)}. \end{eqnarray} It then remains to remark that, by the bootstrap assumptions \eqref{bootf2} and \eqref{bootf3}, \begin{itemize}
\item $\mathbb{E}[h_2](t) \leq \log^{ (i_2+2|\xi_2|+|\beta|)a}(3+t) \mathbb{E}^0_{N-1}[f](t) \lesssim \epsilon \log^{ (i_2+2|\xi_2|+|\beta|)a}(3+t)$, if $|\xi_2| \leq N-2$, or
\item $\mathbb{E}[h_2](t) \leq \log^{a i_2}(3+t) \overline{\mathbb{E}}_{N}[f](t) \lesssim \epsilon (1+t)^{\eta} \log^{a i_2}(3+t)$, if $|\xi_2| = N-1$. \end{itemize} \end{proof}
Let us move now on the expressions of \eqref{eq:cat4}. The ones where $|\gamma|=N$ are the more critical terms and will be treated later. \begin{Pro}\label{M41}
Suppose that $N-2 \leq |\gamma| \leq N-1$. Then, if $|\xi_2| \leq N-2$, \begin{flalign*}
& \hspace{1cm} I_4=\int_0^t \int_{\Sigma_s} \int_v \tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \right| \left| h \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}} \log^{(3+j+|\xi_1|+|\xi_2|+|\beta|)a}(3+t) & \end{flalign*} and $I_4 \lesssim \epsilon^{\frac{3}{2}} (1+t)^{\frac{3}{4} \eta}$ otherwise. \end{Pro} For similar reasons as those given in Remark \ref{nopb}, these bounds are sufficient to close the energy estimates on $\overline{\mathbb{E}}_N[f]$ and $\mathbb{E}^0_{N-1}[f]$.
\begin{proof}
Denoting by $(\alpha, \underline{\alpha}, \rho, \sigma )$ the null decomposition of $\mathcal{L}_{Z^{\gamma}}(\widetilde{F})$ and using $|v^A| \lesssim \sqrt{v^0v^{\underline{L}}}$, we have \begin{eqnarray}
\nonumber \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \right| & \lesssim & |\alpha (\mathcal{L}_{ Z^{\gamma}}(F))|+|\sigma (\mathcal{L}_{ Z^{\gamma}}(F)) |+|\rho (\mathcal{L}_{ Z^{\gamma}}(F)) |+\sqrt{\frac{v^{\underline{L}}}{v^0}}|\underline{\alpha} (\mathcal{L}_{ Z^{\gamma}}(F)) | \\ \nonumber
& \lesssim & |\alpha|+|\rho|+|\sigma|+\sqrt{\frac{ v^{\underline{L}}}{v^0}}|\underline{\alpha}|+\left| \mathcal{L}_{Z^{\gamma}}( \overline{F} ) \right|. \end{eqnarray} and we can then bound $I_4$ by $I_{\alpha,\sigma,\rho}+I_{\underline{\alpha}}+I_{\overline{F}}$ (these quantities will be clearly defined below). Note now that \begin{equation}\label{eq:alphasigma}
\left\| \sqrt{\tau_+} |\alpha| +\sqrt{\tau_+} |\rho|+\sqrt{\tau_+} |\sigma| \right\|^2_{L^2(\Sigma_s)}+\left\| \sqrt{\tau_-} |\underline{\alpha}| \right\|^2_{L^2(\Sigma_s)} \hspace{1mm} \lesssim \hspace{1mm} \mathcal{E}^{Ext}_{N}[\widetilde{F}](s)+ \mathcal{E}_{N-1}[F](s) \hspace{1mm} \lesssim \hspace{1mm} \epsilon \log^{2M}(3+s). \end{equation} Then, using the Cauchy-Schwarz inequality twice (in $(t,x)$ and then in $v$), the estimates \eqref{eq:h1} and \eqref{eq:h11} as well as $a = M+1$, we get \begin{eqnarray}
\nonumber I_{\underline{\alpha}} & := & \int_0^t \int_{\Sigma_s} \tau_+ |\underline{\alpha}| \int_v \sqrt{\frac{ v^{\underline{L}}}{v^0}} |h| \frac{dv}{v^0} dx ds \\ \nonumber
& \lesssim & \left| \int_0^t \frac{\|\sqrt{\tau_-} |\underline{\alpha}| \|^2_{L^2(\Sigma_s)}}{1+s}ds \int_{u=-\infty}^t \int_{C_u(t)} \frac{\tau_+^2(1+s)}{\tau_-} \left| \int_v \sqrt{ \frac{ v^{\underline{L}}}{v^0}} |h|\frac{ dv}{v^0} \right|^2 dC_u(t) du \right|^{\frac{1}{2}} \\ \nonumber
& \lesssim & \sqrt{\epsilon} \log^{M+\frac{1}{2}}(3+t) \left| \int_{u=-\infty}^t \frac{1}{\tau_-} \left\| \tau_+^2(1+s) \int_v |h_1| \frac{dv}{(v^0)^2} \right\|_{L^{\infty}(C_u(t))} \int_{C_u(t)} \int_v \frac{ v^{\underline{L}}}{v^0} |h_2| dv dC_u(t) du \right|^{\frac{1}{2}} \\ \nonumber
& \lesssim & \epsilon^{\frac{3}{2}} \log^{-\frac{1}{2}+\frac{a}{2}(5+i_1+2|\xi^1|+|\beta|)}(3+t) \sqrt{\mathbb{E}[h_2](t)} \left| \int_{u=-\infty}^0 \frac{du}{\tau_-^{\frac{3}{2}}}+\int_{u=0}^t \frac{du}{\tau_-} \right|^{\frac{1}{2}} \\ \nonumber
& \lesssim & \epsilon^{\frac{3}{2}} \log^{\frac{a}{2}(6+i_1+2|\xi^1|+|\beta|)}(3+t) \sqrt{\mathbb{E}[h_2](t)}. \end{eqnarray} Similarly, one has \begin{eqnarray}
\nonumber I_{\alpha, \rho, \sigma} & := & \int_0^t \int_{\Sigma_s} \tau_+ (|\alpha|+|\rho|+|\sigma|) \int_v |h| \frac{dv}{v^0} dx ds \\ \nonumber
& \lesssim & \int_0^t \| \sqrt{\tau_+}|\alpha|+\sqrt{\tau_+} |\rho|+\sqrt{\tau_+} |\sigma| \|_{L^2(\Sigma_s)} \left\| \sqrt{\tau_+} \int_v |h| \frac{dv}{v^0} \right\|_{L^2(\Sigma_s)} ds \\ \nonumber
& \lesssim & \int_0^t \sqrt{\epsilon} \log^M(3+s) \left\| \tau_+ \int_v |h_1|\frac{dv}{(v^0)^2} \right\|_{L^{\infty}(\Sigma_s)}^{\frac{1}{2}} \left\| \int_v |h_2| dv \right\|_{L^1(\Sigma_s)}^{\frac{1}{2}} ds \\ \nonumber
& \lesssim & \epsilon \log^{\frac{a}{2}(6+i_1+2|\xi^1|+|\beta|)} (3+t) \left\| \mathbb{E}[h_2] \right\|_{L^{\infty}([0,t])}^{\frac{1}{2}}. \end{eqnarray}
For the last integral, recall from Propositions \ref{propcharge} and \ref{decayF} that $\overline{F}(t,x)$ vanishes for all $t-|x| \geq -1$ and that $|\mathcal{L}_{Z^{\gamma}}(\overline{F})| \lesssim \epsilon \tau_+^{-2}$. We are then led to bound \begin{eqnarray}
\nonumber I_{\overline{F}} & := & \int_0^t \int_{|x| \geq s+1} \tau_+ |\mathcal{L}_{Z^{\gamma}}(\overline{F})|\int_v |h| \frac{dv}{v^0} dx ds \\ \nonumber
& \lesssim & \int_0^t \frac{\sqrt{\epsilon}}{1+s} \int_{\Sigma_s} \int_v \sqrt{ \left| h_1 h_2 \right| } dv dx ds \hspace{1mm} \lesssim \hspace{1mm} \int_0^t \frac{\sqrt{\epsilon}}{1+s} \left| \int_{\Sigma_s} \int_v \left| h_1 \right| dv dx \int_{\Sigma_s} \int_v \left| h_2 \right| dv dx \right|^{\frac{1}{2}} ds \\ \nonumber
& \lesssim & \sqrt{\epsilon} \log(3+t) \left\| \mathbb{E}[h_1] \right\|_{L^{\infty}([0,t])}^{\frac{1}{2}}\left\| \mathbb{E}[h_2] \right\|_{L^{\infty}([0,t])}^{\frac{1}{2}} . \end{eqnarray}
Thus, as $\left\| \mathbb{E}[h_1] \right\|_{L^{\infty}([0,t])} \lesssim \epsilon \log^{(i_1+2|\xi_1|+|\beta|)a}(3+t)$ and $i_1+i_2=2j$, we have \begin{itemize}
\item $I_4 \lesssim \epsilon^{\frac{3}{2}} (1+t)^{\frac{3}{4} \eta}$ if $|\xi_2|=N-1$, since $\mathbb{E}[h_2](t) \leq \log^{ai_2}(3+t) \overline{\mathbb{E}}_N[f](t) \leq \epsilon (1+t)^{\eta} \log^{ai_2}(3+t) $, and
\item $I_4 \lesssim \epsilon^{\frac{3}{2}} \log^{(3+j+|\xi_1|+|\xi_2|+|\beta|)a}(3+t)$ otherwise, as $\mathbb{E}[h_2] \leq \log^{(i_2+2|\xi^2|+|\beta|)a}(3+t) \mathbb{E}_{N-1}^0[f](t)$ . \end{itemize} \end{proof}
A better pointwise decay estimate on $\int_v |h_1|(v^0)^{-2}dv$ is requiered to bound sufficiently well $I_4$ when $|\gamma|=N$. We will then treat this case below, in the last part of this section. However, note that all the Propositions already proved in this section imply \eqref{improvebootf2}, for $Q=N-1$, and then $\mathbb{E}^0_{N-1}[f] \leq 3\epsilon$ on $[0,T[$.
\subsection{Estimates for $ \mathbb{E}^X_{N-1}[f]$, $ \mathbb{E}^X_{N}[f]$ and obtention of optimal decay near the lightcone for velocity averages}\label{Ximpro}
The purpose of this subsection is to establish that\footnote{Note that we cannot unify these norms because of a lack of weights $z \in \mathbf{k}_1$. As we will apply Proposition \ref{ComuPkp} with $N_0=2N-1$, we cannot propagate more than $2N-2$ weights and avoid in the same time the problematic terms.} $ \mathbb{E}^X_{N-1}[f]$, $ \mathbb{E}^X_{N}[f] \leq 3\epsilon$ on $[0,T[$ and then to deduce optimal pointwise decay estimates on the velocity averages of the particle density. Remark that, according to the energy estimate of Proposition \ref{energyf}, $ \mathbb{E}^X_{N}[f] \leq 3\epsilon$ follows, if $\epsilon$ is small enough, from
\begin{equation}\label{4:eq} \int_0^t \int_{\Sigma_s} \int_v \left| T_F \left( z^q P^X_{\xi} (\Phi ) Y^{\beta} f \right) \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}}\log^{2q} (3+t), \end{equation} \begin{itemize}
\item for all multi-indices $\beta$ and $\xi$ such that $\max(|\beta|+|\xi|,|\xi|+1) \leq N$ and \item for all $z \in \mathbf{k}_1$ and $q \in \mathbb{N}$ such that $q \leq 2N-2-\xi_P-\beta_P$. \end{itemize} Most of the work has already been done. Indeed, the commutation formula of Proposition \ref{ComuPkpX} (applied with $N_0=2N-1$) leads us to bound only terms of \eqref{eq:cat0} and \eqref{eq:cat1} since $q \leq 2N-2-\xi_P-\beta_P$. Note that we control quantities of the form
$$ z^j P_{\xi^1}(\Phi) Y^{\beta^1} f, \hspace{5mm} \text{with} \hspace{5mm} |\xi^1|+|\beta^1| \leq N, \hspace{5mm} |\xi^1| \leq N-1 \hspace{5mm} \text{and} \hspace{5mm} j \leq 2N-1-\xi^1_P-\beta^1_P.$$ Consequently, \eqref{4:eq} ensues from Propositions \ref{M1}, \ref{M2} and \ref{M21}. $\mathbb{E}_{N-1}^X[f]$ can be estimated similarly since we also control quantities such as
$$ z^j P_{\xi^1}(\Phi) P_{\xi^2}(\Phi) Y^{\kappa} f, \hspace{5mm} \text{with} \hspace{5mm} \max(|\xi^1|+|\kappa|,|\xi^2|+|\kappa|) \leq N-1 \hspace{5mm} \text{and} \hspace{5mm} j \leq 2N-1-\xi^1_P-\xi^2_P-\kappa_P.$$
Note that \eqref{4:eq} also provides us, through Theorem \ref{decayopti}, that, for all $\max ( |\xi|+|\beta|, 1 +|\xi| ) \leq N-3$,
$$ \forall \hspace{0.5mm} |x| \leq t < T, \hspace{3mm} z \in \mathbf{k}_1, \hspace{3mm} j \leq 2N-5-\xi_P-\beta_P, \hspace{5mm} \int_v \left|z^j P^X_{\xi} (\Phi) Y^{\beta} f \right| \frac{dv}{(v^0)^2} \lesssim \epsilon \frac{\log^{2j}(3+t)}{\tau_+^3}.$$
For the exterior region, use Proposition \ref{decayopti2} and $ \mathbb{E}^{X}_N[f] \leq 3\epsilon$ to derive, for all $\max ( |\xi|+|\beta|, |\xi|+1 ) \leq N-3$,
$$ \forall \hspace{0.5mm} (t,x) \in V_0(T), \hspace{3mm} z \in \mathbf{k}_1, \hspace{3mm} j \leq 2N-6-\xi_P-\beta_P, \hspace{5mm} \int_v \left|z^j P^X_{\xi} (\Phi) Y^{\beta} f \right| \frac{dv}{(v^0)^2} \lesssim \epsilon \frac{\log^{2(j+1)}(3+t)}{\tau_+^3\tau_-}.$$ We summerize all these results in the following proposition (the last estimate comes from Corollary \ref{KS2}). \begin{Pro}\label{Xdecay}
If $\epsilon$ is small enough, then $ \mathbb{E}^{X}_{N-1}[f] \leq 3 \epsilon$ and $ \mathbb{E}^{X}_{N}[f] \leq 3 \epsilon$ hold on $[0,T]$. Moreover, we have, for all $\max ( |\xi|+|\beta|, |\xi|+1 ) \leq N-3$, $z \in \mathbf{k}_1$ and $j \leq 2N-6 - \xi_P-\beta_P$, \begin{eqnarray}
\nonumber \forall \hspace{0.5mm} (t,x) \in [0,T[ \times \mathbb{R}^3, \hspace{10mm} \int_v \left|z^j P^X_{\xi} (\Phi) Y^{\beta} f \right| \frac{dv}{(v^0)^2} & \lesssim & \epsilon \frac{\log^{2j}(3+t)}{\tau_+^3} \mathds{1}_{t \geq |x|}+ \epsilon\frac{\log^{2(j+1)}(3+t)}{\tau_+^3\tau_-} \mathds{1}_{|x| \geq t}, \\ \nonumber
\forall \hspace{0.5mm} (t,x) \in [0,T[ \times \mathbb{R}^3, \hspace{10mm} \int_v \left|z^j P^X_{\xi} (\Phi) Y^{\beta} f \right| dv & \lesssim & \epsilon \frac{\log^{2j}(3+t)}{\tau_+^2 \tau_-}. \end{eqnarray} \end{Pro}
\subsection{The critical terms}\label{sec86}
We finally bound $I_4$, defined in Proposition \ref{M41}, when $|\gamma|=N$, which concerns only the improvement of the bound of the higher order energy norm $\overline{\mathbb{E}}_N[f]$. We keep the notations introduced in Subsection \ref{ref:L2elec} and we start by precising them. Using the properties of the terms of \eqref{eq:cat4}, we remark that we necessarily have
$$P_{\xi^0}(\Phi)=Y^{\xi^0} \Phi, \hspace{5mm} |\xi^0|=N-1, \hspace{5mm} |\beta^0| \leq 1, \hspace{5mm} |\xi^1|=0, \hspace{5mm} \beta = \beta^0, \hspace{5mm} \gamma_T = \xi^0_T \hspace{5mm} \text{and} \hspace{5mm} j=q.$$ We are then led to prove
$$I_4 = \int_0^t \int_{\Sigma_s} \int_v \tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \right| \left| z^{q} P_{\xi^2}(\Phi) Y^{\beta^0} f \right| \frac{dv}{v^0} dx ds \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} (1+t)^{\eta} \log^{aq}(3+t).$$
If $\gamma_T=\xi^0_T \geq 1$, one can use inequality \eqref{eq:zeta2} of Proposition \ref{ExtradecayLie} and $|v^A| \lesssim \sqrt{v^0 v^{\underline{L}}}$ in order to obtain
$$ \tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \right| \lesssim \left(1+ \frac{\sqrt{ v^{\underline{L}}}\tau_+}{\sqrt{v^0}\tau_-} \right) \sum_{|\gamma_0| \leq N} \left| \nabla_{Z^{\gamma_0}} F \right|+ \frac{\tau_+}{\tau_-} \sum_{|\gamma_0| \leq N} \left| \alpha ( \mathcal{L}_{Z^{\gamma_0}}(F)) \right|+\left| \rho ( \mathcal{L}_{Z^{\gamma_0}}(F)) \right|$$ and then split $I_4$ in four parts and bound them by $\epsilon^{\frac{3}{2}}$ or $\epsilon^{\frac{3}{2}} (1+t)^{\frac{3}{4} \eta}$, as $I^1_1$, $I^3_1$, $I^1_3$ and $I^2_3$ in Propositions \ref{M21} and \ref{M32}. Otherwise, $\xi^0_P=N-1$ and $q \leq N-\xi^2_P-\beta^0_P$ so that we take $i_2 \leq 2N-1-2\xi^2_P-\beta^0_P$ and $i_1 \leq 1-\beta^0_P$. Then, we divide $[0,t] \times \mathbb{R}^3$ in two parts, $V_0(t)$ and its complement. Following the proof of Proposition \ref{M41}, one can prove, as $\mathcal{E}_{N}^{Ext}[\widetilde{F}] \lesssim \epsilon$ and $|\mathcal{L}_{Z^{\gamma}}(\overline{F})| \lesssim \epsilon \tau_+^{-2}$ on $[0,T[$, that
$$ \int_0^t \int_{\overline{\Sigma}^0_s} \int_v \tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \right| \left| z^{q} P_{\xi^2}(\Phi) Y^{\beta^0} f \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}} (1+t)^{\frac{3}{4} \eta}.$$
To lighten the notations, let us denote the null decomposition of $\mathcal{L}_{Z^{\gamma}}(F)$ by $(\alpha, \underline{\alpha}, \rho, \sigma)$. Recall from Lemma \ref{weights1} that $ \tau_+ |v^A| \lesssim v^0 \sum_{w \in \mathbf{k}_1} |w|$ and $\tau_+v^{\underline{L}} \lesssim \tau_-v^0+v^0\sum_{w \in \mathbf{k}_1} |w|$, so that \begin{eqnarray}
\nonumber \tau_+ \left| \frac{v^{\mu}}{v^0} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \right| & \lesssim & \tau_+\left(|\alpha|+|\rho|\right)+\tau_+\frac{v^{\underline{L}}}{v^0} |\underline{\alpha}|+ \tau_+ \frac{|v^A|}{v^0}\left( |\sigma|+|\underline{\alpha}| \right) \\ \nonumber
& \lesssim & \left(\tau_+|\alpha|+\tau_+|\rho|+\tau_-|\underline{\alpha}| \right)+\sum_{w \in \mathbf{k}_1} |w| \left( |\sigma|+|\underline{\alpha}| \right). \end{eqnarray}
We can then split the remaining part of $I_4$ in two integrals. The one associated to $\sum_{w \in \mathbf{k}_1} |w|(|\sigma|+|\underline{\alpha}|)$ can be bounded by $\epsilon^{\frac{3}{2}}$ as $I^1_1$ in Proposition \ref{M21} since $i_1+1 \leq 2N-1-\beta_P^0$. For the one associated to $\left(\tau_+|\alpha|+\tau_+|\rho|+\tau_-|\underline{\alpha}| \right)$, $\overline{I}_4$, we have \begin{eqnarray}
\nonumber \overline{I}_4 & := & \int_0^t \int_{\Sigma^0_s} \left(\tau_+|\alpha|+\tau_+|\rho|+\tau_-|\underline{\alpha}| \right) \int_v \left| z^{q} P_{\xi^2}(\Phi) Y^{\beta^0} f \right| \frac{dv}{v^0} dx ds \\ \nonumber
& \lesssim & \int_0^t \sqrt{\mathcal{E}_N[F](s)} \left\| \sqrt{\tau_+} \int_v \left| z^{q} P_{\xi^2}(\Phi) Y^{\beta^0} f \right| \frac{dv}{v^0} \right\|_{L^2(\Sigma^{0}_s)} ds \\ \nonumber
& \lesssim & \sqrt{\epsilon} \int_0^t \sqrt{\mathcal{E}_N[F](s)} \left\| \tau_+ \int_v \left| z^{i_1} Y^{\beta^0} f \right| \frac{dv}{(v^0)^2} \right\|^{\frac{1}{2}}_{L^{\infty}(\Sigma^0_s)} \left\| \int_v \left| z^{i_2} P_{\xi^2}(\Phi)^2 Y^{\beta^0} f \right| dv \right\|^{\frac{1}{2}}_{L^1(\Sigma^0_s)} ds . \end{eqnarray}
Using the bootstrap assumptions \eqref{bootf3}, \eqref{bootF4} and the pointwise decay estimate on $\int_v \left| z^{i_1} Y^{\beta^0} f \right| \frac{dv}{(v^0)^2}$ given in Proposition \ref{Xdecay}, we finally obtain $$\overline{I}_4 \hspace{2mm} \lesssim \hspace{2mm} \sqrt{\epsilon} \int_0^t (1+s)^{\frac{\eta}{2}} \frac{\sqrt{\epsilon} \log^{i_1}(3+s)}{1+s} \sqrt{\epsilon} (1+s)^{\frac{\eta}{2}} \log^{\frac{a}{2}i_2}(3+s) ds \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} (1+t)^{\eta} \log^{a q}(3+t),$$ which concludes the improvement of the bootstrap assumption \eqref{bootf3}. \begin{Rq} In view of the computations made to estimate $\overline{I}_4$, note that. \begin{itemize}
\item The use of Theorem \ref{decayopti}, instead of \eqref{decayf} combined with $1 \lesssim v^0 v^{\underline{L}}$ and Lemma \ref{weights1}, was necessary. Indeed, for the case $q=0$, a decay rate of $\log^2 (3+t) \tau_+^{-3}$ on $\int_v \left| Y^{\beta^0} f \right| \frac{dv}{(v^0)^2}$ would prevent us from closing energy estimates on $\mathcal{E}_N[F]$ and $\overline{\mathbb{E}}_N[f]$.
\item Similarly, it was crucial to have a better bound on $\mathcal{E}^{Ext}_{N}[G](t)$ than $\epsilon (1+t)^{\eta}$ as the decay rate given by Proposition \ref{Xdecay} on $\int_v \left| Y^{\beta^0} f \right| \frac{dv}{(v^0)^2}$ is weaker, in the $t+r$ direction, outside the light cone. \end{itemize} \end{Rq} Note that Propositions \ref{M2}, \ref{M3}, \ref{MM3}, \ref{M21}, \ref{M32} and \ref{M41} also prove that \begin{equation}\label{Auxenergy}
\mathbb{A}[f](t) := \sum_{i=1}^2 \sum_{\begin{subarray}{} |\xi^i|+|\beta| \leq N \\ |\xi^i| \leq N-2 \end{subarray}}\sum_{\begin{subarray}{} |\zeta^i|+|\beta| \leq N \\ |\zeta^i| \leq N-1 \end{subarray}} \mathbb{E} \left[ P_{\xi^1}(\Phi) P_{\xi^2}(\Phi) Y^{\beta} f \right] (t)+\mathbb{E} \left[ P_{\zeta^1}^X(\Phi) P_{\zeta^2}^X(\Phi) Y^{\beta} f \right] (t) \lesssim \epsilon (1+t)^{\frac{3}{4} \eta}. \end{equation}
Indeed, to estimate this energy norm, we do not have to deal with the critical terms of this subsection (as $|\xi^i| \leq N-2$ and according to Proposition \ref{ComuPkpX}).
\section{$L^2$ decay estimates for the velocity averages of the Vlasov field}\label{sec11}
In view of the commutation formula of Propositions \ref{ComuMaxN} and \ref{CommuFsimple}, we need to prove enough decay on quantities such as $\left\| \sqrt{\tau_-} \int_v | Y^{\beta} f | dv \right\|_{L^2_x}$, for all $|\beta| \leq N$. Applying Proposition \ref{Xdecay}, we are already able to obtain such estimates if $|\beta| \leq N-3$ (see Proposition \ref{estiL2} below). The aim of this section is then to treat the case of the higher order derivatives. For this, we follow the strategy used in \cite{FJS} (Section $4.5.7$). Before exposing the proceding, let us rewrite the system. Let $I_1$, $I_2$ and $I^q_1$, for $N-5 \leq q \leq N$, be the sets defined as \begin{flalign*}
& \hspace{0.7cm} I_1 := \left\{ \beta \hspace{2mm} \text{multi-index} \hspace{2mm} / \hspace{2mm} N-5 \leq |\beta| \leq N \right\} = \{ \beta^1_{1}, \beta^1_{2}, ..., \beta^1_{|I_1|} \}, \hspace{1.3cm} I^q_1 := \left\{ \beta \in I_1 \hspace{2mm} / \hspace{2mm} |\beta| = q \right\}, & \\
& \hspace{0.7cm} I_2 := \left\{ \beta \hspace{2mm} \text{multi-index} \hspace{2mm} / \hspace{2mm} |\beta| \leq N-5 \right\} = \{ \beta^2_{1}, \beta^2_{2}, ..., \beta^2_{|I_2|} \}, & \end{flalign*}
and $R^1$ and $R^2$ be two vector valued fields, of respective length $|I_1|$ and $|I_2|$, such that $$ R^1_j= Y^{\beta^1_{j}}f \hspace{2cm} \text{and} \hspace{2cm} R^2_j= Y^{\beta^2_{j}}f. $$
We will sometimes abusively write $j \in I_i$ instead of $\beta^i_{j} \in I_i$ (and similarly for $j \in I^k_1$). The goal now is to prove $L^2$ estimates on $\int_v |R^1| dv$. Finally, we denote by $\mathbb{V}$ the module over the ring $C^0([0,T[ \times \mathbb{R}^3_x \times \mathbb{R}^3_v)$ engendered by $( \partial_{v^l})_{1 \leq l \leq 3}$. In the following lemma, we apply the commutation formula of Proposition \ref{ComuVlasov} in order to express $T_F(R^1)$ in terms of $R^1$ and $R^2$ and we use Lemma \ref{GammatoYLem} for transforming the vector fields $\Gamma^{\sigma} \in \mathbb{G}^{|\sigma|}$. \begin{Lem}\label{bilanL2}
There exists two matrix functions $A :[0,T[ \times \mathbb{R}^3 \times \mathbb{R}^3_v \rightarrow \mathfrak M_{|I_1|}(\mathbb{V})$ and $B :[0,T[ \times \mathbb{R}^3 \times \mathbb{R}^3_v \rightarrow \mathfrak M_{|I_1|,|I_2|}(\mathbb{V})$ such that $T_F(R^1)+AR^1=B R^2$. Furthermore, if $1 \leq i \leq |I_1|$, $A$ and $B$ are such that $T_F(R^1_i)$ is a linear combination, with good coefficients $c(v)$, of the following terms, where $r \in \{1,2 \}$ and $\beta^r_j \in I_r$. \begin{itemize} \item \begin{equation}\label{eq:com11}
z^d P_{k,p}(\Phi) \frac{v^{\mu}}{v^0}\mathcal{L}_{Z^{\gamma}}(F)_{ \mu \nu} R^r_j, \tag{type 1}
\end{equation}
where \hspace{2mm} $z \in \mathbf{k}_1$, \hspace{2mm} $d \in \{ 0,1 \}$, \hspace{2mm} $\max ( |\gamma|, |k|+|\beta^r_j| ) \leq |\beta^1_i|$, \hspace{2mm} $|k| \leq |\beta^1_i|- 1$, \hspace{2mm} $|k|+|\gamma|+|\beta^r_j| \leq |\beta^1_i|+1$ \hspace{2mm} and \hspace{2mm} $p+k_P+(\beta^r_j)_P+d \leq (\beta^1_i)_P$. \item \begin{equation}\label{eq:com22} P_{k,p}(\Phi) \mathcal{L}_{X Z^{\gamma_0}}(F) \Big( v, \nabla_v \left( c(v) P_{q,s}(\Phi) R^r_j \right) \Big), \tag{type 2} \end{equation}
where \hspace{2mm} $|k|+|q|+|\gamma_0|+|\beta^r_j| \leq |\beta^1_i|-1$, \hspace{2mm} $|q| \leq |\beta^1_i|-2$,\hspace{2mm} $p+s+k_P+q_P+(\beta^r_j)_P \leq (\beta^1_i)_P$ \hspace{2mm} and \hspace{2mm} $p \geq 1$. \item \begin{equation}\label{eq:com44}
P_{k,p}(\Phi) \mathcal{L}_{ \partial Z^{\gamma_0}}(F) \Big( v, \nabla_v \left( c(v) P_{q,s}(\Phi) R^r_j \right) \Big), \tag{type 3}
\end{equation}
where \hspace{2mm} $|k|+|q|+|\gamma_0|+|\beta^r_j| \leq |\beta^1_i|-1$, \hspace{2mm} $|q| \leq |\beta^1_i|-2$, \hspace{2mm} $p+s+|\gamma_0| \leq |\beta^1_i|-1$ \hspace{2mm} and \hspace{2mm} $p+s+k_P+q_P+(\beta^r_j)_P \leq (\beta^1_i)_P$. \end{itemize}
We also impose that $|\beta^2_j| \leq N-6$ on the terms of \eqref{eq:com22}, \eqref{eq:com44} and that $|\beta^1_j| \geq N-4$ on the terms of \eqref{eq:com11}, which is possible since $ \beta \in I_1 \cap I_2$ if $|\beta| = N-5$. \end{Lem} \begin{Rq}\label{rqcondiH}
Note that if $\beta^1_i \in I^{N-5}_1$, then $A^q_i =0$ for all $q \in \llbracket 1, |I_1| \rrbracket$. If $1 \leq n \leq 5$ and $\beta^1_i \in I^{N-5+n}_1$, then the terms composing $A_i^q$ are such that $\max(|k|+1,|\gamma|) \leq n$ or $|k|+|q|+|\gamma_0| \leq n-1$. \end{Rq} Let us now write $R=H+G$, where $H$ and $G$ are the solutions to $$\left\{
\begin{array}{ll}
T_F(H)+AH=0 \hspace{2mm}, \hspace{2mm} H(0,.,.)=R(0,.,.),\\
T_F(G)+AG=BR^2 \hspace{2mm}, \hspace{2mm} G(0,.,.)=0.
\end{array} \right.$$
The goal now is to prove $L^2$ estimates on the velocity averages of $H$ and $G$. As the derivatives of $F$ and $\Phi$ composing the matrix $A$ are of low order, we will be able to commute the transport equation satisfied by $H$ and to bound the $L^1$ norm of its derivatives of order $3$ by estimating pointwise the electromagnetic field and the $\Phi$ coefficients, as we proceeded in Subsection \ref{sec82}. The required $L^2$ estimates will then follow from Klainerman-Sobolev inequalities. Even if we will be lead to modify the form of the equation defining $G$, the idea is to find a matrix $K$ satisfying $G=KR^2$, such that $\mathbb{E}[KKR^2]$ do not grow too fast, and then to take advantage of the pointwise decay estimates on $\int_v |R^2|dv$ in order to obtain the expected decay rate on $\| \int_v |G| dv \|_{L^2_x}$. \begin{Rq}
As in \cite{massless}, we keep the $v$ derivatives in the construction of $H$ and $G$. It has the advantage of allowing us to use Lemma \ref{vradial}. If we had already transformed the $v$ derivatives, as in \cite{dim4}, we would have obtained terms such as $x^{\theta} \partial g$ from $\left( \nabla_v g \right)^r$. Indeed, Lemma \ref{vradial} would have led us to derive coefficients such as $\frac{x^k}{|x|}$ and then to deal, for instance, with factors such as $\frac{t^3}{|x|^3}$ (apply three boost to $\frac{x^k}{|x|}$). We would then have to work with an another commutation formula leading to terms such as $x^{\theta} \frac{v^{\mu}}{v^0}\partial(F)_{\mu \nu} H_j$ and would then need at least a decay rate of $\tau_+^{-\frac{3}{2}}$ on $\rho$, in the $t+r$ direction, in order to close the energy estimates on $H$. This could be obtained by assuming more decay on $F$ initially in order to use the Morawetz vector field $ \overline{K}_0$ or $\tau_-^{-b} \overline{K}_0$ as a multiplier.
However, this creates two technical difficulties compared to what we did in \cite{dim4}. The first one concerns $H$ and will lead us to consider a new hierarchy (see Subsection \ref{subsecH}). The other one concerns $G$ and we will circumvent it by modifying the source term of the transport equation defining it (see Subsecton \ref{subsecG}). \end{Rq} \begin{Rq}
In Subsection \ref{subsecG}, we will consider a matrix $D$ such that $T_F(R^2)=DR^2$ and we will need to estimate pointwise and independently of $M$, in order to improve the bootstrap assumption on $\mathcal{E}_{N-1}[F]$, the derivatives of the electromagnetic field of its components. It explains, in view of Remark \ref{lowderiv}, why we take $I_2$ such as $|\beta^2_j| \leq N-5$. \end{Rq} \subsection{The homogeneous part}\label{subsecH}
The purpose of this subsection is to bound $L^1$ norms of components of $H$ and their derivatives. We will then be able to obtain the desired $L^2$ estimates through Klainerman-Sobolev inequalities. For that, we will make use of the hierarchy between the components of $H$ given by $(\beta^1_i)_P$. However, as, for $N-4 \leq q \leq N$ and $\beta_i^1 \in I^q_1$, we need information on $\| \widehat{Z}^{\kappa} H_j \|_{L^1_{x,v}}$, with $\beta^1_j \in I^{q-1}_1$ and $|\kappa|=4$, in order to close the energy estimate on $\widehat{Z}^{\xi} H_i$, with $|\xi|=3$, we will add a new hierarchy in our energy norms. This leads us to define, for $\delta \in \{ 0,1 \}$,
$$ \mathbb{E}_H^{\delta}(t) := \sum_{ z \in \mathbf{k}_1} \sum_{q=0}^5 \sum_{|\beta| \leq 3+q} \sum_{i \in I^{N-q}_1} \sum_{j=0}^{2N+2+\delta-\beta_P-\beta^1_P} \log^{-j(\delta a+2)}(3+t) \mathbb{E} \left[ z^j \widehat{Z}^{\beta} H_i \right](t).$$ \begin{Lem}\label{Comuhom2}
Let $\widetilde{N} \geq N+3$, $0 \leq q \leq 5$, $i \in I^{N-q}_1$, $|\beta| \leq 3+q$, $ z \in \mathbf{k}_1$ and $j \leq \widetilde{N}-\beta_P-(\beta^1_i)_P$. Then, $T_F( z^j \widehat{Z}^{\beta} H_i)$ can be bounded by a linear combination of the following terms, where $$p \leq 3N, \hspace{5mm} \max(|k|+1,|\gamma|) \leq 8, \hspace{5mm} |\kappa| \leq |\beta|+1, \hspace{5mm} |\beta^1_l| \leq |\beta^1_i| \hspace{5mm} \text{and} \hspace{5mm} |\kappa|+|\beta^1_l| \leq |\beta^1_i|.$$ \begin{itemize} \item \begin{equation}\label{eq:cat0H}
\left| F \left(v, \nabla_v \left( z^j \right) \right) Y^{\beta} H_i \right|. \tag{category $0-H$} \end{equation} \item \begin{equation}\label{eq:cat1H}
\left| P_{k,p}(\Phi) \right| \left| w^{r} Y^{\kappa} H_l \right| \left( \left| \nabla_{Z^{\gamma}} F \right| +\frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right|+\frac{\tau_+}{\tau_-} \sqrt{\frac{v^{\underline{L}}}{v^0}} \left| \sigma \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right| \right), \tag{category $1-H$} \end{equation} where \hspace{2mm} $w \in \mathbf{k}_1$ \hspace{2mm} and \hspace{2mm} $r \leq \widetilde{N} -k_P-\kappa_P-(\beta^1_l)_P$. \item \begin{equation}\label{eq:cat3H}
\hspace{-10mm} \frac{\tau_+}{\tau_-} |\rho \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) | \left|z^{j-1} Y^{\kappa} H_l \right| \hspace{8mm} \text{and} \hspace{8mm} \frac{\tau_+}{\tau_-} \sqrt{\frac{v^{\underline{L}}}{v^0}}\left| \underline{\alpha} \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) \right| \left| z^r Y^{\kappa} H_l \right|, \tag{category $2-H$} \end{equation} where \hspace{2mm} $j-1$, $r=\widetilde{N}-\kappa_P-(\beta^1_l)_P$ and $r \leq j$. \end{itemize} The terms of \eqref{eq:cat3H} can only appear if $j=\widetilde{N}-\beta_P-(\beta^1_i)_P$. \end{Lem} \begin{proof}
We merely sketch the proof as it is very similar to previous computations. One can express $T_F(\widehat{Z}^{\beta} H_i)$ using Lemma \ref{bilanL2} and following what we did in the proof of Proposition \ref{ComuVlasov}. It then remains to copy the proof of Proposition \ref{ComuPkp} with $|\zeta_0|=0$, which explains that we do not have terms of \eqref{eq:cat4}. Note that $\max(|k|+1,|\gamma|) \leq 8$ comes from Remark \ref{rqcondiH} and the fact that $|\kappa|$ can be equal to $|\beta|+1$ ensues from the transformation of the $v$ derivative in the terms obtained from those of \eqref{eq:com22} and \eqref{eq:com44}. \end{proof} \begin{Rq}\label{rqrqrq}
As $|\gamma| \leq 8 \leq N-3$, we have at our disposal pointwise decay estimates on the electromagnetic field (see Proposition \eqref{decayF}). Similarly, as $|k| \leq 7 \leq N-4$, Remark \ref{estiPkp} gives us $|P_{k,p}(\Phi)| \lesssim \log^{M_2}(1+\tau_+)$. \end{Rq}
We are now ready to bound $\mathbb{E}^{\delta}_H$ and then to obtain estimates on $\int_v |z^j H_i|dv$. \begin{Pro}\label{estidecayH}
We have $\mathbb{E}^1_H +\mathbb{E}^0_H \lesssim \epsilon$ on $[0,T[$. Moreover, for $0 \leq q \leq 5$ and $|\beta| \leq q$,
$$\forall \hspace{0.5mm} (t,x) \in [0,T[ \times \mathbb{R}^3, \hspace{3mm} z \in \mathbf{k}_1, \hspace{3mm} i \in I^{N-q}_1, \hspace{3mm} j \leq 2N-1-\beta_P-(\beta^1_i)_P, \hspace{5mm} \int_v |z^j Y^{\beta} H_i| dv \lesssim \epsilon \frac{\log^{2j+M_1}(3+t)}{\tau_+^2 \tau_-}.$$ \end{Pro} \begin{proof}
In the same spirit as Corollary \ref{coroinit} and in view of commutation formula of Lemma \ref{Comuhom2} (applied with $\widetilde{N} = 2N+3$) as well as the assumptions on $f_0$, there exists $C_H >0$ such that $\mathbb{E}^0_H(0) \leq \mathbb{E}^1_H(0) \leq C_H \epsilon$. We can prove that they both stay bounded by $3 C_H \epsilon$ by the continuity method. As it is very similar to what we did previously, we only sketch the proof. Consider $\delta \in \{ 0 , 1 \}$, $0 \leq r \leq 5$, $i \in I^{N-r}_1$, $|\beta| \leq 3+r$, $z \in \mathbf{k}_1$ and $j \leq 2N+2+\delta-\beta_P-(\beta^1_i)_P$. The goal is to prove that
$$ \int_0^t \int_{\Sigma_s} \int_v \left| T_F( z^j H_i ) \right| \frac{dv}{v^0}dxds \lesssim \epsilon^{\frac{3}{2}} \log^{j(\delta a+2)}(3+t).$$ According to Lemma \ref{Comuhom2} (still applied with $\widetilde{N}=2N+3$), it is sufficient to obtain, if $\delta=1$, that the integral over $[0,t] \times \mathbb{R}^3_x \times \mathbb{R}^3_v$ of all terms of \eqref{eq:cat0H}-\eqref{eq:cat3H} are bounded by $\epsilon^{\frac{3}{2}} \log^{j( a+2)}(3+t)$. If $\delta=0$, we only have to deal with terms of \eqref{eq:cat0H} and \eqref{eq:cat1H} and to estimate their integrals by $\epsilon^{\frac{3}{2}} \log^{2j}(3+t)$. In view of Remark \ref{rqrqrq}, we only have to apply (or rather follow the computations of) Propositions \ref{M1}, \ref{M2} and \ref{M3}. The pointwise decay estimates then ensue from the Klainerman-Sobolev inequality of Corollary \ref{KS3}. \end{proof} \begin{Rq} A better decay rate, $\log^{2j}(3+t) \tau_+^{-2} \tau_-^{-1}$, could be proved in the previous proposition by controling a norm analogous to $\mathbb{E}_N^{X}[f]$ but we do not need it to close the energy estimates on $F$. \end{Rq} \begin{Rq}\label{rqgainH} We could avoid any hypothesis on the derivatives of order $N+1$ and $N+2$ of $F^0$ (see Subsection $17.2$ of \cite{FJS3}). \end{Rq} \subsection{The inhomogeneous part}\label{subsecG}
As the matrix $B$ in $T_F(G)+AG=BR^2$ contains top order derivatives of the electromagnetic field, we cannot commute the equation and prove $L^1$ estimates on $\widehat{Z} G$. Let us explain schematically how we will obtain an $L^2$ estimate on $\int_v |G|dv$ by recalling how we proceeded in \cite{dim4}. We did not work with modified vector field and the matrices $A$ and $B$ did not hide $v$ derivatives of $G$. Then we introduced $K$ the solution of $T_F(K)+AK+KD=B$ which initially vanishes and where $T_F(R^2)=DR^2$. Thus $G=KR^2$ and we proved $\mathbb{E}[|K|^2 |R^2|] \leq \epsilon$ so that the expected $L^2$ decay estimate followed from
$$\left\| \int_v |G|dv \right\|_{L^2_x} \lesssim \left\| \int_v |R^2| dv \right\|^{\frac{1}{2}}_{L^{\infty}_x} \mathbb{E}[|K|^2 |R^2|]^{\frac{1}{2}}.$$ The goal now is to adapt this process to our situation. There are two obstacles. \begin{itemize} \item The $v$ derivatives hidden in the matrix $A$ will then be problematic and we need first to transform them. \item The components of the (transformed) matrix $A$ have to decay sufficiently fast. We then need to consider a larger vector valued field than $G$ by including components such as $z^j G_i$ in order to take advantage of the hierarchies in the source terms already used before. \end{itemize} Recall from Definition \ref{orderk1} that we considered an ordering on $\mathbf{k}_1$ and that, if $\kappa$ is a multi-index, we have
$$z^{\kappa}= \prod_{i=1}^{|\kappa|} z_{\kappa_i} \hspace{3mm} \text{and} \hspace{3mm} |z^{\kappa}| \leq \sum_{w \in \mathbf{k}_1} |w|^{|\kappa|}.$$ In this section, we will sometimes have to work with quantities such as $z^{\kappa}$ rather than with $z^j$, where $j \in \mathbb{N}$. \begin{Def} Let $I$ and $I^q$, for $N-5 \leq q \leq N$, be the sets
$$ I := \{ (\kappa, \beta ) \hspace{1mm} / \hspace{1mm} N-5 \leq |\beta| \leq N \hspace{2mm} \text{and} \hspace{2mm} |\kappa| \leq N-\beta_P \} = \{ (\kappa_1,\beta_1),...,(\kappa_{|I|}, \beta_{|I|}) \}, \hspace{5mm} I^q := \{ (\kappa, \beta ) \in I \hspace{1mm} / \hspace{1mm} |\beta|=q \}.$$
Define now $L$, the vector valued fields of length $|I|$, such that
$$L_i = z^{\kappa_i} G_j, \hspace{8mm} \text{with} \hspace{8mm} \beta_j^1 = \beta_i, \hspace{8mm} \text{and} \hspace{8mm} [i]_I:=|\kappa_i|.$$
Moreover, for $Y \in \mathbb{Y}$, $1 \leq j \leq |I_1|$ and $1 \leq i \leq |I|$, we define $j_Y$ and $i_Y$ the indices such that $$R^1_{j_{Y}}=Y Y^{\beta^1_j} f \hspace{8mm} \text{and} \hspace{8mm} L_{i_Y} = z^{\kappa_{i_Y}} G_{j_Y}.$$ \end{Def} The following result will be useful for transforming the $v$ derivatives. \begin{Lem}\label{deriG} Let $Y \in \mathbb{Y}$ and $ \beta^1_i \in I_1 \setminus I^{N}_1$. Then $$Y G_i = G_{i_Y}+H_{i_Y}-Y H_i.$$ \end{Lem} \begin{proof} Recall that $R=H+G$ and remark that $Y R^1_i = Y Y^{\beta^1_i} f= R^1_{i_Y}$. \end{proof} We now describe the source terms of the equations satisfied by the components of $L$. \begin{Pro}\label{bilanG}
There exists $N_1 \in \mathbb{N}^*$, a vector valued field $W$ and three matrix-valued functions $\overline{A} : [0,T[ \times \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathfrak M_{|I|}(\mathbb{R})$, $\overline{B} : [0,T[ \times \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathfrak M_{|I|,N_1}(\mathbb{R})$, $\overline{D} : [0,T[ \times \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathfrak M_{N_1}(\mathbb{R})$ such that
$$T_F(L)+\overline{A}L= \overline{B} W, \hspace{8mm} T_F(W)= \overline{D} W \hspace{8mm} \text{and} \hspace{8mm} \sum_{z \in \mathbf{k}_1} \int_v |z^{2} W| dv \lesssim \epsilon \frac{\log^{3N+M_1} (3+t)}{\tau_+^2 \tau_-}.$$
In order to depict these matrices, we use the quantity $[q]_W$, for $1 \leq q \leq N_1$, which will be defined during the construction of $W$ in the proof. $\overline{A}$ and $\overline{B}$ are such that $T_F(L_i)$ can be bounded, for $1 \leq i \leq |I|$, by a linear combination of the following terms, where \hspace{1mm} $|\gamma| \leq 5$, \hspace{1mm} $1 \leq j,q \leq |I|$ \hspace{1mm} and \hspace{1mm} $1 \leq r \leq N_1$.
\begin{equation}\label{eq:cat0G}
\big( \tau_- \left( |\rho (F) |+|\sigma(F) |+|\underline{\alpha}(F)| \right)+\tau_+ |\alpha(F)| \big) \left| L_j \right|, \hspace{5mm} \text{with} \hspace{5mm} [j]_I = [i]_I-1. \tag{category $0-\overline{A}$} \end{equation}
\begin{equation}\label{eq:cat1G}
\log^{M_1}(3+t) \left| L_j \right| \left( \left| \nabla_{Z^{\gamma}} F \right|+ \frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right|+ \frac{\tau_+}{\tau_-} \sqrt{\frac{v^{\underline{L}}}{v^0}} \left| \sigma \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right| \right) .\tag{category $1-\overline{A}$} \end{equation}
\begin{equation}\label{eq:cat3G}
\frac{\tau_+}{\tau_-} |\rho \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) | \left| L_j \right| + \frac{\tau_+}{\tau_-} \sqrt{\frac{v^{\underline{L}}}{v^0}}\left| \underline{\alpha} \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) \right| \left| L_q \right|, \hspace{5mm} \text{with} \hspace{5mm} [j]_I+1, \hspace{1mm} [q]_I \leq [i]_I. \tag{category $2-\overline{A}$} \end{equation}
\begin{equation}\label{eq:cat1Y}
\left| P_{k,p}(\Phi) \right| \left| W_r \right| \left( \left| \nabla_{Z^{\zeta}} F \right|+ \frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\zeta}}(F) \right) \right|+ \frac{\tau_+}{\tau_-} \sqrt{\frac{v^{\underline{L}}}{v^0}} \left| \sigma \left( \mathcal{L}_{Z^{\zeta}}(F) \right) \right| \right), \tag{category $1-\overline{B}$} \end{equation}
where \hspace{1mm} $p \leq 2N$, \hspace{1mm} $|k| \leq N-1$ \hspace{1mm} and \hspace{1mm} $|k|+|\zeta| \leq N$. Moreover, if $|k| \geq 6$, there exists $\kappa$ and $\beta$ such that \hspace{1mm} $W_r = z^{\kappa} Y^{\beta} f$, \hspace{1mm} $|k|+|\beta| \leq N$ \hspace{1mm} and \hspace{1mm} $|\kappa| \leq N+1-k_P-\beta_P$.
The matrix $\overline{D}$ is such that, for $1 \leq i \leq N_1$, $T_F(W_i)$ is bounded by a linear combination of the following expressions, where \hspace{1mm} $|\gamma| \leq N-5$ \hspace{1mm} and \hspace{1mm} $1 \leq j,q \leq N_1$.
\begin{equation}\label{eq:cat0W}
\left( \tau_- \left( |\rho (F) |+|\sigma(F) |+|\underline{\alpha}(F)| \right)+\tau_+ |\alpha(F)| \right) \left| W_j \right|, \hspace{5mm} \text{with} \hspace{5mm} [j]_W = [i]_W-1. \tag{category $0-\overline{D}$} \end{equation} \begin{equation}\label{eq:cat1W}
\log^{M_1}(3+t) \left| W_j \right| \left( \left| \nabla_{Z^{\gamma}} F \right|+\frac{\tau_+}{\tau_-} \left| \alpha \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right|+ \frac{\tau_+}{\tau_-} \sqrt{\frac{v^{\underline{L}}}{v^0}} \left| \sigma \left( \mathcal{L}_{Z^{\gamma}}(F) \right) \right| \right). \tag{category $1-\overline{D}$} \end{equation} \begin{equation}\label{eq:cat3W}
\frac{\tau_+}{\tau_-} |\rho \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) | \left| W_j \right| + \frac{\tau_+}{\tau_-} \sqrt{\frac{v^{\underline{L}}}{v^0}}\left| \underline{\alpha} \left( \mathcal{L}_{ Z^{\gamma}}(F) \right) \right| \left| W_q \right|, \hspace{5mm} \text{with} \hspace{5mm} [j]_W+1, \hspace{1mm} [q]_W \leq [i]_W. \tag{category $2-\overline{D}$} \end{equation} \end{Pro} \begin{proof} The main idea is to transform the $v$ derivatives in $AG$, following the proof of Lemma \ref{nullG}, and then to apply Lemma \ref{deriG} in order to eliminate all derivatives of $G$ in the source term of the equations. We then define $W$ as the vector valued field, and $N_1$ as its length, containing all the following quantities \begin{itemize}
\item $z^j Y^{\beta} f $, \hspace{1mm} with \hspace{1mm} $z \in \mathbf{k}_1$, \hspace{1mm} $|\beta| \leq N-5$ \hspace{1mm} and \hspace{1mm} $j \leq N+1- \beta_P$, \item $z^j\left(H_{i_Y}-Y H_i \right)$, \hspace{1mm} with \hspace{1mm} $z \in \mathbf{k}_1$, \hspace{1mm} $Y \in \mathbb{Y}$, \hspace{1mm} $\beta_i^1 \in I_1 \setminus I_1^N$ \hspace{1mm} and \hspace{1mm} $j \leq N+3-\left(\beta^1_{i_Y} \right)_P$.
\item $z^j Y^{\beta} H_i$, \hspace{1mm} with \hspace{1mm} $z \in \mathbf{k}_1$, \hspace{1mm} $|\beta|+|\beta^1_i| \leq N$ \hspace{1mm} and \hspace{1mm} $ j \leq N+3-\beta_P-(\beta^1_i)_P$. \end{itemize} Let us make three remarks. \begin{itemize} \item If $1 \leq i \leq N_0$, we can define, in each of the three cases, $[i]_W:=j$. \item Including the terms $z^{N+1- \beta_P} Y^{\beta} f$ and $z^{N+1-\left(\beta^1_{i_Y} \right)_P}\left(H_{i_Y}-Y H_i \right)$ in $W$ allows us to avoid any term of category $2$ related to $\overline{B}$. \item The components such as $z^j Y^{\beta} H_i$ are here in order to obtain an equation of the form $T_F(W)= \overline{D} W$. \end{itemize}
The form of the matrix $\overline{D}$ then follows from Proposition \ref{ComuPkp} if $Y_i = z^j Y^{\beta} f $ and from Lemma \ref{Comuhom2}, applied with $\widetilde{N}=N+3$, otherwise (we made an additional operation on the terms of category $0$ which will be more detailed for the matrix $\overline{A}$). Note that we use Remark \ref{estiPkp} to estimate all quantities such as $P_{k,p}(\Phi)$. The decay rate on $\int_v |z^2 W| dv$ follows from Proposition \ref{Xdecay} and \ref{estidecayH}.
We now turn on the construction of the matrices $\overline{A}$ and $\overline{B}$. Consider then $1 \leq i \leq |I|$ and $1 \leq q \leq |I_1|$ so that $L_i=z^{\kappa_i} G_q$ and $|\kappa_i| \leq N-(\beta^1_q)_P$. Observe that $$T_F(L_i)= T_F(z^{\kappa_i})G_q+z^{\kappa_i} T_F(G_q)= F \left( v, \nabla_v (z^{\kappa_i}) \right)G_q+z^{\kappa_i} T_F(G_q).$$ The first term on the right hand side gives terms of \eqref{eq:cat0G} and \eqref{eq:cat1G} as, following the computations of Proposition \ref{M1}, we have
$$\nabla_v \left( \prod_{r=1}^{|\kappa_i|} z_r \right) = \sum_{p=1}^{|\kappa_i|} \nabla_v(z_p) \prod_{r \neq p} z_r, \hspace{5mm} \left| F(v,\nabla_v z_p) \right| \lesssim \tau_- \left( |\rho (F) |+|\sigma(F) |+|\underline{\alpha}(F)| \right)+\tau_+ |\alpha(F)|+\sum_{w \in \mathbf{k}_1} |w F| .$$ The remaining quantity, $z^{\kappa_i} T_F(G_q)=-z^{\kappa_i} A_q^r G_r+z^{\kappa_i} B_q^r R^2_r$, is described in Lemma \ref{bilanL2}. Express the terms given by $z^{\kappa_i} A_q^r G_r$ in null components and transform the $v$ derivatives\footnote{Note that this is possible since $\partial_v G_r$ can only appear if $\beta^1_r \in I_1 \setminus I^N_1$.} of $G_r$ using Lemma \ref{deriG}, so that, schematically (see \eqref{equ:proof}), \begin{eqnarray} \nonumber v^0\left( \nabla_v G_r \right)^r & = & Y G_r+(t-r) \partial G_r = G_{r_Y}+H_{r_Y}-Y H_r+(t-r)(G_{r_{\partial}}+H_{r_{\partial}}-\partial H_r) \hspace{5mm} \text{and} \hspace{5mm} \\ \nonumber v^0 \partial_{v^b} G_r & = & Y_{0b} G_r+x \partial G_r = G_{r_{Y_{0b}}}+H_{r_{Y_{0b}}}-Y_{0b} H_r+x(G_{r_{\partial}}+H_{r_{\partial}}-\partial H_r) . \end{eqnarray}
By Remark \ref{rqcondiH}, the $\Phi$ coefficients and the electromagnetic field are both derived less than $5$ times. We then obtain, with similar operations as those made in proof of Proposition \ref{ComuPkp}, the matrix $\overline{A}$ and the columns of the matrix $\overline{B}$ hitting the component of $W$ of the form $z^j\left(H_{l_Y}-Y H_l \right)$. For $z^{\kappa_i} B_q^r R^2_r$, we refer to the proof of Proposition \ref{ComuPkp}, where we already treated such terms. \end{proof}
To lighten the notations and since there will be no ambiguity, we drop the index $I$ (respectively $W$) of $[i]_I$ for $1 \leq i \leq |I|$ (respectively $[j]_W$ for $1 \leq j \leq N_1$). Let us introduce $K$ the solution of $T_F(K)+\overline{A}K+K\overline{D}=\overline{B}$, such as $K(0,.,.)=0$. Then, $KY= L$ since they are solution of the same system and they both initially vanish. The goal now is to control $\mathbb{E}[|K|^2|Y|]$. As, for $ 1 \leq i \leq |I|$ and $1 \leq j,p \leq N_1$, \begin{equation}\label{eq:K}
T_F\left( |K^j_i|^2 W_p\right) = |K^j_i |^2\overline{D}^q_pW_q-2\left(\overline{A}^q_i K^j_q +K^q_i \overline{D}^j_q \right) K^j_i W_p+2\overline{B}^j_iK^j_iW_p, \end{equation}
we consider $\mathbb{E}_L$, the following hierarchized energy norm,
$$\mathbb{E}_L(t):= \sum_{ \begin{subarray}{} 1 \leq j,p \leq N_1 \\ \hspace{1mm} 1 \leq i \leq |I| \end{subarray}} \log^{-4[i]-2[p]+4[j]}(3+t) \mathbb{E} \left[\left|K_i^j\right|^2 W_p \right](t). $$ The sign in front of $[j]$ is related to the fact that the hierarchy is inversed on the terms coming from $K \overline{D}$. It prevents us to expect a better estimate than $\mathbb{E}_L(t) \lesssim \log^{4N+12}(3+t)$. \begin{Lem}\label{Inho1} We have, for $M_0 = 4N+12$ and if $\epsilon$ small enough, $\mathbb{E}_L(t) \lesssim \epsilon \log^{M_0}(3+t)$ for all $t \in [0,T[$. \end{Lem} \begin{proof} We use again the continuity method. Let $T_0 \in [0,T[$ be the largest time such that $\mathbb{E}_L(t) \leq 2 \epsilon \log^{M_0}(3+t)$ for all $t \in [0,T_0[$ and let us prove that, if $\epsilon$ is small enough, \begin{equation}\label{thegoal} \forall \hspace{0.5mm} t \in [0,T_0[, \hspace{3mm} \mathbb{E}_L(t) \lesssim \epsilon^{\frac{3}{2}} \log^{M_0}(3+t). \end{equation}
As $T_0 >0$ by continuity ($K$ vanishes initially), we would deduce that $T_0=T$. We fix for the remainder of the proof $1 \leq i \leq |I|$ and $1 \leq j, p \leq N_1$. According to the energy estimate of Proposition \ref{energyf}, \eqref{thegoal} would follow if we prove that \begin{eqnarray}
\nonumber I_{\overline{A}, \overline{D}} & := & \int_0^t \int_{\Sigma_s} \int_v \left| |K^j_i |^2 \overline{D}^q_pW_q-2\left( \overline{A}^k_i K^j_k +K^r_i \overline{D}^j_r \right) K^j_i W_p \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}} \log^{M_0+4[i]+2[p]-4[j]}(3+t), \\ \nonumber I_{\overline{B}} & := & \int_0^t \int_{\Sigma_s} \int_v \left| B_i^j \right| \left| K^j_i W_p \right| \frac{dv}{v^0} dx ds \lesssim \epsilon^{\frac{3}{2}}. \end{eqnarray}
Let us start by $I_{\overline{A}, \overline{D}}$ and note that in all the terms given by Proposition \ref{bilanG}, the electromagnetic field is derived less than $N-5$ times so that we can use the pointwise decay estimates given by Remark \ref{lowderiv}. The terms of \eqref{eq:cat1G} and \eqref{eq:cat1W} can be easily handled (as in Proposition \ref{M2}). We then only treat the following cases, where $|\gamma| \leq N-5$ (the other terms are similar).
$$ \left| \overline{D}^j_r \right| = \tau_- \left( |\rho (F) |+|\sigma(F) |+|\underline{\alpha}(F)| \right)+\tau_+ |\alpha(F)|, \hspace{5mm} \text{with} \hspace{5mm} [j]=[r]-1,$$
$$ \left| \overline{A}^k_i \right| \lesssim \frac{\tau_+ \sqrt{v^{\underline{L}}}}{\tau_- \sqrt{v^0}} |\underline{\alpha} (\mathcal{L}_{Z^{\gamma}}(F))|, \hspace{3mm} \text{with} \hspace{3mm} [k] \leq [i], \hspace{5mm} \text{and} \hspace{5mm} \left| \overline{D}^q_p \right| \lesssim \frac{\tau_+}{\tau_-} |\rho (\mathcal{L}_{Z^{\gamma}}(F))|, \hspace{3mm} \text{with} \hspace{3mm} [q] < [p]. $$ Without any summation on the indices $r$, $k$ and $q$, we have, using Remark \ref{lowderiv}, $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$ and the Cauchy-Schwarz inequality several times, \begin{eqnarray}
\nonumber \int_0^t \int_{\Sigma_s} \int_v \left| K^r_i \overline{D}^j_r K^j_i W_p \right| \frac{dv}{v^0} dx ds & \lesssim & \sqrt{\epsilon} \int_0^t \frac{\log (3+s)}{1+s} \left| \mathbb{E} \left[ \left| K^r_i \right|^2 W_p \right] \hspace{-0.5mm} (s) \hspace{0.5mm} \mathbb{E} \left[ \left| K^j_i \right|^2 W_p \right] \hspace{-0.5mm} (s) \right|^{\frac{1}{2}} ds \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}} \log^{2+M_0+4[i]+2[p]-2[r]-2[j]}(3+t) \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} \log^{M_0+4[i]+2[p]-4[j]}(3+t), \\ \nonumber
\int_0^t \int_{\Sigma_s} \int_v \left|\overline{A}^k_i K^j_k K^j_i W_p\right| \frac{dv}{v^0} dx ds & \lesssim & \sqrt{\epsilon}\int_{u=-\infty}^t \frac{\tau_+}{\tau_+\tau_-^{\frac{3}{2}}} \int_{C_u(t)} \int_v \frac{v^{\underline{L}}}{v^0} \left| K^j_k \right| \left| W_p\right|^{\frac{1}{2}} \left| K^j_i \right| \left| W_p\right|^{\frac{1}{2}} dv dC_u(t) du \\ \nonumber
& \lesssim & \sqrt{\epsilon} \left| \mathbb{E} \left[ \left| K^j_k \right|^2 W_p \right] \hspace{-0.5mm} (t) \hspace{0.5mm} \mathbb{E} \left[ \left| K^j_i \right|^2 W_p \right] \hspace{-0.5mm} (t) \right|^{\frac{1}{2}} \int_{u=-\infty}^{+\infty} \frac{du}{\tau_-^{\frac{3}{2}}} \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}} \log^{M_0+2[k]+2[i]+2[p]-4[j]}(3+t) \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} \log^{M_0+4[i]+2[p]-4[j]}(3+t), \\ \nonumber
\nonumber \int_0^t \int_{\Sigma_s} \int_v \left| K^j_i \right|^2 \left| \overline{D}^q_pW_q \right| \frac{dv}{v^0} dx ds & \lesssim & \sqrt{\epsilon} \int_0^t \int_{\Sigma_s} \int_v \log (3+s) \frac{\sqrt{v^{\underline{L}}v^0}}{\tau_+^{\frac{1}{2}} \tau_-^{\frac{3}{2}}} \left| K^j_i \right|^2 \left| W_q \right| \frac{dv}{v^0} dx ds \\ \nonumber
& \lesssim & \sqrt{\epsilon} \left( \int_0^t \frac{\log(3+s)}{1+s} ds+\log(3+t) \int_{-\infty}^{+\infty} \frac{du}{\tau_-^3} \right) \sup_{[0,t]} \mathbb{E} \left[ \left| K^j_i \right|^2 W_q \right] \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}} \log^{2+M_0+4[i]+2[q]-4[j]}(3+t) \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}} \log^{M_0+4[i]+2[p]-4[j]}(3+t). \end{eqnarray} It remains to study $I_{\overline{B}}$. The form of $\overline{B}^j_i$ is given by Propoposition \ref{bilanG} and the computations are close to the ones of Proposition \ref{M21}. We then only consider the following two cases,
$$ \left| \overline{B}^j_i K_i^j W_p \right| \lesssim \log^{M_1}(1+\tau_+)\frac{\tau_+ \sqrt{v^{\underline{L}}}}{\tau_-\sqrt{v^0}} \left| \sigma (\mathcal{L}_{Z^{\zeta}} (F) ) \right| \left|K_i^j \right| |W_p|, \hspace{5mm} \text{with} \hspace{5mm} |\zeta| \leq N \hspace{5mm} \text{and} \hspace{5mm} $$
$$ \left| \overline{B}^j_i K_i^j W_p \right| \lesssim \left|\Phi^r P_{\xi}(\Phi)|| \nabla_{ Z^{\gamma}} F \right| \left| K_i^j W_p \right|, \hspace{5mm} \text{with} \hspace{5mm} r \leq 2N, \hspace{6mm} |\xi|+|\gamma| \leq N \hspace{5mm} \text{and} \hspace{5mm} 6 \leq |\xi| \leq N-1.$$ In the first case, using the Cauchy-Schwarz inequality twice (in $(t,x)$ and then in $v$), we get \begin{eqnarray}
\nonumber I_{\overline{B}} & \lesssim & \int_{u=-\infty}^t \left\| \sigma (\mathcal{L}_{Z^{\zeta}} (F) ) \right\|_{L^2(C_u(t))} \left| \int_{C_u(t)} \log^{2M_1}(1+\tau_+)\frac{\tau_+^2}{\tau_-^2} \left| \int_v \sqrt{ \frac{v^{\underline{L}}}{v^0}} \left| K^j_i W_p \right| \frac{dv}{v^0} \right|^2 dC_u(t) \right|^{\frac{1}{2}} du \\ \nonumber
& \lesssim & \sqrt{\epsilon} \sum_{q=0}^{+ \infty} \int_{u=-\infty}^t \frac{1}{\tau_-^{\frac{5}{4}}} \left\| \tau_+^{\frac{11}{4}} \int_v \left| W_p \right| \frac{dv}{(v^0)^2} \right\|^{\frac{1}{2}}_{L^{\infty}(C^q_u(t))} \left\| \int_v \frac{v^{\underline{L}}}{v^0} \left| K^j_i \right|^2 \left| W_p \right| dv \right\|_{L^{1}(C^q_u(t))}^{\frac{1}{2}} du \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}} \int_{-\infty}^{+\infty} \frac{du}{\tau_-^{\frac{5}{4}}} \sum_{q=0}^{+\infty} \frac{\log^{\frac{M_0+4[i]+2[p]+3N+M_1}{2}}(3+t_{q+1})}{(1+t_{q})^{\frac{1}{8}}} \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}, \end{eqnarray}
using the bootstrap assumption on $\mathbb{E}_L$ and $\int_v |W_p| \frac{dv}{(v^0)^2} \lesssim \int_v |W_p| \frac{v^{\underline{L}}}{v^0} dv \lesssim \epsilon \log^{3N+M_1}(3+t) \tau_+^{-3}$, which comes from Proposition \ref{bilanG} and Lemma \ref{weights}. For the remaining case, we have $|\gamma| \leq N-6$ and we can then use the pointwise decay estimates on the electromagnetic field given by Proposition \ref{decayF}. Moreover, by Proposition \ref{bilanG}, we have that
$$W_p = z^{\kappa} Y^{\beta} f , \hspace{5mm} \text{with} \hspace{5mm} |\xi|+|\beta| \leq N \hspace{5mm} \text{and} \hspace{5mm} |\kappa| \leq N+1-\beta_P-\xi_P.$$
Suppose first that $|\kappa| \leq 2N-1-\beta_P-2\xi_P$. Then, since $|\Phi|^r | \nabla_{Z^{\gamma}} F| \lesssim \sqrt{\epsilon} \tau_+^{-\frac{3}{4}}\tau_-^{-1}$ and $1 \lesssim \sqrt{v^0 v^{\underline{L}}}$, we get
$$\left| \overline{B}^j_i K^j_i W_p \right| \lesssim \sqrt{\epsilon} \left( \frac{v^0}{\tau_+^{\frac{5}{4}}}+\frac{v^{\underline{L}}}{\tau_+^{\frac{1}{4}} \tau_-^2} \right) \left( \left| z^{\kappa} P_{\xi}(\Phi)^2 Y^{\beta} f \right|+\left|K^j_i \right|^2 \left| W_p \right| \right).$$ Hence, we can obtain $I_{\overline{B}} \lesssim \epsilon^{\frac{3}{2}}$ by following the computations of Proposition \ref{M2}, as, by the bootstrap assumptions on $\overline{\mathbb{E}}_N[f]$ and $\mathbb{E}_L$,
$$\mathbb{E}[ z^{\kappa} P_{\xi}(\Phi)^2 Y^{\beta} f ](t)+\mathbb{E} \left[ \left| K^j_i \right|^2 W_p \right](t) \lesssim \epsilon (1+t)^{\frac{1}{8}}.$$
Otherwise, $|\kappa| = 2N-\beta_P-2\xi_P$ so that $\xi_P=N-1$, $|\beta| \leq 1$ and $|\kappa| = 2-\beta_P$. We can then write $z^{\kappa}=z z^{\kappa_0}$ and find $q \in \llbracket 1,N_1 \rrbracket$ such that $W_q = z^2 z^{\kappa_0} Y^{\beta}f$. It remains to follow the previous case after noticing that
$$\left| \overline{B}^j_i K^j_i W_p \right| \lesssim \sqrt{\epsilon} \left( \frac{v^0}{\tau_+^{\frac{5}{4}}}+\frac{v^{\underline{L}}}{\tau_+^{\frac{1}{4}} \tau_-^2} \right) \left( \left| z^{\kappa_0} P_{\xi}(\Phi)^2 Y^{\beta} f \right|+\left|K^j_i \right|^2 \left| W_q \right| \right) \hspace{5mm} \text{and} \hspace{5mm} |\kappa_0| \leq 2N-1-2\xi_P-\beta_P.$$ \end{proof}
\subsection{$L^2$ estimates on the velocity averages of $f$}
We finally end this section by proving several $L^2$ estimates. The first one is clearly not sharp but is sufficient for us to close the energy estimates for the electromagnetic field. \begin{Pro}\label{estiL2}
Let $z \in \mathbf{k}_1$, $p \leq 3N$, $|k| \leq N-1$ and $\beta$ such that $|k|+|\beta| \leq N$. Then, for all $t \in [0,T[$,
$$ \left\| \frac{1}{\sqrt{\tau_+}} \int_v \left| zP_{k,p}(\Phi) Y^{\beta}f \right| dv \right\|_{L^2 (\Sigma_t)} \lesssim \frac{1}{1+t} \left\| \sqrt{\tau_+} \int_v \left| zP_{k,p}(\Phi) Y^{\beta}f \right| dv \right\|_{L^2 (\Sigma_t)} \lesssim \frac{\epsilon}{(1+t)^{\frac{5}{4}}}$$ \end{Pro} \begin{proof}
The first inequality ensues from $1+t \leq \tau_+$ on $\Sigma_t$. For the other one, we start by the case $|\beta| \leq N-3$. Write $P_{k,p}(\Phi)=\Phi^n P_{\xi}(\Phi)$ and notice that $|\Phi|^n \lesssim \log^{2p}(1+\tau_+)$. Then, using the bootstrap assumption \eqref{bootf3} and Proposition \ref{Xdecay}, \begin{eqnarray}
\nonumber \left\| \sqrt{\tau_+} \int_v \left|z P_{k,p}(\Phi) Y^{\beta}f \right| dv \right\|^2_{L^2 (\Sigma_t)} & \lesssim & \left\| \tau_+ \log^{4p}(1+\tau_+) \int_v \left| P_{\xi}(\Phi)^2 Y^{\beta}f \right| dv \int_v \left|z^2 Y^{\beta}f \right| dv \right\|_{L^{1} (\Sigma_t)} \\ \nonumber
& \lesssim & \left\| \tau_+ \log^{4p}(1+\tau_+) \int_v \left|z^2 Y^{\beta}f \right| dv \right\|_{L^{\infty} (\Sigma_t)} \overline{\mathbb{E}}_N[f](t) \\ \nonumber & \lesssim & \epsilon \frac{\log^{4p+6}(3+t)}{1+t} (1+t)^{\eta} \hspace{2mm} \lesssim \hspace{2mm} \frac{\epsilon^2}{(1+t)^{\frac{3}{4}}}. \end{eqnarray}
Otherwise, $|\beta| \geq N-2$ so that $|k| \leq 2$ and, according to Remark \ref{estiPkp}, $P_{k,p}(\Phi) \lesssim \tau_+^{\frac{1}{8}}$. Moreover, as there exists $i \in \llbracket 1, |I_1| \rrbracket$ such that $\beta=\beta^1_i$, we obtain
$$ \left\|\tau_+^{\frac{1}{2}} \int_v \left| zP_{k,p}(\Phi) Y^{\beta} \right| dv \right\|_{L^2 (\Sigma_t)} \lesssim \left\| \tau_+^{\frac{5}{8}} \int_v \left| z H_i \right| dv \right\|_{L^2 (\Sigma_t)}+\left\| \tau_+^{\frac{5}{8}} \int_v \left| z G_i \right| dv \right\|_{L^2 (\Sigma_t)}.$$ Applying Proposition \ref{estidecayH}, one has
$$\left\| \tau_+^{\frac{5}{8}} \int_v \left| z H_i \right| dv \right\|^2_{L^2 (\Sigma_t)} \lesssim \left\| \tau_+^{\frac{5}{4}} \int_v \left| z^2 H_i \right| dv \right\|_{L^{\infty} (\Sigma_t)}\left\| \int_v \left| H_i \right| dv \right\|_{L^1 (\Sigma_t)} \lesssim \frac{\epsilon^2}{(1+t)^{\frac{1}{2}}}.$$
As there exists $q \in \llbracket 1, |I| \rrbracket$ such that $ G_i=L_q=K_q^j W_j$, we have, using this time Proposition \ref{Inho1} and the decay estimate on $\int_v |z^2 W| dv$ given in Proposition \ref{bilanG}, \begin{eqnarray}
\nonumber \left\| \tau_+^{\frac{5}{8}} \int_v \left| z G_i \right| dv \right\|^2_{L^2 (\Sigma_t)} & = & \left\| \tau_+^{\frac{5}{8}} \int_v \left| z K_q^j W_j \right| dv \right\|^2_{L^2 (\Sigma_t)} \\ \nonumber
& \lesssim & \sum_{j=0}^{N_1} \left\| \tau_+^{\frac{5}{4}} \int_v \left| z^2 W_j \right| dv \right\|_{L^{\infty} (\Sigma_t)} \left\| \int_v \left| K_q^j \right|^2 \left| W_j \right| dv \right\|_{L^1 (\Sigma_t)} \\ \nonumber & \lesssim & \epsilon \frac{\log^{3N+M_1}(3+t)}{(1+t)^{\frac{3}{4}}}\log^{4[q]}(3+t) \mathbb{E}_L(t) \hspace{2mm} \lesssim \hspace{2mm} \frac{\epsilon^2}{(1+t)^{\frac{1}{2}}}. \end{eqnarray} \end{proof} This proposition allows us to improve the bootstrap assumption \eqref{bootL2} if $\epsilon$ is small enough. More precisely, the following result holds. \begin{Cor}
For all $t \in [0,T[$, we have $\sum_{|\beta| \leq N-2} \left\| r^{\frac{3}{2}} \int_v \frac{v^A}{v^0} \widehat{Z}^{\beta} f dv \right\|_{L^2(\Sigma_t)} \lesssim \epsilon$. \end{Cor} \begin{proof}
Let $t \in [0,T[$. Using $\tau_+|v^A| \lesssim v^0 \sum_{z \in \mathbf{k}_1} |z|$ and rewritting $\widehat{Z}^{\beta}$ in terms of modified vector fields through the identity \eqref{lifttomodif}, one has
$$ \sum_{|\beta| \leq N-2} \left\| r^{\frac{3}{2}} \int_v \frac{v^A}{v^0} \widehat{Z}^{\beta} f dv \right\|_{L^2(\Sigma_t)} \hspace{2mm} \lesssim \hspace{2mm} \sum_{z \in \mathbf{k}_1} \sum_{p \leq N-2} \sum_{\begin{subarray}{} |q|+|\kappa| \leq N-2 \\ \hspace{2mm} |q| \leq N-3 \end{subarray}} \left\| \sqrt{r} \int_v \left| P_{q,p}(\Phi) Y^{\kappa} f \right| dv \right\|_{L^2(\Sigma_t)}.$$ It then only remains to apply the previous proposition. \end{proof} The two following estimates are crucial as a weaker decay rate would prevent us to improve the bootstrap assumptions.
\begin{Pro}\label{crucial1}
Let $\beta$ and $\xi$ such that $|\xi|+|\beta| \leq N-1$. Then, for all $t \in [0,T[$, \begin{eqnarray}
\nonumber \left\| \sqrt{\tau_-} \int_v \left| P^X_{\xi}(\Phi) Y^{\beta} f \right| dv \right\|_{L^2(\Sigma_t)} & \lesssim & \epsilon \frac{1}{1+t} \hspace{5mm} \text{if} \hspace{5mm} |\beta| \leq N-3 \\ \nonumber & \lesssim & \epsilon \frac{\log^M(3+t)}{1+t} \hspace{5mm} \text{otherwise}. \end{eqnarray} \end{Pro} \begin{proof}
Suppose first that $|\beta| \leq N-3$. Then, by Proposition \ref{Xdecay}, \begin{eqnarray}
\nonumber \left\| \sqrt{\tau_-} \int_v \left| P^X_{\xi}(\Phi) Y^{\beta} f \right| dv \right\|^2_{L^2(\Sigma_t)} & \lesssim & \left\| \tau_- \int_v \left| Y^{\beta} f \right| dv \right\|_{L^{\infty}(\Sigma_t)} \left\| \int_v \left| P^X_{\xi}(\Phi)^2 Y^{\beta} f \right| dv \right\|_{L^1(\Sigma_t)} \\ \nonumber
& \lesssim & \frac{\epsilon}{(1+t)^2} \mathbb{E}^X_{N-1}[f](t) \hspace{2mm} \lesssim \hspace{2mm} \left| \frac{\epsilon}{1+t} \right|^2 . \end{eqnarray} Otherwise, \begin{itemize}
\item $|\beta| \geq N-2$, so $|\xi| \leq 1$ and then $|P^X_{\xi}(\Phi)| \lesssim \log^{\frac{3}{2}} (1+\tau_+)$ by Proposition \ref{Phi1}.
\item There exists $i \in \llbracket 1, |I_1| \rrbracket$ and $q \in \llbracket 1, |I| \rrbracket$ such that $Y^{\beta} f = H_i+G_i=H_i+L_q$. \end{itemize} Using Proposition \ref{estidecayH} (for the first estimate) and Propositions \ref{bilanG}, \ref{Inho1} (for the second one), we obtain \begin{eqnarray}
\nonumber \nonumber \left\| \sqrt{\tau_-} \int_v \left| P^X_{\xi}(\Phi) H_i \right| dv \right\|^2_{L^2(\Sigma_t)} & \lesssim & \left\| \tau_- \log^{3}(1+\tau_+) \int_v \left| H_i \right| dv \right\|_{L^{\infty}(\Sigma_t)} \left\| \int_v \left| H_i \right| dv \right\|_{L^1(\Sigma_t)} \\ \nonumber
& \lesssim & \left\| \epsilon \frac{\tau_- \log^{3+M_1}(1+\tau_+)}{\tau_+^2 \tau_-} \right\|_{L^{\infty}(\Sigma_t)} \mathbb{E}[H_i](t) \hspace{2mm} \lesssim \hspace{2mm} \epsilon^2 \frac{\log^{3+M_1}(3+t)}{(1+t)^2}, \\ \nonumber
\left\| \sqrt{\tau_-} \int_v \left| P^X_{\xi}(\Phi) L_q \right| dv \right\|^2_{L^2(\Sigma_t)} & = & \left\| \sqrt{\tau_-} \int_v \left| P^X_{\xi}(\Phi) K_q^j W_j \right| dv \right\|^2_{L^2(\Sigma_t)} \\ \nonumber
& \lesssim & \sum_{j=0}^{N_1} \left\| \tau_- \log^{3}(1+\tau_+) \int_v \left| W_j \right| dv \right\|_{L^{\infty}(\Sigma_t)} \left\| \int_v \left|K_q^j \right|^2 |W_j| dv \right\|_{L^1(\Sigma_t)} \\ \nonumber & \lesssim & \epsilon \frac{\log^{3+3N+M_1}(3+t)}{(1+t)^2} \epsilon \log^{M_0+4[q]}(3+t) \hspace{2mm} \lesssim \hspace{2mm} \epsilon^2 \frac{\log^{M_0+M_1+3N+3}(3+t)}{(1+t)^2}, \end{eqnarray} since $[q]=0$. This concludes the proof if $M$ is choosen such that\footnote{Recall from Remark \ref{estiPkp} that $M_1$ is independent of $M$.} $2M \geq M_0+M_1+3N+3$. \end{proof} The following estimates will be needed for the top order energy norm. As it will be used combined with Proposition \ref{CommuFsimple}, the quantity $P_{q,p}(\Phi)$ will contain $\mathbb{Y}_X$ derivatives of $\Phi$. \begin{Pro}\label{crucial2}
Let $\beta$, $q$ and $p$ be such as $|q|+|\beta| \leq N$, $|q| \leq N-1$ and $p \leq q_X+\beta_T$. Then, for all $t \in [0,T[$,
$$\left\| \sqrt{\tau_-} \int_v \left| P_{q,p}(\Phi) Y^{\beta} f \right| dv \right\|_{L^2(\Sigma^0_t)} \lesssim \frac{\epsilon}{(1+t)^{1-\frac{\eta}{2}}}.$$ \end{Pro} \begin{proof}
We consider various cases and, except for the last one, the estimates are clearly not sharp. Let us suppose first that $|\beta| \geq N-2$. Then $|q| \leq 2$ and $|P_{q,p}(\Phi)| \lesssim \log^{M_1}(3+t)$ on $\Sigma^0_t$ by Remark \ref{estiPkp}, so that, using Proposition \ref{crucial1},
$$ \left\| \sqrt{\tau_-} \int_v \left| P_{k,p}(\Phi) Y^{\beta} f \right| dv \right\|_{L^2(\Sigma^0_t)} \hspace{1mm} \lesssim \hspace{1mm} \log^{M_1}(3+t) \left\| \sqrt{\tau_-} \int_v \left| Y^{\beta} f \right| dv \right\|_{L^2(\Sigma^0_t)} \hspace{1mm} \lesssim \hspace{1mm} \epsilon \frac{\log^{M+M_1}(3+t)}{1+t}.$$
Let us write $P_{q,p}(\Phi)=\Phi^r P_{\xi}(\Phi)$ with $r \leq p$ and $(\xi_T,\xi_P,\xi_X)=(q_T,q_P,q_X)$. If $|\beta| \leq N-3$ and $|q| \leq N-2$, then by the Cauchy-Schwarz inequality (in $v$), \eqref{Auxenergy} as well as Propositions \ref{Phi1} and \ref{Xdecay}, \begin{eqnarray}
\nonumber \left\| \sqrt{\tau_-} \int_v \left| P_{k,p}(\Phi) Y^{\beta} f \right| dv \right\|^2_{L^2(\Sigma_t)} & \lesssim & \left\| \tau_- \int_v \left| \Phi^{2r} Y^{\beta} f \right| dv \right\|_{L^{\infty}(\Sigma_t)} \left\| \int_v \left| P_{\xi}(\Phi)^2 Y^{\beta} f \right| dv \right\|_{L^1(\Sigma_t)} \\ \nonumber
& \lesssim & \left\|\tau_-\frac{\epsilon \log^{4r}(1+\tau_+)}{\tau_+^2\tau_-} \right\|_{L^{\infty}(\Sigma_t)}\mathbb{A}[f](t) \hspace{2mm} \lesssim \hspace{2mm} \epsilon^2 \frac{\log^{8N}(3+t)}{(1+t)^{2-\frac{3}{4}\eta}}. \end{eqnarray}
The remaining case is the one where $|q|=N-1$ and $|\beta| \leq 1$. Hence, $p \leq k_X+1$. \begin{itemize}
\item If $p \geq 2$, we have $k_X \geq 1$ and then, schematically, $P_{\xi}(\Phi)=P^X_{\xi^1}(\Phi) P_{\xi^2}(\Phi)$, with $|\xi^1| \geq 1$ and $|\xi^1|+|\xi^2| = N-1$. If $|\xi^2| \geq 1$, we have $\min(|\xi^1|, |\xi^2|) \leq \frac{N-1}{2} \leq N-6$ and one of the two factor can be estimated pointwise, which put us in the context of the case $|k| \leq N-2$ and $|\beta| \leq N-3$. Otherwise, $P_{k,p}(\Phi)= \Phi^r P^X_{\xi^1}(\Phi)$ and, using again \eqref{Auxenergy}, \begin{eqnarray}
\nonumber \left\| \sqrt{\tau_-} \int_v \left| P_{k,p}(\Phi) Y^{\beta} f \right| dv \right\|^2_{L^2(\Sigma_t)} & \lesssim & \left\| \tau_- \int_v \left| \Phi^{2r} Y^{\beta} f \right| dv \right\|_{L^{\infty}(\Sigma_t)} \left\| \int_v \left| P^X_{\xi^1}(\Phi)^2 Y^{\beta} f \right| dv \right\|_{L^1(\Sigma_t)} \\ \nonumber
& \lesssim & \left\|\tau_-\frac{\epsilon \log^{4r}(1+\tau_+)}{\tau_+^2\tau_-} \right\|_{L^{\infty}(\Sigma_t)} \mathbb{A}[f](t) \hspace{2mm} \lesssim \hspace{2mm} \epsilon^2 \frac{\log^{8N}(3+t)}{(1+t)^{2-\frac{3}{4} \eta} }. \end{eqnarray} \item If $p=1$, we have $P_{k,p}(\Phi)=Y^{\kappa} \Phi$ and, using $\overline{\mathbb{E}}_N[f](t) \leq 4 \epsilon (1+s)^{\eta}$, \begin{eqnarray}
\nonumber \left\| \sqrt{\tau_-} \int_v \left| P_{k,p}(\Phi) Y^{\beta} f \right| dv \right\|^2_{L^2(\Sigma_t)} & \lesssim & \left\| \tau_- \int_v \left| Y^{\beta} f \right| dv \right\|_{L^{\infty}(\Sigma_t)} \left\| \int_v \left| Y^{\kappa} \Phi \right|^2 \left| Y^{\beta} f \right| dv \right\|_{L^1(\Sigma_t)} \\ \nonumber
& \lesssim & \left\|\tau_-\frac{\epsilon}{\tau_+^2\tau_-} \right\|_{L^{\infty}(\Sigma_t)} \mathbb{E} \left[ \left| Y^{\kappa} \Phi \right|^2 Y^{\beta} f \right](t) \hspace{2mm} \lesssim \hspace{2mm} \epsilon^2 \frac{(1+t)^{\eta}}{(1+t)^2}. \end{eqnarray} \end{itemize} \end{proof}
\section{Improvement of the energy estimates of the electromagnetic field}\label{sec12} In order to take advantage of the null structure of the system, we start this section by a preparatory lemma. \begin{Lem}\label{calculGFener}
Let $G$ be a $2$-form and $g$ a function, both sufficiently regular and recall that $J(g)^{\nu}= \int_v \frac{v^{\nu}}{v^0} g dv$, $\left| \overline{S}^{L} \right| \lesssim \tau_+$ and $\left| \overline{S}^{\underline{L}} \right| \lesssim \tau_-$. Then, using several times Lemma \ref{weights1} and Remark \ref{rqweights1}, \begin{eqnarray}
\nonumber \left| G_{ 0 \nu} J(g)^{\nu} \right| & \lesssim & |\rho|\int_v |g|dv+(|\alpha_A|+|\underline{\alpha}_A| )\int_v \frac{|v^A|}{v^0}|g|dv \hspace{1.5mm} \lesssim \hspace{1.5mm} |\rho|\int_v |g|dv+\frac{1}{\tau_+}\sum_{w \in \mathbf{k}_1}(|\alpha|+|\underline{\alpha}| )\int_v |wg|dv, \\
\nonumber \left|\overline{S}^{\mu} G_{ \mu \nu} J(g)^{\nu} \right| & \lesssim & \tau_+ |\rho| \int_v \frac{v^{\underline{L}}}{v^0} |g|dv+\tau_- |\rho| \int_v \frac{v^L}{v^0} |g|dv+\tau_+|\alpha| \int_v \frac{|v^A|}{v^0} |g| dv+\tau_-|\underline{\alpha}| \int_v \frac{|v^A|}{v^0}|g| dv \\ \nonumber
& \lesssim & \left( |\alpha|+|\rho|+ \frac{\tau_-}{\tau_+}|\underline{\alpha}| \right) \sum_{z \in \mathbf{k}_1} \int_v |z g | dv \hspace{1cm} \text{if $|x| \geq t$},\\ \nonumber
& \lesssim & |\rho| \int_v \left( \tau_-+\sum_{z \in \mathbf{k}_1} |z| \right) |g| dv+\left( |\alpha|+ \frac{\tau_-}{\tau_+}|\underline{\alpha}| \right) \int_v \sum_{z \in \mathbf{k}_1} |z g | dv \hspace{1cm} \text{otherwise}. \end{eqnarray} \end{Lem} We are now ready to improve the bootstrap assumptions concerning the electromagnetic field. \subsection{For $\mathcal{E}^0_N[F]$} Using Proposition \ref{energyMax1} and commutation formula of Proposition \ref{CommuFsimple}, we have, for all $t \in [0,T]$, \begin{equation}\label{28:eq}
\mathcal{E}^0_N[F](t)-2\mathcal{E}^0_N[F](0) \lesssim \sum_{|\gamma| \leq N} \sum_{\begin{subarray}{l} p \leq |k|+|\beta| \leq N \\ \hspace{3mm} |k| \leq N-1 \end{subarray}} \int_0^t \int_{\Sigma_s} |\mathcal{L}_{Z^{\gamma}}(F)_{\mu 0} J(P_{k,p}(\Phi) Y^{\beta} f)^{\mu} | dx ds. \end{equation}
We fix $|k|+|\beta| \leq N$, $p \leq N$ and $|\gamma| \leq N$. Denoting the null decomposition of $\mathcal{L}_{Z^{\gamma}}(F)$ by $(\alpha, \underline{\alpha}, \rho, \sigma)$, $P_{k,p}(\Phi) Y^{\beta} f$ by $g$ and applying Lemma \ref{calculGFener}, one has
$$ \int_0^t \int_{\Sigma_s} |\mathcal{L}_{Z^{\gamma}}(F)_{\mu 0} J(P_{k,p}(\Phi) Y^{\beta} f)^{\mu} | dx ds \lesssim \int_0^t \int_{\Sigma_s} |\rho| \int_v |g| dv +\left( |\alpha|+|\underline{\alpha}| \right) \sum_{w \in \mathbf{k}_1} \frac{1}{\tau_+} \int_v \left|w g \right|dv dx ds.$$ On the one hand, using Proposition \ref{estiL2}, \begin{eqnarray}
\nonumber \sum_{w \in \mathbf{k}_1} \int_0^t \int_{\Sigma_s} \left( |\alpha|+|\underline{\alpha}| \right) \int_v \frac{1}{\tau_+} \left|w g \right|dv dx ds & \lesssim & \sum_{w \in \mathbf{k}_1} \int_0^t \sqrt{\mathcal{E}^0_N[F](s)} \left\| \frac{1}{\tau_+} \int_v |wg| dv \right\|_{L^2(\Sigma_s)} ds \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}}. \end{eqnarray} On the other hand, as $\rho = \rho (\mathcal{L}_{Z^{\gamma}}(\widetilde{F})) +\rho ( \mathcal{L}_{Z^{\gamma}}(\overline{F}))$ and $\rho ( \mathcal{L}_{Z^{\gamma}}(\overline{F})) \lesssim \epsilon \tau_+^{-2}$ , we have, using Proposition \ref{estiL2} and the bootstrap assumptions \eqref{bootf3}, \eqref{bootext} and \eqref{bootF4}, \begin{eqnarray}
\nonumber \int_0^t \int_{\Sigma_s} |\rho| \int_v |g| dv dx ds & \lesssim & \int_0^t \left\| \sqrt{\tau_+} \rho ( \mathcal{L}_{Z^{\gamma}}(\widetilde{F})) \right\|_{L^2(\Sigma_s)} \left\| \frac{1}{\sqrt{\tau_+}} \int_v |g| dv \right\|_{L^2(\Sigma_s)}+\int_{\Sigma_s} \rho ( \mathcal{L}_{Z^{\gamma}}(\overline{F})) \int_v |g| dv dx ds \\ \nonumber & \lesssim & \int_0^t \sqrt{\mathcal{E}^{Ext}_N[\widetilde{F}](s)+\mathcal{E}_N[F](s)} \frac{\epsilon }{(1+s)^{\frac{5}{4}}}ds+ \int_0^t \frac{\epsilon}{(1+s)^2} \mathbb{E}[g](s) ds \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}. \end{eqnarray} The right-hand side of \eqref{28:eq} is then bounded by $\epsilon^{\frac{3}{2}}$, implying that $\mathcal{E}^0_N[f] \leq 3\epsilon$ on $[0,T[$ if $\epsilon$ is small enough.
\subsection{The weighted norm for the exterior region}
Applying Proposition \ref{energyMax1} and using $\mathcal{E}^{Ext}_N[\widetilde{F}](0) \leq \epsilon$ as well as $\widetilde{F}=F-\overline{F}$, we have, for all $t \in [0,T[$,
$$ \mathcal{E}_N^{Ext}[\widetilde{F}](t) \leq 6\epsilon+\sum_{|\gamma| \leq N} \int_0^t \int_{\overline{\Sigma}^0_s} \left| \overline{S}^{\mu} \mathcal{L}_{Z^{\gamma}}(\widetilde{F})_{\mu \nu} \nabla^{\lambda} {\mathcal{L}_{Z^{\gamma}}(F)_{\lambda} }^{ \nu} \right| dx ds+\int_0^t \int_{\overline{\Sigma}^0_s} \left| \overline{S}^{\mu} \mathcal{L}_{Z^{\gamma}}(\widetilde{F})_{\mu \nu} \nabla^{\lambda} {\mathcal{L}_{Z^{\gamma}}(\overline{F})_{\lambda} }^{ \nu} \right| dx ds .$$
Let us fix $|\gamma| \leq N$ and denote the null decomposition of $\mathcal{L}_{Z^{\gamma}}(\widetilde{F})$ by $(\alpha, \underline{\alpha}, \rho , \sigma)$. As previously, using Proposition \ref{CommuFsimple},
$$\int_0^t \int_{\overline{\Sigma}^0_s} \left| \overline{S}^{\mu} \mathcal{L}_{Z^{\gamma}}(\widetilde{F})_{\mu \nu} \nabla^{\lambda} {\mathcal{L}_{Z^{\gamma}}(F)_{\lambda} }^{ \nu} \right| dx ds \lesssim \sum_{\begin{subarray}{l} p \leq |k|+|\beta| \leq |\gamma| \\ \hspace{3mm} |k| \leq |\gamma|-1 \end{subarray}} \int_0^t \int_{\overline{\Sigma}^0_s} |\overline{S}^{\mu} \mathcal{L}_{Z^{\gamma}}(\widetilde{F})_{\mu \nu} J(P_{k,p}(\Phi) Y^{\beta} f)^{\nu} | dx ds.$$
We fix $|k|+|\beta| \leq N$, $p \leq N$ and $|\gamma| \leq N$ and we denote again $P_{k,p}(\Phi) Y^{\beta} f$ by $g$. Using successively Lemma \ref{calculGFener}, the Cauchy-Schwarz inequality, the bootstrap assumption \eqref{bootext} and Proposition \ref{estiL2}, we obtain \begin{eqnarray}
\nonumber \int_0^t \int_{\overline{\Sigma}^0_s} |\overline{S}^{\mu} \mathcal{L}_{Z^{\gamma}}(\widetilde{F})_{\mu \nu} J(P_{k,p}(\Phi) Y^{\beta} f)^{\nu} | dx ds & \lesssim & \int_0^t \int_{\Sigma_s} \left(|\rho|+ |\alpha|+\frac{\sqrt{\tau_-}}{\sqrt{\tau_+}}|\underline{\alpha}| \right) \sum_{w \in \mathbf{k}_1} \int_v \left|w g \right|dv dx ds. \\ \nonumber
& \lesssim & \sum_{w \in V} \int_0^t \sqrt{\mathcal{E}_N^{Ext}[F](s)} \left\| \frac{1}{\sqrt{\tau+}} \int_v \left|w g \right|dv \right\|_{L^2(\Sigma_s)} ds \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}} \int_0^{+\infty} \frac{ds}{(1+s)^{\frac{5}{4}}} \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}. \end{eqnarray}
Using Proposition \ref{propcharge} and iterating commutation formula of Proposition \ref{basiccom}, we have,
$$\tau_+^2\left| \nabla^{\mu} {\mathcal{L}_{Z^{\gamma}}(\overline{F})_{\mu} }^{ L} \right|(t,x) +\tau_+^4\left| \nabla^{\mu} {\mathcal{L}_{Z^{\gamma}}(\overline{F})_{\mu} }^{ \underline{L}} \right|(t,x) + \tau_+^3 \left| \nabla^{\mu} {\mathcal{L}_{Z^{\gamma}}(\overline{F})_{\mu} }^{ A} \right|(t,x) \lesssim |Q(F)| \mathds{1}_{-2 \leq t-|x| \leq -1}(t,x).$$
Consequently, as $|Q(F)| \leq \| f_0 \|_{L^1_{x,v}} \leq \epsilon$, $\left| \overline{S}^L \right| \lesssim \tau_+$ and $\left| \overline{S}^{\underline{L}} \right| \lesssim \tau_-$,
$$|\overline{S}^{\mu} \mathcal{L}_{Z^{\gamma}}(\widetilde{F})_{\mu \nu} \nabla^{\lambda} {\mathcal{L}_{Z^{\gamma}}(\overline{F})_{\lambda} }^{\nu}| \lesssim \left( \tau_+ |\rho |\frac{\epsilon}{\tau_+^4}+\tau_- |\rho |\frac{\epsilon}{\tau_+^2}+\tau_+ |\alpha |\frac{\epsilon}{\tau_+^3}+\tau_- |\underline{\alpha} | \frac{\epsilon}{\tau_+^3} \right) \mathds{1}_{-2 \leq t-|x| \leq -1}(t,x).$$
Note now that $\tau_- \mathds{1}_{-2 \leq t-|x| \leq -1} \leq \sqrt{5}$, so that, using the bootstrap assumption \eqref{bootext} and the Cauchy-Schwarz inequality, \begin{eqnarray}
\nonumber \int_0^t \int_{\overline{\Sigma}^0_s} \left| S^{\mu} \mathcal{L}_{Z^{\gamma}}(\widetilde{F})_{\mu \nu} \nabla^{\lambda} {\mathcal{L}_{Z^{\gamma}}(\overline{F})_{\lambda} }^{ \nu} \right| dx ds & \lesssim & \int_0^t \frac{\epsilon}{(1+s)^{\frac{5}{2}}} \int_{s+1 \leq |x| \leq s+2} \sqrt{\tau_+}|\rho|+\sqrt{\tau_+}|\alpha|+|\underline{\alpha}| dx ds \\ \nonumber & \lesssim & \int_0^t \frac{\epsilon}{(1+s)^{\frac{5}{2}}} \sqrt{\mathcal{E}_N^{Ext}[\widetilde{F}](s)} \sqrt{s^2+1} ds \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}. \end{eqnarray} Thus, if $\epsilon$ is small enough, we obtain $\mathcal{E}^{Ext}_N[\widetilde{F}] \leq 7 \epsilon$ on $[0,T[$ which improves the bootstrap assumption \eqref{bootext}.
\subsection{The weighted norms for the interior region}
Recall from Proposition \ref{energyMax1} that we have, for $Q \in \{ N-3, N-1, N \}$ and $t \in [0,T[$, \begin{equation}\label{ensource}
\mathcal{E}_Q[F](t) \hspace{2mm} \leq \hspace{2mm} 24\epsilon +\sum_{|\gamma| \leq Q} \int_0^t \int_{\Sigma^0_s} \left| \overline{S}^{\mu} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \nabla^{\lambda} {\mathcal{L}_{Z^{\gamma}}(F)_{\lambda} }^{ \nu} \right| dx ds, \end{equation} since $\mathcal{E}^{Ext}_{N}[\widetilde{F}] \leq 8\epsilon$ on $[0,T[$ by the bootstrap assumption \eqref{bootext}). The remainder of this subsection is divided in two parts. We consider first $Q \in \{ N-3, N-1 \}$ and we end with $Q=N$ as we need to use in that case a worst commutation formula in order to avoid derivatives of $\Phi$ of order $N$, which is the reason of the stronger loss on the top order energy norm.
\subsubsection{The lower order energy norms} Let $Q \in \{ N-3, N-1 \}$. According to commutation formula of Proposition \ref{ComuMaxN}, we can bound the last term of \eqref{ensource} by a linear combination of the following ones. \begin{eqnarray}
\hspace{-4mm} \mathcal{I}_1 \hspace{-1mm} & := & \hspace{-1mm} \int_0^t \int_{|x| \leq s} \left| \overline{S}^{\mu} \mathcal{L}_{ Z^{\gamma}}(F)_{\mu \nu} \int_v \frac{v^{\nu}}{v^0} P^X_{\xi}(\Phi) Y^{\beta} f dv \right| dx ds, \hspace{5mm} \text{with} \hspace{5mm} |\gamma| , \hspace{1mm} |\xi|+|\beta| \leq Q, \label{pevar} \\
\hspace{-4mm} \mathcal{I}_2 \hspace{-1mm} & := & \hspace{-1mm} \int_0^t \int_{|x| \leq s} \left| \overline{S}^{\mu} \mathcal{L}_{ Z^{\gamma}}(F)_{\mu \nu} \int_v \frac{z}{\tau_+} P_{k,p}( \Phi) Y^{\beta} f dv \right| dxds, \hspace{5mm} \text{with} \hspace{5mm} |\gamma| , \hspace{1mm} |k|+|\beta| \leq Q, \hspace{3mm} z \in \mathbf{k}_1, \label{pevarbis} \end{eqnarray}
$0 \leq \nu \leq 3$ and $ p \leq 3N$. Fix $|\gamma| \leq Q$ and denote the null decomposition of $\mathcal{L}_{Z^{\gamma}}(F)$ by $(\alpha, \underline{\alpha}, \rho, \sigma)$. We start by \eqref{pevarbis}, which can be estimated independently of $Q$. Recall that $\left| \overline{S}^{L} \right| \lesssim \tau_+$ and $\left| \overline{S}^{\underline{L}} \right| \lesssim \tau_-$, so that, using Proposition \ref{estiL2} and the bootstrap assumption \eqref{bootF3}, \begin{eqnarray}
\nonumber \mathcal{I}_2 & \lesssim & \int_0^t \int_{\Sigma^0_s} \left( \tau_+ |\rho|+\tau_+ |\alpha|+\tau_- |\underline{\alpha}| \right) \int_v \left| \frac{z}{\tau_+} P_{k,p}( \Phi) Y^{\beta} f \right| dvdxds \\ \nonumber
& \lesssim & \int_0^t \sqrt{\mathcal{E}_{N-1}[F](s)} \left\| \frac{1}{\sqrt{\tau_+}} \int_v \left| z P_{k,p}( \Phi) Y^{\beta} f \right| dv \right\|_{L^2(\Sigma_s)} ds \\ \nonumber & \lesssim & \epsilon^{\frac{3}{2}} \int_0^t \frac{\log^M(3+s)}{(1+s)^{\frac{5}{4}}} ds \hspace{2mm} \lesssim \hspace{2mm} \epsilon^{\frac{3}{2}}. \end{eqnarray}
We now turn on \eqref{pevar} and we then consider $|\xi|+|\beta| \leq Q$. Start by noticing that, by Lemma \ref{calculGFener},
$$ \left| \overline{S}^{\mu} \mathcal{L}_{ Z^{\gamma}}(F)_{\mu \nu} \hspace{-0.3mm} \int_v \frac{v^{\nu}}{v^0} P^X_{\xi}(\Phi) Y^{\beta} f \right| dv \lesssim \tau_-|\rho| \int_v \left| P^X_{\xi}(\Phi) Y^{\beta} f \right| dv+\left(|\rho|+ |\alpha|+\frac{\tau_-}{\tau_+} |\underline{\alpha}| \right) \hspace{-1mm} \sum_{w \in \mathbf{k}_1} \hspace{-0.5mm} \int_v \left| w P^X_{\xi}(\Phi) Y^{\beta} f \right| dv .$$ Consequently, by the bootstrap assumption \eqref{bootF3} and Proposition \ref{estiL2}, \begin{eqnarray}
\nonumber \mathcal{I}_1 & \lesssim & \int_0^t \sqrt{\mathcal{E}_{Q}[F](s)} \left( \left\| \sqrt{\tau_-} \int_v \left| P^X_{\xi}(\Phi) Y^{\beta} f \right| dv \right\|_{L^2(\Sigma_s)}+ \sum_{w \in \mathbf{k}_1} \left\| \frac{1}{\sqrt{\tau_+}} \int_v \left|w P^X_{\xi}(\Phi) Y^{\beta} f \right| dv \right\|_{L^2(\Sigma_s)} \right) ds \\ \nonumber
& \lesssim & \epsilon^{\frac{3}{2}} + \int_0^t \sqrt{\mathcal{E}_{Q}[F](s)} \left\| \sqrt{\tau_-} \int_v \left| P^X_{\xi}(\Phi) Y^{\beta} f \right| dv \right\|_{L^2(\Sigma_s)}ds. \end{eqnarray} The last integral to estimate is the source of the small growth of $\mathcal{E}_Q[F]$. We can bound it, using again the bootstrap assumptions \eqref{bootF2}, \eqref{bootF3} and Proposition \ref{crucial1}, by \begin{itemize} \item $\epsilon^{\frac{3}{2}} \log^2(3+t)$ if $Q=N-3$ and \item $\epsilon^{\frac{3}{2}} \log^{2M}(3+t)$ otherwise. \end{itemize} Hence, combining this with \eqref{ensource} we obtain, for $\epsilon$ small enough, that \begin{itemize} \item $\mathcal{E}_{N-3}[F](t) \leq 25 \epsilon \log^2(3+t)$ for all $t \in [0,T[$ and \item $\mathcal{E}_{N-1}[F](t) \leq 25 \epsilon \log^{2M}(3+t)$ for all $t \in [0,T[$. \end{itemize}
\subsubsection{The top order energy norm} We consider here the case $Q=N$ and we then apply this time the commutation formula of Proposition \ref{CommuFsimple}, so that the last term of \eqref{ensource} can be bounded by a linear combination of terms of the form
$$ \mathcal{I}:= \int_0^t \int_{\Sigma^{0}_s} \left| \overline{S}^{\mu} \mathcal{L}_{Z^{\gamma}}(F)_{\mu \nu} \int_v \frac{v^{\mu}}{v^0} P_{q,p}(\Phi) Y^{\beta} f dv \right| dx ds,$$
with $|\gamma| \leq N$, $|q|+|\beta| \leq N$, $|q| \leq N-1$ and $ p \leq q_X+\beta_T$. Let us fix such parameters. Following the computations made previously to estimate $\mathcal{I}_1$ and using $\mathcal{E}_{N}[F](s) \lesssim \sqrt{\epsilon} (1+s)^{\eta} \lesssim \sqrt{\epsilon} (1+s)^{\frac{1}{8}}$, we get \begin{eqnarray}
\nonumber \mathcal{I}_1 & \lesssim & \int_0^t \sqrt{\mathcal{E}_{N}[F](s)} \left( \left\| \sqrt{\tau_-} \int_v \left| P_{q,p}(\Phi) Y^{\beta} f \right| dv \right\|_{L^2(\Sigma^0_s)}+ \sum_{w \in \mathbf{k}_1} \left\| \frac{1}{\sqrt{\tau_+}} \int_v \left| w P_{q,p}(\Phi) Y^{\beta} f \right| dv \right\|_{L^2(\Sigma_s)} \right) ds \\
& \lesssim & \epsilon^{\frac{3}{2}} + \sqrt{\epsilon} \int_0^t (1+s)^{\frac{\eta}{2}} \left\| \sqrt{\tau_-} \int_v \left| P_{q,p}(\Phi) Y^{\beta} f \right| dv \right\|_{L^2(\Sigma^0_s)}ds. \label{lafin} \end{eqnarray} Applying now Proposition \ref{crucial2}, we can bound \eqref{lafin} by $\epsilon^{\frac{3}{2}}(1+t)^{\eta}$. Thus, if $\epsilon$ is small enough, we obtain $\mathcal{E}_N[F](t) \leq 25 \epsilon (1+t)^{\eta}$ for all $t \in [0,T[$, which concludes the improvement of the bootstrap assumption \eqref{bootF4} and then the proof.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\pagestyle{headings}
\title{Art Gallery Localization\thanks{This work was supported by NSERC.}}
\titlerunning{Art Gallery Localization}
\author{Prosenjit Bose\inst{1} \and Jean-Lou De Carufel\inst{2} \and Alina Shaikhet\inst{1} \and Michiel Smid\inst{1}}
\authorrunning{P. Bose, J.-L. De Carufel, A. Shaikhet and M. Smid}
\institute{School of Computer Science, Carleton University, Ottawa, Canada,\\ \email{\{jit, michiel\}@scs.carleton.ca, [email protected]},\\ \and School of Electrical Engineering and Computer Science, U. of Ottawa, Canada,\\ \email{[email protected]}}
\maketitle
\begin{abstract} We study the problem of placing a set $T$ of broadcast towers in a simple polygon $P$ in order for any point to locate itself in the interior of $P$. Let $V(p)$ denote the \emph{visibility polygon} of a point $p$, as the set of all points $q \in P$ that are visible to $p$. For any point $p \in P$: for each tower $t \in T \cap V(p)$ the point $p$ receives the coordinates of $t$ and the exact Euclidean distance between $t$ and $p$. From this information $p$ can determine its coordinates. We show a tower-positioning algorithm that computes such a set $T$ of size at most $\lfloor 2n/3\rfloor$, where $n$ is the size of $P$. This improves the previous upper bound of $\lfloor 8n/9\rfloor$ towers~\cite{DBLP:conf/cccg/DippelS15}. We also show that $\lfloor 2n/3\rfloor$ towers are sometimes necessary. \keywords{art gallery, trilateration, GPS, polygon partition, localization} \end{abstract}
\section{Introduction} \label{sec:introduction} \pdfbookmark[1]{Introduction}{sec:introduction}
The art gallery problem was introduced in 1973 when Victor Klee asked how many guards are sufficient to \emph{guard} the interior of a simple polygon having $n$ vertices. Although it has been shown by Chv{\'a}tal that $\lfloor n/3\rfloor$ guards are always sufficient and sometimes necessary~\cite{Chvatal197539}, and such a set of guards can be computed easily~\cite{Fisk1978374}, such solutions are usually far from optimal in terms of minimizing the number of guards for a particular input polygon. Moreover, it was shown that determining an optimal number of guards is NP-hard, even for simple polygons~\cite{Lee:1986:CCA:13643.13657}. Determining the actual locations of those guards is even harder~\cite{DBLP:conf/compgeom/AbrahamsenAM17}.
\emph{Trilateration} is the process of determining absolute or relative locations of points by measurement of distances, using the geometry of the environment. In addition to its interest as a geometric problem, trilateration has practical applications in surveying and navigation, including global positioning systems (GPS). Every GPS satellite transmits information about its position and the current time at regular intervals. These signals are intercepted by a GPS receiver, which calculates how far away each satellite is based on how long it took for the messages to arrive. GPS receivers take this information and use trilateration to calculate the user's location.
In our research we combine the art gallery problem with trilateration. We address the problem of placing broadcast \emph{towers} in a simple polygon $P$ in order for a point in $P$ (let us call it an \emph{agent}) to locate itself. Towers can be defined as points, which can transmit their coordinates together with a time stamp to other points in their visibility region. The agent receives messages from all the towers that belong to its visibility region. Given a message from the tower $t$, the agent can determine its distance to $t$. In our context, \emph{trilateration} is the process, during which the agent can determine its absolute coordinates from the messages the agent receives. Receiving a message from one tower only will not be sufficient for the agent to locate itself (unless the agent and the tower are at the the same location). In Euclidean plane, two distinct circles intersect in at most two points. If a point lies on two circles, then the circle centers and the two radii provide sufficient information to narrow the possible locations down to two. Additional information may narrow the possibilities down to one unique location.
In relation to GPS systems, towers can be viewed as GPS satellites, while agents (query points interior to the polygon) can be compared to GPS receivers. Naturally, we would like to minimize the number of towers.
Let $P$ be a simple polygon in general position (no three vertices are collinear) having a total of $n$ vertices on its boundary (denoted by $\partial P$). Note that $\partial P \subset P$. Two points $u, v \in P$ are \emph{visible to each other} if the segment $\overline{uv}$ is contained in $P$. We also say that $u$ \emph{sees} $v$. Note that $\overline{uv}$ may touch $\partial P$ in one or more points. For $u \in P$, we let $V(u)$ denote the \emph{visibility polygon} of $u$, as the set of all points $q \in P$ that are visible to $u$. Notice that $V(u)$ is a star-shaped polygon contained in $P$ and $u$ belongs to its \emph{kernel} (the set of points from which all of $V(u)$ is visible).
\textbf{Problem Definition:} Let $T$ be a set of points (called \emph{towers}) in $P$ satisfying the following properties. For any point $p \in P$: for each $t \in T \cap V(p)$, the point $p$ receives the coordinates of $t$ and can compute the Euclidean distance between $t$ and $p$, denoted $d(t,p)$. From this information, $p$ can determine its coordinates. We consider the following problems: \begin{enumerate} \item Design an algorithm that, on any input polygon $P$ in general position, computes a ``small'' set $T$ of towers. \item Design a localization algorithm. \end{enumerate}
We show how to compute such a set $T$ of size at most $\lfloor 2n/3\rfloor$ by using the polygon partition method introduced by T{\'o}th~\cite{Toth2000121}. T{\'o}th showed that any simple polygon with $n$ vertices can be guarded by $\lfloor n/3\rfloor$ point guards whose range of vision is $180^\circ$. T{\'o}th partitions a polygon into subpolygons, on which he then can apply induction. He cuts along diagonals whenever it is possible, otherwise he cuts along a continuation of some edge of $P$; along a two-line segment made of an extension of two edges of $P$ that intersect inside $P$; or along the bisector of a reflex vertex of $P$. Notice that the three latter types of cuts may introduce new vertices that are not necessarily in general position with the given set of vertices. Succeeding partitions of the subpolygons may create polygons that are not simple. However, T{\'o}th assumed that his partition method creates subpolygons whose vertices are in general position (refer to~\cite{Toth2000121} Section $2$). We lift this assumption and show how to adapt his method to a wider range of polygons, which we define in Section~\ref{sec:partition}. Under the new conditions the partition of $P$ may contain star-shaped polygons whose kernel is a single point. It does not pose an obstacle to T{\'o}th's problem, but it is a severe complication to our problem, because we require a pair of distinct towers in the kernel of each polygon of the partition. We modify T{\'o}th's partition method and show how to use it with respect to our problem. It is important to notice that we assume that the input polygon is in general position, while non-general position may occur for subpolygons of the partition (refer to Definition~\ref{def:polygon}). We show that after the modification each $180^\circ$-guard $g$ can be replaced with a pair of towers \textbf{close} to $g$. We embed the orientation of the $180^\circ$-guard into the coordinates of the towers. That is, we specify to which side of the line $L$ through the pair of towers their primary localization region resides. We do it by positioning the towers at a distance that is an exact rational number. The parity of the numerator of the rational number (in the reduced form) defines which one of the two half-planes (defined by $L$) the pair of towers are responsible for. We call it the \emph{parity trick}. For example, if we want a pair $t_1$, $t_2$ of towers to be responsible for the half-plane to the left of the line through $t_1$ and $t_2$, then we position the towers at a distance, which is a reduced rational number whose numerator is even. The localization algorithm is allowed to use this information.
Our interest in this problem started with the paper by Dippel and Sundaram~\cite{DBLP:conf/cccg/DippelS15}.
They provide the first non-trivial bounds on agent localization in simple polygons, by showing that $\lfloor 8n/9\rfloor$ towers suffice for any non-degenerate polygon of $n$ vertices, and present an $O(n \log n)$ algorithm for the corresponding placement. Their approach is to decompose the polygon into at most $\lfloor n/3\rfloor$ fans. A polygon $P'$ is a \emph{fan} if there exist a vertex $u$, such that for every other vertex $v$ not adjacent to $u$, $\overline{u v}$ is a diagonal of $P'$; the vertex $u$ is called the \emph{center} of the fan. In each fan with fewer than $4$ triangles Dippel and Sundaram position a pair of towers on an edge of the fan; every fan with $4$ or more triangles receives a triple of towers in its kernel. In a classical trilateration, the algorithm for locating an agent knows the coordinates of the towers that can see $p$ together with distances between $p$ and the corresponding towers. However, the localization algorithm presented in~\cite{DBLP:conf/cccg/DippelS15} requires a lot of additional information, such as a complete information about the polygon, its decomposition into fans and the coordinates of \textbf{all} towers.
Our localization algorithm has no information about $P$. It receives as input only the coordinates of the towers that can see $p$ together with their distances to $p$. In addition our algorithm is empowered by the knowledge of the parity trick. When only a pair $t_1$, $t_2$ of towers can see $p$ then the coordinates of the towers together with the distances $d(t_1,p)$ and $d(t_2,p)$ provide sufficient information to narrow the possible locations of $p$ down to two. Refer to Figures~\ref{fig:example2},\ref{fig:example3}. Those two locations are reflections of each other over the line through $t_1$ and $t_2$. In this situation our localization algorithm uses the parity trick. It calculates the distance between the two towers and judging by the parity of this number decides which of the two possible locations is the correct position of $p$.
We show how to position at most $\lfloor 2n/3\rfloor$ towers inside $P$, which is an improvement over the previous upper bound in~\cite{DBLP:conf/cccg/DippelS15}. We also show that $\lfloor 2n/3\rfloor$ towers are sometimes necessary. The comb polygon from the original art gallery problem can be used to show a lower bound. Refer to Fig.~\ref{fig:comb}. No point in the comb can see two different comb spikes. Thus we need at least two towers per spike to localize all of the points in its interior. In addition we need to know the parity trick. Or, alternatively, we need to know $P$, its exact location and orientation. We show in Theorem~\ref{thm:no_map} that without any additional information (such as the parity trick or the complete knowledge about $P$ including its partition) it is \textbf{not} possible to localize an agent in a simple $n$-gon (where $n = 3k + q$, for integer $k \geq 1$ and $q = 0$, $1$ or $2$) with less than $n - q$ towers.
In Section~\ref{sec:preliminaries} we give basic definitions and present some properties and observations. Section~\ref{sec:partition} shows some of our modifications of the polygon partition given by T{\'o}th~\cite{Toth2000121} and its adaptation to our problem. In Section~\ref{sec:localization} we present a localization algorithm.
\section{Preliminaries} \label{sec:preliminaries} \pdfbookmark[1]{Preliminaries}{sec:preliminaries}
Consider a point (an agent) $p$ in the interior of $P$, whose location is unknown. Let $C(x,r)$ denote the circle centered at a point $x$ with radius $r$. If only one tower $t$ can see $p$ then $p$ can be anywhere on $C(t,d(p,t)) \cap V(t)$ (refer to Fig.~\ref{fig:example1}), which may not be enough to identify the location of~$p$. By the \emph{map} of $P$ we mean the complete information about $P$ including the coordinates of all the vertices of $P$ and the vertex adjacency list. Notice that one must know the map of $P$ to calculate $V(t)$. If two towers $t_1$ and $t_2$ can see $p$ then the location of $p$ can be narrowed down to at most two points $C(t_1,d(p,t_1)) \cap C(t_2,d(p,t_2)) \cap V(t_1) \cap V(t_2)$. Refer to Fig.~\ref{fig:example2}. The two points are reflections of each other over the line through $t_1$ and $t_2$. We call this situation the \emph{ambiguity along the line}, because without any additional information we do not know to which one of the two locations $p$ belongs. To avoid this uncertainty we can place both towers on the same edge of $P$. Consider for example Fig.~\ref{fig:example3} where two towers are placed on the line segment $kernel(P) \cap \partial P$. In this example, if the map of $P$ is known (and thus we know $V(t_1)$ and $V(t_2)$) then the intersection $C(t_1,d(p,t_1)) \cap C(t_2,d(p,t_2)) \cap V(t_1) \cap V(t_2)$ is a single point (highlighted in red). Alternatively, if the map of $P$ is unknown, we can place a triple of non-collinear towers in the kernel of $P$ (highlighted in cyan on Fig.~\ref{fig:example3}) to localize any point interior to $P$.
\begin{figure}
\caption{Trilateration example. \textbf{(a)} The point $p$ can be anywhere on $C(t,d(p,t)) \cap V(t)$, which is highlighted in red. \textbf{(b)} Ambiguity along the line $L(t_1, t_2)$. \textbf{(c)} If the map of $P$ is known then the location of $p$ can be identified precisely. The kernel of $P$ is highlighted in cyan.}
\label{fig:example1}
\label{fig:example2}
\label{fig:example3}
\label{fig:example}
\end{figure}
For a simple polygon $P$ in general position, we can partition it into star-shaped polygons $P_1, P_2, \ldots P_l$ such that $kernel(P_i)$, for every $1 \leq i \leq l$, does not degenerate into a single point. In every $P_i$ ($1 \leq i \leq l$) we can position a pair of towers on a line segment in $kernel(P_i) \cap \partial P_i$ (such that the towers belong to the same edge of $P_i$) or a triple of towers in $kernel(P_i)$ if $kernel(P_i) \cap \partial P_i$ is empty or contains a single point. Notice that a pair of towers positioned on the edge of $P_i$ will not necessarily be on the boundary of $P$. Thus, to localize an agent, it is not enough to know the map of $P$. We need to know more, for example, if in addition to the map of $P$ we know the partition of $P$ into star-shaped polygons and which pair of towers is responsible for which subpolygon then the agent can be localized.
In our solution we do not use this extra information or the map of $P$. Moreover, to get a tight bound of $\lfloor 2n/3\rfloor$ towers, we abstain from placing a triple of towers per subpolygon, since some polygons cannot be partitioned into less than $\lfloor n/3\rfloor$ star-shaped subpolygons. The idea is to use a \emph{parity trick}.
\textbf{Parity trick:} Let $L(u, v)$ be the line through points $u$ and $v$. Let $L(u, v)^+$ denote the half plane to the left of $L(u,v)$ (or above, if $L$ is horizontal). Similarly, $L(u, v)^-$ denotes the half plane to the right (or below) of $L$. We embed information about the primary orientation of the pair of towers into their coordinates. If we want a pair $t_1$, $t_2$ of towers to be responsible for $L(t_1, t_2 )^+$ (respectively $L(t_1, t_2 )^-$), then we position the towers at a distance which is a reduced rational number whose numerator is even (respectively odd). In this way, we specify on which side of $L(t_1 , t_2 )$ the primary localization region of $t_1$ and $t_2$ resides. Refer to Sect~\ref{subsec:PartitionAlg} where the parity trick is explained in greater detail.
To achieve localization with at most $\lfloor 2n/3\rfloor$ towers we should partition $P$ into at most $\lfloor n/3\rfloor$ star-shaped polygons $P_1,\ldots, P_l$ such that there exists a line segment $\overline{uv} \in kernel(P_i) \cap \partial P_i$ such that $P_i \in L(u, v )^+$ or $P_i \in L(u, v )^-$ for every $1 \leq i \leq l$ (we assume that $u$ and $v$ are distinct points).
\begin{theorem}[Chv{\'a}tal's Theorem~\cite{Chvatal197539}] \label{thm:chvatal} Every triangulation of a polygon with $n$ vertices can be partitioned into $m$ fans where $m \leq \lfloor n/3\rfloor$. \end{theorem}
The statement of the following lemma may seem trivial, still we provide its proof for completeness.
\begin{lemma} \label{lem:pentagon} Any simple polygon $P$ with $3$, $4$ or $5$ sides is star-shaped and its kernel contains a boundary segment that is not a single point. \end{lemma} \begin{proof} Let $n$ be the number of vertices of $P = v_1 v_2 \ldots v_n$. By Theorem~\ref{thm:chvatal}, $P$ can be partitioned into $\lfloor n/3\rfloor = 1$ fans (since $n = 3$, $4$ or $5$). Notice that a fan is star-shaped by definition, from which $P$ is star-shaped. The kernel of $P$ is an intersection of $n$ half-planes defined by the edges of $P$. Let $r$ be the number of reflex angles of $P$. There are three cases to consider: \begin{enumerate} \item $r = 0$: $P$ is a convex polygon and thus $kernel(P) = P$, implying $\partial P \in kernel(P)$. \item $r = 1$: Let $v_1$ be the vertex of $P$ at the reflex angle. Refer to Fig.~\ref{fig:KernelBoundary1} and~\ref{fig:KernelBoundary2} for possible polygons. The angles at $v_2$ and $v_n$ are smaller than $\pi$ by the simplicity of $P$. The claim of the lemma then is implied by inspection. Consider the edge $\overline{v_3 v_4}$. It contains a line segment that belongs to $kernel(P)$. \begin{figure}
\caption{The kernel of $P$ is highlighted in cyan.}
\label{fig:KernelBoundary1}
\label{fig:KernelBoundary2}
\label{fig:KernelBoundary3}
\label{fig:KernelBoundary4}
\label{fig:KernelBoundary}
\end{figure} \item $r = 2$: In this case $n = 5$. Let $v_1$ be the vertex of $P$ at one of the reflex angles. Refer to Fig.~\ref{fig:KernelBoundary3} and~\ref{fig:KernelBoundary4} for possible polygons. The vertex at the other reflex angle is either adjacent to $v_1$ or not. Consider first the case where the vertices at the reflex angles of $P$ are not adjacent. Without loss of generality let $v_4$ be a vertex of $P$ at reflex angle (refer to Fig.~\ref{fig:KernelBoundary3}). The angles at $v_2$, $v_3$ and $v_5$ are smaller than $\pi$ by the simplicity of $P$. It follows that the edge $\overline{v_2 v_3}$ must contain a line segment that belongs to $kernel(P)$. Consider now the case where the vertices at the reflex angles of $P$ are adjacent. Without loss of generality let $v_5$ be a vertex at reflex angle of $P$ (refer to Fig.~\ref{fig:KernelBoundary4}). The claim of the lemma holds similarly to the discussion presented in the case for $r = 1$, Fig.~\ref{fig:KernelBoundary1}. \end{enumerate} \qed \end{proof}
The problem we study is twofold: \begin{enumerate} \item We are given a simple polygon $P$ of size $n$. Our goal is to position at most $\lfloor 2n/3\rfloor$ towers inside $P$ such that every point $p \in P$ can be localized. \item We want to design a localization algorithm which does not know $P$, but knows that the locations of the towers were computed using the parity trick. For any point $p \in P$, its location can be found by using the coordinates of the towers that \textbf{see} $p$ and the distances from those towers to $p$. \end{enumerate}
It may seem counter-intuitive, but the knowledge of the parity trick is \textbf{stronger} than the knowledge of the map of $P$. Some towers (while still on the boundary of some subpolygon of the partition) may end up in the interior of $P$. This is not a problem when the parity trick is used but may lead to ambiguities when only the map of $P$ is known (refer for example to Fig.~\ref{fig:example2}).
The following theorem shows that additional information like the parity trick or the map of $P$ (including its partition) is necessary to achieve localization with the use of less than $n$ towers.
\begin{theorem} \label{thm:no_map}
Let $P$ be a simple polygon with $n$ vertices. An agent cannot localize itself inside $P$ when less than $n-(n\bmod 3)$ towers are used if the {\em only} information available to the localization algorithm are the coordinates of the towers and the distances of the agent to the towers visible to it. \end{theorem} \begin{proof} Let $P$ be an arbitrary simple polygon with $n=3$ vertices. Assume to the contrary, that $P$ can be trilaterated with $2$ towers. Given the coordinates of the two towers $t_1$ and $t_2$ together with the distances to a query point $p$ one can deduce that $p$ is in one of the two possible locations $C(t_1,d(p,t_1)) \cap C(t_2,d(p,t_2)) = \{p_1, p_2\}$. But without additional information it is impossible to choose one location over another. Refer to Fig.~\ref{fig:no_map}. Similarly, any quadrilateral or pentagon requires at least $3$ towers to trilaterate it.
\begin{figure}
\caption{How to locate $p$ if the map of the polygon is unavailable? \textbf{(a)} Ambiguity along $L(t_1, t_2)$. We know that $p_1$ and/or $p_2$ belongs to $P$. \textbf{(b)} We need additional information to tell if $P$ (and $p$) is above or below $L(t_1, t_2)$.}
\label{fig:no_map1}
\label{fig:no_map2}
\label{fig:no_map}
\end{figure}
Let $P$ be a comb polygon with $n = 3k + q$ vertices (for integer $k \geq 1$ and $q = 0$, $1$ or $2$) such that one of its $k$ spikes contains $q$ extra vertices. Refer to Fig.~\ref{fig:comb_full}. No point of $P$ can see the complete interior of two different spikes. Assume to the contrary that $P$ can be trilaterated with less than $n - q$ towers. In other words, assume that $P$ can be trilaterated with $3k - 1$ towers. It follows that one of the spikes contains less than $3$ towers. We showed for smaller polygons (with $n = 3$, $4$ or $5$) that two towers cannot trilaterate it. Even if we know that the two towers are positioned on the same edge of $P$, the polygon can be mirrored along this edge to avoid unique trilateration. Observe that no polygon or spike can be trilaterated with one or no towers.
\begin{figure}
\caption{Comb polygon with $n = 3k + q$ vertices. \textbf{(a)} One of the $k$ spikes $S$ is shown in cyan. \textbf{(b)} One of the spikes when $q = 1$. \textbf{(c)} One of the spikes when $q = 2$.}
\label{fig:comb}
\label{fig:comb1}
\label{fig:comb2}
\label{fig:comb_full}
\end{figure} \qed \end{proof}
\section{T{\'o}th's Partition} \label{sec:partition} \pdfbookmark[1]{T{\'o}th's Partition}{sec:partition}
A simple polygonal chain $c$ is a \emph{cut} of a polygon $P$ if the two endpoints of $c$ belong to $\partial P$ and all interior points of $c$ are interior points of $P$. A cut $c$ decomposes $P$ into two polygons $P_1$ and $P_2$ such that $P = P_1 \cup P_2$ and $c = P_1 \cap P_2$.
\begin{definition}[Diagonal] A \emph{diagonal} of $P$ is a line segment that connects non-adjacent vertices of $P$ and is contained in or on the boundary of $P$. If $c$ is a cut and a diagonal of $P$ then it is called a \emph{diagonal cut}. \end{definition}
Notice that the above definition of the diagonal is non-standard. We define a diagonal as a line segment that can contain boundary points in its interior, while a diagonal in a standard definition is a line segment whose intersection with the boundary is precisely its two endpoints.
T{\'o}th showed in~\cite{Toth2000121} that any simple polygon in general position of size $n$ can be guarded by $\lfloor n/3\rfloor$ point guards whose range of vision is $180^\circ$ (let us call this result \textbf{T{\'o}th's Theorem}). His approach is to decompose $P$ into subpolygons via \emph{cuts} and to specify the locations of the guards. The cuts are composed of one or two line segments and are not restricted to be diagonal cuts. He uses a constructive induction to show his main result. Let $n_1$ (respectively $n_2$) denote the size of $P_1$ (respectively $P_2$). When $P$ contains a cut that satisfies $\left \lfloor \frac{n_1}{3} \right \rfloor + \left \lfloor \frac{n_2}{3} \right \rfloor \leq \left \lfloor \frac{n}{3} \right \rfloor$, the proof of T{\'o}th's Theorem can be completed by applying induction to both $P_1$ and $P_2$.
T{\'o}th's method heavily relies upon the partitioning of a polygon into subpolygons (on which he can apply induction). He performs diagonal cuts whenever it is possible, otherwise he cuts along a continuation of some edge of $P$; along a two-line segment made of an extension of two edges of $P$ that intersect inside $P$; or along the bisector of a reflex vertex of $P$. Notice that the three latter types of cuts may introduce new vertices that are not necessarily in general position with the given set of vertices.
However, T{\'o}th assumes that every cut produces polygons in general position, which is a very strong assumption. We strengthen the work~\cite{Toth2000121} by lifting this assumption and reproving T{\'o}th's result. We assume that the input polygon is in general position, while non-general position may occur for subpolygons of the partition. Moreover, we found and fixed several mistakes in~\cite{Toth2000121}.
\begin{definition}[Good Cut] Let $n > 5$. A cut is called a \emph{good cut} if it satisfies the following: $\left \lfloor \frac{n_1}{3} \right \rfloor + \left \lfloor \frac{n_2}{3} \right \rfloor \leq \left \lfloor \frac{n}{3} \right \rfloor$. If a good cut is a diagonal then it is called a \emph{good diagonal cut}. \end{definition}
\begin{definition}[Polygon*] \label{def:polygon} \emph{Polygon*} is a weakly simple polygon (as defined in~\cite{DBLP:conf/soda/ChangEX15}) whose vertices are distinct and whose interior angles are strictly bigger than $0$ and strictly smaller than $2\pi$. Notice that polygon* includes simple polygons. \end{definition}
We assume that every polygon of the partition (i.e. subpolygon of $P$) is polygon*. To avoid confusion, let $P'$ refer to a polygon* to which we apply the inductive partitioning. In other words, $P'$ can refer to the input polygon $P$ before any partition was applied as well as to any subpolygon of $P$ obtained during the partitioning of $P$. Recall that $P$ is in general position, while $P'$ may not be in general position.
\begin{definition}[Triangulation] A decomposition of a polygon into triangles by a maximal set of non-intersecting (but possibly overlapping) diagonals is called a \emph{triangulation} of the polygon. Notice that a triangulation of polygon* may contain triangles whose three vertices are collinear. \end{definition}
Notice that the above definition of triangulation is different from the classical one by that that it allows overlapping diagonals and thus permits triangles with three collinear vertices.
If the polygon is not in general position then its triangulation may include triangles whose three vertices are collinear. We call such triangles \emph{degenerate triangles}. Refer to Fig.~\ref{fig:case01}, showing an example of a degenerate triangle $\triangle v_1 v_2 v_3$. The diagonal $\overline{v_1 v_3}$ cannot be a good diagonal cut even if it partitions $P'$ into $P_1$ and $P_2$ such that $\left \lfloor \frac{n_1}{3} \right \rfloor + \left \lfloor \frac{n_2}{3} \right \rfloor \leq \left \lfloor \frac{n}{3} \right \rfloor$, because interior points of a cut cannot contain points of $\partial P'$.
To extend T{\'o}th's partition to polygons* we need to extend the definition of a cut. A simple polygonal chain $c'$ is a \emph{dissection} of $P'$ if the two endpoints of $c'$ belong to $\partial P'$ and all interior points of $c'$ are either interior points of $P'$ or belong to $\partial P'$. A dissection $c'$ decomposes $P'$ into two polygons* $P_1$ and $P_2$ (that are not necessarily in general position) such that $P' = P_1 \cup P_2$ and $c' = P_1 \cap P_2$. If $c'$ is a dissection and a diagonal of $P'$ then it is called a \emph{diagonal dissection}.
\begin{definition}[Good Dissection] Let $n > 5$. A dissection is called a \emph{good dissection} if $\left \lfloor \frac{n_1}{3} \right \rfloor + \left \lfloor \frac{n_2}{3} \right \rfloor \leq \left \lfloor \frac{n}{3} \right \rfloor$. \end{definition}
We extend the results in~\cite{Toth2000121} by removing the assumption that the partitioning produces subpolygons in general position and by thus strengthening the result. In this paper, we need to refer to many Lemmas, Propositions and Claims from~\cite{Toth2000121}. Indeed, we apply some of these results, we fill the gaps in some proofs of these results and we generalize some others. In order for this paper to be self-contained, we write the statements of all these results with their original numbering~\cite{Toth2000121}.
\noindent \textbf{Simplification Step:} If $P'$ has consecutive vertices $v_1$, $v_2$ and $v_3$ along $\partial P'$ that are collinear then: \begin{enumerate} \item If the angle of $P'$ at $v_2$ is $\pi$ then replace $v_2$ together with its adjacent edges by the edge $\overline{v_1 v_3}$. \item If the angle of $P'$ at $v_2$ is $0$ or $2\pi$ then delete $v_2$ together with its adjacent edges and add the edge $\overline{v_1 v_3}$. Assume w.l.o.g. that the distance between $v_3$ and $v_2$ is smaller than the distance between $v_1$ and $v_2$. The line segment $\overline{v_2 v_3}$ will be guarded despite that it was removed from $P'$. Refer to the following subsections for examples. \end{enumerate} In both cases we denote the updated polygon by $P'$; its number of vertices decreased by $1$.
Assume for simplicity that $P'$ does not contain consecutive collinear vertices along $\partial P'$. Let $n > 5$ be the number of vertices of $P'$. Let $k$ be a positive integer and $q \in \{0, 1, 2\}$ such that $n = 3k +q$. Notice that a diagonal dissects $P'$ into $P_1$ (of size $n_1$) and $P_2$ (of size $n_2$) such that $n_1 + n_2 = n+2$.
The proofs of the following three propositions are identical to those in T{\'o}th's paper.
\begin{mydefP} \label{TothProp1} For $P'$ with $n = 3k$ any diagonal is a good dissection. \end{mydefP}
\begin{mydefP} \label{TothProp2} For $P'$ with $n = 3k+1$ there exists a diagonal that represents a good dissection. \end{mydefP}
\begin{mydefP} \label{TothProp3} For $P'$ with $n = 3k+2$ a dissection is a good dissection if it decomposes $P'$ into polygons* $P_1$ and $P_2$ (not necessarily simple polygons) such that $n_1 = 3k_1 + 2$ and $n_2 = 3k_2 + 2$ (for $k_1 +k_2 = k$). \end{mydefP}
\subsection{Case Study} \label{subsec:CaseStudy} \pdfbookmark[2]{Case Study}{subsec:CaseStudy}
\begin{wrapfigure}{r}{0.35\textwidth}
\centering \includegraphics[width=0.3\textwidth]{case01.pdf} \caption{The diagonal $\overline{v_1 v_3}$ contains a vertex $v_2$ of $P'$. $P_2 = P_a \cup P_b$. Notice that $v_2 \notin P_1$.} \label{fig:case01}
\end{wrapfigure} In this subsection we study how Propositions~\ref{TothProp1},~\ref{TothProp2} and~\ref{TothProp3} can be applied to polygons*. Let $\triangle v_1 v_2 v_3$ be a minimum degenerate triangle in $P'$, i.e. it does not contain any vertex of $P'$ other than $v_1$, $v_2$ or $v_3$. Consider the example depicted in Fig.~\ref{fig:case01}. The diagonal $\overline{v_1 v_3}$ partitions $P'$ into two polygons: $P_1$ and $P_2$ ($P_2$ can be further viewed as a union of two subpolygons $P_a$ and $P_b$). Notice that $v_2 \notin P_1$. There is a possibility for $P'$ to have a vertex $v' \notin \{v_1, v_2, v_3\}$ such that $v' \in L_{v_1 v_3}$. However $v'$ cannot belong to the line segment $\overline{v_1 v_3}$, otherwise, the triangle $\triangle v_1 v_2 v_3$ is not minimum, and we can choose $\triangle v_1 v_2 v'$ or $\triangle v' v_2 v_3$ instead. It follows that no vertex of $P_1$ (respectively $P_a$, $P_b$) is located on the edge $\overline{v_1 v_3}$ (respectively $\overline{v_1 v_2}$, $\overline{v_2 v_3}$). Notice that $P'$ may contain an edge $e'$ that contains $\overline{v_1 v_3}$; $P_1$ inherits it (since $v_2 \notin P_1$), but it won't cause any trouble because of the ``simplification'' step described in the beginning of this section. Notice also, that two vertices of $P'$ cannot be at the same location.
Assume that $\overline{v_1 v_3}$ is a good dissection, i.e. it decomposes $P'$ into $P_1$ and $P_2$ such that $\left \lfloor \frac{n_1}{3} \right \rfloor + \left \lfloor \frac{n_2}{3} \right \rfloor \leq \left \lfloor \frac{n}{3} \right \rfloor$. Assume also that $n>5$. Let $n_a$ (respectively $n_b$) be the number of vertices of $P_a$ (respectively $P_b$). Notice that $n_a + n_b = n_2 + 1$ because $v_2$ was counted twice. When it is possible, we prefer to avoid cutting along the diagonal $\overline{v_1 v_3}$. However, when necessary, it can be done in the following way. We consider three cases (refer to Fig.~\ref{fig:case01}):
{\color{red} \textbf{Case 1:}} $n = 3k$. By T{\'o}th's Proposition~\ref{TothProp1} every diagonal is a good dissection. If $P_a$ is \textbf{not} composed of the line segment $\overline{v_1 v_2}$ only, then $\overline{v_1 v_2}$ is a good diagonal dissection, that partitions $P'$ into two polygons $P_a$ and $P_1 \cup P_b$. Otherwise, $\overline{v_2 v_3}$ is a good diagonal dissection, that partitions $P'$ into two polygons $P_b$ and $P_1 \cup P_a$. Notice that $\overline{v_1 v_2}$ and $\overline{v_2 v_3}$ cannot be both edges of $P'$ because of the simplification step we applied to $P'$.
Alternatively, we can dissect $P'$ along $\overline{v_1 v_3}$. The polygon $P_2$ has an edge $\overline{v_1 v_3}$ that contains vertex $v_2$. If a $180^\circ$-guard is located at $v_2$ for any subpolygon $P_2'$ of $P_2$, then we position the towers close to $v_2$ on a line segment that contains $v_2$ (but not necessarily in $kernel(P_2') \cap \partial P_2'$) only if $v_2$ is not a vertex of the original polygon $P$. However, if $v_2 \in P$ then we position our towers in the close proximity to $v_2$ but in the interior of $P_1$, oriented towards $P_2$.
{\color{red} \textbf{Case 2:}} $n = 3k+1$. Since $\overline{v_1 v_3}$ decomposes $P'$ into $P_1$ and $P_2$ such that $\left \lfloor \frac{n_1}{3} \right \rfloor + \left \lfloor \frac{n_2}{3} \right \rfloor \leq \left \lfloor \frac{n}{3} \right \rfloor$ then either $n_1 = 3k_1 + 1$ and $n_2 = 3k_2 + 2$, or $n_1 = 3k_1 + 2$ and $n_2 = 3k_2 + 1$ for $k_1 +k_2 = k$. If $\overline{v_1 v_2}$ is an edge of $P'$ (meaning that $P_a$ consists of a line segment $\overline{v_1 v_2}$ only) then we can dissect $P'$ either along $\overline{v_2 v_3}$ or along $\overline{v_1 v_3}$. In the former case $P'$ will be decomposed into $P_1 \cup \{v_2\}$ and $P_b$ ($v_2$ will be deleted from $P_1 \cup \{v_2\}$ during the simplification step to obtain $P_1$). In the latter case $P'$ will be decomposed into $P_1$ and $P_b \cup \{v_1\}$ ($v_1$ will be removed from $P_b \cup \{v_1\}$ during the simplification step to obtain $P_b$). In either case we do not have to specifically guard the line segment $\overline{v_1 v_2}$. If $P_1$ is guarded then so is $\overline{v_1 v_2}$. Similarly treat the case where $\overline{v_2 v_3}$ is an edge of $P'$.
Assume that $n_a, n_b >2$. Recall, that $n_b = n_2 - n_a + 1$. Several cases are possible: \begin{itemize} \item[$\blacktriangleright$] $n_a = 3k_a + 1$. Then we dissect $P'$ along $\overline{v_1 v_2}$. This partition creates the polygons $P_a$ of size $n_a = 3k_a + 1$ and $P_1 \cup P_b$ of size $3(k_1 + k_2 - k_a) +2$. \item[$\blacktriangleright$] $n_a = 3k_a + 2$. Then we dissect $P'$ along $\overline{v_1 v_2}$. This partition creates the polygons $P_a$ of size $n_a = 3k_a + 2$ and $P_1 \cup P_b$ of size $3(k_1 + k_2 - k_a) +1$. \item[$\blacktriangleright$] $n_a = 3k_a$. Then: \begin{itemize} \item $n_1 = 3k_1 + 2$ and $n_2 = 3k_2 + 1$. Then we dissect $P'$ along $\overline{v_2 v_3}$. This partition creates the polygons $P_b$ of size $n_b = 3(k_2 - k_a)+2$ and $P_1 \cup P_a$ of size $3(k_1 +k_a) +1$. \item $n_1 = 3k_1 + 1$ and $n_2 = 3k_2 + 2$. Then both $\overline{v_1 v_2}$ and $\overline{v_2 v_3}$ are \textbf{not} good diagonal cuts. We cannot avoid dissecting along $\overline{v_1 v_3}$. If a $180^\circ$-guard eventually needs to be positioned at $v_2$ (for any subpolygon of $P_2$) and $v_2$ is a vertex of $P$, then we position our towers close to $v_2$ but in the interior of $P_1$, oriented towards~$P_2$. \end{itemize} \end{itemize}
{\color{red} \textbf{Case 3:}} $n = 3k+2$. In this case $n_1 = 3k_1+2$ and $n_2 = 3k_2+2$. If $n_a = 2$ then either dissect $P'$ along $\overline{v_2 v_3}$ (and later delete $v_2$ from $P_1 \cup v_2$ during the simplification step), or dissect $P'$ along $\overline{v_1 v_3}$ (delete $v_1$ from $P_b \cup v_1$ during the simplification step). Notice that in both cases $\overline{v_1 v_2}$ will be guarded. The case where $n_b = 2$ is similar.
If $n_a, n_b >2$ then we have no choice but to dissect along $\overline{v_1 v_3}$. This creates a polygon $P_2$ whose kernel degenerates into a single point $v_2$. It is not a problem for $180^\circ$-guards but it is a serious obstacle for our problem. If $v_2$ is a vertex of $P$ and a $180^\circ$-guard is located at $v_2$ for any subpolygon of $P_2$ then we position our towers close to $v_2$ but in the interior of $P_1$, and orient those towers towards $P_2$. We consider the general approach to this problem in the following subsection.
\subsection{Point-Kernel Problem} \label{subsec:PointKernel} \pdfbookmark[2]{Point-Kernel Problem.}{subsec:PointKernel}
In Section~\ref{subsec:CaseStudy} we studied simple cases where a diagonal dissection is applied to polygons* (or polygons in non-general position). In this subsection, we show how to circumvent some difficulties that arise when adapting T{\'o}th's partitioning to our problem.
The dissection of $P$ may create subpolygons that are polygons*. This means that the partition of $P$ may contain star-shaped polygons whose kernels degenerate into a single point. While this is not a problem for $180^\circ$-guards, it is a serious obstacle for tower positioning. Indeed, we need at least two distinct points in the kernel of each part of the partition to trilaterate~$P$.
\begin{wrapfigure}{r}{0.42\textwidth} \centering \includegraphics[width=0.38\textwidth]{PointKernel1.pdf} \caption{$\overline{v_1 v_4}$ is a good diagonal dissection that contains two vertices of $P$: $v_2$ and $v_3$. $P_1 = P_a\cup P_b$ and $P_2 = P_c\cup P_d$. Notice that $v_2 \notin P_a$ and $v_3 \notin P_c$.} \label{fig:PointKernel1}
\end{wrapfigure}
Let $\overline{v_1 v_4}$ be a good dissection, i.e. it decomposes $P'$ into $P_1$ and $P_2$ such that $\left \lfloor \frac{n_1}{3} \right \rfloor + \left \lfloor \frac{n_2}{3} \right \rfloor \leq \left \lfloor \frac{n}{3} \right \rfloor$. Assume also that $n>5$. Assume that $\overline{v_1 v_4}$ contains at least $2$ more vertices of $P'$ in its interior. Let $v_2, v_3 \in \overline{v_1 v_4}$. Refer to Fig.~\ref{fig:PointKernel1}. Notice that $v_2$ and $v_3$ belong to subpolygons of $P'$ on the opposite sides of $L(v_1, v_4)$. Recall that the original polygon $P$ is given in general position. Thus at most two vertices of $P$ belong to the same line. If $v_2$ and $v_3$ are not both vertices of $P$ then there is no problem for tower positioning, and the dissection via $\overline{v_1 v_4}$ can be applied directly. We then follow T{\'o}th's partition and replace every $180^\circ$-guard with a pair of towers in the guard's vicinity. However, if $v_2$ and $v_3$ are vertices of $P$ then the visibility of towers can be obstructed. Consider the example of Fig.~\ref{fig:PointKernel1}. Assume that a $180^\circ$-guard is positioned at $v_2$ to observe a subpolygon of $P_c \cup P_d$ (say, pentagon $v_1 v_4 v_c v_2 v_d$ for $v_c \in P_c$ and $v_d \in P_d$). We cannot replace the $180^\circ$-guard with a pair of towers because the kernel of the pentagon degenerates into a single point $v_2$. Our attempt to position towers in the vicinity of $v_2$ but interior to $P_a$ is not successful either, because $v_3$, as a vertex of $P$, blocks visibility with respect to $P_c$. We have to find another dissection. In general, we are looking for a dissection that destroys the diagonal $\overline{v_1 v_4}$.
Recall that in this subsection $P'$ is a subpolygon of $P$ with collinear vertices $v_1$, $v_2$, $v_3$ and $v_4$. Assume that $v_2$ and $v_3$ are vertices of $P$. It follows that there are no vertices of $P$ on $L(v_1, v_4)$ other than $v_2$ and $v_3$. Assume that $\overline{v_1 v_4}$ is a good dissection of $P'$. If one of $\overline{v_1v_3}$, $\overline{v_1v_2}$, $\overline{v_3v_4}$, $\overline{v_2v_4}$ or $\overline{v_2v_3}$ is a good dissection then we cut along it. Only the cut along $\overline{v_2v_3}$ destroys the diagonal $\overline{v_1v_4}$ and eliminates the problem of having towers in the vicinity of $v_2$ (respectively $v_3$) with visibility obstructed by $v_3$ (respectively $v_2$). The cut along $\overline{v_1v_3}$, $\overline{v_1v_2}$, $\overline{v_3v_4}$ or $\overline{v_2v_4}$ only reduces $P'$. So assume that none of the diagonals $\overline{v_1v_3}$, $\overline{v_1v_2}$, $\overline{v_3v_4}$, $\overline{v_2v_4}$ or $\overline{v_2v_3}$ represent a good dissection. It follows from T{\'o}th's Proposition~\ref{TothProp1} that $n \neq 3k$. Consider the following cases:
{\color{red} \textbf{Case 1:}} $n = 3k+1$. By T{\'o}th's Proposition~\ref{TothProp2} either $n_1 = 3k_1 + 2$ and $n_2 = 3k_2 + 1$, or $n_1 = 3k_1 + 1$ and $n_2 = 3k_2 + 2$ for $k_1 +k_2 = k$, where $P_1 = P_a\cup P_b$ and $P_2 = P_c\cup P_d$. Assume w.l.o.g. that $n_1 = 3k_1 + 2$ and $n_2 = 3k_2 + 1$. We assumed that $\overline{v_2v_3}$ is not a good dissection and thus the size of $P_c \cup P_b$ is a multiple of $3$ and the size of $P_d \cup P_a$ is a multiple of $3$. Consider the following three subcases:
{\color{blue} \textbf{Case 1.1:}} $|P_d| = 3k_d$; thus $|P_c| = 3k_c+2$ or $|P_c| = 2$. Since the size of $P_c \cup P_b$ is a multiple of $3$, we have $|P_b| = 3k_b+2$ or $|P_b| = 2$. Notice that $P_b$ and $P_c$ cannot be both of size $2$, otherwise $v_4$ would be deleted during the simplification step. Both diagonals $\overline{v_3v_4}$ and $\overline{v_2v_4}$ represent a good dissection. Thus, if $|P_c| \neq 2$ then dissect along $\overline{v_2v_4}$; if $|P_b| \neq 2$ then dissect along $\overline{v_3v_4}$. The polygon $P_a \cup P_d \cup \triangle v_2 v_3 v_4$ will be simplified and it will loose $\overline{v_3 v_4}$ (which is guarded by $P_b$ or/and $P_c$).
{\color{blue} \textbf{Case 1.2:}} $|P_d| = 3k_d+1$ or $|P_d| = 3k_d+2$. Then $\overline{v_1v_2}$ is a good diagonal dissection.
{\color{blue} \textbf{Case 1.3:}} $|P_d| = 2$. Then $|P_a| = 3k_a +2$ and thus $\overline{v_1v_3}$ is a good diagonal dissection, which creates the polygons $P_a$ and $P_b \cup P_c \cup \{v_1\}$. The vertex $v_1$ will be deleted from $P_b \cup P_c \cup \{v_1\}$ during the simplification step. Notice that the line segment $\overline{v_1v_2}$ will be guarded by $P_a$.
{\color{red} \textbf{Case 2:}} $n = 3k+2$. Since $\overline{v_1 v_4}$ is a good dissection, it partitions $P'$ into $P_1 = P_a\cup P_b$ of size $n_1 = 3k_1 +2$ and $P_2 = P_c\cup P_d$ of size $n_2 = 3k_2 +2$.
{\color{blue} \textbf{Case 2.1:}} $|P_b| = 3k_b+1$; thus $|P_a|=3k_a +2$ or $|P_a| = 2$. We assumed that $\overline{v_1v_3}$ is not a good dissection and thus \begin{wrapfigure}{r}{0.42\textwidth}
\centering \includegraphics[width=0.4\textwidth]{PointKernel2.pdf}
\caption{$\overline{v_1 v_4}$ is a good diagonal dissection that contains two vertices of $P$: $v_2$ and $v_3$. $P_a = \{v_1, v_3\}$, $|P_b| = 3k_b+1$, $|P_c| = 3k_c$ and $|P_d| = 3k_d$.} \label{fig:PointKernel2}
\end{wrapfigure}
$|P_a| \neq3 k_a +2$. It follows that $P_a$ is a line segment $\overline{v_1v_3}$. Symmetrically, if $|P_d| = 3k_d+1$ then $|P_c| = 2$. If $|P_b| = 3k_b+1$ and $|P_d| = 3k_d+1$ then we have a good dissection $\overline{v_2 v_3}$, which is a contradiction.
The case where $|P_b| = 3k_b+1$ and $|P_c| = 3k_c+1$ is not possible because in this case $P_a = \overline{v_1v_3}$ and $P_d = \overline{v_1v_2}$ and thus $v_1$ would not survive the simplification step.
Thus, if $|P_b| = 3k_b+1$ then $|P_c| = 3k_c$ and $|P_d| = 3k_d$. Refer to Fig.~\ref{fig:PointKernel2}. Let $v_2'$ be an immediate neighbour of $v_2$ such that $v_2'$ is a vertex of $P_c$ and $v_2' \neq v_4$. If $v_2'$ and $v_3$ see each other, then $\overline{v_2' v_3}$ is a diagonal of $P'$. It dissects $P'$ into two subpolygons $P_d \cup \triangle v_2v_3v_2'$ of size $3k_d+2$ and $(P_b \cup P_c) \setminus \triangle v_2v_3v_2'$ of size $3k_b+3k_c-1 = 3(k_b+k_c-1)+2$. Assume now that $v_2'$ and $v_3$ do not see each other. Let us give numerical labels to the vertices of $P_c$ as follows: $v_2$ gets label $1$, $v_2'$ gets $2$, and so on, $v_4$ will be labelled $3k_c$. If $v_3$ can see a vertex of $P_c$ with label whose value modulo $3$ equals $2$ then dissect $P'$ along the diagonal that connects $v_3$ and that vertex. If no such vertex can be seen from $v_3$, we do the following. Let $v_3'$ be an immediate neighbour of $v_3$ such that $v_3'$ is a vertex of $P_b$ and $v_3' \neq v_4$. Let $v_3''$ be the point on $\partial P_c$ where the line supporting $\overline{v_3' v_3}$ first hits $\partial P_c$ (refer to Fig.~\ref{fig:PointKernel2}). If $v_3''$ belongs to an edge with vertices whose labels modulo $3$ equal $1$ and $2$ then dissect $P'$ along $\overline{v_3 v_3''}$. This dissection creates polygons $P_d \cup \{1,2, \ldots, 3k'+1, v_3'', v_3\}$ of size $3(k_d+k')+2$ and $(\{v_3'', 3k'+2, \ldots 3k_c\} \cup P_b) \setminus \{v_3\}$ of size $3(k_c-k'+k_b-1)+2$.
\begin{wrapfigure}{r}{0.42\textwidth}
\centering \includegraphics[width=0.36\textwidth]{PointKernel3.pdf}
\caption{$v_2$ and $v_3$ are vertices of $P$. $|P_b| = 3k_b+1$, $|P_c| = 3k_c$ and $|P_d| = 3k_d$. } \label{fig:PointKernel3}
\end{wrapfigure} We are still in case $2.1$ and all its assumptions are still applicable. If the above scenario (about dissecting $P_c$ via a line that contains $v_3$) has not worked, we consider the case that we tried to avoid all along: the dissection of $P'$ along $\overline{v_1 v_4}$ such that the successive partitioning of $P_2$ created a subpolygon with a single-point kernel at $v_2$ whose edge contains $v_3$ in its interior. In other words, we assume that $P_2 = P_c \cup P_d$ has undergone some partitioning that resulted in the creation of the pentagon $v_1' v_4' v_c v_2 v_d$ for $v_c, v_4' \in P_c$, $v_d, v_1' \in P_d$ such that $v_1', v_4' \in L(v_2, v_3)$ and $v_4' \in \overline{v_3 v_4}$ (refer to Fig.~\ref{fig:PointKernel3}). It is impossible to localize an agent in the pentagon $v_1' v_4' v_c v_2 v_d$ with a pair of towers only. However, if $v_4' \in \overline{v_2 v_3}$ or $v_4' = v_3$ then a pair of towers in the vicinity of $v_2$ can oversee the pentagon, because $v_3$ is not causing an obstruction in this case. Despite that $P_a$ is a line segment, the original polygon $P$ has a non-empty interior in the vicinity of $v_2$ to the right of the ray $\overrightarrow{v_3 v_2}$. Notice that $v_1'$ may be equal to $v_1$; $v_4'$ may be equal to $v_4$. Moreover, $v_c$ and $v_4'$ may not be vertices of $P_1$ but were created during the partition. We also assume that $P_1 = P_b \cup \{v_1\}$ was not partitioned yet. Let $P_c'$ be a subpolygon of $P_c$ that inherits the partition of $P_c$ to the side of the dissection $\overline{v_c v_4'}$ that contains $v_4$ (refer Fig.~\ref{fig:PointKernel45}). Notice that $v_c$ and $v_3$ can see each other. If $\overline{v_c v_4'}$ is a diagonal dissection then instead use $\overline{v_c v_3}$. In this case $v_4'$ can be deleted and $v_3$ is added to $P_c'$. The size of $P_c'$ does not change and the number of guards required to guard $P_c'$ does not increase. A pair of towers in the vicinity of $v_2$ can oversee the pentagon $v_1' v_3 v_c v_2 v_d$.
If the dissection $\overline{v_c v_4'}$ is a continuation of an edge of $P_c$ (let $v$ be a vertex that is adjacent to this edge) then two cases are possible. In the first case, the edge $\overline{v_c v}$ that produced the dissection is contained in $\overline{v_c v_4'}$ (refer Fig.~\ref{fig:PointKernel4}). In this case, instead of dissecting along $\overline{v_c v_4'}$ use $\overline{v v_3}$. The size of $P_c'$ does not change and the hexagon $v_1' v_3 v v_c v_2 v_d$ can be guarded by a pair of towers in the vicinity of $v_2$ (because $v_2 v_c v v_3$ is convex). In the second case the edge $\overline{v_c v}$ that produced the dissection $\overline{v_c v_4'}$ is not contained in $\overline{v_c v_4'}$ (refer Fig.~\ref{fig:PointKernel5}). In the worst case the size of $P_c'$ is $3k_{c'}+2$. If instead of dissecting along $\overline{v_c v_4'}$ we use $\overline{v_c v_3}$ then the size of $P_c'$ increases by $1$, which results in an increased number of guards. But, if we dissect along $\overline{v_c v_3}$ then the line segment $\overline{v_1 v_3}$ can be removed from $P_1$ (because it is guarded by the guard of $v_1' v_3 v_c v_2 v_d$) and $P_b$ can be joined with the updated $P_c'$. The size of $(P_b \cup P_c' \cup\{v_c\}) \setminus\{v_4'\}$ is $3(k_b+k_{c'})+2$, so the number of guards of $P'$ does not increase as a result of adjusting the partition. However, the current partition of $P_c'$ may no longer be relevant and may require repartition as part of the polygon $(P_b \cup P_c' \cup\{v_c\}) \setminus\{v_4'\}$.
\begin{figure}
\caption{We avoid dissecting along $\overline{v_c v_4'}$.}
\label{fig:PointKernel4}
\label{fig:PointKernel5}
\label{fig:PointKernel45}
\end{figure}
{\color{blue} \textbf{Case 2.2:}} $|P_b| = 3k_b$. Then $|P_a| = 3k_a$. If $|P_c| = 3k_c$ and thus $|P_d| = 3k_d$ then $\overline{v_2 v_3}$ is a good diagonal dissection, which is a contradiction. Thus either $|P_c| = 3k_c+1$ and $|P_d| = 2$ or $|P_d| = 3k_d+1$ and $|P_c| = 2$. Those cases are symmetrical to the the case 2.1.
\subsection{$n = 3k+2$ and $P'$ has no good diagonal dissection} \label{subsec:Partition} \pdfbookmark[2]{$n = 3k+2$ and $P'$ has no good diagonal dissection}{subsec:Partition}
In this subsection we assume that $n = 3k+2$ and every diagonal of $P'$ decomposes it into polygons of size $n_1 = 3k_1$ and $n_2 = 3k_2+1$, where $k_1 + k_2 = k + 1$. Let $\mathcal{T}$ be a fixed triangulation of $P'$, and let $G(\mathcal{T})$ be the dual graph of $\mathcal{T}$. Notice that $G(\mathcal{T})$ has $n-2$ nodes. (Later, during the algorithm, we may change this triangulation.)
The proofs of the following Propositions~\ref{TothProp4} and~\ref{TothProp5} are identical to those in~\cite{Toth2000121}. We prove the following Lemmas~\ref{TothLem1},~\ref{TothLem2} and~\ref{TothLem3} in this paper.
\begin{mydefP} \label{TothProp4} $G(\mathcal{T})$ has exactly $k+1$ leaves. \end{mydefP}
\begin{mydefL} \label{TothLem1} If $P'$ has $n = 3k+2$ vertices and has no good dissection then $P'$ has at most $k$ reflex angles. \end{mydefL}
\begin{mydefP} \label{TothProp5} If $P'$ has at most $k$ reflex angles, then $k$ $180^\circ$-guards can monitor $P'$. \end{mydefP}
\begin{mydefL} \label{TothLem2} If $P'$ has size $n = 3k+2$ then at least one of the following two statements is true: \begin{enumerate} \item $P'$ has a good dissection. \item For every triangle $\triangle A B C$ in $\mathcal{T}$ that corresponds to a leaf of $G(\mathcal{T})$ with $AC$ being a diagonal of $P'$, either $\angle A < \pi$ or $\angle C < \pi$ in $P'$. \end{enumerate} \end{mydefL}
\begin{mydefL} \label{TothLem3} If a convex angle of $P'$ is associated to two leaves in $G(\mathcal{T})$, then $\left \lfloor \frac{n}{3} \right \rfloor$ $180^\circ$-guards can monitor~$P'$. \end{mydefL}
\subsubsection{Adaptation of T{\'o}th's Lemma~\ref{TothLem3} to our problem} \label{subsubsec:Lemma3} \pdfbookmark[3]{Adaptation of T{\'o}th's Lemma~\ref{TothLem3} to our problem.}{subsubsec:Lemma3}
\\ \\ Let $v_1$ be a vertex at a convex angle of $P'$ associated to two leaves in $G(\mathcal{T})$ (refer to Fig.~\ref{fig:TothLemma3}). Let $P_1$ be the polygon formed by all triangles of $\mathcal{T}$ adjacent to $v_1$. Notice that $P_1$ is a fan; $v_1$ is a center of the fan; and the dual of the triangulation of $P_1$, inherited from $P'$, is a path of nodes. Recall that the center of a fan $P_f$ is a vertex of $P_f$ that can see all other vertices of $P_f$.
\begin{wrapfigure}{r}{0.6\textwidth}
\centering \includegraphics[width=0.58\textwidth]{TothLemma3.pdf} \caption{$P'$ has size $n = 20 = 3k + 2$ for $k = 6$, and does not have good diagonal dissection. $G(\mathcal{T})$ is drawn on top of $\mathcal{T}$. $P_1$ is a fan with dominant point $v_1$ and it is highlighted in cyan.} \label{fig:TothLemma3}
\end{wrapfigure}
The original proof of T{\'o}th's Lemma~\ref{TothLem3} still holds for polygons*. However, this is one of the places where a $180^\circ$-guard is explicitly positioned at $v_1$ to monitor all the triangles of $\mathcal{T}$ adjacent to $v_1$. If $kernel(P_1)$ is not a single point, then the positioning can be reused for our problem. We can put a pair of towers on the same edge of $P'$ in $kernel (P_1) \cap \partial P_1$ to locate an agent in $P_1$. However, since there can be degenerate triangles among those adjacent to $v_1$, the kernel of $P_1$ can degenerate into a single point: $kernel (P_1) = v_1$. In this case, the location of an agent in $P_1$ cannot be determined with a pair of towers only. To overcome this problem we prove the following lemma.
\begin{lemma} \label{lem:TL3} There exist a triangulation of $P'$ such that its part, inherited by $P_1$ (i.e. triangles incident to $v_1$), does not contain any degenerate triangle. \end{lemma} \begin{proof} Let us first observe that a triangle of $\mathcal{T}$, corresponding to a leaf in $G(\mathcal{T})$, cannot be degenerate due to the simplification step performed on $P'$. There are two cases to consider:
{\color{red} \textbf{Case 1:}} Suppose that the triangulation/fan of $P_1$ contains two or more degenerate triangles adjacent to each other. Let $v_1, u_1, u_2, \ldots, u_i$ for some $i > 2$ be the vertices of the degenerate triangles sorted according to their distance from $v_1$. Notice that $v_1, u_1, u_2, \ldots, u_i$ belong to the same line. Those degenerate triangles must be enclosed in $P_1$ between a pair of non-degenerate triangles, let us call them $\triangle_1$ and $\triangle_2$. Since $P'$ does not have angles of size $2\pi$, the diagonal $\overline{v_1 u_1}$ must be shared with one of $\triangle_1$ or $\triangle_2$. Assume, without loss of generality, that $u_1$ is a vertex of $\triangle_1$. Let $u_j$ be a vertex of $\triangle_2$, for some $1 < j \leq i$. It is possible to re-triangulate $P'$ such that $P_1$ will contain only one degenerate triangle $\triangle v_1 u_1 u_j$ between $\triangle_1$ and $\triangle_2$ and other triangles of $P_1$ (that are not between $\triangle_1$ and $\triangle_2$) will not be affected. The shape of $P_1$ may or may not change but its size will decrease.
{\color{red} \textbf{Case 2:}} Suppose that the triangulation/fan of $P_1$ contains one degenerate triangle $\triangle v_1 u_1 u_j$ enclosed between a pair of non-degenerate triangles $\triangle_1$ and $\triangle_2$. Assume, without loss of generality, that $\triangle_2$ shares a diagonal $\overline{v_1 u_j}$ with $\triangle v_1 u_1 u_j$. Let $w$ be the third vertex of $\triangle_2$, so $\triangle_2 = \triangle v_1 u_j w$. Notice that $u_1$ and $w$ see each other, because $u_1$ belongs to the line segment $\overline{v_1 u_j}$. We can flip the diagonal $\overline{v_1 u_j}$ into $\overline{u_1 w}$ in $\mathcal{T}$. As a result, $P_1$ will now contain $\triangle v_1 u_1 w$ instead of $\triangle v_1 u_1 u_j$ and $\triangle_2$. Notice that $\triangle v_1 u_1 w$ is non-degenerate.
We showed that we can obtain a triangulation of $P'$ in which all the triangles incident to $v_1$ are non-degenerate. Thus, $P_1$ will have no degenerate triangles and still contain vertex $v_1$ together with two leaves of $G(\mathcal{T})$ associated with $v_1$. \qed \end{proof}
\subsubsection{Proof of T{\'o}th's Lemma~\ref{TothLem2} and its adaptation to our problem} \label{subsubsec:Lemma2} \pdfbookmark[3]{Proof of T{\'o}th's Lemma~\ref{TothLem2} and its adaptation to our problem.}{subsubsec:Lemma2}
\\ \\ Section~\ref{subsubsec:Lemma2} is devoted to the proof of T{\'o}th's Lemma~\ref{TothLem2}.
T{\'o}th defines two types of elements of $G(\mathcal{T})$. A leaf of $G(\mathcal{T})$ is called a \emph{short leaf} if it is adjacent to a node of degree $3$. If a leaf of $G(\mathcal{T})$ is adjacent to a node of degree $2$ then this leaf is called a \emph{long leaf}. Since we are under the assumption that $n = 3k+2$ and $P'$ has no good diagonal dissection then the node of $G(\mathcal{T})$ adjacent to a long leaf is also adjacent to a node of degree~$3$.
\begin{wrapfigure}{r}{0.32\textwidth}
\centering \includegraphics[width=0.3\textwidth]{TothShortLeaf_1.pdf} \caption{$\triangle ABC$ corresponds to a short leaf in $G(\mathcal{T})$. $\triangle ACD$ can be degenerate.} \label{fig:TothShortLeaf_1}
\end{wrapfigure} In this subsection we keep T{\'o}th's original names and notations to simplify cross-reading.
The angle $\angle ABC$ is the angle that the ray $\overrightarrow{BA}$ makes while rotating counter-clockwise towards the ray $\overrightarrow{BC}$. For example, the angle of $P'$ at $B$ is $\angle CBA$ (refer to Fig.~\ref{fig:TothShortLeaf_1}).
Let $\triangle ABC$ correspond to a short leaf in $G(\mathcal{T})$, where $\overline{AC}$ is a diagonal of $P'$, and let $\triangle ACD$ correspond to a node in $G(\mathcal{T})$ adjacent to the leaf $\triangle ABC$. Refer to Fig.~\ref{fig:TothShortLeaf_1}. Notice that $\triangle ABC$ cannot be degenerate, otherwise the vertex $B$ would be deleted during the simplification step. However, $\triangle ACD$ can be degenerate. The diagonals $\overline{AD}$ and $\overline{CD}$ decompose $P'$ into polygons $P_a$, $ABCD$ and $P_c$, where $A \in P_a$ and $C \in P_c$. Let $n_a$ be size of $P_a$ and $n_c$ be size of $P_c$.
\begin{mydefC} \label{claimT1} $n_a = 3 k_a + 1$ and $n_c = 3 k_c + 1$. \end{mydefC}
\begin{mydefC} \label{claimT2} $ABCD$ is a non-convex quadrilateral, i.e. it has a reflex vertex at $A$ or $C$. \end{mydefC}
The original proof of both claims (refer to~\cite{Toth2000121}) can be reused for polygons*, so we do not present their proofs here. However, T{\'o}th's Claim~\ref{claimT3} (presented below) requires a slightly different proof to hold true for polygons*. In addition, there was a mistake in the original proof of T{\'o}th's Claim~\ref{claimT3} that we fixed here. But first we would like to clarify the framework we work in, to explain tools and assumptions we use.
Assume, without loss of generality, that the reflex vertex of $ABCD$ is at $C$ (which exists by T{\'o}th's Claim~\ref{claimT2}). Assume that $\mathcal{T}$ is the triangulation of $P'$ in which $n_a$ is \textbf{minimal}. This assumption together with the fact that $P'$ does not have a good diagonal dissection implies that there does not exist a vertex of $P_a$ in the interior of a line segment $\overline{DA}$. By T{\'o}th's Claim~\ref{claimT1}, $n_a \geq 4$ (notice that $n_a \neq 1$ because $A$, $D \in P_a$). Let $A_0$ and $D_0$ be vertices of $P_a$ adjacent to $A$ and $D$ respectively (refer to Fig.~\ref{fig:TothShortLeaf_1}). By $\overrightarrow{uv}$ we denote the ray that starts at $u$ and passes through $v$. Let $A'$ be a point such that $A' \in \partial P' \cap \overrightarrow{BA}$ and there exists a point $A'' \in \partial P'$ strictly to the \textbf{right} of $\overrightarrow{BA}$ such that $A'$ and $A''$ belong to the same edge of $P'$ and both visible to $B$ and $A$. Let $D'$ be a point such that $D' \in \partial P' \cap \overrightarrow{CD}$ and there exist a point $D'' \in \partial P'$ strictly to the \textbf{left} of $\overrightarrow{CD}$ such that $D'$ and $D''$ belong to the same edge of $P'$ and both visible to $C$ and $D$. Let $C'$ be a point defined similarly to $D'$ but with respect to the ray $\overrightarrow{BC}$. Notice that if $A=A'$ then $\angle A = \angle B A A_0 < \pi$ in $P'$ and thus the second condition of T{\'o}th's Lemma~\ref{TothLem2} holds. Therefore, assume that $A \neq A'$.
\begin{mydefC} \label{claimT3} The points $A'$ and $D'$ belong to the same edge of $P'$. \end{mydefC} \begin{proof} Let us rotate the ray $\overrightarrow{CA'}$ around $C$ in the direction of $D'$. Notice that in the original proof, T{\'o}th uses $\overrightarrow{CA}$. However, there are polygons (even in general position) for which T{\'o}th's proof does not hold. For example, when the ray $\overrightarrow{CA}$ hits $A_0$, then T{\'o}th claims that $\angle BAA_0 < \pi$. Figure~\ref{fig:TothShortLeaf_1} can serve as a counterexample to this claim, because $A_0$ is indeed the first point hit by the rotating ray $\overrightarrow{CA}$, however $\angle BAA_0 > \pi$.
The structure of the original proof can be used with respect to $\overrightarrow{CA'}$, assuming that $A'$ is visible to $C$. Thus, assume first that $C$ and $A'$ can see each other. We rotate $\overrightarrow{CA'}$ around $C$ in the direction of $D'$. Let $O$ be the first vertex of $P'$ visible from $C$ that was hit by the ray (in case there are several collinear such vertices of $P_a$, then let $O$ be the one that is closest to $C$).
If $O = D$ then $A'$ and $D'$ belong to the same edge of $P'$ (notice that it is possible that $D' = D$), so the claim holds.
If $O = A_0$ then $A=A'$, $\angle A = \angle B A A_0 < \pi$ and thus the second claim in T{\'o}th's Lemma~\ref{TothLem2} holds.
If $O \neq D$ and $O \neq A_0$, then $\overline{AO}$ and $\overline{CO}$ are diagonals of $P'$. Refer to Fig.~\ref{fig:TothShortLeaf_2}. There exists a triangulation of $P'$ that contains $\triangle A C O$ and has $\triangle A B C$ as a short leaf. Consider quadrilateral $ABCO$. It is non-convex by T{\'o}th's Claim~\ref{claimT2}. By construction, $O$ is to the right of $\overrightarrow{BA}$, thus $\angle B A O < \pi$. It follows, that the only possible reflex angle in $ABCO$ is $\angle O C B$, which is a contradiction to the minimality of $P_a$ (recall, we assumed that we consider the triangulation of $P'$ in which $n_a$ is minimal), and thus, such a vertex $O$ does not exist.
\begin{figure}
\caption{$\triangle ABC$ corresponds to a short leaf in $G(\mathcal{T})$; $\triangle ACD$ can be degenerate. $\angle D_0 D C > \pi$.}
\label{fig:TothShortLeaf_2}
\end{figure}
Assume now that $A'$ is \textbf{not} visible to $C$. It follows that some part of $\partial P'$ belongs to the interior of $\triangle A A'C$. Recall, that $A'$ is defined as a point such that $A' \in \partial P' \cap \overrightarrow{BA}$ and there exists a point $A'' \in \partial P'$ strictly to the \textbf{right} of $\overrightarrow{BA}$ such that $A'$ and $A''$ belong to the same edge of $P'$ and both visible to $B$ and $A$. Thus, there is no intersection between the part of $\partial P'$ that belongs to the interior of $\triangle A A'C$ and $\overline{AA'}$. It follows, that $\partial P'$ crossed $\overline{CA'}$ at least twice, and thus there must be a vertex of $P_a$ interior to $\triangle A A'C$. Among all the vertices of $P_a$ that lie in $\triangle A A'C$, let $O'$ be the vertex that is closest to the line passing through $\overline{CA}$. Notice, that $O'$ is visible to $B$ (because $\triangle B A' C$ contains $\triangle A A'C$). Notice also that $\overline{AO'}$ and $\overline{CO'}$ are diagonals of $P'$. One of $\overline{AO'}$, $\overline{BO'}$ or $\overline{CO'}$ is a good diagonal dissection, which is a contradiction to the main assumption of Sect~\ref{subsec:Partition} that $P'$ has no good diagonal dissection. Thus, $A'$ is visible to $C$. \qed \end{proof}
It follows from T{\'o}th's Claim~\ref{claimT3} that the quadrilateral $A'BCD'$ has no common points with $\partial P'$ in its interior, but on the boundary only.
We have derived several properties satisfied by $P'$ and now we are ready to show the existence of good (non-diagonal) dissections that consist of one or two line segments. We discuss the following three cases, that span over Claims $4,5$ in~\cite{Toth2000121}.
{\color{red} \textbf{Case 1:}} $\angle D_0 D C < \pi$. In this case $D = D'$. It follows from T{\'o}th's Claim~\ref{claimT3} that $A', C' \in \overline{D D_0}$. Refer to Fig.~\ref{fig:TothShortLeaf_3}.
Line segment $\overline{CC'}$ represents a good dissection that splits $P'$ into two polygons: $P_c \cup \triangle CC'D$ of size $n_c + 1 = 3 k_c + 2$ and $(P' \setminus P_c) \setminus \triangle CC'D$ of size $n_a + 1 = 3 k_a + 2$. Notice that $\overline{BC'}$ is an edge of the subpolygon of $P'$ to the left of $\overrightarrow{BC}$ and thus $C$ is not a vertex of this subpolygon. If $\triangle ACD$ is degenerate (in which case $C$ is between $A$ and $D$), then $\overline{CC'}$ is still a good dissection. However, if $D_0$ is also collinear with $D$, $C$ and $A$, then $D_0$ is visible to $C$ and $\overline{CD_0}$ represents a good diagonal dissection.
\begin{figure}
\caption{$\triangle ABC$ corresponds to a short leaf in $G(\mathcal{T})$; $\triangle ACD$ can be degenerate. $\angle D_0 D C < \pi$; $\overline{CC'}$ is a good dissection.}
\label{fig:TothShortLeaf_3}
\end{figure}
{\color{red} \textbf{Case 2:}} $\angle D_0 D C = \pi$. In this case $D_0 = D'$ and $\overline{CD}$ is a good diagonal dissection that splits $P'$ into $P_c$ of size $n_c = 3 k_c + 1$ and $P_a \cup ABCD$ of size $n_a + 1= 3 k_a + 2$. Notice that $\overline{CD_0}$ is an edge in $P_a \cup ABCD$ and thus $D$ is not a vertex in $P_a \cup ABCD$.
{\color{red} \textbf{Case 3:}} $\angle D_0 D C > \pi$. Let $D_0'$ be the point closest to $D$ where the ray $\overrightarrow{D_0 D}$ reaches $\partial P'$. If the line segments $\overline{CC'}$ and $\overline{DD_0'}$ intersect inside the quadrilateral $C A A' D'$ at $Q$ (refer to Fig.~\ref{fig:TothShortLeaf_4}), then $\overline{DQ} \cup \overline{QC}$ is a good dissection, that splits $P'$ into polygon $P_c \cup \triangle CDQ$ of size $n_c + 1 = 3 k_c + 2$ and polygon $P_a \cup DQBA$ of size $n_a + 1 = 3 k_a + 2$. However, if $C A A' D'$ degenerates into a line segment (which happens when $\triangle ACD$ is degenerate and there exists an edge $\overline{IJ}$ of $P_a$ that contains $\overline{AD}$) then $Q$ cannot be defined. Refer to Fig.~\ref{fig:TothShortLeaf_5}. \begin{figure}
\caption{$\triangle ABC$ corresponds to a short leaf in $G(\mathcal{T})$; $\angle D_0 D C > \pi$. \textbf{(a)} $\overline{DQ} \cup \overline{QC}$ is a good dissection of $P'$. \textbf{(b)} $\triangle ACD$ is degenerate; the edge $\overline{IJ}$ of $P_a$ contains $\overline{DA}$. $P_a$ is highlighted in pink. Notice that $C$ is not a vertex of $P_a$. $P_a'$ is a subpolygon of $P_a$ that contains $D_0$ and $\overline{ID}$ is its edge.}
\label{fig:TothShortLeaf_4}
\label{fig:TothShortLeaf_5}
\label{fig:TothShortLeaf_45}
\end{figure} In this case we show that $P'$ has a good diagonal dissection. Notice that $\overline{ID}$ is a diagonal of $P_a$; it splits $P_a$ into two subpolygons. Let $P_a'$ be a subpolygon of $P_a$ that contains $D_0$. Let $n_a'$ be the size of $P_a'$. We consider three cases: \begin{itemize} \item[$\blacktriangleright$] $n_a' = 3k_a'$: In this case $\overline{IA}$ is a good diagonal dissection. The size of $P_a' \cup P_c \cup \triangle ABC$ is $3k_a' + 3k_c + 1 + 3 - 2 = 3(k_a' + k_c) +2$. Notice that the ``$-2$'' in the previous formula stands for vertices $D$ and $C$ that were counted twice. \item[$\blacktriangleright$] $n_a' = 3k_a' + 1$: In this case $\overline{CJ}$ is a good diagonal dissection. The size of $P_a' \cup P_c \cup IDCJ$ is $3k_a' + 1 + 3k_c + 1 + 1 - 1 = 3(k_a' + k_c) +2$. \item[$\blacktriangleright$] $n_a' = 3k_a' + 2$: In this case $\overline{ID}$ is a good diagonal dissection. \end{itemize}
In this subsection we assumed that $P'$ has no good diagonal dissection, thus we deduce that $C A A' D'$ cannot degenerate into a line segment. Notice that $Q$ exists even when $\triangle ACD$ is degenerate.
\begin{wrapfigure}{r}{0.42\textwidth}
\centering \includegraphics[width=0.36\textwidth]{TothShortLeaf_6.pdf} \caption{$\triangle ABC$ corresponds to a short leaf in $G(\mathcal{T})$; $\angle D_0 D C > \pi$; $\triangle ACD$ can be degenerate. One of $\overline{DD_0'}$, $\overline{CC'}$ and $\overline{AA'}$ is a good dissection of $P'$.} \label{fig:TothShortLeaf_6}
\end{wrapfigure} If $\overline{CC'}$ and $\overline{DD_0'}$ do not intersect inside $C A A' D'$, then $D_0'$ belongs to the line segment $C'D'$. Refer to Fig.~\ref{fig:TothShortLeaf_6}. T{\'o}th shows in Claim $5$ in~\cite{Toth2000121} that one of the line segments $\overline{DD_0'}$, $\overline{CC'}$ and $\overline{AA'}$ is a good dissection.
It was shown so far that if $\triangle ABC$ is a short leaf in $G(\mathcal{T})$, then either $P'$ has a good dissection or the angle at vertex $A$ or $C$ in $P'$ is convex. It is left to prove that T{\'o}th's Lemma~\ref{TothLem2} is true for long leaves. Notice that if $\triangle ABC$ is a long leaf in $G(\mathcal{T})$ but there exists a triangulation of $P'$ where $\triangle ABC$ is a short leaf then T{\'o}th's Lemma~\ref{TothLem2} is true for $\triangle ABC$.
Let $\triangle ABC$ be a long leaf of $G(\mathcal{T})$ such that there does not exist a triangulation of $P'$ where $\triangle ABC$ is a short leaf. Recall that in this subsection, we assumed that $P'$ has no good diagonal dissection. Thus, the node of $G(\mathcal{T})$ adjacent to a long leaf is also adjacent to a node of degree $3$. We also concluded that $\triangle ABC$ cannot be degenerate.
\stepcounter{mydefC} \stepcounter{mydefC} \begin{mydefC} \label{claimT6} If $\triangle ABC$ is a long leaf of $G(\mathcal{T})$ for every triangulation $\mathcal{T}$ of $P'$ then the node of $G(\mathcal{T})$ adjacent to the node $\triangle ABC$ corresponds to the same triangle for every~$\mathcal{T}$. \end{mydefC}
T{\'o}th's Claim~\ref{claimT6} is true for a triangulation $\mathcal{T}$ that contains degenerate triangles and thus it is true for the $P'$ defined in this paper.
Let $\triangle ACD$ be a triangle adjacent to $\triangle ABC$ in $\mathcal{T}$. The ray $\overrightarrow{CA}$ (respectively $\overrightarrow{CD}$, $\overrightarrow{BC}$, $\overrightarrow{BA}$) reaches $\partial P'$ at $A'$ (respectively $D'$, $C'$, $B'$). Notice that $A'$ is defined differently than in the case with short leaves. Refer to Fig.~\ref{fig:TothShortLeaf_7}. By T{\'o}th's Claim~\ref{claimT6}, $\triangle ACD$ is unique. Notice that $\triangle ACD$ can be degenerate, but because it is unique, it does not contain any other vertex of $P'$. Moreover, there does not exist an edge $\overline{IJ}$ of $P'$ that contains $\triangle ACD$, otherwise $P'$ has a good diagonal dissection since $A$, $C$, $D$, $I$ and $J$ can see each other. T{\'o}th's Claim~\ref{claimT7}, that follows this discussion, is thus true for polygons whose triangulation may contain degenerate triangles.
\begin{figure}
\caption{$\triangle ABC$ corresponds to a long leaf in $G(\mathcal{T})$. $\overline{IJ}$ is an edge of $P'$; it contains $A'$, $B'$, $C'$ and $D'$. One of $\overline{DD'}$, $\overline{CC'}$ or $\overline{AB'}$ is a good dissection of $P'$. $P_1$ is highlighted in cyan.}
\label{fig:TothShortLeaf_7}
\end{figure}
\begin{mydefC} \label{claimT7} The points $A'$ and $D'$ belong to the same edge of $P'$ or the angle of $P'$ at $A$ is convex and thus the second condition of T{\'o}th's Lemma~\ref{TothLem2} holds for $\triangle ABC$. Refer to Fig.~\ref{fig:TothShortLeaf_7}. \end{mydefC}
It follows that $C'$ and $B'$ belong to the same edge as $A'$ and $D'$.
Assume that the angles of $P'$ at $A$ and $C$ are reflex, otherwise the second condition of T{\'o}th's Lemma~\ref{TothLem2} holds for $\triangle ABC$ and our proof is complete.
Consider the angle of $P'$ at $D$. It can be either convex or reflex. Notice that non of the angles of $P'$ equals $\pi$ or $0$ because of the simplification step. We discuss the following two cases, that span over Claims $8,9$ in~\cite{Toth2000121}, and show that in either case $P'$ has a good dissection.
{\color{red} \textbf{Case 1:}} $\angle D_0 D C > \pi$. One of the line segments $\overline{DD'}$, $\overline{CC'}$ or $\overline{AB'}$ is a good dissection of $P'$. Refer to Fig.~\ref{fig:TothShortLeaf_7}. $\overline{DD'}$ partitions $P'$ into two subpolygons. Let $P_1$ be one of them that contains $D_0$ (it is highlighted in cyan on Fig.~\ref{fig:TothShortLeaf_7}). The size of $P_1$ is $n_1 = 3k_1 + q_1$; the size of $P_1 \cup CDD'C'$ is $n_1 + 1$; the size of $P_1 \cup CDD'C' \cup ACC'B'$ is $n_1 + 2$. If $q_1 = 2$ (respectively $q_1 = 1$, $q_1 = 0$) then $\overline{DD'}$ (respectively $\overline{CC'}$, $\overline{AB'}$) is a good dissection of $P'$.
{\color{red} \textbf{Case 2:}} $\angle D_0 D C < \pi$. It follows that $D'=D$, $A' \in \overline{DD_0}$ and $A' \neq D_0$, otherwise $\overline{AD_0}$ is a good diagonal dissection. Refer to Fig.~\ref{fig:TothShortLeaf_89}. Let $A_0'$ be the point where $\overrightarrow{A_0A}$ reaches $\partial P'$. T{\'o}th's Claim~\ref{claimT7} implies that $A_0' \in \overline{CD}$ or $A_0' \in \overline{DB'}$. If $A_0' \in \overline{DB'}$ (refer to Fig.~\ref{fig:TothShortLeaf_8}) then $\overline{AA_0'}$ is a good dissection of $P'$. It creates a pentagon $ABCDA_0'$ and a polygon $P' \setminus ABCDA_0'$ whose size is $n-3 = 3(k-1)-2$ (notice that $\overline{AA_0'}$ is an edge of $P' \setminus ABCDA_0'$ and thus $A \notin P' \setminus ABCDA_0'$). Notice that if $A_0' = D$ then $A_0$ can see $D$ and thus $\overline{A_0D}$ is a good diagonal dissection~of~$P'$.
\begin{figure}
\caption{$\triangle ABC$ corresponds to a long leaf in $G(\mathcal{T})$; $\angle D_0 D C < \pi$. \textbf{(a)} Case where$A_0' \in \overline{DB'}$. $\overline{AA_0'}$ is a good dissection of $P'$. \textbf{(b)} Case where $A_0' \in \overline{CD}$.}
\label{fig:TothShortLeaf_8}
\label{fig:TothShortLeaf_9}
\label{fig:TothShortLeaf_89}
\end{figure}
Assume that $A_0' \in \overline{CD}$. Refer to Fig.~\ref{fig:TothShortLeaf_9}. Let us assign labels to the vertices of $P'$ according to their order around $\partial P'$ as follows: $v_0 = A$, $v_1 = A_0$, $v_2$, $\ldots$, $v_{n-4} = D_0$, $v_{n-3} = D$, $v_{n-2} = C$, $v_{n-1} = B$. $\overline{DA}$ is a diagonal of $P'$ and thus at least some interval of $\overline{AA_0}$ is visible to $D$. Let us rotate $\overrightarrow{DA}$ towards $A_0$. The ray hits $v_i$ for $1 < i \leq v_{n-5}$ (notice that the ray cannot hit $A_0$, otherwise $\overline{DA_0}$ is a good diagonal dissection). Observe that the angle of $P'$ at $v_i$ must be reflex. Let $X_1$ (respectively $X_2$) be a point where $\overrightarrow{v_{i+1} v_i}$ (respectively $\overrightarrow{v_{i-1}v_i}$) reaches $\partial P'$. By construction, $X_1 \in \overline{AA_0}$ or $X_1 \in \overline{CD}$. The same is true for $X_2$. If $i$ is not a multiple of 3 then one of $\overline{v_i X_1}$ or $\overline{v_i X_2}$ is a good dissection. \begin{itemize} \item[$\blacktriangleright$] $i \equiv 1$ mod $3$: \begin{itemize}
\item $X_1 \in \overline{AA_0}$. Then the subpolygon $v_1 v_2 \ldots v_i X_1$ has size $3k'+2$, and the subpolygon $A X_1 v_{i+1} v_{i+2} \ldots v_{n-1}$ has size $3k''+2$ for some $k' +k'' = k$ (recall that $n = 3k+2$).
\item $X_1 \in \overline{CD}$. Then the polygon $C B A v_1 v_2 \ldots v_i X_1$ has size $3k'+2$, and the subpolygon $X_1 v_{i+1} v_{i+2} \ldots v_{n-3}$ has size $3k''+2$ for some $k' +k'' = k$.
\end{itemize}
\item[$\blacktriangleright$] $i \equiv 2$ mod $3$: \begin{itemize}
\item $X_2 \in \overline{AA_0}$. Then the polygon $v_1 v_2 \ldots v_{i-1} X_2$ has size $3k'+2$, and the subpolygon $A X_2 v_{i} v_{i+1} \ldots v_{n-1}$ has size $3k''+2$ for some $k' +k'' = k$.
\item $X_2 \in \overline{CD}$. Then the polygon $C B A v_1 v_2 \ldots v_{i-1} X_2$ has size $3k'+2$, and the subpolygon $X_2 v_{i} v_{i+1} \ldots v_{n-3}$ has size $3k''+2$ for some $k' +k'' = k$. \end{itemize} \end{itemize}
If $i$ is a multiple of $3$ then we repeat the above procedure and rotate the ray $\overrightarrow{Dv_i}$ towards $v_{i+1}$. Notice that if the ray hits $v_{i+1} \neq D_0$ then $\overline{D v_{i+1}}$ is a good diagonal dissection (which is a contradiction to our main assumption). Thus the ray hits $v_j$ for $i+1 < j \leq v_{n-5}$. If $j$ is not a multiple of $3$ then one of the rays $\overrightarrow{v_{j+1} v_j}$ or $\overrightarrow{v_{j-1} v_j}$ contains a good dissection. Observe that those rays reach $\partial P'$ at $\overline{CD}$, $\overline{AA_0}$ or $\overline{v_i v_{i+1}}$. Since we perform counting modulo $3$, those edges are considered to be identical in terms of vertices' indices. It means that we do not have to know where exactly $\overrightarrow{v_{j+1} v_j}$ or $\overrightarrow{v_{j-1} v_j}$ reach $\partial P'$ to decide which dissection to apply. That is, if $j \equiv 1$ mod $3$ then we use $\overrightarrow{v_{j+1} v_j}$; if $j \equiv 2$ mod $3$ then we use $\overrightarrow{v_{j-1} v_j}$.
If $j$ is a multiple of $3$ then the procedure is repeated again. Eventually, the ray spinning around $D$ must hit $D_0$. Recall that $D_0 =v_{n-4}$; $n-4 = 3k+2-4 = 3(k-1)+1$, which is not a multiple of $3$. At this point T{\'o}th comes to a contradiction and states that the angle of $P'$ at $D$ cannot be convex (refer to Claim $9$ in~\cite{Toth2000121}). However there is no contradiction. We show that the situation is possible and discuss how to find a good dissection in this case.
Let $v_z$ be the last vertex hit by the ray spinning around $D$ before it hit $D_0$. Notice that $z$ is a multiple of $3$. Two cases are possible: \begin{enumerate} \item $v_{z+1} \neq D_0$: notice that $D_0$ can see $v_z$. The size of the subpolygon $v_z v_{z+1} \ldots v_{n-4}$ is $3k' + 2$ for some integer $k'>1$ ($k$ is strictly bigger than $1$ because $v_{z+1} \neq D_0$). Therefore, the diagonal $\overline{v_z D_0}$ is a good diagonal dissection, meaning that this case is not possible. \item $v_{z+1} = D_0$: in this case $z = n-5$ and the angle of $P'$ at $D_0$ is convex. Refer to Fig.~\ref{fig:TothShortLeaf_1011}. \begin{itemize} \item[$\blacktriangleright$] If $v_z$ can see $A$ then dissect $P'$ along $\overline{v_z A}$. This dissection creates two subpolygons: $Av_zD_0DCB$ of size $6$ and $Av_1 v_2 \ldots v_z$ of size $n-4 = 3(k-1)+1$. Notice that the hexagon $Av_zD_0DCB$ has a non-empty kernel whose intersection with the boundary of $Av_zD_0DCB$ is $\overline{C'B'}$. One $180^\circ$-guard on $\overline{C'B'}$ can monitor $Av_zD_0DCB$. Similarly, for our problem, two distinct towers on $\overline{C'B'}$ can localise an agent in $Av_zD_0DCB$ (notice that $C' \neq B'$ and thus $\overline{C'B'}$ contains at least two distinct points). \item[$\blacktriangleright$] If $v_z$ cannot see $A$ then consider the ray $\overrightarrow{D_0v_z}$. Let $Z$ be a point where $\overrightarrow{D_0v_z}$ reaches $\partial P$ or $\overline{AA'}$ (whichever happens first). \begin{itemize} \item If $Z \in \overline{AA'}$ (refer to Fig.~\ref{fig:TothShortLeaf_10}) then dissect $P'$ along the two line segments $\overline{AZ} \cup \overline{Zv_z}$. $P'$ falls into two subpolygons: $AZv_zD_0DCB$ of size $7$ (which is technically a hexagon since $Z$, $v_z$ and $D_0$ are collinear) and $ZAv_1 v_2 \ldots v_z$ of size $n-3 = 3(k-1)+2$. Similarly to the hexagon from the previous case, the heptagon $AZv_zD_0DCB$ can be guarded by one $180^\circ$-guard on $\overline{C'B'}$ and our agent can be localised by a pair of distinct towers positioned on $\overline{C'B'}$. \item If $Z \notin \overline{AA'}$ (refer to Fig.~\ref{fig:TothShortLeaf_11}) then there must be a vertex $v_x$ to the left of $\overrightarrow{AA'}$ visible to $D_0$ and to $D$. Recall that for every vertex $v_x$ visible from $D$, the index $x$ is a multiple of $3$.
Thus $\overline{v_xD_0}$ is a good diagonal dissection. It means that this case is not possible. \end{itemize} \end{itemize} \end{enumerate}
\begin{figure}
\caption{$\triangle ABC$ corresponds to a long leaf in $G(\mathcal{T})$; $\angle D_0 D C < \pi$; $A_0' \in \overline{CD}$. \textbf{(a)}Two line segments $\overline{AZ} \cup \overline{Zv_z}$ is a good dissection. \textbf{(b)} Impossible case, because $\overline{v_xD_0}$ is a good diagonal dissection.}
\label{fig:TothShortLeaf_10}
\label{fig:TothShortLeaf_11}
\label{fig:TothShortLeaf_1011}
\end{figure}
This completes the proof of T{\'o}th's Lemma~\ref{TothLem2}.
\subsection{Partition algorithm} \label{subsec:PartitionAlg} \pdfbookmark[2]{Partition algorithm}{subsec:PartitionAlg}
We are given a simple polygon $P$ in general position of size $n = 3k +q$ where $k$ is a positive integer and $q \in \{0, 1, 2\}$. If $P$ has at most $k$ reflex angles then $P$ can be partitioned simply by bisecting $k-1$ of the reflex angles. This creates $k$ star-shaped subpolygons of $P$ that can be watched by $2k$ towers. If the number of reflex angles of $P$ is bigger than $k$ then we look for a good diagonal dissection. In Section~\ref{subsec:PointKernel} we modified T{\'o}th's partition~\cite{Toth2000121} to deal with subpolygons of $P$ (polygons*) that are not in general position. The important difference to notice is that we avoided dissecting along diagonals of $P'$ that contain vertices of $P$ in their interior. If the dissection is unavoidable we showed how to position towers and in the worst case - repartition subpolygons of $P'$.
If $P'$ has no good diagonal dissection then its size is $n=3k+2$. In this case we look for a good dissection via the \emph{short} or \emph{long leaf} approach discussed in Section~\ref{subsubsec:Lemma2}. If no good cut is found then by T{\'o}th's Lemma~\ref{TothLem2} every leaf in $G(\mathcal{T})$ is associated to two convex angles of $P'$. It follows by T{\'o}th's Lemma~\ref{TothLem3} that $P'$ can be monitored by $\left \lfloor \frac{n}{3} \right \rfloor$ $180^\circ$-guards. Refer to Section~\ref{subsubsec:Lemma3} on how to adapt T{\'o}th's Lemma~\ref{TothLem3} for tower positioning. We refer to T{\'o}th's Lemma~\ref{TothLem1} to show the coherence of the algorithm. T{\'o}th's Lemma~\ref{TothLem1} states that if $P'$ has $n = 3k+2$ vertices and has no good diagonal or other dissection then $P'$ has at most $k$ reflex angles and thus $P'$ would be treated during the first step of the algorithm.
The obtained partition together with the locations of $180^\circ$-guards is reused for tower positioning. Every $180^\circ$-guard that guards subpolygon $P'$ is positioned on the boundary of $P'$ (either on an edge or a convex vertex of $P'$) and oriented in such a way that $P'$ completely belongs to the half-plane $H_l$ monitored by the guard. In our problem every $180^\circ$-guard is replaced by a pair of towers $t_1$ and $t_2$ on the same edge of $P'$ in $\partial P' \cap kernel(P')$ and close to the $180^\circ$-guard. The orientation of $180^\circ$-guard is embedded into the tower coordinates via the \textbf{parity trick}. Specifically, given the partition of $P$ we can calculate a line segment $\overline{ab}$ suitable for tower positioning for each polygon $P'$ of the partition. Towers can be anywhere on $\overline{ab}$ as long as the distance between them is a rational number. Notice, that the coordinates of the endpoints of $\overline{ab}$ can be irrational numbers because $a$ and $b$ can be vertices of $P$ or $P'$ or intersection points between lines that support edges of $P$ or $P'$. However, we will show that it is always possible to to find the desired positions for the two towers: We place the tower $t_1$ at the point $a$. The tower $t_2$ will be placed at the point $c = a + \lambda (b-a)$ for an appropriate choice of $\lambda$. Let $s \geq 1$ be the smallest integer such that $1/3^s \leq d(a,b)$. If we take $\lambda = \frac{1}{3^{s+1} d(a,b)}$, then $d(a,c) = d(t_1,t_2) = \frac{1}{3^{s+1}}$, which is a reduced rational number whose numerator is odd. On the other hand, if we take $\lambda = \frac{2}{3^{s+1} d(a,b)}$, then $d(a,c) = d(t_1,t_2) = \frac{2}{3^{s+1}}$, which is a reduced rational number whose numerator is even.
If we want the pair of towers $t_1$ and $t_2$ to be responsible for the half-plane $L(t_1, t_2)^+$ (respectively $L(t_1, t_2)^-$) we position the towers at $\frac{2}{3^{t+1}}$ (respectively $\frac{1}{3^{t+1}}$) distance from each other.
Notice that $L(t_1, t_2)$ is not always parallel to the line that supports $H_l$. If the $180^\circ$-guard is positioned at a convex vertex $v$ of $P'$ then only one tower can be positioned at $v$. Another tower is placed on the edge adjacent to $v$ in $\partial P' \cap kernel(P')$. If $kernel(P')$ is a single point then we position our towers outside of $P'$ and close to $kernel(P')$, such that $L(t_1, t_2)$ is parallel to the line that supports $H_l$. If $L(t_1, t_2)^+$ (respectively $L(t_1, t_2)^-$) contains $H_l$ then we position the towers at a distance, which is a rational number whose numerator is even (respectively odd).
Algorithm~\ref{alg_Partition} partitions $P$ and positions at most $\left \lfloor \frac{2n}{3} \right \rfloor$ towers that can localize an agent anywhere in $P$. The localization algorithm (Algorithm~\ref{alg_Locate}) can be found in Section~\ref{sec:localization}.
\begin{algorithm} \caption{Polygon Partition; Tower Positioning}\label{alg_Partition} \KwIn{$P'$ of size $n = 3k +q$, for positive integer $k$ and $q \in \{0, 1, 2\}$} \KwOut{Set of at most $\left \lfloor \frac{2n}{3} \right \rfloor$ towers in $P'$} \BlankLine If $P'$ has at most $k$ reflex angles then position $2k$ towers by bisecting reflex angles. Halt\; Simplify $P'$\; \eIf{there exists a good diagonal dissection that contains at most one vertex of $P$ in its interior}{apply it and run this algorithm on $P_1$ and $P_2$} (\tcp*[f]{$n = 3k +2$}) { \eIf{there exist a good dissection via a continuation of an edge}{apply it; run this algorithm on $P_1$ and $P_2$}{ \eIf{there exists a good diagonal dissection} {this dissection contains two vertices of $P$ in its interior\; apply it and run this algorithm on $P_2$\; \eIf{pentagon with a pair of vertices of $P$ in the interior of the same edge is created}{Repartition $P_2$ as described in Section~\ref{subsec:PointKernel}}{run this algorithm on $P_1$} } (\tcp*[f]{$n = 3k +2$ and $P'$ has no good diagonal dissection}) { {\If{$P'$ has a good dissection via \emph{short} or \emph{long leaf} approach (refer to~\cite{Toth2000121} and Section~\ref{subsubsec:Lemma2})}{use it; repeat algorithm on $P_1$ and $P_2$}
}
} } } \end{algorithm}
The running time of Algorithm~\ref{alg_Partition} is $O(n^3)$ because of the cases where repartitioning of already partitioned subpolygons is required.
In Section~\ref{sec:partition} we showed how to use the polygon partition method introduced by T{\'o}th~\cite{Toth2000121} for wider range of polygons by lifting his assumption that the partition method creates subpolygons whose vertices are in general position. We reproved T{\'o}th's results and showed how to use his partition method for localization problem. We showed how to compute a set $T$ of size at most $\lfloor 2n/3\rfloor$ that can localize an agent anywhere in $P$. The results of Section~\ref{sec:partition} are summarized in the following theorem.
\begin{theorem} \label{thm:main_result} Given a simple polygon $P$ in general position having a total of $n$ vertices. Algorithm~\ref{alg_Partition} computes a set $T$ of broadcast towers of cardinality at most $\lfloor 2n/3\rfloor$, such that any point interior to $P$ can localize itself. \end{theorem}
\subsection{Counterexample to T{\'o}th's conjecture} \label{sec:counterexample} \pdfbookmark[2]{Counterexample to T{\'o}th's conjecture}{sec:counterexample}
We refute the conjecture given by T{\'o}th in~\cite{Toth2000121}.
\textbf{Conjecture:} Any simple polygon of $n$ sides can be guarded by $\lfloor n/3\rfloor$ $180^\circ$-guards that are located exclusively on the boundary of the polygon.
Figure~\ref{fig:counterexample} shows a simple polygon $P$ with $n=8$ that can be guarded by $2$ general $180^\circ$-guards but requires at least $3$ $180^\circ$-guards that must reside on the boundary of $P$. Observe that $P$ is not a star-shaped polygon. Figure~\ref{fig:counterexample11} shows a possible partition of $P$ into two star-shaped polygons: $v_1 v_2 v_3 v_4 v_5$ and $v_1 v_5 v_6 v_7 v_8$. \begin{figure}
\caption{Counterexample on $n=8$ vertices. Guards are highlighted in red. \textbf{(b)} $P$ can be guarded by $2$ general $180^\circ$-guards\textbf{(c)} $P$ requires at least $3$ $180^\circ$-guards that must reside on the boundary of $P$.}
\label{fig:counterexample1}
\label{fig:counterexample11}
\label{fig:counterexample12}
\label{fig:counterexample}
\end{figure} The polygon $v_1 v_2 v_3 v_4 v_5$ can be guarded by the $180^\circ$-guard $g_2$ located on $\overline{v_1 v_5}$ and oriented upwards (i.e. $g_2$ observes $L(v_1, v_5)^+$). The second $180^\circ$-guard $g_1$ is located on $\overline{v_1 v_8}$ in $kernel(v_1 v_5 v_6 v_7 v_8)$. It is oriented to the right of $L(v_1, v_8)$ (i.e. $g_1$ observes $L(v_1, v_8)^-$) and thus guards $v_1 v_5 v_6 v_7 v_8$. Consider Figure~\ref{fig:counterexample12}. The visibility region, from where the complete interior of $v_1 v_2 v_3 v_4 v_5$ can be seen, is highlighted in magenta. We want to assign a single $180^\circ$-guard that can see both vertices $v_2$ and $v_4$ and be located on $\partial P$. Notice that the intersection of this visibility region with $\partial P$ contains a single point $v_3$. However, the angle of $P$ at $v_3$ is reflex and the guards have a restricted $180^\circ$ field of vision. Thus it is impossible to guard $v_1 v_2 v_3 v_4 v_5$ with a single $180^\circ$-guard located on $\partial P$. Notice that the visibility region of the vertex $v_7$ does not intersect with the visibility region of $v_2$ and the visibility region of $v_4$. Thus it requires an additional guard. It follows that $P$ requires at least $3$ $180^\circ$-guards located on $\partial P$. Notice that $\lfloor n/3\rfloor = \lfloor 8/3\rfloor = 2$. This is a contradiction to the above conjecture.
In general, consider the polygon shown in Fig.~\ref{fig:counterexample2}. It has $n = 5s + 2$ vertices, where $s$ is the number of \emph{double-spikes}. Each spike requires its own guard on the boundary of $P$ (two guards per double-spike), resulting in $2s$ boundary guards in total. This number is strictly bigger than $\lfloor n/3\rfloor = \lfloor 5s + 2/3\rfloor$ for $s \geq 3$.
\begin{figure}
\caption{Counterexample to T{\'o}th's conjecture. The polygon $P$ is in general position.}
\label{fig:counterexample2}
\end{figure}
\section{Localization Algorithm} \label{sec:localization} \pdfbookmark[1]{Localization Algorithm}{sec:localization}
In Section~\ref{sec:partition} we showed how to position at most $\left \lfloor \frac{2n}{3} \right \rfloor$ towers in a given polygon. We used a modification of T{\'o}th's partition method that dissects a polygon into at most $\lfloor n/3\rfloor$ star-shaped polygons each of which can be monitored by a pair of towers. In this section we show how we can localize an agent $p$ in the polygon. Our localization algorithm receives as input only the coordinates of the towers that can see $p$ together with their distances to $p$. In this sense, our algorithm uses the classical trilateration input. In addition, our algorithm knows that the parity trick was used to position the towers. Based on this information alone, and without any additional information about $P$, the agent can be localized. When only a pair of towers $t_1$ and $t_2$ can see the point $p \in P$ then the coordinates of the towers together with the distances $d(t_1,p)$ and $d(t_2,p)$ provide sufficient information to narrow the possible locations of $p$ down to two. Those two locations are reflections of each other over the line through $t_1$ and $t_2$. In this situation our localization algorithm uses the parity trick. It calculates the distance between the two towers and judging by the parity of the numerator of this rational number decides which of the two possible locations is the correct one. Refer to Algorithm~\ref{alg_Locate}.
\begin{algorithm}[h] \caption{Compute the coordinates of point $p$.}\label{alg_Locate} \KwIn{\begin{list}{}{} \item $t_1$, \ldots, $t_k$ -- coordinates of the towers that see $p$. \item $d_1$, \ldots, $d_{\ell}$ -- distances between the corresponding towers and $p$. \end{list}} \KwOut{coordinates of $p$.} \BlankLine \eIf{$\ell \geq 3$}{
$p = C(t_1, d_1) \cap C(t_2, d_2) \cap C(t_3, d_3)$\; }(\tcp*[f]{$\ell = 2$}) {
\eIf{the numerator of $d(t_1,t_2)$ is even}{
$p = C(t_1, d_1) \cap C(t_2, d_2) \cap L(t_1, t_2)^+$\;
}
{
$p = C(t_1, d_1) \cap C(t_2, d_2) \cap L(t_1, t_2)^-$\;
} } Return $p$\; \end{algorithm}
\section{Concluding Remarks} \label{sec:conclusion} \pdfbookmark[1]{Concluding Remarks}{sec:conclusion}
We presented a tower-positioning algorithm that computes a set of size at most $\lfloor 2n/3\rfloor$ towers, which improves the previous upper bound of $\lfloor 8n/9\rfloor$~\cite{DBLP:conf/cccg/DippelS15}. We strengthened the work~\cite{Toth2000121} by lifting the assumption that the polygon partition produces polygons whose vertices are in general position. We reproved T{\'o}th's result. We found and fixed mistakes in claims $2$ and $7$ in~\cite{Toth2000121}.
We believe it is possible to avoid the repartition step (described in Section~\ref{subsec:PointKernel}) and as a consequence bring the running time of Algorithm~\ref{alg_Partition} to $O(n^2)$ instead of $O(n^3)$.
As a topic for future research we would like to show that determining an optimal number of towers for polygon trilateration is NP-hard.
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Complete polynomial vector fields on ${\mathbb C}^2$, {\sc Part I}}
\author{Julio C. Rebelo}
\date{}
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\begin{abstract} In this work, under a mild assumption, we give the classification of the complete polynomial vector fields in two variables up to algebraic automorphisms of ${\mathbb C}^2$. The general problem is also reduced to the study of the combinatorics of certain resolutions of singularities. Whereas we deal with ${\mathbb C}$-complete vector fields, our results also apply to ${\mathbb R}$-complete ones thanks to a theorem of Forstneric \cite{forst}. \end{abstract}
\noindent \hspace{0.9cm} {\small Key-words: line at infinity - vector fields - singular foliations}
\noindent \hspace{0.9cm} {\small AMS-Classification 34A20, 32M25, 32J15}
\section{Introduction}
\hspace{0.4cm} Recall that a {\it holomorphic flow} on ${\mathbb C}^2$ is a holomorphic mapping $ \Phi \, : \; {\mathbb C} \times {\mathbb C}^2 \longrightarrow {\mathbb C}^2 $ satisfying the two conditions below:
\noindent $\bullet$ $\Phi (0, p) = p$ for every $p \in {\mathbb C}^2$;
\noindent $\bullet$ $\Phi (T_1 + T_2 ,p ) = \Phi (T_1 , \Phi (T_2, p))$.
A holomorphic flow $\Phi$ on ${\mathbb C}^2$ induces a {\it holomorphic vector field} $X$ on ${\mathbb C}^2$ by the equation $$
X (p) = \left. \frac{d \Phi (T,p)}{dT} \right|_{T=0} \, . $$ Conversely a holomorphic vector field $X$ on ${\mathbb C}^2$ is said to be {\it complete} if it is associated to a holomorphic flow $\Phi$. Since every polynomial vector field of degree~$1$ is complete, we assume that $X$ has degree~$2$ or greater. A polynomial vector field $X$ can be considered as a meromorphic vector field on ${\mathbb C} {\mathbb P} (2)$ therefore inducing a singular holomorphic foliation ${\mathcal F}_X$ on ${\mathbb C} {\mathbb P} (2)$. The singularities of ${\mathcal F}_X$ lying in the ``line at infinity'' $\Delta$ will be denoted by $p_1 ,\ldots ,p_k$. A singularity $p_i$ as above is called {\it dicritical} if there are infinitely many analytic curves invariant by ${\mathcal F}_X$ and passing through $p_i$. The first result of this paper is the following:
\noindent {\bf Theorem A} {\sl Let $X$ be a complete polynomial vector field on ${\mathbb C}^2$ with degree~$2$ or greater and let ${\mathcal F}_X$ be the singular foliation induced by $X$ on ${\mathbb C} {\mathbb P} (2)$. Assume that ${\mathcal F}_X$ has a dicritical singularity in $\Delta$. Then $X$ is conjugate by a polynomial automorphism to one of the following vector fields: \begin{enumerate}
\item $P(y) x^{\epsilon} \partial /\partial x$, where $P(y)$ is a polynomial in $y$ and $\epsilon=0,\, 1$.
\item $x^ny^m(mx {\partial /\partial x} - ny {\partial /\partial y})$, where ${\rm g.c.d} (m,n) =1$ and $m,n \in {\mathbb N}$;
\end{enumerate} }
In view of Theorem~A, we just need to consider vector fields all of whose singularities belonging to $\Delta$ are not dicritical. Again let $X$ be such a vector field and let ${\mathcal F}_X$ be its associated foliation. Consider a singularity $p_i$ of ${\mathcal F}_X$ in the line at infinity $\Delta$ and a vector field $\widetilde{X}$ obtained through a finite sequence of blowing-ups of $X$ beginning at $p_i$. Denote by ${\mathcal E}$ the corresponding exceptional divisor and by $D_i$, $i=1, \ldots ,l$, its irreducible components which are all rational curves. We say that $X$ has adapted poles at $p_i$ if, for every sequence of blow-ups as above and every irreducible component $D_i$ of the corresponsding exceptional divisor ${\mathcal E}$, either $X$ vanishes identically on $D_i$ or $D_i$ consists of pole of $\widetilde{X}$ (in other words $\widetilde{X}$ is not regular on $D_i$). Clearly the great majority of polynomial vector fields have adapted poles at its singularities at infinity. We then have:
\noindent {\bf Theorem B} {\sl Let $X$ be a complete polynomial vector field on ${\mathbb C}^2$ and denote by $p_i$, $i=1, \ldots ,k$ the singularities of the associated foliation ${\mathcal F}_X$ belonging to the line at infinity $\Delta$. Suppose that $X$ has adapted poles at each $p_i$. Then ${\mathcal F}_X$ possesses a dicritical singularity in the line at infinity $\Delta$.}
There is a good amount of literature devoted to complete vector fields on ${\mathbb C}^2$, in particular our results are complementary to the recent results obtained by Cerveau and Scardua in \cite{cesc}. Note however that the points of view adopted in both papers are almost disjoint. For more information on complete polynomial vector fields the reader can consult the references at the end as well as references in \cite{cesc}.
After Theorems~A and~B, in order to classify all complete polinomial vector fields on ${\mathbb C}^2$ we just have to consider vector fields which do not have adapted poles at one of the singularities $p_1, \ldots , p_k$. In particular we can always assume that none of these singularities is dicritical.
Let us close this Introduction by giving a brief description of the structure of the paper. First we observe that the method developed here may be pushed forward to deal with vector fields which do not have adapted poles. Indeed the assumption that $X$ has adapted poles in $\Delta$ is used only in Section~6. More precisely, from Section~2 to Section~5, the classification of complete polynomial vector fields is reduced to a problem of understanding the possible configurations of rational curves arising from blow-ups of the singularities of ${\mathcal F}_X$ in the line at infinity $\Delta$. The role of our main assumption is to make the ``combinatorics'' of these configurations simpler so as to allow for the complete description given in Section~6. It is reasonable to expect that a more detailed study of these configurations will lead to the general classification of complete polynomial vector fields.
Another feature of our method is its local nature. Indeed most of our results are local and therefore have potential to be applied in other situations (especially to other Stein surfaces). We mention in particular the results of Sections~3 and~5 (cf. Theorem~(\ref{selano}), Proposition~(\ref{prop4.2})). Also the local vector fields $Z_{1,11}, \, Z_{0,12}, \, Z_{1,00}$ introduced in Proposition~(\ref{prop4.2}) might have additional interest.
This paper is also a sequence of \cite{re3} where it was observed, in particular, that the problem of understanding complete polynomial vector fields on ${\mathbb C}^2$ can be unified with classical problems in Complex Geometry through the notion of {\it meromorphic semi-complete vector fields}. The method employed in the proof of our main result relies heavily on this connection. Indeed an important part of the proof the preceding theorems is a discussion of semi-complete singularities of meromorphic vector fields. The study of these singularities was initiated in \cite{re3} but the present discussion is based on a different setting.
\noindent {\bf Acknowledgements}: I am grateful to D. Cerveau and B. Scardua who raised my interest in this problem by sending me their preprint \cite{cesc}.
\section{Basic notions and results}
\hspace{0.4cm} The local orbits of a polynomial vector field $X$ induce a singular holomorphic foliation ${\mathcal F}_X$ on ${\mathbb C}^2$. Besides, considering ${\mathbb C} {\mathbb P} (2)$ as the natural compactification of ${\mathbb C}^2$ obtained by adding the line at infinity $\Delta$, the foliation ${\mathcal F}_X$ extend to a holomorphic foliation, still denoted by ${\mathcal F}_X$, on the whole of ${\mathbb C} {\mathbb P} (2)$. This extension may or may not leave the line at infinity $\Delta$ invariant. On the other hand, the vector field $X$ possesses a {\it meromorphic} extension to ${\mathbb C} {\mathbb P} (2)$, also denoted by $X$, whose {\it pole divisor} coincides with $\Delta$. Note that the meromorphic extension of $X$ to ${\mathbb C} {\mathbb P} (2)$ happens to be holomorphic if and only if the degree of $X$ is $1$ or if $X$ has degree $2$ and the line at infinity $\Delta$ is not invariant by ${\mathcal F}_X$ (for further details cf. below). Let us make these notions more precise.
Recall that a {\it meromorphic}\, vector field $Y$ on a neighborhood $U$ of the origin $(0, \ldots , 0) \in {\mathbb C}^n$ is by definition a vector field of the form $$ Y = F_1 \frac{\partial}{\partial z_1} + \cdots + F_n \frac{\partial}{\partial z_n} \, , $$ where the $F_i$'s are meromorphic functions on $U$ (i.e. $F_i = f_i /g_i$ with $f_i ,g_i$ holomorphic on $U$). Note that $Y$ may not be defined on the whole $U$ even though we consider $\infty$ as a value since $F_i$ may have indeterminacy points. We denote by $D_Y$ the union of the sets $\{ g_i =0 \}$. Of course $D_Y$ is a divisor consisting of poles and indeterminacy points of $Y$ which is called the {\it pole divisor} of $Y$
\begin{defnc} The meromorphic vector field $Y$ is said to be semi-complete on $U$ if and only if there exists a meromorphic map $\Phi_{sg} : \Omega \subseteq {\mathbb C} \times U \rightarrow U$, where $\Omega$ is an open set of ${\mathbb C} \times U$, satisfying the conditions below. \begin{enumerate}
\item $$
\left. \frac{d \Phi_{sg} (T, x)}{dT} \right|_{T=0} = Y(x) \; \, \mbox{for all $x \in U \setminus D_x$;} $$
\item $\Phi_{sg} (T_1 + T_2, x) = \Phi_{sg} (T_1, \Phi_{sg} (T_2 ,x))$ provided that both sides are defined;
\item If $(T_i ,x)$ is a sequence of points in $\Omega$ converging to a point $(\hat{T} ,x)$ in the boundary of $\Omega$, then $\Phi_{sg} (T_i ,x)$ converges to the boundary of $U \setminus D_Y$ in the sense that the sequence leaves every compact subset of $U \setminus D_Y$. \end{enumerate} \end{defnc}
The map ${\Phi_{sg}}$ is called the meromorphic semi-global flow associated to $Y$ (or induced by $Y$).
Assume we are given a meromorphic vector field $Y$ defined on a neighborhood of $(0,0) \in {\mathbb C}^2$. It is easy to see that $Y$ has the form $Y = f Z/g$ where $f,g$ are holomorphic functions and $Z$ is a holomorphic vector field having at most an isolated singularity at the origin. Naturally we suppose that $f,g$ do not have a (non-trivial) common factor, so that $f,g$ and $Z$ are unique (up to a trivial, i.e. inversible, factor). Next, let ${\mathcal F}$ denote the local singular foliation defined by the orbits of $Z$. We call ${\mathcal F}$ the foliation associated to $Y$ and note that either ${\mathcal F}$ is regular at the origin or the origin is an isolated singularity of ${\mathcal F}$. An analytic curve ${\mathcal S}$ passing through the origin and invariant by ${\mathcal F}$ is said to be a {\it separatrix} of ${\mathcal F}$ (or of $Y, Z$).
The rest of this section is devoted to establishing some preliminary results concerning both theorems in the Introduction. Particular attention will be paid to meromorphic semi-complete vector fields which appear when we restrict $X$ to a neighborhood of the line at infinity $\Delta$. As it will be seen, a large amount of information on $X$ arises from a detailed study of this restriction. To begin with, let us recall the notion of {\it time-form} $dT$ of a meromorphic vector field and the basic lemma about integrals of $dT$ over curves. For other general facts about meromorphic semi-complete vector fields the reader is referred to Section~2 of \cite{re3}.
Let $Y$ be a meromorphic vector field defined on an open set $U$ and let ${\mathcal F}$ denote its associated foliation. The regular leaves of ${\mathcal F}$ (after excluding possible punctures corresponding to {\it zeros} or poles of $Y$) are naturally equipped with a {\it foliated} holomorphic $1$-form $dT$ defined by imposing $dT . Y = 1$. As a piece of terminology, whenever the $1$-form $dT$ is involved, the expression ``regular leaf of ${\mathcal F}$'' should be understood as a regular leaf $L$ (in the sense of the foliation ${\mathcal F}$) from which the intersection with the set of {\it zeros} or poles of $Y$ was deleted. Hence the restriction of $dT$ to a regular leaf $L$ is, by this convention, always holomorphic. Note also that $dT$ is ``foliated'' in the sense that it is defined only on the tangent spaces of the leaves. We call $dT$ the {\it time-form} associated to (or induced by) $Y$. Lemma~(\ref{timeform}) below is the most fundamental result about semi-complete vector fields (cf. \cite{re1}, \cite{re3}).
\begin{lema} \label{timeform} Let $Y$, $U$, ${\mathcal F}$ and $dT$ be as above. Consider a regular leaf $L$ of ${\mathcal F}$ and an embedded (open) curve $c: [0,1] \rightarrow L$. If $Y$ is semi-complete on $U$ then the integral of $dT$ over $c$ does not vanish.
\fbox{} \end{lema}
Let us now go back to a complete polynomial vector field $X$ on ${\mathbb C}^2$ whose degree is $d \in {\mathbb N}$. Set \begin{equation} X = X_0 + X_1 + \cdots + X_d \label{equa1} \end{equation} where $X_i$, $i =1 ,\ldots ,d$, stands for the homogeneous component of degree $i$ of $X$. With this notation the vector fields whose associated foliations ${\mathcal F}_X$ do not leave the line at infinity $\Delta$ invariant admit an elementary characterization namely: ${\mathcal F}_X$ does not leave $\Delta$ invariant if and only if $X_d$ has the form $F(x,y) (x {\partial /\partial x} \, + \, y {\partial /\partial y})$, where $F$ is a homogeneous polynomial of degree $d-1$. Furthermore, viewing $X$ as a meromorphic vector field on ${\mathbb C} {\mathbb P} (2)$, a direct inspection shows that the order of the pole divisor $\Delta$ is $d-1$ provided that $\Delta$ is invariant under ${\mathcal F}_X$. If $\Delta$ is not invariant under ${\mathcal F}_X$ then this order is $d-2$. On the other hand, given a point $p \in \Delta$ and a neighborhood $U \subset {\mathbb C} {\mathbb P} (2)$ of $p$, it is clear that $X$ defines a meromorphic semi-complete vector field on $U$.
Our first lemma shows that we can suppose that the line at infinity is invariant by the associated foliation ${\mathcal F}_X$. Whereas the proof is elementary, we give a detailed account since some basic ideas will often be used later on.
\begin{lema} \label{lema2.1} Consider a complete polynomial vector field $X$ on ${\mathbb C}^2$ and denote by ${\mathcal F}_X$ the foliation induced by $X$ on ${\mathbb C} {\mathbb P} (2)$. Assume that the line at infinity $\Delta$ is not invariant under ${\mathcal F}_X$. Then the degree of $X$ is at most $1$. \end{lema}
\noindent {\it Proof}\,: First we set $X = F Z$ where $F$ is a polynomial of degree $0 \leq n \leq d$ and $Z$ is a polynomial vector field of degree $d-n$ and isolated {\it zeros}. In other words, we have $Z = P {\partial /\partial x} + Q {\partial /\partial y}$ where $P, Q$ for polynomials $P,Q$ without non-trivial common factors.
First we suppose for a contradiction that $d$ is strictly greater than $2$. In view of the preceding discussion, the line at infinity $\Delta$ is the polar divisor of $X$ and has order $d-2 \geq 1$. Let ${\mathcal C} \subset {\mathbb C} {\mathbb P} (2)$ be the algebraic curve induced in affine coordinates by $F=0$. Finally consider a ``generic'' point $p \in \Delta$ and a neighborhood $U$ of $p$ such that $U \cap {\mathcal C} = \emptyset$.
Let ${\mathcal F}_X$ be the singular foliation induced by $X$ on ${\mathbb C} {\mathbb P} (2)$ and notice that $p$ is a regular point of ${\mathcal F}_X$. Besides the leaf $L$ containing $p$ is transverse at $p$ to $\Delta$. Thus we can introduce coordinates $(u,v)$ on $U$, identifying $p$ with $(0,0) \in {\mathbb C}^2$, in which $X$ becomes $$ X(u,v) = u^{2-d}. f. \frac{\partial}{\partial u} $$ where $f$ is a holomorphic function such that $f(0,0) \neq 0$ and $\{ u = 0 \} \subset \Delta$ (here we use the fact that $U \cap {\mathcal C} = \emptyset$). The axis $\{ v=0 \}$ is obviously invariant under ${\mathcal F}$ and the time-form $dT_{\{v = 0\}}$ induced on $\{ v =0 \}$ by $X$ is given by $dT_{\{v = 0\}} = u^{d-2} du / f(u,0)$. Since $d \geq 3$ and $f(0,0) \neq 0$, it easily follows the existence of an embedded curve $c:[0,1] \rightarrow \{ v=0 \} \setminus (0,0)$ on which the integral of $dT_{\{v = 0 \}}$ (cf. Remark~(\ref{obsidiota})). The resulting contradiction shows that $d \leq 2$.
It only remains to check the case $d=2$. Modulo a linear change of coordinates, we have $X= X_0 + X_1 + x( x {\partial /\partial x} \, + \, y {\partial /\partial y})$. The above calculation shows that the natural extension of $X$ to ${\mathbb C} {\mathbb P} (2)$ is, in fact, holomorphic. Therefore $X$ is complete on ${\mathbb C} {\mathbb P} (2)$ since ${\mathbb C} {\mathbb P} (2)$ is compact. Besides a generic point $p$ of $\Delta$ is regular for $X$ and has its (local) orbit transverse to $\Delta$. It follows that points in the orbit of $p$ reaches $\Delta$ in finite time. Thus the restriction of $X$ to the affine ${\mathbb C}^2$ cannot be complete as its flow goes off to infinity in finite time.
\fbox{}
In view of Lemma~(\ref{lema2.1}) all complete polynomial vector fields considered from now on will be such that the associated foliation ${\mathcal F}_X$ leaves the line at infinity $\Delta$ invariant. In particular, in the sequel, the extension of $X$ to ${\mathbb C} {\mathbb P} (2)$ is strictly meromorphic. Furthermore the pole divisor is constituted by the line at infinity $\Delta$ and has order $d-1$ (where $d$ is the degree of $X$).
\begin{lema} \label{lema2.2} Let $X$ be as above and let $X_d$ be its top-degree homogeneous component (as in (\ref{equa1})). Then $X_d$ is semi-complete on the entire ${\mathbb C}^2$. \end{lema}
\noindent {\it Proof}\,: For each integer $k \in {\mathbb N}$, we consider the homothety $\Lambda_k (x,y) = (k x , k y)$ of ${\mathbb C}^2$. The vector fields defined by $\Lambda_k^{\ast} X$ are obviously complete, and therefore semi-complete, on ${\mathbb C}^2$. Next we set $X^k = k^{1-d} \Lambda_k^{\ast} X$. Since $X^k$ and $\Lambda_k^{\ast} X$ differ by a multiplicative constant, it is clear that $X^k$ is complete on ${\mathbb C}^2$. Finally, when $k$ goes to infinity, $X^k$ converges uniformly on ${\mathbb C}^2$ towards $X_d$. Since the space of semi-complete vector fields is closed under uniform convergence (cf. \cite{ghre}), it results that $X_d$ is semi-complete on ${\mathbb C}^2$. The lemma is proved.
\fbox{}
The lemma above has an immediate application. In fact, a homogeneous polynomial vector field has a $1$-parameter group of symmetries consisting of homotheties. Hence these vector fields can essentially be integrated. In other words, it is possible to describe all homogeneous polynomial vector fields which are semi-complete on ${\mathbb C}^2$. This classification was carried out in \cite{ghre} and, combined with Lemma~(\ref{lema2.2}), yields:
\begin{coro} \label{lema2.3} Let $X$ and $X_d$ be as in the statement of Lemma~(\ref{lema2.2}). Then, up to a linear change of coordinates of ${\mathbb C}^2$, $X_d$ has one of the following normal forms: \begin{enumerate}
\item $X_d = y^a f(x,y) {\partial /\partial x}$ where $f$ has degree strictly less than~$3$ and $a \in {\mathbb N}$.
\item $X_d = x(x {\partial /\partial x} + ny {\partial /\partial y})$, $n \in N$, $n\neq 1$.
\item $X_d = x^i y^j (mx {\partial /\partial x} - n y {\partial /\partial y})$ where $m,n \in {\mathbb N}^{\ast}$ and $ni-mj = -1, \, 0 \, 1$. We also may have $X_d = (xy)^a (x {\partial /\partial x} - y {\partial /\partial y})$, $a \in {\mathbb N}$.
\item $X_d = x^2 {\partial /\partial x} - y(nx-(n+1)y) {\partial /\partial y}$, $n \in {\mathbb N}$.
\item $X_d = [xy(x-y)]^a [x(x-2y) {\partial /\partial x} + y(y-2x) {\partial /\partial y}]$, $a \in {\mathbb N}$.
\item $X_d = [xy(x-y)^2]^a [x(x-3y) {\partial /\partial x} + y(y-3x) {\partial /\partial y}]$, $a \in {\mathbb N}$.
\item $X_d = [xy^2(x-y)^3]^a [x(2x-5y) {\partial /\partial x} + y(y-4x) {\partial /\partial y}]$, $a \in {\mathbb N}$.
\fbox{} \end{enumerate} \end{coro} As an application of Corollary~(\ref{lema2.3}), we shall prove Lemma~(\ref{lema2.4}) below. This lemma estimates the number of singularities that the foliation ${\mathcal F}_X$ induced by $X$ on ${\mathbb C} {\mathbb P} (2)$ may have in the line at infinity. This estimate will be useful in Section~7. Also recall that the line at infinity is supposed to be invariant by ${\mathcal F}_X$ (cf. Lemma~(\ref{lema2.1})).
\begin{lema} \label{lema2.4} Let $X$ be a complete polynomial vector field on ${\mathbb C}^2$ and denote by ${\mathcal F}_X$ the foliation induced by $X$ on ${\mathbb C} {\mathbb P} (2)$. Then the line at infinity contains at most $3$ singularities of ${\mathcal F}_X$. \end{lema}
\noindent {\it Proof}\,: We consider the change of coordinates $u=1/x$ and $v = y/x$ and note that in the coordinates $(u,v)$ the line at infinity $\Delta$ is represented by $\{ u=0\}$.
First we set $X_d = F. Y_d$ where $F$ is a polynomial of degree $k$ and $Y_d$ is a polynomial vector field of degree $d-k$. Next let us consider the algebraic curve ${\mathcal C} \subset {\mathbb C} {\mathbb P} (2)$ induced on ${\mathbb C} {\mathbb P} (2)$ by the affine equation $F=0$. We also consider the foliation ${\mathcal F}_d$ induced on ${\mathbb C} {\mathbb P} (2)$ by $Y_d$. Finally denote by $\Delta \cap {\rm Sing} ({\mathcal F}_X)$ (resp. $\Delta \cap {\rm Sing} ({\mathcal F}_d)$) the set of singularities of ${\mathcal F}_X$ (resp. ${\mathcal F}_d$) belonging to $\Delta$. An elementary calculation with the coordinates $(u,v)$ shows that $$ \Delta \cap {\rm Sing} ({\mathcal F}_X) \subseteq (\Delta \cap {\rm Sing} ({\mathcal F}_d)) \cup ({\mathcal C} \cap \Delta) \, . $$ Now a direct inspection in the list of Corollary~(\ref{lema2.3}) implies that the set $(\Delta \cap {\rm Sing} ({\mathcal F}_d)) \cup ({\mathcal C} \cap \Delta)$ consists of at most~$3$ points. The proof of the lemma is over.
\fbox{}
\section{Simple semi-complete singularities}
\hspace{0.4cm} In this section we shall begin the study of a certain class of semi-complete singularities. The results obtained here will largely be used in the remaining sections. In the sequel $Y$ stands for a meromorphic vector field defined on a neighborhood of $(0,0) \in {\mathbb C}^2$. We always set $Y = fZ/g$ where $f,g$ are holomorphic functions without common factors and $Z$ is a holomorphic vector field for which the origin is either a regular point or an isolated singularity. Also ${\mathcal F}$ will stand for the foliation associated to $Y$ (or to $Z$). We point out that the decomposition $Y = fZ/g$ is unique up to an inversible factor.
If $Z$ is singular at $(0,0)$, we can consider its eigenvalues at this singularity. Three cases can occur:
\noindent {\bf a-} Both eigenvalues, $\lambda_1 , \lambda_2$ of $Z$ vanish at $(0,0)$.
\noindent {\bf b-} Exactly one eigenvalue, $\lambda_2$, vanishes at $(0,0)$.
\noindent {\bf c-} None of the eigenvalues $\lambda_1 , \lambda_2$ vanishes at $(0,0)$
\noindent Whereas $Z$ is defined up to an inversible factor, all the cases {\bf a}, {\bf b} and {\bf c} are well-defined. In the case {\bf c}, the precise values of $\lambda_1 , \lambda_2$ are not well-defined but so is their ratio $\lambda_1 / \lambda_2$. Following a usual abuse of notation, in the case {\bf c}, we shall say that the foliation ${\mathcal F}$ associated to $Z$ has eigenvalues $\lambda_1 , \lambda_2$ different from {\it zero}. In other words, given a singular holomorphic foliation ${\mathcal F}$, we say that ${\mathcal F}$ has eigenvalues $\lambda_1 ,\lambda_2$ if there exists $Z$ as before having $\lambda_1 ,\lambda_2$ as its eigenvalues at $(0,0)$. The reader will easily check that all the relevant notions discussed depend only on the ratio $\lambda_1 / \lambda_2$. A singularity is said to be {\it simple} if it has at least one eigenvalue different from {\it zero}. A simple singularity possessing exactly one eigenvalue different from {\it zero} is called a {\it saddle-node}.
More generally the {\it order} of ${\mathcal F}$ at $(0,0)$ is defined as the order of $Z$ at $(0,0)$, namely it is the degree of the first non-vanishing homogeneous component of the Taylor series of $Z$ based at $(0,0)$. It is obvious that the order of ${\mathcal F}$ does not depend on the vector field with isolated singularities $Z$ chosen.
\begin{obs} \label{obsidiota} {\rm A useful fact is the non-existence, in dimension~$1$, of {\it strictly meromorphic} semi-complete vector fields. In other words, if $Y = f(x) \partial /\partial x$ with $f$ meromorphic, then $Y$ is not semi-complete on a neighborhood of $0 \in {\mathbb C}$. In fact, fixed a neighborhood $U$ of $0 \in {\mathbb C}$, we have $f(x) = x^{-n}. h(x)$ where $n \geq 1$, $h$ is holomorphic and $h (0) \neq 0$. Thus the corresponding time-form is $dT = x^n dx /h$. It easily follows the existence of an embedded curve $c : [0,1] \rightarrow U \setminus \{ (0,0) \}$ on which the integral the integral of $dT$ vanishes. Similarly we can also prove that $Y$ is not semi-complete provided that $0 \in {\mathbb C}$ is an essential singularity of $f$.
Summarizing the preceding discussion, the fact that $Y$ is semi-complete implies that $f$ is holomorphic at $0 \in {\mathbb C}$. Consider now that $f$ is holomorphic but $f(0) = f'(0) = f''(0) =0$. An elementary estimate (cf. \cite{re1}) shows that in this case $Y$ is not semi-complete. Finally when $Y$ is semi-complete but $f(0) = f'(0) =0$ it is easy to see that $Y$ is conjugate to $x^2 {\partial /\partial x}$ (cf. \cite{ghre}). These elementary results give a complete description of semi-complete singularities in dimension~$1$.} \end{obs}
Let us say that $P = P_{\alpha} /P_{\beta}$ is a {\it homogeneous rational function} if $P_{\alpha}$ and $P_{\beta}$ are homogeneous polynomials (possibly with different degrees). The next lemma is borrowed from \cite{re3}
\begin{lema} \label{le3.1} Consider the linear vector field $Z = x \partial /\partial x + \lambda y \partial /\partial y$. Suppose that $P = P_{\alpha} /P_{\beta}$ is a (non-constant) homogeneous rational function and that $\lambda \not\in {\mathbb R}_{+}$. Suppose also that $P Z$ is semi-complete. Then one has
\noindent 1. $\lambda$ is rational, i.e. $\lambda =-n/m$ for appropriate coprime positive integers $m,n$.
\noindent 2. $P$ has one of the forms below:
\noindent {\bf 2i} $P = x^c y^d$ where $mc-nd=0$ or $\pm 1$.
\noindent {\bf 2ii} If $\lambda =-1$, then $P$ is $x^c y^d$ ($mc-nd=0$ or $\pm 1$) or $P = (x-y)(xy)^a$ for $a \in {\mathbb Z}$.
\fbox{} \end{lema}
\begin{obs} \label{ipc} {\rm Consider a holomorphic vector field of the form $$ x^a y^b h(x,y) [ x(1 + {\rm h.o.t.}) {\partial /\partial x} \; - \; y(1 + {\rm h.o.t.}) {\partial /\partial y} \, , $$ where $a,b \in {\mathbb Z}$ and $h$ is holomorphic with $h(0,0) =0$. Of course we suppose that $h$ is not divisible by $x,y$. Next we assume that $X$ is semi-complete on a neighborhood of $(0,0)$. Denote by $h^k$ the homogeneous component of the first non-trivial jet of $h$ at $(0,0)$. The same argument employed in the proof of Lemma~(\ref{lema2.2}), modulo replacing $k$ by $1/k$, shows that the vector field $x^a y^b h^k (x {\partial /\partial x} - y {\partial /\partial y})$ is semi-complete. From the preceding lemma it then follows that $h^k = x-y$ and $a=b$. However a much stronger conclusion holds: $X$ admits the normal form $$ (xy)^a (x-y) g(x {\partial /\partial x} - y {\partial /\partial y}) \, , $$ where $g$ is a holomorphic function satisfying $g(0,0) \neq 0$. In fact, in order to deduce the normal form above, we just need to check that $x(1 + {\rm h.o.t.}) {\partial /\partial x} \; - \; y(1 + {\rm h.o.t.}) {\partial /\partial y}$ is linearizable. After \cite{mamo}, this amounts to prove that the local holonomy of their separatrizes is the identity. However the integral of time-form on a curve $c$ projecting onto a loop around $0$ in $\{ y=0\}$ is cearly equal to {\it zero}. Since $X$ is semi-complete, such curve must be closed which means that the holonomy in question is trivial.} \end{obs}
Of course the next step is to discuss the case $\lambda >0$. However, at this point, we do not want to consider only linear vector fields. This discussion will naturally lead us to consider singularities having an infinite number of separatrizes. Recall that a singularity of a holomorphic foliation ${\mathcal F}$ is said to be {\it dicritical} if ${\mathcal F}$ possesses infinitely many separatrizes at $p$. Sometimes we also say that a vector field $Y$ defined on a neighborhood of $(0,0) \in {\mathbb C}^2$ is dicritical to say that $(0,0)$ is a dicritical singularity of the foliation associated to $Y$. Let us begin with the following:
\begin{lema} \label{le3.2} Consider a semi-complete meromorphic vector field $Y$ defined on a neighborhood of $(0,0) \in {\mathbb C}^2$ and having the form $$ Y = \frac{f}{g} Z $$ where $f,g$ are holomorphic functions with $f(0,0)g(0,0)=0$ and $Z$ is a holomorphic vector field with an isolated singularity at $(0,0) \in {\mathbb C}^2$ whose eigenvalues are $1$ and $\lambda$. Assume that $\lambda >0$ but neither $\lambda$ nor $1/\lambda$ belongs to ${\mathbb N}$. Then $Z,Y$ are dicritical vector fields. \end{lema}
\noindent {\it Proof}\,: Note that Poincar\'e's linearization theorem \cite{ar} ensures that $Z$ is linearizable. Therefore, in appropriate coordinates, we have $Y =f(x {\partial /\partial x} + \lambda y {\partial /\partial y}) / g$. If $\lambda$ is rational equal to $n/m$, then $Z$ admits the meromorphic first integral $x^n y^{-m}$ and therefore admits an infinite number of separatrizes.
In order to prove the lemma is now sufficient to check that $\lambda$ is rational provided that $Y$ is semi-complete. Let $P_{\alpha}$ (resp. $P_{\beta}$) be the first non-vanishing homogeneous component of the Taylor series of $f$ (resp. $g$) at $(0,0) \in {\mathbb C}^2$. The same argument carried out in the proof of Lemma~(\ref{lema2.2}), modulo replacing $k$ by $1/k$, shows that the vector field $Y^{\rm ho} = P_{\alpha} (x {\partial /\partial x} + \lambda y {\partial /\partial y}) /P_{\beta}$ is semi-complete on ${\mathbb C}^2$. We are going to see that this implies that $\lambda$ is rational.
Suppose for a contradiction that $\lambda$ is not rational. In this case the only separatrizes of $Y^{\rm ho}$ are the axes $\{ x=0 \}$, $\{ y=0\}$. Since in dimension~$1$ there is no meromorphic semi-complete vector field, it follows that the {\it zero set} of $P_{\beta}$ has to be invariant under the foliation ${\mathcal F}_{Y^{\rm ho}}$ associated to $Y^{\rm ho}$. Thus $P_{\beta}$ must have the form $x^a y^b$ for some $a,b \in {\mathbb N}$. Therefore we can write $P$ as $x^cy^d Q(x,y)$ where $Q$ is a homogeneous polynomial.
Observe that the orbit $L$ of $Y^{\rm ho}$ (or ${\mathcal F}_{Y^{\rm ho}}$) passing through the point $(x_1 ,y_1)$ ($x_1 y_1 \neq 0$) is parametrized by $\textsf{A} : \, T \mapsto (x_1 e^T , y_1 e^{\lambda T})$. The restriction to $L$ of the vector field $P Z$ is given in the coordinate $T$ by $P(x_1 e^T , y_1e^{\lambda T}) \partial /\partial T$. Because $\lambda$ is not rational, the parametrization $\textsf{A}$ is an one-to-one map from ${\mathbb C}$ to $L$. It results that the one-dimensional vector field $x_1^cy_1^d e^{(c+\lambda d)T} Q(x_1e^T , y_1^{\lambda T}) \partial /\partial T$ is semi-complete on the entire ${\mathbb C}$. On the other hand the function $T \mapsto e^{(c+\lambda d)T} Q(x_1e^T , y_1^{\lambda T})$ is defined on the whole of ${\mathbb C}$. Since $\lambda$ is not rational and $Q$ is a polynomial, we conclude that this function has an essential singularity at infinity. This contradicts the fact that this function corresponds to a semi-complete vector field (cf. Remark~(\ref{obsidiota})). The lemma is proved.
\fbox{}
Let us now consider the case $\lambda \in {\mathbb N}$ since the case $1/\lambda \in {\mathbb N}$ is analogous. Thus we denote by $1$ and $n \in {\mathbb N}^{\ast}$ the eigenvalues of $Z$ at $(0,0)$. Such $Z$ is either linearizable or conjugate to its Poincar\'e's-Dulac normal form \cite{ar} \begin{equation} (nx + y^n) {\partial /\partial x} \; + \; y {\partial /\partial y} \, . \label{awui} \end{equation} When $Z$ is linearizable, it has infinitely many separatrizes. Thus we are, in fact, interested in the case in which $Z$ is conjugate to the Poincar\'e-Dulac normal form~(\ref{awui}). In particular $\{ y =0 \}$ is the unique separatrix of $Z$ (or $Y$).
\begin{lema} \label{le3.3} Let $Y$ be $Y = fZ/g$ where $Z$ is a vector field as in~(\ref{awui}) and $f,g$ are holomorphic functions satisfying $f(0,0) g (0,0) =0$. Assume that $Y$ is semi-complete and that the regular orbits of $Y$ can contain at most one singular point whose order is necessarily~$1$. Then $n=1$. Furthermore, up to an inversible factor, $g(x,y) = y$ and $\{ f= 0\}$ defines a smooth analytic curve which is not tangent to $\{ y=0\}$. \end{lema}
\noindent {\it Proof}\,: Since the divisor of poles of $Y$ is contained in the union of the separatrizes, it follows that $Y$ has the form $$ Y = y^{k} F(x,y) [(nx + y^n) {\partial /\partial x} \; + \; y {\partial /\partial y}] \, , $$ where $k \in {\mathbb Z}$ and $F$ is a holomorphic function which is not divisible by $y$. Clearly $k < 0$, otherwise the first homogeneous component of $Y$ would not be semi-complete (cf. Corollary~(\ref{lema2.3}) and Remark~(\ref{obsidiota})).
Let us first deal with the case $n=1$. Note that we are going to strongly use Theorem~(\ref{selano}) which is the next result to be proved. This theorem is concerned with the so-called {\it saddle-node} singularities which are those having exactly one eigenvalue different from {\it zero}. Blowing-up the vector field $Y$ we obtain a new vector field $\widetilde{Y}$ defined and semi-complete on a neighborhood of the exceptional divisor $\pi^{-1} (0)$ (where $\pi$ stands for the blow-up map). Denote by $\widetilde{\mathcal F}$ the foliation associated to $\widetilde{Y}$ and note that $\widetilde{\mathcal F}$ has a unique singularity $p \in \pi^{-1} (0)$. More precisely, $\widetilde{Y}$ on a neighborhood of $p$ is given in standard coordinates $(x,t)$ ($\pi (x,t) = (x, tx)$), by $$ H (x,t) [(x(1+t) {\partial /\partial x} \; - \; t^2 \partial /\partial t] \, , $$ where $H$ is meromorphic function. Because of Theorem~(\ref{selano}), we know that the restriction of $\widetilde{Y}$ to the exceptional divisor $\{ x=0\}$ has to be regular i.e. $H$ is not divisible by $x$ or $x^{-1}$. This implies that the order of $F$ at $(0,0) \in {\mathbb C}^2$ is $k$ and, in particular, $F(0,0) =0$.
On the other hand, $H$ has the form $t^k h (x,t)$ where $h$ is holomorphic on a neighborhood of $x=t=0$ and not divisible by $t$ or $t^{-1}$. Again Theorem~(\ref{selano}) shows that $h$ has to be an inversible factor, i.e. $h(0,0) \neq 0$. In other words, the proper transform of the (non-trivial) analytic curve $F=0$ intersects $\pi^{-1} (0)$ at points different from $x=t=0$.
The restriction of $\widetilde{Y}$ to $\pi^{-1} (0)$ is a holomorphic vector field which has a singularity at $\{x=t=0\}$ whose order is $2-k$. In particular $k \in \{ 0, 1,2\}$. The other singularities correspond to the intesection of $\pi^{-1} (0)$ with the proper transform of $\{ F= 0\}$. The statement follows since the regular orbits of $X$ can contain only one singular point whose order is~$1$.
\fbox{}
In the rest of this section we briefly discuss the case of singularities as in {\bf b}, that is, those singularities having exactly one eigenvalue different from {\it zero}. As mentioned they are called {\it saddle-nodes} and were classified in \cite{mara}. A consequence of this classification is the existence of a large moduli space. The subclass of saddle-nodes consisting of those associated to semi-complete holomorphic vector fields was characterized in \cite{re4}. In the sequel we summarize and adapt these results to meromorphic semi-complete vector fields.
To begin with, let $\omega$ be a singular holomorphic $1$-form defining a saddle-node ${\mathcal F}$. According to Dulac \cite{dulac}, $\omega$ admits the normal form $$ \omega (x,y) = [x(1+ \lambda y^p) + yR(x,y)] \, dy \; - \; y^{p+1} \, dx \; , $$ where $\lambda \in {\mathbb C}$, $p \in {\mathbb N}^{\ast}$ and $R(x,0) = o (\mid x \mid^p)$. In particular ${\mathcal F}$ has a (smooth) separatrix given in the above coordinates by $\{ y=0\}$. This separatrix is often referred to as the {\it strong invariant manifold} of ${\mathcal F}$. Furthermore there is also a {\it formal} change of coordinates $(x,y) \mapsto (\varphi (x,y) ,y)$ which brings $\omega$ to the form \begin{equation} \omega (x,y) = x(1 +\lambda y^{p+1}) \, dy \; - y^{p+1} \, dx \; . \label{fnf} \end{equation} The expression in (\ref{fnf}) is said to be the {\it formal normal form} of ${\mathcal F}$. In these formal coordinates the axis $\{ x=0\}$ is invariant by ${\mathcal F}$ and called the {\it weak invariant manifold} of ${\mathcal F}$. Note however that the weak invariant manifold of ${\mathcal F}$ does not necessarily correspond to an actual separatrix of ${\mathcal F}$ since the change of coordinates $(x,y) \mapsto (\varphi (x,y) ,y)$ does not converge in general. Finally it is also known that a saddle-node ${\mathcal F}$ possesses at least one and at most two separatrizes (which are necessarily smooth) depending on whether or not the weak invariant manifold of ${\mathcal F}$ is convergent.
A general remark about saddle-nodes is the following one: denoting by $\pi_2$ the projection $\pi_2 (x,y) = y$, Dulac's normal form implies that the fibers of $\pi_2$, namely the vertical lines, are transverse to the leaves of ${\mathcal F}$ away from $\{ y=0\}$. This allows us to define the monodromy of ${\mathcal F}$ as being the ``first return map'' to a fixed fiber.
As to semi-complete vector fields whose associated foliation ${\mathcal F}$ is a saddle-node, one has:
\begin{teo} \label{selano} Suppose that $Y$ is a meromorphic semi-complete vector field defined around $(0,0) \in {\mathbb C}^2$. Suppose that the foliation ${\mathcal F}$ associated to $Y$ is a saddle-node. Then, up to an inversible factor, $Y$ has one of the following normal forms: \begin{enumerate} \item $Y = x(1+ \lambda y) {\partial /\partial x} + y^2 {\partial /\partial y}$, $\lambda \in {\mathbb Z}$.
\item $Y = y^{-p} [(x(1+ \lambda y^p) + yR(x,y)) {\partial /\partial x} + y^{p+1} {\partial /\partial y}]$.
\item $Y = y^{1-p} [(x(1+ \lambda y^p) + yR(x,y)) {\partial /\partial x} + y^{p+1} {\partial /\partial y}]$ and the monodromy induced by ${\mathcal F}$ is trivial (in particular $\lambda \in {\mathbb Z}$ and the weak invariant manifold of ${\mathcal F}$ is convergent). \end{enumerate} \end{teo}
\noindent {\it Proof}\,: The proof of this theorem relies heavily on the methods introduced in Section~4 of~\cite{re2} and Section~4 of~\cite{re4}. For convenience of the reader we summarize the argument below.
First we set $Y=fZ/g$ where $Z$ is a holomorphic vector field with an isolated singularity at $(0,0)$. Note that when $f(0,0) .g(0,0) \neq 0$ (i.e. when $Y$ is holomorphic with an isolated singularity at $(0,0)$), then $Y$ has the normal form~1. Indeed this is precisely the content of Theorem~4.1 in Section~4 of \cite{re4}.
Next we observe that the pole divisor $\{ g=0\}$ is contained in the strong invariant manifold of ${\mathcal F}$. To verify this assertion we first notice that $\{ g=0\}$ must be invariant under ${\mathcal F}$ as a consequence of the general fact that there is no one-dimensional meromorphic semi-complete vector field (cf. Remark~(\ref{obsidiota})). Thus $\{ g=0\}$ is contained in the union of the separatrizes of ${\mathcal F}$. Next we suppose for a contradiction that the weak invariant manifold of ${\mathcal F}$ is convergent (i.e. defines a separatrix) and part of the pole divisor of $Y$. In this case the technique used in the proof of Proposition~4.2 of \cite{re2} applies word-by-word to show that the resulting vector field $Y$ is not semi-complete. This contradiction implies that $\{ g=0 \}$ must be contained in the strong invariant manifold of ${\mathcal F}$ as desired.
Combining the information above with Dulac's normal form, we conclude that $Y$ possesses the form $$ Y = \frac{f}{y^k} \left[ (x(1+ \lambda y^p) + yR(x,y)) \partx + y^{p+1} \party \right] \, . $$ Now we are going to prove that $f(0,0) \neq 0$ i.e. $f$ is an inversible factor. Hence we assume for a contradiction that $f(0,0) =0$ but $f$ is not divisible by $y$. Still according to the terminology of Section~4 of \cite{re2}, we see that the ``asymptotic order of the divided time-form'' induced by $Y$ on $\{ y=0\}$ is at least $2$ since this form is $dx /(xf(x,0))$ (this is also a consequence of the fact that the index of the strong invariant manifold of a saddle-node is {\it zero}, cf. Section~5). However this order cannot be greater than~$2$ since $Y$ is semi-complete. Furthermore when this order happens to be~$2$, the local holonomy of the separatrix in question must be the identity provided that $Y$ is semi-complete. Nonetheless the local holonomy of the strong invariant manifold of a saddle-node is never the identity. In fact, using Dulac's normal form, an elementary calculation shows that this holonomy has the form $H(z) = z + z^p + \cdots$. We then conclude that $f(0,0) \neq 0$.
Therefore we have so far $$ Y = y^{-k} H(x,y) \left[ (x(1+ \lambda y^p) + yR(x,y)) \partx + y^{p+1} \party \right] \, , $$ where $H$ is holomorphic and satisfies $H(0,0) \neq 0$.
Recall that $\pi_2$ denotes the projection $\pi_2 (x,y) =y$ whose fibers are transverse to the leaves of ${\mathcal F}$ away from $\{ y=0\}$. Let $L$ be a regular leaf of ${\mathcal F}$ and consider an embedded curve $c: [0,1] \rightarrow L$. If $dT_L$ stands for the time-form induced on $L$ by $Y$, we clearly have $$ \int_c \, dT_l \; = \; \int_{\pi_2 (c)} \, h(y) y^{p-k -1} dy \, , $$ where $h (y) = H(0,y)$ so that $h (0) \neq 0$. Since the integral on the left hand side is never {\it zero}, it follows that $p-k -1 =0$ or~$1$. The case $k=p-1$ does not require further comments. On the other hand, if $k=p-2$, then the integral of $dT_L$ over $c$ is {\it zero} provided that $\pi_2 (c)$ is a loop around the origin $0 \in \{ x=0\}$. This implies that $c$ must be closed itself. In other words the monodromy of ${\mathcal F}$ with respect to the fibration induced by $\pi_2$ is trivial. Conversely it is easy to check that the normal forms 1, 2 and 3 in the statement are, in fact, semi-complete. This finishes the proof of the theorem.
\fbox{}
Before closing the section, it is interesting to translate the condition in the item~3 of Theorem~(\ref{selano}) in terms of the classifying space of Martinet-Ramis \cite{mara}. Note however that this translation will not be needed for the rest of the paper.
Fix $p \in {\mathbb N}^{\ast}$ and consider the foliation ${\mathcal F}_{p, \lambda}$ whose leaves are ``graphs'' (over the $y$-axis) of the form $$ x = {\rm const}\, .\, y^{\lambda} \exp (-1/px^p) \, . $$ Given $\lambda \in {\mathbb C}$, the moduli space of saddle-nodes ${\mathcal F}$ having $p, \lambda$ fixed is obtained from the foliation above through the following data: \begin{itemize}
\item $p$ translations $z \mapsto z+ c_i$, $z, c_i \in {\mathbb C}$ denoted by $g_1^+ , \ldots , g_p^+$.
\item $p$ local diffeomorphisms $z \mapsto z + \cdots$, $z \in {\mathbb C}$, tangent to the identity denoted by $g_1^- , \ldots , g_p^-$. \end{itemize}
These diffeomorphisms induce a permutation of (part of) the leaves of ${\mathcal F}_{p, \lambda}$. More precisely the total permutation (after one tour around $0 \in {\mathbb C}$) is given by the composition $$ g_p^- \circ g_p^+ \circ \ldots \circ g_1^- \circ g_1^+ \, . $$ However recall that our saddle-node has trivial monodromy. One easily checks that this cannot happen in the presence of the ramification $y^{\lambda}$. Thus we conclude that $\lambda$ belongs to ${\mathbb Z}$ and, in particular, the model ${\mathcal F}_{p, \lambda}$ introduced above has itself trivial monodromy. Hence the monodromy of ${\mathcal F}$ itself is nothing but $g_p^- \circ g_p^+ \circ \ldots \circ g_1^- \circ g_1^+$. In other words the condition is $$ g_p^- \circ g_p^+ \circ \ldots \circ g_1^- \circ g_1^+ = Id \, . $$ In particular note that, if $p =1$, the above equation implies that $g_1^- = g_1^+ = Id$. This explains why in item~1 of Theorem~(\ref{selano}) the saddle-node in question is analytically conjugate to its formal normal form.
\section{Polynomial vector fields and first integrals}
\hspace{0.4cm} Here we want to specifically consider complete polynomial vector fields whose associated foliation ${\mathcal F}$ has a singularity in the line at infinity which admits infinitely many separatrizes. Recall that such a singularity is said to be dicritical. The main result of the section is Proposition~(\ref{jouano}) below.
\begin{prop} \label{jouano} Let $X$ be a complete polynomial vector field on ${\mathbb C}^2$ and let ${\mathcal F}_X$ denote the foliation induced by $X$ on ${\mathbb C} {\mathbb P} (2)$. Assume that ${\mathcal F}_X$ possesses a dicritical singularity $p$, belonging to the line at infinity $\Delta$. Then ${\mathcal F}_X$ has a meromorphic first integral on ${\mathbb C} {\mathbb P} (2)$. Furthermore, modulo a normalization, the closure of the regular leaves of ${\mathcal F}_X$ is isomorphic to ${\mathbb C} {\mathbb P} (1)$. \end{prop}
We begin with a weakened version of this proposition which is the following lemma.
\begin{lema} \label{prejoa} Let $X ,\; {\mathcal F}_X$ be as in the statement of Proposition~(\ref{jouano}) and denote by $p \in \Delta$ a dicritical singularity of $X$. Then $X$ possesses a meromorphic first integral on ${\mathbb C}^2$. Moreover, if this first integral is not algebraic, then the set of leaves of ${\mathcal F}_X$ that pass through $p$ only once contains an open set. \end{lema}
\noindent {\it Proof}\,: Suppose that ${\mathcal F}_X$ as above possesses a singularity $p \in \Delta$ with infinitely many separatrizes. We consider coordinates $(u,v)$ around $p$ such that $\{ u=0 \} \subset \Delta$. Given a small neighborhood $V$ of $p$, we consider the restriction $X_{\mid_V}$ of $X$ to $V$. Clearly $X_{\mid_V}$ defines a meromorphic semi-complete vector field on $V$.
Obviously only a finite number of separatrizes of ${\mathcal F}_X$ going through $p$ may be contained in the divisor of poles or zeros of $X$. Thus there are (infinitely many) separatrizes ${\mathcal S}$ of ${\mathcal F}_X$ at $p$ which are (local) regular orbits of $X$. We fix one of these separatrizes ${\mathcal S}$. Recall that ${\mathcal S}$ has a Puiseux parametrization $\textsf{A}(t) = (a(t), b(t))$, $\textsf{A}(0) = (0,0)$, defined on a neighborhood $W$ of $0 \in {\mathbb C}$. Furthermore $\textsf{A}$ is injective on $W$ and a diffeomorphism from $W \setminus \{ 0 \}$ onto ${\mathcal S} \setminus \{ (0,0) \}$. Since ${\mathcal S}$ is invariant under $X$, the restriction to ${\mathcal S} \setminus \{ (0,0) \}$ of $X$ can be pulled-back by $\textsf{A}$ to give a meromorphic vector field $Z$ on $W$, i.e. $Z(t) = \textsf{A}^{\ast} \left( X\! \! \mid_{{\mathcal S}} \right)$ where $t \in W \setminus \{ (0,0) \}$ for a sufficiently small neighborhood $W$ of $0 \in {\mathbb C}$. We also have that $Z$ is semi-complete on $W$ since $X_{\mid_V}$ is semi-complete on $V$ and $\textsf{A}$ is injective on $W$. It follows from Remark~(\ref{obsidiota}) that $Z$ admits a holomorphic extension to $0 \in {\mathbb C}$ which is still denoted by $Z$. Moreover, letting $Z(t) = h(t) \partial /\partial t$, we cannot have $h(0) = h'(0) = h'' (0) =0$. However we must have at least $h(0) =0$. Otherwise the semi-global flow of $Z$ would reach the origin $0 \in {\mathbb C}$ in finite time. Since $\textsf{A} (0)$ lies in the line at infinity $\Delta$, it would follow that points in the orbit of $X$ containing ${\mathcal S}$ reach $\Delta$ in finite time. This is impossible since $X$ is complete on ${\mathbb C}^2$.
Therefore we have only two cases left, namely $h(0) =0$ but $h'(0) \neq 0$ and $h(0) = h'(0) =0$ but $h''(0) \neq 0$. Let us discuss them separately. First suppose that $h(0) = h'(0) =0$ but $h'' (0) \neq 0$. Modulo a normalization we can suppose that ${\mathcal S}$ is smooth at $p$. Again we denote by $L$ the global orbit of $X$ containing ${\mathcal S}$. By virtue of the preceding $L$ is a Riemann surface endowed with a complete holomorphic vector field $X_{\mid_L}$ which has a singularity of order~$2$ (where $X_{\mid_L}$ stands for the restriction of $X$ to $L$). It immediately results that $L$ has to be compactified into ${\mathbb C} {\mathbb P} (1)$. In other words the closure of $L$ is a rational curve and, in particular, an algebraic invariant curve of ${\mathcal F}_X$. Since there are infinitely many such curves, Jouanolou's theorem \cite{joa} ensures that ${\mathcal F}_X$ has a meromorphic first integral (alternatively we can also apply Borel-Nishino's theorem, cf. \cite{la}). Furthermore the level curves of this first integral are necessarily rational curves (up to normalization) as it follows from the discussion above.
Suppose now that $h(0) =0$ but $h'(0) \neq 0$. In this case the vector field $Z$ has a non-vanishing residue at $0 \in {\mathbb C}$. We then conclude that ${\mathcal S}$ possesses a {\it period} i.e. there exists a loop $c:[0,1] \rightarrow {\mathcal S}$ ($c(0) = c(1)$) on which the integral of the corresponding time-form is different from {\it zero}. If $L$ is the global orbit of $X$ containing ${\mathcal S}$ the preceding implies that $L$ is isomorphic to ${\mathbb C}^{\ast}$.
On the other hand the characterization of singularities with infinitely many separatrizes obtained through Seidenberg's theorem \cite{sei} (cf. Section~7 for further details) ensures that the set of orbits $L$ as above has positive logarithmic capacity. In fact it contains an open set. Thus Suzuki's results in \cite{suzu1}, \cite{suzu2} apply to provide the existence of a non-constant meromorphic first integral for $X$. If this first integral is not algebraic, then a ``generic'' orbit passes through $p$ once but not twice. Otherwise the leaf would contain two singularities of $X$ and therefore it would be a rational curve (up to normalization). In this case the mentioned Jouanolou's theorem would provide an algebraic first integral for ${\mathcal F}_X$. The proof of the proposition is over.
\fbox{}
The preceding lemma suggests a natural strategy to establish Proposition~(\ref{jouano}). Namely we assume for a contradiction that ${\mathcal F}_X$ does not have an algebraic first integral. Then we have to show that a generic leaf passing through a dicritical singularity $p$ must return and cross $\Delta$ once again (maybe through another dicritical singularity). The resulting contradiction will then complete our proof.
Using Seidenberg's theorem, we reduce the singularities of ${\mathcal F}_X$ in $\Delta$ so that they all will have at least one eigenvalue different from {\it zero}. In particular we obtain a normal crossing divisor ${\mathcal E}$ whose irreducible components are rational curves, one of them being the proper transform of $\Delta$ (which will still be denoted by $\Delta$). The other components were introduced by the punctual blow-ups performed and are denoted by $D_i$, $i =1, \ldots ,s$. The fact that $p$ is a dicritical singularity implies that one of the following assertions necessarily holds: \begin{enumerate}
\item There is a component $D_{i_0}$ of ${\mathcal E}$ which is not invariant by $\widetilde{\mathcal F}_X$ (where $\widetilde{\mathcal F}_X$ stands for the proper transform of ${\mathcal F}_X$).
\item There is a singularity $p_0$ of $\widetilde{\mathcal F}$ in ${\mathcal E}$ which is dicritical and has two eigenvalues different from {\it zero}. \end{enumerate}
We fix a local cross section $\Sigma$ through a point $q$ of $\Delta$ which is regular for $\widetilde{\mathcal F}_X$. Note that a regular leaf $L$ of $\widetilde{\mathcal F}_X$ necessarily meets $\Sigma$ infinitely many times unless $L$ is algebraic. Indeed first we observe that $L$ is properly embedded in the affine part ${\mathbb C}^2$ since ${\mathcal F}_X$ possesses a meromorphic first integral on ${\mathbb C}^2$. Thus all the accumulation points of $L$ are contained in $\Delta$. Obviously if $L$ accumulates a regular point of $\Delta$ then $L$ intersects $\Sigma$ infinitely many times as required. On the other hand, if $L$ accumulates only points of $\Delta$ which are singularities of ${\mathcal F}_X$, then Remmert-Stein theorem shows that the closure of $L$ is algebraic. Summarizing, using Jouanolou's or Borel-Nishino's theorem, we can suppose without loss of generality that all the leaves of $\widetilde{\mathcal F}_X$ intersects $\Sigma$ an infinite number of times (and in fact these intersection points approximate the point $q = \Sigma \cap \Delta$).
To prove that ${\mathcal F}_X$ has infinitely many leaves cutting the exceptional divisor ${\mathcal E}$ more than once, we fix a neighborhood $\mathcal U$ of ${\mathcal E}$. Proposition~(\ref{jouano}) is now a consequence of the next proposition.
\begin{prop} \label{dutrans} Under the above assumptions, there is an open neighborhood $V \subset \Sigma$ of $q$ in $\Sigma$ and an open subset $W \subset V$ of $V$ with the following property: any leaf $L$ passing through a point of $W$ intersects the exceptional divisor ${\mathcal E}$ before leaving the neighborhood $\mathcal U$. \end{prop}
In fact, since the set of leaves meeting ${\mathcal E}$ contains an open set and all of them (with possible exception of a finite number) cross $\Sigma$ and accumulates $\Delta$, Proposition~(\ref{dutrans}) clearly shows the existence of infinitely many leaves (orbits of $X$) intersecting ${\mathcal E}$ more than one time thus providing the desired contradiction.
In order to prove Proposition~(\ref{dutrans}) we keep the preceding setting and notations. We are naturally led to discuss the behavior of the leaves of ${\mathcal F}_X$ on a neighborhood of the point $p_{ij}$ of intersection of the irreducible components $D_i , D_j$ belonging to ${\mathcal E}$.
Now let us fix coordinates $(x,y)$ around $p_{ij}$ ($p_{ij} \simeq (0,0)$) such that $\{ y= 0\} \subseteq D_i$ and $\{ x=0 \} \subseteq D_j$. Without loss of generality we can suppose that the domain of definition of the $(x,y)$-coordinates contains the bidisc of radius~$2$. Next we fix a segment of vertical line $\Sigma_x$ (resp. horizontal line $\Sigma_y$) passing through the point $(1,0)$ (resp. $(0,1)$). We assume that $D_i$ is invariant under $\widetilde{\mathcal F}_X$ but $D_j$ may or may not be invariant under $\widetilde{\mathcal F}_X$. Let us also make the following assumptions: \begin{description}
\item[{\bf A)}] $p_{ij} \simeq (0,0)$ is not a dicritical singularity of $\widetilde{\mathcal F}$.
\item[{\bf B)}] $\widetilde{\mathcal F}_X$ has at least one eigenvalue different from {\it zero} at $p_{ij} \simeq (0,0)$.
\item[{\bf C)}] The vector field $X$ whose associated foliation is $\widetilde{\mathcal F}$ is meromorphic semi-complete in the domain of the coordinates~$(x,y)$. \end{description}
We are going to discuss a variant of the so-called {\it Dulac's transform}, namely if leaves intersecting $\Sigma_x$ necessarily cut $\Sigma_y$. Precisely we fix a neighborhood ${\bf U}$ of $\{ x=0 \} \cup \{ y=0 \}$, we then have:
\begin{lema} \label{passara} Under the preceding assumptions, there is an open neighborhood $V_x \subset \Sigma_x$ of $(1,0)$ in $\Sigma_x$ and an open set $W_x \subset V_x$ with the following property: any leaf $L$ of ${\mathcal F}$ passing through a point of $W_x$ meets $\Sigma_y$ before leaving $U$. In particular, if $D_j$ is not invariant by ${\mathcal F}$, then the leaves of ${\mathcal F}$ passing through points of $W_x$ cross the axis~$\{ x=0 \}$ before leaving ${\bf U}$. In addition, by choosing $V_x$ very small, the ratio between the area of $W_x$ and the area of $V_x$ becomes arbitrarily close to~$1$. \end{lema}
Before proving this lemma, let us deduce the proof of Proposition~(\ref{dutrans}).
\noindent {\it Proof of Proposition~(\ref{dutrans})}\,: Recall that $\widetilde{\mathcal F}_X$, the proper transform of ${\mathcal F}_X$, has only simple singularities in ${\mathcal E}$. In particular, if ${\bf p} \in {\mathcal E}$ is a dicritical singularity of $\widetilde{\mathcal F}_X$, then $\widetilde{\mathcal F}_X$ has~$2$ eigenvalues different from {\it zero} at ${\bf p}$. Hence $\widetilde{\mathcal F}_X$ is linearizable around ${\bf p}$ and, as a consequence, there is a small neighborhood $U_{\bf p}$ of ${\bf p}$ such that any regular leaf $L$ of $\widetilde{\mathcal F}_X$ entering $U_{\bf p}$ must cross ${\mathcal E}$ before leaving $U_{\bf p}$. This applies in particular if ${\bf p}$ coincides with the intersection of two irreducible components $D_i, D_j$ of ${\mathcal E}$.
On the other hand $\Delta$ is invariant by $\widetilde{\mathcal F}_X$ and all but a finite number of leaves of $\widetilde{\mathcal F}_X$ intersect $\Sigma$ in arbitrarily small neighborhoods of $q= \Sigma \cap \Delta$. In particular if $\widetilde{\mathcal F}_X$ has a dicritical singularity on $\Delta$, then the statement follows from the argument above.
Suppose now that $\widetilde{\mathcal F}_X$ does not have a dicritical singularity on $\Delta$. Let $D_1$ be another irreducible component of ${\mathcal E}$ which intersects $\Delta$ at $p_{01}$. Note that $p_{01}$ is not a dicritical singularity of $\widetilde{\mathcal F}_X$ by the preceding discussion. If $D_1$ is not invariant by $\widetilde{\mathcal F}_X$, then Lemma~(\ref{passara}) allows us to find infinitely many leaves of $\widetilde{\mathcal F}_X$ intersecting ${\mathcal E}$. Thus the proposition would follow. On the other hand, if $D_1$ is invariant by $\widetilde{\mathcal F}_X$, then Lemma~(\ref{passara}) still allows us to find a local transverse section $\Sigma_1$ through a regular point $q_1$ of $D_1$ with the desired property namely: excepted for a set of leaves whose volume can be made arbitrarily small (modulo choosing $V_x$ sufficiently small), all leaves of $\widetilde{\mathcal F}_X$ meet $\Sigma_1$ in arbitrarily small neighborhoods of $q_1 = \Sigma_1 \cap D_1$. We then continue the procedure replacing $\Delta$ by $D_1$. Since we eventually will find an irreducible component of ${\mathcal E}$ which is not invariant by $\widetilde{\mathcal F}_X$ or contains a dicritical singularity, the proposition is proved. This also concludes the proof of Proposition~(\ref{jouano}).
\fbox{}
The rest of the section is devoted to the proof of Lemma~(\ref{passara}). Let us begin with the easier case in which $D_j$ is not invariant by $\widetilde{\mathcal F}_X$.
\begin{lema} \label{10.fim1} The vector field $\widetilde{X}$ vanishes with order~$1$ on $D_j$. It also has poles of order~$1$ on $D_i$ and the origin $(0,0)$ is a LJ-singularity of $\widetilde{\mathcal F}_X$. \end{lema}
\noindent {\it Proof}\,: By assumption $D_j$ is contained in ${\mathcal E}$ and is not invariant by $\widetilde{\mathcal F}_X$. In particular $\widetilde{X}$ cannot have poles on $D_j$ since there is no strictly meromorphic semi-complete vector field in dimension~$1$. Neither can $\widetilde{X}$ be regular on $D_j$ otherwise certain points of ${\mathbb C}^2$ would reach the infinity in finite time, thus contradicting the fact that $\widetilde{X}$ is complete. Finally the order of $\widetilde{X}$ on $D_j$ cannot be greater than~$2$, otherwise $\widetilde{X}$ would not be semi-complete. Besides, if this order is~$2$, then infinitely many orbits of $X$ will be compactified into rational curves. This would imply that ${\mathcal F}_X$ has an algebraic first integral which is impossible. This shows that $\widetilde{X}$ vanishes with order~$1$ on $D_j$.
Because $\widetilde{X}$ vanishes on $D_j$ and $D_j$ is not invariant by the associated foliation $\widetilde{\mathcal F}_X$, it follows from Section~4 that either $(0,0)$ is a LJ-singularity of $\widetilde{\mathcal F}_X$ or $\widetilde{\mathcal F}_X$ is linearizable with eigenvalues~$1,-1$ at $(0,0)$. In the latter case the conclusions of Lemma~(\ref{passara}) are obvious. Thus we can suppose that $(0,0)$ is a LJ-singularity. It follows from Lemma~(\ref{le3.3}) that $\widetilde{X}$ must have a pole divisor of order~$1$ on $D_i$.
\fbox{}
\noindent {\it Proof of Lemma~(\ref{passara}) when $D_j$ is not invariant by $\widetilde{\mathcal F}_X$}\,: Modulo blowing-up $(0,0)$, the problem is immediately reduced to the discussion of the Dulac's transform between the strong and the weak invariant manifolds of the saddle-node determined by $$ x(1+y) \, dy \; \, + \; \, y^2 \, dx \; . $$ Thinking of this foliation as a differential equation, we obtain the solution $$ x(T) = \frac{x_0 e^T}{1-y_0 T} \; \; \, {\rm and} \; \; \, y(T) = \frac{y_0}{1-y_0 T} \, . $$ Let us fix $x_0 =1$. Given $y_0$, we search for $T_0$ so that $y(T_0) =1$. Furthermore we also require that the norm of $x(T)$ stays ``small'' during the procedure. This is clearly possible if $y_0$ is real negative (sufficiently close to {\it zero}). Actually we set $T_0 =(1-y_0) /y_0 \in {\mathbb R}_-$ so that $T$ can be chosen real negative during the procedure. Thus the norm of $x(T)$ will remain controlled by that of $y_0$.
On the other hand, let $y_0$ be in the transverse section $\Sigma_x$ and suppose that $y_0$ is not real positive. Then the orbit of $y_0$ under the local holonomy of the strong invariant manifold converges to {\it zero} and is asymptotic to ${\mathbb R}_-$. Indeed this local holonomy is represented by a local diffeomorphism of the form $z \mapsto z + z^2 + {\rm h.o.t.}$ and the local topological dynamics of these diffeomorphisms is simple and well-understood (known as a ``flower'', in the present case this dynamics is also called the parabolic bifurcation). Hence for a sufficiently large iterate of $y_0$ (without leaving $V_x$), the above ``Dulac's transform'' is well-defined. Thus the statement of Lemma~(\ref{passara}) is verified as long as we take $W_x = V_x \setminus {\mathbb R}_-$ in the above coordinates.
\fbox{}
From now on we can suppose that $D_j$ is invariant under ${\mathcal F}_X$. We have three cases to check:
\noindent 1) ${\mathcal F}_X$ has two eigenvalues different from {\it zero} at $(0,0)$ and is locally linearizable.
\noindent 2) ${\mathcal F}_X$ has two eigenvalues different from {\it zero} at $(0,0)$ but is not locally linearizable. In this case the quotient of the eigenvalues is real negative.
\noindent 3) ${\mathcal F}_X$ defines a saddle-node at $(0,0)$.
In the Case~$1$ the verification is automatic and left to the reader. Case~2 follows from \cite{mamo} (note that our convention of signs is opposite to the convention of \cite{mamo}). So we just need to consider the case of saddle-nodes. Of course all the possible saddle-nodes necessarily have a convergent weak invariant manifold. Without loss of generality we can suppose that $D_i$ is the strong invariant manifold so that $D_j$ is the weak invariant manifold (the other possibility is analogous). All the background material about saddle-nodes used in what follows can be found in \cite{mara}.
Thanks to Lemma~(\ref{selano}), we can find coordinates $(x,y)$ as above where the vector field $\widetilde{X}$ becomes $$ \widetilde{X} = y^{-k} [ (x(1+ \lambda y^p) +yR) {\partial /\partial x} \; + \; y^{p+1} {\partial /\partial y} ] \, . $$ Since our problem depends only on the foliation associated to $\widetilde{X}$, we drop the factor $y^{-k}$ in the sequel. We also notice that $\widetilde{X}$ is regular on $D_j$. The argument which is going to be employed here is a generalization of the one employed to deal with the saddle-node appearing after blowing-up the LJ-singularity in the previous case.
Following \cite{mara} we consider open sets $V_i \subset {\mathbb C}$, $i = 0, \ldots ,2p-1$, defined by $V_i = \{ z \in {\mathbb C} \; ; \; (2i+1)\pi /2p - \pi /p < \arg z < (2i+1)\pi /2p + \pi /p \}$. The $V_i$'s, $i=1, \ldots ,2p-1$, define a covering of ${\mathbb C}^{\ast}$ and, besides, each $V_i$ intersects only $V_{i-1}$ and $V_{i+1}$ (where $V_{-1} = V_{2p-1}$). We also let $W_i^+$ (resp. $W_i^-$) be defined by $$ W_i^+ = \left\{ \frac{(4i-1) \pi}{2p} < \arg z < \frac{(4i+1) \pi}{2p} \right\} \, \; \; {\rm and} \, \; \; W_i^- = \left\{ \frac{(4i+1) \pi}{2p} < \arg z < \frac{(4i+3) \pi}{2p} \right\} \, . $$ for $i=0 ,\ldots ,2p-1$. We point out that ${\rm Re} \, (y^p) < 0$ (resp. ${\rm Re} \, (y^p) > 0$) provided that $y \in W_i^-$ (resp. $y \in W_i^+$). In addition we have $W_i^+ = V_{2i} \cap V_{2i+1}$ and $W_i^- = V_{2i+1} \cap V_{2i+2}$ (unless $p=1$ where $V_0 \cap V_1 = W_0^+ \cup W_0^-$). Given $\varepsilon >0$, we set $$
U_{i, V} = \{ (x,y) \in {\mathbb C}^2 \; ; \; \| x \| < \varepsilon
\, , \; \, \| y \| < \varepsilon \; \; {\rm and} \; \; y \in V_i \} \, . $$ According to Hukuara-Kimura-Matuda, there is a bounded holomorphic mapping $\phi_{U_{i, V}} (x,y) = (\varphi_{U_{i,V}} (x,y) , y)$ defined on $U_{i, V}$ which brings the vector field $\widetilde{X}$ to the form \begin{equation} x(1+ \lambda y^p) {\partial /\partial x} \; + \; y^{p+1} {\partial /\partial y} \, . \label{quasela} \end{equation} The vector field in~(\ref{quasela}) can be integrated to give \begin{equation} x (T) = \frac{x_0 e^T}{\sqrt[p]{(1-py_0^pT)^{\lambda}}} \; \; {\rm and} \; \; y(T) = \frac{y_0}{\sqrt[p]{1-p y_0^p T}} \, . \label{intee} \end{equation}
\noindent {\it Proof of Lemma~(\ref{passara})}\,: We keep the preceding notations. On $U_{i,V}$ we consider a normalizing mapping $\phi_{U_{i, V}}$ such that $\widetilde{X}$ is as in~(\ref{quasela}). In this coordinate we fix a vertical line $\Sigma_x$ as before and let $\Sigma_{y,i}$ denote the intersection of the horizontal line through $(0,1)$, $\Sigma_y$, with the sector $V_i$. Again we want to know which leaves of $\widetilde{\mathcal F}$ passing through a point of $\Sigma_x$ will intersect $\Sigma_y$ as well. Thus starting with $x_0 =1$, we search for $y (T_0) =1$. Thanks to equations~(\ref{intee}), it is enough to choose $T_0=(1-y_0^p) /py_0^p$. In particular $T_0 \in {\mathbb R}_-$ provided that $y_0^p \in {\mathbb R}_-$. The formula for $x(T)$ in~(\ref{intee}) shows that $x(T)$ remains in the fixed neighborhood ${\bf U}$ of $\{ x=0 \} \cup \{ y=0 \}$ provided that we keep $T \in {\mathbb R}_-$ during the procedure and choose $y_0$ sufficiently small. More generally if the real part ${\rm Re} \, (y_0^p)$ is negative and the quotient between imaginary and real parts is bounded, then the same argument applies. In other words, if $y_0$ belongs to a compact subsector of $W_{[i+1/2]-1}^-$, the set $W_j^-$ contained in $V_i$, then the Dulac's transform in question is well-defined modulo choosing $y_0$ uniformly small.
The local holonomy associated to $D_i$ (the strong invariant manifold) has the form $\textsf{h} (z) = z + z^{p+1} + {\rm h.o.t.}$ The dynamical picture corresponding to this diffeomorphims is still a ``flower''. However, in general, it is not true that the orbit of a ``generic'' point will intersect a fixed $W_{[i+1/2]-1}^-$ since $\textsf{h}$ may have invariant sectors. However, the above argument can be applied separately for each $i$. Clearly to each $i$ fixed we have a different $\Sigma_y \cap V_i$ associated. Nonetheless they are all equivalent since the weak invariant manifold of the foliation is convergent. It follows that apart from a finite number of curves whose union has empty interior, the leaf through a point of $\Sigma_x$ meets $\Sigma_y$ before leaving the neighborhood ${\bf U}$. As mentioned the case where $D_i$ is the (convergent) weak invariant manifold and $D_j$ is the strong invariant manifold is analogous. The statement follows.
\fbox{}
\section{Arrangements of simple singularities}
\hspace{0.4cm} Now we are going to study the possible arrangements of simple semi-complete singularities over a rational curve of self-intersection~$-1$ which are obtained by blowing-up a semi-complete vector field on a neighborhood of $(0,0) \in {\mathbb C}^2$.
We shall make a number of assumptions which are always satisfied in our cases. We denote by $\widetilde{\mathbb C}^2$ the blow-up of ${\mathbb C}^2$ at the origin and by $\pi : \widetilde{\mathbb C}^2 \rightarrow {\mathbb C}^2$ the corresponding blow-up map. Given a vector field $Y$ (resp. foliation ${\mathcal F}$) defined on a neighborhood $U$ of the origin, $\pi$ naturally induces a vector field $\widetilde{Y}$ (resp. foliation $\widetilde{\mathcal F}$) defined on $\pi^{-1} (U)$. Furthermore $Y$ is semi-complete on $U$ if and only if $\widetilde{Y}$ is semi-complete on $\pi^{-1} (U)$.
Given a meromorphic vector field $Y = fZ/g$ with $f,g, Z$ holomorphic, we call the vector field $fZ$ the {\it holomorphic part}\, of $Y$. In this section it is discussed the nature of a meromorphic semi-complete vector field $Y$ defined on a neighborhood of the origin $(0,0) \in {\mathbb C}^2$ which satisfies the following assumptions: \begin{enumerate} \item $Y = x^{k_1} y^{k_2} f_1 Z $ where $Z$ is a holomorphic vector field having an isolated singularity at $(0,0) \in {\mathbb C}^2$, $f_1$ is holomorphic function and $k_1 , k_2 \in {\mathbb Z}$.
\item The regular orbits of $Y$ contain at most $1$ singular point. Furthermore the order of $Y$ at this singular point is one.
\item The regular orbits of $Y$ contains at most~$1$ period (i.e. there is only one homology class containing loops on which the integral of the time-form is different from zero).
\item The foliation ${\mathcal F}$ associated to $Y$ (or to $Z$) has both eigenvalues equal to {\it zero} at $(0,0) \in {\mathbb C}^2$.
\item The blow-up $\widetilde{\mathcal F}$ of ${\mathcal F}$ is such that every singularity $\tilde{p} \in \pi^{-1} (0)$ of $\widetilde{\mathcal F}$ is simple.
\item ${\mathcal F}$ is not dicritical at $(0,0)$.
\end{enumerate}
Before continuing let us introduce two basic definitions. Assume that ${\mathcal F}$ is a singular holomorphic foliation defined on a neighborhood of a singular point $p$. Let ${\mathcal S}$ be a smooth separatrix of ${\mathcal F}$ at $p$. We want to define the {\it order }\, of ${\mathcal F}$ with respect to ${\mathcal S}$ at $p$, ${\rm ord}_{{\mathcal S}} ({\mathcal F} ,p)$ (also called the multiplicity of ${\mathcal F}$ along ${\mathcal S}$), and the {\it index}\, of ${\mathcal S}$ w.r.t. ${\mathcal F}$ at $p$, ${\rm Ind}_{p} ({\mathcal F} , {\mathcal S})$ (cf. \cite{cama}). In order to do that, we consider coordinates $(x,y)$ where ${\mathcal S}$ is given by $\{ y=0 \}$ and a holomorphic $1$-form $\omega = F(x,y) \, dy - G (x,y)\, dx$ defining ${\mathcal F}$ and having an isolated singularity at $p$. Then we let \begin{eqnarray} {\rm ord}_{{\mathcal S}} ({\mathcal F} , p) & = & {\rm ord} \, (F(x,0)) \; \; \, {\rm at} \; \; \, 0 \in {\mathbb C} \label{ordem1} \; \; \, {\rm and} \\ {\rm Ind}_p ({\mathcal F} ,{\mathcal S} ) & = & {\rm Res} \, \frac{\partial}{\partial y} \left( \frac{G}{F} \right) (x,0) \, dx \; . \label{index1} \end{eqnarray} In the above formulas ${\rm ord} \, (F(x,0))$ stands for the order of the function $x \mapsto F(x,0)$ at $0 \in {\mathbb C}$ and ${\rm Res}$ for the residue of the $1$-form in question.
Let $p_1 ,\ldots ,p_r$ denote the singularities of $\widetilde{\mathcal F}$ belonging to $\pi^{-1} (0)$. Since $\pi^{-1} (0)$ naturally defines a separatrix for every $p_i$, we can consider both ${\rm ord}_{\pi^{_1} (0)} (\widetilde{\mathcal F} , p_i)$ and ${\rm Ind}_{p_i} (\widetilde{\mathcal F} ,\pi^{-1} (0))$. Easy calculations and the Residue Theorem then provides (cf. \cite{mamo} , \cite{cama}): \begin{eqnarray} & & {\rm ord}_{(0,0)} ({\mathcal F}) + 1 = \sum_{i=1}^r {\rm ord}_{\pi^{-1} (0)} (\widetilde{\mathcal F} , p_i) \; , \label{ordem2} \\ & & \sum_{i=1}^r {\rm Ind}_{p_i} (\widetilde{\mathcal F}, \pi^{-1} (0)) = -1 \; . \label{index2} \end{eqnarray}
On the other hand the order of $\pi^{-1} (0)$ as a divisor of zeros or poles of $\widetilde{Y}$ is \begin{equation} {\rm ord}_{\pi^{-1} (0)} \widetilde{Y} = {\rm ord}_{(0,0)} (f) + {\rm ord}_{(0,0)} ({\mathcal F}) -{\rm ord}_{(0,0)} (g) -1 \, . \label{equa2} \end{equation} In particular if this order is {\it zero}, then $\widetilde{Y}$ is regular on $\pi^{-1} (0)$.
To abridge notations, the local singular foliation induced by the linear vector field $$ (x+y) {\partial /\partial x} + y {\partial /\partial y} $$ will be called a LJ-singularity (where LJ stands for linear and in the Jordan form).
To simplify the statement of the main result of this section, namely Proposition~(\ref{prop4.2}), we first introduce~$3$ types, or models, of vector fields. Let us keep the preceding notation. \begin{description}
\item[{\bf Model} $Z_{1,11}$:] Let ${\mathcal F}_{1,11}$ be the foliation associated to $Z_{1,11}$ and $\widetilde{\mathcal F}_{1,11}$ its blow-up. Then $\widetilde{\mathcal F}_{1,11}$ contains~$3$ singularities $p_1 ,p_2 ,p_3$ on $\pi^{-1} (0)$ whose eigenvalues are respectively $1,1$, $-1,1$ and $-1,1$. The singularity $p_1$ is a LJ-singularity and the blow-up $\widetilde{Z}_{1,11}$ has a pole of order~$1$ on $\pi^{-1} (0)$. The separatrix of $p_2$ (resp. $p_3$) transverse $\pi^{-1} (0)$ is a pole divisor of $\widetilde{Z}_{1,11}$ of order~$1$ as well. Finally $\widetilde{Z}_{1,11}$ has a curve of zeros passing through $p_1$ which is not invariant under $\widetilde{\mathcal F}_{1,11}$.
\item[{\bf Model} $Z_{0, 12}$:] With similar notations, $\widetilde{\mathcal F} _{0,12}$ has~$3$ singularities $p_1, p_2 ,p_3$ on $\pi^{-1} (0)$ of eigenvalues equal to $1,0$, $-1,2$ and $-1, 2$. The singularity $p_1$ is a saddle-node with strong invariant manifold contained in $\pi^{-1} (0)$. The separatrix of $p_2$ (resp. $p_3$) transverse to $\pi^{-1} (0)$ is a pole of order~$d\neq 0$ of $\widetilde{Z}_{0,12}$. The $\pi^{-1} (0)$ is a pole of order~$2d-1$ of $\widetilde{Z}_{0,12}$. There is no other component of the divisor of zeros or poles of $\widetilde{Z}_{0,12}$.
\item[{\bf Model} $Z_{1, 00}$:] $\widetilde{\mathcal F}_{1,00}$ still has~$3$ singularities $p_1, p_2 ,p_3$ whose eigenvalues are $-1,1$, $1,0$ and~$1,0$. The singularities $p_2, p_3$ are saddle-nodes with strong invariant manifolds contained in $\pi^{-1} (0)$. The separatrix of $p_1$ transverse to $\pi^{-1} (0)$ is a pole of $\widetilde{Z}_{1,00}$ of order~$d\neq 0$. The exceptional divisor $\pi^{-1} (0)$ is a pole of order~$d-1$ and there is no other component of the divisor of zeros or poles of $\widetilde{Z}_{1,00}$,
\end{description}
Note in particular that Formula~(\ref{ordem2}) implies that ${\mathcal F}_{1,11}$ (resp. ${\mathcal F}_{0,12}$, ${\mathcal F}_{1,00}$) has a singularity of order~$2$ at the origin.
\begin{prop} \label{prop4.2} Let $Y$, $\widetilde{Y}$ be as above. Assume that the order of $\widetilde{Y}$ on $\pi^{-1} (0)$ is different from {\it zero}. Then the structure of the singularities of $\widetilde{\mathcal F}$ on $\pi^{-1} (0)$ is equal to that of one of the models $Z_{1,11}$, $Z_{0,12}$ or $Z_{1,00}$. \end{prop}
\begin{lema} \label{lema4.3} Denote by $\lambda_1^i , \lambda_2^i$ the eigenvalues of $\widetilde{\mathcal F}$ at $p_i$ ($i=1 ,\ldots ,r$). Then one of the following possibilities holds: \begin{description}
\item[($\imath$)] $\lambda_1^i /\lambda_2^i = -n/m$ where $n,m$ belong to ${\mathbb N}^{\ast}$.
\item [($\imath \imath$)] $p_i$ is a saddle-node (i.e. the eigenvalues are $1$ and {\it zero}) whose strong invariant manifold coincides with $\pi^{-1} (0)$.
\item[($\imath \imath \imath $)] $p_i$ is a LJ-singularity. \end{description} Furthermore there may exist at most one LJ-singularity and, when such singularity does exist, all the remaining singularities are as in ($\imath$). \end{lema}
\noindent {\it Proof}\,: First let us suppose that one of the eigenvalues $\lambda_1^i ,\lambda_2^i$ vanishes. In this case $\widetilde{\mathcal F}$ defines a saddle-node at $p_i$. Moreover $p_i$ belongs to the divisor of zeros or poles of $\widetilde{Y}$, so that Theorem~(\ref{selano}) shows that the strong invariant manifold of $\widetilde{\mathcal F}$ at $p_i$ coincides with $\pi^{-1} (0)$ as required.
On the other hand, if both $\lambda_1^i ,\lambda_2^i$ are different from {\it zero}, then they satisfy condition ($\imath$) as a consequence of Lemma~(\ref{le3.1}). Finally it remains only to consider the case where $p_1$ is a LJ-singularity. Thus in local coordinates $(x,t)$, $\{ x=0 \} \subset \pi^{-1} (0)$, around $p_1$, $\widetilde{Y}$ is given by $$ x^{-1} h [ (t+x) \partial /\partial t + x {\partial /\partial x}] \, . $$ Hence the regular orbits of $\widetilde{Y}$ contain a {\it zero} of $\widetilde{Y}$ corresponding to their intersection with $h=0$. Because of condition~2, this implies that only one of the $p_i$'s can be a LJ-singularity. Furthermore, by the same reason, the holonomy of $\pi^{-1} (0) \setminus \{ p_1 ,\ldots ,p_r \}$ has to be trivial. In particular none of the remaining singularities can be a saddle-node.
\fbox{}
Combining the information contained in the preceding lemma with Formula~(\ref{ordem2}) we obtain:
\begin{coro} \label{lema4.1} The order of ${\mathcal F}$ at $(0,0) \in {\mathbb C}$ equals $r-1$, i.e. ${\rm ord}_{(0,0)} ({\mathcal F}) = r-1$.
\fbox{} \end{coro}
The case where $p_1$ is a LJ-singularity is indeed easy to analyse. After the preceding lemma and the fact that the holonomy of $\pi^{-1} (0) \setminus \{ p_1 ,\ldots ,p_r \}$ is trivial, we conclude that all the remaining singularities $p_2 ,\ldots ,p_r$ have eigenvalues $1$ and $-1$. Now using formulas~(\ref{ordem2}) and~(\ref{index2}) we conclude that $\widetilde{Y}$ is as in the model~$Z_{1,11}$.
Hereafter we suppose without loss of generality that none of the $p_i$'s is a LJ-singularity. For $s \leq r$, we denote by $p_1 ,\ldots ,p_r$ the singularities of $\widetilde{\mathcal F}$ where $\widetilde{\mathcal F}$ has two non-vanishing eigenvalues (whose quotient has the form $-n/m$, $m,n \in {\mathbb N}$). The remaining $p_{s+1} , \ldots ,p_r$ singularities are therefore saddle-nodes. Recall that the strong invariant manifolds of these saddle-nodes coincide with $\pi^{-1} (0)$ thanks to Theorem~(\ref{selano}). Next we have:
\begin{lema} \label{lema4.2} At least one of the $p_i$'s is a saddle-node (i.e. $s$ is strictly less than $r$). \end{lema}
\noindent {\it Proof}\,: The proof relies on Section~4 of \cite{re3}. Suppose for a contradiction that none of the $p_i$'s is a saddle-node. Given that there is no $LJ$-singularity, it follows that the quotient $\lambda_1^i /\lambda_2^i$ is negative rational for every $i=1, \ldots ,r$. Hence the hypotheses of Proposition~4.2 of \cite{re3} are verified. It results that $X$ has one of the normal forms indicated in that proposition. As is easily seen, all those vector fields have orbits with~$2$ distinct periods which is impossible in our case. The lemma is proved.
\fbox{}
To complete the proof of Proposition~(\ref{prop4.2}) we proceed as follows. For $i \in \{ 1, \ldots ,s\}$, we consider local coordinates $(x_i ,t_i)$, $\{ x_i =0 \} \subset \pi^{-1} (0)$, around $p_i$. In these coordinates $\widetilde{Y}$ has the form \begin{equation} \widetilde{Y} = x_i^{({\rm ord}_{\pi^{-1} (0)} (\widetilde{Y}))} t_i^{d_i} h_i [ m_i x_i (1 + {\rm h.o.t.}) \partial /\partial x_i - n_i t_i (1 + {\rm h.o.t}) \partial /\partial t_i] \label{forca1} \end{equation} where $d_i \in {\mathbb Z}$, $m_i ,n_i \in {\mathbb N}$ and $h_i$ is holomorphic but not divisible by either $x_i, t_i$. Similarly, around the saddle-nodes singularities $p_{s+1} ,\ldots ,p_r$, we have \begin{equation} \widetilde{Y} = x_i^{({\rm ord}_{\pi^{-1} (0)} (\widetilde{Y}))} h_i[x_i^{p_i+1} \partial /\partial x_i + t_i(1 + {\rm h.o.t}) \partial /\partial t_i] \, . \label{forca2} \end{equation}
We claim that $h_i (0,0) \neq 0$. This is clear in equation~(\ref{forca2}) thanks to Theorem~(\ref{selano}). As to equation~(\ref{forca1}), let us suppose for a contradiction that $h_i (0,0) =0$. Hence the regular leaves of $\widetilde{Y}$ have a zero corresponding to the intersection of these leaves with $\{ h_i =0 \}$. Given condition~2, it results that only one of the $h_i$'s may verify $h_i (0,0) =0$. Without loss of generality we suppose that $h_1 (0,0) =0$. From Lemma~(\ref{le3.1}) and Remark~(\ref{ipc}), it follows that $(x_i ,t_i)$ can be chosen so as to have $\widetilde{Y} = (xy)^a (x-y) (x_i \partial /\partial x_i - t_i \partial /\partial t_i)$. Formula~(\ref{index2}) then shows that all the remaining singularities have to be saddle-nodes since the sum of the indices is~$-1$. Nonetheless, again condition~2, implies that the holonomy of $\pi^{-1} (0) \setminus \{ p_1 ,\ldots , p_r \}$ is trivial. Thus no saddle-node can appear on $\pi^{-1} (0)$. In other words $r$ must be equal to~$1$ which is impossible.
\noindent {\it Proof of Proposition(\ref{prop4.2})}\,: Considering the normal forms~(\ref{forca1}) and~(\ref{forca2}), we can suppose that $h_i (0,0) \neq 0$. Set $\epsilon_i = ({\rm ord_{\pi^{-1} (0)}} (\widetilde{Y})) m_i - n_i d_i$ so that $\epsilon_i \in \{ -1, 0 ,1\}$ (cf. Lemma~(\ref{le3.1})). Alternatively we let $d_i= ({\rm ord_{\pi^{-1} (0)}} (\widetilde{Y})) m_i/n_i - \epsilon_i /n_i$.
On the other hand, Formula~(\ref{index2}), in the present context, becomes $$ \sum_{i=1}^r m_i /n_i = 1 \, , $$ where $m_i=0$ if and only if $p_i$ is a saddle-node and $n_i \neq 0$. Since all $m_i ,n_i$ are non-negative, only one of the $n_i$'s can be equal to~$1$ provided that $m_i \neq 0$. In this case, we must have $m_i=1$ as well and the remaining singularities are saddle-nodes. We claim that this implies that $h_i (0,0)$ in~(\ref{forca1}) is always different from {\it zero}. Indeed if, say $h_1 (0,0) =0$, then $m+1 =n_1 =1$ and the remaining singularities are saddle-nodes. The fact that the holonomy associated to the strong invariant manifold of a saddle-node is has order infinity, implies that this case cannot be produced. The resulting contradiction establishes the claim.
Now the fact that $h_i (0,0) \neq 0$ show that $\sum_{i=1}^r d_i = {\rm ord}_{(0,0)} (f) - {\rm ord}_{(0,0)} (g)$. Therefore \begin{eqnarray*} {\rm ord}_{(0,0)} (f) - {\rm ord}_{(0,0)} & = & \sum_{i=1}^r d_i = ({\rm ord_{\pi^{-1} (0)}} (\widetilde{Y})) (\sum_{i=1}^r m_i /n_i = 1) - \sum_{i=1}^r \epsilon_i /n_i \\
& = & - \sum_{i=1}^r \epsilon_i /n_i + {\rm ord}_{(0,0)} (f) + r-1 -{\rm ord}_{(0,0)} (g) -1 \, . \end{eqnarray*} In other words, one has \begin{equation} \sum_{i=1}^s (1- \epsilon_i /n_i) = 2 + s -r <2\, .\label{2sr=2} \end{equation} As mentioned, only one of the $n_i$'s may be equal to~$1$. In this case the remaining singularities are saddle-nodes and we obtain the model~$Z_{1,00}$.
Next assume that all the $n_i$'s are strictly greater than~$1$. In particular $1-\epsilon_i /n_i \geq 1/2$. The only new possibility is to have $n_1=n_2=2$ and $r-s=1$. Thus we obtain the model~$Z_{0,12}$ completing the proof of our proposition.
\fbox{}
\begin{obs} \label{locallinear} {\rm To complement the description of the Models $Z_{1,11}$, $Z_{0,12}$ and $Z_{1,00}$, we want to point out that excepted for the saddle-nodes, all the singularities appearing in the exceptional divisor after blowing-up are linearizable. Indeed this results from the finiteness of the local holonomies associated to their separatrizes. To check that these holonomies are finite we just have to use an argument analogous to the one employed in Remark~(\ref{ipc}).
As a consequence of the above fact, we conclude that the two saddle-nodes appearing as singularities of $\widetilde{\mathcal F}_{1,00}$ are identical. In particular either both have convergent weak invariant manifold or both have divergent weak invariant manifold. } \end{obs}
\section{The combinatorics of the reduction of singularities}
\hspace{0.4cm} In this last section we are going to prove our main results. Since we are going to work in local coordinates, we can consider a meromorphic semi-complete vector field $Y$ defined around $(0,0) \in {\mathbb C}^2$. As usual let ${\mathcal F}$ be the foliation associated to $Y$. In view of Seidenberg's theorem \cite{sei}, there exists a sequence of punctual blow-ups $\pi_j$ together with singular foliations $\widetilde{\mathcal F}^j$, \begin{equation} {\mathcal F} = \widetilde{\mathcal F}^0 \stackrel{\pi_1}{\longleftarrow} \widetilde{\mathcal F}^1 \stackrel{\pi_2}{\longleftarrow} \cdots \stackrel{\pi_r}{\longleftarrow} \widetilde{\mathcal F}^r \, , \label{resotree} \end{equation} where $\widetilde{\mathcal F}^j$ is the blow-up of $\widetilde{\mathcal F}^{j-1}$, such that all singularities of $\widetilde{\mathcal F}^r$ are simple. Furthermore each $\pi_j$ is centered at a singular point where $\widetilde{\mathcal F}^{j-1}$ has vanishing eigenvalues. The sequence $(\widetilde{\mathcal F}^j , \pi_j)$ is said to be the {\it resolution tree} of ${\mathcal F}$. Fixed $j \in \{ 1, \ldots ,r \}$, we denote by ${\mathcal E}^j$ the total exceptional divisor $(\pi_1 \circ \cdots \circ \pi_j)^{-1} (0,0)$ and by $D^j$ the irreducible component of ${\mathcal E}^j$ introduced by $\pi_j$. Note that $D^j$ is a rational curve given as $\pi_j^{-1} (\tilde{p}^{j-1})$ where $\tilde{p}^{j-1}$ is a singularity of $\widetilde{\mathcal F}^{j-1}$. Finally we identify curves and their proper transforms in the obvious way. Also $\widetilde{Y}^j$ will stand for the corresponding blow-up of $Y$.
Throughout this section $Y, {\mathcal F}$ are supposed to verify the following assumptions:
\noindent {\bf A}. $Y = y^{-k} fZ$ where $k \geq 2$, $Z$ is a holomorphic vector field having an isolated singularity at $(0,0) \in {\mathbb C}^2$ and $f$ is a holomorphic function.
\noindent {\bf B}. Assumptions~2 and~3 of Section~4.
\noindent {\bf C}. The origin is not a dicritical singularity of ${\mathcal F}$.
\noindent It immediately results from the above assumptions that the axis $\{ y=0\}$ is a smooth separatrix of ${\mathcal F}$. Letting $Z= f{\partial /\partial x} + g {\partial /\partial y}$, recall that the multiplicity of ${\mathcal F}$ along $\{ y=0\}$ (or the order of ${\mathcal F}$ w.r.t. $\{y=0\}$ and $(0,0)$) is by definition the order at $0 \in {\mathbb C}$ of the function $x \mapsto f(x,0)$.
The main result of this section is Theorem~(\ref{aque11}) below.
\begin{teo} \label{aque11} Let $Y, {\mathcal F}$ be as above. Suppose that the divisor of zeros/poles of $\widetilde{Y}^r$ contains ${\mathcal E}^r$ (i.e. there is no component $D^j$ of ${\mathcal E}^r$ where $\widetilde{Y}^r$ is regular). Then the multiplicity of ${\mathcal F}$ along $\{ y=0\}$ is at most~$2$ (in particular the order of ${\mathcal F}$ at $(0,0)$ is not greater than~$2$). \end{teo}
As a by-product of our proof, the cases in which the multiplicity of ${\mathcal F}$ along $\{ y=0\}$ is~$2$ are going to be characterized as well. Also note that assumption~{\bf C} ensures that all the components $D^j$ are invariant by ${\mathcal F}^r$. Moreover none of the singularities of $\widetilde{\mathcal F}^r$ is dicritical. In what follows we shall obtain the proof of Theorem~(\ref{aque11}) through a systematic analyse of the structure of the resolution tree of ${\mathcal F}$.
\begin{obs} \label{obs11} {\rm Let ${\mathcal F}$ be a foliation defined on a neighborhood of $(0,0) \in {\mathbb C}^2$ and consider a separatrix ${\mathcal S}$ of ${\mathcal F}$. Denote by $\widetilde{\mathcal F}$ the blow-up of ${\mathcal F}$ and by $\widetilde{{\mathcal S}}$ the proper transform of ${\mathcal S}$. Naturally $\widetilde{{\mathcal S}}$ constitutes a separatrix for some singularity $p$ of $\widetilde{\mathcal F}$. In the sequel the elementary relation \begin{equation} {\rm Ind}_{p} (\widetilde{\mathcal F} , \widetilde{{\mathcal S}}) = {\rm Ind}_{(0,0)} ({\mathcal F} ,{\mathcal S}) -1 \, \label{baixa1} \end{equation} will often be used.} \end{obs}
Consider the resolution tree~(\ref{resotree}) of ${\mathcal F}$. By assumption ${\mathcal E}^r$ contains a rational curve $D^r$ of self-intersection~$-1$. Blowing-down (collapsing) this curve yields a foliation $\widetilde{\mathcal F}^{r_1}$ together with the total exceptional divisor ${\mathcal E}^{r_1}$. If ${\mathcal E}^{r_1}$ contains a rational curve with self-intersection~$-1$ where all the singularities of $\widetilde{\mathcal F}^{r_1}$ are simple, we then continue by blowing-down this curve. Proceeding recurrently in this way, we eventually arrive to a foliation $\widetilde{\mathcal F}^{r_1}$, $1 \leq r_1 < r$, together with an exceptional divisor ${\mathcal E}^{r_1}$ such that every irreducible component of ${\mathcal E}^{r_1}$ with self-intersection~$-1$ contains a singularity of $\widetilde{\mathcal F}^{r_1}$ with vanishing eigenvalues. Let $\widetilde{Y}^{r_1}$ be the vector field corresponding to $\widetilde{\mathcal F}^{r_1}, \; {\mathcal E}^{r_1}$, using Proposition~(\ref{prop4.2}) we conclude the following:
\begin{lema} \label{sequence1} ${\mathcal E}^{r_1}$ contains (at least) one rational curve $D^{r_1}$ of self-intersection~$-1$. Moreover if $p$ is a singularity of $\widetilde{\mathcal F}^{r_1}$ belonging to $D^{r_1}$ then either $p$ is simple for $\widetilde{\mathcal F}^{r_1}$ or $\widetilde{Y}^{r_1}$ has one of the normal forms $Z_{1,11},\; Z_{0,12}, \; Z_{1,00}$ around $p$. Finally there is at least one such singularity $p_1$ which is not simple for $\widetilde{\mathcal F}^{r_1}$.
\fbox{} \end{lema}
The next step is to consider the following description of the models $Z_{1,11}, \; Z_{0,12}, \; Z_{1,00}$ (and their respective associated foliations ${\mathcal F}_{1,11} , \; {\mathcal F}_{0,12}, \; {\mathcal F}_{1,00}$) which results at once from the definition of these models given in Section~5. While this description is slightly less precise than the previous one, it emphasizes the properties more often used in the sequel.
\noindent $\bullet$ $Z_{1,11}, \; {\mathcal F}_{1,11}$\,: ${\mathcal F}_{1,11}$ has exactly 2 separatrizes ${\mathcal S}_1 ,{\mathcal S}_2$ which are smooth, transverse and of index {\it zero}. ${\mathcal F}_{1,11}$ has order~$2$ at the origin. The multiplicity of ${\mathcal F}_{1,11}$ along ${\mathcal S}_1 , \; {\mathcal S}_2$ is~$2$. The vector field $Z_{1,11}$ has poles of order~$1$ on each of the separatrizes ${\mathcal S}_1$, ${\mathcal S}_2$.
\noindent $\bullet$ $Z_{0,12}, \; {\mathcal F}_{0,12}$\,: ${\mathcal F}_{0,12}$ has order~$2$ at the origin and 2 smooth, transverse separatrizes coming from the separatrizes associated to the singularities of eigenvalues~$-1,2$ of $\widetilde{\mathcal F}_{0,12}$ (they are denoted ${\mathcal S}_1 ,{\mathcal S}_2$ and called strong separatrizes of ${\mathcal F}_{0,12}$). The multiplicity of ${\mathcal F}_{0,12}$ along ${\mathcal S}_1 , \; {\mathcal S}_2$ is~$2$ and the corresponding indices are both~$-1$. ${\mathcal F}_{0,12}$ has still a formal third separatrix ${\mathcal S}_3$, referred to as the weak separatrix of ${\mathcal F}_{0,12}$, which may or may not be convergent. The vector field $Z_{0,12}$ has poles of order~$d \in {\mathbb N}^{\ast}$ on both ${\mathcal S}_1 ,{\mathcal S}_2$.
\noindent $\bullet$ $Z_{1,00}, \; {\mathcal F}_{1,00}$\,: ${\mathcal F}_{1,00}$ has order~$2$ and 1 smooth separatrix ${\mathcal S}_1$ coming from the singularity of $\widetilde{\mathcal F}_{1,00}$ whose eigenvalues are~$-1,1$ which will be called the strong separatrix of ${\mathcal F}_{1,00}$. Note that the multiplicity of ${\mathcal F}_{1,00}$ along ${\mathcal S}_1$ is~$2$ and the index of ${\mathcal S}_1$ is {\it zero}. $\widetilde{\mathcal F}_{1,00}$ also has 2 additional formal separatrizes ${\mathcal S}_2 ,{\mathcal S}_3$ coming from the weak invariant manifold of the saddle-nodes singularities of $\widetilde{\mathcal F}_{1,00}$ and, accordingly, called the weak separatrizes of ${\mathcal F}_{1,00}$. Naturally the weak separatrizes of ${\mathcal F}_{1,00}$ may or may not converge. Finally the vector field $Z_{1,00}$ has poles of order~$d \in {\mathbb N}^{\ast}$ on ${\mathcal S}_1$.
\begin{obs} \label{combin} {\rm The content of this remark will not be proved in these notes and therefore will not be formally used either. Nonetheless it greatly clarifies the structure of the combinatorial discussion that follows. Consider a meromorphic vector field $X$ having a smooth separatrix ${\mathcal S}$. Using appropriate coordinates $(x,y)$ we can identify ${\mathcal S}$ with $\{ y=0 \}$ and write $X$ as $y^d (f{\partial /\partial x} + yg {\partial /\partial y})$ where $d \in {\mathbb Z}$ and $f(x,0)$ is a non-trivial meromorphic function. We define the {\it asymptotic order} of $X$ at ${\mathcal S}$ (at $(0,0)$), ${\rm ord\,asy}_{(0,0)} (X ,{\mathcal S})$, by means of the formula \begin{equation} {\rm ord\,asy}_{(0,0)} (X, {\mathcal S}) = {\rm ord}_0 f(x,0) + d. {\rm Ind}_{(0,0)} ({\mathcal F} ,{\mathcal S}) \, \label{defioras} \end{equation} where ${\mathcal F}$ is the foliation associated to $X$. It can be proved that $0 \leq {\rm ord\,asy}_{(0,0)} (X ,{\mathcal S}) \leq 2$ provided that $X$ is semi-complete. Besides, if ${\mathcal S}$ is induced by a global rational curve (still denoted by ${\mathcal S}$) and $p_1 ,\ldots, p_s$ are the singularities of ${\mathcal F}$ on ${\mathcal S}$, then we have \begin{equation} \sum_{i=1}^s {\rm ord\,asy}_{p_i} (X, {\mathcal S}) = 2 \label{somou2} \end{equation} provided that $X$ is semi-complete on a neighborhood of ${\mathcal S}$. Note that the above formula indeed generalizes formula~(\ref{2sr=2}).} \end{obs}
Now we go back to the vector field $\widetilde{Y}^{r_1}$ on a neighborhood of $D^{r_1}$. Again we denote by $p_1 ,\ldots ,p_s$ the singularities of $\widetilde{\mathcal F}^{r_1}$ on $D^{r_1}$ and, without loss of generality, assume that $\widetilde{Y}^{r_1}$ admits one of the normal forms $Z_{1,11},\; Z_{0,12}, \; Z_{1,00}$ around $p_1$.
\begin{lema} \label{sequence2} All the singularities $p_2 ,\ldots ,p_s$ are simple for $\widetilde{\mathcal F}^{r_1}$. \end{lema}
\noindent {\it Proof}\,: We have to prove that it is not possible to exist two singularities with one of the normal forms $Z_{1,11},\; Z_{0,12}, \; Z_{1,00}$ on $D^{r_1}$. Clearly we cannot have two singularities of type $Z_{1,11}$ otherwise a ``generic'' regular orbit of $\widetilde{Y}^{r_1}$ would contain two singular points which contradicts assumption~{\bf B}. Indeed the fact that a ``generic'' regular orbit of $\widetilde{Y}^{r_1}$ effectively intersects the divisor of zeros of both singularities results from the method employed in the preceding section. In the present case the discussion is simplified since the singularity appearing in the intersection of the two irreducible components of the exceptional divisor is linear with eigenvalues~$-1,1$ (cf. description of $Z_{1,11}$).
To prove that the other combinations are also impossible, it is enough to repeat the argument employed in Section~5, in particular using the fact that the order $d_1\neq 0$ of $\widetilde{Y}^{r_1}$ on $D^{r_1}$ does not depend of the singularity $p_i$. If the reader takes for grant Formula~(\ref{somou2}), this verification can easily be explained. In fact, the asymptotic order of $Z_{0,12}$ (resp. $Z_{1,00}$) with respect to its strong separatrizes is already~$2$. Furthermore the asymptotic order of $Z_{1,11}$ with respect to its separatrizes is~$1$. Since the asymptotic order of a semi-complete singularity cannot be negative, it becomes obvious that two such singularities cannot co-exist on $D^{r_1}$ provided that $Y^{r_1}$ is semi-complete.
\fbox{}
Now we analyse each of the three possible cases.
\noindent $\bullet$ The normal form of $\widetilde{Y}^{r_1}$ around $p_1$ is $Z_{1,11}$: First recall that $D^{r_1}$ is an irreducible component of order~$1$ of the pole divisor of $\widetilde{Y}^{r_1}$. Since the number of singularities of regular orbits of $\widetilde{Y}^{r_1}$ is at most~$1$, it follows that none of the remaining singularities $p_2 ,\ldots , p_s$ can be a LJ-singularity for $\widetilde{\mathcal F}^{r_1}$. Otherwise there would be another curve of {\it zeros} of $\widetilde{Y}^{r_1}$ which is not invariant by $\widetilde{\mathcal F}^{r_1}$ so that ``generic'' regular orbits of $\widetilde{Y}^{r_1}$ would have~$2$ singularities (cf. above). By the same reason the holonomy of $D^{r_1} \setminus \{ p_2 ,\ldots ,p_s\}$ with respect to $\widetilde{\mathcal F}^{r_1}$ must be trivial. This implies that none of the singularities $p_2 ,\ldots ,p_s$ is a saddle-node for $\widetilde{\mathcal F}^{r_1}$. In fact, by virtue of Theorem~(\ref{selano}), a saddle-node must have strong invariant manifold contained in $D^{r_1}$ which ensures that the above mentioned holonomy is non-trivial. Then we conclude that all the remaining singularities $p_2 ,\ldots ,p_s$ have eigenvalues $m_i ,-n_i$ ($i=2,\ldots ,s$) with $m_i ,n_i \in {\mathbb N}^{\ast}$. Once again the fact that the holonomy of $D^{r_1} \setminus \{ p_2 ,\ldots ,p_s\}$ is trivial shows that $m_i /n_i \in {\mathbb N}^{\ast}$. Finally Formula~(\ref{index2}) implies that $s=2$ and $m_2=n_2 =1$. An immediate application of Formulas~(\ref{ordem2}) and~(\ref{equa2}) (or yet Formula~(\ref{somou2})) shows that the separatrix of $p_2$ transverse to $D^{r_1}$ is a component of order~$1$ of the pole divisor of $\widetilde{Y}^{r_1}$. Finally we denote by $Z_{1,11}^{(1)}$ (resp. ${\mathcal F}_{1,11}^{(1)}$) the local vector field (resp. holomorphic foliation) resulting from the collapsing of $D^{r_1}$. Summarizing one has:
\noindent $\bullet$ $Z_{1,11}^{(1)}, \; {\mathcal F}_{1,11}^{(1)}$\,: The foliation ${\mathcal F}_{1,11}^{(1)}$ has exactly two separatrizes ${\mathcal S}_1 ,{\mathcal S}_2$ which are smooth, transverse and of indices respectively equal to~$1$ and~$0$. The order of ${\mathcal F}_{1,11}^{(1)}$ at the origin is~$2$ as well as the multiplicity of ${\mathcal F}_{1,11}^{(1)}$ along ${\mathcal S}_1 , {\mathcal S}_2$. The vector field $Z^{(1)}_{1,11}$ has poles of order~$1$ on ${\mathcal S}_1 ,{\mathcal S}_2$.
Now let us discuss the second case.
\noindent $\bullet$ The normal form of $\widetilde{Y}^{r_1}$ around $p_1$ is $Z_{0,12}$\,: Recall that $D^{r_1}$ is an irreducible component of order $d\neq 0$ of the pole divisor of $\widetilde{Y}^{r_1}$. Repeating the argument of the previous section, we see that the singularities $p_2 , \ldots ,p_s$ can be neither saddle-nodes nor LJ-singularities. Again this can directly be seen from Formula~(\ref{somou2}): by virtue of Lemma~(\ref{le3.3}) and Theorem~(\ref{selano}), both types of singularities in question have asymptotic order equal to~$1$. Nonetheless the asymptotic order of $Z_{0,12}$ is already~$2$ which implies the claim. It follows that $\widetilde{\mathcal F}^{r_1}$ has eigenvalues $m_i, -n_i$ at each of the remaining singularities $p_2 , \ldots, p_s$ ($m_i ,n_i \in {\mathbb N}^{\ast}$). In particular the index of $D^{r_1}$ w.r.t. $\widetilde{\mathcal F}^{r_1}$ around each $p_i$ is strictly negative. Hence Formula~(\ref{index2}) shows that $p_1$ is, in fact, the unique singularity of $\widetilde{\mathcal F}^{r_1}$ on $D^{r_1}$. We denote by $Z_{0,12}^{(1)}$ (resp. ${\mathcal F}_{0,12}^{(1)}$) the local vector field (resp. holomorphic foliation) arising from the collapsing of $D^{r_1}$. Thus:
\noindent $\bullet$ $Z_{0,12}^{(1)}, \; {\mathcal F}_{0,12}^{(1)}$\,: The order of ${\mathcal F}_{0,12}^{(1)}$ at the origin is~$1$, besides the linear part of ${\mathcal F}_{0,12}^{(1)}$ is nilpotent. ${\mathcal F}_{0,12}^{(1)}$ has one strong separatrix ${\mathcal S}_1$ obtained through the strong separatrix of ${\mathcal F}_{0,12}$ which is transverse to $D^{r_1}$. This separatrix is smooth and has index {\it zero}, furthermore the multiplicity of ${\mathcal F}_{0,12}^{(1)}$ along ${\mathcal S}_1$ is~$2$. The foliation ${\mathcal F}_{0,12}^{(1)}$ has still another formal weak separatrix which may or may not converge. Finally the vector field $Z_{0,12}^{(1)}$ has poles of order $d\neq 0$ on ${\mathcal S}_1$.
Finally we have:
\noindent $\bullet$ The normal form of $\widetilde{Y}^{r_1}$ around $p_1$ is $Z_{1,00}$\,: Note that $D^{r_1}$ is an irreducible component of order~$d\neq 0$ of the pole divisor of $\widetilde{Y}^{r_1}$ (cf. description of the vector field $Z_{1,00}$). As before the remaining singularities cannot be saddle-nodes or LJ-singularities (the asymptotic order of $Z_{1,00}$ w.r.t. $D^{r_1}$ is already~$2$). It follows that $\widetilde{\mathcal F}^{r_1}$ has eigenvalues $m_i, -n_i$ at the remaining singularities $p_2 ,\ldots ,p_s$ ($m_i, n_i \in {\mathbb N}$). Around each singularity $p_i$ ($i=2,\ldots ,s$), the vector field $\widetilde{Y}^{r_1}$ can be written as $$ x_i^{-d} t_i^{k_i} h_i [m_i x_i (1+ {\rm h.o.t.}) \partial /\partial x_i - n_i t_i (1+ {\rm h.o.t.}) \partial /\partial t_i] $$ where $h_i (0,0) \neq 0$. We just have to repeat the argument of Section~$5$, here we summarize the discussion by using the ``fact'' that the asymptotic order of $\widetilde{Y}^{r_1}$ w.r.t. $D^{r_1}$ has to be {\it zero} at each $p_i$. Indeed, this gives us that $k_i =-1+dm_i /n_i$. Comparing this with Lemma~(\ref{le3.1}), it results that $n_i=1$. Hence Formula~(\ref{index2}) informs us that $s=2$ and $m_2 =n_2 =1$. It also follows that $k_2 =d-1$. Let us denote by $Z_{1,00}^{(1)}$ (resp. ${\mathcal F}_{1,00}^{(1)}$) the local vector field (resp. holomorphic foliation) arising from the collapsing of $D^{r_1}$.
\noindent $\bullet$ $Z_{1,00}^{(1)}, \; {\mathcal F}_{1,00}^{(1)}$\,: The order of ${\mathcal F}_{1,00}^{(1)}$ at the origin is~$2$ and it has one strong separatrix ${\mathcal S}_1$ obtained through the separatrix of $p_2$ which is transverse to $D^{r_1}$. This separatrix is smooth and has index {\it zero}, furthermore the multiplicity of ${\mathcal F}_{1,00}^{(1)}$ along ${\mathcal S}_1$ is~$2$. The foliation ${\mathcal F}_{1,00}^{(1)}$ has still two formal weak separatrizes which may or may not converge. Finally the vector field $Z_{1,00}^{(1)}$ has poles of order $d\neq 0$ on ${\mathcal S}_1$.
Summarizing what precedes, we easily obtain the following analogue of Lemma~(\ref{sequence1}):
\begin{lema} \label{sequence4} ${\mathcal E}^{r_2}$ contains (at least) one rational curve $D^{r_2}$ of self-intersection~$-1$. Moreover if $p$ is a singularity of $\widetilde{\mathcal F}$ belonging to $D^{r_2}$ then either $p$ is simple for $\widetilde{\mathcal F}^{r_2}$ or $\widetilde{Y}^{r_2}$ has one of the normal forms $Z_{1,11},\; Z_{0,12}, \; Z_{1,00}, \; Z_{1,11}^{(1)}, \; Z_{0,12}^{(1)}, \; Z_{1,00}^{(1)}$ around $p$. Finally there is at least one such singularity $p_1$ which is not simple for $\widetilde{\mathcal F}^{r_2}$.
\fbox{} \end{lema}
The argument is now by recurrence, we shall discuss only the next step in details. Again $p_1 ,\ldots ,p_s$ are the singularities of $\widetilde{\mathcal F}^{r_2}$ on $D^{r_2}$ and, without loss of generality, $\widetilde{Y}^{r_2}$ admits one of the normal forms $Z_{1,11},\; Z_{0,12}, \; Z_{1,00}, \;Z_{1,11}^{(1)}, \; Z_{0,12}^{(1)}, \; Z_{1,00}^{(1)}$ around $p_1$.
\begin{lema} \label{sequence5} All the singularities $p_2 ,\ldots ,p_s$ are simple for $\widetilde{\mathcal F}^{r_2}$. \end{lema}
\noindent {\it Proof}\,: The proof is as in Lemma~(\ref{sequence2}). Since no leaf of $\widetilde{\mathcal F}^{r_2}$ can meet the divisor of zeros of $\widetilde{Y}^{r_2}$ in more than one point, it results that we can have at most one singularity $Z_{1,11}$ or $Z_{1,11}^{(1)}$ on $D^{r_2}$. The fact that the models $Z_{0,12}, \; Z_{1,00}, \; Z_{0,12}^{(1)}$ and $Z_{1,00}^{(1)}$ cannot be combined among them or with $Z_{1,11}, \; Z_{1,11}^{(1)}$ follows from the natural generalization of the method of Section~5 (which is again explained by the fact that these $3$ vector fields have asymptotic order~$2$ w.r.t. $D^{r_2}$). The lemma is proved.
\fbox{}
So we have obtained three new possibilities according to the normal form of $\widetilde{Y}^{r_2}$ around $p_1$ is $Z_{1,11}^{(1)}, \; Z_{0,12}^{(1)}$ or $Z_{1,00}^{(1)}$. Let us analyse them separately.
\noindent $\bullet$ The normal form of $\widetilde{Y}^{r_2}$ around $p_1$ is $Z_{1,11}^{(1)}$\,: Note that ${\mathcal F}_{1,11}^{(1)}$ has two separatrizes which may coincide with $D^{r_2}$, one with index {\it zero} and other with index~$1$. In any case, $D^{r_2}$ is an irreducible component of order~$1$ of the divisor of poles of $\widetilde{Y}^{r_2}$. Suppose first that the index of $D^{r_2}$ w.r.t. ${\mathcal F}^{r_2}$ at $p_1$ is {\it zero}. The discussion then goes exactly as in the case of $Z_{1,11}$. We conclude that $s=2$ and that ${\mathcal F}^{r_2}$ has eigenvalues~$-1,1$ at $p_2$. The fact that the holonomy of $D^{r_2} \setminus \{ p_1 ,p_2 \}$ w.r.t. ${\mathcal F}^{r_2}$ is trivial also implies that ${\mathcal F}^{r_2}$ is linearizable at $p_2$. Formulas~(\ref{ordem2}) and~(\ref{equa2}) show that the separatrix of $p_2$ transverse to $D^{r_2}$ is a component with order~$1$ of the divisor of poles of $\widetilde{Y}^{r_2}$. Finally we denote by $Z_{1,11}^{(2)}$ (resp. ${\mathcal F}_{1,11}^{(2)}$) the local vector field (resp. holomorphic foliation) arising from the collapsing of $D^{r_2}$. One has
\noindent $\bullet$ $Z_{1,11}^{(2)}, \; {\mathcal F}_{1,11}^{(2)}$\,: The foliation ${\mathcal F}_{1,11}^{(2)}$ has exactly 2 separatrizes ${\mathcal S}_1 ,{\mathcal S}_2$ which are smooth, transverse and of indices respectively equal to {\it zero} and~$2$. ${\mathcal F}_{1,11}^{(2)}$ has order~$2$ at the origin and its multiplicity along ${\mathcal S}_1 , \; {\mathcal S}_2$ is~$2$. The vector field $Z_{1,11}^{(2)}$ has poles of order~$1$ on each of the separatrizes ${\mathcal S}_1$, ${\mathcal S}_2$.
Now let us prove that the index of $D^{r_2}$ w.r.t. ${\mathcal F}^{r_2}$ at $p_1$ cannot be~$1$. Suppose for a contradiction that this index is~$1$. Again the triviality of the holonomy of the regular leaf contained in $D^{r_2}$ implies that all the singularities $p_2 ,\ldots ,p_s$ are linearizable with eigenvalues~$-1,1$. Hence Formula~(\ref{index2}) ensures that $s=3$. In turn, Formula~(\ref{equa2}) shows that the separatrix of $p_2$ (resp. $p_3$) transverse to $D^{r_2}$ is a component with order~$1$ of the pole divisor of $\widetilde{Y}^{r_2}$. This is however impossible in view of Formula~(\ref{somou2}). Alternate, we can observe that the divisor of poles of the vector field obtained by collapsing $D^{r_2}$ consists of three smooth separatrizes. By virtue of assumptions {\bf A, B, C} one of them must be the proper transform of $\{ y=0\}$. In particular its order as component of the pole divisor should be $k \geq 2$, thus providing the desired contradiction.
Next one has:
\noindent $\bullet$ The normal form of $\widetilde{Y}^{r_2}$ around $p_1$ is $Z_{0,12}^{(1)}$\,: The divisor $D^{r_2}$ constitutes a separatrix of $\widetilde{\mathcal F}^{r_2}$ at $p_1$. Besides $\widetilde{Y}^{r_2}$ has poles of order $d \neq 0$ on $D^{r_2}$ (cf. description of $Z_{0,12}^{(1)}$). Note also that the index of $D^{r_2}$ w.r.t. $\widetilde{\mathcal F}^{r_2}$ at $p_1$ is {\it zero}. Our standard method shows that the remaining singularities cannot be saddle-nodes or LJ-singularities (the asymptotic order of $Z_{0,12}^{(1)}$ w.r.t. $D^{r_2}$ is already~$2$). Thus $\widetilde{\mathcal F}^{r_2}$ has eigenvalues $m_i, -n_i$ at the singularity $p_i$, $i=2,\ldots s$ ($m_i, n_i \in {\mathbb N}$). On a neighborhood of $p_i$, $\widetilde{Y}^{r_2}$ is given in appropriate coordinates by $$ x_i^{-d} t_i^{k_i} h_i [m_i x_i (1+ {\rm h.o.t.}) \partial /\partial x_i - n_i t_i (1+ {\rm h.o.t.}) \partial /\partial t_i] $$ where $h_i (0,0) \neq 0$. It is enough to repeat the discussion of Section~$5$. Using the ``fact'' that the asymptotic order of $\widetilde{Y}^{r_1}$ w.r.t. $D^{r_1}$ has to be {\it zero} at each $p_i$, this can be summarized as follows. The asymptotic order is given by $k_i + 1 -dm_i /n_i$, since it must equal {\it zero}, one has $k_i =-1 +dm_i/n_i$. Comparing with Lemma~(\ref{le3.1}), we conclude that $n_i=1$. Hence Formula~(\ref{index2}) implies that $s=2$ and $m_2=n_2 =1$. In particular $k_2=d-1$. We then denote by $Z_{0,12}^{(2)}$ (resp. ${\mathcal F}_{0,12}^{(2)}$) the local vector field (resp. holomorphic foliation) arising from the collapsing of $D^{r_2}$.
\noindent $\bullet$ $Z_{0,12}^{(2)}, \; {\mathcal F}_{0,12}^{(2)}$\,: The order of ${\mathcal F}_{0,12}^{(2)}$ at the origin is~$2$. ${\mathcal F}_{0,12}^{(2)}$ has one strong separatrix ${\mathcal S}_1$ obtained through the strong separatrix of ${\mathcal F}_{0,12}^{(1)}$ which is transverse to $D^{r_2}$. This separatrix is smooth and has index {\it zero}, furthermore the multiplicity of ${\mathcal F}_{0,12}^{(2)}$ along ${\mathcal S}_1$ is~$2$. The foliation ${\mathcal F}_{0,12}^{(2)}$ has still another formal weak separatrix which may or may not converge. Finally the vector field $Z_{0,12}^{(2)}$ has poles of order $d\neq 0$ on ${\mathcal S}_1$.
Finally let us consider $Z_{1,00}^{(1)}$.
\noindent $\bullet$ The normal form of $\widetilde{Y}^{r_2}$ around $p_1$ is $Z_{1,00}^{(1)}$\,: The discussion is totally analoguous to the case $Z_{1,00}$. After collapsing $D^{r_2}$, we obtain a local vector field $Z_{1,00}^{(2)}$ (resp. holomorphic foliation ${\mathcal F}_{1,00}^{(2)}$) with the following characteristics:
\noindent $\bullet$ $Z_{1,00}^{(2)}, \; {\mathcal F}_{1,00}^{(2)}$\,: The order of ${\mathcal F}_{1,00}^{(2)}$ at the origin is~$2$ and it has one strong separatrix ${\mathcal S}_1$ obtained through the separatrix of $p_2$ which is transverse to $D^{r_2}$. This separatrix is smooth and has index {\it zero}, furthermore the multiplicity of ${\mathcal F}_{1,00}^{(2)}$ along ${\mathcal S}_1$ is~$2$. The foliation ${\mathcal F}_{1,00}^{(2)}$ has still two formal weak separatrizes which may or may not converge. Finally the vector field $Z_{1,00}^{(2)}$ has poles of order $d\neq 0$ on ${\mathcal S}_1$.
Let us inductively define a sequence of vector fields $Z_{1,11} ^{(n)}$ by combining over a rational curve with self-intersection~$-1$ a model $Z_{1,11}^{(n-1)}$ with a linear singularity $p_2$ having eigenvalues~$-1,1$. The rational curve in question induces a separatrix of index {\it zero} for $Z_{1,11}^{(n-1)}$ and both separatrizes of $p_2$ are components having order~$1$ of the pole divisor of the corresponding vector field. The model $Z_{1,11}^{(n)}$ is then obtained by collapsing the mentioned rational curve. Similarly we also define the sequences $Z_{1,00}^{(n)}, \, Z_{0,12}^{(n)}$.
\noindent {\it Proof of Theorem~(\ref{aque11})}\,: Let $Y,\; {\mathcal F}$ be as in the statement of this theorem. Suppose first that the order of ${\mathcal F}$ at $(0,0)$ is greater than one. We consider a resolution tree~(\ref{resotree}) for ${\mathcal F}$ and the recurrent procedure discussed above. Whenever we collapse a rational curve with self-intersection~$-1$ contained in one of the exceptional divisors ${\mathcal E}^j$, it results a singularity which is either simple or belongs to the list $Z_{1,11}, \; Z_{1,11}^{(n)}, \; Z_{1,00}, \; Z_{1,00}^{(n)},\; Z_{0,12}, \; Z_{0,12}^{(n)}$. After a finite number of steps of the above procedure, we arrive to the original vector field $Y$ (resp. foliation ${\mathcal F}$). Therefore $Y$ must admit one of the above indicated normal forms. Since the divisor of poles of $Y$ is constituted by the axis $\{ y=0\}$, the cases $Z_{1,11},\; Z_{1,11}^{(n)}, \; Z_{0,12}$ cannot be produced (their divisor of poles consist of two irreducible components). The case $Z_{0,12}^{(1)}$ cannot be produced either since the order of the associated foliation is supposed to be greater than or equal to~$2$. Thus we conclude that $Y=Z_{1,00}^{(n)}$ or, for $n\geq 2$, $Y=Z_{0,12}^{(n)}$ and the theorem follows in this case.
Next suppose that the order of ${\mathcal F}$ at $(0,0)$ is~$1$. Clearly if the linear part of ${\mathcal F}$ at $(0,0)$ has rank~$2$ (i.e. the corresponding holomorphic vector field with isolated singularities has~$2$ non-vanishing eigenvalues at $(0,0)$), then the conclusion is obvious. Next suppose that ${\mathcal F}$ is a saddle-node. If $\{ y=0\}$ corresponds to the strong invariant manifold of ${\mathcal F}$ then the statement is obvious. On the other hand, $\{ y=0\}$ cannot be the weak invariant manifold thanks to Theorem~(\ref{selano}).
It only remains to check the case where the linear part of ${\mathcal F}$ at $(0,0)$ is nilpotent. The blow-up $\widetilde{Y}$ (resp. $\widetilde{\mathcal F}$) of $Y$ (resp. ${\mathcal F}$) is such that $\widetilde{\mathcal F}$ has a single singularity $p \in \pi^{-1} (0)$. In addition the order of $\widetilde{\mathcal F}$ at $p$ is necessarily~$2$. On a neighborhood of $p$, the vector field $\widetilde{Y}$ is given by $x^{-k} y^{-k} Z$ where $Z$ is as in assumption~{\bf A}. The assumptions~{\bf B}, {\bf C} are clearly verified as well. An inspection in the preceding discussion immediately shows that it applies equally well to this vector field $\widetilde{Y}$. We conclude that $\widetilde{Y}$ is given on a neighborhood of $p$ by the model $Z_{0,12}$. Hence $Y$ is the model $Z_{0,12}^{(1)}$ completing the proof of the theorem.
\fbox{}
\noindent {\bf {\large $\bullet$ Conclusion}}:
\noindent {\it Proof of Theorem~A}\,: Let $X$ be a complete polynomial vector field in ${\mathbb C}^2$ with degree~$2$ or greater. We denote by ${\mathcal F}_X$ the singular holomorphic foliation induced by $X$ on ${\mathbb C} {\mathbb P} (2)$. We know from Lemma~(\ref{lema2.1}) that the line at infinity $\Delta \subset {\mathbb C} {\mathbb P} (2)$ is invariant under ${\mathcal F}_X$. On the other hand there is a dicritical singularity $p_1$ of ${\mathcal F}_X$ belonging to $\Delta$. Hence Proposition~(\ref{jouano}) ensures that ${\mathcal F}_X$ has a meromorphic first integral on ${\mathbb C} {\mathbb P} (2)$. Besides the generic leaves of ${\mathcal F}_X$ in ${\mathbb C} {\mathbb P} (2)$ are, up to normalization, rational curves (i.e. isomorphic to ${\mathbb C} {\mathbb P} (1)$). According to Saito and Suzuki (cf. \cite{suzu2}), up to polynomial automorphisms of ${\mathbb C}^2$, ${\mathcal F}_X$ is given by a first integral $R$ having one of the following forms: \begin{description}
\item[$\imath$)] $R(x,y) = x$;
\item[$\imath \imath$)] $R(x,y) = x^n y^m$, with ${\rm g.c.d} (m,n) =1$ and $m,n \in {\mathbb Z}$;
\item[$\imath \imath \imath$)] $R(x,y)=x^n(x^ly + P(x))^m$, with ${\rm g.c.d} (m,n) =1$ and $m,n \in {\mathbb Z}$, $l\geq 1$. Moreover $P$ is a polynomial of degree at most $l-1$ satisfying $P (0) \neq 0$. \end{description} To each of these first integrals there corresponds the foliations associated to the vector fields $X_1 ={\partial /\partial x}$, $X_2 = mx {\partial /\partial x} - ny {\partial /\partial y}$ and $X_3 = mx^{l+1} {\partial /\partial x} -[(n+lm)x^ly + nP(x) + mxP' (x)] {\partial /\partial y}$. Therefore the original vector field $X$ has the form $X=Q. X_i$, where $Q$ is a polynomial and $i =1,2,3$. If $i=1$ then it follows at once that $Q$ has to be as in the in order to produce a complete vector field $X$. Assume now that $i=2$. Using for instance Lemma~(\ref{le3.1}), we see that $P$ has again the form indicated in the statement unless $X_2 = x {\partial /\partial x} - y {\partial /\partial y}$ in which case we can also have $P = (xy)^a (x-y)$. Nonetheless we immediately check that the resulting vector field $X$ is not complete in this case.
Finally let us assume that $i=3$. It is again easy to see that that the resulting vector field cannot be complete. This follows for example from the fact that $(0,0)$ is a singularity of $X_3$ with trivial eigenvalues (cf. \cite{re4}). The theorem is proved.
\fbox{}
\noindent {\it Proof of Theorem~B}\,: Let us suppose for a contradiction that none of the singularities $p-1 ,\ldots ,p_k$ of ${\mathcal F}_X$ in $\Delta$ is dicritical. We write $X$ as $F.Z$ where $F$ is a polynomial of degree $n\in {\mathbb N}$ and $Z$ is a polynomial vector field of degree $d-n$ and having isolated zeros (where $d$ is the degree of $X$).
We consider the restriction of $X$ to a neighborhood of $p_i$. Clearly this restriction satisfies assumptions~{\bf A}, {\bf B} and~{\bf C} of Section~5. In particular Theorem~(\ref{aque11}) applies to show that the multiplicity of ${\mathcal F}$ along $\Delta$ is at most~$2$. Moreover, if this multiplicity is~$2$, then $X$ admits one of the normal forms $Z_{1,11}, \; Z_{1,11}^{(n)}, \; Z_{1,00}, \; Z_{1,00}^{(n)},\; Z_{0,12}, \; Z_{0,12}^{(n)}$ on a neighborhood of $p_i$.
From Lemma~(\ref{lema2.4}) we know that $k \leq 3$. Since the sum of the multiplicities of ${\mathcal F}$ along $\Delta$ at each $p_i$ is equal to $d-n+1$, it follows that $d-n \leq 5$.
However, if $k=3$, Corollary~(\ref{lema2.3}) shows that the top-degree homogeneous component of $X$ is as in the cases~5, 6, 7 of the corollary in question. Simple calculations guarantees that it is not possible to realize the models $Z_{1,11}, \ldots , Z_{0,12}^{(n)}$ in this way. The case $k=1$ being trivial, we just need to consider the case $k=2$. Now we have $d-n \leq 3$ and again it is very easy to conclude that none of these possibilities lead to a complete polynomial vector field. The resulting contradiction proves the theorem.
\fbox{}
\noindent {\sc Julio C. Rebelo \begin{tabbing} {\rm {\bf Permanent Address}} \hspace{3.8cm} \= {\rm {\bf Current Address}}\\ PUC-Rio \> Institute for Mathematical Sciences\\ R. Marques de S. Vicente 225 \> SUNY at Stony Brook\\ Rio de Janeiro RJ CEP 22453-900 \> Stony Brook NY 11794-3660\\ Brazil \> USA\\ {\it email [email protected]} \> {\it email [email protected]} \end{tabbing}
}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{f On bi-exactness of discrete quantum groups}
\begin{abstract} We define Ozawa's notion of bi-exactness to discrete quantum groups, and then prove some structural properties of associated von Neumann algebras. In particular, we prove that any non amenable subfactor of free quantum group von Neumann algebras, which is an image of a faithful normal conditional expectation, has no Cartan subalgebras. \end{abstract}
\section{\bf Introduction}
A countable discrete group $\Gamma$ is said to be \textit{bi-exact} (or said to be in \textit{class} $\cal S$)
if it is exact and there exists a map $\mu\colon \Gamma \rightarrow \mathrm{Prob}(\Gamma)\subset \ell^1(\Gamma)$ such that $\limsup_{x\rightarrow\infty}\|\mu(sxt)-s\mu(x)\|_1=0$ for any $s,t\in \Gamma$. This notion was introduced and studied by Ozawa in his book with Brown $\cite[\textrm{Section 15}]{BO08}$. In particular, he gave the following two different characterizations of bi-exactness (Lemma 15.1.4 and Proposition 15.2.3(2) in the book): \begin{itemize}
\item[$\rm (i)$] The group $\Gamma$ is exact and the algebra $L\Gamma$ satisfies condition $\rm (AO)^+$ with the dense $C^*$-algebra $C^*_\lambda(\Gamma)$.
\item[$\rm (ii)$] There exists a unital $C^*$-subalgebra ${\cal B}\subset \ell^\infty (\Gamma)$ such that
\begin{itemize}
\item[$\bullet$] the algebra $\cal B$ contains $c_0(\Gamma)$ (so that we can define ${\cal B}_\infty:={\cal B}/c_0(\Gamma)$);
\item[$\bullet$] the left translation action on $\ell^\infty(\Gamma)$ induces an amenable action on ${\cal B}_{\infty}$, and the right one induces the trivial action on ${\cal B}_{\infty}$.
\end{itemize} \end{itemize} Here we recall that a von Neumann algebra $M\subset \mathbb{B}(H)$ with a standard representation satisfies \textit{condition} $\rm (AO)^+$ $\cite[\textrm{Definition 3.1.1}]{Is12_2}$ if there exists a unital $\sigma$-weakly dense locally reflexive $C^*$-subalgebra $A\subset M$ and a unital completely positive (say, u.c.p.) map $\theta\colon A\otimes A^{\rm op} \rightarrow \mathbb{B}(H)$ such that $\theta(a\otimes b^{\rm op})-ab^{\rm op}\in \mathbb{K}(H)$ for any $a,b \in A$. Here $A^{\rm op}$ means the opposite algebra of $A$, which acts canonically on $H$ by right.
These characterizations of bi-exactness have been used in different contexts. For example, condition (i) is very close to condition (AO) introduced in $\cite{Oz03}$. In particular Ozawa's celebrated theorem in the same paper says that von Neumann algebras of bi-exact groups are solid, meaning that any relative commutant of a diffuse amenable subalgebra is still amenable. Condition (ii) is used to show that hyperbolic groups are bi-exact. This follows from the fact that all hyperbolic groups act on their Gromov boundaries as amenable actions. The definition of bi-exactness itself is also interesting since it is closely related to an \textit{array}, which was introduced and studied in $\cite{CS11}$ and $\cite{CSU11}$.
In the present paper, we generalize bi-exactness to discrete quantum groups in two different ways, which correspond to conditions (i) and (ii). This is not a difficult task since condition (i) does not rely on the structure of group $C^*$-algebras (this is a $C^*$-algebraic condition) and all objects in condition (ii) are easily defined for discrete quantum groups. We then study some basic facts on these conditions. In particular, we prove that free products of free quantum groups or quantum automorphism groups are bi-exact, showing that a condition close to bi-exactness is closed under taking free products.
After these observations, we prove some structural properties of von Neumann algebras associated with bi-exact quantum groups. We first give the following theorem which generalizes $\cite[\textrm{Theorem 3.1}]{PV12}$. This is a natural generalization from discrete groups to discrete quantum groups of Kac type. In the proof, we need only slight modifications since the original proof for groups do not rely on the structures of group von Neumann algebras very much. Note that a special case of this theorem was already generalized in $\cite{Is12_2}$ for general $\rm II_1$ factors.
\begin{ThmA}\label{A} Let $\mathbb{G}$ be a compact quantum group of Kac type, whose dual acts on a tracial von Neumann algebra $B$ as a trace preserving action. Write $M:=\hat{\mathbb{G}}\ltimes B$. Let $q$ be a projection in $M$ and $A\subset qMq$ a von Neumann subalgebra, which is amenable relative to $B$ in $M$. Assume that $\hat{\mathbb{G}}$ is weakly amenable and bi-exact. Then we have either one of the following statements: \begin{itemize}
\item[$\rm (i)$] We have $A\preceq_{M} B$.
\item[$\rm (ii)$] The algebra ${\cal N}_{qMq}(A)''$ is amenable relative to $B$ in $M$ (or equivalently, $L^2(qM)$ is left ${\cal N}_{qMq}(A)''$-amenable as a $qMq$-$B$-bimodule.). \end{itemize} \end{ThmA}
The same arguments as in the group case give the following corollary. As we mentioned, free quantum groups and quantum automorphism groups are examples of the corollary (see also Theorem \ref{C}). \begin{CorA} Let $\mathbb{G}$ be a compact quantum group of Kac type. Assume that $\hat{\mathbb{G}}$ is weakly amenable and bi-exact.
\begin{itemize}
\item[$\rm (1)$] The algebra $L^\infty(\mathbb{G})$ is strongly solid. Moreover any non amenable von Neumann subalgebra of $L^\infty(\mathbb{G})$ has no Cartan subalgebras.
\item[$\rm (2)$] If $\hat{\mathbb{G}}$ is non amenable, then $L^\infty(\mathbb{G})\otimes B$ has no Cartan subalgebras for any finite von Neumann algebra $B$.
\end{itemize} \end{CorA}
We also mention that if a non-amenable, weakly amenable and bi-exact discrete quantum group $\hat{\mathbb{G}}$ admits an action on a commutative von Neumann algebra $L^\infty(X,\mu)$, so that $\hat{\mathbb{G}}\ltimes L^\infty(X,\mu)$ is a $\rm II_1$ factor and $L^\infty(X,\mu)$ is a Cartan subalgebra, then $L^\infty(X,\mu)$ is the unique Cartan subalgebra up to unitary conjugacy. However such an example is not known.
Next we investigate similar results on discrete quantum groups of non Kac type. For this, let us recall condition $\rm (AOC)^+$, which is a similar one to condition $\rm (AO)^+$ on continuous cores. We say a von Neumann algebra $M$ and a faithful normal state $\phi$ on $M$ satisfies \textit{condition} $\rm (AOC)^+$ $\cite[\textrm{Definition 3.2.1}]{Is12_2}$ if there exists a unital $\sigma$-weakly dense $C^*$-subalgebra $A\subset M$ such that the modular action of $\phi$ gives a norm continuous action on $A$ (so that we can define $A \rtimes_{\rm r} \mathbb{R}$), $A \rtimes_{\rm r} \mathbb{R}$ is locally reflexive, and there exists a u.c.p.\ map $\theta\colon (A\rtimes_{\rm r}\mathbb{R})\otimes (A\rtimes_{\rm r}\mathbb{R})^{\rm op} \rightarrow \mathbb{B}(L^2(M,\phi)\otimes L^2(\mathbb{R}))$ such that $\theta(a\otimes b^{\rm op})-ab^{\rm op}\in \mathbb{K}(L^2(M)\otimes\mathbb{B}(L^2(\mathbb{R}))$ for any $a,b\in A\rtimes_{\rm r}\mathbb{R}$.
In $\cite{Is12_2}$, we proved that von Neumann algebras of free quantum groups with Haar states satisfy condition $\rm (AOC)^+$ and then deduced that they do not have modular action (of Haar states) invariant Cartan subalgebras if they have the $\rm W^*$CBAP (and they are non-amenable). This is a partial answer for the absence of Cartan subalgebras in two senses: the $\rm W^*$CBAP of free quantum groups were not known, and we could prove only the absence of special Cartan subalgebras under the $\rm W^*$CBAP.
Very recently De Commer, Freslon and Yamashita solved the first problem. In fact, they proved that free quantum groups and quantum automorphism groups are weakly amenable and hence von Neumann algebras of them have the $\rm W^*$CBAP $\cite{DFY13}$. Thus only the second problem remains to be proved.
In this paper, we solve the second problem. In fact, the following theorem is an analogue of Theorem \ref{A} on continuous cores and it gives a complete answer for our Cartan problem (except for amenability). In the theorem below, $C_h(L^\infty(\mathbb{G}))$ means the continuous core of $L^\infty(\mathbb{G})$ with respect to the Haar state $h$. \begin{ThmA}\label{B} Let $\mathbb{G}$ be a compact quantum group, $h$ a Haar state of $\mathbb{G}$ and $(B,\tau_B)$ a tracial von Neumann algebra. Denote $M:= C_h(L^\infty(\mathbb{G}))\otimes B$ and $\mathrm{Tr}_M:=\mathrm{Tr} \otimes\tau_B$, where $\mathrm{Tr}$ is the canonical trace on $C_h(L^\infty(\mathbb{G}))$. Let $q$ be a $\mathrm{Tr}_M$-finite projection in $M$ and $A\subset qMq$ an amenable von Neumann subalgebra. Assume that $L^\infty(\mathbb{G})$ has the $ W^*$CBAP and $(L^\infty(\mathbb{G}),h)$ satisfies condition $\rm (AOC)^+$ with the dense $C^*$-algebra $C_{\rm red}(\mathbb{G})$. Then we have either one of the following statements: \begin{itemize}
\item[$\rm (i)$] We have $A\preceq_{M} L\mathbb{R}\otimes B$.
\item[$\rm (ii)$] The algebra $L^2(qM)$ is left ${\cal N}_{qMq}(A)''$-amenable as a $qMq$-$B$-bimodule. \end{itemize} \end{ThmA}
\begin{CorA} Let $\mathbb{G}$ be a compact quantum group. Assume that $L^\infty(\mathbb{G})$ has the $ W^*$CBAP and $(L^\infty(\mathbb{G}),h)$ satisfies condition $\rm (AOC)^+$ with the dense $C^*$-subalgebra $C_{\rm red}(\mathbb{G})$.
\begin{itemize}
\item[$\rm (1)$]
Any non amenable von Neumann subalgebra of $L^\infty(\mathbb{G})$, which is an image of a faithful normal conditional expectation, has no Cartan subalgebras.
\item[$\rm (2)$] If $L^\infty(\mathbb{G})$ is non amenable, then $L^\infty(\mathbb{G})\otimes B$ has no Cartan subalgebras for any finite von Neumann algebra $B$. \end{itemize} \end{CorA}
As further examples of corollaries above, we give the following theorem. Note that this theorem does not say anything about amenability of $\hat{\mathbb{G}}$ and $L^\infty(\mathbb{G})$.
\begin{ThmA}\label{C} Let $\mathbb{G}$ be one of the following compact quantum groups. \begin{itemize}
\item[$\rm (i)$] A co-amenable compact quantum group.
\item[\rm (ii)] The free unitary quantum group $A_u(F)$ for any $F\in\mathrm{GL}(n,\mathbb{C})$.
\item[\rm (iii)] The free orthogonal quantum group $A_o(F)$ for any $F\in\mathrm{GL}(n,\mathbb{C})$.
\item[\rm (iv)] The quantum automorphism group $A_{\rm aut}(B, \phi)$ for any finite dimensional $C^*$-algebra $B$ and any faithful state $\phi$ on $B$.
\item[\rm (v)] The dual of a bi-exact and weakly amenable discrete group $\Gamma$ with $\Lambda_{\rm cb}(\Gamma)=1$.
\item[\rm (vi)] The dual of a free product $\hat{\mathbb{G}}_1*\cdots *\hat{\mathbb{G}}_n$, where each $\mathbb{G}_i$ is as in from $\rm (i)$ to $\rm (v)$ above.
\end{itemize} Then the dual $\hat{\mathbb{G}}$ is weakly amenable and bi-exact. The associated von Neumann algebra $L^\infty(\mathbb{G})$ has the $W^*$CBAP and satisfies condition $\rm (AOC)^+$ with the Haar state and the dense $C^*$-subalgebra $C_{\rm red}(\mathbb{G})$. \end{ThmA}
\noindent Throughout the paper, we always assume that discrete groups are countable, quantum group $C^*$-algebras are separable, von Neumann algebras have separable predual, and Hilbert spaces are separable.
\noindent {\bf Acknowledgement.} The author would like to thank Yuki Arano, Cyril Houdayer, Yasuyuki Kawahigashi, Narutaka Ozawa, Sven Raum and Stefaan Vaes for fruitful conversations. He was supported by JSPS, Research Fellow of the Japan Society for the Promotion of Science and FMSP, Frontiers of Mathematical Sciences and Physics. This research was carried out while he was visiting the Institut de Math$\rm \acute{e}$matiques de Jussieu. He gratefully acknowledges the kind hospitality of them.
\section{\bf Preliminaries}
\subsection{\bf Compact (and discrete) quantum groups}\label{CQG}
Let $\mathbb{G}$ be a compact quantum group. In the paper, we basically use the following notation. We denote the comultiplication by $\Phi$, Haar state by $h$, the set of all equivalence classes of all unitary corepresentations by $\mathrm{Irred}(\mathbb{G})$, and right and left regular representations by $\rho$ and $\lambda$ respectively. We regard $C_{\rm red}(\mathbb{G}):=\rho(C(\mathbb{G}))$ as our main object and we frequently omit $\rho$ when we see the dense Hopf $*$-algebra. The canonical unitary $\mathbb{V}$ is defined as $\mathbb{V}=\bigoplus_{x\in\mathrm{Irred}\mathbb{G}} U^x$. The GNS representation of $h$ is written as $L^2(\mathbb{G})$ and it has a decomposition $L^2(\mathbb{G})=\sum_{x\in\mathrm{Irred}(\mathbb{G})}\oplus (H_x\otimes H_{\bar{x}})$. Along the decomposition, the modular operator of $h$ is of the form $\Delta^{it}=\sum_{x\in\mathrm{Irred}(\mathbb{G})}\oplus (F_x^{it}\otimes F_{\bar{x}}^{-it})$. The canonical conjugation is denoted by $J$. All dual objects are written with hat (e.g.\ $\hat{\mathbb{G}}, \hat{\Phi},\ldots$). We frequently use the unitary element $U:=J\hat{J}$ which satisfies $U\rho(C(\mathbb{G}))U=\lambda(C(\mathbb{G}))$ and $U\hat{\rho}(c_0(\hat{\mathbb{G}}))U=\hat{\lambda}(c_0(\hat{\mathbb{G}}))$.
{\bf $\bullet$ Crossed products}
For crossed products of quantum groups, we refer the reader to $\cite{BS93}$.
Let $\hat{\mathbb{G}}$ be a discrete quantum group and $A$ a $C^*$-algebra. Recall following notions: \begin{itemize}
\item A (\textit{left}) \textit{action} of $\hat{\mathbb{G}}$ on $A$ is a non-degenerate $*$-homomorphism $\alpha\colon A \rightarrow M(c_0(\hat{\mathbb{G}})\otimes A)$ such that $(\iota\otimes \alpha)\alpha=(\hat{\Phi}\otimes\iota)\alpha$ and $(\hat{\epsilon}\otimes\iota)\alpha(a)=a$ for any $a\in A$, where $\hat{\epsilon}$ is the counit.
\item A \textit{covariant representation} of an action $\alpha$ into a $C^*$-algebra $B$ is a pair $(\theta,X)$ such that
\begin{itemize}
\item $\theta$ is a non-degenerate $*$-homomorphism from $A$ into $M(B)$;
\item $X \in M(c_0(\hat{\mathbb{G}})\otimes B)$ is a unitary representation of $\hat{\mathbb{G}}$;
\item they satisfy the covariant relation $(\iota\otimes \theta)\alpha(a)=X^*(1\otimes \theta(a))X$ ($a\in A$).
\end{itemize} \end{itemize}
Let $\alpha$ be an action of $\hat{\mathbb{G}}$ on $A$. Then for a covariant representation $(\theta,X)$ into $B$, the closed linear span \begin{equation*} \overline{\mathrm{span}}\{\theta(a)(\omega\otimes\iota)(X) \mid a\in A, \omega\in \ell^\infty(\hat{\mathbb{G}})_* \} \subset M(B) \end{equation*} becomes a $C^*$-algebra.
The \textit{reduced crossed product $C^*$-algebra} is the $C^*$-algebra associated with the covariant representation $((\hat{\lambda}\otimes\iota)\alpha, \hat{\mathscr{W}}\otimes 1)$ into ${\cal L}(L^2(\mathbb{G})\otimes A)$, where $\hat{\lambda}$ is the left regular representation, $\hat{\mathscr{W}}:=(\iota\otimes\rho)(\mathbb{V})$, and ${\cal L}(L^2(\mathbb{G})\otimes A)$ is the $C^*$-algebra of all right $A$-module maps on the Hilbert module $L^2(\mathbb{G})\otimes A$. The \textit{full crossed product $C^*$-algebra} is defined as a universal object for all covariant representations. When $A\subset \mathbb{B}(H)$ is a von Neumann algebra, the \textit{crossed product von Neumann algebra} is defined as the $\sigma$-weak closure of the reduced crossed product in $\mathbb{B}(L^2(\mathbb{G})\otimes H)$. We denote them by $\hat{\mathbb{G}}\ltimes_{\rm r}A$, $\hat{\mathbb{G}}\ltimes_{\rm f}A$ and $\hat{\mathbb{G}}\ltimes A$ respectively.
The crossed product von Neumann algebra has a canonical conditional expectation $E_A$ onto $A$, given by \begin{equation*} \hat{\mathbb{G}}\ltimes A \ni \theta(a)(\omega\otimes\iota\otimes\iota)(\hat{\mathscr{W}}\otimes 1) \mapsto a (\omega\otimes h \otimes\iota)(\hat{\mathscr{W}}\otimes 1)\in A, \end{equation*} where $h$ is the Haar state (more explicitly it sends $\theta(a)\rho(u_{i,j}^z)$ to $a h(u_{i,j}^z)$). For a faithful normal state $\phi$ on $A$, we can define a canonical faithful normal state on $\hat{\mathbb{G}}\ltimes A$ by $\tilde{\phi}:=\phi\circ E_A$. When $\phi$ is a trace and the given action $\alpha$ is $\phi$-preserving (i.e.\ $(\iota\otimes\phi)\alpha(a)=\phi(a)$ for $a\in A$), $\tilde{\phi}$ also becomes a trace.
{\bf $\bullet$ Free products}
For free products of discrete quantum groups, we refer the reader to $\cite{Wa95}$.
We first recall fundamental facts of free products of $C^*$-algebras. Let $A_i$ $(i=1,\ldots,n)$ be unital $C^{\ast}$-algebras and $\phi_i$ non-degenerate states on $A_i$ (i.e.\ GNS representations of $\phi_i$ are faithful). Denote GNS-representations of $\phi_i$ by $(\pi_{i},H_{i},\xi_{i})$ and decompose $H_{i}=\mathbb{C}\xi_{i}\oplus H_{i}^{0}$ as Hilbert spaces, where $H_{i}^{0}:=(\mathbb{C}\xi_{i})^{\perp}$. Define two Hilbert spaces by
\begin{eqnarray*} H_1*\cdots*H_n:=\mathbb{C}\Omega \oplus \bigoplus_{k\geq1}\hspace{0.3em}&\bigoplus_{(i_1,\dots,i_k)\in N_k}& H_{i_{1}}^{0}\otimes H_{i_{2}}^{0}\cdots \otimes H_{i_{k}}^{0},\\ H(i):=\mathbb{C}\Omega \oplus \bigoplus_{k\geq1}\hspace{0.3em}&\bigoplus_{(i_1,\dots,i_k)\in N_k, i\neq i_{1}}& H_{i_{1}}^{0}\otimes H_{i_{2}}^{0}\cdots \otimes H_{i_{k}}^{0}, \end{eqnarray*} where $\Omega$ is a fixed norm one vector and $N_k:=\{(i_1,\dots,i_k)\in \{1,\ldots,n\}^k \mid i_l\neq i_{l+1} \textrm{ for all }l=1,\dots,k-1\}$. Write $H:=H_1*\cdots*H_n$. Let $W_i$ be unitary operators given by \begin{eqnarray*} W_{i}\colon H &=& H(i)\oplus(H_{i}^{0}\otimes H(i))\\ &\simeq& (\mathbb{C}\xi_{i}\otimes H(i))\oplus(H_{i}^{0}\otimes H(i))\\ &\simeq&(\mathbb{C}\xi_{i} \oplus H_{i}^{0})\otimes H(i)\\ &=&H_{i}\otimes H(i) \end{eqnarray*} Then a canonical $A_i$-action (more generally $\mathbb{B}(H_i)$-action) on $H$ is given by $\lambda_i(a):=W_i^* (a\otimes 1)W_i$ $(a\in A_i)$. The \textit{free product $C^*$-algebra} $(A_1,\phi_1)*\cdots *(A_n,\phi_n)$ is the $C^*$-algebra generated by $\lambda_i(A_i)$ $(i=1,\ldots, n)$. The vector state of the canonical cyclic vector $\Omega$ is called the free product state and denoted by $\phi_1*\cdots *\phi_n$.
When each $A_i$ is a von Neumann algebra and $\phi_i$ is normal, the \textit{free product von Neumann algebra} is defined as the $\sigma$-weak closure of the free product $C^*$-algebra in $\mathbb{B}(H)$. We denote it by $(A_1,\phi_1)\bar{*}\cdots \bar{*}(A_n,\phi_n)$.
Let $M_i$ $(i=1,\ldots,n)$ be von Neumann algebras and $\phi_i$ be faithful normal states on $M_i$. Denote modular objects of them by $\Delta_i$ and $J_i$. Then modular objects on the free product von Neumann algebra are given by \begin{alignat*}{3} \Delta^{it} &\colon H_{i_{1}}^{0}\otimes \cdots \otimes H_{i_{k}}^{0} \ni \xi_{i_{1}}\otimes \cdots \otimes \xi_{i_{k}} \longmapsto \Delta^{it}_{i_1}\xi_{i_{1}}\otimes \cdots \otimes \Delta^{it}_{i_k}\xi_{i_{k}} &\in& H_{i_{1}}^{0}\otimes \cdots \otimes H_{i_{k}}^{0},\\ J \hspace{0.7em} &\colon H_{i_{1}}^{0}\otimes \cdots \otimes H_{i_{k}}^{0} \ni \xi_{i_{1}}\otimes \cdots \otimes \xi_{i_{k}} \longmapsto J_{i_k}\xi_{i_{k}}\otimes \cdots \otimes J_{i_1}\xi_{i_{1}} &\in& H_{i_{k}}^{0}\otimes \cdots \otimes H_{i_{1}}^{0}. \end{alignat*}
The unitary elements $V_i:=\Sigma(J_i\otimes J|_{JH(i)})W_iJ$, where $\Sigma$ is a flip from $H_i\otimes H(i)$ to $H(i)\otimes H_i$, gives a right $M_i$-action (more generally $\mathbb{B}(H_i)$-action) by $\rho_i(a):=J\lambda_i(a)J=V_i^*(1\otimes J_iaJ_i)V_i$.
Let $\mathbb{G}_i$ $(i=1,\ldots,n)$ be compact quantum groups and $h_i$ Haar states of them. Then the reduced free product $(C_{\rm red}(\mathbb{G}_1), h_1)*\cdots *(C_{\rm red}(\mathbb{G}_n), h_n)$ has a structure of compact quantum group with the Haar state $h_1*\cdots *h_n$. We write its dual as $\hat{\mathbb{G}}_1*\cdots *\hat{\mathbb{G}}_n$ and call it the \textit{free product} of $\hat{\mathbb{G}}_i$ $(i=1,\ldots,n)$. Modular objects $\hat{J}$ and $U:=J\hat{J}$ are given by \begin{alignat*}{3} \hat{J} \hspace{0.5em} &\colon H_{i_{1}}^{0}\otimes \cdots \otimes H_{i_{n}}^{0} \ni \xi_{i_{1}}\otimes \cdots \otimes \xi_{i_{n}} \longmapsto \hat{J}_{i_1}\xi_{i_1}\otimes \cdots \otimes \hat{J}_{i_n}\xi_{i_{n}} &\in& H_{i_{1}}^{0}\otimes \cdots \otimes H_{i_{n}}^{0},\\ U \hspace{0.5em} &\colon H_{i_{1}}^{0}\otimes \cdots \otimes H_{i_{n}}^{0} \ni \xi_{i_{1}}\otimes \cdots \otimes \xi_{i_{n}} \longmapsto U_{i_n}\xi_{i_{n}}\otimes \cdots \otimes U_{i_1}\xi_{i_{1}} &\in& H_{i_{n}}^{0}\otimes \cdots \otimes H_{i_{1}}^{0}. \end{alignat*}
We have a formula $W_i\hat{J}=(\hat{J}_i\otimes \hat{J}|_{H(i)}) W_i$.
By the natural inclusion $(C_{\rm red}(\mathbb{G}_i), h_i)\subset (C_{\rm red}(\mathbb{G}_1), h_1)*\cdots *(C_{\rm red}(\mathbb{G}_n), h_n)$ $(i=1,\ldots,n)$, we can regard all corepresentations of $\mathbb{G}_i$ as those of the dual of $\hat{\mathbb{G}}_1*\cdots *\hat{\mathbb{G}}_n$. A representative of all irreducible representations of the dual of $\hat{\mathbb{G}}_1*\cdots *\hat{\mathbb{G}}_n$ is given by the trivial corepresentation and all corepresentations of the form $w_1\boxtimes w_2\boxtimes \cdots \boxtimes w_n$, where each $w_l$ is in $\mathrm{Irred}(\mathbb{G}_i)$ for some $i$ (and not trivial corepresentation) and no two adjacent factors are taken from corepresentations of the same quantum group.
{\bf $\bullet$ Amenability of actions}
Let $\alpha$ be an action of a discrete quantum $\hat{\mathbb{G}}$ on a unital $C^*$-algebra $A$. Denote by $L^2(\mathbb{G})\otimes A$ the Hilbert module obtained from $L^2(\mathbb{G})\otimes_{\rm alg} A$ with $\langle\xi\otimes a | \eta\otimes b\rangle:=\langle\xi|\eta\rangle b^*a$ for $\xi,\eta\in L^2(\mathbb{G})$ and $a,b\in A$. We say $\alpha$ is \textit{amenable} $\cite[\textrm{Definition 4.1}]{VV05}$ if there exists a sequence $\xi_n\in L^2(\mathbb{G})\otimes A$ satisfying \begin{itemize}
\item[$\rm (i)$] $\langle\xi_n |\xi_n\rangle \rightarrow 1$ in $A$;
\item[$\rm (ii)$] for all $x\in \mathrm{Irred}(\mathbb{G})$, $\|((\iota\otimes \alpha)(\xi_n)-\hat{\mathscr{V}_{12}}(\xi_n)_{13}) (1\otimes p_x \otimes 1)\|\rightarrow 0$;
\item[$\rm (iii)$] $((\hat{\rho}\times\hat{\lambda})\circ\hat{\Phi}\otimes \iota)\alpha(a)\xi_n =\xi_n a$ for any $n\in\mathbb{N}$ and $a\in A$, \end{itemize} where $\hat{\mathscr V}:=(\lambda\otimes \iota)(\mathbb{V}_{21})$ and $\hat{\rho}\times\hat{\lambda}$ is the multiplication map from $\ell^\infty(\hat{\mathbb{G}})\otimes\ell^\infty(\hat{\mathbb{G}})$ into $\mathbb{B}(L^2(\mathbb{G}))$, which is bounded since $\ell^\infty(\hat{\mathbb{G}})$ is amenable. It is clear that $\hat{\mathbb{G}}$ is amenable if and only if any trivial action of $\hat{\mathbb{G}}$ is amenable.
{\bf $\bullet$ Free quantum groups and quantum automorphism groups}
Let $F$ be a matrix in $\mathrm{GL}(n,\mathbb{C})$. The \textit{free unitary quantum group} (resp.\ \textit{free orthogonal quantum group}) of $F$ $\cite{VW96}\cite{Wa95}$ is the $C^*$-algebra $C(A_u(F))$ (resp.\ $C(A_o(F))$) is defined as the universal unital $C^*$-algebra generated by all the entries of a unitary $n$ by $n$ matrix $u=(u_{i,j})_{i,j}$ satisfying that $F(u_{i,j}^*)_{i,j}F^{-1}$ is unitary (resp.\ $F(u_{i,j}^*)_{i,j}F^{-1}=u$).
Recall that a \textit{coaction} of a compact quantum group $\mathbb{G}$ on a unital $C^*$-algebra $B$ is a unital $*$-homomorphism $\beta\colon B\rightarrow B\otimes C(\mathbb{G})$ satisfying $(\beta\otimes \iota)\circ \beta=(\iota\otimes \Phi)\circ \beta$ and $\overline{{\rm span}}\{\beta(B)(1\otimes A)=B\otimes C(\mathbb{G})$. Let $(B,\phi)$ be a pair of a finite dimensional $C^*$-algebra and a faithful state on $B$. Then the \textit{quantum automorphism group} of $(B,\phi)$ $\cite{Wa98_1}\cite{Ba01}$ is the universal compact quantum group $A_{\rm aut}(B,\phi)$ which can be endowed with a coaction $\beta\colon B\rightarrow B\otimes C(A_{\rm aut}(B,\phi))$ satisfying $(\phi\otimes \iota)\circ \beta(x)=\phi(x)1$ for $x\in B$.
\subsection{\bf Weak amenability and the $\rm \bf W^*$CBAP}
Let $\hat{\mathbb{G}}$ be a discrete quantum group. Denote the dense Hopf $*$-algebra by ${\mathscr C}(\hat{\mathbb{G}})$. For any element $a\in\ell^\infty(\hat{\mathbb{G}})$, we can associate a linear map $m_a$ from ${\mathscr C}(\hat{\mathbb{G}})$ to ${\mathscr C}(\hat{\mathbb{G}})$, given by $(m_a\otimes \iota)(u^x)=(1\otimes ap_x)u^x$ for any $x\in \mathrm{Irred}(\mathbb{G})$, where $p_x\in c_0(\hat{\mathbb{G}})$ is the canonical projection onto $x$ component. We we say $\hat{\mathbb{G}}$ is \textit{weakly amenable} if there exist a net $(a_i)_i$ of elements of $\ell^\infty(\hat{\mathbb{G}})$ such that \begin{itemize}
\item each $a_i$ has finite support, namely, $a_ip_x=0$ except for finitely many $x\in \mathrm{Irred}(\mathbb{G})$;
\item $(a_i)_i$ converges to 1 pointwise, namely, $a_ip_x$ converges to $p_x$ in $\mathbb{B}(H_x)$ for any $x\in \mathrm{Irred}(\mathbb{G})$;
\item $\limsup_i\|m_{a_i}\|_{\rm c.b.}$ is finite. \end{itemize}
We also recall that a von Neumann algebra $M$ has the $\it weak^*$ \textit{completely approximation property} (or $\ W^*$\textit{CBAP}, in short)
if there exists a net $(\psi_i)_i$ of completely bounded (say c.b.) maps on $M$ with normal and finite rank such that $\limsup_i\|\psi_i\|_{\rm c.b.}<\infty$ and $\psi_i$ converges to $\mathrm{id}_M$ in the point $\sigma$-weak topology.
Then the \textit{Cowling--Haagerup constant} of $\hat{\mathbb{G}}$ and $M$ are defined as \begin{eqnarray*}
&&\Lambda_{\rm c.b.}(\hat{\mathbb{G}}):=\inf\{\ \limsup_i\|m_{a_i}\|_{\rm c.b.}\mid (a_i)_i \textrm{ satisfies the above condition}\},\\
&&\Lambda_{\rm c.b.}(M):=\inf\{\ \limsup_i\|\psi_i\|_{\rm c.b.}\mid (\psi_i) \textrm{ satisfies the above condition}\}. \end{eqnarray*} It is known that $\Lambda_{\rm c.b.}(\hat{\mathbb{G}})\geq\Lambda_{\rm c.b.}(L^\infty(\mathbb{G}))$.
De Commer, Freslon, and Yamashita recently proved that $\Lambda_{\rm c.b.}(\hat{\mathbb{G}})=1$, where $\mathbb{G}$ is a free quantum group or a quantum automorphism group $\cite{DFY13}$. Note that a special case of this was already solved by Freslon $\cite{Fr12}$.
\subsection{\bf Popa's intertwining techniques and relative amenability}
We first recall Popa's intertwining techniques in both non-finite and semifinite situations.
\begin{Def}\label{popa embed def}\upshape Let $M$ be a von Neumann algebra, $p$ and $q$ projections in $M$, $A\subset qMq$ and $B\subset pMp$ von Neumann subalgebras, and let $E_{B}$ be a faithful normal conditional expectation from $pMp$ onto $B$. Assume $A$ and $B$ are finite. We say $A$ \textit{embeds in $B$ inside} $M$ and denote by $A\preceq_M B$ if there exist non-zero projections $e\in A$ and $f\in B$, a unital normal $\ast$-homomorphism $\theta \colon eAe \rightarrow fBf$, and a partial isometry $v\in M$ such that \begin{itemize} \item $vv^*\leq e$ and $v^*v\leq f$, \item $v\theta(x)=xv$ for any $x\in eAe$. \end{itemize} \end{Def}
\begin{Thm}[\textit{non-finite version, }{\cite{Po01}\cite{Po03}\cite{Ue12}\cite{HV12}}]\label{popa embed} Let $M,p,q,A,B$, and $E_{B}$ be as in the definition above, and let $\tau$ be a faithful normal trace on $B$. Then the following conditions are equivalent. \begin{itemize}
\item[$\rm (i)$]The algebra $A$ embeds in $B$ inside $M$.
\item[$\rm (ii)$]There exists no sequence $(w_n)_n$ of unitaries in $A$ such that $E_{B}(b^*w_n a)\rightarrow 0$ strongly for any $a,b\in qMp$. \end{itemize} \end{Thm}
\begin{Thm}[\textit{semifinite version, }{\cite{CH08}\cite{HR10}}]\label{popa embed2}
Let $M$ be a semifinite von Neumann algebra with a faithful normal semifinite trace $\mathrm{Tr}$, and $B\subset M$ be a von Neumann subalgebra with $\mathrm{Tr}_{B}:=\mathrm{Tr}|_{B}$ semifinite. Denote by $E_{B}$ the unique $\mathrm{Tr}$-preserving conditional expectation from $M$ onto $B$. Let $q$ be a $\mathrm{Tr}$-finite projection in $M$ and $A\subset qMq$ a von Neumann subalgebra. Then the following conditions are equivalent. \begin{itemize}
\item[$\rm (i)$]There exists a non-zero projection $p\in B$ with $\mathrm{Tr}_{B}(p)<\infty$ such that $A\preceq_{eMe}pBp$, where $e:=p\vee q$.
\item[$\rm (ii)$]There exists no sequence $(w_n)_n$ in unitaries of $A$ such that $E_{B}(b^{\ast}w_n a)\rightarrow 0$ strongly for any $a,b\in qM$.
\end{itemize} We use the same symbol $A\preceq_{M}B$ if one of these conditions holds. \end{Thm} \noindent By the proof of the semifinite one, we have that $A\not\preceq_M B$ if and only if there exists a net $(p_j)$ of $\mathrm{Tr}_B$-finite projections in $B$ which converges to 1 strongly and $A\not\preceq_{e_jMe_j}p_jBp_j$, where $e_j:=p_j\vee q$. We also mention that when $A$ is diffuse and $B$ is atomic, then $A\not\preceq_M B$. This follows from the existence of a normal unital $*$-homomorphism $\theta$ in the definition.
We next recall relative amenability introduced in $\cite{OP07}$ and $\cite{PV11}$.
\begin{Def}\upshape Let $(M,\tau)$ be a tracial von Neumann algebra, $q\in M$ a projection, and $B\subset M$ and $A\subset qMq$ be von Neumann subalgebras. We say $A$ is amenable relative to $B$ in $M$ if there exists a state $\phi$ on $q\langle M,e_B\rangle q$ such that $\phi$ is $A$-central and the restriction of $\phi$ on $A$ coincides with $\tau$. \end{Def}
\begin{Def}\upshape Let $(M,\tau)$ and $(B,\tau_B) $ be tracial von Neumann algebras and $A\subset M$ be a von Neumann subalgebra. We say an $M$-$B$-bimodule ${}_M K_B$ is left $A$-amenable if there exists a state $\phi$ on $\mathbb{B}(K)\cap (B^{\rm op})'$ such that $\phi$ is $A$-central and the restriction of $\phi$ on $A$ coincides with $\tau$. \end{Def}
We note that for any $B\subset M$ and $A\subset qMq$, amenability of $A$ relative to $B$ in $M$ is equivalent to left $A$-amenability of $qL^2(M)$ as a $qMq$-B-bimodule, since $q\langle M,e_B\rangle q= q(\mathbb{B}(L^2(M))\cap (B^{\rm op})')q=\mathbb{B}(qL^2(M))\cap (B^{\rm op})'$. We also mention that when $B$ is amenable, then since $\mathbb{B}(K)\cap (B^{\rm op})'$ is amenable, there exists a conditional expectation from $\mathbb{B}(K)$ onto $\mathbb{B}(K)\cap (B^{\rm op})'$. In this case, relative amenability of $A$ (or left $A$-amenability) means amenability of $A$.
\section{\bf Bi-exactness}
\subsection{\bf Two definitions of bi-exactness}
We introduce two notions of bi-exactness on discrete quantum groups. These notions are equivalent for discrete groups as we have seen in Introduction. Recall that $C_{\rm red}(\mathbb{G})=\rho(C(\mathbb{G}))$ and $UC_{\rm red}(\mathbb{G})U=\lambda(C(\mathbb{G}))=C_{\rm red}(\mathbb{G})^{\rm op}$, where $U=J\hat{J}=\hat{J}J$. Basically we use $UC_{\rm red}(\mathbb{G})U$ instead of $C_{\rm red}(\mathbb{G})^{\rm op}$.
\begin{Def}\upshape\label{bi-ex} Let $\hat{\mathbb{G}}$ be a discrete quantum group. We say $\hat{\mathbb{G}}$ is \textit{bi-exact} if it satisfies following conditions: \begin{itemize}
\item[$\rm (i)$] the quantum group $\hat{\mathbb{G}}$ is exact (i.e.\ $C_{\rm red}(\mathbb{G})$ is exact);
\item[$\rm (ii)$] there exists a u.c.p.\ map $\theta\colon C_{\rm red}(\mathbb{G})\otimes C_{\rm red}(\mathbb{G}) \rightarrow \mathbb{B}(L^2(\mathbb{G}))$ such that $\theta(a\otimes b)-aUbU\in \mathbb{K}(L^2(\mathbb{G}))$ for any $a,b \in C_{\rm red}(\mathbb{G})$. \end{itemize} \end{Def} \begin{Def}\upshape\label{st bi-ex} Let $\hat{\mathbb{G}}$ be a discrete quantum group. We say $\hat{\mathbb{G}}$ is \textit{strongly bi-exact} if there exists a unital $C^*$-subalgebra ${\cal B}$ in $\ell^{\infty}(\hat{\mathbb{G}})$ such that \begin{itemize}
\item[$\rm (i)$] the algebra $\cal B$ contains $c_0(\hat{\mathbb{G}})$, and the quotient ${\cal B}_{\infty}:={\cal B}/c _0(\hat{\mathbb{G}})$ is nuclear;
\item[\rm (ii)] the left translation action on $\ell^\infty(\hat{\mathbb{G}})$ induces an amenable action on ${\cal B}_{\infty}$, and the right one induces the trivial action on ${\cal B}_{\infty}$. \end{itemize} \end{Def}
\begin{Rem}\upshape Amenability implies strong bi-exactness. In fact, we can choose ${\cal B}:=c_0(\hat{\mathbb{G}})+\mathbb{C}1$. In this case, both the left and right actions on ${\cal B}_\infty(\simeq \mathbb{C})$ are trivial. \end{Rem}
We first observe relationship between bi-exactness and strong bi-exactness. In (i) of the definition of strong bi-exactness, nuclearity of ${\cal B}_\infty$ is equivalent to that of $\cal B$. Moreover, together with the condition (ii), the $C^*$-subalgebra ${\cal C}_l\subset \mathbb{B}(L^2(\mathbb{G}))$ generated by $\hat{\lambda}({\cal B})$ and $C_{\rm red}(\mathbb{G})(=\rho(C(\mathbb{G})))$ is also nuclear. In fact, the quotient image of ${\cal C}_l$ in $\mathbb{B}(L^2(\mathbb{G}))/\mathbb{K}(L^2(\mathbb{G}))$ is nuclear since there is a canonical surjective map from $\hat{\mathbb{G}} \ltimes_{\rm f} {\cal B}_\infty$ to ${\cal C}_l$ and $\hat{\mathbb{G}} \ltimes_{\rm f} {\cal B}_\infty$ is nuclear by amenability of the action. Then ${\cal C}_l$ is an extension of $\mathbb{K}(L^2(\mathbb{G}))$ by this quotient image, and hence is nuclear. Note that ${\cal C}_l$ contains $\mathbb{K}(L^2(\mathbb{G}))$, since it contains $C_{\rm red}(\mathbb{G})$ and the orthogonal projection from $L^2(\mathbb{G})$ onto $\mathbb{C}\hat{1}$. We put ${\cal C}_r:=U{\cal C}_lU$, where $U:=J\hat{J}$. Triviality of the right action in (ii) implies that all commutators of $UC_{\rm red}(\mathbb{G})U$ and $\hat{\lambda}({\cal B})$ (respectively $C_{\rm red}(\mathbb{G})$ and $U\hat{\lambda}({\cal B})U$) are contained in $\mathbb{K}(L^2(\mathbb{G}))$. This implies that all commutators of ${\cal C}_l$ and ${\cal C}_r$ are also contained in $\mathbb{K}(L^2(\mathbb{G}))$. Thus we obtained the following $\ast$-homomorphism:
\begin{equation*} \nu\colon {\cal C}_l \otimes {\cal C}_r \longrightarrow \mathbb{B}(L^2(\mathbb{G}))/\mathbb{K}(L^2(\mathbb{G})); a\otimes b \longmapsto ab. \end{equation*} This map is an extension of the multiplication map on $C_{\rm red}(\mathbb{G}) \otimes UC_{\rm red}(\mathbb{G})U$, and so this multiplication map is nuclear since so is ${\cal C}_l \otimes {\cal C}_r$. Finally by the lifting theorem of Choi and Effros $\cite{CE76}$ (or see $\cite[\textrm{Theorem C.3}]{BO08}$), we obtain a u.c.p.\ lift $\theta$ of the multiplication map. Thus we observed that strong bi-exactness implies bi-exactness (exactness of $\hat{\mathbb{G}}$ follows from nuclearity of ${\cal C}_l$). The intermediate object ${\cal C}_l$ is important for us, and we will use this algebra in the next subsection.
We summary these observations as follows.
\begin{Lem}\label{intermediate} Strong bi-exactness implies bi-exactness. The following condition is an intermediate condition between bi-exactness and strong bi-exactness: \begin{itemize}
\item There exists a nuclear $C^*$-algebra ${\cal C}_l\subset \mathbb{B}(L^2(\mathbb{G}))$ which contains $C_{\rm red}(\mathbb{G})$ and $\mathbb{K}(L^2(\mathbb{G}))$, and all commutators of ${\cal C}_l$ and ${\cal C}_r(:=U{\cal C}_lU)$ are contained in $\mathbb{K}(L^2(\mathbb{G}))$. \end{itemize} \end{Lem}
Examples of bi-exact quantum groups were first given by Vergnioux $\cite{Ve05}$. He constructed a u.c.p.\ lift directly for free quantum groups. Then he, in a joint work with Vaes $\cite{VV05}$, gave a new proof for bi-exactness of $\hat{A}_o(F)$ and they in fact proved strong bi-exactness. In the proof, they only used the fact that $A_o(F)$ is monoidally equivalent to some $\mathrm{SU}_q(2)$ with $-1<q<1$ and $q\neq0$, seeing some estimates on intertwiner spaces of $\mathrm{SU}_q(2)$ $\cite[\textrm{Lemma 8.1}]{VV05}$. Since the dual of $\mathrm{SO}_q(3)$ is a quantum subgroup of some dual of $\mathrm{SU}_q(2)$, intertwiner spaces of $\mathrm{SO}_q(3)$ have the same estimates. From this fact, we can deduce strong bi-exactness of a dual of a compact quantum group which is monoidally equivalent to $\mathrm{SO}_q(3)$ (by the same argument as that for $\mathrm{SU}_q(2)$). We also mention that strong bi-exactness of $A_u(F)$ was proved by the same argument $\cite{VV08}$.
We summary these observations as follows. \begin{Thm}\label{example st bi-exact} Let $\mathbb{G}$ be a compact quantum group which is monoidally equivalent to $\mathrm{SU}_q(2)$, $\mathrm{SO}_q(3)$, or $A_u(F)$, where $-1<q<1$, $q\neq 0$, $F$ is not a scalar multiple of a $2$ by $2$ unitary. Then the dual $\hat{\mathbb{G}}$ is strongly bi-exact. \end{Thm}
In $\cite{Is12_2}$, we introduced condition $\rm (AOC)^+$, which is similar to condition $\rm (AO)^+$ on continuous cores, and proved that von Neumann algebras of free quantum groups satisfy this condition. In the proof we also used only the fact that $A_o(F)$ is monoidally equivalent to some $\mathrm{SU}_q(2)$ and hence we actually proved the following statement.
\begin{Thm}\label{example bi-exact} Let $\mathbb{G}$ be a compact quantum group which is monoidally equivalent to $\mathrm{SU}_q(2)$, $\mathrm{SO}_q(3)$, or $A_u(F)$, where $-1<q<1$, $q\neq 0$, $F$ is not a scalar multiple of a $2$ by $2$ unitary. Then $L^\infty(\mathbb{G})$ and its Haar state satisfy condition $\rm (AOC)^+$ with the dense $C^*$-algebra $C_{\rm red}(\mathbb{G})$. \end{Thm}
In the proof we gave a sufficient condition to condition $\rm (AOC)^+$, which was formulated for general von Neumann algebras $\cite[\textrm{Lemma 3.2.3}]{Is12_2}$. When we see a quantum group von Neumann algebra, we have a more concrete sufficient condition as follows. To verify this, see Subsection 3.2 in the same paper. In the lemma below, $\pi$ means the canonical $*$-homomorphism from $\mathbb{B}(L^2(\mathbb{G}))$ into $\mathbb{B}(L^2(\mathbb{G})\otimes L^2(\mathbb{R}))$ defined by $(\pi(x)\xi)(t):=\Delta_h^{-it}x\Delta_h^{it}\xi(t)$ for $x\in\mathbb{B}(L^2(\mathbb{G}))$, $t\in\mathbb{R}$, and $\xi\in L^2(\mathbb{G})\otimes L^2(\mathbb{R})$.
\begin{Lem}\label{AOC} Let $\mathbb{G}$ be a compact quantum group and ${\cal C}_l\subset C^*\{C_{\rm red}(\mathbb{G}), \hat{\lambda}(\ell^\infty(\hat{\mathbb{G}})) \}$ a $C^*$-subalgebra which contains $C_{\rm red}(\mathbb{G})$ and $\mathbb{K}(L^2(\mathbb{G}))$. Put ${\cal C}_r:=U{\cal C}_lU$. Assume the following conditions: \begin{itemize}
\item[$\rm (i)$] the algebra ${\cal C}_l$ is nuclear;
\item[$\rm (ii)$] a family of maps $\mathrm{Ad}\Delta_h^{it}$ $(t\in\mathbb{R})$ gives a norm continuous action of $\mathbb{R}$ on ${\cal C}_l$;
\item[$\rm (iii)$] all commutators of $\pi({\cal C}_l)$ and ${\cal C}_r \otimes1$ are contained in $\mathbb{K}(L^2(\mathbb{G}))\otimes\mathbb{B}(L^2(\mathbb{R}))$. \end{itemize} Then $L^\infty(\mathbb{G})$ and its Haar state satisfy condition $\rm (AOC)^+$ with the dense $C^*$-algebra $C_{\rm red}(\mathbb{G})$. \end{Lem}
\begin{Rem}\upshape\label{amenable} When $\hat{\mathbb{G}}$ is amenable, then $L^\infty(\mathbb{G})$ and its Haar state satisfy condition $\rm (AOC)^+$. In fact, we can choose ${\cal B}:=c_0(\hat{\mathbb{G}})+\mathbb{C}1$ and ${\cal C}_l:=C^*\{C_{\rm red}(\mathbb{G}), \hat{\lambda}({\cal B})\}$. In this case, all conditions in this lemma are easily verified. \end{Rem}
\begin{Rem}\upshape\label{group} When $\hat{\mathbb{G}}$ is a strongly bi-exact discrete quantum group of Kac type (possibly bi-exact discrete group) with ${\cal B}\subset \ell^\infty(\hat{\mathbb{G}})$, then since the modular operator is trivial, ${\cal C}_l:=C^*\{C_{\rm red}(\mathbb{G}), \lambda({\cal B})\}$ satisfies these conditions. \end{Rem}
\begin{Rem}\upshape In the proof of $\cite[\textrm{Proposition 3.2.4}]{Is12_2}$, we put ${\cal C}_l=\hat{\mathbb{G}}\ltimes_{\rm r} {\cal B}_\infty$ (here we write it as $\tilde{\cal C}_l$), and in this subsection we are putting ${\cal C}_l=C^*\{C_{\rm red}(\mathbb{G}), \hat{\lambda}({\cal B})\}$. Both of them are nuclear $C^*$-algebras containing $C_{\rm red}(\mathbb{G})$ and do the same work to get condition $\rm (AOC)^+$. The difference of them is that ${\cal C}_l$ is contained in $\mathbb{B}(L^2(\mathbb{G}))$ but $\tilde{\cal C}_l$ is not. Hence ${\cal C}_l$ is more useful and $\tilde{\cal C}_l$ is more general (since $\tilde{\cal C}_l$ produces ${\cal C}_l$). In the previous paper, we preferred the generality and hence used $\tilde{\cal C}_l$ in the proof. \end{Rem}
\subsection{\bf Free products of bi-exact quantum groups}
Free products of bi-exact discrete groups (more generally, free products of von Neumann algebras with condition (AO)) were studied in $\cite{Oz04}\cite[\textrm{Section 4}]{GJ07}\cite[\textrm{Section 15.3}]{BO08}$. In this subsection we will prove similar results on discrete quantum groups. We basically follow the strategy in $\cite{Oz04}$.
\begin{Lem}[{\cite[\rm Lemma\ 2.4]{Oz04}}]\label{nuclear} Let $B_i\subset \mathbb{B}(H_i)$ $(i=1,2)$ be $C^*$-subalgebras with $B_i$-cyclic vectors $\xi_i$ and denote by $\omega_i$ the corresponding vector states (note that each $\omega_i$ is non-degenerate). If each $B_i$ contains $P_i$, the orthogonal projection onto $\mathbb{C}\xi_i$, and is nuclear, then the free product $(B_1,\omega_1)*(B_2,\omega_2)$ is also nuclear. \end{Lem} \begin{Rem}\upshape In this Lemma, each $B_i$ contains $\mathbb{K}(H_i)$ since it contains $P_i$ and the vector $\xi_i$ is $B_i$-cyclic. Projections $\lambda_i(P_i)\in(B_1,\omega_1)*(B_2,\omega_2)$ are orthogonal projections onto $H(i)$ and hence the orthogonal projection onto $\mathbb{C}\Omega$ is of the form $\lambda_1(P_1)+\lambda_2(P_2)-1$, which is contained in $(B_1,\omega_1)*(B_2,\omega_2)$. Since the vector $\Omega$ is cyclic for $(B_1,\omega_1)*(B_2,\omega_2)$, $(B_1,\omega_1)*(B_2,\omega_2)$ contains all compact operators. \end{Rem}
For free product von Neumann algebras, we use the same notation $W_i$, $V_i$, $\Delta$, $\Delta_i$, $J$ and $J_i$ as in the free product part of Subsection \ref{CQG}.
\begin{Lem}[{\cite[\rm Lemma\ 3.1]{Oz04}}] For a free product von Neumann algebra $(M_1,\phi_1)*\cdots*(M_n,\phi_n)$, we have the following equations: \begin{eqnarray*} \lambda_i(a) = J\rho_i(a)J &=&JV_i^*(1_{JH(i)}\otimes J_iaJ_i)V_iJ \\ &=&V_i^*(P_\Omega\otimes a+\lambda_i(a)\mid_{JH(i)\ominus \mathbb{C}\Omega}\otimes 1_{H_i})V_i\\ &=&V_j^*(\lambda_i(a)\mid_{JH(j)}\otimes 1_{H_j})V_j, \end{eqnarray*} for any $a\in \mathbb{B}(H_i)$ and $i\neq j$, where $P_\Omega$ is the orthogonal projection onto $\mathbb{C}\Omega$. \end{Lem} \begin{Rem}\upshape\label{commutator} Simple calculations with the lemma show that $[\lambda_i(\mathbb{B}(H_i)),J\lambda_j(\mathbb{B}(H_j))J]=0$ when $i\neq j$, and that \begin{equation*} [\lambda_i(a), J\lambda_i(b)J]= V_i^* (P_\Omega \otimes [a, J_ibJ_i])V_i \end{equation*} for $a,b\in \mathbb{B}(H_i)$.
Since $V_i=\Sigma(J_i\otimes J|_{JH(i)})W_iJ$, where $\Sigma$ is the flip, this equation means \begin{eqnarray*} [\lambda_i(a), J\lambda_i(b)J] &=& V_i^* (P_\Omega \otimes [a, J_ibJ_i])V_i \\
&=&J^* W_i^* (J_i\otimes J|_{JH(i)})^*\Sigma^* (P_\Omega \otimes [a, J_ibJ_i]) \Sigma(J_i\otimes J|_{JH(i)})W_iJ \\ &=& J^* W_i^*(J_i[a, J_ibJ_i]J_i\otimes P_\Omega) W_iJ. \end{eqnarray*} Hence we get
\begin{equation*} [\lambda_i(a), J\lambda_i(b)J] =W_i^*([a, J_ibJ_i]\otimes P_\Omega) W_i \quad (a,b\in \mathbb{B}(H_i)). \end{equation*} This means the operator $[\lambda_i(a), J\lambda_i(b)J]$ is, as an operator on $H_1*\cdots *H_n$, $[a, J_ibJ_i]$ on $\mathbb{C}\Omega\oplus H_i^0(=H_i)$ and 0 otherwise. \end{Rem}
\begin{Pro}\label{free prod bi-exact} Let $\mathbb{G}_i$ $(i=1,\ldots,n)$ be compact quantum groups. If each $\hat{\mathbb{G}}_i$ satisfies the intermediate condition in \textrm{Lemma $\ref{intermediate}$} with ${\cal C}_l^i$, then the free product $\hat{\mathbb{G}}_1*\cdots*\hat{\mathbb{G}}_n$ satisfies the same condition with the nuclear $C^*$-algebra $({\cal C}_l^1,h_1)*\cdots* ({\cal C}_l^n,h_n)$, where $h_i$ are the vector states of $\hat{1}_{\mathbb{G}_i}$. In particular, $\hat{\mathbb{G}}_1*\cdots *\hat{\mathbb{G}}_n$ is bi-exact if each $\hat{\mathbb{G}}_i$ is strongly bi-exact. \end{Pro} \begin{proof}
We may assume $n=2$. Write $H:=L^2(\mathbb{G}_1)*L^2(\mathbb{G}_2)$. By Lemma \ref{nuclear} and the following remark, ${\cal C}_l^1* {\cal C}_l^2$ is nuclear and contains $\mathbb{K}(H)$. So what to show is that commutators of ${\cal C}_l^1* {\cal C}_l^2$ and $U({\cal C}_l^1* {\cal C}_l^2)U$ are contained in $\mathbb{K}(H)$. We have only to check that $[\lambda_i({\cal C}_l^i),U\lambda_j({\cal C}_l^j)U]$ $(i,j=1,2)$ are contained in $\mathbb{K}(H)$, since $\mathbb{K}(H)$ is an ideal. This is easily verified by Remark \ref{commutator}. \end{proof}
\begin{Pro}\label{free prod AOC^+} Let $\mathbb{G}_i$ $(i=1,\ldots,n)$ be compact quantum groups. If each $\hat{\mathbb{G}}_i$ satisfies conditions in \textrm{Lemma $\ref{AOC}$} with ${\cal C}_l^i$, then the free product $\hat{\mathbb{G}}_1*\cdots*\hat{\mathbb{G}}_n$ satisfies the same condition with the nuclear $C^*$-algebra $({\cal C}_l^1,h_1)*\cdots* ({\cal C}_l^n,h_n)$, where $h_i$ are the vector states of $\hat{1}_{\mathbb{G}_i}$. \end{Pro} \begin{proof} We may assume $n=2$ and write $H:=L^2(\mathbb{G}_1)*L^2(\mathbb{G}_2)$. By the same manner as in the last proposition, ${\cal C}_l^1* {\cal C}_l^2$ is nuclear and contains $\mathbb{K}(H)$. This algebra is contained in $C^*\{C_{\rm red}(\mathbb{G}_1)*C_{\rm red}(\mathbb{G}_2), \hat{\lambda}(\ell^\infty(\hat{\mathbb{G}}_1*\hat{\mathbb{G}}_2)) \}$. Norm continuity of the modular action is trivial since it is continuous on each $\lambda_k({\cal C}_l^k)$. By Remark \ref{commutator}, our commutators in the algebra $\mathbb{B}(H)$ (not the algebra $\mathbb{B}(H\otimes L^2(\mathbb{R}))$) are of the form \begin{equation*} [\lambda_k(a), U\lambda_k(b)U] =W_k^*([a, U_kbU_k]\otimes P_\Omega) W_k \quad (a,b\in {\cal C}^k_l) \end{equation*} (or $[\lambda_k(a), U\lambda_l(b)U]=0$ when $k\neq l$). Modular actions for $a$ is of the form \begin{equation*} [\Delta^{it}\lambda_k(a)\Delta^{-it}, U\lambda_k(b)U] =[\lambda_k(\Delta_k^{it}a\Delta_k^{-it}), U\lambda_k(b)U] =W_k^*([\Delta_k^{it}a\Delta_k^{-it}, U_kbU_k]\otimes P_\Omega) W_k,
\end{equation*} where $\Delta_k$ is the modular operator for $\mathbb{G}_k$. Hence when we see commutators of $\pi({\cal C}_l^k)$ and ${\cal C}_r^l\otimes 1$ in $\mathbb{B}(H\otimes L^2(\mathbb{R}))$, we can first assume $k=l$ since these commutators are zero when $k\neq l$. When we see these commutators for a fixed $k$ (and $k=l$), we actually work on $\mathbb{B}(L^2(\mathbb{G}_k)\otimes L^2(\mathbb{R}))$ with the modular action of $\mathbb{G}_k$, where we regard $L^2(\mathbb{G}_k)\simeq\mathbb{C}\Omega\oplus L^2(\mathbb{G}_k)^0\subset H$. Hence by the assumption on $\mathbb{G}_k$, we get \begin{equation*} [\pi({\cal C}^k_l), {\cal C}^k_r\otimes 1]\subset \mathbb{K}(L^2(\mathbb{G}_k))\otimes \mathbb{B}(L^2(\mathbb{R})) \subset \mathbb{K}(L^2(\mathbb{G}))\otimes \mathbb{B}(L^2(\mathbb{R})). \end{equation*} Thus we get the condition on commutators. \end{proof}
Now we can give new examples of bi-exact quantum groups and von Neumann algebras with condition $\rm (AOC)^+$.
\begin{Cor}\label{free prod AOC} Let $\mathbb{G}_i$ $(i=1,\ldots,n)$ be compact quantum groups. Assume that each $\mathbb{G}_i$ is monoidally equivalent to $\mathrm{SU}_q(2)$, $\mathrm{SO}_q(3)$, or $A_u(F)$, where $-1<q<1$, $q\neq 0$, $F$ is not a scalar multiple of a $2$ by $2$ unitary. Then the free product $\hat{\mathbb{G}}_1*\cdots *\hat{\mathbb{G}}_n$ is bi-exact. The associated von Neumann algebra $(L^\infty(\mathbb{G}_1),h_1)\bar{*}\cdots \bar{*}(L^\infty(\mathbb{G}_n),h_n)$ and its Haar state $h_1*\cdots *h_n$ satisfies condition $\rm (AOC)^+$ with the dense $C^*$-algebra $(C_{\rm red}(\mathbb{G}_1),h_1)*\cdots *(C_{\rm red}(\mathbb{G}_n),h_n)$. \end{Cor}
\subsection{\bf Proof of Theorem \ref{C}}
We first recall some known properties on free quantum groups and quantum automorphism groups. They were originally proved in $\cite{Ba97}\cite{Wa98_2}\cite{BDV05}$ for free quantum groups and $\cite{RV06}\cite{So08}\cite{Br12}$ for quantum automorphism groups. See $\cite[\textrm{Section 4}]{DFY13}$ for the details.
When $F\in\mathrm{GL}(2,\mathbb{C})$ is a scalar multiple of a $2$ by $2$ unitary, then $A_u(F)=A_u(1_2)$ and the dual of $A_u(1_2)$ is a quantum subgroup of $\mathbb{Z}*\hat{A}_o(1_2)$. When $F\in\mathrm{GL}(n,\mathbb{C})$ is any matrix, then the dual of $A_o(F)$ is isomorphic to a free product of some $\hat{A}_o(F_1)$ and $\hat{A}_u(F_1)$ with $F_1\bar{F_1}\in \mathbb{R}\cdot \mathrm{id}$. For such a matrix $F$ as $F\bar{F}=c \cdot\mathrm{id}$ for some $c\in\mathbb{R}$, the quantum group $A_o(F)$ is mononidally equivalent to $\mathrm{SU}_q(2)$, where $-\mathrm{Tr}(FF^*)/c=q+q^{-1}$, $-1\leq q\leq 1$ and $q\neq0$.
When $q=\pm1$, then $\mathrm{dim}_q(u)=|-\mathrm{Tr}(FF^*)/c|=2$, where $u$ is the fundamental representation of $A_o(F)$, and hence $A_o(F)=\mathrm{SU}_{\pm1}(2)$. Thus every $\hat{A}_o(F)$ and $\hat{A}_u(F)$ is a discrete quantum subgroup of a free product of amenable discrete quantum groups and duals of compact quantum groups which are monoidally equivalent to $\mathrm{SU}_q(2)$ or $A_u(F)$, where $-1< q< 1$, $q\neq0$, $F\in\mathrm{GL}(n,\mathbb{C})$ is not a scalar multiple of a $2$ by $2$ unitary.
The quantum automorphism group $A_{\rm aut}(M,\phi)$ is co-amenable if and only if $\mathrm{dim}(M)\leq4$. For any finite dimensional $C^*$-algebra $M$ and any state $\phi$ on $M$, $\hat{A}_{\rm aut}(M,\phi)$ is isomorphic to a free product of duals of quantum automorphism groups with $\delta$-form. Such quantum automorphism groups are co-amenable or monoidally equivalent to $\mathrm{SO}_q(3)$, where $\delta=q+q^{-1}$ and $0< q\leq1$. When $q=1$ and $\delta=2$, since $\mathrm{dim}(M)\leq\delta^2=4$, $A_{\rm aut}(M,\phi)$ is co-amenable. Thus every $\hat{A}_{\rm aut}(M,\phi)$ is a free product of amenable discrete quantum groups and duals of compact quantum groups which are monoidally equivalent to $\mathrm{SO}_q(3)$ for some $q$ with $0<q<1$.
We see the following easy lemma before the proof. \begin{Lem} Let $\mathbb{G}$ and $\mathbb{H}$ be compact quantum groups. Assume that $\hat{\mathbb{H}}$ is a quantum subgroup of $\hat{\mathbb{G}}$. If $\hat{\mathbb{G}}$ is bi-exact (resp.\ $(L^\infty(\mathbb{G}),h)$ satisfies condition $\rm (AOC)^+$ with the $C^*$-algebra $C_{\rm red}(\mathbb{G})$), then $\hat{\mathbb{H}}$ is bi-exact (resp.\ $(L^\infty(\mathbb{H}),h)$ satisfies condition $\rm (AOC)^+$ with the $C^*$-algebra $C_{\rm red}(\mathbb{H})$). \end{Lem} \begin{proof} By assumption there exists the unique Haar state preserving conditional expectation $E_{\mathbb{H}}$ from $L^\infty(\mathbb{G})$ onto $L^\infty(\mathbb{H})$ (and from $C_{\rm red}(\mathbb{G})$ onto $C_{\rm red}(\mathbb{H})$). It extends to a projection $e_{\mathbb{H}}$ from $L^2(\mathbb{G})$ onto $L^2(\mathbb{H})$. Let $\theta$ be a u.c.p.\ map as in the definition of bi-exactness (resp.\ condition $\rm (AOC)^+$). Then a u.c.p.\ map given by $a\otimes b^{\rm op}\mapsto e_{\mathbb{H}}\theta(a\otimes b^{\rm op})e_{\mathbb{H}}$ for $a, b\in C_{\rm red}(\mathbb{H})$ (resp.\ $a\otimes b^{\rm op}\mapsto (e_{\mathbb{H}}\otimes 1)\theta(a\otimes b^{\rm op})(e_{\mathbb{H}}\otimes 1)$ for $a, b\in C_{\rm red}(\mathbb{H})\rtimes_{\rm r}\mathbb{R}$) do the work. Note that modular objects $J$ and $\Delta^{it}$ of the Haar state commute with $e_{\mathbb{H}}$. Local reflexivity of $C_{\rm red}(\mathbb{H})$ (resp.\ $C_{\rm red}(\mathbb{H})\rtimes_{\rm r}\mathbb{R}$) follows from that of $C_{\rm red}(\mathbb{G})$ (resp.\ $C_{\rm red}(\mathbb{G})\rtimes_{\rm r}\mathbb{R}$) since it is a subalgebra. \end{proof}
\begin{proof}[\bf Proof of Theorem \ref{C}] For weak amenablity and the $\rm W^*$CBAP, they are already discussed in $\cite[\textrm{Section 5}]{DFY13}$ (see also $\cite[\textrm{Theorem 4.6}]{Fr11}$). Hence we see only bi-exactness and condition $\rm (AOC)^+$.
Let $\mathbb{G}$ be as in the statement. Then thanks for the observation above, $\hat{\mathbb{G}}$ is a discrete quantum subgroup of $\hat{\mathbb{G}}_1*\cdots *\hat{\mathbb{G}}_n$, where each $\mathbb{G}_i$ is co-amenable, a dual of bi-exact discrete group, or monoidally equivalent to $\mathrm{SU}_q(2)$, $\mathrm{SO}_q(3)$, or $A_u(F)$, where $-1<q<1$, $q\neq 0$, $F$ is not a scalar multiple of a $2$ by $2$ unitary. Hence by Theorems \ref{example st bi-exact} and \ref{example bi-exact}, Remarks \ref{amenable} and \ref{group}, Propositions \ref{free prod bi-exact} and \ref{free prod AOC^+}, and the last lemma, $\hat{\mathbb{G}}$ is bi-exact and $(L^\infty(\mathbb{G}),h)$ satisfies condition $\rm (AOC)^+$ with $C_{\rm red}(\mathbb{G})$. \end{proof}
\section{\bf Proofs of main theorems}
\subsection{\bf Proof of Theorem \ref{A}}
To prove Theorem \ref{A}, we can use the same manner as that in $\cite[\rm Theorem\ 3.1]{PV12}$ except for (i) Proposition 3.2 and (ii) Subsection 3.5 (case 2) in $\cite{PV12}$.
To see (ii) in our situation, we need a structure of quantum group von Neumann algebras, which is weaker than that of group von Neumann algebras but enough to solve our problem. Since we will see a similar (and more general) phenomena in the next subsection (Lemma \ref{case2}), we omit it.
Hence here we give a proof of (i). To do so, we see one proposition which is a quantum analogue of a well known property for crossed products with discrete groups. \begin{Pro} Let $\mathbb{G}$ be a compact quantum group of Kac type, whose dual acts on a tracial von Neumann algebra $(B,\tau_B)$ as a trace preserving action. Write $M:= \hat{\mathbb{G}}\ltimes B$. Let $p$ be a projection in $M$ and $A\subset pMp$ a von Neumann subalgebra. Then the following conditions are equivalent:
\begin{itemize}
\item[$\rm (i)$] $A\not\preceq_M B$;
\item[$\rm (ii)$] there exists a net $(w_n)_n$ of unitaries in $A$ such that $\lim_n \|(w_n)_{i,j}^x \|_{2,\tau_B}=0$ for any $x\in \mathrm{Irred}(\mathbb{G})$ and $i,j$, where $(w_n)_{i,j}^x$ is given by $w_i=\sum_{x\in \mathrm{Irred}(\mathbb{G}), i,j}(w_n)_{i,j}^x u_{i,j}^x$ for a fixed basis $(u_{i,j}^x)$ satisfying $h(u_{i,j}^x u_{k,l}^{y*})=\delta_{i,k}\delta_{j,l}\delta_{x,y}h(u_{i,j}^x u_{i,j}^{x*})$. \end{itemize} \end{Pro} \begin{proof}
We first assume (i). Then by definition, there exists a net $(w_n)_n$ in ${\cal U}(A)$ such that $\lim_n \|E_B(b^*w_na) \|_2=0$ for any $a,b \in M$. Putting $b=1$ and $a=u_{k,l}^{y*}$ and since $E_B(w_n u_{k,l}^{y*})=\sum_{x,i,j}(w_n)_{i,j}^x h(u_{i,j}^x u_{k,l}^{y*})=(w_n)_{k,l}^y h(u_{k,l}^y u_{k,l}^{y*})$, we have \begin{equation*}
\|(w_n)_{k,l}^y\|_2= |h(u_{k,l}^y u_{k,l}^{y*})|^{-1}\|E_B(w_n u_{k,l}^{y*})\|_2\rightarrow 0 \quad (n\rightarrow \infty).
\end{equation*}
Conversely, assume (ii). We will show $\lim_n \|E_B(b^*w_na) \|_2=0$ for any $a,b \in M$. To see this, we may assume $a=u_{\alpha,\beta}^y$ and $b=u_{k,l}^z$. For any $c\in B$, $u_{k,l}^{z*}c$ is a linear combination of $c' u_{k',l'}^{z*}$ for some $c'\in B$ and $k',l'$. When we apply $E_B$ to $u_{k,l}^{z*} c u_{i,j}^x u_{\alpha,\beta}^y$, it does not vanish only if $\bar{z}\otimes x\otimes y$ contains the trivial representation. For fixed $y,z$, the number of such $x$ is finite (since this means $x\in z\otimes \bar{y}$). So we have
\begin{eqnarray*}
\|E_B(b^*w_na) \|_2 &=& \|\sum_{\textrm{finitely many }x, i,j} E_B(b^*(w_n)_{i,j}^x u_{i,j}^xa) \|_2\\
&\leq& \sum_{\textrm{finitely many }x, i,j} \|b^*(w_n)_{i,j}^x u_{i,j}^xa \|_2\\
&\leq& \sum_{\textrm{finitely many }x, i,j} \|b^*\|\|(w_n)_{i,j}^x \|_2 \|u_{i,j}^xa\| \rightarrow 0. \end{eqnarray*} \end{proof}
The following proposition is a corresponding one to (i) Proposition 3.2 in $\cite{PV12}$. \begin{Pro} Theorem $\ref{A}$ is true if it is true for any trivial action. \end{Pro} \begin{proof} Let $\mathbb{G}$, $B$, $M$, $p$, and $A$ be as in Theorem \ref{A}. By Fell's absorption principle, we have the following $*$-homomorphism: \begin{equation*} \Delta\colon M= \hat{\mathbb{G}}\ltimes B \ni bu_{i,j}^x \longmapsto \sum_{k=1}^{n_x} bu_{i,k}^x\otimes u_{k,j}^x\in M\otimes L^\infty(\mathbb{G})=:{\cal M}. \end{equation*} Put ${\cal A}:=\Delta(A)$, $\tilde{q}:=\Delta(q)$ and ${\cal P}:={\cal N}_{\tilde{q}{\cal M}\tilde{q}}({\cal A})''$. Then consider following statements:
\begin{itemize}
\item[(1)] If $A$ is amenable relative to $B$ in $M$, then $\cal A$ is amenable relative to $M\otimes1$ in $\cal M$.
\item[(2)] If $\cal P$ is amenable relative to $M\otimes1$ in $\cal M$, then ${\cal N}_{qMq}(A)''$ is amenable relative to $B$ in $M$.
\item[(3)] If ${\cal A}\preceq_{\cal M}M\otimes 1$, then $A\preceq_M B$. \end{itemize} If one knows them, then (1) and our assumption say that we have either $\cal P$ is amenable relative to $M\otimes1$ or ${\cal A}\preceq_{\cal M}M\otimes 1$. Then (2) and (3) imply our desired conclusion.
To show (1) and (2), we only need the property $\Delta\circ E_B=E_{M\otimes 1}\circ \Delta=E_{B\otimes1}\circ \Delta$. So we can use the same strategy as in the group case. To show (3) we need the previous proposition, and once we accept it, then (3) is easily verified. \end{proof}
\subsection{\bf Proof of Theorem \ref{B}}
We use the same symbol as in the statement of Theorem \ref{B}. We moreover use the following notation: \begin{eqnarray*}
&&P:= {\cal N}_{qMq}(A)'', \ C_h:=C_h(L^\infty(\mathbb{G})),\
\tau:=\mathrm{Tr}_M(q\cdot q),\\ &&L^2(M):=L^2(M,\mathrm{Tr}_M)=L^2(B,\tau_B)\otimes L^2(C_h,\mathrm{Tr}),\\ &&\pi\colon M=B\otimes C_h \ni b\otimes x \mapsto (b\otimes_A q \otimes x) \in \mathbb{B}((L^2(M)q \otimes_A L^2(P))\otimes L^2(C_h,\mathrm{Tr})),\\ &&\theta\colon P^{\rm op} \ni y^{\rm op}\mapsto (q\otimes_A y^{\rm op}) \otimes 1 \in \mathbb{B}((L^2(M)q \otimes_A L^2(P))\otimes L^2(C_h,\mathrm{Tr})),\\ &&N:=W^*\{\pi(B\otimes 1), \theta(P^{\rm op})\},\ {\cal N}:=N\otimes C_h. \end{eqnarray*}
We first recall the following theorem which is a generalization of $\cite[\textrm{Theorem 3.5}]{OP07}$ and $\cite[\textrm{Theorem B}]{Oz10}$. Now we are working on a semifinite von Neumann algebra $\cal N$ but still the theorem is true. To verify this, we need the following observation. If a semifiite von Neumann algebra has the $\rm W^*$CBAP, $(\phi_i)_i$ is an approximation identity, and $(p_j)_j$ is a net of trace-finite projections converging to 1 strongly, then a subnet of $(\psi_j\circ\phi_i)$, where $\psi_j$ is a compression by $p_j$, is an approximation identity. Hence we can find an approximate identity whose images are contained in a trace-finite part of the semifinite von Neumann algebra. As finite rank maps relative to $B$ $\cite[\textrm{Definition 5.3}]{PV11}$, one can take linear spans of \begin{equation*} \psi_{y,z,r,t}\colon qMq\rightarrow qMq; x\mapsto y(\mathrm{id}_B\otimes \mathrm{Tr})(zxr)t, \end{equation*} where $y,z,r,t\in M$ satisfying $y=qy$, $t=tq$, $z=(1\otimes p)z$, and $r=r(1\otimes p')$ for some Tr-finite projections $p,p'\in C_h$. Note that for fixed $p\in C_h$ with $\mathrm{Tr}(p)<\infty$, $\mathrm{Tr}(p)^{-1}(\mathrm{id}_B\otimes \mathrm{Tr})$ is a conditional expectation from $B\otimes pC_hp$ onto $B$.
\begin{Thm}[{\cite[\textrm{Theorem 5.1}]{PV12}}] There exists a net $(\omega_i)_i$ of normal states on $\pi(q){\cal N}\pi(q)$ such that \begin{itemize}
\item[$\rm (i)$] $\omega_i(\pi(x))\rightarrow \tau(x)$ for any $x\in qMq$;
\item[$\rm (ii)$] $\omega_i(\pi(a)\theta(\bar{a}))\rightarrow 1$ for any $a\in {\cal U}(A)$;
\item[$\rm (iii)$] $\|\omega_i\circ\mathrm{Ad}(\pi(u)\theta(\bar{u}))-\omega_i\| \rightarrow 0$ for any $u \in {\cal N}_{qMq}(A)$. \end{itemize} Here $\bar a$ means $(a^{\rm op})^*$. \end{Thm}
Let $H$ be a standard Hilbert space of $N$ with a canonical conjugation $J_N$. From now on, as a standard representation of $C_h$, we use $L^2(C_h):=L^2(C_h,\hat{h})=L^2(\mathbb{G})\otimes L^2(\mathbb{R})$, where $\hat{h}$ is the dual weight of $h$.
Then since $H\otimes L^2(C_h)$ is standard for $\cal N$ with a canonical conjugation ${\cal J}:=J_N\otimes J_{C_h}$, there exists a unique net $(\xi_i)_i$ of unit vectors in the positive cone of $\pi(q){\cal J}\pi(q){\cal J}(H\otimes L^2(C_h))$ such that $\omega_i(\pi(q)x\pi(q))=\langle x \xi_i | \xi_i \rangle$ for $x\in {\cal N}$. Then conditions on $\omega_i$ are translated to the following conditions: \begin{itemize}
\item[$\rm (i)'$] $\langle \pi(x) \xi_i | \xi_i \rangle\rightarrow \tau(qxq)$ for any $x\in M$;
\item[$\rm (ii)'$] $\|\pi(a)\theta(\bar a)\xi_i -\xi_i\|\rightarrow 0$ for any $a\in {\cal U}(A)$;
\item[$\rm (iii)'$] $\|\mathrm{Ad}(\pi(u)\theta(\bar{u}))\xi_i -\xi_i\| \rightarrow 0$ for any $u \in {\cal N}_{qMq}(A)$. \end{itemize} Here $\mathrm{Ad}(x)\xi_i:=x{\cal J}x{\cal J}\xi_i$. To see $\rm (iii)'$, we need a generalized Powers--St$\rm \o$rmer inequality (e.g.\ $\cite[\textrm{Theorem IX.1.2.(iii)}]{Ta2}$).
The following lemma is a very similar statement to that in $\cite[\textrm{Subsection 3.5 (case 2)}]{PV12}$. Since we treat quantum groups and moreover our object $C_h$ is semifinite and twisted (not a tensor product), we need more careful treatment. So here we give a complete proof of the lemma. \begin{Lem}\label{case2} Let $(\xi_i)_i$ be as above. Assume $A\not\preceq_{M} B\otimes L\mathbb{R}$.
Then we have \begin{equation*}
\limsup_i\| (1_{\mathbb{B}(H)}\otimes x \otimes 1_{L\mathbb{R}})\xi_i\|=0 \quad \textrm{for any } x \in \mathbb{K}(L^2(\mathbb{G})). \end{equation*} \end{Lem} \begin{proof} Suppose by contradiction that there exist $\delta>0$ and a finite subset ${\cal F}\subset \mathrm{Irred}(\mathbb{G})$ such that \begin{equation*}
\limsup_i\| (1_{\mathbb{B}(H)}\otimes P_{\cal F} \otimes 1_{L\mathbb{R}})\xi_i\| >\delta, \end{equation*} where $P_{\cal F}$ is the orthogonal projection onto $\sum_{x\in {\cal F}}H_x \otimes H_{\bar x}$. Replacing with a subnet of $(\xi_i)_i$, we may assume that \begin{equation*}
\liminf_i\| (1_{\mathbb{B}(H)}\otimes P_{\cal F} \otimes 1_{L\mathbb{R}})\xi_i\| >\delta. \end{equation*} Our goal is to find a finite set ${\cal F}_1\subset \mathrm{Irred}(\mathbb{G})$ and a subnet of $(\xi_i)_i$ which satisfy \begin{equation*}
\liminf_i\| (1_{\mathbb{B}(H)}\otimes P_{{\cal F}_1} \otimes 1_{L\mathbb{R}})\xi_i\| >2^{1/2}\delta. \end{equation*} Then repeating this argument, we get a contradiction since \begin{equation*}
1\geq\liminf_i\| (1_{\mathbb{B}(H)}\otimes P_{{\cal F}_k} \otimes 1_{L\mathbb{R}})\xi_i\| >2^{k/2}\delta \quad (k\in \mathbb{N}). \end{equation*}
\noindent \textit{{\bf claim 1.} There exists a $\mathrm{Tr}$-finite projection $r\in L\mathbb{R}$ and a subnet of $(\xi_i)_i$ such that } \begin{equation*}
\liminf_i\| (1_{\mathbb{B}(H)}\otimes P_{\cal F} \otimes r)\xi_i\| >\delta. \end{equation*} \begin{proof}[\bf proof of claim 1]
Define a state on $\mathbb{B}(H\otimes L^2(C_h))$ by $\Omega(X):=\mathrm{Lim}_i\langle X\xi_i | \xi_i\rangle$. Then the condition $\rm (i)'$ of $(\xi_i)_i$ says that $\Omega(\pi(x))=\tau(qxq)$ for $x\in M$. Let $r_j$ be a net of $\mathrm{Tr}$-finite projections in $L\mathbb{R}$ which converges to 1 strongly. Then we have \begin{eqnarray*}
|\Omega((1\otimes P_{\cal F}\otimes 1) \pi(1-r_j))|^2&\leq&\Omega((1\otimes P_{\cal F}\otimes 1))\Omega(\pi(1-r_j)) \\ &=&\Omega((1\otimes P_{\cal F}\otimes 1))\tau(q(1-r_j)q) \rightarrow 0 \quad (j\rightarrow \infty). \end{eqnarray*} This implies that \begin{eqnarray*} 0&\leq&
\mathrm{Lim}_i\| (1\otimes P_{\cal F} \otimes 1)\xi_i\| - \mathrm{Lim}_i\| (1\otimes P_{\cal F} \otimes r_j)\xi_i\| \\ &=&\Omega(1\otimes P_{\cal F} \otimes 1) - \Omega( (1\otimes P_{\cal F} \otimes 1)\pi(r_j))\\ &=&\Omega((1\otimes P_{\cal F}\otimes 1) \pi(1-r_j)) \rightarrow 0 \quad (j\rightarrow \infty) \end{eqnarray*}
Hence we can find a $\mathrm{Tr}$-finite projection $r\in L\mathbb{R}$ such that $\mathrm{Lim}_i\| (1\otimes P_{\cal F} \otimes r)\xi_i\|>\delta$. Finally by taking a subnet of $(\xi_i)_i$, we have $\liminf_i\| (1\otimes P_{\cal F} \otimes r)\xi_i\|>\delta$. \end{proof}
We fix a basis $\{u_{i,j}^x \}_{i,j}^x (=:X)$ of the dense Hopf $*$-algebra of $C(\mathbb{G})$ and use the notation \begin{eqnarray*} &&\mathrm{Irred}(\mathbb{G})_x:=\{ u_{i,j}^x\in X \mid i,j=1,\ldots,n_x\} \quad (x\in \mathrm{Irred}(\mathbb{G})),\\ &&\mathrm{Irred}(\mathbb{G})_{\cal E}:=\cup_{x\in{}\cal E}\mathrm{Irred}(\mathbb{G})_x \quad ({\cal E} \subset \mathrm{Irred}(\mathbb{G})). \end{eqnarray*} We may assume that each $\hat{u}_{i,j}^x\in L^2(\mathbb{G})$ is an eigenvector of the modular operator of $h$, namely, for any $u_{i,j}^x$ there exists $\lambda>0$ such that $\Delta_h^{it}\hat{u}_{i,j}^x=\lambda^{it}\hat{u}_{i,j}^x$ $(t\in \mathbb{R})$. In this case we have a formula $\sigma_t^h(u_{i,j}^x)=\lambda^{it}u_{i,j}^x$ and hence $\sigma_t^h(u_{i,j}^{x*} a u_{i,j}^{x})=u_{i,j}^{x*} \sigma_t^h(a) u_{i,j}^{x}$ for any $a\in L^\infty(\mathbb{G})$.
Let $P_e$ be the orthogonal projection from $L^2(\mathbb{G})$ onto $\mathbb{C}\hat 1$. For any $u\in X$, consider a compression map \begin{equation*} \Phi_u(x):=h(u^*u)^{-1}(1\otimes P_eu^* \otimes 1)\pi(x)(1\otimes uP_e \otimes 1) \end{equation*} for $x\in M$, which gives a normal map from $M$ into $B\otimes \mathbb{C}P_e\otimes\mathbb{B}(L^2(\mathbb{R}))\simeq B\otimes \mathbb{B}(L^2(\mathbb{R}))$.
\noindent \textit{{\bf claim 2.} For any $u\in X$, we have} \begin{equation*} \Phi_{u}(b\otimes af)= b\otimes h(u^*u)^{-1}h(u^{*}au)f \quad (b\in B, a\in L^\infty(\mathbb{G}), f\in L\mathbb{R}). \end{equation*} \textit{In particular, $\Phi_u$ is a normal conditional expectation from $M$ onto $B\otimes L\mathbb{R}$.} \begin{proof}[\bf proof of claim 2] Assume $B=\mathbb{C}$ for simplicity. Recall that any element $a\in L^\infty(\mathbb{G})$ in the continuous core is of the form $a=\int\sigma_{-t}^h(a)\otimes e_t \cdot dt$, which means $(a\xi)(s)=\sigma_{-s}^h(a)\xi(s)$ for $\xi \in L^2(\mathbb{G})\otimes L^2(\mathbb{R})$ and $s\in \mathbb{R}$. Hence a simple calculation shows that for any $a\in L^\infty(\mathbb{G})$, \begin{eqnarray*} h(u^*u)\Phi_u(a)&=&(P_eu^* \otimes 1)\int\sigma_{-t}^h(a)\otimes e_t \cdot dt (uP_e \otimes 1)\\ &=&\int P_eu^*\sigma_{-t}^h(a)uP_e\otimes e_t \cdot dt\\
&=&\int h(u^*au)P_e\otimes e_t \cdot dt= h(u^*au)P_e\otimes 1, \end{eqnarray*} where we used $P_eu^*\sigma_{-t}^h(a)uP_e=h(u^*\sigma_{-t}^h(a)u)P_e=h(\sigma_{-t}^h(u^*au))P_e=h(u^*au)P_e$. Thus $\Phi_u$ satisfies our desired condition. \end{proof}
\noindent \textit{{\bf claim 3.} For any $u\in X$ and $x\in M$, we have} \begin{equation*}
\limsup_i\|\pi(x)(1\otimes P_u\otimes r)\xi_i\|\leq h(u^*u)^{-1/2} \|x(1\otimes r)\|_{2,\mathrm{Tr}_M\circ\Phi_u}, \end{equation*} \textit{where $P_u$ is the orthogonal projection from $L^2(\mathbb{G})$ onto $\mathbb{C}\hat{u}$.} \begin{proof}[\bf proof of claim 3] This follows from a direct calculation. Since $P_u=h(u^*u)^{-1} uP_e u^*$, we have \begin{eqnarray*}
&&h(u^*u)\|(\pi(x)(1\otimes P_u\otimes r)\xi_i\|^2\\
&=&h(u^*u)\langle (1\otimes P_u \otimes 1)\pi(\tilde{x}^*\tilde{x}) (1\otimes P_u \otimes 1)\xi_i | \xi_i\rangle \qquad (\tilde x := x(1\otimes r))\\
&=&h(u^*u)^{-1}\langle (1\otimes P_eu^* \otimes 1)\pi(\tilde{x}^*\tilde{x}) (1\otimes uP_e \otimes 1)\tilde{\xi}_i | \tilde{\xi}_i\rangle \qquad (\tilde{\xi}_i :=(1\otimes u^* \otimes 1) \xi_i)\\
&=&\langle (1\otimes P_e \otimes 1)\pi(\Phi_u(\tilde{x}^*\tilde{x})) (1\otimes P_e \otimes 1)\tilde{\xi}_i | \tilde{\xi}_i\rangle\\
&=&\| \pi(y) (1\otimes P_eu^* \otimes 1)\xi_i \|^2 \qquad (y:=\Phi_u(\tilde{x}^*\tilde{x})^{1/2})\\
&\leq& \| \pi(y) \xi_i \|^2 \qquad (\textrm{since $\pi(y)$ and $(1\otimes P_eu^* \otimes 1$) commute})\\ &\rightarrow& \mathrm{Tr}_M(qy^*yq) \leq \mathrm{Tr}_M(y^*y) = \mathrm{Tr}_M(\Phi_u(\tilde{x}^*\tilde{x}))= \mathrm{Tr}_M\circ\Phi_u((1\otimes r)x^*x(1\otimes r)). \end{eqnarray*} \end{proof}
\noindent \textit{{\bf claim 4.} For any $\epsilon>0$ and any finite subset ${\cal E}\subset \mathrm{Irred}(\mathbb{G})$, there exist $a\in {\cal U}(A)$ and $v\in M$ such that \begin{itemize}
\item $v$ is a finite sum of elements of the form $b\otimes u f$, where $b\in B$, $f\in L\mathbb{R}$ and $u\in X \setminus \mathrm{Irred}(\mathrm{G})_{\cal E}$;
\item $h(u^*u)^{-1/2}\|(a-v)(1\otimes r)\|_{2,\mathrm{Tr}_M\circ\Phi_u}<\epsilon$ for any $u\in\mathrm{Irred}(\mathrm{G})_{\cal F}$. \end{itemize} } \begin{proof}[\bf proof of claim 4]
Since $A\not\preceq_{{}_e M_e}B\otimes L\mathbb{R}r$, where $e:=q\vee r$, for any $\epsilon>0$ and any finite subset ${\cal E}\subset \mathrm{Irred}(\mathbb{G})$, there exist $a\in {\cal U}(A)$ s.t\ $\|E_{B\otimes L\mathbb{R}}((1\otimes r)(1\otimes u^*)a(1\otimes r))\|_{2,\mathrm{Tr}_M}<\epsilon$ for any $u\in \mathrm{Irred}(\mathrm{G})_{\cal E}$.
Since the linear span of $\{b\otimes u\lambda_t| b\in B, t\in\mathbb{R},u\in X\}$ is strongly dense in $M$, we can find a bounded net $(z_j)$ of elements in this linear span which converges to $a$ in the strong topology.
Hence we can find $z$ in the linear span such that $\|(a-z)(1\otimes r)\|_{2,\mathrm{Tr}_M}$ and $\|(a-z)(1\otimes r)\|_{2,\mathrm{Tr}_M\circ\Phi_u}$ $(u\in\mathrm{Irred}(\mathrm{G})_{\cal F})$ are very small.
In this case we may assume that $\|E_{B\otimes L\mathbb{R}}((1\otimes r)(1\otimes u^*)z(1\otimes r))\|_{2,\mathrm{Tr}_M}<\epsilon$ for any $u\in \mathrm{Irred}(\mathrm{G})_{\cal E}$.
Write $z=\sum_{\rm finite} b_{i,j}^x\otimes u_{i,j}^x f_{i,j}^x$. Then for any $y\in{\cal E}$ the above inequality implies \begin{eqnarray*}
\epsilon&>&\|E_{B\otimes L\mathbb{R}}((1\otimes r)(1\otimes u^{y*}_{k,l})z(1\otimes r))\|_{2,\mathrm{Tr}_M}\\
&=&\|\sum_{\rm finite} (1\otimes r)(b_{i,j}^x\otimes h(u^{y*}_{k,l}u_{i,j}^x)f_{i,j}^x)(1\otimes r)\|_{2,\mathrm{Tr}_M}\\
&=&\sum_{\rm finite}\delta_{x,y}\delta_{i,k}\delta_{j,l}h(u^{y*}_{k,l}u_{i,j}^x)\|b_{i,j}^x\otimes f_{i,j}^xr\|_{2,\mathrm{Tr}_M}
=\|u^{y}_{k,l}\|^2_{2,h}\|b_{k,l}^y\otimes f_{k,l}^yr\|_{2,\mathrm{Tr}_M}. \end{eqnarray*} Hence if we write $z=\sum_{x\in {\cal E}}b_{i,j}^x\otimes u_{i,j}^x f_{i,j}^x + \sum_{x\not\in {\cal E}}b_{i,j}^x\otimes u_{i,j}^x f_{i,j}^x $, and say $v$ for the second sum, then we have \begin{eqnarray*}
\|(z-v)(1\otimes r)\|_{2,\mathrm{Tr}_M\circ\Phi_u}
&\leq&\sum_{x\in{\cal E},i,j}\| b_{i,j}^x\otimes u_{i,j}^x f_{i,j}^xr \|_{2,\mathrm{Tr}_M\circ\Phi_u} \\
&\leq&\sum_{x\in{\cal E},i,j}\|u_{i,j}^x\| \| b_{i,j}^x\otimes f_{i,j}^xr \|_{2,\mathrm{Tr}_M\circ\Phi_u} \\
&\leq&\sum_{x\in{\cal E},i,j}\| b_{i,j}^x\otimes f_{i,j}^xr \|_{2,\mathrm{Tr}_M} \\ &<&C({\cal E})\cdot\epsilon, \end{eqnarray*}
for any $u\in\mathrm{Irred}(\mathrm{G})_{\cal F}$, where $C({\cal E})>0$ is a constant which depends only on ${\cal E}$. Thus this $v$ is our desired one. \end{proof}
Now we return to the proof. Since $1\otimes P_{\cal F}\otimes r$ is a projection, we have \begin{eqnarray*}
\limsup_i\|\xi_i-(1\otimes P_{\cal F}\otimes r)\xi_i\|^2
&=&\limsup_i(\|\xi_i\|^2-\|(1\otimes P_{\cal F}\otimes r)\xi_i\|^2)\\
&=&1-\liminf_i\|(1\otimes P_{\cal F}\otimes r)\xi_i\|^2<1-\delta^2. \end{eqnarray*}
So there exists $\epsilon>0$ such that $\limsup_i\|\xi_i-(1\otimes P_{\cal F}\otimes r)\xi_i\|<(1-\delta^2)^{1/2}-\epsilon$. We apply claim 4 to $(\sum_{x\in{\cal F}}\mathrm{dim}(H_x)^2)^{-1}\epsilon$ and ${\cal E}:={\cal F}\bar{\cal F}$ $(=\{z| z\in x\otimes \bar{y} \textrm{ for some }x,y\in{\cal F}\})$, and get $a$ and $v$. Then we have \begin{eqnarray*}
&&\limsup_i\|\xi_i-\theta(\bar{a})\pi(v)(1\otimes P_{\cal F}\otimes r)\xi_i\|\\
&\leq&\limsup_i\|\xi_i-\theta(\bar{a})\pi(a)(1\otimes P_{\cal F}\otimes r)\xi_i\|+\limsup_i\|\theta(\bar{a})\pi(a-v)(1\otimes P_{\cal F}\otimes r)\xi_i\|\\
&\leq&\limsup_i\|\theta(a^{\rm op})\pi(a^*)\xi_i-(1\otimes P_{\cal F}\otimes r)\xi_i\| + \sum_{u\in \mathrm{Irred}(\mathbb{G})_{\cal F}} h(u^*u)^{-1/2}\|(a-v)(1\otimes r)\|_{2, \mathrm{Tr}_M\circ\Phi_u} \\
&<&\limsup_i\|\xi_i-(1\otimes P_{\cal F}\otimes r)\xi_i\| + \epsilon < (1-\delta^2)^{1/2}, \end{eqnarray*} where we used condition $\rm (ii)'$ of $(\xi_i)$ and claim 3. By the choice of $v$, it is of the form $\sum_{x\in{\cal S},i,j}b_{i,j}^x\otimes u_{i,j}^x f_{i,j}^x$ for some finite set ${\cal S}\subset \mathrm{Irred}(\mathbb{G})$ with ${\cal S}\cap{\cal F}\bar{\cal F}=\emptyset$. Note that this means ${\cal SF}\cap{\cal F}=\emptyset$. The vector $\theta(\bar{a})\pi(v)(1\otimes P_{\cal F}\otimes r)\xi_i$ is contained in the range of $1\otimes P_{\cal SF}\otimes 1$. This is because the modular operator $\Delta_h^{it}$ commutes with $P_{\cal SF}$ and $P_{\cal F}$ and hence \begin{equation*} (1\otimes P_{{\cal SF}}\otimes 1)\pi(v)(1\otimes P_{\cal F}\otimes 1)=\pi(v)(1\otimes P_{\cal F}\otimes 1). \end{equation*} Then we have \begin{eqnarray*} 1-\delta^2
&>&\limsup_i\|\xi_i-\theta(\bar{a})\pi(v)(1\otimes P_{\cal F}\otimes r)\xi_i\|^2 \\
&\geq&\limsup_i\|(1\otimes P_{\cal SF}\otimes 1)^{\perp}\xi_i\|^2\\
&=&1-\liminf_i\|(1\otimes P_{\cal SF}\otimes 1)\xi_i\|^2. \end{eqnarray*}
This means $\delta<\liminf_i\|(1\otimes P_{\cal SF}\otimes 1)\xi_i\|$. Finally put ${\cal F}_1:={\cal F}\cup{\cal SF}$. Then since ${\cal SF}\cap{\cal F}=\emptyset$, we get \begin{equation*}
\liminf_i\|(1\otimes P_{{\cal F}_1}\otimes 1)\xi_i\|^2
\geq\liminf_i\|(1\otimes P_{\cal F}\otimes 1)\xi_i\|^2 + \liminf_i\|(1\otimes P_{\cal SF}\otimes 1)\xi_i\|^2>2\delta^2. \end{equation*} Thus ${\cal F}_1$ is our desired one and we can end the proof. \end{proof}
Now we start the proof. We follow that of $\cite[\textrm{Subsection 3.4 (case 1)}]{PV12}$. Since this part of the proof does not rely on the structure of group von Neumann algebras, we need only slight modifications, which were already observed in $\cite{Is12_2}$. Hence here we give a rough sketch of the proof. We use the following notation which is used in $\cite{PV12}$. \begin{eqnarray*} &&{\cal D}:=M\odot M^{\rm op} \odot P^{\rm op} \odot P \supset (C_{\rm red}(\mathbb{G})\rtimes_{\rm r}\mathbb{R})\odot (C_{\rm red}(\mathbb{G})\rtimes_{\rm r}\mathbb{R})^{\rm op} \odot P^{\rm op} \odot P =:{\cal D}_0,\\ &&\Psi\colon {\cal D}\rightarrow \mathbb{B}(H\otimes L^2(C_h)\otimes L^2(C_h)), \quad \Theta \colon {\cal D}\rightarrow \mathbb{B}(H\otimes L^2(C_h)),\\ &&\Psi((b_1\otimes x_1) \otimes (b_2\otimes x_2)^{\rm op}\otimes y^{\rm op}\otimes z)=b_1J_Nb_2^*J_Ny^{\rm op}J\bar{z}J\otimes x_1\otimes x_2^{\rm op}, \\ &&\Theta((b_1\otimes x_1) \otimes (b_2\otimes x_2)^{\rm op}\otimes y^{\rm op}\otimes z)=b_1J_Nb_2^*J_Ny^{\rm op}J\bar{z}J\otimes x_1x_2^{\rm op}\\ &&\phantom{\Theta((b_1\otimes x_1) \otimes (b_2\otimes x_2)^{\rm op}\otimes y^{\rm op}\otimes z)}=\pi(b_1\otimes x_1){\cal J}\pi(b_2\otimes x_2)^*{\cal J}\theta(y^{\rm op}){\cal J}\theta(\bar{z}){\cal J}. \end{eqnarray*}
\begin{proof}[\bf Proof of Theorem \ref{B}]
Define a state on $\mathbb{B}(H\otimes L^2(C_h))$ by $\Omega_1(X):=\mathrm{Lim}_i\langle X\xi_i|\xi_i\rangle$. Our condition $\rm (AOC)^+$, together with Lemma \ref{case2}, implies that $|\Omega_1(\Theta(X))|\leq \|\Psi(X)\|$ for any $X\in{\cal D}_0$. We extend this inequality on $\cal D$ by using an approximation identity of $C_h$, which take vales in $C_{\rm red}(\mathbb{G})\rtimes_{\rm r}\mathbb{R}$ (such a net exists, see $\cite[\textrm{Lemma 2.3.1}]{Is12_2}$).
Then a new functional $\Omega_2$ on $C^*(\Psi({\cal D}))$ is defined by $\Omega_2(\Psi(X)):=\Omega_1(\Theta(X))$ $(X\in{\cal D})$.
Its Hahn--Banach extension on $\mathbb{B}(H\otimes L^2(C_h)\otimes L^2(C_h))$ restricts to a $P$-central state on $q(B\otimes \mathbb{B}(C_h))q\otimes \mathbb{C}$. More precisely we have a $P$-central state on $\mathbb{B}(qL^2(M))\cap(B^{\rm op})'(=q(B\otimes \mathbb{B}(C_h))q)$ which restricts to the trace $\tau$ on $qMq$. \end{proof}
\subsection{\bf Proofs of corollaries}
For the Kac type case, the same proofs as in the group case work. So we see only the proof of Corollary \ref{B}.
Let $\mathbb{G}$ be a compact quantum group in the statement of the corollary, $h$ the Haar state of $\mathbb{G}$, and $(B,\tau_B)$ be a tracial von Neumann algebra. Write $M:=B\otimes L^\infty(\mathbb{G})$. Let $(A,\tau_A)$ be an amenable tracial von Neumann subalgebra in $M$ with expectation $E_A$.
Then since modular actions of $\tau_A$ and $\tau_A\circ E_A$ has the relation $\sigma^{\tau_A}=\sigma^{\tau_A\circ E_A}|_A$ (see the proof of $\cite[\rm{Theorem\ IX.4.2}]{Ta2}$), we have an inclusion $A\otimes L\mathbb{R}=C_{\tau_A}(A)\subset C_{\tau_A\circ E_A}(M)$ with a faithful normal conditional expectation $\tilde{E}_A$ given by $\tilde{E}_A(x\lambda_t)=E_A(x)\lambda_t$ for $x\in M$ and $t\in \mathbb{R}$. Since continuous cores are canonically isomorphic with each other, there is a canonical isomorphism from $C_{\tau_A\circ E_A}(M)$ onto $C_{\tau_B\otimes h}(M)=B\otimes C_h(L^\infty(\mathbb{G}))$ $(:=\tilde{M})$. We denote by $\tilde{A}$ the image of $C_{\tau_A}(A)$ in $\tilde{M}$. Put $\tilde{P}:={\cal N}_{\tilde{M}}(\tilde{A})''$. Note that there is a faithful normal conditional expectation $E_{\tilde{P}}$ from $\tilde{M}$ onto $\tilde{P}$, since $\tilde{A}$ is an image of an expectation (use $\cite[\rm{Theorem\ IX.4.2}]{Ta2}$). \begin{Lem} Under the setting above, for any projection $z\in{\cal Z}(A)\cap (1_B\otimes L^\infty(\mathbb{G}))$ (possibly $z=1$) we have either \begin{itemize}
\item[$\rm (i)$] $Az\preceq_M B$;
\item[$\rm (ii)$] there exists a conditional expectation from $z(B\otimes \mathbb{B}(L^2(C_h)))z$ onto $z\tilde{P}z$, where we regard $z\in 1_B\otimes C_h$ by the canonical inclusion $L^\infty(\mathbb{G})\subset C_h$. \end{itemize} \end{Lem} \begin{proof} Suppose $Az\not\preceq_M B$. Then by the same manner as in $\cite[\textrm{Proposition 2.10}]{BHR12}$, we have $\tilde{A}zq\not\preceq_{\tilde{M}}B\otimes L\mathbb{R}r$ for any projections $q\in {\cal Z}(\tilde{A})$ and $r\in L\mathbb{R}$ with $(\tau_B\otimes\mathrm{Tr})(q)<\infty$ and $\mathrm{Tr}(r)<\infty$. By the comment below Theorem \ref{popa embed2}, this means $\tilde{A}zq\not\preceq_{\tilde{M}}B\otimes L\mathbb{R}$ for any projection $q\in {\cal Z}(\tilde{A})$ with $(\tau_B\otimes\mathrm{Tr})(q)<\infty$. We fix such $q$ and assume $z=1$ for simplicity.
Apply Theorem \ref{B} to them and get that $L^2(q\tilde{M})$ is left ${\cal N}_{q\tilde{M}q}(q\tilde{A}q)''$-amenable as a $q\tilde{M}q$-$B$-bimodule. Note that ${\cal N}_{q\tilde{M}q}(q\tilde{A}q)''=q\tilde{P}q$ (e.g.\ $\cite[\rm Lemma\ 2.2]{FSW10}$). By $\cite[\textrm{Proposition 2.4}]{PV12}$ this means that ${}_{q\tilde{M}q}L^2(q\tilde{M}q)_{q\tilde{P}q} \prec {}_{q\tilde{M}q}L^2(q\tilde{M})\otimes_B L^2(\tilde{M}q)_{q\tilde{P}q}$. Let $\nu_q$ (or $\nu$ for $q=1$) be the following canonical multiplication $*$-homomorphism: \begin{alignat*}{5} & \hspace{5em}\mathbb{B}(\tilde{q}K) && \hspace{1em}\mathbb{B}(qq^{\rm op}L^2(\tilde{M}))\\ & \hspace{6em}\cup && \hspace{3.5em}\cup \\ & \textrm{$*$-alg}\{q\tilde{M}q\otimes_B q^{\rm op}, q\otimes_B (q\tilde{P}q)^{\rm op}\} &\quad\xrightarrow{\nu_q}\quad& \textrm{$*$-alg}\{q\tilde{M}q, (q\tilde{P}q)^{\rm op}\}& \end{alignat*} where $K:=L^2(\tilde{M})\otimes_B L^2(\tilde{M})$ and $\tilde{q}:=(q\otimes_B1)(1\otimes_B q^{\rm op})\in\mathbb{B}(K)$. The weak containment above means that $\nu_q$ is bounded. Let $(q_i)_i$ be a net of $(\tau_B\otimes\mathrm{Tr})$-finite projections in ${\cal Z}(\tilde{A})$ which converges to 1 strongly. Then since each $q_i$ satisfies the weak containment above, we have for any $x\in \textrm{$*$-alg}\{\tilde{M}\otimes_B1, 1\otimes_B (\tilde{P})^{\rm op}\}\subset\mathbb{B}(K)$, \begin{equation*}
\|x\|_{\mathbb{B}(K)}=\sup_i\|\tilde{q_i}x\tilde{q_i}\|_{\mathbb{B}(\tilde{q_i}K)}\geq
\sup_i\|\nu_{q}(\tilde{q_i}x\tilde{q_i})\|_{\mathbb{B}(q_iq_i^{\rm op}L^2(\tilde{M}))}=\|\nu(x)\|_{\mathbb{B}(L^2(\tilde{M}))}. \end{equation*} Hence $\nu$ is bounded and we have ${}_{\tilde{M}}L^2(\tilde{M})_{\tilde{P}} \prec {}_{\tilde{M}}L^2(\tilde{M})\otimes_B L^2(\tilde{M})_{\tilde{P}}$. (For a general $z$, we have ${}_{z\tilde{M}z}L^2(z\tilde{M}z)_{z\tilde{P}z} \prec {}_{z\tilde{M}z}L^2(z\tilde{M})\otimes_B L^2(\tilde{M}z)_{z\tilde{P}z}$.)
By Arveson's extension theorem, we extend $\nu$ on $C^*\{\tilde{M}\otimes_B1, 1\otimes_B (B'\cap \mathbb{B}(L^2(\tilde{M})))\}$ as a u.c.p.\ map into $\mathbb{B}(L^2(\tilde{M}))$ and denote it by $\Phi$. Then since $\tilde{M}\otimes 1$ is contained in the multiplicative domain of $\Phi$, the image of $1\otimes_B (B'\cap \mathbb{B}(L^2(\tilde{M})))$ is contained in $\tilde{M}'=\tilde{M}^{\rm op}$. Consider the following composition map: \begin{equation*} B^{\rm op}\otimes \mathbb{B}(L^2(C_h))= (B'\cap\mathbb{B}(L^2(\tilde{M})))\simeq1\otimes_B (B'\cap\mathbb{B}(L^2(\tilde{M})))\xrightarrow{\Phi}\tilde{M}^{\rm op}\xrightarrow{E_{\tilde{P}}^{\rm op}} \tilde{P}^{\rm op}. \end{equation*} (For a general $z$, we have $z^{\rm op}(B^{\rm op}\otimes \mathbb{B}(L^2(C_h)))z^{\rm op}\rightarrow z(z\tilde{M}z)^{\rm op}\simeq (z\tilde{M}z)^{\rm op}\rightarrow (z\tilde{P}z)^{\rm op}.$) Finally composing with $\mathrm{Ad}(J_{\tilde{M}})=\mathrm{Ad}(J_B\otimes J_{C_h})$, we get a conditional expectation from $B\otimes \mathbb{B}(L^2(C_h))$ onto $\tilde{P}$. \end{proof}
Now assume that $L^\infty(\mathbb{G})$ is non amenable and $A\subset M$ is a Cartan subalgebra. Let $w$ be a central projection in $L^\infty(\mathbb{G})$ such that $L^\infty(\mathbb{G})w$ has no amenable direct summand. Write $z=1_B\otimes w$. Then $z\in A$ since $A$ is maximal abelian. Since $\tilde{A}$ is Cartan in $\tilde{M}$ (e.g.\ $\cite[\textrm{Subsection 2.3}]{HR10}$), we have $\tilde{M}=\tilde{P}$. The lemma above says that we have either (i) $Az\preceq_MB$ or (ii) there exists a conditional expectation from $B\otimes \mathbb{B}(wL^2(C_h))$ onto $z\tilde{M}z=B\otimes C_hw$. If (ii) holds, then composing with $\tau_B\otimes \mathrm{id}_{C_h}$, we get a conditional expectation from $\mathbb{B}(wL^2(C_h))$ onto $C_hw$. This is a contradiction since $C_hw=(L^\infty(\mathbb{G})w)\rtimes \mathbb{R}$ is non amenable. If (i) holds, then we have $Az\preceq_{Mz}B\otimes \mathbb{C}w$ since $z\in {\cal Z}(M)$, and hence $(B\otimes \mathbb{C}w)'\cap Mz \preceq_{Mz} A'\cap Mz$ by $\cite[\textrm{Lemma 3.5}]{Va08}$ (this is true for non finite $M$). This means $1_B \otimes L^\infty(\mathbb{G})w\preceq_{Mz} Az$ and hence $L^\infty(\mathbb{G})w$ has an amenable direct summand. Thus we get a contradiction.
Next assume that $B=\mathbb{C}$. Let $N\subset L^\infty(\mathbb{G})(=M)$ be a non-amenable subalgebra with expectation $E_N$ and assume that $A\subset N$ is a Cartan subalgebra with expectation $E_A$. Let $z$ be a central projection in $N$ such that $Nz$ is non-amenable and diffuse. Since $A$ is maximal abelian, $z\in A$ and $Az$ is a Cartan subalgebra in $Nz$. Hence $Az$ is diffuse, which means $Az\not\preceq_{M}\mathbb{C}$. The above lemma implies that there exists a conditional expectation from $\mathbb{B}(zL^2(C_h))$ onto $z{\cal N}_{\tilde{M}}(\tilde{A})''z={\cal N}_{z\tilde{M}z}(\tilde{A}z)''$. Now ${\cal N}_{z\tilde{M}z}(\tilde{A}z)''$ is non amenable and hence a contradiction. In fact, we have inclusions $C_{\tau_A}(A)z\subset C_{\tau_A\circ E_A}(N)z\subset zC_{\tau_A\circ E_A \circ E_N}(M)z$, and since the first inclusion is Cartan, the normalizer of $\tilde{A}z$ in $z\tilde{M}z$ generates a non-amenable subalgebra.
\section{\bf Further remarks}
\subsection{\bf Continuous and discrete cores}
In $\cite[\textrm{Subsection 5.2}]{Is12_2}$, we discussed semisolidity and non solidity of continuous cores of free quantum group $\rm III_1$ factors. Non solidity of them follow from non amenability of the relative commutant of the diffuse algebra $L\mathbb{R}$, which contains a non-amenable centralizer algebra.
Let $\mathbb{G}$ be a compact quantum group as in Theorem \ref{C} and assume that $L^\infty(\mathbb{G})$ is a full $\rm III_1$ factor and the Haar state is $\mathrm{Sd}(L^\infty(\mathbb{G}))$-almost periodic. Denote its discrete core by $D(L^\infty(\mathbb{G}))$ (see $\cite{Co74}\cite{Dy94}$). Then $qD(L^\infty(\mathbb{G}))q$ is always strongly solid for any trace finite projection $q$, since $D(L^\infty(\mathbb{G}))$ is isomorphic to $L^\infty(\mathbb{G})_h\otimes \mathbb{B}(H)$ for some separable Hilbert space $H$ and $L^\infty(\mathbb{G})_h$ is a strongly solid $\rm II_1$ factor (this follows from Theorem \ref{B}). Hence structures of these cores are different. In the subsection, we observe this difference from some viewpoints.
Write as $\Gamma\leq \mathbb{R}^*$ the Sd-invariant of $L^\infty(\mathbb{G})$. Then regarding $\mathbb{R}\subset \hat{\Gamma}$, take the crossed product $L^\infty(\mathbb{G})\rtimes \hat{\Gamma}$ by the extended modular action of the Haar state. This algebra is an explicit realization of the discrete core, namely, $D(L^\infty(\mathbb{G}))\simeq L^\infty(\mathbb{G})\rtimes \hat{\Gamma}$. Since this extended action is the modular action on the dense subgroup $\mathbb{R}$, we can easily verify that $L^\infty(\mathbb{G})\rtimes \hat{\Gamma} (=:M)$ satisfies the same condition as condition $\rm (AOC)^+$ replacing $\mathbb{R}$-action with $\hat{\Gamma}$-action. Under this setting, we can prove Theorem \ref{B}, namely, for any amenable subalgebra $A\subset qMq$ (with trace finite projection $q\in L\hat{\Gamma}$), we have either (i) $A\preceq_{M} L\hat{\Gamma}$ or (ii) ${\cal N}_{qMq}(A)''$ is amenable. This obviously implies strong solidity of $qMq$, since all diffuse subalgebra $A$ automatically satisfies $A\not\preceq_{qMq}L\hat{\Gamma}$ (because $L\hat{\Gamma}$ is atomic).
Roughly speaking, this observation implies that the only obstruction of the solidity of $C_h(L^\infty(\mathbb{G}))(=:C_h)$ is a diffuse subalgebra $L\mathbb{R}$. More precisely, $C_h$ is non solid since the subalgebra $L\mathbb{R}$ is diffuse, and $D(L^\infty(\mathbb{G}))$ is strongly solid since the subalgebra $L\hat{\Gamma}$ is atomic.
The next view point is from property Gamma (or non fullness). It is known that a $\rm II_1$ factor $M$ has the property Gamma if and only if $C^*\{M,M' \}\cap \mathbb{K}(L^2(M))=0$ $\cite{aaa}$. Moreover it is not difficult to see that a von Neumann algebra $M$ satisfies condition (AO) and $C^*\{M,M' \}\cap \mathbb{K}(L^2(M))=0$, then it is amenable. Hence any non amenable $\rm II_1$ factor can not have property Gamma and condition (AO) at the same time. When $L^\infty(\mathbb{G})$ is a full $\rm III_1$ factor, on the one hand, for any $\mathrm{Tr}$-finite projection $p$, $pC_hp$ is non full, since it admits a non trivial central sequence (use the almost periodicity of $h$). Hence we can deduce $C^*\{C_h,C_h' \}\cap \mathbb{K}(L^2(C_h)=0$. In particular $C_h$ (and $pC_hp$ for \textit{any} projection $p$) does not satisfy condition (AO). On the other hand, for the discrete core $D(L^\infty(\mathbb{G}))$, $qD(L^\infty(\mathbb{G}))q$ always satisfies condition $\rm (AO)^+$ for any projection $q\in L\hat{\Gamma}\simeq \ell^\infty(\Gamma)$ with finite support. This is because $D(L^\infty(\mathbb{G}))$ satisfies condition $\rm (AO)^+$ with respect to the quotient $\mathbb{K}(L^2(\mathbb{G}))\otimes \mathbb{B}(\ell^2(\Gamma))$ and hence $qD(L^\infty(\mathbb{G}))q$ satisfies $\rm (AO)^+$ for $\mathbb{K}(L^2(\mathbb{G}))\otimes q\mathbb{B}(\ell^2(\Gamma))q=\mathbb{K}(L^2(\mathbb{G}))\otimes \mathbb{K}(q\ell^2(\Gamma))$. By $\cite[\textrm{Theorem B}]{Is12_2}$, we again get the strong solidity of $qD(L^\infty(\mathbb{G}))q$. We also get fullness of $qD(L^\infty(\mathbb{G}))q$ (since this is non amenable) for any $q\in \ell^\infty(\Gamma)$ with finite support and hence that of $D(L^\infty(\mathbb{G}))$.
\subsection{\bf Primeness of crossed products by bi-exact quantum groups}
In this subsection, we consider only compact quantum groups of Kac type. Let $M=\hat{\mathbb{G}}\ltimes B$ be a finite von Neumann algebra as in the statement of Theorem \ref{A}. Assume that $B$ is amenable. Then we have for any amenable subalgebra $A\subset qMq$ $(q\in M$ is a projection), we have either (i) $A\preceq_M B$ or (ii) ${\cal N}_{qMq(A)''}$ is amenable. This is a sufficient condition to semisolidity when $B$ is abelian, and to semiprimeness, which means that any tensor decomposition has an amenable tensor component, when $B$ is non abelian. Thus we proved that any non amenable von Neumann subalgebra $N\subset p(\hat{\mathbb{G}}\ltimes B)p$ is prime when $B$ is abelian, and semiprime when $B$ is amenable.
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Preference-based Teaching}
\titlerunning{Preference-based Teaching}
\author{Ziyuan Gao$^1$ \and Christoph Ries$^2$ \and Hans Ulrich Simon$^2$ \and Sandra Zilles$^1$} \authorrunning{Z.~Gao, C.~Ries, H.~Simon and S.~Zilles}
\institute{Department of Computer Science\\University of Regina, Regina, SK, Canada S4S 0A2 \\\email{\{gao257,zilles\}@cs.uregina.ca} \and Department of Mathematics, \\ Ruhr-Universit\"{a}t Bochum, D-44780 Bochum, Germany\\\email{\{christoph.ries,hans.simon\}@rub.de}}
\maketitle
\begin{abstract} We introduce a new model of teaching named ``preference-based \linebreak[4]teaching'' and a corresponding complexity parameter---the preference-based \linebreak[4]teaching dimension (PBTD)---representing the worst-case number of examples needed to teach any concept in a given concept class. Although the PBTD coincides with the well-known recursive teaching dimension (RTD) on finite classes, it is radically different on infinite ones: the RTD becomes infinite already for trivial infinite classes (such as half-intervals) whereas the PBTD evaluates to reasonably small values for a wide collection of infinite classes including classes consisting of so-called closed sets w.r.t.~a given closure operator, including various classes related to linear sets over ${\mathbb N}_0$ (whose RTD had been studied quite recently) and including the class of Euclidean half-spaces. On top of presenting these concrete results, we provide the reader with a theoretical framework (of a combinatorial flavor) which helps to derive bounds on the PBTD. \end{abstract}
\begin{keywords} teaching dimension, preference relation, recursive teaching dimension,\linebreak[4] learning halfspaces, linear sets \end{keywords}
\section{Introduction} \label{sec:introduction}
The classical model of teaching~\cite{SM1991,GK1995} formulates the following interaction protocol between a teacher and a student: \begin{itemize} \item Both of them agree on a ``classification-rule system'', formally given by a concept class ${\mathcal{L}}$. \item In order to teach a specific concept $L \in {\mathcal{L}}$, the teacher presents to the student a \emph{teaching set}, i.e., a set $T$ of labeled examples so that $L$ is the only concept in ${\mathcal{L}}$ that is consistent with $T$. \item The student determines $L$ as the unique concept in ${\mathcal{L}}$ that is consistent with $T$. \end{itemize}
Goldman and Mathias~\cite{GM1996} pointed out that this model of teaching is not powerful enough, since the teacher is required to make \emph{any\/} consistent learner successful. A challenge is to model powerful teacher/student interactions without enabling unfair ``coding tricks''. Intuitively, the term ``coding trick'' refers to any form of undesirable collusion between teacher and learner, which would reduce the learning process to a mere decoding of a code the teacher sent to the learner. There is no generally accepted definition of what constitutes a coding trick, in part because teaching an exact learner could always be considered coding to some extent: the teacher presents a set of examples which the learner ``decodes'' into a concept.
In this paper, we adopt the notion of ``valid teacher/learner pair'' introduced by \cite{GM1996}. They consider their model to be intuitively free of coding tricks while it provably allows for a much broader class of interaction protocols than the original teaching model. In particular, teaching may thus become more efficient in terms of the number of examples in the teaching sets. Further definitions of how to avoid unfair coding tricks have been suggested~\cite{ZLHZ2011}, but they were less stringent than the one proposed by Goldman and Mathias. The latter simply requests that, if the learner hypothesizes concept $L$ upon seeing a sample set $S$ of labeled examples, then the learner will still hypothesize $L$ when presented with any sample set $S\cup S'$, where $S'$ contains only examples labeled consistently with $L$. A coding trick would then be any form of exchange between the teacher and the learner that does not satisfy this definition of validity.
The model of recursive teaching~\cite{ZLHZ2011,MGZ2014}, which is free of coding tricks according to the Goldman-Mathias definition, has recently gained attention because its complexity parameter, the recursive teaching dimension (RTD), has shown relations to the VC-dimension and to sample compression~\cite{ChenCT16,DFSZ2014,MSWY2015,SZ2015}, when focusing on finite concept classes. Below though we will give examples of rather simple infinite concept classes with infinite RTD, suggesting that the RTD is inadequate for addressing the complexity of teaching infinite classes.
In this paper, we introduce a model called \emph{preference-based teaching}, in which the teacher and the student do not only agree on a classification-rule system ${\mathcal{L}}$ but also on a preference relation (a strict partial order) imposed on ${\mathcal{L}}$. If the labeled examples presented by the teacher allow for several consistent explanations (= consistent concepts) in ${\mathcal{L}}$, the student will choose a concept $L \in {\mathcal{L}}$ that she prefers most. This gives more flexibility to the teacher than the classical model: the set of labeled examples need not distinguish a target concept $L$ from any other concept in ${\mathcal{L}}$ but only from those concepts $L'$ over which $L$ is not preferred.\footnote{Such a preference relation can be thought of as a kind of bias in learning: the student is ``biased'' towards concepts that are preferred over others, and the teacher, knowing the student's bias, selects teaching sets accordingly.} At the same time, preference-based teaching yields valid teacher/learner pairs according to Goldman and Mathias's definition. We will show that the new model, despite avoiding coding tricks, is quite powerful. Moreover, as we will see in the course of the paper, it often allows for a very natural design of teaching sets.
Assume teacher and student choose a preference relation that minimizes the worst-case number $M$ of examples required for teaching any concept in the class $\mathcal{L}$. This number $M$ is then called the preference-based teaching dimension (PBTD) of $\mathcal{L}$. In particular, we will show the following:
(i) Recursive teaching is a special case of preference-based teaching where the preference relation satisfies a so-called ``finite-depth condition''. It is precisely this additional condition that renders recursive teaching useless for many natural and apparently simple infinite concept classes. Preference-based teaching successfully addresses these shortcomings of recursive teaching, see Section~\ref{sec:rtd}. For finite classes, PBTD and RTD are equal.
(ii) A wide collection of geometric and algebraic concept classes with infinite RTD can be taught very efficiently, i.e., with low PBTD. To establish such results, we show in Section~\ref{sec:closure-operator} that spanning sets can be used as preference-based teaching sets with positive examples only --- a result that is very simple to obtain but quite useful.
(iii) In the preference-based model, linear sets over ${\mathbb N}_0$ with origin 0 and at most $k$ generators can be taught with $k$ positive examples, while recursive teaching with a bounded number of positive examples was previously shown to be impossible and it is unknown whether recursive teaching with a bounded number of positive and negative examples is possible for $k \geq 4$. We also give some almost matching upper and lower bounds on the PBTD for other classes of linear sets, see Section~\ref{sec:linsets}.
(iv) The PBTD of halfspaces in $\mathbbm{R}^d$ is upper-bounded by $6$, independent of the dimensionality $d$ (see Section~\ref{sec:halfspaces}), while its RTD is infinite.
(v) We give full characterizations of concept classes that can be taught with only one example (or with only one example, which is positive) in the preference-based model (see Section~\ref{sec:pbtd1}).
Based on our results and the naturalness of the teaching sets and preference relations used in their proofs, we claim that
preference-based teaching is far more suitable to the study of infinite concept classes than recursive teaching.
Parts of this paper were published in a previous conference version~\cite{GRSZ2016}.
\section{Basic Definitions and Facts} \label{sec:definitions}
${\mathbb N}_0$ denotes the set of all non-negative integers and ${\mathbb N}$ denotes the set of all positive integers. A {\em concept class} ${\mathcal{L}}$ is a family of subsets over a universe ${\mathcal{X}}$, i.e., ${\mathcal{L}} \subseteq 2^{\mathcal{X}}$ where $2^{\mathcal{X}}$ denotes the powerset of ${\mathcal{X}}$. The elements of ${\mathcal{L}}$ are called {\em concepts}. A {\em labeled example} is an element of ${\mathcal{X}} \times\{-,+\}$. We slightly deviate from this notation in Section~\ref{sec:halfspaces}, where our treatment of halfspaces makes it more convenient to use $\{-1,1\}$ instead of $\{-,+\}$, and in Section~\ref{sec:pbtd1}, where we perform Boolean operations on the labels and therefore use $\{0,1\}$ instead of $\{-,+\}$. Elements of ${\mathcal{X}}$ are called {\em examples}. Suppose that $T$ is a set of labeled examples. Let $T^+ = \{x\in{\mathcal{X}} : (x,+) \in T\}$ and $T^- = \{x\in{\mathcal{X}} : (x,-) \in T\}$. A set $L \subseteq{\mathcal{X}}$ is {\em consistent with $T$} if it includes all examples in $T$ that are labeled ``$+$'' and excludes all examples in $T$ that are labeled ``$-$'', i.e, if $T^+ \subseteq L$ and $T^- \cap L = \emptyset$. A set of labeled examples that is consistent with $L$ but not with $L'$ is said to {\em distinguish $L$ from $L'$}. The classical model of teaching is then defined as follows.
\begin{definition}[\cite{SM1991,GK1995}] \label{def:classical-model} A {\em teaching set} for a concept $L \in {\mathcal{L}}$ w.r.t.~${\mathcal{L}}$ is a set $T$ of labeled examples such that $L$ is the only concept in ${\mathcal{L}}$ that is consistent with $T$, i.e., $T$ distinguishes $L$ from any other concept in ${\mathcal{L}}$. Define
$\mathrm{TD}(L,{\mathcal{L}}) = \inf\{|T| : T\mbox{ is a teaching\ }$ $\mbox{set for $L$ w.r.t.~${\mathcal{L}}$}\}$. i.e., $\mathrm{TD}(L,{\mathcal{L}})$ is the smallest possible size of a teaching set for $L$ w.r.t.~${\mathcal{L}}$. If $L$ has no finite teaching set w.r.t.~${\mathcal{L}}$, then $\mathrm{TD}(L,{\mathcal{L}})=\infty$. The number $\mathrm{TD}({\mathcal{L}}) = \sup_{L \in {\mathcal{L}}}\mathrm{TD}(L,{\mathcal{L}}) \in {\mathbb N}_0\cup\{\infty\}$ is called the {\em teaching dimension of ${\mathcal{L}}$}. \end{definition}
For technical reasons, we will occasionally deal with the number $\mathrm{TD}_{min}({\mathcal{L}}) = \inf_{L \in {\mathcal{L}}}\mathrm{TD}(L,$ ${\mathcal{L}})$, i.e., the number of examples needed to teach the concept from ${\mathcal{L}}$ that is easiest to teach.
In this paper, we will examine a teaching model in which the teacher and the student do not only agree on a classification-rule system ${\mathcal{L}}$ but also on a preference relation, denoted as $\prec$, imposed on ${\mathcal{L}}$. We assume that $\prec$ is a {\em strict partial order} on ${\mathcal{L}}$, i.e., $\prec$ is asymmetric and transitive.
The partial order that makes every pair $L \neq L' \in {\mathcal{L}}$ incomparable is denoted by $\prec_\emptyset$. For every $L \in{\mathcal{L}}$, let \[ {\mathcal{L}}_{\prec L} = \{L'\in{\mathcal{L}}: L' \prec L\} \] be the set of concepts over which $L$ is strictly preferred. Note that ${\mathcal{L}}_{\prec_\emptyset L} = \emptyset$ for every $L\in{\mathcal{L}}$.
As already noted above, a teaching set $T$ of $L$ w.r.t.~${\mathcal{L}}$ distinguishes $L$ from any other concept in ${\mathcal{L}}$. If a preference relation comes into play, then $T$ will be exempted from the obligation to distinguish $L$ from the concepts in ${\mathcal{L}}_{\prec L}$ because $L$ is strictly preferred over them anyway.
\begin{definition} \label{def:td-prec-L} A {\em teaching set for $L \subseteq X$ w.r.t.~$({\mathcal{L}},\prec)$} is defined as a teaching set for $L$ w.r.t.~${\mathcal{L}}\setminus{\mathcal{L}}_{\prec L}$. Furthermore define \[
\mathrm{PBTD}(L,{\mathcal{L}},\prec) = \inf\{|T| : T\mbox{ is a teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec$})\} \in {\mathbb N}_0\cup\{\infty\} \enspace . \] The number $\mathrm{PBTD}({\mathcal{L}},\prec) = \sup_{L \in {\mathcal{L}}}\mathrm{PBTD}(L,{\mathcal{L}},\prec) \in {\mathbb N}_0\cup\{\infty\}$ is called the {\em teaching dimension of $({\mathcal{L}},\prec)$}. \end{definition} Definition~\ref{def:td-prec-L} implies that \begin{equation} \label{eq:td-prec-L} \mathrm{PBTD}(L,{\mathcal{L}},\prec) = \mathrm{TD}(L,{\mathcal{L}}\setminus{\mathcal{L}}_{\prec L}) \enspace . \end{equation} Let $L \mapsto T(L)$ be a mapping that assigns a teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec)$ to every $L\in{\mathcal{L}}$. It is obvious from Definition~\ref{def:td-prec-L} that $T$ must be injective, i.e., $T(L) \neq T(L')$ if $L$ and $L$' are distinct concepts from ${\mathcal{L}}$. The classical model of teaching is obtained from the model described in Definition~\ref{def:td-prec-L} when we plug in the empty preference relation $\prec_\emptyset$ for $\prec$. In particular, $\mathrm{PBTD}({\mathcal{L}},\prec_\emptyset)$ $= \mathrm{TD}({\mathcal{L}})$.
\iffalse To avoid unfair coding tricks, \cite{GM1996} required that the student identify a concept $L\in{\mathcal{L}}$ even from any superset $S\supseteq T(L)$ of the set $T(L)$ presented by the teacher, as long as $L$ is consistent with $S$. It is easy to see that preference-based teaching with respect to a fixed preference relation fulfills this requirement. \fi
We are interested in finding the partial order that is optimal for the purpose of teaching and we aim at determining the corresponding teaching dimension. This motivates the following notion: \begin{definition} \label{def:td-best-le} The {\em preference-based teaching dimension of ${\mathcal{L}}$} is given by \[ \mathrm{PBTD}({\mathcal{L}}) = \inf\{\mathrm{PBTD}({\mathcal{L}},\prec) : \mbox{$\prec$ is a strict partial order on ${\mathcal{L}}$}\} \enspace . \] \end{definition}
A relation $R'$ on ${\mathcal{L}}$ is said to be an {\em extension of a relation $R$} if $R \subseteq R'$. The {\em order-extension principle} states that any partial order has a linear extension \cite{Jech-1973}. The following result (whose second assertion follows from the first one in combination with the order-extension principle) is pretty obvious:
\begin{lemma} \label{lem:extension} \begin{enumerate} \item Suppose that $\prec'$ extends $\prec$. If $T$ is a teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec)$, then $T$ is a teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec')$. Moreover $\mathrm{PBTD}({\mathcal{L}},\prec') \le \mathrm{PBTD}({\mathcal{L}},$ $\prec)$. \item $\mathrm{PBTD}({\mathcal{L}}) = \inf\{\mathrm{PBTD}({\mathcal{L}},\prec) : \mbox{$\prec$ is a strict linear order on ${\mathcal{L}}$}\}$. \end{enumerate} \end{lemma}
Recall that Goldman and Mathias \cite{GM1996} suggested to avoid coding tricks by requesting that any superset $S$ of a teaching set for a concept $L$ remains a teaching set, if $S$ is consistent with $L$. This property is obviously satisfied in preference-based teaching. A preference-based teaching set needs to distinguish a concept $L$ from all concepts in ${\mathcal{L}}$ that are preferred over $L$. Adding more labeled examples from $L$ to such a teaching set will still result in a set distinguishing $L$ from all concepts in ${\mathcal{L}}$ that are preferred over $L$.
\paragraph{Preference-based teaching with positive examples only.}
Suppose that ${\mathcal{L}}$ contains two concepts $L,L'$ such that $L \subset L'$. In the classical teaching model, any teaching set for $L$ w.r.t.~${\mathcal{L}}$ has to employ a negative example in order to distinguish $L$ from $L'$. Symmetrically, any teaching set for $L'$ w.r.t.~${\mathcal{L}}$ has to employ a positive example. Thus classical teaching cannot be performed with one type of examples only unless ${\mathcal{L}}$ is an antichain w.r.t.~inclusion. As for preference-based teaching, the restriction to one type of examples is much less severe, as our results below will show.
A teaching set $T$ for $L \in {\mathcal{L}}$ w.r.t.~$({\mathcal{L}},\prec)$ is said to be {\em positive} if it does not make use of negatively labeled examples, i.e., if $T^- = \emptyset$. In the sequel, we will occasionally identify a positive teaching set $T$ with $T^+$. A positive teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec)$ can clearly not distinguish $L$ from a proper superset of $L$ in ${\mathcal{L}}$. Thus, the following holds:
\begin{lemma} \label{lem:ts-pos} Suppose that $L \mapsto T^+(L)$ maps each $L \in {\mathcal{L}}$ to a positive teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec)$. Then $\prec$ must be an extension of $\supset$ (so that proper subsets of a set $L$ are strictly preferred over $L$) and, for every $L \in {\mathcal{L}}$, the set $T^+(L)$ must distinguish $L$ from every proper subset of $L$ in ${\mathcal{L}}$. \end{lemma}
\noindent Define \begin{equation} \label{eq:td-plus-L}
\mathrm{PBTD}^+(L,{\mathcal{L}},\prec) = \inf\{|T| : T\mbox{ is a positive teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec$})\} \enspace . \end{equation} The number $\mathrm{PBTD}^+({\mathcal{L}},\prec) = \sup_{L \in {\mathcal{L}}}\mathrm{PBTD}^+(L,{\mathcal{L}},\prec)$ (possibly $\infty$) is called the {\em positive teaching dimension of $({\mathcal{L}},\prec)$}. The {\em positive preference-based teaching dimension of ${\mathcal{L}}$} is then given by \begin{equation} \label{eq:td-plus-cL} \mathrm{PBTD}^+({\mathcal{L}}) = \inf\{\mathrm{PBTD}^+({\mathcal{L}},\prec) : \mbox{$\prec$ is a strict partial order on ${\mathcal{L}}$}\} \enspace . \end{equation}
\paragraph{Monotonicity.}
A complexity measure $K$ that assigns a number $K({\mathcal{L}}) \in {\mathbb N}_0$ to a concept class ${\mathcal{L}}$ is said to be {\em monotonic} if ${\mathcal{L}}' \subseteq {\mathcal{L}}$ implies that $K({\mathcal{L}}') \le K({\mathcal{L}})$. It is well known (and trivial to see) that $\mathrm{TD}$ is monotonic. It is fairly obvious that $\mathrm{PBTD}$ is monotonic, too:
\begin{lemma} \label{lem:monotonicity} $\mathrm{PBTD}$ and $\mathrm{PBTD}^+$ are monotonic. \end{lemma}
\noindent As an application of monotonicity, we show the following result:
\begin{lemma} \label{lem:lb-tdmin} For every finite subclass ${\mathcal{L}}'$ of ${\mathcal{L}}$, we have $ \mathrm{PBTD}({\mathcal{L}}) \ge \mathrm{PBTD}({\mathcal{L}}') \ge \mathrm{TD}_{min}({\mathcal{L}}') $. \end{lemma}
\begin{proof} The first inequality holds because $\mathrm{PBTD}$ is monotonic. The second inequality follows from the fact that a finite partially ordered set must contain a minimal element. Thus, for any fixed choice of $\prec$, ${\mathcal{L}}'$ must contain a concept $L'$ such that ${\mathcal{L}}'_{\prec L'} = \emptyset$. Hence, \[ \mathrm{PBTD}({\mathcal{L}}',\prec) \ge \mathrm{PBTD}(L',{\mathcal{L}}',\prec) \stackrel{(\ref{eq:td-prec-L})}{=} \mathrm{TD}(L',{\mathcal{L}}'\setminus{\mathcal{L}}'_{\prec L'}) = \mathrm{TD}(L',{\mathcal{L}}') \ge \mathrm{TD}_{min}({\mathcal{L}}') \enspace . \] Since this holds for any choice of $\prec$, we get $\mathrm{PBTD}({\mathcal{L}}') \ge \mathrm{TD}_{min}({\mathcal{L}}')$, as desired. \end{proof}
\section{Preference-based versus Recursive Teaching} \label{sec:rtd}
The preference-based teaching dimension is a relative of the recursive teaching dimension. In fact, both notions coincide on finite classes, as we will see shortly. We first recall the definitions of the recursive teaching dimension and of some related notions~\cite{ZLHZ2011,MGZ2014}.
A {\em teaching sequence for ${\mathcal{L}}$} is a sequence of the form ${\mathcal{S}} = ({\mathcal{L}}_i,d_i)_{i\ge1}$ where ${\mathcal{L}}_1,{\mathcal{L}}_2,{\mathcal{L}}_3,\ldots$ form a partition of ${\mathcal{L}}$ into non-empty sub-classes and, for every $i\ge1$, we have that \begin{equation} \label{eq:rtd} d_i = \sup_{L\in{\mathcal{L}}_i}\mathrm{TD}\left(L,{\mathcal{L}}\setminus\cup_{j=1}^{i-1}{\mathcal{L}}_j\right) \enspace . \end{equation} If, for every $i\ge 1$, $d_i$ is the supremum over all $L \in {\mathcal{L}}_i$ of the smallest size of a \emph{positive teaching set} for $L$ w.r.t.\ $\cup_{j\geq i}{\mathcal{L}}_j$ (and $d_i = \infty$ if some $L \in {\mathcal{L}}_i$ does not have a positive teaching set w.r.t.\ $\cup_{j\ge i}{\mathcal{L}}_j$), then ${\mathcal{S}}$ is said to be a \emph{positive teaching sequence for ${\mathcal{L}}$}. The {\em order} of a teaching sequence or a positive teaching sequence ${\mathcal{S}}$ (possibly $\infty$) is defined as $\mathrm{ord}({\mathcal{S}}) = \sup_{i\ge1}d_i$. The {\em recursive teaching dimension of ${\mathcal{L}}$} (possibly $\infty$) is defined as the order of the teaching sequence of lowest order for ${\mathcal{L}}$. More formally, $\mathrm{RTD}({\mathcal{L}}) = \inf_{{\mathcal{S}}}\mathrm{ord}({\mathcal{S}})$ where ${\mathcal{S}}$ ranges over all teaching sequences for ${\mathcal{L}}$. Similarly, $\mathrm{RTD}^+({\mathcal{L}}) = \inf_{{\mathcal{S}}}\mathrm{ord}({\mathcal{S}})$, where ${\mathcal{S}}$ ranges over all positive teaching sequences for ${\mathcal{L}}$. Note that the following holds for every ${\mathcal{L}}' \subseteq {\mathcal{L}}$ and for every teaching sequence ${\mathcal{S}} = ({\mathcal{L}}_i,d_i)_{i\ge1}$ for ${\mathcal{L}}'$ such that $\mathrm{ord}({\mathcal{S}}) = \mathrm{RTD}({\mathcal{L}}')$: \begin{equation} \label{eq:rtd-tdmin} \mathrm{RTD}({\mathcal{L}}) \ge \mathrm{RTD}({\mathcal{L}}') = \mathrm{ord}({\mathcal{S}}) \ge d_1 = \sup_{L\in{\mathcal{L}}_1}\mathrm{TD}(L,{\mathcal{L}}') \ge \mathrm{TD}_{min}({\mathcal{L}}') \enspace . \end{equation}
Note an important difference between $\mathrm{PBTD}$ and $\mathrm{RTD}$: while $\mathrm{RTD}({\mathcal{L}}) \ge$ $\mathrm{TD}_{min}$ $({\mathcal{L}}')$ for \emph{all}\/ ${\mathcal{L}}'\subseteq{\mathcal{L}}$, in general the same holds for $\mathrm{PBTD}$ only when restricted to finite ${\mathcal{L}}'$, cf.\ Lemma~\ref{lem:lb-tdmin}. This difference will become evident in the proof of Lemma~\ref{lem:huge-gap}.
The {\em depth} of $L \in {\mathcal{L}}$ w.r.t.~a strict partial order imposed on ${\mathcal{L}}$ is defined as the length of the longest chain in $({\mathcal{L}},\prec)$ that ends with the $\prec$-maximal element $L$ (resp.~as $\infty$ if there is no bound on the length of these chains). The recursive teaching dimension is related to the preference-based teaching dimension as follows:
\begin{lemma} \label{lem:rtd-pbtd} $\mathrm{RTD}({\mathcal{L}}) = \inf_{\prec}\mathrm{PBTD}({\mathcal{L}},\prec)$ and $\mathrm{RTD}^+({\mathcal{L}}) = \inf_{\prec}\mathrm{PBTD}^+({\mathcal{L}},\prec)$ where $\prec$ ranges over all strict partial orders on ${\mathcal{L}}$ that satisfy the following ``finite-depth condition'': every $L \in {\mathcal{L}}$ has a finite depth w.r.t.~$\prec$. \end{lemma}
The following is an immediate consequence of Lemma~\ref{lem:rtd-pbtd} and the trivial observation that the finite-depth condition is always satisfied if ${\mathcal{L}}$ is finite:
\begin{corollary} \label{cor:rtd-pb} $\mathrm{PBTD}({\mathcal{L}}) \le \mathrm{RTD}({\mathcal{L}})$, with equality if ${\mathcal{L}}$ is finite. \end{corollary}
\noindent While $\mathrm{PBTD}({\mathcal{L}})$ and $\mathrm{RTD}({\mathcal{L}})$ refer to the same finite number when ${\mathcal{L}}$ is finite, there are classes for which $\mathrm{RTD}$ is finite and yet larger than $\mathrm{PBTD}$, as Lemma~\ref{lem:huge-gap} will show. Generally, for infinite classes, the gap between $\mathrm{PBTD}$ and $\mathrm{RTD}$ can be arbitrarily large:
\begin{lemma} \label{lem:huge-gap} There exists an infinite class ${\mathcal{L}}_\infty$ of VC-dimension $1$ such that $\mathrm{PBTD}^+$ $({\mathcal{L}}_\infty)$ $=1$ and $\mathrm{RTD}({\mathcal{L}}_\infty)=\infty$. Moreover, for every $k\ge1$, there exists an infinite class ${\mathcal{L}}_k$ such that $\mathrm{PBTD}^+({\mathcal{L}}_k)=1$ and $\mathrm{RTD}({\mathcal{L}}_k)=k$. \end{lemma}
\iffalse The complete proof of Lemma~\ref{lem:huge-gap} is given in Appendix~\ref{app:rtd}. Here we only specify the classes ${\mathcal{L}}_\infty$ and ${\mathcal{L}}_k$ that are employed in this proof: \begin{itemize} \item Choose ${\mathcal{L}}_\infty$ as the class of half-intervals $[0,a]$, where $0 \le a < 1$, over the universe $[0,1)$.\footnote{$\mathrm{RTD}({\mathcal{L}}_\infty)=\infty$ had been observed by~\cite{MSWY2015} already.} \item Let ${\mathcal{X}} = [0,2)$. For each $a = \sum_{n \ge 1}\alpha_n2^{-n}\in [0,1)$ and for all $i=1,\ldots,k$, let $1 \le a_i < 2$ be given by $a_i = 1+\sum_{n \ge 0}\alpha_{kn+i}2^{-n}$. Finally, let $I_a = [0,a) \cup \{a_1,\ldots,a_k\} \subseteq {\mathcal{X}}$ and let ${\mathcal{L}}_k = \{I_a: 0 \le a < 1\}$. \end{itemize}. \fi
\begin{proof} We first show that there exists a class of VC-dimension $1$, say ${\mathcal{L}}_\infty$, such that $\mathrm{PBTD}^+({\mathcal{L}}_\infty)=1$ while $\mathrm{RTD}({\mathcal{L}}_\infty)=\infty$. To this end, let ${\mathcal{L}}_\infty$ be the family of closed half-intervals over $[0,1)$, i.e., ${\mathcal{L}}_\infty = \{ [0,a]: 0 \le a < 1\}$. We first prove that $\mathrm{PBTD}^+({\mathcal{L}}_\infty) = 1$. Consider the preference relation given by $[0,b] \prec [0,a]$ iff $a<b$. Then, for each $0 \le a <1$, we have \[ \mathrm{PBTD}([0,a],{\mathcal{L}}_\infty,\prec) \stackrel{(\ref{eq:td-prec-L})}{=} \mathrm{TD}([0,a],\{[0,b]:\ 0 \le b \le a\}) = 1 \] because the single example $(a,+)$ suffices for distinguishing $[0,a]$ from any interval $[0,b]$ with $b<a$.
It was observed by~\cite{MSWY2015} already that $\mathrm{RTD}({\mathcal{L}}_\infty)=\infty$ because every teaching set for some $[0,a]$ must contain an infinite sequence of distinct reals that converges from above to $a$. Thus, using Equation~(\ref{eq:rtd-tdmin}) with ${\mathcal{L}}'={\mathcal{L}}$, we have $\mathrm{RTD}({\mathcal{L}}_\infty) \ge \mathrm{TD}_{min}({\mathcal{L}}_\infty) = \infty$.
Next we show that, for every $k\ge1$, there exists a class, say ${\mathcal{L}}_k$, such that $\mathrm{PBTD}^+$ $({\mathcal{L}}_k)=1$ while $\mathrm{RTD}({\mathcal{L}}_k)=k$. To this end, let ${\mathcal{X}} = [0,2)$. For each $a \in [0,1)$, fix a binary representation $\sum_{n \ge 1}\alpha_n2^{-n}$ of $a$,
where $\alpha_n\in\{0,1\}$ are binary coefficients, and for all $i=1,\ldots,k$, let $1 \le a_i < 2$ be given by $a_i = 1+\sum_{n \ge 0}\alpha_{kn+i}2^{-kn+i}$.\footnote{Note that, for $a=\frac{m}{2^N}$ with $m,N\in\mathbb{N}$, there are two binary representations. We can pick either one to define the $\alpha_n$ and $a_i$ values.} Let $A$ be the set of all $a \in [0,1)$ such that if $\sum_{n \ge 1}\alpha_n2^{-n}$ is the binary representation of $a$ fixed earlier, then for all $i \in \{1,\ldots,k\}$, there is some $n \geq 0$ for which $\alpha_{nk+i} \neq 0$. Finally, let $I_a = [0,a] \cup \{a_1,\ldots,a_k\} \subseteq {\mathcal{X}}$ and let ${\mathcal{L}}_k = \{I_a: 0 \le a < 1 \wedge a \in A\}$.
Clearly $\mathrm{PBTD}^+({\mathcal{L}}_k)=1$ because, using the preference relation given by $I_b \prec I_a$ iff $a<b$, we can teach $I_a$ w.r.t.~${\mathcal{L}}_k$ by presenting the single example $(a,+)$ (the same strategy as for half-intervals). Moreover, note that $I_a$ is the only concept in ${\mathcal{L}}_k$ that contains $a_1,\ldots,a_k$, i.e., $\{a_1,\ldots,a_k\}$ is a positive teaching set for $I_a$ w.r.t.~${\mathcal{L}}_k$. It follows that $\mathrm{RTD}({\mathcal{L}}_k) \le \mathrm{TD}({\mathcal{L}}_k) \le k$. It remains to show that $\mathrm{RTD}({\mathcal{L}}_k) \ge k$. To this end, we consider the subclass ${\mathcal{L}}'_k$ consisting of all concepts $I_a$ such that $a \in A$ and $a$ has only finitely many $1$'s in its binary representation $(\alpha_n)_{n\in\mathbb{N}}$, i.e., all but finitely many of the $\alpha_n$ are zero. Pick any concept $I_a \in {\mathcal{L}}'_k$. Let $T$ be any set of at most $k-1$ examples labeled consistently according to $I_a$. At least one of the positive examples $a_1,\ldots,a_k$ must be missing, say $a_i$ is missing. Let $J_{a,i}$ be the set of indices given by $J_{a,i} = \{n\in{\mathbb N}_0:\ \alpha_{kn+i}=0\}$. The following observations show that there exists some $a' \in {\mathcal{X}}\setminus\{a\}$ such that $I_{a'}$ is consistent with $T$. \begin{itemize} \item When we set some (at least one but only finitely many) of the bits $\alpha_{kn+i}$ with $n \in J_{a,i}$ from $0$ to $1$ (while keeping fixed the remaining bits of the binary representation of $a$), then we obtain a number $a' \neq a$ such that $I_{a'}$ is still consistent with all positive examples in $T$ (including the example $(a,+)$ which might be in $T$). \item Note that $J_{a,i}$ is an infinite set. It is therefore possible to choose the bits that are set from $0$ to $1$ in such a fashion that the finitely many bit patterns represented by the numbers in $T^- \cap [1,2)$ are avoided. \item It is furthermore possible to choose the bits that are set from $0$ to $1$ in such a fashion that the resulting number $a'$ is as close to $a$ as we like so that $I_{a'}$ is also consistent with the negative examples from $T^- \cap[0,1)$ and $a' \in A$. \end{itemize} It follows from this reasoning that no set with less than $k$ examples can possibly be a teaching set for~$I_a$. Since this holds for an arbitrary choice of $a$, we may conclude that $\mathrm{RTD}({\mathcal{L}}_k) \ge \mathrm{RTD}({\mathcal{L}}'_k) \ge \mathrm{TD}_{min}({\mathcal{L}}'_k) = k$. \end{proof}
\section{Preference-based Teaching with Positive Examples Only} \label{sec:closure-operator}
The main purpose of this section is to relate positive preference-based teaching to ``spanning sets" and ``closure operators", which are well-studied concepts in the computational learning theory literature. Let ${\mathcal{L}}$ be a concept class over the universe ${\mathcal{X}}$. We say that $S\subseteq{\mathcal{X}}$ is a {\em spanning set} of $L\in{\mathcal{L}}$ w.r.t.~${\mathcal{L}}$ if $S \subseteq L$ and any set in ${\mathcal{L}}$ that contains $S$ must contain $L$ as well.\footnote{This generalizes the classical definition of a spanning set~\cite{HSW1990}, which is given w.r.t.~intersection-closed classes only.} In other words, $L$ is the unique smallest concept in ${\mathcal{L}}$ that contains $S$. We say that $S\subseteq{\mathcal{X}}$ is a {\em weak spanning set} of $L\in{\mathcal{L}}$ w.r.t.~${\mathcal{L}}$ if $S\subseteq L$ and $S$ is not contained in any proper subset of $L$ in ${\mathcal{L}}$.\footnote{Weak spanning sets have been used in the field of recursion-theoretic inductive inference under the name ``tell-tale sets''~\cite{Ang1980}.} We denote by $I({\mathcal{L}})$ (resp.~$I'({\mathcal{L}})$) the smallest number $k$ such that every concept $L \in {\mathcal{L}}$ has a spanning set (resp.~a weak spanning set) w.r.t.~${\mathcal{L}}$ of size at most $k$. Note that $S$ is a spanning set of $L$ w.r.t.~${\mathcal{L}}$ iff $S$ distinguishes $L$ from all concepts in ${\mathcal{L}}$ except for supersets of $L$, i.e., iff $S$ is a positive teaching set for $L$ w.r.t.~$({\mathcal{L}},\supset)$. Similarly, $S$ is a weak spanning set of $L$ w.r.t.~${\mathcal{L}}$ iff $S$ distinguishes $L$ from all its proper subsets in ${\mathcal{L}}$ (which is necessarily the case when $S$ is a positive teaching set). These observations can be summarized as follows:
\begin{equation} \label{eq:span-pbtd} I'({\mathcal{L}}) \le \mathrm{PBTD}^+({\mathcal{L}}) \le \mathrm{PBTD}^+({\mathcal{L}},\supset) \le I({\mathcal{L}}) \enspace . \end{equation}
The last two inequalities are straightforward. The inequality $I'({\mathcal{L}}) \le \mathrm{PBTD}^+({\mathcal{L}})$ follows from Lemma~\ref{lem:ts-pos}, which implies that no concept $L$ can have a preference-based teaching set $T$ smaller than its smallest weak spanning set. Such a set $T$ would be consistent with some proper subset of $L$, which is impossible by Lemma~\ref{lem:ts-pos}.
Suppose ${\mathcal{L}}$ is intersection-closed. Then $\cap_{L\in{\mathcal{L}}:S \subseteq L}L$ is the unique smallest concept in ${\mathcal{L}}$ containing $S$. If $S \subseteq L_0$ is a weak spanning set of $L_0 \in {\mathcal{L}}$, then $\cap_{L\in{\mathcal{L}}:S \subseteq L}L = L_0$ because, on the one hand, $\cap_{L\in{\mathcal{L}}:S \subseteq L}L \subseteq L_0$ and, on the other hand, no proper subset of $L_0$ in ${\mathcal{L}}$ contains $S$. Thus the distinction between spanning sets and weak spanning sets is blurred for intersection-closed classes:
\begin{lemma} \label{lem:cap-closed} Suppose that ${\mathcal{L}}$ is intersection-closed. Then $I'({\mathcal{L}}) = \mathrm{PBTD}^+({\mathcal{L}}) = I({\mathcal{L}})$. \end{lemma}
\begin{example} Let ${\mathcal{R}}_d$ denote the class of $d$-dimensional axis-parallel hyper-rectangles (= $d$-dimensio- nal boxes). This class is intersection-closed and clearly $I({\mathcal{R}}_d)=2$. Thus $\mathrm{PBTD}^+({\mathcal{R}}_d)=2$. \end{example}
A mapping $\mathrm{cl}:2^{\mathcal{X}} \rightarrow 2^{\mathcal{X}}$ is said to be a {\em closure operator} on the universe ${\mathcal{X}}$ if the following conditions hold for all sets $A,B \subseteq {\mathcal{X}}$: \[ A \subseteq B \Rightarrow \mathrm{cl}(A) \subseteq \mathrm{cl}(B)\ \mbox{ and }\ A \subseteq \mathrm{cl}(A) = \mathrm{cl}(\mathrm{cl}(A)) \enspace . \] The following notions refer to an arbitrary but fixed closure operator. The set $\mathrm{cl}(A)$ is called the {\em closure} of $A$. A set $C$ is said to be {\em closed} if $\mathrm{cl}(C) = C$. It follows that precisely the sets $\mathrm{cl}(A)$ with $A \subseteq {\mathcal{X}}$ are closed. With this notation, we observe the following lemma.
\begin{lemma} \label{lem:span-closure} Let ${\mathcal{C}}$ be the set of all closed subsets of ${\mathcal{X}}$ under some closure operator $\mathrm{cl}$, and let $L\in{\mathcal{C}}$. If $L = \mathrm{cl}(S)$, then $S$ is a spanning set of $L$ w.r.t.~${\mathcal{C}}$. \end{lemma}
\begin{proof} Suppose $L'\in{\mathcal{C}}$ and $S\subseteq L'$. Then $L = \mathrm{cl}(S) \subseteq \mathrm{cl}(L') = L'$. \end{proof}
For every closed set $L \in {\mathcal{L}}$, let $s_{cl}(L)$ denote the size (possibly $\infty$) of the smallest set $S \subseteq {\mathcal{X}}$ such that $\mathrm{cl}(S) = L$. With this notation, we get the following (trivial but useful) result:
\begin{theorem} \label{th:span} Given a closure operator, let ${\mathcal{C}}[m]$ be the class of all closed subsets $C \subseteq {\mathcal{X}}$ with $s_{cl}(C) \le m$. Then $\mathrm{PBTD}^+({\mathcal{C}}[m]) \le \mathrm{PBTD}^+({\mathcal{C}}[m],\supset) \le m$. Moreover, this holds with equality provided that ${\mathcal{C}}[m] \setminus {\mathcal{C}}[m-1] \neq \emptyset$. \end{theorem}
\begin{proof} The inequality $\mathrm{PBTD}^+({\mathcal{C}}[m],\supset) \le m$ follows directly from Equation~(\ref{eq:span-pbtd}) and Lemma~\ref{lem:span-closure}. \\ Pick a concept $C_0 \in {\mathcal{C}}[m]$ such that $s_{cl}(C_0) = m$. Then any subset $S$ of $C_0$ of size less than $m$ spans only a proper subset of $C_0$, i.e., $\mathrm{cl}(S) \subset C_0$. Thus $S$ does not distinguish $C_0$ from $\mathrm{cl}(S)$. However, by Lemma~\ref{lem:ts-pos}, any preference-based learner must strictly prefer $\mathrm{cl}(S)$ over $C_0$. It follows that there is no positive teaching set of size less than $m$ for $C_0$ w.r.t.~${\mathcal{C}}[m]$. \end{proof}
Many natural classes can be cast as classes of the form ${\mathcal{C}}[m]$ by choosing the universe and the closure operator appropriately; the following examples illustrate the usefulness of Theorem~\ref{th:span} in that regard.
\begin{example}\label{exmp:pbtdpluslinset} Let
\[ \mathrm{LINSET}_k = \{\spn{G}: (G \subset {\mathbb N}) \wedge (1 \le |G| \le k)\} \] where $\spn{G} = \left\{\sum_{g \in G}a(g)g: a(g)\in{\mathbb N}_0\right\}$. In other words, $\mathrm{LINSET}_k$ is the set of all non-empty linear subsets of $\mathbb{N}_0$ that are generated by at most $k$ generators. Note that the mapping $G \mapsto \spn{G}$ is a closure operator over the universe ${\mathbb N}_0$. Since obviously $\mathrm{LINSET}_k \setminus \mathrm{LINSET}_{k-1} \neq \emptyset$, we obtain $\mathrm{PBTD}^+(\mathrm{LINSET}_k) = k$. \end{example}
\begin{example} \label{ex:polygons} Let ${\mathcal{X}} = \mathbbm{R}^2$ and let $\mathcal{C}_k$ be the class of convex polygons with at most $k$ vertices. Defining $\mathrm{cl}(S)$ to be the convex closure of $S$, we obtain ${\mathcal{C}}[k]=\mathcal{C}_k$ and thus $\mathrm{PBTD}^+(\mathcal{C}_k) = k$. \end{example}
\begin{example} Let $\mathcal{X}=\mathbb{R}^n$ and let $\mathcal{C}_k$ be the class of polyhedral cones that can be generated by $k$ (or less) vectors in $\mathbbm{R}^n$. If we take $\mathrm{cl}(S)$ to be the conic closure of $S \subseteq\mathbbm{R}^n$, then $\mathcal{C}[k]=\mathcal{C}_k$ and thus $\mathrm{PBTD}^+(\mathcal{C}_k)=k$. \end{example}
\section{A Convenient Technique for Proving Upper Bounds} \label{sec:admissible-mappings}
In this section, we give an alternative definition of the preference-based teaching dimension using the notion of an ``admissible mapping". Given a concept class ${\mathcal{L}}$ over a universe ${\mathcal{X}}$, let $T$ be a mapping $L \mapsto T(L) \subseteq {\mathcal{X}} \times \{-,+\}$ that assigns a set $T(L)$ of labeled examples to every set $L \in {\mathcal{L}}$ such that the labels in $T(L)$ are consistent with $L$. The {\em order} of $T$, denoted as $\mathrm{ord}(T)$, is defined
as $\sup_{L \in {\mathcal{L}}}|T(L)| \in {\mathbb N}\cup\{\infty\}$. Define the mappings $T^+$ and $T^-$ by setting $T^+(L) = \{x : (x,+) \in T(L)\}$ and $T^-(L) = \{x : (x,-) \in T(L)\}$ for every $L\in{\mathcal{L}}$. We say that $T$ is {\em positive} if $T^-(L)=\emptyset$ for every $L\in{\mathcal{L}}$. In the sequel, we will occasionally identify a positive mapping $L \mapsto T(L)$ with the mapping $L \mapsto T^+(L)$. The symbol ``$+$'' as an upper index of $T$ will always indicate that the underlying mapping $T$ is positive.
\noindent The following relation will help to clarify under which conditions the sets $(T(L))_{L\in{\mathcal{L}}}$ are teaching sets w.r.t.~a suitably chosen preference relation: \[ R_T = \{(L,L')\in{\mathcal{L}}\times{\mathcal{L}}:\ (L \neq L') \wedge (\mbox{$L$ is consistent with $T(L')$})\} \enspace . \] The transitive closure of $R_T$ is denoted as $\mathrm{trcl}(R_T)$ in the sequel. The following notion will play an important role in this paper:
\begin{definition} \label{def:admissible} A mapping $L \mapsto T(L)$ with $L$ ranging over all concepts in ${\mathcal{L}}$ is said to be {\em admissible for ${\mathcal{L}}$} if the following holds: \begin{enumerate} \item For every $L \in{\mathcal{L}}$, $L$ is consistent with $T(L)$. \item The relation $\mathrm{trcl}(R_T)$ is asymmetric (which clearly implies that $R_T$ is asymmetric too). \end{enumerate} \end{definition} If $T$ is admissible, then $\mathrm{trcl}(R_T)$ is transitive and asymmetric, i.e., $\mathrm{trcl}(R_T)$ is a strict partial order on ${\mathcal{L}}$. We will therefore use the notation $\prec_T$ instead of $\mathrm{trcl}(R_T)$ whenever $T$ is known to be admissible.
\begin{lemma} Suppose that $T^+$ is a positive admissible mapping for ${\mathcal{L}}$. Then the relation $\prec_{T^+}$ on ${\mathcal{L}}$ extends the relation $\supset$ on ${\mathcal{L}}$. More precisely, the following holds for all $L,L' \in {\mathcal{L}}$: \[ L' \subset L \Rightarrow (L,L') \in R_{T^+} \Rightarrow L \prec_{T^+} L' \enspace . \] \end{lemma}
\begin{proof} If $T^+$ is admissible, then $L'$ is consistent with $T^+(L')$. Thus $T^+(L') \subseteq L' \subset L$ so that $L$ is consistent with $T^+(L')$ too. Therefore $(L,L') \in R_{T^+}$, i.e., $L \prec_{T^+} L'$. \end{proof}
\noindent The following result clarifies how admissible mappings are related to preference-based teaching:
\begin{lemma} For each concept class ${\mathcal{L}}$, the following holds: \[ \mathrm{PBTD}({\mathcal{L}}) = \inf_T \mathrm{ord}(T)\ \mbox{ and }\ \mathrm{PBTD}^+({\mathcal{L}}) = \inf_{T^+} \mathrm{ord}(T^+) \] where $T$ ranges over all mappings that are admissible for ${\mathcal{L}}$ and $T^+$ ranges over all positive mappings that are admissible for ${\mathcal{L}}$. \end{lemma}
\begin{proof} We restrict ourselves to the proof for $\mathrm{PBTD}({\mathcal{L}}) = \inf_T \mathrm{ord}(T)$ because the equation $\mathrm{PBTD}^+({\mathcal{L}}) = \inf_{T^+} \mathrm{ord}(T^+)$ can be obtained in a similar fashion. We first prove that $\mathrm{PBTD}({\mathcal{L}})$ $\le \inf_T \mathrm{ord}(T)$. Let $T$ be an admissible mapping for ${\mathcal{L}}$. It suffices to show that, for every $L\in{\mathcal{L}}$, $T(L)$ is a teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec_T)$. Suppose $L'\in{\mathcal{L}}\setminus\{L\}$ is consistent with $T(L)$. Then $(L',L) \in R_T$ and thus $L' \prec_T L$. It follows that $\prec_T$ prefers $L$ over all concepts $L'\in{\mathcal{L}}\setminus\{L\}$ that are consistent with $T(L)$. Thus $T$ is a teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec_T)$, as desired.
We now prove that $\inf_T \mathrm{ord}(T) \le \mathrm{PBTD}({\mathcal{L}})$. Let $\prec$ be a strict partial order on ${\mathcal{L}}$ and let $T$ be a mapping such that, for every $L\in{\mathcal{L}}$, $T(L)$ is a teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec)$. It suffices to show that $T$ is admissible for ${\mathcal{L}}$. Consider a pair $(L',L) \in R_T$. The definition of $R_T$ implies that $L' \neq L$ and that $L'$ is consistent with $T(L)$. Since $T(L)$ is a teaching set w.r.t.~$({\mathcal{L}},\prec)$, it follows that $L' \prec L$. Thus, $\prec$ is an extension of $R_T$. Since $\prec$ is transitive, it is even an extension of $\mathrm{trcl}(R_T)$. Because $\prec$ is asymmetric, $\mathrm{trcl}(R_T)$ must be asymmetric, too. It follows that $T$ is admissible. \end{proof}
\section{Preference-based Teaching of Linear Sets} \label{sec:linsets}
Some work in computational learning theory \cite{Abe89,GSZ2015,Takada92} is concerned with learning \emph{semi-linear sets}, i.e., unions of linear subsets of $\mathbb{N}^k$ for some fixed $k\ge 1$, where each linear set consists of exactly those elements that can be written as the sum of some constant vector $c$ and a linear combination of the elements of some fixed set of generators, see Example~\ref{exmp:pbtdpluslinset}. While semi-linear sets are of common interest in mathematics in general, they play a particularly important role in the theory of formal languages, due to \emph{Parikh's theorem}, by which the so-called Parikh vectors of strings in a context-free language always form a semi-linear set~\cite{Parikh66}.
A recent study \cite{GSZ2015} analyzed computational teaching of classes of linear subsets of $\mathbb{N}$ (where $k=1$) and some variants thereof, as a substantially simpler yet still interesting special case of semi-linear sets. In this section, we extend that study to preference-based teaching.
Within the scope of this section, all concept classes are formulated over the universe ${\mathcal{X}} = {\mathbb N}_0$. Let $G = \{g_1,\ldots$ $,g_k\}$ be a finite subset of ${\mathbb N}$. We denote by $\spn{G}$ resp.~by $\spn{G}_+$ the following sets: \[ \spn{G} = \left\{\sum_{i=1}^{k}a_ig_i:\ a_1,\ldots,a_k\in{\mathbb N}_0\right\}\ \mbox{ and }\ \spn{G}_+ = \left\{\sum_{i=1}^{k}a_ig_i:\ a_1,\ldots,a_k\in{\mathbb N}\right\} \enspace . \]
We will determine (at least approximately) the preference-based teaching dimension of the following concept classes over ${\mathbb N}_0$: \begin{eqnarray*}
\mathrm{LINSET}_k & = & \{\spn{G}:\ (G \subset {\mathbb N}) \wedge (1 \le |G| \le k)\} \enspace . \\
\mathrm{CF}\mbox{-}\mathrm{LINSET}_k & = & \{\spn{G}:\ (G \subset {\mathbb N}) \wedge (1 \le |G| \le k) \wedge (\gcd(G)=1)\} \enspace . \\
\mathrm{NE}\mbox{-}\mathrm{LINSET}_k & = & \{\spn{G}_+:\ (G \subset {\mathbb N}) \wedge (1 \le |G| \le k) \} \enspace . \\
\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k & = & \{\spn{G}_+:\ (G \subset {\mathbb N}) \wedge (1 \le |G| \le k) \wedge (gcd(G)=1)\} \enspace . \end{eqnarray*}
A subset of ${\mathbb N}_0$ whose complement in ${\mathbb N}_0$ is finite is said to be {\em co-finite}. The letters ``CF'' in $\mathrm{CF}\mbox{-}\mathrm{LINSET}$ mean ``co-finite''. The concepts in $\mathrm{LINSET}_k$ have the algebraic structure of a monoid w.r.t.~addition. The concepts in $\mathrm{CF}\mbox{-}\mathrm{LINSET}_k$ are also known as ``numerical semigroups''~\cite{RG-S2009}. A zero coefficient $a_j=0$ erases $g_j$ in the linear combination $\sum_{i=1}^{k}a_ig_i$. Coefficients from ${\mathbb N}$ are non-erasing in this sense. The letters ``NE'' in ``$\mathrm{NE}\mbox{-}\mathrm{LINSET}$'' mean ``non-erasing''.
The {\em shift-extension} ${\mathcal{L}}'$ of a concept class ${\mathcal{L}}$ over the universe ${\mathbb N}_0$ is defined as follows: \begin{equation} \label{eq:shift-extension} {\mathcal{L}}' = \{c+L:\ (c\in{\mathbb N}_0) \wedge (L\in{\mathcal{L}})\} \enspace . \end{equation}
The following bounds on $\mathrm{RTD}$ and $\mathrm{RTD}^+$ (for sufficiently large values of $k$)\footnote{For instance, $\mathrm{RTD}^+(\mathrm{LINSET}_k)=\infty$ holds for all $k\ge2$ and $\mathrm{RTD}(\mathrm{LINSET}_k) = \mbox{?}$ (where ``?'' means ``unknown'') holds for all $k\ge4$.} are known from~\cite{GSZ2015}: \[
\begin{array}{|l|l|l|} \hline
& \mathrm{RTD}^+ & \mathrm{RTD} \\ \hline \mathrm{LINSET}_k & =\infty & \mbox{?} \\ \mathrm{CF}\mbox{-}\mathrm{LINSET}_k & =k & \in\{k-1,k\} \\ \mathrm{NE}\mbox{-}\mathrm{LINSET}'_k & =k+1 & \in\{k-1,k,k+1\} \\ \hline \end{array} \] Here $\mathrm{NE}\mbox{-}\mathrm{LINSET}'_k$ denotes the shift-extension of $\mathrm{NE}\mbox{-}\mathrm{LINSET}_k$ .
The following result shows the corresponding bounds with PBTD in place of RTD:
\begin{theorem} \label{th:bounds-linset} The bounds in the following table are valid: \[
\begin{array}{|l|l|l|} \hline
& \mathrm{PBTD}^+ & \mathrm{PBTD} \\ \hline \mathrm{LINSET}_k & =k & \in\{k-1,k\} \\ \mathrm{CF}\mbox{-}\mathrm{LINSET}_k & =k & \in\{k-1,k\} \\ \mathrm{NE}\mbox{-}\mathrm{LINSET}_k &
\in \left[k-1:k\right] &
\in \left[\left\lfloor\frac{k-1}{2}\right\rfloor:k\right] \\ \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k &
\in \left[k-1:k\right] &
\in \left[\left\lfloor\frac{k-1}{2}\right\rfloor:k\right] \\ \hline \end{array} \] Moreover \begin{equation} \label{eq:more-bounds} \mathrm{PBTD}^+({\mathcal{L}}') = k+1\ \wedge\ \mathrm{PBTD}({\mathcal{L}}') \in \{k-1,k,k+1\} \end{equation} holds for all ${\mathcal{L}}\in\{\mathrm{LINSET}_k,\mathrm{CF}\mbox{-}\mathrm{LINSET}_k,\mathrm{NE}\mbox{-}\mathrm{LINSET}_k,\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k\}$. \end{theorem}
Note that the equation $\mathrm{PBTD}^+(\mathrm{LINSET}_k) = k$ was already proven in Example~\ref{exmp:pbtdpluslinset}, using the fact that $G\mapsto\spn{G}$ is a closure operator. Since $G\mapsto\spn{G}_+$ is not a closure operator, we give a separate argument to prove an upper bound of $k$ on $\mathrm{PBTD}^+(\mathrm{NE}\mbox{-}\mathrm{LINSET}_k)$ (see Lemma~\ref{lem:nelinset-ub} in Appendix~\ref{app:linsets}). All other upper bounds in Theorem~\ref{th:bounds-linset} are then easy to derive. The lower bounds in Theorem~\ref{th:bounds-linset} are much harder to obtain. A complete proof of Theorem~\ref{th:bounds-linset} will be given in Appendix~\ref{app:linsets}.
\iffalse \section{Hierarchical Preference-based Teaching} \label{sec:hierarchical-pbt}
In this section, we introduce a technique for defining preference relations that we use later on in Section~\ref{sec:halfspaces} to obtain upper bounds on the $\mathrm{PBTD}$ of classes of halfspaces.
Suppose that ${\mathcal{L}}$ is a parametrized concept class in the sense that any concept of ${\mathcal{L}}$ can be fixed by assigning real values $q = (q_1,\ldots,q_d)$ to ``programmable parameters'' $Q = (Q_1,\ldots,Q_d)$. The concept resulting from setting $Q_i = q_i$ for $i=1,\ldots,d$ is denoted as $L_q$. Let ${\mathcal{D}} \subseteq \mathbbm{R}^d$ be a set which makes the representation $q$ of $L_q$ unique, i.e., $L_q = L_{q'}$ with $q,q' \in {\mathcal{D}}$ implies that $q=q'$. We will then identify a preference relation over ${\mathcal{L}}$ with a preference relation over ${\mathcal{D}}$. For every $p \in \{\downarrow,\uparrow\}^d$, let $\prec_p$ be the following algorithmically defined (lexicographic) preference relation: \begin{enumerate} \item Given $q \neq q' \in {\mathcal{D}}$, find the smallest index $i \in [d]$ such that $q_i \neq q'_i$, say $q_i < q'_i$. \item If $p_i = \downarrow$ (resp.~$p_i=\uparrow$), then $q' \prec_p q$ (resp.~$q \prec_p q')$. \end{enumerate} Imagine a student with this preference relation who has seen a collection of labeled examples. The following hierarchical system of Rules $i=1,\ldots,d$ clarifies which value $q'_i$ she should assign to the unknown parameter $Q_i$: \begin{description} \item[Rule i:] With $i$-highest priority do the following. Choose $q'_i$ as small as possible if $p_i=\downarrow$ and as large as possible if $p_i=\uparrow$. Assign the value $q'_i$ to the parameter $Q_i$. \end{description} It is important to understand that this rule system is hierarchical (as expressed by the distinct priorities of the rules): when Rule $i$ becomes active, then the values of the parameters $Q_1,\ldots,Q_{i-1}$ (in accordance with the rules $1,\ldots,i-1$) have been chosen already.
Suppose that $L_q$ with $q \in {\mathcal{D}}$ is the target concept. A teacher who designs a teaching set for $L_q$ w.r.t.~$({\mathcal{L}},\prec_p)$ can proceed in stages $i=1,\ldots,d$ as follows: \begin{description} \item[Stage i:] Suppose that $p_i=\downarrow$ (resp.~$p_i=\uparrow$). Choose the next part $T_i$ of the teaching set so that every hypothesis $L_{q'}$ with $q' \in {\mathcal{D}}$, and $q'_1=q_1,\ldots,q'_{i-1}=q_{i-1}$ satisfies the following condition: if $L_{q'}$ is consistent with $T_1 \cup \ldots \cup T_{i-1} \cup T_i$, then $q'_i \ge q_i$ (resp.~$q'_i \le q_i$). \end{description} In other words, the teacher chooses $T_1$ so that the student with preference relation $\prec_p$ will assign the value $q_1$ to $Q_1$. Given that $Q_1=q_1$, the teacher chooses $T_2$ so that the student will next assign the value $q_2$ to $Q_2$, and so on.
The use of this basic technique can be made more convenient by allowing more than one parameter to be handled in a single stage.
\fi
\section{Preference-based Teaching of Halfspaces} \label{sec:halfspaces}
In this section, we study preference-based teaching of halfspaces. We will denote the all-zeros vector as $\vec{0}$. The vector with $1$ in coordinate $i$ and with $0$ in the remaining coordinates is denoted as $\vec{e}_i$. The dimension of the Euclidean space in which these vectors reside will always be clear from the context. The sign of a real number $x$ (with value $1$ if $x>0$, value $-1$ if $x<0$, and value $0$ if $x=0$) is denoted by $\mathrm{sign}(x)$.
Suppose that $w\in\mathbbm{R}^d\setminus\{\vec0\}$ and $b\in\mathbbm{R}$. The {\em (positive) halfspace induced by $w$ and $b$} is then given by \[ H_{w,b} = \{x\in\mathbbm{R}^d:\ w^\top x + b \ge 0\} \enspace . \] Instead of $H_{w,0}$, we simply write $H_w$. Let ${\mathcal{H}}_d$ denote the class of $d$-dimensional Euclidean half\-spaces: \[ {\mathcal{H}}_d = \{H_{w,b}:\ w\in\mathbbm{R}^d\setminus\{\vec{0}\} \wedge b \in \mathbbm{R}\} \enspace . \] Similarly, ${\mathcal{H}}_d^0$ denotes the class of $d$-dimensional homogeneous Euclidean halfspaces: \[ {\mathcal{H}}_d^0 = \{H_w:\ w\in\mathbbm{R}^d\setminus\{\vec{0}\}\} \enspace . \] Let $S_{d-1}$ denote the $(d-1)$-dimensional unit sphere in $\mathbbm{R}^d$. Moreover $S_{d-1}^+ = \{x\in S_{d-1}: x_d>0\}$ denotes the ``northern hemisphere''. If not stated explicitly otherwise, we will represent homogeneous halfspaces with normalized vectors residing on the unit sphere. We remind the reader of the following well-known fact:
\begin{remark} \label{rem:orthogonal-group} The orthogonal group in dimension $d$ (i.e., the multiplicative group of orthogonal $(d \times d)$-matrices) acts transitively on $S_{d-1}$ and it conserves the inner product. \end{remark}
We now prove a helpful lemma, stating that each vector $w^*$ in the northern hemisphere may serve as a representative for some homogeneous halfspace $H_u$ in the sense that all other elements of $H_u$ in the northern hemisphere have a strictly smaller $d$-th component than $w^*$. This will later help to teach homogeneous halfspaces with a preference that orders vectors by the size of their last coordinate.
\begin{lemma} \label{lem:close2northpole} Let $d\ge2$, let $0<h\le1$ and let $R_{d,h} = \{w \in S_{d-1}: w_d = h\}$. With this notation the following holds. For every $w^* \in R_{d,h}$, there exists $u\in\mathbbm{R}^d\setminus\{\vec{0}\}$ such that \begin{equation} \label{eq:good-choice} (w^* \in H_u) \wedge (\forall w \in (S_{d-1}^+ \cap H_u)\setminus\{w^*\}: w_d < h) \enspace . \end{equation} \end{lemma}
\begin{proof} For $h=1$, the statement is trivial, since $R_{d,1} = \{\vec{e}_d\}$. So let $h<1$.
Because of Remark~\ref{rem:orthogonal-group}, we may assume without loss of generality that the vector $w^* \in R_{d,h}$ equals $(0,\ldots,0,\sqrt{1-h^2},h)$. It suffices therefore to show that, with this choice of $w^*$, the vector $u = (0,\ldots,0,w_d^*,-w_{d-1}^*)$ satisfies~(\ref{eq:good-choice}). Note that $w \in H_{u}$ iff $\spn{u,w} = w_d^*w_{d-1} - w_{d-1}^*w_d \ge 0$. Since $\spn{u,w^*}=0$, we have $w^* \in H_u$. Moreover, it follows that \[ S_{d-1}^+ \cap H_{u} = \left\{w\in S_{d-1}^+: \frac{w_{d-1}}{w_d} \ge \frac{w_{d-1}^*}{w_d^*} > 0 \right\} \enspace . \] It is obvious that no vector $w \in S_{d-1}^+ \cap H_u$ can have a $d$-th component $w_d$ exceeding $w_d^*=h$ and that setting $w_d=h=w_d^*$ forces the settings $w_{d-1}=w_{d-1}^*=\sqrt{1-h^2}$ and $w_1 = \ldots = w_{d-2} = 0$. Consequently,~(\ref{eq:good-choice}) is satisfied, which concludes the proof. \end{proof}
With this lemma in hand, we can now prove an upper bound of 2 for the preference-based teaching dimension of the class of homogeneous halfspaces, independent of the underlying dimension~$d$.
\begin{theorem} \label{th:halfspace0-ub} $\mathrm{PBTD}({\mathcal{H}}_1^0) = \mathrm{TD}({\mathcal{H}}_1^0) = 1$ and, for every $d\ge2$, we have $\mathrm{PBTD}({\mathcal{H}}_d^0)$ $\le 2$. \end{theorem}
\begin{proof} Clearly, $\mathrm{PBTD}({\mathcal{H}}_1^0) = \mathrm{TD}({\mathcal{H}}_1^0) = 1$ since ${\mathcal{H}}_1^0$ consists of the two sets $\{x\in\mathbbm{R}: x\ge0\}$ and $\{x\in\mathbbm{R}: x\le0$\}.
\noindent Suppose now that $d\ge2$. Let $w^*$ be the target weight vector (i.e., the weight vector that has to be taught). Under the following conditions, we may assume without loss of generality that $w_d^*\neq0$: \begin{itemize} \item For any $0<s_1<s_2$, the student prefers any weight vector that ends with $s_2$ zero coordinates over any weight vector that ends with only $s_1$ zero coordinates. \item If the target vector ends with (exactly) $s$ zero coordinates, then the teacher presents only examples ending with (at least) $s$ zero coordinates. \end{itemize} In the sequel, we specify a student and a teacher such that these conditions hold, so that we will consider only target weight vectors $w^*$ with $w_d^*\neq0$.
\noindent The student has the following preference relation: \begin{itemize} \item Among the weight vectors $w$ with $w_d\neq0$,
the student prefers vectors with larger values of $|w_d|$
over those with smaller values of $|w_d|$. \end{itemize}
\noindent The teacher will use two examples. The first one is chosen as \[ \left\{ \begin{array}{ll}
(-\vec{e}_d,-) & \mbox{if $w^*_d>0$} \\
(\vec{e}_d,-) &\mbox{if $w^*_d<0$}
\end{array} \right. \enspace . \] This example reveals whether the unknown weight vector $w^* \in S_{d-1}$ has a strictly positive or a strictly negative $d$-th component. For reasons of symmetry, we may assume that $w_d^*>0$. We are now precisely in the situation that is described in Lemma~\ref{lem:close2northpole}. Given $w^*$ and $h=w_d^*$, the teacher picks as a second example $(u,+)$ where $u\in\mathbbm{R}^d\setminus\{\vec{0}\}$ has the properties described in the lemma. It follows immediately that the student's preferences will make her choose the weight vector $w^*$. \end{proof}
The upper bound of 2 given in Theorem~\ref{th:halfspace0-ub} is tight, as is stated in the following lemma.
\begin{lemma} \label{lem:halfspace-neg} For every $d\ge2$, we have $\mathrm{PBTD}({\mathcal{H}}_d^0)\ge 2$. \end{lemma}
\begin{proof} We verify this lemma via Lemma~\ref{lem:lb-tdmin}, by providing a finite subclass ${\mathcal{F}}$ of ${\mathcal{H}}_2^0$ such that $\mathrm{TD}_{min}({\mathcal{F}})=2$. Let ${\mathcal{F}}=\{H_w : \vec{0} \neq w\in\{-1,0,1\}^2\}$. It is easy to verify that each of the $8$ halfspaces in ${\mathcal{F}}$ has a teaching dimension of 2 with respect to ${\mathcal{F}}$. This example can be extended to higher dimensions in the obvious way. \end{proof}
We thus conclude that the class of homogeneous halfspaces has a preference-based teaching dimension of 2, independent of the dimensionality $d\ge 2$.
\begin{corollary} For every $d\ge2$, we have $\mathrm{PBTD}({\mathcal{H}}_d^0)=2$. \end{corollary}
By contrast, we will show next that the recursive teaching dimension of the class of homogeneous halfspaces grows with the dimensionality.
\begin{theorem}\label{thm:tdrtdhalfspace} For any $d \ge 2$, $\mathrm{TD}(\mathcal{H}^0_d) = \mathrm{RTD}(\mathcal{H}^0_d) = d+1$. \end{theorem}
\begin{proof} Assume by normalization that the target weight vector has norm $1$, i.e., it is taken from $S_{d-1}$. Remark~\ref{rem:orthogonal-group} implies that all weight vectors in $S_{d-1}$ are equally hard to teach. It suffices therefore to show that $\mathrm{TD}(H_{\vec{e}_1},{\mathcal{H}}^0_d) = d+1$.
We first show that $\mathrm{TD}(H_{\vec{e}_1},{\mathcal{H}}^0_d) \le d+1$. Define $u = -\sum_{i=2}^{d}\vec{e}_i$. We claim that $T = \{ (\vec{e}_i,+): 2 \leq i \leq d\} \cup \{(u,+),(\vec{e}_1,+)\}$ is a teaching set for $H_{\vec{e}_1}$ w.r.t.~${\mathcal{H}}^0_d.$ Consider any $w \in S_{d-1}$ such that $H_w$ is consistent with $T$. Note that $w_i = \spn{\vec{e}_i,w} \geq 0$ for all $i \in \{2,\ldots,d\}$ and $\spn{u,w} = -\sum_{i=2}^{d}w_i \geq 0$ together imply that $w_i = 0$ for all $i\in\{2,\ldots,d\}$ and therefore $w = \pm \vec{e}_1$. Furthermore, $w_1 = \spn{w,\vec{e}_1} \geq 0$, and so $w = \vec{e}_1$, as required.
Now we show that $\mathrm{TD}(H_{\vec{e}_1},{\mathcal{H}}^0_d) \geq d+1$ holds for all $d\ge2$. It is easy to see that two examples do not suffice for distinguishing $\vec{e}_1\in\mathbbm{R}^2$ from all weight vectors in $S_1$. In other words, $\mathrm{TD}(H_{\vec{e}_1},{\mathcal{H}}^0_2) \ge3$. Suppose now that $d\ge3$. It is furthermore easy to see that a teaching set $T$ which distinguishes $\vec{e}_1$ from all weight vectors in $S_{d-1}$ must contain at least one positive example $u$ that is orthogonal to $\vec{e}_1$. The inequality $\mathrm{TD}(H_{\vec{e}_1},{\mathcal{H}}^0_d) \geq d+1$ is now obtained inductively because the example $(u,+) \in T$ leaves open a problem that is not easier than teaching $\vec{e}_1$ w.r.t.~the $(d-2)$-dimensional sphere $\{x \in S_{d-1}: x \perp u\}$. \end{proof}
We have thus established that the class of homogeneous halfspaces has a recursive teaching dimension growing linearly with $d$, while its preference-based teaching dimension is constant. In the case of general (i.e., not necessarily homogeneous) $d$-dimensional halfspaces, the difference between $\mathrm{RTD}$ and $\mathrm{PBTD}$ is even more extreme. On the one hand, by generalizing the proof of Lemma~\ref{lem:huge-gap}, it is easy to see that $\mathrm{RTD}({\mathcal{H}}_d)=\infty$ for all $d\ge 1$. On the other hand, we will show in the remainder of this section that $\mathrm{PBTD}({\mathcal{H}}_d) \le 6$, independent of the value of $d$.
We will assume in the sequel (by way of normalization) that an inhomogeneous halfspace has a bias $b\in\{\pm1\}$. We start with the following result:
\begin{lemma} \label{lem:3-examples} Let $w^* \in \mathbbm{R}^d$ be a vector with a non-trivial $d$-th component $w^*_d\neq0$ and let $b^*\in\{\pm1\}$ be a bias. Then there exist three examples labeled according to $H_{w^*,b^*}$ such that the following holds. Every weight-bias pair $(w,b)$ consistent with these examples satisfies $b=b^*$, $\mathrm{sign}(w_d) = \mathrm{sign}(w_d^*)$ and \begin{equation} \label{eq:w-d-constraint} \left\{ \begin{array}{ll}
|w_d| \ge |w_d^*| & \mbox{if $b^*=-1$} \\
|w_d| \le |w_d^*| & \mbox{if $b^*=+1$}
\end{array} \right. \enspace . \end{equation} \end{lemma}
\begin{proof} Within the proof, we use the label ``$1$'' instead of ``$+$'' and the label ``$-1$'' instead of ``$-$''. The pair $(w,b)$ denotes the student's hypothesis for the target weight-bias pair $(w^*,b^*)$. The examples shown to the student will involve the unknown quantities $w^*$ and $b^*$. Each example will lead to a new constraint on $w$ and $b$. We will see that the collection of these constraints reveals the required information. We proceed in three stages: \begin{enumerate} \item The first example is chosen as $(\vec{0},b^*)$. The pair $(w,b)$ can be consistent with this example only if $b = -1$ in the case that $b^*=-1$ and $b\in\{0,1\}$ in the case that $b^*=1$. \item The next example is chosen as $\vec{a}_2 = -\frac{2b^*}{w_d^*}\cdot\vec{e}_d$ and labeled ``$-b^*$''. Note that $\spn{w^*,\vec{a}_2}+b^* = -b^*$. We obtain the following new constraint: \[ \spn{w,\vec{a}_2}+b = \left\{ \begin{array}{ll}
-2\frac{w_d}{w_d^*}+\overbrace{b}^{\in\{0,1\}} < 0 & \mbox{if $b^*=1$} \\
+2\frac{w_d}{w_d^*}+\underbrace{b}_{=-1} \ge 0 & \mbox{if $b^*=-1$}
\end{array} \right. \enspace . \] The pair $(w,b)$ with $b=b^*$ if $b^*=-1$ and $b\in\{0,1\}$ if $b^*=1$ can satisfy the above constraint only if the sign of $w_d$ equals the sign of $w_d^*$. \item The third example is chosen as the example $\vec{a}_3 = -\frac{b^*}{w_d^*}\cdot\vec{e}_d$ with label ``$1$''. Note that $\spn{w^*,\vec{a}_3}^*+b^* = 0$. We obtain the following new constraint: \[ \spn{w,\vec{a}_3} = -\frac{b^*w_d}{w_d^*}+b \ge 0 \enspace . \] Given that $w$ is already constrained to weight vectors satisfying $\mathrm{sign}(w_d)=\mathrm{sign}($ $w_d^*)$, we can safely
replace $w_d/w_d^*$ by $|w_d|/|w_d^*|$. This yields
$|w_d|/|w_d^*| \le b$ if $b^*=1$ and $|w_d|/|w_d^*|\ge -b$
if $b^*=-1$. Since $b$ is already constrained as described in stage 1 above, we obtain $|w_d|/|w_d^*| \le b \in \{0,1\}$
if $b^*=1$ and $|w_d|/|w_d^*|\ge -b = 1$ if $b^*=-1$. The weight-bias pair $(w,b)$ satisfies these constraints only if $b=b^*$ and if~(\ref{eq:w-d-constraint}) is valid. \end{enumerate} The assertion of the lemma is immediate from this discussion. \end{proof}
\begin{theorem} \label{th:halfspace-ub} $\mathrm{PBTD}({\mathcal{H}}_d) \le 6$. \end{theorem}
\begin{proof} As in the proof of Lemma~\ref{lem:3-examples}, we use the label ``$1$'' instead of ``$+$'' and the label ``$-1$'' instead of ``$-$''. As in the proof of Theorem~\ref{th:halfspace0-ub}, we may assume without loss of generality that the target weight vector $w^*\in\mathbbm{R}^d$ satisfies $w_d^* \neq 0$. The proof will proceed in stages. On the way, we specify six rules which determine the preference relation of the student.
{\bf Stage 1} is concerned with teaching homogeneous halfspaces given by $w^*$ (and $b^*=0$). The student respects the following rules: \begin{description} \item[Rule 1:] She prefers any pair $(w,0)$ over any pair $(w',b)$ with $b\neq0$. In other words, any homogeneous halfspace is preferred over any non-homogeneous halfspace. \item[Rule 2:] Among homogeneous halfspaces, her preferences are the same as the ones that were used within the proof of Theorem~\ref{th:halfspace0-ub} for teaching homogeneous halfspaces. \end{description} Thus, if $b^*=0$, then we can simply apply the teaching protocol for homogeneous halfspaces. In this case, $w^*$ can be taught at the expense of only two examples.
Stage~1 reduces the problem to teaching inhomogeneous halfspaces given by $(w^*,$ $b^*)$ with $b^*\neq0$. We assume, by way of normalization, that $b^*\in\{\pm1\}$, but note that $w^*$ can now not be assumed to be of unit (or any other fixed) length.
In {\bf stage 2}, the teacher presents three examples in accordance with Lemma~\ref{lem:3-examples}. It follows that the student will take into consideration only weight-bias pairs $(w,b)$ such that the constraints $b=b^*$, $\mathrm{sign}(w_d) = \mathrm{sign}(w_d^*)$ and~(\ref{eq:w-d-constraint}) are satisfied. The following rule will then induce the constraint $w_d=w_d^*$: \begin{description} \item[Rule 3:] Among the pairs $(w,b)$ such that $w_d\neq0$ and $b\in\{\pm1\}$, the student's preferences are as follows. If $b=-1$ (resp.~$b=1$), then she prefers vectors $w$ with a smaller (resp.~larger) value
of $|w_d|$ over those with a larger (resp.~smaller) value of $|w_d|$. \end{description} Thanks to Lemma~\ref{lem:3-examples} and thanks to Rule 3, we may from now on assume that $b=b^*$ and $w_d=w_d^*$. In the sequel, let $w^*$ be decomposed according to $w^* = (\vec{w}_{d-1}^*,w_d^*) \in \mathbbm{R}^{d-1}\times\mathbbm{R}$. We think of $\vec{w}_{d-1}$ as the student's hypothesis for $\vec{w}_{d-1}^*$.
{\bf Stage~3} is concerned with the special case where $\vec{w}_{d-1}^*=\vec{0}$. The student will automatically set $\vec{w}_{d-1}=\vec{0}$ if we add the following to the student's rule system: \begin{description} \item[Rule 4:] Given that the values for $w_d$ and $b$ have been fixed already (and are distinct from $0$), the student prefers weight-bias pairs with $\vec{w}_{d-1}=\vec{0}$ over any weight-bias pair with $\vec{w}_{d-1}\neq\vec{0}$. \end{description}
Stage 3 reduces the problem to teaching $(w^*,b^*)$ with fixed non-zero values for $w_d$ and $b^*$ (known to the student) and with $\vec{w}_{d-1}^* \neq \vec{0}$. Thus, essentially, only $\vec{w}_{d-1}^*$ has still to be taught. In the next stage, we will argue that the problem of teaching $\vec{w}_{d-1}^*$ is equivalent to teaching a homogeneous halfspace.
In {\bf stage 4}, the teacher will present only examples $a$ such that $a_d = -\frac{b^*}{w_d^*}$ so that the contribution of the $d$-th component to the inner product of $w^*$ and $a$ cancels with the bias $b^*$. Given this commitment for $a_d$, the first $d-1$ components of the examples can be chosen so as to teach the homogeneous halfspace $H_{\vec{w}_{d-1}^*}$. According to Theorem~\ref{th:halfspace0-ub}, this can be achieved at the expense of two more examples. Of course the student's preferences must match with the preferences that were used in the proof of this theorem: \begin{description} \item[Rule 5:] Suppose that the values of $w_d$ and $b$ have been fixed already (and are distinct from $0$) and suppose that $\vec{w}_{d-1}\neq\vec{0}$. Then the preferences for the choice of $\vec{w}_{d-1}$ match with the preferences that were used in the protocol for teaching homogeneous halfspaces. \end{description}
After stage 4, the student takes into consideration only weight-bias pairs $(w,b)$ such that $w_d=w_d^*$, $b=b^*$ and $H_{\vec{w}_{d-1}} = H_{\vec{w}_{d-1}^*}$. However, since we had normalized the bias and not the weight vector, this does not necessarily mean that $\vec{w}_{d-1} = \vec{w}_{d-1}^*$. On the other hand, the two weight vectors already coincide modulo a positive scaling factor, say \begin{equation} \label{eq:scaling} \vec{w}_{d-1} = s \cdot \vec{w}_{d-1}^* \mbox{ for some $s>0$} \enspace . \end{equation} In order to complete the proof, it suffices to teach the $L_1$-norm of $\vec{w}_{d-1}^*$ to the student (because~(\ref{eq:scaling})
and $\|\vec{w}_{d-1}\|_1 = \|\vec{w}_{d-1}^*\|_1$ imply that $\vec{w}_{d-1} = \vec{w}_{d-1}^*$). The next (and final) stage serves precisely this purpose.
As for {\bf stage~5}, we first fix some notation. For $i=1,\ldots,k-1$, let $\beta_i = \mathrm{sign}(w_i^*)$. Note that~(\ref{eq:scaling}) implies
that $\beta_i = \mathrm{sign}(w_i)$. Let $L = \|\vec{w}_{d-1}^*\|_1$ denote the $L_1$-norm of $\vec{w}_{d-1}^*$. The final example is chosen as $\vec{a}_6 = (\beta_1,\ldots,\beta_{d-1},-(L+b^*)/w_d^*)$ and labeled ``$1$''. Note that \[
\spn{w^*,\vec{a_6}} + b^* = |w_1^*|+\ldots+|w_{d-1}^*|-L = 0 \enspace . \] Given that $\beta_i=\mathrm{sign}(w_i)$, $w_d=w_d^*$ and $b=b^*$, the student can derive from $\vec{a_6}$ and its label the following constraint on~$\vec{w}_{d-1}$: \[
\spn{w,\vec{a_6}} + b = |w_1|+\ldots+|w_{d-1}| - L \ge 0 \enspace . \]
In combination with the following rule, we can now force the constraint $\|\vec{w}_{d-1}\|_1 = L$: \begin{description} \item[Rule 6:] Suppose that the values of $w_d$ and $b$ have been fixed already (and are distinct from $0$) and suppose that $H_{\vec{w}_{d-1}}$ has already been fixed. Then, among the vectors representing $H_{\vec{w}_{d-1}}$, the ones with a smaller $L_1$-norm are preferred over the ones with a larger $L_1$-norm. \end{description} An inspection of the six stages reveals that at most six examples altogether were shown to the student (three in stage 2, two in stage 4, and one in stage 5). This completes the proof of the theorem. \\ \mbox{} \end{proof}
Note that Theorems~\ref{th:halfspace0-ub} and~\ref{th:halfspace-ub} remain valid when we allow $w$ to be the all-zero vector, which extends $\mathcal{H}_d^0$ by $\{{\mathbb{R}}^d\}$ and $\mathcal{H}_d$ by $\{{\mathbb{R}}^d,\emptyset\}$. ${\mathbb{R}}^d$ will be taught with a single positive example, and $\emptyset$ with a single negative example. The student will give the highest preference to ${\mathbb{R}}^d$, the second highest to $\emptyset$, and among the remaining halfspaces, the student's preferences stay the same.
\section{Classes with $\mathrm{PBTD}$ or $\mathrm{PBTD}^+$ Equal to One}\label{sec:pbtd1}
In this section, we will give complete characterizations of (i) the concept classes with a positive preference-based teaching dimension of $1$, and (ii) the concept classes with a preference-based teaching dimension of $1$. Throughout this section, we use the label ``$1$'' to indicate positive examples and the label ``$0$'' to indicate negative examples.
Let $I$ be a (possibly infinite) index set. We will consider a mapping $A: I \times I \rightarrow \{0,1\}$ as a binary matrix $A \in \{0,1\}^{I \times I}$. $A$ is said to be {\em lower-triangular} if there exists a linear ordering $\prec$ on $I$ such that $A(i,i')=0$ for every pair $(i,i')$ such that $i \prec i'$.
We will occasionally identify a set $L\subseteq{\mathcal{X}}$ with its indicator function by setting $L(x) = \mathbbm{1}_{[x \in L]}$.
\noindent For each $M \subseteq {\mathcal{X}}$, we define \[ M \oplus L=(L \setminus M) \cup (M \setminus L) \] and \[ M \oplus {\mathcal{L}} = \{M \oplus L: L\in{\mathcal{L}}\} \enspace . \] For $T \subseteq {\mathcal{X}}\times\{0,1\}$, we define similarly \[ M \oplus T = \{(x,\bar y): (x,y) \in T\mbox{ and } x \in M\} \cup \{(x,y) \in T: x \notin M\} \enspace . \] Moreover, given $M\subseteq{\mathcal{X}}$ and a linear ordering $\prec$ on ${\mathcal{L}}$, we define a linear ordering $\prec_M$ on $M \oplus {\mathcal{L}}$ as follows: \[ M \oplus L' \prec_M M \oplus L \Longleftrightarrow \underbrace{M \oplus (M \oplus L')}_{=L'} \prec \underbrace{M \oplus (M \oplus L)}_{=L} \enspace . \]
\begin{lemma} \label{lem:flip} With this notation, the following holds. If the mapping ${\mathcal{L}} \ni L \mapsto T(L) \subseteq{\mathcal{X}}\times\{0,1\}$ assigns a teaching set to $L$ w.r.t.~$({\mathcal{L}},\prec)$, then the mapping $M \oplus {\mathcal{L}} \ni M \oplus L \mapsto M \oplus T(L) \subseteq {\mathcal{X}}\times\{0,1\}$ assigns a teaching set to $M \oplus L$ w.r.t.~$(M\oplus{\mathcal{L}},\prec_M)$. \end{lemma} Since this result is rather obvious, we skip its proof.
We say that ${\mathcal{L}}$ and ${\mathcal{L}}'$ are {\em equivalent} if ${\mathcal{L}}' = M\oplus{\mathcal{L}}$ for some $M\subseteq{\mathcal{X}}$ (and this clearly is an equivalence relation). As an immediate consequence of Lemma~\ref{lem:flip}, we obtain the following result:
\begin{lemma} If ${\mathcal{L}}$ is equivalent to ${\mathcal{L}}'$, then $\mathrm{PBTD}({\mathcal{L}}) = \mathrm{PBTD}({\mathcal{L}}')$. \end{lemma}
The following lemma provides a necessary condition for a concept class to have a preference-based teaching dimension of one.
\begin{lemma} \label{lem:single-occurrence} Suppose that ${\mathcal{L}} \seq2^{\mathcal{X}}$ is a concept class of $\mathrm{PBTD}$ $1$. Pick a linear ordering $\prec$ on ${\mathcal{L}}$ and a mapping ${\mathcal{L}} \ni L \mapsto (x_L,y_L) \in {\mathcal{X}}\times\{0,1\}$ such that, for every $L\in{\mathcal{L}}$, $\{(x_L,y_L)\}$ is a teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec)$. Then \begin{itemize} \item either every instance $x\in{\mathcal{X}}$ occurs at most once in $(x_L)_{L\in{\mathcal{L}}}$ \item or there exists a concept $L^*\in{\mathcal{L}}$ that is preferred over all other concepts in ${\mathcal{L}}$ and $x_{L^*}$ is the only instance from ${\mathcal{X}}$ that occurs twice in $(x_L)_{L\in{\mathcal{L}}}$. \end{itemize} \end{lemma}
\begin{proof} Since the mapping $T$ must be injective, no instance can occur twice in $(x_L)_{L\in{\mathcal{L}}}$ with the same label. Suppose that there exists an instance $x\in{\mathcal{X}}$ and concepts $L \prec L^*$ such that $x = x_L = x_{L^*}$ and, w.l.o.g., $y_L=1$ and $y_{L^*}=0$. Since $\{(x,1)\}$ is a teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec)$, every concept $L' \succ L$ (including the ones that are preferred over $L^*$) must satisfy $L'(x) = 0$. For analogous reasons, every concept $L' \succ L^*$ (if any) must satisfy $L'(x)=1$. A concept $L'\in{\mathcal{L}}$ that is preferred over $L^*$ would have to satisfy $L'(x)=0$ and $L'(x)=1$, which is impossible. It follows that there can be no concept that is preferred over~$L^*$. \end{proof}
The following result is a consequence of Lemmas~\ref{lem:flip} and~\ref{lem:single-occurrence}.
\begin{theorem} \label{th:pbtd1-equivalence} If $\mathrm{PBTD}({\mathcal{L}})=1$, then there exists a concept class ${\mathcal{L}}'$ that is equivalent to ${\mathcal{L}}$ and satisfies $\mathrm{PBTD}({\mathcal{L}}')=\mathrm{PBTD}^+({\mathcal{L}}')=1$. \end{theorem}
\begin{proof} Pick a linear ordering $\prec$ on ${\mathcal{L}}$ and, for every $L\in{\mathcal{L}}$, a pair $(x_L,y_L) \in {\mathcal{X}}\times\{0,1\}$ such that $T(L)=\{(x_L,y_L)\}$ is a teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec)$. \begin{description} \item[Case 1:] Every instance $x\in{\mathcal{X}}$ occurs at most once in $(x_L)_{L\in{\mathcal{L}}}$. \\ Then choose $M = \{x_L: y_L=0\}$ and apply Lemma~\ref{lem:flip}. \item[Case 2:] There exists a concept $L^*\in{\mathcal{L}}$ that is preferred over all other concepts in ${\mathcal{L}}$ and $x_{L^*}$ is the only instance from ${\mathcal{X}}$ that occurs twice in $(x_L)_{L\in{\mathcal{L}}}$. \\ Then choose $M = \{x_L: y_L=0 \wedge L \neq L^*\}$ and apply Lemma~\ref{lem:flip}. With this choice, we obtain $M \oplus T(L) = \{(x_L,1)\}$ for every $L\in{\mathcal{L}}\setminus\{L^*\}$. Since $L^*$ is preferred over all other concepts in ${\mathcal{L}}$, we may teach $L^*$ w.r.t.~$({\mathcal{L}},\prec)$ by the empty set (instead of employing a possibly $0$-labeled example). \end{description} The discussion shows that there is a class ${\mathcal{L}}'$ that is equivalent to ${\mathcal{L}}$ and can be taught in the preference-based model with positive teaching sets of size $1$ (or size $0$ in case of $L^*$). \end{proof}
We now have the tools required for characterizing the concept classes whose positive PBTD equals $1$.
\begin{theorem} \label{th:pbtd-plus1} $\mathrm{PBTD}^+({\mathcal{L}})=1$ if and only if there exists a mapping ${\mathcal{L}} \ni L \mapsto x_L \in {\mathcal{X}}$ such that the matrix $A\in\{0,1\}^{({\mathcal{L}}\setminus\{\emptyset\})\times({\mathcal{L}}\setminus\{\emptyset\})}$ given by $A(L,L') = L'(x_L)$ is lower-triangular. \end{theorem}
\begin{proof} Suppose first that $\mathrm{PBTD}^+({\mathcal{L}})=1$. Pick a linear ordering $\prec$ on ${\mathcal{L}}$ and, for every $L\in{\mathcal{L}}\setminus\{\emptyset\}$, pick $x_L\in{\mathcal{X}}$ such that $\{x_L\}$ is a positive teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec)$.\footnote{Such an $x_L$ always exists, even if $\emptyset$ is a teaching set for $L$, because every superset of a teaching set for $L$ that is still consistent with $L$ is still a teaching set for $L$, cf.\ the discussion immediately after Lemma~\ref{lem:extension}.} If $L \prec L'$ (so that $L'$ is preferred over $L$), we must have $L'(x_L)=0$. It follows that the matrix $A$, as specified in the theorem, is lower-triangular.
Suppose conversely that there exists a mapping ${\mathcal{L}} \ni L \mapsto x_L \in {\mathcal{X}}$ such that the matrix $A\in\{0,1\}^{({\mathcal{L}}\setminus\{\emptyset\})\times({\mathcal{L}}\setminus\{\emptyset\})}$ given by $A(L,L') = L'(x_L)$ is lower-triangular, say w.r.t.~the linear ordering $\prec$ on ${\mathcal{L}}\setminus\{\emptyset\}$. Then, for every $L\in{\mathcal{L}}\setminus\{\emptyset\}$, the singleton $\{x_L\}$ is a positive teaching set for $L$ w.r.t.~$({\mathcal{L}},\prec)$ because it distinguishes $L$ from $\emptyset$ (of course) and also from every concept $L' \in {\mathcal{L}}\setminus\{\emptyset\}$ such that $L' \succ L$. If $\emptyset \in{\mathcal{L}}$, then extend the linear ordering $\prec$ by preferring $\emptyset$ over every other concept from ${\mathcal{L}}$ (so that $\emptyset$ is a positive teaching set for $\emptyset$ w.r.t.~$({\mathcal{L}},\prec)$). \end{proof}
In view of Theorem~\ref{th:pbtd1-equivalence}, Theorem~\ref{th:pbtd-plus1} characterizes every class ${\mathcal{L}}$ with $\mathrm{PBTD}({\mathcal{L}})=1$ up to equivalence.
Let $\mathrm{Sg}({\mathcal{X}}) = \{\{x\}: x \in {\mathcal{X}}\}$ denote the class of singletons over ${\mathcal{X}}$ and suppose that $\mathrm{Sg}({\mathcal{X}})$ is a sub-class of ${\mathcal{L}}$ and $\mathrm{PBTD}({\mathcal{L}})=1$. We will show that only fairly trivial extensions of $\mathrm{Sg}({\mathcal{X}})$ with a preference-based dimension of $1$ are possible.
\begin{lemma} \label{lem:pbtd1-implications} Let ${\mathcal{L}}\seq2^{\mathcal{X}}$ be a concept class of $\mathrm{PBTD}$ $1$ that contains $\mathrm{Sg}({\mathcal{X}})$. Let $T$ be an admissible mapping for ${\mathcal{L}}$ that assigns a labeled example $(x_L,y_L)\in{\mathcal{X}}\times\{0,1\}$ to each $L\in{\mathcal{L}}$. For $b=0,1$, let ${\mathcal{L}}^b = \{L\in{\mathcal{L}}: y_L=b\}$. Similarly, let ${\mathcal{X}}^b = \{x\in{\mathcal{X}}: y_{\{x\}}\in{\mathcal{L}}^b\}$. With this notation, the following holds: \begin{enumerate} \item If $L\in{\mathcal{L}}^1$ and $L \subset L' \in {\mathcal{L}}$, then $L'\in{\mathcal{L}}^1$. \item If $L'\in{\mathcal{L}}^0$ and $L' \supset L \in {\mathcal{L}}$, then $L\in{\mathcal{L}}^0$. \item
$|{\mathcal{X}}^0| \le 2$. Moreover if $|{\mathcal{X}}^0|=2$, then there exist $q \neq q' \in{\mathcal{X}}$ such that ${\mathcal{X}}^0=\{q,q'\}$ and $x_{\{q\}} = q'$. \end{enumerate} \end{lemma}
\begin{proof} Recall that $R_T = \{(L,L')\in{\mathcal{L}}\times{\mathcal{L}}:\ (L \neq L') \wedge (\mbox{$L$ is consistent with $T(L')$})\}$ and that $R_T$ (and even the transitive closure of $R_T$) is asymmetric if $T$ is admissible. \begin{enumerate} \item If $L \in {\mathcal{L}}^1$ and $L \subset L'$, then $y_L=1$ so that $L'$ is consistent with the example $(x_L,y_L)$. It follows that $(L',L) \in R_T$. $L' \in {\mathcal{L}}^0$ would similarly imply that $(L,L') \in R_T$ so that $R_T$ would not be asymmetric. This is in contradiction with the admissibility of $T$. \item The second assertion in the lemma is a logically equivalent reformulation of the first assertion. \item Suppose for the sake of contradiction that ${\mathcal{X}}^0$ contains three distinct points, say $q_1,q_2,q_3$. Since, for $i=1,2,3$, $T$ assigns a $0$-labeled example to $\{q_i\}$, at least one of the remaining two points is consistent with $T(\{q_i\})$. Let $G$ be the digraph with the nodes $q_1,q_2,q_3$ and with an edge from $q_j$ to $q_i$ iff $\{q_j\}$ is consistent with $T(\{q_i\})$. Then each of the three nodes has an indegree of at least $1$. Digraphs of this form must contain a cycle so that $\mathrm{trcl}(R_T)$ is not asymmetric. This is in contradiction with the admissibility of $R_T$.
A similar argument holds if ${\mathcal{X}}^0$ contains only two distinct elements, say $q$ and $q'$. If neither $x_{\{q\}} = q'$ nor $x_{\{q'\}} = q$, then $(\{q'\},\{q\}) \in R_T$ and $(\{q\},\{q'\}) \in R_T$ so that $R_T$ is not asymmetric --- again a contradiction to the admissibility of $R_T$. \end{enumerate} \end{proof}
\noindent We are now in the position to characterize those classes of $\mathrm{PBTD}$ one that contain all singletons.
\begin{theorem} \label{th:pbtd1} Suppose that ${\mathcal{L}}\seq2^{\mathcal{X}}$ is a concept class that contains $\mathrm{Sg}({\mathcal{X}})$. Then $\mathrm{PBTD}({\mathcal{L}})=1$ if and only if the following holds. Either ${\mathcal{L}}$ coincides with $\mathrm{Sg}({\mathcal{X}})$ or ${\mathcal{L}}$ contains precisely one additional concept, which is either the empty set or a set of size $2$. \end{theorem}
\begin{proof} We start with proving ``${\Leftarrow}$''. It is well known that $\mathrm{PBTD}^+({\mathcal{L}})=1$ for ${\mathcal{L}} = \mathrm{Sg}({\mathcal{X}})\cup\{\emptyset\}$: prefer $\emptyset$ over any singleton set, set $T(\emptyset)=\emptyset$ and, for every $x\in{\mathcal{X}}$, set $T(\{x\})=\{(x,1)\}$. In a similar fashion, we can show that $\mathrm{PBTD}({\mathcal{L}})=1$ for ${\mathcal{L}} = \mathrm{Sg}({\mathcal{X}})\cup\{\{q,q'\}\}$ for any choice of $q \neq q' \in {\mathcal{X}}$. Prefer $\{q,q'\}$ over $\{q\}$ and $\{q'\}$, respectively. Furthermore, prefer $\{q\}$ and $\{q'\}$ over all other singletons. Finally, set $T(\{q,q'\})=\emptyset$, $T(\{q\})=\{(q',0)\}$, $T(\{q'\})=\{(q,0)\}$ and, for every $x\in{\mathcal{X}}\setminus\{q,q'\}$, set $T(\{x\}) = \{(x,1)\}$.
As for the proof of ``${\Rightarrow}$'', we make use of the notions $T,x_L,y_L,{\mathcal{L}}^0,{\mathcal{L}}^1,{\mathcal{X}}^0,{\mathcal{X}}^1$ that had been introduced in Lemma~\ref{lem:pbtd1-implications} and we proceed by case analysis.
\begin{description} \item[Case 1:] ${\mathcal{X}}^0 = \emptyset$. \\ Since ${\mathcal{X}}^0 = \emptyset$, we have ${\mathcal{X}}={\mathcal{X}}^1$. In combination with the first assertion in Lemma~\ref{lem:pbtd1-implications}, it follows that ${\mathcal{L}}\setminus\{\emptyset\} = {\mathcal{L}}^1$. We claim that no concept in ${\mathcal{L}}$ contains two distinct elements. Assume for the sake of contradiction that there is a concept $L\in{\mathcal{L}}$
such that $|L|\ge2$. It follows that, for every $q \in L$, $x_{\{q\}}=q$ and $y_{\{q\}}=1$ so that $(L,\{q\}) \in R_T$. Moreover, there exists $q_0 \in L$ such that $x_L=q_0$ and $y_L=1$. It follows that $(\{q_0\},L) \in R_T$, which contradicts the fact that $R_T$ is asymmetric. \item[Case 2:] ${\mathcal{X}}^0 = \{q\}$ for some $q\in{\mathcal{X}}$. \\ Set $q' = x_{\{q\}}$ and note that $y_{\{q\}}=0$. Moreover, since ${\mathcal{X}}^1 = {\mathcal{X}}\setminus\{q\}$, we have $x_{\{p\}}=p$ and $y_{\{p\}}=1$
for every $p\in{\mathcal{X}}\setminus\{q\}$. We claim that ${\mathcal{L}}$ cannot contain a concept $L$ of size at least $2$ that contains an element of ${\mathcal{X}}\setminus\{q,q'\}$. Assume for the sake of contradiction, that there is a set $L$ such that $|L|\ge2$ and $p \in L$ for some $p\in{\mathcal{X}}\setminus\{q,q'\}$. The first assertion in Lemma~\ref{lem:pbtd1-implications} implies that $y_L=1$ (because $y_{\{p\}}=1$ and $\{p\} \subseteq L$). Since all pairs $(x,1)$ with $x \neq q$ are already in use for teaching the corresponding singletons, we may conclude that $q \in L$ and $T(L) = \{(q,1)\}$. This contradicts the fact that $\mathrm{trcl}(R_T)$ is asymmetric, because our discussion implies that $(L,\{p\}) , (\{p\},\{q\}) , (\{q\},L) \in R_T$. We may therefore safely assume that there is no concept of size at least $2$ in ${\mathcal{L}}$ that has a non-empty intersection with ${\mathcal{X}}\setminus\{q,q'\}$. Thus, except for the singletons, the only remaining sets that possibly belong to ${\mathcal{L}}$ are $\emptyset$ and $\{q,q'\}$. We still have to show that not both of them can belong to ${\mathcal{L}}$. Assume for the sake of contradiction that $\emptyset,\{q,q'\}\in{\mathcal{L}}$. Since $\emptyset$ is consistent with $T(\{q\})=\{(q',0)\}$, we have $(\emptyset,\{q\}) \in R_T$. Clearly, $y_\emptyset=0$. Since $\{q\}$ is consistent with every pair $(x,0)$ except for $(q,0)$, we must have $x_\emptyset = q$. (Otherwise, we have $(\{q\},\emptyset) \in R_T$ and arrive at a contradiction.) Let us now inspect the possible teaching sets for $L = \{q,q'\}$. Since $\{q,q'\}$ is consistent with $T(\{q'\}) = \{(q',1)\}$, setting $y_L = 0$ would lead to a contradiction. The example $(q',1)$ is already in use for teaching $\{q'\}$. It is therefore necessary to set $T(L) = \{(q,1)\}$. An inspection of the various teaching sets shows that $(\emptyset,\{q\}) , (\{q\},L) , (L,\{q'\}), (\{q'\},\emptyset) \in R_T$, which contradicts the fact that $\mathrm{trcl}(R_T)$ is asymmetric. \item[Case 3:] ${\mathcal{X}}^0=\{q,q'\}$ for some $q \neq q' \in {\mathcal{X}}$. \\ Note first that $y_{\{q\}}=y_{\{q'\}}=0$ and $y_{\{p\}}=1$ for every $p\in{\mathcal{X}}\setminus\{q,q'\}$. We claim that $\emptyset\notin{\mathcal{L}}$. Assume for the sake of contradiction that $\emptyset\in{\mathcal{L}}$. Then $(\emptyset,\{q\}) , (\emptyset,\{q'\}) \in R_T$ since $\emptyset$ is consistent with the teaching sets for instances from ${\mathcal{X}}^0$. But then, no matter how $x$ in $T(\emptyset) = \{(x,0)\}$ is chosen, at least one of the sets $\{q\}$ and $\{q'\}$ will be consistent with $T(\emptyset)$ so that at least one of the pairs $(\{q\},\emptyset)$ and $(\{q'\},\emptyset)$ belongs to $R_T$. This contradicts the fact that $R_T$ must be asymmetric. Thus $\emptyset\notin{\mathcal{L}}$, indeed. Now it suffices to show that ${\mathcal{L}}$ cannot contain a concept of size at least $2$ that contains an element of ${\mathcal{X}}\setminus\{q,q'\}$. Assume for the sake of contradiction that there is a set $L\in{\mathcal{L}}$
such that $|L|\ge2$ and $p \in L$ for some $p\in{\mathcal{X}}\setminus\{q,q'\}$. Observe that $(L,\{p\}) \in R_T$. Another application of the first assertion in Lemma~\ref{lem:pbtd1-implications} shows that $y_L=1$ (because $y_{\{p\}}=1$ and $p \in L$) and $x_L \in \{q,q'\}$ (because the other $1$-labeled instances are already in use for teaching the corresponding singletons). It follows that one of the pairs $(\{q\},L)$ and $(\{q'\},L)$ belongs to $R_T$. The third assertion of Lemma~\ref{lem:pbtd1-implications} implies that $T(q)=\{(q',0)\}$ or $T(q')=\{(q,0)\}$. For reasons of symmetry, we may assume that $T(q)=\{(q',0)\}$. This implies that $(\{p\},\{q\}) \in R_T$. Let $q''$ be given by $T(q') = \{(q'',0)\}$. Note that either $q''=q$ or $q''\in{\mathcal{X}}\setminus\{q,q'\}$. In the former case, we have that $(\{p\},\{q'\}) \in R_T$ and in the latter case we have that $(\{q\},\{q'\}) \in R_T$. Since $(\{p\},\{q\}) \in R_T$ (which was observed above already), we conclude that in both cases, $(\{p\},\{q\}) , (\{p\},\{q'\}) \in \mathrm{trcl}(R_T)$. Combining this with our observations above that $(L,\{p\} ) \in R_T$ and that one of the pairs $(\{q\},L)$ and $(\{q'\},L)$ belongs to $R_T$, yields a contradiction to the fact that $\mathrm{trcl}(R_T)$ is asymmetric. \end{description} \end{proof}
\begin{corollary} Let ${\mathcal{L}}\seq2^{\mathcal{X}}$ be a concept class that contains $\mathrm{Sg}({\mathcal{X}})$. If $\mathrm{PBTD}({\mathcal{L}})=1$, then $\mathrm{RTD}({\mathcal{L}})=1$. \end{corollary}
\begin{proof} According to Theorem~\ref{th:pbtd1}, either $L$ coincides with $\mathrm{Sg}({\mathcal{X}})$ or ${\mathcal{L}}$ contains precisely one additional concept that is $\emptyset$ or a set of size $2$. The partial ordering $\prec$ on ${\mathcal{L}}$ that is used in the first part of the proof of Theorem~\ref{th:pbtd1} (proof direction ``${\Leftarrow}$'') is easily compiled into a recursive teaching plan of order $1$ for ${\mathcal{L}}$.\footnote{This also follows from Lemma~\ref{lem:rtd-pbtd} and the fact that there are no chains of a length exceeding $2$ in $({\mathcal{L}},\prec)$.} \end{proof}
The characterizations proven above can be applied to certain geometric concept classes.
Consider a class ${\mathcal{L}}$, consisting of bounded and topologically closed objects in the $d$-dimensional Euclidean space, that satisfies the following condition: for every pair $(A,B)\in\mathbbm{R}^d$, there is exactly one object in ${\mathcal{L}}$, denoted as $L_{A,B}$
in the sequel, such that $A,B \in L$ and such that $\|A-B\|$ coincides with the diameter of $L$. This assumption implies
that $|{\mathcal{L}} \setminus \mathrm{Sg}(\mathbbm{R}^d)|=\infty$. By setting $A=B$, it furthermore implies $\mathrm{Sg}(\mathbbm{R}^d) \subseteq {\mathcal{L}}$. Let us prefer objects with a small diameter over objects with a larger diameter. Then, obviously, $\{A,B\}$ is a positive teaching
set for $L_{A,B}$. Because of $|{\mathcal{L}} \setminus \mathrm{Sg}(\mathbbm{R}^d)|=\infty$, ${\mathcal{L}}$ does clearly not satisfy the condition in Theorem~\ref{th:pbtd1}, which is necessary for ${\mathcal{L}}$ to have a PBTD of $1$. We may therefore conclude that $\mathrm{PBTD}({\mathcal{L}}) = \mathrm{PBTD}^+({\mathcal{L}}) = 2$.
The family of classes with the required properties is rich and includes, for instance, the class of $d$-dimensional balls as well as the class of $d$-dimensional axis-parallel rectangles.
\section{Conclusions}
Preference-based teaching uses the natural notion of preference relation to extend the classical teaching model. The resulting model is (i) more powerful than the classical one, (ii) resolves difficulties with the recursive teaching model in the case of infinite concept classes, and (iii) is at the same time free of coding tricks even according to the definition by \cite{GM1996}. Our examples of algebraic and geometric concept classes demonstrate that preference-based teaching can be achieved very efficiently with naturally defined teaching sets and based on intuitive preference relations such as inclusion. We believe that further studies of the PBTD will provide insights into structural properties of concept classes that render them easy or hard to learn in a variety of formal learning models.
We have shown that spanning sets lead to a general-purpose construction for \linebreak[4]preference-based teaching sets of only positive examples. While this result is fairly obvious, it provides further justification of the model of preference-based teaching, since the teaching sets it yields are often intuitively exactly those a teacher would choose in the classroom (for instance, one would represent convex polygons by their vertices, as in Example~\ref{ex:polygons}). It should be noted, too, that it can sometimes be difficult to establish whether the upper bound on PBTD obtained this way is tight, or whether the use of negative examples or preference relations other than inclusion yield smaller teaching sets.
Generally, the choice of preference relation provides a degree of freedom that increases the power of the teacher but also increases the difficulty of establishing lower bounds on the number of examples required for teaching.
\noindent\textbf{Acknowledgements.} Sandra Zilles was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), in the Discovery Grant and Canada Research Chairs programs. We thank the anonymous referees for their numerous thoughtful comments, which greatly helped to improve the presentation of the paper.
\appendix
\iffalse \section{Proof of Lemma~\ref{lem:huge-gap}} \label{app:rtd} \fi
\section{Proof of Theorem~\ref{th:bounds-linset}} \label{app:linsets}
In Section~\ref{subsec:shift-lemma}, we present a general result which helps to verify the upper bounds in Theorem~\ref{th:bounds-linset}. These upper bounds are then derived in Section~\ref{subsec:linset-ubs}. Section~\ref{subsec:linset-lbs} is devoted to the derivation of the lower bounds.
\subsection{The Shift Lemma} \label{subsec:shift-lemma}
In this section, we assume that ${\mathcal{L}}$ is a concept class over a universe ${\mathcal{X}} \in \{{\mathbb N}_0,{\mathbb Q}_0^+,\mathbbm{R}_0^+\}$. We furthermore assume that $0$ is contained in every concept $L \in {\mathcal{L}}$. We can extend ${\mathcal{L}}$ to a larger class, namely the shift-extension ${\mathcal{L}}'$ of ${\mathcal{L}}$, by allowing each of its concepts to be shifted by some constant which is taken from ${\mathcal{X}}$: \[ {\mathcal{L}}' = \{c+L:\ (c\in{\mathcal{X}}) \wedge (L\in{\mathcal{L}})\} \enspace . \] The next result states that this extension has little effect only on the complexity measures $\mathrm{PBTD}$ and $\mathrm{PBTD}^+$:
\begin{lemma} [Shift Lemma]\label{lem:shift} With the above notation and assumptions, the following holds: \[ \mathrm{PBTD}({\mathcal{L}}) \le \mathrm{PBTD}({\mathcal{L}}') \le 1+\mathrm{PBTD}({\mathcal{L}})\ \mbox{ and }\ \mathrm{PBTD}^+({\mathcal{L}}) \le \mathrm{PBTD}^+({\mathcal{L}}') \le 1+\mathrm{PBTD}^+({\mathcal{L}}) \enspace . \] \end{lemma}
\begin{proof} It suffices to verify the inequalities $\mathrm{PBTD}({\mathcal{L}}') \le 1+\mathrm{PBTD}({\mathcal{L}})$ and $\mathrm{PBTD}^+$ $({\mathcal{L}}') \le 1+\mathrm{PBTD}^+({\mathcal{L}})$ because the other inequalities hold by virtue of monotonicity. Let $T$ be an admissible mapping for ${\mathcal{L}}$. It suffices to show that $T$ can be transformed into an admissible mapping $T'$ for ${\mathcal{L}}'$ such that $\mathrm{ord}(T') \le 1+\mathrm{ord}(T)$ and such that $T'$ is positive provided that $T$ is positive. To this end, we define $T'$ as follows: \[ T'(c+L) = \{(c,+)\} \cup \{(c+x,b):\ (x,b) \in T(L)\} \enspace . \] Obviously $\mathrm{ord}(T') \le 1+\mathrm{ord}(T)$. Note that $c \in c+L$ because of our assumption that $0$ is contained in every concept in ${\mathcal{L}}$. Moreover, since the admissibility of $T$ implies that $L$ is consistent with $T(L)$, the above definition of $T'(c+L)$ makes sure that $c+L$ is consistent with $T'(c+L)$. It suffices therefore to show that the relation $\mathrm{trcl}(R_{T'})$ is asymmetric. Consider a pair $(c'+L',c+L) \in R_{T'}$. By the definition of $R_{T'}$, it follows that $c'+L'$ is consistent with $T'(c+L)$. Because of $(c,+) \in T'(c+L)$, we must have $c' \le c$. Suppose that $c'=c$. In this case, $L'$ must be consistent with $T(L)$. Thus $L' \prec_T L$. This reasoning implies that $(c'+L',c+L) \in R_{T'}$ can happen only if either $c'<c$ or $(c'=c) \wedge (L' \prec_T L)$. Since $\prec_T$ is asymmetric, we may now conclude that $\mathrm{trcl}(R_{T'})$ is asymmetric, as desired. Finally note that, according to our definition above, the mapping $T'$ is positive provided that $T$ is positive. This concludes the proof. \end{proof}
\subsection{The Upper Bounds in Theorem~\ref{th:bounds-linset}} \label{subsec:linset-ubs}
We remind the reader that the equality $\mathrm{PBTD}^+(\mathrm{LINSET}_k)=k$ was stated in Example~\ref{exmp:pbtdpluslinset}. We will show in Lemma~\ref{lem:nelinset-ub} that $\mathrm{PBTD}^+(\mathrm{NE}\mbox{-}\mathrm{LINSET}_k)\le k$. In combination with the Shift Lemma, this implies that $\mathrm{PBTD}^+(\mathrm{LINSET}'_k) \le k+1$ and $\mathrm{PBTD}^+(\mathrm{NE}\mbox{-}\mathrm{LINSET}'_k) \le k+1$. All remaining upper bounds in Theorem~\ref{th:bounds-linset} follow now by virtue of monotonicity.
\begin{lemma}\label{lem:nelinset-ub} $\mathrm{PBTD}^+(\mathrm{NE}\mbox{-}\mathrm{LINSET}_k)\le k$. \end{lemma}
\begin{proof} We want to show that there is a preference relation for which $k$ positive examples suffice to teach any concept in $\mathrm{NE}\mbox{-}\mathrm{LINSET}_k$. To this end, let $G=\{g_1,\ldots,g_\ell\}$ be a generator set with $\ell \le k$ where $g_1 < \ldots < g_\ell$. We use $\mathrm{sum}(G) = g_1 + \ldots + g_\ell$ to denote the sum of all generators in $G$. We say that $g_i$ is a {\em redundant generator} in $G$ if $g_i \in \spn{\{g_1,\ldots,g_{i-1}\}}$. Let $G^* = \{g_1^*,\ldots,g^*_{\ell^*}\} \subseteq G$ with $g^*_1 < \ldots < g^*_{\ell^*}$ be the set of non-redundant generators in $G$ and let $\mathrm{tuple}(G) = (g_1^*,\ldots,g_{\ell^*}^*)$ be the corresponding ordered sequence. Then $G^*$ is an independent subset of $G$ generating the same linear set as $G$ when allowing zero coefficients, i.e., we have $\spn{G^*} = \spn{G}$ (although $\spn{G^*}_+ \neq \spn{G}_+$ whenever $G^*$ is a proper subset of $G$).
To define a suitable preference relation, let $G,\widehat G$ be generator sets of size $k$ or less with $\mathrm{tuple}(G) = (g^*_1,\ldots,g^*_{\ell^*})$ and $\mathrm{tuple}(\widehat G) = (\widehat g^*_1,\ldots,\widehat g^*_{\widehat\ell^*})$. Let the student prefer $G$ over $\widehat G$ if any of the following conditions is satisfied: \begin{description} \item[Condition 1:] $\mathrm{sum}(G)>\mathrm{sum}(\widehat G)$. \item[Condition 2:] $\mathrm{sum}(G)=\mathrm{sum}(\widehat G)$ and $\mathrm{tuple}(G)$ is lexicographically greater than \linebreak[4]$\mathrm{tuple}$ $(\widehat G)$ without having $\mathrm{tuple}(\widehat G)$ as prefix. \item[Condition 3:] $\mathrm{sum}(G)=\mathrm{sum}(\widehat G)$ and $\mathrm{tuple}(G)$ is a proper prefix of $\mathrm{tuple}(\widehat G)$. \end{description}
To teach a concept $\spn{G}\in\mathrm{NE}\mbox{-}\mathrm{LINSET}_k$ with $\mathrm{sum}(G)=g$ and $\mathrm{tuple}(G) = (g_1^*,\ldots,g_{\ell^*}^*)$, one uses the teaching set \[ S = \{ (g,+) , (g+g_1^*,+) ,\ldots, (g+g_{h^*}^*,+) \} \] where \begin{equation} \label{eq:def-h} h = \left\{ \begin{array}{ll}
\ell^*-1 & \mbox{if $G^*=G$} \\
\ell^* & \mbox{if $G^* \subset G$}
\end{array} \right. \enspace . \end{equation}
Note that $S$ contains at most $|G| \le k$ examples. Let $\widehat G$ with $\spn{\widehat G}_+ \in \mathrm{NE}\mbox{-}\mathrm{LINSET}_k$ denote the generator set that is returned by the student. Clearly $\spn{\widehat G}$ satisfies $\mathrm{sum}(\widehat G)=g$ since \begin{itemize} \item concepts with larger generator sums are inconsistent with $(g,+)$, and \item concepts with smaller generator sums have a lower preference (compare with Condition~1 above). \end{itemize} It follows that $g+g^*_i \in \spn{\widehat G}_+$ is equivalent to $g^*_i \in \spn{\widehat G} = \spn{\widehat G^*}$. We conclude that the smallest generator in $\mathrm{tuple}(\widehat G)$ equals $g^*_1$ since \begin{itemize} \item a smallest generator in $\mathrm{tuple}(\widehat G)$ that is greater than $g^*_1$ would cause an inconsistency with $(g+g^*_1,+)$, and \item a smallest generator in $\mathrm{tuple}(\widehat G)$ that is smaller than $g^*_1$ would have a lower preference (compare with Condition~2 above). \end{itemize} Assume inductively that the $i-1$ smallest generators in $\mathrm{tuple}(\widehat G)$ are $g^*_1,\ldots,g^*_{i-1}$. Since $g^*_i \notin \spn{\{g^*_1,\ldots,g^*_{i-1}\}}$, we may apply a reasoning that is similar to the above reasoning concerning $g^*_1$ and conclude that the $i$'th smallest generator in $\mathrm{tuple}(\widehat G)$ equals $g^*_i$. The punchline of this discussion is that the sequence $\mathrm{tuple}(\widehat G)$ starts with $g^*_1,\ldots,g^*_{h}$ with $h$ given by~(\ref{eq:def-h}). Let $G' = G \setminus G^*$ be the set of redundant generators in $G$ and note that \[ g - \sum_{i=1}^{h}g^*_i = \left\{ \begin{array}{ll}
g^*_{\ell^*} & \mbox{if $G^* = G$} \\
\sum_{g' \in G'}g' & \mbox{if $G^* \subset G$} \\
\end{array} \right. \enspace . \] Let $\widehat G' = \widehat G \setminus \{g^*_1,\ldots,g^*_h\}$. We proceed by case analysis: \begin{description} \item[Case 1:] $G^*=G$. \\ Since $\widehat G$ is consistent with $(g,+)$, we have $\sum_{g' \in \widehat G'}g' = g_{\ell^*}^*$. Since $g^*_{\ell^*} \notin \langle\{g^*_1,\ldots,$ $g^*_{\ell^*-1}\}\rangle$, the set $\widehat G'$ must contain an element that cannot be generated by $g^*_1,\ldots,$ $g^*_{\ell^*-1}$. Given the preferences of the student (compare with Condition~2), she will choose $\widehat G' = \{g^*_{\ell^*}\}$. It follows that $\widehat G = G$. \item[Case 2:] $G^* \subset G$. \\ Here, we have $\sum_{g' \in \widehat G'}g' = \sum_{g' \in G'}g'$. Given the preferences of the student (compare with Condition~3), she will choose $\widehat G$ such that $\widehat G^* = G^*$ and $\widehat G'$ consists of elements from $\spn{G^*}$ that sum up to $\sum_{g' \in G'}g'$ (with $\widehat G' = \left\{\sum_{g' \in G'}g'\right\}$ among the possible choices). Clearly, $\spn{\widehat G}_+ = \spn{G}_+$. \end{description} Thus, in both cases, the student comes up with the right hypothesis. \end{proof}
\subsection{The Lower Bounds in Theorem~\ref{th:bounds-linset}} \label{subsec:linset-lbs}
The lower bounds in Theorem~\ref{th:bounds-linset} are an immediate consequence of the following result:
\begin{lemma} \label{lem:lbs-linset} The following lower bounds are valid: \begin{eqnarray} \mathrm{PBTD}^+(\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k) & \ge & k+1 \enspace . \label{eq:lb1} \\ \mathrm{PBTD}(\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k) & \ge & k-1 \enspace . \label{eq:lb2} \\ \mathrm{PBTD}(\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k) & \ge & \frac{k-1}{2} \enspace . \label{eq:lb3} \\ \mathrm{PBTD}(\mathrm{CF}\mbox{-}\mathrm{LINSET}_k) & \ge & k-1 \enspace . \label{eq:lb4} \\ \mathrm{PBTD}^+(\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k) & \ge & k-1 \enspace . \label{eq:lb5} \end{eqnarray} \end{lemma}
This lemma can be seen as an extension and a strengthening of a similar result in~\cite{GSZ2015} where the following lower bounds were shown: \begin{eqnarray*} \mathrm{RTD}^+(\mathrm{NE}\mbox{-}\mathrm{LINSET}'_k) & \ge & k+1 \enspace .\\ \mathrm{RTD}(\mathrm{NE}\mbox{-}\mathrm{LINSET}'_k) & \ge & k-1 \enspace .\\ \mathrm{RTD}(\mathrm{CF}\mbox{-}\mathrm{LINSET}_k) & \ge & k-1 \enspace . \end{eqnarray*} The proof of Lemma~\ref{lem:lbs-linset} builds on some ideas that are found in~\cite{GSZ2015} already, but it requires some elaboration to obtain the stronger results.
We now briefly explain why the lower bounds in Theorem~\ref{th:bounds-linset} directly follow from Lemma~\ref{lem:lbs-linset}. Note that the lower bound $k-1$ in~(\ref{eq:more-bounds}) is immediate from~(\ref{eq:lb2}) and a monotonicity argument. This is because $\mathrm{NE}\mbox{-}\mathrm{LINSET}'_k\supseteq\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k$ as well as $\mathrm{LINSET}'_k\supseteq\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k\supseteq\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k$. Note furthermore that $\mathrm{PBTD}^+$ $(\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k) \ge k+1$ because of~(\ref{eq:lb1}) and a monotonicity argument. Then the Shift Lemma implies that $\mathrm{PBTD}^+(\mathrm{CF}\mbox{-}\mathrm{LINSET}_k) \ge k$. Similarly, $\mathrm{PBTD}^+(\mathrm{NE}\mbox{-}\mathrm{LINSET}_k$ $)$ $\geq k-1$ follows from~(\ref{eq:lb5}) and a monotonicity argument. All remaining lower bounds in Theorem~\ref{th:bounds-linset} are obtained from these observations by virtue of monotonicity.
The proof of Theorem~\ref{th:bounds-linset} can therefore be accomplished by proving Lemma~\ref{lem:lbs-linset}. It turns out that the proof of this lemma is quite involved. We will present in Section~\ref{subsubsec:lb0} some theoretical prerequisites. Sections~\ref{subsubsec:lb1} and~\ref{subsubsec:lb234} are devoted to the actual proof of the lemma.
\subsubsection{Some Basic Concepts in the Theory of Numerical Semigroups} \label{subsubsec:lb0}
Recall from Section~\ref{sec:linsets} that $\spn{G} = \left\{\sum_{g \in G}a(g)g: a(g)\in{\mathbb N}_0\right\}$. The elements of $G$ are called {\em generators} of $\spn{G}$. A set $P \subset {\mathbb N}$ is said to be {\em independent} if none of the elements in $P$ can be written as a linear combination (with coefficients from ${\mathbb N}_0$) of the remaining elements (so that $\spn{P'}$ is a proper subset of $\spn{P}$ for every proper subset $P'$ of $P$). It is well known~\cite{RG-S2009} that independence makes generating systems unique, i.e., if $P,P'$ are independent, then $\spn{P} = \spn{P'}$ implies that $P=P'$. Moreover, for every independent set $P$, the following implication is valid: \begin{equation} \label{eq:independent-set} (S \subseteq \spn{P} \wedge P \not\subseteq S)\ \Rightarrow\ (\spn{S} \subset \spn{P}) \enspace . \end{equation}
Let $P = \{a_1,\ldots,a_k\}$ be independent with $a_1 = \min P$. It is well known\footnote{E.g., see~\cite{RG-S2009}} and easy to see that the residues of $a_1,a_2,\ldots,a_k$ modulo $a_1$ must be pairwise distinct (because, otherwise, we would obtain
a dependence). If $a_1$ is a prime and $|P|\ge2$, then the independence of $P$ implies that $\gcd(P)=1$. Thus the following holds:
\begin{lemma} \label{lem:ind-gcd} If $P \subset {\mathbb N}$ is an independent set of cardinality at least $2$ and $\min P$ is a prime, then $\gcd(P)=1$. \end{lemma}
\noindent In the remainder of the paper, the symbols $P$ and $P'$ are reserved for denoting independent sets of generators.
It is well known that $\spn{G}$ is co-finite iff $\gcd(G)=1$~\cite{RG-S2009}. Let $P$ be a finite (independent) subset of ${\mathbb N}$ such that $\gcd(P)=1$. The largest number in ${\mathbb N}\setminus\spn{P}$ is called the {\em Frobenius number of $P$} and is denoted as $F(P)$. It is well known~\cite{RG-S2009} that \begin{equation} \label{eq:frobenius} F(\{p,q\}) = pq-p-q \end{equation} provided that $p,q \ge 2$ satisfy $\gcd(p,q)=1$.
\subsubsection{Proof of~(\ref{eq:lb1})} \label{subsubsec:lb1}
The shift-extension of $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k$ is (by way of definition) the following class: \begin{equation} \label{eq:necflinset-extension} \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k =
\{c+\spn{P}_+:\ (c\in{\mathbb N}_0) \wedge (P\subset{\mathbb N}) \wedge (|P| \le k) \wedge (\gcd(P)=1)\} \enspace . \end{equation} It is easy to see that this can be written alternatively in the form \begin{equation} \label{eq:necflinset-rewritten} \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k = \left\{N+\spn{P}:\
N\in{\mathbb N}_0 \wedge P\subset{\mathbb N} \wedge |P| \le k \wedge \gcd(P)=1 \wedge \sum_{p\in P}p \le N\right\} \end{equation} where $N$ in~(\ref{eq:necflinset-rewritten}) corresponds to $c+\sum_{p \in P}p$ in~(\ref{eq:necflinset-extension}).
For technical reasons, we define the following subfamilies of $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k$. For each $N\ge0$, let \[ \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k[N] = \{N+L : L\in\mathrm{LINSET}_k[N]\} \] where \[ \mathrm{LINSET}_k[N] = \left\{\spn{P}\in\mathrm{LINSET}_k : (\gcd(P)=1) \wedge \left(\sum_{p \in P}p \le N\right)\right\} \enspace . \] In other words, $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k[N]$ is the subclass consisting of all concepts in $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k$ (written in the form~(\ref{eq:necflinset-rewritten})) whose constant is $N$.
\noindent A central notion for proving~(\ref{eq:lb1}) is the following one: \begin{definition} \label{def:special-set} Let $k,N\ge2$ be integers. We say that a set $L\in\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'$ is {\em $(k,N)$-special} if it is of the form $L = N+\spn{P}$ such that the following holds: \begin{enumerate} \item $P$ is an independent set of cardinality $k$ and $\min P$ is a prime (so that $\gcd(P)=1$ according to Lemma~\ref{lem:ind-gcd}, which furthermore implies that $\spn{P}$ is co-finite). \item Let $q(P)$ denote the smallest prime that is greater than $F(P)$ and greater than $\max P$. For $a = \min P$ and $r=0,\ldots,a-1$, let \[ t_r(P) = \min \{s\in\spn{P} : s \equiv r \pmod{a}\}\ \mbox{ and }\ t_{max}(P) = \max_{0 \le r \le a-1}t_r(P) \enspace . \] Then \begin{equation} \label{eq:large-constant} N \ge k(a+t_{max}(P))\ \mbox{ and }\ N \ge q(P)+\sum_{p \in P\setminus\{a\}}p \enspace . \end{equation} \end{enumerate} \end{definition}
We need at least $k$ positive examples in order to distinguish a $(k,N)$-special set from all its proper subsets in $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k[N]$, as the following result shows:
\begin{lemma} \label{lem:special-sets} For all $k\ge2$, the following holds. If $L\in\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'$ is $(k,N)$-special, then $L \in \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'[N]$ and $I'(L,\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k[N]) \ge k$. \end{lemma}
\begin{proof} Suppose that $L = N+\spn{P}$ is of the form as described in Definition~\ref{def:special-set}. Let $P = \{a,a_2\ldots,a_k\}$ with $a = \min P$. For the sake of simplicity, we will write $t_r$ instead of $t_r(P)$ and $t_{max}$ instead of $t_{max}(P)$. The independence of $P$ implies that $t_{a_i \bmod a} = a_i$ for $i=2,\ldots,k$. It follows that $t_{max} \ge \max P$. Since, by assumption, $N \ge k \cdot t_{max}$, it becomes obvious that $L \in \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'[N]$.
Assume by way of contradiction that the following holds: \begin{itemize} \item[(A)] There is a weak spanning set $S$ of size $k-1$ for $L$ w.r.t.~$\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k[N]$. \end{itemize} Since $N$ is contained in any concept from $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k[N]$, we may assume that $N \notin S$ so that $S$ is of the form $S = \{N+x_1,\ldots,N+x_{k-1}\}$ for integers $x_i\ge1$. For $i = 1,\ldots,k-1$, let $r_i = x_i \bmod{a} \in \{0,1,\ldots,a-1\}$. It follows that each $x_i$ is of the form $x_i = q_ia+t_{r_i}$ for some integer $q_i\ge0$. Let $X = \{x_1,\ldots,x_{k-1}\}$. We proceed by case analysis: \begin{description} \item[Case 1:] $X \subseteq \{a_2,\ldots,a_k\}$
(so that, in view of $|X|=k-1$, we even have $X = \{a_2,\ldots,a_k\}$). \\ Let $L' = N+\spn{X}$. Then $S \subseteq L'$. Note that $X \subseteq P$ but $P \not\subseteq X$. We may conclude from~(\ref{eq:independent-set}) that $\spn{X} \subset \spn{P}$ and, therefore, $L' \subset L$. Thus $L'$ is a proper subset of $L$ which contains $S$. Note that~(\ref{eq:large-constant}) implies that $N \ge \sum_{i=2}^{k}a_i = \sum_{i=1}^{k-1}x_i$. If $\gcd(X)=1$, then $L' \in \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}[N]$ and we have an immediate contradiction to the above assumption~(A). Otherwise, if $\gcd(X)\ge2$, then we define $L'' = N+\spn{X\cup\{q(P)\}}$. Note that $S \subseteq L' \subseteq L''$. Since $q(P)>F(P)$, we have $X\cup\{q(P)\} \subseteq \spn{P}$ and, since $q(P) > \max P$, we have $P \not\subseteq X\cup\{q(P)\}$. We may conclude from~(\ref{eq:independent-set}) that $\spn{X\cup\{q(P)\}} \subset \spn{P}$ and, therefore, $L'' \subset L$. Thus, $L''$ is a proper subset of $L$ which contains $S$. Because $X = \{a_2,\ldots,a_k\}$ and $q(P)$ is a prime that is greater than $\max P$, it follows that $\gcd(X\cup\{q(P)\}) = 1$. In combination with~(\ref{eq:large-constant}), it easily follows now that $L'' \in \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}[N]$. Putting everything together, we arrive at a contradiction to the assumption~(A). \item[Case 2:] $X \not\subseteq \{a_2,\ldots,a_k\}$. \\ If $r_i=0$ for $i=1,\ldots,k-1$, then each $x_i$ is a multiple of $a$. In this case, $N+\spn{a,q(P)}$ is a proper subset of $L = N+\spn{P}$ that is consistent with $S$, which yields a contradiction. We may therefore assume that there exists $i' \in\{1,\ldots,k-1\}$ such that $r_{i'}\neq0$. From the case assumption, $X \not\subseteq \{a_2,\ldots,a_k\}$, it follows that there must exist an index $i''\in\{1,\ldots,k-1\}$ such that $q_{i''} \ge 1$ or $t_{r_{i''}}\notin\{a_2,\ldots,a_k\}$. For $i=1,\ldots,k-1$, let $q'_i = \min\{q_i,1\}$ and $x'_i = q'_ia+t_{r_i}$. Note that $q'_{i''}=1$ iff $q_{i''} \ge 1$. Define $L'' = N+\spn{X'}$ for $X' = \{a,x'_1,\ldots,x'_{k-1}\}$ and observe the following. First, the set $L''$ clearly contains $S$. Second, the choice of $x'_1,\ldots,x'_{k-1}$ implies that $X' \subseteq \spn{P}$. Third, it easily follows from $q'_{i''}=1$ or $t_{r_{i''}}\notin\{a_2,\ldots,a_k\}$ that $P \not\subseteq \{a,x'_1,\ldots,x'_{k-1}\}$. We may conclude from~(\ref{eq:independent-set}) that $\spn{X'} \subset \spn{P}$ and, therefore, $L'' \subset L$. Thus, $L''$ is a proper subset of $L$ which contains $S$. Since $r_{i'} \neq 0$ and $a$ is a prime, it follows that $\gcd(a,x'_{i'})=1$ and, therefore, $\gcd(X')=1$. In combination with~(\ref{eq:large-constant}), it easily follows now that $L'' \in \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}[N]$. Putting everything together, we obtain again a contradiction to the assumption~(A). \end{description} \end{proof}
For the sake of brevity, let ${\mathcal{L}} = \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'$. Assume by way of contradiction that there exists a positive mapping $T$ of order $k$ that is admissible for ${\mathcal{L}}_k$. We will pursue the following strategy: \begin{enumerate} \item We define a set $L \in {\mathcal{L}}_k$ of the form $L = N+p+\spn{1}$. \item We define a second set $L' = N+\spn{G} \in {\mathcal{L}}$ that is $(k,N)$-special and consistent with $T^+(L)$. Moreover, $L' \setminus L = \{N\}$. \end{enumerate} If this can be achieved, then the proof will be accomplished as follows: \begin{itemize} \item According to Lemma~\ref{lem:special-sets}, $T^+(L')$ must contain at least $k$ examples (all of which are different from $N$) for distinguishing $L'$ from all its proper subsets in ${\mathcal{L}}_k[N]$. \item Since $L'$ is consistent with $T^+(L)$, the set $T^+(L')$ must contain an example which distinguishes $L'$ from $L$. But the only example which fits this purpose is $(N,+)$. \item The discussion shows that $T^+(L')$ must contain $k$ examples in order to distinguish $L'$ from all its proper subsets in ${\mathcal{L}}_k$ plus one additional example, $N$, needed to distinguish $L'$ from $L$. \item We obtain a contradiction to our initial assumption that $T^+$ is of order $k$. \end{itemize} We still have to describe how our proof strategy can actually be implemented. We start with the definition of $L$. Pick the smallest prime $p \ge k+1$. Then $\{p,p+1,\ldots,p+k\}$ is independent. Let $M = F(\{p,p+1\}) \stackrel{(\ref{eq:frobenius})}{=} p(p+1)-p-(p+1)$. An easy calculation shows that $k\ge2$ and $p \ge k+1$ imply that $M \ge p+k$. Let $I = \{p,p+1,\ldots,M\}$. Choose $N$ large enough so that all concepts of the form
\[ N+\spn{P}\ \mbox{ where}\ |P|=k,\ p = \min P \mbox{ and } P \subseteq I \] are $(k,N)$-special. With these choices of $p$ and $N$, let $L = N+p+\spn{1}$. Note that $N+p,N+p+1 \in T^+(L)$ because, otherwise, one of the concepts $N+p+1+\spn{1},N+p+\spn{2,3} \subset L$ would be consistent with $T^+(L)$ whereas $T^+(L)$ must distinguish $L$ from all its proper subsets in
${\mathcal{L}}_k$. Setting $A = \{x: N+x \in T^+(L)\}$, it follows that $|A| = |T^+(L)| \le k$ and $p,p+1 \in A$. The set $A$ is not necessarily independent but it contains an independent subset $B$ such that $p,p+1 \in B$ and $\spn{A} = \spn{B}$. Since $M = F(\{p,p+1\})$, it follows that any integer greater than $M$ is contained in $\spn{p,p+1}$. Since $B$ is an independent extension of $\{p,p+1\}$, it cannot contain any integer greater than $M$.
It follows that $B \subseteq I$. Clearly, $|B| \le k$ and $\gcd(B)=1$. We would like to transform $B$ into another generating system $G \subseteq I$ such that
\[ \spn{B} \subseteq \spn{G}, \gcd(G) = 1 \mbox{ and } |G|=k \enspace . \]
If $|B|=k$, we can simply set $G=B$. If $|B|<k$, then we make use of the elements in the independent set $\{p,p+1,\ldots,p+k\} \subseteq I$ and add them, one after the other, to $B$ (thereby removing other elements from $B$ whenever their removal leaves $\spn{B}$ invariant) until the resulting set $G$ contains $k$ elements. We now define the set $L'$ by setting $L' = N+\spn{G}$. Since $G \subseteq I = \{p,p+1,\ldots,M\}$, and $p,p+1 \in G$, it follows that $p = \min G$, $\gcd(G)=1$ and $\min(L' \setminus\{N\})$ is $N+p$. Thus, $L' \setminus L = \{N\}$, as desired. Moreover, since $N$ had been chosen large enough, the set $L'$ is $(k,N)$-special. Thus $L$ and $L'$ have all properties that are required by our proof strategy and the proof of~(\ref{eq:lb1}) is complete.
\subsubsection{Proof of~(\ref{eq:lb2}),~(\ref{eq:lb3}),~(\ref{eq:lb4}) and~(\ref{eq:lb5})} \label{subsubsec:lb234}
\noindent We make use of some well known (and trivial) lower bounds on $\mathrm{TD}_{min}$:
\begin{example} \label{ex:powerset} For every $k\in{\mathbb N}$, let $[k] = \{1,2,\ldots,k\}$, let $2^{[k]}$ denote the powerset of $[k]$ and, for all $\ell = 0,1,\ldots,k$, let
\[ {[k] \choose \ell} = \{S \subseteq [k]:\ |S|=\ell\} \] denote the class of those subsets of $[k]$ that have exactly $\ell$ elements. It is trivial to verify that \[ \mathrm{TD}_{min}\left(2^{[k]}\right) = k\ \mbox{ and }\ \mathrm{TD}_{min}\left({[k] \choose \ell}\right) = \min\{\ell,k-\ell\} \enspace . \] \end{example}
In view of $\mathrm{PBTD}^+(\mathrm{LINSET}_k) = k$, the next results
show that negative examples are of limited help only as far as preference-based teaching of concepts from $\mathrm{LINSET}_k$ is concerned:
\begin{lemma} \label{lem:td-min} For every $k\ge1$ and for all $\ell=0,\ldots,k-1$, let \begin{eqnarray*} {\mathcal{L}}_k & = & \{\spn{k,p_1,\ldots,p_{k-1}}:\ p_i \in \{k+i,2k+i\}\} \enspace , \\
{\mathcal{L}}_{k,\ell} & = &\{ \{\spn{k,p_1,\ldots,p_{k-1}}\in{\mathcal{L}}_k:\ |\{i : p_i=k+i\}| = \ell\} \enspace . \end{eqnarray*} With this notation, the following holds: \[ \mathrm{TD}_{min}({\mathcal{L}}_k) \ge k-1\ \mbox{ and }\ \mathrm{TD}_{min}({\mathcal{L}}_{k,\ell}) \ge \min\{\ell,k-1-\ell\} \enspace . \] \end{lemma}
\begin{proof} For $k=1$, the assertion in the lemma is vacuous. Suppose therefore that $k\ge2$. An inspection of the generators $k,p_1,\ldots,p_{k-1}$ with $p_i \in \{k+i,2k+i\}$ shows that \begin{eqnarray*} {\mathcal{L}}_k & = & \{L_{k,S}:\ S \subseteq \{k+1,k+2,\ldots,2k-1\}\} \\
{\mathcal{L}}_{k,\ell} & = & \{L_{k,S}:\ (S \subseteq \{k+1,k+2,\ldots,2k-1\}) \wedge (|S|=\ell)\}\ \end{eqnarray*} where \[ L_{k,S} = \{0,k\} \cup \{2k,2k+1,\ldots\} \cup S \enspace . \] Note that the examples in $\{0,1,\ldots,k\} \cup \{2k,2k+1,\ldots,\}$ are redundant because they do not distinguish between distinct concepts from ${\mathcal{L}}_k$. The only useful examples are therefore contained in the interval $\{k+1,k+2,\ldots,2k-1\}$. From this discussion, it follows that teaching the concepts of ${\mathcal{L}}_k$ (resp.~of ${\mathcal{L}}_{k,\ell}$) is not essentially different from teaching the concepts of $2^{[k-1]}$ $\left(\mbox{resp.~of }{[k-1] \choose \ell}\right)$. This completes the proof of the lemma because we know from Example~\ref{ex:powerset} that $\mathrm{TD}_{min}(2^{[k-1]}) = k-1$ and $\mathrm{TD}_{min}\left({[k-1] \choose \ell}\right) = \min\{\ell,k-1-\ell\}$. \end{proof}
We claim now that the inequalities~(\ref{eq:lb2}),~(\ref{eq:lb3}) and~(\ref{eq:lb4}) are valid, i.e., we claim that the following holds: \begin{enumerate} \item $\mathrm{PBTD}(\mathrm{CF}\mbox{-}\mathrm{LINSET}_k) \ge k-1$. \item $\mathrm{PBTD}(\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k) \ge \lfloor (k-1)/2 \rfloor$. \item $\mathrm{PBTD}(\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k) \ge k-1$. \end{enumerate}
\begin{proof} For $k=1$, the inequalities are obviously valid. Suppose therefore that $k\ge2$. \begin{enumerate} \item Since $\gcd(k,k+1) = \gcd(k,2k+1) = 1$, it follows that ${\mathcal{L}}_k$ is a finite subclass of $\mathrm{CF}\mbox{-}\mathrm{LINSET}_k$. Thus $\mathrm{PBTD}(\mathrm{CF}\mbox{-}\mathrm{LINSET}_k) \ge \mathrm{PBTD}({\mathcal{L}}_k) \ge \mathrm{TD}_{min}({\mathcal{L}}_k) \ge k-1$. \item Define ${\mathcal{L}}_{k}[N] = \{N+L:\ L\in{\mathcal{L}}_k\}$ and ${\mathcal{L}}_{k,\ell}[N] = \{N+L:\ L\in {\mathcal{L}}_{k,\ell}\}$. Clearly $\mathrm{TD}_{min}({\mathcal{L}}_k[N]) = \mathrm{TD}_{min}({\mathcal{L}}_k)$ and $\mathrm{TD}_{min}({\mathcal{L}}_{k,\ell}[N]) = \mathrm{TD}_{min}({\mathcal{L}}_{k,\ell})$ holds for every $N\ge0$. It follows that the lower bounds in Lemma~\ref{lem:td-min} are also valid for the classes ${\mathcal{L}}_k[N]$ and ${\mathcal{L}}_{k,\ell}[N]$ in place of ${\mathcal{L}}_k$ and ${\mathcal{L}}_{k,\ell}$, respectively. Let \begin{equation} \label{eq:special-shift} N(k) = k^2 + (k-1-\lfloor (k-1)/2 \rfloor)k + \sum_{i=1}^{k-1}i = k^2+(k-1-\lfloor (k-1)/2 \rfloor)k+\frac{1}{2}(k-1)k \enspace . \end {equation} It suffices to show that $N(k)+{\mathcal{L}}_{k,\lfloor (k-1)/2 \rfloor}$ is a finite subclass of $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k$. To this end, first note that \[ \spn{k,p_1,\ldots,p_{k-1}}_+ = k+\sum_{i=1}^{k-1}p_i + \spn{k,p_1,\ldots,p_{k-1}} \enspace . \] Call $p_i$ ``light'' if $p_i = k+i$ and call it ``heavy'' if $p_i = 2k+i$. Note that a concept $L$ from $N(k)+{\mathcal{L}}_{k,\ell}$ is of the general form \begin{equation} \label{eq:general-form} L = N(k) + \spn{k,p_1,\ldots,p_{k-1}} \end{equation} with exactly $\ell$ light parameters among $p_1,\ldots,p_{k-1}$. A straightforward calculation shows that, for $\ell = \lfloor (k-1)/2 \rfloor$, the sum $k+\sum_{i=1}^{k-1}p_i$ equals the number $N(k)$ as defined in~(\ref{eq:special-shift}). Thus, the concept $L$ from~(\ref{eq:general-form}) with exactly $\lfloor (k-1)/2 \rfloor$ light parameters among $\{p_1,\ldots,p_{k-1}\}$ can be rewritten as follows: \[ L = N(k) + \spn{k,p_1,\ldots,p_{k-1}} = \spn{k,p_1,\ldots,p_{k-1}}_+ \enspace . \] This shows that $L \in \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k$. As $L$ is a concept from $N(k)+$ \linebreak[4]${\mathcal{L}}_{k,\lfloor (k-1)/2 \rfloor}$ in general form, we may conclude that $N(k)+{\mathcal{L}}_{k,\lfloor (k-1)/2 \rfloor}$ is a finite subclass of $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k$, as desired. \item The proof of the third inequality is similar to the above proof of the second one. It suffices to show that, for every $k\ge2$, there exists $N \in {\mathbb N}$ such that $N+{\mathcal{L}}_k$ is a subclass of $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k$. To this end, we set $N = 3k^2$. A concept $L$ from $3k^2+{\mathcal{L}}_k$ is of the general form \[ L = 3k^2+\spn{k,p_1,\ldots,p_{k-1}} \] with $p_i \in \{k+i,2k+i\}$ (but without control over the number of light parameters). It is easy to see that the constant $3k^2$ is large enough so that $L$ can be rewritten as \[ L = 3k^2 - \left(k+\sum_{i=1}^{k-1}p_i\right) + \spn{k,p_1,\ldots,p_{k-1}}_+ \] where $3k^2 - \left(k+\sum_{i=1}^{k-1}p_i\right) \ge 0$. This shows that $L \in \mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k$. As $L$ is a concept from $3k^2+{\mathcal{L}}_k$ in general form, we may conclude that $3k^2+{\mathcal{L}}_k$ is a finite subclass of $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}'_k$, as desired.
\end{enumerate} \end{proof}
We conclude with the proof of the inequality~(\ref{eq:lb5}).
\begin{lemma} \label{lem:pbtdplusnecflinsetklb} $\mathrm{PBTD}^+(\mathrm{NE}\mbox{-}\mathrm{LINSET}_k) \geq \mathrm{PBTD}^+(\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k) \geq k-1$. \end{lemma}
\begin{proof} The class $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_1$ contains only ${\mathbb N}$, and so $\mathrm{PBTD}^+(\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_1)$ $= 0$. The class $\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_2$ contains at least two members so that $\mathrm{PBTD}^+$ \linebreak[4]$(\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_2)$ $\geq 1$. Now assume $k \geq 3$. Set \[ N = \sum_{i=0}^{k-1} \left(k+i\right) \] and \[ L = \spn{k,k+1,\ldots,2k-1}_+ = N+\spn{k,k+1,\ldots,2k-1}
= \{N\} \cup \{N+k,N+k+1,\ldots\} \enspace . \] Choose and fix an arbitrary set $S \subseteq L$ of size $k-2$. It suffices to show $S$ is not a weak spanning set for $L$ w.r.t.~$\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k$. If $S$ does not contain $N$, then the set \[ L' = \spn{N+k-1,1}_+ = L\setminus\{N\} \] satisfies $S \subset L' \subset L$ so that $S$ cannot be a weak spanning set for $L$. Suppose therefore from now on that $N \in S$. We proceed by case analysis: \begin{description} \item[Case 1:] $k = 3$. \\
Then $N = 12$, $L = 12+\spn{3,4,5} = \{12\} \cup \{15,16,17,\ldots\}$. Moreover $|S|=1$ so that $S=\{12\}$. Now the set $L' = \spn{5,7}_+ = 12+\spn{5,7}$ satisfies $S \subset L' \subset L$ so that $S$ cannot be a weak spanning set for $L$. \item[Case 2:] $k = 4$. \\
Then $N = 22$, $L = 22+\spn{4,5,6,7} = \{22\} \cup \{26,27,28,\ldots\}$. Moreover $|S|=2$ so that $S=\{22\} \cup \{26+x\}$ for some $x\ge0$. Let $a = (x \bmod 4) \in \{0,1,2,3\}$. It is easy to check that the set \[ L' = \left\{ \begin{array}{ll}
22+\spn{4,5,13} & \mbox{if $a\in\{0,1\}$} \\
22+\spn{4,7,11} & \mbox{if $a=3$} \\
22+\spn{5,6,11} & \mbox{if $x=a=2$} \\
22+\spn{4,5,13} & \mbox{if $x>a=2$}
\end{array} \right . \] satisfies $S \subset L' \subset L$ so that $S$ cannot be a weak spanning set for $L$. \item[Case 3:] $k \geq 5$. \\ Then the set $S$ has the form $S = \{N\} \cup \{N+k+x_1,\ldots,N+k+x_{k-3}\}$ for distinct integers $x_1,\ldots,x_{k-3}\ge0$. For $i=1,\ldots,k-3$, let $a_i = (x_i \bmod k) \in \{0,\ldots,k-1\}$. The the set \[ L' = N + \spn{k,k+a_1,\ldots,k+a_{k-3},N-(k-2)k-(a_1+\ldots+a_{k-3})} \] satisfies $S \subset L' \subset L$ so that $S$ cannot be a weak spanning set for $L$. \end{description} In any case, we came to the conclusion that a subset of $L$ with only $k-2$ elements cannot be a weak spanning set for $L$ w.r.t.~$\mathrm{NE}\mbox{-}\mathrm{CF}\mbox{-}\mathrm{LINSET}_k$. \end{proof}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Information propagation through quantum chains with fluctuating disorder}
\author{Christian K.\ \surname{Burrell}} \affiliation{Department of Mathematics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, UK}
\author{Jens \surname{Eisert}} \affiliation{Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany} \affiliation{QOLS, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, UK}
\author{Tobias J.\ \surname{Osborne}} \affiliation{Department of Mathematics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, UK}
\begin{abstract} We investigate the propagation of information through one-dimensional quantum chains in fluctuating external fields. We find that information propagation is suppressed, but in a quite different way compared to the situation with static disorder. We study two settings: (i) a general model where an unobservable fluctuating field acts as a source of decoherence; (ii) the XX model with both observable and unobservable fluctuating fields. In the first setting we establish a noise threshold below which information can propagate ballistically and above which information is localised. In the second setting we find localisation for all levels of unobservable noise, whilst an observable field can yield diffusive propagation of information. \end{abstract}
\maketitle
Quantum lattice models as frequently encountered in the condensed-matter context or in quantum optics obey a kind of locality: once locally excited the excitation will travel through the lattice at a finite velocity. For spin models this speed at which information can propagate is generally limited by the \emph{Lieb-Robinson bound} \cite{LieRob72} which says that there is an effective \emph{``light cone''} (or \emph{``sound cone''}) for correlations, with exponentially decaying tails, whose radius grows \emph{linearly} with time \cite{NacSim05}. The importance of understanding information propagation in interacting quantum systems was only understood recently when it was exploited to establish the Lieb-Schultz-Mattis theorem in higher dimensions \cite{Has04}. In generalising the proof of this breakthrough result an intimate link between the speed of information propagation and the efficient simulation of these systems has been revealed \cite{Osb05,Has08}.
The argument underlying the Lieb-Robinson bound relies only on the ultra-violet cutoff imposed by the lattice structure and is therefore very general. Hence there are some situations where the Lieb-Robinson bound is not the best available: when we know more about the structure of the interactions it should be possible to construct tighter bounds---an intuition which has been borne out in Refs.\ \cite{BurOsb07, ZniProPre07} where it was shown that for the XX model with static disorder there exists an effective ``light cone'' whose radius grows \emph{logarithmically} with time \cite{BurOsb07}. In this example \emph{Anderson localisation} \cite{And57} for the tight-binding model was combined with a Lieb-Robinson type argument to supply the stronger bound on the multiparticle dynamics.
\begin{figure}
\caption{Schematic illustration of the different ``light cones'' for
the different regimes: (a) the (lower) \emph{linear} one
for ordered systems; (b) the (upper) \emph{logarithmic} one
for static disorder; (c) the (middle) \emph{diffusive} one for dynamic disorder. In each regime information is strongly attenuated outside the associated light cone.}
\label{fig:light-cones}
\end{figure}
Interacting quantum lattice systems are also of great interest in quantum information science as they could be used as \emph{quantum wires} to join together different parts of quantum computers or quantum devices \cite{Bur06, Bos08}. When used in this manner quantum spin chains become \emph{quantum channels} and the standard Lieb-Robinson bound then supplies a fundamental lower bound on the time required to transmit information: the receiver must wait for a time at least proportional to the length of the chain before the message arrives. For the disordered XX model mentioned above the bound of Ref.\ \cite{BurOsb07} shows that the receiver must wait for a time which is \emph{exponential} in the chain length before information about the message can be extracted, thus rendering such systems essentially useless as quantum channels.
In this work we obtain bounds on the propagation of information through interacting quantum spin chains in a \emph{fluctuating} disordered field, a setting which is ubiquitous in the condensed matter context \cite{CM}. It will become apparent that---in some situations---information propagates diffusively; in other situations the system becomes localised in the sense that information can propagate effectively by at most a constant number of sites.
We study two models: (i) a spin chain with general nearest-neighbour interactions in a noisy field fluctuating both in strength and direction; (ii) an XX spin chain in a field fluctuating in strength only. To solve the dynamics of both these models we find a master equation which describes the time evolution. We then explicitly calculate an improved Lieb-Robinson bound for the first model before finding various correlation functions for the second model. This allows us to identify several different possible regimes of information propagation: either localised or ballistic propagation for the first model and either localised or diffusive propagation for the second.
{\it Master equation ---} We derive a master equation for each of our models with time-dependent Hamiltonian
\begin{eqnarray}\label{eqn:hamiltonian}
H(\xi,t) = H_0 + \sum_{\alpha,j} \xi^\alpha_j(t) \sigma^\alpha_j
\end{eqnarray} where $\sigma^x, \sigma^y, \sigma^z$ are the Pauli matrices, $H_0$ is the intrinsic system Hamiltonian and the second term describes the interaction with the noisy field. For the first model $\alpha$ runs over the three directions $x,y,z$ whilst for the second model the field direction is fixed ($\alpha=z$); $j$ runs over all lattice sites and $\xi_j^\alpha(t)$ are the field strengths at time $t$, which must be distributed according to a probability distribution with finite second moment (e.g. a Gaussian distribution, in which case $\xi_j^\alpha = dB_j^\alpha$ describes a Brownian motion $B_j^\alpha$).
After time $t$ an initial state $\rho(0)$ evolves to $\rho(\xi,t) = U(\xi,t)\rho(0)U(\xi,t)^\dag$ where the propagator $U(\xi,t)$ is given by the time-ordered exponential $U(\xi,t)=\mathcal{T}\exp{(-i\int_0^t H(\xi,s)ds)}$. We now describe the derivation of the master equation for the \emph{ensemble averaged density operator} $\rho(t) = \mathbb{E}_\xi \rho(\xi,t)$ where $\mathbb{E}_\xi$ represents an average over all possible realisations of the disorder. We split the time evolution up into $M$ small time steps and at each step we couple each site to ancillas or meters---one for each direction $x,y,z$ (only one meter per site is used for the model with fixed field direction). When each meter is initialised in a Gaussian state, one can take the small time step limit $M\to\infty$, following the steps of continuous-measurement theory in \cite{CavMil87}, resulting in the master equation
\begin{equation}\label{eqn:master1}
\partial_t \rho(t) = -i[\rho(t), H_0] - \gamma \sum_{\alpha,j}
[[\rho(t),\sigma^\alpha_j],\sigma^\alpha_j]
\end{equation} where $\gamma = c\mathbb{E}(\xi^\alpha_j)^2$, $c>0$, is a constant measuring the strength of the disorder. As before, for the second model we omit the sum over $\alpha$ and simply set $\alpha=z$. The above derivation is valid when all times scales of interest are longer than the typical fluctuation timescale.
{\it General spin chain model ---} In this section we derive a Lieb-Robinson bound for a general spin chain of length $n$ in an unobservable field whose strength and direction fluctuate independently on each site. The Hamiltonian is given by Eq.\ (\ref{eqn:hamiltonian}) with intrinsic Hamiltonian equal to a sum of general nearest-neighbor interactions $H_0 = \sum_j h_j$, where $h_j$ acts on sites $j$ and $j+1$. Note that this model is general enough to include all the standard quantum lattice systems, including the ferromagnetic and antiferromagnetic Heisenberg models and also frustrated systems.
A Lieb-Robinson bound governs the speed of ``light'' or ``sound'' in the lattice model. It is commonly expressed as an upper bound on the \emph{Lieb-Robinson commutator},
\begin{equation*}
C_B(x,t) = \sup_{A_x} \frac{\|[A_x,B(t)]\|}{\|A_x\|}
\end{equation*}
where $A_x$ is an operator acting non-trivially only on site $x$, $B(t) =\mathbb{E}_\xi U^\dag(\xi,t) B U(\xi,t)$ is an ensemble average over all possible realisations of the operator $B$ in the Heisenberg picture, and $\|\cdot\|$ is the operator norm. While the Lieb-Robinson commutator is not directly observable, any bound for it easily translates to a bound on \emph{all} observable time-dependent two-point correlation functions $\langle A B(t) \rangle - \langle A\rangle \langle B(t)\rangle$. Hence, the Lieb-Robinson bound is a convenient bound to express the decay of correlations.
In the Heisenberg picture $B$ obeys an equation similar to the master equation derived in the previous section, namely $\partial_t B(t) = i[H_0,B(t)] - \gamma \sum_j \mathcal{D}_j(B(t))$ where the decoherence operators are defined as $\mathcal{D}_j(B(t))=\sum_\alpha[\sigma^\alpha_j,[\sigma^\alpha_j,B(t)]]$. Defining $\mathcal{D}(B) = \sum_j \mathcal{D}_j(B)$ and using a Taylor expansion we arrive at
\begin{equation*}
\begin{array}{rcl}
\|[A_x,B(t)]\| & \leq & (1-8\varepsilon\gamma n) \, \|[A_x,B(t)+i\varepsilon[H_0,B(t)]]\|\\
& + & \varepsilon\gamma\|[A_x,8 n B(t)-\mathcal{D}(B(t))]\| + O(\varepsilon^2)
\end{array}
\end{equation*}
We deal with each of the non-negligible terms on the right hand side in turn. Making repeated use of Taylor expansions and the unitary equivalence of operator norm it is easy to see that the first term can bounded as $\|[A_x,B(t)+i\varepsilon[H_0,B(t)]]\| \leq \varepsilon \|[[H_0,A_x],B(t)]\| + \|[A_x,B(t)]\| + O(\varepsilon^2)$. In order to bound the second term, we note the following two identities: (i) $\mathcal{D}_k(\sigma^\alpha_j) = 8\delta_{j,k}\sigma^\alpha_k$ for all $\alpha\neq 0$ and $\mathcal{D}_k(\sigma^0_j) = 0$, where $\sigma^0_k = \mathbb{I}$; (ii) for all $\beta\neq 0$
\begin{equation}
\sum_{\alpha\in\{0,x,y,z\}} \sigma^\alpha_k \sigma^\beta_k
\sigma^\alpha_k = 0,\,
\sum_{\alpha\in\{0,x,y,z\}} \sigma^\alpha_k \sigma^0_k
\sigma^\alpha_k = 4\mathbb{I}
\end{equation} If we define $B_{\bm\alpha} = \sigma^{\alpha_1}_1 \otimes\cdots\otimes \sigma^{\alpha_n}_n$ to be a tensor product operator indexed by the vector $\bm\alpha = (\alpha_1, \ldots, \alpha_n)$, it is straightforward to use the above identities to show that
\begin{equation}\label{eqn:commutator2}
\begin{array}{l}
\|[A_x,8nB_{\bm\alpha} - \mathcal{D}(B_{\bm\alpha})]\| \\
\qquad \leq
\sum_{k\neq x,\beta\in\{0,x,y,z\}} \|[A_x,\sigma^\beta_k B_{\bm\alpha} \sigma^\beta_k]\|\\
\qquad = 8(n-1)\|[A_x,B_{\bm\alpha}]\|
\end{array}
\end{equation}
We can extend this result to general $B$ by linearity: $B(t) = \sum_{\bm\alpha} c_{\bm\alpha}(t)B_{\bm\alpha}$. This implies that the second term obeys the bound (\ref{eqn:commutator2}) with $B_{\bm\alpha}$ replaced by $B(t)$. Combining the above equations allows us to bound the time derivative $\partial_t C_B(x,t) \leq -8\gamma C_B(x,t) + \sup_{A_x} {\|[[H_0,A_x],B(t)]\|}/{\|A_x\|}$. It remains to rewrite the off-diagonal term, the right-most term containing the double commutator. Note that $[H_0,A_x] = 2\|H_0\|\|A_x\|V$ with $V = \sum_{\alpha,\beta\in\{0,x,y,z\}} (u^{\alpha,\beta}_{x-1}\sigma^\alpha_{x-1}\sigma^\beta_x + u^{\alpha,\beta}_x \sigma^\alpha_x\sigma^\beta_{x+1} )$ where $|u^{\alpha,\beta}_y| \leq \|V\| \leq 1$. By defining the vector $\textbf{\emph{C}}_B(t) = (C_B(1,t), \ldots, C_B(n,t))$ and employing the above expansion for $V$ we are able to rewrite the bound on the time-derivative of the Lieb-Robinson commutator as
\begin{equation*}
\partial_t \textbf{\emph{C}}_B(t) \leq (-8(\gamma-8\|H_0\|)\mathbb{I} +32\|H_0\|R)
\textbf{\emph{C}}_B(t)
\end{equation*} where the inequality is to be understood componentwise and $R_{j,k}=\delta_{j,k+1} + \delta_{j+1,k}$. This has solution
\begin{eqnarray}\label{eqn:lrbound}
C_B(x,t) \leq e^{-8(\gamma-8\|H_0\|)t} \sum_j \left(e^{32\|H_0\|Rt}\right)_{x,j} C_B(j,0).
\end{eqnarray} This is the main result of this section: a new Lieb-Robinson bound for our class of models. The key idea here is that the first term can (provided the noise $\gamma$ is large enough) exponentially suppress information propagation, whilst the second term can cause ballistic propagation of information. We now analyse the interplay between these two effects:
{\it Discussion of the new bound ---} Noting that $\|R\|=2$, it is clear that if $\gamma < 16\|H_0\|$ then our bound is exponentially growing and the original Lieb-Robinson bound is superior to ours: it limits us only to ballistic propagation of information. If, however, $\gamma > 16\|H_0\|$ then the right-hand-side of our new bound (\ref{eqn:lrbound}) is negligible if either (i) $|j-k| \geq t\kappa_\varepsilon$ for some constant $\kappa_\varepsilon>0$ or (ii) $t \geq t_\varepsilon =\log{({C}/{\varepsilon})} /\Gamma$ where $C=\sum_j C_B(j,0)$ and $\Gamma=8\gamma + 128\|H_0\|$. Equivalently, the right hand side of the new bound is non-negligible only when $|j-k| < \kappa_\varepsilon t_\varepsilon$, which is to say that non-negligible correlations can propagate by at most a constant number of sites.
\begin{figure}
\caption{Illustration of the dynamics of the correlation function
$c_{j,1}(\xi,t) = \langle\Omega| a_{j}(0) a^\dag_1(t)|\Omega\rangle$ generated by Eq.~(\ref{eq:noisexx}) for a 50 site chain for a specific realisation of the fluctuating field $\xi^z_j(t)$. The horizontal axis is site number $j$ and the vertical axis is the time (for $\gamma = 0.05$, $\gamma = 0.1$, $\gamma = 0.2$ and $\gamma = 0.5$).}
\label{fig:numerics}
\end{figure}
{\it XX spin chain ---} In this section we abandon the Lieb-Robinson commutator in favor of various correlation functions as they are easier to calculate in this specific scenario. In the previous section we studied a noisy field fluctuating in both strength and direction but here we consider a fluctuating field oriented in the $z$-direction. For convenience we study the XX model, although one can expect similar results to hold for other spin chain models with nearest-neighbor interactions and a fixed field direction. We begin by applying the \emph{Jordan Wigner transformation} to map our system of qubits into a system of spinless fermions
\begin{equation}\label{eq:noisexx}
H(\xi,t) = \sum_j \left(\adagop{j}\aop{j+1} + \adagop{j}\aop{j-1} +
\xi^z_j(t)\adagop{j}\aop{j}\right)
\end{equation} The sum is over an infinite chain or a finite ring (in which case we impose periodic boundary conditions). In order to study the propagation of information through the system we introduce \emph{disorder dependent correlation functions} which are the probability amplitude for a fermion to hop from one site to another in a given time interval
\begin{equation*}
c_{j,k}(\xi,t) = \bra{\Omega} \aop{j} U(\xi,t) \adagop{k} \ket{\Omega}
\end{equation*} where $\ket{\Omega}$ is the vacuum. We further define \emph{ensemble averaged correlation functions} $c_{j,k}(t) =\mathbb{E}_\xi c_{j,k}(\xi,t) = \bra{\Omega} \aop{j}(t)\adagop{k}\ket{\Omega}$ where $\aop{j}(t) =\mathbb{E}_\xi U^\dag(\xi,t) \aop{j} U(\xi,t)$ is the ensemble averaged annihilation operator in the Heisenberg picture. Using Eq.\ (\ref{eqn:master1}) we form a differential equation for $\text{tr}[\aop{j}(t)\rho]$, yielding $ \partial_t\aop{j}(t) = -i(\aop{j+1}(t) + \aop{j-1}(t)) - \gamma \aop{j}(t)$. Introducing $\textbf{\emph{a}}(t) = (\aop{1}(t), \aop{2}(t), \ldots)$ allows us to rewrite this as a vector differential equation with solution $\textbf{\emph{a}}(t) = e^{-\gamma t} e^{iRt} \textbf{\emph{a}}(0)$. This in turn implies that the ensemble averaged correlation functions are
\begin{equation*}
c_{j,k}(t) = e^{-\gamma t} ( e^{-iRt} )_{j,k}
\end{equation*}
An analysis similar to that performed previously for the Lieb-Robinson bound reveals that all ensemble averaged correlation functions are exponentially small excepting those for which $|j-k| < \kappa(\varepsilon,\gamma) = \text{const}$. In other words, the information can---on average---propagate by at most a constant number of sites. In parallel with the results of the first model we have localisation of information but in contrast we have no noise threshold. See Fig.\ \ref{fig:numerics}.
It must be noted that this result holds only when we average over the disorder: specific realisations of the disorder could lead to less localised dynamics. In order to see how far the dynamics of a specific realisation of the noise can stray from the averaged dynamics found above we restrict ourselves to the infinite chain. We introduce the discrete \emph{squared position operator} and the \emph{momentum operator}
\begin{equation*}
{x}^2 = \sum_{j=-\infty}^{\infty} j^2 |j\rangle\langle j|,\,
p=\sum_{j} i\left(
|j+1\rangle\langle j| - |j\rangle\langle j+1|
\right)
\end{equation*}
(where $|j\rangle$ denotes the state vector of a single excitation at site $j$). The Heisenberg picture time derivatives obey equations similar to Eq.\ (\ref{eqn:master1}), e.g.,
$\partial_t {x}^2(t) = i[H_0,{x}^2(t)] - \gamma \mathcal{E}({x}^2(t))$ where $ \mathcal{E}(M)/2 = M-\sum_j |j\rangle\langle j| M |j\rangle\langle j|$ eliminates the diagonal entries of $M$. When the initial state is a single particle at site $0$, $\rho(0)=|0\rangle\langle 0|$, then
\begin{equation*}
\langle{x}^2\rangle(t) = \mathbb{E}_\xi\text{tr}[{x}^2(t)\rho(0)]
= \frac{2t}{\gamma} + \frac{e^{-2\gamma t}-1}{\gamma^2} =:
f(\gamma,t)
\end{equation*}
Recalling the trivial bound $\rho_{j,j}(t) \leq 1$, using translational invariance, and noting that $\langle{x}^2\rangle(t) = \sum_j j^2 \rho_{j,j}(t)$ allows us to conclude that when we place the particle initially at site $k$, $\rho(0)=|k\rangle\langle k|$, then
\begin{equation}\label{eqn:variance-bound}
\rho_{j,j}(t) \leq \min \left\{ 1, {f(\gamma,t)}/{|j-k|^2}\right\}
\end{equation} One can define \emph{ensemble variance correlation functions} by
$v_{j,k}(t) =\mathbb{E}_\xi |c_{j,k}(\xi,t)|^2 - |\mathbb{E}_\xi c_{j,k}(\xi,t)|^2$. A little algebra verifies that $\mathbb{E}_\xi |c_{j,k}(\xi,t)|^2$ are precisely the diagonal elements $\rho_{j,j}(t)$ when $\rho(0)=|k\rangle\langle k|$; consequently the $v_{j,k}(t)$ also obey the bound (\ref{eqn:variance-bound}).
This bound implies that it is unlikely that we will stray far from the averaged dynamics given by $c_{j,k}(t)$ for small times $(2t/\gamma)^{1/2}\ll |j-k|$. By employing Chebyshev's inequality which states that for a random variable $X$ with mean $\mu$ and finite variance $\sigma^2$ and for any positive real number $\kappa$ then $\mathbb{P}\left( \left| X-\mu\right| \geq \kappa\sigma\right) \leq {1}/{\kappa^2}$, we can conclude that there is an effective light cone whose radius grows proportionally to ${f(\gamma,t)}^{1/2} \sim (2t/\gamma)^{1/2}$. In other words, information can---on average---propagate by a distance proportional to the square-root of the time elapsed, reminiscent of a classical random walk \cite{TailNote}.
To aid understanding of the above result, we express the averaged motion in the Heisenberg picture to find that {\it wave packets stop} in their motion: since $[H_0,p]=0$ and $\mathcal{E}(p)=2p$, the Heisenberg picture momentum obeys $\langle p\rangle (t)=\mathbb{E}_\xi \text{tr}[p(t)\rho(0)] = e^{-\gamma t} \langle p\rangle (0)$, so an initial excitation ``localizes'' in momentum space exponentially fast.
{\it Mixing properties of generic fluctuating disorder --- } What is the steady state the dynamics converges to on average for very long times? We now finally link the dynamics under disorder to unitary designs \cite{Design,Harrow}, showing that generically one will obtain the maximally mixed state on average. Consider the following master equation for a spin chain (with $\gamma>0$)
\begin{equation}\label{eqn:master2}
\partial_t \rho(t) = -i[\rho(t), H_0] - \gamma \sum_{j}
[[\rho(t),X_j],X_j]
\end{equation} Define the matrix $F$ by $-i [ H_0,B_{\bm\alpha}]= \sum_{\bm\beta} F_{{\bm\alpha},{\bm\beta}} B_{\bm\beta}$, where $B_{\bm\alpha} = \sigma^{\alpha_1}_1 \otimes\cdots\otimes \sigma^{\alpha_n}_n$ as before. The operators $X_j$ can without loss of generality be taken to be of the form $X_j=x\sigma_j^{0}+y\sigma_j^{z}$, and we write $I= \{ {\bm\alpha}: \alpha_j\in \{x,y\} \text{ for some } j\}$. If the operators governing the local disorder have full rank and if the submatrix of $F$ of entries $F_{{\bm\alpha},{\bm\beta}}$ for which ${\bm\alpha}\not\in I$ and ${\bm\beta}\in I$ is a matrix of maximum rank, then $\lim_{t\rightarrow\infty}\rho(t)=\lim_{t\rightarrow\infty} \mathbb{E}_\xi \rho(\xi,t)= (\mathbb{I}/2)^{\otimes n}$ \cite{Proof}, meaning that the state becomes maximally mixed. Almost all Hamiltonians $H_0$ and local fluctuations have this property. Time evolution under fluctuating disorder therefore approximates a unitary $1$-design \cite{Design,Harrow} arbitrarily well for large times and the system ``relaxes'' on average.
{\it Summary and outlook --- } In conclusion we have studied various regimes of fluctuating on-site disorder (noise) in quantum spin chains and we have identified several regimes of information propagation. These have been compared to the propagation regimes for similar models with both static disorder and no disorder. We have found that in some instances (namely those with low noise levels or fixed field direction) that the localisation due to fluctuating disorder is weaker than that caused by static disorder; conversely we have identified other regimes (those with high levels of unobservable noise) for which the localisation is stronger than that caused by static noise. This highlights the complicated nature of disordered quantum systems. Indeed, such localization under time-dependent disorder should also be directly observable in experiments with cold atoms, exploiting similar ideas of fluctuating speckle potentials as used in the seminal experiment of Ref.\ \cite{Aspect}. One could even think of using fluctuating disorder in cold atom systems as a ``disorder filter'', shaping traveling wavepackets in their form or letting them stop.
It is a simple matter to extend the results above by replacing the intrinsic Hamiltonian $H_0$ with a time dependent one $H_0(t)$: the same master equation is obeyed. This would allow us to implement quantum logic gates on neighboring qubits hence providing a means with which to accomplish quantum computation. The important point is that this provides a more realistic model of noise than has been hitherto used by many authors, including Refs.\ \cite{Buh-et-al06}: the usual scheme is to assume perfect gate operation followed by an instantaneous error; our model allows one to analyse systems where the qubits are subject to noise and decoherence at all times and in particular whilst a gate is being applied.
{\it Acknowledgements ---} This work was supported by the EPSRC, the Nuffield Foundation, the EU (QAP, COMPAS), and the EURYI.
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{abstract}
We study the strong instability of standing waves for a system of nonlinear Schr\"odinger equations with quadratic interaction under the mass resonance condition in dimension $d=5$.
\end{abstract}
\maketitle
\section{Introduction}
\label{S:0}
We consider the system NLS equations
\begin{equation} \label{Syst-NLS}
\left\{
\renewcommand*{\arraystretch}{1.3}
\begin{array}{rcl}
i\partial_t u + \frac{1}{2m} \Delta u & = & \lambda v \overline{u}, \\
i\partial_t v + \frac{1}{2M} \Delta v & = & \mu u^2,
\end{array}
\right.
\end{equation}
where $u, v: \mathbb{R} \times \mathbb{R}^d \rightarrow \mathbb{C}$, $m$ and $M$ are positive constants, $\Delta$ is the Laplacian in $\mathbb{R}^d$ and $\lambda, \mu$ are complex constants.
The system \eqref{Syst-NLS} is regarded as a non-relativistic limit of the system of nonlinear Klein-Gordon equations
\[
\left\{
\renewcommand*{\arraystretch}{1.3}
\begin{array}{rcl}
\frac{1}{2c^2m}\partial^2_t u - \frac{1}{2m} \Delta u + \frac{mc^2}{2} u& = & -\lambda v \overline{u}, \\
\frac{1}{2c^2M}\partial^2_t v - \frac{1}{2M} \Delta v + \frac{Mc^2}{2} v& = & -\mu u^2,
\end{array}
\right.
\]
under the mass resonance condition
\begin{align} \label{mas-res}
M=2m.
\end{align}
Indeed, the modulated wave functions $(u_c,v_c):= (e^{itmc^2} u, e^{itMc^2} v)$ satisfy
\begin{align}\label{klei-gord}
\left\{
\renewcommand*{\arraystretch}{1.3}
\begin{array}{rcl}
\frac{1}{2c^2m} \partial^2_t u_c - i\partial_t u_c - \frac{1}{2m} \Delta u_c &=& - e^{itc^2(2m-M)} \lambda v_c \overline{u}_c,\\
\frac{1}{2c^2M} \partial^2_t v_c - i\partial_t v_c - \frac{1}{2M} \Delta v_c &=& - e^{itc^2(M-2m)} \mu u^2_c.
\end{array}
\right.
\end{align}
We see that the phase oscillations on the right hand sides vanish if and only if \eqref{mas-res} holds, and the system \eqref{klei-gord} formally yields \eqref{Syst-NLS} as the speed of light $c$ tends to infinity. The system \eqref{Syst-NLS} also appears in the interaction process for waves propagation in quadratic media (see e.g. \cite{CMS}).
The system \eqref{Syst-NLS} has attracted a lot of interest in past several years. The scattering theory and the asymptotic behavior of solutions have been studied in \cite{HLN, HLN-modi, HLO, OU}. The Cauchy problem for \eqref{Syst-NLS} in $L^2$, $H^1$ and in the weighted $L^2$ space $\langle x \rangle^{-1} L^2 = \mathcal{F}(H^1)$ under mass resonance condition have been studied in \cite{HOT}. The space-time analytic smoothing effect has been studied in \cite{HO-1, HO-2, Hoshino}. The sharp threshold for scattering and blow-up for \eqref{Syst-NLS} under the mass resonance condition in dimension $d=5$ has been studied in \cite{Hamano}. The existence, stability of standing waves and the characterization of finite time blow-up solutions with minimal mass have been studied recently in \cite{Dinh}.
Let us recall the local well-posedness in $H^1$ for \eqref{Syst-NLS} due to \cite{HOT}. To ensure the conservation law of total charge, it is natural to consider the following condition:
\begin{align} \label{mas-con}
\exists ~ c \in \mathbb{R} \backslash \{0\} \ : \ \lambda = c \overline{\mu}.
\end{align}
\begin{proposition}[LWP in $H^1$ \cite{HOT}] \label{prop-lwp-h1}
Let $d\leq 6$ and let $\lambda$ and $\mu$ satisfy \eqref{mas-con}. Then for any $(u_0,v_0) \in H^1\times H^1$, there exists a unique paire of local solutions $(u,v) \in Y(I)\times Y(I)$ of \eqref{Syst-NLS} with initial data $(u(0), v(0))=(u_0,v_0)$, where
\begin{align*}
Y(I) = (C\cap L^\infty)(I,H^1) \cap L^4(I,W^{1,\infty}) &\text{ for } d=1, \\
Y(I) = (C\cap L^\infty)(I,H^1) \cap L^{q_0}(I,W^{1,{r_0}}) &\text{ for } d=2,
\end{align*}
where $0<\frac{2}{q_0}=1-\frac{2}{r_0}<1$ with $r_0$ sufficiently large,
\begin{align*}
Y(I) = (C\cap L^\infty)(I, H^1) \cap L^2(I, W^{1,\frac{2d}{d-2}}) &\text{ for } d\geq 3.
\end{align*}
Moreover, the solution satisfies the conservation of mass and energy: for all $t\in I$,
\begin{align*}
M(u(t),v(t))&:= \|u(t)\|^2_{L^2}+ c\|v(t)\|^2_{L^2} = M(u_0,v_0), \\
E(u(t),v(t))&:= \frac{1}{2m}\|\nabla u(t)\|^2_{L^2} + \frac{c}{4M} \|\nabla v(t)\|^2_{L^2} + \emph{Re} (\lambda \langle v(t), u^2(t) \rangle ) = E(u_0,v_0),
\end{align*}
where $\langle \cdot, \cdot \rangle$ is the scalar product in $L^2$.
\end{proposition}
We now assume that $\lambda$ and $\mu$ satisfy \eqref{mas-con} with $c>0$ and $\lambda \ne 0, \mu \ne 0$. By change of variables
\[
u(t,x) \mapsto \sqrt{\frac{c}{2}} |\mu| u \left(t,\sqrt{\frac{1}{2m}} x \right), \quad v(t,x) \mapsto -\frac{\lambda}{2} v\left( t, \sqrt{\frac{1}{2m}} x\right),
\]
the system \eqref{Syst-NLS} becomes
\begin{equation} \label{cha-Syst}
\left\{
\renewcommand*{\arraystretch}{1.3}
\begin{array}{rcl}
i\partial_t u + \Delta u & = & -2 v \overline{u}, \\
i\partial_t v + \kappa \Delta v & = & - u^2,
\end{array}
\right.
\end{equation}
where $\kappa=\frac{m}{M}$ is the mass ratio. Note that the mass and the energy now become
\begin{align*}
M(u(t),v(t)) &= \|u(t)\|^2_{L^2} + 2 \|v(t)\|^2_{L^2}, \\
E(u(t),v(t)) &= \frac{1}{2} (\|\nabla u(t)\|^2_{L^2} + \kappa \|\nabla v(t)\|^2_{L^2} ) - \text{Re}( \langle v(t), u^2(t)\rangle).
\end{align*}
The local well-posedness in $H^1$ for \eqref{cha-Syst} reads as follows.
\begin{proposition} [LWP in $H^1$] \label{prop-lwp-wor}
Let $d\leq 6$. Then for any $(u_0, v_0) \in H^1 \times H^1$, there exists a unique pair of local solutions $(u,v) \in Y(I) \times Y(I)$ of \eqref{cha-Syst} with initial data $(u(0), v(0))=(u_0,v_0)$. Moreover, the solution satisfies the conservation of mass and energy: for all $t \in I$,
\begin{align*}
M(u(t),v(t)) &:= \|u(t)\|^2_{L^2} + 2 \|v(t)\|^2_{L^2} = M(u_0,v_0), \\
E(u(t),v(t)) &:= \frac{1}{2} (\|\nabla u(t)\|^2_{L^2} + \kappa \|\nabla v(t)\|^2_{L^2}) - \emph{Re} (\langle v(t),u^2(t)\rangle) = E(u_0,v_0).
\end{align*}
\end{proposition}
The main purpose of this paper is to study the strong instability of standing waves for the system \eqref{cha-Syst} under the mass resonance condition $\kappa=\frac{1}{2}$ in dimension $d=5$. Let $d=5$ and consider
\begin{equation} \label{mas-res-Syst}
\left\{
\renewcommand*{\arraystretch}{1.3}
\begin{array}{rcl}
i\partial_t u + \Delta u & = & -2 v \overline{u}, \\
i\partial_t v + \frac{1}{2} \Delta v & = & - u^2,
\end{array}
\right.
\end{equation}
We call a standing wave a solution to \eqref{mas-res-Syst} of the form $(e^{i\omega t} \phi_\omega, e^{i2\omega t} \psi_\omega)$, where $\omega \in \mathbb{R}$ is a frequency and $(\phi_\omega, \psi_\omega) \in H^1 \times H^1$ is a non-trivial solution to the elliptic system
\begin{equation} \label{ell-equ}
\left\{
\renewcommand*{\arraystretch}{1.3}
\begin{array}{rcl} -\Delta \phi_\omega + \omega \phi_\omega & = & 2 \psi_\omega \overline{\phi}_\omega, \\ -\frac{1}{2} \Delta \psi_\omega + 2\omega \psi_\omega & = & \phi_\omega^2.\end{array}
\right.
\end{equation}
We are interested in showing the strong instability of ground state standing waves for \eqref{mas-res-Syst}. Let us first introduce the notion of ground states related to \eqref{mas-res-Syst}. Denote
\[
S_\omega(u,v):= E(u,v) + \frac{\omega}{2} M(u,v) = \frac{1}{2} K(u,v) + \frac{\omega}{2} M(u,v) - P(u,v),
\]
where
\[
K(u,v) = \|\nabla u\|^2_{L^2} + \frac{1}{2} \|\nabla v\|^2_{L^2}, \quad M(u,v) = \|u\|^2_{L^2} + 2 \|v\|^2_{L^2}, \quad P(u,v) = \text{Re} \int \overline{v} u^2 dx.
\]
We also denote the set of non-trivial solutions of \eqref{ell-equ} by
\[
\mathcal{A}_\omega:= \{ (u,v) \in H^1 \times H^1 \backslash \{(0,0)\} \ : \ S'_\omega(u,v) =0 \}.
\]
\begin{definition} \label{def-gro-sta-ins}
A pair of functions $(\phi,\psi) \in H^1 \times H^1$ is called ground state for \eqref{ell-equ} if it is a minimizer of $S_\omega$ over the set $\mathcal{A}_\omega$. The set of ground states is denoted by $\mathcal{G}_\omega$. In particular,
\[
\mathcal{G}_\omega= \{(\phi,\psi) \in \mathcal{A}_\omega \ : \ S_\omega(\phi,\psi) \leq S_\omega(u,v), \forall (u,v) \in \mathcal{A}_\omega \}.
\]
\end{definition}
We have the following result on the existence of ground states for \eqref{ell-equ}.
\begin{proposition} \label{prop-exi-gro-sta-ins}
Let $d=5$, $\kappa = \frac{1}{2}$ and $\omega>0$. Then the set $\mathcal{G}_\omega$ is not empty, and it is characterized by
\[
\mathcal{G}_\omega = \{ (u,v) \in H^1 \times H^1 \backslash \{(0,0)\} \ : \ S_\omega(u,v) = d(\omega), K_\omega(u,v) =0 \},
\]
where
\[
K_\omega(u,v) = \left. \partial_\gamma S_\omega(\gamma u, \gamma v) \right|_{\gamma=1} = K(u,v) + \omega M(u,v) - 3 P(u,v)
\]
is the Nehari functional and
\begin{align} \label{d-ome}
d(\omega) := \inf \{ S_\omega(u,v) \ : \ (u,v) \in H^1 \times H^1 \backslash \{(0,0)\}, K_\omega(u,v) =0 \}.
\end{align}
\end{proposition}
The existence of real-valued ground states for \eqref{ell-equ} was proved in \cite{HOT} (actually for $d\leq 5$ and $\kappa>0$). Here we proved the existence of ground states (not necessary real-valued) and proved its characterization. This characterization plays an important role in the study of strong instability of ground states standing waves for \eqref{mas-res-Syst}. We only state and prove Proposition $\ref{prop-exi-gro-sta-ins}$ for $d=5$ and $\kappa=\frac{1}{2}$. However, it is still available for $d\leq 5$ and $\kappa>0$.
We also recall the definition of the strong instability of standing waves.
\begin{definition} \label{def-str-ins}
We say that the standing wave $(e^{i\omega t} \phi_\omega, e^{i2\omega t} \psi_\omega)$ is strongly unstable if for any ${\varepsilon}>0$, there exists $(u_0,v_0) \in H^1 \times H^1$ such that $\|(u_0,v_0) - (\phi_\omega, \psi_\omega)\|_{H^1 \times H^1} <{\varepsilon}$ and the corresponding solution $(u(t),v(t))$ to \eqref{mas-res-Syst} with initial data $(u(0), v(0))=(u_0,v_0)$ blows up in finite time.
\end{definition}
Our main result of this paper is the following.
\begin{theorem} \label{theo-str-ins}
Let $d=5$, $\kappa=\frac{1}{2}$, $\omega>0$ and $(\phi_\omega, \psi_\omega) \in \mathcal{G}_\omega$. Then the ground state standing waves $(e^{i\omega t} \phi_\omega, e^{i2\omega t} \psi_\omega)$ for \eqref{mas-res-Syst} is strongly unstable.
\end{theorem}
To our knowledge, this paper is the first one addresses the strong instability of standing waves for a system of nonlinear Schr\"odinger equations with quadratic interaction. In \cite{CCO-ins}, Colin-Colin-Ohta proved the instability of standing waves for a system of nonlinear Schr\"odinger equations with three waves interaction. However, they only studied the orbital instability not strong instability by blow-up, and they only consider a special standing wave solution $(0,0,e^{2i\omega t} \varphi)$, where $\varphi$ is the unique positive radial solution to the elliptic equation
\[
-\Delta \varphi + 2 \omega \varphi - |\varphi|^{p-1} \varphi=0.
\]
This paper is organized as follows. In Section $\ref{S:1}$, we show the existence of ground states and its characterization given in Proposition $\ref{prop-exi-gro-sta-ins}$. In Section $\ref{S:2}$, we give the proof of the strong instability of standing waves given in Theorem $\ref{theo-str-ins}$.
\section{Exitence of ground states}
\label{S:1}
We first show the existence of ground states given in Proposition $\ref{prop-exi-gro-sta-ins}$. To do so, we need the following profile decomposition.
\begin{proposition}[Profile decomposition] \label{prop-pro-dec-5D}
Let $d=5$ and $\kappa=\frac{1}{2}$. Le $(u_n,v_n)_{n\geq 1}$ be a bounded sequence in $H^1 \times H^1$. Then there exist a subsequence, still denoted by $(u_n,v_n)_{n\geq 1}$, a family $(x^j_n)_{n\geq 1}$ of sequences in $\mathbb{R}^5$ and a sequence $(U^j, V^j)_{j\geq 1}$ of $H^1\times H^1$-functions such that
\begin{itemize}
\item[(1)] for every $j\ne k$,
\begin{align} \label{ort-pro-dec-5D}
|x^j_n-x^k_n| \rightarrow \infty \text{ as } n\rightarrow \infty;
\end{align}
\item[(2)] for every $l\geq 1$ and every $x \in \mathbb{R}^5$,
\[
u_n(x) = \sum_{j=1}^l U^j(x-x^j_n) + u^l_n(x), \quad v_n(x)= \sum_{j=1}^l V^j(x-x^j_n) + v^l_n(x),
\]
with
\begin{align} \label{err-pro-dec-5D}
\limsup_{n\rightarrow \infty} \|(u^l_n, v^l_n)\|_{L^q\times L^q} \rightarrow 0 \text{ as } l \rightarrow \infty,
\end{align}
for every $q\in (2, 10/3)$.
\end{itemize}
Moreover, for every $l\geq 1$,
\begin{align}
M(u_n,v_n) &= \sum_{j=1}^l M(U^j_n, V^j_n) + M(u^l_n,v^l_n) + o_n(1), \label{mas-pro-dec-5D} \\
K(u_n,v_n) &= \sum_{j=1}^l K(U^j,V^j) + K(u^l_n, v^l_n) + o_n(1), \label{kin-pro-dec-5D} \\
P(u_n,v_n) &= \sum_{j=1}^l P(U^j,V^j) + P(u^l_n, v^l_n) + o_n(1), \label{sca-pro-dec-5D}
\end{align}
where $o_n(1) \rightarrow 0$ as $n\rightarrow \infty$.
\end{proposition}
We refer the reader to \cite[Proposition 3.5]{Dinh} for the proof of this profile decomposition.
The proof of Proposition $\ref{prop-exi-gro-sta-ins}$ is done by several lemmas. To simplify the notation, we denote for $\omega>0$,
\[
H_\omega(u,v):= K(u,v) + \omega M(u,v).
\]
It is easy to see that for $\omega>0$ fixed,
\begin{align} \label{equ-nor}
H_\omega(u,v) \sim \|(u,v)\|_{H^1 \times H^1}.
\end{align}
Note also that
\[
S_\omega(u,v)=\frac{1}{2} K_\omega(u,v)+\frac{1}{2}P(u,v)= \frac{1}{3}K_\omega(u,v)+\frac{1}{6} H_\omega(u,v).
\]
\begin{lemma} \label{lem-pos-d-ome}
$d(\omega)>0$.
\end{lemma}
\begin{proof}
Let $(u,v) \in H^1 \times H^1 \backslash \{(0,0)\}$ be such that $K_\omega(u,v)=0$ or $H(u,v) = 3 P(u,v)$. We have from Sobolev embedding that
\begin{align*}
P(u,v) \leq \int |v| |u|^2 dx \lesssim \|v\|_{L^3} \|u\|^2_{L^3} \lesssim \|\nabla v\|_{L^2} \|\nabla u\|^2_{L^2} \lesssim [H_\omega(u,v)]^3 \lesssim [P(u,v)]^3.
\end{align*}
This implies that there exists $C>0$ such that
\[
P(u,v) \geq \sqrt{\frac{1}{C}}>0.
\]
Thus
\[
S_\omega(u,v) = \frac{1}{2} K(u,v) + \frac{1}{2} P(u,v) \geq \frac{1}{2} \sqrt{\frac{1}{C}}>0.
\]
Taking the infimum over all $(u,v)\in H^1 \times H^1 \backslash \{(0,0)\}$ satisfying $K_\omega(u,v)=0$, we get the result.
\end{proof}
We now denote the set of all minimizers for $d(\omega)$ by
\[
\mathcal{M}_\omega := \left\{ (u,v) \in H^1 \times H^1 \backslash \{(0,0)\} \ : \ K_\omega(u,v) =0, S_\omega(u,v) = d(\omega) \right\}.
\]
\begin{lemma} \label{lem-no-emp-M-ome}
The set $\mathcal{M}_\omega$ is not empty.
\end{lemma}
\begin{proof}
Let $(u_n,v_n)_{n\geq 1}$ be a minimizing sequence for $d(\omega)$, i.e. $(u_n, v_n) \in H^1 \times H^1 \backslash \{(0,0)\}$, $K_\omega(u_n,v_n) =0$ for any $n\geq 1$ and $\lim_{n\rightarrow \infty} S_\omega(u_n,v_n) = d(\omega)$. Since $K_\omega(u_n,v_n) = 0$, we have that $H_\omega(u_n,v_n) = 3 P(u_n,v_n)$ for any $n\geq 1$. We also have that
\[
S_\omega (u_n, v_n) = \frac{1}{3} K_\omega(u_n,v_n) + \frac{1}{6}H_\omega(u_n,v_n) \rightarrow d(\omega) \text{ as } n\rightarrow \infty.
\]
This yields that there exists $C>0$ such that
\[
H_\omega(u_n,v_n) \leq 6 d(\omega) + C
\]
for all $n\geq 1$. By \eqref{equ-nor}, $(u_n,v_n)_{n\geq 1}$ is a bounded sequence in $H^1 \times H^1$. We apply the profile decomposition given in Proposition $\ref{prop-pro-dec-5D}$ to get up to a subsequence,
\[
u_n(x) = \sum_{j=1}^l U^j(x-x^j_n) + u^l_n(x), \quad v_n(x) = \sum_{j=1}^l V^j(x-x^j_n) + v^l_n(x)
\]
for some family of sequences $(x^j_n)_{n\geq 1}$ in $\mathbb{R}^5$ and $(U^j,V^j)_{j\geq 1}$ a sequence of $H^1 \times H^1$-functions satisfying \eqref{err-pro-dec-5D} -- \eqref{sca-pro-dec-5D}. We see that
\[
H_\omega (u_n,v_n)=\sum_{j=1}^l H_\omega (U^j,V^j) + H_\omega(u^l_n, v^l_n) + o_n(1).
\]
This implies that
\begin{align*}
K_\omega(u_n,v_n)&=H_\omega(u_n,v_n)-3P(u_n,v_n)\\
&=\sum_{j=1}^l H_\omega(U^j,V^j) + H_\omega(u^l_n,v^l_n) - 3P(u_n,v_n)+ o_n(1) \\
&=\sum_{j=1}^l K_\omega(U^j,V^j) + 3\sum_{j=1}^l P(U^j,V^j)- 3 P(u_n,v_n) + H_\omega(u^l_n,v^l_n)+o_n(1).
\end{align*}
Since $K_\omega(u_n,v_n)=0$ for any $n\geq 1$, $P(u_n,v_n) \rightarrow 2 d(\omega)$ as $n\rightarrow \infty$ and $H_\omega(u^l_n,v^l_n) \geq 0$ for any $n\geq 1$, we infer that
\[
\sum_{j=1}^l K_\omega(U^j,V^j) + 3 \sum_{j=1}^l P(U^j,V^j) - 6 d(\omega) \leq 0
\]
or
\[
\sum_{j=1}^l H_\omega (U^j,V^j) - 6d(\omega) \leq 0.
\]
By H\"older's inequality and \eqref{err-pro-dec-5D}, it is easy to see that $\limsup_{n\rightarrow \infty} P(u_n^l, v^l_n) =0$ as $l\rightarrow \infty$. Thanks to \eqref{sca-pro-dec-5D}, we have that
\[
2d(\omega) = \lim_{n\rightarrow \infty} P(u_n,v_n) = \sum_{j=1}^\infty P(U^j,V^j).
\]
We thus obtain
\begin{align} \label{pro-dec-app-5D}
\sum_{j=1}^\infty K_\omega(U^j,V^j) \leq 0 \text{ and } \sum_{j=1}^\infty H_\omega(U^j,V^j) \leq 6d(\omega).
\end{align}
We now claim that $K_\omega(U^j,V^j) =0$ for all $j\geq 1$. Indeed, suppose that if there exists $j_0 \geq 1$ such that $K_\omega(U^{j_0},V^{j_0}) <0$, then we see that the equation $K_\omega(\gamma U^{j_0}, \gamma V^{j_0}) = \gamma^2 H_\omega(U^{j_0},V^{j_0}) - 3 \gamma^3 P(U^{j_0},V^{j_0})=0$ admits a unique non-zero solution
\[
\gamma_0 := \frac{H_\omega(U^{j_0},V^{j_0})}{3 P(U^{j_0}, V^{j_0})} \in (0,1).
\]
By the definition of $d(\omega)$, we have
\[
d(\omega) \leq S_\omega(\gamma_0 U^{j_0}, \gamma_0 V^{j_0}) = \frac{1}{6} H_\omega(\gamma_0 U^{j_0}, \gamma_0 V^{j_0}) = \frac{\gamma_0^2}{6} H(U^{j_0},V^{j_0}) <\frac{1}{6} H_\omega(U^{j_0},V^{j_0})
\]
which contradicts to the second inequality in \eqref{pro-dec-app-5D}. We next claim that there exists only one $j$ such that $(U^j,V^j)$ is non-zero. Indeed, if there are $(U^{j_1},V^{j_1})$ and $(U^{j_2},V^{j_2})$ non-zero, then by \eqref{pro-dec-app-5D}, both $H_\omega(U^{j_1},V^{j_1})$ and $H_\omega(U^{j_2},V^{j_2})$ are strictly smaller than $6d(\omega)$. Moreover, since $K_\omega(U^{j_1},V^{j_1}) =0$,
\[
d(\omega) \leq S_\omega(U^{j_1},V^{j_1}) = \frac{1}{6} H_\omega(U^{j_1},V^{j_1}) <d(\omega)
\]
which is absurd. Therefore, without loss of generality we may assume that the only one non-zero profile is $(U^1,V^1)$. We will show that $(U^1,V^1) \in \mathcal{M}_\omega$. Indeed, we have $P(U^1,V^1) = 2d(\omega)>0$ which implies $(U^1,V^1) \ne (0,0)$. We also have
\[
K_\omega(U^1,V^1) =0 \text{ and } S_\omega(U^1,V^1) = \frac{1}{2} P(U^1,V^1) =d(\omega).
\]
This shows that $(U^1,V^1)$ is a minimizer for $d(\omega)$. The proof is complete.
\end{proof}
\begin{lemma} \label{lem-inc-M-G}
$\mathcal{M}_\omega \subset \mathcal{G}_\omega$.
\end{lemma}
\begin{proof}
Let $(\phi,\psi) \in \mathcal{M}_\omega$. Since $K_\omega(\phi,\psi) =0$, we have $H_\omega(\phi,\psi) = 3 P(\phi,\psi)$. On the other hand, since $(\phi,\psi)$ is a minimizer for $d(\omega)$, there exists a Lagrange multiplier $\gamma \in \mathbb{R}$ such that
\[
S'_\omega(\phi,\psi) = \gamma K'_\omega(\phi,\psi).
\]
This implies that
\[
0 = K_\omega(\phi,\psi) = \langle S'_\omega(\phi,\psi), (\phi,\psi)\rangle = \gamma \langle K'_\omega(\phi,\psi), (\phi,\psi)\rangle.
\]
A direct computation shows that
\[
\langle K'_\omega(\phi,\psi), (\phi,\psi)\rangle = 2 K(\phi,\psi) + 2 \omega M(\phi,\psi) - 9 P(\phi,\psi) = 2 H_\omega(\phi,\psi) - 9 P(\phi,\psi) = - 3P(\phi,\psi) <0.
\]
Therefore, $\gamma = 0$ and $S'_\omega(\phi,\psi) =0$ or $(\phi,\psi) \in \mathcal{A}_\omega$. It remains to show that $S_\omega(\phi,\psi) \leq S_\omega(u,v)$ for all $(u,v) \in \mathcal{A}_\omega$. Let $(u,v) \in\mathcal{A}_\omega$. We have $K_\omega(u,v) = \langle S'_\omega(u,v), (u,v) \rangle =0$. By the definition of $d(\omega)$, we get $S_\omega(\phi,\psi) \leq S_\omega(u,v)$. The proof is complete.
\end{proof}
\begin{lemma} \label{lem-inc-G-M}
$\mathcal{G}_\omega \subset \mathcal{M}_\omega$.
\end{lemma}
\begin{proof}
Let $(\phi_\omega, \psi_\omega) \in \mathcal{G}_\omega$. Since $\mathcal{M}_\omega$ is not empty, we take $(\phi,\psi) \in \mathcal{M}_\omega$. We have from Lemma $\ref{lem-inc-M-G}$ that $(\phi,\psi) \in \mathcal{G}_\omega$. Thus, $S_\omega(\phi_\omega,\psi_\omega) = S_\omega(\phi, \psi)=d(\omega)$. It remains to show that $K_\omega(\phi_\omega,\psi_\omega)=0$. Since $(\phi_\omega,\psi_\omega) \in \mathcal{A}_\omega$, $S'_\omega(\phi_\omega,\psi_\omega)=0$. This implies that
\[
K_\omega(\phi_\omega,\psi_\omega) = \langle S'_\omega(\phi_\omega,\psi_\omega), (\phi_\omega,\psi_\omega) \rangle =0.
\]
The proof is complete.
\end{proof}
\noindent \textit{Proof of Proposition $\ref{prop-exi-gro-sta-ins}$.}
The proof of Proposition $\ref{prop-exi-gro-sta-ins}$ follows immediately from Lemmas $\ref{lem-no-emp-M-ome}$, $\ref{lem-inc-M-G}$ and $\ref{lem-inc-G-M}$.
$\Box$
\section{Strong instability of standing waves}
\label{S:2}
We are now able to study the strong instability of standing waves for \eqref{mas-res-Syst}. Note that the local well-posedness in $H^1 \times H^1$ for \eqref{mas-res-Syst} in 5D is given in Proposition $\ref{prop-lwp-wor}$. Let us start with the following so-called Pohozaev's identities.
\begin{lemma} \label{lem-poh-ide}
Let $d=5$, $\kappa=\frac{1}{2}$ and $\omega>0$. Let $(\phi_\omega,\psi_\omega) \in H^1 \times H^1$ be a solution to \eqref{ell-equ}. Then the following identities hold
\[
2 K(\phi_\omega, \psi_\omega) = 5 P(\phi_\omega, \psi_\omega), \quad 2\omega M(\phi_\omega, \psi_\omega) = P(\phi_\omega,\psi_\omega).
\]
\end{lemma}
\begin{proof}
We only make a formal calculation. The rigorous proof follows from a standard approximation argument. Multiplying both sides of the first equation in \eqref{ell-equ} with $\overline{\phi}_\omega$, integrating over $\mathbb{R}^5$ and taking the real part, we have
\[
\|\nabla \phi_\omega\|^2_{L^2} + \omega \|\phi_\omega\|^2_{L^2}= 2 \text{Re} \int \overline{\psi}_\omega \phi_\omega^2 dx.
\]
Multiplying both sides of the second equation in \eqref{ell-equ} with $\overline{\psi}_\omega$, integrating over $\mathbb{R}^5$ and taking the real part, we get
\[
\frac{1}{2} \|\nabla \psi_\omega\|^2_{L^2} + 2 \omega \|\psi_\omega\|^2_{L^2} = \text{Re} \int \overline{\psi}_\omega \phi_\omega^2 dx.
\]
We thus obtain
\begin{align} \label{poh-ide-pro-1}
K(\phi_\omega,\psi_\omega) + 2 \omega M(\phi_\omega,\psi_\omega) = 3 P(\phi_\omega,\psi_\omega).
\end{align}
Multiplying both sides of the first equation in \eqref{ell-equ} with $x \cdot \nabla \overline{\phi}_\omega$, integrating over $\mathbb{R}^5$ and taking the real part, we see that
\[
-\text{Re} \int \Delta \phi_\omega x \cdot \nabla \overline{\phi}_\omega dx + \omega \text{Re} \int \phi_\omega x \cdot \nabla \overline{\phi}_\omega dx = 2 \text{Re} \int \psi_\omega \overline{\phi}_\omega x \cdot \nabla \overline{\phi}_\omega dx.
\]
A direct computation shows that
\begin{align*}
\text{Re} \int \Delta\phi_\omega x \cdot \nabla \overline{\phi}_\omega dx &=\frac{3}{2} \|\nabla \phi_\omega\|^2_{L^2}, \\
\text{Re} \int \phi_\omega x \cdot \nabla \overline{\phi}_\omega dx &= -\frac{5}{2} \|\phi_\omega\|^2_{L^2}, \\
\text{Re} \int \psi_\omega \overline{\phi}_\omega x \cdot \nabla \overline{\phi}_\omega dx &= -\frac{5}{2} \text{Re} \int \overline{\psi}_\omega (\phi_\omega)^2 dx - \frac{1}{2} \text{Re} \int \phi_\omega^2 x\cdot \nabla \overline{\psi}_\omega dx.
\end{align*}
It follows that
\[
-\frac{3}{2} \|\nabla \phi_\omega\|^2_{L^2} - \frac{5}{2} \omega \|\phi_\omega\|^2_{L^2} = - 5 \text{Re} \int \overline{\psi}_\omega \phi^2_\omega dx - \text{Re} \int \phi^2_\omega x \cdot \nabla \overline{\psi}_\omega dx.
\]
Similarly, multiplying both sides of the second equation in \eqref{ell-equ} with $x \cdot \nabla \overline{\psi}_\omega$, integrating over $\mathbb{R}^5$ and taking the real part, we have
\[
-\frac{3}{4} \|\nabla \psi_\omega\|^2_{L^2} - 5 \omega \|\psi_\omega\|^2_{L^2} = \text{Re} \int \phi^2_\omega x\cdot \nabla \overline{\psi}_\omega dx.
\]
We thus get
\begin{align} \label{poh-ide-pro-2}
\frac{3}{2} K(\phi_\omega,\psi_\omega) + \frac{5}{2} \omega M(\phi_\omega,\psi_\omega) = 5 P(\phi_\omega,\psi_\omega).
\end{align}
Combining \eqref{poh-ide-pro-1} and \eqref{poh-ide-pro-2}, we prove the result.
\end{proof}
We also have the following exponential decay of solutions to \eqref{ell-equ}.
\begin{lemma} \label{lem-dec-pro-gro-sta}
Let $d=5$, $\kappa=\frac{1}{2}$ and $\omega>0$. Let $(\phi_\omega,\psi_\omega) \in H^1 \times H^1$ be a solution to \eqref{ell-equ}. Then the following properties hold
\begin{itemize}
\item $(\phi_\omega,\psi_\omega) \in W^{3,p} \times W^{3,p}$ for every $2 \leq p <\infty$. In particular, $(\phi_\omega,\psi_\omega) \in C^2 \times C^2$ and $|D^\beta \phi_\omega(x)| + |D^\beta \psi_\omega (x)| \rightarrow 0$ as $|x| \rightarrow \infty$ for all $|\beta| \leq 2$;
\item
\[
\int e^{|x|} (|\nabla \phi_\omega|^2 + |\phi_\omega|^2) dx <\infty, \quad \int e^{|x|} (|\nabla \psi_\omega|^2 + 4|\psi_\omega|^2) dx <\infty.
\]
In particular, $(|x| \phi_\omega, |x| \psi_\omega) \in L^2 \times L^2$.
\end{itemize}
\end{lemma}
\begin{proof}
The follows the argument of \cite[Theorem 8.1.1]{Cazenave}. Let us prove the first item. We note that if $(\phi_\omega, \psi_\omega) \in L^p \times L^p$ for some $2 \leq p<\infty$, then $\psi_\omega \overline{\phi}_\omega, \phi^2_\omega \in L^{\frac{p}{2}}$. It follows that $(\phi_\omega, \psi_\omega) \in W^{2,\frac{p}{2}} \times W^{2,\frac{p}{2}}$. By Sobolev embedding, we see that
\begin{align} \label{dec-pro-pro-1}
(\phi_\omega,\psi_\omega) \in L^q \times L^q \text{ for some } q \geq \frac{p}{2} \text{ satisfying } \frac{1}{q} \geq \frac{2}{p} - \frac{2}{5}.
\end{align}
We claim that $(\phi_\omega,\psi_\omega) \in L^p \times L^p$ for any $2 \leq p<\infty$. Since $(\phi_\omega,\psi_\omega) \in H^1 \times H^1$, the Sobolev embedding implies that $(\phi_\omega, \psi_\omega) \in L^p \times L^p$ for any $2 \leq p<\frac{10}{3}$. It remains to show the claim for any $p$ sufficiently large. To see it, we define the sequence
\[
\frac{1}{q_n} = 2^n \left( -\frac{1}{15} + \frac{2}{5 \times 2^n} \right).
\]
We have
\[
\frac{1}{q_{n+1}} -\frac{1}{q_n} = -\frac{1}{15} \times 2^n <0.
\]
This implies that $\frac{1}{q_n}$ is decreasing and $\frac{1}{q_n} \rightarrow -\infty$ as $n\rightarrow \infty$. Since $q_0= 3$ (we take $(\phi_\omega, \psi_\omega) \in L^3 \times L^3$ to prove our claim), it follows that there exists $k \geq 0$ such that
\[
\frac{1}{q_n} >0 \text{ for } 0 \leq n \leq k \text{ and } \frac{1}{q_{n+1}} \leq 0.
\]
We will show that $(\phi_\omega, \psi_\omega) \in L^{q_k} \times L^{q_k}$. If $(\phi_\omega, \psi_\omega) \in L^{q_{n_0}} \times L^{q_n}$ for some $0 \leq n_0 \leq k-1$, then by \eqref{dec-pro-pro-1}, $(\phi_\omega,\psi_\omega) \in L^q \times L^q$ for some $q \geq \frac{q_{n_0}}{2}$ satisfying $\frac{1}{q} \geq \frac{2}{q_{n_0}} - \frac{2}{5}$. By the choice of $q_n$, it is easy to check that $\frac{2}{q_{n_0}} - \frac{2}{5} = \frac{2}{q_{n_0+1}}$. In particular, $(\phi_\omega,\psi_\omega) \in L^{q_{n_0+1}} \times L^{q_{n_0+1}}$. By induction, we prove $(\phi_\omega, \psi_\omega) \in L^{q_k} \times L^{q_k}$. Applying again \eqref{dec-pro-pro-1}, we have
\[
(\phi_\omega,\psi_\omega) \in L^q \times L^q \text{ for all } q \geq \frac{q_k}{2} \text{ such that } \frac{1}{q} \geq \frac{1}{q_{k+1}}.
\]
This shows that $(\phi_\omega, \psi_\omega)$ belongs to $L^p \times L^p$ for any $p$ sufficiently large. The claim follows. Using the claim, we have in particular $\psi_\omega \overline{\phi}_\omega, \phi^2_\omega \in L^p$ for any $2 \leq p<\infty$. Hence $(\phi_\omega, \psi_\omega) \in W^{2,p} \times W^{2,p}$ for any $2 \leq p<\infty$. By H\"older's inequality, we see that $\partial_j(\psi_\omega \overline{\phi}_\omega), \partial_j(\phi^2_\omega) \in L^p$ for any $2 \leq p<\infty$ and any $ 1\leq j \leq 5$. Thus $(\partial_j \phi_\omega, \partial_j \psi_\omega) \in W^{2,p} \times W^{2,p}$ for any $2 \leq p<\infty$ and any $1 \leq j \leq 5$, or $(\phi_\omega,\psi_\omega) \in W^{3,p} \times W^{3,p}$ for any $2 \leq p<\infty$. By Sobolev embedding, $(\phi_\omega,\psi_\omega) \in C^{2,\delta} \times C^{2,\delta}$ for all $0<\delta <1$. In particular, $|D^\beta \phi_\omega(x)| + |D^\beta \psi_\omega(x)| \rightarrow 0$ as $|x| \rightarrow \infty$ for all $|\beta| \leq 2$.
To see the second item. Let ${\varepsilon}>0$ and set $\chi_{\varepsilon}(x) := e^{\frac{|x|}{1+{\varepsilon} |x|}}$. For each ${\varepsilon}>0$, the function $\chi_{\varepsilon}$ is bounded, Lipschitz continuous and satisfies $|\nabla \chi_{\varepsilon}| \leq \chi_{\varepsilon}$ a.e. Multiplying both sides of the first equation in \eqref{ell-equ} by $\chi_{\varepsilon} \overline{\phi}_\omega$, integrating over $\mathbb{R}^5$ and taking the real part, we have
\[
\text{Re} \int \nabla \phi_\omega \cdot \nabla (\chi_\omega \overline{\phi}_\omega) dx + \int \chi_{\varepsilon} |\phi_\omega|^2 dx = 2 \text{Re} \int \chi_{\varepsilon} \psi_\omega \overline{\phi}^2_\omega dx.
\]
Since $\nabla(\chi_{\varepsilon} \overline{\phi}_\omega) = \chi_{\varepsilon} \nabla \overline{\phi}_\omega + \nabla \chi_{\varepsilon} \overline{\phi}_\omega$, the Cauchy-Schwarz inequality implies that
\begin{align*}
\text{Re} \int \nabla \phi_\omega \cdot \nabla (\chi_{\varepsilon} \overline{\phi}_\omega) dx &= \int \chi_{\varepsilon} |\nabla \phi_\omega|^2 dx + \text{Re} \int \nabla \chi_{\varepsilon} \nabla \phi_\omega \overline{\phi}_\omega dx \\
&\geq \int \chi_{\varepsilon} |\nabla \phi_\omega|^2 dx - \int |\nabla \chi_{\varepsilon}| |\nabla \phi_\omega| |\phi_\omega| dx \\
& \geq \int \chi_{\varepsilon} |\nabla \phi_\omega|^2 dx - \frac{1}{2} \int \chi_{\varepsilon} (|\nabla \phi_\omega|^2 + |\phi_\omega|^2) dx.
\end{align*}
We thus get
\begin{align} \label{dec-pro-pro-2}
\int \chi_{\varepsilon} (|\nabla \phi_\omega|^2 + |\phi_\omega|^2) dx \leq 4 \text{Re} \int \chi_{\varepsilon} \psi_\omega \overline{\phi}^2_\omega dx.
\end{align}
Similarly, multiplying both sides of the second equation in \eqref{ell-equ} with $\chi_{\varepsilon} \overline{\psi}_\omega$, integrating over $\mathbb{R}^5$ and taking the real part, we get
\begin{align} \label{dec-pro-pro-3}
\int \chi_{\varepsilon}(|\nabla \psi_\omega|^2 + 4 |\psi_\omega|^2) dx \leq \frac{8}{3} \text{Re} \int \chi_{\varepsilon} \overline{\psi}_\omega \phi^2_\omega dx.
\end{align}
By the first item, there exists $R>0$ large enough such that $|v(x)| \leq \frac{1}{8}$ for $|x| \geq R$. We have that
\begin{align*}
4 \text{Re} \int \chi_{\varepsilon} \psi_\omega \overline{\phi}^2_\omega dx & \leq 4 \int \chi_{\varepsilon} |\psi_\omega| |\phi_\omega|^2 dx \\
&=4 \int_{|x| \leq R} \chi_{\varepsilon} |\psi_\omega| |\phi_\omega|^2 dx + \int_{|x| \geq R} \chi_{\varepsilon} |\psi_\omega| |\phi_\omega|^2 dx \\
&\leq 4 \int_{|x| \leq R} e^{|x|} |\psi_\omega| |\phi_\omega|^2 dx + \frac{1}{2} \int \chi_{\varepsilon} |\phi_\omega|^2 dx.
\end{align*}
We thus get from \eqref{dec-pro-pro-2} that
\begin{align} \label{dec-pro-pro-4}
\int \chi_{\varepsilon} (|\nabla \phi_\omega|^2 + |\phi_\omega|^2) dx \leq 8 \int_{|x| \leq R} e^{|x|} |\psi_\omega||\phi_\omega|^2 dx.
\end{align}
Letting ${\varepsilon} \rightarrow 0$, we obtain
\[
\int e^{|x|}( |\nabla \phi_\omega|^2 + |\phi_\omega|^2) dx \leq 8 \int_{|x| \leq R} e^{|x|} |\psi_\omega||\phi_\omega|^2 dx <\infty.
\]
Similarly, by \eqref{dec-pro-pro-3} and \eqref{dec-pro-pro-4},
\begin{align*}
\int \chi_{\varepsilon} (|\nabla \psi_\omega|^2 + 4 |\psi_\omega|^2) dx &\leq \frac{2}{3} \left(4 \int_{|x| \leq R} e^{|x|} |\psi_\omega| |\phi_\omega|^2 dx + \frac{1}{2} \int \chi_{\varepsilon} |\phi_\omega|^2 dx \right) \\
&\leq \frac{16}{3} \int_{|x| \leq R} e^{|x|} |\psi_\omega| |\phi_\omega|^2 dx.
\end{align*}
Letting ${\varepsilon} \rightarrow 0$, we get
\[
\int e^{|x|}( |\nabla \psi_\omega|^2 + 4|\psi_\omega|^2) dx \leq \frac{16}{3} \int_{|x| \leq R} e^{|x|} |\psi_\omega||\phi_\omega|^2 dx <\infty.
\]
The proof is complete.
\end{proof}
We also need the following virial identity related to \eqref{mas-res-Syst}.
\begin{lemma} \label{lem-vir-ide-ins}
Let $d=5$ and $\kappa=\frac{1}{2}$. Let $(u_0,v_0) \in H^1 \times H^1$ be such that $(|x|u_0, |x| v_0) \in L^2 \times L^2$. Then the corresponding solution to \eqref{mas-res-Syst} with initial data $(u(0),v(0)) = (u_0,v_0)$ satisfies
\begin{align*}
\frac{d^2}{dt^2} (\|xu(t)\|^2_{L^2} + 2 \|xv(t)\|^2_{L^2}) = 8 \left(\|\nabla u(t)\|^2_{L^2} + \frac{1}{2} \|\nabla v(t)\|^2_{L^2}\right) - 20 \emph{Re} \int \overline{v}(t) u^2(t) dx.
\end{align*}
\end{lemma}
\begin{proof}
The above identity follows immediately from \cite[Lemma 3.1]{Dinh} with $\chi(x) = |x|^2$.
\end{proof}
Now let us denote for $(u,v) \in H^1 \times H^1 \backslash \{(0,0)\}$,
\[
Q(u,v) := K(u,v) - \frac{5}{2} P(u,v).
\]
It is obvious that
\begin{align} \label{vir-ide-ins}
\frac{d^2}{dt^2} (\|xu(t)\|^2_{L^2} + 2 \|xv(t)\|^2_{L^2}) = 8 Q(u(t),v(t)).
\end{align}
Note that if we take
\begin{align} \label{scaling}
u^\gamma(x) = \gamma^{\frac{5}{2}} u (\gamma x), \quad v^\gamma(x) = \gamma^{\frac{5}{2}} v(\gamma x),
\end{align}
then
\begin{align*}
S_\omega(u^\gamma, v^\gamma) &= \frac{1}{2} K(u^\gamma,v^\gamma) + \frac{\omega}{2} M(u^\gamma,v^\gamma) - P(u^\gamma,v^\gamma) \\
&=\frac{\gamma^2}{2} K(u,v) + \frac{\omega}{2} M(u,v) - \gamma^{\frac{5}{2}} P(u,v).
\end{align*}
It is easy to see that
\[
Q(u,v) = \left. \partial_\gamma S_\omega(u^\gamma, v^\gamma) \right|_{\gamma=1}.
\]
\begin{lemma} \label{lem-cha-gro-sta-5D}
Let $d=5$, $\kappa=\frac{1}{2}$ and $\omega>0$. Let $(\phi_\omega,\psi_\omega) \in \mathcal{G}_\omega$. Then
\[
S_\omega(\phi_\omega,\psi_\omega) = \inf \left\{ S_\omega(u,v) \ : \ (u,v) \in H^1 \times H^1 \backslash \{(0,0)\}, Q(u,v)=0 \right\}.
\]
\end{lemma}
\begin{proof}
Denote $m:= \inf \left\{ S_\omega(u,v) \ : \ (u,v) \in H^1 \times H^1 \backslash \{(0,0)\}, Q(u,v)=0 \right\}$. Since $(\phi_\omega,\psi_\omega)$ is a solution of \eqref{ell-equ}, it follows from Lemma $\ref{lem-poh-ide}$ that $Q(\phi_\omega,\psi_\omega) =K_\omega(\phi_\omega,\psi_\omega)=0$. Thus
\begin{align} \label{cha-gro-sta-5D-pro-1}
S_\omega(\phi_\omega,\psi_\omega) \geq m.
\end{align}
Now let $(u,v) \in H^1 \times H^1 \backslash \{(0,0)\}$ be such that $Q(u,v) =0$. If $K_\omega(u,v) =0$, then by Proposition $\ref{prop-exi-gro-sta-ins}$, $S_\omega(u,v) \geq S_\omega(\phi_\omega,\psi_\omega)$. If $K_\omega(u,v) \ne 0$, we consider $K_\omega(u^\gamma, v^\gamma) = \gamma^2 K(u,v) + \omega M(u,v) - \gamma^{\frac{5}{2}} P(u,v)$, where $(u^\gamma, v^\gamma)$ is as in \eqref{scaling}. Since $\lim_{\gamma \rightarrow 0} K_\omega(u^\gamma, v^\gamma)= \omega M(u,v) >0$ and $\lim_{\gamma \rightarrow \infty} K_\omega(u^\gamma, v^\gamma) = -\infty$, there exists $\gamma_0>0$ such that $K_\omega(u^{\gamma_0},v^{\gamma_0}) =0$. It again follows from Proposition $\ref{prop-exi-gro-sta-ins}$, $S_\omega(u^{\gamma_0},v^{\gamma_0}) \geq S_\omega(\phi_\omega,\psi_\omega)$. On the other hand,
\[
\partial_\gamma S_\omega(u^\gamma,v^\gamma) = \gamma K(u,v) - \frac{5}{2} \gamma^{\frac{3}{2}} P(u,v).
\]
We see that the equation $\partial_\gamma S_\omega(u^\gamma, v^\gamma) =0$ admits a unique non-zero solution
\[
\left( \frac{2K(u,v)}{5P(u,v)} \right)^2=1
\]
since $Q(u,v) =0$. This implies that $\partial_\gamma S_\omega(u^\gamma, v^\gamma)>0$ if $\gamma \in (0,1)$ and $\partial_\gamma S_\omega(u^\gamma,v^\gamma)<0$ if $\gamma \in (1,\infty)$. In particular, $S_\omega(u^\gamma,v^\gamma) \leq S_\omega(u,v)$ for all $\gamma >0$. Hence $S_\omega(u^{\gamma_0},v^{\gamma_0}) \leq S_\omega(u,v)$. We thus obtain $S_\omega(\phi_\omega,\psi_\omega) \leq S_\omega(u,v)$ for any $(u,v) \in H^1 \times H^1 \backslash \{(0,0)\}$ satisfying $Q(u,v)=0$. Therefore,
\begin{align} \label{cha-gro-sta-5D-pro-2}
S_\omega(\phi_\omega,\psi_\omega) \leq m.
\end{align}
Combining \eqref{cha-gro-sta-5D-pro-1} and \eqref{cha-gro-sta-5D-pro-2}, we prove the result.
\end{proof}
Let $(\phi_\omega,\psi_\omega) \in \mathcal{G}_\omega$. Define
\[
\mathcal{B}_\omega:= \left\{ (u,v) \in H^1 \times H^1 \backslash \{(0,0)\} \ : \ S_\omega(u,v) < S_\omega(\phi_\omega,\psi_\omega), Q(u,v) <0 \right\}.
\]
\begin{lemma} \label{lem-inv-set}
Let $d=5$, $\kappa=\frac{1}{2}$, $\omega>0$ and $(\phi_\omega,\psi_\omega) \in \mathcal{G}_\omega$. The set $\mathcal{B}_\omega$ is invariant under the flow of \eqref{mas-res-Syst}.
\end{lemma}
\begin{proof}
Let $(u_0,v_0) \in \mathcal{B}_\omega$. We will show that the corresponding solution $(u(t),v(t))$ to \eqref{mas-res-Syst} with initial data $(u(0),v(0)) = (u_0,v_0)$ satisfies $(u(t),v(t)) \in \mathcal{B}_\omega$ for any $t$ in the existence time. Indeed, by the conservation of mass and energy, we have
\begin{align} \label{inv-set-pro}
S_\omega(u(t),v(t)) = S_\omega(u_0,v_0) < S_\omega (\phi_\omega,\psi_\omega)
\end{align}
for any $t$ in the existence time. It remains to show that $Q(u(t),v(t))<0$ for any $t$ as long as the solution exists. Suppose that there exists $t_0 >0$ such that $Q(u(t_0),v(t_0)) \geq 0$. By the continuity of the function $t\mapsto Q(u(t),v(t))$, there exists $t_1 \in (0,t_0]$ such that $Q(u(t_1),v(t_1)) =0$. It follows from Lemma $\ref{lem-cha-gro-sta-5D}$ that $S_\omega(u(t_1),v(t_1)) \geq S_\omega(\phi_\omega,\psi_\omega)$ which contradicts to \eqref{inv-set-pro}. The proof is complete.
\end{proof}
\begin{lemma} \label{lem-key-lem}
Let $d=5$, $\kappa=\frac{1}{2}$, $\omega>0$ and $(\phi_\omega,\psi_\omega) \in \mathcal{G}_\omega$. If $(u,v) \in \mathcal{B}_\omega$, then
\[
Q(u,v) \leq 2 (S_\omega(u,v) - S_\omega(\phi_\omega,\psi_\omega)).
\]
\end{lemma}
\begin{proof}
Let $(u,v) \in \mathcal{B}_\omega$. Set
\[
f(\gamma):= S_\omega(u^\gamma,v^\gamma) = \frac{\gamma^2}{2} K(u,v) + \frac{\omega}{2} M(u,v) - \gamma^{\frac{5}{2}} P(u,v).
\]
We have
\[
f'(\gamma) = \gamma K(u,v) - \frac{5}{2} \gamma^{\frac{3}{2}} P(u,v) = \frac{Q(u^\gamma, v^\gamma)}{\gamma}.
\]
We see that
\begin{align} \label{key-lem-pro}
(\gamma f'(\gamma))' &= 2\gamma K(u,v) - \frac{25}{4} \gamma^{\frac{3}{2}} P(u,v) \nonumber \\
&= 2 \left(\gamma K(u,v) - \frac{5}{2} \gamma^{\frac{3}{2}} P(u,v) \right) - \frac{5}{4} \gamma^{\frac{3}{2}} P(u,v) \nonumber \\
&\leq 2f'(\gamma)
\end{align}
for all $\gamma >0$. Note that $P(u,v) \geq 0$ which follows from the fact $Q(u,v) <0$. We also note that since $Q(u,v) <0$, the equation $\partial_\gamma S_\omega(u^\gamma, v^\gamma)=0$ admits a unique non-zero solution
\[
\gamma_0 = \left(\frac{2K(u,v)}{5P(u,v)} \right)^2 \in (0,1),
\]
and $Q(u^{\gamma_0},v^{\gamma_0}) = \gamma_0 \times \left.\partial_\gamma S_\omega(u^\gamma,v^\gamma)\right|_{\gamma=\gamma_0} =0$. Taking integration of \eqref{key-lem-pro} over $(\gamma_0,1)$ and using the fact $\gamma f'(\gamma) = Q(u^\gamma,v^\gamma)$, we get
\[
Q(u,v) - Q(u^{\gamma_0},v^{\gamma_0}) \leq 2 (S_\omega(u,v) - S_\omega(u^{\gamma_0},v^{\gamma_0})).
\]
The result then follows from the fact that $S_\omega(\phi_\omega,\psi_\omega) \leq S_\omega(u^{\gamma_0},v^{\gamma_0})$ since $Q(u^{\gamma_0},v^{\gamma_0}) = 0$.
\end{proof}
We are now able to prove the strong instability of standing waves given in Theorem $\ref{theo-str-ins}$.
\noindent \textit{Proof of Theorem $\ref{theo-str-ins}$.}
Let ${\varepsilon}>0$. Since $(\phi^{\gamma}_\omega, \psi^{\gamma}_\omega) \rightarrow (\phi_\omega,\psi_\omega)$ as $\gamma \rightarrow 1$, there exists $\gamma_0>1$ such that $\|(\phi^{\gamma_0}_\omega,\psi^{\gamma_0}_\omega) - (\phi_\omega,\psi_\omega)\|_{H^1 \times H^1} <{\varepsilon}$. We claim that $(\phi^{\gamma_0}_\omega, \psi^{\gamma_0}_\omega) \in \mathcal{B}_\omega$. Indeed, we have
\begin{align*}
S_\omega(\phi^\gamma_\omega, \psi^\gamma_\omega) &= \frac{\gamma^2}{2} K(\phi_\omega,\psi_\omega) +\frac{\omega}{2} M(\phi_\omega,\psi_\omega) -\gamma^{\frac{5}{2}} P(\phi_\omega,\psi_\omega), \\
\partial_\gamma S_\omega(\phi^\gamma_\omega, \psi^\gamma_\omega) &= \gamma K(\phi_\omega, \psi_\omega) - \frac{5}{2} \gamma^{\frac{3}{2}} P(\phi_\omega,\psi_\omega) = \frac{Q(\phi^\gamma_\omega, \psi^\gamma_\omega)}{\gamma}.
\end{align*}
Since $Q(\phi_\omega,\psi_\omega)=0$, the equation $\partial_\gamma S_\omega(\phi^\gamma_\omega, \psi^\gamma_\omega) =0$ admits a unique non-zero solution
\[
\left( \frac{2K(\phi_\omega,\psi_\omega)}{5P(\phi_\omega,\psi_\omega)} \right)^2 =1.
\]
This implies that $\partial_\gamma S_\omega(\phi^\gamma_\omega, \psi^\gamma_\omega) >0$ if $\gamma \in (0,1)$ and $\partial_\gamma S_\omega(\phi^\gamma_\omega, \psi^\gamma_\omega)<0$ if $\gamma \in (1,\infty)$. In particular, $S_\omega(\phi^\gamma_\omega,\psi^\gamma_\omega)<S_\omega(\phi_\omega,\psi_\omega)$ for any $\gamma>0$ and $\gamma \ne 1$. On the other hand, since $Q(\phi^\gamma_\omega,\psi^\gamma_\omega)= \gamma \partial_\gamma S_\omega(\phi^\gamma_\omega, \psi^\gamma_\omega)$, we see that $Q(\phi^\gamma_\omega, \psi^\gamma_\omega) >0$ if $\gamma \in (0,1)$ and $Q(\phi^\gamma_\omega, \psi^\gamma_\omega)<0$ if $\gamma \in (1,\infty)$. Since $\gamma_0>1$, we see that
\[
S_\omega(\phi^{\gamma_0}_\omega, \psi^{\gamma_0}_\omega)< S_\omega(\phi_\omega,\psi_\omega) \text{ and } Q(\phi^{\gamma_0}_\omega,\psi^{\gamma_0}_\omega) <0.
\]
Therefore, $(\phi^{\gamma_0}_\omega, \psi^{\gamma_0}_\omega) \in \mathcal{B}_\omega$ and the claim follows.
By the local well-posedness, there exists a unique solution $(u(t), v(t)) \in C([0,T), H^1 \times H^1)$ to \eqref{mas-res-Syst} with initial data $(u(0),v(0)) = (\phi^{\gamma_0}_\omega, \psi^{\gamma_0}_\omega)$, where $T>0$ is the maximal time of existence. By Lemma $\ref{lem-inv-set}$, we see that $(u(t),v(t)) \in \mathcal{B}_\omega$ for any $t\in [0,T)$. Thus, applying Lemma $\ref{lem-key-lem}$, we get
\[
Q(u(t),v(t)) \leq 2 (S_\omega(u(t),v(t)) - S_\omega (\phi_\omega,\psi_\omega)) = 2(S_\omega(\phi^{\gamma_0},\psi^{\gamma_0}) - S_\omega(\phi_\omega, \psi_\omega)) =- \delta
\]
for any $t\in [0,T)$, where $\delta= 2 (S_\omega(\phi_\omega, \psi_\omega) - S_\omega(\phi^{\gamma_0}_\omega, \psi^{\gamma_0}_\omega)) >0$. Since $(|x|\phi_\omega, |x|\psi_\omega) \in L^2 \times L^2$, it follows that $(|x| \phi^{\gamma_0}_\omega, |x| \psi^{\gamma_0}_\omega) \in L^2 \times L^2$. Thanks to the virial identity \eqref{vir-ide-ins}, we obtain
\[
\frac{d^2}{dt^2} \left( \|xu(t)\|^2_{L^2} + 2 \|xv(t)\|^2_{L^2} \right) = 8 Q(u(t),v(t)) \leq -8 \delta <0,
\]
for any $t\in [0,T)$. The classical argument of Glassey \cite{Glassey} implies that the solution blows up in finite time. The proof is complete.
$\Box$
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{\huge On the usage of lines in $GC_n$ sets} \author{Hakop Hakopian, Vahagn Vardanyan} \date{}
\maketitle
\begin{abstract}
A planar node set $\mathcal X,$ with $|\mathcal X|=\binom{n+2}{2}$ is called $GC_n$ set if each node possesses fundamental polynomial in form of a product of $n$ linear factors. We say that a node uses a line $Ax+By+C=0$ if $Ax+By+C$ divides the fundamental polynomial of the node. A line is called $k$-node line if it passes through exactly $k$-nodes of $\mathcal X.$ At most $n+1$ nodes can be collinear in $GC_n$ sets and an $(n+1)$-node line is called maximal line. The Gasca - Maeztu conjecture (1982) states that every $GC_n$ set has a maximal line. Until now the conjecture has been proved only for the cases $n \le 5.$ Here we adjust and prove a conjecture proposed in the paper - V. Bayramyan, H. H., Adv Comput Math, 43: 607-626, 2017. Namely, by assuming that the Gasca-Maeztu conjecture is true, we prove that for any $GC_n$ set $\mathcal X$ and any $k$-node line $\ell$ the following statement holds:
\noindent Either the line $\ell$ is not used at all, or it is used by exactly $\binom{s}{2}$ nodes of $\mathcal X,$ where $s$ satisfies the condition $\sigma:=2k-n-1\le s\le k.$ If in addition $\sigma \ge 3$ and $\mu(\mathcal X)>3$ then the first case here is excluded, i.e., the line $\ell$ is necessarily a used line. Here $\mu(\mathcal X)$ denotes the number of maximal lines of $\mathcal X.$
At the end, we bring a characterization for the usage of $k$-node lines in $GC_n$ sets when $\sigma=2$ and $\mu(\mathcal X)>3.$ \end{abstract}
{\bf Key words:} Polynomial interpolation, Gasca-Maeztu conjecture, $n$-poised set, $n$-independent set, $GC_n$ set, fundamental polynomial, maximal line.
{\bf Mathematics Subject Classification (2010):} \\ 41A05, 41A63.
\section{Introduction\label{sec:intro}} An $n$-poised set $\mathcal X$ in the plane is a node set for which the interpolation problem with bivariate polynomials of total degree at most $n$ is unisolvent. Node sets with geometric characterization: $GC_n$ sets, introduced by Chang and Yao \cite{CY77}, form an important subclass of $n$-poised sets. In a $GC_n$ set the fundamental polynomial of each node is a product of n linear factors. We say that a node uses a line if the line is a factor of the fundamental polynomial of this node. A line is called $k$-node line if it passes through exactly $k$-nodes of $\mathcal X.$ It is a simple fact that at most $n+1$ nodes can be collinear in $GC_n$ sets. An $(n+1)$-node line is called a maximal line. The conjecture of M. Gasca and J. I. Maeztu \cite{GM82} states that every $GC_n$ set has a maximal line. Until now the conjecture has been proved only for the cases $n \le 5$ (see \cite{B90} and \cite{HJZ14}). For a maximal line $\lambda$ in a $GC_n$ set $\mathcal X$ the following statement is evident: the line $\lambda$ is used by all $\binom{n+1}{2}$ nodes in $\mathcal X\setminus \lambda.$ This immediately follows from the fact that if a polynomial of total degree at most $n$ vanishes at $n+1$ points of a line then the line divides the polynomial (see Proposition \ref{prp:n+1points}, below). Here we consider a conjecture proposed in the paper \cite{BH} by V. Bayramyan and H. H., concerning the usage of any $k$-node line of $GC_n$ set. In this paper we make a correction in the mentioned conjecture and then prove it. Namely, by assuming that the Gasca-Maeztu conjecture is true, we prove that for any $GC_n$ set $\mathcal X$ and any $k$-node line $\ell$ the following statement holds:
\noindent The line $\ell$ is not used at all, or it is used by exactly $\binom{s}{2}$ nodes of $\mathcal X,$ where $s$ satisfies the condition $\sigma=\sigma(\mathcal{X},\ell):=2k-n-1\le s\le k.$ If in addition $\sigma \ge 3$ and $\mu(\mathcal X)>3$ then the first case here is excluded, i.e., the line $\ell$ is necessarily a used line. Here $\mu(\mathcal X)$ denotes the number of maximal lines of $\mathcal X.$ We prove also that the subset of nodes of $\mathcal X$ that use the line $\ell$ forms a $GC_{s-2}$ set if it is not an empty set. Moreover we prove that actually it is a $\ell$-proper subset of $\mathcal X,$ meaning that it can be obtained from $\mathcal X$ by subtracting the nodes in subsequent maximal lines, which do not intersect the line $\ell$ at a node of $\mathcal{X}$ or the nodes in pairs of maximal lines intersecting $\ell$ at the same node of $\mathcal{X}.$ At the last step, when the line $\ell$ becomes maximal, the nodes in $\ell$ are subtracted (see the forthcoming Definition \ref{def:proper}).
At the end, we bring a characterization for the usage of $k$-node lines in $GC_n$ sets when $\sigma=2$ and $\mu(\mathcal X)>3.$
Let us mention that earlier Carnicer and Gasca proved that a $k$-node line $\ell$ can be used by atmost $\binom{k}{2}$ nodes of a $GC_n$ set $\mathcal X$ and in addition there are no $k$ collinear nodes that use $\ell$, provided that GM conjecture is true (see \cite{CG03}, Theorem 4.5).
\subsection{Poised sets}
Denote by $\Pi_n$ the space of bivariate polynomials of total degree at most $n:$ \begin{equation*} \Pi_n=\left\{\sum_{i+j\leq{n}}c_{ij}x^iy^j \right\}. \end{equation*} We have that \begin{equation} \label{N=} N:=\dim \Pi_n=\binom{n+2}{2}. \end{equation} Let $\mathcal{X}$ be a set of $s$ distinct nodes (points): \begin{equation*} {\mathcal X}={\mathcal X}_s=\{ (x_1, y_1), (x_2, y_2), \dots , (x_s, y_s) \} . \end{equation*} The Lagrange bivariate interpolation problem is: for given set of values $\mathcal{C}_s:=\{ c_1, c_2, \dots , c_s \}$ find a polynomial $p \in \Pi_n$ satisfying the conditions \begin{equation}\label{int cond} p(x_i, y_i) = c_i, \ \ \quad i = 1, 2, \dots s. \end{equation}
\begin{definition} A set of nodes ${\mathcal X}_s$ is called \emph{$n$-poised} if for any set of values $\mathcal{C}_s$ there exists a unique polynomial $p \in \Pi_n$ satisfying the conditions \eqref{int cond}. \end{definition} It is an elementary Linear Algebra fact that if a node set $\mathcal{X}_s$ is $n$-poised then $s=N.$ Thus from now on we will consider sets $\mathcal{X}=\mathcal{X}_N$ when $n$-poisedness is studied. If a set $\mathcal{X}$ is $n$-poised then we say that $n$ is the degree of the set $\mathcal{X}.$
\begin{proposition} \label{prp:poised} The set of nodes ${\mathcal X}_N$ is $n$-poised if and only if the following implication holds: $$p \in \Pi_n,\ p(x_i, y_i) = 0, \quad i = 1, \dots , N \Rightarrow p = 0.$$ \end{proposition}
A polynomial $p \in \Pi_n$ is called an \emph{$n$-fundamental polynomial} for a node $ A = (x_k, y_k) \in {\mathcal X}_s,\ 1\le k\le s$, if \begin{equation*} p(x_i, y_i) = \delta _{i k}, \ i = 1, \dots , s , \end{equation*} where $\delta$ is the Kronecker symbol.
Let us denote the $n$-fundamental polynomial of the node $A \in{\mathcal X}_s$ by $p_A^\star=p_{A, {\mathcal X}}^\star.$
A polynomial vanishing at all nodes but one is also called fundamental, since it is a nonzero constant times the fundamental polynomial.
\begin{definition} Given an $n$-poised set $ {\mathcal X}.$ We say that a node $A\in{\mathcal X}$ \emph{uses a line $\ell\in \Pi_1$,} if \begin{equation*}
p_A^\star = \ell q, \ \text{where} \ q\in\Pi_{n-1}. \end{equation*} \end{definition} The following proposition is well-known (see, e.g., \cite{HJZ09b} Proposition 1.3): \begin{proposition}\label{prp:n+1points} Suppose that a polynomial $p \in \Pi_n$ vanishes at $n+1$ points of a line $\ell.$ Then we have that \begin{equation*} p = \ell r, \ \text{where} \ r\in\Pi_{n-1}. \end{equation*} \end{proposition}
\noindent Thus at most $n+1$ nodes of an $n$-poised set $\mathcal{X}$ can be collinear. A line $\lambda$ passing through $n+1$ nodes of the set ${\mathcal X}$ is called a \emph{maximal line}. Clearly, in view of Proposition \ref{prp:n+1points}, any maximal line $\lambda$ is used by all the nodes in $\mathcal{X}\setminus \lambda.$
Below we bring other properties of maximal lines: \begin{corollary}[\cite{CG00}, Prop. 2.1]\label{properties} Let ${\mathcal X}$ be an $n$-poised set. Then we have that
\begin{enumerate} \setlength{\itemsep}{0mm} \item Any two maximal lines of $\mathcal{X}$ intersect necessarily at a node of $\mathcal{X};$ \item Any three maximal lines of $\mathcal{X}$ cannot be concurrent; \item $\mathcal{X}$ can have at most $n+2$ maximal lines. \end{enumerate} \end{corollary}
\section{$GC_n$ sets and the Gasca-Maeztu conjecture \label{ss:GMconj}}
Now let us consider a special type of $n$-poised sets satisfying a geometric characterization (GC) property introduced by K.C. Chung and T.H. Yao: \begin{definition}[\cite{CY77}] An n-poised set ${\mathcal X}$ is called \emph{$GC_n$ set} (or $GC$ set)
if the $n$-fundamental polynomial of each node $A\in{\mathcal X}$ is a product of $n$ linear factors. \end{definition} \noindent Thus, $GC_n$ sets are the sets each node of which uses exactly $n$ lines.
\begin{corollary}[\cite{CG03}, Prop. 2.3] \label{crl:minusmax} Let $\lambda$ be a maximal line of a $GC_n$ set ${\mathcal X}.$ Then the set ${\mathcal X}\setminus \lambda$ is a $GC_{n-1}$ set. Moreover, for any node $A\in \mathcal X\setminus \lambda$ we have that \begin{equation}\label{aaaaa}p_{A, {\mathcal X}}^\star= \lambda p_{A, {\{\mathcal X\setminus \lambda}\}}^\star.\end{equation} \end{corollary}
Next we present the Gasca-Maeztu conjecture, briefly called GM conjecture:
\begin{conjecture}[\cite{GM82}, Sect. 5]\label{conj:GM} Any $GC_n$ set possesses a maximal line. \end{conjecture}
\noindent Till now, this conjecture has been confirmed for the degrees $n\leq 5$ (see \cite{B90}, \cite{HJZ14}). For a generalization of the Gasca-Maeztu conjecture to maximal curves see \cite{HR}.
In the sequel we will make use of the following important result:
\begin{theorem}[\cite{CG03}, Thm. 4.1]\label{thm:CG} If the Gasca-Maeztu conjecture is true for all $k\leq n$, then any $GC_n$ set possesses at least three maximal lines. \end{theorem}
This yields, in view of Corollary \ref{properties} (ii) and Proposition \ref{prp:n+1points}, that each node of a $GC_n$ set $\mathcal{X}$ uses at least one maximal line.
Denote by $\mu:=\mu(\mathcal{X})$ the number of maximal lines of the node set $\mathcal{X}.$
\begin{proposition}[\cite{CG03}, Crl. 3.5]\label{prp:CG-1} Let $\lambda$ be a maximal line of a $GC_n$ set $\mathcal{X}$ such that $\mu(\mathcal{X}\setminus \lambda)\ge 3.$ Then we have that $$\mu(\mathcal{X}\setminus \lambda)=\mu(\mathcal{X})\quad \hbox{or}\quad \mu(\mathcal{X})-1.$$ \end{proposition}
\begin{definition}[\cite{CG01}] Given an $n$-poised set $\mathcal{X}$ and a line $\ell.$ Then $\mathcal{X}_\ell$ is the subset of nodes of $\mathcal{X}$ which use the line $\ell.$ \end{definition} Note that a statement on maximal lines we have mentioned already can be expressed as follows \begin{equation}\label{maxaaa} \mathcal{X}_\ell=\mathcal{X}\setminus \ell, \ \hbox{if $\ell$ is a maximal line}. \end{equation}
Suppose that $\lambda$ is a maximal line of $\mathcal{X}$ and $\ell\neq \lambda$ is any line. Then in view of the relation \eqref{aaaaa} we have that \begin{equation}\label{rep} \mathcal{X}_\ell\setminus \lambda=(\mathcal{X}\setminus \lambda)_\ell. \end{equation}
In the sequel we will use frequently the following two lemmas of Carnicer and Gasca.
Let $\mathcal{X}$ be an $n$-poised set and ${\ell}$ be a line with $|\ell\cap\mathcal{X}|\le n.$ A maximal line $\lambda$ is called $\ell$-\emph{disjoint} if \begin{equation}\label{aaaa} \lambda \cap {\ell} \cap \mathcal{X} =\emptyset. \end{equation}
\begin{lemma}[\cite{CG03}, Lemma 4.4]\label{lem:CG1}
Let $\mathcal{X}$ be an $n$-poised set and ${\ell}$ be a line with $|\ell\cap\mathcal{X}|\le n.$ Suppose also that a maximal line $\lambda$ is $\ell$-disjoint. Then we have that \begin{equation}\label{aaa} \mathcal{X}_{\ell} = {(\mathcal{X} \setminus \lambda)}_{\ell}. \end{equation} Moreover, if ${\ell}$ is an $n$-node line then we have that $\mathcal{X}_{\ell} = \mathcal{X} \setminus (\lambda \cup {\ell}),$ hence $\mathcal{X}_{\ell}$ is an $(n-2)$-poised set. \end{lemma}
Let $\mathcal{X}$ be an $n$-poised set and ${\ell}$ be a line with $|\ell\cap\mathcal{X}|\le n.$ Two maximal lines $\lambda', \lambda''$ are called $\ell$-\emph{adjacent} if \begin{equation}\label{bbbb} \lambda' \cap \lambda''\cap {\ell} \in \mathcal{X}.\end{equation}
\begin{lemma}[\cite{CG03}, proof of Thm. 4.5]\label{lem:CG2}
Let $\mathcal{X}$ be an $n$-poised set and ${\ell}$ be a line with $3\le|\ell\cap\mathcal{X}|\le n.$ Suppose also that two maximal lines $\lambda', \lambda''$ are $\ell$-adjacent. Then we have that \begin{equation}\label{bbb} \mathcal{X}_{\ell} = {(\mathcal{X} \setminus (\lambda' \cup \lambda''))}_{\ell}. \end{equation} Moreover, if ${\ell}$ is an $n$-node line then we have that $\mathcal{X}_{\ell} = \mathcal{X} \setminus (\lambda' \cup \lambda'' \cup {\ell}),$ hence $\mathcal{X}_\ell$ is an $(n-3)$-poised set. \end{lemma}
\noindent Next, by the motivation of above two lemmas, let us introduce the concept of an $\ell$-reduction of a $GC_n$ set. \begin{definition}\label{def:reduct} Let $\mathcal{X}$ be a $GC_n$ set, $\ell$ be a $k$-node line $k\ge 2.$ We say that a set $\mathcal{Y}\subset\mathcal{X}$ is an $\ell$-reduction of $\mathcal{X},$ and briefly denote this by $\mathcal{X}\searrow_\ell\mathcal{Y},$ if $$\mathcal{Y}=\mathcal{X} \setminus \left(\mathcal{C}_0\cup \mathcal{C}_1\cup \cdots \cup\mathcal{C}_k\right),$$ where \begin{enumerate} \item $\mathcal{C}_0$ is an $\ell$-disjoint maximal line of $\mathcal{X},$ or $\mathcal{C}_0$ is the union of a pair of $\ell$-adjacent maximal lines of $\mathcal{X};$ \item $\mathcal{C}_i$ is an $\ell$-disjoint maximal line of the $GC$ set $\mathcal{Y}_i:=\mathcal{X}\setminus (\mathcal{C}_0\cup \mathcal{C}_1\cup \cdots \cup \mathcal{C}_{i-1}),$ or $\mathcal{C}_i$ is the union of a pair of $\ell$-adjacent maximal lines of $\mathcal{Y}_i,\ i=1,\ldots k;$ \item $\ell$ passes through at least $2$ nodes of $\mathcal{Y}.$ \end{enumerate} \end{definition}
Note that, in view of Corollary \ref{crl:minusmax}, the set $\mathcal{Y}$ here is a $GC_m$ set, where $m=n-\sum_{i=0}^{k}\delta_i,$ and $\delta_i =1$ or $2$ if $\mathcal{C}_i$ is an
$\ell$-disjoint maximal line or a union of a pair of $\ell$-adjacent maximal lines, respectively.
We get immediately from Lemmas \ref{lem:CG1} and \ref{lem:CG2} that \begin{equation}\label{abcd}\mathcal{X}\searrow_\ell\mathcal{Y} \Rightarrow \mathcal{X}_\ell=\mathcal{Y}_\ell.\end{equation} Notice that we cannot do any further $\ell$-reduction with the set $\mathcal{Y}$ if the line $\ell$ is a maximal line here. For this situation we have the following \begin{definition}\label{def:proper} Let $\mathcal{X}$ be a $GC_n$ set, $\ell$ be a $k$-node line, $k\ge 2.$ We say that the set $\mathcal{X}_\ell$ is $\ell$-proper $GC_m$ subset of $\mathcal{X}$ if there is a $GC_{m+1}$ set $\mathcal{Y}$ such that \begin{enumerate} \item $\mathcal{X}\searrow_\ell\mathcal{Y};$ \item The line $\ell$ is a maximal line in $\mathcal{Y}.$ \end{enumerate} \end{definition} Note that, in view of the relations \eqref{abcd} and \eqref{maxaaa} here we have that $$\mathcal{X}_\ell = \mathcal{Y}\setminus \ell=\mathcal{X} \setminus \left(\mathcal{C}_0\cup \mathcal{C}_1\cup \cdots \cup\mathcal{C}_k\cup\ell\right),$$ where the sets $\mathcal{C}_i$ satisfy conditions listed in Definition \ref{def:reduct}.
In view of this relation and Corollary \ref{crl:minusmax} we get that infact $\mathcal{X}_\ell$ is a $GC_m$ set, if it is an $\ell$-proper $GC_m$ subset of $\mathcal{X}.$
Note also that the node set $\mathcal{X}_\ell$ in Lemma \ref{lem:CG1} or in Lemma \ref{lem:CG2} is an $\ell$-proper subset of $\mathcal{X}$ if $\ell$ is an $n$-node line.
We immediately get from Definitions \ref{def:reduct} and \ref{def:proper} the following \begin{proposition} \label{proper} Suppose that $\mathcal{X}$ is a $GC_n$ set.
If $\mathcal{X}\searrow_\ell\mathcal{Y}$ and $\mathcal{Y}_\ell$ is a proper $GC_m$ subset of $\mathcal{Y}$ then $\mathcal{X}_\ell$ is a proper $GC_m$ subset of $\mathcal{X}.$ \end{proposition}
\subsection{Classification of $GC_n$ sets\label{ssec:SE}}
Here we will consider the results of Carnicer, Gasca, and God\'es concerning the classification of $GC_n$ sets according to the number of maximal lines the sets possesses. Let us start with
\begin{theorem}[\cite{CGo10}]\label{th:CGo10} Let $\mathcal{X}$ be a $GC_n$ set with $\mu(\mathcal{X})$ maximal lines. Suppose also that $GM$ conjecture is true for the degrees not exceeding $n.$ Then $\mu(\mathcal{X})\in\left\{3, n-1, n,n+1,n+2\right\}.$ \end{theorem}
\noindent \emph{1. Lattices with $n+2$ maximal lines - the Chung-Yao natural lattices.}
Let a set $\mathcal{L}$ of $n+2$ lines be in general position, i.e., no two lines are parallel and no three lines are concurrent, $n\ge 1.$ Then the Chung-Yao set is defined as the set $\mathcal{X}$ of all $\binom{n+2}{2}$ intersection points of these lines. Notice that the $n+2$ lines of $\mathcal{L}$ are maximal for $\mathcal{X}.$ Each fixed node here is lying in exactly $2$ lines and does not belong to the remaining $n$ lines. Observe that the product of the latter $n$ lines gives the fundamental polynomial of the fixed node. Thus $\mathcal{X}$ is a $GC_n$ set.
Let us mention that any $n$-poised set with $n+2$ maximal lines clearly forms a Chung-Yao lattice. Recall that there are no $n$-poised sets with more maximal lines (Proposition \ref{properties}, (iii)).
\noindent \emph{2. Lattices with $n+1$ maximal lines - the Carnicer-Gasca lattices.}
Let a set $\mathcal{L}$ of $n+1$ lines be in general position, $n\ge 2.$ Then the Carnicer-Gasca lattice $\mathcal{X}$ is defined as $\mathcal{X}:=\mathcal{X}^\times\cup\mathcal{X}',$
where $\mathcal{X}^\times$ is the set of all intersection points of these $n+1$ lines and $\mathcal{X}'$ is a set of other $n+1$ non-collinear nodes, called "free" nodes, one in each line, to make the line maximal. We have that $|\mathcal{X}|=\binom{n+1}{2}+(n+1)=\binom{n+2}{2}.$ Each fixed "free" node here is lying in exactly $1$ line. Observe, that the product of the remaining $n$ lines gives the fundamental polynomial of the fixed "free" node. Next, each fixed intersection node is lying in exactly $2$ lines. The product of the remaining $n-1$ lines and the line passing through the two "free" nodes in the $2$ lines gives the fundamental polynomial of the fixed intersection node. Thus $\mathcal{X}$ is a $GC_n$ set. It is easily seen that $\mathcal{X}$ has exactly $n+1$ maximal lines, i.e., the lines of $\mathcal{L}.$
Let us mention that any $n$-poised set with exactly $n+1$ maximal lines clearly forms a Carnicer-Gasca lattice (see \cite{CG00}, Proposition 2.4).
\noindent \emph{3. Lattices with $n$ maximal lines.}
Suppose that a $GC_n$ set $\mathcal{X}$ possesses exactly $n$ maximal lines, $n\ge 3.$ These lines are in a general position and we have that $\binom{n}{2}$ nodes of $\mathcal{X}$ are intersection nodes of these lines. Clearly in each maximal line there are $n-1$ intersection nodes and therefore there are $2$ more nodes, called "free" nodes, to make the line maximal. Thus $N-1=\binom{n}{2}+2n$ nodes are identified. The last node $O,$ called outside node, is outside the maximal lines.
Thus the lattice $\mathcal{X}$ has the following construction \begin{equation}\label{01O}\mathcal{X}:=\mathcal{X}^\times\cup\mathcal{X}''\cup \{O\},\end{equation} where $\mathcal{X}^\times$ is the set of intersection nodes, and $\mathcal{X}''$ is the set of $2n$ "free" nodes.
In the sequel we will need the following characterization of $GC_n$ sets with exactly $n$ maximal lines due to Carnicer and Gasca:
\begin{proposition}[\cite{CG00}, Prop. 2.5]\label{prp:nmax} A set $\mathcal{X}$ is a $GC_n$ set with exactly $n,\ n\ge 3,$ maximal lines $\lambda_1,\ldots,\lambda_n,$ if and only if the representation \eqref{01O} holds with the following additional properties:
\begin{enumerate} \setlength{\itemsep}{0mm} \item There are $3$ lines $\ell_1^o, \ell_2^o,\ell_3^o$ concurrent at the outside node $O: \ O=\ell_1^o\cap \ell_2 ^o\cap \ell_3^o$ such that $\mathcal{X}''\subset \ell_1^o\cup \ell_2^o \cup \ell_3^o;$ \item No line $\ell_i^o,\ i=1,2,3,$ contains $n+1$ nodes of $\mathcal{X}.$ \end{enumerate} \end{proposition}
\noindent \emph{4. Lattices with $n-1$ maximal lines.}
Suppose that a $GC_n$ set $\mathcal{X}$ possesses exactly $n-1$ maximal lines: $\lambda_1,\ldots,\lambda_{n-1},$ where $n\ge 4.$ These lines are in a general position and we have that $\binom{n-1}{2}$ nodes of $\mathcal{X}$ are intersection nodes of these lines. Now, clearly in each maximal line there are $n-2$ intersection nodes and therefore there are $3$ more nodes, called "free" nodes, to make the line maximal. Thus $N-3=\binom{n-1}{2}+3(n-1)$ nodes are identified. The last $3$ nodes $O_1,O_2,O_3$ called outside nodes, are outside the maximal lines. Clearly, the outside nodes are non-collinear. Indeed, otherwise the set $\mathcal{X}$ is lying in $n$ lines, i.e., $n-1$ maximal lines and the line passing through the outside nodes. This, in view of Proposition \ref{prp:poised}, contradicts the $n$-poisedness of $\mathcal{X}.$
Thus the lattice $\mathcal{X}$ has the following construction
\begin{equation}\label{01OOO} \mathcal{X}:=\mathcal{X}^\times\cup\mathcal{X}'''\cup \{O_1,O_2,O_3\},\end{equation} where $\mathcal{X}^\times$ is the set of intersection nodes, and $\\$ $\mathcal{X}''' =\left\{ A_{i}^1, A_i^2,A_i^3\in \lambda_i, : 1\le i\le n-1\right\},$ is the set of $3(n-1)$ "free" nodes.
Denote by $\ell_{i}^{oo},\ 1\le i\le 3,$ the line passing through the two outside nodes $\{O_1,O_2,O_3\}\setminus \{O_i\}.$ We call this lines $OO$ lines.
In the sequel we will need the following characterization of $GC_n$ sets with exactly $n-1$ maximal lines due to Carnicer and God\'es:
\begin{proposition}[\cite{CGo07}, Thm. 3.2]\label{prp:n-1max} A set $\mathcal{X}$ is a $GC_n$ set with exactly $n-1$ maximal lines $\lambda_1,\ldots,\lambda_{n-1},$ where $n\ge 4,$ if and only if, with some permutation of the indexes of the maximal lines and "free" nodes, the representation \eqref{01OOO} holds with the following additional properties:
\begin{enumerate} \setlength{\itemsep}{0mm} \item $\mathcal{X}'''\setminus \{ A_{1}^1, A_2^2,A_3^3\}\subset \ell_{1}^{oo}\cup \ell_{2}^{oo} \cup \ell_{3}^{oo};$ \item Each line $\ell_{i}^{oo}, i=1,2,3,$ passes through exactly $n-2$ "free" nodes (and through $2$ outside nodes). Moreover, $\ell_{i}^{oo}\cap\lambda_i\notin\mathcal{X},\ i=1,2,3;$ \item The triples $\{O_1,A_2^2,A_3^3\},\ \{O_2,A_1^1,A_3^3\},\ \{O_3,A_1^1,A_2^2\}$ are collinear. \end{enumerate} \end{proposition}
\noindent \emph{5. Lattices with $3$ maximal lines - generalized principal lattices.}
A principal lattice is defined as an affine image of the set $$PL_n:=\left\{(i,j)\in{\mathbb N}_0^2 : \quad i+j\le n\right\}.$$ Observe that the following $3$ set of $n+1$ lines, namely $\{x=i:\ i=0,1,\ldots,n+1\},\ \ \{y=j:\ j=0,1,\ldots,n+1\}$ and $\{x+y=k:\ k=0,1,\ldots,n+1\},$ intersect at $PL_n.$ We have that $PL_n$ is a $GC_n$ set. Moreover, clearly the following polynomial is the fundamental polynomial of the node $(i_0,j_0)\in PL_n:$ \begin{equation}\label{aaabbc} p_{i_0j_0}^\star(x,y) =\prod_{0\le i<i_0,\ 0\le j<j_0,\ 0\le k<k_0} (x-i)(y-j)(x+y-n+k),\end{equation} where $k_0=n-i_0-j_0.$
Next let us bring the definition of the generalized principal lattice due to Carnicer, Gasca and God\'es (see \cite{CG05}, \cite{CGo06}): \begin{definition}[\cite{CGo06}] \label{def:GPL} A node set $\mathcal{X}$ is called a generalized principal lattice, briefly $GPL_n$ if there are $3$ sets of lines each containing $n+1$ lines \begin{equation}\label{aaagpl}\ell_i^j(\mathcal{X})_{i\in \{0,1,\ldots,n\}}, \ j=0,1,2,\end{equation} such that the $3n+3$ lines are distinct, $$\ell_i^0(\mathcal{X})\cap \ell_j^1(\mathcal{X}) \cap \ell_k^2(\mathcal{X}) \cap \mathcal{X} \neq \emptyset \iff i+j+k=n$$ and
$$\mathcal{X}=\left\{x_{ijk}\ |\ x_{ijk}:=\ell_i^0(\mathcal{X})\cap \ell_j^1(\mathcal{X}) \cap \ell_k^2(\mathcal{X}), 0\le i,j,k\le n, i+j+k=n\right\}.$$ \end{definition}
Observe that if $0\le i,j,k\le n,\ i+j+k=n$ then the three lines $\ell_i^0(\mathcal{X}), \ell_j^1(\mathcal{X}), \ell_k^2(\mathcal{X})$ intersect at a node $x_{ijk}\in \mathcal{X}.$ This implies that a node of $\mathcal{X}$ belongs to only one line of each of the three sets of $n+1$ lines. Therefore $|\mathcal{X}|=(n+1)(n+2)/2.$
One can find readily, as in the case of $PL_n$, the fundamental polynomial of each node $x_{ijk}\in \mathcal{X},\ i+j+k=n:$ \begin{equation}\label{aaabbc} p_{i_0j_0k_0}^\star =\prod_{0\le i<i_0,\ 0\le j<j_0,\ 0\le k<k_0}\ell_{i}^0(\mathcal{X}) \ell_{j}^1(\mathcal{X}) \ell_{k}^2(\mathcal{X}).\end{equation} Thus $\mathcal{X}$ is a $GC_n$ set.
Now let us bring a characterization for $GPL_n$ set due to Carnicer and God\'es:
\begin{theorem}[\cite{CGo06}, Thm. 3.6]\label{th:CGo06} Assume that GM Conjecture holds for all degrees up to $n-3$. Then the following statements are equivalent:
\begin{enumerate} \setlength{\itemsep}{0mm} \item $\mathcal{X}$ is generalized principal lattice of degree n; \item $\mathcal{X}$ is a $GC_n$ set with exactly $3$ maximal lines. \end{enumerate} \end{theorem}
\section{A conjecture concerning $GC_n$ sets \label{s:conj}}
Now we are in a position to formulate and prove the corrected version of the conjecture proposed in the paper \cite{BH} of V. Bayramyan and H. H.:
\begin{conjecture}[\cite{BH}, Conj. 3.7]\label{mainc} Assume that GM Conjecture holds for all degrees up to $n$. Let $\mathcal{X}$ be a $GC_n$ set, and ${\ell}$ be a $k$-node line, $2\le k\le n+1.$ Then we have that
\begin{equation} \label{1bbaa} \mathcal{X}_{\ell} =\emptyset,\ \hbox{or} \ \mathcal{X}_\ell \ \hbox{is an $\ell$-proper $GC_{s-2}$ subset of $\mathcal{X},$ hence}\ |\mathcal{X}_{\ell}| = \binom{s}{2}, \end{equation} where $\sigma:=2k-n-1\le s \le k.$\\ Moreover, if $\sigma\ge 3$ and $\mu(\mathcal{X}) >3$ then we have that $\mathcal{X}_{\ell}\neq \emptyset.$\\ Furthermore, for any maximal line $\lambda$ we have:
$|\lambda\cap \mathcal{X}_{\ell}| = 0$ or $|\lambda\cap \mathcal{X}_{\ell}| = s-1.$ \end{conjecture}
In the last subsection we characterize constructions of $GC_n$ sets for which there are non-used $k$-node lines with $\sigma=2$ and $\mu(\mathcal{X})>3.$
Let us mention that in the original conjecture in \cite{BH} the possibility that the set $\mathcal{X}_{\ell}$ may be empty in \eqref{1bbaa} was not foreseen. Also we added here the statement that $\mathcal{X}_\ell$ is an $\ell$-proper $GC$ subset.
\subsection{Some known special cases of Conjecture \ref{mainc} \label{s:conj}}
The following theorem concerns the special case $k=n$ of Conjecture \ref{mainc}. It is a corrected version of the original result in \cite{BH}: Theorem 3.3. This result was the first step toward the Conjecture \ref{mainc}.
The corrected version appears in \cite{HV}, Theorem 3.1. \begin{theorem}\label{thm:corrected} \label{th:corrected} Assume that GM Conjecture holds for all degrees up to $n$. Let $\mathcal{X}$ be a $GC_n$ set, $n \ge 1,\ n\neq 3,$ and ${\ell}$ be an $n$-node line. Then we have that
\begin{equation} \label{2bin} |\mathcal{X}_{\ell}| = \binom{n}{2}\quad \hbox{or} \quad \binom{n-1}{2}. \end{equation} Moreover, the following hold:
\begin{enumerate} \setlength{\itemsep}{0mm} \item
$|\mathcal{X}_{\ell}| = \binom{n}{2}$ if and only if there is an $\ell$-disjoint maximal line $\lambda,$ i.e., $\lambda \cap {\ell} \cap \mathcal{X} =\emptyset.$ In this case we have that $\mathcal{X}_{\ell}=\mathcal{X}\setminus (\lambda\cup \ell).$ Hence it is an $\ell$-proper $GC_{n-2}$ set;
\item
$|\mathcal{X}_{\ell}| = \binom{n-1}{2}$ if and only if there is a pair of $\ell$-adjacent maximal lines $\lambda', \lambda'',$ i.e., $\lambda' \cap \lambda'' \cap {\ell} \in \mathcal{X}.$ In this case we have that $\mathcal{X}_{\ell}=\mathcal{X}\setminus (\lambda'\cup \lambda''\cup \ell).$ Hence it is an $\ell$-proper $GC_{n-3}$ set. \end{enumerate} \end{theorem}
Next let us bring a characterization for the case $n=3,$ which is not covered in above Theorem (see \cite{HT}, Corollary 6.1). \begin{proposition}[\cite{HV}, Prop. 3.3]\label{prp:n=3} Let $\mathcal{X}$ be a $3$-poised set and ${\ell}$ be a $3$-node line. Then we have that
\begin{equation} \label{2bin} |\mathcal{X}_{\ell}| = 3,\quad 1,\quad\hbox{or} \quad 0. \end{equation} Moreover, the following hold:
\begin{enumerate} \setlength{\itemsep}{0mm} \item
$|\mathcal{X}_{\ell}| = 3$ if and only if there is a maximal line $\lambda_0$ such that $\lambda_0 \cap {\ell} \cap \mathcal{X} =\emptyset.$ In this case we have that $\mathcal{X}_{\ell}=\mathcal{X}\setminus (\lambda_0\cup \ell).$ Hence it is an $\ell$-proper $GC_{1}$ set. \item
$|\mathcal{X}_{\ell}| = 1$ if and only if there are two maximal lines $\lambda', \lambda'',$ such that $\lambda' \cap \lambda'' \cap {\ell} \in \mathcal{X}.$ In this case we have that $\mathcal{X}_{\ell}=\mathcal{X}\setminus (\lambda'\cup \lambda''\cup \ell).$ Hence it is an $\ell$-proper $GC_{0}$ set.
\item $|\mathcal{X}_{\ell}| = 0$ if and only if there are exactly three maximal lines in $\mathcal{X}$ and they intersect $\ell$ at three distinct nodes. \end{enumerate} \end{proposition} Let us mention that the statement \eqref{2bin} of Proposition \ref{prp:n=3} (without the ``Moreover" part) is valid for $3$-node lines in any $n$-poised set (see \cite{HT}, Corollary 6.1). More precisely the following statement holds:
\begin{equation*} \hbox{If $\mathcal{X}$ is an $n$-poised set and $\ell$ is a $3$-node line then\ } |\mathcal{X}_{\ell}| = 3,\quad 1,\quad\hbox{or} \quad 0. \end{equation*}
Note that this statement contains all conclusions of Conjecture \ref{mainc} for the case of $3$-node lines, except the claim that the set $\mathcal{X}_\ell$ is an $\ell$-proper $GC_n$ subset in the cases $ |\mathcal{X}_{\ell}| = 3,1.$ And for this reason we cannot use it in proving Conjecture \ref{mainc}.
Let us mention that, in view of the relation \eqref{maxaaa}, Conjecture \ref{mainc} is true if the line $\ell$ is a maximal line. Also Conjecture \ref{mainc} is true in the case when $GC_n$ set $\mathcal{X}$ is a Chung -Yao lattice. Indeed, in this lattice the only used lines are the maximal lines. Also for any $k$-node line in $\mathcal{X}$ with $k\le n$ we have that $2k\le n+2,$ since through any node there pass two maximal lines. Thus for these non-used $k$-node lines we have $\sigma\le 1$ (see \cite{BH}).
The following proposition reveals a rich structure of the Carnicer-Gasca lattice. \begin{proposition}[\cite{BH}, Prop. 3.8]\label{CGl} Let $\mathcal{X}$ be a Carnicer-Gasca lattice of degree $n$ and ${\ell}$ be a $k$-node line. Then we have that
\begin{equation} \label{2bba} \mathcal{X}_\ell \ \hbox{is an $\ell$-proper $GC_{s-2}$ subset of $\mathcal{X},$ hence}\ |\mathcal{X}_{\ell}| = \binom{s}{2}, \end{equation} where $\sigma:=2k-n-1\le s \le k.$\\ Moreover, for any maximal line $\lambda$ we have:
$|\lambda\cap \mathcal{X}_{\ell}| = 0$ or $|\lambda\cap \mathcal{X}_{\ell}| = s-1.$\\ Furthermore, for each $n, k$ and $s$ with $\sigma\le s \le k,$ there is a Carnicer-Gasca lattice of degree $n$ and a $k$-node line $\ell$ such that \eqref{2bba} is satisfied. \end{proposition} Note that the phrase ``$\ell$-proper" is not present in the formulation of Proposition in \cite{BH} but it follows readily from the proof there.
Next consider the following statement of Carnicer and Gasca (see \cite{CG03}, Proposition 4.2):
\begin{equation} \label{2nodel}\hbox{If $\mathcal{X}$ is a $GC_n$ set and $\ell$ is a $2$-node line then\ } |\mathcal{X}_{\ell}| = 1\ \ \hbox{or}\ \ 0. \end{equation}
Let us adjoin this with the following statement:
If $\mathcal{X}$ is a $GC_n$ set, $\ell$ is a $2$-node line, and $|\mathcal{X}_\ell|=1,$ then $\mathcal{X}_\ell$ is an $\ell$-proper $GC_0$ subset, provided that $GM$ conjecture is true for all degrees up to $n.$
Indeed, suppose that $\mathcal{X}_\ell=\{A\}$ and $\ell$ passes through the nodes $B,C\in\mathcal{X}.$ The node $A$ uses a maximal $(n+1)$-node line in $\mathcal{X}$ which we denote by $\lambda_0.$ Next, $A$ uses a maximal $n$-node line in $\mathcal{X}\setminus \lambda_0$ which we denote by $\lambda_1.$ Continuing this way we find consecutively the lines $\lambda_2, \lambda_3,\ldots, \lambda_{n-1}$ and obtain that $$\{A\}=\mathcal{X}\setminus (\lambda_0\cup \lambda_1\cup \cdots \cup \lambda_{n-1}).$$ To finish the proof it suffices to show that $\lambda_{n-1}=\ell$ and the remaining lines $\lambda_i, i=0,\ldots,n-2$ are $\ell$-disjoint. Indeed, the node $A$ uses $\ell$ and since it is a $2$-node line it may coincide only with the last maximal line $\lambda_{n-1}.$ Now, suppose conversely that a maximal line $\lambda_k,\ 0\le k\le n-2,$ intersects $\ell$ at a node, say $B.$ Then consider the polynomial of degree $n:$ $$p=\ell_{A,C}\prod_{i\in\{0,\ldots,n-1\}\setminus\{k\}}\lambda_i,$$ where $\ell_{A,C}$ is the line through $A$ and $C.$ Clearly $p$ passes through all the nodes of $\mathcal{X}$ which contradicts Proposition \ref{prp:poised}.
Now, in view of the statement \eqref{2nodel} and the adjoint statement, we conclude that Conjecture \ref{mainc} is true for the case of $2$-node lines in any $GC_n$ sets.
It is worth mentioning that the statement \eqref{2nodel} is true also for any $n$-poised set $\mathcal{X}$ (see \cite{BH}, relation (1.4), due to V. Bayramyan).
\subsection{Some preliminaries for the proof of Conjecture \ref{mainc} \label{s:conj}}
Here we prove two propositions. The following proposition shows that Conjecture \ref{mainc} is true if the node set $\mathcal{X}$ has exactly $3$ maximal lines.
\begin{proposition}\label{prop:3max} Assume that GM Conjecture holds for all degrees up to $n-3$. Let $\mathcal{X}$ be a $GC_n$ set with exactly $3$ maximal lines, and ${\ell}$ be an $m$-node line, $2\le m\le n+1.$ Then we have that
\begin{equation} \label{1bin3} \mathcal{X}_{\ell}=\emptyset,\ \hbox{or} \ \mathcal{X} \ \hbox{is a proper $GC_{m-2}$ subset of $\mathcal{X},$ hence}\ |\mathcal{X}_{\ell}| = \binom{m}{2}. \end{equation} Moreover, if $\mathcal{X}_{\ell}\neq\emptyset$ and $m\le n$ then for a maximal line $\lambda_1$ of $\mathcal{X}$ we have:
\begin{enumerate} \setlength{\itemsep}{0mm} \item
$\lambda_1 \cap {\ell} \notin \mathcal{X}$ and $|\lambda_1\cap \mathcal{X}_{\ell}| = 0.$
\item
$|\lambda\cap \mathcal{X}_{\ell}| = m-1\ $ for the remaining two maximal lines. \end{enumerate} Furthermore, if the line $\ell$ intersects each maximal line at a node then $\mathcal{X}_\ell=\emptyset.$ \end{proposition} \begin{proof} According to Theorem \ref{th:CGo06} the set $\mathcal{X}$ is a generalized principal lattice of degree n with some three sets of $n+1$ lines \eqref{aaagpl}. Then we obtain from \eqref{aaabbc} that the only used lines in $\mathcal{X}$ are the lines $\ell_{s}^r(\mathcal{X}),$ where $0\le s< n,\ r=0,1,2.$ Therefore the only used $m$-node lines are the lines $\ell_{n-m+1}^r(\mathcal{X}),\ r=0,1,2.$ Consider the line, say with $r=0,$ i.e., $\ell\equiv \ell_{n-m+1}^0(\mathcal{X}).$ It is used by all the nodes $x_{ijk}\in \mathcal{X}$ with $i>n-m+1,$ i.e., $i=n-m+2, n-m+3,\ldots,n.$ Thus, $\ell$ is used by exactly $\binom{m}{2}=(m-1)+(m-2)+\cdots+1$ nodes. This implies also that $\mathcal{X}_\ell= \mathcal{X}\setminus (\ell_0\cup \ell_1\cup\cdots\cup \ell_{n-m+1}).$ Hence $\mathcal{X}$ is a proper $GC_{m-2}$ subset of $\mathcal{X}.$ The part ``Moreover" also follows readily from here. Now it remains to notice that the part ``Furthermore" is a straightforward consequence of the part ``Moreover". \end{proof}
The following statement on the presence and usage of $(n-1)$-node lines in $GC_n$ sets with exactly $n-1$ maximal lines will be used in the sequel (cf. Proposition 4.2, \cite{HV}). \begin{proposition}\label{prp:n-1} Let $\mathcal{X}$ be a $GC_n$ set with exactly $n-1$ maximal lines and $\ell$ be an $(n-1)$-node line, where $n\ge 4.$ Assume also that through each node of $\ell$ there passes exactly one maximal line. Then we have that either $n=4$ or $n=5.$ Moreover, in both these cases we have that $\mathcal{X}_\ell=\emptyset.$ \end{proposition} \begin{proof} Consider a $GC_n$ set with exactly $n-1$ maximal lines. In this case we have the representation \eqref{01OOO}, i.e., $\mathcal{X}:=\mathcal{X}^\times\cup\mathcal{X}'''\cup \{O_1,O_2,O_3\},$ satisfying the conditions of Proposition \ref{prp:n-1max}. Here $\mathcal{X}^\times$ is the set of all intersection nodes of the maximal lines, $\mathcal{X}'''$ is the set of the remaining ``free" nodes in the maximal lines, and $O_1,O_2,O_3$ are the three nodes outside the maximal lines. Let $\ell$ be an $(n-1)$-node line.
First notice that, according to the hypothesis of Proposition, all the nodes of the line $\ell$ are ``free" nodes. Therefore $\ell$ does not coincide with any $OO$ line, i.e., line passing through two outside nodes.
From Proposition \ref{prp:n-1max} we have that all the ``free" nodes except the three special nodes $A_1^1,A_2^2,A_3^3,$ which we will call here $(s)$ nodes, belong to the three $OO$ lines. We have also, in view of Proposition \ref{prp:n-1max}, (iii), that the nodes $A_1^1,A_2^2,A_3^3$ are not collinear. Therefore there are three possible cases:
\begin{enumerate} \setlength{\itemsep}{0mm} \item $\ell$ does not pass through any $(s)$ node, \item $\ell$ passes through two $(s)$ nodes, \item $\ell$ passes through one $(s)$ node. \end{enumerate} In the first case $\ell$ may pass only through nodes lying in three $OO$ lines. Then it may pass through at most three nodes, i.e., $n\le 4.$ Therefore, in view of the hypothesis $n\ge 4,$ we get $n=4.$ Since $\mu(\mathcal{X})=3$ we get, in view of Proposition \ref{prop:3max}, part ``Furthermore", that $\mathcal{X}_\ell=\emptyset.$
Next, consider the case when $\ell$ passes through two $(s)$ nodes. Then, according to Proposition \ref{prp:n-1max}, (iii), it passes through an outside $O$ node. Recall that this case is excluded since $\ell$ passes through ``free" nodes only.
Finally, consider the third case when $\ell$ passes through exactly one $(s)$ node. Then it may pass through at most three other ``free" nodes lying in $OO$ lines. Therefore $\ell$ may pass through at most four nodes.
First suppose that $\ell$ passes through exactly $3$ nodes. Then again we obtain that $n=4$ and $\mathcal{X}_\ell=\emptyset.$
Next suppose that $\ell$ passes through exactly $4$ nodes. Then we have that $n=5.$ Without loss of generality we may assume that the (s) node $\ell$ passes through is, say, $A_1^1.$
Next let us show first that $|\mathcal{X}_\ell|\le 1.$ Here we have exactly $4=n-1$ maximal lines. Consider the maximal line $\lambda_4$ for which, in view of Proposition \ref{prp:n-1max}, the intersection with each $OO$ line is a node in $\mathcal{X}$ (see Fig. \ref{pic1}). Denote $B:=\ell\cap \lambda_4.$ Assume that the node $B$ belongs to the line $\in \ell_i^{oo},\ 1\le i\le 3,$ i.e., the line passing through $\{O_1, O_2,O_3\}\setminus \{O_i\}$ ($i=2$ in Fig. \ref{pic1}). According to the condition (ii) of Proposition \ref{prp:n-1max} we have that $\ell_i^{oo}\cap \lambda_i\notin \mathcal{X}$. Denote $C:=\lambda_i\cap \lambda_4.$ \begin{figure}
\caption{The node set $\mathcal{X}$ for $n=5.$}
\label{pic1}
\end{figure}
Now let us prove that $\mathcal{X}_\ell\subset \{C\}$ which implies $|\mathcal{X}_\ell|\le 1.$
Consider the $GC_4$ set $\mathcal{X}_i=\mathcal{X}\setminus \lambda_i.$ Here we have two maximal lines: $\lambda_4$ and $\ell_i^{oo},$ intersecting at the node $B\in \ell.$ Therefore we conclude from Lemma \ref{lem:CG2} that no node from these two maximal lines uses $\ell$ in $\mathcal{X}_2.$ Thus, in view of \eqref{rep}, no node from $\lambda_4,$ except possibly $C,$ uses $\ell$ in $\mathcal{X}.$ Now consider the $GC_4$ set $\mathcal{X}_4=\mathcal{X}\setminus \lambda_4.$ Observe, on the basis of the characterization of Proposition \ref{prp:n-1max}, that $\mathcal{X}_4$ has exactly $3$ maximal lines. On the other hand here the line $\ell$ intersects each maximal line at a node. Therefore, in view of Proposition \ref{prop:3max}, part "Furthermore", we have that ${(\mathcal{X}_4)}_\ell=\emptyset.$ Hence, in view of \eqref{rep}, we conclude that $\mathcal{X}_\ell\subset \{C\}.$
Now, to complete the proof it suffices to show that the node $C$ does not use $\ell.$ Let us determine which lines uses $C.$ Since $C=\lambda_i\cap \lambda_4$ first of all it uses the two maxmal lines $\{\lambda_1,\lambda_2,\lambda_3\}\setminus \{\lambda_i\}.$ It is easily seen that the next two lines $B$ uses are $OO$ lines: $\{\ell_1^{oo},\ell_2^{oo},\ell_3^{oo}\}\setminus\{\ell_i^{oo}\}.$ Now notice that, the two nodes, except $C$, which do not belong to the four used lines are $B$ and the (s) node $A_i^i.$ Hence the fifth line $B$ uses is the line passing through the latter two nodes. This line coincides with $\ell$ if and only if $i=1.$
In the final and most interesting part of the proof we will show that the case $i=1,$ when the node $B$ uses the line $\ell,$ i.e., the case when the (s) node $\ell$ passes is $A_1^1$ and $B\in \ell_1^{oo},$ where $B=\ell\cap\lambda_4,$ is impossible. \begin{figure}
\caption{The case $i=1.$}
\label{pic12}
\end{figure} More precicely, we will do the following (see Fig. \ref{pic12}). By assuming that \begin{enumerate} \item the maximal lines $\lambda_i, i=1,\ldots,4,$ are given, the three outside nodes $O_1,O_2,O_3$ are given, \item the two $OO$ lines $\ell_2^{oo},\ell_3^{oo}$ are intersecting the three maximal lines at $6$ distinct points, and \item the three conditions in Proposition \ref{prp:n-1max}, part ``Moreover" are satisfied, i.e., the line through the two special nodes $\{A_1^1,A_2^2,A_3^3\}\setminus A_i^i\}$ passes through the outside node $O_i$ for each $i=1,2,3,$ \end{enumerate} we will prove that the third $OO$ line $\ell_1^{oo}$ passes necessarily through the the node $D,$ i.e., intersection node of the two maximal lines $\lambda_2$ and $\lambda_3.$ Since this makes the number of all nodes of the set $\mathcal{X}$ only $19$ instead of $21,$ we may conclude that the abovementioned case is impossible.
For this end, to simplify the Fig. \ref{pic12} let us delete from it the maximal lines $\lambda_1$ and $\lambda_4$ to obtain the following Fig. \ref{pic15}. \begin{figure}
\caption{The $(9_3)$ configuration with $3$ triangles.}
\label{pic15}
\end{figure} Let us now apply the well-known Pappus hexagon theorem for the pair of triple collinear nodes here $$A_1^1,\ \ E,\ \ F;$$ $$O_1,\ \ A_2^2,\ \ A_3^3.$$ Now observe that $$\ell(A_1^1,A_2^2) \cap \ell(E,O_1)=O_3,\ \ell(E,A_3^3) \cap \ell(F,A_2^2)=D,\ \ell(A_1^1,A_3^3) \cap \ell(F,O_1)=O_2,$$ where $\ell(A,B)$ denotes the line passing through the points $A$ and $B.$ Thus, according to the Pappus theorem we get that the triple of nodes $D,O_2,O_3$ is collinear. \end{proof}
Notice that in Fig. \ref{pic15} we have a $(9_3)$ configuration of 9 lines and 9 points (three triangles) that occurs with each line meeting 3 of the points and each point meeting 3 lines (see \cite{HCV}, Chapter 3, Section 17). \begin{figure}
\caption{A non-used $4$-node line $\ell_4^*$ in a $GC_5$ set $\mathcal{X}^*$}
\label{pic11}
\end{figure} \begin{remark} \label{rm} Let us show that the case of non-used $4$-node line in Figure \ref{pic1} is possible nevertheless. The problem with this is that we have to confirm that the three conditions in Proposition \ref{prp:n-1max}, are satisfied. More precicely: \begin{enumerate} \item $\ell_i^{oo}\cap \lambda_i=\emptyset$ for each $i=1,2,3;$ \item The line through the two special nodes $\{A_1^1,A_2^2,A_3^3\}\setminus A_i^i\}$ passes through the outside node $O_i$ for each $i=1,2,3.$ \end{enumerate} Let us outline how one can get a desired figure (see Fig. \ref{pic11}). Let us start the figure with the three maximal lines (non-concurrent) $\lambda_1, \lambda_2,\lambda_3.$ Then we choose two $OO$ lines lines $\ell_1^{oo}, \ell_3^{oo}$ through the outside node $O_2,$ which intersect the three maximal lines at $6$ distinct points. Next we get the line $\ell_4^*$ which passes through the points $B$ and $C,$ where $B=\ell_1^{oo}\cap \lambda_3$ and $C=\ell_3^{oo}\cap \lambda_2.$ Now we find the point $A_1^1:=\ell_4^*\cap \lambda_1.$ By intersecting the line through $O_2$ and $A_1^1$ with the maximal line $\lambda_3$ we get the node $A_3^3.$ Then we choose a ``free node" $A_2^2$ on the maximal line $\lambda_2.$ This enables to determine the two remaining outside nodes: $O_1, O_3.$ Namely we have that $O_1$ is the intersection point of the line through $A_2^2$ and $A_3^3$ with the line $\ell_3^{oo}$ and $O_3$ is the intersection point of the line through $A_2^2$ and $A_1^1$ with the line $\ell_1^{oo}.$ Thus we get the line $\ell_2^{oo}$ which passes through the outside nodes $O_1, O_3.$ Next we get the points of intersection of the line $\ell_2^{oo}$ with the three maximal lines as well as the point of intersection with the line $\ell_4^*$ denoted by $D.$ Now, we choose the maximal line $\lambda_4$ passing through $D$ and intersecting the remaining maximal lines and the three $OO$ lines at $6$ new distinct nodes. Finally, all the specified intersection points in Fig. \ref{pic11} we declare as the nodes of $\mathcal{X}^*.$\end{remark}
\section{Proof of Conjecture \ref{mainc} \label{ss:conj}}
Let us start the proof with a list of the major cases in which Conjecture \ref{mainc} is true.
\noindent \emph{Step 1.} The Conjecture \ref{mainc} is true in the following cases:
\begin{enumerate} \setlength{\itemsep}{0mm} \item The line $\ell$ is a maximal line.
Indeed, as we have mentioned already, in this case we have $\mathcal{X}_\ell=\mathcal{X}\setminus \ell$ and all the conclusions of Conjecture can be readily verified.
\item The line $\ell$ is an $n$-node line, $n\in \mathbb N$.
In this case Conjecture \ref{mainc} is valid by virtue of Theorem \ref{th:corrected} (for $n\in \mathbb N\setminus \{3\}$) and Proposition \ref{prp:n=3} (for $n=3$).
\item The line $\ell$ is a $2$-node line.
In this case Conjecture \ref{mainc} follows from the statement \eqref{2nodel} and the adjoint statement next to it. \end{enumerate}
Now, let us prove Conjecture by complete induction on $n$ - the degree of the node set $\mathcal{X}.$ Obviously Conjecture is true in the cases $n=1,2.$ Note that this follows also from Step 1 (i) and (ii).
Assume that Conjecture is true for any node set of degree not exceeding $n-1.$ Then let us prove that it is true for the node set $\mathcal{X}$ of degree $n.$ Suppose that we have a $k$-node line $\ell.$
\noindent \emph{Step 2:} Suppose additionally that there is an $\ell$-disjoint maximal line $\lambda.$ Then we get from Lemma \ref{lem:CG1} that \begin{equation}\label{abb}\mathcal{X}_{\ell} = {(\mathcal{X} \setminus \lambda)}_{\ell}. \end{equation} Therefore by using the induction hypothesis for the $GC_{n-1}$ set $\mathcal{X}':=\mathcal{X} \setminus \lambda$ we get the relation \eqref{1bbaa}, where $\sigma':=\sigma(\mathcal{X}',\ell)\le s \le k$ and $\sigma'=2k-(n-1)-1=2k-n=\sigma+1.$ Here we use also Proposition \ref{proper} in checking that $\mathcal{X}_\ell$ is an $\ell$-proper subset of $\mathcal{X}.$
Now let us verify the part "Moreover". Suppose that $\sigma=2k-n-1\ge 3,$ i.e. $2k\ge n+4,$ and $\mu(\mathcal{X})>3.$ For the line $\ell$ in the $GC_{n-1}$ set $\mathcal{X}_0$ we have $\sigma'=\sigma+1\ge 4.$ Thus if $\mu(\mathcal{X}')>3$ then, by the induction hypothesis, we have that $(\mathcal{X}')_\ell\neq\emptyset.$ Therefore we get, in view of \eqref{abb}, that $\mathcal{X}_\ell\neq\emptyset.$ It remains to consider the case $\mu(\mathcal{X}')=3.$ In this case, in view of Proposition \ref{prp:CG-1}, we have that $\mu(\mathcal{X})=4,$ which, in view of Theorem \ref{th:CGo10}, implies that $4\in \{n-1,n,n+1,n+2\},$ i.e., $2\le n\le 5.$
The case $n=2$ was verified already. Now, since $2k\ge n+4$ we deduce that either $k\ge 4$ if $n=3, 4,$ or $k\ge 5$ if $n=5.$ All these cases follow from Step 1 (i) or (ii).
The part "Furthermore" follows readily from the relation \eqref{abb}.
\noindent \emph{Step 3:} Suppose additionally that there is a pair of $\ell$-adjacent maximal lines $\lambda', \lambda''.$ Then we get from Lemma \ref{lem:CG2} that \begin{equation}\label{bbac} \mathcal{X}_{\ell} = {(\mathcal{X} \setminus (\lambda' \cup \lambda''))}_{\ell}. \end{equation}
Therefore by using the induction hypothesis for the $GC_{n-2}$ set $\mathcal{X}'':=\mathcal{X} \setminus (\lambda' \cup \lambda''))$ we get the relation \eqref{1bbaa}, where $\sigma'':=\sigma(\mathcal{X}'',\ell)\le s \le k-1$ and $\sigma''=2(k-1)-(n-2)-1=\sigma.$ Here we use also Proposition \ref{proper} to check that $\mathcal{X}_\ell$ is an $\ell$-proper subset of $\mathcal{X}.$
Let us verify the part "Moreover". Suppose that $\sigma=2k-n-1\ge 3$ and $\mu(\mathcal{X})>3.$ The line $\ell$ is $(k-1)$-node line in the $GC_{n-2}$ set $\mathcal{X}''$ and we have that $\sigma''=\sigma\ge 3.$ Thus if $\mu(\mathcal{X}'')>3$ then, by the induction hypothesis, we have that $(\mathcal{X}'')_\ell\neq\emptyset$ and therefore we get, in view of \eqref{bbac}, that $\mathcal{X}_\ell\neq\emptyset.$ It remains to consider the case $\mu(\mathcal{X}'')=3.$ Then, in view of Proposition \ref{prp:CG-1}, we have that $\mu(\mathcal{X})=4$ or $5,$ which, in view of Theorem \ref{th:CGo10}, implies that $4\ \hbox{or}\ 5\in \{n-1,n,n+1,n+2,\}$ i.e., $2\le n\le 6.$
The cases $2\le n\le 5$ were considered in the previous step. Thus suppose that $n=6.$ Then, since $2k\ge n+4,$ we deduce that $k\ge 5.$ In view of Step 1, (ii), we may suppose that $k=5.$
Now the set $\mathcal{X}''$ is a $GC_4$ and the line $\ell$ is a $4$-node line there. Thus, in view of Step 1 (ii) we have that $(\mathcal{X}'')_\ell\neq \emptyset.$ Therefore we get, in view of \eqref{bbac}, that
$\mathcal{X}_\ell\neq \emptyset.$
The part "Furthermore" follows readily from the relation \eqref{bbac}.
\noindent \emph{Step 4.} Now consider any $k$-node line $\ell$ in a $GC_n$ set $\mathcal{X}.$ In view of Step 1 (iii) we may assume that $k\ge 3.$ In view of Theorem \ref{th:CGo10} and Proposition \ref{prop:3max} we may assume also that $\mu(\mathcal{X})\ge n-1.$
Next suppose that $k\le n-2.$ Since then $\mu(\mathcal{X})>k$ we necessarily have either the situation of Step 2 or Step 3.
Thus we may assume that $k\ge n-1.$ Then, in view of Step 1 (i) and (ii), it remains to consider the case $k =n-1,$ i.e., $\ell$ is an $(n-1)$-node line. Again if $\mu(\mathcal{X})\ge n$ then we necessarily have either the situation of Step 2 or Step 3. Therefore we may assume also that $\mu(\mathcal{X})=n-1.$ By the same argument we may assume that each of the $n-1$ nodes of the line $\ell$ is an intersection node with one of the $n-1$ maximal lines.
Therefore the conditions of Proposition \ref{prp:n-1} are satisfied and we arrive to the two cases: $n=4, k=3, \sigma=1$ or $n=5, k=4, \sigma=2.$ In both cases we have that $\mathcal{X}_\ell=\emptyset.$ Thus in this case Conjecture is true.
$\square$
\subsection{The \label{ss:count} characterization of the case $\sigma=2,\ \mu(\mathcal{X}) >3$}
Here we bring, for each $n$ and $k$ with $\sigma=2k-n-1 =2,$ two constructions of $GC_n$ set and a nonused $k$-node line there. At the end (see forthcoming Proposition \ref{charact}) we prove that these are the only constructions with the mentioned property.
Let us start with a counterexample in the case $n=k=3$ (see \cite{HV}, Section 3.1), i.e., with a $GC_3$ set $\mathcal{Y}^*$ and a $3$-node line $\ell_3^*$ there which is not used.
\begin{figure}
\caption{A non-used $3$-node line $\ell_3^*$ in a $GC_3$ set $\mathcal{Y}^*$}
\label{pic2}
\end{figure}
Consider a $GC_3$ set $\mathcal{Y}^*$ of $10$ nodes with exactly three maximal lines: $\lambda_1,\lambda_2,\lambda_3$ (see Fig. \ref{pic2}). This set has construction \eqref{01O} and satisfies the conditions listed in Proposition \ref{prp:nmax}. Now observe that the $3$-node line $\ell_3^*$ here intersects all the three maximal lines at nodes. Therefore, in view of Proposition \ref{prp:n=3}, (iii), the line $\ell_3^*$ cannot be used by any node in $\mathcal{Y}^*,$ i.e., $$(\mathcal{Y}^*)_{\ell_3^*}=\emptyset.$$
Let us outline how one can get Fig. \ref{pic2}. We start the mentioned figure with the three lines $\ell_1^o, \ell_2^o,\ell_3^o$ through $O,$ i.e., the outside node. Then we choose the maximal lines $\lambda_1,\lambda_2,$ intersecting $\ell_1^o, \ell_2^o$ at $4$ distinct points. Let $A_i$ be the intersection point $\lambda_i\cap\ell_i^o,\ i=1,2.$ We chose the points $A_1$ and $A_2$ such that the line through them: $\ell_3^*$ intersects the line $\ell_3^o$ at a point $A_3.$ Next we choose a third maximal line $\lambda_3$ passing through $A_3.$ Let us mention that we chose the maximal lines such that they are not concurrent and intersect the three lines through $O$ at nine distinct points. Finally, all the specified intersection points in Fig. \ref{pic2} we declare as the nodes of $\mathcal{Y}^*.$
In the general case of $\sigma=2k-n-1=2$ we set $k=m+3$ and obtain $n=2m+3,$ where $m=0,1,2,\ldots$. Let us describe how the previous $GC_3$ node set $\mathcal{Y}^*$ together with the $3$-node line $\ell_3^*$ can be modified to $GC_n$ node set $\bar{\mathcal{X}}^*$ with a $k$-node line $\bar{\ell}_3^*$ in a way to fit the general case. That is to have that $(\bar{\mathcal{Y}^*})_{\bar{\ell_3}}=\emptyset.$ \begin{figure}\label{pic3}
\end{figure}
For this end we just leave the line $\ell_3^*$ unchanged, i.e., $\bar{\ell}_3^*\equiv\ell_3^*$ and extend the set $\mathcal{Y}^*$ to a $GC_n$ set $\bar{\mathcal{Y}}^*$ in the following way. We fix $m$ points: $B_i, i=1,\ldots,m,$ in $\ell_3^*$ different from $A_1,A_2,A_3$ (see Fig. \ref{pic3}). Then we add $m$ pairs of (maximal) lines $\lambda_i',\lambda_i'', i=1,\ldots,m,$ intersecting at these $m$ points respectively: $$\lambda_i'\cap \lambda_i'' = B_i,\ i=1,\ldots,m.$$
We assume that the following condition is satisfied: \begin{enumerate} \item The $2m$ lines $\lambda_i',\lambda_i'', i=1,\ldots,m,$ together with $\lambda_1,\lambda_2,\lambda_3$ are in general position, i.e., no two lines are parallel and no three lines are concurrent; \item The mentioned $2m+3$ lines intersect the lines $\ell_1^o, \ell_2^o,\ell_3^o$ at distinct $3(2m+3)$ points. \end{enumerate} Now all the points of intersections of the $2m+3$ lines $\lambda_i',\lambda_i'', i=1,\ldots,m,$ together with $\lambda_1,\lambda_2,\lambda_3$ are declared as the nodes of the set $\bar{\mathcal{Y}}^*.$ Next for each of the lines $\lambda_i',\lambda_i'', i=1,\ldots,m,$ also two from the three intersection points with the lines $\ell_1,\ell_2,\ell_3,$ are declared as (``free") nodes. After this the lines $\lambda_1,\lambda_2,\lambda_3$ and the lines $\lambda_i',\lambda_i'', i=1,\ldots,m,$ become $(2m+4)$-node lines, i.e., maximal lines.
Now one can verify readily that the set $\bar{\mathcal{Y}}^*$ is a $GC_n$ set since it satisfies the construction \eqref{01O} with $n=2m+3$ maximal lines: $\lambda_i',\lambda_i'', i=1,\ldots,m,$ together with $\lambda_1,\lambda_2,\lambda_3,$ and satisfies the conditions listed in Proposition \ref{prp:nmax}.
Finally, in view of Lemma \ref{lem:CG2} and the relation \eqref{bbb}, applied $m$ times with respect to the pairs $\lambda_i',\lambda_i'', i=1,\ldots,m,$ gives: ${(\bar{\mathcal{Y}}^*)}_{\bar{\ell}_3^*}=(\mathcal{Y}^*)_{\ell_3^*}=\emptyset.$
Let us call the set $\bar{\mathcal{Y}}^*$ an $m$-modification of the set $\mathcal{Y}^*.$ In the same way we could define $\bar{\mathcal{X}}^*$ as an $m$-modification of the set $\mathcal{X}^*$ from Fig. \ref{pic1}, with the $4$-node non-used line $\ell_4^*$ (see Remark \ref{rm}). The only differences from the previous case here are: \begin{enumerate} \item
Now $k=m+4,\ n=2m+5, m=0,1,2,\ldots$ (again $\sigma=2$); \item We have $2n+4$ maximal lines: $\lambda_i',\lambda_i'', i=1,\ldots,m,$ and the lines $\lambda_1,\lambda_2,\lambda_3,\lambda_4;$ \item Instead of the lines $\ell_1^o, \ell_2^o,\ell_3^o$ we have the lines $\ell_{1}^{oo}, \ell_{2}^{oo}, \ell_{3}^{oo};$ \item For each of the lines $\lambda_i',\lambda_i'', i=1,\ldots,m,$ all three intersection points with the lines $\ell_1^{oo},\ell_2^{oo},\ell_3^{oo},$ are declared as (``free") nodes. \end{enumerate} Now one can verify readily that the set $\bar{\mathcal{X}}^*$ is a $GC_n$ set since it satisfies the construction \eqref{01OOO} with $n=2m+4$ maximal lines,
and satisfies the conditions listed in Proposition \ref{prp:n-1max}.
Thus we obtain another construction of non-used $k$-node lines in $GC_n$ sets, with $\sigma=2,$ where $k=m+4, n=2m+5.$
At the end let us prove the following \begin{proposition}\label{prp:sigma2}\label{charact} Let $\mathcal{X}$ be a $GC_n$ set and $\ell$ be a $k$-node line with $\sigma:=2k-n-1=2$ and $\mu(\mathcal{X})>3.$ Suppose that the line $\ell$ is a non-used line. Then we have that either $\mathcal{X}=\bar{\mathcal{X}}^*, \ \ell={\bar{\ell}_4^*},$ or $\mathcal{X}=\bar{\mathcal{Y}}^*, \ \ell={\bar{\ell}_3^*}.$ \end{proposition} \begin{proof} Notice that $n$ is an odd number and $n\ge 3.$ In the case $n=3$ Proposition \ref{prp:sigma2} follows from Proposition \ref{prp:n=3}.
Thus suppose that $n\ge 5.$ Since $\mu(\mathcal{X})>3,$ we get, in view of Theorem \ref{th:CGo10}, that \begin{equation}\label{n-1}\mu(\mathcal{X})\ge n-1. \end{equation}
Next, let us prove that there is no $\ell$-disjoint maximal line $\lambda$ in $\mathcal{X}.$
Suppose conversely that $\lambda$ is a maximal line with $\lambda\cap \ell\notin\mathcal{X}.$ Denote by $\mathcal{X}':=\mathcal{X}\setminus \lambda.$ Since $\mathcal{X}_\ell=\emptyset$ therefore, by virtue of the relation \eqref{rep}, we obtain that $(\mathcal{X}')_\ell=\emptyset.$ Then we have that $\sigma':=\sigma(\mathcal{X}',\ell)=2k-(n-1)-1=3.$ By taking into account the latter two facts, i.e., $(\mathcal{X}')_\ell=\emptyset$ and $\sigma'=3,$ we conclude from Conjecture \ref{mainc}, part ``Moreover", that $\mu(\mathcal{X}')=3.$ Next, by using \eqref{n-1} and Proposition \ref{prp:CG-1}, we obtain that that $\mu(\mathcal{X})=4.$ By applying again \eqref{n-1} we get that $4\ge n-1,$ i.e., $n\le 5.$ Therefore we arrive to the case: $n=5.$ Since $\sigma=2$ we conclude that $k=4.$ Then observe that the line $\ell$ is $4$-node line in the $GC_4$ set $\mathcal{X}'$. By using Theorem \ref{th:corrected} we get that $(\mathcal{X}')_\ell\ne\emptyset,$ which is a contradiction.
Now let us prove Proposition in the case $n=5.$ As we mentioned above then $k=4.$ We have that there is no $\ell$-disjoint maximal line. Suppose also that there is no pair of $\ell$-adjacent maximal lines. Then, in view of \eqref{n-1}, we readily get that $\mu(\mathcal{X})= 4$ and through each of the four nodes of the line $\ell$ there passes a maximal line. Therefore, Proposition \ref{prp:n-1max} yields that $\mathcal{X}$ coincides with $\mathcal{X}^*$ (or, in other words, $\mathcal{X}$ is a $0$-modification of $\mathcal{X}^*$) and $\ell$ coincides with $\ell_4^*.$
Next suppose that there is a pair of $\ell$-adjacent maximal lines: $\lambda',\lambda''.$ Denote by $\mathcal{X}'':=\mathcal{X}\setminus (\lambda'\cup\lambda'').$ Then we have that $\ell$ is a $3$-node non-used line in $\mathcal{X}''.$ Thus we conclude readily that $\mathcal{X}$ coincides with $\bar{\mathcal{Y}}^*$ and $\ell$ coincides with ${\bar\ell}_3^*$ (with $m=1$).
Now let us continue by using induction on $n.$ Assume that Proposition is valid for all degrees up to $n-1.$ Let us prove it in the case of the degree $n.$ We may suppose that $n\ge 7.$ It suffices to prove that there is a pair of $\ell$-adjacent maximal lines: $\lambda',\lambda''.$ Indeed, in this case we can complete the proof just as in the case $n=5.$
Suppose by way of contradiction that there is no pair of $\ell$-adjacent maximal lines. Also we have that there is no $\ell$-disjoint maximal line. Therefore we have that $\mu(\mathcal{X}) \le k.$ Now, by using \eqref{n-1}, we get that $k\ge n-1.$ Therefore $2=\sigma=2k-n-1\ge 2n-2-n-1=n-3.$ This implies that $n-3\le 2,$ i.e., $n\le 5,$ which is a contradiction. \end{proof}
\noindent \emph{Hakop Hakopian}
\noindent{Department of Informatics and Applied Mathematics\\ Yerevan State University\\ A. Manukyan St. 1\\ 0025 Yerevan, Armenia}
\noindent E-mail: $<[email protected]$>$
\noindent \emph{Vahagn Vardanyan}
\noindent{Institute of Mathematics\\ National Academy of Sciences of Armenia\\ 24/5 Marshal Baghramian Ave. \\ 0019 Yerevan, Armenia}
\noindent E-mail:$<[email protected]$>$
\end{document}
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\begin{document}
\title{Topological Protection of Coherence in Noisy Open Quantum Systems} \author{Yu Yao$^{*}$} \affiliation{Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484} \author{Henning Schlömer$^{*}$} \affiliation{Institute for Theoretical Solid State Physics, RWTH Aachen University, Otto-Blumenthal-Str. 26, D-52056 Aachen, Germany} \author{Zhengzhi Ma} \affiliation{Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484} \author{Lorenzo Campos Venuti} \affiliation{Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484} \author{Stephan Haas} \affiliation{Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484} \date{\today} \begin{abstract} We consider topological protection mechanisms in dissipative quantum systems in the presence of quenched disorder, with the intent to prolong coherence times of qubits. The physical setting is a network of qubits and dissipative cavities whose coupling parameters are tunable, such that topological edge states can be stabilized. The evolution of a fiducial qubit is entirely determined by a non-Hermitian Hamiltonian which thus emerges from a bona-fide physical process. It is shown how even in the presence of disorder winding numbers can be defined and evaluated in real space, as long as certain symmetries are preserved. Hence we can construct the topological phase diagrams of noisy open quantum models, such as the non-Hermitian disordered Su-Schrieffer-Heeger dimer model and a trimer model that includes longer-range couplings. In the presence of competing disorder parameters, interesting re-entrance phenomena of topologically non-trivial sectors are observed. This means that in certain parameter regions, increasing disorder drastically increases the coherence time of the fiducial qubit.
\end{abstract} \maketitle \global\long\def*{*} \footnotetext{These authors contributed equally to this work}
\section{Introduction} Due to their characteristic protection against environmental noise, topological phases of matter \citep{Fu2007,Fu2007_2, Kobayashi2013} are considered to be promising candidates for the realization of noise-resilient quantum computers \citep{Sankar2008, Sau2010, Sarma2015, Stern2013, Alicea2011, Freedman2006, Freedman2003, Akhmerov2010}. Furthermore, it was shown in \cite{Venuti2017} that the presence of topological edge states can preserve quantum mechanical features, e.g. coherence of a fiducial qubit, in the presence of dissipation (see also \citep{man_cavity-based_2015, campos_venuti_long-distance_2006, campos_venuti_qubit_2007} for other works in a similar spirit). In that work, dissipative one-dimensional (1D) quantum optical qubit-cavity architectures were analyzed, where effective non-Hermitian Hamiltonians of the form of a tight-binding chain with diagonal complex entries were derived. The time evolution of the boundary-qubit coherence, driven by such non-Hermitian Hamiltonians, was extensively studied for choices of hopping parameters that admit symmetry-protected topological states localized at the edges of the system. It was found that (quasi-)dark modes, i.e., boundary states with exponentially small (in system size) imaginary parts, protect the edge qubits from decoherence effects via photon leakage through cavities.
Moreover, disordered as well as non-Hermitian generalizations of 1D topological insulators, such as the Su-Schrieffer-Heeger (SSH) model \citep{Su1980, Heeger1988}, have been studied theoretically \citep{Mondragon2014,Luo2019}, where the real space winding number was analyzed for different disorder strengths on the hopping parameters.
Here, we build on the work presented in \cite{Venuti2017}, addressing the role of quenched disorder in the qubit-cavity arrays. We fully characterize the disordered, non-Hermitian system's topology by computing the winding number in real space in the parameter space spanned by the coupling amplitude and the disorder strength. From this characterization, accurate predictions for the behavior of a fiducial qubit's coherence can be made for long times, therefore expanding the discussion of the quantum optical systems to a broader physical context, considering both quenched disorder and dissipation.
The remainder of this paper is organized as follows. In Sec.~\ref{sec:setup}, we derive the effective non-Hermitian Hamiltonian describing qubit-cavity arrays using the Lindblad formalism. In Sec.~\ref{sec:real_space_method}, the topological characterization of dissipative, disordered systems is illustrated, which is then applied to non-Hermitian dimer and trimer models in Sec~\ref{sec:main}. Special focus is put on the coherence of the qubit located at the boundary, whose faith can be accurately predicted from the phase diagrams. We then briefly discuss possible applications in quantum computation via dark-state braiding in Sec.~\ref{sec:quantum_computation} and conclude in Sec.~\ref{sec:conclusion}.
\section{The Setup}
\label{sec:setup} We consider a network consisting of qubits coupled to dissipative cavities in a Jaynes-Cumming fashion. Specifically, we study networks of $M$ qubits and $K$ dissipative cavities, as illustrated in Fig. \ref{fig:network} for $M=4$ and $K=5$. The Hamiltonian of the system has the following form \begin{eqnarray} H & = & \sum_{l,m=1}^{K}J_{l,m}(a_{l}^{\dagger}a_{m}+\text{h.c.})\nonumber \\
& + & \sum_{i=1}^{M}\sum_{l=1}^{K}\tilde{J}_{l,i}(a_{l}^{\dagger}\sigma_{i}^{-}+\text{h.c.}),\label{hamiltonian} \end{eqnarray} where $a_{l}^{\dagger}$ and $a_{l}$ are the bosonic creation and annihilation operators for cavity mode $l$, and $\sigma_{i}^{\pm}$ are the ladder operators for qubit $i$. We consider a Lindblad master equation $\dot{\rho}=\mathcal{L}[\rho]$, where $\mathcal{L}=\mathcal{K}+\mathcal{D}$, the coherent part is $\mathcal{K}=-i\left[H,\bullet\right]$, whereas the dissipative part is \begin{equation} \mathcal{D}[\rho]=\sum_{l=1}^{K}\Gamma_{l}\Big[2a_{l}\rho a_{l}^{\dagger}-\{a_{l}^{\dagger}a_{l},\rho\}\Big].\label{lindbladian} \end{equation} Such Lindbladian description is accurate at sufficiently low temperature in particular in circuit QED experiments.
\begin{figure}
\caption{Example of a network of qubits (filled circles) interacting with lossy cavities (hollow circles). Wavy lines indicate coherent hopping matrix elements $\tilde{J}_{i,j}$, connecting cavity $i$ with qubit $j$, and $J_{i,j}$, connecting cavity $i$ with cavity $j$. Arrows indicate incoherent decay $\Gamma_{i}$ in cavity $i$. }
\label{fig:network}
\end{figure}
The Lindbladian can also be written as $\mathcal{L}=\mathcal{K}'+\mathcal{D}'$, where $\mathcal{K}'(\rho)=-i(H'\rho-\rho{H'}^{\dagger})$ defines the non-Hermitian Hamiltonian $H'$: \begin{equation} H'=H-i\sum_{l=1}^{K}\Gamma_{l}a_{l}^{\dagger}a_{l}, \end{equation} and \begin{equation} \mathcal{D}'(\rho)=\sum_{l=1}^{K}2\Gamma_{l}a_{l}\rho a_{l}^{\dagger}. \end{equation}
Consider the Fock space of the system $\mathcal{F}=\oplus_{n=0}^{\infty}\mathcal{H}_{n}$
where $\mathcal{H}_{n}$ is the Hilbert space of $n$ particles (at this level the distinction between spins and bosons is unimportant). In the space of operators on $\mathcal{F}$ we define the Hilbert-Schmidt scalar product $\langle\!\langle x|y\rangle\!\rangle=\operatorname{Tr}(x^{\dagger}y)$. Using the isomorphism $\mathcal{B}_{HS}(\mathcal{F})\simeq\mathcal{F}\otimes\mathcal{F}^{*}$ ($\mathcal{B}_{HS}(\mathcal{F})$ is the space of bounded Hilbert-Schmidt operators on $\mathcal{F}$), the space of operators $\mathcal{B}_{HS}(\mathcal{F})$ can be identified with \begin{equation} \mathcal{B}_{HS}(\mathcal{F})\simeq\bigoplus_{i,j=0}^{\infty}\mathcal{H}_{i}\otimes\mathcal{H}_{j}^{*}. \end{equation} In simpler terms, $\mathcal{B}_{HS}(\mathcal{F})$ has a block structure with two labels $(i,j)$ each label being a particle number. The non-Hermitian Hamiltonian $H'$ preserves the number of particles and correspondigly $\mathcal{K}'$ is block-diagonal in $(i,j)$. Instead, $\mathcal{D}'$ connects the sector $(i,j)$ with the sector $(i-1,j-1)$, i.e.~it decreases the number of particles by one.
In this paper we will be mostly interested in the coherence of a fiducial qubit, that, without loss of generality we place at site 1. In the standard basis, the coherence of a qubit in state $\rho$, can be defined as $\mathcal{C}=2\vert\rho_{\downarrow,\uparrow}\vert$ \citep{Baumgratz2014}. We initialize the system such that all cavities are empty and qubits are in the lowest state ($|\downarrow\rangle$), while on the fiducial qubit the state is $|\psi\rangle=\alpha|\downarrow\rangle+\beta|\uparrow\rangle$. We further fix $|\alpha\beta|=1/2$ which means that at the beginning the coherence assumes its maximal value one. We are interested in the evolution of the coherence as a function of time.
Let $|0\rangle$ be the vacuum state with no excitation on any qubit or cavity, while $|j\rangle$ denotes a single excitation on the $j$th site, describing either an excited qubit or a cavity hosting a photon. We use the notation $|j\rangle\!\rangle\leftrightarrow|0\rangle\langle j|$. It can be shown \cite{Venuti2017} that the evolution of the coherence at later time is given by \begin{eqnarray}
\mathcal{C}(t)=2|\rho_{\downarrow,\uparrow}(t)| & = & 2|\langle0|\rho(t)|1\rangle|=2|\operatorname{Tr}|1\rangle\langle0|\rho(t)|\nonumber \\
& = & 2|\langle\!\langle1|\rho(t)\rangle\!\rangle|=2|\langle\!\langle1|e^{t\mathcal{L}}|\rho(0)\rangle\!\rangle|.\label{coh} \end{eqnarray}
Because of the block structure of the Lindbladian, $\langle\!\langle1|e^{t\mathcal{L}}|\rho(0)\rangle\!\rangle=\langle\!\langle1|e^{t\tilde{\mathcal{L}}}|\tilde{\rho}(0)\rangle\!\rangle$, where $\tilde{X}$ is the operator $X$ restricted to the linear space
$\mathcal{V}_{0,1}=\mathrm{Span}(|0\rangle\langle j|),j=1,2,\ldots,N$. In particular, $\tilde{\rho}(0)=\overline{\alpha}\beta|0\rangle\langle1|=(1/2)|0\rangle\langle1|$. For what regard the restriction of the Lindbladian we have $\tilde{\mathcal{L}}=\tilde{\mathcal{K}'}$. Moreover, in $\mathcal{V}_{0,1}$, \begin{align}
{\mathcal{K}'}_{l,m} & =\operatorname{Tr}\left(|l\rangle\langle0|{\mathcal{K}'}(|0\rangle\langle m|)\right)\\
& =i\langle m|{H'}^{\dagger}|l\rangle\\
& =i\overline{\langle l|H'|m\rangle}. \end{align}
In other words, the evolution of the coherence of the fiducial qubit is entirely determined by the non-Hermitian Hamiltonian $-\overline{H'}$ in the one-particle sector. Calling $\mathsf{H}:=-\left.\overline{H'}\right\vert _{\mathrm{one\ particle}}$ we have finally
\begin{equation}
\mathcal{C}(t)=\left\vert \langle\!\langle1|e^{-it\mathsf{H}}|1\rangle\!\rangle\right\vert .\label{eq:coherence_final} \end{equation}
We would like to stress that, in this setting, a non-Hermitian Hamiltonian emerges from a genuine \emph{bona-fide} quantum evolution whereas in most current proposals non-Hermitian Hamiltonians are simulated in classical dissipative wave-guides via the analogy between Helmoltz and Schrödinger equation (see e.g.~\citep{rudner_topological_2009}).
In order to prolong the coherence Eq.~\eqref{eq:coherence_final}, one seeks a (non-Hermitian) Hamiltonian $\mathsf{H}$ that admits i) long-lived states, i.e.~eigenstates of $\mathsf{H}$ with small (negative) imaginary part; and ii) that also have large amplitude on the site
$|1\rangle\!\rangle$ (conventionally placed at the beginning of the chain).
Interestingly, both these requirement are satisfied to a large degree in one dimensional topological systems which admit edge states with the required properties in the non-trivial phase. The classification of such dissipative, non-Hermitian, topological chains has been done in \citep{Levitov2016} and utilized to prolong quantum coherence for the first time in \citep{Venuti2017}. Here we extend the analysis of \citep{Venuti2017} to disordered systems where translational invariance is broken. The phase diagrams of topological dissipative chains will tell us which parameter regions and models can be used to prolong the quantum coherence of the fiducial qubit.
\begin{comment}
Let us focus on the single excitation sector of this model. By initializing the first qubit in a state $\alpha_{0}|\uparrow\rangle+\beta_{0}|\downarrow\rangle$
and restricting all other qubits and cavities to be in the $|\downarrow\rangle$ and empty state, respectively, there exists at most one excitation in the entire system during the Lindbladian evolution. Within this super-one-particle sector, the corresponding Hilbert space $\mathcal{H}$
is spanned by $\{|0\rangle$,$|j\rangle$; $j=1,2,\cdots,N=M+K\}$. Here, the $|0\rangle$ state represents the vacuum with no excitation on any qubit or cavity. The remaining states $|j\rangle$ correspond to a single excitation on the $j$th site, i.e., they describe either an excited qubit or a cavity hosting a photon. As a result, the system's density matrix is of the form \begin{equation}
\rho=\rho_{0,0}|0\rangle\langle0|+\sum_{j=1}^{N}(\rho_{0,j}|0\rangle\langle j|+\text{h.c.})+\sum_{i,j=1}^{N}\rho_{i,j}|i\rangle\langle j|,\label{rho_restricted} \end{equation}
with the matrix basis $\{|i\rangle\langle j|;i,j=0,1,2,\cdots,N\}$ spanning the superoperator Hilbert space $\mathcal{H}^{\otimes2}$. Intending to exploit topological edge state protection, we are interested in the coherence of the qubit located at one of the edges of the system, for which we can find the reduced density matrix by tracing out all other degrees of freedom, i.e., \begin{eqnarray}
\operatorname{Tr}_{\text{rest}}\rho & = & \Big(\rho_{0,0}+\sum_{i=2}^{N}\rho_{i,i}\Big)|\downarrow\rangle\langle\downarrow|+\rho_{1,1}|\uparrow\rangle\langle\uparrow|\nonumber \\
& + & (\rho_{\downarrow,\uparrow}|\downarrow\rangle\langle\uparrow|+\text{h.c.}). \end{eqnarray} A coherence measure can now be defined by the sum over all off-diagonal elements of the qubit density matrix \citep{Baumgratz2014}, such that \begin{eqnarray}
\mathcal{C}(t)=2|\rho_{\downarrow,\uparrow}(t)| & = & |\langle0|\rho(t)|1\rangle|=|\operatorname{Tr}|1\rangle\langle0|\rho(t)|\nonumber \\
& = & |\langle\!\langle1|\rho(t)\rangle\!\rangle|=|\langle\!\langle1|e^{t\mathcal{L}}|\rho(0)\rangle\!\rangle|,\label{coh} \end{eqnarray}
where $|1\rangle\!\rangle\leftrightarrow|0\rangle\langle1|$ and $\langle\!\langle x|y\rangle\!\rangle=\operatorname{Tr}(x^{\dagger}y)$. It is easy to show that the Lindblad operator $\mathcal{L}$ only mixes $|1\rangle\!\rangle\leftrightarrow|0\rangle\langle1|$ with
$|j\rangle\!\rangle\leftrightarrow|0\rangle\langle j|$ in the superoperator Hilbert space $\mathcal{H}^{\otimes2}$, i.e. it has a block-diagonal structure, such that we can restrict our considerations to the representation of $\mathcal{L}$ within the subspace $\mathcal{V}_{0,1}$ spanned by the basis $\{|j\rangle\!\rangle;j=1,2,\cdots,N\}$. Therefore, by restricting $\mathcal{L}$ to $\tilde{\mathcal{L}}=\mathcal{L}|_{\mathcal{V}_{0,1}}$
and projecting the initial density matrix to $\tilde{\rho}(0)=\rho(0)|_{\mathcal{V}_{0,1}}$, the fiducial edge qubit's coherence can be evaluated via \begin{equation}
C(t)=2\big|\langle\!\langle1|e^{t\tilde{\mathcal{L}}}|\tilde{\rho}(0)\rangle\!\rangle\big|. \end{equation}
The problem therefore boils down to studying the spectrum of the restricted Lindblad operator $\tilde{\mathcal{L}}=-i\mathsf{H}$. While the coherent part of $\mathsf{H}$ inherits the real and hermitian tight binding character of the original Hamiltonian \eqref{hamiltonian}, the dissipative part contributes complex diagonal elements, given by $i\mathcal{D}(|0\rangle\langle j|)=-i\frac{\Gamma_{j}}{2}|0\rangle\langle j|$, where $j$ labels a cavity. As the imaginary diagonal elements make $\mathsf{H}$ a non-Hermitian operator, the defined $\ell^{2}$ scalar products do not represent physical probabilities anymore. The coherence measure in Eq.~\eqref{coh}, however, is exactly given by such a non-Hermitian time evolution. \end{comment}
\section{Topological invariant of dissipative systems in real space}
\label{sec:real_space_method}
We now briefly recall the topological classification of non-Hermitian quantum systems provided by Rudner \textit{et al.} in Ref.~\citep{Levitov2016}. For non-Hermitian quantum systems hosting dissipative sites, the topological invariant can be defined as the winding number around the dark-state manifold in the Hamiltonian parameter space. A non-trivial phase in dissipative systems corresponds to long-lived edge modes with infinite or exponential large lifetimes.
In previous work \citep{Venuti2017}, the topological classification of non-Hermitian models was formulated within the framework of Bloch theory, which we briefly outline here for comparison with the real-space approach to be introduced. Consider a one-dimensional periodic non-Hermitian chain with $n$ sites per unit cell. In the thermodynamic limit, the Hamiltonian is given by $\mathsf{H}=\oint dk/(2\pi)\sum_{\alpha,\beta=1}^{n}H_{\alpha,\beta}(k)|k,\alpha\rangle\langle k,\beta|$. We shall only focus on the cases with one leaky site per unit cell, as the topological characterization is trivial in all other cases if no additional constraints are imposed \citep{Levitov2016}. The Bloch Hamiltonian of any such system is an $n\times n$ matrix, which can be written as \begin{equation} H(k)=\left(\begin{array}{cc} h(k) & v_{k}\\ v_{k}^{\dagger} & \Delta(k)-i\Gamma \end{array}\right), \end{equation} where $h(k)$ is an $(n-1)\times(n-1)$ Hermitian matrix, $v_{k}$ is a $(n-1)$-dimensional vector and $\Delta(k)-i\Gamma$ is a complex number. The Hamiltonian can be further decomposed in the following manner \begin{equation} H(k)=\left(\begin{array}{cc} U(k) & 0\\ 0 & 1 \end{array}\right)\left(\begin{array}{cc} \tilde{h}(k) & \tilde{v}_{k}\\ \tilde{v}_{k}^{\dagger} & \Delta(k)-i\Gamma \end{array}\right)\left(\begin{array}{cc} U(k)^{\dagger} & 0\\ 0 & 1 \end{array}\right),\label{eqn:Hk} \end{equation} where $U(k)$ is a $(n-1)\times(n-1)$ unitary matrix whose columns are the eigenvectors of $h(k)$, and $\tilde{h}(k)$ is the $(n-1)\times(n-1)$ diagonal matrix of the corresponding eigenvalues. The phases of the eigenvectors are fixed by making all entries of the $(n-1)$-dimensional vector $\tilde{v}_{k}$ real and positive. Any $U(k)$ satisfying the above criteria can be chosen without affecting the final result. Since $U(k)$ is the only component parametrizing the Hamiltonian that can lead to non-trivial topology \citep{Levitov2016}, the winding number of $H(k)$ reduces to the one of $U(k)$, which is given by \begin{equation} W=\oint\frac{dk}{2\pi i}\partial_{k}\ln\operatorname{det}\{U(k)\}.\label{eqn:Wk} \end{equation} We now construct a real-space representation of the winding number that remains well defined when translation invariance is destroyed by e.g.~the presence of disorder. Consider a chain with $n$ sites in each cell and $M$ number of unit cells. For what we said previously, we consider only one leaky site per unit cell, which, without loss of generality, we place at the final site of the cell.
The one-particle (non-Hermitian) Hamiltonian can be written as
\begin{equation}
\mathsf{H}=\sum_{i,j=1}^{M}\sum_{\alpha,\beta=1}^{n}H_{\alpha,\beta}^{i,j}|i,\alpha\rangle\langle j,\beta|\label{eq:H_one} \end{equation}
Generally one thinks of the chain as being made of $M$ cells with $n$ sites each, but one may as well think of $n$ sections with $M$ sites each. In other words, we rearrange Eq.~\eqref{eq:H_one} according to the following block structure \begin{eqnarray} \mathsf{H}=\left(\begin{array}{ccccc} H_{1,1} & H_{1,2} & H_{1,3} & \dots & H_{1,n}\\ H_{2,1} & H_{2,2} & H_{2,3} & \dots & H_{2,n}\\ \vdots & \vdots & \vdots & & \vdots\\ H_{n,1} & H_{n,2} & H_{n,3} & \dots & H_{n,n} \end{array}\right), \end{eqnarray} where each $H_{\alpha,\beta}$ is a $M\times M$ matrix. The matrices $H_{\alpha,\alpha}$ $\alpha=1,\ldots,(n-1)$ are diagonal with chemical potentials on the diagonal. Since we put the leaky site at position $\alpha=n$, the matrix $H_{n,n}=\epsilon_{n}-i\Gamma\leavevmode{\rm 1\ifmmode\mkern-4.8mu \else\kern-0.3em \fi I}$, where $\epsilon_{n}$ is a diagonal matrix of chemical potentials and for simplicity we set the leakage to have value $\Gamma$ on each site.
Recalling the approach used in $k$-space, we first write the real-space Hamiltonian as \begin{align} H & =\left(\begin{array}{cc} \Lambda & V\\ V^{\dagger} & \epsilon_{n}-i\Gamma\leavevmode{\rm 1\ifmmode\mkern-4.8mu \else\kern-0.3em \fi I} \end{array}\right)\nonumber \\
& =\left(\begin{array}{cc} U & 0\\ 0 & \leavevmode{\rm 1\ifmmode\mkern-4.8mu \else\kern-0.3em \fi I} \end{array}\right)\left(\begin{array}{cc} \tilde{\Lambda} & \tilde{V}\\ \tilde{V}^{\dagger} & \epsilon_{n}-i\Gamma\leavevmode{\rm 1\ifmmode\mkern-4.8mu \else\kern-0.3em \fi I} \end{array}\right)\left(\begin{array}{cc} U^{\dagger} & 0\\ 0 & \leavevmode{\rm 1\ifmmode\mkern-4.8mu \else\kern-0.3em \fi I} \end{array}\right).\label{eqn:Hr} \end{align}
$\Lambda$ is a $(n-1)M\times(n-1)M$ Hermitian matrix, while $V$ is a $(n-1)M\times M$ real matrix describing the hopping between decaying and non-decaying sites. $\tilde{\Lambda}$ is a $(n-1)L\times(n-1)L$ diagonal matrix with real eigenvalues of $\Lambda$, and $U$ is a $(n-1)L\times(n-1)L$ unitary matrix that diagonalizes $\Lambda$. The degrees of freedom for the choice of $U$ are fixed by making each $L\times L$ submatrix in $\tilde{V}$ positive-definite, analogous to the procedure in reciprocal space.
With these preparations, the winding number of the unitary matrix $U$ in real space can be evaluated with the prescription of \citep{Kitaev2006} and further elaborations of Refs.~\citep{Mondragon2014,Wang2019,Luo2019}. In particular, $\int_{0}^{2\pi}(dk/2\pi)\times\operatorname{tr}\{\}$ and $\partial_{k}$ become trace per volume and the commutator $-i[X,]$ ($X$ being the position operator), respectively. Note that $X$ is the $M$-sized cell position operator, i.e.~$X=\operatorname{diag}(1,2,\ldots,M,1,2,\dots,M,\ldots,M-1,M)$.
Thus, Eq.~(\ref{eqn:Wk}) in real space can be written as \begin{equation} W=\frac{1}{L^{\prime}}\operatorname{tr}^{\prime}(U^{\dagger}[X,U]).\label{eqn:Wr} \end{equation} Here, $\operatorname{tr}^{\prime}$ stands for trace with truncation. Specifically, we take the trace over the middle interval of length $M^{\prime}$ and leave out $\ell$ sites on each side (total length $M=M^{\prime}+2\ell$).
With Eq.~(\ref{eqn:Wr}), we can explore topological phases in presence of dissipation and disorder.
Note that in the model that we will consider, the matrix $\Lambda$ is not noisy. In general, the model supports a non-trivial topological phase as long as a certain (chiral) symmetry is preserved. Disorder on the elements of $\Lambda$ destroys the symmetry and consequently the system becomes topologically trivial.
\section{Disordered non-Hermitian Systems}
\label{sec:main}
We now apply the real space formalism to investigate topological features in two explicit network geometries, namely the disordered non-Hermitian SSH dimer model and a disordered non-Hermitian trimer model.
\subsection{Disordered non-Hermitian SSH Dimer Model} \label{sec:ssh} \begin{figure}
\caption{Non-Hermitian Su-Schrieffer-Heeger (SSH) model with open boundary conditions. The qubit-cavity and cavity-qubit couplings are given by $J_{1}$ and $J_{2}$ respectively. Off-diagonal disorder is controlled via a uniform distribution function from which the hopping parameters are drawn. }
\label{fig:ssh}
\end{figure}
This model describes an open quantum system of coupled qubits and optical cavities which are arranged in an alternating manner, as shown in Fig.~\ref{fig:ssh}. In the super-one-particle sector, the corresponding restricted Hamiltonian $\mathsf{H}$ in the presence of disorder is given by \begin{eqnarray}
\mathsf{H} & = & \sum_{j=1}^{M}\epsilon_{A,j}|j,A\rangle\langle j,A|+(\epsilon_{B,j}-i\Gamma)|j,B\rangle\langle j,B|\nonumber \\
& + & \sum_{j=1}^{M}(J_{1,j}|j,B\rangle\langle j,A|+\mathrm{h.c.})\nonumber \\
& + & \sum_{j=1}^{M-1}(J_{2,j}|j+1,A\rangle\langle j,B|+\mathrm{h.c.}).\label{2siteH} \end{eqnarray}
Due to the chiral symmetry and the pseudo-anti-hermiticity of the non-dissipative and dissipative model, respectively \citep{Venuti2017,Lieu2018}, the topological states are expected to be robust against the chiral symmetry preserving off-diagonal disorder, i.e., noise in the hopping parameters. In contrast, disorder in the on-site potentials breaks the symmetries and is thus expected to quickly diminish topological features. Indeed, diagonal disorder leads to a unit cell as large as the system, thus having more than one dissipative site per unit cell and hence preventing the existence of topological dark states according to the argument in \citep{Levitov2016}. We therefore restrict the randomness to act on the hopping parameters, i.e., $J_{1,j}\equiv J_{1}+\mu_{1}\omega_{1,j}$ and $J_{2,j}\equiv J_{2}+\mu_{2}\omega_{2,j}$, where $\omega_{\alpha,j}$ are independent random variables with uniform distribution in the range $[-1,+1]$.
\begin{figure}
\caption{Complex density of states of the restricted Hamiltonian $\mathsf{H}$. (a)\&(b) Topologically trivial regime for the clean and disordered ($\mu=1$) case, respectively. (c)\&(d) Topologically non-trivial phase for clean and disordered ($\mu=1$) systems, respectively. Results are averaged over 1000 diagonalizations. Here, $N=20$, $\Gamma=0.5$, $J_{2}=1$ and $J_{1}=1.5$ ($J_{1}=0.5$) for the topologically trivial (non-trivial) configurations.}
\label{fig:dos_offdiag}
\end{figure}
The effect of off-diagonal disorder on the spectrum of the restricted Hamiltonian $\mathsf{H}$ is illustrated in Figure \ref{fig:dos_offdiag}, where the density of states is plotted in the complex plane. In the topologically trivial regime of the clean system, Fig.~\ref{fig:dos_offdiag}~(a), all eigenvalues have imaginary part $-\Gamma/2$. When disorder is introduced, they mainly wash out on axis $\mathrm{Im}(E)=-\Gamma/2$, as seen in Fig. \ref{fig:dos_offdiag}(b) . There is, however, a notable non-vanishing density of states emerging in the vicinity of $\operatorname{Re}(E)=0$. In the topologically non-trivial regime, Figs.~\ref{fig:dos_offdiag}~(c)\&(d), a dark state with corresponding $\operatorname{Im}(E)=0$ can be found. Its topological protection against off-diagonal disorder manifests itself in its eigenvalue being left almost unchanged when disorder disturbs the system, while the bulk states featuring eigenvalues with imaginary part $-\Gamma/2$ blur out. The protected dark state corresponds to an edge state having support only on the non-dissipative sites, thus not decaying through the cavities. Another state emerging in the non-trivial phase lives, on the contrary, only on the dissipative sites, with eigenvalue satisfying $\mathrm{Im}(E)=-\Gamma$, as also seen in Fig.~ \ref{fig:dos_offdiag}~(c). The mentioned destructive character of on-site potential disorder is discussed in the Appendix, Sec.~\ref{sec:diagonal_disorder}, where the density of states for diagonal disorder is analyzed, see Figs~\ref{fig:diag_disorder}~(a)-(d).
We now turn to the computation of the winding number. In absence of disorder we can go to reciprocal space and realize that the unitary $U(k)$ in Eq.~\eqref{eqn:Hk} is simply given by the phase of $J_{1}+J_{2}e^{-ik}$. The winding number of the dissipative system is thus the same as the winding number of the closed, Hermitian SSH-chain, resulting in \begin{equation}
W=\Theta(|J_{2}|-|J_{1}|),\label{eqn:Wssh} \end{equation} where $\Theta$ is the Heaviside function ($\Theta(x)=1$ for $x>0$ and $\Theta(x)=0$ for $x<0$). In order to compute the winding number in real space for the non-Hermitian SSH model, we follow the steps described in Sec.~\ref{sec:real_space_method}. First, the Hamiltonian is written in the order of sublattices and divided into four blocks, as in Eq.~\eqref{eqn:Hr}. In this case, $\Lambda=H_{1,1}=\epsilon_{A}\leavevmode{\rm 1\ifmmode\mkern-4.8mu \else\kern-0.3em \fi I}$, and $V=H_{1,2}$. From Eq.~\eqref{eqn:Hr}, we get $U\tilde{V}=V$, where $U$, $V$ and $\tilde{V}$ are all of dimension $M\times M$. To determine the unitary matrix $U$, we need to fulfill two requirements: i) the columns of $U$ need to be eigenvectors of $\Lambda$ and ii) $\tilde{V}$ needs to be positive definite. Since $\Lambda\propto\leavevmode{\rm 1\ifmmode\mkern-4.8mu \else\kern-0.3em \fi I}$, the first requirement is satisfied for any vector. In order to satisfy the second requirement, we recall that the polar decomposition of an invertible square matrix $V$ is a factorization of the form $V=U\tilde{V}$, where $U$ is a unitary matrix and $\tilde{V}$ is a positive-definite Hermitian matrix. $\tilde{V}$ is uniquely determined by $\tilde{V}=(V^{\dagger}V)^{1/2}$. As a result, $U$ can be written as \begin{equation} U=V(V^{\dagger}V)^{-1/2}. \end{equation}
Finally, the winding number $W$ can be calculated via Eq.~\eqref{eqn:Wr}. From here on, we set the on-site potentials to be zero, i.e., $\epsilon_{A}=\epsilon_{B}=0$.
\begin{figure}
\caption{Phase diagram of the disordered, dissipative SSH model for $N=1000$, $\Gamma=0.5$ and $J_2=1$. Results are averaged over 40 random realizations. (a) isotropic disorder ($\mu_{1}=\mu_{2}=\mu$), (b) anisotropic disorder ($\mu_{1}=2\mu_{2}=\mu$). White lines indicate points of diverging localization length in the thermodynamic limit. }
\label{fig:pd_ssh}
\end{figure} Fig.~\ref{fig:pd_ssh} presents the phase diagrams of the disordered dissipative SSH model as a function of coupling and disorder strength. In Fig.~\ref{fig:pd_ssh}~(a), the disorder is isotropic, i.e.~$\mu_{1}=\mu_{2}=\mu$ and $J_{2}=1$, while in Fig.~\ref{fig:pd_ssh}~(b), we consider anisotropic disorder with $\mu_{1}=2\mu_{2}=\mu$ and $J_{2}=1$. The exact location of the phase transition, illustrated by the white lines in Fig.~\ref{fig:pd_ssh}, can be obtained analytically by studying loci of the divergences in the localization length of the edge modes
\citep{Mondragon2014,book:Palyi2016}, as elucidated in more detail in the Appendix, Sec.~\ref{sec:analytic_contour}. In Fig.~\ref{fig:pd_ssh}~(a), the phase transition occurs at $|J_{2}/J_{1}|=1$ for all disorder strengths as for the clean case.
Fig.~\ref{fig:pd_ssh}~(b) shows a non-trivial \emph{topology by disorder} effect. Namely, for fixed value of $|J_{1}|>1$ close to one, one enters the topologically non-trivial region by increasing the disorder strength $\mu$, before transitioning into the topologically trivial regime after further increasing the noise. This widening of the topological phase boundary is observed for any kind of anisotropic disorder $\mu_{1}\neq\mu_{2}$.
As already mentioned, the exact phase transition points can be evaluated from the divergence of the localization length. In particular, the phase boundary of the disordered SSH model is given by the equation
$\mathbb{E}(\log|J_{1,j}|)=\mathbb{E}(\log|J_{2,j}|)$, where $\mathbb{E}(\bullet)$
denotes average over disorder (see Eq.~\eqref{eq:loc_ssh}). We first discuss the widening at small disorder strengths observed in Fig.~\ref{fig:pd_ssh}~(b). The second order Taylor expansion of $\mathbb{E}(\log|X|)$ in $\mu_{i}/J_{i}$ reads $\mathbb{E}[\log|X|]\simeq\log(\mathbb{E}[X])-\frac{\mathbb{V}[X]}{2\mathbb{E}[X]^{2}}$ \citep{NIPS2006_3113}, resulting in the following approximation of the phase boundary equation,
\begin{equation}
\log|J_{1}|-\frac{\mu_{1}^{2}}{6J_{1}^{2}}\simeq\log|J_{2}|-\frac{\mu_{2}^{2}}{6J_{2}^{2}}.\label{eq:phaseb_approx} \end{equation}
Fixing $J_{2}$ and $\mu_{2}$ such that the right hand side of Eq.~(\ref{eq:phaseb_approx}) is constant, we see that the function $\log|J_{1}|-\frac{\mu_{1}^{2}}{6J_{1}^{2}}$ is monotonically increasing in $J_{1}$ and decreasing in $\mu_{1}$. Hence, if $\mu_{1}$ increases, $J_{1}$
needs to grow as well in order to compensate. This corresponds to a widening of the topologically non-trivial region for small increasing noise. In the opposite, strong disorder limit, we can expand $\mathbb{E}(\log|J_{\alpha,i}|)$
in $J_{\alpha}/\mu_{\alpha}$, obtaining $\mathbb{E}(\log|J_{\alpha,i}|)=\log|\mu_{\alpha}|-1+O(J_{\alpha}/\mu_{\alpha})$. The phase boundary equation in this regime becomes
\begin{equation}
\log|\mu_{1}|\simeq\log|\mu_{2}|. \end{equation} Hence, for strong disorder, the phase boundary is roughly independent of $J_{1}$ accounting for the horizontal boundary in Fig.~\ref{fig:pd_ssh}~(b).
Similar disorder-induced topological characteristics were also recently discussed in the context of other non-Hermitian models \citep{Luo2019,Zhang2020}. \\
\begin{figure}
\caption{Coherence of the first qubit in the disordered, dissipative SSH model for $N=100$, $\Gamma=0.5$ and $J_2=1$. Results are averaged over 40 random realizations. (a) isotropic disorder ($\mu_{1}=\mu_{2}=\mu$) along the vertical line where $J_1=0$ with $\mu=0.1$ (orange), $\mu=0.5$ (purple), $\mu=1.0$ (red), and in the topologically trivial regime $J_1=1.5$ with $\mu=0.5$ (black). (b) anisotropic disorder ($\mu_{1}=2\mu_{2}=\mu$) along the vertical line where $J_1=1.2$ with $\mu=0.5$ (purple), $\mu=1.5$ (red), $\mu=2.5$ (black), and for $J_1=1.5$ with $\mu=0.5$ (orange). Solid black lines indicate the asymptotic prediction $\mathbb{E}(1-x^2)$ Eq.~(\ref{eq:cinf_approx}) valid for small disorders. }
\label{fig:coh_ssh}
\end{figure}
For each phase diagram, we now fix $J_2=1$ and choose four characteristic parameter configurations in order to get representative coherence time evolutions for the different topological sectors, depicted Fig.~\ref{fig:coh_ssh}. For isotropic disorder, Fig.~\ref{fig:coh_ssh}~(a), we choose three points along the vertical $J_1=0$ with $\mu=0.1,0.5,1.0$ as well as the configuration $J_1=1.5, \mu=0.5$, representing the disordered topologically non-trivial and trivial regime, respectively. The coherence decays to a non-zero (respectively zero) value at large times in the topologically non-trivial (respectively trivial) sector, thus matching the phase diagram Fig.~\ref{fig:pd_ssh}~(a). In the topologically non-trivial regime, increasing disorder leads to a smaller asymptotic value of the coherence. Similarly, for anisotropic disorder, Fig.~\ref{fig:coh_ssh}~(b), we choose three points along the vertical $J_1=1.2$ with $\mu=0.5,1.5,2.5$ as well as $J_1=0,\mu=0.5$. The former three parameter pairs lie on a vertical line cutting through the broadening of the topologically non-trivial regime, thus representing the reentrance phenomenon into a higher topological phase. It can be seen that a finite coherence of the first qubit is present at large times only for $\mu=1.5$, being in consent with the corresponding phase diagram Fig.~\ref{fig:pd_ssh}~(b). For $J_1=0$ and $\mu=0.5$, a similar behavior as for the isotropic disordered chain can be observed, with a large asymptotic coherence value. In previous work \citep{Venuti2017}, it was shown that for large chains in the topologically non-trivial regime, the coherence saturates to approximately \[ \mathcal{C}(t\rightarrow\infty)\approx 1-x^2, \]
where $x=J_1/J_2$, with $|x|<1$. It is thus natural to assume that the expectation value of the asymptotic coherence including disorder is given by \begin{gather} \mathbb{E}[\mathcal{C}(t\rightarrow\infty)]\approx \mathbb{E}(1-x^2) = \nonumber \\ \frac{1}{4\mu_1 \mu_2} \int_{-\mu_2}^{\mu_2}\int_{-\mu_1}^{\mu_1} 1- \Big( \frac{J_1+\mu_1}{J_2+\mu_2} \Big)^2 d\mu_1 d\mu_2 = \label{eq:cinf_approx} \\ 1-\frac{3J_1^2 + \mu_1^2}{3J_2^2-3\mu_2^2} \, . \nonumber \end{gather}
Note that this only holds for weak to moderate disorder such that no change of topological phase can be generated randomly, i.e., $\mu_1 + \mu_2 < |J_2| - |J_1|$.
In Fig.~\ref{fig:coh_ssh}, the prediction Eq.~(\ref{eq:cinf_approx}) is illustrated by black solid lines for disorder strengths falling into the discussed regime. For large disorder, random phase changes result in a decrease of the mean coherence in the simulation, and Eq.~(\ref{eq:cinf_approx}) breaks down. \\ \\ It is important to note that, even though the phase diagram is the same as those found in previous works \citep{Mondragon2014,Luo2019}, the physical interpretation is different, as our models include dissipation. Edge states do not correspond to actual electronic states located at one of the boundaries of the chain, but rather describe the physics of the projected density matrix introduced in Sec.~\ref{sec:setup}. A non-trivial topological phase, resulting in quasi-dark states of the restricted Hamiltonian, leads to having an exponentially long (in system size) coherence time of the edge qubit. In the topologically trivial regime, the decoherence of the edge qubit is governed by dissipation, leading to a finite coherence time.
\subsection{Disordered non-Hermitian Trimer Model}
\label{sec:3site}
\begin{figure}
\caption{Non-Hermitian trimer model. Here, the nearest-neighbor couplings $J_{1},J_{2},J_{3}$ alternate cyclically, building a unit cell with three sites. Next-nearest-neighbor couplings $J$ link the first and third site in each unit cell, thus enabling three distinct winding numbers $W=0,1,2$.}
\label{fig:3site}
\end{figure}
Next, we consider a trimer chain with nearest-neighbor as well as next-nearest-neighbor couplings, as depicted in Fig.~\ref{fig:3site}. The corresponding non-Hermitian Hamiltonian, derived from the restricted Lindbladian, is given by \begin{eqnarray}
\mathsf{H} & = & \sum_{j=1}^{M}(J_{1,j}|j,B\rangle\langle j,A|+\text{h.c.})\nonumber \\
& + & \sum_{j=1}^{M-1}(J_{2,j}|j,C\rangle\langle j,B|+\text{h.c.})\nonumber \\
& + & \sum_{j=1}^{M-1}(J_{j}|j,C\rangle\langle j,A|+\text{h.c.}) \\
& + & \sum_{j=1}^{M-1}(J_{3,j}|j+1,A\rangle\langle j,C|+\text{h.c.})\nonumber \\
& + & \sum_{j=1}^{M}\epsilon_{A}|j,A\rangle\langle j,A|+\sum_{j=1}^{N}\epsilon_{B}|j,B\rangle\langle j,B|\nonumber \\
& + & \sum_{j=1}^{M}(\epsilon_{C,j}-i\Gamma)|j,C\rangle\langle j,C|.\label{3site_H}\nonumber \end{eqnarray}
It has been demonstrated that robust chiral edge modes exist in non-dissipative trimer chains, even in the absence of inversion symmetry \citep{Alvarez2019}. It has been argued that their topological character is inherited through a mapping of a higher-dimensional model, namely the commensurate off-diagonal Aubry-André-Harper model, which is topologically equivalent to a two dimensional tight-binding lattice pierced by a magnetic flux \citep{Kraus2012}.
The topological classification by Rudner \textit{et al.}~including dissipation, however, imposes only translational symmetry. In fact, it turns out that the winding number in Eq.~\eqref{eqn:Wr} can be used as a reliable predictor for the number of (quasi)-dark states located on the edge of the trimer chain with open boundary conditions. In previous work \citep{Venuti2017}, it was found that in the clean case, the presence of next-nearest-neighbor couplings enable winding numbers $W=0,1,2$. Concretely, $W$ is given by \begin{align}
W & =\Theta\left(\left|J_{3}\right|-\left|J+J_{2}\tan(\vartheta/2)\right|\right)\nonumber \\
& +\Theta\left(\left|J_{3}\right|-\left|J-J_{2}\cot(\vartheta/2)\right|\right),\label{windingeq_3site} \end{align} where $\vartheta=\arccos\left[\left(\epsilon_{A}-\epsilon_{B}\right)/\sqrt{4J_{1}^{2}+\left(\epsilon_{A}-\epsilon_{B}\right)^{2}}\right].$ We further verify the above equation in the Appendix, Sec.~\ref{sec:appendix}, by solving the system analytically for a convenient system size and counting the number of dark states localized on one edge of the chain. \begin{figure}
\caption{Topology and coherence for the clean, non-disordered trimer model. (a) Winding number and (b)-(d) time dependent coherence for three parameter configurations corresponding to the three topological sectors. The dotted lines indicate the theoretically predicted asymptotic coherence as $t\rightarrow\infty$. Dissipation is set to $\Gamma=0.5$ and a chain with $N=300$, $J_{1}=1,J_{2}=2$ and $J=1$ is considered. The three time evolutions of the coherence in the topological sectors $W=0,1,2$ correspond to parameter choices $J_{3}=0.5,2.0,3.5$, respectively. }
\label{fig:clean_3site}
\end{figure} In order to calculate the winding number using the the real-space approach, we first rewrite the Hamiltonian with respect to its sublattices and decompose it as in Eq.~\eqref{eqn:Wr}. In this case, the matrices $\Lambda$ and (respectively $V$) with dimensions $2M\times2M$ (respectively $2M\times M$) are given by \begin{equation} \Lambda=\begin{pmatrix}\epsilon_{A}\mathbb{1} & H_{AB}\\ H_{BA} & \epsilon_{B}\mathbb{1} \end{pmatrix} ;\qquad V=\begin{pmatrix}H_{AC}\\ H_{BC} \end{pmatrix}. \end{equation} Here, $H_{AB}=J_{1}\mathbb{1}$. Due to the symmetry of $\Lambda$, $U$ from Eq.~\eqref{eqn:Hr} can be written as \begin{equation} U=\begin{pmatrix}-\cos(\vartheta/2)U_{-} & \sin(\vartheta/2)U_{+}\\ \sin(\vartheta/2)U_{-} & \cos(\vartheta/2)U_{+} \end{pmatrix},\label{eq:U_vsUpm} \end{equation} where $U_{\pm}$ are two $M\times M$ so far unspecified unitaries and $\vartheta$ has been given above. From Eq.~\eqref{eqn:Hr}, we further get $U\tilde{V}=V$, which gives \begin{equation} \begin{pmatrix}-\cos(\vartheta/2)U_{-} & \sin(\vartheta/2)U_{+}\\ \sin(\vartheta/2)U_{-} & \cos(\vartheta/2)U_{+} \end{pmatrix}\begin{pmatrix}\tilde{V}_{-}\\ \tilde{V}_{+} \end{pmatrix}=\begin{pmatrix}H_{AC}\\ H_{BC} \end{pmatrix}, \end{equation} where $\tilde{V}:=(\tilde{V}_{-},\tilde{V}_{+})^{T}$. From the above equation we find \begin{align} U_{+}\tilde{V}_{+} & =\frac{1}{2}(\cos(\vartheta/2)H_{AC}+\sin(\vartheta/2)H_{BC}),\nonumber \\ U_{-}\tilde{V}_{-} & =\frac{1}{2}(-\sin(\vartheta/2)H_{AC}+\cos(\vartheta/2)H_{BC}).\label{eq:3site_decomp} \end{align} Recall that we must fix the gauge freedom in $U$ by requiring the submatrices $\tilde{V}_{\pm}$ to be positive definite. Consequently, $U_{\pm}$ can be determined by polar decomposition of the right hand side of Eq.~\eqref{eq:3site_decomp}, after which the unitary matrix $U$ is obtained using Eq.~\eqref{eq:U_vsUpm}. Finally, the winding number is computed via Eq.~\eqref{eqn:Wr}. Interestingly, it can be shown that the winding number of $U$ is nothing more than the sum of the winding numbers of $U_{+}$ and $U_{-}$.
\begin{figure}
\caption{Phase diagram of the disordered, dissipative trimer model for $N=1500$, $\Gamma=0.5$ , $J_1=2$, $J_2=2$, $J_3=3$. Results are averaged over 40 random realizations. In (a), $\mu_{2}=\mu_{J}=\mu_{3}=\mu$, whereas (b) describes disorder with $2\mu_{J}=2\mu_{2}=\mu_{3}=\mu$. White lines indicate the loci of diverging localization lengths in the thermodynamic limit. }
\label{fig:pd_3site}
\end{figure}
For simplicity, we again limit our considerations to the case of vanishing the on-site chemical potentials, i.e., $\epsilon_{A}=\epsilon_{B}=\epsilon_{C}=0$. Using the real-space winding number approach for the clean trimer model results in Fig.~\ref{fig:clean_3site}~(a), matching Eq.~\eqref{windingeq_3site}. Figs~\ref{fig:clean_3site}~(b)-(d) show the typical behavior of the coherence in the three distinct topological sectors $W=0,1,2$ in the clean trimer model, respectively. In the topologically trivial regime, no dark states are present, driving decoherence of the first qubit. For $W=1$, the dark state manifold is one-dimensional, leading to a saturation of the coherence at infinite times. For $W=2$, the existence of two dark states result in Rabi like oscillations of the first qubit's coherence. The asymptotic solution, Eq.~\eqref{theo_coh}, is also featured in Figs.~\ref{fig:clean_3site}~(b)-(d).
Because of the $J_{1}$ dependence of the dark states, disorder in $J_{1}$ is expected to quickly destroy the topological features of the system. This is further suggested by the degree of freedom of the matrix $U$, Eq.~\eqref{eqn:Hr}, which collapses as soon as $J_{1}$ becomes disordered, leading to an immediate collapse of a well-defined winding number. Therefore, we shall from now on focus on the analysis of the disordered regime where only $J_{2},J_{3},J$ are exposed to noise, which we control via additive random noise drawn from a uniform distribution. Concretely, if $j$ labels the unit cell and $\{\omega_{1}\}$, $\{\omega_{2}\}$, $\{\omega\}$ are sets of independent, uniformly distributed random variables $\in[-1,1]$, $J_{i,j}=J_{i}+\mu_{i}\omega_{i,j}$ for $i=2,3$, $J_{j}=J+\mu_{J}\omega_{j}$, and $J_{1,j}=J_{1}$ for all $j$. Looking at the density of states for the different disorder types, depicted in Figure \ref{fig:trimer_dos}, the selection rules for the type of disorder under which topological dark states are stable is further underlined.
As for the disordered non-Hermitian SSH model, the full phase diagram for different disorder strengths can be constructed, shown in Fig.~\ref{fig:pd_3site}. Again, the exact phase transition points in the thermodynamic limit are depicted by white lines, which are derived via the dark state localization length considering disorder in the Appendix, Sec.~\ref{sec:appendix}. The phase diagram features rich structures, presenting widenings of topologically non-trivial phases for moderate (high) disorder strengths in the chain with equal (different) disorder amplitudes. Note that the system with different distributions on the disordered parameters, $2\mu_{2}=2\mu_{J}=\mu_{3}=\mu$, is more similar to what we called anisotropic disorder in the SSH model, being due
to the competition between $|J_{2}\pm J|$ and $J_{3}$ deciding the topological phase for the trimer model Eq.~\eqref{windingeq_3site}. When computing the localization length, the disorder amplitudes of
$J_{2}$ and $J$ hence add up, as is explicitly seen in Eq.~\eqref{eq:3siteLL}. Note, however, that the effective disorder on $|J_{2}\pm J|$ is $\mu/\sqrt{2}<\mu$, which results in having a widening of the non-topological phases in the large disorder regime. Analogously, the trimer system having equal disorder on all hopping parameters resembles the case $\mu_{1}>\mu_{2}$ of the SSH-model, featuring a widening of the topologically non-trivial regimes for small disorders. \\ \\ \begin{figure}
\caption{Coherence of the first qubit in the disordered, dissipative trimer model for $N=300$, $\Gamma=0.5$, $J_1=1$, $J_2=2$ and $J_3=3$. Results are averaged over 40 random realizations. (a) For $\mu_{J}=\mu_{2}=\mu_{3}=\mu$, we show the coherence for $(J,\mu)=(0,1)$ (orange), $(J,\mu)=(3,1)$ (purple), and $(J,\mu)=(0,7)$ (black), corresponding to $W=2,1,0$, respectively. (b) For $2\mu_{J}=2\mu_{2}=\mu_{3}=\mu$, we highlight the reentrance into a higher topological phase along the vertical line $J=1.2$, with $\mu=1$ (purple) and $\mu=7$ (orange), corresponding to $W=1,2$, respectively. We further show the trivial regime by evaluating the coherence for $(J,\mu)=(6,1)$ (black). The observable broadening of the curves is due to the error of the mean, pictured by error bars for every data point. }
\label{fig:coh_3site}
\end{figure} We shall again pick three points in each phase diagram and illustrate the corresponding time evolution of the first qubits coherence, seen in Fig.~\ref{fig:coh_3site}. For $\mu_{J}=\mu_{2}=\mu_{3}=\mu$, Fig.~\ref{fig:coh_3site}~(a), we choose the parameter pairs $(J,\mu) = (0,1), (3,1), (0,7)$, belonging to winding numbers $W=2,1,0$, respectively (cf. Fig.~\ref{fig:pd_3site}). For all configurations, we find that the asymptotic behavior of the coherence the one of the clean case, namely a decrease to zero for $W=0$, a convergence to a constant larger than zero for $W=1$, and an oscillation for $W=2$. For different disorder strengths $2\mu_{J}=2\mu_{2}=\mu_{3}=\mu$, we focus on the reentrance phenomenon $W=1\rightarrow 2$ by computing the coherence for $(J,\mu)=(1.2,1), (1.2,7)$. Indeed, we find that for large enough disorder, an oscillating behavior emerges, signaling the change of topological phase. For completeness, we also include $(J,\mu)=(6,1)$ representing the trivial sector, where a vanishing coherence can be observed at large times.
\section{Application to Quantum Computation} \label{sec:quantum_computation} Ever since Kitaev's proposal \citep{Kitaev2006} to braid anyons in order to realize non-trivial quantum gates, the field of topological quantum computation has been an exceptionally active field of research \citep{Sankar2008, Sau2010, Sarma2015, Stern2013, Alicea2011, Freedman2006, Freedman2003, Akhmerov2010}. This is mainly due to the promising protection against environmental noise governed by the non-locality of the state manifold used for braiding \citep{Lahtinen2017}. Spinless p-wave superconductor wires hosting non-Abelian Majorana fermions bound to topological defects have been of particular interest \citep{Fisher2010}, as the intrinsic particle-hole symmetry of the BdG-Hamiltonian promises a realizable topological protection. Recently, the SSH model has been analyzed in terms of its applicability to quantum computation \citep{Andras2019}, where it was found that the non-trivial braiding statistics of the topological edge modes can be used to build quantum gates via Y-junctions. However, as for all quantum gates based on symmetry protected topological states, the set of quantum gates is not universal \citep{Lahtinen2017}. Nevertheless, studying the braiding statistics for our concrete open disordered models seems like an exciting and promising work for future projects.
\section{Conclusions} \label{sec:conclusion} We have analyzed and topologically classified disordered dissipative qubit-cavity dimer and trimer architectures, with special focus on topological protection mechanisms of the coherence measure in a fiducial qubit. The evolution of the coherence's qubit is exactly given by a non-Hermitian Hamiltonian which thus emerges from a bona-fide physical system. We demonstrated the use of a real-space topological invariant $W$, which accurately predicts the number of non-trivial (quasi-)dark modes in disordered, non-Hermitian models, as long as certain symmetries are preserved by the disorder operators. We then computed the phase diagrams of dimer and trimer chains in the parameter space spanned by the tunneling amplitude and the disorder strength, predicting the faith of the fiducial qubit's coherence at long times, i.e., decay to zero, a constant value or oscillatory behavior for winding numbers $W=0,1,2$, respectively. For certain choices of disorder strengths or the hopping parameters, reentrance phenomena into topological phases with higher winding numbers were observed, leading to an increase of coherence times (exponentially large in system size) when introducing higher noise levels. Possible applications in topological quantum computing via braiding of dark modes were briefly discussed, opening up interesting questions for future research. Furthermore, generalizations of the classification to larger numbers of sites per unit cell and systems of higher dimension would be of great interest.
\textbf{Acknowledgements:} We would like to thank Hubert Saleur for useful discussions. This work was supported by the US Department of Energy under grant number DE-FG03-01ER45908. L.C.V. acknowledges partial support from the Air Force Research Laboratory award no. FA8750-18-1- 0041. The research is based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office contract W911NF-17-C-0050. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.
\onecolumngrid
\appendix
\section{Dark states in the dissipative trimer model}
\label{sec:appendix}
\label{ds3site}
We here derive an exact form of the asymptotic coherence dynamics and the topological phase transition in the trimer model by studying the dark states, i.e., by finding all states that obey $\mathsf{H}|\psi\rangle\!\rangle=E|\psi\rangle\!\rangle$ with $E\in\mathbb{R}$. For the sake of convenience, the following considerations assume chain lengths $N\mod3=2$, as the system then hosts exact dark states with vanishing imaginary part. For all other system sizes the states are quasi-dark, as they have an imaginary part exponentially small in the system size. Of course, in the thermodynamic limit, these differences vanish, and the dynamics is exactly described by the result below. The ansatz is to look for possible dark states with energies $E=\pm J_{1}$, i.e., to find the kernel of the matrix \begin{equation} \mathsf{H}\mp\mathbb{1}J_{1}=\begin{pmatrix}\mp J_{1} & J_{1} & J & 0 & 0 & 0\\ J_{1} & \mp J_{1} & J_{2} & 0 & 0 & 0\\ J & J_{2} & \mp J_{1}-i\Gamma & J_{3} & 0 & 0\\ 0 & 0 & J_{3} & \mp J_{1} & J_{1} & J\\ 0 & 0 & 0 & J_{1} & \mp J_{1} & \ddots\\ 0 & 0 & 0 & J & \ddots & \ddots \end{pmatrix}.\label{dseq} \end{equation} For $N\mod3=2$, solutions of \eqref{dseq} are of the form \begin{eqnarray} v_{+} & = & \big(1,1,0,-\delta_{+},-\delta_{+},0,(-\delta_{+})^{2},(-\delta_{+})^{2},0,...,(-\delta_{+})^{\frac{N-2}{3}},(-\delta_{+})^{\frac{N-2}{3}}\big)^{T},\nonumber \\ v_{-} & = & \big(1,-1,0,\delta_{-},-\delta_{-},0,\delta_{-}^{2},-\delta_{-}^{2},0,...,\delta_{-}^{\frac{N-2}{3}},-\delta_{-}^{\frac{N-2}{3}}\big)^{T}.\label{ds} \end{eqnarray}
These solutions are intuitive and analogous to the open SSH model \citep{Venuti2017}, in the sense that they disappear on all dissipative sites. The condition $E=\pm J_{1}$ signals the equivalence of the first two sites of each unit cell up to a sign factor. Eq.~\eqref{ds} leads to \begin{equation}
\delta_{\pm}=\frac{|J\pm J_{2}|}{J_{3}}, \end{equation}
where the sign of the solution is fixed without loss of generality by assuming $\delta_{\pm}$ to be positive. The winding number classification is illustrated in the corresponding vectors, as we find zero, one, or two dark states localized at the outer left qubit for different topological sectors, i.e., $W=\Theta(J_{3}>|J-J_{2}|)+\Theta(J_{3}>J+J_{2})$. Taking into account the normalization factor of the solutions, \begin{equation} A_{\pm}^{-2}=2\sum_{k=0}^{\frac{N-2}{3}}\delta_{\pm}^{2k}=2\frac{1-\delta_{\pm}^{\frac{2N-4}{3}}}{1-\delta_{\pm}^{2}}, \end{equation} the time dependent coherence can be approximated for large times $t\gg1/\Gamma$, \begin{align}
\mathcal{C}(t) & =|\langle\langle1|e^{-i\mathsf{H}t}|1\rangle\rangle|\approx|e^{-iJ_{1}t}A_{+}^{2}+e^{iJ_{1}t}A_{-}^{2}|\nonumber \\
& =|A_{+}^{4}+A_{-}^{4}+2A_{+}^{2}A_{-}^{2}\cos2J_{1}t|.\label{theo_coh} \end{align}
\section{Analytical Determination of Critical Phase Transition Contours} \label{sec:analytic_contour}
In \citep{Mondragon2014}, the critical phase transition surface was derived for the Hermitian SSH model, using the numerical transfer matrix method and level-spacing statistics analysis. The analytical critical phase transition contour for non-Hermitian models can be calculated in a similar manner. To see this, consider the non-Hermitian SSH model. Here, the dark edge state is exactly at zero energy and only lives on the non-decaying sublattice. We consider the critical phase transition in the thermodynamic limit, such that the results for the linear chain of odd length coincide with the results of even length. Now recall that the edge state of the disordered Hermitian SSH model is also supported entirely by one sublattice or the other. Its zero energy edge state on sublattice A ($\psi_{n,B}=0$) can be written as \begin{equation}
\psi_{n,A}=i^{n-1}\prod_{j=1}^{n}\left|\frac{J_{1j}}{J_{2j}}\right|\psi_{1,A},\label{eq:wf_phasetrans} \end{equation} where $J_{1}j$ and $J_{2}j$ are the two perturbed hopping parameters in the $j$th unit cell. The edge states in the two systems share an identical distribution in the clean limit. Consequently, in this case, the non-Hermitian problem follows the same localization length and phase transition as a one-dimensional Hermitian SSH model.
With the exact wave function distribution as in Eq.~\eqref{eq:wf_phasetrans}, the inverse localization length of a edge mode can be obtained by \begin{eqnarray}
\Lambda^{-1} & = & -\lim_{n\rightarrow\infty}\frac{1}{n}\log\left|\psi_{n,A}\right|\nonumber \\
& = & \left|\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=1}^{n}\left(\ln\left|J_{1j}\right|-\ln\left|J_{2j}\right|\right)\right| \end{eqnarray} An analytical result can be obtained by taking the ensemble average of the last expression. The limit of the sum turns into an integration for independent and identically distributed disorder, \begin{equation}
\Lambda^{-1}=\frac{1}{4}\left|\int_{-1}^{1}d\omega\int_{-1}^{1}d\omega^{\prime}\left(\ln\left|J_{1}+\mu_{1}\omega\right|-\ln\left|J_{2}+\mu_{2}\omega^{\prime}\right|\right)\right|,\label{eq:loc_ssh} \end{equation} where $J_{1}$ and $J_{2}$ are the unperturbed hopping parameters. $\mu_{1}$ and $\mu_{2}$ control the strength of disorder in $J_{1}$ and $J_{2}$ respectively. The random variables $\omega$ and $\omega^{\prime}$ are both drawn from a uniform distribution in the range $[-1,1]$, leading to a normalization prefactor $1/4$. The analytic solution to this integral has been obtained in \citep{Mondragon2014}, \begin{multline}
\Lambda^{-1}=\frac{1}{4\mu_{1}}\left[\left(J_{1}+\mu_{1}\right)\log\left|J_{1}+\mu_{1}\right|-\left(J_{1}-\mu_{1}\right)\log\left|J_{1}-\mu_{1}\right|\right]\\
-\frac{1}{4\mu_{2}}\left[\left(J_{2}+\mu_{2}\right)\log\left|J_{2}+\mu_{2}\right|-\left(J_{2}-\mu_{2}\right)\log\left|J_{2}-\mu_{2}\right|\right]\label{eq:2siteLL} \end{multline}
For small disorder, $\mu_{1},\mu_{2}\ll J_{2},J_{1}$, the localization length Eq.~\eqref{eq:loc_ssh} can be approximated by \begin{eqnarray}
\Lambda^{-1} & \propto & \int_{-1}^{1}\int_{-1}^{1}d\omega_{1}d\omega_{2}\ln|J_{1}+\omega_{1}\mu_{1}|-\ln|J_{2}+\omega_{2}\mu_{2}|\nonumber \\
& = & \int_{-1}^{1}\int_{-1}^{1}d\omega_{1}d\omega_{2}\ln|J_{1}|+\frac{\omega_{1}\mu_{1}}{J_{1}}-\frac{1}{2}\Big(\frac{\omega_{1}\mu_{1}}{J_{1}}\Big)^{2}-\Big[\ln|J_{2}|+\frac{\omega_{2}\mu_{2}}{J_{2}}-\frac{1}{2}\Big(\frac{\omega_{2}\mu_{2}}{J_{2}}\Big)^{2}\Big]\nonumber \\
& + & \mathcal{O}\Big(\Big(\frac{\mu_{1}}{J_{1}}\Big)^{3}\Big)+\mathcal{O}\Big(\Big(\frac{\mu_{2}}{J_{2}}\Big)^{3}\Big). \end{eqnarray}
Performing the integration up to order $\mathcal{O}\big(\big(\frac{\mu_{1}}{J_{1}}\big)^{3}\big)$ and $\mathcal{O}\big(\big(\frac{\mu_{2}}{J_{2}}\big)^{3}\big)$, one finds that the localization length diverges for \begin{equation}
|J_{1}|(\mu_{1},\mu_{2})=|J_{2}|\exp\Big(\frac{J_{2}^{2}\mu_{1}^{2}-J_{1}^{2}\mu_{2}^{2}}{J_{1}^{2}J_{2}^{2}}\Big), \end{equation} which, up to leading order in the expansion of the exponential function, reduces to \begin{equation}
|J_{1}|(\mu_{1},\mu_{2})=|J_{2}|\exp\Big(\frac{\mu_{1}^{2}-\mu_{2}^{2}}{J_{2}^{2}}\Big). \end{equation} We thus arrive at the conclusion that the value of $J_{1}$ where the non-trivial$\leftrightarrow$trivial transition occurs increases (decreases) compared to the clean case for small disorder strengths if $\mu_{2}<\mu_{1}$ ($\mu_{2}>\mu_{1}$). This corresponds to the \textit{topology by disorder} effect discussed in the main text and can be nicely seen in Fig.~\ref{fig:pd_ssh}(b). For $\mu_{1}=\mu_{2}$, the phase transition always occurs at $J_{1}=J_{2}$, as observed in Fig.~\ref{fig:pd_ssh}(a). Now we continue to generalize the result to the non-Hermitian trimer model. In Sec.~\ref{ds3site}, we have shown that the trimer model of length $N\mod{3}$ can host two dark edge modes with energies $E=\pm J_{1}$. These edge modes are supported purely by non-decaying sublattices. The wave functions of the two dark states for disordered three-site model is given by \begin{equation}
\psi_{n,ds\pm}=(-1)^{n-1}\prod_{j=1}^{n}\left|\frac{J_{2j}\pm J_{j}}{J_{3j}}\right|\psi_{1,ds\pm}, \end{equation} where $J_{j}$, $J_{2j}$ and $J_{3j}$ are perturbed hopping parameters in the $j$th unit cell. Since there exist two different edge modes, we would expect two disjoint localization lengths, \begin{align}
\Lambda_{ds\pm}^{-1} & =-\lim_{n\rightarrow\infty}\frac{1}{n}\log\left|\psi_{n,ds\pm}\right|\\
& =\left|\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=1}^{n}\left(\ln\left|J_{2j}\pm J_{j}\right|-\ln\left|J_{3j}\right|\right)\right|. \end{align} Again, we take the ensemble average, and the summation turns into an integration, which gives \begin{equation}
\Lambda_{ds\pm}^{-1}=\frac{1}{8}\left|\int_{-1}^{1}d\omega\int_{-1}^{1}d\omega^{\prime}\int_{-1}^{1}d\omega^{\prime\prime}\left(\ln\left|(J+\mu\omega)\pm(J_{2}+\mu_{2}\omega^{\prime})\right|-\ln\left|J_{3}+\mu_{3}\omega^{\prime\prime}\right|\right)\right|. \end{equation} Here $J$, $J_{2}$ and $J_{3}$ are unperturbed hopping parameters. $\mu$ $\mu_{2}$ and $\mu_{3}$ define the amplitudes of disorder. $\omega$, $\omega^{\prime}$ and $\omega^{\prime\prime}$ are three independent and identically distributed random variables in the range of $[-1,1]$. After performing the integration explicitly, we arrive at \begin{eqnarray}
\Lambda_{ds\pm}^{-1} & = & {2\mu\mu_{2}}\Big\{\left(J\pm J_{2}-\mu-\mu_{2}\right)^{2}\log\left(\left|J\pm J_{2}-\mu-\mu_{2}\right|\right)-\left(J\pm J_{2}+\mu-\mu_{2}\right)^{2}\log\left(\left|J\pm J_{2}+\mu-\mu_{2}\right|\right)\nonumber \\
& - & \left(J\pm J_{2}-\mu+\mu_{2}\right)^{2}\log\left(\left|J\pm J_{2}-\mu+\mu_{2}\right|\right)+\left(J\pm J_{2}+\mu+\mu_{2}\right)^{2}\log\left|J\pm J_{2}+\mu+\mu_{2}\right|\Big\}\nonumber \\
& - & \frac{1}{\mu_{3}}\Big\{\left(J_{3}+\mu_{3}\right)\log\left(\left|J_{3}+\mu_{3}\right|\right)-\left(J_{3}-\mu_{3}\right)\log\left(\left|J_{3}-\mu_{3}\right|\right)\Big\}-4.\label{eq:3siteLL} \end{eqnarray} Eq. \eqref{eq:2siteLL} and Eq. \eqref{eq:3siteLL} allow us to trace the exact critical phase transition contours in the non-Hermitian SSH dimer and trimer models.
\section{Diagonal Disorder} \label{sec:diagonal_disorder} \begin{figure}
\caption{Eigenspectrum density of states of the restricted Hamiltonian $\mathsf{H}$ in the topologically non-trivial regime for (a) the clean system, and diagonal disorder strengths (b) $\mu=0.5$, (c) $\mu=1$, (d) $\mu=1.5$. Results are averaged over 1000 diagonalizations. Here, $N=20$, $\Gamma=0.5$, $J_{1}=0.5$, $J_{2}=1$. (e) Coherence time of the edge qubit as a function of the intra-cell hopping strength $J_{1}$ and diagonal disorder $\mu$. In this setting, $J_{2}=1$, $N=100$, and the time evolution of the coherence is disorder averaged over 50 realizations. }
\label{fig:diag_disorder}
\end{figure}
As argued in the main text, diagonal disorder destroys the protective chiral symmetry of the SSH model, making it collapse to a topologically trivial phase. This effect can be nicely seen when considering the eigenspectrum density of states of the restricted Hamiltonian, as already introduced in the main text for symmetry conserving disorder. In analogy to off-diagonal disorder, the on-site potentials $\epsilon_{A,i}$ and $\epsilon_{B,i}$ are chosen to be uniformly distributed between $[-\mu,\mu]$.
Fig.~\ref{fig:diag_disorder}~(a)-(d) illustrates how the topological dark states appearing in the clean system quickly wash out, joining the non-topological bulk state manifold. This is in in stark contrast to a finite symmetry conserving off-diagonal disorder, where the topological dark states were almost unaffected by the noise, cf. Figure \ref{fig:dos_offdiag}.
To underline the destructive effect further, the edgequbit's coherence is inspected. As soon as disorder on the on-site potentials is introduced, the coherence time is not infinite anymore, but it is reduced to a finite value $\tau$. By assuming an exponential decay in time, i.e., $\mathcal{C}(t)=\mathcal{C}(t_{0})e^{-(t-t_{0})/\tau}$ for some $t_{0}\gg1/\Gamma$, we can extract $\tau$ by integrating over the time evolution of the coherence, i.e., \begin{equation} I:=\int_{t_{0}}^{t_{1}}\mathcal{C}(t)dt=\int_{t_{0}}^{t_{1}}\mathcal{C}(t_{0})e^{-(t-t_{0})/\tau}dt=\tau(\mathcal{C}(t_{1})-\mathcal{C}(t_{0})). \end{equation}
Numerical integration leads to the results depicted in Fig.~\ref{fig:diag_disorder}~(e), where a sharp drop of the coherence time away from the fully dimerized, clean limit can be observed (notice the logarithmic scaling on the z-axis). For the trimer model, very similar behavior is being observed, for disorder acting on either on-site potentials or the coupling parameter $J_{1}$, see Fig.~\ref{fig:trimer_dos} for the DOS.
\begin{figure}
\caption{Density of states for the trimer model. (a)-(c) Clean density of states for $W=2,1,0$, respectively. (d)-(f) Diagonal disorder $\mu=1$. (g)-(i) Off diagonal disorder on $J_{2},J_{3},J$ with $\mu=1$. Here, $N=21$, $J_{1}=J_{2}=2$, $J_{3}=3$, and $J=0,3,6$ for the topological phases $W=2,1,0$, respectively. It is seen how $W$ quasi-dark states with energies $E=\pm J_{1}$ exist in the clean system, being unstable (stable) for the considered diagonal (off-diagonal) disorder. }
\label{fig:trimer_dos}
\end{figure}
\end{document}
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\begin{document}
\title{ f The record method for two and three dimensional parameters random fields}
\abstract Let $S$ be a regular set of $\R^d$ and $X : S\rightarrow \R$ be Gaussian field with regular paths. In order to give bound to the tail of the distribution of the maximum, we use the record method of Mercadier. We present some new form in dimension 2 and extend it to dimension 3 using the result of the expectation of the absolute value of quadratic forms by Li and Wei. Comparison with other methods is conducted.
\textbf{Key-words:} Stochastic processes, Gaussian fields, Rice formula, distribution of the maximum.\\ \indent \textbf{Classifications:} 60G15, 60G60, 60G70. \section{Introduction} The problem of computing the tail of the maximum has a lot of applications in spatial statistics, image processing, oceanography, genetics etc ..., see for example Cressie and Wikle \cite{7}. It is exactly solved only for about ten processes with parameter of dimension 1, see Aza\"\i s and Wschebor \cite{5} p4 for a complete list. In the other cases, one has to use some approximations. Several methods have been used, in particular \begin{itemize}
\item The tube method, Sun \cite{17}.
\item Double sum method, Piterbarg \cite{14}.
\item Euler characteristic method see, for example, Adler and Taylor \cite{1}.
\item Rice or direct method, Aza\"\i s and Delmas \cite{3}, Aza\"\i s and Wschebor \cite{5}.
\end{itemize} With respect to these methods, the record method which is the main subject of this paper and which is detailed in Section \ref{se2} has the advantage of simplicity and also the advantage of giving a bound which is non asymptotic: it is true for every level and not for large $u$ only.
It has been introduced for one-parameter random processes by Rychlik \cite{16} and extended to two-parameter random fields by Mercadier \cite{13} to study the tail of the maximum of smooth Gaussian random fields on rather regular sets.\\
It has two version, one is an exact implicit formula : Theorem 2 in \cite{13} that is interesting for numerical purpose and that will not be considered here; the other form is a bound for the tail, see inequality (\ref{re1}) hereunder.
This bound has the advantage of its simplicity. In particular it avoids the computation of the expectation of the absolute value of the Hessian determinant as in the direct method of \cite{4} but it works only dimension 2. \\
For practical applications, the dimensions 2 and 3 (for the parameter set) are the most relevant so there is a need of an extension to dimension 3 and this is done in Section 3 using results on quadratic forms by Li and Wei \cite{11}.\\ The bound also has the drawback of demanding a parameterization of the boundary. For example, if we consider the version of Aza\"\i s and Wschebor (\cite{5}, Theorem 9.5 ) of the result of Mercadier, under some mild conditions on the set $S \subset \R^2$ and on the Gaussian process $X$, we have \begin{align}\label{re1}
\PP \{M_S \geq u \} \leq & \PP\{Y(O)\geq u\} + \int_0^L \E(|Y'(l)|\mid Y(l)=u)p_{Y(l)}(u)\, dl \notag \\
& + \int_S \E(|X''_{11}(t)^-X'_2(t)^+|\mid X(t)=u,\, X'_1(t)=0) p_{X(t),X'_1(t)}(u,0)\, dt, \end{align} where \begin{itemize} \item $M_S $ is the maximum of $X(t)$ on the set $S$. \item
$Y(l)=X(\rho (l))$ with $\rho: \; [0,L]\rightarrow \partial S$ is a parameterization of the boundary $\partial S $ by its length. \item $\displaystyle X''_{ij}=\frac{\partial^2 X}{\partial x_i \partial x_j}$. \item $p_Z(x)$: the value of the density function of random vector $Z$ at point $x$. \item $x^+=\sup (x,0)$, \; $x^-=\sup (-x,0)$. \end{itemize} The proof is based on considering the point with minimal ordinate (second coordinate) on the level curve. As we will see, this point can be considered as a ``record point''. \\ So the second direction of generalizations is to propose nicer and stronger forms of the inequality (\ref{re1}). This is done in Section \ref{se2}. The result on quadratic form is presented in Section \ref{se4} and some numerical experiment is presented in Section \ref{se5}.
\subsection*{Notation}
\begin{itemize} \item $S$ is some rather regular set included in $\R^2$ or $\R^3$. $\partial S$ is its boundary; $\overset{\circ}{S}$ is its interior. \item $M_S=\underset{s\in S}{\max}\, X(s)$ where $X(s)$ is some rather regular process. \item $\sigma_i$ is the surface measure of dimension $i$. It can be defined as a Hausdorff measure. \item $X', \; X''$ are the first and second derivatives of the process $ X(t)$. In particular if $\alpha $ is some direction then $X'_{\alpha}$ is the derivative along the direction $\alpha$. \item $M \preceq 0$ means that the square matrix $M$ is semi-definite negative. \item $S^{ +\epsilon}$ is the tube around $S$, i.e $$S^{+\epsilon}=\{s\in \R^2:\, \mbox{dist}(s,S)\leq \epsilon\}.$$
\item $d_H$ is the Hausdorff distance between sets, defined by $$d_H(S,T)=\inf \{\epsilon:\; S \subset T^{+\epsilon},\, T\subset S^{+\epsilon}\}.$$ \item $\varphi(x)$ and $\Phi(x)$ are the density and distribution function of a
standard normal variable. \\$\overline{\Phi}(x)=1-\Phi(x)$. \end{itemize}
\section{The record method in dimension 2 revisited}\label{se2} We will work essentially under the following assumption: \\
\textbf{Assumption 1:} $\{X(t),\; t \in NS \subset \R^2\}$ is a Gaussian stationary field, defined in a neighborhood $NS$ of $S$ with $\mathcal{C}^1$ paths and such that there exists some direction, that will be assumed (without loss of generality) to be the direction
of the first coordinate, in which the second derivative $X''_{11}(t)$ exists.\\
We assume moreover the following normalizing conditions that can always be obtained by a scaling $$ \E(X(t))=0, \; \Var(X(t))=1, \; \Var {X'(t)}=I_2 .
$$
Finally we assume that $ \Var(X''_{11}(t))>1$ which is true as soon as the spectral measure of the process restricted to the first axis is not concentrated on two opposite atoms. \\
In some cases we will assume in addition \\ \textbf{Assumption 2:}
$X(t)$ is isotropic, i.e $\textnormal{Cov}(X(s),X(t))=\rho(\|t-s\|^2)$, with $\mathcal{C}^2$ paths and $S$ is a convex polygon. \\
Under Assumption 1 and 2 plus some light additional hypotheses, the Euler Characteritic (EC) method \cite{1} gives $$ \PP\{M_S \geq u \} = \PP_E (u) +\mbox{Rest}, $$
with
$$
\PP_E (u) =
\overline{\Phi}(u)+\frac{\sigma_1(\partial S)}{2\sqrt{2\pi}} \varphi (u)+\frac{\sigma_2(S)}{2\pi}u\varphi (u),
$$ where the rest is super exponentially small.\\ The direct method gives \cite{4} \begin{align} \PP\{M_S \geq u \} & \leq \PP_M (u) = \overline{\Phi}(u)\displaystyle +\frac{\sigma_1(\partial S)}{2\sqrt{2\pi}} \int_u^{\infty} \left[ c\varphi (x\textrm{/}c)+x\Phi (x\textrm{/c})\right]\varphi (x)dx \notag \\ &\displaystyle + \frac{\sigma_2(S)}{2\pi } \int_u^{\infty} \left[x^2-1+\frac{\displaystyle (8\rho''(0) )^{3/2}\exp(-x^2.(24\rho''(0) -2)^{-1})}{ \sqrt{24\rho''(0) -2}}\right]\varphi(x)dx, \label{bou} \end{align} where $c=\sqrt{\textnormal{Var} (X''_{11})-1}=\sqrt{12\rho''(0)-1}$.\\ The record method gives \cite{13} $$\PP\{M_S \geq u \} \leq \overline{\Phi}(u)+\frac{\sigma_1(\partial S)}{\sqrt{2\pi}} \varphi (u)+\frac{\sigma_2(S)}{2\pi }\left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/c})\right]\varphi (u). $$
A careful examination of these equations shows that the main terms are almost the same except that in the record method the coefficient of $\sigma_1(\partial S)$ is twice too large. When $S$ is a rectangle $[0,T_1]\times[0,T_2]$, it is easy to prove that this coefficient 2 can be removed, see for example Exercise 9.2 in \cite{5}.
\\ The goal of this section is to extend the result above to more general sets and to fields satisfying Assumption 1 only. The main result of this section is the following
\begin{theorem}\label{th1} Let $X$ satisfy the Assumption 1 and suppose that $S$ is the Hausdorff limit of connected polygons $S_n$. Then, \begin{equation}\label{bou1} \PP \{M_S \geq u\} \leq \overline{\Phi}(u)+ \frac{\liminf_n \sigma_1(\partial S_n) \varphi (u)}{2\sqrt{2\pi}}+\frac{\sigma_2(S)}{2\pi }\left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/c})\right]\varphi (u), \end{equation} where $c=\sqrt{\textnormal{Var} (X''_{11})-1}$. \end{theorem}
{\bf Remark:} the choice of the direction of ordinates is arbitrary and is a consequence of the arbitrary choice of the the second derivative $X''_{11}$. When the process $X(t)$ admits derivative in all direction, the choice that gives the sharpest bound consists in chosing as first axis, the direction $\alpha$ such that $ \Var(X''_{\alpha \alpha})$ is minimum.
Unfortunately the proof it is based on an exotic topological property of the set $S$ that will be called ''emptyable". \begin{definition} The compact set $S$ is emptyable if there exists a point $O\in S$ which has minimal ordinate,
and such that for every $s \in S$ there exists a continuous path inside $S$ from $O$ to $s$ with non decreasing ordinate. \end{definition} In other word, suppose that $S$ is filled with water and that gravity is in the usual direction; $S$ is emptyable if after making a small hole at $O$, all the water will empty out, see Figure \ref{fig1}. \\ \begin{figure}
\caption{Example of non-emptyable set. The non-emptyable part is displayed in black. }
\label{fig1}
\end{figure} \begin{proof} Step1 : Suppose for the moment that $X$ has $\mathcal{C}^{\infty}$ paths and that $S$ is an emptyable polygon. Considering the event $\{M_S \geq u\}$, we have \begin{equation} \label{pt1} \PP \{M_S \geq u\} = \PP \{ X(O) \geq u\} + \PP \{ X(O) <u,M_S \geq u\}. \end{equation} It is clear that if $X(O)< u$ and $M_S\geq u$, because $S$ is connected, the level curve $$ \mathcal{C}_u=\{t \in S: \; X(t)=u\} $$ is not empty, and there is at least one point $T$ on $\mathcal{C}_u$ with minimal ordinate. There are two possibilities: \begin{itemize} \item $T$ is in the interior of $S$. In that case, suppose that there exists a point $s \in S$ with smaller ordinate than $T$ ($s_2<T_2$), such that $X(s)>u$. Then, due to the emptyable property, on the continuous path from $O$ to $s$ there would exist one point $s'$ with smaller ordinate than $T$, and with $X(s') =u$.
This is in contradiction with the definition of $T$. So we have proved that for every $s \in S$, $s_2<T_2$ we have $X(s) \leq u$. It is in the sense that $T$ can be considered as a record point. It implies that
$$ \{ X'_1(T)=0, \; X'_2(T) \geq 0, \; X''_{11}(T) \leq 0 \}.
$$
The probability that there exists such a point is clearly bounded, by the Markov inequality, by
$$
\E\left(\textnormal{card}\{t\in S:\; X(t)=u, \; X'_1(t)=0, \; X'_2(t) \geq 0, \; X''_{11}(t) \leq 0\}\right). \notag \\
$$
Applying the Rice formula to the field $Z=(X,X'_1)$ from $\R^2$ to $\R^2$, we get that \begin{align} &\quad \PP \{\exists \, t \in \overset{\circ}{S}: \, X(t)=u, \; t \; \mbox{has minimal ordinate on} \; \mathcal{C}_u \} \notag \\
\leq & \quad \displaystyle \int_S \E\left(|\det(Z'(t))| \mathbb{I}_{X'_2(t) \geq 0} \mathbb{I}_{X''_{11}(t) \leq 0}\mid Z(t)=(u,0)\right) \times p_{Z(t)}(u,0) \; dt \notag \\
= & \quad \displaystyle \sigma_2(S) \frac{\varphi(u)}{\sqrt{2\pi}}\; \E\left(X''^-_{11}(t)X^{\prime +}_2(t) \mid X(t)=u, X'_1(t)=0\right) \notag \\
= & \quad \displaystyle \sigma_2(S) \frac{\varphi(u)}{\sqrt{2\pi}} \; \E\left(X^{\prime +}_2(t)\right) \; \E\left(X''^-_{11}(t) \mid X(t)=u, X'_1(t)=0\right) \notag \\ = &\displaystyle \quad \sigma_2(S) \frac{\varphi (u)}{2\pi }\left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/}c)\right]. \label{pt2} \end{align} Note that the validity of the Rice formula holds true because the paths are of class $\cal{C}^\infty $ and that $X(t)$ and $X'_1(t)$ are independent. The computations above use some extra independences that are a consequence of the normalization of the process. The main point is that, under the conditioning $$ \det( Z'(t) ) = X''_{11}(t) X^{\prime}_2(t).$$
\item $T$ is on the boundary of $S$ that is the union of the edges $(F_1,\ldots, F_n)$. It is with probability 1 not located on a vertex. Suppose that, without loss of generality, it belongs to $F_1$. Using the reasoning we have done in the preceding case, because of the emptyable property, it is easy to see that $$ \{X(T)=u, \; X'_{\alpha}(T) \geq 0, \; X'_{\beta}(T) \leq 0\}, $$ where $\alpha$ is the upward direction on $F_1$ and $\beta$ is the inward horizontal direction. Then, apply the Markov inequality and Rice formula in the edge $F_1$,
\begin{displaymath} \begin{array}{rl} & \PP \{\exists \, t \in F_1: \; X(t)=u, \; t\; \mbox{has minimal ordinate on} \; \mathcal{C}_u\} \\ \leq & \PP\{\exists \, t \in F_1:\; X(t)=u, \; X'_{\alpha}(t) \geq 0, \; X'_{\beta}(t) \leq 0\}\\ \leq & \E\left(\textnormal{card}\{t\in F_1:\; X(t)=u, \; X'_{\alpha}(t) \geq 0, \; X'_{\beta}(t) \leq 0\}\right) \\
= & \displaystyle \int_{F_1} \E\left(|X'_{\alpha}(t))| \mathbb{I}_{X'_{\alpha}(t) \geq 0}\mathbb{I}_{X'_{\beta}(t) \leq 0}\mid X(t)=(u)\right) \times p_{X(t)}(u) \; dt \\ = & \sigma_1(F_1) \varphi(u) \, \E\left(X'^+_{\alpha}(t)\, \mathbb{I}_{X'_{\beta}(t) \leq 0}\right). \end{array} \end{displaymath} Denote by $\theta_1$ the angle $(\alpha,\beta)$. $X'_{\beta}$ can be expressed as $$ \cos\theta_1\, X'_{\alpha}+\sin\theta_1\,Y, $$ with $Y$ is a standard normal variable that is independent with $X'_{\alpha}$. Then \begin{displaymath} \begin{array}{rl}
&\E(X'^+_{\alpha}(t)\, \mathbb{I}_{X'_{\beta}(t) \leq 0})\\ =&\E(X'^+_{\alpha}\, \mathbb{I}_{\cos \theta_1 X'_{\alpha}+\sin \theta_1 Y \leq 0})\\ =&\displaystyle \frac{1-\cos\theta_1}{2\sqrt{2\pi}}. \end{array} \end{displaymath} Summing up, the term corresponding to the boundary of $S$ is at most equal to \begin{equation}\label{pt3} \varphi(u) \sum_{i=1}^n \frac{(1-\cos\theta_i)\sigma_1(F_i)}{2\sqrt{2\pi}}=\frac{\varphi(u)\sigma_1(\partial S)}{2\sqrt{2\pi}}, \end{equation} since $\displaystyle \sum_{i=1}^n \sigma_1(F_i) \, \cos\theta_i $ is just the length of the oriented projection of the boundary of $S$ on the $x$ -axis, so it is zero. \end{itemize} Hence, summing up (\ref{pt2}), (\ref{pt3}) and substituting into (\ref{pt1}), we obtain the desired upper-bound in our particular case.
Step 2: Suppose now that $S$ is a general connected polygon such that the vertex $O$ with minimal ordinate is unique. We define $S_1$ as the maximal emptyable subset of $S$ that contains $O$. It is easy to prove that $S_1$ is still a polygon with some horizontal edges and that $S\backslash S_1$ consists of several polygons with horizontal edges, say $S_2^1,\ldots,S_2^{n_2}$, see Figure \ref{fig2}.\\ \begin{figure}
\caption{ Example on construction of $S_1$. }
\label{fig2}
\end{figure} \\ So we write \begin{equation}\label{pt4} \PP\{M_S \geq u\}\leq \PP\{X(O)\geq u\}+\PP\{M_{S_1}\geq u,\; X(O)<u \}+\underset{i=1}{\overset{n_2}{\sum}}\PP\{M_{S_1}< u, \; M_{S_2^i}\geq u\}. \end{equation} Suppose for the moment that all the $S_2^i,\; i=1,\ldots,n$ are emptyable. Then, to give bounds to the event $$\{M_{S_1}< u, \; M_{S_2^i}\geq u\},$$ we can apply the reasoning of the preceding proof but inverting the direction: in $S_2^i$, we search points on the level curve with {\bf maximum ordinate}. Let $E$ be the common edge of $S_1$ and $S_2^i$. Clearly, when $\{M_{S_1}< u, \; M_{S_2^i}\geq u\},$ the level curve is non empty and by the same arguments as in Theorem \ref{th1}, there exists $t\in S_2^i$ satisfying whether (except events with zero probability) \begin{itemize} \item $t$ is in the interior of $S^i_2$ and
$$\{ X(t)=u, \; X'_1(t)=0, \; X'_2(t) \leq 0, \; X''_{11}(t) \leq 0 \}.$$
From Markov inequality and Rice formula, this probability is at most equal to \begin{equation}\label{pt5}
\frac{\varphi (u)\sigma_2(S_2^i)}{2\pi }\left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/c})\right].
\end{equation} \item $t$ lies on some edges of $S^i_2$. Note that $t$ can not belong to $E$. Then, as in Theorem \ref{th1}, we consider the event $t$ is on each edge and sum up the bounds to obtain \begin{equation}\label{pt6} \begin{array}{rl} & \PP \left( \{\exists \, t \in \partial S_2^i: \, X(t)=u, \, t \, \mbox{has maximal second ordinate on the level curve} \, \} \cap \{M_{S_1}<u\}\right)\\ \leq &\displaystyle \frac{\varphi(u)[\sigma_1(\partial S^i_2)-2\sigma_1(E)]}{2\sqrt{2\pi}}. \end{array} \end{equation} \end{itemize} From (\ref{pt5}) and (\ref{pt6}) we have \begin{equation}\label{pt7} \displaystyle \PP\{M_{S_1}< u, \; M_{S_2^i}\geq u\} \leq \frac{\varphi (u)\sigma_2(S_2^i)}{2\pi }\left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/c})\right]+\frac{\varphi(u)[\sigma_1(\partial S^i_2)-2\sigma_1(E)]}{2\sqrt{2\pi}}. \end{equation} Summing up all the bounds as in (\ref{pt7}), considering the upper bound for $\PP\{X(O)<u,\, M_{S_1}\geq u\}$ as in Theorem \ref{th1} and substituting into (\ref{pt4}), we get the result.\\ In the general case, when some $S_2^i$ is not emptyable, we can decompose $S_2^i$ as we did for $S$ and by induction. Since the number of vertices is decreasing, we get the result.
Step 3: Passing to the limit. The extension to process with non $\mathcal{C}^\infty$ paths is direct by an approximation argument. Let $\overline{X}_\epsilon(t)$ be the Gaussian field obtained by convolution of $X(t)$ with a size $\epsilon $ convolution kernel (for example a Gaussian density with variance $ \epsilon^2 I_2$). We can apply the preceding bound to the process $$ X_ \epsilon (t) := \frac 1{\sqrt{\Var (\overline{X}_\epsilon(t)) }} \overline{X}_\epsilon\left( \Sigma ^{-1/2} _ \epsilon t\right) , $$ where $ \Sigma _ \epsilon = \Var(\overline{X}'_\epsilon(t))$. Since $ \Var (\overline{X}_\epsilon(t)) \to 1$ and $ \Sigma _ \epsilon \to I_2$ , $\max_{t \in S } X_\epsilon (t) \to M_S$ and we are done.
The passage to the limit for $S_n$ tending to $ S$ is direct. \end{proof}
\subsection*{Some examples} \begin{itemize} \item If $S$ is compact convex with non-empty interior then it is easy to construct a sequence of polygons $S_n$ converging to $S$ and such that $\liminf_n \sigma_1(\partial S_n)=\sigma_1(\partial S)$, giving \begin{equation}\label{f:bound} \PP\{M_S\geq u\}\leq \PP_R(u)=\overline{\Phi}(u)+ \frac{\sigma_1(\partial S)}{2\sqrt{2\pi}}\varphi(u)+ \frac{\sigma_2(S)}{2\pi }\left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/c})\right]\varphi (u). \end{equation} \item More generaly, if $S$ is compact and has a boundary that is piecewise-$\mathcal{C}^2$ except for a finite number of points and the closure of the interior of $S$ equals to $S$, we get (\ref{f:bound}) by the same tools.
\item Let us now get rid of the condition $\overline{\overset{\circ}{S}}=S$ but still assuming the piecewise-$\mathcal{C}^2$ condition. Define the ``outer Minkowski content" of a closed subset $S \subset \R^2$ as (see \cite{8}) $$\rm{OMC}(S)= \underset{\epsilon \rightarrow 0}{\lim} \frac{\sigma_2(S^{+\epsilon}\setminus S)}{\epsilon},$$ whenever the limit exists (for more treatment in this subject, see \cite{2}). This definition of the perimeter differs from the quantity $\sigma_1(\partial S)$. A simple counter-example is a set corresponding to the preceding example with some ``whisker'' added. Using approximation by polygons, we get \begin{equation} \label{star} \PP\{M_S\geq u\}\leq \PP_R(u)=\overline{\Phi}(u)+ \frac{\rm{OMC}( S)}{2\sqrt{2\pi}}\varphi(u)+ \frac{\sigma_2(S)}{2\pi }\left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/c})\right]\varphi (u). \end{equation} \item The next generalization concerns compact $r$-convex sets with a positive $r$ in the sense of \cite{8}. These sets satisfy $$S=\underset{\overset{\circ}{B}(x,r) \cap S=\emptyset}{\bigcap}\R^2\setminus\overset{\circ}{B}(x,r). $$ This condition is slightly more general than the condition of having positive reach in the sense of Federer \cite{9}. Suppose in addition that $S$ satisfies the interior local connectivity property: there exists $\alpha_0 >0$ such that for all $0<\alpha<\alpha_0$ and for all $x\in S,\; \mbox{int}\left(B(x,\alpha)\cap S\right)$ is a non-empty connected set. Then we can construct a sequence of approximating polygons in the following way.
Let $X_1,X_2,\ldots ,X_n$ be a random sample drawn from a uniform distribution on $S$ and $S_n$ be the r-convex hull of this sample, i.e $$S_n=\underset{\overset{\circ}{B}(x,r) \cap \{X_1,X_2,\ldots , X_n\}=\emptyset}{\bigcap}\R^2\setminus\overset{\circ}{B}(x,r),$$ which can be approximated by polygons with an arbitrary error. By Theorem 6 of Cuevas et al \cite{8}, $S_n$ is a fully consistent estimator of $S$, it means that $d_H(S_n,S)$ and $d_H(\partial S_n,\partial S)$ tend to 0 as $n$ tends to infinity. This implies $\sigma_2(S_n)\rightarrow \sigma_2(S)$ and $\rm{OMC}(S_n)\rightarrow \rm{OMC}(S)$. Hence, we obtain (\ref{star}).
\item A complicated case: a ``Swiss cheese''. Here, we consider an unit square and inside it, we remove a sequence of disjoint disks of radius $r_i$ such that $\displaystyle \pi \underset{i=1} {\overset{\infty}{\sum}}r_i^2 <1$ to obtain the set $S$. When $\displaystyle \underset{i=1}{\overset{\infty}{\sum}}r_i <\infty$ the bound (\ref{bou1}) makes sense directly. But examples can be constructed from the Sierpinski carpet (see Figure \ref{fig3}) such that $\displaystyle \underset{i=1}{\overset{\infty}{\sum}}r_i =\infty$ :
divide the square into 9 subsquares of the same size and instead of removing the central square, remove the disk inscribed in this square and do the same procedure for the remaining 8 subsquares, ad infinitum.\\ \begin{figure}
\caption{ Sierpinski carpet (source: Wikipedia).}
\label{fig3}
\end{figure}\\
In our case, $$\sum_{i=1}^{\infty}r_i^2=\frac{1}{4}\sum_{i=1}^{\infty}\frac{8^{i-1}}{3^{2i}} =\frac{1}{4}$$ This proves that the obtained set $S$ has positive Lebesgue measure and is not fractal. We have on the other hand
$$
\sum_{i=1}^{\infty}r_i=\frac{1}{2}\sum_{i=1}^{\infty}\frac{8^{i-1}}{3^{i}} = \infty.
$$ Let $S_n$ be the set obtained after removing the n-th disk. Since $S \subset S_n$, $$\PP \{M_S \geq u\} \leq \PP \{M_{S_n } \geq u\} \leq \overline{\Phi}(u)+ \frac{ \varphi (u)}{2\sqrt{2\pi}}(4+2\pi\underset{i=1}{\overset{n}{\sum}}r_i)+(1-\pi\underset{i=1}{\overset{n}{\sum}}r_i^2)\left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/c})\right]\varphi(u)/(2\pi).$$ Hence, $$\PP \{M_S \geq u\} \leq \overline{\Phi}(u)+ \underset{n}{\min}\left[\frac{ \varphi (u)}{2\sqrt{2\pi}}(4+2\pi\underset{i=1}{\overset{n}{\sum}}r_i)+(1-\pi\underset{i=1}{\overset{n}{\sum}}r_i^2)\left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/c})\right]\varphi(u)/(2\pi)\right].$$ \end{itemize}
{\bf Remarks}: \begin{itemize}
\item[1.] In comparison with other results, all the examples considered here are new. Firstly the conditions on the process are minimal and weaker than the ones of the other methods. Secondly the considered sets are not covered by any other methods. Even for the first example, because we do not assume that the number of irregular points is finite, which is needed, for example, for the convex set to be a stratified manifold as in \cite{1}. \item[2.] Theorem \ref{th1} can be extended directly to non connected sets using sub-additivity
$$
\PP\{M_{S_1\cup S_2}\geq u \} \leq \PP\{M_{S_1}\geq u \} + \PP\{M_{ S_2}\geq u \}.
$$
This implies that the coefficient of $\overline{\Phi}(u)$ in (\ref{bou1}) must be the number of components.
\end{itemize}
\subsection*{Is the bound sharp?} \begin{itemize} \item Under Assumption 2, Adler and Taylor \cite{1} show that
$$\underset{u\rightarrow +\infty}{\liminf} -2u^2\log |\PP\{M_S\geq u\} -\PP_E(u)|\geq 1+1/c^2.$$ From $$0\leq \PP_R(u)-\PP_E(u)=\frac{\sigma_2(S)}{2\pi}\varphi(u)\left[c\varphi (u\textrm{/}c)-u\overline{\Phi} (u\textrm{/}c)\right]$$ and the elementary inequality for $x>0$, $$\varphi(x)\left(\frac{1}{x}-\frac{1}{x^3}\right)<\overline{\Phi}(x)<\varphi(x)\left(\frac{1}{x}-\frac{1}{x^3}+\frac{3}{x^5}\right),$$ it is easy to see that $$\underset{u\rightarrow +\infty}{\lim\inf} -2u^2\log (\PP_R(u)-\PP_E(u)) \geq 1+1/c^2.$$
So the upper bound $\PP_R(u)$ is as sharp as $\PP_E(u)$ .
\item Let $S$ be a compact and simply connected domain in $\R^2$ having a $\mathcal{C}^3$-piecewise boundary. Assume that all the discontinuity point are convex, in the sense that if we parametrize the boundary in the direction of positive rotation, then at each discontinuity point, the angle of the tangent has a positive discontinuity. Then, it is easy to see that the quantity
$$\kappa (S)= \underset{t\in S}{\sup} \underset{s\in S,\; s\neq t}{\sup} \frac{\rm{dist}(s-t,\mathcal{C}_t)}{\|s-t\|^2}$$ is finite, where $\rm{dist}$ is the Euclidean distance and $C_t$ is the cone generated by the set of directions
$$\Biggl\{ \lambda \in \R^2 :\; \|\lambda\|=1,\, \exists s_n \in S \; \rm{such\; that}\; s_n\rightarrow t \; \rm{and} \; \frac{s_n-t}{\|s_n-t\|}\rightarrow \lambda \Biggr\}.$$ In order to apply the Theorem 8.12 in \cite{5}, besides the Assumption 1, we make some additional assumptions on the field $X$ such that it satisfies the conditions (A1)-(A5) page 185 in \cite{5}. Assume that \begin{itemize} \item $X$ has $\mathcal{C}^3$ paths.
\item The covariance function $r(t)$ satisfies $|r(t)|\neq 1$ for all $t\neq 0$. \item For all $s\neq t$, the distribution of $(X(s),X(t),X'(s),X'(t))$ does not degenerate. \end{itemize} With these hypotheses, we can see that \begin{itemize} \item The conditions (A1)-(A3) are easily verified. \item The condition (A4) which states that the maximum is attained at a single point, can be deduced from Proposition 6.11 in \cite{5} since for $s\neq t$, $(X(s),X(t),X'(s),X'(t))$ has a nondegenerate distribution. \item The condition (A5) which states that almost surely there is no point $t\in S$ such that $X'(t)=0$ and $\det(X''(t))=0$, can be deduced from Proposition 6.5 in \cite{5} applied to the process $X'(t)$. \end{itemize} Since all the required conditions are met, by Theorem 8.12 in \cite{5}, we have \begin{equation}\label{f:1} \underset{x\rightarrow +\infty}{\liminf}-2x^2 \log \big[\PP_M(x)-\PP\{M_S\geq x\}\big] \geq 1+ \underset{t\in S}{\inf}\frac{1}{\sigma_t^2+\kappa_t^2}>1, \end{equation} where $$\sigma_t^2=\underset{s\in S\setminus\{t\}}{\sup}\frac{\rm{Var}\left(X(s)\mid X(t),X'(t)\right)}{(1-r(s,t))^2}$$ and $$\kappa_t=\underset{s\in S\setminus\{t\}}{\sup}\frac{\rm{dist}\left(\frac{\partial}{\partial t} \it{r}(s,t),C_t\right)}{1-r(s,t)}.$$ Note that the condition $\kappa(S)$ is finite implies that $\kappa(t)$ is also finite for every $t\in S$. (\ref{f:1}) is true also for $\PP_R$, since as $x\rightarrow +\infty$, $\PP_R(x)$ is smaller than $\PP_M(x)$ (see Section \ref{se5} for the easy proof). As a consequence $\PP_R$ is super exponentially sharp.
\item Suppose that $S$ is a circle in $\R^2$. Then $\{X(t)\, :\; t\in S\}$ can be viewed as a periodic process on the line. In that case, it is easy to show, see for example Exercise 4.2 in \cite{5}, that as $u\rightarrow \infty$ $$ \PP(M_S\geq u)=\frac{\sigma_1(S)}{\sqrt{2\pi}}\varphi(u) + O(\varphi(u(1+\delta))=\frac{\rm{OMC}(S)}{2\sqrt{2\pi}}\varphi(u) + O(\varphi(u(1+\delta)) $$ for some $\delta >0$; while Theorem \ref{th1} gives with a standard approximation of the circle by polygons $$\PP(M_S\geq u) \leq P_R(u)=\overline{\Phi}(u)+ \frac{\rm{OMC}(S)}{2\sqrt{2\pi}}\varphi(u), $$ which is too large. This shows that the bound $P_R$ is not always super exponentially sharp. \end{itemize}
\section{The record method in dimension 3}\label{se3}
For example, with the direct method, some difficulties arise in dimension 3 because we need to compute
$$\E | \det (X'' (t)|,$$
under some conditional law. This can be conducted only in the isotropic case using random matrices theory, see \cite{4} and even in this case the result is complicated. In dimension 2, the record method is a trick that permits to spare a dimension in the size of the determinant we have to consider because the conditioning implies a factorization. For example in equation (\ref{pt2}) we have used the fact that
$$ \det( Z'(t) ) = X''_{11}(t) X^{\prime }_2(t),$$
under the condition. In this section we will use the same kind of trick to pass from a 3,3 matrix to a 2,2 matrix and then a 2,2 determinant is just a quadratic form so we can use, to compute the expectation of its absolute value, the Fourier method of Berry and Dennis \cite{6} or Li and Wei \cite{11}. This computation is detailed in Section 4 and is one of the main contributions of the paper. \\
Before stating the main theorem of this section, we recall the following lemma (see Chapter 5 of Prasolov and Sharygin \cite{15})
\begin{lemma}\label{lema}
Let $Oxyz$ be a trihedral. Denote by $a,\, b$ and $c$ the plane angles $\widehat{xOy},\, \widehat{yOz}$ and $\widehat{zOx}$, respectively. Denote by $A, \,B$ and $ C$ the angles between two faces containing the line $Oz, \, Ox$ and $ Oy$, respectively. Then, \begin{itemize} \item[a.] $\sin a : \sin A=\sin b : \sin B=\sin c : sin C$. \item[b.] $\cos a=\cos b\cos c+\sin b\sin c \cos A$. \end{itemize} \end{lemma} Our main result is the following
\begin{theorem}\label{th3} Let $S$ be a compact and convex subset of $\R^3$ with non-empty interior and let $X$ satisfy Assumption 1. Suppose, in addition that
$X$ is isotropic with respect to the first and second coordinate, i.e
$$\textnormal{Cov}(X(t_1,t_2,t_3);X(s_1,s_2,t_3))=\rho((t_1-s_1)^2+(t_2-s_2)^2) \mbox{ with}\; \rho \, \mbox{ of class}\; \mathcal{C}^2.$$
Then, for every real $u$, \begin{displaymath} \begin{array}{rl} \PP \{ M\geq u \} \leq & 1 -\Phi (u) + \displaystyle \frac{2\lambda(S)}{\sqrt{2\pi }} \varphi (u) + \frac{\sigma_2(S) \varphi (u)}{4\pi } \left[ \sqrt{12\rho''(0) -1}\varphi \left(\frac{u}{\sqrt{12\rho''(0) -1}} \right) + u \Phi \left(\frac{u}{\sqrt{12\rho''(0) -1}} \right)\right]\\ &+ \displaystyle \frac{\sigma_3(S)\varphi (u)}{(2\pi)^{3/2} } \left[u^2-1+\frac{\displaystyle (8\rho''(0))^{3/2}\exp\left(-u^2.(24\rho''(0)-2)^{-1}\right)}{\displaystyle \sqrt{24\rho''(0)-2}}\right], \end{array} \end{displaymath} where $\lambda$ is the caliper diameter. \end{theorem} \begin{proof} By the same limit argument as in Theorem \ref{th1}, we can assume that $X(t) $ has $\mathcal{C}^{\infty}$ paths and that $S$ is a convex polyhedron. Let $O$ be the vertex of $S$ that has minimal third coordinate, we can assume also that this vertex is unique. It is clear that if $X(O) < u$ and $M_S>u$ then the level set $$\mathcal{C}(u)=\{t\in S: \; X(t)=u\}$$ is non empty and there exists at least one point $T$ having minimal third coordinate on this set. Then, \begin{equation} \begin{array}{rcl} \PP\{M_S\geq u\}& = & \PP \{ X(O)\geq u \}+\PP\{X(O)<u,\, M_S\geq u\}\\ &\leq & \PP \{ X(O)\geq u \}+\PP\{\exists \, T\in S: \, X(T)=u,\, T \, \mbox{has minimal third coordinate on}\; \mathcal{C}_u\}. \end{array} \end{equation} Now, we consider three possibilities:\\ $\bullet $ Firstly, if $T$ is in the interior of $S$, then by the same arguments as in Theorem \ref{th1}, for all the point $s\in S$ with the third coordinate smaller than the one of $T$, $X(s)<X(T)$; it means that, at $T$, $X(t)$ has a local maximum
with respect to the first and second coordinates and is non-decreasing with respect to the third coordinate. Therefore, setting
\begin{displaymath} A(t)= \left( \begin{array}{cc} X''_{11}(t) & X''_{12}(t) \\ X''_{12}(t) & X''_{22}(t) \end{array} \right), \end{displaymath} we have
$$
\{X(T)=u, \;X'_1(T)=0, \; X'_2(T)=0,\; A(T) \preceq 0,\; X'_3(T) \geq 0\}.
$$ Then, apply the Rice formula to the field $Z=(X,X'_1,X'_2)$ and the Markov inequality, \begin{displaymath} \begin{array}{rl} &\displaystyle\PP\{\exists \, T\in \overset{\circ}{S}: \, X(T)=u,\, T \, \mbox{has minimal third coordinate on}\; \mathcal{C}_u\}\\
\leq &\displaystyle \PP\{\exists \, t \in \overset{\circ}{S}:\;X(t)=u, \; X'_1(t)=0, \; X'_2(t)= 0, \; X'_3(t)\geq 0,\; A(t) \preceq 0\}\\ \leq &\displaystyle \E\big(\textnormal{card}\{t\in \overset{\circ}{S}:\; X(t)=u, \; X'_1(t)=0, \; X'_2(t)= 0, \; X'_3(t)\geq 0,\; A(t) \preceq 0\}\big) \\ = &\displaystyle \E\big(\textnormal{card}\{t\in \overset{\circ}{S}:\;Z(t)=(u,0,0), \; X'_3(t) \geq 0, \; A(t) \preceq 0\}\big)\\
=& \displaystyle \int_{\overset{\circ}{S}} \E\left(|\det(Z'(t))|\mathbb{I}_{X'_3(t) \leq 0}\mathbb{I}_{A(t)\preceq 0} \mid Z(t)=(u,0,0)\right) \times p_{Z(t)}(u,0,0)\; dt.\\
\end{array} \end{displaymath} Under the condition $Z(t)=(u,0,0)$, it is clear that $\det(Z'(t))=X'_3(t)\, \det(A(t))$. So, we obtain the bound
$$ \sigma_3(S) \, \frac{\varphi(u)}{2\pi} \, \E\left(|\det(A(t))|.\mathbb{I}_{A(t) \preceq 0}X'^+_3(t) \mid Z(t)=(u,0,0)\right). $$ From Corollary \ref{cor} of Section \ref{se4}, we know that $$
\E\left(|\det(A(t))|. \mathbb{I}_{A(t)\preceq 0}\mid Z(t)=(u,0,0) \right) \leq u^2-1+\frac{\displaystyle (8\rho''(0) )^{3/2}\exp\left(-u^2.(24\rho''(0) -2)^{-1}\right)}{\displaystyle \sqrt{24\rho''(0) -2}}.$$ Hence, \begin{align} &\PP\{\exists \, T\in \overset{\circ}{S}: \, X(T)=u,\, T \, \mbox{has minimal third coordinate on}\; \mathcal{C}_u\} \notag \\ \leq \;\;& \label{t1}
\frac{\sigma_3(S)\varphi (u)}{(2\pi)^{3/2} } \left[u^2-1+\frac{\displaystyle (8\rho''(0))^{3/2}\exp\left(-u^2.(24\rho''(0)-2)^{-1}\right)}{\displaystyle \sqrt{24\rho''(0)-2}}\right].
\end{align} $\bullet $ Secondly, if $T$ is in the interior of a face $S_1$, then, in this face, we choose the base $\overrightarrow{\alpha},\, \overrightarrow{\beta}$ such that $\overrightarrow{\alpha}$ is in the horizontal plane $ 0\ t_1t_2$ such that along this vector, the second coordinate is not decreasing. Let us denote vector $\overrightarrow{\gamma}$ in the horizontal plane that is perpendicular to $\alpha$ and goes into $S$. It is easy to see that $$\{ X(T)=u, \; X'_ {\alpha}(T) =0,\; X'_{\beta}(T) \geq 0,\; X'_{\gamma}(T) \leq 0,\; X''_{\alpha}(T) \leq 0 \}.$$ Apply Markov inequality and Rice formula to the field $Y(t)=(X(t),X'_{\alpha}(t))$, \begin{displaymath} \begin{array}{rl} &\displaystyle \PP\{\exists \, T\in \overset{\circ}{S_1}: \, X(T)=u,\, T \, \mbox{has the minimal third ordinate on}\; \mathcal{C}_u\}\\
\leq & \PP \{\exists \, t \in \overset{\circ}{S_1}: \; X(t)=u, \; X'_ {\alpha}(t) =0,\; X'_{\beta}(t) \geq 0,\; X'_{\gamma}(t) \leq 0,\; X''_{\alpha} \leq 0 \}\\ \leq & \E (\textnormal{card}\{ t \in \overset{\circ}{S_1}: \; X(t)=u, \; X'_ {\alpha}(t) =0,\; X'_{\beta}(t) \geq 0,\; X'_{\gamma}(t) \leq 0,\; X''_{\alpha} \leq 0 \})\\
= & \displaystyle \int_{\overset{\circ}{S_1}} \E\left( |\det(Y'(t))| \mathbb{I}_{X'_{\beta}(t) \geq 0}\mathbb{I}_{X'_{\gamma}(t) \leq 0}\mathbb{I}_{X''_{\alpha}(t) \leq 0}\mid Y(t)=(u,0)\right)\, p_{Y(t)}(u,0)\, dt\\
= &\displaystyle \frac{\sigma_2(S_1)\varphi(u)}{\sqrt{2\pi}} \E\left( |X''^-_{\alpha}(t)|X'^+_{\beta}(t)\mathbb{I}_{X'_{\gamma}(t) \leq 0} \mid Y(t)=(u,0)\right). \end{array} \end{displaymath} As in Theorem \ref{th1}, it is clear that \begin{align*}
\E\big( |X''^-_{\alpha}(t)| \mid Y(t)=(u,0)\big) & =\, \sqrt{12\rho''(0) -1}\varphi \left(\frac{u}{\sqrt{12\rho''(0) -1}} \right) + u \Phi \left(\frac{u}{\sqrt{12\rho''(0) -1}} \right),\\
\E\big( X'^+_{\beta}(t)\mathbb{I}_{X'_{\gamma}(t) \leq 0} \mid Y(t) &=(u,0)\big)=\frac{1-\cos(\beta,\gamma)}{2\sqrt{2\pi}}. \end{align*} Observe that the angle between $\beta$ and $\gamma$ is the angle $\theta_1$ between the face $S_1$ and the horizontal plane, then the probability that there exists one point with minimal third coordinate on the level set and in the interior of the face $S_1$ is at most equal to $$ \frac{\sigma_2(S_1) \varphi (u)(1-\cos\theta_1)}{4\pi } \left[ \sqrt{12\rho''(0) -1}\varphi \left(\frac{u}{\sqrt{12\rho''(0) -1}} \right) + u \Phi \left(\frac{u}{\sqrt{12\rho''(0) -1}} \right)\right] .$$ Taking the sum of all the bounds at each faces, observing that $$\sum_{i=1}^n \sigma_2(S_i)\cos\theta_i=0,$$ we have the following upper bound for the probability of having a point $T$ with minimal third coordinate on the level set and belonging to a face: \begin{equation}\label{t2}
\frac{\sigma_2(S) \varphi (u)}{4\pi } \left[ \sqrt{12\rho''(0) -1}\varphi \left(\frac{u}{\sqrt{12\rho''(0) -1}} \right) + u \Phi \left(\frac{u}{\sqrt{12\rho''(0) -1}} \right)\right] . \end{equation}
$\bullet $ Thirdly, when $T$ belongs to one edge, for example $F_1$. Let us define $\overrightarrow{\eta}$ is the upward direction on this edge, i.e such that along this vector, the third coordinate is not decreasing, and $\overrightarrow{\alpha}$ and $\overrightarrow{\beta}$ are the two horizontal directions that go inside two faces containing the edge. Then, $$ \{X(T)=u,\; X'_{\eta}(T) \geq 0,\; X'_{\alpha}(T) \leq 0,\; X'_{\beta}(T) \leq 0\} .$$ By Rice formula, the expectation of the number of the points in $F_1$ satisfying this condition is \begin{displaymath} \begin{array}{rl} & \displaystyle \int_{F_1} \E\left(X'^+_{\eta}(t) \mathbb{I}_{X'_{\alpha}(t) \leq 0}\mathbb{I}_{X'_{\beta}(t) \leq 0} \mid X(t)=u\right)\times p_{X(t)}(u) \; dt\\ =& \displaystyle \sigma_1(F_1) \; \varphi(u) \; \E\left(X'^+_{\eta}(t) \mathbb{I}_{X'_{\alpha}(t) \leq 0}\mathbb{I}_{X'_{\beta}(t) \leq 0}\right). \end{array} \end{displaymath} Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors in two faces containing the edge $F_1$ and perpendicular to $\overrightarrow{\eta}$; $\theta_1$ be the angle between $\overrightarrow{\alpha}$ and $\overrightarrow{\eta}$; $\theta_2$ be the angle between $\overrightarrow{\beta}$ and $\overrightarrow{\eta}$. It is clear that $$X'_{\alpha}(t)=\cos\theta_1 \, X'_{\eta}(t)+ \sin\theta_1 \, X'_a(t) ,$$ $$ X'_{\beta}(t)=\cos\theta_2 \, X'_{\eta}(t)+\sin\theta_2 \, X'_b(t), $$ and $\textnormal{cov}(X'_a(t),X'_b(t))=\cos\theta_3$, where $\theta_3$ is the angle between two faces containing the edge $F_1$. Then, \begin{displaymath} \begin{aligned} &\E\left(X'^+_{\eta} \mathbb{I}_{X'_{\alpha}(t) \leq 0}\mathbb{I}_{X'_{\beta}(t) \leq 0}\right)\\ =& \E\left(X'^+_{\eta} \mathbb{I}_{\{\cos\theta_1 \, X'_{\eta}(t)+ \sin\theta_1 \, X'_a(t) \leq 0\} }\mathbb{I}_{\{\cos\theta_2 \, X'_{\eta}(t)+\sin\theta_2 \, X'_b(t) \leq 0\}}\right)\\ =& \displaystyle \int_0^{\infty} x \, \varphi(x) \, F(x) \, dx,\\ \end{aligned} \end{displaymath} where \begin{displaymath} \begin{aligned} F(x)= & \E\left( \mathbb{I}_{\{\cos\theta_1 \, X'_{\eta}(t)+ \sin\theta_1 \, X'_a(t) \leq 0\}}\mathbb{I}_{\{\cos\theta_2 \, X'_{\eta}(t)+\sin\theta_2 \, X'_b(t) \leq 0\}} \mid X'_{\eta}(t)=x\right)\\ = & \displaystyle \int_{-\infty}^{-\cot\theta_1 .x} \varphi(y) \, \Phi\left(\frac{- \cot\theta_2 .x - \cos\theta_3 .y}{\sin \theta_3}\right) \, dy.\\ \end{aligned} \end{displaymath} So, \begin{displaymath} \begin{aligned} F'(x)=&-\cot \theta_2 \; \varphi(-\cot\theta_2 \, .x) \, \Phi\left(\frac{-\cot\theta_1 \,.x+\cos \theta_3 \, \cot \theta_2\, . x}{\sin \theta_3}\right)\\ &-\cot \theta_1 \; \varphi(-\cot (\theta_1 \, x)) \, \Phi\left(\frac{-\cot\theta_2\, . x+\cos \theta_3 \, \cot \theta_1\, . x}{\sin \theta_3}\right). \end{aligned} \end{displaymath} By integration by parts, \begin{displaymath} \begin{aligned} \displaystyle \int_0^{\infty} x\, \varphi(x)\, F(x)\, dx & = -\int_0^{\infty} F(x) \, d(\varphi(x))\\ &= \displaystyle F(0)\varphi(0)+ \int_0^{\infty} \varphi(x)\, F'(x)\, dx. \end{aligned} \end{displaymath} It is easy to check that
$$\int_0^{\infty} \varphi(x)\, \Phi(mx) \, dx=\frac{1}{4}+\frac{\arctan (m)}{2\pi},$$ $$\int_{-\infty}^0 \varphi(x)\, \Phi(mx) \, dx=\frac{1}{4}-\frac{\arctan (m)}{2\pi}.$$ From the above results, we have \begin{displaymath} \begin{aligned} F(0)\varphi(0)& = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^0 \varphi(y)\Phi\left( \frac{-\cos\theta_3 \, }{\sin\theta_3}y\right) \, dy\\ &= \frac{1}{(2\pi)^{3/2}}\left(\pi-\theta_3\right). \end{aligned} \end{displaymath} and \begin{displaymath} \begin{aligned} \int_0^{\infty} \varphi(x)\, F'(x)\, dx&= -\cot \theta_2 \,\int_0^{\infty} \varphi(x) \varphi(-\cot \theta_2\, . x) \, \Phi\left(\frac{-\cot \theta_1\, . x+\cos \theta_3 \, \cot \theta_2\, . x}{\sin \theta_3}\right) \, dx\\ &\quad -\cot \theta_1 \,\int_0^{\infty} \varphi(x) \varphi(-\cot \theta_1\, . x) \, \Phi\left(\frac{-\cot \theta_2\, .x+\cos \theta_3 \, \cot \theta_1\, . x}{\sin \theta_3}\right)\,dx\\ &= \frac{-\cos\theta_2}{\sqrt{2\pi}}\left(\frac{1}{4}+\frac{1}{2\pi}\arctan \left(\frac{-\sin\theta_2.\cot\theta_1+\cos\theta_3\cos\theta_2}{\sin\theta_3}\right)\right)\\ &\quad + \frac{-\cos\theta_1}{\sqrt{2\pi}}\left(\frac{1}{4}+\frac{1}{2\pi}\arctan \left(\frac{-\sin\theta_1\cot\theta_2+\cos\theta_3\cos\theta_1}{\sin\theta_3}\right)\right). \end{aligned} \end{displaymath} Therefore, the probability that there exists one point with minimal third coordinate on the level set $\mathcal{C}(u)$ and belonging to $F_1$ is at most equal to \begin{displaymath} \begin{array}{rl}
\displaystyle \frac{\sigma_1(F_1)(\pi-\theta_3)\varphi(u)}{(2\pi)^{3/2}} +\sigma_1(F_1)& \displaystyle \left[\frac{-\cos\theta_2}{\sqrt{2\pi}} \left( \frac{1}{4}+\frac{1}{2\pi}\arctan \left(\frac{-\sin\theta_2\cot\theta_1+\cos\theta_3\cos\theta_2}{\sin\theta_3}\right)\right) \right.\\
&\displaystyle +\left. \frac{-\cos\theta_1}{\sqrt{2\pi}}\left(\frac{1}{4}+\frac{1}{2\pi}\arctan \left(\frac{-\sin\theta_1\cot\theta_2+\cos\theta_3\cos\theta_1}{\sin\theta_3}\right)\right)\right]. \end{array} \end{displaymath} Summing up all the terms at all the edges, we obtain the bound \begin{displaymath} \begin{aligned} \displaystyle \varphi(u)\sum_{i=1}^n \frac{\sigma_1(F_i)(\pi-\theta_{3i})}{(2\pi)^{3/2}} & +\displaystyle \varphi(u)\sum_{i=1}^n \sigma_1(F_i)\left[ \frac{-\cos\theta_{2i}}{\sqrt{2\pi}}\left( \frac{1}{4}+\frac{1}{2\pi}\arctan \left(\frac{-\sin\theta_{2i}\cot\theta_{1i}+\cos\theta_{3i}\cos\theta_{2i}}{\sin\theta_{3i}}\right)\right) \right.\\ &\displaystyle \qquad + \left. \frac{-\cos\theta_{1i}}{\sqrt{2\pi}}\left(\frac{1}{4}+\frac{1}{2\pi}\arctan \left(\frac{-\sin\theta_{1i}\cot\theta_{2i}+\cos\theta_{3i}\cos\theta_{1i}}{\sin\theta_{3i}}\right)\right)\right]. \end{aligned} \end{displaymath} By definition, $$\sum_{i=1}^n \sigma_1(F_i) (\pi-\theta_{3i})=4\pi\lambda(S).$$ Now, we prove \begin{displaymath} \begin{aligned} I = & \displaystyle \sum_{i=1}^n l_i \left[ \frac{\cos\theta_{2i}}{\sqrt{2\pi}}\left(\frac{1}{4}+\frac{1}{2\pi}\arctan \left(\frac{-\sin\theta_{2i}\cot\theta_{1i}+\cos\theta_{3i}\cos\theta_{2i}}{\sin\theta_{3i}}\right)\right) \right.\\ &\qquad + \left. \frac{\cos\theta_{1i}}{\sqrt{2\pi}}\left(\frac{1}{4}+\frac{1}{2\pi}\arctan \left(\frac{-\sin\theta_{1i}\cot\theta_{2i}+\cos\theta_{3i}\cos\theta_{1i}}{\sin\theta_{3i}}\right)\right) \right]=0. \end{aligned} \end{displaymath} Indeed, from Lemma \ref{lema}, we have $$ \frac{-\sin\theta_1\cot\theta_2+\cos\theta_3\cos\theta_1}{\sin\theta_3}=\frac{-\cos h}{\sin h}, $$ where $h$ is the dihedral angle at $\overrightarrow{\alpha}$, i.e, the angle between the horizontal plane and the face containing $\overrightarrow{\alpha}$ and $\overrightarrow{\eta}$. Since $h$ is constant for each face, $$ I= \sum_{S \in \{S_1,\ldots,S_k\}} \sum_{l \subset S} l\cos \theta_1 \left(\frac{1}{4}+\frac{1}{2\pi}\arctan\left(\frac{-\cos h}{\sin h}\right)\right)=0. $$ Therefore, we have the following upper bound for the probability of having a point $T$ with minimal third coordinate on the level set and belonging to an edge: \begin{equation}\label{t3} \frac{2\lambda(S)\varphi(u)}{(2\pi)^{1/2}}. \end{equation} From (\ref{t1}), (\ref{t2}), (\ref{t3}) and the fact that $\PP\{X(O)>u\}=\overline{\Phi}(u)$, the result follows. \end{proof} \section{Computation of the absolute value of the determinant of the Hessian matrices}\label{se4} As we see in the proof of Theorem \ref{th3}, we deal with the following
$$ \E(|\det(X''(t))|\mathbb{I}_{X''(t)\preceq 0} \mid X(t)=u,\; X'_1(t)=0,\;X'_2(t)=0). $$ To evaluate this quantity, we have the following statement that is one of our main results in this paper: \begin{theorem} Let $X$ be a standard stationary isotropic centered two-dimensional Gaussian field. One has \begin{equation}\label{eth3}
\E\left(|\det(X''(t))|\mid (X,X'_1,X'_2)(t)=(u,0,0) \right) \, = u^2-1+2\frac{\displaystyle (8\rho''(0) )^{3/2}\exp(-u^2.(24\rho''(0) -2)^{-1})}{\displaystyle \sqrt{24\rho''(0) -2}}. \end{equation} \end{theorem} \begin{proof} Under the condition, the vector $(X''_{11},X''_{12},X''_{22})$ has the same distribution with $(Y_1,Y_2,Y_3)+(-u,0,-u)$, where $(Y_1,Y_2,Y_3)$ is a centered Gaussian vector with the covariance matrix:\\
$$\Sigma = \left( \begin{array}{ccc}
12\rho''(0) -1 & 0 & 4\rho''(0) -1\\
0 & 4\rho''(0) & 0 \\
4\rho''(0) -1& 0 & 12\rho''(0) -1 \\ \end{array} \right).$$ Then, the LHS in (\ref{eth3}) can be written as\\ $$\begin{array}{ll}
& \E(|X''_{11}(t)X''_{22}(t)-X''_{12}(t)^2| \, \mid \, (X,X'_1,X'_2)(t)=(u,0,0) ) \\
=& \E(|(Y_1-u)(Y_3-u)-Y^2_2|)\\
=& \E(|Y_1Y_3-Y^2_2-u(Y_1+Y_3)+u^2|)\\
=& \E(|<Y,AY>+<b,Y>+u^2|), \end{array}$$ where $A=\left(\begin{array}{ccc} 0 & 0 & \frac{1}{2}\\ 0 & -1 & 0 \\ \frac{1}{2} & 0 & 0 \\ \end{array}\right)$ and $b=\left(\begin{array}{c} -u \\ 0 \\ -u \\ \end{array}\right)$.\\ Here, from Theorem 2.1 of \cite{11}, the expectation is equal to
$$\E(|<Y,AY>+<b,Y>+u^2|)=\frac{2}{\pi } \int_0^{\infty }t^{-2}(1-F(t)-\overline{F}(t))dt,$$ where $$F(t)=\frac{\exp(itu^2-2^{-1}t^2<b,(I-2it\Sigma A)^{-1}\Sigma b>)}{2\det(I-2it\Sigma A)^{1/2}}.$$ It is clear that $$F(t)=\frac{\exp(itu^2[1-it(16\rho''(0) -2)]^{-1})}{2(1+8it\rho''(0) )[1-it(16\rho''(0) -2)]^{1/2}},$$ and $$\bar {F}(t)=\frac{\exp(-itu^2[1+it(16\rho''(0) -2)]^{-1})}{2(1-8it\rho''(0) )[1+it(16\rho''(0) -2)]^{1/2}}=F(-t).$$ So, the expectation is equal to $$\begin{array}{rl} \displaystyle \frac{2}{\pi} \int_0^{\infty }\frac{1}{t^2}(1-F(t)-\overline{F}(t))dt=& \displaystyle \textnormal{Re}( \frac{1}{\pi} \int_{-\infty}^{\infty }\frac{1}{t^2}(1-2.F(t))dt)\\
=& \displaystyle \textnormal{Re}(\frac{1}{\pi} \int_{-\infty}^{\infty }\frac{1}{t^2}(1-\frac{\displaystyle \exp(itu^2[1-it(16\rho''(0) -2)]^{-1})}{\displaystyle (1+8it\rho''(0) )[1-it(16\rho''(0) -2)]^{1/2}})dt). \end{array}$$ Here, we apply the residue theorem to compute $$\begin{array}{ll} & \displaystyle \frac{1}{\pi} \int_{-\infty}^{\infty }\frac{1}{t^2}(1-\frac{\displaystyle \exp(itu^2[1-it(16\rho''(0) -2)]^{-1})}{\displaystyle (1+8it\rho''(0) )[1-it(16\rho''(0) -2)]^{1/2}})dt\\ =& 2i . \left(\text{sum of residues in upper half plane}\right) + i. \left(\text{sum of residues on x-axis} \right). \end{array}$$ The residues come from two poles at $i.(8\rho''(0) )^{-1}$ and $0$ and we see that:\\ The residue at $0$ is equal to
$$\frac{d}{dt}\left( 1-\frac{\exp(itu^2[1-it(16\rho''(0) -2)]^{-1})}{(1+8it\rho''(0) )[1-it(16\rho''(0) -2)]^{1/2}}\right) \bigg|_{t=0}=-i.u^2+i.$$ And the residue at $i.(8\rho''(0) )^{-1}$ is equal to \begin{displaymath} \begin{array}{l}
\displaystyle \frac{(1+8it\rho''(0) )[1-it(16\rho''(0) -2)]^{}-\exp(itu^2[1-it(16\rho''(0) -2)]^{-1})}{t^2.8i\rho''(0) .[1-it(16\rho''(0) -2)]^{1/2}} \bigg|_{t=i.(8\rho''(0) )^{-1}}\\ = \displaystyle \frac{(8\rho''(0) )^{3/2}\exp(-u^2.(24\rho''(0) -2)^{-1})}{\sqrt{24\rho''(0) -2}.i}. \end{array} \end{displaymath} These two residues imply the result. \end{proof} We have the corollary \begin{corollary} \label{cor} Let $X$ be a standard stationary isotropic centered Gaussian field. One has
$$\E(|\det(X''(t))|. \mathbb{I}_{X''(t)\preceq 0}\mid (X,X'_1,X'_2)(t)=(u,0,0) ) \leq u^2-1+\frac{\displaystyle (8\rho''(0) )^{3/2}\exp(-u^2.(24\rho''(0) -2)^{-1})}{\displaystyle \sqrt{24\rho''(0) -2}}.$$ \end{corollary} \begin{proof} The result follows from two observations \begin{itemize}
\item $\displaystyle |\det(X''(t))|. \mathbb{I}_{X''(t)\preceq 0} \leq \frac{|\det(X''(t))|+\det(X''(t))}{2}.$ \item $\E(\det(X''(t))\mid (X,X'_1,X'_2)(t)=(u,0,0)) \; = \; u^2-1.$ \end{itemize} \end{proof}
\section{Numerical comparison}\label{se5}
In this section, we compare the upper bounds given by the direct method and record method with the approximation given by the EC method. For simplicity we limit our attention to the case where $S$ is the square $[0,T]^2$ and $X$ is a standard stationary isotropic centered Gaussian field with covariance function $\rho(\|s-t\|^2)$. Note that only $\rho''(0)$ plays a role, the exact form of $\rho$ does not need to be specified. More precisely, we consider
\begin{itemize}
\item[1.] the approximation given by the EC method
$$
\PP_E (u) =
\overline{\Phi}(u)+\frac{2T}{\sqrt{2\pi}} \varphi (u)+\frac{T^2}{2\pi}u\varphi (u);
$$ \item[2.] and the upper bound given by the direct method \begin{align}
\PP_M (u) = \quad \overline{\Phi}(u)\displaystyle &+\frac{2T}{\sqrt{2\pi}} \int_u^{\infty} \left[ c\varphi (x\textrm{/}c)+x\Phi (x\textrm{/c})\right]\varphi (x)dx \notag \\ &\displaystyle + \frac{T^2}{2\pi } \int_u^{\infty} \left[x^2-1+\left(\frac{2(c^2+1)}{3}\right)^{3/2} \sqrt{\pi}\frac{ \varphi(x/c)}{c}\right]\varphi(x)dx, \notag \end{align} where $c=\sqrt{12\rho''(0)-1}$,\\
\item[3.] and the one given by the record method $$\PP_R(u) = \overline{\Phi}(u)+\frac{2T}{\sqrt{2\pi}} \varphi (u)+\frac{T^2}{2\pi }\left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/c})\right]\varphi (u). $$ \end{itemize} It is easy to see that $\PP_E$ is always less than $\PP_R$ and $\PP_M$. We will prove that $\PP_R(u) $ is smaller than $\PP_M (u) $ as $u$ is large. Indeed, if we compare the "dimension 1 terms" (corresponding to $\sigma_1(\partial S)$), we have
\begin{displaymath} \begin{array}{rl} & \int_u^{\infty} \left[ c\varphi (x\textrm{/}c)+x\Phi (x\textrm{/c})\right]\varphi (x)dx - \varphi (u)\\ = & \int_u^{\infty} \left[ c\varphi (x\textrm{/}c)+x\Phi (x\textrm{/c})\right]\varphi (x)dx - \int_u^{\infty} x\varphi (x)dx\\ =& \int_u^{\infty} \left[ c\varphi (x\textrm{/}c)-x\overline{\Phi} (x\textrm{/c})\right]\varphi (x)dx \geq 0, \end{array} \end{displaymath} since when $x\geq 0$, $$\frac{\varphi(x)}{x}\geq \overline{\Phi}(x).$$ So the term in the direct method is always larger when $u\geq 0$.
\\
Let us consider now the two terms corresponding to $\sigma_2(S)$:
\begin{itemize} \item $A_d = u\varphi(u) + \int_u^{\infty} \left[ \left(\frac{2(c^2+1)}{3}\right)^{3/2} \sqrt{\pi}\frac{ \varphi(x/c)}{c}\right]\varphi(x)dx = u \varphi(u) + \overline{A}_d.$
\item $A_r = \left[ c\varphi (u\textrm{/}c)+u\Phi (u\textrm{/c})\right]\varphi (u) = u\varphi(u) +\overline{A}_r.$ \end{itemize} It is easy to show that , as $u\to +\infty$, $$
\overline{A}_d= \int _u ^{\infty} \varphi \bigg( \frac x c\bigg) \varphi(x) dx = (const) \overline{\Phi} \bigg( u \sqrt{ \frac{1+ c^2}{ c^2}}\bigg) \simeq (const) u^{-1} \varphi \bigg( u \sqrt{ \frac{1+ c^2}{ c^2}}\bigg).
$$
and that
$$
\overline{A}_r \simeq (const) u^{-2} \varphi \bigg( u \sqrt{ \frac{1+ c^2}{ c^2}}\bigg).
$$ This shows that for $u$ sufficiently large $ A_r$ is smaller than $ A_d$. \\
The numerical comparison is performed in Figure 4 for six different situations. It shows that the record method is always better than the direct method. EC method and record method are very close, but it is not possible to identify the better among those two since $\PP_E$ can be smaller than the true value.
\begin{figure}\end{figure}
[email protected]\\ [email protected]
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{$\Omega$-theorem for short trigonometric sum}
\author{Jan Moser}
\address{Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA}
\email{[email protected]}
\keywords{Riemann zeta-function}
\begin{abstract} We obtain in this paper new application of the classical E.C. Titchmarsh' discrete method (1934) in the theory of the Riemann $\zf$ - function. Namely, we shall prove the first localized $\Omega$-theorem for short trigonometric sum. This paper is the English version of the work of reference \cite{4}. \end{abstract}
\maketitle
\section{Result}
\subsection{}
In this paper we shall study the following short trigonometric sum \begin{equation} S(t,T,K)=\sum_{e^{-1/K}P_0<n<P_0}\cos(t\ln n),\ P_0=\sqrt{\frac{T}{2\pi}},\ t\in [T,T+U], \end{equation} where \begin{equation} U=T^{1/2}\psi\ln T,\ \psi\leq K\leq T^{1/6}\ln^2T,\ \psi <\ln T, \end{equation} and $\psi=\psi(T)$ stands for arbitrary slowly increasing function unbounded (from above). For example \begin{displaymath} \psi(T)=\ln\ln T,\ \ln\ln\ln T,\ \dots \end{displaymath} Let \begin{displaymath} \{ t_\nu\} \end{displaymath} be the Gram-Titchmarsh sequence defined by the formula \begin{displaymath} \vartheta(t_\nu)=\pi\nu,\quad \nu=1,2,\dots \end{displaymath} (see \cite{6}, pp. 221, 329) where \begin{displaymath} \vartheta(t)=-\frac t2\ln\pi+\mbox{Im}\left\{\ln\Gamma\left(\frac 14+i\frac t2\right)\right\}. \end{displaymath} Next, we denote by \begin{displaymath} G(T,K,\psi) \end{displaymath} the number of such $t_\nu$ that (see (1.1)) obey the following
\begin{equation} t_\nu\in [T,T+U] \ \wedge \ |S(t_\nu,T,K)|>\frac 12 \sqrt{\frac{P_0}{K}}=AT^{1/4}K^{-1/2}. \end{equation} The following theorem holds true.
\begin{mydef1} There are \begin{displaymath} T_0(K,\psi)>0, \ A>0 \end{displaymath} such that \begin{equation} G(T,K,\psi)>AT^{1/6}K^{-1}\psi \ln^2T,\ T\geq T_0(K,\psi). \end{equation} \end{mydef1}
\subsection{}
Let us remind the following estimate by Karatsuba (see \cite{1}, p. 89) \begin{displaymath} \overset{*}{S}(x)=\sum_{1\leq n\leq x}n^{it}=\mathcal{O}(\sqrt{x}t^{\epsilon}),\ 0<x<t \end{displaymath} holds true on the Lindel\" of hypothesis ($0<\epsilon$ is an arbitrary small number in this). Of course, \begin{equation} \begin{split} & \overset{*}{S}(t,T,K)=\sum_{e^{-1/K}P_0\leq n\leq P_0}n^{it}=\mathcal{O}(\sqrt{P_0}T^\epsilon)=\mathcal{O}(T^{1/4+\epsilon}), \\ & t\in [T,T+U], \end{split} \end{equation} and \begin{displaymath}
|\overset{*}{S}(t,T,K)|\geq |S(t,T,K)|. \end{displaymath}
\begin{remark}
Since (see (1.1), (1.3), (1.4)) the inequality
\begin{displaymath}
|S(t,T,\ln\ln T)|> A\frac{T^{1/4}}{\sqrt{\ln\ln T}},\ T\to\infty
\end{displaymath}
is fulfilled for arbitrary big $t$, then the Karatsuba's estimate (1.5) is an almost exact estimate. \end{remark}
\subsection{}
With regard to connection between short trigonometric sum and the theory of the Riemann zeta function see our paper \cite{3}.
\section{Main lemmas and proof of Theorem}
Let \begin{equation} w(t,T,K)=\sum_{e^{-1/K}P_0<n<P_0}\frac{1}{\sqrt{n}}\cos(t\ln n), \end{equation} \begin{equation} w_1(t,T,K)=\sum_{e^{-1/K}P_0<n<P_0}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{P_0}}\right)\cos(t\ln n). \end{equation} The following lemmas hold true.
\begin{mydef5A}
\begin{equation}
\sum_{T\leq t_\nu\leq T+U}w^2(t_\nu,T,K)=\frac{1}{4\pi}UK^{-1}\ln\frac{T}{2\pi}+\mathcal{O}(K^{-1}\sqrt{T}\ln^2T).
\end{equation} \end{mydef5A}
\begin{mydef5B}
\begin{equation}
\sum_{T\leq t_\nu\leq T+U}w_1^2(t_\nu,T,K)=\frac{1}{48\pi}UK^{-3}\ln\frac{T}{2\pi}+\mathcal{O}(K^{-3}\sqrt{T}\ln^2T).
\end{equation} \end{mydef5B}
\begin{remark}
Of course, the formulae (2.3), (2.4) are asymptotic ones (see (1.2)). \end{remark}
Now we use the main lemmas A and B for completion of the Theorem. Since (see \cite{2}, (23)) \begin{equation} Q_0=\sum_{T\leq t_\nu\leq T+U}1=\frac{1}{2\pi}U\ln\frac{T}{2\pi}+\mathcal{O}\frac{U^2}{T}, \end{equation} then we obtain (see (2.3), (2.4)) that \begin{equation} \frac{1}{Q_0}\sum_{T\leq t_\nu\leq T+U}w^2(t_\nu,T,K)\sim \frac{1}{2K},\ T\to\infty, \end{equation} \begin{equation} \frac{1}{Q_0}\sum_{T\leq t_\nu\leq T+U}w_1^2(t_\nu,T,K)\sim \frac{1}{24K^3}, \end{equation} \begin{equation} \frac{1}{Q_0}\sum_{T\leq t_\nu\leq T+U}w\cdot w_1=\mathcal{O}\left(\frac{1}{K^2}\right), \end{equation} (we used the Schwarz inequality in (2.8)). Next, we have (see (1.1), (2.1), (2.2)) that \begin{equation} w(t_\nu,T,K)=\frac{1}{\sqrt{P_0}}S(t_\nu,T,K)+w_1(t_\nu,T,K). \end{equation} Consequently we obtain (see (2.6) -- (2.9)) the following
\begin{mydef6}
\begin{equation}
\frac{1}{Q_0}\sum_{T\leq t_\nu\leq T+U}S^2\sim \frac{P_0}{2K},\ T\to\infty.
\end{equation} \end{mydef6} Next, we denote by $Q_1$ the number of such values \begin{displaymath} t_\nu\in [T,T+U], \end{displaymath} that fulfill the inequality (see (1.3)) \begin{equation}
|S|>\frac{1}{2}\sqrt{\frac{P_0}{K}};\quad Q_1=G(T,K,\psi), \end{equation} and \begin{displaymath} Q_0-Q_1=Q_2 \end{displaymath} (see (2.5)). Since (see (1.1) and \cite{6}, p. 92) \begin{displaymath}
|S(t,T,K)|<A \sqrt{P_0}T^{1/6},\ t\in [T,T+U], \end{displaymath} then we have (see (2.10), (2.11)) that \begin{displaymath} \frac{1}{3K}< AT^{1/3}\frac{Q_1}{Q_0}+\frac{1}{4K}\frac{Q_2}{Q_0}<A T^{1/3}\frac{Q_1}{Q_0}+\frac{1}{4K}, \end{displaymath} i. e. \begin{equation} AQ_1>\frac{1}{12}Q_0T^{-1/3}K^{-1}. \end{equation} Consequently, we obtain (see (1.2), (2.5), (2.11), (2.12)) the following estimate \begin{displaymath} Q_1=G>A T^{1/6}K^{-1}\psi \ln^2T \end{displaymath} that is required result (1.4).
\section{Lemma 1}
Let \begin{equation} w_2=\ssum_{e^{-1/K}P_0<n<m<P_0}\frac{1}{\sqrt{nm}}\cos\left( t_\nu\ln\frac nm\right). \end{equation}
The following lemma holds true.
\begin{mydef51}
\begin{equation}
\sum_{T\leq t_\nu\leq T+U}w_2=\mathcal{O}(K^{-1}\sqrt{T}\ln^2T).
\end{equation} \end{mydef51}
\begin{proof}
The following inner sum (comp. \cite{5}, p. 102; $t_{\nu+1}\longrightarrow t_\nu$)
\begin{equation}
w_{21}=\sum_{T\leq T_\nu\leq T+U}\cos\{ 2\pi\psi_1(\nu)\},
\end{equation}
where
\begin{displaymath}
\psi_1(\nu)=\frac{1}{2\pi}t_\nu\ln\frac nm
\end{displaymath}
applies to our sum (3.2).
Now we obtain by method \cite{5}, pp. 102-103 the following estimate
\begin{equation}
w_{21}=\mathcal{O}\left(\frac{\ln T}{\ln\frac nm}\right).
\end{equation} Since (see (1.2)) \begin{displaymath} e^{-1/K}>1-\frac 1K\geq 1-\frac{1}{\psi}>\frac 12, \end{displaymath} then \begin{displaymath} 2m>2e^{-1/K}P_0>P_0,\quad m\in (e^{-1/K}P_0,P_0), \end{displaymath} i. e. in our case (see (3.1)) we have that \begin{displaymath} 2m>n. \end{displaymath} Consequently, the method \cite{6}, p. 116, $\sigma=\frac 12,\ m=n-r$ gives the estimate \begin{equation} \begin{split}
& \ssum_{e^{-1/K}P_0<n<m<P_0}\frac{1}{\sqrt{mn}\ln\frac nm}< \\
& <
A\sum_{e^{-1/K}P_0<n<P_0}\sum_{r\leq n/2}\frac 1r< AK^{-1}P_0\ln P_0<AK^{-1}\sqrt{T}\ln T, \end{split} \end{equation} where \begin{equation} \sum_{e^{-1/K}P_0<n<P_0} 1\sim \frac{P_0}{K}. \end{equation} Now, required result (3.2) follows from (3.1), (3.3) -- (3.5). \end{proof}
\section{Lemma 2}
Let \begin{equation} w_3=\ssum_{e^{-1/K}P_0<m<n<P_0}\frac{1}{\sqrt{mn}}\cos\{ t_\nu\ln(mn)\}. \end{equation} The following lemma holds true. \begin{mydef52}
\begin{equation}
\sum_{T\leq t_\nu\leq T+U}w_3=\mathcal{O}(K^{-1}\sqrt{T}\ln^2T).
\end{equation} \end{mydef52}
\begin{proof}
The following inner sum (comp. \cite{5}, p. 103; $t_{\nu+1}\longrightarrow t_\nu$)
\begin{displaymath}
w_{31}=\sum_{T\leq t_\nu\leq T+U}\cos\{ 2\pi \chi(\nu)\},
\end{displaymath}
where
\begin{displaymath}
\chi(\nu)=\frac{1}{2\pi}t_\nu\ln(nm)
\end{displaymath}
applies to our sum (4.2). Next, the method \cite{5}, pp. 103-104 gives us that
\begin{equation} \begin{split}
& w_{31}=\int_{\chi'(x)<1/2}\cos\{ 2\pi\chi(x)\}{\rm d}x+ \\
& + \int_{\chi'(x)>1/2}\cos[2\pi\{ \chi(x)-x\}]{\rm d}x+\mathcal{O}(1)=J_1+J_2+\mathcal{O}(1), \end{split} \end{equation} where \begin{displaymath} J_1=\mathcal{O}\left(\frac{\ln T}{\ln n}\right)=\mathcal{O}(1),\ n\in (e^{-1/K}P_0,P_0), \end{displaymath} and ($m<n<2m,\ n=m+r$) \begin{displaymath} J_2=\mathcal{O}\left( \frac{m\ln(m+1)}{r}\right). \end{displaymath} Now, the term $J_1$ contributes to the sum (4.2), (comp. (3.6)) as \begin{equation} \begin{split}
& \mathcal{O}\left(\ssum_{e^{-1/K}P_0<m<n<P_0}\frac{1}{\sqrt{mn}}\right) = \\
& = \mathcal{O}\left( \frac{1}{P_0}\ssum_{e^{-1/K}P_0<m<n<P_0} 1\right)= \\
& = \mathcal{O}\left(\frac{1}{P_0}\frac{P_0^2}{K^2}\right)=\mathcal{O}(K^{-2}\sqrt{T}), \end{split} \end{equation} and the same contribution corresponds to the term $\mathcal{O}(1)$ in (4.3), while the contribution of the term $J_2$ is \begin{equation} \begin{split}
& \mathcal{O}\left(\sum_{e^{-1/K}P_0<m<P_0}\frac{1}{\sqrt{m}}\sum_{r=1}^m\frac{1}{\sqrt{m}}\frac{m\ln(m+1)}{r}\right)= \\
& = \mathcal{O}\left(\frac{P_0}{K}\ln^2P_0\right)=\mathcal{O}(K^{-1}\sqrt{T}\ln^2T). \end{split} \end{equation} Now, the required result (4.2) follows from (4.1), (4.4), (4.5). \end{proof}
\section{Lemma 3}
Let \begin{equation} w_4=\sum_{e^{-1/K}P_0<n<P_0}\frac 1n. \end{equation} The following lemma holds true.
\begin{mydef53} \begin{equation} \sum_{T\leq t_\nu\leq T+U}w_4=\mathcal{O}(K^{-1}\sqrt{T}\ln^2T). \end{equation} \end{mydef53}
\begin{proof} The following inner sum \begin{displaymath} w_{41}=\sum_{T\leq t_\nu\leq T+U}\cos\{ 2\pi\chi_1(\nu)\}, \end{displaymath} where \begin{displaymath} \chi_1(\nu)=\frac{1}{\pi}t_\nu\ln n. \end{displaymath} applies to our sum (5.2). Since (comp. \cite{5}, p. 103) \begin{displaymath} \chi_1'(\nu)=\frac{\ln n}{\vartheta'(t_\nu)}, \end{displaymath} next, (comp. \cite{5}, p. 100) \begin{displaymath} \begin{split}
& \vartheta'(t_\nu)=\frac 12\ln\frac{t_\nu}{2\pi}+\mathcal{O}\left(\frac{1}{t_\nu}\right)=
\frac 12\ln\frac{T}{2\pi}+\mathcal{O}\left(\frac UT\right)+\mathcal{O}\left(\frac 1T\right)\sim \\
& \sim \ln P_0, \end{split} \end{displaymath} and \begin{displaymath} \ln P_0-\frac 1K<\ln n<\ln P_0,\ n\in (e^{-1/K}P_0,P_0), \end{displaymath} then \begin{displaymath} \chi_1'(\nu)\sim 1. \end{displaymath} Since, for example, \begin{displaymath} \frac 12<\chi_1'(\nu)<\frac 32, \end{displaymath} then we have (comp. \cite{5}, p. 104) that \begin{displaymath} w_{41}=\int\cos[2\pi\{\chi_1(x)-x\}]{\rm d}x+\mathcal{O}(1)=J_3+\mathcal{O}(1). \end{displaymath} Now, (comp. \cite{5}, p. 104) \begin{displaymath} \chi_1''(\nu)<-A\frac{\ln n}{T\ln^3T}<-\frac{B}{T\ln^2T},\ n\in (e^{-1/K}P_0,P_0), \end{displaymath} and \begin{displaymath} J_3=\mathcal{O}(\sqrt{T}\ln T),\quad w_{41}=\mathcal{O}(\sqrt{T}\ln T). \end{displaymath} Consequently, we get the required result (5.2) \begin{displaymath} \sum_{T\leq t_\nu\leq T+U}w_4=\mathcal{O}\left(\sqrt{T}\ln T\sum_{e^{-1/K}P_0<n<P_0}\frac 1n\right)= \mathcal{O}(K^{-1}\sqrt{T}\ln T), \end{displaymath} where \begin{displaymath} \sum_{e^{-1/K}P_0<n<P_0}\frac 1n\sim \frac 1K \end{displaymath} by the well-known Euler's formula \begin{displaymath} \sum_{1\leq n<x}\frac 1n=\ln x+c+\mathcal{O}\left(\frac 1x\right), \end{displaymath} where $c$ is the Euler's constant. \end{proof}
\section{Lemmas A and B}
\subsection{Proof of Lemma A}
First of all, we have (see (2.1)) that \begin{equation} \begin{split} & w^2(t_\nu,T,K)=\\ & = \ssum_{e^{-1/K}P_0<m,n<P_0}\frac{1}{\sqrt{mn}}\cos( t_\nu\ln m)\cos( t_\nu\ln n)= \\ & = \frac 12 \sum_n \frac 1n+\ssum_{m<n}\frac{1}{\sqrt{mn}}\cos\left( t_\nu\ln\frac nm\right)+ \\ & + \ssum_{m<n}\frac{1}{\sqrt{mn}}\cos\{ t_\nu\ln (mn)\}+\frac 12\sum_{n}\frac 1n\cos( 2t_\nu\ln n)= \\ & = \frac{1}{2K}+\mathcal{O}\left(\frac{1}{\sqrt{T}}\right)+w_2+w_3+w_4, \end{split} \end{equation} (see (5.3), (3.1), (4.1), (5.1)). Consequently, we obtain the required result (2.3) from (6.1) by (2.5), (3.2), (4.2), (5.2).
\subsection{Proof of Lemma B} First of all we have (see (2.2)) that \begin{displaymath} w_1(t_\nu,T,K)=\sum_{e^{-1/K}P_0<n<P_0}\frac{\alpha(n)}{\sqrt{n}}\cos(t_\nu\ln n), \end{displaymath} where \begin{displaymath} \alpha(n)=1-\sqrt{\frac{n}{P_0}}. \end{displaymath} Of course, $\alpha(n)$ is decreasing and \begin{equation} 0<\alpha(n)<\frac 1K,\quad n\in (e^{-1/K}P_0,P_0). \end{equation} Next, (comp. (6.1)) \begin{equation} \begin{split} & w_1^2(t_\nu,T,K)= \\ & = \frac 12\sum_n\frac{\alpha^2(n)}{n}+\ssum_{m<n}\frac{\alpha(m)\alpha(n)}{\sqrt{mn}}\cos\left( t_\nu\ln\frac nm\right)+ \\ & + \ssum_{m<n}\frac{\alpha(m)\alpha(n)}{\sqrt{mn}}\cos\left( t_\nu\ln(mn)\right)+\frac 12\sum_n \frac{\alpha^2(n)}{n}\cos( 2t_\nu\ln n)= \\ & = \frac 12\bar{w}_1+\bar{w}_2+\bar{w}_3+\frac 12\bar{w}_4. \end{split} \end{equation} Since (see (6.2)) \begin{displaymath} \alpha(m)\alpha(n)<K^{-2} \end{displaymath} then we obtain by a similar way as in the case of the estimates (3.2), (4.2) and (5.2) that \begin{equation} \sum_{T\leq t_\nu\leq T+U}\left\{ \bar{w}_2+\bar{w}_3+\frac 12\bar{w}_4\right\}= \mathcal{O}(K^{-3}\sqrt{T}\ln^2T). \end{equation} In the case of the sum \begin{equation} \frac 12\sum_{T\leq t_\nu\leq T+U}\bar{w}_1 \end{equation} we use the following summation formula (see \cite{6}, p. 13) \begin{displaymath} \begin{split} & \sum_{a\leq n<b}\varphi(n)=\int_a^b \varphi(x){\rm d}x+\int_a^b \left( x-[x]-\frac 12\right)\varphi'(x){\rm d}x+ \\ & + \left( a-[a]-\frac 12\right)\varphi(a)-\left( b-[b]-\frac 12\right)\varphi(b) \end{split} \end{displaymath} in the case \begin{displaymath} a=e^{-1/K}P_0, b=P_0, \varphi(x)=\frac{\alpha^2(x)}{x}=\frac 1x-\frac{2}{\sqrt{P_0x}}+\frac{1}{P_0}. \end{displaymath} Hence \begin{displaymath} \begin{split} & \int_{e^{-1/K}P_0}^{P_0}\frac{\alpha^2(x)}{x}{\rm d}x=\frac{1}{K}-4\left( 1-e^{-1/(2K)}\right)+1-e^{-1/K}= \\ & = \frac{1}{12K^3}+\mathcal{O}(K^{-4}), \end{split} \end{displaymath} and \begin{displaymath} \varphi'(x)=\mathcal{O}(P_0^{-2}),\ \varphi(e^{-1/K}P_0)=\mathcal{O}\left(\frac{1}{x^2P_0}\right),\ \varphi(P_0)=0. \end{displaymath} Consequently, we have (see (6.5)) \begin{displaymath} \frac 12\bar{w}_1=\frac{1}{24K^3}+\mathcal{O}\left(\frac{1}{K^4}\right), \end{displaymath} and (see (1.2), (2.5)) \begin{equation} \begin{split} & \frac 12\sum_{T\leq t_\nu\leq T+U}\bar{w}_1=\frac{1}{48\pi}UK^{-3}\ln\frac{T}{2\pi}+ \mathcal{O}\left(\frac{U\ln T}{K^4}\right)+\mathcal{O}\left(\frac{U^2}{K^3T}\right)= \\ & = \frac{1}{48\pi}UK^{-3}\ln\frac{T}{2\pi}+\mathcal{O}(K^{-3}\sqrt{T}\ln T). \end{split} \end{equation} Finally, we obtain the required result (2.4) from (6.3) by (6.4) -- (6.6).
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Mean value formulas for classical solutions to uniformly parabolic equations in divergence form} \author{{\sc{Emanuele Malagoli} \thanks{Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit\`{a} degli Studi di Modena e Reggio Emilia, via Campi 213/b, 41125 Modena (Italy). E-mail: [email protected]}, \ \sc{Diego Pallara} \thanks{Dipartimento di Matematica e Fisica ``Ennio De Giorgi'', Universit\`{a} del Salento and INFN, Sezione di Lecce, Ex Collegio Fiorini - Via per Arnesano - Lecce (Italy). E-mail: [email protected]} \ \sc{Sergio Polidoro} \thanks{Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit\`{a} degli Studi di Modena e Reggio Emilia, Via Campi 213/b, 41125 Modena (Italy). E-mail: [email protected]} }}
\date{ }
\maketitle
\begin{abstract} We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in divergence form. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with {\em almost $C^1$-boundary}, the other stated for sets with finite perimeter. \end{abstract}
\setcounter{equation}{0} \section{Introduction}\label{secIntro} Let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$. We consider classical solutions $u$ to the equation $\L u = f$ in $\Omega$, where $\L$ is a parabolic operator in divergence form defined for $z=(x,t) \in \mathbb{R}^{N+1}$ as follows \begin{equation} \label{e-L}
\L u (z) := \sum_{i,j=1}^{N}\tfrac{\partial}{\partial x_i} \left( a_{ij} (z) \tfrac{\partial u}{\partial x_j}(z) \right) +
\sum_{i=1}^{N} b_{i} (z) \tfrac{\partial u}{\partial x_i}(z) + c(z) u(z) - \, \tfrac{\partial u}{\partial t}(z). \end{equation} In the following we use the notation $A(z) := \left( a_{ij}(z) \right)_{i,j=1,\dots, N}, b(z) := \left( b_{1} (z), \dots, b_{N} (z) \right)$ and we write $\L u $ in the short form \begin{equation} \label{e-LL}
\L u (z) := \div \left( A (z) \nabla_x u(z) \right) + \langle b(z), \nabla_x u(z)\rangle + c(z) u(z) -
\, \tfrac{\partial u}{\partial t}(z). \end{equation} Here $\div, \nabla_x$ and $\langle \, \cdot \, , \, \cdot \, \rangle$ denote the divergence, the gradient and the inner product in $\mathbb{R}^N$, respectively. We assume that the matrix $A(z)$ is symmetric and that the coefficients of the operator $\L$ are H\"older continuous functions with respect to the parabolic distance. This means that there exist two constants $M>0$ and $\alpha \in ]0,1]$, such that \begin{equation} \label{e-hc}
|c (x,t) - c (y,s)| \le M \left(|x-y|^\alpha + |t-s|^{\alpha/2}\right), \end{equation} for every $(x,t), (y,s) \in \mathbb{R}^{N+1}$. We require that the above condition is satisfied not only by $c$, but also by $a_{ij}, \frac{\partial a_{ij}}{\partial x_i}, b_{i}, \frac{\partial b_{i}}{\partial x_i}$, for $i, j = 1, \dots, N$, with the same constants $M$ and $\alpha$. We finally assume that the coefficients of $\L$ are bounded and that $\L$ is uniformly parabolic, \emph{i.e.}, there exist two constants $\lambda, \Lambda$, with $0 < \lambda < \Lambda$, such that \begin{equation} \label{e-up}
\lambda |\xi|^2 \le \langle A(z) \xi, \xi \rangle \le \Lambda |\xi|^2,
\quad \left|\tfrac{\partial a_{ij}}{\partial x_i}\right| \le \Lambda,
\quad |b_i(z)| \le \Lambda, \quad |c(z)| \le \Lambda, \end{equation} for every $\xi \in \mathbb{R}^N$, for every $z \in \mathbb{R}^{N+1}$, and for $i,j=1, \dots, N$. Under the above assumptions, the classical parametrix method provides us with the existence of a fundamental solution $\Gamma$. In Section \ref{secFundSol} we shall quote from the monograph of Friedman \cite{Friedman} the results we need for our purposes.
The main achievements of this note are some mean value formulas for the solutions to $\L u = f$ that are written in terms of the level and super-level sets of the fundamental solution $\Gamma$. We extend previous results of Fabes and Garofalo \cite{FabesGarofalo} and Garofalo and Lanconelli \cite{GarofaloLanconelli-1989} in that we weaken the regularity requirement on the coefficients of $\L$ that in \cite{FabesGarofalo, GarofaloLanconelli-1989} are assumed to be $C^\infty$ smooth. As applications of the mean value formulas we give an elementary proof of the parabolic strong maximum principle.
We note that the conditions on the functions $\frac{\partial a_{ij}}{\partial x_i}$'s are needed in order to deal with classical solutions to the adjoint equation $\L^* v = 0$, as the mean value formulas rely on the divergence theorem applied to the function $(\x,\tau) \mapsto \Gamma(x,t,\xi,\tau)$.
We introduce some notation in order to state our main results. For every $z_0=(x_0, t_0) \in \mathbb{R}^{N+1}$ and for every $r>0$, we set \begin{equation} \label{e-Psi} \begin{split}
\psi_r(z_0) & := \left\{ z \in \mathbb{R}^{N+1} \mid \Gamma(z_0; z ) = \tfrac{1}{r^N} \right\},
\\
\Omega_r(z_0) & := \left\{ z \in \mathbb{R}^{N+1} \mid \Gamma(z_0; z) > \tfrac{1}{r^N} \right\}. \end{split} \end{equation}
\begin{center} \begin{tikzpicture} \clip (-.51,7.21) rectangle (6.31,1.99); \path[draw,thick] (-.5,7.2) rectangle (6.3,2); \begin{axis}[axis x line=middle, axis y line=middle,
xtick=\empty,ytick=\empty,
ymin=-1.2, ymax=1.2, xmin=-.2,xmax=1.7, samples=121, rotate= -90] \addplot [ddddblue,line width=.7pt,domain=.001:.1111] {sqrt(- 3 * x * ln(9*x)}; \addplot [ddddblue,line width=.7pt,domain=.001:.1111] {- sqrt(- 3 * x * ln(9*x))}; \addplot [dddblue,line width=.7pt,domain=.001:.25] {sqrt(- 2 * x * ln(4*x)}; \addplot [dddblue,line width=.7pt,domain=.001:.25] {- sqrt(- 2 * x * ln(4*x))}; \addplot [ddblue,line width=.7pt, domain=.001:1] {sqrt(- 2 * x * ln(x))}; \addplot [ddblue,line width=.7pt,domain=.001:1] {- sqrt(- 2 * x * ln(x))}; \addplot [black,line width=1pt, domain=-.01:.01] {sqrt(.0001 - x * x)} node[above] {\quad $z_0$}; \addplot [black,line width=1pt, domain=-.01:.01] {-sqrt(.0001 - x * x)} node[below]{\qquad \qquad \qquad \qquad \quad \quad \quad \quad {\color{ddddblue} $\psi_r(z_0)$}}; \end{axis} \draw [<-,line width=.4pt] (2.8475,7) -- (2.8475,2); \end{tikzpicture}
{{\sc Fig.1}} - $\psi_r(z_0)$ for three different values of $r$. \end{center}
Similarly to the elliptic case, we call $\psi_r(z_0)$ and $\Omega_r(z_0)$ respectively the \emph{parabolic sphere} and the \emph{parabolic ball} with radius $r$ and ``center'' at $(x_0,t_0)$. Note that, unlike the elliptic setting, $z_0$ belongs to the topological boundary of $\Omega_r(z_0)$. Because of the properties of the fundamental solution of uniformly parabolic operators, the parabolic balls $\Omega_r(z_0)$ are bounded sets and shrink to the center $z_0$ as $r \to 0$. We finally introduce the following kernels \begin{equation} \label{e-kernels} \begin{split}
K (z_0; z) & :=
\frac{\langle A(z) \nabla_x \Gamma(z_0; z), \nabla_x \Gamma(z_0; z) \rangle }
{|\nabla_{(x,t)}\Gamma(z_0;z)|},
\\
M(z_0; z) & :=
\frac{\langle A(z) \nabla_x \Gamma(z_0; z), \nabla_x \Gamma(z_0; z) \rangle }
{\Gamma(z_0; z)^2}. \end{split} \end{equation}
Here $\nabla_x \Gamma(z_0; z)$ and $|\nabla_{(x,t)}\Gamma(z_0; z)|$ denote the gradient with respect to the space variable $x$ and the norm of the gradient with respect to the variables $(x,t)$ of $\Gamma$, respectively. Moreover, we agree to set $K (z_0; z) = 0$ whenever $\nabla_{(x,t)}\Gamma(z_0;z)=0$. In the following, $\H^{N}$ denotes the $N$-dimensional Hausdorff measure. The first achievements of this note are the following mean value formulas.
\begin{theorem} \label{th-1}
Let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$, $f\in C(\Omega)$ and let $u$ be a classical solution to $\L u = f$ in $\Omega$. Then, for every $z_0 \in \Omega$ and for almost every $r>0$ such that $\overline{\Omega_r(z_0)} \subset \Omega$ we have
\begin{align*}
u(z_0) = \int_{\psi_r(z_0)} K (z_0; z) u(z) \, d \H^{N} (z)
+ & \int_{\Omega_r(z_0)} f (z) \left( \tfrac{1}{r^N} - \Gamma(z_0; z) \right)\ dz \\
+ & \frac{1}{r^N} \int_{\Omega_r(z_0)} \left( \div \, b(z) - c(z) \right) u(z) \ dz,
\\
u(z_0) = \frac{1}{r^N} \int_{\Omega_r(z_0)} \!\!\!\!\! M (z_0; z) u(z) \, dz
+ \frac{N}{r^N} & \int_0^{r} \left(\r^{N-1}
\int_{\Omega_\r (z_0)} f (z) \left( \tfrac{1}{\r^N} - \Gamma(z_0;z) \right) dz \right) d \r \\
+ & \frac{N}{r^N} \int_0^{r} \left(\frac{1}{\r}
\int_{\Omega_\r (z_0)} \left( \div \, b(z) - c(z) \right) u(z) \, dz \right) d \r. \end{align*} The second statement holds for \emph{every} $r>0$ such that ${\Omega_r(z_0)} \subset \Omega$. \end{theorem}
Note that $\tfrac{1}{r^N} - \Gamma(z_0; z) < 0$ in the set $\Omega_r(z_0)$, because of its very definiton \eqref{e-Psi}. This fact, together with the non-negativity of the kernels \eqref{e-kernels} will be used in the sequel to obtain the strong maximum principle from Theorem \ref{th-1}.
We next put Theorem \ref{th-1} in its context. It restores the mean value formulas first proved by Pini in \cite{Pini1951} for the heat equation $\partial_t u = \partial_x^2 u$, then by Watson in \cite{Watson1973} for the heat equation in several space variables. We also recall the mean value formulas first proved by Fabes and Garofalo in \cite{FabesGarofalo} for the equation $\L u = 0$, then extended by Garofalo and Lanconelli \cite{GarofaloLanconelli-1989} to the equation $\L u = f$, where the operator $\L$ has the form \eqref{e-L} and its coefficients are assumed to be $C^\infty$ smooth. This extra regularity assumption on the coefficients of $\L$ is due to the fact that the mean value formula relies on the divergence theorem applied to the parabolic ball $\Omega_r(z_0)$. Since the explicit epression of the fundamental solution $\Gamma$ is not available when the coefficients of $\L$ are variable, the authors of \cite{FabesGarofalo} and \cite{GarofaloLanconelli-1989} rely on the Sard theorem (see \cite{Sard}) which guarantees that $\psi_r(z_0)$ is a manifold for \emph{almost every} positive $r$, provided that the fundamental solution $\Gamma$ is $N+1$ times differentiable. The smoothness of the coefficients of the operator $\L$ is used in \cite{FabesGarofalo} and \cite{GarofaloLanconelli-1989} in order to have the needed regularity on $\Gamma$.
The main goal of this note is the restoration of natural regularity hypotheses for the existence of classical solutions to $\L u =f$. These assumptions can be further weakened, since the existence of a fundamental solution has been proved for operators with Dini continuous coefficients. We prefer to keep our treatment in the usual setting of H\"older continuous functions for the sake of simplicity. The unnecessary regularity conditions on the coefficients of $\L$ can be removed in two ways. Following an approach close to the classical one, it is possible to rely on a result due to Dubovicki\u{\i} \cite{Dubovickii} (see also Bojarski, Ha{j\l}asz, and Strzelecki \cite{BojHajStrz}) which allows to reduce the regularity requirement on $\Gamma$ in order to apply a generalized divergence theorem for {\em sets with almost $C^1$ boundary}. This is presented in Section \ref{SectionDivergence} and applied in Section \ref{sectionProof}. The other approach relies on geometric measure theory and is presented in the last section: we show how the proof of Theorem \ref{th-1} can be modified relying on the generalized divergence theorem proved by De Giorgi \cite{DeGiorgi2,DeGiorgi3} in the framework of finite perimeter sets. As said before, this deep theory is not necessary in the present context, but it is more flexible and its generalization to Carnot groups (where the analogue of Dubovicki\u{\i}'s Theorem is not available) will allow us to extend the results of the present paper to {\em degenerate} parabolic operators. We have presented the application to uniformly parabolic operators to pave the way to this generalization, which will be the subject of a forthcoming paper.
The mean value formulas stated in Theorem \ref{th-1} provide us with a simple proof of the strong maximum (minimum) principle for the operator $\L$ when $c = 0$. Note that, in this case, the constant function $u(x,t) = 1$ is a solution to $\L u = 0$, so that the mean value formula gives $\frac{1}{r^N}\int_{\Omega_r(z_0)}M(z_0;z)dz=1$. In order to state this result we first introduce the notion of \emph{attainable set}. We say that a curve $ \g: [0,T] \rightarrow \mathbb{R}^{N+1}$ is \emph{$\L$-admissible} if it is absolutely continuous and \begin{equation*}
\dot{\g}(s) = \left( \dot {x}_1(s), \dots, \dot {x}_N(s), -1 \right) \end{equation*} for almost every $s \in [0,T]$, with $\dot{x}_1, \dots, \dot{x}_N \in L^{2}([0,T])$.
\begin{definition} \label{def-prop-set} Let $\O$ be any open subset of $\mathbb{R}^{N+1}$, and let $z_0 \in \O$. The \emph{attainable set} is \begin{equation*}
\AS ( \O ) =
\begin{Bmatrix}
z \in \O \mid \hspace{1mm} \text{\rm there exists an} \ \L - \text{\rm admissible curve}
\ \g : [0,T] \rightarrow \O \hspace{1mm} \\
\text{\rm such that} \ \g(0) = z_0 \hspace{1mm} {\rm and}
\hspace{1mm} \g(T) = z
\end{Bmatrix}. \end{equation*} Whenever there is no ambiguity on the choice of the set $\O$ we denote $\AS = \AS ( \O )$. \end{definition}
\begin{proposition} \label{prop-smp} Let $\O$ be any open subset of $\mathbb{R}^{N+1}$, and suppose that $c = 0$. Let $z_0=(x_0,t_0) \in \O$ and let $u$ be a classical solution to $\L u = f$. If $u (z_0) = \max_\Omega u$ and $f \ge 0$ in $\Omega$, then \begin{equation*} u(z) = u(z_0) \quad \text{and} \quad f(z) = 0 \qquad \text{for every} \ z \in \overline {\AS ( \O )}. \end{equation*} The analogous result holds true if $u (z_0) = \min_\Omega u$ and $f \le 0$ in $\Omega$. \end{proposition}
If we remove the assumption $c =0$ we obtain the following weaker result.
\begin{proposition} \label{prop-smp1} Let $\O$ be any open subset of $\mathbb{R}^{N+1}$. Let $u \le 0$ ($u \ge 0$, respectively) be a classical solution to $\L u = f$ with $f \ge 0$ ($f \le 0$, respectively) in $\Omega$. If $u (z_0) = 0$ for some $z_0 \in \Omega$, then \begin{equation*} u(z) = 0 \quad \text{and} \quad f(z) = 0 \qquad \text{for every} \ z \in \overline {\AS ( \O )}. \end{equation*}
\end{proposition}
In the remaning part of this intoduction we focus on some modified mean value formulas useful in the proof of parabolic Harnack inequality. As already noticed, the main difficulty one encounters in the proof of the Harnack inequality is due to the unboundedness of the kernels introduced in \eqref{e-kernels}. In order to overcome this issue, we can rely on the idea introduced by Kupcov in \cite{Kupcov4}, and developed by Garofalo and Lanconelli in \cite{GarofaloLanconelli-1989} in the case of parabolic operators with smooth coefficients. This method provides us with some bounded kernels and gives us a useful tool for a direct proof of the Harnack inequality. We outline here the procedure. Let $m$ be a positive integer, and let $u$ be a solution to $\L u = f$ in $\mathbb{R}^{N+1}$. We set
\begin{equation*}
\widetilde u(x,y,t) := u(x,t), \qquad \widetilde f(x,y,t) := f(x,t), \qquad (x,y,t) \in \mathbb{R}^{N}\times \mathbb{R}^{m} \times \mathbb{R}, \end{equation*} and we note that \begin{equation*}
\widetilde \L \ \widetilde u(x,y,t) = \widetilde f(x,y,t) \qquad \widetilde \L
= \L + \sum_{j=1}^{m}\tfrac{\partial^2}{\partial y^2_j} = \L + \Delta_y. \end{equation*} Moreover, if $\Gamma$ and $K_m$ denote fundamental solutions of $\L$ and of the heat equation in $\mathbb{R}^m$, respectively, then the function \begin{equation*}
\widetilde \Gamma(\xi, \eta, \tau ;x,y,t) = \Gamma (\xi, \tau ;x,t) K_m (\eta,\tau ;y,t) \end{equation*} is a fundamental solution of $\widetilde \L$. Then, integrating with respect to $y$ in the mean value formulas of Theorem \ref{th-1}, applied to $\widetilde u$ and to the operator $\widetilde \L$, gives new kernels, that are bounded whenever $m>2$. We intoduce further notations. \begin{equation} \label{e-Omegam} \begin{split}
\Omega^{(m)}_r(z_0) := & \left\{ z \in \mathbb{R}^{N+1} \mid (4 \pi (t_0-t))^{-m/2}\Gamma(z_0; z) >
\tfrac{1}{r^{N+m}} \right\}, \\
N_r(z_0;z) := & 2 \sqrt{t_0-t}\sqrt{\log\left(\tfrac{r^{N+m}}{ (4 \pi (t_0-t))^{m/2}} \Gamma(z_0;z) \right)},
\\
M_r^{(m)} (z_0;z):= & \omega_m N_r^m(z_0;z) \left( M(z_0;z) + \frac{m}{m+2} \cdot \frac {N_r^2(z_0;z)}{4(t_0-t)^2} \right),\\
W_r^{(m)} (z_0;z):= & \frac{\omega_m}{r^{N+m}} N_r^m(z_0;z) - \frac{m}{2} \cdot \frac{\omega_m}{(4 \pi)^{m/2}} \,
\Gamma(z_0,z) \cdot \widetilde \gamma \left(\frac{m}{2}; \frac {N_r^2(z_0;z)}{4(t_0-t)} \right), \end{split} \end{equation} where $M(z_0;z)$ is the kernel introduced in \eqref{e-kernels}, $\omega_m$ denotes the volume of the $m$-dimensional unit ball
and $\widetilde \gamma$ is the lower incomplete gamma function \begin{equation*}
\widetilde \gamma(s;w) := \int_0^w \tau^{s-1} e^{- \tau} d \tau. \end{equation*} Note that the function $N(z_0,z)$ is well defined for every $z \in \Omega^{(m)}_r(z_0)$, as the argument of the logarithm is positive, and that we did not point out the dependence of $N_r$ on the space dimension $m$ to avoid a possible confusion with its powers appearing in the definitions of $M_r^{(m)}$ and $W_r^{(m)}$. \begin{proposition} \label{prop-2}
Let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$, and let $u$ be a classical solution to $\L u = f$ in $\Omega$. Then, for every $z_0 \in \Omega$ and for every $r>0$ such that ${\Omega^{(m)}_r(z_0)} \subset \Omega$ we have
\begin{equation*}
\begin{split}
u(z_0) = & \frac{1}{r^{N+m}} \int_{\Omega^{(m)}_r(z_0)} \! \! \! \! \! \! M_r^{(m)} (z_0; z) u(z) \, dz \,
\\
& + \frac{N+m}{r^{N+m}} \int_0^{r} \left( \r^{N+m-1} \int_{\Omega_\r^{(m)} (z_0)}
\! \! \! \! W_\r^{(m)} (z_0;z) f (z) \, dz \right) d \r \\
& + \frac{N+m}{r^{N+m}} \int_0^{r} \left( \frac{\omega_m}{\r} \int_{\Omega_\r^{(m)} (z_0)} \! \! \! \! N_\r^m(z_0;z)
\left( \div \, b(z) - c(z) \right) u(z) \, dz \right) d \r.
\end{split} \end{equation*} \end{proposition}
We conclude this introduction with two statements of the parabolic Harnack inequality. The first one is given in terms of the parabolic ball $\Omega_{r}^{(m)}(z_0)$, the second one is the usual invariant parabolic Harnack inequality. We emphasize that our proof is elementary, as it is based on the mean value formula, however some accurate estimates of the fundamental solution are needed in order to control the Harnack constant and the size of the cylinders appearing in its statement. For every $z_0 = (x_0,t_0) \in \mathbb{R}^{N+1}, r >0$, and $m \in \mathbb{N}$ we set \begin{equation} \label{e-Kr}
K^{(m)}_{r}(z_0) := \overline{\Omega_{r}^{(m)}(z_0)} \cap \Big\{ t \le t_0 - \frac{1}{4 \pi \lambda^{N/(N+m)}} \,r^2 \Big\}. \end{equation} We note that, as a consequence of Lemma \ref{lem-localestimate*} below, for every sufficiently small $r$ the compact set $K^{(m)}_{r}(z_0)$ is non empty.
\begin{proposition} \label{prop-Harnack} For every $m \in \mathbb{N}$ with $m > 2$, there exist two positive constants $r_0$ and $C_K$, only depending on $\L$ and $m$, such that the following inequality holds. Let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$. For every $z_0 \in \Omega$ and for every positive $r$ such that $r \le r_0$ and $\Omega_{5r}^{(m)}(z_0) \subset \Omega$ we have that
\begin{equation} \label{e-H0}
\sup_{K^{(m)}_{r}(z_0)} u \le C_K u(z_0) \end{equation} for every $u \ge 0$ solution to $\L u = 0$ in $\Omega$. \end{proposition}
We introduce some further notation in order to state an \emph{invariant} Harnack inequality. For every $z_0 = (x_0,t_0) \in \mathbb{R}^{N+1}$ and for every $r>0$ we set \begin{equation} \label{e-Q}
\Q_{r}(z_0) := B_r(x_0) \times ]t_0- r^2,t_0[, \end{equation} where $B_r(x_0)$ denotes the Euclidean ball with center at $x_0$ and radius $r$. Moreover, for $0 < \iota < \kappa < \mu < 1$ and $0 < \vartheta < 1$ we set \begin{equation} \label{e-QPM}
\Q^-_{r}(z_0) := B_{\vartheta r}(x_0) \times ]t_0 - \kappa r^2,t_0 - \mu r^2[, \qquad
\Q^+_{r}(z_0) := B_{\vartheta r}(x_0) \times ]t_0 - \iota r^2,t_0[. \end{equation} We have
\begin{theorem} \label{th-Harnack-inv} Choose positive constants $R_0$ and $\iota, \kappa, \mu, \vartheta$ as above and let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$. Then there exists a positive constant $C_H$, only depending on $\L$, on $R_0$ and on the constant that define the cylinders $\Q, \Q^+, \Q^-$, such that the following inequality holds. For every $z_0 \in \Omega$ and for every positive $r$ such that $r \le R_0$ and ${\Q_{r}(z_0)} \subset \Omega$ we have that
\begin{equation} \label{e-H1}
\sup_{\Q^-_{r}(z_0)} u \le C_H \inf_{\Q^+_{r}(z_0)} u \end{equation} for every $u \ge 0$ solution to $\L u = 0$ in $\Omega$. \end{theorem}
\begin{center} \begin{tikzpicture} \path[draw,thick] (-1,.7) rectangle (9,6.8); \filldraw [fill=blue!20!white, draw=blue, line width=.6pt] (1.5,6) rectangle node {{\color{blue} $Q^+_r(x_0,t_0)$}} node [below=.7cm,right=1.95cm] {{\color{blue} $t_0 - \iota r^2$}} (5.5,4.5); \filldraw [fill=red!20!white, draw=red, line width=.6pt] (1.5,2) rectangle node {{\color{red} $Q^-_r(x_0,t_0)$}} node [above=.7cm,right=1.95cm] {{\color{red} $t_0 - \kappa r^2$}} node [below=.7cm,right=1.95cm] {{\color{red} $t_0 - \mu r^2$}} (5.5,3.5); \draw [line width=.6pt] (0,6) rectangle node [above=2cm,right=3.6cm] {$Q_r(x_0,t_0)$} node [below=2.2cm,right=3.6cm] {$t_0-r^2$} (7,1); \draw [line width=.6pt] (3.5,6) circle (.6pt) node[above] {$(x_0,t_0)$} node[above= 7pt, right=1.2cm] {{\color{blue}$\vartheta r$}} node[above=7pt, right=2.8cm] {$r$}; \end{tikzpicture}
{\sc \qquad Fig.2} - The set $Q_r(x_0,t_0)$. \end{center}
We conclude this introduction with some comments about our main results. Mean value formulas don't require the uniqueness of the fundamental solution $\Gamma$. In Section \ref{secFundSol} we recall the main results we need on the existence of a fundamental solution together with some known facts about its uniqueness. We also recall in Proposition \ref{prop-localestimate} an asymptotic bound of $\Gamma$ which allows us to use a direct procedure in a part of the proof of the mean value formulas stated in Theorem \ref{th-1}. We point out that recent progresses on mean value formulas and their applications can be found e.g. in \cite{CMPCS}. Moreover, an alternative and more general approach has been introduced by Cupini and Lanconelli in \cite{CupiniLanconelli2021}, where a wide family of differential operators with smooth coefficients is considered. We continue the outline of this article. Section \ref{SectionDivergence} contains the statement of a generalized divergence theorem for {sets with almost $C^1$ boundary} that is used in Section \ref{sectionProof} for the proof of the mean values formulas. Section \ref{sectionHarnack} is devoted to the proof of the Harnack inequality. In the last section we present an alternative approach for the mean value formula that relies on geometric measure theory.
We finally remark that our method also applies to uniformly elliptic equations. Moreover, mean value formulas and Harnack inequality are fundamental tools in the development of the Potential Theory for the operator $\L$.
\setcounter{equation}{0} \section{Fundamental solution}\label{secFundSol}
In this Section we recall some notations and some known results on the classical theory of uniformly parabolic equations that will be used in the sequel. Points of $\mathbb{R}^{N+1}$ are denoted by $z=(x,t), \z = (\x,\t)$ and $\Omega$ denotes an open subset of $\mathbb{R}^{N+1}$.
Let $u$ be a real valued function defined on $\Omega$. We say that $u$ belongs to $C^{2,1}(\O)$ if $u, \frac{\partial u}{\partial x_j}, \frac{\partial^2 u}{\partial x_i\partial x_j}$ for $i,j= 1, \dots, N$ and $\frac{\partial u}{\partial t}$ are continuous functions, it belongs to $C^{2+ \alpha,1 + \alpha/2}(\O)$ if $u$ and all the derivatives of $u$ listed above belong to the space $C^{\alpha}(\O)$ of the H\"older continuous functions defined by \eqref{e-hc}. A function $u$ belongs to $C^\alpha_\loc(\Omega)$ ($C^{2+ \alpha,1 + \alpha/2}_\loc(\O)$, respectively) if it belongs to $C^\alpha(K)$ (resp. $C^{2+ \alpha,1 + \alpha/2}(K)$) for every compact set $K \subset \O$. Let $f$ be a continuous function defined on $\O$. We say that $u \in C^{2,1}(\O)$ is a classical solution to $\L u = f$ in $\Omega$ if the equation \eqref{e-L} is satisfied at every point $z \in \O$.
According to Friedman \cite{Friedman}, we say that a fundamental solution $\G$ for the operator $\L$ is a function $\G = \G(z; \z)$ defined for every $(z; \z) \in \mathbb{R}^{N+1} \times \mathbb{R}^{N+1}$ with $t> \tau$, which satisfies the following contitions: \begin{enumerate}
\item For every $\z = (\xi, \t) \in \mathbb{R}^{N+1}$ the function $\G( \, \cdot \, ; \z)$ belongs to
$C^{2,1}(\mathbb{R}^{N} \times ]\t, + \infty[)$ and is a classical solution to
$\L \, \G(\cdot \,; \z) = 0$ in $\mathbb{R}^{N} \times ]\t, + \infty[$;
\item for every $\phi \in C_c(\mathbb{R}^N)$ the function $$
u(z)=\int_{\mathbb{R}^{N}}\Gamma(z;\xi, \t)\phi(\x)d \x, $$ is a classical solution to the Cauchy problem \begin{equation*} \left\{
\begin{array}{ll}
\L u = 0, & \hbox{$z \in \mathbb{R}^{N} \times ]\t, + \infty[$} \\
u( \cdot,\t) = \varphi & \hbox{in} \ \mathbb{R}^N.
\end{array} \right. \end{equation*} \end{enumerate}
Note that $u$ is defined for $t>\t$, then the above identity is understood as follows: for every $\x \in \mathbb{R}^N$ we have $\lim_{(x,t) \to (\x,\t)}u(x,t) = \varphi (\x)$. We also point out that the two above conditions do not guarantee the uniqueness of the fundamental solution. However, as we shall see in the following, estimates \eqref{upper-bound} and \eqref{e-deriv-bds} hold for the fundamental solution $\Gamma$ built by the parametrix method and the fundamental solution verifying such estimates is unique. Indeed, it follows from the proof of Theorem 15 in Ch.1 of \cite{Friedman} that there is only one fundamental solution under the further assumptions that $\Gamma(x,t;\x,\t) \to 0$ as $|x| \to + \infty$ and $|\partial_{x_j}\Gamma(x,t;\x,\t)| \to 0$ as $|x| \to + \infty$, for $j=1, \dots,N$, uniformly with respect to $t$ varying in bounded intervals of the form $]\tau, \tau + T]$.
We outline here the parametrix method for the construction of a fundamental solution $\G$ of $\L$. We first note that, if the matrix $A$ in the operator $\L$ is constant, then the fundamental solution of $\L$ is explicitly known \begin{equation} \label{eq-fundsol-A}
\G_A(z;\z) = \frac{1}{\sqrt{(4 \pi (t-\t))^{N}\det A}}
\exp \left( - \frac{\langle A^{-1}(x-\x),x-\x \rangle}{4(t-\t)} \right), \end{equation} and moreover the \emph{reproduction property} holds: \begin{equation} \label{eq-rep-prop-A}
\G_A(z;\z) = \int_{\mathbb{R}^N} \G_A(x,t;y,s) \G_A(y,s;\x,\t) d y, \end{equation} for every $z=(x,t), \z=(\x,\t) \in \mathbb{R}^{N+1}$ and $s \in \mathbb{R}$ with $\t < s < t$. A direct computation shows that, for every $T>0$, and $\Lambda^+ > \Lambda$ as in \eqref{e-up}, there exists a positive constant $C^+= C^+(\lambda, \Lambda, \Lambda^+,T)$ such that \begin{equation} \label{eq-fundsol-bd-2} \begin{split}
\left| \frac{\partial \G_A}{\partial {x_j}}(z;\z) \right| \le \frac{C^+}{\sqrt{t-\t}} \G^+(z;\z),
\qquad
\left| \frac{\partial^2 \G_A}{\partial{x_i x_j}}(z;\z) \right| \le \frac{C^+}{t-\t} \G^+(z;\z) \end{split} \end{equation} for any $i,j= 1, \dots, N$ and for every $z, \z \in \mathbb{R}^{N+1}$ such that $ 0 < t - \t \le T$. Here the function \begin{equation} \label{eq-fundsol+}
\G^+(z;\z) = \frac{1}{(\Lambda^+4\pi (t-\t))^{N/2}} \exp \left( - \frac{|x- \x|^2}{4 \Lambda^+(t-\t)} \right), \end{equation} is the fundamental solution of $\Lambda^+ \Delta - \frac{\partial}{\partial t}$. The \emph{parametrix} $Z$ for $\L$ is defined as \begin{equation} \label{eq-fundsol-Z}
Z(z;\z) := \G_{A(\z)}(z;\z) =
\frac{1}{\sqrt{(4 \pi (t-\t))^{N}\det A(\z)}} \exp \left( -
\frac{\langle A(\z)^{-1}(x-\x),x-\x \rangle}{4(t-\t)} \right). \end{equation} More specifically, for every fixed $(\x,\tau) \in \mathbb{R}^{N+1}, Z( \, \cdot \, ; \x,\t)$ is the fundamental solution of the operator $\L_{\z}$ obtained by \emph{freezing} the coefficients $a_{ij}$'s of the operator $\L$ at the point $\z$: \begin{equation} \label{e-L0}
\L_{\z} := \div \left( A (\z) \nabla_x \right) - \, \tfrac{\partial }{\partial t}. \end{equation} Note that \begin{equation} \label{eq-LZ}
\L Z(z;\z) := \div \left[\left( A(z) - A(\z) \right) \nabla_x Z(z;\z) \right], \end{equation} which vanishes as $z \to \z$, by the continuity of the matrix $A$. The fundamental solution $\G$ for $\L$ is obtained from $Z$ by an iterative procedure. We define the sequence of functions $(\L Z)_1 (z;\z) := \L Z(z;\z)$, \begin{equation} \label{eq-LkZ}
(\L Z)_{k+1}(z;\z) := \int_\t^t \bigg( \int_{\mathbb{R}^N}(\L Z)_{k} (x,t;y,s) \L Z(y,s;\x,\t) d y \bigg) d s,
\qquad k \in \mathbb{N}. \end{equation} Note that estimates \eqref{eq-fundsol-bd-2} also apply to $Z$ then, by using the H\"older continuity of the coefficients of $\L$, we obtain \begin{equation*}
\left|\L Z(z;\z) \right| \le \frac{\widetilde C}{(t-\t)^{1 - \alpha/2}} \G^+(z;\z), \end{equation*} for a positive constant $\widetilde C$ depending on $\lambda, \Lambda, \Lambda^+, T$ and on the constant $M$ in \eqref{e-hc}. This inequality and the reproduction property \eqref{eq-rep-prop-A} applied to $\Gamma^+$ imply that, for every $k \ge 2$, the integral that defines $(\L Z)_{k}$ converges and \begin{equation*}
\left| (\L Z)_{k}(z;\z) \right| \le \frac{ (\Gamma_E(\alpha/2) \widetilde C)^k }{\Gamma_E(\alpha k /2)(t-\t)^{1 - k\alpha/2}}
\G^+(z;\z), \qquad k \in \mathbb{N},
\end{equation*} were $\Gamma_E$ denotes the Euler's Gamma function. Theorem 8 in \cite[Chapter 1]{Friedman} states that, under the assumption that the coefficients $a_{ij}, \frac{\partial a_{ij}}{\partial x_i}, b_{i}, \frac{\partial b_{i}}{\partial x_i}$, for $i, j = 1, \dots, N$ and $c$ belong to the space $C^\alpha(\mathbb{R}^N \times ]T_0, T_1[)$ with $T_0 < T_1$ and satisfy \eqref{e-up}, the series \begin{equation} \label{eq-Gamma}
\G (z;\z) := Z(z;\z) + \sum_{k=1}^{\infty} \int_\t^t \bigg( \int_{\mathbb{R}^N}Z(x,t;y,s) (\L Z)_k (y,s;\x,\t) d y \bigg) d s \end{equation} converges in $\mathbb{R}^N \times ]T_0, T_1[$ and it turns out that its sum $\G$ is a fundamental solution for $\L$. We next list some properties of the function $\G$ defined in \eqref{eq-Gamma}. We mainly refer to Chapter I in the monograph \cite{Friedman} by Friedman. \begin{enumerate} \item Theorem 8 in \cite{Friedman}: for every $\z \in \mathbb{R}^{N+1}$ the function $\G(\cdot \, ; \z)$ belongs to $C^{2,1}(\mathbb{R}^N \times ]\t, + \infty[)$ and it is a classical solution to $\L \, \G = 0$ in $\mathbb{R}^N \times ]\t, + \infty[$. \item Theorem 9 in \cite{Friedman}: for every bounded functions $\phi \in C(\mathbb{R}^N)$ and $f \in C^\alpha(\mathbb{R}^N \times ]\t, T_1[)$, with $T_0 < \t < T_1$, the function $$
u(z)=\int_{\mathbb{R}^{N}}\Gamma(z;\z)\phi(\x)d \x - \int_\t^t \bigg( \int_{\mathbb{R}^{N}}\Gamma(x,t;\x, s) f(\x, s)d \x \bigg) d s $$ is a classical solution to the Cauchy problem \begin{equation} \label{cauchyproblem} \left\{
\begin{array}{ll}
\L u = f, & \hbox{$z \in \mathbb{R}^{N} \times ]\t, + \infty[$} \\
u( \cdot,\t) = \varphi & \hbox{in} \ \mathbb{R}^N.
\end{array} \right. \end{equation} \item Theorem 15 in \cite{Friedman}: The function $\G^*(z;\z) := \G(\z;z)$ is the fundamental solution of the transposed operator $\L^*$ acting on a suitably smooth function $v$ as follows \begin{equation} \label{e-L*}
\L^* v (z) := \div \left( A (z) \nabla_x v(z) \right) - \langle b(z), \nabla_x v(z)\rangle +
(c(z) - \div \, b(z)) v(z) + \, \tfrac{\partial u}{\partial t}(z). \end{equation} \item Inequalities (6.10) and (6.11) in \cite{Friedman}: for every positive $T$ and $\Lambda^+ > \Lambda$ there exists a positive constant $C^{+}$ such that \begin{equation} \label{upper-bound} \G(z; \z) \le C^{+} \, \G^{+} (z; \z), \end{equation} for every $z = (x,t), \z = (\x, \t) \in \mathbb{R}^{N+1}$ with $0 < t- \t < T$. Moreover, the following bounds for the derivatives hold \begin{equation} \label{e-deriv-bds} \begin{split}
\left| \frac{\partial \G}{\partial {x_j}}(z;\z) \right| & \le \frac{C^+}{\sqrt{t-\t}} \G^+(z;\z), \quad
\left| \frac{\partial^2 \G}{\partial{x_i x_j}}(z;\z) \right| \le \frac{C^+}{t-\t} \G^+(z;\z),
\\
\left| \frac{\partial \G}{\partial {\x_j}}(z;\z) \right| & \le \frac{C^+}{\sqrt{t-\t}} \G^+(z;\z), \quad
\left| \frac{\partial^2 \G}{\partial{\x_i \x_j}}(z;\z) \right| \le \frac{C^+}{t-\t} \G^+(z;\z), \end{split} \end{equation} for any $i,j=1, \dots, N$ and for every $z, \z \in \mathbb{R}^{N+1}$ with $0 < t- \t < T$. \end{enumerate} We recall that the monograph \cite{Friedman} also contains an existence and uniqueness result for the Cauchy problem under the assumptions that the functions $\varphi$ and $f$ in the Cauchy problem \eqref{cauchyproblem} do satisfy the following growth condition: \begin{equation*}
| \varphi(x)| + |f(z)| \le C_0 \exp \left( h |x|^2 \right) \qquad \text{for every} \ x\in \mathbb{R}^N \ \text{and} \ t \in ]\tau, T_1], \end{equation*} for some positive constants $C_0$ and $h$. The reproduction property \eqref{eq-rep-prop-A} for $\G$ holds as a direct consequence of the uniqueness of the solution to the Cauchy problem. We also have \begin{equation*}
e^{-\Lambda(t-\t)} \le \int_{\mathbb{R}^{N}} \G(x,t;\x, \t) \; d \x \le e^{\Lambda(t-\t)} \end{equation*} for every $(x, t), (\x, \t) \in \mathbb{R}^{N+1}$ with $\t < t$, where $\Lambda$ is the constant introduced in \eqref{e-up}.
We conclude this section by quoting a statement on the asymptotic behavior of fundamental solutions, which in the stochastic theory is referred to as \emph{large deviation principle}. In our setting it is useful in the description of the \emph{parabolic ball} $\Omega_r(z_0)$ introduced in \eqref{e-Psi}. The first large deviation theorem is due to Varhadhan \cite{Varadhan1967behavior, Varadhan1967diffusion}, who considers parabolic operators $\L$ whose coefficients only depend on $x$ and are H\"older continuous. It states that \begin{equation}
4 (t-\tau) \log (\Gamma(x,t;\xi,\tau)) \longrightarrow - d^2(x,\xi) \quad \text{as} \ t \to \tau, \end{equation} uniformly with respect to $x,\xi$ varying on compact sets. Here $d(x,\xi)$ denotes the Riemannian distance (induced by the matrix $A$) of $x$ and $\xi$. Several extensions of the large deviation principle are available in literature, under different assumption on the regularity of the coefficients of $\L$. Azencott considers in \cite{Azencott84} operators with smooth coefficients and proves more accurate estimates for the asymptotic behavior of $\log\big(\Gamma(x,t;\x,\t)\big)$. Garofalo and Lanconelli prove an analogous result by using purely PDEs methods in \cite{GarofaloLanconelli-1989}. We recall here a version of this result which is suitable for our purposes.
\begin{proposition} \label{prop-localestimate} {\sc [Theorem 1.2 in \cite{Polidoro2}]} \ For every $\eta \in ]0,1[$ there exists $C_\eta>0$ such that \begin{equation} \label{eq:approximation}
(1 - \eta) Z(z;\z) \le \G(z;\z) \le (1 + \eta) Z(z;\z) \end{equation} for every $z, \z \in \mathbb{R}^{N+1}$ such that $Z(z;\z) > C_\eta $. \end{proposition}
We finally prove a simple consequence of Proposition \ref{prop-localestimate} that will be used in the following. We introduce some further notation in order to give its statement. We first note that the function $\Gamma^*$ can be built by using the parametrix method, starting from the expression of the parametrix relevant to $\L^*$, that is \begin{equation} \label{eq-fundsol-Z*}
Z^*(z;\z) := \G^*_{A(\z)}(z;\z) = \frac{1}{\sqrt{(4 \pi (\t-t))^{N}\det A(\z)}} \exp \left( -
\frac{\langle A(\z)^{-1}(x-\x),x-\x \rangle}{4(\t-t)} \right). \end{equation} We set \begin{equation} \label{eq-Omega_r*}
\Omega_r^*(z_0) := \bigg\{ z \in \mathbb{R}^{N+1} \mid Z^*(z;z_0) \ge \frac{2}{r^N} \bigg\}, \end{equation} and we point out that its explicit expression is: \begin{equation} \label{eq-Omega_r*-exp} \begin{split}
\Omega_r^*(z_0) = & \Big\{ (x,t) \in \mathbb{R}^{N+1} \mid \langle A^{-1}(z_0)(x-x_0), x-x_0 \rangle \le \\
& \qquad - 4 (t_0 - t) \left( \log \big( \tfrac{2}{r^N} \big) + \tfrac{1}{2} \log (\text{det} A(z_0) )+ \tfrac{N}{2} \log (4 \pi (t_0 - t)) \right) \Big\}. \end{split} \end{equation} We have
\begin{lemma} \label{lem-localestimate*} There exists a positive constant $r^*$, only depending on the operator $\L$, such that \begin{equation*}
\Omega_r^*(z_0) \subset \Omega_r(z_0) \subset \Omega_{3r}^*(z_0) \end{equation*} for every $z_0 \in \mathbb{R}^{N+1}$ and $r \in ]0,r^*]$. \end{lemma}
\begin{proof} As said before, the function $\Gamma^*$ can be built by using the parametrix $Z^*$ defined in \eqref{eq-fundsol-Z*}. In particular, Proposition \ref{prop-localestimate} applies to $\Gamma^*$. Then, if we apply the estimate \eqref{eq:approximation} with $\eta = \frac12$ and we use \eqref{e-L*}, we find that there exists $C^*>0$ such that \begin{equation*}
\frac12 Z^*(\z;z_0) \le \G(z_0;\z) \le \frac32 Z^*(\z;z_0) \end{equation*} for every $z_0, \z \in \mathbb{R}^{N+1}$ such that $Z^*(\z;z_0) > C^*$. The claim then follows from \eqref{e-Psi} and \eqref{eq-Omega_r*} by choosing $r^*:=\left(\frac{2}{C^*}\right)^{1/N}$. \end{proof}
We conclude this section with a further result useful in the proof of the Harnack inequality.
\begin{lemma} \label{lem-localestimategradient} {\sc [Proposition 5.3 in \cite{Polidoro2}]} \ Let $r^*$ be the constant appearing in Lemma \ref{lem-localestimate*}. There exists a positive constants $C$, only depending on the operator $\L$, such that \begin{equation*}
\left| \partial_{x_j} \Gamma(z_0, z) \right| \le C \left( \frac{|x_0-x|}{t_0-t} + 1 \right) \Gamma(z_0, z),
\qquad j=1, \dots, N, \end{equation*} for every $z_0 \in \mathbb{R}^{N+1}$ and $z \in \Omega_r(z_0)$ with $r \in ]0,r^*]$. \end{lemma}
\setcounter{equation}{0} \section{A generalized divergence theorem}\label{SectionDivergence}
Let $\Omega$ be an open subset of $\mathbb{R}^n$, and let $\Phi \in C^1 \pr{\Omega;\mathbb{R}^{n}}$. The classical divergence formula reads \begin{equation} \label{eq-div}
\int_{ E } \mathrm{div}\,\Phi\ dz =-\int_{ \partial E} \scp{\nu,\Phi}\ d \H^{n-1}, \end{equation} where $E$ is a bounded set such that $\overline E \subset \Omega$ and its boundary is $C^1$.
We are interested in the situation in which $E$ is the \emph{super-level} set of a real valued function $F \in C^1\pr{\O}$, that is $E = \left\{F>y\right\}$ for some $y \in \mathbb{R}$. At every point $z\in \partial E$ such that $\nabla F(z)\neq 0$ the \emph{inner} unit normal vector $\nu = \nu(z)$
appearing in \eqref{eq-div} is defined as $\nu(z) = \frac{1}{|\nabla F(z)|}\nabla F(z)$ and $\partial E$ is a $C^1$ manifold in a neighborhood of $z$. But, if we denote $$
\mathrm{Crit} \pr{F}:=\left\{z \in \mathbb{R}^n : \nabla F =0\right\}, $$ the set of \emph{critical points} and $F \pr{\mathrm{Crit}\pr{F}}$ the set of \emph{critical values} of $F$, under our hypotheses we cannot apply the classical Sard theorem to state that ``for almost every $y \in \mathbb{R}$ the level set $\{F=y\}$ is globally a $C^1$ manifold''. Indeed, Whitney proves in \cite{whitney1935} that there exist functions $F \in C^1\pr{\O}$ having the property that $\{F=y\} \cap \mathrm{Crit} \pr{F}$ is not empty for every $y$. Therefore, the purpose of this section is to discuss a version of \eqref{eq-div} when the boundary of $E$ is $C^1$ up to a closed set of null Hausdorff measure and to see how it can be applied in our framework. We first introduce the class of sets with the relevant regularity and state the corresponding divergence formula. We draw this definition and the following theorem from \cite[Section 9.3]{Maggi}.
\begin{definition}\label{def-almost-C1} An open set $E\subset \mathbb{R}^n$ has {\em almost $C^1$-boundary} if there is a closed set $M_0\subset \partial E$ with $\H^{n-1}(M_0)=0$ such that, for every $z_0\in M = \partial E \setminus M_0$ there exist $s > 0$ and $F \in C^1 (B(z_0, s))$ with the property that \begin{align*} B(z_0,s) \cap E & = \{z\in B(z_0,s):\ F(z) > 0\} , \\ B(z_0,s) \cap \partial E & = \{z\in B(z_0,s):\ F(z) = 0\} \end{align*} and $\nabla F(z)\neq 0$ for every $z\in B(z_0,s)$. We call $M$ the {\em regular part} of $\partial E$ (note that $M$ is a $C^1$-hypersurface). The inner unit normal to $E$ is the continuous vector field $\nu\in C^0 (M;{\mathbb S}^{n-1})$ given by \[
\nu(z) = \frac{\nabla F(z)}{|\nabla F(z)|}, \quad z \in B(z_0,s) \cap M . \] \end{definition}
Let us state the divergence theorem for sets with almost $C^1$-boundary.
\begin{theorem}\label{gen-div-thm} If $E\subset\mathbb{R}^n$ is an open set with almost $C^1$-boundary and $M$ is the regular part of its boundary, then for every $\Phi\in C^1_c(\mathbb{R}^n;\mathbb{R}^n)$ the following equality holds \begin{equation}\label{eq-div-almost}
\int_{ E } \mathrm{div}\,\Phi\ dz =-\int_{M} \scp{\nu,\Phi}\ d \H^{n-1} . \end{equation} \end{theorem}
If $F \in C^1\pr{\O}$ and $E = \left\{F>y\right\}$ for some $y \in \mathbb{R}$, we can apply Theorem \ref{gen-div-thm} thanks to the following result due to A.~Ya.~Dubovicki\v{\i} \cite{Dubovickii}, that generalizes Sard's theorem.
\begin{theorem}[\sc Dubovicki\v{\i}] \label{th-Dubo} Assume that $\mathbb{N}^n$ and $\M^m$ are two smooth Riemannian manifolds of dimension $n$ and $m$, respectively. Let $F:\mathbb{N}^n \rightarrow \M^m$ be a function of class $C^k$. Set $s=n-m-k+1$, then for $\H^m-$a.e.~$y \in \M^m$ \begin{equation}\label{e-Du} \H^s \pr{\left\{F=y \right\} \cap \mathrm{Crit} \pr{F}}=0. \end{equation} \end{theorem} Notice that if $m=k=1$ and $\M^m=\mathbb{R}$, then $s=n-1$ and for $\H^1-$a.e.~$y \in \mathbb{R}$ the critical part of $\left\{F=y \right\}$ is an $\H^{n-1}$ null set, while its regular part is an $\pr{n-1}-$manifold of class $C^1$. In other words, $\{F=y\}$ is a set with almost $C^1$-boundary and we cannot apply the classical divergence theorem \eqref{eq-div}, but rather Theorem \ref{gen-div-thm}. Summarizing, we have the following result, that immediately follows from the above discussion.
\begin{proposition} \label{prop-1} Let $\Omega$ be an open subset of $\mathbb{R}^{n}$ and let $F \in C^1 \pr{{\Omega};\mathbb{R}}$. Then, for $\H^1$-almost every $y \in \mathbb{R}$, we have: $$
\int_{\left\{F>y\right\}} \mathrm{div}\,\Phi\ dz =
-\int_{\left\{F=y\right\}\setminus \mathrm{Crit} \pr{F}} \scp{\nu,\Phi}\ d \H^{n-1},
\quad \forall \: \Phi \in C_c^1 \pr{\Omega;\mathbb{R}^{n}}, $$ were $\nu=\tfrac{\nabla F}{\abs{\nabla F}}$. \end{proposition} \begin{proof} By Dubovicki\v{\i} Theorem \ref{th-Dubo} for $\H^1-$almost every $y \in \mathbb{R}$ the set $\{F>y\}$ has almost $C^1$-boundary, hence Theorem \ref{eq-div-almost} applies. Moreover, as $F$ is continuous, for any such $y$ we have $\partial\{F>y\}\subset\{F=y\}$, $\H^{n-1}(\{F=y\}\setminus\partial\{F>y\})=0$ and the regular part of $\partial\{F>y\}$ is $\{F=y\}\setminus\{\nabla F=0\}$ and has full $\H^{n-1}$ measure. \end{proof}
In order to prove Theorem \ref{th-1} we apply Proposition \ref{prop-1} to the super-level set $\Omega_r(z_0)$ of the fundamental solution $\Gamma(z_0,\cdot)$ of $\L$. Then, as explained in the Introduction, we have to cut at a time less than $t_0$ to avoid the singularity of the kernels at $z_0$. Therefore, we specialize Proposition \ref{prop-1} as follows.
\begin{proposition}\label{prop-1bis} Let $G \in C^1 \pr{\mathbb{R}^{N+1} \setminus \left\{ \pr{x_0,t_0} \right\};\mathbb{R}}$. Then for $\H^1-$almost every $w, \varepsilon \in \mathbb{R}$ $$ \int_{\left\{ G > w \right\} \cap \left\{ t<t_0-\varepsilon \right\}} \!\!\!\!\!\!\mathrm{div}\Phi\, dz =-\int_{(\{ G = w \} \setminus \mathrm{Crit}\pr{G}) \cap \{ t<t_0-\varepsilon \}} \!\!\!\!\!\!\scp{\nu,\Phi}d \H^{N} +\int_{\left\{ G > w \right\} \cap \left\{ t=t_0-\varepsilon \right\}} \!\!\!\!\!\!\scp{e,\Phi}d\H^{N}, $$ for every $\Phi \in C^1_c \pr{\Omega;\mathbb{R}^{N+1}}$, where $\nu=\tfrac{\nabla G }{\abs{\nabla G}}$ and $e=\pr{0,\ldots,0,1}$. \end{proposition} \begin{proof} Notice that for $\H^1-$a.e. $w\in\mathbb{R}$ the level set $\{G>w\}$ has almost-$C^1$ boundary and fix such a value. Let $S$ be the $\H^N$-negligible singular set of $\partial\{G>w\}$: by Fubini theorem, for $\H^1-$a.e. $\varepsilon>0$ the set $S\cap\{t=t_0-\varepsilon\}$ is in turn $\H^{N-1}-$negligible, and out of this set the unit normal is given $\H^N-$a.e. by $\nu$ in $\{ G = w \} \setminus \mathrm{Crit}\pr{G} \cap \{ t<t_0-\varepsilon \}$ and by $e$ in $\{ G > w \} \cap \{ t=t_0-\varepsilon \}$. Therefore, Proposition \ref{prop-1} applies with $n=N+1$, $\Omega=\mathbb{R}^{N+1}\setminus \left\{ \pr{x_0,t_0} \right\}$, \[ F(x,t)= (G(x,t)- w)\wedge (t-t_0+\varepsilon) , \] $y=0$ and the set \begin{equation*} \Sigma=(\partial\{ G > w \} \cap \{ t<t_0-\varepsilon \} \cap \mathrm{Crit}\pr{G} \bigr) \cup \bigl(\{ G = w \} \cap \{ t=t_0-\varepsilon \}\bigr) \end{equation*} is $\H^N$-negligible. \end{proof}
The last result we need to prove Theorem \ref{th-1} is the coarea formula for Lipschitz functions. We refer to \cite[3.2.12]{federer1969geometric} or \cite{AmbrosioFuscoPallara}, Theorem 2.93 and formula (2.74) for the proof.
\begin{theorem}[\sc Coarea formula for Lipschitz functions] \label{th-co} Let $G:\mathbb{R}^n \rightarrow \mathbb{R}$ be a Lipschitz function, and let $g$ be a non-negative measurable function. Then \begin{equation} \label{e-co} \int_{\mathbb{R}^n} g\pr{z}\abs{\nabla G \pr{z}}dz= \int_{\mathbb{R}}\pr{\int_{\left\{ G=y \right\}}g\pr{z}d\H^{n-1} \pr{z}}dy. \end{equation} \end{theorem}
\setcounter{equation}{0} \section{Proof of the mean value formulas and maximum principle}\label{sectionProof}
In this Section we give the proof of the mean value formulas and of the strong maximun principle.
\begin{proof} {\sc of Theorem \ref{th-1}.} Let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$, and let $u$ be a classical solution to $\L u = f$ in $\Omega$. Let $z_0=(x_0,t_0) \in \Omega$ and let $r_0>0$ be such that $\overline{\Omega_{r_0}(z_0)} \subset \Omega$. We prove our claim by applying Proposition \ref{prop-1bis} with $G(z) = \Gamma(z_0; z)$ and $w = \frac{1}{r^N}$, where $r \in ]0,r_0]$ is such that the statement of Proposition \ref{prop-1bis} holds true with $w = \frac{1}{r^N}$, and $\varepsilon := \varepsilon_k$ for some monotone sequence $\big(\varepsilon_k\big)_{k \in \mathbb{N}}$ such that $\varepsilon_k \to 0$ as $k \to + \infty$ (see Figure 3).
\begin{center} \begin{tikzpicture} \clip (-.5,7.5) rectangle (6.7,2); \shadedraw [top color=blue!10] (-2,6) rectangle (7,1); \begin{axis}[axis y line=middle, axis x line=middle,
xtick=\empty,ytick=\empty,
ymin=-1.1, ymax=1.1,
xmin=-.2,xmax=1.8, samples=101, rotate= -90] \addplot [black,line width=.7pt, domain=-.01:.01] {sqrt(.0001 - x * x)} node[above] {\hskip12mm $(x_0,t_0)$}; \addplot [black,line width=.7pt, domain=-.01:.01] {-sqrt(.0001 - x * x)}; \addplot [blue,line width=.7pt, domain=.001:1] {sqrt(- 2 * x * ln(x))}; \addplot [blue,line width=.7pt,domain=.001:1] {- sqrt(- 2 * x * ln(x))} node[below] { \hskip20mm $\Omega_r(x_0,t_0)$};
\end{axis} \draw [<-,line width=.4pt] (2.8475,7) -- (2.8475,2); \draw [red, line width=.6pt,] (-1,6) -- (6.7,6) node[below] { \hskip-18mm $t = t_0 - \varepsilon_k$}; \path[draw,thick] (-.49,7.49) rectangle (6.69,2.01); \end{tikzpicture}
{\sc Fig.3} - The set $\Omega_r(x_0,t_0) \cap \big\{ t < t_0 - \varepsilon_k \big\}$. \end{center}
For this choice of $r$, we set $v(z) := \Gamma(z_0;z) - \frac{1}{r^N}$, and we note that \begin{equation} \label{eq-div-L*} \begin{split}
u(z) \L^* v(z) - v(z) \L u(z) =
& \div_x \big( u(z) A(z) \nabla_x v(z) - v(z) A(z) \nabla_x u(z) \big) - \\
& \div_x \big( u(z) v(z) b(z) \big) + \partial_t (u(z)v(z)) \end{split} \end{equation} for every $z \in \Omega \backslash \big\{z_0 \big\}.$ We then recall that $\L^* v = \frac{1}{r^N} \left( \div \, b - c \right)$ and $\L u = f$ in $\Omega \backslash \big\{z_0 \big\}$. Then \eqref{eq-div-L*} can be written as follows \begin{equation*}
\frac{1}{r^N} \left( \div \, b(z) - c(z) \right) u(z) - v(z) f(z) =
\div \, \Phi (z), \qquad \Phi(z) := \big( u A \nabla_x v - v A \nabla_x u - uv b, uv \big)(z). \end{equation*} We then apply Proposition \ref{prop-1bis} to the set $\Omega_r(z_0) \cap \left\{ t<t_0-\varepsilon_k \right\}$ and we find \begin{equation} \label{eq-div-k} \begin{split}
\int_{\Omega_r(z_0) \cap \left\{ t<t_0-\varepsilon_k \right\}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \left( \tfrac{1}{r^N} \left( \div \, b(z) - c(z) \right) u(z) - v(z) f(z)\right) dz = & \\ - \int_{\psi_r(z_0)\setminus \mathrm{Crit}\pr{\Gamma} \cap \left\{ t<t_0-\varepsilon_k \right\}} \!\!\scp{\nu,\Phi} & d \H^{N} + \int_{\Omega_r(z_0) \cap \left\{ t=t_0-\varepsilon_k \right\}} \!\!\scp{e,\Phi}d\H^{N}, \end{split} \end{equation} where $\nu(z)=\tfrac{\nabla_{(x,t)} \Gamma(z_0,z) }{\abs{\nabla_{(x,t)} \Gamma(z_0,z)}}$ and $e=\pr{0,\ldots,0,1}$. We next let $k \to + \infty$ in the above identity. As $f$ is continuous on $\overline{\Omega_r(z_0)}$ and $v \in L^1({\Omega_r(z_0)})$, we find \begin{equation} \label{eq-div-1} \begin{split}
\lim_{k \to + \infty} \int_{\Omega_r(z_0) \cap \left\{ t<t_0-\varepsilon_k \right\}}
&\left( \tfrac{1}{r^N} \left( \div \, b(z) - c(z) \right) u(z) - v(z) f(z)\right) dz = \\ & \int_{\Omega_r(z_0)} \left( \tfrac{1}{r^N} \left( \div \, b(z) - c(z) \right) u(z) - v(z) f(z)\right) dz. \end{split} \end{equation} We next consider the last integral in the right hand side of \eqref{eq-div-k}. We have $\scp{e,\Phi} (z) = u(z) v(z)$, then \begin{equation} \label{eq-psir}
\int_{\Omega_r(z_0) \cap \left\{ t=t_0-\varepsilon_k \right\}} \!\!\scp{e,\Phi}d\H^{N} =
\int_{\mathcal{I}_r^k (z_0)} \!\! \!\!
u(x,t_0- \varepsilon_k) \left( \Gamma(x_0, t_0; x,t_0 - \varepsilon_k) - \frac{1}{r^N} \right) d x, \end{equation} where we have denoted \begin{equation*}
\mathcal{I}_r^k (z_0) := \left\{ x \in \mathbb{R}^N \mid (x,t_0-\varepsilon_k) \in \overline{\Omega_r (z_0)} \right\}. \end{equation*} We next prove that the right hand side of \eqref{eq-psir} tends to $u(z_0)$ as $k \to + \infty$. Since $\Gamma$ is the fundamental solution to $\L$ we have \begin{equation*}
\lim_{k \to + \infty} \int_{\mathbb{R}^N} \!\! \!\! \Gamma(x_0, t_0; x,t_0 - \varepsilon_k)
u(x,t_0- \varepsilon_k) d x = u(x_0,t_0), \end{equation*} then, being $u$ continuous on $\overline{\Omega_r (z_0)}$, we only need to show that \begin{equation} \label{eq-claim-2} \lim_{k \to + \infty} \H^{N} \left( \mathcal{I}_r^k (z_0) \right) = 0, \qquad
\lim_{k \to + \infty} \int_{\mathbb{R}^N \backslash \mathcal{I}_r^k (z_0)} \!\! \!\!
\Gamma(x_0, t_0; x,t_0 - \varepsilon_k) d x = 0. \end{equation} With this aim, we note that the upper bound \eqref{upper-bound} and \eqref{eq-fundsol+} imply \begin{equation*}
\mathcal{I}_r^k (z_0) \subset \left\{ x \in \mathbb{R}^N \mid | x - x_0|^2 \le 4 \Lambda^+ \varepsilon_k
\left( \log \left(C^+ r^N \right) - \tfrac{N}{2} \log (4 \pi \Lambda^+ \varepsilon_k) \right) \right\}. \end{equation*} The first assertion of \eqref{eq-claim-2} is then a plain consequence of the above inclusion. In order to prove the second statement in \eqref{eq-claim-2}, we rely on Lemma \ref{lem-localestimate*}. We let $r_0 := \min (r, r^*)$, so that \begin{equation*}
\Omega_{r_0}^*(z_0) \subset \Omega_{r_0}(z_0) \subset \Omega_r(z_0), \end{equation*} thus \begin{equation*} \begin{split}
\mathbb{R}^N \backslash \mathcal{I}_r^k (z_0) \subset & \left\{ x \in \mathbb{R}^N \mid Z^*(x,t_0- \varepsilon_k;x_0,t_0) \le \tfrac{2}{r_0^N} \right\} \\
= & \Big\{ x \in \mathbb{R}^N \mid \langle A(z_0) (x - x_0), x-x_0 \rangle \\
& \quad \ge - 4 \varepsilon_k \left( \log \left( \tfrac{2}{r_0^N} \right) + \tfrac{1}{2} \log (\det A(z_0) ) + \tfrac{N}{2} \log (4 \pi \varepsilon_k) \right) \Big\}. \end{split} \end{equation*} By using again \eqref{upper-bound}, the above inclusion, and the change of variable $x = x_0 + 2 \sqrt{\Lambda^+ \varepsilon_k} \, \xi$, we find \begin{equation*} \begin{split}
\int_{\mathbb{R}^N \backslash \mathcal{I}_r^k (z_0)} & \!\! \!\! \Gamma(x_0, t_0; x,t_0 - \varepsilon_k) d x \le C^+ \int_{\mathbb{R}^N \backslash \mathcal{I}_r^k (z_0)} \!\! \!\! \Gamma^+ (x_0, t_0; x,t_0 - \varepsilon_k) d x \\
& \le \frac{C^+}{\pi^{N/2}} \int_{ \left\{ \langle A(z_0) \xi, \xi \rangle \ge - \tfrac{1}{\Lambda^+} \left( \log \left( \tfrac{2}{r_0^N} \right) + \tfrac{1}{2} \log (\det A(z_0) ) + \tfrac{N}{2} \log (4 \pi \varepsilon_k) \right) \right\} }
\!\! \exp\left( - |\xi|^2 \right) d \xi. \end{split} \end{equation*} The second assertion of \eqref{eq-claim-2} then follows. Thus, we have shown that \begin{equation} \label{eq-div-2}
\lim_{k \to + \infty} \int_{\Omega_r(z_0) \cap \left\{ t=t_0-\varepsilon_k \right\}} \scp{e,\Phi} d\H^{N} = u(z_0). \end{equation} We are left with the first integral in the right hand side of \eqref{eq-div-k}. We preliminarily note that its limit, as $k \to + \infty$, does exist. Moreover, for every $z \in \psi_r(z_0)$ we have $v(z) = 0$, then $\Phi(z) = \big( u (z) A(z) \nabla_x v(z), 0 \big)$, so that \begin{equation*}
\int_{\psi_r(z_0) \setminus \mathrm{Crit}\pr{\Gamma} \cap \left\{ t<t_0-\varepsilon_k \right\}} \!\!\scp{\nu,\Phi}d \H^{N} = \int_{\psi_r(z_0) \setminus \mathrm{Crit}\pr{\Gamma} \cap \left\{ t<t_0-z\varepsilon_k \right\}} \!\! u(x,t) K (z_0;z) d \H^{N}, \end{equation*} where \begin{equation*}
K (z_0; z) = \frac{\langle A(z) \nabla_x \Gamma(z_0;z), \nabla_x \Gamma(z_0;z) \rangle }
{|\nabla_{(x,t)}\Gamma(z_0; z)|} \end{equation*} is the kernel defined in \eqref{e-kernels}. Note that $K$ is non-negative and, if we consider the function $u = 1$ and we let $k \to + \infty$, we find \begin{equation*}
\lim_{k \to + \infty} \int_{\psi_r(z_0) \setminus \mathrm{Crit}\pr{\Gamma} \cap
\left\{ t<t_0-\varepsilon_k \right\}} \!\! K (z_0; z) d \H^{N} = \int_{\psi_r(z_0)\setminus \mathrm{Crit}\pr{\Gamma}} \!\!\! K (z_0;z) d \H^{N} < + \infty. \end{equation*} Thus, if $u$ is a classical solution to $\L u = 0$, we obtain \begin{equation} \label{eq-div-3i}
\lim_{k \to + \infty} \int_{\psi_r(z_0) \setminus \mathrm{Crit}\pr{\Gamma} \cap
\left\{ t<t_0-\varepsilon_k \right\}} \!\!\scp{\nu,\Phi}d \H^{N} = \int_{\psi_r(z_0)\setminus \mathrm{Crit}\pr{\Gamma}} \!\!\! K (z_0;z) u(z) d \H^{N}. \end{equation} We recall that Dubovicki\v{\i}'s theorem implies that $\H^{N} \left(\psi_r(z_0) \cap \mathrm{Crit}\pr{\Gamma}\right) = 0$ for $\H^{1}$ almost every $r$, so that we can equivalently write \begin{equation} \label{eq-div-3}
\lim_{k \to + \infty} \int_{\psi_r(z_0) \setminus \mathrm{Crit}\pr{\Gamma} \cap
\left\{ t<t_0-\varepsilon_k \right\}} \!\!\scp{\nu,\Phi}d \H^{N} = \int_{\psi_r(z_0)} \!\!\! K (z_0;z) u(z) d \H^{N}. \end{equation} The proof of the first assertion of Theorem \ref{th-1} then follows by using \eqref{eq-div-1}, \eqref{eq-div-2} and \eqref{eq-div-3} in \eqref{eq-div-k}.
The proof of the second assertion of Theorem \ref{th-1} is a direct consequence of the first one and of the coarea formula stated in Theorem \ref{th-co}. Indeed, fix a positive $r$ as above, multiply by $\frac{N}{r^N}$ and integrate over $]0,r[$. We find \begin{equation} \label{e-meanvalue-step1} \begin{split}
\frac{N}{r^N} \int_0^r \varrho^{N-1} u(z_0) d \varrho = &
\frac{N}{r^N} \int_0^r \varrho^{N-1} \bigg(\int_{\psi_\varrho(z_0)}
K (z_0;z) u(z) \, d \H^{N} (x,t) \bigg) d \varrho\,
\\
& + \frac{N}{r^N} \int_0^r \varrho^{N-1} \bigg( \int_{\Omega_\r(z_0)} f (z)
\left( \tfrac{1}{r^N} - \Gamma(z_0;z) \right) dz \bigg) d \varrho
\\
& + \frac{N}{r^N} \int_0^r \frac{1}{\varrho} \bigg(
\int_{\Omega_\r (z_0)} \left( \div \, b(z) - c(z) \right) u(z) \, dz \bigg) d \varrho. \end{split} \end{equation} The left hand side of the above equality equals $u(z_0)$, while the last two terms agree with the last two terms appearing in the statement of Theorem \ref{th-1}. In order to conclude the proof we only need to show that \begin{equation} \label{e-meanvalue-step2} \begin{split}
\int_0^r \varrho^{N-1} \bigg(\int_{\left\{\Gamma(z_0;z) =
\tfrac{1}{\varrho^N}\right\} } & K (z_0;z) u(z) \, d \H^{N} (z) \bigg) d \varrho
\\
& = \frac{1}{N} \int_0^{r} \r^{N-1}\int_{\Omega_\r (z_0)} M (z_0;z) u(z) dz. \end{split} \end{equation} With this aim, we substitute $y = \frac{1}{\varrho^N}$ in the left hand side of \eqref{e-meanvalue-step2} and we recall the definition of the kernel $K$. We find \begin{equation} \label{e-meanvalue-step3} \begin{split}
\int_0^r \varrho^{N-1} & \bigg( \int_{\left\{\Gamma(z_0;z) = \tfrac{1}{\varrho^N}\right\} }
\frac{\langle A(z) \nabla_x \Gamma(z_0;z), \nabla_x \Gamma(z_0;z) \rangle }
{|\nabla_{(x,t)}\Gamma(z_0;z)|} u(z) \, d \H^{N} (z) \bigg) d \varrho \\
& = \frac{1}{N}
\int_{\frac{1}{r^N}}^{+ \infty} \frac{1}{y^{2}} \bigg(\int_{\left\{\Gamma(z_0;z) = y\right\} }
\frac{\langle A(z) \nabla_x \Gamma(z_0; z), \nabla_x \Gamma(z_0;z) \rangle }
{|\nabla_{(x,t)}\Gamma(z_0;z)|}u(z) \, d \H^{N} (z) \bigg) d y \\
& =\frac{1}{N} \int_{\frac{1}{r^N}}^{+ \infty} \bigg(\int_{\left\{\Gamma(z_0;z) = y\right\} }
\frac{\langle A(z) \nabla_x \Gamma(z_0;z), \nabla_x \Gamma(z_0;z) \rangle }
{\Gamma^2(z_0;z) {|\nabla_{(x,t)}\Gamma(z_0;z)|}} u(z) \, d \H^{N} (z) \bigg) d y. \end{split} \end{equation} We conclude the proof of \eqref{e-meanvalue-step2} by applying the coarea formula stated in Theorem \ref{th-co}. \end{proof}
\begin{proof} {\sc of Proposition \ref{prop-smp}.} We prove our claim under the additional assumption $\div \, b \ge 0$. At the end of the proof we show that this assumption is not restrictive.
We first note that, as a direct consequence of our assumption $c=0$, we have that $\L \, 1 = 0$, then Theorem \ref{th-1} yields
\begin{equation*}
\frac{1}{\varrho^N} \int_{\Omega_\varrho(z_1)} M (z_1; z) \, dz +
\frac{N}{\r^N} \int_0^{\r} \Big(\frac{1}{s} \int_{\Omega_s (z_1)} \div \, b(z) \, dz \Big) d s = 1 \end{equation*} for every $z_1 \in \Omega$ and $\r >0$ such that $\overline{\Omega_\varrho(z_1)} \subset \Omega$.
We claim that, if $u(z_1) = \max_\Omega u$, then \begin{equation} \label{eq-claim-smp}
u(z) = u(z_1) \qquad \text{for every} \quad z \in \overline{\Omega_\varrho(z_1)}. \end{equation} By using again Theorem \ref{th-1} and the above identity we obtain
\begin{align*}
0 = & \frac{1}{\varrho^N} \int_{\Omega_\varrho(z_1)} M (z_1; z) \big((u(z)- u(z_1)\big) \, dz
\\
& + \frac{N}{\r^N} \int_0^{\r} \Big(\frac{1}{s} \int_{\Omega_s (z_1)} \div \, b(z)
\big((u(z)- u(z_1)\big) \, dz \Big) d s
\\
& + \frac{N}{\r^N} \int_0^{\r} \left(s^{N-1}
\int_{\Omega_s (z_1)} f (z) \left( \tfrac{1}{s^N} - \Gamma(z_1;z) \right) dz \right) d s \le 0, \end{align*} since $f \ge 0$, $\div \, b \ge 0$ and $u(z) \le u(z_1)$, being $u(z_1) = \max_{\Omega} u$. We have also used the fact that $M(z_1;z) \ge 0$ and $\Gamma(z_1;z) \ge \tfrac{1}{s^N}$ for every $z \in \Omega_s(z_1)$. Hence, $M (z_1; z) \big((u(z)- u(z_1)\big)=0$ for $\H^{N+1}$ almost every $z \in \Omega_\r(z_1)$. As already noticed, Dubovicki\v{\i}'s theorem implies that $\H^{N} \left( \psi_s (z_1) \cap \mathrm{Crit}\pr{\Gamma} \right) = 0$, for almost every $s \in ]0,\r]$, then $M (z_1; z) \ne 0$ for $\H^{N+1}$ almost every $z \in \Omega_\r(z_1)$. As a consequence $u(z)= u(z_1)$ for $\H^{N+1}$ almost every $z \in \Omega_\r(z_1)$, and the claim \eqref{eq-claim-smp} follows from the continuity of $u$.
We are in position to conclude the proof of Proposition \ref{prop-smp}. Let $z$ be a point of $\AS ( \O )$, and let $\gamma: [0,T] \to \O$ be an $\L$--admissible path such that $\g(0)= z_0$ and $\g(T) = z$. We will prove that $u(\gamma(t)) = u(z_0)$ for every $t \in [0,T]$. Let \begin{equation*}
I := \big\{ t \in [0,T] \mid u(\gamma(s)) = u(z_0) \ \text{for every} \ s \in[0,t] \big\}, \qquad \overline t := \sup I. \end{equation*} Clearly, $I \ne \emptyset$ as $0 \in I$. Moreover $I$ is closed, because of the continuity of $u$ and $\gamma$, then $\overline t \in I$. We now prove by contradiction that $\overline t = T$.
Suppose that $\overline t < T$. Let $z_1 := \gamma(\overline t)$ and note that $z_1 \in \Omega$, $u(z_1) = \max_\Omega u$. We aim to show that there exist positive constants $r_1$ and $s_1$ such that $\overline{\Omega_{r_1}(z_1)} \subset \O$ and \begin{equation} \label{eq-claim2-smp}
\gamma(\overline t + s) \in \Omega_{r_1}(z_1)\quad \text{for every} \quad s \in [0, s_1[. \end{equation} As a consequence of \eqref{eq-claim-smp} we obtain $u(\gamma(\overline t + s)) = u(z_1) = u(z_0)$ for every $s \in [0, s_1[$, and this contradicts the assumption $\overline t < T$.
The proof of \eqref{eq-claim2-smp} is a consequence of Lemma \ref{lem-localestimate*}. It is not restrictive to assume that $r_1 \le r^*$, then it is sufficient to show that there exists a positive $s_1$ such that \begin{equation} \label{eq-claim4-smp}
\gamma(\overline t + s) \in \Omega^*_{r_1}(z_1) \quad \text{for every} \quad s \in [0, s_1[.
\end{equation} Recall the definition of $\gamma(\overline t + s) = (x(\overline t + s), t(\overline t + s))$. We have $\gamma(\overline t) = z_1 = (x_1, t_1), t(\overline t + s) = t_1-s$ and, for every positive $s$ \begin{equation*} \begin{split}
|x(s + \overline t) - x_1 | & = \left| \int_0^s \dot x(\overline t + \sigma) d \sigma \right| \le \int_0^s \left| \dot x(\overline t + \sigma) \right| d \sigma \\
& \le \left( \int_0^s \left| \dot x(\overline t + \sigma) \right|^2 d\sigma \right)^{1/2} s^{1/2} \le \|\dot x \|_{L^2([0,T])} \sqrt{s}, \end{split} \end{equation*} then \begin{equation*} \langle A^{-1}(z_1 )(x(\overline t + s)-x_1), x(\overline t + s)-x_1 \rangle \le
s \cdot \| A^{-1}(z_1)\| \cdot \|\dot x \|_{L^2([0,T])}^2. \end{equation*} By using the above inequality in \eqref{eq-Omega_r*-exp} we see that there exists a positive constant $s_1$ such that \eqref{eq-claim4-smp} holds. This proves \eqref{eq-claim2-smp}, and then $u(z) = u(z_0)$ for every $z \in \AS ( \O )$. By the continuity of $u$ we conclude that $u(z) = u(z_0)$ for every $z \in \overline{\AS ( \O )}$. Eventually, since $u$ is constant in $\overline{\AS ( \O )}$ and $c= 0$, we conclude that $\L u = 0$.
We finally prove that the additional assumption $\div \, b \ge 0$ is not restrictive. Let $k$ be any given constant such that $k>\Lambda$, where $\Lambda$ is the quantity appearing in \eqref{e-up}, recall that $z_0 = (x_0,t_0)$ and define the function \begin{equation}
v(y,t):= u\left( e^{-k(t-t_0)}y, t\right), \qquad (y,t) \in \widehat \Omega, \end{equation} where $(y,t) \in \widehat \Omega$ if, and only if $\left( e^{-k(t-t_0)}y, t\right) \in \Omega$. Then $v$ is a solution to \begin{equation*}
\widehat \L v (y,t) :=
\div \left( \widehat A (y,t) \nabla_y v(y,t) \right) + \langle \widehat b(y,t) + k y, \nabla_y v(y,t)\rangle -
\, \tfrac{\partial v}{\partial t}(y,t) = f\left( e^{-k(t-t_0)}y, t\right), \end{equation*}
where $\widehat A (y,t) = \left( \widehat a_{ij} (y,t) \right)_{i,j=1, \dots, N}$, $\widehat b(y,t) = \left( \widehat b_1(y,t), \dots, \widehat b_N(y,t) \right)$, are defined as $\widehat a_{ij} (y,t) = e^{-2k(t-t_0)} a_{ij} \left( e^{-k(t-t_0)}y, t\right), \widehat b_{j} (y,t) = e^{-k(t-t_0)} b_{j} \left( e^{-k(t-t_0)}y, t\right)$, for $i,j=1, \dots, N$. Note that from the assumption \eqref{e-up} it follows that $|\div \, b| \le N \Lambda$, then \begin{equation*}
\div \left( \widehat b(y,t) + k y \right) \ge N \left(k - \Lambda e^{-2k(t-t_0)} \right). \end{equation*} In particular, there exists a positive $\delta$, depending on $k$ and $\Lambda$, such that the right hand side of above expression is non-negative as $t \ge t_0 - \delta$. Then, if we set $\widehat t_0 := t_0- \delta$, we have \begin{equation*}
\div \left( \widehat b(y,t) + k y \right) \ge 0 \qquad \text{for every} \quad (y,t) \in \widehat \Omega
\cap \big\{t \ge \widehat t_0 \big\}. \end{equation*} We also note that $\widehat \L$ satisfies the same assumptions as $\L$, with a possibily different constant $\widehat \Lambda$, in every set of the form $\widehat \Omega \cap \big( \mathbb{R}^N \times I \big)$, where $I$ is any bounded open interval of $\mathbb{R}$. We then apply the above argument to prove that, if $v$ reaches its maximum at some point $(y_0,t_0) \in \widehat \Omega$, then it is constant in its propagation set in $\widehat \Omega \cap \big\{t \ge \widehat t_0 \big\}$. Note that $u$ reaches its maximum at some point $(x,t)$ if, and only if, $v$ reaches its maximum at $\left( e^{k(t-t_0)}x, t\right)$. Moreover, $(x(s), t-s)$ is an admissible curve for $\L$ if, and only if, $(e^{-k(t-s-t_0)}x(s), t-s)$ is an admissible curve for $\widehat \L$. We conclude that \begin{equation*} u(z) = u(z_0) \qquad \text{for every} \ z \in \overline {\AS ( \O )} \cap \big\{t \ge \widehat t_0 \big\}. \end{equation*} We then repeat the above argument. Assume that $u$ reaches its maximum at some point $\left( \widehat x_0, \widehat t_0\right)$, we define a new function $\widehat v(y,t):= u\left( e^{-k(t-\hat t_0)}y, t\right)$ and we find a new constant $\widehat t_1 := t_0 - 2 \delta$ such that \begin{equation*} u(z) = u(z_0) \qquad \text{for every} \ z \in \overline {\AS ( \O )} \cap \big\{t \ge \widehat t_1 \big\}. \end{equation*} As we can use the same constant $\delta$ at every iteration, we conclude that the above identity holds for every $z \in \overline {\AS ( \O )}$. \end{proof}
\begin{proof} {\sc of Proposition \ref{prop-smp1}.} Let $k$ be a constant such that $\div \, b - c -k \ge 0$ and note that the function $v(x,t) := e^{k t}u(x,t)$ is a non negative solution to the equation \begin{equation*}
\L v(x,t) + k v(x,t) = e^{k t}f(x,t), \qquad (x,t) \in \Omega. \end{equation*} Then Theorem \ref{th-1} yields
\begin{align*}
0 = & \frac{1}{\varrho^N} \int_{\Omega_\varrho(z_1)} M (z_1; z) v(z) \, dz
\\
& + \frac{N}{\r^N} \int_0^{\r} \Big(\frac{1}{s} \int_{\Omega_s (z_1)} \big(\div \, b(z) - c(z) - k \big)
v(z) \, dz \Big) d s
\\
& + \frac{N}{\r^N} \int_0^{\r} \left(s^{N-1}
\int_{\Omega_s (z_1)} e^{k t} f (z) \left( \tfrac{1}{s^N} - \Gamma(z_1;z) \right) dz \right) d s \le 0, \end{align*} for every $z_1 \in \Omega$ and $\r >0$ such that $\overline{\Omega_\varrho(z_1)} \subset \Omega$. Here we have sued the facts that $f \ge 0$, and $\div \, b - c -k \ge 0$. By following the same argument used in the proof of Proposition \ref{prop-smp} we find that $v \ge 0$ in $\overline {\AS ( \O )}$. This concludes the proof of Proposition \ref{prop-smp1}. \end{proof}
\begin{proof} {\sc of Proposition \ref{prop-2}.} Let $m$ be a positive integer, and let $u$ be a solution to $\L u = f$ in $\Omega \subset \mathbb{R}^{N+1}$. As said in the Introduction, we set \begin{equation*}
\widetilde u(x,y,t) := u(x,t), \qquad \widetilde f(x,y,t) := f(x,t), \end{equation*} for every $(x,y,t) \in \mathbb{R}^{N}\times \mathbb{R}^{m} \times \mathbb{R}$ such that $(x,t) \in \Omega$, and we note that \begin{equation*}
\widetilde \L \ \widetilde u(x,y,t) = \widetilde f(x,y,t)
\qquad \widetilde \L := \L + \sum_{j=1}^{m}\tfrac{\partial^2}{\partial y_j^2}. \end{equation*} Moreover, the function \begin{equation} \label{eq-tildegamma}
\widetilde \Gamma(x_0, y_0, t_0 ;x,y,t) := \Gamma (x_0, t_0 ;x,t) \cdot
\frac{1}{(4 \pi (t_0-t))^{m/2}}\exp \left( \frac{-|y_0-y|^2}{4(t_0-t)} \right) \end{equation} is a fundamental solution of $\widetilde \L$. We then use $\widetilde \Gamma$ to represent the solution $u$ in accordance with Theorem \ref{th-1} as follows \begin{equation*} \begin{split}
u(z_0) = & \, \widetilde u(x_0,y_0,t_0) = \frac{1}{r^{N+m}} \int_{\widetilde\Omega_r(x_0,y_0,t_0)} \! \! \! \! \widetilde M(x_0,y_0,t_0; x,y,t) u(x,t) \, dx \, dy\, dt \\
& +\frac{N+m}{r^{N+m}} \int_0^{r} \left(\r^{N+m-1}
\int_{\widetilde\Omega_\r(x_0,y_0,t_0)} \! \! \! \!
f (x,t) \left( \tfrac{1}{\r^{N+m}} - \widetilde \Gamma(x_0,y_0,t_0;x,y,t) \right) \, dx \, dy\, dt \right) d \r \\
& + \frac{N+m}{r^{N+m}} \int_0^{r} \left(\frac{1}{\r}
\int_{\widetilde\Omega_\r(x_0,y_0,t_0)} \! \! \! \!
\left( \div \, b(x,t) - c(x,t) \right) u(x,t) \, dx \, dy\, dt \right) d \r. \end{split} \end{equation*} where $\widetilde\Omega_r(x_0,y_0,t_0)$ is the parabolic ball relevant to $\widetilde \Gamma$ and \begin{equation*}
\widetilde M(x_0,y_0,t_0; x,y,t) = M(x_0,t_0;x,t) + \frac{|y_0-y|^2}{4(t_0-t)^2}. \end{equation*} The proof is accomplished by integrating the above identity with respect to the variable $y$. \end{proof}
\setcounter{equation}{0} \section{Proof of the Harnack inequalities}\label{sectionHarnack}
In this Section we use the mean value formula stated in Proposition \ref{prop-2} to give a simple proof of the parabolic Harnack inequality.
\begin{proof} {\sc of Proposition \ref{prop-Harnack}.} We first prove our claim under the additional assumption that $\div \, b - c = 0$. This assumption simplifies the proof as in this case we only need to use the first integral in the representation formula given in Proposition \ref{prop-2}. It will be removed at the end of the proof.
Let $m \in \mathbb{N}$ with $m > 2$, let $\Omega$ be an open subset of $\mathbb{R}^{N+1}, z_0 \in \Omega$ and $r>0$ such that $\Omega_{4r}^{(m)}(z_0) \subset \Omega$. We claim that there exist four positive constants $r_0, \vartheta, M^+, m^-$ such that the following assertions hold for every $r \in ]0,r_0]$. \begin{description} \item[{\it i)}] $K^{(m)}_{r}(z_0) \ne \emptyset$; \item[{\it ii)}] $M_{\vartheta r}^{(m)} (z; \zeta) \le M^+$ for every $\zeta \in \Omega_{\vartheta r}^{(m)}(z)$; \item[{\it iii)}] $\Omega_{\vartheta r}^{(m)}(z) \subset \Omega_{4r}^{(m)}(z_0) \cap \big\{\tau \le t_0 - \frac{r^2}{4 \pi \lambda^{N/(N+m)}}\big\}$ for every $z \in K^{(m)}_{r}(z_0)$; \item[{\it iv)}] $M_{5 r}^{(m)} (z_0; \zeta) \ge m^-$ for every $\zeta = (\xi, \tau) \in \Omega_{4 r}^{(m)}(z_0)$ such that $\tau \le t_0 - \frac{r^2}{4 \pi \lambda^{N/(N+m)}}$. \end{description} By using Proposition \ref{prop-2} and the above claim it follows that, for every $z \in K^{(m)}_{r}(z_0)$, it holds \begin{equation} \label{e-core-h} \begin{split}
u(z) = & \frac{1}{(\vartheta r)^{N+m}} \int_{\Omega^{(m)}_{\vartheta r}(z)}
\! \! \! \! \! \! M_{\vartheta r}^{(m)} (z; \zeta) u(\zeta) \, d\zeta \\
& (\text{by {\it ii}}) \le \frac{M^+}{(\vartheta r)^{N+m}}
\int_{\Omega^{(m)}_{\vartheta r}(z)} \!\!\! u(\zeta) \, d\zeta \\ & (\text{by {\it iii}}) \le \frac{M^+}{(\vartheta r)^{N+m}}
\int_{\Omega_{4r}^{(m)}(z_0) \cap \big\{\tau \le t_0 - \frac{r^2}{4 \pi \lambda^{N/(N+m)}}\big\}}
\!\!\! u(\zeta) \, d\zeta \\
& (\text{by {\it iv}}) \le \frac{M^+}{m^- (\vartheta r)^{N+m}}
\int_{\Omega_{5r}^{(m)}(z_0)} \! \! \! \! \! \! M_{5 r}^{(m)} (z_0; \zeta)u(\zeta) \, d\zeta =
\frac{5^{N+m}M^+}{\vartheta^{N+m} m^-} u(z_0). \end{split} \end{equation} This proves Proposition \ref{prop-Harnack} with $C_K := \frac{5^{N+m}M^+}{ \vartheta^{N+m}m^-}$.
We are left with the proof of our claims. We mainly rely on Lemma \ref{lem-localestimate*}, applied to the function $\widetilde \Gamma$ introduced in \eqref{eq-tildegamma}. In the sequel we let $r^*$ be the constant appearing in Lemma \ref{lem-localestimate*} and relative to $\widetilde \Gamma$, and in accordance with \eqref{eq-Omega_r*}, \begin{equation} \label{eq-Omega_rm*}
\Omega_r^{(m)*}(z_0) := \bigg\{ z \in \mathbb{R}^{N+1} \mid (4 \pi (t_0-t))^{-m/2}Z^*(z;z_0) \ge \frac{2}{r^{N+m}} \bigg\}. \end{equation} Moreover, we choose $r_0 := r^*/2$.
\begin{center} \begin{tikzpicture} \clip (-.52,7.02) rectangle (6.52,1.78); \path[draw,thick] (-.5,7) rectangle (6.5,1.8); \begin{axis}[axis y line=none, axis x line=none,
xtick=\empty,ytick=\empty,
ymin=-1.1, ymax=1.1,
xmin=-.2,xmax=1.8, samples=101, rotate= -90] \addplot [black,line width=1pt, domain=-.01:.01] {sqrt(.0001 - x * x)} node[above] {$z_0$}; \addplot [black,line width=1pt, domain=-.01:.01] {-sqrt(.0001 - x * x)}; \addplot [lblue,line width=.7pt,domain=.001:.051] {sqrt(- 3 * x * ln(9*x)}; \addplot [lblue,line width=.7pt,domain=.001:.051] {- sqrt(- 3 * x * ln(9*x))}; \addplot [blue,line width=.7pt, domain=.001:1] {sqrt(- 2 * x * ln(x))}; \addplot [blue,line width=.7pt,domain=.001:1] {- sqrt(- 2 * x * ln(x))} node[below=-10pt] { \hskip35mm $\Omega^{(m)}_{4r}(z_0)$}; \addplot [ddddblue,line width=.7pt,domain=.051:.1111] {sqrt(- 3 * x * ln(9*x)}; \addplot [ddddblue,line width=.7pt,domain=.051:.1111] {- sqrt(- 3 * x * ln(9*x))}; \addplot [red,line width=.7pt,domain=.0001:.0625,below=15pt,right=9pt] {sqrt(- 4 * x * ln(16*x)}; \addplot [red,line width=.7pt,domain=.0001:.0625,below=15pt,right=9pt] {- sqrt(- 4 * x * ln(16*x))}; \addplot [black,line width=1pt, domain=-.01:.01,below=15pt,right=9pt] {sqrt(.0001 - x * x)} node[below=-3pt] {$z$}; \addplot [black,line width=1pt, domain=-.01:.01,below=15pt,right=9pt] {-sqrt(.0001 - x * x)} node[below=15pt,right=10pt] {\hskip-20mm \color{red} $\Omega^{(m)}_{\vartheta r}(z)$}; \end{axis} \draw [ddddblue,line width=.7pt] (1.92,6) -- node [below=5pt] { \hskip25mm $K^{(m)}_r(z_0)$} (3.75,6); \end{tikzpicture}
{\sc \qquad Fig.4} - The inclusion \emph{(iii)}. \end{center}
\noindent {\it Proof of i)} \ By the definition \eqref{e-Kr} of $K^{(m)}_{r}(z_0)$ we only need to show that there exists at least a point $(x,t) \in \overline{\Omega^{(m)}_{r}(z_0)}$ with $t \le t_0 - \frac{1}{4 \pi \lambda^{N/(N+m)}} \,r^2$. From Lemma \ref{lem-localestimate*} it follows that $\Omega_r^{(m)*}(z_0) \subset \Omega^{(m)}_r(z_0)$, then we only need to show that that the point $\big(x_0, t_0 - \frac{1}{4 \pi \lambda^{N/(N+m)}} \,r^2\big)$ belongs to $\overline{\Omega^{(m)}_{r}(z_0)}$. In view of \eqref{eq-fundsol-Z*}, this is equivalent to $\det A(z_0)\ge \lambda^N/4$, which directly follows from the parabolicity assumption \eqref{e-up}.
\noindent {\it Proof of ii)} \ We first note that \eqref{eq-fundsol+} and the defintion of $N_{\vartheta r}$ directly give \begin{equation} \label{eq-boundN}
N_{\vartheta r} (z;\z) \le 2 \sqrt{t-\tau}\sqrt{\log\left(
\tfrac{C^+ (\vartheta r)^{N+m}}{ (\Lambda^+)^{N/2}(4 \pi (t-\tau))^{(N+m)/2}} \right)} = 2 \sqrt{t-\tau}
\sqrt{C_1 + \tfrac{N+m}{2}\log\left( \tfrac{\vartheta^2 \, r^2 }{t-\tau} \right)}, \end{equation} where $C_1$ is a positive constant that only depends on $\L$. Moreover, Lemma \ref{lem-localestimategradient} implies that there exists a positive constant $M_0$, only depending on the operator $\L$, such that \begin{equation*}
M(z,\z) \le M_0 \left( \frac{|x-\xi|^2}{(t-\tau)^2} + 1 \right), \qquad \text{for every} \
\zeta \in \Omega_{\vartheta r}^{(m)}(z). \end{equation*} Moreover, Lemma \ref{lem-localestimate*} implies that there exists another positive constant $M_1$ such that \begin{equation} \label{eq-boundM}
M(z,\z) \le M_1 \left( \frac{1}{t-\tau} \log\left( \frac{\vartheta^2 \, r^2}{t-\tau} \right) + 1 \right),
\qquad \text{for every} \
\zeta \in \Omega_{\vartheta r}^{(m)}(z). \end{equation} We point out that the constants $C_1$ and $M_1$ depend neither on the choice of $\vartheta \in ]0,1[$, that will be specified in the following proof of the point {\it iii)}, nor on the choice of $r \in ]0,r_0[$. By using \eqref{eq-boundN} and \eqref{eq-boundM} we conclude that there exists a positive constant $M_2$, that only depends on $\L$ and on $m$, such that \begin{equation*}
M_{\vartheta r}^{(m)} (z; \zeta) \le M_2 (t-\tau)^{m/2}
\left( 1 + \left| \log\left( \tfrac{\vartheta^2 \, r^2 }{t-\tau} \right) \right| \right)^{m/2}
\left( 1 + \tfrac{1}{t-\tau} \left| \log\left( \tfrac{\vartheta^2 \, r^2 }{t-\tau} \right) \right| \right) \end{equation*} for every $\zeta \in \Omega_{\vartheta r}^{(m)}(z)$. The right hand side of the above inequality is bounded whenever $m>2$, uniformly with respect to $r \in ]0,r_0[$ and $\vartheta \in ]0,1[$. This concludes the proof of {\it ii)}.
\noindent {\it Proof of iii)} \ We prove the existence of a constant $\vartheta \in ]0,1[$ as claimed by using a compactness argument and the parabolic scaling. We first observe that Lemma \ref{lem-localestimate*} implies that \begin{equation*}
K_r^{(m)}(z_0) \subset \overline{\Omega^{(m)*}_{3r}(z_0)} \cap
\Big\{ t \le t_0 - \tfrac{1}{4 \pi \lambda^{N/(N+m)}} \,r^2 \Big\}, \end{equation*} which is a compact subset of $\Omega^{(m)*}_{4r}(z_0)$. We now show that there exists $\vartheta \in ]0,1[$ such that \begin{equation} \label{eq-claim*}
\Omega_{3 \vartheta r}^{(m)*}(z) \subset \Omega_{4r}^{(m)*}(z_0)
\cap \Big\{\tau \le t_0 - \tfrac{r^2}{4 \pi \lambda^{N/(N+m)}}\Big\} \end{equation} for every $z \in \overline{\Omega^{(m)*}_{3r}(z_0)} \cap \Big\{ t \le t_0 - \frac{1}{4 \pi \lambda^{N/(N+m)}} \,r^2 \Big\}$. Our claim {\it iii)} will follow from \eqref{eq-claim*} and from Lemma \ref{lem-localestimate*}.
We next prove \eqref{eq-claim*} by using the parabolic scaling. We note that \begin{equation} \label{eq-parabolicscaling}
(x_0 + r \xi,t_0 + r^2 \tau) \in {\Omega^{(m)*}_{k r}(z_0)} \quad \Longleftrightarrow \quad
(x_0+\xi,t_0 + \tau) \in {\Omega^{(m)*}_{k}(z_0)}, \end{equation} for every positive $k$. We will need to use $k = 3, 4$ and $\vartheta$.
We next show that \eqref{eq-claim*} holds for $r=1$. The result for every $r \in ]0,r_0[$ will follow from \eqref{eq-parabolicscaling}. Let $\delta(z_0)$ be the distance of the compact set $\overline{\Omega_{3}^{(m)*}(z_0)} \cap \big\{t \le t_0 - \frac{1}{4 \pi \lambda^{N/(N+m)}}\big\}$ from the boundary of $\Omega_{4}^{(m)*}(z_0)$. We have that $\delta$ is a strictly positive function which depends continuously on $z_0$ through the coefficients of the matrix $A(z_0)$. Moreover, the condition \eqref{e-up} is satisfied, then there exists a positive constant $\delta_0$, only depending on $\lambda, \Lambda$ and $N$, such that \begin{equation*}
\delta(z_0) \ge \delta_0 \quad \text{for every} \ z_0 \in \Omega. \end{equation*} On the other hand, the diameter of the set $\Omega^{(m)*}_{1}(z)$ is bounded by a constant that doesn't depend on $z$. Then, by \eqref{eq-parabolicscaling}, it is possible to find $\vartheta \in ]0,1[$ such that the diameter of $\Omega^{(m)*}_{\vartheta}(z)$ is not greater than $\delta_0$. This concludes the proof of \eqref{eq-claim*} in the case $r=1$. As said above, the case $r \in ]0,r_0[$ does follow from \eqref{eq-parabolicscaling}. This concludes the proof of {\it iii)}.
\noindent {\it Proof of iv)} \ From \eqref{e-Omegam} it directly follows that \begin{equation*}
M_{5r}^{(m)} (z_0;\z) \ge \frac{m \, \omega_m }{m+2} \cdot \frac {N_{5r}^{m+2}(z_0;\z)}{4(t_0-\t)^2} \ge
\frac{m \, \omega_m}{m+2} (2(t_0-\t))^{m-2} \left( (N+m) \log(5/4)\right)^{(m+2)/2}. \end{equation*} The last claim then follows by choosing \begin{equation*}
m^-:= \lambda^{- N(m-2)/(N+m)}\frac{m \, \omega_m}{m+2} \frac{r^{2m-4}}{(2 \pi)^{m-2} } \left( (N+m) \log(5/4)\right)^{(m+2)/2}. \end{equation*}
We eventually remove the assumption $\div \, b - c = 0$. We mainly rely on the steps in the display \eqref{e-core-h} and we point out the needed changes for this more difficult situation. With this aim, we recall that $| \div \, b(z) - c(z)| \le k := (N+1) \Lambda $ because of \eqref{e-up}. We next introduce two auxiliary functions \begin{equation*}
\widehat u(x,t) := e^{k(t-t_0)} u(x,t), \qquad \widetilde u(x,t) := e^{-k(t-t_0)} u(x,t). \end{equation*} Note that, as $\L u = 0$, we have that \begin{equation*}
\widehat \L \, \widehat u : = \L \widehat u + k \widehat u = 0, \qquad
\widetilde \L \, \widetilde u : = \L \widetilde u - k \widetilde u = 0. \end{equation*} Note that $\widehat \L, \widetilde \L$ satisfy the condition \eqref{e-up} with $\Lambda$ replaced by $k$. In particular, the statemets \emph{i)}-\emph{iv)} hold for $\L, \widehat \L$ and $\widetilde \L$ with the same constants $r_0, \vartheta, M^+, m^-$.
We denote by $\widehat M^{(m)}, \widetilde M^{(m)}$ the kernels relative to $\widehat \L, \widetilde \L$, respectively, and $\widehat \Omega^{(m)}, \widetilde \Omega^{(m)}$ the superlevel sets we use in the representation formulas appearing in \eqref{e-core-h}. As we did before, we let $r^*$ be the constant appearing in Lemma \ref{lem-localestimate*} and we choose $r_0 := r^*/2$. As a direct consequence of the definiton of $\widehat u$ and $\widetilde u$, there exist two positive constants $\widehat c$ and $\widetilde c$ such that \begin{equation} \label{e-bounds-u}
\widehat c u(z) \le \widehat u(z) \le u(z), \qquad u(z) \le \widetilde u(z) \le \widetilde c u(z), \end{equation} for every $z \in \Omega_{5r}^{(m)}(z_0)$.
We are now in position to conclude the of proof Proposition \ref{prop-Harnack}. Let's consider the first two lines of \eqref{e-core-h}. Since $\widehat u$ is a solution to $\widehat \L \, \widehat u = 0$, for every $z \in K^{(m)}_{r}(z_0)$, it holds \begin{equation*}
\widehat u(z) \le \frac{1}{(\vartheta r)^{N+m}} \int_{\widehat \Omega^{(m)}_{\vartheta r}(z)}
\! \! \! \! \! \! \widehat M_{\vartheta r}^{(m)} (z; \zeta) \widehat u(\zeta) \, d\zeta \le
\frac{M^+}{(\vartheta r)^{N+m}}
\int_{\widehat \Omega^{(m)}_{\vartheta r}(z)} \!\!\! \widehat u(\zeta) \, d\zeta. \end{equation*} The first inequality follows from the fact that $\div \, b(\zeta) - c(\zeta) - k \le 0$ for every $\zeta$. From \eqref{e-bounds-u} it then follows that \begin{equation*}
u(z) \le \frac{M^+}{\widehat c \, (\vartheta r)^{N+m}}
\int_{\widehat \Omega^{(m)}_{\vartheta r}(z)} \!\!\! u(\zeta) \, d\zeta. \end{equation*} Continuing along the next lines of \eqref{e-core-h}, we note that {\it iii)} also holds in this form: $\widehat \Omega_{\vartheta r}^{(m)}(z) \subset \widetilde \Omega_{4r}^{(m)}(z_0) \cap \big\{\tau \le t_0 - \frac{r^2}{4 \pi \lambda^{N/(N+m)}}\big\}$ for every $z \in K^{(m)}_{r}(z_0)$, so that \begin{equation*}
u(z) \le \frac{M^+}{\widehat c \, (\vartheta r)^{N+m}}
\int_{\widetilde \Omega_{4r}^{(m)}(z_0) \cap \big\{\tau \le t_0 - \frac{r^2}{4 \pi \lambda^{N/(N+m)}}\big\}}
\!\! u(\zeta) \, d\zeta. \end{equation*} On the other hand, using the fact that $\widetilde \L \, \widetilde u = 0$, and $\div \, b(\zeta) - c(\zeta) + k \ge 0$, we find \begin{equation*}
\frac{m^-}{(5 r)^{N+m}} \int_{\widetilde \Omega_{4r}^{(m)}(z_0) \cap \big\{\tau \le t_0 - \frac{r^2}{4 \pi \lambda^{N/(N+m)}}\big\}}
\!\!\! \widetilde u(\zeta) \, d\zeta \le
\frac{1}{(5 r)^{N+m}} \int_{\widetilde \Omega_{5r}^{(m)}(z_0)} \! \! \! \! \! \! \widetilde M_{5 r}^{(m)} (z_0; \zeta) \widetilde u(\zeta) \, d\zeta \le
\widetilde u(z_0). \end{equation*} Thus, recalling that $u \le \widetilde u$ and $u (z_0) = \widetilde u(z_0)$, we conclude that \begin{equation*}
u(z) \le
\frac{5^{N+m} M^+}{\widehat c \, \vartheta^{N+m} m^-} u(z_0), \end{equation*} for every $z \in K^{(m)}_{r}(z_0)$. This concludes the proof of Proposition \ref{prop-Harnack}. \end{proof}
As a simple consequence of Proposition \ref{prop-Harnack} we obtain the following result.
\begin{corollary} \label{cor-Harnack-inv} There exist four positive constants $r_1, \kappa_1, \vartheta_1$ and $C_D$, with $\kappa_1, \vartheta_1 < 1$, such that the following inequality holds. For every $z_0 \in \Omega$ and for every positive $r$ such that $r \le r_1$ and ${\Q_{r}(z_0)} \subset \Omega$ we have that
\begin{equation} \label{e-HD}
\sup_{D_{r}(z_0)} u \le C_D u(z_0) \end{equation} for every $u \ge 0$ solution to $\L u = 0$ in $\Omega$. Here
\begin{equation*}
D_{r}(z_0) := B_{\vartheta_1 r}(x_0) \times \{t_0 - \kappa_1 r^2\}. \end{equation*} \end{corollary}
\begin{center} \begin{tikzpicture} \clip(-.52,6.82) rectangle (7.02,1.48); \path[draw,thick] (-.5,6.8) rectangle (7,1.5); \begin{axis}[axis y line=none, axis x line=none,
xtick=\empty,ytick=\empty,
ymin=-1.1, ymax=1.1,
xmin=-.2,xmax=1.8, samples=101, rotate= -90] \addplot [black,line width=.7pt, domain=-.01:.01] {sqrt(.0001 - x * x)} node[above] {$z_0$}; \addplot [black,line width=.7pt, domain=-.01:.01] {-sqrt(.0001 - x * x)}; \addplot [ddblue,line width=.7pt, domain=.001:1] {sqrt(- 2 * x * ln(x))}; \addplot [ddblue,line width=.7pt,domain=.001:1] {- sqrt(- 2 * x * ln(x))} node[below] { \hskip30mm {\color{dddblue} $\Omega_r(z_0)$}}; \addplot [bblue,line width=.7pt,domain=.001:.1111] {sqrt(- 3 * x * ln(9*x)}; \addplot [bblue,line width=.7pt,domain=.001:.1111] {- sqrt(- 3 * x * ln(9*x))}; \end{axis} \draw [line width=.6pt] (0,6.165) rectangle node [above=1.5cm,right=2.8cm] {$Q_r(z_0)$} (5.65,2); \draw [red,line width=1.2pt] (2.1,5.9) -- node [below=5pt] { \hskip20mm $D_r(z_0)$} (3.6,5.9); \end{tikzpicture}
{\sc \qquad Fig.5} - The set $D_r(z_0)$. \end{center}
The above assertion follows from the fact that there exists a positive constant $\delta_1$ such that $\Omega_{r}^{(m)}(z_0) \subset \Q_{\delta_1 r}(z_0)$ for every $r \in 0], r_0[$ and that $D_{r}(z_0) \subset K^{(m)}_{r}(z_0)$, for some positive $\kappa_1, \vartheta_1$. Note that Corollary \ref{cor-Harnack-inv} and Theorem \ref{th-Harnack-inv} differ in that, unlike the cylinders $\Q^+_{r}(z_0)$ and $\Q^-_{r}(z_0)$, the set $D_{r}(z_0)$ is not arbitrary. We next prove Theorem \ref{th-Harnack-inv} by using iteratively the Harnack inequality proved in Corollary \ref{cor-Harnack-inv}.
\begin{proof} {\sc of Theorem \ref{th-Harnack-inv}.} As a first step we note that, up to the change of variable $v(x,t) := u(x_0 + r t, t_0 +r^2 t)$, it is not restrictive to assume that $z_0 = 0$ and $r = 1$. Indeed, the function $v$ is a solution to an equation $\widehat \L v = 0$, where the coefficients of the operator $\widehat \L$ are $\widehat a_{ij}(x,t) = a_{ij}(x_0 + r t, t_0 +r^2 t)$ satisfy all the assumptions made for $\L$, with the constants $M$ and $\Lambda$ appearing in \eqref{e-hc} and \eqref{e-up} replaced by $r^\alpha M$ and $r^\alpha \Lambda$, respectively, and the same constant $\lambda$ in \eqref{e-up}. Then, as $r \in ]0, R_0]$, the H\"older constant in \eqref{e-hc} of $\widehat \L$ is $R_0^{\, \alpha} M$ and the parabolicity constants in \eqref{e-up} are $\lambda$ and $R_0^{\, \alpha} \Lambda$, for every $r \in ]0, R_0]$. In the following we then assume that $z_0 = 0$ and $r=1$. Moreover, $r_1$ denotes the constant appearing in Corollary \ref{cor-Harnack-inv} and relative to $\widehat \L$, which depends on the constants $M, \lambda, \Lambda$ and $R_0$. We then choose four positive constants $\iota, \kappa, \mu, \vartheta$ with $0 < \iota < \kappa < \mu < 1$ and $0 < \vartheta < 1$ and we consider the cylinders $\Q^+ := \Q_1^+(0)$ and $\Q^- := \Q_1^-(0)$ as defined in \eqref{e-QPM}. We let \begin{equation*}
r_0 := \min \big\{r_1, 1 - \vartheta, \sqrt{1 - \mu} \big\} \end{equation*} and we note that $\Q_r(z) \subset \Q_1(0)$ whenever $z \in B(0,\vartheta) \times ]- \mu, 0[$ and $0 < r < r_0$.
We next choose any $z^- = (x^-,t^-) \in \Q^-, z^+ = (x^+,t^+) \in \Q^+$ and we rely on Corollary \ref{cor-Harnack-inv} to construct a \emph{Harnack chain}, that is a finite sequence $w_0, w_1, \dots, w_m$ in $\Q_1(0)$ such that \begin{equation} \label{eq-HC}
w_0 = z^+, \qquad w_k = z^-, \qquad u(w_j) \le C_D u(w_{j-1}), \quad j=1, \dots, m. \end{equation}
\begin{center} \begin{tikzpicture} \path[draw,thick] (-1,.7) rectangle (8.7,6.5); \filldraw [fill=black!4!white, line width=.6pt] (1.5,6) rectangle node[right=2cm] {$Q^+_r(z_0)$} (5.5,4.5); \filldraw [fill=black!4!white, line width=.6pt] (1.5,2) rectangle node[right=2cm] {$Q^-_r(z_0)$} (5.5,3.5); \draw [line width=.6pt] (0,6) rectangle node[right=3.5cm] {$Q_r(z_0)$}(7,1); \draw [line width=.6pt] (3.5,6) circle (1pt) node[above] {$z_0$}; \foreach \x in {0,1,...,5} \draw [xshift=\x*.4 cm,yshift=-\x*.2 cm,bblue,line width=.6pt] (.4,3.6) rectangle (3.2,4.6); \foreach \x in {0,1,...,5} \draw [xshift=\x*.4 cm,yshift=-\x*.2 cm,red,line width=1.2pt] (1.4,4.4) -- (2.2,4.4); \foreach \x in {0,1,...,5} \draw [xshift=\x*.4 cm,yshift=-\x*.2 cm,line width=1pt] (2.2,4.4) circle (1pt); \draw [line width=1pt] (1.8,4.6) circle (1pt) node[above] {$z^+$}; \draw [line width=1pt] (4.2,3.4) circle (1pt) node[above=2pt] {$\, z^-$}; \end{tikzpicture}
{\sc \qquad Fig.6} - A Harnack chain. \end{center}
We build a Harnack chain as follows. For a positive integer $m$ that will be fixed in the sequel, we choose a positive $r$ and the vector $y \in \mathbb{R}^N$ satisfying \begin{equation} \label{eq-y}
m \kappa_1 r^2 = t^+-t^-, \qquad m r y = x^+ - x^-. \end{equation} Let $\kappa_1,\ \vartheta_1$ be the constants in Corollary \ref{cor-Harnack-inv}. We define \begin{equation} \label{eq-wj}
w_j := (x^+ + j r y, t^+ - j \kappa_1 r^{2}), \qquad j=0,1, \dots, m. \end{equation}
Clearly, if $r \le r_0$, then $\Q_r(w_j) \subset \Q_1(0)$ for $j=0, 1, \dots, m$. If moreover $|y| \le \vartheta_1$, then $w_{j} \in D_{r}(w_j-1)$ for $j=1, \dots, m$. This proves that \eqref{eq-HC} holds, and we conclude that \begin{equation} \label{eq-H+-}
u(z^-) \le C_D^{\, m} u(z^+). \end{equation}
We next choose $m$ in order to have both condtions $r \le r_0$ and $|y| \le \vartheta_1$ satisfied.
The choice of $m$ is different in the case $|x^+ - x^-|$ is \emph{small} or \emph{large} with respect to $t^+-t^-$. If \begin{equation} \label{eq-case1}
\frac{|x^+-x^-|}{t^+-t^-} \le \frac{\vartheta_1}{\kappa_1 r_0}, \end{equation} we let $m$ be the positive integer satisfying \begin{equation} \label{eq-mm}
(m-1) \kappa_1 r_0^{\, 2} < t^+ - t^- \le m \kappa_1 r_0^{\, 2}, \end{equation}
and, in accordance with \eqref{eq-y}, we choose $r$ as the unique positive number satisfying $m \kappa_1 r^2 = t^+-t^-$. From \eqref{eq-mm} it directly follows $r \le r_0$, while from \eqref{eq-mm} and \eqref{eq-case1} we obtain $|y| \le \vartheta_1$.
Suppose now that \begin{equation} \label{eq-case2}
\frac{|x^+-x^-|}{t^+-t^-} > \frac{\vartheta_1}{\kappa_1 r_0}. \end{equation} In view of \eqref{eq-y}, in this case we choose $m$ as the integer satisfying
\begin{equation} \label{eq-m}
m - 1 < \frac{\kappa_1 |x^+-x^-|^2}{\vartheta_1^{\, 2} (t^+-t^-)} \le m, \end{equation} and we let $y$ be the vector parallel to $x^- - x^+$ and such that \begin{equation*}
m |y| = \frac{\kappa_1 |x^+-x^-|^2}{{\vartheta_1 (t^+-t^-)}}. \end{equation*}
Clearly, $|y| \le \vartheta_1$, and \eqref{eq-case2} implies $r \le r_0$.
We next find a bound for the integer $m$, which is uniform with respect to $z^- \in \Q^-$ and $z^+ \in \Q^+$, and we rely on \eqref{eq-H+-} to conclude the proof. In the first case \eqref{eq-case1} we obtain from \eqref{eq-mm} that $m \le \frac{t^+-t^-}{\kappa_1 r_0^{\, 2}}$. In the second case \eqref{eq-case2} we rely on \eqref{eq-m} and we note that $t^+-t^- \ge \kappa- \iota$, by our choiche of $\Q^-$ and $\Q^+$. Then in this case we have $m < \frac{ 4\kappa_1}{\vartheta_1^{\, 2} (\kappa - \iota)}$ Summarizing, we have proved that the inequality \eqref{e-H1} holds with \begin{equation*}
C_H := \exp \left( \max \big\{ \tfrac{1}{\kappa_1 r_0^{\, 2}},
\tfrac{4\kappa_1}{\vartheta_1^{\, 2} (\kappa - \iota)} \big\} \log C_D \right). \end{equation*} \end{proof}
\setcounter{equation}{0} \section{An approach relying on sets of finite perimeter}\label{SectionBV}
In this section we present another approach to the generalized divergence theorem, relying on De Giorgi's theory of perimeters, see \cite{DeGiorgi2,DeGiorgi3} or \cite{AmbrosioFuscoPallara,Maggi}, and we show how this leads to a slightly different proof of Theorem \ref{th-1}. This approach requires more prerequisites than that used in Section \ref{SectionDivergence}, but, as explained in the Introduction, is more flexible and avoids the Dubovicki\v{\i} theorem. In this section, if $\mu$ is a Borel measure and $E$ is a Borel set, we use the notation $\mu\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} E(B)=\mu(E\cap B)$. As before, $C^1_c \pr{\Omega}$ denotes the set of $C^1$ functions compactly supported in the open set $\Omega \subset \mathbb{R}^n$.
\begin{definition}[\sc $BV$ Functions] Let $u \in L^1 \pr{\Omega}$; we say that $u$ is a function of bounded variation in $\Omega$ if its distributional derivative $Du=\pr{D_1 u,\ldots,D_n u}$ is an $\mathbb{R}^n$-valued Radon measure in $\Omega$, {\em i.e.}, if \[
\int_{\Omega}u \frac{\partial \varphi}{\partial z_i}\ dz=-\int_{\Omega}\varphi \, dD_i u,
\quad \forall \: \varphi \in C_c^1 \pr{\Omega}, \; i=1,\ldots,n \] or, in vectorial form, \begin{equation} \label{e-bv-div}
\int_{\Omega}u\, \mathrm{div}\,\Phi\ dz=-\sum_{i=1}^n \int_{\Omega}\Phi_i\ d D_i u
= -\int_{\Omega}\langle\Phi, Du\rangle , \quad \forall \: \Phi \in C_c^1 \pr{\Omega;\mathbb{R}^n}. \end{equation} The vector space of all functions of bounded variation in $\Omega$ is denoted by $BV \pr{\Omega}$. The variation $V \pr{u,\Omega}$ of $u$ in $\Omega$ is defined by: $$
V \pr{u,\Omega}:=\sup \left\{ \int_{\Omega}u\, \mathrm{div}\,\Phi\ dz:
\Phi \in C^1_c \pr{\Omega;\mathbb{R}^{n}},\, \norm{\Phi}_{\infty} \leq 1 \right\}. $$ \end{definition} We recall that ${V \pr{u,\Omega}=\abs{Du}\pr{\Omega}}<\infty$ for any $u \in BV \pr{\Omega}$, where $\abs{Du}$ denotes the total variation of the measure $Du$. We also recall that if $u \in C^1 \pr{\Omega}$ then $$ V \pr{u,\Omega}=\int_{\Omega} \abs{\nabla u} dz. $$
When the function $u$ is the characteristic functions $\chi_E$ of some measurable set, its variation is said \emph{ perimeter of $E$}. \begin{definition}[\sc Sets of finite perimeter] Let $E$ be an $\mathcal{L}^n-$measurable subset of $\mathbb{R}^n$. For any open set $\Omega \subset \mathbb{R}^n$ the perimeter of $E$ in $\Omega$ is denoted by $P \pr{E,\Omega}$ and it is the variation of $\rchi_{E}$ in $\Omega$, {\em i.e.}, \begin{equation*} P \pr{E,\Omega}:=\sup \left\{\int_{E}\mathrm{div}\, \Phi\ dz:\Phi \in C^1_c \pr{\Omega;\mathbb{R}^n},\, \norm{\Phi}_{\infty}\leq 1 \right\}. \end{equation*} We say that $E$ is a set of finite perimeter in $\Omega$ if $P \pr{E,\Omega}<\infty$. \end{definition}
Obviously, several properties of the perimeter of $E$ can be stated in terms of the variation of $\chi_E$. In particular, if $\mathcal{L}^n(E \cap \Omega)$ is finite, then $\rchi_{E} \in L^1 \pr{\Omega}$ and $E$ has finite perimeter in $\Omega$ if and only if $\rchi_{E}\in BV \pr{\Omega}$ and $P \pr{E,\Omega} = \abs{D{\rchi_{E}}}\pr{\Omega}$. Both the notations $|D\chi_E|(B)$ and $P(E,B)$, $B$ Borel, are used to denote the total variation measure of $\chi_E$ on a Borel set $B$ and we say that $E$ is a set of locally finite perimeter in $\Omega$ if $P(E,K)<\infty$ for every compact set $K \subset \Omega$. Finally, formula \eqref{e-bv-div} looks like a divergence theorem: \begin{equation} \label{e-per-div}
\int_{E}\, \mathrm{div}\, \Phi\ dz=- \int_{\Omega}\langle\Phi,D\chi_E\rangle, \quad \forall \: \Phi \in C_c^1 \pr{\Omega;\mathbb{R}^n}, \end{equation} but it becomes more readable if some precise information is given on the set where the measure $D\chi_E$ is concentrated. Therefore, we introduce the notions of {\em reduced boundary} and of {\em density} and recall the structure theorem for sets with finite perimeter due to E. De Giorgi, see \cite{DeGiorgi3} and \cite[Theorem 3.59]{AmbrosioFuscoPallara}, and the characterization due to H. Federer.
\begin{definition}[\sc Reduced boundary] Let $\Omega$ be an open subset of $\mathbb{R}^n$ and let $E$ be a set of locally finite perimeter in $\Omega$.
We say that $z \in \Omega$ belongs to the reduced boundary ${\mathcal F}E$ of $E$ if $|D\chi_E|(B_\r(z))>0$ for every $\r>0$ and the limit $$
\nu_E\pr{z}:=\lim_{\r \to 0^+}\frac{D{\rchi_{E}}\pr{B_{\r}\pr{z}}}{\abs{D{\rchi_{E}}}\pr{B_{\r}\pr{z}}} $$ exists in $\mathbb{R}^n$ and satisfies $\abs{\nu_E \pr{z}}=1$. The function $\nu_E:{\mathcal F}E \rightarrow {\mathbb S}^{n-1}$ is Borel continuous and it is called the {\em generalized (or measure-theoretic) inner normal} to $E$. \end{definition} Notice that the reduced boundary is a subset of the topological boundary. The Besicovitch differentiation theorem, see e.g. \cite[Theorem 2.22]{AmbrosioFuscoPallara}, yields $D\rchi_{E} = \nu_E\abs{D\rchi_{E}}$, and $\abs{D\rchi_{E}}(\Omega\setminus{\mathcal F}E)=0$, hence \eqref{e-per-div} becomes \begin{equation} \label{e-3} \int_{E}\mathrm{div}\, \Phi\ dz=-\int_{{\mathcal F}E}\scp{\nu_E,\Phi}\ d \abs{D\rchi_{E}}, \quad \forall \: \Phi \in C^1_c \pr{\Omega;\mathbb{R}^N}. \end{equation} \begin{comment} The following celebrated \emph{De Giorgi’s structure theorem} gives a further improvement of the above identity. Let us recall that a set $S\subset\mathbb{R}^n$ is {\em countably $(n-1)-$rectifiable} if there exist countably many Lipschitz functions $\psi_j:\mathbb{R}^{n-1}\to\mathbb{R}^n$ such that \[ {\cal H}^{n-1}\left(S\setminus\bigcup_{j=0}^\infty \psi_j(\mathbb{R}^{n-1})\right)=0 \] and ${\cal H}^{n-1}(S)<\infty$.
\begin{theorem}[\sc De Giorgi] Let $E$ be an $\mathcal{L}^n-$measurable subset of $\mathbb{R}^n$. Then ${\mathcal F}E$ is countably $\pr{n-1}-$rectifiable and $\abs{D{\rchi_{E}}}=\H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \mathcal{F}E$. In addition, for any $z_0 \in {\mathcal F}E$ the following statements hold: \begin{description} \item[$(a)$] the sets $\frac{1}{\r}\pr{E-z_0}$ locally converge in measure in $\mathbb{R}^n$ as $\r \to 0^+$ to the half-space $H$ orthogonal to $\nu_E \pr{z_0}$ and containing $\nu_E \pr{z_0}$; \item[$(b)$] $\mathrm{Tan}^{n-1} \pr{\H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} {\mathcal F} E,z_0}=\H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \nu_E^\perp \pr{z_0}$, {\em i.e.}, \[ \lim_{\varrho\to 0+}\frac{1}{\varrho^{n-1}} \int_{{\mathcal F}E}\phi\Bigl(\frac{z-z_0}{\varrho}\Bigr) d\H^{n-1}(z) =\int_{\nu_E^\perp(z_0)} \Phi(z)\ d\H^{n-1}(z). \] In particular, \begin{equation*} \lim_{\r \to 0^+}\frac{\H^{n-1}\pr{{\mathcal F}E \cap B_{\r}\pr{z_0}}}{\omega_{n-1}\r^{n-1}}=1, \end{equation*} where $\omega_{n-1}$ is the Lebesgue measure of the unit ball in $\mathbb{R}^{n-1}$. \end{description} \end{theorem} Here $\mathrm{Tan}^{n-1}\pr{\mu,z}$ denotes the \emph{approximate tangent space} to a Radon measure $\mu$ at point $z$ while $\mu \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} E$ denotes the \emph{restriction} of a measure $\mu$ to $E$ (see Definitions 1.65 and 2.79 in \cite{AmbrosioFuscoPallara}). \end{comment}
The relation between the topological boundary and the reduced boundary can be further analyzed by introducing the notion of {\em density} of a set at a given point.
\begin{definition}[\sc Points of density $\alpha$] For every $\alpha \in \left[0,1\right]$ and every $\mathcal{L}^n-$measurable set $E \subset \mathbb{R}^n$ we denote by $E^{\pr{\alpha}}$ the set $$ E^{\pr{\alpha}}=\left\{ z \in \mathbb{R}^n: \lim_{\r \to 0^+} \frac{\mathcal{L}^n(E \cap B_\r \pr{z})}{\mathcal{L}^n(B_\r \pr{z})}=\alpha\right\}. $$ \end{definition} Thus $E^{\pr{\alpha}}$, which turns out to be a Borel set, is the set of all points where $E$ has density $\alpha$. The sets $E^{\pr{0}}$ and $E^{\pr{1}}$ are called \emph{the measure-theoretic exterior} and \emph{interior} of $E$ and, in general, strictly contain the topological exterior and interior of the set $E$, respectively. We recall the well known \emph{Lebesgue's density theorem}, that asserts that for every $\mathcal{L}^n-$measurable set $E \subset \mathbb{R}^n$ $$ \mathcal{L}^n(E \triangle E^{\pr{1}})=0, \quad \mathcal{L}^n(\pr{\mathbb{R}^n \setminus E} \triangle E^{\pr{0}})=0, $$ {\em i.e.}, the density of $E$ is $0$ or $1$ at $\mathcal{L}^n-$almost every point in $\mathbb{R}^n$. This notion allows to introduce the {\em essential} or {\em measure-theoretic} boundary of $E$ as $\partial^*E=\mathbb{R}^n\setminus (E^{\pr{0}}\cup E^{\pr{1}})$, which is contained in the topological boundary and contains the reduced boundary. Finally, the De Giorgi structure theorem says that
$|D\rchi_E|=\H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}{\mathcal F}E$ and a deep result due to Federer (see \cite[4.5.6]{federer1969geometric} or \cite[Theorem 3.61]{AmbrosioFuscoPallara}) states that if $E$ has finite perimeter in $\mathbb{R}^n$ then \[ \mathcal{F}E\subset E^{\pr{1/2}}\subset \partial^*E \quad \text{and}\quad \H^{n-1}(\mathbb{R}^n\setminus (E^{\pr{0}}\cup\mathcal{F}E\cup E^{\pr{1}}))=0 \] hence, in particular, $\nu_E$ is defined $\H^{n-1}-$a.e. in $\partial^*E$. Notice also (see \cite[Theorem 3.62]{AmbrosioFuscoPallara}) that if $\H^{n-1}(\partial E)<\infty$ then $E$ has finite perimeter. The results of De Giorgi and Federer imply that if $E$ is a set of finite perimeter in $\Omega$ then $D \rchi_E=\nu_E \H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \mathcal{F}E$ and the divergence theorem \eqref{e-3} can be rewritten in the form: \begin{equation}\label{e-rem-1} \int_E \mathrm{div}\, \Phi\ dz= -\int_{{\mathcal F}E}\scp{\nu_E,\Phi}\ d \H^{n-1}= -\int_{\partial^*E}\scp{\nu_E,\Phi}\ d \H^{n-1}, \quad \forall \: \Phi \in C^1_c \pr{\Omega;\mathbb{R}^n}, \end{equation} much closer to the classical formula \eqref{eq-div}. Indeed, the only difference is that the inner normal and the boundary are understood in a measure-theoretic sense and not in the topological one; in particular, for a generic set of finite perimeter, ${\mathcal F}E$ needs not to be closed and $\nu_E$ needs not to be continuous. Moreover, $\nu_E$ is defined $\H^{n-1}-$a.e. in $\partial^*E$.
Let us see now how we can rephrase the results of Section \ref{SectionDivergence} in terms of perimeters and how we can modify the proof of Theorem \ref{th-1}. We first recall the Fleming--Rischel formula (see \cite{fleming} or \cite[Theorem 3.40]{AmbrosioFuscoPallara}), {\em i.e.}, the coarea formula for $BV$ functions. \begin{theorem}[\sc Coarea formula in $BV$] \label{th-cobv} For any open set $\Omega \subset \mathbb{R}^n$ and $G \in L^1_{\mathrm{loc}}\pr{\Omega}$ one has $$ V \pr{G,\Omega}=\int_{\mathbb{R}}P\pr{\{z \in \Omega:G \pr{z}>y\},\Omega}\ dy. $$ In particular, if $G \in BV \pr{\Omega}$ the set $\left\{G >y \right\}$ has finite perimeter in $\Omega$ for $\H^1-$a.e.~$y \in \mathbb{R}$ and $$ \abs{DG}\pr{B}=\int_{\mathbb{R}}\abs{D{\rchi_{\left\{G>y\right\}}}}\pr{B}\ dy, \quad DG\pr{B}=\int_{\mathbb{R}}D{\rchi_{\left\{G>y\right\}}}\pr{B}\ dy, \quad \forall \: B \in \mathcal{B}\pr{\Omega}. $$ \end{theorem}
Now we are ready to state the analogue of Proposition \ref{prop-1} and to prove Theorem \ref{th-1} again.
\begin{proposition} \label{prop-1BV} Let $\Omega$ be an open subset of $\mathbb{R}^{n}$ and let $F \in BV \pr{{\Omega};\mathbb{R}}\cap C \pr{{\Omega};\mathbb{R}}$. Then, for $\H^1-$almost every $y \in \mathbb{R}$, we have: \begin{equation}\label{cobv}
\int_{\left\{F>y\right\}} \mathrm{div}\, \Phi\ dz =
-\int_{\partial^*\{F>y\}} \scp{\nu,\Phi}\ d \H^{n-1},
\quad \forall \: \Phi \in C_c^1 \pr{\Omega;\mathbb{R}^{n}}, \end{equation} were $\nu$ is the generalized inner normal to $\{F>y\}$. \end{proposition}
\begin{proof} By Theorem \ref{th-cobv}, the set $\left\{F>y\right\}$ has finite perimeter in $\Omega$ for $\H^1-$a.e. $y \in \mathbb{R}$, hence we may apply \eqref{e-rem-1} with $E=\{F>y\}$ and conclude. \end{proof}
As in Section \ref{SectionDivergence}, we have to cut the integration domain: therefore, we study the intersection between the super-level set of a generic function $G \in BV\pr{\Omega;\mathbb{R}}\cap C\pr{\Omega;\mathbb{R}}$ and a half-space $H_t=\left\{ x \in \mathbb{R}^n: \scp{x,e}<t \right\}$, for some $e \in {\mathbb S}^{n-1}$, $t \in \mathbb{R}$. First, we present a general formula that characterizes the intersection of two sets of finite perimeter for which we refer to Maggi's book, see \cite[Theorem 16.3] {Maggi}.
\begin{theorem}[\sc Intersection of sets of finite perimeter] \label{th-int} If $A$ and $B$ are sets of locally finite perimeter in $\Omega$, and we let $$ \left\{ \nu_A = \nu_B \right \} = \left\{ x \in {\mathcal F}A \cap {\mathcal F}B: \nu_A \pr{x}=\nu_B \pr{x} \right\}, $$ then $A \cap B$ is a set of locally finite perimeter in $\Omega$, with \begin{equation} \label{e-9} D \rchi_{A \cap B}=D \rchi_A \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} B^{\pr{1}}+D \rchi_B \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} A^{\pr{1}} + \nu_A \H^{n-1} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \left\{ \nu_A = \nu_B \right\}. \end{equation} \end{theorem} In the case in which $B$ is a half-space, formula \eqref{e-9} can be greatly simplified; indeed we can prove the following corollary.
\begin{corollary}[\sc Intersections with a half-space]\label{cor-1} Let $E$ be a set of locally finite perimeter in $\Omega$ and let $H_t=\left\{ z \in \mathbb{R}^n : \scp{z,e} < t\right\}$ for some $e \in {\mathbb S}^{n-1}$, $t \in \mathbb{R}$. Then, for every $t \in \mathbb{R}$, $E \cap H_t$ is a set of locally finite perimeter in $\Omega$ and moreover, for $\H^{1}-$almost every $t \in \mathbb{R}$, $$ D \rchi_{E \cap H_t}=D \rchi_E \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} H_t-e\H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \pr{E \cap \left\{ \scp{x,e}=t\right\}}. $$ \end{corollary} \begin{proof} The half-space $H_t$ is clearly a set of locally finite perimeter in $\Omega$ for every $t\in\mathbb{R}$, and for every $t \in \mathbb{R}$ we have, $H_t^{\pr{1}}=H_t$, $\mathcal{F}H_t=\partial H_t =\left\{ \scp{x,e}=t \right\}$ and $\nu_{H_t} \equiv -e$. Then, applying Theorem \ref{th-int} we see that $E \cap H_t$ is a set of locally finite perimeter in $\Omega$ for every $t \in \mathbb{R}$ and \eqref{e-9} reads \[ D \rchi_{E \cap H_t}=D \rchi_E \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} H_t - e \H^{n-1} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} (E^{\pr{1}}\cup \left\{ \nu_E = \nu_{H_t} \right\}) . \] Since by Fubini theorem \[ 0=\mathcal{L}^n (E\triangle E^{(1)}) = \int_{\mathbb{R}} \H^{n-1}\pr{(E\triangle E^{(1)}) \cap \left\{ \scp{x,e}=t \right\}}\ dt , \] for $\H^1-$a.e. $t \in \mathbb{R}$ we have \[ \H^{n-1}\pr{E \triangle E^{\pr{1}} \cap \left\{ \scp{x,e}=t \right\}}=0. \] Therefore, \[ D \rchi_{H_t} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} E = D \rchi_{H_t} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} E^{\pr{1}} = -e\H^{n-1} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} (E\cap\{\scp{x,e}=t\}) = -e\H^{n-1} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} (E\cap\{\nu_E=\nu_{H_t}\}) \] for $\H^1-$a.e. $t \in \mathbb{R}$ and the thesis follows. \end{proof}
The following corollary allows us to perform (with some modifications) the last part of the proof of our main result.
\begin{corollary}\label{prop-2-BV} Let $\Omega=\mathbb{R}^{N+1} \setminus \left\{ \pr{z_0} \right\}$, $G \in BV\pr{\Omega;\mathbb{R}} \cap C \pr{\Omega;\mathbb{R}}$, $H_t=\left\{ z \in \mathbb{R}^{N+1} : \scp{z,e} < t\right\}$ for some $e \in {\mathbb S}^N$, $t \in \mathbb{R}$. Then, for $\H^{1}-$almost every $w \in \mathbb{R}$ and for every $t < t_0$ the set $E \cap H_t$ has locally finite perimeter in $\Omega$ and $$ \int_{\left\{ G > w \right\} \cap H_t}\mathrm{div}\, \Phi\ dz= -\int_{\partial^*\{ G > w \} \cap H_t}\scp{\nu,\Phi}\ d \H^N +\int_{\left\{ G >w \right\} \cap \left\{ \scp{x,e}=t \right\}} \scp{e,\Phi}\ d\H^N, $$ for every $\Phi \in C^1_c \pr{\Omega;\mathbb{R}^n}$, where $\nu$ is the generalized inner normal to $\partial^*(\{G>w\})$. In particular, if $e=\pr{0,\ldots,0,1}$, for every $\varepsilon > 0$ $$ \int_{\left\{ G > w \right\} \cap \left\{ t<t_0-\varepsilon \right\}} \!\!\!\mathrm{div}\, \Phi\ dz =-\int_{\partial^*\{ G>w \} \cap \left\{ t<t_0-\varepsilon \right\}} \!\!\scp{\nu,\Phi}d \H^{N} +\int_{\left\{ G>w \right\} \cap \left\{ t=t_0-\varepsilon \right\}} \!\!\scp{e,\Phi}d\H^{N}. $$ \end{corollary}
Notice that the difference between Proposition \ref{prop-1bis} and Corollary \ref{prop-2-BV} is that in the former we can exclude the set of critical points of $G$ from the surface integral, thanks to Dubovicki\v{\i} theorem, and we know that $\nu$ is given by the normalized gradient of $G$ {\em everywhere} in the integration set, whereas in the latter we don't need to know any estimate on the size of ${\rm Crit}(G)$ and $\nu$ is defined $\H^N-${\em a.e.} on the integration set (still coinciding with the normalized gradient of $G$ out of ${\rm Crit}(G)$, of course). First, notice that we apply Corollary \ref{prop-2-BV} to $G(z)=\Gamma(z_0;z)$, which is $C^1(\Omega)$, hence Lipschitz on bounded sets. As a consequence, $\partial\{G>w\}\subseteq\{G=w\}$ and comparing the coarea formulas \eqref{e-co} and \eqref{cobv}, we deduce that $\H^N(\{G=w\}\setminus\partial^*\{G>w\})=0$ for $\H^{1}$-a.e. $w$. Let us see how this entails modifications of the proof of Theorem \ref{th-1}: the proof goes in the same vein until \eqref{eq-div-3i}, \eqref{eq-div-3}, which in the present context are replaced by \begin{align*}
\lim_{k \to +\infty} \int_{\psi_r(z_0) \cap \left\{ t<t_0-\varepsilon_k \right\}} \scp{\nu,\Phi}d \H^{N} & = \int_{\psi_r(z_0)} K (z_0;z) u(z) d \H^{N} \\ & = \int_{\psi_r(z_0)\setminus \mathrm{Crit}\pr{\Gamma}} K (z_0;z) u(z) d \H^{N} \end{align*} where the first equality follows from Corollary \ref{prop-2-BV}, as explained, and the last equality follows from the fact that the kernel $K$ vanishes in ${\rm Crit}(\Gamma)$. The rest of the proof needs no modifications.
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\begin{document}
\title{Maximum-Likelihood-Estimate Hamiltonian learning via efficient and robust quantum likelihood gradient} \author{Tian-Lun Zhao} \affiliation{International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China} \author{Shi-Xin Hu} \affiliation{International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China} \author{Yi Zhang} \email{[email protected]} \affiliation{International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China}
\date{Today}
\begin{abstract} Given the recent developments in quantum techniques, modeling the physical Hamiltonian of a target quantum many-body system is becoming an increasingly practical and vital research direction. Here, we propose an efficient strategy combining maximum likelihood estimation, gradient descent, and quantum many-body algorithms. Given the measurement outcomes, we optimize the target model Hamiltonian and density operator via a series of descents along the quantum likelihood gradient, which we prove is negative semi-definite with respect to the negative-log-likelihood function. In addition to such optimization efficiency, our maximum-likelihood-estimate Hamiltonian learning respects the locality of a given quantum system, therefore, extends readily to larger systems with available quantum many-body algorithms. Compared with previous approaches, it also exhibits better accuracy and overall stability toward noises, fluctuations, and temperature ranges, which we demonstrate with various examples. \end{abstract}
\maketitle
\section{Introduction}
Understanding the quantum states and the corresponding properties of a given quantum Hamiltonian is a crucial problem in quantum physics. Many powerful numerical and theoretical tools have been developed for such purposes and made compelling progress \cite{Lanczos, SteveWhite1992, dmrg, QMC, rgf}. On the other hand, with the rapid experimental developments of quantum technology, e.g., near-term quantum computation \cite{nielsen2002, Nayak2008} and simulation \cite{Buluta2009, Georgescu2014, Barthelemy2013, Browaeys2020, Scholl2021, Ebadi2022, Bluvstein2021}, it is also vital to explore the inverse problem, e.g., Hamiltonian learning - optimize a model Hamiltonian characterizing a quantum system with respect to the measurement results. Given the knowledge and assumption of a target system, researchers have achieved many resounding successes modeling quantum Hamiltonians with physical pictures and phenomenological approaches \cite{tbg, bcs}. However, such subjective perspectives may risk biases and are commonly insufficient on detailed quantum devices. Therefore, the explorations for objective Hamiltonian learning strategies have attracted much recent attention \cite{Qi2019, FETH, Corr1, RN571, Corr2, Entangle, dynamic1, Wenjunyu, Hsinyuan}.
There are mainly two categories of Hamiltonian-learning strategies, based upon either quantum measurements on a large number of (identical copies of) quantum states, e.g., Gibbs states or eigenstates \cite{Qi2019, FETH, Corr1, RN571, Corr2, Entangle}, or initial states' time evolution dynamics \cite{dynamic1, Wenjunyu, Hsinyuan, TNmethod}, corresponding to the target quantum system. For example, given the measurements of the correlations of a set of local operators, the kernel of the resulting correlation matrix offers a candidate model Hamiltonian \cite{Qi2019, FETH, Corr1}. On the other hand, while established theoretically, most approaches suffer from elevated costs and are limited to small systems in experiments or numerical simulations \cite{Corr1, RN571, RN572, Zoller}. Besides, there remains much room for improvements in stability towards noises and temperature ranges.
Maximum likelihood estimation (MLE) is a powerful tool that parameterizes and then optimizes the probability distribution of a statistical model so that the given observed data is most probable. MLE's intuitive and flexible logic makes it a prevailing method for statistical inference. Adding to its wide range of applications, MLE has been applied successfully to quantum state tomography\cite{QSE, PhysRevA.75.042108, Lvovsky_2004, PhysRevA.85.042317,PhysRevA.63.020101}, providing the most probable quantum states given the measurement outputs.
Inspired by MLE's successes in quantum problems, we propose a general MLE Hamiltonian learning protocol: given finite-temperature measurements of the target quantum system in thermal equilibrium, we optimize the model Hamiltonian towards the MLE step-by-step via a ``quantum likelihood gradient". We show that such quantum likelihood gradient, acting collectively on all presenting operators, is negative semi-definite with respect to the negative-log-likelihood function and thus provides efficient optimization. In addition, our strategy may take advantage of the locality of the quantum system, therefore allowing us to extend studies to larger quantum systems with tailored quantum many-body ansatzes such as Lanczos, quantum Monte Carlo (QMC), density matrix renormalization group (DMRG), and finite temperature tensor network (FTTN) \cite{FTTN, ITensor} algorithms in suitable scenarios. We also demonstrate that MLE Hamiltonian learning is more accurate, less restrictive, and more robust against noises and broader temperature ranges. Further, we generalize our protocol to measurements on pure states, such as the target quantum systems' ground states or quantum chaotic eigenstates. Therefore, MLE Hamiltonian learning enriches our arsenal for cutting-edge research and applications of quantum devices and experiments, such as quantum computation, quantum simulation, and quantum Boltzmann machines \cite{QBM}.
We organize the rest of the paper as follows: In Sec. II, we review the MLE context and introduce the MLE Hamiltonian learning protocol; especially, we show explicitly that the corresponding quantum likelihood gradient leads to a negative semi-definite change to the negative-log-likelihood function. Via various examples in Sec. III, we demonstrate our protocol's capability, especially its robustness against noises and temperature ranges. We generalize the protocol to quantum measurements of pure states in Sec. IV and Appendix D, with consistent results for exotic quantum systems such as quantum critical and topological models. We summarize our studies in Sec. V with a conclusion on our protocol's advantages (and limitations), potential applications, and future outlooks.
\section{Maximum-likelihood-estimate Hamiltonian learning}
To start, we consider an unknown target quantum system $\hat H_s = \sum_{j} \mu_{j} \hat{O}_{j}$ in thermal equilibrium, and measurements of a set of observables $\{\hat{O}_{i}\}$ on its Gibbs state $\hat{\rho}_{s}=\exp(-\beta\hat{H}_s)/\mbox{tr}[\exp(-\beta\hat{H}_s)]$, where $\beta$ is the inverse temperature. Given a sufficient number $N_i$ of measurements of the operator $\hat{O}_{i}$, the occurrence time $f_{\lambda_i}$ of the $\lambda_i^{th}$ eigenvalue $o_{\lambda_i}$ approaches: \begin{equation} f_{\lambda_i}= p_{\lambda_{i}}N_i\approx\mbox{tr}[\hat{\rho}_{s}\hat{P}_{\lambda_{i}}] N_i, \end{equation} where $p_{\lambda_i}=f_{\lambda_i}/N_i$ denotes the statistics of the outcome $o_{\lambda_i}$, and $\hat{P}_{\lambda_i}$ is the corresponding projection operator to the $o_{\lambda_i}$ sector. Our goal is to locate the model Hamiltonian $\hat H_s$ for the quantum system, which commonly requires the presence of all $\hat{H}_s$'s terms in the measurement set $\{\hat{O}_i\}$.
Following previous MLE analysis \cite{QSE, PhysRevA.75.042108, Lvovsky_2004, PhysRevA.85.042317,PhysRevA.63.020101}, the statistical weight of any given state $\hat \rho$ is: \begin{equation}
\mathcal{L}(\hat{\rho})\propto \prod_{i, \lambda_i}\{ \mbox{tr}[\hat{\rho}\hat{P}_{\lambda_i}]^{\frac{f_{\lambda_i}}{N_{tot}}}\}^{N_{tot}},
\label{LHF} \end{equation} upto a trivial factor, where $N_{tot}=\sum_{i}N_i$ is the total number of measurements. For Hamiltonian learning, we search for (the set of parameters ${\mu_j}$ of) the MLE Hamiltonian $\hat{H}$, whose Gibbs state $\hat \rho$ maximizes the likelihood function in Eq. \ref{LHF}. The maximum condition for Eq. \ref{LHF} can be re-expressed as: \begin{eqnarray}
&&\hat{R}(\hat{\rho}) \hat{\rho} = \hat{\rho}, \nonumber \\
&&\hat{R}(\hat{\rho}) = \sum_{i,\lambda_i}\frac{f_{\lambda_i}}{N_{tot}} \frac{\hat{P}_{\lambda_i}}{\mbox{tr} [\hat \rho \hat P_{\lambda_i}]}, \label{eq:mlerho} \end{eqnarray} see Appendix A for a detailed review. Solving Eq. \ref{eq:mlerho} is a nonlinear and nontrivial problem, for which many algorithms have been proposed \cite{ Lvovsky_2004, PhysRevA.75.042108, PhysRevA.85.042317,PhysRevA.63.020101}. For example, we can employ iterative updates $\hat{\rho}_{k+1} \propto \hat{R}(\hat{\rho}_k)\hat{\rho}_k\hat{R}(\hat{\rho}_k)$ until Eq. \ref{eq:mlerho} is fulfilled \cite{Lvovsky_2004}. These algorithms mostly center around the parameterization and optimization of a quantum state $\hat \rho$, whose cost is exponential in the system size. Besides, such iterative updates do not guarantee that the quantum state $\hat{\rho}$ remains a Gibbs form, especially when the measurements are insufficient to uniquely determine the state (e.g., large noises or small numbers of measurements and there are many quantum states satisfying Eq. \ref{eq:mlerho}). Consequently, extracting $\hat{H}\propto-\frac{1}{\beta}\ln\hat{\rho}$ from $\hat \rho$ further adds up to the inconvenience.
Considering that the operator $\hat{R}(\hat{\rho})$ has the same operator structure as the Hamiltonian, we take an alternative stance for the Hamiltonian learning task and update the candidate Hamiltonian $\hat{H}_k$, i.e., the model parameters, collectively and iteratively. In particular, we integrate the corrections to the Hamiltonian coefficients to the operator $\hat{R}(\hat{\rho})$, which offers such a quantum likelihood gradient (Fig. \ref{fig:algorithm}): \begin{eqnarray}
\hat H_{k+1} &=& \hat H_k -\gamma \hat R_k, \nonumber\\
\hat{\rho}_{k+1}&=&\frac{e^{-\beta\hat{H}_{k+1}}}{\mbox{tr}[e^{-\beta\hat{H}_{k+1}}]}=\frac{e^{-\beta(\hat{H}_{k}-\gamma\hat{R}_{k})}}{\mbox{tr}[e^{-\beta(\hat{H}_{k}-\gamma\hat{R}_{k})}]}, \label{eq:iter} \end{eqnarray} where $\gamma>0$ is the learning rate - a small parameter controlling the step size. We denote $\hat{R}_k \equiv \hat{R}(\hat{\rho}_k)$ for short here afterwards. Compared with previous Hamiltonian extractions from MLE quantum state tomography, the update in Eq. \ref{eq:iter} possesses several advantages in Hamiltonian learning. First, we can utilize the Hamiltonian structure (e.g., locality) to choose suitable numerical tools (e.g., QMC and FTTN) and even calculate within the subregions - we circumvent the costly parametrization of the quantum state $\hat{\rho}$. Also, the update guarantees a state in its Gibbs form. Last but not least, we will show that for $\gamma \ll 1$, such a quantum likelihood gradient in Eq. \ref{eq:iter} yields a negative semi-definite contribution to the negative-log-likelihood function, guaranteeing the MLE Hamiltonian (upto a trivial constant) at its convergence and an efficient optimization toward it.
\begin{figure}
\caption{An illustration of the MLE Hamiltonian learning algorithm: given the quantum measurements on the Gibbs state $\hat{\rho}_s$ of the target quantum system, we update the candidate Hamiltonian iteratively until the negative-log-likelihood function (or relative entropy) converges below a given threshold $\epsilon$, after which the output yields the MLE Hamiltonian. Within each iterative step, we evaluate the operator expectation values with respect to the Gibbs state $\hat{\rho}_k = \exp(-\beta\hat H_k) / \mbox{tr}[\exp(-\beta\hat H_k)]$, which directs the model update $\Delta \hat{H}_k=-\gamma\hat{R}_k$ for the next iterative step.}
\label{fig:algorithm}
\end{figure}
\textbf{Theorem:} For $\gamma\ll 1$, $\gamma>0$, the quantum likelihood gradient in Eq. \ref{eq:iter} yields a negative semi-definite contribution to the negative-log-likelihood function $ M(\hat{\rho}_{k+1})=-\frac{1}{N_{tot}}\log\mathcal{L}(\hat{\rho}_{k+1})$.
\textbf{Proof:} We note that upto linear order in $\gamma \ll 1$: \begin{eqnarray}
e^{-\beta\hat{H}_{k+1}}&=&e^{-\beta\hat{H}_k}\prod_{n=0}^{\infty} \exp\left[\frac{(-1)^{n}}{(n+1)!}{\rm ad}_{-\beta\hat{H}_k}^{n}(\beta\gamma\hat{R}_k)+o(\gamma^2)\right] \nonumber\\
&\approx& e^{-\beta\hat{H}_k}\left[1+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+1)!}{\rm ad}_{-\beta\hat{H}_k}^{n}(\beta\gamma\hat{R}_k)\right]\nonumber\\
&=&e^{-\beta\hat{H}_k}(1+\beta\gamma \int_{0}^{1}e^{\beta s\hat{H}_k}\hat{R}_k e^{-\beta s\hat{H}_k}{\rm d}s), \label{eq:linear_gamma} \end{eqnarray} where ${\rm ad}_{\hat{A}}^{j}\hat{B}=[\hat{A},{\rm ad}_{\hat{A}}^{j-1}\hat{B}]$ and ${\rm ad}_{\hat{A}}^{0}\hat{B}=\hat{B}$ are the adjoint action of the Lie algebra. The first and third lines are based on the Zassenhaus formula\cite{zassen} and the Baker-Hausdorff formula \cite{zassen}, respectively, while the second line neglects terms above the linear order of $\gamma$.
Following this, we can re-express the quantum state in Eq. \ref{eq:iter} as: \begin{equation}
\hat{\rho}_{k+1}=\hat{\rho}_k\frac{1+\beta\gamma \int_{0}^{1}e^{\beta s\hat{H}_k}\hat{R}_k e^{-\beta s\hat{H}_k}{\rm d}s}{1+\beta\gamma},
\label{appr_iter} \end{equation} where we have used $\mbox{tr}[\hat{\rho}_k \hat{R}_k]=1$ as a direct consequence of $R_k$'s definition in Eq. \ref{eq:mlerho}.
Subsequently, after introducing the quantum likelihood gradient, the negative-log-likelihood function becomes: \begin{eqnarray}
M(\hat{\rho}_{k+1})&=&-\frac{1}{N_{tot}}\log\mathcal{L}(\hat{\rho}_{k+1}) \nonumber\\
&=&-\sum_{i,\lambda_i}\frac{f_{\lambda_i}}{N_{tot}}\log \mbox{tr}[\hat{\rho}_{k+1}\hat{P}_{\lambda_i}] \nonumber\\
&\approx& M(\hat{\rho}_k)+\beta\gamma(1-\Delta_k), \label{MLE_iter} \end{eqnarray} where we keep terms upto linear order of $\gamma$ in the $\log$ expansion.
On the other hand, we can establish the following inequality: \begin{eqnarray}
\Delta_k &=& \mbox{tr}[\hat{\rho}_k\int_{0}^{1}e^{\beta s\hat{H}_k}\hat{R}_k e^{-\beta s\hat{H}_k}\hat{R}_k{\rm d}s] \nonumber\\
&=&\int_{0}^{1}\mbox{tr}[\hat{\rho}_k e^{\beta s\hat{H}_k}\hat{R}_k e^{-\beta s\hat{H}_k}\hat{R}_k]\mbox{tr}[\hat{\rho}_k]{\rm d}s \nonumber\\
&=&\int_{0}^{1}||e^{-\beta s\hat{H}_{k}/2}\hat{R}_{k}e^{-\beta(1-s)\hat{H}_{k}/2}||_{F}^{2}||e^{-\beta\hat{H}_{k}/2}||_{F}^{2}\frac{{\rm d}s}{Z_{k}^2} \nonumber\\
&\ge& \int_{0}^{1}\mbox{tr}[\hat{\rho}_{k}\hat{R}_{k}]^2{\rm d}s=1, \label{eq:deltagt1} \end{eqnarray}
where $Z_{k}=\mbox{tr}[e^{-\beta\hat{H}_{k}}]$ is the partition function, $||A||_{F}=\sqrt{\mbox{tr}[A^{\dagger}A]}$ is the Frobenius norm of matrix $A$, and the non-negative definiteness of $\hat \rho_k$ allows $\hat{\rho}_{k}=(\hat{\rho}_{k}^{1/2})^2=(e^{-\beta\hat{H}_{k}/2})^2/Z_{k}$. The inequality in the fourth line follows the Cauchy-Schwarz inequality.
We note that the equality - the convergence criteria of our MLE Hamiltonian learning protocol - is established if and only if: \begin{equation}
e^{-\beta s\hat{H}_{k}/2}\hat{R}_{k}e^{-\beta(1-s)\hat{H}_{k}/2}=e^{-\beta\hat{H}_{k}/2}, \end{equation} which implies the conventional MLE optimization target $\hat{R}\hat{\rho}=\hat{\rho}$ in Eq. \ref{eq:mlerho}. We can also establish such consistency from our iterative convergence \footnote{In practice, given sufficient measurements, we have $\hat{R}_k\sim \hat{I}$ dictating the quantum likelihood gradient at the iteration's convergence.} following Eq. \ref{eq:iter}: \begin{equation}
\hat \rho_{k+1} =\frac{e^{-\beta(\hat{H}_k-\gamma\hat{R}_k)}} {\mbox{tr}[e^{-\beta(\hat{H}_{k}-\gamma \hat{R}_k)}]} = \frac{e^{\beta \gamma\hat{R}_k }\hat{\rho}_k}{\mbox{tr}[e^{\beta \gamma\hat{R}_k}\hat{\rho}_k]}=\hat \rho_{k} , \label{eq:extre2} \end{equation} where we have used the commutation relation $[\hat{R}_k,\hat{H}_k]=0$ between the Hermitian operators $\hat{R}_k$ and $\hat{H}_k$ following $[\hat{R}_k, \hat{\rho}_k] = [\hat{R}_k, e^{-\beta\hat{H}_k}] = 0$.
Finally, combining Eq. \ref{MLE_iter} and Eq. \ref{eq:deltagt1}, we have shown that $M(\hat{\rho}_{k+1})-M(\hat{\rho}_{k}) \le 0 $ is a negative semi-definite quantity, which proves the theorem.
We conclude that the quantum likelihood gradient in Eq. \ref{eq:iter} offers an efficient and collective optimization towards the MLE Hamiltonian, modifying all model parameters simultaneously. For each step of quantum likelihood gradient, the most costly calculation is on $\hat{\rho}_{k+1}$, or more precisely, the expectation value $\mbox{tr}[\hat{\rho}_{k+1}\hat{P}_{\lambda_i}]$ from $\hat{H}_{k+1}$. Fortunately, this is a routine calculation in quantum many-body physics and condensed matter physics with various tailored candidate algorithms under different scenarios. For example, we may resort to the FTTN, or the QMC approaches, which readily apply to much larger systems than brute-force exact diagonalization. Thus, we emphasize that MLE Hamiltonian learning works with evaluations of the expectation values of quantum states instead of the more expensive quantum states themselves in their entirety.
Interestingly, MLE Hamiltonian learning also allows a more local stance. For a given Hamiltonian, the necessary expectation value of its Gibbs state $\mbox{tr}[\hat \rho \hat P_{\lambda_i}]$ takes the form: \begin{eqnarray} \mbox{tr}[\hat \rho \hat P_{\lambda_i}] &=& \mbox{tr}[\hat \rho^A_{eff} \hat P_{\lambda_i}], \nonumber \\ \hat \rho^A_{eff} = \mbox{tr}_{\bar A} [\hat \rho] &=& \frac{e^{-\beta \hat H^{A}_{eff}}}{\mbox{tr}[e^{-\beta \hat H^{A}_{eff}}]}, \label{eq:subregion} \end{eqnarray}
where $\hat \rho^A_{eff}$ is the reduced density operator defined upon a relatively local subsystem $A$ still containing $\hat P_{ \lambda_i}$. The effective Hamiltonian $\hat{H}_{eff}^{A}=\hat{H}_A+\hat{V}_{eff}^{A}$ of the subregion $A$ contains the existing terms $\hat{H}_A$ within the subsystem and the effective interacting terms $\hat{V}_{eff}^{A}$ from the trace operation \cite{QBP2}. According to the conclusions of the quantum belief propagation theory \cite{QBP1, QBP2}, the locality of the interaction in the latter term $||\hat{V}_{eff}^{A}(l_A)||_F\propto (\beta/\beta_c)^{l_A/r}$, where $\beta$ ($\beta_c$) denotes the current (model-dependent critical) inverse temperature, $l_A$ is the distance between a specific site in the bulk of $A$ and the boundary of the subregion $A$, and $r$ is the maximum acting distance(diameter) of a single operator in the original Hamiltonian(similar to the k-local in the next section). Thus, when $\beta<\beta_c$(especially when $\beta\ll \beta_c$), $\hat{V}_{eff}^{A}$ is exponentially localized around the boundary of $A$, and the effective Hamiltonian in the bulk of $A$ remains the same as that of the original $\hat{H}_A$ of the entire system. Therefore, we may further boost the efficiency of MLE Hamiltonian learning by redirecting the expectation-value evaluations of the global quantum system to that of a series of local patches, as we will show in the next section.
In summary, given the quantum measurements of a thermal (Gibbs) state: $\{\hat{O}_{i}\}$, $N_i$, and $f_{\lambda_i}$, we can perform MLE Hamiltonian learning to obtain the MLE Hamiltonian via the following steps (Fig. \ref{fig:algorithm}): \begin{itemize}
\item [1)] Initialization/Update:
For initialization, start with a random model Hamiltonian $\hat{H}_0$:
\begin{equation}
\hat{H}_0= \sum_{i}\mu_{i}\hat{O}_{i},
\end{equation}
or an identity Hamiltonian.
For update, carry out the quantum likelihood gradient:
\begin{equation}
\hat{H}_{k+1} = \hat{H}_k - \gamma \hat{R}_k,
\end{equation}
where $\hat{R}_k$ is defined in Eq. \ref{eq:mlerho} or Eq. \ref{eq:hiter}.
\item [2)] Evaluate the properties $\mbox{tr}[\hat \rho_k \hat P_{\lambda_i}]$ of the quantum state:
\begin{equation}
\hat{\rho}_{k}=\frac{e^{-\beta\hat{H}_{k}}}{\mbox{tr}[e^{-\beta\hat{H}_{k}}]}, \label{eq:iter3}
\end{equation}
with suitable numerical methods.
\item [3)] Check for convergence: loop back to step 1) to update, $k\rightarrow k+1$, if the relative entropy $M(\hat{\rho}_{k})-M_0 \ge \epsilon$ is above a given threshold $\epsilon$; otherwise, terminate the process, and the final $\hat{H}_k$ is the result for the MLE Hamiltonian. Here, $M_0$ is the theoretical minimum of the negative-log-likelihood function:
\begin{equation}
M_0=-\sum_{i,\lambda_i}\frac{N_i}{N_{tot}}p_{\lambda_i}\log{p_{\lambda_i}}.
\end{equation} \end{itemize}
In practice, $\hat R_k$ in Eq. \ref{eq:iter} is singular for small values of $\mbox{tr}[\hat \rho_k \hat P_{\lambda_i}]$ and may become numerically unstable, which requires a minimal or dynamical learning rate $\gamma$ to maintain the range of quantum likelihood gradient properly. Instead, we may employ a re-scaled version of $\hat R_k$: \begin{equation}
\tilde{\hat{R}}_{k} = \sum_{i,\lambda_i} \frac{N_i}{N_{tot}} f_{g}(p_{\lambda_i}/ \mbox{tr}[\hat \rho_k \hat{P}_{\lambda_i}]) \hat{P}_{\lambda_i}, \label{eq:hiter} \end{equation} where $f_{g}$ is a monotonic tuning-function: \begin{equation}
f_{g}(x)=\frac{gx}{x+g-1}, g>1,g\in\mathbb{N},
\label{eq:fg} \end{equation} which maps its argument in $(0,\infty)$ to a finite range $(0,g)$. Such a re-scaled $\tilde{\hat{R}}_k$ regularizes the quantum likelihood gradient and allows a simple yet relatively larger learning rate $\gamma$ for more efficient MLE Hamiltonian learning. We also have $f_g(1)=1$, therefore $\tilde{\hat{R}}_k \rightarrow \hat{R}_k$ as we approach convergence. We will mainly employ $\tilde{\hat{R}}_{k}$ for our examples in the following sections.
In addition to the negative-log-likelihood function $M(\hat{\rho}_k)$, we also consider the Hamiltonian distance as another criterion on the quality of Hamiltonian learning: \begin{equation}
\Delta{\vec{\mu}}_k = \frac{||\vec{\mu}_s-\vec{\mu}_{k}||_{2}}{||\vec{\mu}_s||_2},
\label{eq:hamdis} \end{equation} where $\vec{\mu}_s$ and $\vec{\mu}_{k}$ are the (vectors of) coefficients \footnote{We typically perform MLE Hamiltonian learning and update the model Hamiltonian on the projection-operator basis; therefore, we transform the Hamiltonian back to the original, ordinary operator basis before $\Delta \vec{\mu}_k$ evaluations.} of the target Hamiltonian and the learned Hamiltonian after $k$ iterations, respectively. However, we do not recommend $\Delta{\vec{\mu}}_k$($\Delta \mu$ for short) as convergence criteria as $\vec{\mu}_s$ is generally unknown aside from benchmark scenarios \cite{Zoller}.
\section{Example models and results}
In this section, we demonstrate the performance of the MLE Hamiltonian learning protocol. For better numerical simulations, we consider the $k$-local Hamiltonians, with operators acting non-trivially on no more than $k$ contiguous sites in each direction. For example, for a 1-dimensional spin-$\frac{1}{2}$ system, a $k$-local operator for $k=2$ takes the form $\hat{S}_{i}^{\alpha}\hat{S}_{i+1}^{\beta}$ or $\hat{S}_{i}^{\alpha}$, $\alpha, \beta\in\{x,y,z\}$, where $\hat{S}_{i}^{\alpha}$ denotes the spin operator. In particular, we focus on general 1D quantum spin chains with $k=2$, taking the following form: \begin{equation}
\hat{H}_s = \sum_{i,\alpha,\beta}^{L-1}J_{i}^{\alpha\beta}\hat{S}_{i}^{\alpha}\hat{S}_{i+1}^{\beta}+\sum_{i,\alpha}^{L}h_{i}^{\alpha}\hat{S}_{i}^{\alpha},
\label{XZ} \end{equation}
where $\hat{S}_{i}^{\alpha}$ denotes the spin operator on site $i$, $\alpha,\beta,\in\{x,y,z\}$. There are $12L-9$ 2-local operators under the open boundary condition, where $L$ is the system size. We generate the model parameters $\vec{\mu}_s = \{ J_{i}^{\alpha\beta}, h_{i}^{\alpha} \}$ randomly following a uniform distribution in $[-1,1]$. This Hamiltonian $\hat{H}_s$, specifically the model parameters $\vec{\mu}_s$, will be our target for MLE Hamiltonian learning. As the protocol's inputs, we simulate quantum measurements of all 2-local operators $\{\hat{O}_i\}$ on the Gibbs states of $\hat{H}_s$ numerically via exact diagonalization on small systems and FTTN for large systems. For the latter, we use a tensor network ansatz called the ``ancilla" method \cite{FTTN}, where we purify a Gibbs state with some auxiliary qubits $\hat\rho_s = \mbox{tr}_{aux}|\psi_s\rangle\langle \psi_s|$, and obtain $\ket{\psi_s}=e^{-\beta\hat{H}_s/2}\ket{\psi_0}$ from a maximally-entangled state $\ket{\psi_0}$ via imaginary time evolution. In addition, given a large number $n$ of Trotter steps, the imaginary time evolution operator $e^{-\beta\hat{H}_s/2}$ is decomposed into Trotter gates' product as $(\Pi_{i}e^{-\mu_i\hat{O}_{i} \beta/2n})^{n}+O(\beta^2/n)$. Here, we set the Trotter step $\delta t=\beta/n \in [0.01,0.1]$, for which the Trotter errors of order $O(\beta^2/n)$ show little impact on our protocol's accuracy. Without loss of generality, we employ the integrated FTTN algorithm in the ITensor numerical toolkit \cite{ITensor}, and set the number of measures $N_i=N$ for all operators in our examples for simplicity.
\begin{figure}
\caption{Both the Hamiltonian distance $\Delta\mu$ defined in Eq. \ref{eq:hamdis} and the negative-log-likelihood function $M(\hat{\rho}_{k+1})$ (or relative entropy $M(\hat{\rho}_{k+1})-M_0$) show successful convergence of the iterations in MLE Hamiltonian learning, albeit a variety of system sizes and temperature ranges. We simulate the target Hamiltonian and the iteration process by FTTN with Trotter step $\delta t=0.1$. Each curve is averaged on 10 trials of random $\hat{H}_0$ initializations. We set the learning rate $\gamma=0.1$. The maximum number of iterations here is 1000.}
\label{Gibbs_differ}
\end{figure}
As we demonstrate in Fig. \ref{Gibbs_differ}, MLE Hamiltonian learning obtains the target Hamiltonians with high accuracy and efficiency under various settings of system sizes and inverse temperatures $\beta$. Besides, instead of the original quantum likelihood gradient in Eq. \ref{eq:mlerho}, we may obtain a faster convergence with the re-scaled $\tilde{\hat{R}}_k$ in Eq. \ref{eq:hiter} and a larger learning rate, as we discuss in Appendix B. In the following numerical examples, we use the re-scaled quantum likelihood gradient $\tilde{\hat{R}}_k$ and set $g=2$ for the tuning function in Eq. \ref{eq:fg}. Within the given iterations, not only have we achieved results (Hamiltonian distance $\Delta\mu\sim O(10^{-12})$ and relative entropy $M(\hat{\rho}_k)-M_0\sim O(10^{-16})$) comparable to, if not exceeding, previous methods \cite{Corr1} for $L=10$ systems and $\beta=1$ straightforwardly, but we have also achieved satisfactory consistency ($\Delta\mu\sim O(10^{-2})$ and $M(\hat{\rho}_k)-M_0\sim O(10^{-9})$) for large systems $L=100$ and low temperatures $\beta=3$ that were previously inaccessible.
\begin{figure}
\caption{The performance of MLE Hamiltonian learning maintains relatively well against noises and, especially, broader temperature ranges. Left: the Hamiltonian distance versus the inverse temperature $\beta$ shows a broader applicable temperature range. Each data point contains 10 trials. Right: the performance (left figure's data averaged over temperature) versus the noise strength $\delta$ shows the impact of noises and the protocol's relative robustness against them. The slope of the straight line is $\sim 1$, indicating a linear relationship between $\Delta\mu$ and $\delta$. Note the log scale $\log(\Delta\mu)$ for the vertical axis. We set $L=10$ for the system size, and learning rate $\gamma=1$.}
\label{Gibbs_temp}
\end{figure}
MLE Hamiltonian learning is also relatively robust against temperature and noises, two key factors impacting accuracy in Hamiltonian learning. For illustration, we include random errors $\delta \langle \hat O_i \rangle$ following Gaussian distribution with zero mean and standard deviation $\delta$ to all quantum measurements: $\langle \hat O_i \rangle \rightarrow \langle \hat O_i \rangle + \delta \langle \hat O_i \rangle$. We note that such $\delta$ may also depict the quantum fluctuations \cite{Corr1, Corr2} from a finite number of measurements $\delta \propto N_i^{-1/2}$. We also focus on smaller systems with $L=10$ and employ exact diagonalization to avoid confusion from potential Trotter error of the FTTN ansatz\cite{FTTN}. We summarize the results in Fig. \ref{Gibbs_temp}.
Most previous algorithms on Hamiltonian learning have a rather specific applicable temperature range. For example, the high-temperature expansion of $e^{-\beta\hat{H}}$ only works in the $\beta\ll 1$ limit \cite{ogl, PhysRevA.92.052322}. Besides, gradient descent on the log partition function, despite a convex optimization, performs well in a narrow temperature range \cite{RN571}. The gradient of this algorithm is proportional to the inverse temperature, so the algorithm's convergence slows at high temperatures. Also, the gradient descent algorithm cannot extend to the $\beta\rightarrow\infty$ limit - the ground state, while our protocol is directly applicable to the ground states of quantum systems, as we will generalize and justify later.
MLE Hamiltonian learning is also more robust to noises, with an accuracy of Hamiltonian distance $\Delta\mu \sim O(10^{-11})$ across a broad temperature range at noise strength $\delta \sim O(10^{-12})$. Such noise level is hard to realize in practice; nevertheless, it is necessary to safeguard the correlation matrix method \cite{Qi2019, Corr1, sbhl}. Even so, due to the uncontrollable spectral gap, the correlation matrix method is susceptible to high temperature, and its accuracy drastically decreases to $\Delta\mu \sim O(10^{-3})$ at $\beta=0.01$. In comparison, MLE Hamiltonian learning is more versatile, with an approximately linear dependence between its accuracy $\Delta\mu$ and the noise strength $\delta$ across a broad range of temperatures and noise strengths, saturating the previous bound \cite{RN571}; see the right panel of Fig. \ref{Gibbs_temp}. We also provide more detailed comparisons between the algorithms in Appendix C.
\begin{figure}
\caption{Upper: MLE Hamiltonian learning's need for evaluations of local observables can be satisfied among local patches for thermal states at sufficiently high temperatures, where the effective potential $V^A_{eff}$ ($V^{B}_{eff}$) is weak and localized on the boundaries of subregion $A$ ($B$) within a cut-off range $\Lambda_A$ ($\Lambda_B$). Consequently, for $\hat{P}_{\lambda_i}$ defined sufficiently deep inside $A$ ($B$), we can estimate its expectation value via $\mbox{tr}[\hat{\rho}^{A}\hat{P}_{\lambda_i}]$ ($\mbox{tr}[\hat{\rho}^{B}\hat{P}_{\lambda_i}]$). Lower: the Hamiltonian distance $\Delta\mu$ of the results after MLE Hamiltonian learning indicates better validity of the local-patch approximation at higher temperatures. The total system size is $L=100$, while the local patches are of sizes $L_A=10, 16$ with cut-offs $\Lambda=1, 2, 4$, respectively. Each data point contains 10 trials. We set $\delta t=0.1$ for the Trotter step in FTTN ansatz and learning rate $\gamma=1$. }
\label{subregion}
\end{figure}
Despite efficient quantum likelihood gradient and applicable quantum many-body ansatz, the computational cost of MLE Hamiltonian learning still increases rapidly with the system size $L$. Fortunately, as stated above in Eq. \ref{eq:subregion}, we may resort to calculations on local patches, especially for low dimensions and high temperatures due to their quasi-Markov property. In particular, when $\beta<\beta_c$ ($T>T_c$), the difference between the cutoff Hamiltonian $\hat{H}_A$ and the effective Hamiltonian $\hat{H}_{eff}^{A}$ in a local subregion $A$, $V_{eff}^{A}$, should be weak, short-ranged, and localized at $A$'s boundary \cite{QBP1, QBP2}; therefore, for those operators $\hat{P}_{\lambda_i}$ adequately deep inside $A$, we can use $\hat \rho^A$, the Gibbs state defined by $\hat{H}_A$, to estimate the corresponding $\mbox{tr}[\hat{\rho}\hat{P}_{\lambda_i}]$; see illustration in Fig. \ref{subregion} upper panel.
For example, we apply MLE Hamiltonian learning on $L=100$ systems, where we iteratively calculate the necessary expectation values on different local patches of size $L_A=10, 16$. We also choose different cut-offs $\Lambda$, and evaluate $\mbox{tr}[\hat{\rho}^{A}\hat{P}_{\lambda_i}]$ for those operators at least $\Lambda$ away from the boundaries and sufficiently deep inside the subregion $A$, so that the effective potential $V^{A}_{eff}$ may become negligible. We also employ a sufficient number of local patches to guarantee full coverage of necessary observables - operators outside $A$ or in $\Lambda_A$ are obtainable from another local patch $B$, as shown in the upper panel of Fig. \ref{subregion}, and so on so forth. Both the $L=100$ target system and the local patches for MLE Hamiltonian learning are simulated via FTTN. We have no problem achieving convergence, and the resulting Hamiltonians' accuracy, the Hamiltonian distance $\Delta\mu$ versus the inverse temperature $\beta$, is summarized in the lower panel of Fig. \ref{subregion}. Indeed, the local-patch approximation is more reliable at higher temperatures, as well as with larger subsystems and cutoffs, albeit with rising costs. We also note that we can achieve much larger systems with the local patches than $L=100$ we have demonstrated.
\section{MLE Hamiltonian learning for pure eigenstates}
In addition to the Gibbs states, MLE Hamiltonian learning also applies to measurements of certain eigenstates of target quantum systems:
1. The ground states are essentially the $\beta \rightarrow \infty$ limit of the Gibbs states. However, due to the order-of-limit issue, the $\gamma \rightarrow 0$ requirement of the theorem on Gibbs states forbids a direct extension to the ground states. In the Appendix D, we offer rigorous proof of the effectiveness of quantum likelihood gradient based on ground-state measurements, along with several nontrivial MLE Hamiltonian learning examples on quantum critical and topological ground states. We note that Ref. \onlinecite{wjb} offers preliminary studies on pure-state quantum state tomography, inspiring this work.
2. A highly-excited eigenstate of a (non-integrable) quantum chaotic system $\hat{H}_s$ is believed to obey the eigenstate thermalization hypothesis (ETH), that its density operator $\hat{\rho}_{s}=\ket{\psi_s}\bra{\psi_s}$ behaves locally indistinguishable from a Gibbs state $\hat \rho_{s, A}$ in thermal equilibrium \cite{Tarun}: \begin{equation}
\hat{\rho}_{s,A}=\mbox{tr}_{\bar{A}}[\hat{\rho}_s]\approx \frac{e^{-\beta_s\hat{H}_A}}{\mbox{tr}[e^{-\beta_s\hat{H}_A}]},
\label{eq:eth} \end{equation}
where $\beta_s$ is an effective temperature determined by the energy expectation value $\braket{\psi_s|\hat{H}_s|\psi_s}=\frac{\mbox{tr}[e^{-\beta_s\hat{H}_s}\hat{H}_s]}{\mbox{tr}[e^{-\beta_s\hat{H}_s}]}$. As MLE Hamiltonian learning only engages local operators, its applicability directly generalizes to such eigenstates $\ket{\psi_s}$ following ETH.
3. In general, ETH applies to eigenstates in the center of the spectrum of quantum chaotic systems, while low-lying eigenstates are too close to the ground state to exhibit ETH \cite{PhysRevE.90.052105}. However, in the rest of the section, we demonstrate numerically that MLE Hamiltonian learning still works well for low-lying eigenstates.
We consider the 1D longitudinal-transverse-field Ising model \cite{Tarun, PhysRevE.90.052105} as our target quantum system: \begin{equation}
\hat{H}_s=J\sum_{j}^{L-1}\hat{\sigma}_{j}^{z}\hat{\sigma}_{j+1}^{z}+g_{z}\sum_{j}^{L}\hat{\sigma}_{j}^{z}+g_{x}\sum_{j}^{L}\hat{\sigma}_{j}^{x},
\label{ETH_ham} \end{equation} where the system size is $L=80$. We set $J=1$, $g_{x}=0.9045$, and $g_{z}=0.8090$. The quantum system is strongly non-integrable under such settings. Previous studies mainly focused on eigenstates in the middle of the energy spectrum. In contrast, we pick the first excited state - a typical low-lying eigenstate considered asymptotically integrable and ETH-violating \cite{PhysRevE.90.052105} - for quantum measurements (via DMRG) and then MLE Hamiltonian learning for its candidate Hamiltonian (via FTTN).
\begin{figure}
\caption{The coefficients obtained via MLE Hamiltonian learning (green columns) compare well with those of the target Hamiltonian $\hat{H}_s$ even though the quantum measurements are based upon a low-lying (first) excited state. The red columns denote the coefficients of $\hat{H}_s$ in Eq. \ref{ETH_ham} multiply the effective (inverse) temperature $\beta_s=4$. The error bars demonstrate the variances over the lattices and trials. We set the system size $L=80$, learning rate $\gamma=0.1$, and the Trotter step $\delta t=0.1$.}
\label{ETH}
\end{figure}
We summarize the results in Fig. \ref{ETH}. Further, the model Hamiltonian we established is approximately equivalent to the target quantum Hamiltonian at an (inverse) temperature $\beta_s\approx 4$ \cite{Tarun}, which we have absorbed into the unit of our $\hat{H}_k$. Therefore, we have accurately established the model Hamiltonian and derived the effective temperature consistent with previous results \cite{Tarun} for a low-lying excited eigenstate not necessarily following ETH. The physical reason for quantum likelihood gradient applicability in such states is an interesting problem that deserves further studies.
\section{Discussions}
We have proposed a novel MLE Hamiltonian learning protocol to achieve the model Hamiltonian of the target quantum system based on quantum measurements of its Gibbs states. The protocol updates the model Hamiltonian iteratively with respect to the negative-log-likelihood function from the measurement data. We have theoretically proved the efficiency and convergence of the corresponding quantum likelihood gradient and demonstrated it numerically on multiple non-trivial examples, which show more accuracy, better robustness against noises, and less temperature dependence. Indeed, the accuracy is almost linear to the imposed noise amplitude, thus inverse proportional to the square root of the number of samples, the asymptotic upper bound\cite{RN571}. Further, MLE Hamiltonian learning directly rests on the Hamiltonians and their physical properties instead of direct and costly access to the quantum many-body states. Consequently, we can resort to various quantum many-body ansatzes in our systematic quantum toolbox and even local-patch approximation when the situation allows. These advantages allow applications to larger systems and lower temperatures with better accuracy than previous approaches. On the other hand, while our protocol is generally applicable for learning any Hamiltonian, its advantages are most apparent for local Hamiltonians, where various quantum many-body ansatzes and local-patch approximation shine. Despite such limitations, we note that the physical systems are characterized by local Hamiltonians in a significant proportion of scenarios.
In addition to the Gibbs states, we have generalized the applicability of MLE Hamiltonian learning to eigenstates of the target quantum states, including ground states, ETH states, and even selected cases of low-lying excited states. We have also provided theoretical proof of quantum likelihood gradient rigor and convergence in the Appendix D, along with several other numerical examples.
Our strategy may apply to the entanglement Hamiltonians and the tomography of the quantum states under the maximum-likelihood-maximum-entropy assumption \cite{PhysRevLett.107.020404}. Besides, our algorithm may also provide insights into the quantum Boltzmann machine \cite{QBM} - a quantum version of the classical Boltzmann machine with degrees of freedom that obey the distribution of a target quantum Gibbs state. Instead of brute-force calculations of the loss function derivatives with respect to the model parameters or approximations with the gradients' upper bounds, our protocol provides an efficient optimization that updates the model parameters collectively.
\emph{Acknowledgement:}- We thank insightful discussions with Jia-Bao Wang. We acknowledge support from the National Key R\&D Program of China (No.2021YFA1401900) and the National Science Foundation of China (No.12174008 \& No.92270102). The calculations of this work are supported by HPC facilities at Peking University.
\appendix
\section{Maximum condition for MLE}
In this appendix, we review the derivation of the maximum condition \cite{PhysRevA.75.042108} in Eq. \ref{eq:mlerho} in the main text.
A general quantum state takes the form of a density operator: \begin{equation}
\hat{\rho}=\sum_{j}p_j\ket{\psi_j}\bra{\psi_j},
\label{general} \end{equation} where $p_j\ge 0$, $\sum_j p_j=1$, and $\ket{\psi_j}$ is a set of orthonormal basis. The search for the quantum state that maximizes the likelihood function: \begin{equation}
\mathcal{L}(\hat{\rho}) = \prod_{i, \lambda_i}\{\mbox{tr}[\hat{\rho}\hat{P}_{\lambda_i}]^{\frac{f_{\lambda_i}}{N_{tot}}}\}^{N_{tot}}, \end{equation} can be converted to the optimization problem: \begin{equation}
\begin{split}
&\min\limits_{\hat{\rho}\in\mathcal{D}}\quad M(\hat{\rho})=-\frac{1}{N_{tot}}\log[\mathcal{L}(\hat{\rho})]\\
&{\rm subject\enspace to}\quad \hat{\rho}\succeq 0, \mbox{tr}[\hat{\rho}]=1.\\
\label{eq:optprob}
\end{split} \end{equation} It is hard to solve this semi-definite programming problem directly and numerically. Instead, forgoing the non-negative definiteness, we adopt the Lagrangian multiplier method: \begin{equation}
\frac{\partial}{\partial{\bra{\psi_j}}}\{M(\hat{\rho})+\lambda \mbox{tr}[\hat{\rho}]\}=0, \end{equation} where $\lambda$ is a Lagrangian multiplier. Given Eq. \ref{general}, we obtain the following solution: \begin{equation} \begin{split}
&\hat{R}\ket{\psi_j}=\ket{\psi_j}\\
&\hat{R} = \sum_{i,\lambda_i}\frac{f_{\lambda_i}}{N_{tot}} \frac{\hat{P}_{\lambda_i}}{\mbox{tr} [\hat \rho \hat P_{\lambda_i}]},\\ \end{split} \label{extre_appendix} \end{equation} and $\lambda=1$. Combining Eq. \ref{general} and Eq. \ref{extre_appendix}, we obtain the maximum condition: \begin{equation}
\hat{R}\hat{\rho}=\hat{\rho}.
\label{eq:exteme} \end{equation} We note that Eq. \ref{extre_appendix} does not guarantee the positive semi-definiteness of the density operator. Instead, one may search within the density-operator space (the space of positive semi-definite matrix with unit trace) to locate the MLE quantum state fulfilling Eq. \ref{extre_appendix} or Eq. \ref{eq:exteme}. For the Hamiltonian learning task in this work, the search space is naturally the space of Gibbs states (under selected quantum many-body ansatz).
\section{MLE Hamiltonian learning with rescaling function}
In this appendix, we compare the MLE Hamiltonian learning with the quantum likelihood gradient $\hat{R}_k$ and the re-scaled counterpart $\tilde{\hat{R}}_k$. As we state in the main text, $\tilde{\hat{R}}_k$ regularizes the gradient, allowing us to employ a larger learning rate $\gamma=1$, which leads to a faster convergence (Fig. \ref{fig:rescaled_FIG2}) and a higher accuracy (Tab. \ref{table}) given identical number of iterations.
\begin{figure}\label{fig:rescaled_FIG2}
\end{figure}
\begin{table} \renewcommand\arraystretch{1.6} \resizebox{80mm}{!}{
\begin{tabular}{|c|c|c|c|r|l|} \hline
& $L=10$,$\beta=1$ & $L=50$,$\beta=2$ & $L=100$,$\beta=3$ \\ \hline $\hat{R}_k$ &$O(10^{-6})$ & $O(10^{-2})$ & $O(10^{-1})$ \\ \hline $\tilde{\hat{R}}_k$ & $O(10^{-12})$ & $O(10^{-3})$ & $O(10^{-2})$ \\ \hline \end{tabular}} \caption{The algorithm's accuracy (Hamiltonian distance $\Delta\mu$) further improves with a re-scaled quantum likelihood gradient $\tilde{\hat{R}}_k$ under various system sizes $L$ and (inverse) temperatures $\beta$. } \label{table} \end{table}
\section{Comparisons between Hamiltonian learning algorithms}
In this appendix, we compare different Hamiltonian learning algorithms, including the correlation matrix (CM) method \cite{Qi2019, Corr1}, the gradient descent (GD) method \cite{RN571}, and the MLE Hamiltonian learning (MLEHL) algorithm, by looking into some of their numerical results and performances. We consider general 2-local Hamiltonians in Eq. \ref{XZ} in the main text for demonstration and measurements $\{\hat{O}_i\}$ over all the 2-local operators (instead of all 4-local operators as in Ref. \cite{Corr1}).
We summarize the results in Fig. \ref{fig:compare}: the accuracy of CM is unstable and highly sensitive to temperature; while GD performs similarly to the proposed MLEHL algorithm at low temperatures, its descending gradient becomes too small at high temperatures to allow a satisfactory convergence within the given maximum iterations.
\begin{figure}
\caption{The performances (logarithm of Hamiltonian distances) of different algorithms versus the inverse temperature $\beta$ show the advantages of the MLEHL algorithm. Each data point contains 10 trials of random Hamiltonians. We also include noises following a normal distribution with zero means and $O(10^{-12})$ standard deviation. For both GD and MLEHL algorithms, we employ a learning rate $\gamma = 1$ and a maximum number of iterations of 7000. The system size is $L=7$.}
\label{fig:compare}
\end{figure}
We also compare the convergence rates of the MLEHL and GD algorithms with the same learning rate. As in Fig. \ref{fig:compare_GD_MLEHL}, the MLEHL algorithm exhibits a faster convergence and a smaller computational cost, which is similar under both algorithms for each iteration.
\begin{figure}
\caption{The logarithm of the Hamiltonian distances versus the number of iterations shows a faster convergence under the proposed MLEHL algorithm. We also include noises following a normal distribution with zero means and $O(10^{-12})$ standard deviation. We employ a learning rate of $\gamma = 1$ for both GD and MLEHL. We set the system size $L=7$ and the (inverse) temperature $\beta=1$.}
\label{fig:compare_GD_MLEHL}
\end{figure}
\section{Hamiltonian learning from ground state}
In this appendix, we prove the effectiveness of the quantum likelihood gradient based on measurements of the target quantum system's ground state and provide several nontrivial numerical examples, including 1D quantum critical states and 2D topological states.
\subsection{Proof for ground-state-based quantum likelihood gradient}
Given a sufficient number $N_i$ measurements of the operator $\hat{O}_i$ on the non-degenerate ground state $\ket{\psi_s}$ of a target system $\hat{H}_s$, we obtain a number of outcomes as the $\lambda_i^{th}$ eigenvalue of $\hat{O}_i$ as: \begin{equation} f_{\lambda_i}=p_{\lambda_i}N_i\approx\bra{\psi_s}\hat{P}_{\lambda_i}\ket{\psi_s}N_i, \end{equation} where $p_{\lambda_i}=f_{\lambda_i}/N_i$, and $\hat{P}_{\lambda_i}$ is the projection operator of the eigenvalue $o_{\lambda_i}$.
Our MLE Hamiltonian learning follows the iterations: \begin{equation}
\begin{split}
&\hat{H}_{k+1}=\hat{H}_k-\gamma\hat{R}_k,\\
&\hat{R}_k = \sum_{i,\lambda_i}\frac{f_{\lambda_i}}{N_{tot}}\frac{\hat{P}_{\lambda_i}}{\bra{\psi_{k}^{gs}}\hat{P}_{\lambda_i}\ket{\psi_{k}^{gs}}},\\
\end{split}
\label{iter_gs} \end{equation} where $\ket{\psi_k^{gs}}$ is the non-degenerate ground state of $\hat{H}_k$.
\textbf{Theorem:} For $\gamma\ll1,\gamma>0$, the quantum likelihood gradient in Eq. \ref{iter_gs} yields a negative semi-definite contribution to the negative-log-likelihood function $M(\ket{\psi_{k+1}^{gs}})=-\frac{1}{N_{tot}}\log\mathcal{L}(\ket{\psi_{k+1}^{gs}})$ following Eq. \ref{LHF} in the main text.
\textbf{Proof:} At the linear order in $\gamma$, we may treat the addition of $-\gamma\hat{R}_k$ to $\hat{H}_k$ at the $k^{th}$ iteration as a perturbation: \begin{equation}
\ket{\psi_{k+1}^{gs}}=\ket{\psi_{k}^{gs}}-\gamma\hat{G}_{k}\hat{R}_{k}\ket{\psi_{k}^{gs}}+O(\gamma^2),\label{eq:itergs_pertb} \end{equation} where $\hat{G}_k$ is the Green's function in the $k_{th}$ iteration: \begin{equation}
\hat{G}_k =\hat{Q}_k \frac{1}{E_{k}^{gs}-\hat{H}_k}\hat{Q}_k, \end{equation} where $\hat{Q}_k=I-\ket{\psi_{k}^{gs}}\bra{\psi_{k}^{gs}}$ is the projection operator orthogonal to the ground space $\ket{\psi_{k}^{gs}}\bra{\psi_{k}^{gs}}$, and $E_{k}^{gs}$ is the ground state energy. Keeping terms upto the linear order of $\gamma$ in the log expansion of the negative-log-likelihood function, we have: \begin{equation}
\begin{split}
M(\ket{\psi_{k+1}^{gs}})&=-\frac{1}{N_{tot}}\log\mathcal{L}(\ket{\psi_{k+1}^{gs}})\\
&=-\sum_{i,\lambda_i}\frac{f_{\lambda_i}}{N_{tot}}\log\bra{\psi_{k+1}^{gs}}\hat{P}_{\lambda_i}\ket{\psi_{k+1}^{gs}},\\
&\approx M(\ket{\psi_k^{gs}})+2\gamma\Delta_k.
\end{split}
\label{nlf_gs} \end{equation} where difference takes the form: \begin{equation}
\begin{split}
\Delta_k &= \bra{\psi_{k}^{gs}}\hat{R}_k\hat{G}_k\hat{R}_k\ket{\psi_{k}^{gs}}\\
&=\sum_{l\neq gs}\frac{|\bra{\psi_{k}^{gs}}\hat{R}_{k}\ket{\psi_{k}^{l}}|^2}{E_{k}^{gs}-E_{k}^{l}}\le 0.
\end{split}
\label{nlf_gs2} \end{equation} Here, $E_{k}^{l}>E_{k}^{gs}$ because $E_{k}^{l}$ denotes the energy for eigenstates other than the ground state. Our iteration converges when the equality in Eq. \ref{nlf_gs2} is established. This happens when $\ket{\psi_k^{gs}}$ is an eigenstate of $\hat{R}_k$, consistent with the MLE condition $\hat{R}\ket{\psi}=\ket{\psi}$ (or $\hat{R}\hat{\rho}=\hat{\rho}$).
Finally, combining Eq. \ref{nlf_gs} and Eq. \ref{nlf_gs2}, we have shown that $M(\ket{\psi_{k+1}^{gs}})-M(\ket{\psi_{k}^{gs}})$ is a negative semi-definite quantity, which proves the theorem.
One potential complication to the proof is that Eq. \ref{eq:itergs_pertb} needs to assume there is no ground-state level crossing or degeneracy after adding the quantum likelihood gradient. A potential remedy is to keep some low-lying excited states together with the ground state and compare them for maximum likelihood, especially for steps with singular behaviors. Otherwise, we can only hope such transitions are sparse, especially near convergence, and they establish a new line of iterations heading toward the same convergence. A more detailed discussion is available in Ref. \onlinecite{wjb}.
\subsection{Example: $c=\frac{3}{2}$ CFT ground state of Majorana fermion chain}
Here, we consider the spinless 1D Majorana fermion chain model of length $2L$ as an example \cite{PhysRevB.92.235123}: \begin{equation} \hat{H}_s=\sum_{j}it\hat{\gamma}_{j}\hat{\gamma}_{j+1}+g\hat{\gamma}_{j}\hat{\gamma}_{j+1}\hat{\gamma}_{j+2}\hat{\gamma}_{j+3}, \label{major} \end{equation} where $\hat{\gamma}_{j}$ is the Majorana fermion operator obeying: \begin{equation}
\hat{\gamma}_{j}^{\dagger}=\hat{\gamma}_{j}, \{\hat{\gamma}_{i}, \hat{\gamma}_{j}\}=\delta_{ij}, \end{equation} and $t$ and $g=-1$ are model parameters. This model presents a wealth of nontrivial quantum phases under different $t/g$. We focus on the model parameters in $t/g\in(-2.86, -0.28)$, where the ground state of Eq. \ref{major} is a $c=\frac{3}{2}$ CFT composed of a critical Ising theory ($c=\frac{1}{2}$) and a Luttinger liquid ($c=1$).
\begin{figure}\label{fig:majorana}
\end{figure}
Through the definition of the complex fermions followed by the Jordan-Wigner transformation: \begin{eqnarray}
\hat{c}_{j}&=&\frac{\hat{\gamma}_{2j}+i\hat{\gamma}_{2j+1}}{2}, \nonumber\\
\hat{\sigma}_{j}^{z}&=&2\hat{n}_{j}-1, \\
\hat{\sigma}_{j}^{+}&=&e^{-i\pi\sum_{i<j}\hat{n}_{i}}\hat{c}_{j}^{\dagger}, \nonumber \end{eqnarray} where $\hat{n}_{j}=\hat{c}_{j}^{\dagger}\hat{c}_{j}$ is the complex fermion number operator, we map Eq. \ref{major} to a 3-local spin chain of length $L$: \begin{equation} \begin{split}
\hat{H}_s=&t\sum_{j}\hat{\sigma}_{j}^{z}-t\sum_{j}\hat{\sigma}_{i}^{x}\hat{\sigma}_{i+1}^{x} \\
-&g\sum_{j}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}-g\sum_{j}\hat{\sigma}_{i}^{x}\hat{\sigma}_{i+2}^{x}.\\ \end{split} \end{equation}
We employ quantum measurements on the ground state $\ket{\psi_s}$ of this Hamiltonian, based on which we carry out our MLE Hamiltonian learning protocol. Here, we evaluate the ground-state properties via exact diagonalization. The numerical results for two cases of $t=0.5, 1.5$ are in Fig. \ref{fig:majorana}. We achieve successful convergence and satisfactory accuracy on the target Hamiltonian. The relative entropy's instabilities are mainly due to the ground state's level crossing and degeneracy.
\subsection{Example: alternative Hamiltonian for ground state}
We have seen that MLE Hamiltonian learning can retrieve the unknown target Hamiltonians via quantum measurements of its Gibbs states, even its ground states. For pure states, however, one interesting byproduct is that the relation between Hamiltonian and eigenstates is essentially many-to-one. Therefore, it is possible to obtain various candidate Hamiltonians $\hat{H}_k$ sharing the same ground state as the original target $\hat{H}_s$, especially by controlling the operator/observable set. Here, we show such numerical examples.
\begin{figure}\label{XZ}
\end{figure}
As our target quantum system, we consider the transverse field Ising model (TFIM) of length $L=15$: \begin{equation} \hat{H}_s=J\sum_{j}\hat{S}_{j}^{z}\hat{S}_{j+1}^{z}+g\sum_{j}\hat{S}_{j}^{x}, \end{equation} at its critical point $J=g=1$. Its ground state is $\ket{\psi_s}$. However, instead of the operators presenting in $\hat{H}_s$, we employ a different operator set for $\ket{\psi_s}$'s quantum measurements: \begin{equation}
\{\hat{O}_{i}\} = \{\hat{S}_{i}^{z}\hat{S}_{i+1}^{z}, \hat{S}_{i}^{x}\hat{S}_{i+1}^{x}\}.
\label{eq:op_set} \end{equation} We evaluate the ground-state properties via DMRG.
The subsequent MLE Hamiltonian learning results are in Fig. \ref{XZ}. Since we obtain a candidate Hamiltonian with the operators in Eq. \ref{eq:op_set} and destined to differ from $\hat{H}_s$, the Hamiltonian distance is no longer a viable measure of its accuracy. Instead, we introduce the ground-state fidelity $f_{gs}=\braket{\psi_s|\psi_{k}^{gs}}$, where $\ket{\psi_s}$ ($\ket{\psi_{k}^{gs}}$) is the ground state of $\hat{H}_s$ ($\hat{H}_k$). Interestingly, while the relative entropy shows full convergence, the fidelity $f_{gs}$ jumps between $\sim 99.5\%$ and $\sim 10^{-3}\%$. This is understandable, as the quantum system is gapless, and the ground and low-lying excited states have similar properties under quantum measurements.
\subsection{Example: two-dimensional topological states}
\begin{figure}\label{CSLConstruct}
\end{figure}
Here, we consider MLE Hamiltonian learning on two-dimensional topological quantum systems. In particular, we consider the chiral spin liquid (CSL) on a triangular lattice: \begin{equation}
\hat{H}_s =J_1\sum_{\langle ij\rangle}\Vec{S}_i\cdot\Vec{S}_j+J_2\sum_{\langle\langle ij\rangle\rangle}\Vec{S}_i\cdot\Vec{S}_j+K\sum_{i,j,k\in \bigtriangledown/\bigtriangleup}\Vec{S}_i\cdot\left(\Vec{S}_j\times\Vec{S}_k\right), \end{equation}
where the first and second terms are Heisenberg interactions, and the last term is a three-spin chiral interaction. Previous DMRG studies have established $\hat{H}_s$'s ground state as a CSL under the model parameters $J_1 = 1.0$, $J_2=0.1$, and $K=0.2$\cite{PhysRevB.96.075116}, which we set as the parameters of the target Hamiltonian. Here, we employ exact diagonalization on a $4 \times 4$ system. Based upon entanglement studies of the lowest-energy eigenstates, we verify that both the modular $SU$ matrix corresponding to $C_6$ rotations and the entanglement entropy fit well with a CSL topological phase\cite{PhysRevB.85.235151}. Subsequently, we perform MLE Hamiltonian learning based on quantum measurements of the ground state, focusing on the operators presenting in $\hat{H}_s$. We summarize the results in Fig. \ref{CSLConstruct}. The Hamiltonian distance indicates a stable converging accuracy, yet the relative entropy and the fidelity $f_{gs}=\braket{\psi_s | \psi_{k}^{gs}}$ witness certain instabilities. Indeed, being a topological phase means ground-state degeneracy - competing low-energy eigenstates with global distinctions yet similar local properties.
\end{document}
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\begin{document}
\title{Extending a valuation centered in a local domain to the formal
completion.}
\section{Introduction} \label{In}
All the rings in this paper will be commutative with 1.
Let $(R,m,k)$ be a local noetherian domain with field of fractions
$K$ and $R_\nu$ a valuation ring, dominating $R$ (not necessarily birationally). Let $\nu|_K:K^*\twoheadrightarrow\Gamma$ be the restriction of $\nu$ to $K$; by definition, $\nu|_K$ is centered at $R$. Let $\hat R$ denote the $m$-adic completion of $R$. In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace $R$ by its $m$-adic completion $\hat R$ and $\nu$ by a suitable extension $\hat\nu_-$ to $\frac{\hat R}P$ for a suitably chosen prime ideal $P$, such that $P\cap R=(0)$ (one specific application we have in mind has to do with the approaches to proving the Local Uniformization Theorem in arbitrary characteristic such as \cite{Spi2} and \cite{Te}). The first reason is that the ring $\hat R$ is not in general an integral domain, so that we can only hope to extend $\nu$ to a \textit{pseudo-valuation} on $\hat R$, which means precisely a valuation $\hat\nu_-$ on a quotient $\frac{\hat R}P$ as above. The prime ideal $P$ is called the \textit{support} of the pseudo-valuation. It is well known and not hard to prove that such extensions $\hat\nu_-$ exist for some minimal prime ideals $P$ of $\hat R$. Although, as we shall see, the datum of a valuation $\nu$ determines a unique minimal prime of $\hat R$ when $R$ is excellent, in general there are many possible primes $P$ as above and for a fixed $P$ many possible extensions $\hat\nu_-$. This is the second reason to study extensions $\hat\nu_-$.
The purpose of this paper is to give, assuming that $R$ is excellent, a systematic description of all such extensions $\hat\nu_-$ and to identify certain classes of extensions which are of particular interest for applications. In fact, the only assumption about $R$ we ever use in this paper is a weaker and more natural condition than excellence, called the G condition, but we chose to talk about excellent rings since this terminology seems to be more familiar to most people. For the reader's convenience, the definitions of excellent and G-rings are recalled in the Appendix. Under this assumption, we study \textbf{extensions to (an
integral quotient of) the completion $\hat R$ of a valuation $\nu$
and give descriptions of the
valuations with which such extensions are composed. In particular we give criteria
for the uniqueness of the extension if certain simple data on these
composed valuations are fixed}. \noindent
We conjecture (see statement 5.19 in \cite{Te} and Conjecture \ref{teissier} below for a stronger and more precise statement) that\par\noindent
\textbf{given an excellent local ring $R$ and a valuation $\nu$ of
$R$ which is positive on its maximal ideal $m$, there exists a prime ideal
$H$ of the $m$-adic completion $\hat R$ such that $H\bigcap R=(0)$
and an extension of $\nu$ to $\frac{\hat R}{H}$ which has the same
value group as $\nu$.}
When studying extensions of $\nu$ to the completion of $R$, one is led to the study of its extensions to the henselization $\tilde R$ of $R$ as a natural first step. This, in turn, leads to the study of extensions of $\nu$ to finitely generated local strictly \'etale extensions $R^e$ of $R$. We therefore start out by letting $\sigma:R\rightarrow R^\dag$ denote one of the three operations of completion, (strict) henselization, or a finitely generated local strictly \'etale extension: \begin{eqnarray} R^\dag&=&\hat R\quad\mbox{ or}\label{eq:hatR}\\ R^\dag&=&\tilde R\quad\mbox{ or}\label{eq:tildeR}\\ R^\dag&=&R^e.\label{eq:Re} \end{eqnarray} The ring $R^\dag$ is local; let $m^\dag$ denote its maximal ideal. The homomorphisms $$ R\rightarrow\tilde R\quad\text{ and }\quad R\rightarrow R^e $$ are regular for any ring $R$; by definition, if $R$ is an excellent ring then the completion homomorphism is regular (in fact, regularity of the completion homomorphism is precisely the defining property of G-rings; see the Appendix for the definition of regular homomorphism).
Let $r$ denote the (real) rank of $\nu$. Let $(0)=\Delta_r\subsetneqq \Delta_{r-1}\subsetneqq\dots\subsetneqq \Delta_0=\Gamma$ be the isolated subgroups of $\Gamma$ and $P_0=(0)\subsetneqq P_1\subseteq\dots\subseteq P_r=m$ the prime valuation ideals of $R$, which need not, in general, be distinct. In this paper, we will assume that $R$ is excellent. Under this assumption, we will canonically associate to $\nu$ a chain $H_1\subset H_3\subset\dots\subset H_{2r+1}=mR^\dag$ of ideals of $R^\dag$, numbered by odd integers from 1 to $2r+1$, such that $H_{2\ell+1}\cap R=P_\ell$ for $0\le \ell\le r$. We will show that all the ideals $H_{2\ell+1}$ are prime. We will define $H_{2\ell}$ to be the unique minimal prime ideal of $P_\ell R^\dag$, contained in $H_{2\ell+1}$ (that such a minimal prime is unique follows from the regularity of the homomorphism $\sigma$).
We will thus obtain, in the cases (\ref{eq:hatR})--(\ref{eq:Re}), a chain of $2r+1$ prime ideals $$ H_0\subset H_1\subset\dots\subset H_{2r}=H_{2r+1}=mR^\dag, $$ satisfying $H_{2\ell}\cap R=H_{2\ell+1}\cap R=P_\ell$ and such that $H_{2\ell}$ is a minimal prime of $P_\ell R^\dag$ for $0\le \ell\le r$. Moreover, if $R^\dag=\tilde R$ or $R^\dag=R^e$, then $H_{2\ell}=H_{2\ell+1}$. We call $H_i$ the $i${\bf-th implicit prime
ideal} of $R^\dag$, associated to $R$ and $\nu$. The ideals $H_i$ behave well under local blowing ups along $\nu$ (that is, birational local homomorphisms $R\to R'$ such that $\nu$ is centered in $R'$), and more generally under \textit{$\nu$-extensions} of $R$ defined below in subsection \ref{trees}. This means that given any local blowing up along $\nu$ or $\nu$-extension $R\rightarrow R'$, the $i$-th implicit prime ideal $H'_i$ of ${R'}^\dag$ has the property that $H'_i\cap R^\dag=H_i$. This intersection has a meaning in view of Lemma \ref{factor} below.
For a prime ideal $P$ in a ring $R$, $\kappa(P)$ will denote the residue field $\frac{R_P}{PR_P}$.
Let $(0)\subsetneqq \mathbf{m}_1\subsetneqq\dots\subsetneqq \mathbf{m}_{r-1}\subsetneqq\mathbf{m}_r=\mathbf{m}_\nu$ be the prime ideals of the valuation ring $R_\nu$. By definitions, our valuation $\nu$ is a composition of $r$ rank one valuations $\nu=\nu_1\circ\nu_2\dots\circ\nu_r$, where $\nu_\ell$ is a valuation of the field $\kappa(\mathbf{m}_{\ell-1})$, centered at $\frac{(R_\nu)_{\mathbf{m}_\ell}}{\mathbf{m}_{\ell -1}}$ (see \cite{ZS}, Chapter VI, \S10, p. 43 for the definition of composition of valuations; more information and a simple example of composition is given below in subsection \ref{trees}, where we interpret each $\mathbf{m}_\ell$ as the limit of a tree of ideals).
If $R^\dag=\tilde R$, we will prove that there is a unique extension $\tilde\nu_-$ of $\nu$ to $\frac{\tilde R}{H_0}$. If $R^\dag=\hat R$, the situation is more complicated. First, we need to discuss the behaviour of our constructions under $\nu$-extensions.
\subsection{Local blowings up and trees.}\label{trees}
We consider \textit{extensions} $R\rightarrow R'$ of local rings, that is, injective morphisms such that $R'$ is an $R$-algebra essentially of finite type and $m'\cap R=m$. In this paper we consider only extensions with respect to $\nu$; that is, both $R$ and $R'$ are contained in a fixed valuation ring $R_\nu$. Such extensions form a direct system $\{R'\}$. We will consider many direct systems of rings and of ideals indexed by $\{R'\}$; direct limits will always be taken with respect to the direct system $\{R'\}$. Unless otherwise specified, we will assume that \begin{equation} \lim\limits_{\overset\longrightarrow{R'}}R'=R_\nu.\label{eq:ZariskiRiemann} \end{equation} Note that by the fundamental properties of valuation rings (\cite{ZS}, \S VI), assuming the equality (\ref{eq:ZariskiRiemann}) is equivalent to assuming that $\lim\limits_{\overset\longrightarrow{R'}}K'=K_\nu$, where $K'$ stands for the field of fractions of $R'$ and $K_\nu$ for that of $R_\nu$, and that $\lim\limits_{\overset\longrightarrow{R'}}R'$ is a valuation ring. \begin{definition} A \textbf{tree} of $R'$-algebras is a direct system
$\{S'\}$ of rings, indexed by the directed set $\{R'\}$, where $S'$ is an $R'$-algebra. Note that the maps are not necessarily injective. A morphism $\{S'\}\to\{T'\}$ of trees is the datum of a map of $R'$-algebras $S'\to T'$ for each $R'$ commuting with the tree morphisms for each map $R'\to R''$. \end{definition} \begin{lemma}\label{factor} Let $R\rightarrow R'$ be an extension of
local rings. We have:\par \noindent 1) The ideal
$N:=m^\dag\otimes_R1+1\otimes_Rm'$ is maximal in the $R$-algebra
$R^\dag\otimes_RR'$.\par \noindent 2) The natural map of completions (resp. henselizations) $R^\dag\to{R'}^\dag$ is injective. \end{lemma} \begin{proof} 1) follows from that fact that $R^\dag/m^\dag =R/m$. The proof of 2) relies on a construction which we shall use often: the map $R^\dag\to{R'}^\dag$ can be factored as \begin{equation} R^\dag\to\left(R^\dag\otimes_RR'\right)_N\to{R'}^\dag,\label{eq:iota} \end{equation} where the first map sends $x$ to $x\otimes 1$ and the second is determined by $x\otimes x'\mapsto \hat b(x).c(x')$ where $\hat b$ is the natural map $R^\dag\to {R'}^\dag$ and $c$ is the canonical map $R'\to{R'}^\dag$. The first map is injective because $R^\dag$ is a flat $R$-algebra and it is obtained by tensoring the injection $R\to R'$ by the $R$-algebra $R^\dag$; furthermore, elements of $R^\dag$ whose image in $R^\dag\otimes_RR'$ lie outside of $N$ are precisely units of $R^\dag$, hence they are not zero divisors in $R^\dag\otimes_RR'$ and $R^\dag$ injects in every localization of $R^\dag\otimes_RR'$.\par \noindent Since $m'\cap R=m$, we see that the inverse image by the natural map of $R'$-algebras $$ \iota\colon R'\to(R^\dag\otimes_RR')_N, $$ defined by $x'\mapsto1\otimes_Rx'$, of the maximal ideal $M=(m^\dag\otimes_R1+1\otimes_Rm')(R^\dag\otimes_RR')_N$ of $(R^\dag\otimes_RR')_N$ is the ideal $m'$ and that $\iota$ induces a natural isomorphism $\frac{R'}{{m'}^i}\overset\sim\rightarrow\frac{(R^\dag\otimes_RR')_N}{M^i}$ for each $i$. From this it follows by the universal properties of completion and henselization that the second map in the sequence (\ref{eq:iota}) is the completion (resp. the henselization inside the completion) of $R^\dag\otimes_RR'$ with respect to the ideal $M$. It is therefore also injective. \end{proof}
\begin{definition} Let $\{S'\}$ be a tree of $R'$-algebras. For each $S'$,
let $I'$ be an ideal of $S'$. We say that $\{I'\}$ is a tree of
ideals if for any arrow $b_{S'S''}\colon S'\rightarrow S''$ in our
direct system, we have $b^{-1}_{S'S''}I''=I'$. We have the obvious
notion of inclusion of trees of ideals. In particular, we may speak
about chains of trees of ideals. \end{definition}
\begin{examples} The maximal ideals of the local rings of our system $\{R'\}$ form a tree of ideals.
For any non-negative element $\beta\in\Gamma$, the valuation ideals $\mathcal{P}'_\beta\subset R'$ of value $\beta$ form a tree of ideals of $\{R'\}$. Similarly, the $i$-th prime valuation ideals $P'_i\subset R'$ form a tree. If $rk\ \nu=r$, the prime valuation ideals $P'_i$ give rise to a chain \begin{equation} P'_0=(0)\subsetneqq P'_1\subseteq\dots\subseteq P'_r=m'\label{eq:treechain'} \end{equation} of trees of prime ideals of $\{R'\}$. \end{examples}
We discuss this last example in a little more detail and generality in order to emphasize our point of view, crucial throughout this paper: the data of a composite valuation is equivalent to the data of its components. Namely, suppose we are given a chain of trees of ideals as in (\ref{eq:treechain'}), where we relax our assumptions of the $P'_i$ as follows. We no longer assume that the chain (\ref{eq:treechain'}) is maximal, nor that $P'_i\subsetneqq P'_{i+1}$, even for $R'$ sufficiently large; in particular, for the purposes of this example we momentarily drop the assumption that $rk\ \nu=r$. We will still assume, however, that $P'_0=(0)$ and that $P'_r=m'$.
Taking the limit in (\ref{eq:treechain'}), we obtain a chain \begin{equation} (0)=\mathbf{m}_0\subsetneqq\mathbf{m}_1\subseteqq\dots\subseteqq \mathbf{m}_r=\mathbf{m}_\nu\label{eq:treechainlim} \end{equation} of prime ideals of the valuation ring $R_\nu$.
Similarly, for each $1\leq\ell\leq r$ one has the equality $$ \lim\limits_{\overset\longrightarrow{R'}}{\frac{R'}{P'_\ell}}=\frac{R_\nu}{\bf
m_\ell}. $$ Then \textbf{specifying the valuation $\nu$ is equivalent to specifying valuations $\nu_0,\nu_1$, \dots, $\nu_r$, where $\nu_0$ is the trivial valuation of $K$ and, for $1\le \ell\le r$, $\nu_\ell$ is a valuation of the residue field $k_{\nu_{\ell-1}}=\kappa(\mathbf{m}_{\ell-1})$, centered at the local ring $\lim\limits_{\longrightarrow}\frac{R'_{P'_\ell}}{P'_{\ell-1}R'_{P'_\ell}}= \frac{(R_\nu)_{\mathbf{m}_\ell}}{\mathbf{m}_{\ell-1}}$ and taking its values in the totally ordered group $\frac{\Delta_{\ell-1}}{\Delta_\ell}$.}
The relationship between $\nu$ and the $\nu_\ell$ is that $\nu$ is the composition \begin{equation} \nu=\nu_1\circ\nu_2\circ\dots\circ\nu_r.\label{eq:composition1} \end{equation} For example, the datum of the valuation $\nu$, or of its valuation ring $R_\nu$, is equivalent to the datum of the valuation ring $\frac{R_\nu}{\mathbf{m}_{r-1}}\subset \frac{(R_\nu)_{\mathbf{m}_{r-1}}}{\mathbf{m}_{r-1}(R_\nu)_{\mathbf{m}_{r-1}}}=\kappa (\mathbf{m}_{r-1})$ of the valuation $\nu_r$ of the field $\kappa(\mathbf{m}_{r-1})$ and the valuation ring $(R_\nu)_{\mathbf{m}_{r-1}}$. If we assume, in addition, that for $R$ sufficiently large the chain (\ref{eq:treechain'}) (equivalently, (\ref{eq:treechainlim})) is a maximal chain of distinct prime ideals then $rk\ \nu=r$ and $rk\ \nu_\ell=1$ for each $\ell$.\par
\begin{remark}\label{composite} Another way to describe the same property of valuations is that, given a prime ideal $H$ of the local integral domain $R$ one builds all valuations centered in $R$ having $H$ as one of the $P_\ell$ by choosing a valuation $\nu_1$ of $R$ centered at $H$, so that $\mathbf{m}_{\nu_1}\cap R=H$ and choosing a valuation subring $\overline R_{\overline\nu}$ of the field $\frac{R_{\nu_1}}{\mathbf{m}_{\nu_1}}$ centered at $R/H$. Then $\nu=\nu_1\circ \overline\nu$.\par \noindent Note that choosing a valuation of $R/H$ determines a valuation of its field of fractions $\kappa(H)$, which is in general much smaller than $\frac{R_{\nu_1}}{\mathbf{m}_{\nu_1}}$. Given a valuation of $R$ with center $H$, in order to determine a valuation of $R$ with center $m$ inducing on $R/H$ a given valuation $\mu$ we must choose an extension $\overline\nu$ of $\mu$ to $\frac{R_{\nu_1}}{\mathbf{m}_{\nu_1}}$, and there are in general many possibilities.\par This will be used in the sequel. In particular, it will be applied to the case where a valuation $\nu$ of $R$ extends uniquely to a valuation $\hat\nu_-$ of $\frac{\hat R}{H}$ for some prime $H$ of $\hat R$. Assuming that $\hat R$ is an integral domain, this determines a unique valuation of $\hat R$ only if the height ${\operatorname{ht}}\ H$ of $H$ in $\hat R$ is at most one. In all other cases the dimension of $\hat R_H$ is at least $2$ and we have infinitely many valuations with which to compose $\hat\nu_-$. This is the source of the height conditions we shall see in \S\ref{extensions}. \end{remark}
\begin{example} Let $k_0$ be a field and $K=k_0((u,v))$ the field of fractions of the complete local ring $R=k_0[[u,v]]$. Let $\Gamma=\mathbf Z^2$ with lexicographical ordering. The isolated subgroups of $\Gamma$ are $(0)\subsetneqq(0)\oplus\mathbf Z\subsetneqq\mathbf Z^2$. Consider the valuation $\nu:K^*\rightarrow\mathbf Z^2$, centered at $R$, given by \begin{eqnarray} \nu(v)&=&(0,1)\\ \nu(u)&=&(1,0)\\ \nu(c)&=&0\quad\text{ for any }c\in k_0^*. \end{eqnarray} This information determines $\nu$ completely; namely, for any power series $$ f=\sum\limits_{\alpha,\beta}c_{\alpha\beta}u^\alpha v^\beta \in k_0[[u,v]], $$ we have $$
\nu(f)=\min\{(\alpha,\beta)\ |\ c_{\alpha\beta}\ne 0\}. $$ We have $rk\ \nu=rat.rk\ \nu=2$. Let $\Delta=(0)\oplus\mathbf Z$. Let $\Gamma_+$ denote the semigroup of all the non-negative elements of $\Gamma$. Let $k_0[[\Gamma_+]]$ denote the $R$-algebra of power series $\sum c_{\alpha,\beta}u^\alpha v^\beta$ where $c_{\alpha,\beta}\in k_0$ and the exponents $(\alpha ,\beta)$ form a well ordered subset of $\Gamma_+$. By classical results (see \cite{Kap1}, \cite{Kap2}), it is a valuation ring with maximal ideal generated by all the monomials $u^\alpha v^\beta$, where $(\alpha,\beta)>(0,0)$ (in other words, either $\alpha>0,\beta\in\mathbf Z$ or $\alpha=0,\beta>0$). Then $$ R_\nu=k_0[[\Gamma_+]]\bigcap k_0((u,v)) $$ is a valuation ring of $K$, and contains $k[[u,v]]$; it is the valuation ring of the valuation $\nu$. The prime ideal $\mathbf{m}_1$ is the ideal of $R_\nu$ generated by all the $uv^\beta$, $\beta\in\mathbf Z$. The valuation $\nu_1$ is the discrete rank 1 valuation of $K$ with valuation ring $$ (R_\nu)_{\mathbf{m}_1}=k_0[[u,v]]_{(u)} $$ and $\nu_2$ is the discrete rank 1 valuation of $k_0((v))$ with valuation ring $\frac{R_\nu}{\mathbf{m}_1}\cong k_0[[v]]$. \end{example}
\begin{example} To give a more interesting example, let $k_0$ be a field of characteristic zero and $$ K=k_0(x,y,z) $$ a purely transcendental extension of $k_0$ of degree 3. Let $w$ be an independent variable and put $k=\bigcup\limits_{j=1}^\infty k_0\left(w^{\frac 1j}\right)$. Let $\Gamma=\mathbf Z\oplus\mathbf{Q}$ with the lexicographical ordering and $\Delta=(0)\oplus\mathbf{Q}$ the non-trivial isolated subgroup of $\Gamma$. Let $u,v$ be new variables and let $\mu_1:k((u,v))\rightarrow\mathbf{Z}^2_{lex}$ be the valuation of the previous example. Let $\mu_2$ denote the $x$-adic valuation of $k$ and put $\mu=\mu_1\circ\mu_2$. Consider the map $\iota:k_0[x,y,z]\rightarrow k[[u,v]]$ which sends $x$ to $w$, $y$ to $v$ and $z$ to
$u-\sum\limits_{j=1}^\infty w^{\frac 1j}v^j$. Let $\nu_1=\left.\mu_1\right|_K$ and
$\nu=\mu|_K$.
The valuation $\nu:K^*\rightarrow\Gamma$ is centered at the local ring $R=k_0[x,y,z]_{(x,y,z)}$; we have \begin{eqnarray} \nu(x)&=&(0,1)\\ \nu(y)&=&(1,0),\\ \nu(z)&=&(1,1). \end{eqnarray} Write as a composition of two rank 1 valuations: $\nu=\nu_1\circ\nu_2$. We have natural inclusions $R_{\nu_1}\subset R_{\mu_1}$ and $k_{\nu_1}\subset k_{\mu_1}=k$. We claim that $k_{\nu_1}$ is not finitely generated over $k_0$. Indeed, if this were not the case then there would exist a prime number $p$ such that $w^{\frac1p}\mbox{$\in$ \hspace{-.8em}/} k_{\nu_1}$. Let $k'=k_0\left(x^{\frac1{(p-1)!}}\right)$. Let $L= k'(y,z)$. Consier the tower of field extensions $K\subset L\subset k[[u,v]]$ and let $\nu'$ denote the restriction of $\mu$ to $L$. Let $\Gamma'$ be the value group of $\nu'$ and $k_{\nu'}$ the residue field of its valuation ring. Now, $L$ contains the element $z_p:=z-\sum\limits_{j=1}^{p-1}x^{\frac1j}y^j$ as well as $\frac{z_p}{y^p}$. We have \begin{equation} \nu'(z_p)=\left(p,\frac1p\right), \end{equation} $\nu'\left(\frac{z_p}{y^p}\right)=0$ and the natural image of $\frac{z_p}{y^p}$ in $k_{\mu_1}=k$ is $w^{\frac1p}$. Now,
$p\not|\ [L:K]$, $\left.[\Gamma':\Gamma]\ \right|\
[L:K]$ and $\left.[k_{\nu'}:k_\nu]\ \right|\ [L:K]$. This implies that $z_p\in L$ and $w^{\frac1p}\in k_{\nu_1}$, which gives the desired contradicion.
It is not hard to show that for each $j$, there exists a local blowing up $R\rightarrow R'$ of $R$ such that, in the notation of (\ref{eq:treechain'}), we have $\kappa(P'_1)=k_0\left(w^{\frac1{j!}}\right)$ and that $\kappa(\mathbf{m}_1)=\lim\limits_{j\to\infty}\kappa(P'_1)=k$. The first one is the blowing up of the ideal $(y,z)R$, localized at the point $y=0, z/y=x$. Then one blows up the ideal $(z/y-x, y)$, and so on.
Another way to see the valuation $\nu=\nu_1\circ\nu_2$ is to note that $\nu_1$ is the restriction to $K$ of the $v$-adic valuation under the inclusion of fields deduced from the inclusion of rings $$ k_0[[x,y,z]]_{(y,z)}\hookrightarrow k\left[\left[v^{{\mathbf Z}_+}\right]\right] $$ which sends $x$ to $w$, $y$ to $v$ and $z$ to $\sum\limits_{j=1}^\infty w^{\frac1j}v^j$. Recall that the ring on the right is made of power series with non negative rational exponents whose set of exponents is well ordered. We have $k_{\nu_1}=k$. \end{example} \begin{remark} The point of the last example is to show that, given a
composed valuation as in (\ref{eq:composition1}), $\nu_\ell$ is a
valuation of the field $k_{\nu_{\ell-1}}$, which may
\textbf{properly} contain $\kappa(P'_{\ell-1})$ for \textbf{every}
$R'\in\mathcal{T}$. This fact will be a source of complication
later on and we prefer to draw attention to it from the beginning. \end{remark} Coming back to the implicit prime ideals, we will see that the implicit prime ideals $H'_i$ form a tree of ideals of $R^\dag$.
We will show that if $\nu$ extends to a valuation of $\hat\nu_-$ centered at $\frac{\hat R}P$ with $P\cap R=(0)$ then the prime $P$ must contain the minimal prime $H_0$ of $\hat R$. We will then show that specifying an extension $\hat\nu_-$ of $\nu$ as above is equivalent to specifying a chain of prime valuation ideals \begin{equation} \tilde H'_0\subset\tilde H'_1\subset\dots\subset\tilde H'_{2r}=m'\hat R'\label{eq:chaintree} \end{equation} of $\hat R'$ such that $H'_\ell\subset\tilde H'_\ell$ for all $\ell\in\{0,\dots,2r\}$, and valuations $\hat\nu_1,\hat\nu_2,\dots,\hat\nu_{2r}$, where $\hat\nu_i$ is a valuation of the field $k_{\hat\nu_{i-1}}$ (the residue field of the valuation ring $R_{\hat\nu_{i-1}}$), arbitrary when $i$ is odd and satisfying certain conditions, coming from the valuation $\nu_{\frac
i2}$, when $i$ is even.
The prime ideals $H_i$ are defined as follows.\par \noindent Recall that given a valued ring $(R,\nu)$, that is a subring $R\subseteq R_\nu$ of the valuation ring $R_\nu$ of a valuation with value group $\Gamma$, one defines for each $\beta\in \Gamma$ the valuation ideals of $R$ associated to $\beta$: $$ \begin{array}{lr} {\cal P}_\beta (R)=&\{x\in R/\nu(x)\geq\beta\}\cr {\cal P}^+_\beta (R)=&\{x\in R/\nu(x)>\beta\}\end{array} $$ and the associated graded ring $$ \hbox{\rm gr}_\nu R=\bigoplus_{\beta\in \Gamma}\frac{{\cal P}_\beta (R)} {{\cal P}^+_\beta (R)}=\bigoplus_{\beta\in \Gamma_+}\frac{{\cal P}_\beta (R)}{{\cal P}^+_\beta (R)}. $$ The second equality comes from the fact that if $\beta\in\Gamma_-\setminus\{0\}$, we have ${\cal P}^+_\beta (R)={\cal
P}_\beta (R)=R$. If $R\to R'$ is an extension of local rings such that $R\subset R'\subset R_\nu$ and $m_\nu\cap R'=m'$, we will write ${\cal P}'_\beta$ for ${\cal P}_\beta(R')$.
Fix a valuation ring $R_\nu$ dominating $R$, and a tree ${\cal
T}=\{R'\}$ of n\oe therian local $R$-subalgebras of $R_\nu$, having the following properties: for each ring $R'\in\cal{T}$, all the birational $\nu$-extensions of $R'$ belong to $\cal{T}$. Moreover, we assume that the field of fractions of $R_\nu$ equals $\lim\limits_{\overset\longrightarrow{R'}}K'$, where $K'$ is the field of fractions of $R'$. The tree $\cal{T}$ will stay constant throughout this paper. In the special case when $R$ happens to have the same field of fractions as $R_\nu$, we may take $\cal{T}$ to be the tree of all the birational $\nu$-extensions of $R$.
\begin{notation} For a ring $R'\in\cal T$, we shall denote by ${\cal T}(R')$ the subtree of $\cal T$ consisting of all the $\nu$-extensions $R''$ of $R'$. \end{notation}
We now define \begin{equation} H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}} \left(\left(\lim\limits_{\overset\longrightarrow{R'}}{\cal P}'_\beta
{R'}^\dag\right)\bigcap R^\dag\right),\ 0\le\ell\le r-1\label{eq:defin} \end{equation} (in the beginning of \S\ref{basics} we provide some motivation for this definition and give several elementary examples of $H'_i$ and $\tilde H'_i$).
The questions answered in this paper originally arose from our work on the Local Uniformization Theorem, where passage to completion is required in both the approaches of \cite{Spi2} and \cite{Te}. In \cite{Te}, one really needs to pass to completion for valuations of arbitrary rank. One of the main intended applications of the theory of implicit prime ideals is the following conjecture. Let \begin{equation} \Gamma\hookrightarrow\hat\Gamma\label{eq:extGamma} \end{equation} be an extension of ordered groups of the same rank. Let \begin{equation} (0)=\Delta_r\subsetneqq\Delta_{r-1}\subsetneqq\dots\subsetneqq\Delta_0=\Gamma \label{eq:isolated} \end{equation} be the isolated subgroups of $\Gamma$ and $$ (0)=\hat\Delta_r\subsetneqq\hat\Delta_{r-1}\subsetneqq\dots\subsetneqq \hat\Delta_0=\hat\Gamma $$ the isolated subgroups of $\hat\Gamma$, so that the inclusion (\ref{eq:extGamma}) induces inclusions \begin{eqnarray} \Delta_\ell&\hookrightarrow&\hat\Delta_\ell\quad\text{ and}\\ \frac{\Delta_\ell}{\Delta_{\ell+1}}&\hookrightarrow& \frac{\hat\Delta_\ell}{\hat\Delta_{\ell+1}}. \end{eqnarray} Let $G\hookrightarrow\hat G$ be an extension of graded algebras without zero divisors, such that $G$ is graded by $\Gamma_+$ and $\hat G$ by $\hat\Gamma_+$. The graded algebra $G$ is endowed with a natural valuation with value group $\Gamma$ and similarly for $\hat G$ and $\hat\Gamma$. These natural valuations will both be denoted by $ord$. \begin{definition} We say that the extension $G\hookrightarrow\hat G$ is
\textbf{scalewise birational} if for any $x\in\hat G$ and
$\ell\in\{1,\dots,r\}$ such that $ord\ x\in\hat\Delta_\ell$ there
exists $y\in G$ such that $ord\ y\in\Delta_\ell$ and $xy\in G$. \end{definition} Of course, scalewise birational implies birational and also that $\hat\Gamma=\Gamma$.\par \noindent While the main result of this paper is the primality of the implicit ideals associated to a valuation, and the subsequent description of the extensions of the valuation to the completion, the main conjecture stated here is the following:\par \begin{conjecture}\label{teissier} Assume that $\dim\ R'=\dim\ R$ for all
$R'\in\mathcal{T}$. Then there exists a tree of prime ideals $H'$
of $\hat R'$ with $H'\cap R'=(0)$ and a valuation $\hat\nu_-$,
centered at $\lim\limits_\to\frac{\hat R'}{H'}$ and having the
following property:\par\noindent For any $R'\in\cal{T}$ the graded
algebra $\mbox{gr}_{\hat\nu_-}\frac{\hat R'}{H'}$ is a scalewise birational
extension of $\mbox{gr}_\nu R'$. \end{conjecture} The example given in remark 5.20, 4) of \cite{Te} shows that the morphism of associated graded rings is not an isomorphism in general.
The approach to the Local Uniformization Theorem taken in \cite{Spi2} is to reduce the problem to the case of rank 1 valuations. The theory of implicit prime ideals is much simpler for valuations of rank 1 and takes only a few pages in Section \ref{archimedian}.
The paper is organized as follows. In \S\ref{basics} we define the odd-numbered implicit ideals $H_{2\ell+1}$ and prove that $H_{2\ell+1}\cap R=P_\ell$. We observe that by their very definition, the ideals $H_{2\ell+1}$ behave well under $\nu$-extensions; they form a tree. Proving that $H_{2\ell+1}$ is indeed prime is postponed until later sections; it will be proved gradually in \S\ref{technical}--\S\ref{prime}. In the beginning of \S\ref{basics} we will explain in more detail the respective roles played by the odd-numbered and the even-numbered implicit ideals, give several examples (among other things, to motivate the need for taking the limit with respect to $R'$ in (\ref{eq:defin})) and say one or two words about the techniques used to prove our results.
In \S\ref{technical} we prove the primality of the implicit prime ideals assuming a certain technical condition, called \textbf{stability}, about the tree $\cal T$ and the operation ${\ }^\dag$. It follows from the noetherianity of $R^\dag$ that there exists a specific $R'$ for which the limit in (\ref{eq:defin}) is attained. One of the main points of \S\ref{technical} is to prove properties of stable rings which guarantee that this limit is attained whenever $R'$ is stable. We then use the excellence of $R$ to define the even-numbered implicit prime ideals: for $i=2\ell$ the ideal $H_{2\ell}$ is defined to be the unique minimal prime of $P_\ell R^\dag$, contained in $H_{2\ell+1}$ (in the case $R^\dag=\hat R$ it is the excellence of $R$ which implies the uniqueness of such a minimal prime). We have $$ H_{2\ell}\cap R=P_\ell $$ for $\ell\in\{0,\dots,r\}$. The results of \S\ref{technical} apply equally well to completions, henselizations and other local \'etale extensions; to complete the proof of the primality of the implicit ideals in various contexts such as henselization or completion, it remains to show the existence of stable $\nu$-extensions in the corresponding context.
In \S\ref{Rdag} we describe the set of extensions $\nu^\dag_-$ of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{P'{R'}^\dag}$, where $P'$ is a tree of prime ideals of ${R'}^\dag$ such that $P'\cap R'=(0)$. We show (Theorem \ref{classification}) that specifying such a valuation $\nu^\dag_-$ is equivalent to specifying the following data:
(1) a chain (\ref{eq:chaintree}) of trees of prime ideals $\tilde H'_i$ of ${R'}^\dag$ (where $\tilde H'_0=P'$), such that $H'_i\subset\tilde H'_i$ for each $i$ and each $R'\in\mathcal{T}$, satisfying one additional condition (we will refer to the chain (\ref{eq:chaintree}) as the chain of trees of ideals, \textbf{determined by} the extension $\nu^\dag_-$)
(2) a valuation $\nu^\dag_i$ of the residue field $k_{\nu^\dag_{i-1}}$ of $\nu^\dag_{i-1}$, whose restriction to the field $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$ is centered at the local ring $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$.
If $i=2\ell$ is even, the valuation $\nu^\dag_i$ must be of rank 1 and its restriction to $\kappa(\mathbf{m}_{\ell-1})$ must coincide with $\nu_\ell$.
Notice the recursive nature of this description of $\nu^\dag_-$: in order to describe $\nu^\dag_i$ we must know $\nu^\dag_{i-1}$ in order to talk about its residue field $k_{\nu^\dag_{i-1}}$.
In \S\ref{extensions} we address the question of uniqueness of $\nu^\dag_-$. We describe several classes of extensions $\nu^\dag_-$ which are particularly useful for the applications: \textbf{minimal} and \textbf{evenly minimal} extensions, and also those $\nu^\dag_-$ for which, denoting by ${\operatorname{ht}}\ I$ the height of an ideal, we have \begin{equation} {\operatorname{ht}}\ \tilde H'_{2\ell+1}- {\operatorname{ht}}\ \tilde H'_{2_\ell} \le 1\quad\text{ for }0\le\ell\le r;\label{eq:odd=even4} \end{equation} in fact, the special case of (\ref{eq:odd=even4}) which is of most interest for the applications is \begin{equation} \tilde H'_{2\ell}=\tilde H'_{2\ell+1}\quad\text{ for }1\le\ell\le r.\label{eq:odd=even} \end{equation} We prove some necessary and some sufficient conditions under which an extension $\nu^\dag_-$ whose corresponding ideals $\tilde H'_i$ satisfy (\ref{eq:odd=even}) is uniquely determined by the ideals $\tilde H'_i$. We also give sufficient conditions for the graded algebra $gr_\nu R'$ to be scalewise birational to $gr_{\hat\nu_-}\hat R'$ for each $R'\in\cal{T}$. These sufficient conditions are used in \S\ref{locuni1} to prove some partial results towards Conjecture \ref{teissier}.
In \S\ref{henselization} we show the existence of $\nu$-extensions in $\cal T$, stable for henselization, thus reducing the proof of the primality of $H_{2\ell+1}$ to the results of \S\ref{technical}. We study the extension of $\nu$ to $\tilde R$ modulo its first prime ideal and prove that such an extension is unique.
In \S\ref{prime} we use the results of \S\ref{henselization} to prove the existence of $\nu$-extensions in $\cal T$, stable for completion. Combined with the results of \S\ref{technical} this proves that the ideals $H_{2\ell+1}$ are prime.
In \S\ref{locuni1} we describe a possible approach and prove some partial results towards constructing a chain of trees (\ref{eq:chaintree}) of prime ideals of $\hat R'$ satisfying (\ref{eq:odd=even}) and a corresponding valuation $\hat\nu_-$ which satisfies the conclusion of Conjecture \ref{teissier}. We also prove a necessary and a sufficient condition for the uniqueness of $\hat\nu_-$, assuming Conjecture \ref{teissier}.
We would like to acknowledge the paper \cite{HeSa} by Bill Heinzer and Judith Sally which inspired one of the authors to continue thinking about this subject, as well as the work of S.D. Cutkosky, S. El Hitti and L. Ghezzi: \cite{CG} (which contains results closely related to those of \S\ref{archimedian}) and \cite{CE}.
\section{Extending a valuation of rank one centered in a local domain to its formal completion.} \label{archimedian}
Let $(R,M,k)$ be a local noetherian domain, $K$ its field of fractions, and $\nu:K\rightarrow{\Gamma}_+\cup\{\infty\}$ a rank one valuation, centered at $R$ (that is, non-negative on $R$ and positive on $M$).\par Let $\hat R$ denote the formal completion of $R$. It is convenient to extend $\nu$ to a valuation centered at $\frac{\hat R}H$, where $H$ is a prime ideal of $\hat R$ such that $H\cap R=(0)$. In this section, we will assume that $\nu$ is of rank one, so that the value group ${\Gamma}$ is archimedian. We will explicitly describe a prime ideal $H$ of $\hat R$, canonically associated to $\nu$, such that $H\cap R=(0)$ and such that $\nu$ has a unique extension $\hat\nu_-$ to $\frac{\hat R}H$.
Let ${\Phi}=\nu(R\setminus (0))$, let $\mathcal{P}_\beta$ denote the $\nu$-ideal of $R$ of value $\beta$ and $\mathcal{P}_{\beta}^+$ the greatest $\nu$-ideal, properly contained in $\mathcal{P}_\beta$. We now define the main object of study of this section. Let \begin{equation} H:=\bigcap\limits_{\beta\in{\Phi}}(\mathcal{P}_\beta\hat R).\label{tag51} \end{equation}
\begin{remark}\label{Remark51} Since $R$ is noetherian, we have $\nu
(M)>0$ and since the ordered group $\Gamma$ is archimedian, for
every $\beta\in{\Phi}$ there exists $n\in\mathbf N$ such that
$M^n\subset \mathcal{P}_\beta$. In other words, the $M$-adic topology on $R$
is finer than (or equal to) the $\nu$-adic topology. Therefore an
element $x\in\hat R$ lies in $\mathcal{P}_\beta\hat R\iff$ there exists a
Cauchy sequence $\{x_n\}\subset R$ in the $M$-adic topology,
converging to $x$, such that $\nu(x_n)\ge\beta$ for all $n\iff$ for
{\it every} Cauchy sequence $\{x_n\}\subset R$, converging to $x$,
$\nu(x_n)\ge\beta$ for all $n\gg0$. By the same token, $x\in H\iff$
there exists a Cauchy sequence $\{x_n\}\subset R$, converging to
$x$, such that $\lim\limits_{n\to\infty}\nu(x_n)=\infty\iff$ for
{\it every} Cauchy sequence $\{x_n\}\subset R$, converging to $x$,
$\lim\limits_{n\to\infty}\nu(x_n)=\infty$. \end{remark}
\begin{example} Let $R=k[u,v]_{(u,v)}$. Then $\hat R=k[[u,v]]$. Consider an element $w=u-\sum\limits_{i=1}^\infty c_iv^i\in\hat R$, where $c_i\in k^*$ for all $i\in\mathbf N$, such that $w$ is transcendental over $k(u,v)$. Consider the injective map $\iota:k[u,v]_{(u,v)}\rightarrow k[[t]]$ which sends $v$ to $t$ and $u$ to $\sum\limits_{i=1}^\infty c_it^i$. Let $\nu$ be the valuation induced from the $t$-adic valuation of $k[[t]]$ via $\iota$. The value group of $\nu$ is $\mathbf Z$ and ${\Phi}=\mathbf N_0$. For each $\beta\in\mathbf N$, $\mathcal{P}_\beta=\left(v^\beta,u-\sum\limits_{i=1}^{\beta-1}c_iv^i\right)$. Thus $H=(w)$.
We come back to the general theory. Since the formal completion homomorphism $R\rightarrow\hat R$ is faithfully flat, \begin{equation} \mathcal{P}_\beta\hat R\cap R=\mathcal{P}_\beta\quad\text{for all }\beta\in{\Phi}.\label{tag52} \end{equation} Taking the intersection over all $\beta\in{\Phi}$, we obtain \begin{equation} H\cap R=\left(\bigcap\limits_{\beta\in{\Phi}}\left(\mathcal{P}_\beta\hat R\right)\right)\cap R=\bigcap\limits_{\beta\in{\Phi}}\mathcal{P}_\beta=(0),\label{tag53} \end{equation}
In other words, we have a natural inclusion $R\hookrightarrow\frac{\hat R}H$. \end{example} \begin{theorem}\label{th53} \begin{enumerate} \item $H$ is a prime ideal of $\hat R$. \item $\nu$ extends uniquely to a valuation $\hat\nu_-$, centered at $\frac{\hat R}H$. \end{enumerate} \end{theorem}
\begin{proof} Let $\bar x\in\frac{\hat R}H\setminus\{0\}$. Pick a representative $x$ of $\bar x$ in $\hat R$, so that $\bar x= x\ {\operatorname{mod}}\ H$. Since $x\mbox{$\in$ \hspace{-.8em}/} H$, we have $ x\mbox{$\in$ \hspace{-.8em}/} \mathcal{P}_\alpha\hat R$ for some $\alpha\in{\Phi}$. \begin{lemma}\label{lemma36} {\rm (See \cite{ZS}, Appendix 5, lemma 3)} Let $\nu$ be a valuation of rank one centered in a local noetherian domain $(R,M,k)$. Let $$ {\Phi}=\nu(R\setminus (0))\subset{\Gamma}. $$ Then ${\Phi}$ contains no infinite bounded sequences. \end{lemma}
\begin{proof} An infinite ascending sequence $\alpha_1<\alpha_2<\dots$ in ${\Phi}$, bounded above by an element $\beta\in{\Phi}$, would give rise to an infinite descending chain of ideals in $\frac R{\mathcal{P}_\beta}$. Thus it is sufficient to prove that $\frac R{\mathcal{P}_\beta}$ has finite length.
Let $\delta:=\nu(M)\equiv\min({\Phi}\setminus\{0\})$. Since ${\Phi}$ is archimedian, there exists $n\in\mathbf N$ such that $\beta\le n\delta$. Then $M^n\subset \mathcal{P}_\beta$, so that there is a surjective map $\frac R{M^n}\twoheadrightarrow\frac R{\mathcal{P}_\beta}$. Thus $\frac R{\mathcal{P}_\beta}$ has finite length, as desired. \end{proof}
By Lemma \ref{lemma36}, the set $\{\beta\in{\Phi}\ |\ \beta<\alpha\}$ is finite. Hence there exists a unique $\beta\in{\Phi}$ such that \begin{equation} x\in \mathcal{P}_\beta\hat R\setminus \mathcal{P}_{\beta}^+\hat R.\label{tag54} \end{equation} Note that $\beta$ depends only on $\bar x$, but not on the choice of the representative $ x$. Define the function $\hat\nu_-:\frac{\hat R}H\setminus\{0\}\rightarrow{\Phi}$ by \begin{equation} \hat\nu_-(\bar x)=\beta.\label{tag55} \end{equation} By (\ref{tag52}), if $x\in R\setminus\{0\}$ then \begin{equation} \hat\nu_-(x)=\nu(x). \end{equation} It is obvious that \begin{equation} \hat\nu_-(x+y)\ge\min\{\hat\nu_-(x),\hat\nu_-(y)\}\label{tag57} \end{equation} \begin{equation} \hat\nu_-(xy)\ge\hat\nu_-(x)+\hat\nu_-(y)\label{tag58} \end{equation} for all $x,y\in\frac{\hat R}H$. The point of the next lemma is to show that $\frac{\hat R}H$ is a domain and that $\hat\nu_-$ is, in fact, a valuation (i.e. that the inequality (\ref{tag58}) is, in fact, an equality).
\begin{lemma}\label{lemma54} For any non-zero $\bar x,\bar
y\in\frac{\hat R}H$, we have $\bar x\bar y\ne0$ and $\hat\nu_-(\bar
x\bar y)=\hat\nu_-(\bar x)+\hat\nu_-(\bar y)$. \end{lemma}
\begin{proof} Let $\alpha=\hat\nu_-(\bar x)$, $\beta=\hat\nu_-(\bar y)$. Let $ x$ and $ y$ be representatives in $\hat R$ of $\bar x$ and $\bar y$, respectively. We have $M\mathcal{P}_\alpha\subset \mathcal{P}_{\alpha}^+$, so that \begin{equation} \frac{\mathcal{P}_\alpha}{\mathcal{P}_{\alpha}^+}\cong\frac{\mathcal{P}_\alpha}{\mathcal{P}_{\alpha}^++M\mathcal{P}_\alpha}\cong \frac{\mathcal{P}_\alpha}{\mathcal{P}_{\alpha}^+}\otimes_Rk\cong\frac{\mathcal{P}_\alpha}{\mathcal{P}_{\alpha}^+} \otimes_R\frac{\hat R}{M\hat R}\cong\frac{\mathcal{P}_\alpha\hat R}{(\mathcal{P}_{\alpha}^++M\mathcal{P}_\alpha)\hat R}\cong\frac{\mathcal{P}_\alpha\hat R}{\mathcal{P}_{\alpha}^+\hat R},\label{tag59} \end{equation} and similarly for $\beta$. By (\ref{tag59}) there exist $z\in \mathcal{P}_\alpha$, $w\in \mathcal{P}_\beta$, such that $z\equiv x\ {\operatorname{mod}}\ \mathcal{P}_{\alpha}^+\hat R$ and $w\equiv y\ {\operatorname{mod}}\ \mathcal{P}_{\beta}^+\hat R$. Then \begin{equation}
xy\equiv zw\ {\operatorname{mod}}\ \mathcal{P}_{\alpha+\beta}^+\hat R.\label{tag510} \end{equation} Since $\nu$ is a valuation, $\nu(zw)=\alpha+\beta$, so that $zw\in \mathcal{P}_{\alpha+\beta}\setminus \mathcal{P}_{\alpha+\beta}^+$. By (\ref{tag52}) and (\ref{tag510}), this proves that $xy\in \mathcal{P}_{\alpha+\beta}\hat R\setminus \mathcal{P}_{\alpha+\beta}^+\hat R$. Thus $xy\mbox{$\in$ \hspace{-.8em}/} H$ (hence $\bar x\bar y\ne0$ in $\frac{\hat R}H$) and $\hat\nu_-(\bar x\bar y)=\alpha+\beta$, as desired. \end{proof}
By Lemma \ref{lemma54}, $H$ is a prime ideal of $\hat R$. By (\ref{tag57}) and Lemma \ref{lemma54}, $\hat\nu_-$ is a valuation, centered at $\frac{\hat R}H$. To complete the proof of Theorem \ref{th53}, it remains to prove the uniqueness of $\hat\nu_-$. Let $x$, $\bar x$, the element $\alpha\in{\Phi}$ and \begin{equation} z\in \mathcal{P}_\alpha\setminus \mathcal{P}_{\alpha}^+\label{tag511} \end{equation} be as in the proof of Lemma \ref{lemma54}. Then there exist \begin{equation} \begin{array}{rl} u_1,\ldots , u_n&\in \mathcal{P}_{\alpha}^+\text{ and}\\
v_1,\ldots ,v_n&\in\hat R \end{array}\label{tag512} \end{equation} such that $ x=z+\sum\limits_{i=1}^nu_i v_i$. Letting $\bar v_i:=v_i\ {\operatorname{mod}}\ H$, we obtain $\bar x=\bar z+\sum\limits_{i=1}^n\bar u_i\bar v_i$ in $\frac{\hat R}H$. Therefore, by (\ref{tag511})--(\ref{tag512}), for any extension of $\nu$ to a valuation $\hat\nu '_-$, centered at $\frac{\hat R}H$, we have \begin{equation} \hat\nu '_-(\bar x)=\alpha=\hat\nu_-(\bar x),\label{tag513} \end{equation} as desired. This completes the proof of Theorem \ref{th53}. \end{proof}
\begin{definition}\label{deft55} The ideal $H$ is called the {\bf implicit prime ideal} of $\hat R$, associated to $\nu$. When dealing with more than one ring at a time, we will sometimes write $H(R,\nu)$ for $H$. \end{definition}
More generally, let $\nu$ be a valuation centered at $R$, not necessarily of rank one. In any case, we may write $\nu$ as a composition $\nu=\mu_2\circ\mu_1$, where $\mu_2$ is centered at a non-maximal prime ideal $P$ of $R$ and $\mu_1\left|_{\frac RP}\right.$
is of rank one. The valuation $\mu_1\left|_{\frac RP}\right.$ is centered at $\frac RP$. We define the {\bf implicit prime ideal} of
$R$ with respect to $\nu$, denoted $H(R,\nu)$, to be the inverse image in $\hat R$ of the implicit prime ideal of $\frac{\hat R}P$ with respect to $\mu_1\left|_{\frac RP}\right.$. For the rest of this section, we will continue to assume that $\nu$ is of rank one.
\begin{remark}\label{Remark56} By (\ref{tag59}), we have the following natural isomorphisms of graded algebras: $$ \begin{array}{rl} \mbox{gr}_\nu R&\cong\mbox{gr}_{\hat\nu_-}\frac{\hat R}H\\ G_\nu&\cong G_{\hat\nu_-}. \end{array} $$ \end{remark}
\par
We will now study the behaviour of $H$ under local blowings up of $R$ with respect to $\nu$ and, more generally, under local homomorphisms. Let $\pi:(R,M)\rightarrow(R',M')$ be a local homomorphism of local noetherian domains. Assume that $\nu$ extends to a rank one valuation $\nu':R'\setminus\{0\}\rightarrow{\Gamma}'$, where ${\Gamma}'\supset{\Gamma}$. The homomorphism $\pi$ induces a local homomorphism $\hat\pi:\hat R\rightarrow\hat R'$ of formal completions. Let ${\Phi}'=\nu'(R'\setminus\{0\})$. For $\beta\in{\Phi}'$, let $\mathcal{P}'_\beta$ denote the $\nu'$-ideal of $R_{\nu'}$ of value $\beta$, as above. Let $H'=H(R',\nu')$.
\begin{lemma}\label{lemma58} Let $\beta\in{\Phi}$. Then \begin{equation} \left(\mathcal{P}'_\beta\hat R'\right)\cap\hat R=\mathcal{P}_\beta\hat R.\label{tag516} \end{equation} \end{lemma}
\begin{proof} Since by assumption $\nu'$ extends $\nu$ we have $\mathcal{P}'_\beta\cap R=\mathcal{P}_\beta$ and the inclusion \begin{equation} \left(\mathcal{P}'_\beta\hat R'\right)\cap\hat R\supseteq \mathcal{P}_\beta\hat R.\label{tag517} \end{equation} We will now prove the opposite inclusion. Take an element $x\in\left(\mathcal{P}'_\beta\hat R'\right)\cap\hat R$. Let $\{x_n\}\subset R$ be a Cauchy sequence in the $M$-adic topology, converging to $x$. Then $\{\pi(x_n)\}$ converge to $\hat\pi(x)$ in the $M'$-adic topology of $\hat R'$. Applying remark \ref{Remark51} to $R'$, we obtain \begin{equation} \nu(x_n)\equiv\nu'(\pi(x_n))\ge\beta\quad\text{for }n\gg0.\label{tag518} \end{equation} By (\ref{tag518}) and Remark \ref{Remark56}, applied to $R$, we have $x\in \mathcal{P}_\beta\hat R$. This proves the opposite inclusion in (\ref{tag517}), as desired. \end{proof}
\begin{corollary}\label{Corollary59} We have $$ H'\cap\hat R=H. $$ \end{corollary}
\begin{proof} Since $\nu'$ is of rank one, ${\Phi}$ is cofinal in ${\Phi}'$. Now the Corollary follows by taking the intersection over all $\beta\in{\Phi}$ in (\ref{tag516}). \end{proof}
Let $J$ be a non-zero ideal of $R$ and let $R\rightarrow R'$ be the local blowing up along $J$ with respect to $\nu$. Take an element $f\in J$, such that $\nu(f)=\nu(J)$. By the {\bf strict transform} of $J$ in $\hat R'$ we will mean the ideal $$ J^{\text{str}}:=\bigcup\limits_{i=1}^\infty\left(\left(J\hat R'\right):f^i\right)\equiv\left(J\hat R'_f\right)\cap\hat R'. $$ If $g$ is another element of $J$ such that $\nu(g)=\nu(J)$ then $\nu\left(\frac fg\right)=0$, so that $\frac fg$ is a unit in $R'$. Thus the definition of strict transform is independent of the choice of $f$.
\begin{corollary}\label{Corollary510} $H^{\text{str}}\subset H'$. \end{corollary}
\begin{proof} Since $H\hat R'\subset H'$, we have $H^{\text{str}}=\left(H\hat R'_f\right)\cap\hat R'\subset\left(H'\hat
R'_f\right)\cap\hat R'=H'$, where the last equality holds because $H'$ is a prime ideal of $\hat R'$, not containing $f$. \end{proof}
Using Zariski's Main Theorem, it can be proved that $H^{\text{str}}$ is prime. Since this fact is not used in the sequel, we omit the proof. \begin{corollary}\label{Corollary511} Let the notation and assumptions be as in corollary \ref{Corollary510}. Then \begin{equation} {\operatorname{ht}}\ H'\ge{\operatorname{ht}}\ H.\label{tag519} \end{equation} In particular, \begin{equation} \dim\frac{\hat R'}{H'}\le\dim\frac{\hat R}H.\label{tag520} \end{equation} \end{corollary}
\begin{proof} Let $\bar R:=\left(\hat
R\otimes_RR'\right)_{M'\hat R'\cap(\hat R\otimes_RR')}$. Let $\phi$ denote the natural local homomorphism $$ \bar R\rightarrow\hat R'. $$ Let $\bar H:=H'\cap\bar R$. Now, take $f\in J$ such that $\nu(f)=\nu(J)$. Then $f\mbox{$\in$ \hspace{-.8em}/} H'$ and, in particular, $f\mbox{$\in$ \hspace{-.8em}/}\bar H$. Since $R'_f\cong R_f$, we have $\hat R_f=\bar R_f$. In view of Corollary \ref{Corollary59}, we obtain $H\hat R_f\cong\bar H\bar R_f$, so \begin{equation} {\operatorname{ht}} \ H={\operatorname{ht}}\ \bar H.\label{tag521} \end{equation} Now, $\bar R$ is a local noetherian ring, whose formal completion is $\hat R'$. Hence $\phi$ is faithfully flat and therefore satisfies the going down theorem. Thus we have ${\operatorname{ht}}\ H'\ge{\operatorname{ht}}\ \bar H$. Combined with (\ref{tag521}), this proves (\ref{tag519}). As for the last statement of the Corollary, it follows from the well known fact that dimension does not increase under blowing up (\cite{Spi1}, Lemma 2.2): we have $\dim\ R'\le\dim\ R$, hence $$ \dim\ \hat R'=\dim R'\le\dim\ R=\dim\ \hat R, $$ and (\ref{tag520}) follows from (\ref{tag519}) and from the fact that complete local rings are catenarian. \end{proof}
It may well happen that the containment of corollary \ref{Corollary510} and the inequality in (\ref{tag519}) are strict. The possibility of strict containement in corollary \ref{Corollary510} is related to the existence of subanalytic functions, which are not analytic. We illustrate this statement by an example in which $H^{\text{str}}\subsetneqq H'$ and ${\operatorname{ht}}\ H<{\operatorname{ht}}\ H'$.
\begin{example} Let $k$ be a field and let $$ \begin{array}{rl} R&=k[x,y,z]_{(x,y,z)},\\ R'&=k[x',y',z']_{(x',y',z')}, \end{array} $$ where $x'=x$, $y'=\frac yx$ and $z'=z$. We have $K=k(x,y,z)$, $\hat R=k[[x,y,z]]$, $\hat R'=k[[x',y',z']]$. Let $t_1,t_2$ be auxiliary variables and let $\sum\limits_{i=1}^\infty c_it_1^i$ (with $c_i\in k$) be an element of $k[[t_1]]$, transcendental over $k(t_1)$. Let $\theta$ denote the valuation, centered at $k[[t_1,t_2]]$, defined by $\theta(t_1)=1$, $\theta(t_2)=\sqrt2$ (the value group of $\theta$ is the additive subgroup of $\mathbf R$, generated by 1 and $\sqrt2$). Let $\iota:R'\hookrightarrow k[[t_1,t_2]]$ denote the injective map defined by $\iota(x')=t_2$, $\iota(y')=t_1$, $\iota(z')=\sum\limits_{i=1}^\infty c_it_1^i$. Let $\nu$ denote the restriction of $\theta$ to $K$, where we view $K$ as a subfield of $k((t_1,t_2))$ via $\iota$. Let ${\Phi}=\nu(R\setminus\{0\})$; ${\Phi}'=\nu(R'\setminus\{0\})$. For $\beta\in{\Phi}'$, $P'_\beta$ is generated by all the monomials of the form ${x'}^\alpha{y'}^\gamma$ such that $\sqrt2\alpha+\gamma\ge\beta$, together with $z'-\sum\limits_{j=1}^ic_j{y'}^j$, where $i$ is the greatest non-negative integer such that $i<\beta$.
Let $w':=z'-\sum\limits_{i=1}^\infty c_i{y'}^i$. Then $H'=(w')$, but $H=H'\cap\hat R=(0)$, so that $H^{\text{str}}=(0)\subsetneqq H'$ and ${\operatorname{ht}}\ H=0<1={\operatorname{ht}}\ H'$. \end{example} Recall the following basic result of the theory of G-rings: \begin{proposition}\label{Corollary57} Assume that $R$ is a reduced G-ring. Then $\hat R_H$ is a regular local ring. \end{proposition}
\begin{proof} Let $K=R_{\mathcal{P}_\infty}=\kappa(\mathcal{P}_\infty)$ (here we are using that $R$ is reduced and that $\mathcal{P}_\infty$ is a minimal prime of $R$). By definition of G-ring, the map $R\rightarrow\hat R$ is a regular homomorphism. Then by (\ref{tag53}) $\hat R_H$ is geometrically regular over $K$, hence regular. \end{proof} \begin{remark} Having extended in a unique manner the valuation $\nu$ to
a valuation $\hat\nu_-$ of $\frac{\hat R}{H}$, we see that if $R$ is
a G-ring, by Proposition \ref{Corollary57} there is a unique minimal
prime $\hat\mathcal{P}_\infty$ of $\hat R$ contained in $H$, corresponding to
the ideal $(0)$ in $\hat R_H$. Since $H\cap R=(0)$, we have the
equality $\hat\mathcal{P}_\infty\cap R=(0)$. Choosing a valuation $\mu$ of
the fraction field of $\frac{\hat R_H}{\hat\mathcal{P}_\infty\hat R_H}$
centered at $\frac{\hat R_H}{\hat\mathcal{P}_\infty\hat R_H}$ whose value
group $\Psi$ is a free abelian group produces a composed valuation
$\hat\nu_-\circ\mu$ on $\frac{\hat R}{\hat\mathcal{P}_\infty}$ with value group
$\Psi\bigoplus\Gamma$ ordered lexicographically, as
follows:\par\noindent Given $x\in \frac{\hat R}{\hat\mathcal{P}_\infty}$, let
$\psi=\mu(x)$ and blow up in $R$ the ideal $\mathcal{P}_\psi$ along our
original valuation, obtaining a local ring $R'$. According to what
we have seen so far in this section, in its completion $\hat R'$ we
can write $x=ye$ with $\mu(e)=\psi$ and $y\in \hat R'\setminus
H'$. The valuation $\nu$ on $R'$ extends uniquely to a valuation of
$\frac{\hat R'}{H'}$, which we may still denote by $\hat\nu_-$
because it induces $\hat\nu_-$ on $\frac{\hat R}{H}$. Let us
consider the image $\overline y$ of $y$ in $\frac{\hat
R'}{H'}$. Setting $(\hat\nu_-\circ\mu)(x)=\psi\bigoplus
\hat\nu_-(\overline y)\in \Psi\bigoplus\Gamma$ determines a
valuation of $\frac{\hat R}{\hat\mathcal{P}_\infty}$ as
required.\par
If we drop the assumption that $\Psi$ is a free abelian group, the above construction still works, but the value group $\hat\Gamma$ of $\hat\nu_-\circ\mu$ need not be isomorphic to the direct sum $\Psi\bigoplus\Gamma$. Rather, we have an exact sequence $0\rightarrow\Gamma\rightarrow\bar\Gamma\rightarrow\Psi\rightarrow0$, which need not, in general, be split; see {\rm\cite{V1}, Proposition 4.3}.
\noindent In the sequel we shall reduce to the case where $\hat R$ is an integral domain, so that $\hat\mathcal{P}_\infty=(0)$ and we will have constructed a valuation of $\hat R$. \end{remark}
\section{Definition and first properties of implicit ideals.} \label{basics}
Let the notation be as above. Before plunging into technical details, we would like to give a brief and informal overview of our constructions and the motivation for them. Above we recalled the well known fact that if $rk\ \nu=r$ then for every $\nu$-extension $R\rightarrow R'$ the valuation $\nu$ canonically determines a flag (\ref{eq:treechain'}) of $r$ subschemes of $\mbox{Spec}\ R'$. This paper shows the existence of subschemes of $\mbox{Spec}\ \hat R$, determined by $\nu$, which are equally canonical and which become explicit only after completion. To see what they are, first of all note that the ideal $P'_l\hat R'$, for $R'\in\cal{T}$ and $0\le\ell\le r-1$, need not in general be prime (although it is prime whenever $R'$ is henselian). Another way of saying the same thing is that the ring $\frac{R'}{P'_\ell}$ need not be analytically irreducible in general. However, we will see in \S\ref{prime} (resp. \S\ref{henselization}) that the valuation $\nu$ picks out in a canonical way one of the minimal primes of $P'_l\hat R'$ (resp. $P'_l\tilde R'$). We call this minimal prime $H'_{2\ell}$ for reasons which will become apparent later. By the flatness of completion (resp. henselization), we have $H'_{2\ell}\cap R'=P'_\ell$. We will show that the ideals $H'_{2\ell}$ form a tree.
Let \begin{equation} (0)=\Delta_r\subsetneqq\Delta_{r-1}\subsetneqq\dots\subsetneqq\Delta_0= \Gamma\label{eq:isolated1} \end{equation} be the isolated subgroups of $\Gamma$. There are other ideals of $\hat R$, apart from the $H_{2\ell}$, canonically associated to $\nu$, whose intersection with $R$ equals $P_\ell$, for example, the ideal $\bigcap\limits_{\beta\in\Delta_\ell}\mathcal{P}_\beta\hat R$. The same is true of the even larger ideal \begin{equation} H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_\ell} \left(\left(\lim\limits_{\overset\longrightarrow{R'}}{\cal
P}'_\beta\hat R'\right)\bigcap\hat R\right),\label{eq:defin2} \end{equation} (that $H_{2\ell+1}\cap R=P_\ell$ is easy to see and will be shown later in this section, in Proposition \ref{contracts}). While the examples below show that the ideal $\bigcap\limits_{\beta\in\Delta_\ell}\mathcal{P}_\beta\hat R$ need not, in general, be prime, the ideal $H_{2\ell+1}$ always is (this is the main theorem of this paper; it will be proved in \S\ref{prime}). The ideal $H_{2\ell+1}$ contains $H_{2\ell}$ but is not, in general equal to it. To summarize, we will show that the valuation $\nu$ picks out in a canonical way a generic point $H_{2\ell}$ of the formal fiber over $P_\ell$ and also another point $H_{2\ell+1}$ in the formal fiber, which is a specialization of $H_{2\ell}$.
The main technique used to prove these results is to to analyze the set of zero divisors of $\frac{{R'}^\dag}{P'_\ell{R'}^\dag}$ (where $R^\dag$ stands for either $\hat R$, $\tilde R$, or a finite type \'etale extension $R^e$ of $R$), as follows. We show that the reducibility of $\frac{{R'}^\dag}{P'_\ell{R'}^\dag}$ is related to the existence of non-trivial algebraic extensions of $\kappa(P_\ell)$ inside $\kappa(P_\ell)\otimes_RR^\dag$. More precisely, in the next section we define $R$ to be \textbf{stable} if $\frac{R^\dag}{P_{\ell
+1}R^\dag}$ is a domain and there does not exist a non-trivial algebraic extension of $\kappa(P_{\ell+1})$ which embeds both into $\kappa(P_{\ell +1})\otimes_RR^\dag$ and into $\kappa(P'_{\ell+1})$ for some $R'\in\mathcal{T}$. We show that if $R$ is stable then $\frac{{R'}^\dag}{P'_{\ell+1}{R'}^\dag}$ is a domain for all $R'\in\cal T$. For $\overline\beta\in\frac\Gamma{\Delta_{\ell+1}}$, let \begin{equation}
\mathcal{P}_{\overline\beta}=\left\{x\in R\ \left|\ \nu(x)\mod\Delta_{\ell+1}\ge\overline\beta\right.\right\}\label{eq:pbetamodl} \end{equation} If $\Phi$ denotes the semigroup $\nu(R\setminus\{0\})\subset\Gamma$, which is well ordered since $R$ is noetherian (see [ZS], Appendix 4, Proposition 2), and $$
\beta(\ell)=\min\{\gamma\in\Phi\ |\ \beta-\gamma\in\Delta_{\ell+1}\} $$ then $\mathcal{P}_{\overline\beta}=\mathcal{P}_{\beta(l)}$.\par\noindent
We have the inclusions $$ P_\ell\subset \mathcal{P}_{\overline\beta}\subset P_{\ell+1}, $$ and $\mathcal{P}_{\overline\beta}$ is the inverse image in $R$ by the canonical map $R\to \frac{R}{P_\ell}$ of a valuation ideal $\overline \mathcal{P}_{\overline\beta}\subset \frac{R}{P_\ell}$ for the rank one valuation $\frac{R}{P_\ell}\setminus\{0\}\to \frac{\Delta_\ell}{\Delta_{\ell+1}}$ induced by $\nu_{\ell+1}$.\par We will deduce from the above that if $R$ is stable then for each $\overline\beta\in\frac{\Delta_\ell}{\Delta_{\ell+1}}$ and each $\nu$-extension $R\rightarrow R'$ we have $\mathcal{P}'_{\overline\beta}{R'}^\dag\cap R^\dag=\mathcal{P}_{\overline\beta} R^\dag$, which gives us a very good control of the limit in the definition of $H_{2\ell+1}$ and of the $\nu$-extensions $R'$ for which the limit is attained.
We then show, separately in the cases when $R^\dag=\tilde R$ (\S\ref{henselization}) and $R^\dag=\hat R$ (\S\ref{prime}), that there always exists a stable $\nu$-extension $R'\in\cal{T}$.
We are now ready to go into details, after giving several examples of implicit ideals and the phenomena discussed above.
Let $0\le\ell\le r$. We define our main object of study, the $(2\ell+1)$-st implicit prime ideal $H_{2\ell+1}\subset R^\dag$, by \begin{equation} H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_\ell} \left(\left(\lim\limits_{\overset\longrightarrow{R'}}{\cal P}'_\beta
{R'}^\dag\right)\bigcap R^\dag\right),\label{eq:defin1} \end{equation} where $R'$ ranges over $\mathcal{T}$. As usual, we think of (\ref{eq:defin1}) as a tree equation: if we replace $R$ by any other $R''\in{\cal T}$ in (\ref{eq:defin1}), it defines the corresponding ideal $H''_{2\ell +1}\subset\hat R''^\dag$. Note that for $\ell=r$ (\ref{eq:defin1}) reduces to $$ H_{2r+1}=mR^\dag. $$ We start by giving several examples of the ideals $H'_i$ (and also of $\tilde H'_i$, which will appear a little later in the paper).
\begin{example}{\label{Example31}} Let $R=k[x,y,z]_{(x,y,z)}$. Let $\nu$ be the valuation with value group $\Gamma=\mathbf Z^2_{lex}$, defined as follows. Take a transcendental power series $\sum\limits_{j=1}^\infty c_ju^j$ in a variable $u$ over $k$. Consider the homomorphism $R\hookrightarrow k[[u,v]]$ which sends $x$ to $v$, $y$ to $u$ and $z$ to $\sum\limits_{j=1}^\infty c_ju^j$. Consider the valuation $\nu$, centered at $k[[u,v]]$, defined by $\nu(v)=(0,1)$ and $\nu(u)=(1,0)$; its restriction to $R$ will also be denoted by $\nu$, by abuse of notation. Let $R_\nu$ denote the valuation ring of $\nu$ in $k(x,y,z)$ and let $\mathcal{T}$ be the tree consisting of all the local rings $R'$ essentially of finite type over $R$, birationally dominated by $R_\nu$. Let ${}^\dag=\hat{\ }$ denote the operation of formal completion. Given $\beta=(a,b)\in\mathbf Z^2_{lex}$, we have ${\mathcal{P}}_\beta=x^b\left(y^a,z-c_1y-\dots-c_{a-1}y^{a-1}\right)$. The first isolated subgroup $\Delta_1=(0)\oplus\mathbf Z$. Then $\bigcap\limits_{\beta\in(0)\oplus\mathbf Z}\left({\cal P}_\beta\hat
R\right)=(y,z)$ and $\bigcap\limits_{\beta\in\Gamma=\Delta_0}\left({\cal P}_\beta\hat
R\right)=\left(z-\sum\limits_{j=1}^\infty c_jy^j\right)$. It is not hard to show that for any $R'\in\mathcal{T}$ we have $H'_1=\left(z-\sum\limits_{j=1}^\infty c_jy^j\right)\hat R'$ and that $H_3=(y,z)\hat R$. It will follow from the general theory developed in \S\ref{extensions} that $\nu$ admits a unique extension $\hat\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\hat R'$. This extension has value group $\hat\Gamma=\mathbf Z^3_{lex}$ and is defined by $\hat\nu(x)=(0,0,1)$, $\hat\nu(y)=(0,1,0)$ and $\hat\nu\left(z-\sum\limits_{j=1}^\infty c_jy^j\right)=(1,0,0)$. For each $R'\in\mathcal{T}$ the ideal $H'_1$ is the prime valuation ideal corresponding to the isolated subgroup $(0)\oplus\mathbf Z^2_{lex}$ of $\hat\Gamma$ (that is, the ideal whose elements have values outside of $(0)\oplus\mathbf Z^2_{lex}$) while $H'_3$ is the prime valuation ideal corresponding to the isolated subgroup $(0)\oplus(0)\oplus\mathbf Z$. \end{example}
\begin{example}{\label{Example32}} Let $R=k[x,y,z]_{(x,y,z)}$, $\Gamma=\mathbf Z^2_{lex}$, the power series $\sum\limits_{j=1}^\infty c_ju^j$ and the operation ${}^\dag=\hat{\ }$ be as in the previous example. This time, let $\nu$ be defined as follows. Consider the homomorphism $R\hookrightarrow k[[u,v]]$ which sends $x$ to $u$, $y$ to $\sum\limits_{j=1}^\infty c_ju^j$ and $z$ to $v$. Consider the valuation $\nu$, centered at $k[[u,v]]$, defined by $\nu(v)=(1,0)$ and $\nu(u)=(0,1)$; its restriction to $R$ will be also denoted by $\nu$. Let $R_\nu$ denote the valuation ring of $\nu$ in $k(x,y,z)$ and let $\mathcal{T}$ be the tree consisting of all the local rings $R'$ essentially of finite type over $R$, birationally dominated by $R_\nu$. Given $\beta=(a,b)\in\mathbf Z^2_{lex}$, we have $\mathcal{P}_\beta=z^a\left(x^b,y-c_1x-\dots-c_{b-1}x^{b-1}\right)$. The first isolated subgroup $\Delta_1=(0)\oplus\mathbf Z$. Then $\bigcap\limits_{\beta\in(0)\oplus\mathbf Z}\left({\mathcal{P}}_\beta\hat
R\right)=\left(y-\sum\limits_{j=1}^\infty c_jx^j,z\right)$ and $\bigcap\limits_{\beta\in\Gamma=\Delta_0}\left({\cal P}_\beta\hat
R\right)=(0)$. It is not hard to show that for any $R'\in\cal{T}$ we have $H'_1=(0)$ and that $H_3=\left(y-\sum\limits_{j=1}^\infty
c_jx^j,z\right)\hat R'$. In this case, the extension $\hat\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\hat R'$ is not unique. Indeed, one possible extension $\hat\nu^{(1)}$ has value group $\hat\Gamma=\mathbf Z^3_{lex}$ and is defined by $\hat\nu^{(1)}(x)=(0,0,1)$, $\hat\nu^{(1)}\left(y-\sum\limits_{j=1}^\infty c_jx^j\right)=(0,1,0)$ and $\hat\nu^{(1)}(z)=(1,0,0)$. In this case, for any $R'\in\cal{T}$ the ideal $H'_3$ is the prime valuation ideal corresponding to the isolated subgroup $(0)\oplus(0)\oplus\mathbf Z$ of $\hat\Gamma$.
Another extension $\hat\nu^{(2)}$ of $\nu$ is defined by $\hat\nu^{(2)}(x)=(0,0,1)$, $\hat\nu^{(2)}\left(y-\sum\limits_{j=1}^\infty c_jx^j\right)=(1,0,0)$ and $\hat\nu^{(2)}(z)=(0,1,0)$. In this case, the tree of ideals corresponding to the isolated subgroup $(0)\oplus(0)\oplus\mathbf Z$ is $H'_3$ (exactly the same as for $\hat\nu^{(1)}$) while that corresponding to $(0)\oplus\mathbf Z^2_{lex}$ is $\tilde H'_1=\left(y-\sum\limits_{j=1}^\infty c_jx^j\right)$. The tree $\tilde H'_1$ of prime $\hat\nu^{(2)}$-ideals determines the extension $\hat\nu^{(2)}$ completely. \end{example}
The following two examples illustrate the need for taking the limit over the tree $\mathcal{T}$.
\begin{example}{\label{Example33}} Let us consider the local domain $S=\frac{k[x,y]_{(x,y)}}{(y^2-x^2-x^3)}$. There are two distinct valuations centered in $(x,y)$. Let $a_i\in k,\ i\geq 2$ be such that $$ \left(y+x+\sum_{i\geq 2}a_ix^i\right)\left(y-x-\sum_{i\geq 2}a_ix^i\right)=y^2-x^2-x^3. $$ We shall denote by $\nu_+$ the rank one discrete valuation defined by $$ \nu_+(x)=\nu_+(y)=1, $$ $$ \nu_+(y+x)=2, $$ $$ \nu_+\left(y+x+\sum_{i= 2}^{b-1}a_ix^i\right)=b. $$ Now let $R=\frac{k[x,y,z]_{(x,y,z)}}{(y^2-x^2-x^3)}$. Let $\Gamma =\mathbf Z^2$ with the lexicographical ordering. Let $\nu$ be the composite valuation of the $(z)$-adic one with $\nu_+$, centered in $\frac R{(z)}$. The point of this example is to show that $$ H^*_{2\ell+1}=\bigcap_{\beta\in\Delta_{\ell}}\mathcal{P}_{\beta}{\hat R} $$ does not work as the definition of the $(2\ell+1)$-st implicit prime ideal because the resulting ideal $H^*_{2\ell+1}$ is not prime. Indeed, as $\mathcal{P}_{(a,0)}=(z^a)$, we have $$ H_1^*=\bigcap_{(a,b)\in\mathbf{Z}^2}\mathcal{P}_{(a,b)}{\hat R}=(0). $$ Let $f=y+x+\sum\limits_{i\geq 2}a_ix^i,g=y-x-\sum\limits_{i\geq
2}a_ix^i\in\hat R$. Clearly $f,g\mbox{$\in$ \hspace{-.8em}/} H^*_1=(0)$, but $f\cdot g=0$, so the ideal $H^*_1$ is not prime.
One might be tempted (as we were) to correct this problem by localizing at $H^*_{2\ell+3}$. Indeed, if we take the new definition of $H^*_{2\ell+1}$ to be, recursively in the descending order of $\ell$, \begin{equation} H^*_{2\ell +1}=\left(\bigcap_{\beta\in\Delta_{\ell}}\mathcal{P}_{\beta}{\hat R}_{H^*_{2\ell +3}}\right)\cap{\hat R},\label{eq:localization} \end{equation} then in the present example the resulting ideals $H^*_3=(z,f)$ and $H^*_1=(f)$ are prime. However, the next example shows that the definition (\ref{eq:localization}) also does not, in general, give rise to prime ideals. \end{example}
\begin{example}{\label{Example34}} Let $R=\frac{k[x,y,z]_{(x,y,z)}}{(z^2-y^2(1+x))}$. Let $\Gamma=\mathbf Z^2$ with the lexicographical ordering. Let $t$ be an independent variable and let $\nu$ be the valuation, centered in $R$, induced by the $t$-adic valuation of $k\left[\left[t^\Gamma\right]\right]$ under the injective homomorphism $\iota:R\hookrightarrow k\left[\left[t^\Gamma\right]\right]$, defined by $\iota(x)=t^{(0,1)}$, $\iota(y)=t^{(1,0)}$ and $\iota(z)=t^{(1,0)}\sqrt{1+t^{(0,1)}}$. The prime $\nu$-ideals of $R$ are $(0)\subsetneqq P_1\subsetneqq m$, with $P_1=(y,z)$. We have $\bigcap\limits_{\beta\in\Delta_1}\mathcal{P}_\beta\hat R=(y,z)\hat R=P_1\hat R$ and $\bigcap\limits_{\beta\in\Gamma}\mathcal{P}_\beta\hat R_{(y,z)}=\bigcap\limits_{\beta\in\Gamma}\mathcal{P}_\beta\hat R=(0)$. Note that the ideal $(0)$ is not prime in $\hat R$. Now, let $R'=R\left[\frac zy\right]_{m'}$, where $m'=\left(x,y,\frac
zy-1\right)$ is the center of $\nu$ in $R\left[\frac zy\right]$. We have $z-y\sqrt{1+x}\in\hat R\setminus\mathcal{P}_{(2,0)}\hat R$. On the other hand, $z-y\sqrt{1+x}=y\left(\frac zy-\sqrt{1+x}\right)=0$ in $\hat R'$; in particular, $z-y\sqrt{1+x}\in\bigcap\limits_{\beta\in\Gamma}\mathcal{P}'_\beta\hat R'$. Thus this example also shows that the ideals $\mathcal{P}_\beta\hat R$, $\bigcap\limits_{\beta\in\Delta_\ell}\mathcal{P}_\beta\hat R$ and $\bigcap\limits_{\beta\in\Delta_\ell}\mathcal{P}_\beta\hat R_{H_{2\ell+3}}$ do not behave well under blowing up. \end{example}
Note that both Examples \ref{Example33} and \ref{Example34} occur not only for the completion $\hat R$ but also for the henselization $\tilde R$.
We come back to the general theory of implicit ideals. \begin{proposition}\label{contracts} We have $H_{2\ell+1}\cap R=P_\ell$. \end{proposition}
\begin{proof} Recall that $P_\ell=\left\{x\in R\ \left|\ \
\nu(x)\mbox{$\in$ \hspace{-.8em}/} \Delta_\ell\right.\right\}$. If $x\in P_\ell$ then, since $\Delta_\ell$ is an isolated subgroup, we have $x\in {\cal
P}_\beta$ for all $\beta\in \Delta_\ell$. The same inclusion holds for the same reason in all extensions $R'\subset R_\nu$ of $R$, and this implies the inclusion $P_\ell\subseteq H_{2\ell+1}\cap R$. Now let $x$ be in $H_{2\ell+1}\cap R$ and assume $x\mbox{$\in$ \hspace{-.8em}/} P_\ell$. Then there is a $\beta\in \Delta_\ell$ such that $x\mbox{$\in$ \hspace{-.8em}/}{\cal
P}_\beta$. By faithful flatness of $R^\dag$ over $R$ we have ${\cal
P}_\beta R^\dag\cap R={\cal P}_\beta$. This implies that $x\mbox{$\in$ \hspace{-.8em}/}{\cal P}_\beta R^\dag$, and the same argument holds in all the extensions $R'\in\mathcal{T}$, so $x$ cannot be in $H_{2\ell+1}\cap R$. This contradiction shows the desired equality. \end{proof}
\begin{proposition}\label{behavewell} The ideals $H'_{2\ell+1}$ behave well
under $\nu$-extensions $R\rightarrow R'$ in $\mathcal{T}$. In other
words, let $R\rightarrow R'$ be a $\nu$-extension in $\mathcal{T}$
and let $H'_{2\ell+1}$ denote the $(2\ell+1)$-st implicit prime
ideal of $\hat R'$. Then $H_{2\ell+1}=H'_{2\ell+1}\cap R^\dag$. \end{proposition} \begin{proof} Immediate from the definitions. \end{proof}
To study the ideals $H_{2\ell+1}$, we need to understand more explicitly the nature of the limit appearing in (\ref{eq:defin1}). To study the relationship between the ideals $P_\beta R^{\dag}$ and $P'_\beta {R'}^\dag\bigcap R^\dag$, it is useful to factor the natural map $R^\dag\rightarrow{R'}^\dag$ as $R^\dag\rightarrow(R^\dag\otimes_RR')_{M'}\overset\phi\rightarrow{R'}^\dag$ as we did in the proof of Lemma \ref{factor}. In general, the ring $R^\dag\otimes_RR'$ is not local (see the above examples), but it has one distinguished maximal ideal $M'$, namely, the ideal generated by $mR^\dag\otimes1$ and $1\otimes m'$, where $m'$ denotes the maximal ideal of $R'$. The map $\phi$ factors through the local ring $\left(R^\dag\otimes_RR'\right)_{M'}$ and the resulting map $\left(R^\dag\otimes_RR'\right)_{M'}\rightarrow{R'}^\dag$ is either the formal completion or the henselization; in either case, it is faithfully flat. Thus $P'_\beta{R'}^\dag\cap\left(R^\dag\otimes_RR'\right)_{M'}= P'_\beta\left(R^\dag\otimes_RR'\right)_{M'}$. This shows that we may replace ${R'}^\dag$ by $\left(R^\dag\otimes_RR'\right)_{M'}$ in (\ref{eq:defin1}) without affecting the result, that is, \begin{equation} H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_\ell}\left( \left(\lim\limits_{\overset\longrightarrow{R'}} {\cal P}'_\beta\left(R^\dag\otimes_RR'\right)_{M'}\right)\bigcap R^\dag\right).\label{eq:defin3} \end{equation} From now on, we will use (\ref{eq:defin3}) as our working definition of the implicit prime ideals. One advantage of the expression (\ref{eq:defin3}) is that it makes sense in a situation more general than the completion and the henselization. Namely, to study the case of the henselization $\tilde R$, we will need to consider local \'etale extensions $R^e$ of $R$, which are contained in $\tilde R$ (particularly, those which are essentially of finite type). The definition (\ref{eq:defin3}) of the implicit prime ideals makes sense also in that case.
\section{Stable rings and primality of their implicit ideals.} \label{technical}
Let the notation be as in the preceding sections. As usual, $R^\dag$ will denote one of $\hat R$, $\tilde R$ or $R^e$ (a local \'etale $\nu$-extension essentially of finite type). Take an $R'\in\mathcal{T}$ and $\overline\beta\in\frac{\Delta_\ell}{\Delta_{\ell+1}}$. We have the obvious inclusion of ideals \begin{equation} \mathcal{P}_{\overline\beta} R^\dag\subset\mathcal{P}_{\overline\beta}{R'}^\dag\cap R^\dag\label{eq:stab1} \end{equation} (where $\mathcal{P}_{\overline\beta}$ is defined in (\ref{eq:pbetamodl})). A useful subtree of $\mathcal{T}$ is formed by the $\ell$-stable rings, which we now define. An important property of stable rings, proved below, is that the inclusion (\ref{eq:stab1}) is an equality whenever $R'$ is stable. \begin{definition}\label{stable} A ring $R'\in\mathcal{T}(R)$ is said to be
$\ell$-\textbf{stable} if the following two conditions hold:
(1) the ring \begin{equation} \kappa\left(P'_\ell\right)\otimes_R\left(R'\otimes_RR^\dag\right)_{M'} \label{eq:extension1} \end{equation} is an integral domain and
(2) there do not exist an $R''\in\mathcal{T}(R')$ and a non-trivial algebraic extension $L$ of $\kappa(P'_\ell)$ which embeds both into $\kappa\left(P'_\ell\right)\otimes_R\left(R'\otimes_RR^\dag\right)_{M'}$ and $\kappa(P''_\ell)$.
We say that $R$ is \textbf{stable} if it is $\ell$-stable for each $\ell\in\{0,\dots,r\}$. \end{definition} \begin{remark}\label{interchanging} (1) Rings of the form
(\ref{eq:extension1}) will be a basic object of study in this
paper. Another way of looking at the same ring, which we will often
use, comes from interchanging the order of tensor product and
localization. Namely, let $T'$ denote the image of the
multiplicative system $\left(R'\otimes_RR^\dag\right)\setminus M'$
under the natural map
$R'\otimes_RR^\dag\rightarrow\kappa\left(P'_\ell\right)\otimes_RR^\dag$. Then
the ring (\ref{eq:extension1}) equals the localization
$(T')^{-1}\left(\kappa\left(P'_\ell\right)\otimes_RR^\dag\right)$.
(2) In the special case $R'=R$ in Definition \ref{stable}, we have $$ \kappa\left(P'_\ell\right)\otimes_R\left(R'\otimes_RR^\dag\right)_{M'}= \kappa\left(P_\ell\right)\otimes_RR^\dag. $$ If, moreover, $\frac R{P_\ell}$ is analytically irreducible then the hypothesis that $\kappa\left(P_\ell\right)\otimes_RR^\dag$ is a domain holds automatically; in fact, this hypothesis is equivalent to analytic irreducibility of $\frac R{P_\ell}$ if $R^\dag=\hat R$ or $R^\dag=\tilde R$.
(3) Consider the special case when $R'$ is Henselian and ${\ }^\dag=\hat{\ }$. Excellent Henselian rings are algebraically closed inside their formal completions, so both (1) and (2) of Definition \ref{stable} hold automatically for this $R'$. Thus excellent Henselian local rings are always stable. \end{remark} In this section we study $\ell$-stable rings. We prove that if $R$ is $\ell$-stable then so is any $R'\in\mathcal{T}(R)$ (justifying the name ``stable''). The main result of this section, Theorem \ref{primality1}, says that if $R$ is stable then the implicit ideal $H'_{2\ell+1}$ is prime for each $\ell\in\{0,\dots,r\}$ and each $R'\in\mathcal{T}(R)$. \begin{remark}\label{hypotheses} In the next two sections we will show
that there exist stable rings $R'\in\cal T$ for both $R^\dag=\hat R$
and $R^\dag=R^e$. However, the proof of this is different depending
on whether we are dealing with completion or with an \'etale
extension, and will be carried out separately in two separate
sections: one devoted to henselization, the other to completion. \end{remark} \begin{proposition}\label{largeR1} Fix an integer $\ell$, $0\le\ell\le
r$. Assume that $R'$ is $\ell$-stable and let
$R''\in\mathcal{T}(R')$. Then $R''$ is $\ell$-stable. \end{proposition} \begin{proof} We have to show that (1) and (2) of Definition \ref{stable} for $R'$ imply (1) and (2) of Definition \ref{stable} for $R''$. The ring \begin{equation} \kappa\left(P''_\ell\right)\otimes_R\left(R''\otimes_RR^\dag\right)_{M''} \label{eq:extension} \end{equation} is a localization of $\kappa\left(P''_\ell\right)\otimes_R\left(\kappa\left(P'_\ell\right) \otimes_R\left(R'\otimes_RR^\dag\right)_{M'}\right)$. Hence (1) and (2) of Definition \ref{stable}, applied to $R'$, imply that $\kappa\left(P''_\ell\right)\otimes_R\left(R''\otimes_RR^\dag\right)_{M''}$ is an integral domain, so (1) of Definition \ref{stable} holds for $R''$. Replacing $R'$ by $R''$ clearly does not affect the hypotheses about the non-existence of the extension $L$, so (2) of Definition \ref{stable} also holds for $R''$. \end{proof}
Next, we prove a technical result on which much of the rest of the paper is based. For $\overline\beta\in\frac\Gamma{\Delta_{\ell+1}}$, let \begin{equation}
\mathcal{P}_{\overline\beta+}=\left\{x\in R\ \left|\ \nu(x)\mod\Delta_{\ell+1}>\overline\beta\right.\right\}.\label{eq:pbetamodl+} \end{equation} As usual, $\mathcal{P}'_{\overline\beta+}$ will stand for the analogous notion, but with $R$ replaced by $R'$, etc. \begin{proposition}\label{largeR2} Assume that $R$ itself is $(\ell+1)$-stable and let $R'\in\mathcal{T}(R)$. \begin{enumerate} \item For any $\overline\beta\in\frac{\Delta_\ell}{\Delta_{\ell+1}}$ \begin{equation} \mathcal{P}'_{\overline\beta}{R'}^\dag\cap R^\dag=\mathcal{P}_{\overline\beta} R^\dag.\label{eq:stab} \end{equation} \item For any $\overline\beta\in\frac\Gamma{\Delta_{\ell+1}}$ the natural map
\begin{equation}\label{eq:gammaversion}
\frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\rightarrow \frac{\mathcal{P}'_{\overline\beta}{R'}^\dag}{\mathcal{P}'_{\overline\beta+}{R'}^\dag}
\end{equation}
is injective. \end{enumerate} \end{proposition} \begin{proof} As explained at the end of the previous section, since ${R'}^\dag$ is faithfully flat over the ring $\left(R^\dag\otimes_RR'\right)_{M'}$, we may replace ${R'}^\dag$ by $\left(R^\dag\otimes_RR'\right)_{M'}$ in both 1 and 2 of the Proposition.
\noi\textbf{Proof of 1 of the Proposition:} It is sufficient to prove that \begin{equation} \mathcal{P}'_{\overline\beta}\left(R^\dag\otimes_RR'\right)_{M'}\bigcap R^\dag=\mathcal{P}_{\overline\beta} R^\dag.\label{eq:stab2} \end{equation} One inclusion in (\ref{eq:stab2}) is trivial; we must show that \begin{equation} \mathcal{P}'_{\overline\beta}\left(R^\dag\otimes_RR'\right)_{M'}\bigcap R^\dag\subset\mathcal{P}_{\overline\beta} R^\dag.\label{eq:stab3} \end{equation} \begin{lemma}\label{injectivity} Let $T'$ denote the image of the multiplicative set $\left(R'\otimes_RR^\dag\right)\setminus M'$ under the natural map of $R$-algebras $R'\otimes_RR^\dag\rightarrow\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta} R'_{P'_{\ell+1}}}\otimes_RR^\dag$. Then the map of $R$-algebras \begin{equation} \bar\pi:\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline\beta}R_{P_{\ell+1}}}\otimes_RR^\dag \rightarrow(T')^{-1}\left(\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta} R'_{P'_{\ell+1}}}\otimes_RR^\dag\right) \label{eq:inclusion3} \end{equation} induced by $\pi:R\rightarrow R'$ is injective. \end{lemma} \begin{proof}\textit{(of Lemma \ref{injectivity})} We start with the field extension $$ \kappa(P_{\ell+1})\hookrightarrow\kappa(P'_{\ell+1}) $$ induced by $\pi$. Since $R^\dag$ is flat over $R$, the induced map $\pi_1:\kappa(P_{\ell+1})\otimes_RR^\dag\rightarrow\kappa(P'_{\ell+1})\otimes_R R^\dag$ is also injective. By (1) of Definition \ref{stable}, $\kappa(P'_{\ell+1})\otimes_RR^\dag$ is a domain. In particular, \begin{equation} \kappa\left(P'_{\ell+1}\right)\otimes_RR^\dag= \left(\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}\otimes_R R^\dag\right)_{red}. \label{eq:reduced1} \end{equation} The local ring $\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}$ is artinian because it can be seen as the quotient of $\frac{R'_{P'_{\ell+1}}}{P'_\ell R'_{P'_{\ell+1}}}$ by a valuation ideal corresponding to a rank one valuation. Since the ring is noetherian the valuation of the maximal ideal is positive, and since the group is archimedian, a power of the maximal ideal is contained in the valuation ideal.\par Therefore, its only associated prime is its nilradical, the ideal $\frac{P'_{\ell+1}
R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}$; in particular, the $(0)$ ideal in this ring has no embedded components. Since $R^\dag$ is flat over $R$, $\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}\otimes_RR^\dag$ is flat over $\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}$ by base change. Hence the $(0)$ ideal of $\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}\otimes_RR^\dag$ has no embedded components. In particular, since the multiplicative system $T'$ is disjoint from the nilradical of $\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}\otimes_RR^\dag$, the set $T'$ contains no zero divisors, so localization by $T'$ is injective.\par By the definition of $\mathcal{P}_{\overline\beta}$, the map $\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline\beta}R_{P_{\ell+1}}}\rightarrow \frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}$ is injective, hence so is $$ \frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline\beta}R_{P_{\ell+1}}}\otimes_RR^\dag\rightarrow \frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}\otimes_RR^\dag $$ by the flatness of $R^\dag$ over $R$. Combining this with the injectivity of the localization by $T'$, we obtain that $\bar\pi$ is injective, as desired. Lemma \ref{injectivity} is proved. \end{proof}
Again by the definition of $\mathcal{P}_{\overline\beta}$, the localization map $\frac R{\mathcal{P}_{\overline\beta}}\rightarrow\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline\beta} R_{P_{\ell+1}}}$ is injective, hence so is the map \begin{equation} \frac {R}{\mathcal{P}_{\overline\beta}}\otimes_RR^\dag\rightarrow \frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline\beta}R_{P_{\ell+1}}}\otimes_RR^\dag \label{eq:injective} \end{equation} by the flatness of $R^\dag$ over $R$. Combining this with Lemma \ref{injectivity}, we see that the composition \begin{equation} \frac R{\mathcal{P}_{\overline\beta}}\otimes_RR^\dag\rightarrow (T')^{-1}\left(\frac{R'_{P'_\ell}}{\mathcal{P}'_{\overline\beta}
R'_{P'_\ell}}\otimes_RR^\dag\right) \label{eq:injective1} \end{equation} of (\ref{eq:injective}) with $\bar\pi$ is also injective. Now, (\ref{eq:injective1}) factors through $\left(\frac{R'}{\mathcal{P}'_{\overline\beta}}\otimes_RR^\dag\right)_{M'}$ (here we are guilty of a slight abuse of notation: we denote the natural image of $M'$ in $\frac{R'}{\mathcal{P}'_{\overline\beta}}\otimes_RR^\dag$ also by $M'$). Hence the map \begin{equation} \frac R{\mathcal{P}_{\overline\beta}}\otimes_RR^\dag\rightarrow \left(\frac{R'}{\mathcal{P}'_{\overline\beta}}\otimes_RR^\dag\right)_{M'} \label{eq:injective2} \end{equation} is injective. Since $\frac R{\mathcal{P}_{\overline\beta}}\otimes_RR^\dag\cong\frac{R^\dag}{\mathcal{P}_{\overline\beta}
R^\dag}$ and $\left(\frac{R'}{\mathcal{P}'_{\overline\beta}}\otimes_RR^\dag\right)_{M'}\cong \frac{\left(R'\otimes_RR^\dag\right)_{M'}}{\mathcal{P}'_{\overline\beta} \left(R'\otimes_RR^\dag\right)_{M'}}$, the injectivity of (\ref{eq:injective2}) is the same as (\ref{eq:stab3}). This completes the proof of 1 of the Proposition.
\noi\textbf{Proof of 2 of the Proposition:} We start with the injective homomorphism \begin{equation} \frac{\mathcal{P}_{\overline\beta}}{\mathcal{P}_{\overline\beta+}}\otimes_R\kappa(P_{\ell+1}) \rightarrow\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_R \kappa(P'_{\ell+1})\label{eq:vspaces} \end{equation} of $\kappa(P_{\ell+1})$-vector spaces. Since $R^\dag$ is flat over $R$, tensoring (\ref{eq:vspaces}) produces an injective homomorphism \begin{equation} \frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\otimes_R \kappa(P_{\ell+1})\rightarrow\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}} \otimes_R\kappa(P'_{\ell+1})\otimes_RR^\dag\label{eq:Rdagmodules} \end{equation} of $R^\dag$-modules. Now, the $\kappa(P_{\ell+1})$-vector space $\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_R\kappa(P'_{\ell+1})$ is, in particular, a torsion-free $\kappa(P_{\ell+1})$-module. Since $\kappa(P_{\ell+1})\otimes_RR^\dag$ is a domain by definition of $(\ell+1)$-stable and by the flatness of $R^\dag\otimes_R\kappa(P_{\ell+1})$ over $\kappa(P_{\ell+1})$, the $R^\dag\otimes_R\kappa(P_{\ell+1})$-module $\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_R\kappa(P'_{\ell+1}) \otimes_RR^\dag$ is also torsion-free; in particular, its localization map by any multiplicative system is injective. Let $S'$ denote the image of the multiplicative set $\left(R'\otimes_RR^\dag\right)\setminus M'$ under the natural map of $R$-algebras $R'\otimes_RR^\dag\rightarrow\kappa(P'_{\ell+1})\otimes_RR^\dag$. By the above, the composition \begin{equation} \frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\otimes_R \kappa(P_{\ell+1})\rightarrow(S')^{-1} \left(\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_R \kappa(P'_{\ell+1})\otimes_RR^\dag\right)\label{eq:Rdagmodules1} \end{equation} of (\ref{eq:Rdagmodules}) with the localization by $S'$ is injective.
By the definition of $\mathcal{P}_{\overline\beta}$, the localization map $\frac{\mathcal{P}_{\overline\beta}}{\mathcal{P}_{\overline\beta+}}\rightarrow \frac{\mathcal{P}_{\overline\beta}}{\mathcal{P}_{\overline\beta+}}\otimes_R\kappa(P_{\ell+1})$ is injective, hence so is the map \begin{equation} \frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}= \frac{\mathcal{P}_{\overline\beta}}{\mathcal{P}_{\overline\beta+}}\otimes_RR^\dag\rightarrow \frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\otimes_R \kappa(P_{\ell+1})\label{eq:injectivegamma} \end{equation} by the flatness of $R^\dag$ over $R$. Combining this with the injectivity of (\ref{eq:Rdagmodules1}), we see that the composition \begin{equation} \frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\rightarrow (S')^{-1}\left(\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_R \kappa(P'_{\ell+1})\otimes_RR^\dag\right) \label{eq:injective1gamma} \end{equation} of (\ref{eq:injectivegamma}) with (\ref{eq:Rdagmodules1}) is also injective. Now, (\ref{eq:injective1gamma}) factors through $\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'} \left(R'\otimes_RR^\dag\right)_{M'}$. Hence the map \begin{equation} \frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\rightarrow \frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'} \left(R'\otimes_RR^\dag\right)_{M'}\label{eq:injective2gamma} \end{equation} is injective. Since $\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'}{R'}^\dag\cong \frac{\mathcal{P}'_{\overline\beta}{R'}^\dag}{\mathcal{P}'_{\overline\beta+}{R'}^\dag}$ and by faithful flatness of ${R'}^\dag$ over $\left(R'\otimes_RR^\dag\right)_{M'}$, the injectivity of (\ref{eq:injective2gamma}) implies the injectivity of the map (\ref{eq:gammaversion}) required in 2 of the Proposition.
This completes the proof of the Proposition. \end{proof}
\begin{corollary}\label{stableimplicit} Take an integer
$\ell\in\{0,\dots,r-1\}$ and assume that $R$ is
$(\ell+1)$-stable. Then \begin{equation} H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_\ell}{\cal P}_\beta R^\dag.\label{eq:defin5} \end{equation} \end{corollary} \begin{proof} By Lemma 4 of Appendix 4 of \cite{ZS}, the ideals $\mathcal{P}_{\overline\beta}$ are cofinal among the ideals $\mathcal{P}_\beta$ for $\beta\in \Delta_\ell$. \end{proof}
\begin{corollary}\label{stablecontracts} Assume that $R$ is stable. Take an
element $\beta\in\Gamma$. Then $\mathcal{P}'_\beta{R'}^\dag\cap
R^\dag=\mathcal{P}_\beta$. \end{corollary} \begin{proof} It is sufficient to prove that for each $\ell\in\{0,\dots,r-1\}$ and $\bar\beta\in\frac\Gamma{\Delta_{\ell+1}}$, we have \begin{equation} \mathcal{P}'_{\bar\beta}{R'}^\dag\cap R^\dag=\mathcal{P}_{\bar\beta};\label{eq:scalewisebeta} \end{equation} the Corollary is just the special case of (\ref{eq:scalewisebeta}) when $\ell=r-1$. We prove (\ref{eq:scalewisebeta}) by contradiction. Assume the contrary and take the smallest $\ell$ for which (\ref{eq:scalewisebeta}) fails to be true. Let $\Phi'=\nu(R'\setminus\{0\})$. We will denote by $\frac\Phi{\Delta_{\ell+1}}$ the image of $\Phi$ under the composition of natural maps $\Phi\hookrightarrow\Gamma\rightarrow\frac\Gamma{\Delta_{\ell+1}}$ and similarly for $\frac{\Phi'}{\Delta_{\ell+1}}$. Clearly, if (\ref{eq:scalewisebeta}) fails for a certain $\bar\beta$, it also fails for some $\bar\beta\in\frac{\Phi'}{\Delta_{\ell+1}}$; take the smallest $\bar\beta\in\frac{\Phi'}{\Delta_{\ell+1}}$ with this property. If we had $\bar\beta=\min\left\{\left.\tilde\beta\in\frac{\Phi'}{\Delta_{\ell+1}}\
\right|\ \tilde\beta-\bar\beta\in\Delta_\ell\right\}$, then (\ref{eq:scalewisebeta}) would also fail with $\bar\beta$ replaced by $\bar\beta\mod\Delta_\ell\in\frac\Gamma{\Delta_\ell}$, contradicting the minimality of $\ell$. Thus \begin{equation}\label{eq:notminimum} \bar\beta>\min\left\{\left.\tilde\beta\in\frac{\Phi'}{\Delta_{\ell+1}}\
\right|\ \tilde\beta-\bar\beta\in\Delta_\ell\right\}. \end{equation} Let $\bar\beta-$ denote the immediate predecessor of $\bar\beta$ in $\frac{\Phi'}{\Delta_{\ell+1}}$. By (\ref{eq:notminimum}), we have $\bar\beta-\bar\beta-\in\Delta_\ell$. By the choice of $\bar\beta$, we have $\mathcal{P}_{\bar\beta-}=\mathcal{P}'_{\bar\beta-}{R'}^\dag\cap R^\dag$ but $\mathcal{P}_{\bar\beta}\subsetneqq\mathcal{P}'_{\bar\beta}{R'}^\dag\cap R^\dag$. This contradicts Proposition \ref{largeR2}, applied to $\bar\beta-$. The Corollary is proved. \end{proof}
Now we are ready to state and prove the main Theorem of this section. \begin{theorem}\label{primality1} (1) Fix an integer
$\ell\in\{1,\dots,r+1\}$. Assume that there exists
$R'\in\mathcal{T}(R)$ which is $(\ell+1)$-stable. Then the ideal
$H_{2\ell+1}$ is prime.
(2) Let $i=2\ell+2$. There exists an extension $\nu^\dag_{i0}$ of $\nu_{\ell+1}$ to $\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$, with value group \begin{equation} \Delta_{i-1,0}=\frac{\Delta_\ell}{\Delta_{\ell+1}},\label{eq:groupequal2} \end{equation} whose valuation ideals are described as follows. For an element $\overline\beta\in\frac{\Delta_\ell}{\Delta_{\ell+1}}$, the $\nu^\dag_{i0}$-ideal of $\frac{R^\dag}{H_{i-1}}$ of value $\overline\beta$, denoted by $\mathcal{P}^\dag_{\overline\beta\ell}$, is given by the formula \begin{equation} \mathcal{P}^\dag_{\overline\beta,\ell+1}=\left(\lim\limits_{\overset\longrightarrow{R'}} \frac{\mathcal{P}'_{\overline\beta}{R'}^\dag}{H'_{i-1}}\right)\cap\frac{R^\dag}{H_{i-1}}. \label{eq:valideal1} \end{equation} \end{theorem} \begin{remark} Once the even-numbered implicit prime ideals $H'_{2\ell}$
are defined below, we will show that $\nu^\dag_{i0}$ is the unique
extension of $\nu_{\ell+1}$ to
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$,
centered in the local ring
$\lim\limits_{\overset\longrightarrow{R'}} \frac{R'^\dag_{H'_{2\ell+2}}}{H'_{2\ell+1}R'^\dag_{H'_{2\ell+2}}}$. \end{remark} \begin{proof}\textit{(of Theorem \ref{primality1})} Let $R'$ be a stable ring in $\mathcal{T}(R)$. Once Theorem \ref{primality1} is proved for $R'$, the same results for $R$ will follow easily by intersecting all the ideals of ${R'}^\dag$ in sight with $R^\dag$. Therefore from now on we will replace $R$ by $R'$, that is, we will assume that $R$ itself is stable.
Let $\Phi_\ell$ denote the image of the semigroup $\nu(R\setminus\{0\})$ in $\frac{\Gamma}{\Delta_{\ell+1}}$. As we saw above, $\Phi_\ell$ is well ordered. For an element $\overline\beta\in\Phi_\ell$, let $\overline\beta+$ denote the immediate successor of $\overline\beta$ in $\Phi_\ell$.
Take any element $x\in R^\dag\setminus H_{i-1}$. By Corollary \ref{stableimplicit}, there exists (a unique) $\overline\beta\in\Phi_\ell\cap\frac{\Delta_\ell}{\Delta_{\ell+1}}$ such that \begin{equation} x\in{\cal P}_{\overline\beta} R^\dag\setminus{\cal
P}_{\overline\beta+}R^\dag\label{eq:xinbeta} \end{equation} (where, of course, we allow $\overline\beta=0$). Let $\bar x$ denote the image of $x$ in $\frac{R^\dag}{H_{i-1}}$. We define $$ \nu^\dag_{i0}(\bar x)=\overline\beta. $$ Next, take another element $y\in R^\dag\setminus H_{2\ell+1}$ and let $\gamma\in\Phi_\ell\cap\frac{\Delta_\ell}{\Delta_{\ell+1}}$ be such that \begin{equation} y\in{\cal P}_{\overline\gamma}R^\dag\setminus{\cal P}_{\overline\gamma+}R^\dag.\label{eq:yingamma} \end{equation} Let $(a_1,...,a_n)$ be a set of generators of ${\cal P}_{\overline\beta}$ and $(b_1,...,b_s)$ a set of generators of ${\cal P}_{\overline\gamma}$, with $\nu_{\ell+1}(a_1)=\overline\beta$ and $\nu_{\ell+1}(b_1)=\overline\gamma$. Let $R'$ be a local blowing up along $\nu$ such that $R'$ contains all the fractions $\frac{a_i}{a_1}$ and $\frac{b_j}{b_1}$. By Proposition \ref{largeR1} and Definition \ref{stable} (1), the ideal
$P'_{\ell+1}{R'}^\dag$ is prime. By construction, we have $a_1\ |\ x$ and $b_1\ |\ y$ in ${R'}^\dag$. Write $x=za_1$ and $y=wb_1$ in ${R'}^\dag$. The equality (\ref{eq:stab}), combined with (\ref{eq:xinbeta}) and (\ref{eq:yingamma}), implies that $z,w\mbox{$\in$ \hspace{-.8em}/} P'_{\ell+1}{R'}^\dag$, hence \begin{equation} zw\mbox{$\in$ \hspace{-.8em}/} P'_{\ell+1}{R'}^\dag\label{eq:zwnotin} \end{equation} by the primality of $P'_{\ell+1}{R'}^\dag$. We obtain \begin{equation} xy=a_1b_1zw.\label{eq:xyzw} \end{equation} Since $\nu$ is a valuation on $R'$, we have $\left({\cal P}'_{\overline\beta+\overline\gamma+}:(a_1b_1)R'\right)\subset P'_{\ell +1}$. By faithful flatness of ${R'}^\dag$ over $R'$ we obtain \begin{equation} \left({\cal P}'_{\overline\beta+\overline\gamma+}{R'}^\dag:(a_1b_1){R'}^\dag\right)\subset P'_{\ell+1}{R'}^\dag. \end{equation} Combining this with (\ref{eq:zwnotin}) and (\ref{eq:xyzw}), we obtain \begin{equation} xy\mbox{$\in$ \hspace{-.8em}/}{\cal P}_{\overline\beta+\overline\gamma+}R^\dag,\label{eq:xynotin} \end{equation} in particular, $xy\mbox{$\in$ \hspace{-.8em}/} H_{2\ell+1}$. We started with two arbitrary elements $x,y\in R^\dag\setminus H_{2\ell+1}$ and showed that $xy\mbox{$\in$ \hspace{-.8em}/} H_{2\ell+1}$. This proves (1) of the Theorem.
Furthermore, (\ref{eq:xynotin}) shows that $\nu^\dag_{i0}(\bar x\bar y)=\overline\beta+\overline\gamma$, so $\nu^\dag_{i0}$ induces a valuation of $\kappa(H_{i-1})$ and hence also of $\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$. Equality (\ref{eq:groupequal2}) holds by definition and (\ref{eq:valideal1}) by the assumed stability of $R$. \end{proof}
Next, we define the even-numbered implicit prime ideals $H'_{2\ell}$. The only information we need to use to define the prime ideals $H'_{2\ell}\subset H'_{2\ell+1}$ and to prove that $H'_{2\ell-1}\subset H'_{2\ell}$ are the facts that $H_{2\ell+1}$ is a prime lying over $P_\ell$ and that the ring homomorphism $R'\rightarrow{R'}^\dag$ is regular. \begin{proposition}\label{H2l} There exists a unique minimal prime ideal $H_{2\ell}$ of $P_\ell R^\dag$, contained in $H_{2\ell+1}$. \end{proposition} \begin{proof} Since $H_{2\ell+1}\cap R=P_\ell$, $H_{2\ell+1}$ belongs to the fiber of the map $Spec\ R^\dag\rightarrow Spec\ R$ over $P_\ell$. Since $R$ was assumed to be excellent, $S:={R^\dag}\otimes_R\kappa(P_\ell)$ is a regular ring (note that the excellence assumption is needed only in the case $R^\dag=\hat R$; the ring homomorphism $R\rightarrow R^\dag$ is automatically regular if $R^\dag=\tilde R$ or $R^\dag=R^e$). Hence its localization $\bar S:=S_{H_{2\ell+1}S}\cong\frac{R^\dag_{H_{2\ell+1}}}{P_\ell R^\dag_{H_{2\ell+1}}}$ is a regular {\em local} ring. In particular, $\bar S$ is an integral domain, so $(0)$ is its unique minimal prime ideal. The set of minimal prime ideals of $\bar S$ is in one-to-one correspondence with the set of minimal primes of $P_\ell$, contained in $H_{2\ell+1}$, which shows that such a minimal prime $H_{2\ell}$ is unique, as desired. \end{proof}
We have $P_\ell\subset H_{2\ell}\cap R\subset H_{2\ell+1}\cap R=P_\ell$, so $H_{2\ell}\cap R\subset P_\ell$. \begin{proposition} We have $H_{2\ell-1}\subset H_{2\ell}$. \end{proposition} \begin{proof} Take an element $\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$ and a stable ring $R'\in\cal T$. Then $\mathcal{P}'_\beta\subset P'_\ell$, so \begin{equation} H'_{2\ell-1}\subset\mathcal{P}'_\beta{R'}^\dag\subset P'_\ell{R'}^\dag\subset H'_{2\ell}.\label{eq:inclusion} \end{equation} Intersecting (\ref{eq:inclusion}) back with $R^\dag$ we get the result. \end{proof}
In \S\ref{henselization} we will see that if $R^\dag=\tilde R$ or $R^\dag=R^e$ then $H_{2\ell}=H_{2\ell+1}$ for all $\ell$.
Let the notation be the same as in Theorem \ref{primality1}. \begin{proposition}\label{nu0unique} The valuation $\nu_{i0}^\dag$ is the unique extension of $\nu_\ell$ to a valuation of $\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$, centered in the local ring $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}$. \end{proposition} \begin{proof} As usual, without loss of generality we may assume that $R$ is stable. Take an element $x\in R^\dag\setminus H_{2\ell-1}$. Let $\beta=\nu^\dag_{i0}(\bar x)$ and let $R'$ be the blowing up of the ideal $\mathcal{P}_\beta=(a_1,\dots,a_n)$, as in the proof of Theorem \ref{primality1}. Write \begin{equation} x=za_1\label{eq:xza} \end{equation} in $R'$. We have $z\in{R'}^\dag\setminus P'_\ell{R'}^\dag$, hence \begin{equation} \bar z\in\frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}\setminus\frac{P'_\ell{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}= \frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}\setminus\frac{H'_{2\ell}{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}. \label{eq:znotinPl} \end{equation} If $\nu^*$ is any other extension of $\nu_\ell$ to $\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$, centered in $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}$, then $\nu^*(\bar a_1)=\beta$, $\nu^*(z)=0$ by (\ref{eq:znotinPl}), so $\nu^*(\bar x)=\beta=\nu_{i0}^\dag(\bar x)$. This completes the proof of the uniqueness of $\nu_{i0}^\dag$. \end{proof} \begin{remark}\label{sameresfield} If $R'$ is stable, we have a natural isomorphism of graded algebras $$ \mbox{gr}_{\nu^\dag_{i0}}\frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}\cong \mbox{gr}_{\nu_\ell}\frac{R'_{P'_\ell}}{P'_{\ell-1}R'_{P'_\ell}}\otimes_{R'}\kappa(H'_{2\ell}). $$ In particular, the residue field of $\nu^\dag_{i0}$ is $k_{\nu^\dag_{i0}}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{2\ell})$. \end{remark}
\section{A classification of extensions of $\nu$ to $\hat R$.} \label{Rdag}
The purpose of this section is to give a systematic description of all the possible extensions $\nu^\dag_-$ of $\nu$ to a quotient of $R^\dag$ by a minimal prime as compositions of $2r$ valuations, \begin{equation} \nu^\dag_-=\nu^\dag_1\circ\dots\circ\nu^\dag_{2r},\label{eq:composition} \end{equation} satisfying certain conditions. One is naturally led to consider the more general problem of extending $\nu$ not only to rings of the form $\frac{R^\dag}P$ but also to the ring $\lim\limits_\to\frac{{R'}^\dag}{P'}$, where $P'$ is a tree of prime ideals of ${R'}^\dag$, such that $P'\cap R'=(0)$. We deal in a uniform way with all the three cases $R^\dag=\hat R$, $R^\dag=\tilde R$ and $R^\dag=R^e$, in order to be able to apply the results proved here to all three later in the paper. However, the reader should think of the case $R^\dag=\hat R$ as the main case of interest and the cases $R^\dag=\tilde R$ and $R^\dag=R^e$ as auxiliary and slightly degenerate, since, as we shall see, in these cases the equality $H_{2\ell}=H_{2\ell+1}$ is satisfied for all $\ell$ and the extension $\nu^\dag_-$ will later be shown to be unique.
We will associate to each extension $\nu^\dag_-$ of $\nu$ to $R^\dag$ a chain \begin{equation} \tilde H'_0\subset\tilde H'_1\subset\dots\subset\tilde H'_{2r}=m'{R'}^\dag\label{eq:chaintree''} \end{equation} of prime $\nu^\dag_-$-ideals, corresponding to the decomposition (\ref{eq:composition}) and prove some basic properties of this chain of ideals.
Now for the details. We wish to classify all the pairs $\left(\left\{\tilde H'_0\right\},\nu^\dag _+\right)$, where $\left\{\tilde H'_0\right\}$ is a tree of prime ideals of ${R'}^\dag$, such that $\tilde H'_0\cap R'=(0)$, and $\nu^\dag_+$ is an extension of $\nu$ to the ring $\lim\limits_\to\frac{{R'}^\dag}{\tilde H'_0}$.
Pick and fix one such pair $\left(\left\{\tilde
H'_0\right\},\nu^\dag_+\right)$. We associate to it the following collection of data, which, as we will see, will in turn determine the pair $\left(\left\{\tilde H'_0\right\},\nu^\dag_+\right)$.
First, we associate to $\left(\left\{\tilde
H'_0\right\},\nu^\dag_-\right)$ a chain (\ref{eq:chaintree''}) of $2r$ trees of prime $\nu^\dag_-$-ideals. Let $\Gamma^\dag$ denote the value group of $\nu^\dag_-$. Defining (\ref{eq:chaintree''}) is equivalent to defining a chain \begin{equation} \Gamma^\dag=\Delta^\dag_0\supset\Delta^\dag_1\supset\dots\supset\Delta^\dag_{2r}= \Delta^\dag_{2r+1}=(0)\label{eq:groups} \end{equation} of $2r$ isolated subroups of $\Gamma^\dag$ (the chain (\ref{eq:groups}) will not, in general, be maximal, and $\Delta^\dag_{2\ell +1}$ need not be distinct from $\Delta^\dag_{2\ell}$).
We define the $\Delta^\dag_i$ as follows. For $0\le\ell\le r$, let $\Delta^\dag_{2\ell}$ and $\Delta^\dag_{2\ell +1}$ denote, respectively, the greatest and the smallest isolated subgroups of $\Gamma^\dag$ such that \begin{equation} \Delta^\dag_{2\ell}\cap\Gamma=\Delta^\dag_{2\ell
+1}\cap\Gamma=\Delta_\ell.\label{eq:Delta} \end{equation} \begin{lemma}\label{rank1} We have \begin{equation} rk\ \frac{\Delta^\dag_{2\ell-1}}{\Delta^\dag_{2\ell}}=1\label{eq:rank1} \end{equation} for $1\le \ell\le r$. \end{lemma} \begin{proof} Since by construction $\Delta^\dag_{2\ell}\neq\Delta^\dag_{2\ell-1}$, equality (\ref{eq:rank1}) is equivalent to saying that there is no isolated subgroup $\Delta^\dag$ of $\Gamma^\dag$ which is properly contained in $\Delta^\dag_{2\ell-1}$ and properly contains $\Delta^\dag_{2\ell}$. Suppose such an isolated subgroup $\Delta^\dag$ existed. Then \begin{equation} \Delta_\ell=\Delta^\dag_{2\ell}\cap\Gamma\subsetneqq\Delta^\dag\cap\Gamma \subsetneqq\Delta^\dag_{2\ell-1}\cap\Gamma=\Delta_{\ell-1}, \label{eq:noninclusion} \end{equation} where the first inclusion is strict by the maximality of $\Delta^\dag_{2\ell}$ and the second by the minimality of $\Delta^\dag_{2\ell -1}$. Thus $\Delta^\dag\cap\Gamma$ is an isolated subgroup of $\Gamma$, properly containing $\Delta_\ell$ and properly contained in $\Delta_{l-1}$, which is impossible since $rk\ \frac{\Delta_{\ell-1}}{\Delta_\ell}=1$. This is a contradiction, hence $rk\ \frac{\Delta^\dag_{2\ell -1}}{\Delta^\dag_{2\ell}}=1$, as desired. \end{proof} \begin{definition} Let $0\le i\le 2r$. The $i$-th prime ideal
\textbf{determined} by $\nu^\dag_-$ is the prime $\nu^\dag_-$-ideal
$\tilde H'_i$ of ${R'}^\dag$, corresponding to the isolated subgroup
$\Delta^\dag_i$ (that is, the ideal $\tilde H'_i$ consisting of all
the elements of ${R'}^\dag$ whose values lie outside of
$\Delta^\dag_i$). The chain of trees (\ref{eq:chaintree''}) of prime
ideals of ${R'}^\dag$ formed by the $\tilde H'_i$ is referred to as
the chain of trees \textbf{determined} by $\nu^\dag_-$. \end{definition} The equality (\ref{eq:Delta}) says that \begin{equation} \tilde H'_{2\ell}\cap R'=\tilde H'_{2\ell+1}\cap R'=P'_\ell\label{eq:tildeHcapR} \end{equation} By definitions, for $1\le i\le2r$, $\nu^\dag_i$ is a valuation of the field $k_{\nu^\dag_{i-1}}$. In the sequel, we will find it useful to talk about the restriction of $\nu^\dag_i$ to a smaller field, namely, the field of fractions of the ring $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$; we will denote this restriction by $\nu^\dag_{i0}$. The field of fractions of $\frac{{R'}^\dag_{\tilde H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde
H'_i}}$ is $\kappa(\tilde H'_{i-1})$, hence that of $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$ is $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$, which is a subfield of $k_{\nu^\dag_{i-1}}$. The value group of $\nu^\dag_{i0}$ will be denoted by $\Delta_{i-1,0}$; we have $\Delta_{i-1,0}\subset\frac{\Delta^\dag_{i-1}}{\Delta^\dag_i}$. If $i=2\ell$ is even then $\frac{R'_{P'_l}}{P'_{l-1}R'_{P'_l}}<\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$, so $\lim\limits_{\overset\longrightarrow{R'}}\frac{R'_{P'_l}}{P'_{l-1}R'_{P'_l}}< \lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$. In this case $rk\ \nu^\dag_i=1$ and $\nu^\dag_i$ an extension of the rank 1 valuation $\nu_\ell$ from $\kappa(P_{\ell-1})$ to $k_{\nu^\dag_{i-1}}$; we have \begin{equation} \frac{\Delta_{\ell-1}}{\Delta_\ell}\subset\Delta_{i-1,0}\subset \frac{\Delta^\dag_{i-1}}{\Delta^\dag_i}.\label{eq:deltalindelati-1} \end{equation} \begin{proposition}\label{necessary} Let $i=2\ell$. As usual, for an element $\overline\beta\in\left(\frac{\Delta_\ell}{\Delta_{\ell+1}}\right)_+$, let $\mathcal{P}_{\overline\beta}$ (resp. $\mathcal{P}'_{\overline\beta}$) denote the preimage in $R$ (resp. in $R'$) of the $\nu_{\ell+1}$-ideal of $\frac R{P_\ell}$ (resp. $\frac{R'}{P'_\ell}$) of value greater than or equal to $\overline\beta$. Then \begin{equation} \bigcap\limits_{\overline\beta\in\left(\frac{\Delta_\ell}{\Delta_{\ell+1}}\right)_+} \lim\limits_{\overset\longrightarrow{R'}} \left(\mathcal{P}'_{\overline\beta}{R'}^\dag+\tilde
H'_{i+1}\right){R'}^\dag_{\tilde H'_{i+2}}\cap{R}^\dag\subset\tilde H_{i+1}. \label{eq:restriction} \end{equation} \end{proposition} The inclusion (\ref{eq:restriction}) should be understood as a condition on the tree of ideals. In other words, it is equally valid if we replace $R'$ by any other ring $R''\in\cal T$.
\begin{proof}\textit{(of Proposition \ref{necessary})} Since $rk\frac{\Delta^\dag_{i+1}}{\Delta^\dag_{i+2}}=1$ by Lemma \ref{rank1}, $\frac{\Delta_\ell}{\Delta_{\ell+1}}$ is cofinal in $\frac{\Delta^\dag_{i+1}}{\Delta^\dag_{i+2}}$. Then for any $x\in\bigcap\limits_{\overline\beta\in\left(\frac{\Delta_\ell} {\Delta_{\ell+1}}\right)_+}\lim\limits_{\overset \longrightarrow{R'}}\left(\mathcal{P}'_{\overline\beta}{R'}^\dag+\tilde H'_{i+1}\right){R'}^\dag_{\tilde H'_{i+2}}\cap{R}^\dag$ we have $\nu^\dag_-(x)\mbox{$\in$ \hspace{-.8em}/}\Delta^\dag_i$, hence $x\in\tilde H_{i+1}$, as desired. \end{proof}
From now to the end of \S\ref{extensions}, we will assume that $\cal T$ contains a stable ring $R'$, so that we can apply the results of the previous section, in particular, the primality of the ideals $H'_i$. \begin{proposition}\label{Hintilde} We have \begin{equation} H'_i\subset\tilde H'_i\qquad\text{ for all }i\in\{0,\dots,2r\}.\label{eq:Hintilde} \end{equation} \end{proposition} \begin{proof} For $\beta\in\Gamma^\dag$ and $R'\in\cal T$, let ${\mathcal{P}'_\beta}^\dag$ denote the $\nu^\dag_-$-ideal of ${R'}^\dag$ of value $\beta$. Fix an integer $\ell\in\{0,\dots,r\}$. For each $R'\in\cal T$, each $\beta\in\Delta_\ell$ and $x\in\mathcal{P}'_\beta$ we have $\nu^\dag_-(x)=\nu(x)\ge\beta$, hence \begin{equation} \mathcal{P}'_\beta{R'}^\dag\subset{\mathcal{P}'_\beta}^\dag.\label{eq:PinPdag} \end{equation} Taking the inductive limit over all $R'\in\cal T$ and the intersection over all $\beta\in\Delta_\ell$ in (\ref{eq:PinPdag}), and using the cofinality of $\Delta_\ell$ in $\Delta^\dag_{2\ell+1}$ and the fact that $\bigcap\limits_{\beta\in\Delta^\dag_{2\ell}} \left(\lim\limits_{\overset\longrightarrow{R'}}{\mathcal{P}'_\beta}^\dag\right)= \lim\limits_{\overset\longrightarrow{R'}}\tilde H'_{2\ell+1}$, we obtain the inclusion (\ref{eq:Hintilde}) for $i=2\ell+1$. To prove (\ref{eq:Hintilde}) for $i=2\ell$, note that $\tilde H'_{2\ell}\cap R'=\tilde H'_{2\ell+1}\cap R'=P_\ell$. By the same argument as in Proposition \ref{H2l}, excellence of $R'$ implies that there is a unique minimal prime $H^*_{2\ell}$ of $P'_\ell{R'}^\dag$, contained in $\tilde H'_{2\ell+1}$ and a unique minimal prime $H^{**}_{2\ell}$ of $P'_\ell{R'}^\dag$, contained in $\tilde H'_{2\ell}$. Now, Proposition \ref{H2l} and the facts that $H'_{2\ell+1}\subset\tilde H'_{2\ell+1}$ and $\tilde H'_{2\ell}\subset\tilde H'_{2\ell+1}$ imply that $H'_{2\ell}=H^*_{2\ell}=H^{**}_{2\ell}$, hence $H'_{2\ell}=H^{**}_{2\ell}\subset\tilde H'_{2\ell}$, as desired. \end{proof} \begin{definition} A chain of trees (\ref{eq:chaintree''}) of prime ideals
of ${R'}^\dag$ is said to be \textbf{admissible} if
$H'_i\subset\tilde H'_i$ and (\ref{eq:tildeHcapR}) and
(\ref{eq:restriction}) hold. \end{definition} Equalities (\ref{eq:tildeHcapR}), Proposition \ref{necessary} and Proposition \ref{Hintilde} say that a chain of trees (\ref{eq:chaintree''}) of prime ideals of ${R'}^\dag$, determined by $\nu^\dag_-$, is admissible.
Summarizing all of the above results, and keeping in mind the fact that specifying a composition of $2r$ valuation is equivalent to specifying all of its $2r$ components, we arrive at one of the main theorems of this paper: \begin{theorem}\label{classification} Specifying the valuation $\nu^\dag_-$
is equivalent to specifying the following data. The data will be
described recursively in $i$, that is, the description of
$\nu^\dag_i$ assumes that $\nu^\dag_{i-1}$ is already defined:
(1) An admissible chain of trees (\ref{eq:chaintree''}) of prime ideals of ${R'}^\dag$.
(2) For each $i$, $1\le i\le 2r$, a valuation $\nu^\dag_i$ of $k_{\nu^\dag_{i-1}}$ (where $\nu^\dag_0$ is taken to be the trivial valuation by convention), whose restriction to $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$ is centered at the local ring $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$.
The data $\left\{\nu^\dag_i\right\}_{1\le i\le 2r}$ is subject to the following additional condition: if $i=2\ell$ is even then $rk\ \nu^\dag_i=1$ and $\nu^\dag_i$ is an extension of $\nu_\ell$ to $k_{\nu^\dag_{i-1}}$ (which is naturally an extension of $k_{\nu_{\ell-1}}$). \end{theorem} In particular, note that such extensions $\nu^\dag_-$ always exist, and usually there are plenty of them. The question of uniqueness of $\nu^\dag_-$ and the related question of uniqueness of $\nu^\dag_i$, especially in the case when $i$ is even, will be addressed in the next section.
\section{Uniqueness properties of $\nu^\dag_-$.} \label{extensions}
In this section we address the question of uniqueness of the extension $\nu^\dag_-$. One result in this direction, which will be very useful here, was already proved in \S\ref{technical}: Proposition \ref{nu0unique}. We give some necessary and some sufficient conditions both for the uniqueness of $\nu^\dag_-$ once the chain (\ref{eq:chaintree''}) of prime ideals determined by $\nu^\dag_-$ has been fixed, and also for the unconditional uniqueness of $\nu^\dag_-$. In \S\ref{henselization} we will use one of these uniqueness criteria to prove uniqueness of $\nu^\dag_-$ in the cases $R^\dag=\tilde R$ and $R^\dag=R^e$. At the end of this section we generalize and give a new point of view of an old result of W. Heinzer and J. Sally (Proposition \ref{HeSal}), which provides a sufficient condition for the uniqueness of $\nu^\dag_-$; see also \cite{Te}, Remarks 5.22.
For a ring $R'\in\cal T$ let $K'$ denote the field of fractions of $R'$. For some results in this section we will need to impose an additional condition on the tree $\cal T$: we will assume that there exists $R_0\in\cal T$ such that for all $R'\in{\cal T}(R_0)$ the field $K'$ is algebraic over $K_0$. This assumption is needed in order to be able to control the height of all the ideals in sight. Without loss of generality, we may take $R_0=R$. \begin{proposition}\label{htstable} Assume that for all $R'\in\cal T$ the
field $K'$ is algebraic over $K$. Consider a ring homomorphism
$R'\rightarrow R''$ in $\cal T$. Take an $\ell\in\{0,\dots,r\}$. We
have \begin{equation} {\operatorname{ht}}\ H''_{2\ell}\le {\operatorname{ht}}\ H'_{2\ell}.\label{eq:odddecreases} \end{equation} If equality holds in (\ref{eq:odddecreases}) then \begin{equation} {\operatorname{ht}}\ H''_{2\ell+1}\ge{\operatorname{ht}}\ H'_{2\ell+1}.\label{eq:odddecreases1} \end{equation} \end{proposition} \begin{proof} We start by recalling a well known Lemma (for a proof see \cite{ZS}, Appendix 1, Propositions 2 and 3, p. 326): \begin{lemma}\label{idealheight} Let $R\hookrightarrow R'$ be an
extension of integral domains, essentially of finite type. Let $K$
and $K'$ be the respective fields of fractions of $R$ and
$R'$. Consider prime ideals $P\subset R$ and $P'\subset R'$ such
that $P=P'\cap R$. Then \begin{equation} {\operatorname{ht}}\ P'+tr.deg.(\kappa(P')/\kappa(P))\le\ {\operatorname{ht}}\ P+tr.deg.(K'/K).\label{eq:heightdrops} \end{equation} Moreover, equality holds in (\ref{eq:heightdrops}) whenever $R$ is universally catenarian. \end{lemma} Apply the Lemma to the rings $R'$ and $R''$ and the prime ideals $P'_\ell\subset R'$ and $P''_\ell\subset R''$. In the case at hand we have $tr.deg.(K''/K')=0$ by assumption. Hence \begin{equation} {\operatorname{ht}}\ P''_\ell\le\ {\operatorname{ht}}\ P'_\ell.\label{eq:htP} \end{equation} Since $H'_{2\ell}$ is a minimal prime of $P'_\ell{R'}^\dag$ and ${R'}^\dag$ is faithfully flat over $R'$, we have $ht\ P'_\ell=ht\ H'_{2\ell}$. Similarly, ${\operatorname{ht}}\ P''_\ell={\operatorname{ht}}\ H''_{2\ell}$, and (\ref{eq:odddecreases}) follows. Furthermore, equality in (\ref{eq:odddecreases}) is equivalent to equality in (\ref{eq:htP}).
To prove (\ref{eq:odddecreases1}), let $\bar R=(R''\otimes_{R'}{R'}^\dag)_{M''}$, where $M''=(m''\otimes1+1\otimes m'{R'}^\dag)$ and let $\bar m$ denote the maximal ideal of $\bar R$. We have the natural maps ${R'}^\dag\overset\iota\rightarrow\bar R\overset\sigma\rightarrow{R''}^\dag$. The homomorphism $\sigma$ is nothing but the formal completion of the local ring $\bar R$; in particular, it is faithfully flat. Let \begin{equation} \bar H=H''_{2\ell+1}\cap\bar R,\label{eq:barH} \end{equation} $\bar H_0=H''_0\cap\bar R$. Since $H''_0$ is a minimal prime of ${R''}^\dag$ and $\sigma$ is faithfully flat, $\bar H_0$ is a minimal prime of $\bar R$.
Assume that equality holds in (\ref{eq:odddecreases}) (and hence also in (\ref{eq:htP})). Since equality holds in (\ref{eq:htP}), by Lemma \ref{idealheight} (applied to the ring extension $R'\rightarrow R''$) the field $\kappa(P'')$ is algebraic over $\kappa(P')$.
Apply Lemma \ref{idealheight} to the ring extension $\frac{{R'}^\dag}{H'_0}\hookrightarrow\frac{\bar R}{\bar H_0}$ and the prime ideals $\frac{H'_{2\ell+1}}{H'_0}$ and $\frac{\bar H}{\bar
H_0}$. Since $K''$ is algebraic over $K'$, $\kappa(\bar H_0)$ is algebraic over $\kappa(H'_0)$. Since $\kappa(P'')$ is algebraic over $\kappa(P')$, $\kappa(\bar H)$ is algebraic over $\kappa(H'_{2\ell+1})$. Finally, $\hat R'$ is universally catenarian because it is a complete local ring. Now in the case ${\ }^\dag=\hat{\ }$ Lemma \ref{idealheight} says that ${\operatorname{ht}}\ \frac{H'_{2\ell+1}}{H'_0}={\operatorname{ht}}\ \frac{\bar H}{\bar H_0}$. Since both $\hat R'$ and $\bar R$ are catenarian, this implies that \begin{equation} {\operatorname{ht}}\ H'_{2\ell+1}={\operatorname{ht}}\ \bar H.\label{eq:htequal} \end{equation} In the case where ${\ }^\dag$ stands for henselization or a finite \'etale extension, (\ref{eq:htequal}) is an immediate consequence of (\ref{eq:barH}). Thus (\ref{eq:htequal}) is true in all the cases. Since $\sigma$ is faithfully flat and in view of (\ref{eq:barH}), ${\operatorname{ht}}\bar H\le {\operatorname{ht}}\ H''_{2\ell+1}$. Combined with (\ref{eq:htequal}), this completes the proof. \end{proof} \begin{corollary}\label{htstable1} For each $i$, $0\le i\le 2r$, the
quantity ${\operatorname{ht}}\ H'_i$ stabilizes for $R'$ sufficiently far out in $\cal T$. \end{corollary} The next Proposition is an immediate consequence of Theorem \ref{classification}. \begin{proposition}\label{uniqueness2} Suppose given an admissible chain of
trees (\ref{eq:chaintree''}) of prime ideals of ${R'}^\dag$. For
each $\ell\in\{0,\dots,r-1\}$, consider the set of all $R'\in\cal T$
such that \begin{equation} {\operatorname{ht}}\ \tilde H'_{2\ell+1}- {\operatorname{ht}}\ \tilde H'_{2_\ell} \le 1\qquad\text{
for all even }i\label{eq:odd=even3} \end{equation} and, in case of equality, the 1-dimensional local ring $\frac{{R'}^\dag_{\tilde H'_{2\ell+1}}}{\tilde
H'_{2\ell}{R'}^\dag_{\tilde H'_{2\ell+1}}}$ is unibranch (that is, analytically irreducible). Assume that for each $\ell$ the set of such $R'$ is cofinal in $\cal T$. Let $\nu^\dag_{2\ell+1,0}$ denote the unique valuation centered at $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_{2\ell+1}}}{\tilde H'_{2\ell}{R'}^\dag_{\tilde H'_{2\ell+1}}}$.
Assume that for each even $i=2\ell$, $\nu_\ell$ admits a unique extension $\nu^\dag_{i0}$ to a valuation of $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$, centered in $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$. Then specifying the valuation $\nu^\dag_-$ is equivalent to specifying for each odd $i$, $2<i<2r$, an extension $\nu_i^\dag$ of the valuation $\nu^\dag_{i0}$ of $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$ to its field extension $k_{\nu^\dag_{i-1}}$ (in particular, such extensions $\nu^\dag_-$ always exist). If for each odd $i$, $2<i<2r$, the field extension $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})\rightarrow k_{\nu^\dag_{i-1}}$ is algebraic and the extension $\nu_i^\dag$ of $\nu^\dag_{i0}$ to $k_{\nu^\dag_{i-1}}$ is unique then there is a unique extension $\nu^\dag_-$ of $\nu$ such that the $\tilde H'_i$ are the prime ideals, determined by $\nu^\dag_-$.
Conversely, assume that $K'$ is algebraic over $K$ and that there exists a unique extension $\nu^\dag_-$ of $\nu$ such that the $\tilde H'_i$ are the prime $\nu^\dag_-$-ideals, determined by $\nu^\dag_-$. Then for each $\ell\in\{0,\dots,r-1\}$ and for all $R'$ sufficiently far out in $\cal T$ the inequality (\ref{eq:odd=even3}) holds. For each even $i=2\ell$, $\nu_\ell$ admits a unique extension $\nu^\dag_{i0}$ to a valuation of $\lim\limits_{\overset\longrightarrow{R'}}\kappa\left(\tilde
H'_{i-1}\right)$, centered in $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$; we have $rk\ \nu^\dag_{i0}=1$. For each odd $i$, the ring $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$ is a valuation ring of a (not necessarily discrete) rank 1 valuation. For each odd $i$, $1\le i<2r$, the field extension $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})\rightarrow k_{\nu^\dag_{i-1}}$ is algebraic and the extension $\nu_i^\dag$ of $\nu^\dag_{i0}$ to $k_{\nu^\dag_{i-1}}$ is unique. \end{proposition} \begin{remark} We do not know of a simple criterion to decide when, given
an algebraic field extension $K\hookrightarrow L$ and a valuation
$\nu$ of $K$, is the extension of $\nu$ to $L$ unique. See
\cite{HOS}, \cite{V2} for more information about this question and an
algorithm for arriving at the answer using MacLane's key
polynomials. \end{remark} Next we describe three classes of extensions of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}{R'}^\dag$, which are of particular interest for applications, and which we call \textbf{minimal}, \textbf{evenly minimal} and \textbf{tight} extensions. \begin{definition} Let $\nu^\dag_-$ be an extension of $\nu$ to
$\lim\limits_{\overset\longrightarrow{R'}}{R'}^\dag$ and let the
notation be as above. We say that $\nu^\dag_-$ is \textbf{evenly
minimal} if whenever $i=2\ell$ is even, the following two conditions hold:
(1) \begin{equation} \Delta_{i-1,0}=\frac{\Delta_{\ell-1}}{\Delta_\ell}.\label{eq:groupequal} \end{equation}
(2) For an element $\overline\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$, the $\nu^\dag_{i0}$-ideal of $\frac{R^\dag_{\tilde H_i}}{\tilde H_{i-1}R^\dag_{\tilde H_i}}$ of value $\overline\beta$, denoted by $\mathcal{P}^\dag_{\overline\beta ,\ell}$, is given by the formula \begin{equation} \mathcal{P}^\dag_{\overline\beta,\ell}=\left(\lim\limits_{\overset\longrightarrow{R'}} \frac{\mathcal{P}'_{\overline\beta}{R'}^\dag_{\tilde H'_i}}{\tilde
H'_{i-1}{R'}^\dag_{\tilde H'_i}}\right)\cap\frac{R^\dag_{\tilde
H_i}}{\tilde H_{i-1}R^\dag_{\tilde H_i}}.\label{eq:valideal} \end{equation} We say that $\nu^\dag_-$ is \textbf{minimal} if $\tilde H'_i=H'_i$ for each $R'$ and each $i\in\{0,\dots,2r+1\}$. We say that $\nu^\dag_-$ is \textbf{tight} if it is evenly minimal and \begin{equation} \tilde H'_i=\tilde H'_{i+1}\qquad\text{ for all even }i.\label{eq:odd=even2} \end{equation} \end{definition} \begin{remark} (1) The valuation $\nu^\dag_{i0}$ is uniquely determined by
conditions (\ref{eq:groupequal}) and (\ref{eq:valideal}). Recall
also that if $i=2\ell$ is even and we have: \begin{eqnarray} \tilde H'_i&=&H'_i\qquad\text{ and}\label{eq:tilde=H1}\\ \tilde H'_{i-1}&=&H'_{i-1}\label{eq:tilde=H2} \end{eqnarray} then $\nu^\dag_{i0}$ is uniquely determined by $\nu_\ell$ by Proposition \ref{nu0unique}. In particular, if $\nu^\dag_-$ is minimal (that is, if (\ref{eq:tilde=H1})--(\ref{eq:tilde=H2}) hold for all $i$) then $\nu^\dag_-$ is evenly minimal.
(2) The definition of evenly minimal extensions can be rephrased as follows in terms of the associated graded algebras of $\nu_\ell$ and $\nu^\dag_{i0}$. First, consider a homomorphism in $\mathcal T$ and the following diagram, composed of natural homomorphisms: \begin{equation}\label{eq:CDevminimal} \xymatrix{{\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'}{R'}^\dag_{\tilde H'_i}}\ar[r]^-{\lambda'}&{\frac{\mathcal{P}'_{\overline\beta}{R'}^\dag_{\tilde H'_i}}{\mathcal{P}'_{\overline\beta+}{R'}^\dag_{\tilde H'_i}}} \ar[r]^-{\phi'}& {\frac{{\mathcal{P}'_{\overline\beta}}^\dag}{{\mathcal{P}'}_{\overline\beta+}^\dag}}\\ {\ }&{\ }&{\frac{\mathcal{P}_{\overline\beta}^\dag}{\mathcal{P}_{\overline\beta+}^\dag}}\ar[u]_-{\psi'}} \end{equation} It follows from Nakayama's Lemma that equality (\ref{eq:valideal}) is equivalent to saying that \begin{equation}\label{eq:equalgraded} \frac{\mathcal{P}_{\overline\beta}^\dag}{\mathcal{P}_{\overline\beta+}^\dag}=\lim\limits_{\overset\longrightarrow{R'}}{\psi'}^{-1}\left((\phi'\circ\lambda')\left(\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'}\kappa\left(\tilde H'_i\right)\right)\right). \end{equation} Taking the direct sum in (\ref{eq:equalgraded}) over all $\overline\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$ and passing to the limit on the both sides, we see that the extension $\nu^\dag_-$ is evenly minimal if and only if we have the following equality of graded algebras: $$ \mbox{gr}_{\nu_\ell}\left(\lim\limits_{\overset\longrightarrow{R'}}\frac{R'}{P'_\ell}\right)\otimes_{R'}\kappa\left(\tilde H'_i\right)=\mbox{gr}_{\nu^\dag_{i0}}\left(\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde H'_i}}{\tilde
H'_{i-1}{R'}^\dag_{\tilde H'_i}}\right). $$ \end{remark} \begin{examples} The extension $\hat\nu$ of Example \ref{Example31} (page \pageref{Example31}) is minimal, but not tight. The valuation $\nu$ admits a unique tight extension $\hat\nu_2\circ\hat\nu_3$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{H'_1}$; the valuation $\hat\nu$ is the composition of the discrete rank 1 valuation $\hat\nu_1$, centered in $\lim\limits_{\overset\longrightarrow{R'}}\hat R'_{H'_1}$ with $\hat\nu_2\circ\hat\nu_3$.
The extension $\hat\nu^{(1)}$ of Example \ref{Example32} (page \pageref{Example32}) is minimal. The extension $\hat\nu^{(2)}$ is evenly minimal but not minimal. Neither $\hat\nu^{(1)}$ nor $\hat\nu^{(2)}$ is tight. The valuation $\nu$ admits a unique tight extension $\hat\nu_2\circ\hat\nu_3$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{R'}{\tilde H'_1}$, where $\tilde H'_1=\left(y-\sum\limits_{j=1}^\infty c_jx^j\right)$; the valuation $\hat\nu^{(2)}$ is the composition of the discrete rank 1 valuation $\hat\nu_1$, centered in $\lim\limits_{\overset\longrightarrow{R'}}\hat R'_{\tilde H'_1}$ with $\hat\nu_2\circ\hat\nu_3$. \end{examples} \begin{remark} As of this moment, we do not know of any examples of
extensions $\hat\nu_-$ which are not evenly minimal. Thus, formally,
the question of whether every extension $\hat\nu_-$ is evenly minimal
is open, though we strongly suspect that counterexamples do
exist. \end{remark} \begin{proposition}\label{resmin} Let $i=2\ell$ be even and let
$\nu^\dag_{i0}$ be the extension of $\nu_\ell$ to
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$,
centered at the local ring
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$, defined by
(\ref{eq:valideal}). Then \begin{equation} k_{\nu^\dag_{i0}}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_i).\label{eq:resfield} \end{equation} \end{proposition} \begin{proof} Take two elements $x,y\in\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$, such that $\nu_{i0}^\dag(x)=\nu_{i0}^\dag(y)$. We must show that the image of $\frac xy$ in $k_{\nu^\dag_{i0}}$ belongs to $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_i)$. Without loss of generality, we may assume that $x,y\in\frac{R^\dag_{\tilde H_i}}{\tilde H_{i-1}{R}^\dag_{\tilde
H_i}}$. Let $\beta=\nu_{i0}^\dag(x)=\nu_{i0}^\dag(y)$. Choose $R'\in\cal T$ sufficiently far out in the direct system so that $x,y\in\frac{\mathcal{P}'_\beta{R'}^\dag_{\tilde H'_i}}{\tilde
H'_{i-1}{R'}^\dag_{\tilde H'_i}}$. Let $R'\rightarrow R''$ be the blowing up of the ideal $\mathcal{P}'_\beta R'$. Then in $\frac{\mathcal{P}''_\beta{R''}^\dag_{\tilde H''_i}}{\tilde
H''_{i-1}{R''}^\dag_{\tilde H''_i}}$ we can write \begin{eqnarray} x&=&az\qquad\text{ and }\label{eq:xaz}\\ y&=&aw, \end{eqnarray} where $\nu_{i0}^\dag(a)=\beta$ and $\nu_{i0}^\dag(z)=\nu_{i0}^\dag(w)=0$. Let $\bar z$ be the image of $z$ in $\kappa(\tilde H''_i)$ and similarly for $\bar w$. Then the image of $\frac xy$ in $k_{\nu^\dag_{i0}}$ equals $\frac{\bar z}{\bar
w}\in\kappa(\tilde H''_i)$, and the result is proved.
\end{proof} \begin{remark}\label{minimalexist} Theorem \ref{classification} and the existence of the
extension $\nu^\dag_{2\ell,0}$ of $\nu_\ell$ in the case when
$\tilde H'_{2\ell}=H'_{2\ell}$ and $\tilde
H'_{2\ell-1}=H'_{2\ell-1}$ guaranteed by Theorem \ref{primality1}
(2) allow us to give a fairly explicit description of the totality
of minimal extensions as compositions of $2r$ valuations and, in
particular, to show that they always exist. Indeed, minimal
extensions $\nu^\dag_-$ can be constructed at will, recursively in
$i$, as follows. Assume that the valuations
$\nu^\dag_1,\dots,\nu^\dag_{i-1}$ are already constructed. If $i$ is
odd, let $\nu^\dag_i$ be an arbitrary valuation of the residue field
$k_{\nu^\dag_{i-1}}$ of the valuation ring $R_{\nu^\dag_{i-1}}$. If
$i=2\ell$ is even, let $\nu^\dag_{i0}$ be the extension of $\nu_\ell$ to
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$,
centered at the local ring
$\lim\limits_{\overset\longrightarrow{R'}} \frac{{R'}^\dag_{H'_i}}{H'_{i-1}{R'}^\dag_{H'_i}}$, whose existence and uniqueness are guaranteed by Theorem \ref{primality1} (2) and Proposition \ref{nu0unique}, respectively. Let $\nu^\dag_i$ be an arbitrary extension of $\nu^\dag_{i0}$ to the field $k_{\nu^\dag_{i-1}}$. It is clear that all the minimal extensions $\nu^\dag_-$ of $\nu$ are obtained in this way.
In the next section we will use this remark to show that if $R^\dag=\tilde R$ or $R^\dag=R^e$ then $\nu$ admits a unique extension to $\frac{R^\dag}{H_0}$, which is necessarily minimal. \end{remark} We end this section by giving some sufficient conditions for the uniqueness of $\nu^\dag_-$. \begin{proposition}\label{uniqueness1} Suppose given an admissible chain of
trees (\ref{eq:chaintree''}) of prime ideals of ${R'}^\dag$. For
each $\ell\in\{0,\dots,r-1\}$, consider the set of all $R'\in\cal T$ such that \begin{equation} ht\ \tilde H'_{2\ell+1}- ht\ \tilde H'_{2\ell} \le 1\qquad\text{ for
all even }i\label{eq:odd=even1} \end{equation} and, in case of equality, the 1-dimensional local ring $\frac{{R'}^\dag_{\tilde
H'_{2\ell+1}}}{\tilde H'_{2\ell}{R'}^\dag_{\tilde H'_{2\ell+1}}}$ is unibranch (that is, analytically irreducible). Assume that for each $\ell$ the set of such $R'$ is cofinal in $\cal T$.
Let $\nu^\dag_-$ be an extension of $\nu$ such that the $\tilde H'_i$ are prime $\nu^\dag_-$-ideals. Assume that $\nu^\dag_-$ is evenly minimal. Then there is at most one such extension $\nu^\dag_-$ and exactly one such $\nu^\dag_-$ if \begin{equation} \tilde H'_i=H'_i\quad\text{ for all }i.\label{eq:tildeH=H} \end{equation} (in the latter case $\nu^\dag_-$ is minimal by definition). \end{proposition} \begin{proof} By Theorem \ref{primality1} (2) and Proposition \ref{nu0unique}, if (\ref{eq:tildeH=H}) holds then $\nu^\dag_-$ is minimal and for each even $i$ the extension $\nu^\dag_{i0}$ exists and is unique. Therefore we may assume that in all the cases $\nu^\dag_-$ is evenly minimal and that $\nu^\dag_{i0}$ exists whenever (\ref{eq:tildeH=H}) holds.
The valuation $\nu^\dag_-$, if it exists, is a composition of $2r$ valuations: $\nu^\dag_-=\nu^\dag_1\circ\nu^\dag_2\circ\dots\circ\nu^\dag_{2r}$, subject to the conditions of Theorem \ref{classification}. We prove the uniqueness of $\nu^\dag_-$ by induction on $r$. Assume the result is true for $r-1$. This means that there is at most one evenly minimal extension $\nu^\dag_3\circ\nu^\dag_4\circ\dots\circ\nu^\dag_{2r}$ of $\nu_2\circ\nu_3\circ\dots\circ\nu_r$ to $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_2)$, and exactly one in the case when (\ref{eq:tildeH=H}) holds. To complete the proof of uniqueness of $\nu^\dag_-$, it is sufficient to show that both $\nu_1^\dag$ and $\nu_2^\dag$ are unique and that the residue field of $\nu_2^\dag$ equals $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_2)$.
We start with the uniqueness of $\nu_1^\dag$. If (\ref{eq:odd=even2}) holds then $\nu_1^\dag$ is the trivial valuation. Suppose, on the other hand, that equality holds in (\ref{eq:odd=even1}). Then the restriction of $\nu_1^\dag$ to each $R'\in\cal T$ such that the local ring $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'_{\tilde
H'_1}}{\tilde H'_0\hat R'_{\tilde H'_1}}$ is one-dimensional and unibranch is the unique divisorial valuation centered in that ring (in particular, its residue field is $\kappa(\tilde H'_1)$). By the assumed cofinality of such $R'$, the valuation $\nu^\dag_1$ is unique and its residue field equals $\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_1)$. Thus, regardless of whether or not the inequality in (\ref{eq:odd=even1}) is strict, $\nu^\dag_1$ is unique and we have the equality of residue fields \begin{equation} k_{\nu_1^\dag}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_1)\label{eq:eqres} \end{equation} This equality implies that $\nu_2^\dag=\nu^\dag_{20}$. Now, the valuation $\nu^\dag_2=\nu^\dag_{20}$ is uniquely determined by the conditions (\ref{eq:groupequal}) and (\ref{eq:valideal}), and its residue field is \begin{equation} k_{\nu^\dag_2}=k_{\nu^\dag_{20}}=\lim\limits_{\overset\longrightarrow{R'}} \kappa(\tilde H'_2).\label{eq:resfield1} \end{equation} by Proposition \ref{resmin}. Furthermore, by Theorem \ref{primality1} exactly one such $\nu_2^\dag$ exists whenever (\ref{eq:tildeH=H}) holds. This proves that there is at most one possibility for $\nu^\dag_-$: the composition of $\nu_1^\dag\circ\nu^\dag_2$ with $\nu^\dag_3\circ\nu^\dag_4\circ\dots\circ\nu^\dag_{2r}$, and exactly one if (\ref{eq:tildeH=H}) holds. \end{proof} \begin{proposition}\label{tight=scalewise} The extension $\nu^\dag_-$ is tight if and only if for each $R'$ in our direct system the natural graded algebra extension $\mbox{gr}_\nu R'\rightarrow\mbox{gr}_{\nu^\dag_-}{R'}^\dag$ is scalewise birational. \end{proposition} \begin{remark}\label{rephrasing} Proposition \ref{tight=scalewise} allows us
to rephrase Conjecture \ref{teissier} as follows: the valuation $\nu$
admits at least one tight extension $\nu^\dag_-$. \end{remark} \begin{proof}\textit{(of Proposition \ref{tight=scalewise})} ``If'' Assume that for each $R'$ in our direct system the natural graded algebra extension $\mbox{gr}_\nu R'\rightarrow\mbox{gr}_{\nu^\dag_-}{R'}^\dag$ is scalewise birational. Then \begin{equation}\label{eq:gamma=dag} \Gamma^\dag=\Gamma. \end{equation} Together with (\ref{eq:Delta}) this implies that for each $l\in\{1,\dots,r+1\}$ we have $\Delta^\dag_{2\ell-2}=\Delta^\dag_{2\ell-1}=\Delta_{\ell-1}$ under the identification (\ref{eq:gamma=dag}). Then $\frac{\Delta^\dag_{2\ell-1}}{\Delta^\dag_{2\ell}}=(0)$, so for all odd $i$ the valuation $\nu^\dag_i$ is trivial. This proves the equality (\ref{eq:odd=even2}) in the definition of tight. It remains to show that $\nu^\dag_-$ is evenly minimal. We will prove the even minimality in the form of equality (\ref{eq:equalgraded}) for each $\bar\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$. The right hand side of (\ref{eq:equalgraded}) is trivially contained in the left hand side; we must prove the opposite inclusion. To do that, take a non-zero element $x\in\frac{\mathcal{P}_{\overline\beta}^\dag}{\mathcal{P}_{\overline\beta+}^\dag}$. By scalewise birationaliy, there exist non-zero elements $\bar y,\bar z\in\mbox{gr}_\nu R$, with $ord\ \bar y,ord\ \bar z\in\Delta_\ell$, such that $x\bar y=\bar z$. Let $y$ be a representative of $\bar y$ in $R$, and similarly for $z$. Let $R\rightarrow R'$ be the local blowing up with respect to $\nu$ along the ideal $(y,z)$. Then, in the notation of (\ref{eq:equalgraded}), we have \begin{equation}\label{eq:equalgraded1} x={\psi'}^{-1}\left(\left(\phi'\circ\lambda'\right)\left(\frac{\bar z}{\bar y}\otimes_{R'}1\right)\right)\in{\psi'}^{-1}\left((\phi'\circ\lambda')\left(\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'}\kappa\left(\tilde H'_i\right)\right)\right). \end{equation} This proves (\ref{eq:equalgraded}). ``If'' is proved.
``Only if''. Assume that $\nu^\dag_-$ is tight (that is, it is evenly minimal and (\ref{eq:odd=even2}) holds) and take $R'\in\cal T$. Then the valuation $\nu_{2\ell+1}$ is trivial for all $\ell$, so $\nu^\dag_-=\nu^\dag_2\circ\nu^\dag_4\circ\dots\circ\nu^\dag_{2r}$. We must show that the graded algebra extension $\mbox{gr}_\nu R'\rightarrow\mbox{gr}_{\nu^\dag_-}{R'}^\dag$ is scalewise birational. Again, we use induction on $r$. Take an element $x\in{R'}^\dag$. If $\nu^\dag_-(x)\in\Delta_1$ then $\mbox{in}_{\nu^\dag_-}x\in\mbox{gr}_{\nu^\dag_4\circ\dots\circ\nu^\dag_{2r}} \frac{{R'}^\dag}{\tilde H'_2}$, hence by the induction assumption there exists $y\in R'$ with $\nu^\dag_-(x)\in\Delta_1$ and $\mbox{in}_{\nu^\dag_-}(xy)\in\mbox{gr}_\nu\frac{R'}{P_1}$. In this case, there is nothing more to prove. Thus we may assume that $\nu^\dag_-(x)\mbox{$\in$ \hspace{-.8em}/}\Delta_1$. It remains to show that there exists $y\in R'$ such that $\mbox{in}_{\nu^\dag_-}(xy)\in\mbox{gr}_\nu R'$. Since the natural map sending each element of the ring to its image in the graded algebra behaves well with respect to multiplication and division, local blowings up induce birational transformations of graded algebras, and it is enough to find a local blowing up $R''\in{\cal T}(R')$ and $y\in R''$ such that $\mbox{in}_{\nu^\dag_-}(xy)\in\mbox{gr}_\nu R''$.
Now, Proposition \ref{resmin} shows that there exists a local blowing up $R'\rightarrow R''$ such that $x=az$ (\ref{eq:xaz}), with $z\in R''$ and $\nu^\dag_2(a)=\nu^\dag_{2,0}(a)=0$. The last equality means that $\nu^\dag_-(a)\in\Delta_1$, and the result follows from the induction assumption, applied to $a$. \end{proof}
The argument above also shows the following. Let ${\Phi'}^\dag=\nu^\dag_-\left({R'}^\dag\setminus\{0\}\right)$, take an element $\beta\in{\Phi'}^\dag$ and let ${\cal P'}^\dag_\beta$ denote the $\nu^\dag_-$-ideal of ${R'}^\dag$ of value $\beta$. \begin{corollary}\label{blup1} Take an element $x\in{\cal P'}^\dag_\beta$. There exists a local blowing up $R'\rightarrow R''$ such that $\beta\in\nu(R'')\setminus\{0\}$ and $x\in{\cal P}''_\beta{R''}^\dag$. \end{corollary} The next Proposition gives a sufficient condition for the uniqueness of $\nu^\dag_-$ (this result is due to Heinzer and Sally \cite{HeSa}). \begin{proposition}\label{HeSal} Assume that $K'$ is algebraic over $K$ for
all $R'\in\cal T$ and that the following conditions hold:
(1) $ht\ H'_1\le1$
(2) $ht\ H'_1+rat.rk\ \nu=\dim\ R'$, where $R'$ is taken to be sufficiently far out in the direct system.
Let $\nu^\dag_-$ be an extension of $\nu$ to a ring of the form $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{\tilde
H'_0}$. Then either \begin{eqnarray} \tilde H'_0&=&H'_0\qquad\text{ or }\label{eq:tildeH=H0}\\ \tilde H'_0&=&H'_1.\label{eq:tildeH=H1} \end{eqnarray} The valuation $\nu$ admits a unique extension to $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{H'_0}$ and a unique extension to $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{H'_1}$. The first extension is minimal and the second is tight. \end{proposition} \begin{proof} For $1\le\ell\le r$, let $r_\ell$ denote the rational rank of $\nu_\ell$. Let $\nu^\dag_-$ be an extension of $\nu$ to a ring of the form $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{\tilde
H'_0}$, where $\tilde H'_0$ is a tree of prime ideals of ${R'}^\dag$ such that $\tilde H'_0\cap R'=(0)$.
By Corollary \ref{htstable1} $ht\ H'_i$ stabilizes for $1\le i\le 2r$ and $R'$ sufficiently far out in the direct system. From now on, we will assume that $R'$ is chosen sufficiently far so that the stable value of $ht\ H'_i$ is attained. Now, let $i=2\ell$. The valuation $\nu^\dag_{i0}$ is centered in the local noetherian ring $\frac{{R'}^\dag_{\tilde H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde
H'_i}}$, hence by Abhyankar's inequality \begin{equation} rat.rk\ \nu^\dag_{i0}\le\dim\frac{{R'}^\dag_{\tilde H'_i}}{\tilde
H'_{i-1}{R'}^\dag_{\tilde H'_i}}\le ht\ \tilde H'_i-ht\ \tilde H'_{i-1}.\label{eq:abhyankar} \end{equation} Since this inequality is true for all even $i$, summing over all $i$ we obtain: \begin{equation} \begin{array}{rcl} \dim R'&=&\dim\ {R'}^\dag=\sum\limits_{i=1}^{2r}(ht\ \tilde H'_i-ht\ \tilde H'_{i-1})\ge ht\ \tilde H'_1+\sum\limits_{\ell=1}^r(ht\ \tilde H'_{2\ell}-ht\ \tilde H'_{2\ell-1})\ge\\ &\ge&ht\ H'_1+\sum\limits_{\ell=1}^rrat.rk\ \nu^\dag_{2\ell,0}\ge ht\ H'_1+\sum\limits_{\ell=1}^rr_\ell=ht\ H'_1+rat.rk\ \nu=\dim R'.\label{eq:inequalities} \end{array} \end{equation} Hence all the inequalities in (\ref{eq:abhyankar}) and (\ref{eq:inequalities}) are equalities. In particular, we have $$ ht\ \tilde H'_1=ht\ H'_1; $$ combined with Proposition \ref{Hintilde} this shows that \begin{equation} \tilde H'_1=H'_1. \end{equation} Together with the hypothesis (1) of the Proposition, this already proves that at least one of (\ref{eq:tildeH=H0})--(\ref{eq:tildeH=H1}) holds. Furthermore, equalities in (\ref{eq:abhyankar}) and (\ref{eq:inequalities}) prove that $$ ht\ \tilde H'_i=ht\ \tilde H'_{i-1} $$ for all odd $i>1$, so that \begin{equation} \tilde H'_i=\tilde H'_{i-1}\qquad\text{whenever $i>1$ is
odd}\label{eq:oddiseven} \end{equation} and that \begin{equation} r_i=ht\ \tilde H'_i-ht\ \tilde H'_{i-1}\label{heightfixed} \end{equation} whenever $i$ is even.
Now, consider the special case when $\tilde H'_i=H'_i$ for $i\ge1$ and $\tilde H'_0$ is as in (\ref{eq:tildeH=H0})--(\ref{eq:tildeH=H1}). According to Proposition \ref{nu0unique} for each even $i=2\ell$ there exists a unique extension $\nu^\dag_{i0}$ of $\nu_l$ to a valuation of $\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$, centered in the local ring $\lim\limits_{\overset\longrightarrow{R'}} \frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}}$. Moreover, we have \begin{equation} k_{\nu^\dag_{2\ell,0}}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{2\ell}) \label{eq:knudag2l} \end{equation} by Remark \ref{sameresfield}. By Theorem \ref{classification}, there exists an extension $\nu^\dag_-$ of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{\tilde
H'_0}$ such that the $\{\tilde H'_i\}$ as above is the chain of trees of prime ideals, determined by $\nu^\dag_-$. In particular, (\ref{eq:oddiseven}) and (\ref{heightfixed}) hold with $\tilde H'_i$ replaced by $H'_i$.
Now (\ref{heightfixed}) and Proposition \ref{Hintilde} imply that for \textit{any} extension $\nu^\dag_-$ we have $\tilde H'_i=H'_i$ for $i>0$, so that the special case above is, in fact, the only case possible. Furthermore, by (\ref{eq:oddiseven}) we have $H'_{2\ell+1}=H'_{2\ell}$ for all $\ell\in\{1,\dots,r\}$. This implies that for all such $\ell$ the valuation $\nu^\dag_{2\ell+1,0}$ is the trivial valuation of $\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{2\ell})$; in particular, \begin{equation} k_{\nu^\dag_{2\ell+1,0}}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{2\ell}) \label{eq:knudag2l+1} \end{equation} for all $\ell\in\{1,\dots,r-1\}$.
If $\tilde H'_0=H'_1=\tilde H'_1$ then the only possibility for $\nu^\dag_{10}=\nu^\dag_1$ is the trivial valuation of $\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_1)$; we have \begin{equation} k_{\nu^\dag_1}=k_{\nu^\dag_{10}}=\lim\limits_{\overset\longrightarrow{R'}} \kappa(H'_1).\label{eq:knudag10} \end{equation} If $\tilde H'_0=H'_0$ then by the hypothesis (1) of the Proposition and the excellence of $R$ the ring $\frac{{R'}^\dag_{H'_1}}{H'_0{R'}^\dag_{H'_1}}$ is a regular one-dimensional local ring (in particular, unibranch), hence the valuation $\nu^\dag_1=\nu^\dag_{10}$ centered at $\lim\limits_{\overset\longrightarrow{R'}} \frac{{R'}^\dag_{H'_1}}{H'_0{R'}^\dag_{H'_1}}$ is unique and (\ref{eq:knudag10}) holds also in this case.
By induction on $i$, it follows from (\ref{eq:knudag2l}), (\ref{eq:knudag2l+1}), the uniqueness of $\nu^\dag_{2\ell,0}$ and the triviality of $\nu^\dag_{2\ell+1,0}$ for $\ell\ge1$ that $\nu_i^\dag$ is uniquely determined for all $i$ and $k_{\nu^\dag_i}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_i)$. This proves that in both cases (\ref{eq:tildeH=H0}) and (\ref{eq:tildeH=H1}) the valuation $\nu^\dag_-=\nu^\dag_1\circ\dots\circ\nu^\dag_{2r}$ is unique. The last statement of the Proposition is immediate from definitions. \end{proof}
A related necessary condition for the uniqueness of $\nu^\dag_-$ will be proved in \S\ref{locuni1}.
\section{Extending a valuation centered in an excellent local domain
to its henselization.} \label{henselization}
Let $\tilde R$ denote the henselization of $R$, as above. The completion homomorphism $R\rightarrow\hat R$ factors through the henselization: $R\rightarrow\tilde R\rightarrow\hat R$. In this section, we will show that $H_1$ is a minimal prime of $\tilde R$, that $\nu$ extends uniquely to a valuation $\tilde\nu_-$ of rank $r$ centered at $\frac{\tilde R}{H_1}$, and that $H_1$ is the unique prime ideal $P$ of $\tilde R$ such that $\nu$ extends to a valuation of $\frac{\tilde R}P$. Furthermore, we will prove that $H_{2\ell+1}$ is a minimal prime of $P_\ell\tilde R$ for all $\ell$ and that these are precisely the prime $\tilde\nu$-ideals of $\tilde R$.
Studying the implicit prime ideals of $\tilde R$ and the extension of $\nu$ to $\tilde R$ is a logical intermediate step before attacking the formal completion, for the following reason. As we will show in the next section, if $R$ is already henselian in (\ref{eq:defin1}) then $\mathcal{P}'_\beta\hat R'_{H'_{2\ell+1}}\cap\hat R=\mathcal{P}_\beta\hat R$ for all $\beta$ and $R'$ and thus we have $H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}}\left({\cal
P}_\beta\hat R\right)$.
We state the main result of this section. In the case when $R^e$ is an \'etale extension of $R$, contained in $\tilde R$, we use (\ref{eq:defin3}) with $R^\dag=R^e$ as our definition of the implicit prime ideals. \begin{theorem}\label{hensel0} Let $R^e$ be a local \'etale extension of
$R$, contained in $\tilde R$. Then:
(1) The ideal $H_{2\ell+1}$ is prime for $0\le l\le r$; it is a minimal prime of $P_\ell R^e$. In particular, $H_1$ is a minimal prime of $R^e$. We have $H_{2\ell}=H_{2\ell+1}$ for $0\le l\le r$.
(2) The ideal $H_1$ is the unique prime $P$ of $R^e$ such that there
exists an extension $\nu^e_-$ of $\nu$ to $\frac{R^e}P $; the
extension $\nu^e_-$ is unique. The graded algebra
$\mbox{gr}_{\nu^e_-}\frac{R^e}{H_1}$ is scalewise birational to $\mbox{gr}_\nu
R$; in particular, $rk\ \nu^e_-=r$.
(3) The ideals $H_{2\ell+1}$ are precisely the prime $\nu^e_-$-ideals of
$R^e$. \end{theorem} \begin{proof} By assumption, the ring $R^e$ is a direct limit of local, strict \'etale extensions of $R$ which are essentially of finite type. All the assertions (1)--(3) behave well under taking direct limits, so it is sufficient to prove the Theorem in the case when $R^e$ is essentially of finite type over $R$. From now on, we will restrict attention to this case.
The next step is to describe explicitly those local blowings up $R\rightarrow R'$ for which $R'$ is $\ell$-stable. Their interest to us is that, according to Proposition \ref{largeR2}, if $R'$ is $\ell$-stable then for all $R''\in{\cal T}(R')$ and all $\beta\in\frac{\Delta_\ell}{\Delta_{\ell+1}}$, we have the equality \begin{equation} {\cal P}''_\beta(R''\otimes_RR^e)\cap R^e={\cal P}_\beta R^e;\label{eq:contracts} \end{equation} in particular, the limit in (\ref{eq:defin3}) is attained, that is, we have the equality \begin{equation} H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_\ell}\left(\left({\cal P}'_\beta\left(R^e\otimes_RR'\right)_{M'}\right) \bigcap R^e\right).\label{eq:defin4} \end{equation} \begin{lemma}\label{lift} Let $\frac{R}{P_\ell}\to T$ be a finitely generated extension of $\frac {R}{P_\ell}$, contained in $\frac{R_\nu}{\bf m_\ell}$. Let $$ {\bf q}=\frac{\bf m_\nu}{\bf m_\ell}\cap T. $$ There exists a $\nu$-extension $R\to R'$ of $R$ such that $\frac{R'}{P'_\ell}=T_{\bf q}$. \end{lemma}
\begin{proof} Write $T=\frac R{P_\ell}\left[\overline a_1,\ldots,\overline a_k\right]$, with $\overline a_i\in\frac{R_\nu}{\bf m_\ell}$, that is, $\nu_{\ell+1}\left(\overline a_i\right)\geq 0,\ 1\leq i\leq k$. We can lift the $\overline a_i$ to elements $a_i$ in $R_\nu$ such that $\nu\left(a_i\right)\geq0$. Let us consider the ring $R''=R\left[a_1,\ldots,a_k\right]\subset R_\nu$ and its localization $R'=R''_{{\bf m}_\nu\cap R''}$. The ideal $P'_\ell$ is the kernel of the natural map $R'\rightarrow\frac{R_\nu}{\bf m_\ell}$. Thus both $\frac{R'}{P'_\ell}$ and $T_{\bf q}$ are equal to the $\frac R{P_l}$-subalgebra of $\frac{R_\nu}{\bf m_\ell}$, obtained by adjoining $\overline a_1,\ldots,\overline a_k$ to $\frac R{P_l}$ inside $\frac{R_\nu}{\bf m_\ell}$ and then localizing at the preimage of the ideal $\frac{\bf m_\nu}{\bf m_\ell}$. This proves the Lemma. \end{proof}
Let us now go back to our \'etale extension $R\to R^e$. \begin{lemma}\label{anirred1} Fix an integer $l\in\{0,\dots,r\}$. There exists a
local blowing up $R\rightarrow R'$ along $\nu$ having the following
property: let $P'_\ell$ denote the $\ell$-th prime $\nu$-ideal of $R'$. Then
the ring $\frac{R'}{P'_\ell}$ is analytically irreducible; in particular, $\frac{R'}{P'_\ell}\otimes_R R^e$ is an integral
domain. \end{lemma} \begin{remark} We are not claiming that there exists $R'\in\cal T$ such that $\frac{R'}{P'_\ell}$ is analytically irreducible for all $\ell$ (and we do not know how to prove such a claim), only that for each $\ell$ there exists an $R'$, which may depend on $\ell$, such that $\frac{R'}{P'_\ell}$ is analytically irreducible. On the other hand, below we will prove that there exists an $\ell$-stable $R'\in\cal T$. According to Definition \ref{stable} (2) and Proposition \ref{largeR1}, such a stable $R'$ has the property that $\kappa\left(P''_\ell\right)\otimes_R\left(R''\otimes_RR^e\right)_{M''}$ is a domain for all $R''\in{\cal T}(R')$. For a given $R''$, this property is weaker than the analytic irreducibility of $R''/P''_\ell$. The latter is equivalent to saying that $\kappa(P''_\ell)\otimes_R(R''\otimes_RR^\sharp)_{M''}$ is a domain for every local \'etale extension $R^\sharp$ of $R''$. \end{remark} \begin{proof}\textit{(of Lemma \ref{anirred1})} Since $R$ is an excellent local ring, every homomorphic image of $R$ is Nagata \cite{Mat} (Theorems 72 (31.H), 76 (33.D) and 78 (33.H)). Let $\pi:\frac R{P_\ell}\rightarrow S$ be the normalization of $\frac {R}{P_\ell}$. Then $S$ is a finitely generated $\frac {R}{P_\ell}$-algebra contained in $\frac{R_\nu}{\bf m_\ell}$, to which we can apply Lemma \ref{lift}. We obtain a $\nu$-extension $R\to R'$ such that the ring $\frac{R'}{P'_\ell}\cong\frac{R'}{P_\ell R'}$ is a localization of $S$ at a prime ideal, hence it is an excellent normal local ring. In particular, it is analytically irreducible (\cite{Nag}, Theorem (43.20), p. 187 and Corollary (44.3), p. 189), as desired. \end{proof}
Next, we fix $\ell\in\{0,\dots,r\}$ and study the ring $(T')^{-1}(\kappa(P'_\ell)\otimes_RR^e)$, in particular, the structure of the set of its zero divisors, as $R'$ runs over ${\cal T}(R)$ (here $T'$ is as in Remark \ref{interchanging}). Since $R^e$ is separable algebraic, essentially of finite type over $R$, the ring $(T')^{-1}(\kappa(P'_\ell)\otimes_RR^e)$ is finite over $\kappa(P'_\ell)$; this ring is reduced, but it may contain zero divisors. In fact, it is a direct product of fields which are finite separable extensions of $\kappa(P'_\ell)$ because $R^e$ is separable and essentially of finite type over $R$.\par
Consider a chain $R\rightarrow R'\rightarrow R''$ of $\nu$-extensions in $\cal T$. Let \begin{eqnarray} \kappa(P_\ell)\otimes_RR^e&=&\prod\limits_{j=1}^nK_j\\ (T')^{-1}\left(\kappa\left(P'_\ell\right)\otimes_RR^e\right)&=&\prod\limits_{j=1}^{n'}K'_j\\ (T'')^{-1}\left(\kappa\left(P''_\ell\right)\otimes_RR^e\right)&=&\prod\limits_{j=1}^{n''}K''_j \end{eqnarray} be the corresponding decompositions as products of finite field extensions of $\kappa(P_\ell)$ (resp. $\kappa(P'_\ell)$, resp. $\kappa(P''_\ell)$). We want to compare $(T')^{-1}\left(\kappa\left(P'_\ell\right)\otimes_RR^e\right)$ with $(T'')^{-1}\left(\kappa\left(P''_\ell\right)\otimes_RR^e\right)$. \begin{remark} The ring $\kappa\left(P'_\ell\right)\otimes_RR^e$ is itself a direct product of finite extensions of $\kappa\left(P'_\ell\right)$; say $\kappa\left(P'_\ell\right)=\prod\limits_{j\in S'}K'_j$ for a certain set $S'$. In this situation, localization is the same thing as the natural projection to the product of the $K'_j$ over a certain subset $\{1,\dots,n'\}$ of $S'$. Thus the passage from $(T')^{-1}\left(\kappa\left(P'_\ell\right)\otimes_RR^e\right)$ to $(T'')^{-1}\left(\kappa\left(P''_\ell\right)\otimes_RR^e\right)$ can be viewed as follows: first, tensor each $K'_j$ with $\kappa\left(P''_\ell\right)$ over $\kappa\left(P'_\ell\right)$; then, in the resulting direct product of fields, remove a certain number of factors. \end{remark} Let $\bar K'_1,\dots,\bar K'_{\bar n'}$ be the distinct isomorphism classes of finite extensions of $\kappa\left(P'_\ell\right)$ appearing among $K'_1,\dots,K'_{n'}$, arranged in such a way that $\left[\bar K'_j:\kappa\left(P'_\ell\right)\right]$ is non-increasing with $j$, and similarly for $\bar K''_1,\dots,\bar K''_{\bar n''}$. \begin{lemma}\label{decrease} We have the inequality \begin{equation} \left(\left[\bar K''_1:\kappa\left(P''_\ell\right)\right],\dots,\left[\bar K''_{\bar n''}:\kappa\left(P''_\ell\right)\right], n''\right)\le \left(\left[\bar K'_1:\kappa\left(P'_\ell\right)\right],\dots,\left[\bar K'_{\bar n'}:\kappa\left(P'_\ell\right)\right], n'\right)\label{eq:lex} \end{equation} in the lexicographical ordering. Furthermore, either $R'$ is $\ell$-stable or there exists $R''\in\cal T$ such that strict inequality holds in (\ref{eq:lex}). \end{lemma} \begin{proof} Fix a $q\in\{1,\dots,\bar n'\}$ and consider the tensor product $\bar K'_q\otimes_R\kappa\left(P''_\ell\right)$. Since $\bar K'_q$ is separable over $\kappa\left(P'_\ell\right)$, the ring $\bar K'_q\otimes_R\kappa\left(P''_\ell\right)=\prod\limits_{j\in S''_q}K''_j$ is a product of fields. Moreover, two cases are possible:
(a) there exists a non-trivial extension $L$ of $\kappa\left(P'_\ell\right)$ which embeds both into $\kappa\left(P''_\ell\right)$ and $\bar K'_q$. In this case \begin{equation} \left[K''_j:\kappa\left(P''_\ell\right)\right]<\left[\bar K'_q:\kappa\left(P'_\ell\right)\right]\quad\text{ for all }j\in S''_q.\label{eq:strict} \end{equation} (b) there is no field extension $L$ as in (a). In this case $\bar K'_q\otimes_R\kappa\left(P''_\ell\right)$ is a field, so \begin{equation} \#S''_q=1\label{eq:card1} \end{equation} and \begin{equation} \left[K''_j:\kappa\left(P''_\ell\right)\right]=\left[\bar K'_q:\kappa\left(P'_\ell\right)\right]\quad\text{ for }j\in S''_q.\label{eq:equal} \end{equation} Now, if there exists $q\in\{1,\dots,\bar n'\}$ for which (a) holds, take the smallest such $q$. Then (\ref{eq:strict})--(\ref{eq:equal}) imply that strict inequality holds in (\ref{eq:lex}). On the other hand, if (b) holds for all $q\in\{1,\dots,\bar n'\}$ then (\ref{eq:card1}) and (\ref{eq:equal}) imply that \begin{equation} \left(\left[\bar K''_1:\kappa\left(P''_\ell\right)\right],\dots,\left[\bar K''_{\bar n''}:\kappa\left(P''_\ell\right)\right] \right)= \left(\left[\bar K'_1:\kappa\left(P'_\ell\right)\right],\dots,\left[\bar K'_{\bar n'}:\kappa\left(P'_\ell\right)\right] \right)\label{eq:lex1} \end{equation} and $n''\le n'$, so again (\ref{eq:lex}) holds.
Finally, assume that $R'$ is not $\ell$-stable. If there exists $R''\in\cal T$ and $q\in\{1,\dots,\bar n'\}$ for which (a) holds, then by the above we have strict inequality in (\ref{eq:lex}) and there is nothing more to prove. Assume there are no such $R''$ and $q$. Then $(T')^{-1}(\kappa(P'_\ell)\otimes_RR^e)$ is not a domain, so $n'>1$.
Take $R''\in{\cal T}(R')$ such that $\left(\frac{R''}{P''_l}\otimes_RR^e\right)_{M''}$ is an integral domain; such an $R''$ exists by Lemma \ref{anirred1}. Then $n''=1<n'$, as desired. \end{proof} \begin{corollary}\label{anirred2} There exists a stable $R'\in\cal T$. The limit in (\ref{eq:defin3}) is attained for this $R'$. \end{corollary} \begin{proof} In view of Proposition \ref{largeR1}, it is sufficient to prove that there exists $R'\in\cal T$ which which is $\ell$-stable for all $\ell\in\{0,1,\dots,r\}$. First, we fix $\ell\in\{0,1,\dots,r\}$. Lemma \ref{decrease} implies that there exists $R'\in\mathcal{T}(R)$ which is $\ell$-stable.
By Proposition \ref{largeR1}, repeating the procedure above for each $\ell$ we can successively enlarge $R'$ in such a way that it becomes stable.
The last statement follows from Proposition \ref{largeR2}. \end{proof}
We are now in the position to prove Theorem \ref{hensel0}.
By Theorem \ref{primality1} (1), $H_{2\ell-1}$ is prime. By Proposition \ref{contracts}, $H_{2\ell+1}$ maps to $P_\ell$ under the map $\pi^e:\mbox{Spec}\ R^e\rightarrow\mbox{Spec}\ R$. Since this map is \'etale, its fibers are zero-dimensional, which shows that $H_{2\ell+1}$ is a minimal prime of $P_\ell$. This proves (1) of Theorem \ref{hensel0}.
By Proposition \ref{Hintilde}, for $0\le i\le2r$, $\tilde H_i$ is a prime ideal of $R^e$, containing $H_i$. Since the fibers of $\pi^e$ are zero-dimensional, we must have $\tilde H_i=H_i$, so $\tilde H_{2\ell}=\tilde H_{2\ell+1}=H_{2\ell}=H_{2\ell+1}$ for $0\le\ell\le r$. In particular, $\tilde H_0=H_1$. This shows that the unique prime $\tilde H_0$ of $R^e$ such that there exists an extension $\nu^e_-$ of $\nu$ to $\frac{R^e}{\tilde H_0}$ is $\tilde H_0=H_1$. Now (2) of the Theorem is given by Proposition \ref{uniqueness1}.
(3) of Theorem \ref{hensel0} is now immediate. This completes the proof of Theorem \ref{hensel0}. \end{proof}
We note the following corollary of the proof of (2) of Theorem \ref{hensel0} and Corollary \ref{blup1}. Let $\Phi^e=\nu^e_-(R^e\setminus\{0\})$, take an element $\beta\in\Phi^e$ and let ${\cal P}^e_\beta$ denote the $\nu^e_-$-ideal of $R^e$ of value $\beta$. \begin{corollary}\label{blup} Take an element $x\in {\cal P}^e_\beta$. There exists a local blowing up $R\rightarrow R'$ such that $\beta\in\nu(R')\setminus\{0\}$ and $x\in {\cal P}'_\beta{R'}^e$. \end{corollary}
\section{The Main Theorem: the primality of implicit ideals.} \label{prime}
In this section we study the ideals $H_j$ for $\hat R$ instead of $\tilde R$. The main result of this section is \begin{theorem}\label{primality} The ideal $H_{2\ell-1}$ is prime. \end{theorem} \begin{proof} For the purposes of this proof, let $H_{2\ell-1}$ denote the implicit ideals of $\hat R$ and $\tilde H_{2\ell-1}$ the implicit prime ideals of the henselization $\tilde R$ of $R$.
Let $S$ be a local domain. By \cite{Nag} (Theorem (43.20), p. 187) there exists bijective maps between the set of minimal prime ideals of the henselization $\tilde S$ and the maximal ideals of the normalization $S^n$. If, in addition, $S$ is excellent, the two above sets also admit a natural bijection to the set of minimal primes of $\hat S$ \cite{Nag} (Corollary (44.3), p. 189). If $S$ is a henselian local domain, its only minimal prime is the (0) ideal, hence by the above the same is true of $\hat S$. Thus $\hat S$ is also a domain.
This shows that any excellent henselian local domain is analytically irreducible, hence $\tilde H_{2\ell-1}\hat R$ is prime for all $\ell\in\{1,\dots,r+1\}$. Let $\tilde\nu_-$ denote the unique extension of $\nu$ to $\frac{\tilde R}{\tilde H_1}$, constructed in the previous section. Let $H^*_{2\ell-1}\subset\frac{\tilde R}{\tilde H_1}$ denote the implicit ideals associated to the henselian ring $\frac{\tilde R}{\tilde H_1}$ and the valuation $\tilde\nu_-$.
\noi\textit{Claim.} We have $H^*_{2\ell-1}=\frac{H_{2\ell-1}}{\tilde H_1}$.
\noi\textit{Proof of the claim:} For $\beta\in\Gamma$, let $\tilde P_\beta$ denote the $\tilde\nu_-$-ideal of $\frac{\tilde R}{\tilde H_1}$ of value $\beta$. For all $\beta$, we have $\frac{P_\beta}{\tilde H_1}\subset\tilde P_\beta$, and the same inclusion holds for all the local blowings up of $R$, hence $\frac{H_{2\ell-1}}{\tilde H_1}\subset H^*_{2\ell-1}$. To prove the opposite inclusion, we may replace $\tilde R$ by a finitely generated strict \'etale extension $R^e$ of $R$. Now let $\Phi^e=\nu^e_-\left(R^e\setminus\{0\}\right)$ and take an element $\beta\in\Phi^e\cap\Delta_{\ell -1}$. By Corollary \ref{blup}, there exists a local blowing up $R\rightarrow R'$ such that $x\in P'_\beta{R'}^e$. Letting $\beta$ vary over $\Phi^e\cap\Delta_{\ell-1}$, we obtain that if $x\in H^*_{2\ell-1}$ then $x\in\frac{H_{2\ell-1}}{\tilde H_1}$, as desired. This completes the proof of the claim.
The Claim shows that replacing $R$ by $\frac{\tilde R}{\tilde H_1}$ in Theorem \ref{primality} does not change the problem. In other words, we may assume that $R$ is a henselian domain and, in particular, that $\hat R$ is also a domain. Similarly, the ring $\frac R{P_\ell}\otimes_R\hat R\cong\frac{\hat R}{P_\ell}$ is a domain, hence so is its localization $\kappa(P_\ell)\otimes_R\hat R$.
Since $R$ is a henselian excellent ring, it is algebraically closed in $\hat R$ (\cite{Nag}, Corollary (44.3), p. 189 and Corollary \ref{notnormal} of the Appendix); of course, the same holds for $\frac R{P_\ell}$ for all $\ell$. Then $\kappa(P_\ell)$ is algebraically closed in $\kappa(P_\ell)\otimes_R\hat R$. This shows that the ring $R$ is stable. Now the Theorem follows from Theorem \ref{primality1}. This completes the proof of Theorem \ref{primality}. \end{proof}
\section{Towards a proof of Conjecture \ref{teissier}, assuming local
uniformization in lower dimension} \label{locuni1}
Let the notation be as in the previous sections. In this section, we assume that the Local Uniformization Theorem holds and propose an approach to proving Conjecture \ref{teissier}. We prove a Corollary of Conjecture \ref{teissier} which gives a sufficient condition for $\hat\nu_-$ to be unique, which also turns out to be necessary under the additional assumption that $\hat\nu_-$ is minimal. We will assume that all the $R'\in\mathcal T$ are birational to each other, so that all the fraction fields $K'=K$ and the homomorphisms $R'\rightarrow R''$ are local blowings up with respect to $\nu$. Finally, we assume that $R$ contains a field $k_0$ and a local $k_0$-subalgebra $S$ essentailly of finite type, over which $R$ is strictly \'etale. In particular, all the rings in sight are equicharacteristic.
First, we state the Local Uniformization Theorem in the precise form in which we are going to use it. \begin{definition}\label{lut} We say that
\textbf{the embedded Local Uniformization theorem holds in }
$\mathcal T$ if the following conditions are satisfied.
Take an integer $\ell\in\{1,\dots,r-1\}$. Let $\mu_{\ell+1}:=\nu_{\ell+1}\circ\nu_{\ell+2}\circ\dots\circ\nu_r$. Consider a tree $\{H'\}$ of prime ideals of $\frac{\hat R'}{P'_\ell}$ such that $H'\cap\frac{R'}{P'_\ell}=(0)$ and a tight extension $\hat\mu_{2\ell+2}$ of $\mu_{\ell+1}$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{H'}$.
(1) There exists a local blowing up $\pi:R\rightarrow R'$ in $\mathcal T$, which induces an isomorphism at the center of $\nu_\ell$, such that $\frac{R'}{P'_\ell}$ is a regular local ring.
(2) Assume that $\frac{R'}{P'_\ell}$ is a regular local ring. Then there exists in $\mathcal T$ a sequence $\pi:R\rightarrow R'$ of local blowings up along non-singular centers not containing the center of $\nu_l$ such that $\frac{\hat R'}{H'}$ is a regular local ring. \end{definition} It is well known (\cite{A}, \cite{L}, \cite{Z}) that the embedded Local Uniformization theorem holds if $R$ is an excellent local domain such that either $char\ k=0$ or $\dim\ R\le3$ (to be precise, (1) of Definition \ref{lut} is well known and (2) is an easy consequence of known results). While the Local Uniformization theorem in full generality is still an open problem, it is widely believed to hold for arbitrary quasi-excellent local domains. Proving this is an active field of current research in algebraic geometry. Proving local uniformization for rings of arbitrary characteristic is one of the intended applications of Conjecture \ref{teissier}. Note that in Definition \ref{lut} we require only local uniformization of rings of dimension strictly less than $\dim\ R$; the idea is to use induction on $\dim\ R$ to prove local uniformization of rings of dimension $\dim\ R$.
We begin by stating a strengthening of Conjecture \ref{teissier} (using Remark \ref{rephrasing}): \begin{conjecture}\label{teissier1} The valuation $\nu$ admits at least one
tight extension $\hat\nu_-$. This tight extension $\hat\nu_-$ can be
chosen to have the following additional property: for rings $R'$
sufficiently far in the tree $\mathcal T$ we have the equality of
semigroups $\hat\nu_-\left(\frac{\hat R'}{\tilde
H'_0}\setminus\{0\}\right)=\nu(R'\setminus\{0\})$ and for
$\beta\in\nu(R'\setminus\{0\})$ the $\hat\nu_-$-ideal of value $\beta$
is $\frac{\mathcal P_\beta\hat R'}{\tilde H'_0}$. In particular, we
have the equality of graded algebras $\mbox{gr}_\nu
R'=\mbox{gr}_{\hat\nu_-}\frac{\hat R'}{\tilde H'_0}$. \end{conjecture} Below, we give an explicit construction of a valuation $\hat\nu_-$ whose existence is asserted in the Conjecture by describing the trees of ideals $\tilde H'_i$, $0\le i\le 2r$ and, for each $i$, a valuations $\hat\nu_i$ of the residue field $k_{\nu_{i-1}}$, such that $\hat\nu_-=\hat\nu_1\circ\dots\hat\nu_{2r}$. More precisely, for $\ell\in\{0,\dots,r-1\}$, we will construct, recursively in the descending order of $\ell$, a tree $J'_{2\ell+1}$ of prime ideals of $\frac{\hat R'}{H'_{2\ell}}$, $R'\in\mathcal{T}$, such that $J'_{2\ell+1}\cap\frac{R'}{P_\ell}=(0)$, and an extension $\hat\mu_{2\ell+2}$ of $\mu_{\ell+1}$ to $\lim\limits_{\overset\longrightarrow{R'\in\mathcal{T}}}\frac{\hat R'}{J'_{2\ell+1}\hat R'}$; the valuation $\hat\mu_2$ will be our candidate for the desired tight extension $\hat\nu_-$ of $\mu_1=\nu$. Unfortunately, two steps in this construction still remain conjectural, namely, proving that $\hat\mu_{2\ell+2}$ is, indeed, a valuation, and that it is tight (this is essentially the content of Conjectures \ref{strongcontainment} and \ref{containment} below). Once these conjectures are proved, our recursive construction will be complete and Conjecture \ref{teissier1} will follow by setting $\hat\nu_-=\hat\mu_2$.
Let us now describe the construction in detail. According to Corollary \ref{htstable1}, we may assume that $ht\ H'_i$ is constant for each $i$ after replacing $R$ by some other ring sufficiently far in $\mathcal T$. From now on, we will make this assumption without always stating it explicitly.
By (1) of Definition \ref{lut}, applied successively to the trees of ideals $$ P'_\ell\subset R',\quad\ell\in\{1,\dots,r-1\}, $$ there exists $R''\in\cal T$ such that $\frac{R''}{P''_\ell}$ is regular for all $\ell\in\{1,\dots,r-1\}$. Without loss of generality, we may also assume that $R''$ is stable.
For $\ell\in\{1,\dots,r-1\}$ and $R''\in\mathcal{T}$, let $\mathcal{T}_\ell(R'')$ denote the subtree of $\mathcal T$, consisting of all the local blowings up of $R''$ along ideals not contained in $P''_\ell$ (such local blowings up induce an isomorphism at the point $P''_\ell\in\mbox{Spec}\ R''$). Below, we will sometimes work with trees of rings and ideals indexed by $\mathcal{T}_\ell(R'')$ for suitable $\ell$ and $R''$ (instead of trees indexed by all of $\cal T$); the precise tree with which we are working will be specified in each case.
For $\ell=r-1$, we define $J'_{2r-1}:=H'_{2r-1}$ and $\hat\mu_{2r}:=\hat\nu_{2r,0}$; according to Proposition \ref{nu0unique}, $\hat\nu_{2r,0}=\hat\nu_{2r}$ is the unique extension of $\nu_r$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{H'_{2r-1}\hat R'}$.
Next, assume that $\ell\in\{1,\dots,r-1\}$, that the tree $J'_{2\ell+1}$ of prime ideals of $\frac{\hat R'}{H'_{2\ell}\hat R'}$ and a tight extension $\hat\mu_{2\ell+2}$ of $\mu_{\ell+1}$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{J'_{2\ell+1}}$ are already constructed for $R'\in\mathcal{T}$ and that $J'_{2\ell+1}\cap\frac{R'}{P_\ell}=(0)$. It remains to construct the ideals $J'_{2\ell-1}\subset\frac{\hat R'}{H'_{2\ell-2}\hat R'}$ and a tight extension $\hat\mu_{2\ell}$ of $\mu_\ell$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{J'_{2\ell-1}}$ for $R'\in\mathcal{T}$.
We will assume, inductively, that for all $R'\in\mathcal{T}$ the quantity $ht\ J'_{2\ell+1}$ is constant and the following conditions hold: \begin{enumerate} \item We have the equality of semigroups $\hat\mu_{2\ell+2}\left(\frac{\hat R'}{J'_{2\ell+1}}\setminus\{0\}\right)\cong \mu_{\ell+1}\left(\frac{R'}{P'_\ell}\setminus\{0\}\right)$. \item For all $\beta\in\mu_{\ell+1}\left(\frac{R'}{P'_\ell}\setminus\{0\}\right)$ the $\hat\mu_{2\ell+2}$-ideal of $\frac{\hat R'}{J'_{2\ell+1}}$ of value $\beta$ is the extension to $\frac{\hat R'}{J'_{2\ell+1}}$ of the $\mu_{\ell+1}$-ideal of $\frac{R'}{P'_\ell}$ of value $\beta$. \item In particular, we have a canonical isomorphism $gr_{\hat\mu_{2\ell+2}}\frac{\hat R'}{J'_{2\ell+1}}\cong gr_{\mu_{\ell+1}}\frac{R'}{P'_\ell}$ of graded algebras. \end{enumerate} By (2) of Definition \ref{lut} applied to the prime ideals $J'_{2\ell+1}\subset\frac{\hat R'}{H'_{2\ell}\hat R'}$, there exists $R'\in\cal T$ such that both $\frac{R'}{P'_\ell}$ and $\frac{\hat R'}{J'_{2\ell+1}}$ are regular. The fact that $\frac{R'}{P'_\ell}$ is regular implies that so is $\frac{\hat R'}{P'_\ell\hat R'}$. In particular, $\frac{\hat R'}{P'_\ell\hat R'}$ is a domain, so $H'_{2\ell}=P'_\ell\hat R'$. Take a regular system of parameters $$ \bar u'=(\bar u'_1,\dots,\bar u'_{n_\ell}) $$ of $\frac{R'}{P'_\ell}$. Let $k'$ denote the common residue field of $R'$, $\frac{R'}{P'_{\ell-1}}$ and $\frac{R'}{P'_\ell}$. Fix an isomorphism $\frac{R'}{P'_\ell}\cong k'[[\bar u']]$. Renumbering the variables, if necessary, we may assume that there exists $s_\ell\in\{1,\dots,n_\ell\}$ such that $\bar u'_1,\dots,\bar u'_{s_\ell}$ are $k'$-linearly independent modulo $({m'}^2+J'_{2\ell+1})\frac{\hat R'}{H'_{2\ell}}$. Since $\frac{\hat R'}{J'_{2\ell+1}}$ is regular, the ideal $J'_{2\ell+1}$ is generated by a set of the form $\bar v'=(\bar v'_{s_\ell+1},\dots,\bar v'_{n_\ell})$, where $$ \bar v'_j=\bar u'_j-\bar\phi_j(\bar u'_1,\dots,\bar u'_{s_\ell}),\ \bar\phi_j(\bar u'_1,\dots,\bar u'_{s_\ell})\in k'[[\bar u'_1,\dots,\bar u'_{s_\ell}]]. $$ Let $\bar w'=(\bar w'_1,\dots,\bar w'_{s_\ell})=(\bar u'_1,\dots,\bar u'_{s_\ell})$. Let $z'$ be a minimal set of generators of $\frac{P'_\ell}{P'_{\ell-1}}$. Let $k'_0$ be a quasi-coefficient field of $R'$ (that is, a subfield of $R'$ over which $k'$ is formally \'etale; such a quasi-coefficient field exists by \cite{Mat}, moreover, since $R'$ is algebraic over a finite type algebra over a field by hypotheses, $k'$ is finite over $k'_0$). By the hypotheses on $R$ and since $\frac{R'}{P'_\ell}$ is a regular local ring and $\bar u'$ is a minimal set of generators of its maximal ideal $\frac{m'}{P'_\ell}$, there exists an ideal $I\subset k'_0[z']$ such that $\frac{R'}{P'_{\ell-1}}$ is an \'etale extension of $\frac{k'_0[z',\bar u']_{(z',\bar u')}}I$. By assumptions, we have $ht\ P'_{\ell-1}<ht\ P'_\ell$, so $0<ht\ P'_\ell-ht\ P'_{\ell-1}=ht(z')-ht\ I$, in other words, \begin{equation} ht\ I<ht(z').\label{eq:Inotmaximal} \end{equation} Next, we prove two general lemmas about ring extensions.
\begin{notation} Let $k_0$ be a field and $(S,m,k)$ a local noetherian $k_0$-algebra. For a field extension \begin{equation} k_0\hookrightarrow L\label{eq:k0inktilde} \end{equation} such that $k\otimes_{k_0}L$ is a domain, let $S(L)$ denote the localization of the ring $S\otimes_{k_0}L$ at the prime ideal $m(S\otimes_{k_0}L)$. \end{notation}
\begin{lemma}\label{noetherian} Let $k_0$, $(S,m,k)$ and $L$ be as above. The ring $S(L)$ is noetherian. \end{lemma} \begin{proof} If the field extension (\ref{eq:k0inktilde}) is finitely generated, the Lemma is obvious. In the general case, write $L=\lim\limits_{\overrightarrow{i}}L_i$ as a direct limit of its finitely generated subextensions. For each $L_i$, let $k_i$ denote the residue field of $S(L_i)$; $k_i$ is nothing but the field of fractions of $k\otimes_{k_0}L_i$. Write $\hat S=\frac{k[[x]]}H$, where $x$ is a set of generators of $m$ and $H$ a certain ideal of $k[[x]]$. Then $\widehat{S(L_i)}\cong\frac{k_i[[x]]}{Hk_i[[x]]}$. Given two finitely generated extensions $L_i\subset L_j$ of $k_0$, contained in $L$, we have a commutative diagram $$ \begin{matrix} S(L_j)&\overset{\pi_j}\rightarrow&\widehat{S(L_j)}&\cong&\frac{k_j[[x]]}{Hk_j[[x]]}&\\ \psi_{ij}\uparrow&\ &\uparrow&\ &\uparrow\phi_{ij}\\ S(L_i)&\overset{\pi_i}\rightarrow&\widehat{S(L_i)}&\cong&\frac{k_i[[x]]}{Hk_i[[x]]} \end{matrix} $$ where $\phi_{ij}$ is the map induced by the natural inclusion $k_i\hookrightarrow k_j$ and the identity map of $x$ to itself. Let $k_\infty=\lim\limits_{\overrightarrow{i}}k_i$. Then, for each $i$, we have the obvious faithfully flat map $\rho_i:\widehat{S(L_i)}\rightarrow\frac{k_\infty[[x]]}{Hk_\infty[[x]]}$, defined by the natural inclusion $k_i\hookrightarrow k_\infty$ and the identity map of $x$ to itself; the maps $\rho_i$ commute with the $\phi_{ij}$. Thus, we have constructed a faithfully flat map $\rho_i\circ\pi_i$ from each element of the direct system $S(L_i)$ to the fixed noetherian ring $\frac{k_\infty[[x]]}{Hk_\infty[[x]]}$; moreover, the maps $\rho_i\circ\pi_i$ are compatible with the homomorphisms $\psi_{ij}$ of the direct system. This implies that the ring $S(L)=\lim\limits_{\overrightarrow{i}}S(L_i)$ is noetherian. \end{proof}
\begin{lemma}\label{IS(t)} Let $(S,m,k)$ be a local noetherian ring. Let $t$ be an arbitrary collection of independent variables. Consider the rings $S[t]$ and $S(t):=S[t]_{mS[t]}$. Let $I$ be an ideal of $S$. Then \begin{equation} IS(t)\cap S[t]=IS[t].\label{eq:IcapSt=I} \end{equation} \end{lemma} \begin{proof} First, assume the collection $t$ consists of a single variable. Consider elements $f,g\in S[t]$ such that \begin{equation} f\mbox{$\in$ \hspace{-.8em}/} mS[t]\label{eq:fnotinmSt} \end{equation}
and
\begin{equation}
fg\in IS[t].\label{eq:fginISt}
\end{equation}
Proving the equation (\ref{eq:IcapSt=I}) amounts to proving that \begin{equation} g\in IS[t].\label{eq:ginISt} \end{equation} We prove (\ref{eq:ginISt}) by contradiction. Assume that $g\mbox{$\in$ \hspace{-.8em}/} IS[t]$. Then there exists $n\in\mathbb N$ such that $g\mbox{$\in$ \hspace{-.8em}/}(I+m^n)S[t]$. Take the smallest such $n$, so that \begin{equation} g\in\left(I+m^{n-1}\right)S[t]\setminus(I+m^n)S[t].\label{eq:setminus} \end{equation} Write $f=\sum\limits_{j=0}^qa_jt^j$ and $g=\sum\limits_{j=0}^lb_jt^j$. Let \begin{eqnarray}
l_0:&=&\max\{j\in\{0,\dots,l\}\ |\ b_j\mbox{$\in$ \hspace{-.8em}/} I+m^n\}\quad\text{ and}\\
q_0:&=&\max\{j\in\{0,\dots,q\}\ |\ a_j\mbox{$\in$ \hspace{-.8em}/} m\}. \end{eqnarray} Let $c_{l_0+q_0}$ denote the $(l_0+q_0)$-th coefficient of $fg$. We have $$ c_{l_0+q_0}=\sum\limits_{i+j=l_0+q_0}a_ib_j=a_{q_0}b_{l_0}+\sum\limits_{\begin{array}{c}i+j=l_0+q_0\\ i>q_0\end{array}}a_ib_j+\sum\limits_{\begin{array}{c}i+j=l_0+q_0\\ j>l_0\end{array}}a_ib_j. $$ By definition of $l_0$ and $q_0$ and (\ref{eq:setminus}) we have: \begin{eqnarray} a_{q_0}b_{l_0}&\mbox{$\in$ \hspace{-.8em}/}&I+m^n\quad\text{and}\\ \sum\limits_{\begin{array}{c}i+j=l_0+q_0\\ i>q_0\end{array}}a_ib_j+\sum\limits_{\begin{array}{c}i+j=l_0+q_0\\ j>l_0\end{array}}a_ib_j&\in&I+m^n. \end{eqnarray} Hence $c_{l_0+q_0}\mbox{$\in$ \hspace{-.8em}/} I+m^n$, which contradicts (\ref{eq:fginISt}). This completes the proof of Lemma \ref{IS(t)} in the case when $t$ is a single variable. The case of a general $t$ now follows by transfinite induction on the collection $t$. \end{proof}
\begin{lemma}\label{contractsto0} There exist sets of representatives $$ u'=(u'_1,\dots,u'_{n_\ell}) $$ of $\bar u'$ and $\phi_j$ of $\bar\phi_j$, $s_\ell<j\le n_\ell$, in $\frac{\hat R'}{H'_{2\ell-2}\hat R'}$, having the following properties. Let \begin{eqnarray} w'=(w'_1,\dots,w'_{s_\ell})&=&(u'_1,\dots,u'_{s_\ell}),\\ v'=(v'_{s_\ell+1},\dots,v'_{n_\ell})&=& (u'_{s_\ell+1}-\phi_{s_\ell+1},\dots,u'_{n_\ell}-\phi_{n_\ell})\label{eq:defv}. \end{eqnarray} Let $J'_{2\ell-1}=\frac{H'_{2\ell-1}}{H'_{2\ell-2}}+(v')\subset\frac{\hat R'}{H'_{2\ell-2}}$. Then \begin{equation} w'\subset\frac{R'}{P'_{\ell-1}}\label{eq:winR} \end{equation} and \begin{equation} J'_{2\ell-1}\cap\frac{R'}{P'_{\ell-1}}=(0).\label{eq:contractsto0} \end{equation} \end{lemma} \begin{proof}\textit{(of Lemma \ref{contractsto0})} There is no problem choosing $w'$ to satisfy (\ref{eq:winR}).
As for (\ref{eq:contractsto0}), we first prove the Lemma under the assumption that $k$ is countable. We choose the representatives $u'$ arbitrarily and let $\bar\phi_j(u')\in k[[u']]$ denote the formal power series obtained by substituting $u'$ for $\bar u'$ in $\bar\phi_j$. Any representative $\phi_j$ of $\bar\phi_j$, $s_\ell<j\le n_\ell$ has the form $\phi_j=\bar\phi_j(u')+h_j$ with $h_j\in(z')\frac{\hat R'}{H'_{2\ell-2}}$. We define the $h_j$ required in the Lemma recursively in $j$. Take $j\in\{s_\ell+1,\dots,n_\ell\}$. Assume that $h_{s_{\ell+1}},\dots,h_{j-1}$ are already defined and that \begin{equation} (v'_{s_\ell+1},\dots,v'_{j-1})\cap\frac{R'}{P'_{\ell-1}}=(0),\label{eq:j-1inter0} \end{equation} where we view $\frac{R'}{P'_{\ell-1}}$ as a subring of $\frac{\hat R'}{H'_{2\ell-1}}$. Since the ring $\frac{R'}{P'_{\ell-1}}$ is countable, there are countably many ideals in $\frac{\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$, not contained in $\frac{(z')\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$, which are minimal primes of ideals of the form $(f)\frac{\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$, where $f$ is a non-zero element of $\frac{m'}{P'_{\ell-1}}$. Let us denote these ideals by $\{I_q\}_{q\in\mathbb N}$; we have \begin{equation} ht\ I_q=1\quad\text{ for all }q\in\mathbb N.\label{eq:ht=1} \end{equation} We note that \begin{equation} \frac{(z')\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}\not\subset I_q\quad\text{ for all }q\in\mathbb N.\label{eq:znotin} \end{equation} Indeed, by (\ref{eq:Inotmaximal}) and (\ref{eq:j-1inter0}) we have $ht\ \frac{(z')\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}\ge1$. In view of (\ref{eq:ht=1}), containment in (\ref{eq:znotin}) would imply equality, which contradicts the definition of $I_q$.
Since $H'_{2\ell-1}\subsetneqq H'_{2\ell}$ and $J'_{2\ell+1}\subsetneqq\frac{m'\hat R'}{H'_{2\ell}}$, we have $$ \dim\frac{\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}\ge(ht\ H'_{2\ell}-ht\ H'_{2\ell-1})+ht\ \frac{m'\hat R'}{H'_{2\ell}}-(j-s_\ell-1)\ge $$ \begin{equation} (ht\ H'_{2\ell}-ht\ H'_{2\ell-1})+ht\ \frac{m'\hat R'}{H'_{2\ell}}-ht\ J'_{2\ell+1}+1\ge3. \end{equation} Let $\tilde u_j$ denote the image of $u'_j-\bar\phi_j(u')$ in $\frac{\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$. Next, we construct an element $\tilde h_j\in\frac{(z')\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$ such that \begin{equation} \tilde u_j-\tilde h_j\mbox{$\in$ \hspace{-.8em}/}\bigcup\limits_{q=1}^\infty I_q.\label{eq:notinIq} \end{equation} The element $\tilde h_j$ will be given as the sum of an infinite series $\sum\limits_{t=0}^\infty h_{jt}^t$ in $(z')\frac{\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$, convergent in the $m'$-adic topology, which we will now construct recursively in $t$. Put $h_{j0}=0$. Assume that $t>0$, that $h_{j0},\dots,h_{j,t-1}$ are already defined and that for $q\in\{1,\dots,t-1\}$ we have $u'_j-\bar\phi_j(u')-\sum\limits_{l=0}^qh_{jl}\mbox{$\in$ \hspace{-.8em}/}\bigcup\limits_{l=1}^qI_l$ and $h_{jq}\in(z')\bigcap\left(\bigcap\limits_{l=1}^{q-1}I_l\right)$. If $u'_j-\bar\phi_j(u')-\sum\limits_{l=0}^{t-1}h_{jl}\mbox{$\in$ \hspace{-.8em}/} I_t$, put $h_{jt}=0$. If $u'_j-\bar\phi_j(u')-\sum\limits_{l=0}^{t-1}h_{jl}\in I_t$, let $h_{jt}$ be any element of $(z')\bigcap\left(\bigcap\limits_{l=1}^{t-1}I_l\right)\setminus I_t$ (such an element exists because $I_t$ is prime, in view of (\ref{eq:znotin})). This completes the definition of $\tilde h_j$. Let $h_j$ be an arbitrary representative of $\tilde h_j$ in $\frac{\hat R'}{H'_{2\ell-2}}$.
We claim that \begin{equation} \left(H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_j)\right)\cap\frac{R'}{P'_{\ell-1}}=(0).\label{eq:jinter0} \end{equation} Indeed, suppose the above intersection contained a non-zero element $f$. Then any minimal prime $\tilde I$ of the ideal $(v'_j)\frac{\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$ is also a minimal prime of $(f)\frac{\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$. Since $v_j\mbox{$\in$ \hspace{-.8em}/}\frac{(z')\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$, we have $\tilde I\not\subset\frac{(z')\hat R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$. Hence $\tilde I=I_q$ for some $q\in\mathbb N$. Then $v_j\in I_q$, which contradicts (\ref{eq:notinIq}).
Carrying out the above construction for all $j\in\{s_\ell+1,\dots,n_\ell\}$ produces the elements $\phi_j$ required in the Lemma. This completes the proof of Lemma \ref{contractsto0} in the case when $k$ is countable.
Next, assume that $k$ is uncountable. Let $u'$ be chosen as above.
By assumption, $\frac{R'}{P'_{\ell-1}}$ contains a $k_0$-subalgebra $S$ essentially of finite type, over which $\frac{R'}{P'_{\ell-1}}$ is strictly \'etale. Take a countable subfield $L_1\subset k_0$ such that the algebra $S$ is defined already over $L_1$ (this means that $S$ has the form \begin{equation} S=(S'_1\otimes_{L_1}k_0)_{m'_1(S'_1\otimes_{L_1}k_0)},\label{eq:R'/P'} \end{equation} where $(S'_1,m'_1,k'_1)$ is a local $L_1$-algebra essentially of finite type). Next, let $L_1\subset L_2\subset...$ be an increasing chain of finitely generated field extensions of $L_1$, contained in $k_0$, having the following property. Let $(S'_q,m'_q,k'_q)$ denote the localization of $S'_1\otimes_{k'_1}L_q$ at the maximal ideal $m'_1(S'_1\otimes_{k'_1}L_q)$. We require that $$ k'_\infty:=\bigcup\limits_{q=1}^\infty k'_q $$ contain all the coefficients of all the formal power series $\bar\phi_{s_\ell+1},\dots,\bar\phi_{n_\ell}$ and such that the ideal $\frac{H'_{2\ell-1}}{H'_{2\ell-2}}$ is generated by elements of $\frac{k'_\infty[[z']]}{I_\infty}[[u']]$, where $I_\infty$ is the kernel of the natural homomorphism $k'_\infty[[z']]\rightarrow\frac{\hat R'}{H'_{2\ell-2}}$. Let $H'_{2\ell-1,\infty}=\frac{H'_{2\ell-1}}{H'_{2\ell-2}}\cap \frac{k'_\infty[[z']]}{I_\infty}[[u']]$. We have constructed an increasing chain $S'_1\subset S'_2\subset...$ of local $L_1$-algebras essentially of finite type such that $k'_q$ is the residue field of $S'_q$. Then $S'_\infty:=\bigcup\limits_{q=1}^\infty S'_q$ is a local noetherian ring whose completion is $\frac{k'_\infty[[z',u']]}{\left(P'_{\ell-1}\cap S'_\infty\right)k'_\infty[[z',u']]}$. Let $m'_\infty$ denote the maximal ideal of $S'_\infty$. The above argument in the countable case shows that there exist representatives $\phi_{s_\ell+1},\dots,\phi_{n_\ell}$ of $\bar\phi_{s_\ell+1},\dots,\bar\phi_{n_\ell}$ in $\frac{\hat S'_\infty}{H'_{2\ell-2}\cap\hat S'_\infty}$ such that, defining $v'=(v'_{s_\ell+1},\dots,v'_{n_\ell})$ as in (\ref{eq:defv}), we have \begin{equation} \left((v')+H'_{2\ell-1,\infty}\right)\cap S'_\infty=(0).\label{eq:S'infty0} \end{equation} Let $L_\infty=\bigcup\limits_{q=1}^\infty L_q$ and let $t$ denote a transcendence base of $k_0$ over $L_\infty$. Let the notation be as in Lemma \ref{noetherian} with $k_0$ replaced by $L_\infty$. For example, $S'_\infty(L_\infty(t))$ will denote the localization of the ring $S'_\infty\otimes_{L_\infty}L_\infty(t)$ at the prime ideal ideal $m'_\infty(S'_\infty\otimes_{L_\infty}L_\infty(t))$.
By (\ref{eq:S'infty0}), \begin{equation} \left((v')+H'_{2\ell-1,\infty}\right)\hat S'_\infty[t]\cap S'_\infty[t]=(0).\label{eq:S'infty0t} \end{equation} Now Lemma \ref{IS(t)} and the fact that $S'_\infty[t]$ is a domain imply that \begin{equation} \left((v')+H'_{2\ell-1,\infty}\right)\hat S'_\infty(L_\infty(t))\cap S'_\infty(L_\infty(t))=(0).\label{eq:capbarS=0} \end{equation} Next, let $\tilde L$ be a finite extension of $L_\infty(t)$, contained in $k_0$; then $S'_\infty(\tilde L)$ is finite over $S'_\infty(L_\infty(t))$. Since $\hat S'_\infty(\tilde L)$ is faithfully flat over $\hat S'_\infty(L_\infty(t))$ and in view of (\ref{eq:capbarS=0}), we have $$ \left(\left((v')+H'_{2\ell-1,\infty}\right)\hat S'_\infty(\tilde L)\cap S'_\infty(\tilde L)\right)\cap S'_\infty(L_\infty(t))=(0). $$ Hence $ht\ \left((v')+H'_{2\ell-1,\infty}\right)\hat S'_\infty(\tilde L)\cap S'_\infty(\tilde L)=0$. Since $S'_\infty(\tilde L)$ is a domain, this implies that \begin{equation} \left((v')+H'_{2\ell-1,\infty}\right)\hat S'_\infty(\tilde L)\cap S'_\infty(\tilde L)=(0).\label{eq:tildek=0} \end{equation} Since $k_0$ is algebraic over $L_\infty(t)$, it is the limit of the direct system of all the finite extensions of $L_\infty(t)$ contained in it. We pass to the limit in (\ref{eq:tildek=0}). By (\ref{eq:R'/P'}), we have $S=S'_\infty(k_0)$; we also note that $\hat S=\frac{\hat R'}{H'_{2\ell-2}}$.
Since the natural maps $\hat S'_\infty(\tilde L)\rightarrow\hat S'_\infty(k_0)$ are all faithfully flat, we obtain \begin{equation} \left((v')+H'_{2\ell-1,\infty}\right)\hat S'_\infty(k_0)\cap S=(0).\label{eq:capS=0} \end{equation} Since $\hat S=\frac{\hat R'}{H'_{2\ell-2}}$ is also the formal completion of $\hat S'_\infty(k_0)$, it is faithfully flat over $\hat S'_\infty(k_0)$. Hence \begin{equation} J'_{2\ell-1}\cap\hat S'_\infty(k_0)=\left((v')+H'_{2\ell-1,\infty}\right)\frac{\hat R'}{H'_{2\ell-2}}\cap\hat S'_\infty(k_0)=\left((v')+H'_{2\ell-1,\infty}\right)\hat S'_\infty(k_0).\label{eq:capbarS} \end{equation} Combining this with (\ref{eq:capS=0}), we obtain \begin{equation} J'_{2\ell-1}\cap S=(0).\label{eq:capS(t)=0} \end{equation} Thus the ideal $J'_{2\ell-1}\cap\frac{R'}{P'_{\ell-1}}$ contracts to $(0)$ in $S$. Since $\frac{R'}{P'_{\ell-1}}$ is \'etale over $S$, this implies the desired equality (\ref{eq:contractsto0}). This completes the proof of Lemma \ref{contractsto0}. \end{proof}
Since $\frac{\hat R'}{H'_{2\ell}\hat R'}$ is a complete regular local ring and $(w',v')$ is a set of representatives of a minimal set of generators of its maximal ideal $\frac{m'\hat R'}{H'_{2\ell}}$, there exists a complete local domain $R'_\ell$ (not necessarily regular) such that $\frac{\hat R'}{H'_{2\ell-1}}\cong R'_\ell[[w',v']]$. Consider the ring homomorphism \begin{equation} R'_\ell[[w',v']]\rightarrow R'_\ell[[w']],\label{eq:homomorphisms} \end{equation} obtained by taking the quotient modulo $(v')$. By (\ref{eq:homomorphisms}), the quotient of $\frac{\hat R'}{H'_{2\ell-2}}$ by $J'_{2\ell-1}$ is the integral domain $R'_\ell[[w']]$, hence $J'_{2\ell-1}$ is prime.
Consider a local blowing up $R'\rightarrow R''$ in $\mathcal T$. Because of the stability assumption on $R$, the ring $\frac{\hat R''}{H''_{2\ell-2}}\otimes_R\kappa(P''_{l-1})$ is finite over $\frac{\hat R'}{H'_{2\ell-2}}\otimes_R\kappa(P'_{l-1})$; hence the ring $\lim\limits_{\overset\longrightarrow{R''\in\mathcal T}}\left(\frac{\hat R''}{H''_{2\ell-2}}\otimes_R\kappa(P''_{l-1})\right)$ is integral over $\frac{\hat R'}{H'_{2\ell-2}}\otimes_R\kappa(P'_{l-1})$. In particular, there exists a prime ideal in $$ \lim\limits_{\overset\longrightarrow{R''\in\mathcal T}}\left(\frac{\hat R''}{H''_{2\ell-2}}\otimes_R\kappa(P''_{l-1})\right), $$ lying over $J'_{2\ell-1}\frac{\hat R'}{H'_{2\ell-2}}\otimes_R\kappa(P'_{l-1})$. Pick and fix one such prime ideal. Intersecting this ideal with $\frac{\hat R''}{H''_{2\ell-2}}$ for each $R''\in\mathcal T$, we obtain a tree $J''_{2\ell-1}$ of prime ideals of $\frac{\hat R''}{H''_{2\ell-2}}$, $R''\in\mathcal T$.
Our next task is to define the restriction of the valuation $\hat\mu_{2\ell}$ to the ring $\frac{\hat R'}{J'_{2\ell-1}}$. By the induction assumption, $\hat\mu_{2\ell+2}$ is already defined on $\lim\limits_{\overset\longrightarrow{R'\in\mathcal{T}}}\frac{\hat R'}{J'_{2\ell+1}\hat R'}$. For all \textit{stable} $R''\in\mathcal{T}$ we have the isomorphism $gr_{\hat\mu_{2\ell+2}}\frac{\hat R''}{J''_{2\ell+1}}\cong gr_{\mu_{\ell+1}}\frac{R''}{P''_\ell}$ of graded algebras (in particular, $gr_{\hat\mu_{2\ell+2}}\frac{\hat R''}{J''_{2\ell+1}}$ is scalewise birational to $gr_{\mu_{\ell+1}}\frac{R''}{P''_\ell}$ for any $R''\in\mathcal{T}$ and $\hat\mu_{2\ell+2}$ has the same value group $\Delta_\ell$ as $\mu_{\ell+1}$).
Define the prime ideals $\tilde H''_{2\ell-2}=\tilde H''_{2\ell-1}$ to be equal to the preimage of $J''_{2\ell-1}$ in $\hat R''$ and $\tilde H''_{2\ell}=\tilde H''_{2\ell+1}$ the preimage of $J''_{2\ell+1}$ in $\hat R''$. By definition of tight extensions, the valuation $\hat\nu_{2\ell+1}$ must be trivial. It remains to describe the valuation $\hat\mu_{2\ell}$ on $\frac{\hat R''}{J''_{2\ell-1}}$, $R''\in\mathcal T$. We will first define $\hat\nu_{2\ell}$ and then put $\hat\mu_{2\ell}=\hat\nu_{2\ell}\circ\hat\mu_{2\ell+2}$.
By definition of tight extensions, the value group of $\hat\mu_{2\ell}$ must be equal to $\Delta_{\ell-1}$ and that of $\hat\nu_{2\ell}$ to $\frac{\Delta_{\ell-1}}{\Delta_\ell}$. For a positive element $\bar\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$, define the candidate for $\hat\nu_{2\ell}$-ideal of $\frac{\hat R''_{\tilde H''_{2\ell}}}{\tilde H''_{2\ell-1}\hat R''_{\tilde H''_{2\ell}}}$ of value $\bar\beta$, denoted by $\hat{\mathcal P}''_{\beta\ell}$, by the formula \begin{equation} \hat{\mathcal P}''_{\bar\beta\ell}=\frac{\mathcal{P}''_{\bar\beta}\hat R''_{\tilde H''_{2\ell}}}{\tilde H''_{2\ell-1}\hat R''_{\tilde H''_{2\ell}}}.\label{eq:validealmain} \end{equation} \begin{conjecture}\label{strongcontainment} The elements $\phi_j$ of Lemma \ref{contractsto0} can be chosen in such a way that the following condition holds. For each positive element $\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$ and each tree morphism $R'\rightarrow R''$ in $\mathcal T$, we have $$ \hat{\mathcal P}''_{\beta\ell}\cap\hat R'_{\tilde H'_{2\ell}}=\hat{\mathcal P}'_{\beta\ell}. $$ \end{conjecture} \begin{conjecture}\label{containment} The elements $\phi_j$ of Lemma \ref{contractsto0} can be chosen in such a way that \begin{equation} \bigcap\limits_{\bar\beta\in\left(\frac{\Delta_{\ell-1}}{\Delta_\ell}\right)_+} \left(\mathcal{P}'_{\bar\beta}+\tilde H'_{2\ell-1}\right)\hat R'_{\tilde H'_{2\ell}}\subset\tilde H'_{2\ell-1}. \label{eq:restrictionmain} \end{equation} \end{conjecture} For the rest of this section assume that Conjectures \ref{strongcontainment} and \ref{containment} are true.
For all $\bar\beta\in\left(\frac{\Delta_{\ell-1}}{\Delta_\ell}\right)_+$, we have the natural isomorphism $$ \lambda_{\bar\beta}:\frac{\mathcal{P}'_{\bar\beta}}{\mathcal{P}'_{\bar\beta+}}\otimes_{\kappa(P'_{\ell-1})}\kappa(\tilde H'_{2\ell})\longrightarrow\frac{\hat{\mathcal P}'_{\bar\beta}}{\hat{\mathcal P}'_{\bar\beta+}} $$ of $\kappa(\tilde H'_{2\ell})$-vector spaces. The following fact is an easy consequences of Conjecture \ref{strongcontainment}: \begin{corollary}\label{integraldomain}\textbf{(conditional on Conjecture \ref{strongcontainment})} If the elements $\phi_j$ of Lemma \ref{contractsto0} can be chosen as in Conjecture \ref{strongcontainment} then the graded algebra $$ \mbox{gr}_{\nu_\ell}\frac{R'_{P'_\ell}}{P'_{\ell-1}R'_{P'_\ell}}\otimes_{\kappa(P'_{\ell-1})}\kappa(\tilde H'_{2\ell})\cong\bigoplus\limits_{\bar\beta\in\left(\frac{\Delta_{\ell-1}}{\Delta_\ell}\right)_+}\frac{\hat{\mathcal P}'_{\bar\beta}}{\hat{\mathcal P}'_{\bar\beta+}} $$ is an integral domain. \end{corollary}
For a non-zero element $x\in\frac{\hat R'_{\tilde H'_{2\ell}}}{\tilde H'_{2\ell-1}}$, let $\operatorname{Val}_\ell(x)=\left\{\left.\beta\in\nu_\ell\left(\frac{R'}{P'_{\ell-1}}\setminus\{0\}\right)\ \right|\ x\in\hat{\mathcal P}'_{\beta\ell}\right\}$. We define $\hat\nu_{2\ell}$ by the formula \begin{equation}\label{eq:mu2l} \hat\nu_{2\ell}(x)=\max\ \operatorname{Val}_\ell(x). \end{equation} Since $\nu_\ell$ is a rank 1 valuation, centered in a local noetherian domain $\frac{R'}{P'_{\ell-1}}$, the semigroup $\nu_\ell\left(\frac{R'}{P'_{\ell-1}}\setminus\{0\}\right)$ has order type $\mathbb N$, so by (\ref{eq:restrictionmain}) the set $Val_\ell(x)$ contains a maximal element. This proves that the valuation $\hat\nu_{2\ell}$ is well defined by the formula (\ref{eq:mu2l}), and that we have a natural isomorphism of graded algebras $$ \mbox{gr}_{\nu_\ell}\frac{R'_{P'_\ell}}{P'_{\ell-1}R'_{P'_\ell}}\otimes_{\kappa(P'_{\ell-1})}\kappa(\tilde H'_{2\ell})\cong\mbox{gr}_{\hat\nu_{2\ell}}\frac{\hat R_{\tilde H'_{2\ell}}}{\tilde H'_{2\ell-1}}. $$ Since the above construction is valid for all $R\in\mathcal T$, $\hat\nu_{2\ell}$ extends naturally to a valuation centered in the ring $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R''}{\tilde H''_{2\ell-1}\hat R''}$ (by abuse of notation, this extension will also be denoted by $\hat\nu_{2\ell}$).
The extension $\hat\mu_{2\ell}$ of $\mu_\ell$ to $\lim\limits_{\overset\longrightarrow{R''\in\mathcal{T}(R')}}\frac{\hat R''}{\tilde H''_{2\ell-1}\hat R'}$ is defined by $\hat\mu_{2\ell}=\hat\nu_{2\ell}\circ\hat\mu_{2\ell+2}$.
This completes the proof of Conjecture \ref{teissier1} (assuming Conjectures \ref{strongcontainment} and \ref{containment}) by descending induction on $\ell$.
$\Box$
The next Corollary of Conjecture \ref{teissier1} gives necessary conditions for $\hat\nu_-$ to be uniquely determined by $\nu$; it also shows that the same conditions are sufficient for $\hat\nu_-$ to be the unique minimal extension of $\nu$, that is, to satisfy \begin{equation} \tilde H'_i=H'_i,\quad0\le i\le 2r.\label{eq:tilde=nothing} \end{equation} Suppose given a tree $\left\{\tilde H'_0\right\}$ of minimal prime ideals of $\hat R'$ (in particular, $R'\cap\tilde H'_0=(0)$). If the valuation $\nu$ admits an extension to a valuation $\hat\nu_-$ of $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{\tilde H'_0}$, then $\tilde H'_0$ is the 0-th prime ideal of $\hat R'$, determined by $\hat\nu_-$. Since $\tilde H'_0$ is assumed to be a \textit{minimal} prime, we have $\tilde H'_0=H'_0$ by Proposition \ref{Hintilde}. \begin{remark} Let the notation be as in Conjecture \ref{teissier1}. Denote the tree of prime ideals $\{\tilde H'_0\}$ by $\{H'\}$ for short. Consider a homomorphism \begin{equation} R'\rightarrow R''\label{eq:R'toR''} \end{equation} in $\mathcal T$. Assume that the local rings $R'$ and $\frac{\hat R'}{H'}$ are regular, and let $V=(V_1,\dots,V_s)$ be a minimal set of generators of $H'$. Then $V$ can be extended to a regular system of parameters for $\hat R'$. We have an isomorphism $\hat R'\cong\frac{\hat R'}{H'}[[V]]$. The morphism (\ref{eq:R'toR''}) induces an isomorphism $\hat R'_{H'}\cong \hat R''_{H''}$, so that $V$ induces a regular system of parameters of $\hat R''_{H''}$. In particular, the $H''$-adic valuation of $\hat R''_{H''}$ coincides with the $H'$-adic valuation of $\hat R'_{H'}$. On the other hand, we do not know, assuming that $R''$ and $\frac{\hat R''}{H''}$ are regular and $ht\ H''=ht\ H'$, whether $V$ induces a minimal set of generators of $H''$; we suspect that the answer is ``no''. \end{remark} \begin{corollary}\label{uncond1}\textbf{(conditional on Conjecture \ref{teissier1})} If the valuation $\nu$ admits a unique extension to a valuation $\hat\nu_-$ of $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{H'_0}$, then the following conditions hold:
(1) $ht\ H'_1\le 1$
(2) $H'_i=H'_{i-1}$ for all odd $i>1$.
Moreover, this unique extension $\hat\nu_-$ is minimal.
Conversely, assume that (1)--(2) hold. Then there exists a unique minimal extension $\hat\nu_-$ of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{H'_0}$. \end{corollary} \begin{proof} The fact that conditions (1), (2) and equations (\ref{eq:tilde=nothing}) determine $\hat\nu_-$ uniquely is nothing but Proposition \ref{uniqueness1}. Conversely, assume that there exists a unique extension $\hat\nu_-$ of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{H'_0}$. By Remark \ref{minimalexist}, there exist minimal extensions of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{H'_0}$, hence $\hat\nu_-$ must be minimal.
Next, by Conjecture \ref{teissier1}, there exists a tree of prime ideals $\tilde H'$ with $H'\cap R'=(0)$ and a tight extension $\hat\mu_-$ of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{H'}$. The ideals $H'$ are both the the 0-th and the 1-st ideals determined by $\hat\mu_-$; in particular, we have \begin{equation} H'_0\subset H'_1\subset H'\label{eq:inH'} \end{equation} by Proposition \ref{Hintilde}. Now, take any valuation $\theta$, centered in the regular local ring $\frac{R'_{H'}}{H'_0}$, such that the residue field $k_\theta=\kappa(H')$. Then the composition $\hat\mu_-\circ\theta$ is an extension of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{\hat R'}{H'_0}$, hence \begin{equation} \hat\mu_-\circ\theta=\hat\nu_-\label{eq:mucirctheta} \end{equation} by uniqueness. For $i\ge1$, the $i$-th prime ideal, determined by $\hat\mu_-\circ\theta=\hat\nu_-$ coincides with that determined by $\hat\mu_-$. Since $\nu$ is minimal and $\hat\mu_-$ is tight, we obtain condition (2) of the Corollary. Finally, if we had $ht\ H'>1$, there would be infinitely many choices for $\theta$, contradicting \ref{eq:mucirctheta} and the uniqueness of $\hat\nu_-$. Thus $ht\ H'\le1$. Combined with \ref{eq:inH'}, this proves (1) of the Corollary. This completes the proof of Corollary (\ref{uncond1}), assuming Conjecture \ref{teissier1}. \end{proof}
\appendix{Regular morphisms and G-rings.}
In this Appendix we recall the definitions of regular homomorphism, G-rings and excellent and quasi-excellent rings. We also summarize some of their basic properties used in the rest of the paper. \begin{definition}\label{regmor} (\cite{Mat}, Chapter 13, (33.A), p. 249) Let $\sigma:A\rightarrow B$ be a homomorphism of noetherian rings. We say that $\sigma$ is {\bf regular} if it is flat, and for every prime ideal $P\subset A$, the ring $B\otimes_A\kappa(P)$ is geometrically regular over $\kappa(P)$ (this means that for any finite field extension $\kappa(P)\rightarrow k'$, the ring $B\otimes_Ak'$ is regular). \end{definition} \begin{remark} If $\kappa(P)$ is perfect, the ring $B\otimes_A\kappa(P)$ is geometrically regular over $\kappa(P)$ if and only if it is regular. \end{remark} \begin{remark} It is known that a morphism of finite type is regular in the above sense if and only if it is smooth (that is, formally smooth in the sense of Grothendieck with respect to the discrete topology), though we do not use this fact in the present paper. \end{remark} Regular morphisms come up in a natural way when one wishes to pass to the formal completion of a local ring: \begin{definition}\label{Gring} (\cite{Mat}, (33.A) and (34.A)) Let $R$ be a noetherian ring. For a maximal ideal $m$ of $R$, let $\hat R_m$ denote the $m$-adic completion of $R$. We say that $R$ is a {\bf G-ring} if for every maximal ideal $m$ of $R$, the natural map $R\rightarrow\hat R_m$ is a regular homomorphism. \end{definition} The property of being a G-ring is preserved by localization and passing to rings essentially of finite type over $R$. \begin{definition}\label{quasiexcellent} (\cite{Mat}, Definition 2.5, (34.A), p. 259) Let $R$ be a noetherian ring. We say that $R$ is {\bf quasi-excellent} if the following two conditions hold:
(1) $R$ is J-2, that is, for any scheme $X$, which is reduced and of finite type over $\mbox{Spec}\ R$, $Reg(X)$ is open in the Zariski topology.
(2) For every maximal ideal $m\subset R$, $R_m$ is a G-ring. \end{definition} It is known \cite{Mat} that a \textit{local} G-ring is automatically J-2, hence automatically quasi-excellent. Thus for local rings ``G-ring'' and ``quasi-excellent'' are one and the same thing. A ring is said to be \textbf{excellent} if it is quasi-excellent and universally catenary, but we do not need the catenary condition in this paper.
Both excellence and quasi-excellence are preserved by localization and passing to rings of finite type over $R$ (\cite{Mat}, Chapter 13, (33.G), Theorem 77, p. 254). In particular, any ring essentially of finite type over a field, $\mathbf Z$, $\mathbf Z_{(p)}$, $\mathbf Z_p$, the Witt vectors or any other excellent Dedekind domain is excellent. See \cite{Nag} (Appendix A.1, p. 203) for some examples of non-excellent rings.
Rings which arise from natural constructions in algebra and geometry are excellent. Complete and complex-analytic local rings are excellent (see \cite{Mat}, Theorem 30.D) for a proof that any complete local ring is excellent).
Finally, we remark that the category of quasi-excellent rings is a natural one for doing algebraic geometry, since it is the largest reasonable class of rings for which resolution of singularities can hold. Namely, let $R$ be a noetherian ring. Grothendieck (\cite{EGA}, IV.7.9) proves that if all of the irreducible closed subschemes of $\mbox{Spec}\ R$ and all of their finite purely inseparable covers admit resolution of singularities, then $R$ must be quasi-excellent. Grothendieck's result means that the largest {\it class} of noetherian rings, closed under homomorphic images and finite purely inseparable extensions, for which resolution of singularities could possibly exist, is {\it quasi-excellent} rings.
We now summarize the specific uses we make of quasi-excellence in the present paper. We begin by recalling three results from \cite{Mat} and \cite{Nag}. As a point of terminology, we note that Nagata's ``pseudo-geometric'' rings are now commonly known as Nagata rings. Quasi-excellent rings are Nagata (\cite{Mat}, (33.H), Theorem 78). \begin{theorem}\label{annormal}(\cite{Mat}, (34.C), Theorem 79) Let $R$ be an excellent normal local ring. Then $R$ is analytically normal (this means that its formal completion $\hat R$ is normal). \end{theorem} \begin{theorem}(\cite{Nag}, (43.20), p. 187) Let $R$ be a local integral domain, $\tilde R$ its Henselization and $R'$ its normalization. There is a natural one-to-one correspondence between the minimal primes of $\tilde R$ and the maximal ideals of $R'$. \end{theorem} \begin{proposition}\label{anirred}(\cite{Nag}, Corollary (44.3), p. 189) Let $R$ be a quasi-excellent analytically normal local ring. Then its Henselization $\tilde R$ is analytically irreducible and is algebraically closed in its formal completion. \end{proposition} From the above results we deduce \begin{corollary}\label{notnormal} Let $(R,\mathbf m)$ be a Henselian excellent local domain. Then $R$ is analytically irreducible and is algebraically closed in $\hat R$. \end{corollary} \begin{proof} If, in addition, we assume $R$ to be normal, the result follows from Theorems \ref{annormal} and \ref{anirred}. In the general case, let $R'$ denote the normalization of $R$. Then $R'$ is a Henselian normal quasi-excellent local ring, so it satisfies the conclusions of the Corollary. Consider the commutative diagram \begin{equation}\label{eq:CD} \xymatrix{R\ar[d]_-\psi \ar[r]^-\phi & {\hat R}\ar[d]_-{\hat\psi}\\ R'\ar[r]^-{\phi'}&{\hat R'}} \end{equation} where $\hat R'$ stands for the formal completion of $R'$. Since $R$ is Nagata, $R'$ is a finite $R$-module. Thus $\phi'$ coincides with the $\mathbf m$-adic completion of $R'$, viewed as an $R$-module. Hence $\hat R'\cong R'\otimes_R\hat R$. Since $\psi$ is injective and $\hat R$ is flat over $r$, the map $\hat\phi$ is also injective. Since $R'$ is analytically irreducible, $\hat R'$ is a domain, and therefore so is its subring $\hat R$. This proves that $R$ is analytically irreducible.
To prove that $R$ is algebraically closed in $\hat R$, take an element $x\in\hat R$, algebraic over $R$. Since all the maps in \ref{eq:CD} are injective, let us view all the rings involved as subrings of $\hat R'$. Since $R'$ is algebraically closed in $\hat R'$, we have $x\in R'$, in particular, we may write $x=\frac ab$ with $a,b\in R$. Now, since $(a)\hat R\subset(b)\hat R$ and $\hat R$ is faithfully flat over $R$, we have $(a)\subset(b)$ in $R$, so $x=\frac ab\in R$. This proves that $R$ is algebraically closed in $\hat R$. The Corollary is proved. \end{proof}
Next we summarize, in a more specific manner, the way in which these results are applied in the present paper. The main applications are as follows.
(1) Let $R$ be an excellent local domain, $P$ a prime ideal of $R$ and $H_i\subset H_{i+1}$ two prime ideals of $\hat R$ such that \begin{equation} H_i\cap R=H_{i+1}\cap R=P.\label{eq:HicontractstoP} \end{equation} Then $\frac RP$ is also excellent. Definitions \ref{regmor}, \ref{Gring} and \ref{quasiexcellent} imply that the ring $\hat R\otimes_R\kappa(P)$ is geometrically regular over $P$, in particular, regular. Moreover, (\ref{eq:HicontractstoP}) implies that the ideal $\frac{H_{i+1}}{P\hat R}$ is a prime ideal of $\frac{\hat R}{P\hat R}$, disjoint from the natural image of $R\setminus P$ in $\frac{\hat R}{P\hat R}$. Thus the local ring $\frac{\hat R_{H_{i+1}}}{P\hat R_{H_{i+1}}}$ is a localization of $\hat R\otimes_R\kappa(P)$ at the prime ideal $H_{i+1}(\hat R\otimes_R\kappa(P))$ and so is a local ring, geometrically regular over $\kappa(P)$, in particular, a regular local ring and, in particular, a domain.
(2) Assume, in addition, that $H_i$ is a minimal prime of $P\hat R$. Since $\frac{\hat R_{H_{i+1}}}{P\hat R_{H_{i+1}}}$ is a domain, $H_i$ is the only minimal prime of $P\hat R$, contained in $H_{i+1}$. We have $P\hat R_{H_{i+1}}=H_i\hat R_{H_{i+1}}$.
\end{document}
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\begin{document}
\title[Unitary groups and ramified extensions]{Unitary groups and ramified extensions}
\author{J. Cruickshank} \address{School of Mathematics, Statistics and Applied Mathematics , National University of Ireland, Galway, Ireland} \email{[email protected]}
\author{F. Szechtman} \address{Department of Mathematics and Statistics, Univeristy of Regina, Canada} \email{[email protected]} \thanks{The second author was supported in part by an NSERC discovery grant}
\subjclass[2010]{15A21, 15A63, 11E39, 11E57, 20G25}
\keywords{unitary group; skew-hermitian form; local ring}
\begin{abstract} We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in question is finite. \end{abstract}
\maketitle
\section{Introduction}
More than half a century ago \cite{D} offered a systematic study of unitary groups, as well as other classical groups, over fields and division rings. Thirty five years later, \cite{HO} expanded this study to fairly general classes of rings. In particular, the normal structure, congruence subgroup, and generation problems for unitary (as well as other classical) groups are addressed in \cite{HO} in great generality. In contrast, the problem of determining the order of a unitary group appears in \cite{HO} only in the classical case of finite fields, as found in \cite[\S 6.2]{HO}.
General formulae for the orders of unitary groups defined over a finite ring where 2 is unit were given later in \cite{FH}. The proofs in \cite{FH} are fairly incomplete and, in fact, the formulae in \cite[Theorem 3]{FH} are incorrect when the involutions induced on the given residue fields are not the identity (even the order of the classical unitary groups defined over finite fields is wrong in this case). The argument in \cite[Theorem 3]{FH} is primarily based on a reduction homomorphism, stated without proof to be surjective.
Recently, the correct orders of unitary groups defined over a finite local ring where 2 is invertible were given in \cite{CHQS}, including complete proofs. It should be noted that the forms underlying these groups were taken to be hermitian, which ensured the existence of an orthogonal basis. Moreover, the unitary groups from \cite{CHQS} were all extensions of orthogonal or unitary groups defined over finite fields.
In the present paper we study the unitary group $U_n(A)$ associated to a non-degenerate skew-hermitian form $h:V\times V\to A$ defined on a free right $A$-module $V$ of finite rank $n$, where $A$ is a local ring, not necessarily commutative, endowed with an involution $*$ that satisfies $a-a^*\in{\mathfrak r}$ for all $a\in A$, and ${\mathfrak r}$ is the Jacobson radical of $A$. It is also assumed that ${\mathfrak r}$ is nilpotent and $2\in U(A)$, the unit group of $A$. These conditions occur often, most commonly when dealing with ramified quadratic extensions of quotients of local principal ideal domains with residue field of characteristic not 2 (see Example \ref{tresdos} for more details). A distinguishing feature of this case, as opposed to that of \cite{CHQS}, is that $h(v,v)$ is a non-unit for every $v\in V$. In particular, $V$ lacks an orthogonal basis. As the existence of an orthogonal basis is the building block of the theory developed in \cite{CHQS}, virtually all arguments from \cite{CHQS} become invalid under the present circumstances. Moreover, it turns out that now $n=2m$ must be even and, when $A$ is finite, $U_{2m}(A)$ is an extension of the
\emph{symplectic} group ${\mathrm {Sp}}_{2m}(q)$ defined over the residue field $F_q=A/{\mathfrak r}$. In view of the essential differences between the present case and that of \cite{CHQS}, we hereby develop, from the beginning, the tools required to compute $|U_{2m}(A)|$ when $A$ is finite and the above hypotheses apply. In particular, a detailed and simple proof that the reduction homomorphism is surjective is given.
The paper is essentially self-contained and its contents are as follows. It is shown in \S\ref{s1} that $n=2m$ must be even, and that $V$ admits a basis relative to which the Gram matrix of $h$ is equal to $$ J=\left(
\begin{array}{cc}
0 & 1 \\
-1 & 0 \\
\end{array}
\right), $$ where all blocks have size $m\times m$. Thus, $U_{2m}(A)$ consists of all $X\in {\mathrm{ GL}}_{2m}(A)$ such that $$ X^*JX=J, $$ where $(X^*)_{ij}=(X_{ji})^*$. We prove in \S\ref{s2} that $U_{2m}(A)$ acts transitively on basis vectors of $V$ having the same length. An important tool is found in \S\ref{s3}, namely the fact that the canonical reduction map $U_{2m}(A)\to U_{2m}(A/{\mathfrak i})$ is surjective, where ${\mathfrak i}$ is a $*$-invariant ideal of $A$ (the proof of the corresponding result from \cite{CHQS} makes extensive use of the fact that an orthogonal basis exists). The surjectivity of the reduction map allows us to compute, in \S\ref{s4}, the order of $U_{2m}(A)$ when $A$ is finite, by means of a series of reductions (a like method was used in \cite{CHQS}). We find that \begin{equation}\label{or1}
|U_{2m}(A)|=|{\mathfrak r}|^{2m^2-m}|{\mathfrak m}|^{2m} |{\mathrm {Sp}}_{2m}(q)|, \end{equation} where ${\mathfrak m}=R\cap{\mathfrak r}$ and $R$ is the additive group of all $a\in A$ such that $a^*=a$. We also obtain in~\S\ref{s4} the order of the kernel, say $U_{2m}({\mathfrak i})$, of the reduction map $U_{2m}(A)\to U_{2m}(A/{\mathfrak i})$. Here $U_{2m}({\mathfrak i})$ consists of all $1+X\in U_{2m}(A)$ such that $X\in M_{2m}({\mathfrak i})$. When ${\mathfrak i}\neq A$, we obtain \begin{equation}\label{or3}
U_{2m}({\mathfrak i})=|{\mathfrak i}|^{2m^2-m}|{\mathfrak i}\cap{\mathfrak m}|^{2m}. \end{equation}
A totally independent way of computing $|U_{2m}(A)|$ is offered in \S\ref{s5}, where we show that \begin{equation}\label{or2}
|U_{2m}(A)|=\frac{|{\mathfrak r}|^{m(m+1)}|A|^{m^2} (q^{2m}-1)(q^{2(m-1)}-1)\cdots (q^2-1)}{|S|^{2m}}. \end{equation} Here $S$ stands for the additive group of all $a\in A$ such that $a^*=-a$. It should be noted that (\ref{or1})-(\ref{or2}) are valid even when $A$ is neither commutative nor principal. We also prove in \S\ref{s5} that the number of basis vectors of $V$ of any given length is independent of this length and equal~to $$
(|A|^{2m}-|{\mathfrak r}|^{2m})/|S|. $$ In this regard, in \S\ref{s6} we demonstrate that the order of the stabilizer, say $S_v$, in $U_{2m}(A)$ of a basis vector $v$ is independent of $v$ and its length, obtaining $$
|S_v|=|U_{2(m-1)}(A)|\times |A|^{2m-1}/|S|. $$ We end the paper in \S\ref{s7}, where a refined version of (\ref{or1}) and (\ref{or2}) is given when $A$ is commutative and principal.
Virtually all the above material will find application in a forthcoming paper on the Weil representation of $U_{2m}(A)$.
\section{Non-degenerate skew-hermitian forms}\label{s1}
Let $A$ be a ring with $1\neq 0$. The Jacobson radical of $A$ will be denoted by ${\mathfrak r}$ and the unit group of $A$ by $U(A)$. We assume that $A$ is endowed with an involution $*$, which we interpret to mean an antiautomorphism of order $\leq 2$. Note that if $*=1_A$ then $A$ is commutative. Observe also that ${\mathfrak r}$ as well as all of its powers are $*$-invariant ideals of $A$.
We fix a right $A$-module $V$ and we view its dual $V^*$ as a right $A$-module via $$ (\alpha a)(v)=a^* \alpha(v),\quad v\in V,a\in A,\alpha\in V^*. $$ We also fix a skew-hermitian form on $V$, that is, a function $h:V\times V\to A$ that is linear in the second variable and satisfies $$ h(v,u)=-h(u,v)^*,\quad u,v\in V. $$ In particular, we have $$ h(u+v,w)=h(u,w)+h(v,w)\text{ and }h(ua,v)=a^*h(u,v),\quad u,v,w\in V,a\in A. $$ Associated to $h$ we have an $A$-linear map $T_h:V\to V^*$, given by $$ T_h(v)=h(v,-),\quad v\in V. $$ We assume that $h$ is non-degenerate, in the sense that $T_h$ is an isomorphism.
\begin{lemma}\label{deco} Suppose $V=U\perp W$, where $U,W$ are submodules of $V$. Then the restriction $h_U$ of $h$ to $U\times U$ is non-degenerate. \end{lemma}
\begin{proof} Suppose $T_{h_U}(u)=0$ for some $u\in U$. Since $h(u,W)=0$ and $V=U+W$, it follows that $h(u,V)=0$, so $u=0$ by the non-degeneracy of $h$. This proves that $T_{h_U}$ is injective.
Suppose next that ${\alpha}\in U^*$. Extend ${\alpha}$ to ${\beta}\in V^*$ via ${\beta}(u+w)={\alpha}(u)$. Since $h$ is non-degenerate, there is $v\in V$ such that ${\beta}=T_h(v)$. Now $v=u+w$ for some $u\in U$ and $w\in W$. We claim that ${\alpha}=T_{h_U}(u)$. Indeed, given any $z\in U$, we have $$ [T_{h_U}(u)](z)=h_U(u,z)=h(u,z)=h(u+w,z)=h(v,z)={\beta}(z)={\alpha}(z). $$ \end{proof}
The Gram matrix $M\in M_k(A)$ of a list of vectors $v_1,\dots,v_k\in V$ is defined by $M_{ij}=(v_i,v_j)$.
\begin{lemma}\label{gram} Suppose the Gram matrix of $u_1,\dots,u_k\in A$, say $M$, is invertible. Then $u_1,\dots,u_k$ are linearly independent. \end{lemma}
\begin{proof} Suppose $u_1a_1+\cdots+u_ka_k=0$. Then $h(u_i,u_1)a_1+\cdots+h(u_i,u_k)a_k=0$ for every $1\leq i\leq k$. Since $M$ is invertible, we deduce that $a_1=\cdots=a_k=0$. \end{proof}
We make the following assumptions on $A$ for the remainder of the paper:
(A1) $A$ is a local ring (this means that $A/{\mathfrak r}$ is a division ring or, alternatively, that every element of $A$ is either in $U(A)$ or in ${\mathfrak r}$).
(A2) $2\in U(A)$.
(A3) If $a\in A$ and $a^*=-a$ then $a\in{\mathfrak r}$; in particular, $b-b^*\in{\mathfrak r}$ for all $b\in A$.
(A4) ${\mathfrak r}$ is nilpotent; the nilpotency degree of ${\mathfrak r}$ will be denoted by $e$.
\begin{exa}\label{tresdos}{\rm Let $B$ be a local commutative ring with nilpotent Jacobson radical ${\mathfrak b}$. Suppose $2\in U(B)$ and let ${\sigma}$ be an automorphism of $B$ of order $\leq 2$. Consider the twisted polynomial ring $C=B[t; {\sigma}]$. Then $C$ has a unique involution $*$ that sends $t$ to $-t$ and fixes every $b\in B$. Thus $$ (b_0+b_1t+b_2 t^2+b_3 t^3+\cdots)^*=b_0-b_1^{\sigma} t+b_2 t^2-b_3^{\sigma} t^3+\cdots{\rm ( finite\; sum)} $$ Given any $b\in{\mathfrak b}$, the ideal $(t^2-b)$ of $C$ is $*$-invariant, so the quotient ring $A=C/(t^2-b)$ inherits an involution from $C$ and satisfies all our requirements. Note that if ${\sigma}\neq 1_B$ then $A$ is not commutative. Two noteworthy special cases are the following:
$\bullet$ Let $D$ be a local principal ideal domain with finite residue field of odd characteristic and let $B$ be a quotient of $D$ by a positive power of its maximal ideal; take ${\sigma}=1_B$ and let $b$ be a generator of ${\mathfrak b}$. Then $A$ is a finite, commutative, principal ideal, local ring.
$\bullet$ Let $B$ be a finite, commutative, principal ideal, local ring with Jacobson radical ${\mathfrak b}$. Suppose $2\in U(B)$ and let ${\sigma}\neq 1$ be an automorphism of $B$ of order 2 (as an example, take $A$ and its involution, as in the previous case). Take $b$ to be a generator of ${\mathfrak b}$ and let $a=t+(t^2-b)\in A$. Then $Aa=aA$ is the Jacobson radical of $A$. Moreover, every left (resp. right) ideal of $A$ is a power of $Aa$ (and hence an ideal). Furthermore, note that $A/Aa\cong B/Bb$.} \end{exa}
We will also make the following assumption on $V$ for the remainder of the paper:
(A5) $V$ is a free $A$-module of finite rank $n>0$ (reducing $V$ modulo ${\mathfrak r}$, we see the rank of $A$ is well-defined).
In what follows we write $(u,v)$ instead of $h(u,v)$.
\begin{lemma}\label{basis} Any linearly independent list of vectors from $V$ is part of a basis; if the list has $n$ vectors, it is already a basis, and no list has more than $n$ vectors. \end{lemma}
\begin{proof} Suppose $u_1,\dots,u_k\in V$ are linearly independent and $v_1,\dots,v_n$ is a basis of $V$. By (A4) there is some $r\neq 0$ in ${\mathfrak r}^{e-1}$ such that ${\mathfrak r} r=0$. Write $u_1=v_1a_1+\cdots+v_na_n$, where $a_i\in A$. If all $a_i\in{\mathfrak r}$ then $u_1r=0$, contradicting linear independence. By (A1), we may assume without loss of generality that $a_1\in U(A)$, whence $u_1,v_2,\dots,v_n$ is a basis of $V$. Next write $u_2=u_1b_1+v_2b_2+\cdots+v_nb_n$, where $b_i\in A$. If $b_i\in{\mathfrak r}$ for all $i>1$ then $u_1(-b_1r)+u_2r=0$, contradicting linear independence. This process can be continued and the result follows. \end{proof}
\begin{cor}\label{bas} If $v_1,\dots,v_n$ is a basis of $V$, ${\mathfrak i}$ is a proper ideal of $A$, and $v_i\equiv u_i\mod V{\mathfrak i}$ for all $1\leq i\leq n$, then $u_1,\dots,u_n$ is also a basis of $V$. \end{cor}
\begin{proof} Let $M$ (resp. $N$) be the Gram matrix of $v_1,\dots,v_n$ (resp. $u_1,\dots,u_n$). Then, by assumption, $N=M+P$, for some $P\in M_n({\mathfrak i})$, so $P\in M_n({\mathfrak r})$ by (A1). It is well-known (see \cite[Theorem 1.2.6]{H}) that $M_n({\mathfrak r})$ is the Jacobson radical of $M_n(A)$. On the other hand, by assumption and Lemma \ref{gram}, $M\in{\mathrm{ GL}}_n(A)$. It follows that $N\in{\mathrm{ GL}}_n(A)$ as well. \end{proof}
\begin{lemma}\label{basin} Let $v_1,\dots,v_n$ be a basis of $V$. Then, given any $1\leq i\leq n$, there exists $1\leq j\leq n$ such that $j\neq i$ and $(v_i,v_j)\in U(A)$. \end{lemma}
\begin{proof} Suppose, if possible, that $(v_i,v_j)\notin U(A)$ for all $j\neq i$. Then by (A1), $(v_i,v_j)\in {\mathfrak r}$ for all $j\neq i$. Since $(v_i,v_i)\in{\mathfrak r}$ by (A3), it follows that $(v_i,V)\in{\mathfrak r}$. As $v_1,\dots,v_n$ span $V$ and $T_h:V\to V^*$ is surjective, every linear functional $V\to A$ has values in ${\mathfrak r}$, a contradiction. \end{proof}
\begin{cor}\label{basincor} If $v\in V$ is a basis vector, there is some $w\in V$ such that $(v,w)=1$.\qed \end{cor}
\begin{lemma}\label{uno} Let ${\mathfrak i}$ be a proper ideal of $A$. Suppose $v\in V$ is a basis vector such that $(v,v)\in {\mathfrak i}$. Then there is a basis vector $z\in V$ such that $v\equiv z\mod V{\mathfrak i}$ and $(z,z)=0$. \end{lemma}
\begin{proof} It follows from (A1) and (A4) that ${\mathfrak i}$ is nilpotent, and we argue by induction on the nilpotency degree, say $f$, of ${\mathfrak i}$. If $f=1$ then ${\mathfrak i}=0$ and we take $z=v$. Suppose $f>1$ and the result is true for ideals of nilpotency degree less than $f$. By Corollary \ref{basincor}, there is $w\in V$ such that $(v,w)=1$. Observing that the Gram matrix of $v,w$ is invertible, Lemmas \ref{gram} and \ref{basis} imply that $v,w$ belong to a common basis of $V$. Then, for any $b\in A$, $v+wb$ is also a basis vector and, moreover, $$ (v+wb,v+wb)=(v,v)+b^*(w,w)b+b-b^*. $$ Thanks to (A2), we may take $b=-(v,v)/2\in {\mathfrak i}$. Then $b^*=-b$, so $b-b^*=-(v,v)$ and $$ (v+wb,v+wb)=b^*(w,w)b=-b(w,w)b\in {\mathfrak i}^2. $$ As the nilpotency degree of ${\mathfrak i}^2$ is less than $f$, by induction hypothesis there is a basis vector $z\in V$ such that $v+wb\equiv z\mod V{\mathfrak i}^2$ and $(z,z)=0$. Since $v\equiv v+wb\equiv z\mod V{\mathfrak i}$, the result follows. \end{proof}
\begin{lemma}\label{tres} Let ${\mathfrak i}$ be a proper ideal of $A$. Suppose $u,v\in V$ satisfy $(u,v)\equiv 1\mod {\mathfrak i}$. Then there is $z\in V$ such that $z\equiv v\mod V{\mathfrak i}$ and $(u,z)=1$. \end{lemma}
\begin{proof}
Since $(u,v) \equiv 1 \mod {\mathfrak i}$, $(u,v)$ must be a unit. Moreover,
$(u,v)^{-1} \equiv 1 \mod {\mathfrak i}$, so we can take $z = v(u,v)^{-1}$. \end{proof}
\begin{lemma}\label{dos} Let ${\mathfrak i}$ be an ideal of $A$. Suppose $u,v\in V$ satisfy $(u,u)=0$, $(u,v)=1$ and $(v,v)\in {\mathfrak i}$. Then there is $z\in V$ such that $z\equiv v\mod V{\mathfrak i}$, $(u,z)=1$ and $(z,z)=0$. \end{lemma}
\begin{proof} Set $b=(v,v)/2\in {\mathfrak i}$ and $z=ub+v$. Then $b^*=-b$, so that $b^*-b=-(v,v)$ and $$ (z,z)=(ub+v,ub+v)=b^*-b+(v,v)=0,\quad (u,z)=(u,u)b+(u,v)=1. $$ \end{proof}
A symplectic basis of $V$ is a basis $u_1,\dots,u_m,v_1,\dots,v_m$ such that $(u_i,u_j)=0=(v_i,v_j)$ and $(u_i,v_j)=\delta_{ij}$. A pair of vectors $u,v$ of $V$ is symplectic if $(u,v)=1$ and $(u,u)=0=(v,v)$.
\begin{lemma}\label{exte} Suppose the Gram matrix, say $M$, of $v_1,\dots,v_s$ is invertible. If $s<n$ then there is a basis $v_1,\dots,v_s,w_1,\dots,w_t$ of $V$ such that $(v_i,w_j)=0$ for all $i$ and $j$. \end{lemma}
\begin{proof} By Lemmas \ref{gram} and \ref{basis}, there is a basis $v_1,\dots,v_s,u_1,\dots,u_t$ of $V$. Given $1\leq i\leq t$, we wish to find $a_1,\dots,a_s$ so that $w_i=u_i-(v_1a_1+\cdots+v_s a_s)$ is orthogonal to all $v_j$. This means $$ (v_j,v_1)a_1+\cdots+(v_j,v_s)a_s=(v_j,u_i),\quad 1\leq j\leq s. $$ This linear system can be solved by means of $M^{-1}$. Since $v_1,\dots,v_s,w_1,\dots,w_t$ is a basis of $V$, the result follows. \end{proof}
\begin{prop}\label{sp} $V$ has a symplectic basis; in particular $n=2m$ is even. \end{prop}
\begin{proof} Let $w_1,\dots,w_n$ be a basis of $V$, whose existence is guaranteed by (A5). We argue by induction on $n$.
By (A3), $(w_1,w_1)\in{\mathfrak r}$. We infer from Lemma \ref{uno} the existence of a basis vector $u\in V$ such that $(u,u)=0$. By Corollary \ref{basincor}, there is $w\in V$ such that $(u,w)=1$. By Lemma \ref{dos}, there is $v\in V$ such that $(v,v)=0$ and $(u,v)=1$.
It follows from Lemmas \ref{gram} and \ref{basis} that $u,v$ is part of a basis of $V$. If $n=2$ we are done. Suppose $n>2$ and the result is true for smaller ranks. By Lemma \ref{exte} there is basis $u,v,w_1,\dots,w_{n-2}$ of $V$ such that every $w_i$ is orthogonal to both $u$ and $v$. Let $U$ be the span of $u,v$ and let $W$ be the span of $w_1,\dots,w_{n-2}$. By Lemma \ref{deco}, the restriction of $h$ to $W$ is non-degenerate. By inductive assumption, $n-2$ is even, say $n-2=2(m-1)$, and $W$ has a symplectic basis, say $u_2,\dots,u_m,v_2,\dots,v_m$. It follows that $u_1,u_2,\dots,u_m,v,v_2,\dots,v_m$ is a symplectic basis of $V$. \end{proof}
\begin{cor} All non-degenerate skew-hermitian forms on $V$ are equivalent.\qed \end{cor}
\section{The unitary group acts transitively on basis vectors of the same length}\label{s2}
By definition, the unitary group associated to $(V,h)$ is the subgroup, say $U(V,h)$, of ${\mathrm{ GL}}(V)$ that preserves~$h$. Thus, $U(V,h)$ consists of all $A$-linear automorphisms $g:V\to V$ such that $h(gu,gv)=h(u,v)$ for all $u,v\in V$.
\begin{theorem}\label{actra} Let $u,v\in V$ be basis vectors satisfying $(u,u)=(v,v)$. Then there exists $g\in U(V,h)$ such that $gu=v$. \end{theorem}
\begin{proof} As $u$ is a basis vector, there is a vector $u'\in V$ such that $(u,u')=1$. Let $W$ be the span of $u,u'$. By Lemma \ref{exte}, $V=W\oplus W^\perp$. The restrictions of $h$ to $W$ and $W^\perp$ are non-degenerate by Lemma \ref{deco}. A like decomposition exists for $v$. Thus, by means of Proposition \ref{sp}, we may restrict to the case $n=2$.
By Proposition \ref{sp}, there is a symplectic basis $x,y$ of $V$ and we have $u=xa+yb$ for some $a,b\in A$. Since $u$ is a basis vector, one of these coefficients, say $a$ is a unit. Replacing $x$ by $xa$ and $y$ by $y(a^*)^{-1}$, we may assume that $u=x+yb$ for some $b\in A$, where $x,y$ is still a symplectic basis of $V$. We have $b-b^*=(u,u)$. Likewise, there is a symplectic basis $w,z$ of $V$ such that $v=w+zc$, where $c-c^*=(v,v)=(u,u)=b-b^*$. It follows that $c=b+r$ for some $r\in A$ such that $r^*=r$.
Replace $w,z$ by $w-zr,z$. This basis of $V$ is also symplectic, since $(w-zr,w-zr)=0$ (because $r-r^*=0$). Moreover, $v=(w-zr)+z(c+r)=(w-rz)+zb$. Thus, $u$ and $v$ have exactly the same coordinates, namely $1,b$ relative to some symplectic bases of $V$. Let $g\in U$ map one symplectic basis into the other one. Then $gu=v$, as required. \end{proof}
\section{Reduction modulo a $*$-invariant ideal}\label{s3}
\begin{lemma}\label{util} Let ${\mathfrak i}$ be a proper ideal of $A$. Suppose $w_1,\dots,w_m,z_1,\dots,z_m\in V$ satisfy $$ (w_i,z_j)\equiv\delta_{ij}\mod{\mathfrak i},\; 1\leq i\leq m,\quad (w_i,w_j)\equiv 0\equiv (z_i,z_j)\mod{\mathfrak i},\; 1\leq i,j\leq m. $$ Then there exists a symplectic basis $w'_1,\dots,w'_m,z'_1,\dots,z'_m$ of $V$ such that $$ w_i\equiv w'_i\mod V{\mathfrak i},\quad z_i\equiv z'_i\mod V{\mathfrak i},\; 1\leq i\leq m. $$ \end{lemma}
\begin{proof} By induction on $m$. Successively applying Lemmas \ref{uno}, \ref{tres} and \ref{dos} we obtain $w'_1,z'_1\in V$ such that $$ w_1\equiv w'_1 \mod V{\mathfrak i},\quad z_1\equiv z'_1 \mod V{\mathfrak i},\quad (w'_1,w'_1)=0=(z'_1,z'_1),\quad (w'_1,z'_1)=1. $$ If $m=1$ we are done. Suppose $m>1$ and the result is true for smaller ranks. Applying Corollary \ref{bas}, we see that $w'_1,z'_1,w_2\dots,w_m,z_2,\dots,z_m$ is a basis of $V$. Applying the procedure of Lemma \ref{exte} we obtain a basis $w'_1,z'_1,w^0_2\dots,w^0_m,z^0_2,\dots,z^0_m$ of $V$ such that $w'_1,z'_1$ are orthogonal to all other vectors in this list. Since $(x,y)\in{\mathfrak i}$ when $x\in\{w'_1,z'_1\}$ and $y\in\{w_2\dots,w_m,z_2,\dots,z_m\}$, the proof of Lemma \ref{exte} shows that $$ w^0_i\equiv w_i\mod V{\mathfrak i},\quad z^0_i\equiv z_i\mod V{\mathfrak i},\quad 2\leq i\leq m. $$ Therefore $$ (w^0_i,z^0_j)\equiv\delta_{ij}\mod{\mathfrak i},\; 2\leq i\leq m,\quad (w^0_i,w^0_j)\equiv 0\equiv (z^0_i,z^0_j)\mod{\mathfrak i},\; 2\leq i,j\leq m. $$ By Lemma \ref{deco}, the restriction of $h$ to the span, say $W$, of $w^0_2\dots,w^0_m,z^0_2,\dots,z^0_m$ is non-degenerate. By induction hypothesis, there is a symplectic basis $w'_2,\dots,w'_m,z'_2,\dots,z'_m$ of $W$ such that $$ w'_i\equiv w^0_i\mod V{\mathfrak i}, \quad z'_i\equiv z^0_i\mod V{\mathfrak i},\; 2\leq i\leq m. $$ Then $w'_1,\dots,w'_m,z'_1,\dots,z'_m$ is a basis of $V$ satisfying all our requirements. \end{proof}
Let ${\mathfrak i}$ be a $*$-invariant ideal of $A$. Then $\overline{A}=A/{\mathfrak i}$ inherits an involution, also denoted by $*$, from $A$ by declaring $(a+{\mathfrak i})^*=a^*+{\mathfrak i}$. This is well-defined, since ${\mathfrak i}$ is $*$-invariant. Set $\overline{V}=V/V{\mathfrak i}$ and consider the skew-hermitian form $\overline{h}:\overline{V}\times \overline{V}\to \overline{A}$, given by $\overline{h}(u+V{\mathfrak i},v+V{\mathfrak i})=h(u,v)+{\mathfrak i}$. We see that $\overline{h}$ is well-defined and non-degenerate. We then have a group homomorphism $U(V,h)\to U(\overline{V},\overline{h})$, given by $g\mapsto \overline{g}$, where $\overline{g}(u+V{\mathfrak i})=g(u)+V{\mathfrak i}$.
\begin{theorem}\label{sur} Let ${\mathfrak i}$ be a proper $*$-invariant ideal of $A$. Then the canonical group homomorphism $U(V,h)\to U(\overline{V},\overline{h})$ is surjective. \end{theorem}
\begin{proof} Let $f\in U(\overline{V},\overline{h})$. By Proposition \ref{sp}, $V$ has a symplectic basis $u_1,\dots,u_m,v_1,\dots,v_m$. We have $f(u_i+V{\mathfrak i})=w_i+V{\mathfrak i}$ and $f(v_i+V{\mathfrak i})=z_i+V{\mathfrak i}$ for some $w_i,z_i\in V$ and $1\leq i\leq m$. Since $f$ preserves $\overline{h}$, we must have $$ (w_i,z_j)\equiv\delta_{ij}\mod{\mathfrak i},\; 1\leq i\leq m,\quad (w_i,w_j)\equiv 0\equiv (z_i,z_j)\mod{\mathfrak i},\; 1\leq i,j\leq m. $$ By Lemma \ref{util} there is a symplectic basis $w'_1,\dots,w'_m,z'_1,\dots,z'_m$ of $V$ such that $$ w_i\equiv w'_i\mod V{\mathfrak i},\quad z_i\equiv z'_i\mod V{\mathfrak i},\; 1\leq i\leq m. $$ Let $g\in U(V,h)$ map $u_1,\dots,u_m,v_1,\dots,v_m$ into $w'_1,\dots,w'_m,z'_1,\dots,z'_m$. Then $\overline{g}=f$. \end{proof}
\section{Computing the order of the $U_{2m}(A)$ by successive reductions}\label{s4}
We refer to an element $a$ of $A$ as hermitian (resp. skew-hermitian) if $a=a^*$ (resp. $a=-a^*$). Let $R$ (resp. $S$) be the subgroup of the additive group of $A$ of all hermitian (resp. skew-hermitian) elements. We know by (A3) that \begin{equation} \label{sr} S\subseteq {\mathfrak r}. \end{equation} Moreover, it follows from (A2) that \begin{equation} \label{ars} A=R\oplus S. \end{equation} Letting $${\mathfrak m}=R\cap{\mathfrak r},$$ we have a group imbedding $R/{\mathfrak m}\hookrightarrow A/{\mathfrak r}$. In fact, we deduce from (\ref{sr}) and (\ref{ars}) that \begin{equation} \label{ars2} A/{\mathfrak r}\cong R/{\mathfrak m}. \end{equation}
\begin{lem} \label{cuad}
Suppose ${\mathfrak i}$ is a $*$-invariant ideal of $A$ satisfying ${\mathfrak i}^2=0$. Let $\{v_1,\dots,v_n\}$ be a basis of $V$ and let $J$ be the Gram matrix of $v_1,\dots,v_n$. Then, relative to $\{v_1,\dots,v_n\}$, the kernel of the canonical epimorphism $U(V,h)\to U(\overline{V},\overline{h})$ consists of all matrices $1+M$, such that $M\in M_n({\mathfrak i})$ and \begin{equation} \label{anr} M^*J+JM=0. \end{equation} \end{lem}
\begin{proof} By definition the kernel of $U(V,h)\to U(\overline{V},\overline{h})$ consists of all matrices of the form $1+M$, where $M\in M_n({\mathfrak i})$ and $$ (1+M)^* J(1+M)=J. $$ Expanding this equation and using ${\mathfrak i}^2=0$ yields (\ref{anr}). \end{proof}
Let $\{u_1,\dots,u_m,v_1,\dots,v_m\}$ be a symplectic basis of $V$. We write $U_{2m}(A)$ for the image of $U(V,h)$ under the group isomorphism $GL(V)\to{\mathrm{ GL}}_{2m}(A)$ relative to $\{u_1,\dots,u_m,v_1,\dots,v_m\}$.
We make the following assumption on $A$ for the remainder of the paper:
(A6) $A$ is a finite ring.
We deduce from (\ref{ars}) that
$$|A|=|R||S|.$$ On the other hand, it follows from (A1), (A2) and (A6) that $F_q=A/{\mathfrak r}$ is a finite field of odd characteristic. By (A3), $a+{\mathfrak r}=a^* +{\mathfrak r}$ for all $a\in A$, so the involution that $*$ induces on $F_q$ is the identity. Taking ${\mathfrak i}={\mathfrak r}$ in Theorem \ref{sur}, we have ${U}_{2m}(\overline{A})={\mathrm {Sp}}_{2m}(q)$, the symplectic group of rank $2m$ over $F_q$. Recall \cite[Chapter 8]{T} that $$
|{\mathrm {Sp}}_{2m}(q)|=(q^{2m}-1)q^{2m-1}(q^{2(m-1)}-1)q^{2m-3}\cdots (q^2-1)q=q^{m^2}(q^{2m}-1)(q^{2(m-1)}-1)\cdots (q^{2}-1). $$
\begin{cor} \label{cuad2} Suppose ${\mathfrak i}$ is a $*$-invariant ideal of $A$ satisfying ${\mathfrak i}^2=0$. Then the kernel of
$U(V,h)\to U(\overline{V},\overline{h})$ has order $|{\mathfrak i}|^{2m^2-m}|{\mathfrak i}\cap {\mathfrak m}|^{2m}$. \end{cor}
\begin{proof} Let $\{u_1,\dots,u_m,v_1,\dots,v_m\}$ be a symplectic basis of $V$. Thus, the Gram matrix, say $J\in M_{2m}(A)$, of $u_1,\dots,u_m,v_1,\dots,v_m$ is $$ J=\left(
\begin{array}{cc}
0 & 1 \\
-1 & 0 \\
\end{array}
\right), $$ where all blocks are in $M_{m}(A)$. According to Lemma \ref{cuad}, the kernel of $U(V,h)\to U(\overline{V},\overline{h})$ consists of all $1+M$, where $$ M=\left(
\begin{array}{cc}
P & Q \\
T & S \\
\end{array}
\right), $$ and $S=-P^*$, $Q=Q^*$ and $T=T^*$, which yields the desired result. \end{proof}
\begin{thm} \label{zxz} Let $A$ be a finite local ring, not necessarily commutative, with Jacobson radical~${\mathfrak r}$ and residue field $F_q$ of odd characteristic. Suppose $A$ has an involution $*$ such that $a-a^*\in{\mathfrak r}$ for all $a\in A$. Let ${\mathfrak m}$ be the group of all $a\in{\mathfrak r}$ such that $a=a^*$. Then $$
|U_{2m}(A)|=|{\mathfrak r}|^{2m^2-m}|{\mathfrak m}|^{2m} q^{m^2}(q^{2m}-1)(q^{2(m-1)}-1)\cdots (q^2-1). $$ \end{thm}
\begin{proof} Consider the rings $$ A=A/{\mathfrak r}^{e}, A/{\mathfrak r}^{e-1},\dots,A/{\mathfrak r}^2,A/{\mathfrak r}. $$ Each of them is a factor of $A$, so is local and inherits an involution from $*$. Each successive pair is of the form $C=A/{\mathfrak r}^k,D=A/{\mathfrak r}^{k-1}$, where the kernel of the canonical epimorphism $C\to D$ is ${\mathfrak j}={\mathfrak r}^{k-1}/{\mathfrak r}^k$, so that ${\mathfrak j}^2=0$. We may thus apply Theorem \ref{sur} and Corollary \ref{cuad2} $e-1$ times to obtain the desired result, as follows. We have $$
|{\mathfrak r}|=|{\mathfrak r}^{e-1}/{\mathfrak r}^e|\cdots |{\mathfrak r}/{\mathfrak r}^2| $$ and $$
|{\mathfrak m}|=|{\mathfrak m}\cap{\mathfrak r}^{e-1}/{\mathfrak m}\cap {\mathfrak r}^e|\cdots |{\mathfrak m}\cap{\mathfrak r}^{k-1}/{\mathfrak m}\cap {\mathfrak r}^{k}|\cdots |{\mathfrak m}\cap {\mathfrak r}/{\mathfrak m}\cap {\mathfrak r}^2|, $$ where the group of hermitian elements in the kernel of $C\to D$
has $|{\mathfrak m}\cap{\mathfrak r}^{k-1}/{\mathfrak m}\cap {\mathfrak r}^{k}|$ elements. Indeed, these elements are those $a+{\mathfrak r}^k$ such that $a\in {\mathfrak r}^{k-1}$ and $a-a^*\in {\mathfrak r}^k$. But $a+a^*$ is hermitian, so $a+a^*\in{\mathfrak m}\cap{\mathfrak r}^{k-1}$. Thus $$a=(a-a^*)/2+(a+a^*)/2\in {\mathfrak r}^k+{\mathfrak m}\cap{\mathfrak r}^{k-1}.$$ Hence the group of hermitian elements in the kernel of $C\to D$ is $$ ({\mathfrak m}\cap{\mathfrak r}^{k-1}+{\mathfrak r}^k)/{\mathfrak r}^k\cong {\mathfrak m}\cap{\mathfrak r}^{k-1}/({\mathfrak m}\cap{\mathfrak r}^{k-1}\cap {\mathfrak r}^k)\cong {\mathfrak m}\cap{\mathfrak r}^{k-1}/{\mathfrak m}\cap{\mathfrak r}^{k}. $$ \end{proof}
\begin{thm} Given a $*$-invariant proper ideal ${\mathfrak i}$ of $A$, the kernel of
$U(V,h)\to U(\overline{V},\overline{h})$ has order $|{\mathfrak i}|^{2m^2-m}|{\mathfrak i}\cap{\mathfrak m}|^{2m}$. \end{thm}
\begin{proof} By Theorems \ref{sur} and \ref{zxz}, the alluded kernel has order $$
\frac{|U(V,h)|}{|U(\overline{V},\overline{h})|}=\frac{|{\mathfrak r}|^{2m^2-m}|{\mathfrak m}|^{2m}}{|{\mathfrak r}/{\mathfrak i}|^{2m^2-m}|({\mathfrak m}+{\mathfrak i})/{\mathfrak i}|^{2m}}=|{\mathfrak i}|^{2m^2-m}|{\mathfrak i}\cap{\mathfrak m}|^{2m}. $$ \end{proof} \section{Computing the order of $U_{2m}(A)$ by counting symplectic pairs}\label{s5}
The following easy observation will prove useful. Given $s\in S$ and $y\in A$, we have \begin{equation} \label{yx} y-y^*=s\text{ if and only if }y\in s/2+R. \end{equation}
By the length of a vector $v\in V$ we understand the element $(v,v)\in S$. Given $s\in S$, the number of basis vectors of $V$ of length $s$ will be denoted by $N(m,s)$.
\begin{lemma}\label{numbers} Given $s\in S$, we have $$
N(1,s)=(|A|-|{\mathfrak r}|)(|R|+|{\mathfrak m}|)=(|A|^2-|{\mathfrak r}|^2)/|S|,\quad s\in S. $$ In particular, $N(1,s)$ is independent of $s$. \end{lemma}
\begin{proof} Let $u,v$ be a symplectic basis of $V$. Given $(a,b)\in A^2$, the length of $w=ua+vb$ is $$ (w,w)=a^* b-b^* a. $$ Thus, we need to count the number of pairs $(a,b)\in A^2\setminus {\mathfrak r}^2$ such that \begin{equation} \label{lens} a^* b-b^* a=s. \end{equation} For this purpose, suppose first $a\in U(A)$. Setting $y=a^* b$ and using (\ref{yx}), we see that (\ref{lens}) holds if and only if $b\in (a^*)^{-1}(s/2+R)$. Thus, the number of solutions $(a,b)\in A^2$ to (\ref{lens}) such that $a\in U(A)$ is
$(|A|-|{\mathfrak r}|)|R|$. Suppose next that $a\notin U(A).$ Then $b \in U(A)$. Rewriting (\ref{lens}) in the form \begin{equation} \label{lens2} b^* a-a^* b=-s \end{equation} and setting $y=b^* a$, we see as above that (\ref{lens2}) holds if and only if $a\in (b^*)^{-1}(-s/2+R)$. Recalling that $a\in{\mathfrak r}$, we are thus led to calculating $$
|[(b^*)^{-1}(-s/2+R)]\cap{\mathfrak r}|=|(-s/2+R)\cap b^*{\mathfrak r}|=|(-s/2+R)\cap {\mathfrak r}|=|R\cap{\mathfrak r}|, $$ the last two equalities holding because $b\in U(R)$ and $s\in{\mathfrak r}$. Recalling that ${\mathfrak m}=R\cap{\mathfrak r}$, it follows that
$N(1,s)=(|A|-|{\mathfrak r}|)(|R|+|{\mathfrak m}|)$. Since this is independent of $s$, we infer $N(1,s)=(|A|^2-|{\mathfrak r}|^2)/|S|$. \end{proof}
Note that the identity $(|A|-|{\mathfrak r}|)(|R|+|{\mathfrak m}|)=(|A|^2-|{\mathfrak r}|^2)/|S|$ also follows from $|R|=|A|/|S|$ and $|{\mathfrak m}|=|{\mathfrak r}|/|S|$.
\begin{prop}\label{igual} Given $s\in S$, we have $$
N(m,s)=(|A|^{2m}-|{\mathfrak r}|^{2m})/|S|,\quad s\in S. $$ In particular, $N(m,s)$ is independent of $s$. \end{prop}
\begin{proof} The first assertion follows from the second. We prove the latter by induction on $m$. The case $m=1$ is done in Lemma
\ref{numbers}. Suppose that $m>1$ and $N(m-1,s)$ is independent of $s$. Set $N=N(1,0)$ and $M=N(m-1,0)$. Decompose $V$ as $U\perp W$ where $U$ has rank 2. Thus $N(m,s)$ is the number of pairs $(u,w) \in U \times W$ such that either $w$ is an arbitrary element of $W$ and $u$ is a basis vector of $U$ of length $s-(w,w)$, or, $u$ is an arbitrary non basis vector of $U$ and $w$ is a basis vector of $W$ of length $s - (u,u)$. These two possibilities are mutually exclusive. It follows, using the inductive hypothesis, that $$
N(m,s)=N|A|^{2(m-1)}+|{\mathfrak r}|^2 M, $$ which is independent of $s$.
\end{proof}
\begin{cor}\label{nusy} The number of symplectic pairs in $V$ is $$
\frac{(|A|^{2m}-|{\mathfrak r}|^{2m})|A|^{2m-1}}{|S|^2}. $$ \end{cor}
\begin{proof} By Proposition \ref{igual}, the number of basis vectors of length 0 is
$(|A|^{2m}-|{\mathfrak r}|^{2m})/|S|$. Given any such vector, say $u$, Lemma \ref{dos} ensures the existence of a vector $v\in V$ of length 0 such that $(u,v)=1$. Then, a vector $w\in V$ satisfies $(u,w)=1$ if and only if $w=au+v+z$, where $z$ is orthogonal to $u,v$. Moreover, given any such $z$, we see that $w$ has length 0 if and only if $$ 0=(w,w)=(ua+v,ua+v)+(z,z)=a^*-a+(z,z). $$
It follows from (\ref{yx}) that the number of solutions $a \in A$ to this equation is $|R|$. We infer that the number of symplectic pairs is $$
\frac{(|A|^{2m}-|{\mathfrak r}|^{2m})}{|S|}\times |A|^{2m-2}|R|=\frac{(|A|^{2m}-|{\mathfrak r}|^{2m})|A|^{2m-1}}{|S|^2}. $$ \end{proof}
It follows from Lemma \ref{exte} and Proposition \ref{sp} that $U_{2m}(A)$ acts transitively on symplectic pairs. Moreover, we readily see that the stabilizer of a given symplectic pair is isomorphic to $U_{2(m-1)}(A)$. We infer from Corollary \ref{nusy} that $$
|U_{2m}(A)|=\frac{(|A|^{2m}-|{\mathfrak r}|^{2m})|A|^{2m-1}}{|S|^2}\times \frac{(|A|^{2(m-1)}-|{\mathfrak r}|^{2(m-1)})|A|^{2m-3}}{|S|^2}\times\cdots\times
\frac{(|A|^{2}-|{\mathfrak r}|^{2})|A|}{|S|^2}. $$ We have proven the following result. \begin{thm} \label{zxz2} Let $A$ be a finite local ring, not necessarily commutative, with Jacobson radical~${\mathfrak r}$ and residue field $F_q$ of odd characteristic. Suppose $A$ has an involution $*$ such that $a-a^*\in{\mathfrak r}$ for all $a\in A$. Let $S$ be the group of all $a\in A$ such that $a=-a^*$. Then $$
|U_{2m}(A)|=\frac{|{\mathfrak r}|^{m(m+1)}|A|^{m^2} (q^{2m}-1)(q^{2(m-1)}-1)\cdots (q^2-1)}{|S|^{2m}}.\qed $$ \end{thm}
\begin{note}{\rm We readily verify, by means of (\ref{ars}) and (\ref{ars2}), the equivalence of the formulae given in Theorems \ref{zxz} and \ref{zxz2}.} \end{note}
\section{The order of the stabilizer of a basis vector}\label{s6}
\begin{thm}\label{lasta} Let $v\in V$ be a basis vector and let $S_v$ be the stabilizer of $v$ in $U(V,h)$. Then $$
|S_v|=|U_{2(m-1)}(A)|\times |A|^{2m-1}/|S|. $$ In particular, the order of $S_v$ is independent of $v$ and its length. \end{thm}
\begin{proof} By Theorem \ref{actra}, the number of basis vectors of length $(v,v)$ is equal to $|U_{2m}(A)|/|S_v|$. It follows from Proposition \ref{igual} that \begin{equation}\label{idw}
|U_{2m}(A)|/|S_v|=(|A|^{2m}-|{\mathfrak r}|^{2m})/|S|. \end{equation} On the other hand, the above discussion shows that \begin{equation}\label{idw2}
|U_{2m}(A)|=\frac{(|A|^{2m}-|{\mathfrak r}|^{2m})|A|^{2m-1}}{|S|^2}\times |U_{2(m-1)}(A)|. \end{equation} Combining (\ref{idw}) and (\ref{idw2}) we obtain the desired result. \end{proof}
\section{The case when $A$ is commutative and principal}\label{s7}
We make the following assumptions on $A$ until further notice:
(A7) There is $a\in{\mathfrak r}$ such that $Aa=aA={\mathfrak r}$.
(A8) The elements of $R$ commute among themselves.
Using (A7), we see that $|A|=q^e$, $|{\mathfrak r}|=q^{e-1}$. Moreover, from $Aa=aA$, we get $a^*A=Aa^*$. Since $a=(a-a^*)/2+(a+a^*)/2$, not both $a-a^*$ and $a+a^*$ can be in ${\mathfrak r}^2$. Thus, ${\mathfrak r}$ has a generator $x$ that is hermitian or skew-hermitian and satisfies $Ax=xA$. In any case, $x^2$ is hermitian. We claim that $$ A=R+Rx. $$ Note first of all that, because of (A8), $R$ is a subring of $A$. Clearly, $R$ is a local ring with maximal ideal ${\mathfrak m}=R\cap{\mathfrak r}$ and residue field $R/{\mathfrak m}\cong A/{\mathfrak r}$. Secondly, from $A = R+S$ and $S\subseteq{\mathfrak r} = Ax$, we deduce \begin{equation} \label{der} A=R+Ax. \end{equation} Repeatedly using (\ref{der}) as well as (A8), we obtain $$ \begin{aligned} A &=R+(R+Ax)x=R+Rx+Ax^2=R+Rx+(R+Ax)x^2\\ &=R+Rx+Ax^3=R+Rx+(R+Ax)x^3=R+Rx+Ax^4=\dots=R+Rx. \end{aligned} $$ If $*=1_A$ then $A=R$ and ${\mathfrak r}={\mathfrak m}$ has $q^{e-1}$ elements. We make the following assumptions on $A$ until further notice:
(A9) $*\neq 1_A$.
(A10) $R\cap Rx=(0)$.
It follows from (A9) and $A=R+Rx$ that $x$ cannot be hermitian. Therefore $x$ is skew-hermitian. Note that $R$ is a principal ring with maximal ideal ${\mathfrak m}=Rx^2$, since $$ {\mathfrak m}=R\cap Ax=R\cap (R+Rx)x=R\cap (Rx+Rx^2)=Rx^2+(R\cap Rx)=Rx^2. $$
\begin{lemma}\label{evod} The group epimorphism $f:R\to Rx$, given by $f(r)=rx$, is injective if $e$ is even, whereas the kernel of $f$ is $Rx^{e-1}$ and has $q$ elements if $e$ is odd. \end{lemma}
\begin{proof} Note that every non-zero element of $A$ is of the form $cx^i$ for some unit $c\in U(A)$ and a unique $0\leq i<e$. It follows that the annihilator of $x$ in $A$ is equal to $Ax^{e-1}$. From $A=R+Rx$, we infer $Ax^{e-1}=Rx^{e-1}$. Thus, the kernel of $f$ is $R\cap Rx^{e-1}$. If $e$ is even then $R\cap Rx^{e-1}\subseteq R\cap Rx=(0)$, while if $e$ is odd $$R\cap Rx^{e-1}=Rx^{e-1}=Ax^{e-1}$$ is a 1-dimensional vector space over $F_q=A/{\mathfrak r}$. \end{proof}
\begin{cor} We have $$
|A|=|R|^2\text{ if }e\text{ is even and }|A|=\frac{|R|^2}{q}\text{ if }e\text{ is odd.} $$ Thus, either $e=2\ell$ is even and $$
|{\mathfrak r}|=q^{2\ell-1},\;|{\mathfrak m}|=q^{\ell-1} $$ or $e=2\ell -1$ is odd and $$
|{\mathfrak r}|=q^{2\ell-2},\; |{\mathfrak m}|=q^{\ell-1}. $$ \end{cor}
\begin{proof} This follows from $A=R\oplus Rx$, Lemma \ref{evod} and the group isomorphism $R/{\mathfrak m}\cong F_q$. \end{proof}
We now resume the general discussion and note that if $A$ is a commutative, principal ideal ring and $*\neq 1_A$ then conditions (A7)-(A10) are automatically satisfied, for in this case we have $R\cap Rx\subseteq R\cap S=(0)$. It is clear that $A\cong R[t]/(t^2-x^2)$ if $e=2\ell$ is even, and $A\cong R[t]/(t^2-x^2,t^{2\ell-1})$ if $e=2\ell-1$ is odd. Using part of the above information together with Theorem \ref{zxz}, we obtain the following result.
\begin{thm} \label{ords} Let $A$ be a finite, commutative, principal, local ring with Jacobson radical~${\mathfrak r}$ and residue field $A/{\mathfrak r}\cong F_q$ of odd characteristic. Let $e$ be the nilpotency degree of ${\mathfrak r}$. Suppose $A$ has an involution $*$ such that $a-a^*\in{\mathfrak r}$ for all $a\in A$.
(a) If $*=1_A$ then $$
|U_{2m}(A)|=|{\mathrm {Sp}}_{2m}(A)|=q^{(e-1)(2m^2+m)+m^2}(q^{2m}-1)(q^{2(m-1)}-1)\cdots (q^2-1). $$ (b) If $*\neq 1_A$ and $e=2\ell$ is even then $$
|U_{2m}(A)|=q^{(2\ell-1)(2m^2-m)}q^{2(\ell-1)m} q^{m^2}(q^{2m}-1)(q^{2(m-1)}-1)\cdots (q^2-1). $$ (b) If $*\neq 1_A$ and $e=2\ell-1$ is odd then $$
|U_{2m}(A)|=q^{(2\ell-2)(2m^2-m)}q^{2(\ell-1)m} q^{m^2}(q^{2m}-1)(q^{2(m-1)}-1)\cdots (q^2-1).\qed $$ \end{thm}
\begin{note}{\rm Our initial conditions on $A$ do not force $R$ to be a subring of $A$ or $R\cap Rx=(0)$. Indeed, let $A$ be as indicated in the parenthetical remark of the second case of Example \ref{tresdos}, and set $x=a$, $r=b$. Then $rx\in R\cap Rx$, so $R\cap Rx\neq (0)$, and $rxr=-r^2x$ with $(-r^2 x)^*=r^2 x$, so $R$ is not a subring of $A$. It is also clear that $A$ need not be principal, even if so is $R$, as can be seen by taking $b=0$ and $B$ not a field in the general construction of Example \ref{tresdos} (e.g. $A=Z_{p^2}[t]/(t^2)$).} \end{note}
\noindent{\bf Acknowledgement.} We are very grateful to the referee for a thorough reading of the paper and valuable suggestions.
\end{document}
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\begin{document}
\title[]{ON STRONG $r$-HELIX SUBMANIFOLDS AND SPECIAL CURVES} \author{Evren Z\i plar} \address{Department of Mathematics, Faculty of Science, University of Ankara, Tando\u{g}an, Turkey} \email{[email protected]} \urladdr{} \author{Ali \c{S}enol} \address{Department of Mathematics, Faculty of Science, \c{C}ank\i r\i\ Karatekin University, \c{C}ank\i r\i , Turkey} \email{[email protected]} \author{Yusuf Yayl\i } \address{Department of Mathematics, Faculty of Science, University of Ankara, Tando\u{g}an, Turkey} \email{[email protected]} \thanks{} \urladdr{} \date{} \subjclass[2000]{ \ 53A04, 53B25, 53C40, 53C50.} \keywords{Strong $r$-helix submanifold; Line of curvature; Geodesic curve; Slant helix.\\ Corresponding author: Evren Z\i plar, e-mail: [email protected]} \thanks{}
\begin{abstract} In this paper, we investigate special curves on a strong $r$-helix submanifold in Euclidean $n$-space $E^{n}$. Also, we give the important relations between strong $r$-helix submanifolds and the special curves such as line of curvature, geodesic and slant helix. \end{abstract}
\maketitle
\section{Introduction}
In differential geometry of manifolds, an helix submanifold of $IR^{n}$ with respect to a fixed direction $d$ in $IR^{n}$ is defined by the property that tangent planes make a constant angle with the fixed direction $d$ (helix direction) in [5]. Di Scala and Ruiz-Hern\'{a}ndez have introduced the concept of these manifolds in [5]. Besides, the concept of strong $r$-helix submanifold of $IR^{n}$ was introduced in [4]. Let $M\subset IR^{n}$ be a submanifold and let $H(M)$ be the set of helix directions of $M$. We say that $M$ is a strong $r$-helix if the set $H(M)$ is a linear subspace of $ IR^{n}$ of dimension greater or equal to $r$ in [4].
Recently, M. Ghomi worked out the shadow problem given by H.Wente. And, He mentioned the shadow boundary in [8]. Ruiz-Hern\'{a}ndez investigated that shadow boundaries are related to helix submanifolds in [12].
Helix hypersurfaces has been worked in nonflat ambient spaces in [6,7]. Cermelli and Di Scala have also studied helix hypersurfaces in liquid cristals in [3].
The plan of this paper is as follows. In section 2, we mention some basic facts in the general theory of strong $r$-helix, manifolds and curves. And, in section 3, we give the important relations between strong $r$-helix submanifolds and some special curves such as line of curvature, geodesic and slant helix.
\section{PRELIMINARIES}
\begin{definition} Let $M\subset IR^{n}$ be a submanifold of a euclidean space. A unit vector $ d\in IR^{n}$ is called a helix direction of $M$ if the angle between $d$ and any tangent space $T_{p}M$ is constant. Let $H(M)$ be the set of helix directions of $M$. We say that $M$ is a strong $r$-helix if $H(M)$ is a $r$ -dimensional linear subspace of $IR^{n}$ [4]. \end{definition}
\begin{definition} A submanifold $M\subset IR^{n}$ is a strong $r$-helix if the set $H(M)$ is a linear subspace of $IR^{n}$ of dimension greater or equal to $r$ [4]. \end{definition}
\begin{definition} A unit speed curve $\alpha :I\rightarrow E^{n}$ is called a slant helix if its unit principal normal $V_{2}$ makes a constant angle with a fixed direciton $U$ [1]. \end{definition}
\begin{definition} Let the $(n-k)$-manifold $M$ be submanifold of the Riemannian manifold $ \overline{M}=E^{n}$ and let $\overline{D}$ be the Riemannian connexion on $ \overline{M}=E^{n}$. For $C^{\infty \text{ }}$fields $X$ and $Y$ with domain $A$ on $M$ (and tangent to $M$), define $D_{X}Y$ and $V(X,Y)$ on $A$ by decomposing $\overline{D}_{X}Y$ into unique tangential and normal components, respectively; thus, \begin{equation*} \overline{D}_{X}Y=D_{X}Y+V(X,Y)\text{. } \end{equation*} Then, $D$ is the Riemannian connexion on $M$ and $V$ is a symmetric vector-valued 2-covariant $C^{\infty \text{ }}$tensor called the second fundamental tensor. The above composition equation is called the Gauss equation [9]. \end{definition}
\begin{definition} Let the $(n-k)$-manifold $M$ be submanifold of the Riemannian manifold $ \overline{M}=E^{n}$ , let $\overline{D}$ be the Riemannian connexion on $ \overline{M}=E^{n}$ and let $D$ be the Riemannian connexion on $M$. Then, the formula of Weingarten \begin{equation*} \overline{D}_{X}N=-A_{N}(X)+D_{X}^{\bot }N \end{equation*} for every $X$ and $Y$ tangent to $M$ and for every $N$ normal to $M$. $A_{N}$ is the shape operator associated to $N$ also known as the Weingarten operator corresponding to $N$ and $D^{\bot }$ is the induced connexion in the normal bundle of $M$ ($A_{N}(X)$ is also the tangent component of $- \overline{D}_{X}N$ and will be denoted by $A_{N}(X)$ $=$tang($-\overline{D} _{X}N$)). Specially, if $M$ is a hypersurface in $E^{n}$, we have $ \left\langle V(X,Y),N\right\rangle =\left\langle A_{N}(X),Y\right\rangle $ for all $X$, $Y$ tangent to $M$. So, \begin{equation*} V(X,Y)=\left\langle V(X,Y),N\right\rangle N=\left\langle A_{N}(X),Y\right\rangle N \end{equation*} and we obtain \begin{equation*} \overline{D}_{X}Y=D_{X}Y+\left\langle A_{N}(X),Y\right\rangle N\text{ .} \end{equation*} For this definition 2.5, note that the shape operator $A_{N}$ is defined by the map $A_{N}:\varkappa (M)\rightarrow \varkappa (M)$, where $\varkappa (M)$ is the space of tangent vector fields on $M$ and if $p\in M$, the shape operator $A_{N}$ is defined by the map $A_{p}:T_{p}(M)\rightarrow T_{p}(M)$ .The eigenvalues of $A_{p}$ are called the principal curvatures (denoted by $ \lambda _{i}$) and the eigenvectors of $A_{p}$ are called the principal vectors [10,11]. \end{definition}
\begin{definition} If $\alpha $ is a (unit speed) curve in $M$ with $C^{\infty \text{ }}$unit tangent $T$, then $V(T,T)$ is called normal curvature vector field of $ \alpha $ and $k_{T}=\left\Vert V(T,T)\right\Vert $ is called the normal curvature of $\alpha $ [9]. \end{definition}
\section{Main Theorems}
\begin{theorem} \textbf{\ }Let $M$ be a strong $r$-helix hypersurface and $H(M)\subset E^{n}$ be the set of helix directions of $M$. If $\alpha :I\subset IR\rightarrow M$ is a (unit speed) line of curvature (not a line) on $M$, then $d_{j}\notin Sp\left\{ N,T\right\} $ along the curve $\alpha $ for all $d_{j}\in H(M)$, where $T$ is the tangent vector field of $\alpha $ and $N$ is a unit normal vector field of $M$. \end{theorem}
\begin{proof} We assume that $d_{j}\in Sp\left\{ N,T\right\} $ along the curve $\alpha $ for any $d_{j}\in H(M)$. Then, along the curve $\alpha $, since $M$ is a strong $r$-helix hypersurface, we can decompose $d_{j}$ in tangent and normal components: \begin{equation} d_{j}=\cos (\theta _{j})N+\sin (\theta _{j})T \end{equation} where $\theta _{j}$ is constant. From (3.1),by taking derivatives on both sides along the curve $\alpha $, we get: \begin{equation} 0=\cos (\theta _{j})N^{
{\acute{}}
}+\sin (\theta _{j})T^{
{\acute{}}
}\text{ } \end{equation} Moreover, since $\alpha $ is a line of curvature on $M$, \begin{equation} N
{\acute{}}
=\lambda \alpha
{\acute{}}
\text{ } \end{equation} along the curve $\alpha $. By using the equations (3.2) and (3.3), we deduce that the system $\left\{ \alpha
{\acute{}}
,T^{
{\acute{}}
}\right\} $ is linear dependent. But, the system $\left\{ \alpha
{\acute{}}
,T^{
{\acute{}}
}\right\} $ is never linear dependent. This is a contradiction. This completes the proof. \end{proof}
\begin{theorem} Let $M$ be a submanifold with $(n-k)$ dimension in $E^{n}$. Let $\overline{D} $ be Riemannian connexion (standart covariant derivative) on $E^{n}$ and $D$ be Riemannian connexion on $M$. Let us assume that $M\subset E^{n}$ be a strong $r$-helix submanifold and $H(M)\subset E^{n}$ be the space of the helix directions of $M$. If $\alpha :I\subset IR\rightarrow M$ is a (unit speed) geodesic curve on $M$ and if $\left\langle V_{2},\xi _{j}\right\rangle $ is a constant function along the curve $\alpha $, then $ \alpha $ is a slant helix in $E^{n}$, where $V_{2}$ is the unit principal normal of $\alpha $ and $\xi _{j}$ is the normal component of a direction $ d_{j}\in H(M)$. \end{theorem}
\begin{proof} Let $T$ be the unit tangent vector field of $\alpha $. Then, from the formula Gauss in Definition (2.4), \begin{equation} \overline{D}_{T}T=D_{T}T+V(T,T) \end{equation} According to the Theorem, since $\alpha $ is a geodesic curve on $M$, \begin{equation} D_{T}T=0 \end{equation} So, by using (3.4),(3.5) and Frenet formulas, we have: \begin{equation*} \overline{D}_{T}T=k_{1}V_{2}=V(T,T) \end{equation*} That is, the vector field $V_{2}\in \vartheta (M)$ along the curve $\alpha $ , where $\vartheta (M)$ is the normal space of $M$. On the other hand, since $M$ is a strong $r$-helix submanifold, we can decompose any $d_{j}\in H(M)$ in its tangent and normal components: \begin{equation} d_{j}=\cos (\theta _{j})\xi _{j}+\sin (\theta _{j})T_{j} \end{equation} where $\theta _{j}$ is constant. Moreover, according to the Theorem, $ \left\langle V_{2},\xi _{j}\right\rangle $ is a constant function along the curve $\alpha $ for the normal component $\xi _{j}$ of \ a direction $ d_{j}\in H(M)$. Hence, doing the scalar product with $V_{2}$ in each part of the equation (3.6), we obtain: \begin{equation} \left\langle d_{j},V_{2}\right\rangle =\cos (\theta _{j})\left\langle V_{2},\xi _{j}\right\rangle +\sin (\theta _{j})\left\langle V_{2},T_{j}\right\rangle \end{equation} Since $\cos (\theta _{j})\left\langle V_{2},\xi _{j}\right\rangle =$constant and $\left\langle V_{2},T_{j}\right\rangle =0$ ( $V_{2}\in \vartheta (M)$) along the curve $\alpha $, from (3.7) we have: \begin{equation*} \left\langle d_{j},V_{2}\right\rangle =\text{constant.} \end{equation*} along the curve $\alpha $. Consequently, $\alpha $ is a slant helix in $ E^{n} $. \end{proof}
\begin{theorem} Let $M$ be a submanifold with $(n-k)$ dimension in $E^{n}$. Let $\overline{D} $ be Riemannian connexion (standart covariant derivative) on $E^{n}$ and $D$ be Riemannian connexion on $M$. Let us assume that $M\subset E^{n}$ be a strong $r$-helix submanifold and $H(M)\subset E^{n}$ be the space of the helix directions of $M$. If $\alpha :I\subset IR\rightarrow M$ is a (unit speed) curve on $M$ with the normal curvature function $k_{T}=0$ and if $ \left\langle V_{2},T_{j}\right\rangle $ is a constant function along the curve $\alpha $, then $\alpha $ is a slant helix in $E^{n}$, where $V_{2}$ is the unit principal normal of $\alpha $ and $T_{j}$ is the tangent component of a direction $d_{j}\in H(M)$. \end{theorem}
\begin{proof} Let $T$ be the unit tangent vector field of $\alpha $. Then, from the formula Gauss in Definition (2.4), \begin{equation} \overline{D}_{T}T=D_{T}T+V(T,T) \end{equation} According to the Theorem, since the normal curvature $k_{T}=0$, \begin{equation} V(T,T)=0 \end{equation} So, by using (3.8),(3.9) and Frenet formulas, we have: \begin{equation*} \overline{D}_{T}T=k_{1}V_{2}=D_{T}T\text{.} \end{equation*} That is, the vector field $V_{2}\in T_{\alpha (t)}M$, where $T_{\alpha (t)}M$ is the tangent space of $M$. On the other hand, since $M$ is a strong $r$ -helix submanifold, we can decompose any $d_{j}\in H(M)$ in its tangent and normal components: \begin{equation} d_{j}=\cos (\theta _{j})\xi _{j}+\sin (\theta _{j})T_{j} \end{equation} where $\theta _{j}$ is constant. Moreover, according to the Theorem, $ \left\langle V_{2},T_{j}\right\rangle $ is a constant function along the curve $\alpha $ for the tangent component $T_{j}$ of \ a direction $d_{j}\in H(M)$. Hence, doing the scalar product with $V_{2}$ in each part of the equation (3.10), we obtain: \begin{equation} \left\langle d_{j},V_{2}\right\rangle =\cos (\theta _{j})\left\langle V_{2},\xi _{j}\right\rangle +\sin (\theta _{j})\left\langle V_{2},T_{j}\right\rangle \end{equation} Since $\sin (\theta _{j})\left\langle V_{2},T_{j}\right\rangle =$constant and $\left\langle V_{2},\xi _{j}\right\rangle =0$ ($V_{2}\in T_{\alpha (t)}M$ ) along the curve $\alpha $, from (3.11) we have: \begin{equation*} \left\langle d_{j},V_{2}\right\rangle =\text{constant.} \end{equation*} along the curve $\alpha $. Consequently, $\alpha $ is a slant helix in $ E^{n} $. \end{proof}
\begin{definition} Given an Euclidean submanifold of arbitrary codimension $M\subset IR^{n}$. A curve $\alpha $ in $M$ is called a line of curvature if its tangent $T$ is a principal vector at each of its points. In other words, when $T$ (the tangent of $\alpha $) is a principal vector at each of its points, for an arbitrary normal vector field $N\in \vartheta (M)$, the shape operator $ A_{N} $ associated to $N$ says $A_{N}(T)=$tang$(-$ $\overline{D} _{T}N)=\lambda _{j}T$ along the curve $\alpha $, where $\lambda _{j}$ is a principal curvature and $\overline{D}$ be the Riemannian connexion(standart covariant derivative) on $IR^{n}$ [2]. \end{definition}
\begin{theorem} Let $M$ be a submanifold with $(n-k)$ dimension in $E^{n}$ and let $ \overline{D}$ be Riemannian connexion (standart covariant derivative) on $ E^{n}$. Let us assume that $M\subset E^{n}$ be a strong $r$-helix submanifold and $H(M)\subset E^{n}$ be the space of the helix directions of $ M$. If $\alpha :I\rightarrow M$ is a line of curvature with respect to the normal component $N_{j}\in \vartheta (M)$ of a direction $d_{j}\in H(M)$ and if $N_{j}^{
{\acute{}}
}\in \varkappa (M)$ along the curve $\alpha $, then $d_{j}\in Sp\left\{ T\right\} ^{\bot }$ along the curve $\alpha $, where $T$ \ is the unit tangent vector field of $\alpha $. \end{theorem}
\begin{proof} We assume that $\alpha :I\rightarrow M$ is a line of curvature with respect to the normal component $N_{j}\in \vartheta (M)$ of a direction $d_{j}\in H(M)$. Since $M$ is a strong $r$-helix submanifold, we can decompose $ d_{j}\in H(M)$ in its tangent and normal components: \begin{equation*} d_{j}=\cos (\theta _{j})N_{j}+\sin (\theta _{j})T_{j}\text{ } \end{equation*} where $\theta _{j}$ is constant. So, $\left\langle N_{j},d_{j}\right\rangle = $constant and by taking derivatives on both sides along the curve $\alpha $ , we get $\left\langle N_{j}^{
{\acute{}}
},d_{j}\right\rangle =0$. On the other hand, since $\alpha :I\rightarrow M$ is a line of curvature with respect to the $N_{j}\in \vartheta (M)$, \begin{equation*} A_{N_{j}}(T)\text{=tang}(-\overline{D}_{T}N_{j})=\text{tang}(-N_{j}^{
{\acute{}}
})=\lambda _{j}T\text{ } \end{equation*} along the curve $\alpha $. According to this Theorem, $N_{j}^{
{\acute{}}
}\in \varkappa (M)$ along the curve $\alpha $. Hence, \begin{equation} \text{tang}(-N_{j}^{
{\acute{}}
})=-N_{j}^{
{\acute{}}
}=\lambda _{j}T\text{ } \end{equation} Therefore, by using the equalities $\left\langle N_{j}^{
{\acute{}}
},d_{j}\right\rangle =0$ and (3.12), we obtain: \begin{equation*} \left\langle T,d_{j}\right\rangle =0 \end{equation*} along the curve $\alpha $. This completes the proof. \end{proof}
\begin{theorem} Let $M$ be a submanifold with $(n-k)$ dimension in $E^{n}$ and let $ \overline{D}$ be Riemannian connexion (standart covariant derivative) on $ E^{n}$. Let us assume that $M\subset E^{n}$ be a strong $r$-helix submanifold and $H(M)\subset E^{n}$ be the space of the helix directions of $ M$. If $\alpha :I\rightarrow M$ is a curve in $M$ and if the system $\left\{ T_{j}^{
{\acute{}}
},T\right\} $ is linear dependent along the curve $\alpha $, where $T_{j}^{
{\acute{}}
}$ is the derivative of the tangent component $T_{j}$ of a direction $ d_{j}\in H(M)$ and $T$ the tangent to the curve $\alpha $, then $\alpha $ is a line of curvature in $M$. \end{theorem}
\begin{proof} Since $M$ is a strong $r$-helix submanifold, we can decompose $d_{j}\in H(M)$ in its tangent and normal components: \begin{equation} d_{j}=\cos (\theta _{j})N_{j}+\sin (\theta _{j})T_{j}\text{ } \end{equation} where $\theta _{j}$ is constant. If we take derivative in each part of the equation (3.13) along the curve $\alpha $, we obtain: \begin{equation} 0=\cos (\theta _{j})N_{j}^{
{\acute{}}
}+\sin (\theta _{j})T_{j}^{
{\acute{}}
}\text{ } \end{equation} From (3.14), we can write \begin{equation} N_{j}^{
{\acute{}}
}=-\tan (\theta _{j})T_{j}^{
{\acute{}}
}\text{ } \end{equation} So, for the tangent component of $-N_{j}^{
{\acute{}}
}$, from (3.15) we can write: \begin{equation} A_{N_{j}}(T)\text{=tang}(-\overline{D}_{T}N_{j})=\text{tang}(-N_{j}^{
{\acute{}}
})=\text{tang}(\tan (\theta _{j})T_{j}^{
{\acute{}}
})\text{ } \end{equation} along the curve $\alpha $. According to the hypothesis, the system $\left\{ T_{j}^{
{\acute{}}
},T\right\} $ is linear dependent along the curve $\alpha $. Hence, we get $ T_{j}^{
{\acute{}}
}=\lambda _{j}T$. And, by using the equation (3.16), we have: \begin{equation*} A_{N_{j}}(T)=\text{tang}(\tan (\theta _{j})T_{j}^{
{\acute{}}
})=\text{tang}(\tan (\theta _{j})\lambda _{j}T) \end{equation*} and \begin{equation} A_{N_{j}}(T)=\text{tang}(\tan (\theta _{j})\lambda _{j}T)\text{ } \end{equation} Moreover, since $T\in \varkappa (M)$, tang$(\tan (\theta _{j})\lambda T)=(\tan (\theta _{j})\lambda _{j})T=k_{j}T$. Therefore, from (3.17), we have: \begin{equation*} A_{N_{j}}(T)=k_{j}T\text{.} \end{equation*} It follows that $\alpha $ is a line of curvature in $M$ for $N_{j}\in \vartheta (M)$. This completes the proof. \end{proof}
\noindent \textbf{Acknowledgment.} The authors would like to thank two anonymous referees for their valuable suggestions and comments that helped to improve the presentation of this paper.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{On the failure of the Gorenstein property\
for Hecke algebras of prime weight}
\begin{abstract}
In this article we report on extensive calculations concerning the
Gorenstein defect for Hecke algebras of spaces of modular forms of
prime weight~$p$ at maximal ideals of residue characteristic~$p$
such that the attached mod~$p$ Galois representation is unramified
at~$p$ and the Frobenius at~$p$ acts by scalars. The results lead us
to the ask the question whether the Gorenstein defect and the
multplicity of the attached Galois representation are always equal
to~$2$. We review the literature on the failure of the Gorenstein
property and multiplicity one, discuss in some detail a very
important practical improvement of the modular symbols algorithm
over finite fields and include precise statements on the
relationship between the Gorenstein defect and the multiplicity of
Galois representations.
MSC Classification: 11F80 (primary), 11F33, 11F25 (secondary). \end{abstract}
\tableofcontents
\section{Introduction} \label{section-introduction}
In Wiles' proof of Fermat's Last Theorem (see \cite{wiles}) an essential step was to show that certain Hecke algebras are Gorenstein rings. Moreover, the Gorenstein property of Hecke algebras is equivalent to the fact that Galois representations appear on certain Jacobians of modular curves precisely with multiplicity one. This article is concerned with the Gorenstein property and with the multiplicity one question. We report previous work and exhibit many new examples where multiplicity one and the Gorenstein property fail. We compute the multiplicty in these cases. Moreover, we ask the question suggested by our computations whether it is always equal to two if it fails.
We first have to introduce some notation. For integers $N \ge 1$ and $k \ge 2$ and a Dirichlet character $\chi: (\mathbb{Z}/N\mathbb{Z})^\times \to \mathbb{C}^\times$ we let $S_k(\Gamma_1(N))$ be the $\mathbb{C}$-vector space of holomorphic cusp forms on $\Gamma_1(N)$ of weight~$k$ and $S_k(N,\chi)$ the subspace on which the diamond operators act through the character~$\chi$. We now introduce some extra notation for Hecke algebras over specified rings. \begin{notation}[Notation for Hecke algebras] Whenever $S \subseteq R$ are rings and $M$ is an $R$-module on which the Hecke and diamond operators act, we let $\mathbb{T}_S(M)$ be the $S$-subalgebra inside the $R$-endomorphism ring of~$M$ generated by the Hecke and the diamond operators. If $\phi: S \to S'$ is a ring homomorphism, we let $\mathbb{T}_\phi (M) := \mathbb{T}_S(M) \otimes_S S'$ or with $\phi$ understood $\mathbb{T}_{S \to S'} (M)$. \end{notation} We will mostly be dealing with the Hecke algebras $\mathbb{T}_\mathbb{Z}(S_k(\Gamma_1(N)))$ and $\mathbb{T}_{\mathbb{Z}[\chi]}(S_k(N,\chi))$, their completions $\mathbb{T}_{\mathbb{Z} \to \mathbb{Z}_p}(S_k(\Gamma_1(N)))$ and $\mathbb{T}_{\mathcal{O}\to \widehat{\mathcal{O}}}(S_k(N,\chi))$, as well as their reductions $\mathbb{T}_{\mathbb{Z} \to \mathbb{F}_p}(S_k(\Gamma_1(N)))$ and $\mathbb{T}_{\mathcal{O} \to \mathbb{F}}(S_k(N,\chi))$. Here, $p$ is a prime and $\mathcal{O} = \mathbb{Z}[\chi]$ is the smallest subring of~$\mathbb{C}$ containing all values of~$\chi$, $\widehat{\mathcal{O}}$ is a completion and $\mathcal{O} \twoheadrightarrow \mathbb{F}$ is a reduction modulo a prime above~$p$. In Section~\ref{modular-symbols} the reduced Hecke algebras are identified with Hecke algebras of mod~$p$ modular forms, which are closely related to Hecke algebras of Katz modular forms over finite fields (see Section~\ref{section-relation}).
We choose a holomorphic cuspidal Hecke eigenform as the starting point of our discussion and treatment. So let $f \in S_k(N,\chi) \subseteq S_k(\Gamma_1(N))$ be an eigenform for all Hecke and diamond operators. It (more precisely, its Galois conjugacy class) corresponds to minimal ideals, both denoted by $\mathfrak{p}_f$, in each of the two Hecke algebras $\mathbb{T}_\mathbb{Z}(S_k(\Gamma_1(N)))$ and $\mathbb{T}_{\mathbb{Z}[\chi]}(S_k(N,\chi))$. We also choose maximal ideals $\mathfrak{m}=\mathfrak{m}_f$ containing~$\mathfrak{p}_f$ of residue characteristic~$p$ again in each of the two. Note that the residue fields are the same in both cases.
By work of Shimura and Deligne, one can associate to~$f$ (more precisely, to~$\mathfrak{m}$) a continuous odd semi-simple Galois representation \[ \rho_f = \rho_{\mathfrak{m}_f} = \rho_\mathfrak{m}: \Gal({\overline{\QQ}}/\mathbb{Q}) \rightarrow \mathrm{GL}_2(\hecke{N}/\mathfrak{m}) \] unramified outside $Np$ satisfying~$\Tr(\rho_\mathfrak{m}(\Frob_l)) \equiv T_l \mod \mathfrak{m}$ and $\Det(\rho_m(\Frob_l)) \equiv l^{k-1}\chi(l) \mod \mathfrak{m}$ for all primes~$l \nmid Np$. In the case of weight~$k=2$ and level~$N$, the representation~$\rho_m$ can be naturally realised on the $p$-torsion points of the Jacobian of the modular curve $X_1(N)_\mathbb{Q}$. The algebra $\mathbb{T}_{\mathbb{Z} \to \mathbb{F}_p}(S_2(\Gamma_1(N)))$ acts naturally on $J_1(N)_\mathbb{Q}({\overline{\QQ}})[p]$ and we can form the Galois module $J_1(N)_\mathbb{Q} ({\overline{\QQ}})[\mathfrak{m}] = J_1(N)_\mathbb{Q}({\overline{\QQ}})[p][\widetilde{\mathfrak{m}}]$ with $\widetilde{\mathfrak{m}}$ the maximal ideal of $\mathbb{T}_{\mathbb{Z} \to \mathbb{F}_p}(S_2(\Gamma_1(N)))$ which is the image of~$\mathfrak{m}$ under the natural projection. Supposing that $\rho_m$ is absolutely irreducible, the main result of~\cite{blr} shows that the Galois representation $J_1(Np)_\mathbb{Q}({\overline{\QQ}})[\mathfrak{m}]$ is isomorphic to a direct sum of $r$~copies of~$\rho_m$ for some integer $r \ge 1$, which one calls the {\em multiplicity of~$\rho_m$} (cf.~\cite{ribet-stein}). One says that $\rho_m$ is a {\em multiplicity one representation} or {\em satisfies multiplicity one}, if $r=1$. See \cite{mazur} for a similar definition for $J_0(N)$ and Prop.~\ref{nulleins} for a comparison.
The notion of multiplicity can be naturally extended to Galois representations attached to eigenforms~$f$ of weights $3 \le k \le p+1$ for $p \nmid N$. This is accomplished by a result of Serre's which implies the existence of a maximal ideal $\mathfrak{m}_2 \subset \heckeoneNk{Np}{2}$ such that $\rho_{\mathfrak{m}_f} \cong \rho_{\mathfrak{m}_2}$ (see Prop.~\ref{totwo}). One hence obtains the notion of multiplicity (on $J_1(Np)$) for the representation~$\rho_{m_f}$ by defining it as the multiplicity of~$\rho_{\mathfrak{m}_2}$. Moreover, when allowing twists by the cyclotomic character, it is even possible to treat arbitrary weights. The following theorem summarises results on when the multiplicity in the above sense is known to be one.
\begin{thm}[Mazur~\cite{mazur}, Edixhoven~\cite{edixhoven-serre},
Gross~\cite{gross}, Buzzard~\cite{buzzard_appendix}] \label{gorenstein-theorem} Let~$\rho_\mathfrak{m}$ be a representation associated with a modular cuspidal eigenform~$f \in S_{k}(N,\chi)$ and let~$p$ be the residue characteristic of~$\mathfrak{m}$. Suppose that $\rho_\mathfrak{m}$ is absolutely irreducible and that $p$ does not divide~$N$. If either \begin{enumerate} \itemsep=0cm plus 0pt minus 0pt \item $2 \le k \le p-1$, or
\item $k=p$ and $\rho_\mathfrak{m}$ is ramified at~$p$, or
\item $k=p$ and $\rho_\mathfrak{m}$ is unramified at~$p$ and~$\rho_\mathfrak{m}(\Frob_p)$ is not scalar,
\end{enumerate} then the multiplicity of~$\rho_\mathfrak{m}$ is one. \end{thm}
This theorem is composed of Lemma~15.1 from Mazur~\cite{mazur}, Theorem~9.2 from Edixhoven~\cite{edixhoven-serre}, Proposition~{12.10} from Gross~\cite{gross} and Theorem~6.1 from Buzzard~\cite{buzzard_appendix}.
The following theorem by the second author (\cite{nongor}, Corollary~4.4) tells us when the multiplicity is not one.
\begin{thm}\label{nongor}
Let $\rho_\mathfrak{m}$ as in the previous theorem. Suppose $k=p$ and that
$\rho_\mathfrak{m}$ is unramified at~$p$. If $p=2$, assume also that a Katz
cusp form over~$\mathbb{F}_2$ of weight~$1$ on~$\Gamma_1(N)$ exists which
gives rise to~$\rho_\mathfrak{m}$.
If $\rho_\mathfrak{m}(\Frob_p)$ is a scalar matrix, then the multiplicity
of~$\rho_\mathfrak{m}$ is bigger than~$1$. \end{thm}
In Section~\ref{section-relation} we explain how the Galois representation $J_1(Np)_\mathbb{Q}({\overline{\QQ}})[\mathfrak{m}]$ is related to the different Hecke algebras evoked above and see in many cases of interest a precise relationship between the geometrically defined term of {\em multiplicity} and the {\em Gorenstein defect} of these algebras. The latter can be computed explicitly, which is the subject of the present article. We now give the relevant definitions.
\begin{defi}[The Gorenstein property]
Let~$A$ be a local Noetherian ring with maximal ideal~$\mathfrak{m}$.
Suppose first that the Krull dimension of~$A$ is zero, i.e.\ that
$A$ is Artinian. We then define the \emph{Gorenstein defect} of~$A$
to be the minimum number of $A$-module generators of the annihilator
of~$\mathfrak{m}$ (i.e.\ $A[\mathfrak{m}]$) minus~1; equivalently, this is
the~$A/\mathfrak{m}$-dimension of the annihilator of~$\mathfrak{m}$ minus~1. We say that
$A$ is \emph{Gorenstein} if its Gorenstein defect is~0, and
\emph{non-Gorenstein} otherwise.
If the Krull dimension of~$A$ is positive, we inductively call~$A$
{\em Gorenstein}, if there exists a non-zero-divisor $x \in \mathfrak{m}$
such that $A/(x)$ is a Gorenstein ring of smaller Krull dimension
(see~\cite{Eisenbud}, p.~532; note that our definition implies that
$A$ is Cohen-Macaulay).
A (not necessarily local) Noetherian ring is said to be
\emph{Gorenstein} if and only if all of its localisations at its
maximal ideals are Gorenstein. \end{defi}
We will for example be interested in the Gorenstein property of $\mathbb{T}_\mathbb{Z}(S_k(\Gamma_1(N)))_\mathfrak{m}$. Choosing $x=p$ in the definition, we see that this is equivalent to the Gorenstein defect of the finite dimensional $\mathbb{F}_p$-algebra $\mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_k(\Gamma_1(N)))_\mathfrak{m}$ being zero. Whenever we refer to the Gorenstein defect of the former algebra (over~$\mathbb{Z}$), we mean the one of the latter. Our computations will concern the Gorenstein defect of $\mathbb{T}_{\mathcal{O}\to\mathbb{F}}(S_k(\Gamma_1(N),\chi))_\mathfrak{m}$. See Section~\ref{section-relation} for a comparison with the one not involving a character. It is important to remark that the Gorenstein defect of a local Artin algebra over a field does not change after passing to a field extension and taking any of the conjugate local factors.
We illustrate the definition by an example. The algebra $k[x,y,z]/(x^2,y^2,z^2,xy,xz,yz)$ for a field~$k$ is Artinian and local with maximal ideal $\mathfrak{m}:=(x,y,z)$ and the annihilator of~$\mathfrak{m}$ is~$\mathfrak{m}$ itself, so the Gorenstein defect is~$3-1=2$. We note that this particular case does occur in nature; a localisation~$\mathbb{T}_{\mathbb{Z}\to\mathbb{F}_2}(S_2(\Gamma_0(431)))_\mathfrak{m}$ at one maximal ideal is isomorphic to this, with~$k=\mathbb{F}_2$ (see~\cite{emerton}, the discussion just before Lemma~6.6). This example can also be verified with the algorithm presented in this paper.
We now state a translation of Theorem~\ref{gorenstein-theorem} in terms of Gorenstein defects, which is immediate from the propositions in Section~\ref{section-relation}.
\begin{thm}\label{thgor} Assume the set-up of Theorem~\ref{gorenstein-theorem} and that one of 1.,\ 2.,\ or 3.\ is satisfied. We use notations as in the discussion of multiplicities above.
If $k=2$, then $\mathbb{T}_\mathbb{Z}(S_2(\Gamma_1(N)))_\mathfrak{m}$ is a Gorenstein ring.
If $k\ge3$, then $\mathbb{T}_\mathbb{Z}(S_2(\Gamma_1(Np)))_{\mathfrak{m}_2}$ is, too. Supposing in addition that $\mathfrak{m}$ is ordinary (i.e.\ $T_p \not\in \mathfrak{m}$), then also $\mathbb{T}_\mathbb{Z}(S_k(\Gamma_1(N)))_\mathfrak{m}$ is Gorenstein. If, moreover, $p \ge 5$ or $\rho_\mathfrak{m}$ is not induced from $\mathbb{Q}(\sqrt{-1})$ (if $p=2$) or $\mathbb{Q}(\sqrt{-3})$ (if $p=3$), then $\mathbb{T}_{\mathcal{O}\to\mathbb{F}}(S_k(N,\chi))_\mathfrak{m}$ is Gorenstein as well. \hspace* {.5cm}
$\Box$ \end{thm}
We now turn our attention to computing the Gorenstein defect and the multiplicity in the case when it is known not to be one.
\begin{cor}\label{corfailure} Let~$\rho_\mathfrak{m}$ be a representation associated with a cuspidal eigenform~$f \in S_{p}(N,\chi)$ with~$p$ the residue characteristic of~$\mathfrak{m}$. Assume that $\rho_\mathfrak{m}$ is absolutely irreducible, unramified at~$p$ such that $\rho_\mathfrak{m}(\Frob_p)$ is a scalar matrix. Let $r$ be the multiplicity of~$\rho_\mathfrak{m}$ and $d$ the Gorenstein defect of any of $\mathbb{T}_{\mathcal{O}\to\mathbb{F}}(S_k(N,\chi))_\mathfrak{m}$, $\mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_k(\Gamma_1(N)))_\mathfrak{m}$ or $\mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_2(\Gamma_1(Np)))_{\mathfrak{m}_2}$.
Then the relation $d = 2r-2$ holds. \end{cor}
{\bf Proof. } The equality of the Gorenstein defects and the relation with the multiplicity are proved in Section~\ref{section-relation}, noting that $\mathfrak{m}$ is ordinary, as $a_p(f)^2 = \chi(p) \neq 0$ by \cite{gross}, p.~487. \hspace* {.5cm}
$\Box$
\subsection{Previous results on the failure of multiplicity one or the Gorenstein property} \label{previous-results}
Prior to the present work and the article~\cite{nongor}, there have been some investigations into when Hecke algebras fail to be Gorenstein. In~\cite{kilford-nongorenstein}, the first author showed, using {\sc Magma}~\cite{magma}, that the Hecke algebras~$\heckeNktwo{431}$, $\heckeNktwo{503}$ and~$\heckeNktwo{2089}$ are not Gorenstein by explicit computation of the localisation of the Hecke algebra at a suitable maximal ideal above~2, and in~\cite{ribet-stein}, it is shown that~$\heckeNktwo{2071}$ is not Gorenstein in a similar fashion. These examples were discovered by considering elliptic curves~$E/\mathbb{Q}$ such that in the ring of integers of~$\mathbb{Q}(E[2])$ the prime ideal~$(2)$ splits completely, and then doing computations with {\sc Magma}.
There are also some results in the literature on the failure of multiplicity one within the torsion of certain Jacobians. In~\cite{agashe-ribet-stein}, the following theorem is proved: \begin{thm}[Agashe-Ribet-Stein~\cite{agashe-ribet-stein}, Proposition~5.1]
Suppose that~$E$ is an optimal elliptic curve over~$\mathbb{Q}$ of
conductor~$N$, with congruence number~$r_E$ and modular degree~$m_E$
and that~$p$ is a prime such that~$p | r_E$ but~$p \nmid m_E$.
Let~$\mathfrak{m}$ be the annihilator in~$\heckeNktwo{N}$ of~$E[p]$.
Then multiplicity one fails for~$\mathfrak{m}$. \end{thm} They give a table of examples; for instance, $\heckeNktwo{54}$ does not satisfy multiplicity one at some maximal ideal above~$3$. It is not clear whether this phenomenon occurs infinitely often.
In~\cite{ribet-festschrift}, it is shown that the mod~$p$ multiplicity of a certain representation in the Jacobian of the Shimura curve derived from the rational quaternion algebra of discriminant~$11 \cdot 193$ is~2; this result inspired the calculations in~\cite{kilford-nongorenstein}.
Let us finally mention that for $p=2$ precisely the Galois representations~$\rho$ with image equal to the dihedral group~$D_3$ come from an elliptic curve over~$\mathbb{Q}$. For, on the one hand, we only need observe that $D_3=\mathrm{GL}_2(\mathbb{F}_2)$. On the other hand, any $S_3$-extension~$K$ of the rationals can be obtained as the splitting field of an irreducible integral polynomial $f=X^3+aX+b$. The $2$-torsion of the elliptic curve $E: Y^2=f$ consists precisely of the three roots of~$f$ and the point at infinity. So, the field generated over $\mathbb{Q}$ by the $2$-torsion of~$E$ is~$K$.
\subsection{New results}
Using the modular symbols algorithm over finite fields with an improved stop criterion (see Section~\ref{modular-symbols}), we performed computations in {\sc Magma} concerning the Gorenstein defect of Hecke algebras of cuspidal modular forms of prime weights~$p$ at maximal ideals of residue characteristic~$p$ in the case of Theorem~\ref{nongor}. All of our 384 examples have Gorenstein defect equal to~$2$ and hence their multiplicity is~$2$.
We formulate part of our computational findings as a theorem.
\begin{thm}
For every prime $p < 100$ there exists a prime $N \neq p$ and a
Dirichlet character~$\chi$ such that the Hecke algebra
$\mathbb{T}_{\mathbb{Z}[\chi]\to\mathbb{F}}(S_p(N,\chi))$ has Gorenstein defect~$2$ at
some maximal ideal~$\mathfrak{m}$ of residue characteristic~$p$. The
corresponding Galois representation~$\rho_\mathfrak{m}$ appears with
multiplicity two on the $p$-torsion of the Jacobian $J_1(Np)_\mathbb{Q}({\overline{\QQ}})$
if $p$ is odd, respectively $J_1(N)_\mathbb{Q}({\overline{\QQ}})$ if~$p=2$. \end{thm}
Our computational results are discussed in more detail in Section~\ref{computational-results}.
\subsection{A question}
\begin{question}\label{ourquestion} Let $p$ be a prime. Let $f$ be a normalised cuspidal modular eigenform of weight~$p$, prime level $N \neq p$ for some Dirichlet character $\chi$. Let $\rho_f : G_\mathbb{Q} \to \mathrm{GL}_2({\overline{\FF}_p})$ be the modular Galois representation attached to~$f$. We assume that $\rho_f$ is irreducible and unramified at~$p$ and that $\rho_f(\Frob_p)$ is a scalar matrix.
Write $\mathbb{T}_{\mathbb{F}}$ for $\mathbb{T}_{\mathbb{Z}[\chi]\to\mathbb{F}}(S_p(N,\chi))$. Recall that this notation stands for the tensor product over $\mathbb{Z}[\chi]$ of a residue field $\mathbb{F}/\mathbb{F}_p$ of $\mathbb{Z}[\chi]$ by the $\mathbb{Z}[\chi]$-algebra generated inside the endomorphism algebra of~$S_p(N,\chi)$ by the Hecke operators and by the diamond operators. Let $\mathfrak{m}$ be the maximal ideal of $\mathbb{T}_{\mathbb{F}}$ corresponding to~$f$.
Is the Gorenstein defect of the Hecke algebra~$\mathbb{T}_{\mathbb{F}}$ localised at ${\mathfrak{m}}$, denoted by $\mathbb{T}_{\mathfrak{m}}$, always equal to~$2$?
Equivalently, is the multiplicity of the Galois representation attached to~$f$ always equal to~$2$? \end{question}
This question was also raised both by Kevin Buzzard and James Parson in communications to the authors.
\section{Relation between multiplicity and Gorenstein defect} \label{section-relation}
In this section we collect results, some of which are well-known, on the multiplicity of Galois representations, the Gorenstein defect and relations between the two. Whereas the mod~$p$ modular symbols algorithm naturally computes mod~$p$ modular forms (see Section~\ref{modular-symbols}), this rather geometrical section uses (mostly in the references) the theory of Katz modular forms over finite fields (see e.g.~\cite{edixhoven-boston}). If $N \ge 5$ and $k\ge 2$, the Hecke algebra $\mathbb{T}_{\mathbb{Z} \to \mathbb{F}_p}(S_k(\Gamma_1(N)))$ is both the Hecke algebra of mod~$p$ cusp forms of weight~$k$ on~$\Gamma_1(N)$ and the Hecke algebra of the corresponding Katz cusp forms over~$\mathbb{F}_p$. However, in the presence of a Dirichlet character $\mathbb{T}_{\mathbb{Z}[\chi] \to \mathbb{F}}(S_k(N,\chi))$ only has an interpretation as the Hecke algebra of the corresponding mod~$p$ cusp forms and there may be differences with the respective Hecke algebra for Katz forms (see Carayol's Lemma, Prop.~1.10 of~~\cite{edixhoven-boston}).
We start with the well-known result in weight~$2$ (see e.g.\ \cite{mazur}, Lemma 15.1) that multiplicity one implies that the corresponding local Hecke factor is a Gorenstein ring.
\begin{prop}
Let $\mathfrak{m}$ be a maximal ideal of~$\mathbb{T} := \heckeoneNk{N}{2}$ of residue
characteristic~$p$ which may divide~$N$. Denote by $\widetilde{\mathfrak{m}}$
the image of $\mathfrak{m}$ in
$\mathbb{T}_{\mathbb{F}_p} := \mathbb{T} \otimes_\mathbb{Z} \mathbb{F}_p = \mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_2(\Gamma_1(N)))$.
Suppose that the Galois representation~$\rho_\mathfrak{m}$ is irreducible and
satisfies multiplicity one (see Section~\ref{section-introduction}).
Then as $\mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}}$-modules one has
$$ J_1(N)_\mathbb{Q}({\overline{\QQ}})[p]_{\widetilde{\mathfrak{m}}} \cong
\mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}} \oplus \mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}}$$
and the localisations $\mathbb{T}_\mathfrak{m}$ and $\mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}}$
are Gorenstein rings. Similar results
hold if one replaces $\Gamma_1(N)$ and $J_1(N)$ by $\Gamma_0(N)$
and~$J_0(N)$. \end{prop}
{\bf Proof. } For the proof we have to pass to $\mathbb{T}_{\mathbb{Z}_p} = \mathbb{T}_\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z}_p$. We also denote by~$\mathfrak{m}$ the maximal ideal in~$\mathbb{T}_{\mathbb{Z}_p}$ that corresponds to~$\mathfrak{m}$. Let $V$ be the $\mathfrak{m}$-part of the $p$-Tate module of $J_1(N)_\mathbb{Q}$. Multiplicity one implies that $V/\mathfrak{m} V$ is a $2$-dimensional $\mathbb{T}/\mathfrak{m} = \mathbb{T}_{\mathbb{Z}_p}/\mathfrak{m} = \mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}}/\widetilde{\mathfrak{m}}$-vector space, since $$ V/\mathfrak{m} V \cong (V/pV)/\widetilde{\mathfrak{m}}
\cong (J_1(N)_\mathbb{Q}({\overline{\QQ}})[p]) / \widetilde{\mathfrak{m}}
\cong (J_1(N)_\mathbb{Q}({\overline{\QQ}})[p])^\vee / \widetilde{\mathfrak{m}}
\cong (J_1(N)_\mathbb{Q}({\overline{\QQ}})[\mathfrak{m}])^\vee,$$ where the self-duality comes from the modified Weil pairing which respects the Hecke action (see e.g.\ \cite{gross}, p.~485). Nakayama's Lemma hence implies that $V$ is a $\mathbb{T}_{\mathbb{Z}_p,\mathfrak{m}}$-module of rank~$2$. As one knows that $V \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$ is a $\mathbb{T}_\mathfrak{m} \otimes \mathbb{Q}_p$-module of rank~$2$, it follows that $V$ is a free $\mathbb{T}_{\mathbb{Z}_p,\mathfrak{m}}$-module of rank~$2$, whence $J_1(N)_\mathbb{Q}({\overline{\QQ}})[p]_{\widetilde{\mathfrak{m}}}$ is a free $\mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}}$-module of rank~$2$. Taking the $\widetilde{\mathfrak{m}}$-kernel gives $J_1(N)_\mathbb{Q}({\overline{\QQ}})[\mathfrak{m}]=(\mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}}[\widetilde{\mathfrak{m}}])^2$, whence the Gorenstein defect is zero. In the $\Gamma_0$-situation, the same proof holds. \hspace* {.5cm}
$\Box$
In the so-called ordinary case, we have the following precise relationship between the multiplicity and the Gorenstein defect, which was suggested to us by Kevin Buzzard. The proof can be found in~\cite{nongor}.
\begin{prop}\label{gormult}
Suppose $p \nmid N$ and let $M=N$ or $M=Np$. Let $\mathfrak{m}$ be a maximal
ideal of $\heckeoneNk M2$ of residue characteristic~$p$ and assume
that $\mathfrak{m}$ is ordinary,
i.e.\ that the $p$-th Hecke operator~$T_p$ is not in~$\mathfrak{m}$. Assume also
that $\rho_\mathfrak{m}$ is irreducible. Denote
by $\widetilde{\mathfrak{m}}$ the image of~$\mathfrak{m}$ in
$\mathbb{T}_{\mathbb{F}_p} := \mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_2(\Gamma_1(M)))$.
Then the following statements hold: \begin{enumerate}[(a)] \item
There is the exact sequence
$$ 0 \to \mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}}
\to J_1(M)({\overline{\QQ}})[p]_{\widetilde{\mathfrak{m}}}
\to \mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}}^\vee
\to 0$$
of $\mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}}$-modules, where the dual is the
$\mathbb{F}_p$-linear dual.
\item
If $d$ is the Gorenstein defect of $\mathbb{T}_{\mathbb{F}_p,\widetilde{\mathfrak{m}}}$
and $r$ is the multiplicity of $\rho_\mathfrak{m}$, then the relation
$$ d = 2r-2$$
holds. \end{enumerate} \end{prop}
We now establish a relation between mod~$p$ Hecke algebras of weights $3 \le k \le p+1$ for levels~$N$ not divisible by~$p$ with Hecke algebras of weight~$2$ and level~$Np$. It is needed in order to compare the Hecke algebras in higher weight to those acting on the $p$-torsion of Jacobians and thus to make a link to the multiplicity of the attached Galois representations.
\begin{prop}\label{totwo}
Let $N \ge 5$, $p \nmid N$ and $3 \le k \le p+1$. Let $\mathfrak{m}$ be a maximal
ideal of the mod~$p$ Hecke algebra $\mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_k(\Gamma_1(N))$.
Then there exists a maximal ideal $\mathfrak{m}_2$
of $\mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_2(\Gamma_1(Np))$ and a natural surjection
$$\mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_2(\Gamma_1(Np))_{\mathfrak{m}_2} \twoheadrightarrow
\mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_k(\Gamma_1(N))_\mathfrak{m}.$$
If $\mathfrak{m}$ is ordinary ($T_p \not\in \mathfrak{m}$), this surjection is an
isomorphism. \end{prop}
{\bf Proof. } From Sections~5 and~6 of \cite{Faithful}, whose notation we adopt for this proof, one obtains without difficulty the commutative diagram of Hecke algebras: $$ \xymatrix@=1.0cm{ \mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_2(\Gamma_1(Np))_{\mathfrak{m}_2} \ar@{->>}[r] \ar@{=}[d] & \mathbb{T}_{\mathbb{F}_p}(J_1(Np)_\mathbb{Q}({\overline{\QQ}})[p])_{\mathfrak{m}_2} \ar@{->>}[r] \ar@{->>}[d] & \mathbb{T}_{\mathbb{F}_p} (H^1(\Gamma_1(N),V_{k-2}(\mathbb{F}_p)))_\mathfrak{m} \\ \mathbb{T}_{\mathbb{Z}_p \to \mathbb{F}_p} (L)_{\mathfrak{m}_2} \ar@{->>}[r] & \mathbb{T}_{\mathbb{F}_p} (\overline{L})_{\mathfrak{m}_2} \ar@{->>}[r] & \mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_k(\Gamma_1(N))_\mathfrak{m}. \ar@{->>}[u]}$$ The claimed surjection can be read off. In the ordinary situation, Proposition~\ref{gormult} shows that the upper left horizontal arrow is in fact an isomorphism. That also the upper right horizontal arrow is an isomorphism is explained in~\cite{Faithful}. The result follows immediately. \hspace* {.5cm}
$\Box$
In the next proposition we compare Hecke algebras for spaces of modular forms on $\Gamma_1(N)$ to those of the same level and weight, but with a Dirichlet character.
\begin{prop}
Let $N \ge 5$, $k \ge 2$ and let $\chi: (\mathbb{Z}/N\mathbb{Z})^\times \to
\mathbb{C}^\times$ be a Dirichlet character. Let $f \in S_k(N,\chi)
\subset S_k(\Gamma_1(N))$ be a normalised Hecke eigenform. Let
further $\mathfrak{m}_{\overline{\chi}}$ be the maximal ideal in $\mathbb{T}_\mathbb{F}^{\overline{\chi}} :=
\mathbb{T}_{\mathbb{Z}[\chi]\to\mathbb{F}}(S_k(N,\chi))$ and $\mathfrak{m}$ the one in
$\mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_k(\Gamma_1(N)))$ of residue characteristic~$p$
for $p \nmid N$ belonging to~$f$. If $k=2$, suppose additionally
that $\rho_\mathfrak{m}$ is irreducible. If $p=2$, suppose that $\rho_\mathfrak{m}$ is
not induced from $\mathbb{Q}(\sqrt{-1})$, and if $p=3$, suppose that
$\rho_\mathfrak{m}$ is not induced from $\mathbb{Q}(\sqrt{-3})$.
Then the Gorenstein defects of $\mathbb{T}_{\mathbb{Z}[\chi]\to\mathbb{F}}(S_k(N,\chi))_{\mathfrak{m}_{\overline{\chi}}}$
and $\mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_k(\Gamma_1(N)))_\mathfrak{m}$ are equal. \end{prop}
{\bf Proof. } Write $\Delta := (\mathbb{Z}/N\mathbb{Z})^\times$ and let $\Delta_p$ be its $p$-Sylow subgroup. Let ${\overline{\chi}}: \Delta \to \mathbb{F}^\times$ be the reduction of~$\chi$ obtained by composing~$\chi$ with~$\mathbb{Z}[\chi]\to\mathbb{F}$. As the Gorenstein defect is invariant under base extension, it is no loss to work with $\mathbb{T}_\mathbb{F} := \mathbb{T}_{\mathbb{Z}\to\mathbb{F}}(S_k(\Gamma_1(N)))$. We still write~$\mathfrak{m}$ for the maximal ideal in $\mathbb{T}_\mathbb{F}$ belonging to~$f$. Note that $\langle \delta \rangle - {\overline{\chi}}(\delta) \in \mathfrak{m}$ for all~$\delta \in \Delta$.
We let $\Delta$ act on $\mathbb{T}_\mathbb{F}$ via the diamond operators and we let $\mathbb{F}^{\overline{\chi}}$ be a copy of~$\mathbb{F}$ with $\Delta$-action through the inverse of~${\overline{\chi}}$. We have $$ (\mathbb{T}_{\mathbb{F},\mathfrak{m}} \otimes_\mathbb{F} \mathbb{F}^{\overline{\chi}})_\Delta
= (\mathbb{T}_{\mathbb{F},\mathfrak{m}} \otimes_\mathbb{F} \mathbb{F}^{\overline{\chi}})/(1-\delta | \delta \in \Delta)
\cong \mathbb{T}_{\mathbb{F},\mathfrak{m}_{\overline{\chi}}}^{\overline{\chi}},$$ which one obtains by considering the duals, identifying Katz cusp forms with mod~$p$ ones on~$\Gamma_1(N)$ and applying Carayol's Lemma (\cite{edixhoven-boston}, Prop.~1.10). For the case $k=2$, we should point the reader to the correction at the end of the introduction to~\cite{edixhoven}. However, the statement still holds after localisation at maximal ideals corresponding to irreducible representations. Moreover, the equality $ (\mathbb{T}_{\mathbb{F},\mathfrak{m}} \otimes_\mathbb{F} \mathbb{F}^{\overline{\chi}})^\Delta
= \mathbb{T}_{\mathbb{F},\mathfrak{m}}[\langle \delta \rangle - {\overline{\chi}}(\delta) | \delta \in \Delta ]$ holds by definition.
Now Lemma~7.3 of~\cite{Faithful} tells us that the localisation at~$\mathfrak{m}$ of the $\mathbb{F}$-vector space of Katz cusp forms of weight~$k$ on $\Gamma_1(N)$ over~$\mathbb{F}$ is a free $\mathbb{F}[\Delta_p]$-module. Note that the standing hypothesis $k \le p+1$ of Section~7 of~\cite{Faithful} is not used in the proof of that lemma and see also \cite{Faithful},~Remark~7.5. From an elementary calculation one now obtains that $N_\Delta = \sum_{\delta \in \Delta} \delta$ induces an isomorphism $$ (\mathbb{T}_{\mathbb{F}_\mathfrak{m}} \otimes_\mathbb{F} \mathbb{F}^{\overline{\chi}})_\Delta
\xrightarrow{N_\Delta}
(\mathbb{T}_{\mathbb{F}_\mathfrak{m}} \otimes_\mathbb{F} \mathbb{F}^{\overline{\chi}})^\Delta.$$ We now take the $\mathfrak{m}_{\overline{\chi}}$-kernel on both sides and obtain $$ \mathbb{T}_{\mathbb{F},\mathfrak{m}_{\overline{\chi}}}^{\overline{\chi}} [\mathfrak{m}_{\overline{\chi}}]
\cong
(\mathbb{T}_{\mathbb{F},\mathfrak{m}} \otimes_\mathbb{F} \mathbb{F}^{\overline{\chi}})_\Delta [\mathfrak{m}_{\overline{\chi}}]
\cong
(\mathbb{T}_{\mathbb{F},\mathfrak{m}} \otimes_\mathbb{F} \mathbb{F}^{\overline{\chi}})_\Delta [\mathfrak{m}]
\cong
(\mathbb{T}_{\mathbb{F},\mathfrak{m}} \otimes_\mathbb{F} \mathbb{F}^{\overline{\chi}})^\Delta [\mathfrak{m}]
=
\mathbb{T}_{\mathbb{F},\mathfrak{m}}[\mathfrak{m}].$$ This proves that the two Gorenstein defects are indeed equal. \hspace* {.5cm}
$\Box$
The Gorenstein defect that we calculate on the computer is the number~$d$ of the following corollary, which relates it to the multiplicity of a Galois representation.
\begin{cor}
Let $p$ be a prime, $N \ge 5$ an integer such that $p \nmid N$, $k$
an integer satisfying $2 \le k \le p$ and $\chi: (\mathbb{Z}/N\mathbb{Z})^\times
\to \mathbb{C}^\times$ a character. Let $f \in S_k(N,\chi)$ be a
normalised Hecke eigenform. Let further $\mathfrak{m}$ denote the maximal
ideal in $\mathbb{T}_{\mathbb{Z}[\chi]\to\mathbb{F}}(S_k(N,\chi))$ belonging to~$f$.
Suppose that $\mathfrak{m}$ is ordinary and that $\rho_\mathfrak{m}$ is irreducible and
not induced from $\mathbb{Q}(\sqrt{-1})$ (if $p=2$) and not induced from
$\mathbb{Q}(\sqrt{-3})$ (if $p=3$). We define~$d$ to be the Gorenstein
defect of $\mathbb{T}_{\mathbb{Z}[\chi]\to\mathbb{F}}(S_k(N,\chi))_\mathfrak{m}$ and~$r$ to be the
multiplicity of~$\rho_\mathfrak{m}$.
Then the equality $d = 2r-2$ holds.
\hspace* {.5cm}
$\Box$ \end{cor}
We include the following proposition because it establishes equality of the two different notions of multiplicities of Galois representations in the case of the trivial character.
\begin{prop}\label{nulleins}
Let $N \ge 1$ and $p \nmid N$ and $f \in S_2(\Gamma_0(N))
\subseteq S_2(\Gamma_1(N))$ be a normalised Hecke eigenform
belonging to maximal ideals~$\mathfrak{m}_0 \subseteq \mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_2(\Gamma_0(N)))$
and $\mathfrak{m}_1 \subseteq \mathbb{T}_{\mathbb{Z}\to\mathbb{F}_p}(S_2(\Gamma_1(N)))$ of residue characteristic~$p$.
Suppose that $\rho_{\mathfrak{m}_0} \cong \rho_{\mathfrak{m}_1}$ is irreducible.
Then the multiplicity of $\rho_{\mathfrak{m}_1}$ on
$J_1(N)_\mathbb{Q}({\overline{\QQ}})[p]$ is equal to the multiplicity of~$\rho_{\mathfrak{m}_0}$ on
$J_0(N)_\mathbb{Q}({\overline{\QQ}})[p]$. Thus, if $p>2$, this multiplicity is equal
to one by Theorem~\ref{gorenstein-theorem}. \end{prop}
{\bf Proof. } Let $\Delta := (\mathbb{Z}/N\mathbb{Z})^\times$. We first remark that one has the isomorphism $$ J_0(N)_\mathbb{Q}({\overline{\QQ}})[p]_{\mathfrak{m}_0} \cong \big((J_1(N)_\mathbb{Q}({\overline{\QQ}})[p])^\Delta\big)_{\mathfrak{m}_0},$$ which one can for example obtain by comparing with the parabolic cohomology with $\mathbb{F}_p$-coefficients of the modular curves $Y_0(N)$ and $Y_1(N)$. Taking the $\mathfrak{m}_0$-kernel yields $$ J_0(N)_\mathbb{Q}({\overline{\QQ}})[\mathfrak{m}_0] \cong J_1(N)_\mathbb{Q}({\overline{\QQ}})[\mathfrak{m}_1],$$ since $\mathfrak{m}_1$ contains $\langle \delta \rangle - 1 $ for all~$\delta \in \Delta$. \hspace* {.5cm}
$\Box$
\section{Modular Symbols and Hecke Algebras} \label{modular-symbols}
The aim of this section is to present the algorithm that we use for the computations of local factors of Hecke algebras of mod~$p$ modular forms. It is based on mod~$p$ modular symbols which have been implemented in {\sc Magma} \cite{magma} by William Stein.
The bulk of this section deals with proving the main advance, namely a stop criterion (Corollary~\ref{stopcor}), which in practice greatly speeds up the computations in comparison with ``standard'' implementations, as it allows us to work with many fewer Hecke operators than indicated by the theoretical Sturm bound (Proposition~\ref{propsturm}). We shall list results proving that the stop criterion is attained in many cases. However, the stop criterion does not depend on them, in the sense that it being attained is equivalent to a proof that the algebra it outputs is equal to a direct factor of a Hecke algebra of mod~$p$ modular forms.
Whereas for Section~\ref{section-relation} the notion of Katz modular forms seems the right one, the present section works entirely with mod~$p$ modular forms, the definition of which is also recalled. This is very natural, since all results in this section are based on a comparison with the characteristic zero theory.
\subsection{Mod~$p$ modular forms and modular symbols}
\subsubsection*{Mod~$p$ modular forms}
Let us for the time being fix integers $N \ge 1$ and $k \ge 2$, as well as a character $\chi: (\mathbb{Z}/N\mathbb{Z})^\times \to \mathbb{C}^\times$ such that $\chi(-1) = (-1)^k$. Let $M_k(N,\chi)$ be the space of holomorphic modular forms for $\Gamma_1(N)$, Dirichlet character $\chi$, and weight~$k$. It decomposes as a direct sum (orthogonal direct sum with respect to the Petersson inner product) of its cuspidal subspace $S_k(N,\chi)$ and its Eisenstein subspace ${\rm Eis}_k(N,\chi)$. As before, we let $\mathcal{O} = \mathbb{Z}[\chi]$. Moreover, we let $\mathfrak{P}$ be a maximal ideal of~$\mathcal{O}$ above~$p$ with residue field~$\mathbb{F}$ and $\widehat{\mathcal{O}}$ be the completion of~$\mathcal{O}$ at~$\mathfrak{P}$. Furthermore, let $K = \mathbb{Q}_p(\chi)$ be the field of fractions of~$\widehat{\mathcal{O}}$ and $\bar{\chi}$ be $\chi$ followed by the natural projection $\mathcal{O} \twoheadrightarrow \mathbb{F}$.
Denote by $M_k(N,\chi\,;\,\mathcal{O})$ the sub-$\mathcal{O}$-module generated by those modular forms whose (standard) $q$-expansion has coefficients in~$\mathcal{O}$. It follows from the $q$-expansion principle that $$M_k(N,\chi\,;\,\mathcal{O}) \cong
{\rm Hom}_{\mathcal{O}} \big(\mathbb{T}_{\mathcal{O}}(M_k(N,\chi)),\mathcal{O}\big)$$ and that hence $M_k(N,\chi\,;\,\mathcal{O}) \otimes_{\mathcal{O}} \mathbb{C} \cong M_k(N,\chi)$. We put $$ M_k(N,\bar{\chi}\,;\,\mathbb{F}) :=
M_k(N,\chi\,;\,\mathcal{O}) \otimes_{\mathcal{O}} \mathbb{F} \cong
{\rm Hom}_{\mathbb{F}} \big(\mathbb{T}_{\mathcal{O}}(M_k(N,\chi)),\mathbb{F}\big) $$ and call the elements of this space {\em mod~$p$ modular forms}. The Hecke algebra $\mathbb{T}_{\mathcal{O}}(M_k(N,\chi))$ acts naturally and it follows that $\mathbb{T}_{\mathcal{O}\to\mathbb{F}}(M_k(N,\chi)) \cong \mathbb{T}_{\mathbb{F}}(M_k(N,\chi\,;\,\mathbb{F}))$. Similar statements hold for the cuspidal and the Eisenstein subspaces and we use similar notations.
We call a maximal ideal $\mathfrak{m}$ of $\mathbb{T}_{\mathcal{O}\to\mathbb{F}} (M_k(N,\chi\,;\,\mathcal{O})$ (respectively, the corresponding maximal ideal of $\mathbb{T}_{\mathcal{O}\to\widehat{\mathcal{O}}} (M_k(N,\chi\,;\,\mathcal{O}))$) {\em non-Eisenstein} if and only if $$ S_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m} \cong M_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m}.$$ Otherwise, we call $\mathfrak{m}$ {\em Eisenstein}.
We now include a short discussion of minimal and maximal primes, in view of Proposition~\ref{commalg}. Write $\mathbb{T}_{\widehat{\mathcal{O}}}$ for $\mathbb{T}_{\mathcal{O} \to \widehat{\mathcal{O}}} (S_k(N,\chi))$. Let $\mathfrak{m}$ be a maximal ideal of $\mathbb{T}_{\widehat{\mathcal{O}}}$. It corresponds to a $\Gal({\overline{\FF}_p}/\mathbb{F})$-conjugacy class of normalised eigenforms in $S_k(N,\bar{\chi}\,;\,\mathbb{F})$. That means for each $n\in \mathbb{N}$ that the minimal polynomial of $T_n$ acting on $S_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m}$ is equal to a power of the minimal polynomial of the coefficient~$a_n$ of each member of the conjugacy class. Moreover, $\mathfrak{p}$ corresponds precisely as above to a $\Gal({\overline{\QQ}_p}/K)$-conjugacy class of normalised eigenforms in $S_k(N,\chi\,;\,\mathcal{O}) \otimes_\mathcal{O} K$.
Suppose that $\mathfrak{m}$ contains minimal primes $\mathfrak{p}_i$ for $i=1,\dots,r$. Then the normalised eigenforms corresponding to the~$\mathfrak{p}_i$ are congruent to one another modulo a prime above~$p$. Conversely, any congruence arises in this way. Thus, a maximal ideal $\mathfrak{m}$ of $\mathbb{T}_{\widehat{\mathcal{O}}}$ is Eisenstein if and only if it contains a minimal prime corresponding to a conjugacy class of Eisenstein series. As it is the reduction of a reducible representation, the mod~$p$ Galois representation corresponding to a non-Eisenstein prime is reducible. It should be possible to show the converse.
\subsubsection*{Modular symbols}
We now recall the modular symbols formalism and prove two useful results on base change and torsion. The main references for the definitions are \cite{SteinBook} and~\cite{HeckeMS}.
Let $R$ be a ring, $\Gamma \le \mathrm{SL}_2(\mathbb{Z})$ a subgroup and $V$ a left $R[\Gamma]$-module. Recall that $\mathbb{P}^1(\mathbb{Q}) = \mathbb{Q} \cup \{\infty\}$ is the set of cusps of $\mathrm{SL}_2(\mathbb{Z})$, which carries a natural $\mathrm{SL}_2(\mathbb{Z})$-action via fractional linear transformations. We define the $R$-modules
$$ \mathcal{M}_R := R[\{\alpha,\beta\}| \alpha,\beta \in \mathbb{P}^1(\mathbb{Q})]/ \langle \{\alpha,\alpha\}, \{\alpha,\beta\} + \{\beta,\gamma\} +
\{\gamma,\alpha\} | \alpha,\beta,\gamma \in \mathbb{P}^1(\mathbb{Q})\rangle$$ and $\mathcal{B}_R := R[\mathbb{P}^1(\mathbb{Q})]$. They are connected via the {\em boundary map}
$\delta: \mathcal{M}_R \to \mathcal{B}_R$ which is given by $\{\alpha,\beta\} \mapsto \beta - \alpha.$ Both are equipped with the natural left $\Gamma$-actions. Also let $\mathcal{M}_R(V) := \mathcal{M}_R \otimes_R V$ and $\mathcal{B}_R(V) := \mathcal{B}_R \otimes_R V$ with the left diagonal $\Gamma$-action. We call the $\Gamma$-coinvariants $$ \mathcal{M}_R (\Gamma,V) := \mathcal{M}_R(V)_\Gamma =
\mathcal{M}_R(V)/ \langle (x - g x) | g \in \Gamma, x \in \mathcal{M}_R(V) \rangle$$ {\em the space of $(\Gamma,V)$-modular symbols.} Furthermore, {\em the space of $(\Gamma,V)$-boundary symbols} is defined as the $\Gamma$-coinvariants $$ \mathcal{B}_R(\Gamma,V) := \mathcal{B}_R(V)_\Gamma =
\mathcal{B}_R(V)/ \langle (x - g x) | g \in \Gamma, x \in \mathcal{B}_R(V) \rangle.$$ The boundary map $\delta$ induces the {\em boundary map} $\mathcal{M}_R(\Gamma,V) \to \mathcal{B}_R(\Gamma,V)$. Its kernel is denoted by $\mathcal{CM}_R(\Gamma,V)$ and is called {\em the space of cuspidal $(\Gamma,V)$-modular symbols.}
Let now $N \ge 1$ and $k \ge 2$ be integers and
$\chi: (\mathbb{Z}/N\mathbb{Z})^\times \to R^\times$ be a character, i.e.\ a group homomorphism, such that
$\chi(-1) = (-1)^k$ in~$R$. Write $V_{k-2}(R)$ for the homogeneous polynomials of degree $k-2$ over~$R$ in two variables, equipped with the natural $\Gamma_0(N)$-action. Denote by $V_{k-2}^\chi (R)$ the tensor product
$V_{k-2}(R) \otimes_R R^\chi$ for the diagonal $\Gamma_0(N)$-action which on $R^\chi$ comes from the isomorphism
$\Gamma_0(N) / \Gamma_1(N) \cong (\mathbb{Z}/N\mathbb{Z})^\times$ given by sending $\mat abcd$ to~$d$ followed by~$\chi^{-1}$.
We use the notation $\mathcal{M}_k(N,\chi\,;\,R)$ for $\mathcal{M}(\Gamma_0(N),V_{k-2}^\chi(R))$, as well as similarly for the boundary and the cuspidal spaces. The natural action of the matrix $\eta = \mat {-1}001$ gives an involution on all of these spaces. We will denote by the superscript ${}^+$ the subspace invariant under this involution, and by the superscript ${}^-$ the anti-invariant one. On all modules discussed so far one has Hecke operators $T_n$ for all $n \in \mathbb{N}$ and diamond operators. For a definition see~\cite{SteinBook}.
\begin{lem}\label{basechange}
Let $R$, $\Gamma$ and $V$ as above and let $R \to S$ be a ring
homomorphism. Then
$$\mathcal{M}(\Gamma,V) \otimes_R S \cong \mathcal{M}(\Gamma,V \otimes_R S).$$ \end{lem}
{\bf Proof. } This follows immediately from the fact that tensoring and taking coinvariants are both right exact. \hspace* {.5cm}
$\Box$
\begin{prop}\label{torsion} Let $R$ be a local integral domain of characteristic zero with principal maximal ideal $\mathfrak{m} = (\pi)$ and residue field $\mathbb{F}$ of characteristic~$p$. Also let $N \ge 1$, $k \ge 2$ be integers and $\chi: (\mathbb{Z}/N\mathbb{Z})^\times \to R^\times$ a character such that $\chi(-1) = (-1)^k$. Suppose
(i) that $p \ge 5$ or
(ii) that $p = 2$ and $N$ is divisible by a prime which is
$3$ modulo~$4$ or by~$4$ or
(iii) that $p = 3$ and $N$ is divisible by a prime which is $2$
modulo~$3$ or by~$9$. Then the following statements hold: \begin{enumerate}[(a)] \item If $k \ge 3$, then $\mathcal{M}_k(N,\chi\,;\,R)[\pi] = \big(V_{k-2}^\chi(\mathbb{F})\big)^{\Gamma_0(N)}$. \item If $k=2$ or if $3 \le k \le p+2$ and $p \nmid N$, then $\mathcal{M}_k(N,\chi\,;\,R)[\pi] = 0$. \end{enumerate} \end{prop}
{\bf Proof. } The conditions assure that the group $\Gamma_0(N)$ does not have any stabiliser of order $2p$ for its action on the upper half plane. Hence, by \cite{HeckeMS}, Theorem~6.1, the modular symbols space $\mathcal{M}_k(N,\chi\,;\,R)$ is isomorphic to $H^1(\Gamma_0(N),V_{k-2}^\chi(R))$. The arguments are now precisely those of the beginning of the proof of \cite{Faithful}, Proposition~2.6. \hspace* {.5cm}
$\Box$
\subsubsection*{Hecke algebras of modular symbols and the Eichler-Shimura isomorphism}
From Lemma~\ref{basechange} one deduces a natural surjection \begin{equation}\label{eqms}
\mathbb{T}_{\mathcal{O} \to \mathbb{F}} (\mathcal{M}_k(N,\chi\,;\,\mathcal{O}))
\twoheadrightarrow
\mathbb{T}_{\mathbb{F}}(\mathcal{M}_k(N,\bar{\chi}\,;\,\mathbb{F})). \end{equation} In the same way, one also obtains \begin{equation}\label{eqmseins}
\mathbb{T}_{\mathcal{O}} (\mathcal{M}_k(N,\chi\,;\,\mathcal{O})) \twoheadrightarrow
\mathbb{T}_{\mathcal{O}} (\mathcal{M}_k(N,\chi\,;\,\mathcal{O})/\text{torsion})
\cong \mathbb{T}_{\mathcal{O}} (\mathcal{M}_k(N,\chi\,;\,\mathbb{C})), \end{equation} where one uses for the isomorphism that the Hecke operators are already defined over~$\mathcal{O}$. A similar statement holds for the cuspidal subspace.
We call a maximal prime $\mathfrak{m}$ of $\mathbb{T}_{\mathcal{O}\to\widehat{\mathcal{O}}} (\mathcal{M}_k(N,\chi\,;\,\mathcal{O}))$ (respectively the corresponding prime of $\mathbb{T}_{\mathcal{O}\to\mathbb{F}} (\mathcal{M}_k(N,\chi\,;\,\mathcal{O}))$) {\em non-torsion} if $$ \mathcal{M}_k(N,\chi\,;\,\widehat{\mathcal{O}})_\mathfrak{m} \cong
(\mathcal{M}_k(N,\chi\,;\,\widehat{\mathcal{O}})/\text{torsion})_\mathfrak{m}.$$ This is equivalent to the height of $\mathfrak{m}$ being~$1$. Proposition~\ref{torsion} tells us some cases in which all primes are non-torsion.
\begin{thm}[Eichler-Shimura]\label{thmes} There are isomorphisms respecting the Hecke operators \begin{enumerate}[(a)] \item $M_k(N,\chi) \oplus S_k(N,\chi)^\vee
\cong \mathcal{M}_k(N,\chi\,;\,\mathbb{C}),$ \item $S_k(N,\chi) \oplus S_k(N,\chi)^\vee
\cong \mathcal{CM}_k(N,\chi\,;\,\mathbb{C}),$ \item $S_k(N,\chi) \cong \mathcal{CM}_k(N,\chi\,;\,\mathbb{C})^+.$ \end{enumerate} \end{thm}
{\bf Proof. } Parts~(a) and (b) are \cite{DiamondIm}, Theorem 12.2.2, together with the comparison of \cite{HeckeMS}, Theorem~6.1. We use that the space of anti-holomorphic cusp forms is dual to the space of holomorphic cusp forms. Part~(c) is a direct consequence of~(b). \hspace* {.5cm}
$\Box$
\begin{cor}\label{cores} There are isomorphisms $$\mathbb{T}_{\mathcal{O}}(S_k(N,\chi)) \cong \mathbb{T}_{\mathcal{O}} (\mathcal{CM}_k(N,\chi\,;\,\mathbb{C}))
\cong \mathbb{T}_{\mathcal{O}} (\mathcal{CM}_k(N,\chi\,;\,\mathbb{C})^+),$$ given by sending $T_n$ to $T_n$ for all positive~$n$. \hspace* {.5cm}
$\Box$ \end{cor}
\subsection{The stop criterion}
Although it is impossible to determine a priori the dimension of the local factor of the Hecke algebra associated with a given modular form mod~$p$, Corollary~\ref{stopcor} implies that the computation of Hecke operators can be stopped when the algebra generated has reached a certain dimension that is computed along the way. This criterion has turned out to be extremely useful and has made possible some of our computations which would not have been feasible using the Hecke bound naively. See Section~\ref{computational-results} for a short discussion of this issue.
\subsubsection*{Some commutative algebra}
We collect some useful statements from commutative algebra, which will be applied to Hecke algebras in the sequel.
\begin{prop}\label{commalg} Let $R$ be an integral domain of characteristic zero which is a finitely generated $\mathbb{Z}$-module. Write $\widehat{R}$ for the completion of $R$ at a maximal ideal of~$R$ and denote by $\mathbb{F}$ the residue field and by $K$ the fraction field of~$\widehat{R}$. Let furthermore $A$ be a commutative $R$-algebra which is finitely generated as an $R$-module. For any ring homomorphism $R\to S$ write $A_S$ for $A \otimes_R S$. Then the following statements hold.
\begin{enumerate}[(a)] \item The Krull dimension of $A_{\widehat{R}}$ is less than or equal
to~$1$. The maximal ideals of $A_{\widehat{R}}$ correspond
bijectively under taking pre-images to the maximal ideals of
$A_\mathbb{F}$. Primes $\mathfrak{p}$ of height $0$ which are contained in a prime
of height $1$ of $A_{\widehat{R}}$ are in bijection with primes of
$A_K$ under extension (i.e.\ $\mathfrak{p} A_K$), for which the notation
$\mathfrak{p}^e$ will be used.
Under these correspondences, one has
$A_{\mathbb{F},\mathfrak{m}} \cong A_{\widehat{R},\mathfrak{m}} \otimes_{\widehat{R}} \mathbb{F}$
and
$A_{K,\mathfrak{p}^e} \cong A_{\widehat{R},\mathfrak{p}}$.
\item The algebra $A_{\widehat{R}}$ decomposes as
$$ A_{\widehat{R}} \cong \prod_\mathfrak{m} A_{\widehat{R},\mathfrak{m}},$$
where the product runs over the maximal ideals $\mathfrak{m}$ of
$A_{\widehat{R}}$.
\item The algebra $A_\mathbb{F}$ decomposes as
$$ A_\mathbb{F} \cong \prod_\mathfrak{m} A_{\mathbb{F},\mathfrak{m}},$$
where the product runs over the maximal ideals $\mathfrak{m}$ of $A_\mathbb{F}$.
\item The algebra $A_K$ decomposes as
$$ A_K \cong \prod_\mathfrak{p} A_{K,\mathfrak{p}^e} \cong \prod_\mathfrak{p} A_{\widehat{R},\mathfrak{p}},$$
where the products run over the minimal prime ideals $\mathfrak{p}$ of
$A_{\widehat{R}}$ which are contained in a prime ideal of
height~$1$. \end{enumerate} \end{prop}
{\bf Proof. } As $A_{\widehat{R}}$ is a finitely generated $\widehat{R}$-module, $A_{\widehat{R}}/\mathfrak{p}$ with a prime $\mathfrak{p}$ is an integral domain which is a finitely generated $\widehat{R}$-module. Hence, it is either a finite field or a finite extension of $\widehat{R}$. This proves that the height of $\mathfrak{p}$ is less than or equal to~$1$. The correspondences and the isomorphisms of Part~(a) are easily verified. The decompositions in Parts~(b) and~(c) are \cite{Eisenbud}, Corollary~7.6. Part~(d) follows by tensoring~(b) over $\widehat{R}$ with~$K$. \hspace* {.5cm}
$\Box$
Similar decompositions for $A$-modules are derived by applying the idempotents of the decompositions of Part~(b).
\begin{prop}\label{dimension} Assume the set-up of Proposition~\ref{commalg} and let $M,N$ be $A$-modules which as $R$-modules are free of finite rank. Suppose that \begin{enumerate}[(a)] \item $M \otimes_R \mathbb{C} \cong N \otimes_R \mathbb{C}$ as $A \otimes_R \mathbb{C}$-modules, or \item $M \otimes_R \bar{K} \cong N \otimes_R \bar{K}$ as $A \otimes_R \bar{K}$-modules. \end{enumerate} Then for all prime ideals $\mathfrak{m}$ of $A_\mathbb{F}$ corresponding to height $1$ primes of $A_{\widehat{R}}$ the equality $$ \dim_\mathbb{F} (M \otimes_R \mathbb{F})_\mathfrak{m} = \dim_\mathbb{F} (N \otimes_R \mathbb{F})_\mathfrak{m}$$ holds. \end{prop}
{\bf Proof. } As for $A$, we also write $M_K$ for $M \otimes_R K$ and similarly for $N$ and $\widehat{R}$, $\mathbb{F}$, etc. By choosing an isomorphism $\mathbb{C} \cong \bar{K}$, it suffices to prove Part~(b). Using Proposition~\ref{commalg}, Part~(d), the isomorphism $M \otimes_R \bar{K} \cong N \otimes_R \bar{K}$ can be rewritten as $$ \bigoplus_{\mathfrak{p}} (M_{K, \mathfrak{p}^e} \otimes_K \bar{K}) \cong \bigoplus_{\mathfrak{p}} (N_{K, \mathfrak{p}^e} \otimes_K \bar{K}),$$ where the sums run over the minimal primes $\mathfrak{p}$ of $A_{\widehat{R}}$ which are properly contained in a maximal prime. Hence, an isomorphism $M_{K, \mathfrak{p}^e} \otimes_K \bar{K} \cong N_{K, \mathfrak{p}^e} \otimes_K \bar{K}$ exists for each~$\mathfrak{p}$. Since for each maximal ideal $\mathfrak{m}$ of $A_{\widehat{R}}$ of height $1$ we have by Proposition~\ref{commalg} $$ M_{\widehat{R}, \mathfrak{m}} \otimes_{\widehat{R}} K \cong
\bigoplus_{\mathfrak{p} \subseteq \mathfrak{m} \text{ min.}} M_{K, \mathfrak{p}^e}$$ and similarly for $N$, we get \begin{align*}
\dim_\mathbb{F} M_{\mathbb{F}, \mathfrak{m}}
=& {\rm rk}_{\widehat{R}} M_{\widehat{R}, \mathfrak{m}}
= \sum_{\mathfrak{p} \subseteq \mathfrak{m} \text{ min.}} \dim_K M_{K, \mathfrak{p}^e} \\
=& \sum_{\mathfrak{p} \subseteq \mathfrak{m} \text{ min.}} \dim_K N_{K, \mathfrak{p}^e}
= {\rm rk}_{\widehat{R}} N_{\widehat{R}, \mathfrak{m}}
= \dim_\mathbb{F} N_{\mathbb{F}, \mathfrak{m}}. \end{align*} This proves the proposition. \hspace* {.5cm}
$\Box$
\subsubsection*{The stop criterion}
\begin{prop}\label{propdim} Let $\mathfrak{m}$ be a maximal ideal of $\mathbb{T}_{\mathcal{O}\to\mathbb{F}}(\mathcal{M}_k(N,\chi\,;\,\mathcal{O}))$ which is non-torsion and non-Eisenstein. Then the following statements hold: \begin{enumerate}[(a)] \item $\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m} \cong \mathcal{M}_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m}$. \item $2 \cdot \dim_{\mathbb{F}} S_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m} =
\dim_{\mathbb{F}} \mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m}$. \item If $p \neq 2$, then
$\dim_{\mathbb{F}} S_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m} =
\dim_{\mathbb{F}} \mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})^+_\mathfrak{m}$. \end{enumerate} \end{prop}
{\bf Proof. } Part~(c) follows directly from Part~(b) by decomposing $\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})$ into a direct sum of its plus- and its minus-part. Statements~(a) and~(b) will be concluded from Proposition~\ref{dimension}. More precisely, it allows us to derive from Theorem~\ref{thmes} that \begin{align*}
&\dim_{\mathbb{F}}\big((\mathcal{M}_k(N,\chi\,;\,\mathcal{O})/\text{torsion})
\otimes_{\mathcal{O}} \mathbb{F}\big)_\mathfrak{m}\\ = &\dim_{\mathbb{F}}\big({\rm Eis}_k (N,\bar{\chi}\,;\,\mathbb{F})
\oplus S_k (N,\bar{\chi}\,;\,\mathbb{F})
\oplus S_k (N,\bar{\chi}\,;\,\mathbb{F})^\vee \big)_\mathfrak{m} \end{align*} and $$ \dim_{\mathbb{F}}\big((\mathcal{CM}_k(N,\chi\,;\,\mathcal{O})/\text{torsion})
\otimes_{\mathcal{O}} \mathbb{F}\big)_\mathfrak{m} = 2 \cdot \dim_{\mathbb{F}}\big(S_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m}.$$ The latter proves Part~(b), since $\mathfrak{m}$ is non-torsion. As by the definition of a non-Eisenstein prime ${\rm Eis}_k (N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m} = 0$ and again since $\mathfrak{m}$ is non-torsion, it follows that $$\dim_{\mathbb{F}}\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m} = \dim_{\mathbb{F}}\mathcal{M}_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m},$$ which implies Part~(a). \hspace* {.5cm}
$\Box$
We will henceforth often regard non-Eisenstein non-torsion primes as in the proposition as maximal primes of $\mathbb{T}_{\mathbb{F}}(S_k(N,\bar{\chi}\,;\,\mathbb{F})) = \mathbb{T}_{\mathcal{O}\to\mathbb{F}}(S_k(N,\chi))$.
\begin{cor}[Stop Criterion]\label{stopcor} Let $\mathfrak{m}$ be a maximal ideal of $\mathbb{T}_{\mathbb{F}}(S_k(N,\bar{\chi}\,;\,\mathbb{F}))$ which is non-Eisen\-stein and non-torsion. \begin{enumerate}[(a)] \item One has $\dim_{\mathbb{F}} \mathcal{M}_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m} = 2 \cdot \dim_{\mathbb{F}} \mathbb{T}_{\mathbb{F}} \big(\mathcal{M}_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m}$ if and only if $$\mathbb{T}_{\mathbb{F}} \big(S_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m} \cong \mathbb{T}_{\mathbb{F}} \big(\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m}.$$ \item One has $\dim_{\mathbb{F}} \mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})_\mathfrak{m} = 2 \cdot \dim_{\mathbb{F}} \mathbb{T}_{\mathbb{F}} \big(\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m}$ if and only if $$\mathbb{T}_{\mathbb{F}} \big(S_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m} \cong \mathbb{T}_{\mathbb{F}} \big(\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m}.$$ \item Assume $p \neq 2$. One has $\dim_{\mathbb{F}} \mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})^+_\mathfrak{m} = \dim_{\mathbb{F}} \mathbb{T}_{\mathbb{F}} \big(\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m}$ if and only if $$\mathbb{T}_{\mathbb{F}} \big(S_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m} \cong \mathbb{T}_{\mathbb{F}} \big(\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})^+\big)_\mathfrak{m}.$$ \end{enumerate} \end{cor}
{\bf Proof. } We only prove (a), as (b) and (c) are similar. From Part~(b) of Proposition~\ref{propdim} and the fact that the $\mathbb{F}$-dimension of the algebra $\mathbb{T}_{\mathbb{F}} \big(S_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m}$ is equal to the one of $S_k(N,\bar{\chi}\,;\,\mathbb{F})$, as they are dual to each other, it follows that $$ 2 \cdot \dim_\mathbb{F} \mathbb{T}_{\mathbb{F}} \big(S_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m} = \dim_\mathbb{F} \big(\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m}.$$ The result is now a direct consequence of Equations~\ref{eqms} and~\ref{eqmseins} and Corollary~\ref{cores}. \hspace* {.5cm}
$\Box$
Note that the first line of each statement only uses modular symbols and not modular forms, but it allows us to make statements involving modular forms. This is the aforementioned stop criterion; the computation of Hecke operators can be stopped if this equality is reached.
We now list some results concerning the validity of the equivalent statements of Corollary~\ref{stopcor}. The reader can also consult \cite{EPW} for general results in the ordinary and distinguished case.
\begin{prop} Let $p \ge 5$ be a prime, $k \ge 2$ and $N \ge 5$ with $p \nmid N$ integers, $\mathbb{F}$ a finite extension of~$\mathbb{F}_p$, $\bar{\chi}: (\mathbb{Z}/N\mathbb{Z})^\times \to \mathbb{F}^\times$ a character and $\mathfrak{m}$ a maximal ideal of $\mathbb{T}_\mathbb{F}(S_k(N,\bar{\chi}\,;\,\mathbb{F}))$ which is non-Eisenstein and non-torsion. Suppose (i) that $2 \le k \le p-1$ or (ii) that $k \in \{p,p+1\}$ and $\mathfrak{m}$ is ordinary. Then $$ \mathbb{T}_{\mathbb{F}} \big(S_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m} \cong
\mathbb{T}_{\mathbb{F}} \big(\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})\big)_\mathfrak{m} \cong
\mathbb{T}_{\mathbb{F}} \big(\mathcal{CM}_k(N,\bar{\chi}\,;\,\mathbb{F})^+\big)_\mathfrak{m}.$$ \end{prop}
{\bf Proof. } Using the comparison with group cohomology of \cite{HeckeMS}, Theorem~6.1, the result follows under Assumption~(i) from \cite{edixhoven}, Theorem~5.2, and is proved under Assumption~(ii) in \cite{Faithful}, Corollary~6.9, for the case of the group $\Gamma_1(N)$ and no Dirichlet character. The passage to a character is established by \cite{Faithful}, Theorem~7.4 and the remark following it. One identifies the mod~$p$ modular forms appearing with corresponding Katz forms using Carayol's Lemma (\cite{edixhoven-boston}, Prop.~1.10). \hspace* {.5cm}
$\Box$
We end this section by stating the so-called Sturm bound (also called the Hecke bound), which gives the best a priori upper bound for how many Hecke operators are needed to generate all the Hecke algebra. We only need it in our algorithm in cases in which it is theoretically not known that the stop criterion will be reached. This will enable the algorithm to detect if the Hecke algebra on modular symbols is not isomorphic to the corresponding one on cuspidal modular forms.
\begin{prop}[Sturm bound]\label{propsturm} The Hecke algebra $\mathbb{T}_\mathbb{C}(S_k(N,\chi))$ can be generated as an algebra by the Hecke operators $T_l$ for all primes~$l$ smaller than or equal to $\frac{kN}{12}\prod_{q \mid N, q \text{ prime}} (1+\frac{1}{q})$. \end{prop}
{\bf Proof. } This is discussed in detail in Chapter~11 of \cite{SteinBook}. \hspace* {.5cm}
$\Box$
\subsection{Algorithm}
In this section we present a sketch of the algorithm that we used for our computations. The {\sc Magma} code can be downloaded from the second author's webpage and a manual is included as Appendix~\ref{section-manual}.
\noindent {\bf\underline{Input:}} Integers $N \ge 1$, $k \ge 2$, a finite field $\mathbb{F}$, a character $\chi: (\mathbb{Z}/N\mathbb{Z})^\times \to \mathbb{F}^\times$ and for each prime~$l$ less than or equal to the Sturm bound an irreducible polynomial $f_l \in \mathbb{F}[X]$.\\ {\bf \underline{Output:}} An $\mathbb{F}$-algebra. \begin{itemize} \itemsep=0cm plus 0pt minus 0pt \item $M \leftarrow \mathcal{CM}_k(N,\chi\,;\,\mathbb{F})$, $l \leftarrow 1$, $L \leftarrow $ empty list. \item repeat \begin{itemize} \itemsep=0cm plus 0pt minus 0pt \item $l \leftarrow $ next prime after $l$. \item Compute $T_l$ on $M$ and append it to the list $L$. \item $M \leftarrow $ the restriction of $M$ to the $f_l$-primary subspace for $T_l$, i.e.\ to the biggest subspace of $M$ on which the minimal polynomial of $T_l$ is a power of $f_l$. \item $A \leftarrow $ the $\mathbb{F}$-algebra generated by the restrictions to $M$ of $T_2, T_3, \dots, T_l$. \end{itemize} \item until $2 \cdot \dim (A) = \dim (M)$ \emph{[the stop criterion]} or $l > $ Sturm bound. \item return $A$. \end{itemize}
The $f_l$ should, of course, be chosen as the minimal polynomials of the coefficients $a_l(f)$ of the normalised eigenform $f \in S_k(N,\chi\,;\,\bar{\mathbb{F}})$ whose local Hecke algebra one wants to compute. Suppose the algorithm stops at the prime~$q$. If $q$ is bigger than the Sturm bound, the equivalent conditions of Corollary~\ref{stopcor} do not hold. In that case the output should be disregarded. Otherwise, $A$ is isomorphic to a direct product of the form $\prod_\mathfrak{m} \mathbb{T}(S_k(N,\chi\,;\,\mathbb{F}))_\mathfrak{m}$ where the $\mathfrak{m}$ are those maximal ideals such that the minimal polynomials of $T_2, T_3, \dots, T_q$ on $\mathbb{T}(S_k(N,\chi\,;\,\mathbb{F}))_\mathfrak{m}$ are equal to powers of $f_2, f_3, \dots, f_q$. It can happen that $A$ consists of more than one factor. Hence, one should still decompose $A$ into its local factors. Alternatively, one can also replace the last line but one in the algorithm by \begin{itemize} \item until $\big((2 \cdot \dim (A) = \dim (M))$ and $A$ is local$\big)$ or $l > $ Sturm bound, \end{itemize} which ensures that the output is a local algebra. In practice, one modifies the algorithm such that not for every prime~$l$ a polynomial~$f_l$ need be given, but that the algorithm takes each irreducible factor of the minimal polynomial of~$T_l$ if no $f_l$ is known. It is also useful to choose the order how $l$ runs through the primes. For example, one might want to take $l=p$ at an early stage with $p$ the characteristic of~$\mathbb{F}$, if one knows that this operator is needed, which is the case in all computations concerning Question~\ref{ourquestion}.
\section{Computational results} \label{computational-results}
In view of Question~\ref{ourquestion}, we produced 384 examples of odd irreducible continuous Galois representations $\Gal({\overline{\QQ}}/\mathbb{Q}) \to \mathrm{GL}_2({\overline{\FF}_p})$ that are completely split at~$p$. The results are documented in the accompanying tables (Appendix~\ref{section-tables}). The complete data can be downloaded from the second author's webpage.
The Galois representations were created either by class field theory or from an irreducible integer polynomial whose Galois group embeds into $\mathrm{GL}_2({\overline{\FF}_p})$. All examples but one are dihedral; the remaining one is icosahedral. For each of these an eigenform was computed giving rise to it. The Gorenstein defect of the corresponding local Hecke algebra factor turned out always to be~$2$, supporting Question~\ref{ourquestion}.
The authors preferred to proceed like this, instead of computing all Hecke algebras mod $p$ in weight $p$ for all ``small'' primes~$p$ and all ``small'' levels, since non-dihedral examples in which the assumptions of Question~\ref{ourquestion} are satisfied are very rare.
\subsection{Table entries}
For every computed local Hecke algebra enough data are stored to recreate it as an abstract algebra and important characteristics are listed in the tables of Appendix~\ref{section-tables}. A sample table entry is the following.
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline 5939 & 5 & 3 & 12 & 3 & 5 & 2 & 5 & 366 & $D_{7}$ \\ \hline \end{longtable}
Each entry corresponds to the Galois conjugacy class of an eigenform $f$ mod~$p$ with associated local Hecke algebra~$A$. The first and the second column indicate the level and the weight of~$f$. The latter is in all examples equal to the characteristic of the base field $k$ (a finite extention of~$\mathbb{F}_p$) of the algebra. Let $\mathfrak{m}_A$ denote the maximal ideal of~$A$. Then ResD stands for the degree of $K = A/\mathfrak{m}_A$ over~$\mathbb{F}_p$. Let us consider $A \otimes_k K$. It decomposes into a direct product of a local $K$-algebra $B$ and its $\Gal(K/k)$-conjugates. The $K$-dimension of $B$ (which is equal to the $k$-dimension of~$A$) is recorded in the fourth column.
Let $\mathfrak{m}_B$ be the maximal ideal of~$B$. The {\em embedding dimension} EmbDim is the $K$-dimension of $\mathfrak{m}_B/\mathfrak{m}_B^2$. By Nakayama's Lemma this is the minimal number of $B$-generators for~$\mathfrak{m}_B$. The {\em nilpotency order} NilO is the maximal integer~$n$ such that $\mathfrak{m}_B^n$ is not the zero ideal. The column GorDef contains the Gorenstein defect of~$B$ (which is the same as the Gorenstein defect of~$A$).
By \#Ops it is indicated how many Hecke operators were used to generate the algebra~$A$, applying the stop criterion (Corollary~\ref{stopcor}). This is contrasted with the number of primes smaller than the Sturm bound (Proposition~\ref{propsturm}, it is also called the Hecke bound), denoted by \#(p$<$HB). One immediately observes that the stop criterion is very efficient. Whereas the Sturm bound is roughly linear in the level, in 365 of the 384 calculated examples, less than 10 Hecke operators sufficed, in 252 examples even 5 were enough.
The final column contains the image of the mod~$p$ Galois representation attached to~$f$ as an abstract group.
\subsection{Dihedral examples}
All Hecke algebras except one in our tables correspond to eigenforms whose Galois representations are dihedral, since these are by far the easiest to obtain explicitly as one can use class field theory. This is explained now.
Let $p$ be a prime and $d$ a square-free integer which is $1$ mod~$4$ and not divisible by~$p$. We denote by $K$ the quadratic field~$\mathbb{Q}(\sqrt{d})$. Further we consider an unramified character $\chi: \Gal({\overline{\QQ}}/K) \to {\overline{\FF}_p}^\times$ of order $n \ge 3$. We assume that its inverse $\chi^{-1}$ is equal to $\chi$ conjugated by $\sigma$, denoted $\chi^\sigma$, for $\sigma$ (a lift of) the non-trivial element of $\Gal(K/\mathbb{Q})$. The induced representation $$\rho_\chi := {\rm Ind}_{\Gal({\overline{\QQ}}/K)}^{\Gal({\overline{\QQ}}/\mathbb{Q})}(\chi) :
\Gal({\overline{\QQ}}/\mathbb{Q}) \to \mathrm{GL}_2({\overline{\FF}_p})$$ is irreducible and its image is the dihedral group $D_n$ of order~$2n$. If $l$ is a prime not dividing $2d$, we have $\rho_\chi(\Frob_l) = \mat 0110$ if $\big(\frac{d}{l}\big)=-1$, and $\rho_\chi(\Frob_l) = \mat {\chi(\Frob_\Lambda)}00 {\chi^\sigma(\Frob_\Lambda)}$ if $\big(\frac{d}{l}\big)=1$ and $l \mathcal{O}_K = \Lambda \sigma(\Lambda)$. This explicit description makes it obvious that the determinant of $\rho_\chi$ is the Legendre symbol $l \mapsto \big(\frac{d}{l}\big)$.
Since the kernel of $\chi$ corresponds to a subfield of the Hilbert class field of~$K$, simple computations in the class group of~$K$ allow one to determine which primes split completely. These give examples satisfying the assumptions of Question~\ref{ourquestion} (the Frobenius at~$p$ is the identity) if $\rho_\chi$ is odd, i.e.\ if $p=2$ or $d < 0$.
We remark that for characters $\chi$ of odd order~$n$ the assumption $\chi^{-1} = \chi^\sigma$ is not a big restriction, since any character can be written as $\chi = \chi_1 \chi_2$ with $\chi_1^\sigma = \chi_1^{-1}$ and $\chi_2^\sigma = \chi_2$, hence the latter descends to a character of $\Gal({\overline{\QQ}}/\mathbb{Q})$ and the representation $\rho_\chi$ is isomorphic to $\rho_{\chi_1} \otimes \chi_2$.
All dihedral representations are known to come from eigenforms in the minimal possible weight with level equal to the (outside of $p$) conductor of the representation (see \cite{Dihedral}, Theorem~1).
In the tables we computed the Hecke algebras of odd dihedral representations as above in the following ranges. For each prime~$p$ less than~$100$ and each prime $l$ less than or equal to the largest level occuring in the table for~$p$, we chose $d$ as plus or minus~$l$ such that $d$ is $1$ mod $4$ and we let $H$ run through all non-trivial cyclic quotients of the class group of $\mathbb{Q}(\sqrt{d})$ of order coprime to~$p$. For each $H$ we chose (unramified) characters~$\chi$ of the absolute Galois group of $\mathbb{Q}(\sqrt{d})$ corresponding to~$H$, up to Galois conjugacy and up to replacing $\chi$ by its inverse. Then $\chi$ is not the restriction of a character of $\Gal({\overline{\QQ}}/\mathbb{Q})$. By genus theory the order of $\chi$ is odd, as the class number is, so we necessarily have $\chi^{-1} = \chi^\sigma$. We computed the local factor of $\mathbb{T}_{\mathbb{F}_p}(S_p(d,\big(\frac{d}{\cdot}\big) \,;\, \mathbb{F}_p))$ corresponding to~$\rho_\chi$ if $\rho_\chi$ is odd and $p$ is completely split. For the prime $p=2$ we also allowed square-free integers~$d$ which are $1$ mod~$4$ and whose absolute value is less than~$5000$.
\subsection{Icosahedral example}
With the help of a list of polynomials provided by Gunter Malle (\cite{malle}) a Galois representation of $\Gal({\overline{\QQ}}/\mathbb{Q})$ with values in $\mathrm{GL}_2({\overline{\FF}_2})$ which is of prime conductor, completely split at~$2$ and thus satisfies the assumptions of Question~\ref{ourquestion} and whose image is isomorphic to the icosahedral group~$A_5$ could be described explicitly. The modular forms in weight $2$ predicted by Serre's conjecture were found and the corresponding Hecke algebra turned out to have Gorenstein defect equal to~$2$.
Let $f \in \mathbb{Z}[X]$ be an irreducible polynomial of degree~$5$ whose Galois group, i.e.\ the Galois group of the normal closure $L$ of $K = \mathbb{Q}[X]/(f)$, is isomorphic to~$A_5$. We assume that $K$ is unramified at $2$, $3$ and~$5$. We have the Galois representation $$ \rho_f : \Gal({\overline{\QQ}}/\mathbb{Q}) \twoheadrightarrow
\Gal(L/\mathbb{Q}) \cong A_5 \cong \mathrm{SL}_2(\mathbb{F}_4).$$ We now determine its conductor and its traces. Let $p$ be a ramified prime. As the ramification is tame, the image of the inertia group $\rho_f(I_p)$ at $p$ is cyclic of order $2$, $3$ or~$5$. In the first case, the image of a decomposition group $\rho_f(D_p)$ at $p$ is either equal to $\rho_f(I_p)$ or equal to $\mathbb{Z}/2\mathbb{Z} \times \rho_f(I_p)$. If the order of $\rho_f(I_p)$ is odd and $\rho_f(I_p) = \rho_f(D_p)$, then any completion of $L$ at the unique prime above~$p$ is totally ramified and cyclic of degree $\#\rho_f(I_p)$, hence contained in $\mathbb{Q}_p(\zeta_p)$ for $\zeta_p$ a primitive $p$-th root of unity. It follows that $p$ is congruent to $1$ mod $\#\rho_f(I_p)$. If the order of $\rho_f(I_p)$ is odd, but $\rho_f(I_p)$ is not equal to $\rho_f(D_p)$, then $\rho_f(D_p)$ is a dihedral group and the completion of $L$ at a prime above $p$ has a unique unramified quadratic subfield~$S$. Thus, we have the exact sequence $$ 0 \to \rho_f(I_p) \to \rho_f(D_p) \to \Gal(S/\mathbb{Q}_p) \to 0.$$ On the one hand, it is well-known that the conjugation by a lift of the Frobenius element of $\Gal(S/\mathbb{Q}_p)$ acts on $\rho_f(I_p)$ by raising to the $p$-th power. On the other hand, as the action is non-trivial it also corresponds to inversion on $\rho_f(I_p)$, since the only elements of order~$2$ in $(\mathbb{Z}/3\mathbb{Z})^\times$ and $(\mathbb{Z}/5\mathbb{Z})^\times$ are~$-1$. As a consequence, $p$ is congruent to $-1$ mod $\#\rho_f(I_p)$ in this case.
We hence have the following cases. \begin{enumerate}[(1)] \item Suppose $p \mathcal{O}_K = \mathfrak{P}^5$. Then $p \equiv \pm 1 \mod 5$. \begin{enumerate}[(a)]
\item If $p \equiv 1 \mod 5$, then $\rho_f|_{I_p} \sim \mat \chi00{\chi^{-1}}$ with $\chi$ a totally ramified character of $\Gal({\overline{\QQ}_p}/\mathbb{Q}_p)$ of order~$5$. \item If $p \equiv -1 \mod 5$, then $\rho_f(D_p)$ is the dihedral group with $10$ elements. \end{enumerate}
\item Suppose $p \mathcal{O}_K = \mathfrak{P}^3 \mathfrak{Q} \mathfrak{R}$ or $p \mathcal{O}_K = \mathfrak{P}^3 \mathfrak{Q}$. \begin{enumerate}[(a)]
\item If $p \equiv 1 \mod 3$, then $\rho_f|_{I_p} \sim \mat \chi00{\chi^{-1}}$ with $\chi$ a totally ramified character of $\Gal({\overline{\QQ}_p}/\mathbb{Q}_p)$ of order~$3$. \item If $p \equiv -1 \mod 3$, then $\rho_f(D_p)$ is the dihedral group with $6$ elements. \end{enumerate}
\item Suppose that $p$ is ramified, but that we are neither in Case~(1)
nor in Case~(2). Then $\rho_f|_{I_p} \sim \mat 1101$. \end{enumerate}
By the definition of the conductor at $p$ it is clear that it is $p^2$ in Cases (1) and~(2) and $p$ in Case~(3). However, in Cases (1)(a) and (2)(a) one can choose a character $\epsilon$ of $\Gal({\overline{\QQ}}/\mathbb{Q})$ of the same order as~$\chi$ whose restriction to $D_p$ gives the character~$\chi$. If one twists the representation $\rho_f$ by $\epsilon$ one finds also in these cases that the conductor at~$p$ is~$p$.
An inspection of the conjugacy classes of the group $\mathrm{SL}_2(\mathbb{F}_4)$ shows that the traces of $\rho_f$ twisted by some character $\epsilon$ of $\Gal({\overline{\QQ}}/\mathbb{Q})$ are as follows. Let $l$ be an unramified prime. \begin{itemize} \item If the order of $\Frob_l$ is $5$, then the trace at $\Frob_l$ is~$\epsilon(\Frob_l) w$ where $w$ is a root of the polynomial $X^2+X+1$ in $\mathbb{F}_2[X]$. \item If the order of $\Frob_l$ is $3$, then the trace at $\Frob_l$ is~$\epsilon(\Frob_l)$. \item If the order of $\Frob_l$ is $1$ or $2$, then the trace at $\Frob_l$ is~$0$. \end{itemize}
These statements allow the easy identification of the modular form belonging to an icosahedral representation.
We end this section with some remarks on our icosahedral example. It was obtained using the polynomial $x^5 - x^4 - 79 x^3 + 225 x^2 + 998 x - 3272$. The corresponding table entry is:
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline 89491 & 2 & 2 & 12 & 4 & 3 & 2 & 4 & 1746 & $A_5$ \\ \hline \end{longtable}
Hence, in level $89491$ and weight~$2$ there is a single eigenform~$g$ mod~$2$ up to Galois conjugacy whose first couple of $q$-coefficients agree with the traces of a twist of the given icosahedral Galois representation. From this one can deduce that the Galois representation $\rho_g$ of~$g$ has an icosahedral image and is only ramified at~$89491$. As weight and level lowering are not known in our case, we cannot prove that $\rho_g$ coincides with a twist of the given one. It might, however, be possible to exclude the existence of two distinct icosahedral extensions of the rationals inside~$\mathbb{C}$ that ramify only at~$89491$ by consulting tables. According to Malle, the icosahedral extension used has smallest discriminant among all totally real $A_5$-extensions of the rationals in which $2$ splits completely.
\section{Further results and questions} \label{further-questions}
In this section we present some more computational observations for Hecke algebras under the assumptions of Question~\ref{ourquestion}, which lead us to ask some more questions.
\subsection*{On the dimension of the Hecke algebra}
From the data, we see that many even integers appear as dimensions of the~$\mathbb{T}_\mathfrak{m}$. We know that the dimension must be at least~4, as this is the dimension of the smallest non-Gorenstein algebra which can appear in our case. This extends the results of~\cite{kilford-nongorenstein}, where the dimensions of the Hecke algebras~$\mathbb{T}_{\mathbb{Z} \to \mathbb{F}_2}(S_2(\Gamma_0(431)))$ and~$\mathbb{T}_{\mathbb{Z} \to \mathbb{F}_2}(S_2(\Gamma_0(503)))$ localised at the non-Gorenstein maximal ideals are shown to be~4.
In this table we see exactly how many times each dimension appears in our data. We observe that every even integer between~4 and~32 appears, and that the largest dimension is~60. The most common dimension is~4, which appears about half of the time. However, as the dimension of the Hecke algebra attached to~$S_k(\Gamma_1(N))$ increases with~$N$ and with~$k$, this may be an artifact of the data being collected for ``small'' levels~$N$ and primes~$p$.
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|} \hline Dimension & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & 22 \\ \hline Number of algebras &206 & 58& 25 & 3 & 24 & 6 & 20 & 3 & 12 & 3 \\ \hline \hline Dimension& 24 & 26 & 28 & 30 & 32 & 36 & 40 & 46 & 56 & 60\\ \hline Number of algebras &5 & 4 & 2 & 1 & 2 & 2 & 4 & 1 & 2 & 1\\ \hline \end{tabular}
It seems reasonable that there should be infinitely many cases with dimension~4, and plausible that every even integer greater than or equal to~4 should appear as a dimension infinitely many times. From the tables, we see that dimension~4 algebras appear at very high levels, so they do not appear to be becoming rare as the dimension increases, but this may, of course, be an artifact of our data.
We note that not every example that arises from an elliptic curve in characteristic $p=2$ has Hecke algebra with dimension~$4$; for example the algebra $\mathbb{T}_{\mathbb{Z} \to \mathbb{F}_2}(S_2(\Gamma_0(2089)))$ localised at its non-Gorenstein maximal ideal has dimension~18. In level~$18097$ there is a dimension~$36$ example arising from an elliptic curve.
\subsection*{On the residue degree}
We will now solve an easy aspect of the question of the possible structures of non-Gorenstein local algebras occurring as local Hecke algebras. We assume for the couple of lines to follow the Generalised Riemann Hypothesis (GRH).
We claim that then the residue degrees of $\mathbb{T}_{\mathfrak{m}}$ (in the notation of Question~\ref{ourquestion}) are unbounded, if we let~$p$ and $N$ run through the primes such that $p \neq N$ and $N$ is congruent to~$3$ modulo~$4$.
For, class groups of imaginary quadratic fields $\mathbb{Q}(\sqrt{-N})$ have arbitrarily large cyclic factors of odd order, as the exponent of these class groups is known to go to infinity as $N$ does, by the main result of~\cite{boyd}, which assumes GRH. So the discussion on dihedral forms in Section~\ref{computational-results} immediately implies the claim.
\subsection*{On the embedding dimension}
One can ask whether the embedding dimension of the local Hecke algebras in the situation of Question~\ref{ourquestion} is bounded, if we allow $p$ and~$N$ to vary. This, however, seems to be a difficult problem. The embedding dimensions occuring in our tables are $3$ (299 times), $4$ (78 times) and~$5$ (7 times).
The embedding dimension~$d$ is related to the number of Hecke operators needed to generate the local Hecke algebra, in the sense that at least~$d$ Hecke operators are needed. Probably, $d$ Hecke operators can be found that do generate, but they need not be the first~$d$ prime Hecke operators, of course. However, as our tables suggest, in most cases the actual computations were done using very few operators, and there are 99 of the 384 cases when the computation already finished after $d$ operators.
\noindent \begin{tabular}{p{8cm}p{6cm}} L.~J.~P.~Kilford & Gabor Wiese \\ Mathematics Institute & NWF 1-Mathematik \\ Oxford University & Universität Regensburg \\ 24--29 St Giles' & D-93040 Regensburg \\ Oxford OX1 3LB & Germany \\ United Kingdom & \\
& {\tt [email protected]} \\ {\tt [email protected]} & {\tt http://maths.pratum.net} \end{tabular}
\appendix
\section{The {\sc Magma} package HeckeAlgebra (by Gabor Wiese)} \label{section-manual}
\abstract{This is a short manual for the {\sc Magma} package {\tt HeckeAlgebra}, which can be downloaded from the author's webpage. The author would like to thank Lloyd Kilford for very helpful suggestions.}
\subsection{Example}
{\setlength{\parindent}{0pt}
The following example explains the main functions of the package. Let us suppose that the file {\tt HeckeAlgebra.mg} is stored in the current path. We first attach the package.\\ \magma{> Attach("HeckeAlgebra.mg");}
We want the package to be silent, so we put:\\ \magma{> SetVerbose ("HeckeAlgebra",false);}
If we would like more information on the computations being performed, we should have put the value \magma{true}. Since we want to store the data to be computed in a file, we now create the file.
\magma{> my\_file := "datafile";\\ > CreateStorageFile(my\_file);}
Next, we would like to compute the Hecke algebras of the dihedral eigenforms of level $2039$ over extensions of $\mathbb{F}_2$. First, we create a list of such forms.\\ \magma{> dih := DihedralForms(2039 : ListOfPrimes := [2], completely\_split := false);}\\ Now, we compute the corresponding Hecke algebras, print part of the computed data in a human readable format, and finally save the data to our file.\\ \magma{> for f in dih do\\ for> ha := HeckeAlgebras(f);\\ for> HeckeAlgebraPrint1(ha);\\ for> StoreData(my\_file, ha);\\ for> end for;}
\magmaout{ Level 2039\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 1\\ Number of operators used 3\\ Primes lt Hecke bound 68\\ Residue degree 2\\ ---------------------------\\ Level 2039\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 2\\ Dimension 6\\ Number of operators used 4\\ Primes lt Hecke bound 68\\ Residue degree 2\\ ---------------------------\\ Level 2039\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 1\\ Number of operators used 3\\ Primes lt Hecke bound 68\\ Residue degree 6\\ ---------------------------\\ Level 2039\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 1\\ Number of operators used 3\\ Primes lt Hecke bound 68\\ Residue degree 4\\ ---------------------------\\ Level 2039\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 1\\ Number of operators used 3\\ Primes lt Hecke bound 68\\ Residue degree 4\\ ---------------------------\\ Level 2039\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 1\\ Number of operators used 3\\ Primes lt Hecke bound 68\\ Residue degree 12\\ ---------------------------\\ Level 2039\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 1\\ Number of operators used 3\\ Primes lt Hecke bound 68\\ Residue degree 12\\ --------------------------- }
With the function \magma{DihedralForms} one may also compute exclusively representations that are completely split in the characteristic. The default is \magma{completely\_split := true}. By the option \magma{bound} we indicate primes up to which bound should be used as the characteristic. The following example illustrates this.\\ \magma{> dih1 := DihedralForms (431 : bound := 20);\\ > for f in dih1 do\\ for> ha := HeckeAlgebras(f);\\ for> HeckeAlgebraPrint1(ha);\\ for> StoreData(my\_file, ha);\\ for> end for;}
\magmaout{ Level 431\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 2\\ Dimension 4\\ Number of operators used 6\\ Primes lt Hecke bound 20\\ Residue degree 1\\ ---------------------------\\ Level 431\\ Weight 11\\ Characteristic 11\\ Gorenstein defect 2\\ Dimension 4\\ Number of operators used 5\\ Primes lt Hecke bound 77\\ Residue degree 3\\ --------------------------- }
One can also compute icosahedral modular forms over extensions of $\mathbb{F}_2$, starting from an integer polynomial with Galois group $A_5$, as follows.\\ \magma{> R<x> := PolynomialRing(Integers());\\ > pol := x\^{}5-x\^{}4-780*x\^{}3-1795*x\^{}2+3106*x+344;\\ > f := A5Form(pol);}
With this kind of icosahedral examples one has to pay attention to the conductor, as it can be huge. This polynomial has prime conductor. But conductors need not be square-free, in general.\\ \magma{> print Modulus(f`Character);}
\magmaout{1951}
So it's reasonable. We do the computation.\\ \magma{> ha := HeckeAlgebras(f);\\ > HeckeAlgebraPrint1(ha);}
\magmaout{ Level 1951\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 3\\ Number of operators used 3\\ Primes lt Hecke bound 66\\ Residue degree 4\\ ---------------------------\\ Level 1951\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 6\\ Number of operators used 3\\ Primes lt Hecke bound 66\\ Residue degree 4\\ --------------------------- }
There are two forms, which is okay, since they come from a weight one form in two different ways and this case is not exceptional. We now save them, as always.\\ \magma{> StoreData(my\_file, ha);}
It is also possible to compute all forms at a given character and weight.\\ \magma{> eps := DirichletGroup(229,GF(2)).1;\\ > ha := HeckeAlgebras(eps,2);\\ > HeckeAlgebraPrint1(ha);}
\magmaout{ Level 229\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 1\\ Number of operators used 12\\ Primes lt Hecke bound 12\\ Residue degree 1\\ ---------------------------\\ Level 229\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 2\\ Number of operators used 12\\ Primes lt Hecke bound 12\\ Residue degree 2\\ ---------------------------\\ Level 229\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 4\\ Number of operators used 12\\ Primes lt Hecke bound 12\\ Residue degree 1\\ ---------------------------\\ Level 229\\ Weight 2\\ Characteristic 2\\ Gorenstein defect 0\\ Dimension 2\\ Number of operators used 12\\ Primes lt Hecke bound 12\\ Residue degree 5\\ --------------------------- }
\magma{> StoreData(my\_file,ha);}
Next, we illustrate how one reloads what has been saved. One would like to type: \magma{load my\_file;} but that does not work. One has to do it as follows.\\ \magma{> load "datafile";\\ > mf := RecoverData(LoadIn,LoadInRel);}
Now, \magma{mf} contains a list of all algebra data computed before. There's a rather concise printing function, displaying part of the information, namely \magma{HeckeAlgebraPrint(mf);}.
One can also create a LaTeX longtable. The entries can be chosen in quite a flexible way. The standard usage is the following.\\ \magma{> HeckeAlgebraLaTeX(mf,"table.tex");}
A short LaTeX file displaying the table is the following:\\ {\tt {$\backslash$}documentclass[11pt]\{article\}\\ {$\backslash$}usepackage\{longtable\}\\ {$\backslash$}begin\{document\}\\ {$\backslash$}input\{table\}\\ {$\backslash$}end\{document\}}
The table we created is this one:
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 2039 & 2 & 2 & 1 & 0 & 0 & 0 & 3 & 68 & $D_{3}$ \\ 2039 & 2 & 2 & 6 & 3 & 2 & 2 & 4 & 68 & $D_{5}$ \\ 2039 & 2 & 6 & 1 & 0 & 0 & 0 & 3 & 68 & $D_{9}$ \\ 2039 & 2 & 4 & 1 & 0 & 0 & 0 & 3 & 68 & $D_{15}$ \\ 2039 & 2 & 4 & 1 & 0 & 0 & 0 & 3 & 68 & $D_{15}$ \\ 2039 & 2 & 12 & 1 & 0 & 0 & 0 & 3 & 68 & $D_{45}$ \\ 2039 & 2 & 12 & 1 & 0 & 0 & 0 & 3 & 68 & $D_{45}$ \\ 431 & 2 & 1 & 4 & 3 & 1 & 2 & 6 & 20 & $D_{3}$ \\ 431 & 11 & 3 & 4 & 3 & 1 & 2 & 5 & 77 & $D_{7}$ \\ 1951 & 2 & 4 & 3 & 1 & 2 & 0 & 3 & 66 & $A_5$ \\ 1951 & 2 & 4 & 6 & 2 & 3 & 0 & 3 & 66 & $A_5$ \\ 229 & 2 & 1 & 1 & 0 & 0 & 0 & 12 & 12 & $$ \\ 229 & 2 & 2 & 2 & 1 & 1 & 0 & 12 & 12 & $$ \\ 229 & 2 & 1 & 4 & 1 & 3 & 0 & 12 & 12 & $$ \\ 229 & 2 & 5 & 2 & 1 & 1 & 0 & 12 & 12 & $$ \\ \end{longtable}
In the examples of level $229$ the image of the Galois representation as an abstract group is not know. That is due to the fact that we created these examples without specifying the Galois representation in advance.
It is possible to compute arbitrary Hecke operators on the local Hecke factors generated by \magma{HeckeAlgebras($\cdot$)}, as the following example illustrates.\\ \magma{> A,B,M,C := HeckeAlgebras(DirichletGroup(253,GF(2)).1,2 : over\_residue\_field := true);}
Suppose that we want to know the Hecke operator $T_{17}$ on the $4$th local factor.\\ \magma{> i := 4;\\ > T := BaseChange(HeckeOperator(M,17),C[i]);}
The coefficients are the eigenvalues (only one):\\ \magma{> Eigenvalues(T);}
\magmaout{\{ $<$ \$ . 1 $\hat{ }$ 5, 8 $>$ \}}
Let us remember the eigenvalue.\\ \magma{> e := SetToSequence(Eigenvalues(T))[1][1];}
In order to illustrate the option \magma{over\_residue\_field}, we also compute the following:\\ \magma{> A1,B1,M1,C1 := HeckeAlgebras(DirichletGroup(253,GF(2)).1,2 : over\_residue\_field := false);\\ > T1 := BaseChange(HeckeOperator(M1,17),C1[i]);\\ > Eigenvalues(T1);}
\magmaout{\{\}}
The base field is strictly smaller than the residue field in this example and the operator \magma{T1} cannot be diagonalised over the base field. We check that \magma{e} is nevertheless a zero of the minimal polynomial of \magma{T1}.\\ \magma{> Evaluate(MinimalPolynomial(T1),e);}
\magmaout{0}
The precise usage of the package is described in the following sections. }
\subsection{Hecke algebra computation}
\subsubsection{The modular form format}\label{secmodforms}
In the package, modular forms are often represented by the following record.\\ \noindent\magma{ModularFormFormat := recformat <} \begin{longtable}{ll} \magma{Character} &\magma{: GrpDrchElt,}\\ \magma{Weight} &\magma{: RngIntElt,}\\ \magma{CoefficientFunction} &\magma{: Map, }\\ \magma{ImageName} &\magma{: MonStgElt,}\\ \magma{Polynomial} &\magma{: RngUPolElt } \end{longtable} \magma{>;}
The fields \magma{Character} and \magma{Weight} have the obvious meaning. Sometimes, the image of the associated Galois representation is known as an abstract group. Then that name is recorded in \magma{ImageName}, e.g.\ \magma{A\_5} or \magma{D\_3}. In some cases, a polynomial is known whose splitting field is the number field cut out by the Galois representation. Then the polynomial is stored in \magma{Polynomial}. The cases in which polynomials are known are usually icosahedral ones. The \magma{CoefficientFunction} is a function from the integers to a polynomial ring. For all primes $l$ different from the characteristic and not dividing the level of the modular form (i.e.\ the modulus of the \magma{Character}), the coefficient function should return the minimal polynomial of the $l$-th coefficient in the $q$-expansion of the modular form in question.
\subsubsection{Dihedral modular forms}\label{secdih}
Eigenforms whose associated Galois representations take dihedral groups as images provide an important source of examples, in many contexts. These eigenforms are called {\em dihedral}. The big advantage is that their Galois representation, and hence their $q$-coefficients, can be computed using class field theory. That enables one to exhibit Galois representations in the context of modular forms with certain number theoretic properties. The property for which these functions were initially created is that the representations should be unramified in the characteristic, say $p$, and that $p$ is completely split in the number field cut out by the representation.
We consider dihedral representations whose determinant is the Legendre symbol of a quadratic field $\mathbb{Q}(\sqrt{N})$. The representations produced by the functions to be described are obtained by induction of an unramified character $\chi$ of~$\mathbb{Q}(\sqrt{N})$ whose conjugate by the non-trivial element of the Galois group of $\mathbb{Q}(\sqrt{N})$ over $\mathbb{Q}$ is assumed to be $\chi^{-1}$.
\intr{intrinsic GetLegendre (N :: RngIntElt, K :: FldFin ) -> GrpDrchElt} For an odd positive integer \magma{N}, this function returns the element of \magma{DirichletGroup(Abs(N),K)} (with \magma{K} a finite field of characteristic different from $2$) which corresponds to the Legendre symbol $p \mapsto \left(\frac{\pm N}{p}\right)$. If \magma{N} is $1$ mod $4$ the sign is $+1$, and $-1$ otherwise.
\intr{intrinsic DihedralForms (N :: RngIntElt : \\ \hspace*{2cm} ListOfPrimes := [],
bound := 100,
odd\_only := true,
quad\_disc := 0, \\ \hspace*{2cm} completely\_split := true,
all\_conjugacy\_classes := true
) -> Rec} This function computes all modular forms (in the sense of Section~\ref{secmodforms}) of level \magma{N} and weight~$p$ over a finite field of characteristic $p$ that come from dihedral representations whose determinant is the Legendre symbol of the quadratic field $K=\mathbb{Q}(\sqrt{\pm \text{\magma{quad\_disc}}})$ and which are obtained by induction of an unramified character of~$K$. If \magma{quad\_disc} is $1$ mod $4$ the sign is $+1$, and $-1$ otherwise. If \magma{quad\_disc} is $0$, the value of \magma{N} is used. If the option \magma{completely\_split} is set, only those representations are returned which are completely split at~$p$. If the option \magma{ListOfPrimes} is assigned a non-empty list of primes, only those primes are considered as the characteristic. If it is the empty set, all primes $p$ up to the \magma{bound} are taken into consideration. If the option \magma{odd\_only} is true, only odd Galois representations are returned. If the option \magma{all\_conjugacy\_classes} is true, each unramified character as above up to Galois conjugacy and up to taking inverses is used. Otherwise, a single choice is made. That there may be non-conjugate characters cutting out the same number field is due to the fact that there may be non-conjugate elements of the same order in the multiplicative group of a finite field.
\subsubsection{Icosahedral modular forms}\label{secicosa}
Eigenforms whose attached Galois representations take the group $A_5$ as projective images are called {\em icosahedral}. Since extensive tables of $A_5$-extensions of the rationals are available, one can consider icosahedral Galois representations which one knows very well. That allows one to test certain conjectures concerning modular forms on icosahedral ones.
We note the isomorphism $A_5 \cong \mathrm{SL}_2(\mathbb{F}_4)$. Thus, $A_5$-extentions of the rationals give rise to icosahedral Galois representations in characteristic $2$ which (should) come from modular forms mod $2$. It would also be possible to use certain other primes, but this has not been implemented.
\intr{intrinsic A5Form (f :: RngUPolElt) -> Rec} Returns the icosahedral form in characteristic $2$ and weight $2$ of smallest predicted level corresponding to the polynomial \magma{f} which is expected to be of degree~$5$ and whose Galois group is supposed to be~$A_5$. No checks about \magma{f} are performed.
\subsubsection{The Hecke algebra format}
The data concerning the Hecke algebra of an eigenform that is computed by the function \magma{HeckeAlgebras} is a record of the following form.
\noindent\magma{AlgebraData := recformat <} \begin{longtable}{ll}
\magma{Level} &\magma{: RngIntElt,}\\
\magma{Weight} &\magma{: RngIntElt,}\\
\magma{Characteristic} &\magma{: RngIntElt,}\\
\magma{BaseFieldDegree} &\magma{: RngIntElt,}\\
\magma{CharacterOrder} &\magma{: RngIntElt,}\\
\magma{CharacterConductor} &\magma{: RngIntElt,}\\
\magma{CharacterIndex} &\magma{: RngIntElt,}\\
\magma{AlgebraFieldDegree} &\magma{: RngIntElt,}\\
\magma{ResidueDegree} &\magma{: RngIntElt,}\\
\magma{Dimension} &\magma{: RngIntElt,}\\
\magma{GorensteinDefect} &\magma{: RngIntElt,}\\
\magma{EmbeddingDimension} &\magma{: RngIntElt,}\\
\magma{NilpotencyOrder} &\magma{: RngIntElt,}\\
\magma{Relations} &\magma{: Tup,}\\
\magma{NumberGenUsed} &\magma{: RngIntElt,}\\
\magma{ImageName} &\magma{: MonStgElt,}\\
\magma{Polynomial} &\magma{: RngUPolElt} \end{longtable} \magma{>;}
\magma{Level} and \magma{Weight} have the obvious meaning. Let $K$ be the base field for the space of modular symbols used. It is (expected to be) a finite field. Then \magma{Characteristic} is the characteristic of $K$ and \magma{BaseFieldDegree} is the degree of $K$ over its prime field. The entries \magma{CharacterOrder}, \magma{CharacterConductor} and \magma{CharacterIndex} concern the Dirichlet character for which the modular symbols have been computed. The latter field is the index of the character in \magma{Elements(DirichletGroup($\cdot$))}. Note that that might change between different versions of {\sc Magma}. The fields \magma{ResidueDegree} (over the prime field), \magma{Dimension} and \magma{GorensteinDefect} have their obvious meaning for the Hecke algebra in question. The tuple \begin{quote} \magma{<AlgebraFieldDegree, EmbeddingDimension, NilpotencyOrder, Relations>} \end{quote} are data from which \magma{AffineAlgebra} can recreate the Hecke algebra up to isomorphism. \magma{NumberGenUsed} indicates the number of generators used by the package for the computation of the Hecke algebra. This number is usually much smaller than the Sturm bound. \magma{ImageName} and \magma{Polynomial} have the same meaning as in the record \magma{ModularFormFormat}.
\subsubsection{Hecke algebras}
\intr{intrinsic HeckeAlgebras (eps :: GrpDrchElt, weight :: RngIntElt :\\ \hspace*{2cm}
UserBound := 0,
first\_test := 3,
test\_interval := 1,
when\_test\_p := 3, \\ \hspace*{2cm}
when\_test\_bad := 4,
test\_sequence := [],
dimension\_factor := 2,\\ \hspace*{2cm}
ms\_space := 0,
cuspidal := true,
DegreeBound := 0,
OperatorList := [], \\ \hspace*{2cm}
over\_residue\_field := true,
try\_minimal := true,
force\_local := false, \\ \hspace*{.5cm} ) -> SeqEnum, SeqEnum, ModSym, Tup, Tup\\ intrinsic HeckeAlgebras ( t :: Rec :\\ \hspace*{2cm}
UserBound := 0,
first\_test := 3,
test\_interval := 1,
when\_test\_p := 3, \\ \hspace*{2cm}
when\_test\_bad := 4,
test\_sequence := [],
dimension\_factor := 2,\\ \hspace*{2cm}
ms\_space := 0,
cuspidal := true,
DegreeBound := 0,
OperatorList := [], \\ \hspace*{2cm}
over\_residue\_field := true,
try\_minimal := true,
force\_local := false, \\ \hspace*{.5cm} ) -> SeqEnum, SeqEnum, ModSym, Tup, Tup}
These functions compute all local Hecke algebras (up to Galois conjugacy) in the specified \magma{weight} for the given Dirichlet character \magma{eps}, respectively those corresponding to the modular form \magma{t} given by a record of type \magma{ModularFormFormat}. The functions return 5 values \magma{A,B,C,D,E}. \magma{A} contains a list of records of type \magma{AlgebraData} describing the local Hecke algebra factors. \magma{B} is a list containing the local Hecke algebra factors as matrix algebras. \magma{C} is the space of modular symbols used in the computations. \magma{D} is a tuple containing the base change tuples describing the local Hecke factors. We need to know \magma{D} in order to compute matrices representing Hecke operators for the local factor. Finally, \magma{E} contains a tuple consisting of all Hecke operators computed so far for each local factor of the Hecke algebra.
The usage in practice is described in the example at the beginning of this manual. We now explain the different options in detail.
The modular symbols space to be used in the computations can be determined as follows. The option \magma{ms\_space} can be set to the values $1$ (the plus-space), $-1$ (the minus-space) and $0$ (the full space). Whether the restriction to the cuspidal subspace is taken, is determined by \magma{cuspidal}. It is not necessary to pass to the cuspidal subspace, for example, if a cusp form is given by a coefficient function (see the description of the record \magma{ModularFormFormat}).
In some cases, a list of Hecke operators on the modular symbols space in question may already have been computed. In order to prevent {\sc Magma} from redoing their computations, they may be passed on to the function using the option \magma{OperatorList}.
Often, one wants to compute the local Hecke algebra of a modular form whose degree of the coefficient field over its prime field is known, e.g.\ in the case of an icosahedral form in characteristic $2$ for the trivial Dirichlet character the coefficient field is $\mathbb{F}_4$. By assigning a positive value to the option \magma{DegreeBound} the function will automatically discard any systems of eigenvalues beyond that bound, which speeds up the computations. One must be a bit careful with this option, as there may be cases when the bound may not be respected at ``bad primes''. But it usually suffices to take twice the degree of the coefficient field, e.g.\ one chooses \magma{DegreeBound := 4} in the icosahedral example just described. If no system of eigenvalues should be discarded for degree reasons, one must set \magma{DegreeBound := 0}.
All of the options \magma{first\_test}, \magma{test\_interval}, \magma{when\_test\_p},\magma{when\_test\_bad}, \magma{test\_sequence}, \magma{force\_local}, \magma{dimension\_factor} and \magma{UserBound} concern the stop criterion. Theoretically, the Sturm bound (see \magma{HeckeBound}) tells us up to which bound Hecke operators must be computed in order to be sure that they generate the whole Hecke algebra. In practice, however, the algorithm can often determine itself when enough Hecke operators have been computed to generate the algebra. That number is usually much smaller than the Sturm bound. The Sturm bound can be overwritten by assigning a positive number to \magma{UserBound}.
The stop criterion is the following. Let $M$ be the modular symbols space used and $S$ the set of Hecke operators computed so far. Then $M = \bigoplus_{i=1}^r M_i$ (for some $r$) such that each $M_i$ is respected by the Hecke operators and the minimal polynomial of each $T \in S$ restricted to $M_i$ is a power of an irreducible polynomial
(i.e.\ each $M_i$ is a primary space for the action of the algebra generated by all elements of $S$). Let $A_i$ be the algebra generated by $T|_{M_i}$ for all $T \in S$. One knows (in many cases, and in all cases of interest) that $A_i$ is equal to a direct product of local Hecke algebras if one has the equality $$ f \times \dim (A_i) = \text{ dimension of } M_i.$$ Here, $f$ is given by \magma{dimension\_factor} and should be $1$ if the plus-space or the minus space of modular symbols are used, and $2$ otherwise. The correct assignment of \magma{dimension\_factor} must be made by hand, whence experimentations are possible. If the stop criterion is not reached, the algorithm terminates at the Hecke bound.
It may happen that, when the stop criterion is reached, one $A_i$ is isomorphic to a direct product of more than one local Hecke algebras. If in that case the option \magma{force\_local} is \magma{true}, the computation of Hecke operators is continued until each $A_i$ is isomorphic to a single Hecke factor. If \magma{force\_local} is \magma{false}, then a fast localisation algorithm is applied to each $A_i$. The option is useful, when one expects only a single local Hecke algebra factor, for example, when a modular form is given.
In many cases of interest the Hecke operator $T_p$ with $p$ the characteristic is needed in order to generate the whole Hecke algebra. The option \magma{when\_test\_p} tells the algorithm at which step to compute $T_p$. It is very advisable to choose a small number. In practice, the stop criterion is reached after very few steps, e.g.\ 5 steps, when $T_p$ is computed early. Otherwise, the algorithm often has to continue until $T_p$ is computed, although most of the operators before did not change the generated algebra. The option \magma{when\_test\_bad} has a similar meaning for the $T_l$ for primes $l$ dividing the level. However, paying attention to them is only required when the modular form is old at~$l$. Moreover, one can assign a list of primes to \magma{test\_sequence}. The algorithm will then start with the Hecke operators indicated by that sequence, and then continue with the others.
The option \magma{first\_test} tells the algorithm at which step the first test for the stop criterion is to be performed. The next test is then carried out after \magma{test\_interval} many steps, and so on. These numbers should be chosen small, too, unless the dimension test takes much time, which is rare, so that one wants to perform it less often, meaning that possibly more Hecke operators than necessary are computed (time consuming).
The option \magma{over\_residue\_field} tells the algorithm whether at the end of the computation the local Hecke factors should be base changed to their residue field. If that is done, only one of the conjugate local factors of the base changed algebra is retained.
Finally, the option \magma{try\_minimal} is passed on to \magma{AffineAlgebra}, when the output is generated. Calling that function with the option set \magma{true} can sometimes be very time consuming, but makes the output much shorter.
\subsubsection{Storage functions}
The package provides functions to store a list whose elements are records of type \magma{AlgebraData} in a file, and to re-read it. The usage of these functions is explained in the example at the beginning of this manual.
\intr{intrinsic CreateStorageFile ( filename :: MonStgElt )} This function prepares the file \magma{filename} for storing the data.
\intr{intrinsic StoreData (filename :: MonStgElt, forms :: SeqEnum)} This functions appends the list \magma{forms} of Hecke algebra data to the file \magma{filename}. That file must have been created by \magma{CreateStorageFile}.
\intr{intrinsic StoreData (filename :: MonStgElt, form :: Rec)} This function appends the Hecke algebra data \magma{form} to the file \magma{filename}. That file must have been created by \magma{CreateStorageFile}.
\intr{intrinsic RecoverData (LoadIn :: SeqEnum, LoadInRel :: Tup ) -> SeqEnum} In order to read Hecke algebra data from file \magma{``name''}, proceed as follows:\\ \magma{\hspace*{.5cm}> load ``name'';\\ \hspace*{.5cm}> readData := RecoverData(LoadIn,LoadInRel).}\\ Then \magma{readData} will contain a list whose elements are records of type \magma{AlgebraData}.
\subsubsection{Output functions}
\intr{intrinsic HeckeAlgebraPrint (ha :: SeqEnum)\\ intrinsic HeckeAlgebraPrint1 (ha :: SeqEnum)} These functions print part of the data stored in the list \magma{ha} of records of type \magma{AlgebraData} in a human readable format.
\intr{intrinsic GetLevel (a :: Rec) -> Any\\ intrinsic GetWeight (a :: Rec) -> Any\\ intrinsic GetCharacteristic (a :: Rec) -> Any\\ intrinsic GetResidueDegree (a :: Rec) -> Any\\ intrinsic GetDimension (a :: Rec) -> Any\\ intrinsic GetGorensteinDefect (a :: Rec) -> Any\\ intrinsic GetEmbeddingDimension (a :: Rec) -> Any\\ intrinsic GetNilpotencyOrder (a :: Rec) -> Any\\ intrinsic GetHeckeBound (a :: Rec) -> Any\\ intrinsic GetPrimesUpToHeckeBound (a :: Rec) -> Any\\ intrinsic GetNumberOperatorsUsed (a :: Rec) -> Any\\ intrinsic GetPolynomial (a :: Rec) -> Any\\ intrinsic GetImageName (a :: Rec) -> Any} These functions return the property of the record \magma{a} of type \magma{AlgebraData} specified by the name of the function. If the corresponding attribute is not assigned, the empty string is returned.
\intr{intrinsic HeckeAlgebraLaTeX (ha :: SeqEnum, filename :: MonStgElt : which := [ \\ \hspace*{2cm}
<GetLevel,"Level">,
<GetWeight,"Wt">,
<GetResidueDegree,"ResD">,\\ \hspace*{2cm}
<GetDimension,"Dim">,
<GetEmbeddingDimension,"EmbDim">,\\ \hspace*{2cm}
<GetNilpotencyOrder,"NilO">,
<GetGorensteinDefect,"GorDef">,\\ \hspace*{2cm}
<GetNumberOperatorsUsed,"\#Ops">,\\ \hspace*{2cm}
<GetPrimesUpToHeckeBound,"\#(p$<$HB)">,
<GetImageName,"Gp">
] )} This function creates the LaTeX file \magma{filename} containing a longtable consisting of certain properties of the objects in \magma{ha} which are supposed to be records of type \magma{AlgebraData}. The properties to be written are indicated by the list given in the option \magma{which} consisting of tuples \magma{<f, name>}. Here \magma{f} is a function that evaluates a record of type \magma{AlgebraData} to some Magma object which is afterwards transformed into a string using \magma{Sprint}. Examples for \magma{f} are the functions \magma{GetLevel} etc., which are described above. The \magma{name} will appear in the table header. For a sample usage, see the example at the beginning of this manual.
\subsubsection{Other functions}
\intr{intrinsic HeckeBound ( N :: RngIntElt, k :: RngIntElt ) -> RngIntElt\\ intrinsic HeckeBound ( eps :: GrpDrchElt, k :: RngIntElt ) -> RngIntElt} These functions compute the Hecke bound for weight \magma{k} and level \magma{N}, respectively Dirichelt character \magma{eps}. Note that the Hecke bound is also often called the Sturm bound.
\subsection{Algebra handling}
\subsubsection{Affine algebras}
Let $A$ be a commutative local Artin algebra with maximal ideal $\mathfrak{m}$ over a finite field $k$. The residue field $K=A/\mathfrak{m}$ is a finite extension of~$k$. By base changing to $K$ and taking one of the conjugate local factors, we now assume that $k=K$. The {\em embedding
dimension} $e$ is the $k$-dimension of $\mathfrak{m}/\mathfrak{m}^2$. By Nakayama's Lemma, this is the minimal number of generators for $\mathfrak{m}$. The name comes from the fact that there is a surjection $$ \pi: k[x_1, \dots, x_e] \twoheadrightarrow A.$$ Its kernel is called the {\em relations ideal}. By the {\em nilpotency order} we mean the maximal integer $n$ such that $m^n$ is not the zero ideal. (As the algebra is local and Artin, its maximal ideal is nilpotent.) We know that the ideal $$ J^{n+1} \text{ with } J := (x_1, \dots, x_e) $$ is in the kernel of $\pi$. So, in order to store $\pi$, we only need to store the kernel $R$ of the linear map between two finite dimensional $k$-vector spaces $$ \pi_1: k[x_1, \dots, x_e]/J^{n+1} \twoheadrightarrow A.$$
From the tuple $<k,e,n,R>$ the algebra can be recreated (up to isomorphism). Let us point out, however, that from the tuple it is not obvious whether two algebras are isomorphic. That would have to be tested after recreating the algebras.
These functions are used in order to store the Hecke algebras computed by \magma{HeckeAlgebras} in a way that does not use much memory, but retains the algebra up to isomorphism.
\intr{intrinsic AffineAlgebra (A :: AlgMat : try\_minimal := true) -> RngMPolRes\\ intrinsic AffineAlgebra (A :: AlgAss : try\_minimal := true) -> RngMPolRes} This function turns the local commutative algebra \magma{A} into an affine algebra over its residue field. In fact, the algebra is first base changed to its residue field, then for one of the conjugate local factors an affine presentation is computed. If the option \magma{try\_minimal} is true, the number of relations will in general be smaller, but the computation time may be longer.
\intr{intrinsic AffineAlgebraTup (A :: AlgMat : try\_minimal := true ) -> Tup\\ intrinsic AffineAlgebraTup (A :: AlgAss : try\_minimal := true) -> Tup} Given a commutative local Artin algebra \magma{A}, this function returns a tuple \magma{<k,e,n,R>}, consisting of the residue field \magma{k} of \magma{A}, the embedding dimension \magma{e}, the nilpotency order \magma{n} and relations \magma{R}. From these data, an affine algebra can be recreated which is isomorphic to one of the local factors of $A$ base changed to its residue field. If the option \magma{try\_minimal} is true, the number of relations will in general be smaller, but the computation time may be longer.
\intr{intrinsic AffineAlgebra (form :: Rec) -> RngMPolRes} Given a record of type \magma{AlgebraData}, this function returns the corresponding Hecke algebra as an affine algebra.
\intr{intrinsic AffineAlgebra (A :: Tup) -> RngMPolRes} This function turns a tuple \magma{<k,e,n,R>}, as above consisting of a field \magma{k}, two integers \magma{e}, \magma{n} (the embedding dimension and the nilpotency order) and relations \magma{R}, into an affine algebra.
\subsubsection{Matrix algebra functions}
\intr{intrinsic MatrixAlgebra ( L :: SeqEnum ) -> AlgMat} Given a list of matrices \magma{L}, this function returns the matrix algebra generated by the members of~\magma{L}.
\intr{intrinsic RegularRepresentation ( A :: AlgMat ) -> AlgMat} This function computes the regular representation of the commutative matrix algebra \magma{A}.
\intr{intrinsic CommonLowerTriangular ( A :: AlgMat ) -> AlgMat} Given a local commutative matrix algebra \magma{A}, this function returns an isomorphic matrix algebra whose matrices are all lower triangular, after a scalar extension to the residue field and taking one of the Galois conjugate factors.
\noindent{\bf Base change}
\intr{intrinsic BaseChange ( S :: Tup, T :: Tup ) -> Tup} This function computes the composition of the base change matrices \magma{T = <C,D>}, followed by those in \magma{S = <E,F>}.
\intr{intrinsic BaseChange ( M :: Mtrx, T :: Tup ) -> Mtrx} Given a matrix \magma{M} and a tuple \magma{T = <C,D>} of base change matrices (for a subspace), this function computes the matrix of \magma{M} with respect to the basis corresponding to \magma{T}.
\intr{intrinsic BaseChange ( M :: AlgMat, T :: Tup ) -> AlgMat} Given a matrix algebra \magma{M} and a tuple \magma{T = <C,D>} of base change matrices (for a subspace), this function computes the matrix algebra of \magma{M} with respect to the basis corresponding to \magma{T}.
\noindent{\bf Decomposition}
\intr{intrinsic Decomposition ( M :: Mtrx : DegBound := 0 ) -> Tup\\ intrinsic DecompositionUpToConjugation ( M :: Mtrx : DegBound := 0 ) -> Tup} Given a matrix \magma{M}, these functions compute a decomposition of the standard vector space such that \magma{M} acts as multiplication by a scalar on each summand. The output is a tuple consisting of base change tuples \magma{<C,D>} corresponding to the summands. With the second usage, summands conjugate under the absolute Galois group only appear once.
\intr{intrinsic Decomposition ( L :: SeqEnum : DegBound := 0 ) -> Tup\\ intrinsic DecompositionUpToConjugation ( L :: SeqEnum : DegBound := 0) -> Tup} Given a sequence \magma{L} of commuting matrices, these functions compute a decomposition of the standard vector space such that each matrix in \magma{L} acts as multiplication by a scalar on each summand. The output is a tuple consisting of base change tuples \magma{<C,D>} corresponding to the summands. With the second usage, summands conjugate under the absolute Galois group only appear once.
\intr{intrinsic Decomposition ( A :: AlgMat : DegBound := 0 ) -> Tup\\ intrinsic DecompositionUpToConjugation ( A :: AlgMat : DegBound := 0 ) -> Tup} Given a commutative matrix algebra \magma{A}, these functions compute a decomposition of the standard vector space such that each element in \magma{A} acts as multiplication by a scalar on each summand. The output is a tuple consisting of base change tuples \magma{<C,D>} corresponding to the summands. With the second usage, summands conjugate under the absolute Galois group only appear once.
\intr{intrinsic AlgebraDecomposition ( A :: AlgMat : DegBound := 0 ) -> SeqEnum\\ intrinsic AlgebraDecompositionUpToConjugation ( A :: AlgMat : DegBound := 0 )\\ \hspace*{2cm} -> SeqEnum} Given a matrix algebra \magma{A} over a finite field, these functions return a local factor of \magma{A} after scalar extension to the residue field. With the second usage, factors conjugate under the absolute Galois group only appear once.
\intr{intrinsic ChangeToResidueField ( A :: AlgMat ) -> SeqEnum} This function is identical to \magma{AlgebraDecompositionUpToConjugation}.
\noindent{\bf Localisations}
\intr{intrinsic Localisations ( L :: SeqEnum ) -> Tup, Tup\\ intrinsic Localisations ( A :: AlgMat ) -> Tup, Tup} Given a list \magma{L} of commuting matrices or a commutative matrix algebra \magma{A}, this function computes two tuples \magma{C}, \magma{D}, where \magma{C} contains a tuple consisting of the localisations of \magma{A}, respectively of the matrix algebra generated by \magma{L}, and \magma{D} consists of the corresponding base change tuples.
\subsubsection{Associative algebras}
\intr{intrinsic Localisations ( A :: AlgAss ) -> SeqEnum} This function returns a list of all localisations of the Artin algebra \magma{A}, which is assumed to be commutative. The output is a list of associative algebras.
\subsubsection{Gorenstein defect}
Let $A$ be a local Artin algebra over a field with unique maximal ideal $\mathfrak{m}$. We define the {\em Gorenstein defect} of $A$ to be $(\dim_{A/\mathfrak{m}} A[\mathfrak{m}]) -1$, which is equal to the number of $A$-module generators of the annihilator of the maximal ideal minus one. The algebra is said to be {\em Gorenstein} if its Gorenstein defect is equal to~$0$.
\intr{intrinsic GorensteinDefect ( A :: RngMPolRes) -> RngIntElt\\ intrinsic GorensteinDefect ( A :: AlgAss) -> RngIntElt\\ intrinsic GorensteinDefect ( A :: AlgMat ) -> RngIntElt} These functions return the Gorenstein defect of the local commutative algebra \magma{A}.
\intr{intrinsic IsGorenstein ( M :: RngMPolRes ) -> BoolElt\\ intrinsic IsGorenstein ( M :: AlgAss ) -> BoolElt\\ intrinsic IsGorenstein ( M :: AlgMat ) -> BoolElt} These functions test whether the commutative local algebra \magma{M} is Gorenstein.
\section{Tables of Hecke algebras} \label{section-tables}
\subsection*{Characteristic $p=2$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 431 {}\footnote{See \cite{kilford-nongorenstein}.} & 2 & 1 & 4 & 3 & 1 & 2 & 6 & 20 & $D_{3}$ \\ 503 {}\footnote{See \cite{kilford-nongorenstein}.} & 2 & 1 & 4 & 3 & 1 & 2 & 3 & 23 & $D_{3}$ \\ 1319 & 2 & 2 & 4 & 3 & 1 & 2 & 6 & 47 & $D_{5}$ \\ 1439 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 52 & $D_{3}$ \\ 1559 & 2 & 1 & 4 & 3 & 1 & 2 & 7 & 55 & $D_{3}$ \\ 1607 & 2 & 1 & 4 & 3 & 1 & 2 & 3 & 56 & $D_{3}$ \\ 1759 & 2 & 1 & 4 & 3 & 1 & 2 & 5 & 62 & $D_{3}$ \\ 1823 & 2 & 2 & 4 & 3 & 1 & 2 & 3 & 62 & $D_{5}$ \\ 1879 & 2 & 1 & 16 & 4 & 5 & 2 & 6 & 65 & $D_{3}$ \\ 1951 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 66 & $D_{3}$ \\ 1999 & 2 & 1 & 4 & 3 & 1 & 2 & 5 & 67 & $D_{3}$ \\ 2039 & 2 & 2 & 6 & 3 & 2 & 2 & 4 & 68 & $D_{5}$ \\ 2089 {}\footnote{See \cite{kilford-nongorenstein}.} & 2 & 1 & 18 & 4 & 7 & 2 & 5 & 70 & $D_{3}$ \\ 2351 & 2 & 1 & 6 & 3 & 2 & 2 & 6 & 77 & $D_{3}$ \\ 3407 & 2 & 1 & 16 & 4 & 5 & 2 & 7 & 103 & $D_{3}$ \\ 3527 & 2 & 2 & 4 & 3 & 1 & 2 & 3 & 107 & $D_{5}$ \\ 3623 & 2 & 1 & 6 & 3 & 2 & 2 & 3 & 110 & $D_{3}$ \\ 3967 & 2 & 1 & 14 & 4 & 4 & 2 & 4 & 121 & $D_{3}$ \\ 4231 & 2 & 1 & 4 & 3 & 1 & 2 & 10 & 126 & $D_{3}$ \\ 4481 & 2 & 1 & 8 & 4 & 2 & 2 & 7 & 132 & $D_{3}$ \\ 4799 & 2 & 1 & 12 & 4 & 3 & 2 & 5 & 139 & $D_{3}$ \\ 4943 & 2 & 2 & 6 & 3 & 2 & 2 & 4 & 143 & $D_{5}$ \\ 5167 & 2 & 1 & 6 & 3 & 2 & 2 & 5 & 149 & $D_{3}$ \\ 5591 & 2 & 1 & 12 & 4 & 3 & 2 & 8 & 158 & $D_{3}$ \\ 5591 & 2 & 3 & 4 & 3 & 1 & 2 & 5 & 158 & $D_{9}$ \\ 5791 & 2 & 1 & 8 & 3 & 3 & 2 & 8 & 162 & $D_{3}$ \\ 6199 & 2 & 1 & 16 & 4 & 5 & 2 & 7 & 174 & $D_{3}$ \\ 6287 & 2 & 1 & 6 & 3 & 2 & 2 & 4 & 175 & $D_{3}$ \\ 6343 & 2 & 1 & 12 & 4 & 3 & 2 & 5 & 177 & $D_{3}$ \\ 6551 & 2 & 1 & 6 & 3 & 2 & 2 & 7 & 182 & $D_{3}$ \\ 6823 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 189 & $D_{3}$ \\ 6911 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 190 & $D_{3}$ \\ 6967 & 2 & 1 & 12 & 4 & 3 & 2 & 8 & 191 & $D_{3}$ \\ 7057 & 2 & 1 & 16 & 4 & 4 & 2 & 6 & 193 & $D_{3}$ \\ 7103 & 2 & 3 & 4 & 3 & 1 & 2 & 3 & 194 & $D_{7}$ \\ 7151 & 2 & 2 & 4 & 3 & 1 & 2 & 4 & 195 & $D_{5}$ \\ 7351 & 2 & 1 & 12 & 4 & 3 & 2 & 9 & 200 & $D_{3}$ \\ 7487 & 2 & 2 & 6 & 3 & 2 & 2 & 3 & 203 & $D_{5}$ \\ 7583 & 2 & 1 & 4 & 3 & 1 & 2 & 6 & 205 & $D_{3}$ \\ 7951 & 2 & 1 & 4 & 3 & 1 & 2 & 6 & 216 & $D_{3}$ \\ 8111 & 2 & 5 & 4 & 3 & 1 & 2 & 5 & 217 & $D_{11}$ \\ 8167 & 2 & 1 & 6 & 3 & 2 & 2 & 4 & 218 & $D_{3}$ \\ 8191 & 2 & 2 & 6 & 3 & 2 & 2 & 5 & 218 & $D_{5}$ \\ 8623 & 2 & 1 & 4 & 3 & 1 & 2 & 8 & 227 & $D_{3}$ \\ 8713 & 2 & 1 & 16 & 4 & 4 & 2 & 4 & 231 & $D_{3}$ \\ 9127 & 2 & 1 & 16 & 4 & 5 & 2 & 4 & 240 & $D_{3}$ \\ 9281 & 2 & 1 & 12 & 4 & 3 & 2 & 8 & 243 & $D_{3}$ \\ 9439 & 2 & 1 & 8 & 3 & 3 & 2 & 4 & 248 & $D_{3}$ \\ 9623 & 2 & 2 & 6 & 3 & 2 & 2 & 5 & 252 & $D_{5}$ \\ 9967 & 2 & 1 & 6 & 3 & 2 & 2 & 4 & 260 & $D_{3}$ \\ 10079 & 2 & 1 & 12 & 4 & 3 & 2 & 8 & 263 & $D_{3}$ \\ 10103 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 263 & $D_{3}$ \\ 10391 & 2 & 1 & 6 & 3 & 2 & 2 & 9 & 269 & $D_{3}$ \\ 10391 & 2 & 3 & 4 & 3 & 1 & 2 & 8 & 269 & $D_{9}$ \\ 10487 & 2 & 1 & 10 & 3 & 4 & 2 & 3 & 272 & $D_{3}$ \\ 10567 & 2 & 1 & 12 & 3 & 5 & 2 & 5 & 274 & $D_{3}$ \\ 10639 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 274 & $D_{3}$ \\ 10663 & 2 & 1 & 6 & 3 & 2 & 2 & 9 & 275 & $D_{3}$ \\ 10687 & 2 & 1 & 6 & 3 & 2 & 2 & 4 & 275 & $D_{3}$ \\ 10799 & 2 & 1 & 14 & 3 & 6 & 2 & 8 & 278 & $D_{3}$ \\ 11159 & 2 & 3 & 4 & 3 & 1 & 2 & 4 & 283 & $D_{7}$ \\ 11321 & 2 & 1 & 8 & 4 & 2 & 2 & 9 & 289 & $D_{3}$ \\ 11743 & 2 & 1 & 4 & 3 & 1 & 2 & 5 & 297 & $D_{3}$ \\ 13063 & 2 & 1 & 6 & 3 & 2 & 2 & 5 & 326 & $D_{3}$ \\ 13487 & 2 & 1 & 8 & 3 & 3 & 2 & 6 & 334 & $D_{3}$ \\ 13999 & 2 & 3 & 4 & 3 & 1 & 2 & 4 & 345 & $D_{7}$ \\ 14303 & 2 & 1 & 4 & 3 & 1 & 2 & 5 & 354 & $D_{3}$ \\ 14543 & 2 & 3 & 4 & 3 & 1 & 2 & 3 & 360 & $D_{7}$ \\ 14639 & 2 & 2 & 4 & 3 & 1 & 2 & 5 & 361 & $D_{5}$ \\ 15121 & 2 & 2 & 8 & 4 & 2 & 2 & 6 & 369 & $D_{5}$ \\ 15193 & 2 & 1 & 16 & 4 & 6 & 2 & 4 & 370 & $D_{3}$ \\ 15271 & 2 & 1 & 6 & 3 & 2 & 2 & 8 & 372 & $D_{3}$ \\ 15383 & 2 & 2 & 6 & 3 & 2 & 2 & 3 & 375 & $D_{5}$ \\ 15391 & 2 & 1 & 6 & 3 & 2 & 2 & 7 & 375 & $D_{3}$ \\ 15551 & 2 & 1 & 4 & 3 & 1 & 2 & 11 & 377 & $D_{3}$ \\ 15607 & 2 & 1 & 6 & 3 & 2 & 2 & 8 & 378 & $D_{3}$ \\ 15641 & 2 & 1 & 32 & 4 & 7 & 2 & 8 & 378 & $D_{3}$ \\ 15919 & 2 & 1 & 26 & 4 & 7 & 2 & 6 & 383 & $D_{3}$ \\ 15991 & 2 & 1 & 12 & 4 & 3 & 2 & 9 & 386 & $D_{3}$ \\ 16127 & 2 & 3 & 4 & 3 & 1 & 2 & 3 & 390 & $D_{7}$ \\ 16369 & 2 & 1 & 24 & 4 & 6 & 2 & 6 & 398 & $D_{3}$ \\ 16487 & 2 & 1 & 6 & 3 & 2 & 2 & 4 & 400 & $D_{3}$ \\ 16649 & 2 & 1 & 16 & 4 & 4 & 2 & 8 & 403 & $D_{3}$ \\ 17471 & 2 & 1 & 6 & 3 & 2 & 2 & 11 & 421 & $D_{3}$ \\ 18047 & 2 & 1 & 30 & 4 & 6 & 2 & 4 & 431 & $D_{3}$ \\ 18097 & 2 & 1 & 36 & 5 & 6 & 2 & 9 & 432 & $D_{3}$ \\ 18127 & 2 & 1 & 12 & 4 & 3 & 2 & 5 & 433 & $D_{3}$ \\ 18257 & 2 & 1 & 12 & 4 & 4 & 2 & 4 & 436 & $D_{3}$ \\ 19079 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 449 & $D_{3}$ \\ 19079 & 2 & 3 & 4 & 3 & 1 & 2 & 4 & 449 & $D_{9}$ \\ 19441 & 2 & 1 & 16 & 4 & 4 & 2 & 7 & 457 & $D_{3}$ \\ 19543 & 2 & 2 & 4 & 3 & 1 & 2 & 4 & 460 & $D_{5}$ \\ 19583 & 2 & 1 & 4 & 3 & 1 & 2 & 7 & 461 & $D_{3}$ \\ 19751 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 462 & $D_{3}$ \\ 19919 & 2 & 1 & 16 & 4 & 5 & 2 & 7 & 467 & $D_{3}$ \\ 19927 & 2 & 1 & 6 & 3 & 2 & 2 & 4 & 467 & $D_{3}$ \\ 20183 & 2 & 2 & 4 & 3 & 1 & 2 & 7 & 474 & $D_{5}$ \\ 20599 & 2 & 1 & 6 & 3 & 2 & 2 & 6 & 481 & $D_{3}$ \\ 20759 & 2 & 1 & 18 & 4 & 6 & 2 & 9 & 483 & $D_{3}$ \\ 20887 & 2 & 2 & 6 & 3 & 2 & 2 & 4 & 487 & $D_{5}$ \\ 21319 & 2 & 3 & 4 & 3 & 1 & 2 & 9 & 497 & $D_{7}$ \\ 21647 & 2 & 1 & 6 & 3 & 2 & 2 & 6 & 504 & $D_{3}$ \\ 21737 & 2 & 1 & 24 & 5 & 4 & 2 & 7 & 507 & $D_{3}$ \\ 21839 & 2 & 2 & 4 & 3 & 1 & 2 & 7 & 509 & $D_{5}$ \\ 22159 & 2 & 2 & 4 & 3 & 1 & 2 & 9 & 515 & $D_{5}$ \\ 22511 & 2 & 1 & 4 & 3 & 1 & 2 & 6 & 522 & $D_{3}$ \\ 22567 & 2 & 3 & 4 & 3 & 1 & 2 & 4 & 523 & $D_{7}$ \\ 22751 & 2 & 3 & 4 & 3 & 1 & 2 & 4 & 526 & $D_{7}$ \\ 23159 & 2 & 1 & 20 & 4 & 6 & 2 & 6 & 535 & $D_{3}$ \\ 23159 & 2 & 3 & 6 & 3 & 2 & 2 & 5 & 535 & $D_{9}$ \\ 23279 & 2 & 1 & 8 & 3 & 3 & 2 & 4 & 537 & $D_{3}$ \\ 23321 & 2 & 1 & 12 & 4 & 4 & 2 & 7 & 538 & $D_{3}$ \\ 23417 & 2 & 1 & 26 & 4 & 7 & 2 & 10 & 539 & $D_{3}$ \\ 23567 & 2 & 1 & 4 & 3 & 1 & 2 & 3 & 544 & $D_{3}$ \\ 23687 & 2 & 3 & 4 & 3 & 1 & 2 & 3 & 548 & $D_{7}$ \\ 23743 & 2 & 2 & 4 & 3 & 1 & 2 & 4 & 548 & $D_{5}$ \\ 24151 & 2 & 2 & 4 & 3 & 1 & 2 & 5 & 556 & $D_{5}$ \\ 24281 & 2 & 1 & 16 & 4 & 4 & 2 & 5 & 557 & $D_{3}$ \\ 24439 & 2 & 2 & 6 & 3 & 2 & 2 & 7 & 561 & $D_{5}$ \\ 24847 & 2 & 2 & 4 & 3 & 1 & 2 & 8 & 570 & $D_{5}$ \\ 25031 & 2 & 1 & 8 & 3 & 3 & 2 & 7 & 573 & $D_{3}$ \\ 25111 & 2 & 1 & 6 & 3 & 2 & 2 & 8 & 574 & $D_{3}$ \\ 25247 & 2 & 1 & 6 & 3 & 2 & 2 & 10 & 575 & $D_{3}$ \\ 25409 & 2 & 1 & 8 & 4 & 2 & 2 & 8 & 580 & $D_{3}$ \\ 25439 & 2 & 1 & 4 & 3 & 1 & 2 & 6 & 580 & $D_{3}$ \\ 25447 & 2 & 1 & 6 & 3 & 2 & 2 & 11 & 581 & $D_{3}$ \\ 25793 & 2 & 1 & 16 & 4 & 4 & 2 & 5 & 590 & $D_{3}$ \\ 26431 & 2 & 2 & 6 & 3 & 2 & 2 & 4 & 599 & $D_{5}$ \\ 26839 & 2 & 3 & 4 & 3 & 1 & 2 & 5 & 607 & $D_{7}$ \\ 26959 & 2 & 1 & 6 & 3 & 2 & 2 & 23 & 610 & $D_{3}$ \\ 27143 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 615 & $D_{3}$ \\ 27143 & 2 & 3 & 4 & 3 & 1 & 2 & 3 & 615 & $D_{9}$ \\ 27647 & 2 & 1 & 26 & 4 & 10 & 2 & 7 & 623 & $D_{3}$ \\ 27673 & 2 & 1 & 8 & 4 & 2 & 2 & 7 & 623 & $D_{3}$ \\ 27743 & 2 & 3 & 4 & 3 & 1 & 2 & 3 & 624 & $D_{7}$ \\ 28031 & 2 & 2 & 6 & 3 & 2 & 2 & 5 & 631 & $D_{5}$ \\ 28031 & 2 & 1 & 8 & 3 & 3 & 2 & 5 & 631 & $D_{3}$ \\ 28031 & 2 & 4 & 4 & 3 & 1 & 2 & 7 & 631 & $D_{15}$ \\ 28279 & 2 & 1 & 8 & 3 & 3 & 2 & 4 & 635 & $D_{3}$ \\ 28279 & 2 & 1 & 26 & 4 & 8 & 2 & 14 & 635 & $D_{3}$ \\ 28279 & 2 & 1 & 4 & 3 & 1 & 2 & 11 & 635 & $D_{3}$ \\ 28279 & 2 & 1 & 6 & 3 & 2 & 2 & 6 & 635 & $D_{3}$ \\ 28703 & 2 & 1 & 20 & 4 & 5 & 2 & 5 & 642 & $D_{3}$ \\ 28759 & 2 & 1 & 8 & 3 & 3 & 2 & 5 & 645 & $D_{3}$ \\ 29023 & 2 & 1 & 4 & 3 & 1 & 2 & 8 & 650 & $D_{3}$ \\ 29287 & 2 & 2 & 8 & 3 & 3 & 2 & 4 & 653 & $D_{5}$ \\ 29311 & 2 & 3 & 6 & 3 & 2 & 2 & 5 & 653 & $D_{7}$ \\ 29399 & 2 & 1 & 6 & 3 & 2 & 2 & 7 & 654 & $D_{3}$ \\ 29567 & 2 & 1 & 6 & 3 & 2 & 2 & 6 & 657 & $D_{3}$ \\ 29879 & 2 & 2 & 4 & 3 & 1 & 2 & 5 & 666 & $D_{5}$ \\ 29959 & 2 & 1 & 6 & 3 & 2 & 2 & 4 & 668 & $D_{3}$ \\ 29959 & 2 & 3 & 4 & 3 & 1 & 2 & 4 & 668 & $D_{9}$ \\ 30223 & 2 & 1 & 8 & 3 & 3 & 2 & 5 & 674 & $D_{3}$ \\ 30367 & 2 & 2 & 6 & 3 & 2 & 2 & 5 & 677 & $D_{5}$ \\ 30431 & 2 & 2 & 4 & 3 & 1 & 2 & 4 & 677 & $D_{5}$ \\ 30559 & 2 & 3 & 4 & 3 & 1 & 2 & 9 & 680 & $D_{7}$ \\ 30727 & 2 & 1 & 6 & 3 & 2 & 2 & 9 & 685 & $D_{3}$ \\ 30911 & 2 & 2 & 4 & 3 & 1 & 2 & 5 & 686 & $D_{5}$ \\ 31079 & 2 & 2 & 4 & 3 & 1 & 2 & 5 & 690 & $D_{5}$ \\ 31159 & 2 & 1 & 16 & 4 & 5 & 2 & 6 & 691 & $D_{3}$ \\ 31247 & 2 & 6 & 4 & 3 & 1 & 2 & 3 & 692 & $D_{13}$ \\ 31271 & 2 & 1 & 6 & 3 & 2 & 2 & 9 & 693 & $D_{3}$ \\ 31321 & 2 & 3 & 8 & 4 & 2 & 2 & 7 & 693 & $D_{7}$ \\ 31513 & 2 & 1 & 16 & 4 & 4 & 2 & 4 & 697 & $D_{3}$ \\ 31543 & 2 & 2 & 4 & 3 & 1 & 2 & 5 & 697 & $D_{5}$ \\ 31847 & 2 & 5 & 4 & 3 & 1 & 2 & 3 & 703 & $D_{11}$ \\ 32009 & 2 & 1 & 12 & 4 & 3 & 2 & 9 & 706 & $D_{3}$ \\ 32143 & 2 & 3 & 4 & 3 & 1 & 2 & 4 & 708 & $D_{7}$ \\ 32183 & 2 & 3 & 4 & 3 & 1 & 2 & 3 & 708 & $D_{7}$ \\ 32327 & 2 & 1 & 16 & 4 & 5 & 2 & 5 & 710 & $D_{3}$ \\ 32327 & 2 & 3 & 4 & 3 & 1 & 2 & 3 & 710 & $D_{9}$ \\ 32353 & 2 & 1 & 20 & 4 & 6 & 2 & 5 & 711 & $D_{3}$ \\ 32401 & 2 & 1 & 20 & 4 & 6 & 2 & 6 & 712 & $D_{3}$ \\ 32479 & 2 & 5 & 4 & 3 & 1 & 2 & 4 & 714 & $D_{11}$ \\ 32647 & 2 & 3 & 4 & 3 & 1 & 2 & 4 & 719 & $D_{7}$ \\ 32687 & 2 & 5 & 4 & 3 & 1 & 2 & 3 & 720 & $D_{11}$ \\ 32719 & 2 & 2 & 4 & 3 & 1 & 2 & 14 & 721 & $D_{5}$ \\ 32887 & 2 & 2 & 6 & 3 & 2 & 2 & 6 & 724 & $D_{5}$ \\ 32983 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 725 & $D_{3}$ \\ 33223 & 2 & 1 & 4 & 3 & 1 & 2 & 9 & 732 & $D_{3}$ \\ 33343 & 2 & 1 & 4 & 3 & 1 & 2 & 5 & 733 & $D_{3}$ \\ 33679 & 2 & 1 & 4 & 3 & 1 & 2 & 5 & 738 & $D_{3}$ \\ 33767 & 2 & 3 & 4 & 3 & 1 & 2 & 3 & 739 & $D_{7}$ \\ 34351 & 2 & 2 & 4 & 3 & 1 & 2 & 8 & 753 & $D_{5}$ \\ 34471 & 2 & 2 & 12 & 4 & 3 & 2 & 6 & 756 & $D_{5}$ \\ 34487 & 2 & 1 & 14 & 4 & 4 & 2 & 7 & 756 & $D_{3}$ \\ 34591 & 2 & 2 & 4 & 3 & 1 & 2 & 5 & 757 & $D_{5}$ \\ 34679 & 2 & 1 & 6 & 3 & 2 & 2 & 4 & 758 & $D_{3}$ \\ 34679 & 2 & 3 & 4 & 3 & 1 & 2 & 4 & 758 & $D_{9}$ \\ 34721 & 2 & 1 & 12 & 4 & 4 & 2 & 9 & 759 & $D_{3}$ \\ 34847 & 2 & 1 & 16 & 4 & 5 & 2 & 6 & 762 & $D_{3}$ \\ 35401 & 2 & 1 & 56 & 5 & 8 & 2 & 10 & 776 & $D_{3}$ \\ 35591 & 2 & 9 & 4 & 3 & 1 & 2 & 7 & 779 & $D_{19}$ \\ 35759 & 2 & 3 & 6 & 3 & 2 & 2 & 5 & 781 & $D_{7}$ \\ 35839 & 2 & 2 & 4 & 3 & 1 & 2 & 5 & 781 & $D_{5}$ \\ 35977 & 2 & 2 & 8 & 4 & 2 & 2 & 5 & 783 & $D_{5}$ \\ 36191 & 2 & 1 & 46 & 5 & 7 & 2 & 11 & 786 & $D_{3}$ \\ 36791 & 2 & 1 & 12 & 4 & 3 & 2 & 8 & 799 & $D_{3}$ \\ 36871 & 2 & 2 & 6 & 3 & 2 & 2 & 5 & 801 & $D_{5}$ \\ 37087 & 2 & 1 & 32 & 4 & 11 & 2 & 5 & 804 & $D_{3}$ \\ 37199 & 2 & 1 & 18 & 4 & 4 & 2 & 8 & 806 & $D_{3}$ \\ 37607 & 2 & 1 & 4 & 3 & 1 & 2 & 3 & 814 & $D_{3}$ \\ 37831 & 2 & 2 & 4 & 3 & 1 & 2 & 5 & 820 & $D_{5}$ \\ 37879 & 2 & 3 & 4 & 3 & 1 & 2 & 5 & 821 & $D_{7}$ \\ 37993 & 2 & 2 & 8 & 4 & 2 & 2 & 5 & 824 & $D_{5}$ \\ 38047 & 2 & 2 & 4 & 3 & 1 & 2 & 9 & 825 & $D_{5}$ \\ 38167 & 2 & 1 & 14 & 4 & 4 & 2 & 10 & 829 & $D_{3}$ \\ 38231 & 2 & 1 & 4 & 3 & 1 & 2 & 4 & 830 & $D_{3}$ \\ 38287 & 2 & 1 & 4 & 3 & 1 & 2 & 5 & 832 & $D_{3}$ \\ 38303 & 2 & 1 & 4 & 3 & 1 & 2 & 3 & 832 & $D_{3}$ \\ 38593 & 2 & 1 & 20 & 4 & 6 & 2 & 9 & 836 & $D_{3}$ \\ 38959 & 2 & 1 & 10 & 3 & 4 & 2 & 9 & 842 & $D_{3}$ \\ 38977 & 2 & 1 & 20 & 4 & 6 & 2 & 13 & 842 & $D_{3}$ \\ 39023 & 2 & 1 & 40 & 5 & 7 & 2 & 10 & 842 & $D_{3}$ \\ 39199 & 2 & 2 & 4 & 3 & 1 & 2 & 7 & 844 & $D_{5}$ \\ 39631 & 2 & 1 & 16 & 4 & 5 & 2 & 11 & 853 & $D_{3}$ \\ 39679 & 2 & 1 & 6 & 3 & 2 & 2 & 12 & 854 & $D_{3}$ \\ 39679 & 2 & 3 & 4 & 3 & 1 & 2 & 4 & 854 & $D_{9}$ \\ \end{longtable}
\subsection*{Characteristic $p=2$ and non-prime square-free levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 1055 & 2 & 1 & 24 & 4 & 9 & 2 & 5 & 47 & $D_{3}$ \\ 1727 & 2 & 1 & 16 & 4 & 5 & 2 & 6 & 65 & $D_{3}$ \\ 2071 {}\footnote{First found by W.\ Stein.} & 2 & 1 & 8 & 4 & 2 & 2 & 5 & 73 & $D_{3}$ \\ 2631 & 2 & 1 & 40 & 4 & 17 & 2 & 6 & 106 & $D_{3}$ \\ 2991 & 2 & 1 & 40 & 4 & 17 & 2 & 4 & 121 & $D_{3}$ \\ 3095 & 2 & 1 & 40 & 4 & 17 & 2 & 4 & 114 & $D_{3}$ \\ 3431 & 2 & 1 & 24 & 5 & 4 & 2 & 6 & 107 & $D_{3}$ \\ 3471 & 2 & 1 & 16 & 5 & 3 & 2 & 5 & 146 & $D_{3}$ \\ 3639 & 2 & 1 & 28 & 4 & 11 & 2 & 5 & 140 & $D_{3}$ \\ 4031 & 2 & 1 & 16 & 4 & 4 & 2 & 6 & 125 & $D_{3}$ \\ 4087 & 2 & 1 & 8 & 4 & 2 & 2 & 6 & 126 & $D_{3}$ \\ 4119 & 2 & 1 & 12 & 4 & 3 & 2 & 4 & 156 & $D_{3}$ \\ 4415 & 2 & 1 & 8 & 4 & 2 & 2 & 6 & 153 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=2$, icosahedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 89491 & 2 & 2 & 12 & 4 & 3 & 2 & 4 & 1746 & $A_5$ \\ \end{longtable}
\subsection*{Characteristic $p=3$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 1031 & 3 & 2 & 4 & 3 & 1 & 2 & 4 & 55 & $D_{5}$ \\ 1511 & 3 & 3 & 4 & 3 & 1 & 2 & 9 & 74 & $D_{7}$ \\ 2087 & 3 & 2 & 4 & 3 & 1 & 2 & 3 & 98 & $D_{5}$ \\ 4259 & 3 & 2 & 4 & 3 & 1 & 2 & 3 & 179 & $D_{5}$ \\ 4799 & 3 & 3 & 22 & 4 & 9 & 2 & 9 & 196 & $D_{7}$ \\ 5939 & 3 & 2 & 4 & 3 & 1 & 2 & 4 & 235 & $D_{5}$ \\ 6899 & 3 & 2 & 4 & 3 & 1 & 2 & 3 & 269 & $D_{5}$ \\ 6959 & 3 & 2 & 4 & 3 & 1 & 2 & 4 & 270 & $D_{5}$ \\ 7523 & 3 & 2 & 4 & 3 & 1 & 2 & 4 & 289 & $D_{5}$ \\ 7559 & 3 & 2 & 4 & 3 & 1 & 2 & 6 & 290 & $D_{5}$ \\ 7583 & 3 & 3 & 20 & 3 & 9 & 2 & 6 & 290 & $D_{7}$ \\ 8219 & 3 & 2 & 4 & 3 & 1 & 2 & 6 & 310 & $D_{5}$ \\ 8447 & 3 & 5 & 20 & 3 & 9 & 2 & 3 & 318 & $D_{11}$ \\ 8699 & 3 & 2 & 6 & 3 & 2 & 2 & 9 & 326 & $D_{5}$ \\ 9431 & 3 & 3 & 4 & 3 & 1 & 2 & 4 & 350 & $D_{7}$ \\ 9743 & 3 & 2 & 8 & 3 & 3 & 2 & 3 & 360 & $D_{5}$ \\ 9887 & 3 & 2 & 8 & 3 & 3 & 2 & 3 & 365 & $D_{5}$ \\ 10079 & 3 & 2 & 60 & 3 & 29 & 2 & 5 & 368 & $D_{5}$ \\ 10247 & 3 & 2 & 10 & 4 & 3 & 2 & 5 & 375 & $D_{5}$ \\ 10847 & 3 & 3 & 22 & 4 & 9 & 2 & 9 & 395 & $D_{7}$ \\ 12011 & 3 & 2 & 4 & 3 & 1 & 2 & 3 & 431 & $D_{5}$ \\ 12119 & 3 & 2 & 56 & 3 & 27 & 2 & 8 & 434 & $D_{5}$ \\ 12263 & 3 & 2 & 8 & 3 & 3 & 2 & 3 & 438 & $D_{5}$ \\ 12959 & 3 & 5 & 20 & 3 & 9 & 2 & 4 & 457 & $D_{11}$ \\ 13907 & 3 & 2 & 22 & 4 & 9 & 2 & 8 & 487 & $D_{5}$ \\ 14699 & 3 & 2 & 4 & 3 & 1 & 2 & 6 & 513 & $D_{5}$ \\ 14783 & 3 & 3 & 20 & 3 & 9 & 2 & 3 & 515 & $D_{13}$ \\ 14783 & 3 & 3 & 20 & 3 & 9 & 2 & 3 & 515 & $D_{13}$ \\ \end{longtable}
\subsection*{Characteristic $p=5$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 419 & 5 & 1 & 4 & 3 & 1 & 2 & 3 & 40 & $D_{3}$ \\ 439 & 5 & 1 & 14 & 4 & 5 & 2 & 10 & 42 & $D_{3}$ \\ 491 & 5 & 1 & 4 & 3 & 1 & 2 & 3 & 46 & $D_{3}$ \\ 751 & 5 & 1 & 12 & 3 & 5 & 2 & 3 & 65 & $D_{3}$ \\ 839 & 5 & 1 & 6 & 3 & 2 & 2 & 6 & 70 & $D_{3}$ \\ 1231 & 5 & 1 & 4 & 3 & 1 & 2 & 3 & 97 & $D_{3}$ \\ 2579 & 5 & 1 & 4 & 3 & 1 & 2 & 3 & 180 & $D_{3}$ \\ 2699 & 5 & 1 & 14 & 4 & 5 & 2 & 8 & 188 & $D_{3}$ \\ 3299 & 5 & 1 & 4 & 3 & 1 & 2 & 6 & 220 & $D_{3}$ \\ 3359 & 5 & 1 & 4 & 3 & 1 & 2 & 6 & 222 & $D_{3}$ \\ 4111 & 5 & 1 & 4 & 3 & 1 & 2 & 3 & 267 & $D_{3}$ \\ 4219 & 5 & 1 & 20 & 3 & 6 & 2 & 5 & 274 & $D_{3}$ \\ 4931 & 5 & 3 & 12 & 3 & 5 & 2 & 3 & 310 & $D_{7}$ \\ 5011 & 5 & 1 & 4 & 3 & 1 & 2 & 5 & 316 & $D_{3}$ \\ 5639 & 5 & 1 & 6 & 3 & 2 & 2 & 5 & 348 & $D_{3}$ \\ 5939 & 5 & 3 & 12 & 3 & 5 & 2 & 5 & 366 & $D_{7}$ \\ 6079 & 5 & 1 & 6 & 3 & 2 & 2 & 5 & 370 & $D_{3}$ \\ 6271 & 5 & 1 & 4 & 3 & 1 & 2 & 3 & 379 & $D_{3}$ \\ 6571 & 5 & 1 & 12 & 3 & 5 & 2 & 5 & 399 & $D_{3}$ \\ 6691 & 5 & 1 & 4 & 3 & 1 & 2 & 5 & 405 & $D_{3}$ \\ 6779 & 5 & 1 & 6 & 3 & 2 & 2 & 6 & 410 & $D_{3}$ \\ 7459 & 5 & 1 & 12 & 3 & 5 & 2 & 7 & 443 & $D_{3}$ \\ 7759 & 5 & 3 & 4 & 3 & 1 & 2 & 3 & 457 & $D_{7}$ \\ 8779 & 5 & 1 & 12 & 3 & 5 & 2 & 12 & 511 & $D_{3}$ \\ 8819 & 5 & 3 & 4 & 3 & 1 & 2 & 3 & 513 & $D_{7}$ \\ 9011 & 5 & 1 & 4 & 3 & 1 & 2 & 6 & 522 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=7$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 199 & 7 & 1 & 4 & 3 & 1 & 2 & 3 & 30 & $D_{3}$ \\ 839 & 7 & 1 & 4 & 3 & 1 & 2 & 6 & 93 & $D_{3}$ \\ 1259 & 7 & 1 & 4 & 3 & 1 & 2 & 3 & 130 & $D_{3}$ \\ 1291 & 7 & 1 & 4 & 3 & 1 & 2 & 4 & 133 & $D_{3}$ \\ 1319 & 7 & 1 & 4 & 3 & 1 & 2 & 6 & 136 & $D_{3}$ \\ 1399 & 7 & 1 & 4 & 3 & 1 & 2 & 3 & 141 & $D_{3}$ \\ 1559 & 7 & 1 & 4 & 3 & 1 & 2 & 7 & 155 & $D_{3}$ \\ 1567 & 7 & 1 & 8 & 3 & 4 & 2 & 3 & 156 & $D_{3}$ \\ 1823 & 7 & 1 & 6 & 3 & 2 & 2 & 4 & 179 & $D_{3}$ \\ 1823 & 7 & 3 & 4 & 3 & 1 & 2 & 4 & 179 & $D_{9}$ \\ \end{longtable}
\subsection*{Characteristic $p=11$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 431 & 11 & 3 & 4 & 3 & 1 & 2 & 5 & 77 & $D_{7}$ \\ 563 & 11 & 1 & 4 & 3 & 1 & 2 & 3 & 97 & $D_{3}$ \\ 1187 & 11 & 1 & 6 & 3 & 2 & 2 & 6 & 181 & $D_{3}$ \\ 1223 & 11 & 3 & 4 & 3 & 1 & 2 & 4 & 187 & $D_{7}$ \\ 1231 & 11 & 1 & 4 & 3 & 1 & 2 & 3 & 189 & $D_{3}$ \\ 1231 & 11 & 3 & 4 & 3 & 1 & 2 & 3 & 189 & $D_{9}$ \\ 1327 & 11 & 1 & 4 & 3 & 1 & 2 & 3 & 199 & $D_{5}$ \\ 1327 & 11 & 1 & 4 & 3 & 1 & 2 & 4 & 199 & $D_{5}$ \\ 1583 & 11 & 1 & 24 & 3 & 11 & 2 & 4 & 230 & $D_{3}$ \\ 1619 & 11 & 1 & 4 & 3 & 1 & 2 & 3 & 235 & $D_{3}$ \\ 1823 & 11 & 1 & 4 & 3 & 1 & 2 & 4 & 263 & $D_{3}$ \\ 2243 & 11 & 1 & 4 & 3 & 1 & 2 & 3 & 310 & $D_{3}$ \\ 2351 & 11 & 1 & 4 & 3 & 1 & 2 & 6 & 325 & $D_{3}$ \\ 2351 & 11 & 3 & 4 & 3 & 1 & 2 & 6 & 325 & $D_{9}$ \\ 2503 & 11 & 1 & 4 & 3 & 1 & 2 & 3 & 341 & $D_{3}$ \\ 2591 & 11 & 1 & 4 & 3 & 1 & 2 & 5 & 351 & $D_{3}$ \\ 2647 & 11 & 1 & 4 & 3 & 1 & 2 & 3 & 360 & $D_{3}$ \\ 2767 & 11 & 1 & 4 & 3 & 1 & 2 & 3 & 370 & $D_{3}$ \\ 2791 & 11 & 1 & 4 & 3 & 1 & 2 & 3 & 375 & $D_{3}$ \\ 3011 & 11 & 1 & 4 & 3 & 1 & 2 & 5 & 402 & $D_{3}$ \\ 3119 & 11 & 1 & 4 & 3 & 1 & 2 & 5 & 415 & $D_{3}$ \\ 3299 & 11 & 1 & 4 & 3 & 1 & 2 & 4 & 434 & $D_{3}$ \\ 3299 & 11 & 3 & 4 & 3 & 1 & 2 & 4 & 434 & $D_{9}$ \\ 3571 & 11 & 1 & 4 & 3 & 1 & 2 & 4 & 462 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=13$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 367 & 13 & 1 & 4 & 3 & 1 & 2 & 3 & 78 & $D_{3}$ \\ 439 & 13 & 2 & 4 & 3 & 1 & 2 & 4 & 91 & $D_{5}$ \\ 563 & 13 & 1 & 4 & 3 & 1 & 2 & 3 & 111 & $D_{3}$ \\ 971 & 13 & 2 & 4 & 3 & 1 & 2 & 4 & 177 & $D_{5}$ \\ 1223 & 13 & 2 & 4 & 3 & 1 & 2 & 4 & 216 & $D_{5}$ \\ 1427 & 13 & 1 & 4 & 3 & 1 & 2 & 5 & 243 & $D_{3}$ \\ 1439 & 13 & 1 & 28 & 3 & 13 & 2 & 5 & 246 & $D_{3}$ \\ 1823 & 13 & 1 & 4 & 3 & 1 & 2 & 4 & 298 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=17$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 59 & 17 & 1 & 6 & 3 & 2 & 2 & 3 & 23 & $D_{3}$ \\ 239 & 17 & 2 & 4 & 3 & 1 & 2 & 5 & 68 & $D_{5}$ \\ 1327 & 17 & 1 & 4 & 3 & 1 & 2 & 3 & 289 & $D_{3}$ \\ 1427 & 17 & 2 & 4 & 3 & 1 & 2 & 3 & 306 & $D_{5}$ \\ 1951 & 17 & 1 & 4 & 3 & 1 & 2 & 4 & 402 & $D_{3}$ \\ 2503 & 17 & 1 & 4 & 3 & 1 & 2 & 3 & 497 & $D_{3}$ \\ 2687 & 17 & 1 & 36 & 3 & 17 & 2 & 4 & 529 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=19$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 439 & 19 & 1 & 6 & 3 & 2 & 2 & 3 & 125 & $D_{3}$ \\ 751 & 19 & 1 & 4 & 3 & 1 & 2 & 5 & 195 & $D_{5}$ \\ 751 & 19 & 1 & 4 & 3 & 1 & 2 & 5 & 195 & $D_{5}$ \\ 1427 & 19 & 1 & 6 & 3 & 2 & 2 & 3 & 335 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=23$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 83 & 23 & 1 & 4 & 3 & 1 & 2 & 3 & 37 & $D_{3}$ \\ 503 & 23 & 3 & 4 & 3 & 1 & 2 & 4 & 162 & $D_{7}$ \\ 971 & 23 & 2 & 4 & 3 & 1 & 2 & 4 & 284 & $D_{5}$ \\ 1259 & 23 & 1 & 4 & 3 & 1 & 2 & 3 & 358 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=29$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 107 & 29 & 1 & 4 & 3 & 1 & 2 & 3 & 55 & $D_{3}$ \\ 199 & 29 & 1 & 4 & 3 & 1 & 2 & 3 & 92 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=31$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 367 & 31 & 1 & 4 & 3 & 1 & 2 & 3 & 161 & $D_{3}$ \\ 743 & 31 & 3 & 4 & 3 & 1 & 2 & 4 & 293 & $D_{7}$ \\ \end{longtable}
\subsection*{Characteristic $p=37$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 139 & 37 & 1 & 4 & 3 & 1 & 2 & 4 & 83 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=41$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 83 & 41 & 1 & 4 & 3 & 1 & 2 & 3 & 61 & $D_{3}$ \\ 139 & 41 & 1 & 4 & 3 & 1 & 2 & 4 & 92 & $D_{3}$ \\ 431 & 41 & 1 & 4 & 3 & 1 & 2 & 5 & 233 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=43$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline 419 & 43 & 1 & 4 & 3 & 1 & 2 & 3 & 239 & $D_{3}$ \\ \hline \end{longtable}
\subsection*{Characteristic $p=47$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 31 & 47 & 1 & 4 & 3 & 1 & 2 & 3 & 30 & $D_{3}$ \\ 107 & 47 & 1 & 4 & 3 & 1 & 2 & 3 & 82 & $D_{3}$ \\ 139 & 47 & 1 & 4 & 3 & 1 & 2 & 4 & 101 & $D_{3}$ \\ 179 & 47 & 2 & 4 & 3 & 1 & 2 & 3 & 126 & $D_{5}$ \\ \end{longtable}
\subsection*{Characteristic $p=53$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 131 & 53 & 2 & 4 & 3 & 1 & 2 & 3 & 106 & $D_{5}$ \\ 211 & 53 & 1 & 4 & 3 & 1 & 2 & 4 & 159 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=59$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 23 {}\footnote{First found by K.\ Buzzard, unpublished.} & 59 & 1 & 4 & 3 & 1 & 2 & 4 & 30 & $D_{3}$ \\ 211 & 59 & 1 & 4 & 3 & 1 & 2 & 4 & 175 & $D_{3}$ \\ 227 & 59 & 1 & 4 & 3 & 1 & 2 & 3 & 187 & $D_{5}$ \\ 227 & 59 & 1 & 4 & 3 & 1 & 2 & 3 & 187 & $D_{5}$ \\ 367 & 59 & 1 & 4 & 3 & 1 & 2 & 3 & 279 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=61$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 239 & 61 & 1 & 4 & 3 & 1 & 2 & 4 & 199 & $D_{5}$ \\ 239 & 61 & 1 & 4 & 3 & 1 & 2 & 4 & 199 & $D_{5}$ \\ 431 & 61 & 1 & 4 & 3 & 1 & 2 & 5 & 327 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=67$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 31 & 67 & 1 & 4 & 3 & 1 & 2 & 6 & 41 & $D_{3}$ \\ 239 & 67 & 2 & 4 & 3 & 1 & 2 & 5 & 217 & $D_{5}$ \\ \end{longtable}
\subsection*{Characteristic $p=71$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 59 & 71 & 1 & 4 & 3 & 1 & 2 & 3 & 71 & $D_{3}$ \\ 239 & 71 & 1 & 4 & 3 & 1 & 2 & 5 & 223 & $D_{3}$ \\ 283 & 71 & 1 & 4 & 3 & 1 & 2 & 5 & 263 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=73$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 211 & 73 & 1 & 4 & 3 & 1 & 2 & 4 & 209 & $D_{3}$ \\ 283 & 73 & 1 & 4 & 3 & 1 & 2 & 5 & 269 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=79$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 307 & 79 & 1 & 4 & 3 & 1 & 2 & 5 & 307 & $D_{3}$ \\ \end{longtable}
\subsection*{Characteristic $p=83$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 47 & 83 & 2 & 4 & 3 & 1 & 2 & 4 & 67 & $D_{5}$ \\ 79 & 83 & 2 & 4 & 3 & 1 & 2 & 3 & 101 & $D_{5}$ \\ 107 & 83 & 1 & 6 & 3 & 2 & 2 & 3 & 132 & $D_{3}$ \\ 211 & 83 & 1 & 4 & 3 & 1 & 2 & 4 & 232 & $D_{3}$ \\ 251 & 83 & 1 & 4 & 3 & 1 & 2 & 3 & 271 & $D_{7}$ \\ 251 & 83 & 1 & 4 & 3 & 1 & 2 & 3 & 271 & $D_{7}$ \\ 251 & 83 & 1 & 4 & 3 & 1 & 2 & 3 & 271 & $D_{7}$ \\ \end{longtable}
\subsection*{Characteristic $p=89$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 131 & 89 & 1 & 4 & 3 & 1 & 2 & 3 & 165 & $D_{5}$ \\ 131 & 89 & 1 & 4 & 3 & 1 & 2 & 3 & 165 & $D_{5}$ \\ \end{longtable}
\subsection*{Characteristic $p=97$, prime levels, dihedral}
\begin{longtable}{||c|c|c|c|c|c|c|c|c|c||} \hline Level & Wt & ResD & Dim & EmbDim & NilO & GorDef & \#Ops & \#(p$<$HB) & Gp \\ \hline\endhead\hline\endfoot\hline\hline\endlastfoot 307 & 97 & 1 & 4 & 3 & 1 & 2 & 5 & 367 & $D_{3}$ \\ \end{longtable}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{On tempered representations} \author[David Kazhdan and Alexander Yom Din]{David Kazhdan and Alexander Yom Din}
\begin{abstract}
Let $G$ be a unimodular locally compact group. We define a property of irreducible unitary $G$-representations $V$ which we call c-temperedness, and which for the trivial $V$ boils down to F{\o}lner's condition (equivalent to the trivial $V$ being tempered, i.e. to $G$ being amenable). The property of c-temperedness is a-priori stronger than the property of temperedness.
We conjecture that for semisimple groups over local fields temperedness implies c-temperedness. We check the conjecture for a special class of tempered $V$'s, as well as for all tempered $V$'s in the cases of $G := SL_2 ({\mathbb R})$ and of $G = PGL_2 (\Omega)$ for a non-Archimedean local field $\Omega$ of characteristic $0$ and residual characteristic not $2$. We also establish a weaker form of the conjecture, involving only $K$-finite vectors.
In the non-Archimedean case, we give a formula expressing the character of a tempered $V$ as an appropriately-weighted conjugation-average of a matrix coefficient of $V$, generalizing a formula of Harish-Chandra from the case when $V$ is square-integrable. \end{abstract}
\maketitle
\setcounter{tocdepth}{1} \tableofcontents
\section{Introduction}\label{sec intro}
\subsection{}
Throughout the paper, we work with a unimodular second countable locally compact group $G$, and fix a Haar measure $dg$ on it. In the introduction, in \S\ref{ssec intro 1} - \S\ref{ssec intro 2.5} $G$ is assumed semisimple over a local field, while in \S\ref{ssec intro 3} - \S\ref{ssec intro 4} there is no such assumption. After the introduction, in \S\ref{sec Kfin} - \S\ref{sec proof of SL2R} $G$ is assumed semisimple over a local field, while in \S\ref{sec bi tempered} - \S\ref{sec ctemp is temp} there is no such assumption. Unitary representations of $G$ are pairs $(V , \pi)$, but for lightness of notation we denote them by $V$, keeping $\pi$ implicit.
\subsection{}\label{ssec intro 1}
Assume that $G$ is a semisimple group over a local field\footnote{So, for example, $G$ can be taken $SL_n ({\mathbb R})$ or $SL_n ({\mathbb Q}_p)$.}. The characterization of temperedness of irreducible unitary $G$-representations in terms of the rate of decrease of $K$-finite matrix coefficients is well-studied (see for example \cite{Wa,CoHaHo,Be}). Briefly, fixing a maximal compact subgroup $K \subset G$, an irreducible unitary $G$-representation $V$ is tempered if and only if for every two $K$-finite vectors $v_1 , v_2 \in V$ there exists $C>0$ such that $$ |\langle gv_1 , v_2 \rangle | \leq C \cdot \Xi_G (g)$$ for all $g \in G$, where $\Xi_G : G \to {\mathbb R}_{\ge 0}$ is Harish-Chandra's $\Xi$-function (see \S\ref{ssec V1 and Xi} for a reminder on the definition of $\Xi_G$). When considering matrix coefficients of more general vectors, differentiating between tempered and non-tempered irreducible unitary $G$-representations becomes more problematic, as the following example shows.
\begin{example}[see Claim \ref{clm counterexample}]\label{rem counterexample}
Let $G := PGL_2 (\Omega)$, $\Omega$ a local field. Denote by $A \subset G$ the subgroup of diagonal matrices. Given a unitary $G$-representation $V$ let us denote $$ {\mathcal M}_V (A) := \left\{ a \mapsto \langle a v_1 , v_2 \rangle \right\}_{v_1 , v_2 \in V} \subset C(A),$$ i.e. the set of matrix coefficients of $V$ restricted to $A$. Let us also denote $$ \widehat{L^1} (A) := \left\{ a \mapsto \int_{\hat{A}} \chi (a) \cdot \phi (\chi) \cdot d \chi \right\}_{\phi \in L^1 (\hat{A})} \subset C(A),$$ i.e. the set of Fourier transforms of $L^1$-functions on $\hat{A}$. Then for any non-trivial irreducible unitary $G$-representation $V$ we have $$ {\mathcal M}_V (A) = \widehat{L^1} (A).$$ \end{example}
The remedy proposed in this paper is that, instead of analysing the pointwise growth of matrix coefficients, we analyse their ``growth in average", i.e. the behaviour of integrals of norm-squared matrix coefficients over big balls.
\subsection{}
We fix a norm\footnote{In the non-Archimedean case, norms on finite-dimensional vector spaces are discussed, for example, in \cite[Chapter II, \S 1]{We}.} $|| - ||$ on the vector space ${\mathfrak g} := {\rm Lie} (G)$ and consider also the induced operator norm $|| - ||$ on $\textnormal{End} ({\mathfrak g})$. We define the ``radius" function $\mathbf{r} : G \to {\mathbb R}_{\ge 0}$ by $$ \mathbf{r} (g) := \log \left( \max \{ || \textnormal{Ad} (g) ||, || \textnormal{Ad} (g^{-1}) || \} \right)$$ where $\textnormal{Ad} : G \to \textnormal{Aut} ({\mathfrak g})$ is the adjoint representation. We denote then by $G_{<r} \subset G$ the subset of elements $g$ for which $\mathbf{r} ( g) < r$.
\begin{conjecture}[``asymptotic Schur orthogonality relations"]\label{main conj red exact}
Let $V$ be a tempered irreducible unitary $G$-representation. There exist $\mathbf{d}(V) \in {\mathbb Z}_{\ge 0}$ and $\mathbf{f} (V) \in {\mathbb R}_{>0}$ such that for all $v_1 , v_2 , v_3 , v_4 \in V$ we have $$ \lim_{r \to +\infty} \frac{\int_{G_{< r}} \langle g v_1 , v_2 \rangle \overline{\langle gv_3 , v_4 \rangle } \cdot dg}{ r^{\mathbf{d} (V)}} = \frac{1}{\mathbf{f} (V)} \cdot \langle v_1 , v_3 \rangle \overline{\langle v_2 , v_4 \rangle}.$$ \end{conjecture}
\begin{remark}[see Claim \ref{clm doesnt depend on norm}]\label{rem doesnt depend on norm}
The validity of Conjecture \ref{main conj red exact}, as well as the resulting invariants $\mathbf{d} (V)$ and $\mathbf{f} (V)$ (and of other similar results/conjectures below - see the formulation of Claim \ref{clm doesnt depend on norm}), do not depend on the choice of the norm $|| - ||$ on ${\mathfrak g}$ (used to construct the subsets $G_{<r}$). \end{remark}
\begin{remark}[see Remark \ref{rem ctemp red is temp}]
An irreducible unitary $G$-representation $V$ for which the condition of Conjecture \ref{main conj red exact} is verified is tempered. \end{remark}
\begin{remark}
In the notation of Conjecture \ref{main conj red exact}, $\mathbf{d} (V) = 0$ if and only if $V$ is square-integrable. In that case, $\mathbf{f} (V)$ is the well-known formal degree of $V$. \end{remark}
\begin{remark}[following from Proposition \ref{prop c-tempered orth rel cross}]
Let $V$ and $W$ be two tempered irreducible unitary $G$-representations for which Conjecture \ref{main conj red exact} holds, and which are non-isomorphic. Then for all $v_1 , v_2 \in V$ and $w_1 , w_2 \in W$ one has $$ \lim_{r \to +\infty} \frac{\int_{G_{<r}} \langle g v_1 , v_2 \rangle \overline{\langle g w_1 , w_2 \rangle} \cdot dg}{r^{(\mathbf{d} (V) + \mathbf{d} (W))/2}} = 0.$$ \end{remark}
\subsection{}\label{ssec intro 1.25}
We show the following statement, weaker than Conjecture \ref{main conj red exact}:
\begin{theorem}[see \S\ref{sec Kfin}]\label{thm main Kfin}
Let $V$ be a tempered irreducible unitary $G$-representation and $K \subset G$ a maximal compact subgroup. There exists $\mathbf{d}(V) \in {\mathbb Z}_{\ge 0}$ such that:
\begin{enumerate}
\item if $G$ is non-Archimedean, there exists $\mathbf{f} (V) \in {\mathbb R}_{>0}$ such that for all $K$-finite\footnote{When $G$ is non-Archimedean $K$-finite is the same as smooth (in particular does not depend on $K$).} $v_1 , v_2 , v_3 , v_4 \in V$ we have $$ \lim_{r \to +\infty} \frac{\int_{G_{< r}} \langle g v_1 , v_2 \rangle \overline{\langle gv_3 , v_4 \rangle } \cdot dg}{ r^{\mathbf{d} (V)}} = \frac{1}{\mathbf{f} (V)} \cdot \langle v_1 , v_3 \rangle \overline{\langle v_2 , v_4 \rangle}.$$
\item If $G$ is Archimedean, for any given non-zero $K$-finite vectors $v_1 , v_2 \in V$ there exists $C(v_1 , v_2) >0$ such that $$ \lim_{r \to +\infty} \frac{\int_{G_{< r}} | \langle g v_1 , v_2 \rangle |^2 \cdot dg}{ r^{\mathbf{d} (V)}} = C (v_1 , v_2).$$
\end{enumerate} \end{theorem}
\begin{remark}
We expect that it should not be very difficult to establish the statement of item $(1)$ of Theorem \ref{thm main Kfin} also in the Archimedean case, instead of the weaker statement of item $(2)$. \end{remark}
Concentrating on the non-Archimedean case for simplicity, Theorem \ref{thm main Kfin} has as a corollary the following proposition, a generalization (from the square-integrable case to the tempered case) of a formula of Harish-Chandra (see \cite[Theorem 9]{Ha2}), expressing the character as a conjugation-average of a matrix coefficient.
\begin{definition}
Assume that $G$ is non-Archimedean. We denote by $C^{\infty} (G)$ the space of (complex-valued) smooth functions on $G$ and by $D_c^{\infty} (G)$ the space of smooth distributions on $G$ with compact support. We denote by $C^{-\infty} (G)$ the dual to $D_c^{\infty} (G)$, i.e. the space of generalized functions on $G$ (thus we have an embedding $C^{\infty} (G) \subset C^{-\infty} (G)$). Given an admissible unitary $G$-representation $V$, we denote by $\Theta_V \in C^{-\infty} (G)$ the character of $V$. \end{definition}
\begin{proposition}[see \S\ref{ssec proof of prop formula ch}]\label{prop formula ch}
Let $V$ be a tempered irreducible unitary $G$-representation. Let $v_1 , v_2 \in V$ be smooth vectors. Denote by $m_{v_1 , v_2} \in C^{\infty}(G) \subset C^{-\infty} (G)$ the matrix coefficient $m_{v_1 , v_2} (g) := \langle g v_1 , v_2 \rangle$. Denoting $({}^g m) (x) := m(g^{-1} x g)$, the limit $$\lim_{r \to +\infty} \frac{\int_{G_{<r}} {}^g m_{v_1 , v_2} \cdot dg}{r^{\mathbf{d} (V)}}$$ exists in $C^{-\infty}(G)$, in the sense of weak convergence of generalized functions (i.e. convergence when paired against every element in $D_c^{\infty} (G)$), and is equal to $$ \frac{\langle v_1 , v_2 \rangle}{\mathbf{f} (V)} \cdot \Theta_V.$$ \end{proposition}
\subsection{}\label{ssec intro 1.5}
We are able to verify Conjecture \ref{main conj red exact} in some cases.
\begin{theorem}[see Theorem \ref{thm V1 c temp}]\label{thm intro slowest}
Conjecture \ref{main conj red exact} is true for the principal series irreducible unitary representation of ``slowest decrease", i.e. the unitary parabolic induction of the trivial character via a minimal parabolic subgroup. \end{theorem}
Here is the main result of the paper:
\begin{theorem}[see \S\ref{sec proof of SL2R}]\label{thm intro SL2R}
Conjecture \ref{main conj red exact} is true for all tempered irreducible unitary representations of $G := SL_2 ({\mathbb R})$ and of $G := PGL_2 (\Omega)$, where $\Omega$ is a non-Archimedean field of characteristic $0$ and of residual characteristic not equal to $2$. \end{theorem}
\subsection{}\label{ssec intro 2}
The proposition that follows shows that a seemingly weaker property implies that of Conjecture \ref{main conj red exact}.
\begin{definition}
Given a unitary $G$-representation $V$ and vectors $v_1 , v_2 \in V$ we define $$ \Ms{v_1}{v_2}{r} := \int_{ G_{<r}} |\langle g v_1 , v_2 \rangle |^2 \cdot dg.$$ \end{definition}
\begin{proposition}[see \S\ref{ssec clms red 1}]\label{prop from folner to exact}
Let $V$ be an irreducible unitary $G$-representation. Let $v_0 \in V$ be a unit vector such that the following holds:
\begin{enumerate}
\item For any vectors $v_1 , v_2 \in V$ we have $$ \underset{r \to +\infty}{\limsup} \frac{ \Ms{v_1}{v_2}{r} }{ \Ms{v_0}{v_0}{r} } < +\infty.$$
\item For any vectors $v_1 , v_2 \in V$ and $r^{\prime} > 0$ we have $$ \lim_{r \to +\infty} \frac{ \Ms{v_1}{v_2}{r + r^{\prime}} - \Ms{v_1}{v_2}{r - r^{\prime}} }{ \Ms{v_0}{v_0}{r} } = 0.$$
\end{enumerate}
Then Conjecture \ref{main conj red exact} holds for $V$.
\end{proposition}
\begin{question}
Does item $(1)$ of Proposition \ref{prop from folner to exact} hold for arbitrary irreducible unitary $G$-representations? \end{question}
\begin{remark}[see Proposition \ref{clm Prop tempered is c-tempered reductive holds}]\label{rem ctemp red is temp}
An irreducible unitary $G$-representation for which there exists a unit vector $v_0 \in V$ such that conditions $(1)$ and $(2)$ of Proposition \ref{prop from folner to exact} are satisfied is tempered. \end{remark}
\subsection{}\label{ssec intro 2.5}
After finishing writing the current paper, we have found previous works \cite{Mi} and \cite{An}. Work \cite{Mi} intends at giving an asymptotic Schur orthogonality relation for tempered irreducible unitary representations, but we could not understand its validity; on the first page the author defines a seminorm $||-||_p^2$ on $C^{\infty} (G)$ by a limit, but this limit clearly does not always exist. Work \cite{An} (which deals with the more general setup of a symmetric space) provides an asymptotic Schur orthogonality relation for $K$-finite vectors in a tempered irreducible unitary $G$-representation, in the case when $G$ is real and under a regularity assumption on the central character. This work also seems to provide an interpretation of what we have denoted as $\mathbf{f} (V)$ in terms of the Plancherel density (but it would be probably good to work this out in more detail).
\subsection{}\label{ssec intro 3}
Let now $G$ be an arbitrary unimodular second countable locally compact group. We formulate a property of irreducible unitary $G$-representations which we call c-temperedness (see Definition \ref{def c-temp}). The property of c-temperedness is, roughly speaking, an abstract version of properties (1) and (2) of Proposition \ref{prop from folner to exact}. Here $G_{< r} \subset G$ are replaced by a sequence $\{ F_n \}_{n \ge 0}$ of subsets of $G$, which we call a F{\o}lner sequence, whose existence is part of the definition (so that we speak of a representation c-tempered with F{\o}lner sequence $\{ F_n \}_{n \ge 0}$), while the condition replacing property (2) of Proposition \ref{prop from folner to exact} generalizes, in some sense, the F{\o}lner condition for a group to be amenable (i.e. for the trivial representation to be tempered).
We show in Corollary \ref{cor c-temp are temp} that any c-tempered irreducible unitary $G$-representation is tempered and pose the question:
\begin{question}\label{main question}
For which groups $G$ every tempered irreducible unitary $G$-representation is c-tempered with some F{\o}lner sequence? \end{question}
As before, c-tempered irreducible unitary $G$-representations enjoy a variant of asymptotic Schur orthogonality relations (see Proposition \ref{prop c-tempered orth rel}): \begin{equation}\label{eq orth rel} \lim_{n \to +\infty} \frac{\int_{F_n} \langle g v_1 , v_3 \rangle \overline{\langle gv_2 , v_4 \rangle} \cdot dg }{\int_{F_n} |\langle g v_0 , v_0 \rangle |^2 \cdot dg} = \langle v_1 , v_2 \rangle \overline{\langle v_3 , v_4 \rangle}\end{equation} for all $v_1 , v_2 , v_3 , v_4 \in V$ and all unit vectors $v_0 \in V$. Also, we have a variant for a pair of non-isomorphic representations (see Proposition \ref{prop c-tempered orth rel cross}).
\begin{definition}
Let us say that two irreducible unitary $G$-representations are twins if their closures in $\hat{G}$ (w.r.t. the Fell topology) coincide. \end{definition}
\begin{question}
Let $V_1$ and $V_2$ be irreducible unitary $G$-representations and assume that $V_1$ and $V_2$ are twins. Suppose that $V_1$ is c-tempered with F{\o}lner sequence $\{ F_n \}_{n \ge 0}$.
\begin{enumerate}
\item Is it true that $V_2$ is also c-tempered with F{\o}lner sequence $\{ F_n \}_{n \ge 0}$?
\item If so, is it true that for unit vectors $v_1 \in V_1$ and $v_2 \in V_2$ we have $$ \lim_{n \to +\infty} \frac{\int_{F_n} |\langle g v_1 , v_1 \rangle |^2 \cdot dg}{\int_{F_n} |\langle g v_2 , v_2 \rangle|^2 \cdot dg} = 1?$$
\end{enumerate} \end{question}
\subsection{}\label{ssec intro 4}
For many groups there exist tempered representations with the slowest rate of decrease of matrix coefficients. For such representations it is often much easier to prove analogs of c-temperedness or of orthogonality relation (\ref{eq orth rel}) than for other representations - as exemplified by Theorem \ref{thm intro slowest} above. See \cite{BoGa} for hyperbolic groups.
\subsection{} Alexander Yom Din's research was supported by the ISRAEL SCIENCE FOUNDATION (grant No 1071/20). David Kazhdan's research was partially supported by ERC grant No 669655. We would like to thank Pavel Etingof for great help with the proof of Claim \ref{clm SL2 3} in the case $G = SL_2 ({\mathbb R})$, which was present in a prior draft of the paper, before we encountered the work \cite{BrCoNiTa}. We would like to thank Vincent Lafforgue for a very useful discussion. We thank Erez Lapid for useful correspondence.
\subsection{}\label{ssec notation} Throughout the paper, $G$ is a unimodular second countable locally compact group. We fix a Haar measure $dg$ on $G$, as well as Haar measures on the other unimodular groups we encounter ($dk$ on the group $K$, etc.). We denote by $\textnormal{vol}_G (-)$ the volume with respect to $dg$.
All unitary $G$-representations are on separable Hilbert spaces.
Given a unitary $G$-representation $V$, vectors $v_1 , v_2 \in V$ and a measurable subset $F \subset G$, we denote $$ \Mss{v_1}{v_2}{F} := \int_{F} |\langle gv_1 , v_2 \rangle |^2 \cdot dg.$$ So in the case of a semisimple group over a local field as above, we have set $$ \Ms{v_1}{v_2}{r} := \Mss{v_1}{v_2}{G_{<r}}.$$
We write $L^2 (G) := L^2 (G , dg)$, considered as a unitary $G$-representation via the right regular action.
Given Hilbert spaces $V$ and $W$, we denote by ${\mathcal B} (V; W)$ the space of bounded linear operators from $V$ to $W$, and write ${\mathcal B} (V) := {\mathcal B} (V; V)$.
We write $F_1 \smallsetminus F_2$ for set differences and $F_1 \triangle F_2 := (F_1\smallsetminus F_2) \cup (F_2 \smallsetminus F_1)$ for symmetric set differences.
\section{Notion of c-temperedness}\label{sec bi tempered}
In this section, let $G$ be a unimodular second countable locally compact group. We introduce the notion of a c-tempered (with a given F{\o}lner sequence) irreducible unitary $G$-representation.
\subsection{}
The following definition aims at a generalization of the hypotheses of Proposition \ref{prop from folner to exact}, so as to make them suitable for a general group.
\begin{definition}\label{def c-temp}
Let $V$ be an irreducible unitary $G$-representation. Let $F_0 , F_1 , \ldots \subset G$ be a sequence of measurable pre-compact subsets all containing a neighbourhood of $1$. We say that $V$ is \textbf{c-tempered\footnote{``c" stands for ``matrix coefficients".} with F{\o}lner sequence $F_0 , F_1 , \ldots$} if there exists a unit vector $v_0 \in V$ such that the following two conditions are satisfied:
\begin{enumerate}
\item For all $v_1 , v_2 \in V$ we have\footnote{The notation $\Mss{-}{-}{-}$ is introduced in \S\ref{ssec notation}.} $$ \limsup_{n \to +\infty} \frac{ \Mss{v_1}{v_2}{F_n} }{ \Mss{v_0}{v_0}{F_n} } < +\infty.$$
\item For all $v_1 , v_2 \in V$ and all compact subsets $K \subset G$ we have $$ \lim_{n \to +\infty} \frac{\sup_{g_1 , g_2 \in K} \Mss{v_1}{v_2}{F_n \triangle g_2^{-1} F_n g_1} }{ \Mss{v_0}{v_0}{F_n} } = 0.$$
\end{enumerate} \end{definition}
\begin{example}
The trivial unitary $G$-representation is c-tempered with F{\o}lner sequence $F_0 , F_1 , \ldots$ if for any compact $K \subset G$ we have \begin{equation}\label{eq folner} \lim_{n \to +\infty} \sup_{g_1 , g_2 \in K} \frac{\textnormal{vol}_G (F_n \triangle g_2^{-1} F_n g_1)}{\textnormal{vol}_G (F_n)} = 0.\end{equation} By \textbf{F{\o}lner's condition}, the existence of such a sequence is equivalent\footnote{When stating F{\o}lner's condition for the amenability of $G$ it is more usual to consider $g_2^{-1} F_n$ rather than $g_2^{-1} F_n g_1$ in (\ref{eq folner}), i.e. to shift only on one side. However, using, for example, \cite[Theorem 4.1]{Gr} applied to the action of $G \times G$ on $G$, we see that the above stronger ``two-sided" condition also characterizes amenability.} to the trivial irreducible unitary $G$-representation being tempered, i.e. to $G$ being \textbf{amenable}. \end{example}
\subsection{} Irreducible unitary $G$-representations which are c-tempered satisfy ``asymptotic Schur orthogonality relations":
\begin{proposition}\label{prop c-tempered orth rel}
Let $V$ be an irreducible unitary $G$-representation. Assume that $V$ is c-tempered with F{\o}lner sequence $F_0 , F_1 , \ldots$ and let $v_0 \in V$ be a unit vector for which the conditions (1) and (2) of Definition \ref{def c-temp} are satisfied. Then for all $v_1 , v_2 , v_3 , v_4 \in V$ we have \begin{equation}\label{eq approx schur bi} \lim_{n \to +\infty} \frac{\int_{F_n} \langle g v_1 , v_2 \rangle \overline{\langle g v_3 , v_4 \rangle} \cdot dg}{ \Mss{v_0}{v_0}{F_n} } = \langle v_1 , v_3 \rangle \overline{\langle v_2 , v_4 \rangle}.\end{equation} \end{proposition}
\begin{proof}
First, notice that in order to show that the limit in (\ref{eq approx schur bi}) holds, it is enough to show that for every sub-sequence there exists a further sub-sequence of it on which the limit holds. Replacing our sequence by the sub-sequence, it is therefore enough to show simply that there exists a sub-sequence on which the limit holds - which is what we will do.
Define bilinear maps\footnote{Recall that $L^2 (G)$ denotes $L^2 (G , dg)$, viewed as a unitary $G$-representation via the right regular action.} $$ S_0 , S_1 , \ldots : V \times \overline{V} \to L^2 (G)$$ by $$ S_n (v_1 , v_2) (g) := \begin{cases} \frac{1}{\sqrt{ \Mss{v_0}{v_0}{F_n} }} \cdot \langle g v_1 , v_2 \rangle, & g \in F_n \\ 0, & g \notin F_n \end{cases}.$$ Clearly those are bounded.
\begin{itemize}
\item The bilinear maps $S_n$ are jointly bounded, i.e. there exists $C>0$ such that $||S_n||^2 \leq C$ for all $n$.
\end{itemize}
Indeed, by condition (1) of Definition \ref{def c-temp}, for any fixed $v_1 , v_2 \in V$ there exists $C>0$ such that $||S_n (v_1 , v_2)||^2 \leq C$ for all $n$. By the Banach-Steinhaus theorem, there exists $C>0$ such that $||S_n ||^2 \leq C$ for all $n$.
Next, define quadlinear forms $$ \Phi_1 , \Phi_2 , \ldots : V \times \overline{V} \times \overline{V} \times V \to {\mathbb C}$$ by $$ \Phi_n (v_1 , v_2 , v_3 , v_4) := \langle S_n (v_1 , v_2) , S_n (v_3 , v_4) \rangle$$
\begin{itemize}
\item The quadlinear forms $\Phi_n$ are jointly bounded, in fact $|| \Phi_n || \leq C$ for all $n$.
\end{itemize}
This follows immediately from the above finding $|| S_n ||^2 \leq C$ for all $n$.
\begin{itemize}
\item For all $g_1 , g_2 \in G$ and $v_1 , v_2 , v_3 , v_4 \in V$ we have \begin{equation}\label{eq approx GG-inv} \lim_{n \to +\infty } \left( \Phi_n (g_1 v_1 , g_2 v_2 , g_1 v_3, g_2 v_4 ) - \Phi_n (v_1 , v_2 , v_3 , v_4) \right)= 0.\end{equation}
\end{itemize}
Indeed, $$ | \Phi_n (g_1 v_1 , g_2 v_2 , g_1 v_3 , v_2 v_4) - \Phi_n (v_1 , v_2 , v_3 , v_4 ) | = $$ $$ = \frac{ | \int_{F_n} \langle g g_1 v_1 , g_2 v_2 \rangle \overline{\langle g g_1 v_3 , g_2 v_4 \rangle} \cdot dg - \int_{F_n} \langle g v_1 , v_2 \rangle \overline{\langle g v_3 , v_4 \rangle} \cdot dg |}{ \Mss{v_0}{v_0}{F_n} } \leq $$ $$ \leq \frac{ \int_{F_n \triangle g_2^{-1} F_n g_1 } |\langle g v_1 , v_2 \rangle | \cdot | \langle g v_3 , v_4 \rangle | \cdot dg}{ \Mss{v_0}{v_0}{F_n} } \leq $$ $$ \leq \sqrt{\frac{ \Mss{v_1}{v_2}{F_n \triangle g_2^{-1} F_n g_1} }{ \Mss{v_0}{v_0}{F_n} }} \cdot \sqrt{\frac{ \Mss{v_3}{v_4}{F_n \triangle g_2^{-1} F_n g_1} }{ \Mss{v_0}{v_0}{F_n} }}$$ and the last expression tends to $0$ as $n \to +\infty$ by condition (2) of Definition \ref{def c-temp}.
\begin{itemize}
\item There exists a sub-sequence $0 \leq m_0 < m_1 < \ldots$ such that $$ \lim_{n \to +\infty} \Phi_{m_n} (v_1 , v_2 , v_3 , v_4) = \langle v_1 , v_3 \rangle \overline{ \langle v_2 , v_4 \rangle } $$ for all $v_1 , v_2 , v_3 , v_4 \in V$.
\end{itemize}
By the sequential Banach-Alaouglu theorem (which is applicable since $V$ is separable), we can find a sub-sequence $0 \leq m_0 < m_1 < \ldots$ and a bounded quadlinear form $$ \Phi : V \times \overline{V} \times \overline{V} \times V \to {\mathbb C}$$ such that $\lim_{n \to +\infty}^{\textnormal{weak-}*} \Phi_{m_n} = \Phi$. Passing to the limit in equation (\ref{eq approx GG-inv}) we obtain that for all $g_1 , g_2 \in G$ and all $v_1 , v_2 , v_3 , v_4 \in V$ we have $$ \Phi (g_1 v_1 , g_2 v_2 , g_1 v_3 , g_2 v_4) = \Phi (v_1 , v_2 , v_3 , v_4).$$ Fixing $v_2 , v_4$, we obtain a bounded bilinear form $\Phi (- , v_2 , - , v_4) : V \times \overline{V} \to {\mathbb C}$ which is $G$-invariant, and hence by Schur's lemma is a multiple of the form $\langle - , - \rangle$, i.e. we have a uniquely defined $c_{v_2 , v_4} \in {\mathbb C}$ such that $$ \Phi (v_1 , v_2 , v_3 , v_4) = c_{v_2 , v_4} \cdot \langle v_1 , v_3 \rangle$$ for all $v_1 , v_3 \in V$. Similarly, fixing $v_1 , v_3$ we see that we have a uniquely defined $d_{v_1 , v_3} \in {\mathbb C}$ such that $$ \Phi (v_1 , v_2 , v_3 , v_4) = d_{v_1 , v_3} \cdot \overline{\langle v_2 , v_4 \rangle}$$ for all $v_2, v_4 \in {\mathbb C}$. Since $\Phi (v_0 , v_0, v_0, v_0) = 1$, plugging in $(v_1 , v_2 , v_3 ,v_4) := (v_0 , v_0 , v_0, v_0)$ in the first equality we find $c_{v_0 , v_0} = 1$. Then plugging in $(v_1 , v_2 , v_3 , v_4) := (v_1 , v_0 , v_3 , v_0)$ in both equalities and comparing, we find $d_{v_1 , v_3} = \langle v_1 , v_3 \rangle$. Hence we obtain $$ \Phi (v_1 , v_2 , v_3 , v_4) = \langle v_1 , v_3 \rangle \overline{\langle v_2 , v_4 \rangle} $$ for all $v_1 , v_2 , v_3 , v_4 \in V$.
Now, writing explicitly $\Phi_{m_n} (v_1 , v_2 , v_3 , v_4)$, we see that the limit in (\ref{eq approx schur bi}) is valid on our sub-sequence, so we are done, as we explained in the beginning of the proof.
\end{proof}
\subsection{} If one unit vector $v_0$ satisfies conditions (1) and (2) of Definition \ref{def c-temp} then all unit vectors do:
\begin{proposition}\label{prop c-tempered all are temp}
Let $V$ be an irreducible unitary $G$-representation. Assume that $V$ is c-tempered with F{\o}lner sequence $F_0 , F_1 , \ldots$ and let $v_0 \in V$ be a unit vector for which the conditions (1) and (2) of Definition \ref{def c-temp} are satisfied. Then for any unit vector $v'_0 \in V$ the conditions (1) and (2) of Definition \ref{def c-temp} are satisfied. \end{proposition}
\begin{proof}
Let $v'_0 \in V$ be a unit vector. From (\ref{eq approx schur bi}) we get $$ \lim_{n \to +\infty} \frac{ \Mss{v'_0}{v'_0}{F_n} }{ \Mss{v_0}{v_0}{F_n} } = 1.$$ This makes the claim clear.
\end{proof}
\subsection{}
We also have the following version of ``asymptotic Schur orthogonality relations" for a pair of non-isomorphic irreducible representations:
\begin{proposition}\label{prop c-tempered orth rel cross}
Let $V$ and $W$ be irreducible unitary $G$-representations. Assume that $V$ and $W$ are c-tempered with the same F{\o}lner sequence $F_0 , F_1 , \ldots$ and let $v_0 \in V$ and $w_0 \in W$ be unit vectors for which the conditions (1) and (2) of Definition \ref{def c-temp} are satisfied. Then for all $v_1 , v_2 \in V$ and $w_1 , w_2 \in W$ we have \begin{equation}\label{eq approx schur bi cross} \lim_{n \to +\infty} \frac{\int_{F_n} \langle g v_1 , v_2 \rangle \overline{\langle g w_1 , w_2 \rangle } \cdot dg}{ \sqrt{\Mss{v_0}{v_0}{F_n}} \sqrt{\Mss{w_0}{w_0}{F_n}} } = 0.\end{equation} \end{proposition}
\begin{proof}
We proceed similarly to the proof of Proposition \ref{prop c-tempered orth rel}. Namely, again it is enough to find a sub-sequence on which the limit holds. We define quadlinear forms $$ \Phi_1 , \Phi_2 , \ldots : V \times \overline{V} \times \overline{W} \times W \to {\mathbb C}$$ by $$ \Phi_n (v_1 , v_2 , w_1 , w_2) := \frac{\int_{F_n} \langle g v_1 , v_2 \rangle \overline{ \langle g w_1 , w_2 \rangle} \cdot dg}{\sqrt{\Mss{v_0}{v_0}{F_n}} \sqrt{\Mss{w_0}{w_0}{F_n}} }.$$ We see that these are jointly bounded, and that for all $g_1 , g_2 \in G$ and $v_1 , v_2 \in V$ and $w_1 , w_2 \in W$ we have $$ \lim_{n \to +\infty} \left( \Phi_n (g_1 v_1 , g_2 v_2 , g_1 w_1 , g_2 w_2) - \Phi_n (v_1 , v_2 , w_1 , w_2) \right) = 0.$$ We then find a bounded quadlinear form $$ \Phi : V \times \overline{V} \times \overline{W} \times W \to {\mathbb C}$$ and a sub-sequence $0 \leq m_0 < m_1 < \ldots$ such that $\lim_{n \to +\infty}^{\textnormal{weak-}*} \Phi_{m_n} = \Phi$. We get, for all $g_1 , g_2 \in G$ and $v_1 , v_2 \in V$ and $w_1 , w_2 \in W$: $$ \Phi (g_1 v_1 , g_2 v_2 , g_1 w_1 , g_2 w_2) = \Phi (v_1 , v_2 , w_1 , w_2).$$ By Schur's lemma we obtain $\Phi = 0$, giving us the desired. \end{proof}
\subsection{}
It is easy to answer Question \ref{main question} in the case of square-integrable representations:
\begin{proposition}\label{prop sq int}
Let $V$ be a square-integrable irreducible unitary $G$-representation. Then $V$ is c-tempered with F{\o}lner sequence any increasing sequence $F_0 , F_1 , \ldots $ of open pre-compact subsets in $G$, such that $1 \in F_0$ and $\cup_{n \ge 0} F_n = G$. \end{proposition}
\begin{proof}
Recall, that matrix coefficients of a square-integrable irreducible representation are square integrable. Let $v_0 \in V$ be a unit vector. Let $F_0 , F_1 , \ldots $ be any increasing sequence of open pre-compact subsets in $G$ whose union is $G$ and with $1 \in F_0$. Let $v_1,v_2 \in V$. Condition (1) of Definition \ref{def c-temp} holds because we have $$ \Mss{v_1}{v_2}{F_n} \leq \Mss{v_1}{v_2}{G} \leq \left( \frac{ \Mss{v_1}{v_2}{G} }{ \Mss{v_0}{v_0}{F_1} } \right) \cdot \Mss{v_0}{v_0}{F_n} .$$ As for condition (2) of Definition \ref{def c-temp}, let $\epsilon > 0$ and let $K \subset G$ be compact. There exists $n_0 \ge 0$ such that $$ \Mss{v_1}{v_2}{G \smallsetminus F_{n_0}} \leq \epsilon \cdot \Mss{v_0}{v_0}{F_1} .$$ There exists $n_1 \ge n_0$ such that $K F_{n_0} K^{-1} \subset F_{n_1}$. Let $n \ge n_1$ and let $g_1 , g_2 \in K$. Notice that $ ( F_n \triangle g_2^{-1} F_n g_1 ) \cap F_{n_0} = \emptyset$. Thus we have $$ \Mss{v_1}{v_2}{F_n \triangle g_2^{-1} F_n g_1} \leq \Mss{v_1}{v_2}{G \smallsetminus F_{n_0}} \leq \epsilon \cdot \Mss{v_0}{v_0}{F_1} \leq $$ $$ \leq \epsilon \cdot \Mss{v_0}{v_0}{F_n}.$$ \end{proof}
\section{c-Tempered irreps are tempered}\label{sec ctemp is temp}
In this section, let $G$ be a unimodular second countable locally compact group. We introduce some intermediate concepts, with the goal of showing that c-tempered irreducible unitary $G$-representations are tempered (Corollary \ref{cor c-temp are temp}).
\subsection{}
Let us recall some standard definitions and statements regarding weak containment.
\begin{definition}
Let $V$ and $W$ be unitary $G$-representations.
\begin{enumerate}
\item $V$ is \textbf{weakly contained} in $W$ if for every $v \in V$, compact $K \subset G$ and $\epsilon > 0$ there exist $w_1 , \ldots , w_r \in W$ such that $$ | \langle gv , v \rangle - \sum_{1 \leq i \leq r} \langle g w_i , w_i \rangle | \leq \epsilon$$ for all $g \in K$.
\item $V$ is \textbf{Zimmer-weakly contained}\footnote{or ``weakly contained in the sense of Zimmer", following \cite[Remark F.1.2.(ix)]{BeHaVa}.} in $W$ if for every $v_1 , \ldots , v_r \in V$, compact $K \subset G$ and $\epsilon > 0$ there exist $w_1 , \ldots , w_r \in W$ such that $$ |\langle gv_i , v_j \rangle - \langle g w_i , w_j \rangle | \leq \epsilon $$ for all $1 \leq i,j \leq r$ and $g \in K$.
\end{enumerate}
\end{definition}
To facilitate the formulation of the next lemma, let us also give the following intermediate definition:
\begin{definition}
Let $V$ and $W$ be unitary $G$-representations. Let us say that $V$ is \textbf{strongly-weakly contained} in $W$ if for every $v \in V$, compact $K \subset G$ and $\epsilon > 0$ there exists $w \in W$ such that $$ | \langle g v , v \rangle - \langle gw , w \rangle | \leq \epsilon$$ for all $g \in K$. \end{definition}
\begin{lemma}\label{lem prop of weak cont}
Let $V$ and $W$ be unitary $G$-representations.
\begin{enumerate}
\item If $V$ is Zimmer-weakly contained in $W$ then $V$ is strongly-weakly contained in $W$, and if $V$ is strongly-weakly contained in $W$ then $V$ is weakly contained in $W$.
\item If $V$ is weakly contained in $W$ then $V$ is strongly-weakly contained in\footnote{Here, $W^{\oplus \infty}$ stands for the Hilbert direct sum of countably many copies of $W$.} $W^{\oplus \infty}$.
\item If $V$ is weakly contained in $W^{\oplus \infty}$ then $V$ is weakly contained in $W$.
\item If $V$ is irreducible and $V$ is weakly contained in $W$ then $V$ is strongly-weakly contained in $W$.
\item If $V$ is cyclic (in particular, if $V$ is irreducible) and $V$ is strongly-weakly contained in $W$ then $V$ is Zimmer-weakly contained in $W$.
\item If $V$ is strongly-weakly contained in $W$ then $V$ is Zimmer-weakly contained in $W^{\oplus \infty}$.
\end{enumerate} \end{lemma}
\begin{proof}
Statements $(1)$, $(2)$ and $(3)$ are straight-forward. For statement $(4)$ see, for example, \cite[Proposition F.1.4]{BeHaVa}. For statement $(5)$ see \cite[proof of $(iii)\implies(iv)$ of Proposition 2.2]{Ke}. For statement $(6)$, again see \cite[proof of $(iii)\implies(iv)$ of Proposition 2.2]{Ke} (one writes $V$ as a Hilbert direct sum of countably many cyclic unitary $G$-representations, and uses item $(5)$). \end{proof}
\begin{corollary}\label{cor temp Zimmer temp}
Let $V$ and $W$ be unitary $G$-representations.
\begin{enumerate}
\item $V$ is weakly contained in $W$ if and only if $V$ is Zimmer-weakly contained in $W^{\oplus \infty}$.
\item If $V$ is irreducible, $V$ is weakly contained in $W$ if and only if $V$ is Zimmer-weakly contained in $W$.
\end{enumerate}
\end{corollary}
The following definition of temperedness is classical:
\begin{definition}
A unitary $G$-representation $V$ is said to be \textbf{tempered} if $V$ is weakly contained in\footnote{Recall that $L^2 (G)$ denotes $L^2 (G , dg)$, viewed as a unitary $G$-representation via the right regular action.} $L^2 (G)$. \end{definition}
\begin{remark}\label{rem irr zimmer} Notice that an irreducible unitary $G$-representation is tempered if and only if it is Zimmer-weakly contained in $L^2 (G)$, by part $(2)$ of Corollary \ref{cor temp Zimmer temp}. \end{remark}
\subsection{}
The next definitions are related to the idea that one representation is weakly contained in another if there ``almost" exists a $G$-intertwining isometric embedding from the one to the other.
\begin{definition}\label{def asymp emb}
Let $V$ and $W$ be unitary $G$-representations. A sequence $\{ S_n \}_{n \ge 0} \subset {\mathcal B} (V ; W)$ is an \textbf{asymptotic embedding} if the following conditions are satisfied:
\begin{enumerate}
\item The operators $\{ S_n \}_{n \ge 0}$ are jointly bounded, i.e. there exists $C>0$ such that $|| S_n ||^2 \leq C$ for all $n \ge 0$.
\item Given $v_1 , v_2 \in V$ and a compact $K \subset G$ we have $$ \lim_{n \to +\infty} \ \sup_{g \in K} | \langle (S_n g - g S_n) v_1 , S_n v_2 \rangle | = 0.$$
\item Given $v_1 , v_2 \in V$, we have $$ \lim_{n \to +\infty} \ \langle S_n v_1 , S_n v_2 \rangle = \langle v_1 , v_2 \rangle.$$
\end{enumerate} \end{definition}
\begin{definition} Let $V$ and $W$ be unitary $G$-representations.
\begin{enumerate}
\item We say that $V$ is \textbf{o-weakly contained}\footnote{``o" stands for ``operator".} in $W$ if there exists an asymptotic embedding $\{ S_n \}_{n \ge 0} \subset {\mathcal B} (V ; W)$.
\item We say that $V$ is \textbf{o-tempered} if it is o-weakly contained in $L^2 (G)$.
\end{enumerate} \end{definition}
\begin{lemma}\label{lem asymp emb uniform}
In the context of Definition \ref{def asymp emb}, if conditions $(1)$ and $(2)$ of Definition \ref{def asymp emb} are satisfied then given compacts $L_1 , L_2 \subset V$ and a compact $K \subset G$ we have $$ \lim_{n \to +\infty} \sup_{v_1 \in L_1 , v_2 \in L_2 , g \in K} | \langle (S_n g - g S_n) v_1 , S_n v_2 \rangle | = 0,$$ and if conditions $(1)$ and $(3)$ of Definition \ref{def asymp emb} are satisfied then given compacts $L_1, L_2 \subset V$ we have $$ \lim_{n \to +\infty} \sup_{v_1 \in L_1 , v_2 \in L_2} |\langle S_n v_1 , S_n v_2 \rangle - \langle v_1 , v_2 \rangle| = 0.$$ \end{lemma}
\begin{proof}
This follows from the well-known fact from functional analysis that pointwise convergence coincides with compact convergence on equi-continuous subsets, see \cite[Proposition 32.5]{Tr}. \end{proof}
\begin{lemma}\label{lem asymp emb irrep}
In the context of Definition \ref{def asymp emb}, assume that $V$ is irreducible. If conditions $(1)$ and $(2)$ of Definition \ref{def asymp emb} are satisfied then there exists a sub-sequence $0 \leq m_0 < m_1 < \ldots$ and $c \in {\mathbb R}_{\ge 0}$ such that for all $v_1 , v_2 \in V$ we have \begin{equation}\label{eq limit c} \lim_{n \to +\infty} \ \langle S_{m_n} v_1 , S_{m_n} v_2 \rangle = c \cdot \langle v_1 , v_2 \rangle.\end{equation} In particular, if there exists $v \in V$ such that $\liminf_{n \to +\infty} || S_n v ||^2 > 0$ then there exists $d \in {\mathbb R}_{> 0}$ (in fact, $d^{-2} = \lim_{n \to +\infty} || S_{m_n} v ||^2 / || v ||^2$) such that $\{ d S_{m_n} \}_{n \ge 0}$ satisfies condition $(3)$ of Definition \ref{def asymp emb}, i.e. is an asymptotic embedding. \end{lemma}
\begin{proof}
By the sequential Banach–Alaoglu theorem (applicable as $V$ is separable, and $\{ S_n^* S_n \}_{n \ge 0}$ are jointly bounded by condition $(1)$), there exists a sub-sequence $1 \leq m_0 < m_1 < \ldots$ such that $\{ S_{m_n}^* S_{m_n} \}_{n \ge 0}$ converges in the weak operator topology to some $S \in {\mathcal B} (V)$.
Let us first check that $S$ is $G$-invariant. For $g \in G$ and $v_1 , v_2 \in V$ we have $$ |\langle S_n^* S_n g v_1 , v_2 \rangle - \langle S_n^* S_n v_1 , g^{-1} v_2 \rangle | = | \langle S_n g v_1 , S_n v_2 \rangle - \langle S_n v_1 , S_n g^{-1} v_2 \rangle | \leq $$ $$ \leq |\langle (S_n g - g S_n) v_1 , S_n v _2 \rangle | + |\langle S_n v_1 , (g^{-1} S_n - S_n g^{-1}) v_2 \rangle |$$ and both summands in the last expression converge to $0$ as $n \to +\infty$ by condition $(2)$. Therefore $$ | \langle Sg v_1 , v_2 \rangle - \langle g S v_1 , v_2 \rangle | = \lim_{n \to +\infty} |\langle S_{m_n}^* S_{m_n} g v_1 , v_2 \rangle - \langle S_{m_n}^* S_{m_n} v_1 , g^{-1} v_2 \rangle | = 0 $$ i.e. $\langle S g v_1 , v_2 \rangle = \langle g S v_1 , v_2 \rangle$. Thus, since $v_1$ and $v_2$ were arbitrary, $S g = g S$. This holds for all $g \in G$, i.e. $S$ is $G$-invariant.
By Schur's lemma, we deduce $S = c \cdot \textnormal{Id}_V$ for some $c \in {\mathbb C}$. This translates precisely to (\ref{eq limit c}). The last claim is then straight-forward.
\end{proof}
\begin{remark}\label{rem alt cond}
Using Lemma \ref{lem asymp emb uniform}, it is straight-forward that, assuming condition $(1)$ of Definition \ref{def asymp emb}, conditions $(2)$ and $(3)$ in Definition \ref{def asymp emb} are equivalent to the one condition that for $v_1 , v_2 \in V$ and a compact $K \subset G$ one has $$ \lim_{n \to +\infty} \sup_{g \in K} | \langle g S_n v_1 , S_n v_2 \rangle - \langle g v_1 , v_2 \rangle | = 0.$$ Indeed, let us write \begin{equation}\label{eq alt cond} \langle g S_n v_1 , S_n v_2 \rangle - \langle gv_1 , v_2 \rangle = \langle (g S_n - S_n g) v_1 , S_n v_2 \rangle + ( \langle S_n g v_1 , S_n v_2 \rangle - \langle gv_1 , v_2 \rangle).\end{equation} The current condition gives condition $(3)$ by plugging in $g = 1$, and then (\ref{eq alt cond}) gives condition $(2)$, using the uniformity provided by Lemma \ref{lem asymp emb uniform}. Conversely, (\ref{eq alt cond}) shows immediately (again taking into consideration Lemma \ref{lem asymp emb uniform}) that conditions $(2)$ and $(3)$ imply the current condition. \end{remark}
\subsection{}
The concept of o-weak containment in fact coincides with that of Zimmer-weak containment:
\begin{proposition}\label{prop o cont is weakly cont}
Let $V$ and $W$ be unitary $G$-representations. Then $V$ is o-weakly contained in $W$ if and only if $V$ is Zimmer-weakly contained in $W$. \end{proposition}
\begin{proof}
Let $\{ S_n \}_{n \ge 0} \subset {\mathcal B} (V ; W)$ be an asymptotic embedding. Given $v_1 , \ldots , v_r \in V$, by Remark \ref{rem alt cond}, given any compact $K \subset G$ we have $$ \lim_{n \to +\infty} \sup_{g \in K} |\langle g S_n v_i , S_n v_j \rangle - \langle g v_i , v_j \rangle | = 0$$ for all $1 \leq i,j \leq r$, and thus $$ \lim_{n \to +\infty} \sup_{g \in K} \sup_{1 \leq i,j \leq r} |\langle g S_n v_i , S_n v_j \rangle - \langle g v_i , v_j \rangle | = 0.$$ Thus by definition $V$ is Zimmer-weakly contained in $W$.
Conversely, suppose that $V$ is Zimmer-weakly contained in $W$. Let $\{ e_n \}_{n \ge 0}$ be an orthonormal basis for $V$. Let $\{ K_n \}_{n \ge 0}$ be an increasing sequence of compact subsets in $G$, with $1 \in K_0$ and with the property that for any compact subset $K \subset G$ there exists $n \ge 0$ such that $K \subset K_n$. As $V$ is Zimmer-weakly contained in $W$, given $n \ge 0$, let us find $w^n_0 , \ldots , w^n_n \in W$ such that $$ \sup_{g \in K_n} |\langle g e_i , e_j \rangle - \langle g w^n_i , w^n_j \rangle | \leq \frac{1}{n+1}$$ for all $0 \leq i,j \leq n$. Define $S_n : V \to W$ by $$ S_n \left( \sum_{i \ge 0} c_i \cdot e_i \right) := \sum_{0 \leq i \leq n} c_i \cdot w^n_i.$$ We want to check that $\{ S_n \}_{n \ge 0}$ is an asymptotic embedding. As for condition $(1)$, notice that $$ \left\Vert S_n \left( \sum_{i \ge 0} c_i e_i \right) \right\Vert^2 = \left\Vert \sum_{0 \leq i \leq n} c_i w^n_i \right\Vert^2 = \left| \sum_{0 \leq i,j \leq n} c_i \overline{c_j} \cdot \langle w^n_i , w^n_j \rangle \right| \leq $$ $$ \leq \left| \sum_{0 \leq i,j \leq n} c_i \overline{c_j} \cdot \langle e_i , e_j \rangle \right| + \left| \sum_{0 \leq i,j \leq n } c_i \overline{c_j} \cdot \left( \langle w_i^n , w_j^n \rangle - \langle e_i , e_j \rangle \right)\right| \leq $$ $$ \leq \sum_{0 \leq i \leq n} |c_i|^2 + \frac{1}{n+1} \left( \sum_{0 \leq i \leq n} |c_i| \right)^2 \leq2 \sum_{0 \leq i \leq n} |c_i|^2 \leq 2 \cdot \left\Vert \sum_{i \ge 0} c_i e_i \right\Vert^2,$$ showing that $|| S_n ||^2 \leq 2$ for all $n \ge 0$. It is left to show the condition as in Remark \ref{rem alt cond}. Let us thus fix a compact $K \subset G$. Notice that it is straight-forward to see that it is enough to check the condition for vectors in a subset of $V$, the closure of whose linear span is equal to $V$. So it is enough to check that $$ \lim_{n \to +\infty} \sup_{g \in K} |\langle g S_n e_i , S_n e_j \rangle - \langle g e_i , e_j \rangle | = 0$$ for any given $i,j \ge 0$. Taking $n$ big enough so that $K \subset K_n$ and $n \ge \max \{ i,j \}$, we have $$ \sup_{g \in K} |\langle g S_n e_i , S_n e_j \rangle - \langle g e_i , e_j \rangle | = \sup_{g \in K} |\langle g w^n_i , w^n_j \rangle - \langle g e_i , e_j \rangle | \leq \frac{1}{n+1},$$ giving the desired. \end{proof}
\begin{corollary}\label{cor o temp is temp}
An irreducible unitary $G$-representation is o-tempered if and only if it is tempered. \end{corollary}
\begin{proof}
This is a special case of Proposition \ref{prop o cont is weakly cont}, taking into account Remark \ref{rem irr zimmer}. \end{proof}
\subsection{}
Here we give a weaker version of c-temperedness, which is technically convenient to relate to other concepts of this section.
\begin{definition}\label{def right-c-temp}
Let $V$ be an irreducible unitary $G$-representation. Let $F_0 , F_1 , \ldots \subset G$ be a sequence of measurable pre-compact subsets all containing a neighbourhood of $1$. We say that $V$ is \textbf{right-c-tempered with F{\o}lner sequence $F_0 , F_1 , \ldots$} if there exists a unit vector $v_0 \in V$ such that the following two conditions are satisfied:
\begin{enumerate}
\item For all $v \in V$ we have $$ \limsup_{n \to +\infty} \frac{ \Mss{v}{v_0}{F_n} }{ \Mss{v_0}{v_0}{F_n} } < +\infty.$$
\item For all $v \in V$ and all compact subsets $K \subset G$ we have $$ \lim_{n \to +\infty} \frac{\sup_{g \in K} \Mss{v}{v_0}{F_n \triangle F_n g} }{ \Mss{v_0}{v_0}{F_n} } = 0.$$
\end{enumerate} \end{definition}
\subsection{}
Finally, we can show that c-tempered irreducible unitary $G$-representations are tempered.
\begin{proposition}\label{prop right-c is op}
Let $V$ be an irreducible unitary $G$-representation. Assume that $V$ is right-c-tempered (with some F{\o}lner sequence). Then $V$ is o-tempered. More precisely, suppose that $V$ is right-c-tempered with F{\o}lner sequence $F_0 , F_1 , \ldots$ and let $v_0 \in V$ be a unit vector for which the conditions (1) and (2) of Definition \ref{def right-c-temp} are satisfied. Then the sequence of operators $$ S_0 , S_1 , \ldots : V \to L^2 (G)$$ given by $$ S_n (v) (x) := \begin{cases} \frac{1}{\sqrt{ \Mss{v_0}{v_0}{F_n} }} \cdot \langle x v , v_0 \rangle , & x \in F_n \\ 0, & x \notin F_n \end{cases}$$ admits a sub-sequence which is an asymptotic embedding. \end{proposition}
\begin{corollary}\label{cor c-temp are temp}
Every c-tempered irreducible unitary $G$-representation (with some F{\o}lner sequence) is tempered. \end{corollary}
\begin{proof}
It is clear that c-temperedness implies right-c-temperedness, Proposition \ref{prop right-c is op} says that right-c-temperedness implies o-temperedness, and Corollary \ref{cor o temp is temp} says that o-temperedness is equivalent to temperedness. \end{proof}
\begin{proof}[Proof (of Proposition \ref{prop right-c is op}).]
Clearly each $S_n$ is bounded. By condition (2) of Definition \ref{def right-c-temp}, for any fixed $v \in V$ there exists $C>0$ such that $||S_n (v)||^2 \leq C$ for all $n$. By the Banach-Steinhaus theorem, this implies that the operators $S_0 , S_1 , \ldots$ are jointly bounded, thus condition $(1)$ of Definition \ref{def asymp emb} is verified.
To verify condition $(2)$ of Definition \ref{def asymp emb}, fix $v \in V$ and a compact $K \subset G$. Given $g \in K$ and a function $f \in L^2 (G)$ of $L^2$-norm one, we have $$ | \langle S_n (g v) - g S_n (v) , f \rangle | = \frac{ \left| \int_{F_n} \langle x g v , v_0 \rangle \overline{f(x)} \cdot dx - \int_{F_n g^{-1}} \langle xg v , v_0 \rangle \overline{f(x)} \cdot dx \right| }{\sqrt{ \Mss{v_0}{v_0}{F_n} }} \leq $$ $$ \leq \frac{ \int_{F_n \triangle F_n g^{-1}} \left| \langle xg v , v_0 \rangle \overline{f(x)} \right| \cdot dx }{\sqrt{ \Mss{v_0}{v_0}{F_n} }} \leq \sqrt{\frac{\int_{F_n \triangle F_n g^{-1}} |\langle xg v , v_0 \rangle |^2 \cdot dx}{ \Mss{v_0}{v_0}{F_n} }} \cdot \sqrt{\int_{G} |f(x)|^2 \cdot dx} = $$ $$ = \sqrt{\frac{ \Mss{v}{v_0}{F_n g \triangle F_n} }{ \Mss{v_0}{v_0}{F_n} }}.$$ Since $f$ was arbitrary, this implies $$ || S_n (gv) - g S_n (v) || \leq \sqrt{\frac{ \Mss{v}{v_0}{F_n g \triangle F_n} }{ \Mss{v_0}{v_0}{F_n} }}$$ for $g \in K$. By condition (2) of Definition \ref{def right-c-temp}, this tends to $0$ as $n \to +\infty$, uniformly in $g \in K$, and hence the desired.
Now, using Lemma \ref{lem asymp emb irrep} we see that some sub-sequence will satisfy condition $(3)$ of Definition \ref{def asymp emb}, once we notice that $|| S_n v_0 ||^2 = 1$ for all $n$ by construction.
\end{proof}
\section{The case of $K$-finite vectors}\label{sec Kfin}
In this section $G$ is a semisimple group over a local field. We continue with notations from \S\ref{sec intro}. The purpose of this section is to prove Theorem \ref{thm main Kfin}.
\subsection{}
Let us first show that, when $G$ is non-Archimedean, it is enough to establish condition (2) of Theorem \ref{thm main Kfin}, and condition (1) will then follow. So we assume condition (2) and use the notation $C(v_1 , v_2)$ therein.
Let us denote by $\underline{V} \subset V$ the subspace of $K$-finite (i.e. smooth) vectors. By the polarization identity, it is clear that for all $v_1 , v_2 , v_3 , v_4 \in \underline{V}$ the limit $$ \lim_{r \to +\infty} \frac{\int_{G_{<r}} \langle gv_1 , v_2 \rangle \overline{\langle gv_3 , v_4 \rangle} \cdot dg}{r^{\mathbf{d} (V)}}$$ exists, let us denote it by $D(v_1 , v_2 , v_3 , v_4)$, and $D$ is a quadlinear form $$D : \underline{V} \times \overline{\underline{V}} \times \overline{\underline{V}} \times \underline{V} \to {\mathbb C}.$$
Next, we claim that for all $v_1 , v_2 , v_3 , v_4 \in \underline{V}$ and all $g_1 , g_2 \in G$ we have $$ D(g_1 v_1, g_2 v_2 , g_1 v_3 , g_2 v_4) = D(v_1 , v_2 , v_3 , v_4).$$ Indeed, again by the polarization identity, it is enough to show that for all $v_1 , v_2 \in \underline{V}$ and all $g_1 , g_2 \in G$ we have \begin{equation}\label{eq Kfin nonArch C is G inv} C(g_1 v_1 , g_2 v_2) = C(v_1 , v_2).\end{equation} There exists $r_0 \ge 0$ such that $$ G_{<r-r_0} \subset g_2^{-1} G_{<r} g_1 \subset G_{<r+r_0}.$$ We have: $$ \int_{G_{<r}} |\langle g g_1 v_1 , g_2 v_2 \rangle |^2 \cdot dg = \int_{g_2^{-1} G_{<r} g_1} |\langle gv_1 , v_2 \rangle |^2 \cdot dg$$ and therefore $$ \int_{G_{<r-r_0}} |\langle g v_1 , v_2 \rangle |^2 \cdot dg \leq \int_{G_{<r}} |\langle g g_1 v_1 , g_2 v_2 \rangle |^2 \cdot dg \leq \int_{G_{<r+r_0}} |\langle gv_1 , v_2 \rangle |^2 \cdot dg.$$ Dividing by $r^{\mathbf{d} (V)}$ and taking the limit $r \to +\infty$ we obtain (\ref{eq Kfin nonArch C is G inv}).
Now, by Schur's lemma (completely analogously to the reasoning with $\Phi$ in the proof of Proposition \ref{prop c-tempered orth rel}), we obtain that for some $C >0$ we have $$ D(v_1 , v_2 , v_3 , v_4) = C \cdot \langle v_1 , v_ 3 \rangle \overline{\langle v_2 , v_4 \rangle}$$ for all $v_1 , v_2 , v_3 , v_4 \in \underline{V}$.
\subsection{}
Thus, we aim at establishing condition (2) of Theorem \ref{thm main Kfin} in either the non-Archimedean or the Archimedean cases. Since a complex group can be considered as a real group and the formulation of the desired theorem will not change, we assume that we are either in the real case or in the non-Archimedean case.
Also, notice that to show Theorem \ref{thm main Kfin} for all maximal compact subgroups it is enough to show it for one maximal compact subgroup (in the non-Archimedean case because the resulting notion of $K$-finite vectors does not depend on the choice of $K$ and in the real case since all maximal compact subgroups are conjugate).
\subsection{}
Let us fix some notation. We choose a maximal split torus $A \subset G$ and a minimal parabolic $P \subset G$ containing $A$. We denote $$ {\mathfrak a} := {\textnormal{Hom}}_{{\mathbb Z}} (X^* (A) , {\mathbb R}).$$ We let $L \subset {\mathfrak a}$ to be ${\mathfrak a}$ itself in the real case and the lattice in ${\mathfrak a}$ corresponding to $X_* (A)$ in the non-Archimedean case. We let $\exp : L \to A$ be the exponential map constructed in the usual way: \begin{itemize}
\item If $G$ is real, we let $\exp$ to be the composition $L = {\mathfrak a} \cong {\rm Lie} (A) \to A$ where the last map is the exponential map from the Lie algebra to the Lie group, while the isomorphism is the identification resulting from the map $X^* (A) \to {\rm Lie} (A)^*$ given by taking the differential at $1 \in A$.
\item If $G$ is non-Archimedean, we let $\exp$ be the composition $L \cong X_* (A) \to A$ where the last map is given by sending $\chi$ to $\chi (\varpi^{-1})$, where $\varpi$ is a uniformizer. \end{itemize}
We denote by $$ \Delta \subset \widetilde{\Delta} \subset X^* (A) \subset {\mathfrak a}^*$$ the set of simple roots $\Delta$ and the set of positive roots $\widetilde{\Delta}$ (resulting from the choice of $P$). We identify ${\mathfrak a}$ with ${\mathbb R}^{\Delta}$ in the clear way. We set $$ {\mathfrak a}^+ := \{ x \in {\mathfrak a} \ | \ \alpha (x) \ge 0 \ \forall \alpha \in \Delta \}$$ and $L^+ := L \cap {\mathfrak a}^+$.
Let us in the standard way choose a maximal comapct subgroup $K \subset G$ ``in good relative position" with $A$. In the real case this means ${\rm Lie} (A)$ sitting in the $(-1)$-eigenspace of a Cartan involution whose $1$-eigenspace is ${\rm Lie} (K)$ and in the non-Archimedean case it is as in \cite[V.5.1., Th{\' e}or{\` e}me]{Re}. In the non-Archimedean case let us also, to simplify notation, assume that $G = K \exp (L^+) K$ (in general there is a finite subset $S \subset Z_G (A)$ such that $G = \coprod_{s \in S} K \exp (L^+) s K$ and one proceeds with the obvious modifications).
Let us denote $\rho := \frac{1}{2}\sum_{\alpha \in \widetilde{\Delta}} \mu_{\alpha} \cdot \alpha \in {\mathfrak a}^*$ where $\mu_{\alpha} \in {\mathbb Z}_{\ge 1}$ is the multiplicity of the root $\alpha$.
Fixing Haar measures, especially denoting by $dx$ a Haar measure on $L$, we have a uniquely defined continuous $\omega : L^+ \to {\mathbb R}_{\ge 0}$ such that the following integration formula holds: $$ \int_{G} f(g) \cdot dg = \int_{K \times K} \left( \int_{L^+} \omega (x) f(k_1 \exp (x) k_2) \cdot dx \right) \cdot dk_1 dk_2.$$ Regarding the behaviour of $\omega (x)$, we can use \cite[around Lemma 1.1]{Ar} as a reference. In the real case there exists $C>0$ such that \begin{equation}\label{eq omega is like rho Arch} \frac{\omega (x)}{e^{2\rho (x)}} = C \cdot \prod_{\alpha} \left( 1 - e^{-2\alpha (x)} \right) \end{equation} where $\alpha$ runs over $\widetilde{\Delta}$ according to multiplicities $\mu_{\alpha}$. In the non-Archimedean case, for every $\Theta \subset \Delta$ there exists $C_{\Theta} > 0$ such that \begin{equation}\label{eq omega is like rho nonArch} \frac{\omega (x)}{e^{2\rho (x)}} = C_{\Theta} \end{equation} for all $x \in L^+$ satisfying $\alpha (x) = 0$ for all $\alpha \in \Theta$ and $\alpha (x) \neq 0$ for all $\alpha \in \Delta \smallsetminus \Theta$.
Since, by Claim \ref{clm doesnt depend on norm}, we are free in our choice of the norm $||-||$ on ${\mathfrak g}$, let us choose $||-||$ to be a supremum norm in coordinates gotten from an $A$-eigenbasis. Then\footnote{Recall the notation $\mathbf{r}$ from \S\ref{sec intro}.} $$ \mathbf{r} (\exp (x)) = \log q \cdot \max_{\alpha \in \widetilde{\Delta} } |\alpha (x)|$$ where $q$ is the residual cardinality in the non-Archimedean case and $q := e$ in the real case. Let us denote $$ {\mathfrak a}_{<r} := \{ x \in {\mathfrak a} \ | \ | \alpha (x)| < \tfrac{r}{\log q} \ \forall \alpha \in \widetilde{\Delta} \}$$ and ${\mathfrak a}^+_{<r} := {\mathfrak a}_{<r} \cap {\mathfrak a}^+$ and similarly $L_{<r} := L \cap {\mathfrak a}_{<r}$, $L^+_{<r} := L^+ \cap L_{<r}$. Then $ L_{<r} = \exp^{-1} (G_{<r})$. Hence there exists $r_0 \ge 0$ such that \begin{equation}\label{eq Gr is like Lr} K \exp (L^+_{<r-r_0}) K \subset G_{<r} \subset K \exp(L^+_{<r+r_0}) K. \end{equation}
\subsection{}
Let now $V$ be a tempered irreducible unitary $G$-representation. Let us denote by $\underline{V} \subset V$ the subspace of $K$-finite vectors. Given $v_1 , v_2 \in V$, we will denote by $f_{v_1 , v_2}$ the continuous function on $L^+$ given by $$f_{v_1 , v_2} (x) := e^{\rho (x)} \langle \exp (x) v_1 , v_2 \rangle.$$ We have $$ \Ms{v_1}{v_2}{r} = \int_{G_{<r}} |\langle gv_1 , v_2 \rangle |^2 \cdot dg = $$ $$= \int_{K \times K} \left( \int_{L^+ \cap \exp^{-1} (k_2 G_{<r} k_1^{-1})} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{k_1 v_1 , k_2 v_2} (x)|^2 \cdot dx \right) \cdot dk_1 dk_2.$$ In view of (\ref{eq Gr is like Lr}), in order to prove Theorem \ref{thm main Kfin} it is enough to show:
\begin{claim}\label{clm Kfin ess}
There exists $\mathbf{d} (V) \in {\mathbb Z}_{\ge 0}$ such that for every non-zero $v_1 , v_2 \in \underline{V}$ there exists $C(v_1 , v_2)>0$ such that $$ \lim_{r \to +\infty} \frac{\int_{K \times K} \left( \int_{L^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}}| f_{k_1 v_1 , k_2 v_2} (x) |^2 \cdot dx \right) \cdot dk_1 dk_2}{r^{\mathbf{d} (V)}} = C(v_1 , v_2).$$ \end{claim}
\subsection{} We have the following:
\begin{claim}\label{clm growth exists}\
\begin{enumerate}
\item Given $v_1 , v_2 \in \underline{V}$, either $f_{v_1 , v_2} = 0$ in which case we set $\mathbf{d} (v_1 , v_2) := -\infty$, or there exist $\mathbf{d} (v_1 , v_2) \in {\mathbb Z}_{\ge 0}$ and $C(v_1 , v_2)>0$ such that $$ \lim_{r \to +\infty} \frac{1}{r^{\mathbf{d} (v_1 , v_2)}} \int_{L^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{v_1 , v_2} (x)|^2 \cdot dx = C(v_1 , v_2).$$
\item In the real case, we have $\mathbf{d} (v_1 , X v_2) \leq \mathbf{d} (v_1 , v_2)$ for all $v_1 , v_2 \in \underline{V}$ and $X \in {\mathfrak g}$.
\item Denoting $\mathbf{d}(V) := \sup_{v_1 , v_2 \in \underline{V}} \mathbf{d} (v_1 , v_2)$, we have neither $\mathbf{d} (V) = -\infty$ nor $\mathbf{d} (V) =+\infty$ (i.e. $\mathbf{d} (V) \in {\mathbb Z}_{\ge 0}$).
\end{enumerate} \end{claim}
Let us establish Claim \ref{clm Kfin ess} given Claim \ref{clm growth exists}:
\begin{proof}[Proof (of Claim \ref{clm Kfin ess} given Claim \ref{clm growth exists}).]
Let us first handle the non-Archimedean case. Let us notice that we can replace $v_1$ and $v_2$ by $g_1 v_1$ and $g_2 v_2$ for any $g_1,g_2 \in G$. Indeed, for some $r_0 \ge 0$ we have $$g_2^{-1} G_{<r-r_0} g_1 \subset G_{<r} \subset g_2^{-1} G_{<r+r_0} g_1$$ and thus $$ \int_{G_{<r-r_0}} | \langle g g_1 v_1 , g_2 v_2 \rangle |^2 \cdot dg \leq \int_{G_{<r}} |\langle g v_1 , v_2 \rangle |^2 \cdot dg \leq \int_{G_{<r+r_0}} | \langle g g_1 v_1 , g_2 v_2 \rangle |^2 \cdot dg,$$ from which the claim clearly follows. Since $G \cdot v_1$ spans $\underline{V}$ and $G \cdot v_2$ spans $\underline{V}$, we deduce that by replacing $v_1$ and $v_2$ we can assume that $\mathbf{d} (v_1 , v_2) = \mathbf{d} (V)$. Now, since the integral $$ \int_{K \times K} \left( \int_{L^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{k_1 v_1, k_2 v_2} (x)|^2 \cdot dx \right) \cdot dk_1 dk_2$$ over $K \times K$ is simply a finite linear combination the claim is clear.
Let us now handle the real case. First, we would like to see that for some $k_1 , k_2 \in K$ we have $\mathbf{d} (k_1 v_1 , k_2 v_2) = \mathbf{d} (V)$. To that end, let us denote by ${\mathfrak n}$ and ${\mathfrak n}^-$ the Lie algebras of $N$ and $N^-$ (the unipotent radicals of $P$ and of $P^-$, the opposite to $P$ with respect to $A$) and identify ${\mathfrak a}$ with the Lie algebra of $A$ as before. Since ${\mathcal U} ({\mathfrak n}^-) {\mathcal U} ({\mathfrak a}) K v_1$ spans $\underline{V}$ and ${\mathcal U} ({\mathfrak n}) {\mathcal U} ({\mathfrak a}) K v_2$ spans $\underline{V}$, we can find $k_1 , k_2 \in K$ and some elements $v'_1 \in {\mathcal U} ({\mathfrak n}^-) {\mathcal U} ({\mathfrak a}) k_1 v_1$ and $v'_2\in {\mathcal U} ({\mathfrak n}) {\mathcal U} ({\mathfrak a}) k_2 v_2$ such that $\mathbf{d} (v'_1, v'_2) = \mathbf{d} (V)$. By Claim \ref{clm growth exists}(2) this forces $\mathbf{d} (k_1 v_1 , k_2 v_2) = \mathbf{d} (V)$.
Next, given two continuous functions $f_1 , f_2$ on ${\mathfrak a}^+$ and $d \in {\mathbb Z}_{\ge 0}$ let us denote $$ \langle f_1 , f_2 \rangle_d := \lim_{r \to +\infty} \frac{ \int_{{\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} f_1(x) \overline{f_2 (x)} \cdot dx}{r^d}$$ if the limit exists, and $|| f ||^2_d := \langle f , f \rangle_d$.
We claim that the function $(k_1 , k_2) \mapsto || f_{k_1 v_1 , k_2 v_2} ||^2_{\mathbf{d} (V)}$ on $K \times K$ is continuous and that $$ \lim_{r \to +\infty} \frac{\int_{K \times K} \left( \int_{{\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{k_1 v_1 , k_2 v_2} (x)|^2 \cdot dx \right) \cdot dk_1 dk_2}{r^{\mathbf{d} (V)}} = \int_{K \times K} || f_{k_1 v_1 , k_2 v_2 } ||^2_{\mathbf{d} (V)} \cdot dk_1 dk_2.$$ Then the right hand side is non-zero since we have seen that $\mathbf{d} (k_1 v_1 , k_2 v_2) = \mathbf{d} (V)$ for some $k_1 , k_2 \in V$, and we are done.
Let $(v_1^i)$ be a basis for the ${\mathbb C}$-span of $\{ k v_1 \}_{k \in K}$ and let $(v_2^j)$ be a basis for the ${\mathbb C}$-span of $\{ k v_2 \}_{k \in K}$. Let us write $k v_1 = \sum_i c_i(k) v_1^i$ and $k v_2 = \sum_j d_j (k) v_2^j$, so that $c_i$ and $d_j$ are continuous ${\mathbb C}$-valued functions of $K$. Then $$ \int_{{\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{k_1 v_1 , k_2 v_2} (x)|^2 \cdot dx = $$ $$ = \sum_{i_1 , i_2 , j_1 , j_2} c_{i_1} (k_1) \overline{c_{i_2} (k_1)} \overline{d_{j_1} (k_2)} d_{j_2} (k_2) \int_{{\mathfrak a}^+_{<r} } \frac{\omega (x)}{e^{2 \rho (x)}} \cdot f_{v_1^{i_1} , v_2^{j_1}} (x) \cdot \overline{f_{v_1^{i_2} , v_2^{j_2}} (x)} \cdot dx.$$ Therefore $$ || f_{k_1 v_1 , k_2 v_2}||^2_{\mathbf{d} (V)} = \sum_{i_1 , i_2 , j_1 , j_2} c_{i_1} (k_1) \overline{c_{i_2} (k_1)} \overline{d_{j_1} (k_2)} d_{j_2} (k_2) \langle f_{v_1^{i_1} , v_2^{j_1}} , f_{v_1^{i_2} , v_2^{j_2}} \rangle_{\mathbf{d} (V)} $$ so $(k_1 , k_2) \mapsto || f_{k_1 v_1 , k_2 v_2} ||^2_{\mathbf{d} (V)}$ is indeed continuous. Also, it is now clear that we have $$ \lim_{r \to +\infty} \frac{\int_{K \times K} \left( \int_{{\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{k_1 v_1 , k_2 v_2} (x)|^2 \cdot dx \right) \cdot dk_1 dk_2}{r^{\mathbf{d} (V)}} = $$ $$ = \sum_{i_1 , i_2 , j_1 , j_2} \langle f_{v_1^{i_1} , v_2^{j_1}} , f_{v_1^{i_2} , v_2^{j_2}} \rangle_{\mathbf{d} (V)} \int_{K \times K} c_{i_1} (k_1) \overline{c_{i_2} (k_1)} \overline{d_{j_1} (k_2)} d_{j_2} (k_2) \cdot dk_1 dk_2 = $$ $$ = \int_{K \times K} || f_{k_1 v_1 , k_2 v_2} ||^2_{\mathbf{d} (V) } \cdot dk_1 dk_2$$ \end{proof}
\subsection{}
Let us now explain Claim \ref{clm growth exists} in the case when $G$ is non-Archimedean. Let $v_1 , v_2 \in \underline{V}$. Let us choose a positive integer $k$ large enough so that $k \cdot {\mathbb Z}_{\ge 0}^{\Delta} \subset L$. By enlarging $k$ even more if necessary, by \cite[Theorem 4.3.3.]{Ca} for every $\Theta \subset \Delta$ and every $y \in ( {\mathbb R}_{\ge 0}^{\Theta} \times {\mathbb R}_{>k}^{\Delta \smallsetminus \Theta}) \cap L^+ $ the function $$ f_{v_1 , v_2 , \Theta , y} : k \cdot {\mathbb Z}_{\ge 0}^{\Delta \smallsetminus \Theta} \to {\mathbb C}$$ given (identifying ${\mathbb R}^{\Delta \smallsetminus \Theta}$ with a subspace of ${\mathbb R}^{\Delta}$ in the clear way) by $x \mapsto f_{v_1 , v_2}(y+x)$, can be written as $$ \sum_{1 \leq i \leq p} c_i \cdot e^{\lambda_i (x_{\Delta \smallsetminus \Theta})} q_i (x_{\Delta \smallsetminus \Theta})$$ where $c_i \in {\mathbb C} \smallsetminus \{ 0\}$, $\lambda_i$ is a complex-valued functional on ${\mathbb R}^{\Delta \smallsetminus \Theta}$, and $q_i$ is a monomial on ${\mathbb R}^{\Delta \smallsetminus \Theta}$. Here $x_{\Delta \smallsetminus \Theta}$ is the image of $x$ under the natural projection ${\mathbb R}^{\Delta} \to {\mathbb R}^{\Delta \smallsetminus \Theta}$. We can assume that the couples in the collection $\{ (\lambda_i , q_i) \}_{1 \leq i \leq p}$ are pairwise different. Since $V$ is tempered, by ``Casselman's criterion" we in addition have that for every $1 \leq i \leq p$, ${\rm Re} (\lambda_i)$ is non-negative on ${\mathbb R}_{\ge 0}^{\Delta \smallsetminus \Theta}$.
By Claim \ref{clm appb lat}, either $p = 0$, equivalently $f_{v_1 , v_2 , \Theta , y} = 0$ (in which case we set $d_{v_1 , v_2 , \Theta , y} := -\infty$), or there exists $d_{v_1 , v_2 , \Theta , y} \in {\mathbb Z}_{\ge 0}$ such that the limit $$ \lim_{r \to +\infty} \frac{1}{r^{d_{v_1 , v_2 , \Theta , y}}} \sum_{x \in (k \cdot {\mathbb Z}_{\ge 0}^{\Delta \smallsetminus \Theta}) \cap L^+_{<r}} |f_{v_1 , v_2}(y+x)|^2$$ exists and is strictly positive.
Now, given $y \in L^+$ let us denote $\Theta_y := \{ \alpha \in \Delta \ | \ y_{\alpha} \leq k \}$ where by $y_{\alpha}$ we denote the coordinate of $y \in {\mathbb R}^{\Delta}$ at the $\alpha$-place. Let $Y \subset L^+$ be the subset of $y \in L^+$ for which $y_{\alpha} \leq 2k$ for all $\alpha \in \Delta$. Then $Y$ is a finite set, and we have \begin{equation}\label{eq break L into subsets} L^+ = \coprod_{y \in Y} \left( y + k \cdot {\mathbb Z}_{\ge 0}^{\Delta \smallsetminus \Theta_y} \right).\end{equation} Notice also that $\omega (x) / e^{2 \rho (x)}$ is a positive constant on each one of the subset of which we take union in (\ref{eq break L into subsets}). We set $\mathbf{d} (v_1 , v_2) := \max_{y \in Y} d_{v_1 , v_2 , \Theta_y , y}$. We see that either $f_{v_1 , v_2} = 0$ (then $\mathbf{d} (v_1 , v_2) = -\infty$) or the limit $$ \lim_{r \to +\infty} \frac{1}{r^{\mathbf{d} (v_1 , v_2)}} \sum_{x \in L^+_{<r} } \frac{\omega (x)}{e^{2 \rho (x)}} |f_{v_1 , v_2} (x)|^2$$ exists and is strictly positive.
That $\mathbf{d} (V)$ is finite follows from $\mathbf{d} (v_1 , v_2)$ being controlled by finitely many Jacquet modules, with the finite central actions on them.
\subsection{}
Let us now explain Claim \ref{clm growth exists} in the case when $G$ is real. Using \cite{CaMi} we know that, fixing $k >0$, given $\Theta \subset \Delta$ the restriction of $\frac{\omega (x)^{1/2}}{e^{\rho (x)}} f_{v_1 , v_2} (x)$ to ${\mathbb R}_{\ge k}^{\Delta \smallsetminus \Theta} \times [0,k]^{\Theta}$ can be written as $$ \sum_{1 \leq i \leq p} e^{\lambda_i (x_{\Delta \smallsetminus \Theta})} q_i (x_{\Delta \smallsetminus \Theta}) \phi_i (x) $$ where the notation is as follows. First, $\lambda_i$ is a complex-valued functional on ${\mathbb R}^{\Delta \smallsetminus \Theta}$. Next, $q_i$ is a monomial on ${\mathbb R}^{\Delta \smallsetminus \Theta}$. The couples $(\lambda_i , q_i)$, for $1 \leq i \leq p$, are pairwise distinct. The function $\phi_i$ is expressible as a composition $$ [0,k]^{\Theta} \times {\mathbb R}_{\ge k}^{\Delta \smallsetminus \Theta} \xrightarrow{{\rm id} \times {\rm ei}} [0,k]^{\Theta} \times ({\mathbb C}_{|-|<1})^{\Delta \smallsetminus \Theta} \xrightarrow{\phi_i^{\circ}} {\mathbb C}$$ where ${\rm ei}$ is the coordinate-wise application of $x \mapsto e^{-x}$ and $\phi_{i}^{\circ}$ is a continuous function such that, for every $b \in [0,k]^{\Theta}$, the restriction of $\phi_i^{\circ}$ via $({\mathbb C}_{|-|<1})^{\Delta \smallsetminus \Theta} \xrightarrow{z \mapsto (b,z)} [0,k]^{\Theta} \times ({\mathbb C}_{|-|<1})^{\Delta \smallsetminus \Theta}$ is holomorphic. Lastly, the function $b \mapsto \phi_{i}^{\circ} (b , \{0\}^{\Delta \smallsetminus \Theta} ) $ on $[0,k]^{\Theta}$ is not identically zero. Since $V$ is tempered, by ``Casselman's criterion" we in addition have that for every $1 \leq i \leq p$, ${\rm Re} (\lambda_i)$ is non-negative on ${\mathbb R}_{\ge 0}^{\Delta \smallsetminus \Theta}$.
If $p = 0$, we set $d_{v_1 , v_2 , \Theta} := -\infty$. Otherwise, Claim \ref{clm appb} provides a number $d_{v_1 , v_2 , \Theta} \in {\mathbb Z}_{\ge 0}$, described concretely in terms of $\{ (\lambda_i , q_{i} , \phi_{i} ) \}_{1 \leq i \leq p}$, such that the limit $$ \lim_{r \to +\infty} \frac{1}{r^{d_{v_1 , v_2 , \Theta}}} \int_{ ( [0,k]^{\Theta} \times {\mathbb R}_{\ge k}^{\Delta \smallsetminus \Theta} ) \cap {\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{v_1 , v_2} (x)|^2 \cdot dx $$ exists and is strictly positive. We set $\mathbf{d} (v_1 , v_2) := \max_{\Theta \subset \Delta} d_{v_1 , v_2 , \Theta}$. Then either $f_{v_1 , v_2} = 0$ or $\mathbf{d} (v_1 , v_2) > 0$ and the limit $$ \lim_{r \to +\infty} \frac{1}{r^{\mathbf{d} (v_1 , v_2)}} \int_{{\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{v_1 , v_2} (x)|^2 \cdot dx$$ exists and is strictly positive. That $\mathbf{d} (V)$ is finite follows from $\mathbf{d} (v_1 , v_2)$ being controlled by finitely many data, as in \cite{CaMi}. Part (2) of Claim \ref{clm growth exists} follows easily from the concrete description of $d_{v_1 , v_2 , \Theta}$ in Claim \ref{clm appb}.
\section{Proofs for Remark \ref{rem counterexample}, Remark \ref{rem doesnt depend on norm} , Proposition \ref{prop formula ch}, Proposition \ref{prop from folner to exact} and Remark \ref{rem ctemp red is temp}.}\label{sec clms red}
In this section, $G$ is a semisimple group over a local field. We continue with notations from \S\ref{sec intro}. We explain Remark \ref{rem counterexample} (in Claim \ref{clm counterexample}), explain Remark \ref{rem doesnt depend on norm} (in Claim \ref{clm doesnt depend on norm}), prove Proposition \ref{prop formula ch} (in \S\ref{ssec proof of prop formula ch}), prove Proposition \ref{prop from folner to exact} (in \S\ref{ssec clms red 1}) and explain Remark \ref{rem ctemp red is temp} (in Claim \ref{clm Prop tempered is c-tempered reductive holds}).
\subsection{}\label{ssec clms red 1}
\begin{lemma}\label{lem red temp is ctemp}
Let $V$ be an irreducible unitary $G$-representation and suppose that there exists a unit vector $v_0 \in V$ satisfying properties (1) and (2) of Proposition \ref{prop from folner to exact}. Let $0 < r_0 < r_1 < \ldots$ be a sequence such that $\lim_{n \to +\infty} r_n = +\infty$. Then $V$ is c-tempered with F{\o}lner sequence $G_{< r_0} , G_{<r_1}, \ldots$. \end{lemma}
\begin{proof}
Property (1) of Definition \ref{def c-temp} is immediate from property (1) of Proposition \ref{prop from folner to exact}. Let us check property (2) of Definition \ref{def c-temp}. Thus, let $v_1 , v_2 \in V$ and let $K \subset G$ be a compact subset. Fix $r^{\prime} \ge 0$ big enough so that $K \subset G_{< r^{\prime}}$ and $K^{-1} \subset G_{< r^{\prime}}$. We then have, for all $r>0$ and all $g_1 , g_2 \in K$: $$G_{< r} \triangle g_2^{-1} G_{< r} g_1 \subset G_{< r + 2 r^{\prime}} \smallsetminus G_{< r - 2 r^{\prime}}.$$ Therefore, using property (2) of Proposition \ref{prop from folner to exact}, $$ \limsup_{r \to +\infty} \frac{\sup_{g_1 , g_2 \in K} \Mss{v_1}{v_2}{G_{< r} \triangle g_2^{-1} G_{< r} g_1}}{ \Mss{v_0}{v_0}{G_{<r}} } \leq \limsup_{r \to +\infty} \frac{ \Mss{v_1}{v_2}{G_{< r + 2 r^{\prime}} \smallsetminus G_{\leq r - 2 r^{\prime} }} }{ \Mss{v_0}{v_0}{G_{<r}}} = 0$$ and therefore also $$ \lim_{n \to +\infty} \frac{\sup_{g_1 , g_2 \in K} \Mss{v_1}{v_2}{G_{< r_n} \triangle g_2^{-1} G_{< r_n} g_1} }{ \Mss{v_0}{v_0}{G_{< r_n}}} = 0.$$ \end{proof}
\begin{proof}[Proof (of Proposition \ref{prop from folner to exact}).]
Let us fix a $K$-finite unit vector $v'_0 \in V$, for some maximal compact subgroup $K \subset G$. Let $0 < r_0 < r_1 < \ldots$ be a sequence such that $\lim_{n \to +\infty} r_n = +\infty$. By Lemma \ref{lem red temp is ctemp} $V$ is c-tempered with F{\o}lner sequence $G_{<r_0} , G_{<r_1} , \ldots$ and hence by Proposition \ref{prop c-tempered orth rel} we obtain $$ \lim_{n \to +\infty} \frac{\int_{g \in G_{<r_n}} \langle gv_1 , v_2 \rangle \overline{\langle gv_3 , v_4 \rangle} \cdot dg}{\Ms{v'_0}{v'_0}{r_n}} = \langle v_1 , v_3 \rangle \overline{\langle v_2 , v_4 \rangle}$$ for all $v_1 , v_2 , v_3 , v_4 \in V$. Since this holds for any such sequence $\{ r_n \}_{n \ge 0}$, we obtain \begin{equation}\label{eq formula again} \lim_{r \to +\infty} \frac{\int_{g \in G_{<r}} \langle gv_1 , v_2 \rangle \overline{\langle gv_3 , v_4 \rangle} \cdot dg}{\Ms{v'_0}{v'_0}{r}} = \langle v_1 , v_3 \rangle \overline{\langle v_2 , v_4 \rangle} \end{equation} for all $v_1 , v_2 , v_3 , v_4 \in V$. By Theorem \ref{thm main Kfin} we have $$ \lim_{r \to +\infty} \frac{\Ms{v'_0}{v'_0}{r}}{r^{\mathbf{d}(V)}} = C$$ for some $C>0$. This enables to rewrite (\ref{eq formula again}) as $$ \lim_{r \to +\infty} \frac{\int_{g \in G_{<r}} \langle gv_1 , v_2 \rangle \overline{\langle gv_3 , v_4 \rangle} \cdot dg}{r^{\mathbf{d} (V)}} = C \cdot \langle v_1 , v_3 \rangle \overline{\langle v_2 , v_4 \rangle} $$ for all $v_1 , v_2 , v_3 , v_4 \in V$, as desired.
\end{proof}
\subsection{}
\begin{claim}\label{clm doesnt depend on norm}
The validity of Conjecture \ref{main conj red exact}, as well as the resulting invariants $\mathbf{d} (V)$ and $\mathbf{f} (V)$, of Theorem \ref{thm main Kfin} as well as the resulting invariants $\mathbf{d} (V)$ and $\mathbf{f} (V)$ (the latter in the non-Archimedean case), and of Proposition \ref{prop from folner to exact}, do not depend on the choice of the norm $|| - ||$ on ${\mathfrak g}$. \end{claim}
\begin{proof}
Let $|| - ||^{\prime}$ be another norm on ${\mathfrak g}$, let $\mathbf{r}^{\prime} : G \to {\mathbb R}_{\ge 0}$ be the resulting function, and let $G_{<r}^{\prime} \subset G$ be the resulting subsets. There exists $r_0 \ge 0$ such that $$ e^{-r_0} \cdot || X || \leq ||X ||^{\prime} \leq e^{r_0} \cdot || X||, \quad \forall X \in {\mathfrak g}$$ and therefore $$ e^{-2r_0} \cdot ||{\rm Ad} (g)|| \leq || {\rm Ad} (g) ||' \leq e^{2 r_0} \cdot || {\rm Ad} (g) ||, \quad \forall g \in G.$$ Then $$ G^{\prime}_{< r} \subset G_{<r + 2r_0}, \quad \forall r\ge 0$$ and $$ G_{< r} \subset G^{\prime}_{<r + 2r_0}, \quad \forall r\ge 0.$$ These ``sandwich" relations readily imply the independence claims.
\end{proof}
\subsection{}
\begin{claim}\label{clm Prop tempered is c-tempered reductive holds}
An irreducible unitary $G$-representation for which there exists a unit vector $v_0 \in V$ such that conditions (1) and (2) of Proposition \ref{prop from folner to exact} are satisfied is tempered. \end{claim}
\begin{proof}
Clear from Lemma \ref{lem red temp is ctemp} coupled with Corollary \ref{cor c-temp are temp}. \end{proof}
\subsection{}
\begin{claim}\label{clm counterexample}
Let $G := PGL_2 (\Omega)$, $\Omega$ a local field. Let $A \subset G$ be the subgroup of diagonal matrices. Then, for every non-trivial irreducible unitary $G$-representation $V$, the set of matrix coefficients of $V$ restricted to $A$ is equal to the set of function on $A$ of the form $$ a \mapsto \int_{\hat{A}} \chi (a) \cdot \phi (\chi) \cdot d\chi$$ as $\phi$ runs over $ L^1 (\hat{A})$. \end{claim}
\begin{proof}
Denote by $B \subset G$ the subgroup of upper-triangular matrices and by $N \subset B$ its unipotent radical.
Let us recall that, by Mackey theory, there is a unique (up to isomorphism) infinite-dimensional irreducible unitary $B$-representation $W$, and the rest of irreducible unitary $B$-representations are killed by $N$. The restriction ${\rm Res}^B_A W$ is isomorphic to the right regular unitary $A$-representation $L^2 (A)$.
Let now $V$ be a non-trivial irreducible unitary $G$-representation. Recall that by the Howe-Moore theorem (or by a step in one of its usual proofs) $V$ does not contain non-zero $N$-invariant vectors. By decomposing the restriction ${\rm Res}^G_B V$ into a direct integral of irreducible unitary $B$-representations, and using the fact that $V$ admits no non-zero $N$-invariant vectors, we see that ${\rm Res}^G_B V$ is a multiple of $W$. Hence, we deduce that ${\rm Res}^G_A A$ is a multiple of the right regular unitary $A$-representation $L^2 (A)$.
Now, the matrix coefficients of a multiple of the right regular unitary $A$-representation $L^2 (A)$ are easily seen to be the functions on $A$ of the form $$ a \mapsto \int_{\hat{A}} \chi (a) \cdot \phi (\chi) \cdot d\chi$$ where $\phi \in L^1 (\hat{A})$.
\end{proof}
\subsection{}\label{ssec proof of prop formula ch}
\begin{proof}[Proof (of Proposition \ref{prop formula ch}).]
Fix $d \in D_c^{\infty} (G)$. Let $K \subset G$ be an open compact subgroup such that $d$ is invariant under $K$ both on left and on right. Let us denote by $e_1 , \ldots , e_n$ an orthonormal basis of $V^K$, and let us denote by $\pi_K : V \to V^K$ the orthonormal projection. Let us denote by $[-,-] : C^{-\infty} (G) \times D_c^{\infty} (G) \to {\mathbb C}$ the canonical pairing. We have $$ [{}^g m_{v_1 , v_2} , d] = [m_{g v_1 , g v_2} , d] = \langle d g v_1 , g v_2 \rangle = \langle d \pi_K (g v_1) , \pi_K (g v_2) \rangle = $$ $$ = \sum_{1 \leq i,j \leq n} \langle g v_1 , e_i \rangle \overline{\langle g v_2 , e_j \rangle} \langle d e_i , e_j \rangle. $$
Hence $$ \frac{\int_{G_{<r}} [{}^g m_{v_1 , v_2} , d] \cdot dg}{r^{\mathbf{d} (V)}} = \sum_{1 \leq i,j \leq n} \langle d e_i , e_j \rangle \cdot \frac{\int_{G_{<r}} \langle g v_1 , e_i \rangle \overline{\langle g v_2 , e_j \rangle} \cdot dg}{r^{\mathbf{d} (V)}}$$ and therefore $$ \lim_{r \to +\infty} \frac{\int_{G_{<r}} [{}^g m_{v_1 , v_2} , d] \cdot dg}{r^{\mathbf{d} (V)}} = \frac{1}{\mathbf{f} (V)} \sum_{1 \leq i,j \leq n} \langle d e_i , e_j \rangle \cdot \langle v_1 , v_2 \rangle \overline{\langle e_i , e_j \rangle} = $$ $$ = \frac{1}{\mathbf{f} (V)} \sum_{1 \leq i \leq n} \langle d e_i , e_i \rangle \cdot \langle v_1 , v_2 \rangle = \frac{\langle v_1 , v_2 \rangle}{\mathbf{f} (V)} \Theta_V (d).$$ \end{proof}
\section{The case of the principal series representation $V_1$ of slowest decrease}\label{sec proof of V1}
In this section $G$ is a semisimple group over a local field. We continue with notations from \S\ref{sec intro}. Our goal is to prove Theorem \ref{thm intro slowest} (restated as Theorem \ref{thm V1 c temp} below).
\subsection{}\label{ssec V1 and Xi}
We fix a minimal parabolic $P \subset G$ and a maximal compact subgroup $K \subset G$ such that $G = PK$. We consider the principal series unitary $G$-representation $V_1$ consisting of functions $f : G \to {\mathbb C}$ satisfying $$ f(pg) = \Delta_P (p)^{1/2} \cdot f(g) \quad \forall p \in P , g \in G$$ where $\Delta_P : P \to {\mathbb R}^{\times}_{>0}$ is the modulus function of $P$. The $G$-invariant inner product on $V_1$ can be taken to be $$ \langle f_1 , f_2 \rangle = \int_K f_1 (k) \cdot \overline{f_2 (k)} \cdot dk$$ (where we normalize the Haar measure on $K$ to have total mass $1$). Recall that $V_1$ is irreducible. We denote by $f_0 \in V_1$ the spherical vector, determined by $f_0 (k) = 1$ for all $k \in K$. We also write $$ \Xi_G (g) := \langle g f_0 , f_0 \rangle.$$
\begin{lemma}\label{lem growth Xi}
Given $r' \ge 0$ we have $$ \lim_{r \to +\infty} \frac{\int_{G_{<r+r'} \smallsetminus G_{<r-r'}} \Xi_G (g)^2 \cdot dg}{\int_{G_{<r}} \Xi_G (g)^2 \cdot dg} = 0$$ and $$ \lim_{r \to +\infty} \frac{\int_{G_{<r+r'}} \Xi_G (g)^2 \cdot dg}{\int_{G_{<r}} \Xi_G (g)^2 \cdot dg} = 1.$$ \end{lemma}
\begin{proof}
The second equality follows from the first, and the first is immediately implied by Theorem \ref{thm main Kfin}. \end{proof}
\subsection{} The main result of this section is:
\begin{theorem}\label{thm V1 c temp}
Let $V$ be an irreducible tempered unitary $G$-representation. Suppose that there exist a unit vector $v_0 \in V$ such that \begin{equation}\label{eq good vector} \underset{r \to +\infty}{\limsup} \frac{ \int_{G_{<r}} \Xi_G (g)^2 \cdot dg }{ \Ms{v_0}{v_0}{r} } < +\infty. \end{equation} Then Conjecture \ref{main conj red exact} holds for $V$. In particular, Conjecture \ref{main conj red exact} holds for $V_1$. \end{theorem}
\subsection{}
We will prove Theorem \ref{thm V1 c temp} using the following result:
\begin{claim}\label{clm est using Xi}
Let $V$ be a tempered unitary $G$-representation. Then for all unit vectors $v_1,v_2 \in V$ and all measurable $K$-biinvariant subsets $S \subset G$ we have $$ \int_S |\langle gv_1 , v_2 \rangle |^2 \cdot dg \leq \int_S \Xi_G (g)^2 \cdot dg.$$ \end{claim}
\begin{proof}[Proof (of Theorem \ref{thm V1 c temp} given Claim \ref{clm est using Xi}).]
To show that Conjecture \ref{main conj red exact} holds for $V$ we will use Proposition \ref{prop from folner to exact}, applied to our $V$ and our $v_0$.
There exists $r_0 \ge 0$ such that $K G_{<r} K \subset G_{< r + r_0}$ for all $r \ge 0$.
Let us verify condition (1) of Proposition \ref{prop from folner to exact}. For unit vectors $v_1 , v_2 \in V$ we have $$ \frac{\Ms{v_1}{v_2}{r}}{\Ms{v_0}{v_0}{r}} \leq \frac{\int_{G_{<r+r_0}} \Xi_G (g)^2 \cdot dg}{\Ms{v_0}{v_0}{r}}$$ and therefore condition (1) of Proposition \ref{prop from folner to exact} follows from (\ref{eq good vector}) and Lemma \ref{lem growth Xi}. Let us now verify condition (2) of Proposition \ref{prop from folner to exact}. For unit vectors $v_1 , v_2 \in V$ and $r' \ge 0$ we have $$ \frac{ \Ms{v_1}{v_2}{r+r'} - \Ms{v_1}{v_2}{r-r'}}{\Ms{v_0}{v_0}{r}} \leq \frac{\int_{G_{<r+r'+r_0} \smallsetminus G_{<r-(r'+r_0)}} \Xi _G(g)^2 \cdot dg}{\int_{G_{<r}} \Xi_G (g)^2 \cdot dg} \cdot \frac{\int_{G_{<r}} \Xi_G (g)^2 \cdot dg}{\Ms{v_0}{v_0}{r}}$$ and therefore condition (2) of Proposition \ref{prop from folner to exact} follows from (\ref{eq good vector}) and Lemma \ref{lem growth Xi}. \end{proof}
\subsection{}
We will prove Claim \ref{clm est using Xi} using the following result:
\begin{claim}\label{clm norm of conv}
Let $\phi \in L^2 (G)$ be zero outside of a measurable $K$-biinvariant subset $S \subset G$ of finite volume. Denote by $T_{\phi} : L^2 (G) \to L^2 (G)$ the operator of convolution $\psi \mapsto \phi \star \psi$. Then\footnote{Here $|| \phi ||$ stands for the $L^2$-norm of $\phi$.} $$ || T_{\phi} ||^2 \leq \left( \int_{S} \Xi_G (g)^2 \cdot dg \right) \cdot || \phi ||^2.$$ \end{claim}
\begin{proof}[Proof (of Claim \ref{clm est using Xi} given Claim \ref{clm norm of conv}).]
We can clearly assume that $S$ has finite volume. Let us denote $$ \phi (g) := {\rm ch}_{S} (g) \cdot \overline{\langle gv_1 , v_2 \rangle},$$ where ${\rm ch}_S$ stands for the characteristic function of $S$. Let us denote by $S_{\phi} : V \to V$ the operator $$ v \mapsto \int_G \phi (g) \cdot gv \cdot dg.$$ Since $V$ is tempered, we have $ ||S_{\phi}|| \leq || T_{\phi}||$. Therefore $$ \int_{S} |\langle gv_1 , v_2 \rangle |^2 \cdot dg = \int_G \phi (g) \cdot \langle gv_1 , v_2 \rangle \cdot dg = \langle S_{\phi} v_1 , v_2 \rangle \leq || S_{\phi} || \leq || T_{\phi} || \leq $$ $$ \leq \left( \sqrt{\int_S \Xi_G (g)^2 \cdot dg} \right) \cdot || \phi || = \left( \sqrt{\int_S \Xi_G (g)^2 \cdot dg} \right) \cdot \left( \sqrt{\int_S |\langle gv_1 , v_2 \rangle |^2 \cdot dg} \right)$$ thus $$\int_S |\langle gv_1 , v_2 \rangle |^2 \cdot dg \leq \int_S \Xi_G (g)^2 \cdot dg$$ as desired. \end{proof}
\subsection{}
Finally, let us prove Claim \ref{clm norm of conv}, following \cite{ChPiSa}.
\begin{proof}[Proof (of Claim \ref{clm norm of conv}).]
By\footnote{In the lemma we refer to it is assumed that $\phi$ is continuous but the arguments there apply to our $\phi$ without any modification.} \cite[Lemma 3.5]{ChPiSa} we can assume that $\phi$ is $K$-biinvariant and non-negative. By \cite[Proposition 4.3]{ChPiSa} we have $$ || T_{\phi} || = \int_G \Xi_G (g) \cdot \phi (g) \cdot dg.$$ Applying the Cauchy-Schwartz inequality, we obtain $$ ||T_{\phi} ||^2 \leq \left( \int_S \Xi_G (g)^2 \cdot dg\right) \cdot|| \phi||^2,$$ as desired. \end{proof}
\section{The proof of Theorem \ref{thm intro SL2R}}\label{sec proof of SL2R}
In this section we let $G$ be either $SL_2 ({\mathbb R})$ or $PGL_2 (\Omega)$, where $\Omega$ is a non-Archimedean local field of characteristic $0$ and residual characteristic not equal to $2$. We prove Theorem \ref{thm intro SL2R}.
\subsection{}
If $G = PGL_2 (\Omega)$, we denote by $\varpi$ a uniformizer in $\Omega$, by ${\mathcal O}$ the ring of integers in $\Omega$, by $p$ the residual characteristic of $\Omega$ and $q:= |{\mathcal O} / \varpi {\mathcal O} |$.
\subsection{}\label{ssec SL2 notation}
We denote by $A \subset G$ the subgroup of diagonal matrices and by $U \subset G$ the subgroup of unipotent upper-triangular matrices. If $G = SL_2 ({\mathbb R})$ we define the isomorphism $$ \mathbf{a} : {\mathbb R}^{\times} \to A, \quad t \mapsto \mtrx{t}{0}{0}{t^{-1}}$$ and if $G = PGL_2 (\Omega)$ we define the isomorphism $$ \mathbf{a} : \Omega^{\times} \to A, \quad t \mapsto \mtrx{t}{0}{0}{1}.$$ We denote $A^+ := \{ a \in A \ | \ |\mathbf{a}^{-1} (a) | \ge 1 \}$.
If $G = SL_2 ({\mathbb R})$ then we can (and will) take $||- ||$ on ${\mathfrak g}$ to be such that $$\mathbf{r} \left( k_1 \mtrx{t}{0}{0}{t^{-1}} k_2 \right) = \log \max \{ |t|^2 , |t|^{-2} \}$$ where $t\in {\mathbb R}^{\times}$ and $k_1 , k_2 \in SO (2)$. If $G = PGL_2 (\Omega)$ then we can (and will) take $|| - ||$ on ${\mathfrak g}$ to be such that $$\mathbf{r} \left( k_1 \mtrx{t}{0}{0}{s} k_2 \right) = \log \max \{ |t/s| , |s/t| \}$$ where $t,s \in \Omega^{\times}$ and $k_1 , k_2 \in PGL_2 ({\mathcal O})$. Let us denote $A^+_{<r} := A^+ \cap G_{<r}$.
If $G = SL_2 ({\mathbb R})$ we set $K := SO (2) \subset G$. If $G = PGL_2 (\Omega)$ we choose a non-square $\zeta \in {\mathcal O}^{\times}$ and set $K \subset G$ to be the subgroup of elements of the form $\mtrx{a}{\zeta b}{b}{a}$, $(a,b) \in \Omega^2 \smallsetminus \{ (0,0) \}$ (so $K$ is a closed compact subgroup in $G$, but not open, and in particular not maximal).
We set $\omega : A^+ \to {\mathbb R}_{\ge 0}$ to be given by $\omega (\mathbf{a} (t)) := |t^2 - t^{-2} |$ if $G = SL_2 ({\mathbb R})$ and $\omega (\mathbf{a} (t)) := |t - t^{-1} |$ if $G = PGL_2 (\Omega)$. Then, taking the Haar measure on $K$ to have total mass $1$ and appropriately normalizing the Haar measure on $A$, for all non-negative-valued measurable functions $f$ on $G$ we have $$ \int_G f(g) \cdot dg = \int_{A^+} \omega (a) \left( \int_{K \times K} f(k_2 a k_1) \cdot dk_1 dk_2 \right) da.$$
Given a unitary $G$-representation $V$, vectors $v_1 , v_2 \in V$ and $a \in A^+$, we write $$ \ms{v_1}{v_2}{a} := \int_{K \times K} | \langle k_2 a k_1 v_1 , v_2 \rangle |^2 \cdot dk_1 dk_2.$$ We have \begin{equation}\label{eq M in term of Mcirc} \Ms{v_1}{v_2}{r} := \int_{A^+_{<r}} \omega (a) \cdot \ms{v_1}{v_2}{a} \cdot da \end{equation} (where $M_{v_1 , v_2} (r)$ was already defined in \S\ref{sec intro}).
Given a unitary character\footnote{${\rm U} (1)$ denotes the subgroup of ${\mathbb C}^{\times}$ consisting of complex numbers with absolute value $1$.} $\chi: A \to {\rm U}(1) $ we consider the principal series unitary $G$-representation $V_{\chi}$, consisting of functions $f : G \to {\mathbb C}$ satisfying $$ f(u a g) = \chi (a) \cdot \Delta (a)^{1/2} \cdot f(g) \quad \forall a \in A, u \in U , g \in G,$$ where $\Delta (a) = |\mathbf{a}^{-1} (a)|^2$ if $G = SL_2 ({\mathbb R})$ and $\Delta (a) = |\mathbf{a}^{-1} (a)|$ if $G = PGL_2 (\Omega)$. Here $G$ acts by $(g^{\prime}f)(g) := f(gg^{\prime})$. The $G$-invariant inner product on $V_{\chi}$ can be expressed as $$ \langle f_1 , f_2 \rangle = \int_{K} f_1 (k) \cdot \overline{f_2 (k)} \cdot dk.$$ For $\theta \in \hat{K}$, let $h^{\chi}_{\theta} \in V_{\chi}$ denote the unique vector determined by $h^{\chi}_{\theta} (k) = \theta (k)$ for $k \in K$, if it exists, and write $\textnormal{types} (V_{\chi}) \subset \hat{K}$ for the subset of $\theta$'s for which it exists. Thus $(h^{\chi}_{\theta})_{\theta \in \textnormal{types} (V_{\chi})}$ is a Hilbert basis for $V_{\chi}$.
\subsection{}
Let us now give several preparatory remarks.
First, we do not establish Conjecture \ref{main conj red exact} directly but, rather, establish conditions (1) and (2) of Proposition \ref{prop from folner to exact} (which suffices by that proposition).
Second, for a square-integrable irreducible unitary $G$-representation $V$, establishing conditions (1) and (2) of Proposition \ref{prop from folner to exact} with any unit vector $v_0 \in V$ is straight-forward (see the proof of Proposition \ref{prop sq int} for a spelling-out). As is well-known, a tempered irreducible unitary $G$-representation which is not square-integrable is a direct summand in some $V_{\chi}$. Therefore, we establish conditions (1) and (2) of Proposition \ref{prop from folner to exact} for irreducible direct summands in $V_{\chi}$.
Third, if $\chi = 1$ when $G = SL_2 ({\mathbb R})$ or if $\chi^2 = 1$ when $G = PGL_2 (\Omega)$, $V_{\chi}$ satisfies Conjecture \ref{main conj red exact} by Theorem \ref{thm V1 c temp}. So we assume throughout:
\begin{equation}\label{eq condition} \chi \neq 1 \textnormal{ if } G=SL_2 ({\mathbb R}), \quad \quad \quad \chi^2 \neq 1 \textnormal{ if } G = PGL_2 (\Omega). \end{equation}
\subsection{}
We reduce Conjecture \ref{main conj red exact} for an irreducible summand in $V_{\chi}$ to the following two claims.
\begin{claim}\label{clm SL2 2} Fix $\chi$ satisfying (\ref{eq condition}). Let $V$ be an irreducible direct summand in $V_{\chi}$. There exist $f \in V$, $r_0 \ge 0$ and $D>0$ such that for all $r \ge r_0$ we have \begin{equation}\label{eq from below} \Ms{f}{f}{r} \geq D \cdot r \end{equation} \end{claim}
\begin{claim}\label{clm SL2 2b}
Fix $\chi$ satisfying (\ref{eq condition}). There exist $r_0 > 0$ and $C>0$ (depending on $\chi$) such that for all $a \in A^+ \smallsetminus A^+_{<r_0}$ we have \begin{equation}\label{eq conj SL2 2} \ms{f_1}{f_2}{a} \leq C \cdot \omega (a)^{-1} \cdot ||f_1||^2 \cdot ||f_2||^2 \quad \quad \forall f_1, f_2 \in V_{\chi}.\end{equation} \end{claim}
\begin{proof}[Proof (of Conjecture \ref{main conj red exact} for summands in $V_{\chi}$ given Claim \ref{clm SL2 2} and Claim \ref{clm SL2 2b} for $\chi$).]
Let $V$ be an irreducible direct summand in $V_{\chi}$. Let $f$, $r_0$, $D$ and $C$ be as in Claim \ref{clm SL2 2} and as in Claim \ref{clm SL2 2b} (taking $r_0$ to be the maximum of the values from the two statements).
In order to verify Conjecture \ref{main conj red exact} for $V$, we will verify the conditions (1) and (2) of Proposition \ref{prop from folner to exact}, where for $v_0$ we take our $f$.
Using (\ref{eq conj SL2 2}) we obtain the existence of $E,E'>0$ such that for all $r_0 \leq r_1 < r_2$ we have $$ \Ms{f_1}{f_2}{r_2} - \Ms{f_1}{f_2}{r_1} \leq E \cdot \textnormal{vol}_A ( A^+_{<r_2} \smallsetminus A^+_{<r_1} ) \cdot || f_1||^2 \cdot || f_2 ||^2 \leq $$ $$ \leq E' \cdot (1 + (r_2 - r_1)) \cdot || f_1||^2 \cdot || f_2 ||^2.$$ From this and (\ref{eq from below}) the conditions (1) and (2) of Proposition \ref{prop from folner to exact} are immediate.
\end{proof}
\subsection{}\label{sec proof of clm SL2 2}
Let us prove Claim \ref{clm SL2 2}.
\begin{proof}[Proof (of Claim \ref{clm SL2 2}).]
Let $V$ be an irreducible direct summand of $V_{\chi}$.
Let us first treat the case $G = PGL_2 (\Omega)$. We use the (normalized) Jacquet $A$-module $J(-)$ with respect to $G \hookleftarrow AU \twoheadrightarrow A$. We denote by $\underline{V} \subset V$ the subspace of smooth vectors. $J(\underline{V})$ is isomorphic to ${\mathbb C}_{\chi} \oplus {\mathbb C}_{\chi^{-1}}$. We consider $v \in \underline{V}$ whose projection under the canonical $\underline{V} \twoheadrightarrow J(\underline{V})$ is non-zero and is an $A$-eigenvector with eigencharacter $\chi$. By Casselman's canonical pairing theory there exists a non-zero $\alpha \in J(\underline{V})^*$ which is $A$-eigenvector with eigencharacter $\chi^{-1}$ such that $\langle a v , v \rangle = |a |^{-1/2} \alpha (a v)$ whenever $a \in A^+ \smallsetminus A^+_{<r_0}$, for large enough $r_0 \ge 0$. Since we have $\alpha (a v) = \chi (a) \cdot \alpha (v)$ and $\alpha (v) \neq 0$, we deduce that for some $C>0$ we have $|\langle a v , v \rangle|^2 = C \cdot |a|^{-1} $ for $a \in A^+ \smallsetminus A^+_{<r_0}$. Let $K_v \subset K$ be an open compact subgroup, small enough so that $K_v v = v$. We have, again for $a \in A^+ \smallsetminus A^+_{< r_0}$: $$ \ms{v}{v}{a} = \int_{K \times K} |\langle k_2 a k_1 v , v \rangle |^2 \cdot dk_1 dk_2 \ge $$ $$ \ge \int_{K_v \times K_v} |\langle k_2 a k_1 v , v \rangle |^2 \cdot dk_1 dk_2 = C^{\prime} \cdot |\langle a v , v \rangle |^2$$ for some $C^{\prime}>0$ and so $\ms{v}{v}{a} \ge C^{\prime \prime} \cdot |a|^{-1}$ for some $C^{\prime \prime}>0$. From this we obtain the desired.
Let us now treat the case $G = SL_2 ({\mathbb R})$. Fix any $\theta \in {\rm types} (V_{\chi})$. The leading asymptotic of $K$-finite vectors are well-known, and can be computed from explicit expressions in terms of the hypergeometric function (see \cite[\S 6.5]{KlVi}). In the case $\chi|_{\mathbf{a} ({\mathbb R}^{\times}_{>0})} \neq 1$, denoting by $0 \neq s \in {\mathbb R}$ the number for which $\chi (\mathbf{a} (t)) = t^{is}$ for all $t \in {\mathbb R}^{\times}_{>0}$, we have $$ \langle \mathbf{a} (e^x) h^{\chi}_{\theta} , h^{\chi}_{\theta} \rangle \sim e^{-x} \cdot (E_1 \cdot e^{-isx} + E_2 \cdot e^{isx} + o(1)) \quad\quad (x \to +\infty)$$ for some non-zero $E_1$ and $E_2$ and so $$ |\langle \mathbf{a} (e^x) h^{\chi}_{\theta} , h^{\chi}_{\theta} \rangle|^2 \sim e^{-2x} \left( D + E_3 \cdot e^{-2isx} + E_4 \cdot e^{2isx} + o(1) \right) \quad\quad (x \to +\infty)$$ for some $D>0$, $E_3$ and $E_4$. From this we obtain the desired. In the case $\chi|_{\mathbf{a} ({\mathbb R}^{\times}_{>0})} = 1$, and so $\chi (\mathbf{a} (-1)) = - 1$, we have $$ \langle \mathbf{a} (e^x) h_{\theta}^{\chi} , h_{\theta}^{\chi} \rangle \sim E \cdot e^{-x} \quad\quad (x \to +\infty) $$ for some non-zero $E$ and so $$| \langle \mathbf{a} (e^x) h_{\theta}^{\chi} , h_{\theta}^{\chi} \rangle |^2 \sim D \cdot e^{-2x} \quad\quad (x \to +\infty) $$ for some $D>0$. From this we obtain the desired.
\end{proof}
\subsection{}
We further reduce Claim \ref{clm SL2 2b}.
\begin{claim}\label{clm SL2 3}
Fix $\chi$ satisfying (\ref{eq condition}). There exist $r_0 > 0$ and $C>0$ (depending on $\chi$) such that for all $\theta,\eta \in \textnormal{types} (V_{\chi})$ and all $a \in A^+ \smallsetminus A^+_{<r_0}$ we have \begin{equation}\label{eq thm SL2R 3} |\langle a h^{\chi}_{\theta} , h^{\chi}_{\eta} \rangle|^2 \leq C \cdot \omega(a)^{-1} .\end{equation} \end{claim}
\begin{proof}[Proof (of Claim \ref{clm SL2 2b} for $\chi$ given Claim \ref{clm SL2 3} for $\chi$).]
Let $f_1 , f_2 \in V_{\chi}$ and write $$ f_1 = \sum_{\theta \in \textnormal{types} (V_{\chi})} c_{\theta} \cdot h^{\chi}_{\theta}, \quad f_2 = \sum_{\theta \in \textnormal{types} (V_{\chi})} d_{\theta} \cdot h^{\chi}_{\theta}$$ with $c_{\theta} , d_{\theta} \in {\mathbb C}$. Using Fourier expansion of the function $(k_1 , k_2) \mapsto \langle a k_1 f_1 , k_2 f_2 \rangle$ on $K \times K$ we have, for $a \in A^+ \smallsetminus A^+_{<r_0}$: $$ \ms{f_1}{f_2}{a} = \int_{K\times K} \left| \langle a k_1 f_1 , k_2 f_2 \rangle \right|^2 \cdot dk_1 dk_2 = $$ $$ = \sum_{\theta,\eta \in \textnormal{types} (V_{\chi})} |c_{\theta}|^2 \cdot |d_{\eta}|^2 \cdot \left| \langle a h^{\chi}_{\theta} , h^{\chi}_{\eta} \rangle \right|^2 \leq C \cdot \omega (a)^{-1} \cdot ||f_1||^2 \cdot ||f_2||^2.$$
\end{proof}
\subsection{}
Let us now establish Claim \ref{clm SL2 3} in the case $G = SL_2 ({\mathbb R})$:
\begin{proof}[Proof (of Claim \ref{clm SL2 3} in the case $G = SL_2 ({\mathbb R})$).]
In the case $\chi |_{\mathbf{a} ({\mathbb R}^{\times}_{>0})} \neq 1$, this is the contents of \cite[Theorem 2.1]{BrCoNiTa} (which contains a stronger claim, incorporating $\chi$ into the inequality). Let us therefore assume $\chi|_{\mathbf{a} ({\mathbb R}^{\times}_{>0})} = 1$ and so $\chi (\mathbf{a} (-1)) = -1$. For $n \in {\mathbb Z}$, let us denote by $\theta_n$ the character of $K$ given by $\mtrx{c}{-s}{s}{c} \mapsto (c+is)^n$. We want to see that $$ \cosh (x) \left| \langle \mathbf{a} (e^x) h_{\theta_n}^{\chi} , h_{\theta_m}^{\chi} \rangle \right|$$ is bounded as we vary $x \in [0,+\infty)$ and $m,n \in 1 + 2 {\mathbb Z}$. We have $V_{\chi} = V_{\chi}^- \oplus V_{\chi}^+$, where $V_{\chi}^-$ and $V_{\chi}^+$ are irreducible unitary $G$-representations, ${\rm types} (V_{\chi}^-) = \{ \theta_n \ : \ n \in -1 - 2 {\mathbb Z}_{\ge 0}\}$ and ${\rm types} (V_{\chi}^+) = \{ \theta_n \ : \ n \in 1 + 2 {\mathbb Z}_{\ge 0}\}$. Since the matrix coefficient in question vanishes when $m \in -1 - 2 {\mathbb Z}_{\ge 0}$ and $n \in 1 + 2 {\mathbb Z}_{\ge 0}$ or vice versa, and since the matrix coefficient in question does not change when we replace $(n ,m)$ by $(-n , -m)$, we can assume $m,n \in 1 + 2 {\mathbb Z}_{ \ge 0}$. Furthermore, by conjugating by $\mtrx{0}{1}{-1}{0}$ it is straight-forward to see that we can assume that $m \ge n$. We denote $k := \tfrac{m-n}{2}$. Let us use the well-known concrete expression of matrix coefficients in terms of the hypergeometric function (see \cite[\S 6.5]{KlVi}): $$ \cosh (x) \cdot \langle \mathbf{a} (e^x) h_{\theta_n}^{\chi} , h_{\theta_m}^{\chi} \rangle = \tanh (x)^k \cdot \frac{(1+\frac{n-1}{2})_k}{k!} \cdot {}_2 F_1 \left( \frac{n-1}{2}+k+1 , \ -\frac{n-1}{2} , \ k+1 , \ \tanh (x)^2 \right).$$ We want to show that this expression is bounded as we vary $x \in [0,+\infty)$, $n \in 1+2 {\mathbb Z}_{\ge 0}$ and $k \in {\mathbb Z}_{\ge 0}$. Performing a change of variables $\frac{1-t}{2} := \tanh (x)^2$, denoting $r := \tfrac{n-1}{2}$ and interpreting in terms of Jacobi polynomials, we can rewrite this last expression as: $$Q^k_r (t) := (\tfrac{1-t}{2})^{k/2} \cdot P^{(k,0)}_{r} (t).$$ We want to see that $Q^k_r (t)$ is bounded as we vary $t \in [-1,1]$, $r \in {\mathbb Z}_{\ge 0}$ and $k \in {\mathbb Z}_{\ge 0}$. But, it is known that under a suitable interpretation of the variable $t$, $Q^k_r (t)$ is equal to a matrix coefficient of unit vectors in an irreducible unitary representation of ${\rm SU} (2)$, see, for example, \cite[\S 6.3]{KlVi}. Therefore, we have $ |Q^k_r (t)| \leq 1$ for all $t \in [-1,1]$, $r \in {\mathbb Z}_{\ge 0}$ and $k \in {\mathbb Z}_{\ge 0}$, as desired. See also \cite[Equation (20)]{HaSc} for a direct proof of this last inequality.
\end{proof}
\subsection{}
Finally, we want to establish Claim \ref{clm SL2 3} in the case $G = PGL_2 (\Omega)$. We will use the following proposition:
\begin{proposition}\label{prop padic estimate}
There exists $C>0$ such that the following holds. Let $\psi_1$ and $\psi_2$ be unitary characters of $\varpi {\mathcal O}$. Let $\alpha_1 , \alpha_2 \in {\mathcal O}^{\times} x + x ^2\Omega [[x ]] \subset \Omega [[x]]$ be power series. Let $\chi$ be a non-trivial unitary character of $\Omega^{\times}$. Denote by $c(\chi)$ the number $0$ if $\chi|_{{\mathcal O}^{\times}} = 1$ and otherwise the smallest number $c \in {\mathbb Z}_{\ge 1}$ for which $\chi|_{1+\varpi^c {\mathcal O}} = 1$. Also, denote by $d(\chi)$ the number $1/|1-\chi (\varpi)|$ if $c(\chi) = 0$ and the number $0$ if $c(\chi) \neq 0$. Let $0 < m_1 \leq m_2 \leq n$ be integers. Then $$ \left| \int_{\varpi^{m_1} {\mathcal O} \smallsetminus \varpi^{m_2} {\mathcal O}} \psi_1 (\alpha_1 (x)) \psi_2 (\alpha_2 (\varpi^n x^{-1})) \chi (x) \cdot \frac{dx}{|x|} \right| \leq C(c(\chi)+d(\chi) + 1).$$ \end{proposition}
\begin{proof}[Proof (of Claim \ref{clm SL2 3} in the case $G = PGL_2 (\Omega)$ given Proposition \ref{prop padic estimate}).]
Let us calculate more concretely the inner product appearing in Claim \ref{clm SL2 3}. We can normalize the inner product on $V_{\chi}$ so that for $f_1, f_2 \in V_{\chi}$ we have $$ \langle f_1 , f_2 \rangle = \int_{\Omega} f_1 \mtrx{1}{0}{x}{1} \overline{f_2 \mtrx{1}{0}{x}{1}} \cdot dx.$$ We then calculate $$ \left\langle \mtrx{t}{0}{0}{1} h^{\chi}_{\theta} , h^{\chi}_{\mu} \right\rangle = |t|^{-1/2} \chi (t) \int_{\Omega} h^{\chi}_{\theta} \mtrx{1}{0}{x}{1} \overline{h^{\chi}_{\mu} \mtrx{1}{0}{t^{-1} x}{1}} \cdot dx $$ and thus we want to see that $$ I_{\theta , \mu} (t) := \int_{\Omega} h^{\chi}_{\theta} \mtrx{1}{0}{x}{1} \overline{h^{\chi}_{\mu} \mtrx{1}{0}{t^{-1} x}{1}} \cdot dx $$ is bounded independently of $\theta , \mu \in \hat{K}$ and $t \in F$ satisfying $|t| \ge 1$. In general, for $\theta \in \hat{K}$ and $x \in \Omega$, we have $$ h^{\chi}_{\theta} \mtrx{1}{0}{x}{1} = \frac{\chi^{-1} (1 - \zeta x^2)}{|1 - \zeta x^2|^{1/2}} \theta \mtrx{1}{\zeta x}{x}{1}.$$ And so we obtain \begin{equation}\label{eq for I} I_{\theta , \mu} (t) = \int_{\Omega} \frac{\chi^{-1} (1 - \zeta x^2)}{|1 - \zeta x^2|^{1/2}} \frac{\chi (1 - \zeta t^{-2} x^2)}{|1 - \zeta t^{-2} x^2|^{1/2}} \theta \mtrx{1}{\zeta x}{x}{1} \mu^{-1} \mtrx{1}{\zeta t^{-1} x}{t^{-1} x}{1} \cdot dx. \end{equation} If for $D \subset F$ we denote by $I^D_{\theta , \mu} (t)$ the same expression as that for $I_{\theta , \mu} (t)$ in (\ref{eq for I}) but where integration is performed over $D$, we see $$ | I^{\varpi {\mathcal O}}_{\theta , \mu} (t) | \leq \int_{\varpi {\mathcal O}} dx = 1/q$$ and thus this is bounded. Furthermore, $$ |I^{\Omega \smallsetminus t {\mathcal O}}_{\theta , \mu} (t)| \leq \int_{\Omega \smallsetminus t {\mathcal O}} \frac{dx}{|x| \cdot |t^{-1} x|} = 1/q $$ and thus this is bounded. Finally, for any specific $- \log_q |t| \leq m \leq 0$, we have $$ |I^{\varpi^{m} {\mathcal O}^{\times}}_{\theta , \mu} (t)| \leq \int_{\varpi^{m} {\mathcal O}^{\times}} \frac{dx}{|x|} =1-1/q.$$ Therefore, denoting by $k \in {\mathbb Z}_{\ge 1}$ a number such that $\chi|_{1+\varpi^{2k} {\mathcal O}} = 1$, it is enough to bound $I^{\varpi^k t {\mathcal O} \smallsetminus \varpi^{-k} {\mathcal O}}_{\theta , \mu} (t)$. We have $$I^{\varpi^k t {\mathcal O} \smallsetminus \varpi^{-k} {\mathcal O}}_{\theta , \mu} (t) = $$ $$ = \chi^{-1} (- \zeta) \theta \mtrx{0}{\zeta}{1}{0} \int_{\varpi^k t {\mathcal O} \smallsetminus \varpi^{-k} {\mathcal O}} \chi^{-2} (x) \theta \mtrx{1}{x^{-1}}{\zeta^{-1} x^{-1} }{1} \mu^{-1} \mtrx{1}{\zeta t^{-1} x}{t^{-1} x}{1} \cdot \frac{dx}{|x|}.$$ Let us denote by $K^{\prime} \subset K$ the subgroup consisting of $\mtrx{x}{\zeta y}{y}{x}$ for which $|y| \leq |x| \cdot |p|$. We have an isomorphism of topological groups $$ e : p {\mathcal O} \to K^{\prime}$$ given by $$ e (y) := \exp \mtrx{0}{\zeta y}{y}{0}.$$ Let us denote by $\alpha : p {\mathcal O} \to p {\mathcal O}$ the map given by $\alpha (y) := e^{-1} \mtrx{1}{\zeta y}{y}{1}$. We have a power series expansion $\alpha (y) = y + \zeta y^3 / 3 + \zeta^2 y^5 / 5 + \ldots$. Let us now denote by $\widetilde{\theta}$ the unitary character of $p {\mathcal O}$ satisfying $\theta|_{K'} \circ e = \widetilde{\theta}$ and by $\widetilde{\mu}$ the unitary character of $p {\mathcal O}$ satisfying $\mu|_{K'} \circ e = \widetilde{\mu}$. Returning to our integral, we can take $k$ big enough so that $\varpi^k \in p {\mathcal O}$. Substituting $x^{-1}$ in place of $x$ in the integral we have, we see that we need to show that $$ \int_{\varpi^{k+1} {\mathcal O} \smallsetminus \varpi^{-k+1} t^{-1} {\mathcal O}} \widetilde{\theta} (\alpha (\zeta^{-1} x)) \widetilde{\mu}^{-1} (\alpha (t^{-1} x^{-1})) \cdot \chi^2 (x) \cdot \frac{dx}{|x|}$$ is bounded independently of $\widetilde{\theta} , \widetilde{\mu} \in \widehat{p {\mathcal O}}$ and $t \in F$ satisfying $|t| \ge 1$. This is implied by Proposition \ref{prop padic estimate}. \end{proof}
\begin{remark}
We see from the proof that we have a more precise version of Claim \ref{clm SL2 3}: There exists $C>0$ such that for $\chi$ satisfying (\ref{eq condition}), $\theta , \mu \in {\rm types} (V_{\chi})$ and $t \in \Omega^{\times}$ satisfying $|t| \ge 1$, we have $$ \left| \left\langle \mtrx{t}{0}{0}{1} h^{\chi}_{\theta} , h^{\chi}_{\mu} \right\rangle \right| \leq C \cdot |t|^{-1/2} \cdot (c(\chi) +d(\chi)+1).$$ Here $c(\chi)$ and $d(\chi)$ are as in the formulation of Proposition \ref{prop padic estimate} (when we identify $\chi$ with a character of $\Omega^{\times}$ via the isomorphism ${\boldsymbol{a}} : \Omega^{\times} \to A$). \end{remark}
Let us prove Proposition \ref{prop padic estimate}.
\begin{proof}[Proof (of Proposition \ref{prop padic estimate}).]
We can assume that the $1$-th coefficients of $\alpha_1$ and $\alpha_2$ are equal to $1$. Let us fix a unitary character $\psi$ of $\Omega$ satisfying $\psi|_{{\mathcal O}} = 1$ and $\psi|_{\varpi^{-1} {\mathcal O}} \neq 1$. For $i \in \{ 1 , 2 \}$, let $a_i \in \Omega$ be such that $\psi (a_i x) = \psi_i (x)$ for all $x \in \varpi {\mathcal O}$ ($a_i$ are defined up to addition of elements in $\varpi^{-1} {\mathcal O}$, so in particular we can assume that $a_i \neq 0$). Given $0 < m < n$ we define $$ J^m := \int_{\varpi^m {\mathcal O}^{\times}} \psi \left( a_1 \alpha_1 (x) + a_2 \alpha_2 (\varpi^n x^{-1}) \right) \cdot \chi (x) \cdot \frac{dx}{|x|} = $$ $$ = \chi (\varpi)^{m} \int_{ {\mathcal O}^{\times} } \psi \left( a_1 \alpha_1 (\varpi^m x) + a_2 \alpha_2 (\varpi^{n-m} x^{-1}) \right) \cdot \chi (x) \cdot dx,$$ so that the integral in question equals $\sum_{m_1 \leq m < m_2} J^m$. Let us abbreviate $b := (a_2 \varpi^{n-m}) / (a_1 \varpi^m)$. As we vary $m$, let us divide into cases.
\begin{enumerate}
\item Assume that $ |\varpi^m | < q^{-c (\chi)}$ and that $|b| < q^{-c(\chi)}$. Set $$\beta (x) := \varpi^{-m} \alpha_1 (\varpi^m x) + b \cdot \varpi^{m-n} \alpha_2 (\varpi^{n-m} x^{-1}).$$ Then $\beta$ gives a well-defined invertible analytic map ${\mathcal O}^{\times} \to {\mathcal O}^{\times}$, whose derivative is everywhere a unit. Moreover, if ${c(\chi)} > 0$, we have $\beta^{-1} (x_0 (1 + \varpi^{c(\chi)} {\mathcal O})) = x_0 (1 + \varpi^{c(\chi)} {\mathcal O})$ for any $x_0 \in {\mathcal O}^{\times}$. Hence $$ J^m = \chi (\varpi)^{m} \int_{ {\mathcal O}^{\times}} \psi (a_1 \varpi^m \beta (x)) \cdot \chi (x) \cdot dx = $$ $$ = \chi (\varpi)^{m} \int_{ {\mathcal O}^{\times} } \psi (a_1 \varpi^m x) \cdot \chi (x) \cdot dx.$$ Therefore:
\begin{enumerate}
\item Suppose that $|a_1 \varpi^m| \leq 1$. Then $J^m = \chi (\varpi)^{m} (1-1/q)$ if $\chi$ is unramified and $J^m = 0$ if $\chi$ is ramified.
\item Suppose that $|a_1 \varpi^m \varpi^{c(\chi)} | > q^{-1}$. Then $J^m = 0$ if $\chi$ is unramified. If $\chi$ is ramified, we write $$ J^m = \chi (\varpi)^{m} \sum_{x_0 \in {\mathcal O}^{\times} / (1+\varpi^{c(\chi)} {\mathcal O})} \psi (a_1 \varpi^m x_0) \chi (x_0) \int_{\varpi^{c(\chi)} {\mathcal O}} \psi (a_1 \varpi^m x) \cdot dx$$ and each integral here is equal to $0$, so that also in that case we obtain $J^m = 0$.
\item The case when neither of these two cases is satisfied corresponds to only finitely many values of $m$, whose number is linearly bounded in terms of $c(\chi)+1$, and so we can be content with the crude estimate $|J^m| \leq 1$ in this case. \end{enumerate}
\item Assume that $|\varpi^{n-m}| < q^{-c(\chi)}$ and $|b^{-1}| < q^{-c ( \chi)}$. This case is dealt with analogously to the previous one; one denotes $$ \beta (x) := \omega^{m-n} \alpha_2 (\varpi^{n-m} x^{-1}) + b^{-1} \varpi^{-m} \alpha_1 (\varpi^m x)$$ and gets $$ J^m = \chi (\varpi)^m \int_{{\mathcal O}^{\times}} \psi (a_2 \varpi^{n-m} x) \cdot \chi (x) \cdot dx.$$ And thus:
\begin{enumerate}
\item Suppose that $|a_2 \varpi^{n-m}| \leq 1$. Then $J^m = \chi (\varpi)^{m} (1-1/q)$ if $\chi$ is unramified and $J^m = 0$ if $\chi$ is ramified.
\item Suppose that $|a_2 \varpi^{n-m} \varpi^{c(\chi)} | > q^{-1}$. Then $J^m = 0$ if $\chi$ is unramified. If $\chi$ is ramified, we write $$ J^m = \chi (\varpi)^{m} \sum_{x_0 \in {\mathcal O}^{\times} / (1+\varpi^{c(\chi)} {\mathcal O})} \psi (a_2 \varpi^{n-m} x_0) \chi (x_0) \int_{\varpi^{c(\chi)} {\mathcal O}} \psi (a_2 \varpi^{n-m} x) \cdot dx$$ and each integral here is equal to $0$, so that also in that case we obtain $J^m = 0$.
\item The case when neither of these two cases is satisfied corresponds to only finitely many values of $m$, whose number is linearly bounded in terms of $c(\chi)+1$, and so we can be content with the crude estimate $|J^m| \leq 1$ in this case. \end{enumerate}
\item The case when neither of these two cases is satisfied corresponds to only finitely many values of $m$, whose number is linearly bounded in terms of $c(\chi)$, and so we can be content with the crude estimate $|J^m| \leq 1$ in this case.
\end{enumerate}
As our integral in question is equal to $\sum_{m_1 \leq m < m_2} J^m$, it is straight-forward that the findings above give the boundedness as desired. \end{proof}
\appendix
\section{Auxiliary claims regarding polynomial growth of exponential integrals and sums}
\subsection{Some notation}
We denote $[n] := \{1 , 2, \ldots , n \}$. We denote $${\mathbb C}_{\leq 0} := \{ z \in {\mathbb C} \ | \ {\rm Re} (z) \leq 0\}, \quad D := \{ z \in {\mathbb C} \ | \ |z| \leq 1\}.$$ Given $x = (x_1 , \ldots , x_n) \in {\mathbb R}_{\ge 0}^n$ and $m = (m_1 , \ldots , m_n) \in {\mathbb Z}_{\ge 0}^n$, we write $x^m := x_1^{m_1} \ldots x_n^{m_n}$. Given $\lambda \in {\mathbb C}_{\leq 0}^n$ we denote $$J_{\lambda} := \{ 1 \leq j \leq n \ | \ {\rm Re} (\lambda_j) = 0\}.$$ Given $(\lambda , m) \in {\mathbb C}_{\leq 0}^n \times {\mathbb Z}_{\ge 0}^n$, we denote $d(\lambda , m) := \sum_{j \in J_{\lambda}} (1+m_j)$. Given $J \subset [n]$ and some set $X$, let us denote by ${\rm res}_{J} : X^n \to X^{J}$ the natural restriction and by ${\rm ext}^{J} : X^{J} \to X^{n}$ the natural extension by zero.
We fix a finite set ${\mathcal I} \subset {\mathbb R}_{\ge 0}^n$ with the property that given $j \in [n]$ there exists $v \in {\mathcal I}$ such that $\langle v , e_j \rangle \neq 0$, where $e_j$ the $j$-th standard basis vector. We denote $$ P_{< r} := \{ x \in {\mathbb R}_{\ge 0}^n \ | \ \langle v , x \rangle < r \ \forall v \in {\mathcal I} \}.$$ Given $J \subset [n]$, we denote by $P_J \subset {\mathbb R}_{\ge 0}^J$ the convex pre-compact subset $\{ y \in {\mathbb R}_{\ge 0}^J \ | \ {\rm ext}^{J} (y) \in P_{<1}\}$.
In \S\ref{ssec growth integral} we will also use the following notations. We consider a compact space $B$ equipped with a nowhere vanishing Radon measure $db$. Let us say that a function $\phi : B \times {\mathbb R}_{\ge 0}^n \to {\mathbb C}$ is \textbf{nice} if it is expressible as $$ B \times {\mathbb R}_{\ge 0}^n \xrightarrow{{\rm id}_B \times {\rm ei}} B \times D^n \xrightarrow{\phi^{\circ}} {\mathbb C}$$ where ${\rm ei}(x_1 , \ldots , x_n) := (e^{-x_1} , \ldots , e^{-x_n})$ and $\phi^{\circ}$ is continuous and holomorphic in the second variable (in the sense that when we fix the variable in $B$ it is the restriction of a holomorphic function on a neighbourhood of $D^n$). Given $J \subset [n]$ we denote by ${\rm res}_{J} \phi : B \times {\mathbb R}_{\ge 0}^J \to {\mathbb C}$ the function given by ${\rm res}_{J} \phi (b , y) := \phi^{\circ} (b , {\rm ext}^{J} ({\rm ei} (y)))$. We also write $\phi (b , +\infty)$ for $\phi^{\circ} (b , 0)$ etc. \subsection{Growth - the case of summation over a lattice}
\begin{lemma}\label{lem appb lat 1}
Let $\lambda := (\lambda_1 , \ldots , \lambda_n) \in {\mathbb C}_{\leq 0}^n$ and $m := (m_1 , \ldots , m_n) \in {\mathbb Z}_{\ge 0}$. Let $K \subset {\mathbb R}_{\ge 0}^n$ be a compact subset. Assume that ${\rm Re} (\lambda) = 0$ and $\lambda \notin (2\pi i) {\mathbb Z}^n$. We have $$ \sup_{Q \subset K} \left| \frac{1}{r^n} \sum_{x \in \frac{1}{r} {\mathbb Z}_{\ge 0}^n \cap Q} x^m e^{r \langle \lambda , x \rangle} \right| = O( r^{-1} )$$ as $r \to +\infty$, where $Q$ denote convex subsets. \end{lemma}
\begin{proof}
Let us re-order the variables, assuming that $\lambda_1 \notin 2\pi i {\mathbb Z}$. Let us write $x = (x_1 , x')$ where $x' = (x_2 , \ldots , x_n)$ and analogously write $m'$ et cetera. Given a convex subset $Q \subset K$ and $x' \in {\mathbb R}_{\ge 0}^{n-1}$ let us denote by $Q^{x'} \subset {\mathbb R}_{\ge 0}$ the subset consisting of $x_1$ for which $(x_1 , x') \in Q$ (it is an interval). Let us enlarge $K$ for convenience, writing it in the form $K = K_1 \times K'$ where $K_1 \subset {\mathbb R}_{\ge 0}$ is a closed interval and $K' \subset {\mathbb R}_{\ge 0}^{n-1}$ is the product of closed intervals.
We have $$ \sum_{x \in \frac{1}{r} {\mathbb Z}_{\ge 0}^n \cap Q} x^m e^{r\langle \lambda , x \rangle} = \sum_{ x' \in \frac{1}{r} {\mathbb Z}_{\ge 0}^{n-1} \cap K' } (x')^{m'} e^{r \langle \lambda' , x' \rangle }\left( \sum_{ x_1 \in \frac{1}{r} {\mathbb Z}_{\ge 0} \cap Q^{x'}} x_1^{m_1} e^{r \lambda_1 x_1} \right).$$ We have $Q^{x'} \subset K'$ and it is elementary to see that $$ \sup_{R \subset K'} \left| \sum_{x_1 \in \frac{1}{r} {\mathbb Z}_{\ge 0} \cap R} x_1^{m_1} e^{r \lambda_1 x_1} \right| = O(1)$$ as $r \to +\infty$, where $R$ denote intervals. Therefore we obtain, for some $C>0$ (not depending on $Q$) and all $r \ge 1$: $$ \left| \frac{1}{r^n} \sum_{x \in \frac{1}{r} {\mathbb Z}_{\ge 0}^n \cap Q} x^m e^{r\langle \lambda , x \rangle} \right| \leq C \left( \frac{1}{r^{n-1}} \sum_{ x' \in \frac{1}{r} {\mathbb Z}_{\ge 0}^{n-1} \cap K' } (x')^{m'} \right) r^{-1}.$$ Since the expression in brackets is clearly bounded independently of $r$, we are done. \end{proof}
\begin{lemma}\label{lem appb lat 2} Let $(\lambda , m ) \in {\mathbb C}_{\leq 0}^n \times {\mathbb Z}_{\ge 0}^n$. Then the limit $$ \lim_{r \to +\infty} \frac{1}{r^{d(\lambda , m)}} \sum_{x \in {\mathbb Z}_{\ge 0}^n \cap P_{<r}} x^m e^{\langle \lambda , x \rangle} dx$$ exists, equal to $0$ if ${\rm res}_{J_{\lambda}} (\lambda) \notin 2\pi i \cdot {\mathbb Z}^{J_{\lambda}}$ and otherwise equal to $$\left( \int_{P_{J_{\lambda}}} y^{{\rm res}_{J_{\lambda}} (m)} dy \right) \left( \sum_{z \in {\mathbb Z}_{\ge 0}^{J_{\lambda}^c}} z^{{\rm res}_{J_{\lambda}^c} (m)} e^{\langle {\rm res}_{J_{\lambda}^c} (\lambda) , z \rangle } \right)$$ (the sum converging absolutely). \end{lemma}
\begin{proof}
Let us abbreviate $J := J_{\lambda}$. Let us denote $\lambda' := {\rm res}_J (\lambda)$ and $\lambda'' := {\rm res}_{J^c} (\lambda)$, and similarly for $m$. Given $x'' \in {\mathbb Z}_{\ge 0}^{J_c}$ let us denote by $P_{(<r)}^{x''} \subset {\mathbb R}_{\ge 0}^{J}$ the subset consisting of $y'$ for which ${\rm ext}^{J} (r y') + {\rm ext}^{J^c} (x'') \in P_{<r}$.
We have $$ \sum_{x \in {\mathbb Z}_{\ge 0}^n \cap P_{<r}} x^m e^{\langle \lambda , x \rangle} = r^{d(\lambda , m) - |J|} \sum_{x'' \in {\mathbb Z}_{\ge 0}^{J^c}} (x'')^{m''} e^{\langle \lambda'' , x'' \rangle} \sum_{y' \in \frac{1}{r} {\mathbb Z}_{\ge 0}^{J} \cap P^{x''}_{(<r)}} (y')^{m'} e^{r \langle \lambda' , y' \rangle} := \triangle.$$
Let us assume first that $\lambda' \notin 2 \pi i \cdot {\mathbb Z}^{J}$. Then by Lemma \ref{lem appb lat 1} there exists $C>0$ such that for all convex subsets $Q \subset P_J$ and all $r \ge 1$ we have $$ \left| \frac{1}{r^{|J|}} \sum_{y' \in \frac{1}{r} {\mathbb Z}_{\ge 0}^{J} \cap Q} (y')^{m'} e^{r \langle \lambda' , y' \rangle} \right| \leq C \cdot r^{-1}.$$ Therefore $$ | \triangle | \leq C r^{d(\lambda , m) - 1} \sum_{x'' \in {\mathbb Z}_{\ge 0}^{J^c}} (x'')^{m''} e^{\langle {\rm Re} (\lambda'') , x'' \rangle}, $$ giving the desired.
Now we assume $\lambda' \in 2 \pi i \cdot {\mathbb Z}^J$. It is not hard to see that $$ \lim_{r \to +\infty} \frac{1}{r^{|J|}} \sum_{y' \in \frac{1}{r} {\mathbb Z}_{\ge 0}^{J} \cap P^{x''}_{(<r)}} (y')^{m'} = \int_{P_J} (y')^{m'} dy'.$$ Hence we have (by dominated convergence) $$ \lim_{r \to +\infty} \frac{1}{r^{d(\lambda,m)}} \triangle = \sum_{x'' \in {\mathbb Z}_{\ge 0}^{J^c}} (x'')^{m''} e^{\langle \lambda'' , x'' \rangle} \int_{P_J} (y')^{m'} dy' .$$ \end{proof}
\begin{claim}\label{clm appb lat}
Let $p \ge 1$, let $\{ ( \lambda^{(\ell)} , m^{(\ell)}) \}_{\ell \in [p]} \subset {\mathbb C}_{\leq 0}^{n} \times {\mathbb Z}_{\ge 0}^n$ be a collection of pairwise different couples and let $\{ c^{(\ell)} \}_{\ell \in [p]} \subset {\mathbb C} \smallsetminus \{ 0 \}$ be a collection of non-zero scalars. Denote $d := \max_{\ell \in [p]} d(2{\rm Re} (\lambda^{(\ell)}) , 2 m^{(\ell)})$.
The limit $$ \lim_{r \to +\infty} \frac{1}{r^d} \sum_{x \in {\mathbb Z}_{\ge 0}^n \cap P_{<r}} \left| \sum_{\ell \in [p]} c^{(\ell)} x^{m^{(\ell)}} e^{\langle \lambda^{(\ell)} , x \rangle } \right|^2 $$ exists and is strictly positive. \end{claim}
\begin{proof}
Let us break the integrand into a sum following $$\left| \sum_{\ell \in [p]} A_{\ell} \right|^2 = \sum_{\ell_1 , \ell_2 \in [p]} A_{\ell_1} \overline{A_{\ell_2}}.$$ Using Lemma \ref{lem appb lat 2} we see the that resulting limit breaks down as a sum, over $(\ell_1 ,\ell_2) \in [p]^2$, of limits which exist, so the only thing to check is that the resulting limit is non-zero. It is easily seen that the limit at the $(\ell_1 , \ell_2)$ place is zero unless $d(\lambda^{(\ell_1)} , m^{(\ell_1)} ) = d$, $d(\lambda^{(\ell_2)} , m^{(\ell_2)} ) = d$, $J_{\lambda^{(\ell_1)}} = J_{\lambda^{(\ell_2)}}$ and ${\rm res}_{J^{(\ell_1)}} (\lambda^{(\ell_2)}) - {\rm res}_{J^{(\ell_1)}} (\lambda^{(\ell_1)}) \in 2\pi i \cdot {\mathbb Z}^J$. We thus can reduce to the case when, for a given $J \subset [n]$, we have $J_{\lambda^{(\ell)}} = J$ for all $\ell \in [p]$, we have $d(\lambda^{(\ell)} , m^{(\ell)}) = d$ for all $\ell \in [p]$, and we have ${\rm res}_{J} (\lambda^{(\ell_2)}) - {\rm res}_{J} (\lambda^{(\ell_1)}) \in 2\pi i \cdot {\mathbb Z}^J$ for all $\ell_1 , \ell_2 \in [p]$. We then obtain, using Lemma \ref{lem appb lat 2}, that our overall limit equals $$ \sum_{z \in {\mathbb Z}_{\ge 0}^{J^c}} \int_{P_J} \left| \sum_{\ell \in [p]} c^{(\ell)} y^{{\rm res}_J (m^{(\ell)})} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\langle {\rm res}_{J^c} (\lambda^{(\ell)}) , z \rangle } \right|^2 dy.$$ It is therefore enough to check that $$ \sum_{\ell \in [p]} c^{(\ell)} y^{{\rm res}_J (m^{(\ell)})} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\langle {\rm res}_{J^c} (\lambda^{(\ell)}) , z \rangle }, $$ a function in $(z,y ) \in {\mathbb Z}_{\ge 0}^{J^c} \times P_J$, is not identically zero. By the local linear independence of powers of $y$, we can further assume that ${\rm res}_J (m^{(\ell)})$ is independent of $\ell \in [p]$, and want to check that $$\sum_{\ell \in [p]} c^{(\ell)} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\langle {\rm res}_{J^c} (\lambda^{(\ell)}) , z \rangle }, $$ a function in $z \in {\mathbb Z}_{\ge 0}^{J^c}$, is not identically zero. Notice that, by our assumptions, the elements in the collection $ \{ ( {\rm res}_{J^c} (\lambda^{(\ell)}) , {\rm res}_{J^c} (m^{(\ell)}) ) \}_{\ell \in [p]}$ are pairwise different. Thus the non-vanishing of our sum is clear (by linear algebra of generalized eigenvectors of shift operators on ${\mathbb Z}^{J^c}$). \end{proof}
\subsection{Growth - the case of an integral}\label{ssec growth integral}
\begin{lemma}\label{lem appb 1}
Let $\lambda := (\lambda_1 , \ldots , \lambda_n) \in {\mathbb C}_{\leq 0}^n$ and $m := (m_1 , \ldots , m_n) \in {\mathbb Z}_{\ge 0}$. Let $K \subset {\mathbb R}_{\ge 0}^n$ be a compact subset. Assume that ${\rm Re} (\lambda) = 0$ and $\lambda \neq 0$. We have $$ \sup_{Q \subset K} \left| \int_Q x^m e^{r \langle \lambda , x \rangle} dx \right| = O( r^{-1} )$$ as $r \to +\infty$, where $Q$ denote convex subsets. \end{lemma}
\begin{proof}
Let us re-order the variables, assuming that $\lambda_1 \neq 0$. Let us write $x = (x_1 , x')$ where $x' = (x_2 , \ldots , x_n)$ and analogously write $m'$ etcetera. Given a convex subset $Q \subset K$ and $x' \in {\mathbb R}_{\ge 0}^{n-1}$ let us denote by $Q^{x'} \subset {\mathbb R}_{\ge 0}$ the subset consisting of $x_1$ for which $(x_1 , x') \in Q$ (it is an interval). Let us enlarge $K$ for convenience, writing it in the form $K = K_1 \times K'$ where $K_1 \subset {\mathbb R}_{\ge 0}$ is a closed interval and $K' \subset {\mathbb R}_{\ge 0}^{n-1}$ is the product of closed intervals.
Using Fubini's theorem $$ \int_Q x^m e^{r\langle \lambda , x \rangle} dx = \int_{ K' } (x')^{m'} e^{r \langle \lambda' , x' \rangle }\left( \int_{Q^{x'}} x_1^{m_1} e^{r \lambda_1 x_1} dx_1 \right) dx'.$$ We have $Q^{x'} \subset K'$ and it is elementary to see that $$ \sup_{R \subset K'} \left| \int_{R} x_1^{m_1} e^{r \lambda_1 x_1} dx_1 \right| = O(r^{-1})$$ as $r \to +\infty$, where $R$ denote intervals. Therefore we obtain, for some $C>0$ and all $r \ge 1$: $$ \left| \int_Q x^m e^{r\langle \lambda , x \rangle} dx \right| \leq C \left( \int_{K'} (x')^{m'} dx' \right) r^{-1},$$ as desired. \end{proof}
\begin{lemma}\label{lem appb 2} Let $(\lambda , m ) \in {\mathbb C}_{\leq 0}^n \times {\mathbb Z}_{\ge 0}^n$ and let $\phi : B \times {\mathbb R}_{\ge 0}^n \to {\mathbb C}$ be a nice function. Then the limit $$ \lim_{r \to +\infty} \frac{1}{r^{d(\lambda , m)}} \int_B \int_{P_{<r}} x^m e^{\langle \lambda , x \rangle} \phi (b,x)dxdb$$ exists, equal to $0$ if ${\rm res}_{J_{\lambda}} \lambda \neq 0$ and otherwise equal to $$\left( \int_{P_{J_{\lambda}}} y^{{\rm res}_{J_{\lambda}} (m)} dy \right) \left( \int_B \int_{{\mathbb R}_{\ge 0}^{J_{\lambda}^c}} z^{{\rm res}_{J_{\lambda}^c} (m)} e^{\langle {\rm res}_{J_{\lambda}^c} (\lambda) , z\rangle} {\rm res}_{J_{\lambda}^c} \phi (b , z) dz db \right)$$ (the double integral converging absolutely). \end{lemma}
\begin{proof}
Let us re-order the variables, assuming that $J := J_{\lambda} = [k]$. Let us write $x = (x',x'')$ where $x'$ consists of the first $k$ components and $x''$ consists of the last $k$ components. Let us write analogously $m' , \lambda'$ etc.
First, let us notice that if $k \neq 0$, we can write $$ \phi (b , x ) = e^{-x_1} \phi_0 (b , x) + \phi_1 (b , x)$$ where $\phi_0 , \phi_1 : B \times {\mathbb R}_{\ge 0}^n \to {\mathbb C}$ are nice functions and $\phi_1$ does not depend on $x_1$. Dealing with $ e^{-x_1} \phi_0 (b,x)$ instead of $\phi (b,x)$ makes us consider $\lambda$ with smaller set $J_{\lambda}$ and thus $(\lambda , m)$ with a smaller $d(\lambda , m)$ and from this, reasoning inductively, we see that we can assume that $\phi$ only depends on $(b,x'')$. Let us write $\phi'' := {\rm res}_{J^c} \phi$.
Let us perform a change of variables $y' := \frac{1}{r} x'$. Let $P_{(<r)} \subset {\mathbb R}_{\ge 0}^n$ denote the transform of $P_{<r}$ under this changes of variables (i.e. $(x',x'') \in P_{<r}$ if and only if $(y',x'') \in P_{(<r)}$). We obtain $$ \int_B \int_{P_{<r}} x^m e^{\langle \lambda , x \rangle} \phi (b , x) dx db = $$ $$ = r^{d} \int_B \int_{P_{(<r)}} (y')^{m'} e^{r \langle \lambda' , y' \rangle } (x'')^{m''} e^{\langle \lambda'' , x'' \rangle} \phi'' (b , x'') dy' dx'' db =: \triangle.$$
Given $x'' \in {\mathbb R}_{\ge 0}^{J^c}$, let us denote by $P_{(<r)}^{x''} \subset {\mathbb R}_{\ge 0}^{J}$ the set consisting of $y'$ for which $(y',x'') \in P_{(<r)}$. Notice that $P_{(<r_1)}^{x''} \subset P_{(<r_2)}^{x''}$ for $r_1 < r_2$ and $\cup_{r} P_{(<r)}^{x''} = P_{J}$. Using Fubini's theorem
$$ \triangle = r^{d} \int_B \int_{ {\mathbb R}_{\ge 0}^{J^c}} (x'')^{m''} e^{\langle \lambda'' , x'' \rangle } \phi'' (b , x'') \left( \int_{ P_{(<r)}^{x''}} (y')^{m'} e^{r \langle \lambda' , y' \rangle } dy' \right) dx'' db.$$
If $\lambda' \neq 0$, by Lemma \ref{lem appb 1} there exists $C>0$ such that for all convex subsets $Q \subset P_J$ and all $r \ge 1$ we have $$ \left| \int_Q (y')^{m'} e^{r \langle \lambda' , y' \rangle } dy' \right| \leq C \cdot r^{-1}.$$ We have therefore $$ \left| \triangle \right| \leq C \cdot r^{d - 1} \cdot \int_B \int_{ {\mathbb R}_{\ge 0}^{J^c}} (x'')^{m''} e^{\langle {\rm Re} (\lambda'' ) , x'' \rangle } | \phi'' (b , x'')| dx'' db $$ and thus indeed the desired limit is equal to $0$.
Now we assume $\lambda' = 0$. Using Lebesgue's dominated convergence theorem we have $$ \lim_{r \to +\infty} \frac{1}{r^d} \triangle = \lim_{r \to +\infty} \int_B \int_{{\mathbb R}_{\ge 0}^{J^c}} (x'')^{m''} e^{\langle \lambda'' , x'' \rangle } \phi'' (b , x'') \left( \int_{ P_{(<r)}^{x''}} (y')^{m'} dy' \right) dx'' db = $$ $$ = \int_B \int_{ {\mathbb R}_{\ge 0}^{J^c}} (x'')^{m''} e^{\langle \lambda'' , x'' \rangle } \phi'' (b , x'') \left( \int_{P_J} (y')^{m'} dy' \right) dx'' db$$ as desired. \end{proof}
\begin{claim}\label{clm appb}
Let $\{ ( \lambda^{(\ell)} , m^{(\ell)}) \}_{\ell \in [p]} \subset {\mathbb C}_{\leq 0}^{n} \times {\mathbb Z}_{\ge 0}^n$ be a collection of pairwise different couples. Let $\{ \phi^{(\ell)} \}_{\ell \in [p]}$ be a collection of nice functions $B \times {\mathbb R}_{\ge 0}^n \to {\mathbb C}$, such that for every $\ell \in [p]$ the function $b \mapsto \phi^{(\ell)} (b , +\infty)$ on $B$ is not identically zero. Denote $d := \max_{\ell \in [p]} d(2{\rm Re} (\lambda^{(\ell)}) , 2 m^{(\ell)})$.
The limit $$ \lim_{r \to +\infty} \frac{1}{r^d} \int_B \int_{P_{<r}} \left| \sum_{\ell \in [p]} x^{m^{(\ell)}} e^{\langle \lambda^{(\ell)} , x \rangle } \phi^{(\ell)} (b , x)\right|^2 dx db$$ exists and is strictly positive. \end{claim}
\begin{proof}
Let us break the integrand into a sum following $$\left| \sum_{\ell \in [p]} A_{\ell} \right|^2 = \sum_{\ell_1 , \ell_2 \in [p]} A_{\ell_1} \overline{A_{\ell_2}}.$$ Using Lemma \ref{lem appb 2} we see the that resulting limit breaks down as a sum, over $(\ell_1 ,\ell_2) \in [p]^2$, of limits which exist, so the only thing to check is that the resulting limit is non-zero. It is easily seen that the limit at the $(\ell_1 , \ell_2)$ place is zero unless $d(\lambda^{(\ell_1)} , m^{(\ell_1)} ) = d$, $d(\lambda^{(\ell_2)} , m^{(\ell_2)} ) = d$, $J_{\lambda^{(\ell_1)}} = J_{\lambda^{(\ell_2)}}$ and ${\rm res}_{J^{(\ell_1)}} (\lambda^{(\ell_1)}) = {\rm res}_{J^{(\ell_1)}} (\lambda^{(\ell_2)})$. We thus can reduce to the case when, for a given $J \subset [n]$, we have $J_{\lambda^{(\ell)}} = J$ for all $\ell \in [p]$, we have $d(\lambda^{(\ell)} , m^{(\ell)}) = d$ for all $\ell \in [p]$, and we have ${\rm res}_{J} (\lambda^{(\ell_1)}) = {\rm res}_{J} (\lambda^{(\ell_2)})$ for all $\ell_1 , \ell_2 \in [p]$. We then obtain, using Lemma \ref{lem appb 2}, that our overall limit equals $$ \int_B \int_{{\mathbb R}_{\ge 0}^{J^c}} \int_{P_J} \left| \sum_{\ell \in [p]} y^{{\rm res}_J (m^{(\ell)})} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\langle {\rm res}_{J^c} (\lambda^{(\ell)}) , z\rangle} {\rm res}_{J^c} \phi^{(\ell)} (b , z) \right|^2 dy dz db.$$ It is therefore enough to check that $$ \sum_{\ell \in [p]} y^{{\rm res}_J (m^{(\ell)})} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\langle {\rm res}_{J^c} (\lambda^{(\ell)}) , z\rangle} {\rm res}_{J^c} \phi^{(\ell)} (b , z),$$ a function in $(b,z,y ) \in B \times {\mathbb R}_{\ge 0}^{J^c} \times P_J$, is not identically zero. By the local linear independence of powers of $y$, we can further assume that ${\rm res}_J (m^{(\ell)})$ is independent of $\ell \in [p]$, and want to check that $$ \sum_{\ell \in [p]} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\langle {\rm res}_{J^c} (\lambda^{(\ell)}) , z\rangle} {\rm res}_{J^c} \phi^{(\ell)} (b , z),$$ a function in $(b,z) \in B \times {\mathbb R}_{\ge 0}^{J^c}$, is not identically zero. Notice that, by our assumptions, the elements in the collection $ \{ ( {\rm res}_{J^c} (\lambda^{(\ell)}) , {\rm res}_{J^c} (m^{(\ell)}) ) \}_{\ell \in [p]}$ are pairwise different and for every $\ell \in [p]$, the function $b \mapsto \phi^{(\ell)} (b , {\rm ext}^{J^c} (+\infty))$ on $B$ is not identically zero. Considering the partial order on ${\mathbb C}^{J^c}$ given by $\mu_1 \leq \mu_2$ if $\mu_2 - \mu_1 \in {\mathbb Z}_{\ge 0}^{J^c}$, we can pick $\ell \in [p]$ for which ${\rm res}_{J^c} (\lambda^{(\ell)})$ is maximal among the $\{ {\rm res}_{J^c} (\lambda^{(\ell')}) \}_{\ell' \in [p]}$. We can then pick $b \in B$ such that $\phi^{(\ell)} (b , {\rm ext}^{J^c} (+\infty)) \neq 0$. We then boil down to Lemma \ref{lem appb 4} that follows.
\end{proof}
In the end of the proof of Claim \ref{clm appb} we have used the following:
\begin{lemma}\label{lem appb 4}
Let $\{ (\lambda^{(\ell)} , m^{(\ell)}) \}_{\ell \in [p]} \subset {\mathbb C}^n \times {\mathbb Z}_{\ge 0}^n$ be a collection of pairwise different couples. Let $\{ \phi^{(\ell)}\}_{\ell \in [p]}$ be a collection of nice functions ${\mathbb R}_{\ge 0}^n \to {\mathbb C}$ (so here $B = \{ 1 \}$). Suppose that $\phi^{(\ell)} (+\infty) \neq 0$ for some $\ell \in [p]$ for which $\lambda^{(\ell)}$ is maximal among the $\{ \lambda^{(\ell')} \}_{\ell' \in [p]}$ with respect to the partial order $\lambda_1 \leq \lambda_2$ if $\lambda_2 - \lambda_1 \in {\mathbb Z}_{\ge 0}^n$. Then the function $$ x \mapsto \sum_{\ell \in [p]} x^{m^{(\ell)}} e^{\langle \lambda^{(\ell)} , x \rangle} \phi^{(\ell)} (x)$$ on ${\mathbb R}_{\ge 0}^n$ is not identically zero. \end{lemma}
\begin{proof}
We omit the proof - one develops the $\phi^{(\ell)}$ into power series in $e^{-x_1} , \ldots , e^{-x_n}$ and uses separation by generalized eigenvalues of the partial differentiation operators $\partial_{x_1} , \ldots , \partial_{x_n}$. \end{proof}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\author{Ciprian Demeter} \address{Department of Mathematics, Indiana University, Bloomington IN} \email{demeterc@@indiana.edu} \author{Shaoming Guo} \address{Department of Mathematics, Indiana University, Bloomington IN} \email{shaoguo@@indiana.edu}
\thanks{The first author is partially supported by the NSF grant DMS-1161752}
\title[Schr\"odinger maximal function]{Schr\"odinger maximal function estimates via the pseudoconformal transformation}
\begin{abstract} We present an alternative way to recover the recent result from \cite{LR} using the pseudoconformal transformation. \end{abstract} \maketitle
\section{Introduction} Recall that the solution of the Schr\"odinger equation \begin{equation} \label{e2} i\partial_t u(x,t)+\Delta u(x,t)=0,\;x\in\R^n,\;t\ge 0 \end{equation} with initial data $u_0\in L^2(\R^n)$ is given by
$$u(x,t)=e^{it\Delta}u_0(x)=\int_{\R^n}\widehat{u}_0(\xi)e^{2\pi ix\cdot\xi-4\pi^2it|\xi|^2}d\xi.$$ A fundamental open question for $n\ge 2$ is identifying the smallest Sobolev index $s>0$ for which $$\lim_{t\to 0}u(x,t)=u_0(x)\; a.e., \text{ for each }u_0\in H^s(\R^n).$$
The main goal of this note is to give an alternative argument for the following recent result of Luc\`a and Rogers, which proves a lower bound on the Sobolev regularity index $s$. \begin{theorem}\label{main}
Let $n\ge 2$ and $s<\frac{n}{2(n+2)}$. Then there exist $R_k\to\infty$ and $f_k\in L^2(\R^n)$ with $\widehat{f_k}$ supported in the annulus $|\xi|\sim R_k$ such that \begin{equation} \label{e1}
\lim_{k\to\infty}\frac{\|\sup_{0<t\lesssim 1}|e^{it\Delta}f_k(x)|\|_{L^2(B(0,1))}}{R_k^{s}\|f_k\|_{L^2(\R^n)}}=\infty. \end{equation} \end{theorem}
We use the pseudoconformal symmetry, according to which, if $u(x,t)$ solves \eqref{e2} then so does
$$v(x,t)=\frac1{t^{n/2}}\bar{u}(\frac{x}{t},\frac1{t})e^{i\frac{|x|^2}{4t}}.$$ Moreover, the initial data of the two solutions will have comparable $L^2$ norms. See the Appendix. We will start with a solution $u$ (the same as the one in \cite{LR}) that is big on a cartesian set $X\times T$ of $(x,t)$. The set $X$ will be a small neighborhood of a rescaled copy of ${\mathbb Z}^n$ inside $[-1,1]^n$, while $T$ will be a discrete lattice inside $t\sim 1$. The measure of $X$ will be significantly smaller than 1, of order $R_k^{-\alpha_n}$, for some $\alpha_n>0$. The property that our construction exploits is that the set $Y=\frac{X}{T}$ can be made much larger than $X$, in fact it can be made to have measure comparable to 1. Note that the new solution $v$ will now be big for each $x\in Y$ (for some $t$ depending on $x$). This will be enough to prove Theorem \ref{main}.
Let us compare our approach with other recent ones. Luc\`a and Rogers \cite{LR} use the Galilean symmetry, according to which if $u(x,t)$ solves \eqref{e2} then so does
$$v(x,t)=u(x-t\theta,t)e^{it\frac{|\theta|^2}{4}}e^{i\frac{x\cdot\theta}2}$$ for arbitrary $\theta\in\R^n$. Moreover, the initial data of the two solutions will have comparable $L^2$ norms. As mentioned before, they start with the same $u$, and thus have the same $X,T$. Their observation is that, for appropriate $\theta$, the set $Y=X-\theta T$ will have measure comparable to 1.
Bourgain \cite{Bo} constructs a solution $u$ which has two attributes. On the one hand, it is big on a cartesian product $X\times \{0\}$. So $T=\{0\}$. The second property of $u$ is that it is very symmetrical, almost invariant under a certain space-time translation. More precisely, for an appropriate $\nu\in\R^{n+1}$ \begin{equation} \label{e3} u(x,t)\approx u((x,t)+s\nu) \end{equation} will hold for all $x\in B(0,1)$ and all $t,s\sim 1$. The original small set $X$ where $u$ was large gets amplified from the fact that the $x$ projection of the set $$Y=(X\times \{0\})+[\frac1{10},10]\nu$$ has measure comparable to $1$.
In both our example and the one from \cite{LR}, the Fourier transform $\widehat{u}_0$ of the initial data is essentially the characteristic function of a small neighborhood of a rescaled (and truncated) copy of ${\mathbb Z}^n$. In Bourgain's construction, the mass lives on a small portion of this set, where lattice points are restricted to a sphere. The key is that the lift of this sphere to the paraboloid $(\xi,|\xi|^2)$ is a collection of points that live in a hyperplane $H\subset \R^{n+1}$. The existence of a nonzero vector $\nu\in H^{\perp}$ is what makes the remarkable symmetry \eqref{e3} possible.
In terms of the actual mathematics that is involved in proving that the enhanced set $Y$ has measure comparable to 1, the three methods described above are at least superficially different. Luc\`a and Rogers derive a quantitative version of the ergodic theorem involving the Funk-Hecke theorem. Bourgain uses a bit of Fourier analysis but his argument also has diophantine flavor. Our argument elaborates on a quantitative version of the multidimensional Dirichlet principle, which in its simplest form can be stated as follows. \begin{lemma} \label{8} Given $y_1,\ldots,y_n\in [0,1]$ and a real number $N\ge 1$, there is $1\le p\le N+2$ such that
$$\max_{1\le i\le n}\|py_i\|\le \frac1{N^{1/n}}.$$ \end{lemma}
Here and in the following, $\|x\|$ will denote the distance of $x$ to ${\mathbb Z}$. The proof of this lemma is an immediate application of pigeonholing.
It is hard to conjecture what the optimal $s$ in Theorem \ref{main} should be. The authors feel that the likeliest possibility is $s=\frac{n}{2(n+1)}$. If one runs a multilinear type Bourgain--Guth argument for this problem (as was done in \cite{Bo}), the $n+1$ linear term has a favorable estimate consistent with this value of $s$. Another interesting question is whether the optimal $s$ is the same for a larger class of curved hyper-surfaces $(\xi,\varphi(\xi))$ generalizing the paraboloid $(\xi,|\xi|^2)$. It is worth mentioning that Bourgain exhibits a surface
$$\varphi(\xi)=\langle A\xi,\xi\rangle+O(|\xi|^3)$$ with $A$ positive definite, for which a stronger result is proved: Theorem \ref{main} will hold even with $s<\frac{n-1}{2n}$ ($n\ge 3$) and $s<\frac{5}{16}$ ($n=2$).
\begin{ack*} The authors thank J. Bennett, R. Luc\`a, K. Rogers and A. Vargas for a few interesting discussions on this topic. \end{ack*}
\section{The main construction and the proof of Theorem \ref{main}} Via rescaling, Theorem \ref{main} will follow from the following result. \begin{theorem}\label{main1}
Let $n\ge 2$ and $s<\frac{n}{2(n+2)}$. Then there exist $R_k\to\infty$ and $v_k\in L^2(\R^n)$ with $\widehat{v_k}$ supported in the annulus $|\xi|\sim 1$ such that \begin{equation} \label{e25}
\lim_{k\to\infty}\frac{\|\sup_{0<t\lesssim R_k}|e^{it\Delta}v_k(x)|\|_{L^2(B(0,R_k))}}{R_k^{s}\|v_k\|_{L^2(\R^n)}}=\infty. \end{equation} \end{theorem} We will prove this in the end of the section, using some elementary number theoretical results derived in Section \ref{50}.
For $0<u<v$ define the annuli
$${\mathbb A}}\def\W{{\mathcal W}_{u,v}=\{x\in\R^n:\;u<|x|<v\}.$$ Fix $\sigma<\frac1{n+2}$. Fix a Schwartz function $\theta$ on $\R^n$ whose Fourier transform is supported inside ${\mathbb A}}\def\W{{\mathcal W}_{4^{-n-3},4\sqrt{n}}$ and equals 1 on ${\mathbb A}}\def\W{{\mathcal W}_{4^{-n-2},2\sqrt{n}}$. The next three lemmas are used to align the phases of an exponential sum so that the absolute value of the sum is comparable to the number of exponentials in the sum. \begin{lemma} \label{4} There exists $\epsilon_1>0$ so that for each $R$ large enough, the following holds:
For each $x\in {\mathbb A}}\def\W{{\mathcal W}_{4^{-n-2},2\sqrt{n}}$, each $t\in (0,1)$ and each $\xi'\in\R^n$ with $|\xi'|\le \epsilon_1R$ we have
$$|\int e^{2\pi i[(x-\frac{2t\xi'}{R})\cdot \xi-\frac{t}{R}|\xi|^2]}\theta(\xi)d\xi-1|<\frac12.$$ \end{lemma} \begin{proof}
Let $$\psi(\xi)=-2\pi [\frac{2t\xi'}{R}\cdot \xi+\frac{t}{R}|\xi|^2].$$ Use the fact that $$\int e^{2\pi ix\cdot \xi}\theta(\xi)d\xi=1.$$
Then estimate $$|\int e^{2\pi ix\cdot \xi}\theta(\xi)[e^{i\psi(\xi)}-1]d\xi|\le 2\int_{|\xi|>C}|\theta(\xi)|d\xi+2\sup_{|\xi|\le C}|\psi(\xi)|\int_{\R^n}|\theta|.$$
Choose first $C$ so that $$\int_{|\xi|>C}|\theta(\xi)|d\xi<\frac14.$$
Then note that $$\sup_{|\xi|\le C}|\psi(\xi)|\le 2\pi(\frac{C^2}{R}+2\epsilon_1C).$$
Choose $\epsilon_1$ so small that $$4\pi(\frac{C^2}{R}+2\epsilon_1C)\int_{\R^n}|\theta|<\frac14.$$ for all $R$ large enough. \end{proof} The following lemma is rather trivial. \begin{lemma} \label{5} Let $\Omega$ be a finite set. Consider $a_{\xi'},b_{\xi'}\in{\mathbb C}$ for $\xi'\in\Omega$ such that
$$\max|a_{\xi'}-1|\le \delta_1$$
$$\max|b_{\xi'}-1|\le \delta_2.$$ Then
$$|\sum_{\xi'\in \Omega}a_{\xi'}b_{\xi'}-|\Omega||\le |\Omega|(\delta_1\max|b_{\xi'}|+\delta_2\max|a_{\xi'}|+\delta_1\delta_2).$$ \end{lemma}
For $\epsilon_2>0$ small enough (depending only on $\theta$, as revealed in the proof of Proposition \ref{40}), define $$X:=(R^{\sigma-1}{\mathbb Z}^n+B(0,\frac{\epsilon_2}R))\cap {\mathbb A}}\def\W{{\mathcal W}_{4^{-n-2},2\sqrt{n}}$$ $$T:=(R^{2\sigma-1}{\mathbb Z})\cap (4^{-n-1},1),$$ and $$\Omega:=(R^{1-\sigma}{\mathbb Z}^n)\cap B(0,\epsilon_1R).$$ Define also the Fourier transform of the initial data $$\widehat{u_0}(\xi)=\sum_{\xi'\in\Omega}\theta(\xi-\xi').$$ Note that \begin{equation} \label{24}
\|u_0\|_2\sim R^{\frac{\sigma n}{2}}. \end{equation} and \begin{equation} \label{41} {\operatorname{supp}}}\def\det{{\operatorname{det}}}\def\MLK{{\textbf{MLK}}\; u_0\subset {\mathbb A}}\def\W{{\mathcal W}_{4^{-n-3},4\sqrt{n}}. \end{equation} The following is essentially proved in (3.2) from \cite{LR}. \begin{proposition} \label{40} We have the following lower bound for each $(x,t)\in X\times T$ \begin{equation} \label{20}
|e^{i\frac{t}{2\pi R}\Delta}u_0(x)|\gtrsim R^{\sigma n}. \end{equation} \end{proposition} \begin{proof} Note first that
$$e^{i\frac{t}{2\pi R}\Delta}u_0(x)=\sum_{\xi'\in\Omega}e^{2\pi i[x\cdot \xi'-\frac{t}{R}|\xi'|^2]}\int e^{2\pi i[(x-\frac{2t\xi'}{R})\cdot \xi-\frac{t}{R}|\xi|^2]}\theta(\xi)d\xi$$ One easily checks that for $(x,t)\in X\times T$ and $\xi'\in\Omega$ we have
$$x\cdot \xi'-\frac{t}{R}|\xi'|^2\in {\mathbb Z}+B(0,\epsilon_3),$$ where $\epsilon_3$ can be chosen as small as desired by choosing $\epsilon_2$ small enough. In particular, we can make sure that
$$|e^{2\pi i[x\cdot \xi'-\frac{t}{R}|\xi'|^2]}-1|<\min\{\frac1{100}, \frac{1}{100}\int|\theta|\}.$$
It suffices now to combine this with Lemma \ref{4} and Lemma \ref{5}, once we also note that $$|\Omega|\sim R^{\sigma n}.$$ \end{proof}
Recall that $u_0$ depends on $R$, so we might as well write $u_0=u_{0,R}$. Let now $u_{0, R}(x,t)=e^{it\Delta}u_{0,R}(x)$ and let
$$v_R(x,t)=\frac1{t^{n/2}}\bar{u}_R(\frac{x}{t},\frac1{t})e^{i\frac{|x|^2}{4t}}$$ be its pseudoconformal transformation. The proposition in the Appendix shows that $v_R$ solves the Schr\"odinger equation with some initial data that we call $v_{0,R}$.
We record the properties of $v_R$ in the following proposition. \begin{proposition} \label{11} We have for each large enough $R$ such that $R^{\sigma}$ is an integer \begin{equation}
|\{x\in B(0,R):\sup_{0<t\lesssim R}|v_R(x,t)|\gtrsim R^{\sigma n-\frac{n}2}\}|\gtrsim R^n \end{equation} \begin{equation}
\|v_{0,R}\|_2\sim R^{\frac{\sigma n}2} \end{equation} \begin{equation}
\frac{\|\sup_{0<t\lesssim R}|v_R(x,t)|\|_{L^2(B_R)}}{\|v_{0,R}\|_2}\gtrsim R^{\frac{\sigma n}2} \end{equation} \begin{equation} {\operatorname{supp}}}\def\det{{\operatorname{det}}}\def\MLK{{\textbf{MLK}} \;\widehat{v_{0,R}}\subset 4\pi ({\operatorname{supp}}}\def\det{{\operatorname{det}}}\def\MLK{{\textbf{MLK}} \;u_{0,R})\subset {\mathbb A}}\def\W{{\mathcal W}_{4^{-n-2}\pi, 16\pi\sqrt{n}}. \end{equation} \end{proposition} \begin{proof} The first property follows from \eqref{20} and \eqref{21}. The second one follows from \eqref{24} and \eqref{25}. The third one is a consequence of the first two. The fourth one also follows from \eqref{25} and \eqref{41}. \end{proof}
Let now $s<\frac{n}{2(n+2)}$. The proof of Theorem \ref{main1} for this $s$ will now immediately follow by choosing $\sigma<\frac1{n+2}$ such that $\frac{\sigma n}{2}>s$, and by using $v_k=v_{0, R_k}$ from Proposition \ref{11}, with $R_k^\sigma$ an integer that grows to infinity with $k$.
\section{Number theoretical considerations} \label{50} \begin{lemma} \label{7}
For each large enough real number $N$ there is $S=S_N\subset [0,1]^n$ with $|S|\ge \frac34$ such that for each $(y_1,\ldots,y_n)\in S$ there exists $p\in [4^{-n-1}N,N+2]$ satisfying \begin{equation} \label{9}
\max_{1\le i\le n}\|py_i\|\le \frac1{N^{\frac1n}}. \end{equation} \end{lemma} \begin{proof} Using Lemma \ref{8}, we know that \eqref{9} holds for each $(y_1,\ldots,y_n)\in [0,1]^n$, if we allow $p\in [1,N+2]$. We need an upper bound for those $(y_1,\ldots,y_n)$ corresponding to $1\le p\le 4^{-n-1}N$. For each $p$ define
$$A_p=\{(y_1,\ldots,y_n)\in [0,1]^n:\;\max_{1\le i\le n}\|py_i\|\le \frac1{N^{\frac1n}}\}=$$
$$=\bigcup_{0\le k_i\le p}\{(y_1,\ldots,y_n)\in [0,1]^n:\;\max_{1\le i\le n}|py_i-k_i|\le \frac1{N^{\frac1n}}\}.$$ The crude estimate
$$|A_p|\le (p+1)^n(\frac{2}{N^{\frac1n}p})^n<\frac{4^n}{N}$$ leads to
$$|\bigcup_{1\le p\le 4^{-n-1}N}A_p|\le \frac14.$$ \end{proof} \begin{proposition} Assume $R^{\sigma}$ is a large enough integer. Let $$\frac{XR}{T}=\{\frac{xR}{t}:x\in X, t\in T\}.$$ Then \begin{equation} \label{21}
|\frac{XR}{T}\cap B(0,R)|\gtrsim R^n. \end{equation} \end{proposition} \begin{proof} It suffices to prove that
$$|\frac{X}{T}\cap [0,1]^n|\ge \frac12.$$
This can be written as $|U|\ge \frac12$ where
$$U=\{x\in[0,1]^n:\max_{1\le i\le n}|x_i-R^{-\sigma}\frac{k_i}{p}|\le \frac{\epsilon_2}R,\text{ with }$$$$4^{-n-2}R^{1-\sigma}\le |k|\le 2\sqrt{n}R^{1-\sigma},\;4^{-n-1}R^{1-2\sigma}\le p\le R^{1-2\sigma} \}.$$ This will follow if we prove that the larger set (we have dropped the restriction on $k_i$)
$$V=\{x\in[0,1]^n:\max_{1\le i\le n}|x_i-R^{-\sigma}\frac{k_i}{p}|\le \frac{\epsilon_2}R,\text{ with }4^{-n-1}R^{1-2\sigma}\le p\le R^{1-2\sigma} \}$$ satisfies
$|V|\ge \frac34.$ Indeed, note first that the restriction
$$|k|\le 2\sqrt{n}R^{1-\sigma}$$is redundant, as the inequality $\max|k_i|\le R^{1-\sigma}+1\le 2R^{1-\sigma}$ is forced by the combination of $x\in[0,1]^n$, $\max_{1\le i\le n}|x_i-R^{-\sigma}\frac{k_i}{p}|\le \frac{\epsilon_2}R$ and $p\le R^{1-2\sigma}$. Second, note that
$$W=\{x\in[0,1]^n:\max_{1\le i\le n}|x_i-R^{-\sigma}\frac{k_i}{p}|\le \frac{\epsilon_2}R,\text{ with }$$$$|k|\le 4^{-n-2}R^{1-\sigma},\;4^{-n-1}R^{1-2\sigma}\le p\le R^{1-2\sigma} \}$$
satisfies $W\subset[0,\frac12]^n$ and thus $|W|<\frac14$.
We next focus on showing that $|V|\ge \frac34$. The inequality $$\max_{1\le i\le n}|x_i-R^{-\sigma}\frac{k_i}{p}|\le \frac{\epsilon_2}R$$ can be written as
$$\max_{1\le i\le n}\|p\{R^{\sigma}x_i\}\|\le \frac{\epsilon_2pR^{\sigma}}R,$$ where $\{z\}$ is the fractional part of $z$. Note that given the lower bound $p\ge 4^{-n-1}R^{1-2\sigma}$ and that $\sigma<\frac1{n+2}$, for $R$ large enough we have $$\frac{\epsilon_2pR^{\sigma}}R\ge \frac1{(R^{1-2\sigma}-2)^{\frac1n}}.$$ Let $N:=R^{1-2\sigma}-2$. Then $V_0\subset V$ where
$$V_0=\{x\in[0,1]^n:\max_{1\le i\le n}\|p\{R^{\sigma}x_i\}\|\le \frac1{N^{\frac1n}},\text{ for some }4^{-n-1}N\le p\le N+2 \}.$$ Since $R^{\sigma}$ is an integer, the map $$(x_1,\ldots,x_n)\mapsto (\{R^{\sigma}x_1\},\ldots, \{R^{\sigma}x_n\})$$
is a measure preserving transformation on $[0,1]^n$. The fact that $|V_0|\ge \frac34$ now follows from Lemma \ref{7}. \end{proof}
\section{Appendix} We record below the following classical result. See for example \cite{Ta}, page 72. \begin{proposition} If $u(x,t)$ solves \eqref{e2} then so does its pseudoconformal transformation
$$v(x,t)=\frac1{t^{n/2}}\bar{u}(\frac{x}{t},\frac1{t})e^{i\frac{|x|^2}{4t}}.$$ Moreover, the initial data $u_0(x)=u(x,0)$, $v_0(x)=v(x,t)$ are related by the formula \begin{equation} \label{25} \widehat{v_0}(y)=Cu_0(4\pi y). \end{equation}
In particular, $$\|u_0\|_2\sim\|v_0\|_2.$$ \end{proposition}
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Existence of weak solutions to
stochastic heat equations driven
by truncated $lpha$-stable white noises with
non-Lipschitz coefficients}
\begin{abstract}
We consider a class of stochastic heat equations driven by truncated $\alpha$-stable white noises for $\alpha\in(1,2)$ with noise coefficients that are continuous but not necessarily Lipschitz and satisfy globally linear growth conditions. We prove the existence of weak solution, taking values in two different forms under different conditions, to such an equation using a weak convergence argument on solutions to the approximating stochastic heat equations. More precisely, for $\alpha\in(1,2)$ there exists a measure-valued weak solution. However, for $\alpha\in(1,5/3)$ there exists a function-valued weak solution,
and in this case we further show that for $p\in(\alpha,5/3)$ the uniform $p$-th moment in $L^p$-norm of the weak solution is finite, and that the weak solution is uniformly stochastic continuous in $L^p$ sense.
\noindent\textbf{Keywords:} Non-Lipschitz noise coefficients; Stochastic heat equations; Truncated $\alpha$-stable white noises; Uniform $p$-th moment; Uniform stochastic continuity.
\noindent{{\bf MSC Classification (2020):} Primary: 60H15; Secondary: 60F05, 60G17} \end{abstract}
\section{Introduction} \label{sec1} In this paper we study the existence of weak solution to the following non-linear stochastic heat equation \begin{equation} \label{eq:originalequation1} \left\{\begin{array}{lcl} \dfrac{\partial u(t,x)}{\partial t}=\dfrac{1}{2} \dfrac{\partial^2u(t,x)}{\partial x^2}+ \varphi(u(t-,x))\dot{L}_{\alpha}(t,x), && (t,x)\in (0,\infty) \times(0,L),\\[0.3cm] u(0,x)=u_0(x),&&x\in[0,L],\\[0.3cm] u(t,0)=u(t,L)=0,&& t\in[0,\infty), \end{array}\right. \end{equation} where $L$ is an arbitrary positive constants, $\dot{L}_{\alpha}$ denotes a truncated $\alpha$-stable white noise on $[0,\infty)\times[0,L]$ with $\alpha\in(1,2)$, the noise coefficient $\varphi:\mathbb{R}\rightarrow \mathbb{R}$ satisfies the hypotheses given below, and the initial function $u_0$ is random and measurable.
Before studying the equation of particular form (\ref{eq:originalequation1}), we first consider a general stochastic heat equation \begin{equation} \label{GeneralSPDE} \dfrac{\partial u(t,x)}{\partial t}=\dfrac{1}{2}\dfrac{\partial^2u(t,x)}{\partial x^2}+G(u(t,x))+H(u(t,x))\dot{F}(t,x), \quad t\geq0, x\in\mathbb{R}, \end{equation} in which $G:\mathbb{R}\rightarrow \mathbb{R}$ is Lipschitz continuous, $H:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and $\dot{F}$ is a space-time white noise.
When $\dot{F}$ is a Gaussian white noise, there is a growing literature on stochastic partial differential equations (SPDEs for short) related to (\ref{GeneralSPDE})
such as the stochastic Burgers equations (see, e.g., Bertini and Cancrini \cite{Bertini:1994}, Da Prato et al. \cite{Daprato:1994}), SPDEs with reflection (see, e.g., Zhang \cite{Zhang:2016}), Parabolic Anderson Model (see, e.g., {G\"{a}rtner and Molchanov \cite{Gartner:1990}), etc. In particular, such a SPDE arises from super-processes (see, e.g., Konno and Shiga \cite{Konno:1988}, Dawson \cite{Dawson:1993} and Perkins \cite{P1991} and references therein). For $G\equiv 0$ and $H(u)=\sqrt{u}$, the solution to (\ref{GeneralSPDE}) is the density field of a one-dimensional super-Brownian motion. For $H(u)=\sqrt{u(1-u)}$ (stepping-stone model in population genetics), Bo and Wang \cite{Bo:2011} considered a stochastic interacting model consisting of equations (\ref{GeneralSPDE}) and proved the existence of weak solution to the system by using a weak convergence argument.
In the case that $\dot{F}$ is a Gaussian colored noise that is white in time and colored in space, for continuous function $H$ satisfying the linear growth condition, Sturm \cite{Sturm:2003} proved the existence of a pair $(u,F)$ satisfying (\ref{GeneralSPDE}), the so-called weak solution, by first establishing the existence and uniqueness of lattice systems of SDEs driven by correlated Brownian motions with non-Lipschitz diffusion coefficients that describe branching particle systems in random environment in which the motion process has a discrete Laplacian generator and the branching mechanism is affected by a colored Gaussian random field, and then applying an approximation procedure. Xiong and Yang \cite{Xiong:2023} proved the existence of weak solution $(u,F)$ to (\ref{GeneralSPDE}) in a finite spatial domain with different boundary conditions by considering the weak limit of a sequence of approximating SPDEs of (\ref{GeneralSPDE}). They further proved the existence and uniqueness of the strong solution under additional H\"{o}lder continuity assumption on $H$.
If $\dot{F}$ is a L\'{e}vy space-time white noise with Lipschitz continuous coefficient $H$, Albeverio et al. \cite{Albeverio:1998} first proved the existence and uniqueness of the solution when $\dot{F}$ is a Poisson white noise. Applebaum and Wu \cite{Applebaum:2000} extended the results to a general L\'{e}vy space-time white noise. For a stochastic fractional Burgers type non-linear equation that is similar to equation (\ref{GeneralSPDE}) and driven by L\'{e}vy space-time white noise on multidimensional space variables, we refer to Wu and Xie \cite{Wu:2012} and references therein.
In particular, when $\dot{F}$ is an $\alpha$-stable white noise for $\alpha\in (0,1)\cup(1,2)$, Balan \cite{Balan:2014} studied SPDE (\ref{GeneralSPDE}) with $G\equiv 0$ and Lipschitz coefficient $H$ on a bounded domain in $\mathbb{R}^d$ with zero initial condition and Dirichlet boundary, and proved the existence of random field solution $u$ for the given noise $\dot{F}$ (the so-called strong solution). The approach in \cite{Balan:2014} is to first solve the equation with truncated noise (by removing the big jumps, the jumps size exceeds a fixed value $K$, from $\dot{F}$), yielding a solution $u_K$, and then show that for $N\geq K$ the solutions $u_N=u_K$ on the event $t\leq\tau_K$, where $\{\tau_K\}_{K\geq1}$ is a sequence of stopping times which tends to infinity as $K$ tends to infinity. Such a localization method which is also applied in Peszat and Zabczyk \cite{Peszat:2006} to show the existence of weak Hilbert-space valued solution.
For $\alpha\in(1,2)$, Wang et al. \cite{Wang:2023} studied the existence and pathwise uniqueness of strong function-valued solution of (\ref{GeneralSPDE}) with Lipschitz coefficient $H$ using a localization method, and showed a comparison principle of solutions to such equation with different initial functions and drift coefficients. Yang and Zhou \cite{Yang:2017} found sufficient conditions on pathwise uniqueness of solutions to a class of SPDEs (\ref{GeneralSPDE}) driven by $\alpha$-stable white noise without negative jumps and with non-decreasing H\"{o}lder continuous noise coefficient $H$. But the existence of weak solution to (\ref{GeneralSPDE}) with general non-decreasing H\"{o}lder continuous noise coefficient is left open. For stochastic heat equations driven by general heavy-tailed noises with Lipschitz noise coefficients, we refer to Chong \cite{Chong:2017} and references therein.
When $G=0$, $H(u)=u^{\beta}$ with $0<\beta<1$ (non-Lipschitz continuous) in (\ref{GeneralSPDE}) and $\dot{F}$ is an $\alpha$-stable ($\alpha\in(1,2)$) white noise on $[0,\infty)\times\mathbb{R}$ without negative jumps, it is shown in Mytnik \cite{Mytnik:2002} that for $0<\alpha\beta<3$ there exists a weak solution $(u,F)$ satisfying (\ref{GeneralSPDE}) by constructing a sequence of approximating processes that is tight with its limit solving the associated martingale problem, and that in the case of $\alpha\beta=1$ the weak uniqueness of solution to (\ref{GeneralSPDE}) holds. The pathwise uniqueness is shown in \cite{Yang:2017} for $\alpha\beta=1$ and $1<\alpha<\sqrt{5}-1$.
For $\alpha$-stable colored noise $\dot{F}$ without negative jumps and with H\"{o}lder continuous coefficient $H$, Xiong and Yang \cite{Xiong:2019} proved the existence of week solution $(u,F)$ to (\ref{GeneralSPDE}) by showing the weak convergence of solutions to SDE systems on rescaled lattice with discrete Laplacian and driven by common stable random measure, which is similar to \cite{Sturm:2003}. In both \cite{Sturm:2003} and \cite{Xiong:2019} the dependence of colored noise helps with establishing the existence of weak solution.
Inspired by work in the above mentioned literature, we are interested in the stochastic heat equation (\ref{eq:originalequation1}) in which the noise coefficient $\varphi$ satisfies the following more general hypothesis: \begin{hypothesis} \label{Hypo} $\varphi:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and globally linear growth, and there exists a sequence of Lipschitz continuous functions $\varphi^n:\mathbb{R}\rightarrow \mathbb{R}$ such that \begin{itemize} \item[(i)] $\varphi^n$ uniformly converges to $\varphi$ as $n\rightarrow\infty$;
\item[(ii)] for each $n\geq1$, there exists a constant $C_n$ such that $$\vert\varphi^n(x)-\varphi^n(y)\vert \leq C_n\vert x-y\vert,\,\,\forall x,y\in\mathbb{R};$$ \end{itemize} \end{hypothesis}
The main contribution of this paper is to prove the existence and regularity of weak solutions to equation (\ref{eq:originalequation1}) under Hypothesis \ref{Hypo}. To this end, we consider two types of weak solutions that are measure-valued and function-valued, respectively. In addition, we also study the uniform $p$-moment and uniform stochastic continuity
of the weak solution to equation (\ref{eq:originalequation1}). In the case that $\varphi$ is Lipschitz continuous, the existence of the solution can be usually obtained by standard Picard iteration (see, e.g., Dalang et al. \cite{Dalang:2009}, Walsh \cite{Walsh:1986}) or Banach fixed point principle (see, e.g., Truman and Wu \cite{Truman:2003}, Bo and Wang \cite{Bo:2006}). We thus mainly consider the case that $\varphi$ is non-Lipschitz continuous.
Since the classical approaches of Picard iteration and Banach fixed point principle fail for SPDE (\ref{eq:originalequation1}) with non-Lipschitz $\varphi$, to prove the existence of a weak solution $(u,L_{\alpha})$ to (\ref{eq:originalequation1}), we first construct an approximating SPDE sequence with Lipschitz continuous noise coefficients $\varphi^n$, and prove the existence and uniqueness of strong solutions to the approximating SPDEs. We then proceed to show that the sequence of solution is tight in appropriate spaces. Finally, we prove that there exists a weak solution of (\ref{eq:originalequation1}) by using a weak convergence procedure.
The rest of this paper is organized as follows. In the next section, we introduce some notation and the main theorems on the existence, uniform $p$-moment and uniform stochastic continuity of weak solution to (\ref{eq:originalequation1}). Section \ref{sec3} is devoted to the proof of the existence of measure-valued weak solution to (\ref{eq:originalequation1}). In Section \ref{sec4}, for $\alpha\in(1,5/3)$ we prove that there exists a weak solution to (\ref{eq:originalequation1}) as an $L^p$-valued process with $p\in(\alpha,5/3)$, and that the weak solution has the finite uniform $p$-th moment and the uniform stochastic continuity in the $L^p$ norm with $p\in(\alpha,5/3)$.
\section{Notation and main results} \label{sec2} \subsection{Notation} \label{sec2.1} Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq0}, \mathbb{P})$ be a complete probability space with filtration $(\mathcal{F}_t)_{t\geq0}$ satisfying the usual conditions, and let $ N(dt,dx,dz): [0,\infty)\times[0,L]\times \mathbb{R}\setminus\{0\}\rightarrow \mathbb{N} \cup\{0\}\cup\{\infty\} $ be a Poisson random measure on $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq0}, \mathbb{P})$ with intensity measure $dtdx\nu_{\alpha}(dz)$, where $dtdx$ denotes the Lebesgue measure on $[0,\infty)\times[0,L]$ and the jump size measure $\nu_{\alpha}(dz)$ for $ \alpha\in(1,2) $ is given by \begin{align} \label{eq:smalljumpsizemeasure} \nu_{\alpha}(dz):=(c_{+}z^{-\alpha-1}1_{(0,K]}(z)+c_{-}(-z)^{-\alpha-1}1_{[-K,0)}(z) )dz, \end{align} where $c_{+}+c_{-}=1$ and $K>0$ is an arbitrary constant. Define \begin{align*} \tilde{N}(dt,dx,dz):= N(dt,dx,dz)-dtdx\nu_{\alpha}(dz). \end{align*} Then $\tilde{N}(dt,dx,dz)$ is the compensated Poisson random measure (martingale measure) on $[0,\infty)\times[0,L]\times \mathbb{R}\setminus\{0\}$. As in Balan \cite[Section 5]{Balan:2014}, define a martingale measure \begin{align} \label{def:stablenoise} L_{\alpha}(dt,dx):= \int_{\mathbb{R}\setminus\{0\}}z\tilde{N}(dt,dx,dz) \end{align} for $(t, x)\in [0,\infty)\times[0,L]$. Then the corresponding distribution-valued derivative $\{\dot{L}_{\alpha}(t,x):t\in[0,\infty),x\in[0,L]\}$ is a truncated $\alpha$-stable white noise. Write $\mathcal{G}^{\alpha}$ for the class of almost surely $\alpha$-integrable random functions defined by \begin{align*} \mathcal{G}^{\alpha}:=\left\{f\in\mathbb{B}: \int_0^t\int_0^L\vert f(s,x)\vert ^{\alpha}dxds<\infty, \mathbb{P}\text{-a.s.}\,\,\text{for all} \,\,t\in[0,\infty)\right\}, \end{align*} where $\mathbb{B}$ is the space of progressively measurable functions on $[0,\infty)\times[0,L]\times\Omega$. Then it holds by Mytnik \cite[Section 5]{Mytnik:2002} that the stochastic integral with respect to $\{L_{\alpha}(dx,ds)\}$ is well defined for all $f\in \mathcal{G}^{\alpha}$.
Throughout this paper, $C$ denotes the arbitrary positive constant whose value might vary from line to line. If $C$ depends on some parameters such as $p,T$, we denote it by $C_{p,T}$.
Let $G_t(x,y)$ be the fundamental solution of heat equation $\frac{\partial u}{\partial t} =\frac{1}{2}\frac{\partial^2 u}{\partial x^2}$ on the domain $[0,\infty)\times[0,L]\times[0,L]$ with Dirichlet boundary conditions (the subscript $t$ is not a derivative but a variable). Its explicit formula (see, e.g., Feller \cite[Page 341]{Feller:1971}) is given by \begin{equation*} G_t(x,y)=\dfrac{1}{\sqrt{2\pi t}}\sum_{k=-\infty}^{+\infty}\left\{ \exp\left(-\dfrac{(y-x+2kL)^2}{2t}\right) -\exp\left(-\dfrac{(y+x+2kL)^2}{2t} \right)\right\} \end{equation*} for $t\in(0,\infty),x,y\in[0,L]$; and $\lim_{t\downarrow0}G_t(x,y)=\delta_y(x)$,
where $\delta$ is the Dirac delta distribution. Moreover, it holds by Xiong and Yang \cite[Lemmas 2.1-2.3]{Xiong:2023} that for $s,t\in[0,\infty)$ and $x,y,z\in[0,L]$ \begin{equation} \label{eq:Greenetimation0} G_t(x,y)=G_t(y,x),\,\,
\int_0^L|G_t(x,y)|dy+\int_0^L|G_t(x,y)|dx\leq C,
\end{equation} \begin{equation} \label{eq:Greenetimation1}
\int_0^LG_s(x,y)G_t(y,z)dy=G_{t+s}(x,z), \end{equation} \begin{equation} \label{eq:Greenetimation2}
\int_0^L\vert G_t(x,y)\vert ^pdy\leq Ct^{-\frac{p-1}{2}},\,\, p\geq1. \end{equation}
Given a topological space $V$, let $D([0,\infty),V)$ be the space of c\`{a}dl\`{a}g paths from $[0,\infty)$ to $V$ equipped with the Skorokhod topology. For given $p\geq1$ we denote by $v_t\equiv\{v(t,\cdot),t\in[0,\infty)\}$ the $L^p([0,L])$-valued process equipped with norm \begin{equation*} \vert\vert v_t\vert\vert_{p}=\left(\int_0^L\vert v(t,x)\vert^pdx\right)^{\frac{1}{p}}. \end{equation*} For any $p\geq1$ and $T>0$ let $L_{loc}^p([0,\infty)\times[0,L])$ be the space of measurable functions $f$ on $[0,\infty)\times[0,L])$ such that \begin{equation*} \vert \vert f\vert \vert _{p,T}=\left(\int_0^T\int_0^L \vert f(t,x)\vert ^pdxdt\right)^{\frac{1}{p}}<\infty,\,\, \forall\,\, 0<T<\infty. \end{equation*}
Let $B([0,L])$ be the space of all Borel functions on $[0,L]$, and let $\mathbb{M}([0,L])$ be the space of finite Borel measures on $[0,L]$ equipped with the weak convergence topology. For any $f\in B([0,L])$ and $\mu\in\mathbb{M}([0,L])$ define $ \langle f,\mu\rangle:=\int_0^L f(x)\mu(dx) $ whenever it exists. With a slight abuse of notation, for any $f,g\in B([0,L])$ we also denote by $ \langle f,g\rangle=\int_0^L f(x)g(x)dx. $
\subsection{Main results} \label{sec2.2} By a solution to equation (\ref{eq:originalequation1}) we mean a process $u_t\equiv\{u(t,\cdot),t\in[0,\infty)\}$ satisfying the following weak (variational) form equation: \begin{align} \label{eq:variationform} \langle u_t,\psi\rangle &=\langle u_0,\psi\rangle+\dfrac{1}{2}\int_0^t \langle u_s,\psi{''}\rangle ds +\int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}} \varphi(u(s-,x))\psi(x)z\tilde{N}(ds,dx,dz) \end{align} for all $t\in[0,\infty)$ and for any $\psi\in C^{2}([0,L])$ with $\psi(0)=\psi(L)=\psi^{'}(0)=\psi^{'}(L)=0$ or equivalently satisfying the following mild form equation: \begin{align} \label{eq:mildform} u(t,x)= \int_0^LG_t(x,y)u_0(y)dy +\int_0^{t+}\int_0^L \int_{\mathbb{R}\setminus\{0\}}G_{t-s}(x,y) \varphi(u(s-,y))z\tilde{N}(ds,dy,dz) \end{align} for all $t\in [0, \infty)$ and for a.e. $x\in [0,L]$, where the last terms in above equations follow from (\ref{def:stablenoise}). For the equivalence between the weak form (\ref{eq:variationform}) and mild form (\ref{eq:mildform}), we refer to Walsh \cite{Walsh:1986} and references therein.
We first give the definition (see also in Mytnik \cite{Mytnik:2002}) of a weak solution to stochastic heat equation (\ref{eq:originalequation1}). \begin{definition} Stochastic heat equation (\ref{eq:originalequation1}) has a weak solution with initial function $u_0$ if there exists a pair $(u,L_{\alpha})$ defined on some filtered probability space such that ${L}_{\alpha}$ is a truncated $\alpha$-stable martingale measure on $[0,\infty)\times[0,L]$ and $(u,L_{\alpha})$ satisfies either equation (\ref{eq:variationform}) or equation (\ref{eq:mildform}). \end{definition}
We now state the main theorems in this paper. The first theorem is on the existence of weak solution in $D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$ with $p\in(\alpha,2]$
that was first considered in
Mytnik \cite{Mytnik:2002}. \begin{theorem} \label{th:mainresult} If the initial function $u_0$ satisfies $\mathbb{E}[\vert\vert u_0\vert\vert_p^p]<\infty$ for some $p\in(\alpha,2]$, then under
{\rm Hypothesis \ref{Hypo}} there exists a weak solution $(\hat{u}, {\hat{L}}_{\alpha})$ to equation (\ref{eq:originalequation1}) defined on a filtered probability space $(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}_t\}_{t\geq0}, \hat{\mathbb{P}})$ such that \begin{itemize} \item[\rm (i)] $\hat{u}\in D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$; \item[\rm (ii)] ${\hat{L}}_{\alpha}$ is a truncated $\alpha$-stable martingale measure with the same distribution as ${L}_{\alpha}$. \end{itemize} Moreover, for any $T>0$ we have \begin{equation} \label{eq:momentresult} \hat{\mathbb{E}}\left[\vert\vert\hat{u}\vert\vert_{p,T}^p\right]= \hat{\mathbb{E}}\left[\int_0^T\vert\vert\hat{u}_t\vert\vert_p^pdt\right]<\infty. \end{equation} \end{theorem}
The proof of Theorem \ref{th:mainresult} is deferred to Section \ref{sec3}.
Under additional assumption on $\alpha$, we can show that there exists a weak solution in $D([0,\infty),L^p([0,L]))$, $p\in(\alpha,5/3)$
with better regularity. \begin{theorem} \label{th:mainresult2} Suppose that $\alpha\in (1,5/3)$. If the initial function $u_0$
satisfies $\mathbb{E}[||u_0||_p^p]<\infty$ for some $p\in(\alpha,5/3)$, then under {\rm Hypothesis \ref{Hypo}} there exists a weak solution $(\hat{u}, {\hat{L}}_{\alpha})$ to equation (\ref{eq:originalequation1}) defined on a filtered probability space $(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}_t\}_{t\geq0}, \hat{\mathbb{P}})$ such that \begin{itemize} \item[\rm (i)] $\hat{u}\in D([0,\infty),L^p([0,L]))$; \item[\rm (ii)] ${\hat{L}}_{\alpha}$ is a truncated $\alpha$-stable martingale measure with the same distribution as ${L}_{\alpha}$. \end{itemize} Furthermore, for any $T>0$ we have the following uniform $p$-moment and uniform stochastic continuity, that is,
\begin{equation} \label{eq:momentresult2} \hat{\mathbb{E}}\left[\sup_{0\leq t\leq T}\vert\vert\hat{u}_t\vert\vert_p^p\right]<\infty,
\end{equation} and that for each $0\leq h\leq\delta$ \begin{equation} \label{eq:timeregular} \lim_{\delta\rightarrow0}\hat{\mathbb{E}}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\vert\vert \hat{u}_{t+h}-\hat{u}_t\vert\vert_p^p\right]=0.
\end{equation} \end{theorem}
The proof of Theorem \ref{th:mainresult2} is deferred to Section \ref{sec4}.
We present a specific stochastic heat equation to illustrate our results in the following example. \begin{example} Given $0<\beta<1$, consider the equation (\ref{eq:originalequation1}) with $\varphi(u)=\vert u\vert^{\beta}$ for $u\in\mathbb{R}$, that is, \begin{equation*} \label{eq:example} \left\{\begin{array}{lcl} \dfrac{\partial u(t,x)}{\partial t}=\dfrac{1}{2} \dfrac{\partial^2u(t,x)}{\partial x^2}+ \vert u(t-,x)\vert^{\beta}\dot{L}_{\alpha}(t,x), && (t,x)\in (0,\infty) \times(0,L),\\[0.3cm] u(0,x)=u_0(x),&&x\in[0,L],\\[0.3cm] u(t,0)=u(t,L)=0,&& t\in[0,\infty). \end{array}\right. \end{equation*} It is clear that $\vert u \vert^{\beta}$ is a non-Lipschitz continuous function with globally linear growth. We can construct a sequence of Lipschitz continuous functions $(\varphi^n)_{n\geq1}$ of the form \begin{equation*} \varphi^n(u)=(\vert u \vert\vee\varepsilon_n)^{\beta}, u\in\mathbb{R},
\end{equation*} where $\varepsilon_n\downarrow0$ as $n\uparrow\infty$, such that $\varphi^n$ satisfies {\rm Hypothesis \ref{Hypo}}. One can then apply {\rm Theorems \ref{th:mainresult} and \ref{th:mainresult2}} to establish the existence of weak solutions to the above stochastic heat equation. \end{example}
Finally, we provide some discussions on our main results in the following remarks.
\begin{remark} Note that the globally linear growth of $\varphi$ in {\rm Hypothesis \ref{Hypo}} guarantees the global existence of weak solutions. One can remove this condition if one only needs the existence of a weak solution up to the explosion time. On the other hand, the uniqueness of the solution to equation (\ref{eq:originalequation1}) is still an open problem because $\varphi$ is non-Lipschitz continuous. \end{remark}
\begin{remark} The weak solutions of equation (\ref{eq:originalequation1}) in {\rm Theorems \ref{th:mainresult}} and {\rm\ref{th:mainresult2}} are proved by showing the tightness of the approximating solution sequence $(u^n)_{n\geq1}$ of equation (\ref{eq:approximatingsolution}); see {\rm Propositions \ref{th:tightnessresult}} and {\rm\ref{prop:tightnessresult2}} in {\rm Sections \ref{sec3}} and {\rm \ref{sec4}}, respectively. To show that the equation (\ref{eq:originalequation1}) has a function-valued weak solution, it is necessary to restrict $\alpha\in(1,5/3)$ due to a technical reason that Doob's maximal inequality can not be directly applied to show the uniform $p$-moment estimate of $(u^n)_{n\geq1}$ that is key to the proof of the tightness for $(u^n)_{n\geq1}$. To this end, we apply the factorization method in {\rm Lemma \ref{lem:uniformbound}} for transforming the stochastic integral such that the uniform $p$-moment of $(u^n)_{n\geq1}$ can be obtained. In order to remove this restriction and consider the case of $\alpha\in(1,2)$, we apply another tightness criteria, i.e., {\rm Lemma \ref{lem:tightcriterion0}}, to show the tightness of $(u^n)_{n\geq1}$. However, the weak solution of equation (\ref{eq:originalequation1}) is a measure-valued process in this situation. We also note that the existence of function-valued weak solution of equation (\ref{eq:originalequation1}) in the case of $\alpha\in[5/3,2)$ is still an unsolved problem. \end{remark}
\begin{remark} If we remove the restriction of the bounded jumps for the $\alpha$-stable white noise $\dot{L}_{\alpha}$ in equation (\ref{eq:originalequation1}), the jump size measure $\nu_{\alpha}(dz)$ in (\ref{eq:smalljumpsizemeasure}) becomes \begin{align*} \nu_{\alpha}(dz)=(c_{+}z^{-\alpha-1}1_{(0,\infty)}(z)+c_{-}(-z)^{-\alpha-1}1_{(-\infty,0)}(z) )dz \end{align*} for $\alpha\in(1,2)$ and $c_{+}+c_{-}=1$. As in Wang et al. \cite[Lemma 3.1]{Wang:2023} we can construct a sequence of truncated $\alpha$-stable white noise $\dot{L}^K_{\alpha}$ with the jumps size measure given by (\ref{eq:smalljumpsizemeasure}) and a sequence of stopping times $(\tau_K)_{K\geq1}$ such that \begin{equation} \label{eq:stoppingtimes1} \lim_{K\rightarrow+\infty} \tau_K=\infty,\,\,\mathbb{P}\text{-a.s.}. \end{equation} Similar to equation (\ref{eq:originalequation1}), for given $K\geq1$, we can consider the following non-linear stochastic heat equation \begin{equation} \label{eq:truncatedSHE} \left\{\begin{array}{lcl} \dfrac{\partial u_K(t,x)}{\partial t}=\dfrac{1}{2} \dfrac{\partial^2u_K(t,x)}{\partial x^2}+ \varphi(u_K(t-,x))\dot{L}^K_{\alpha}(t,x), && (t,x)\in (0,\infty) \times(0,L),\\[0.3cm] u_K(0,x)=u_0(x),&&x\in[0,L],\\[0.3cm] u_K(t,0)=u_K(t,L)=0,&& t\in[0,\infty). \end{array}\right. \end{equation}
If $\varphi$ is Lipschitz continuous, similar to the proof of {\rm Proposition \ref{th:Approximainresult}} in Wang et al. \cite[Proposition 3.2]{Wang:2023} one can show that there exists a unique strong solution $u_K=\{u_K(t,\cdot),t\in[0,\infty)\}$ to equation (\ref{eq:truncatedSHE}) by using the Banach fixed point principle. On the other hand, by Wang et al. \cite[Lemma 3.4]{Wang:2023}, it holds for each $K\leq N$ that \begin{align*} u_K=u_N \,\,\mathbb{P}\text{-} \,a.s. \,\,\text{on}\,\{t<\tau_K\}. \end{align*} By setting \begin{align*} u=u_K,\,0\leq t<\tau_K, \end{align*} and by the fact (\ref{eq:stoppingtimes1}), we obtain the strong (weak) solution $u$ to equation (\ref{eq:originalequation1}) with noise of unbounded jumps via letting $K\uparrow+\infty$.
If $\varphi$ is non-Lipschitz continuous, for any $K\geq1$, {\rm Theorem \ref{th:mainresult}} or {\rm Theorem \ref{th:mainresult2}} shows that there exists a weak solution $(\hat{u}_K, {\hat{L}}^K_{\alpha})$ to equation (\ref{eq:truncatedSHE}) defined on a filtered probability space $(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}_t\}_{t\geq0}, \hat{\mathbb{P}})_K$. However, we can not show that for each $K\leq N$ \begin{align*} (\hat{u}_K, {\hat{L}}^K_{\alpha})=(\hat{u}_N, {\hat{L}}^N_{\alpha}) \,\,\mathbb{P}\text{-} \,a.s. \,\,\text{on}\,\,\{t<\tau_K\} \end{align*} due to the non-Lipschitz continuity of $\varphi$. Therefore, we do not know whether there exists a common probability space $(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}_t\}_{t\geq0}, \hat{\mathbb{P}})$ on which all of the weak solutions $((\hat{u}_K, {\hat{L}}^K_{\alpha}))_{K\geq1}$ are defined. Hence, the localization method in Wang et al. \cite{Wang:2023} becomes invalid, and the existence of the weak solution to equation (\ref{eq:originalequation1}) with untruncated $\alpha$-stable noise remains an unsolved problem. \end{remark}
\section{Proof of Theorem \ref{th:mainresult}}\label{sec3} The proof of Theorem \ref{th:mainresult} proceeds in the following three steps.
We first construct a sequence of the approximating SPDEs with globally Lipschitz continuous noise coefficients $(\varphi^n)_{n\geq1}$ satisfying Hypothesis \ref{Hypo}, and show that for each fixed $n\geq1$ there exists a unique strong solution $u^n$ in $D([0,\infty),L^p([0,L])$ with $p\in(\alpha,2]$ of the approximating SPDE; see Proposition \ref{th:Approximainresult}. We then prove that the approximating solution sequence $(u^n)_{n\geq1}$ is tight in both $D([0,\infty),\mathbb{M}([0,L]))$ and $L_{loc}^p([0,\infty)\times[0,L])$ for all $p\in(\alpha,2]$; see Proposition \ref{th:tightnessresult}. Finally, we proceed to show that there exists a weak solution $(\hat{u},\hat{L}_{\alpha})$ to equation (\ref{eq:originalequation1}) defined on another probability space $(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}_t\}_{t\geq0}, \hat{\mathbb{P}})$ by applying a weak convergence argument.
For each fixed $n\geq1$, we construct the approximate SPDE of the form \begin{equation} \label{eq:approximatingsolution} \left\{\begin{array}{lcl} \dfrac{\partial u^n(t,x)}{\partial t}=\dfrac{1}{2}\dfrac{\partial^2u^n(t,x)} {\partial x^2}+\varphi^n(u^n(t-,x))\dot{L}_{\alpha}(t,x),&& (t,x)\in (0,\infty) \times (0,L),\\[0.3cm] u^n(0,x)=u_0(x),&&x\in [0,L], \\[0.3cm] u^n(t,0)=u^n(t,L)=0,&&t\in[0,\infty), \end{array}\right. \end{equation} where the coefficient $\varphi^n$ satisfies Hypothesis \ref{Hypo}.
Given $n\geq1$, by a solution to equation (\ref{eq:approximatingsolution}) we mean a process $u^n_t\equiv\{u^n(t,\cdot), t\in[0,\infty)\}$ satisfying the following weak form equation: \begin{align} \label{eq:approxivariationform} \langle u^n_t,\psi\rangle &=\langle u_0,\psi\rangle+\dfrac{1}{2}\int_0^t \langle u^n_s,\psi{''}\rangle ds
+\int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}} \varphi^n(u^n(s-,x))\psi(x)z\tilde{N}(ds,dx,dz) \end{align} for all $t\in[0,\infty)$ and for any $\psi\in C^{2}([0,L])$ with $\psi(0)=\psi(L)=\psi^{'}(0)=\psi^{'}(L)=0$ or equivalently satisfying the following mild form equation: \begin{align} \label{mildformapproxi0} u^n(t,x)&=\int_0^LG_t(x,y)u_0(y)dy
+\int_0^{t+}\int_0^L \int_{\mathbb{R}\setminus\{0\}}G_{t-s}(x,y) \varphi^n(u^n(s-,y))z\tilde{N}(ds,dy,dz) \end{align} for all $t\in [0, \infty)$ and for a.e. $x\in [0, L]$.
We now present the definition (see also in Wang et al. \cite{Wang:2023}) of a strong solution to stochastic heat equation (\ref{eq:approximatingsolution}). \begin{definition} Given $p\geq 1$, the stochastic heat equation (\ref{eq:approximatingsolution}) has a strong solution in $D([0,\infty),L^p([0,L]))$
with initial function $u_0$ if for a given truncated $\alpha$-stable martingale measure $L_{\alpha}$ there exists a process $u^n_t\equiv\{u^n(t,\cdot),t\in[0,\infty)\}$ in $D([0,\infty),L^p([0,L]))$ such that either equation (\ref{eq:approxivariationform}) or equation (\ref{mildformapproxi0}) holds. \end{definition}
Note that for each $n\geq1$ the noise coefficient $\varphi^n$ is not only Lipschitz continuous but also of globally linear growth. Indeed, for a given $\epsilon>0$ and $n_0\in\mathbb{N}$ large enough, Hypothesis \ref{Hypo} (i) and the globally linear growth of $\varphi$ imply that \begin{equation} \label{eq:glo-lin-growth}
|\varphi^n(x)|\leq|\varphi^n(x)-\varphi(x)|
+|\varphi(x)|\leq\epsilon+C(1+|x|),\,\, \forall n\geq n_0,\, \forall x\in\mathbb{R}. \end{equation} Therefore, we can use the classical Banach fixed point principle to show the existence and pathwise uniqueness of the strong solution to equation (\ref{eq:approximatingsolution}). Since the proof is standard, we just state the main result in the following proposition. For more details of the proof, we refer to Wang et al. \cite[Proposition 3.2]{Wang:2023} and references therein. Also note that the same method was applied in Truman and Wu \cite{Truman:2003} and in Bo and Wang \cite{Bo:2006} where the stochastic Burgers equation and the stochastic {C}ahn-{H}illiard equation driven by L\'{e}vy space-time white noise were studied, respectively.
\begin{proposition} \label{th:Approximainresult} Given any $n\geq1$, if the initial function $u_0$
satisfies $\mathbb{E}[||u_0||_p^p]<\infty$ for some $p\in(\alpha,2]$, then under {\rm Hypothesis \ref{Hypo}} there exists a pathwise unique strong solution $u^n_t\equiv\{u^n(t,\cdot),t\in[0,\infty)\}$ to equation (\ref{eq:approximatingsolution}) such that for any $T>0$ \begin{equation} \label{eq:Approximomentresult} \sup_{n\geq1}\sup_{0\leq t\leq T}\mathbb{E}\left[\vert \vert u^n_t\vert \vert _p^p\right]<\infty.
\end{equation} \end{proposition}
\begin{remark} By {\rm Hypothesis \ref{Hypo} (ii)}, (\ref{eq:glo-lin-growth}) and estimate (\ref{eq:Approximomentresult}), the stochastic integral on the right-hand side of (\ref{mildformapproxi0}) is well defined. \end{remark}
We are going to prove that the approximating solution sequence $(u^n)_{n\geq1}$ is tight in both $D([0,\infty),\mathbb{M}([0,L]))$ and $L_{loc}^p([0,\infty)\times[0,L])$ for all $p\in(\alpha,2]$ by using the following tightness criteria; see, e.g., Xiong and Yang \cite[Lemma 2.2]{Xiong:2019}. Note that this tightness criteria can be obtained by Ethier and Kurtz \cite[Theorems 3.9.1, 3.9.4 and 3.2.2]{Ethier:1986}. \begin{lemma} \label{lem:tightcriterion0} Given a complete and separable metric space $E$, let $(X^n=\{X^n(t),t\in[0,\infty)\})_{n\geq1}$ be a sequence of stochastic processes with sample paths in $D([0,\infty),E)$, and let $C_a$ be a subalgebra and dense subset of $C_b(E)$ (the bounded continuous functions space on $E$). Then the sequence $(X^n)_{n\geq1}$ is tight in $D([0,\infty),E)$ if both of the following conditions hold: \begin{itemize} \item[\rm (i)] For every $\varepsilon>0$ and $T>0$ there exists a compact set $\Gamma_{\varepsilon,T}\subset E$ such that \begin{equation} \label{eq:tightcriterion1} \inf_{n\geq1}\mathbb{P}[X^n(t)\in\Gamma_{\varepsilon,T}\,\, \text{for all}\,\, t\in[0,T] ]\geq1-\varepsilon. \end{equation} \item[\rm (ii)] For each $f\in C_a$, there exists a process $g_n\equiv\{g_n(t),t\in[0,\infty)\}$ such that \begin{equation*} f(X^n(t))-\int_0^tg_n(s)ds \end{equation*} is an $(\mathcal{F}_t)$-martingale and \begin{align} \label{eq:tight-moment} \sup_{0\leq t\leq T}\mathbb{E}\left[\vert f(X^n(t))\vert +\vert g_n(t)\vert \right]<\infty \end{align} and \begin{align} \label{eq:tight-moment2} \sup_{n\geq1}\mathbb{E}\left[\left(\int_0^T\vert g_n(t)\vert ^qdt\right)^{\frac{1}{q}}\right]<\infty \end{align} for each $n\geq1, T>0$ and $q>1$. \end{itemize} \end{lemma}
Before showing the tightness of solution sequence $(u^n)_{n\geq1}$, we first find a uniform moment estimate in the following lemma. \begin{lemma} \label{le:uniformbounded} For each $n\geq1$ let $u^n$ be the strong solution to equation (\ref{eq:approximatingsolution}) given by {\rm Proposition \ref{th:Approximainresult}}. Then for given $T>0$ and $\psi\in C^{2}([0,L])$ with $\psi(0)=\psi(L)=\psi^{'}(0)=\psi^{'}(L)=0$ and $\vert \psi^{''}(x)\vert \leq C\psi(x),x\in[0,L]$, we have for $p\in(\alpha,2]$ that \begin{align} \label{eq:p-uniform} \sup_{n\geq1}\mathbb{E}\left[\sup_{0\leq t\leq T} \left\vert \int_{0}^Lu^n(t,x)\psi(x)dx\right\vert ^p\right]<\infty. \end{align} \end{lemma} \begin{proof} By (\ref{eq:approxivariationform}), it holds that for each $n\geq1$ \begin{align*} \mathbb{E}\left[\sup_{0\leq t\leq T} \left\vert \int_{0}^Lu^n(t,x)\psi(x)dx\right\vert ^p\right]\leq C_p(A_1+A_2+A_3), \end{align*} where \begin{align*} A_1&=\mathbb{E}\left[\left\vert \int_0^Lu_0(x)\psi(x)dx\right\vert ^p\right], \\ A_2&=\mathbb{E}\left[\sup_{0\leq t\leq T} \left\vert \int_0^{t}\int_0^Lu^n(s,x)\psi^{''}(x)dxds \right\vert ^p\right], \\ A_3&=\mathbb{E}\left[\sup_{0\leq t\leq T} \left\vert \int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}\varphi^n(u^n(s-,x))\psi(x)z\tilde{N}(ds,dx,dz) \right\vert ^p\right]. \end{align*}
For $p\in(\alpha,2]$ we separately estimate $A_1$, $A_2$ and $A_3$ as follows. For $A_1$, it holds by H\"{o}lder's inequality that \begin{align*} A_1&\leq C_p\left(\int_0^L\vert \psi(x)\vert ^{\frac{p}{p-1}}dx\right)^{\frac{p(p-1)}{p}} \mathbb{E}\left[\int_0^L\vert u_0(x)\vert ^pdx\right] \leq C_p\mathbb{E}[\vert \vert u_0\vert \vert _p^p] \leq C_p \end{align*} due to $\psi\in C^{2}([0,L])$ and $\mathbb{E}[\vert \vert u_0\vert \vert _p^p]<\infty$.
For $A_2$, it holds by $\vert \psi^{''}(x)\vert \leq C\psi(x),x\in[0,L]$ and H\"{o}lder's inequality that \begin{align*} A_2&\leq C_p\mathbb{E}\left[\sup_{0\leq t\leq T} \left\vert \int_0^{t}\int_0^Lu^n(s,x)\psi(x)dxds \right\vert ^p\right] \leq C_{p,T}\int_0^{T} \mathbb{E}\left[\sup_{0\leq r\leq s} \left\vert \int_0^Lu^n(r,x)\psi(x)dx \right\vert ^p\right]ds. \end{align*}
For $A_3$, the Doob maximal inequality and the Burkholder-Davis-Gundy inequality
imply that \begin{align*}
A_3&\leq C_p \mathbb{E}\left[\left|\int_0^{T}\int_0^L\int_{\mathbb{R}\setminus\{0\}}\vert \varphi^n(u^n(s-,x))
\psi(x)z\vert^2N(ds,dx,dz)\right|^{\frac{p}{2}}\right] \\ &\leq C_p \mathbb{E}\left[\int_0^{T}\int_0^L\int_{\mathbb{R}\setminus\{0\}}\vert \varphi^n(u^n(s-,x)) \psi(x)z\vert ^pN(ds,dx,dz)\right] \\ &=C_p \mathbb{E}\left[\int_0^{T}\int_0^L\int_{\mathbb{R}\setminus\{0\}}\vert \varphi^n(u^n(s,x)) \psi(x)z\vert ^pdsdx\nu_{\alpha}(dz)\right], \end{align*} where the second inequality follows from the fact that \begin{align} \label{eq:element-inequ} \left\vert \sum_{i=1}^ka_i^2\right\vert ^{\frac{q}{2}}\leq \sum_{i=1}^{k}\vert a_i\vert ^q \end{align} for $a_i\in\mathbb{R}, k\geq1$, and $q\in(0,2]$. By (\ref{eq:smalljumpsizemeasure}), it holds that for $p>\alpha$ \begin{align} \label{eq:jumpestimate} \int_{\mathbb{R}\setminus\{0\}}\vert z\vert ^p\nu_{\alpha} (dz)=c_{+}\int_0^Kz^{p-\alpha-1}dz+c_{-}\int_{-K}^0(-z)^{p-\alpha-1}dz =\frac{K^{p-\alpha}}{p-\alpha}, \end{align} then there exists a constant $C_{p,K,\alpha}$ such that \begin{align*} A_3\leq C_{p,K,\alpha} \mathbb{E}\left[\int_0^{T}\int_0^L\vert \varphi^n(u^n(s,x))\psi(x)\vert ^pdsdx\right]. \end{align*}
By (\ref{eq:glo-lin-growth}),
$\psi\in C^{2}([0,L])$ and (\ref{eq:Approximomentresult}) in Proposition \ref{th:Approximainresult},
it is easy to see that \begin{align*} A_3&\leq C_{p,K,\alpha,T}\left(1+\sup_{0\leq s\leq T} \mathbb{E}[\vert \vert u^n_s\vert \vert _p^p]\right)\leq C_{p,K,\alpha,T}. \end{align*} Combining the estimates $A_1,A_2$ and $A_3$, we have \begin{align*} \mathbb{E}&\left[\sup_{0\leq t\leq T} \left\vert \int_{0}^Lu^n(t,x)\psi(x)dx\right\vert ^p\right] \leq C_{p,K,\alpha,T}+C_{p,T}\int_0^{T}\mathbb{E}\left[\sup_{0\leq r\leq s} \left\vert \int_0^Lu^n(r,x)\psi(x)dx\right\vert ^p\right]ds \end{align*} Therefore, it holds by Gronwall's lemma that for $p\in(\alpha,2]$ \begin{align*} \sup_{n\geq1}\mathbb{E}\left[\sup_{0\leq t\leq T} \left\vert \int_{0}^Lu^n(t,x)\psi(x)dx\right\vert ^p\right]<\infty, \end{align*} which completes the proof.
$\Box$
\end{proof}
Note that for any function $v\in L^q[0,L]$ with $q\geq1$, we can identify $L^q([0,L])$ as a subset of $\mathbb{M}([0,L])$ by using the following correspondence $$v(x)\mapsto v(x)dx.$$ Then for each $n\geq1$ we can identify the $D([0,\infty),L^p([0,L]))$-valued random variable $u^n$ as a $D([0,\infty),\mathbb{M}([0,L]))$-valued random variable (still denoted by $u^n$). We now show the tightness of $(u^n)_{n\geq1}$ in the following proposition. \begin{proposition} \label{th:tightnessresult} The solution sequence $(u^n)_{n\geq1}$ to equation (\ref{eq:approximatingsolution}) given by {\rm Proposition \ref{th:Approximainresult}} is tight in both $D([0,\infty),\mathbb{M}([0,L]))$ and $L_{loc}^p([0,\infty)\times [0,L])$ for $p\in(\alpha,2]$. Let $u$ be an arbitrary limit point of $u^n$. Then \begin{align} \label{eq:tightresult} u\in D([0,\infty),\mathbb{M}([0,L])) \cap L_{loc}^p([0,\infty)\times [0,L]) \end{align} for $p\in(\alpha,2]$. \end{proposition} \begin{proof} For each $n\geq1, t\geq0$ and $\psi\in C^{2}([0,L])$ with $\psi(0)=\psi(L)=\psi^{'}(0)=\psi^{'}(L)=0$ and $\vert \psi^{''}(x)\vert \leq C\psi(x),x\in[0,L]$, let us define \begin{align*} \langle u^n,\psi\rangle:=\langle u_t^n,\psi\rangle=\int_0^Lu^n(t,x)\psi(x)dx. \end{align*} We first prove that the sequence $(\langle u^n,\psi\rangle)_{n\geq1}$ is tight in $D([0,\infty),\mathbb{R})$ by using Lemma \ref{lem:tightcriterion0}. It is easy to see that the condition (i) in Lemma \ref{lem:tightcriterion0} can be verified by Lemma \ref{le:uniformbounded} . In the following we mainly verify the condition (ii) in Lemma \ref{lem:tightcriterion0}.
For each $f\in C_b^2(\mathbb{R})$ ($f,f^{'},f^{''}$ are bounded and uniformly continuous) with compact supports, it holds by (\ref{eq:approxivariationform}) and It\^{o}'s formula that \begin{align} \label{eq:ito} f(\langle u_t^n,\psi\rangle)&=f(\langle u_0^n,\psi\rangle) +\int_0^tf^{'}(\langle u_s^n,\psi\rangle)\langle u_s^n,\psi^{''}\rangle) ds \nonumber\\ &\quad+\int_0^t\int_0^L\int_{\mathbb{R}\setminus\{0\}} \mathcal{D}(\langle u_s^n,\psi\rangle,\varphi^n(u^n(s,x))\psi(x)z) dsdx\nu_{\alpha}(dz)+\text{mart.}, \end{align} where $ \mathcal{D}(u,v)=f(u+v)-f(u)-vf^{'}(u) $ for $u,v\in\mathbb{R}$. Since $f,f^{'},f^{''}$ are bounded and $\vert \psi^{''}(x)\vert \leq C\psi(x),x\in[0,L]$, then \begin{align} \label{eq:estimate1} \vert f^{'}(\langle u_s^n,\psi\rangle)\langle u_s^n,\psi^{''}\rangle\vert \leq C\left\vert \int_0^Lu^n({s},x)\psi(x)dx\right\vert . \end{align} By Taylor's formula, one can show that $\vert \mathcal{D}(u,v)\vert \leq C(\vert v\vert\wedge \vert v\vert ^2)$, which also implies that $\vert \mathcal{D}(u,v)\vert \leq C(\vert v\vert \wedge \vert v\vert ^p)$ for $p\in(\alpha,2]$. Thus we have for $p\in(\alpha,2]$, \begin{align} \label{eq:estimate2} &\int_0^L\int_{\mathbb{R}\setminus\{0\}} \vert \mathcal{D}(\langle u_s^n,\psi\rangle,\varphi^n(u^n(s,x))\psi(x)z)\vert dx\nu_{\alpha}(dz) \nonumber\\ &\leq C\left(\int_{\mathbb{R}\setminus\{0\}}\vert z\vert \wedge \vert z\vert ^p\nu_{\alpha}(dz)\right) \int_0^L(\vert \varphi^n(u^n(s,x))\psi(x)\vert +\vert \varphi^n(u^n(s,x))\psi(x)\vert ^p)dx \nonumber\\ &\leq C_{p,K,\alpha}\int_0^L(\vert \varphi^n(u^n(s,x))\psi(x)\vert +\vert \varphi^n(u^n(s,x))\psi(x)\vert ^p)dx, \end{align} where by (\ref{eq:smalljumpsizemeasure}), \begin{align*} \int_{\mathbb{R}\setminus\{0\}}\vert z\vert \wedge \vert z\vert ^p\nu_{\alpha}(dz) &=c_{+}\int_0^1z^{p-\alpha-1}dz +c_{-}\int_{-1}^0(-z)^{p-\alpha-1}dz +c_{+}\int_1^Kz^{-\alpha}dz \\ &\quad +c_{-}\int_{-K}^{-1}(-z)^{-\alpha}dz \\ &=\frac{1}{p-\alpha}+\dfrac{1-K^{1-\alpha}}{\alpha-1}\leq C_{p,K,\alpha}. \end{align*}
For given $n\geq1$ let us define \begin{align*} g_n(s)&:=f^{'}(\langle u_s^n,\psi\rangle)\langle u_s^n,\psi^{''}\rangle) +\int_0^L\int_{\mathbb{R}\setminus\{0\}} \mathcal{D}(\langle u_s^n,\psi\rangle,\varphi^n(u^n(s,x))\psi(x)z) dx\nu_{\alpha}(dz). \end{align*} By (\ref{eq:ito}), it is easy to see that \begin{align*} f(\langle u_t^n,\psi\rangle)-\int_0^{t}g_n(s)ds \end{align*} is an $(\mathcal{F}_t)$-martingale.
Now we verify the moment estimates (\ref{eq:tight-moment}) and (\ref{eq:tight-moment2}) of the condition (ii) in Lemma \ref{lem:tightcriterion0}. For each $t\in[0,T]$, it holds by the boundedness of $f$, estimates
(\ref{eq:estimate1})-(\ref{eq:estimate2}) and (\ref{eq:glo-lin-growth}) that \begin{align*} \mathbb{E}\left[\vert f(\langle u_t^n,\psi\rangle)\vert +\vert g_n(t)\vert \right] &\leq C\left(1+\mathbb{E}\left[\left\vert \int_0^Lu^n(t,x)\psi(x)dx\right\vert \right]\right) \\ &\quad +C_{p,K,\alpha}\mathbb{E}\left[\int_0^L(\vert \psi(x)\vert +\vert u^n(t,x)\psi(x)\vert )dx\right] \\ &\quad+C_{p,K,\alpha}\mathbb{E}\left[\int_0^L(\vert \psi(x)\vert ^p+ \vert u^n(t,x)\psi(x)\vert ^p)dx\right]. \end{align*} Since $\psi\in C^2([0,L])$ implies that $\psi$ is bounded, then it holds by H\"{o}lder's inequality and (\ref{eq:p-uniform}) that for $p\in(\alpha,2]$
\begin{align*} \mathbb{E}\left[\vert f(\langle u_t^n,\psi\rangle)\vert +\vert g_n(t)\vert \right] &\leq C_{p,K,\alpha}\left(1+\left(\sup_{0\leq t\leq T} \mathbb{E}[\vert \vert u^n_t\vert \vert_p^p]\right)^{\frac{1}{p}} +\sup_{0\leq t\leq T} \mathbb{E}[\vert \vert u^n_t\vert \vert_p^p]\right), \end{align*} and so by (\ref{eq:Approximomentresult}), \begin{align*} \sup_{0\leq t\leq T}\mathbb{E}\left[\vert f(\langle u_t^n,\psi\rangle)\vert +\vert g_n(t)\vert \right]<\infty, \end{align*} which verifies the estimate (\ref{eq:tight-moment}).
To verify (\ref{eq:tight-moment2}), it suffices to show that for each $n\geq1$ \begin{align*} \mathbb{E}\left[\int_0^T\vert g_n(t)\vert ^qdt\right]<\infty \end{align*} for some $q>1$. By the estimates (\ref{eq:estimate1})-(\ref{eq:estimate2}) and (\ref{eq:glo-lin-growth}), we have \begin{align*} \mathbb{E}\left[\int_0^T\vert g_n(t)\vert ^qdt\right] &\leq C_{q}\mathbb{E}\left[\int_0^T\left\vert \int_0^Lu^n(t,x)\psi(x)dx \right\vert ^qdt\right] \\ &\quad+C_{p,q,K,\alpha}\mathbb{E}\left[\int_0^T\left\vert \int_0^L(\vert \psi(x)\vert + \vert u^n(t,x)\psi(x)\vert )dx \right\vert ^qdt\right] \\ &\quad+C_{p,q,K,\alpha}\mathbb{E}\left[\int_0^T\left\vert \int_0^L(\vert \psi(x)\vert ^p+ \vert u^n(t,x)\psi(x)\vert ^p)dx \right\vert ^qdt\right]. \end{align*} Taking $1<q<2/p$, the H\"{o}lder inequality and boundedness of $\psi$ imply that \begin{align*} \mathbb{E}\left[\int_0^T\vert g_n(t)\vert ^qdt\right] &\leq C_{p,q,K,\alpha,T}\left(1+\sup_{0\leq t\leq T}\mathbb{E}[\vert \vert u^n_t\vert\vert_q^q] +\sup_{0\leq t\leq T}\mathbb{E} \left[\vert \vert u^n_t\vert \vert_{pq}^{pq}\right]\right), \end{align*} and so by (\ref{eq:Approximomentresult}), \begin{align*} \sup_{n\geq1}\mathbb{E}\left[\int_0^T\vert g_n(t)\vert ^qdt\right]<\infty, \end{align*} which verifies the estimate (\ref{eq:tight-moment2}).
Therefore, for each $\psi\in C^{2}([0,L])$ with $\psi(0)=\psi(L)=\psi^{'}(0)=\psi^{'}(L)=0$ and $\vert \psi^{''}(x)\vert \leq C\psi(x),x\in[0,L]$, the sequence $(\langle u^n,\psi\rangle)_{n\geq1}$ is tight in $D([0,\infty),\mathbb{R})$, and so it holds by Mitoma's theorem (see, e.g., Walsh \cite[pp.361--365]{Walsh:1986}) that $(u^n)_{n\geq1}$ is tight in $D([0,\infty),\mathbb{M}([0,L]))$.
On the other hand, by (\ref{eq:Approximomentresult}) we have for each $T>0$ \begin{align*} \sup_{n\geq1}\mathbb{E} \left[\int_0^T\int_0^L\vert u^n(t,x)\vert ^pdxdt\right] \leq C_T\sup_{n\geq1}\sup_{0\leq t\leq T} \mathbb{E}[\vert \vert u^n_t\vert \vert _p^p]<\infty \end{align*} for $p\in(\alpha,2]$. The Markov's inequality implies that for each $\varepsilon>0,T>0$ there exists a constant $C_{\varepsilon,T}$ such that \begin{align*} \sup_{n\geq1}\mathbb{P}\left[\int_0^T\int_0^L\vert u^n(t,x)\vert ^pdxdt>C_{\epsilon,T}\right]<\varepsilon \end{align*} for $p\in(\alpha,2]$. Therefore, the sequence $(u^n)_{n\geq1}$ is also tight in $L^p_{loc}([0,\infty)\times[0,L])$ for $p\in(\alpha,2]$, and the conclusion (\ref{eq:tightresult}) holds.
$\Box$
\end{proof}
\begin{Tproof}\textbf{~of Theorem \ref{th:mainresult}.} We are going to prove Theorem \ref{th:mainresult} by applying weak convergence arguments. For each $n\geq1$, let $u^n$ be the strong solution
of equation (\ref{eq:approximatingsolution}) given by Proposition \ref{th:Approximainresult}.
It can also be regarded as an element in
$D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$
with $p\in(\alpha,2]$.
By Proposition \ref{th:tightnessresult}, there exists
a $D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$-valued random variable $u$ such that $u^n$ converges to $u$ in distribution in
$D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$ for $p\in(\alpha,2]$.
On the other hand, the Skorokhod Representation Theorem (see, e.g., Either and Kurtz \cite[Theorem 3.1.8]{Ethier:1986}) yields that there exists another filtered probability space $(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}}_t)_{t\geq0},\hat{\mathbb{P}})$ and on it a further subsequence
$(\hat{u}^n)_{n\geq1}$ and $\hat{u}$ which have the same distribution as $(u^n)_{n\geq1}$ and $u$, so that $\hat{u}^n$ almost surely converges to $\hat{u}$ in $D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$ for $p\in(\alpha,2]$.
For each $t\geq0,n\geq1$ and any test function $\psi\in C^{2}([0,L])$ with $\psi(0)=\psi(L)=0$ and $\psi^{'}(0)= \psi^{'}(L)=0$, let us define \begin{align*} \hat{M}^n_{t}(\psi)&:=\int_0^L\hat{u}^n(t,x) \psi(x)dx-\int_0^L\hat{u}_0(x) \psi(x)dx -\frac{1}{2}\int_0^{t} \int_0^L\hat{u}^n(s,x)\psi^{''}(x)dxds. \end{align*} Since $\hat{u}^n$ almost surely converges to $\hat{u}$ in the Skorokhod topology as $n\rightarrow\infty$, then \begin{align} \label{eq:M^n_t} \hat{M}^n_{t}(\psi)& \overset{\mathbf{\hat{P}}\text{-a.s.}}{\longrightarrow} \int_0^L\hat{u}(t,x)\psi(x)dx- \int_0^L\hat{u}_0(x)\psi(x)dx -\frac{1}{2}\int_0^{t} \int_0^L\hat{u}(s,x) \psi^{''}(x) dxds \end{align} in the Skorokhod topology as $n\rightarrow\infty$.
By (\ref{eq:approxivariationform}) and the fact that $\hat{u}^n$ has the same distribution as $u^n$ for each $n\geq1$, we have \begin{align*} \hat{M}^n_{t}(\psi) \overset{D}=& \int_0^Lu^n(t,x) \psi(x)dx-\int_0^Lu_0(x)\psi(x)dx -\frac{1}{2}\int_0^{t}\int_0^L u^n(s,x)\psi^{''}(x)dxds \nonumber\\ =&\int_0^{t+} \int_0^L\int_{\mathbb{R}\setminus\{0\}}\psi(x) \varphi^n(u^n(s-,x))z\tilde{N}(ds,dx,dz), \end{align*} where $\overset{D}=$ denotes the identity in distribution. The Burkholder-Davis-Gundy inequality, (\ref{eq:element-inequ})-(\ref{eq:jumpestimate}) and (\ref{eq:glo-lin-growth}) imply that for $p\in(\alpha,2]$ \begin{align*} \hat{\mathbb{E}}[\vert \hat{M}^n_{t}(\psi)\vert ^p] &=\mathbb{E} \left[\left\vert \int_0^{t+} \int_0^L\int_{\mathbb{R}\setminus\{0\}}\psi(x) \varphi^n(u^n(s-,x))z\tilde{N}(ds,dx,dz) \right\vert ^p\right] \\ &\leq C_p\mathbb{E} \left[\int_0^t\int_0^L \int_{\mathbb{R}\setminus\{0\}} \vert \psi(x)\vert ^p(1+\vert u^n(s,x)\vert )^p \vert z\vert ^pdsdx\nu_{\alpha}(dz) \right] \\ &\leq C_{p,K,\alpha,T} \left(\int_0^L\vert \psi(x)\vert ^pdx+\bigg\vert \sup_{x\in[0,L]}\psi(x)\bigg\vert ^p \sup_{0\leq t\leq T}\mathbb{E}\left[\vert \vert u^n_t\vert \vert _p^p\right]\right). \end{align*} Then by $\psi\in C^{2}([0,L])$ and (\ref{eq:Approximomentresult}), we have for each $T>0$ \begin{align*} \sup_{n\geq1}\sup_{0\leq t\leq T} \hat{\mathbb{E}}[\vert \hat{M}^n_{t}(\psi)\vert ^p]<\infty. \end{align*} Therefore, it holds by (\ref{eq:M^n_t}) that there exists an $(\hat{\mathcal{F}}_t)$-martingale $\hat{M}_{t}(\psi)$ such that $\hat{M}^n_{t}(\psi)$ converges weakly to $\hat{M}_{t}(\psi)$ as $n\rightarrow\infty$, and for each $t\geq0$ \begin{align} \label{martingle1} \hat{M}_{t}(\psi)&= \int_0^L\hat{u}(t,x)\psi(x)dx- \int_0^L \hat{u}_0(x)\psi(x)dx-\frac{1}{2}\int_0^{t} \int_0^L\hat{u}(s,x) \psi^{''}(x)dxds. \end{align}
By Hypothesis \ref{Hypo} (i), the quadratic variation of $\{\hat{M}^n_{t}(\psi),t\in[0,\infty)\}$ satisfies that \begin{align*} \langle \hat{M}^n(\psi), \hat{M}^n(\psi) \rangle_t&=\int_0^{t}\int_0^L \int_{\mathbb{R}\setminus\{0\}}\varphi^n({u}^n(s,x))^2 \psi(x)^2z^2dsdx\nu_{\alpha}(dz) \\ &\overset{D}=\int_0^{t}\int_0^L \int_{\mathbb{R}\setminus\{0\}}\varphi^n(\hat{u}^n(s,x))^2 \psi(x)^2z^2dsdx\nu_{\alpha}(dz) \\ &\overset{\mathbb{P}-a.s.} \rightarrow\int_0^{t}\int_0^L \int_{\mathbb{R}\setminus\{0\}}\varphi(\hat{u}(s,x))^2 \psi(x)^2z^2dsdx\nu_{\alpha}(dz),\,\,t\in[0,T], \end{align*}
as $n\rightarrow\infty$. We denote by $\{\langle \hat{M}(\psi),\hat{M}(\psi)\rangle_t,t\in[0,\infty)\}$ the quadratic variation process \begin{align*} \langle \hat{M}(\psi),\hat{M}(\psi)\rangle_t=\int_0^{t} \int_0^L\int_{\mathbb{R}\setminus\{0\}}\varphi(\hat{u}(s,x))^2 \psi(x)^2z^2dsdx\nu_{\alpha}(dz),\,\,t\geq0. \end{align*} Similar to Konno and Shiga \cite[Lemma 2.4]{Konno:1988}, $\langle \hat{M}(\psi),\hat{M}(\psi)\rangle_t$ corresponds to an orthonormal martingale measure $\hat{M}(dt,dx,dz)$ defined on the filtered probability space $(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}_t})_{t\geq0}, \hat{\mathbb{P}})$ in the sense of Walsh \cite[Chapter 2]{Walsh:1986} whose quadratic measure is given by \begin{align*} \varphi(\hat{u}(t,x))^2z^2dtdx\nu_{\alpha}(dz). \end{align*} Let $\{\dot{\bar{L}}_{\alpha}(t,x):t\in[0,\infty),x\in[0,L]\}$ be another truncated $\alpha$-stable white noise, defined possibly on $(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}_t})_{t\geq0}, \hat{\mathbb{P}})$, independent of $\hat{M}(dt,dx,dz)$ and define \begin{align*} \hat{L}_{\alpha}(t,\psi)&:= \int_0^{t+}\int_{0}^L\int_{\mathbb{R}\setminus\{0\}} \dfrac{1}{\varphi(\hat{u}(s-,x))} 1_{\{\varphi(\hat{u}(s-,x))\neq0\}}\psi(x)z\hat{M}(ds,dx,dz) \\ &\quad+\int_0^{t+}\int_{0}^L \psi(x) 1_{\{\varphi(\hat{u}(s-,x))=0\}} \bar{L}_{\alpha}(ds,dx). \end{align*} Then $\{\hat{L}_{\alpha}(t,\psi): \,t\in[0,\infty),\, \psi\in C^2([0, L]), \psi(0)=\psi(L)=0, \psi^{'}(0)=\psi^{'}(L)=0\}$ determines a truncated $\alpha$-stable white noise $\dot{\hat{L}}_{\alpha}(t,x)$ on $(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}_t})_{t\geq0}, \hat{\mathbb{P}})$ with the same distribution as $\dot{L}_{\alpha}(t,x)$ such that \begin{align*} \hat{M}_t(\psi)&=\int_0^{t+} \int_0^L\varphi(\hat{u}(s-,x)) \psi(x) \hat{L}_{\alpha}(ds,dx) =\int_0^{t+} \int_0^L\int_{\mathbb{R}\setminus\{0\}}\varphi(\hat{u}(s-,x)) \psi(x)z\widetilde{\hat{N}}(ds,dx,dz), \end{align*} where $\widetilde{\hat{N}}(dt,dx,dz)$ denotes the compensated Poisson random measure associated to the truncated $\alpha$-stable martingale measure $\hat{L}_{\alpha}(t,x)$. Hence, it holds by (\ref{martingle1}) that $(\hat{u},\hat{L}_{\alpha})$ is a weak solution to (\ref{eq:originalequation1}) defined on $(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}_t})_{t\geq0}, \hat{\mathbb{P}})$.
On the other hand, since $\hat{u}^n$ has the same distribution as $u^n$ for each $n\geq1$, then the moment estimates (\ref{eq:Approximomentresult}) in Proposition \ref{th:Approximainresult} can be replaced by \begin{equation*} \sup_{n\geq1}\sup_{0\leq t\leq T} \hat{\mathbb{E}}\left[\vert \vert \hat{u}^n_t\vert \vert _p^p\right]<\infty \end{equation*} for $p\in(\alpha,2]$. For moment estimate (\ref{eq:momentresult}), the Fatou's Lemma implies that for $p\in(\alpha,2]$ \begin{align*} \hat{\mathbb{E}}\left[\vert \vert \hat{u}\vert \vert _{p,T}^p\right]&= \hat{\mathbb{E}}\left[\int_0^T\vert \vert \hat{u}_t\vert \vert _p^pdt\right] \leq\liminf_{n\rightarrow\infty}C_T \sup_{0\leq t\leq T}\hat{\mathbb{E}}\left[\vert \vert \hat{u}^n_t\vert \vert _p^p\right]<\infty, \end{align*} which completes the proof.
$\Box$
\end{Tproof}
\section{Proof of Theorem \ref{th:mainresult2}}\label{sec4} The proof of Theorem \ref{th:mainresult2} is similar to that of Theorem \ref{th:mainresult}. The main difference between them is that in the current proof we need to prove the solution sequence $(u^n)_{n\geq1}$ to equation (\ref{eq:approximatingsolution}), obtained from Proposition \ref{th:Approximainresult}, is tight in $D([0,\infty),L^p([0,L]))$ for $p\in(\alpha,5/3)$. To this end, we need the following tightness criteria; see, e.g., Ethier and Kurtz \cite[Theorem 3.8.6 and Remark (a)] {Ethier:1986}. Note that the same criteria was also applied in Sturm \cite {Sturm:2003} with Gaussian colored noise setting.
\begin{lemma} \label{lem:tightcriterion} Given a complete and separable metric space $(E,\rho)$,
let $(X^n)$ be a sequence of stochastic processes with sample paths in $D([0,\infty),E)$. The sequence is tight in $D([0,\infty),E)$ if the following conditions hold: \begin{itemize} \item[\rm (i)] For every $\varepsilon>0$ and rational $t\in[0,T]$, there exists a compact set $ \Gamma_{\varepsilon,T}\subset E$ such that \begin{equation} \label{eq:tightcriterion1} \inf_{n}\mathbb{P}[X^n(t)\in\Gamma_{\varepsilon,T}] \geq1-\varepsilon. \end{equation} \item[\rm (ii)] There exists $p>0$ such that \begin{equation} \label{eq:tightcriterion2} \lim_{\delta\rightarrow0}\sup_n\mathbb{E} \left[\sup_{0\leq t\leq T}\sup_{0\leq u\leq \delta}(\rho(X^n_{t+u},X^n_t)\wedge1)^p\right]=0. \end{equation} \end{itemize} \end{lemma} To verify condition (i) of Lemma \ref{lem:tightcriterion}, we need the following characterization of the relatively compact set in $L^p({[0,L]}),p\geq1$; see, e.g., Sturm \cite[Lemma 4.3]{Sturm:2003}. \begin{lemma} \label{lem:compactcriterion} A subset $\Gamma\subset L^p({[0,L]})$ for $p\geq1$ is relatively compact if and only if the following conditions hold: \begin{itemize} \item[\rm (a)] $\sup_{f\in\Gamma} \int_0^L\vert f(x)\vert ^pdx<\infty$, \item[\rm (b)] $\lim_{y\rightarrow0}\int_0^L\vert f(x+y)-f(x)\vert ^pdx=0$ uniformly for all $f\in\Gamma$, \item[\rm (c)] $\lim_{\gamma\rightarrow\infty} \int_{(L-\frac{L}{\gamma},L]}\vert f(x)\vert ^pdx=0$ for all $f\in\Gamma$. \end{itemize} \end{lemma}
The proof of the tightness of $(u^n)_{n\geq1}$ is accomplished by verifying conditions (i) and (ii) in Lemma \ref{lem:tightcriterion}. To this end, we need some estimates on $(u^n)_{n\geq1}$, that is, the uniform bound estimate in Lemma \ref{lem:uniformbound}, the temporal difference estimate in Lemma \ref{lem:temporalestimation} and the spatial difference estimate in Lemma \ref{lem:spatialestimation}, respectively.
\begin{lemma} \label{lem:uniformbound} Suppose that $\alpha\in(1,5/3)$ and for each $n\geq1$ $u^n$ is the solution to equation (\ref{eq:approximatingsolution}) given by {\rm Proposition \ref{th:Approximainresult}}. Then for given $T>0$ there exists a constant $C_{p,K,\alpha,T}$ such that \begin{equation} \label{eq:unformlybounded} \sup_n\mathbb{E}\left[\sup_{0\leq t\leq T}\vert \vert u^n_t\vert \vert _p^p\right]\leq C_{p,K,\alpha,T},\,\,\, \text{for}\,\,\,p\in(\alpha,5/3). \end{equation} \end{lemma}
\begin{proof} For each $n\geq 1$, by (\ref{mildformapproxi0}) it is easy to see that $$\mathbb{E}\left[\sup_{0\leq t\leq T}\vert \vert u^n_t\vert \vert _p^p\right]\leq C_p(A_1+A_2),$$ where \begin{align*} A_1&=\mathbb{E}\left[\sup_{0\leq t\leq T}\Bigg\vert \Bigg\vert \int_0^LG_{t}(\cdot,y)u_0(y)dy\Bigg\vert \Bigg\vert _p^p\right],\\ A_2&=\mathbb{E}\left[\sup_{0\leq t\leq T}\Bigg\vert \Bigg\vert \int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}} G_{t-s}(\cdot,y)\varphi^n(u^n(s-,y))z \tilde{N}(ds,dy,dz)\Bigg\vert \Bigg\vert _p^p\right]. \end{align*} We separately estimate $A_1$ and $A_2$ as follows. For $A_1$, it holds by Young's convolution inequality and (\ref{eq:Greenetimation0}) that \begin{align*} \label{eq:A_1} A_1
&\leq C\mathbb{E}\left[\int_0^L
\sup_{0\leq t\leq T}\left(\int_0^L|G_{t}
(x,y)|dx\right) \vert u_0(y)\vert ^pdy\right]\leq C_T\mathbb{E}[\vert\vert u_0\vert \vert _p^p]. \end{align*} By Proposition \ref{th:Approximainresult}, we have $\mathbb{E}[\vert \vert u_0\vert \vert _p^p]<\infty$ for $p\in(\alpha,2]$, and so there exists a constant $C_{p,T}$ such that $A_1\leq C_{p,T}$.
For $A_2$, we use the factorization method; see, e.g., Da Prato et al. \cite{Prato:1987}, which is based on the fact that for $0<\beta<1$ and $\,0\leq s\leq t$, \begin{equation*} \int_s^t(t-r)^{\beta-1}(r-s)^{-\beta}dr=\dfrac{\pi}{\sin(\beta\pi)}. \end{equation*} For any function $v: [0,\infty)\times {[0,L]} \rightarrow \mathbb{R}$ define
\begin{align*} &\mathcal{J}^{\beta}v(t,x) :=\dfrac{\sin(\beta\pi)}{\pi}\int_0^t\int_0^L(t-s)^{\beta-1}G_{t-s}(x,y)v(s,y)dyds,\\ &\mathcal{J}^n_{\beta}v(t,x):=\int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}} (t-s)^{-\beta}G_{t-s}(x,y)\varphi^n(v(s-,y))z\tilde{N}(ds,dy,dz). \end{align*} By the stochastic Fubini Theorem and (\ref{eq:Greenetimation1}), we have \begin{align*} \mathcal{J}^{\beta}\mathcal{J}^n_{\beta}u^n(t,x) &=\dfrac{\sin(\beta\pi)}{\pi}\int_0^t\int_0^L (t-s)^{\beta-1}G_{t-s}(x,y)\Bigg(\int_0^{s+} \int_0^L\int_{\mathbb{R}\setminus\{0\}} (s-r)^{-\beta}\\ &\quad\quad\times G_{s-r}(y,m) \varphi^n(u^n(r-,m))z\tilde{N}(dr,dm,dz)\Bigg)dyds \\ &=\dfrac{\sin(\beta\pi)}{\pi}\int_0^{t+} \int_0^L\int_{\mathbb{R}\setminus\{0\}} \Bigg[\int_r^{t}(t-s)^{\beta-1}(s-r)^{-\beta}\\ &\quad\quad\times\Bigg(\int_0^LG_{t-s}(x,y) G_{s-r}(y,m)dy\Bigg)ds\Bigg] \varphi^n(u^n(r-,m))z\tilde{N}(dr,dm,dz) \\ &=\int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}G_{t-s}(x,y)\varphi^n(u^n(s-,y))z\tilde{N}(ds,dy,dz). \end{align*} Thus, $$A_2=\mathbb{E}\left[\sup_{0\leq t\leq T}\vert \vert \mathcal{J}^{\beta}\mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p\right].$$
Until the end of the proof we fix a $0<\beta<1$ satisfying \begin{align} \label{eq:factorization} 1-\frac{1}{p}<\beta<\frac{3}{2p}-\frac{1}{2}, \end{align} which requires that $$\frac{3}{2p}-\frac{1}{2}-(1-\frac{1}{p})>0.$$
Therefore, we need the assumption $p<5/3$ for this lemma.
Back to our main proof, to estimate $A_2$ we first estimate $\mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]$.
For $p\in(\alpha,5/3)$, the Burkholder-Davis-Gundy inequality, (\ref{eq:element-inequ})-(\ref{eq:jumpestimate}) and (\ref{eq:glo-lin-growth}) imply that \begin{align*} \mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p] =&\int_0^L\mathbb{E}\left[\left\vert \int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}} (t-s)^{-\beta}G_{t-s}(x,y)\varphi^n(u^n(s-,y))z\tilde{N}(ds,dy,dz)\right\vert ^p\right]dx\nonumber\\ \leq&C_p\int_0^L\int_0^{t}\int_0^L\int_{\mathbb{R}\setminus\{0\}} \mathbb{E}\left[\vert (t-s)^{-\beta}G_{t-s}(x,y)\varphi^n(u^n(s,y))z\vert ^p\right]\nu_{\alpha}(dz)dydsdx\nonumber\\ \leq&C_{p,K,\alpha}\int_0^L\int_0^{t}\int_0^L \mathbb{E}\left[1+\vert u^n(s,y)\vert ^p\right]\vert t-s\vert ^{-\beta p}\vert G_{t-s}(x,y)\vert ^pdydsdx. \end{align*}
Combine (\ref{eq:Greenetimation2}), we have \begin{align*} \mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]\leq&C_{p,K,\alpha}\left(L+\mathbb{E}\left[\sup_{0\leq s\leq t}\vert \vert u^n_s\vert \vert _p^p\right]\right) \int_0^Ts^{-(\frac{p-1}{2}+\beta p)}ds. \end{align*} For $p<5/3$, by (\ref{eq:factorization}) we have $$\int_0^Ts^{-(\frac{p-1}{2}+\beta p)}ds<\infty.$$ Therefore, there exists a constant $C_{p,K,\alpha,T}$ such that \begin{equation} \label{eq:momentestimation} \mathbb{E}\left[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p\right]\leq C_{p,K,\alpha,T}\left(1+\mathbb{E}\left[\sup_{0\leq s\leq t}\vert \vert u^n_s\vert \vert _p^p\right]\right). \end{equation}
We now estimate $A_2=\mathbb{E}[\sup_{0\leq t\leq T}\vert \vert \mathcal{J}^{\beta}\mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]$. The Minkowski inequality implies that \begin{align} \label{eq:A_2_1} A_2 =&\mathbb{E}\left[\sup_{0\leq t\leq T}\dfrac{\sin(\pi\beta)}{\pi}\Bigg\vert \Bigg\vert \int_0^{t}\int_0^L(t-s)^{\beta-1} G_{t-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dyds \Bigg\vert \Bigg\vert _p^p\right]\nonumber\\ \leq&\dfrac{\sin(\pi\beta)}{\pi}\mathbb{E}\left[\sup_{0\leq t\leq T}\left(\int_0^{t}(t-s)^{\beta-1} \Bigg\vert \Bigg\vert \int_0^LG_{t-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dy\Bigg\vert \Bigg\vert _pds\right)^p\right]. \end{align} By the H\"{o}lder inequality and (\ref{eq:Greenetimation0}), we have \begin{align} \label{eq:A_2_2} &\Bigg\vert \Bigg\vert \int_0^LG_{t-s}(\cdot-y)\mathcal{J}^n_{\beta}u^n(s,y)dy\Bigg\vert \Bigg\vert _p\nonumber\\ &\quad=\left(\int_0^L\left\vert \int_0^L\vert G_{t-s}(x,y)\vert ^{\frac{p-1}{p}}\vert G_{t-s}(x,y)\vert ^{\frac{1}{p}}\mathcal{J}^n_{\beta}u^n(s,y) dy\right\vert ^pdx\right)^{\frac{1}{p}}\nonumber\\ &\quad\leq\left(\int_0^L\left\vert \left(\int_0^L\vert G_{t-s}(x,y)\vert dy\right)^{\frac{p-1}{p}} \left(\int_0^LG_{t-s}(x,y)\vert \mathcal{J}^n_{\beta}u^n(s,y)\vert ^pdy\right)^{\frac{1}{p}}\right\vert ^pdx\right)^{\frac{1}{p}}\nonumber\\
&\quad\leq\left(\sup_{x\in[0,L]}\int_0^L|G_{t-s}(x,y)|dy\right)^{\frac{p-1}{p}}
\left(\int_0^L\int_0^L|G_{t-s}(x,y)|\vert \mathcal{J}^n_{\beta}u^n(s,y)\vert ^pdxdy\right)^{\frac{1}{p}}\nonumber\\ &\quad\leq C_T\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p. \end{align} Therefore, it follows from (\ref{eq:A_2_1}), (\ref{eq:A_2_2}), and the H\"{o}lder inequality that \begin{align*} \label{eq:A_2} A_2\leq&\dfrac{\sin(\pi\beta)C_{p,T}}{\pi}\mathbb{E}\left[\sup_{0\leq t\leq T}\left(\int_0^t(t-s)^{\beta-1}\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _pds\right)^p\right]\nonumber\\ \leq&\dfrac{\sin(\pi\beta)C_{p,T}}{\pi}\mathbb{E}\left[\sup_{0\leq t\leq T}\left(\int_0^{t}1^{\frac{p}{p-1}}ds\right)^{p-1} \left(\int_0^{t}(t-s)^{(\beta-1)p}\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p^pds\right)\right] \nonumber\\ \leq&\dfrac{\sin(\pi\beta)C_{p,T}}{\pi}\int_0^{T} (T-s)^{(\beta-1)p} \mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p^p]ds. \end{align*} By (\ref{eq:momentestimation}), it also holds that \begin{align} A_2 \leq&\dfrac{\sin(\pi\beta)C_{p,K,\alpha,T}}{\pi} \int_0^{T}(T-s)^{(\beta-1)p}\left(1+\mathbb{E} \left[\sup_{0\leq r\leq s}\vert \vert u^n_r\vert \vert _p^p\right]\right)ds\nonumber\\ \leq&\dfrac{\sin(\pi\beta)C_{p,K,\alpha,T}}{\pi} \left(1+\int_0^{T}(T-s)^{(\beta-1)p}\mathbb{E} \left[\sup_{0\leq r\leq s}\vert \vert u^n_r \vert \vert _p^p\right]ds\right). \end{align} Combining (\ref{eq:A_2}) and the estimate for $A_1$, we have for each $T>0$, \begin{align*} &\mathbb{E}\left[\sup_{0\leq t \leq T}\vert \vert u^n_t\vert \vert _p^p\right]\leq C_{p,T} +\dfrac{\sin(\pi\beta)C_{p,K,\alpha,T}}{\pi} \int_0^{T}(T-s)^{(\beta-1)p}\mathbb{E} \left[\sup_{0\leq r\leq s}\vert \vert u^n_r\vert \vert _p^p\right]ds. \end{align*} Since $\beta>1-1/p$, applying a generalized Gronwall's Lemma (see, e.g., Lin \cite[Theorem 1.2]{Lin:2013}), we have \begin{align*}
\sup_{n}\mathbb{E}\left[\sup_{0\leq t \leq T}\vert \vert u^n_t\vert \vert _p^p\right]\leq C_{p,K,\alpha,T}, \,\,\,\text{for}\,\,\,p\in(\alpha,5/3), \end{align*} which completes the proof.
$\Box$
\end{proof}
\begin{lemma} \label{lem:temporalestimation} Suppose that $\alpha\in(1,5/3)$ and for each $n\geq1$ $u^n$ is the solution to equation (\ref{eq:approximatingsolution}) given by {\rm Proposition \ref{th:Approximainresult}}. Then for given $T>0$, $0\leq h\leq\delta$ and $p\in(\alpha,5/3)$ \begin{equation} \label{eq:temporalestimation} \lim_{\delta\rightarrow0}\sup_n\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\vert \vert u^n_{t+h}-u^n_t\vert \vert _p^p\right]=0. \end{equation} \end{lemma} \begin{proof}
For each $n\geq 1$, by the factorization method in the proof of Lemma \ref{lem:uniformbound}, we have $$\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\vert \vert u^n_{t+h}-u^n_t\vert \vert _p^p\right]\leq C_p(B_1+B_2),$$ where \begin{align*} B_1&=\mathbb{E}\left[\sup_{0\leq t\leq T} \sup_{0\leq h\leq \delta}\Bigg\vert \Bigg\vert \int_0^L(G_{t+h}(\cdot-y)-G_{t}(\cdot- y))u_0(y)dy\Bigg\vert \Bigg\vert _p^p\right],\\ B_2&=\mathbb{E}\left[\sup_{0\leq t\leq T} \sup_{0\leq h\leq \delta}\vert \vert \mathcal{J}^{\beta} \mathcal{J}^n_{\beta}u^n_{t+h}-\mathcal{J} ^{\beta}\mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p\right]. \end{align*} For $B_1$, Young's convolution inequality and (\ref{eq:Greenetimation0}) imply that \begin{align*}
B_1
&\leq \mathbb{E}\left[\int_0^L\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\left(\int_0^L(|G_{t+h}(x,y)|+|G_{t}(x,y)|)dx\right)\vert u_0(y)\vert ^pdy\right] \leq C_T\mathbb{E}[\vert \vert u_0\vert \vert _p^p]<\infty. \end{align*} Therefore, it holds by Lebesgue's dominated convergence theorem that $B_1$ converges to 0 as $\delta\rightarrow0$.
For $B_2$, it is easy to see that $$B_2\leq \frac{\sin(\beta\pi)C_p}{\pi} (B_{2,1}+B_{2,2}+B_{2,3}),$$ where \begin{align*} B_{2,1}&=\mathbb{E}\Bigg[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\Bigg\vert \Bigg\vert \int_0^{t}\int_0^L(t-s)^{\beta-1}
(G_{t+h-s}(\cdot,y)-G_{t-s}(\cdot,y)) \mathcal{J}^n_{\beta}u^n(s,y)dyds\Bigg\vert \Bigg\vert _p^p\Bigg],\\ B_{2,2}&=\mathbb{E}\Bigg[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\Bigg\vert \Bigg\vert \int_0^{t}\int_0^L((t+h-s)^{\beta-1}-(t-s)^{\beta-1})G_{t+h-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dyds\Bigg\vert \Bigg\vert _p^p\Bigg],\\ B_{2,3}&=\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\Bigg\vert \Bigg\vert \int_{t}^{t+h}\int_0^L(t+h-s)^{\beta-1}G_{t+h-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dyds\Bigg\vert \Bigg\vert _p^p\right]. \end{align*} By the assumption $p\in(\alpha,5/3)$ of this lemma we can choose a $0<\beta<1$ satisfying $1-1/p<\beta<3/2p-1/2$. By Lemma \ref{lem:uniformbound} and (\ref{eq:momentestimation}), there exists a constant $C_{p,K,\alpha,T}$ such that \begin{equation} \label{ineq:0} \sup_{0\leq t\leq T}\mathbb{E}\left[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p\right]\leq C_{p,K,\alpha,T}. \end{equation}
To estimate $B_{2,1}$, we set $G^h_t(x,y)=G_{t+h}(x,y)-G_{t}(x,y)$. Similar to the estimates for (\ref{eq:A_2_1}) and (\ref{eq:A_2_2}) in the proof of Lemma \ref{lem:uniformbound}, we have \begin{align*} B_{2,1}\leq&\mathbb{E}\Bigg[\sup_{0\leq t\leq T} \sup_{0\leq h\leq \delta}\Bigg(\int_0^{t}(t-s)^{\beta-1} \Bigg(\sup_{x\in [0,L]} \int_0^LG^h_{t-s}(x,y)dy\Bigg)^{\frac{p-1}{p}} \vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _pds\Bigg)^p\Bigg]. \end{align*} It also follows from the H\"{o}lder inequality and (\ref{ineq:0}) that \begin{align*} B_{2,1}\leq&C_{p,T}\sup_{0\leq t\leq T}\mathbb{E}\left[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p\right] \sup_{0\leq h\leq \delta} \left(\int_0^{T}s^{(\beta-1)p}\left(\sup_{x\in [0,L]}\int_0^LG_{t-s}^h(x,y)dy\right)^{p-1}ds\right)\nonumber\\ \leq&C_{p,K,\alpha,T}\sup_{0\leq h\leq \delta} \left(\int_0^{T}s^{(\beta-1)p}\left(\sup_{x\in [0,L]}\int_0^LG^h_{t-s}(x,y)dy\right)^{p-1}ds\right). \end{align*} Moreover, since $\beta>1-1/p$, it holds by (\ref{eq:Greenetimation0}) that \begin{align*} &\int_0^{T}s^{(\beta-1)p}\left(\sup_{x\in [0,L]}\int_0^LG_{t-s}^h(x,y)dy\right)^{p-1}ds\\ &\quad\leq\int_0^{T}s^{(\beta-1)p}\left(\sup_{x\in
[0,L]}\left(\int_0^L|G_{t+h-s}(x,y)|dy+\int_0^L|G_{t-s}(x,y)|dy\right)\right)^{p-1}ds\\ &\quad\leq C_{p,T}\int_0^{T}s^{(\beta-1)p}ds<\infty. \end{align*} Thus, Lebesgue's Dominated Convergence Theorem implies that $B_{2,1}$ converges to 0 as $\delta\rightarrow0$.
For $B_{2,2}$, the Minkowski inequality and Young's convolution inequality imply that
\begin{align*}
B_{2,2}\leq&\mathbb{E}\Bigg[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta} \Bigg(\int_0^{t}((t+h-s)^{\beta-1}-(t-s)^{\beta-1}) \Bigg\vert \Bigg\vert \int_0^LG_{t+h-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dy\Bigg\vert \Bigg\vert _p ds\Bigg)^p\Bigg]\nonumber\\ \leq&C_{p,T}\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\left(\int_0^{t}((t+h-s)^{\beta-1}-(t-s)^{\beta-1}) \vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p ds\right)^p\right].\nonumber \end{align*} By the H\"{o}lder inequality and (\ref{ineq:0}) we have for $\beta>1-1/p$, \begin{align*} B_{2,2}\leq&C_{p,T}\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta} \int_0^{t}\vert (t+h-s)^{\beta-1}-(t-s)^{\beta-1}\vert ^{p} \vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p^p ds\right]\nonumber\\ \leq&C_{p,T}\sup_{0\leq t\leq T}\mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]\int_0^{T}\vert (s+\delta)^{\beta-1}-s^{(\beta-1)}\vert ^pds\nonumber\\ \leq&C_{p,K,\alpha,T}\int_0^{T}\vert (s+\delta)^{\beta-1}-s^{(\beta-1)}\vert ^pds<\infty. \end{align*}
Therefore, by Lebesgue's dominated convergence theorem, we know that $B_{2,2}$ converges to 0 as $\delta\rightarrow0$.
For $B_{2,3}$, similar to $B_{2,2}$, we get \begin{align} \label{ineq:B_{2_3}} B_{2,3}\leq&\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta} \left(\int_{t}^{t+h}(t+h-s)^{\beta-1}\Big\vert \Big\vert \int_0^LG_{t+h-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dy\Big\vert \Big\vert _p ds\right)^p\right]\nonumber\\ \leq&C_{p,T}\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\left( \int_{t}^{t+h}(t+h-s)^{\beta-1}\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p ds\right)^p\right]\nonumber\\ \leq&C_{p,T}\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta} \int_{t}^{t+h}\vert (t+h-s)^{\beta-1}\vert ^p\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p^p ds\right]\nonumber\\ \leq&C_{p,T}\sup_{0\leq t\leq T}\mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]\sup_{0\leq h\leq \delta} \int_{t}^{t+h}\vert (t+h-s)^{\beta-1}\vert \leq C_{p,K,\alpha,T}\int_0^{\delta}s^{(\beta-1)p}ds. \end{align} Since $\beta>1-1/p$, we can conclude that the right-hand side of (\ref{ineq:B_{2_3}}) converges to 0 as $\delta\rightarrow0$. Therefore, by the estimates of $B_{2,1}, B_{2,2}, B_{2,3}$ and $B_1$, the desired result (\ref{eq:temporalestimation}) holds, which completes the proof.
$\Box$
\end{proof}
\begin{lemma} \label{lem:spatialestimation} For each $n\geq1$ let $u^n$ be the solution to equation (\ref{eq:approximatingsolution}) given by {\rm Proposition \ref{th:Approximainresult}}. Then for given $t\in[0,\infty)$, $0\leq\vert x_1\vert \leq\delta$ and $p\in(\alpha,2]$ \begin{equation} \label{eq:spatialestimation} \lim_{\delta\rightarrow 0}\sup_n\mathbb{E}\left[ \sup_{\vert x_1\vert \leq \delta}\vert \vert u^n(t,\cdot+x_1)-u^n(t, \cdot)\vert \vert _p^p\right]=0. \end{equation} \end{lemma} \begin{proof} Since the shift operator is continuous in $L^p([0,L])$, then for each $n\geq1$ and $\delta>0$ there exists a pathwise $x_1^{n,\delta}(t)\in\mathbb{R}$ such that $\vert x_1^{n,\delta}(t)\vert \leq\delta$ and $$\sup_{\vert x_1\vert \leq \delta}\vert \vert u^n(t,\cdot+x_1)-u^n(t,\cdot)\vert \vert _p^p=\vert \vert u^n(t,\cdot+x_1^{n,\delta}(t))-u^n(t,\cdot)\vert \vert _p^p.$$ As before, it is easy to see that $$\mathbb{E}[\vert \vert u^n(t,\cdot+x_1^{n,\delta}(t))-u^n(t,\cdot) \vert \vert _p^p]\leq C_p(C_1+C_2),$$ where \begin{align*} C_1&=\mathbb{E}\left[ \bigg\vert \bigg\vert \int_0^L (G_{t}(\cdot+x_1^{n,\delta}(t),y) -G_{t}(\cdot,y))u_0(y)dy\bigg\vert \bigg\vert _p^p \right], \\ C_2&=\mathbb{E}\bigg[\bigg\vert \bigg\vert \int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}} (G_{t-s}(\cdot+{x_1^{n,\delta}(t)},y) -G_{t-s}(\cdot,y)) \varphi^n(u^n(s-,y)) z\tilde{N}(dz,dy,ds)\bigg\vert \bigg\vert _p^p\bigg]. \end{align*} For $C_1$, Young's convolution inequality and (\ref{eq:Greenetimation0}) imply that \begin{align*} C_1&\leq\mathbb{E}\left[\int_0^L
\left(\int_0^L(|G_{t}(x+x_1^{n,
\delta}(t),y)|+|G_{t}(x,y))|dx\right) \vert u_0(y)\vert ^pdy\right] \leq C_T\mathbb{E}[\vert \vert u_0\vert \vert _p^p]<\infty. \end{align*} Thus, the Lebesgue dominated convergence theorem implies that $C_1$ converges to 0 as $\delta\rightarrow0$.
For $C_2$, it follows from the Burkholder-Davis-Gundy inequality, (\ref{eq:element-inequ})-(\ref{eq:jumpestimate}), (\ref{eq:glo-lin-growth}) and (\ref{eq:Greenetimation2}) that for $p\in(\alpha,2]$ \begin{align*}
C_2 =&\int_0^L\mathbb{E}\bigg[\bigg\vert \int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}} (G_{t-s}(x+{x_1^{n,\delta}(t)},y)-G_{t-s}(x,y))\varphi^n(u^n(s-,y))z \tilde{N}(ds,dy,dz)\bigg\vert ^p\bigg]dx\nonumber\\ \leq&C_p\int_0^L\int_0^{t}\int_0^L\int_{\mathbb{R}\setminus\{0\}} \mathbb{E}[\vert (G_{t-s}(x+{x_1^{n,\delta}(t)},y)-G_{t-s}(x,y))\varphi^n(u^n(s,y))z\vert ^p]\nu_{\alpha}(dz)dydsdx\nonumber\\ \leq&C_{p,K,\alpha}\int_0^L\int_0^{t}\int_0^L\mathbb{E}[(1+\vert u^n(s,y)\vert )^p] \vert (G_{t-s}(x+{x_1^{n,\delta}(t)},y)-G_{t-s}(x,y))\vert ^pdydsdx\nonumber\\ \leq&C_{p,K,\alpha}\left(\int_0^L\int_0^{t}\vert G_{t-s}(x+{x_1^{n,\delta}(t)},y)-G_{t-s}(x,y)\vert ^pdsdx\right) \left(L+\sup_{0\leq s\leq t}\mathbb{E}\left[\vert \vert u^n_s\vert \vert _p^p\right]\right) \\ \leq& C_{p,K,\alpha}\left(\int_0^L\int_0^{t}(\vert G_{t-s}(x+{x_1^{n,\delta}(t)},y)\vert ^p+\vert G_{t-s}(x,y)\vert ^p)dsdx\right) \left(L+\sup_{0\leq s\leq t}\mathbb{E}\left[\vert \vert u^n_s\vert \vert _p^p\right]\right) \\ \leq& C_{p,K,\alpha}\left(\int_0^t(t-s)^{-\frac{p-1}{2}}ds\right) \left(L+\sup_{0\leq s\leq t}\mathbb{E}\left[\vert \vert u^n_s\vert \vert _p^p\right]\right) \end{align*}
Therefore, it holds by (\ref{eq:Approximomentresult}) and Lebesgue's dominated convergence theorem that $C_2$ converges to 0 as $\delta\rightarrow0$. Hence, by the estimates of $C_1$ and $C_2$, we obtain \begin{align*} \lim_{\delta\rightarrow 0}\sup_n\mathbb{E}\bigg[\sup_{\vert x_1\vert \leq \delta}\vert \vert u^n(t,\cdot+x_1)-u^n(t,\cdot)\vert \vert _p^p\bigg]&=0, \end{align*} which completes the proof.
$\Box$
\end{proof}
\begin{proposition} \label{prop:tightnessresult2} Suppose that $\alpha\in(1,5/3)$. The sequence of solutions $(u^n)_{n\geq1}$ to equation (\ref{eq:approximatingsolution}) given by {\rm Proposition \ref{th:Approximainresult}} is tight in $D([0,\infty),L^p([0,L]))$ for $p\in(\alpha,5/3)$. \end{proposition} \begin{proof} From (\ref{eq:Approximomentresult}) and Markov's inequality, for each $\varepsilon>0$, $p\in(\alpha,2]$ and $T>0$ there exists a $N\in\mathbb{N}$ such that \begin{align*} \sup\limits_n\mathbb{P}\left[\vert \vert u^n_t\vert \vert _p^p>N\right]\leq\dfrac{\varepsilon}{3},\quad t\in [0,T]. \end{align*} Let $\Gamma^1_{\varepsilon,T}$ be a closed set defined by \begin{align} \label{eq:Gamma1} \Gamma^1_{\varepsilon,T}:=\{v_t\in L^p([0,L]): \vert \vert v_t\vert \vert _p^p\leq N,t\in[0,T]\}. \end{align}
By Lemma \ref{lem:spatialestimation} and Markov's inequality, it holds that for each $\varepsilon>0$, $p\in(\alpha,2]$ and $T>0$ \begin{equation*} \lim_{\delta\rightarrow 0}\sup_n\mathbb{P}\left[ \sup_{\vert x_1\vert \leq \delta}\vert \vert u^n(t,\cdot+x_1)-u^n(t,\cdot)\vert \vert _p^p>\varepsilon\right]=0,\quad t\in[0,T]. \end{equation*} Then for $k\in\mathbb{N}$ we can choose a sequence $(\delta_k)_{k\geq1}$ with $\delta_k\rightarrow0$ as $k\rightarrow\infty$ such that \begin{align*} \sup\limits_n\mathbb{P}\left[\sup_{\vert x_1\vert \leq \delta_k}\vert \vert u^n(t,\cdot+x_1)-u^n(t,\cdot)\vert \vert _p^p>\frac{1}{k} \right]\leq\dfrac{\varepsilon}{3}2^{-k},\quad t\in[0,T]. \end{align*} Let $\Gamma^2_{\varepsilon,T}$ be a closed set defined by \begin{align} \label{eq:Gamma2} \Gamma^2_{\varepsilon,T}:=\bigcap_{k=1}^{\infty} \left\{v_t\in L^p([0,L]): \sup_{\vert x_1\vert \leq \delta_k}\vert \vert v(t,\cdot+x_1)-v(t,\cdot)\vert \vert _p^p\leq\frac{1}{k},t\in[0,T] \right\}. \end{align}
We next prove that for each $\varepsilon>0$ and $p\in(\alpha,2]$ , \begin{equation} \label{eq:compactset1} \lim_{\gamma\rightarrow \infty}\sup_n\mathbb{P}\left[\int_{(L-\frac{L}{\gamma},L]}\vert u^n(t,x)\vert ^pdx>\varepsilon\right]=0. \end{equation} It is easy to see that \begin{align*} \mathbb{E}\bigg[\int_{(L-\frac{L}{\gamma},L]}\vert u^n(t,x)\vert ^pdx\bigg] &=\mathbb{E}\bigg[\int_0^L \vert u^n(t,x)\vert ^p1_{(L-\frac{L}{\gamma},L]}(x)dx\bigg] \leq C_p(D_1+D_2), \end{align*}
where \begin{align*}
D_1&=\int_0^L\mathbb{E}\left[\left\vert \int_0^LG_{t} (x,y)u_0(y)dy\right\vert ^p\right] 1_{(L-\frac{L}{\gamma},L]}(x)dx, \\ D_2&=\int_0^L\mathbb{E}\Bigg[\Bigg\vert \int_0^{t+} \int_0^L\int_{\mathbb{R}\setminus\{0\}} G_{t-s}(x,y) \varphi^n(u^n(s-,y))z \tilde{N}(ds,dy,dz) \Bigg\vert ^p\Bigg]1_{(L-\frac{L}{\gamma},L]}(x)dx. \end{align*} It is easy to prove that $D_1$ converges to 0 as $\gamma\rightarrow\infty$ by using Young's convolution inequality and Lebesgue's dominated convergence theorem. For $D_2$, it holds by the Burkholder-Davis-Gundy inequality, (\ref{eq:element-inequ})-(\ref{eq:jumpestimate}) and (\ref{eq:glo-lin-growth}) that for $p\in(\alpha,2]$ \begin{align*} D_2 \leq &C_p\int_0^L\int_0^{t} \int_0^L\int_{\mathbb{R}\setminus\{0\}} \mathbb{E}[\vert G_{t-s}(x,y)\varphi^n(u^n(s,y))z\vert ^p] 1_{(L-\frac{L}{\gamma},L]}(x)\nu_{\alpha}(dz)dydsdx \nonumber\\ \leq&C_{p,K,\alpha}\int_0^L\int_0^{t} \int_0^L\mathbb{E}(1+\vert u^n(s,y)\vert )^p\vert G_{t-s}(x,y)\vert ^p1_{(L-\frac{L}{\gamma},L]}(x)dydsdx \nonumber\\ \leq&C_{p,K,\alpha,T}\left(L+\sup_{0\leq t\leq T}\mathbb{E} \left[\vert \vert u^n_t\vert \vert _p^p\right]\right) \int_0^L\int_0^{t}(t-s)^{-\frac{p-1}{2}} 1_{(L-\frac{L}{\gamma},L]}(x)dsdx. \end{align*} Since $p\leq2$, it holds that \begin{align*} D_2\leq C_{p,K,\alpha,T} \left(\int_0^L1_{(L-\frac{L}{\gamma},L]}(x)dx\right)\left(L+\sup_{0\leq t\leq T}\mathbb{E} \left[\vert \vert u^n_t\vert \vert _p^p\right]\right). \end{align*} By (\ref{eq:Approximomentresult}), $D_2$ converges to 0 as $\gamma\rightarrow\infty$. Therefore, (\ref{eq:compactset1}) is obtained from the estimates of $D_1$ and $D_2$ and Markov's inequality.
For any $k\in\mathbb{N}$ and $T>0$ we can choose a sequence $(\gamma_k)_{k\geq1}$ with $\gamma_k\rightarrow\infty$ as $k\rightarrow\infty$ such that \begin{align*} \sup\limits_n\mathbb{P}\left [\int_{(L-\frac{L}{\gamma_k},L]}\vert u^n(t,x)\vert ^pdx>\frac{1}{k}\right] \leq\dfrac{\varepsilon}{3}2^{-k},\quad t\in[0,T]. \end{align*} Let $\Gamma^3_{\varepsilon,T}$ be a closed set defined by \begin{align} \label{eq:Gamma3} \Gamma^3_{\varepsilon,T}:=\bigcap_{k=1}^{\infty} \left\{v_t\in L^p([0,L]): \int_{(L-\frac{L}{\gamma_k},L]}\vert v(t,x)\vert ^pdx\leq\frac{1}{k}, t\in[0,T] \right\}. \end{align}
Combining (\ref{eq:Gamma1}), (\ref{eq:Gamma2}) and (\ref{eq:Gamma3}) to define \begin{align*} \Gamma_{\varepsilon,T}:=\Gamma^1_{\varepsilon,T} \cap\Gamma^2_{\varepsilon,T} \cap\Gamma^3_{\varepsilon,T}, \end{align*}
then $\Gamma_{\varepsilon,T}$ is a closed set in $L^p([0,L]),p\in(\alpha,2]$. For any function $f\in\Gamma_{\varepsilon,T}$ the definition of $\Gamma_{\varepsilon,T}$ implies that the conditions (a)-(c) in Lemma \ref{lem:compactcriterion} hold, and so $\Gamma_{\varepsilon,T}$ is a relatively compact set in $L^p([0,L]),p\in(\alpha,2]$. Combing the closeness and relatively compactness, we know that $\Gamma_{\varepsilon,T}$ is a compact set in $L^p([0,L]),p\in(\alpha,2]$. Moreover, the definition of $\Gamma_{\varepsilon,T}$ implies that \begin{align*} \inf_n\mathbb{P} [u^n_t\in\Gamma_{\varepsilon,T}]&\geq1- \frac{\varepsilon}{3}\left(1+2\sum_{k=1}^{\infty} 2^{-k}\right)=1-\varepsilon, \end{align*} which verifies condition (i) of Lemma \ref{lem:tightcriterion}. Condition (ii) of Lemma \ref{lem:tightcriterion} is verified by Lemma \ref{lem:temporalestimation} with $p\in(\alpha,5/3)$. Therefore, $(u^n)_{n\geq1}$ is tight in $D([0,\infty),L^p([0,L]))$ for $p\in(\alpha,5/3)$, which completes the proof.
$\Box$
\end{proof}
\begin{Tproof}\textbf{~of Theorem \ref{th:mainresult2}.} According to Proposition \ref{prop:tightnessresult2}, there exists a $D([0,\infty),L^p([0,T]))$-valued random variable $u$ such that $u^n$ converges to $u$ in distribution in the Skorohod topology. The Skorohod Representation Theorem yields that there exists another filtered probability space $(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}}_t)_{t\geq0},\hat{\mathbb{P}})$ and on it a further subsequence
$(\hat{u}^n)_{n\geq1}$ and $\hat{u}$ which have the same distribution as $(u^n)_{n\geq1}$ and $u$, so that $\hat{u}^n$ almost surely converges to $\hat{u}$ in the Skorohod topology. The rest of the proofs, including the construction of a truncated $\alpha$-stable measure $\hat{L}_{\alpha}$ such that $(\hat{u},\hat{L}_{\alpha})$ is a weak solution to equation (\ref{eq:originalequation1}), is same as the proof of Theorem \ref{th:mainresult} and we omit them.
Since $\hat{u}^n$ has the same distribution as $u^n$ for each $n\geq1$, the moment estimate (\ref{eq:unformlybounded}) in Lemma \ref{lem:uniformbound} can be written as $$ \sup_{n\geq1}\hat{\mathbb{E}}\left[\sup_{0\leq t\leq T}\vert \vert \hat{u}^n_t\vert \vert _p^p\right]\leq C_{p,K,\alpha,T}. $$ Hence, by Fatou's Lemma, $$ \hat{\mathbb{E}}\left[\sup_{0\leq t\leq T}\vert \vert \hat{u}_t\vert \vert _p^p\right]\leq\liminf_{n\rightarrow\infty} \hat{\mathbb{E}}\left[\sup_{0\leq t\leq T}\vert \vert \hat{u}^n_t\vert \vert _p^p\right]<\infty. $$ This yields the uniform $p$-moment estimate (\ref{eq:momentresult2}). Similarly, we can obtain the uniform stochastic continuity (\ref{eq:timeregular}) by Lemma \ref{lem:temporalestimation}.
$\Box$
\end{Tproof}
\noindent \textbf{Acknowledgements} This work is supported by the National Natural Science Foundation of China (NSFC) (Nos. 11631004, 71532001), Natural Sciences and Engineering Research Council of Canada (RGPIN-2021-04100).
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{abstract} We propose an approximate solver for multi-medium Riemann problems with materials described by a family of general Mie-Gr{\"u}neisen equations of state, which are widely used in practical applications. The solver provides the interface pressure and normal velocity by an iterative method. The well-posedness and convergence of the solver is verified with mild assumptions on the equations of state. To validate the solver, it is employed in computing the numerical flux on phase interfaces of a numerical scheme on Eulerian grids that was developed recently for compressible multi-medium flows. Numerical examples are presented for Riemann problems, air blast and underwater explosion applications. \end{abstract}
\begin{keyword} Multi-medium Riemann problem, approximate Riemann solver, Mie-Gr{\"u}neisen equation of state, multi-medium flow \end{keyword} \title{{f An Approximate Solver for Multi-medium Riemann Problem
with Mie-Gr{\"u}
\section{Introduction} Numerical simulations of compressible multi-medium flow are of great interest in practical applications, such as mechanical engineering, chemical industry, and so on. Many conservative Eulerian algorithms perform very well in single-medium compressible flows. However, when such an algorithm is employed to compute multi-medium flows, numerical inaccuracies may occur at the material interfaces \cite{Abgrall1996, Saurel2007, Liu2003}, due to the great discrepancy of densities and equations of state across the interface. The simulation of compressible multi-medium flow in an Eulerian framework requires special attention in describing the interface connecting distinct fluids, especially for the problems that involve highly nonlinear equations of state. Several techniques have been taken to treat the multi-medium flow interactions. See \cite{Liu2003, Abgrall2001, Karni1994, Arienti2004, Shyue2001, Saurel1999, Price2015} for instance.
A typical procedure of multi-medium compressible flows in Eulerian grids mainly consists of two steps, the interface capture and the interaction between different fluids. There are mainly two different approaches in literatures, the diffuse interface method (DIM) and the sharp interface method (SIM). The former \cite{Abgrall1996, Abgrall2001, Saurel1999, Saurel2009, Petitpas2009, Ansari2013} assumes the interface as a diffuse zone, and smears out the interface over a certain number of grid cells to avoid pressure oscillations. Diffuse interfaces correspond to artificial mixtures created by numerical diffusion, and the key issue is to fulfill interface conditions within this artificial mixture. The latter assumes the interface to be a sharp contact discontinuity, and different fluids are immiscible. Several approaches such as the volume of fluid (VOF) method \cite{Scardovelli1999, Noh1976}, level set method \cite{Sethian2001, Sussman1994}, moment of fluid (MOF) method \cite{Ahn2007, Dyadechko2008, Anbarlooei2009} and front-tracking method \cite{Glimm1998, Tryggvason2001} are used extensively to capture the interface. A key element for both diffuse and sharp interface methods, is to determine the interface states. The accurate prediction of the interface states can be used to stabilize the numerical diffusion in diffuse interface methods, and to compute the numerical flux and interface motion in sharp interface methods. One common approach is to solve a multi-medium Riemann problem which contains the fundamentally physical and mathematical properties of the governing equations. Indeed, the Riemann problem plays a key role in understanding the wave structures, since a general fluid flow may be interpreted as a nonlinear superpositions of the Riemann solutions.
The solution of a multi-medium Riemann problem depends not only on the initial states at each side of the interface, but also on the forms of equations of state. For some simple equations of state, such as ideal gas, covolume gas or stiffened gas, the solution of the Riemann problem can be achieved to any desired accuracy with an exact solver. While the Riemann problems for the above equations of state have been fully investigated in \cite{Godunov1976, Plohr1988, Gottlieb1988, Toro2008} for instance, there exist some difficulties in the cases of general nonlinear equations of state due to their high nonlinearity. A variety of methods to solve the corresponding Riemann problems have then been proposed. For example, Larini \textit{et al.} \cite{Larini1992} developed an exact Riemann solver and applied their methods to a fifth-order virial equation of state (EOS), which is particularly suited to the gaseous detonation products of high explosive compounds. Shyue \cite{Shyue2001} developed a Roe's approximate Riemann solver for the Mie-Gr{\"u}neisen EOS with variable Gr{\"u}neisen coefficient. Quartapelle \textit{et al.} \cite{Quartapelle2003} proposed an exact Riemann solver by applying the Newton-Raphson iteration to the system of two nonlinear equations imposing the equality of pressure and of velocity, assuming as unknowns the two values of the specific volume at each side of the interface, and implemented it for the van der Waals gas. Arienti \textit{et al.} \cite{Arienti2004} applied a Roe-Glaster solver to compute the equations combining the Euler equations involving chemical reaction with the Mie-Gr{\"u}neisen EOS. More recently, Rallu \cite{Rallu2009} and Farhat \textit{et al.} \cite{Farhat2012} utilized a sparse grid technique to tabulate the solutions of exact multi-medium Riemann problems. Lee \textit{et al.} \cite{Lee2013} developed an exact Riemann solver for the Mie-Gr{\"u}neisen EOS with constant Gr{\"u}neisen coefficient, where the integral terms are evaluated using an iterative Romberg algorithm. Banks \cite{Banks2010} and Kamm \cite{Kamm2015} developed a Riemann solver for the convex Mie-Gr{\"u}neisen EOS by solving a nonlinear equation for the density increment involved in the numerical integration of rarefaction curves, and chose the JWL (Jones-Wilkins-Lee) EOS as a representative case.
In this paper, we propose an approximate multi-medium Riemann solver for a family of general Mie-Gr{\"u}neisen equations of state in an iterative manner, which provides a strategy to reproduce the physics of interaction between two mediums across the interface. Several mild conditions on the coefficients of Mie-Gr{\"u}neisen EOS are assumed to ensure the convexity of equations of state and the well-posedness of our Riemann solver. The algebraic equation of the Riemann problem is solved by an inexact Newton method \cite{Dembo1982}, where the function and its derivative are evaluated approximately depending on the wave configuration. And the convergence of our Riemann solver is analyzed. To validate the proposed solver, we employed it in the computation of two-medium compressible flows with Mie-Gr{\"u}neisen EOS, as an extension of the numerical scheme that was developed recently for two-medium compressible flows with ideal gases \cite{Guo2016}.
The rest of this paper is arranged as follows. In Section \ref{sec:riemann}, a solution strategy for the multi-medium Riemann problem with arbitrary Mie-Gr{\"u}neisen equations of state is presented. In Section \ref{sec:aps}, the procedures of our approximate Riemann solver are outlined, and the well-posedness and convergence are analyzed. In Section \ref{sec:model}, the application of our Riemann solver in two-medium compressible flow calculations \cite{Guo2016} is briefly introduced. In Section \ref{sec:num}, several classical Riemann problems and applications for air blast and underwater explosions are carried out to validate the accuracy and robustness of our solver. Finally, a short conclusion is drawn in Section \ref{sec:conclusion}.
\section{Multi-medium Riemann Problem} \label{sec:riemann}
Let us consider the following one-dimensional multi-medium Riemann problem of the compressible Euler equations \begin{subequations} \begin{equation}
\dfrac{\partial}{\partial\tau}
\begin{bmatrix}
\rho \\ \rho u \\ E
\end{bmatrix}
+\dfrac{\partial}{\partial\xi}
\begin{bmatrix}
\rho u \\ \rho u^2+p \\ (E+p)u
\end{bmatrix}
=\bm 0,\quad
E=\rho e+\dfrac{1}{2}\rho u^2. \end{equation} Here $\tau$ is time and $\xi$ is spatial coordinate, and $\rho$, $u$, $p$, $E$ and $e$ are the density, velocity, pressure, total energy and specific internal energy, respectively. The system has initial values \begin{equation}
[\rho,u,p]^\top (\xi,\tau=0) =
\begin{cases}
[\rho_l,u_l,p_l]^\top, & \xi<0, \\
[\rho_r,u_r,p_r]^\top, & \xi>0.
\end{cases} \end{equation} \label{system:oneriemann} \end{subequations}
Here the equations of state for both mediums under our consideration can be classified into the family known as \emph{the Mie-Gr\"uneisen EOS}. The Mie-Gr\"uneisen EOS can be used to describe a lot of real materials, for example, the gas, water and gaseous products of high explosives \cite{Arienti2004, Liu2003, Kamm2015}, which is particularly useful in those practical applications we are studying now. The general form of the Mie-Gr\"uneisen EOS is given by \begin{equation} p(\rho, e)=\varGamma(\rho)\rho e + h(\rho), \label{eq:particulareos} \end{equation} where $\varGamma(\rho)$ is the Gr{\"u}neisen coefficient, and $h(\rho)$ is a reference state associated with the cold contribution resulting from the interactions of atoms at rest \cite{Heuze2012}. Thus the EOS of the multi-medium Riemann problem \eqref{system:oneriemann} is given by \[ p(\rho, e) = \begin{cases} \varGamma_l(\rho)\rho e+h_l(\rho), \\ \varGamma_r(\rho)\rho e+h_r(\rho), \end{cases} \] for the medium on the left and the right sides, respectively. For the ease of our analysis, we impose on $\varGamma(\rho)$ and $h(\rho)$ the following assumptions:
\textbf{(C1)} $\varGamma'(\rho) \le 0,~(\rho\varGamma(\rho))' \ge 0,~(\rho\varGamma(\rho))''\ge 0$;
\textbf{(C2)} $\lim\limits_{\rho\rightarrow+\infty}\varGamma(\rho)=
\varGamma_\infty>0,~\varGamma(\rho)\le \varGamma_{\infty}+2$;
\textbf{(C3)} $h'(\rho)\ge 0,~h''(\rho)\ge 0$.
\begin{remark}
An immediate consequence is that the Gr{\"u}neisen coefficient
$\varGamma(\rho)$ must be nonnegative since
$\varGamma(\rho) \ge-\varGamma'(\rho)\rho \ge 0$ by the condition
\textbf{(C1)}. \end{remark} A lot of equations of state of our interests fulfill these assumptions. Particularly, we collect some equations of state in Appendix A which are used in our numerical tests as examples. These examples include ideal gas EOS, stiffened gas EOS, polynomial EOS, JWL EOS, and Cochran-Chan EOS. The coefficients $\varGamma(\rho)$, $h(\rho)$ and their derivatives for these equations of state are all provided therein.
The Riemann problem for general convex equations of state have been fully analyzed, for example, in \cite{Smith1979, Menikoff1989}. Here the problem is more specific, thus we can present the structure of the solution in a straightforward way. The property of EOS is essential on the wave structures in the solution of Riemann problems. It is pointed out that the wave structures are composed solely of elementary waves \cite{Menikoff1989} if the \emph{fundamental derivative} \cite{Thompson1971} \[
\mathscr G=\dfrac{1}{c}\cdot\left.\dfrac{\partial\rho c}{\partial\rho}\right|_s=
1+\dfrac{\rho}{2c^2}\cdot\left.\dfrac{\partial c^2}{\partial \rho}\right|_s, \] keeps positive \footnote{
When the positivity condition $\mathscr G>0$ is violated, other configurations
of waves may occur, such as composite waves, split waves, expansive shock
waves or compressive rarefaction waves \cite{Menikoff1989}. The above
anomalous wave structures are common issues in phase transitions. For further
discussions on these anomalous wave structures, we refer the readers to
\cite{Weyl1949, Thompson1973, Liu1975, Liu1976, Pego1986, Bethe1998,
Bates2002, Voss2005, Muller2006, Fossati2014} for instance.}, where $s$ is the specific entropy and $c$ is the speed of sound. For the Mie-Gr{\"u}neisen EOS \eqref{eq:particulareos}, the speed of sound can be expressed as \[
c(p,\rho)=\sqrt{\left.\dfrac{\partial p}{\partial \rho}\right|_e
+\dfrac{p}{\rho^2}
\left.\dfrac{\partial p}{\partial e}\right|_{\rho}}= \sqrt{\left(\dfrac{1}{\rho} +
\dfrac{\varGamma_k'(\rho)}{\varGamma_k(\rho)}\right)(p-h_k(\rho))
+\dfrac{p}{\rho}\varGamma_k(\rho) + h_k'(\rho)}, \] and a tedious calculus gives us that the fundamental derivative is \begin{equation} \begin{split} \mathscr G = &\dfrac{1}{c^2}\left( \dfrac12\left(\rho(\rho\varGamma_k(\rho))'' +{(\rho\varGamma_k(\rho))'}(2+\varGamma_k(\rho))\right)e+\dfrac12\rho h_k''(\rho) \right. \\ &\left.+\dfrac{p}{2\rho}(\varGamma_k^2(\rho)+2(\rho\varGamma_k(\rho))') +\dfrac12(2+\varGamma_k(\rho))h_k'(\rho)\right). \end{split} \label{eq:fundd} \end{equation} We conclude that \begin{lemma}
The solution of the multi-medium Riemann problem
\eqref{system:oneriemann} consists of only elementary waves if the
conditions \textbf{(C1), (C2)} and \textbf{(C3)} are fulfilled. \end{lemma} \begin{proof}
With the conditions \textbf{(C1), (C2)} and \textbf{(C3)}, a direct
check on the terms in \eqref{eq:fundd} gives us that
\[
\mathscr G > 0,
\]
then the result in \cite{Menikoff1989} is applied to give us the
conclusion. \end{proof} Precisely, the solution of \eqref{system:oneriemann} consists of four constant regions connected by a linearly degenerate wave and two genuinely nonlinear waves (either shock wave or rarefaction wave, depending on the initial states), as is shown schematically in Fig. \ref{fig:Riemann-Fluid}. The linearly degenerate wave is actually the phase interface.
\begin{figure}
\caption{Typical wave structure of the Riemann problem in $\xi - \tau$ space.}
\label{fig:Riemann-Fluid}
\end{figure}
Following the convention on notations, we denote the pressure and the velocity by $p^*$ and $u^*$ in the star region, respectively, which have the same value crossing over the phase interface. This allows us to derive a single nonlinear algebraic equation for $p^*$. Then we will solve this algebraic equation by an iterative method. At the beginning of each step of the iterative method, a provisional value of $p^*$ determines the wave structures from four possible configurations. The wave structures then prescribe the specific formation of the algebraic equation. Based on the residual of the algebraic equation, the value of $p^*$ is updated, which closes a single loop of the iterative method.
Below let us give the details of the plan above. Firstly we need to study the relations of the solution across a nonlinear wave, saying a shock wave or a rarefaction wave. For convenience, we use the subscript $k = l$ or $r$ standing for either the left initial state $l$ or the right initial state $r$ hereafter.
\begin{itemize} \item[-] {\bf Solution through a shock wave}
If $p^*>p_k$, the corresponding nonlinear wave is a shock wave, and the star region state $\bm U_k^*$ is connected to the adjacent initial state $\bm U_k$ through a Hugoniot locus. The Rankine-Hugoniot jump conditions \cite{Toro2008} yield
\begin{equation} \mp(u^*-u_k) = \left((p^*-p_k)\left(\dfrac{1}{\rho_k}-\dfrac{1}{\rho^*_k} \right)\right)^{1/2}, \label{eq:diffushock} \end{equation} and \begin{equation} e(p^*,\rho^*_k) - e(p_k,\rho_k) + \dfrac{1}{2}(p^*+p_k) \left(\dfrac{1}{\rho^*_k} -\dfrac{1}{\rho_k} \right) = 0, \label{eq:eose} \end{equation} where \[ e(p,\rho)=\dfrac{p-h_k(\rho)}{\varGamma_k(\rho)\rho}. \] Multiplying both sides of the equality \eqref{eq:eose} by $\rho_k\rho_k^*\varGamma_k(\rho_k)\varGamma_k(\rho_k^*)$ gives rise to \[ \begin{split} &\varGamma_k(\rho_k)\rho_k (p^*-h_k(\rho_k^*)) - \varGamma_k(\rho_k^*) \rho_k^* (p_k-h_k(\rho_k)) \\ &-\dfrac{1}{2}\varGamma_k(\rho_k^*)\varGamma_k(\rho_k)(p^*+p_k) (\rho_k^*-\rho_k)=0. \end{split} \] For convenience we introduce the \emph{Hugoniot function} as follows \begin{equation} \begin{split} \varPhi_k(p,\rho) &:=\varGamma_k(\rho_k)\rho_k (p-h_k(\rho)) - \varGamma_k(\rho) \rho (p_k-h_k(\rho_k))\\ &-\dfrac{1}{2}\varGamma_k(\rho_k)(p+p_k)\varGamma_k(\rho) (\rho-\rho_k), \end{split} \label{eq:phi} \end{equation} then the relation \eqref{eq:eose} boils down to the algebraic equation $\varPhi_k(p^*,\rho_k^*)=0$. The derivative of $\varPhi_k$ with respect to the density is found to be \begin{equation} \begin{split} \dfrac{\partial\varPhi_k}{\partial\rho}(p,\rho) &= -\varGamma_k(\rho_k)\rho_k \left(h_k'(\rho)-\dfrac{p+p_k}{2}\varGamma_k'(\rho)\right)\\ &-\varGamma_k(\rho_k)(\varGamma_k(\rho)+\rho\varGamma_k'(\rho)) \left(\rho_ke_k+\dfrac{p+p_k}{2}\right). \end{split} \label{eq:phide} \end{equation} As a result, the slope of the Hugoniot locus can be found by the method of implicit differentiation, namely, \[
\chi(p,\rho) := \left.\dfrac{\partial p}{\partial \rho}\right|_{\varPhi_k}= -\dfrac{2\partial\varPhi_k(p,\rho)/\partial\rho}{ \varGamma_k(\rho_k)(2\rho_k-{\varGamma_k(\rho)}(\rho-\rho_k))}. \] Before we discuss the properties of the Hugoniot function $\varPhi_k(p,\rho)$, let us introduce the compressive limit of the density $\rho_{\max}$ such that $\chi(p,\rho_{\max})=\infty$. By definition $\rho_{\max}$ solves the algebraic equation $2\rho_k-\varGamma_k(\rho)(\rho-\rho_k)=0$. This quantity is uniquely defined since the function \[ W(\rho):=\left(\dfrac{\rho}{\rho_{k}}-1\right)\varGamma_k(\rho)-2, \] is monotonically increasing in the interval $(\rho_k,+\infty)$ by the condition \textbf{(C1)}, and \[ W(\rho_k)=-2,\quad \lim_{\rho\rightarrow+\infty}W(\rho) \ge \lim_{\rho\rightarrow+\infty} \left(\dfrac{\rho}{\rho_{k}}-1\right)\varGamma_{\infty}-2=+\infty. \] We have the following results on the function $\varPhi_k(p,\rho)$ \begin{lemma}\label{thm:phi}
Assume that the functions $\varGamma_k(\rho)$ and $h_k(\rho)$ satisfy the conditions \textbf{(C1), (C2)} and \textbf{(C3)}. Then $\varPhi_k(p,\rho)$ defined in \eqref{eq:phi} satisfies the following properties:
\begin{itemize}
\item[(1).] $\varPhi_k(p,\rho_k)>0$;
\item[(2).] $\varPhi_k(p,\rho_{\max})<0$;
\item[(3).] ${\partial\varPhi_k}(p,\rho)/{\partial \rho}<0$;
\item[(4).] ${\partial^2\varPhi_k}(p,\rho)/{\partial\rho^2}<0$ if
$\varGamma''_k(\rho)=0$.
\end{itemize} \end{lemma} \begin{proof}
(1). By definition \eqref{eq:phi} we have
\[
\begin{split}
\varPhi_k(p,\rho_k)&=\varGamma_k(\rho_k)\rho_k (p-h_k(\rho_k))-
\varGamma_k(\rho_k) \rho_k (p_k-h_k(\rho_k)) \\
&= \varGamma_k(\rho_k)\rho_k(p-p_k) > 0.
\end{split}
\]
(2). Since the compressive limit of the density $\rho_{\max}$ satisfies
the relation $(\rho_{\max}-\rho_k)\varGamma_k(\rho_{\max})=2\rho_k$, we have
\begin{equation}
\begin{split}
&\varPhi_k(p,\rho_{\max})\\
=&\varGamma_k(\rho_k)\rho_k
(p-h_k(\rho_{\max}))-
\varGamma_k(\rho_{\max}) \rho_{\max} (p_k-h_k(\rho_k)) \\
-&
\dfrac{1}{2}\varGamma_k(\rho_{\max})(\rho_{\max}
-\rho_k)\varGamma_k(\rho_k)(p+p_k)\\
=&-\varGamma_k(\rho_k)\rho_k(p_k+h_k(\rho_{\max})) -
(\varGamma_k(\rho_{\max})+2)\rho_k(p_k-h_k(\rho_k)).
\end{split}
\label{eq:phimax}
\end{equation}
Obviously $\varPhi_k(p,\rho_{\max})<0$ if $h_k(\rho_{\max})\ge 0$. On
the other hand, suppose that $h_k(\rho_{\max})< 0$. Rewriting
\eqref{eq:phimax} as
\[
\begin{split}
\varPhi_k(p,\rho_{\max})&=
-(\varGamma_k(\rho_k)+\varGamma_k(\rho_{\max})+2)\rho_kp_k\\
&-\rho_k(\varGamma_k(\rho_k)h_k(\rho_{\max})
-(\varGamma_k(\rho_{\max})+2)h_k(\rho_k)),
\end{split}
\]
and using the inequality resulting from the conditions \textbf{(C2)} and \textbf{(C3)}
\[
\varGamma_k(\rho_k)h_k(\rho_{\max})
\ge \varGamma_k(\rho_k)h_k(\rho_k)\ge
(\varGamma_\infty+2)h_k(\rho_k)\ge
(\varGamma_k(\rho_{\max})+2)h_k(\rho_k),
\]
we conclude that $\varPhi_k(p,\rho_{\max})<0$.
(3). This is an obvious result from the expression \eqref{eq:phide}.
(4). The second derivative of $\varPhi_k(p,\rho)$ with respect to the density is
\[
\begin{split}
\dfrac{\partial^2\varPhi_k}{\partial\rho^2}(p,\rho)&= -\varGamma_k(\rho_k)\rho_k
\left(h_k''(\rho)-\dfrac{p+p_k}{2}\varGamma_k''(\rho)\right)\\
&-\varGamma_k(\rho_k)(2\varGamma_k'(\rho)
+\rho\varGamma_k''(\rho))
\left(\rho_ke_k+\dfrac{p+p_k}{2}\right),
\end{split}
\]
which is negative if $\varGamma''_k(\rho)=0$. This completes the proof of the whole theorem. \end{proof} Instantly by Lemma \ref{thm:phi}, the density $\rho$ can be uniquely determined from the equation $\varPhi_k(p,\rho)=0$ on the interval $(\rho_k,\rho_{\max})$ for any fixed $p$. Also the Hugoniot curve is monotonic due to $\chi>0$. Since the equation \eqref{eq:eose} uniquely defines the interface density $\rho_k^*$ for a given value of $p^*$, the right hand side of \eqref{eq:diffushock} can be regarded as a function of the interface pressure $p^*$ alone, formally written as \[ f_k(p^*)=\left((p^*-p_k)\left(\dfrac{1}{\rho_k}-\dfrac{1}{\rho^*_k} \right)\right)^{1/2},\quad p^*>p_k. \]
\item[-] {\bf Solution through a rarefaction wave}
If, on the other hand, $p^*\le p_k$, then the corresponding nonlinear wave is a rarefaction wave, and the interface state $\bm U_k^*$ is connected to the adjacent initial state $\bm U_k$ through a rarefaction curve. Since the Riemann invariant \[ u\pm \int \dfrac{1}{\rho c}\mathrm d p, \] is constant along the right-facing (left-facing) rarefaction curve, we have \begin{equation} \mp(u^*-u_k)=\int_{p_k}^{p^*} \dfrac{1}{\rho c}\mathrm d p, \label{eq:diffu} \end{equation} where the density $\rho$ is expressed in terms of $p$ by solving the isentropic relation \begin{equation} \dfrac{\partial p}{\partial \rho}=c^2(p,\rho). \label{eq:isenr} \end{equation} Similarly, the right hand side of \eqref{eq:diffu} can be expressed as a function of $p^*$ alone. Formally we define \[ f_k(p^*) = \int^{p^*}_{p_k} \dfrac{1}{\rho c}\mathrm d p,\quad p^*\le p_k. \]
\end{itemize}
Collecting both cases above, we have that \[ u^*-u_l=-f_l(p^*) \quad \text{ and } \quad u^*-u_r=f_r(p^*). \] Therefore, the interface pressure $p^*$ is the zero of the following \emph{pressure function} \[ f(p) := f_l(p)+f_r(p) + u_r - u_l. \] And the interface velocity $u^*$ can be determined by \[ u^*=\dfrac{1}{2}(u_l+u_r+f_r(p^*)-f_l(p^*)). \] Recall that the formula of the function $f_k(p)$ is given by \[ f_k(p)= \begin{cases} \displaystyle\int^{p}_{p_k} \dfrac{1}{\rho c}\mathrm d p, & p\le p_k,\\ \left((p-p_k)\left(\dfrac{1}{\rho_k}-\dfrac{1}{\rho}\right) \right)^{1/2}, & p>p_k, \end{cases} \] where $\rho$ is determined through either the Hugoniot relation \eqref{eq:eose} or the isentropic relation \eqref{eq:isenr} for a given $p$. We claim on $f_k(p)$ that \begin{lemma}\label{thm:propf}
Assume that the conditions \textbf{(C1), (C2)} and \textbf{(C3)}
hold for $\varGamma_k(\rho)$ and $h_k(\rho)$, the function $f_k(p)$ is
monotonically increasing and concave, i.e.
\[
f_k'(p)>0 \quad \text{ and } \quad f_k''(p)<0,
\]
if the Hugoniot function is concave with respect to the density,
i.e. ${\partial^2 \varPhi_k(p,\rho)}/{\partial \rho^2}<0$. \end{lemma} \begin{proof}
The first and second derivatives of $f_k(p)$ can be found to be
\[
f_k'(p)=
\begin{cases}
\dfrac{1}{\rho c}, & p\le p_k, \\
\dfrac{1}{2f_k(p)} \left(\dfrac{1}{\rho_k}-\dfrac{1}{\rho}
+\dfrac{p-p_k}{\rho^2\chi}\right), & p>p_k,
\end{cases}
\]
and
\[
f_k''(p)=
\begin{cases}
-\dfrac{\mathscr G}{\rho^2c^3}, &p\le p_k, \\
\begin{aligned}
&-\dfrac{1}{4f_k^3(p)}\left(
\dfrac{2(p-p_k)^2}{\rho^2\chi^2}\left(\dfrac{1}{\rho_k}-\dfrac{1}{\rho
}\right)
\left(\dfrac{2}{\rho}+\left.\dfrac{\partial\chi}{\partial p}\right|_{\varPhi_k}\right)
\right.\\
&+
\left.
\left(\dfrac{1}{\rho_k}-\dfrac{1}{\rho}-\dfrac{p-p_k}{\rho^2\chi}
\right)^2
\right)
,
\end{aligned}
& p>p_k,
\end{cases}
\]
where $\mathscr G$ is the fundamental derivative \eqref{eq:fundd} and
\[
\left.\dfrac{\partial\chi}{\partial p}\right|_{\varPhi_k}
=
\left(\dfrac{\partial\varPhi_k}{\partial\rho}(p,\rho)\right)^{-1}
\left(
\dfrac{\partial^2\varPhi_k}{\partial\rho^2}(p,\rho)+\chi\varGamma_k(\rho_k)
(\rho_k\varGamma_k'(\rho)-(\rho\varGamma_k(\rho))')
\right).
\]
The result then follows by direct observation. \end{proof} The behavior of $f_k(p)$ is related to the existence and uniqueness of the solution of the Riemann problem. The existence and uniqueness of the Riemann solution for gas dynamics under appropriate conditions have been established by Liu \cite{Liu1975} and by Smith \cite{Smith1979}. It is easy to show that the conditions \textbf{(C1), (C2)} and \textbf{(C3)} imply Smith's ``strong'' condition $\partial e(p,\rho)/\partial \rho<0$. However, for completeness we provide a short proof of the results for the case of Mie-Gr{\"u}neisen EOS in the following theorem. \begin{theorem}\label{thm:unique}
Assume that the Mie-Gr{\"u}neisen EOS \eqref{eq:particulareos}
satisfies the conditions \textbf{(C1), (C2)} and \textbf{(C3)}. The
Riemann problem \eqref{system:oneriemann} admits a unique solution
(in the class of admissible shocks, interfaces and rarefaction waves
separating constant states) if and only if the initial states
satisfy the constraint
\begin{equation}
u_r-u_l<\int_{0}^{p_l}\dfrac{\mathrm dp}{\rho c}+
\int_{0}^{p_r}\dfrac{\mathrm dp}{\rho c}.
\label{eq:vacuum}
\end{equation} \end{theorem} \begin{proof}
By definition and Lemma \ref{thm:propf} we know that the pressure
function $f(p)$ is monotonically increasing. Next we study the
behavior of $f(p)$ when $p$ tends to infinity. Let $\tilde\rho$
represents the density such that $\varPhi_k(2p_k,\tilde\rho)=0$.
When the pressure $p>2p_k$, we have $\rho>\tilde{\rho}$, and thus
\[
f_k^2(p)=(p-p_k)\left(\dfrac{1}{\rho_k}-\dfrac{1}{\rho}\right)
>(p-p_k)\left(\dfrac{1}{\rho_k}-\dfrac{1}{\tilde\rho}\right).
\]
As a result, $f_k(p)$ tends to positive infinity as
$p\rightarrow +\infty$ and so does $f(p)$.
Based on the behavior of the function $f(p)$, a necessary and
sufficient condition for the interface pressure $p^*>0$ such that
$f(p^*)=0$ to be uniquely defined is given by
\[
f(0)=f_l(0)+f_r(0)+u_r-u_l<0,
\]
or equivalently, the constraint given by \eqref{eq:vacuum}. This
completes the proof. \end{proof}
\begin{remark}
When the initial states violate the constraint \eqref{eq:vacuum},
the Riemann problem \eqref{system:oneriemann} has no solution in the
above sense. One can yet define a solution by introducing a
\emph{vacuum}. However, we are not going to address this issue
since it is beyond the scope of our current interests. \end{remark}
\section{Approximate Riemann Solver} \label{sec:aps}
The solution of the Riemann problem \eqref{system:oneriemann} is obtained by finding the unique zero $p^*$ of the function $f(p)$. A first try to this problem is to use the Newton-Raphson iteration as \begin{equation} p_{n+1}=p_n-\dfrac{f(p_n)}{f'(p_n)} =p_n-\dfrac{f_l(p_n)+f_r(p_n) + u_r - u_l}{f_l'(p_n)+f_r'(p_n)}, \label{eq:iterp1} \end{equation} with a suitable initial estimate which we may choose as, for example, the acoustic approximation \cite{Godunov1976} \[ p_0 = \dfrac{\rho_lc_lp_r+\rho_rc_rp_l+\rho_lc_l\rho_rc_r(u_l-u_r)} {\rho_lc_l+\rho_rc_r}. \] The concavity of the pressure function $f(p)$ leads to the following global convergence of the Newton-Raphson iteration \begin{corollary}
The Newton-Raphson iteration for \eqref{eq:iterp1} converges if
$\varGamma_l''(\rho)=\varGamma_r''(\rho)=0$. \end{corollary}
Unfortunately, there is no closed-form expression for the pressure function $f(p)$ or its derivative $f'(p)$ for equations of state such as polynomial EOS, JWL EOS or Cochran-Chan EOS. Therefore, we have to implement the iteration \eqref{eq:iterp1} approximately. Here we adopt the \emph{inexact Newton method} \cite{Dembo1982} instead, which is formulated as \begin{equation} \left\{ \begin{aligned} p_{n+1}&=p_n-\dfrac{F_n}{F_n'}=p_n-\dfrac{F_{n,l}+F_{n,r}+u_r-u_l} {F'_{n,l}+F'_{n,r}},\\ u_{n}&=\dfrac{1}{2}(u_l+u_r+F_{n,r}-F_{n,l}), \end{aligned}\right. \label{eq:iterp2} \end{equation} where $F_{n,k}$ and $F'_{n,k}$ approximate $f_k(p_{n})$ and $f_k'(p_{n})$, respectively.
To specify the sequences $\{F_{n,k}\}$ and $\{F'_{n,k}\}$, we solve the Hugoniot loci through the numerical iteration and the isentropic curves by the numerical integration. It is natural to expect that the sequences $p_n$ and $u_n$ will tend to $p^*$ and $u^*$ respectively whenever the evaluation errors
$|F_{n,k}-f_k(p_n)|$ and $|F'_{n,k}-f'_k(p_n)|$ are sufficiently small. As a preliminary result, let us introduce the following Lemma \ref{the:conv_p}, which states the local convergence of inexact Newton iterates \begin{lemma}
\label{the:conv_p}
If the initial iterate $p_0$ is sufficiently close to $p^*$, and the
evaluation errors of $f(p_n)$ and $f'(p_n)$ satisfy the following
constraint
\[
2|F_n-f(p_n)|+|\Delta p_n|\cdot|F'_n-f'(p_n)|\le \eta |F_n|,
\]
where $\Delta p_n=p_{n+1}-p_n=-F_n/F'_n$ denotes the step increment
and $\eta\in (0,1)$ is a fixed constant, then the sequence of
inexact Newton iterates $p_n$ defined by
\[
p_{n+1}=p_n-\dfrac{F_n}{F'_n},
\]
converges to $p^*$. Moreover, the convergence is linear in the sense
that $|p_{n+1}-p^*|\le \zeta |p_n-p^*|$ for $\zeta \in (\eta,1)$. \end{lemma} \begin{proof}
Since $\eta<\zeta$, there exists $\gamma>0$ sufficiently small that
$(1+\gamma)((\eta+2)\gamma+\eta)\le \zeta$.
Choose $\epsilon>0$ sufficiently small that
\begin{itemize}
\item[(1).] $|f'(p)-f'(p^*)|\le \gamma |f'(p^*)|$;
\item[(2).] $|f'(p)^{-1}-f'(p^*)^{-1}|\le \gamma |f'(p^*)|^{-1}$;
\item[(3).] $|f(p)-f(p^*)-f'(p^*)(p-p^*)|\le \gamma |f'(p^*)||p-p^*|$,
\end{itemize}
whenever $|p-p^*|\le \epsilon$. Now we prove the convergence rate by
induction. Let the initial solution satisfy $|p_0-p^*|\le
\epsilon$. Suppose that $|p_n-p^*|\le \epsilon$, then
\[
\begin{split}
|f(p_n)|&=|f(p_n)-f(p^*)-f'(p^*)(p_n-p^*)+f'(p^*)(p_n-p^*)| \\
&\le (1+\gamma)|f'(p^*)||p_n-p^*|.
\end{split}
\]
The error of $(n+1)$-th iterate can be written as
\[
\begin{split}
p_{n+1}-p^*&= f'(p_n)^{-1}
(r_n+(f'(p_n)-f'(p^*))(p_n-p^*)\\
&-(f(p_n)-f(p^*)-f'(p^*)(p_n-p^*))),
\end{split}
\]
where the residual $r_n=f'(p_n)\Delta p_n+f(p_n)$ satisfies
\[
\begin{split}
|r_n|&=|(f'(p_n)-F_n')\Delta p_n+f(p_n)-F_n|\\
&=|(f'(p_n)-F_n')\Delta p_n+(1+\eta)(f(p_n)-F_n)
-\eta(f(p_n)-F_n)|
\le \eta |f(p_n)|.
\end{split}
\]
Therefore
\[
\begin{split}
|p_{n+1}-p^*|&\le (1+\gamma)|f'(p^*)|^{-1}
(\eta|f(p_n)|+\gamma |f'(p^*)||p_n-p^*| \\
& + \gamma
|f'(p^*)||p_n-p^*|)\\
&\le (1+\gamma)(\eta(1+\gamma)+2\gamma)|p_n-p^*|\\
&\le \zeta |p_n-p^*|,
\end{split}
\]
and hence $|p_{n+1}-p^*|\le \epsilon$.
It follows that $p_n$ converges linearly to $p^*$. \end{proof}
Recall that $f(p)$ and $f'(p)$ can be decomposed as \[ f(p)=f_l(p)+f_r(p)+u_r-u_l \text{~~and~~} f'(p)=f_l'(p)+f_r'(p). \] The following Theorem \ref{the:conv_pu} tells us that if $f_k$ and $f_k'$ are evaluated accurately enough, the convergences of iterates $p_n$ and $u_n$ are ensured. \begin{theorem}\label{the:conv_pu}
Suppose that the initial estimate $p_0$ is sufficiently close to
$p^*$. If the evaluation errors of $f_k(p_n)$ and $f'_k(p_n)$
satisfy
\[
|F_{n,k}-f_k(p_n)|\le \dfrac{1}{6}\eta |F_n|
\text{~~and~~}
|F'_{n,k}-f_k'(p_n)|\le \dfrac{1}{6}\eta |F_n'|,
\]
where $\eta\in (0,1)$ is a constant. Moreover, assume that the
sequence $\{F_n'\}$ is bounded. Then the sequences of pressure and
velocity defined in \eqref{eq:iterp2} converge, namely
\[
p_n\rightarrow p^*\text{~~and~~} u_n\rightarrow u^*.
\] \end{theorem} \begin{proof}
Since
\begin{align*}
|F_n-f(p_n)|\le |F_{n,l}-f_l(p_n)|+ |F_{n,r}-f_r(p_n)|
\le \dfrac{1}{3}\eta |F_n|,\\
|F'_n-f'(p_n)|\le |F'_{n,l}-f'_l(p_n)|+
|F'_{n,r}-f'_r(p_n)|
\le \dfrac{1}{3}\eta |F'_n|,
\end{align*}
we have
\[
2|F_n-f(p_n)|+|\Delta p_n|\cdot|F'_n-f'(p_n)|\le \dfrac{2}{3}\eta
|F_n|+ \left|\dfrac{F_n}{F'_n}\right|\cdot
\dfrac{1}{3}\eta|F'_n|=\eta |F_n|.
\]
From Lemma \ref{the:conv_p} we conclude that $p_n\rightarrow p^*$.
Due to the continuity of $f_k$ we have
$f_k(p_n)\rightarrow f_k(p^*)$. From $F_n=(p_n-p_{n+1})F_n'$ and
the boundness of $\{F'_n\}$ we know that $F_n\rightarrow 0$. It
follows that
\[
|F_{n,k}-f_k(p^*)|\le
|F_{n,k}-f_k(p_n)|+|f_k(p_n)-f_k(p^*)|\rightarrow 0.
\]
Finally, by the definition of $u_n$ and $u^*$ we have
\[
|u_n-u^*|=\dfrac{1}{2}\left|
(F_{n,r}-f_r(p^*)) - (F_{n,l} - f_l(p^*))
\right|\rightarrow 0,
\]
or $u_n\rightarrow u^*$. \end{proof}
By Theorem \ref{the:conv_pu}, the convergence is guaranteed by \textit{a posteriori} control on the evaluation errors of $f_k(p_n)$ and $f'_k(p_n)$. The evaluation errors of $f_k(p_n)$ and $f'_k(p_n)$ depend on the residual of the algebraic equation as well as the truncation error of the ordinary differential equation. Therefore, to achieve better accuracy one may effectively reduce the residual term and the truncation error. To achieve a trade-off between the accuracy and efficiency in a practical implementation, we apply the Newton-Raphson method to solve the Hugoniot loci, and the Runge-Kutta method to solve the isentropic curves.
Precisely, for the given $n$-th iterator $p_n$, if $p_n>p_k$, then we solve the following algebraic equation \begin{equation} \varPhi_k(p_n,\tilde\rho_{n,k})=0, \label{eq:algphi} \end{equation} to obtain $\tilde\rho_{n,k}$ to a prescribed tolerance with the Newton-Raphson method \[ \rho_{n,k,m+1}=\rho_{n,k,m}- \dfrac{\varPhi_k(p_n,\rho_{n,k,m})} {\partial\varPhi_k(p_n,\rho_{n,k,m})/\partial\rho}. \] By Lemma \ref{thm:phi}, we arrive at the following convergence result \begin{corollary}
The Newton-Raphson iteration for \eqref{eq:algphi} must converge if
$\varGamma''_k(\rho)=0$. \end{corollary} The values of $F_{n,k}$ and $F'_{n,k}$ for the shock branch are thus taken as \begin{align} \label{eq:shockf}
F_{n,k}&=\left((p_n-p_k)\left(\dfrac{1}{\rho_k}-\dfrac{1}
{\tilde\rho_{n,k}}\right)\right)^{1/2},\\ \label{eq:shockdf}
F'_{n,k}&=\dfrac{1}{2F_{n,k}}\left(
\dfrac{1}{\rho_k}-\dfrac{1}{\tilde\rho_{n,k}}
+\dfrac{p_n-p_k}{\rho_{n,k}^2\chi(p_n,\tilde\rho_{n,k})}
\right). \end{align}
If, on the other hand, $p_n\le p_k$, then we solve the following initial value problem
\begin{align*}
\dfrac{\mathrm d f_k}{\mathrm d p}=\dfrac{1}{\rho c}, \\
f_k|_{p=p_k}=0, \end{align*} coupled with the initial value problem of the isentropic relations
\begin{align*}
\dfrac{\mathrm d\rho}{\mathrm d p}=\dfrac{1}{c^2},\\
\rho|_{p=p_k}=\rho_k,
\end{align*} backwards until $p=p_n$ using fourth-order Runge-Kutta method. The values of $F_{n,k}$ and $F'_{n,k}$ are then taken as \begin{align}
\label{eq:raref}
F_{n,k} &= -\dfrac{1}{6}(p_k-p_n)
\left(\dfrac{1}{Z_1}+\dfrac{2}{Z_2}
+\dfrac{2}{Z_3}+\dfrac{1}{Z_4}
\right), \\
\label{eq:raredf}
F'_{n,k} &=\dfrac{1}{\tilde\rho_{n,k} c(p_n,\tilde\rho_{n,k})}, \end{align} where \[
\tilde\rho_{n,k} = \rho_k-\dfrac{1}{6}(p_k-p_n) \left(\dfrac{1}{c_1^2}+\dfrac{2}{c_2^2}+\dfrac{2}{c_3^2}
+\dfrac{1}{c_4^2}\right),\\ \] and the intermediate states are given by \[ \begin{aligned}
c_1&=c(p_k,\rho_k),& Z_1&=\rho_kc_1,\\
c_2&=c\left(\dfrac{p_k+p_n}{2},\rho_k-\dfrac{p_k-p_n}{2c_1^2}\right),
& Z_2&= \left(\rho_k-\dfrac{p_k-p_n}{2c_1^2}\right)c_2,\\
c_3&=c\left(\dfrac{p_k+p_n}{2},\rho_k-\dfrac{p_k-p_n}{2c_2^2}\right),
& Z_3&= \left(\rho_k-\dfrac{p_k-p_n}{2c_2^2}\right)c_3, \\
c_4&=c\left(p_n,\rho_k-\dfrac{p_k-p_n}{c_3^2}\right), & Z_4&=
\left(\rho_k-\dfrac{p_k-p_n}{c_3^2}\right)c_4. \end{aligned} \] Finally the procedure of the approximate Riemann solver for \eqref{system:oneriemann} is listed below. \begin{mdframed}
\setlength{\parindent}{0pt}
\textbf{Step 1} Provide initial estimates of the interface
pressure
\[
p_0 =
\dfrac{\rho_lc_lp_r+\rho_rc_rp_l+\rho_lc_l\rho_rc_r(u_l-u_r)}
{\rho_lc_l+\rho_rc_r}.
\]
\textbf{Step 2} Assume that the $n$-th iterator $p_n$ is
obtained. Determine the types of left and right nonlinear
waves.
(i) If $p_n>\max\{p_l,p_r\}$, then both nonlinear waves
are shocks.
(ii) If $\min\{p_l,p_r\}\le p_n \le \max\{p_l,p_r\}$, then
one of the two nonlinear waves is a shock, and the other is a
rarefaction wave.
(iii) If $p_n<\min\{p_l,p_r\}$, then both nonlinear waves
are
rarefaction waves.
\textbf{Step 3} Evaluate $F_{n,k}$ and $F'_{n,k}$ through
\eqref{eq:shockf}~\eqref{eq:shockdf} or
\eqref{eq:raref}~\eqref{eq:raredf}
according to the wave structures and update the interface
pressure
through
\[
p_{n+1}=p_n-\dfrac{F_{n,l}+F_{n,r} +
u_r - u_l}{F'_{n,l}+F'_{n,r}}.
\]
\textbf{Step 4} Terminate whenever the relative change of the
pressure reaches the prescribed tolerance. The sufficiently accurate estimate
$p_n$ is then taken as the approximate interface pressure $p^*$.
Otherwise return to Step 2.
\textbf{Step 5} Compute the interface velocity
$u^*$ through
\[
u^*=\dfrac{1}{2}\left(u_l+u_r+F_{n,r} - F_{n,l}\right).
\] \end{mdframed}
In the above algorithm the tolerances of the outer iteration for pressure function and the inner iteration for Hugoniot function are both set as $10^{-8}$. Although the evaluation errors of $f_k(p_n)$ and $f'_k(p_n)$ for the isentropic branch is not effectively controlled in our implementation without applying a more accurate numerical quadrature rule, we have never encountered any failure in the convergence of inexact Newton iteration in the numerical tests.
\section{Application on Two-medium Flows} \label{sec:model} We are considering the two-medium fluid flow described by an immiscible model in the domain $\Omega$. Two fluids are separated by a sharp interface $\Gamma(t)$ characterized by the zero of the level set function $\phi(\bm x,t)$ which satisfies \begin{equation} \dfrac{\partial \phi}{\partial t} +
\tilde u|\nabla\phi| = 0. \label{eq:levelset:eov} \end{equation} Here $\tilde u$ denotes the normal velocity of the interface, which is determined by the solution of multi-medium Riemann problem. And the normal direction is chosen as the gradient of the level set function. \iffalse \begin{equation} \dfrac{\partial \phi}{\partial t} + \bm{\tilde u} \cdot \nabla\phi = 0, \label{eq:levelset:eov} \end{equation} where $\bm{\tilde u}$ denotes the velocity of interface. Its normal component $\tilde u_n$ is determined by the solution of multi-medium Riemann problem, while the tangent component $\tilde u_\tau$ is directly copied from the local velocity of fluid, where \[ \tilde u_n = \bm{\tilde u} \cdot \bm n, \quad \tilde u_\tau = \bm{\tilde u} - \tilde u_n \bm n, \]
and $\bm n = \left. \frac{\nabla \phi}{|\nabla \phi|}
\right|_{\phi=0}$ is the unit normal direction of the interface. \fi The region occupied by each fluid can be expressed in terms of the level set function $\phi(\bm x,t)$ \[
\Omega^+(t):=\{\bm x\in\Omega~|~\phi(\bm x,t)> 0\} \text{~~and~~}
\Omega^-(t):=\{\bm x\in\Omega~|~\phi(\bm x,t)< 0\}. \] And the fluid in each region is governed by the Euler equations \begin{equation}
\dfrac{\partial \bm{U}}{\partial t} +
\nabla\cdot \bm F(\bm U)=
\bm 0,\quad \bm x\in \Omega^\pm (t),
\label{eq:euler} \end{equation} where \[ \bm{U}= \begin{bmatrix} \rho \\ \rho \bm u \\ E \end{bmatrix}\text{~~and~~} \bm{F(\bm U)}= \begin{bmatrix} \rho \bm u^\top \\ \rho \bm u\otimes \bm u+p\mathbf I\\ (E+p)\bm u^\top \end{bmatrix}. \] \iffalse Here $\rho$ stands for the density, $\bm u$ stands for the velocity, $p$ stands for the pressure and $E$ stands for the total energy per unit volume such that \[
E = \rho e + \frac{1}{2} \rho \|\bm u\|^2, \] where $e$ is the specific internal energy. \fi Here $\bm u$ stands for the velocity vector, and other variables represent the same as that in \eqref{system:oneriemann}. To provide closure for the above Euler equations, we need to specify the equation of state for each fluid, i.e. \[ p= \begin{cases} P^+(\rho,e), & \phi>0,\\ P^-(\rho,e), & \phi<0. \end{cases} \] The specific forms of the corresponding equations of state are the Mie-Gr\"uneisen EOS discussed in Section \ref{sec:riemann}.
The level set equation \eqref{eq:levelset:eov} and the Euler equations \eqref{eq:euler} are coupled in the sense that the level set equation provides the boundary geometry for the Euler equations, whereas the Euler equations provide the interface motion for the level set equation. To solve the coupled system, we use the numerical scheme following Guo \textit{et al.} \cite{Guo2016}, which is implemented on an Eulerian grid. For completeness, however, we briefly sketch the main steps of numerical scheme for the two-medium flow therein. The approximate Riemann solver we proposed is applied to calculate the numerical flux at the interface in the numerical scheme.
As a numerical scheme on an Eulerian grid, the whole domain $\Omega$ is divided into a conforming mesh with simplex cells, and may be refined or coarsened during the computation based on the $h$-adaptive mesh refinement strategy of Li and Wu \cite{Li2013}. The whole scheme is divided into three steps:
\begin{itemize} \item[(1).] {\bf Evolution of interface}
The level set function is approximated by a continuous piecewise-linear function. The discretized level set function \eqref{eq:levelset:eov} is updated through the characteristic line tracking method once the motion of the interface is given. Due to the nature of level set equation, it remains to specify the normal velocity $\tilde u$ within a narrow band near the interface. This can be achieved by firstly solving a multi-medium Riemann problem across the interface and then extending the velocity field to the nearby region using the harmonic extension technique of Di \textit{et al.} \cite{Di2007}. The solution of the multi-medium Riemann problem has been elaborated in Section \ref{sec:riemann}.
In order to keep the property of signed distance function, we solve the following reinitialization equation \[ \begin{cases}
\dfrac{\partial \psi}{\partial \tau} =
\sgn(\psi_0)\cdot\left(1-\left|\nabla \psi \right|\right), \\
\psi(\bm x,0)=\psi_0=\phi(\bm x, t), \end{cases} \] until steady state using the explicitly positive coefficient scheme \cite{Di2007}.
Once the level set function is updated until $n$-th time level, we can obtain the discretized phase interface $\Gamma_{h,n}$. A cell $K_{i,n}$ is called an \emph{interface cell} if the intersection of $K_{i,n}$ and $\Gamma_{h,n}$, denoted as $\Gamma_{K_{i,n}}$, is nonempty. Since the level set function is piecewise-linear and the cell is simplex, $\Gamma_{K_{i,n}}$ must be a linear manifold in $K_{i,n}$. The interface $\Gamma_{h,n}$ further cuts the cell $K_{i,n}$ and one of its boundaries $S_{ij,n}$ into two parts, which are represented as $K_{i,n}^\pm$ and $S_{ij,n}^\pm$ respectively (may be an empty set). The unit normal of $\Gamma_{K_{i,n}}$, pointing from $K_{i,n}^-$ to $K_{i,n}^+$, is denoted as $\bm n_{K_{i,n}}$. These quantities can be readily computed from the geometries of the interface and cells. See Fig. \ref{fig:twophasemodel} for an illustration.
\begin{figure}
\caption{Illustration of the two-medium fluid model.}
\label{fig:twophasemodel}
\end{figure}
\item[(2).] {\bf Numerical flux}
The numerical flux for the two-medium flow is composed of two parts: the cell edge flux and the interface flux. Below we explain the flux contribution towards any given cell $K_{i,n}$. We introduce two sets of flow variables at $n$-th time level \[ \bm U_{K_{i,n}}^\pm = \left[\begin{array}{c} \rho_{K_{i,n}}^\pm \\ \rho_{K_{i,n}}^\pm\bm u_{K_{i,n}}^\pm \\ E_{K_{i,n}}^\pm \end{array}\right], \] which refer to the constant states in the cell $K_{i,n}^\pm$. Note that the flow variables vanish if there is no corresponding fluid in a given cell.
\begin{itemize} \item {\bf Cell edge flux}
The cell edge flux is the exchange of flux between the same fluid across the cell boundary. For any edge $S_{ij}$ of the cell $K_{i,n}$, let $\bm n_{ij,n}$ be the unit normal pointing from $K_{i,n}$ into the adjacent cell $K_{j,n}$. The cell edge flux across $S_{ij,n}^\pm$ is calculated as \begin{equation}\label{eq:cell_bound_flux}
\hat{\bm F}_{ij,n}^\pm = \Delta t_n
\left|S_{ij,n}^\pm \right|
\hat{\bm F} \left(
\bm U_{K_{i,n}}^\pm, \bm U_{K_{j,n}}^\pm;
\bm n_{ij,n} \right), \end{equation} where $\Delta t_n$ denotes the current time step length, and $\hat{\bm F}(\bm U_l, \bm U_r; \bm n)$ is a consistent monotonic numerical flux along $\bm n$. Here we adopt the local Lax-Friedrich flux \[ \hat{\bm F}(\bm U_l, \bm U_r; \bm n) = \dfrac{1}{2} \left(\bm F(\bm U_l) +
\bm F(\bm U_r)\right) \cdot \bm n - \dfrac{1}{2}\lambda (\bm U_r - \bm U_l), \] where $\lambda$ is the maximal signal speed over $\bm U_l$ and $\bm U_r$.
\item {\bf Interface flux} The interface flux is the exchange of the flux between two fluids due to the interaction of fluids at the interface. If $K_{i,n}$ is an interface cell, then the flux across the interface can be approximated by \begin{equation} \hat{\bm F}_{K_{i,n}}^{\pm}=
\Delta t_n\left|\Gamma_{K_{i,n}}\right| \begin{bmatrix} 0 \\ p_{K_{i,n}}^*\bm n_{K_{i,n}} \\ p_{K_{i,n}}^*u_{K_{i,n}}^* \end{bmatrix}. \label{eq:phaseflux} \end{equation} Here $p_{K_{i,n}}^*$ and $u_{K_{i,n}}^*$ are the interface pressure and normal velocity, which are obtained by applying the approximate solver we proposed in Section \ref{sec:aps} to a local one-dimensional Riemann problem in the normal direction of the interface with initial states \[ \begin{split} [\rho_l,u_l,p_l]^\top= [\rho_{K_{i,n}}^-,
\bm u_{K_{i,n}}^-\cdot \bm n_{K_{i,n}},
p_{K_{i,n}}^-]^\top,\\ [\rho_r,u_r,p_r]^\top= [\rho_{K_{i,n}}^+,
\bm u_{K_{i,n}}^+\cdot \bm n_{K_{i,n}},
p_{K_{i,n}}^+]^\top.
\end{split} \] Here the pressure $p_{K_{i,n}}^\pm$ in the initial states are given through the corresponding equations of state, namely \[ p_{K_{i,n}}^\pm=P^\pm (\rho_{K_{i,n}}^\pm,e_{K_{i,n}}^\pm),\quad e_{K_{i,n}}^\pm=\dfrac{E_{K_{i,n}}^\pm}{\rho_{K_{i,n}}^\pm}
-\dfrac{1}{2}\|\bm u_{K_{i,n}}^\pm\|^2. \] \end{itemize}
\item[(3).] {\bf Update of conservative variables}
Once the edge flux \eqref{eq:cell_bound_flux} and interface flux \eqref{eq:phaseflux} are computed, the conservative variables at $(n+1)$-th time level is thus assigned as \[ \bm U^{\pm}_{K_{i,n+1}} \!=\! \left\{\begin{array}{ll} \bm 0, & K^\pm_{i,n+1} \!=\! \varnothing, \\
\dfrac{1}{|K^\pm_{i,n+1}|} \!
\left(|K^\pm_{i,n}| \bm U^\pm_{K_{i,n}} \!+\! \displaystyle\sum_{S_{ij}^\pm\subseteq \partial K_{i,n}^\pm} \hat{\bm F}_{ij,n}^\pm \!+\! \hat{\bm F}^\pm_{K_{i,n}}\right), & K^\pm_{i,n+1} \!\neq\! \varnothing. \end{array}\right. \] \end{itemize}
Basically, the steps we presented above may close the numerical scheme, while there are more details in the practical implementation to gurantee the stability of the scheme. Please see \cite{Guo2016} for those details.
\section{Numerical Examples}\label{sec:num}
In this section we present several numerical examples to validate our schemes, including one-dimensional Riemann problems, spherically symmetric problems and multi-dimensional shock impact problems. One-dimensional simulations are carried out on uniform interval meshes, while two and three-dimensional simulations are carried out on unstructured triangular and tetrahedral meshes respectively.
\subsection{One-dimensional problems}
In this part, we present some numerical examples of one-dimensional Riemann problems. The computational domain is $[0,1]$ with $400$ cells. And the reference solution, if mentioned, is computed on a very fine mesh with $10^4$ cells.
\subsubsection{Shyue shock tube problem}
This is a single-medium JWL Riemann problem used by Shyue \cite{Shyue2001}. The parameters of JWL EOS \eqref{eq:jwl} therein are given by $A_1=\unit[8.545\times10^{11}]{Pa}, A_2=\unit[2.050\times10^{10}]{Pa}$, $\omega=0.25$, $R_1=4.6$, $R_2=1.35$ and $\rho_0=\unit[1840]{kg/m^3}$. The initial conditions are \[
[\rho, u, p]^\top = \left\{
\begin{array}{ll}
[1700, ~0, ~10^{12}]^\top, &x<0.5,\\ [2mm]
[1000, ~0, ~5.0\times 10^{10}]^\top, & x > 0.5.
\end{array}
\right. \] The simulation terminates at $t=1.2\times 10^{-5}$. In Fig. \ref{res:rm_shyue}, the top panel shows the results of our numerical methods, while the bottom panel contains the results of Shyue \cite{Shyue2001}. Our results agree well with the highly resolved solution shown in a solid line given by Shyue.
\begin{figure}
\caption{Shyue shock tube problem (top panel: our results, bottom panel:
results from Shyue \cite{Shyue2001}).}
\label{res:rm_shyue}
\end{figure}
\subsubsection{Saurel shock tube problem}
In this problem, we consider the advection of a density discontinuity of the liquid nitromethane described by Cochran-Chan EOS \eqref{eq:CC} in a uniform flow \cite{Saurel2007,Lee2013}. The parameters are given by $A_1=\unit[8.192\times10^8]{Pa}$, $A_2=\unit[1.508\times10^9]{Pa}, \omega=1.19$, $R_1=4.53$, $R_2=1.42$ and $\rho_0=\unit[1134]{kg/m^3}$. The initial value is \[
[\rho, u, p]^\top = \left\{
\begin{array}{ll}
[1134, ~1000, ~2.0\times 10^{10}]^\top, &x<0.5,\\ [2mm]
[500, ~1000, ~2.0\times 10^{10}]^\top, & x>0.5.
\end{array}
\right. \] The simulation terminates at $t=4.0\times 10^{-5}$. We use this Riemann problem to assess the performance of our methods against highly nonlinear equations of state. Fig. \ref{res:rm_saurel} displays the results of our numerical scheme and that of Saurel \textit{et al.} \cite{Saurel2007}, where we can see that there is no non-physical pressure and velocity across the contact discontinuity in our numerical scheme.
\begin{figure}
\caption{Saurel shock tube problem (top row: our results, bottom row: results from Saurel \textit{et al.} \cite{Saurel2007}).}
\label{res:rm_saurel}
\end{figure}
\subsubsection{Ideal gas-water Riemann problem}
In this example we simulate the ideal gas-water Riemann problems. The water is modeled by either the stiffened gas EOS \eqref{eq:stiffened} or the polynomial EOS \eqref{eq:poly}. The initial density, velocity and pressure are both assigned with \[ [\rho, u, p]^\top = \left\{ \begin{array}{ll} [1630, ~0, ~7.0\times 10^9]^\top, &x<0.5,\\ [2mm] [1000, ~0, ~1.0\times 10^5]^\top, & x > 0.5. \end{array} \right. \]
The adiabatic exponent is $\gamma=2.0$ for the ideal gas EOS. The parameters of water are $\gamma=7.15$ and $p_\infty=\unit[3.31\times10^8]{Pa}$ for the stiffened gas EOS \cite{Rallu2009, Wang2008}, and $\rho_0=\unit[1000]{kg/m^3}$, $A_1=\unit[2.20\times10^9]{Pa}$, $A_2=\unit[9.54\times10^9]{Pa}$, $A_3=\unit[1.45\times10^{10}]{Pa}$, $B_0=B_1=0.28$, $T_1=\unit[2.20\times10^9]{Pa}$ and $T_2=0$ for the polynomial EOS \cite{Jha2014}, respectively. The same parameters are chosen in the remaining numerical examples for the water.
The results with distinct equations of state at $t=8.0\times 10^{-5}$ are shown in Fig. \ref{res:rm_gm_water}, where we can observe that the numerical results agree well with the corresponding reference solutions. The comparison between these two graphs also shows the discrepancies due to the choices of the equations of state. It is observed that the shock wave in the stiffened gas EOS propagates faster than that in the polynomial EOS.
\subsubsection{JWL-polynomial Riemann problem}
This example concerns the JWL-polynomial Riemann problem. The initial states are \[ [\rho, u, p]^\top = \left\{ \begin{array}{ll} [1630, ~0, ~8.3\times 10^9]^\top, &x<0.5,\\ [2mm] [1000, ~0, ~1.0\times 10^5]^\top, & x > 0.5. \end{array} \right. \] We use the following values to describe the TNT \cite{Smith1999}: $A_1=\unit[3.712\times10^{11}]{Pa}$, $A_2=\unit[3.230\times10^9]{Pa}$, $\omega=0.30$, $R_1=4.15$, $R_2=0.95$ and $\rho_0=\unit[1630]{kg/m^3}$. The result at $t=8.0\times 10^{-5}$ is shown in Fig. \ref{res:rm_jwl_pol}, where we can observe that both the interface and shock are captured well without spurious oscillation.
\begin{figure}
\caption{Ideal gas-water Riemann problem (top row: stiffened gas, bottom row: polynomial).}
\label{res:rm_gm_water}
\caption{JWL-polynomial Riemann problem.}
\label{res:rm_jwl_pol}
\end{figure}
\subsection{Spherically symmetric problems}
In this part we present two spherically symmetric problems, where the governing equations are formulated as follows \begin{equation} \dfrac{\partial}{\partial t} \begin{bmatrix} r^2\rho \\ r^2\rho u \\ r^2E \end{bmatrix} +\dfrac{\partial}{\partial r} \begin{bmatrix} r^2\rho u \\ r^2(\rho u^2+p) \\ r^2(E+p)u \end{bmatrix} =\begin{bmatrix} 0 \\ 2rp \\ 0 \end{bmatrix}. \label{eq:spheq} \end{equation} The source term in \eqref{eq:spheq} is discretized using an explicit Euler method.
\subsubsection{Air blast problem}\label{problem:blast}
The shock wave that propagates through the air as a consequence of the nuclear explosion is commonly referred to as the \emph{blast wave}. In this example we simulate the blast wave from one kiloton nuclear charge. The explosion products and air are modeled by the ideal gas EOS with adiabatic exponents $\gamma=1.2$ and $1.4$ respectively. The initial density and pressure are $\unit[618.935]{kg/m^3}$ and $\unit[6.314 \times 10^{12}]{Pa}$ for the explosion products, and $\unit[1.29]{kg/m^3}$ and $\unit[1.013\times 10^5]{Pa}$ for the air. The initial interface is located at $r=\unit[0.3]{m}$ initially. To effectively capture the wave propagation we use a computational domain of radius $\unit[5000]{m}$.
It is known that the destructive effects of the blast wave can be measured by its \emph{overpressure}, i.e., the amount by which the static pressure in the blast wave exceeds the ambient pressure ($\unit[1.013\times 10^5]{Pa}$). The overpressure increases rapidly to a peak value when the blast wave arrives, followed by a roughly exponential decay. The integration of the overpressure over time is called \emph{impulse}. See Fig. \ref{rm::air_blast} (a) for an illustration of these terminologies. Fig. \ref{rm::air_blast} (b) -- (d) show the peak overpressure, impulse and shock arrival time at different radii. The results are compared with the point explosion solutions in Qiao \cite{Qiao2003}, which confirm the accuracy of our methods in the air blast applications.
\subsubsection{Underwater explosion problem}
We use this example to simulate the underwater explosion problem where a TNT of one hundred kilograms explodes in the water. The high explosives and water are characterized by the JWL EOS and polynomial EOS, respectively. The radii of the computational domain and the initial interface are $\unit[15]{m}$ and $\unit[0.245]{m}$ respectively. The initial pressure of the high explosives is $\unit[9.5\times 10^9]{Pa}$. The same problem has been simulated in \cite{Jia2007} using ANSYS/AUTODYN. Fig. \ref{rm::udex} shows the computed peak overpressure and impulse at different radii. The results agree well with the empirical law provided in \cite{Cole1948}.
\begin{figure}
\caption{Shock wave parameters for air blast problem.}
\label{rm::air_blast}
\end{figure}
\begin{figure}
\caption{Shock wave parameters for underwater explosion problem.}
\label{rm::udex}
\end{figure}
\subsection{Two-dimensional problems}
In this part, we present a few two-dimensional cylindrically symmetric flows in engineering applications. The Euler equations for this configuration are formulated as a cylindrical form \[ \dfrac{\partial}{\partial t} \begin{bmatrix} r\rho \\ r\rho u \\ r\rho v\\ rE \end{bmatrix} +\dfrac{\partial}{\partial r} \begin{bmatrix} r\rho u \\ r(\rho u^2+p) \\ r\rho uv \\ r(E+p)u \end{bmatrix} +\dfrac{\partial}{\partial z} \begin{bmatrix} r\rho v \\ r\rho uv \\ r(\rho v^2+p)\\ r(E+p)v \end{bmatrix} =\begin{bmatrix} 0 \\ p \\ 0 \\ 0 \end{bmatrix}. \] To improve the efficiency of the simulation, the $h$-adaptive mesh method is adopted here \cite{Li2013}. Roughly speaking, more elements will be distributed in the region where the jump of pressure is sufficiently large.
\subsubsection{Nuclear air blast problem}
In this example we simulate the nuclear air blast in the computational domain $0\le r,z\le \unit[2000]{m}$. The initial states of the explosion products and air in this example are the same as that in Section \ref{problem:blast}, except that the bottom edge $z=0$ is now a rigid ground. The explosive center is located at the height $z=\unit[100]{m}$. And the radius of the initial interface is $\unit[0.3]{m}$ at $t=0$. Fig. \ref{rm::air_blast2} shows the pressure contours and adaptive meshes at $t=\unit[0.09]{s}$ and $\unit[0.3]{s}$. When the blast wave produced by the nuclear explosion arrives at the ground, it will be reflected firstly and propagate along the rigid ground simultaneously. When the incident angle exceeds the limit, the reflective wave switches from regular to irregular, and a Mach blast wave occurs. The peak overpressure and impulse at different radii are shown in Fig. \ref{rm::air_blast_ref2}. Our numerical results agree well the the reference data interpolated from the given standard data in \cite{Glasstone1977}.
\begin{figure}
\caption{Pressure contours and adaptive meshes for nuclear
air blast problem.}
\caption{Shock wave parameters for nuclear air blast problem.}
\label{rm::air_blast2}
\label{rm::air_blast_ref2}
\end{figure}
\subsubsection{TNT explosion in air}\label{problem:tntair}
We use this example to assess the isotropic behavior of TNT explosion in a computational domain $0\le r\le \unit[20]{m},0\le z\le \unit[40]{m}$. The initial density and pressure are $\unit[1630]{kg/m^3}$ and $\unit[9.5\times 10^9]{Pa}$ for high explosives, and $\unit[1.29]{kg/m^3}$ and $\unit[1.013\times 10^5]{Pa}$ for the air. The initial interface is a sphere of radius $\unit[0.0527]{m}$ centered at the height $z=\unit[10]{m}$. The results of shock produced by the high explosives are shown in Fig. \ref{rm::tnt1}. The shock parameters are shown in Fig. \ref{rm::tnt2}, in comparison with the experimental data in \cite{Baker1973} and \cite{Crowl1969}.
\begin{figure}
\caption{Pressure contours and adaptive meshes
for TNT explosion in air.}
\caption{Shock wave parameters for TNT explosion in air. The reference
solution 1 is taken from Baker \cite{Baker1973}, while the
reference solution 2 is taken from Crowl \cite{Crowl1969}.}
\label{rm::tnt1}
\label{rm::tnt2}
\end{figure}
\subsection{Three-dimensional shock impact problem}
In the last example, we present a three-dimensional shock impact problem in practical applications. The computational domain is a cross-shaped confined space $E_1\bigcup E_2$ where $E_1=[-2,2]\times [-2,2]\times [-6,6]$ and $E_2=[-6,6]\times [-2,2]\times[-2,2]$ in meters. The sphere of radius $\unit[0.4]{m}$ centered at the origin is filled with high explosives, while the remaining region is filled with air. The initial states are exactly the same as that of the problem in Section \ref{problem:tntair}. Due to the symmetry we only compute the problem in the first octant, namely an L-shaped region. All the boundaries are reflective walls. Again we use the $h$-adaptive technique to capture the shock front. The total number of cells is about $0.8$ -- $1$ million. To accelerate the computation we perform the parallel computing with eight processors based on the classical domain decomposition methods.
The numerical results of shock wave produced by the high explosives at $t=\unit[2.4\times 10^{-4}]{s}$ and $t=\unit[9.2\times 10^{-4}]{s}$ are displayed in Fig. \ref{rm::3d1}. From here we can see that the shock wave propagates as an expansive spherical surface in earlier period. When the spherical shock wave impinges on the rigid surface, shock strength increases and shock reflection occurs. It is also observed in the slice plots of Fig. \ref{rm::3d2} that the wave structures are much more complicated in the three-dimensional confined space. The numerical results here also show the capability of our methods in resolving fully three-dimensional flows.
\begin{figure}
\caption{Pressure contours and adaptive meshes for three-dimensional
shock impact problem. The plots (c) and (d) are turned over in the
$y$-direction in order to display the shock reflection.}
\caption{Pressure slices at $t=9.2\times 10^{-4}$s.}
\label{rm::3d1}
\label{rm::3d2}
\end{figure}
\section{Conclusions}\label{sec:conclusion}
We extend the numerical scheme in Guo \textit{et al.} \cite{Guo2016} to the fluids that obey a general Mie-Gr{\"u}neisen equations of state. The algorithm of the multi-medium Riemann problem is elaborated, which is a fundamental element of the two-medium fluid flow. A variety of preliminary numerical examples and application problems validate the effectiveness and robustness of our methods. In our future work, we will generalize the framework to fluid-solid coupling problems.
\iffalse \section*{Acknowledgments}
The authors appreciate the financial supports provided by the National Natural Science Foundation of China (Grant No. 91630310, 11421110001, and 11421101). \fi
\appendix
\section{Collections of equations of state}
In this appendix we elaborate the equations of state that are mentioned in the numerical results. For convenience, we also collect the expression of coefficients and their derivatives at the end of this part.
\subsubsection*{\bf Ideal gas EOS}
Most of gases can be modeled by the ideal gas law \begin{equation} p = (\gamma-1) \rho e, \label{eq:ideal} \end{equation} where $\gamma>1$ is the adiabatic exponent.
\subsubsection*{\bf Stiffened gas EOS}
When considering water under high pressures, the following stiffened gas EOS is often used \cite{Rallu2009,Wang2008}: \begin{equation} p = (\gamma-1) \rho e-\gamma p_\infty, \label{eq:stiffened} \end{equation} where $\gamma>1$ is the adiabatic exponent, and $p_\infty$ is a constant.
\subsubsection*{\bf Polynomial EOS}
The polynomial EOS \cite{Jha2014} can be used to model various materials \begin{equation} p = \begin{cases}
A_1\mu + A_2\mu^2 + A_3\mu^3 + (B_0+B_1\mu)\rho_0 e, &
\mu>0, \\
T_1 \mu + T_2\mu^2 + B_0 \rho_0e, &\mu \le 0, \end{cases} \label{eq:poly} \end{equation} where $\mu = {\rho}/{\rho_0} -1$ and $A_1,A_2,A_3,B_0,B_1, T_1,T_2,\rho_0$ are positive constants. In this paper, we take an alternative formulation in the tension branch \cite{Autodyn2003}, where $p=T_1 \mu + T_2\mu^2 + (B_0+B_1\mu)\rho_0e$ for $\mu\le0$, to ensure the continuity of the speed of sound at $\mu=0$. Such a formulation avoids the occurance of anomalous waves in the Riemann problem, which does not exist in real physics. When $B_1\le B_0\le B_1+2$ and $T_1\ge 2T_2$, the polynomial EOS satisfies the conditions \textbf{(C1)} and \textbf{(C3)}. In addition, if the density $\rho\ge {B_0\rho_0}/{(B_1+2)}$, then the polynomial EOS also satisfies the condition \textbf{(C2)}.
\subsubsection*{\bf JWL EOS}
Various detonation products of high explosives can be characterized by the JWL EOS \cite{Smith1999} \begin{equation} p = A_1\left(1-\frac{\omega\rho}{R_1\rho_0}\right)
\exp\left(-\frac{R_1\rho_0}{\rho}\right) +
A_2\left(1-\frac{\omega\rho}{R_2\rho_0}\right) \exp\left(-\frac{
R_2\rho_0}{\rho}\right) + \omega\rho e, \label{eq:jwl} \end{equation} where $A_1,A_2,\omega,R_1,R_2$ and $\rho_0$ are positive constants.
Obviously the JWL EOS \eqref{eq:jwl} satisfies the conditions \textbf{(C1)} and \textbf{(C2)}. To enforce the condition \textbf{(C3)} we first notice that \[ \lim_{\rho\rightarrow 0^+} h'(\rho) = 0. \] Then it suffices to ensure that $h''(\rho)\ge 0$, which is equivalent to the following inequality in terms of $\nu=\rho_0/\rho$: \[ R_1\nu-2-\omega \ge G(\nu):=\dfrac{A_2R_2}{A_1R_1}(2+\omega -R_2\nu) \exp((R_1-R_2)\nu). \] A simple calculus shows that the maximum value of the function $G(\nu)$ above is given by \[ \alpha = \dfrac{A_2R_2^2}{A_1R_1(R_1-R_2)} \exp\left(\dfrac{(2+\omega)(R_1-R_2)-R_2}{R_2}\right). \] Therefore a sufficient condition for \textbf{(C3)} is that the density satisfies \[ \rho\le \dfrac{R_1}{2+\omega+\alpha}\rho_0, \] which is valid for most cases.
\subsubsection*{\bf Cochran-Chan EOS}
The Cochran-Chan EOS is commonly used to describe the reactants of condensed phase explosives \cite{Saurel2007,Lee2013}, which can be formulated as \begin{equation} p = \dfrac{A_1(R_1-1-\omega)}{R_1-1}
\left(\dfrac{\rho}{\rho_0}\right)^{R_1}-
\dfrac{A_2(R_2-1-\omega)}{R_2-1}
\left(\dfrac{\rho}{\rho_0}\right)^{R_2}+
\omega \rho e, \label{eq:CC} \end{equation} where $A_1,A_2,\omega,R_1,R_2$ and $\rho_0$ are positive constants. The Cochran-Chan EOS satisfies the conditions \textbf{(C1), (C2)} and \textbf{(C3)} if $1<R_2\le 1+\omega \le R_1$.
\setlength{\tabcolsep}{2pt} \setlength{\rotFPtop}{0pt plus 1fil} \begin{sidewaystable}
\centering
\caption*{List of coefficients and their derivatives for several equations of state.}
\small \begin{tabular}{cccccc} \toprule & \multicolumn{1}{c}{\bf Ideal}&
\multicolumn{1}{c}{\bf Stiffened} &
\multicolumn{1}{c}{\bf Polynomial} &
\multicolumn{1}{c}{\bf JWL}&
\multicolumn{1}{c}{\bf Cochran-Chan} \\ \midrule
$\varGamma(\rho)$ & $\gamma-1$ & $\gamma-1$ & $B_1+{(B_0-B_1)\rho_0}/{\rho}$ & $\omega$ & $\omega$ \\ [10mm]
$\varGamma'(\rho)$ & $0$ & $0$ & $-{(B_0-B_1)\rho_0}/{\rho^2}$ & $0$ & $0$ \\ [10mm]
$\varGamma''(\rho)$ & $0$ & $0$ & ${2(B_0-B_1)\rho_0}/{\rho^3}$ & $0$ & $0$ \\ [10mm]
$h(\rho)$ & $0$ & $-\gamma p_\infty$ & $\begin{cases} A_1\mu + A_2 \mu^2 + A_3 \mu^3, & \rho > \rho_0 \\ T_1 \mu+T_2\mu^2, & \rho \le \rho_0
\end{cases}$ &
\bgroup
\footnotesize \begin{tabular}{r} $A_1\left(1-\dfrac{\omega\rho}{R_1\rho_0}\right)
\exp\left(-\dfrac{R_1\rho_0}{\rho}\right)$ \\ $+A_2\left(1-\dfrac{\omega\rho}{R_2\rho_0}\right)
\exp\left(-\dfrac{R_2\rho_0}{\rho}\right)$ \end{tabular} \egroup & \bgroup
\footnotesize \begin{tabular}{r}
$\dfrac{A_1(R_1-1-\omega)}{R_1-1}\left(\dfrac{\rho}{\rho_0}\right)^{R_1}$ \\
$-\dfrac{A_2(R_2-1-\omega)}{R_2-1}\left(\dfrac{\rho}{\rho_0}\right)^{R_2}$ \end{tabular} \egroup \\ [10mm]
$h'(\rho)$ & $0$ & $0$ & $\begin{cases} (A_1+2A_2\mu+3A_3\mu^2)/\rho_0, & \rho > \rho_0 \\ (T_1+2T_2\mu)/\rho_0, & \rho \le \rho_0
\end{cases}$ & \bgroup
\footnotesize \begin{tabular}{r} $A_1\left(\dfrac{R_1\rho_0}{\rho^2}-\dfrac{\omega}{\rho} -\dfrac{\omega}{R_1\rho_0}\right) \exp\left(-\dfrac{R_1\rho_0}{\rho}\right)$ \\ $+A_2\left(\dfrac{R_2\rho_0}{\rho^2}-\dfrac{\omega}{\rho} -\dfrac{\omega}{R_2\rho_0}\right) \exp\left(-\dfrac{R_2\rho_0}{\rho}\right)$ \end{tabular} \egroup & \bgroup
\footnotesize \begin{tabular}{r}
$\dfrac{A_1R_1(R_1-1-\omega)}{R_1-1}\left(\dfrac{\rho}{\rho_0}\right)^{R_1-1}$ \\
$-\dfrac{A_2R_2(R_2-1-\omega)}{R_2-1}\left(\dfrac{\rho}{\rho_0}\right)^{R_2-1}$ \end{tabular} \egroup \\ [10mm]
$h''(\rho)$ & $0$ & $0$ & $\begin{cases} (2A_2+6A_3\mu)/\rho_0^2, & \rho > \rho_0 \\ 2T_2/\rho_0^2, & \rho \le \rho_0 \end{cases}$ & \bgroup
\footnotesize \begin{tabular}{r}
$\dfrac{A_1R_1\rho_0}{\rho^3}
\left(\dfrac{R_1\rho_0}{\rho}-2-\omega\right)
\exp\left(-\dfrac{R_1\rho_0}{\rho}\right)$ \\
$+\dfrac{A_2R_2\rho_0}{\rho^3}
\left(\dfrac{R_2\rho_0}{\rho}-2-\omega\right)
\exp\left(-\dfrac{R_2\rho_0}{\rho}\right)$ \end{tabular} \egroup & \bgroup
\footnotesize \begin{tabular}{r}
${A_1R_1(R_1-1-\omega)}\left(\dfrac{\rho}{\rho_0}\right)^{R_1-2}$ \\
$-{A_2R_2(R_2-1-\omega)}\left(\dfrac{\rho}{\rho_0}\right)^{R_2-2}$ \end{tabular} \egroup \\ \bottomrule
\end{tabular} \end{sidewaystable}
\section*{\refname}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Positive weight function and classification of g-frames} \begin{abstract} Given a positive weight function and an isometry map on a Hilbert spaces ${\mathbb H}$, we study a class of linear maps which is a $g$-frame, $g$-Riesz basis and a $g$-orthonormal basis for ${\mathbb H}$ with respect to $\C$ in terms of the weight function. We apply our results to study the frame for shift-invariant subspaces on the Heisenberg group. \end{abstract}
\section{Introduction and preliminaries}
A frame for a Hilbert space is a countable set of overcompleted vectors such that each vector in the Hilbert space can be represented in a non-unique way in terms of the frame elements. The redundancy and the flexibility in the representation of a Hilbert space vector by the frame elements make the frames a useful tool in mathematics as well as in interdisciplinary fields such as sigma-delta quantization \cite{ben06}, neural networks \cite{can99}, image processing \cite{can05}, system modelling \cite{dud98}, quantum measurements \cite{eld02}, sampling theory \cite{fei94}, wireless communications \cite{str03} and many other well known fields.
Given a Hilbert space $\mathcal H$, a countable family of vectors $\{x_j\}_{j\in J}\subset \mathcal H$ is called a {\it frame} for $\mathcal H$ if there are positive constants $A$ and $B$ such that for any $x\in \mathcal H$,
$$A\|x\|^2 \leq \sum_{j\in J} |\langle x, x_j\rangle|^2 \leq B \|x\|^2 .$$
The frames were introduced for the first time by Duffin and Schaeffer \cite{duf52}, in the context of nonharmonic Fourier series \cite{you01}. The notion of a frame extended to g-frame by Sun \cite{sun06} in 2006 to generalize all the existing frames such as bounded quasi-projectors \cite{for04}, frames of subspaces \cite{cas04}, pseudo-frames \cite{li04}, oblique frames \cite{chr04}, outer frames \cite{ald04}, and time-frequency localization operators \cite{dor06}. Here, we recall the definition of a g-frame.
\begin{definition}
Let $\mathcal H$ be a Hilbert space and $\{\mathcal K_j\}_{j\in J}$ be a countable family of Hilbert spaces with associated norm $\| \cdot\|_{\mathcal K_j}$. A countable family of linear and bounded operators $\{\Lambda_j: {\mathbb H} \to \K_j\}_{j \in J}$ is called a {\it $g$-frame} for $\mathcal H$ with respect to $\{\mathcal K_j\}_{j\in J}$ if there are two positive constants $A$ and $B$ such that for any $f\in \mathcal H$ we have \begin{equation}\label{eq01}
A\|f\|_{\mathcal H}^2 \leq \sum_{j\in J} \|\Lambda_j(f)\|_{\mathcal K_j}^2 \leq B \|f\|_{\mathcal H}^2. \end{equation} \end{definition}
The constants $A$ and $B$ are called {\it $g$-frame lower and upper bounds}, respectively. If $A=B=1$, then it is called a {\it Parseval $g$-frame}. For example, by the Riesz representation theorem, every $g$-frame is a frame if $\mathcal K_j= \Bbb C$ for all $j\in J$. And, every frame is a $g$-frame with respect to $\Bbb C$. If the right-hand side of (\ref{eq01}) holds, it is said to be a {\it $g$-Bessel sequence} with bound $B$. The family $\{\Lambda_j \}_{j \in J}$ is called {\it $g$-complete}, if for any vector $f\in \mathcal H$ with $\Lambda_j(f)=0$ for $j \in J$, we have $f=0$. If $\{\Lambda_j \}_{j \in J}$ is $g$-complete and there are positive constants $A$ and $B$ such that for any finite subset $J_1 \subset J$ and $g_j \in \K_j, \; j \in J_1$,
\[ A \sum_{j \in J_1} \|g_j \|^2 \leq \bigg\|\sum_{j \in J_1} \Lambda_j^* (g_j) \bigg\|^2 \leq B \sum_{j \in J_1}\|g_j \|^2, \] then $\{\Lambda_j \}_{j \in J}$ is called a {\it $g$-Riesz basis} for ${\mathbb H}$ with respect to $\{\K_j\}_{j \in J}$. Here, $\Lambda_j^*$ denotes the adjoint operator. We say $\{\Lambda_j \}_{j \in J}$ is a $g$-orthonormal basis for ${\mathbb H}$ with respect to $\{\K_j\}_{j \in J}$ if it satisfies the following: \begin{align}\label{onb1} \langle \Lambda^*_{i} g, \Lambda^*_{j} h \rangle &= 0 \quad \forall g \in \K_i, \; h \in \K_j, i\neq j \\\label{onb2}
\| \Lambda^*_{i} g\|^2 &= \|g \|^2, \quad \forall i, \; g \in \K_i \\\label{onb3}
\sum_{j \in J}\|\Lambda_jf \|^2 &=\|f \|^2, \quad \forall f \in {\mathbb H}. \end{align}
Before we state the main results of this paper, let us consider the following example. For a function $\phi\in L^2(\Bbb R^d)$, $d\geq 1$ and $m, n\in \Bbb Z^d$, the modulation and translation of $\phi$ by multi-integers $m$ and $n$ are defined by
$$M_m\phi(x) = e^{2\pi i \langle m, x\rangle} \phi(x), \quad T_n\phi(x) =\phi(x-n) .$$ The Gabor (Weil-Heisenberg) system generated by $\phi$ is
$$\mathcal G(\phi):=\{M_m T_n\phi: \ m, n\in \Bbb Z^d\} .$$
It is well-known that the ``basis\rq\rq{} property of the Gabor system $\mathcal G(\phi)$ for its spanned vector space can be studied by the Zak transform of $\phi$
$$Z\phi(x, \xi) = \sum_{k\in \Bbb Z^d} \phi(x+k) e^{2\pi i \langle \xi, k\rangle} .$$
For example, the Gabor system $\mathcal G(\phi)$ is a Riesz basis for the spanned vector space if and only if there are positive constants $A$ and $B$ such that $A\leq |Z\phi(x, \xi)|\leq B$ a.e. $(x,\xi)\in [0,1]^d\times [0,1]^d$ (\cite{HSWW}). The purpose of this paper is to show that the above result is a particular case of a more general theory involving $g$-frames.
For the rest of the paper we shall assume the following. $\Omega$ is a measurable set with measure $dx$. We assume that $\Omega$ has finite measure $|\Omega|$ and $|\Omega|=1$. We let $w:\Omega \to (0,\infty)$be a measurable map with $\int_\Omega w(x)dx<\infty$. Let $\U$ be a Hilbert space over the field $\Bbb F$ ($\Bbb R$ or $\Bbb C$) with associated inner product $\langle \cdot , \cdot \rangle_{\U} $. We denote by $L^2_w(\Omega, \U)$ the weighted Hilbert space of all measurable functions $f:\Omega\to \U$ such that
$$\|f\|_{L^2_w(\Omega, \U)}^2: = \int_\Omega \|f(x)\|_\U^2 w(x) dx<\infty .$$
The associated inner product of any two functions $f, g$ in $L_w^2(\Omega, \U)$ is then given by $$\langle f, g\rangle_{L_w^2(\Omega, \U)} = \int_\Omega \langle f(x), g(x)\rangle_\U w(x) dx. $$
A countable family of unit vectors $\{f_k\}_{k\in K}$ in ${L_w^2(\Omega, \U)}$ constitute an ONB (orthonormal basis) for ${L_w^2(\Omega, \U)}$ with respect to the weight function $w$ if the family is orthogonal and for any $g\in {L_w^2(\Omega, \U)}$ the Parseval identity holds:
$$\| g \|^2 = \sum_{k\in K} \left|\langle g, f_k\rangle_{L_w^2(\Omega, \U)}\right|^2 =\sum_{k\in K}\left|
\int_\Omega \langle g(x), f_k(x)\rangle_{\U} w(x) dx\right|^2 .$$
To avoid any confusion, in the sequel, we shall use subscripts for all inner products and associated norms for Hilbert spaces, when necessary. For the rest, we assume that $S: \mathcal H \to L^2_w(\Omega, \U)$ is a linear and unitary map. Thus for any $f\in \mathcal H$
$$ \|f\|^2 = \int_\Omega \|Sf(x)\|_{\U}^2 w(x) dx. $$
We fix an ONB $\{f_n\}_{n\in K}$ for $L^2(\Omega)$ and ONB $\{g_m\}_{m\in J}$ for the Hilbert space $\U$, and define $$G_{m,n}(x): = f_n(x) g_m, \quad \forall x\in \Omega, \ (m,n)\in K\times J.$$ And, $$\tilde \Lambda_{(m,n)}(f)(x) = \langle S(f)(x), G_{(m,n)}(x) \rangle_{\U} \quad \forall \ f\in \mathcal H, x\in \Omega .$$
Our main results are the following. \begin{theorem}\label{th1}
Let $\{f_n\}_{n\in K}\subset L^2(\Omega)$, $\{g_m\}_{m\in J} \subset \U$ and $\{\tilde \Lambda_{m,n}\}$ be as in above. Assume that $|f_n(x)|=1$ for a.e. $x\in \Omega$. Then the following hold: \begin{itemize} \item [(a)] $\{\tilde\Lambda_{(m,n)}\}_m $ is a Parseval $g$-frame for $\mathcal H$. Thus $\{\Lambda_{(m,n)}\}_m$ is a Bessel sequence. \item [(b)] For any ${(m,n)}$, the linear map $\Lambda_{m,n}: \mathcal H \to \C$ defined by $$\Lambda_{m,n}(f) =\int_\Omega \tilde\Lambda_{m,n}(f)(x) \ w(x) dx $$ is well-defined. And, $\{\Lambda_{m,n}\}$ is a frame for ${\mathbb H}$ if and only if there are positive finite constants $A$ and $B$ such that $A\leq w(x)\leq B$ for a.e. $x\in \Omega$. \item [(c)] The family $\{\Lambda_{m,n}\}$ is a Riesz basis for ${\mathbb H}$ if and only if there are positive finite constants $A$ and $B$ such that $A\leq w(x)\leq B$ for a.e. $x\in \Omega$. \end{itemize} \end{theorem}
\begin{corollary}\label{main result}
Let $\{\lambda_k\}_{k\in J}$ be an orthonormal basis (or a Parseval frame) for $L^2(\Omega)$ such that $|\lambda_k(x)|=1$ for all $x\in \Omega$. Assume that $S: \mathcal H \to L^2_w(\Omega)$ is a unitary map. Then the sequence of operators $\{\Lambda_k\}_{k\in J}$ defined by $$\Lambda_k(f) =\int_\Omega S(f)(x) \overline{ \lambda_k(x) } w(x) dx $$ is a frame for ${\mathbb H}$ if and only if there are positive finite constants $A$ and $B$ such that $A\leq w(x)\leq B$ for a.e. $x\in \Omega$. \end{corollary}
\begin{theorem}\label{th3}
Let $\{f_n\}_{n\in K}\subset L^2(\Omega)$, $\{g_m\}_{m\in J} \subset \U$ and $\{\tilde \Lambda_{m,n}\}$ be as in above. Assume that $|f_n(x)|=1$ for a.e. $x\in \Omega$. The family $\{\Lambda_{m,n}\}$ is an ONB for ${\mathbb H}$ if and only if $w(x)=1$ for a.e. $x\in \Omega$. \end{theorem}
\section{Proof of Theorem \ref{th1}} First we prove the following lemmas which we need for the proof of Theorem \ref{th1}.
\begin{lemma}\label{technical lemma} Let $\{f_n\}$ be an ONB for the weighted Hilbert space $L_w^2(\Omega)$. Let $\{g_m\}$ be an ONB for a Hilbert space $\U$. Define $G_{m,n}(x) = f_n(x) g_m$. Then the family $\{G_{m,n}\}_{J\times K}$ is an ONB for $L_w^2(\Omega, \U)$. \end{lemma}
In order to prove the lemma, we shall recall the following result from \cite{iosevich-mayeli-14} and prove it here for the sake of completeness.
\begin{lemma}\label{mixed orthonormal bases} Let $(X,\mu)$ be a measurable space, and $\{f_n\}_n$ be an orthonormal basis for $L^2(X):=L^2(X, d\mu)$. Let $Y$ be a Hilbert space and $\{g_m\}_m$ be a family in $Y$. For any $m, n$ and $x\in X$ define $G_{m,n}(x) := f_n(x) g_m$. Then $\{G_{m,n}\}_{m,n}$ is an orthonormal basis for the Hilbert space $L^2(X,Y,d\mu)$ if and if $\{g_m\}_m$ is an orthonormal basis for $Y$. \end{lemma}
\begin{proof} For any $m, n$ and $m\rq{}, n\rq{}$ we have \begin{align}\label{orthogonality-relation} \langle G_{m,n}, G_{m\rq{},n\rq{}} \rangle &= \int_X \langle f_m(x) g_n, f_{m\rq{}}(x) g_{n\rq{}} \rangle_Y \ d\mu(x)\\\notag & = \langle f_m,f_{m\rq{}}\rangle_{L^2(X)} \langle g_n, g_{n\rq{}}\rangle_Y \\\notag &= \delta_{m,m\rq{}} \langle g_n, g_{n\rq{}}\rangle_Y. \end{align}
This shows that the orthogonality of $\{G_{m,n}\}_{m,n}$ is equivalent to the orthogonality of $\{g_m\}_m$. And, $\|G_{m,n}\| =1$ if and only if $\|g_n\|=1$.
Let $\{g_m\}_m$ be an orthonormal basis for $Y$. To prove the completeness of $\{G_{m,n}\}$ in $L^2(X,Y,d\mu)$, let $F\in L^2(X,Y,d\mu)$ such that $\langle F, G_{m,n}\rangle =0$, $\forall \ m, n$. We claim $F=0$. By the definition of the inner product we have \begin{align}\label{inner-product} 0=\langle F, G_{m,n}\rangle &=\int_X \langle F(x), G_{m,n}(x)\rangle_Y d\mu(x)\\\notag &= \int_X \langle F(x), f_m(x) g_n\rangle_Y d\mu(x) \\\notag &= \int_X \langle F(x), g_n\rangle_Y \overline{f_m(x)} d\mu(x) \\\notag &= \langle A_n, f_m\rangle \end{align} where $$A_n: X\to \C; \ \ x \mapsto \langle F(x), g_n\rangle_Y. $$
$A_n$ is a measurable function and lies in $L^2(X)$ with $\|A_n\|\leq \|F\|$. Since $\langle A_n, f_m\rangle_{L^2(X)}=0$ for all $m$, then $A_n=0$ by the completeness of $\{f_m\}$. On the other hand, by the definition of $A_n$ we have $\langle F(x), g_n\rangle_{Y}=0$ for a.e. $x\in X$. Since $\{g_n\}$ is complete in $Y$, then $F(x)=0$ for a.e. $x\in X$. This proves the claim.
Conversely, assume that $\{G_{m,n}\}_{m,n}$ is an orthonormal basis for the Hilbert space $ L^2(X,Y,d\mu)$. Therefore by (\ref{orthogonality-relation}), $\{g_m\}$ is an orthonormal set. We prove that if for $g\in Y$ and $\langle g, g_m\rangle=0$ for all $m$, then $g$ must be identical to zero. For this, for any $n$ define the map $$B_n: X\to Y; \ \ x\mapsto f_n(x)g.$$
Then $B_n$ is measurable and it belongs to $ L^2(X,Y,d\mu)$ and $\|B_n\| = \|g\|_Y$. Thus \begin{align} B_n&=\sum_{n\rq{},m} \langle B_n, G_{m,n\rq{}}\rangle_{L^2(X,Y,d\mu)} G_{m,n\rq{}}\\\notag &= \sum_{n\rq{},m} \langle f_n, f_{n\rq{}}\rangle_{L^2(X)} \langle g, g_m\rangle_Y G_{m,n\rq{}}\\\notag &= \sum_{m} \langle g, g_m\rangle_Y G_{n,m}. \end{align} By the assumption that $ \langle g, g_m\rangle_Y=0$ for all $m$, we get $B_n=0$. This implies that $B_n(x)= f_n(x) g=0$ for a.e. $x$. Since, $f_n\neq 0$, then $g$ must be a zero vector, and hence we are done. \end{proof}
\begin{lemma}\label{boundedness}
$\tilde \Lambda_{(m,n)}: \mathcal H\to L^2_w(\Omega)$ is a bounded operator and $\|\tilde \Lambda_{(m,n)}(f)\|_{L^2_w(\Omega)}\leq \|f\|$. \end{lemma}
\begin{proof} Let $f\in {\mathbb H}$. Then for any $m\in J$ and $n\in K$, \begin{align*}
\| \tilde\Lambda_{m,n}(f)\|_{L^2_w(\Omega)}^2 &= \int_\Omega |\tilde\Lambda_{m,n}(f)(x)|^2 w(x) dx \\
&= \int_\Omega |\langle Sf(x), f_n(x)g_m\rangle_{\U}|^2 w(x) dx\\
&= \int_\Omega |\langle Sf(x), g_m\rangle_{\U}|^2 w(x) dx . \end{align*} Using the Cauchy--Schwartz inequality in the preceding line, we get \begin{align*}
\| \tilde\Lambda_{m,n}(f)\|_{L^2_w(\Omega)}^2 &\leq \int_\Omega \|Sf(x)\|^2 w(x) dx = \|f\|^2 . \end{align*} \end{proof}
Here, we first calculate the adjoint of $\tilde\Lambda_{m,n}$: $S$ is a unitary map. Then for any $f\in {\mathbb H}$ and $h\in L_w^2(\Omega,\U)$ we have \begin{align} \int_\Omega \langle Sf(x), h(x)\rangle_{\U} w(x)dx= \langle Sf, h\rangle_{L_w^2(\Omega,\U)} =\langle f, S^{-1}h\rangle . \end{align} Therefore for any $\phi\in L_w^2(\Omega)$ we get \begin{align}\label{adjoint}
\langle \tilde \Lambda_{m,n} f, \phi\rangle = \langle f, S^{-1}((f_n \phi)g_m)\rangle , \end{align} where $(f_n \phi)g_m \in L_w^2(\Omega, \U)$ and $(f_n \phi)g_m(x) = f_n(x)\phi(x) g_m$. The relation (\ref{adjoint}) indicates that $$\tilde \Lambda_{m,n}^*(\phi) = S^{-1}((f_n \phi)g_m).$$ Notice, for any $f\in {\mathbb H}$, $$\Lambda_{m,n} f = \langle f, S^{-1}(f_ng_m)\rangle_{{\mathbb H}} .$$ Thus $\Lambda_{m,n}^*: \C \to {\mathbb H}$ is given by $c\to cS^{-1}(f_ng_m)$.
\begin{proof}[Proof of Theorem \ref{th1}]
$(a)$: Observe that $|\tilde \Lambda_{m,n}(f)(x)|\leq \| Sf(x)\|$ and $S$ is an isometry map. For any $f \in {\mathbb H}$ and $n\in K$ we have \begin{align*}
\sum_m \| \tilde \Lambda_{m,n} (f)\|^2_{L^2_w(\Omega)} & =\sum_m \int_{\Omega} | \tilde \Lambda_{m,n} (f)(x)|^2 w(x) dx \\
& = \sum_m \int_{\Omega} | \langle G_{m,n}(x) , S(f)(x)\rangle_{\U} |^2 w(x) dx \\
& = \int_{\Omega} \sum_m| \langle G_{m,n}(x) , S(f)(x)\rangle_{\U} |^2 w(x) dx \\
& = \int_{\Omega} \sum_m| \langle g_m , w^{1/2}S(f)(x)\rangle_{\U} |^2 dx. \end{align*} By the assumptions of the theorem, for a.e. $x\in \Omega$, the sequence $\{g_m\}_m$ is an ONB for $\U$. Invoking this along the isometry property of $S$ in the last equation above, we get \begin{align}\label{parseval-proprty}
\sum_m \| \tilde \Lambda_{m,n} (f)\|^2_{L^2_w(\Omega)}
= \int_{\Omega} \| S(f)(x) \|^2_{\U} \; w(x) dx
= \| f \|^2. \end{align} Therefore, $\{\tilde \Lambda_{m,n}\}_m$ is a Parseval $g$-frame for ${\mathbb H}$ with respect to $L^2_w(\Omega)$. To prove that $\{\Lambda_{m,n}\}_m$ is a Bessel sequence, note that by the H{\"o}lder's inequality for in the weighted Hilbert space $L^2_w(\Omega)$ we can write \begin{align*}
|\Lambda_{m,n}(f)| \leq \int_\Omega |\tilde \Lambda_{m,n} (f)(x)| w(x) dx & \leq \bigg( \int_\Omega |\tilde \Lambda_{m,n} (f)(x)|^2 w(x) dx \bigg)^{\frac{1}{2}} \bigg( \int_\Omega w(x) dx \bigg)^{\frac{1}{2}}.
\end{align*} By Lemma \ref{boundedness}, the first integral on the right is finite. Therefore by summing the square of the terms over $m$ we get
\[ \sum_{m \in J} |\Lambda_{m,n}(f)|^2 \leq C \sum_{m \in J} \int_\Omega |\tilde \Lambda_{m,n} (f)(x) |^2 w(x) dx =C \sum_{m \in J} \| \tilde \Lambda_{m,n} (f) \|_{L^2_w(\Omega)}^2 =C\| f \|^2, \] where $C:= \int_\Omega w(x) dx$ is a non-zero constant. Thus $\{\Lambda_{m,n}\}_{m \in J}$ is a Bessel sequence for ${\mathbb H}$ with bound $C$. Notice, in the last equality we used (\ref{parseval-proprty}).
$(b)$ The map $\Lambda_{m,n}:{\mathbb H} \to \C$ is linear, well-defined and bounded. Indeed, for any $f\in {\mathbb H}$, \begin{align*}
\int_\Omega |\tilde \Lambda_{m,n} (f)(x)| w(x) dx &\leq \left(\int_\Omega w(x) dx\right)^{1/2} \left(\int_\Omega\|Sf(x)\|^2 w(x) dx\right)^{1/2} \\
&= \|f\| \left(\int_\Omega w(x) dx\right)^{1/2}. \end{align*} Assume that $A\leq w(x)\leq B$ for almost every $x\in \Omega$. Let $f\in \mathcal H$. Then \begin{align}\notag
\sum_{m,n} |\Lambda_{m,n}(f)|^2 &= \sum_{m,n} \left| \int_\Omega \langle G_{m,n}(x) , S(f)(x)\rangle_{\U} w(x) dx\right|^2 \\\notag
&= \sum_{m,n} \left| \int_\Omega \langle G_{m,n}(x) , S(f)(x)w(x)\rangle_{\U} dx\right|^2 \\\label{eq1}
&=\sum_{m,n} \left| \langle G_{m,n} , S(f)w\rangle_{L^2(\Omega, \U)}\right|^2 . \end{align} Since the countable family $\{ G_{m,n}\}_{m,n}$ is an ONB for $L^2(\Omega, \U)$. Thus \begin{align}\notag (\ref{eq1})&= \Vert S(f) w \Vert^2_{L^2(\Omega, \U)} \\\label{equ2} & = \int_\Omega \Vert S(f)(x)\Vert^2_{\U} \ w(x)^2 dx. \end{align} By invoking the assumption that $w(x)\leq B$ for a.e. $x\in \Omega$ in (\ref{equ2}) we obtain \begin{align}\notag (\ref{equ2})
\leq B \int_\Omega \Vert S(f)(x)\Vert^2_{\U} \ w(x) dx
= B \|S(f)\|^2_{L^2_w(\Omega, \U)}= B \| f\|^2. \end{align} This proves that the sequence $\{\Lambda_{m,n}\}_{m,n}$ is a Bessel sequence for ${\mathbb H}$. An analogues argument also proves the frame lower bound condition for $\{\Lambda_{m,n}\}_{m,n}$.
For the converse, assume that $\{\Lambda_{m,n}\}_{m,n}$ is a frame for $\mathcal H$ with the frame bounds $0<A\leq B<\infty$. Therefore for any $f\in \mathcal H$
$$A\|f\|^2 \leq \sum_{m,n} |\Lambda_{m,n}(f)|^2 \leq B \|f\|^2.$$ Assume that there is a set $E\subset \Omega$ with positive measure such that $w(x)<A$ for all $x\in E$. We will prove that there exits a function in $\mathcal H$ for which the lower frame condition dose not hold. To this end, let $e_0 \in \U$ be a unit vector and let $\vec{0}$ denote the zero vector in $\U$. Define $\chi_E(x):= 1_E(x) e_0$. By the assumption, $w \in L^1(E)$. Thus $\chi_E \in L^2_w(\Omega, \U)$. $S$ is unitary, therefore there is a function $\phi_E\in {\mathbb H}$ such that $S(\phi_E)=\chi_E$, and we have \begin{align}\label{iso}
\| S(\phi_E) \|_{L^2_w(\Omega, \U)}= \|\phi_E \|_{{\mathbb H}}=\| \chi_E \|_{L^2_w(\Omega, \U)}. \end{align} On the other hand, the sequence $\{G_{m,n}\}_{m,n}$ is an ONB for $L^2(\Omega, \U)$. Thus \begin{align*}
\sum_{m,n} |\Lambda_{m,n}(\phi_E)|^2 &= \sum_{m,n} \left| \langle G_{m,n}, S(\phi_E)w \rangle_{L^2(\Omega, \U)} \right|^2 \\ &= \Vert \chi_E w \Vert^2_{L^2(\Omega, \U)} \\ & = \int_\Omega \Vert \chi_E(x) \Vert^2_{\U} \ w(x)^2 dx \\ &< A \int_\Omega \Vert \chi_E(x) \Vert^2_{\U} \ w(x) dx \\
& = A \|\chi_E\|^2_{L_w^2(\Omega,\U)} \\
&= A \| \phi_E \|^2_{{\mathbb H}} \hspace{1in} \text{by \ (\ref{iso})}. \end{align*} The preceding calculation shows that the lower frame bound condition fails for $\phi_E$. This contradicts our assumption that $\{\Lambda_{m,n}\}_{m,n}$ is a frame for ${\mathbb H}$, therefore $w(x)\geq A$ a.e. $x\in \Omega$. The argument for the upper bound for $w$ follows similarly.
$(c)$ Assume that $A\leq w(x)\leq B$ for almost every $x\in \Omega$. Let $\{c_{m,n}\}_{m,n}$ be any finite sequence in $\C$. Then \begin{align*}
\left\| \sum_{m,n} \Lambda_{m,n}^*(c_{m,n}) \right\|_{\mathbb H}^2 &= \left\| \sum_{m,n} c_{m,n} S^{-1}(f_ng_m) \right\|_{\mathbb H}^2 \\
& = \left\| S^{-1}\left(\sum_{m,n} c_{m,n} f_ng_m\right) \right\|_{\mathbb H}^2 \\
& = \left\| \sum_{m,n} c_{m,n} f_ng_m\right\|_{L^2_w(\Omega,\U)}^2 & \text{($S$ is unitary)}\\
&= \int_\Omega \left\| \sum_{m,n} c_{m,n} f_n(x)g_m\right\|_{\U}^2 w(x) dx\\
& = \int_\Omega \sum_{m,n} |c_{m,n}|^2 w(x) dx & \text{(by orthogonality of $g_m$)}\\
&\leq B \sum_{m,n} |c_{m,n}|^2 &\text{(since $w(x)\leq B$ a.e. $x\in \Omega$).} \end{align*} We also have \begin{align*}
\left\| \sum_{m,n} \Lambda_{m,n}^*(c_{m,n}) \right\|_{\mathbb H}^2 & =
\int_\Omega \sum_{m,n} |c_{m,n}|^2 w(x) dx \geq A \sum_{m,n} |c_{m,n}|^2 &\text{(since $w(x) \geq A$ a.e. $x\in \Omega$).} \end{align*} These show that $\{\Lambda_{m,n}\}_{K\times J}$ is a Riesz basis for ${\mathbb H}$ with lower and upper Riesz bounds $A$ and $B$, respectively.
Now assume that $\{\Lambda_{m,n}\}_{K\times J}$ is a Riesz basis for ${\mathbb H}$ with Riesz bounds $A$ and $B$. Therefore, for any sequence $\{c_{m,n}\}_{m,n}$ the inequalities hold: \begin{align}\label{Riesz inequality}
A\sum_{m,n} |c_{m,n}|^2 \leq \left\| \sum_{m,n}\Lambda_{m,n}^*(c_{m,n})\right\|^2 \leq B \sum_{m,n} |c_{m,n}|^2. \end{align}
We show that there are positive constants $A$ and $B$ such that $A\leq w(x)\leq B$ for a.e. $x\in \Omega$. In contrary, without loss of generality, we assume then there is a measurable subset $E\subset \Omega$ with positive measure such that $w(x)<A$ for all $x\in E$. Let $e$ be any unitary vector in the Hilbert space $\U$ and define the function ${\bf 1}_E(x) = e$ if $x\in E$ and otherwise ${\bf 1}_E(x)=0$. It is clear that ${\bf 1}_E\in L^2(\Omega, \U)$. Thus, there are coefficients $\{c_{m,n}\}_{K\times J}$ such that ${\bf 1}_E= \sum_{m,n} c_{m,n} f_n g_m$ and $\|{\bf 1}_E\|^2= \sum_{m,n}|c_{m,n}|^2$. Then $\|{\bf 1}_E(x)\|=1$ for all $x\in E$ and we get \begin{align*}
\left\| \sum_{m,n}\Lambda_{m,n}^*(c_{m,n})\right\|^2 &= \left\| \sum_{m,n} c_{m,n} f_ng_m\right\|_{L^2_w(\Omega,\U)}^2 \\
&= \huge\int_\Omega \left\| \sum_{m,n} c_{m,n} f_n(x)g_m\right\|_{\U}^2 w(x) dx\\
&= \huge\int_\Omega \left\| {\bf 1}_E(x)\right\|_{\U}^2 w(x) dx\\ &= \huge\int_E w(x) dx\\
&\leq A \int_\Omega \left\| {\bf 1}_E(x)\right\|_{\U}^2 dx \quad \text{(by the assumption $w(x)<A$ for all $x\in E$)}\\
&= A \int_\Omega \left\| \sum_{m,n} c_{m,n} f_n(x)g_m\right\|_{\U}^2 dx \\
&= A \int_\Omega \sum_{m,n} |c_{m,n}|^2 dx \\
&= A \sum_{m,n} |c_{m,n}|^2 . \end{align*} This is contrary to the lower bound condition in (\ref{Riesz inequality}). \end{proof}
\section{Proof of Corollary \ref{main result}}
\begin{proof} Assume that $A\leq w(x)\leq B$ for almost every $x\in \Omega$. Let $f\in \mathcal H$. Then \begin{align}\label{first line}
\sum_{k \in J} |\Lambda_k(f)|^2 &= \sum_{k \in J} \left| \int_\Omega S(f)(x) \overline{\lambda_k(x)} w(x) dx\right|^2. \end{align} By the fact that $\{\lambda_k\}_{k \in J}$ is an orthonormal basis for $L^2(\Omega)$, using the Plancherel\rq{}s theorem we continue as follows: \begin{align*}
(\ref{first line})&= \int_\Omega |S(f)(x) w(x)|^2 dx
\leq B \int_\Omega |S(f)(x)|^2 w(x) dx
= B \|f\|^2. \end{align*} The frame boundedness from below by $A$ also follows with a similar calculation.
For the converse, we shall use a contradiction argument. Assume that $\{\Lambda_k\}_{k \in J}$ is a frame for $\mathcal H$ with the frame bounds $0<A\leq B<\infty$. Therefore for any $f\in \mathcal H$ we have
$$A\|f\|^2 \leq \sum_{k \in J} |\Lambda_k(f)|^2 \leq B \|f\|^2.$$
Assume that $E\subset \Omega$ be a measurable set with positive measure such that $w(x)< A$ for all $x\in E$. We shall show that the lower bound condition for the frame $\{\Lambda_k\}_{k \in J}$ must then fail for the lower bound $A$.
By the assumptions, $w \in L^1(E)$. Thus $1_E\in L^2_w(\Omega)$. Since $S$ is an onto map, assume that $\phi_E$ is the pre-image of $1_E$ in $\mathcal H$. Therefore $S(\phi_E) = 1_E$ and $\|\phi_E\|_\mathcal H = \|1_E\|_{L^2_w(\Omega)}$ and we have \begin{align}\label{E1}
\sum_{k \in J} |\Lambda_k(\phi_E)|^2 = \sum_{k \in J} \left|\int_\Omega S(\phi_E)(x) \overline{\lambda_k(x)} w(x)dx \right|^2
= \int_\Omega |1_E(x)|^2 |w(x)|^2 dx
= \int_E |w(x)|^2 dx. \end{align} Since $w(x)<A$ for all $x\in E$, then from the last integral we obtain the following: \begin{align*} (\ref{E1}) \leq A \int_E w(x) dx
= A \int_\Omega |1_E(x)|^2 w(x) dx
= A\int_\Omega |S(\phi_E)(x)|^2 w(x) dx
= A\|\phi_E\|^2. \end{align*} The preceding calculation shows that the frame lower bound condition fails for $\phi_E$. This contradicts our assumption that $\{\Lambda_k\}_{k \in J}$ is a frame, therefore $w(x)\geq A$ a.e. $x\in \Omega$. The argument for the upper bound follows similarly. \end{proof}
\section{Proof of Theorem \ref{th3}} \begin{proof} Assume that $w(x)=1$ a.e. $x\in \Omega$. By the equations (\ref{eq1}) and (\ref{equ2}), for any $f\in {\mathbb H}$ we have \begin{align}\notag
\sum_{m,n} |\Lambda_{m,n}(f)|^2
&=\sum_{m,n} \left| \langle G_{m,n} , S(f)\rangle_{L^2(\Omega, \U)}\right|^2 = \| Sf\|^2 = \|f\|^2 . \end{align} This proves (\ref{onb3}). Let $c_1, c_2\in \C$. For any $m,m\rq{}\in J$ and $n, n\rq{}\in K$ we have
\begin{align*}
\langle \Lambda_{m,n}^*(c_1),\Lambda_{m\rq{},n\rq{}}^*(c_2)\rangle &= c_1\overline{c_2} \langle S^{-1}(f_ng_m), S^{-1}(f_{n\rq{}}g_{m\rq{}})\rangle_{{\mathbb H}} \\
& = c_1\overline{c_2} \langle f_ng_m, f_{n\rq{}}g_{m\rq{}}\rangle_{L^2(\Omega, \U)} \\
&= c_1\overline{c_2} \delta_{m,m\rq{}}\delta_{n,n\rq{}}.
\end{align*} This proves the relations (\ref{onb1}) and (\ref{onb2}).
To prove the converse, we assume contrary. We assume that there is a subset $E\subset \Omega$ of positive measure for which $w(x)<1$. As in the proof of Theorem \ref{th1}, one can show the existence of a function $\phi_E$ for which with an analogous calculation following the relation (\ref{iso}) the following holds:
$$\sum_{m,n} |\Lambda_{m,n}(\phi_E)|^2 \leq \|\phi_E\|^2 .$$ This indicates that the relation (\ref{onb3}) does not hold for $\phi_E$, which contradicts the assumption. \end{proof}
\section{Examples} \begin{example}\label{example 1} Let $\Omega=D$ be a fundamental domain in $\Bbb R^d$ with Lebesgue measure one. Assume that $\Gamma=M\Bbb Z^d$ where $M$ is an $d\times d$ invertible matrix and the exponentials $\{e_n(x):= e^{2\pi i \langle n,x\rangle}: \ n\in \Gamma\}$ form an orthonormal basis for $L^2(D)$.
For $0\neq\phi\in L^2(\Bbb R^d)$, define ${\mathbb H}:= \overline{\text{span}\{\phi(\cdot-n): n\in \Bbb Z^d\}}$ and the weight function $w$ by $w(x) := \sum_{n\in \Gamma^\perp} |\hat\phi(x+n)|^2$, a.e. $x\in D$. We claim that $\int_D w(x) dx= \|\phi\|^2$, thus is finite. To this end, notice $D$ is a fundamental domain for the lattice $\Gamma$. By a result by Fuglede \cite{Fug74}, $D$ tiles $\Bbb R^d$ by the dual lattice $\Gamma^\perp=M^{-t}\Bbb Z^d$, $M^{-t}$ the inverse transpose of $M$. Therefore, we have \begin{align*}
\int_D w(x) dx &= \int_D \sum_{n\in \Gamma^\perp} |\hat\phi(x+n)|^2 dx
= \int_{\mathbb R^d} |\hat\phi(x)|^2 dx
= \|\hat \phi\|^2
= \|\phi\|^2 . \end{align*} Let $E_w:= \{x\in D: w(x)>0\}$ and for any $f\in {\mathbb H}$ define $$S(f)(x):= 1_{E_w}(x) {w(x)}^{-1} \sum_{n\in \Gamma^\perp} \hat f(x+n) \overline{\hat \phi(x+n)} \quad \text{a.e.}\ x\in E_w.$$ Then $S$ is an unitary map from ${\mathbb H}$ onto the weighted Hilbert space $L_w^2(0,1)$ with $\U=\C$. Note that $Sf(x)= 0$ a.e. $x\in D\setminus E_w$ (Theorem 3.1 (i) \cite{HSWW}).
For $k\in \Gamma$ define $$\Lambda_k(f):= \int_{E_w} \left(\sum_{n\in \Gamma^\perp} \hat f(x+n)\overline{\hat\phi(x+n)}\right) e^{-2\pi i \langle x, k\rangle } dx.$$
By Corollary \ref{main result}, the operators $\{\Lambda_k\}_{k\in \Gamma}$ constitute a frame for ${\mathbb H}$ if there are positive constants $A$ and $B$ such that $A\leq \sum_{n\in \Gamma^\perp} |\hat\phi(x+n)|^2\leq B$ a.e. $x\in E$. By the well-known periodization method, it is obvious that $\Lambda_k(f) = \langle T_k \phi, f\rangle$ for any $f\in {\mathbb H}$ and $k\in \Gamma$, with $T_k\phi(x)= \phi(x-k)$. Thus, $\{\Lambda_k\}_\Gamma$ is the translation family $\{T_k \phi\}_\Gamma$.
For example, if $\phi\in L^2(\mathbb R)$ such that $\hat\phi= 1_{[0,1]}$, the indicator function of the unit interval, then the inequalities for $w$ holds for $A=1$ and $B=2$, hence $\{\Lambda_k\}_{k\in \Gamma}$ is a frame with lower and upper frame bounds $1$ and $2$, respectively. \end{example}
\section{Application: Frames for shift-invariant subspaces on the Heisenberg group}
In this section we shall revisit the example of a function in $L^2(\Bbb H^d)$ that was introduced in \cite{BHM14} and exploit our current results to study the frame and Riesz property of the lattice translations of the function for a shift-invariant subspace of $L^2(\Bbb H^d)$.
\subsection{The Heisenberg group} The $d$-dimensional Heisenberg group $\Bbb H^d$ is identified with $\mathbb R^d \times \mathbb R^d \times \mathbb R$ and the noncommutative group law is given by \begin{equation}\label{equ:Hlaw} (p,q,t) (p',q',t') = (p + p', q + q', t + t' + p\cdot q'). \end{equation} The inverse of an element is given by $(p,q,t)^{-1}= (-p,-q, -t+p\cdot q)$. Here, $x\cdot y$ is the inner product of two vectors in $\mathbb R^d$. The Haar measure of the group is the Lebesgue measure on $\Bbb R^{2d+1}$.
The class of non-zero measure irreducible representations of $\Bbb H^d$ is identified by non-zero elements $\lambda \in \mathbb R^*:=\mathbb R \setminus \{0\}$ (see \cite{F1995}). Indeed, for any $\lambda\neq 0$, the associated irreducible representation $\rho_\lambda$ of the Heisenberg group is equivalent to Schr\"odinger representation into the class of unitary operators on $L^2(\Bbb R^d)$, such that for any $(p,q,t)\in \Bbb H^d$ and $f\in L^2(\Bbb R^d)$ \begin{align}\label{definition-of-schroedinger-representation} \rho_\lambda(p,q,t)f(x) = e^{2\pi i t \lambda} e^{-2\pi i \lambda \langle q\cdot x\rangle} f(x-p) . \end{align}
Notice $\rho_\lambda(p,q,0)f(x) = M_{\lambda q} T_p f(x)$ is the unitary frequency-translation operator, where $M_x$ and $T_y$ are the modulation and translation operators, respectively. For $\varphi\in L^2(\Bbb H^d)$, we denote by $\hat \varphi$ the operator valued Fourier transform of $\varphi$ which is defined by \begin{equation}\label{equ:Hfourier} \hat \varphi(\lambda) = \int_\HD \varphi(x) \rho_\lambda(x) dx \quad \forall \lambda\in \mathbb R \setminus \{0\} . \end{equation}
The operator $\hat \varphi(\lambda)$ is a Hilbert-Schmidt operators on $L^2(\Bbb R^d)$ such that for any $f\in L^2(\Bbb R^d)$ \begin{equation}\notag \hat \varphi(\lambda)f(y) = \int_\HD \varphi(x) \rho_\lambda(x)f(y) \ dx \quad \forall \lambda\in \mathbb R \setminus \{0\} , \end{equation} and the equality is understood in $L^2$-norm sense.
For any $\psi$ and $\varphi$ in $L^2(\Bbb H^d)$ and $\lambda \in \mathbb R \setminus \{0\}$, the Hilbert-Schmidt inner product $\langle \hat \varphi(\lambda), \hat\psi(\lambda)\rangle_{\HS}$ is the trace of an operator. Indeed, \begin{equation}\label{eq:HS} \langle\hat\varphi(\lambda), \hat\psi(\lambda)\rangle_{\HS} = \textnormal{trace}_{L^2(\mathbb R^d)} \left(\hat\varphi (\lambda) \hat\psi(\lambda)^*\right) . \end{equation} (Here, $\hat\psi(\lambda)^*$ denotes the $L^2(\mathbb R^d)$ adjoint of the operator $\hat\psi(\lambda)$.) It is easy to see that $\hat\varphi (\lambda) \hat\psi(\lambda)^*$ is a kernel operator. Thus $\langle\hat\varphi(\lambda), \hat\psi(\lambda)\rangle_{\HS}$ is trace of a kernel operator (\cite{F1995}). The Plancherel formula for the Heisenberg group is given by \begin{equation}\label{eq:Plancherel}
\langle \varphi, \psi\rangle_{L^2(\HD)} = \int_\mathbb R \langle\hat\varphi(\lambda), \hat\psi(\lambda)\rangle_{\HS} |\lambda|^d d\lambda . \end{equation}
The measure $|\lambda|^d d\lambda$ is the Plancherel measure on the non-zero measure class of irreducible representations of the Heisenberg group (\cite{F1995}) and $d\lambda$ is the Lebesgue measure on $\Bbb R^*$. By the periodization method, the integral in (\ref{eq:Plancherel}) is can be equivalently written as \begin{equation}\notag
\int_\mathbb R \langle\hat\varphi(\lambda), \hat\psi(\lambda)\rangle_{\HS} |\lambda|^d d\lambda = \int_0^1
\sum_{j\in \Bbb Z} \langle\hat\varphi(\alpha+j), \hat\psi(\alpha+j)\rangle_{\HS} |\alpha+j|^d d\alpha . \end{equation} Thus, for any $\varphi\in L^2(\Bbb H^d)$, by the Plancherel formula we deduce the following: \begin{equation}\label{periodization}
\|\varphi\|^2 = \int_0^1 \sum_{j\in \Bbb Z} \|\hat \varphi(\alpha+j)\|^2_{\HS} |\alpha+j|^d d\alpha . \end{equation}
\subsection{Frames for a shift-invariant subspace} Let $u={\bf 1}_{[0,1]^d}\in L^2(\Bbb R^d)$ be the indicator function of the unit cube $[0,1]^d$. For $\alpha\neq 0$, define the $L^2$-unitary dilation of $u$ with respect to $\alpha$ by
$u_\alpha(x)= |\alpha|^{-d/2}u(\alpha x)$. Let $a$ and $b$ be two real numbers such that $0\neq ab\in \Bbb Z$. Then the family consisted of translations and modulations of $u_\alpha$ by $a\Bbb Z^d$ and $b \Bbb Z^d$, respectively, is then given by
$$ \left\{|\alpha|^{d/2}e^{-2\pi i\alpha b\langle m, x\rangle} {\bf 1}_{(0,\frac{1}{\alpha})^d}(x-an): \ \ m, n\in \Bbb Z^d\right\} .$$ It is known that the family is an orthonormal basis for $L^2(\Bbb R^d)$ and is called orthonormal Gabor or Weyl-Heisenberg basis for $L^2(\Bbb R^d)$ with the window function $u_\alpha$.
Fix $0<\epsilon<1$ (for the rest of the paper) and define the projector map $\Psi_\epsilon$ from $(0,1)$ into the class of Hilbert-Schmidt operators of rank one on $L^2(\Bbb R^d)$ by $$\Psi_\epsilon(\alpha) := (u_\alpha\otimes u_\alpha) 1_{(\epsilon,1]}(\alpha),$$ where for any $f, g, h\in L^2(\Bbb R^d)$ we have $(f\otimes g)h:= \langle h, g\rangle f$. By the definition of the Hilbert-Schmidt norm, we then have \begin{align}\label{value of HS-norm}
\|\Psi_\epsilon(\alpha)\|_{\mathcal{HS}} = 1_{(\epsilon,1]}(\alpha) . \end{align}
Thus $\|\Psi_\epsilon\|^2 = (d+1)^{-1}(1-\epsilon^{d+1})$. This implies that $\Psi_\epsilon\in L^2(\mathbb R^*,\HS(L^2(\mathbb R^d)),|\lambda|^d d\lambda)$. Therefore, by the inverse Fourier transform for the Heisenberg group, there is a function in $L^2(\Bbb H^d)$ whose Fourier transform is identical to $\Psi_\epsilon$ in $L^2$-norm. We let $\psi_\epsilon$ denote this function $ L^2(\Bbb H^d)$.
Let $\Bbb A$ and $\Bbb B$ be any $d\times d$ matrices in $GL(\Bbb R, d)$ such that $\Bbb A\Bbb B^t\in GL(\Bbb Z,d)$. Define $\Gamma :=\Bbb A\Bbb Z^d \times \Bbb B \Bbb Z^d\times\Bbb Z$. Then $\Gamma$ is a lattice subgroup of the Heisenberg group, a discrete and co-compact subgroup. For any $\gamma=(p,q,t)\in \Gamma$, we denote by $T_\gamma \psi_{\epsilon}$ the $\gamma$-translation of $\psi_\epsilon$ which is given by $$T_\gamma \psi_{\epsilon}(x,y,z)= \psi_{\epsilon}(\gamma^{-1}(x,y,z)).$$ Our goal here is to employ the current results and study the frame property of the family $\{T_\gamma\psi_{\epsilon}\}_{\gamma\in \Gamma}$ for its spanned vector space $V_{\Gamma, \psi_\epsilon}= \overline{\text{span}\{T_\gamma \psi_\epsilon: \gamma\in \Gamma\}}$. It is obvious that $V_{\Gamma, \psi_\epsilon}$ is $\Gamma$-translation-invariant subspace of $L^2(\Bbb H^d)$.
For fixed $\epsilon$, define $w_\epsilon$ on $\Bbb R$ by
$$w_\epsilon(\alpha) = \sum_{j \in \Z} \| \Psi_\epsilon (\alpha + j)\|^2_{\HS(L^2(\mathbb R^d))}|\alpha + j|^d.$$ The function $w_\epsilon$ is a positive and periodic function. Let $E_{w_\epsilon}:=\{\alpha\in (0,1): \ w_\epsilon(\alpha)>0\}$. The definition of $\Psi_\epsilon$ along (\ref{value of HS-norm}) yields the following result.
\begin{lemma}\label{lem:example1} For any $\alpha \in E_{w_\epsilon}$ \begin{align}\label{inequality for the toy weight} \epsilon^d \leq w_\epsilon(\alpha) \leq 1 . \end{align} \end{lemma}
Let $k:=(0,0,k) \in \Bbb Z$ and $T_k\psi_\epsilon$ denote the translation of $\psi_\epsilon$ at the center direction of the Heisenberg: $$T_k \psi_\epsilon(p,q,t) =\psi_\epsilon (p,q,t-k), \quad (p,q,t)\in \Bbb H^d .$$
Let ${\mathbb H}= \overline{\text{span}\{T_k \psi_\epsilon: \ k\in \Bbb Z\}}$ and $f\in {\mathbb H}$. For any $\alpha\in (0,1)$ define \begin{align}\label{isometry S}
S(f)(\alpha) := 1_{E_{w_\epsilon}}(\alpha) w_\epsilon(\alpha)^{-1} \sum_{j \in \Z} \langle \hat \psi_\epsilon (\alpha + j), \hat f(\alpha+j)\rangle_{\HS(L^2(\mathbb R^d))}|\alpha + j|^d. \end{align}
\begin{lemma}\label{isometry lemma} The map $S: {\mathbb H} \to L_{w_\epsilon}^2(0,1)$ defined in (\ref{isometry S}) is an unitary map. \end{lemma} \begin{proof} First we prove that $S$ is a bounded map on $L^2(\Bbb H^d)$. Let $f\in L^2(\Bbb H^d)$. Then \begin{align}\notag
& \int_0^1 |Sf(\alpha)|^2 w_\epsilon(\alpha)d\alpha \\\notag
& = \int_0^1 1_{E_{w_\epsilon}}(\alpha) w_\epsilon(\alpha)^{-2} \left(\sum_{j \in \Z} |\langle \hat \psi_\epsilon (\alpha + j), \hat f(\alpha+j)\rangle_{\HS(L^2(\mathbb R^d))}||\alpha + j|^d \right)^2 w_\epsilon(\alpha) d\alpha \\\notag
& \leq \int_0^1 1_{E_{w_\epsilon}}(\alpha) w_\epsilon(\alpha)^{-1} \left(\sum_{j \in \Z} \|\hat \psi_\epsilon (\alpha + j)\|_{\HS} \|\hat f(\alpha+j)\|_{\HS} |\alpha + j|^d \right)^2 d\alpha \\\notag &\leq \int_0^1 1_{E_{w_\epsilon}}(\alpha) w_\epsilon(\alpha)^{-1} B_{\psi_\epsilon}(\alpha) B_f(\alpha) d\alpha\\\notag &= \int_0^1 1_{E_{w_\epsilon}}(\alpha) B_f(\alpha) d\alpha \end{align}
where $B_g(\alpha) := \sum_j \|\hat g(\alpha+j)\|^2 |\alpha+j|^d$ for $g\in L^2(\Bbb H^d)$ and $\alpha\in (0,1)$. By the definition of $w_\epsilon$, it is immediate that $w_\epsilon(\alpha)^{-1} B_{\psi_\epsilon}(\alpha) = 1$ for a.e. $\alpha\in (0,1)$. Therefore \begin{align}\notag
\int_0^1 |Sf(\alpha)|^2 w_\epsilon(\alpha)d\alpha \leq
\int_0^1 1_{E_{w_\epsilon}}(\alpha) B_f(\alpha) d\alpha
= \| f\|^2 \quad \quad \text{by (\ref{periodization})}. \end{align} This proves that $S$ is a bounded operator. Next we prove that $S$ is an isometry map on $\mathcal{H}$. Assume that $f= \sum_k a_k T_k \psi_\epsilon$ for any finite linear combination of $T_k\psi_\epsilon$, $k\in\Bbb Z$. Thus \begin{align}\notag
& \int_0^1 |Sf(\alpha)|^2 w_\epsilon(\alpha)d\alpha \\\notag
& = \int_0^1 1_{E_{w_\epsilon}}(\alpha) w_\epsilon(\alpha)^{-2} \left|\sum_{j \in \Z} \langle \hat \psi_\epsilon (\alpha + j), \hat f(\alpha+j)\rangle_{\HS(L^2(\mathbb R^d))}|\alpha + j|^d \right|^2 w_\epsilon(\alpha) d\alpha \\\notag
& =\int_0^1 1_{E_{w_\epsilon}}(\alpha) w_\epsilon(\alpha)^{-1} \left|\sum_{j \in \Z} \langle \hat \psi_\epsilon (\alpha + j), \sum_k a_k \widehat{T_k \psi_\epsilon}(\alpha+j)\rangle_{\HS(L^2(\mathbb R^d))}|\alpha + j|^d \right|^2 d\alpha \\\notag & =\int_0^1 1_{E_{w_\epsilon}}(\alpha) w_\epsilon(\alpha)^{-1}
\left|\sum_k a_k e^{-2\pi i k \alpha}\right|^2 \left|\sum_{j \in \Z} \langle \hat \psi_\epsilon (\alpha + j), \hat\psi_\epsilon(\alpha+j)\rangle_{\HS(L^2(\mathbb R^d))}|\alpha + j|^d \right|^2 d\alpha\\\notag
& =\int_0^1 1_{E_{w_\epsilon}}(\alpha) \left|\sum_k a_k e^{-2\pi i k \alpha}\right|^2 w_\epsilon(\alpha) d\alpha \quad \text{(by the definition of $w_\epsilon$).}\\\notag \end{align} Notice we can write
$$\left|\sum_k a_k e^{-2\pi i k \alpha}\right|^2 w_\epsilon(\alpha)= \sum_j \left\| \sum_k a_k \widehat{T_k \psi_\epsilon}(\alpha+j) \right\|_{\mathcal{HS}}^2 |\alpha + j|^d .$$ Applying this in above we get \begin{align}\notag
\int_0^1 |Sf(\alpha)|^2 w_\epsilon(\alpha)d\alpha
&= \int_0^1 1_{E_{w_\epsilon}}(\alpha) \sum_j \left\| \sum_k a_k \widehat{T_k \psi_\epsilon}(\alpha+j)\right\|_{\mathcal{HS}}^2 |\alpha + j|^d d\alpha\\\notag
&= \int_0^1 1_{E_{w_\epsilon}}(\alpha) \sum_j \| \hat f(\alpha+j)\|_{\mathcal{HS}}^2 |\alpha + j|^d d\alpha\\\notag
&= \|f\|^2 \quad \quad \text{by (\ref{periodization})}. \end{align} This completes the proof of the lemma. \end{proof}
\begin{theorem} For any $k\in \Bbb Z$ and $f\in {\mathbb H}$ define
$$\Lambda_k(f) = \int_{E_{w_\epsilon}} \sum_{j \in \Z} \langle \hat \psi_\epsilon (\alpha + j), \hat f(\alpha+j)\rangle_{\HS(L^2(\mathbb R^d))}|\alpha + j|^d e^{-2\pi i \alpha k} d\alpha .$$ Then $\Lambda_k (f) = \langle T_k \psi_{\epsilon}, f\rangle $ and $\{\Lambda_k\}_{k\in\Z}$ is a frame for ${\mathbb H}$ with respect to $\Bbb C$. \end{theorem} \begin{proof} The equation $\Lambda_k (f)= \langle T_k \psi_{\epsilon}, f\rangle $ is a result of the Parseval identity. The family $\{\Lambda_k\}_{k\in\Z}$ is a frame for ${\mathbb H}$ with respect to $\Bbb C$ by Lemmas \ref{lem:example1}, \ref{isometry lemma} and Theorem \ref{th1} (b). \end{proof}
\section*{Acknowledgments} The author is deeply indebted to Dr. Azita Mayeli for several fruitful discussions and generous comments. The author wishes to thank the anonymous referees for their helpful comments and suggestions that helped to improve the quality of the paper.
\end{document}
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arXiv
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\begin{document}
\title{\bf Efficient coordination mechanisms\\for unrelated machine scheduling\thanks{A preliminary version of the results of this paper appeared in {\em Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)}
\begin{abstract} We present three new coordination mechanisms for scheduling $n$ selfish jobs on $m$ unrelated machines. A coordination mechanism aims to mitigate the impact of selfishness of jobs on the efficiency of schedules by defining a local scheduling policy on each machine. The scheduling policies induce a game among the jobs and each job prefers to be scheduled on a machine so that its completion time is minimum given the assignments of the other jobs. We consider the maximum completion time among all jobs as the measure of the efficiency of schedules. The approximation ratio of a coordination mechanism quantifies the efficiency of pure Nash equilibria (price of anarchy) of the induced game.
Our mechanisms are deterministic, local, and preemptive in the sense that the scheduling policy does not necessarily process the jobs in an uninterrupted way and may introduce some idle time. Our first coordination mechanism has approximation ratio $\Theta(\log m)$ and always guarantees that the induced game has pure Nash equilibria to which the system converges in at most $n$ rounds. This result improves a bound of $O(\log^2 m)$ due to Azar, Jain, and Mirrokni and, similarly to their mechanism, our mechanism uses a global ordering of the jobs according to their distinct IDs. Next we study the intriguing scenario where jobs are anonymous, i.e., they have no IDs. In this case, coordination mechanisms can only distinguish between jobs that have different load characteristics. Our second mechanism handles anonymous jobs and has approximation ratio $O\left(\frac{\log m}{\log \log m}\right)$ although the game induced is not a potential game and, hence, the existence of pure Nash equilibria is not guaranteed by potential function arguments. However, it provides evidence that the known lower bounds for non-preemptive coordination mechanisms could be beaten using preemptive scheduling policies. Our third coordination mechanism also handles anonymous jobs and has a nice ``cost-revealing'' potential function. We use this potential function in order, not only to prove the existence of equilibria, but also to upper-bound the price of stability of the induced game by $O(\log m)$ and the price of anarchy by $O(\log^2m)$. Our third coordination mechanism is the first that handles anonymous jobs and simultaneously guarantees that the induced game is a potential game and has bounded price of anarchy. In order to obtain the above bounds, our coordination mechanisms use $m$ as a parameter. Slight variations of these mechanisms in which this information is not necessary achieve approximation ratios of $O\left(m^{\epsilon}\right)$, for any constant $\epsilon>0$. \end{abstract}
\section{Introduction} We study the classical problem of {\em unrelated machine scheduling}. In this problem, we have $m$ parallel machines and $n$ independent jobs. Job $i$ induces a (possibly infinite) positive processing time (or load) $w_{ij}$ when processed by machine $j$. The load of a machine is the total load of the jobs assigned to it. The quality of an assignment of jobs to machines is measured by the makespan (i.e., the maximum) of the machine loads or, alternatively, the maximum completion time among all jobs. The optimization problem of computing an assignment of minimum makespan is a fundamental APX-hard problem, quite well-understood in terms of its offline \cite{LST93} and online approximability \cite{AAFPW93,ANR95}.
The approach we follow in this paper is both algorithmic and game-theoretic. We assume that each job is owned by a selfish agent. This gives rise to a {\em selfish scheduling} setting where each agent aims to minimize the completion time of her job with no regard to the global optimum. Such a selfish behaviour can lead to inefficient schedules from which no agent has an incentive to unilaterally deviate in order to improve the completion time of her job. From the algorithmic point of view, the designer of such a system can define a {\em coordination mechanism} \cite{CKN04}, i.e., a {\em scheduling policy} within each machine in order to ``coordinate'' the selfish behaviour of the jobs. Our main objective is to design coordination mechanisms that guarantee that the assignments reached by the selfish agents are {\em efficient}.
\noindent {\bf The model.} A scheduling policy simply defines the way jobs are scheduled within a machine and can be either {\em non-preemptive} or {\em preemptive}. Non-preemptive scheduling policies process jobs uninterruptedly according to some order. Preemptive scheduling policies do not necessarily have this feature and can also introduce some idle time (delay). Although this seems unnecessary at first glance, as we show in this paper, it is a very useful tool in order to guarantee coordination. A coordination mechanism is a set of scheduling policies running on the machines. In the sequel, we use the terms coordination mechanisms and scheduling policies interchangeably.
A coordination mechanism defines (or induces) a game with the job owners as players. Each job has all machines as possible {\em strategies}. We call an {\em assignment} (of jobs to machines) or {\em state} any set of strategies selected by the players, with one strategy per player. Given an assignment of jobs to machines, the cost of a player is the completion time of her job on the machine it has been assigned to; this completion time depends on the scheduling policy on that machine and the characteristics of all jobs assigned to that machine. Assignments in which no player has an incentive to change her strategy in order to decrease her cost given the assignments of the other players are called {\em pure Nash equilibria}. The global objective that is used in order to assess the efficiency of assignments is the {\em maximum completion time} over all jobs. A related quantity is the {\em makespan} (i.e., the maximum of the machine loads). Notice that when preemptive scheduling policies are used, these two quantities may not be the same (since idle time contributes to the completion time but not to the load of a machine). However, the optimal makespan is a lower bound on the optimal maximum completion time. The {\em price of anarchy} \cite{P01} is the maximum over all pure Nash equilibria of the ratio of the maximum completion time among all jobs over the optimal makespan. The {\em price of stability} \cite{ADKTWR04} is the minimum over all pure Nash equilibria of the ratio of the maximum completion time among all jobs over the optimal makespan. The {\em approximation ratio} of a coordination mechanism is the maximum of the price of anarchy of the induced game over all input instances.
Four natural coordination mechanisms are the {\sf Makespan}, {\sf Randomized}, {\sf LongestFirst}, and {\sf ShortestFirst}. In the {\sf Makespan} policy, each machine processes the jobs assigned to it ``in parallel'' so that the completion time of each job is the total load of the machine. {\sf Makespan} is obviously a preemptive coordination mechanism. In the {\sf Randomized} policy, the jobs are scheduled non-preemptively in random order. Here, the cost of each player is the expected completion time of her job. In the {\sf ShortestFirst} and {\sf LongestFirst} policies, the jobs assigned to a machine are scheduled in non-decreasing and non-increasing order of their processing times, respectively. In case of ties, a {\em global ordering} of the jobs according to their distinct IDs is used. This is necessary by any deterministic non-preemptive coordination mechanism in order to be well-defined. Note that no such information is required by the {\sf Makespan} and {\sf Randomized} policies; in this case, we say that they handle {\em anonymous jobs}. According to the terminology of \cite{AJM08}, all these four coordination mechanisms are {\em strongly local} in the sense that the only information required by each machine in order to compute a schedule are the processing times of the jobs assigned to it. A {\em local} coordination mechanism may use all parameters (i.e., the load vector) of the jobs assigned to the same machine.
Designing coordination mechanisms with as small approximation ratio as possible is our main concern. But there are other issues related to efficiency. The price of anarchy is meaningful only in games where pure Nash equilibria exist. So, the primary goal of the designer of a coordination mechanism should be that the induced game {\em always has} pure Nash equilibria. Furthermore, these equilibria should be {\em easy to find}. A very interesting class of games in which the existence of pure Nash equilibria is guaranteed is that of {\em potential} games. These games have the property that a {\em potential function} can be defined on the states of the game so that in any two states differing in the strategy of a single player, the difference of the values of the potential function and the difference of the cost of the player have the same sign. This property guarantees that the state with minimum potential is a pure Nash equilibrium. Furthermore, it guarantees that, starting from any state, the system will reach (converge to) a pure Nash equilibrium after a finite number of {\em selfish moves}. Given a game, its {\em Nash dynamics} is a directed graph with the states of the game as nodes and edges connecting two states differing in the strategy of a single player if that player has an incentive to change her strategy according to the direction of the edge. The Nash dynamics of potential games do not contain any cycle. Another desirable property here is {\em fast} convergence, i.e., convergence to a pure Nash equilibrium in a polynomial number of selfish moves. A particular type of selfish moves that have been extensively considered in the literature \cite{AAEMS08,CMS06,FFM08,MV04} is that of {\em best-response} moves. In a best-response move, a player having an incentive to change her strategy selects the strategy that yields the maximum decrease in her cost.
Potential games are strongly related to {\em congestion games} introduced by Rosenthal \cite{R73}. Rosenthal presented a potential function for these games with the following property: in any two states differing in the strategy of a single player, the difference of the values of the potential function {\em equals} the difference of the cost of the player. Monderer and Shapley \cite{MS96} have proved that each potential game having this property is isomorphic to a congestion game. We point out that potential functions are not the only way to guarantee the existence of pure Nash equilibria. Several generalizations of congestion games such as those with player-specific latency functions \cite{M96} are not potential games but several subclasses of them provably have pure Nash equilibria.
\noindent {\bf Related work.} The study of the price of anarchy of games began with the seminal work of Koutsoupias and Papadimitriou \cite{KP99} and has played a central role in the recently emerging field of Algorithmic Game Theory \cite{NRTV07}. Several papers provide bounds on the price of anarchy of different games of interest. Our work follows a different direction where the price of anarchy is the {\em objective to be minimized} and, in this sense, it is similar in spirit to studies where the main question is how to change the rules of the game at hand in order to improve the price of anarchy. Typical examples are the introduction of taxes or tolls in congestion games \cite{CKK06,CDR06,FJM04,KK04,S07}, protocol design in network and cost allocation games \cite{CRV08,KLO95}, Stackelberg routing strategies \cite{KS06,KLO97,KM02,R04,S07}, and network design \cite{R06}.
Coordination mechanisms were introduced by Christodoulou, Koutsoupias, and Nanavati in \cite{CKN04}. They study the case where each player has the same load on each machine and, among other results, they consider the {\sf LongestFirst} and {\sf ShortestFirst} scheduling policies. We note that the {\sf Makespan} and {\sf Randomized} scheduling policies were used in \cite{KP99} as models of selfish behaviour in scheduling, and since that paper, the {\sf Makespan} policy has been considered as standard in the study of selfish scheduling games in models simpler than the one of unrelated machines and is strongly related to the study of congestion games (see \cite{V07,R07} and the references therein). Immorlica et al. \cite{ILMS05} study these four scheduling policies under several scheduling settings including the most general case of unrelated machines. They prove that the {\sf Randomized} and {\sf ShortestFirst} policies have approximation ratio $O(m)$ while the {\sf LongestFirst} and {\sf Makespan} policies have unbounded approximation ratio. Some scheduling policies are also related to earlier studies of local-search scheduling heuristics. So, the fact that the price of anarchy of the induced game may be unbounded follows by the work of Schuurman and Vredeveld \cite{SV01}. As observed in \cite{ILMS05}, the equilibria of the game induced by {\sf ShortestFirst} correspond to the solutions of the {\sf ShortestFirst} scheduling heuristic which is known to be $m$-approximate \cite{IK77}. The {\sf Makespan} policy is known to induce potential games \cite{EKM03}. The {\sf ShortestFirst} policy also induces potential games as proved in \cite{ILMS05}. In Section \ref{sec:b-cm}, we present examples showing that the scheduling policies {\sf LongestFirst} and {\sf Randomized} do not induce potential games.\footnote{After the appearance of the conference version of the paper, we became aware of two independent proofs that {\sf Longest-First} may induce games that do not have pure Nash equilibria \cite{DT09,FRS10}.}
Azar et al. \cite{AJM08} study non-preemptive coordination mechanisms for unrelated machine scheduling. They prove that any local non-preemptive coordination mechanism is at least $\Omega(\log m)$-approximate\footnote{The corresponding proof of \cite{AJM08} contained a error which has been recently fixed by Fleischer and Svitkina \cite{FS10}.} while any strongly local non-preemptive coordination mechanism is at least $\Omega(m)$-approximate; as a corollary, they solve an old open problem concerning the approximation ratio of the {\sf ShortestFirst} heuristic. On the positive side, the authors of \cite{AJM08} present a non-preemptive local coordination mechanism (henceforth called {\sf AJM-1}) that is $O(\log m)$-approximate although it may induce games without pure Nash equilibria. The extra information used by this scheduling policy is the {\em inefficiency} of jobs (defined in the next section). They also present a technique that transforms this coordination mechanism to a preemptive one that induces potential games with price of anarchy $O(\log^2 m)$. In their mechanism, the players converge to a pure Nash equilibrium in $n$ rounds of best-response moves. We will refer to this coordination mechanism as {\sf AJM-2}. Both {\sf AJM-1} and {\sf AJM-2} use the IDs of the jobs.
\noindent {\bf Our results.} We present three new coordination mechanisms for unrelated machine scheduling. Our mechanisms are deterministic, preemptive, and local. The schedules in each machine are computed as functions of the characteristics of jobs assigned to the machine, namely the load of jobs on the machine and their inefficiency. In all cases, the functions use an integer parameter $p\geq 1$; the best choice of this parameter for our coordination mechanisms is $p=O(\log m)$. Our analysis is heavily based on the convexity of simple polynomials and geometric inequalities for Euclidean norms.
\begin{table} \centerline{\footnotesize
\begin{tabular}{|l||c|c|c|c|l|}
\hline
Coordination & & & & & \\
mechanism & PoA & Pot. & PNE & IDs & Characteristics \\\hline\hline
{\sf ShortestFirst} & $\Theta(m)$ & Yes & Yes & Yes & Strongly local, non-preemptive\\\hline
{\sf LongestFirst} & unbounded & No & No & Yes & Strongly local, non-preemptive\\\hline
{\sf Makespan} & unbounded & Yes & Yes & No & Strongly local, preemptive\\\hline
{\sf Randomized} & $\Theta(m)$ & No & ? & No & Strongly local, non-preemptive\\\hline
{\sf AJM-1} & $\Theta(\log m)$ & No & No & Yes & Local, non-preemptive\\\hline
{\sf AJM-2} & $O(\log^2m)$ & Yes & Yes & Yes & Local, preemptive, uses $m$ \\\hline\hline
{\sf ACOORD} & $\Theta(\log m)$ & Yes & Yes & Yes & Local, preemptive, uses $m$ \\
& $O(m^\epsilon)$ & Yes & Yes & Yes & Local, preemptive \\\hline
{\sf BCOORD} & $O\left(\frac{\log m}{\log\log m}\right)$ & No & ? & No & Local, preemptive, uses $m$ \\
& $O(m^\epsilon)$ & No & ? & No & Local, preemptive \\\hline
{\sf CCOORD} & $O(\log^2m)$ & Yes & Yes & No & Local, preemptive, uses $m$ \\
& $O(m^\epsilon)$ & Yes & Yes & No & Local, preemptive \\\hline \end{tabular}} \label{tab:comparison} \caption{Comparison of our coordination mechanisms to previously known ones with respect to the price of anarchy of the induced game (PoA), whether they induced potential games or not (Pot.), the existence of pure Nash equilibria (PNE), and whether they use the job IDs or not.} \end{table}
Motivated by previous work, we first consider the scenario where jobs have distinct IDs. Our first coordination mechanism {\sf ACOORD} uses this information and is superior to the known coordination mechanisms that induce games with pure Nash equilibria. The game induced is a potential game, has price of anarchy $\Theta(\log m)$, and the players converge to pure Nash equilibria in at most $n$ rounds. Essentially, the equilibria of the game induced by {\sf ACOORD} can be thought of as the solutions produced by the application of a particular online algorithm, similar to the greedy online algorithm for minimizing the $\ell_p$ norm of the machine loads \cite{AAFPW93,C08}. Interestingly, the local objective of the greedy online algorithm for the $\ell_p$ norm may not translate to a completion time of jobs in feasible schedules; the online algorithm implicit by {\sf ACOORD} uses a different local objective that meets this constraint. The related results are presented in Section \ref{sec:a-cm}.
Next we address the case where no ID information is associated to the jobs (anonymous jobs). This scenario is relevant when the job owners do not wish to reveal their identities or in large-scale settings where distributing IDs to jobs is infeasible. Definitely, an advantage that could be used for coordination is lost in this way but this makes the problem of designing coordination mechanisms more challenging. In Section \ref{sec:b-cm}, we present our second coordination mechanism {\sf BCOORD} which induces a simple congestion game with player-specific polynomial latency functions of a particular form. The price of anarchy of this game is only $O\left(\frac{\log m}{\log \log m}\right)$. This result demonstrates that preemption may be useful in order to beat the $\Omega(\log m)$ lower bound of \cite{AJM08} for non-preemptive coordination mechanisms. On the negative side, we show that the game induced may not be a potential game by presenting an example where the Nash dynamics have a cycle.
Our third coordination mechanism {\sf CCOORD} is presented in Section \ref{sec:c-cm}. The scheduling policy on each machine uses an interesting function on the loads of the jobs assigned to the machine and their inefficiency. The game induced by {\sf CCOORD} is a potential game; the associated potential function is ``cost-revealing'' in the sense that it can be used to upper-bound the cost of equilibria. In particular, we show that the price of stability of the induced game is $O(\log m)$ and the price of anarchy is $O(\log^2 m)$. The coordination mechanism {\sf CCOORD} is the first that handles anonymous jobs and simultaneously guarantees that the induced game is a potential game and has bounded price of anarchy. Table \ref{tab:comparison} compares our coordination mechanisms to the previously known ones.
Observe that the dependence of the parameter $p$ on $m$ requires that our mechanisms use the number of machines as input. By setting $p$ equal to an appropriately large constant, our mechanisms achieve price of anarchy $O(m^\epsilon)$ for any constant $\epsilon>0$. In particular, the coordination mechanisms {\sf ACOORD} and {\sf CCOORD} are the first ones that do not use the number of machines as a parameter, induce games with pure Nash equilibria, and have price of anarchy $o(m)$.
We remark that the current paper contains several improvements compared to its conference version. There, the three coordination mechanisms had the restriction that a job with inefficiency more than $m$ on some machine has infinite completion time when assigned to that machine. Here, we have removed this restriction and have adapted the analysis accordingly. A nice consequence of the new definition is that the coordination mechanisms can now be defined so that they do not use the number of machines as a parameter. Furthermore, the definition of {\sf ACOORD} has been significantly simplified. Also, the analysis of the price of anarchy of the coordination mechanism {\sf BCOORD} in the conference version used a technical lemma which is implicit in \cite{STZ04}. In the current version, we present a different self-contained proof that is based on convexity properties of polynomials and Minkowski inequality; the new proof has a similar structure with the analysis of the price of anarchy of mechanism {\sf ACOORD}.
We begin with preliminary technical definitions in Section \ref{sec:prelim} and conclude with interesting open questions in Section \ref{sec:open}.
\section{Preliminaries}\label{sec:prelim} In this section, we present our notation and give some statements that will be useful later. We reserve $n$ and $m$ for the number of jobs and machines, respectively, and the indices $i$ and $j$ for jobs and machines, respectively. Unless specified otherwise, the sums $\sum_i$ and $\sum_j$ run over all jobs and over all machines, respectively. Assignments are denoted by $N$ or $O$. With some abuse in notation, we use $N_j$ to denote both the set of jobs assigned to machine $j$ and the set of their loads on machine $j$. We use the notation $L(N_j)$ to denote the load of machine $j$ under the assignment $N$. More generally, $L(A)$ denotes the sum of the elements for any set of non-negative reals $A$. For an assignment $N$ which assigns job $i$ to machine $j$, we denote the completion time of job $i$ under a given scheduling policy by ${\cal P}(i,N_j)$. Note that, besides defining the completion times, we do not discuss the particular way the jobs are scheduled by the scheduling policies we present. However, we require that {\em feasible} schedules are computable efficiently. A natural sufficient and necessary condition is the following: for any job $i\in N_j$, the total load of jobs with completion time at most ${\cal P}(i,N_j)$ is at most ${\cal P}(i,N_j)$.
Our three coordination mechanisms use the inefficiency of jobs in order to compute schedules. We denote by $w_{i,\min}$ the minimum load of job $i$ over all machines. Then, its inefficiency $\rho_{ij}$ on machine $j$ is defined as $\rho_{ij}=w_{ij}/w_{i,\min}$.
Our proofs are heavily based on the convexity of simple polynomials such as $z^k$ for $k\geq 1$ and on the relation of Euclidean norms of the machine loads and the makespan. Recall that the $\ell_k$ norm of the machine loads for an assignment $N$ is $\left(\sum_j{L(N_j)^k}\right)^{1/k}$. The proof of the next lemma is trivial. \begin{lemma}\label{lem:lp-norm} For any assignment $N$, $\max_j{L(N_j)} \leq \left(\sum_j{L(N_j)^k}\right)^{1/k} \leq m^{1/k} \max_j{L(N_j)}$. \end{lemma} In some of the proofs, we also use the Minkowski inequality (or the triangle inequality for the $\ell_p$ norm).
\begin{lemma}[Minkowski inequality]\label{lem:minkowski} $\left(\sum_{t=1}^s{(a_t+b_t)^k}\right)^{1/k} \leq \left(\sum_{t=1}^s{a_t^k}\right)^{1/k}+\left(\sum_{t=1}^s{b_t^k}\right)^{1/k}$, for any $k\geq 1$ and $a_t,b_t\geq 0$. \end{lemma}
The following two technical lemmas are used in some of our proofs. We include them here for easy reference.
\begin{lemma}\label{lem:convexity} Let $r\geq 1$, $t\geq 0$ and $a_i\geq 0$, for $i=1, ..., k$. Then, \[\sum_{i=1}^k{\left(\left(t+a_i\right)^r-t^r\right)}\leq \left(t+\sum_{i=1}^k{a_i}\right)^r-t^r\] \end{lemma}
\begin{proof} The case when $a_i=0$ for $i=1, ..., k$ is trivial. Assume otherwise and let $\xi=\sum_{i=1}^{k}{a_i}$ and $\xi_i=a_i/\xi$. Clearly, $\sum_{i=1}^k{\xi_i}=1$. By the convexity of function $z^r$ in $[0,\infty)$, we have that \begin{eqnarray}\nonumber (t+a_i)^r &= & \left((1-\xi_i)t+\xi_i\left(t+\sum_{i=1}^k{a_i}\right)\right)^r\\\label{eq:convexity} &\leq & (1-\xi_i)t^r+\xi_i\left(t+\sum_{i=1}^k{a_i}\right)^r \end{eqnarray} for $i=1, ..., k$. Using (\ref{eq:convexity}), we obtain \begin{eqnarray*} \sum_{i=1}^k{\left((t+a_i)^r-t^r\right)} &\leq & t^r\left(\sum_{i=1}^k{(1-\xi_i)}-k\right)+\left(t+\sum_{i=1}^k{a_i}\right)^r\sum_{i=1}^k{\xi_i}\\ &=& \left(t+\sum_{i=1}^k{a_i}\right)^r-t^r \end{eqnarray*} \qed\end{proof}
\begin{lemma}\label{lem:slope} For any $z_0\geq 0$, $\alpha\geq 0$, and $p\geq 1$, it holds \[(p+1)\alpha z_0^p \leq (z_0+\alpha)^{p+1}-z_0^{p+1}\leq (p+1)\alpha (z_0+\alpha)^p.\] \end{lemma} \begin{proof} The inequality trivially holds if $\alpha=0$. If $\alpha>0$, the inequality follows since, due to the convexity of the function $z^{p+1}$, the slope of the line that crosses points $(z_0,z_0^{p+1})$ and $(z_0+\alpha,(z_0+\alpha)^{p+1})$ is between its derivative at points $z_0$ and $z_0+\alpha$.\qed\end{proof}
We also refer to the multinomial and binomial theorems. \cite{HLP52} provides an extensive overview of the inequalities we use and their history (see also {\tt wikipedia.org} for a quick survey).
\section{The coordination mechanism {\sf ACOORD}}\label{sec:a-cm} The coordination mechanism {\sf ACOORD} uses a global ordering of the jobs according to their distinct IDs. Without loss of generality, we may assume that the index of a job is its ID. Let $N$ be an assignment and denote by $N^i$ the restriction of $N$ to the jobs with the $i$ smallest IDs. {\sf ACOORD} schedules job $i$ on machine $j$ so that it completes at time \[{\cal P}(i,N_j) = \left(\rho_{ij}\right)^{1/p} L(N^i_j).\] Since $\rho_{ij}\geq 1$, the schedules produced are always feasible.
Consider the sequence of jobs in increasing order of their IDs and assume that each job plays a best-response move. In this case, job $i$ will select that machine $j$ so that the quantity $\left(\rho_{ij}\right)^{1/p} L(N^i_j)$ is minimized. Since the completion time of job $i$ depends only on jobs with smaller IDs, no job will have an incentive to change its strategy and the resulting assignment is a pure Nash equilibrium. The following lemma extends this observation in a straightforward way.
\begin{lemma}\label{lem:a-cm-potential-convergence} The game induced by the coordination mechanism {\sf ACOORD} is a potential game. Furthermore, any sequence of $n$ rounds of best-response moves converges to a pure Nash equilibrium. \end{lemma}
\begin{proof} Notice that since a job does not affect the completion time of jobs with smaller IDs, the vector of completion times of the jobs (sorted in increasing order of their IDs) decreases lexicographically when a job improves its cost by deviating to another strategy and, hence, it is a potential function for the game induced by the coordination mechanism {\sf ACOORD}.
Now, consider $n$ rounds of best-response moves of the jobs in the induced game such that each job plays at least once in each round. It is not hard to see that after round $i$, the job $i$ will have selected that machine $j$ so that the quantity $\left(\rho_{ij}\right)^{1/p} L(N^i_j)$ is minimized. Since the completion time of job $i$ depends only on jobs with smaller IDs, job $i$ has no incentive to move after round $i$ and, hence, no job will have an incentive to change its strategy after the $n$ rounds. So, the resulting assignment is a pure Nash equilibrium. \qed \end{proof}
The sequence of best-response moves mentioned above can be thought of as an online algorithm that processes the jobs in increasing order of their IDs. The local objective is slightly different that the local objective of the greedy online algorithm for minimizing the $\ell_{p+1}$ norm of the machine loads \cite{AAG+95,C08}; in that algorithm, job $i$ is assigned to a machine $j$ so that the quantity $(L(N^{i-1}_j)+w_{ij})^{p+1}-L(N^{i-1}_j)^{p+1}$ is minimized. Here, we remark that we do not see how the local objective of that algorithm could be simulated by a scheduling policy that always produces feasible schedules. This constraint is trivially satisfied by the coordination mechanism {\sf ACOORD}. The next lemma bounds the maximum completion time at pure Nash equilibria in terms of the $\ell_{p+1}$ norm of the machine loads and the optimal makespan.
\begin{lemma}\label{lem:completion-acoord} Let $N$ be a pure Nash equilibrium of the game induced by the coordination mechanisms {\sf ACOORD} and let $O$ be an optimal assignment. Then \[\max_{j,i\in N_j}{{\cal P}(i,N_j)} \leq \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}.\] \end{lemma}
\begin{proof} Let $i^*$ be the job that has the maximum completion time in assignment $N$. Denote by $j_1$ the machine $i^*$ uses in $N$ and let $j_2$ be a machine such that $\rho_{i^*j_2}=1$. If $j_1=j_2$, the definition of the coordination mechanism {\sf ACOORD} yields \begin{eqnarray*} \max_{j,i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\ &=& L(N_{j_1}^{i^*})\\ &\leq & L(N_{j_1})\\ &\leq & \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}. \end{eqnarray*} Otherwise, since player $i^*$ has no incentive to use machine $j_2$ instead of $j_1$, we have \begin{eqnarray*} \max_{j,i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\ &\leq & {\cal P}(i^*,N_{j_2}\cup \{w_{i^*j_2}\})\\ &=& L(N_{j_2}^{i^*})+w_{i^*j_2}\\ &\leq & L(N_{j_2})+\min_j{w_{i^*j}}\\ &\leq & \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}. \end{eqnarray*} \qed \end{proof}
Next we show that the approximation ratio of {\sf ACOORD} is $O(\log m)$ (for well-selected values of the parameter $p$). The analysis borrows and extends techniques from the analysis of the greedy online algorithm for the $\ell_p$ norm in \cite{C08}.
\begin{theorem}\label{thm:a-cm-poa} The price of anarchy of the game induced by the coordination mechanism {\sf ACOORD} with $p=\Theta(\log m)$ is $O(\log m)$. Also, for every constant $\epsilon\in (0,1/2]$, the price of anarchy of the game induced by the coordination mechanism {\sf ACOORD} with $p=1/\epsilon-1$ is $O\left(m^\epsilon\right)$. \end{theorem}
\begin{proof} Consider a pure Nash equilibrium $N$ and an optimal assignment $O$. Since no job has an incentive to change her strategy from $N$, for any job $i$ that is assigned to machine $j_1$ in $N$ and to machine $j_2$ in $O$, by the definition of {\sf ACOORD} we have that \begin{eqnarray*}\left(\rho_{ij_1}\right)^{1/p} L(N_{j_1}^i) &\leq & \left(\rho_{ij_2}\right)^{1/p} \left(L(N_{j_2}^{i-1})+w_{ij_2}\right). \end{eqnarray*} Equivalently, by raising both sides to the power $p$ and multiplying with $w_{i,\min}$, we have that \begin{eqnarray*}w_{ij_1} L(N_{j_1}^i)^p &\leq & w_{ij_2} \left(L(N_{j_2}^{i-1})+w_{ij_2}\right)^p.\end{eqnarray*}
Using the binary variables $x_{ij}$ and $y_{ij}$ to denote whether job $i$ is assigned to machine $j$ in the assignment $N$ ($x_{ij}=1$) and $O$ ($y_{ij}=1$), respectively, or not ($x_{ij}=0$ and $y_{ij}=0$, respectively), we can express this last inequality as follows. \begin{eqnarray*} \sum_j{x_{ij}w_{ij}L(N_{j}^i)^p} &\leq &\sum_j{y_{ij}w_{ij}\left(L(N_{j}^{i-1})+w_{ij}\right)^{p}} \end{eqnarray*} By summing over all jobs and multiplying with $(e-1)(p+1)$, we have \begin{eqnarray}\nonumber & & (e-1)(p+1) \sum_i\sum_j{x_{ij}w_{ij}L(N_j^i)^p}\\\nonumber &\leq & (e-1)(p+1) \sum_i\sum_j{y_{ij}w_{ij}\left(L(N_j^{i-1})+w_{ij}\right)^p}\\\nonumber &\leq & (e-1)(p+1) \sum_j\sum_i{y_{ij}w_{ij}\left(L(N_j)+w_{ij}\right)^p}\\\nonumber &= & (e-1)(p+1) \sum_j\sum_i{y_{ij}w_{ij}\left(L(N_j)+y_{ij}w_{ij}\right)^p}\\\nonumber &\leq & \sum_j\sum_i{\left(\left(L(N_j)+ey_{ij}w_{ij}\right)^{p+1}-\left(L(N_j)+y_{ij}w_{ij}\right)^{p+1}\right)}\\\nonumber &\leq & \sum_j\sum_i{\left(\left(L(N_j)+ey_{ij}w_{ij}\right)^{p+1}-L(N_j)^{p+1}\right)}\\\nonumber &\leq & \sum_j{\left(\left(L(N_j)+e\sum_i{y_{ij}w_{ij}}\right)^{p+1}-L(N_j)^{p+1}\right)}\\\nonumber &=& \sum_j{\left(L(N_j)+eL(O_j)\right)^{p+1}}-\sum_j{L(N_j)^{p+1}}\\\label{eq:a-minkowski} &\leq & \left(\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}+e\left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\right)^{p+1}-\sum_j{L(N_j)^{p+1}}. \end{eqnarray} The second inequality follows by exchanging the sums and since $L(N_j^{i-1})\leq L(N_j)$, the first equality follows since $y_{ij}\in \{0,1\}$, the third inequality follows by applying Lemma \ref{lem:slope} with $\alpha=(e-1)y_{ij}w_{ij}$ and $z_0=L(N_j)+y_{ij}w_{ij}$, the fourth inequality is obvious, the fifth inequality follows by Lemma \ref{lem:convexity}, the second equality follows since the definition of the variables $y_{ij}$ implies that $L(O_j)=\sum_i{y_{ij}w_{ij}}$, and the last inequality follows by Minkowski inequality (Lemma \ref{lem:minkowski}).
Now, we will relate the $\ell_{p+1}$ norm of the machines loads of assignments $N$ and $O$. We have \begin{eqnarray*} (e-1)\sum_j{L(N_j)^{p+1}} &=& (e-1)\sum_j{L(N_j^n)^{p+1}}\\ &=& (e-1)\sum_{i=1}^n{\sum_j{\left(L(N_j^i)^{p+1}-L(N_j^{i-1})^{p+1}\right)}}\\ &=& (e-1)\sum_{i=1}^n{\sum_j{\left(L(N_j^i)^{p+1}-(L(N_j^{i})-x_{ij}w_{ij})^{p+1}\right)}}\\ &\leq & (e-1)(p+1)\sum_i\sum_j{x_{ij}w_{ij}L(N_j^i)^p}\\ &\leq & \left(\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}+e\left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\right)^{p+1}-\sum_j{L(N_j)^{p+1}}. \end{eqnarray*} The first two equalities are obvious (observe that $L(N_j^0)=0$), the third one follows by the definition of variables $x_{ij}$, the first inequality follows by applying Lemma \ref{lem:slope} with $\alpha=x_{ij}w_{ij}$ and $z_0=L(N_j^{i})-x_{ij}w_{ij}$, and the last inequality follows by inequality (\ref{eq:a-minkowski}).
So, the above inequality yields \begin{eqnarray*} \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} &\leq & \frac{e}{e^{\frac{1}{p+1}}-1} \left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\\ &\leq & e(p+1) \left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\\ &\leq & e(p+1)m^{\frac{1}{p+1}} \max_j{L(O_j)}. \end{eqnarray*} The second inequality follows since $e^z\geq z+1$ for $z\geq 0$ and the third one follows by Lemma \ref{lem:lp-norm}.
Now, using Lemma \ref{lem:completion-acoord} and this last inequality, we obtain that \begin{eqnarray*} \max_{j,i\in N_j}{{\cal P}(i,N_j)} &\leq &\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}\\ &\leq & \left(e(p+1)m^{\frac{1}{p+1}}+1\right) \max_j{L(O_j)}. \end{eqnarray*} The desired bounds follow by setting $p=\Theta(\log m)$ and $p=1/\epsilon-1$, respectively. \qed \end{proof} Our logarithmic bound is asymptotically tight; this follows by the connection to online algorithms mentioned above and the lower bound of \cite{ANR95}.
\section{The coordination mechanism {\sf BCOORD}}\label{sec:b-cm} We now turn our attention to coordination mechanisms that handle anonymous jobs. We define the coordination mechanism {\sf BCOORD} by slightly changing the definition of {\sf ACOORD} so that the completion time of a job does not depend on its ID. So, {\sf BCOORD} schedules job $i$ on machine $j$ so that it finishes at time $${\cal P}(i,N_j)=\left(\rho_{ij}\right)^{1/p}L(N_j).$$ Since $\rho_{ij}\geq 1$, the schedules produced are always feasible. The next lemma bounds the maximum completion time at pure Nash equilibria (again in terms of the $\ell_{p+1}$ norm of the machine loads and the optimal makespan).
\begin{lemma}\label{lem:completion-bcoord} Let $N$ be a pure Nash equilibrium of the game induced by the coordination mechanisms {\sf BCOORD} and let $O$ be an optimal assignment. Then \[\max_{j,i\in N_j}{{\cal P}(i,N_j)} \leq \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}.\] \end{lemma}
\begin{proof} The proof is almost identical to the proof of Lemma \ref{lem:completion-acoord}; we include it here for completeness. Let $i^*$ be the job that has the maximum completion time in assignment $N$. Denote by $j_1$ the machine $i^*$ uses in $N$ and let $j_2$ be a machine such that $\rho_{i^*j_2}=1$. If $j_1=j_2$, the definition of the coordination mechanism {\sf BCOORD} yields \begin{eqnarray*} \max_{j,i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\ &=& L(N_{j_1})\\ &\leq & \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}. \end{eqnarray*} Otherwise, since player $i^*$ has no incentive to use machine $j_2$ instead of $j_1$, we have \begin{eqnarray*} \max_{j,i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\ &\leq & {\cal P}(i^*,N_{j_2}\cup \{w_{i^*j_2}\})\\ &=& L(N_{j_2})+w_{i^*j_2}\\ &= & L(N_{j_2})+\min_j{w_{i^*j}}\\ &\leq & \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}. \end{eqnarray*}\qed\end{proof}
We are ready to present our upper bounds on the price of anarchy of the induced game.
\begin{theorem} The price of anarchy of the game induced by the coordination mechanism {\sf BCOORD} with $p=\Theta(\log m)$ is $O\left(\frac{\log m}{\log \log m}\right)$. Also, for every constant $\epsilon\in (0,1/2]$, the price of anarchy of the game induced by the coordination mechanism {\sf BCOORD} with $p=1/\epsilon-1$ is $O\left(m^\epsilon\right)$. \end{theorem}
\begin{proof} Consider a pure Nash equilibrium $N$ and an optimal assignment $O$. Since no job has an incentive to change her strategy from $N$, for any job $i$ that is assigned to machine $j_1$ in $N$ and to machine $j_2$ in $O$, we have that \begin{eqnarray*}(\rho_{ij_1})^{1/p}L(N_{j_1}) &\leq & (\rho_{ij_2})^{1/p}(L(N_{j_2})+w_{ij_2}). \end{eqnarray*} Equivalently, by raising both sides to the power $p$ and multiplying both sides with $w_{i,\min}$, we have that \begin{eqnarray*}w_{ij_1}L(N_{j_1})^p &\leq &w_{ij_2}(L(N_{j_2})+w_{ij_2})^p.\end{eqnarray*} Using the binary variables $x_{ij}$ and $y_{ij}$ to denote whether job $i$ is assigned to machine $j$ in the assignments $N$ ($x_{ij}=1$) and $O$ ($y_{ij}=1$), respectively, or not ($x_{ij}=0$ and $y_{ij}=0$, respectively), we can express this last inequality as follows: \begin{eqnarray*} \sum_j{x_{ij}w_{ij}L(N_j)^p} &\leq & \sum_j{y_{ij}w_{ij}(L(N_j)+w_{ij})^p}. \end{eqnarray*} By summing over all jobs and multiplying with $p$, we have \begin{eqnarray}\nonumber & & p\sum_i{\sum_j{x_{ij}w_{ij}L(N_j)^p}}\\\nonumber &\leq & p\sum_i{\sum_j{y_{ij}w_{ij}(L(N_j)+w_{ij})^p}}\\\nonumber &=& p\sum_j{\sum_i{y_{ij}w_{ij}(L(N_j)+y_{ij}w_{ij})^p}}\\\nonumber &\leq & \sum_j\sum_i{\left(\left(L(N_j)+\frac{2p+1}{p+1}y_{ij}w_{ij}\right)^{p+1}-\left(L(N_j)+y_{ij}w_{ij}\right)^{p+1}\right)}\\\nonumber &\leq & \sum_j\sum_i{\left(\left(L(N_j)+\frac{2p+1}{p+1}y_{ij}w_{ij}\right)^{p+1}-L(N_j)^{p+1}\right)}\\\nonumber &\leq & \sum_j{\left(\left(L(N_j)+\frac{2p+1}{p+1}\sum_i{y_{ij}w_{ij}}\right)^{p+1}-L(N_j)^{p+1}\right)}\\\nonumber &=& \sum_j{\left(\left(L(N_j)+\frac{2p+1}{p+1}L(O_j)\right)^{p+1}-L(N_j)^{p+1}\right)}\\\label{eq:b-minkowski} &\leq& \left(\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}+\frac{2p+1}{p+1}\left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\right)^{p+1}-\sum_j{L(N_j)^{p+1}} \end{eqnarray} The first equality follows by exchanging the sums and since $y_{ij}\in \{0,1\}$, the second inequality follows by applying Lemma \ref{lem:slope} with $\alpha=\frac{p}{p+1}y_{ij}w_{ij}$ and $z_0=L(N_j)+y_{ij}w_{ij}$, the third inequality is obvious, the fourth inequality follows by applying Lemma \ref{lem:convexity}, the second equality follows since the definition of variables $y_{ij}$ implies that $L(O_j)=\sum_i{y_{ij}w_{ij}}$, and the last inequality follows by applying Minkowski inequality (Lemma \ref{lem:minkowski}).
Now, we relate the $\ell_{p+1}$ norm of the machine loads of assignments $N$ and $O$. We have \begin{eqnarray*} (p+1)\sum_j{L(N_j)^{p+1}} &=& p\sum_j{L(N_j)^{p+1}}+ \sum_j{L(N_j)^{p+1}}\\ &=& p\sum_i{\sum_j{x_{ij}w_{ij}L(N_j)^p}} + \sum_j{L(N_j)^{p+1}}\\ &\leq & \left(\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}+\frac{2p+1}{p+1}\left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\right)^{p+1}. \end{eqnarray*} The first equality is obvious, the second one follows by the definition of variables $x_{ij}$ and the inequality follows by inequality (\ref{eq:b-minkowski}).
So, the above inequalities yield \begin{eqnarray*} \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} &\leq & \frac{2p+1}{p+1} \frac{1}{(p+1)^{\frac{1}{p+1}}-1} \left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\\ &\leq & \frac{2p+1}{\ln{(p+1)}} \left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\\ &\leq & \frac{2p+1}{\ln{(p+1)}} m^{\frac{1}{p+1}} \max_j{L(O_j)}. \end{eqnarray*} The second inequality follows since $e^z\geq z+1$ for $z\geq 0$ and the third one follows by Lemma \ref{lem:lp-norm}.
Now, using Lemma \ref{lem:completion-bcoord} and our last inequality we have \begin{eqnarray*} \max_{j,i\in N_j}{{\cal P}(i,N_j)} &\leq & \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}\\ &\leq & \left(1+\frac{2p+1}{\ln{(p+1)}}m^{\frac{1}{p+1}}\right)\max_j{L(O_j)}. \end{eqnarray*} The desired bounds follows by setting $p=\Theta(\log m)$ and $p=1/\epsilon-1$, respectively.\qed \end{proof}
Note that the game induced by {\sf BCOORD} with $p=1$ is the same with the game induced by the coordination mechanism {\sf CCOORD} (with $p=1$) that we present in the next section. As such, it also has a potential function (also similar to the potential function of \cite{FKS05} for linear weighted congestion games) as we will see in Lemma \ref{lem:psi-potential}. In this way, we obtain a coordination mechanism that induces a potential game, handles anonymous jobs, and has aproximation ratio $O(\sqrt{m})$. Unfortunately, the next theorem demonstrates that, for higher values of $p$, the Nash dynamics of the game induced by {\sf BCOORD} may contain a cycle.
\begin{theorem}\label{thm:no-potential} The game induced by the coordination mechanism {\sf BCOORD} with $p=2$ is not a potential game. \end{theorem}
Before proving Theorem \ref{thm:no-potential}, we show that the games induced by the coordination mechanisms {\sf LongestFirst} and {\sf Randomized} may not be potential games either. All the instances presented in the following consist of four machines and three basic jobs $A$, $B$, and $C$. In each case, we show that the Nash dynamics contain a cycle of moves of the three basic jobs.
First consider the {\sf LongestFirst} policy and the instance depicted in the following table.
\
\centerline{
\begin{tabular}{|c|c|c|c|}
\hline
& A & B & C \\
$1$ & $14$ & $\infty$ & $5$ \\
$2$ & $\infty$ & $10$ & $\infty$ \\
$3$ & $3$ & $9$ & $10$ \\
$4$ & $7$ & $8$ & $9$ \\
\hline \end{tabular}}
\
The cycle is defined on the following states: \begin{eqnarray*} & & (C,\underline{B},A,) \rightarrow(C,,\underline{A}B,)\rightarrow(C,,\underline{B},A)\rightarrow(C,,,\underline{A}B)\rightarrow (A\underline{C},,,B)\rightarrow\\ & & (\underline{A},,C,B)\rightarrow(,,A\underline{C},B)\rightarrow(,,A,\underline{B}C)\rightarrow (,B,A,\underline{C})\rightarrow(C,B,A,). \end{eqnarray*} Notice that the first and last assignment are the same. In each state, the player that moves next is underlined. Job $B$ is at machine $2$ in the first assignment and has completion time $10$. Hence, it has an incentive to move to machine $3$ (second assignment) where its completion time is $9$. Job $A$ has completion time $12$ in the second assignment since it is scheduled after job $B$ which has higher load on machine $3$. Moving to machine $4$ (third assignment), it decreases its completion time to $7$. The remaining moves in the cycle can be verified accordingly.
The instance for the {\sf Randomized} policy contains four additional jobs $D$, $E$, $F$, and $G$ which are always scheduled on machines $1$, $2$, $3$, and $4$, respectively (i.e., they have infinite load on the other machines). It is depicted in the following table.
\
\centerline{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
& A & B & C & D & E & F & G\\
$1$ & $80$ & $\infty$ & $100$ & $2$ & $\infty$ & $\infty$ & $\infty$ \\
$2$ & $\infty$ & $171$ & $\infty$ & $\infty$ & $2$ & $\infty$ & $\infty$ \\
$3$ & $2$ & $154$ & $124$ & $\infty$ & $\infty$ & $32$ & $\infty$ \\
$4$ & $2$ & $76$ & $10$ & $\infty$ & $\infty$ & $\infty$ & 184\\
\hline \end{tabular}}
\
The cycle is defined by the same moves of the basic jobs as in the case of {\sf LongestFirst}: \begin{eqnarray*} & & (CD,\underline{B}E,AF,G) \rightarrow(CD,E,\underline{A}BF,G)\rightarrow(CD,E,\underline{B}F,AG)\rightarrow(CD,E,F,\underline{A}BG)\rightarrow\\ & & (A\underline{C}D,E,F,BG)\rightarrow(\underline{A}D,E,CF,BG)\rightarrow(D,E,A\underline{C}F,BG)\rightarrow(D,E,AF,\underline{B}CG)\rightarrow\\ & & (D,BE,AF,\underline{C}G)\rightarrow(CD,BE,AF,G). \end{eqnarray*} Recall that (see \cite{ILMS05,KP99}) the expected completion time of a job $i$ which is scheduled on machine $j$ in an assignment $N$ is $\frac{1}{2}(w_{ij}+L(N_j))$ when the {\sf Randomized} policy is used. In each state, the player that moves next is underlined. It can be easily verified that each player in this cycle improves her cost by exactly $1$. For example, job $B$ has expected completion time $\frac{1}{2}(171+171+2)=172$ at machine $2$ in the first assignment and, hence, an incentive to move to machine $3$ in the second assignment where its completion time is $\frac{1}{2}(154+2+154+32)=171$.
\paragraph{Proof of Theorem \ref{thm:no-potential}.} Besides the three basic jobs, the instance for the {\sf BCOORD} policy with $p=2$ contains two additional jobs $D$ and $E$ which are always scheduled on machines $3$ and $4$, respectively. The instance is depicted in the following table.
\
\centerline{
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& A & B & C & D & E \\
$1$ & $4.0202$ & $\infty$ & $4.0741$ & $\infty$ & $\infty$ \\
$2$ & $\infty$ & $8.2481$ & $\infty$ & $\infty$ & $\infty$ \\
$3$ & $0.0745$ & $0.6302$ & $0.3078$ & $29.1331$ & $\infty$ \\
$4$ & $2.4447$ & $5.1781$ & $2.4734$ & $\infty$ & $2.7592$ \\
\hline \end{tabular}}
\
The cycle is defined by the same moves of the basic jobs as in the previous cases: \[(C,\underline{B},AD,E)\rightarrow (C,,\underline{A}BD,E)\rightarrow(C,,\underline{B}D,AE)\rightarrow(C,,D,\underline{A}BE)\rightarrow(A\underline{C},,D,BE)\rightarrow\] \[(\underline{A},,CD,BE)\rightarrow(,,A\underline{C}D,BE)\rightarrow(,,AD,\underline{B}CE)\rightarrow(,B,AD,\underline{C}E)\rightarrow(C,B,AD,E).\] Notice that, instead of considering the completion time $\left(\rho_{ij}\right)^{1/p}L(N_j)$ of a job $i$ on machine $j$ in an assignment $N$, it is equivalent to consider its cost as $w_{ij}L(N_j)^p$. In this way, we can verify that in any of the moves in the above cycle, the job that moves improves its cost. For example, job $B$ has cost $8.2481^3=561.127758090641$ on machine $2$ in the first assignment and cost $0.6302(0.0745+0.6302+29.1331)^2=561.063473430968$ on machine $3$ in the second assignment. \qed
\section{The coordination mechanism {\sf CCOORD}}\label{sec:c-cm} In this section we present and analyze the coordination mechanism {\sf CCOORD} that handles anonymous jobs and guarantees that the induced game has pure Nash equilibria, price of anarchy at most $O(\log^2 m)$, and price of stability $O(\log m)$. In order to define the scheduling policy, we first define an interesting family of functions.
\begin{defn} For integer $k\geq 0$, the function $\Psi_k$ mapping finite sets of reals to the reals is defined as follows: $\Psi_k(\emptyset)=0$ for any integer $k\geq 1$, $\Psi_0(A)=1$ for any (possibly empty) set $A$, and for any non-empty set $A=\{a_1, a_2, ..., a_n\}$ and integer $k\geq 1$, \[\Psi_k(A)=k! \sum_{1\leq d_1 \leq ... \leq d_k \leq n}{\prod_{t=1}^{k}{a_{d_t}}}.\] \end{defn} So, $\Psi_k(A)$ is essentially the sum of all possible monomials of total degree $k$ on the elements of $A$. Each term in the sum has coefficient $k!$. Clearly, $\Psi_1(A)=L(A)$. For $k\geq 2$, compare $\Psi_k(A)$ with $L(A)^k$ which can also be expressed as the sum of the same terms, albeit with different coefficients in $\{1, ..., k!\}$, given by the multinomial theorem.
The coordination mechanism {\sf CCOORD} schedules job $i$ on machine $j$ in an assignment $N$ so that its completion time is $${\cal P}(i,N_j) = \left(\rho_{ij}\Psi_p(N_j)\right)^{1/p}.$$
Our proofs extensively use the properties in the next lemma; its proof is given in appendix. The first inequality implies that the schedule defined by {\sf CCOORD} is always feasible.
\begin{lemma}\label{lem:properties} For any integer $k\geq 1$, any finite set of non-negative reals $A$, and any non-negative real $b$ the following hold: \[\begin{array}{l l} \mbox{a. } L(A)^k \leq \Psi_k(A) \leq k! L(A)^k & \mbox{d. } \Psi_k(A\cup\{b\}) -\Psi_k(A) = k b\Psi_{k-1}(A\cup \{b\})\\ \mbox{b. } \Psi_{k-1}(A)^{k} \leq \Psi_{k}(A)^{k-1} & \mbox{e. } \Psi_k(A) \leq kL(A)\Psi_{k-1}(A)\\ \mbox{c. } \Psi_k(A\cup\{b\}) = \sum_{t=0}^k{\frac{k!}{(k-t)!}b^t\Psi_{k-t}(A)} &\mbox{f. } \Psi_k(A\cup \{b\}) \leq \left(\Psi_k(A)^{1/k}+\Psi_k(\{b\})^{1/k}\right)^k \end{array}\] \end{lemma}
The second property implies that $\Psi_k(A)^{1/k} \leq \Psi_{k'}(A)^{1/k'}$ for any integer $k'\geq k$. The third property suggests an algorithm for computing $\Psi_k(A)$ in time polynomial in $k$ and $|A|$ using dynamic programming.
A careful examination of the definitions of the coordination mechanisms {\sf BCOORD} and {\sf CCOORD} and property (a) in the above lemma, reveals that {\sf CCOORD} makes the completion time of a job assigned to machine $j$ dependent on the approximation $\Psi_p(N_j)^{1/p}$ of the load $L(N_j)$ of the machine instead of its exact load as {\sf BCOORD} does. This will be the crucial tool in order to guarantee that the induced game is a potential game without significantly increasing the price of anarchy. The next lemma defines a potential function on the states of the induced game that will be very useful later.
\begin{lemma}\label{lem:psi-potential} The function $\Phi(N)=\sum_j{\Psi_{p+1}(N_j)}$ is a potential function for the game induced by the coordination mechanism {\sf CCOORD}. Hence, this game always has a pure Nash equilibrium. \end{lemma}
\begin{proof} Consider two assignments $N$ and $N'$ differing in the strategy of the player controlling job $i$. Assume that job $i$ is assigned to machine $j_1$ in $N$ and to machine $j_2\not=j_1$ in $N'$. Observe that $N_{j_1}=N'_{j_1}\cup \{w_{ij_1}\}$ and $N'_{j_2}=N_{j_2}\cup \{w_{ij_2}\}$. By Lemma \ref{lem:properties}d, we have that $\Psi_{p+1}(N_{j_1})-\Psi_{p+1}(N'_{j_1})=(p+1)w_{ij_1}\Psi_p(N_{j_1})$ and $\Psi_{p+1}(N'_{j_2})-\Psi_{p+1}(N_{j_2})=(p+1)w_{ij_2}\Psi_p(N'_{j_2})$. Using these properties and the definitions of the coordination mechanism {\sf CCOORD} and function $\Phi$, we have \begin{eqnarray*} \Phi(N)-\Phi(N') &=& \sum_j{\Psi_{p+1}(N_j)-\sum_j{\Psi_{p+1}(N'_j)}}\\ &=& \Psi_{p+1}(N_{j_1})+\Psi_{p+1}(N_{j_2})-\Psi_{p+1}(N'_{j_1})-\Psi_{p+1}(N'_{j_2})\\ &=& (p+1)w_{ij_1}\Psi_{p}(N_{j_1})-(p+1)w_{ij_2}\Psi_{p}(N'_{j_2})\\ &=& (p+1)w_{i,\min} \left({\cal P}(i,N_{j_1})^p-{\cal P}(i,N'_{j_2})^p\right) \end{eqnarray*} which means that the difference of the potentials of the two assignments and the difference of the completion time of player $i$ have the same sign as desired. \qed \end{proof}
The next lemma relates the maximum completion time of a pure Nash equilibrium to the optimal makespan provided that their potentials are close. \begin{lemma}\label{lem:completion-time} Let $O$ be an optimal assignment and let $N$ be a pure Nash equilibrium of the game induced by the coordination mechanism {\sf CCOORD} such that $\left(\Phi(N)\right)^{\frac{1}{p+1}}\leq \gamma\left(\Phi(O)\right)^{\frac{1}{p+1}}$. Then, \[\max_{j,i\in N_j}{{\cal P}(i,N_j)} \leq \left(\gamma (p+1)m^{\frac{1}{p+1}}+p\right)\max_j{L(O_j)}.\] \end{lemma}
\begin{proof} Let $i^*$ be the job that has the maximum completion time in $N$. Denote by $j_1$ the machine $i^*$ uses in assignments $N$ and let $j_2$ be a machine such that $\rho_{i^*j_2}=1$. If $j_1=j_2$, the definition of the coordination mechanism {\sf CCOORD} and Lemma \ref{lem:properties}b yield \begin{eqnarray}\nonumber \max_{j, i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\\nonumber &= & \Psi_p(N_{j_1})^{1/p}\\\nonumber &\leq & \Psi_{p+1}(N_{j_1})^{\frac{1}{p+1}}\\\label{eq:c-same-machines} &\leq & \left(\sum_j{\Psi_{p+1}(N_j)}\right)^{\frac{1}{p+1}}. \end{eqnarray} Otherwise, since player $i$ has no incentive to use machine $j_2$ instead of $j_1$, we have \begin{eqnarray}\nonumber \max_{j, i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\\nonumber &\leq & {\cal P}(i^*, N_{j_2} \cup \{w_{i^*j_2}\}) \\\nonumber &=& \Psi_p(N_{j_2}\cup\{w_{i^*j_2}\})^{1/p}\\\nonumber &\leq & \Psi_p(N_{j_2})^{1/p}+\Psi_p(\{w_{i^*j_2}\})^{1/p}\\\nonumber &\leq & \Psi_{p+1}(N_{j_2})^{\frac{1}{p+1}}+(p!)^{1/p}w_{i^*j_2}\\\nonumber &= & \Psi_{p+1}(N_{j_2})^{\frac{1}{p+1}}+(p!)^{1/p}\min_j{w_{i^*j}}\\\label{eq:c-diff-machines} &\leq & \left(\sum_j{\Psi_{p+1}(N_j)}\right)^{\frac{1}{p+1}}+p\max_j{L(O_j)}. \end{eqnarray} The first two equalities follows by the definition of {\sf CCOORD}, the first inequality follows since player $i^*$ has no incentive to use machine $j_2$ instead of $j_1$, the second inequality follows by Lemma \ref{lem:properties}f, the third inequality follows by Lemma \ref{lem:properties}b and the definition of function $\Psi_p$, the third equality follows by the definition of machine $j_2$ and the last inequality is obvious.
Now, observe that the term in parenthesis in the rightmost side of inequalities (\ref{eq:c-same-machines}) and (\ref{eq:c-diff-machines}) equals the potential $\Phi(N)$. Hence, in any case, we have \begin{eqnarray*} \max_{j, i\in N_j}{{\cal P}(i,N_j)} &\leq& (\Phi(N))^{\frac{1}{p+1}}+p\max_j{L(O_j)}\\ &\leq & \gamma (\Phi(O))^{\frac{1}{p+1}}+p\max_j{L(O_j)}\\ &=& \gamma\left(\sum_j{\Psi_{p+1}(O_j)}\right)^{\frac{1}{p+1}}+p\max_j{L(O_j)}\\ &\leq &\gamma\left((p+1)! \sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}+p\max_j{L(O_j)}\\ &\leq & \left(\gamma (p+1)m^{\frac{1}{p+1}}+p\right)\max_j{L(O_j)}. \end{eqnarray*} The second inequality follows by the inequality on the potentials of assignments $N$ and $O$, the equality follows by the definition of the potential function $\Phi$, the third inequality follows by Lemma \ref{lem:properties}a and the last one follows by Lemma \ref{lem:lp-norm}. \qed \end{proof}
A first application of Lemma \ref{lem:completion-time} is in bounding the price of stability of the induced game.
\begin{theorem}\label{lem:stability} The game induced by the coordination mechanism {\sf CCOORD} with $p=\Theta(\log m)$ has price of stability at most $O(\log m)$. \end{theorem}
\begin{proof} Consider the optimal assignment $O$ and the pure Nash equilibrium $N$ of minimum potential. We have $\left(\Phi(N)\right)^{\frac{1}{p+1}}\leq \left(\Phi(O)\right)^{\frac{1}{p+1}}$ and, using Lemma \ref{lem:completion-time}, we obtain that the maximum completion time in $N$ is at most $(p+1)m^{\frac{1}{p+1}}+p$ times the makespan of $O$. Setting $p=\Theta(\log m)$, the theorem follows. \qed \end{proof}
A second application of Lemma \ref{lem:completion-time} is in bounding the price of anarchy. In order to apply it, we need a relation between the potential of an equilibrium and the potential of an optimal assignment; this is provided by the next lemma.
\begin{lemma}\label{lem:equilibrium} Let $O$ be an optimal assignment and $N$ be a pure Nash equilibrium of the game induced by the coordination mechanism {\sf CCOORD}. Then, \[\left(\Phi(N)\right)^{\frac{1}{p+1}} \leq \frac{p+1}{\ln{2}} \left(\Phi(O)\right)^{\frac{1}{p+1}}.\] \end{lemma}
\begin{proof} Consider a pure Nash equilibrium $N$ and an optimal assignment $O$. Since no job has an incentive to change her strategy from $N$, for any job $i$ that is assigned to machine $j_1$ in $N$ and to machine $j_2$ in $O$, we have that \begin{eqnarray*}\left(\rho_{ij_1}\Psi_p(N_{j_1})\right)^{1/p} &\leq & \left(\rho_{ij_2}\Psi_p(N_{j_2}\cup \{w_{ij_2}\})\right)^{1/p}. \end{eqnarray*} Equivalently, by raising both sides to the power $p$ and multiplying both sides with $w_{i,\min}$, we have that \begin{eqnarray*} w_{ij_1}\Psi_p(N_{j_1}) &\leq & w_{ij_2}\Psi(N_{j_2}\cup \{w_{ij_2}\}). \end{eqnarray*} Using the binary variables $x_{ij}$ and $y_{ij}$ to denote whether job $i$ is assigned to machine $j$ in the assignment $N$ ($x_{ij}=1$) and $O$ ($y_{ij}=1$) or not ($x_{ij}=0$ and $y_{ij}=0$, respectively), we can express the last inequality as follows: \begin{eqnarray*} \sum_j{x_{ij}w_{ij}\Psi_p(N_{j})} &\leq & \sum_j{y_{ij}w_{ij}\Psi(N_{j}\cup \{w_{ij}\})} \end{eqnarray*} By summing over all jobs, we have \begin{eqnarray*} \sum_i{\sum_j{x_{ij}w_{ij}\Psi_p(N_{j})} } &\leq & \sum_i{\sum_j{y_{ij}w_{ij}\Psi(N_{j}\cup \{w_{ij}\})}} \end{eqnarray*} By exchanging the double sums and since $\sum_i{x_{ij}w_{ij}}=L(N_j)$, we obtain \begin{eqnarray}\label{ineq:delta} \sum_j{L(N_j)\Psi_p(N_j)} &\leq & \sum_j{\sum_{i}{y_{ij}w_{ij}\Psi_p(N_j \cup \{w_{ij}\})}} \end{eqnarray}
We now work with the potential of assignment $N$. We have \begin{eqnarray*} 2\Phi(N) &= & \Phi(N)+\sum_j{\Psi_{p+1}(N_j)}\\ &\leq & \Phi(N)+(p+1)\sum_j{L(N_j)\Psi_p(N_j)}\\ &\leq & \Phi(N)+ (p+1)\sum_j{\sum_i{y_{ij}w_{ij}}\Psi_p(N_j\cup \{w_{ij}\})}\\ &=& \Phi(N)+(p+1)\sum_j{\sum_i{y_{ij}w_{ij}}\sum_{t=0}^{p}{\frac{p!}{(p-t)!}\Psi_{p-t}(N_j)w_{ij}^t}}\\ &=& \Phi(N)+\sum_j{\sum_{t=0}^{p}{\frac{(p+1)!}{(p-t)!}\Psi_{p-t}(N_j)\sum_i{y_{ij}w_{ij}^{t+1}}}}\\ &\leq & \Phi(N)+\sum_j{\sum_{t=0}^{p}{\frac{(p+1)!}{(p-t)!(t+1)!}\Psi_{p-t}(N_j)\Psi_{t+1}(O_j)}}\\ &=& \Phi(N)+\sum_j{\sum_{t=1}^{p+1}{\left(\begin{array}{c}p+1\\t\end{array}\right)\Psi_{p+1-t}(N_j)\Psi_{t}(O_j)}}\\ &\leq & \Phi(N)+\sum_j{\sum_{t=1}^{p+1}{\left(\begin{array}{c}p+1\\t\end{array}\right)\Psi_{p+1}(N_j)^{\frac{p+1-t}{p+1}}\Psi_{p+1}(O_j)^{\frac{t}{p+1}}}}\\ &=& \Phi(N)+\sum_j{\left(\left(\Psi_{p+1}(N_j)^{\frac{1}{p+1}}+\Psi_{p+1}(O_j)^{\frac{1}{p+1}}\right)^{p+1}-\Psi_{p+1}(N_j)\right)}\\ &=& \Phi(N)+\sum_j{\left(\Psi_{p+1}(N_j)^{\frac{1}{p+1}}+\Psi_{p+1}(O_j)^{\frac{1}{p+1}}\right)^{p+1}}-\sum_j{\Psi_{p+1}(N_j)}\\ &\leq & \left(\left(\sum_j{\Psi_{p+1}(N_j)}\right)^{\frac{1}{p+1}}+\left(\sum_j{\Psi_{p+1}(O_j)}\right)^{\frac{1}{p+1}}\right)^{p+1}\\ &=& \left(\left(\Phi(N)\right)^{\frac{1}{p+1}}+\left(\Phi(O)\right)^{\frac{1}{p+1}}\right)^{p+1} \end{eqnarray*} The first inequality follows by Lemma \ref{lem:properties}e, the second inequality follows by inequality (\ref{ineq:delta}), the second equality follows by Lemma \ref{lem:properties}c, the third equality follows by exchanging the sums, the third inequality follows since the jobs $i$ assigned to machine $j$ are those for which $y_{ij}=1$ and by the definition of function $\Psi_{t+1}$ which yields that $\Psi_{t+1}(O_j)\geq (t+1)! \sum_i{y_{ij}w_{ij}^{t+1}}$, the fourth equality follows by updating the limits of the sum over $t$, the fourth inequality follows by Lemma \ref{lem:properties}b, the fifth equality follows by the binomial theorem, the sixth equality is obvious, the fifth inequality follows by Minkowski inequality (Lemma \ref{lem:minkowski}) and by the definition of the potential $\Phi(N)$, and the last equality follows by the definition of the potentials $\Phi(N)$ and $\Phi(O)$.
By the above inequality, we obtain that \begin{eqnarray*} \left(\Phi(N)\right)^{\frac{1}{p+1}} &\leq & \frac{1}{2^{\frac{1}{p+1}}-1} \left(\Phi(O)\right)^{\frac{1}{p+1}}\leq \frac{p+1}{\ln{2}} \left(\Phi(O)\right)^{\frac{1}{p+1}} \end{eqnarray*} where the last inequality follows using the inequality $e^z\geq z+1$. \qed \end{proof} We are now ready to bound the price of anarchy. \begin{theorem}\label{thm:c-cm-poa} The price of anarchy of the game induced by the coordination mechanism {\sf CCOORD} with $p=\Theta(\log m)$ is $O\left(\log^2 m\right)$. Also, for every constant $\epsilon\in (0,1/2]$, the price of anarchy of the game induced by the coordination mechanism {\sf CCOORD} with $p=1/\epsilon-1$ is $O\left(m^\epsilon\right)$. \end{theorem}
\begin{proof} Consider a pure Nash equilibrium $N$ and let $O$ be the optimal assignment. Using Lemma \ref{lem:equilibrium}, we have that $\left(\Phi(N)\right)^{\frac{1}{p+1}} \leq \frac{p+1}{\ln{2}} \left(\Phi(O)\right)^{\frac{1}{p+1}}$. Hence, by Lemma \ref{lem:completion-time}, we obtain that the maximum completion time in $N$ is at most $\frac{(p+1)^2}{\ln 2}m^{\frac{1}{p+1}}+p$ times the makespan of $O$. By setting $p=\Theta(\log m)$ and $p=1/\epsilon-1$, respectively, the theorem follows. \qed \end{proof}
\section{Discussion and open problems}\label{sec:open} Our focus in the current paper has been on pure Nash equilibria. It is also interesting to generalize the bounds on the price of anarchy of the games induced by our coordination mechanisms for mixed Nash equilibria. Recently, Roughgarden \cite{R09} defined general smoothness arguments that can be used to bound the price of anarchy of games having particular properties. Bounds on the price of anarchy over pure Nash equilibria that are proved using smoothness arguments immediately imply that the same bounds on the price of anarchy hold for mixed Nash equilibria as well. We remark that the arguments used in the current paper in order to prove our upper bounds are not smoothness arguments. At least in the case of the coordination mechanism {\sf BCOORD}, smoothness arguments cannot be used to prove a bound on the price of anarchy as small as $O\left(\frac{\log m}{\log\log m}\right)$ since the price of anarchy over mixed Nash equilibria is provably higher in the case. We demonstrate this using the following construction. Czumaj and Voecking \cite{CV02} present a game induced by the {\sf Makespan} policy on related machines which has price of anarchy over mixed Nash equilibria at least $\Omega\left(\frac{\log m}{\log \log\log m}\right)$. The instance used in \cite{CV02} consists of $n$ jobs and $m$ machines. Each machine $j$ has a speed $\alpha_j\geq 1$ with $\alpha_1=1$ and each job $i$ has a weight $w_i$. The processing time of job $i$ on machine $j$ is $w_{ij}=\alpha_j w_i$ (i.e., the inefficiencies of the jobs are the same on the same machine). Now, consider the game induced by the coordination mechanism {\sf BCOORD} for the instance that consists of the same machines and jobs in which the processing time of job $i$ on machine $j$ is defined by $w'_{ij}=\alpha_j^{\frac{p}{p+1}} w_i$, i.e., the inefficiency of any job on machine $j$ is $\alpha_j^{\frac{p}{p+1}}$. Here, $p$ is the parameter used by {\sf BCOORD}. By the definition of {\sf BCOORD}, we can easily see that the game induced is identical with the game induced by {\sf Makespan} on the original instance of \cite{CV02}. Also note that, in our instance, the processing time of the jobs is not increased (and, hence, the optimal makespan is not larger than that in the original instance of \cite{CV02}). Hence, the lower bound of \cite{CV02} implies a lower bound on the price of anarchy over mixed Nash equilibria of the game induced by the coordination mechanism {\sf BCOORD}.
Our work reveals several other interesting questions. First of all, it leaves open the question of whether coordination mechanisms with constant approximation ratio exist. In particular, is there any coordination mechanism that handles anonymous jobs, guarantees that the induced game has pure Nash equilibria, and has constant price of anarchy? Based on the lower bounds of \cite{AJM08,FS10}, such a coordination mechanism (if it exists) must use preemption. Alternatively, is the case of anonymous jobs provably more difficult than the case where jobs have IDs? Investigating the limits of non-preemptive mechanisms is still interesting. Notice that {\sf AJM-1} is the only non-preemptive coordination mechanism that has approximation ratio $o(m)$ but it does not guarantee that the induced game has pure Nash equilibria; furthermore, the only known non-preemptive coordination mechanism that induces a potential game with bounded price of anarchy is {\sf ShortestFirst}. So, is there any non-preemptive (deterministic or randomized) coordination mechanism that is simultaneously $o(m)$-approximate and induces a potential game? We also remark that Theorem \ref{thm:no-potential} does not necessarily exclude a game induced by the coordination mechanism {\sf BCOORD} from having pure Nash equilibria. Observe that the examples in the proof of Theorem \ref{thm:no-potential} do not consist of best-response moves and, hence, it is interesting to investigate whether best-response moves converge to pure Nash equilibria in such games.
Furthermore, we believe that the games induced by the coordination mechanism {\sf CCOORD} are of independent interest. We have proved that these games belong to the class {\sf PLS} \cite{JPY88}. Furthermore, the result of Monderer and Shapley \cite{MS96} and the proof of Lemma \ref{lem:psi-potential} essentially show that each of these games is isomorphic to a congestion game. However, they have a beautiful definition as games on parallel machines that gives them a particular structure. What is the complexity of computing pure Nash equilibria in such games? Even in case that these games are {\sf PLS}-complete (informally, this would mean that computing a pure Nash equilibrium is as hard as finding any object whose existence is guaranteed by a potential function argument) like several variations of congestion games that were considered recently \cite{ARV06,FPT04,SV08}, it is still interesting to study the convergence time to efficient assignments. A series of recent papers \cite{AAEMS08,CMS06,FFM08,MV04} consider adversarial rounds of best-response moves in potential games so that each player is given at least one chance to play in each round (this is essentially our assumption in Lemma \ref{lem:a-cm-potential-convergence} for the coordination mechanism {\sf ACOORD}). Does the game induced by the coordination mechanism {\sf CCOORD} converges to efficient assignments after a polynomial number of adversarial rounds of best-response moves? Although it is a potential game, it does not have the particular properties considered in \cite{AAEMS08} and, hence, proving such a statement probably requires different techniques.
Finally, recall that we have considered the maximum completion time as the measure of the efficiency of schedules. Other measures such as the weighted sum of completion times that is recently studied in \cite{CGM10} are interesting as well. Of course, considering the application of coordination mechanisms to settings different than scheduling is an important research direction.
\small\paragraph{Acknowledgments.} I would like to thank Chien-Chung Huang for helpful comments on an earlier draft of the paper.
\small
\appendix
\normalsize\section{Proof of Lemma \ref{lem:properties}} The properties clearly hold if $A$ is empty or $k=1$. In the following, we assume that $k\geq 2$ and $A=\{a_1, ..., a_n\}$ for integer $n\geq 1$. \paragraph{a.} Clearly, \begin{eqnarray*} L(A)^k &=& \left(\sum_{t=1}^k{a_t}\right)^k = \sum_{1\leq d_1 \leq ... \leq d_k \leq n}{\zeta(d_1, ..., d_k)\prod_{t=1}^{k}{a_{d_t}}} \end{eqnarray*} where $\zeta(d_1, ..., d_k)$ are multinomial coefficients on $k$ and, hence, belong $\{1, ..., k!\}$. The property then follows by the definition of $\Psi_k(A)$.
\paragraph{b.} We can express $\Psi_{k-1}(A)^k$ and $\Psi_k(A)^{k-1}$ as follows: \begin{eqnarray*} \Psi_{k-1}(A)^k &=& \left((k-1)!\right)^k \left(\sum_{1\leq d_1 \leq ... \leq d_k \leq n}{\prod_{t=1}^{k}{a_{d_t}}}\right)^k \\ &= & \left((k-1)!\right)^k \sum_{1\leq d_1 \leq ... \leq d_{k(k-1)} \leq n}{\zeta_1(d_1, ..., d_{k(k-1)})\prod_{t=1}^{k(k-1)}{a_{d_t}}}.\\ \Psi_k(A)^{k-1} &=& \left(k!\right)^{k-1} \left(\sum_{1\leq d_1 \leq ... \leq d_k \leq n}{\prod_{t=1}^{k}{a_{d_t}}}\right)^{k-1}\\ &= & \left(k!\right)^{k-1} \sum_{1\leq d_1 \leq ... \leq d_{k(k-1)} \leq n}{\zeta_2(d_1, ..., d_{k(k-1)})\prod_{t=1}^{k(k-1)}{a_{d_t}.}} \end{eqnarray*} So, both $\Psi_{k-1}(A)^k$ and $\Psi_k(A)^{k-1}$ are sums of all monomials of degree $k(k-1)$ over the elements of $A$ with different coefficients. The coefficient $\zeta_1(d_1, ..., d_{k(k-1)})$ is the number of different ways to partition the multiset $D=\{d_1, ..., d_{k(k-1)}\}$ of size $k(k-1)$ into $k$ disjoint ordered multisets each of size $k-1$ so that the union of the ordered multisets yields the original multiset. We refer to these partitions as $(k,k-1)$-partitions. The coefficient $\zeta_2(d_1, ..., d_{k(k-1)})$ is the number of different ways to partition $D$ into $k-1$ disjoint ordered multisets each of size $k$ (resp. $(k-1,k)$-partitions). Hence, in order to prove the property, it suffices to show that for any multiset $\{d_1, ..., d_{k(k-1)}\}$, \begin{eqnarray}\label{ineq:partitions} \frac{\zeta_1(d_1, ..., d_{k(k-1)})}{\zeta_2(d_1, ..., d_{k(k-1)})} \leq \frac{k^{k-1}}{(k-1)!}. \end{eqnarray} Assume that some element of $D$ has multiplicity more than one and consider the new multiset $D'=\{d_1, ..., d'_i, ..., d_{k(k-1)}\}$ that replaces one appearance $d_i$ of this element with a new element $d'_i$ different than all elements in $D$. Then, in order to generate all $(k,k-1)$-partitions of $D'$, it suffices to consider the $(k,k-1)$-partitions of $D$ and, for each of them, replace $d_i$ with $d'_i$ once for each of the ordered sets in which $d_i$ appears. Similarly, we can generate all $(k-1,k)$-partitions of $D'$ using the $(k-1,k)$-partitions of $D$. Since the number of sets in $(k,k-1)$-partitions is larger than the number of sets in $(k-1,k)$-partitions, we will have that \begin{eqnarray*} \frac{\zeta_1(d_1, ..., d'_i, ..., d_{k(k-1)})}{\zeta_2(d_1, ..., d'_i, ..., d_{k(k-1)})} \geq \frac{\zeta_1(d_1, ..., d_{k(k-1)})}{\zeta_2(d_1, ..., d_{k(k-1)})} . \end{eqnarray*}
By repeating this argument, we obtain that the ratio at the left-hand side of inequality (\ref{ineq:partitions}) is maximized when all $d_i$'s are distinct. In this case, both $\zeta_1$ and $\zeta_2$ are given by the multinomial coefficients \[\zeta_1(d_1, ..., d_{k(k-1)}) = \left(\begin{array}{c}k(k-1)\\\underbrace{k-1, ..., k-1}_{\mbox{\tiny $k$ times}}\end{array}\right) = \frac{(k(k-1))!}{((k-1)!)^{k}}\] and \[\zeta_1(d_1, ..., d_{k(k-1)}) = \left(\begin{array}{c}k(k-1)\\\underbrace{k, ..., k}_{\tiny \mbox{$k-1$ times}}\end{array}\right) = \frac{(k(k-1))!}{(k!)^{k-1}}\] and their ratio is exactly the one at the right-hand side of the inequality (\ref{ineq:partitions}).
\paragraph{c.} The property follows easily by the definition of function $\Psi_k$ by observing that all the monomials of degree $k$ over the elements of $A$ that contain $b^t$ are generated by multiplying $b^t$ with the terms of $\Psi_{k-t}(A)$.
\paragraph{d.} By property (c), we have \begin{eqnarray*} \Psi_k(A\cup\{b\}) -\Psi_k(A) &=& \sum_{t=1}^k{\frac{k!}{(k-t)!}b^t\Psi_{k-t}(A)}\\ &=& kb\sum_{t=1}^{k}{\frac{(k-1)!}{(k-t)!}b^{t-1}\Psi_{k-t}(A)}\\ &=& kb\sum_{t=0}^{k-1}{\frac{(k-1)!}{(k-1-t)!}b^{t}\Psi_{k-1-t}(A)}\\ &=& kb\Psi_{k-1}(A\cup\{b\}). \end{eqnarray*}
\paragraph{e.} Working with the right-hand side of the inequality and using the definitions of $L$ and $\Psi_{k-1}$, we have \begin{eqnarray*} k L(A)\Psi_{k-1}(A) &=& k!\left(\sum_{t=1}^n{a_t}\right)\cdot \sum_{1\leq d_1 \leq ... \leq d_{k-1}\leq n}{\prod_{t=1}^{k-1}{a_{d_t}}}\\ &\geq & k!\sum_{1\leq d_1 \leq ... \leq d_{k}\leq n}{\prod_{t=1}^{k}{a_{d_t}}}\\ &=& \Psi_{k}(A). \end{eqnarray*} The equalities follow obviously by the definitions. To see why the inequality holds, observe that the multiplication of the sum of all monomials of degree $1$ with the sum of all monomials of degree $k-1$ will be a sum of all monomials of degree $k$, each having coefficient at least $1$. \qed
\paragraph{f.} The proof follows by the derivation below in which we use property (c), the fact that $\Psi_t(\{b\})=t! b^t$ by the definition of function $\Psi_t$, property (b), and the binomial theorem. We have \begin{eqnarray*} \Psi_k(A\cup\{b\}) &=& \sum_{t=0}^k{\frac{k!}{(k-t)!}b^t\Psi_{k-t}(A)}\\ &=& \sum_{t=0}^k{\left(\begin{array}{c}k\\t\end{array}\right)\Psi_{k-t}(A)\Psi_t(\{b\})}\\ &\leq & \sum_{t=0}^k{\left(\begin{array}{c}k\\t\end{array}\right)\Psi_{k}(A)^\frac{k-t}{k}\Psi_k(\{b\})^\frac{t}{k}}\\ &=& \left(\Psi_k(A)^{1/k}+\Psi_k(\{b\})^{1/k}\right)^k. \end{eqnarray*} \qed \end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\newcommand{\begin{equation}}{\begin{equation}} \newcommand{\end{equation}}{\end{equation}}
\begin{center} {\Large {\bf Experiment on interaction-free measurement in neutron interferometry}}
Meinrad Hafner$^a$\footnote{Email: [email protected]} and Johann Summhammer$^b$\footnote{Email: [email protected]}
$^a$ Institute for Astronomy
University of Vienna
Tuerkenschanzstrasse 17
A-1180 Vienna, Austria
$^b$ Atominstitut der \"Osterreichischen Universit\"aten
Sch\"uttelstrasse 115
A-1020 Vienna, Austria
\end{center}
\begin{abstract} A neutron interferometric test of interaction-free detection of the presence of an absorbing object in one arm of a neutron interferometer has been performed. Despite deviations from the ideal performance characteristics of a Mach-Zehnder interferometer it could be shown that information is obtained without interaction. \end{abstract}
\section{Introduction}
In classical mechanics the interaction between a measuring device and the object is treated as arbitrarily small and can therefore be neglected. This is fundamentally different in quantum physics, as examplified in the Heisenberg-microscope: A measurement of the position of an electron inevitably leads to a disturbance of its momentum by the scattering of a photon. It seems that no measurement can be performed without the interaction with a particle. Even in the case investigated by Dicke \cite{Dicke}, where knowledge about the position of an atom is gained by ascertaining where it is not, it had to be concluded, that photon scattering still is responsible for providing the information. However, the scattering remains unresolvable, because the change in the photon's momentum is within the momentum uncertainty of the illuminating beam of light. But recently Elitzur and Vaidman (EV) proposed an experiment where the presence of a perfectly absorbing classical object in one arm of a Mach-Zehnder interferometer can be detected \cite{EV1}, \cite{EV2}, \cite{Bennet}, \cite{Vaidman.quant-ph}. EV symbolized the object by a bomb which explodes when the measuring particle hits it. Their scheme is shown in Fig.1. Here, no interaction is involved. For, had there been an interaction, the test particle would have been absorbed and could not have reached any of the detectors at the outputs of the interferometer. The drawback of this scheme is that even when adjusting the transmissivity of the beam splitters of the interferometer, the probability of recognizing the object without interaction does not exceed 50\%. This disadvantage could be overcome by Zeilinger et al., who introduced a multi-loop interferometer. In the limit of infinitely many loops, one can obtain a probability of 100\% of determining the presence of the absorbing object without interaction \cite{Zeilinger.idea}. Paul and Pavicic incorporated this idea into a proposal employing an optical resonator \cite{Paul.Pavicic}. A first experiment with polarized photons has already been reported \cite{Zeilinger.exp}, \cite{Kwiat.review}. Due to imperfections of the optical devices only around 50\% of interaction-free detections of the object were possible. The theoretical analysis of these situations has also been extended to absorbing {\it quantum} objects, where it is no longer possible to assume a perfect absorber. But even then, a sizable amount of knowledge is gained without interaction \cite{Krenn}, \cite{Karlsson}.
In this paper we report on the first experiment with neutrons. The original scheme of EV was implemented in a single crystal neutron interferometer. Although such interferometers are not ideal Mach-Zehnder interferometers, it was possible to ascertain the presence of an absorbing object, which was realized as a neutron detector, with higher probability than classical physics would permit, so that one must conclude that some knowledge is obtained without interaction. Before dealing with the details of this experiment, it is useful to recall the ideal case of EV (Fig.1).
The two interferometric paths have equal length and the beam splitters at the entrance and at the exit have the same reflectivity of 50\%. When the interferometer is empty the probability amplitudes leading to P1 interfere constructively, while those leading to P2 interfere destructively. Therefore a particle will always be directed towards P1 (Fig.1a). If a perfectly absorbing object --- from now on called a black object --- is inserted into one path of the interferometer, only one beam can reach the exit ports, because the wavefunction behind the object vanishes. No interference can occur. With semitransparent mirrors at the entrance and exit of the interferometer, there is a probability of 50\% that the particle will be absorbed in the object, and a probability of 25\% each that it will reach either P1 or P2.
In order to clarify why interaction-free measurement is involved here, consider a collection of identically looking objects, some perfectly absorbing, some perfectly transparent (i.e. not even inducing a phase shift) for the kind of particles used in the interferometer. Will it be possible to separate the objects into the two classes when testing them in the interferometer? Classically there is no way. Without permitting interaction we could only randomly pick out objects and put them into the one or the other class. But quantum mechanically a separation is possible. This is illustrated in Fig.2. Suppose we place one such object, of which it is not known whether it is black or transparent, into one path of the interferometer and send one particle. Three different outcomes can occur:
\begin{description} \item[i.] The particle is detected at port P1. This can happen in either case, so no information is gained and another particle may be used. \item[ii.] The particle is detected at port P2. This is impossible with a perfectly transparent object, so we know now, that a black object is in the interferometer. Only one particle entered the interferometer and it was not absorbed, but was detected at P2. So it cannot have interacted with the object, and we got information of the absorbing property of the object {\it without interaction}. \item[iii.] The particle is detected neither at P1 nor at P2. Here we conclude, that the particle was absorbed by a black object, which, clearly, involved an interaction.
\end{description}
The result in ii is a nice example of the wave-particle-duality. The wave description determines the probabilities of the detection at the ports P1 and P2 by interference of the wavefunctions of the two paths. The particle description lets us conclude, that the particle {\it did not interact} with the black object when it was detected at P2: If it had been at the site of the black object, it would have been caught there, and could not have reached P2.
The three outcomes permit putting the tested objects into three groups as sketched in Fig.2. Groups ii and iii will only contain black objects. All transparent objects are accumulated in group i, but also some black objects, because in a single test the probability for a transparent object to be placed there is 100\%, while for a black object it is 25\%. With repeated tests of the objects of this group, the probability of still having a black object there will approach zero. Simultaneously, the number of objects in group ii will reach one third of the originally available black objects. Two thirds of the black objects will end up in group iii, because they interacted with the test particle and absorbed it.
The purpose of the present experiment was to perform such a selection procedure with a real neutron interferometer. The black objects were realized as a neutron detector. In this manner the three possible outcomes of an individual test could be observed. For transparent objects the interferometric paths were left empty.
\section{Experiment and Results}
The experiment was performed with the perfect-silicon-crystal neutron interferometer installed at the north beam port of the 250kW TRIGA MARK II reactor of the Atominstitut. This interferometer separates the paths by about 4 cm and recombines them again. The action of semitransparent mirrors is accomplished through Bragg-diffraction in Laue geometry at the 220-planes of the four crystal slabs. As can be seen in Fig.3, the basic layout is similar to a Mach-Zehnder interferometer and therefore well suited to test the EV-scheme. The main difference to the ideal case is that the two output beams are not equivalent. Output beam {\bf 1} can have a contrast of 100\% (i.e. be fully modulated), but output beam {\bf 2} cannot \cite{Petrascheck}, because the number of reflections and transmissions at the crystal slabs is not the same for the two paths making up output beam {\bf 2}. But due to crystal imperfections, even output beam {\bf 1} never reaches a contrast of 100\%. A contrast of 40\% is obtained with a cross section of 4 mm (width) times 7 mm (height) of the incident beam, and a wavelength of $\lambda = 1.8 \AA$ with a spread of $\Delta \lambda / \lambda \approx 1\%$. When normalizing the average intensity at output {\bf 1} to 1, the intensities of our set up are, as a function of a phase shift $\phi$ between paths {\bf I} and {\bf II} (setup as shown in Fig.3, but without the Cd sheet and without an object in path {\bf I}): \begin{equation} I_1(\phi) = 1 + 0.4 \cos(\phi) \end{equation} and \begin{equation} I_2(\phi) = 1.7 - 0.4 \cos(\phi). \end{equation} There is no phase shift $\phi$ at which one of the output beams is completely dark. This is contrary to the crucial assumption in the EV scheme. Consequently, a statement of the sort "if a neutron is detected at P2, a black object is with certainty in one of the arms of the intereferometer" was no longer possible. Nevertheless each test had to lead to a decision into which group the tested object should be put. We kept the three groups outlined above. But the certain identification of black objects had to be replaced by a probabilistic one, and it was attempted to achieve an enrichment of transparent objects in one group and of black objects in the two other groups. For this purpose theoretical simulations were undertaken, using the performance characteristics of the actual interferometer \cite{Diplom.Meinrad}. The questions to be answered were, whether the probability of successful interaction-free identification of black objects could be increased by
\begin{description} \item[(a)] attenuating the intensity of beam {\bf I}, into which the test objects were to be placed, with an additional partial absorber, and \item[(b)] introducing an additional constant phase shift of $\pi$, such that a registration of a neutron at P1 rather than at P2 could be taken to indicate the presence of a black object. \end{description}
It turned out that the optimal conditions could be obtained when attenuating beam {\bf I} to $t=16\%$ of its original intensity. This would reduce the amplitude of the intensity oscillations at P1 and P2 as a function of $\phi$ by a factor $\sqrt{t}=0.40$. The attenuation was done with a Cd-sheet of 125$\mu m$ thickness, which was placed in front of the second crystal slab, and parallel to this slab, as shown in Fig.3. The actual transmittance of this sheet was 0.158(4). The simulation also showed, that the phase shift between beams {\bf I} and {\bf II} should remain zero, just as in the EV-scheme, such that the count rate at P1 is at the {\it maximum} when a transparent object, or none, is present at the test position. But this, still, made it necessary to set the aluminum phase shifter to a certain value, because contrary to the ideal case, the actual interferometer has an internal phase offset due to inaccuracies of slab thicknesses and distances. In the experiment the registration of a neutron at P1 was interpreted as the presence of a transparent object, while registration at P2 was taken to indicate a black object.
The probabilities of the possible results were determined by measuring the intensities of the neutron beams at the detectors P1 and P2, and, when applicable, at the detector D, which constituted the black object. The measured intensities of a typical run are listed in Table 1 below. Averaging over several runs was not done, because they showed different contrast, depending on the vibrational level of the building. A run lasted for approximately four hours, during which measurements with and without the black object in path {\bf I}, as well as background measurements and normal interference scan measurements had to be taken \cite{Meinrad.p61}. Without any object in the paths, but with the attenuating Cd-sheet in place (Fig.3), the total intensity of P1 {\it plus} P2, including background, was 1.25 counts/sec.
\noindent {\bf Table 1:} Observed neutron counts with transparent (=none) or black object in the test position.
\begin{tabular}{|c||r|r||r|} \hline detector & transparent object & black object & background \\ \hline \hline P1 & 3561 & 2073 & 215 \\ \hline P2 & 793 & 999 & 59 \\ \hline D & --- & 2253 & 1422 \\ \hline \end{tabular}
The background was determined by rotating the interferometer crystal around the vertical axis away from the Bragg condition by some 3 degrees, such that the incident neutron beam went straight through the first and second crystal slabs. The background at D is due to its position close to the incident beam, whose intensity is three orders of magnitude larger than the intensity which the Bragg condition selects for the interferometer. The background at D is in part also due to the gamma radiation which is created when neutrons are absorbed in the Cd sheet in a nuclear reaction. The detectors P1 and P2 were standard $BF_3$-filled (atmospheric pressure) cylindrical neutron detectors with a diameter of 5 cm and a length of around 40 cm, and were hit axially by the respective beams. Their efficiencies exceeded 97\%. The detector D, representing the black object, had an efficiency of only 65\%. Its outside diameter was 2.54 cm, and its length some 30 cm. When at the test position, beam {\bf I} hit it perpendicular to the axis of the cylinder.
It was filled with four atmospheres of $He^3$. The capture cross section of a thermal neutron at a $He^3$ nucleus is 5330 barns. This results in a probability of 0.76 that a neutron will be captured in the gas. The reduction to 0.65 is due to averaging over the beam cross section and to electronic discrimination, which was needed to suppress as much as possible the unwanted gamma-background. If the neutron went through detector D unaffected, it was certainly absorbed by the shielding material behind the detector. Thereby it was ensured, that the black object really was black.
From the data in table 1 it is possible to infer whether an actual interaction-free identification of black objects is possible. In so doing we will assume 100\% efficiency of detectors P1 and P2. Since this is not very different from their actual efficiency the resulting error will be small. Detector D of the black object is not needed for the analysis. This detector only served for a check of the consistency of the results.
After subtracting the background counts, which is permissible, because they can in principle be made arbitrarily low with improved shielding and electronics, the probabilities for an object to be put into one of the three groups after a test with only a single particle, can be calculated as given in table 2:
\noindent {\bf Table 2:} Probabilities of detection of neutron at one of the detectors with black or transparent object in test position. Standard deviations include only Poisson statistics of the counts of table 1. Numbers in rectangular brackets are probabilities for ideal Mach-Zehnder interferometer.
\begin{tabular}{|c||c|c|c|} \hline object & detection at P1 (i) & detection at P2 (ii) & absorption in object (iii) \\ \hline \hline black & .455$\pm$.014 [.25] & .231$\pm$.009 [.25] & .314$pm$.018 [.50] \\ \hline transparent & .820$\pm$.020 [1] & .180$\pm$.008 [0] & --- \\ \hline \end{tabular}
It is noteworthy, that the probability that the neutron gets absorbed in the black object is significantly lower than in an ideal Mach-Zehnder interferometer. This improved performance is also retained in the limit of infinitely many tests of the objects remaining in group i. In such a procedure, the probability for a black object {\it not to interact} with the neutron and ultimately to be put into group ii is given by \begin{equation} p_{black}^{(ii)} \sum_{n=0}^{\infty}\left[ p_{black}^{(i)}\right]^n \approx .424 , \end{equation} where we have inserted from table 2, $p_{black}^{(i)} = .455$ and $p_{black}^{(ii)} = .231$. Correspondingly, the probability that the black object ultimately absorbs a particle in this procedure and thus is put into group iii is $1-.424=.576$. But, unfortunately, in such repeated tests the transparent objects, too, will accumulate in group ii. The reason is, that their probability of being put into group i in a single test is less than 1, namely $p_{trans}^{(i)}=.820$, according to table 2. Therefore, when testing objects in group i again and again, until either P1 or D fires, all transparent objects and 42.4\% of the black objects will end up in group ii.
A closer analysis reveals, that with our real neutron interferometer, the best separation of black and transparent objects is obtained when testing each object {\it only once}. Then one can obtain a separation as shown in Fig.4. The x-axis represents the fraction of black objects in the original ensemble. The full curves going from the lower left to the upper right corner represent the fraction of black objects ending up in group ii, which is the group where the tested object is put when the particle is detected at P2. The thick line is the most likely value, the two thin ones delimit its standard deviation. The dashed line indicates "no separation" of black and transparent objects. One notes that an enrichment of black objects in group ii does happen, because their fraction there is always larger than their fraction in the original ensemble. For instance, for a fraction of .5 of black objects in the original ensemble, their fraction in the final ensemble ii will be .562$\pm$.014. Therefore, even with a neutron interferometer, whose performance is far from an ideal Mach-Zehnder interferometer, and in fact far from the best available neutron interferometers, {\it some information} is obtained in an interaction-free manner. For the sake of completeness Fig.4 also shows the fraction of transparent objects in group i after a single test. These are the curves extending from the upper left to the lower right corner. As expected, transparent objects are accumulated in group i.
It is also interesting to ask whether for the {\it real} neutron interferometer there exists a method, different from the EV-method, which permits arbitrarily good separation of transparent and black objects. This is indeed possible, albeit at the cost of only a tiny fraction of the originally available black objects remaining. The method consists in repeated tests of the objects in {\it group ii}. As before, let $p_{black}^{(ii)}$ and $p_{trans}^{(ii)}$ denote the probabilities that a black, respectively transparent, object will be put into group ii after a single test. According to table 2 we have, again neglecting experimental uncertainties, $p_{black}^{(ii)} = .231$ and $p_{trans}^{(ii)} = .180$. When testing objects of group ii a further $N-1$ times, a black object of the original ensemble is therefore $\left( p_{black}^{(ii)} / p_{trans}^{(ii)} \right)^N = (1.283)^N$ times more likely to remain in group ii as compared to a transparent object of the original ensemble. Arbitrarily good purification of group ii is therefore possible {\it without interaction.} The number of black objects group ii contains in the end, will however only be $\left( p_{black}^{(ii)} \right)^N$ times the number of black objects in the original ensemble.
\section{Conclusion}
We have performed a test of interaction-free measurement with a neutron interferometer of the Mach-Zehnder type. In the unobstructed interferometer the probability of a neutron to be found in the path of the test position for the black and the transparent objects to be identified was around .3, and correspondingly the probability to find it in the other path was around .7. The visibility contrast at the exit port P1 only reached around 40\%. With such strong deviations from the characteristics of an ideal Mach-Zehnder interferometer it was nevertheless possible to show that an original ensemble with unknown proportions of black and transparent objects can be separated into two groups by testing each object with essentially only one neutron (about 1.04 neutrons on average), which must not be absorbed in the test object. Then one of these groups is guaranteed to contain a higher proportion of black objects than the original ensemble. The black objects are laid out as a neutron detector plus absorptive shielding. A neutron interacting with a black object would certainly be absorbed by it. Therefore, the result shows interaction-free measurement at work.
\section{Acknowledgment}
We would like to thank Professor H. Rauch for permission to use his neutron interferometry setup and to adapt it to the needs of this experiment.
\begin{thebibliography}{99}
\bibitem{Dicke} R. H. Dicke, Am. J. Phys. {\bf 49}, 925 (1981). \bibitem{EV1} A. C. Elitzur and L. Vaidman, Found. Phys. {\bf 23}, 987 (1993). \bibitem{EV2} L. Vaidman, Quantum Opt. {\bf 6}, 119 (1984). \bibitem{Bennet} C. H. Bennet, Nature {\bf 371}, 479 (1994). \bibitem{Vaidman.quant-ph} L. Vaidman, http://xxx.lanl.gov/abs/quant-ph/9610033. \bibitem{Zeilinger.idea} P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. Kasevich, Ann. N.Y. Acad. Sci. {\bf 755}, 383 (1995). \bibitem{Paul.Pavicic} H. Paul and M. Pavicic, J. Opt. Soc. Am. {\bf B 14}, 1275 (1997); M. Pavicic, Phys. Lett. {\bf A 223}, 241 (1996). \bibitem{Zeilinger.exp} P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. Kasevich, Phys. Rev. Lett. {\bf 74}, 4763 (1995). \bibitem{Kwiat.review} P. Kwiat, H. Weinfurter and A. Zeilinger, Scientific American, Nov. 1996, p. 52. \bibitem{Krenn} G. Krenn, J. Summhammer, and K. Svozil, Phys. Rev. {\bf A53}, 1228 (1996). \bibitem{Karlsson} A. Karlsson, G. Bj\"ork, E. Forsberg, http://xxx.lanl.gov/abs/quant-ph/9705006. \bibitem{Petrascheck} D. Petrascheck, Phys. Rev. {\bf B 35}, 6549 (1987), and references therein. \bibitem{Diplom.Meinrad} M. Hafner, Diploma thesis, Faculty of Natural Sciences, Technical University of Vienna, 1996 (in German; unpublished). \bibitem{Meinrad.p61} ibid., p.61, ff. \end {thebibliography}
\pagebreak {\bf Figure Captions}
\noindent {\bf Fig.1: (a)} The test particle will exit the interferometer at P1 when there is no object, or a perfectly transparent one, in one path of the interferometer. {\bf (b)} With the perfectly absorbing object D in one path, the test particle, if it is not absorbed, can exit at P1 as well as at P2.
\noindent {\bf Fig.2:} An original ensemble of black and transparent objects can be separated into three groups by a test in the interferometer. The black objects in group ii get there without having interacted with the test particle.
\noindent {\bf Fig.3:} Experimental scheme. The base area of the four plate interferometer is 144mm $\times$ 100mm. Crystal slabs, phase shifter, black object (= detector D plus shielding) and beam widths are drawn to scale. The wide incident beam is collimated by a boron carbide (BC) slit. The transmitted part of beam I after the second crystal slab, and that of beam II after the third crystal slab, leave the interferometer unused. Detectors at exit beams P1 and P2 not to scale.
\noindent {\bf Fig.4:} Results of experiment. Full lines (mean, and plus minus one standard deviation): In a single test of each object, a black object is more likely to be put into group ii, unless the neutron is absorbed by it in an interaction, while a transparent object is more likely to be put into group i. Dashed lines: What could be achieved when randomly putting objects of the original ensemble into different groups.
\end{document}
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\begin{document}
\title*{Understanding the Dynamics of Collision and Near-Collision Motions in the $N$-Body Problem} \titlerunning{Collisions and Near-Collisions} \author{Lennard F. Bakker} \institute{Lennard F. Bakker \at Department of Mathematics, Brigham Young University, USA \email{[email protected]}}
\maketitle
\abstract {Although rare, collisions of two or more bodies in the $N$-body problem are apparent obstacles at which Newton's Law of Gravity ceases to make sense. Without understanding the nature of collisions, a complete understanding of the N-body problem can not be achieved. Historically, several methods have been developed to probe the nature of collisions in the $N$-body problem. One of these methods removes, or regularizes, certain types of collisions entirely, thereby relating not only analytically, but also numerically, the dynamics of such collision motions with their near collision motions. By understanding the dynamics of collision motions in the regularized setting, a better understanding of the dynamics of near-collision motions is achieved.}
\section{Introduction} \label{sec:1}
For ages, humankind has observed the regular and predicable motion of the planets and other bodies in the solar system and asked, will the motion of the bodies in the solar system continue for ever as they are currently observed? This philosophical question is the object of the mathematical notion of stability. A difficulty in applying the notion of stability to the motion of the solar system is that of collision and near-collision motions of bodies in the solar system. Collision and near-collision motions do occur in the solar system. Section \ref{sec:2} recounts a few of these that have been observed or predicted.
The standard mathematical model for understanding the motion of planets and other bodies in the solar system is the Newtonian $N$-Body problem, presented in Section \ref{sec:3}. In included here are some of the basic features and mathematical theory of the Newtonian $N$-Body Problem, its integrals or constants of motion, special solutions such as periodic solutions, and the notions of stability and linear stability of periodic solutions and their relationship.
The notions and basic theory of collisions and singularities in the Newtonian $N$-Body Problem is presented in Section \ref{sec:4}. This includes a discussion of the probabilities of collisions, and the regularization or the lack thereof for collisions. A collision motion is rare in that is has a probability of zero of occurring, whereas a near collision motion has a positive probability of occurring. Regularization is a mathematical technique that removes the collision singularities from the Newtonian $N$-Body Problem and enables an analysis of near-collision motions in terms of collision motions through the continuous dependence of motions on initial conditions. This regularization is illustrated in the collinear $2$-Body Problem, the simplest of all the $N$-Body Problems.
Recent results are presented in Section \ref{sec:5} on the analytic and numerical existence and numerical stability and linear stability of periodic motions with regularizable collisions in various $N$-Body Problems with $N=3$ and $N=4$. Although fictitious, these periodic motions with regularizable collisions provide a view of their near-collision motions which could be motions of the bodies in the $N$-Body Problem that are collision-free and bounded for all time.
\section{Phenomenon} \label{sec:2}
Collisions and near-collisions of two or more solar system bodies are apparent obstacles at which Newton's Law of Gravity becomes problematic. Velocities of colliding bodies become infinite at the moment of collision, while velocities of near-colliding bodies become very large as they pass by each other. Both of these situations present problems for numerical estimates of the motion of such bodies.
Although collisions are rare, historical evidence of collisions of solar system bodies is viewable on the surface of the Earth and the Moon \cite{Ce}. Only recently have collisions of solar systems bodies actually been observed. As the comet Shoemaker-Levy 9 approached Jupiter it was torn apart into fragments by tidal forces. In July of 1994, at least 21 discernible fragments of Shoemaker-Levy 9 collided with Jupiter. These were the {\it first} ever observed collisions of solar system bodies. An animation of some of the fragments of Shoemaker-Levy 9 colliding with Jupiter can be found at www2.jpl.nasa.gov/sl9/anim.html.
Near-collision motion are less rare than collisions. As of March 2012, there are nearly 9000 known near-Earth asteroids\footnote{See http://neo.jpl.nasa.gov/stats/}, of which 1306 are potentially hazardous to Earth\footnote{See http://neo.jpl.nasa.gov/neo/groups.html}. One of these potentially hazardous asteroids, named 2012 DA14, was discovered in 2012. This asteroid will pass by Earth on February 15, 2013, coming closer to the Earth than satellites in geostationary orbit\footnote{See article about 2012 DA14 posted March 6, 2012 on MSNBC.com}. How close will 2012 DA14 pass by Earth? A mere 17000 miles (27000 km)\footnote{See article about 2012 DA14 posted March 8, 2012 on Earthsky.org}. In cosmic terms, this close shave of 2012 DA14 with Earth in 2013 is a near-collision motion.
\section{The $N$-Body Problem} \label{sec:3}
To model collision and near-collision motions we make some simplifying assumptions and use Newton's Inverse Square Law of Gravity. We assume that all the bodies are idealized as particles with zero volume (i.e., as points), that no particle is torn apart by tidal forces, that the mass of each particle never changes, and that besides Newton's Law of Gravity there are no other forces acting on the bodies. Under these assumptions we would think of Shoemaker-Levy 9 as not being torn apart by tidal forces, but as colliding with Jupiter as a whole.
\subsection{The Equations} \label{sec:3.1}
The particles modeling the bodies move in three-dimensional Euclidean space which we denote by ${\mathbf R}^3$. For a positive integer $N\geq 2$, suppose there are $N$ particles with positions ${\mathbf q}_j\in{\mathbf R}^3$ and masses $m_j>0$, $j=1,\dots,N$. The distance between two of the particles is denoted by \[ r_{jk} = \vert {\mathbf q}_j-{\mathbf q}_k\vert, \ \ j\ne k,\] which is the standard Euclidean distance between two points in ${\mathbf R}^3$. The Newtonian $N$-Body Problem is the system of second-order nonlinear differential equations \[ m_j {\mathbf q}_j^{\prime\prime} = \sum_{k\ne j} \frac{Gm_jm_k({\mathbf q}_k-{\mathbf q}_j)}{r_{jk}^3},\ \ j=1,\dots,N,\] where ${}^\prime = d/dt$ for a time variable $t$ and $G=6.6732\times 10^{-11}\ {\rm m}^2/{\rm s}^2{\rm kg}$. By an appropriate choice of units of the ${\mathbf q}_j$, we will assume that $G=1$ because we are investigating the qualitative or geometric, rather than the quantitative, behavior of collision and near-collision motions.
By the standard existence, uniqueness, and extension theory in differential equations (see \cite{Ch} for example), the initial value problem \begin{equation}\label{IVP} m_j {\mathbf q}_j^{\prime\prime} = \sum_{k\ne j} \frac{m_jm_k({\mathbf q}_k-{\mathbf q}_j)}{r_{jk}^3}, \ \ {\mathbf q}_j(t_0) = {\mathbf q}_j^0, \ \ {\mathbf q}_j^{\prime}(t_0) = {\mathbf q}_j^{\prime 0}, \end{equation} has a unique solution \[ {\mathbf q}(t) = ( {\mathbf q}_1(t),\dots,{\mathbf q}_N(t))\] defined on a maximal interval of definition $(t^-,t^+)$ as long as $r_{jk}\ne 0$ for all $j\ne k$ at $t=t_0$. Such a solution ${\mathbf q}(t)$ describes a motion of the $N$ particles.
Not every initial value problem (\ref{IVP}) will have a solution ${\mathbf q}(t)$ with $t^-=-\infty$ and $t^+=\infty$. A solution with either $t^->-\infty$ or $t^+<\infty$ experiences a singularity at the finite endpoint of its maximal interval of definition. The notion of a singularity is addressed in Section \ref{sec:4.1}.
\subsection{Integrals} \label{sec:3.2}
An integral of motion of the Newtonian $N$-Body Problem is a differentiable function $F$ of the position ${\mathbf q}$ and/or the velocity ${\mathbf q}^\prime$ and/or the masses ${\mathbf m}=(m_1,\dots,m_N)$ such that \[ \frac{d}{dt} F({\mathbf q}(t),{\mathbf q}^\prime(t),{\mathbf m}) = 0, \ \ t\in (t^-,t^+).\] Along a solution ${\mathbf q}(t)$ an integral $F$ of motion satisfies \[ F({\mathbf q}(t),{\mathbf q}^\prime(t),{\mathbf m}) = F({\mathbf q}(t_0),{\mathbf q}^\prime(t_0),{\mathbf m}),\ \ t^-<t<t^+,\] i.e., the value of $F$ is constant along the solution. The Newtonian $N$-Body Problem has ten known integrals of motion.
The translation invariance of the equations of the Newtonian $N$-Body Problem give rise to $6$ integrals of motion. With $M=\sum_{j=1}^N m_j$, three of these are given by the components of the center of mass vector \[ {\mathbf C} = \frac{1}{M} \sum_{j=1}^m m_j {\mathbf q}_j,\] and three more are given by the components of the linear momentum vector \[ {\mathbf L} = \frac{1}{M} \sum_{j=1}^N m_j{\mathbf q}_j^\prime.\] Typically, both of these are set to ${\mathbf 0}$ so that the {\it relative motion} of the $N$ particles is emphasized.
The rotational symmetry of the equations of the Newtonian $N$-Body Problem give rise to $3$ more integrals of motion. These integrals are given by the components of the angular momentum vector, \[ {\mathbf A} = \sum_{j=1}^N m_j {\mathbf q}_j\times {\mathbf q}^\prime_j.\] The angular momentum plays a key role in understanding collisions in the $N$-Body Problem, as we will see later on.
There is one more integral of motion of the Newtonian $N$-Body Problem. The {\it self-potential} (or negative of the potential energy) is \[ U = \sum_{j<k} \frac{m_j m_k}{r_{ik}^2}.\] The {\it kinetic energy} is \[ K = \frac{1}{2} \sum_{j=1}^N m_j {\mathbf q}^\prime_j\cdot{\mathbf q}^\prime_j.\] The {\it total energy} \[ H = K-U\] is an integral of motion for the Newtonian $N$-Body Problem.
In the late 1800's, the mathematical strategy for ``solving'' the Newtonian $N$-Body Problem was to find enough ``independent'' integrals of motion \cite{Sa}. This would implicitly give each solution as the curve of intersection of the hypersurfaces corresponding to the integrals of motion. Each solution ${\mathbf q}(t)$ is a curve in ${\mathbf R}^{6N}$. However, the intersection of the hypersurfaces of the ten integrals of motion gives a $6N-10>1$ dimension hypersurface in ${\mathbf R}^{6N}$, which is not a curve! The ten known integrals of motion are independent of each other (one is not a function of the others), and are algebraic functions of positions, velocities, and masses. Are there any more algebraic integrals of motion? This was answered a long time ago in 1887-1888 by Bruns \cite{Br}.
\begin{theorem} There are no algebraic integrals of motion independent of the ten known integrals of motion. \end{theorem}
\noindent Consequently, new integrals of motion, if any, can not be algebraic! In 1893, Newcomb \cite{Ne} lamented that no additional integrals had been found to enable the implicit solution of the $3$-Body Problem. It is well-known that the Newtonian $2$-Body Problem can be solved implicitly\footnote{See en.wikipedia.org/wiki/Gravitational\_two-body\_problem}, but all attempts to solve the $N$-Body Problem with $N\geq 3$ have been futile\footnote{Karl Sundman did solve the $3$-Body Problem when ${\mathbf A}\ne 0$ by convergent power series defined for all time, but the series converge too slowly to be of any theoretic or numerical use \cite{Sa}.}.
Typically then the solution ${\mathbf q}(t)$ of the initial value problem (\ref{IVP}) is estimated numerically. From the constant total energy $H$ along a solution ${\mathbf q}(t)$, we observe that if any of the distances $r_{jk}$ get close to $0$, i.e., at least two of the particles are near collision, the self-potential becomes large and the kinetic energy becomes large too. The latter implies that the velocity of at least one of the particles becomes large, and the linear momentum ${\mathbf L}$ along ${\mathbf q}(t)$ implies that the velocity of at least two particles becomes large. In particular, from the equations of the Newtonian $N$-Body Problem, the particles that are near collision are the one with the large velocities. These large velocities presents problems for the numerical estimates of such a solution.
\subsection{Special Solutions} \label{sec:3.3}
Rather than solving the $N$-Body Problem for all of its solutions by finding enough independent integrals of motion, it is better to examine special solutions with particular features. The simplest solutions to find are equilibrium solutions, where the position ${\mathbf q}_j(t)$ of each particle is constant for all time. But the Newtonian $N$-Body Problem has none of these (see p.29 in \cite{MHO}). The next simplest solutions are periodic solutions, i.e., there exist $T>0$ such that ${\mathbf q}(t+T) = {\mathbf q}(t)$ for all $t\in{\mathbf R}$. These are part of the larger collection of solutions ${\mathbf q}(t)$ with $t^-=-\infty$ and $t^+=\infty$ that are bounded. Such solutions must have a particular total energy (see p. 160 in \cite{Sa}).
\begin{theorem} If a solution ${\mathbf q}(t)$ of the Newtonian $N$-Body Problem exists for all time and is bounded, then the total energy $H<0$. \end{theorem}
\noindent Consequently, any periodic solution ${\mathbf q}(t)$ of the Newtonian $N$-Body Problem must has negative total energy. This is why in the search for periodic solutions the total energy is always assigned a negative value.
\subsection{Stability} \label{sec:3.4}
A periodic solution ${\mathbf q}(t)$ of the Newtonian $N$-Body Problem gives a predictable future: we know with certainty what the positions of the $N$ particles will be at any time $t>0$. But what if our measurements of the initial conditions ${\mathbf q}(0)$ and ${\mathbf q}^\prime(0)$ are slightly off? A solution $\tilde{\mathbf q}(t)$ with initial conditions near ${\mathbf q}(0)$ and ${\mathbf q}^\prime(0)$ will stay close to ${\mathbf q}(t)$ for a time, by a property of solutions of initial value problems called continuity of solutions with respect to initial conditions (see \cite{Ch}). But if it stays close for all $t>0$, we think of ${\mathbf q}(t)$ as being stable.
To {\it quantify} this notion of stability for a periodic solution, we use a Poincar\'e section which is a hyperplane $S$ containing the point $({\mathbf q}(0),{\mathbf q}^\prime(0))$ that is transverse to the curve $({\mathbf q}(t),{\mathbf q}^\prime(t))$. If ${\mathbf x} = (\tilde{\mathbf q}(0),\tilde{\mathbf q}^\prime(0))$ is a point on $S$ near the $({\mathbf q}(0),{\mathbf q}^\prime(0))$, then $P({\mathbf x})$ is next point where the the curve $(\tilde{\mathbf q}(t),\tilde{\mathbf q}^\prime(t))$ intersects $S$\footnote{For an illustration of this see en.wikipedia.org/wiki/Poincar\'e\_map}, and $P^2({\mathbf x})$ is the next point, and so on. The initial condition ${\mathbf x}^0 = ({\mathbf q}(0),{\mathbf q}^\prime(0))$ is a {\it fixed point} of this Poincar\'e map $P$ from $S$ to $S$, i.e., $P({\mathbf x}^0)={\mathbf x}^0$.
\begin{definition}\label{stability} The periodic solution ${\mathbf q}(t)$ is stable if for every real $\epsilon>0$ there exist a real $\delta>0$ such that $\vert P^k({\mathbf x}) - {\mathbf x}^0\vert<\epsilon$ for all $k=1,2,3,\dots,$ whenever $\vert {\mathbf x} - {\mathbf x}^0\vert<\delta$. \end{definition}
\noindent When ${\mathbf q}(t)$ is not stable, there are solutions which start nearby but eventually move away from ${\mathbf q}(t)$, and we say that ${\mathbf q}(t)$ is {\it unstable}.
Showing directly that ${\mathbf q}(t)$ is stable or unstable is very difficult. Instead, the related concept of linearized stability is investigated, at least numerically. The derivative of the Poincar\'e map at the fixed point ${\mathbf x}^0$ is a square matrix $DP({\mathbf x}^0)$.
\begin{definition}\label{linearstability} A periodic solution ${\mathbf q}(t)$ is \begin{enumerate} \item spectrally stable\footnote{There is a more restrictive notion of spectral stability known as linear stability that requires additional technical conditions on the square matrix $DP({\mathbf x}^0)$.} if all the eigenvalues of $DP({\mathbf x}^0)$ have modulus one, and is \item linearly unstable if any eigenvalue of $DP({\mathbf x}^0)$ has modulus bigger than one. \end{enumerate} \end{definition}
\noindent In 1907, Liapunov \cite{Li} established a connection between the stability of Definition \ref{stability} and the linearized stability of Definition \ref{linearstability}.
\begin{theorem} If a periodic solution ${\mathbf q}(t)$ is stable then it is spectrally stable, and if ${\mathbf q}(t)$ is linearly unstable, then it is unstable. \end{theorem}
\noindent If a periodic solution is shown numerically to be linearly unstable, then by Theorem 3 the periodic solution is unstable. On the other hand, if a periodic solution is shown numerically to be spectrally stable, it may be stable or unstable. Examples exist with spectrally stable fixed points of maps like $P$ that are unstable (see \cite{SM}).
The notion of stability for a non-periodic solution, such as the motion of the sun and planets in the solar system, is harder to grasp. Here is a sampling of the history and opinions on this stability problem. In 1891, Poincar\'e commented that the stability of the solar system had at that time already preoccupied much time and attention of researchers (see p. 147 in \cite{DH}). In 1971, Siegel and Moser lamented that a resolution of the stability problem for the $N$-Body Problem would probably be in the distant future (see p. 219 in \cite{SM}). In 1978, Moser noted that the answer to the stability of the solar system was still not known (see p. 127 in \cite{DH}). In 2005, Saari stated that a still unresolved problem for the $N$-Body Problem is that of stability (see p. 132 in \cite{Sa}). Meyer, Hall, and Offin commented how little is known about the stability problem and how difficult it was to get (see p. 229 in \cite{MHO}).
In 1996, Diacu and Holmes suggested that the solar system should be considered stable (in a weak sense) if no collisions occur among the sun and the planets, and no planet ever escape from the solar system (see. p.129 in \cite{DH}). In this weak sense of stability, the solar system is stable for the next few billion years according to numerical work of Hayes \cite{Ha} in 2007. Much longer-term numerical studies of the solar system by Batygin and Laughlin \cite{BL} in 2008 using small changes in the initial conditions suggest that Mercury could fall into the sun in 1.261Gyr\footnote{Gyr means giga-year or 1,000,000,000 years}, or that Mercury and Venus could collide in 862Myr\footnote{Myr means mega-year or 1,000,000 years} and Mars could escape from the solar system in 822Myr. The Newtonian $N$-Body Problem thus suggests that in the near future, the Solar System should be free of collisions of planets and the Sun, with no planets escaping the solar system. But this still leaves open the possibility that smaller objects, such as asteroids and comets, could collide with any of the planets in the short and long term. Recall that there are nearly 9000 of those near-Earth asteroids to consider, with 2012 DA14 making its near-collision approach with Earth on February 15 of 2013.
\section{Collisions} \label{sec:4}
Either in the short term of the long term, collisions put a {\it wrench} into the question of any notion of stability. Why should a solution or any nearby solution of the Newtonian $N$-Body Problem be defined for all time? Remember that Shoemaker-Levy 9 has $t^+<\infty$!
\subsection{Singularities} \label{sec:4.1}
Collisions are one of the {\it two} kinds of singularities in the Newtonian $N$-Body Problem. The solution ${\mathbf q}(t)$ of initial value problem (\ref{IVP}) is real analytic (i.e., a convergent power series) on an interval $(t_0-\delta,t_0+\delta)$ for some $\delta>0$, as long as $r_{jk}\ne 0$ for all $j\ne k$ at $t_0$. By a process called analytic continuation (see for example \cite{MaH}), the interval $(t_0-\delta,t_0+\delta)$ can be extended to the maximal interval $(t^-,t^+)$.
\begin{definition} A {\it singularity} of the Newtonian $N$-Body Problem is a time $t=t^+$ or $t^-$ when $t^+<\infty$ or $t^->-\infty$. \end{definition}
In 1897, Painlev\'e \cite{Pa} characterized a singularity of the Newtonian $N$-Body Problem, using the quantity \[ r_{\rm min}(t) = \min_{j\ne k} r_{jk}(t)\] determined by a solution ${\mathbf q}(t)$.
\begin{theorem} A singularity for the Newtonian $N$-body Problem occurs at time $t=t^*$ if and only if $r_{\rm min}(t)\to 0$ as $t\to t^*$. \end{theorem}
\noindent An understanding what this means is obtained by considering the {\it collision set} \[ \Delta = \bigcup_{j\ne k} \{ {\mathbf q} : {\mathbf q}_j={\mathbf q}_k\}\subset ({\mathbf R}^3)^N,\] which is the set of points where two or more of the $N$-particles occupy the same position. Painlev\'e's characterization means that ${\mathbf q}(t)$ approaches the collison set, i.e., \[ {\mathbf q}(t)\to \Delta\ \ {\rm as}\ \ t\to t^*\] when $t^*$ is a singularity of the Newtonian $N$-Body Problem. Painlev\'e's chararterization introduces two classes of singularities.
\begin{definition} A singularity $t^*$ of the Newtonian $N$-Body Problem is a collision singularity when $q(t)$ approaches a specific point of $\Delta$ as $t\to t^*$. Otherwise the singularity $t^*$ is a non-collision singularity. \end{definition}
Only collision singularities can occur in the Newtonian $2$-Body Problem because it can be implicitly solved. In 1897, Painlev\'e \cite{Pa} showed that only one other Newtonian $N$-Body problem has only collision singularities.
\begin{theorem} In the $3$-Body Problem, all singularities are collision singularities. \end{theorem}
\noindent Unable to extend his result to more than $3$ bodies, Painlev\'e conjectured that there exist non-collision singularities in the Newtonian $4$ or larger Body Problem. In 1992, Xia \cite{Xi} mostly confirmed Painlev\'e's conjecture, giving an example in the Newtonian $5$-Body Problem.
\begin{theorem} There exist non-collision singularities in the $N$-Body Problem for $N\geq 5$. \end{theorem}
\noindent That leaves unresolved the question of the existence of non-collision singularities in the Newtonian $4$-Body Problem.
An understanding of what a non-collision singular looks like is obtained by considering one-half of the {\it polar moment of inertia} of the Newtonian $N$-Body Problem: \[ I = \frac{1}{2} \sum_{j=1}^N m_j {\mathbf q}_j\cdot{\mathbf q}_j.\] This scalar quantity measures the ``diameter'' of the $N$ particles in the Newtonian $N$-Body Problem. In 1908, von Zeipel \cite{Ze} characterized a collision singularity in terms of the polar moment of inertia.
\begin{theorem}\label{vonZeipel} A singularity of the Newtonian $N$-Body Problem at $t=t^*$ is a collision if and only if $I$ is bounded as $t\to t^*$. \end{theorem}
\noindent This implies that for a non-collision singularity, at least one of the $N$-particles has to achieve an infinite distance from the origin in just a finite time. This is a rather strange thing for Newton's Law of Gravity to predict. On the other hand, by Theorem \ref{vonZeipel}, for a collision singularity, all of the positions of the $N$ particles remain bounded at the moment of the singularity.
A total collapse is an example of a collision singularity in the $N$-Body Problem for which all $N$ particles collide at the same point at the singularity $t^*$. For a solution ${\mathbf q}(t)$, the quantity \[ r_{\rm max} = \max_{j\ne k} r_{jk}(t)\] characterizes a total collapse: a total collapse occurs at $t^*$ if and only if \[ r_{\rm max}(t) \to 0 {\rm\ as\ } t\to t^*.\] There is a relationship between total collapse and the angular momentum that was known by Weierstrass and established by Sundman (see \cite{Sa}).
\begin{theorem}\label{angular} If ${\mathbf A}\ne 0$, then $r_{\rm max}(t)$ is bounded away from zero. \end{theorem}
\noindent This does not preclude the collision of less than $N$ particles when ${\mathbf A}\ne 0$, as will be illustrated for certain Newtonian $N$-Body Problems in Section \ref{sec:5}.
\subsection{Improbability} \label{sec:4.2}
Recall that there are 1306 potentially hazardous near-Earth asteroids. What are the chances that Earth will be hit by a near-Earth asteroid, or Jupiter will be hit by another comet? Well, it depends on the arrangement of the particles.
\begin{definition} A solution ${\mathbf q}(t)$ is called collinear if the $N$ particles always move on the same fixed line in ${\mathbf R}^3$. Otherwise it is called non-collinear. \end{definition}
\noindent Every collinear solution has zero angular momentum because ${\mathbf q}_j(t)$ is parallel with ${\mathbf q}_j^\prime(t)$ for all $t\in (t^-,t^+)$. In 1971 and 1973, Saari \cite{Sa71,Sa73} established the probability of collisions.
\begin{theorem}\label{collisionprobability}The probability that a non-collinear solution ${\mathbf q}(t)$ will have a collision is zero. Every collinear solution ${\mathbf q}(t)$ has a collision. \end{theorem}
With collision singularities being rare for a non-collinear $N$-Body Problem, why bother to study them? Diacu and Holmes (see p. 84 and p. 103 in \cite{DH}) argue for the study of collision singularities because without such a study, a complete understanding of the Newtonian $N$-Body Problem could not be achieved. In particular, solutions near collision singularities could behave strangely, and the probability of a solution coming close to a collision singularity is positive and thus can not be ignored. Understanding then the collision singularities enables an understanding of the near-collision solutions.
\subsection{Regularization} \label{sec:4.3}
Regularization is one method by which we can get an understanding of a collision singularity. To {\it regularize} a collision means to extend the solution beyond the collision through an elastic bounce without loss or gain of total energy in such a way that all of the solutions nearby have continuity with respect to initial conditions, i.e., they look like the extended collision solution for a time (see p. 104 and p. 107 in \cite{DH}). Regularization is typically done by a Levi-Civita type change of the dependent variables, and a Sundman type change of the independent variable (see \cite{Ce}), that together {\it removes} the collision singularity from the equations. We illustrate this regularization in the simplest of the $N$-Body Problems.
In the Collinear $2$-Body Problem (or Col2BP for short), the positions of the two particles are the scalar quantities $q_1$ and $q_2$. If $x=q_2-q_1$ is the distance between the particle with mass $m_1$ at $q_1$ and the particle with mass $m_2$ at $q_2>q_1$, then the Col2BP takes the form \begin{equation}\label{Col2BP} x^{\prime\prime} = - \frac{m_1+m_2}{x^2}, \ x>0,\end{equation} and the total energy takes the form \begin{equation}\label{totalenergy} H = \frac{m_1m_2}{2(m_1+m_2)} (x^\prime)^2 - \frac{m_1m_2}{x}.\end{equation} As $x\to 0$ the two particles approach collision, and the total energy implies that the two particles collide with an infinite velocity, \[ (x^\prime)^2\to\infty.\] To regularize the {\it binary collision} (or total collapse) in this problem, define a new independent variable $s$ and a new dependent variable $w$ by \[ \frac{ds}{dt} = \frac{1}{x}, \ \ w^2 = x,\] where the former is the Sundman type change of the independent variable, and the latter is the Levi-Civita type change of the dependent variable. If $\ \dot{}= d/ds$, the second-order equation (\ref{Col2BP}) becomes \begin{equation}\label{regularizedODE} w^2\big[ 2w\ddot w - 2\dot w^2 + (m_1+m_2)\big]=0,\end{equation} and the total energy (\ref{totalenergy}) becomes \begin{equation}\label{Renergy} Hw^2 = \frac{2m_1m_2}{m_1+m_2}\dot w^2 - m_1m_2.\end{equation} As $w\to 0$, the second-order equation (\ref{regularizedODE}) makes sense (no dividing by zero), and the total energy (\ref{Renergy}) implies that \[ (\dot w)^2 \to \frac{m_1+m_2}{2},\] which is a finite nonzero velocity! The collision singularity has been regularized.
The regularized nonlinear second-order equation (\ref{regularizedODE}) can actually be solved! Solving the total energy (\ref{Renergy}) for $2(\dot w)^2$ and substituting this into the second-order equation (\ref{regularizedODE}) gives \begin{equation}\label{linearODE} 2w^3\left[ \ddot w - \frac{(m_1+m_2)H}{2m_1m_2} w \right] = 0.\end{equation} This makes sense when $w=0$, i.e., the moment of collision! For negative $H$, the linear second-order equation\footnote{This is a simple harmonic oscillator for $H<0$ whose solutions are in terms of cosine and sine.} inside the square brackets in (\ref{linearODE}) solves to give a real analytic stable periodic solution $w(s)$ which experiences a collision every half period in terms of the regularized time variable $s$. The corresponding solution $x(t)$ is periodic and experiences a collision once a period in terms of the original time variable $t$. This doubling of the number of collisions per period is because the change of dependent variable $w^2=x$ has $w(s)$ ``doubling'' $x(t)$ in that $w(s)$ passes through $0$ twice a period, going from positive to negative and then negative to positive, while $x(t)$ is positive except at collision where it is zero.
The binary collision singularity in the Newtonian $2$-Body Problem can be regularized in a similar but more complicated way than what was done above for the Col2BP (see \cite{Sa}). By Theorem \ref{angular}, a solution of the $2$-Body Problem with nonzero angular momentum does not experience a collision or total collapse. A nonzero angular momentum near-collision solution looks like the zero angular momentum collision solution\footnote{Binary Star Systems are known to exist in the Universe. The Newtonian $2$-Body Problem predicts stability for a Binary Star System, a collision-free solution that is bounded for all time.}. The regularized $2$-Body Problem provides good numerical estimates of the motion because there are no infinite velocities!
\subsection{McGehee} \label{sec:4.4}
What about regularization of a triple collision, when three of the particles meet? In 1974, McGehee \cite{McG} showed that regularization of a triple collision is in general not possible\footnote{This is achieved by ``blowing-up'' the triple collision singularity and slowing down the motion as the particles approach a triple collision. This setting does allow for good numerical estimates of near-triple collisions.}. Starting close together, two solutions that approach a near triple collision can describe {\it radically different} motions after the near triple collision. This kind of behavior is known as ``sensitive dependence on initial conditions,'' and is an antithesis of stability. Triple collisions present a numerical nightmare! By extension, collisions with four or more particles present the same nightmare! So the only regularizable collisions are those that are essentially a binary collision.
\section{Results} \label{sec:5}
Spectrally stable periodic solutions have been found in Newtonian $N$-Body Problems with regularizable collisions for $N\geq 3$. Three of these situations discussed here are the Collinear $3$-Body Problem (or Col3BP), the Collinear Symmetric $4$-Body Problem (or ColS4BP), and the Planar Pairwise Symmetric $4$-Body Problem (or PPS4BP). There are other Newtonian $N$-Body Problems where periodic solutions with regularizable collisions whose existence has been given analytically \cite{Ya1,Ya2,Sh}, some of whose stability (in the sense of Definition \ref{stability}) and linear stability (as defined in Definition \ref{linearstability}) has been numerically determined \cite{BS,Wa,Ya1,Ya2}.
\subsection{Col3BP} \label{sec:5.1}
As a subproblem of the Newtonian $3$-Body Problem, the Col3BP requires that the three particles always lie on the same line through the origin. The positions of the three particles in the Col3BP are the scalars $q_1$, $q_2$, and $q_3$ which can be assumed to satisfy \[ q_1\leq q_2\leq q_3.\] By Theorem \ref{collisionprobability}, collisions always occur in the Col3BP. Because the three particles are collinear for all time, their angular is zero, and by Theorem \ref{angular} a total collapse is possible\footnote{Initial conditions leading to total collapse in the equal mass Col3BP are easy to realize: set $q_1=-1$, $q_2=0$, and $q_3=1$ with the initial velocity of each particle set to $0$.} in the Col3BP. In 1974, S.J. Aareth and Zare \cite{AZ} showed that any two of the three possible binary collisions in the $3$-Body Problem are regularizable\footnote{A good numerical model for the Sun-Jupiter-Shoemaker-Levy 9 or Earth-Moon-2012DA14 situation is regularized $3$-Body Problem of Aarseth and Zare.}. In 1993, Hietarinta and Mikkola \cite{HM} used Aarseth and Zare's regularization \cite{AZ} to regularize the binary collisions $q_1=q_2$ and $q_2=q_3$ in the Col3BP.
In 1956, Schubart \cite{Sc} numerically found a periodic orbit in the equal mass Col3BP of negative total energy in which the inner particle oscillates between binary collisions with the outer particles. In 1977, H\'enon \cite{He} numerically extended Schubart's periodic solution to arbitrary masses and investigated their linear stability. In 1993, Hietarinta and Mikkola \cite{HM} also numerically investigated the linear stability of Schubart's periodic solution for arbitrary masses. Together they showed that Schubart's periodic solution is spectrally stable for certain masses, and linearly unstable for the remaining masses. Hietarinta and Mikkola \cite{HM} further numerically investigated the Poincar\'e section for Schubart's periodic solution for arbitrary masses, showing when there is stability as described in Definition \ref{stability}. In 2008, Moeckel \cite{Mo} and Venturelli \cite{Ve} separately proved the analytic existence of Schubart's solution when $m_1=m_3$ and $m_2$ is arbitrary. Only recently, in 2011, did Shibayama \cite{Sh} analytically prove the existence of Schubart's periodic solution for arbitrary masses in the Col3BP.
Schubart's period solution for the Col3BP is also a periodic solution of the $3$-Body Problem, where in the latter the continuity with respect to initial conditions can be see for near-collision solutions. For example, Schubart's periodic solution for the nearly equal masses \[ m_1=0.333333,\ m_2=0.333334,\ m_3=0.333333\] is spectrally stable. Considered in $3$-Body Problem, Schubart's periodic solution for these mass values remains spectrally stable \cite{He}, and numerically the near-collision solutions in the Newtonian $3$-Body Problem behave like Schubart's periodic solution. It is therefore possible that in the $3$-Body Problem, there are solutions near Schubart's periodic solution that are free of collisions and bounded for all time. Imagine, as did H\'enon \cite{He}, of Newton's Law of Gravity predicting a triple star system that is free of collisions and bounded for all time!
\subsection{ColS4BP} \label{sec.5.2}
As a subproblem of the Newtonian $4$-Body Problem, the ColS4BP requires that the four particles always lie on the same line through the origin. The positions of the four particles are the scalars $q_1$, $q_2$, $q_3$, and $q_4$ that satisfy \[ q_4=-q_1,\ q_3=-q_2,\ q_1\geq 0,\ q_2\geq 0,\] and \[ -q_1 \leq -q_2\leq 0\leq q_2\leq q_1\] with masses \[ m_1=1,\ m_2=m>0,\ m_3=m,\ m_4=1.\] The angular momentum for all solutions of the ColS4BP is zero because of the collinearity, and so by Theorem \ref{angular} a total collapse is possible. There are two kinds of non-total collapse collisions in the ColS4BP: the binary collision of the inner pair of particles of mass $m$ each, i..e, $q_2=0$, and the {\it simultaneous} binary collision of the two outer pairs of particles, i.e., $q_1=q_2>0$. In 2002 and 2006, Sweatman \cite{SW1,SW2} showed, by adapting the regularization of Aarseth and Zare \cite{AZ}, that these non-total collapse collisions in the ColS4BP are regularizable.
Sweatman \cite{SW1,SW2} numerically found a Schubart-like periodic solution in the ColS4BP with negative total energy for arbitrary $m$ where the outer pairs collide in a simultaneous binary collision at one moment and then the inner pair collides at another moment. He determined numerically that this Schubart-like periodic solution is spectrally stable when \[ 0<m<2.83\ {\rm and}\ m>35.4,\] and is otherwise linearly unstable. In 2010, Bakker et al \cite{BOYSR} verified Sweatman's linear stability for the Schubart-like periodic solution using a different technique. In 2011-2012, Ouyang and Yan \cite{OY2}, Shibayama \cite{Sh}, and Huang \cite{Hu} proved separately the analytic existence of the Schubart-like periodic solution in the ColS4BP.
\subsection{PPS4BP} \label{sec:5.3}
The PPS4BP has two particles of mass $1$ located at the planar locations \[ {\mathbf q}_1 {\rm\ and\ }{\mathbf q}_3=-{\mathbf q}_1,\] and two particles of mass $0<m\leq 1$ located at the planar locations \[ {\mathbf q}_2 {\rm\ and\ } {\mathbf q}_4=-{\mathbf q}_2.\] The four particles in the PPS4BP need not be collinear, so that the angular momentum need not be $0$. Unlike the ColS4BP, total collapse can be avoided in the PPS4BP by Theorem \ref{angular} when the angular momentum is not zero. Like the ColS4BP, there are two kinds of non-total collapse collisions in the PPS4BP: simultaneous binary collisions when ${\mathbf q}_1={\mathbf q}_2$ and ${\mathbf q}_3={\mathbf q}_4$, or when ${\mathbf q}_1 = {\mathbf q}_4$ and ${\mathbf q}_2={\mathbf q}_3$; and binary collisions when ${\mathbf q}_1=0$ or when ${\mathbf q}_2=0$. In 2010, Sivasankaran, Steves, and Sweatman \cite{SSS} showed that these non-total collapse collisions in the PPS4BP are regularizable.
The Schubart-like periodic solution in the ColS4BP is also a periodic solution of the PPS4BP, where in the latter the continuity with respect to initial conditions can be observed for near-collision solutions. However, as shown by Sweatman \cite{SW2}, in the PPS4BP the Schubart-like periodic solution of the ColS4BP becomes linearly unstable for \[ 0<m<0.406, {\rm \ and\ } 0.569<m<1.02\] as well as $2.83<m<35.4$, while it remains spectrally stable for \[ 0.407<m<0.567{\rm\ and\ } m>35.4.\] By long-term numerical integrations for the Schubart-like periodic solution as a solution of the PPS4BP, Sweatman \cite{SW2} showed that stability in the sense of Definition \ref{stability} is possible when $0.407<m<0.567$ and when $m>35.4$. It is therefore possible for these values of $m$ that near Schubart's periodic solution there are collision-free solutions of the PPS4BP that are bounded for all time.
In 2011, adapting the regularization of Aarseth and Zare \cite{AZ} to simultaneous binary collisions, Bakker, Ouyang, Yan, and Simmons \cite{BOYS} proved the analytic existence of a non-collinear periodic solution in the equal mass PPS4BP. This periodic solution has zero angular momentum, negative total energy, and alternates between a simultaneous binary collision of the symmetric pairs in the first and third quadrant where ${\mathbf q}_1={\mathbf q}_2$ and ${\mathbf q}_3={\mathbf q}_4$, and the simultaneous binary collision of the symmetric pairs in the second and fourth quadrants where ${\mathbf q}_1={\mathbf q}_4$ and ${\mathbf q}_2={\mathbf q}_3$. Bakker, Ouyang, Yan and Simmons \cite{BOYS} then numerically extended this non-collinear periodic simultaneous binary collision solution to unequal masses $0<m<1$. In 2012, Bakker, Mancuso, and Simmons \cite{BMS} have numerically determined that the non-collinear periodic simultaneous binary collision solution is spectrally stable when \[ 0.199<m<0.264{\rm\ and\ } 0.538<m\leq 1\] and is linearly unstable for the remaining values of $m$. Long-term numerical integrations of the regularized equations done by Bakker, Ouyang, Yan, and Simmons \cite{BOYS} suggest instability when $0.199<m<0.264$ and stability when $0.538<m\leq 1$ in the sense of Definition \ref{stability}. For these latter values of $m$ could the near-collision solutions in the PPS4BP that look like the non-collinear periodic simultaneous binary collision solution be collision-free and bounded for all time?
\section{Future Work?} \label{sec:6}
Both the ColS4BP and the PPS4BP are subproblems of the Newtonian $4$-Body Problem, where the non-total collapse collisions in the former two problems are regularizable. What is not known is how to, if possible, regularize binary collisions and simultaneous binary collisions in the Newtonian $4$-Body Problem within one coordinate system \footnote{During the Special Session on Celestial Mechanics at the American Mathematical Society's Sectional Conference in April 2011 at the College of the Holy Cross, Worcester, Massachusetts, Rick Moeckel put forth the problem of finding an elegant coordinate system for the Newtonian $4$-Body Problem in which regularizes binary collisions and simultaneous binary collisions and blows up all triple collisions and total collapse. The regularization of binary collisions and simultaneous binary collisions can be achieved within multiple coordinate systems, with one coordinate system for each regularizable collision.}. If such a regularization is possible, then all of the periodic solutions thus known in the ColS4BP and PPS4BP would also be periodic solutions of the Newtonian $4$-Body Problem and the investigation of their stability and linear stability in the Newtonian $4$-Body Problem could begin. With more possible perturbations of initial conditions in the Newtonian $4$-Body Problem as compared with the PPS4BP, a loss of spectral stability could indeed happen as it did with going from the ColS4BP to the PPS4BP. But some of the spectral stability might survive passage from the PPS4BP to the Newtonian $4$-Body Problem, giving the possibility of near-collision solutions that are collision-free and bounded for all time.
\end{document}
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\begin{document}
\title{Diameter two properties and the Radon-Nikod\'ym property in Orlicz spaces} \keywords{Banach function space, Orlicz space, Daugavet property, (local, strong) diameter two property, Radon-Nikod\'ym property, octahedral norm, uniformly non-$\ell_1^2$ points} \subjclass[2010]{46B20, 46E30, 47B38}
\author{Anna Kami\'{n}ska} \address{Department of Mathematical Sciences, The University of Memphis, TN 38152-3240} \email{[email protected]}
\author{Han Ju Lee} \address{Department of Mathematics Education, Dongguk University - Seoul, 04620 (Seoul), Republic of Korea} \email{[email protected]}
\author{Hyung Joon Tag} \address{Department of Mathematical Sciences, The University of Memphis, TN 38152-3240} \email{[email protected]}
\date{\today}
\thanks{ The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology [NRF-2020R1A2C1A01010377].}
\begin{abstract} Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikod\'ym property. Consequently it is shown that an Orlicz function space $L_\varphi$ over a non-atomic $\sigma$-finite measure space $(\Omega, \Sigma,\mu)$, not necessarily separable, has the Radon-Nikod\'ym property if and only if $\varphi$ is an $N$-function at infinity and satisfies the appropriate $\Delta_2$ condition. For an Orlicz sequence space $\ell_\varphi$, it has the Radon-Nikod\'ym property if and only if $\varphi$ satisfies the $\Delta_2^0$ condition. In the second part a relationship between uniformly $\ell_1^2$ points of the unit sphere of a Banach space and the diameter of the slices are studied. Using these results, a quick proof is given that an Orlicz space $L_\varphi$ has the Daugavet property only if $\varphi$ is linear, so when $L_\varphi$ is isometric to $L_1$. Another consequence is that Orlicz spaces equipped with the Orlicz norm generated by $N$-functions never have the local diameter two property, while it is well-known that when equipped with the Luxemburg norm, it may have that property. Finally, it is shown that the local diameter two property, the diameter two property, and the strong diameter two property are equivalent in Orlicz function and sequence spaces with the Luxemburg norm under appropriate conditions on $\varphi$. \end{abstract}
\maketitle
\begin{center}{Dedicated to the memory of Professor W.A.J. Luxemburg}\end{center}
\section{Introduction}
The objective of this paper is to study geometrical properties in real Banach spaces, in particular in Banach function spaces and Orlicz spaces. A Banach space $(X, \|\cdot\|)$ is said to have the {\it Daugavet property} if every rank one operator $T: X\to X$ satisfies the equation \[
\|I + T\| = 1 + \|T\|. \]
It is well-known that $C[0,1]$ has the Daugavet property. Also, a rearrangement invariant space $X$ over a finite non-atomic measure space with the Fatou property satisfies the Daugavet property if the space is isometrically isomorphic to either $L_1$ or $L_{\infty}$ \cite{AKM, AKM2}. If a rearrangement invariant space $X$ over an infinite non-atomic measure space is uniformly monotone, then it is isometrically isomorphic to $L_1$ \cite{AKM2}. Furthermore, the only separable rearrangement invariant space over $[0,1]$ with the Daugavet property is $L_1[0,1]$ with the standard $L_1$-norm \cite{KMMW}. In \cite{KK}, a characterization of Musielak-Orlicz spaces with the Daugavet property has been provided. We refer to \cite{KSSW, W} for further information on the Daugavet property.
Let $S_X$ and $B_X$ be the unit sphere and the unit ball of a Banach space $X$ and let $X^*$ be the dual space of $X$. A slice of $B_X$ determined by $x^*\in S_{X^*}$ and $\epsilon>0$ is defined by the set \[ S(x^*; \epsilon) = \{x\in B_X : x^*(x) > 1 - \epsilon\}. \] Analogously, for $x\in S_X$ and $\epsilon >0$, a weak$^*$-slice $S(x, \epsilon)$ of $B_{X^*}$ is defined by the set \[ S(x; \epsilon) = \{x^*\in B_{X^*}: x^*(x) > 1 - \epsilon\}. \] There are several geometrical properties related to slices and weak$^*$-slices. We say that $X$ has
\begin{enumerate}[{\rm(i)}] \item the {\it local diameter two property} (LD2P) if every slice of $B_X$ has the diameter two. \item the {\it diameter two property} (D2P) if every non-empty relatively weakly open subset of $B_X$ has the diameter two. \item the {\it strong diameter two property} (SD2P) if every finite convex combination of slices of $B_X$ has the diameter two. \item the {\it weak$^*$-local diameter two property} (weak$^{*}$-LD2P) if every weak$^*$-slice of $B_{X^*}$ has the diameter two. \item the {\it Radon-Nikod\'ym property} (RNP) if there exist slices of $B_X$ with arbitrarily small diameter. \end{enumerate}
A few remarks are in order now. Condition $\rm(v)$ is a geometrical interpretation of the classical Radon-Nikod\'ym property \cite[Theorem 3, p. 202]{DU}. By the definitions, we see that properties (i), (ii) and (iii) are on the opposite spectrum of $\rm(v)$. It is clear that $\rm(ii) \implies \rm(i)$, and the implication $\rm(iii) \implies \rm(ii)$ results from
\cite[Lemma II.1 p. 26]{GGMS}.
It is also well-known that these three properties are not equivalent in general {\cite{ALN, BLR}, see also introductory remarks in \cite{HLP}. A Banach space $X$ with the Daugavet property satisfies the SD2P \cite[Theorem 4.4]{ALN}.
After Preliminaries, in Section 3, we show first that if a Banach function space $X$ over a $\sigma$-finite measure space has the RNP then it must be order continuous. The opposite implication is not true in general. However we prove it under additional assumptions when $X$ satisfies the Fatou property, and when the subspaces of order continuous elements and the closure of simple functions coincide in its K\"othe dual space $X'$. We will provide some examples to see that this assumption is necessary in order to show the converse. Applying the obtained results further, we conclude the section with the necessary and sufficient condition for the RNP in Orlicz spaces. There is a well-known criterion for the RNP in Orlicz spaces $L_\varphi$ over a separable complete non-atomic measure space $(\Omega, \Sigma, \mu)$, generated by an $N$-function $\varphi$. Here we drop the assumption of separability of a measure space and show that necessary and sufficient conditions for the RNP is that $\varphi$ satisfies appropriate $\Delta_2$ condition and that $\varphi$ is an $N$-function at infinity. In sequence spaces $\ell_\varphi$ we drop the assumption that $\varphi$ is an $N$-function.
In section 4 the Daugavet and various diameter two properties are studied. In the first main theorem we give a local characterization of uniformly $\ell_1^2$ points $x\in S_X$, where $X$ is a Banach space, and the diameter of the weak$^*$-slice $S(x;\epsilon)$ generated by $x$. Analogously we describe a relationship between $x^*\in S_{X^*}$ and the diameter of the slice $S(x^*,\epsilon)$ given by $x^*$. Consequently, we obtain a description of global properties of $X$ or $X^*$ being locally octahedral and $X^*$ or $X$ respectively, having (weak$^*$) local diameter two property. We also obtain relationships among the Daugavet property of $X$ and the D2P of $X$ and weak$^*$-LD2P of $X^*$. In Theorem \ref{th:KamKub} we provide sufficient conditions for the existence of uniformly non-$\ell_1^2$ points in $L_\varphi$ and $\ell_\varphi$ both equipped with the Luxemburg norms. Combining this with the previous general facts we recover instantly that the only Orlicz space $L_\varphi$ generated by a finite function $\varphi$ with the Daugavet property must coincide with $L_1$ as sets with equivalent norm. The other consequences are that large class of the Orlicz spaces $L_\varphi^0$ and $\ell_\varphi^0$ equipped with the Orlicz norm, does not have the LD2P, in the striking opposition to the same Orlicz spaces equipped with the Luxemburg norm. In the final result we show that the LD2P, D2P, SD2P and the appropriate condition $\Delta_2$ are equivalent in $L_\varphi$ and $\ell_\varphi$.
\section{Preliminaries}
Let $(\Omega, \Sigma, \mu)$ be a measure space with a $\sigma$-finite complete measure $\mu$ and let $L^0(\Omega)$ be the set of all equivalence classes of $\mu$-measurable functions $f:\Omega\to \mathbb{R}$ modulo a.e. equivalence. We denote by $L^0=L^0(\Omega)$ if $\Omega$ is a non-atomic measure space and by $ \ell^0=L^0(\mathbb{N})$ if $\Omega = \mathbb{N}$ with the counting measure $\mu$. That is, $\ell^0$ consists of all real-valued sequences $x = \{x(n)\}$. A Banach space $(X, \|\cdot\|) \subset L^0(\Omega)$ is called a {\it Banach function lattice} if for $f\in L^0(\Omega)$ and $g \in X$, $0 \leq f \leq g$ implies $f \in X$ and $\|f\| \leq \|g\|$. We call $X$ a {\it Banach function space} if $\Omega$ is a non-atomic measure space and a {\it Banach sequence space} if $\Omega = \mathbb{N}$ with the counting measure $\mu$. A Banach function lattice $(X, \| \cdot \|)$ is said to have the {\it Fatou property} if whenever a sequence $(f_n) \subset X$ satisfies $\sup_n \|f_n\| < \infty$ and $f_n \uparrow f\in L^0(\Omega)$ a.e., we have $f \in X$ and $\|f_n\| \uparrow \|f\|$. An element $f\in X$ is said to be {\it order continuous} if for every $(f_n) \subset L^0(\Omega)$ such that $0\le f_n \le f$, $f_n \downarrow 0$ a.e. implies $\|f_n\| \downarrow 0$. The set of all order continuous elements in $X$ is denoted by $X_a$, and the closure in $X$ of all simple functions belonging to $X$ is denoted by $X_b$. If $X=X_a$ then we say that $X$ is order continuous. In this paper, a simple function is a finitely many valued function whose support is of finite measure. It is well-known that $X_a \subset X_b$ \cite[Theorem 3.11, p. 18]{BS}\cite{ Lux}.
The {\it K\"{o}the dual space}, denoted by $X^{\prime}$, of a Banach function lattice $X$ is a set of $x \in L^0(\Omega)$, such that \[
\|x\|_{X^{\prime}}= \sup\left\{\int_\Omega xy : \|y\| \leq 1\right\} < \infty. \]
The space $X^{\prime}$, equipped with the norm $\| \cdot \|_{X^{\prime}}$, is a Banach function lattice satisfying the Fatou property. It is well known that $X = X''$ if and only if $X$ satisfies the Fatou property \cite[Theorem 1, Ch. 15, p. 71]{Z}.
We say $f,g \in X$ are {\it equimeasurable}, denoted by $f \sim g$, if $\mu\{t: |f(t)| > \lambda\} = \mu\{t: |g(t)| > \lambda\}$ for every $\lambda > 0$. A Banach function lattice $(X, \|\cdot\|)$ is said to be {\it rearrangement invariant} (r.i.) if $f \sim g$ ($f,g \in X$) implies $\|f\| = \|g\|$. The Lebesgue, Orlicz and Lorentz spaces are classical examples of r.i. spaces. The fundamental function of a r.i. space $X$ over a non-atomic measure space is defined by $\phi_X(t) = \|\chi_{E}\|_X$, $t\ge 0$, where $E\in \Sigma$ is such that $\mu(E) = t$. It is known that $\lim_{t \rightarrow 0^+} \phi_X(t) = 0$ if and only if $X_a = X_b$ \cite[Theorem 5.5, p. 67]{BS}. For the purely atomic case where each atom has the same measure,}$X_a = X_b$ is always true \cite[Theorem 5.4, p. 67]{BS}.
Recall that a measure space $(\Omega, \Sigma, \mu)$ is said to be {\it separable} if there is a countable family $\mathcal{T}$ of measurable subsets such that for given $\epsilon>0$ and for each $E\in \Sigma$ of finite measure there is $A\in \mathcal{T}$ such that $\mu(A\Delta E)<\epsilon$, where $A\Delta E$ is the symmetric difference of $A$ and $E$. It is easy to check that if $\Sigma$ is a $\sigma$-algebra generated by countable subsets then $(\Omega, \Sigma, \mu)$ is separable \cite{Hal}.
A function $\varphi:\mathbb{R}_+\to [0,\infty]$ is called an {\it Orlicz function} if $\varphi$ is convex, $\varphi(0)=0$, and $\varphi$ is left-continuous, not identically zero nor infinite on $(0,\infty)$. The complementary function $\varphi_{\ast}$ to $\varphi$ is defined by \[ \varphi_{\ast}(u) = \sup_{v \geq 0}\{uv- \varphi(v) \}, \ \ \ u\ge 0. \] The complementary function $\varphi_*$ is also an Orlicz function and $\varphi_{**} = \varphi$.
An Orlicz function $\varphi$ is an {\it $N$-function at zero} if $\lim_{u \rightarrow 0^+} \frac{\varphi(u)}{u} = 0$ and {\it at infinity} if $\lim_{u \rightarrow \infty} \frac{\varphi(u)}{u} = \infty$. If $\varphi$ is an $N$-function at both zero and infinity then we say that $\varphi$ is an {\it $N$-function}. A function $\varphi$ is an $N$-function if and only if $\varphi_*$ is an $N$-function.
An Orlicz function $\varphi$ satisfies the {\it $\Delta_2$ condition} if there exists $K>2$ such that $\varphi(2u) \leq K \varphi(u)$ for all $u\geq 0$, the $\Delta_2^\infty$ condition if there exist $K>2$ and $u_0\ge 0$ such that $\varphi(u_0) < \infty$ and for all $u\geq u_0$, $\varphi(2u) \leq K \varphi(u)$, and the $\Delta_2^0$ condition if there exist $K>2$ and $u_0$ such that $\infty > \varphi(u_0) > 0$ and for all $0\le u \leq u_0$, $\varphi(2u) \leq K \varphi(u)$. When we use the term {\it the appropriate $\Delta_2$ condition}, it means $\Delta_2$ in the case of a non-atomic measure $\mu$ with $\mu(\Omega) = \infty$, $\Delta_2^\infty$ for a non-atomic measure $\mu$ with $\mu(\Omega) < \infty$, and $\Delta_2^0$ for $\Omega = \mathbb{N}$ with the counting measure i.e. $\mu\{n\} =1$ for every $n\in \mathbb{N}$.
The Orlicz space $L_\varphi(\Omega)$ is a collection of all $f\in L^0(\Omega)$ such that for some $\lambda > 0$, \[
I_\varphi(\lambda f):= \int_\Omega\varphi(\lambda |f(t)|)\,d\mu(t) = \int_\Omega\varphi(\lambda |f|)\,d\mu < \infty. \] The Orlicz spaces are equipped with either the Luxemburg norm \[
\|f\|_\varphi= \inf\left\{\epsilon > 0: I_\varphi\left(\frac{f}{\epsilon}\right) \le 1\right\}, \] or the Orlicz (or Amemiya) norm \[
\|f\|_\varphi^0= \sup\left\{\int_\Omega fg : I_{\varphi_*} (g)\le 1\right\} = \inf_{k>0} \frac{1}{k}(1 + I_\varphi(kf)). \]
It is well-known that $\|f\|_\varphi \le \|f\|_\varphi^0 \le 2\|f\|_\varphi$ for $f\in L_\varphi(\Omega)$. By $L_\varphi(\Omega)$ we denote an Orlicz space equipped with the Luxemburg norm and by $L_\varphi^0(\Omega)$ with the Orlicz norm. The Orlicz spaces with either norms are rearrangement invariant spaces and have the Fatou property.
If $\varphi$ is finite, i.e. $\varphi(u)< \infty$ for all $u>0$, then $(L_\varphi(\Omega))_a \ne \{0\}$ and it contains all simple functions. Therefore \[ (L_\varphi(\Omega))_a = (L_\varphi(\Omega))_b = \{x\in L^0: I_\varphi(\lambda x) < \infty \ \ \text{for all} \ \ \lambda > 0\}. \] It is also well-known that $L_\varphi(\Omega) = (L_\varphi(\Omega))_a$ if and only if $\varphi$ satisfies the appropriate $\Delta_2$ condition. The K\"othe duals of $L_\varphi(\Omega)$ and $L_\varphi^0(\Omega)$ are described by Orlicz spaces induced by $\varphi_*$ \cite{Chen, BS}. In fact, \[
(L_\varphi(\Omega))' = L_{\varphi_*}^0(\Omega)\ \ \text{ and} \ \ \ (L_\varphi^0(\Omega))' = L_{\varphi_*}(\Omega). \]
In the case of non-atomic measure (resp., counting measure), we use the symbols $L_\varphi$ and $L_\varphi^0$, (resp., $\ell_\varphi$ and $\ell_{\varphi}^0$) for Orlicz spaces equipped with the Luxemburg and the Orlicz norm, respectively. For complete information on Orlicz spaces we refer the reader to the monographs \cite{BS, Chen, KR, LT1, LT2, Lux}.
\section{The Radon-Nikod\'ym property}
We start with a general result on the Radon-Nikod\'ym property in Banach function spaces. \begin{Theorem}\label{th:RNKothe} Let $X$ be a Banach function space over a complete $\sigma$-finite measure space $(\Omega, \Sigma, \mu)$. \begin{itemize} \item[(i)] If $X$ has the RNP then $X$ is order continuous. \item[(ii)] Assume that $X$ has the Fatou property and $(X')_a = (X')_b$. Then if $X$ is order continuous then $X$ has the RNP. \end{itemize} \end{Theorem}
\begin{proof} (i) If $X$ is not order continuous then it contains an order isomorphic copy of $\ell_\infty$ \cite[Theorem 14.4, p.220]{AB}. Since $\ell_\infty$ does not have the RNP, $X$ does not have this property either.
(ii) Suppose that $(X')_a = (X')_b$ and that $X$ is order continuous with the Fatou property. It is well-known that every separable dual space possesses the RNP \cite{DU}. Since $((X')_a)' = ((X')_b)'$, $((X')_a)'=(X')' = X''$ and $((X')_a)^*\simeq ((X')_a)'$ by Corollary 1.4.2 in \cite{BS}. It follows by the Fatou property that $X'' = X$. Therefore \[ ((X')_a)' \simeq ((X')_a)^* \simeq X ''=X. \] Hence $X$ is the dual space of $(X')_a$. If the measure space $(\Omega, \Sigma,\mu)$ is separable, then the order continuous space $X$ is also separable by Theorem 2.5.5 in \cite{BS}. Thus in this case, $X$ has the RNP.
Now, suppose that $(\Omega, \Sigma,\mu)$ is not separable and we show that $X$ still has the RNP. We will use the fact that a Banach space $X$ satisfies the RNP if and only if every separable closed subspace $Y\subset X$ has the RNP \cite[Theorem 2, p. 81]{DU}.
Since $X$ is order continuous $X=X_a=X_b$. Let $Y\subset X$ be a closed separable subspace of $X$. Then there exists a dense and countable set $\mathcal{Y} \subset Y$. For every $y\in \mathcal{Y} \subset X=X_b$, there exists a sequence of simple functions $(y_n) \subset X$ with supports of finite measure and such that $\|y-y_n\|_X \to 0$. Each $y_n$ can be expressed as $y_n = \sum_{i=1}^{m_n} a_i^{(n)} \chi_{A_i^{(n)}}$, where $a_i^{(n)}\in \mathbb{R}$, $A_i^{(n)} \in \Sigma$ with $\mu(A_i^{(n)}) < \infty$, so $y \in \overline{span} \{\chi_{A_i^{(n)}}, i=1,\dots,m_n, \ n\in \mathbb{N}\}$. Letting $\mathcal{A}_y = \{A_i^{(n)}: i=1,\dots,m_n, n\in\mathbb{N}\}$ and $\mathcal{A} = \cup_{ y\in \mathcal{Y}}\mathcal{A}_y$, the family $\mathcal{A}$ is countable.
For our convenience, let $\mathcal{A} = \{E_i: i\in \mathbb{N}\}$. For each $i\in \mathbb{N}$ we have $\mu(E_i) < \infty$. Then we have \[ Y = \overline{\mathcal{Y}}\subset \overline{span} \{\chi_{E_i}, \ E_i \in \mathcal{A}\} \subset X. \]
Let $\widetilde\Omega = \cup_{i=1}^\infty E_i$, $\sigma(\mathcal{A})$ be the smallest $\sigma$-algebra of $\Omega$ containing $\mathcal{A}$, $\widetilde{\Sigma} = \{\widetilde{\Omega} \cap E: E \in \sigma(\mathcal{A})\}$ and $\widetilde\mu = \mu|_{\widetilde\Sigma}$ the measure $\mu$ restricted to $\widetilde{\Sigma}$. In fact, it is easy to show that $\widetilde{\Sigma} = \sigma(\mathcal{A})$. Hence $\widetilde{\Sigma}$ is generated by a countable set, namely $\mathcal{A}$, so the measure space $(\widetilde\Omega, \widetilde\Sigma,\widetilde\mu)$ is separable \cite[Theorem B, p. 168]{Hal}). Now define the set \[ \widetilde X = \{x\chi_{\widetilde\Omega}: x\in X, \ x \ \text{is} \ \widetilde{\Sigma} - \text{measurable}\}. \] It is straightforward to check that $\widetilde X$ is a closed subspace of $X$ such that it is an order continuous Banach function space on $(\widetilde\Omega,\widetilde\Sigma, \widetilde\mu)$ with the Fatou property. So $\widetilde X$ is separable. The K\"othe dual of $\widetilde{X}$ is \[ \widetilde X' := (\widetilde X)' = \{y\chi_{\widetilde\Omega}: y\in X', \ y \ \text{is} \ \widetilde{\Sigma} - \text{measurable}\}. \] Clearly $\widetilde X' \subset L^0(\widetilde\Omega, \widetilde\Sigma, \widetilde\mu)$. From the assumption we have $(\widetilde X')_a = (\widetilde X')_b$. Hence $\widetilde X = \widetilde{X}'' \simeq ((\widetilde{X}')_a)^*$ by Corollary 1.4.2 in \cite{BS} again. Therefore, $\widetilde X$ is a separable dual space such that $Y \subset \overline{span} \{\chi_{E_i}, \ E_i\in \mathcal{A}\}\subset \widetilde X$, which implies that $\widetilde X$ and hence $Y$ has the RNP. Since the choice of $Y$ was arbitrary, $X$ has the RNP. \end{proof}
\begin{Remark} $(1)$ The Fatou property in (ii) is a necessary assumption. For example, take $X=c_0$. This space does not have the RNP \cite[p. 61]{DU} and clearly does not satisfy the Fatou property. However, since $(c_0)'= \ell_1$ and $\ell_1$ is order continuous we have $((c_0)')_a = (\ell_1)_a = (\ell_1)_b = ((c_0)')_b$, which is the second assumption of the theorem.
$(2)$ The assumption $(X')_a = (X')_b$ in (ii) is also necessary. Consider $X=L_1[0,1]$ that is clearly order continuous. Moreover $(X')_a = (L_\infty[0,1])_a = \{0\}$ and $(X')_b = (L_\infty[0,1])_b = L_\infty[0,1]$. Hence $(X')_a \ne (X')_b$ and it is well-known that $L_1[0,1]$ does not have the RNP \cite[p. 60]{DU}.
\end{Remark}
\begin{Proposition}\label{pro1} Let $\mu$ be a non-atomic measure and $\varphi$ be a finite Orlicz function. If $\varphi$ is not an $N$-function at infinity, then $L_\varphi$ contains a subspace isomorphic to $L_1[0,1]$. \end{Proposition} \begin{proof}
Suppose $\varphi$ is not an $N$-function at infinity. We will show that given $A\in\Sigma$ with $\mu(A) < \infty$, the space $L_\varphi(A) = \{x\chi_A: x\in L_\varphi\}$ is equal to $ L_1(A)$ with equivalent norms. By the assumption $\lim_{u\to\infty} \varphi(u)/u =K< \infty$, and by the fact that
the function $\varphi(u)/u$ is increasing, there exist $M> 0$ and $u_0 >0$ such that
\begin{equation}\label{eq:31}
\varphi(u) \le Ku \ \ \text{for} \ \ u\ge 0, \ \ \ \text{and} \ \ \ \varphi(u)\ge M u \ \ \ \text{for} \ \ u \geq u_0.
\end{equation}
Let $f\in L_\varphi(A)$ with $\|f\|_\varphi = 1$. Then $\supp f\subset A$ and $I_\varphi(f) \le 1$. Set $A_1 = \{t\in A: |f(t)| \le u_0\}$. Thus in view of the second part of inequality (\ref{eq:31}) we get
\[
\|f\|_1 = \int_{A_1} |f| \, d\mu + \int_{A\setminus A_1} |f|\, d\mu \le u_0 \mu(A) + \frac1M I_\varphi(f) \le C,
\]
where $C = u_0 \mu(A) + \frac1M.$
Therefore for any $f$ from $L_\varphi(A)$, $\|f\|_1 \le C \|f\|_\varphi$.
On the other hand, by the second part of inequality (\ref{eq:31}), for any $E\subset A$ we have \[ \int_A\varphi\left(\frac{\chi_E}{K\mu(E)}\right)\, d\mu = \int_E\varphi\left(\frac{1}{K\mu(E)}\right) \, d\mu \le 1. \]
It follows that $\|\chi_E\|_\varphi \le K\mu(E) =K \|\chi_E\|_1$. Hence for any simple function $x = \sum_{i=1}^n a_i \chi_{E_i}\in L_\varphi(A)$ with $E_i \cap E_j = \emptyset$ for $i\ne j$, \[
\|x\|_\varphi \le \sum_{i=1}^n |a_i| \|\chi_{E_i}\|_\varphi \le K \sum_{i=1}^n |a_i| \mu(E_i) = K \|x\|_1. \]
Now by the Fatou property of $L_1(A)$ and $L_\varphi(A)$, $\|x\|_\varphi \le K \|x\|_1$ for every $x \in L_1(A)$.
Hence $L_\varphi(A) = L_1(A)$ with equivalent norms, and the proof is completed since $L_1(A)$ contains a subspace isomorphic to $L_1[0,1]$ (see \cite[p. 127, Theorem 9 (1)]{Lac}). \end{proof}
Recall that an Orlicz function is said to be finite if its range does not contain the infinity. \begin{Lemma}\label{lem:finite} \rm{(a)} A finite Orlicz function $\varphi$ is an $N$-function at infinity if and only if $\varphi_*$ is finite.
\rm{(b)} Let $\mu$ be a non-atomic measure. If $\varphi$ is a finite Orlicz function then $\|\chi_A\|_\varphi = 1/\varphi^{-1}(1/t)$, where $t>0$, $\mu(A) = t$. Consequently \[ \lim_{t\to 0+} \phi_{L_\varphi}(t) = \lim_{t\to 0+} 1/\varphi^{-1}(1/t) = 0. \] \end{Lemma}
\begin{proof} (a) Suppose $\varphi$ is not an $N$-function at infinity. Then there exists $K>0$ such that for every $u >0$, $\varphi(u) \leq Ku$. Hence \[ \varphi_*(v) = \sup_{u >0}\{uv - \varphi(u)\} \geq \sup_{u >0}\{(v-K)u\}. \] Therefore if $v > K$ then $\varphi_*(v) =\sup_{u>0}\{(v-K)u\} = \infty$.
Conversely, suppose there exists $K > 0$ such that for every $v > K$, $\varphi_*(v) = \infty$. Then \[ \varphi(u) = \sup\{uv - \varphi_*(v) : v \in (0,K)\}. \] By $\frac{\varphi(u)}{u} = \sup\{v - \frac{\varphi_*(v)}{u}: v \in (0, K)\}$, we have $\lim_{u \rightarrow \infty} \frac{\varphi(u)}{u} \leq K < \infty$, which shows that $\varphi$ is not an $N$-function at infinity.
(b) Let $a_\varphi=\sup\{t: \varphi(t)=0\}$, and let $A\in \Sigma$, $\mu(A) = t$, $t > 0$. Then $I_\varphi(\chi_A/\epsilon) = 0$ if $\epsilon \ge 1/a_\varphi$, and $I_\varphi(\chi_A/\epsilon) = \varphi(1/\epsilon) t$ if $ \epsilon < 1/a_\varphi$. By the latter condition if $I_\varphi(\chi_A/\epsilon) = 1$, we get that $\|\chi_A\|_\varphi = \epsilon = 1/\varphi^{-1}(1/t)$. Clearly for $t\to 0+$ we get that $1/\varphi^{-1}(1/t) \to 0$. \end{proof}
The next result provides a criterion of the Radon-Nikod\'ym property of Orlicz spaces over non-atomic measure spaces. We do not need the assumption of separability of the measure space \cite[Theorem 3.32]{Chen}.
\begin{Theorem}\label{th:OrRN-funct} Let $\mu$ be a complete $\sigma$-finite, non-atomic measure on $\Sigma$ and $\varphi$ be a finite Orlicz function. Then the Orlicz spaces $L_\varphi$ {\rm (}and $L_\varphi^0${\rm )} over $(\Omega, \Sigma, \mu)$ have the Radon-Nikod\'ym property if and only if $\varphi$ is an $N$-function at infinity and satisfies the appropriate $\Delta_2$ condition. \end{Theorem} \begin{proof} Since the Luxemburg and Orlicz norms are equivalent we consider only $L_\varphi$ equipped with the Luxemburg norm. By the assumption that $\varphi$ is an $N$-function at infinity and Lemma \ref{lem:finite}(a) we get that $\varphi_*$ is finite on $(0,\infty)$. Applying now Lemma \ref{lem:finite}(b) to the function $\varphi_*$ we get that $\phi_{L_{\varphi_*}}(t) \to 0$ if $t\to 0+$. Hence $\lim_{t\to 0+} \phi_{L^0_{\varphi_*}}(t) = 0$. Applying now Theorem 2.5.5 in \cite{BS} we get $(L_{\varphi_*}^0)_a = (L_{\varphi_*}^0)_b$ and in view of $(L_\varphi)' = L_{\varphi_*}^0$ \cite[Corollary 8.15, p. 275]{BS} \cite{KR}, we have $((L_\varphi)')_a = ((L_\varphi)')_b$.
It is well-known that $L_\varphi$ has the Fatou property and that $L_{\varphi}$ is order continuous if and only if $\varphi$ satisfies the appropriate $\Delta_2$ condition. Therefore, by Theorem \ref{th:RNKothe}(ii) the Orlicz space $L_\varphi$ has the RNP.
For the converse, assume that $L_\varphi$ has the RNP. Since $L_1[0,1]$ does not have the RNP, $\varphi$ needs to be an $N$-function at infinity by Proposition~\ref{pro1}. If $\varphi$ does not satisfy the appropriate $\Delta_2$ condition, then $L_\varphi$ is not order continuous, and by Theorem \ref{th:RNKothe}(i) it does not have the RNP. \end{proof}
By Theorem 2.5.4 in \cite{BS}, $X_a = X_b$ holds for every rearrangement invariant sequence space $X$. Consequently we obtain a characterization of the RNP in Orlicz sequence spaces $\ell_\varphi$ as a consequence of Theorem \ref{th:RNKothe}. This result is well-known for $\varphi$ being an $N$-function \cite[Theorem 3.32]{Chen}.
\begin{Theorem} \label{th:RNP-ORseq} Let $\varphi$ be a finite Orlicz function. An Orlicz sequence space $\ell_{\varphi}$ has the Radon-Nikod\'ym property if and only if $\varphi$ satisfies the $\Delta_2^0$ condition. \end{Theorem}
\begin{proof}
Since any Orlicz sequence space is an r.i. space with the Fatou property, we always have $((\ell_{\varphi})')_a = (\ell_{\varphi_*}^0)_a = (\ell_{\varphi_*}^0)_b = ((\ell_{\varphi})')_b$. Moreover it is well-known that $\ell_{\varphi}$ is order continuous if and only if $\varphi$ satisfies the $\Delta_2^0$ condition \cite[Proposition 4.a.4]{LT1}. Hence, $\ell_{\varphi}$ has the RNP by Theorem \ref{th:RNKothe}.
Conversely, suppose that $\ell_{\varphi}$ has the RNP. Then $\ell_{\varphi}$ is order continuous by Theorem \ref{th:RNKothe}. This implies that $\varphi$ satisfies the $\Delta_2^0$ condition.
\end{proof}
\section{Locally octahedral norm, uniformly non-$\ell_1^2$ points, diameter two properties and the Daugavet property}
In this section, we first examine the relationship between locally octahedral norms and the Daugavet property. \begin{Definition}\cite{G, BLR2, HLP}
A Banach space $X$ is locally octahedral if for every $x \in X$ and $\epsilon >0$, there exists $y \in S_X$ such that $\|\lambda x + y\| \geq (1 - \epsilon) (|\lambda| \|x\| + \|y\|)$ for all $\lambda \in \mathbb{R}$. \end{Definition}
A point $x\in S_X$ is called a \emph{uniformly non-$\ell_1^2$} point if there exists $\delta>0$ such that $\min\{\|x + y\|, \|x - y\|\} \leq 2-\delta$ for all $y\in S_X$. Motivated by this, we introduce the following.
\begin{Definition}
A point $x\in S_X$ is called a \emph{uniformly $\ell_1^2$} point if, given $\delta>0$, there is $y\in S_X$ such that $\min\{\|x + y\|, \|x - y\|\} > 2-\delta$. \end{Definition}
By Proposition 2.1 in \cite{HLP} we get immediately the following corollary.
\begin{Corollary}\label{cor:octah-non} Every point $x \in S_X$ is a uniformly $\ell_1^2$ point if and only if the Banach space $X$ is locally octahedral. \end{Corollary}
\begin{Lemma}\cite{HLP}\label{lem:aux}
If $x, y \in S_X$ satisfy $\|x \pm y\| > 2 - \delta$ and $\alpha, \beta \in \mathbb{R}$, then
\[
(1-\delta)(|\alpha| + |\beta|) < \|\alpha x \pm \beta y\| \leq |\alpha| + |\beta|.
\] \end{Lemma}
\begin{proof} See the proof of the implication from (iii) to (ii) in Proposition 2.1 in \cite{HLP}.
\end{proof}
In the next theorem we give a local characterization of uniformly $\ell_1^2$ points $x\in S_X$ (resp. $x^*\in S_{X^*}$) and the diameter of the slice $S(x;\epsilon)$ (resp. the diameter of the weak$^{*}$-slice $S(x^*,\epsilon)$). The techniques used in the proof are somewhat similar to the proof of Theorem 3.1 in \cite{HLP}, but the key ideas are more subtle emphasizing the local nature of discussed properties.
It follows a corollary on relationships between global properties of local diameter two property in $X$ and of $X^*$ being locally octahedral, as well as between the weak$^*$-local diameter two property of $X^*$ and $X$ being locally octahedral.
\begin{Theorem}\label{th:unif} \rm(a) An element $x \in S_{X}$ is a uniformly $\ell_1^2$ point if and only if the diameter of a weak$^{*}$-slice $S(x;\epsilon)$ is two for every $\epsilon > 0$.
\rm(b) An element $x^* \in S_{X^*}$ is a uniformly $\ell_1^2$ point if and only if ${\rm{diam}}\,S(x^*;\epsilon)=2$ for every $\epsilon > 0$. \end{Theorem} \begin{proof} We will prove only (a) since (b) follows analogously. Suppose that for all $0<\epsilon < 1$, ${\rm{diam}} \, S(x; \epsilon) = 2$. Then there exist $x_1^*, x_2^*\in S(x;\epsilon)$ such that
\begin{equation}\label{eq:11}
x_1^*(x)> 1-\epsilon, \ \ \ x_2^* (x) > 1 - \epsilon, \ \ \|x_1^* - x_2^*\| > 2- \epsilon.
\end{equation}
Hence we can find $y \in S_X$ with $(x_1^* - x_2^*)(y) > 2 - \epsilon$. Thus
\[
2 \ge x_1^*(y) - x_2^*(y) > 2-\epsilon \ \ \ \text{and} \ \ \ x_1^*(y)\le1,\ -x_2^*(y) \le 1,
\]
and so $x_1^*(y) > 1-\epsilon$ and $-x_2^*(y) > 1-\epsilon$. Combining this with (\ref{eq:11}) we get that $x_1^*(x+y) > 2-2\epsilon$, $x_2^*(x-y) > 2 - 2\epsilon$, and so $\|x + y\| > 2 - 2\epsilon$ and $\|x - y\| > 2 -2\epsilon$. We showed that for every $0< \epsilon< 1$ there exists $y\in S_X$ such that
\[
\min\{\|x + y\|, \|x - y\|\} > 2 -2\epsilon,
\]
which means that $x$ is a uniformly $\ell_1^2$ point.
Conversely, suppose that $x \in S_X$ is a uniformly $\ell_1^2$ point. Then for any $\epsilon>0$, there exists $y \in S_X$ such that $\|x \pm y\| > 2 - \epsilon$. Define bounded linear functionals $x_1^*, x_2^*$ on the subspace ${\rm span}\{x,y\}$ such that
\[
x_1^*(x) = 1,\ \ x_1^*(y) = 0,\ \ x_2^*(x) = 0 \ \ \text{and} \ \ x_2^*(y) = 1.
\]
\noindent Note that $\|x_1^*\| \geq 1$ and $\|x_2^*\| \geq 1$. By Lemma \ref{lem:aux}, for $\alpha, \beta \in \mathbb{R}$ we have
\[
|(x_1^* \pm x_2^*)(\alpha x + \beta y)| = |\alpha \pm \beta| \leq |\alpha| + |\beta| \leq (1-\epsilon)^{-1}\|\alpha x + \beta y\|,
\]
\noindent so $\|x_1^* \pm x_2^*\| \leq (1-\epsilon)^{-1}$.
Now, let $\widetilde{x_1}^* = \frac{x_1^* + x_2^*}{\|x_1^* + x_2^*\|}$ and $\widetilde{x_2}^* = \frac{x_1^* - x_2^*}{\|x_1^* - x_2^*\|}$. Then
\[
\|\widetilde{x_1}^* - (x_1^* + x_2^*)\| = |\|x_1^* + x_2^*\| - 1| \leq \left|\frac{1}{1-\epsilon} - 1 \right| = \frac{\epsilon}{1 - \epsilon}.
\]
Similarly,
\[
\|\widetilde{x_2}^* - (x_1^* - x_2^*)\| \leq \frac{\epsilon}{1 - \epsilon}.
\]
Since $(x_1^* \pm x_2^*)(x)=1$, we have $\widetilde{x_1}^*(x) = \frac{1}{\|x_1^* + x_2^*\|} \geq 1 - \epsilon$ and $\widetilde{x_2}^*(x) = \frac{1}{\|x_1^* - x_2^*\|} \geq 1 - \epsilon$. Hence $\widetilde{x_1}^*, \widetilde{x_2}^* \in S(x,\epsilon)$. Furthermore,
\begin{eqnarray*}
\|\widetilde{x_1}^* - \widetilde{x_2}^*\| &=& \|\widetilde{x_1}^* + (x_1^* + x_2^*) - (x_1^* + x_2^*) + (x_1^* - x_2^*) - (x_1^* - x_2^*) - \widetilde{x_2}^*\| \\
&\geq& 2 \|x_2^*\| -\|\widetilde{x_1}^* - (x_1^* + x_2^*)\| -\|\widetilde{x_2}^* - (x_1^* - x_2^*)\| \geq 2 - \frac{2\epsilon}{1-\epsilon}.
\end{eqnarray*} \noindent Since $\epsilon > 0$ is arbitrary, ${\rm{diam}}\, S(x, \epsilon) = 2$. Finally by the Hahn-Banach theorem, we can extend the bounded linear functionals $x_1^*$ and $x_2^*$ from ${\rm span}\{x,y\}$ to $X$ and the proof is completed. \end{proof}
Combining Corollary \ref{cor:octah-non} and Theorem \ref{th:unif} we obtain the following result proved earlier in \cite{HLP}.
\begin{Corollary} \cite[Theorem 3.2, 3.4]{HLP}\label{HLP} Let $X$ be a Banach space. Then the following hold.
\begin{enumerate}
\item[$(1)$] $X$ is locally octahedral if and only if $X^*$ satisfies the weak$^{*}$ local diameter two property.
\item[$(2)$] $X^*$ is locally octahedral if and only if $X$ satisfies the local diameter two property.
\end{enumerate}
\end{Corollary}
Recall the equivalent geometric interpretation of the Daugavet property.
\begin{Lemma}\cite[Lemma 2.2]{KSSW}
\label{lem:Daug}
The following are equivalent.
\begin{enumerate}[{\rm(i)}]
\item A Banach space $(X,\|\cdot\|)$ has the Daugavet property,
\item\label{Daugii} For every slice $S = S(x^*,\epsilon)$ where $x^*\in S_{X^*}$, every $x \in S_X$ and every $\epsilon>0$, there exists $y\in S_{X}\cap S$ such that $\|x+y\|>2-\epsilon$,
\item\label{Daugiii} For every weak$^{*}$-slice $S^* = S(x,\epsilon)$ where $x\in S_{X}$, every $x^* \in S_{X^*}$ and every $\epsilon>0$, there exists $y^*\in S_{X^*}\cap S^*$ such that $\|x^*+y^*\|>2-\epsilon$,
\end{enumerate} \end{Lemma}
The next result is known in a stronger form \cite[Theorem 4.4]{ALN} \cite[Corollary 2.5]{BLR2} \cite{HLP}, namely, if $X$ has the Daugavet property then it has the SD2P as well as its dual $X^*$ has the weak$^*$-SD2P.
\begin{Proposition}\label{prop:slice} If a Banach space $X$ has the Daugavet property, then $X$ has the local diameter two property and $X^*$ has the weak$^*$-local diameter two property. \end{Proposition}
\begin{proof} Let $x^*\in S_{X^*}$ and $S(x^*; \epsilon)$ be a slice of $B_X$. Then there is $x\in S_X$ such that $-x^*(x) > 1- \epsilon$. By (iii) of Lemma \ref{lem:Daug} we find $y\in S_X$ with $x^*(y) > 1-\epsilon$ and $\|x + y\| > 2 - 2\epsilon$. Clearly $-x, y \in S(x^*;\epsilon)$, and so ${\rm diam} \, S(x;\epsilon) = 2$.
Now let $x\in S_X$ and $S(x; \epsilon)$ be a weak$^*$-slice of $B_{X^*}$. There exists $y^*\in S_{X^*}$ such that $-y^*(x) > 1 - \epsilon$. By (ii) of Lemma \ref{lem:Daug} there is $x^* \in S_{X^*}$ with $x^*(x) > 1 - \epsilon$ and $\|x^* + y^*\| > 2 - 2\epsilon$. Since both $x^*, -y^* \in S(x;\epsilon)$, $\epsilon > 0$ is arbitrary, we have that ${\rm diam} \, S(x,\epsilon) = 2$.
\end{proof}
The next result is an instant corollary of Theorem \ref{th:unif} and Proposition \ref{prop:slice}.
\begin{Corollary}\cite[Proposition 4.4]{KK}
\label{prop}
If $(X,\|\cdot\|)$ has the Daugavet property, then all elements in $S_X$ and $S_{X^*}$ are uniformly $\ell_1^2$ points. \end{Corollary}
Next we shall consider Orlicz spaces $L_{\varphi}, \,\ell_\varphi$ and $L_{\varphi}^0, \, \ell_\varphi^0$. Let us define first the following numbers related to Orlicz function $\varphi:\mathbb{R}_+ \to [0,\infty]$. Recall the Orlicz function $\varphi$ is called a {\it linear function} if $\varphi(u)= ku$ on $\mathbb{R}_+$ for some $k>0$. Set
\[
d_\varphi =\sup\{u: \varphi(u)\ \ \text{is linear}\}, \ \ c_\varphi = \sup\{u: \varphi(u) \le 1\},\ \ \
b_\varphi = \sup\{u: \varphi(u) < \infty\}.
\]
\begin{Lemma} \cite[Lemma 4.1]{KK} \label{lem:1} Let $\varphi$ be an Orlicz function For every closed and bounded inteval $I \subset (d_{\varphi}, b_{\varphi})$ there is a constant $\sigma \in (0,1)$ such that $2 \varphi(u/2)/\varphi(u) \leq \sigma$ for $u \in I$. Moreover, if $\varphi(b_\varphi) < \infty$ then the same statement holds true for closed intervals $I \subset (d_\varphi, b_\varphi]$. \end{Lemma}
\begin{Theorem}\label{th:KamKub} \begin{enumerate}[{\rm(1)}]
\item[\rm(i)] Let $\mu$ be non-atomic. Let $\varphi$ be an Orlicz function such that $\varphi(b_\varphi)\mu(\Omega) >1$ and $d_\varphi < b_\varphi$. Then there exists $a> 0$ and $A \in \Sigma$ such that $x = a \chi_A, \|x\|_{\varphi} =1$ and $x$ is uniformly non-$\ell_1^2$ point in $L_{\varphi}$. If $b_\varphi = \infty$ then $x\in (L_\varphi)_a$.
\item[\rm(ii)] Let $\mu$ be the counting measure on $\mathbb{N}$ and $\varphi$ be an Orlicz function such that $d_\varphi < c_\varphi$ and $\varphi(c_\varphi) = 1$. Then there exist $a > 0$ and $A \subset \mathbb{N}$ such that $x = a \chi_A, \|x\|_{\varphi} =1$ and $x$ is uniformly non-$\ell_1^2$ point in $\ell_{\varphi}$. If $b_\varphi = \infty$ then $x\in (\ell_\varphi)_a$. \end{enumerate} \end{Theorem}
\begin{proof}
(i): By the assumptions on $\varphi$ and non-atomicity of $\mu$, there exist $A\in \Sigma$ and $a\in (d_\varphi,b_\varphi)$ such that $\varphi(a) \mu(A) = 1$.
Letting $x=a \chi_A$, we get $I_{\varphi}(x) =1$, and $\|x\|_{\varphi} = 1$. Clearly $x\in (L_\varphi)_a$ if $b_\varphi = \infty$.
Let $y \in S_{L_\varphi}$ be arbitrary. Hence for a.e. $t\in \Omega$, $|y(t)| < b_\varphi$ if $b_\varphi=\infty$ or $|y(t)| \le b_\varphi$ if $b_\varphi<\infty$. Then, for any $\lambda >1$, $I_{\varphi}({y}/{\lambda}) \leq 1$. We claim that \begin{equation}\label{cond1}
\text{there exist }\, d\in (a,b_\varphi) \,\, \text{and} \,\, B=\{t \in \Omega : |y(t)| \leq d \chi_A (t)\} \,\, \text{such that} \,\, \mu(A \cap B) >0. \end{equation}
Indeed, let first $b_\varphi = \infty$. Define $B_k = \{t \in \Omega : |y(t)| \leq k \chi_A (t)\}$ for $k \in \mathbb{N}$. The sequence of sets $\{B_k\}$ is increasing, and so $0 < \mu(A) = \mu (A \cap (\cup_{k=1}^{\infty} B_k) = \lim_{k \rightarrow \infty} \mu (A \cap B_k)$, and this implies that there exists $m \in \mathbb{N}$ such that $2a <m$, $\mu (A \cap B_{m}) >0$. Letting $B = B_{m}$, $d=m$, we get (\ref{cond1}).
Let now $b_\varphi < \infty$. Define $C_k = \{t \in \Omega : |y(t)| \leq (b_\varphi - 1/k) \chi_A (t)\}$ for $k \in \mathbb{N}$. Like before, $\{C_k\}$ is increasing and $\lim_{k \rightarrow \infty} \mu(A \cap C_k)>0$. So there exists $m$ such that $b_\varphi - 1/m > a$. Let now $d = b_\varphi - 1/m$ and $B = C_m$, and so (\ref{cond1}) is satisfied.
Set \begin{equation}\label{eq:111}
\gamma = I_{\varphi} (a \chi_{A \setminus B}).
\end{equation}
Clearly $\gamma\in [0,1)$. For any $\delta>0$, there exists $1>\epsilon>0$ such that $I_{\varphi}((1+ \epsilon)x) = \varphi((1+\epsilon)a)\mu(A) \leq 1 + \delta$. We can choose $\epsilon$ so small that we also have $(1+\epsilon)a < d$. Let $z = (1+\epsilon) x = (1 + \epsilon)a \chi_A$. Thus \begin{equation}\label{eq:11} I_\varphi(z) \le 1+\delta. \end{equation} Define \[ D = \{t \in A \cap B : x(t)y(t) \geq 0 \},\,\, E = (A \cap B) \setminus D. \]
For $t \in A \cap B$, $\max \{ |z(t)|, |y(t)| \} = \max \{ |(1+\epsilon)a|, |y(t)| \} \in [a,d]$.
Since $D \subset A \cap B$, we have $|z(t) - y(t)|/2 \le \max\{|z(t)|, |y(t)|\}$ for $t\in D$. Moreover by Lemma \ref{lem:1}, there exists $\sigma \in (0,1)$ such that $2\varphi(u/2)/\varphi(u) \le \sigma$ for $u\in [a,d] \subset (d_{\varphi}, b_{\varphi})$. Therefore \begin{eqnarray*}
I_{\varphi}\left(\frac{z-y}{2} \chi_D \right) \leq I_{\varphi}\left(\frac{\max\{|z|, |y| \}}{2} \chi_D \right) &\leq& \frac{\sigma}{2} I_{\varphi}(\max\{|z|, |y| \}\chi_D)\\ &\leq& \frac{\sigma}{2}(I_{\varphi}(z \chi_D)+ I_{\varphi}(y \chi_D)). \end{eqnarray*}
\noindent Analogously we can also show that \[ I_{\varphi} \left(\frac{z+y}{2} \chi_E \right) \leq \frac{\sigma}{2}(I_{\varphi}(z \chi_E)+ I_{\varphi}(y \chi_E)). \]
Then, by the convexity of $\varphi$ and $A \cap B = D\cup E$, \begin{eqnarray*} &\,&I_{\varphi}\left(\frac{z-y}{2} \chi_{A \cap B} \right) + I_{\varphi} \left(\frac{z+y}{2} \chi_{A \cap B} \right)\\ &=& I_{\varphi} \left(\frac{z-y}{2} \chi_D \right) + I_{\varphi} \left(\frac{z+y}{2} \chi_D \right) + I_{\varphi} \left(\frac{z-y}{2} \chi_E \right) + I_{\varphi} \left(\frac{z+y}{2} \chi_E \right)\\ &\leq& \frac{\sigma}{2}(I_{\varphi}(z\chi_D) + I_{\varphi}(y\chi_D)) + \frac{1}{2}(I_{\varphi}(z\chi_D) + I_{\varphi}(y\chi_D))+ \frac{1}{2}(I_{\varphi}(z\chi_E) + I_{\varphi}(y\chi_E)) \\ &+& \frac{\sigma}{2}(I_{\varphi}(z\chi_E) + I_{\varphi}(y\chi_E) = \frac{1+ \sigma}{2}(I_{\varphi}(z \chi_{A \cap B}) + I_{\varphi}(y \chi_{A \cap B})). \end{eqnarray*}
Now, choose $\delta \in (0, \frac{(1-\sigma)(1- \gamma)}{2})$. By the assumption $I_{\varphi}(y) \leq 1$ and by (\ref{eq:111}), (\ref{eq:11}) we have \[ 2+ \delta \geq I_{\varphi}(y) + 1 + \delta \geq I_{\varphi}(y) + I_{\varphi}(z), \] and so \begin{eqnarray*} 2+ \delta - I_{\varphi}\left(\frac{z+y}{2} \right) - I_{\varphi}\left(\frac{z-y}{2}\right) &\geq& I_{\varphi}(y)+ I_{\varphi}(z) - I_{\varphi}\left(\frac{z+y}{2}\right) - I_{\varphi}\left(\frac{z-y}{2}\right)\\ &\geq& I_{\varphi}(y)+ I_{\varphi}(z) - \frac{1+ \sigma}{2}(I_{\varphi}(z \chi_{A \cap B}) + I_{\varphi}(y \chi_{A \cap B}))\\ &\geq& \frac{1- \sigma}{2}(I_{\varphi}(z \chi_{A \cap B}) + I_{\varphi}(y \chi_{A \cap B}))\\ &\geq& \frac{1- \sigma}{2}I_{\varphi}(a \chi_{A \cap B}) = \frac{(1- \sigma)(1-\gamma)}{2}, \end{eqnarray*}
\noindent which implies that \[ I_{\varphi}\left(\frac{z+y}{2} \right) + I_{\varphi}\left(\frac{z-y}{2} \right) \leq 2+ \delta - \frac{(1- \sigma)(1-\gamma)}{2} \le 2. \] It follows \[ \min \left \{I_{\varphi}\left(\frac{z+y}{2} \right), I_{\varphi}\left(\frac{z-y}{2} \right) \right\} \leq 1. \]
If $I_{\varphi}\left(\frac{z+y}{2}\right) \leq 1$, then $\left \| \frac{z+y}{2} \right \|_{\varphi} \leq 1$, and so $\left \| \frac{x+(y/(1+\epsilon))}{2} \right \|_{\varphi} \leq \frac{1}{1+\epsilon}$. Moreover,
\begin{equation*}
\left | \left\| \frac{x+y}{2}\right \|_{\varphi} - \left \|\frac{x+(y/(1+\epsilon))}{2} \right \|_{\varphi} \right | \leq \left \| \frac{x+y}{2} - \frac{x+(y/(1+\epsilon))}{2} \right \|_{\varphi} = \frac{\epsilon}{2(1+\epsilon)}. \end{equation*} Hence \[
\left \| \frac{x+y}{2} \right \|_{\varphi} \leq \left \|\frac{x+(y/(1+\epsilon))}{2} \right \|_{\varphi} + \frac{\epsilon}{2(1+\epsilon)} \leq \frac{1}{1+ \epsilon} + \frac{\epsilon}{2(1+ \epsilon)} = 1 - \frac{\epsilon}{2(1+\epsilon)}. \]
In a similar way, if $I_{\varphi} \left(\frac{z-y}{2} \right) \leq 1$, then $\left \| \frac{x-y}{2} \right \|_{\varphi} \leq 1 - \frac{\epsilon}{2(1+\epsilon)}$. Thus, we just showed that for any $y \in S_{L_\varphi}$, $\min \left\{ \left \| \frac{x+y}{2} \right \|_{\varphi}, \left \| \frac{x-y}{2} \right \|_{\varphi} \right \} \leq 1 - \frac{\epsilon}{2(1+\epsilon)}$, which means
\begin{equation*}
\min \{ \| x+y\|_{\varphi}, \| x-y\|_{\varphi}\} \leq 2 - \frac{\epsilon}{1+ \epsilon} < 2. \end{equation*}
Therefore, $x= a \chi_A$, $\|x\|_{\varphi} = 1$ is a uniformly non-$\ell_1^2$ point in $L_\varphi$.
(ii): If $x \in S_{\ell_{\varphi}}$, then $I_\varphi(x) = \sum_{i=1}^{\infty} \varphi(|x(i)|) \leq 1$. So for every $i \in \mathbb{N}$, $\varphi(|x(i)|) \leq 1$. Hence for any element of $S_{\ell_\varphi}$, we only consider $u \geq 0$ such that $\varphi(u)\leq 1$. Then by the assumptions $1= \frac{1}{\varphi(c_\varphi)}< \frac{1}{\varphi(d_\varphi)}$, there exist $a \in (d_{\varphi}, c_\varphi]$ and $A \subset \mathbb{N}$ such that $\varphi(a) = 1/\mu(A)$. Let $x = a\chi_A$. Then $I_\varphi(x) = \varphi(a)\mu(A) = 1$ and $\|x\|_\varphi = 1$. If $b_\varphi = \infty$ then $x \in (\ell_\varphi)_a$.
Now for $y \in S_{\ell_\varphi}$, we want to show that there exists $d \in (a, c_\varphi)$ and $B = \{i \in \mathbb{N} : |y(i)| \leq d\}$ such that $\mu(A \cap B) > 0$, which corresponds to (\ref{cond1}) in function case.
Since $y$ is in the unit ball of $\ell_\varphi$, for each $i\in \mathbb{N}$, $|y(i)| \le c_\varphi$.
Define $C_k = \{i \in \mathbb{N} : |y(i)| \leq (c_\varphi - 1/k) \chi_A (i)\}$ for $k \in \mathbb{N}$. The sequence $\{C_k\}$ is increasing and
\[
0 < \mu (A) = \mu (A \cap (\cup_{k=1}^{\infty} C_k) = \lim_{k \rightarrow \infty} \mu (A \cap C_k).
\]
So there exists $m$ such that $c_\varphi - 1/m > a$. Let now $d = c_\varphi - 1/m$ and $B = C_m$. Then
$d\in (a,c_\varphi)$, $|y(i)| \le d \chi_A(i)$ for $i\in A\cap B$ and $\mu(A\cap B) > 0$.
Further we proceed analogously as in the proof for function spaces starting from (\ref{eq:111}). We apply Lemma \ref{lem:1} for the interval $I=[a,d] \subset (d_\varphi, c_\varphi)\subset (d_\varphi, b_\varphi)$. \end{proof}
Concerning the Daugavet property we will consider only the case of non-atomic measure since it is not difficult to show that any rearrangement invariant sequence space never has the Daugavet property. In \cite{AKM2}, it was shown that an Orlicz space $L_\varphi$ generated by a finite Orlicz function $\varphi$ has the Daugavet property if and only if the space is isometrically isomorphic to $L_1$. Similar result
can be derived also from \cite{KK} where it was proved for Musielak-Orlicz spaces. Below the given proof for Orlicz spaces $L_\varphi$ is much simpler than those in \cite{AKM2, KK}. In fact it is a direct corollary of Theorem \ref{th:KamKub}(i).
\begin{Theorem}\label{thm:DaugOrlicz}
Let $\mu$ be a non-atomic measure.
If $\varphi$ is a finite Orlicz function then the only Orlicz space $L_\varphi$ having the Daugavet property correspond to a linear function $\varphi$, that is $L_\varphi = L_1$ isometrically.
\end{Theorem}
\begin{proof}
If $L_\varphi = L_1$ isometrically, clearly the Orlicz space has the Daugavet property.
Supposing $L_\varphi$ has the Daugavet property, by Corollary \ref{prop}, every point of the unit sphere of $L_\varphi$ is a uniformly $\ell_1^2$ point. Applying now Theorem \ref{th:KamKub}(i), $d_\varphi = b_\varphi$, where $b_\varphi = \infty$ by the assumption that $\varphi$ assumes finite values. Therefore $\varphi(u) = ku$, for some $k>0$ and all $u\ge 0$. Consequently, $L_\varphi = L_1$ and $\|\cdot\|_\varphi = k\|\cdot\|_1$. \end{proof}
\begin{Theorem}\label{thm:Orlicznormdiam} \rm(i) Let $\mu$ be a non-atomic measure. If $d_{\varphi_*} < b_{\varphi_*}$ and $\varphi_*(b_{\varphi_*}) \mu(\Omega) > 1$ then $L_\varphi^0$ does not have the local diameter two property.
\rm(ii) Let $\mu$ be the counting measure on $\mathbb{N}$. If $d_{\varphi_*} < c_{\varphi_*}$ and $\varphi_*(c_{\varphi_*}) =1$
then $\ell_\varphi^0$ does not have the local diameter two property. \end{Theorem}
\begin{proof} We will show only (i), since the sequence case is proved similarly. By the assumptions in view of Theorem \ref{th:KamKub}(i), the space $L_{\varphi_*}$ has a uniformly non-$\ell_1^2$ point. In view of Theorem \ref{th:unif} it is equivalent to that the dual space $(L_{\varphi_*})^*$ does not have the weak$^*$-star local diameter two property. It is well-known that the dual space to Orlicz space $L_\varphi$ is isometrically isomorphic to the direct sum $L_{\varphi_*}^0 \oplus_1 \mathcal{S}$, where $\mathcal{S}$ is a set of the singular functionals on $L_\varphi$ (\cite{Chen}).
Therefore the dual space $ (L_{\varphi_*})^*$ is isometrically isomorphic to $L_\varphi^0 \oplus_1 \mathcal{S}$ due to $\varphi_{**} = \varphi$ \cite{KR}. By Theorem \ref{th:KamKub}(i), there exists a uniformly non-$\ell_1^2$ point $x\in S_{L_{\varphi_*}}$ of a unit ball in $L_{\varphi_*}$. Hence in view of Proposition \ref{th:unif}, there exists $\epsilon > 0$ such that ${\rm diam}\, S(x;\epsilon) < 2$ where $S(x;\epsilon) = \{x^* \in B_{(L_{\varphi_*})^*} : x^*(x) > 1 - \epsilon\}$ is a weak$^*$-slice. Now, let $J: L_{\varphi_*} \rightarrow (L_{\varphi_*})^{**}$ be the canonical mapping so that $J(x)(x^*) = x^*(x)$. Letting $i: L_{\varphi}^0 \rightarrow (L_{\varphi_*})^*$ be isometric embedding, $T: = J(x) \circ i\in B_{(L_\varphi^0)^*}$ and $S(T;\epsilon) = \{y \in B_{L_{\varphi}^0} : T(y) > 1 - \epsilon\} $ is a slice of the unit ball in $L_\varphi^0$. Moreover,
\[ S(T;\epsilon) \subset \{x^* \in B_{(L_{\varphi_*})^*} : J(x)(x^*) > 1 - \epsilon\} = \{x^* \in B_{(L_{\varphi_*})^*} : x^*(x) > 1 - \epsilon\} = S(x; \epsilon).
\]
Therefore, ${\rm diam} \,S(T;\epsilon)<2$, and the space $L_\varphi^0$ does not have the local diameter two property. \end{proof}
In \cite{AKM2} it has been proved that if $\varphi$ does not satisfy the appropriate $\Delta_2$ condition then $L_\varphi$ or $\ell_\varphi$ has the local diameter two property. This result was generalized later to Orlicz-Lorentz spaces \cite{KT}. For the Orlicz spaces equipped with the Orlicz norm the situation is different. As shown below, for a large class of finite Orlicz functions the spaces $L_\varphi^0$ or $\ell_\varphi^0$ have no local diameter two property.
\begin{Corollary}\label{Cor:Orlicznormdiam}
Let $\mu$ be a non-atomic measure on $\Sigma$ or $\mu$ be the counting measure on $\mathbb{N}$. Let $\varphi$ be a finite $N$-function at infinity. Then there exists a slice of $B_{L_\varphi^0}$, respectively of $B_{\ell_\varphi^0}$, with diameter less than two. Consequently, the Orlicz spaces $L_\varphi^0$ or $\ell_\varphi^0$ equipped with the Orlicz norm do not have the local diameter two property. \end{Corollary}
\begin{proof}
Since $\varphi$ is an $N$-function at infinity then in view of Lemma \ref{lem:finite}, $\varphi_*$ is a finite function and so $b_{\varphi_*} = \infty$. We also have that $d_{\varphi_*} <\infty$. Indeed, if for a contrary $d_{\varphi_*} =\infty$ then $\varphi_*(v) = kv$ for some $k>0$ and all $v\ge 0$. Then it is easy to show that $\varphi = \varphi_{**}$ assumes only zero or infinity values, which contradicts the assumption that $\varphi$ is a finite Orlicz function. We complete the proof by application of Theorem \ref{thm:Orlicznormdiam}.\end{proof}
We conclude this paper by showing that the SD2P, D2P and LD2P are equivalent in $L_\varphi$ or $\ell_\varphi$ when $\varphi$ does not satisfy the appropriate $\Delta_2$ condition. Recall a subspace $Y$ of $X^*$ is said to be {\it norming} if for every $x \in X$, \[
\|x\| = \sup\{|x^*(x)| : \|x^*\|_{X^*} \leq 1, x^* \in Y\}. \]
\begin{Proposition}\cite[Proposition 1.b.18]{LT2}, \label{LT2}
If $X$ is a Banach function space with the Fatou property, then the K\"othe dual space $X'$ is order isometric to a norming subspace of $X^*$. \end{Proposition}
We say a closed subspace $Y$ is an \emph{$M$-ideal} in $X$ if $Y^\perp$ is the range of the bounded projection $P: X^* \rightarrow X^*$ such that $\|x^*\| = \|Px^*\| + \|(I-P)x^*\|$, that is $X^* = Y^{\perp} \oplus_1 Z$ for some subspace $Z$ of $X^*$. In fact, there is a connection between $M$-ideals and the SD2P.
\begin{Theorem} \cite[Theorem 4.10]{ALN}\label{SD2P}
Let $Y$ be a proper subspace of $X$ and let $Y$ be an $M$-ideal in $X$ i.e. $X^* = Y^{\perp} \oplus_1 Z$. If $Z$ is a norming subspace of $X^*$, then both $X$ and $Y$ have the strong diameter two property. \end{Theorem}
\begin{Corollary}\label{th:Mideal} Let $\mu$ be a non-atomic measure on $\Sigma$ or the counting measure on $\mathbb{N}$. Given a finite Orlicz function $\varphi$ which does not satisfy the appropriate $\Delta_2$ condition, the spaces $L_{\varphi}$ or $\ell_\varphi$ and their proper subspaces $(L_{\varphi})_a\ne \{0\}$ or $(\ell_\varphi)_a\ne \{0\}$ have the strong diameter two property.
\end{Corollary}
\begin{proof} Let $\mu$ be non-atomic. By the assumption that $\varphi$ is finite, the subspace $(L_{\varphi})_a$ is non-trivial. Moreover it is well-known that it is an $M$-ideal in $L_{\varphi}$ \cite{HWW}. It is a proper subspace if $(L_{\varphi})_a\ne L_{\varphi}$, which is equivalent to that $\varphi$ does not satisfy the appropriate $\Delta_2$ condition. By Proposition \ref{LT2}, $(L_{\varphi})' \simeq ((L_{\varphi})_a)^*$ is a norming subspace of $(L_\varphi)^*$. Hence by Theorem \ref{SD2P}, both $(L_{\varphi})_a$ and $L_{\varphi}$ have the strong diameter two property. The proof in sequence case is similar. \end{proof}
The $M$-ideal property of the order continuous subspace of an Orlicz-Lorentz space has been studied \cite{KLT}. In our final result, we obtain full characterization of (local, strong) diameter two properties in Orlicz spaces equipped with the Luxemburg norm. It is completion and extension of Theorems 2.5 and 2.6 from \cite{AKM}, where it was shown that $L_\varphi$ or $\ell_\varphi$ have the D2P whenever $\varphi$ does not satisfy appropriate condition $\Delta_2$.
\begin{Theorem}\label{OReq} Let $\mu$ be a non-atomic measure on $\Sigma$ or the counting measure on $\mathbb{N}$ and let $\varphi$ be a finite Orlicz function. Consider the following properties.
\begin{itemize}
\item[(i)] $L_\varphi$ or $\ell_\varphi$ has the local diameter two property.
\item[(ii)] $L_\varphi$ or $\ell_\varphi$ has the diameter two property.
\item[(iii)] $L_\varphi$ or $\ell_\varphi$ has the strong diameter two property.
\item[(iv)] $\varphi$ does not satisfy the appropriate $\Delta_2$ condition.
\end{itemize} Then $\rm(iii) \implies (ii) \implies (i)$. For the sequence space $\ell_\varphi$ all properties $\rm(i)-(iv)$ are equivalent. If in addition $\varphi$ is $N$-function at infinity then all $\rm(i)-(iv)$ are also equivalent for the function space $L_\varphi$. \end{Theorem}
\begin{proof}
The fact $\rm(iii) \implies (ii) \implies (i)$ is well-known in general Banach spaces \cite{ALN, GGMS}. The implication $\rm(iv) \implies (iii)$ follows from Corollary \ref{th:Mideal}. If $L_\varphi$ has the local diameter two property then the space can not have the RNP. Thus from Theorems \ref{th:OrRN-funct} and \ref{th:RNP-ORseq}, (i) $\implies$ (iv). \end{proof}
\end{document}
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\begin{document}
\begin{abstract} We study the translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces. Such geometric structures naturally appear when studying compactifications of the strata of the moduli space of Abelian differentials.
We compute the number of connected components of the strata of the moduli space of meromorphic differentials. We show that in genus greater than or equal to two, one has up to three components with a similar description as the one of Kontsevich and Zorich for the moduli space of Abelian differentials. In genus one, one can obtain an arbitrarily large number of connected components that are distinghished by a simple topological invariant. \end{abstract}
\maketitle
\setcounter{tocdepth}{1} \tableofcontents
\section{Introduction} A nonzero holomorphic one-form ({Abelian differential}) on a compact Riemann surface naturally defines a flat metric with conical singularities on this surface. Geometry and dynamics on such flat surfaces, in relation to geometry and dynamics on the corresponding moduli space of Abelian differentials is a very rich topic and has been widely studied in the last 30 years. It is related to interval exchange transformations, billards in polygons, Teichmüller dynamics.
A noncompact translation surface corresponds to a a one form on a noncompact Riemann surface. The dynamics and geometry on some special cases of noncompact translation surfaces have been studied more recently. For instance, dynamics on $\mathbb{Z}^d$ covers of compact translation surfaces (see \cite{HW,HLT, DHL}), infinite square tiled surfaces (see \cite{Hubert:Schmithuesen}), or general noncompact surfaces (see \cite{Bowman, BV,PSV}).
In this paper, we investigate the case of translation surfaces that come from meromorphic differentials defined on compact Riemann surfaces. In this case, we obtain infinite area surfaces, with ``finite complexity''. Dynamics of the geodesic flow on a generic direction on such surface is trivial any infinite orbit converges to the poles. Also, $SL_2(\mathbb{R})$ action doesn't seem as rich as in the Abelian case (see Appendix~A).
However, it turns out that such structures naturaly appear when studying compactifications of strata of the moduli space of Abelian differentials. Eskin, Kontsevich and Zorich show in a recent paper \cite{EKZ}
that when a sequence of Abelian differentials $(X_i,\omega_i)$ converges to a boundary point in the Deligne-Munford compactification, then subsets $(Y_{i,j},\omega_{i,j})$ corresponding to thick components of the $X_i$, after suitable rescaling converge to meromorphic differentials (see \cite{EKZ}, Theorem~10). Smillie, in a work to appear, constructs a geometric compactification of the strata of the moduli space of Abelian differentials, by using only flat geometry, and where flat structures defined by meromorphic differentials are needed.
The connected components of the moduli space of Abelian differentials were described by Kontsevich and Zorich in \cite{KoZo}. They showed that each stratum has up to three connected component, which are described by two invariants: hyperellipticity and the parity of the spin structure, that arise under some conditions on the set of zeroes.
Later, Lanneau has described the connected components of the moduli space of quadratic differentials.
The main goal of the paper is to describe connected components of the moduli space of meromorphic differentials with prescribed poles and zeroes. It is well known that each stratum of the moduli space of genus zero meromorphic differentials is connected. We show that when the genus is greater than, or equal to two, there is an analogous classification as the one of Kontsevich and Zorich, while in genus one, there can be an arbitrarily large number of connected components.
In this paper, we will call \emph{translation surface with poles} a translation surface that comes from a meromorphic differential on a punctured Riemann surface, where poles correspond to the punctured points. We describe in Section~\ref{sec:flat:merom} the local models for neighborhoods of poles. Similarly to the Abelian case, we denote by $\mathcal{H}(n_1,\dots ,n_r,-p_1,\dots ,-p_s)$ the moduli space of translation surfaces with poles that corresponds to meromorphic differentials with zeroes of degree $n_1,\dots ,n_r$ and poles of degree $p_1,\dots ,p_s$. It will be called \emph{stratum of the moduli space of meromorphic differentials}. We will always assume that $s>0$. A strata is nonempty as soon as $\sum_{i} n_i-\sum_j p_j=2g-2$, for some nonnegative integer $g$ and $\sum_{j}p_j>1$.
For a genus one translation surface $S$ with poles, we describe the connected components by using a geometric invariant easily computable in terms of the flat metric, that we call the \emph{rotation number of a surface}. As we will see in Section~\ref{genus1}, in the stratum $\mathcal{H}(n_1,\dots ,n_r,-p_1,\dots ,-p_s)$, the rotation number is a positive integer that divides all the $n_i,p_j$.
\begin{theorem}\label{MT1} Let $\mathcal{H}(n_1,\dots ,n_r,-p_1,\dots ,-p_s)$, with $n_i,p_j>0$ and $\sum_{j}p_j>1$ be a stratum of genus one meromorphic differentials. Denote by $c$ be the number of positive divisors of $\gcd(n_1,\dots ,n_r,p_1,\dots ,p_s)$. The number of connected components of the stratum is: \begin{itemize} \item $c-1$ if $r=s=1$. In this case $n_1=p_1=\gcd(n_1,p_1)$ and each connected component corresponds to a a rotation number that divides $n_1$ and is not $n_1$. \item $c$ otherwise. In this case each connected component corresponds to a rotation number that divides $\gcd(n_1,\dots ,n_r, p_1,\dots ,p_s)$. \end{itemize}
\end{theorem} A consequence of the previous theorem is that, contrary to the case of Abelian differentials, there can be an arbitrarily large number of connected components for a stratum of meromorphic differentials (in genus~1). For instance, the stratum $\mathcal{H}(24,-24)$ has 7 connected components since the positive divisors of 24 that are not 24 are $1,2,3,4,6,8$ and $12$.
The general classification uses analogous criteria as for Abelian differentials. We recall that in this case, the connected components are distinguished by the following (up to a few exception in low genera): \begin{itemize} \item \emph{hyperellipticity}: if there is only one singularity or two singularities of equal degree, there is a component that consists only of hyperelliptic Riemann surfaces. For each translation surface, the hyperelliptic involution is an isometry. Slightly abusing with terminology, we usually call this component the \emph{hyperelliptic component}. \item \emph{the parity of the spin structure}: If all singularities are of even degree, there are two connected component (none of which is the hyperelliptic component) distinguished by a topological invariant easily computable in terms of the flat metric. \end{itemize}
In Section~\ref{invariants}, we define in our context the notion of hyperelliptic component and spin structure.
In the next theorem, we say that the set of poles and zeroes is: \begin{itemize} \item of \emph{hyperelliptic type} if the degree of zeroes are or the kind $\{2n\}$ or $\{n,n\}$, for some positive integer $n$, and if the degree of the poles are of the kind $\{-2p\}$ or $\{-p,-p\}$, for some positive integer $p$. \item of \emph{even type} if the degrees of zeroes are all even, and if the degrees of the poles are either all even, or are $\{-1,-1\}$. \end{itemize}
\begin{theorem}\label{MT2} Let $\mathcal{H}=\mathcal{H}(n_1,\dots ,n_r,-p_1,\dots ,-p_s)$, with $n_i,p_j>0$ be a stratum of genus $g\geq 2$ meromorphic differentials. We have the following. \begin{enumerate} \item If $\sum_{i} p_i$ is odd and greater than two, then $\mathcal{H}$ is nonempty and connected. \item If $\sum_i p_i=2$ and $g=2$, then: \begin{itemize} \item if the set of poles and zeroes is of hyperelliptic type, then there are two connected components, one hyperelliptic, the other not (in this case, these two components are also distinghished by the parity of the spin structure)
\item otherwise, the stratum is connected. \end{itemize} \item If $\sum_i p_i>2$ or if $g>2$, then: \begin{itemize} \item if the set of poles and zeroes is of hyperelliptic type, there is exactly one hyperelliptic connected component, and one or two nonhyperelliptic components that are discribed below. Otherwise, there is no hyperelliptic component.
\item if the set of poles and zeroes is of even type, then $\mathcal{H}$ contains exactly two nonhyperelliptic connected components that are distinguished by the parity of the spin structure. Otherwise $\mathcal{H}$ contains exactly one nonhyperelliptic component. \end{itemize} \end{enumerate} \end{theorem}
From the previous theorem, we see that there are at most three connected component in genus greater than or equal to two. For instance, the stratum $\mathcal{H}(4,4,-1,-1)$ contains a hyperelliptic connected component (zeroes and poles are of hyperelliptic type) and two nonhyperelliptic components (the zeroes are even and the poles are $\{-1,-1\}$). So it has three components. The stratum $\mathcal{H}(2,4,-1,-1,-2)$ is connected, since it does not have a hyperelliptic connected component and the poles and zeroes are not of even type.
The structure of the paper is the following: \begin{itemize} \item In Section~\ref{sec:flat:merom}, we describe general facts about the metric defined by a meromorphic differential and define a topology on the moduli space. \item In Section~\ref{sec:tools}, we present three tools that are needed in the proof. The first two ones appear already in the paper of Kontsevich and Zorich, and in the paper of Lanneau. The third one is a version of the well known Veech construction for the case of translation surfaces with poles. \item In Section~\ref{genus1}, we describe the connected components in the genus one case. Some of the results in genus one will be very useful for the general genus. \item In Section \ref{invariants}, we describe the topological invariants for the general genus case, \emph{i.e.} hyperelliptic connected components and the parity of the spin structure. \item In Section \ref{sec:minimal}, we compute the connected components for the minimal strata, which are the strata with only one conical singularity (and possibly several poles). \item In Section \ref{sec:nonminimal}, we compute the connected components for the general case. \end{itemize}
\subsubsection*{Acknowledgments} I thank Martin Moeller for many discussions about meromorphic differentials and about spin structures. I thank John Smillie for motivating the work on this paper and interesting discussions. I am also thankful to Pascal Hubert and Erwan Lanneau for the frequent discussions during the development of this paper. This work is partially suported by the ANR Project "GeoDym".
\section{Flat structures defined by meromorphic differentials.}\label{sec:flat:merom}
\subsection{Holomorphic one forms and flat structures} Let $X$ be a Riemann surface and let $\omega$ be a holomorphic one form. For each $z_0\in X$ such that $\omega(z_0)\neq 0$, integrating $\omega$ in a neighborhood of $z_0$ gives local coordinates whose corresponding transition functions are translations, and therefore $X$ inherits a flat metric, on $X\backslash \Sigma$, where $\Sigma$ is the set of zeroes of $\omega$.
In a neighborhood of an element of $\Sigma$, such metric admits a conical singularity of angle $(k+1)2\pi$, where $k$ is the degree of the corresponding zero of $\omega$. Indeed, a zero of degree $k$ is given locally, in suitable coordinates by $\omega=(k+1)z^k dz$. This form is precisely the preimage of the constant form $dz$ by the ramified covering $z\to z^{k+1}$. In terms of flat metric, it means that the flat metric defined locally by a zero of order $k$ appear as a connected covering of order $k+1$ over a flat disk, ramified at zero.
When $X$ is compact, the pair $(X,\omega)$, seen as a smooth surface with such translation atlas and conical singularities, is usually called a \emph{translation surface}.
If $\omega$ is a meromorphic differential on a compact Riemann $\overline{X}$, we can consider the translation atlas defined defined by $\omega$ on $X=\overline{X}\backslash \Sigma'$, where $\Sigma'$ is the set of poles of $\omega$. We obtain a translation surface with infinite area. We will call such surface \emph{translation surface with poles}.
\begin{convention} When speaking of a translation surface with poles $S=(X,\omega)$. The surface $S$ equipped with the flat metric is noncompact. The underlying Riemann surface $X$ is a punctured surface and $\omega$ is a holomorphic one-form on $X$. The corresponding closed Riemann surface is denoted by $\overline{X}$, and $\omega$ extends to a meromorphic differential on $\overline{X}$ whose set of poles is precisely $\overline{X}\backslash X$. \end{convention}
Similarly to the case of Abelian differentials. A \emph{saddle connection} is a geodesic segment that joins two conical singularities (or a conical singularity to itself) with no conical singularities on its interior.
We also recall that it is well known that $\sum_{i=1}^r n_i-\sum_{j=1}^s p_j=2g-2$, where $\{n_1,\dots ,n_r\}$ is the set (with multiplicities) of degree of zeroes of $\omega$ and $\{p_1,\dots ,p_s\}$ is the set (with multiplicities) of degree of the poles of $\omega$.
\subsection{Local model for poles}
The neighborhood of a pole of order one is an infinite cylinder with one end. Indeed, up to rescaling, the pole is given in local coordinates by $\omega=\frac{1}{z}dz$. Writing $z=e^{z'}$, we have $\omega=dz'$, and $z'$ is in a infinite cylinder.
Now we describe the flat metric in a neighborhood of a pole of order $k\geq 2$ (see also \cite{strebel}). First, consider the meromorphic 1-form on $\mathbb{C}\cup \{\infty \}$ defined on $\mathbb{C}$ by $\omega=z^k dz$. Changing coordinates $w=1/z$, we see that this form has a pole $P$ of order $k+2$ at $\infty $, with zero residue. In terms of translation structure, a neighborhood of the pole is obtained by taking an infinite cone of angle $(k+1)2\pi$ and removing a compact neighborhood of the conical singularity. Since the residue is the only local invariant for a pole of order k, this gives a local model for a pole with zero residue.
Now, define $U_R=\{z\in \mathbb{C}| |z|>R\}$ equipped with the standard flat metric. Let $V_R$ be the Riemann surface obtained after removing from $U_R$ the ${\pi}$--neighborhood of the real half line $\mathbb{R}^-$, and identifying by the translation $z\to z+\imath 2\pi$ the lines $-\imath {\pi}+\mathbb{R}^-$ and $\imath {\pi}+\mathbb{R}^-$. The surface $V_R$ is naturally equipped with a holomorphic one form $\omega$ coming from $dz$ on $V_R$. We claim that this one form has a pole of order 2 at infinity and residue -1. Indeed, start from the one form on $U_{R'}$ defined by $(1+1/z)dz$ and integrate it. Choosing the usual determination of $\ln(z)$ on $\mathbb{C}\backslash \mathbb{R}^-$, one gets the map $z\to z+\ln(z)$ from $U_{R'}\backslash \mathbb{R}^-$ to $\mathbb{C}$, which extends to a injective holomorphic map $f$ from $U_{R'}$ to $V_R$, if $R'$ is large enough. Furthermore, the pullback of the form $\omega$ on $V_R$ gives $(1+1/z)dz$. Then, the claim follows easily after the change of coordinate $w=1/z$
Let $k\geq 2$. The pullback of the form $(1+1/z)dz$ by the map $z\to z^{k-1}$ gives $((k-1)z^{k-2}+(k-1)/z)dz$, \emph{i.e.} we get at infinity a pole of order $k$ with residue $-(k-1)$. In terms of flat metric, a neighborhood of a pole of order $k$ and residue $-(k-1)$ is just the natural cyclic $(k-1)$--covering of $V_R$. Then, suitable rotating and rescaling gives the local model for a pole of order $k$ with a nonzero residue.
For flat geometry, it will be convenient to forget the term $2\imath \pi$ when speaking of residue, hence we define the \emph{flat residue} of a pole $P$ to be $\int_{\gamma_P} \omega$, where $\gamma_P$ is a small closed path that turns around a pole counterclokwise.
\subsection{Moduli space}
If $(X,\omega)$ and $(X',\omega')$ are such that there is a biholomorphism $f:X\to X'$ with $f^* \omega'=\omega$, then $f$ is an isometry for the metrics defined by $\omega$ and $\omega'$. Even more, for the local coordinates defined by $\omega,\omega'$, the map $f$ is in fact a translation.
As in the case of Abelian differentials, we consider the moduli space of meromorphic differentials, where $(X,\omega)\sim (X',\omega')$ if there is a biholomorphism $f:X\to X'$ such that $f^* \omega'=\omega$. A stratum corresponds to prescribed degree of zeroes and poles. We denote by $\mathcal{H}(n_1,\dots ,n_r,-p_1,\dots ,-p_s)$ the \emph{stratum} that corresponds to meromorphic differentials with zeroes of degree $n_1,\dots ,n_r$ and poles of degree $p_1,\dots ,p_s$. Such stratum is nonempty if and only if $\sum_{i=1}^r n_i-\sum_{j=1}^s p_j=2g-2$ for some integer $g\geq 0$ and if $\sum_{j=1}^s p_j>1$ (\emph{i.e.} if there is not just one simple pole.). A \emph{minimal stratum} is a stratum with $r=1$, \emph{i.e.} which corresponds to surfaces with only one conical singularity and possibly several poles.
We define the topology on this space in the following way: a small neighborhood of $S$, with conical singularities $\Sigma$, is defined to be the equivalent classes of surfaces $S'$ for which there is a differentiable injective map $f:S\backslash V(\Sigma)\to S'$ such that $V(\Sigma)$ is a (small) neighborhood of $\Sigma$, $Df$ is close the identity in the translation charts, and the complement of the image of $f$ is a union on disks. One can easily check that this topology is Hausdorff.
\section{Tools}\label{sec:tools}
\subsection{Breaking up a singularity: local construction}\label{bzero:loc} Here we describe a sur\-ge\-ry, introduced by Eskin, Masur and Zorich (see \cite{EMZ}, Section~8.1) for Abelian differentials, that ``break up'' a singularity of degree $k_1+k_2\geq 2$ into two singularities of degree $k_1\geq 1$ and $k_2\geq 1$ respectively. This surgery is local, since the metric is modified only in a neighborhood of the singularity of degree $k_1+k_2$. In particular, it is also valid for the flat structure defined by a meromorphic differential.
\begin{figure}
\caption{Breaking up a zero, after Eskin, Masur and Zorich}
\label{bzero}
\end{figure}
We start from a singularity of degree $k_1+k_2$. A neighborhood of such singularity is obtained by gluing $(2k_1+2k_2+2)$ Euclidean half disks in a cyclic order. The singularity breaking procedure consists in changing continuously the way these half disks are glued together, as in Figure~\ref{bzero}. This breaks the singularity of degree $k_1+k_2$ into singularities of degree $k_1$ and $k_2$ respectively, and with a small saddle connection joining them.
\subsection{Bubbling a handle}\label{bubbling} The following surgery was introduced by Kontsevich and Zorich in \cite{KoZo}. Since it is a local construction, it is also valid for meromorphic differentials. As before, we start from a singularity of degree $k_1+k_2$ on a surface $S$. We first apply the previous surgery to get a pair of singularities of degree $k_1$ and $k_2$ respectively, and with a small saddle connection $\gamma$ joining them. Then, we cut the surface along $\gamma$ and obtain a flat surface with a boundary component that consists of two geodesic segments $\gamma_1,\gamma_2$. We identify their endpoints and the corresponding segments are now closed boundary geodescis $\gamma_1',\gamma_2'$. Then, we consider a cylinder with two boundary components isometric to $\gamma_i'$, and glue each of these component to $\gamma_i'$. The angle between $\gamma_1'$ and $\gamma_2'$ is $(k_1+1)2\pi$ (and $(k_2+1)2\pi$)
Using a notation similar to the one introduced by Lanneau in \cite{La:cc}, we will denote by $S\oplus (k_1+1)$ the resulting surface for an arbitrary choice of continuous parameters. Different choices of continuous parameters lead to the same connected component and from a path $(S_t)_{t\in [0,1]}$, one can easily deduce a continuous path $S_t\oplus (k_1+1)$. Hence, as in \cite{La:cc}, the connected component of $S\oplus s$ only depends on $s$ and on the connected component of $S$. So, if $S$ is in a connected component $\mathcal{C}$ of a stratum of Abelian (resp. meromorphic) differential with only one singularity, $\mathcal{C}\oplus s$ is the connected component of a stratum of Abelian (resp. meromorphic) differentials obtained by the construction.
\begin{remark} {The notation $\oplus$ slightly differs to the one introduced by Lanneau: since he manipulates \emph{quadratic differentials}, the angles can be any multiples of $\pi$, while in our case, we only have multiples of $2\pi$. So the surface we obtain would have been written $S\oplus 2(k_1+1)$ with the notation of Lanneau.} \end{remark}
The following lemma is Proposition~2.9 in the paper of Lanneau \cite{La:cc}, written in our context. The ideas behind this proposition were also in the paper of Kontsevich and Zorich \cite{KoZo}. \begin{lemma}\label{lemme:lanneau} Let $\mathcal{C}$ be connected component a minimal stratum of the moduli space of meromorphic differentials, of the form $\mathcal{H}(n,-p_1,\dots ,-p_r)$. Then, the following statements hold. \begin{enumerate} \item $ \mathcal{C}\oplus s_1 \oplus s_2=\mathcal{C}\oplus s_2 \oplus s_1$ if $1\leq s_1,s_2\leq n+1$ and either $s_1\neq \frac{n}{2}+1$ or $s_2 \neq \frac{n+2}{2}+1$. \item $ \mathcal{C}\oplus s_1 \oplus s_2=\mathcal{C}\oplus s_2-1 \oplus s_1+1$ if $1\leq s_1\leq n+1$ and $2\leq s_2\leq n+3$. \item $\mathcal{C}\oplus s_1 \oplus s_2=\mathcal{C}\oplus s_2-2 \oplus s_1$ if $1\leq s_1\leq n+1$ and $1\leq s_2\leq n+3$ and $s_2-s_1\geq 2$. \item $\mathcal{C}\oplus s=\mathcal{C}\oplus (n+2-s)$ for all $s\in \{1,\dots, n+1\}$ \end{enumerate} \end{lemma} \begin{remark} There is a small mistake in the statement of Lanneau: the condition ``either $s_1\neq \frac{n}{2}+1$ or $s_2 \neq \frac{n+2}{2}+1$'' does not appear while it is necessary.
This leads to a gap in the proof of Lanneau's Lemma~6.13 in \cite{La:cc}, but this gap is easily solved by using Lemma~A.2 of the same paper. \end{remark}
\subsection{The infinite zippered rectangle construction} In this section, we describe a construction of translation surfaces with poles which is analogous to the well known Veech zippered rectangle construction. We will call this construction the \emph{infinite zippered rectangles construction}.
We first recall Veech's construction.
\subsubsection{The Veech construction of a translation surface}\label{veech:constr} The {Veech construction}, or {zippered rectangle construction} is usually seen as a way to define a suspension over an interval exchange map (see \cite{Veech82}). We can also see it as a easy way to define (almost any) translation surface. Consider a finite alphabet $\mathcal{A}=\{\alpha_1,\dots ,\alpha_d\}$, and a pair on one to one maps $\pi_t,\pi_b:\mathcal{A}\to \{1,\dots ,d\}$. Let $\zeta\in \mathbb{C}^\mathcal{A}$ be a vector for which each entry has positive real part.
The Veech construction can be seen in two (almost) equivalent ways. One with a $2d$ sided polygon, and one with $d$ rectangles that are identified on their boundary.
We present the first one, which is simpler but not as general as the second one. Consider the broken line $L_t$ on $\mathbb{C}=\mathbb R^2$ defined by concatenation of the vectors $\zeta_{\pi_t^{-1}(j)}$ (in this order) for $j=1,\dots,d$ with starting point at the origin. Similarly, we consider the broken line $L_b$ defined by concatenation of the vectors $\zeta_{\pi_b^{-1}(j)}$ (in this order) for $j=1,\dots,d$ with starting point at the origin.
We assume that $\zeta$ is such that the vertices of $L_t$ are always above the real line, except possibly the foremost right (and of course the one at the origin), and that similarly, the vertices of $L_b$ are below the real line. Such $\zeta$ is called \emph{suspension datum} (see \cite{MMY}), and exists under a combinatorial condition on $(\pi_t,\pi_b)$ usually called ``irreducibility''.
If the lines $L_t$ and $L_b$ have no intersections other than the endpoints, we can construct a translation surface $X$ by identifying each side $\zeta_j$ on $L_t$ with the side $\zeta_j$ on $L_b$ by a translation. The resulting surface is a translation surface endowed with the form $\omega=dz$ (see Figure~\ref{Veech:construction}).
\begin{figure}
\caption{Veech's construction of a translation surface}
\label{Veech:construction}
\end{figure}
\begin{remark} The surface constructed in this way can also be seen as a union of rectangles whose dimensions are easily deduced from $\pi_t,\pi_b$ and $\zeta$, and that are ``zippered'' on their boundary. One can define $S$ directely in this way: the construction works also if $L_t,L_b$ have other intersection points. This is the \emph{zippered rectangle construction}, due to Veech (\cite{Veech82}). This construction coincides with the first one in the previous case. \end{remark}
\subsubsection{Basic domains}\label{basic:domains} Now we generalize the previous construction to obtain a flat surface that corresponds to a compact Riemann surface with a meromorphic differential. Instead of having a polygon with pairwise identification on its boundary, we will have a finite union of some ``basic domains'' which are half-planes and infinite cylinders with polygonal boundaries (see Figure~\ref{fig:basic:domains}).
Let $n\geq 0$. Let $\zeta\in \mathbb{C}^n$ be a complex vector whose entries have positive real part.
Consider the broken line $L$ on $\mathbb{C}$ defined by concatenation of the following: \begin{itemize} \item the half-line $l_1$ that corresponds to $\mathbb{R}^-$, \item the broken line $L_t$ defined as above, \emph{i.e.} the concatenation of the segment defined by the vectors $\zeta_{j}$ (in this order) for $j=1,\dots,n$ with starting point at the origin, \item the horizontal half line $l_2$ starting from the right end of $L_t$, and going to the right. \end{itemize} We consider the subset $D^+(z_1,\dots ,z_n)$ (resp. $D^-(z_1,\dots ,z_n)$) as the set of complex numbers that are above $L$. The line $l_1$ will be refered to as the \emph{left} half-line, and $l_2$ will be refered to as the \emph{right} half-line. We will sometime write such domains $D^{+}$ or $D^-$ for short. The sets $D^\pm$ are kinds of half-planes with polygonal boundaries. Note that $n$ might be equal to $0$, and in this case, $D^+$ (resp. $D^-$) is just a half-plane with a marked point on its (horizontal) boundary.
Similarly, if $n\geq 1$, we can define the subset $C^+(z_1,\dots ,z_n)$ (resp. $C^-(z_1,\dots ,z_n)$) as the set of complex numbers that are above $L_t$. Its boundary consists of two infinite vertical half-lines joined by the broken line $L_t$. The two infinite half-lines will be identified in the resulting construction, hence $C^\pm$ is just an infinite half-cylinder with a polygonal boundary.
\begin{figure}
\caption{A domain $D^+(\zeta_1,\zeta_2,\zeta_3)$ and a domain $C^+(\zeta_1,\zeta_2,\zeta_3)$}
\label{ex:D:C}
\label{fig:basic:domains}
\end{figure}
\subsubsection{An example: a surface with a single pole of order 2.} The idea of the construction is to glue by translation the basic domains together in order to get a noncompact translation surface with no boundary components. Since the formal description is rather technical, we first present a simple version of the construction.
Let $\mathcal{A}$ be a finite alphabet and $\pi_t,\pi_b:\mathcal{A}\to \{1,\dots ,d\}$ be one-to-one maps. Let $\zeta\in \mathbb{C}^\mathcal{A}$ be such that $Re(\zeta_\alpha)>0$ for all $\alpha\in \mathcal{A}$.
We define a flat surface $S$ as the disjoint union of the two half-planes $D^+=D^+(\zeta_{\pi_t^{-1}(1)},\dots ,\zeta_{\pi_t^{-1}(n)})$ and $D^-=D^-(\zeta_{\pi_b^{-1}(1)},\dots ,\zeta_{\pi_b^{-1}(n)})$ glued on their boundary by translation: the left half line of $D^+$ being glued to the left half-line of $D^-$ and similarly with the right half-lines, and a segment in $D^+$ corresponding to some $\zeta_i$ is glued to the corresponding one of $D^-$.
Note that contrary to the case of compact translation surfaces, there is no ``suspension data condition'' on $\zeta$, hence, no combinatorial condition on $\pi$. The only condition that we require is that $Re(\zeta_i)>0$ for all $i$. Note also that we can have $n=0$, in this case $S=\mathbb{C}$.
\begin{figure}
\caption{Construction of a translation surface with a degree 2 pole.}
\label{ex:surf:av:pole2}
\end{figure}
\subsubsection{General case} \label{inf:zipp:rect}
We can generalize the above construction in order to have several poles of any order. Instead of considering two half-planes $D^+,D-$, we will do the same construction starting from $2d$ half-planes, $s^++s^-$ infinite cylinders, and define identification on their boundary. More precisely:
Let $\zeta\in \mathbb{C}^n$ with positive real part. We consider the following combinatorial data: \begin{itemize} \item A collection {$\bf n^+$} of integers $0=n_0^+\leq n_1^+\leq \dots n_d^+ < \dots <n_{d+s^+}^+=n$ \item A collection {$\bf n^-$} of integers $0=n_0^-\leq n_1^-\leq \dots n_d^- < \dots <n_{d+s^-}^-=n$ \item A pair of maps $\pi_t,\pi_b: \mathcal{A}\to \{1,\dots ,n\}$ \item A collection {$\bf d$} of integers $0=d_0< d_1 < d_2 <\dots < d_{r}=d$. \end{itemize} The resulting surface will have $r$ poles of order greater than or equal to two, and $s^++s^-$ poles of order 1.
For each $i \in \{0,\dots ,d-1\}$, we consider the basic domains as defined before $D_i^+(\zeta_{\pi_{t}^{-1}(n_i^++1)},\dots ,\zeta_{\pi_{t}^{-1}(n_{i+1}^+)})$ and $D_i^-(\zeta_{\pi_b^{-1}(n_i^-+1)},\dots ,\zeta_{\pi_b^{-1}(n_{i+1}^-}))$.
For $i\in\{d,\dots ,d+s^+-1\}$, we define $C_i^+(\zeta_{\pi_{t}^{-1}(n_i^++1)},\dots ,\zeta_{\pi_{t}^{-1}(n_{i+1}^+)})$.
For $i\in\{d,\dots ,d+s^--1\}$, we define $C_i^-(\zeta_{\pi_b^{-1}(n_i^-+1)},\dots ,\zeta_{\pi_b^{-1}(n_{i+1}^-}))$.
\begin{figure}
\caption{Construction of a translation surface with a degree 3 pole.}
\label{ex:surf:pole:3}
\end{figure}
\begin{figure}
\caption{Construction of a translation surface with two poles of degree $2$}
\label{ex:surf:2pole:2}
\end{figure}
Then, we glue these domains together on their boundary by translations: \begin{itemize} \item each segment corresponding to a parameter $\zeta_i$ in a $``+''$ domain is glued to the unique corresponding one in a $``-''$ domain. \item each left line of a domain $D_i^+$ is glued to the left line of the domain $D_i^-$. \item For each $i\in \{1,\dots ,d\} \backslash \{d_1,\dots , d_r \}$ the right line of the domain $D_i^-$ is glued to one of the domain $D_{i+1}^+$. \item For each $i=d_k$, $k>0$, the right line of the domain $D_i^-$ is glued to one of the domain $D_{d_{k-1}+1}^+$. \item For each $C_i^+$ and $C_i^-$, the two vertical lines are glued together. \end{itemize}
The resulting surface $S$ has no more boundary components, and is a flat surface with poles and conical singularities. It might not be connected for a given combinatorial data. We will consider only those that that give connected surfaces.
Note that such surface is easily decomposes into a finite union of vertical strips and half-planes with vertical boundary (\emph{i.e.} ``infinite rectangle''), that are ``zippered'' on their boundary.
\begin{example} Figure~\ref{ex:surf:pole:3} shows an example with $d=2$, $s^+=s^-=0$, ${n^+}=(0,2,4)$, ${n^-}=(0,2,4)$, $\pi_t=Id$, $(\pi_b^{-1}(1),\dots ,\pi_b^{-1}(n))=(2,3,4,1)$ and ${\bf d}=(0,2)$. One gets a surface in $\mathcal{H}(-3,5)$.
Figure~\ref{ex:surf:2pole:2} shows an example with the same data, except that ${\bf d}=(0,1,2)$. One gets a surface in $\mathcal{H}(-2,-2,2,2)$. \end{example}
\begin{lemma}\label{lem:dim} Let $S$ be a genus $g$ surface in $\mathcal{H}(n_1,\dots,n_r,-p_1,\ldots,-p_s)$, obtained by the infinite zippered rectangle construction with a continuous parameter $\zeta\in \mathbb{C}^n$. Then $$n=2g+r+s-2$$ \end{lemma}
\begin{proof} By construction, the surface (pole included) is obtained by gluing $s$ disks on their boundary. The resulting surfaces admits a decompositions into cells, with $s$ faces, $n$ edges, and $r$ vertices. So, we have $2-2g=s-n+r$, and the result follows. \end{proof}
The following proposition will be very useful in the remaing of the paper. It is analogous to the well-known fact that that any translation surface with no vertical saddle connection is obtained by the Veech construction. \begin{proposition}\label{no:vertical} Any translation surface with poles with no vertical saddle connection is obtained by the infinite zippered rectangle construction. \end{proposition}
\begin{proof} According to the book of Strebel \cite{strebel} Section~11.4, when there are no vertical saddle connections the vertical trajectories are of the following two types: \begin{enumerate} \item lines that converge to a pole in each direction. \item half-lines starting (or ending) from a conical singularity and converging to a pole on their infinite direction. \end{enumerate} Furthermore, the set of non-singular vertical trajectories is a disjoint union of half-planes and of vertical strips $\sqcup_i \mathcal{P}_i \sqcup_j \mathcal{S}_j$. The half-planes have one vertical boundary component, and the strips have two vertical boundary component. We choose these half-planes or strips as large as possible, so each boundary component necessarily contains a conical singularity. This singularity is unique for each connected component, otherwise there would be a vertical saddle connection on the surface. This number of half-planes and strips is necessarily finite, since there is only a finite number of conical singularities, and each conical singularities is adjacent to a finite number of half-plane or strip.
We cut each half-plane $\mathcal{P}_i$ in the following way: the boundary of $\mathcal{P}_i$ consists of a union of two vertical half-lines starting from the conical singularity. We consider the unique horizontal half-line starting from this singularity and cut $\mathcal{P}_i$ along this half line. It decomposes $\mathcal{P}_i$ into two components $\mathcal{P}_i^+$ (the upper one) and $\mathcal{P}_i^-$ (the lower one).
We cut each strip $\mathcal{S}_j$ in the following way: the boundary of $\mathcal{S}_i$ has two components, each consists of a union of two vertical half-lines starting from a conical singularity. There is a unique geodesic segment joining these two boundary singularities. We cut $\mathcal{S}_j$ along this segment and obtain two components $\mathcal{S}_j^+$ and $\mathcal{S}_j^-$.
The surface $S$ is obtained as the disjoint union of the $\mathcal{S}_j^\pm$ and $\mathcal{P}_i^\pm$, glued to each other by translation on their boundary components. Now we remark that the $\mathcal{P}_i^+$ and $\mathcal{S}_j^+$ are necessarily glued to each other along their vertical boundary components. Under this identification, $\sqcup \mathcal{P}_i^+ \sqcup_j \mathcal{S}_j^+$ is a union of subsets of the type $D^+$ and $S^+$, as in the previous construction.
Similarly, gluing together along vertical sides the union of the $\mathcal{S}_j^-$ and $\mathcal{P}_i^-$, one obtains a union of $D^-$ and $C^-$ type subsets.
So the surface is obtained by the infinite zippered rectangle construction. \end{proof}
\begin{remark}\label{rq:param} Note that the parameters $(\zeta_i)_i$ are uniquely defined (once the suitable vertical direction is fixed) and the infinite zipperered rectangle construction defines a triangulation of the surface for which the $(\zeta_i)$ are the local parameters for the strata. Hence, map $S\to (\zeta_i)_i$ is a local homeomorphism. The corresponding saddle connections form a basis of the relative homology $H_1(S, \Sigma,\mathbb{Z})$, where $\Sigma$ is the set of conical singularities of $S$. \end{remark}
Note that for any translation surface with poles, the set of saddle connections is at most countable, so it is always possible to rotate the surface in such way that there are no vertical saddle connections. Hence the previous theorem gives a representation for \emph{any} translation surface with poles. One important consequence this theorem is Proposition~\ref{adj:minimal}, which is the analogous version of a key argument in \cite{KoZo}.
\section{Genus one case}\label{genus1} \subsection{Connected components} We first recall the following result in algebraic geometry, which is a consequence of Abel's theorem.
\begin{NNthm} Let $\overline{X}=\mathbb{C}/\Gamma$ be a torus and let $D=\sum_i \alpha_i P_i$ be a divisor. Then there exists a meromorphic differential whose divisor is $D$ if and only if $\sum {z_i}\in \Gamma$, where for each $i$, ${z_i}$ is a representative in $\mathbb{C}$ of $P_i$. \end{NNthm}
Now we use this theorem to describe the connected components in the genus one case.
\begin{theorem} \label{th:g1:complex} Let $\mathcal{H}=\mathcal{H}(n_1,\dots ,n_r,-p_1,\dots ,-p_s) $ be a stratum of genus one meromorphic differentials. Let $d=\gcd(n_1,\dots ,n_r,p_1,\dots p_s)$, and let $c$ be the number of positive divisors of $d$. Then the number of connected components of $\mathcal{H}$ is: \begin{itemize} \item $c$ if $r+s\geq 3$. \item $c-1$ if $r=s=1$. \end{itemize} \end{theorem}
\begin{proof}
We first assume that $r+s=2$. Then $\mathcal{H}=\mathcal{H}(n,-n)$, for some $n\geq 2$ (the stratum $\mathcal{H}(1,-1)$ is empty). We have $d=n$. An element in $\mathcal{H}$ is given, up to a constant multiple, by a pair $(\overline{X},D)$, were $\overline{X}$ is a torus and $D=-nP+nQ$ is a divisor on $\overline{X}$. One can assume that $P=0$, and from the previous theorem, there is only a finite number of possibilities for $Q$, depending only on $n$. Hence, the map $(\overline{X},-n0+nQ)\mapsto (\overline{X},0)$ defines a covering from
$P\mathcal{H}$ to $\mathcal{M}_{1,1}$, where $\mathcal{M}_{1,1}$ denotes the moduli space of genus one Riemann surfaces with a marked point.
Fix $\overline{X}_0\in \mathcal{M}_{1,1}$ a regular point, and choose $v_1,v_2$ such that $\overline{X}_0=\mathbb{C}/(v_1 \mathbb{Z}+v_2\mathbb{Z})$. An element $(\overline{X}_0,\omega)\in \mathcal{H}$ is uniquely defined by the coordinates of $Q$, which are given in a unique way by a complex number of the form $\frac{p}{n}v_1+\frac{q}{n}v_2$, with $(p,q)\neq (0,0)$ and $0\leq p,q<n$. Since $\overline{X}_0$ is taken regular, there is a one to one correspondance between such pairs $(p,q)$ of integers and the elements of $\mathcal{H}$ whose underlying Riemann surface is $\overline{X}_0$.
The monodromy of the covering $P\mathcal{H}\to \mathcal{M}_{1,1}$ is generated by the two maps $\phi_1:(p,q)\to (p+q,q) \mod n$ and $\phi_2:(p,q)\to (p,q+p) \mod n$. We remark that $d'=\gcd(p,q,n)$ is invariant by this action and the condition on $(p,q)$ implies that $0<d'<n$. Hence $d'$ is an invariant of the connected components of $\mathcal{H}$. The number of possible $d'$ is $c-1$. We claim that each $(p,q)$ has a (unique) representative modulo these actions of the kind $(d',0)$. To prove the claim, we start from an element $(p,q)$ and do an algorithm similar to Euclid's algorithm. Without loss of generality, one can assume that $p\neq 0$ and $q\neq 0$. Applying $\phi_1^r$ for some suitably chosen $r$, we can obtain $(p',q)$ with $0<p'\leq \gcd(q,n)$. Similarly, we can obtain $(p,q')$ with $0<q<\gcd(p,n)$. So if either $\gcd(q,n)<p$ or $\gcd(p,n)<q$, we obtain $(p',q')$ with $p'+q'<p+q$. Otherwise $p\leq \gcd(q,n)\leq q$ and $q\leq \gcd(p,n)\leq p$. This implies $p=q=d'$, and the result follows.
Now we assume $r+q\geq 3$. We proceed in a similar way as before: we fix $\overline{X}_0=\mathbb{C}/\Gamma$ and a basis $v_1,v_2$ of $\Gamma$. Then a meromorphic differential is given by a a vector $(z_1,\dots z_{r},z_1',\dots ,z_{s}')\in \mathbb{C}^{r+s}$ with pairwise different entries (modulo $\Gamma$), and satisfying the linear equality $\sum_{i=1}^r n_i z_i -\sum_{i=1}^s p_i z_i=p v_1+ q v_2$. One can remark that: \begin{itemize} \item For each $(p,q)$ the set of $(z_i)_i,(z_j')_j$ satisfying the previous condition is nonempty and connected. \item If we choose other representatives $z_i,z_j'$ for the same differential $\omega$, this changes $(p,q)$ by $(p+\sum_i \alpha_i n_i+\sum_j \beta_j p_j,q+ \sum_{i} \alpha_i' n_i+\sum_j \beta_j' p_j)$, where $(\alpha_i,\beta_j, \alpha_i',\beta_j')$ can be any integers. \item The action of the two generators of the modular group changes $(p,q)$ by $(p+q,q)$ and $(p,q+p)$ respectively. \end{itemize}
Then, by a proof very similar to the previous one, one can see that $d'=\gcd(p,q,n_1,\dots ,n_r,p_1,\dots ,p_s)$ is an invariant of connected component and one can find representative in each connected component satisfying $(p,q)=(d',0)$. So the number of connected components is precisely $c$.
Note that the difference with the first case is that any pair $(p,q)\in \mathbb{Z}^2$ is possible. \end{proof}
\subsection{Flat point of view: rotation number} The previous section classifies the connected component of the moduli space of meromorphic differentials in the genus one case from a complex analytic point of view.
But the invariant which is given is not easy to describe in terms of flat geometry. The next theorem gives an interpretation in terms of flat geometry.
Let $\gamma$ be a simple closed curve parametrized by the arc length on a translation surface that avoids the singularities. Then $t\to \gamma'(t)$ defines a map from $\mathbb{S}^1$ to $\mathbb{S}^1$. We denote by $Ind(\gamma)$ the index of this map.
\begin{definition} Let $S=(X,\omega)\in \mathcal{H}(n_1,\dots ,n_r,-p_1,\dots -p_s)$ be a genus one translation surface with poles. Let $(a,b)$ be a symplectic basis of the homology the underlying compact Riemann surface $\overline{X}$ and $\gamma_a,\gamma_b$ be arc-length representatives of $a,b$, with no self intersections, and that avoid the zeroes and poles of $\omega$. We define the \emph{rotation number} of $S$ to by: $$rot(S)=\gcd(Ind(\gamma_a),Ind(\gamma_b),n_1,\dots n_r,p_1,\dots ,p_s)$$ \end{definition}
\begin{theorem} Let $\mathcal{H}=\mathcal{H}(n_1,\dots ,n_r,-p_1,\dots -p_s)$ be a stratum of genus 1 meromorphic differentials. The rotation number is an invariant of connected components of $\mathcal{H}$.
Any positive integer $d$ which divides $\gcd(n_1,\dots ,n_r,p_1,\dots p_s)$ is realised by a unique connected component of $\mathcal{H}$, except for the case $\mathcal{H}=\mathcal{H}(n,-n)$ where $d=n$ doesn't occur. \end{theorem}
\begin{proof} Let $(a,b)$ be a symplectic basis of $H_1(\overline{X},\mathbb{Z})$. Let $\gamma_a, \gamma_a'$ be representatives of $a$ that are simple closed curves and don't contain a singularity. Since $\overline{X}$ is a torus, $\gamma_a$ and $\gamma_a'$ are homotopic as curves defined on $\overline{X}$. The index of $\gamma_a$ doesn't change while we deform $\gamma_a$ without crossing a pole or a zero. It is easy to see that when crossing a singularity of order $k\in \mathbb{Z}$, the index is changed by adding $\pm k$. Hence the rotation number only depend on the homology class of $a$ and $b$.
If $\gamma_a$ and $\gamma_b$ intersects in one point, then there is a standard way to construct a simple closed curve representing $a\pm b$. Its index is $Ind(\gamma_a)\pm Ind(\gamma_b)$, and we obtain representatives of the symplectic basis $(a\pm b,b)$ (or $(a,a\pm b)$). The rotation number doesn't change by this operation. With this procedure, we can obtain any other symplectic basis of $\overline{X}$.
Hence the rotation number is well defined for a given element of $\mathcal{H}$. Also, it is invariant by deforming $(\overline{X},\omega)$ inside the ambiant stratum, since by continuous deformation, we can keep track of a pair of representatives of a basis, and the indices are constant under continuous deformations.
To prove the last part of the theorem, we remark that a surface in $\mathcal{H}(n,-p_1,\dots ,-p_s)$ obtained from $\mathcal{H}(n-2,-p_1,\dots ,-p_s)$ by bubbling a handle with parameter $k\in \{1,\dots ,n-1\}$ has a rotation number equal to $\gcd(k,p_1,\dots ,p_s)$ by a direct computation. Since $k<n$, we have $\gcd(k,p_1,\dots ,p_s)<n$ so $n$ is never a rotation number. Now we break up the singularity of order $n$ to get $r$ singularities of order $n_1,\dots ,n_r$. Since this doesn't change the metric outside a small neighborhood of the singularity of order $n$, we obtain a rotation number equal to $\gcd(k,n_1,\dots ,n_r,p_1,\dots ,p_s)$.
The previous construction gives at least as many connected component as the number given by Theorem~\ref{th:g1:complex}. So, we see each rotation number is realized by a unique component, and that this component is realized by the bubbling a handle construction. \end{proof}
Note that the last two paragraphs of the proof of the last theorem gives the following description of the connected components of the minimal strata in genus one.
\begin{proposition} \label{g1:cyl} Let $\mathcal{H}=\mathcal{H}(n,-p_1,\dots ,-p_s)$ be a minimal stratum of genus one meromorphic differentials. Any connected component of $\mathcal{H}$ is of the form $\mathcal{H}_0 \oplus k$, for some $1\leq k \leq n-1 $, where $\mathcal{H}_0$ is the connected stratum $\mathcal{H}(n-2,-p_1,\dots ,-p_s)$.
Also, for $1\leq k_1,k_2 \leq n-1$ we have: $$\mathcal{H}_0\oplus k_1=\mathcal{H}_0\oplus k_2$$ if and only if $\gcd(k_1,p_1,\dots ,p_s)=\gcd(k_2,p_1,\dots ,p_s)$. \end{proposition}
\begin{remark} It is shown in the appendix that there are some translation surface with pole that do not contain any closed geodesic. \end{remark}
\section{Spin structure and hyperelliptic components}\label{invariants} Recall that in the classification of the connected components of strata of the moduli space of Abelian differentials \cite{KoZo}, the connected components are distinghished by two invariants. \begin{itemize} \item ``Hyperelliptic components'': there are some connected components whose corresponding translation surfaces all have an extra symmetry. \item ``The parity of the spin structure'', which is a complex invariant that can be expressed in terms of the flat geometry by a simple formula. \end{itemize} \subsection{Hyperelliptic components} \begin{definition} A translation surface with poles $S$ is said to be \emph{hyperelliptic} if there exists an isometric involution $\tau:S\to S$ such that $S/\tau$ is a sphere. Equivalently, the underlying Riemann surface $\overline{X}$ is hyperelliptic and the hyperelliptic involution $\tau$ satisfies $\tau^* \omega=-\omega$. \end{definition} \begin{remark} In the case of Abelian differentials, if the underlying Riemann surface is hyperelliptic, then the translation surface is hyperelliptic since there are no nonzero holomorphic one forms on the sphere. In our case, similarly to the case of quadratic differentials, the underlying Riemann surface might be hyperelliptic, while the corresponding translation surface is not. \end{remark}
\begin{proposition}\label{list:hyp} Let $n,p$ be positive integers with $n\geq p$. The following strata admit a connected component that consists only of hyperelliptic translation surfaces. \begin{itemize} \item $\mathcal{H}(2n,-2p)$ \item $\mathcal{H}(2n,-p,-p)$ \item $\mathcal{H}(n,n,-2p)$ \item $\mathcal{H}(n,n,-p,-p)$ \end{itemize} Furthermore, any stratum that contains an open set of flat surfaces with a nontrivial isometric involution is in the previous list for some $n\geq p\geq 1$.
\end{proposition}
\begin{proof} Let $\mathcal{H}$ be a stratum and $\mathring{\mathcal{H}}^{hyp}\subset \mathcal{H}$ the interior of the set of elements of $\mathcal{H}$ that admit a nontrivial isometric involution.
Given a combinatoria datum $\sigma=(\mathbf{n}^+,\mathbf{n}^-,\pi_t,\pi_b,\mathbf{d})$ that defines an infinite zippered rectangle construction, we denote by $\mathcal{C}_\sigma$ the set of flat surfaces that are obtained by this construction with parameter $\sigma$, up to a rotation. Clearly, $\mathcal{C}_\sigma$ is open and connected.
We claim that for each $\sigma$, the intersection between $\mathcal{C}_\sigma$ and $\mathring{\mathcal{H}}^{hyp}$ is either $\mathcal{C}_\sigma$ or empty. Indeed, choose a generic parameter $\zeta$ for the infinite zippered rectangle construction, such that the corresponding surface $S(\sigma,\zeta)$ is in $\mathring{\mathcal{H}}^{hyp}$. Let $D^+(z_1,\dots ,z_k)\subset S(\sigma,\zeta)$ be a half-plane of the construction. Then, $\zeta$ being generic, an isometric involution $\tau$ will necessarily send the segment corresponding to $z_i$ to itself. Hence if $\tau$ is not the identity, it is easy to see that the set $D^+(z_1,\dots ,z_k)$ will be sent to $D^-(z_k,z_{k-1},\dots , z_1)$, and therefore, we can define a similar involution for \emph{any} value of $z_1,\dots ,z_k$. Since this argument is valid for any $D^{\pm}$ and $C^{\pm}$ components, we see that all flat surfaces obtained by the infinite zippered rectangle construction with combinatorial datum $\sigma$ have a nontrivial isometric involution. This proves the claim.
Now we remark that, by Proposition~\ref{no:vertical}, $\mathcal{H}=\cup_\sigma \mathcal{C}_\sigma$, where the union is taken on all $\sigma$ that corresponds to $\mathcal{H}$. The previous claim implies that $\mathring{\mathcal{H}}^{hyp}$ and its complement in $\mathcal{H}$ are both unions of some $\mathcal{C}_\sigma$, so if $\mathring{\mathcal{H}}^{hyp}$ is nonempty, it is a connected component of $\mathcal{H}$.
Now we check that if $\mathring{\mathcal{H}}^{hyp}$ is not empty, then the stratum $\mathcal{H}$ is in the given list, \emph{i.e.} there is either one even degree zero (resp. pole) or two zeroes (res. poles) of equal degree. Let $\zeta_1,\dots ,\zeta_n$ be the continuous data in the infinite zippered rectangle construction for an element $S$ in $\mathring{\mathcal{H}}^{hyp}$. The above condition implies that for each $\zeta_i$, the middle of the corresponding segment in the surface is a fixed point for the involution $\tau$. So, there are at least $n$ fixed points. Let $r$ be the number of conical singularities, let $s$ be the number of poles and let $g'$ be the genus of $S/\tau$. We must have $\#(Fix(\tau))=2g+2-4g'$, and $2g+r+s-2=n\leq \#(Fix(\tau))$ (see Lemma~\ref{lem:dim}). Since $r,s>1$, this implies $g'=0$, so $S$ is hyperelliptic, and $\#(Fix(\tau))-n=4-r-s$. The fixed points of $\tau$ in $\overline{X}$ that do not correspond to the middle of a $z_i$ segment are necessarily either conical singularities or pole.
The above combinatorial condition on the infinite zippered rectangle construction implies that $S$ has either two equal degree poles that are interchanged by $\tau$ or one pole of even degree that is preserved by $\tau$. So the condition $\#(Fix(\tau))-n=4-r-s$ implies that either there is one conical singularity which is fixed by $\tau$, or there are two singularities $P_1, P_2$ that are not fixed by $\tau$. By a similar argument as in the proof of Proposition~\ref{adj:minimal}, $P_1,P_2$ are the endpoints of a saddle connection corresponding to a parameter $\zeta_i$, so they are interchanged by $\tau$, hence they are of the same degree. Therefore, the stratum is necessarily one of the given list.
The last step of the proof is to check that for the strata given in the statement, $\mathring{\mathcal{H}}^{hyp}$ is nonempty. This is an elementary check by using the infinite zippered rectangle construction that satisfies the previous condition.
\end{proof}
\subsection{The parity of the spin structure for translation surfaces with even singularities}\label{subsec:spin} \subsubsection{Spin structures on a surface} There are two equivalent definitions of spin structure for a compact Riemann surface $X$ commonly used.
The first one is topological: let $P$ be the $\mathbb{S}^1$ bundle of directions of nonzero tangent vectors to $X$. A \emph{spin structure} on $X$ is a two-to-one covering $Q\to P$, whose restriction to a $\mathbb{S}^1$ fiber is the usual double covering $\mathbb{S}^1\to \mathbb{S}^1$. It is equivalent to a morphism $\xi:H_1(P,\mathbb{Z}/2\mathbb{Z})\to \mathbb{Z}/2\mathbb{Z}$ such that the image of the cycle $z$ corresponding to a fiber is one. Indeed, in this case the monodromy $\pi_1(P)\to \mathbb{Z}/2\mathbb{Z}$ factors to a map $\xi: H_1(P, \mathbb{Z}/2\mathbb{Z})\to \mathbb{Z}/2\mathbb{Z}$.
The second equivalent definition comes from algebraic geometry (see \cite{Ati}). A \emph{theta characteristic}, is a solution of the equation $2D=K$ in the divisor class group, where $K$ is the canonical divisor. Equivalently, it is a complex line bundle $L$ such that $L\otimes L\sim \Omega_1(X)$. For such $L$, Atiyah and Munford showed independently \cite{Ati,Mum} that the dimension modulo 2 of the vector space of holomorphic sections of $L$ is invariant by deformation of the complex structure. This is the \emph{parity of the spin structure}.
In \cite{Johnson}, Johnson provides a topological way to compute this invariant. He first constructs a lift $C\mapsto \widetilde{C}$ from $H_1(X,\mathbb{Z}/2\mathbb{Z})$ to $H_1(P,\mathbb{Z}/2\mathbb{Z})$. We refer \cite{Johnson} for details on the construction of the lift. In our case, it is enough to observe that when $C=[\gamma]$ is the class of a simple closed curve, the lift of $C$ is $\widetilde{C}=[\ha{\gamma}]+z$ where $\ha{\gamma}$ is the natural lift obtained by framing $\gamma$ with its speed vector $\gamma'$, and $z$ is the class of a $\mathbb{S}^1$ fiber.
The composition of this lift with the map $\xi$ gives a quadratic form $\Omega:H_1(X,\mathbb{Z}/2\mathbb{Z}) \to \mathbb{Z}/2\mathbb{Z}$, $\Omega(C):=\xi(\widetilde{C})$. Johnson then shows that the parity of the spin structure is equal to the Arf invariant of $\Omega$, \emph{i.e.} for a symplectic basis $(a_1,b_1),\dots ,(a_g,b_g)$ of $H_1(X,\mathbb{Z}/2\mathbb{Z})$, the parity of the spin structure is $$\sum_{i=1}^g \Omega(a_i)\Omega(b_i).$$
\subsubsection{The parity of the spin structure for translation surfaces with even singularities}
A translation surface $(X,\omega)$ with even poles and zeroes naturally gives a spin structure on $X$ in the following way: let $(\omega)=\sum_i 2n_i N_i-\sum_j 2p_jP_j $ the divisor associated to $\omega$. Then $D=\sum n_i N_i-\sum_j P_j$ satisfies $2D=K$. From the results of Atiyah and Munford in \cite{Ati, Mum}, it follows that the parity of the spin structure is a locally constant function on the strata where it is defined. Hence, it is an invariant of connected components, for strata with even poles and zeroes.
The parity of the spin structure can be easily computed by using Johnson's construction. Following \cite{KoZo}, it is easy to see that the corresponding map $\Omega$ satisfies, for $\gamma$ a simple closed curve, $\Omega([\gamma])=ind(\gamma)+1$. Hence, the parity of the spin structure for a translation surface (with poles) is: $$\sum_{i=1}^g (ind(a_i)+1)(ind(b_i)+1).$$
\subsection{The parity of the spin structure for translation surfaces with only two simple poles}\label{spin:poles:simples}
Let $S=(X,\omega)\in \mathcal{H}(2n_1,\dots ,2n_r,-1,-1)$ be a translation surface with zeroes of even order and a pair of simple poles (and no other poles). Since there are odd degree singularities, $\omega$ does not define a spin structure on $X$. However, one can still define a topological invariant, that will be the parity of the spin structure on another surface $S'$.
Recall that a neigborhood a simple pole is an infinite cylinder. Choose a pair of waist curves $\gamma_1,\gamma_2$ on each cylinder associated to the two simple poles. Since there are no other poles than the pair of simple poles, the two have opposite residues by Stokes' theorem, hence $\gamma_1,\gamma_2$ are isometric. Now we cut the surface $S$ along $\gamma_1$ and $\gamma_2$. We obtain a compact translation surface with two geodesic boundary components. The condition on the residues implies that gluing together these boundary components by a translation gives a translation surface $S'$, where the pair of infinite cylinders in $S$ corresponds to a finite cylinder $C\subset S'$. The surface $S'$ belongs to the stratum $\mathcal{H}(2n_1,\dots ,2n_r)$ where the spin structure was defined by Kontsevich and Zorich. Note that other choices for $\gamma_1,\gamma_2$, and for the gluing operation only change the length and twist of $C$, hence gives the same connected component of $\mathcal{H}(2n_1,\dots ,2n_r)$. We will call \emph{the parity of the spin structure of S} the parity of the spin structure of the corresponding translation surface $S'$.
\begin{remark} Note that one can also define the parity of the spin structure of $S=(X,\omega)$ in a more algebro-geometric way: we consider the stable curve $\overline{X}$ obtained by gluing together the two poles.
In \cite{Cornalba}, Cornalba extends to a large class of stable curves (including this case) the notion of spin structure, and shows that the parity of the spin structure is invariant by deformations. The one form $\omega$ on $X$ can then be used to define on $\overline{X}$ a spin structure. \end{remark}
\section{Higher genus case: minimal stratum} \label{sec:minimal}
Recall that a minimal stratum correspond to the case where there is only one conical singularity (and possibly several poles). As in the papers of Kontsevich-Zorich \cite{KoZo} and Lanneau \cite{La:cc}, we first describe the connected components of minimal strata. The idea is similar: show that each such strata is obtained by bubbling $g$ cylinders and compute the connected components in this case.
The first step is to find a surface obtained by bubbling a handle. In \cite{KoZo} and in \cite{La:cc} is used a rather combinatorial argument. A similar approach is possible in our case by using the infinite zippered rectangle construction, but this is quite technical. Another possibility is to reduce the problem to the genus one case for which it was proven in Section~\ref{genus1} that any minimal stratum contains a surface obtained by bubbling a handle.
\begin{proposition} \label{exists:cyl} Let $\mathcal{C}$ be a connected component of the stratum $\mathcal{H}(n,-p1,\ldots,-p_s)$. We assume that the genus $g$ is nonzero. Then, there exists a flat surface in $\mathcal{C}$ which is obtained by bubbling a handle from a genus $g-1$ flat surface. \end{proposition}
\begin{proof} We start from a surface in $\mathcal{C}$ obtained by the infinite zippered rectangle construction. It is defined by a combinatorial data and a continous parameter $\zeta\in\mathbb{C}^n$, with $n=2g+s-1$.
Each $\zeta_i$ defines a closed geodesic path $\gamma_i$ joining the conical singularity to itself. The intersection number between any two such paths is $0$ or $\pm 1$. The genus is higher than zero and $\{\gamma_1,\ldots,\gamma_n\}$ generates the whole homology space $H_1(S,\mathbb{Z})$ since the complement is a union of punctured disks. Hence, there is a pair $\gamma_i,\gamma_j$ whose intersection number is one.
Now we shrink $\zeta_i,\zeta_j$ until they are very small compared to all the other parameters. Then, we observe that a neighborhood of $\gamma_i,\gamma_j$ is isometric to the complement of a neighborhood of a pole for a surface in $\mathcal{H}(n,-n)$. Then, deforming suitably the surface, using Proposition~\ref{g1:cyl}, one obtains the desired result. \end{proof}
We recall the notation introduced in Section~\ref{bubbling}. Let $\mathcal{C}$ is a connected component of a minimal stratum $\mathcal{H}(n,-p_1,\dots ,-p_s)$. Let $s\in \{1,\dots ,n+1\}$. The set $\mathcal{C}\oplus s$ is the connected component of the stratum $\mathcal{H}(n+2,-p_1,\dots ,-p_s)$ obtained by bubbling a handle after breaking the singularity of order $n$ into two singularities of order $(s-1)$ and $(n+1-s)$.
The proposition that follows uses roughly the same arguments as in \cite{KoZo} and \cite{La:cc}. The only difference is the case when $n$ is odd, which does not occur for Abelian or quadratic differentials.
\begin{proposition} \label{min:strat:upper:bound} Let $\mathcal{H}(n,-p_1,\dots ,-p_s)$ be a stratum of meromorphic differentials genus $g\geq 2$ surfaces, and denote by $\mathcal{C}_0$ the unique component of $\mathcal{H}(n-2g,-p_1,\dots ,-p_s)$. The following holds: \begin{itemize} \item If $n$ is odd, the stratum $\mathcal{H}(n,-p_1,\dots ,-p_s)$ is connected. \item If $n$ is even, the stratum $\mathcal{H}(n,-p_1,\dots ,-p_s)$ has at most three connected components which are in the following list: \begin{itemize} \item $\mathcal{C}_0\oplus (\frac{n-2g}{2}+1) \oplus (\frac{n-2g}{2}+2)\oplus\dots \oplus (\frac{n-2g}{2}+g)$ \item $\mathcal{C}_0\oplus 1\oplus\dots \oplus 1\oplus 1$ \item $\mathcal{C}_0\oplus 1\oplus\dots \oplus 1\oplus 2$ \end{itemize}
\end{itemize} \end{proposition} \begin{proof} Let $\mathcal{C}$ be a connected component of $\mathcal{H}(n,-p_1,\dots ,-p_s)$. By proposition \ref{exists:cyl}, there exist integers $s_1,\dots ,s_g$, such that: $$\mathcal{C}=\mathcal{C}_0\oplus s_1\oplus\dots \oplus s_g$$ and for each $i\in \{1,\dots ,g\}$, $1\leq s_i\leq n-2g-2+2i+1$, since at Step~$i$, the handle corresponding to $s_i$ is bubbled on a zero of degree $n-2g+2(i-1)$.
We assume for simplicity that $g=2$, and $(s_1,s_2)\neq ( \frac{n-2g}{2}+1, \frac{n-2g}{2}+2)$. Using operations $(1)$ and $(3)$ of Lemma~\ref{lemme:lanneau}, one can assume that $1\leq s_1\leq s_2\leq s_1+1$. Then, if $1\neq s_1$, using operations $(1)$, $(2)$, $(3)$ and $(1)$ (in this order), we have $\mathcal{C}_0\oplus s_1\oplus s_2=\mathcal{C}_0\oplus (s_1-1)\oplus (s_2-1)$. Repeating the same sequence of operations, we see that $\mathcal{C}$ is one of the following: \begin{itemize} \item $\mathcal{C}_0\oplus (\frac{n-2g}{2}+1) \oplus (\frac{n-2g}{2}+2)$ \item $\mathcal{C}_0\oplus 1\oplus 1$ \item $\mathcal{C}_0\oplus 1\oplus 2$ \end{itemize}
If $n$ is odd, then the first case doesn't appear. By operation $(4)$ of Lemma~\ref{lemme:lanneau}, we have $$\mathcal{C}_0\oplus s_1\oplus s_2=\mathcal{C}_0\oplus s_1\oplus ((n-2g+2)+2-s_2)$$ so we can assume that $s_1$ and $s_2$ are of the same parity. Then, using the previous argument, we have: $$\mathcal{C}=\mathcal{C}_0\oplus 1\oplus 1$$ The case $g>2$ easily follows. \end{proof}
The above proposition uses purely local constructions in a neighborhood of a singularity. The next proposition explains why the existence of suitable poles (at infinity) will ``kill'' some components.
\begin{proposition} \label{min:strat:odd:poles:or:non:hyp} Let $\mathcal{H}(n,-p_1,\dots ,-p_s)$ be a stratum of meromorphic differentials on surfaces of genus $g\geq 2$ with $n$ even and $s\geq 2$, and denote by $\mathcal{C}_0$ the unique component of $\mathcal{H}(n-2g,-p_1,\dots ,-p_s)$. The following holds: \begin{enumerate} \item If there is a odd degree pole and $\sum_i p_i>2$, then: $$\mathcal{C}_0\oplus 1\oplus\dots \oplus 1=\mathcal{C}_0\oplus 1\oplus\dots \oplus 1 \oplus 2 $$ \item If $s> 2$ or $p_1\neq p_2$, then: $$\mathcal{C}_0\oplus \left(\frac{n-2g}{2}+1\right) \oplus \dots \oplus \left(\frac{n-2g}{2}+g\right) = \mathcal{C}_0 \oplus 1\oplus\dots \oplus 1\oplus s$$ for some $s\in \{1,2\}$. \end{enumerate} \end{proposition}
\begin{proof}
Case (1).\\ Note that $s\geq 2$ implies that we necessarily have $\sum_{i} p_i\geq 2$.
From Proposition~\ref{g1:cyl}, $\mathcal{C}_0\oplus 2=\mathcal{C} _0\oplus k$ if and only if $\gcd(k,p_1,\dots ,p_s)=\gcd(2,p_1,\dots ,p_s)$. So, if there is an odd degree pole, $\gcd(2,p_1,\dots ,p_s)=1=\gcd(1,p_1,\dots ,p_s)$, hence $$(\mathcal{C}_0\oplus 1)\oplus 1\dots \oplus 1=(\mathcal{C}_0\oplus 2)\oplus 1\dots \oplus 1= \mathcal{C}_0\oplus 1\dots \oplus 1 \oplus 2,$$ which concludes the proof. Note that $\mathcal{C}_0\oplus 2$ is well defined because $\sum_{i}p_i>2$.
\noindent Case (2).\\ As before, we use the classification in genus one.
Since $n-2g-\sum_{i} p_i=-2$, we have $\frac{n-2g}{2}+1=\frac{\sum_i p_i}{2}$. If $s> 2$ or $p_1\neq p_2$, then there exists $i\in \{1,\dots ,s\}$ such that $\frac{n-2g}{2}+1> p_i$, so $\gcd(\frac{n-2g}{2}+1,p_1,\dots ,p_s)< \frac{n-2g}{2}+1$, hence there exists $k<\frac{n-2g}{2}+1$ such that $\mathcal{C}_0\oplus (\frac{n-2g}{2}+1)=\mathcal{C}_0\oplus k$. So we have $$\mathcal{C}_0\oplus (\frac{n-2g}{2}+1)\oplus (\frac{n-2g}{2}+2)\oplus \dots = \mathcal{C}_0 \oplus k \oplus (\frac{n-2g}{2}+2) \oplus \dots $$ Then, as in the proof of Proposition~\ref{min:strat:upper:bound}, $$\mathcal{C}_0 \oplus k \oplus (\frac{n-2g}{2}+2) \dots \oplus (\frac{n-2g}{2}+g) = \mathcal{C}_0 \oplus 1\oplus \dots \oplus 1\oplus s $$ for some $s\in \{1,2\}$.
\end{proof}
Putting together the last two propositions and the invariants, we have the following theorem.
\begin{theorem}\label{th:str:min} Let $\mathcal{H}=\mathcal{H}(n,-p_1,\dots ,-p_s)$ be a minimal stratum of meromorphic differentials on genus $g\geq 2$ surfaces. We have: \begin{enumerate} \item If $n$ is even and $s=1$, then $\mathcal{H}$ has two connected components if $g=2$ and $p_s=2$, three otherwise. \item If $\mathcal{H}=\mathcal{H}(n,-p,-p)$, with $p$ even, then $\mathcal{H}$ has three connected components. \item If $\mathcal{H}=\mathcal{H}(n,-1,-1)$, then $\mathcal{H}$ has three connected components for $g>2$, two otherwise. \item If $\mathcal{H}=\mathcal{H}(n,-p,-p)$, with $p\neq 1$ odd, then $\mathcal{H}$ has two connected components. \item If all poles are of even degree and we are not in the previous case, then $\mathcal{H}$ has two connected components. \item In the remaining cases, $\mathcal{H}$ is connected. \end{enumerate}
\end{theorem}
\begin{proof} From Proposition~\ref{min:strat:upper:bound}, when $n$ is odd, which is part of Case~(6), $\mathcal{H}$ is connected. So we can assume that $n$ is even. Let $\mathcal{C}$ be a connected component of $\mathcal{H}$. Let $\mathcal{C}_0$ be the (connected) genus 0 stratum $\mathcal{H}(n-2g,-p_1,\dots ,-p_s)$. From Proposition~\ref{min:strat:upper:bound}, we have one of the three following possibilities. \begin{enumerate} \item[a)] $\mathcal{C}=\mathcal{C}_0\oplus (\frac{n-2g}{2}+1) \oplus (\frac{n-2g}{2}+2)\oplus\dots \oplus \frac{n}{2} $ \item[b)] $\mathcal{C}=\mathcal{C}_0\oplus 1 \oplus \dots \oplus 1 \oplus 1 $ \item[c)] $\mathcal{C}=\mathcal{C}_0\oplus 1 \oplus \dots \oplus 1\oplus 2$ \end{enumerate}
When $\mathcal{H}=\mathcal{H}(n,-p)$ or $\mathcal{H}=\mathcal{H}(n,-p,-p)$, it is easy to see that case $a)$ corresponds to a hyperelliptic connected component, while case $b)$ does not, and neither $c)$ (except for the case $n-2g=0$ and $g=2$, where $a)$ and $c)$ are the same).
When all degree of zeroes (and poles) are even, then Lemma~11 in \cite{KoZo} shows that cases b) and c) correspond to different spin structures, so are a different connected components. This is also true for $\mathcal{H}(n,-1,-1)$ by Section~\ref{spin:poles:simples}.
The arguments of the two previous paragraphs proves the result for Cases (1), (2) and (3). Remark that $(n-2g)-\sum_i p_i=-2$.
For Case $(4)$, Proposition~\ref{min:strat:odd:poles:or:non:hyp} shows that there are at most two connected components. Since $n-2g=2p-2>0$, Case~$a)$ corresponds to a hyperelliptic component while $b)$and $c)$ do not correspond to a hyperelliptic component. So there are at least two components. Since there are odd degree poles, $b)$ and $c)$ correspond to the same component by Proposition~\ref{min:strat:odd:poles:or:non:hyp}. So there are two components.
For Case $(5)$, Proposition~\ref{min:strat:odd:poles:or:non:hyp} shows that $a)$ is in the same connected component as $b)$ or $c)$, while Lemma~11 in \cite{KoZo} shows that $b)$ and $c)$ have different spin structures.
For Case $(6)$, with $n$ is even: this corresponds to having at least one odd pole, and either at least three poles or two poles of different degree. Then a direct application of Proposition~\ref{min:strat:odd:poles:or:non:hyp} shows that $a)$, $b)$ and $c)$ are the same connected component.
This concludes the proof.
\end{proof}
\section{Higher genus case: nonminimal strata} \label{sec:nonminimal} The remaining part of the paper uses similar arguments as in Sections~5.2--5.4 in \cite{KoZo}. We quickly recall the three main steps. \begin{itemize} \item Each stratum is adjacent to a minimal stratum, and we can bound the number of connected components of a stratum by the number of connected components of the corresponding minimal one. \item We construct paths in suitable strata with two conical singularities that join the different connected components of the minimal stratum. \item We deduce from the previous arguments upper bounds on the number of connected component of a stratum, lower bounds are given by the topological invariants. \end{itemize}
The following proposition is analogous to Corollary~4 in \cite{KoZo}. It is proven there by constructing surfaces with a one cylinder decomposition. Such surfaces never exist in our case, we use the infinite zippered rectangle construction instead.
\begin{proposition}\label{adj:minimal} Any connected component of a stratum of meromorphic differentials is adjacent to the minimal stratum obtained by collapsing together all the zeroes. \end{proposition}
\begin{proof} Let $S$ be in a stratum $\mathcal{H}$ of meromorphic differentials. We prove the result by induction on the number of conical singularities of $S$. We can assume that $S$ is obained by the previous construction. By connectivity of $S$, there is a $D^\pm$ component or a $C^\pm$ component that contains two different conical singularities on its boundary, hence, there is a parameter $\zeta_i$ whose corresponding segment on that component joins two different conical singularities. The segment is on the boundary of two components. Assume for instance, that it is a $D^+$ and a $C^-$ component. Now we just need to check that the surface obtained by shrinking $\zeta_i$ to zero is nondegenerate. Hence it will correspond to an element in a stratum with one conical singularity less. The set $D'^+$ obtained by shrinking $\zeta_i$ to zero from $D^+$ is still a domain as defined in Section~\ref{basic:domains}. The set $C'^-$ obtained by shrinking $\zeta_i$ to zero from $C^-$ is also a domain as defined in Section~\ref{basic:domains} except if we have $C^-=C^-(\zeta_i)$. But in this case, since the two vertical lines of $C^-$ are identified together, the two endpoints of the segment defined by $\zeta_i$ are necessarily the same singularity, contradicting the hypothesis.
So, in any case, we obtain a surface $S'$ with fewer conical singularities. \end{proof}
The following proposition is analogous to Corollary~2 in \cite{KoZo}, and is the first step of the proof described in the beginning of this section. The proof of Kontsevich and Zorich uses a deformation theory argument. We propose a proof that uses only flat geometry.
\begin{proposition} \label{prop:upper:bound} The number of connected component of a stratum is smaller than or equal to the number of connected component of the corresponding minimal stratum. \end{proposition}
\begin{proof} From the previous proposition, any connected component of a stratum $\mathcal{H}=\mathcal{H}(k_1,\dots ,k_r,-p_1,\dots, -p_s)$ is adjacent to a minimal stratum $\mathcal{H}^{min}= \mathcal{H}(k_1+,\dots +k_r,-p_1,\dots ,-p_s)$ by collapsing zeroes. It is enough to show that if $(S_n),\ (S'_n)$ are two sequences in $\mathcal{H}$ that converge to a surface $S\in \mathcal{H}^{min}$, then $S_n$ and $S_n'$ are in the same connected component of $\mathcal{H}$ for $n$ large enough.
By definition of the topology on the moduli space of meromorphic differentials , for $n$ large enough,
the conical singularities of $S_n$ (resp. $S_n'$) are all in a small disk $D_n$ (resp. $D_n'$) which is embedded in the surface $S_n$ (resp. $S_n'$), and whose boundary is a covering of a metric circle.
Note that $D_n$ and $D_n'$ can be chosen arbitrarily small if $n$ is large enough, and we can assume that they have isometric boundaries. Replacing $D_n$ by a disk with a single singularity, one obtains a translation surface $\widetilde{S}_n$ which is very near to $S$, hence in the same connected component, and similarly for $S_n'$.
Now we want to deform $D_n$ to obtain $D_n'$. It is obtained in the following way: $D_n$ can be seen as a subset of a genus zero translation surface $S_1$ in the stratum $\mathcal{H}(k_1,\dots ,k_r,-2-\sum_{i=1}^r k_i)$: we just ``glue'' a neighborhood of a pole to the boundary of the disk $D_n$. We proceed similarly with the disk $D_n'$ and obtain a translation surface $S_2$ in the same stratum as $S_1$. This stratum is connected since the genus is zero. Hence we deduce a continuous transformation that deform $D_n$ to $D_n'$.
From the last two paragraphs, we easily deduce a continuous path from $S_n$ to $S_n'$, which proves the proposition. \end{proof}
The following proposition is the second step of the proof. It is the analogous of Proposition~5 and Proposition~6 in \cite{KoZo}. Our proof is also valid for the Abelian case, and gives an interesting alternative proof.
\begin{proposition}\label{join:cc} \begin{enumerate} \item Let $\mathcal{H}=\mathcal{H}(n,-p_1,\dots ,-p_s)$ be a genus $g\geq 2$ minimal stratum whose poles are all even or the pair $(-1,-1)$. For any $n_1,n_2$ odd such that $n_1+n_2=n$, there is a path $\gamma(t)\in \overline{\mathcal{H}(n_1,n_2,-p_1,\dots ,-p_s)}$ such that $\gamma(0),\gamma(1)\in \mathcal{H}$ and have different parities of spin structures. \item Let $\mathcal{H}=\mathcal{H}(n,-p_1,\dots ,-p_s)$ be a genus $g\geq 2$ minimal stratum that contains a hyperelliptic connected component. For any $n_1\neq n_2$ such that $n_1+n_2=n$, there is a path $\gamma(t)\in \overline{\mathcal{H}(n_1,n_2,-p_1,\dots ,-p_s)}$ such that $\gamma(0)$ is in a hyperelliptic component of $\mathcal{H}$ and $\gamma(1)$ is in a nonhyperelliptic component of $\mathcal{H}$. \end{enumerate} \end{proposition}
\begin{proof} \emph{Case (1)}\\ Let $\mathcal{C}_0=\mathcal{H}(n-2g,-p_1,\dots ,-p_s)$. The connected components of $\mathcal{H}$ given by $\mathcal{C}_0\oplus 1\dots \oplus 1 \oplus 1$ and $ \mathcal{C}_0\oplus 1\dots \oplus 1\oplus 2$ have different parities of spin structures. We can rewrite these components as $ \mathcal{C}\oplus 1$ and $\mathcal{C}\oplus 2$, where $\mathcal{C}= \mathcal{C}_0\oplus 1\dots \oplus 1$.
Fix $S_{g-1}\in \mathcal{C}$. For a surface $S_1\in \mathcal{H}(n,-n)$, one can get a surface $S$ in $\mathcal{H}(n,-p_1,\dots ,-p_s)$ by the following surgery: \begin{itemize} \item Cut $S_{g-1}$ along a small metric circle that turns around the singularity of degree $n-2$, and remove the disk bounded by this circle \item Cut $S_1$ along a large circle that turns around the pole of order $n$, and rescale $S_1$ such that this circle is isometric to the previous one. Remove the neighborhood of the pole of order $n$ bounded by this circle. \item Glue the two remaning surfaces along these circle, to obtain a surface $S\in \mathcal{H}(n,-p_1,\dots ,-p_s)$. \end{itemize} All choices in previous construction lead to the same connected component of $\mathcal{H}(n,-p_1,\dots ,-p_s)$, once $S_{g-1}, S_1$ are fixed. Similarly, we can do the same starting from a surface in $S_1\in\mathcal{H}(n_1,n_2,-n)$ and get a surface in $\mathcal{H}(n_1,n_2,-p_1,\dots ,-p_s)$.
Now we start from a surface $S_{1,1} \in \mathcal{H}(n,-n)$ obtained by bubbling a handle with angle $2\pi$, \emph{i.e.} $S_{1,1}\in \mathcal{H}(n-2,-n)\oplus 1$. The rotation number of this surface is $\gcd(1,n)=1$. Breaking up the singularity into two singularities of order $n_1,n_2$, the rotation number is still $1$. Similarly, start from $S_{1,2}\in \mathcal{H}(n-2,-n)\oplus 2$. Its rotation number is $\gcd(2,n)=2$. Breaking up the singularity into two singularities of order $n_1,n_2$, the rotation number becomes $\gcd(2,n_1,n_2)=1$ since $n_1,n_2$ are odd. Hence there is a path in $\overline{\mathcal{H}(n_1,n_2,-n)}$ that joins $S_{1,1}\in\mathcal{H}(n-2,-n)\oplus 1$ to $S_{1,2}\in \mathcal{H}(n-2,-n)\oplus 2$. From this path, we deduce a path in $\overline{\mathcal{H}(n_1,n_2,-p_1,\dots ,-p_s)}$ that joins $\mathcal{C}\oplus 1$ to $\mathcal{C}\oplus 2$. So Part~$(1)$ of the proposition is proven.
\noindent Case (2)\\ The proof is similar as the previous one: the hyperelliptic component of $\mathcal{H}(n,-p_1,\dots ,-p_s)$ is of the kind $\mathcal{C}\oplus \frac{n}{2}$, for some component $\mathcal{C}$. Any component of the kind $\mathcal{C}\oplus k$, with $k\neq \frac{n}{2}$ is nonhyperelliptic. As before, we use the case of genus one strata. A surface in $\mathcal{H}(n-2,-n)\oplus \frac{n}{2}$ is of rotation number $\gcd(\frac{n}{2},n)=\frac{n}{2}$. Breaking up the singularity of degree $n$ into two singularities of degree $n_1,n_2$, one obtains surface in $\mathcal{H}(n_1,n_2,-n)$ of rotation number $\gcd(\frac{n}{2},n_1,n_2)$. Since $n_1+n_2=n$ and $n_1\neq n_2$, this rotation number is not $\frac{n}{2}$, but some integer $k\in \{1,\dots ,\frac{n}{2}-1\}$. Hence there is a path in $\overline{\mathcal{H}(n_1,n_2,-n)}$ that joins $\mathcal{H}(n-2,-n)\oplus \frac{n}{2}$ to $\mathcal{H}(n-2,-n)\oplus k$. From this, we deduce the required path in $\overline{\mathcal{H}(n_1,n_2,-p_1,\dots ,-p_s)}$. \end{proof}
Now we have all the intermediary results to prove Theorem~\ref{MT2}.
\begin{proof}[Proof of Theorem~\ref{MT2}] Let $\mathcal{H}=\mathcal{H}(n_1,\dots ,n_r,-p_1,\dots ,-p_s)$ be a stratum of genus $g\geq 2$ surfaces. Denote by $\mathcal{H}_{min}$ the minimal stratum obtained by collapsing all zeroes. Recall that by Proposition~\ref{prop:upper:bound}, the number of connected components of $\mathcal{H}$ is smaller than, or equal to the number of connected components of $\mathcal{H}_{min}$.
If $\sum_i p_i$ is odd, then the minimal stratum is connected and therefore the stratum is connected. So we can assume that $\sum_i p_i$ is even.
Assume that $\sum p_i>2$ or $g>2$. From Theorem~\ref{th:str:min}, $\mathcal{H}_{min}$, hence $\mathcal{H}$ has at most three components.
We fix some vocabulary: we say that the set of degree of zeroes (resp. poles) is of \emph{hyperelliptic type} if this set is $\{n,n\}$ or $\{2n\}$ (resp. $\{-p,-p\}$ or $\{-2p\}$), \emph{i.e.} it is the set of degree of zeroes or poles of a hyperelliptic component. Note that the set of degree of poles are of hyperelliptic type if and only if the corresponding minimal stratum contains a hyperelliptic connected component. We will also say that the set of degree of poles is of \emph{even type} if the degrees of the zeroes are all even or if they are $\{-1,-1\}$. This means that the underlying minimal stratum has two nonhyperelliptic components distinghished by the parity of the spin structure.
\begin{itemize} \item If the stratum is $\mathcal{H}(n,n,-2p)$ or $\mathcal{H}(n,n,-p,-p)$. There is a hyperelliptic connected component. The corresponding minimal stratum $\mathcal{H}(2n,*)$ has one hyperelliptic component and at least one nonhypereliptic component. It is easy to see that breaking up the singularity of degree $2n$ into two singularities of degree $n$, from a nonhyperelliptic translation surface gives a surface in a nonhyperelliptic connected component. So, the stratum $\mathcal{H}(n,n,*)$ has one hyperelliptic connected component and at least one nonhyperelliptic connected component. \item If the set of degrees of poles and zeroes is of even type, we know from Theorem~\ref{th:str:min} that the minimal stratum has two nonhyperelliptic components (and possibly one hyperelliptic). Breaking up the singularity into even degree singularities preserves the spin structure, which therefore gives at least two nonhyperelliptic components in the stratum. \end{itemize} From the above description, we obtain lower bounds on the number of connected components. In particular, we see that if the degrees of zeroes and poles are both of hyperelliptic and even type, $\mathcal{H}$ as at least, so exactly, three connected components. Also, if the set of degrees of zeroes and poles is of hyperelliptic or even type, $\mathcal{H}$ has at least two connected components.
Now we give upper bounds. \begin{enumerate} \item Assume that the poles are of hyperelliptic and even type, \emph{i.e.} the minimal stratum has three connected components. Denote respectively by $\mathcal{C}^{hyp}, \mathcal{C}^{odd}$ and $\mathcal{C}^{even}$ the connected components of $\mathcal{H}$ that are adjacent respectively to the three connected components of $\mathcal{H}_{min}$, $\mathcal{H}_{min}^{hyp},\mathcal{H}_{min}^{odd}$ and $\mathcal{H}_{min}^{even}$. For any $j\in \{1,\dots ,r\}$, the stratum $\mathcal{H}(n_j,\sum_{i\neq j} n_i,-p_1,\dots ,-p_s)$ is adjacent to $\mathcal{H}_{min}$. \begin{itemize} \item If the zeroes are not of hyperelliptic type, we can choose, $n_i$ so that $n_i\neq \sum_{i\neq j} n_i$, and by Proposition~\ref{join:cc}
there is a path in $\mathcal{H}(n_j,\sum_{i\neq j} n_i,-p_1,\dots ,-p_s)$ joining the hyperelliptic component of $\mathcal{H}_{min}$ to a nonhyperelliptic connected component. Breaking up the singularity of order $\sum_{i\neq j}^r n_i$ along this path into singularities of order $(n_i)_{i\neq j}$, we obtain a path in $\mathcal{H}$ that joins a neighborhood of $\mathcal{H}_{min}^{hyp}$ to a neighborhood of a nonhyperelliptic component of $\mathcal{H}_{min}$. Hence, we necessarily have $\mathcal{C}^{hyp}=\mathcal{C}^{odd}$ or $\mathcal{C}^{hyp}=\mathcal{C}^{even}$.
\item
If the zeroes are not even, we conclude similarly that $\mathcal{C}^{odd}=\mathcal{C}^{even}$
\item Note that if the zeroes are neither of hyperelliptic type nor of even type, then $\mathcal{C}^{even}=\mathcal{C}^{odd}=\mathcal{C}^{hyp}$, so there is only one component for $\mathcal{H}$. \end{itemize}
\item Assume that the poles are of hyperelliptic type but not of even type. The minimal stratum has two connected components, so there are at most two connected components for $\mathcal{H}$. If the zeroes are of hyperelliptic type, we have already seen that there are two components.
Assume the zeroes are not of hyperelliptic type. Denote respectively by $\mathcal{C}^{hyp}, \mathcal{C}^{nonhyp}$ the connected components of $\mathcal{H}$ that are adjacent respectively to the hyperelliptic and the nonhyperelliptic component of $\mathcal{H}_{min}$. By the same argument as in $(1)$, using Proposition~\ref{join:cc} we have $\mathcal{C}^{hyp}=\mathcal{C}^{nonhyp}$, so $\mathcal{H}$ is connected.
\item Assume that the poles are of even type but not of hyperelliptic type. The minimal stratum has two connected components distinguished by the parity of the spin structure. So there are at most two components for $\mathcal{H}$. If the zeroes are of even type, there are exactly two connected component for $\mathcal{H}$, that are distinguished by the parity of the spin structure.
If the zeroes are not of even type, denote respectively by $\mathcal{C}^{odd}, \mathcal{C}^{even}$ the connected components of $\mathcal{H}$ that are adjacent respectively to the two components of $\mathcal{H}_{min}$. By the same argument as in $(1)$, using Proposition~\ref{join:cc} we have $\mathcal{C}^{odd}=\mathcal{C}^{even}$.
\item Assume that the poles are neither of hyperelliptic nor of even type, then the minimal stratum is connected, so $\mathcal{H}$ is connected. \end{enumerate}
It remains to prove the theorem when $g=2$ and $\sum_{i} p_i=2$. The minimal stratum has two connected components. In this case, it is equivalent to say that the zeroes are of hyperelliptic type or to say that they are of even type. If $\mathcal{H}=\mathcal{H}(2,2,*)$ or $\mathcal{H}(4,*)$, the stratum has at least two components, so exactly two. Otherwise, the stratum is adjacent to $\mathcal{H}(3,1,*)$, which connects $\mathcal{H}_{min}^{odd}$ to $\mathcal{H}_{min}^{even}$, hence $\mathcal{H}$ is connected. \end{proof}
\appendix \section{Negative results for meromorphic differentials} In this section, we quickly give some examples to show that many well known results for the dynamics on translation surfaces are false in the case of translation surfaces with poles. \subsection{Dynamics of the geodesic flow} On a standard translation surface, the geodesic flow is uniquely ergodic for almost any directions. From the result of Proposition~\ref{no:vertical}, for almost any direction on a translation surface with poles, all infinite orbits for the geodesic flow converge to a pole.
\subsection{Cylinders and closed geodesics} On a standard translation surface, always exists infinitely many closed geodesics (hence cylinders). For the case of translation surface with poles, one can consider the following example. Take the plane $\mathbb{C}$ and remove the inside of a square, and glue together by translation the corresponding opposite sides. One gets a surface in $\mathcal{H}(-2,2)$. It is easy to see that there are exactly two saddle connections joining the conical singularity to itself and no closed geodesic. A similar example in $\mathcal{H}(-2,1,1)$ obtained by removing a regular hexagon gives an example without a single saddle connection joining a conical singularity to itself.
\subsection{$SL_2(\mathbb{R})$ action}
The $SL_2(\mathbb{R})$ action on the strata of the moduli space of Abelian differentials is ergodic. It is not the case for the moduli space of meromorphic differentials if we consider the (infinite) volume form defined by the flat local coordinates. Indeed, consider the stratum $\mathcal{H}(-2,2)$, which is connected. Consider the set of surfaces obtained with the infinite zippered rectangle construction, by gluing together the set $D^+(z_1,z_2)$ and the set $D^-(z_2,z_1)$. It is easy to see that if $Im(z_2)<0<Im(z_1)$, there are no cylinders on the surface while if $Im(z_2)>0>Im(z_1)$, there is a cylinder on the surface. These two cases form two nointersecting open subsets of $\mathcal{H}(-2,2)$. Considering $SL_2(\mathbb{R})$ orbits, we obtain two disjoints $SL_2(\mathbb{R})$-invariants open subsets of a connected stratum.
\nocite*
\end{document}
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\begin{document}
\date{} \title{
Distributed Optimization Over Dependent Random Networks} \sloppy
\begin{abstract} We study the averaging-based distributed optimization solvers over random networks. We show a general result on the convergence of such schemes using weight-matrices that are row-stochastic almost surely and column-stochastic in expectation for a broad class of dependent weight-matrix sequences. In addition to implying many of the previously known results on this domain, our work shows the robustness of distributed optimization results to link-failure. Also, it provides a new tool for synthesizing distributed optimization algorithms. {To prove our main theorem, we establish new results on the rate of convergence analysis of averaging dynamics over (dependent) random networks. These secondary results, along with the required martingale-type results to establish them, might be of interest to a broader research endeavors in distributed computation over random networks. } \end{abstract} \section{Introduction}
Distributed optimization has received increasing attention in recent years due to its applications in
distributed control of robotic networks \cite{bullo2009distributed}, study of opinion dynamics in social networks \cite{hegselmann2002opinion}, distributed estimation and signal processing \cite{rabbat2004distributed,cattivelli2010diffusion,stankovic2011decentralized}, and power networks \cite{dominguez2011distributed,dominguez2012decentralized,cherukuri2015distributed}.
In distributed optimization, we are often interested in finding an optimizer of a decomposable function $F({z})=\sum_{i=1}^{n}f_{i}({z})$ such that $f_i(\cdot)$s are distributed through a network of $n$ agents, i.e., agent $i$ only knows $f_i(\cdot)$, and we are seeking to solve this problem without sharing the local objective function $f_i(z)$. Therefore, the goal is to find distributed dynamics over time-varying networks that, asymptotically, all the nodes agree on an optimizer of $F(\cdot)$.
The most well-know algorithm that achieves this is, what we refer to as, the averaging-based distributed optimization solver where each node maintains an estimate of an optimal point, and at each time step, each node computes the average of the estimates of its neighbors and performs (sub-)gradient descent on its local objective function \cite{nedic2009distributed}. However, in order for such an algorithm to converge, the corresponding weight matrices should be doubly stochastic. While making a row-stochastic or a column-stochastic matrix is easy, this is not the case for doubly stochastic matrices. Therefore, it is often assumed that such a doubly stochastic matrix sequence is given.
A solution to overcome this challenge is to establish more complicated distributed algorithms that effectively \textit{reconstruct} the average-state distributively. The first algorithm in this category was proposed in \cite{rabbat}, which is called subgradient-push (or push-sum), and later was extended for time-varying networks \cite{nedic2014distributed}. In this scheme, the weight matrices are assumed to be column-stochastic, and through the use of auxiliary state variables the approximate average state is reconstructed.
Another scheme in this category that works with row-stochastic matrices, but does not need the column-stochastic assumption, is proposed in \cite{mai2016distributed,xi2018linear}. However, to use this scheme, every node needs to be assigned and know its unique label. Assigning those labels distributively is also another challenge in this respect. In addition, both these schemes invoke division operation which results in theoretical challenges in establishing their stability in random networks \cite{rezaeinia2019push,rezaeinia2020distributed}.
{Another solution to address this challenge is to use gossip \cite{7165636} and broadcast gossip \cite{5585721} algorithms over random networks. The weight matrices of gossip algorithms are row-stochastic and in-expectation column-stochastic. This fact was generalized in \cite{7849184}, where it is proven that it is sufficient to have row-stochastic weight matrices, that are column-stochastic in-expectation. In all the above works on distributed optimization over random networks, all weight matrices are assumed to be independent and identically distributed (i.i.d.). In \cite{lobel2011distributed}, the broader class of random networks, which is Markovian networks was studied in distributed optimization; however, weight matrices were assumed to be doubly stochastic almost surely. Also, our work is closely related to the existing works on distributed averaging on random networks \cite{consensus1,consensus2,consensus3,touri2014endogenous}.
In this paper, we study distributed optimization over random networks, where the randomness is not only time-varying but also, possibly, dependent on the past. Under the standard assumptions on the local objective functions and step-size sequences for the gradient descent algorithm, we show that the averaging-based distributed optimization solver at each node converges to a global optimizer almost surely if the weight matrices are row-stochastic almost surely, column-stochastic {in-expectation}, and satisfy certain connectivity assumptions. }
The paper is organized as follows: we conclude this section by introducing mathematical notations that will be used subsequently. In Section \ref{sec:ProblemStatement}, we formulate the problem of interest and state the main result of this work and discuss some of its immediate consequences. To prove the main result, first we study the behavior of the distributed averaging dynamics over random networks in Section~\ref{sec:ConsensusWithoutPerturbation}. Then, in Section \ref{sec:ConsensusWithPerturbation}, we extent this analysis to the dynamics with arbitrary control inputs. Finally, the main result is proved in Section \ref{sec:Convergence}. We conclude this work in Section~\ref{Conclusion}.
\textbf{Notation and Basic Terminology:} The following notation will be used throughout the paper. We let $[n]\triangleq\{1,\ldots,n\}$. We denote the space of real numbers by $\mathbb{R}$ and natural (positive integer) numbers by $\mathbb{N}$. We denote the space of $n$-dimensional real-valued vectors by $\mathbb{R}^n$. In this paper, all vectors are assumed to be column vectors. The transpose of a vector $x\in \mathbb{R}^n$ is denoted by $x^T$. For a vector $x\in \mathbb{R}^n$, $x_i$ represents the $i$th coordinate of $x$. We denote the all-one vector in $\mathbb{R}^n$ by $e^n=[1,1,\ldots,1]^T$. We drop the superscript $n$ in $e^n$ whenever the dimension of the space is understandable from the context.
A non-negative matrix $A$ is a row-stochastic (column-stochastic) matrix if $Ae=e$ ($e^TA= e^T$).
Let $(\Omega,\mathcal{F},\text{Pr})$ be a probability space and let $\{W(t)\}$ be a chain of random
matrices, i.e., for all $t\geq 0$ and $i,j\in[n]$, $w_{ij}(t):\Omega\to\mathbb{R}$ is a Borel-measurable function.
For random vectors (variables) { ${\bf x}(1),\ldots,{\bf x}(t)$, we denote the sigma-algebra generated by these random variables by $\sigma({\bf x}(0),\ldots,{\bf x}(t))$.} We say that $\{\mathcal{F}(t)\}$ is a filtration for $(\Omega,\mathcal{F})$ if $\mathcal{F}(0)\subseteq \mathcal{F}(1)\subseteq \cdots \subseteq \mathcal{F}$. Further, we say that a random process $\{V(t)\}$ (of random variables, vectors, or matrices) is adapted to $\{\mathcal{F}(t)\}$ if $V(t)$ is measurable with respect to $\mathcal{F}(t)$.
Throughout this paper we mainly deal with directed graphs. A directed graph $\mathcal{G}=([n],\mathcal{E})$ (on $n$ vertices) is defined by a vertex set (identified by) $[n]$ and an edge set $\mathcal{E}\subset [n]\times [n]$. {A graph $\mathcal{G}=([n],\mathcal{E})$ has a spanning directed rooted tree if it has a vertex $r\in [n]$ as \textit{a} root such that there exists a (directed) path from $r$ to every other vertex in the graph.} For a matrix $A=[a_{ij}]_{n\times n}$, the associated directed graph with parameter $\gamma>0$ is the graph $\mathcal{G}^\gamma(A)=([n],\mathcal{E}^\gamma(A))$ with the edge set $\mathcal{E}^\gamma(A)=\{(j,i)\mid i,j\in [n],a_{ij}>\gamma \}$. Later, we fix the value $0<\gamma<1$ throughout the paper and hence, unless otherwise stated, for notational convenience, we use $\mathcal{G}(A)$ and $\mathcal{E}(A)$ instead of $\mathcal{G}^\gamma(A)$ and $\mathcal{E}^\gamma(A)$.
{ The function $f:\mathbb{R}^m\to\mathbb{R}$ is convex if for all $x,y\in\mathbb{R}^m$ and all $\theta\in[0,1]$, \begin{align*}
f(\theta x+(1-\theta)y)\leq \theta f(x)+(1-\theta)f(y). \end{align*} We say that $g\in\mathbb{R}^m$ is a subgradient of the function $f(\cdot)$ at $\hat{x}$ if for all $x\in\mathbb{R}^m$,
$f(x)-f(\hat{x})\geq \lrangle{g,x-\hat{x}},$
where $\lrangle{u_1,u_2}=u_1^Tu_2$ is the standard inner product in $\mathbb{R}^m$.}
The set of all subgradients of $f(\cdot)$ at $x$ is denoted by $\nabla f(x)$. For a convex function $f(\cdot)$, $\nabla f(x)$ is not empty for all $x\in\mathbb{R}^m$ (see e.g., Theorem 3.1.15 in \cite{nesterov2018lectures}). Finally, for convenience and due to the frequent use of $\ell_\infty$ norm in the study of averaging dynamics, we use $\|\cdot\|$ to denote the $\ell_\infty$ norm $\|x\|\triangleq \max_{i\in [m]}|x_i|$. \section{Problem Formulation and Main Result}\label{sec:ProblemStatement} In this section, we discuss the main problem and the main result of this work. The proof of the result is provided in the subsequent sections. \subsection{General Framework} Consider a communication network with $n$ nodes or agents such that node $i$ has the cost function $f_i:\mathbb{R}^m\to\mathbb{R}$. Let $F({z})\triangleq\sum_{i=1}^{n}f_{i}({z})$. The goal of this paper is to solve \begin{align}\label{eqn:mainProblem}
\arg\min_{{z}\in\mathbb{R}^m}F({z}) \end{align} distributively with the following assumption on the objective function. \begin{assumption}[Assumption on the Objective Function]\label{assump:AssumpOnFunction} We assume that: \begin{enumerate}[(a)]
\item All $f_i({z})$ are convex functions over $\mathbb{R}^m$.
\item The optimizer set $\mathcal{Z}\triangleq\arg\min_{{z}\in\mathbb{R}^m}F({z})$ is non-empty.
\item The subgradients $f_i({z})$s are uniformly upper bounded, i.e., for all $g\in \nabla f_i(z)$, $\|g\|\leq L_i$ for all $z\in \mathbb{R}^m$ and all $i\in[n]$. {We let $L\triangleq\sum_{i=1}^nL_i$.} \end{enumerate} \end{assumption}
In this paper, we are dealing with the dynamics of the $n$ agents estimates of an optimizer $z^*\in \mathcal{Z}$ which we denote them by ${\bf x}_i(t)$ for all $i\in[n]$. Therefore, we view ${\bf x}(t)$ as a \textbf{vector} of $n$ elements in the vector space $\mathbb{R}^m$.
A distributed solution of \eqref{eqn:mainProblem} was first proposed in \cite{nedic2009distributed} using the following deterministic dynamics \begin{align*}
{{\bf x}}_i(t+1)&=\sum_{j=1}^n w_{ij}(t+1){{\bf x}}_j(t)-\alpha(t){{\bf g}}_i(t) \end{align*} for $t\geq t_0$ for an initial time $t_0\in \mathbb{N}$, initial conditions ${\bf x}_i(t_0)\in \mathbb{R}^m$ for all $i\in [n]$, where ${\bf g}_i(t)\in\mathbb{R}^m$ is a subgradient of $f_{i}(z)$ at $z={\bf x}_{i}(t)$ for $i\in[n]$, and $\{\alpha(t)\}$ is a step-size sequence (in \cite{nedic2009distributed} the constant step-sizes variation of this dynamics was studied). {We simply refer to this dynamics as the averaging-based distributed optimization solver}. We can compactly write the above dynamics as \begin{align}\label{eqn:MainLineAlgorithm}
{{\bf x}}(t+1)&=W(t+1){{\bf x}}(t)-\alpha(t){{\bf g}}(t), \end{align} where, ${{\bf g}}(t)=[{\bf g}_1(t),\ldots,{\bf g}_n(t)]^T$ is the vector of the sub-gradient vectors and matrix multiplication should be understood over the vector-field $\mathbb{R}^m$, i.e., \[[W(t+1){{\bf x}}(t)]_i\triangleq\sum_{j=1}^nw_{ij}(t+1){\bf x}_j(t).\]
In distributed optimization, the goal is to find distributed dynamics ${\bf x}_i(t)$s such that $\lim_{t\to\infty}{\bf x}_i(t)=z$ where $z\in \mathcal{Z}$ for all $i\in[n]$.
\subsection{Our Contribution}
In the paper, we consider the random variation of \eqref{eqn:MainLineAlgorithm}, i.e., when $\{W(t)\}$ is a chain of random matrices. This random variation was first studied in \cite{lobel2010distributed} where to ensure the convergence, it was assumed that this sequence is {doubly stochastic almost surely} and i.i.d.. This was generalized to random networks that is Markovian in \cite{lobel2011distributed}. The dynamics~\eqref{eqn:MainLineAlgorithm} with i.i.d.\ weight matrices that are row-stochastic almost surely and column-stochastic \textit{in-expectation} was studied in \cite{7849184}. A special case of \cite{7849184} is the asynchronous gossip algorithm that were introduced in \cite{5585721}. In this work, we provide an overarching framework for the study of~\eqref{eqn:MainLineAlgorithm} with random weight matrices that are row-stochastic almost surely and column-stochastic in-expectation, that are not even independent in general. More precisely, we have following assumptions. \begin{assumption}[Stochastic Assumption]\label{assump:AssumpOnStochasticity} We assume that the weight matrix sequence $\{W(t)\}$, adapted to a filtration $\{\mathcal{F}(t)\}$, satisfies \begin{enumerate}[(a)]
\item For all $t\geq t_0$, $W(t)$ is row-stochastic almost surely.
\item For every $t> t_0$, $\mathbb{E}[W(t)\mid \mathcal{F}(t-1)]$ is column-stochastic (and hence, doubly stochastic) almost surely. \end{enumerate} \end{assumption}
Similar to other works in this domain, our goal is to ensure that $\lim_{t\to\infty}{\bf x}_i(t)=z$ {\it almost surely} for some optimal $z\in \mathcal{Z}$ for all $i\in[n]$. To reach such a consensus value, we need to ensure enough flow of information between the agents, i.e., the associated graph sequence of $\{W(t)\}$ satisfies some form of connectivity over time. More precisely, we assume the following connectivity conditions. \begin{assumption}[Conditional $B$-Connectivity Assumption]\label{assump:AssumpOnConnectivity} We assume that for all $t \geq t_0$ \begin{enumerate}[(a)]
\item \label{cond:connecta} Every node in $\mathcal{G}(W(t))$ has a self-loop, almost surely.
\item \label{cond:bconnectivity}There exists an integer $B>0$ such that the random graph $\mathcal{G}_B(t)=([n],\mathcal{E}_{B}(t))$ where
\begin{align*}
\mathcal{E}_{B}(t)=\bigcup_{\tau=tB+1}^{(t+1)B}\mathcal{E}(\mathbb{E}[W(\tau)|\mathcal{F}(tB)])
\end{align*}
has a spanning rooted tree almost surely. \end{enumerate} \end{assumption}
Note that Assumption~\ref{assump:AssumpOnConnectivity}-\eqref{cond:bconnectivity} is satisfied if the random graph with vertex set $[n]$ and the edge set
$\bigcup_{\tau=tB+1}^{(t+1)B}\mathcal{E}(\mathbb{E}[W(\tau)|\mathcal{F}(\tau-1)])$
has a spanning rooted tree almost surely.
Finally, we assume the following standard condition on the step-size sequence $\{\alpha(t)\}$. \begin{assumption}[Assumption on Step-size]\label{assump:AssumpOnStepSize} For the step-size sequence $\{\alpha(t)\}$, we assume that $0<\alpha(t)\leq Kt^{-\beta}$ for some $K,\beta>0$ and all $t\geq t_0$, $\lim_{t\to\infty}\frac{\alpha(t)}{\alpha(t+1)}=1$, and \begin{align}\label{eqn:assumptionOnAlpha}
\sum_{t=t_0}^{\infty}\alpha(t)=\infty~~\mbox{ and }~~\sum_{t=t_0}^{\infty}\alpha^2(t)<\infty. \end{align} \end{assumption} The main result of this paper is the following theorem. \begin{theorem}\label{thm:MainTheoremDisOpt} Under the Assumptions \ref{assump:AssumpOnFunction}-\ref{assump:AssumpOnStepSize} on the model and the dynamics~\eqref{eqn:MainLineAlgorithm}, {$\lim_{t\to\infty}{\bf x}_i(t)=z^*$ almost surely for all $i\in[n]$ and all initial conditions ${\bf x}_i(t_0)\in \mathbb{R}^m$, where $z^*$ is a random vector that is supported on the optimal set $\mathcal{Z}$. } \end{theorem}
Before continuing with the technical details of the proof, let us first discuss some of the higher-level implications of this result:
\textbf{1. Gossip-based sequential solvers}: Gossip algorithms, which were originally studied in \cite{boyd2006randomized,aysal2009broadcast}, have been used in solving distributed optimization problems \cite{5585721,7165636}. In gossip algorithms, at each round, a node randomly wakes up and shares its value with all or some of its neighbors. However, it is possible to leverage Theorem \ref{thm:MainTheoremDisOpt} to synthesize algorithms that do not require choosing a node {independently and uniformly at random} or use other coordination methods to update information at every round. An example of such a scheme is as follows: \begin{example} Consider a connected {\it undirected} network\footnote{The graphs do not need to be time-invariant, and this example can be extended to processes over underlying time-varying graphs.} $\mathcal{G}=([n],E)$. Consider a token that is handed sequentially in the network and initially it is handed to an arbitrary agent $\ell(0)\in[n]$ in the network. If at time $t\geq 0$, agent $\ell(t)\in[n]$ is in the possession of the token, it chooses one of its neighbors $s(t+1)\in [n]$ randomly and by flipping a coin, i.e., with probability $\frac{1}{2}$ shares its information to $s(t+1)$ and passes the token and with probability $\frac{1}{2}$ keeps the token and asks for information from $s(t+1)$. It means \begin{align*}
\ell(t+1)=\begin{cases}\ell(t),& \mbox{with probability $\frac{1}{2}$}\\
s(t+1),& \mbox{with probability $\frac{1}{2}$}\end{cases}. \end{align*} Finally, the agent $\ell(t+1)$, who has the token at time $t+1$ and is receiving the information, does \begin{align*}
{\bf x}_{\ell(t+1)}(t+1)=\frac{1}{2}({\bf x}_{s(t+1)}(t)+{\bf x}_{\ell(t)}(t))-\alpha(t){\bf g}_{\ell(t+1)}(t). \end{align*} For the other agents $i\not=\ell(t+1)$, we set \begin{align*}
{\bf x}_i(t+1)={\bf x}_i(t)-\alpha(t){\bf g}_i(t). \end{align*}
Let $\mathcal{F}(t)=\sigma({\bf x}(0),\ldots,{\bf x}(t),\ell(t))$, and the weight matrix $W(t)=[w_{ij}(t)]$ be \begin{align*}
w_{ij}(t)=\begin{cases}\frac{1}{2},& i=j=\ell(t)\\
\frac{1}{2},& i=\ell(t),j\in\{s(t),\ell(t-1)\}\setminus\{\ell(t)\}\\
1,& i=j\not=\ell(t)\\
0,& \mbox{otherwise}
\end{cases}, \end{align*}
which is the weight matrix of this scheme. Note that $\mathbb{E}[W(t)|\mathcal{F}(t-1)]=V(\ell(t-1))$ where $V(h)=[v_{ij}(h)]$ with \begin{align*}
v_{ij}(h)=\begin{cases}\frac{3}{4},& i=j=h\\
\frac{1}{4\delta_i},& i=h,(i,j)\in E \\
\frac{1}{4\delta_i},& j=h,(i,j)\in E\\
1,& i=j\not=h\\
0,& \mbox{otherwise}
\end{cases}, \end{align*}
where $\delta_i$ is the degree of the node $i$. Note that the matrix $\mathbb{E}[W(t)|\mathcal{F}(t-1)]$ is doubly stochastic, satisfies Assumption \ref{assump:AssumpOnConnectivity}-(\ref{cond:connecta}), and only depends on $\ell(t-1)$. Now, we need to check whether $\{W(t)\}$ satisfies Assumption \ref{assump:AssumpOnConnectivity}-(\ref{cond:bconnectivity}). We have \begin{align*}
\mathbb{E}[W(t+n)|\mathcal{F}(t)]
&\stackrel{}{=}\mathbb{E}[\mathbb{E}[W(t+n)\mid\mathcal{F}(t+n-1)]\mid\mathcal{F}(t)]\cr
&\stackrel{}{=}\mathbb{E}[V(\ell(t+n-1))\mid \mathcal{F}(t)]\cr
&=\mathbb{E}\left[\sum_{i=1}^nV(\ell(t+n-1))1_{\{\ell(t+n-1)=i\}}\bigg{|}\mathcal{F}(t)\right]\cr
&\stackrel{}{=}\sum_{i=1}^{n}\mathbb{E}[V(i)1_{\{\ell(t+n-1)=i\}}\mid\mathcal{F}(t)]\cr
&\stackrel{}{=}\sum_{i=1}^{n}V(i)\mathbb{E}[1_{\{\ell(t+n-1)=i\}}\mid\mathcal{F}(t)]. \end{align*}
If the network is connected, starting from any vertex, after $n-1$ steps, the probability of reaching any other vertex is at least $(2\Delta)^{-(n-1)}>0$, where $\Delta\triangleq \max_{i\in [n]}\delta_i$. Therefore, we have $\mathbb{E}[1_{\{\ell(t+n-1)=i\}}|\mathcal{F}(t)]>0$ for all $i\in[n]$ and $t$, and hence, Assumption \ref{assump:AssumpOnConnectivity}-(\ref{cond:bconnectivity}) is satisfied with $B=n$. \end{example}
\textbf{2. Robustness to link-failure}:
Our result shows that \eqref{eqn:MainLineAlgorithm} is robust to random link-failures. Note that the results such as \cite{lobel2010distributed} will not imply the robustness of the algorithms to link failure as it assumes that the resulting weight matrices remain doubly stochastic. To show the robustness of averaging-based solvers, suppose that we have a deterministic doubly stochastic sequence $\{A(t)\}$, and suppose that each link at any time $t$ fails with some probability $p(t)>0$. More precisely, let $B(t)$ be a failure matrix where $b_{ij}(t)=0$ if a failure on link $(i,j)$ occurs at time $t$ and otherwise $b_{ij}(t)=1$ and we have
\begin{align}\label{eqn:E1bijtuFt1begincases}
\mathbb{E}[b_{ij}(t)| \mathcal{F}(t-1)]=1-p(t),
\end{align}
for $i,j\in[n]$.
For example, if $B(t)$ is independent and identically distributed, i.e.,
\[b_{ij}(t)=\begin{cases} 0,&\text{ with probability $p$}\cr
1,& \text{ with probability $1-p$}\end{cases},\]
then $B(t)$ satisfies \eqref{eqn:E1bijtuFt1begincases}.
{Define $W(t)=[w_{ij}(t)]$ as follows
\begin{align*}
w_{ij}(t)\triangleq\begin{cases}
a_{ij}(t)b_{ij}(t),&i\not=j\cr
1-\sum_{j\not=i}a_{ij}(t)b_{ij(t)},& i=j
\end{cases}.
\end{align*}
Note that $W(t)$ is row-stochastic, and since $A(t)$ is column-stochastic, $\mathbb{E}[W(t)|\mathcal{F}(t-1)]$ is column-stochastic. Thus, Theorem \ref{thm:MainTheoremDisOpt}, using $W(t)$, translates to a theorem on robustness of the distributed dynamics \eqref{eqn:MainLineAlgorithm}: as long as the connectivity conditions of Theorem~\ref{thm:MainTheoremDisOpt} holds, the dynamics will reach a minimizer of the distributed problem almost surely. For example, if the link failure probability satisfies $p(t)\leq \bar{p}$ for all $t$ and some $\bar{p}<1$, our result implies that the result of Proposition 4 in \cite{do1} (for unconstrained case) would still hold under the above link-failure model. It is worth mentioning that if $\{A(t)\}$ is time-varying, then $\mathbb{E}[W(t)]$ would be time-varying and hence, the previous results on distributed optimization using i.i.d.\ row-stochastic weight matrices that are column-stochastic in-expectation \cite{7849184} would not imply such a robustness result.
}
\section{Autonomous Averaging Dynamics}\label{sec:ConsensusWithoutPerturbation} To prove Theorem \ref{thm:MainTheoremDisOpt}, we need to study the time-varying distributed averaging dynamics with a particular control input (gradient-like dynamics). To do this, first we study the autonomous averaging dynamics (i.e., without any input) and then, we use the established results to study the controlled dynamics.
For this, consider the time-varying distributed averaging dynamics \begin{align}\label{eqn:unperturbed}
{\bf x}(t+1)=W(t+1){\bf x}(t), \end{align} where $\{W(t)\}$ satisfying Assumption \ref{assump:AssumpOnConnectivity}. Defining transition matrix \begin{align*}
\Phi(t,\tau)\triangleq W(t)\cdots W(\tau+1), \end{align*} and $\Phi(\tau,\tau)=I$, we have ${\bf x}(t)=\Phi(t,\tau){\bf x}(\tau)$. {Note that since $W(t)$s are row-stochastic matrices (a.s.) and the set of row-stochastic matrices is a semi-group (with respect to multiplication), the transition matrices $\Phi(t,\tau)$ are all row-stochastic matrices (a.s.). }
{We say that a chain $\{W(t)\}$ achieves \textit{consensus} for the initial time $t_0 \in \mathbb{N}$ if for all $i$,
$\lim_{t\rightarrow\infty }\|{\bf x}_i(t)-\tilde{x}\|=0$ almost surely, for all choices of initial condition ${\bf x}(t_0)\in (\mathbb{R}^m)^n$ in \eqref{eqn:unperturbed} and some random vector $\tilde{x}=\tilde{x}_{{\bf x}(t_0)}$.} It can be shown that an equivalent condition for consensus (for time $t_0$) that $\lim_{t\rightarrow\infty}\Phi(t,t_0)=e \pi^T(t_0)$ for a random stochastic vector $\pi(t_0,\omega)\in \mathbb{R}^n$, almost surely where $\omega\in \Omega$ is a sample point.
For a matrix $A=[a_{ij}]$, let \begin{align*}
{\rm diam}(A)=\max_{i,j\in[n]}\frac{1}{2}\sum_{\ell=1}^n |a_{i\ell}-a_{j\ell}|, \end{align*} and the mixing parameter \begin{align*}
\Lambda(A)=\min_{i,j\in[n]}\sum_{\ell=1}^n \min\{a_{i\ell},a_{j\ell}\}. \end{align*} Note that for a row-stochastic matrix $A$, ${\rm diam}(A)\in [0,1]$. For a vector ${\bf x}=[{\bf x}_{i}]$ where ${\bf x}_i\in \mathbb{R}^m$ for all $i$, let \begin{align*}
{\rm d}({\bf x})=\max_{i,j\in[n]}\|{\bf x}_i-{\bf x}_j\|. \end{align*}
Note that
${\rm d}({\bf x})\leq 2\max_{i\in[n]}\|{\bf x}_i\|$.
Also, if we have consensus, then $\lim_{t\to\infty}{\rm d}({\bf x}(t))=0$ and $\lim_{t\to\infty}{\rm diam}(\Phi(t,t_0))=0$ and in fact, the reverse implications are true \cite{chatterjee1977towards}, i.e., a chain achieves consensus (for time $t_0$) if and only if $\lim_{t\to\infty}{\rm d}({\bf x}(t))=0$ for all ${\bf x}(t_0)\in (\mathbb{R}^{m})^n$ or $\lim_{t\to\infty}{\rm diam}(\Phi(t,t_0))=0$.
The following results relating the above quantities are useful for our future discussions. \begin{lemma}[\cite{hajnal1958weak,shen2000geometric}]\label{lem:lambdadiam} For $n\times n$ row-stochastic matrices $A,B$, we have \begin{align*}
{\rm diam}(AB)\leq(1-\Lambda(A)){\rm diam}(B). \end{align*}
\end{lemma} \begin{lemma}\label{lem:DiamBehrouz} For any $n\times n$ row-stochastic matrices $A,B$, we have \begin{enumerate}[(a)]
\item ${\rm d}(A{\bf x})\leq{\rm diam}(A){\rm d}({\bf x})$ for all ${\bf x} \in(\mathbb{R}^m)^n$,
\item ${\rm d}({\bf x}+{\bf y})\leq{\rm d}({\bf x})+{\rm d}({\bf y})$ for all ${\bf x},{\bf y}\in (\mathbb{R}^m)^n$,
\item ${\rm diam}(A)=1-\Lambda(A)$,
\item ${\rm diam}(AB)\leq {\rm diam}(A){\rm diam}(B)$, and
\item $\left\|{\bf x}_i-\sum_{j=1}^{n}\pi_j{\bf x}_j\right\|\leq {\rm d}({\bf x})$ for all $i\in[n]$, ${\bf x}\in (\mathbb{R}^m)^n$, and any stochastic vector $\pi\in [0,1]^n$ (i.e., $\sum_{i=1}^n\pi_i=1$).
\end{enumerate} \end{lemma} \pf The proof is provided in Appendix. \qed
The main goal of this section is to obtain an exponentially decreasing upper bound (in terms of $t_1-\tau_1$ and $t_2-\tau_2$) on $\mathbb{E}[{\rm diam}(\Phi(t_2,\tau_2)){\rm diam}(\Phi(t_1,\tau_1))\mid \mathcal{F}(\tau_1)]$.
Using this result and the connectivity assumption \ref{assump:AssumpOnConnectivity}, we can show that the transition matrices $\Phi(t,s)$ become mixing \textit{in-expectation} for large enough $t-s$. \begin{lemma}\label{lem:ExpectationLambda} Under Assumption \ref{assump:AssumpOnConnectivity} (Connectivity), there exists a parameter $\theta>0$ such that for every $s\geq t_0$, we have almost surely \begin{align*}
\mathbb{E}[\Lambda(\Phi((n^2+s)B,sB))\mid\mathcal{F}(sB)]\geq\theta. \end{align*} \end{lemma} \pf
Fix $s\geq 0$. Let $\mathbb{T}$ be the set of all collection of edges $E$ such that the graph $([n],E)$ has a spanning rooted tree, and for $k\in[n^2]$, \begin{align*}
\mathcal{E}_{B}^{}(k)\triangleq\bigcup\limits_{\tau=(s+k-1)B+1}^{(s+k)B}\mathcal{E}^{}(\mathbb{E}[W(\tau)|\mathcal{F}((s+k-1)B)]). \end{align*}
For notational simplicity, denote $\mathcal{F}(sB)$ by $\mathcal{F}$ and $\mathcal{F}((s+k)B)$ by $\mathcal{F}_k$ for $k\in[n^2]$. Let $V=\{\omega\mid\forall k~\mathcal{E}_{B}^{}(k)\in\mathbb{T}\}$. From Assumption \ref{assump:AssumpOnConnectivity}, we have $P(V)=1$. For $\omega\in V$ and $k\geq 1$, define the random graph $([n],\mathcal{T}_k)$ on $n$ vertices by \begin{align*}
\mathcal{T}_k&=\begin{cases}
\mathcal{T}_{k-1},& \mbox{if } \mathcal{T}_{k-1}\in \mathbb{T}\\
\mathcal{T}_{k-1}\cup\{u_k\},& \mbox{if } \mathcal{T}_{k-1}\not\in \mathbb{T}
\end{cases}, \end{align*} with $ \mathcal{T}_0=\emptyset$, where \begin{align}\label{eqn:ukinEEBkcap}
u_k\in \mathcal{E}_{B}^{}(k)\cap\overline{\mathcal{T}}_{k-1}, \end{align} and $\overline{\mathcal{T}}_{k}$ is the edge-set of the complement graph of $([n],{\mathcal{T}}_{k})$. Note that since $\mathcal{E}_{B}^{}(k)$ has a spanning rooted tree, if $\mathcal{T}_{k-1}\not\in \mathbb{T}$, then $\mathcal{E}_B(k)$ should contain an edge that does not belong to ${\mathcal{T}}_{k-1}$, which we identify it as $u_k$ in \eqref{eqn:ukinEEBkcap}. Hence, $\mathcal{T}_k$ is well-defined. Since there are at most $n(n-1)$ potential edges in a graph on $n$ vertices, $\mathcal{T}_{n^2}$ has a spanning rooted tree for $\omega\in V$.
For $k\in[n^2]$, let \begin{align*}
\mathcal{D}_{B}^{}(k)\triangleq\bigcup\limits_{\tau=(s+k-1)B+1}^{(s+k)B}\mathcal{E}^{\nu}(W(\tau)), \end{align*} for some fixed $0<\nu<\gamma$, and \[\mathcal{H}_{}^{}(k)\triangleq\cup_{\tau=1}^{k}\mathcal{D}_{B}^{}(\tau).\] Consider the sequences of events $\{U_k\}$ defined by $U_k\triangleq\left\{\omega\in V \mid
\mathcal{T}_{k}\subset\mathcal{H}_{}^{}(k)\right\}$, for $k\geq 1$, and $U_0=V$.
Note that if $\mathcal{T}_{k-1}\in \mathbb{T}$, then $\mathcal{T}_{k-1}\subset\mathcal{H}_{}^{}(k-1)$ implies $\mathcal{T}_{k}\subset\mathcal{H}_{}^{}(k)$, and if $\mathcal{T}_{k-1}\not\in \mathbb{T}$, then $\mathcal{T}_{k-1}\subset\mathcal{H}_{}^{}(k-1)$ and $u_k\in\mathcal{D}_{B}^{}(k)$ imply $\mathcal{T}_{k}\subset\mathcal{H}_{}^{}(k)$. Hence, for $k\geq 1$ \begin{align}\label{eqn:DefinitionofUk}
1_{\{U_k\}}\geq1_{\{U_{k-1}\}}1_{\{\mathcal{T}_{k-1}\not\in \mathbb{T}\}}1_{\{u_k\in\mathcal{D}_{B}^{}(k)\}}+1_{\{U_{k-1}\}}1_{\{\mathcal{T}_{k-1}\in \mathbb{T}\}}. \end{align}
On the other hand, from Tower rule, we have
\begin{align}\label{eqn:E1Uk1mathcalTk-1notinmathbbT}
\mathbb{E}[1_{\{U_{k-1}\}}&1_{\{\mathcal{T}_{k-1}\not\in \mathbb{T}\}}1_{\{u_k\in\mathcal{D}_{B}^{}(k)\}}\mid \mathcal{F}]\\
&=\mathbb{E}[1_{\{U_{k-1}\}}1_{\{\mathcal{T}_{k-1}\not\in \mathbb{T}\}}\mathbb{E}[1_{\{u_k\in\mathcal{D}_{B}^{}(k)\}}|\mathcal{F}_{k-1}]\mid\mathcal{F}].\nonumber \end{align}
Let $u_k(\omega)=(j_k(\omega),i_k(\omega))$.
Since $u_k\in \mathcal{E}_{B}(k)$, there exists $(s+k-1)B<\tau_k\leq (s+k)B$ such that
\[u_k\in\mathcal{E}^{}(\mathbb{E}[W(\tau_k)|\mathcal{F}_{k-1}]),\]
and, we have \begin{align}\label{eqn:E1nu11wikjkkgeq}
\mathbb{E}[(1-\nu) 1_{\{1-w_{i_kj_k}(\tau_k)\geq 1-\nu\}}|\mathcal{F}_{k-1}]\nonumber
&\leq \mathbb{E}[1-w_{i_kj_k}(\tau_k)|\mathcal{F}_{k-1}]
\leq 1-\gamma. \end{align} Therefore, \begin{align*}
\mathbb{E}[1_{\{u_k\in\mathcal{D}_{B}^{}(k)\}}\mid\mathcal{F}_{k-1}]&\geq\mathbb{E}[1_{\{u_k\in\mathcal{E}^{\nu}(W(\tau_k))\}}\mid\mathcal{F}_{k-1}]\cr
&=\mathbb{E}[ 1_{\{w_{i_kj_k}(\tau_k)> \nu\}}\mid\mathcal{F}_{k-1}]\cr
&=1-\mathbb{E}[ 1_{\{w_{i_kj_k}(\tau_k)\leq \nu\}}\mid\mathcal{F}_{k-1}]\cr
&=1-\mathbb{E}[ 1_{\{1-w_{i_kj_k}(\tau_k)\geq 1-\nu\}}\mid\mathcal{F}_{k-1}]\cr
&\geq 1-\frac{1-\gamma}{1-\nu}\triangleq{p}>0, \end{align*} which holds as $\nu<\gamma$.
This inequality and \eqref{eqn:E1Uk1mathcalTk-1notinmathbbT} imply that \begin{align*}
\mathbb{E}[1_{\{U_{k-1}\}}1_{\{\mathcal{T}_{k-1}\not\in \mathbb{T}\}}&1_{\{u_k\in\mathcal{D}_{B}^{}(k)\}}\mid \mathcal{F}]\geq
p\mathbb{E}[1_{\{U_{k-1}\}}1_{\{\mathcal{T}_{k-1}\not\in \mathbb{T}\}}\mid\mathcal{F}]. \end{align*} Therefore, \eqref{eqn:DefinitionofUk} implies \begin{align*}
\mathbb{E}[1_{\{U_k\}}|\mathcal{F}]
&\geq p\mathbb{E}[1_{\{U_{k-1}\}}1_{\{\mathcal{T}_{k-1}\not\in \mathbb{T}\}}|\mathcal{F}]+\mathbb{E}[1_{\{U_{k-1}\}}1_{\{\mathcal{T}_{k-1}\in \mathbb{T}\}}|\mathcal{F}]\cr
&\geq p\left(\mathbb{E}[1_{\{U_{k-1}\}}1_{\{\mathcal{T}_{k-1}\not\in \mathbb{T}\}}|\mathcal{F}]+\mathbb{E}[1_{\{U_{k-1}\}}1_{\{\mathcal{T}_{k-1}\in \mathbb{T}\}}|\mathcal{F}]\right)\cr
&=p\mathbb{E}[1_{\{U_{k-1}\}}|\mathcal{F}], \end{align*}
and hence $\mathbb{E}[1_{\{U_k\}}|\mathcal{F}]\geq p^k$. Finally, since $\mathcal{T}_{n^2}$ has a spanning rooted tree, from Lemma~1 in \cite{lobel2010distributed}, we have \begin{align*}
\Lambda(W(n^2B,\omega)\cdots W(n(n-1)B+n-1,\omega)\cdots W(1,\omega))\geq \nu^{n^2B}, \end{align*} for $\omega\in U_{n^2}$. Therefore, we have \begin{align*}
\mathbb{E}[\Lambda(\Phi((n^2+s)B,sB))\mid\mathcal{F}(sB)]&\geq \nu^{n^2B}\mathbb{E}[1_{\{U_{n^2}\}}\mid \mathcal{F}]\geq \nu^{n^2B}{p}^{n^2}\triangleq\theta>0, \end{align*} which completes the proof.\qed Finally, we need the following result, which is proved in Appendix, to prove the main result of this section. \begin{lemma}\label{lem:ProductwithHistory} For a random process $\{Y(k)\}$, adapted to a filtration $\{\mathcal{F}(k)\}$, let
\[\mathbb{E}[Y(k)|\mathcal{F}(k-1)]\leq a(k)\] for $K_1\leq k\leq K_2$ almost surely, where $K_1\leq K_2$ are arbitrary positive integers and $a(k)$s are (deterministic) scalars. Then, we have almost surely \begin{align*}
\mathbb{E}\left[\prod_{k=K_1}^{K_2}Y(k)\bigg{|}\mathcal{F}(K_1-1)\right]\leq \prod_{k=K_1}^{K_2}a(k). \end{align*} \end{lemma}
Now, we are ready to prove the main result for the convergence rate of the autonomous random averaging dynamics. \begin{lemma}\label{lem:ExpectationDiam} Under Assumption \ref{assump:AssumpOnConnectivity} (Connectivity), there exist $0<C$ and $0\leq \lambda<1$ such that for every $t_0\leq \tau_1\leq t_1$ and $t_0\leq \tau_2\leq t_2$ with $\tau_1\leq\tau_2$, we have almost surely \begin{align*}
\mathbb{E}[{\rm diam}(\Phi(t_2,\tau_2)){\rm diam}(\Phi(t_1,\tau_1))|\mathcal{F}(\tau_1)]\leq C\lambda^{t_1-\tau_1}\lambda^{t_2-\tau_2}. \end{align*} \end{lemma} \pf First, we prove \begin{align}\label{eqn:EdiammPhittauFtau}
\mathbb{E}[{\rm diam}(\Phi(t,\tau))|\mathcal{F}(\tau)]\leq \tilde{C}\tilde{\lambda}^{t-\tau}, \end{align} for some $0<\tilde{C}$ and $0\leq \tilde{\lambda}<1$. Let $s\triangleq\lceil \frac{\tau}{B}\rceil$ and $K\triangleq\lfloor\frac{t-sB}{n^2B}\rfloor$. We have \begin{align*}
\mathbb{E}[{\rm diam}(\Phi(t,\tau))|\mathcal{F}(\tau)]
&= \mathbb{E}\Bigg{[}{\rm diam}\Bigg{(}\Phi(t,sB+Kn^2B)\left[\prod_{k=1}^{K}\!\Phi(sB+kn^2B,sB+(k-1)n^2B)\!\right]\!\Phi(sB,\tau)\!\Bigg{)}\Bigg{|}\mathcal{F}(\tau)\Bigg{]}\cr
&\stackrel{(a)}{\leq} \!\mathbb{E}\left[\prod_{k=1}^{K}\!(1-\Lambda(\Phi(sB+kn^2B,sB+(k-1)n^2B))\Bigg{|}\mathcal{F}(\tau)\right]\cr
&\stackrel{(b)}{\leq} (1-\theta)^{K}\cr
&\leq \tilde{C}(1-\theta)^{\frac{t-\tau}{n^2B}}, \end{align*} where $\tilde{C}=(1-\theta)^{-1-\frac{1}{n^2}}$ and $(a)$ follows from Lemma~\ref{lem:lambdadiam}, Lemma~\ref{lem:DiamBehrouz}-(d), and {${\rm diam}(\Phi(.,.))\leq1$}, and $(b)$ follows from Lemma \ref{lem:ExpectationLambda} and \ref{lem:ProductwithHistory}. Since $\theta>0$, we have $\tilde{\lambda}\triangleq (1-\theta)^{\frac{1}{n^2B}}<1$.
To prove the main statement, we consider two cases: \begin{enumerate}[(i)]
\item intervals $(\tau_1,t_1]$ and $(\tau_2,t_2]$ do not have an intersection, and
\item $(\tau_1,t_1]$ and $(\tau_2,t_2]$ intersect. \end{enumerate} For case (i), since the two intervals do not overlap, we have $t_1\leq\tau_2$, and hence \begin{align*}
\mathbb{E}[{\rm diam}(\Phi(t_2,\tau_2)){\rm diam}(\Phi(t_1,\tau_1))|\mathcal{F}(\tau_1)]
&=\mathbb{E}\bigg{[}\mathbb{E}[{\rm diam}(\Phi(t_2,\tau_2))|\mathcal{F}(\tau_2)]{\rm diam}(\Phi(t_1,\tau_1))\bigg{|}\mathcal{F}(\tau_1)\bigg{]}\cr
&\leq \tilde{C}\tilde{\lambda}^{t_1-\tau_1}\tilde{C}\tilde{\lambda}^{t_2-\tau_2}, \end{align*} which follows from \eqref{eqn:EdiammPhittauFtau}. For case (ii), let's write the union of the intervals $(\tau_1,t_1]$ and $(\tau_2,t_2]$ as disjoint union of three intervals: \[(\tau_1,t_1]\cup(\tau_2,t_2]=(s_1,s_2]\cup (s_2,s_3]\cup (s_3,s_4],\] for $s_1\leq s_2\leq s_3$ where $(s_2,s_3]\triangleq(\tau_1,t_1]\cap(\tau_2,t_2]$, $(s_1,s_2]\cup(s_3,s_4]\triangleq(\tau_1,t_1]\triangle(\tau_2,t_2]$.
{Using this, it can be verified that} \begin{align*}
\mathbb{E}[{\rm diam}(\Phi(t_2,\tau_2)){\rm diam}(\Phi(t_1,\tau_1))|\mathcal{F}(\tau_1)]
&\stackrel{(a)}{\leq}\!\!\mathbb{E}[{\rm diam}(\Phi(s_4,s_3)){\rm diam}^2(\Phi(s_3,s_2)){\rm diam}(\Phi(s_2,s_1))|\mathcal{F}(\tau_1)]\cr
&\stackrel{(b)}{\leq}\!\!\mathbb{E}[{\rm diam}(\Phi(s_4,s_3)){\rm diam}(\Phi(s_3,s_2)){\rm diam}(\Phi(s_2,s_1))|\mathcal{F}(\tau_1)]\cr
&\stackrel{(c)}{\leq} \tilde{C}\tilde{\lambda}^{s_2-s_1}\tilde{C}\tilde{\lambda}^{s_3-s_2}\tilde{C}\tilde{\lambda}^{s_4-s_3}\cr
&= \tilde{C}\tilde{\lambda}^{s_2-s_1}2\tilde{C}\sqrt{\tilde{\lambda}}^{2(s_3-s_2)}\tilde{C}\tilde{\lambda}^{s_4-s_3}\cr
&\leq \tilde{C}^3\sqrt{\tilde{\lambda}}^{t_1-\tau_1}\sqrt{\tilde{\lambda}}^{t_2-\tau_2}, \end{align*} {where $(a)$ follows from Lemma \ref{lem:DiamBehrouz}-(d) , $(b)$ follows from {${\rm diam}(A)\leq1$} for all row-stochastic matrices $A$, and $(c)$ follows from \eqref{eqn:EdiammPhittauFtau} and Lemma \ref{lem:ProductwithHistory}}. Letting $C\triangleq \max\{\tilde{C}^2, \tilde{C}^3\}$ and $\lambda\triangleq \sqrt{\tilde{\lambda}}$, we arrive at the conclusion. \qed \section{Averaging Dynamics with Gradient-Flow Like Feedback }\label{sec:ConsensusWithPerturbation} In this section, we study the controlled linear time-varying dynamics \begin{align}\label{eqn:EpsilonAlgorithm}
{{\bf x}}(t+1)&=W(t+1){{\bf x}}(t)+{\bf u}(t). \end{align} Note that the feedback ${\bf u}(t)=-\alpha(t) {\bf g}(t)$ leads to the dynamics \eqref{eqn:MainLineAlgorithm}. The goal of this section is to establish bounds on the convergence-rate of ${\rm d}({\bf x})$ (to zero) in-expectation and almost surely for a class of regularized input ${\bf u}(t)$.
We start with the following two lemmas.
\begin{lemma}\label{lem:d_x_t} For dynamics \eqref{eqn:EpsilonAlgorithm} and every $t_0\leq \tau\leq t$, we have \begin{align}\label{lem:lemma5}
{\rm d}({\bf x}(t))&
\stackrel{}{\leq}{\rm diam}(\Phi(t,\tau)){\rm d}({\bf x}(\tau))+\sum_{s=\tau}^{t-1}{\rm diam}(\Phi(t,s+1)){\rm d}({\bf u}(s)). \end{align} \end{lemma}
\pf Note that the general solution for the dynamics \eqref{eqn:EpsilonAlgorithm} is given by \begin{align}\label{eqn:PhiPerturbation}
{\bf x}(t)=\Phi(t,\tau){\bf x}(\tau)+\sum_{s=\tau}^{t-1}\Phi(t,s+1){\bf u}(s). \end{align}
Therefore, using the sub-linearity property of ${\rm d}(\cdot)$ (Lemma \ref{lem:DiamBehrouz}-(b)), we have \begin{align*}
{\rm d}({\bf x}(t))&{\leq}{\rm d}(\Phi(t,\tau){\bf x}(\tau))+\sum_{s=\tau}^{t-1}{\rm d}(\Phi(t,s+1){\bf u}(s))\cr
&{\leq}{\rm diam}(\Phi(t,\tau)){\rm d}({\bf x}(\tau))+\sum_{s=\tau}^{t-1}{\rm diam}(\Phi(t,s+1)){\rm d}({\bf u}(s)), \end{align*} where the last inequality follows from Lemma \ref{lem:DiamBehrouz}-(a). \qed
\begin{lemma}\label{lem:approxfunction} Let $\{\beta(t)\}$ be a positive (scalar) sequence such that $\lim_{t\to\infty}\frac{\beta(t)}{\beta(t+1)}=1$. Then for any $\theta\in[0,1)$, there exists some $M>0$ such that \begin{align*}
\sum_{s=\tau}^{t-1}\beta(s)\theta^{t-s}\leq M\beta(t), \end{align*}
for all $t\geq \tau\geq t_0$. \end{lemma} \pf The proof is provided in Appendix.\qed
To prove the main theorem, we need to study how fast $\mathbb{E}[{\rm d}({\bf x}(t))]$ and $\mathbb{E}[{\rm d}^2({\bf x}(t))]$ approach to zero when the diameter of the control input ${\rm d}({\bf u}(t))$ goes to zero. Since $\mathbb{E}[{\rm d}^2({\bf x}(t))]\geq \mathbb{E}^2[{\rm d}({\bf x}(t))]$, it suffice to study convergence rate of $\mathbb{E}[{\rm d}^2({\bf x}(t))]$.
\begin{lemma}\label{lem:ConsensusConveregnseVariance} Under Assumptions \ref{assump:AssumpOnStochasticity}, \ref{assump:AssumpOnConnectivity}, and \ref{assump:AssumpOnStepSize}, if almost surely ${\rm d}({\bf u}(t))<q\alpha(t)$ for some $q>0$, then we have,
\[\frac{\mathbb{E}[{\rm d}^2({\bf x}(t))]}{\alpha^2(t)}\leq \hat{M}\]
for some $\hat{M}>0$ and all $t\geq t_0$. \end{lemma} \pf
Taking the square of both sides of \eqref{lem:lemma5}, for $t>\tau\geq t_0$, we have \begin{align*}
{\rm d}^2({\bf x}(t))
&\stackrel{}{\leq}{\rm diam}^2(\Phi(t,\tau)){\rm d}^2({\bf x}(\tau))
+2{\rm diam}(\Phi(t,\tau)){\rm d}({\bf x}(\tau))\sum_{s=\tau}^{t-1}{\rm diam}(\Phi(t,s+1)){\rm d}({\bf u}(s))\cr
&\qquad\qquad+\sum_{s=\tau}^{t-1}\sum_{\ell=\tau}^{t-1}{\rm diam}(\Phi(t,s+1)){\rm d}({\bf u}(s)){\rm diam}(\Phi(t,\ell+1)){\rm d}({\bf u}(\ell)). \end{align*} Taking the expectation of both sides of the above inequality, and using ${\rm d}({\bf u}(t))<q\alpha(t)$ almost surely, we have \begin{align*}
\mathbb{E}[{\rm d}^2({\bf x}(t))]
&\stackrel{}{\leq}\mathbb{E}[{\rm diam}^2(\Phi(t,\tau)){\rm d}^2({\bf x}(\tau))]
+2\sum_{s=\tau}^{t-1}\mathbb{E}\big{[}{\rm diam}(\Phi(t,\tau)){\rm d}({\bf x}(\tau)){\rm diam}(\Phi(t,s+1)){\rm d}({\bf u}(s))\big{]}\cr
&\qquad\qquad+\sum_{s=\tau}^{t-1}\sum_{\ell=\tau}^{t-1}\mathbb{E}\big{[}{\rm diam}(\Phi(t,s+1)){\rm d}({\bf u}(s)){\rm diam}(\Phi(t,\ell+1)){\rm d}({\bf u}(\ell))\big{]}\cr
&\stackrel{}{\leq}\mathbb{E}\bigg{[}\mathbb{E}[{\rm diam}^2(\Phi(t,\tau))|\mathcal{F}(\tau)]{\rm d}^2({\bf x}(\tau))\bigg{]}+2\sum_{s=\tau}^{t-1}\mathbb{E}\bigg{[}\mathbb{E}[{\rm diam}(\Phi(t,\tau)){\rm diam}(\Phi(t,s+1))|\mathcal{F}(\tau)]{\rm d}({\bf x}(\tau))\bigg{]}\alpha(s)q\cr
&\qquad\qquad+\sum_{s=\tau}^{t-1}\sum_{\ell=\tau}^{t-1}\mathbb{E}[{\rm diam}(\Phi(t,s+1)){\rm diam}(\Phi(t,\ell+1))]\alpha(s)\alpha(\ell)q^2. \end{align*} Therefore, from Lemma \ref{lem:ExpectationDiam}, we have \begin{align*}
\mathbb{E}[{\rm d}^2({\bf x}(t))]
&\stackrel{}{\leq}C\lambda^{2(t-\tau)}\mathbb{E}[{\rm d}^2({\bf x}(\tau))]+\frac{2Cq}{\lambda}\lambda^{t-\tau}\mathbb{E}[{\rm d}({\bf x}(\tau))]\sum_{s=\tau}^{t-1}\lambda^{t-s}\alpha(s)+\frac{Cq^2}{\lambda^2}\sum_{s=\tau}^{t-1}\sum_{\ell=\tau}^{t-1}\alpha(s)\alpha(\ell)\lambda^{t-s}\lambda^{t-\ell}\cr
&\stackrel{}{\leq}C\lambda^{2(t-\tau)}\mathbb{E}[{\rm d}^2({\bf x}(\tau))]+\frac{2CqM}{\lambda}\lambda^{t-\tau}\mathbb{E}[{\rm d}({\bf x}(\tau))]\alpha(t)+\frac{Cq^2M^2}{\lambda^2}\alpha^2(t), \end{align*} where the last inequality follows from Lemma \ref{lem:approxfunction} and the fact that \begin{align*}
\sum_{s=\tau}^{t-1}\sum_{\ell=\tau}^{t-1}\alpha(s)\alpha(\ell)\lambda^{t-s}\lambda^{t-\ell}=\left(\sum_{s=\tau}^{t-1}\alpha(s)\lambda^{t-s}\right)^2. \end{align*} Dividing both sides of the above inequality by $\alpha^2(t)$ and noting $\frac{\alpha(\tau)}{\alpha(t)}\lambda^{t-\tau}=\prod_{\kappa=\tau}^{t-1}\frac{\alpha(\kappa)}{\alpha(\kappa+1)}\lambda$, we have \begin{align*}
&\frac{\mathbb{E}[{\rm d}^2({\bf x}(t))]}{\alpha^2(t)}
\stackrel{}{\leq}C\frac{\mathbb{E}[{\rm d}^2({\bf x}(\tau))]}{\alpha^2(\tau)}\left(\prod_{\kappa=\tau}^{t-1}\frac{\alpha(\kappa)}{\alpha(\kappa+1)}\lambda\right)^2+2\frac{CqM}{\lambda}\frac{\mathbb{E}[{\rm d}({\bf x}(\tau))]}{\alpha(\tau)}\left(\prod_{\kappa=\tau}^{t-1}\frac{\alpha(\kappa)}{\alpha(\kappa+1)}\lambda\right)+\frac{Cq^2M^2}{\lambda^2}. \end{align*} Since $\lim_{\tau\to\infty}\frac{\alpha(\tau)}{\alpha(\tau+1)}=1$, for any $\hat{\lambda}\in (\lambda,1)$, there exists $\hat{\tau}$ such that for $\tau\geq \hat{\tau}$, we have $\frac{\mu(\tau)}{\mu(\tau+1)}\lambda\leq \hat{\lambda}$. Therefore, \begin{align*}
\frac{\mathbb{E}[{\rm d}^2({\bf x}(t))]}{\alpha^2(t)}
&\stackrel{}{\leq}C\frac{\mathbb{E}[{\rm d}^2({\bf x}(\tau))]}{\alpha^2(\tau)}\hat{\lambda}^{2(t-\tau)}+\frac{2CqM}{\lambda}\frac{\mathbb{E}[{\rm d}({\bf x}(\tau))]}{\alpha(\tau)}\hat{\lambda}^{t-\tau}+\frac{Cq^2M^2}{\lambda^2}. \end{align*} Taking the limit of the above inequality, we get \begin{align*}
\limsup_{t\to\infty}\frac{\mathbb{E}[{\rm d}^2({\bf x}(t))]}{\alpha^2(t)}
&\stackrel{}{\leq}\lim_{t\to\infty}C\frac{\mathbb{E}[{\rm d}^2({\bf x}(\tau))]}{\alpha^2(\tau)}\hat{\lambda}^{2(t-\tau)}+\lim_{t\to\infty}\frac{2CqM}{\lambda}\frac{\mathbb{E}[{\rm d}({\bf x}(\tau))]}{\alpha(\tau)}\hat{\lambda}^{t-\tau}+\frac{Cq^2M^2}{\lambda^2}\cr
&=\frac{Cq^2M^2}{\lambda^2}. \end{align*} As a result, there exists an $\hat{M}>0$ such that $\frac{\mathbb{E}[{\rm d}^2({\bf x}(t))]}{\alpha^2(t)}\leq \hat{M}$. \qed
To prove the main theorem, we also need to show that ${\rm d}({\bf x}(t))$ converges to zero almost surely (as will be proved in Lemma~\ref{lem:ConsensusConveregnse}). To do so, using the previous results, we will show (in Lemma~\ref{lem:ConsensusConveregnse}) that \begin{align*}
\mathbb{E}[{\rm d}({\bf x}(t))|\mathcal{F}(\tau)]\leq\mathbb{E}[{\rm diam}(\Phi(t,\tau))|\mathcal{F}(\tau)]{\rm d}({\bf x}(\tau))+K\tau^{-\gamma}, \end{align*}
for some $K,\gamma>0$. However, since $\sum_{\tau=t_0}^\infty \tau^{-\gamma}$ is not necessarily summable, we cannot use the standard Siegmund-Robbins Theorem \cite{ROBBINS1971233} to argue ${\rm d}({\bf x}(t))\to 0$ based on this inequality. We will use the facts that $\mathbb{E}[{\rm diam}(\Phi(t,\tau))|\mathcal{F}(\tau)]<1$ if $t-\tau$ is large enough, and ${\rm diam}(\Phi(t,\tau))\leq 1$ for all $t\geq \tau \geq t$. This leads us to prove a martingale-type result in Lemma \ref{lem:Dooblike}. To prove this lemma, we first show the following.
\begin{lemma}\label{lem:SLLNlike} Consider a non-negative random process $a(t)$ such that \[\mathbb{E}[a(t+1)\mid \mathcal{F}(t)]\leq\tilde{\lambda}\]
for some $\tilde{\lambda}<1$ and all $t\geq 0$. For $\lambda$ satisfying $\tilde{\lambda}<{\lambda}<1$, define the sequence of stopping-times $\{t_s\}_{s\geq 0}$ by \[t_s\triangleq\inf\{t>t_{s-1}|a(t)\leq \lambda\},\] with $t_0=0$. Then, $\lim_{s\to\infty}(t_{s+1}-t_s)t_s^{-\beta}=0$ almost surely for all $\beta>0$. \end{lemma} \pf Let us define the martingale $S(t)$ by \begin{align*}
S(t)=S(t-1)+\left(1_{\{a(t)> \lambda\}}-\mathbb{E}[1_{\{a(t)>\lambda\}}|\mathcal{F}(t-1)]\right), \end{align*} where $S(0)=0$.
Noting $|S(t+1)-S(t)|\leq 1$, from Azuma's inequality, we have \begin{align}\label{eqn:AzumaForSSigam}
P(S(t+\sigma)-S(t)>\sigma\rho)\leq \exp\left(-\frac{\sigma^2\rho^2}{2\sigma}\right), \end{align} for all $\sigma\in \mathbb{N}$ and $\rho\in(0,1)$. For $\theta>0$, let the sequences of events \begin{align*}
A_{\theta}(t)\triangleq\left\{\omega\big{|}S(t+\lfloor \theta t^\beta \rfloor)-S(t)>\lfloor \theta t^\beta \rfloor\rho\right\}. \end{align*} From \eqref{eqn:AzumaForSSigam}, we have \begin{align*}
P(A_{\theta}(t))\leq\exp\left(-\frac{1}{2}(\theta t^\beta-1)\rho^2\right), \end{align*}
implying $\sum_{t=1}^{\infty}P(A_{\theta}(t))<\infty$ as $\exp\left({-t^\beta}\right)\leq \frac{M}{t^2}$ for sufficiently large $M$ (depending on $\beta$). Therefore, the Borel–Cantelli Theorem implies that $P(\{A_{\theta}(t)\quad \text{i.o.}\})=0$ for all $\theta>0$.
For $\theta>0$, let the sequences of events \begin{align*}
B_{\theta}(t)\triangleq\left\{\omega\bigg{|}\frac{\hat{t}-t}{t^{\beta}}>\theta \mbox{ where } \hat{t}=\inf\{\tau>t|a(\tau)\leq \lambda\}\right\}. \end{align*} We show that $B_{\theta}(t)\subset A_{\theta}(t)$ for all $t,\theta$. Fix a constant $\rho\in (0,1)$ such that $1-\frac{{\tilde{\lambda}}}{\lambda}>\rho$. Since $\mathbb{E}[a(\tau)\mid \mathcal{F}(\tau-1)]\leq \tilde{\lambda}$, we have \begin{align*}
\mathbb{E}[\lambda 1_{\{a(\tau)\geq \lambda\}}\mid \mathcal{F}(\tau-1)]&\leq \mathbb{E}[a(\tau)\mid \mathcal{F}(\tau-1)]\cr
&\leq \tilde{\lambda}<\lambda(1-\rho) \end{align*} and hence, \begin{align}\label{eqn:rhoE1ataugeqlambdamidFtau}
\mathbb{E}[ 1_{\{a(\tau)\geq \lambda\}}\mid \mathcal{F}(\tau-1)]<1-\rho. \end{align} Let $\sigma(t)\triangleq\lfloor \theta t^\beta \rfloor$. If $\hat{t}-t>\theta t^\beta$, then \begin{align*}
S(t+\sigma(t))-S(t)&= \sigma(t)-\sum_{\tau=t+1}^{t+\sigma(t)} \mathbb{E}[1_{\{a(\tau)>\lambda\}}|\mathcal{F}(\tau-1)]\cr
&> \sigma(t)-\sigma(t)(1-\rho)=\sigma(t)\rho, \end{align*} which follows from \eqref{eqn:rhoE1ataugeqlambdamidFtau}. Therefore, we have $B_{\theta}(t)\subset A_{\theta}(t)$, and hence, $P(\{B_{\theta}(t)\quad \text{i.o.}\})=0$ for all $\theta>0$.
Finally, by contradiction, we show that $\lim_{s\to\infty}(t_{s+1}-t_s)t_s^{-\beta}=0$. Since, if $\lim_{s\to\infty}(t_{s+1}-t_s)t_s^{-\beta}\not=0$ almost surely, then $\limsup_{s\to\infty}(t_{s+1}-t_s)t_s^{-\beta}>0$ almost surely, and hence, $P\left(\limsup_{s\to\infty}(t_{s+1}-t_s)t_s^{-\beta}>\epsilon\right)>0$ for some $\epsilon>0$. Therefore, $P(\{B_{\epsilon}(t)\quad \text{i.o.}\})>0$, which is a contradiction. \qed
\begin{lemma}\label{lem:Dooblike} Suppose that $\{D(t)\}$ is a non-negative random (scalar) process such that
\begin{align}\label{eqn:dooblike}
D(t+1)\leq a(t+1)D(t)+b(t),\quad \mbox{almost surely}
\end{align}
where $\{b(t)\}$ is a deterministic sequence and $\{a(t)\}$ is an adapted process (to $\{\mathcal{F}(t)\}$), such that $a(t)\in[0,1]$ and \[\mathbb{E}[a(t+1)\mid \mathcal{F}(t)]\leq\tilde{\lambda},\] almost surely for some $\tilde{\lambda}<1$ and all $t\geq 0$. Then, if \[0\leq b(t)\leq{K}{t^{-\tilde{\beta}}}\] for some $K,\tilde{\beta}>0$, we have $\lim_{t\to\infty}D(t)t^\beta=0$, almost surely, for all $\beta<\tilde{\beta}$. \end{lemma}
\pf Let $t_s\triangleq\inf\{t>t_{s-1}|a(t)\leq \lambda\}$ and $t_0=0$ for some $\tilde{\lambda}<\lambda<1$, and $c(s)\triangleq \sum_{\tau=t_s+1}^{t_{s+1}-1}b(\tau)$ . Also, define \begin{align*}
A\triangleq\left\{\omega\bigg{|}\lim_{s\to\infty}\frac{t_{s+1}-t_s}{t_s^{\min\{\tilde{\beta}-\beta,1\}}}=0\right\}. \end{align*} Note that Lemma \ref{lem:SLLNlike} implies $P(A)=1$. On the other hand, using \eqref{eqn:dooblike}, we have \begin{align*}
D(t_{s+1})&\leq D(t_{s})\prod_{\ell=t_{s}+1}^{t_{s+1}}a(\ell)+\sum_{\tau=t_s}^{t_{s+1}-1}b(\tau)\prod_{\ell=\tau+2}^{t_{s+1}}a(\ell)\cr
&\leq D(t_{s})\lambda+c(s), \end{align*} where the last inequality follows from $a(t)\in [0,1]$ and $a(t_{s+1})\leq \lambda$. Letting $R(t)=D(t)t^{\beta}$, we have \begin{align*}
R(t_{s+1})\leq \left(\frac{t_{s+1}}{t_s}\right)^\beta R(t_s)\lambda+c(s)t_{s+1}^{\beta}. \end{align*} Note that, for $\omega\in A$, we have
\begin{align}\label{eqn:lim_{stoinfty}frac{t_{s+1}}{t_{s}}}
\lim_{s\to\infty}\frac{t_{s+1}}{t_{s}}=\lim_{s\to\infty}1+\frac{t_{s+1}-t_s}{t_{s}}=1. \end{align} As a result, for any $\hat{\lambda}\in (\lambda,1)$, there exists $\hat{s}$ such that for $s\geq \hat{s}$, we have $\left(\frac{t_{s+1}}{t_s}\right)^\beta\lambda\leq \hat{\lambda}$, and hence \begin{align*}
R(t_{s+1})\leq R(t_s)\hat{\lambda}+c(s)t_{s+1}^{\beta}. \end{align*} Therefore, \begin{align*}
R(t_s)\leq \hat{\lambda}^{s-\hat{s}} R(t_{\hat{s}})+\sum_{\tau=\hat{s}}^{s-1}c(\tau)t_{\tau+1}^{\beta}\hat{\lambda}^{s-\tau-1}. \end{align*} Taking the limits of the both sides, we have \begin{align*}
\limsup_{s\to\infty}R(t_s)&\leq \limsup_{s\to\infty}\hat{\lambda}^{s-\hat{s}} R(t_{\hat{s}})+\sum_{\tau=\hat{s}}^{s-1}c(\tau)t_{\tau+1}^{\beta}\hat{\lambda}^{s-\tau-1}\cr
&=\lim_{s\to\infty}c(s)t_{s+1}^{\beta}, \end{align*} which is implied by Lemma 3.1-(a) in \cite{ram2010distributed}. For $\omega\in A$, we have \begin{align}\label{eqn:lim_{stoinfty}c(s)t_{s+1}^{gamma}}
\lim_{s\to\infty}c(s)t_{s+1}^{\beta}
&\stackrel{(a)}{\leq} \lim_{s\to\infty}\frac{K(t_{s+1}-t_s)}{t_s^{\tilde{\beta}}}t_{s+1}^{\beta}\cr
&= \lim_{s\to\infty}\frac{K(t_{s+1}-t_s)}{t_s^{\tilde{\beta}-\beta}}\frac{t_{s+1}^{\beta}}{t_{s}^{\beta}}\cr
&\stackrel{(b)}{=} \lim_{s\to\infty}\frac{K(t_{s+1}-t_s)}{t_s^{\tilde{\beta}-\beta}}=0. \end{align} where $(a)$ follows from $b(t)\leq Kt^{-\tilde{\beta}}$, and $(b)$ follows from \eqref{eqn:lim_{stoinfty}frac{t_{s+1}}{t_{s}}}. Therefore, $\lim_{s\to\infty}R(t_s)=0$.
Now for any $t>0$ with $t_s\leq t<t_{s+1}$, let $\sigma(t)=s$. By the definition of $R(t)$, we have $R(t)\leq \left(\frac{t}{\sigma(t)}\right)^\beta R(t_{\sigma(t)})+c(\sigma(t))t_{\sigma(t)+1}^{\beta}$. Therefore, $\lim_{s\to\infty}R(t_s)=0$ and Inequality \eqref{eqn:lim_{stoinfty}c(s)t_{s+1}^{gamma}} imply \begin{align*}
\limsup_{t\to\infty}R(t)&\leq\lim_{t\to\infty}\left(\frac{t}{\sigma(t)}\right)^\beta R(t_{\sigma(t)})+c(\sigma(t))t_{\sigma(t)+1}^{\beta}=0, \end{align*} which is the desired conclusion as $R(t)=D(t)t^\beta$. \qed
Finally, we are ready to show the almost sure convergence $\lim_{t\to\infty}{\rm d}({\bf x}(t))=0$ (and more) under our connectivity assumption and a regularity condition on the input ${\bf u}(t)$ for the controlled averaging dynamics \eqref{eqn:EpsilonAlgorithm}. \begin{lemma}\label{lem:ConsensusConveregnse} Suppose that $\{W(t)\}$ satisfies Assumption \ref{assump:AssumpOnConnectivity}. Then, if ${\rm d}({\bf u}(t))<qt^{-\tilde{\beta}}$ almost surely for some $q\geq 0$, we have
$\lim_{t\to\infty} {{\rm d}({\bf x}(t))}{t^{\beta}}= 0$, {almost surely}, for $\beta<\tilde{\beta}$. \end{lemma}
\pf From inequality \eqref{lem:lemma5}, we have \begin{align}\label{eqn:diambxk}
{\rm d}({\bf x}(k))
&\leq{\rm diam}(\Phi(k,\tau)){\rm d}({\bf x}(\tau))+\sum_{s=\tau}^{k-1}{\rm diam}(\Phi(\tau,s+1)){\rm d}({\bf u}(s))\cr
&\stackrel{(a)}{\leq}{\rm diam}(k,\tau)){\rm d}({\bf x}(\tau))
+\sum_{s=\tau}^{k-1}qs^{-\tilde{\beta}}\cr
&\leq{\rm diam}(\Phi(k,\tau)){\rm d}({\bf x}(\tau))
+(k-\tau)q\tau^{-\tilde{\beta}}, \end{align}
where $(a)$ follows from ${\rm diam}(\Phi(.,.))\leq 1$. Let $C>0$ and $\lambda\in [0,1)$ be the constants satisfying the statement of Lemma~\ref{lem:ExpectationDiam}. Since ${\lambda}<1$, for $T=\lceil|\frac{\log C}{\log {\lambda}}|\rceil+1$, we have $\tilde{\lambda}\triangleq C{\lambda}^T< 1$. Then, Lemma \ref{lem:ExpectationDiam} implies that \begin{align}\label{eqn:EdiammPhihnut1hnutFhnut}
\mathbb{E}[{\rm diam}(\Phi(T(t+1),Tt))|\mathcal{F}(Tt)]\leq C{\lambda}^T= \tilde{\lambda}<1. \end{align} Let $D(t)\triangleq {\rm d}({\bf x}(Tt))$.
From inequality \eqref{eqn:diambxk}, for $\tau=Tt$ and $k=T(t+1)$, we have \begin{align*}
D(t+1)
\leq{\rm diam}(\Phi(T(t+1),Tt))D(t)
+T^{1-\tilde{\beta}}qt^{-\tilde{\beta}}. \end{align*}
Taking conditional expectation of both sides of the above inequality given $\mathcal{F}(Tt)$, we have \begin{align*}
&\mathbb{E}[D(t+1)|\mathcal{F}(Tt)]\leq\mathbb{E}[{\rm diam}(\Phi(T(t+1),Tt))|\mathcal{F}(Tt)]D(t)
+T^{1-\tilde{\beta}}qt^{-\tilde{\beta}}. \end{align*}
By letting $a(t+1)\triangleq {\rm diam}(\Phi(T(t+1),Tt))$ and $b(t)\triangleq Tqt^{-\tilde{\beta}}$, we are in the setting of Lemma \ref{lem:Dooblike}. Therefore, by Inequality \eqref{eqn:EdiammPhihnut1hnutFhnut} and ${\rm diam}(\Phi(T(t+1),Tt))\leq 1$, the conditions of Lemma~\ref{lem:Dooblike} hold, and hence, \begin{align}\label{eqn:Dttlimit}
\lim_{t\to\infty}D(t)t^\beta=0
\end{align}
almost surely.
On the other hand, letting $\tau=T\left\lfloor \frac{k}{T}\right\rfloor$ in \eqref{eqn:diambxk} we have
\begin{align*}
{\rm d}({\bf x}(k))\leq D\left(\left\lfloor \frac{k}{T}\right\rfloor\right)
+T^{1-\tilde{\beta}}q\left\lfloor \frac{k}{T}\right\rfloor^{-\tilde{\beta}}.
\end{align*}
Therefore,
\begin{align*}
\lim_{k\to\infty}{\rm d}({\bf x}(k))k^{\beta}
&\leq\lim_{k\to\infty} D\left(\left\lfloor \frac{k}{T}\right\rfloor\right)k^{\beta}
+T^{1-\tilde{\beta}}q\left\lfloor \frac{k}{T}\right\rfloor^{-\tilde{\beta}}k^{\beta}\cr
&\leq\lim_{k\to\infty} D\left(\left\lfloor \frac{k}{T}\right\rfloor\right)\left(T\left(\left\lfloor \frac{k}{T}\right\rfloor+1\right)\right)^{\beta}
+Tq(k-T)^{-\tilde{\beta}}k^{\beta}\cr&=0, \end{align*} where the last equality follows from \eqref{eqn:Dttlimit} and $\tilde{\beta}>\beta$. \qed
\section{Convergence Analysis of the Main Dynamics}\label{sec:Convergence} Finally, in this section, we will study the main dynamics \eqref{eqn:MainLineAlgorithm}, i.e., the dynamics \eqref{eqn:EpsilonAlgorithm} with the feedback policy ${\bf u}_i(t)=-\alpha(t){\bf g}_i(t)$ where ${\bf g}_i(t)\in \nabla f_i({\bf x}_i(t))$. Throughout this section, we let $\bar{{\bf x}}\triangleq \frac{1}{n}e^T{\bf x}$ for a vector ${\bf x}\in (\mathbb{R}^m)^n$,
First, we prove an inequality (Lemma~\ref{lem:FirstLyapunovDisOpt}) which plays a key role in the proof of Theorem \ref{thm:MainTheoremDisOpt} and to do so, we make use of the following result which is proven as a part of the proof of Lemma 8 (Equation (27)) in \cite{nedic2014distributed}. \begin{lemma}[\cite{nedic2014distributed}]\label{lem:AngeliaLemma} Under Assumption \ref{assump:AssumpOnFunction}, for all $v\in\mathbb{R}^m$, we have \begin{align*}
n\lrangle{\bar{{\bf g}}(t),\bar{{\bf x}}(t)-{ v}}\geq F(\bar{{\bf x}}(t))-F({ v})-2\sum_{i=1}^nL_i\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|. \end{align*} \end{lemma} \begin{lemma}\label{lem:FirstLyapunovDisOpt} For the dynamics \eqref{eqn:MainLineAlgorithm}, under Assumption \ref{assump:AssumpOnFunction}, for all ${ v}\in \mathbb{R}^m$, we have \begin{align*}
\mathbb{E}[\|\bar{{\bf x}}(t+1)-{ v}\|^2|\mathcal{F}(t)]
&\leq\|\bar{{\bf x}}(t)-{ v}\|^2+\alpha^2(t)\frac{L^2}{n^2}+\sum_{i=1}^{n}\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|^2\\
&\qquad\qquad-\frac{2\alpha(t)}{n}(F(\bar{{\bf x}}(t))-F({ v}))+\frac{4\alpha(t)}{n}\sum_{i=1}^nL_i\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|. \end{align*} \end{lemma} \pf Multiplying $\frac{1}{n}e^T$ from left to both sides of \eqref{eqn:MainLineAlgorithm}, we have \begin{align*}
\bar{{\bf x}}(t+1)&=\overline{W}(t+1){{\bf x}}(t)-\alpha(t)\bar{{\bf g}}(t)\\
&=\bar{{\bf x}}(t)-\alpha(t)\bar{{\bf g}}(t)+\overline{W}(t+1){{\bf x}}(t)-\bar{{\bf x}}(t), \end{align*} where $\overline{W}(t)\triangleq\frac{1}{n}e^TW(t)$. Therefore, we can write \begin{align*}
\|\bar{{\bf x}}(t+1)-v\|^2
&=\|\bar{{\bf x}}(t)-{{ v}}-\alpha(t)\bar{{\bf g}}(t)+\overline{W}(t+1){{\bf x}}(t)-\bar{{\bf x}}(t)\|^2\\
&=\|\bar{{\bf x}}(t)-{{ v}}\|^2+\|\alpha(t)\bar{{\bf g}}(t)\|^2+\|\overline{W}(t+1){{\bf x}}(t)-\bar{{\bf x}}(t)\|^2-2\alpha(t)\lrangle{\bar{{\bf g}}(t),\overline{W}(t+1){{\bf x}}(t)-{{ v}}}\\
&\qquad\qquad+2\lrangle{\bar{{\bf x}}(t)-{{ v}},\overline{W}(t+1){{\bf x}}(t)-\bar{{\bf x}}(t)}. \end{align*} Taking conditional expectation of both sides of { the above equality given $\mathcal{F}(t)$,} we have \begin{align*}
\mathbb{E}[\|\bar{{\bf x}}(t+1)-{{ v}}\|^2|\mathcal{F}(t)]
&=\|\bar{{\bf x}}(t)-{{ v}}\|^2+\|\alpha(t)\bar{{\bf g}}(t)\|^2+\mathbb{E}\left[\|\overline{W}(t+1){{\bf x}}(t)-\bar{{\bf x}}(t)\|^2|\mathcal{F}(t)\right]\\
&\quad-2\alpha(t)\lrangle{\bar{{\bf g}}(t),\mathbb{E}\left[\overline{W}(t+1){{\bf x}}(t)-{{ v}}|\mathcal{F}(t)\right]}+2\lrangle{(\bar{{\bf x}}(t)-{{ v}}),\mathbb{E}\left[\overline{W}(t+1){{\bf x}}(t)-\bar{{\bf x}}(t)|\mathcal{F}(t)\right]}\\
&{=}\|\bar{{\bf x}}(t)-{{ v}}\|^2+\|\alpha(t)\bar{{\bf g}}(t)\|^2+\mathbb{E}\left[\|\overline{W}(t+1){{\bf x}}(t)-\bar{{\bf x}}(t)\|^2|\mathcal{F}(t)\right]-2\lrangle{\alpha(t)\bar{{\bf g}}(t),\bar{{\bf x}}(t)-{{ v}}}. \end{align*} The last equality follows from the assumption that, $W(t+1)$ is doubly stochastic in-expectation and hence,
\[\mathbb{E}[\overline{W}(t+1)|\mathcal{F}(t)]=\frac{1}{n}e^T,\]
which implies \[\lrangle{\bar{{\bf g}}(t),\mathbb{E}\left[\overline{W}(t+1){{\bf x}}(t)-{{ v}}|\mathcal{F}(t)\right]}=\lrangle{\alpha(t)\bar{{\bf g}}(t),\bar{{\bf x}}(t)-{{ v}}},\] and
\[\mathbb{E}\left[\overline{W}(t+1){{\bf x}}(t)-\bar{{\bf x}}(t)|\mathcal{F}(t)\right]=0.\]
Note that $\overline{W}(t+1)$ is a stochastic vector (almost surely), therefore, due to the convexity of norm-square $\|\cdot\|^2$, we get \begin{align*}
\|\overline{W}(t+1){{\bf x}}(t)-\bar{{\bf x}}(t)\|^2&\leq \sum_{i=1}^n\overline{W}_i(t+1)\|{{\bf x}}_i(t)-\bar{{\bf x}}(t)\|^2\cr
&\leq \sum_{i=1}^n\|{{\bf x}}_i(t)-\bar{{\bf x}}(t)\|^2, \end{align*} {as $\overline{W}_i(t+1)\leq 1$ for all $i\in[n]$. Therefore, } \begin{align*}
\mathbb{E}[\|\bar{{\bf x}}(t+1)-{{ v}}\|^2|\mathcal{F}(t)]&{=}
\|\bar{{\bf x}}(t)-{{ v}}\|^2+\|\alpha(t)\bar{{\bf g}}(t)\|^2+\mathbb{E}\left[\|\overline{W}(t+1){{\bf x}}(t)-\bar{{\bf x}}(t)\|^2|\mathcal{F}(t)\right]-2\lrangle{\alpha(t)\bar{{\bf g}}(t),\bar{{\bf x}}(t)-{{ v}}}\cr
&\leq \|\bar{{\bf x}}(t)-{{ v}}\|^2+\|\alpha(t)\bar{{\bf g}}(t)\|^2+\sum_{i=1}^{n}\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|^2-2\lrangle{\alpha(t)\bar{{\bf g}}(t),\bar{{\bf x}}(t)-{{ v}}}. \end{align*} Finally, Lemma \ref{lem:AngeliaLemma} and the fact that \begin{align*}
\|\bar{{\bf g}}(t)\|^2=\frac{1}{n^2}\left\|\sum_{i=1}^n{\bf g}_i(t)\right\|^2\leq \frac{1}{n^2}\left(\sum_{i=1}^n\left\|{\bf g}_i(t)\right\|\right)^2\leq \frac{L^2}{n^2}, \end{align*} complete the proof.\qed
{\it Proof of Theorem \ref{thm:MainTheoremDisOpt}:} Recall that Siegmund and Robbins Theorem \cite{ROBBINS1971233} states that for a non-negative random process $\{V(t)\}$ (adapted to the filtration $\{\mathcal{F}(t)\}$) satisfying \begin{align}\label{eqn:RS}
\mathbb{E}&\left[V(t+1)|\mathcal{F}(t)\right]=(1+a(t))V(t)-b(t)+c(t), \end{align} where $a(t),b(t),c(t)\geq 0$ for all $t$, and $\sum_{t=0}^{\infty}a(t)<\infty$ and $\sum_{t=1}^{\infty}c(t)<\infty$ almost surely, $\lim_{t\to\infty}V(x(t))$ exists and $\sum_{t=1}^{\infty}b(t)<\infty$ almost surely. In order to utilize this result and Lemma \ref{lem:FirstLyapunovDisOpt}, for all $t\geq 0$, let \begin{align*}
V(t)&\triangleq\|\bar{{\bf x}}(t)-z\|^2, \cr
a(t)&\triangleq 0, \cr b(t)&\triangleq-\frac{2\alpha(t)}{n}(F(\bar{{\bf x}}(t))-F(z)),\text{ and }\cr c(t)&\triangleq\alpha^2(t)\frac{L^2}{n^2}+\sum_{i=1}^{n}\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|^2+\frac{4\alpha(t)}{n}\sum_{i=1}^nL_i\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|, \end{align*} where $z\in\mathcal{Z}$. First, note that $a(t),b(t),c(t)\geq 0$ for all $t$. To invoke the Siegmund and Robbins result \eqref{eqn:RS}, we need to to prove that $\sum_{t=0}^{\infty}c(t)<\infty$, almost surely. Since $\sum_{t=0}^{\infty}\alpha^2(t)<\infty$, it is enough to show that
\[\sum_{t=0}^{\infty}\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|^2<\infty\] and
\[\sum_{t=0}^\infty\alpha(t)\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|<\infty,\] almost surely, for all $i\in[n]$. From Lemma~\ref{lem:DiamBehrouz}-(e) and Lemma~\ref{lem:ConsensusConveregnseVariance}, we have
{\[\mathbb{E}\left[ \frac{\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|}{\alpha(t)}\right]\leq \frac{\mathbb{E}\left[{\rm d}({\bf x}(t))\right]}{\alpha(t)}\leq \sqrt{\hat{M}}<\infty\]} for some $\hat{M}>0$. Therefore, we have
\begin{align*}
\lim_{T\to\infty}\mathbb{E}\left[\sum_{t=0}^{T}\alpha(t)\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|\right]
&=\lim_{T\to\infty}\mathbb{E}\left[ \sum_{t=0}^{T}\alpha^2(t)\frac{\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|}{\alpha(t)}\right]\cr
&=\lim_{T\to\infty}\sum_{t=0}^{T}\alpha^2(t)\mathbb{E}\left[ \frac{\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|}{\alpha(t)}\right]\cr
&\stackrel{}{\leq} \sqrt{\hat{M}}\sum_{t=0}^{\infty}{\alpha^2(t)}<\infty.
\end{align*}
{Similarly, using Lemma~\ref{lem:ConsensusConveregnseVariance}, there exists some $\hat{M}>0$ such that
\begin{align*}
\frac{\mathbb{E}\left[\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|^2\right]}{\alpha^2(t)}\leq \frac{\mathbb{E}\left[{\rm d}^2({\bf x}(t))\right]}{\alpha^2(t)}\leq\hat{M},
\end{align*}
for all $t\geq 0$, where the first inequality follows from Lemma~\ref{lem:DiamBehrouz}~-~(e).} Therefore,
\begin{align*}
\lim_{T\to\infty}\mathbb{E}\left[\sum_{t=0}^{T}\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|^2\right]
&= \lim_{T\to\infty}\mathbb{E}\left[\sum_{t=0}^{T}\alpha^2(t)\frac{\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|^2}{\alpha^2(t)}\right]\cr
&\stackrel{}{\leq} \hat{M}\sum_{t=0}^{\infty}\alpha^2(t)<\infty.
\end{align*}
Therefore, using Monotone Convergence Theorem, we have
\begin{align*}
\mathbb{E}\left[\sum_{t=0}^{\infty}\alpha(t)\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|\right]&<\infty,\text{ and }\cr
\mathbb{E}\left[\sum_{t=0}^{\infty}\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|^2\right]&<\infty,
\end{align*}
which implies $\sum_{t=0}^{\infty}\alpha(t)\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|<\infty$ and $\sum_{t=0}^{\infty}\|{\bf x}_i(t)-\bar{{\bf x}}(t)\|^2<\infty$, almost surely.
Now that we showed that $c(t)$ is almost surely a summable sequence, Siegmund and Robbins Theorem implies that almost surely \[\lim_{t\to\infty}V(t)=\lim_{t\to\infty}\|\bar{{\bf x}}(t)-z\|^2\text{ exists},\]
and
\[\sum_{t=1}^{\infty}{\alpha(t)}(F(\bar{{\bf x}}(t))-F(z))<\infty.\]
For $z\in \mathcal{Z}$, let's define
\begin{align*}
\Omega_{z}\triangleq\left\{\omega\Bigg{|}\begin{array}{cc}
\lim\limits_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-z\| \text{ exists,} \\
\sum\limits_{t=1}^{\infty}{\alpha(t)}(F(\bar{{\bf x}}(t,\omega))-F^*)<\infty
\end{array}\right\},
\end{align*}
where $F^*\triangleq\min_{z\in\mathbb{R}^m}F(z)$. Per Sigmund-Robbins result, we know that $P(\Omega_{z})=1$. Now, let $\mathcal{Z}_d\subset \mathcal{Z}$ be a countable dense subset of $\mathcal{Z}$ and let
\begin{align*}
\Omega_d\triangleq\bigcap_{z\in \mathcal{Z}_d}\Omega_z.
\end{align*}
Since $\mathcal{Z}_d$ is a countable set, we have $P(\Omega_d)=1$ and for $\omega\in \Omega_d$, since $\sum_{t=1}^{\infty}{\alpha(t)}(F(\bar{{\bf x}}(t,\omega))-F^*)<\infty $ and $\alpha(t)$ is not summable, we have
\[\liminf_{t\to\infty}F(\bar{{\bf x}}(t))=F^*.\]
This fact and the fact that $F(\cdot)$ is a continuous function
implies that for all $\omega\in \Omega_d$, we have $\liminf_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-z^*(\omega)\|=0$ for some $z^*(\omega)\in\mathcal{Z}$.
{To show this, let $\{\bar{{\bf x}}({t_k})\}$ be a sub-sequence that $\lim_{k\to\infty}F(\bar{{\bf x}}(t_k,\omega))=F^*$ (such a sub-sequence depends on the sample path $\omega$).
Since $\omega\in \Omega_d$ and \[\lim\limits_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-\hat{z}\|\quad \text{exists}\] for some $\hat{z}\in \mathcal{Z}_d$, we conclude that $\{\bar{{\bf x}}(t,\omega)\}$ is a bounded sequences. Therefore, $\{\bar{{\bf x}}(t_k,\omega)\}$ is also bounded and it has an accumulation point $z^*\in \mathbb{R}^m$ and hence, there is a sub-sequence $\{\bar{{\bf x}}(t_{k_\tau},\omega)\}_{\tau\geq 0}$ of $\{\bar{{\bf x}}(t_k,\omega)\}_{k\geq 0}$ such that \[\lim_{\tau\to\infty}\bar{{\bf x}}(t_{k_\tau},\omega)=z^*.\] As a result of continuity of $F(\cdot)$, we have \[\lim_{\tau\to\infty}F(\bar{{\bf x}}(t_{k_\tau}))=F(z^*)=F^*\] and hence, $z^*\in \mathcal{Z}$. Note that the point $z^*=z^*(\omega)$ depends on the sample path $\omega$. }
Since $\mathcal{Z}_d\subseteq \mathcal{Z}$ is dense, there is a sequence $\{q^*(s,\omega)\}_{s\geq 0}$ in $\mathcal{Z}_d$ such that $\lim_{s\to\infty}\|q^*(s,\omega)-z^*(\omega)\|=0$. Note that since $\omega\in \Omega_d$, $\lim_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-q^*(s,\omega)\|$ exists for all $s\geq 0$ and we have
{\begin{align*}
\lim_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-q^*(s,\omega)\|
&=\lim_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-z^*(\omega)+z^*(\omega)-q^*(s,\omega)\|\cr
&\leq\liminf_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-z^*(\omega)\|+\|q^*(s,\omega)-z^*(\omega)\|\cr
&=\|q^*(s,\omega)-z^*(\omega)\|.
\end{align*}}
Therefore, we have
\begin{align}\label{eqn:limqstar}
\lim_{s\to\infty}\lim_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-q^*(s,\omega)\|=0.
\end{align}
On the other hand, we have
\begin{align*}
\limsup_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-z^*(\omega)\|
&=\limsup_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-q^*(s,\omega)+q^*(s,\omega)-z^*(\omega)\|\cr
&\leq\limsup_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-q^*(s,\omega)\|+\|q^*(s,\omega)-z^*(\omega)\|\cr
&=\left(\lim_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-q^*(s,\omega)\|\right)+\|q^*(s,\omega)-z^*(\omega)\|.
\end{align*}
Therefore,
\begin{align}\label{eqn:FINALLY}
\limsup_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-z^*(\omega)\|
&=\lim_{s\to\infty}\limsup_{t\to\infty }\|\bar{{\bf x}}(t,\omega)-z^*(\omega)\|\cr
&\leq\lim_{s\to\infty}\lim_{t\to\infty}\|\bar{{\bf x}}(t,\omega)-q^*(s,\omega)\|+\lim_{s\to\infty}\|q^*(s,\omega)-z^*(\omega)\|\cr
&=0,
\end{align}
where the last equality follows by combining \eqref{eqn:limqstar} and $\lim_{s\to\infty}\|q^*(s,\omega)-z^*(\omega)\|=0$. Note that \eqref{eqn:FINALLY}, implies that almost surely (i.e., for all $\omega\in\Omega_d$), we have
\[\lim_{t\to\infty}\bar{{\bf x}}(t)=z^*(\omega)\]
exists and it belongs to $\mathcal{Z}$.
{Finally, according to Assumption \ref{assump:AssumpOnFunction} and \ref{assump:AssumpOnStepSize}, we have
\[{\rm d}(\alpha(t){\bf g}(t))\leq 2{K}t^{-\beta}\max_{i\in[n]}L_i.\] Therefore, from lemma \ref{lem:ConsensusConveregnse}, we conclude that $\lim_{t\to\infty}{\rm d}({\bf x}(t))=0$ almost surely, and hence \[\lim_{t\to\infty}\|\bar{{\bf x}}(t)-{\bf x}_i(t)\|=0\quad \text{almost surely}.\]
Since we almost surely have $\lim_{t\to\infty}\bar{{\bf x}}(t)=z^*$ for a random vector $z^*$ supported in $\mathcal{Z}$, we have $\lim_{t\to\infty}{\bf x}_i(t)=z^*$ for all $i\in[n]$ almost surely and the proof is complete. \qed}
\section{Conclusion and Future Research}\label{Conclusion} In this work, we showed that the averaging-based distributed optimization solving algorithm over dependent random networks converges to an optimal random point under the standard conditions on the objective function and network formation that is conditionally $B$-connected. To do so, we established a rate of convergence for the second moment of the autonomous averaging dynamics over such networks and used that to study the convergence of the sample-paths and second moments of the controlled variation of those dynamics.
Further extensions of the current work to non-convex settings, accelerated algorithms, and distributed online learning algorithms are of interest for future consideration on this topic. \appendix
{\it Proof of Lemma \ref{lem:DiamBehrouz}}:
For the proof of part (a), let ${\bf x}_{i}^{(k)}$ be the $k$th coordinate of ${\bf x}_i$, and define the vector $y=[y^{(1)},\ldots,y^{(m)}]^T$ where \[y^{(k)}=\frac{1}{2}({u}^{(k)}+{U}^{(k)}),\] with ${u}^{(k)}=\min_{i\in [n]}{\bf x}_i^{(k)}$ and ${U}^{(k)}=\max_{i\in [n]}{\bf x}_i^{(k)}$.
Therefore, for $\ell\in[n]$, we have \begin{align}\label{eqn:ykmax}
\|{\bf x}_\ell-y\|&=\max_{k\in[m]}|{\bf x}_{\ell}^{(k)}-y^{(k)}|\cr
&\stackrel{}{=}\max_{k\in[m]}\left|{\bf x}_{\ell}^{(k)}-\frac{1}{2}(u^{(k)}+U^{(k)})\right|\cr
&\stackrel{(a)}\leq\max_{k\in[m]}\frac{1}{2}\left|u^{(k)}-U^{(k)}\right|\cr
&=\frac{1}{2}{\rm d}({\bf x}), \end{align} where $(a)$ follows from $u^{(k)}\leq {\bf x}_{\ell}^{(k)}\leq U^{(k)}$.
Also, we have \begin{align*}
{\rm d}(A{\bf x})&=\max_{i,j\in[n]}\Bigg{\|}\sum_{\ell=1}^na_{i\ell}{\bf x}_\ell-\sum_{\ell=1}^na_{j\ell}{\bf x}_\ell\Bigg{\|}\cr
&=\max_{i,j\in[n]}\Bigg{\|}\sum_{\ell=1}^na_{i\ell}({\bf x}_\ell-{y})-\sum_{\ell=1}^na_{j\ell}({\bf x}_\ell-{y})+\sum_{\ell=1}^n(a_{j\ell}-a_{i\ell}){y}\Bigg{\|}\cr
&=\max_{i,j\in[n]}\Bigg{\|}\sum_{\ell=1}^n(a_{i\ell}-a_{j\ell})({\bf x}_\ell-{y})\Bigg{\|} \end{align*} where the last equality holds as $A$ is a row-stochastic matrix and hence, $\sum_{\ell=1}^n(a_{j\ell}-a_{i\ell})=0$. Therefore, \begin{align*}
{\rm d}(A{\bf x})&=\max_{i,j\in[n]}\Bigg{\|}\sum_{\ell=1}^n(a_{i\ell}-a_{j\ell})({\bf x}_\ell-{y})\Bigg{\|}\cr
&\stackrel{(a)}{\leq}\max_{i,j\in[n]}\sum_{\ell=1}^n|a_{i\ell}-a_{j\ell}|{\|}{\bf x}_\ell-{y}{\|}\cr
&\stackrel{(b)}{\leq}\max_{i,j\in[n]}\frac{1}{2}{\rm d}({\bf x})\sum_{\ell=1}^n|a_{i\ell}-a_{j\ell}|\cr
&\leq{\rm diam}(A){\rm d}({\bf x}), \end{align*} where $(a)$ follows from the triangle inequality, and $(b)$ follow from \eqref{eqn:ykmax}. For the part (b), we have \begin{align*}
{\rm d}({\bf x}+{\bf y})&=\max_{i,j\in[n]}\|({\bf x}+{\bf y})_i-({\bf x}+{\bf y})_j\|\cr
&=\max_{i,j\in[n]}\|{\bf x}_i-{\bf x}_j+{\bf y}_i-{\bf y}_j\|\cr
&\stackrel{(b)}{\leq}\max_{i,j\in[n]}\left(\|{\bf x}_i-{\bf x}_j\|+\|{\bf y}_i-{\bf y}_j\|\right)\cr
&\leq\max_{i,j\in[n]}\|{\bf x}_i-{\bf x}_j\|+\max_{i,j\in[n]}\|{\bf y}_i-{\bf y}_j\|\cr
&={\rm d}({\bf x})+{\rm d}({\bf y}), \end{align*} where $(b)$ follows from the triangle inequality.
For the proof of part (c), we have \begin{align*}
{\rm diam}(A)&=\max_{i,j\in[n]}\sum_{\ell=1}^n \frac{1}{2}|a_{i\ell}-a_{j\ell}|\cr
&=\max_{i,j\in[n]}\sum_{\ell=1}^n \left(\frac{1}{2}(a_{i\ell}+a_{j\ell})-\min\{a_{i\ell},a_{j\ell}\}\right)\cr
&\stackrel{(a)}{=}\max_{i,j\in[n]}1-\sum_{\ell=1}^n \min\{a_{i\ell},a_{j\ell}\}\cr
&=1-\min_{i,j\in[n]}\sum_{\ell=1}^n \min\{a_{i\ell},a_{j\ell}\}\cr
&=1-\Lambda(A), \end{align*} where $(a)$ follows from the fact that $A$ is row-stochastic. The proof of part (d) follows from part (c) and Lemma \ref{lem:lambdadiam}.
For the part (e), due to the convexity of $\|\cdot\|$, we have \begin{align*}
\left\|{\bf x}_i-\sum_{j=1}^{n}\pi_j{\bf x}_j\right\|\leq \sum_{j=1}^{n}\pi_j\left\|{\bf x}_i-{\bf x}_j\right\|\leq \sum_{j=1}^{n}\pi_j{\rm d}({\bf x})={\rm d}({\bf x}). \end{align*} \qed
{\it Proof of Lemma \ref{lem:ProductwithHistory}}: We prove by induction on $K_2$. By the assumption, the lemma is true for $K_2=K_1$. For $K_2>K_1$, we have \begin{align*}
\mathbb{E}\left[\prod_{k=K_1}^{K_2+1}Y(k)\bigg{|}\mathcal{F}(K_1-1)\right]
&=\mathbb{E}\left[\mathbb{E}\left[\prod_{k=K_1}^{K_2+1}Y(k)\bigg{|}\mathcal{F}(K_2)\right]\bigg{|}\mathcal{F}(K_1-1)\right]\cr
&=\mathbb{E}\left[\mathbb{E}\left[Y(K_2+1){|}\mathcal{F}(K_2)\right]\prod_{k=K_1}^{K_2}Y(k)\bigg{|}\mathcal{F}(K_1-1)\right]\cr
&\leq\mathbb{E}\left[a(K_2+1)\prod_{k=K_1}^{K_2}Y(k)\bigg{|}\mathcal{F}(K_1-1)\right]\cr
&\leq \prod_{k=K_1}^{K_2+1}a(k). \end{align*} \qed
{\it Proof of Lemma \ref{lem:approxfunction}}:
Consider $\hat{\tau}\geq t_0$ such that $\hat{\theta}\triangleq\sup_{t\geq\hat{\tau}}\frac{\beta({t})}{\beta({t}+1)}\theta<1$ and let $D(t)\triangleq \sum_{s=\tau}^{t-1}\beta(s)\theta^{t-s}$. Dividing both sides by $\beta(t)>0 $, for $t> \hat{\tau}$, we have \begin{align*}
\frac{D(t)}{\beta(t)}&=\sum_{s=\tau}^{t-1}\frac{\beta(s)}{\beta(t)}\theta^{t-s}\cr
&=\sum_{s=\tau}^{t-1}\prod_{\kappa=s}^{t-1}\frac{\beta(\kappa)}{\beta(\kappa+1)}\theta\cr
&\leq\sum_{s=\tau}^{\hat{\tau}-1}\prod_{\kappa=s}^{t-1}\frac{\beta(\kappa)}{\beta(\kappa+1)}\theta+\sum_{s=\hat{\tau}}^{t-1}\hat{\theta}^{t-s}\cr
&=\sum_{s=\tau}^{\hat{\tau}-1}\prod_{\kappa=s}^{t-1}\frac{\beta(\kappa)}{\beta(\kappa+1)}\theta+\sum_{k=1}^{t-\hat{\tau}}\hat{\theta}^{k}\cr
&\leq \sum_{s=\tau}^{\hat{\tau}-1}\prod_{\kappa=s}^{t-1}\frac{\beta(\kappa)}{\beta(\kappa+1)}\theta+\frac{\hat{\theta}}{1-\hat{\theta}}. \end{align*} Let \[M_1\triangleq \sup_{t>\hat{\tau}}\sup_{\hat{\tau}\geq \tau\geq t_0}\sum_{s=\tau}^{\hat{\tau}-1}\prod_{\kappa=s}^{t-1}\frac{\beta(\kappa)}{\beta(\kappa+1)}\theta+\frac{\hat{\theta}}{1-\hat{\theta}}.\] Note that \begin{align*}
\lim_{t\to\infty}\prod_{\kappa=s}^{t-1}\frac{\beta(\kappa)}{\beta(\kappa+1)}\theta\leq\lim_{t\to\infty}\hat{\theta}^{t-\hat{\tau}}\prod_{\kappa=s}^{\hat{\tau}-1}\frac{\beta(\kappa)}{\beta(\kappa+1)}\theta=0. \end{align*} Therefore, $\sup_{t\geq\tau}\prod_{\kappa=s}^{t-1}\frac{\beta(\kappa)}{\beta(\kappa+1)}\theta<\infty$, and hence, $M_1<\infty$. Thus, $D(t)\leq \max\{M_1,M_2\}\beta(t)$, where $M_2\triangleq \max_{\hat{\tau}\geq t\geq \tau\geq t_0}\sum_{s=\tau}^{t-1}\frac{\beta(s)}{\beta(t)}\theta^{t-s}$, and the proof is complete.\qed
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Long-time behavior of micropolar fluid equations in cylindrical domains} \author{Bernard Nowakowski} \thanks{The author is partially supported by Polish KBN grant N N201 393137} \address{Bernard Nowakowski\\ Institute of Mathematics\\ Polish Academy of Sciences\\ \'Snia\-deckich 8\\ 00-956 Warsaw\\ Poland} \email{[email protected]}
\subjclass[2000]{35Q30, 76D05}
\date{November, 2011}
\begin{abstract}
In this paper we investigate the existence of $H^1$-uniform attractor and long-time behavior of solutions to non-autonomous micropolar fluid equations in three dimensional cylindrical domains. In our considerations we take into account that existence of global and strong solutions is proved under the assumption on smallness of change of the initial and the external data along the axis of the cylinder. Therefore, we refine the concept of uniform attractor by adopting the idea which was proposed by J.W. Cholewa and T. DÅ‚otko in \cite{chol1}. \end{abstract}
\keywords{micropolar fluids, cylindrical domains, qualitative properties of solutions, uniform restricted attractor} \maketitle
\section{Introduction}
The study of attractors is an important part of examining dynamical systems. It was thoroughly investigated in many works (see e.g. \cite{Hale:1988fk}, \cite{Ladyzhenskaya:1991uq}, \cite{Temam:1997vn}, \cite{chol}, \cite{che}, \cite{Sell:2002kx}). These classical results cover many both autonomous and non-autonomous equations of mathematical physics. Roughly speaking, the general abstract theory justifying attractors' existence may be applied when the equations describing autonomous systems posses unique, strong and global solutions. For non-autonomous systems some additional uniformity for external data is required. Further obstacles arise when unbounded domains are examined (see e.g. \cite{luk5} and \cite{zhao} and references therein).
In three dimensions the situation for nonlinear evolution problems is even more complex. In case of micropolar equations we lack information about the regularity of weak solutions at large. Thus, results concerning the existence of uniform attractors are merely conditional (see e.g. \cite{che}). In this article we present some remedy which is based on \cite{chol1}. It takes into account that global and strong solutions exist if some smallness of $L_2$-norms on the rate of change of the external and the initial data is assumed (see \cite{2012arXiv1205.4507N}). This leads to a restriction of the uniform attractor to a proper phase space.
Introduced by A.~Eringen in \cite{erin}, the micropolar fluid equations form a useful generalization of the classical Navier-Stokes model in the sense that they take into account the structure of the media they describe. With many potential applications (see e.g. \cite{pop1}, \cite{pop2}, \cite{ari}, \cite{luk1}) they became an interesting and demanding area of interest for mathematicians and engineers (see e.g. \cite{erin}, \cite{pad}, \cite{gald}, \cite{ort}, \cite{luk1}, \cite{luk2} et al.)
In this article we study the following initial-boundary value problem \begin{equation}
\begin{aligned}\label{p1}
&v_{,t} + v\cdot \nabla v - (\nu + \nu_r)\triangle v + \nabla p = 2\nu_r\Rot\omega + f & &\text{in } \Omega^{\infty} := \Omega\times(0,\infty),\\
&\Div v = 0 & &\text{in } \Omega^{\infty},\\
&\begin{aligned}
&\omega_{,t} + v\cdot \nabla \omega - \alpha \triangle \omega - \beta\nabla\Div\omega + 4\nu_r\omega \\
&\mspace{60mu} = 2\nu_r\Rot v + g
\end{aligned}& &\text{in } \Omega^{\infty}, \\
&v\cdot n = 0 & &\text{on $S^{\infty} := S\times(0,\infty)$}, \\
&\Rot v \times n = 0 & &\text{on $S^{\infty}$}, \\
&\omega = 0 & &\text{on $S_1^{\infty}$}, \\
&\omega' = 0, \qquad \omega_{3,x_3} = 0 & &\text{on $S_2^{\infty}$}, \\
&v\vert_{t = 0} = v(0), \qquad \omega\vert_{t = 0} = \omega(0) & &\text{in $\Omega$}.
\end{aligned} \end{equation} By $\Omega \subset \mathbb{R}^3$ we mean a cylindrical type domain, i.e. \begin{equation*}
\Omega = \left\{\left(x_1,x_2\right)\in\mathbb{R}^2\colon \varphi\left(x_1,x_2\right) \leq c_0\right\} \times \left\{ x_3\colon -a \leq x_3 \leq a\right\}, \end{equation*} where $a > 0$ and $c_0$ are real constants, $\varphi$ is a closed curve of class $\mathcal{C}^2$. The functions $v\colon\mathbb{R}^3\to\mathbb{R}^3$, $\omega\colon\mathbb{R}^3\to\mathbb{R}^3$ and $p\colon\mathbb{R}^3\to\mathbb{R}$ denote the velocity field, the micropolar field and the pressure. The external data (forces and momenta) are represented by $f$ and $g$ respectively. The viscosity coefficients $\nu$ and $\nu_r$ are real and positive.
The choice of the boundary conditions for $v$ and $\omega$ was thoroughly described in \cite{2012arXiv1205.4046N}. Note that usually the zero Dirichlet condition is assumed for both the velocity and micropolar field. From physical point of view this is not suitable for some classes of fluids (see e.g. \cite{bou}). In fact, the explanation of particles behavior at the boundary is far more complex and was extensively discussed in \cite{con}, \cite{all3}, \cite{mig} and \cite{eri6}.
The article is organized as follows: in the next Section we present an overview of the current state of the art. Subsequently we introduce notation and give some preliminary results. In Section \ref{sec4} the main results are formulated. Section \ref{sec5} is entirely devoted to the basics of semi-processes. In Sections \ref{sec6}, \ref{sec7} and \ref{sec8} we prove the existence of the restricted uniform attractor and analyze the behavior of the micropolar fluid flow for the large viscosity $\nu$. We demonstrate that for time independent data the global and unique solution $(v(t),\omega(t))$ converges to the solutions of the stationary problem and if $\nu_r \approx 0$ then the trajectories $(v(t),\omega(t))$ of micropolar flow and $u(t)$ of the Navier-Stokes flow differ indistinctively.
\section{State-of-the-art}
The study of the uniform attractors has mainly been focused on the the case of unbounded domains (see \cite{ukaszewicz:2004kx} and \cite{Dong:2006uq} and the references therein) which is barely covered by general abstract theory introduced in \cite{che}. Note that both these studies cover only two dimensional case. In three dimensions no results have been obtained so far due to lack of regularity of weak solutions. Another obstacle arises from relaxing the assumption of certain uniformity of the external data. To avoid these difficulties, the concepts of trajectory and pull-back attractors have been suggested and applied (see \cite{Chen:2007zr}, \cite{Chen:2009ys}, \cite{luk7}, \cite{Kapustyan:2010vn} and \cite{tar2}).
\section{Notation and preliminary results}\label{sec3}
In this paper we use the following function spaces: \begin{enumerate}
\item[$\bullet$] $W^m_p(\Omega)$, where $m \in \mathbb{N}$, $p \geq 1$, is the closure of $\mathcal{C}^{\infty}(\Omega)$ in the norm
\begin{equation*}
\norm{u}_{W^m_p(\Omega)} = \left(\sum_{\abs{\alpha} \leq m} \norm{\Ud^{\alpha} u}_{L_p(\Omega)}^p\right)^{\frac{1}{p}},
\end{equation*}
\item[$\bullet$] $H^k(\Omega)$, where $k \in \mathbb{N}$, is simply $W^k_2(\Omega)$,
\item[$\bullet$] $W^{2,1}_p(\Omega^t)$, where $p \geq 1$, is the closure of $\mathcal{C}^{\infty}(\Omega\times(t_0,t_1))$ in the norm
\begin{equation*}
\norm{u}_{W^{2,1}_p(\Omega^t)} = \left(\int_{t_0}^{t_1}\!\!\!\int_{\Omega} \abs{u_{,xx}(x,s)}^p + \abs{u_{,x}(x,s)}^p + \abs{u(x,s)}^p + \abs{u_{,t}(x,s)}^p\, \ud x\, \ud s\right)^{\frac{1}{p}},
\end{equation*}
\item[$\bullet$] $\mathcal{V}$ is the set of all smooth solenoidal functions whose normal component vanishes on the boundary
\begin{equation*}
\mathcal{V} = \left\{u\in\mathcal{C}^\infty(\Omega)\colon \Div u = 0 \text{ in } \Omega, \ u\cdot n\vert_S = 0 \right\},
\end{equation*}
\item[$\bullet$] $H$ is the closure of $\mathcal{V}$ in $L_2(\Omega)$,
\item[$\bullet$] $\widehat H = H \times L_2(\Omega)$,
\item[$\bullet$] $V$ is the closure of $\mathcal{V}$ in $H^1(\Omega)$,
\item[$\bullet$] $\widehat V = V\times H^1(\Omega)$,
\item[$\bullet$] $V^k_2(\Omega^t)$, where $k \in \mathbb{N}$, is the closure of $\mathcal{C}^{\infty}(\Omega\times(t_0,t_1))$ in the norm
\begin{equation*}
\norm{u}_{V^k_2(\Omega^t)} = \underset{t\in (t_0,t_1)}{\esssup}\norm{u}_{H^k(\Omega)} \\
+\left(\int_{t_0}^{t_1}\norm{\nabla u}^2_{H^{k}(\Omega)}\, \ud t\right)^{1/2}.
\end{equation*} \end{enumerate}
To shorten the energy estimates we use: \begin{equation}\label{eq14}
\begin{aligned}
E_{v,\omega}(t) &:= \norm{f}_{L_2(t_0,t;L_{\frac{6}{5}}(\Omega))} + \norm{g}_{L_2(t_0,t;L_{\frac{6}{5}}(\Omega))} + \norm{v(t_0)}_{L_2(\Omega)} + \norm{\omega(t_0)}_{L_2(\Omega)}, \\
E_{h,\theta}(t) &:= \norm{f_{,x_3}}_{L_2(t_0,t;L_{\frac{6}{5}}(\Omega))} + \norm{g_{,x_3}}_{L_2(t_0,t;L_{\frac{6}{5}}(\Omega))} + \norm{h(t_0)}_{L_2(\Omega)} + \norm{\theta(t_0)}_{L_2(\Omega)}.
\end{aligned} \end{equation} The following function is of particular interest \begin{equation}\label{eq26}
\delta(t) := \norm{f_{,x_3}}^2_{L_2(\Omega^t)} + \norm{g_{,x_3}}^2_{L_2(\Omega^t)} + \norm{\Rot h(t_0)}^2_{L_2(\Omega)} + \norm{h(t_0)}^2_{L_2(\Omega)} + \norm{\theta(t_0)}^2_{L_2(\Omega)}. \end{equation} It expresses the smallness assumption which has to be made in order to prove the existence of regular solutions (see \cite{2012arXiv1205.4046N}. It contains no $L_2$-norms of the initial or the external data but only $L_2$-norms of their derivatives alongside the axis of the cylinder. In other words the data need not to be small but small must be their rate of change.
With the above notation the following Theorem was proved in \cite{2012arXiv1205.4507N}. It is fundamental for further considerations. \begin{thm}\label{thm0}
Let $0 < T < \infty$ be sufficiently large and fixed. Suppose that $v(0), \omega(0) \in H^1(\Omega)$ and $\Rot h(0) \in L_2(\Omega)$. In addition, let the external data satisfy $f_3\vert_{S_2} = 0$, $g'\vert_{S_2} = 0$,
\begin{equation*}
\begin{aligned}
&\norm{f(t)}_{L_2(\Omega)} \leq \norm{f(kT)}_{L_2(\Omega)}e^{-(t - kT)}, & & &\norm{f_{,x_3}(t)}_{L_2(\Omega)} &\leq \norm{f_{,x_3}(kT)}_{L_2(\Omega)}e^{-(t - kT)},\\
&\norm{g(t)}_{L_2(\Omega)} \leq \norm{g(kT)}_{L_2(\Omega)}e^{-(t - kT)}, & & &\norm{g_{,x_3}(t)}_{L_2(\Omega)} &\leq \norm{g_{,x_3}(kT)}_{L_2(\Omega)}e^{-(t - kT)}
\end{aligned}
\end{equation*}
and
\begin{equation*}
\sup_k \left\{f(kT),f_{,x_3}(kT),g(kT),g_{,x_3}(kT)\right\} < \infty.
\end{equation*}
Then, for sufficiently small $\delta(T)$ there exists a unique and regular solution to problem \eqref{p1} on the interval $(0,\infty)$. Moreover,
\begin{multline*}
\norm{v}_{W^{2,1}_2(\Omega^{\infty})} + \norm{\omega}_{W^{2,1}_2(\Omega^{\infty})} + \norm{\nabla p}_{L_2(\Omega^{\infty})} \leq \sup_k \Big(\norm{f}_{L_2(\Omega^{kT})} + \norm{f_{,x_3}}_{L_2(\Omega^{kT})} \\
+ \norm{g}_{L_2(\Omega^{kT})} + \norm{g_{,x_3}}_{L_2(\Omega^{kT})} + \norm{v(0)}_{H^1(\Omega)} + \norm{\omega(0)}_{H^1(\Omega)} + 1\Big)^3.
\end{multline*} \end{thm}
\section{Main results}\label{sec4}
The main result of this paper reads: \begin{thm}\label{thm1}
Let the assumption from Theorem \ref{thm0} be satisfied. Suppose that
\begin{equation*}
\norm{u\big((k + 1)T + t\big) - u(kT + t)}_{L_2(\Omega)} < \epsilon,
\end{equation*}
for any $t \in [0,T]$ and $k \in \mathbb{N}$, where $u$ is any element of the set $\{f,g,f_{,x_3},g_{,x_3}\}$. Then, the family of semi-processes $\{U_\sigma(t,\tau)\colon t\geq\tau\geq 0\}$, $\sigma \in \Sigma^{\epsilon}_{\sigma_0}$ corresponding to problem \eqref{p1} has an uniform attractor $\mathcal{A}_{\Sigma^{\epsilon}_{\sigma_0}}$ which coincides with the uniform attractor $\mathcal{A}_{\omega(\Sigma^{\epsilon}_{\sigma_0})}$ of the family of semiprocesses $\{U_\sigma(t,\tau)\colon t\geq\tau\geq 0\}$, $\sigma \in \omega\left(\Sigma^{\epsilon}_{\sigma_0}\right)$, i.e.
\begin{equation*}
\mathcal{A}_{\Sigma^{\epsilon}_{\sigma_0}} = \mathcal{A}_{\omega\big(\Sigma^{\epsilon}_{\sigma_0}\big)}.
\end{equation*} \end{thm} For explanatory notation we refer to Section \ref{sec5}.
The second result deals with problem \eqref{p1} when the external data are time independent and the viscosity $\nu$ is large enough. It is expected that in such a case the solutions tend to stationary solutions.
\begin{thm}[convergence to stationary solutions]\label{thm3}
Suppose that the external data $f$ and $g$ do not depend on time. Then, there exists a positive constant $\nu_*$ such that for all $\nu > \nu_*$ the stationary problem
\begin{equation*}
\begin{aligned}
&v_{\infty}\cdot \nabla v_{\infty} - (\nu + \nu_r)\triangle v_{\infty} + \nabla p_{\infty} = 2\nu_r\Rot\omega_{\infty} + f & &\text{in } \Omega,\\
&\Div v_{\infty} = 0 & &\text{in } \Omega,\\
&\begin{aligned}
&v_{\infty}\cdot \nabla \omega_{\infty} - \alpha \triangle \omega_{\infty} - \beta\nabla\Div\omega_{\infty} + 4\nu_r\omega_{\infty} \\
&\mspace{60mu} = 2\nu_r\Rot v_{\infty} + g
\end{aligned}& &\text{in } \Omega, \\
&v_{\infty}\cdot n = 0 & &\text{on $S$}, \\
&\Rot v_{\infty} \times n = 0 & &\text{on $S$}, \\
&\omega_{\infty} = 0 & &\text{on $S_1$}, \\
&\omega'_{\infty} = 0, \qquad \omega_{\infty 3,x_3} = 0 & &\text{on $S_2$}
\end{aligned}
\end{equation*}
has a unique solution $(v_{\infty},\omega_{\infty})$ for which
\begin{equation}\label{eq43}
\norm{v_\infty}_{H^2(\Omega)} + \norm{p_{\infty}}_{H^1(\Omega)} + \norm{\omega_{\infty}}_{H^2(\Omega)} \leq F\big(\norm{f}_{L_2(\Omega)},\norm{g}_{L_2(\Omega)}\big)
\end{equation}
holds, where $F\colon[0,\infty)\times[0,\infty]\to[0,\infty)$, $F(0,0) = 0$ is a continuous and increasing function. Moreover, under assumptions of Theorem \ref{thm1} there exists a unique solution $(v(t),\omega(t))$ to problem \eqref{p1} converging to the the stationary solution $(v_{\infty},\omega_{\infty})$ as $t\to\infty$ and satysfying
\begin{equation*}
\norm{v(t) - v_{\infty}}^2_{L_2(\Omega)} + \norm{\omega(t) - \omega_{\infty}}^2_{L_2(\Omega)} \leq \left(\norm{v(0) - v_{\infty}}^2_{L_2(\Omega)} + \norm{\omega(0) - \omega_{\infty}}^2_{L_2(\Omega)}\right)e^{-\Delta(\nu)t},
\end{equation*}
where
\begin{equation*}
\Delta(\nu) = c_1(\nu) - \frac{3c_2}{\nu}F^4\big(\norm{f_\infty}_{L_2(\Omega)},\norm{g_\infty}_{L_2(\Omega)}\big)
\end{equation*}
and
\begin{equation*}
\begin{aligned}
c_1(\nu) &= \frac{\min\{\nu,\alpha\}}{c_{\Omega}}, \\
c_2 &= \frac{c_{I,\Omega}}{\min\{1,\alpha\}}.
\end{aligned}
\end{equation*} \end{thm}
The last result of this articles establishes certain relation between the trajectories of the standard Navier-Stokes equations and micropolar equations.
\begin{thm}\label{thm2}
Let the pair $(u,\Theta)$ be a solution to the following initial-boundary value problem
\begin{equation}\label{p11}
\begin{aligned}
&u_{,t} - \nu\triangle u + (u\cdot \nabla) u + \nabla q = f & &\text{in $\Omega^t$}, \\
&\Div u = 0 & &\text{in $\Omega^t$}, \\
&\Theta_{,t} - \alpha \triangle \Theta - \beta \nabla \Div\Theta + (u\cdot \nabla)\Theta = g & &\text{in $\Omega^t$}, \\
&\Rot u \times n = 0 & &\text{on $S^t$}, \\
&u \cdot n = 0 & &\text{on $S^t$}, \\
&\Theta = 0 & &\text{on $S_1^t$}, \\
&\Theta' = 0, \qquad \Theta_{3,x_3} = 0 & &\text{on $S_2^t$}, \\
&u\vert_{t = t_0} = u(t_0), \qquad \Theta\vert_{t = t_0} = \Theta(t_0) & &\text{in $\Omega\times\{t = t_0\}$}.
\end{aligned}
\end{equation}
Suppose that $\nu > 0$ is sufficiently large. Finally, let the assumptions of Theorem \ref{thm1} hold. Then, for any $(u(t_0),\Theta(t_0)) \in B_{(v(t_0),\omega(t_0))}(R)$ (i.e. ball centered at $(v(t_0),\omega(t_0))$ with radius $R$), where $R > 0$, there exists $t^* = t^*(R)$ such that for all $t \geq t^*$ the trajectory $(v(t),\omega(t))$ lies in the $\epsilon$-neighborhood of the trajectory $(u(t),\Theta(t))$. \end{thm}
\section{Basics of semiprocesses}\label{sec5}
We begin with recalling a few facts from \cite[Ch. 1]{chol} and \cite[Ch. 2]{che}.
Let $\{T(t)\}$ be a semigroup acting on a complete metric or Banach space $X$. Denote by $\mathcal{B}(X)$ the set of all bounded sets in $X$ with respect to metric in $X$. We say that $P \subset X$ is an \emph{attracting set} for $\{T(t)\}$ if for any $B \in \mathcal{B}(X)$ \begin{equation*}
\dist_X\big(T(t)B,P\big)\xrightarrow[t\to\infty]{} 0. \end{equation*}
Now we may define an attractor: \begin{defi}
A set $\mathcal{A}$ is called \emph{global attractor} for the semigroup $\{T(t)\}$, if it satisfies:
\begin{itemize}
\item $\mathcal{A}$ is compact in $X$,
\item $\mathcal{A}$ is an attracting set for $\{T(t)\}$,
\item $\mathcal{A}$ is strictly invariant with respect to $\{T(t)\}$, i.e. $T(t)\mathcal{A} = \mathcal{A}$ for all $t \geq 0$.
\end{itemize} \end{defi}
\begin{defi}
For any $B \in \mathcal{B}(X)$ the set
\begin{equation*}
\omega(B) = \bigcap_{\tau \geq 0} \overline{\bigcup_{t\geq \tau}T(t) B}^{X}
\end{equation*}
is called an \emph{$\omega$-limit set} for $B$. \end{defi}
The existence of the global attractor in ensured by the following Proposition: \begin{prop}\label{prop2}
Suppose that $\{T(t)\}$ is a continuous semigroup in a complete metric space $X$ having a compact attracting set $K \Subset X$. Then the semigroup $\{T(t)\}$ has a global attractor $\mathcal{A}$ ($\mathcal{A} \Subset K$). The attractor coincides with $\omega(K)$, i.e. $\mathcal{A} = \omega(K)$. \end{prop}
\begin{proof}
We refer the reader to \cite[Ch. 2, \S 3, Theorem 3.1]{che}. \end{proof}
To give the proof of Theorem \ref{thm1} in the next Section we introduce the notions and concept of semi-processes. For this purpose we make use of \cite[Ch. $7$]{che}.
Let us rewrite equation \eqref{p1}$_1$ in the abstract form \begin{equation}\label{eq37}
(v_{,t},\omega_{,t}) = A(v,\omega,t) = A_{\sigma(t)}(v,\omega), \qquad t\in \mathbb{R}^+, \end{equation} where the right-hand side depends directly on the time-dependent forces and momenta, which is indicated by the presence of function $\sigma(t) = (f(t),g(t))$. The function $\sigma(t)$ is referred as the \emph{time symbol} (or the \emph{symbol}).
By $\Psi$ we denote a Banach space, which contains the values of $\sigma(t)$ for almost all $t \in \mathbb{R}_+$. In our case $\Psi = L_{\frac{6}{5}}(\Omega)\cap L_2(\Omega)\times L_{\frac{6}{5}}(\Omega)\cap L_2(\Omega)$ (although we could write $L_2(\Omega)\times L_2(\Omega)$ only since $\Omega$ is bounded but we want to point out that in certain cases a weaker assumption on forces can be imposed). Moreover we assume that $\sigma(t)$, as a function of $t$, belongs to a topological space $\Xi_+ := \{\xi(t), t \in\mathbb{R}_+\colon \xi(t) \in \Psi \ \text{for a.e. $t\in\mathbb{R}_+$}\}$. It is tempting to write $\Xi_+ = L_{2,loc}(0,\infty;\Psi)$ but we must take into account certain restrictions, which were imposed on the data (see Theorem \ref{thm1}). Thus, we describe $\Xi_+$ in greater detail below.
Consider then the family of equations of the form of \eqref{eq37}, where $\sigma(t) \in \Sigma \subseteq \Xi_+$. The space $\Sigma$ is called \emph{symbol space} and is assumed to contain, along with $\sigma(t)$, all translations, i.e. $\sigma(t + s) = T(s)\sigma(t) \in \Sigma$ for all $s\geq 0$, where $T(s)$ is a translation operator acting on $\Xi_+$. Furthermore, we fix $\sigma_0(t)$, $t \geq 0$. Consider the closure in $\Xi_+$ of the set \begin{equation*}
\left\{T(s)\sigma_0(t)\colon s \geq 0\right\} = \left\{\sigma_0(t + s)\colon s\geq 0\right\}. \end{equation*} We call this closure the \emph{hull} of the symbol $\sigma_0(t)$ and denote by $\mathcal{H}_+(\sigma_0)$, \begin{equation*}
\mathcal{H}_+(\sigma_0) = \overline{\left\{T(s)\sigma_0\colon s \geq 0\right\}}^{\Xi_+}. \end{equation*} \begin{defi}
The symbol $\sigma_0(t) \in \Xi_+$ is called \emph{translation compact} in $\Xi_+$ if the hull $\mathcal{H}_+(\sigma_0)$ is compact in $\Xi_+$. \end{defi}
In view of the above definition we would like to set $\Sigma = \mathcal{H}_+(\sigma_0)$. However for this purpose we need to determine how to choose $\Sigma$ to ensure its compactness in $\Xi_+$. This is crucial for the proof of the existence of the uniform attractor because we use: \begin{prop}\label{prop1}
Let $\sigma_0(s)$ be a translation compact function in $\Xi_+$. Let the family of semi-processes $\{U(t,\tau)\colon t\geq \tau\geq 0\}$, $\sigma \in \mathcal{H}_+(\sigma_0)$ be uniformly asymptotically compact and $\big(E\times \mathcal{H}_+(\sigma_0),E\big)$-continuous. Let $\omega(\mathcal{H}_+(\sigma_0))$ be the attractor of the translation semi-group $\{T(t)\}$ acting on $\mathcal{H}_+(\sigma_0)$. Consider the corresponding family of semi-processes $\{U_\sigma(t,\tau)\colon t\geq\tau\geq 0\}$, $\sigma \in \mathcal{H}_+(\sigma_0)$. Then, the uniform attractor $\mathcal{A}_{\mathcal{H}_+(\sigma_0)}$ of the family of semi-processes $\{U_\sigma(t,\tau)\colon t\geq\tau\geq 0\}$, $\sigma \in \mathcal{H}_+(\sigma_0)$ exists and coincides with the uniform attractor $\mathcal{A}_{\omega(\mathcal{H}_+(\sigma_0))}$ of the family of semi-processes $\{U_\sigma(t,\tau)\colon t\geq\tau\geq 0\}$, $\sigma \in \omega(\mathcal{H}_+(\sigma_0))$, i.e.
\begin{equation*}
\mathcal{A}_{\mathcal{H}_+(\sigma_0)} = \mathcal{A}_{\omega(\mathcal{H}_+(\sigma_0))}.
\end{equation*} \end{prop}
\begin{proof}
For the proof we refer the reader to \cite[Ch. 7, \S 3]{che}. \end{proof}
The compactness of $\Sigma$ will be established in the framework of almost periodic functions and Stepanow spaces. We need a few more definitions (see \cite{Guter:1966ys} and \cite[Ch. 7, \S 5]{che}): \begin{defi}
We call the expression
\begin{equation*}
\ud_{S^p_l}\big(f(s),g(s)\big) = \sup_{t \geq 0} \left(\frac{1}{l}\int_{t}^{t + l}\norm{f(s) - g(s)}_{\Psi}^p\, \ud s \right)^{\frac{1}{p}}
\end{equation*}
the $S^p_l$-distance of the order $p \geq 1$ corresponding to the length $l$. The space of all $p$-power locally integrable functions with values in $\Psi$ and equipped with the norm generated by $\ud_{S^p_l}$ we call the \emph{Stepanow space} and denote by $L_{p}^S(\mathbb{R}^+;\Psi)$. \end{defi}
In view of the above definition we set \begin{equation*}
\Xi_+ = L^S_{2,loc}(\mathbb{R}^+,\Psi). \end{equation*} Also note, that in the above definition we can put $l = 1$. Thus, we will write $S^p$ instead of $S^p_l$ or $S^p_1$. In addition, all $S_l^p$-spaces are complete.
\begin{defi}
A function $f \in L_p^S(\mathbb{R}^+;\Psi)$ we call \emph{$S^p$-asymptotically almost periodic} if for any $\epsilon \geq 0$ there exists a number $l = l(\epsilon)$ such that for each interval $(\alpha,\alpha + l)$, $\alpha \geq 0$ there exists a point $\tau \in (\alpha,\alpha + l)$ such that
\begin{equation*}
\ud_{S^p}\big(f(s + \tau),f(s)\big) = \sup_{t\geq 0} \left(\int_{t}^{t + 1} \norm{f(s + \tau) - f(s)}^p_{\Psi}\, \ud t\right)^{\frac{1}{p}} < \epsilon.
\end{equation*} \end{defi}
\begin{defi}
We say that a function $f\in L_p^S(\mathbb{R}^+;\Psi)$ is
\emph{$S^p$-normal} if the family of translations
\begin{equation*}
\left\{ f^h(s) = f(h + s)\colon h \in \mathbb{R}^+\right\}
\end{equation*}
is precompact in $L_p^S(\mathbb{R}^+;\Psi)$ with respect to the norm
\begin{equation*}
\left(\sup_{t \geq 0} \int_t^{t + 1} \norm{f(s)}^p_{\Psi}\, \ud s\right)^{\frac{1}{p}}
\end{equation*}
induced by $\ud_{S^p}$. \end{defi}
From our point of view the following Proposition will play a crucial role: \begin{prop}\label{prop3}
A function $f$ is $S^p$-normal if and only if $f$ is $S^p$-asymptotically almost periodic. \end{prop}
Now we define \begin{equation*}
\mathcal{H}_+(\sigma_0) = \overline{\left\{\sigma^h_0(s) = \sigma_0(s + h)\colon h \geq 0\right\}}^{\Xi_+}. \end{equation*} Note, that the set $\mathcal{H}_+(\sigma_0)$ does not contain any information on the smallness assumption on the external data. Therefore, for any given $\sigma_0$ we additionally introduce \begin{equation*}
\sigma^{\epsilon}_{0,x_3} = \big(f_{0,x_3},g_{0,x_3}\big), \end{equation*} where \begin{equation*}
\norm{f_{0,x_3}}_{L_2(\Omega^t)}^2 + \norm{g_{0,x_3}}_{L_2(\Omega^t)}^2 < \epsilon \end{equation*} and define \begin{equation*}
\omega(\sigma_{0,x_3}^{\epsilon}) = \bigcap_{\tau \geq 0} \overline{\left\{\sigma_{0,x_3}^{\epsilon,h}(s)\colon h \geq \tau\right\}}^{\Xi_+}. \end{equation*} Then we set \begin{equation*}
\Sigma^\epsilon_{\sigma_0} = \mathcal{H}_+(\sigma_0) \cap \omega(\sigma_{0,x_3}^{\epsilon}). \end{equation*}
\begin{rem}\label{rem8}
In view of the above Proposition, the hull $\mathcal{H}_+(\sigma_0)$ is compact if we take $\sigma_0$ $S^2$-asymptotically almost periodic. But recall that we obtained the global existence of regular solutions under the assumption on the external data that they decay exponentially on every time interval of the form $[kT,(k + 1)T]$ (see Theorem \ref{thm0}). Thus, if we assume that
\begin{equation*}
\norm{f\big((k + 1)T + t\big) - f(kT + t)}_{\Psi} < \epsilon
\end{equation*}
(and so for $g$ and their derivatives with respect to $x_3$) then the external data become $S^2$-asympto\-tically almost periodic. In other words, we take almost the same external data on every time interval of the form $[kT,(k + 1)T]$. On the contrary, in the proof of the global existence of regular solutions, the external data could differ from interval to interval without any restrictions. \end{rem}
We can finally introduce the following Definition: \begin{defi}
Let $E$ be a Banach space (or complete metric or a closed subset of $E$). Let a two parameter family of operators
\begin{equation*}
\left\{U_{\sigma}(t,\tau)\colon t\geq\tau\geq 0\right\}, \qquad U_{\sigma}(t,\tau)\colon E\to E
\end{equation*}
be given. We call it \emph{semi-process} in $E$ with the time symbol $\sigma \in \Sigma$ if
\begin{description}
\item[$(\mathbf{U})_1$] $U_{\sigma}(t,\tau) = U_{\sigma}(t,s)U_{\sigma}(s,\tau)$ for all $t\geq s\geq \tau \geq 0$,
\item[$(\mathbf{U})_2$] $U_{\sigma}(\tau,\tau) = \Id$ for all $\tau \geq 0$.
\end{description} \end{defi}
Observe that if we had a global and unique solution to problem \eqref{p1}, we could associate it with certain semi-process $\left\{U_{\sigma}(t,\tau)\colon t\geq\tau\geq 0\right\}$ defined on $\widehat H$. In three dimensions, likewise for the standard Navier-Stokes equations, we only know that such a semi-process would exist on $[0,t_{\max}]$, where $t_{\max} = t_{\max}(v_0,\omega_0,f,g)$.
Following the idea from \cite[\S 2.3]{chol1} we can define a semi-process on certain subspace of $\widehat H$. Let $B^{\epsilon} \in \widehat H$ be such that \begin{equation*}
B^{\epsilon} = \left\{\big(v(0),\omega(0)\big) \in \widehat H\cap \widehat V\colon \norm{v(0)_{,x_3}}_{L_2(\Omega)}^2 + \norm{\Rot v(0)_{,x_3}}_{L_2(\Omega)}^2 + \norm{\omega(0)_{,x_3}}_{L_2(\Omega)}^2 < \epsilon\right\}. \end{equation*} Define \begin{equation*}
\omega_{\tau,\Sigma^{\epsilon}_{\sigma_0}}(B^{\epsilon}) := \bigcap_{t\geq\tau}\overline{\bigcup_{\sigma \in \Sigma^{\epsilon}_{\sigma_0}}\bigcup_{s\geq t}U_{\sigma_0}(s,\tau)B^{\epsilon}}^{\widehat H} \end{equation*} and introduce the sets \begin{equation*}
\begin{aligned}
&\widehat H^{\epsilon} = \widehat H\cap \omega_{\tau,\Sigma^{\epsilon}_{\sigma_0}}(B^{\epsilon}), \\
&\widehat V^{\epsilon} = \widehat V\cap \omega_{\tau,\Sigma^{\epsilon}_{\sigma_0}}(B^{\epsilon}).
\end{aligned} \end{equation*}
\begin{defi}
The family of semiprocesses $\left\{U_{\sigma}(t,\tau)\right\}_{t\geq \tau\geq 0}$, $\sigma \in \Sigma^{\epsilon}_{\sigma_0}$ is said to be $(\widehat H^{\epsilon}\times \Sigma^{\epsilon}_{\sigma_0},\widehat H^{\epsilon})$-continuous if for any fixed $t,\tau \in \mathbb{R}_+$, $t \geq \tau$ the mapping $\big((v,\omega),\sigma\big)\mapsto U_{\sigma}(t,\tau)(v,\omega)$ is continuous from $\widehat H^{\epsilon}\times \Sigma^{\epsilon}_{\sigma_0}$ to $\widehat H^{\epsilon}$. \end{defi}
Let us justify the $(\widehat H^{\epsilon}\times \Sigma^{\epsilon}_{\sigma_0},\widehat H^{\epsilon})$-continuity of $U_{\sigma}(v,\omega)_{t\geq\tau\geq 0}$. We have \begin{lem}\label{lem35}
The family $\left\{U_{\sigma}(t,\tau)\right\}_{t\geq\tau\geq 0}$, $\sigma \in \Sigma^{\epsilon}_{\sigma_0}$ of semiprocesses is $(\widehat H^{\epsilon}\times \Sigma^{\epsilon}_{\sigma_0},\widehat H^{\epsilon})$-continuous. \end{lem}
\begin{proof}
Let $\sigma_1 = (f^1,g^1)$ and $\sigma_2 = (f^2,g^2)$ be two different symbols. Consider two corresponding solutions $(v^1,\omega^1)$ and $(v^2,\omega^2)$ to problem \eqref{p1} with the initial conditions $(v^1_\tau,\omega^1_\tau)$ and $(v^2_\tau,\omega^2_\tau)$, respectively. Denote $V = v^1 - v^2$, $\Theta = \omega^1 - \omega^2$, $P = p^1 - p^2$, $F = f^1 - f^2$ and $G = g^1 - g^2$. Then the pair $(V,\Theta)$,
\begin{equation*}
(V,\Theta) = U_{\sigma_1}(t,\tau)(v^1_\tau,\omega^1_\tau) - U_{\sigma_2}(t,\tau)(v^2_\tau,\sigma^2_\tau),
\end{equation*}
is a solution to a following problem
\begin{equation*}
\begin{aligned}
&V_{,t} - (\nu + \nu_r)\triangle V + \nabla P = - V\cdot \nabla v^1 - v^2\cdot \nabla V + 2\nu_r\Rot\Theta + F& &\text{in } \Omega^t, \\
&\Div V = 0 & &\text{in } \Omega^t,\\
&\begin{aligned}
&\Theta_{,t} - \alpha \triangle \Theta - \beta\nabla\Div\Theta + 4\nu_r \Theta \\
&\mspace{60mu} = -v^1\cdot \nabla \Theta - V\cdot \nabla \omega^2 + 2\nu_r\Rot V + G
\end{aligned}& &\text{in } \Omega^t,\\
&\Rot V \times n = 0, \qquad V\cdot n = 0 & &\text{on } S^t, \\
&\Theta = 0 & &\text{on } S^t_1, \\
&\Theta' = 0, \qquad \Theta_{3,x_3} = 0 & &\text{on } S^t_2, \\
&V\vert_{t = \tau} = V_\tau, \qquad \Theta\vert_{t = \tau} = \Theta_\tau & &\text{on } \Omega\times\{t = \tau\}.
\end{aligned}
\end{equation*}
With $F$, $G$ and the initial conditions set to zero, we studied the above problem in \cite[see Lemma 11.1]{2012arXiv1205.4046N}. Thus, by analogous calculations we obtain the inequality
\begin{multline*}
\frac{1}{2}\Dt \int_{\Omega} \abs{V}^2 + \abs{\Theta}^2\, \ud x + \frac{\nu}{2c_{\Omega}}\norm{\Rot V}^2_{L_2(\Omega)} \\
\leq \frac{c_{I,\Omega}}{\nu} \left(\norm{\nabla v^1}^2_{L_3(\Omega)} \norm{V}^2_{L_2(\Omega)} + \norm{\nabla \omega^2}^2_{L_3(\Omega)} \norm{\Theta}^2_{L_2(\Omega)} + \norm{F}^2_{L_{\frac{6}{5}}(\Omega)} + \norm{G}^2_{L_{\frac{6}{5}}(\Omega)}\right)
\end{multline*}
Application of the Gronwall inequality shows
\begin{multline*}
\norm{V(t)}^2_{L_2(\Omega)} + \norm{\Theta(t)}^2_{L_2(\Omega)} \leq c_{\nu,I,\Omega}\exp\left(\norm{\nabla v^1}^2_{L_2(\tau,t;L_3(\Omega))} + \norm{\nabla \omega^2}^2_{L_2(\tau,t;L_3(\Omega))}\right) \\
\cdot \left(\norm{V(\tau)}^2_{L_2(\Omega)} + \norm{\Theta(\tau)}^2_{L_2(\Omega)} + \norm{F}^2_{L_2(\tau,t;L_{\frac{6}{5}}(\Omega))} + \norm{G}^2_{L_2(\tau,t;L_{\frac{6}{5}}(\Omega))}\right).
\end{multline*}
Since $H^2(\Omega) \hookrightarrow W^1_6(\Omega) \hookrightarrow W^1_3(\Omega)$ we see that
\begin{align*}
\norm{\nabla v^1}^2_{L_2(\tau,t;L_3(\Omega))} &\leq c_{\Omega} \norm{v^1}^2_{W^{2,1}_2(\Omega\times(\tau,t))}, \\
\norm{\nabla \omega^2}^2_{L_2(\tau,t;L_3(\Omega))} &\leq c_{\Omega}\norm{\omega^2}^2_{W^{2,1}_2(\Omega\times(\tau,t))},
\end{align*}
By Theorem \ref{thm0} we get
\begin{multline*}
\norm{V(t)}^2_{L_2(\Omega)} + \norm{\Theta(t)}^2_{L_2(\Omega)} \\
\leq c_{data}\left(\norm{V(\tau)}^2_{L_2(\Omega)} + \norm{\Theta(\tau)}^2_{L_2(\Omega)} + \norm{F}^2_{L_2(\tau,t;L_{\frac{6}{5}}(\Omega))} + \norm{G}^2_{L_2(\tau,t;L_{\frac{6}{5}}(\Omega))}\right),
\end{multline*}
which ends the proof. \end{proof}
\section{Existence of the uniform attractor}\label{sec6} In this section we prove the existence of the uniform attractor to problem \eqref{p1} restricted to $\widehat H^{\epsilon}$. By $\mathcal{B}(\widehat H)$ we denote the family of all bounded sets of $\widehat H$. We begin by introducing some fundamental definitions: \begin{defi}
A family of processes $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma$ is said to be \emph{uniformly bounded} if for any $B \in \mathcal{B}(\widehat H)$ the set
\begin{equation*}
\bigcup_{\sigma\in\Sigma}\bigcup_{\tau\in\mathbb{R}^+}\bigcup_{t\geq\tau}U_\sigma(t,\tau)B\in\mathcal{B}(\widehat H).
\end{equation*} \end{defi}
\begin{defi}
A set $B_0\in \widehat H^{\epsilon}$ is said to be \emph{uniformly absorbing} for the family of processes $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma$, if for any $\tau\in\mathbb{R}^+$ and for every $B\in\mathcal{B}(\widehat H)$ there exists $t_0 = t_0(\tau,B)$ such that $\bigcup_{\sigma\in\Sigma}U_\sigma(t,\tau)B\subseteq B_0$ for all $t\geq t_0$. If the set $B_0$ is compact, we call the family of processes \emph{uniformly compact}. \end{defi}
\begin{defi}\label{def6.3}
A set $P$ belonging to $\widehat H$ is said to be \emph{uniformly attracting} for the family of processes $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma$, if for an arbitrary fixed $\tau \in \mathbb{R}^+$
\begin{equation*}
\lim_{t\to\infty} \left(\sup_{\sigma\in\Sigma}\dist_E \big(U_\sigma(t,\tau)B,P\big)\right) = 0.
\end{equation*}
If the set $P$ is compact, we call the family of processes \emph{uniformly asymptotically compact}. \end{defi}
\begin{defi}
A closed set $\mathcal{A}_{\Sigma} \subset \widehat H$ is said to be the \emph{uniform attractor} of the family of processes $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma$, if it is uniformly attracting set and it is contained in any closed uniformly attracting set $\mathcal{A}'$ of the family processes $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma$, $\mathcal{A}_{\Sigma} \subseteq \mathcal{A}'$. \end{defi}
To prove the existence of the uniform attractor we need two technical lemmas:
\begin{lem}\label{lem33}
Let the assumptions of Theorem \ref{thm0} hold. Then, there exists a bounded and absorbing set in $\widehat H^{\epsilon}$ for the family of semiprocesses $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma^{\epsilon}_{\sigma_0}$. \end{lem} \begin{proof}
Under the assumption of Theorem \ref{thm0} and in view assertion $(\mathbf{B})$ of \cite[Lemma 5.1]{2012arXiv1205.4507N} we see that for $t = (k + 1)T$, $t_0 = kT$ and sufficiently large $T > 0$
\begin{equation*}
\norm{v\big((k + 1)T\big)}_{L_2(\Omega)}^2 + \norm{\omega\big((k + 1)T\big)}_{L_2(\Omega)}^2 \leq \norm{v(kT)}_{L_2(\Omega)}^2 + \norm{\omega(kT)}_{L_2(\Omega)}^2
\end{equation*}
holds. By the same Lemma
\begin{multline}\label{eq55}
\limsup_{t\to\infty}\left(\norm{v(t)}_{L_2(\Omega)}^2 + \norm{\omega(t)}^2_{L_2(\Omega)}\right) \\
\leq c_{\nu,\alpha,\Omega}\left(\sup_{k\in\mathbb{N}}\left(\norm{f(kT)}^2_{L_2(\Omega)} + \norm{g(kT)}^2_{L_2(\Omega)}\right) + \norm{v(0)}_{L_2(\Omega)}^2 + \norm{\omega(0)}_{L_2(\Omega)}^2\right) =: R_1.
\end{multline}
Thus, for every $(v_0,\omega_0)\in \widehat H^{\epsilon}$ there exists $t_0 > 0$ such that
\begin{equation*}
(v(t),\omega(t)) \in B(0,\rho_1) \qquad \forall_{t\geq t_0},
\end{equation*}
where $B(0,\rho_1)$ is the ball in $\widehat H^{\epsilon}$ centered at $0$ with radius $\rho_1 > R_1$. If $B(0,r)\subset \widehat H^{\epsilon}$ is any ball such that $(v_0,\omega_0) \in B(0,r)$ then there exists $t_0 =t_0(r)$ such that \eqref{eq55} holds. This proves the existence of bounded absorbing sets in $\widehat H^{\epsilon}$ for the semiprocess $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma^{\epsilon}_{\sigma_0}$. \end{proof}
\begin{lem}\label{lem34}
Let the assumptions of Theorem \ref{thm0} hold. Then, there exists a bounded and absorbing set in $\widehat V^{\epsilon}$ for the family of semiprocesses $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma^{\epsilon}_{\sigma_0}$. \end{lem}
\begin{proof}
From the assumptions of Theorem \ref{thm0} and from \cite[Lemma 5.2]{2012arXiv1205.4507N}, where we set $t_1 = (k + 1)T$, $t_0 = kT$, we infer that
\begin{multline}\label{eq12}
\limsup_{t \to \infty} \left(\norm{v(t)}^2_{H^1(\Omega)} + \norm{v(t)}^2_{H^1(\Omega)}\right) \\
\leq c_{\nu,\alpha,\beta,I,P,\Omega}\left(\sup_{k\in\mathbb{N}}\left(\norm{f(kT)}^2_{L_2(\Omega)} + \norm{g(kT)}^2_{L_2(\Omega)}\right) + \norm{v(0)}_{H^1(\Omega)}^2 + \norm{\omega(0)}_{H^1(\Omega)}^2\right) =: R_2.
\end{multline}
Likewise in previous Lemma, for every $(v_0,\omega_0)\in \widehat V^{\epsilon}$ there exists $t_0 > 0$ such that
\begin{equation*}
(v(t),\omega(t)) \in B(0,\rho_2) \qquad \forall_{t\geq t_0},
\end{equation*}
where $B(0,\rho_2)$ is the ball in $\widehat V^{\epsilon}$ centered at $0$ with radius $\rho_2 > R_2$. For any $(v_0,\omega_0) \in B(0,r) \subset \widehat V^{\epsilon}$ there exists $t_0 =t_0(r)$ such that \eqref{eq12} holds. Therefore there exist bounded absorbing sets in $\widehat V^{\epsilon}$ for the semiprocess $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma^{\epsilon}_{\sigma_0}$. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm1}]
To prove the existence of the uniform attractor we make use of Proposition \ref{prop1}. We need to check the assumptions:
\begin{itemize}
\item From Remark \ref{rem8} and by assumption on the data it follows that $\sigma_0 = (f_0,g_0) \in \Xi_+$ is translation compact.
\item From Definition \ref{def6.3} it follows that the family of semiprocesses $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma^{\epsilon}_{\sigma_0}$ is uniformly asymptotically compact, if there exists a set $P \in \widehat H^{\epsilon}$, which is uniformly attracting and compact. Lemmas \ref{lem33} and \ref{lem34} and the Sobolev embedding theorem ensure the existence of such set $P$.
\item The $(\widehat H^{\epsilon}\times \Sigma^{\epsilon}_{\sigma_0},\widehat H^{\epsilon})$-continuity of the family of semiprocesses $\{U_\sigma(t,\tau)\}_{t\geq \tau \geq 0}$, $\sigma \in \Sigma^{\epsilon}_{\sigma_0}$ was proved in Lemma \ref{lem35}.
\item To check that $\omega(\Sigma_{\sigma_0}^{\epsilon})$ is the global attractor for the translation semigroup $\{T(t)\}_{t\geq 0}$ acting on $\Sigma_{\sigma_0}^{\epsilon}$ we recall that $\Sigma_{\sigma_0}^{\epsilon} \Subset \Xi_+$ is compact metric space. We can therefore apply Proposition \ref{prop2}. Thus,
\begin{align*}
\dist_{\Xi_+} \big(T(t)\Sigma_{\sigma_0}^{\epsilon},\omega(\Sigma_{\sigma_0}^{\epsilon})\big) \xrightarrow[t\to 0]{}0
\intertext{and}
T(t)\omega(\Sigma_{\sigma_0}^{\epsilon}) = \omega(\Sigma_{\sigma_0}^{\epsilon}) \qquad \forall_{t\geq 0}.
\end{align*}
\end{itemize}
As we see all assumptions are satisfied. The application of Proposition \ref{prop1} ends the proof. \end{proof}
\section{Convergence to stationary solutions for large $\nu$}\label{sec7}
In this section we prove Theorem \ref{thm3}.
\begin{proof}
The existence of solutions for large $\nu$ and theirs estimate follows, after slight modifications, from \cite{luk0} and \cite[Ch. 2, \S 1, Theorems 1.1.1, 1.1.2 and 1.1.3]{luk1}.
Let $V = v(t)-v_\infty$, $\Theta = \omega(t) - \omega_{\infty}$ and $P = p(t) - p_{\infty}$. Then $(V,\Theta)$ satisfies
\begin{equation}\label{eq38}
\begin{aligned}
&V_{,t} - (\nu + \nu_r)\triangle V + \nabla P = -v\cdot\nabla V - V\cdot\nabla v_\infty + 2\nu_r \Rot \Theta & &\text{in $\Omega^t$}, \\
&\Div V = 0 & &\text{in $\Omega^t$}, \\
&\begin{aligned}
&\Theta_{,t} - \alpha \triangle \Theta - \beta\nabla\Div\Theta + 4\nu_r \Theta \\
&\mspace{60mu} = - v\cdot\nabla \Theta - V\cdot \nabla \omega_{\infty} + 2\nu_r \Rot V
\end{aligned} & &\text{in $\Omega^t$}, \\
&\Rot V\times n = 0 & &\text{on $S^t$}, \\
&V\cdot n = 0 & &\text{on $S^t$}, \\
&\Theta = 0 & &\text{on $S_1^t$}, \\
&\Theta' = 0, \qquad \Theta_{3,x_3} = 0 & &\text{on $S_2^t$}, \\
&V\vert_{t = 0} = v(0) - v_\infty, \qquad \Theta\vert_{t = 0} = \omega(0) - \omega_{\infty} & &\text{in $\Omega\times\{t = 0\}$}.
\end{aligned}
\end{equation}
Multiplying the first and the third equation by $V$ and $\Theta$ respectively and integrating over $\Omega$ yields
\begin{multline*}
\frac{1}{2}\Dt \norm{V}^2_{L_2(\Omega)} + \left(\nu + \nu_r\right)\norm{\Rot V}_{L_2(\Omega)}^2 + \int_S \Rot V\times n \cdot V\, \ud S \\
= -\int_{\Omega} V\cdot \nabla v_{\infty} \cdot V\, \ud x + 2\nu_r\int_{\Omega} \Rot V \cdot \Theta\, \ud x + \int_S \Theta \times n \cdot V\, \ud S
\end{multline*}
and
\begin{multline*}
\frac{1}{2}\Dt \norm{\Theta}^2_{L_2(\Omega)} + \alpha\norm{\Rot\Theta}^2_{L_2(\Omega)} + (\alpha + \beta)\norm{\Div \Theta}^2_{L_2(\Omega)} + 4\nu_r\norm{\Theta}^2_{L_2(\Omega)} \\
= -\int_{\Omega} V\cdot \nabla \omega_{\infty}\cdot \Theta\, \ud x + 2\nu_r\int_{\Omega} \Rot V \cdot \Theta\, \ud x,
\end{multline*}
because in view of \eqref{eq38}$_{2,5}$ and \eqref{p1}$_{2,4}$ we see that
\begin{equation*}
\begin{aligned}
&\int_{\Omega}\nabla P\cdot V\, \ud x = \int_S P (V\cdot n)\, \ud S = 0, \\
&-\int_{\Omega}v\cdot \nabla V\cdot V\, \ud x = -\frac{1}{2}\int_{\Omega}v\cdot \nabla\abs{V}^2\, \ud x = -\int_S \abs{V}^2 (v\cdot n)\, \ud S = 0, \\
&-\int_{\Omega}v\cdot \nabla \Theta\cdot \Theta\, \ud x = -\frac{1}{2}\int_{\Omega}v\cdot \nabla\abs{\Theta}^2\, \ud x = -\int_S \abs{\Theta}^2 (v\cdot n)\, \ud S = 0.
\end{aligned}
\end{equation*}
Adding both equalities gives
\begin{multline}\label{eq39}
\frac{1}{2}\Dt \left(\norm{V}^2_{L_2(\Omega)} + \norm{\Theta}^2_{L_2(\Omega)}\right) + \left(\nu + \nu_r\right)\norm{\Rot V}_{L_2(\Omega)}^2 + \alpha\norm{\Rot\Theta}^2_{L_2(\Omega)} \\
+ (\alpha + \beta)\norm{\Div \Theta}^2_{L_2(\Omega)} + 4\nu_r\norm{\Theta}^2_{L_2(\Omega)} \\
= 4\nu_r\int_{\Omega} \Rot V \cdot \Theta\, \ud x -\int_{\Omega} V\cdot \nabla v_{\infty} \cdot V\, \ud x -\int_{\Omega} V\cdot \nabla \omega_{\infty}\cdot \Theta\, \ud x =: I_1 + I_2 + I_3
\end{multline}
Utilizing the H\"older and the Young inequalities on the right-hand side we have for $I_1$
\begin{equation*}
I_1 \leq 4\nu_r \norm{\Rot V}_{L_2(\Omega)}\norm{\Theta}_{L_2(\Omega)} \leq 4\nu_r\epsilon_1\norm{\Rot V}_{L_2(\Omega)} + \frac{\nu_r}{\epsilon_1}\norm{\Theta}_{L_2(\Omega)}.
\end{equation*}
To estimate $I_2$ and $I_3$ we also make use of the interpolation inequality for $L_p$-spaces and the embedding theorem:
\begin{equation*}
\begin{aligned}
\norm{u}_{L_4(\Omega)} \leq \norm{u}_{L_2(\Omega)}^{\frac{1}{4}}\norm{u}_{L_6(\Omega)}^{\frac{3}{4}} \leq c_I \norm{u}_{L_2(\Omega)}^{\frac{1}{4}}\norm{u}_{H^1(\Omega)}^{\frac{3}{4}}, \\
\norm{u}_{L_3(\Omega)} \leq \norm{u}_{L_2(\Omega)}^{\frac{1}{2}}\norm{u}_{L_6(\Omega)}^{\frac{1}{2}} \leq c_I \norm{u}_{L_2(\Omega)}^{\frac{1}{2}}\norm{u}_{H^1(\Omega)}^{\frac{1}{2}}.
\end{aligned}
\end{equation*}
Thus
\begin{multline*}
I_2 \leq \norm{V}_{L_4(\Omega)}\norm{\nabla v_{\infty}}_{L_2(\Omega)}\norm{V}_{L_4(\Omega)} \leq c_I\norm{V}_{L_2(\Omega)}^{\frac{1}{2}} \norm{V}_{H^1(\Omega)}^{\frac{3}{2}}\norm{v_{\infty}}_{H^1(\Omega)} \\
\leq \epsilon_2c_I \norm{V}_{H^1(\Omega)}^2 + \frac{c_I}{4\epsilon_2} \norm{V}_{L_2(\Omega)}^2\norm{v_{\infty}}_{H^1(\Omega)}^4
\end{multline*}
and
\begin{multline*}
I_3 \leq \norm{V}_{L_6(\Omega)} \norm{\nabla \omega_{\infty}}_{L_2(\Omega)} \norm{\Theta}_{L_3(\Omega)} \leq c_I\norm{V}_{H^1(\Omega)}\norm{\omega_{\infty}}_{H^1(\Omega)} \norm{\Theta}_{L_2(\Omega)}^{\frac{1}{2}} \norm{\Theta}_{H^1(\Omega)}^{\frac{1}{2}} \\
\leq c_I \epsilon_{31}\norm{V}_{H^1(\Omega)}^2 + \frac{c_I}{4\epsilon_{31}}\norm{\omega_{\infty}}_{H^1(\Omega)}^2\norm{\Theta}_{L_2(\Omega)}\norm{\Theta}_{H^1(\Omega)} \\
\leq c_I \epsilon_{31}\norm{V}_{H^1(\Omega)}^2 + \frac{c_I\epsilon_{32}}{4\epsilon_{31}} \norm{\Theta}^2_{H^1(\Omega)} + \frac{c_I}{16\epsilon_{31}\epsilon_{32}}\norm{\omega_{\infty}}_{H^1(\Omega)}^4\norm{\Theta}_{L_2(\Omega)}^2.
\end{multline*}
Now we set $\epsilon_1 = \frac{1}{4}$. By \cite[Lemma 6.7]{2012arXiv1205.4046N} it follows that
\begin{equation*}
\nu\norm{\Rot V}_{L_2(\Omega)}^2 + \alpha\norm{\Rot\Theta}^2_{L_2(\Omega)} + (\alpha + \beta)\norm{\Div \Theta}^2_{L_2(\Omega)} \geq \frac{\nu}{c_{\Omega}} \norm{V}_{H^1(\Omega)}^2 + \frac{\alpha}{c_{\Omega}}\norm{\Theta}^2_{H^1(\Omega)}.
\end{equation*}
It enables us to put $\epsilon_2 c_I = \epsilon_{31} c_I = \frac{\nu}{4c_{\Omega}}$ and $\epsilon_{32} \frac{c_I}{4\epsilon_{31}} = \frac{\alpha}{2c_{\Omega}}$. Hence
\begin{align*}
\epsilon_2 = \frac{\nu}{4c_{I,\Omega}}, & &\epsilon_{31} = \frac{\nu}{4c_{I,\Omega}}, & &\epsilon_{32} = \frac{\alpha\nu}{3c_{I,\Omega}}, & \\
\frac{c_I}{4\epsilon_2} = \frac{c_{I,\Omega}}{\nu}, & & & &\frac{c_I}{16\epsilon_{31}\epsilon_{32}} = \frac{3c_{I,\Omega}}{4\alpha\nu^2}. &
\end{align*}
Therefore, from \eqref{eq39} we obtain
\begin{multline*}
\frac{1}{2}\Dt\left(\norm{V}^2_{L_2(\Omega)} + \norm{\Theta}_{L_2(\Omega)}^2\right) + \frac{\nu}{2c_{\Omega}}\norm{V}_{H^1(\Omega)}^2 + \frac{\alpha}{2c_{\Omega}}\norm{\Theta}^2_{H^1(\Omega)} \\
\leq \frac{c_{I,\Omega}}{\nu}\norm{V}_{L_2(\Omega)}^2\norm{v_\infty}^4_{H^1(\Omega)} + \frac{3c_{I,\Omega}}{4\alpha\nu^2}\norm{\Theta}_{L_2(\Omega)}^2\norm{\omega_\infty}^4_{H^1(\Omega)}.
\end{multline*}
Let
\begin{equation*}
\begin{aligned}
c_1(\nu) &= \frac{\min\{\nu,\alpha\}}{c_{\Omega}}, \\
c_2 &= \frac{c_{I,\Omega}}{\min\{1,\alpha\}}
\end{aligned}
\end{equation*}
and define
\begin{equation*}
\Delta(\nu) = c_1(\nu) - \frac{3c_2}{\nu}F^4\big(\norm{f_\infty}_{L_2(\Omega)},\norm{g_\infty}_{L_2(\Omega)}\big).
\end{equation*}
Observe, that $\Delta(\nu) \to \frac{\alpha}{c_{\Omega}} > 0$ as $\nu \to \infty$. In particular there exists $\nu_* > 0$ such that $\Delta(\nu) > 0$ for any $\nu > \nu_*$ holds. The last inequality implies
\begin{equation*}
\Dt\left(\norm{V}^2_{L_2(\Omega)} + \norm{\Theta}_{L_2(\Omega)}^2\right) + \Delta(\nu)\left(\norm{V}_{L_2(\Omega)}^2 + \norm{\Theta}^2_{L_2(\Omega)}\right) \leq 0,
\end{equation*}
which is equivalent to
\begin{equation*}
\Dt\left(\left(\norm{V}^2_{L_2(\Omega)} + \norm{\Theta}_{L_2(\Omega)}^2\right)e^{\Delta(\nu)t}\right) \leq 0.
\end{equation*}
Integration with respect to $t$ yields
\begin{equation*}
\left(\norm{V(t)}^2_{L_2(\Omega)} + \norm{\Theta(t)}_{L_2(\Omega)}^2\right)e^{\Delta(\nu)t} \leq \norm{V(0)}^2_{L_2(\Omega)} + \norm{\Theta(0)}_{L_2(\Omega)}^2,
\end{equation*}
which implies
\begin{equation*}
\norm{V(t)}^2_{L_2(\Omega)} + \norm{\Theta(t)}_{L_2(\Omega)}^2 \leq \left(\norm{V(0)}^2_{L_2(\Omega)} + \norm{\Theta(0)}_{L_2(\Omega)}^2\right)e^{-\Delta(\nu)t}.
\end{equation*}
This concludes the proof. \end{proof}
\section{Continuous dependence on modeling}\label{sec8}
In this Section we examine the difference between $(v,\omega)$ and the solution $(u,\Theta)$ to problem \eqref{p11}.
Observe, that \eqref{p11} is the same as \eqref{p1} with $\nu_r = 0$. Therefore, for $\nu_r$ close to zero we may measure in some sense the deviation of the flows of micropolar fluids from that of modeled by the Navier-Stokes equations. This problem was considered locally in \cite{paye} and globally in \cite[\S 5]{luk2}.
\begin{proof}[Proof of Theorem \ref{thm2}]
Let us denote $V(t) = v(t) - u(t)$, $Z(t) = \omega(t) - \Theta(t)$. Then the pair $(V(t),Z(t))$ is a solution to the problem
\begin{equation*}
\begin{aligned}
&V_{,t} - (\nu + \nu_r)\triangle V - \nu_r \triangle u + \nabla (p - q) = -v\cdot\nabla V - V\cdot\nabla u + 2\nu_r \Rot \omega & &\text{in $\Omega^t$}, \\
&\Div V = 0 & &\text{in $\Omega^t$}, \\
&\begin{aligned}
&Z_{,t} - \alpha \triangle Z - \beta\nabla\Div Z + 4\nu_r \omega \\
&\mspace{60mu} = - v\cdot\nabla Z - V\cdot \nabla \Theta + 2\nu_r \Rot v
\end{aligned} & &\text{in $\Omega^t$}, \\
&\Rot V\times n = 0 & &\text{on $S^t$}, \\
&V\cdot n = 0 & &\text{on $S^t$}, \\
&Z = 0 & &\text{on $S_1^t$}, \\
&Z' = 0, \qquad Z_{3,x_3} = 0 & &\text{on $S_2^t$}, \\
&V\vert_{t = t_0} = v(t_0) - u(t_0), \qquad Z\vert_{t = t_0} = \omega(t_0) - \Theta(t_0) & &\text{in $\Omega\times\{t = t_0\}$}.
\end{aligned}
\end{equation*}
Multiplying the first equation and the third by $V$ and $Z$, respectively and integrating over $\Omega$ yields
\begin{multline*}
\frac{1}{2}\Dt \int_{\Omega} \abs{V}^2\, \ud x + (\nu + \nu_r)\int_{\Omega}\abs{\Rot V}^2\, \ud x + (\nu + \nu_r)\int_S \Rot V \times n \cdot V\, \ud S + \nu_r\int_{\Omega} \Rot u \cdot \Rot V\, \ud x \\
+ \nu_r \int_S \Rot u\times n \cdot V\, \ud S + \int_S (p - q) V \cdot n\, \ud S \\
= -\int_{\Omega} v\cdot \nabla V\cdot V\, \ud x - \int_{\Omega} V\cdot \nabla u \cdot V\, \ud x + 2\nu_r \int_{\Omega} \Rot \omega\cdot V\, \ud x
\end{multline*}
and
\begin{multline*}
\frac{1}{2}\Dt \int_{\Omega} \abs{Z}^2\, \ud x + \alpha\int_{\Omega} \abs{\Rot Z}^2\, \ud x + (\alpha + \beta)\int_{\Omega} \abs{\Div Z}^2\, \ud x - \alpha\int_S Z\times n \cdot \Rot Z\, \ud S + 4\nu_r \int_{\Omega}\omega\cdot Z\, \ud x \\
= -\int_{\Omega} v\cdot \nabla Z\cdot Z\, \ud x - \int_{\Omega} V \cdot \nabla \Theta \cdot Z\, \ud x + 2\nu_r\int_{\Omega} \Rot v\cdot Z\, \ud x.
\end{multline*}
All boundary integrals vanish due to the boundary conditions. In addition
\begin{equation*}
\begin{aligned}
-\int_{\Omega} v\cdot \nabla V\cdot V\, \ud x &= -\frac{1}{2}\int_{\Omega} v\cdot \nabla \abs{V}^2\, \ud x = -\frac{1}{2}\int_S \abs{V}^2 v \cdot n \, \ud S = 0, \\
-\int_{\Omega} v\cdot \nabla Z\cdot Z\, \ud x &= -\frac{1}{2}\int_{\Omega} v\cdot \nabla \abs{Z}^2\, \ud x = -\frac{1}{2}\int_S \abs{Z}^2 v \cdot n \, \ud S = 0.
\end{aligned}
\end{equation*}
We also have
\begin{equation*}
2\nu_r\int_{\Omega} \Rot \omega\cdot V\, \ud x = 2\nu_r\int_{\Omega} \omega\cdot \Rot V\, \ud x + 2\nu_r\int_S \omega\times n \cdot V\, \ud S = 2\nu_r\int_{\Omega} \omega\cdot \Rot V\, \ud x,
\end{equation*}
which follows from integration by parts and the boundary conditions. Thus, we get
\begin{multline*}
\frac{1}{2}\Dt\left(\norm{V}^2_{L_2(\Omega)} + \norm{Z}^2_{L_2(\Omega)}\right) + (\nu + \nu_r)\norm{\Rot V}^2_{L_2(\Omega)} + \alpha\norm{\Rot Z}^2_{L_2(\Omega)} + (\alpha + \beta)\norm{\Div Z}^2_{L_2(\Omega)} \\
= -\nu_r\int_{\Omega} \Rot u \cdot \Rot V\, \ud x- \int_{\Omega} V\cdot \nabla u \cdot V\, \ud x + 2\nu_r \int_{\Omega} \omega\cdot \Rot V\, \ud x \\
- 4\nu_r \int_{\Omega}\omega\cdot Z\, \ud x - \int_{\Omega} V \cdot \nabla \Theta \cdot Z\, \ud x + 2\nu_r\int_{\Omega} \Rot v\cdot Z\, \ud x.
\end{multline*}
Next we estimate the first and the third term on the right-hand side by the means of the H\"older and the Young inequalities. We have
\begin{equation*}
\begin{aligned}
-\nu_r\int_{\Omega} \Rot u \cdot \Rot V\, \ud x &\leq \nu_r\norm{\Rot u}_{L_2(\Omega)}\norm{\Rot V}_{L_2(\Omega)} \leq \epsilon_1\nu_r\norm{\Rot V}_{L_2(\Omega)}^2 + \frac{\nu_r}{4\epsilon_1}\norm{\Rot u}_{L_2(\Omega)}, \\
2\nu_r \int_{\Omega} \omega\cdot \Rot V\, \ud x &\leq 2\nu_r\norm{\omega}_{L_2(\Omega)}\norm{\Rot V}_{L_2(\Omega)} \leq 2\epsilon_2\nu_r\norm{\Rot V}_{L_2(\Omega)}^2 + \frac{\nu_r}{2\epsilon_2}\norm{\omega}_{L_2(\Omega)}.
\end{aligned}
\end{equation*}
We set $\epsilon_1 \nu_r = 2\epsilon_2 \nu_r = \frac{\nu_r}{2}$. Thus, $\epsilon_1 = \frac{1}{2}$ and $\epsilon_2 = \frac{1}{4}$. We get
\begin{multline*}
\frac{1}{2}\Dt\left(\norm{V}^2_{L_2(\Omega)} + \norm{Z}^2_{L_2(\Omega)}\right) + \nu\norm{\Rot V}^2_{L_2(\Omega)} + \alpha\norm{\Rot Z}^2_{L_2(\Omega)} + (\alpha + \beta)\norm{\Div Z}^2_{L_2(\Omega)} \\
\leq - \int_{\Omega} V\cdot \nabla u \cdot V\, \ud x - 4\nu_r \int_{\Omega}\omega\cdot Z\, \ud x - \int_{\Omega} V \cdot \nabla \Theta \cdot Z\, \ud x + 2\nu_r\int_{\Omega} \Rot v\cdot Z\, \ud x.
\end{multline*}
Utilizing \cite[Lemma 6.7]{2012arXiv1205.4046N} on the left-hand side yields
\begin{multline*}
\frac{1}{2}\Dt\left(\norm{V}^2_{L_2(\Omega)} + \norm{Z}^2_{L_2(\Omega)}\right) + \frac{\nu}{c_{\Omega}}\norm{V}^2_{H^1(\Omega)} + \frac{\alpha}{c_{\Omega}}\norm{Z}^2_{H^1(\Omega)} \\
\leq - \int_{\Omega} V\cdot \nabla u \cdot V\, \ud x - 4\nu_r \int_{\Omega}\omega\cdot Z\, \ud x - \int_{\Omega} V \cdot \nabla \Theta \cdot Z\, \ud x + 2\nu_r\int_{\Omega} \Rot v\cdot Z\, \ud x.
\end{multline*}
For the first term we have
\begin{multline*}
- \int_{\Omega} V\cdot \nabla u \cdot V\, \ud x \leq \norm{V}_{L_4(\Omega)}^2\norm{\nabla u}_{L_2(\Omega)} \leq c_I\norm{V}_{L_6(\Omega)}^{\frac{3}{2}}\norm{V}_{L_2(\Omega)}^{\frac{1}{2}}\norm{u}_{H^1(\Omega)} \\
\leq \epsilon_1 c_I\norm{V}_{H^1(\Omega)}^2 + \frac{c_I}{4\epsilon_1} \norm{V}_{L_2(\Omega)}^2\norm{u}_{H^1(\Omega)}^4,
\end{multline*}
where we used the H\"older inequality, the interpolation inequality between $L_2(\Omega)$ and $L_6(\Omega)$ and the Young inequality.
For the second term we get
\begin{equation*}
- 4\nu_r \int_{\Omega}\omega\cdot Z\, \ud x \leq 4\nu_r\norm{\omega}_{L_2(\Omega)} \norm{Z}_{L_2(\Omega)} \leq 4\nu_r\epsilon_2\norm{\omega}_{L_2(\Omega)}^2 + \frac{\nu_rc_I}{\epsilon_2}\norm{Z}_{H^1(\Omega)}^2.
\end{equation*}
The third term we estimate in the following way
\begin{equation*}
- \int_{\Omega} V \cdot \nabla \Theta \cdot Z\, \ud x \leq \norm{V}_{L_6(\Omega)}\norm{\nabla \Theta}_{L_3(\Omega)}\norm{Z}_{L_2(\Omega)} \leq c_I\epsilon_3\norm{V}_{H^1(\Omega)}^2 + \frac{1}{4\epsilon_3}\norm{\nabla \Theta}_{L_3(\Omega)}^2\norm{Z}_{L_2(\Omega)}^2.
\end{equation*}
Finally, for the fourth term we have
\begin{equation*}
2\nu_r \int_{\Omega}\Rot v\cdot Z\, \ud x \leq 2\nu_r\norm{\Rot v}_{L_2(\Omega)} \norm{Z}_{L_2(\Omega)} \leq 2\nu_r\epsilon_4\norm{\Rot v}_{L_2(\Omega)}^2 + \frac{\nu_rc_I}{2\epsilon_4}\norm{Z}_{H^1(\Omega)}^2.
\end{equation*}
We set $\epsilon_1 c_I = \epsilon_3 c_I = \frac{\nu}{4c_{\Omega}}$ and $\frac{\nu_rc_I}{\epsilon_2} = \frac{\nu_rc_I}{2\epsilon_4} = \frac{\alpha}{4c_{\Omega}}$. Hence
\begin{align*}
\frac{c_I}{4\epsilon_1} &= \frac{c_{I,\Omega}}{\nu}, & & & 4\nu_r\epsilon_2 &= \frac{16\nu_r^2c_{I,\Omega}}{\alpha}, & & & \frac{1}{4\epsilon_3} &= \frac{c_{I,\Omega}}{\nu}, & & & 4\nu_r\epsilon_4 &= \frac{8\nu_r^2c_{I,\Omega}}{\alpha}
\end{align*}
and we obtain
\begin{multline*}
\frac{1}{2}\Dt\left(\norm{V}^2_{L_2(\Omega)} + \norm{Z}^2_{L_2(\Omega)}\right) + \frac{\nu}{2c_{\Omega}}\norm{V}^2_{H^1(\Omega)} + \frac{\alpha}{2c_{\Omega}}\norm{Z}^2_{H^1(\Omega)} \\
\leq \frac{c_{I,\Omega}}{\nu}\norm{V}_{L_2(\Omega)}^2\norm{u}_{H^1(\Omega)}^4 + \frac{16\nu_r^2c_{I,\Omega}}{\alpha}\norm{\omega}_{L_2(\Omega)}^2 \\
+ \frac{c_{I,\Omega}}{\nu}\norm{\nabla \Theta}_{L_3(\Omega)}^2\norm{Z}_{L_2(\Omega)}^2 + \frac{8\nu_r^2c_{I,\Omega}}{\alpha}\norm{\Rot v}_{L_2(\Omega)}^2.
\end{multline*}
We introduce
\begin{equation}\label{eq44}
\begin{aligned}
\Delta_1(\nu) &= \frac{\nu}{c_{\Omega}} - \frac{2c_{I,\Omega}}{\nu}\norm{u}_{H^1(\Omega)}^4, \\
\Delta_2(\nu) &= \frac{\alpha}{c_{\Omega}} - \frac{2c_{I,\Omega}}{\nu}\norm{\nabla \Theta}_{L_3(\Omega)}^2, \\
\Delta(\nu) &= \min\left\{\Delta_1(\nu),\Delta_2(\nu)\right\}.
\end{aligned}
\end{equation}
Thus, the last inequality implies
\begin{multline*}
\Dt\left(\norm{V}^2_{L_2(\Omega)} + \norm{Z}^2_{L_2(\Omega)}\right) + \Delta(\nu)\left(\norm{V}^2_{L_2(\Omega)} + \norm{Z}^2_{L_2(\Omega)}\right) \\
\leq \frac{32\nu_r^2c_{I,\Omega}}{\alpha}\norm{\omega}_{L_2(\Omega)}^2 + \frac{16\nu_r^2c_{I,\Omega}}{\alpha}\norm{\Rot v}_{L_2(\Omega)}^2,
\end{multline*}
which is equivalent to
\begin{equation*}
\Dt\left(\left(\norm{V}^2_{L_2(\Omega)} + \norm{Z}^2_{L_2(\Omega)}\right)e^{\Delta(\nu)t}\right) \leq \nu_r^2c_{\alpha,I,\Omega}\left(\norm{\omega}_{L_2(\Omega)}^2 + \norm{\Rot v}_{L_2(\Omega)}^2\right)e^{\Delta(\nu)t}.
\end{equation*}
Integrating with respect to $t \in (t_0,t_1)$ yields
\begin{multline*}
\norm{V(t)}^2_{L_2(\Omega)} + \norm{Z(t)}^2_{L_2(\Omega)} \\
\leq \nu_r^2c_{\alpha,I,\Omega}\left(\norm{\omega}_{L_2(\Omega^t)}^2 + \norm{\Rot v}_{L_2(\Omega^t)}^2\right) + \left(\norm{V(t_0)}^2_{L_2(\Omega)} + \norm{Z(t_0)}^2_{L_2(\Omega)}\right)e^{-\Delta(\nu)t}.
\end{multline*}
In view of the energy estimates (see e.g. \cite[Lemma 8.1]{2012arXiv1205.4046N}) we obtain
\begin{equation*}
\norm{\omega}_{L_2(\Omega^t)}^2 + \norm{\Rot v}_{L_2(\Omega^t)}^2 < E_{v,\omega}(t).
\end{equation*}
On the other hand, under slightly weaker assumption than in Theorem \ref{thm0} it follows from \cite[Theorem B]{wm6} that
\begin{equation}\label{eq10}
\norm{u}_{W^{2,1}_2(\Omega^t)} \leq \varphi(\nu,\text{data})
\end{equation}
for any $t \in [0,\infty]$, where $\varphi$ is a positive and non-decreasing function dependent on the viscosity coefficient and the initial and the external data. By the Embedding Theorem for anisotropic Sobolev spaces (see e.g. \cite[Ch. 2, \S 3, Lemma 3.3]{lad} it follows that $W^{2,1}_2(\Omega^t) \hookrightarrow L_{10}(\Omega^t)$ and
\begin{equation*}
\norm{u}_{L_{10}(\Omega^t)} \leq c_{\Omega} \norm{u}_{W^{2,1}_2(\Omega^t)} \leq \varphi(\text{data})
\end{equation*}
holds. In light of the maximal regularity for parabolic systems (see e.g. \cite[Lemma 6.9]{2012arXiv1205.4046N}) applied to \eqref{p11}$_3$ we deduce that $\Theta \in W^{2,1}_{\frac{5}{3}}(\Omega^t)$ and
\begin{equation*}
\norm{\Theta}_{W^{2,1}_{\frac{5}{3}}(\Omega^t)} \leq c_{\text{data},\alpha,\beta,\Omega} \left(\norm{g}_{L_{\frac{5}{3}}(\Omega^t)} + \norm{u}_{L_{10}(\Omega^t)} + \norm{\Theta(t_0)}_{W^{\frac{4}{5}}_{\frac{5}{3}}(\Omega)}\right).
\end{equation*}
From \eqref{eq10}, the above inequality and the maximal regularity for the Stokes and parabolic system (see e.g. \cite[Lemmas 6.9 and 6.11]{2012arXiv1205.4046N}) it follows that we can improve the regularity for $(u,\Theta)$ as we need. In particular
\begin{equation*}
\begin{aligned}
\sup_{t \geq 0} \norm{u(t)}^4_{H^1(\Omega)} &\leq c_{\text{data}}, \\
\sup_{t \geq 0} \norm{\nabla \Theta(t)}^2_{L_3(\Omega)} &\leq c_{\text{data}}.
\end{aligned}
\end{equation*}
In consequence, $\Delta(\nu)$ becomes positive and finite for $\nu$ sufficiently large. This ends the proof. \end{proof}
\begin{rem}
Observe, that for $u(t_0) = v(t_0)$ and $\Theta(t_0) = \omega(t_0)$ we obtain
\begin{equation*}
\norm{V(t)}^2_{L_2(\Omega)} + \norm{Z(t)}^2_{L_2(\Omega)} \leq \nu_r^2c_{\alpha,I,\Omega}\left(\norm{\omega}_{L_2(\Omega^t)}^2 + \norm{\Rot v}_{L_2(\Omega^t)}^2\right).
\end{equation*}
This estimate implies that as $\nu_r \to 0$ the velocity of the micropolar fluid model converges uniformly with respect to time on $[0,\infty)$ in $L_2$ to the velocity field of standard Navier-Stokes model. \end{rem}
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Reply to Comment on `Spin Decoherence in Superconducting Atom Chips'}
\author{Bo-Sture K. Skagerstam}\email{[email protected]} \affiliation{Complex Systems and Soft Materials Research Group, Department of Physics,
The Norwegian University of Science and Technology, N-7491 Trondheim, Norway}
\author{Ulrich Hohenester} \affiliation{Institut f\"ur Physik, Karl-Franzens-Universit\"at Graz, Universit\"atsplatz 5, A-8010 Graz, Austria}
\author{Asier Eiguren} \affiliation{Institut f\"ur Physik, Karl-Franzens-Universit\"at Graz, Universit\"atsplatz 5, A-8010 Graz, Austria} \author{Per Kristian Rekdal} \affiliation{Institut f\"ur Physik, Karl-Franzens-Universit\"at Graz, Universit\"atsplatz 5, A-8010 Graz, Austria}
\pacs{03.65.Yz, 03.75.Be, 34.50.Dy, 42.50.Ct}
\maketitle
In a recent paper \cite{skagerstam_06} we investigate spin decoherence in superconducting atom chips,
and predict a lifetime enhancement by more than five orders of magnitude in comparison to
normal-conducting atom chips. Scheel, Hinds, and Knight (SHK) \cite{scheel_06} cast doubt on these
results as they are seemingly an artifact of the two-fluid model used for the description of the superconductor,
and estimate a lifetime enhancement by a factor of ten instead. In this reply we show that this criticism
is unwarranted since neither our central result relies on the two-fluid model, nor the predictions of our
model strongly disagree with experimental data.
In Ref.~\cite{skagerstam_06} we employ a dielectric description of the superconductor based on a parameterization
of the complex optical conductivity $\sigma(T) \equiv \sigma_1(T) + i \sigma_2(T)$, viz.
\begin{equation} \label{sigma_eq}
\sigma(T) = 2/\omega\mu_0\delta^2(T)+i/\omega\mu_0\lambda_L^2(T) \, , \end{equation}
\noi
with $\delta(T)$ and $\lambda_L(T)$ the temperature dependent skin and London penetration depth, respectively.
The spin lifetime for $\lambda_L(T) \ll \delta(T)$ is then obtained by matching the electromagnetic fields at
the vacuum-superconductor interface, to arrive at our central result for the spin lifetime
$\tau \propto \sigma^{3/2}_2(T)/\sigma_1(T) \propto \delta^2(T)/\lambda_L^3(T)$. As our analysis is only based
on Maxwell's theory with appropriate boundary conditions, it is valid for $\delta(T)$ and $\lambda_L(T)$
values obtained from either a microscopic model description or from experimental data on $\sigma(T)$.
The specific choice of parameterization in \eq{sigma_eq} is motivated by the
two-fluid model, even though this model is not needed to justify it.
In order to obtain an estimate of the lifetime for niobium, we make use of the experimental value $\sigma_1(T_c) = \sigma_n$
\cite{casalbuoni_06} and consider for $T \leq T_c$ the Gorter-Casimir temperature dependence
$\sigma_1(T) = (T/T_c)^4 \sigma_n$ and $\sigma_2(T) = ( 1 - (T/T_c)^4 ) \, \sigma_2(0)$,
where $\sigma_2(0) = 1/\omega \mu_0 \lambda_L^2(0)$.
A decrease in temperature from $T_c$ to $T_c/2$ as considered in
Ref.~\cite{scheel_06}, then results in a reduction of $\sigma_1(T_c/2)$ by approximately one order of magnitude.
SHK correctly note that the modification of the quasi-particle dispersion in the superconducting state might
give rise to a coherence Hebel-Schlichter peak of $\sigma_1(T)$ below $T_c$ \cite{lifetime}.
To estimate the importance of this peak, we have computed $\sigma_1(T)$ using the Mattis-Bardeen formula.
At the atomic transition frequency of 560 kHz we obtain a peak height of approximately $5 \, \sigma_n$,
not hundred $\sigma_n$ \cite{scheel_05} as used by SHK.
From the literature it is well-known that this coherence peak becomes substantially reduced if lifetime effects
of the quasi-particles are considered \cite{lifetime,lifetime_2} and even disappears in the clean superconductor limit \cite{marsiglio:91}.
As a fair and conservative estimate we correspondingly assign an uncertainty of one order of magnitude to our spin lifetimes.
On the other hand, the major contribution of the $\tau$ enhancement in the superconducting state is due to the additional
$\lambda_L^3(T)$ contribution accounting for the efficient magnetic field screening in superconductors.
This factor is not considered by SHK and appears to be the main reason for the discrepancy between our results and those
of Ref.~\cite{scheel_06}.
The London length $\lambda_L(0) = 35$ nm as used in
\cite{skagerstam_06} corresponds to $\omega \sigma_2(T_c/2) \approx 6.1 \times 10^{20} \, ( \Omega \, $m$ \, $s$ )^{-1}$
and is in agreement with the experimental data of Ref.~\cite{casalbuoni_06}.
In conclusion, modifications of $\sigma(T)$ introduced by the details of the quasi-particle dispersion in the
superconducting state (BCS or Eliashberg theory) are expected to modify the estimated lifetime values by at
most one order of magnitude, but will by no means change the essentials of our findings, which only rely on
generic superconductor properties. Hence, our prediction for a lifetime enhancement by more than five orders
of magnitude prevails. Whether such high lifetimes can be obtained in superconducting atom chips
will have to be determined experimentally.
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Layer potentials for general linear elliptic systems}
\author{Ariel Barton} \address{Ariel Barton, Department of Mathematical Sciences,
309 SCEN,
University of Ar\-kan\-sas,
Fayetteville, AR 72701} \email{[email protected]}
\subjclass[2010]{Primary
35J58, Secondary
31B10,
31B20
}
\begin{abstract} In this paper we construct layer potentials for elliptic differential operators using the Lax-Milgram theorem, without recourse to the fundamental solution; this allows layer potentials to be constructed in very general settings. We then generalize several well known properties of layer potentials for harmonic and second order equations, in particular the Green's formula, jump relations, adjoint relations, and Verchota's equivalence between well-posedness of boundary value problems and invertibility of layer potentials. \end{abstract}
\keywords{Higher order differential equation, layer potentials, Dirichlet problem, Neumann problem}
\maketitle
\tableofcontents
\section{Introduction}
There is by now a very rich theory of boundary value problems for Laplace's operator, and more generally for second order divergence form operators $-\Div \mat A\nabla$. The Dirichlet problem \begin{equation*}-\Div \mat A\nabla u=0 \text{ in }\Omega,\quad u=f \text{ on }\partial\Omega, \quad \doublebar{u}_\XX\leq C\doublebar{f}_\DD\end{equation*} and the Neumann problem \begin{equation*}-\Div \mat A\nabla u=0 \text{ in }\Omega,\quad \nu\cdot\mat A\nabla u=g \text{ on }\partial\Omega, \quad \doublebar{u}_\XX\leq C\doublebar{g}_\NN\end{equation*} are known to be well-posed for many classes of coefficients $\mat A$ and domains~$\Omega$, and with solutions in many spaces $\XX$ and boundary data in many boundary spaces $\DD$ and~$\NN$.
A great deal of current research consist in extending these well posedness results to more general situations, such as operators of order $2m\geq 4$ (for example, \cite{MazMS10,KilS11B,MitMW11, MitM13A, BreMMM14, BarHM17pC}; see also the survey paper \cite{BarM16B}), operators with lower order terms (for example, \cite{BraBHV12,Tao12,Fel16,PanT16,DavHM16p}) and operators acting on functions defined on manifolds (for example, \cite{MitMT01,MitMS06,KohPW13}).
Two very useful tools in the second order theory are the double and single layer potentials given by \begin{align} \label{eqn:introduction:D} \D^{\mat A}_\Omega f(x) &= \int_{\partial\Omega} \overline{\nu\cdot \mat A^*(y)\nabla_{y} E^{L^*}(y,x)} \, f(y)\,d\sigma(y) ,\\ \label{eqn:introduction:S} \s^L_\Omega g(x) &= \int_{\partial\Omega}\overline{E^{L^*}(y,x)} \, g(y)\,d\sigma(y) \end{align} where $\nu$ is the unit outward normal to~$\Omega$ and where $E^L(y,x)$ is the fundamental solution for the operator~$L=-\Div \mat A\nabla$, that is, the formal solution to $L E^L(\,\cdot\,,x)=\delta_x$. These operators are inspired by a formal integration by parts \begin{align*}u(x) &= \int_\Omega \overline{L^*E^{L^*}(\,\cdot\,,x)}\,u \\&=- \int_{\partial\Omega}\!\! \overline{\nu\cdot \mat A^*\nabla E^{L^*}(\,\cdot\,,x)} \, u\,d\sigma +\int_{\partial\Omega}\!\!\overline{E^{L^*}(\,\cdot\,,x)} \, \nu\cdot \mat A\nabla u\,d\sigma +\int_\Omega \overline{E^{L^*}(\,\cdot\,,x)}\,Lu\end{align*} which gives the Green's formula \begin{equation*}u(x) = -\D^{\mat A}_\Omega (u\big\vert_{\partial\Omega})(x) + \s^L_\Omega (\nu\cdot \mat A\nabla u)(x)\quad\text{if $x\in\Omega$ and $Lu=0$ in $\Omega$}\end{equation*} at least for relatively well-behaved solutions~$u$.
Such potentials have many well known properties beyond the above Green's formula, including jump and adjoint relations. In particular, by a clever argument of Verchota \cite{Ver84} and some extensions in \cite{BarM13,BarM16A}, well posedness of the Dirichlet problem is equivalent to invertibility of the operator $g\mapsto \s^L_\Omega g\big\vert_{\partial\Omega}$, and well posedness of the Neumann problem is equivalent to invertibility of the operator $f\mapsto \nu\cdot\mat A\nabla\D^{\mat A}_\Omega f$.
This equivalence has been used to solve boundary value problems in many papers, including \cite{FabJR78,Ver84,DahK87,FabMM98,Zan00} in the case of harmonic functions (that is, the case $\mat A=\mat I$ and $L=-\Delta$) and \cite{AlfAAHK11, Bar13,Ros13, HofKMP15B, HofMayMou15,HofMitMor15, BarM16A} in the case of more general operators under various assumptions on the coefficients~$\mat A$. Layer potentials have been used in other ways in \cite{PipV92,KenR09,Rul07,Mit08,Agr09,MitM11,BarM13,AusM14}. Boundary value problems were studied using a functional calculus approach in \cite{AusAH08,AusAM10,AusA11, AusR12, AusM14, AusS14p, AusM14p}; in \cite{Ros13} it was shown that certain operators arising in this theory coincided with layer potentials.
Thus, it is desirable to extend layer potentials to more general situations. One may proceed as in the homogeneous second order case, by constructing the fundamental solution, formally integrating by parts, and showing that the resulting integral operators have appropriate properties. In the case of higher order operators with constant coefficients, this has been done in \cite{Agm57,CohG83, CohG85, Ver05, MitM13A, MitM13B}. However, all three steps are somewhat involved in the case of variable coefficient operators (although see \cite{DavHM16p,Bar16} for fundamental solutions, for second order operators with lower order terms, and for higher order operators without lower order terms, respectively).
An alternative, more abstract construction is possible. The fundamental solution for various operators was constructed in \cite{HofK07,Bar16,DavHM16p} as the kernel of the Newton potential, which may itself be constructed very simply using the Lax-Milgram theorem. It is possible to rewrite the formulas \eqref{eqn:introduction:D} and~\eqref{eqn:introduction:S} for the second order layer potential directly in terms of the Newton potential, without mediating by the fundamental solution, and this construction generalizes very easily. It is this approach that was taken in \cite{BarHM15p,BarHM17pA}.
In this paper we will provide the details of this construction in a very general context. Roughly, this construction is valid for all differential operators $L$ that may be inverted via the Lax-Milgram theorem, and all domains $\Omega$ for which suitable boundary trace operators exist. We will also show that many properties of traditional layer potentials are valid in the general case.
The organization of this paper is as follows. The goal of this paper is to construct layer potentials associated to an operator~$L$ as bounded linear operators from a space $\DD_2$ or $\NN_2$ to a Hilbert space $\HH_2$ given certain conditions on $\DD_2$, $\NN_2$ and $\HH_2$. In Section~\ref{sec:dfn} we will list these conditions and define our terminology. Because these properties are somewhat abstract, in Section~\ref{sec:example} we will give an example of spaces $\HH_2$, $\DD_2$ and $\NN_2$ that satisfy these conditions in the case where $L$ is a higher order differential operator in divergence form without lower order terms.
This is the context of the paper \cite{BarHM17pC}; we intend to apply the results of the present paper therein to solve the Neumann problem with boundary data in $L^2$ for operators with $t$-independent self-adjoint coefficients.
In Section~\ref{sec:D:S} of this paper we will provide the details of the construction of layer potentials. We will prove the higher order analogues for the Green's formula, adjoint relations, and jump relations in Section~\ref{sec:properties}.
Finally, in Section~\ref{sec:invertible} we will show that the equivalence between well posedness of boundary value problems and invertibility of layer potentials of \cite{Ver84,BarM13,BarM16A} extends to the higher order case.
\section{Terminology} \label{sec:dfn}
We will construct layer potentials $\D^\B_\Omega$ and $\s^L_\Omega$ using the following objects.
\begin{itemize}
\item Two Hilbert spaces $\HH_1$ and $\HH_2$. \item Six vector spaces $\widehat\HH_1^\Omega$, $\widehat\HH_1^\CC$, $\widehat\HH_2^\Omega$, $\widehat\HH_2^\CC$, $\widehat\DD_1$ and $\widehat\DD_2$.
\item Bounded bilinear functionals $\B:\HH_1\times\HH_2\mapsto \C$, $\B^\Omega:\HH_1^\Omega\times\HH_2^\Omega\mapsto \C$, and $\B^\CC:\HH_1^\CC\times\HH_2^\CC\mapsto \C$. (We will define the spaces $\HH_j^\Omega$, $\HH_j^\CC$ momentarily.) \item Bounded linear operators $\Tr_1:\HH_1\mapsto\widehat\DD_1$ and $\Tr_2:\HH_2\mapsto\widehat\DD_2$. \item Bounded linear operators from $\HH_j$ to $\widehat\HH_j^\Omega$ and $\widehat\HH_j^\CC$; we shall denote these operators $\big\vert_\Omega$ and~$\big\vert_\CC$.
\end{itemize}
We will work not with the spaces $\widehat \HH_j^\Omega$, $\widehat\HH_j^\CC$ and $\widehat\DD_j$, but with the spaces $ \HH_j^\Omega$, $\HH_j^\CC$ and $\DD_j$ defined as follows. \begin{gather} \HH_j^\Omega=\{F\big\vert_\Omega:F\in\HH_j\}/\sim\text{ with norm }\doublebar{f}_{\HH^\Omega_j} = \inf\{\doublebar{F}_{\HH_j}: F\big\vert_\Omega=f\} ,\\ \HH_j^\CC=\{F\big\vert_\CC : F\in\HH_j\}/\sim\text{ with norm }\doublebar{f}_{\HH^\CC_j} = \inf\{\doublebar{F}_{\HH_j}: F\big\vert_\CC=f\} ,\\ \DD_j=\{\Tr_j F:F\in\HH_j\}/\sim\text{ with norm }\doublebar{f}_{\DD_j} = \inf\{\doublebar{F}_{\HH_j}: \Tr_j F=f\} \end{gather} where $\sim$ denotes the equivalence relation $f\sim g$ if $\doublebar{f-g}=0$.
We impose the following conditions on the given function spaces and operators. We require that there is some $\lambda>0$ such that for every $u\in\HH_1$, $v\in \HH_2$ and $\varphi$,~$\psi\in \HH_j$ for $j=1$ or $j=2$, the following conditions are valid. \begin{gather} \label{cond:coercive} \sup_{w\in \HH_1\setminus\{0\}} \frac{\abs{\B(w,v)}}{\doublebar{w}_{\HH_1}}\geq \lambda \doublebar{v}_{\HH_2},\quad
\sup_{w\in \HH_2\setminus\{0\}} \frac{\abs{\B(u,w)}}{\doublebar{w}_{\HH_2}}\geq \lambda \doublebar{u}_{\HH_1}. \\ \label{cond:local} \B(u,v) = \B^\Omega(u\big\vert_{\Omega},v\big\vert_\Omega) +\B^\CC(u\big\vert_{\CC}, v\big\vert_{\CC}). \\
\label{cond:trace:extension} {\text{If $\Tr_j \varphi=\Tr_j \psi$, then there is a $w\in\HH_j$ with}} \qquad\qquad \qquad\qquad\qquad \qquad \\\nonumber \qquad\qquad \qquad\qquad\qquad \qquad {\text{ $w\big\vert_\Omega=\varphi\big\vert_\Omega$, $w\big\vert_{\CC}=\psi\big\vert_{\CC}$ and $\Tr_j w= \Tr_j\varphi=\Tr_j\psi$.}} \end{gather}
We now introduce some further terminology.
We will define the linear operator $L$ as follows. If $u\in \HH_2$, let $Lu$ be the element of the dual space $\HH_1^*$ to $\HH_1$ given by \begin{equation}\label{dfn:L}\langle\varphi,Lu\rangle = \B(\varphi,u).\end{equation} Notice that $L$ is bounded $\HH_2\mapsto\HH_1^*$.
If $u\in \HH_2^\Omega$, we let $(Lu)\big\vert_\Omega$ be the element of the dual space to $\{\varphi\in\HH_1:\Tr_1 \varphi=0\}$ given by \begin{equation}\label{dfn:L:interior} \langle\varphi,(Lu)\big\vert_\Omega\rangle = \B^\Omega(\varphi\big\vert_\Omega,u)\quad\text{for all $\varphi\in\HH_1$ with $\Tr_1\varphi=0$} .\end{equation} If $u\in\HH_2$, we will often use $(Lu)\big\vert_\Omega$ as shorthand for $(L(u\big\vert_\Omega))\big\vert_\Omega$. We will primarily be concerned with the case $(Lu)\big\vert_\Omega=0$.
Let \begin{equation}\NN_2=\DD_1^*, \qquad \NN_1=\DD_2^*\end{equation} denote the dual spaces to $\DD_1$ and $\DD_2$. We will now define the Neumann boundary values of an element $u$ of $\HH_2^\Omega$ that satisfies $(Lu)\big\vert_\Omega=0$. If $\Tr_1\varphi=\Tr_1\psi$ and $(Lu)\big\vert_\Omega=0$, then $\B^\Omega(\varphi\big\vert_\Omega-\psi\big\vert_\Omega,u)=0$ by definition of $(Lu)\big\vert_\Omega$. Thus, $\B^\Omega(\varphi\big\vert_\Omega,u)$ depends only on~$\Tr_1\varphi$, not on~$\varphi$. Thus, $\M_\Omega^\B u$ defined as follows is a well defined element of $\NN_2$. \begin{equation}\label{eqn:Neumann}\langle \Tr_1\varphi,\M_\Omega^\B u \rangle = \B^\Omega(\varphi\big\vert_\Omega,u)\quad\text{for all $\varphi\in \HH_1$}.\end{equation}
We may compute \begin{equation*}\abs{\langle \arr f,\M_\Omega^\B u \rangle} \leq \doublebar{\B^\Omega} \inf\{\doublebar{\varphi}_{\HH_1}:\Tr_1\varphi=\arr f\} \doublebar{u}_{\HH_2^\Omega} = \doublebar{\B^\Omega} \doublebar{\arr f}_{\DD_1}\doublebar{u}_{\HH_2^\Omega}\end{equation*} and so we have the bound $\doublebar{\M_\Omega^\B u }_{\NN_2}\leq \doublebar{\B^\Omega}\doublebar{u}_{\HH_2^\Omega}$.
If $(Lu)\big\vert_\Omega\neq 0$, then the linear operator given by $\varphi\mapsto B^\Omega(\varphi\big\vert_\Omega,u)$ is still of interest. We will denote this operator $L(u\1_\Omega)$; that is, if $u\in\HH_2^\Omega$ (or $u\in\HH_2$ as before), then $L(u\1_\Omega)\in \HH_1^*$ is defined by \begin{equation}\label{dfn:L:singular} \langle\varphi,L(u\1_\Omega)\rangle = \B^\Omega(\varphi\big\vert_\Omega,u)\quad\text{for all $\varphi\in\HH_1$} .\end{equation}
\section{An example: higher order differential equations} \label{sec:example}
In this section, we provide an example of a situation in which the terminology of Section~\ref{sec:dfn} and the construction and properties of layer potentials of Sections~\ref{sec:D:S} and~\ref{sec:properties} may be applied. We remark that this is the situation of \cite{BarHM17pC}, and that we will therein apply the results of this paper.
Let $m\geq 1$ be an integer, and let $L$ be an elliptic differential operator of the form \begin{equation} \label{eqn:L} Lu=(-1)^m\sum_{\abs\alpha=\abs\beta= m} \partial^\alpha(A_{\alpha\beta} \partial^\beta u)\end{equation} for some bounded measurable coefficients~$\mat A$ defined on $\R^d$. Here $\alpha$ and $\beta$ are multiindices in $\N_0^d$, where $\N_0$ denotes the nonnegative integers. As is standard in the theory, we say that $Lu=0$ in an open set $\Omega$ in the weak sense if \begin{equation}\label{eqn:weak}\int_\Omega \sum_{\abs\alpha=\abs\beta= m} \partial^\alpha\varphi \,A_{\alpha\beta}\, \partial^\beta u=0 \quad\text{for all $\varphi\in C^\infty_0(\Omega)$}.\end{equation}
We impose the following ellipticity condition: we require that for some $\lambda>0$, \begin{equation*}\Re \sum_{{\abs\alpha=\abs\beta= m}} \int_{\R^d} \overline{\partial^\alpha\varphi}\,A_{\alpha\beta}\,\partial^\beta\varphi\geq \lambda \doublebar{\nabla^m\varphi}_{L^2(\R^d)}^2 \quad \text{for all $\varphi\in\dot W^2_m(\R^d)$.} \end{equation*}
Let $\Omega\subset\R^d$ be a Lipschitz domain with connected boundary, and let $\CC=\R^d\setminus\bar\Omega$ denote the interior of its complement. Observe that $\partial\Omega=\partial\CC$.
The following function spaces and linear operators satisfy the conditions of Section~\ref{sec:dfn}. \begin{itemize} \item $\HH_1=\HH_2=\HH$ is the homogeneous Sobolev space $\dot W^2_m(\R^d)$ of locally integrable functions~$\varphi$ (or rather, of equivalence classes of functions modulo polynomials of degree $m-1$) with weak derivatives of order $m$, and such that the $\HH$-norm given by $\doublebar{\varphi}_\HH=\doublebar{\nabla^m\varphi}_{L^2(\R^d)}$ is finite. This space is a Hilbert space with inner product $\langle \varphi,\psi\rangle =\sum_{\abs\alpha=m} \int_{\R^d} \overline{\partial^\alpha\varphi}\,\partial^\alpha\psi$. \item $\widehat\HH^\Omega$ and $\widehat\HH^\CC$ are the Sobolev spaces $\widehat\HH^\Omega=\dot W^2_m(\Omega)=\{\varphi:\nabla^m\varphi\in L^2(\Omega)\}$ and $\widehat\HH^\CC=\dot W^2_m(\CC)=\{\varphi:\nabla^m\varphi\in L^2(\CC)\}$ with the obvious norms.
\item $\widehat \DD$ denotes the (vector-valued) Besov space $ \dot B^{2,2}_{1/2}(\partial\Omega)$ of locally integrable functions modulo constants with norm \begin{equation*}\doublebar{f}_{\dot B^{2,2}_{1/2}(\partial\Omega)} = \biggl(\int_{\partial\Omega}\int_{\partial\Omega} \frac{\abs{f(x)-f(y)}^2}{\abs{x-y}^d}\,d\sigma(x)\,d\sigma(y)\biggr)^{1/2}.\end{equation*}
\item In \cite{Bar16pA,BarHM15p,BarHM17pC}, $\Tr$ is the linear operator defined on $ \HH$ by $\Tr u=\Trace^\Omega \nabla^{m-1}u\big\vert_\Omega$, where $\Trace^\Omega$ is the standard boundary trace operator of Sobolev spaces. (Given a suitable modification of the trace space $\DD$, it is also possible to choose $\Tr u = \{\Trace^\Omega \partial^\gamma u\}_{\abs\gamma\leq m-1}$, or more concisely $\Tr u = (\Trace^\Omega u,\partial_\nu u,\dots,\partial_\nu^{m-1} u)$, where $\nu$ is the unit outward normal, so that the boundary derivatives of $u$ of all orders are recorded. See, for example, \cite{
PipV95B,She06B,
Agr07, MazMS10,MitM13A}.)
\item $\B$ is the bilinear operator on $\HH\times\HH$ given by \begin{equation*}\B(\psi,\varphi) = \sum_{\abs\alpha=\abs\beta= m}\int_{\R^d} \overline{\partial^\alpha\psi}\,A_{\alpha\beta}\,\partial^\beta\varphi.\end{equation*} $\B^\Omega$ and $\B^\CC$ are defined analogously to~$\B$, but with the integral over $\R^d$ replaced by an integral over $\Omega$ or~$\CC$. \end{itemize}
$\B$, $\B^\Omega$ and $\B^\CC$ are clearly bounded and bilinear, and the restriction operators $\big\vert_\Omega:\HH\mapsto\widehat\HH^\Omega$, $\big\vert_\CC:\HH\mapsto\widehat\HH^\CC$ are bounded and linear.
The trace operator $\Tr$ is linear. If $\Omega=\R^d_+$ is the half-space, then boundedness of $\Tr:\HH\mapsto\DD$ was established in \cite[Section~5]{Jaw77}; this extends to the case where $\Omega$ is the domain above a Lipschitz graph via a change of variables. If $\Omega$ is a bounded Lipschitz domain, then boundedness of $\Tr:W\mapsto\widehat\DD$, where $W$ is the inhomogeneous Sobolev space with norm $\sum_{k=0}^m \doublebar{\nabla^k\varphi}_{L^2(\R^d)}$, was established in \cite[Chapter~V]{JonW84}. Then boundedness of $\Tr:\HH\mapsto\widehat\DD$ follows by the Poincar\'e inequality.
By assumption, the coercivity condition~\eqref{cond:coercive} is valid. If $\partial\Omega$ has Lebesgue measure zero, then Condition~\eqref{cond:local} is valid. A straightforward density argument shows that if $\Tr$ is bounded, then Condition~\eqref{cond:trace:extension} is valid.
Thus, the given spaces and operators satisfy the conditions imposed at the beginning of Section~\ref{sec:dfn}.
We now comment on a few of the other quantities defined in Section~\ref{sec:dfn}.
If $u\in \HH$, and if $Lu=0$ in $\Omega$ in the weak sense of formula~\eqref{eqn:weak}, then by density $\B^\Omega(\varphi,u)=0$ for all $\varphi\in\HH$ with $\Tr\varphi=0$; that is, $(Lu)\big\vert_\Omega$ as defined in Section~\ref{sec:dfn} satisfies $(Lu)\big\vert_\Omega=0$.
For many classes of domains there is a bounded extension operator from $\widehat\HH^\Omega$ to $\HH$, and so $\HH^\Omega=\widehat\HH^\Omega=\dot W^2_m(\Omega)$ with equivalent norms. (If $\Omega$ is a Lipschitz domain then this is a well known result of Calder\'on \cite{Cal61} and Stein \cite[Theorem~5, p.~181]{Ste70}; the result is true for more general domains, see for example \cite{Jon81}.)
As mentioned above, if $\Omega\subset\R^d$ is a Lipschitz domain, then $\Tr$ is a bounded operator $\HH \mapsto \widehat \DD$. If $\partial\Omega$ is connected, then $\Tr$ moreover has a bounded right inverse. (See \cite{JonW84} or \cite[Proposition~7.3]{MazMS10} in the inhomogeneous case, and \cite{Bar16pB} in the present homogeneous case.) Thus, the norm in $\DD$ is comparable to the Besov norm. Furthermore, $\{\nabla^{m-1}\varphi\big\vert_{\partial\Omega}:\varphi\in C^\infty_0(\R^d)\}$ is dense in $\DD$. Thus, if $m=1$ then $\DD=\widehat\DD=\dot B^{2,2}_{1/2}(\partial\Omega)$. If $m\geq 2$ then $\DD$ is a closed {proper} subspace of $\widehat\DD$, as the different partial derivatives of a common function must satisfy certain compatibility conditions. In this case $\DD$ is the Whitney-Sobolev space used in many papers, including \cite{AdoP98, MazMS10, MitMW11, MitM13A, MitM13B, BreMMM14, Bar16pA}.
If $m=1$, then by an integration by parts argument we have that $\M_\B^\Omega u = \nu\cdot \mat A\nabla u$, where $\nu$ is the unit outward normal to~$\Omega$, whenever $u$ is sufficiently smooth. The weak formulation of Neumann boundary values of formula~\eqref{eqn:Neumann} coincides with the formulation of higher order Neumann boundary data of \cite{BarHM15p,Bar16pA,BarHM17pC} given the above choice of $\Tr=\Trace^\Omega \nabla^{m-1}$, and with that of \cite{ Ver05,Agr07, MitM13A} if we instead choose $\Tr u=(\Trace^\Omega u, \partial_\nu u,\dots,\partial_\nu^{m-1} u)$ or $\Tr u = \{\Trace^\Omega \partial^\gamma u\}_{\abs\gamma\leq m-1}$.
\section{Construction of layer potentials} \label{sec:D:S}
We will now use the Babu\v{s}ka-Lax-Milgram theorem to construct layer potentials. This theorem may be stated as follows. \begin{thm}[{\cite[Theorem~2.1]{Bab70}}] \label{thm:lax-milgram} Let $\HH_1$ and $\HH_2$ be two Hilbert spaces, and let $\B$ be a bounded bilinear form on $\HH_1\times \HH_2$ that is coercive in the sense that for some fixed $\lambda>0$, formula~\eqref{cond:coercive} is valid for every $u\in\HH_1$ and $v\in\HH_2$.
Then for every linear functional $T$ defined on ${\HH_1}$ there is a unique $u_T\in {\HH_2}$ such that $\B(v,u_T)=\overline{T(v)}$. Furthermore, $\doublebar{u_T}_{\HH_2}\leq \frac{1}{\lambda}\doublebar{T}_{\HH_1\mapsto\C}$. \end{thm}
We construct layer potentials as follows.
Let $\arr g\in\NN_2$. Then the operator $T_{\arr g}\varphi = \langle \arr g,\Tr_1\varphi\rangle$ is a bounded linear operator on~$\HH_1$. By the Lax-Milgram lemma, there is a unique $u_T=\s^L_\Omega\arr g\in \HH_2$ such that \begin{equation}\label{eqn:S} \B( \varphi, \s^L_\Omega\arr g ) = \langle \Tr \varphi,\arr g\rangle \quad\text{for all $\varphi\in\HH_1$}.\end{equation} We will let $\s^L_\Omega\arr g$ denote the single layer potential of~$\arr g$. Observe that the dependence of $\s^L_\Omega$ on the parameter $\Omega$ consists entirely of the dependence of the trace operator on~$\Omega$, and the connection between $\Tr_2$ and $\Omega$ is given by formula~\eqref{cond:trace:extension}. This formula is symmetric about an interchange of $\Omega$ and $\CC$, and so $\s^L_\Omega \arr g=\s^L_\CC\arr g$.
The double layer potential is somewhat more involved. We begin by defining the Newton potential.
Let $H$ be an element of the dual space $\HH_1^*$ to~$\HH_1$. By the Lax-Milgram theorem, there is a unique element $\PP^L H$ of $\HH_2$ that satisfies \begin{equation} \label{eqn:newton} \B(\varphi,\PP^L H) = \langle \varphi,H\rangle\quad\text{for all $\varphi\in\HH_1$}.\end{equation} We refer to $\PP^L$ as the Newton potential. In some applications, it is easier to work with the Newton potential rather than the single layer potential directly; we remark that \begin{equation}\s^L_\Omega \arr g = \PP^L (T_{\arr g}) \quad\text{where } \langle T_{\arr g},\varphi \rangle = \langle \arr g,\Tr_1\varphi\rangle.\end{equation}
We now return to the double layer potential. Let $\arr f\in\DD_2$. Then there is some $F\in \HH_2$ such that $\Tr_2 F=\arr f$. Let \begin{equation} \label{eqn:D:+}\D_\Omega^\B \arr f = -F\big\vert_\Omega + \PP^L (L(\1_\Omega F))\big\vert_\Omega \qquad\text{if $\Tr_2 F=\arr f$}.\end{equation} Notice that $\D_\Omega^\B \arr f$ is an element of $\HH_2^\Omega$, not of $\HH_2$.
We will conclude this section by showing that $\D^\B_\Omega\arr f$ is well defined, that is, does not depend on the choice of $F$ in formula~\eqref{eqn:D:+}. We will also establish that layer potentials are bounded operators.
\begin{lem}\label{lem:potentials:bounded} The double layer potential is well defined. Furthermore, we have the bounds \begin{gather*}\doublebar{\D^\B_\Omega\arr f}_{\HH_2^\Omega} \leq \frac{\doublebar{B^\CC}}{\lambda}\doublebar{\arr f}_{\DD_2}, \quad \doublebar{\D^\B_\CC\arr f}_{\HH_2^\CC} \leq \frac{\doublebar{B^\Omega}}{\lambda}\doublebar{\arr f}_{\DD_2}, \quad \doublebar{\s^L_\Omega\arr g}_{\HH_2} \leq \frac{1}{\lambda}\doublebar{\arr g}_{\NN_2}. \end{gather*} \end{lem}
\begin{proof} By Theorem~\ref{thm:lax-milgram}, we have that \begin{equation*}\doublebar{\s^L_\Omega \arr g}_{\HH_2} \leq \frac{1}{\lambda} \doublebar{T_{\arr g}}_{\HH_1\mapsto\C} \leq \frac{1}{\lambda} \doublebar{\Tr_1}_{\HH_1\mapsto\DD_1}\doublebar{\arr g}_{\DD_1\mapsto\C}. \end{equation*} By definition of $\DD_1$ and $\NN_2$, $\doublebar{\Tr_1}_{\HH_1\mapsto\DD_1}=1$ and $\doublebar{\arr g}_{\DD_1\mapsto\C}=\doublebar{\arr g}_{\NN_2}$, and so $\s^L_\Omega:\NN_2\mapsto\HH_2$ is bounded with operator norm at most $1/\lambda$.
We now turn to the double layer potential. We will begin with a few properties of the Newton potential. By definition of~$L$, if $\varphi\in\HH_1$ then $\langle \varphi, LF\rangle = \B(\varphi, F)$. By definition of~$\PP^L$, $\B(\varphi,\PP^L (LF)) = \langle\varphi, LF\rangle$. Thus, by coercivity of~$\B$, \begin{equation}F=\PP^L(LF)\quad\text{for all }F\in\HH_2.\end{equation}
By definition of $\B^\Omega$, $\B^\CC$ and $L(\1_\Omega F)$, \begin{align*} \langle\varphi,LF\rangle &= \B(\varphi,F) =\B^\Omega(\varphi\big\vert_\Omega,F\big\vert_\Omega) +\B^\CC(\varphi\big\vert_{\CC},F\big\vert_{\CC})& \\&= \langle\varphi, L(F\1_\Omega)\rangle + \langle \varphi, L(F\1_{\CC})\rangle &\text{for all $\varphi\in\HH_1$}.\end{align*} Thus, $LF=L(F\1_\Omega)+L(F\1_{\CC})$ and so \begin{align} \label{eqn:D:alternate:extensions} -F + \PP^L (L(F\1_\Omega)) &= -F + \PP^L (LF) - \PP^L (F\1_{\CC}) \\\nonumber&= - \PP^L (L(F\1_{\CC})) .\end{align}
In particular, suppose that $\arr f=\Tr_2 F=\Tr_2 F'$. By Condition~\eqref{cond:trace:extension}, there is some $F''\in \HH_2$ such that $F''\big\vert_\Omega=F\big\vert_\Omega$ and $F''\big\vert_{\CC}=F'\big\vert_{\CC}$. Then \begin{align*} -F\big\vert_\Omega + \PP^L (L(\1_\Omega F))\big\vert_\Omega &= -F''\big\vert_\Omega + \PP^L (L(\1_\Omega F''))\big\vert_\Omega = - \PP^L (L(F''\1_{\CC}))\big\vert_\Omega \\&= - \PP^L (L(F'\1_{\CC}))\big\vert_\Omega = -F'\big\vert_\Omega + \PP^L (L(\1_\Omega F'))\big\vert_\Omega \end{align*} and so $\D_\Omega^\B \arr f$ is well-defined, that is, depends only on $\arr f$ and not the choice of function $F$ with $\Tr_2 F=\arr f$.
Furthermore, we have the alternative formula \begin{equation}\label{eqn:D:alternate} \D_\Omega^\B \arr f = - \PP^L (L(\1_{\CC} F))\big\vert_\Omega \qquad\text{if $\Tr_2 F=\arr f$} .\end{equation}
Thus, \begin{equation*}\doublebar{\D_\Omega^B\arr f}_{\HH_2^\Omega} \leq \inf_{\Tr_2 F=\arr f} \doublebar{\PP^L (L(\1_{\CC} F))\big\vert_\Omega}_{\HH_2^\Omega} \leq \inf_{\Tr F=\arr f} \doublebar{\PP^L (L(\1_{\CC} F))}_{\HH_2}. \end{equation*} by definition of the $\HH_2^\Omega$-norm.
By Theorem~\ref{thm:lax-milgram} and definition of $\PP^L$, we have that \begin{equation*}\doublebar{\PP^L (L(\1_{\CC} F))}_{\HH_2}\leq \frac{1}{\lambda} \doublebar{L(\1_{\CC} F)}_{\HH_1\mapsto\C}.\end{equation*} Since $L(\1_{\CC} F)(\varphi) = \B^\CC(\varphi\big\vert_{\CC}, F\big\vert_{\CC})$, we have that \begin{equation*}\doublebar{L(\1_{\CC} F)}_{\HH_1\mapsto\C}\leq \doublebar{\B^\CC}\doublebar{F\big\vert_{\CC}}_{\HH_2^\CC} \leq \doublebar{\B^\CC}\doublebar{F}_{\HH_2}\end{equation*} and so \begin{equation*}\doublebar{\D_\Omega^B\arr f\big\vert_\Omega}_{\HH_2^\Omega} \leq \inf_{\Tr F=\arr f} \frac{1}{\lambda} \doublebar{\B^\CC}\doublebar{F}_{\HH_2} =\frac{1}{\lambda} \doublebar{\B^\CC} \doublebar{\arr f}_{\DD_2} \end{equation*} as desired. \end{proof}
\section{Properties of layer potentials} \label{sec:properties}
We will begin this section by showing that layer potentials are solutions to the equation $(Lu)\big\vert_\Omega=0$ (Lemma~\ref{lem:potentials:solutions}). We will then prove the Green's formula (Lemma~\ref{lem:green}), the adjoint formulas for layer potentials (Lemma~\ref{lem:adjoint}), and conclude this section by proving the jump relations for layer potentials (Lemma~\ref{lem:jump}).
\begin{lem}\label{lem:potentials:solutions} Let $\arr f\in\DD$, $\arr g\in\NN$, and let $u=\D^\B_\Omega\arr f$ or $u=\s^L_\Omega\arr g\big\vert_\Omega$. Then $(Lu)\big\vert_\Omega=0$. \end{lem} \begin{proof} Recall that $(Lu)\big\vert_\Omega=0$ if $\B^\Omega(\varphi_+\big\vert_\Omega,u)=0$ for all $\varphi_+\in\HH_1$ with $\Tr_1\varphi_+=0$. If $\Tr_1\varphi_+=0=\Tr_1 0$, then by Condition~\eqref{cond:trace:extension} there is some $\varphi\in\HH_1$ with $\varphi\big\vert_\Omega=\varphi_+$, $\varphi\big\vert_{\CC} = 0$ and $\Tr_1\varphi=0$.
By the definition~\eqref{eqn:S} of the single layer potential, \begin{equation*}0=\B(\varphi,\s^L\arr g) =\B^\Omega(\varphi\big\vert_\Omega,\s^L_\Omega\arr g\big\vert_\Omega) +\B^\CC(\varphi\big\vert_{\CC},\s^L_\Omega\arr g\big\vert_{\CC}) =\B^\Omega(\varphi_+\big\vert_\Omega,\s^L_\Omega\arr g\big\vert_\Omega) \end{equation*} as desired.
Turning to the double layer potential, if $\varphi\in\HH_1$, then by the definition~\eqref{eqn:D:+} of $\D_\Omega^\B$, formula~\eqref{eqn:D:alternate} for~$\D_\CC^\B$ and linearity of~$\B^\Omega$, \begin{align*} \B^\Omega(\varphi\big\vert_\Omega,\D^\B_\Omega \arr f) &= -\B^\Omega\bigl(\varphi\big\vert_\Omega, F\big\vert_\Omega\bigr) +\B^\Omega\bigl(\varphi\big\vert_\Omega, \PP^L(L(\1_\Omega F))\big\vert_\Omega\bigr) ,\\ \B^\CC(\varphi\big\vert_\CC,\D^\B_\CC \arr f\big\vert_{\CC}) &=-\B^\CC\bigl(\varphi\big\vert_{\CC}, \PP^L(L(\1_\Omega F))\big\vert_{\CC}\bigr) .\end{align*} Subtracting and applying Condition~\eqref{cond:local}, \begin{align*} \B^\Omega(\varphi\big\vert_\Omega,\D^\B_\Omega \arr f) -\B^\CC(\varphi\big\vert_\CC,\D^\B_\CC \arr f\big\vert_{\CC}) &= -\B^\Omega\bigl(\varphi\big\vert_\Omega, F\big\vert_\Omega\bigr) +\B\bigl(\varphi, \PP^L(L(\1_\Omega F))\bigr) .\end{align*}
By the definition~\eqref{eqn:newton} of $\PP^L$, \begin{equation*}\B\bigl(\varphi, \PP^L(L(\1_\Omega F))\bigr) = \langle \varphi, L(\1_\Omega F)\rangle\end{equation*} and by the definition~\eqref{dfn:L:singular} of $L(\1_\Omega F)$, \begin{equation*}\B\bigl(\varphi, \PP^L(L(\1_\Omega F))\bigr) = \B^\Omega(\varphi\big\vert_\Omega,F\big\vert_\Omega).\end{equation*}
Thus, \begin{align}\label{eqn:D:solution} \B^\Omega(\varphi\big\vert_\Omega,\D^\B_\Omega \arr f) -\B^\CC(\varphi\big\vert_\CC,\D^\B_\CC \arr f) &= 0 \quad\text{for all $\varphi\in\HH_1$.} \end{align} In particular, as before if $\Tr_1 \varphi_+=0$ then there is some $\varphi$ with $\varphi\big\vert_\Omega=\varphi_+\big\vert_\Omega$, $\varphi\big\vert_\CC=0$ and so $\B^\Omega(\varphi\big\vert_\Omega,\D^\B_\Omega \arr f)=0$. This completes the proof. \end{proof}
\begin{lem}\label{lem:green} If $u\in\HH_2^\Omega$ and $(Lu)\big\vert_\Omega=0$, then \begin{equation*}u = -\D^\B_\Omega (\Tr_2 U) + \s^L_\Omega (\M^\B_\Omega u)\big\vert_\Omega,\quad 0 = \D^\B_\CC (\Tr_2 U) + \s^L_\CC (\M^\B_\Omega u)\big\vert_{\CC}\end{equation*} for any $U\in\HH_2$ with $U\big\vert_\Omega=u$. \end{lem}
\begin{proof} By definition~\eqref{eqn:D:+} of the double layer potential, \begin{equation*} -\D_\Omega^\B (\Tr_2 U) = U\big\vert_\Omega - \PP^L (L(\1_\Omega U))\big\vert_\Omega = u - \PP^L (L(\1_\Omega u))\big\vert_\Omega \end{equation*} and by formula~\eqref{eqn:D:alternate} \begin{equation*}\D_\CC^\B (\Tr_2 U) = -\PP^L (L(\1_\Omega u))\big\vert_{\CC}.\end{equation*} It suffices to show that $\PP^L(L(\1_\Omega u))=\s^L_\Omega(\M^\B_\Omega u)$.
Let $\varphi\in\HH_1$. By formulas~\eqref{eqn:S} and~\eqref{eqn:Neumann}, \begin{equation*}\B(\varphi,\s^L_\Omega (\M^\B_\Omega u)) =\langle \Tr_1 \varphi,\M^\B_\Omega u\rangle =\B^\Omega(\varphi\big\vert_\Omega,u) .\end{equation*} By formula~\eqref{eqn:newton} for the Newton potential and by the definition~\eqref{dfn:L:singular} of $L(\1_\Omega u)$, \begin{equation*} \B(\varphi, \PP^L(L(\1_\Omega u))) = \langle \varphi, L(\1_\Omega u)\rangle =\B^\Omega(\varphi\big\vert_\Omega,u) .\end{equation*} Thus, $\B(\varphi, \PP^L(L(\1_\Omega u))) = \B(\varphi,\s^L_\Omega (\M^\B_\Omega u))$ for all $\varphi\in\HH$; by coercivity of $\B$, we must have that $\PP^L(L(\1_\Omega u))=\s^L_\Omega(\M^\B_\Omega u)$. This completes the proof. \end{proof}
Let $\B^*(\varphi,\psi)=\overline{\B(\psi,\varphi)}$ and define $\B^\Omega_*$, $\B^\CC_*$ analogously. Then $\B^*$ is a bounded and coercive operator $\HH_2\times \HH_1\mapsto\C$, and so we can define the double and single layer potentials $\D^{\B^*}_\Omega:\DD_1\mapsto \HH_1^\Omega$, $\s^{L^*}_\Omega:\NN_1\mapsto \HH_1$.
We then have the following adjoint relations.
\begin{lem}\label{lem:adjoint} We have the adjoint relations \begin{align} \label{eqn:neumann:D:dual} \langle \arr \varphi, \M_\B^{\Omega} \D^{\B}_\Omega \arr f\rangle &= \langle\M_{\B^*}^{\Omega} \D^{\B^*}_\Omega \arr \varphi, \arr f\rangle ,\\ \label{eqn:dirichlet:S:dual} \langle \arr \gamma, \Tr_2 \s^L_\Omega \arr g\rangle &= \langle \Tr_1 \s^{L^*}_\Omega \arr \gamma, \arr g\rangle \end{align} for all $\arr f\in \DD_2$, $\arr \varphi\in\DD_1$, $\arr g\in\NN_2$ and $\arr\gamma\in\NN_1$.
If we let $\Tr_2^\Omega \D^{\B}_\Omega \arr f = -\Tr_2 F + \Tr_2 \PP^L(L(\1_\Omega F))$, where $F$ is as in formula~\eqref{eqn:D:+}, then $\Tr_2^\Omega \D^{\B}_\Omega \arr f $ does not depend on the choice of $F$, and we have the duality relations \begin{align}\label{eqn:dirichlet:D:dual} \langle \arr \gamma, \Tr_2^\Omega \D^{\B}_\Omega \arr f\rangle &= \langle-\arr \gamma+\M_{\B^*}^{\Omega}\s^{L^*}_\Omega\arr \gamma, \arr f\rangle .\end{align} \end{lem}
\begin{proof} By formula~\eqref{eqn:S}, \begin{gather*}\langle \Tr_1 \s^{L^*}_\Omega \arr \gamma, \arr g\rangle =\B(\s^{L^*}_\Omega\arr \gamma,\s^L_\Omega\arr g\rangle, \\ \langle \Tr_1 \s^{L}_\Omega \arr g, \arr \gamma\rangle =\B^*(\s^{L}_\Omega\arr g,\s^{L^*}_\Omega\arr \gamma\rangle\end{gather*} and so formula~\eqref{eqn:dirichlet:S:dual} follows by definition of~$\B^*$.
Let $\Phi\in\HH_1$ and $F\in\HH_2$ with $\Tr_1\Phi=\arr\varphi$, $\Tr_2F=\arr f$.
Then by formulas \eqref{eqn:Neumann} and~\eqref{eqn:D:+}, \begin{align*}\langle \arr \varphi, \M_\B^{\Omega} \D^{\B}_\Omega \arr f\rangle &=
\B^\Omega(\Phi\big\vert_\Omega, \D^{\B}_\Omega \arr f) =
-\B^\Omega(\Phi\big\vert_\Omega, F\vert_\Omega)
+\B^\Omega(\Phi\big\vert_\Omega, \PP^L(L(\1_\Omega F))\big\vert_\Omega) \\&=
-\overline{\B^\Omega_*(F\vert_\Omega, \Phi\big\vert_\Omega)}
+\overline{\B^\Omega_*(\PP^L(L(\1_\Omega F))\big\vert_\Omega, \Phi\big\vert_\Omega)} .\end{align*} By formula~\eqref{dfn:L:singular}, \begin{equation*}\B^\Omega_*(\PP^L(L(\1_\Omega F))\big\vert_\Omega, \Phi\big\vert_\Omega) = \langle \PP^L(L(\1_\Omega F)), L^*(\1_\Omega\Phi)\rangle.\end{equation*} By formula~\eqref{eqn:newton}, \begin{equation*}\B^\Omega_*(\PP^L(L(\1_\Omega F))\big\vert_\Omega, \Phi\big\vert_\Omega) = \B^\Omega_*( \PP^L(L(\1_\Omega F)), \PP^{L^*}(L^*(\1_\Omega\Phi))).\end{equation*} Thus, \begin{align*}\langle \arr \varphi, \M_\B^{\Omega} \D^{\B}_\Omega \arr f\rangle &=
-\overline{\B^\Omega_*(F\vert_\Omega, \Phi\big\vert_\Omega)}
+\overline{\B^\Omega_*( \PP^L(L(\1_\Omega F)), \PP^{L^*}(L^*(\1_\Omega\Phi)))} .\end{align*} By the same argument \begin{align*}\langle \arr f,\M_{\B^*}^{\Omega} \D^{\B^*}_\Omega \arr \varphi\rangle &=
-\overline{\B^\Omega(\Phi\vert_\Omega, F\big\vert_\Omega)}
+\overline{\B^\Omega( \PP^{L^*}(L^*(\1_\Omega \Phi)), \PP^{L}(L(\1_\Omega F)))} \end{align*} and by definition of $\B^\Omega_*$ formula~\eqref{eqn:neumann:D:dual} is proven.
Finally, by definition of $\Tr_2^\Omega \D^{\B}_\Omega$, \begin{equation*}\langle \arr \gamma, \Tr_2^\Omega \D^{\B}_\Omega \arr f\rangle = -\langle \arr \gamma, \Tr_2 F\rangle + \langle \arr \gamma, \Tr_2 \PP^L(L(\1_\Omega F)) \rangle .\end{equation*} By the definition~\eqref{eqn:S} of the single layer potential, \begin{equation*} \langle \arr \gamma, \Tr_2 \PP^L(L(\1_\Omega F)) \rangle = \overline{\B^*( \PP^L(L(\1_\Omega F)) ,\s^{L^*}_\Omega\arr \gamma)} .\end{equation*} By definition of $\B^*$ and the definition~\eqref{eqn:newton} of the Newton potential, \begin{equation*}\overline{\B^*( \PP^L(L(\1_\Omega F)) ,\s^{L^*}_\Omega\arr \gamma)} = {\langle \s^{L^*}_\Omega\arr \gamma, L(\1_\Omega F) \rangle} \end{equation*} and by the definition~\eqref{dfn:L:singular} of $L(\1_\Omega F)$, \begin{equation*}{\langle \s^{L^*}_\Omega\arr \gamma, L(\1_\Omega F) \rangle} = \B^\Omega (\s^{L^*}_\Omega\arr \gamma\big\vert_\Omega, F\big\vert_\Omega).\end{equation*} By the definition~\eqref{eqn:Neumann} of Neumann boundary values, \begin{equation*}\overline{\B^\Omega_*(F,\s^{L^*}_\Omega\arr \gamma)} = \overline{\langle \Tr_2 F, \M^\Omega_{\B^*}(\s^{L^*}_\Omega\arr\gamma\big\vert_\Omega)\rangle}\end{equation*} and so \begin{equation*}\langle \arr \gamma, \Tr_2^\Omega \D^{\B}_\Omega \arr f\rangle = -\langle\arr\gamma,\arr f\rangle + {\langle \M^\Omega_{\B^*}(\s^{L^*}_\Omega\arr\gamma\big\vert_\Omega), \arr f\rangle} \end{equation*} for any choice of $F$. Thus $\Tr_2^\Omega \D^{\B}_\Omega $ is well-defined and formula~\eqref{eqn:dirichlet:D:dual} is valid. \end{proof}
\begin{lem}\label{lem:jump} Let $\Tr_2^\Omega \D^{\B}_\Omega $ be as in Lemma~\ref{lem:adjoint}. If $\arr f\in\DD$ and $\arr g\in\NN$, then we have the jump and continuity relations \begin{align} \label{eqn:D:jump} \Tr_2^\Omega\D^\B_\Omega\arr f +\Tr_2^\CC\D^\B_\CC\arr f
&=-\arr f ,\\ \label{eqn:S:jump}
\M_\B^\Omega (\s^L_\Omega \arr g\big\vert_\Omega) +\M_\B^{\CC} (\s^L_\Omega\arr g\big\vert_{\CC})
&=\arr g ,\\ \label{eqn:D:cts} \M_\B^{\Omega} (\D^\B_\Omega\arr f) - \M_\B^{\CC} (\D^\B_\CC\arr f )
&=0 .\end{align}
If there are bounded operators $\Tr_2^\Omega:\HH_2^\Omega\mapsto\DD_2$ and $\Tr_2^\CC:\HH_2^\CC\mapsto\DD_2$ such that $\Tr_2 F = \Tr_2^\Omega (F\big\vert_\Omega)= \Tr_2^\CC (F\big\vert_\CC)$ for all $F\in\HH_2$, then in addition \begin{align} \label{eqn:S:cts} \Tr_2^\Omega(\s^L_\Omega \arr g\big\vert_\Omega) -\Tr_2^\CC(\s^L_\Omega \arr g\big\vert_{\CC})
&=0 .\end{align} \end{lem}
The given condition formula for $\Tr_2^\Omega$, $\Tr_2^\CC$ is very natural if $\Omega\subset\R^d$ is an open set, $\CC=\R^d\setminus\bar\Omega$ and $\Tr_2$ denotes a trace operator restricting functions to the boundary~$\partial\Omega$. Observe that if such operators $\Tr_2^\Omega$ and $\Tr_2^\CC$ exist, then by the definition~\eqref{eqn:D:+} of the double layer potential and by the definition of $\Tr_2^\Omega \D^\B_\Omega$ in Lemma~\ref{lem:adjoint}, $\Tr_2^\Omega (\D^\B_\Omega \arr f) = (\Tr_2^\Omega \D^\B_\Omega)\arr f$ and so there is no ambiguity of notation.
\begin{proof}[Proof of Lemma~\ref{lem:jump}] We first observe that by the definition~\eqref{eqn:Neumann} of Neumann boundary values, if $u_+\in\HH_2^\Omega$ and $u_-\in\HH_2^\CC$ with $(Lu_+)\big\vert_\Omega=0$ and $(Lu_-)\big\vert_\CC=0$, then \begin{equation*} \M_\B^\Omega u_+ +\M_\B^{\CC} u_-=\arr \psi \text{ if and only if } \langle\Tr_1\varphi,\arr\psi\rangle = \B^\Omega(\varphi\big\vert_\Omega,u_+) +\B^\CC(\varphi\big\vert_{\CC},u_-) \end{equation*} for all $\varphi\in\HH_1$.
The continuity relation \eqref{eqn:D:cts} follows from formula~\eqref{eqn:D:solution}.
The jump relation~\eqref{eqn:D:jump} follows from the definition of $\Tr_2^\Omega\D^\B_\Omega$ and by using formula~\eqref{eqn:D:alternate:extensions} to rewrite $\Tr_2^\CC\D^\B_\CC$.
The jump relation \eqref{eqn:S:jump} follows from the definition~\eqref{eqn:S} of the single layer potential.
The continuity relation~\eqref{eqn:S:cts} follows because $\s^L_\Omega\arr g\in \HH_2$ and by the definition of $\Tr_2^\Omega$, $\Tr_2^\CC$. \end{proof}
\section{Layer potentials and boundary value problems} \label{sec:invertible}
If $\HH_2^\Omega$ and $\DD_2$ are as in Section~\ref{sec:example}, then by the Lax-Milgram lemma there is a unique solution to the Dirichlet aed and Neumann boundary value problems \begin{equation*}\left\{\begin{aligned} (Lu)\big\vert_\Omega &= 0,\\ \Tr^\Omega_2 u &= \arr f,\\ \doublebar{u}_{\HH_2^\Omega} &\leq C \doublebar{\arr f}_{\DD_2}, \end{aligned}\right.\qquad \left\{\begin{aligned} (Lu)\big\vert_\Omega &= 0,\\ \M_\B^\Omega u &= \arr g,\\ \doublebar{u}_{\HH^\Omega_2} &\leq C \doublebar{\arr g}_{\NN_2} .\end{aligned}\right. \end{equation*} We routinely wish to establish existence and uniqueness to the Dirichlet and Neumann boundary value problems \begin{equation*} (D)^{\widehat L}_\XX\left\{\begin{aligned} (\widehat Lu)\big\vert_\Omega &= 0,\\ \WTr_\XX^\Omega u &= \arr f,\\ \doublebar{u}_{\XX^\Omega} &\leq C \doublebar{\arr f}_{\DD_\XX}, \end{aligned}\right.\qquad (N)^\B_\XX\left\{\begin{aligned} (\widehat Lu)\big\vert_\Omega &= 0,\\ \MM_\B^\Omega u &= \arr g,\\ \doublebar{u}_{\XX^\Omega} &\leq C \doublebar{\arr g}_{\NN_\XX} \end{aligned}\right. \end{equation*} for some constant $C$ and some other solution space $\XX$ and spaces of Dirichlet and Neumann boundary data $\DD_\XX$ and $\NN_\XX$. For example, if $L$ is a second-order differential operator, then as in \cite{JerK81B,KenP93,KenR09,DinPR13p} we might wish to establish well-posedness with $\DD_\XX=\dot W_1^p(\partial\Omega)$, $\NN_\XX=L^p(\partial\Omega)$ and $\XX=\{u:\widetilde N(\nabla u)\in L^p(\partial\Omega)\}$, where $\widetilde N$ is the nontangential maximal function introduced in \cite{KenP93}.
The classic method of layer potentials states that if layer potentials, originally defined as bounded operators $\D^\B_\Omega:\DD_2\mapsto\HH_2$ and $\s^L_\Omega:\NN_2\mapsto\HH_2$, may be extended to operators $\D^\B_\Omega:\DD_\XX\mapsto\XX$ and $\s^L_\Omega:\NN_\XX\mapsto\XX$, and if certain of the properties of layer potentials of Section~\ref{sec:properties} are preserved by that extension, then well posedness of boundary value problems are equivalent to certain invertibility properties of layer potentials.
In this section we will make this notion precise.
As in Sections~\ref{sec:dfn}, \ref{sec:D:S} and~\ref{sec:properties}, we will work with layer potentials and function spaces in a very abstract setting.
\subsection{From invertibility to well posedness} \label{sec:invertible:well-posed}
In this section we will need the following objects. \begin{itemize} \item Quasi-Banach spaces $\XX^\Omega$, $\DD_\XX$ and $\NN_\XX$. \item A linear operator $u\mapsto(\widehat L u)\big\vert_\Omega$ acting on~$\XX^\Omega$. \item Linear operators $\WTr_\XX^\Omega:\{u\in\XX^\Omega:(\widehat L u)\big\vert_\Omega =0\}\mapsto\DD_\XX$ and $\MM_\B^\Omega:\{u\in\XX^\Omega:(\widehat L u)\big\vert_\Omega =0\}\mapsto\NN_\XX$. \item Linear operators $\widehat\D^\B_\Omega:\DD_\XX\mapsto \XX^\Omega$ and $\widehat\s^L_\Omega:\NN_\XX\mapsto \XX^\Omega$. \end{itemize} \begin{rmk} Recall that $\s^L_\Omega=\s^L_\CC$ is defined in terms of a ``global'' Hilbert space $\HH_2$. If $\XX^\Omega=\HH^\Omega$, then $\widehat \s^L_\Omega\arr g = \s^L_\Omega\arr g\big\vert_\Omega$. In the general case, we do not assume the existence of a global quasi-Banach space $\XX$ whose restrictions to $\Omega$ lie in~$\XX^\Omega$, and thus we will let $\widehat \s^L_\Omega\arr g$ be an element of $\XX^\Omega$ without assuming a global extension. \end{rmk}
In applications it is often useful to define $\Tr^\Omega$, $\M_\B^\Omega$, $L$, $\D^\B_\Omega$ and $\s^L_\Omega$ in terms of some Hilbert spaces $\HH_j$, $\HH_j^\Omega$ and to extend these operators to operators with domain or range $\XX^\Omega$ by density or some other means. See, for example, \cite{BarHM17pC}. We will not assume that the operators $\WTr_\XX^\Omega$, $\MM_\B^\Omega$, $\widehat L$, $\widehat\D^\B_\Omega$ and $\widehat\s^L_\Omega$ arise by density; we will merely require that they satisfy certain properties similar to those established in Section~\ref{sec:properties}.
Specifically, we will often use the following conditions; observe that if $\XX^\Omega=\HH_2^\Omega$ for some $\HH_2^\Omega$ as in Section~\ref{sec:dfn}, these properties are valid. \begin{description} \item[\hypertarget{ConditionT}{\condTname}] $\WTr_\XX^\Omega$ is a bounded operator $\{u\in\XX^\Omega:(\widehat L u)\big\vert_\Omega =0\}\mapsto \DD_\XX$. \item[\hypertarget{ConditionM}{\condMname}] $\MM_\B^\Omega$ is a bounded operator $\{u\in\XX^\Omega:(\widehat L u)\big\vert_\Omega =0\}\mapsto \NN_\XX$. \item[\hypertarget{ConditionS}{\condSname}] The single layer potential $\widehat\s^L_\Omega$ is bounded $\NN_\XX\mapsto \XX^\Omega$, and if $\arr g\in \NN_\XX$ then $(\widehat L (\widehat\s^L_\Omega\arr g))\big\vert_\Omega=0$. \item[\hypertarget{ConditionD}{\condDname}] The double layer potential $\widehat\D^\B_\Omega$ is bounded $\DD_\XX\mapsto\XX^\Omega$, and if $\arr f\in \DD_\XX$ then $(\widehat L (\widehat\D^\B_\Omega\arr f))\big\vert_\Omega=0$. \item[\hypertarget{ConditionG}{\condGname}] If $u \in \XX^\Omega$ and $(\widehat L u)\big\vert_\Omega =0$, then we have the Green's formula \begin{equation*}u = -\widehat\D^\B_\Omega (\WTr_\XX^\Omega u) + \widehat\s^L_\Omega (\MM_\B^\Omega u).\end{equation*} \end{description}
We remark that the linear operator $u\mapsto(\widehat L u)\big\vert_\Omega$ is used only to characterize the subspace $\XX^\Omega_{ L}=\{u\in\XX^\Omega:(\widehat L u)\big\vert_\Omega =0\}$. We could work directly with $\XX^\Omega_{ L}$; however, we have chosen to use the more cumbersome notation $\{u\in\XX^\Omega:(\widehat L u)\big\vert_\Omega =0\}$ to emphasize that the following arguments, presented here only in terms of linear operators, are intended to be used in the context of boundary value problems for differential equations.
The following theorem is straightforward to prove and is the core of the classic method of layer potentials. \begin{thm}\label{thm:surjective:existence} Let $\XX$, $\DD_\XX$, $\NN_\XX$, $\widehat\D^\B_\Omega$, $\widehat\s^L_\Omega$, $\WTr_\XX^\Omega$ and $\MM_\B^\Omega$ be quasi-Banach spaces and linear operators with domains and ranges as above.
Suppose that Conditions~\condT\ and \condS\ are valid, and that $\WTr_\XX^\Omega \widehat\s^L_\Omega: \NN_\XX \mapsto \DD_\XX$ is surjective. Then for every $\arr f\in \DD_\XX$, there is some $u$ such that \begin{equation}\label{eqn:Dirichlet:weak} (\widehat L u)\big\vert_\Omega = 0, \quad \WTr_\XX^\Omega u = \arr f, \quad u\in\XX^\Omega.\end{equation} Suppose in addition $\WTr_\XX^\Omega \widehat\s^L_\Omega: \NN_\XX \mapsto \DD_\XX$ has a bounded right inverse, that is, there is a constant $C_0$ such that if $\arr f\in\DD_\XX$, then there is some preimage $\arr g$ of $\arr f$ with $\doublebar{\arr g}_{\NN_\XX}\leq C_0\doublebar{\arr f}_{\DD_\XX}$. Then there is some constant $C_1$ depending on $C_0$ and the implicit constants in Conditions~\condT\ and~\condS\ such that if $\arr f\in\DD_\XX$, then there is some $u\in\XX^\Omega$ such that \begin{equation} \label{eqn:Dirichlet:strong} (\widehat L u)\big\vert_\Omega = 0, \quad \WTr_\XX^\Omega u = \arr f, \quad \doublebar{u}_{\XX^\Omega}\leq C_1\doublebar{\arr f}_{\DD_\XX}.\end{equation}
Suppose that Conditions~\condM\ and \condD\ are valid, and that $\MM_\B^\Omega\widehat\D^\B_\Omega: \DD_\XX \mapsto \NN_\XX$ is surjective. Then for every $\arr g\in \NN_\XX$, there is some $u$ such that \begin{equation}\label{eqn:Neumann:weak}(\widehat L u)\big\vert_\Omega = 0, \quad \MM_\B^\Omega u = \arr g, \quad u\in\XX^\Omega.\end{equation} If $\MM_\B^\Omega\widehat\D^\B_\Omega: \DD_\XX \mapsto \NN_\XX$ has a bounded right inverse, then there is some constant $C_1$ depending on the bound on that inverse and the implicit constants in Conditions~\condM\ and~\condD\ such that if $\arr g\in\NN_\XX$, then there is some $u\in\XX^\Omega$ such that \begin{equation}\label{eqn:Neumann:strong}(\widehat L u)\big\vert_\Omega = 0, \quad \MM_\B^\Omega u = \arr g, \quad \doublebar{u}_{\XX^\Omega}\leq C_1\doublebar{\arr g}_{\NN_\XX}.\end{equation}
\end{thm}
Thus, surjectivity of layer potentials implies of solutions to for boundary value problems.
We may also show that injectivity of layer potentials implies uniqueness of solutions to boundary value problems. This argument appeared first in \cite{BarM16A} and is the converse to an argument of \cite{Ver84}.
\begin{thm}\label{thm:injective:unique} Let $\XX$, $\DD_\XX$, $\NN_\XX$, $\widehat\D^\B_\Omega$, $\widehat\s^L_\Omega$, $\WTr_\XX^\Omega$ and $\MM_\B^\Omega$ be quasi-Banach spaces and linear operators with domains and ranges as above. Suppose that Conditions~\condT, \condM, \condD, \condS\ and~\condG\ are all valid.
Suppose that the operator $\WTr_\XX^\Omega \widehat\s^L_\Omega: \NN_\XX \mapsto \DD_\XX$ is one-to-one. Then for each $\arr f\in\DD_\XX$, there is at most one solution $u$ to the Dirichlet problem \begin{equation*}(\widehat L u)\big\vert_\Omega = 0, \quad \WTr_\XX^\Omega u = \arr f, \quad u\in\XX^\Omega.\end{equation*} If $\WTr_\XX^\Omega \widehat\s^L_\Omega: \NN_\XX \mapsto \DD_\XX$ has a bounded left inverse, that is, there is a constant $C_0$ such that the estimate $\doublebar{\arr g}_{\NN_\XX}\leq C_0 \doublebar{\WTr_\XX^\Omega \widehat\s^L_\Omega \arr g}_{\DD_\XX}$ is valid, then there is some constant $C_1$ such that every $u\in\XX^\Omega$ with $(\widehat L u)\big\vert_\Omega=0$ satisfies $\doublebar{u}_{\XX^\Omega}\leq C_1\doublebar{\Tr_\XX^\Omega u}_{\DD_\XX}$ (that is, if $u$ satisfies the Dirichlet problem~\eqref{eqn:Dirichlet:weak} then $u$ must satisfy the Dirichlet problem~\eqref{eqn:Dirichlet:strong}).
Similarly, if the operator $\MM_\B^\Omega \widehat\D^\B_\Omega:\DD_\XX \mapsto \NN_\XX$ is one-to-one, then for each $\arr g\in\NN_\XX$, there is at most one solution $u$ to the Neumann problem \begin{equation*}(\widehat L u)\big\vert_\Omega = 0, \quad \MM_\B^\Omega u = \arr g, \quad u\in\XX^\Omega.\end{equation*} If $\MM_\B^\Omega \widehat\D^\B_\Omega:\DD_\XX \mapsto \NN_\XX$ has a bounded left inverse, then there is some constant $C_1$ such that every $u\in\XX^\Omega$ with $(\widehat L u)\big\vert_\Omega=0$ satisfies $\doublebar{u}_{\XX^\Omega}\leq C_1\doublebar{\MM_\B^\Omega u}_{\DD_\XX}$. \end{thm}
\begin{proof} We present the proof only for the Neumann problem; the argument for the Dirichlet problem is similar.
Suppose that $u$, $v\in\XX^\Omega$ with $(\widehat Lu)\big\vert_\Omega=(\widehat Lv)\big\vert_\Omega=0$ in~$\Omega$ and $\MM_\B^\Omega u=\arr g=\MM_\B^\Omega v$. By Condition~\condG, \begin{align*} u = -\widehat\D^\B_\Omega (\WTr_\XX^\Omega u) + \widehat\s^L_\Omega (\MM_\B^\Omega u) &= -\widehat\D^\B_\Omega (\WTr_\XX^\Omega u) + \widehat\s^L_\Omega \arr g ,\\ v= -\widehat\D^\B_\Omega (\WTr_\XX^\Omega v) + \widehat\s^L_\Omega (\MM_\B^\Omega v) &= -\widehat\D^\B_\Omega (\WTr_\XX^\Omega v) + \widehat\s^L_\Omega \arr g .\end{align*} In particular, $\MM_\B^\Omega \widehat\D^\B_\Omega (\WTr_\XX^\Omega u) = \MM_\B^\Omega \widehat\D^\B_\Omega (\WTr_\XX^\Omega v)$. If $\MM_\B^\Omega \widehat\D^\B_\Omega$ is one-to-one, then $\WTr_\XX^\Omega u = \WTr_\XX^\Omega v$. Another application of Condition~\condG\ yields that $u=v$.
Now, suppose that we have the estimate $\doublebar{\arr f}_{\DD_\XX}\leq C \doublebar{\MM_\B^\Omega \widehat\D^\B_\Omega \arr f}_{\NN_\XX}$. (This implies injectivity of $\MM_\B^\Omega \widehat\D^\B_\Omega$.) Let $u\in {\XX^\Omega}$ with $(\widehat L u)\big\vert_\Omega=0$; we want to show that $\doublebar{u}_{\XX^\Omega}\leq C \doublebar{\MM_\B^\Omega u}_{\DD_\XX}$.
By Condition~\condG, and because $\XX^\Omega$ is a quasi-Banach space, \begin{equation*}\doublebar{u}_{\XX^\Omega}\leq C\doublebar{\widehat\D^\B_\Omega (\WTr_\XX^\Omega u)}_{\XX^\Omega} + C\doublebar{\widehat\s^L_\Omega (\MM_\B^\Omega u)}_{\XX^\Omega}.\end{equation*} By Conditions~\condD\ and~\condS, \begin{equation*}\doublebar{u}_\XX\leq C\doublebar{\WTr_\XX^\Omega u}_{\DD_\XX} + C\doublebar{\MM_\B^\Omega u}_{\NN_\XX}.\end{equation*} Applying our estimate on $\MM_\B^\Omega \widehat\D^\B_\Omega$, we see that \begin{equation*}\doublebar{u}_\XX\leq C\doublebar{\MM_\B^\Omega \widehat\D^\B_\Omega\WTr_\XX^\Omega u}_{\NN_\XX} + C\doublebar{\MM_\B^\Omega u}_{\NN_\XX}.\end{equation*} By Condition~\condG, $\widehat\D^\B_\Omega(\WTr_\XX^\Omega u) =\widehat\s^L_\Omega(\MM_\B^\Omega u) - u $, and so \begin{equation*}\doublebar{u}_\XX\leq C\doublebar{\MM_\B^\Omega \widehat \s^L_\Omega\MM_\B^\Omega u}_{\NN_\XX} + C\doublebar{\MM_\B^\Omega u}_{\NN_\XX}.\end{equation*} Another application of Condition~\condS\ and of Condition~\condM\ completes the proof. \end{proof}
\subsection{From well posedness to invertibility} \label{sec:well-posed:invertible}
We are now interested in the converse results. That is, we have shown that results for layer potentials imply results for boundary value problems; we would like to show that results for boundary value problems imply results for layer potentials.
Notice that the above results were built on the Green's formula~\condG. The converse results will be built on jump relations, as in Lemma~\ref{lem:jump}. Recall that jump relations treat the interplay between layer potentials in a domain and in its complement; thus we will need to impose conditions in both domains.
In this section we will need the following spaces and operators. \begin{itemize} \item Quasi-Banach spaces $\XX^\UU$, $\XX^\VV$, $\DD_\XX$ and $\NN_\XX$. \item Linear operators $u\mapsto(\widehat L u)\big\vert_\UU$ and $u\mapsto(\widehat L u)\big\vert_\VV$ acting on~$\XX^\UU$ and~$\XX^\VV$. \item Linear operators $\WTr_\XX^\UU $, $\MM_\B^\UU$, $\WTr_\XX^\VV $, and $\MM_\B^\VV$ acting on $\{u\in\XX^\UU:(\widehat L u)\big\vert_\UU =0\}$ or $\{u\in\XX^\VV:(\widehat L u)\big\vert_\VV =0\}$. \item Linear operators $\widehat\D^\B_\UU$, $\widehat\D^\B_\VV$ acting on $\DD_\XX$ and $\widehat\s^L_\UU$, $\widehat\s^L_\VV$ acting on $\NN_\XX$. \end{itemize} In the applications $\UU$ is an open set in $\R^d$ or in a smooth manifold, and $\VV=\R^d\setminus\overline\UU$ is the interior of its complement. The space $\XX^\VV$ is then a space of functions defined in~$\VV$ and is thus a different space from $\XX^\UU$. However, we emphasize that we have defined only one space $\DD_\XX$ of Dirichlet boundary values and one space $\NN_\XX$ of Neumann boundary values; that is, the traces from both sides of the boundary must lie in the same spaces.
We will often use the following conditions. \begin{description} \item[\hypertarget{ConditionTT}{\condTTname}, \hypertarget{ConditionMM}{\condMMname}, \hypertarget{ConditionSS}{\condSSname}, \hypertarget{ConditionDD}{\condDDname}, \hypertarget{ConditionGG}{\condGGname}] Condition \condT, \condM, \condS, \condD, or \condG\ holds for both $\Omega=\UU$ and $\Omega=\VV$. \item[\hypertarget{ConditionJScts}{\condJSctsname}] If $\arr g\in\NN_\XX$, then we have the continuity relation \begin{align*}
\WTr_\XX^\UU (\widehat\s^L_\UU \arr g) -\WTr_\XX^\VV (\widehat\s^L_\VV \arr g)
&=0 .\end{align*} \item[\hypertarget{ConditionJDcts}{\condJDctsname}] If $\arr f\in\DD_\XX$, then we have the continuity relation \begin{align*}
\MM_\B^\UU (\widehat\D^\B_\UU\arr f) - \MM_\B^\VV (\widehat\D^\B_\VV\arr f )
&=0 .\end{align*} \item[\hypertarget{ConditionJSjump}{\condJSjumpname}] If $\arr g\in\NN_\XX$, then we have the jump relation \begin{align*}
\MM_\B^\UU (\widehat\s^L_\UU ) +\MM_\B^\VV (\widehat\s^L_\VV\arr g)
&=\arr g .\end{align*} \item[\hypertarget{ConditionJDjump}{\condJDjumpname}] If $\arr f\in\DD_\XX$, then we have the jump relation \begin{align*}
\WTr_\XX^\UU (\widehat\D^\B_\UU\arr f) +\WTr_\XX^\VV (\widehat\D^\B_\VV\arr f)
&=-\arr f .\end{align*} \end{description}
We now move from well posedness of boundary value problems to invertibility of layer potentials.
The following theorem uses an argument of Verchota from \cite{Ver84}.
\begin{thm}\label{thm:unique:injective} Assume that Conditions~\condMM, \condSS, \condJScts, and \condJSjump\ are valid. Suppose that, for any $\arr f\in\DD_\XX$, there is at most one solution $u_+$ or $u_-$ to each of the two Dirichlet problems \begin{gather*} (\widehat L u_+)\big\vert_\UU = 0, \quad \WTr_\XX^\UU u_+ = \arr f, \quad u_+\in\XX^\UU ,\\ (\widehat L u_-)\big\vert_\VV = 0, \quad \WTr_\XX^\VV u_- = \arr f, \quad u_-\in\XX^\VV .\end{gather*} Then $\Tr_\XX^{\UU}\widehat\s^L_\UU:\NN_\XX\mapsto\DD_\XX$ is one-to-one.
If in addition there is a constant $C_0$ such that every $u_+\in\XX^\UU$ and $u_-\in\XX^\VV$ with $(\widehat L u_+)\big\vert_\UU =0$ and $(\widehat L u_+)\big\vert_\VV = 0$ satisfies \begin{equation*}\doublebar{u_+}_{\XX^\UU}\leq C_0\doublebar{\WTr_\XX^\UU u}_{\DD_\XX}, \quad \doublebar{u_-}_{\XX^\VV}\leq C_0\doublebar{\WTr_\XX^\VV u}_{\DD_\XX}, \end{equation*} then there is a constant $C_1$ such that the bound $\doublebar{\arr g}_{\NN_\XX} \leq C_1 \doublebar{\Tr_\XX^{\UU}\widehat\s^L_\UU\arr g}_{\DD_\XX}$ is valid for all $\arr g\in\NN_\XX$.
Similarly, assume that Conditions~ \condTT, \condDD, \condJDcts, and \condJDjump\ are valid. Suppose that for any $\arr g\in\NN_\XX$, there is at most one solution $u_\pm$ to each of the two Neumann problems \begin{gather*} (\widehat L u_+)\big\vert_\UU = 0, \quad \MM_\B^\UU u_+ = \arr g, \quad u_+\in\XX^\UU ,\\ (\widehat L u_-)\big\vert_\VV = 0, \quad \MM_\B^\VV u_- = \arr g, \quad u_-\in\XX^\VV .\end{gather*} Then $\MM_\B^\UU\widehat\D^\B_\UU:\DD_\XX\mapsto\NN_\XX$ is one-to-one.
If there is a constant $C_0$ such that every $u_+\in\XX^\UU$ and $u_-\in\XX^\VV$ with $(\widehat L u_+)\big\vert_\UU =0$ and $(\widehat L u_+)\big\vert_\VV = 0$ satisfies \begin{equation*}\doublebar{u_+}_{\XX^\UU}\leq C_0\doublebar{\MM_\B^\UU u}_{\DD_\XX}, \quad \doublebar{u_-}_{\XX^\VV}\leq C_0\doublebar{\MM_\B^\VV u}_{\DD_\XX}, \end{equation*} then there is a constant $C_1$ such that the bound $\doublebar{\arr f}_{\DD_\XX} \leq C_1 \doublebar{\MM_\B^\UU\widehat\D^\B_\UU\arr f}_{\NN_\XX}$ is valid for all $\arr f\in\DD_\XX$. \end{thm}
\begin{proof} As in the proof of Theorem~\ref{thm:injective:unique}, we will consider only the relationship between the Neumann problem and the double layer potential.
Let $\arr f$, $\arr h\in\DD_\XX$. By Condition~\condDD, $u_+=\widehat\D^\B_\UU\arr f\in\XX^\UU$ and $v_+=\widehat\D^\B_\UU \arr h\in\XX^\UU$. If $\MM_\B^\UU \widehat\D^\B_\UU \arr f = \MM_\B^\UU \widehat\D^\B_\UU \arr h$, then $\MM_\B^\UU u_+=\MM_\B^\UU v_+$. Because there is at most one solution to the Neumann problem, we must have that $u_+=v_+$, and in particular $\WTr_\XX^\UU \widehat\D^\B_\UU \arr f = \WTr_\XX^\UU \widehat\D^\B_\UU \arr h$.
By Condition~\condJDcts, we have that $\MM_\B^\VV \widehat\D^\B_\VV \arr f = \MM_\B^\VV \widehat\D^\B_\VV \arr h$. By Condition~\condDD\ and uniqueness of solutions to the $\VV$-Neumann problem, $\WTr_\XX^\VV \widehat\D^\B_\VV \arr f = \WTr_\XX^\VV \widehat\D^\B_\VV \arr h$. By \condJDjump, we have that \begin{equation*}\arr f = \WTr_\XX^\UU \widehat\D^\B_\UU \arr f + \WTr_\XX^\VV \widehat\D^\B_\VV \arr f = \WTr_\XX^\UU \widehat\D^\B_\UU \arr h + \WTr_\XX^\VV \widehat\D^\B_\VV \arr h =\arr h\end{equation*} and so $\MM_\B^\UU \widehat\D^\B_\UU$ is one-to-one.
Now assume the stronger condition, that is, that $C_0<\infty$. Because $\DD_\XX$ is a quasi-Banach space, if $\arr f\in\DD_\XX$ then by Condition~\condJDjump, \begin{equation*}\doublebar{\arr f}_{\DD_\XX} \leq C\doublebar{\WTr_\XX^\UU \widehat\D^\B_\UU \arr f}_{\DD_\XX}+\doublebar{\WTr_\XX^\VV \widehat\D^\B_\VV \arr f}_{\DD_\XX}.\end{equation*} By Condition~\condDD, $\widehat\D^\B_\UU \arr f\in\XX^\UU$ with $(\widehat L(\widehat\D^\B_\UU \arr f))\big\vert_\UU=0$. Thus by Condition~\condTT, $\doublebar{\WTr_\XX^\UU \widehat\D^\B_\UU \arr f}_{\NN_\XX}\leq C\doublebar{\widehat\D^\B_\UU \arr f}_{\XX^\UU}$. Thus, \begin{equation*}\doublebar{\arr f}_{\DD_\XX} \leq C\doublebar{\widehat\D^\B_\UU \arr f}_{\XX^\UU}+\doublebar{\widehat\D^\B_\VV \arr f}_{\XX^\VV}.\end{equation*} By definition of~$C_0$, \begin{equation*}\doublebar{\widehat\D^\B_\UU \arr f}_{\XX^\UU}\leq C_0\doublebar{\MM_\B^\UU\widehat\D^\B_\UU \arr f}_{\NN_\XX}\quad\text{and}\quad\doublebar{\widehat\D^\B_\VV \arr f}_{\XX^\VV}\leq C_0\doublebar{\MM_\B^\VV\widehat\D^\B_\VV \arr f}_{\NN_\XX}.\end{equation*} By Condition~\condJDcts, $\MM_\B^\VV\widehat\D^\B_\VV \arr f=\MM_\B^\UU\widehat\D^\B_\UU \arr f$ and so \begin{equation*}\doublebar{\arr f}_{\DD_\XX} \leq 2CC_0\doublebar{\MM_\B^\UU\widehat\D^\B_\UU \arr f}_{\NN_\XX}\end{equation*} as desired. \end{proof}
Finally, we consider the relationship between existence and surjectivity. The following argument appeared first in \cite{BarM13}.
\begin{thm} \label{thm:existence:surjective} Assume that Conditions~\condMM, \condGG,
\condJScts, and \condJDjump\ are valid. Suppose that, for any $\arr f\in\DD_\XX$, there is at least one pair of solutions $u_\pm$ to the pair of Dirichlet problems \begin{equation}\label{eqn:Dirichlet:ES} (\widehat L u_+)\big\vert_\UU = (\widehat L u_-)\big\vert_\VV = 0, \>\>\> \WTr_\XX^\UU u_+ = \WTr_\XX^\VV u_- = \arr f, \>\>\> u_+\in\XX^\UU , \>\>\> u_-\in\XX^\VV .\end{equation} Then $\Tr_\XX^{\UU}\widehat\s^L_\UU:\NN_\XX\mapsto\DD_\XX$ is onto.
Suppose that there is some $C_0<\infty$ such that, if $\arr f\in\DD_\XX$, there is some pair of solutions $u^\pm$ to the problem~\eqref{eqn:Dirichlet:ES} with \begin{equation*}\doublebar{u_+}_{\XX^\UU}\leq C_0\doublebar{\arr f}_{\DD_\XX}, \quad \doublebar{u_-}_{\XX^\VV}\leq C_0\doublebar{\arr f}_{\DD_\XX}. \end{equation*} Then there is a constant $C_1$ such that for any $\arr f\in\DD_\XX$, there is a $\arr g\in\NN_\XX$ such that ${\Tr_\XX^{\UU}\widehat\s^L_\UU\arr g}=\arr f$ and $\doublebar{\arr g}_{\NN_\XX} \leq C_1 \doublebar{\arr f}_{\DD_\XX}$.
Similarly, assume that Conditions~\condTT, \condGG,
\condJDcts, and \condJSjump\ are valid. Suppose that for any $\arr g\in\NN_\XX$, there is at least one pair of solutions $u_\pm$ to the pair of Neumann problems \begin{equation}\label{eqn:Neumann:ES} (\widehat L u_+)\big\vert_\UU = (\widehat L u_-)\big\vert_\VV = 0, \>\>\> \MM_\B^\UU u_+ = \MM_\B^\VV u_- = \arr g, \>\>\> u_+\in\XX^\UU , \>\>\> u_-\in\XX^\VV .\end{equation} Then $\MM_\B^\UU\widehat\D^\B_\UU:\DD_\XX\mapsto\NN_\XX$ is onto.
Suppose that there is some $C_0<\infty$ such that, if $\arr g\in\NN_\XX$, there is some pair of solutions $u^\pm$ to the problem~\eqref{eqn:Neumann:ES} with \begin{equation}\label{eqn:Neumann:bound}\doublebar{u_+}_{\XX^\UU}\leq C_0\doublebar{\arr g}_{\NN_\XX}, \quad \doublebar{u_-}_{\XX^\VV}\leq C_0\doublebar{\arr g}_{\NN_\XX}. \end{equation} Then there is a constant $C_1$ such that for any $\arr g\in\NN_\XX$, there is an $\arr f\in\DD_\XX$ such that ${\MM_\B^\UU\widehat\D^\B_\UU\arr f}=\arr g$ and $\doublebar{\arr f}_{\DD_\XX} \leq C_1 \doublebar{\arr g}_{\NN_\XX}$. \end{thm}
\begin{proof} As usual we present the proof for the Neumann problem. Choose some $\arr g\in\NN_\XX$ and let $u_+$ and $u_-$ be the solutions to the problem~\eqref{eqn:Neumann:ES} assumed to exist. (If $C_0<\infty$ we further require that the bound~\eqref{eqn:Neumann:bound} be valid.)
By Condition~\condTT, $\arr f_+=\WTr_\XX^\UU u_+$ and $\arr f_-=\WTr_\XX^\VV u_-$ exist and lie in~$\DD_\XX$. By Condition~\condGG, \begin{align*} 2\arr g &= \MM_\B^\UU u_+ + \MM_\B^\VV u_- \\&= \MM_\B^\UU (-\widehat\D^\B_\UU \arr f_+ + \widehat\s^L_\UU \arr g) + \MM_\B^\VV (-\widehat\D^\B_\VV \arr f_- + \widehat\s^L_\VV \arr g) .\end{align*} By Conditions~\condJDcts\ and \condJSjump\ and linearity of the operators $\MM_\B^\UU$, $\MM_\B^\VV$, we have that \begin{align*} 2\arr g &= -\MM_\B^\UU\widehat\D^\B_\UU \arr f_+ + \MM_\B^\UU\widehat\s^L_\UU \arr g -\MM_\B^\UU\widehat\D^\B_\UU \arr f_- +\arr g- \MM_\B^\UU\widehat\s^L_\UU \arr g \\&= \arr g -\MM_\B^\UU \widehat\D^\B_\UU (\arr f_+ + \arr f_-) .\end{align*} Thus, $\MM_\B^\UU \widehat\D^\B_\UU$ is surjective. If $C_0<\infty$, then because $\DD_\XX$ is a quasi-Banach space and by Condition~\condTT, \begin{equation*}\doublebar{\arr f_+ + \arr f_-}_{\DD_\XX} \leq C C_0\doublebar{\arr g}_{\NN_\XX} \end{equation*} as desired. \end{proof}
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\begin{document}
\preprintTitle{Time-space adaptive discontinuous Galerkin method for advection-diffusion equations with non-linear reaction mechanism} \preprintAuthor{B\"{u}lent Karas\"{o}zen \footnote{Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey, \textit{email}: [email protected] }, Murat Uzunca \footnote{Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey, \textit{email}: [email protected] }}
\preprintAbstract{\small In this work, we apply a time-space adaptive discontinuous Galerkin method using the elliptic reconstruction technique with a robust (in P\'{e}clet number) elliptic error estimator in space, for the convection dominated parabolic problems with non-linear reaction mechanisms. We derive a posteriori error estimators in the $L^{\infty}(L^2)+L^2(H^1)$-type norm using backward Euler in time and discontinuous Galerkin (symmetric interior penalty Galerkin (SIPG)) in space. Numerical results for advection dominated reactive transport problems in homogeneous and heterogeneous media demonstrate the performance of the time-space adaptive algorithm.}
\preprintKeywords{Non-linear diffusion-convection reaction, Discontinuous Galerkin, Time-space adaptivity, Elliptic reconstruction, A posteriori error estimates}
\preprintDate{August 2014} \preprintNo{2014-6} \makePreprintCoverPage
\section{Introduction} \label{intro}
Advection-diffusion-reaction (ADR) equations are widely used for systems in chemical reaction problems. In the linear convection or advection dominated case, stabilized continuous finite elements and discontinuous Galerkin (DG) methods are capable of handling the nonphysical oscillations. On the other hand, in the non-linear stationary case, the non-linear reaction term produces sharp layers in addition to the spurious oscillations due to the convection. Thus, an accurate and efficient numerical resolution of such layers is a challenge as the exact location of the layers are not known a priori. In the non-stationary case, the resolution of such layers is more critical since the nature of the sharp layers may vary as time progresses. Recently, several stabilization and shock/discontinuity capturing techniques were developed for linear and non-linear steady-state problems \cite{bause12ash}. In contrast to the stabilized continuous Galerkin finite element methods, DG methods produce stable discretizations without the need for stabilization parameters. The DG combines the best properties of the finite volume and continuous finite elements methods. Finite volume methods can only use lower degree polynomials, and continuous finite elements methods require higher regularity due to the continuity requirements. The DG method is in particular suitable for non-matching grids and hp (space and order) adaptivity, detecting sharp layers and singularities. They are easily adapted locally for nonconforming finite elements requiring less regularity. Higher order DG approximation can be easily used by hierarchical bases \cite{burger09pkd}, and the convergence rates are limited by the consistency error which makes the DG suitable for complex fluid flows. The DG methods are robust with respect to the variation of the physical parameters like diffusion constant and permeability. The stability of the DG approximation retained by adjusting the penalty parameter to penalize the jumps at the interface of the elements. Various choices of the penalty parameter is suggested in the literature \cite{castillo12pdg,dobrev08psi,epshteyn07epp,slingerland14fls}. A unified analysis of the interior penalty DG methods for elliptic PDEs are given in \cite{arnold02uad}. Other advantages of the DG methods are conservation of mass and fluxes and parallelization. Moreover, DG methods are better suited for adaptive strategies which are based on (residual-based) a posteriori error estimation. The disadvantages are the resulting larger and denser matrices and ill-conditioning with increasing degree of the discontinuous polynomials.
The main tool in this paper is the adaptivity applied to the DG discretized (in space) systems using (residual-based) a posteriori error estimates. The aim of the a posteriori estimates is to derive bounds between the known numerical solution and the unknown exact solution. For elliptic diffusion-convection-reaction equations, there are various studies applying adaptivity using a posteriori error estimates in the literature \cite{pietro14rra,houston02dhp,klieber06ast,schotzau09rae}, which are well-understood. In this paper, we derive a posteriori error estimates for semi-linear ADR problems using the well-know elliptic a posteriori estimates, by which in contrast to the standard energy techniques we do not need to try to adapt the estimates case by case in order to compare the exact solution with numerical solution directly. For this reason, we use the {\it elliptic reconstruction} technique in \cite{makridakis03era} which allows us to utilize available a posteriori estimates derived for elliptic equations to control the main part of the spatial error. The idea of the elliptic reconstruction technique is to construct an auxiliary solution whose difference to the numerical solution can be estimated by a known (elliptic) a posteriori estimate, and the constructed auxiliary solution satisfies a variant of the given problem with a right hand side which can be controlled in an optimal way. By this way, we are able to obtain results being optimal order in both $L^2(H^1)$ and $L^{\infty}(L^2)$-type norms, while the results obtained by the standard energy methods are only optimal order in $L^2(H^1)$-type norms, but sub-optimal order in $L^{\infty}(L^2)$-type norms. In \cite{cangiani13adg}, a posteriori error estimates in the $L^{\infty}(L^2)+L^2(H^1)$-type norm are derived for linear parabolic diffusion-convection-reaction equations using backward Euler in time and discontinuous Galerkin in space utilizing the elliptic reconstruction technique. In this paper, we extend the study in \cite{cangiani13adg} in a similar way by deriving and implementing a posteriori error estimates in the $L^{\infty}(L^2)+L^2(H^1)$-type norm using backward Euler in time and discontinuous Galerkin (symmetric interior penalty (SIPG)) in space for the convection dominated parabolic problems with non-linear reaction mechanisms. To derive the a posteriori error estimates, we use the modification of the robust (in P\'{e}clet number) a posteriori error estimator in \cite{schotzau09rae} for linear steady-state diffusion-convection-reaction equations to the steady-state diffusion-convection equations with non-linear reaction mechanisms utilizing the elliptic reconstruction technique \cite{makridakis03era}.
Application of the adaptive discontinuous Galerkin methods and a posteriori error estimates to the problems in geoscience are reviewed recently in \cite{pietro14rra}. Most of the applications of DG methods in geoscience concern reactive transport with advection \cite{bastian11udg,klieber06ast,sun05lna} and strong permeability contrasts such as layered reservoirs \cite{slingerland14fls} or vanishing and varying diffusivity posing challenges in computations \cite{proft09dgm}. The permeability in heterogeneous porous and fractured media varies over orders of magnitude in space, which results in highly variable flow field, where the local transport is dominated by advection or diffusion \cite{tambue10eia}. Accurate and efficient numerical solution of the ADR equations to predict the macroscopic mixing, anomalous transport of the solutes and contaminants for a wide range of parameters like permeability and P\'{e}clet numbers, different flow velocities and reaction rates and reaction rates are challenging problems \cite{tambue10eia}. In order to resolve the complex flow patterns accurately, higher order time stepping methods like exponential time stepping methods are used \cite{tambue10eia}. We show here using time-space adaptive first order backward Euler and DG in space, the same results can be obtained.
The rest of this paper is organized as follows. In the next section, we introduce the function spaces and related norms which are used in our analysis, and the model problem with the assumptions to have unique solution. In Section \ref{SIPG}, we give the symmetric discontinuous interior penalty Galerkin discretized semi-discrete system, and fully-discrete system using Backward Euler in time. The derivation of a posteriori error estimator is given in Section~\ref{aposteriori}, first for steady-state problems (with proofs), then for parabolic problems utilizing the elliptic reconstruction technique. In Section~\ref{algorithm}, we state the adaptive algorithm procedure using the a posteriori error estimators derived in Section~\ref{aposteriori}. The solution procedure of the fully-discrete system by Newton method and the structures of the arising matrix and vectors are discussed in Section~\ref{newton}. After demonstrating the performance of the algorithm by some numerical studies in Section~\ref{numeric}, the paper ends with conclusions.
\section{Model problem} \label{secmodel}
Let $\Omega \subset\mathbb{R}^2$ be a bounded, open and convex domain with boundary $\partial\Omega$. For a Banach space $X$, define the spaces $L^p(0,T;X)$ \begin{subequations}\label{model} \begin{align}
\| v\|_{L^p(0,T;X)} &= \left( \int_0^T \| v(t)\|_X^p dt \right)^{1/p}<\infty \; , \qquad \text{for } 1\leq p < +\infty \nonumber \\
\| v\|_{L^{\infty}(0,T;X)} &= \underset{0\leq t\leq T}{\text{esssup}} \|v(t)\|_X <\infty \; , \qquad \text{for } p = +\infty \nonumber \end{align} \end{subequations} Also define the space $$
H^1(0,T;X)=\{ v\in L^2(0,T;X)| \; v_t\in L^2(0,T;X)\}. $$ We denote by $C(0,T;X)$ and $C^{0,1}(0,T;X)$, the spaces of continuous and Lipschitz-continuous functions $v:[0,T]\mapsto X$, respectively, equipped with the norms \begin{subequations} \begin{align}
\| v\|_{C(0,T;X)} &= \underset{0\leq t\leq T}{\text{max}} \|v(t)\|_X <\infty \nonumber \\
\| v\|_{C^{0,1}(0,T;X)} &= \text{max} \left\{ \| v\|_{C(0,T;X)}, \| v_t\|_{C(0,T;X)}\right\} <\infty \nonumber \end{align} \end{subequations} We consider the system of semi-linear diffusion-convection-reaction equations \begin{equation}\label{org}
\frac{\partial u_i}{\partial t}-\nabla\cdot (\epsilon_i\nabla u_i) + \vec{\beta}_i\cdot\nabla u_i + r_i(\vec{u}) = f_i \;\; , \quad i=1,2,\ldots , J \end{equation} in $\Omega\times (0,T]$ for the vector of unknowns $\vec{u}=(u_1,u_2,\ldots , u_J)^T$ with appropriate boundary and initial conditions. We assume that the source functions $f_i\in C(0,T;L^2(\Omega))$, and the velocity fields $\vec{\beta }_i\in C\left(0,T;W^{1,\infty}(\Omega)\right)^2$ either given or computed. For flow in heterogeneous media in Section~\ref{ex4}, the symmetric dispersion tensors $\epsilon_i$ are taken of the form $$ \epsilon_i= \begin{bmatrix} D_i^1 & 0 \\ 0 & D_i^2 \end{bmatrix} $$ with $0<D_i^1,D_i^2\ll 1$. Moreover, we assume that the non-linear reaction terms are bounded, locally Lipschitz continuous and monotone, i.e. satisfies for any $s, s_1, s_2\ge 0$, $s,s_1, s_2 \in \mathbb{R}$ the following conditions \cite{uzunca14adg} \begin{subequations} \begin{align}
|r_i(s)| &\leq C_s , \quad C_s>0 \label{nonl1}\\
\| r_i(s_1)-r_i(s_2)\|_{L^2(\Omega)} &\leq L\| s_1-s_2\|_{L^2(\Omega)} , \quad L>0\label{nonl2} \\
r_i\in C^1(\mathbb{R}_0^+), \quad r_i(0) =0, &\quad r_i'(s)\ge 0. \label{nonl3} \end{align} \end{subequations} We also assume that there are $\kappa , \kappa_*\geq 0$ satisfying for $i=1,2,\ldots ,J$ \begin{equation} \label{ass}
-\frac{1}{2}\nabla\cdot\vec{\beta }_i (x) \geq \kappa, \qquad \| -\nabla\cdot\vec{\beta }_i\|_{C(0,T;L^{\infty}(\Omega))}\leq \kappa^*\kappa, \end{equation} The first inequality in (\ref{ass}) ensures the well-posedness of the problem, and the letter is used in the a posteriori error analysis.
The weak formulation of the system (\ref{org}) reads as: for any $v\in H_0^1(\Omega )$, find $u_i\in L^2(0,T;H_0^1(\Omega))\cap H^1(0,T;L^2(\Omega))$, $i=1,2,\ldots , J$, such that for all $t\in (0,T]$
\begin{equation} \label{weak} \int_{\Omega}\frac{\partial u_i}{\partial t}vdx + a(t;u_i,v)+b_i(t;\vec{u},v)=l_i(v) \end{equation} \begin{subequations} \begin{align} a(t;u_i, v)=& \int_{\Omega}(\epsilon_i\nabla u_i\cdot\nabla v+\vec{\beta }_i\cdot\nabla u_iv)dx , \label{weaka} \\ b_i(t;\vec{u}, v) =& \int_{\Omega}r_i(\vec{u})v dx , \\ l_i( v)=& \int_{\Omega}f_iv dx \end{align} \end{subequations} which have unique solutions $\{ u_i\}_{i=1}^J$ in the space $C(0,T;L^2(\Omega))$ under the given regularity assumptions and the conditions \eqref{nonl1}-\eqref{nonl3}.
In the sequel, for simplicity, we just consider a single equation of the system \eqref{org} (J=1) without any subscript to construct the discontinuous Galerkin discretization in the next section and thereafter, and we also continue with a homogeneous dispersion tensor leading to a simple diffusivity constant $0<\epsilon\ll 1$. We, furthermore, take into account the homogeneous Dirichlet boundary conditions to simplify the notations. It can be proceeded with heterogeneous dispersion tensor and other type of boundary conditions in a standard way.
\section{Discontinuous Galerkin discretization} \label{SIPG}
For space discretization of a single equation of (\ref{org}), we use the symmetric discontinuous interior penalty Galerkin (SIPG) method \cite{arnold02uad,riviere08dgm} with the upwinding for the convection part \cite{ayuso09dgm,houston02dhp}.
Let $\{\xi_h\}$ be a family of shape regular meshes with the elements (triangles) $K_i\in\xi_h$ satisfying $\overline{\Omega}=\cup \overline{K}$ and $K_i\cap K_j=\emptyset$ for $K_i$, $K_j$ $\in\xi_h$. Let us denote by $\Gamma^0$ and $\Gamma^{\partial}$ the set of interior and Dirichlet boundary edges, respectively, so that $\Gamma =\Gamma^0\cup\Gamma^{\partial}$ forms the skeleton of the mesh. For any $K\in\xi_h$, let $\mathbb{P}_k(K)$ be the set of all polynomials of degree at most $k$ on $K$. Then, set the finite dimensional space $$
V_h(\xi_h)=\left\{ v\in L^2(\Omega ) : v|_{K}\in\mathbb{P}_k(K) ,\; \forall K\in \xi_h \right\}\not\subset H_0^1(\Omega). $$ The functions in $V_h(\xi_h)$, in contrast to the standard (continuous) finite elements, are discontinuous along the inter-element boundaries causing that along an interior edge, there are two different traces from the adjacent elements sharing that edge. In the light of this fact, let us first introduce some notations before giving the SIPG formulation. Let $K_i$, $K_j\in\xi_h$ ($i<j$) be two adjacent elements sharing an interior edge $e=K_i\cap K_j\subset \Gamma_0$ (see Fig.\ref{jump}). We denote the trace of a scalar function $v$ from inside $K_i$ by $v_{i}$ and from inside $K_j$ by $v_{j}$, and then we set the jump and average values of $v$ on the edge $e$ $$ [v]= v_{i}\vec{n}_e- v_{j}\vec{n}_e , \quad \{ v\}=\frac{1}{2}(v_{i}+ v_{j}). $$ Here $\vec{n}_e$ denotes the unit normal to the edge $e$ oriented from $K_i$ to $K_j$. Similarly, we set the jump and average values of a vector valued function $\vec{q}$ on e $$ [\vec{q}]= \vec{q}_{i}\cdot \vec{n}_e- \vec{q}_{j}\cdot \vec{n}_e , \quad \{ \vec{q}\}=\frac{1}{2}(\vec{q}_{i}+ \vec{q}_{j}), $$ Observe that $[v]$ is a vector for a scalar function $v$, while, $[\vec{q}]$ is scalar for a vector valued function $\vec{q}$. On the other hand, along any boundary edge $e=K_i\cap \partial\Omega$, we set $$ [v]= v_{i}\vec{n} , \quad \{ v\}=v_{i}, \quad [\vec{q}]=\vec{q}_{i}\cdot \vec{n}, \quad \{ \vec{q}\}=\vec{q}_{i} $$ where $\vec{n}$ is the unit outward normal to the boundary at $e$. We define the inflow and outflow boundary parts at a time $t$ by $$
\Gamma_t^-=\{ x\in \partial\Omega | \; \vec{\beta}(\vec{x},t)\cdot \vec{n}(x)<0\}\; , \qquad \Gamma_t^+=\partial\Omega\setminus\Gamma_t^- $$ At a time $t$, the inflow and outflow parts of an element $K$ are defined by $$
\partial K_t^-=\{ x\in \partial K | \; \vec{\beta}(\vec{x},t)\cdot \vec{n}_K(x)<0\}\; , \qquad \partial K_t^+=\partial K\setminus\partial K_t^- $$ where $\vec{n}_K(x)$ denotes the outward unit normal to the boundary of the element $K$ at $x$. \begin{figure}
\caption{Two adjacent elements sharing an edge (left); an element near to domain boundary (right)}
\label{jump}
\end{figure}
Under the given definitions, the semi-discrete problem reads as: for $t=0$, set $u_h(0)\in V_h(\xi_h)$ as the projection (orthogonal $L^2$-projection) of $u_0$ onto $V_h(\xi_h)$; for each $t\in (0,T]$, for all $v_h\in V_h(\xi_h)$, find $u_h\in C^{0,1}(0,T;V_h(\xi_h))$ such that
\begin{equation} \label{dg} \int_{\Omega}\frac{\partial u_{h}}{\partial t}v_{h}dx + a_{h}(t;u_{h},v_{h}) + K_{h}(u_{h},v_{h})+b_{h}(t;u_{h}, v_{h})=l_{h}(v_{h}) , \end{equation} \begin{subequations} \begin{align} a_{h}(t;u_{h}, v_{h})=& \sum \limits_{K \in {\xi}_{h}} \int_{K} \epsilon \nabla u_{h}\cdot\nabla v_{h} dx + \sum \limits_{K \in {\xi}_{h}} \int_{K} \vec{\beta } \cdot \nabla u_{h} v_{h} dx \label{dga} \\ &+ \sum \limits_{K \in {\xi}_{h}}\int_{\partial K_t^-\setminus\partial\Omega } \vec{\beta }\cdot \vec{n}_K (u_{h}^{out}-u_{h}) v_{h} ds - \sum \limits_{K \in {\xi}_{h}} \int_{\partial K_t^-\cap \Gamma_t^{-}} \vec{\beta }\cdot \vec{n}_K u_{h} v_{h} ds \nonumber \\ &+ \sum \limits_{ e \in \Gamma}\frac{\sigma \epsilon}{h_{e}} \int_{e} [u_{h}]\cdot[v_{h}] ds, \nonumber \\ K_{h}(u_{h},v_{h}) =& - \sum \limits_{ e \in \Gamma} \int_{e} ( \{\epsilon \nabla v_{h} \}\cdot[u_{h}] + \{\epsilon \nabla u_{h} \}\cdot [v_{h}] )ds \label{kh} \\ b_{h}(t;u_{h}, v_{h}) =& \sum \limits_{K \in {\xi}_{h}} \int_{K} r(u_{h}) v_{h} dx, \\ l_{h}( v_{h})=& \sum \limits_{K \in {\xi}_{h}} \int_{K} f_h v_{h} dx \end{align} \end{subequations} where $u_{h}^{out}$ denotes the value on an edge from outside of an element $K$. The parameter $\sigma\in\mathbb{R}_0^+$ is called the penalty parameter which should be sufficiently large; independent of the mesh size $h$ and the diffusion coefficient $\epsilon$ \cite{riviere08dgm} [Sec. 2.7.1]. We choose the penalty parameter $\sigma$ for the SIPG method depending on the polynomial degree $k$ as $\sigma=3k(k+1)$ on interior edges and $\sigma=6k(k+1)$ on boundary edges.
Upon integration by parts on the convective term, bilinear form (\ref{dga}) will be \begin{subequations} \begin{align} a_{h}(t;u_{h}, v_{h})=& \sum \limits_{K \in {\xi}_{h}} \int_{K} \epsilon \nabla u_{h}\cdot\nabla v_{h} dx - \sum \limits_{K \in {\xi}_{h}} \int_{K} ( \vec{\beta } u_h\cdot\nabla v_h + \nabla\cdot\vec{\beta } u_hv_h) dx \label{bilbyp} \\ &+ \sum \limits_{K \in {\xi}_{h}}\int_{\partial K_t^+\setminus\partial\Omega } \vec{\beta }\cdot \vec{n}_K u_{h}(v_h-v_{h}^{out}) ds + \sum \limits_{K \in {\xi}_{h}} \int_{\partial K_t^+\cap \Gamma_t^{+}} \vec{\beta }\cdot \vec{n}_K u_{h} v_{h} ds \nonumber \\ &+ \sum \limits_{ e \in \Gamma}\frac{\sigma \epsilon}{h_{e}} \int_{e} [u_{h}]\cdot[v_{h}] ds \nonumber \end{align} \end{subequations} Note that the bilinear form $a_{h}(t;u,v)$ is well-defined for the functions $u,v\in H_0^1(\Omega )$, and equal to $$ a_{h}(t;u, v)=\int_{\Omega}(\epsilon\nabla u\cdot\nabla v + \vec{\beta}\cdot\nabla uv)dx $$ Thus, the weak formulation (\ref{weaka}) can be rewritten for any $t\in (0,T]$ as \begin{equation}\label{eqweak} \int_{\Omega}\frac{\partial u}{\partial t}vdx + a_h(t;u,v)+b(t;u,v)=l(v) \; , \qquad \forall v\in H_0^1(\Omega ) \end{equation}
For the fully discrete system, consider a subdivision of $[0,T]$ into $n$ time intervals $I_k=(t^{k-1},t^{k}]$ of length $\tau_k$, $k=1,2,\ldots , n$. Set $t^0=0$ and for $n\geq 1$, $t^k=\tau_1+\tau_2+\cdots + \tau_k$. Denote by $\xi_h^0$ an initial mesh and by $\xi_h^k$ the mesh associated to the $k^{th}$ time step for $k>0$, which is obtained from $\xi_h^{k-1}$ by locally refining/coarsening. Moreover, we assign the finite element space $V_h^k=V_h(\xi_h^k)$ to each mesh $\xi_h^k$. Then, the fully discrete problem, backward Euler in time, reads as: for $t=0$, set $u_h^0\in V_h^0$ as the projection (orthogonal $L^2$-projection) of $u_0$ onto $V_h^0$; for $k=1,2,\ldots , n$, find $u_h^k\in V_h^{k}$ such that for all $v_h^k\in V_h^{k}$ \begin{equation}\label{fullydisc} \int_{\Omega}\frac{u_{h}^{k}-u_{h}^{k-1}}{\tau_k}v_{h}^k dx + a_{h}(t^k;u_{h}^k,v_{h}^k)+ K_{h}(u_{h}^k,v_{h}^k)+b_{h}(t^k;u_{h}^k, v_{h}^k)=\int_{\Omega}f^{k}v_{h}^k dx \end{equation}
\section{A posteriori error analysis} \label{aposteriori}
The problems concerned in this paper are convection/reaction dominated non-stationary models which often produce internal/boundary layers where the solutions have steep gradients. Thus, an efficient solution of such equations are needed. The most popular technique to solve the convection/reaction dominated problems is the adaptivity which requires an estimator to locate the regions where the error is too large. In this work, we construct the residual-based, robust (in P\'{e}clet number) a posteriori error estimators for non-stationary convection domianted diffusion-convection equations with non-linear reaction mechanisms. To construct the a posteriori error estimators, we extend the a posteriori error estimators for linear non-stationary problems constructed in \cite{cangiani13adg} which uses the a posteriori error estimators for linear stationary models constructed in \cite{schotzau09rae} utilizing the elliptic reconstruction technique \cite{makridakis03era} to make connection between the stationary and non-stationary error. As in \cite{cangiani13adg}, first we construct and prove the a posteriori error bounds for non-linear stationary models in Section~\ref{stationary}. Then, in Section~\ref{semidiscerror}, we give the a posteriori error bounds for the semi-discrete system of the non-stationary problems with non-linear reaction term. Finally, in Section~\ref{fullydiscerror}, we give the a posteriori error bounds for the fully-discrete system of the non-stationary problems with non-linear reaction term. The main contribution to the error analysis lies in the construction of the error bounds for non-linear stationary models in Section~\ref{stationary}. The remaining utilize the constructions in \cite{cangiani13adg}.
\subsection{A posteriori error bounds for non-linear stationary system} \label{stationary}
We consider first the stationary problem, i.e. for a given $t\in (0,T]$, let $u^s \in H_0^1(\Omega )$ be the unique solution to the weak problem \begin{equation} \label{stweak} a(t;u^s,v)+b(t;u^s,v)=l(v) \; , \qquad \forall v\in H_0^1(\Omega ) \end{equation} and let $u_h^s\in V_h(\xi_h)$ be the unique solution to the discrete problem \begin{equation} \label{stdg} a_h(t;u_h^s,v_h)+K_{h}(u_{h}^s,v_{h})+b_h(t;u_h^s,v_h)=l(v_h) \; , \qquad \forall v\in V_h(\xi_h) \end{equation} In order to measure the spatial error, we use the energy norm
$$
||| v |||^2=\sum \limits_{K \in {\xi}_{h}}(\| \epsilon\nabla v\|_{L^2(K)}^2 +\kappa \| v\|_{L^2(K)}^2) + \sum \limits_{e \in \Gamma}\frac{\epsilon\sigma}{h_{e}}\| [v]\|_{L^2(e)}^2, $$ and the semi-norm \begin{equation} \label{semin}
|v|_C^2=|\vec{\beta }v|_*^2+\sum\limits_{e \in \Gamma}(\kappa h_e+ \frac{h_e}{\epsilon})\| [v]\|_{L^2(e)}^2, \end{equation} where $$
|u|_*=\mathop{\text{sup}}_{w\in H_0^1(\Omega )\setminus \{ 0\}}\frac{\int_{\Omega}u\cdot \nabla w dx}{||| w|||}, $$ For each element $K \in {\xi}_{h}$, we define the local error indicators $\eta_K^2$ as
\begin{eqnarray} \label{res} \eta_K^2= \eta_{R_K}^2 + \eta_{E_K}^2 + \eta_{J_K}^2, \end{eqnarray} where \begin{eqnarray}
\eta_{R_K}^2&=&\rho_K^2\| f_h +\epsilon\Delta u_h^s-\vec{\beta }_h\cdot\nabla u_h^s-r(u_h^s)\|_{L^2(K)}^2, \nonumber \\
\eta_{E_K}^2 &=& \frac{1}{2} \sum \limits_{e \in \partial K\setminus\Gamma^{\partial} }\epsilon^{-\frac{1}{2}}\rho_e\| [\epsilon\nabla u_h^s]\|_{L^2(e)}^2, \nonumber \\
\eta_{J_K}^2 &=& \frac{1}{2}\sum \limits_{e \in \partial K\setminus\Gamma^{\partial} }\left( \frac{\epsilon\sigma}{h_e}+\kappa h_e+\frac{h_e}{\epsilon}\right) \| [u_h^s]\|_{L^2(e)}^2 +\sum \limits_{e \in \partial K\cap\Gamma^{\partial}}\left( \frac{\epsilon\sigma}{h_e}+\kappa h_e+\frac{h_e}{\epsilon}\right) \| u_h^s\|_{L^2(e)}^2, \nonumber \\ \end{eqnarray} with the weights $\rho_K$ and $\rho_e$, on an element $K$, given by $$ \rho_{K}=\min\{h_{K}\epsilon^{-\frac{1}{2}}, \kappa^{-\frac{1}{2}}\}, \; \rho_{e}=\min\{h_{e}\epsilon^{-\frac{1}{2}}, \kappa^{-\frac{1}{2}}\}, $$ for $\kappa \neq 0$. When $\kappa =0$, we take $\rho_{K}=h_{K}\epsilon^{-\frac{1}{2}}$ and $\rho_{e}=h_{e}\epsilon^{-\frac{1}{2}}$. Then, our a posteriori error indicator is given by \begin{equation} \label{errind} \eta=\left( \sum \limits_{K\in{\xi}_{h}}\eta_K^2\right)^{1/2}. \end{equation} We also introduce the data approximation terms, $$
\Theta_K^2 =\rho_K^2(\| f-f_h\|_{L^2(K)}^2 + \| (\vec{\beta } -\vec{\beta }_h)\cdot\nabla u_h^s\|_{L^2(K)}^2). $$ and the data approximation error \begin{equation} \label{data} \Theta=\left( \sum \limits_{K\in{\xi}_{h}}\Theta_K^2\right)^{1/2}. \end{equation} Then, we have the a posteriori error bounds
\begin{eqnarray}
\| u^s-u_h^s\|_{DG} \lesssim \eta + \Theta \qquad &(\text{reliability}), \label{rel} \\[0.2cm]
\eta \lesssim \| u^s-u_h^s\|_{DG} + \Theta \qquad &(\text{efficiency}). \label{eff} \end{eqnarray} with $$
\| v\|_{DG}= ||| v||| + | v|_C $$
\subsubsection{Proof of a posteriori error bounds}
The proof for the bounds of spatial error estimates are analogous to the ones in \cite{schotzau09rae} for the linear problems. Therefore, only the proofs for non-linear reaction term are stated explicitly . In the following, we use the symbols $\lesssim$ and $\gtrsim$ to denote the bounds that are valid up to positive constants independent of the local mesh size $h$, the diffusion coefficient $\epsilon$ and the penalty parameter $\sigma$.
The spatial error $\| u^s-u_h^s\|_{DG}$ is not well-defined, since $u^s\in H_0^1(\Omega)$ and $u_h^s\in V_h(\xi_h)\nsubseteq H_0^1(\Omega)$. Therefore, we split the stationary SIPG solution $u_h^s$ as
$$ u_h^s=u_h^c+u_h^r $$ with $u_h^c\in H_0^1(\Omega)\cap V_h(\xi_h)$ being the conforming part of the solution and $u_h^r\in V_h(\xi_h)$ is the remainder term. In this way, we have $u_h^s\in H_0^1(\Omega)+V_h(\xi_h)$, and from the triangular inequality $$
\| u^s-u_h^s\|_{DG}\leq \| u^s-u_h^c\|_{DG}+\| u_h^r\|_{DG} $$ Now, both the terms on the right hand side are well-defined norms, and our aim is to find bounds for them. Next, we introduce the following auxiliary forms:
\begin{subequations} \begin{align} D_{h}(u, v)=& \sum \limits_{K \in {\xi}_{h}} \int_{K} \left( \epsilon \nabla u\cdot\nabla v -\nabla\cdot \vec{\beta }uv\right) dx \label{d} \\ O_h(u,v)=& -\sum \limits_{K \in {\xi}_{h}} \int_{K}\vec{\beta }u\cdot\nabla v dx +\sum \limits_{K \in {\xi}_{h}} \int_{\partial K^+\cap\Gamma^+}\vec{\beta }\cdot\vec{n}_K uv ds \nonumber \\ & +\sum \limits_{K \in {\xi}_{h}} \int_{\partial K^+\setminus\partial\Omega }\vec{\beta }\cdot\vec{n}_Ku(v-v^{out}) ds \label{o} \\ J_h(u,v) =& \sum \limits_{ e \in \Gamma_{0}\cup\Gamma }\frac{\sigma \epsilon}{h_{e}} \int_{e} [u]\cdot [v] ds. \label{j} \end{align} \end{subequations} We note that, for a specific $t\in (0,T]$, the SIPG bilinear form (\ref{bilbyp}) satisfies $$ a_h(t;u,v)=D_h(u,v)+O_h(u,v)+J_h(u,v) $$ and is well-defined on $H_0^1(\Omega )+V_h(\xi_h)$. Using the first identity in (\ref{ass}), we can easily have for any $u\in H_0^1(\Omega)$ $$
{a}_h(t;u,v) \geq ||| u|||^2 $$ Moreover, the auxiliary forms are continuous \cite{schotzau09rae}[Lemma 4.2]: \begin{eqnarray}
|D_h(u,v)| &\lesssim& |||u|||\; |||v||| \; , \qquad u,v\in H_0^1(\Omega )+V_h(\xi_h) ,\label{cd} \\
|O_h(u,v)| &\lesssim& |\vec{\beta }u|_* \;|||v||| \; , \qquad u\in H_0^1(\Omega )+V_h(\xi_h), v\in H_0^1(\Omega) ,\label{co} \\
|J_h(u,v)| &\lesssim& |||u|||\; |||v||| \; , \qquad u,v\in H_0^1(\Omega )+V_h(\xi_h) \label{cj} , \end{eqnarray} and for $u\in V_h(\xi_h)$, $v\in V_h(\xi_h)\cap H_0^1(\Omega )$ \cite{schotzau09rae}[Lemma 4.3] \begin{eqnarray}
|K_h(u,v)| &\lesssim& \gamma^{-1/2}\left( \sum_{e\in \Gamma_0\cup\Gamma^D} \frac{\gamma\epsilon}{h_e}\| [u]\|_{L^2(e)}\right)^{1/2} |||v|||.\label{ck} \end{eqnarray} We also have for the non-linear form $b_h(t;u,v)$, for a specific time $t$, using the boundedness assumption (\ref{nonl1}), \begin{eqnarray}
|b_h(t;u,v)| &\lesssim& |||v||| \; , \qquad u,v\in H_0^1(\Omega )+V_h(\xi_h). \label{cb} \end{eqnarray} Now, we give some auxiliary results and conditions which are used in the proofs. \begin{itemize}
\item {\bf Inf-sup condition:} \cite{schotzau09rae}[Lemma 4.4] For a nonzero $u\in H_0^1(\Omega )$, for a constant $C>0$, we have \begin{eqnarray}
|||u|||+|\vec{\beta }u|_* \lesssim \underset{v\in H_0^1(\Omega ) \setminus \{ 0\}}{\text{sup}}\frac{{a}_h(t;u,v)}{|||v|||}. \label{infsup} \end{eqnarray}
\item {\bf Approximation operator:} Let $V_h^c=V_h(\xi_h)\cap H_0^1(\Omega )$ be the conforming subspace of $V_h(\xi_h)$. For any $u\in V_h(\xi_h)$, there exists an approximation operator $A_h:\; V_h(\xi_h) \mapsto V_h^c$ satisfying \begin{eqnarray}
\sum_{K\in\xi}\| u-A_hu\|_{L^2(K)}^2 \lesssim \sum_{e\in \Gamma_0\cup\Gamma^D}\int_e h_e|[u]|^2ds, \label{app} \\
\sum_{K\in\xi}\| \nabla (u-A_hu)\|_{L^2(K)}^2 \lesssim \sum_{e\in \Gamma_0\cup\Gamma^D}\int_e \frac{1}{h_e}|[u]|^2ds. \label{dapp} \end{eqnarray}
\item {\bf Interpolation operator:} For any $u\in H_0^1(\Omega )$, there exists an interpolation operator $$
I_h:\; H_0^1(\Omega ) \mapsto \{ w\in C(\overline{\Omega}): \; w|_K\in\mathbb{P}_1(K), \forall K\in\xi, w=0 \; \text{on }\Gamma \} $$ that satisfies \begin{eqnarray}
|||I_h u||| \lesssim |||u||| , \label{int1} \\
\left( \sum_{K\in\xi}\rho_K^{-2}\| u-I_hu\|_{L^2(K)}^2\right)^{1/2} \lesssim |||u|||, \label{int2} \\
\left( \sum_{e\in\Gamma_0\cup\Gamma^D}\epsilon^{1/2}\rho_e^{-1}\| u-I_hu\|_{L^2(K)}^2\right)^{1/2} \lesssim |||u|||. \label{int3} \end{eqnarray} Now, consider the splitting of the stationary solution $u_h^s=u_h^c+u_h^r$ as $u_h^c=A_h u_h^s\in H_0^1(\Omega )\cap V_h(\xi_h)$ with $A_h$ is the approximation operator and $u_h^r=u_h^s-u_h^c\in V_h$.\\
\item {\bf Bound for remainder term:} \cite{schotzau09rae}[Lemma 4.7] For the remainder term, we have the bound \begin{eqnarray} \label{rem}
\| u_h^r\|_{DG} \lesssim \eta \end{eqnarray} where $\eta$ is our a posteriori error indicator (\ref{errind}).\\ \end{itemize}
\noindent {\bf Lemma:} For a given $t\in (0,T]$ and for any $v\in H_0^1(\Omega )$, we have \begin{eqnarray}
\int_{\Omega } f(v-I_hv)dx-{a}_h(t;u_h^s,v-I_hv)-b_h(t;u_h^s,v-I_hv) \lesssim (\eta +\Theta )|||v||| \label{lemma} \end{eqnarray} where $I_h$ is the interpolation operator.\\
\noindent {\it Proof:} Let $$ T=\int_{\Omega } f(v-I_hv)dx-{a}_h(t;u_h^s,v-I_hv)-b_h(t;u_h^s,v-I_hv). $$ Integration by parts yields \begin{eqnarray*} T &=& \sum \limits_{K \in {\xi}_{h}} \int_K (f+\epsilon\Delta u_h^s-\vec{\beta }\cdot\nabla u_h^s-r(u_h^s))(v-I_hv)dx\\ & & -\sum \limits_{K \in {\xi}_{h}} \int_{\partial K}\epsilon\nabla u_h^s\cdot \vec{n}_K(v-I_hv)ds\\ & & +\sum \limits_{K \in {\xi}_{h}} \int_{\partial K^-\setminus \partial\Omega }\vec{\beta }\cdot \vec{n}_k(u_h^s-u_h^{s,out})(v-I_hv)ds\\ &=&T_1+T_2+T_3. \end{eqnarray*} Adding and substracting the data approximation terms into the term $T_1$ \begin{eqnarray*} T_1 &=& \sum \limits_{K \in {\xi}_{h}} \int_K (f_h +\epsilon\Delta u_h^s-\vec{\beta }_h\cdot\nabla u_h^s-r(u_h^s))(v-I_hv)dx\\ & & +\sum \limits_{K \in {\xi}_{h}} \int_K ((f-f_h)-(\vec{\beta }-\vec{\beta }_h)\cdot\nabla u_h^s)(v-I_hv)dx. \end{eqnarray*} Using the Cauchy-Schwarz inequality and interpolation operator identity (\ref{int2}) \begin{eqnarray*}
T_1 &\lesssim & \left( \sum \limits_{K \in {\xi}_{h}} \eta_{R_K}^2\right)^{1/2} \left( \sum \limits_{K \in {\xi}_{h}} \rho_{K}^{-2}\| v-I_hv\|_{L^2(K)}^2\right)^{1/2} \\
&& +\left( \sum \limits_{K \in {\xi}_{h}} \Theta_{K}^2\right)^{1/2} \left( \sum \limits_{K \in {\xi}_{h}} \rho_{K}^{-2}\| v-I_hv\|_{L^2(K)}^2\right)^{1/2} \\
&\lesssim & \left( \sum \limits_{K \in {\xi}_{h}} (\eta_{R_K}^2+\Theta_K^2)\right)^{1/2}|||v|||. \end{eqnarray*} For the terms $T_2$ and $T_3$, we have \cite{schotzau09rae}[Lemma 4.8]
$$
T_2 \lesssim \left( \sum \limits_{K \in {\xi}_{h}} \eta_{E_K}^2 \right)^{1/2}|||v||| $$ $$
T_3 \lesssim \left( \sum \limits_{K \in {\xi}_{h}} \eta_{J_K}^2 \right)^{1/2}|||v||| $$ which finishes the proof.\\
\noindent {\bf Bound to the conforming part of the error:} For a given $t\in (0,T]$, there holds \begin{eqnarray}
\| u^s-u_h^c\|_{DG}\lesssim \eta + \Theta . \label{conf} \end{eqnarray}
\noindent {\it Proof:} Since $u^s-u_h^c\in H_0^1(\Omega )$, we have $|u^s-u_h^c|_C=|\vec{\beta} (u^s-u_h^c)|_*$. Then, from the inf-sup condition (\ref{infsup}) $$
\| u^s-u_h^c\|_{DG} = |||u^s-u_h^c||| +|u^s-u_h^c|_C \lesssim \underset{v\in H_0^1(\Omega ) \setminus \{ 0\}}{\text{sup}}\frac{{a}_h(t;u^s-u_h^c,v)}{|||v|||} . $$ So, we need to bound the term ${a}_h(t;u^s-u_h^c,v)$. Using that $u^s-u_h^c\in H_0^1(\Omega )$, we have \begin{eqnarray*} {a}_h(t;u^s-u_h^c,v) &=& {a}_h(t;u^s,v)-{a}_h(t;u_h^c,v) \\
&=& \int_{\Omega }fv dx -b_h(t;u^s,v)-{a}_h(t;u_h^c,v)\\ &=& \int_{\Omega }fv dx -b_h(t;u^s,v)-D_h(u_h^c,v)-J_h(u_h^c,v)-O_h(u_h^c,v) \\ &=& \int_{\Omega }fv dx -b_h(t;u_h^s,v)+b_h(t;u_h^s,v)-b_h(t;u^s,v)\\ && \quad -{a}_h(t;u_h^s,v)+D_h(u_h^r,v)+J_h(u_h^r,v)+O_h(u_h^r,v). \end{eqnarray*} We also have from the SIPG scheme \begin{eqnarray*} \int_{\Omega }fI_hv dx &=& a_h(t;u_h^s,I_hv)+K_h(u_h^s,I_hv)+b_h(t;u_h^s,I_hv)\\ \end{eqnarray*} Hence, we obtain $$ a(t;u^s-u_h^c,v)=T_1+T_2+T_3+T_4 $$ \begin{eqnarray*} T_1 &=& \int_{\Omega }f(v-I_hv) dx -{a}_h(t;u_h^s,v-I_hv)-b_h(t;u_h^s,v-I_hv)\\ T_2 &=& D_h(u_h^r,v)+J_h(u_h^r,v)+O_h(u_h^r,v)\\ T_3 &=& K_h(u_h^s,I_hv) \\ T_4 &=& b_h(t;u_h^s,v)-b_h(t;u^s,v)\\ \end{eqnarray*} From the inequality (\ref{lemma}), we have $$
T_1 \lesssim (\eta +\Theta )|||v||| $$ The continuity results (\ref{cd}-\ref{cj}) and the bound to remainder term (\ref{rem}) yields $$
T_2 \lesssim (|||u_h^r|||+|\vec{\beta} u_h^r|_*)|||v||| \leq \eta |||v||| $$ Moreover, using the identities (\ref{ck}) and (\ref{int1}), we get $$
T_3 \lesssim \gamma^{-1/2}\left( \sum_{K\in \xi} \eta_{J_K}^2\right)^{1/2} |||I_hv||| \lesssim \gamma^{-1/2}\left( \sum_{K\in \xi} \eta_{J_K}^2\right)^{1/2} |||v|||. $$ Finally, using Cauchy-Shwarz inequality and the boundedness property (\ref{cb}), we get \begin{eqnarray*} T_4 &=& b_h(t;u_h^s,v)-b_h(t;u^s,v) = \int_{\Omega }r(u_h^s)v dx-\int_{\Omega }r(u^s)v dx \\
& \leq & C_1 \| v\|_{L^2(\Omega )} - C_2 \| v\|_{L^2(\Omega )} \\
& \lesssim & |||v|||. \end{eqnarray*} This finishes the proof.\\
\noindent {\bf Proof to the reliability:} Combining the bounds (\ref{rem}) and (\ref{conf}) to the remainder and the conforming parts of the error, respectively, we obtain \begin{eqnarray*}
\|u^s-u_h^s \|_{DG} &\leq& \|u^s-u_h^c \|_{DG} +\|u_h^r \|_{DG}\\ &\leq& \eta + \Theta + \eta \\ &\lesssim & \eta + \Theta \end{eqnarray*}
\noindent {\bf Proof to the efficiency:} The proof of the efficiency is similar to Theorem 3.3 in \cite{schotzau09rae}. We only use the boundedness property (\ref{nonl1}) of the non-linear reaction term to bound the terms occurring in the procedure in \cite{schotzau09rae}.
\subsection{A posteriori error bounds for the semi-discrete system} \label{semidiscerror}
In order to measure the error for the semi-discrete problem, we use the $L^{\infty}(L^2)+L^2(H^1)$-type norm $$
\| v\|_*^2 = \| v\|_{L^{\infty}(0,T;L^2(\Omega ))}^2 + \int_0^T |||v|||^2dt $$ We make use the elliptic reconstruction technique in \cite{makridakis03era}: the elliptic reconstruction $w\in H_0^1(\Omega )$ is defined as the unique solution of the problem \begin{equation}\label{cont_elliptic} a(t;w,v)+b_h(t;w,v)=\int_{\Omega}\left( f-\frac{\partial u_h}{\partial t}\right)vdx \; , \quad \forall v\in H_0^1(\Omega ). \end{equation} The SIPG discretization, on the other hand, of the above system reads as: for each $t\in (0,T]$, find $w_h\in C^{0,1}(0,T;V_h(\xi_h))$ such that for all $v_h\in V_h(\xi_h)$ $$ a_h(t;w_h,v_h)+K_h(w_h,v_h)+b_h(t;w_h,v_h)= \int_{\Omega}\left( f-\frac{\partial u_h}{\partial t}\right)v_hdx $$
which implies that $w_h=u_h$. Hence, the error bound to the term $\| w-u_h\|_{DG}$ can be found using the a posteriori error bound (\ref{rel}) for non-linear stationary problem.
We give the a posteriori error bounds for the semi-discrete system developed as in \cite{cangiani13adg}. For the error $e(t)=u(t)-u_h(t)$ of the semi-discrete problem, we set the decomposition $e(t)=\mu (t)+\nu (t)$ with $\mu (t)= u(t)-w(t)$ and $\nu = w(t)-u_h(t)$. Then, for any $t\in (0,T]$, using (\ref{eqweak}) and (\ref{cont_elliptic}), we obtain that $$ \int_{\Omega}\frac{\partial e}{\partial t}v dx + a_h(t;\mu , v)+ b_h(t;\mu , v)=0 $$ which yields the error bound \cite{cangiani13adg}[Theorem 5.4] $$
\| e\|_* \lesssim \tilde{\eta} $$ where the error estimator $\tilde{\eta}$ is defined by \begin{eqnarray*}
\tilde{\eta}^2 = \| e(0)\|^2 + \int_0^T \tilde{\eta}_{S_1}^2dt + \text{min}\left\{ \left( \int_0^T \tilde{\eta}_{S_2}^2 dt \right)^2 , \rho_T^2\int_0^T \tilde{\eta}_{S_2}^2dt \right\} + \underset{0\leq t\leq T}{\text{max}}\tilde{\eta}_{S_3}^2 \end{eqnarray*} with \begin{eqnarray*}
\tilde{\eta}_{S_1}^2 &=& \sum_{K\in{\xi}_{h}} \rho_K^2\left\| f- \frac{\partial u_h}{\partial t} +\epsilon\Delta u_h - \vec{\beta}\cdot\nabla u_h - r(u_h)\right\|_{L^2(K)}^2 + \sum_{e\in\Gamma_0} \epsilon^{-1/2}\rho_e\| [ \epsilon\nabla u_h ] \|_{L^2(e)}^2 \\
&& + \sum_{e\in\Gamma_0\cup\Gamma^D}\left( \frac{\epsilon\sigma}{h_e}+\kappa h_e+\frac{h_e}{\epsilon}\right)\| [ u_h ] \|_{L^2(e)}^2\\
\tilde{\eta}_{S_2}^2 &=& \sum_{e\in\Gamma_0\cup\Gamma^D} h_e\left\| \left[ \frac{\partial u_h}{\partial t}\right]\right\|_{L^2(e)}^2 \\
\tilde{\eta}_{S_3}^2 &=& \sum_{e\in\Gamma_0\cup\Gamma^D} h_e\| [ u_h]\|_{L^2(e)}^2 \end{eqnarray*} and the weight $\rho_T=\text{min}(\epsilon^{-\frac{1}{2}}, \kappa^{-\frac{1}{2}})$.\\
\subsection{A posteriori error bounds for the fully-discrete system} \label{fullydiscerror}
For the fully discrete case, we consider the solutions at discrete time instances. For this reason, let $A^k\in V_h^k$ be the unique solution of the stationary system $$ a_h(t^k; u_h^k,v_h^k) + K_h(u_h^k,v_h^k) + b_h(t^k ; u_h^k,v_h^k) = \int_{\Omega} A^k v_h^k dx \; , \quad \forall v_h^k \in V_h^k $$ Note that for $k\geq 1$, $A^k=I_h^kf^k - (u_h^k - I_h^ku_h^{k-1})/\tau_k$ with $I_h^k$ being the $L^2$-projection operator onto the space $V_h^k$. Then, the elliptic reconstruction $w^k\in H_0^1(\Omega )$ is defined as the unique solution of the stationary system \begin{equation}\label{disc_elliptic} a(t^k; w^k,v) + b(t^k; w^k,v)=\int_{\Omega} A^k v dx \; , \qquad \forall v\in H_0^1(\Omega ). \end{equation}
Next, we take, as in \cite{cangiani13adg}, the discrete solution $u_h(t)$ as a piecewise continuous function so that on each interval $(t^{k-1},t^k]$, $u_h(t)$ is the linear interpolation of the values $u_h^{k-1}$ and $u_h^k$ given by $$ u_h(t)= l_{k-1}(t)u_h^{k-1} + l_{k}(t)u_h^{k} $$ with the linear Lagrange interpolation basis $l_{k-1}$ and $l_{k}$ defined on $[t^{k-1},t^k]$. Then, for the error $e=u-u_h$, using (\ref{eqweak}) and (\ref{disc_elliptic}), we obtain for all $v\in H_0^1(\Omega)$ \begin{eqnarray*} \int_{\Omega} \frac{\partial e}{\partial t} v dx + a_h(t; e,v) + b_h(t; e,v) &=& \int_{\Omega}(f-f^k)v dx + \int_{\Omega}\left( f^k - \frac{\partial u_h}{\partial t}\right) v dx \\ && - a_h(t;u_h,v) - b_h(t; u_h,v) \end{eqnarray*} which yields the error bound \cite{cangiani13adg}[Theorem 6.5] $$
\| e\|_*^2 \lesssim \eta_S^2 + \eta_T^2 $$ where $\eta_S$ is the spatial estimator given by \cite{cangiani13adg}
\begin{eqnarray*}
\eta_S^2 &=& \| e(0)\|^2 + \frac{1}{3}\sum_{k=1}^n \tau_k (\eta_{S_{1,k-1}}^2+\eta_{S_{1,k}}^2) +\sum_{k=1}^n \tau_k \eta_{S_{2,k}}^2 \\ && + \underset{0\leq k\leq n}{\text{max}}\eta_{S_{3,k}}^2 + \text{min} \left\{ \left( \sum_{k=1}^n \tau_k \eta_{S_{4,k}}\right)^2 , \rho_T^2 \sum_{k=1}^n \tau_j \eta_{S_{4,k}}^2 \right\} \end{eqnarray*} with \begin{eqnarray}
{\eta}_{S_{1,k}}^2 &=& \sum_{K\in{\xi}_{h}^{k-1}\cup {\xi}_{h}^{k}} \rho_K^2\left\| A^k +\epsilon\Delta u_h^k - \vec{\beta}^k\cdot\nabla u_h^k - r(u_h^k)\right\|_{L^2(K)}^2 + \sum_{e\in\Gamma_0} \epsilon^{-1/2}\rho_e\| [ \epsilon\nabla u_h^k ] \|_{L^2(e)}^2 \nonumber \\
&& + \sum_{e\in\Gamma}\left( \frac{\epsilon\sigma}{h_e}+\kappa h_e+\frac{h_e}{\epsilon}\right)\| [ u_h^k ] \|_{L^2(e)}^2 \label{spatialest} \\
{\eta}_{S_{2,k}}^2 &=& \sum_{K\in{\xi}_{h}^{k-1}\cup {\xi}_{h}^{k}} \rho_K^2 \left\| f^k - I_h^kf^k + \frac{u_h^{k-1}-I_h^ku_h^{k-1}}{\tau_k}\right\|_{L^2(K)}^2 \nonumber \\
{\eta}_{S_{3,k}}^2 &=& \sum_{e\in\Gamma} h_e\| [ u_h^k]\|_{L^2(e)}^2 \nonumber \\
{\eta}_{S_{4,k}}^2 &=& \sum_{e\in\Gamma} h_e \left\| \left[ \frac{u_h^k-u_h^{k-1}}{\tau_k}\right]\right\|_{L^2(e)}^2 \nonumber \end{eqnarray} and $\eta_T$ is the temporal estimator given by \cite{cangiani13adg}
\begin{eqnarray} \label{timeest} \eta_T^2 = \sum_{k=1}^n\int_{t_{k-1}}^{t_k}\eta_{T_1,k}^2 dt + \text{min} \left\{ \left( \sum_{k=1}^n \int_{t_{k-1}}^{t_k} \eta_{T_{2,k}}dt\right)^2 , \rho_T^2 \sum_{k=1}^n \int_{t_{k-1}}^{t_k} \eta_{T_{2,k}}^2dt \right\} \end{eqnarray} with \begin{eqnarray*}
\eta_{T_{1,k}}^2 &=& \sum_{K\in {\xi}_{h}^{k-1}\cup {\xi}_{h}^{k}}\epsilon^{-1} \| l_k(\vec{\beta}^k-\vec{\beta})u_h^k + l_{k-1}(\vec{\beta}^{k-1}-\vec{\beta})u_h^{k-1}\|_{L^2(K)}^2 \\
\eta_{T_{2,k}}^2 &=& \sum_{K\in {\xi}_{h}^{k-1}\cup {\xi}_{h}^{k}} \| f-f^k + l_{k-1}(A^k - A^{k-1}) + l_k(\nabla\cdot\vec{\beta}^k-\nabla\cdot \vec{\beta})u_h^k + l_k(\nabla\cdot\vec{\beta}^{j-1}-\nabla\cdot \vec{\beta})u_h^{k-1}\|_{L^2(K)}^2 \end{eqnarray*}
\section{Adaptive algorithm} \label{algorithm}
The time-space adaptive algorithm for linear convection-diffusion problems in \cite{cangiani13adg} is modified for semi-linear problems of type \eqref{model}, see Fig.~\ref{chart}. It is based on the residual-based a posteriori error estimators given in the previous sections. The algorithm starts with an initial uniform mesh in space and with a given initial time step. At each time step, the space and time-step are adaptively arranged according to the user defined tolerances $\mathbf{ttol}$ for time-step refinement, and $\mathbf{stol^+}$ and $\mathbf{stol^-}$ for spatial mesh, former corresponding to refinement and latter corresponding to the coarsening procedures in space. Note that we do not need a temporal tolerance corresponding to time-step coarsening, since we start, in our problems, with a uniform equispaced distribution of $[0,T]$ having sufficiently large time-steps. Thus, it is enough just bisecting the time intervals not satisfying the temporal tolerance $\mathbf{ttol}$. Both the refinement and coarsening processes in space are determined by the indicator $\eta_{S_{1,k}}$ (\ref{spatialest}) appearing in the spatial estimator. Since the temporal estimator $\eta_T$ (\ref{timeest}) is not easy to compute, the adaptive refinement of the time-steps are driven by the modified temporal-indicator \cite{cangiani13adg} $$ \tilde{\eta}_{T_k}^2=\int_{t_{k-1}}^{t_k} \eta_{T_{1,k}}^2dt + \min \{\rho_T,T\}\int_{t_{k-1}}^{t_k} \eta_{T_{2,k}}^2dt $$ sum of which gives a bound for the temporal estimator $\eta_T^2$.
\begin{figure}
\caption{Adaptive algorithm chart}
\label{chart}
\end{figure}
Although the adaptive algorithm, Fig.~\ref{chart}, stands for a single equation of the system \eqref{org}, it is not difficult to extend the algorithm to the coupled systems. For this, say we have a system of two equations for the unknowns $u_1$ and $u_2$, the temporal-indicators $\tilde{\eta}_{T_k}^1$, $\tilde{\eta}_{T_k}^2$ and the spatial-indicators $\eta_{S_{1,k}}^1$, $\eta_{S_{1,k}}^2$ corresponding to the unknowns $u_1$ and $u_2$, respectively, are computed. To adapt the time-step size, we ask the temporal condition for the both temporal-indicators, i.e. $\tilde{\eta}_{T_k}^1\leq \mathbf{ttol}$ and $\tilde{\eta}_{T_k}^2\leq \mathbf{ttol}$. On the other hand, to select the elements to be refined, we take the set of elements which is the union of the sets of the elements satisfying $\eta_{S_{1,k}}^1> \mathbf{stol^+}$ and $\eta_{S_{1,k}}^2> \mathbf{stol^+}$, and similar procedure to select the elements to be coarsened, but not including any elements which are selected to be refined. Numerical studies demonstrate that the adaptive algorithm is capable of resolving the layers in space as the time progresses.
\section{Solution of the fully-discrete system} \label{newton}
In this section, we discuss the solution of the fully-discrete system \eqref{fullydisc} on an arbitrary $k^{th}$ time-step, which is solved for all $k=1,2,\ldots , n$. In order to not be confused about the notations, let us consider the system \eqref{fullydisc} on an arbitrary $k^{th}$ time-step without the superscript for the time-step of the form
\begin{equation}\label{singleeq} \int_{\Omega}\frac{u_{h}-w_{h}}{\tau}v_{h} dx + a_{h}(u_{h},v_{h})+ K_{h}(u_{h},v_{h})+b_{h}(u_{h}, v_{h})=\int_{\Omega}fv_{h} dx \; , \quad \forall v_h\in V_h \end{equation} where we set $u_{h}:=u_{h}^k$, $w_h:=u_{h}^{k-1}$, $v_{h}:=v_{h}^k$, $f:=f^k$, $\tau:=\tau_k$, $a_{h}(u_{h},v_{h}):=a_{h}(t^k;u_{h}^k,v_{h}^k)$, $b_{h}(u_{h},v_{h}):=b_{h}(t^k;u_{h}^k,v_{h}^k)$ and $V_h:=V_h^k$. The approximate solution $u_h$ and the known solution (from the previous time-step) $w_h$ of \eqref{singleeq} have the form \begin{equation}\label{comb} u_h=\sum_{i=1}^{Nel}\sum_{l=1}^{Nloc}u_l^i\phi_l^i \; , \quad w_h=\sum_{i=1}^{Nel}\sum_{l=1}^{Nloc}w_l^i\phi_l^i \end{equation} where $\phi_l^i$'s are the basis polynomials spanning the space $V_h$, $u_l^i$'s are the unknown coefficients to be found, $w_l^i$'s are the known coefficients. $Nel$ denotes the number of triangles and $Nloc$ is the number of local dimension depending on the degree of polynomials $k$ (in 2D, $Nloc=(k+1)(k+2)/2$). In DG methods, we choose the piecewise basis polynomials $\phi_l^i$'s in such a way that each basis function has only one triangle as a support, i.e. we choose on a specific triangle $K_e$, $e\in\{ 1,2,\ldots , Nel\}$, the basis polynomials $\phi_l^e$ which are identically zero outside the triangle $K_e$, $l=1,2,\ldots , Nloc$. By this construction, the stiffness and mass matrices obtained by DG methods has a block structure, each of which related to a triangle (there is no overlapping as in continuous FEM case). The product $dof:=Nel*Nloc$ gives the degree of freedom in DG methods. Inserting the linear combinations \eqref{comb} of $u_h$ and $w_h$ in \eqref{singleeq} and choosing the test functions $v_h=\phi_l^i$, $l=1,2,\ldots , Nloc$, $i=1,2,\ldots , Nel$, the discrete residual of the system (\ref{singleeq}) in matrix vector form is given by \begin{equation}\label{residualsystem} Res(\vec{u})=M\vec{u} - M\vec{w} +\tau( S\vec{u} + \vec{b}(\vec{u}) - \vec{f}) = 0 \end{equation} where $\vec{u}, \vec{w}\in\mathbb{R}^{dof}$ are the vector of unknown and known coefficients $u_l^i$'s and $w_l^i$'s, respectively, $M\in\mathbb{R}^{dof\times dof}$ is the mass matrix, $S\in\mathbb{R}^{dof\times dof}$ is the stiffness matrix corresponding to the bilinear form $\tilde{a}_h(u_h,v_h):=a_h(u_h,v_h)+K_h(u_h,v_h)$, $\vec{b}\in\mathbb{R}^{dof}$ is the vector function of $\vec{u}$ related to the non-linear form $b_h(u_h,v_h)$ and $\vec{f}\in\mathbb{R}^{dof}$ is the vector to the linear form $\int_{\Omega}fv_{h}dx$. The explicit definitions are given by $$ S= \begin{bmatrix} S_{11} & S_{12} & \cdots & S_{1,Nel} \\ S_{21} & S_{22} & & \vdots \\ \vdots & & \ddots & \\ S_{Nel,1} & \cdots & & S_{Nel,Nel} \end{bmatrix} \; , \quad M= \begin{bmatrix} M_{11} & M_{12} & \cdots & M_{1,Nel} \\ M_{21} & M_{22} & & \vdots \\ \vdots & & \ddots & \\ M_{Nel,1} & \cdots & & M_{Nel,Nel} \end{bmatrix} $$
$$
\vec{u}= \begin{bmatrix}
\vec{u}_1 \\ \vec{u}_2 \\ \vdots \\ \vec{u}_{Nel} \end{bmatrix} \; , \quad \vec{w}= \begin{bmatrix}
\vec{w}_1 \\ \vec{w}_2 \\ \vdots \\ \vec{w}_{Nel} \end{bmatrix} \; , \quad \vec{b}(\vec{u})= \begin{bmatrix}
\vec{b}_1(\vec{u}) \\ \vec{b}_2(\vec{u}) \\ \vdots \\ \vec{b}_{Nel}(\vec{u}) \end{bmatrix} \; , \quad \vec{f}= \begin{bmatrix}
\vec{f}_1 \\ \vec{f}_2 \\ \vdots \\ \vec{f}_{Nel} \end{bmatrix} $$ where all the blocks have dimension $Nloc$: $$ S_{ji}= \begin{bmatrix} \tilde{a}_h(\phi_1^i,\phi_1^j) & \tilde{a}_h(\phi_2^i,\phi_1^j) & \cdots & \tilde{a}_h(\phi_{Nloc}^i,\phi_1^j) \\ \tilde{a}_h(\phi_1^i,\phi_2^j) & \tilde{a}_h(\phi_2^i,\phi_2^j) & & \vdots \\ \vdots & & \ddots & \\ \tilde{a}_h(\phi_1^i,\phi_{Nloc}^j) & \cdots & & \tilde{a}_h(\phi_{Nloc}^i,\phi_{Nloc}^j) \end{bmatrix} $$
$$ M_{ji}= \begin{bmatrix} \int_{\Omega}\phi_1^i\phi_1^jdx & \int_{\Omega}\phi_2^i\phi_1^jdx & \cdots & \int_{\Omega}\phi_{Nloc}^i\phi_1^jdx \\ \int_{\Omega}\phi_1^i\phi_2^jdx & \int_{\Omega}\phi_2^i\phi_2^jdx & & \vdots \\ \vdots & & \ddots & \\ \int_{\Omega}\phi_1^i\phi_{Nloc}^jdx & \cdots & & \int_{\Omega}\phi_{Nloc}^i\phi_{Nloc}^jdx \end{bmatrix} $$
$$
\vec{u}_i= \begin{bmatrix}
u_1^i \\ u_2^i \\ \vdots \\ u_{Nloc}^i \end{bmatrix} \; , \quad \vec{w}_i= \begin{bmatrix}
w_1^i \\ w_2^i \\ \vdots \\ w_{Nloc}^i \end{bmatrix} \; , \quad \vec{b}_i(\vec{u})= \begin{bmatrix}
b_h(u_h,\phi_1^i) \\ b_h(u_h,\phi_2^i) \\ \vdots \\ b_h(u_h,\phi_{Nloc}^i) \end{bmatrix} \; , \quad \vec{f}_i= \begin{bmatrix}
\int_{\Omega}f\phi_1^idx \\ \int_{\Omega}f\phi_2^idx \\ \vdots \\ \int_{\Omega}f\phi_{Nloc}^idx \end{bmatrix} . $$ The block structure of DG methods causes to the increase of the condition number of the obtained matrices by the degree $k$ of basis polynomials, which is a drawback of DG methods comparing to the classical (continuous) finite elements. However, taking into account the locality, a valuable property, of DG methods, this drawback can be handled by various preconditioners developed for DG discretized schemes in the literature. Besides, the condition number of the stiffness matrix $S$ increases linearly by the penalty parameter $\sigma$, as well. Therefor, the penalty parameter should not be chosen too large. On the other hand, it should be selected sufficiently large to ensure the coercivity of the bilinear form \cite{riviere08dgm}[Sec. 27.1], which is needed for the stability of the convergence of the DG method. It ensures that the stiffness matrix arising from the DG discretization is symmetric positive definite. In the literature, several choices of the penalty parameter are suggested. In \cite{epshteyn07epp}, computable lower bounds are derived, and in \cite{dobrev08psi}, the penalty parameter is chosen depending on the diffusion coefficient $\epsilon$. The effect of the penalty parameter on the condition number was discussed in detail for the DG discretization of the Poisson equation in \cite{castillo12pdg} and in \cite{slingerland14fls} for layered reservoirs with strong permeability contrasts. In our study, we select the penalty parameter $\sigma$ depending only on the polynomial degree $k$, as $\sigma =3k(k+1)$ on interior edges and $\sigma =6k(k+1)$ on boundary edges. The reason of the coupling of the penalty parameter $\sigma$ on boundary edges is sufficient penalization of the solution on the boundary due to the non-homogeneous Dirichlet boundary conditions which are imposed weakly in DG methods.
Next, we solve the system \eqref{residualsystem} by Newton method. In the sequel, we start with an initial guess $\vec{u}^{(0)}$ (most possibly $\vec{u}^{(0)}=\vec{w}$, i.e. the known solution from the previous time-step) and we solve the system \begin{eqnarray*} J^s\delta\vec{u}^{(s)} &=& -Res(\vec{u}^{(s)}) \\ \vec{u}^{(s+1)} &=& \vec{u}^{(s)} + \delta\vec{u}^{(s)} \; , \quad s=0,1,\ldots \end{eqnarray*} where $J^s=M+\tau (S+J^s_{\vec{b}})$ is the Jacobian matrix of the system at the iteration $\vec{u}^{(s)}$, and $J^s_{\vec{b}}$ denotes the Jacobian matrix to the non-linear vector $\vec{b}(\vec{u})$ at the iteration $\vec{u}^{(s)}$.
\section{Numerical studies} \label{numeric}
In this section, we give the numerical studies demonstrating the performance of the time-space adaptive algorithm. All the computations are implemented on MATLAB-R2014a. In the problems, by the very coarse initial mesh, we mean an initial mesh which is formed, for instance on $\Omega =(0,1)^2$, by dividing the domain with $\Delta x_1=\Delta x_2=0.5$ leading to 8 triangular elements and 48 DoFs for quadratic elements.
As the first example, Example~\ref{ex1}, we give a test example with polynomial type non-linearity having a non-moving internal layer to figure out the benchmark of the algorithm by using different tolerances and diffusion parameters $\epsilon$: rates of error, spatial and temporal estimators, and effectivity indices (proportion of the estimator to the error). We expect that the effectivity indices lie in a small band for different diffusion parameters meaning that our estimators are robust in the system P\'{e}clet number. Moreover, to demonstrate the mentioned properties, we use the weighted DoFs as in \cite{cangiani13adg} $$ \text{Weighted DoFs} = \frac{1}{T} \sum_{k=1}^n \tau_k\lambda_k $$ where $\lambda_k$ denotes the total number of DoFs on the union mesh $\xi_h^{k-1}\cup\xi_h^{k}$. Since the first example has a non-moving internal layer, a monotonic increase in the DoFs is expected by the time progresses. Conversely, we give problems having moving layers by the time progresses in Example~\ref{ex2}-\ref{ex3}. In this case, we expect that the refinement and coarsening procedures in space work simultaneously leading to oscillations in time vs DoFs plots. By Example~\ref{ex2}, we also test the performance of our algorithm for a coupled system. As the final example, Example~\ref{ex4}, we consider an important real life problem representing a reaction in porous media having internal layers at different locations due to the high-permeability rocks.
\subsection{Example with polynomial type non-linearity (benchmark of the algorithm)} \label{ex1}
The first example is taken from \cite{bause12ash} with a polynomial non-linear term $$ u_t+\vec{\beta }\cdot\nabla u-\epsilon\Delta u+r(u)=f \quad \text{in } \; \Omega =(0,1)^2, $$ with the convection field $\vec{\beta }(x,y)=(2,3)^T$, diffusion coefficient $\epsilon =10^{-6}$, the non-linear reaction term $r(u)=u^4$. The source function $f$ and the Dirichlet boundary condition are chosen so that the exact solution is given by \begin{eqnarray*} u(\vec{x},t)&=& 16\sin (\pi t)x_1(1-x_1)x_2(1-x_2) \\ && [ 0.5 + \pi^{-1}\arctan (2\epsilon^{-1/2}(0.25^2-(x_1-0.5)^2-(x_2-0.5)^2)) ]. \end{eqnarray*} We start by demonstrating the decrease of the errors by uniform time-space refinement using linear DG elements. In Fig.~\ref{ex1:uniform}, the expected first order convergence in space and time is shown.
\begin{figure}
\caption{Example \ref{ex1}: Decays of estimators and errors for uniform time-space }
\label{ex1:uniform}
\end{figure}
\begin{figure}
\caption{Example \ref{ex1}: Error/spatial estimator for $\epsilon =10^{-2}$ (left) and $\epsilon =10^{-4}$ (right) }
\label{ex1:errors}
\end{figure} For the time-space adaptive solution, we use quadratic DG elements. We investigate the performance of the spatial estimator by fixing the temporal time-step $\tau =0.005$ so that the temporal error dominated by the spatial error. We reduce the spatial estimator tolerance $\mathbf{stol^+}$ from $10^{-1}$ to $10^{-6}$. The rates of the error and the spatial estimator are similar as illustrated in Fig.~\ref{ex1:errors} for $\epsilon =10^{-2}$ and $\epsilon =10^{-4}$. Fig.~\ref{ex1:spatial} shows the spatial effectivity indices and the decrease of the spatial estimators for the diffusion constant $\epsilon$. The effectivity indices do not exceed 7 which are acceptable as in \cite{cangiani13adg} for linear convection-diffusion problems.
\begin{figure}
\caption{Example \ref{ex1}: Spatial effectivity indices (left) and estimators (right)}
\label{ex1:spatial}
\end{figure} To investigate the performance of the temporal estimator, we fix a sufficiently fine spatial mesh so that the the spatial error dominated by the temporal error, and then we reduce the temporal estimator tolerance $\mathbf{ttol}$ in the range $10^{-1}-10^{-6}$. In Fig.~\ref{ex1:temporal}, the temporal effectivity indices and the decrease of the temporal estimators are not affected by $\epsilon$, and effectivity indices are almost the same within the band 1-2.
\begin{figure}
\caption{Example \ref{ex1}: Temporal effectivity indices (left) and estimators (right)}
\label{ex1:temporal}
\end{figure} Finally, we apply the time-space adaptive algorithm with the tolerances $\mathbf{ttol}=10^{-3}$, $\mathbf{stol^+}=3\times 10^{-4}$ and $\mathbf{stol^-}=3\times 10^{-7}$. Firstly, we prepare an initial mesh starting from a very coarse spatial mesh and a uniform partition of the time interval $[0,0.5]$ with the step-size $\tau =0.25$ until the the user defined tolerances $\mathbf{ttol}$ and $\mathbf{stol^+}$ are satisfied. The adaptive mesh at the final time $T=0.5$ is shown in Fig.~\ref{ex1:mesh}. In Fig.~\ref{ex1:dofdt} on the right, the change of the time-steps is shown, whereas the change in the DoFs is illustrated in Fig.~\ref{ex1:dofdt} on the left. Since the layers in the problem do not move as the time progresses, the number of DoFs increases monotonically by the spatial grid refinement. In Fig.~\ref{ex1:plot}, it is shown that all the oscillations are damped out by adaptive algorithm using less DoFs compared to the uniform one.
\begin{figure}
\caption{Example \ref{ex1}: Adaptive mesh }
\label{ex1:mesh}
\end{figure}
\begin{figure}
\caption{Example \ref{ex1}: Uniform (left) and adaptive (right) solutions at T=0.5 }
\label{ex1:plot}
\end{figure}
\begin{figure}
\caption{Example \ref{ex1}: Evolution of DoFs (left) and time-steps $\tau$ (right) }
\label{ex1:dofdt}
\end{figure}
\subsection{Coupled example with polynomial type non-linearity} \label{ex2}
The next example is a coupled non-linear problem taken from \cite{bause13hof}.
\begin{eqnarray*} \frac{\partial u_i}{\partial t}- \epsilon\Delta u_i+\vec{\beta }_i\cdot\nabla u_i+u_1u_2 &=& f_i , \quad i=1,2 \end{eqnarray*} on $\Omega =(0,1)^2$ with the convection fields $\vec{\beta }_1=(1,0)^T$ and $\vec{\beta }_2=(-1,0)^T$, and the diffusion constant $\epsilon=10^{-5}$. The Dirichlet boundary conditions, initial conditions and the load functions $f_i$ are chosen so that the exact solutions are $$ u_1(\vec{x},t)=\frac{1}{2}\left( 1-\tanh \frac{2x_1-0.2t-0.8}{\sqrt{5\epsilon }}\right) $$ $$ u_2(\vec{x},t)=\frac{1}{2}\left( 1+\tanh \frac{2x_1+0.2t-0.9}{\sqrt{5\epsilon }}\right) $$ We use again quadratic DG elements. We prepare an initial mesh, Fig.~\ref{ex2:mesh} on the left, starting with a very coarse spatial mesh and a uniform partition of the time interval $[0,1]$ with the step-size $\tau =0.1$ until the user defined tolerances $\mathbf{ttol}=10^{-3}$ and $\mathbf{stol^+}=10^{-1}$ are satisfied. Here, two sharp fronts move towards to each other and then mix directly after the time $t=0.1$, Fig.~\ref{ex2:cross}. The movement of the fronts are also visible in Fig.~\ref{ex2:mesh} claiming that refinement/coarsening of the adaptive algorithm works well. We see that the sharp fronts in the cross-wind direction $x_2=0.5x_1+0.75$ are almost damped out. Moreover, Fig.~\ref{ex2:mesh}-\ref{ex2:dtdof} show that both the spatial and temporal estimators catch the time where the two sharp fronts mix.
\begin{figure}
\caption{Example \ref{ex2}: Cross-section plots in the cross-wind direction at $t=0.1$ (left) and $t=1$ (right)}
\label{ex2:cross}
\end{figure}
\begin{figure}
\caption{Example \ref{ex2}: Adaptive meshes at $t=0$, $t=0.1$ and $t=1$ (from left to right)}
\label{ex2:mesh}
\end{figure}
\begin{figure}
\caption{Example \ref{ex2}: Evolution of DoFs (left) and time-steps $\tau$ (right)}
\label{ex2:dtdof}
\end{figure}
\subsection{Non-linear ADR equation in homogeneous porous media} \label{ex3}
We consider the advection-diffusion-reaction (ADR) equation in \cite{tambue10eia} with polynomial type non-linear reaction \begin{eqnarray*} \frac{\partial u}{\partial t}- \epsilon\Delta u + \vec{\beta}\cdot\nabla u+\gamma u^2(u-1) &=& 0 \quad \text{in } \; \Omega\times (0,T] \end{eqnarray*} on $\Omega =(0,1)^2$. We take as in \cite{tambue10eia} the homogeneous dispersion tensor as $\epsilon =10^{-4}$, the velocity field $\vec{\beta}=(-0.01,-0.01)^T$ and $\gamma =100$. The initial and boundary conditions are chosen by the exact solution $$ u(\vec{x},t)=[ 1+\exp ( a(x_1+x_2-bt)+a(b-1)) ]^{-1} $$ with $a=\sqrt{\gamma /(4\epsilon)}$ and $b=-0.02+\sqrt{\gamma\epsilon}$. The problem is a transport of a front in homogeneous porous media. We simulate the given problem for the final time $T=1$, and with quadratic DG elements. We begin by preparing an initial mesh starting from a very coarse spatial mesh and a uniform partition of the time interval $[0,1]$ with the step-size $\tau =0.25$ until the user defined tolerances $\mathbf{ttol}=3\times 10^{-3}$ and $\mathbf{stol^+}=10^{-3}$ are satisfied. In Fig.~\ref{ex3:meshsol}, the adaptive meshes and solution profiles are shown at the times $t=\{ 0.2,0.6,1\}$ where the movement of the front can be seen. The time vs DoFs and time vs time step-size plots in Fig.~\ref{ex3:dtdof} indicate clearly the oscillations in DoFs and time-steps due to the movement of the front.
\begin{figure}
\caption{Example \ref{ex3}: Adaptive meshes (top) and solution profiles (bottom) at $t=0.2$, $t=0.6$ and $t=1$ (from left to right)}
\label{ex3:meshsol}
\end{figure}
\begin{figure}
\caption{Example \ref{ex3}: Evolution of DoFs (left) and time-steps $\tau$ (right)}
\label{ex3:dtdof}
\end{figure}
\subsection{Non-linear ADR in deterministic heterogeneous porous media} \label{ex4}
We consider the ADR equation in \cite{tambue10eia} with Monod or Langmuir isotherm type non-linear reaction \begin{eqnarray*} \frac{\partial u}{\partial t}- \nabla\cdot (\epsilon\nabla u) + \vec{\beta}(x)\cdot\nabla u+\frac{u}{1+u} &=& 0 \quad \text{in } \; \Omega\times (0,T]\\ u(x,t) &=& 1 \quad \text{on } \; \Gamma^D\times [0,T] \\ -\epsilon\nabla u(x,t)\cdot\vec{n} &=& 0 \quad \text{on } \; (\partial\Omega\setminus\Gamma^D)\times [0,T] \\ u(x,0) &=& 0 \quad \text{in } \; \Omega \end{eqnarray*} with $\Omega =(0,3)\times (0,2)$ and $\Gamma^D=\{ 0\}\times [0,2]$. The problem represents a reaction in porous media, for instance, transport in a highly idealized fracture pattern. Here $\epsilon$ stands for the heterogeneous dispersion tensor given by $$ \epsilon = \begin{bmatrix} 10^{-3} & 0\\ 0 & 10^{-4} \end{bmatrix} $$ The velocity field $\vec{\beta}(x)$ is determined via the Darcy's law $$ \vec{\beta} = -\frac{k(x)}{\mu}\nabla p $$ where $p$ is the fluid pressure, $\mu$ is the fluid viscosity and $k(x)$ is the permeability of the porous medium. Using the mass conservation property $\nabla\cdot\vec{\beta}(x)=0$ under the assumption that rock and fluids are incompressible, the velocity field $\vec{\beta}(x)$ is computed by solving \begin{eqnarray*} \nabla\cdot\left( \frac{k(x)}{\mu}\nabla p \right) &=& 0 \quad \text{in } \; \Omega\\ p &=& 1 \quad \text{on } \; \{ 0\}\times [0,2] \\ p &=& 0 \quad \text{on } \; \{ 3\}\times [0,2] \\ -k(x)\nabla p\cdot \vec{n} &=& 0 \quad \text{on } \; (0,3)\times \{ 0,2\} \end{eqnarray*}
We simulate the given problem for the final time $T=1$ using linear DG elements. We take the fluid viscosity $\mu =0.1$, and the permeability field as in \cite{tambue10eia} with three parallel streaks the permeability of which are 100 times greater than the permeability of the surrounding domain, see Fig.~\ref{ex4:perm} on the left, by which the flow is canalized from the lower-permeability rocks into the high-permeability ones, Fig.~\ref{ex4:perm} on the right. For the adaptive procedure, we prepare an initial mesh starting from a very coarse spatial mesh and a uniform partition of the time interval $[0,1]$ with the step-size $\tau =0.05$ until the user defined tolerances $\mathbf{ttol}=10^{-3}$ and $\mathbf{stol^+}=3\times 10^{-4}$ are satisfied. Fig.~\ref{ex4:plot03}-\ref{ex4:plot1} show the adaptive meshes and concentrations at $t=0.3$ and $t=1$, where we can clearly see the flow-focusing due to the high-permeability.
\begin{figure}
\caption{Example \ref{ex4}: Permeability field (left) and velocity streamlines (right)}
\label{ex4:perm}
\end{figure}
\begin{figure}
\caption{Example \ref{ex4}: Adaptive mesh (left) and concentration (right) at $t=0.3$}
\label{ex4:plot03}
\end{figure}
\begin{figure}
\caption{Example \ref{ex4}: Adaptive mesh (left) and concentration (right) at $t=1$}
\label{ex4:plot1}
\end{figure}
Time vs DoFs and time vs time step-size plots are given in Fig.~\ref{ex4:dtdof}. We see that initially small time steps are used and then it reaches a steady time step, Fig.~\ref{ex4:dtdof} on the right. The number of DoFs increases (refinement dominates coarsening) monotonically after the meet of first high-permeability rock until the meet of third high-permeability rock and then the increase stops, Fig.~\ref{ex4:dtdof} on the left. This is meaningful since there is no sharp flow canalization after the third high-permeability rock.
\begin{figure}
\caption{Example \ref{ex4}: Evolution of DoFs (left) and time-steps $\tau$ (right)}
\label{ex4:dtdof}
\end{figure}
\section{Conclusion}
We implemented a time-space adaptive algorithm for non-linear ADR equations based on utilizing the elliptic reconstruction technique to be able to use the elliptic a posteriori error estimator for the convection dominated parabolic problems with non-linear reaction mechanisms. We derived a posteriori error estimator in the $L^{\infty}(L^2) + L^2(H^1)$-type norm using backward Euler in time and SIPG in space. We demonstrated the performance of the algorithm by numerical studies.
\end{document}
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\begin{document}
\def\spacingset#1{\renewcommand{\baselinestretch} {#1}\small\normalsize} \spacingset{1}
\if00 {
\title{\bf Functional Outlier Detection by a Local Depth with Application to NO$_{x}$ Levels}
\author{
\textbf{Carlo Sguera}\\
\normalsize Department of Statistics, Universidad Carlos III de Madrid\\
\normalsize 28903 Getafe (Madrid), Spain\\
\normalsize(\texttt{[email protected]})\\
\textbf{Pedro Galeano} \\
\normalsize Department of Statistics, Universidad Carlos III de Madrid\\
\normalsize 28903 Getafe (Madrid), Spain\\
\normalsize(\texttt{[email protected]})\\
and \\
\textbf{Rosa E. Lillo} \\
\normalsize Department of Statistics, Universidad Carlos III de Madrid\\
\normalsize 28903 Getafe (Madrid), Spain\\
\normalsize(\texttt{[email protected]})\\ }
\date{}
\maketitle } \fi
\if10 {
\begin{center}
{\LARGE\bf Functional Outlier Detection by a Local Depth with Application to NO$_{x}$ Levels} \end{center}
} \fi
\begin{abstract} This paper proposes methods to detect outliers in functional data sets and the task of identifying atypical curves is carried out using the recently proposed kernelized functional spatial depth (KFSD). KFSD is a local depth that can be used to order the curves of a sample from the most to the least central, and since outliers are usually among the least central curves, we present a probabilistic result which allows to select a threshold value for KFSD such that curves with depth values lower than the threshold are detected as outliers. Based on this result, we propose three new outlier detection procedures. The results of a simulation study show that our proposals generally outperform a battery of competitors. We apply our procedures to a real data set consisting in daily curves of emission levels of nitrogen oxides (NO$_{x}$) since it is of interest to identify abnormal NO$_{x}$ levels to take necessary environmental political actions. \end{abstract}
\noindent {\it Keywords:} Functional depths; Functional outlier detection; Kernelized functional spatial depth; Nitrogen oxides; Smoothed resampling.
\spacingset{1.5}
\section{INTRODUCTION} \label{sec:intro} The accurate identification of outliers is an important aspect in any statistical data analysis. Nowadays there are well-established outlier detection techniques in the univariate and multivariate frameworks (for a complete review of the topic, see for example \citeauthor{BarLew1994} \citeyear{BarLew1994}). In recent years, new types of data have become available and tractable thanks to the evolution of computing resources, e.g., big multivariate data sets having more variables than observations (high-dimensional multivariate data) or samples composed of repeated measurements of the same observation taken over an ordered set of points that can be interpreted as realizations of stochastic processes (functional data). In this paper we focus on functional data, which are usually studied with the tools provided by functional data analysis (FDA). For overviews on FDA methods, see \cite{RamSil2005}, \cite{FerVie2006}, \cite{HorKok2012} or \cite{Cue2014}. For environmental statistical problems tackled using FDA techniques, see for example \cite{IgnEtAl2015}, \cite{MenEtAl2014} and \cite{RuiEsp2012}.
As in univariate or multivariate analysis, the detection of outliers is also fundamental in FDA. According to \citeauthor{FebGalGon2007}\ (\citeyear{FebGalGon2007}, \citeyear{FebGalGon2008}), a functional outlier is a curve generated by a stochastic process with a different distribution than the one of normal curves. This definition covers many types of outliers, e.g., magnitude outliers, shape outliers and partial outliers, i.e., curves having atypical behaviors only in some segments of the domain. Shape and partial outliers are typically harder to detect than magnitude outliers (in the case of high magnitude, outliers can even be recognized by simply looking at a graph), and therefore entail more challenging outlier detection problems. In this paper we focus on samples contaminated by low magnitude, shape or partial outliers.
Specifically, we consider a real data set consisting in nitrogen oxides (NO$_{x}$) emission daily levels measured in the Barcelona area (see \citeauthor{FebGalGon2008} \citeyear{FebGalGon2008} for a first analysis of this data set). Since NO$_{x}$ represent one of the most important pollutants, cause ozone formation and contribute to global warning, it is of interest the identification of days with abnormally large NO$_{x}$ emissions to allow the implementation of actions able to control their causes, which are primarily the combustion processes generated by motor vehicles and industries.
We propose to detect functional outliers using the notion of functional depth. A functional depth is a measure providing a $P$-based center-outward ordering criterion for observations of a functional space $\mathbb{H}$, where $P$ is a probability distribution on $\mathbb{H}$. When a sample of curves is available, a functional depth orders the curves from the most to the least central according to their depth values and, if any outlier is in the sample, its depth is expected to be among the lowest values. Therefore, it is reasonable to build outlier detection methods that use functional depths.
\indent In this paper we enlarge the number of available functional outlier detection procedures by presenting three new methods based on a specific depth, the kernelized functional spatial depth (KFSD, \citeauthor{SguGalLil2014}\ \citeyear{SguGalLil2014}). KFSD is a local-oriented depth, that is, a depth which orders curves looking at narrow neighborhoods and giving more weight to close than distant curves. Its approach is opposite to what global-oriented depths do. Indeed, any global depth makes depend the depth of a given curve on the whole rest of observations, with equal weights for all of them. This is the case of a global-oriented depth such as the functional spatial depth (FSD, \citeauthor{ChaCha2014} \citeyear{ChaCha2014}), of which KFSD is its local version. A local depth such as KFSD may result useful to analyze functional samples having a structure deviating from unimodality or symmetry. Moreover, the local approach behind KFSD proved to be a good strategy in supervised classification problems with groups of curves not extremely clear-cut (see \citeauthor{SguGalLil2014} \citeyear{SguGalLil2014}). Alternatively, we illustrate that KFSD ranks well low magnitude, shape or partial outliers, that is, their corresponding KFSD values are in general lower than those of normal curves. Then, we propose different procedures to select a threshold for KFSD to distinguish between normal curves and outliers. These procedures employ smoothing resampling techniques and are based on a theoretical result which allows to obtain a probabilistic upper bound on a desired false alarm probability of detecting normal curves as outliers. Note that the probabilistic foundations of the proposed methods represent a novelty in FDA outlier detection problems. We study the performances of our procedures in a simulation study and analyzing the NO$_{x}$ data set. We show this data set in Figure \ref{fig:NOxW}, where it is already possible to appreciate that the presence of partial outliers is an issue.
\begin{figure}
\caption{NO$_{x}$ levels measured in $\mu g/m^{3}$ every hour of 76 working days between 23/02/2005 and 26/06/2005 in Poblenou, Barcelona.}
\label{fig:NOxW}
\end{figure}
We compare our methods with some alternative outlier detection procedures: \cite{FebGalGon2008} proposed to label as outliers those curves with depth values lower than a certain threshold. As functional depths, they considered the Fraiman and Muniz depth (\citeauthor{FraMun2001} \citeyear{FraMun2001}), the h-modal depth (\citeauthor{CueFebFra2006} \citeyear{CueFebFra2006}) and the integrated dual depth (\citeauthor{CueFra2009} \citeyear{CueFra2009}). To determine the depth threshold, they proposed two different bootstrap procedures based on depth-based trimmed or weighted resampling, respectively; \cite{SunGen2011} introduced the functional boxplot, which is constructed using the ranking of curves provided by the modified band depth (\citeauthor{LopRom2009} \citeyear{LopRom2009}). The proposed functional boxplot detects outliers using a rule that is similar to the one of the standard boxplot; \cite{HynSha2010} proposed to reduce the outlier detection problem from functional to multivariate data by means of functional principal component analysis (FPCA), and to use two alternative multivariate techniques on the scores to detect outliers, i.e., the bagplot and the high density region boxplot, respectively.
The remainder of the article is organized as follows. In Section \ref{sec:kfsd} we recall the definition of KFSD. In Section \ref{sec:odp} we consider the functional outlier detection problem. In Theorem \ref{th:inPaper01} we present the result on which are based three new outlier detection methods which employ KFSD as depth function. In Section \ref{sec:simStudy} we report the results of our simulation study, whereas in Section \ref{sec:NOx} we perform outlier detection on the NO$_{x}$ data set. In Section \ref{sec:conc} we draw some conclusions. Finally, in the Appendix we report a sketch of the proof of Theorem \ref{th:inPaper01}.
\section{THE KERNELIZED FUNCTIONAL SPATIAL DEPTH} \label{sec:kfsd} In functional spaces a depth measure has the purpose of measuring the degree of centrality of curves relative to the distribution of a functional random variable. Various functional depths have been proposed following two alternative approaches: a global approach, which implies that the depth of an observation depends equally on all the observations allowed by $P$ on $\mathbb{H}$, and a local approach, which instead makes depend the depth of an observation more on close than distant observations. Among the existing global-oriented depths there is the Fraiman and Muniz depth (FMD, \citeauthor{FraMun2001} \citeyear{FraMun2001}), the random Tukey depth (RTD, \citeauthor{CueNie2008} \citeyear{CueNie2008}), the integrated dual depth (IDD, \citeauthor{CueFra2009} \citeyear{CueFra2009}), the modified band depth (MBD, \citeauthor{LopRom2009} \citeyear{LopRom2009}) or the functional spatial depth (FSD, \citeauthor{ChaCha2014} \citeyear{ChaCha2014}). Proposals of local-oriented depths are instead the h-modal depth (HMD, \citeauthor{CueFebFra2006}\ \citeyear{CueFebFra2006}) or the kernelized functional spatial depth (KFSD, \citeauthor{SguGalLil2014}\ \citeyear{SguGalLil2014}).
In this paper we focus on KFSD. Before giving its definition, we recall the definition of the functional spatial depth (FSD, \citeauthor{ChaCha2014} \citeyear{ChaCha2014}). Let $\mathbb{H}$ be an infinite-dimensional Hilbert space, then for $x \in \mathbb{H}$ and the functional random variable $Y \in \mathbb{H}$, FSD of $x$ relative to $Y$ is given by
\begin{equation*}
FSD(x,Y) = 1 - \left\|\mathbb{E}\left[\frac{x-Y}{\left\|x-Y\right\|}\right]\right\|, \end{equation*}
\noindent where $\|\cdot\|$ is the norm inherited from the usual inner product in $\mathbb{H}$. For a $n$-size random sample of $Y$, i.e., $Y_{n} = \left\{y_{1}, \ldots, y_{n}\right\}$, the sample version of FSD has the following form:
\begin{equation} \label{eq:sampleFSD}
FSD(x,Y_{n}) = 1 - \frac{1}{n}\left\|\sum_{i=1}^{n}\frac{x-y_{i}}{\left\|x-y_{i}\right\|}\right\|. \end{equation}
As mentioned before, FSD is a global-oriented depth and KFSD is a local version of it. KFSD is obtained writing \eqref{eq:sampleFSD} in terms of inner products and then replacing the inner product function with a positive definite and stationary kernel function. This replacement exploits the relationship
\begin{equation}\label{eq:kappa_phi} \kappa(x,y) = \langle\phi(x), \phi(y)\rangle, \quad x, y \in \mathbb{H}, \end{equation}
\noindent where $\kappa$ is the kernel $\kappa: \mathbb{H} \times \mathbb{H} \rightarrow \mathbb{R}$, $\phi$ is the embedding map $\phi: \mathbb{H} \rightarrow \mathbb{F}$ and $\mathbb{F}$ is a feature space. Indeed, a definition of KFSD in terms of $\phi$ can be given, that is,
\begin{equation} \label{eq:popKFSD}
KFSD(x,Y) = 1 - \left\|\mathbb{E}\left[\frac{\phi(x)-\phi(Y)}{\left\|\phi(x)-\phi(Y)\right\|}\right]\right\|, \end{equation}
\noindent and it can be interpreted as a recoded version of $FSD(x,Y)$ since $KFSD(x,Y)=$ \linebreak $FSD(\phi(x),\phi(Y))$. The sample version of \eqref{eq:popKFSD} is given by
\begin{equation*}
KFSD(x,Y_{n}) = 1 - \frac{1}{n}\left\|\sum_{i=1}^{n}\frac{\phi(x)-\phi(y_{i})}{\left\|\phi(x)-\phi(y_{i})\right\|}\right\|. \end{equation*}
\noindent Then, standard calculations (see Appendix) and \eqref{eq:kappa_phi} allow to provide an alternative expression of $KFSD(x,Y_{n})$, in this case in terms of $\kappa$:
\begin{gather} KFSD(x, Y_{n}) = 1 - \nonumber \\ \label{eq:sampleKFSD} \frac{1}{n} \left(\sum_{\substack{i,j=1; \\ y_{i} \neq x; y_{j} \neq x}}^{n}\frac{\kappa(x,x)+\kappa(y_{i},y_{j})-\kappa(x,y_{i})-\kappa(x,y_{j})}{\sqrt{\kappa(x,x)+\kappa(y_{i},y_{i})-2\kappa(x,y_{i})}\sqrt{\kappa(x,x)+\kappa(y_{j},y_{j})-2\kappa(x,y_{j})}}\right)^{1/2}, \end{gather}
\noindent Note that \eqref{eq:sampleKFSD} only requires the choice of $\kappa$, and not of $\phi$, which can be left implicit. As $\kappa$ we use the Gaussian kernel function given by
\begin{equation} \label{eq:ker}
\kappa(x,y) = \exp\left(-\frac{\|x-y\|^2}{\sigma^2}\right), \end{equation}
\noindent where $x, y \in \mathbb{H}$. In turn, \eqref{eq:ker} depends on the norm function inherited by the functional Hilbert space where data are assumed to lie, and on the bandwidth $\sigma$. Regarding $\sigma$, we initially consider 9 different $\sigma$, each one equal to 9 different percentiles of the empirical distribution of $\left\{\|y_{i}-y_{j}\|, y_{i}, y_{j} \in Y_{n}\right\}$. The first percentile is 10\%, and by increments of 10 we obtain the ninth percentile, i.e., 90\%. Note that the lower $\sigma$, the more local the approach, and therefore the percentiles that we use cover different degrees of KFSD-based local approaches: strongly (e.g., 20\%), moderately (e.g., 50\%) and weakly (e.g., 80\%) local approaches. In Section \ref{sec:simStudy} we present a method to select $\sigma$ in outlier detection problems.
In general, since any functional depth measures the degree of centrality or extremality of a given curve relative to a distribution or a sample, outliers are expected to have low depth values. More in particular, in presence of low magnitude, shape or partial outliers, an approach based on the use of a local depth like KFSD may help in detecting outliers. To illustrate this fact, we present the following example: first, we generated 100 data sets of size 50 from a mixture of two stochastic processes, one for normal curves and one for high magnitude outliers, with the probability that a curve is an outlier equal to 0.05. Second, we generated a group of 100 data sets from a mixture which produces shape outliers. Finally, we generated a group of 100 data sets from a mixture which produces partial outliers. In Figure \ref{fig:globVSloc} we report a contaminated data set for each mixture.
\begin{figure}
\caption{Examples of contaminated data sets: high magnitude contamination (top), shape contamination (middle) and partial contamination (bottom). The solid curves are normal curves and the dashed curves are outliers}
\label{fig:globVSloc}
\end{figure}
Let $n_{out,j}, j=1, \ldots, 100$, be the number of outliers generated in the $j$th data set. For each data set and functional depth, it is desirable to assign the $n_{out,j}$ lowest depth values to the $n_{out,j}$ generated outliers. For each mixture and generated data set, we recorded how many times the depth of an outlier is among the $n_{out,j}$ lowest values. As depth functions, we considered five global depths (FMD, RTD, IDD, MBD and FSD) and two local depths (HMD and KFSD). The results reported in Table \ref{tab:example} show that for all the functional depths the ranking of high magnitude outliers is an easier task than the ranking of shape and partial outliers. However, while the ranking of high magnitude outliers is reasonably good in different cases, e.g., for the local KFSD (94.87\%) and the global RTD (90.17\%), the ranking of shape and partial outliers is markedly better with local depths (shape: 86.72\% for KFSD and 85.47\% for HMD; partial: 82.03\% for KFSD and 81.25\% for HMD) than with the best global depths (shape: FSD with 39.06\%; partial: FSD with 46.48\%). These results suggest that, selecting a proper threshold, KFSD can isolate well outliers.
\begin{table}[!htbp] \captionsetup{justification=justified,width=0.5\textwidth} \caption{Percentages of times a depth assigns a value among the $n_{out,j}$ lowest ones to an outlier. Types of outliers: high magnitude, shape and partial.} \centering \scalebox{.75}{
\begin{tabular}{l|ccccc|cc} \hline type of depths & \multicolumn{5}{c}{global depths} \vline & \multicolumn{2}{c}{local depths} \\ \hline depths & FMD & RTD & IDD & MBD & FSD & HMD & KFSD \\ \hline high magnitude outliers & 86.32 & 90.17 & 81.62 & 69.23 & 68.80 & 85.47 & 94.87 \\ shape outliers & 7.81 & 33.59 & 38.67 & 12.11 & 39.06 & 85.94 & 86.72 \\ partial outliers & 18.75 & 44.53 & 34.77 & 19.14 & 46.48 & 81.25 & 82.03\\ \hline \end{tabular}} \label{tab:example} \end{table}
\section{OUTLIER DETECTION FOR FUNCTIONAL DATA} \label{sec:odp} The outlier detection problem can be described as follows: let $Y_{n}= \left\{y_{1}, \ldots, y_{n}\right\}$ be a sample generated from a mixture of two functional random variables in $\mathbb{H}$, one for normal curves and one for outliers, say $Y_{nor}$ and $Y_{out}$, respectively. Let $Y_{mix}$ be a mixture, i.e.,
\begin{equation} \label{eq:mix} Y_{mix} = \left\{
\begin{array}{ll}
Y_{nor}, & \mbox{with probability } 1-\alpha, \\
Y_{out}, & \mbox{with probability } \alpha,
\end{array} \right. \end{equation}
\noindent where $\alpha \in [0,1]$ is the contamination probability (usually, a value rather close to 0). The curves composing $Y_{n}$ are all unlabeled, and the goal of the analysis is to decide whether each curve is a normal curve or an outlier.\\ \indent KFSD is a functional extension of the kernelized spatial depth for multivariate data (KSD) proposed by \cite{CheDanPenBar2009}, who also proposed a KSD-based outlier detector that we generalize to KFSD: for a given data set $Y_{n}$ generated from $Y_{mix}$ and $t \in [0,1]$, the KFSD-based outlier detector for $x \in \mathbb{H}$ is given by
\begin{equation} \label{eq:g_KFSD_no_b} g(x,Y_{n}) = \left\{
\begin{array}{ll}
1, & \mbox{if } KFSD(x,Y_{n}) \leq t, \\
0, & \mbox{if } KFSD(x,Y_{n}) > t,
\end{array} \right. \end{equation}
\noindent where $t$ is a threshold which allows to discriminate between outliers (i.e., $g(x,Y_{n})=1$) and normal curves (i.e., $g(x,Y_{n})=0$), and it is a parameter that needs to be set.
For the multivariate case, KSD-based outlier detection is carried under different scenarios. One of them consists in an outlier detection problem where two samples are available and the threshold $t$ is selected by controlling the probability that normal observations are classified as outliers, i.e., the false alarm probability (FAP). The selection criterion is based on a result providing a KSD-based probabilistic upper bound on the FAP which depends on $t$. Then, the threshold for KSD is provided by the maximum value of $t$ such that the upper bound does not exceed a given desired FAP. We extend this result to KFSD:
\begin{theorem}\label{th:inPaper01} Let $Y_{n_{Y}} = \left\{y_{i}, \ldots, y_{n_{Y}}\right\}$ and $Z_{n_{Z}} = \left\{z_{i}, \ldots, z_{n_{Z}}\right\}$ be i.\ i.\ d.\ samples generated from the unknown mixture of random variables $Y_{mix} \in \mathbb{H}$ described by (\ref{eq:mix}), with $\alpha > 0$. Let $g(\cdot,Y_{n_{Y}})$ be the outlier detector defined in (\ref{eq:g_KFSD_no_b}). Fix $\delta \in (0,1)$ and suppose that $\alpha \leq r$ for some $r \in [0,1]$. For a new random element $x$ generated from $Y_{nor}$, the following inequality holds with probability at least $1-\delta$:
\begin{equation} \label{eq:ineThe01}
\mathbb{E}_{x | Y_{n_{Y}}}\left[g(x,Y_{n_{Y}})\right] \leq \frac{1}{1-r} \left[\frac{1}{n_{Z}} \sum_{i=1}^{n_{Z}} g\left(z_{i},Y_{n_{Y}}\right) + \sqrt{\frac{\ln{1/\delta}}{2 n_{Z}}} \right], \end{equation}
\noindent where $\mathbb{E}_{x | Y_{n_{Y}}}$ refers to the expected value of $x$ for a given $Y_{n_{Y}}$. \end{theorem}
The proof of Theorem \ref{th:inPaper01} is presented in the Appendix. Recall that the FAP is the probability that a normal observation $x$ is classified as outlier. For the elements of Theorem \ref{th:inPaper01}, $\mathrm{Pr}_{x | Y_{n_{Y}}}\left(g(x,Y_{n_{Y}})=1\right)$ is the FAP. Moreover,
\begin{equation*}
\mathrm{Pr}_{x | Y_{n_{Y}}}\left(g(x,Y_{n_{Y}})=1\right) = \mathbb{E}_{x | Y_{n_{Y}}}\left[g(x,Y_{n_{Y}})\right]. \end{equation*}
\noindent Therefore, the probabilistic upper bound of Theorem \ref{th:inPaper01} applies also to the FAP.
It is worth noting that the application of Theorem \ref{th:inPaper01} requires to observe two samples, circumstance rather uncommon in classical outlier detection problems, in which usually a single sample generated from an unknown mixture of random variables is available. For this reason, we propose a solution which allows to use Theorem \ref{th:inPaper01} in presence of a unique sample. Note that the general idea behind holds also in the multivariate framework, and therefore it would enable to perform KSD-based outlier detection when only a $\mathbb{R}^{d}$-sample is available.
In the functional context, our solution consists in setting $Y_{n_{Y}} = Y_{n}$ and in obtaining $Z_{n_{Z}}$ by resampling with replacement from $Y_{n}$. Note that by doing this, and for sufficiently large values of $n_{Z}$, we also obtain that the effect of $\delta$ on the probabilistic upper bound drastically reduces. Concerning $r$, that is the upper bound for the unknown contamination probability $\alpha$, a true range between 0 and 0.1 appears to be appropriate to cover most of the situations found in practice. Regarding the resampling procedure to obtain $Z_{n_{Z}}$, we consider three different schemes, all of them with replacement. Since we deal with potentially contaminated data sets, besides simple resampling, we also consider two robust KFSD-based resampling procedures inspired by the work of \cite{FebGalGon2008}. The three resampling schemes that we consider are:
\begin{compactenum} \item\label{it:sim} Simple resampling.
\item\label{it:tri} KFSD-based trimmed resampling: once $KFSD(y_{i},Y_{n}), i=1,\ldots,n$ are obtained, it is possible to identify the $\ceil*{\alpha_{T}}\%$ least deepest curves, for a certain $0 < \alpha_{T} < 1$ usually close to 0, that we advise to set equal to $r$. These least deep curves are deleted from the sample, and simple resampling is carried out with the remaining curves.
\item\label{it:wei} KFSD-based weighted resampling: once $KFSD(y_{i},Y_{n}), i=1,\ldots,n$ are obtained, weighted resampling is carried out with weights $w_{i} = KFSD(y_{i},Y_{n})$. \end{compactenum}
\noindent All the above procedures generate samples with some repeated curves. However, in a preliminary stage of our study we observed that it is preferable to work with $Z_{n_{Z}}$ composed of non-repeated curves. To obtain such samples, we add a common smoothing step to the previous three resampling schemes.
To describe the smoothing step, first recall that each curve in $Y_{n}$ is in practice observed at a discretized and finite set of domain points, and that the sets may differ from one curve to another. For this reason, the estimation of $Y_{n}$ at a common set of $m$ equidistant domain points may be required. Let $\left(y_{i}(s_{1}), \ldots,y_{i}(s_{m})\right)$ be the observed or estimated $m$-dimensional equidistant discretized version of $y_{i}$, $\Sigma_{Y_{n}}$ be the covariance matrix of the discretized form of $Y_{n}$ and $\gamma$ be a smoothing parameter. Consider a zero-mean Gaussian process whose discretized form has $\gamma\Sigma_{Y_{n}}$ as covariance matrix. Let $\left(\zeta(s_{1}), \ldots, \zeta(s_{m})\right)$ be a discretized realization of the previous Gaussian process. Consider any of the previous three resampling procedures and assume that at the $j$th trial, $j=1, \ldots, n_{Z}$, the $i$th curve in $Y_{n}$ has been sampled. Then, the discretized form of the $j$th curve in $Z_{n_{Z}}$ would be given by $\left(z_{j}(s_{1}), \ldots, z_{j}(s_{m})\right) = \left(y_{i}(s_{1})+\zeta(s_{1}), \ldots, y_{i}(s_{m})+\zeta(s_{m})\right)$, or, in functional form, by $z_{j} = y_{i} + \zeta$. Therefore, combining each resampling scheme with this smoothing step, we provide three different approximate ways to obtain $Z_{n_{Z}}$, and we refer to them as $smo$, $tri$ and $wei$, respectively. Then, for fixed $\delta$, $r$ and desired FAP, the threshold $t$ for \eqref{eq:g_KFSD_no_b} is selected as the maximum value of $t$ such that the right-hand side of \eqref{eq:ineThe01} does not exceed the desired FAP. Let $t^{*}$ be the selected threshold, which is then used in \eqref{eq:g_KFSD_no_b} to compute $g\left(y_{i}, Y_{n}\right)$, $i=1, \ldots, n$. If $g\left(y_{i}, Y_{n}\right) = 1$, $y_{i}$ is detected as outlier. To summarize, we provide three KFSD-based outlier detection procedures and we refer to them as KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$ depending on how $Z_{n_{Z}}$ is obtained ($smo$, $tri$ and $wei$, respectively; recall that $Y_{n_{Y}}=Y_{n}$). As competitors of the proposed procedures, we consider the methods mentioned in Section \ref{sec:intro} that we now describe.\\ \indent \cite{SunGen2011} proposed a depth-based functional boxplot and an associated outlier detection rule based on the ranking of the sample curves that MBD provides. The ranking is used to define a sample central region, that is, the smallest band containing at least half of the deepest curves. The non-outlying region is defined inflating the central region by 1.5 times. Curves that do not belong completely to the non-outlying region are detected as outliers. The original functional boxplot is based on the use of MBD as depth, but clearly any functional depth can be used. Another contribution of this paper is the study of the performances of the outlier detection rule associated to the functional boxplot (from now on, FBP) when used together with the battery of functional depths mentioned in Section \ref{sec:kfsd}.\\ \indent \cite{FebGalGon2008} proposed two depth-based outlier detection procedures that select a threshold for FMD, HMD or IDD by means of two alternative robust smoothed bootstrap procedures whose single bootstrap samples are obtained using the above described $tri$ and $wei$, respectively. At each bootstrap sample, the 1\% percentile of empirical distribution of the depth values is obtained, say $p_{0.01}$. If $B$ is the number of bootstrap samples, $B$ values of $p_{0.01}$ are obtained. Each method selects as cutoff $c$ the median of the collection of $p_{0.01}$ and, using $c$ as threshold, a first outlier detection is performed. If some curves are detected as outliers, they are deleted from the sample, and the procedure is repeated until no more outliers are found (note that $c$ is computed only in the first iteration). We refer to these methods as B$_{tri}$ and B$_{wei}$, and also in this case we evaluate these procedures using all the functional depths mentioned in Section \ref{sec:kfsd}.\\ \indent Finally, we also consider two procedures proposed by \cite{HynSha2010} that are not based on the use of a functional depth. Both are based on the first two robust functional principal components scores and on two different graphical representations of them. The first proposal is the outlier detection rule associated to the functional bagplot (from now on, FBG), which works as follows: obtain the bivariate robust scores and order them using the multivariate halfspace depth (\citeauthor{Tuk1975} \citeyear{Tuk1975}). Define an inner region by considering the smallest region containing at least the 50\% of the deepest scores, and obtain a non-outlying region by inflating the inner region by 2.58 times. FBG detects as outliers those curves whose scores are outside the non-outlying region. Note that the scores-based regions and outliers allow to draw a bivariate bagplot, which produces a functional bagplot once it is mapped onto the original functional space. The second proposal is related to a different graphical tool, the high density region boxplot (from now on, we refer to its associated outlier detection rule as FHD). In this case, once obtained the scores, perform a bivariate kernel density estimation. Define the $(1-\beta)$-high density region (HDR), $\beta \in (0,1)$, as the region of scores with coverage probability equal to $(1-\beta)$. FHD detects as outliers those curves whose scores are outside the $(1-\beta)$-HDR. In this case, it is possible to draw a bivariate HDR boxplot which can be mapped onto a functional version, thus providing the functional HDR boxplot.
\section{SIMULATION STUDY} \label{sec:simStudy} After introducing KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$, their competitors (FBP, B$_{tri}$, B$_{wei}$, FBG and FHD), as well as seven different functional depths (FMD, HMD, RTD, IDD, MBD, FSD and KFSD), in this section we carry out a simulation study to evaluate the performances of the different methods. For FBP, B$_{tri}$ and B$_{wei}$, we use the notation procedure+depth: for example, FBP+FMD refers to the method obtained by using FBP together with FMD.\\ \indent To perform our simulation study, we consider six models: all of them generate curves according to the mixture of random variables $Y_{mix}$ described by \eqref{eq:mix}. The first three mixture models (MM1, MM2 and MM3) share $Y_{nor}$, with curves generated by
\begin{equation} \label{eq:Y_nor_123} y(s) = 4s + \epsilon(s), \end{equation}
\noindent where $s \in [0,1]$ and $\epsilon(s)$ is a zero-mean Gaussian component with covariance function given by
\begin{equation*} \mathbb{E}(\epsilon(s),\epsilon(s^{\prime})) = 0.25 \exp{(-(s-s^{\prime})^2)}, \quad s,s^{\prime} \in [0, 1]. \end{equation*}
\noindent Also the remaining three mixture models (MM4, MM5 and MM6) share $Y_{nor}$, but, in this case, the curves are generated by
\begin{equation} \label{eq:Y_nor_456} y(s) = u_{1}\sin s + u_{2}\cos s, \end{equation}
\noindent where $s \in [0,2\pi]$ and $u_{1}$ and $u_{2}$ are observations from a continuous uniform random variable between 0.05 and 0.15.\\ \indent MM1, MM2 and MM3 differ in their $Y_{out}$ components. Under MM1, the outliers are generated by
\begin{equation*} y(s) = 8s - 2 + \epsilon(s), \end{equation*}
\noindent which produces outliers of both shape and low magnitude nature. Under MM2, the outliers are generated by adding to \eqref{eq:Y_nor_123} an observation from a $N(0,1)$, and as result outliers are more irregular than normal curves. Finally, under MM3, the outliers are generated by
\begin{equation*} y(s) = 4\exp(s) + \epsilon(s), \end{equation*}
\noindent which produces curves that are normal in the first part of the domain, but that become exponentially outlying.\\ \indent Similarly, MM4, MM5 and MM6 differ in their $Y_{out}$ components. Under MM4, the outliers are generated replacing $u_{2}$ with $u_{3}$ in \eqref{eq:Y_nor_456}, where $u_{3}$ is an observation from a continuous uniform random variable between 0.15 and 0.17. This change produces partial low magnitude outliers in the first and middle part of the domain of the curves. Under MM5, the outliers are generated by adding to \eqref{eq:Y_nor_456} an observation from a $N(0,\left(\frac{0.1}{2}\right)^{2})$, and they turn out to be more irregular curves. Finally, under MM6, the outliers are generated by
\begin{equation} \label{eq:Y_out_6} y(s) = u_{1}\sin s + \exp\left(\frac{0.69s}{2\pi}\right) u_{4} \cos s, \end{equation}
\noindent where $u_{4}$ is an observation from a continuous uniform random variable between 0.1 and 0.15. As MM3, MM6 allows outliers that are normal in the first part of the domain and become outlying with an exponential pattern. In Figure \ref{fig:simStudy} we report a simulated data set with at least one outlier for each mixture model.
\begin{figure}
\caption{Examples of contaminated functional data sets generated by MM1, MM2, MM3, MM4, MM5 and MM6. Solid curves are normal curves and dashed curves are outliers.}
\label{fig:simStudy}
\end{figure}
The details of the simulation study are the following: for each mixture model, we generated 100 data sets, each one composed of 50 curves. As mentioned above, for each single samples Theorem \ref{th:inPaper01} cannot be directly applied, and therefore KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$ represent practical alternatives. Two values of the contamination probability $\alpha$ were considered: 0.02 and 0.05. All curves were generated using a discretized and finite set of 51 equidistant points in the domain of each mixture model ($[0,1]$ for MM1, MM2 and MM3; $[0, 2\pi]$ for MM4, MM5 and MM6) and the discretized versions of the functional depths were used.\\ \indent In relation with the methods and the functional depths that we consider in the study, their specifications are described next:
\begin{compactenum}
\item FBP when used with FMD, HMD, RTD, IDD, MBD, FSD and KFSD: regarding FBP, as reported in Section \ref{sec:odp}, the central region is built considering the 50\% deepest curves and the non-outlying region by inflating by 1.5 times the central region. Regarding the depths, for HMD, we follow the recommendations in \cite{FebGalGon2008}, that is, $\mathbb{H}$ is the $L^2$ space, $\kappa(x,y) = \frac{2}{\sqrt{2\pi}}\exp\left(-\frac{\|x-y\|^2}{2h^2}\right)$ and $h$ is equal to the 15\% percentile of the empirical distribution of $\left\{\|y_{i}-y_{j}\|, y_{i}, y_{j} \in Y_{n}\right\}$. For RTD and IDD, we work with 50 projections in random Gaussian directions. For MBD, we consider bands defined by two curves. For FSD and KFSD, we assume that the curves lie in the $L^{2}$ space. Moreover, in KFSD we set $\sigma$ equal to a moderately local percentile (50\%) of the empirical distribution of $\left\{\|y_{i}-y_{j}\|, y_{i}, y_{j} \in Y_{n}\right\}$.
\item B$_{tri}$ and B$_{wei}$ when used with FMD, HMD, RTD, IDD, MBD, FSD and KFSD: $\gamma=0.05$, $B=200$, $\alpha_{T}=\alpha$. Regarding the depths, we use the specifications reported for FBP.
\item FBG: as reported in Section \ref{sec:odp}, the central region is built considering the 50\% deepest bivariate robust functional principal component scores and the non-outlying region by inflating by 2.58 times the central region.
\item FHD: $\beta = \alpha$.
\item KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$: $n_{Y}=n=50$ (since $Y_{n_{Y}}=Y_{n}$), $\gamma=0.05$, $\alpha_{T}=\alpha$ (only for KFSD$_{tri}$), $n_{Z}=6n$, $\delta=0.05$, $r=\alpha$, desired FAP = 0.10. Moreover, as introduced in Section \ref{sec:kfsd}, for these methods we consider 9 percentiles to set $\sigma$ in KFSD. The way in which we propose to choose the most suitable percentile for outlier detection is presented below. \end{compactenum}
In supervised classification, the availability of training curves with known class memberships makes possible the definition of some natural procedures to set $\sigma$ for KFSD, such as cross-validation. However, in an outlier detection problem, it is common to have no information whether curves are normal or outliers. Therefore, training procedures are not immediately available.\\ \indent We propose to overcome this drawback by obtaining a ``training sample of peripheral curves", and then choosing the percentile that ranks better these peripheral curves as final percentile for KFSD in KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$. We now describe this procedure, which is based on $J$ replications. Let $Y_{n}$ be the functional data set on which outlier detection has to be done and let $Y_{(n)} = \left\{y_{(1)}, \ldots, y_{(n)}\right\}$ be the depth-based ordered version of $Y_{n}$, where $y_{(1)}$ and $y_{(n)}$ are the curves with minimum and maximum depth, respectively. The steps to obtain a set of peripheral curves are the following:
\begin{compactenum}[I.] \item\label{ite:i} Let $\left\{p_{1},\ldots,p_{K}\right\}$ be the set of percentiles in use (in our case, as explained in Section \ref{sec:kfsd}, $p_{k}=(10k)\%$, $k \in \left\{1,\ldots,K=9\right\})$, and choose randomly a percentile from the set. For the $j$th replication, $j \in \left\{1,\ldots,J\right\}$, denote the selected percentile as $p^{j}$. We use $J=20$ in the rest of the paper.
\item\label{ite:ii} Using $p^{j}$, compute $KFSD_{p^{j}}(y_{i},Y_{n})$, $i=1, \ldots, n$, where the notation $KFSD_{p^{j}}(\cdot,\cdot)$ is used to describe what percentile is used. For the $j$th replication, denote the KFSD-based ordered curves as $y_{(1),j}, \ldots, y_{(n),j}$.
\item\label{ite:iii} Take $y_{(1),j}, \ldots, y_{(l_{j}),j}$, where $l_{j} \sim Bin(n,\frac{1}{n})$. Apply the smoothing step described in Section \ref{sec:odp} to these curves. For the smoothing step, we use $\Sigma_{Y_{n}}$ and $\gamma=0.05$. For the $j$th replication, denote the peripheral and smoothed curves as $y_{(1),j}^{*}, \ldots, y_{(l_{j}),j}^{*}$.
\item\label{ite:iv} Repeat $J$ times steps \ref{ite:i}.-\ref{ite:iii}. to obtain a collection of $L=\sum_{j=1}^{J}l_{j}$ peripheral curves, say $Y_{L}$ (for an example, see Figure \ref{fig:trainPeri}). \end{compactenum}
\begin{figure}
\caption{Example of a training sample of peripheral curves for a contaminated data set generated by MM1 with $\alpha=0.05$. The solid and shaded curves are the original curves (both normal and outliers). The dashed curves are the peripheral curves to use as training sample.}
\label{fig:trainPeri}
\end{figure}
Next, $Y_{L}$ acts as training sample according to the following steps: for each $y_{(i),j}^{*} \in Y_{L}$, ($i \leq l_{j}$), and $p_{k} \in \left\{p_{1},\ldots,p_{K}\right\}$, compute $KFSD_{p_{k}}(y_{(i),j}^{*},Y_{-(i),j})$, where $Y_{-(i),j} = Y_{n} \setminus \left\{y_{(i),j}\right\}$. At the end, a $L \times K$ matrix is obtained, say $D_{LK}=\left\{d_{lk}\right\}_{\substack{l=1,\ldots,L\\k=1,\ldots,K}}$, whose $k$th column is composed of the KFSD values of the $L$ training peripheral curves when the $k$th percentile is employed in KFSD. Next, let $r_{lk}$ be the rank of $d_{lk}$ in the vector $\left(KFSD_{p_{k}}(y_{1},Y_{n}),\ldots,KFSD_{p_{k}}(y_{n},Y_{n}),\right.$ $\left. d_{lk} \right)$, e.g., $r_{lk}$ is equal to 1 or $n+1$ if $d_{lk}$ is the minimum or the maximum value in the vector, respectively. Let $R_{LK}$ be the result of this transformation of $D_{LK}$, and sum the elements of each column, obtaining a $K$-dimensional vector, say $\mathbf{R}_{K}$. Since the goal is to assign ranks as low as possible to the peripheral curves, choose the percentile associated to the minimum value of $\mathbf{R}_{K}$. When a tie is observed, we break it randomly.\\ \indent The comparison among methods is performed in terms of both correct and false outlier detection percentages, which are reported in Tables \ref{tab:MM01_I}-\ref{tab:MM06_I}. To ease the reading of the tables, for each model and $\alpha$, we report in bold the 5 best methods in terms of correct outlier detection percentage (c).\footnote{In presence of tie, the method with lower false outlier detection percentage (f) is preferred.} For each model, if a method is among the 5 best ones for both contamination probabilities $\alpha$, we report its label in bold.
\begin{table}[!htbp] \parbox{.475\textwidth}{ \captionsetup{justification=justified,width=0.475\textwidth} \caption{MM1, $\alpha=\left\{0.02,0.05\right\}$. Correct (c) and false (f) outlier detection percentages of FBP, B$_{tri}$, B$_{wei}$, FBG, FHD, KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$.} \centering \scalebox{.60}{
\begin{tabular}{l|cc|cc} \hline
& \multicolumn{2}{c}{$\alpha=0.02$} & \multicolumn{2}{|c}{$\alpha=0.05$}\\ \hline & c & f & c & f\\ \hline FBP+FMD & 44.34 & 1.23 & 43.86 & 0.73\\ FBP+HMD & \textbf{74.53} & 0.94 & 72.81 & 0.61\\ FBP+RTD & 61.32 & 0.57 & 63.16 & 0.31\\ FBP+IDD & 55.66 & 0.61 & 61.84 & 0.34\\ FBP+MBD & 49.06 & 1.33 & 50.44 & 0.69\\ FBP+FSD & 62.26 & 0.67 & 61.84 & 0.40\\ FBP+KFSD & 66.04 & 0.86 & \textbf{74.12} & 0.44\\ \hline B$_{tri}$+FMD & 0.00 & 0.98 & 0.00 & 1.82 \\ B$_{tri}$+HMD & 66.98 & 1.45 & 57.89 & 1.47 \\ B$_{tri}$+RTD & 10.38 & 1.78 & 14.91 & 1.76 \\ B$_{tri}$+IDD & 10.38 & 1.55 & 11.84 & 1.74 \\ B$_{tri}$+MBD & 0.00 & 0.51 & 0.00 & 1.49 \\ B$_{tri}$+FSD & 2.83 & 0.76 & 5.26 & 1.17 \\ B$_{tri}$+KFSD & 70.75 & 1.43 & 58.77 & 1.40 \\ \hline B$_{wei}$+FMD & 0.00 & 1.29 & 0.00 & 1.49 \\ B$_{wei}$+HMD & 71.70 & 1.02 & 47.37 & 0.65 \\ B$_{wei}$+RTD & 13.21 & 2.04 & 13.60 & 1.78 \\ B$_{wei}$+IDD & 17.92 & 1.82 & 10.53 & 1.55 \\ B$_{wei}$+MBD & 0.00 & 1.08 & 0.00 & 1.40 \\ B$_{wei}$+FSD & 2.83 & 1.39 & 3.95 & 1.07 \\ B$_{wei}$+KFSD & 61.32 & 0.88 & 55.26 & 0.48 \\ \hline \textbf{FBG} & \textbf{100.00} & 2.27 & \textbf{97.81} & 2.37\\ \hline FHD & 48.11 & 1.00 & 73.68 & 2.77\\ \hline \textbf{KFSD$_{smo}$} & \textbf{89.62} & 4.50 & \textbf{85.09} & 2.58 \\ \textbf{KFSD$_{tri}$} & \textbf{89.62} & 4.92 & \textbf{92.11} & 4.40 \\ \textbf{KFSD$_{wei}$} & \textbf{97.17} & 9.44 & \textbf{96.93} & 6.54 \\ \hline \end{tabular}} \label{tab:MM01_I} }
\parbox{.475\textwidth}{ \captionsetup{justification=justified,width=0.475\textwidth} \caption{MM2, $\alpha=\left\{0.02,0.05\right\}$. Correct (c) and false (f) outlier detection percentages of FBP, B$_{tri}$, B$_{wei}$, FBG, FHD, KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$.} \centering \scalebox{.60}{
\begin{tabular}{l|cc|cc} \hline
& \multicolumn{2}{c}{$\alpha=0.02$} & \multicolumn{2}{|c}{$\alpha=0.05$}\\ \hline & c & f & c & f\\ \hline FBP+FMD & 99.09 & 1.08 & \textbf{96.39} & 0.84\\ FBP+HMD & 96.36 & 0.96 & 96.39 & 0.88\\ FBP+RTD & \textbf{99.09} & 0.61 & 94.78 & 0.25\\ FBP+IDD & 99.09 & 0.70 & 95.18 & 0.38\\ FBP+MBD & 99.09 & 1.06 & \textbf{96.39} & 0.82\\ FBP+FSD & \textbf{99.09} & 0.57 & 94.78 & 0.36\\ FBP+KFSD & 98.18 & 0.63 & 93.98 & 0.36\\ \hline B$_{tri}$+FMD & 0.00 & 1.06 & 0.00 & 1.96 \\ B$_{tri}$+HMD & 95.45 & 1.51 & \textbf{96.79} & 1.68 \\ B$_{tri}$+RTD & 1.82 & 1.92 & 6.83 & 2.61 \\ B$_{tri}$+IDD & 5.45 & 1.60 & 7.63 & 1.94 \\ B$_{tri}$+MBD & 0.00 & 0.98 & 0.40 & 2.10 \\ B$_{tri}$+FSD & 4.55 & 1.06 & 5.22 & 1.62 \\ B$_{tri}$+KFSD & 97.27 & 1.60 & 95.18 & 1.52 \\ \hline B$_{wei}$+FMD & 0.00 & 1.27 & 0.00 & 1.52 \\ B$_{wei}$+HMD & 95.45 & 1.02 & 86.35 & 0.36 \\ B$_{wei}$+RTD & 5.45 & 2.21 & 8.43 & 2.84 \\ B$_{wei}$+IDD & 7.27 & 1.49 & 9.64 & 2.36 \\ B$_{wei}$+MBD & 0.00 & 1.27 & 0.40 & 1.49 \\ B$_{wei}$+FSD & 8.18 & 1.39 & 4.02 & 1.37 \\ B$_{wei}$+KFSD & 95.45 & 0.96 & 79.52 & 0.51 \\ \hline FBG & 8.18 & 3.07 & 4.42 & 2.95\\ \hline FHD & 7.27 & 1.88 & 12.45 & 5.66\\ \hline KFSD$_{smo}$ & \textbf{100.00} & 3.91 & 95.18 & 2.76 \\ \textbf{KFSD$_{tri}$} & \textbf{100.00} & 5.19 & \textbf{97.99} & 4.84 \\ \textbf{KFSD$_{wei}$} & \textbf{100.00} & 9.20 & \textbf{99.60} & 6.48 \\ \hline \end{tabular}} \label{tab:MM02_I} } \end{table}
\begin{table}[!htbp] \parbox{.475\textwidth}{ \captionsetup{justification=justified,width=0.475\textwidth} \caption{MM3, $\alpha=\left\{0.02,0.05\right\}$. Correct (c) and false (f) outlier detection percentages of FBP, B$_{tri}$, B$_{wei}$, FBG, FHD, KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$.} \centering \scalebox{.60}{
\begin{tabular}{l|cc|cc} \hline
& \multicolumn{2}{c}{$\alpha=0.02$} & \multicolumn{2}{|c}{$\alpha=0.05$}\\ \hline & c & f & c & f\\ \hline FBP+FMD & 65.69 & 0.92 & 49.19 & 0.97\\ \textbf{FBP+HMD} & \textbf{89.22} & 0.57 & \textbf{85.89} & 0.63\\ FBP+RTD & 86.27 & 0.45 & 76.61 & 0.34\\ FBP+IDD & 79.41 & 0.51 & 70.56 & 0.38\\ FBP+MBD & 74.51 & 0.88 & 59.27 & 0.84\\ FBP+FSD & 79.41 & 0.51 & 73.79 & 0.42\\ \textbf{FBP+KFSD} & \textbf{89.22} & 0.57 & \textbf{83.06} & 0.59\\ \hline B$_{tri}$+FMD & 2.94 & 0.73 & 5.24 & 1.22 \\ B$_{tri}$+HMD & 57.84 & 1.57 & 53.63 & 1.56 \\ B$_{tri}$+RTD & 15.69 & 1.76 & 21.37 & 1.81 \\ B$_{tri}$+IDD & 20.59 & 1.65 & 20.56 & 1.70 \\ B$_{tri}$+MBD & 0.98 & 1.06 & 3.23 & 1.54 \\ B$_{tri}$+FSD & 16.67 & 1.14 & 17.34 & 1.22 \\ B$_{tri}$+KFSD & 57.84 & 1.63 & 49.19 & 1.52 \\ \hline B$_{wei}$+FMD & 2.94 & 1.10 & 3.63 & 0.84 \\ B$_{wei}$+HMD & 60.78 & 1.25 & 42.74 & 0.76 \\ B$_{wei}$+RTD & 15.69 & 1.92 & 17.34 & 1.73 \\ B$_{wei}$+IDD & 23.53 & 1.33 & 14.52 & 1.22 \\ B$_{wei}$+MBD & 0.98 & 1.29 & 2.82 & 1.14 \\ B$_{wei}$+FSD & 15.69 & 1.16 & 12.10 & 0.84 \\ B$_{wei}$+KFSD & 56.86 & 1.12 & 41.53 & 0.67 \\ \hline FBG & 86.27 & 2.65 & \textbf{78.63} & 1.73\\ \hline FHD & 49.02 & 1.02 & 65.73 & 2.88\\ \hline KFSD$_{smo}$ & \textbf{89.22} & 3.90 & 73.79 & 2.95 \\ \textbf{KFSD$_{tri}$} & \textbf{90.20} & 4.63 & \textbf{83.47} & 4.71 \\ \textbf{KFSD$_{wei}$} & \textbf{97.06} & 8.96 & \textbf{90.32} & 6.50 \\ \hline \end{tabular}} \label{tab:MM03_I} }
\parbox{.475\textwidth}{ \captionsetup{justification=justified,width=0.475\textwidth} \caption{MM4, $\alpha=\left\{0.02,0.05\right\}$. Correct (c) and false (f) outlier detection percentages of FBP, B$_{tri}$, B$_{wei}$, FBG, FHD, KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$.} \centering \scalebox{.60}{
\begin{tabular}{l|cc|cc} \hline
& \multicolumn{2}{c}{$\alpha=0.02$} & \multicolumn{2}{|c}{$\alpha=0.05$}\\ \hline & c & f & c & f\\ \hline FBP+FMD & 1.02 & 0.00 & 0.00 & 0.00\\ FBP+HMD & 6.12 & 0.00 & 1.60 & 0.02\\ FBP+RTD & 0.00 & 0.00 & 0.00 & 0.00\\ FBP+IDD & 0.00 & 0.00 & 0.00 & 0.00\\ FBP+MBD & 0.00 & 0.00 & 0.00 & 0.00\\ FBP+FSD & 0.00 & 0.00 & 0.00 & 0.00\\ FBP+KFSD & 2.04 & 0.00 & 0.80 & 0.00\\ \hline B$_{tri}$+FMD & 60.20 & 0.16 & \textbf{47.60} & 0.11 \\ B$_{tri}$+HMD & 41.84 & 0.04 & 18.80 & 0.17 \\ B$_{tri}$+RTD & 54.08 & 1.16 & 34.80 & 0.82 \\ B$_{tri}$+IDD & 55.10 & 1.02 & 37.20 & 0.59 \\ \textbf{B$_{tri}$+MBD} & \textbf{64.29} & 0.14 & \textbf{46.40} & 0.13 \\ B$_{tri}$+FSD & \textbf{68.37} & 0.14 & 45.60 & 0.08 \\ B$_{tri}$+KFSD & 58.16 & 0.20 & 28.00 & 0.13 \\ \hline B$_{wei}$+FMD & 51.02 & 0.12 & 23.60 & 0.00 \\ B$_{wei}$+HMD & 38.78 & 0.06 & 10.80 & 0.02 \\ B$_{wei}$+RTD & 37.76 & 0.49 & 25.20 & 0.15 \\ B$_{wei}$+IDD & 43.88 & 0.67 & 28.00 & 0.42 \\ B$_{wei}$+MBD & 56.12 & 0.10 & 25.20 & 0.02 \\ B$_{wei}$+FSD & 63.27 & 0.06 & 29.20 & 0.00 \\ B$_{wei}$+KFSD & 58.16 & 0.12 & 21.20 & 0.00 \\ \hline FBG & 9.18 & 0.53 & 6.80 & 1.09\\ \hline FHD & 51.02 & 1.02 & 37.60 & 4.34\\ \hline \textbf{KFSD$_{smo}$} & \textbf{87.76} & 2.16 & \textbf{50.00} & 1.24 \\ \textbf{KFSD$_{tri}$} & \textbf{91.84} & 3.00 & \textbf{64.80} & 2.91 \\ \textbf{KFSD$_{wei}$} & \textbf{95.92} & 5.08 & \textbf{62.00} & 3.35 \\ \hline \end{tabular}} \label{tab:MM04_I} } \end{table}
\begin{table}[!htbp] \parbox{.475\textwidth}{ \captionsetup{justification=justified,width=0.475\textwidth} \caption{MM5, $\alpha=\left\{0.02,0.05\right\}$. Correct (c) and false (f) outlier detection percentages of FBP, B$_{tri}$, B$_{wei}$, FBG, FHD, KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$.} \centering \scalebox{.60}{
\begin{tabular}{l|cc|cc} \hline
& \multicolumn{2}{c}{$\alpha=0.02$} & \multicolumn{2}{|c}{$\alpha=0.05$}\\ \hline & c & f & c & f\\ \hline FBP+FMD & 55.56 & 0.00 & 54.00 & 0.00\\ FBP+HMD & 66.67 & 0.00 & 68.40 & 0.04\\ FBP+RTD & 57.58 & 0.00 & 54.40 & 0.00\\ FBP+IDD & 52.53 & 0.00 & 56.00 & 0.00\\ FBP+MBD & 55.56 & 0.00 & 55.20 & 0.00\\ FBP+FSD & 55.56 & 0.00 & 55.60 & 0.00\\ FBP+KFSD & 60.61 & 0.00 & 59.20 & 0.00\\ \hline B$_{tri}$+FMD & 3.03 & 0.18 & 2.80 & 0.44 \\ \textbf{B$_{tri}$+HMD} & \textbf{97.98} & 0.12 & \textbf{92.40} & 0.11 \\ B$_{tri}$+RTD & 16.16 & 1.06 & 20.00 & 1.03 \\ B$_{tri}$+IDD & 18.18 & 1.06 & 16.00 & 1.07 \\ B$_{tri}$+MBD & 2.02 & 0.16 & 3.20 & 0.32 \\ B$_{tri}$+FSD & 29.29 & 0.18 & 27.20 & 0.23 \\ B$_{tri}$+KFSD & 93.94 & 0.24 & \textbf{92.40} & 0.21 \\ \hline B$_{wei}$+FMD & 3.03 & 0.29 & 2.40 & 0.23 \\ B$_{wei}$+HMD & \textbf{93.94} & 0.08 & 73.60 & 0.00 \\ B$_{wei}$+RTD & 15.15 & 1.06 & 17.60 & 1.12 \\ B$_{wei}$+IDD & 25.25 & 0.98 & 20.00 & 0.99 \\ B$_{wei}$+MBD & 2.02 & 0.20 & 3.60 & 0.21 \\ B$_{wei}$+FSD & 29.29 & 0.14 & 21.60 & 0.13 \\ B$_{wei}$+KFSD & 83.84 & 0.08 & 72.00 & 0.04 \\ \hline FBG & 0.00 & 1.02 & 0.40 & 0.04\\ \hline FHD & 4.04 & 1.96 & 12.80 & 5.64\\ \hline \textbf{KFSD$_{smo}$} & \textbf{98.99} & 1.82 & \textbf{94.00} & 0.44 \\ \textbf{KFSD$_{tri}$} & \textbf{98.99} & 2.61 & \textbf{98.00} & 2.11 \\ \textbf{KFSD$_{wei}$} & \textbf{100.00} & 4.61 & \textbf{98.40} & 2.11 \\ \hline \end{tabular}} \label{tab:MM05_I} }
\parbox{.475\textwidth}{ \captionsetup{justification=justified,width=0.475\textwidth} \caption{MM6, $\alpha=\left\{0.02,0.05\right\}$. Correct (c) and false (f) outlier detection percentages of FBP, B$_{tri}$, B$_{wei}$, FBG, FHD, KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$.} \centering \scalebox{.60}{
\begin{tabular}{l|cc|cc} \hline
& \multicolumn{2}{c}{$\alpha=0.02$} & \multicolumn{2}{|c}{$\alpha=0.05$}\\ \hline & c & f & c & f\\ \hline FBP+FMD & 48.42 & 0.00 & 44.19 & 0.00 \\ FBP+HMD & 60.00 & 0.18 & \textbf{62.92} & 0.00\\ FBP+RTD & 55.79 & 0.00 & 54.68 & 0.00\\ FBP+IDD & 46.32 & 0.00 & 40.07 & 0.00\\ FBP+MBD & 48.42 & 0.00 & 45.69 & 0.00\\ FBP+FSD & 52.63 & 0.00 & 52.43 & 0.00\\ FBP+KFSD & 57.89 & 0.00 & 56.93 & 0.00\\ \hline B$_{tri}$+FMD & 29.47 & 0.22 & 33.71 & 0.32 \\ B$_{tri}$+HMD & \textbf{71.58} & 0.24 & 45.69 & 0.15 \\ B$_{tri}$+RTD & 35.79 & 0.82 & 31.09 & 0.51 \\ B$_{tri}$+IDD & 38.95 & 0.37 & 35.96 & 0.74 \\ B$_{tri}$+MBD & 29.47 & 0.24 & 31.09 & 0.32 \\ B$_{tri}$+FSD & 52.63 & 0.20 & 43.82 & 0.19 \\ B$_{tri}$+KFSD & \textbf{71.58} & 0.22 & 50.56 & 0.21 \\ \hline B$_{wei}$+FMD & 23.16 & 0.24 & 19.48 & 0.08 \\ B$_{wei}$+HMD & 68.42 & 0.12 & 35.96 & 0.00 \\ B$_{wei}$+RTD & 38.95 & 0.69 & 24.34 & 0.51 \\ B$_{wei}$+IDD & 33.68 & 0.59 & 25.09 & 0.40 \\ B$_{wei}$+MBD & 24.21 & 0.18 & 19.85 & 0.13 \\ B$_{wei}$+FSD & 47.37 & 0.16 & 27.72 & 0.08 \\ B$_{wei}$+KFSD & 66.32 & 0.12 & 44.19 & 0.06 \\ \hline FBG & 17.89 & 0.02 & 14.98 & 0.06\\ \hline FHD & 52.63 & 1.02 & \textbf{61.80} & 2.85\\ \hline \textbf{KFSD$_{smo}$} & \textbf{91.58} & 2.08 & \textbf{71.16} & 0.95 \\ \textbf{KFSD$_{tri}$} & \textbf{93.68} & 2.69 & \textbf{82.02} & 2.49 \\ \textbf{KFSD$_{wei}$} & \textbf{96.84} & 4.69 & \textbf{83.15} & 2.75 \\ \hline \end{tabular}} \label{tab:MM06_I} } \end{table}
The results in Tables \ref{tab:MM01_I}-\ref{tab:MM06_I} show that:
\begin{compactenum} \item KFSD$_{tri}$ and KFSD$_{wei}$ are always among the 5 best methods. KFSD$_{smo}$ is among the 5 best methods 10 times over 12, but when its performance is not among the 5 best, it is neither extremely far from the fifth method (MM2, $\alpha=0.05$: 95.18\% against 96.79\%; MM3, $\alpha=0.05$: 73.79\% against 78.63\%). The rest of the methods are among the 5 best procedures at most 4 times over 12 (FBP+HMD and B$_{tri}$+HMD).
\item Regarding MM5 and MM6, our procedures are clearly the best options in terms of correct detection (c), and in the following order: KFSD$_{wei}$, KFSD$_{tri}$ and KFSD$_{smo}$. In general, this pattern is observed overall the simulation study. Note that for MM6 and $\alpha=0.02$ we observe the best relative performances of KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$, i.e., 91.58\%, 93.68\% and 96.84\%, respectively, against 71.58\% of the fourth best method (B$_{wei}$+KFSD), that is, we observe at least 20\% differences.
\item About MM3, KFSD$_{wei}$ is clearly the best method in terms of correct detection, however at the price of having a greater false detection (f). This is in general the main weak point of KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$. As for correct detection, we observe a overall pattern in our methods in false detection, but in an opposite way, indicating therefore a trade-off between c and f. Relative high false detection percentages are however something expected in KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$ since these methods are based on the definition of a desired false alarm probability, which is equal to 10\% in this study. Concerning MM2, we observe similar results to MM3, but in this case the performances of the best methods in terms of correct detection (KFSD$_{smo}$, KFSD$_{tri}$, KFSD$_{wei}$, FBP-based methods and B$_{tri}$ when used with local depths) are closer to each other.
Finally, there are only 2 cases in which a competitor outperforms all our methods, and it is FBAG under MM1 and both $\alpha$. However, this procedure does not show a behavior as stable as KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$ do. Indeed, FBAG shows poor performances under other models, e.g., MM2. \end{compactenum}
In summary, the above results and remarks show that the proposed KFSD-based procedures are the best methods in detecting outliers for the considered models. Moreover, KFSD$_{tri}$ seems the most reasonable choice to balance the mentioned trade-off between c and f. In terms of correct detection, KFSD$_{wei}$ slightly outperforms KFSD$_{tri}$, which however shows very good and stable performances when compared with the remaining methods. In terms of false detection, KFSD$_{tri}$ considerably improves on KFSD$_{wei}$, especially under some models (e.g., see MM2).
\begin{figure}
\caption{Boxplots of the percentiles selected in the training steps of the simulation study for KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$.}
\label{fig:simBoxPlot}
\end{figure}
In Figure \ref{fig:simBoxPlot} we report a series of boxplots summarizing which percentiles have been selected in the training steps for KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$, and the following general remarks can be made. First, MM6 is the mixture model for which lower percentiles have been selected, and it is also a scenario in which our methods considerably outperform their competitors. The need for a more local approach for MM6-data may explain the two observed facts about this mixture model. Second, lower and more local percentiles have been chosen for mixture models with nonlinear mean functions (MM4, MM5 and MM6) than for mixture models with linear mean functions (MM1, MM2 and MM3). Finally, the percentiles selected by means of the proposed training procedure seem to vary among data sets. However, except for MM3 and $\alpha=0.02$, at least for half of the data sets a percentile not greater than the median has been chosen, which implies at most a moderately local approach.
\section{REAL DATA STUDY: NITROGEN OXIDES (NO$_{x}$) DATA} \label{sec:NOx} Besides simulated data, we consider a real data set which consists in nitrogen oxides (NO$_{x}$) emission level daily curves measured every hour close to an industrial area in Poblenou (Barcelona) and is available in the \texttt{R} package \texttt{fda.usc} (\citeauthor{FebOvi2012} \citeyear{FebOvi2012}). Outlier detection on this data set was first performed by \cite{FebGalGon2008} where these authors proposed B$_{tri}$ and B$_{wei}$. We carry on their study considering more methods and depths.
NO$_{x}$ are one of the most important pollutants, and it is important to identify outlying trajectories because these curves may compromise any statistical analysis or be of special interest for further analysis and to implement environmental political countermeasures. The NO$_{x}$ levels that we consider were measured in $\mu g/m^{3}$ every hour of every day for the period 23/02/2005-26/06/2005. Only for 115 days of the period are available the 24 measurements, and these are the days that compose the final NO$_{x}$ data set. Moreover, following \cite{FebGalGon2008}, since the NO$_{x}$ data set includes working as well as nonworking days, it seems more appropriate to consider a first sample of 76 working day curves (from now on, W) and a second sample of 39 nonworking day curves (from now on, NW). Both W and NW are showed in Figure \ref{fig:NOx}, where it is possible to appreciate at least two facts that justify the split of the original data set. First, the W curves have in general higher values than NW curves, which can be explained by the greater activity of motor vehicles and industries in a city like Barcelona during working days. Second, both data sets contain curves with peaks, but for W curves the peaks occur roughly around 7-8 a.m. and during many days, whereas for NW curves the peaks occur later and during few days, which again can be explained by the differences between Barcelona's economic activity of working and nonworking days.
\begin{figure}
\caption{NO$_{x}$ data: working (top) and non working (bottom) day curves.}
\label{fig:NOx}
\end{figure}
At first glance, each data set may contain outliers, especially partial outliers in the form of abnormal peaks, and therefore a local depth approach by means of KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$ appears to be a good strategy to detect outliers. Besides them, we do outlier detection with all the methods used in Section \ref{sec:simStudy}. For all the procedures we use the same specifications as in Section \ref{sec:simStudy}, and we assume $\alpha=0.05$. For each method, we report the labels of the curves detected as outliers in Table \ref{tab:NOx} and we highlight these curves in Figure \ref{fig:NOxIC}.
\begin{table}[!htbp] \captionsetup{justification=justified,width=0.5\textwidth} \caption{NO$_{x}$ data, Working and Nonworking data sets. Curves detected as outliers by FBP, B$_{tri}$, B$_{wei}$, FBG, FHD, KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$.} \centering \scalebox{.64}{
\begin{tabular}{l|c|c} \hline & working days & nonworking w days\\ \hline & \multicolumn{2}{c}{detected outliers}\\ \hline FBP+FMD & - & -\\ FBP+HMD & 12, 16, 37 & 5, 7, 20, 21\\ FBP+RTD & 37 & 20\\ FBP+IDD & - & 5, 7, 20\\ FBP+MBD & - & -\\ FBP+FSD & 37 & -\\ FBP+KFSD & 12, 16, 37 & 5, 7, 20, 21\\ \hline B$_{tri}$+FMD & 16, 37 & 7\\ B$_{tri}$+HMD & 14, 16, 37 & 7, 20\\ B$_{tri}$+RTD & 16 & 7, 20\\ B$_{tri}$+IDD & 16, 37 & 7, 20\\ B$_{tri}$+MBD & 16, 37 & 7\\ B$_{tri}$+FSD & 14, 16, 37 & -\\ B$_{tri}$+KFSD & 12, 14, 16, 37 & 7, 20\\ \hline B$_{wei}$+FMD & 16 & 7, 20\\ B$_{wei}$+HMD & 16, 37 & 7, 20\\ B$_{wei}$+RTD & 16 & -\\ B$_{wei}$+IDD & 16, 37 & 20\\ B$_{wei}$+MBD & 16 & 7\\ B$_{wei}$+FSD & 16, 37 & -\\ B$_{wei}$+KFSD & 16, 37 & 7, 20\\ \hline FBG & 16, 37 & -\\ \hline FHD & 12, 14, 16, 37 & 7, 20\\ \hline KFSD$_{smo}$ & 14, 16, 37 & 7, 20, 21\\ KFSD$_{tri}$ & 12, 14, 16, 37 & 7, 20, 21\\ KFSD$_{wei}$ & 11, 12, 13, 14, 15, 16, 37, 38 & 7, 20, 21\\ \hline \end{tabular}} \label{tab:NOx} \end{table}
\begin{figure}
\caption{NO$_{x}$ data set, curves detected as outliers in Table \ref{tab:NOx}: working (top) and nonworking (bottom) days.}
\label{fig:NOxIC}
\end{figure}
Concerning W, most of the methods detect as outlier day 37, the Friday at the beginning of the long weekend due to Labor's day in 2005 and whose curve shows a partial outlying behavior before noon and at the end of the day. Another day detected as outlier by many methods is day 16, another Friday before a long weekend, Easter holidays in 2005, and whose curve has the highest morning peak. In addition to curves 16 and 37, KFSD$_{smo}$ detects as outlier curve 14, as other nine methods do, recognizing a seemingly outlying pattern in early hours of the day. Additionally, KFSD$_{tri}$ includes among the outliers also day 12, which may be atypical because of its behavior in early afternoon. Note that both day 12 and 14 are in the week before the above-mentioned Easter holidays. Finally, KFSD$_{wei}$ detects as outliers the greatest number of curves. This last result may appear exaggerated, but all the curves that are outliers according to KFSD$_{wei}$ seem to have some partial deviations from the majority of curves. For example, day 13, whose curve is considered normal by the rest of the procedures, shows a peak at end of the day. Similar peaks can be observed also in other curves detected as outliers by other methods (e.g., days 16 and 37), which means that it may be occurring a masking effect to day 13's detriment, and only KFSD$_{wei}$ points out this possibly outlying feature of the curve. Regarding the training step for KFSD to set $\sigma$, it gives as result the 70\% percentile. Observing the first graph of Figure \ref{fig:NOx}, it can be noticed that some curves have a likely outlying behavior, and this may be the reason why a weakly local approach for KFSD may be adequate enough.
In the case of NW, some methods detect no curves as outliers (e.g., all the FSD-based methods), exclusively three FBP-based methods flag day 5 as outlier, whereas days 7, 20 and 21 are detected as outliers by our methods as well as others. Note that day 7 is the Saturday before Easter and days 20 and 21 are Labor's day eve and the same Labor's day. Days 7 and 20, which have two peaks, at the beginning and end of the day, are also flagged by other twelve and eight methods, respectively, while day 21, which shows a single peak in the first hours of the day, is considered atypical by only two other methods, which happen to be local (FBP+HMD and FBP+KFSD). This last result may be connected with what has been observed at the KFSD training step for selecting the percentile, i.e., the selection of the 30\% percentile. Therefore, KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$ work with a strongly local percentile, and their results partially resemble the ones of the previously mentioned local techniques.
\section{CONCLUSIONS} \label{sec:conc} This paper proposes to tackle outlier detection in functional samples using the kernelized functional spatial depth as a tool. In Theorem \ref{th:inPaper01} we presented a probabilistic result allowing to set a KFSD-threshold to identify outliers, but in practice it is necessary to observe two samples to apply Theorem \ref{th:inPaper01}. To overcome this practical limitation, we proposed KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$ which are methods that can be applied when a unique functional sample is available and are based on both a probabilistic approach and smoothed resampling techniques.
We also proposed a new procedure to set the bandwidth $\sigma$ of KFSD that is based on obtaining training samples by means of smoothed resampling techniques. The general idea behind this procedure can be applied to other functional depths or methods with parameters that need to be set.
We investigated the performances of KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$ by means of a simulation study. We focused on challenging scenarios with low magnitude, shape and partial outliers instead of high magnitude outliers. The results support our proposals. Along the simulation study, KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$ attained the largest correct detection performances in most of the analyzed setups, but in some cases they paid a price in terms of false detection. However, KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$ work with a given desired false alarm probability, and therefore higher false detection percentages than their competitors are due to the inherent structure of the methods. We also observed a trade-off between c and f for KFSD$_{smo}$, KFSD$_{tri}$ and KFSD$_{wei}$, and a clear pattern. For these reasons in our opinion KFSD$_{tri}$ should be preferred to KFSD$_{smo}$ or KFSD$_{wei}$ since it performs extremely well in terms of correct detection, while it has lower false detection percentages than KFSD$_{wei}$. Concerning the remaining methods, there are competitors that in few scenarios outperformed our methods. However, in these few cases the differences are not great, and in addition these competitors are not stable across the considered scenarios.
Furthermore, we also showed that our procedures can be applied in environmental contexts with an example where the goal was to detect outlying NO$_{x}$ curves to identify days possibly characterized by abnormal pollution levels.
To conclude, we present two possible future research lines. First, since KFSD is a depth whose local approach is in part based on the choice of the kernel function, it would be interesting to explore how the choice of different kernels affects the behavior of KFSD. Moreover, each kernel will depend on a bandwidth and a norm. For the selection of the bandwidth, we used a criterion based on the study of the empirical distribution of the sample distances, but alternatives should be investigated, for example an adaptation of the so-called Silverman's rule (\citeauthor{Sil1986} \citeyear{Sil1986}) for selecting the bandwidth of a kernel-based functional depth such as KFSD. For the choice of the norm, a sensitivity study would help in understanding how important is the functional space assumption. Second, since outlier detection can be seen as a special case of cluster analysis (it is a cluster problem with maximum two clusters, and one of them with size much smaller than the other,even 0), a natural step ahead in our research may be the definition of KFSD-based cluster analysis procedures.
{ \section*{ACKNOWLEDGMENTS} The authors would like to thank the editor in chief, the associate editor and an anonymous referee for their helpful comments. This research was partially supported by Spanish Ministry of Science and Innovation grant ECO2011-25706 and by Spanish Ministry of Economy and Competition grant ECO2012-38442. }
\appendix \section{Appendix}
\subsection{From $FSD(x, Y_{n})$ to $KFSD(x, Y_{n})$} \label{sec:app01} To show how to pass from $FSD(x, Y_{n})$ in \eqref{eq:sampleFSD} to $KFSD(x, Y_{n})$ in \eqref{eq:sampleKFSD}, we first show that $FSD(x, Y_{n})$ can be expressed in terms of inner products. We present this result for $n=2$. The norm in \eqref{eq:sampleFSD} can be written as
\begin{equation*} \begin{array}{ll}
\left\|\sum_{i}^{2}\frac{x-y_{i}}{\|x-y_{i}\|}\right\|^{2} &= \left\|\frac{x-y_{1}}{\|x-y_{1}\|}+\frac{x-y_{2}}{\|x-y_{2}\|}\right\|^{2}\\
&= \left\|\frac{x-y_{1}}{\sqrt{\langle x,x\rangle+\langle y_{1},y_{1}\rangle-2\langle x,y_{1}\rangle}}+\frac{x-y_{2}}{\sqrt{\langle x,x\rangle+\langle y_{2},y_{2}\rangle-2\langle x,y_{2}\rangle}}\right\|^{2} \end{array} \end{equation*}
\noindent Let $\delta_{1}=\sqrt{\langle x,x\rangle+\langle y_{1},y_{1}\rangle-2\langle x,y_{1}\rangle}$ and $\delta_{2}=\sqrt{\langle x,x\rangle+\langle y_{2},y_{2}\rangle-2\langle x,y_{2}\rangle}$. Then,
\begin{equation*} \begin{array}{ll}
\left\|\sum_{i}^{2}\frac{x-y_{i}}{\|x-y_{i}\|}\right\|^{2} &= \left\|\frac{x-y_{1}}{\delta_{1}}+\frac{x-y_{2}}{\delta_{2}}\right\|^{2} \\
&= \left\|\frac{x-y_{1}}{\delta_{1}}\right\|+\left\|\frac{x-y_{2}}{\delta_{2}}\right\| + \frac{2}{\delta_{1}\delta_{2}}\langle x-y_{1},x-y_{2}\rangle \\ &= 2 + \frac{2}{\delta_{1}\delta_{2}}(\langle x,x\rangle+\langle y_{1},y_{2}\rangle-\langle x,y_{1}\rangle-\langle x,y_{2}\rangle)\\ &= \sum_{i,j=1}^{2}\frac{\langle x,x\rangle+\langle y_{i},y_{j}\rangle-\langle x,y_{i}\rangle-\langle x,y_{j}\rangle}{\delta_{i}\delta_{j}}, \end{array} \end{equation*}
\noindent and apply the embedding map $\phi$ to all the observations of the last expression. According to \eqref{eq:kappa_phi}, this is equivalent to substitute the inner product function with a positive definite and stationary kernel function $\kappa$, which explains the definition of $KFSD(x, Y_{n})$ in \eqref{eq:sampleKFSD} for $n=2$. The generalization of this result to $n>2$ is straightforward.
\subsection{Proof of Theorem \ref{th:inPaper01}} \label{sec:app02} As explained in Section \ref{sec:odp}, Theorem \ref{th:inPaper01} is a functional extension of a result derived by \cite{CheDanPenBar2009} for KSD, and since they are closely related, next we report a sketch of the proof of Theorem \ref{th:inPaper01}. The proof for KSD is mostly based on an inequality known as \citeauthor{Mcd1989}'s inequality (\citeauthor{Mcd1989} \citeyear{Mcd1989}), which also applies to general probability spaces, and therefore to functional Hilbert spaces. We report this inequality in the next lemma:
\begin{lemma}\label{le:01} \textbf{(\citeauthor{Mcd1989} \citeyear{Mcd1989} [1.2])} Let $\Omega_{1}, \ldots, \Omega_{n}$ be probability spaces. Let $\mathbf{\Omega} = \prod_{j=1}^{n} \Omega_{j}$ and let $X: \mathbf{\Omega} \rightarrow \mathbb{R}$ be a random variable. For any $j \in \left\{1, \ldots, n\right\}$, let $(\omega_{1}, \ldots, \omega_{j}, \ldots,$ $\omega_{n})$ and $\left(\omega_{1}, \ldots, \hat{\omega}_{j}, \ldots, \omega_{n}\right)$ be two elements of $\mathbf{\Omega}$ that differ only in their $j$th coordinates. Assume that $X$ is uniformly difference-bounded by $\{c_j\}$, that is, for any $j \in \left\{1, \ldots, n\right\}$,
\begin{equation} \label{eq:lem01_c_j}
\left|X\left(\omega_{1}, \ldots, \omega_{j}, \ldots, \omega_{n}\right)-X\left(\omega_{1}, \ldots, \hat{\omega}_{j}, \ldots, \omega_{n}\right)\right| \leq c_{j}. \end{equation}
\noindent Then, if $\mathbb{E}[X]$ exists, for any $\tau > 0$
\begin{equation*} \mathrm{Pr}\left(X-\mathbb{E}[X] \geq \tau \right) \leq \exp \left(\frac{-2\tau^2}{\sum_{j=1}^{n} c_{j}^{2}}\right). \end{equation*}
\end{lemma}
In order to apply Lemma \ref{le:01} to our problem, define
\begin{equation} \label{eq:lem01_X_omega}
X(z_{1},\ldots,z_{n_{Z}}) = - \frac{1}{n_{Z}}\sum_{i=1}^{n_{Z}}g(z_{i},Y_{n_{Y}}|Y_{n_{Y}}), \end{equation}
\noindent whose expected value is given by
\begin{equation} \label{eq:lem01_EX}
\mathbb{E}[X] = \mathbb{E}_{z_{i}|Y_{n_{Y}}}\left[- \frac{1}{n_{Z}}\sum_{i=1}^{n_{Z}}g(z_{i},Y_{n_{Y}}|Y_{n_{Y}})\right] = - \mathbb{E}_{z_{1}|Y_{n_{Y}}}\left[g(z_{1},Y_{n_{Y}}|Y_{n_{Y}})\right]. \end{equation}
Now, for any $j \in \left\{1, \ldots, n_{Z}\right\}$ and $\hat{z}_{j} \in \mathbb{H}$, the following inequality holds
\begin{equation*}
\left|X(z_{1},\ldots,z_{j},\ldots,z_{n_{Z}}) - X(z_{1},\ldots,\hat{z}_{j},\ldots,z_{n_{Z}})\right| \leq \frac{1}{n_{Z}}, \end{equation*}
\noindent and it provides assumption \eqref{eq:lem01_c_j} of Lemma \ref{le:01}. Therefore, for any $\tau > 0$
\begin{equation*}
\mathrm{Pr}\left(\mathbb{E}_{z_{1}|Y_{n_{Y}}}\left[g(z_{1},Y_{n_{Y}}|Y_{n_{Y}})\right] - \frac{1}{n_{Z}}\sum_{i=1}^{n_{Z}}g(z_{i},Y_{n_{Y}}|Y_{n_{Y}}) \geq \tau\right) \leq \exp\left(-2n_{Z}\tau^{2}\right), \end{equation*}
\noindent and by the law of total probability
\begin{equation*} \begin{array}{l}
\mathbb{E}\left[\mathrm{Pr}\left(\mathbb{E}_{z_{1}|Y_{n_{Y}}}\left[g(z_{1},Y_{n_{Y}}|Y_{n_{Y}})\right] - \frac{1}{n_{Z}}\sum_{i=1}^{n_{Z}}g(z_{i},Y_{n_{Y}}|Y_{n_{Y}}) \geq \tau\right)\right] \\
= \mathrm{Pr}\left(\mathbb{E}_{z_{1}|Y_{n_{Y}}}\left[g(z_{1},Y_{n_{Y}})\right] - \frac{1}{n_{Z}}\sum_{i=1}^{n_{Z}}g(z_{i},Y_{n_{Y}}) \geq \tau\right) \leq \exp\left(-2n_{Z}\tau^{2}\right)\\ \end{array} \end{equation*}
Next, setting $\delta = \exp\left(-2n_{Z}\tau^{2}\right)$, and solving for $\tau$, the following result is obtained:
\begin{equation*} \tau = \sqrt{\frac{\ln 1/\delta}{2n_{Z}}}. \end{equation*}
\noindent Therefore,
\begin{equation} \label{eq:pr_z1}
\mathrm{Pr}\left(\mathbb{E}_{z_{1}|Y_{n_{Y}}}\left[g(z_{1},Y_{n_{Y}})\right] \leq \frac{1}{n_{Z}}\sum_{i=1}^{n_{Z}}g(z_{i},Y_{n_{Y}}) + \sqrt{\frac{\ln 1/\delta}{2n_{Z}}}\right) \geq 1-\delta. \end{equation}
However, Theorem \ref{th:inPaper01} provides a probabilistic upper bound for $\mathbb{E}_{x|Y_{n_{Y}}}\left[g(x,Y_{n_{Y}})\right]$. First, recall that $z_{1} \sim Y_{mix}$ and note that
\begin{equation*}
\mathbb{E}_{(z_{1} \sim Y_{mix})|Y_{n_{Y}}}\left[g\left(z_{1},Y_{n_{Y}}\right)\right] = (1-\alpha)\mathbb{E}_{(z_{1} \sim Y_{nor})|Y_{n_{Y}}}\left[g\left(z_{1},Y_{n_{Y}}\right)\right] + \alpha\mathbb{E}_{(z_{1} \sim Y_{out})|Y_{n_{Y}}}\left[g\left(z_{1},Y_{n_{Y}}\right)\right]. \end{equation*}
\noindent Then, since $\mathbb{E}_{(z_{1} \sim Y_{nor})|Y_{n_{Y}}}\left[g\left(z_{1},Y_{n_{Y}}\right)\right] = \mathbb{E}_{x|Y_{n_{Y}}}\left[g\left(x,Y_{n_{Y}}\right)\right]$, for $\alpha>0$,
\begin{equation} \label{eq:nor_mix_mix}
\mathbb{E}_{x|Y_{n_{Y}}}\left[g\left(x,Y_{n_{Y}}\right)\right] \leq \frac{1}{1-\alpha}\mathbb{E}_{(z_{1} \sim Y_{mix})|Y_{n_{Y}}}\left[g\left(z_{1},Y_{n_{Y}}\right)\right]. \end{equation}
\noindent Consequently, combining \eqref{eq:pr_z1} and \eqref{eq:nor_mix_mix}, and for $r \geq \alpha$, we obtain
\begin{equation*}
\mathrm{Pr}\left(\mathbb{E}_{x|Y_{n_{Y}}}\left[g(x,Y_{n_{Y}})\right] \leq \frac{1}{1-r}\left[\frac{1}{n_{Z}}\sum_{i=1}^{n_{Z}}g(z_{i},Y_{n_{Y}}) + \sqrt{\frac{\ln 1/\delta}{2n_{Z}}}\right]\right) \geq 1-\delta, \end{equation*}
\noindent which completes the proof.
\begin{flushright} $\blacksquare$ \end{flushright}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{A Class of Continued Radicals} \author{Costas J. Efthimiou} \date{} \maketitle
\begin{abstract} We compute the limits of a class of continued radicals extending the results of a previous note in which only periodic radicals of the class were considered. \end{abstract}
\section{Introduction.}
In \cite{Efthimiou} the author discussed the values for a class of periodic continued radicals of the form {\footnotesize \begin{equation}
a_0\sqrt{2+a_1\sqrt{2+a_2\sqrt{2+a_3\sqrt{2+\cdots}}}} ~, \label{eq:OurRadical} \end{equation} } where for some positive integer $n$, $$
a_{n+k} ~=~ a_k~,~~~k=0,1,2,\dots~, $$ and $$
a_k\in\{-1,+1\}~,~~~k=0,1,\dots,n-1~. $$ It was also shown that the radicals given by equation \eqref{eq:OurRadical} have limits two times the fixed points of the Chebycheff polynomials $T_{2^n}(x)$, thus unveiling an interesting relation between these topics.
In \cite{ZH}, the authors defined the set $S_2$ of all continued radicals of the form \eqref{eq:OurRadical} (with $a_0=1$) and they
investigated some of their properties by assuming that the limit of the radicals exists. In particular, they showed that all elements
of $S_2$ lie between 0 and 2, any two radicals cannot be equal to each other, and $S_2$ is uncountable.
My previous note hence partially bridged this gap but left unanswered the question `\textit{what are the limits if the radicals are not periodic?}' I answer the question in this note. The result is easy to establish, but I realized it only as I was reading the proof of my previous note. Such is the working of the mind!
\section{The Limits.}
Towards the desired result, I present the following lemma from \cite{Shklarsky}, also used in the periodic case, which is an extension of the well known trigonometric formulas of the angles $\pi/2^n$. \begin{lemma} For $a_i\in\{-1,1\}$, with $i=0,1,\dots,n-1$, we have that {\footnotesize $$
2\, \sin \left\lbrack \left( a_0+{a_0a_1\over2}+\dots+{a_0a_1\cdots a_{n-1}\over2^{n-1}} \right) {\pi\over4} \right\rbrack
~=~
a_0\sqrt{2+a_1\sqrt{2+a_2\sqrt{2+\dots+a_{n-1}\sqrt{2}}}}~. $$ } \end{lemma} The lemma is proved in \cite{Shklarsky} using induction.
According to this lemma, the partial sums of the continued radical \eqref{eq:OurRadical} are given by $$
x_n ~=~ 2\sin \left\lbrack \left( a_0+{a_0a_1\over2}+\dots+{a_0a_1\cdots a_{n-1}\over2^{n-1}} \right) {\pi\over4} \right\rbrack~. $$ The series \begin{equation*}
a_0+{a_0a_1\over2}+\dots+{a_0a_1\cdots a_{n-1}\over2^{n-1}} +\cdots \end{equation*} is absolutely convergent and thus it converges to some number $a$. Therefore, the original continued radical
converges to the real number $$
x ~=~ 2\sin{a\pi\over4}~. $$ We can find a concise formula for $x$. For this calculation it is more useful to use the products $$
P_m ~=~ \prod_{k=0}^{m} a_k~,~~~\text{for } m=0,1,2,\dots~, $$ which take the values $\pm1$. We will refer to these as partial parities. (When the pattern is periodic of period $n$ only the first $n$ parities $P_0, P_1, \dots, P_{n-1}$ are independent.) Using the notation with the partial parities, set \begin{eqnarray*}
a &=& P_0 + {P_1\over2} + {P_2\over 2^2} + \cdots+ {P_{n-1}\over 2^{n-1}} + {P_n\over 2^{n}} + \cdots ~. \end{eqnarray*}
We now define $$
Q_m ~=~ {1+P_m\over2}~. $$ Since $P_m\in\{-1,1\}$, it follows that $Q_m\in\{0,1\}$. Inversely, $P_m=2Q_m-1$. Thus \begin{equation*}
a ~=~ \sum_{m=0}^\infty {P_m\over 2^m}
~=~ \sum_{m=0}^\infty {Q_m\over 2^{m-1}} - \sum_{m=0}^\infty {1\over 2^m}
~=~ 4\, \sum_{m=0}^\infty {Q_m\over 2^{m+1}} - 2~. \end{equation*} Notice that the sum $$
Q ~=~ \sum_{m=0}^\infty {Q_m\over 2^{m+1}} $$
in the previous equation is the number $Q$ whose binary expression is $0.Q_0Q_1\cdots Q_{n-3}Q_{n-2}\cdots$. Therefore $a=4Q-2$.
In \cite{ZH}, the authors noticed that all continued radicals of the form \eqref{eq:OurRadical} (with $a_0=1$) are in one-to-one
correspondence with the set of decimals between 0 and 1 as written in binary notation (and that's how they determined that the set
$S_2$ is uncountable). But, with the above calculation, this correspondence is made deeper. It gives the limit of the radical
\eqref{eq:OurRadical} as follows $$
x ~=~ -2\cos \left( Q\pi \right) ~. $$ For example, if $a_k=1$ for all $k$, then also $Q_k=1$ for all $k$ and the number $Q=0.111111111\cdots$ written in the binary system is the number $Q=1$ in the decimal system; hence $x=2$. We thus recover the well known result {\footnotesize \begin{equation*}
2 ~=~ \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}} ~. \end{equation*} }
\section{Conclusion.}
Having found the limit of \eqref{eq:OurRadical}, the next obvious question is to determine the limit of the radical {\footnotesize \begin{equation*}
a_0\sqrt{y+a_1\sqrt{y+a_2\sqrt{y+a_3\sqrt{y+\cdots}}}} ~, \end{equation*} } for values of the variable $y$ that make the radical (and the limit) well defined. However, a direct application of the above method fails and so far a convenient variation has been elusive. Therefore, the limit of the last radical in the general case remains an open problem although it is known in at least two cases \cite{ZH}.
\noindent\textit{ Department of Physics, University of Central Florida, Orlando, FL 32816 \\ [email protected] }
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{On embedded hidden Markov models and\\particle Markov chain Monte Carlo methods} \author{Axel Finke\\Department of Statistical Science, University College London, UK.\\Arnaud Doucet\\Department of Statistics, Oxford University, UK.\\Adam M.~Johansen\\Department of Statistics, University of Warwick, UK.} \maketitle
\begin{abstract}
\noindent{}The \gls{EHMM} sampling method is a \gls{MCMC} technique for state inference in non-linear non-Gaussian state-space models which was proposed in \citet{Neal2003,Neal2004} and extended in \citet{ShestopaloffNeal2016}. An extension to Bayesian parameter inference was presented in \citet{ShestopaloffNeal2013}. An alternative class of \gls{MCMC} schemes addressing similar inference problems is provided by \gls{PMCMC} methods \citep{AndrieuDoucetHolenstein2009,AndrieuDoucetHolenstein2010}. All these methods rely on the introduction of artificial extended target distributions for multiple state sequences which, by construction, are such that one randomly indexed sequence is distributed according to the posterior of interest. By adapting the \glsdesc{MH} algorithms developed in the framework of \gls{PMCMC} methods to the \gls{EHMM} framework, we obtain novel \gls{PF}-type algorithms for state inference and novel \gls{MCMC} schemes for parameter and state inference. In addition, we show that most of these algorithms can be viewed as particular cases of a general \gls{PF} and \gls{PMCMC} framework. We compare the empirical performance of the various algorithms on low- to high-dimensional state-space models. We demonstrate that a properly tuned conditional \gls{PF} with `local' \gls{MCMC} moves proposed in \citet{ShestopaloffNeal2016} can outperform the standard conditional \gls{PF} significantly when applied to high-dimensional state-space models while the novel \gls{PF}-type algorithm could prove to be an interesting alternative to standard \glspl{PF} for likelihood estimation in some lower-dimensional scenarios. \end{abstract}
\glsresetall \glsunset{MCMC}
\section{Introduction}
Throughout this work, for concreteness, we will describe both \gls{PMCMC} and \gls{EHMM} methods in the context of performing inference in non-linear state-space models. However, we stress that those methods can be used to perform inference in other contexts.
Non-linear non-Gaussian state-space models constitute a popular class of time series models which can be described in the time-homogeneous case as follows --- throughout this paper we consider the time-homogeneous case, noting that the generalisation to time-inhomogeneous models is straightforward but notationally cumbersome. Let $\{x_t\}_{t \geq 1}$ be an $\mathcal{X}$-valued latent Markov process satisfying \begin{equation}
x_1 \sim \mu_{\theta}(\,\cdot\,) \quad \text{and} \quad x_{t}\vert (x_{t-1} = x) \sim f_{\theta}(\,\cdot\,| x), \text{ for $t \geq 2$}. \label{eq:modtransition} \end{equation} and let $\{y_t\}_{t \geq 1}$ be a sequence of $\mathcal{Y}$-valued observations which are conditionally independent given $\{x_t\}_{t \geq 1}$ and which satisfy \begin{equation}
y_t \vert (x_1,\dotsc, x_t = x, x_{t+1}, \dotsc )
\sim g_{\theta}(\,\cdot\,| x), \text{ for
$t \geq 1$}. \label{eq:modobs} \end{equation} Here $\theta\in\varTheta$ denotes the vector of parameters of the model.
Let $z_{i:j}$ denotes the components $(z_i,z_{i+1},\dotsc,z_j) $ of a generic sequence $\{z_t\}_{t \geq 1} $.\ Assume that we have access to a realization of the observations $Y_{1:T}=y_{1:T}$. If $\theta$ is known, inference about the latent states $x_{1:T}$ relies upon \begin{equation}
p_{\theta}(x_{1:T} \vert y_{1:T})
= \frac{p_{\theta}(x_{1:T}, y_{1:T})}{p_{\theta}(y_{1:T})}, \end{equation} where \begin{equation}
p_{\theta} (x_{1:T}, y_{1:T})
= \mu_{\theta}(x_1)
\prod_{t=2}^{T} f_{\theta}(x_{t}\vert x_{t-1})
\prod_{t=1}^{T} g_{\theta}(y_t \vert x_t) . \end{equation} When $\theta$ is unknown, to conduct Bayesian inference a prior density $p(\theta)$ is assigned to the parameters and inference proceeds via the joint posterior density \begin{equation}
p(x_{1:T},\theta \vert y_{1:T})
= p(\theta \vert y_{1:T}) p_{\theta}(x_{1:T}\vert y_{1:T}), \end{equation} where the marginal posterior distribution of the parameter satisfies \begin{equation}
p(\theta \vert y_{1:T}) \propto p(\theta) p_{\theta}(y_{1:T}), \end{equation} the likelihood $ p_{\theta}(y_{1:T})$ being given by \begin{equation}
p_{\theta}(y_{1:T}) =\int p_{\theta} (x_{1:T}, y_{1:T}) \mathrm{d}x_{1:T}. \end{equation}
Many algorithms have been proposed over the past twenty-five years to perform inference for this class of models; see \citet{Kantas2015} for a recent survey. We focus here on the \gls{EHMM} algorithm introduced in \citet{Neal2003,Neal2004} and on \gls{PMCMC} introduced in \cite{AndrieuDoucetHolenstein2009,AndrieuDoucetHolenstein2010}. Both classes of methods are fairly generic and do not require the state-space model under consideration to possess additional structural properties beyond \eqref{eq:modtransition} and \eqref{eq:modobs}. The \gls{EHMM} method has been recently extended in \citet{ShestopaloffNeal2013,ShestopaloffNeal2016} while extensions of \gls{PMCMC} have also been proposed in, among other works, \citet{Whiteley2010} and \citet{LindstenJordanSchon2014}. In particular, \citet{Whiteley2010} combined the conditional \gls{PF} algorithm of \cite{AndrieuDoucetHolenstein2009,AndrieuDoucetHolenstein2010} with a backward sampling step. We will denote the resulting algorithm as the conditional \gls{PF} with \gls{BS}.
Both \gls{EHMM} and \gls{PMCMC} methods rely upon sampling a population of $N$ particles for the state $x_{t}$ and introducing an extended target distribution over the resulting $N^{T}$ potential sequences $x_{1:T}$ such that one of the sequences selected uniformly at random is at equilibrium by construction. It was observed in \citet[p.~116]{LindstenSchon2013} that conditional \gls{PF} with \gls{BS} is reminiscent of the \gls{EHMM} method proposed in \citet{Neal2003,Neal2004} and some connections were made between some simple \gls{EHMM} methods and \gls{PMCMC} methods in \citet[pp.~82--87]{Finke2015} who also showed that both methods can be viewed as special cases of a much more general construction. However, to the best of our knowledge, the connections between the two classes of methods have never been investigated thoroughly. Indeed, such an analysis was deemed of interest in \citet{ShestopaloffNeal2014}, where we note that \gls{EHMM} methods are sometimes alternatively referred to as \emph{ensemble} \gls{MCMC} methods: \begin{quote}
``It would \ldots{} be interesting to compare the performance of the ensemble \gls{MCMC} method with the [\gls{PMCMC}]-based methods of \cite{AndrieuDoucetHolenstein2010} and also to see whether techniques used to improve [particle \gls{MCMC}] methods can be used to improve ensemble methods and vice versa.'' \end{quote} In this work, we characterize this relationship and show that it is possible to exploit the similarities between these methods to derive new inference algorithms. The relationship between the various classes of algorithms discussed in this work is shown in Figure~\ref{fig:relationship_between_algorithms}. The remainder of the paper is organized as follows.
Section \ref{sec:pmcmc} reviews some \gls{PMCMC} schemes, including the \gls{PMMH} algorithm and \gls{PG} samplers. We recall how the validity of these algorithms can be established by showing that they are standard \gls{MCMC} algorithms sampling from an extended target distribution. In particular, the \gls{PMMH} algorithm can be thought of as a standard \gls{MH} algorithm sampling from this extended target using a \gls{PF} proposal for the states. Likewise, the theoretical validity of the conditional \gls{PF} with \gls{BS} can be established by showing that it corresponds to a (``partially collapsed'' -- see \citet{VanDyk2008}) Gibbs sampler \citep{Whiteley2010}.
Section~\ref{sec:embeddedHMM} is devoted to the `original' \gls{EHMM} from \citet{Neal2003,Neal2004}. At the core of this methodology is an extended target distribution which shares common features with the \gls{PMCMC} target. We show that the \gls{EHMM} method can be reinterpreted as a collapsed Gibbs sampling procedure for this target. This provides an alternative proof of validity of this algorithm. More interestingly, it is possible to come up with an original \gls{MH} scheme to sample from this extended target distribution reminiscent of \gls{PMMH}. However, whereas the \gls{PMMH} algorithm relies on \gls{PF} estimates of the likelihood $p_{\theta}(y_{1:T})$, this \gls{MH} version of \gls{EHMM} relies on an estimate of $p_{\theta}(y_{1:T})$ computed using a finite-state \gls{HMM}, the cardinality of the state-space being $N$. The computational cost of both of these original \gls{EHMM} methods is $O(N^2T)$ in contrast to the $O(NT)$-cost of \gls{PMCMC} methods.
The high computational cost of the original \gls{EHMM} method has partially motivated the development of a novel class of alternative \gls{EHMM} methods which bring the computational complexity down to $O(NT)$. As described in Section~\ref{sec:embeddedHMMnewversion}, this is done by introducing a set of auxiliary variables playing the same r\^ole as the ancestor indices generated in the resampling step of a standard \gls{PF}. This leads to the extended target distribution introduced in \citet{ShestopaloffNeal2016}. We show that this target coincides in a special case with the extended target of \gls{PMCMC} when one uses the \gls{FAAPF} \citep{PittShephard1999} and the resulting \gls{EHMM} coincides with the conditional \gls{FAAPF} with \gls{BS} in this scenario. We show once more that the validity of this novel \gls{EHMM} method can be established by using a collapsed Gibbs sampler.
In Section~\ref{sec:novel_methodology}, we derive several novel, practical extensions to the alternative \gls{EHMM} method. First, we show that the alternative \gls{EHMM} framework can also be used to derive an \gls{MH} algorithm which, once again, is very similar to the \gls{PMMH} algorithm except that $p_{\theta}(y_{1:T})$ is estimated unbiasedly using a novel \gls{PF} type algorithm relying on local \gls{MCMC} moves. Second, we derive additional bootstrap \gls{PF} and general \gls{APF} type variants of the alternative \gls{EHMM} method.
In Section~\ref{sec:general_pmcmc}, we describe a general, unifying \gls{PMCMC} framework which admits all variants of standard \gls{PMCMC} methods and all variants of alternative \gls{EHMM} discussed in this work as special cases. This also allows us to generalize the \glsdesc{AS} scheme from \citet{LindstenJordanSchon2014}.
In Section~\ref{sec:simulations}, we empirically compare the performance of all the algorithms mentioned above. Our results indicate that, as suggested in \citet{ShestopaloffNeal2016}, a properly tuned version of the conditional \gls{PF} (and hence \gls{PG} sampler) using \gls{MCMC} moves proposed in \citet{ShestopaloffNeal2016} can outperform existing methods in high dimensions while the (`non-conditional') \glspl{PF} using \gls{MCMC} moves are a potentially interesting alternative to standard \glspl{PF} for likelihood and state estimation for lower-dimensional models.
\begin{figure}
\caption{Relationship between the various classes of algorithms discussed in this work. A general construction admitting all of these as special cases can be found in \citet[Section~1.4]{Finke2015}. Novel methodology introduced in this work is highlighted in bold.}
\label{fig:relationship_between_algorithms}
\end{figure}
\section{Particle Markov chain Monte Carlo methods} \label{sec:pmcmc}
This section reviews \gls{PMCMC} methods. For transparency, we first restrict ourselves in this section to the scenario in which the underlying \gls{PF} used is the bootstrap \gls{PF}, and then discuss the \glsdesc{FAAPF} before finally considering the case of general \glsdescplural{APF}.
\subsection{Extended target distribution}
Let $N$ be an integer such that $N \geq 2$. \gls{PMCMC} methods rely on the following extended target density on $\varTheta \times \mathcal{X}^{NT}\times \{1,\dotsc,N\}^{N{(T-1)}+1}$ \begin{equation}
\tilde{\pi}{\bigl(\theta,b_{1:T},\mathbf{x}_{1:T},\mathbf{a}_{1:T-1}^{-b_{2:T}}\bigr)}
\coloneqq
\frac{1}{N^T} \times \underbrace{\pi{\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr)}}_{\mathclap{\text{\footnotesize{target}}}} \times \underbrace{\phi_\theta{\bigl(\mathbf{x}_{1:T}^{-b_{1:T}},\mathbf{a}_{1:T-1}^{-b_{2:T}}|x_{1:T}^{b_{1:T}},b_{1:T}\bigr)} }_{\mathclap{\text{\footnotesize{law of conditional \gls{PF}}}}}, \label{eq:PMCMCtarget} \end{equation}
where $\smash{\pi(\theta,x_{1:T}) \coloneqq p(x_{1:T},\theta| y_{1:T})}$ represents the posterior distribution of interest. In addition, the \emph{particles} $\mathbf{x}_t \coloneqq \{ x_t^1,\dotsc,x_t^N \} \in\mathcal{X}^N$, \emph{ancestor indices} $\mathbf{a}_t \coloneqq \{a_t^1,\dotsc,a_t^N\} \in \{1,\dotsc,N\}^N$ and \emph{particle indices} $b_{1:T} \coloneqq \{ b_1,\dotsc,b_T\}$ are related as \begin{equation}
\mathbf{x}_t^{-b_t}=\mathbf{x}_t\backslash x_t^{b_t}, \quad \mathbf{x}_{1:T}^{-b_{1:T}}=\bigl\{ \mathbf{x}_1^{-b_1},\dotsc,\mathbf{x}_T^{-b_T}\bigr\}, \quad \mathbf{a}_{t-1}^{-b_t}=\mathbf{a}_{t-1}\backslash a_{t-1}^{b_t}, \quad \mathbf{a}_{1:T-1}^{-b_{2:T}}=\bigl\{ \mathbf{a}_1^{-b_{2}},\dotsc,\mathbf{a}_{T-1}^{-b_T}\bigr\}. \end{equation} In particular, given $b_T$, the particle indices $b_{1:T-1}$ are deterministically related to the ancestor indices by the recursive relationship \begin{equation}
b_t = a_t^{b_{t+1}}, \quad \text{for $t = T-1, \dotsc, 1$.} \end{equation} Finally, for any $\smash{(x_{1:T}^{b_{1:T}},b_{1:T}) \in\mathcal{X}^N \times \bigl\{1,\dotsc,N\}^N}$, $\phi_\theta$ denotes a conditional distribution induced by an algorithm referred to as a \gls{CPF} \begin{equation}
\phi_\theta{\bigl(\mathbf{x}_{1:T}^{-b_{1:T}}, \mathbf{a}_{1:T-1}^{-b_{2:T}}\big| x_{1:T}^{b_{1:T}},b_{1:T}\bigr)}
\coloneqq
\prod_{\substack{\mathllap{i}=\mathrlap{1}\\\mathllap{i} \neq \mathrlap{b_1}}}^N
\mu_\theta{\bigl( x_1^i\bigr)} \prod_{t=2}^T \prod_{\substack{\mathllap{i}=\mathrlap{1}\\\mathllap{i} \neq \mathrlap{b_t}}}^N w_{\theta,t-1}^{a_{t-1}^i}\,f_\theta{\bigl( x_t^i\big|x_{t-1}^{a_{t-1}^i}\bigr)}, \label{eq:CPF} \end{equation} where \begin{equation}
w_{\theta,t}^i
\coloneqq \frac{g_\theta(y_t |x_t^i)}{\sum_{j=1}^N g_\theta(y_t | x_t^j)} \label{eq:multinomialresampling} \end{equation} represents the normalised weight associated with the $i$th particle at time~$t$.
The key feature of this high-dimensional target is that by construction it ensures that $\smash{( \theta,x_{1:T}^{b_{1:T}})}$ is distributed according to the posterior of interest. \gls{PMCMC} methods are \gls{MCMC} algorithms which sample from this extended target, hence from the posterior of interest.
\subsection{Particle marginal Metropolis--Hastings} \label{subsec:pmmh}
\glsreset{PMMH} \glsreset{MH}
The \emph{\gls{PMMH}} algorithm is a \gls{MH} algorithm targeting $\tilde{\pi}( \theta,b_{1:T},\mathbf{x}_{1:T},\mathbf{a}_{1:T-1}^{-b_{2:T}}) $ defined through \eqref{eq:PMCMCtarget}, \eqref{eq:CPF} and \eqref{eq:multinomialresampling} using a proposal of the form \begin{equation}
q{(\theta,{\theta'})} \times
\underbrace{\Psi_{{\theta'}}( \mathbf{x}_{1:T},\mathbf{a}_{1:T-1})}_{\mathclap{\text{\footnotesize{law of \gls{PF}}}}} \times \underbrace{w_{{\theta'},T}^{b_T}}_{\mathclap{\text{\parbox{1.3cm}{\centering\footnotesize{path selection}}}}}, \label{eq:proposalparticlefilter} \end{equation} where $b_{1:T}$ is again obtained via the reparametrisation $\smash{b_t = a_t^{b_{t+1}}}$ for $t = T-1, \dotsc,1$ and $\Psi_\theta(\mathbf{x}_{1:T},\mathbf{a}_{1:T-1})$ is the law induced by a bootstrap \gls{PF} \begin{equation}
\Psi_\theta( \mathbf{x}_{1:T},\mathbf{a}_{1:T-1})
\coloneqq
\prod_{i=1}^N \mu_\theta( x_1^i)
\prod_{t=2}^T \prod_{i=1}^N w_{\theta,t-1}^{a_{t-1}^i} f_\theta\bigl(x_t^i \big| x_{t-1}^{a_{t-1}^i}\bigr). \label{eq:distributionPF} \end{equation} The resulting \gls{MH} acceptance probability is of the form \begin{equation}
1 \wedge
\frac{\hat{p}_{{\theta'}}(y_{1:T})p({\theta'})}{\hat{p}_\theta(y_{1:T})p(\theta)}
\frac{q({\theta'},\theta)}{q(\theta,{\theta'})}, \label{eq:PMMHratio} \end{equation} where \begin{equation}
\hat{p}_\theta(y_{1:T})
\coloneqq \prod_{t=1}^T \biggl[\frac{1}{N} \sum_{i=1}^N g_\theta(y_t|
x_t^i) \biggr]\label{eq:likelihoodestimatorPF} \end{equation} is well known to be an unbiased estimate of $p_\theta(y_{1:T})$; see \cite{DelMoral2004}. We stress that the unbiased estimates appearing in the numerator and denominator of \eqref{eq:PMMHratio} each depends upon the particles (and ancestor indices) generated in distinct \glspl{PF} but we suppress this dependence to keep the notation as simple as is possible. The validity of the expression in \eqref{eq:PMMHratio} follows directly by noting that: \begin{align}
\frac{\tilde{\pi}( \theta,b_{1:T},\mathbf{x}_{1:T},\mathbf{a}_{1:T-1}^{-b_{2:T}}) }{\Psi_\theta( \mathbf{x}_{1:T},\mathbf{a}_{1:T-1}) w_{\theta,T}^{b_T}}
& = \frac{1}{N^T} \frac{\pi( \theta,x_{1:T}^{b_{1:T}})}{\mu_\theta(x_1^{b_1}) \bigl[\prod_{t=2}^T w_{\theta,t-1}^{b_{t-1}} f_\theta(x_t^{b_t} | x_{t-1}^{b_{t-1}}) \bigr] w_{\theta,T}^{b_T}}\\
& = \frac{p( \theta| y_{1:T}) }{N^T}\frac
{\mu_\theta( x_1^{b_1}) g_\theta( y_1| x_1^{b_1}) \prod_{t=2}^T f_\theta( x_t^{b_t}| x_{t-1}^{b_{t-1}})
g_\theta( y_t| x_t^{b_t}) }{\mu_\theta( x_1^{b_1})
\bigl[\prod_{t=2}^T w_{\theta,t-1}^{b_{t-1}} f_\theta(x_t^{b_t} | x_{t-1}^{b_{t-1}}) \bigr] w_{\theta,T}^{b_T}}\\
& = p(\theta | y_{1:T}) \frac{\hat{p}_\theta( y_{1:T}) }{p_\theta(y_{1:T})}\\
& \propto\hat{p}_\theta(y_{1:T}) p(\theta), \label{eq:PMCMCratiotargetproposal} \end{align}
where we have again used that $b_t = a_t^{b_{t+1}}$, for $t = T-1,\dotsc,1$ and that $p(\theta|y_{1:T})/p_\theta(y_{1:T}) = p(\theta)/p(y_{1:T})$; see also \cite[Theorem 2]{AndrieuDoucetHolenstein2010}.
\subsection{Particle Gibbs samplers} \label{subsec:cpf} \glsreset{PG} \glsreset{BS} \glsreset{AS}
To sample from $\pi(\theta,x_{1:T})$, one can use the \emph{\gls{PG}} sampler. The \gls{PG} sampler mimics the block Gibbs sampler iterating draws from $\pi(\theta | x_{1:T})$ and $\pi(x_{1:T}| \theta) $. As sampling from $\pi(x_{1:T} | \theta) $ is typically impossible, we can use a so called conditional \gls{PF} kernel with \gls{BS} to emulate sampling from it. Given a current value of $x_{1:T}$, we perform the following steps (see \citet{AndrieuDoucetHolenstein2009}, \citet[Section~4.5]{AndrieuDoucetHolenstein2010}); \begin{enumerate}
\item Sample $b_{1:T}$ uniformly at random and set $x_{1:T}^{b_{1:T}}\leftarrow x_{1:T}$.
\item Run the conditional \gls{PF}, i.e.\ sample from $\phi_\theta(\mathbf{x}_{1:T}^{-b_{1:T}},\mathbf{a}_{1:T-1}^{-b_{2:T}} | x_{1:T}^{b_{1:T}},b_{1:T})$.
\item\label{alg:simple_pg_last_step} Sample $b_T$ according to $\Pr(b_T=m) = w_\theta^{m}$ and set $b_t=a_t^{b_{t+1}}$ for $t = T-1, \dotsc, 1$. \end{enumerate} It was noticed in \citet{Whiteley2010} that it is possible to improve Step~\ref{alg:simple_pg_last_step}: for $t=T-1,\dotsc,1$, instead of deterministically setting $b_t=a_t^{b_{t+1}}$, one can use a backward sampling step which samples \begin{equation}
\Pr{\bigl(b_t=m\bigr)} \propto w_{\theta,t}^{m}f_\theta{\bigl(x_{t+1}^{b_{t+1}}\bigr| x_t^{m}\bigr)}. \label{eq:backwardsampling} \end{equation}
To establish the validity of this procedure (i.e.\ of the conditional \gls{PF} with \gls{BS}), it was shown that this procedure is a (partially) collapsed Gibbs sampler of invariant distribution $\smash{\tilde{\pi}(b_{1:T},\mathbf{x}_{1:T},\mathbf{a}_{1:T-1} | \theta)}$, sampling recursively from $\smash{\tilde{\pi}(b_t | \theta,\mathbf{x}_{1:t},\mathbf{a}_{1:t-1},x_{t+1:T}^{b_{t+1:T}},b_{t+1:T})}$, for $t=T,T-1,\dotsc,1$. Indeed, we have \begin{align}
\MoveEqLeft \tilde{\pi}{\bigl(b_t\bigr| \theta,\mathbf{x}_{1:t},\mathbf{a}_{1:t-1},x_{t+1:T}^{b_{t+1:T}},b_{t+1:T}\bigr)}\\
& \propto \sum_{b_{1:t-1}}\sum_{\mathbf{a}_{t:T-1}} \idotsint
\frac{\pi{\bigl( \theta,x_{1:T}^{b_{1:T}}\bigr)} }{N^T} \prod_{\substack{\mathllap{i}=\mathrlap{1}\\\mathllap{i} \neq \mathrlap{b_1}}}^N \mu_\theta{\bigl( x_1^i\bigr)} \smashoperator{\prod_{n=2}^T} \prod_{\substack{\mathllap{i}=\mathrlap{1}\\\mathllap{i} \neq \mathrlap{b_n}}}^N
w_{\theta,n-1}^{a_{n-1}^i}f_\theta{\bigl(x_n ^i\bigr| x_{n-1}^{a_{n-1}^i}\bigr)}\,\mathrm{d} \mathbf{x}_{t+1:T}^{-b_{t+1:T}}\\
& \propto \smashoperator{\sum_{b_{1:t-1}}} \pi{\bigl( \theta,x_{1:T}^{b_{1:T}}\bigr)}
\prod_{\substack{\mathllap{i}=\mathrlap{1}\\\mathllap{i} \neq \mathrlap{b_1}}}^N \mu_\theta{\bigl( x_1^i\bigr)} \prod_{n=2}^t
\prod_{\substack{\mathllap{i}=\mathrlap{1}\\\mathllap{i} \neq \mathrlap{b_n}}}^N w_{\theta,n-1}^{a_{n-1}^i}f_\theta{\bigl(x_n^i\bigr| x_{n-1}^{a_{n-1}^i}\bigr)} \\
& = \smashoperator{\sum_{b_{1:t-1}}} \pi{\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr)} \frac{\prod_{i=1}^N \mu_\theta{\bigl( x_1^i\bigr)} \prod_{n=2}^t \prod_{i=1}^N w_{\theta,n-1}^{a_{n-1}^i}f_\theta{\bigl(x_n^i\bigr| x_{n-1}^{a_{n-1}^i}\bigr)} }{\mu_\theta{\bigl(x_1^{b_1}\bigr)} \prod_{n=2}^t w_{\theta,n-1}^{b_{n-1}}f_\theta{\bigl(x_n^{b_n}\bigr| x_{n-1}^{b_{n-1}}\bigr)} }, \quad \text{as $\smash{a_{n-1}^{b_n} = b_{n-1}}$,}\label{eq:CPFBS}\\
& \propto \smashoperator{\sum_{b_{1:t-1}}} f_\theta{\bigl(x_{t+1}^{b_{t+1} }\bigr| x_t^{b_t}\bigr)} w_{\theta,t}^{b_t}\\
& \propto f_\theta{\bigl(x_{t+1}^{b_{t+1}}\bigr| x_t^{b_t}\bigr)} w_{\theta,t}^{b_t}, \end{align} where we have used that the numerator of the ratio appearing in \eqref{eq:CPFBS} is independent of $b_{1:t-1}$.
\subsection{Extension to the fully-adapted auxiliary particle filter} \label{Section:perfectadaptationPF}
\glsreset{FAAPF}
It is straightforward to employ a more general class of \glspl{PF} in a \gls{PMCMC} context. One such \gls{PF} is the \emph{\gls{FAAPF}} \citep{PittShephard1999} whose incorporation within \gls{PMCMC} was explored in \citet{Pitt2012}. It is described in this subsection.
When it is possible to sample from $p_\theta(x_1|y_1) \propto \mu_\theta(x_1) g_\theta(y_1| x_1) $ and $p_\theta(x_t| x_{t-1},y_t) \propto f_\theta(x_t| x_{t-1}) g_\theta(y_t|x_t)$ and to compute $p_\theta(y_1) = \int \mu_\theta(x_1) g_\theta(y_1|x_1) \mathrm{d}x_1$ and $p_\theta(y_t|x_{t-1}) = \int f_\theta(x_t|x_{t-1}) g_\theta(y_t| x_t) \mathrm{d} x_t$, it is possible to\ define the target distribution $\smash{\tilde{\pi}(\theta,b_{1:T},\mathbf{x}_{1:T},\mathbf{a}_{1:T-1}^{-b_{2:T}})}$ using an alternative conditional \gls{PF} -- the conditional \gls{FAAPF} -- in \eqref{eq:PMCMCtarget} (more precisely, in these circumstances one can implement the associated \gls{PF}): \begin{equation}
\phi_\theta{\bigl(\mathbf{x}_{1:T}^{-b_{1:T}},\mathbf{a} _{1:T-1}^{-b_{2:T}}\bigr| x_{1:T}^{b_{1:T}},b_{1:T}\bigr)}
= \prod_{\substack{\mathllap{i}=\mathrlap{1}\\\mathllap{i} \neq \mathrlap{b_1}}}^N p_\theta{\bigl(x_1^i\bigr| y_1\bigr)} \prod_{t=2}^T \prod_{\substack{\mathllap{i}=\mathrlap{1}\\\mathllap{i} \neq \mathrlap{b_t}}}^N w_{\theta,t-1}^{a_{t-1}^i}p_\theta{\bigl(x_t^i\bigr| x_{t-1}^{a_{t-1}^i},y_t\bigr)}, \label{eq:CPFperfectadaptation} \end{equation} where \begin{equation}
w_{\theta,t}^{i} \coloneqq \frac{p_\theta(y_{t+1} | x_{t}^{i})}{\sum_{j=1}^Np_\theta(y_{t+1}|
x_{t}^j)}. \label{eq:perfectadaptationweight} \end{equation} In this case, we can target the extended distribution $\smash{\tilde{\pi}(\theta,b_{1:T},\mathbf{x}_{1:T},\mathbf{a}_{1:T-1}^{-b_{2:T}})}$ defined through \eqref{eq:PMCMCtarget}, \eqref{eq:CPFperfectadaptation} and \eqref{eq:perfectadaptationweight} using a \gls{MH} algorithm with proposal \begin{equation}
q{\bigl( \theta,{\theta'}\bigr)} \times \underbrace{\Psi_{{\theta'}}{\bigl(\mathbf{x}_{1:T},\mathbf{a}_{1:T-1}\bigr)}}_{\mathclap{\text{\footnotesize{law of \gls{FAAPF}}}}} \times \underbrace{\frac{1}{N}}_{\mathclap{\text{\parbox{1.3cm}{\centering\footnotesize{path selection}}}}}, \end{equation} i.e.\ we pick $b_T$ uniformly at random, then set $\smash{b_t=a_t^{b_{t+1}}}$ for $t=T-1,\dotsc,1$ and $\Psi_\theta{\bigl(\mathbf{x}_{1:T},\mathbf{a}_{1:T-1}\bigr)}$ is the distribution associated with the \gls{FAAPF} instead of the bootstrap \gls{PF} \begin{equation}
\Psi_\theta{\bigl(\mathbf{x}_{1:T},\mathbf{a}_{1:T-1}\bigr)}
= \prod_{i=1}^N p_\theta{\bigl(x_1^i\bigr| y_1\bigr)} \prod_{t=2}^T \prod_{i=1}^N w_{\theta,t-1}^{a_{t-1}^i}p_\theta{\bigl(x_t^i\bigr| x_{t-1}^{a_{t-1}^i},y_t\bigr)} .
\label{eq:distributionperfectadaptation} \end{equation} It is easy to check that the resulting \gls{MH} acceptance probability is also of the form given in \eqref{eq:PMMHratio} but with \begin{equation}
\hat{p}_\theta(y_{1:T})
= p_\theta(y_1) \prod_{t=2}^T \biggl[\frac{1}{N} \sum_{i=1}^N p_\theta(y_t|x_{t-1}^i)\biggr]. \label{eq:likelihoodestimatorperfectadaptationPF} \end{equation} The conditional \gls{FAAPF} with \gls{BS} proceeds by first running the conditional \gls{FAAPF} defined in \eqref{eq:CPFperfectadaptation}, then sampling $b_T$ uniformly at random and finally sampling $b_{T-1},\dotsc,b_1$ backwards using \begin{equation}
\tilde{\pi}\bigl(b_t \big| \theta,\mathbf{x}_{1:t},\mathbf{a}_{1:t-1}, x_{t+1:T}^{b_{t+1:T}},b_{t+1:T}\bigr)
\propto f_\theta\bigl(x_{t+1}^{b_{t+1}} \big| x_t^{b_t}\bigr), \label{eq:backwardperfectadaptation} \end{equation} where the expression in \eqref{eq:backwardperfectadaptation} is obtained using calculations similar to those in \eqref{eq:CPFBS}.
\subsection{Extension to general auxiliary particle filters} \label{subsec:apf} \glsreset{APF}
The previous section demonstrated that the \gls{FAAPF} leads straightforwardly to valid \gls{PMCMC} algorithms and will allow natural connections to be made to certain \gls{EHMM} methods. Here, we show that as was established in \citet[Appendix 8.2]{Pitt2012}, \emph{any} general \emph{\gls{APF}} can be employed in this context and will lead to natural extensions of these methods.
To facilitate later developments, an explicit representation of the associated extended target distribution and related quantities is useful. Viewing the \gls{APF} as a sequential importance resampling algorithm for an appropriate sequence of target distributions as described in \citet{JohansenDoucet2008}, it is immediate that the density associated with such an algorithm is simply: \begin{align}
\Psi_\theta^{\mathbf{q}_\theta}(\mathbf{x}_{1:T},\mathbf{a}_{1:T-1}) =
\prod_{i=1}^N q_{\theta,1}(x_1^i) \prod_{t=2}^T w_{\theta,t-1}^{a_{t-1}^i} q_{\theta,t}\bigl(x_t^i \big|x_{t-1}^{a_{t-1}^i}\bigr), \end{align} where $\mathbf{q}_\theta = \{q_{\theta, t}\}_{t=1}^T$ and $q_{\theta,t}$ denotes the proposal distribution employed at time $t$ (with dependence of this distribution upon the observation sequence suppressed from the notation) and $\smash{w_{\theta,t}^i = v_{\theta,t}^i / \sum_{j=1}^N v_{\theta,t}^j}$ with: \begin{align}
v_{\theta,t}^i =
\begin{cases}
\dfrac{\mu_\theta{\bigl(x_1^i\bigr)} g_\theta{\bigl(y_1|x_1^i\bigr)} \tilde{p}_\theta{\bigl(y_2|x_1^i\bigr)}}{q_{\theta,1}{\bigl(x_1^i\bigr)}}, & \text{if $t = 1$,}\\
\dfrac{f_\theta{\bigl(x_t^i|x_{t-1}^{a_{t-1}^i}\bigr)} g_\theta{\bigl(y_t|x_t^i\bigr)} \tilde{p}_\theta{\bigl(y_{t+1}|x_t^i\bigr)}}{q_{\theta,t}{\bigl(x_t^i | x_{t-1}^{a_{t-1}^i}\bigr)} \tilde{p}_\theta{\bigl(y_t|x_{t-1}^{a_{t-1}^i}\bigr)}}, & \text{if $1 < t < T$,}\\
\dfrac{f{\bigl(x_T^i|x_{T-1}^{a_{T-1}^i}\bigr)} g_\theta{\bigl(y_T|x_T^i\bigr)}}{q_{\theta,T}{\bigl(x_T^i|x_{t-1}^{a_{T-1}^i}\bigr)} \tilde{p}_\theta{\bigl(y_T|x_{T-1}^{a_{T-1}^i}\bigr)}}, & \text{if $t = T$,}
\end{cases}\label{eq:apfweights} \end{align}
and $\tilde{p}_\theta(y_{t+1}|x_t^i)$ denoting the approximation of the predictive likelihood employed within the weighting of the \gls{APF}. Note that $\tilde{p}_\theta(y_{t+1}|x_t)$ can be any positive function of $x_t$ and the simpler sequential importance resampling \gls{PF} is recovered by setting $\tilde{p}_\theta(y_{t+1}|x_t) \equiv 1$, with the bootstrap \gls{PF} emerging as a particular case thereof when $q_{\theta, t}(x_t|x_{t-1}) = f_\theta(x_t|x_{t-1})$.
Associated with the \gls{APF} is a conditional \gls{PF} of the form: \begin{equation}
\phi^{\mathbf{q}_\theta}_\theta{\bigl(\mathbf{x}_{1:T}^{-b_{1:T}},\mathbf{a}_{1:T-1}^{-b_{2:T}}\bigr| x_{1:T}^{b_{1:T}},b_{1:T}\bigr)}
= \prod_{\substack{\mathllap{i}=\mathrlap{1}\\\mathllap{i} \neq \mathrlap{b_1}}}^N q_{\theta,1}{\bigl(x_1^i\bigr)} \prod_{t=2}^T \prod_{\substack{\mathllap{i}=\mathrlap{1}\\\mathllap{i} \neq \mathrlap{b_t}}}^N w_{\theta,t-1}^{a_{t-1}^i}\,q_{\theta,t}{\bigl(x_t^i\bigr|x_{t-1}^{a_{t-1}^i}\bigr)}. \label{eq:CAPF} \end{equation} A \gls{PMCMC} algorithm is arrived at by employing the extended target distribution, \begin{equation}
\tilde{\pi}^{\mathbf{q}_\theta}{\bigl(\theta,b_{1:T},\mathbf{x}_{1:T},\mathbf{a} _{1:T-1}^{-b_{2:T}}\bigr)}
= \frac{1}{N^T} \times \underbrace{\pi{\bigl( \theta,x_{1:T}^{b_{1:T}}\bigr)}}_{\mathclap{\text{\footnotesize{target}}}}
\times \underbrace{\phi^{\mathbf{q}_\theta}_\theta{\bigl(\mathbf{x}_{1:T}^{-b_{1:T}},\mathbf{a}_{1:T-1}^{-b_{2:T}}\bigr| x_{1:T}^{b_{1:T}},b_{1:T}\bigr)} }_{\mathclap{\text{\footnotesize{law of conditional \gls{APF}}}}}, \label{eq:APMCMCtarget} \end{equation} and proposal distribution, \begin{equation}
q{(\theta,{\theta'})}
\times \underbrace{\Psi_{{\theta'}}^{\mathbf{q}_\theta'}{(\mathbf{x}_{1:T},\mathbf{a}_{1:T-1})}}_{\mathclap{\text{\footnotesize{law of \gls{APF}}}}}
\times \underbrace{w_{{\theta'},T}^{b_T}}_{\mathclap{\text{\parbox{1.3cm}{\centering\footnotesize{path selection}}}}}. \label{eq:proposalauxiliaryparticlefilter} \end{equation} One can straightforwardly verify that this leads to a \gls{MH} acceptance probability of the form stated in \eqref{eq:PMMHratio} but using the natural unbiased estimator of the normalising constant associated with the \gls{APF}, \begin{equation}
\hat{p}_\theta(y_{1:T}) = \prod_{t=1}^T \biggl[\frac{1}{N} \sum_{i=1}^N v_{\theta,t}^i\biggr]. \end{equation} We conclude this section by noting that although the constructions developed above were presented for simplicity with multinomial resampling employed during every iteration of the algorithm, it is straightforward to incorporate more sophisticated, adaptive resampling schemes within this framework.
\section{Original embedded hidden Markov models} \label{sec:embeddedHMM}
\glsreset{EHMM} \glsreset{HMM}
\subsection{Extended target distribution}
The \emph{\gls{EHMM}} method of \cite{Neal2003,Neal2004} is based on the introduction of a target distribution on $\varTheta \times \mathcal{X}^{NT}\times \{ 1,\dotsc,N \}^N$ of the form \begin{align}
\tilde{\pi}(\theta,b_{1:T},\mathbf{x}_{1:T})
= \frac{1}{N^T} \times \underbrace{\pi{\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr)}}_{\mathclap{\text{\footnotesize{target}}}} \times \underbrace{\prod_{t=1}^T \;\; \Bigl\{ \smashoperator{\prod_{i=b_t -1}^1} \widetilde{R}_{\theta,t}{\bigl(x_t^i\bigr| x_t ^{i+1}\bigr)} \cdot \smashoperator{\prod_{i=b_t+1}^N} R_{\theta,t}{\bigl(x_t^i\bigr| x_t^{i-1}\bigr)} \Bigr\}}_{\mathclap{\text{\footnotesize{law of conditional random grid generation}}}}, \label{eq:embeddedHMM2003target} \end{align}
where $R_{\theta,t}$ is a $\rho_{\theta,t}$-invariant Markov transition kernel, i.e.\ $\smash{\int\rho_{\theta,t}(x) R_{\theta,t}(x^\prime| x) \mathrm{d} x = \rho_{\theta,t}(x^\prime)}$, and $\smash{\widetilde{R}_{\theta,t}}$ is its reversal, i.e.\ $\smash{\widetilde{R}_{\theta,t}(x^\prime|x) = \rho_{\theta,t}(x^\prime) R_{\theta,t}(x|x^\prime) / \rho_{\theta,t}(x)}$ (for $\rho_{\theta,t}$-almost every $x$ and $x^\prime$).
Similarly to the \gls{PMCMC} extended target distribution, the key feature of $\tilde{\pi}(\theta,b_{1:T},\mathbf{x}_{1:T})$ is that, by construction, it ensures that the associated marginal distribution of $\smash{(\theta,x_{1:T}^{b_{1:T}})}$ is the posterior of interest.
\subsection{Metropolis--Hastings algorithm} \label{subsec:ehmm-mh}
As detailed in the next section, the algorithm proposed in \citet{Neal2003} can be reinterpreted as a Gibbs sampler targeting $\tilde{\pi}(b_{1:T},\mathbf{x}_{1:T}|\theta)$. We present here an alternative, original \gls{MH} algorithm to sample from $\tilde{\pi}(\theta,b_{1:T},\mathbf{x}_{1:T})$. It relies on a proposal of the form \begin{equation}
q{\bigl( \theta,{\theta'}\bigr)}
\times \underbrace{\Psi_{{\theta'}}{\bigl(\mathbf{x}_{1:T}\bigr)} }_{\mathclap{\parbox{2.3cm}{\text{\parbox{2.3cm}{\centering\footnotesize{law of random grid generation}}}}}} \times \underbrace{q_{{\theta'}}{\bigl(b_{1:T}\bigr| \mathbf{x}_{1:T}\bigr)}}_{\mathclap{\text{\parbox{2cm}{\centering\footnotesize{path selection}}}}}, \label{eq:proposalembeddedHMM2003} \end{equation} where \begin{equation}
\Psi_\theta{\bigl( \mathbf{x}_{1:T}\bigr)}
\coloneqq \frac{1}{N^T} \prod_{t=1}^T \Bigl\{ \rho_{\theta,t}{\bigl(x_t^1\bigr)} \smashoperator{\prod_{i=2}^N} R_{\theta,t}{\bigl(x_t^i\bigr| x_t^{i-1}\bigr)} \Bigr\} \label{eq:distributionrandomHMMfilter} \end{equation} is sometimes referred to as the \emph{ensemble base measure} \citep{Neal2011} and \begin{equation}
q_\theta(b_{1:T} | \mathbf{x}_{1:T})
\coloneqq \frac{\tilde{p}_\theta{\bigl( x_{1:T}^{b_{1:T}},y_{1:T}\bigr)}}{\sum_{b_{1:T}^\prime}\tilde{p}_\theta{\bigl( x_{1:T}^{b_{1:T}^\prime},y_{1:T}\bigr)}}
= \frac{1}{N^T}\frac{\tilde{p}_\theta{\bigl(x_{1:T}^{b_{1:T}},y_{1:T}\bigr)}}{\tilde{p}_\theta(y_{1:T})}. \label{eq:pathselectionEHMM} \end{equation} In this expression, we have (where we note that this is no longer a probability density with respect to Lebesgue measure) \begin{equation}
\tilde{p}_\theta( x_{1:T},y_{1:T})
\coloneqq \frac{\mu_\theta(x_1) g_\theta(y_1|x_1)}{\rho_{\theta,1}(x_1)} \prod_{t=2}^T \frac{f_\theta(x_t|x_{t-1}) g_\theta(y_t|x_t)}{\rho_{\theta,t}(x_t)} \label{eq:modifiedjointposterior} \end{equation} and \begin{equation} \tilde{p}_\theta(y_{1:T}) \coloneqq \frac{1}{N^T}\sum _{b_{1:T}^\prime}\tilde{p}_\theta{\bigl( x_{1:T}^{b_{1:T}^\prime },y_{1:T}\bigr)} . \label{eq:likelihoodestimatorembeddedHMM} \end{equation}
To sample from $\Psi_\theta( \mathbf{x}_{1:T})$, we sample $\smash{x_t^{1}\sim\rho_{\theta,t}}(x_t^{1})$ and $\smash{\mathbf{x}_t^{-1}}\sim\smash{\prod_{i=2}^N R_{\theta,t}(x_t^i| x_t^{i-1})}$ for $t=1,...,T$. Hence, at time $t$ all of the particles are marginally distributed according to $\rho_{\theta,t}$. When $\smash{R_{\theta,t}(x^\prime| x) = \rho_{\theta,t}(x^\prime)}$, this corresponds to the algorithm proposed in \citet{LinChen2005}. Sampling from the high-dimensional discrete distribution $q_\theta(b_{1:T}| \mathbf{x}_{1:T})$ can be performed in $O(N^2T)$ operations with the finite state-space \gls{HMM} filter using the $N$ states $(x_t^i)$ at time $t$, transition probabilities proportional to $f_\theta(x_t^j| x_{t-1}^i)$ and conditional probabilities of the observations proportional to $g_\theta(y_t| x_t^i) / \rho_{\theta,t}(x_t^i)$. We also obtain as a by-product $\tilde{p}_\theta(y_{1:T})$, which is an unbiased estimate of $p_\theta(y_{1:T})$.
The resulting \gls{MH} algorithm targeting the extended distribution given in \eqref{eq:embeddedHMM2003target} with the proposal given in \eqref{eq:proposalembeddedHMM2003} admits an acceptance probability of the form \begin{equation}
1 \wedge
\frac{\tilde{p}_{{\theta'}}(y_{1:T})p({\theta'})}{\tilde{p}_\theta(y_{1:T})
p(\theta)}
\frac{q({\theta'},\theta)}{q(\theta,{\theta'})},
\label{eq:embeddedHMMratio} \end{equation} i.e.\ it looks very much like the \gls{PMMH} algorithm, except that instead of having likelihood terms estimated by a particle filter, these likelihood terms are estimated using a finite state-space \gls{HMM} filter.
To establish the correctness of the acceptance probability given in \eqref{eq:embeddedHMMratio}, we note that \begin{align}
\frac{\tilde{\pi}(\theta,b_{1:T},\mathbf{x}_{1:T})}{\Psi_\theta(\mathbf{x}_{1:T}) q_\theta(b_{1:T}| \mathbf{x}_{1:T}) }
& = \frac{N^{-T}\pi(\theta,x_{1:T}^{b_{1:T}}) \prod_{t=1}^T
\bigl\{ \prod_{i=b_t -1}^1 \widetilde{R}_{\theta,t}(x_t^i| x_t^{i+1}) \cdot \prod_{i=b_t+1}^N
R_{\theta,t}(x_t^i| x_t^{i-1}) \bigr\} }{
\prod_{t=1}^T \bigl\{ \rho_{\theta,t}( x_t^{1}) \cdot \prod_{i=2}^N R_{\theta,t}(x_t^i| x_t^{i-1}) \bigr\}
N^{-T}\frac{\tilde{p}_\theta( x_{1:T}^{b_{1:T}},y_{1:T}) }{\tilde{p}_\theta( y_{1:T}) }}\\
& =\frac{p_\theta( x_{1:T}^{b_{1:T}},y_{1:T}) / p_\theta(y_{1:T}) }{\prod_{t=1}^T \rho_{\theta,t}(x_t^{b_t})}
\biggl[ \frac{p_\theta(x_{1:T}^{b_{1:T}},y_{1:T})}{\tilde{p}_\theta(y_{1:T}) \prod_{t=1}^T \rho_{\theta,t}( x_t^{b_t})}\biggr]^{-1}\\
& = p(\theta| y_{1:T}) \frac{\tilde{p}_\theta(y_{1:T})}{p_\theta(y_{1:T})} \propto \tilde{p}_\theta(y_{1:T}) p(\theta), \label{eq:embeddedHMMratiotargetproposal} \end{align} where we have used that \begin{equation}
\frac{\tilde{p}_\theta(x_{1:T},y_{1:T})}{\tilde{p}_\theta{\bigl( y_{1:T}\bigr)}}
=\frac{p_\theta(x_{1:T},y_{1:T})}{\tilde{p}_\theta(y_{1:T}) \prod_{t=1}^T \rho_{\theta,t}{\bigl(x_t^{b_t}\bigr)}}. \end{equation} In addition, we have used the following identity which we will also exploit in the next section: if $R$ is a $\rho$-invariant Markov kernel and $\widetilde{R}$ the associated reversal, then for any $b,c \in \{1,\dotsc,N\}$, \begin{align}
\smashoperator{\prod_{i=b-1}^1} \widetilde{R}{\bigl(x^i\bigr| x^{i+1}\bigr)} \cdot \rho{\bigl(x^{b}\bigr)} \cdot \smashoperator{\prod_{i=b+1}^N} R{\bigl(x^i\bigr| x^{i-1}\bigr)} = \smashoperator{\prod_{i=c-1}^1} \widetilde{R}{\bigl(x^i\bigr| x^{i+1}\bigr)} \cdot \rho{\bigl(x^c\bigr)} \cdot \smashoperator{\prod_{i=c+1}^N} R{\bigl(x^i\bigr| x^{i-1}\bigr)}. \label{eq:useful_reversal_kernel_identity} \end{align}
\subsection{Interpretation as a collapsed Gibbs sampler} \label{subsec:ehmm-gibbs}
Consider the following Gibbs sampler type algorithm to sample from $\pi(x_{1:T}|\theta)$: \begin{enumerate}
\item \label{enum:ehmm_gibbs:1} Sample $b_{1:T}$ uniformly at random on $\smash{\{1,\dotsc,N\}^T}$ and set $\smash{x_{1:T}^{b_{1:T}}\leftarrow x_{1:T}}$;
\item \label{enum:ehmm_gibbs:2} Sample $\smash{\tilde{\pi}(\mathbf{x}_{1:T}^{-b_{1:T}}| \theta,b_{1:T},x_{1:T}^{b_{1:T}})}$;
\item \label{enum:ehmm_gibbs:3} Sample $\smash{b_T\sim\tilde{\pi}(b_T|\theta,\mathbf{x}_{1:T})}$ then $\smash{b_{T-1}\sim\tilde{\pi}(b_{T-1}| \theta,\mathbf{x}_{1:T-1},x_T^{b_T},b_T)}$ and so on. \end{enumerate} It is obvious that Steps~\ref{enum:ehmm_gibbs:1} and \ref{enum:ehmm_gibbs:2} coincide with the first steps of the \gls{EHMM} algorithm described in \citet{Neal2003}. For Step~\ref{enum:ehmm_gibbs:3}, we note that \begin{align}
\MoveEqLeft \tilde{\pi}{\bigl(b_t\bigr| \theta,\mathbf{x}_{1:t},x_{t+1:T}^{b_{t+1:T}},b_{t+1:T}\bigr)} \\
& \propto \smashoperator{\sum_{b_{1:t-1}}} \idotsint
\pi{\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr)}
\prod_{n=1}^T \;\;
\Bigl\{
\smashoperator{\prod_{i=b_n -1}^1}
\widetilde{R}_{\theta,n}{\bigl(x_n^i\bigr| x_n^{i+1}\bigr)} \cdot
\smashoperator{\prod_{i=b_n+1}^N} R_{\theta,n}{\bigl(x_n^i\bigr| x_n^{i-1}\bigr)} \Bigr\}\,
\mathrm{d}\mathbf{x}_{n+1}^{-b_{n+1}}\cdots \mathrm{d} \mathbf{x}_T^{-b_T}\\
& =\smashoperator{\sum_{b_{1:t-1}}} \pi{\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr)}
\smashoperator{\prod_{n=1}^t} \;\;
\Bigl\{\smashoperator{\prod_{i=b_n -1}^1}
\widetilde{R}_{\theta,n}{\bigl(x_n^i \bigr| x_n^{i+1}\bigr)} \cdot \smashoperator{\prod_{i=b_n+1}^N}
R_{\theta,n}{\bigl(x_n^i\bigr| x_n^{i-1}\bigr)} \Bigr\}\\
& = \smashoperator{\sum_{b_{1:t-1}}} \pi{\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr)} \prod_{n=1}^t \frac{
\prod_{i=b_n -1}^1 \widetilde{R}_{\theta,n}{\bigl(x_n^i\bigr| x_n^{i+1}\bigr)} \cdot \rho_{\theta,n}{\bigl( x_n^{b_n}\bigr)} \cdot \prod_{i=b_n+1}^N R_{\theta,n}{\bigl(x_n^i\bigr| x_n^{i-1}\bigr)}
}{\rho_{\theta,n}{\bigl( x_n^{b_n}\bigr)} }\\
& \propto\smashoperator{\sum_{b_{1:t-1}}}\,\frac{\pi{\bigl( \theta,x_{1:T}^{b_{1:T}}\bigr)} }{
\prod_{n=1}^t \rho_{\theta,n}{\bigl( x_n^{b_n}\bigr)} }, \end{align} where by \eqref{eq:useful_reversal_kernel_identity}, the numerator in the penultimate line is independent of $b_n$. Since \begin{align}
\frac{\pi{\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr)}}{\prod_{n=1}^t \rho_{\theta,n}{\bigl(x_n^{b_n}\bigr)}}
& \propto \frac{p_{\theta
}{\bigl( x_{1:T}^{b_{1:T}},y_{1:T}\bigr)} }{\prod_{n=1}^t \rho_{\theta,n}{\bigl(x_n^{b_n}\bigr)}}\\
& \propto
\underbrace{\prod_{n=1}^t \frac{f_\theta\bigl(x_n^{b_n}\bigr| x_{n-1}^{b_{n-1}}\bigr)g_\theta{\bigl(y_n\bigr| x_n^{b_n}\bigr)}}{\rho_{\theta,n}{\bigl(x_n^{b_n}\bigr)} }}_{\mathclap{\text{\footnotesize{modified posterior $\tilde{p}_\theta(x_{1:t}^{b_{1:t}}| y_{1:t})$}}}} \cdot \smashoperator{\prod_{n=t+1}^T} f_\theta{\bigl(x_n^{b_n}\bigr| x_{n-1}^{b_{n-1}}\bigr)} g_\theta{\bigl(y_n\bigr| x_n^{b_n}\bigr)}, \end{align}
we can compute the marginal $\smash{\tilde{p}_\theta(x_t^{b_t}| y_{1:t}) \coloneqq \sum_{b_{1:t-1}} \tilde{p}_\theta(x_{1:t}^{b_{1:t}}| y_{1:t})}$ using the same (finite state-space) \gls{HMM} filter discussed in the previous section and so \begin{equation}
\tilde{\pi}{\bigl(b_t\bigr| \theta,\mathbf{x}_{1:t},x_{t+1:T}^{b_{t+1:T}},b_{t+1:T}\bigr)} \propto\tilde{p}_\theta{\bigl(x_t^{b_t}\bigr| y_{1:t}\bigr)} f_\theta{\bigl(x_{t+1}^{b_t+1}\bigr| x_t^{b_t}\bigr)} \end{equation} coinciding with the expression obtained in \citet{Neal2003}. This is an alternative proof of validity of the algorithm. The present derivation is more complex than that in \citet{Neal2003} which relies on a simple detailed balance argument. One potential benefit of our approach is that it can be extended systematically to any extended target admitting a similar structure; see for example \citet[p.~116]{LindstenSchon2013} for extensions to the non-Markovian case. Finally, we note that this algorithm may be viewed as a special case of the framework proposed in \citet{Tjelmeland2004} and simplifies to Barker's kernel \citep{Barker1965} if $N=2$ and $T=1$.
\section{Alternative embedded hidden Markov models} \label{sec:embeddedHMMnewversion}
\glsreset{MCMCPF} \glsreset{MCMCFAAPF} \glsreset{MCMCAPF}
In its original version, the \gls{EHMM} method has a computational cost per iteration of order $O(N^2T)$ compared to $O(NT)$ for \gls{PMCMC} methods and it samples particles independently across time which can be inefficient if the latent states are strongly correlated. The new version of \gls{EHMM} methods, which was proposed in \citet{ShestopaloffNeal2016}, resolves both of these limitations. It can be viewed as a \gls{PMCMC}-type algorithm making use of a new type of \gls{PF} that we term the \emph{\gls{MCMCFAAPF}} given its connection to the \gls{FAAPF} which we detail below.
\subsection{Extended target distribution}
This version of the \gls{EHMM}, henceforth referred to as the \emph{alternative} \gls{EHMM} method, relies on the extended target distribution \begin{equation}
\tilde{\pi}{\bigl( \theta,b_{1:T},\mathbf{x}_{1:T}, \mathbf{a}_{1:T-1}^{-b_{2:T}}\bigr)} = \frac{1}{N^T} \times \underbrace{\pi{\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr)}}_{\mathclap{\text{\footnotesize{target}}}} \times \underbrace{\phi_\theta\bigl(\mathbf{x}_{1:T}^{-b_{1:T}},\mathbf{a}_{1:T-1}^{-b_{2:T}}\bigr|x_{1:T}^{b_{1:T}},b_{1:T}\bigr)}_{\mathclap{\text{\footnotesize{law of conditional \gls{MCMCFAAPF}}}}}, \end{equation} where we will refer to the algorithm inducing the following distribution as the conditional \gls{MCMCFAAPF} for reasons which are made clear below: \begin{align}
\MoveEqLeft \phi_\theta{\bigl( \mathbf{x}_{1:T}^{-b_{1:T}},\mathbf{a}_{1:T-1}^{-b_{2:T}}\bigr| x_{1:T}^{b_{1:T}},b_{1:T}\bigr)}\\*
& =
\smashoperator{\prod_{\smash{i=b_1-1}}^1}\widetilde{R}_{\theta,1}{\bigl(x_1^i\bigr| x_1^{i+1}\bigr)} \cdot \smashoperator{\prod_{\smash{i=b_1+1}}^N} R_{\theta,1}{\bigl(x_1^i\bigr| x_1^{i-1}\bigr)} \label{eq:conditionalPFnovelEHMMtarget}\\
& \quad \times \smashoperator{\prod_{t=2}^{\smash{T}}} \;\; \Bigl\{\smashoperator{\prod_{i=b_t -1}^{\smash{1}}} \widetilde{R}_{\theta,t}{\bigl(x_t^i,a_{t-1}^i\bigr| x_t^{i+1},a_{t-1}^{i+1};\mathbf{x}_{t-1}\bigr)} \smashoperator{\prod_{i=b_t+1}^{\smash{N}}} R_{\theta,t}{\bigl(x_t^{i+1},a_{t-1}^{i+1}\bigr| x_t^i,a_{t-1}^i;\mathbf{x}_{t-1}\bigr)} \Bigr\}, \end{align} with $\smash{b_t=a_t^{b_{t+1}}}$ as for \gls{PMCMC} methods.
Here $R_{\theta,1}$ is invariant with respect to $\rho_{\theta,1}(x_1) = p_\theta(x_1 | y_1)$ whereas, for $t=2,\dotsc,T$, $R_{\theta,t}(\,\cdot\,|\,\cdot\,; \mathbf{x}_{t-1})$ is invariant w.r.t.\ \begin{align}
\rho_{\theta,t}{\bigl(x_t, a_{t-1}\bigr| \mathbf{x}_{t-1}\bigr)}
& =\frac{g_\theta{\bigl(y_t\bigr| x_t\bigr)} f_\theta{\bigl(x_t\bigr| x_{t-1}^{a_{t-1}}\bigr)}} {\sum_{i=1}^Np_\theta{\bigl(y_t\bigr| x_{t-1}^i\bigr)}}
=\frac{p_\theta{\bigl(y_t\bigr| x_{t-1}^{a_{t-1}}\bigr)}}{\sum_{i=1}^Np_\theta{\bigl(y_t\bigr| x_{t-1}^i\bigr)}}p_\theta{\bigl(x_t\bigr| y_t,x_{t-1}^{a_{t-1}}\bigr)} , \end{align}
while, for $t=1,\dotsc,T$, $\widetilde{R}_{\theta,t}(\,\cdot\,|\,\cdot\,; \mathbf{x}_{t-1})$ denotes the reversal of the kernel $R_{\theta,t}(\,\cdot\,|\,\cdot\,; \mathbf{x}_{t-1})$ with respect to its invariant distribution.
Note that if $R_{\theta,1}(x_1^\prime|x_1) =\rho_{\theta,1}( x_1^\prime)$ and $R_{\theta,t}(x_t^\prime, a_{t-1}^\prime|x_t, a_{t-1};\mathbf{x}_{t-1}) = \rho_{\theta,t}(x_t^\prime, a_{t-1}^\prime| \mathbf{x}_{t-1})$, the extended target $\tilde{\pi}( \theta,b_{1:T},\mathbf{a}_{1:T-1}^{-b_{2:T}}, \mathbf{x}_{1:T})$ coincides exactly with the extended target associated with the \gls{FAAPF} described in Section~\ref{Section:perfectadaptationPF}. As explored in the following two sections, this allows us to understand this \gls{EHMM} approach as the incorporation of a slightly more general class of \glspl{PF} within a \gls{PMCMC} framework and ultimately suggests further generalisations of these algorithms.
\subsection{Metropolis--Hastings algorithm} \label{subsec:mcmc-fa-apf}
We now consider the following \gls{MH} algorithm to sample from $\smash{\tilde{\pi}(\theta,b_{1:T}, \mathbf{x}_{1:T}, \mathbf{a}_{1:T-1}^{-b_{2:T}})}$. It relies on a proposal of the form \begin{equation}
q{\bigl( \theta,{\theta'}\bigr)} \times \underbrace{\Psi_{{\theta'}}{\bigl(\mathbf{x}_{1:T}, \mathbf{a}_{1:T-1}\bigr)} }_{\mathclap{\text{\parbox{2cm}{\centering\footnotesize{law of \gls{MCMCFAAPF}}}}}} \times \underbrace{\frac{1}{N}}_{\mathclap{\text{\parbox{1.3cm}{\centering\footnotesize{path selection}}}}}, \label{eq:proposalnovelEHMM} \end{equation} i.e.\ to sample $b_{1:T}$, we pick $b_T$ uniformly at random, then set $\smash{b_t=a_t^{b_{t+1}}}$ for $t=T-1,\dotsc,1$. Moreover, \begin{align}
\Psi_\theta{\bigl( \mathbf{x}_{1:T}, \mathbf{a}_{1:T-1}\bigr)}
& = \rho_{\theta,1}{\bigl( x_1^1\bigr)} \smashoperator{\prod_{\smash{i=2}}^N} R_{\theta,1}{\bigl(x_1^i\bigr| x_1^{i-1}\bigr)}\\
& \quad \times \smashoperator{\prod_{t=2}^{\smash{T}}} \; \Bigl\{
\rho_{\theta,t}{\bigl( x_t^1, a_{t-1}^1\bigr| \mathbf{x}_{t-1}\bigr)} \smashoperator{\prod_{i=2}^{\smash{N}}} R_{\theta,t}{\bigl(x_t^{i},a_{t-1}^{i}\bigr| x_t^{i-1},a_{t-1}^{i-1};\mathbf{x}_{t-1}\bigr)} \Bigr\} \label{eq:novelPFforEHMM} \end{align} is the law of a novel \gls{PF} type algorithm, which we refer to as the \gls{MCMCFAAPF}; again the reason for this terminology should become clear below.
The \gls{MCMCFAAPF} proceeds as follows. \begin{enumerate}
\item At time $1$, sample $x_1^1\sim\rho_{\theta,1}(x_1^1)$ and then $\mathbf{x}_1^{-1}\sim \prod_{i=2}^N R_{\theta,1}(x_1^i| x_1^{i-1})$.
\item At time $t=2,\dotsc,T$, sample
\begin{enumerate}
\item $(x_t^1, a_{t-1}^1) \sim\rho_{\theta,t}(x_t^1, a_{t-1}^1| \mathbf{x}_{t-1})$,
\item $(\mathbf{x}_t^{-1},\mathbf{a}_{t-1}^{-1}) \sim \prod_{i=2}^N R_{\theta,t}(x_t^{i},a_{t-1}^{i}| x_t^{i-1},a_{t-1}^{i-1};\mathbf{x}_{t-1})$.
\end{enumerate} \end{enumerate}
If $R_{\theta,1}(x_1^\prime\bigr| x_1) = \rho_{\theta,1}(x_1^\prime)$ and $R_{\theta,t}(x_t^\prime, a_{t-1}^\prime\bigr|x_t, a_{t-1};\mathbf{x}_{t-1}) = \rho_{\theta,t}(x_t^\prime, a_{t-1}^\prime\bigr| \mathbf{x}_{t-1})$, this corresponds to the standard \gls{FAAPF}.
The resulting \gls{MH} algorithm targeting the extended distribution defined in \eqref{eq:embeddedHMM2003target} and using the proposal defined in \eqref{eq:proposalembeddedHMM2003} admits an acceptance probability of the form \begin{equation}
1 \wedge \frac{\hat{p}_{{\theta'}}(y_{1:T}) p({\theta'})}{\hat{p}_\theta(y_{1:T}) p(\theta)}\frac{q({\theta'},\theta)}{q(\theta,{\theta'})}, \label{eq:alternative_ehmm_acceptance_probability} \end{equation} i.e.\ it looks very much like the \gls{PMMH}, except that here $\hat{p}_\theta(y_{1:T})$ is given by the expression in \eqref{eq:likelihoodestimatorperfectadaptationPF} with particles generated via \eqref{eq:novelPFforEHMM}. Note that this estimate is unbiased.
The validity of the acceptance probability in \eqref{eq:alternative_ehmm_acceptance_probability} can be established by calculating \begin{align}
\MoveEqLeft \frac{\tilde{\pi}{\bigl( \theta,b_{1:T}, \mathbf{x}_{1:T}, \mathbf{a}_{1:T-1}^{-b_{2:T}}\bigr)}}{\Psi_\theta{\bigl( \mathbf{a}_{1:T-1},\mathbf{x}_{1:T}\bigr)} \frac{1}{N}}\\
& = N\pi{\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr)} \frac{\prod_{i=b_1-1}^1 \widetilde{R}_{\theta,1}{\bigl(x_1^i\bigr| x_1^{i+1}\bigr)} \cdot \prod_{i=b_1+1}^N R_{\theta,1}{\bigl(x_1^i\bigr| x_1^{i-1}\bigr)}}{\rho_{\theta,1}{\bigl( x_1^1\bigr)} \prod_{i=2}^N R_{\theta,1}{\bigl(x_1^j\bigr| x_1^{i-1}\bigr)}}\\
& \quad \times \prod_{t=2}^T \frac{\prod_{i=b_t -1}^1
\widetilde{R}_{\theta,t}{\bigl(x_t^i,a_{t-1}^i\bigr|x_t^{i+1},a_{t-1}^{i+1};\mathbf{x}_{t-1}\bigr)} \cdot \prod_{i=b_t+1}^N R_{\theta,t}{\bigl(x_t^{i+1},a_{t-1}^{i+1}\bigr| x_t^i,a_{t-1}^i;\mathbf{x}_{t-1}\bigr)}}{\rho_{\theta,t}{\bigl(x_t^{1}, a_{t-1}^{1}\bigr| \mathbf{x}_{1:t-1}\bigr)} \prod_{i=2}^N R_{\theta,t}{\bigl(x_t^{i+1},a_{t-1}^{i+1}\bigr| x_t^i,a_{t-1}^j;\mathbf{x}_{t-1}\bigr)}}\\
& = \frac{N^{T-1}\pi{\bigl( \theta,x_{1:T}^{b_{1:T}}\bigr)} }{\rho_{\theta,1}{\bigl( x_1^{b_1}\bigr)}
\prod_{t=2}^T \rho_{\theta,t}{\bigl(x_t^{b_t}, a_{t-1}^{b_t}\bigr|\mathbf{x}_{1:t-1}\bigr)} }\\
& = \frac{N^{T-1}\pi{\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr)}}{\frac{p_\theta(x_1^{b_1}, y_1)}{p_\theta(y_1)} \prod_{t=2}^T \frac{f_\theta(x_t^{b_t}| x_{t-1}^{b_{t-1}}) g(y_t| x_t^{b_t})}{\sum_{i=1}^N p_\theta(y_t| x_{t-1}^i) } }
= p(\theta | y_{1:T}) \frac{\hat{p}_\theta(y_{1:T})}{p_\theta(y_{1:T})}. \end{align} We have again used identity \eqref{eq:useful_reversal_kernel_identity} and additionally that $\smash{b_t=a_t^{b_{t+1}}}$, for $t = T-1,\dotsc, 1$.
\subsection{Gibbs sampler} \label{subsec:gibbs_mcmc-fa-apf}
The \gls{EHMM} method of \cite{ShestopaloffNeal2016} can be reinterpreted as a collapsed Gibbs sampler to sample from the extended target distribution $\tilde{\pi}(\theta,b_{1:T},\mathbf{x}_{1:T}, \mathbf{a}_{1:T-1}^{-b_{2:T}})$. Given a current value of $x_{1:T}$, the algorithm proceeds as follows. \begin{enumerate}
\item Sample $b_{1:T}$ uniformly at random and set $x_{1:T}^{b_{1:T}}\leftarrow x_{1:T}$.
\item Run the conditional \gls{MCMCFAAPF}, i.e.\ sample from $\phi_\theta(\mathbf{x}_{1:T}^{-b_{1:T}},\mathbf{a}_{1:T-1}^{-b_{2:T}}| x_{1:T}^{b_{1:T}},b_{1:T})$.
\item Sample $b_T$ according to $\Pr(b_T=m) = 1/N$ and then, for $t=T-1,\dotsc,1$, sample $b_t$ according to a distribution proportional to $f_\theta(x_{t+1}^{b_{t+1}}| x_t^{b_t})$. \end{enumerate} The validity of the algorithm is established using a detailed balance argument in \citet{ShestopaloffNeal2016}. Alternatively, we can show using simple calculations similar to the ones presented earlier that \begin{equation}
\tilde{\pi}\bigl( b_t\big| \theta,\mathbf{x}_{1:t}, x_{t+1:T}^{b_{t+1:T}}, b_{t+1:T}\bigr)
\propto f_\theta\bigl(x_{t+1}^{b_{t+1}} \big| x_t^{b_t}\bigr) . \end{equation}
In the standard conditional \gls{PF}, the particles are conditionally independent given the previously sampled values. The conditional \gls{MCMCFAAPF} allows for conditional dependence between all the particles (and ancestor indices) generated in one time step. Indeed, we can choose the kernels $R_{\theta, t}^{\mathbf{q}_\theta}(\,\cdot\,|\,\cdot\,;\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2})$ such that they induce only small, local moves. This can improve the performance of \gls{PG} samplers in high dimensions: as with standard \gls{MCMC} schemes, less ambitious local moves are much more likely to be accepted. Of course, as with any local proposal one could not expect such a strategy to work well with strongly multi-modal target distributions without further refinements.
\section{Novel practical extensions} \label{sec:novel_methodology}
Motivated by the connections identified above, we now develop extensions based upon the more general \gls{PMCMC} algorithms described above, in particular considering constructions based around general \glspl{APF}. In particular, we relax the requirement in the \gls{MH} algorithm from Section~\ref{subsec:mcmc-fa-apf} that it is possible to sample from the proposal distribution of the \gls{FAAPF} (which is possible in only a small number of tractable models) and to compute its associated importance weight.
\subsection{MCMC APF}
Generalising the \gls{MCMCFAAPF} in the same manner as the \gls{APF} generalises the \gls{FAAPF} leads us to propose a (general) \emph{\gls{MCMCAPF}}. Set \begin{align}
\rho_{\theta,t}^{\mathbf{q}_\theta}(x_t,a_{t-1}|\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2})
& =
\begin{cases}
q_{\theta,1}(x_1), & \text{if $t = 1$,}\\
\dfrac{v_{\theta,t-1}^{a_{t-1}}}{\sum_{i=1}^N v_{\theta,t-1}^i} q_{\theta,t}(x_t|x_{t-1}^{a_{t-1}}), & \text{if $t> 1$,}
\end{cases} \end{align}
where $v_{\theta,t-1}^i$ are as defined in \eqref{eq:apfweights}, and is responsible for the dependence upon $a_{t-1}$ and $x_{t-2}$ in particular, and we allow $\smash{R_t^{\mathbf{q}_\theta}(\,\cdot\,|\,\cdot\,;\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2})}$ and $\smash{\widetilde{R}_{t}^{\mathbf{q}_\theta}(\,\cdot\,|\,\cdot\,;\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2})}$ to respectively denote a $\rho_{\theta,t}^{\mathbf{q}_\theta}(\,\cdot\,|\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2})$-invariant Markov kernel and the associated reversal kernel. Although this expression superficially resembles the mixture proposal of the \emph{marginalised} \gls{APF} \citep{Klass2005}, by explicitly including the ancestry variables it avoids incurring the $O(N^2)$ cost and allows an approximation of smoothing distributions. We then define the law of the \gls{MCMCAPF} via: \begin{align}
\Psi^{\mathbf{q}_\theta}_\theta{\bigl(\mathbf{x}_{1:T}, \mathbf{a}_{1:T-1}\bigr)}
& \coloneqq \rho^{\mathbf{q}_\theta}_{\theta,1}{\bigl( x_1^1\bigr)}
\smashoperator{\prod_{\smash{i=2}}^N} R^{\mathbf{q}_\theta}_{\theta,1}{\bigl(x_1^i\bigr| x_1^{i-1}\bigr)}\\
& \quad \times
\smashoperator{\prod_{t=2}^{\smash{T}}}\;\Bigl\{\rho^{\mathbf{q}_\theta}_{\theta,t}{\bigl(x_t^1, a_{t-1}^1\bigr| \mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2}\bigr)}
\bigr. \smashoperator{\prod_{i=2}^{\smash{N}}}
R^{\mathbf{q}_\theta}_{\theta,t}{\bigl(x_t^{i},a_{t-1}^{i}\bigr| x_t^{i-1},a_{t-1}^{i-1};\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2}\bigr)} \Bigr\}. \end{align} The corresponding extended \gls{PMCMC} target distribution is simply: \begin{equation}
\tilde{\pi}^{\mathbf{q}_\theta}{\bigl( \theta,b_{1:T},\mathbf{x}_{1:T}, \mathbf{a}_{1:T-1}^{-b_{2:T}}\bigr)}
= \frac{1}{N^T} \times \underbrace{\pi\bigl(\theta,x_{1:T}^{b_{1:T}}\bigr) }_{\text{\footnotesize{target}}} \times \underbrace{\phi^{\mathbf{q}_\theta}_{\theta}\bigl( \mathbf{x}_{1:T}^{-b_{1:T}},\mathbf{a}_{1:T-1}^{-b_{2:T}}\big\vert x_{1:T}^{b_{1:T}},b_{1:T}\bigr)}_{\text{\footnotesize{law of conditional \gls{MCMCAPF}}}}, \label{eq:novelEHMMtarget} \end{equation} where, as might be expected: \begin{align}
\MoveEqLeft \phi^{\mathbf{q}_\theta}_\theta{\bigl(\mathbf{x}_{1:T}^{-b_{1:T}},\mathbf{a}_{1:T-1}^{-b_{2:T}}\bigr| x_{1:T}^{b_{1:T}},b_{1:T}\bigr)}\\*
& = \smashoperator{\prod_{\smash{i=b_1-1}}^1}
\widetilde{R}^{\mathbf{q}_\theta}_{\theta,1}{\bigl(x_1^i\bigr| x_1^{i+1}\bigr)} \cdot \smashoperator{\prod_{\smash{i=b_1+1}}^N} R^{\mathbf{q}_\theta}_{\theta,1}{\bigl(x_1^i\bigr| x_1^{i-1}\bigr)}\\
& \quad \times \smashoperator{\prod_{t=2}^{\smash{T}}} \;\; \Bigl\{\smashoperator{\prod_{i=b_t-1}^{\smash{1}}} \widetilde{R}^{\mathbf{q}_\theta}_{\theta,t}{\bigl(x_t^i,a_{t-1}^i\bigr| x_t^{i+1},a_{t-1}^{i+1};\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2}\bigr)} \bigr. \smashoperator{\prod_{i=b_t+1}^{\smash{N}}} R^{\mathbf{q}_\theta}_{\theta,t}{\bigl(x_t^{i},a_{t-1}^{i}\bigr| x_t^{i-1},a_{t-1}^{i-1};\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2}\bigr)} \Bigr\}. \end{align}
Note that the \gls{MCMCFAAPF} can be viewed as a special case of the \gls{MCMCAPF} in much the same way that the \gls{FAAPF} from Section~\ref{Section:perfectadaptationPF} can be viewed as a special case of the (general) \gls{APF} from Section~\ref{subsec:apf}.
\subsection{Metropolis--Hastings algorithms}
We arrive at a \gls{PMMH}-type algorithm based around the \gls{MCMCAPF} by considering proposal distributions of the form: \begin{equation}
q{\bigl(\theta,{\theta'}\bigr)} \times \underbrace{\Psi^{\mathbf{q}_\theta'}_{{\theta'}}{\bigl(\mathbf{x}_{1:T}, \mathbf{a}_{1:T-1}\bigr)}}_{\mathclap{\text{\parbox{2.3cm}{\centering\footnotesize{law of \gls{MCMCAPF}}}}}} \times \underbrace{w_{{\theta'},T}^{b_T}}_{\mathclap{\text{\parbox{1.3cm}{\centering\footnotesize{path selection}}}}}, \end{equation} where, as in Section~\ref{subsec:apf}, $\smash{w_{\theta,T}^i = v_{\theta,T}^i / \sum_{j=1}^N v_{\theta,T}^j}$ and $\smash{\hat{p}_\theta(y_{1:T}) = \prod_{t=1}^T N^{-1} \sum_{i=1}^N v_{\theta,t}^i}$ is again an unbiased estimate of the marginal likelihood.
Note that the \gls{PMMH}-type variant of the \gls{MCMCFAAPF} cannot often be used in realistic scenarios because it requires sampling from $p_\theta(x_t|x_{t-1}, y_t)$ and evaluating $x_{t-1} \mapsto p_\theta(y_t|x_{t-1})$ in order to implement the \gls{FAAPF} in \eqref{eq:mcmc_fa-apf_transition_density}. To circumvent this problem, we can define a special case of the \gls{MCMCAPF} algorithm which requires neither sampling from $p_\theta(x_t|x_{t-1}, y_t)$ nor evaluating $x_{t-1} \mapsto p_\theta(y_t|x_{t-1})$. This algorithm, obtained by setting $\tilde{p}_\theta(y|x) \equiv 1$, will be called \emph{\gls{MCMCPF}} as it represents an analogue of the (bootstrap) \gls{PF}. At time~$1$, the \gls{MCMCPF} uses the \gls{MCMC} kernels $\overline{R}_{\theta,1}$ which are invariant w.r.t.\ $\bar{\rho}_{\theta,1}(x_1) \coloneqq \mu_\theta(x_1)$. At time~$t$, $t > 1$, the \gls{MCMCPF} uses the kernels $\overline{R}_{\theta,t}(\,\cdot\,|\,\cdot\,;\mathbf{x}_{t-1})$ which are invariant w.r.t.\ \begin{align}
\bar{\rho}_{\theta,t}(x_t,a_{t-1}|\mathbf{x}_{t-1}) \coloneqq \frac{g_\theta(y_{t-1}|x_{t-1}^{a_{t-1}})}{\sum_{i=1}^N g_\theta(y_{t-1}|x_{t-1}^i)} f_\theta(x_t|x_{t-1}^{a_{t-1}}). \label{eq:mcmc_fa-apf_transition_density} \end{align} The \gls{PMMH}-type variant of the \gls{MCMCPF} may be useful if the \gls{PMMH}-type variant of the \gls{MCMCFAAPF} cannot be implemented.
\subsection{Gibbs samplers}
Given the extended target construction of the \gls{MCMCAPF} algorithm, it is straightforward to implement \gls{PG} algorithms \gls{BS} (or similarly with \gls{AS} -- see Section~\ref{subsec:generalised_PGS}) which target it.
However, Gibbs samplers based around the (conditional) \gls{MCMCPF} do not appear useful as they might be expected to perform less well than the Gibbs sampler based around the \gls{MCMCFAAPF} and are no more easy to implement: in contrast to the \gls{PMMH}-type algorithms, the Gibbs sampler based around the (conditional) \gls{MCMCFAAPF} does \emph{not} generally require sampling from $p_\theta(x_t|x_{t-1}, y_t)$ and it only requires evaluation of the unnormalised density $p_\theta(y_t|x_t)f_\theta(x_t|x_{t-1})$ in the transition density of the \gls{FAAPF} in \eqref{eq:mcmc_fa-apf_transition_density}.
\section{General particle Markov chain Monte Carlo methods} \label{sec:general_pmcmc}
In this section, we describe a slight generalisation of \gls{PMCMC} methods which admits both the standard \gls{PMCMC} methods from Section~\ref{sec:pmcmc} as well as the alternative \gls{EHMM} methods from Section~\ref{sec:embeddedHMMnewversion} as special cases. In addition, we derive both the \glsdesc{BS} and \glsdesc{AS} recursions for this algorithm. We note that this section is necessarily slightly more abstract than the previous sections. As the details developed below are not required for understanding the remainder of this work, this section may be skipped on a first reading.
\subsection{Extended target distribution} We define $\mathbf{z}_1 \coloneqq \mathbf{x}_1$ and $\mathbf{z}_t \coloneqq (\mathbf{x}_t, \mathbf{a}_{t-1})$. For notational brevity, also define $\smash{\mathbf{z}_1^{-i} \coloneqq \mathbf{z}_1 \setminus x_1^i}$, $\smash{\mathbf{z}_t^{-i} \coloneqq \mathbf{z}_t \setminus (x_t^i, a_{t-1}^i)}$ as well as $\smash{\mathbf{z}_{1:t}^{-b_{1:t}} = (\mathbf{z}_{1}^{-b_1}, \dotsc, \mathbf{z}_{t}^{-b_t})}$. We note that further auxiliary variables could be included in $\mathbf{z}_t$ without changing anything in the construction developed below. The law of a general \gls{PF} is given by \begin{align}
\Psi_\theta(\mathbf{z}_{1:T}) \coloneqq \psi_{\theta,1}(\mathbf{z}_1) \prod_{t=2}^T \psi_{\theta,t}(\mathbf{z}_t|\mathbf{z}_{1:t-1}). \end{align} With this notation, general \gls{PMCMC} methods target the following extended distribution: \begin{align}
\tilde{\pi}(\theta, \mathbf{z}_{1:T}, b_T) \coloneqq \frac{1}{N^T} \times \underbrace{\pi(\theta, x_{1:T}^{b_{1:T}})}_{\text{\footnotesize{target}}} \times \underbrace{\phi_\theta(\mathbf{z}_{1:T}^{-b_{1:T}} | x_{1:T}^{b_{1:T}}, b_{1:T})}_{\text{\parbox{2.3cm}{\centering\footnotesize{law of conditional general \gls{PF}}}}}, \label{eq:general_pmcmb_target_distribution} \end{align} where the law of the conditional general \gls{PF} is given by \begin{align}
\phi_\theta(\mathbf{z}_{1:T}^{-b_{1:T}} | x_{1:T}^{b_{1:T}}, b_{1:T}) \coloneqq \psi_{\theta,1}^{-b_1}(\mathbf{z}_1^{-b_1}) \prod_{t=2}^T \psi_{\theta,t}^{-b_t}(\mathbf{z}_t^{-b_t}|\mathbf{z}_{1:t-1}, x_t^{b_t}), \end{align} with \begin{align}
\psi_{\theta,t}^{-i}(\mathbf{z}_t^{-i}|\mathbf{z}_{1:t-1}, x_t^i)
& \coloneqq \frac{\psi_{\theta,t}(\mathbf{z}_t|\mathbf{z}_{1:t-1})}{\psi_{\theta,t}^i(x_t^i, a_{t-1}^i|\mathbf{z}_{1:t-1}, x_t^i)}. \end{align}
Here, $\psi_{\theta,t}^i(\,\cdot\,|\mathbf{z}_{1:t-1})$ denotes the marginal distribution of the $i$th components of $\mathbf{x}_t$ and $\mathbf{a}_{t-1}$ under the distribution $\psi_{\theta,t}(\,\cdot\,|\mathbf{z}_{1:t-1})$. Finally, for any $t \in \{1,\dotsc,T\}$, we define the following unnormalised weight
\begin{align}
\tilde{v}_{\theta, t}^{b_t} \coloneqq \frac{1}{N^t} \frac{\gamma_{\theta,t}(x_{1:t}^{b_{1:t}})}{\psi_{\theta,1}^{b_1}(x_1^{b_1}) \prod_{n=2}^t \psi_{\theta,n}^{b_n}(x_n^{b_n}, a_{n-1}^{b_n}|\mathbf{z}_{1:n-1})}, \end{align} where $b_{1:t-1}$ on the r.h.s.\ are to be interpreted as functions of $b_t$ and the ancestry variables via the usual recursion $\smash{b_t = a_t^{b_{t+1}}}$. Here, $\gamma_{\theta,t}(x_{1:t})$ is the unnormalised density targeted at the $t$th step of the general \gls{PF} -- for all the algorithms discussed in this work, we will state these densities explicitly in Appendix~\ref{subsec:special_cases}; in particular, \begin{equation}
\gamma_{\theta,T}(x_{1:T}) = p_\theta(x_{1:T}, y_{1:T}). \end{equation}
We make the following minimal assumption to ensure the validity of the (general) \gls{PMCMC} algorithms.
\begin{assumption}[absolute continuity] \label{as:absolute_continuity}
For any $t \in \{1,\dotsc,T\}$, any $i \in \{1,\dotsc,N\}$ and any $\mathbf{z}_{1:t-1}$, the support of $(x_t, b_{t-1}) \mapsto \psi_{\theta,t}^i(x_t, b_{t-1}|\mathbf{z}_{1:t-1})$ includes the support of $(x_t, b_{t-1}) \mapsto \gamma_{\theta,t}(x_{1:t-1}^{b_{1:t-1}}, x_{t})$. \end{assumption}
We also make the following assumption which requires that all marginals of the conditional distributions $\psi_{\theta,t}(\,\cdot\,|\mathbf{z}_{1:t-1})$ are identical.
\begin{assumption}[identical marginals] \label{as:exchangeability}
For any $(i,j) \in \{1,\dotsc, N\}^2$ and any $t \in \{1,\dotsc,T\}$, $\psi_{\theta,t}^i = \psi_{\theta,t}^j$. \end{assumption}
\begin{remark}\label{rem:identical_marginals_assumption}
Assumption~\ref{as:exchangeability} can be easily dropped in favour of selecting a suitable (non-uniform) distribution for the particle indices $b_{1:T}$ in \eqref{eq:general_pmcmb_target_distribution}. Indeed, more elaborate constructions could be used to justify resampling schemes which, unlike multinomial resampling, are not exchangeable in the sense of \citet[Assumption~2]{AndrieuDoucetHolenstein2010} (unless one permutes the particle indices uniformly at random at the end of each step as mentioned in \citet{AndrieuDoucetHolenstein2010}). Similarly, such more general constructions would allow us to view the use of more sophisticated \glspl{PF}, such as the \emph{discrete particle filter} of \citet{Fearnhead1998}, with \gls{PMCMC} schemes as special cases of this framework as shown in \citet[Section~2.3.4]{Finke2015}. \end{remark}
In Examples~\ref{ex:antithetic} and \ref{ex:sqmc}, we show how \glspl{APF} with \emph{antithetic variables} \citep{BizjajevaOlsson2008} and (randomised) \emph{\gls{SQMC} methods} \citep{GerberChopin2015} can be considered as special cases of the framework described in this section even though these methods cannot easily be viewed as conventional \glspl{PF} because the particles are not sampled conditionally independently at each step.
\begin{example}[\glspl{APF} with antithetic variables] \label{ex:antithetic}
The \glspl{APF} with antithetic variables from \citet{BizjajevaOlsson2008} aim to improve the performance of \glspl{APF} by introducing negative correlation into the particle population. To that end, the $N$ particles are divided into $M$ groups of $K$ particles; the particles in each group then share the same ancestor index and given the ancestor particle, they are sampled in such a way that they are negatively correlated.
Assume that there exists $K, M \in \mathbb{N}$ such that $N = K M$ and for $\tilde{x}_t \coloneqq (\tilde{x}_t^1, \dotsc, \tilde{x}_t^K) \in \mathcal{X}^K$ let $\tilde{q}_{\theta,t}(\tilde{x}_t| x_{t-1})$ denote some joint proposal kernel for $K$ particles such that if $(\tilde{x}_t^1, \dotsc, \tilde{x}_t^K) \sim \tilde{q}_{\theta,t}(\,\cdot\,| x_{t-1})$ then
\begin{enumerate*}
\item $\tilde{x}_t^1, \dotsc, \tilde{x}_t^K$ are (pairwise) negatively correlated,
\item marginally, $\smash{\tilde{x}_t^k \sim q_{\theta,t}(\,\cdot\,| x_{t-1})}$ for all $k \in \{1,\dotsc,K\}$.
\end{enumerate*}
Given $\mathbf{z}_{1:t-1}$, the \gls{APF} with antithetic variables generates $\mathbf{z}_t = (\mathbf{a}_{t-1}, \mathbf{x}_t)$ as follows (we use the convention that any action prescribed for some $m$ is to be performed for all $m \in \{1,\dotsc,M\}$).
\begin{enumerate}
\item \label{ex:antithetic:step:1} Set $\smash{a_{t-1}^{(m-1)K+1} = i}$ w.p.\ proportional to $v_{\theta, t-1}^i$.
\item Set $\smash{a_{t-1}^{(m-1)K+k} \coloneqq a_{t-1}^{(m-1)K+1}}$ for all $k \in \{2,\dotsc,K\}$.
\item \label{ex:antithetic:step:2} Sample $\smash{\bigl(x_t^{(m-1)K+k}\bigr)_{k \in \{1,\dotsc,K\}} \sim \tilde{q}_{\theta,t}\bigl(\,\cdot\,\big| x_{t-1}^{a_{t-1}^{(m-1)K+1}}\bigr)}$.
\item \label{ex:antithetic:step:3} Permute the particle indices on $\mathbf{z}_t^1, \dotsc, \mathbf{z}_t^N$ uniformly at random.
\end{enumerate}
\end{example}
\begin{example}[sequential quasi Monte Carlo]\label{ex:sqmc}
Let $\mathcal{X} = \mathbb{R}^d$. Randomised \gls{SQMC} algorithms are general \glspl{PF} which stratify sampling of the ancestor indices and particles $\mathbf{z}_t = (\mathbf{a}_{t-1}, \mathbf{x}_t)$ by computing them as a deterministic transformation of a set of randomised quasi Monte Carlo points $\mathbf{u}_t \coloneqq (u_t^1, \dotsc, u_t^N) \in [0,1)^{(d+1)N}$. By construction,
\begin{enumerate*}
\item \label{ex:sqmc:property:1} the set $\mathbf{u}_t = (u_t^1, \dotsc, u_t^N)$ has a low discrepancy,
\item \label{ex:sqmc:property:2} for each $i \in \{1,\dotsc,N\}$, $u_t^i$ is (marginally) uniformly distributed on the $(d+1)$-dimensional hypercube.
\end{enumerate*}
Write $u_t^i = (\tilde{u}_t^i, \tilde{v}_t^i)$ with $\tilde{u}_t^i \in [0,1)$ and $\tilde{v}_t^i \in [0,1)^{d}$. Given $\mathbf{z}_{1:t-1}$, the algorithm \citep[Algorithm~3]{GerberChopin2015} transforms $\mathbf{u}_t \to \mathbf{z}_t = (\mathbf{a}_{t-1}, \mathbf{x}_t)$ as follows (using the convention that any action mentioned for some $i$ is to be performed for all $i \in \{1,\dotsc,N\}$).
\begin{enumerate}
\item \label{ex:sqmc:step:1} Find a suitable permutation $\sigma_{t-1}\colon \{1,\dotsc, N\} \to \{1,\dotsc,N\}$ such that $x_{t-1}^{\sigma_{t-1}(1)} \leq \dotsc \leq x_{t-1}^{\sigma_{t-1}(N)}$, if $d=1$; if $d > 1$, the permutation $\sigma_{t-1}$ is obtained by mapping the particles to the hypercube $[0,1)^{d}$ and projecting them onto $[0,1)$ using the pseudo-inverse of the Hilbert space-filling curve. These projections are then ordered as for $d=1$ (see \citet{GerberChopin2015} for details).
\item \label{ex:sqmc:step:2} Set $\smash{a^i \coloneqq F^{-1}(\tilde{u}_t^i)}$, where $F^{-1}$ denotes the generalised inverse of the \gls{CDF} $F\colon \{1,\dotsc,N\} \to [0,1]$, defined by $\smash{F(i) \coloneqq \sum_{j=1}^i v_{\theta,t-1}^{\sigma_{t-1}(j)} / \sum_{j=1}^N v_{\theta,t-1}^j}$.
\item \label{ex:sqmc:step:3} Set $\smash{a_{t-1}^i \coloneqq \sigma_{t-1}(a^i)}$ and $\smash{x_t^i \coloneqq \varGamma_{\theta,t}(x_{t-1}^{a_{t-1}^i}, \tilde{v}_t^i)}$. Here, if $d=1$, the function $\varGamma_{\theta,t}(x_{t-1}, \,\cdot\,)$ is the (generalised) inverse of the \gls{CDF} associated with $q_{\theta,t}(\,\cdot\,|x_{t-1})$; if $d > 1$, this can be generalised via the Rosenblatt transform.
\item \label{ex:sqmc:step:4} Permute the particle indices on $\mathbf{z}_t^1, \dotsc, \mathbf{z}_t^N$ uniformly at random.
\end{enumerate} \end{example}
While the joint kernel $\psi_{\theta,t}(\mathbf{z}_t|\mathbf{z}_{1:t-1})$ is potentially intractable in both examples, the random permutation of the particle indices (i.e.\ Step~\ref{ex:antithetic:step:3} in Example~\ref{ex:antithetic} and also Step~\ref{ex:sqmc:step:4} in Example~\ref{ex:sqmc}) ensures that Assumption~\ref{as:exchangeability} is satisfied. Indeed, it can be easily verified that in both examples, for any $(i,j) \in \{1,\dotsc,N\}^2$,
\begin{align}
\psi_{\theta,t}^i(x_t,a_{t-1}|\mathbf{z}_{1:t-1})
= \rho_{\theta,t}^{\mathbf{q}_\theta}(x_t, a_{t-1}|\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2})
= \psi_{\theta,t}^j(x_t, a_{t-1}|\mathbf{z}_{1:t-1}).
\end{align}
As pointed out in Remark~\ref{rem:identical_marginals_assumption}, Assumption~\ref{as:exchangeability} is not actually necessary and can be easily dropped in favour of a slightly more general construction of the extended target distribution which is implicitly employed by \citet{BizjajevaOlsson2008, GerberChopin2015} (who therefore do not require the random permutation of the particle indices).
\subsection{General particle marginal Metropolis--Hastings}
In this section, we use the general \gls{PMCMC} framework to derive a general \gls{PMMH} algorithm. All \gls{PMMH} algorithms and \gls{MH} versions of the alternative \gls{EHMM} methods can then be seen as special cases of this general scheme as shown in Appendix~\ref{subsec:special_cases}. As with the standard \gls{PMMH}, we may use an \gls{MH} algorithm to target the extended distribution $\tilde{\pi}(\theta, \mathbf{z}_{1:T}, b_T)$ using a proposal of the form \begin{equation}
q(\theta,{\theta'}) \times \underbrace{\Psi_{{\theta'}}(\mathbf{z}_{1:T})}_{\text{\footnotesize{\parbox{1.5cm}{\centering law of general \gls{PF}}}}} \times \underbrace{q_{\theta'}(b_T|\mathbf{z}_{1:T})}_{\text{\footnotesize{path selection}}}, \label{eq:proposalGeneralisedPF} \end{equation} where we have defined the selection probability \begin{align}
q_\theta(b_T|\mathbf{z}_{1:T}) \coloneqq \frac{\tilde{v}_{\theta, T}^{b_T}}{\sum_{i=1}^N \tilde{v}_{\theta, T}^i}. \end{align} Define the usual unbiased estimate of the marginal likelihood \begin{equation}
\hat{p}_\theta(y_{1:T}) \coloneqq \sum_{i=1}^N \tilde{v}_{\theta, T}^i. \end{equation} Then we obtain the following general \gls{PMMH} algorithm (Algorithm~\ref{alg:general_pmmh}) the validity of which can be established by checking that indeed, \begin{align}
\frac{\tilde{\pi}(\theta, \mathbf{z}_{1:T}, b_T)}{\Psi_\theta(\mathbf{z}_{1:T}) q_\theta(b_T|\mathbf{z}_{1:T})}
= p(\theta|y_{1:T}) \frac{\hat{p}_\theta(y_{1:T})}{p_\theta(y_{1:T})}. \end{align}
\noindent\parbox{\textwidth}{ \begin{flushleft} \begin{framedAlgorithm}[general \gls{PMMH} algorithm] \label{alg:general_pmmh} Given $(\theta, \mathbf{z}_{1:T}, b_T) \sim \tilde{\pi}(\theta, \mathbf{z}_{1:T}, b_T)$ with associated likelihood estimate $\hat{p}_{\theta}(y_{1:T})$.
\begin{enumerate}
\item Propose $\theta' \sim q(\theta, \theta')$, $\mathbf{z}_{1:T}' \sim \Psi_{\theta'}(\mathbf{z}_{1:T}')$ and $b_T' \sim q_{\theta'}(b_T'|\mathbf{z}_{1:T}')$.
\item Compute likelihood estimate $\hat{p}_{\theta'}(y_{1:T})$ based on $\mathbf{z}_{1:T}'$.
\item Set $(\theta, \mathbf{z}_{1:T}, b_T) \leftarrow (\theta', \mathbf{z}_{1:T}', b_T')$ w.p.\
$
1 \wedge
\dfrac{\smash{\hat{p}_{{\theta'}}(y_{1:T})p({\theta'})}}{\hat{p}_{\theta}(y_{1:T})p(\theta)}
\dfrac{\smash{q({\theta'},\theta)}}{q(\theta,{\theta'})}
. \label{eq:generalisedPMMHratio}
$
\end{enumerate} \end{framedAlgorithm} \end{flushleft} }
\subsection{General particle Gibbs samplers} \label{subsec:generalised_PGS} \glsreset{AS} \glsreset{BS}
In this section, we use the general \gls{PMCMC} framework to derive a general \gls{PG} sampler. We also derive \gls{BS} \citep{Whiteley2010} and \gls{AS} \citep{LindstenJordanSchon2014} recursions and prove that they leave the target distribution of interest invariant. As before, all \gls{PG} samplers and Gibbs versions of the alternative \gls{EHMM} method can then be seen as special cases of this general scheme as shown in Appendix~\ref{subsec:special_cases}. Set \begin{equation}
\gamma_\theta(x_{t+1:T}|x_{1:t}) \coloneqq \frac{\gamma_{\theta,T}(x_{1:T})}{\gamma_{\theta,t}(x_{1:t})}. \end{equation} We are then ready to state both (general) \gls{PG} samplers. For the remainder of this section, we let $\tilde{x}_{1:t}^i$ denote the $i$th particle lineage at time~$t$, i.e.\ $\smash{\tilde{x}_{1:t}^i = x_{1:t}^{i_{1:t}}}$, where $i_t = i$ and $\smash{i_n = a_n^{i_{n+1}}}$, for $n = t-1, \dotsc, 1$.
\noindent\parbox{\textwidth}{ \begin{flushleft} \begin{framedAlgorithm}[general \gls{PG} sampler with \gls{BS}] \label{alg:general_pg_bs} Given $(\theta, x_{1:T}) \sim \pi$, obtain $(\theta', x_{1:T}') \sim \pi$ as follows.
\begin{enumerate}
\item Sample $\theta'$ via some $\pi(\,\cdot\,|x_{1:T})$-invariant \gls{MCMC} kernel.
\item For $t = 1,\dotsc,T$, perform the following steps.
\begin{enumerate}
\item If $t=1$, sample $b_1$ uniformly on $\{1,\dotsc,N\}$, set $x_1^{b_1} \coloneqq x_1$ and sample $\mathbf{z}_1^{-b_1} \sim \psi_{\theta',1}^{-b_t}(\mathbf{z}_1^{-b_1}|x_1^{b_1})$.
\item If $t>1$, sample $b_t$ uniformly on $\{1,\dotsc,N\}$, set $x_t^{b_t} \coloneqq x_t$, $a_{t-1}^{b_t} \coloneqq b_{t-1}$ and sample $\mathbf{z}_t^{-b_t} \sim \psi_{{\theta'},t}^{-b_t}(\mathbf{z}_t^{-b_t}|\mathbf{z}_{1:t-1}, x_t^{b_t})$.
\end{enumerate}
\item Sample $b_T \sim q_{\theta'}(b_T|\mathbf{z}_{1:T})$ and for $t = T-1, \dotsc, 1$, set $b_t=i$ w.p.\ proportional to $\tilde{v}_{{\theta'}, t}^{i}\gamma_{\theta'}(x_{t+1:T}^{b_{t+1:T}}|\tilde{x}_{1:t}^i)$.
\item Set $x_{1:T}' \coloneqq x_{1:T}^{b_{1:T}}$.
\end{enumerate} \end{framedAlgorithm} \end{flushleft} }
\noindent\parbox{\textwidth}{ \begin{flushleft} \begin{framedAlgorithm}[general \gls{PG} sampler with \gls{AS}] \label{alg:general_pg_as} Given $(\theta, x_{1:T}) \sim \pi$, obtain $(\theta', x_{1:T}') \sim \pi$ as follows.
\begin{enumerate}
\item Sample $\theta'$ via some $\pi(\,\cdot\,|x_{1:T})$-invariant \gls{MCMC} kernel.
\item For $t = 1,\dotsc,T$, perform the following steps.
\begin{enumerate}
\item If $t=1$, sample $b_1$ uniformly on $\{1,\dotsc,N\}$, set $x_1^{b_1} \coloneqq x_1$ and sample $\mathbf{z}_1^{-b_1} \sim \psi_{\theta',1}^{-b_t}(\mathbf{z}_1^{-b_1}|x_1^{b_1})$.
\item If $t>1$, sample $b_t$ uniformly on $\{1,\dotsc,N\}$, set $x_t^{b_t} \coloneqq x_t$, set $a_{t-1}^{b_t} = i$ w.p.\ proportional to $\tilde{v}_{{\theta'}, t-1}^{i}\gamma_{\theta'}(x_{t:T}^{b_{t:T}}|\tilde{x}_{1:t-1}^{i})$ and sample $\mathbf{z}_t^{-b_t} \sim \psi_{{\theta'},t}^{-b_t}(\mathbf{z}_t^{-b_t}|\mathbf{z}_{1:t-1}, x_t^{b_t})$.
\end{enumerate}
\item Sample $b_T \sim q_{\theta'}(b_T|\mathbf{z}_{1:T})$ and for $t = T-1, \dotsc, 1$, set $b_t \coloneqq a_t^{b_{t+1}}$.
\item Set $x_{1:T}' \coloneqq x_{1:T}^{b_{1:T}}$.
\end{enumerate} \end{framedAlgorithm} \end{flushleft} }
As in previous sections, the \gls{BS} recursion in Algorithm~\ref{alg:general_pg_bs} may be justified via appropriate partially-collapsed Gibbs sampler arguments by noting that \begin{align}
\tilde{\pi}(b_t|\theta, \mathbf{z}_{1:t}, x_{t+1:T}^{b_{t+1:T}})
& \propto \tilde{v}_{{\theta}, t}^{b_t}\gamma_{\theta}(x_{t+1:T}^{b_{t+1:T}}|\tilde{x}_{1:t}^{b_t}). \end{align} The \gls{AS} steps in Algorithm~\ref{alg:general_pg_as} follows similarly since $a_t^{b_{t+1}} = b_t$, by construction.
Alternatively -- without invoking partially-collapsed Gibbs sampler arguments -- the validity of \gls{BS} can be established by even further extending the space to include the new particle indices generated via \gls{BS}. As shown in \citet[Chapter~3.4.3]{Finke2015}, this construction also proves a particular duality of \gls{BS} and \gls{AS}.
\section{Empirical study} \label{sec:simulations}
In this section, we empirically compare the performance of some of the algorithms described in this work on a $d$-dimensional linear-Gaussian state-space model.
\subsection{Model}
The model considered throughout this section is given by \begin{align}
\mu_\theta(x_1) & = \mathrm{Normal}(x_1; m_0, C_0),\\
f_\theta(x_t|x_{t-1}) & = \mathrm{Normal}(x_t; A x_{t-1}, \sigma^2 \mathbf{I}_d), \quad \text{for $t > 1$,}\\
g_\theta(y_t|x_t) & = \mathrm{Normal}(y_t; x_t, \tau^2 \mathbf{I}_d), \quad \text{for $t \geq 1$,} \end{align} where $x_t, y_t \in \mathbb{R}^d$, $\sigma, \tau > 0$, $\mathbf{I}_d$ denotes the $(d,d)$-dimensional identity matrix and $A$ is the $(d,d)$-dimensional symmetric banded matrix with upper and lower bandwidth $1$, with entries $a_0 \in \mathbb{R}$ on the main diagonal, and with entries $a_1 \in \mathbb{R}$ on the remaining bands, i.e.\ \begin{equation}
A =
\begin{bmatrix}
a_0 & a_1 & 0 & \dotsc & 0\\
a_1 & a_0 & a_1 & \ddots & \vdots\\
0 & a_1 & \ddots & \ddots & 0\\
\vdots & \ddots & \ddots & a_0 & a_1\\
0 & \dotsc & 0 & a_1 & a_0
\end{bmatrix}. \end{equation} For simplicity, we assume that the initial mean $m_0 \coloneqq \mathbf{0}_d \in \mathbb{R}^d$ (where $\mathbf{0}_d$ denotes a vector of zeros of length~$d$) and the initial $(d,d)$-dimensional covariance matrix $C_0 = \mathbf{I}_d$ are known. Thus, the task is to approximate the posterior distribution of the remaining parameters $\theta \coloneqq (a_0, a_1, \sigma, \tau)$. The true values of these parameters, i.e.\ the values used for simulating the data are $(0.5, 0.2, 1, 1)$. As prior distributions, we take uniform distributions on $(-1,1)$ for $a_0$ and $a_1$ and inverse-gamma distributions on $\sigma$ and $\tau$ each with shape parameter $1$ and scale parameter $0.5$. All parameters are assumed to be independent a priori. In all algorithms, we propose new values ${\theta'}$ for $\theta$ via a simple Gaussian random-walk kernel, i.e.\ we use $q(\theta, {\theta'}) \coloneqq \mathrm{Normal}({\theta'};\theta, (100 d_\theta d T)^{-1} \mathbf{I}_{d_\theta})$, where $d_\theta$ is the dimension of the parameter vector $\theta$, i.e.\ $d_\theta = 4$.
\subsection{Algorithms} \label{subsec:algorithms}
In this subsection, we detail the specific algorithms whose empirical performance we compare in our simulation study.
\begin{description}[leftmargin=0pt, labelindent=0pt]
\item[Standard \gls{PMCMC}.] We implement the (bootstrap) \gls{PF} and the \gls{FAAPF} using multinomial resampling at every step. Though we note that more sophisticated resampling schemes, e.g.\ adaptive systematic resampling, could easily be employed. As described above, we can implement both \gls{MH} algorithms (i.e.\ the \gls{PMMH}) and Gibbs samplers based around these standard \glspl{PF}. For the latter, we make use of \gls{AS} in the conditional \glspl{PF}.
\item[Original \gls{EHMM}.] We implement the algorithms with $\rho_{\theta,t}(x) = \mathrm{Normal}(x; \mu, \varSigma)$, where $\mu$ and $\varSigma$ represent the mean and covariance matrix associated with the stationary distribution of the latent Markov chain $(X_t)_{t \in \mathbb{N}}$. We compare two different options for constructing the kernels $R_{\theta,t}$ which leave this distribution invariant.
\begin{enumerate}[label=(\Roman*), ref=\Roman*]
\item \label{as:original_EHMM_kernel_a} The kernel $R_{\theta,t}$ generates \gls{IID} samples from its invariant distribution, i.e.\ $\smash{R_{\theta,t}(x_t'|x_t) = \rho_{\theta,t}(x_t')}$.
\item \label{as:original_EHMM_kernel_b} The kernel $R_{\theta,t}$ is a standard \gls{MH} kernel which proposes a value $x_t^\star$ using the Gaussian random-walk proposal $\smash{\mathrm{Normal}(x_t^\star; x_t, d^{-1}\mathbf{I}_d)}$.
\end{enumerate}
\item[Alternative \gls{EHMM}.] We compare four different versions of the \gls{MCMCPF} and \gls{MCMCFAAPF} methods outlined above. Again, we implement both \gls{MH} algorithms and Gibbs samplers (with \gls{AS}) based around these methods. Below, we describe the specific versions which we are comparing. The kernels $\overline{R}_{\theta, t}(\,\cdot\,|\,\cdot\,;\mathbf{x}_{t-1})$ employed in the \gls{MCMCPF} and the kernels $R_{\theta, t}(\,\cdot\,|\,\cdot\,;\mathbf{x}_{t-1})$ employed in the \gls{MCMCFAAPF} are all taken to be \gls{MH} kernels which, given $(x_t, a_{t-1})$, propose a new value $(x_t^\star, a_{t-1}^\star)$ using a proposal of the following form
\begin{equation}
\frac{v_{\theta,t-1}^{a_{t-1}^\star}}{\sum_{i=1}^N v_{\theta,t-1}^{i}} s_{\theta,t}(x_t^\star|x_t;\mathbf{x}_{t-1}, a_{t-1}^\star).
\end{equation}
We compare two different approaches for generating a new value for the particle, $x_t^\star$.
\begin{enumerate}[label=(\Roman*), ref=\Roman*]
\item \label{as:particle_proposal_a} The first proposal uses a simple Gaussian random-walk kernel, i.e.\
\begin{equation}
s_{\theta,t}(x_t^\star|x_t;\mathbf{x}_{t-1}, a_{t-1}^\star) = \mathrm{Normal}(x_t^\star; x_t, d^{-1}\mathbf{I}_d), \label{eq:rw_proposal}
\end{equation}
where the scaling of the covariance matrix is motivated by existing results on optimal scaling for such random-walk proposal kernels \citep{GelmanRobertsGilks1996,RobertsGelmanGilks1997}.
\item \label{as:particle_proposal_b} The second proposal uses the \emph{autoregressive} proposal employed by \cite{ShestopaloffNeal2016}, i.e.\
\begin{equation}
s_{\theta,t}(x_t^\star|x_t;\mathbf{x}_{t-1}, a_{t-1}^\star) = \mathrm{Normal}\bigl(x_t^\star;\mu + \sqrt{1-\varepsilon^2} (x_t - \mu), \varepsilon^2\varSigma\bigr), \label{eq:ar_proposal}
\end{equation}
where $\mu$ and $\varSigma$ denote the mean and covariance matrix of $f_\theta(x_t|x_{t-1}) = \mathrm{Normal}(x_t; \mu, \varSigma)$, i.e.\ $\varSigma=\sigma^2\mathbf{I}$ and $\mu=A x_{t-1}$. To scale the covariance matrix of this proposal with the dimension $d$, we set $\varepsilon \coloneqq \sqrt{d^{-1}}$.
\end{enumerate}
\item[Idealised.] We also implement the algorithms which the above-mentioned algorithms seek to mimic. The idealised Gibbs sampler, is a (Metropolis-within-)Gibbs algorithm which updates the latent states $x_{1:T}$ as one block by sampling them from their full conditional posterior distribution. The idealised marginal \gls{MH} algorithm analytically evaluates the marginal likelihood $p_\theta(y_{1:T})$ via the Kalman filter. \end{description}
\subsection{Results for general PMMH algorithms} In this subsection, we empirically compare the performance of various \gls{PMMH} type samplers. First, we fix $\theta$ in order to assess the variability of the estimates of the marginal likelihood, $\hat{p}_\theta(y_{1:T})$, which is a key ingredient in (general) \gls{PMMH} algorithms. Then, we perform inference about $\theta$.
Recall that in order to implement the \gls{MH} version of the \gls{MCMCFAAPF}, we need to sample at least one particle from $p_\theta(x_t|x_{t-1}, y_t)$ at each time~$t$ and we need to be able to evaluate the function $x_{t-1} \mapsto p_\theta(y_t|x_{t-1})$. In other words, whenever we can implement this algorithm we can also implement a standard \gls{PMMH} algorithm based around the \gls{FAAPF}.
Figure~\ref{fig:likelihood_estimates} shows the relative estimates of the marginal likelihood obtained from the various algorithms described in this work for various model dimensions. Unsurprisingly, the \gls{PF}, resp.\ \gls{FAAPF}, provides lower variance estimates than its corresponding \gls{MCMCPF}, resp.\ \gls{MCMCFAAPF} counterparts. However, more interestingly, the \gls{MCMCFAAPF} can provide lower variance estimates than the standard \gls{PF} and could prove useful in more realistic scenarios where it is computationally very expensive to run the \gls{FAAPF}. As expected, the original \gls{EHMM} method described in Section~\ref{sec:embeddedHMM} breaks down very quickly as the dimension $d$ increases.
\begin{figure}
\caption{$d=2$.}
\label{fig:likelihood_estimates_in_dimension_2}
\caption{$d=5$.}
\label{fig:likelihood_estimates_in_dimension_5}
\caption{$d=10$.}
\label{fig:likelihood_estimates_in_dimension_10}
\caption{$d=25$.}
\label{fig:likelihood_estimates_in_dimension_25}
\caption{Relative estimates of the marginal likelihood $p_\theta(y_{1:T})$. Based on $1\,000$ independent runs of each algorithm (and writing $\hat{p}_\theta(y_{1:T}) = \tilde{p}_\theta(y_{1:T})$ in the case of the original \gls{EHMM} method) with each run using a different data sequence of length $T=10$ simulated from the model. The number of particles was $N=1\,000$ for the $\mathrm{O}(N)$ methods and $N=100$ for the $\mathrm{O}(N^2)$ methods.}
\label{fig:likelihood_estimates}
\end{figure}
The right panel of Figure~\ref{fig:pmmh_acf_parameters} shows kernel-density plots of the estimates of parameter~$a_0$ obtained from various \gls{PMMH}-type algorithms. Clearly, the \gls{PMMH}-type algorithms based around the (bootstrap) \gls{PF} or the \gls{MCMCPF} were unable to obtain sensible parameter estimates within the number of iterations that we fixed. The left panel of Figure~\ref{fig:pmmh_acf_parameters} shows the corresponding empirical autocorrelation. The results are consistent with the efficiency of the likelihood estimates illustrated in Figure~\ref{fig:likelihood_estimates}. That is, at least in this setting, the standard \gls{MH} version of the alternative \gls{EHMM} method does not outperform standard \gls{PMMH} algorithms. The estimates of the other parameters behaved similarly and the results for $(a_1, \sigma, \tau)$ are therefore omitted.
\begin{figure}
\caption{Autocorrelation (left panel) and kernel-density estimate (right panel) of the estimates of Parameter~$a_0$ for model dimension~$d=25$ and with $T=10$ observations. Obtained from $10^6$ iterations (of which the initial $10$~\% were discarded as burn-in) of standard \gls{PMMH} algorithms and \gls{MH} versions of the alternative \gls{EHMM} method using $N=1000$ particles. The autocorrelations shown on the r.h.s.\ are averages over four independent runs of each algorithm. \textit{Note:} the \gls{PMMH} algorithms based on the (bootstrap) \gls{PF} and based on the \gls{MCMCPF} failed to yield meaningful approximations of the posterior distribution and the corresponding kernel-density estimates are therefore suppressed.}
\label{fig:pmmh_acf_parameters}
\end{figure}
\subsection{Results for general particle Gibbs samplers}
In this subsection, we compare empirically the performance of various \gls{PG} type samplers (all using \gls{AS}). Gibbs samplers based on the original \gls{EHMM} method failed to yield meaningful estimates for the model dimensions considered in this subsection and at a similar computational cost as the other algorithms. We therefore do not show results for the original \gls{EHMM} method in the figures below.
Recall that in order to implement the conditional \gls{MCMCFAAPF}, we do not need to sample from $p_\theta(x_t|x_{t-1}, y_t)$ nor evaluate the function $x_{t-1} \mapsto p_\theta(y_t|x_{t-1})$. In other words, we can implement the conditional \gls{MCMCFAAPF} in many situations in which implementing a standard conditional \gls{FAAPF} is impossible.
Figure~\ref{fig:gibbs_acf_state_components_dimX_100} shows the autocorrelation of estimates of the first component of $x_1$ obtained from various \gls{PG} samplers for model dimension $d=100$. For the moment, we have kept $\theta$ fixed to the true values. It appears that in high dimensions, the conditional \glspl{PF} with \gls{MCMC} moves are able to outperform standard conditional \glspl{PF}. Note that although, unsurprisingly, the best performance is obtained with the \gls{MCMCFAAPF}, the simpler \gls{MCMCPF} is able to substantially outperform the approach based upon a standard \gls{PF}. This is supported by Figure~\ref{fig:gibbs_ess_dimX_100} which shows that the conditional \glspl{PF} with \gls{MCMC} moves lead to a higher estimated \gls{ESS} in this setting. The acceptance rates associated with the \gls{MH} kernels are shown in Figure~\ref{fig:gibbs_acceptance_rates_dimX_100}.
\begin{figure}
\caption{Autocorrelation (left panel) and kernel-density estimate (right panel) of the estimates of the first component of $x_{1}$ for the state-space model in dimension~$d=100$ with $T=10$ observations. Obtained from three independent runs of each of the various Gibbs samplers comprising $500\,000$ iterations (of which the initial $10$~\% were discarded as burn-in) and using $N=100$ particles. Here, $\theta$ was fixed to the true parameters throughout each run. The autocorrelations shown on the r.h.s.\ are averages over the three independent runs of each algorithm. \textit{Note:} the conditional (bootstrap) \gls{PF} almost never managed to update the states: the corresponding kernel-density estimates were therefore not meaningful and are hence suppressed.}
\label{fig:gibbs_acf_state_components_dimX_100}
\end{figure}
\begin{figure}
\caption{Average \gls{ESS} for the same setting and colour-coding as in Figure~\ref{fig:gibbs_acf_state_components_dimX_100}. The results are averaged over three independent runs of each algorithm. It is worth noting that the \gls{ESS} does not take the autocorrelation of the state-estimates (over iterations of the (particle) \gls{PMCMC} chain) into account and so may flatter \glspl{MCMCPF} to an extent but does illustrate the lessening of weight degeneracy within the particle set which they achieve.}
\caption{Average acceptance rates for the \gls{MH} kernels $\overline{R}_{\theta, t}(\,\cdot\,|\,\cdot\,;\mathbf{x}_{t-1})$ and $R_{\theta, t}(\,\cdot\,|\,\cdot\,;\mathbf{x}_{t-1})$ for same setting and colour-coding as in Figure~\ref{fig:gibbs_acf_state_components_dimX_100}. Note that standard \glspl{PF} can always be interpreted as using a \gls{MH} kernel that proposes \gls{IID} samples from its invariant distribution so that the acceptance rate is always $1$ in this case. Again the acceptance rates are averaged over three independent runs of each algorithm.}
\label{fig:gibbs_ess_dimX_100}
\label{fig:gibbs_acceptance_rates_dimX_100}
\end{figure}
We conclude this section by showing (in Figure~\ref{fig:gibbs_acf_parameters}) simulation results for the estimates of Parameter~$a_0$ obtained from the various \gls{PG} samplers. The \gls{MH} kernel which updates $\theta$ was employed $100$ times per iteration, i.e. $100$ times between each conditional \gls{PF} update of the latent states as the former is relatively computationally cheap compared to the latter.
Note that as indicated by the kernel-density estimates in the right panel of Figure~\ref{fig:gibbs_acf_parameters}, the Gibbs sampler based around the \gls{PF} did not manage to sufficiently explore the support of the posterior distribution within the number of iterations that we fixed. This lack of convergence also caused the comparatively low empirical autocorrelation of the \gls{PG} chains based around the (bootstrap) \gls{PF} in the left panel of Figure~\ref{fig:gibbs_acf_parameters}: as the chain did not sufficiently traverse support of the target distribution -- due to poor mixing of the state-updates as illustrated in Figure~\ref{fig:gibbs_acf_state_components_dimX_100} -- the empirical autocorrelation shown in Figure~\ref{fig:gibbs_acf_parameters} is a poor estimate of the (theoretical) autocorrelation of the chain. More specifically, the former greatly underestimates the latter.
The estimates of the other parameters behaved similarly and the results for $(a_1, \sigma, \tau)$ are therefore omitted.
\begin{figure}
\caption{Model dimension~$d=25$.}
\caption{Model dimension~$d=100$.}
\caption{Autocorrelation (left panel) and kernel-density estimate (right panel) of the estimates of Parameter~$a_0$ for $T=10$ observations. Obtained one run of the various Gibbs samplers comprising $10^6$ iterations (of which the initial $10$~\% were discarded as burn-in) and using $N=100$ particles. The autocorrelations shown on the l.h.s.\ are averages over the two independent runs of each algorithm.}
\label{fig:gibbs_acf_parameters}
\end{figure}
\section{Discussion} \glsresetall
In this work, we have discussed the connections between the \gls{PMCMC} and \gls{EHMM} methodologies and have obtained novel Bayesian inference algorithms for state and parameter estimation in state-space models. We have compared the empirical performance of the various \gls{PMCMC} and \gls{EHMM} algorithms on a simple high-dimensional state-space model. We have found that a properly tuned conditional \gls{PF} which employs local \glsdesc{MH} moves proposed in \citet{ShestopaloffNeal2016} can dramatically outperform the standard conditional \glspl{PF} in high dimensions. Additionally, by formally establishing that \gls{PMCMC} and the (alternative) \gls{EHMM} methods can be viewed as a special case of a general \gls{PMCMC} framework, we have derived both \glsdesc{BS} and \glsdesc{AS} for this general framework. This provides a promising strategy for extending the range of applicability of \glsdesc{PG} algorithms as well as providing a novel class of \glspl{PF} which might be useful.
There are numerous other potential extensions of these ideas. For instance, many existing extensions of standard \gls{PMCMC} methods could also be considered for the alternative \gls{EHMM} methods, e.g.\ incorporating gradient-information into the parameter proposals $q(\theta, {\theta'})$ or exploiting correlated pseudo-marginal ideas \citep{Deligiannidis2015}. Clearly, further generalisation of the target distribution and associated algorithms introduced here are possible. Many other processes for simulating from an extended target admitting a single random trajectory with the correct marginal distribution are possible, e.g.\ along the lines of \citet{LindstenJohansenNaesseth2014}.
\appendix
\section{Special cases of the general PMCMC algorithm} \label{subsec:special_cases} \glsunset{MCMC} \glsunset{PF} \glsunset{FAAPF} \glsunset{APF} \glsunset{MCMCPF} \glsunset{MCMCFAAPF} \glsunset{MCMCAPF}
In this appendix, we show that all \gls{PMCMC} and alternative \gls{EHMM} methods described this work can be recovered as special cases of the general \gls{PMCMC} framework from Section~\ref{sec:general_pmcmc}. For completeness, we explicitly derive all algorithms as special cases of the general framework even though \gls{PMCMC} methods based around the (bootstrap) \gls{PF} and \gls{FAAPF} were already shown to be special cases of \gls{PMCMC} methods based around the general \gls{APF} and even though, alternative \gls{EHMM} methods based around the \gls{MCMCPF} and \gls{MCMCFAAPF} were already shown to be special cases of alternative \gls{EHMM} methods based around the \gls{MCMCAPF}.
\begin{description}[leftmargin=0pt, labelindent=0pt]
\item[(Bootstrap) \gls{PF}.]
In this case, $\psi_{\theta,1}(\mathbf{z}_1) = \prod_{i=1}^N \mu_\theta(x_1^i) = \prod_{i=1}^N \bar{\rho}_{\theta, 1}(x_1^i)$, and, for $t>1$,
\begin{equation}
\psi_{\theta,t}(\mathbf{z}_t|\mathbf{z}_{1:t-1}) = \prod_{i=1}^N \frac{g_\theta(y_{t-1}|x_{t-1}^{a_{t-1}^i})}{\sum_{j=1}^N g_\theta(y_{t-1}|x_{t-1}^j)} f_\theta(x_t^i|x_{t-1}^{a_{t-1}^i}) = \prod_{i=1}^N \bar{\rho}_{\theta, t}(x_t^i, a_{t-1}^i|\mathbf{x}_{t-1}),
\end{equation}
while $\gamma_{\theta,t}(x_{1:t}) \coloneqq p_\theta(x_{1:t}, y_{1:t})$, for any $t \leq T$. This implies that $\tilde{v}_{\theta,t}^{i} = \frac{1}{N} g_\theta(y_t|x_t^{i}) \prod_{n=1}^{t-1} \frac{1}{N} \sum_{j=1}^N g_\theta(y_n|x_n^j)$, so that we obtain $q_\theta(i|\mathbf{z}_{1:T}) = g_\theta(y_T|x_T^i) / \sum_{j=1}^N g_\theta(y_T|x_T^j)$ and $\hat{p}_\theta(y_{1:T}) = \prod_{t=1}^{T} \frac{1}{N} \sum_{i=1}^N g_\theta(y_t|x_t^i)$, as stated in Section~\ref{sec:pmcmc}.
\item [\gls{FAAPF}.] In this case, $\psi_{\theta,1}(\mathbf{z}_1) = \prod_{i=1}^N p_\theta(x_1^i|y_1) = \prod_{i=1}^N \rho_{\theta,1}(x_1^i)$, and, for $t>1$,
\begin{equation}
\psi_{\theta,t}(\mathbf{z}_t|\mathbf{z}_{1:t-1}) = \prod_{i=1}^N \frac{g_\theta(y_{t}|x_{t-1}^{a_{t-1}^i})}{\sum_{j=1}^N g_\theta(y_{t}|x_{t-1}^j)} p_\theta(x_t^i|x_{t-1}^{a_{t-1}^i}, y_t) = \prod_{i=1}^N \rho_{\theta,t}(x_t^i, a_{t-1}^i|\mathbf{x}_{t-1}),
\end{equation}
while $\gamma_{\theta,t}(x_{1:t}) \coloneqq p_\theta(x_{1:t}, y_{1:t}) p_\theta(y_{t+1}|x_t)$, for $t < T$, and $\gamma_{\theta,T}(x_{1:T}) \coloneqq p_\theta(x_{1:T}, y_{1:T})$. This implies that $\tilde{v}_{\theta,t}^{i} = \frac{1}{N} p_\theta(y_1) \prod_{n=2}^t \frac{1}{N} \sum_{j=1}^N p_\theta(y_n|x_{n-1}^j)$, so that we obtain the selection probability $q_\theta(i|\mathbf{z}_{1:T}) = 1/N$ and the marginal-likelihood estimate $\hat{p}_\theta(y_{1:T}) = p_\theta(y_1) \prod_{t=2}^T \frac{1}{N} \sum_{i=1}^N p_\theta(y_t|x_{t-1}^i)$, as stated in Section~\ref{sec:pmcmc}.
\item[General \gls{APF}.] In this case, $\psi_{\theta,1}(\mathbf{z}_1) = \prod_{i=1}^N q_{\theta,1}(x_1^i) = \prod_{i=1}^N \rho_{\theta,1}^{\mathbf{q}_\theta}(x_1^i)$, and, for $t>1$,
\begin{equation}
\psi_{\theta,t}(\mathbf{z}_t|\mathbf{z}_{1:t-1}) = \prod_{i=1}^N \frac{v_{\theta, t-1}^{a_{t-1}^i}}{\sum_{j=1}^N v_{\theta, t-1}^j} q_{\theta,t}(x_t^i|x_{t-1}^{a_{t-1}^i}) = \prod_{i=1}^N \rho_{\theta,t}^{\mathbf{q}_\theta}(x_t^i, a_{t-1}^i|\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2}),
\end{equation}
while $\gamma_{\theta,t}(x_{1:t}) \coloneqq p_\theta(x_{1:t}, y_{1:t}) \tilde{p}_\theta(y_{t+1}|x_t)$, for $t < T$, and $\gamma_{\theta,T}(x_{1:T}) \coloneqq p_\theta(x_{1:T}, y_{1:T})$. This implies that $\tilde{v}_{\theta,t}^{i} = \frac{1}{N} v_{\theta,t}^i \prod_{n=1}^{t-1} \frac{1}{N} \sum_{j=1}^N v_{\theta,n}^j$, so that we obtain the selection probability $q_\theta(i|\mathbf{z}_{1:T}) = v_{T,\theta}^i / \sum_{j=1}^N v_{T,\theta}^j$ and the marginal-likelihood estimate $\hat{p}_\theta(y_{1:T}) = \prod_{t=1}^T \frac{1}{N} \sum_{i=1}^N v_{t,\theta}^i$, as stated in Section~\ref{sec:pmcmc}.
\item[\gls{MCMCPF}.]
In this case, $\psi_{\theta,1}(\mathbf{z}_1) = \bar{\rho}_{\theta,1}(x_1^1) \prod_{i=2}^N \overline{R}_{\theta,1}(x_1^i|x_1^{i-1})$, and, for $t>1$,
\begin{equation}
\psi_{\theta,t}(\mathbf{z}_t|\mathbf{z}_{1:t-1}) = \bar{\rho}_{\theta,t}(x_t^1, a_{t-1}^1|\mathbf{x}_{t-1}) \prod_{i=2}^N \overline{R}_{\theta,t}(x_t^i, a_{t-1}^i|x_t^{i-1}, a_{t-1}^{i-1}; \mathbf{x}_{t-1}),
\end{equation}
while $\gamma_{\theta,t}(x_{1:t})$, $q_\theta(b_T|\mathbf{z}_{1:T})$ and $\hat{p}_\theta(y_{1:T})$ are the same as for \gls{PMCMC} methods using the bootstrap \gls{PF}.
\item[\gls{MCMCFAAPF}.] In this case, $\psi_{\theta,1}(\mathbf{z}_1) = \rho_{\theta,1}(x_1^1) \prod_{i=2}^N R_{\theta,1}(x_1^i|x_1^{i-1})$, and, for $t>1$,
\begin{equation}
\psi_{\theta,t}(\mathbf{z}_t|\mathbf{z}_{1:t-1}) = \rho_{\theta,t}(x_t^1, a_{t-1}^1|\mathbf{x}_{t-1}) \prod_{i=2}^N R_{\theta,t}(x_t^i, a_{t-1}^i|x_t^{i-1}, a_{t-1}^{i-1}; \mathbf{x}_{t-1}),
\end{equation}
while $\gamma_{\theta,t}(x_{1:t})$, $q_\theta(b_T|\mathbf{z}_{1:T})$ and $\hat{p}_\theta(y_{1:T})$ are the same as for \gls{PMCMC} methods using the \gls{FAAPF}.
\item[\gls{MCMCAPF}.] In this case, $\psi_{\theta,1}(\mathbf{z}_1) = \rho_{\theta,1}^{\mathbf{q}_\theta}(x_1^1) \prod_{i=2}^N R_{\theta,1}^{\mathbf{q}_\theta}(x_1^i|x_1^{i-1})$, and, for $t>1$,
\begin{equation}
\psi_{\theta,t}(\mathbf{z}_t|\mathbf{z}_{1:t-1}) = \rho_{\theta,t}^{\mathbf{q}_\theta}(x_t^1, a_{t-1}^1|\mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2}) \prod_{i=2}^N R_{\theta,t}^{\mathbf{q}_\theta}(x_t^i, a_{t-1}^i|x_t^{i-1}, a_{t-1}^{i-1}; \mathbf{x}_{t-2:t-1}, \mathbf{a}_{t-2}),
\end{equation}
while $\gamma_{\theta,t}(x_{1:t})$, $q_\theta(b_T|\mathbf{z}_{1:T})$ and $\hat{p}_\theta(y_{1:T})$ are the same as for \gls{PMCMC} methods using the general APF. \end{description}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\begin{abstract} We show that MA$_{\kappa}$ implies that each collection of ${P}_{\mathfrak c}$-points of size at most $\kappa$ which has a $P_{\mathfrak c}$-point as an $RK$ upper bound also has a ${P}_{\mathfrak c}$-point as an $RK$ lower bound. \end{abstract}
\maketitle
\section{Introduction}
The Rudin-Keisler ($RK$) ordering of ultrafilters has received considerable attention since its introduction in the 1960s. For example, one can take a look at \cite{rudin,Rudin:66,mer,blass,kn,RS,RK}, or \cite{RV}.
Recall the definition of the Rudin-Keisler ordering.
\begin{definition}
Let $\mathcal U$ and $\mathcal V$ be ultrafilters on $\omega$. We say that $\mathcal U\le_{RK}\mathcal V$ if there is a function $f$ in $\omega^{\omega}$ such that $A\in \mathcal U$ if and only if $f^{-1}(A)\in\mathcal V$ for every $A\subseteq \omega$. \end{definition}
When $\mathcal U$ and $\mathcal V$ are ultrafilters on $\omega$ and $\mathcal U\le_{RK}\mathcal V$, we say that $\mathcal U$ is \emph{Rudin-Keisler (RK) reducible} to $\mathcal V$, or that $\mathcal U$ is \emph{Rudin-Keisler (RK) below} $\mathcal V$. In case $\mathcal U\le_{RK}\mathcal V$ and $\mathcal V\le_{RK}\mathcal U$ both hold, then we say that $\mathcal U$ and $\mathcal V$ are \emph{Rudin-Keisler equivalent}, and write $\mathcal U\equiv_{RK}\mathcal V$.
Very early in the investigation of this ordering of ultrafilters, it was noticed that the class of P-points is particularly interesting. Recall that an ultrafilter $\mathcal U$ on $\omega$ is called a \emph{P-point} if for any $\set{a_n:n<\omega}\subseteq\mathcal U$ there is an $a\in\mathcal U$ such that $a\subseteq^* a_n$ for every $n<\omega$, i.e. the set $a\setminus a_n$ is finite for every $n<\omega$. P-points were first constructed by Rudin in \cite{rudin}, under the assumption of the Continuum Hypothesis. The class of P-points forms a downwards closed initial segment of the class of all ultrafilters. In other words, if $\mathcal U$ is a P-point and $\mathcal V$ is any ultrafilter on $\omega$ with $\mathcal V \; {\leq}_{RK} \; \mathcal U$, then $\mathcal V$ is also a P-point. Hence understanding the order-theoretic structure of the class of P-points can provide information about the order-theoretic structure of the class of all ultrafilters on $\omega$. One of the first systematic explorations of the order-theoretic properties of the class of all ultrafilters, and particularly of the class of P-points, under ${\leq}_{RK}$ was made by Blass in \cite{blassphd} and \cite{blass}, where he proved many results about this ordering under the assumption of Martin's Axiom (MA).
Let us note here that it is not possible to construct P-points in ZFC only, as was proved by Shelah (see \cite{proper}). Thus some set-theoretic assumption is needed to ensure the existence of P-points. The most commonly used assumption when studying the order-theoretic properties of the class of P-points is MA. Under MA every ultrafilter has character $\mathfrak c$. Therefore, the ${P}_{\mathfrak c}$-points are the most natural class of P-points to focus on under MA. Again, the ${P}_{\mathfrak c}$-points form a downwards closed subclass of the P-points. \begin{definition} \label{def:pc}
An ultrafilter $\mathcal U$ on $\omega$ is called a $P_{\mathfrak c}$\emph{-point} if for every $\alpha<\mathfrak c$ and any $\set{a_i:i<\alpha}\subseteq \mathcal U$ there is an $a\in\mathcal U$ such that $a\subseteq^* a_i$ for every $i<\alpha$. \end{definition} In Theorem 5 from \cite{blass}, Blass proved in ZFC that if $\set{\mathcal U_n:n<\omega}$ is a countable collection of P-points and if there is a P-point $\mathcal V$ such that $\mathcal U_n\le_{RK}\mathcal V$ for every $n<\omega$, then there is a P-point $\mathcal U$ such that $\mathcal U\le_{RK}\mathcal U_n$ for every $n<\omega$. In other words, if a countable family of P-points has an upper bound, then it also has a lower bound.
The main result of this paper generalizes Blass' theorem to families of ${P}_{\mathfrak c}$-points of size less than $\mathfrak c$ under MA. More precisely, if MA holds and a family of ${P}_{\mathfrak c}$-points of size less than $\mathfrak c$ has an $RK$ upper bound which is a ${P}_{\mathfrak c}$-point, then the family also has an RK lower bound.
Blass proved his result via some facts from \cite{blassmodeltheory} about non-standard models of complete arithmetic. In order to state these results, we introduce a few notions from \cite{blassmodeltheory}. The language $L$ will consist of symbols for all relations and all functions on $\omega$. Let $N$ be the standard model for this language, its domain is $\omega$ and each relation or function denotes itself. Let $M$ be an elementary extension of $N$, and let ${}^*R$ be the relation in $M$ denoted by $R$, and let ${}^*f$ be the function in $M$ denoted by $f$. Note that if $a\in M$, then the set $\set{{}^*f(a):f:\omega\to\omega}$ is the domain of an elementary submodel of $M$. Submodel like this, i.e.\@ generated by a single element, will be called \emph{principal}. It is not difficult to prove that a principal submodel generated by $a$ is isomorphic to the ultrapower of the standard model by the ultrafilter ${\mathcal U}_{a} = \{X \subseteq \omega: a \in {}^{\ast}{X}\}$. If $A,B\subseteq M$, we say that they are \emph{cofinal with each other} iff $(\forall a\in A)(\exists b\in B)\ a\ {}^*\!\!\le b$ and $(\forall b\in B)(\exists a\in A)\ b\ {}^*\!\!\le a$. Finally, we can state Blass' theorem.
\begin{theorem}[Blass, Theorem 3 in \cite{blassmodeltheory}]\label{t:blassmodel}
Let $M_i$ ($i<\omega$) be countably many pairwise cofinal submodels of $M$.
Assume that at least one of the $M_i$ is principal.
Then $\bigcap_{i<\omega}M_i$ is cofinal with each $M_i$, in fact it contains a principal submodel cofinal with each $M_i$. \end{theorem}
After proving this theorem, Blass states that it is not known to him whether Theorem \ref{t:blassmodel} can be extended to larger collections of submodels. The proof of our main result clarifies this, namely in Theorem \ref{theorem3} below we prove that under MA it is possible to extend it to collections of models of size less than $\mathfrak c$ provided that there is a principal model that is isomorphic to an ultrapower induced by a ${P}_{\mathfrak c}$-point. Then we proceed and use this result to prove Theorem \ref{maintheorem} where we extend Theorem 5 from \cite{blass} to collections of fewer than $\mathfrak c$ many ${P}_{\mathfrak c}$-points.
Recall that MA$_{\alpha}$ is the statement that for every partial order $P$ which satisfies the countable chain condition and for every collection $\mathcal{D} = \{{D}_{i}: i < \alpha\}$ of dense subsets of $P$, there is a filter $G\subseteq P$ such that $G\cap {D}_{i}\neq\emptyset$ for every $i < \alpha$.
\section{The lower bound}
In this section we prove the results of the paper. We begin with a purely combinatorial lemma about functions.
\begin{definition}\label{closedness}
Let $\alpha$ be an ordinal, let $\mathcal F=\set{f_i:i<\alpha}\subseteq \omega^{\omega}$ be a family of functions, and let $A$ be a subset of $\alpha$.
We say that a set $F\subseteq \omega$ is ($A,\mathcal F$)\emph{-closed} if $f_i^{-1}(f_i''F)\subseteq F$ for each $i\in A$. \end{definition}
\begin{remark}
Notice that if $F$ is ($A,\mathcal F$)-closed, then $f_i^{-1}(f_i''F)=F$ for each $i\in A$. \end{remark}
\begin{lemma}\label{finiteunion}
Let $\alpha$ be an ordinal, let $\mathcal F=\set{f_i:i<\alpha}\subseteq \omega^{\omega}$ be a family of functions, and let $A$ be a subset of $\alpha$.
Suppose that $m<\omega$, and that $F_k$ is ($A,\mathcal F$)-closed subset of $\omega$, for each $k<m$.
Then the set $F=\bigcup_{k<m}F_k$ is ($A,\mathcal F$)-closed. \end{lemma}
\begin{proof}
To prove that $F$ is ($A,\mathcal F$)-closed take any $i\in A$, and $n\in f_i^{-1}(f_i''F)$.
This means that there is some $n'\in F$ such that $f_i(n)=f_i(n')$.
Let $k<m$ be such that $n'\in F_k$.
Then $n\in f_i^{-1}(f_i''F_k)$.
Since $F_k$ is ($A,\mathcal F$)-closed, $n\in f_i^{-1}(f_i''F_k)\subseteq F_k$.
Thus $n\in F_k\subseteq F$. \end{proof}
\begin{lemma}\label{ensuringcondition}
Let $\alpha<\mathfrak c$ be an ordinal.
Let $\mathcal F=\set{f_i:i<\alpha}\subseteq \omega^{\omega}$ be a family of finite-to-one functions.
Suppose that for each $i,j<\alpha$ with $i<j$, there is $l<\omega$ such that $f_j(n)=f_j(m)$ whenever $f_{i}(n)=f_{i}(m)$ and $n,m\ge l$.
Then for each finite $A\subseteq \alpha$, and each $n<\omega$, there is a finite $(A,\mathcal F)$-closed set $F$ such that $n\in F$. \end{lemma}
\begin{proof}
First, if $A$ is empty, then we can take $F=\set{n}$.
So fix a non-empty finite $A\subseteq\alpha$, and $n<\omega$.
For each $i,j\in A$ such that $i<j$, by the assumption of the lemma, take $l_{ij}<\omega$ such that for each $n,m\ge l_{ij}$, if $f_i(n)=f_i(m)$, then $f_j(n)=f_j(m)$.
Since $A$ is a finite set, there is $l=\max\set{l_{ij}:i,j\in A,\ i<j}$.
So $l$ has the property that for every $i,j\in A$ with $i<j$, if $f_i(n)=f_i(m)$ and $n,m\ge l$, then $f_j(n)=f_j(m)$.
Let $i_0=\max(A)$.
Clearly, $f_i''l$ is finite for each $i\in A$, and since each $f_i$ is finite-to-one the set ${f}^{-1}_{i}\br{f_i''l}$ is finite for every $i\in A$.
Since the set $A$ is also finite, there is $l'<\omega$ such that $\bigcup_{i\in A}f_i^{-1}\br{f_i''l}\subseteq l'$.
Again, since $f_{i_0}$ is finite-to-one there is $l''<\omega$ such that $f^{-1}_{i_0}\br{f_{i_0}''l'}\subseteq l''$.
Note that by the definition of numbers $l'$ and $l''$, we have $l''\ge l'\ge l$.
\begin{claim}\label{closedset}
For all $k<\omega$, if $k\ge l''$, then the set $f_{i_0}^{-1}(f_{i_0}''\set{k})$ is ($A,\mathcal F$)-closed.
\end{claim}
\begin{proof}
Fix $k\ge l''$ and let $X=f_{i_0}^{-1}(f_{i_0}''\set{k})$.
First observe that $X\cap l'=\emptyset$.
To see this suppose that there is $m\in X\cap l'$.
Since $m\in X$, $f_{i_0}(m)=f_{i_0}(k)$.
Together with $m\in l'$, this implies that $k\in f_{i_0}^{-1}(f_{i_0}''\set{m})\subseteq f_{i_0}^{-1}(f_{i_0}''l')\subseteq l''$.
Thus $k<l''$ contradicting the choice of $k$.
Secondly, observe that if $m<l$ and $k'\in X$, then $f_i(m)\neq f_i(k')$ for each $i\in A$.
To see this, fix $m<l$ and $k'\in X$, and suppose that for some $i\in A$, $f_i(m)=f_i(k')$.
This means that $k'\in f_i^{-1}(f_i''\set{m})\subseteq f_i^{-1}(f_i''l)\subseteq l'$ contradicting the fact that $X\cap l'=\emptyset$.
Now we will prove that $X$ is ($A,\mathcal F$)-closed.
Take any $i\in A$ and any $m\in f_i^{-1}(f_i''X)$.
We should prove that $m\in X$.
Since $m\in f_i^{-1}(f_i''X)$, $f_i(m)\in f_i''X$ so there is some $k'\in X$ such that $f_i(m)=f_i(k')$.
By the second observation, $m\ge l$.
By the first observation $k'\ge l'\ge l$.
By the assumption of the lemma, since $m,k'\ge l$, and $f_i(m)=f_i(k')$, it must be that $f_{i_0}(m)=f_{i_0}(k')$.
Since $k'\in X=f_{i_0}^{-1}(f_{i_0}''\set{k})$, it must be that $f_{i_0}(k)=f_{i_0}(k')=f_{i_0}(m)$.
This means that $m\in f_{i_0}^{-1}(f_{i_0}''\set{k})=X$ as required.
Thus $f_{i_0}^{-1}(f_{i_0}''\set{k})$ is $(A,\mathcal F)$-closed.
\end{proof}
Now we inductively build a tree $T\subseteq \omega^{<\omega}$ we will be using in the rest of the proof.
Fix a function $\Phi:\omega\to\omega^{<\omega}$ so that $\Phi^{-1}(\sigma)$ is infinite for each $\sigma\in \omega^{<\omega}$.
For each $m<\omega$ let $u_m=\Phi(m)\br{\abs{\Phi(m)}-1}$, i.e. $u_m$ is the last element of the sequence $\Phi(m)$.
Let $T_0=\set{\emptyset,\seq{n}}$ (recall that $n$ is given in the statement of the lemma).
Suppose that $m\ge 1$, and that $T_m$ is given.
If $\Phi(m)$ is a leaf node of $T_m$, then let
\[\textstyle
Z_m=\left(\bigcup_{i\in A}f_i^{-1}(f_i''\set{u_m})\right)\setminus\left(\bigcup_{\eta\in T_m}\operatorname{range}(\eta)\right),
\]
and $T_{m+1}=T_m\cup\set{\Phi(m)^{\frown}\seq{k}:k\in Z_m}$.
If $\Phi(m)$ is not a leaf node of $T_m$, then $T_{m+1}=T_m$.
Finally, let $T=\bigcup_{m<\omega}T_m$ and $F=\bigcup_{\eta\in T}\operatorname{range}(\eta)$.
\begin{claim}\label{subclaimtree}
If $\sigma$ is a non-empty element of the tree $T$, then there is $m_0\ge 1$ such that $\sigma$ is a leaf node of $T_{m_0}$, that $\sigma=\Phi(m_0)$ and that $$\textstyle\bigcup_{i\in A}f_i^{-1}(f_i''\set{u_{m_0}})\subseteq \bigcup_{\eta\in T_{m_0+1}}\operatorname{range}(\eta).$$
\end{claim}
\begin{proof}
Fix a non-empty $\sigma$ in $T$.
Let $m_1=\min\set{k<\omega:\sigma\in T_k}$.
Since $\abs{\sigma}>0$, $\sigma$ is a leaf node of $T_{m_1}$.
Consider the set $W=\set{m\ge m_1:\Phi(m)=\sigma}$.
Since the set $\set{m<\omega:\Phi(m)=\sigma}$ is infinite, $W$ is non-empty subset of positive integers, so it has a minimum.
Let $m_0=\min W$.
Note that if $m_0=m_1$, then $\sigma$ is a leaf node of $T_{m_0}$.
If $m_0>m_1$, by the construction of the tree $T$, since $\Phi(k)\neq \sigma$ whenever $m_1\le k<m_0$, it must be that $\sigma$ is a leaf node of every $T_k$ for $m_1<k\le m_0$.
Thus $\sigma$ is a leaf node of $T_{m_0}$ and $\Phi(m_0)=\sigma$.
Again by the construction of the tree $T$, we have $T_{m_0+1}=T_{m_0}\cup \set{\sigma^{\frown}\seq{k}:k\in Z_{m_0}}$.
This means that $$\textstyle\bigcup_{\eta\in T_{m_0+1}}\operatorname{range}(\eta)=Z_{m_0}\cup \bigcup_{\eta\in T_{m_0}}\operatorname{range}(\eta).$$
Finally, the definition of $Z_{m_0}$ implies that
\[\textstyle
\bigcup_{i\in A}f_i^{-1}(f_i''\set{u_{m_0}})\subseteq Z_{m_0}\cup\bigcup_{\eta\in T_{m_0}}\operatorname{range}(\eta)=\bigcup_{\eta\in T_{m_0+1}}\operatorname{range}(\eta),
\]
as required.
\end{proof}
\begin{claim}\label{closedtree}
The set $F$ is ($A,\mathcal F$)-closed, and contains $n$ as an element.
\end{claim}
\begin{proof}
Since $\seq{n}\in T_0$, $n\in F$.
To see that $F$ is ($A,\mathcal F$)-closed, take any $j\in A$, and any $w\in f_j^{-1}\br{f_j''F}$.
We have to show that $w\in F$.
Since $w\in f_j^{-1}\br{f_j''F}$, there is $m\in F$ such that $f_j(w)=f_j(m)$.
Since $m\in F=\bigcup_{\eta\in T}\operatorname{range}(\eta)$, there is $\sigma$ in $T$ such that $\sigma(k)=m$ for some $k<\omega$.
Consider $\sigma\upharpoonright(k+1)$.
Since $\sigma\upharpoonright(k+1)\in T$, by Claim \ref{subclaimtree} there is $m_0\ge 1$ such that $\Phi(m_0)=\sigma\upharpoonright(k+1)$, that $\sigma\upharpoonright(k+1)$ is a leaf node of $T_{m_0}$ and that (note that $u_{m_0}=\sigma(k)=m$)
\[\textstyle
\bigcup_{i\in A}f_i^{-1}(f_i''\set{m})\subseteq \bigcup_{\eta\in T_{m_0+1}}\operatorname{range}(\eta)\subseteq \bigcup_{\eta\in T}\operatorname{range}(\eta)=F.
\]
So $w\in f_j^{-1}(f_j''\set{m})\subseteq F$ as required.
\end{proof}
\begin{claim}\label{finitetree}
The tree $T$ is finite.
\end{claim}
\begin{proof}
First we prove that each level of $T$ is finite.
For $k<\omega$ let $T_{(k)}$ be the $k$-th level of $T$, i.e. $T_{(k)}=\set{\sigma\in T: \abs{\sigma}=k}$.
Clearly $T_{(0)}$ and $T_{(1)}$ are finite.
So suppose that $T_{(k)}$ is finite.
Let $T_{(k)}=\set{\sigma_0,\sigma_2,\dots,\sigma_t}$ be enumeration of that level.
For $s\le t$ let $m_s$ be such that $\Phi(m_s)=\sigma_s$ and that $\sigma_s$ is a leaf node of $T_{m_s}$.
Note that by the construction of the tree $T$ all nodes at the level $T_{(k+1)}$ are of the form $\sigma_s^{\frown}\seq{r}$ where $s\le t$ and $r\in Z_{m_s}$.
Since the set $A$ is finite and all functions $f_i$ (for $i\in A$) are finite-to-one, $Z_{m_s}$ is finite for every $s\le t$.
Thus there are only finitely many nodes of the form $\sigma_s^{\frown}\seq{r}$ where $s\le t$ and $r\in Z_{m_s}$, hence the level $T_{(k+1)}$ must also be finite.
This proves by induction that each level of $T$ is finite.
Suppose now that $T$ is infinite.
By K\"{o}nig's lemma, since each level of $T$ is finite, $T$ has an infinite branch $b$.
By definition of the sets $Z_m$ ($m<\omega$), each node of $T$ is 1-1 function, so $b$ is also an injection from $\omega$ into $\omega$.
In particular, the range of $b$ is infinite.
Let $k=\min\set{m<\omega:b(m)\ge l''}$, and let $\sigma=b\upharpoonright(k+1)$.
Clearly, $\sigma\in T$.
By Claim \ref{subclaimtree}, there is $m_0<\omega$ such that $\sigma$ is a leaf node of $T_{m_0}$, that $\Phi(m_0)=\sigma$, and that $\bigcup_{i\in A}f_i^{-1}(f_i''\set{\sigma(k)})\subseteq \bigcup_{\eta\in T_{m_0+1}}\operatorname{range}(\eta)$.
Since $\sigma(k)=b(k)\ge l''$, Claim \ref{closedset} implies that the set $Y=f_{i_0}^{-1}(f_{i_0}''\set{\sigma(k)})$ is $(A,\mathcal F)$-closed.
By the construction $T_{m_0+1}=T_{m_0}\cup\set{\sigma^{\frown}\seq{m}:m\in Z_{m_0}}$.
Since $b$ is an infinite branch, there is $m'\in Z_{m_0}$ such that $b(k+1)=m'$.
Now $m'\in Z_{m_0}\subseteq \bigcup_{i\in A}f_i^{-1}\br{f_i''\set{\sigma(k)}}$, the fact that $\sigma(k)\in Y$, and the fact that $Y$ is ($A,\mathcal F$)-closed, together imply that
$$\textstyle m'\in \bigcup_{i\in A}f_i^{-1}\br{f_i''\set{\sigma(k)}}\subseteq \bigcup_{i\in A}f_i^{-1}\br{f_i''Y}\subseteq Y.$$
Consider the node $\tau=\sigma^{\frown}\seq{m'}=b\upharpoonright(k+2)$.
Since $b$ is an infinite branch, it must be that $\tau^{\frown}\seq{b(k+2)}\in T$.
By Claim \ref{subclaimtree}, there is $m_1$ such that $\tau$ is a leaf node of $T_{m_1}$ and that $\Phi(m_1)=\tau$.
Clearly, $m_1>m_0$ and $\tau^{\frown}\seq{b(k+2)}\in T_{m_1+1}$.
Recall that we have already shown that $\bigcup_{i\in A}f_i^{-1}(f_i''\set{\sigma(k)})\subseteq \bigcup_{\eta\in T_{m_0+1}}\operatorname{range}(\eta)$.
Thus $Y\subseteq\bigcup_{\eta\in T_{m_0+1}}\operatorname{range}(\eta)$.
This, together with the fact that $\tau(k+1)=m'\in Y$, that $Y$ is $(A,\mathcal F)$-closed, and $m_1>m_0$ jointly imply that
\[\textstyle
\bigcup_{i\in A}f_i^{-1}(f_i''\set{\tau(k+1)})\subseteq Y\subseteq\bigcup_{\eta\in T_{m_0}+1}\operatorname{range}(\eta)\subseteq \bigcup_{\eta\in T_{m_1}}\operatorname{range}(\eta).
\]
This means that
\[\textstyle
b(k+2)\in Z_{n_1}=\bigcup_{i\in A}f_i^{-1}(f_i''\set{\tau(k+1)})\setminus \bigcup_{\eta\in T_{m_1}}\operatorname{range}(\eta)=\emptyset,
\]
which is clearly impossible.
Thus, $T$ is not infinite.
\end{proof}
To finish the proof, note that by Claim \ref{closedtree} the set $F$ is $(A,\mathcal F)$-closed and contains $n$ as an element, while by Claim \ref{finitetree} the set $F$ is finite. So $F$ satisfies all the requirements of the conclusion of the lemma. \end{proof}
The following lemma is the main application of Martin's Axiom. Again, it does not directly deal with ultrafilters, but with collections of functions. \begin{lemma}[MA$_{\alpha}$]\label{mainlemma}
Let $\mathcal F=\set{f_i:i<\alpha}\subseteq \omega^{\omega}$ be a family of finite-to-one functions.
Suppose that for each non-empty finite set $A\subseteq\alpha$, and each $n<\omega$, there is a finite ($A,\mathcal F$)-closed set $F$ containing $n$ as an element.
Then there is a finite-to-one function $h\in\omega^{\omega}$, and a collection $\set{e_i:i<\alpha}\subseteq \omega^{\omega}$ such that for each $i<\alpha$, there is $l<\omega$ such that $h(n)=e_i(f_i(n))$ whenever $n\ge l$. \end{lemma}
\begin{proof}
We will apply MA$_{\alpha}$, so we first define the poset we will be using. Let $\mathcal P$ be the set of all $p=\seq{g_p,h_p}$ such that
\begin{enumerate}[label=(\Roman*)]
\item\label{uslov1} $h_p:N_p\to \omega$ where $N_p$ is a finite subset of $\omega$,
\item\label{uslov2} $g_p=\seq{g^i_p:i\in A_p}$ where $A_p\in [\alpha]^{<\aleph_0}$, and $g^i_p:f_i''N_p\to \omega$ for each $i\in A_p$,
\item\label{uslov3} $N_p$ is ($A_p,\mathcal F$)-closed.
\end{enumerate}
Define the ordering relation $\le$ on $\mathcal P$ as follows: $q\le p$ iff
\begin{enumerate}[resume,label=(\Roman*)]
\item\label{ext1} $N_p\subseteq N_q$,
\item\label{ext2} $A_p\subseteq A_q$,
\item\label{ext3} $h_q\upharpoonright N_p=h_p$,
\item\label{ext4} $g_q^i\upharpoonright f_i''N_p=g_p^i$ for each $i\in A_p$,
\item\label{ext5} $h_q(n)>h_q(m)$ whenever $m\in N_p$ and $n\in N_q\setminus N_p$,
\item\label{ext6} $h_q(n)=g_q^i(f_i(n))$ for each $n\in N_q\setminus N_p$ and $i\in A_p$.
\end{enumerate}
It is clear that $\seq{\mathcal P,\le}$ is a partially ordered set.
\begin{claim}\label{addingn}
Let $p\in \mathcal P$, $n_0<\omega$, and suppose that $A\subseteq \alpha$ is finite such that $A_p\subseteq A$.
Then there is $q\le p$ such that $n_0\subseteq N_q$ and that $N_q$ is ($A,\mathcal F$)-closed.
\end{claim}
\begin{proof}
Applying the assumption of the lemma to the finite set $A$, and each $k\in n_0\cup N_p$, we obtain sets $F_k$ ($k\in n_0\cup N_p$) such that $k\in F_k$ and $f_i^{-1}(f_i''F_k)\subseteq F_k$ for each $k\in n_0\cup N_p$ and $i\in A$.
Let $N_q=\bigcup_{k\in n_0\cup N_p}F_k$, let $A_q=A_p$, and let $t=\max\set{h_p(k)+1:k\in N_p}$.
Finally, define
\[
h_q(n)=\left\{\begin{array}{l} h_p(n),\ \mbox{if}\ n\in N_p\\
t,\ \mbox{if}\ n\in N_q\setminus N_p\end{array}\right.\ \mbox{and}\ g^i_q(k)=\left\{\begin{array}{l} g^i_p(k),\ \mbox{if}\ k\in f_i''N_p\\
t,\ \mbox{if}\ k\in f_i''N_q\setminus f_i''N_p\end{array}\right.\ \mbox{for}\ i\in A_q.
\]
Let $g_q$ denote the sequence $\seq{g^i_q:i\in A_q}$.
Clearly $n_0\subseteq N_q$.
By Lemma \ref{finiteunion}, $N_q$ is ($A,\mathcal F$)-closed.
We still have to show that $q=\seq{g_q,h_q}\in \mathcal P$ and $q\le p$.
Since $h_q$ is defined on $N_q$ and $N_q$ finite, since $A_q=A_p$ and $g_p^i$ is defined on $f_i''N_q$ for $i\in A_p$, and since $N_q$ is $(A_q,\mathcal F)$-closed, conditions \ref{uslov1}-\ref{uslov3} are satisfied by $q$.
Thus $q\le p$.
Next we show $q\le p$.
Conditions \ref{ext1}-\ref{ext4} are obviously satisfied by the definition of $q$.
Since $h_q(n)=t>h_p(k)=h_q(k)$ for each $n\in N_q\setminus N_p$ and $k\in N_p$, conditions \ref{ext4} is also satisfied.
So we still have to check \ref{ext6}.
Take any $i\in A_p$ and $n\in N_q\setminus N_p$.
By the definition of $h_q$, we have $h_q(n)=t$.
Once we prove that $f_i(n)\in f_i''N_q\setminus f_i''N_p$, we will be done because in that case the definition of $g^i_p$ implies that $g^i_p(f_i(n))=t$ as required.
So suppose the contrary, that $f_i(n)\in f_i''N_p$.
Since $p$ is a condition and $A_q=A_p$, it must be that $n\in f_i^{-1}(f_i''N_p)\subseteq N_p$.
But this contradicts the choice of $n$.
Thus condition \ref{ext6} is also satisfied and $q\le p$.
\end{proof}
\begin{claim}\label{addingj}
Let $p\in\mathcal P$, and $j_0<\alpha$. Then there is $q\le p$ such that $j_0\in A_q$.
\end{claim}
\begin{proof}
Let $A_q=A_p\cup\set{j_0}$.
Applying Claim \ref{addingn} to $A_q$ and $n=0$, we obtain a condition $p'\le p$ such that $N_{p'}$ is $(A_q,\mathcal F)$-closed.
Let $N_q=N_{p'}$, $h_{q}=h_{p'}$, and $g^i_q=g^i_{p'}$ for $i\in A_p$.
Define $g^{j_0}_q(k)=0$ for each $k\in f_{j_0}''N_{p'}$, and let $g_p$ denote the sequence $\seq{g^i_q:i\in A_q}$.
Since $j_0\in A_q$, to finish the proof of the claim it is enough to show that $q=\seq{g_q,h_q}\in \mathcal P$, and that $q\le p'$.
Conditions \ref{uslov1}-\ref{uslov3} are clear from the definition of $q$ because $p'\in\mathcal P$ and $N_q=N_{p'}$, $h_q=h_{p'}$, $g_q^{j_0}:f_{j_0}''N_q\to\omega$, and $g_q^i=g_p^i$ for $i\in A_q\setminus\set{j_0}$.
Conditions \ref{ext1}-\ref{ext4} are clear by the definition of $h_q$ and $g_q$.
Conditions \ref{ext5} and \ref{ext6} are vacuously true because $N_{p'}=N_q$.
Thus the claim is proved.
\end{proof}
\begin{claim}\label{merging}
If $p,q\in \mathcal P$ are such that $h_p=h_q$ and $g^i_p=g^i_q$ for $i\in A_p\cap A_q$, then $p$ and $q$ are compatible in $\mathcal P$.
\end{claim}
\begin{proof}
We proceed to define $r\le p,q$.
Let $N=N_p=N_q$.
Let $$t=\max\set{h_p(n)+1:n\in N}.$$
Applying the assumption of the lemma to $A=A_p\cup A_q$, and each $k\in N$, we obtain ($A,\mathcal F$)-closed sets $F_k$ ($k\in N$).
By Claim \ref{finiteunion}, the set $N_r=\bigcup_{k\in N}F_k$ is ($A,\mathcal F$)-closed.
Let $A_r=A$, and define
\[
h_r(n)=\left\{\begin{array}{l} h_p(n),\mbox{ if}\ n\in N\\[1mm]
t,\mbox{ if}\ n\in N_r\setminus N\end{array}\right.\hskip-1.5mm\mbox{and }g^i_r(k)=\left\{\begin{array}{l} g^i_p(k),\mbox{ if}\ i\in A_p\mbox{ and }k\in f_i''N\\[1mm]
g^i_q(k),\mbox{ if}\ i\in A_q\mbox{ and }k\in f_i''N\\[1mm]
t,\ \mbox{ if}\ k\in f_i''N_r\setminus f_i''N\end{array}\right.\hskip-1.5mm,
\]
for $i\in A_r$. Let $g_r$ denote the sequence $\seq{g^i_r:i\in A_r}$.
As we have already mentioned, $r=\seq{h_r,g_r}$ satisfies \ref{uslov3}, and it clear that \ref{uslov1} and \ref{uslov2} are also true for $r$.
To see that $r\le p$ and $r\le q$ note that conditions \ref{ext1}-\ref{ext5} are clearly satisfied.
We will check that $r$ and $p$ satisfy \ref{ext6} also.
Take any $n\in N_r\setminus N$ and $i\in A_p$.
By the definition of $h_r$, $h_r(n)=t$.
By the definition of $g_r^i$, if $f_i(n)\in f_i''N_r\setminus f_i''N$, then $g^i_r(f_i(n))=t=h_r(n)$.
So suppose this is not the case, i.e. that $f_i(n)\in f_i''N$.
This would mean that $n\in f_i^{-1}(f_i''N)$, which is impossible because $n\notin N$ and $N$ is $(A,\mathcal F)$-closed because $p\in \mathcal P$.
Thus we proved $r\le p$.
In the same way it can be shown that $r\le q$.
\end{proof}
\begin{claim}\label{ccc}
The poset $\mathcal P$ satisfies the countable chain condition.
\end{claim}
\begin{proof}
Let $\seq{p_{\xi}:\xi<\omega_1}$ be an uncountable set of conditions in $\mathcal P$.
Since $h_{p_{\xi}}\in [\omega\times\omega]^{<\omega}$ for each $\xi<\omega_1$, there is an uncountable set $\Gamma\subseteq\omega_1$, and $h\in [\omega\times\omega]^{<\omega}$ such that $h_{p_{\xi}}=h$ for each $\xi\in \Gamma$.
Consider the set $\seq{A_{p_{\xi}}:\xi\in\Gamma}$.
By the $\Delta$-system lemma, there is an uncountable set $\Delta\subseteq\Gamma$, and a finite set $A\subseteq\alpha$ such that $A_{p_{\xi}}\cap A_{p_{\eta}}=A$ for each $\xi,\eta\in\Delta$.
Since $\Delta$ is uncountable and $g_{p_{\xi}}^i\in [\omega\times\omega]^{<\omega}$ for each $i\in A$ and $\xi\in\Delta$, there is an uncountable set $\Theta\subseteq\Delta$ and $g_i$ for each $i\in A$, such that $g^i_{p_{\xi}}=g_i$ for each $\xi\in\Theta$ and $i\in A$.
Let $\xi$ and $\eta$ in $\Theta$ be arbitrary.
By Claim \ref{merging}, $p_{\xi}$ and $p_{\eta}$ are compatible in $\mathcal P$.
\end{proof}
Consider sets $D_{j}=\set{p\in \mathcal P: j\in A_p}$ for $j\in\alpha$, and $D_{m}=\set{p\in \mathcal P:m\in N_p}$ for $m\in\omega$.
By Claim \ref{addingj} and Claim \ref{addingn}, these sets are dense in $\mathcal P$.
By MA$_{\alpha}$ there is a filter $\mathcal G\subseteq \mathcal P$ intersecting all these sets.
Clearly, $h=\bigcup_{p\in \mathcal G}h_p$ and $e_i=\bigcup_{p\in\mathcal G}g_p^i$, for each $i\in \alpha$, are functions from $\omega$ into $\omega$.
We will prove that these functions satisfy the conclusion of the lemma.
First we will prove that $h$ is finite-to-one.
Take any $m\in \omega$ and let $k=h(m)$.
By the definition of $h$, there is $p\in \mathcal G$ such that $h_p(m)=k$.
Suppose that $h^{-1}(\set{k})\not\subseteq N_p$.
This means that there is an integer $m_0\notin N_p$ such that $h(m_0)=k$.
Let $q\in \mathcal G$ be such that $h_q(m_0)=k$.
Now for a common extension $r\in\mathcal G$ of both $p$ and $q$, it must be that $h_r(m_0)=h_p(m)$, contradicting the fact that $r\le p$, in particular condition \ref{ext5} is violated in this case.
We still have to show that for each $i\in \alpha$, there is $l\in\omega$ such that $h(n)=e_i(f_i(n))$ whenever $n\ge l$.
So take $i\in\alpha$.
By Claim \ref{addingj}, there is $p\in\mathcal G$ such that $i\in A_p$.
Let $l=\max(N_p)+1$.
We will prove that $l$ is as required.
Take any $n\ge l$.
By Claim \ref{addingn}, there is $q\in\mathcal G$ such that $n\in q$.
Let $r\in \mathcal G$ be a common extension of $p$ and $q$.
Since $n\notin N_p$ and $r\le p$, it must be that $h_r(n)=g_r^i(f_i(n))$, according to condition \ref{ext6}.
Hence $h(n)=e_i(f_i(n))$, as required. \end{proof}
Before we move to the next lemma let us recall that if $c$ is any element of the model $M$, then $\mathcal U=\set{X\subseteq \omega: c\in {}^*X}$ is ultrafilter on $\omega$.
\begin{lemma}\label{ensuringensuring}
Let $\alpha<\mathfrak c$ be an ordinal.
Let $\seq{M_i:i<\alpha}$ be a $\subseteq$-decreasing sequence of principal submodels of $M$, i.e. each $M_i$ is generated by a single element $a_i$ and $M_j\subseteq M_i$ whenever $i<j<\alpha$.
Let each $M_i$ ($i<\alpha$) be cofinal with $M_0$.
Suppose that $\mathcal U_0=\set{X\subseteq \omega: a_0\in \prescript{*}{}X}$ is a $P_{\mathfrak c}$-point.
Then there is a family $\set{f_i:i<\alpha}\subseteq \omega^{\omega}$ of finite-to-one functions such that $\prescript{*}{}f_i(a_0)=a_i$ for $i<\alpha$, and that for $i,j<\alpha$ with $i<j$, there is $l<\omega$ such that $f_j(n)=f_j(m)$ whenever $f_i(n)=f_i(m)$ and $m,n\ge l$. \end{lemma}
\begin{proof}
Let $i<j<\alpha$. Since $M_j\subseteq M_i$, and $M_i$ is generated by $a_i$, there is a function $\varphi_{ij}:\omega\to\omega$ such that $\prescript{*}{}\varphi_{ij}(a_i)=a_j$.
Since $M_j$ is cofinal with $M_i$, by Lemma in \cite[page 104]{blassmodeltheory}, if $i<j<\alpha$, then there is a set $Y_{ij}\subseteq\omega$ such that $a_i\in \prescript{*}{}Y_{ij}$ and that $\varphi_{ij}\upharpoonright Y_{ij}$ is finite-to-one.
For $i<\alpha$ let $g_i:\omega\to\omega$ be defined as follows: if $n<\omega$, then for $n\notin Y_{0i}$ let $g_i(n)=n$, while for $n\in Y_{0i}$ let $g_i(n)=\varphi_{0i}(n)$.
Note that $\prescript{*}{}g_i(a_0)=a_i$, and that $g_i$ is finite-to-one.
The latter fact follows since $g_i$ is one-to-one on $\omega\setminus Y_{0i}$ and on $Y_{0i}$ it is equal to $\varphi_{0i}$, which is finite-to-one on $Y_{0i}$.
Now by the second part of Lemma on page 104 in \cite{blassmodeltheory}, for $i<j<\alpha$ there is a finite-to-one function $\pi_{ij}:\omega\to\omega$ such that $\prescript{*}{}\varphi_{ij}(a_i)=\prescript{*}{}\pi_{ij}(a_i)$.
Note that this means that $\prescript{*}{}g_j(a_0)=\prescript{*}{}\pi_{ij}(\prescript{*}{}g_i(a_0))$ for $i<j<\alpha$, i.e. the set $X_{ij}=\set{n\in\omega:g_j(n)=\pi_{ij}(g_i(n))}$ is in $\mathcal U_0$.
Since $\alpha<\mathfrak c$ and $\mathcal U_0$ is a $P_{\mathfrak c}$-point, there is a set $X\subseteq \omega$ such that $X\in\mathcal U_0$ and that the set $X\setminus X_{ij}$ is finite whenever $i<j<\alpha$.
Consider the sets $W_i=g_i''X$ for $i<\alpha$.
For each $i<\alpha$, let $W_i=W_i^0\cup W_i^1$ where $W_i^0\cap W_i^1=\emptyset$ and both $W_i^0$ and $W_i^1$ are infinite.
Fix $i<\alpha$ for a moment.
We know that
\[\textstyle
X=\left(X\cap \bigcup_{n\in W_i^0}g_i^{-1}(\set{n})\right)\cup \left(X\cap \bigcup_{n\in W_i^1}g_i^{-1}(\set{n})\right).
\]
Since $X\in\mathcal U_0$ and $\mathcal U_0$ is an ultrafilter, we have that either $X\cap \bigcup_{n\in W_i^0}g_i^{-1}(\set{n})\in\mathcal U_0$ or $X\cap \bigcup_{n\in W_i^1}g_i^{-1}(\set{n})\in\mathcal U_0$.
Suppose that $Y=X\cap \bigcup_{n\in W_i^0}g_i^{-1}(\set{n})\in \mathcal U_0$ (the other case would be handled similarly).
Note that by the definition of $\mathcal U_0$ we know that $a_0\in {}^*Y$.
Define $f_i:\omega\to\omega$ as follows: for $n\in Y$ let $f_i(n)=g_i(n)$, while for $n\notin Y$ let $f_i(n)=W_i^1(n)$.
Now that functions $f_i$ are defined, unfix $i$.
We will prove that $\mathcal F=\set{f_i:i<\alpha}$ has all the properties from the conclusion of the lemma.
Since each $g_i$ ($i<\alpha$) is finite-to-one, it is clear that $f_i$ is also finite-to-one.
Again, this is because $g_i$ is finite-to-one on $\omega$, and outside of $Y$ the function $f_i$ is defined so that it is one-to-one.
Since $\prescript{*}{}g_i(a_0)=a_i$ and $a_0\in {}^*Y$, it must be that $\prescript{*}{}f_i(a_0)=a_i$ for each $i<\alpha$.
Now we prove the last property.
Suppose that $i<j<\alpha$.
Since the set $X\setminus X_{ij}$ is finite and $Y\subseteq X$, there is $l<\omega$ so that $Y\setminus l\subseteq X_{ij}$.
Take $m,n\ge l$, and suppose that $f_i(n)=f_i(m)$.
There are three cases.
First, $n,m\notin Y$.
In this case, $f_i(n)=f_i(m)$ implies that $W_i^1(n)=W_i^1(m)$, i.e. $n=m$.
Hence $f_j(n)=f_j(m)$.
Second, $m\in Y$ and $n\notin Y$.
Since $m\in Y$, $g_i(m)=f_i(m)=f_i(n)$ so $f_i(n)\in W_i^0$.
On the other hand, since $n\notin Y$, $f_i(n)=W_i^1(n)$.
Thus we have $f_i(n)\in W_i^0\cap W_i^1$ which is in contradiction with the fact that $W_i^0\cap W_i^1=\emptyset$.
So this case is not possible.
Third, $m,n\in Y$.
In this case $f_i(n)=f_i(m)$ implies that $g_i(n)=g_i(m)$.
Since $m,n\in Y\setminus l\subseteq X_{ij}$ it must be that $f_j(n)=g_j(n)=\pi_{ij}(g_i(n))=\pi_{ij}(g_i(m))=g_j(m)=f_j(m)$ as required.
Thus the lemma is proved. \end{proof}
\begin{lemma}[MA$_{\alpha}$]\label{ensuringtheorem3}
Let $\seq{M_i:i<\alpha}$ be a $\subseteq$-decreasing sequence of principal, and pairwise cofinal submodels of $M$.
Suppose that $\mathcal U_0=\set{X\subseteq \omega: a_0\in \prescript{*}{}X}$ is a $P_{\mathfrak c}$-point, where $a_0$ generates $M_0$.
Then there is an element $c\in\bigcap_{i<\alpha}M_i$ which generates a principal model cofinal with all $M_i$ ($i<\alpha$). \end{lemma}
\begin{proof}
Let $a_i$ for $i<\alpha$ be an element generating $M_i$.
By Lemma \ref{ensuringensuring} there is a family $\mathcal F=\set{f_i:i<\alpha}\subseteq \omega^{\omega}$ of finite-to-one functions such that $\prescript{*}{}f_i(a_0)=a_i$ for $i<\alpha$, and that for $i,j<\alpha$ with $i<j$, there is $l<\omega$ such that $f_j(n)=f_j(m)$ whenever $f_i(n)=f_i(m)$ and $m,n\ge l$.
By Lemma \ref{ensuringcondition}, for each finite $A\subseteq \alpha$, and each $n<\omega$, there is a finite ($A,\mathcal F$)-closed set containing $n$ as an element.
Now using MA$_{\alpha}$, Lemma \ref{mainlemma} implies that there is a finite-to-one function $h\in\omega^{\omega}$, and a collection $\set{e_i:i<\alpha}\subseteq\omega^{\omega}$ such that for each $i<\alpha$ there is $l<\omega$ such that $h(n)=e_i(f_i(n))$ whenever $n\ge l$.
Let $c=\prescript{*}{}h(a_0)$, and let $M_{\alpha}$ be a model generated by $c$.
By Lemma in \cite[pp. 104]{blassmodeltheory}, $M_{\alpha}$ is cofinal with $M_0$.
Thus $M_{\alpha}$ is a principal model cofinal with all $M_i$ ($i<\alpha$).
To finish the proof we still have to show that $c\in \bigcap_{i<\alpha}M_i$.
Fix $i<\alpha$.
Let $l<\omega$ be such that $h(n)=e_i(f_i(n))$ for $n\ge l$.
If $a_0<l$, then $M_j=M$ for each $j<\alpha$ so the conclusion is trivially satisfied. So $a_0\prescript{*}{}\ge l$.
Since the sentence $(\forall n) [n\ge l\Rightarrow h(n)=e_i(f_i(n))]$ is true in $M$, it is also true in $M_0$.
Thus $c=\prescript{*}{}h(a_0)=\prescript{*}{}e_i(\prescript{*}{}f_i(a_0))=\prescript{*}{}e_i(a_i)\in M_i$ as required. \end{proof}
\begin{theorem}[MA$_{\alpha}$]\label{theorem3}
Let $M_i$ ($i<\alpha$) be a collection of pairwise cofinal submodels of $M$.
Suppose that $M_{0}$ is principal, and that $\mathcal U_0=\set{X\subseteq \omega: a_0\in \prescript{*}{}X}$ is a $P_{\mathfrak c}$-point, where $a_0$ generates $M_0$.
Then $\bigcap_{i<\alpha}M_i$ contains a principal submodel cofinal with each $M_i$. \end{theorem}
\begin{proof}
We define models $M'_i$ for $i\le\alpha$ as follows. $M'_0=M_0$. If $M'_i$ is defined, then $M'_{i+1}$ is a principal submodel of $M'_i\cap M_{i+1}$ cofinal with $M'_i$ and $M_{i+1}$. This model exists by Corollary in \cite[pp. 105]{blassmodeltheory}. If $i\le \alpha$ is limit, then the model $M'_i$ is a principal model cofinal with all $M'_j$ ($j<i$). This model exists by Lemma \ref{ensuringtheorem3}. Now the model $M'_{\alpha}$ is as required in the conclusion of the lemma. \end{proof}
\begin{theorem}[MA$_{\alpha}$]\label{maintheorem}
Suppose that $\set{\mathcal U_i:i<\alpha}$ is a collection of P-points.
Suppose moreover that $\mathcal U_0$ is a P$_{\mathfrak c}$-point such that $\mathcal U_i\le_{RK}\mathcal U_0$ for each $i<\alpha$.
Then there is a P-point $\mathcal U$ such that $\mathcal U\le_{RK} \mathcal U_i$ for each $i$. \end{theorem}
\begin{proof}
By Theorem 3 of \cite{blass}, $\omega^{\omega}/\mathcal U_i$ is isomorphic to an elementary submodel $M_i$ of $\omega^{\omega}/\mathcal U_0$.
Since all $\mathcal U_i$ ($i<\alpha$) are non-principal, each model $M_i$ ($i<\alpha$) is non-standard.
By Corollary in \cite[pp. 150]{blass}, each $M_i$ ($i<\alpha$) is cofinal with $M_0$.
This implies that all the models $M_i$ ($i<\alpha$) are pairwise cofinal with each other.
By Theorem \ref{theorem3} there is a principal model $M'$ which is a subset of each $M_i$ ($i<\alpha$) and is cofinal with $M_0$.
Since $M'$ is principal, there is an element $a$ generating $M'$.
Let $\mathcal U=\set{X\subseteq \omega:a\in \prescript{*}{}X}$.
Then $\omega^{\omega}/\mathcal U\cong M'$.
Since $M'$ is cofinal with $M_0$, $M'$ is not the standard model.
Thus $\mathcal U$ is non-principal.
Now $M'\prec M_i$ ($i<\alpha$) implies that $\mathcal U\le_{RK}\mathcal U_i$ (again using Theorem 3 of \cite{blass}).
Since $\mathcal U$ is Rudin-Keisler below a $P$-point, $\mathcal U$ is also a $P$-point. \end{proof} \begin{corollary}[MA] \label{cor:lower}
If a collection of fewer than $\mathfrak c$ many ${P}_{\mathfrak c}$-points has an upper bound which is a ${P}_{\mathfrak c}$-point, then it has a lower bound. \end{corollary} \begin{corollary}[MA] \label{cor:closed}
The class of ${P}_{\mathfrak c}$-points is downwards $< \mathfrak c$-closed under ${\leq}_{RK}$.
In other words, if $\alpha < \mathfrak c$ and $\langle {\mathcal U}_{i}: i < \alpha \rangle$ is a sequence of ${P}_{\mathfrak c}$-points such that $\forall i < j < \alpha\left[{\mathcal U}_{j} \: {\leq}_{RK} \: {\mathcal U}_{i}\right]$, then there is a ${P}_{\mathfrak c}$-point $\mathcal U$ such that $\forall i < \alpha\left[ \mathcal U \: {\leq}_{RK} \: {\mathcal U}_{i} \right]$. \end{corollary}
\end{document}
|
arXiv
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